id	sid	tid	token	lemma	pos
cana-3491	1	1	communications	communication	NOUN
cana-3491	1	2	on	on	ADP
cana-3491	1	3	applied	apply	VERB
cana-3491	1	4	nonlinear	nonlinear	ADJ
cana-3491	1	5	analysis	analysis	NOUN
cana-3491	1	6	issn	issn	NOUN
cana-3491	1	7	:	:	PUNCT
cana-3491	1	8	1074	1074	NUM
cana-3491	1	9	-	-	PUNCT
cana-3491	1	10	133x	133x	NUM
cana-3491	1	11	vol	vol	NOUN
cana-3491	1	12	32	32	NUM
cana-3491	1	13	no	no	NOUN
cana-3491	1	14	.	.	PUNCT
cana-3491	2	1	7s	7	NOUN
cana-3491	2	2	(	(	PUNCT
cana-3491	2	3	2025	2025	NUM
cana-3491	2	4	)	)	PUNCT
cana-3491	2	5	856	856	NUM
cana-3491	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	2	7	elliptic	elliptic	ADJ
cana-3491	2	8	curves	curve	NOUN
cana-3491	2	9	over	over	ADP
cana-3491	2	10	𝒑-adic	𝒑-adic	ADJ
cana-3491	2	11	field	field	NOUN
cana-3491	2	12	𝑸𝒑	𝑸𝒑	PROPN
cana-3491	2	13	and	and	CCONJ
cana-3491	3	1	its	its	PRON
cana-3491	3	2	𝒑-adic	𝒑-adic	ADJ
cana-3491	3	3	point	point	NOUN
cana-3491	3	4	addition	addition	NOUN
cana-3491	3	5	p.	p.	NOUN
cana-3491	3	6	anuradha	anuradha	PROPN
cana-3491	4	1	kameswari1	kameswari1	NOUN
cana-3491	4	2	and	and	CCONJ
cana-3491	4	3	t.	t.	PROPN
cana-3491	4	4	sai	sai	PROPN
cana-3491	5	1	tejaswini2	tejaswini2	PROPN
cana-3491	6	1	1,2department	1,2department	NUM
cana-3491	6	2	of	of	ADP
cana-3491	6	3	mathematics	mathematic	NOUN
cana-3491	6	4	,	,	PUNCT
cana-3491	6	5	andhra	andhra	PROPN
cana-3491	6	6	university	university	PROPN
cana-3491	6	7	,	,	PUNCT
cana-3491	6	8	visakhapatnam	visakhapatnam	PROPN
cana-3491	6	9	530003	530003	NUM
cana-3491	6	10	,	,	PUNCT
cana-3491	6	11	andhra	andhra	PROPN
cana-3491	6	12	pradesh	pradesh	PROPN
cana-3491	6	13	,	,	PUNCT
cana-3491	6	14	india	india	PROPN
cana-3491	6	15	.	.	PROPN
cana-3491	6	16	,	,	PUNCT
cana-3491	6	17	panuradhakameswari@yahoo.in	panuradhakameswari@yahoo.in	PROPN
cana-3491	6	18	,	,	PUNCT
cana-3491	6	19	tejutanniru89@gmail.com	tejutanniru89@gmail.com	PROPN
cana-3491	6	20	.	.	PROPN
cana-3491	6	21	article	article	PROPN
cana-3491	6	22	history	history	NOUN
cana-3491	6	23	:	:	PUNCT
cana-3491	6	24	received	receive	VERB
cana-3491	6	25	:	:	PUNCT
cana-3491	6	26	28	28	NUM
cana-3491	6	27	-	-	SYM
cana-3491	6	28	10	10	NUM
cana-3491	6	29	-	-	PUNCT
cana-3491	6	30	2024	2024	NUM
cana-3491	6	31	revised:12	revised:12	NOUN
cana-3491	6	32	-	-	PUNCT
cana-3491	6	33	11	11	NUM
cana-3491	6	34	-	-	PUNCT
cana-3491	6	35	2024	2024	NUM
cana-3491	6	36	accepted:19	accepted:19	VERB
cana-3491	6	37	-	-	PUNCT
cana-3491	6	38	12	12	NUM
cana-3491	6	39	-	-	PUNCT
cana-3491	6	40	2024	2024	NUM
cana-3491	6	41	abstract	abstract	NOUN
cana-3491	6	42	:	:	PUNCT
cana-3491	6	43	public	public	ADJ
cana-3491	6	44	-	-	PUNCT
cana-3491	6	45	key	key	NOUN
cana-3491	6	46	cryptosystems	cryptosystem	NOUN
cana-3491	6	47	with	with	ADP
cana-3491	6	48	the	the	DET
cana-3491	6	49	elliptic	elliptic	ADJ
cana-3491	6	50	curve	curve	NOUN
cana-3491	6	51	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	6	52	)	)	PUNCT
cana-3491	6	53	over	over	ADP
cana-3491	6	54	finite	finite	ADJ
cana-3491	6	55	field	field	NOUN
cana-3491	6	56	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	6	57	are	be	AUX
cana-3491	6	58	an	an	DET
cana-3491	6	59	alternative	alternative	NOUN
cana-3491	6	60	to	to	ADP
cana-3491	6	61	rsa	rsa	NOUN
cana-3491	6	62	with	with	ADP
cana-3491	6	63	finite	finite	ADJ
cana-3491	6	64	fields	field	NOUN
cana-3491	6	65	.	.	PUNCT
cana-3491	7	1	in	in	ADP
cana-3491	7	2	the	the	DET
cana-3491	7	3	context	context	NOUN
cana-3491	7	4	of	of	ADP
cana-3491	7	5	improving	improve	VERB
cana-3491	7	6	the	the	DET
cana-3491	7	7	efficiency	efficiency	NOUN
cana-3491	7	8	of	of	ADP
cana-3491	7	9	cryptosystems	cryptosystem	NOUN
cana-3491	7	10	with	with	ADP
cana-3491	7	11	elliptic	elliptic	ADJ
cana-3491	7	12	curves	curve	NOUN
cana-3491	7	13	,	,	PUNCT
cana-3491	7	14	the	the	DET
cana-3491	7	15	study	study	NOUN
cana-3491	7	16	of	of	ADP
cana-3491	7	17	elliptic	elliptic	ADJ
cana-3491	7	18	curves	curve	NOUN
cana-3491	7	19	𝐸(𝑄𝑝	𝐸(𝑄𝑝	ADJ
cana-3491	7	20	)	)	PUNCT
cana-3491	7	21	over	over	ADP
cana-3491	7	22	p	p	ADJ
cana-3491	7	23	-	-	PUNCT
cana-3491	7	24	adic	adic	ADJ
cana-3491	7	25	number	number	NOUN
cana-3491	7	26	field	field	NOUN
cana-3491	7	27	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	7	28	was	be	AUX
cana-3491	7	29	consequential	consequential	ADJ
cana-3491	7	30	.	.	PUNCT
cana-3491	8	1	in	in	ADP
cana-3491	8	2	this	this	DET
cana-3491	8	3	paper	paper	NOUN
cana-3491	8	4	we	we	PRON
cana-3491	8	5	first	first	ADV
cana-3491	8	6	obtain	obtain	VERB
cana-3491	8	7	all	all	DET
cana-3491	8	8	the	the	DET
cana-3491	8	9	points	point	NOUN
cana-3491	8	10	in	in	ADP
cana-3491	8	11	the	the	DET
cana-3491	8	12	elliptic	elliptic	ADJ
cana-3491	8	13	curve	curve	NOUN
cana-3491	8	14	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	8	15	)	)	PUNCT
cana-3491	8	16	over	over	ADP
cana-3491	8	17	p	p	ADJ
cana-3491	8	18	-	-	PUNCT
cana-3491	8	19	adic	adic	ADJ
cana-3491	8	20	number	number	NOUN
cana-3491	8	21	field	field	NOUN
cana-3491	9	1	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	9	2	as	as	ADP
cana-3491	9	3	lifts	lift	NOUN
cana-3491	9	4	of	of	ADP
cana-3491	9	5	the	the	DET
cana-3491	9	6	points	point	NOUN
cana-3491	9	7	in	in	ADP
cana-3491	9	8	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	9	9	)	)	PUNCT
cana-3491	9	10	to	to	ADP
cana-3491	9	11	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	9	12	𝑝2	𝑝2	NOUN
cana-3491	9	13	)	)	PUNCT
cana-3491	9	14	and	and	CCONJ
cana-3491	9	15	then	then	ADV
cana-3491	9	16	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	9	17	𝑝2	𝑝2	NOUN
cana-3491	9	18	)	)	PUNCT
cana-3491	9	19	to	to	ADP
cana-3491	9	20	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADV
cana-3491	9	21	𝑝3	𝑝3	ADV
cana-3491	9	22	)	)	PUNCT
cana-3491	9	23	and	and	CCONJ
cana-3491	9	24	so	so	ADV
cana-3491	9	25	on	on	ADV
cana-3491	9	26	and	and	CCONJ
cana-3491	9	27	then	then	ADV
cana-3491	9	28	to	to	PART
cana-3491	9	29	evaluate	evaluate	VERB
cana-3491	9	30	the	the	DET
cana-3491	9	31	arithmetic	arithmetic	NOUN
cana-3491	9	32	on	on	ADP
cana-3491	9	33	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	9	34	)	)	PUNCT
cana-3491	9	35	,	,	PUNCT
cana-3491	9	36	we	we	PRON
cana-3491	9	37	first	first	ADV
cana-3491	9	38	proceed	proceed	VERB
cana-3491	9	39	to	to	PART
cana-3491	9	40	describe	describe	VERB
cana-3491	9	41	the	the	DET
cana-3491	9	42	implementation	implementation	NOUN
cana-3491	9	43	of	of	ADP
cana-3491	9	44	the	the	DET
cana-3491	9	45	arithmetic	arithmetic	NOUN
cana-3491	9	46	of	of	ADP
cana-3491	9	47	points	point	NOUN
cana-3491	9	48	on	on	ADP
cana-3491	9	49	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	9	50	)	)	PUNCT
cana-3491	9	51	to	to	PART
cana-3491	9	52	points	point	NOUN
cana-3491	9	53	on	on	ADP
cana-3491	9	54	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	9	55	𝑝2	𝑝2	NOUN
cana-3491	9	56	)	)	PUNCT
cana-3491	9	57	and	and	CCONJ
cana-3491	9	58	then	then	ADV
cana-3491	9	59	give	give	VERB
cana-3491	9	60	the	the	DET
cana-3491	9	61	algorithms	algorithm	NOUN
cana-3491	9	62	for	for	ADP
cana-3491	9	63	the	the	DET
cana-3491	9	64	computations	computation	NOUN
cana-3491	9	65	.	.	PUNCT
cana-3491	10	1	keywords	keyword	NOUN
cana-3491	10	2	:	:	PUNCT
cana-3491	10	3	elliptic	elliptic	ADJ
cana-3491	10	4	curves	curve	NOUN
cana-3491	10	5	,	,	PUNCT
cana-3491	10	6	𝑝-adic	𝑝-adic	ADJ
cana-3491	10	7	field	field	NOUN
cana-3491	10	8	,	,	PUNCT
cana-3491	10	9	arithmetic	arithmetic	NOUN
cana-3491	10	10	of	of	ADP
cana-3491	10	11	𝑝-adic	𝑝-adic	ADJ
cana-3491	10	12	numbers	number	NOUN
cana-3491	10	13	,	,	PUNCT
cana-3491	10	14	elliptic	elliptic	ADJ
cana-3491	10	15	curve	curve	NOUN
cana-3491	10	16	over	over	ADP
cana-3491	10	17	a	a	DET
cana-3491	10	18	field	field	NOUN
cana-3491	10	19	𝜅.	𝜅.	NOUN
cana-3491	10	20	1	1	NUM
cana-3491	10	21	.	.	PUNCT
cana-3491	11	1	introduction	introduction	NOUN
cana-3491	11	2	elliptic	elliptic	ADJ
cana-3491	11	3	curve	curve	NOUN
cana-3491	11	4	cryptography	cryptography	NOUN
cana-3491	11	5	is	be	AUX
cana-3491	11	6	one	one	NUM
cana-3491	11	7	of	of	ADP
cana-3491	11	8	majorly	majorly	ADV
cana-3491	11	9	using	use	VERB
cana-3491	11	10	cryptosystems	cryptosystem	NOUN
cana-3491	11	11	in	in	ADP
cana-3491	11	12	the	the	DET
cana-3491	11	13	current	current	ADJ
cana-3491	11	14	century	century	NOUN
cana-3491	11	15	which	which	PRON
cana-3491	11	16	provides	provide	VERB
cana-3491	11	17	higher	high	ADJ
cana-3491	11	18	security	security	NOUN
cana-3491	11	19	and	and	CCONJ
cana-3491	11	20	greater	great	ADJ
cana-3491	11	21	efficiency	efficiency	NOUN
cana-3491	11	22	.	.	PUNCT
cana-3491	12	1	it	it	PRON
cana-3491	12	2	provides	provide	VERB
cana-3491	12	3	considerable	considerable	ADJ
cana-3491	12	4	security	security	NOUN
cana-3491	12	5	with	with	ADP
cana-3491	12	6	smaller	small	ADJ
cana-3491	12	7	key	key	ADJ
cana-3491	12	8	size	size	NOUN
cana-3491	12	9	compared	compare	VERB
cana-3491	12	10	to	to	ADP
cana-3491	12	11	any	any	DET
cana-3491	12	12	other	other	ADJ
cana-3491	12	13	public	public	ADJ
cana-3491	12	14	key	key	ADJ
cana-3491	12	15	cryptosystems	cryptosystem	NOUN
cana-3491	12	16	.	.	PUNCT
cana-3491	13	1	in	in	ADP
cana-3491	13	2	the	the	DET
cana-3491	13	3	context	context	NOUN
cana-3491	13	4	of	of	ADP
cana-3491	13	5	improving	improve	VERB
cana-3491	13	6	the	the	DET
cana-3491	13	7	efficiency	efficiency	NOUN
cana-3491	13	8	of	of	ADP
cana-3491	13	9	cryptosystems	cryptosystem	NOUN
cana-3491	13	10	with	with	ADP
cana-3491	13	11	elliptic	elliptic	ADJ
cana-3491	13	12	curves	curve	NOUN
cana-3491	13	13	,	,	PUNCT
cana-3491	13	14	the	the	DET
cana-3491	13	15	study	study	NOUN
cana-3491	13	16	of	of	ADP
cana-3491	13	17	cryptosystems	cryptosystem	NOUN
cana-3491	13	18	with	with	ADP
cana-3491	13	19	elliptic	elliptic	ADJ
cana-3491	13	20	curves	curve	NOUN
cana-3491	13	21	𝐸(𝑄𝑝	𝐸(𝑄𝑝	ADJ
cana-3491	13	22	)	)	PUNCT
cana-3491	13	23	over	over	ADP
cana-3491	13	24	𝑝-adic	𝑝-adic	ADJ
cana-3491	13	25	field	field	NOUN
cana-3491	13	26	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	13	27	was	be	AUX
cana-3491	13	28	consequential	consequential	ADJ
cana-3491	13	29	and	and	CCONJ
cana-3491	13	30	the	the	DET
cana-3491	13	31	arithmetic	arithmetic	NOUN
cana-3491	13	32	of	of	ADP
cana-3491	13	33	points	point	NOUN
cana-3491	13	34	in	in	ADP
cana-3491	13	35	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	13	36	)	)	PUNCT
cana-3491	13	37	is	be	AUX
cana-3491	13	38	implemented	implement	VERB
cana-3491	13	39	to	to	PART
cana-3491	13	40	points	point	NOUN
cana-3491	13	41	on	on	ADP
cana-3491	13	42	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	13	43	𝑝2	𝑝2	NOUN
cana-3491	13	44	)	)	PUNCT
cana-3491	13	45	over	over	ADP
cana-3491	13	46	𝑝	𝑝	ADP
cana-3491	13	47	adic	adic	ADJ
cana-3491	13	48	number	number	NOUN
cana-3491	13	49	field	field	NOUN
cana-3491	14	1	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	14	2	.	.	PUNCT
cana-3491	15	1	in	in	ADP
cana-3491	15	2	this	this	DET
cana-3491	15	3	paper	paper	NOUN
cana-3491	15	4	,	,	PUNCT
cana-3491	15	5	we	we	PRON
cana-3491	15	6	obtain	obtain	VERB
cana-3491	15	7	all	all	DET
cana-3491	15	8	the	the	DET
cana-3491	15	9	points	point	NOUN
cana-3491	15	10	in	in	ADP
cana-3491	15	11	the	the	DET
cana-3491	15	12	elliptic	elliptic	ADJ
cana-3491	15	13	curve	curve	NOUN
cana-3491	15	14	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	15	15	𝑝2	𝑝2	NOUN
cana-3491	15	16	)	)	PUNCT
cana-3491	15	17	over	over	ADP
cana-3491	15	18	p	p	ADJ
cana-3491	15	19	-	-	PUNCT
cana-3491	15	20	adic	adic	ADJ
cana-3491	15	21	number	number	NOUN
cana-3491	15	22	field	field	NOUN
cana-3491	16	1	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	16	2	as	as	ADP
cana-3491	16	3	lifts	lift	NOUN
cana-3491	16	4	of	of	ADP
cana-3491	16	5	the	the	DET
cana-3491	16	6	points	point	NOUN
cana-3491	16	7	in	in	ADP
cana-3491	16	8	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	16	9	)	)	PUNCT
cana-3491	16	10	then	then	ADV
cana-3491	16	11	describe	describe	VERB
cana-3491	16	12	the	the	DET
cana-3491	16	13	implementation	implementation	NOUN
cana-3491	16	14	of	of	ADP
cana-3491	16	15	the	the	DET
cana-3491	16	16	arithmetic	arithmetic	NOUN
cana-3491	16	17	of	of	ADP
cana-3491	16	18	points	point	NOUN
cana-3491	16	19	on	on	ADP
cana-3491	16	20	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	16	21	)	)	PUNCT
cana-3491	16	22	to	to	PART
cana-3491	16	23	points	point	NOUN
cana-3491	16	24	on	on	ADP
cana-3491	16	25	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	16	26	𝑝2	𝑝2	NOUN
cana-3491	16	27	)	)	PUNCT
cana-3491	16	28	and	and	CCONJ
cana-3491	16	29	then	then	ADV
cana-3491	16	30	give	give	VERB
cana-3491	16	31	the	the	DET
cana-3491	16	32	algorithms	algorithm	NOUN
cana-3491	16	33	for	for	ADP
cana-3491	16	34	the	the	DET
cana-3491	16	35	computations	computation	NOUN
cana-3491	16	36	.	.	PUNCT
cana-3491	17	1	for	for	ADP
cana-3491	17	2	this	this	PRON
cana-3491	17	3	,	,	PUNCT
cana-3491	17	4	in	in	ADP
cana-3491	17	5	section	section	NOUN
cana-3491	17	6	2	2	NUM
cana-3491	17	7	,	,	PUNCT
cana-3491	17	8	we	we	PRON
cana-3491	17	9	describe	describe	VERB
cana-3491	17	10	the	the	DET
cana-3491	17	11	𝑝-adic	𝑝-adic	ADJ
cana-3491	17	12	field	field	NOUN
cana-3491	17	13	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	17	14	and	and	CCONJ
cana-3491	17	15	arithmetic	arithmetic	ADJ
cana-3491	17	16	operations	operation	NOUN
cana-3491	17	17	on	on	ADP
cana-3491	17	18	𝑝-adic	𝑝-adic	ADJ
cana-3491	17	19	numbers	number	NOUN
cana-3491	17	20	and	and	CCONJ
cana-3491	17	21	in	in	ADP
cana-3491	17	22	section	section	NOUN
cana-3491	17	23	3	3	NUM
cana-3491	17	24	,	,	PUNCT
cana-3491	17	25	we	we	PRON
cana-3491	17	26	describe	describe	VERB
cana-3491	17	27	elliptic	elliptic	ADJ
cana-3491	17	28	curve	curve	NOUN
cana-3491	17	29	over	over	ADP
cana-3491	17	30	𝑝-adic	𝑝-adic	ADJ
cana-3491	17	31	field	field	NOUN
cana-3491	17	32	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	17	33	and	and	CCONJ
cana-3491	17	34	obtain	obtain	VERB
cana-3491	17	35	the	the	DET
cana-3491	17	36	points	point	NOUN
cana-3491	17	37	in	in	ADP
cana-3491	17	38	elliptic	elliptic	ADJ
cana-3491	17	39	curve	curve	NOUN
cana-3491	17	40	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	17	41	𝑝2	𝑝2	NOUN
cana-3491	17	42	)	)	PUNCT
cana-3491	17	43	over	over	ADP
cana-3491	17	44	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	17	45	by	by	ADP
cana-3491	17	46	considering	consider	VERB
cana-3491	17	47	the	the	DET
cana-3491	17	48	lift	lift	NOUN
cana-3491	17	49	of	of	ADP
cana-3491	17	50	points	point	NOUN
cana-3491	17	51	in	in	ADP
cana-3491	17	52	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	17	53	)	)	PUNCT
cana-3491	17	54	to	to	ADP
cana-3491	17	55	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	17	56	𝑝2	𝑝2	NOUN
cana-3491	17	57	)	)	PUNCT
cana-3491	17	58	.	.	PUNCT
cana-3491	18	1	in	in	ADP
cana-3491	18	2	section	section	NOUN
cana-3491	18	3	4	4	NUM
cana-3491	18	4	,	,	PUNCT
cana-3491	18	5	we	we	PRON
cana-3491	18	6	implement	implement	VERB
cana-3491	18	7	the	the	DET
cana-3491	18	8	arithmetic	arithmetic	NOUN
cana-3491	18	9	of	of	ADP
cana-3491	18	10	points	point	NOUN
cana-3491	18	11	on	on	ADP
cana-3491	18	12	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	18	13	)	)	PUNCT
cana-3491	18	14	to	to	PART
cana-3491	18	15	points	point	NOUN
cana-3491	18	16	on	on	ADP
cana-3491	18	17	elliptic	elliptic	ADJ
cana-3491	18	18	curve	curve	NOUN
cana-3491	18	19	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	18	20	𝑝2	𝑝2	NOUN
cana-3491	18	21	)	)	PUNCT
cana-3491	18	22	over	over	ADP
cana-3491	18	23	𝑝-adic	𝑝-adic	ADJ
cana-3491	18	24	field	field	NOUN
cana-3491	19	1	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	19	2	.	.	PUNCT
cana-3491	19	3	1.1	1.1	NUM
cana-3491	19	4	elliptic	elliptic	ADJ
cana-3491	19	5	curves	curve	NOUN
cana-3491	19	6	over	over	ADP
cana-3491	19	7	a	a	DET
cana-3491	19	8	field	field	NOUN
cana-3491	19	9	𝜿	𝜿	NOUN
cana-3491	19	10	definition	definition	NOUN
cana-3491	19	11	1.1	1.1	NUM
cana-3491	19	12	(	(	PUNCT
cana-3491	19	13	elliptic	elliptic	ADJ
cana-3491	19	14	curve	curve	NOUN
cana-3491	19	15	equation	equation	NOUN
cana-3491	19	16	over	over	ADP
cana-3491	19	17	field	field	NOUN
cana-3491	19	18	𝜅	𝜅	ADP
cana-3491	19	19	with	with	ADP
cana-3491	19	20	𝐶ℎ𝑎𝑟	𝐶ℎ𝑎𝑟	PROPN
cana-3491	19	21	𝜅	𝜅	PRON
cana-3491	19	22	≠	≠	PROPN
cana-3491	19	23	2,3	2,3	NUM
cana-3491	19	24	)	)	PUNCT
cana-3491	19	25	.	.	PUNCT
cana-3491	20	1	for	for	ADP
cana-3491	20	2	any	any	DET
cana-3491	20	3	field	field	NOUN
cana-3491	20	4	𝜅	𝜅	ADP
cana-3491	20	5	with	with	ADP
cana-3491	20	6	characteristic	characteristic	ADJ
cana-3491	20	7	𝜅	𝜅	PRON
cana-3491	20	8	≠	≠	PROPN
cana-3491	20	9	2,3	2,3	NUM
cana-3491	20	10	the	the	DET
cana-3491	20	11	elliptic	elliptic	ADJ
cana-3491	20	12	curve	curve	NOUN
cana-3491	20	13	𝐸	𝐸	PROPN
cana-3491	20	14	over	over	ADP
cana-3491	20	15	𝜅	𝜅	NUM
cana-3491	20	16	is	be	AUX
cana-3491	20	17	denoted	denote	VERB
cana-3491	20	18	by	by	ADP
cana-3491	20	19	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	20	20	)	)	PUNCT
cana-3491	20	21	and	and	CCONJ
cana-3491	20	22	is	be	AUX
cana-3491	20	23	given	give	VERB
cana-3491	20	24	as	as	ADP
cana-3491	20	25	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	20	26	)	)	PUNCT
cana-3491	20	27	=	=	NOUN
cana-3491	20	28	{	{	PUNCT
cana-3491	20	29	(	(	PUNCT
cana-3491	20	30	𝛼	𝛼	PROPN
cana-3491	20	31	,	,	PUNCT
cana-3491	20	32	𝛽	𝛽	NOUN
cana-3491	20	33	)	)	PUNCT
cana-3491	20	34	∈	∈	PROPN
cana-3491	20	35	𝜅	𝜅	PRON
cana-3491	20	36	×	×	NOUN
cana-3491	20	37	𝜅/𝛽2	𝜅/𝛽2	NOUN
cana-3491	20	38	=	=	PUNCT
cana-3491	20	39	𝛼3	𝛼3	NOUN
cana-3491	21	1	+	+	CCONJ
cana-3491	21	2	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	21	3	+	+	CCONJ
cana-3491	21	4	𝐵	𝐵	NOUN
cana-3491	21	5	}	}	PUNCT
cana-3491	21	6	∪	∪	VERB
cana-3491	21	7	{	{	PUNCT
cana-3491	21	8	𝒪	𝒪	NOUN
cana-3491	21	9	}	}	PUNCT
cana-3491	21	10	where	where	SCONJ
cana-3491	21	11	{	{	PUNCT
cana-3491	21	12	𝒪	𝒪	NOUN
cana-3491	21	13	}	}	PUNCT
cana-3491	21	14	is	be	AUX
cana-3491	21	15	the	the	DET
cana-3491	21	16	point	point	NOUN
cana-3491	21	17	at	at	ADP
cana-3491	21	18	infinity	infinity	NOUN
cana-3491	21	19	and	and	CCONJ
cana-3491	21	20	𝐴,𝐵	𝐴,𝐵	NOUN
cana-3491	21	21	∈	∈	PROPN
cana-3491	22	1	𝜅	𝜅	ADP
cana-3491	22	2	such	such	ADJ
cana-3491	22	3	that	that	SCONJ
cana-3491	22	4	the	the	DET
cana-3491	22	5	discriminant	discriminant	NOUN
cana-3491	22	6	𝛥	𝛥	PROPN
cana-3491	22	7	=	=	SYM
cana-3491	22	8	−(4𝐴3	−(4𝐴3	PROPN
cana-3491	22	9	+	+	CCONJ
cana-3491	22	10	27𝐵2	27𝐵2	NUM
cana-3491	22	11	)	)	PUNCT
cana-3491	22	12	≠	≠	PROPN
cana-3491	22	13	0	0	NUM
cana-3491	22	14	.	.	PUNCT
cana-3491	23	1	the	the	DET
cana-3491	23	2	equation	equation	NOUN
cana-3491	23	3	𝛽2	𝛽2	PROPN
cana-3491	23	4	=	=	PROPN
cana-3491	23	5	𝛼3	𝛼3	NOUN
cana-3491	23	6	+	+	CCONJ
cana-3491	23	7	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	23	8	+	+	CCONJ
cana-3491	23	9	𝐵	𝐵	NOUN
cana-3491	23	10	is	be	AUX
cana-3491	23	11	called	call	VERB
cana-3491	23	12	weierstrass	weierstrass	NOUN
cana-3491	23	13	equation	equation	NOUN
cana-3491	23	14	.	.	PUNCT
cana-3491	24	1	communications	communication	NOUN
cana-3491	24	2	on	on	ADP
cana-3491	24	3	applied	apply	VERB
cana-3491	24	4	nonlinear	nonlinear	ADJ
cana-3491	24	5	analysis	analysis	NOUN
cana-3491	24	6	issn	issn	NOUN
cana-3491	24	7	:	:	PUNCT
cana-3491	24	8	1074	1074	NUM
cana-3491	24	9	-	-	PUNCT
cana-3491	24	10	133x	133x	NUM
cana-3491	24	11	vol	vol	NOUN
cana-3491	24	12	32	32	NUM
cana-3491	24	13	no	no	NOUN
cana-3491	24	14	.	.	PUNCT
cana-3491	25	1	7s	7	NOUN
cana-3491	25	2	(	(	PUNCT
cana-3491	25	3	2025	2025	NUM
cana-3491	25	4	)	)	PUNCT
cana-3491	25	5	857	857	NUM
cana-3491	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	25	7	definition	definition	NOUN
cana-3491	25	8	1.2	1.2	NUM
cana-3491	25	9	(	(	PUNCT
cana-3491	25	10	elliptic	elliptic	ADJ
cana-3491	25	11	curve	curve	NOUN
cana-3491	25	12	equation	equation	NOUN
cana-3491	25	13	over	over	ADP
cana-3491	25	14	field	field	NOUN
cana-3491	25	15	𝜅	𝜅	ADP
cana-3491	25	16	with	with	ADP
cana-3491	25	17	𝐶ℎ𝑎𝑟	𝐶ℎ𝑎𝑟	PROPN
cana-3491	25	18	𝜅	𝜅	ADP
cana-3491	25	19	=	=	SYM
cana-3491	25	20	3	3	NUM
cana-3491	25	21	)	)	PUNCT
cana-3491	25	22	.	.	PUNCT
cana-3491	26	1	for	for	ADP
cana-3491	26	2	any	any	DET
cana-3491	26	3	field	field	NOUN
cana-3491	26	4	𝜅	𝜅	ADP
cana-3491	26	5	with	with	ADP
cana-3491	26	6	characteristic	characteristic	ADJ
cana-3491	26	7	𝜅	𝜅	X
cana-3491	26	8	=	=	SYM
cana-3491	26	9	3	3	NUM
cana-3491	26	10	the	the	DET
cana-3491	26	11	elliptic	elliptic	ADJ
cana-3491	26	12	curve	curve	NOUN
cana-3491	26	13	𝐸	𝐸	PROPN
cana-3491	26	14	over	over	ADP
cana-3491	26	15	𝜅	𝜅	NUM
cana-3491	26	16	is	be	AUX
cana-3491	26	17	denoted	denote	VERB
cana-3491	26	18	by	by	ADP
cana-3491	26	19	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	26	20	)	)	PUNCT
cana-3491	26	21	and	and	CCONJ
cana-3491	26	22	is	be	AUX
cana-3491	26	23	given	give	VERB
cana-3491	26	24	as	as	ADP
cana-3491	26	25	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	26	26	)	)	PUNCT
cana-3491	26	27	=	=	NOUN
cana-3491	26	28	{	{	PUNCT
cana-3491	26	29	(	(	PUNCT
cana-3491	26	30	𝛼	𝛼	PROPN
cana-3491	26	31	,	,	PUNCT
cana-3491	26	32	𝛽	𝛽	NOUN
cana-3491	26	33	)	)	PUNCT
cana-3491	26	34	∈	∈	PROPN
cana-3491	27	1	𝜅	𝜅	PRON
cana-3491	27	2	×	×	NOUN
cana-3491	27	3	𝜅/𝛽2	𝜅/𝛽2	NOUN
cana-3491	27	4	=	=	PUNCT
cana-3491	27	5	𝛼3	𝛼3	NOUN
cana-3491	27	6	+	+	CCONJ
cana-3491	27	7	𝐴𝛼2	𝐴𝛼2	NOUN
cana-3491	28	1	+	+	X
cana-3491	28	2	𝐵𝛼	𝐵𝛼	PROPN
cana-3491	28	3	+	+	CCONJ
cana-3491	28	4	𝐶	𝐶	PROPN
cana-3491	28	5	}	}	PUNCT
cana-3491	28	6	∪	∪	X
cana-3491	28	7	{	{	PUNCT
cana-3491	28	8	𝒪	𝒪	NOUN
cana-3491	28	9	}	}	PUNCT
cana-3491	28	10	where	where	SCONJ
cana-3491	28	11	{	{	PUNCT
cana-3491	28	12	𝒪	𝒪	NOUN
cana-3491	28	13	}	}	PUNCT
cana-3491	28	14	is	be	AUX
cana-3491	28	15	the	the	DET
cana-3491	28	16	point	point	NOUN
cana-3491	28	17	at	at	ADP
cana-3491	28	18	infinity	infinity	NOUN
cana-3491	28	19	and	and	CCONJ
cana-3491	28	20	𝐴,𝐵	𝐴,𝐵	NOUN
cana-3491	28	21	,	,	PUNCT
cana-3491	28	22	𝐶	𝐶	PROPN
cana-3491	28	23	∈	∈	PROPN
cana-3491	28	24	𝜅	𝜅	ADP
cana-3491	28	25	such	such	ADJ
cana-3491	28	26	that	that	SCONJ
cana-3491	28	27	the	the	DET
cana-3491	28	28	discriminant	discriminant	NOUN
cana-3491	28	29	𝛥	𝛥	PROPN
cana-3491	28	30	=	=	SYM
cana-3491	28	31	−4𝐴3𝐶	−4𝐴3𝐶	PROPN
cana-3491	29	1	+	+	CCONJ
cana-3491	29	2	𝐴2𝐵2	𝐴2𝐵2	NOUN
cana-3491	29	3	+	+	CCONJ
cana-3491	29	4	18𝐴𝐵𝐶	18𝐴𝐵𝐶	PROPN
cana-3491	29	5	−	−	NOUN
cana-3491	29	6	4𝐵3	4𝐵3	NOUN
cana-3491	30	1	−	−	NOUN
cana-3491	30	2	27𝐶2	27𝐶2	NUM
cana-3491	30	3	≠	≠	PROPN
cana-3491	30	4	0	0	NUM
cana-3491	30	5	.	.	PUNCT
cana-3491	31	1	definition	definition	NOUN
cana-3491	31	2	1.3	1.3	NUM
cana-3491	31	3	(	(	PUNCT
cana-3491	31	4	elliptic	elliptic	ADJ
cana-3491	31	5	curve	curve	NOUN
cana-3491	31	6	equation	equation	NOUN
cana-3491	31	7	over	over	ADP
cana-3491	31	8	field	field	NOUN
cana-3491	31	9	𝜅	𝜅	ADP
cana-3491	31	10	with	with	ADP
cana-3491	31	11	𝐶ℎ𝑎𝑟	𝐶ℎ𝑎𝑟	PROPN
cana-3491	31	12	𝜅	𝜅	ADP
cana-3491	31	13	=	=	SYM
cana-3491	31	14	2	2	NUM
cana-3491	31	15	)	)	PUNCT
cana-3491	31	16	.	.	PUNCT
cana-3491	32	1	for	for	ADP
cana-3491	32	2	any	any	DET
cana-3491	32	3	field	field	NOUN
cana-3491	32	4	𝜅	𝜅	ADP
cana-3491	32	5	with	with	ADP
cana-3491	32	6	characteristic	characteristic	ADJ
cana-3491	32	7	𝜅	𝜅	X
cana-3491	32	8	=	=	SYM
cana-3491	32	9	2	2	NUM
cana-3491	32	10	the	the	DET
cana-3491	32	11	elliptic	elliptic	ADJ
cana-3491	32	12	curve	curve	NOUN
cana-3491	32	13	𝐸	𝐸	PROPN
cana-3491	32	14	over	over	ADP
cana-3491	32	15	𝜅	𝜅	NUM
cana-3491	32	16	is	be	AUX
cana-3491	32	17	denoted	denote	VERB
cana-3491	32	18	by	by	ADP
cana-3491	32	19	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	32	20	)	)	PUNCT
cana-3491	32	21	and	and	CCONJ
cana-3491	32	22	is	be	AUX
cana-3491	32	23	given	give	VERB
cana-3491	32	24	as	as	ADP
cana-3491	32	25	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	32	26	)	)	PUNCT
cana-3491	32	27	=	=	NOUN
cana-3491	32	28	{	{	PUNCT
cana-3491	32	29	(	(	PUNCT
cana-3491	32	30	𝛼	𝛼	PROPN
cana-3491	32	31	,	,	PUNCT
cana-3491	32	32	𝛽	𝛽	NOUN
cana-3491	32	33	)	)	PUNCT
cana-3491	32	34	∈	∈	PROPN
cana-3491	32	35	𝜅	𝜅	PRON
cana-3491	32	36	×	×	NOUN
cana-3491	32	37	𝜅/𝛽2	𝜅/𝛽2	NOUN
cana-3491	32	38	+	+	CCONJ
cana-3491	33	1	𝛼𝛽	𝛼𝛽	NOUN
cana-3491	33	2	=	=	PUNCT
cana-3491	33	3	𝛼3	𝛼3	NOUN
cana-3491	34	1	+	+	CCONJ
cana-3491	34	2	𝑎2′𝛼	𝑎2′𝛼	PROPN
cana-3491	34	3	2	2	NUM
cana-3491	34	4	+	+	CCONJ
cana-3491	34	5	𝑎6′	𝑎6′	X
cana-3491	34	6	}	}	PUNCT
cana-3491	34	7	∪	∪	NOUN
cana-3491	34	8	{	{	PUNCT
cana-3491	34	9	𝒪	𝒪	NOUN
cana-3491	34	10	}	}	PUNCT
cana-3491	34	11	𝑜𝑟	𝑜𝑟	X
cana-3491	34	12	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	34	13	)	)	PUNCT
cana-3491	34	14	=	=	PRON
cana-3491	34	15	{	{	PUNCT
cana-3491	34	16	(	(	PUNCT
cana-3491	34	17	𝛼	𝛼	PROPN
cana-3491	34	18	,	,	PUNCT
cana-3491	34	19	𝛽	𝛽	NOUN
cana-3491	34	20	)	)	PUNCT
cana-3491	34	21	∈	∈	PROPN
cana-3491	34	22	𝜅	𝜅	PRON
cana-3491	34	23	×	×	NOUN
cana-3491	34	24	𝜅/𝛽2	𝜅/𝛽2	NOUN
cana-3491	34	25	+	+	CCONJ
cana-3491	34	26	𝑎3′𝛽	𝑎3′𝛽	X
cana-3491	34	27	=	=	SYM
cana-3491	34	28	𝛼	𝛼	PRON
cana-3491	34	29	3	3	NUM
cana-3491	34	30	+	+	CCONJ
cana-3491	34	31	𝑎2′𝛼	𝑎2′𝛼	PROPN
cana-3491	34	32	2	2	NUM
cana-3491	34	33	+	+	CCONJ
cana-3491	34	34	𝑎6′	𝑎6′	X
cana-3491	34	35	}	}	PUNCT
cana-3491	34	36	∪	∪	NOUN
cana-3491	34	37	{	{	PUNCT
cana-3491	34	38	𝒪	𝒪	NOUN
cana-3491	34	39	}	}	PUNCT
cana-3491	34	40	where	where	SCONJ
cana-3491	34	41	{	{	PUNCT
cana-3491	34	42	𝒪	𝒪	NOUN
cana-3491	34	43	}	}	PUNCT
cana-3491	34	44	is	be	AUX
cana-3491	34	45	the	the	DET
cana-3491	34	46	point	point	NOUN
cana-3491	34	47	at	at	ADP
cana-3491	34	48	infinity	infinity	NOUN
cana-3491	34	49	and	and	CCONJ
cana-3491	34	50	𝑎2′	𝑎2′	NOUN
cana-3491	34	51	,	,	PUNCT
cana-3491	34	52	𝑎3′	𝑎3′	ADJ
cana-3491	34	53	,	,	PUNCT
cana-3491	34	54	𝑎4′	𝑎4′	ADJ
cana-3491	34	55	,	,	PUNCT
cana-3491	34	56	𝑎6′	𝑎6′	NOUN
cana-3491	34	57	∈	∈	PROPN
cana-3491	34	58	𝜅	𝜅	ADP
cana-3491	34	59	such	such	ADJ
cana-3491	34	60	that	that	DET
cana-3491	34	61	𝑎3′	𝑎3′	ADJ
cana-3491	34	62	≠	≠	PROPN
cana-3491	34	63	0	0	NUM
cana-3491	34	64	and	and	CCONJ
cana-3491	34	65	𝑎6′	𝑎6′	NOUN
cana-3491	34	66	≠	≠	PROPN
cana-3491	34	67	0	0	NUM
cana-3491	34	68	.	.	PUNCT
cana-3491	34	69	remark	remark	PROPN
cana-3491	34	70	1	1	NUM
cana-3491	34	71	.	.	PUNCT
cana-3491	35	1	the	the	DET
cana-3491	35	2	discriminant	discriminant	NOUN
cana-3491	35	3	𝛥	𝛥	PROPN
cana-3491	35	4	≠	≠	PROPN
cana-3491	35	5	0	0	NUM
cana-3491	35	6	assures	assure	VERB
cana-3491	35	7	that	that	SCONJ
cana-3491	35	8	the	the	DET
cana-3491	35	9	roots	root	NOUN
cana-3491	35	10	of	of	ADP
cana-3491	35	11	the	the	DET
cana-3491	35	12	cubic	cubic	ADJ
cana-3491	35	13	equation	equation	NOUN
cana-3491	35	14	𝛽2	𝛽2	NOUN
cana-3491	35	15	=	=	PRON
cana-3491	35	16	𝛼3	𝛼3	NOUN
cana-3491	35	17	+	+	CCONJ
cana-3491	36	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	36	2	+	+	CCONJ
cana-3491	36	3	𝐵	𝐵	NOUN
cana-3491	36	4	are	be	AUX
cana-3491	36	5	distinct	distinct	ADJ
cana-3491	36	6	as	as	ADP
cana-3491	36	7	𝛥	𝛥	PROPN
cana-3491	36	8	=	=	SYM
cana-3491	36	9	(	(	PUNCT
cana-3491	36	10	(	(	PUNCT
cana-3491	36	11	𝑒1	𝑒1	NOUN
cana-3491	36	12	−	−	NOUN
cana-3491	36	13	𝑒2)(𝑒2	𝑒2)(𝑒2	NOUN
cana-3491	36	14	−	−	NOUN
cana-3491	37	1	𝑒3)(𝑒3	𝑒3)(𝑒3	VERB
cana-3491	38	1	−	−	NOUN
cana-3491	38	2	𝑒1	𝑒1	NOUN
cana-3491	38	3	)	)	PUNCT
cana-3491	38	4	)	)	PUNCT
cana-3491	38	5	2	2	NUM
cana-3491	38	6	for	for	ADP
cana-3491	38	7	𝑒1	𝑒1	NOUN
cana-3491	38	8	,	,	PUNCT
cana-3491	38	9	𝑒2	𝑒2	PROPN
cana-3491	38	10	,	,	PUNCT
cana-3491	38	11	𝑒3	𝑒3	PROPN
cana-3491	38	12	are	be	AUX
cana-3491	38	13	the	the	DET
cana-3491	38	14	cube	cube	NOUN
cana-3491	38	15	roots	root	NOUN
cana-3491	38	16	and	and	CCONJ
cana-3491	38	17	basing	base	VERB
cana-3491	38	18	on	on	ADP
cana-3491	38	19	this	this	DET
cana-3491	38	20	point	point	NOUN
cana-3491	38	21	the	the	DET
cana-3491	38	22	arithmetic	arithmetic	NOUN
cana-3491	38	23	on	on	ADP
cana-3491	38	24	elliptic	elliptic	ADJ
cana-3491	38	25	curve	curve	NOUN
cana-3491	38	26	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	38	27	)	)	PUNCT
cana-3491	38	28	is	be	AUX
cana-3491	38	29	established	establish	VERB
cana-3491	38	30	.	.	PUNCT
cana-3491	39	1	example	example	NOUN
cana-3491	39	2	1.1	1.1	NUM
cana-3491	39	3	.	.	PUNCT
cana-3491	40	1	find	find	VERB
cana-3491	40	2	the	the	DET
cana-3491	40	3	points	point	NOUN
cana-3491	40	4	on	on	ADP
cana-3491	40	5	an	an	DET
cana-3491	40	6	elliptic	elliptic	ADJ
cana-3491	40	7	curve	curve	NOUN
cana-3491	40	8	𝐸	𝐸	PROPN
cana-3491	40	9	:	:	PUNCT
cana-3491	40	10	𝑦2	𝑦2	PROPN
cana-3491	40	11	=	=	PUNCT
cana-3491	40	12	𝑥3	𝑥3	PROPN
cana-3491	40	13	+	+	CCONJ
cana-3491	40	14	𝑥	𝑥	X
cana-3491	41	1	+	+	NUM
cana-3491	41	2	1	1	NUM
cana-3491	41	3	over	over	ADP
cana-3491	41	4	𝐹5	𝐹5	NOUN
cana-3491	41	5	.	.	PUNCT
cana-3491	42	1	in	in	ADP
cana-3491	42	2	finding	find	VERB
cana-3491	42	3	points	point	NOUN
cana-3491	42	4	on	on	ADP
cana-3491	42	5	𝐸	𝐸	PROPN
cana-3491	42	6	,	,	PUNCT
cana-3491	42	7	we	we	PRON
cana-3491	42	8	first	first	ADV
cana-3491	42	9	consider	consider	VERB
cana-3491	42	10	all	all	DET
cana-3491	42	11	the	the	DET
cana-3491	42	12	possible	possible	ADJ
cana-3491	42	13	values	value	NOUN
cana-3491	42	14	of	of	ADP
cana-3491	42	15	𝑥	𝑥	PRON
cana-3491	42	16	which	which	PRON
cana-3491	42	17	are	be	AUX
cana-3491	42	18	0,1,2,3,4	0,1,2,3,4	NUM
cana-3491	42	19	and	and	CCONJ
cana-3491	42	20	then	then	ADV
cana-3491	42	21	find	find	VERB
cana-3491	42	22	𝑦	𝑦	PRON
cana-3491	42	23	which	which	PRON
cana-3491	42	24	is	be	AUX
cana-3491	42	25	a	a	DET
cana-3491	42	26	square	square	NOUN
cana-3491	42	27	of	of	ADP
cana-3491	42	28	𝑥3	𝑥3	NOUN
cana-3491	42	29	+	+	CCONJ
cana-3491	42	30	𝑥	𝑥	PROPN
cana-3491	42	31	+	+	NUM
cana-3491	42	32	1(𝑚𝑜𝑑5	1(𝑚𝑜𝑑5	NUM
cana-3491	42	33	)	)	PUNCT
cana-3491	42	34	and	and	CCONJ
cana-3491	42	35	the	the	DET
cana-3491	42	36	following	follow	VERB
cana-3491	42	37	table	table	NOUN
cana-3491	42	38	represents	represent	VERB
cana-3491	42	39	the	the	DET
cana-3491	42	40	points	point	NOUN
cana-3491	42	41	in	in	ADP
cana-3491	42	42	𝐸(𝐹5	𝐸(𝐹5	PROPN
cana-3491	42	43	)	)	PUNCT
cana-3491	42	44	.	.	PUNCT
cana-3491	43	1	𝒙	𝒙	NUM
cana-3491	43	2	𝒙𝟑	𝒙𝟑	NOUN
cana-3491	43	3	+	+	CCONJ
cana-3491	43	4	𝒙𝟐	𝒙𝟐	NOUN
cana-3491	43	5	+	+	CCONJ
cana-3491	43	6	𝟏	𝟏	NUM
cana-3491	43	7	𝒚	𝒚	NOUN
cana-3491	43	8	points	point	VERB
cana-3491	43	9	on	on	ADP
cana-3491	43	10	𝑬(𝑭𝟓	𝑬(𝑭𝟓	ADJ
cana-3491	43	11	)	)	PUNCT
cana-3491	43	12	0	0	NUM
cana-3491	43	13	1	1	NUM
cana-3491	43	14	1,4	1,4	NUM
cana-3491	43	15	(	(	PUNCT
cana-3491	43	16	0,4	0,4	NOUN
cana-3491	43	17	)	)	PUNCT
cana-3491	43	18	,	,	PUNCT
cana-3491	43	19	(	(	PUNCT
cana-3491	43	20	0,1	0,1	NOUN
cana-3491	43	21	)	)	PUNCT
cana-3491	43	22	1	1	NUM
cana-3491	43	23	3	3	NUM
cana-3491	43	24	2	2	NUM
cana-3491	43	25	1	1	NUM
cana-3491	43	26	1,4	1,4	NUM
cana-3491	43	27	(	(	PUNCT
cana-3491	43	28	2,1	2,1	NUM
cana-3491	43	29	)	)	PUNCT
cana-3491	43	30	,	,	PUNCT
cana-3491	43	31	(	(	PUNCT
cana-3491	43	32	2,4	2,4	NUM
cana-3491	43	33	)	)	PUNCT
cana-3491	43	34	3	3	NUM
cana-3491	43	35	1	1	NUM
cana-3491	43	36	1,4	1,4	NUM
cana-3491	43	37	(	(	PUNCT
cana-3491	43	38	3,1	3,1	NUM
cana-3491	43	39	)	)	PUNCT
cana-3491	43	40	,	,	PUNCT
cana-3491	43	41	(	(	PUNCT
cana-3491	43	42	3,4	3,4	X
cana-3491	43	43	)	)	PUNCT
cana-3491	43	44	𝒪	𝒪	NOUN
cana-3491	43	45	𝒪	𝒪	NOUN
cana-3491	43	46	𝒪	𝒪	NOUN
cana-3491	43	47	𝒪	𝒪	NOUN
cana-3491	43	48	the	the	DET
cana-3491	43	49	points	point	NOUN
cana-3491	43	50	in	in	ADP
cana-3491	43	51	𝐸(𝐹5	𝐸(𝐹5	NOUN
cana-3491	43	52	)	)	PUNCT
cana-3491	43	53	are	be	AUX
cana-3491	43	54	{	{	PUNCT
cana-3491	43	55	(	(	PUNCT
cana-3491	43	56	0,1	0,1	NUM
cana-3491	43	57	)	)	PUNCT
cana-3491	43	58	,	,	PUNCT
cana-3491	43	59	(	(	PUNCT
cana-3491	43	60	0,4	0,4	NOUN
cana-3491	43	61	)	)	PUNCT
cana-3491	43	62	,	,	PUNCT
cana-3491	43	63	(	(	PUNCT
cana-3491	43	64	2,1	2,1	NUM
cana-3491	43	65	)	)	PUNCT
cana-3491	43	66	,	,	PUNCT
cana-3491	43	67	(	(	PUNCT
cana-3491	43	68	2,4	2,4	NUM
cana-3491	43	69	)	)	PUNCT
cana-3491	43	70	,	,	PUNCT
cana-3491	43	71	(	(	PUNCT
cana-3491	43	72	3,1	3,1	NUM
cana-3491	43	73	)	)	PUNCT
cana-3491	43	74	,	,	PUNCT
cana-3491	43	75	(	(	PUNCT
cana-3491	43	76	3,4)(4,2	3,4)(4,2	NUM
cana-3491	43	77	)	)	PUNCT
cana-3491	43	78	,	,	PUNCT
cana-3491	43	79	(	(	PUNCT
cana-3491	43	80	4,3	4,3	NUM
cana-3491	43	81	)	)	PUNCT
cana-3491	43	82	}	}	PUNCT
cana-3491	43	83	∪	∪	X
cana-3491	43	84	{	{	PUNCT
cana-3491	43	85	𝒪	𝒪	NOUN
cana-3491	43	86	}	}	PUNCT
cana-3491	43	87	.	.	PUNCT
cana-3491	44	1	1.2	1.2	NUM
cana-3491	44	2	arithmetic	arithmetic	ADJ
cana-3491	44	3	on	on	ADP
cana-3491	44	4	elliptic	elliptic	ADJ
cana-3491	44	5	curve	curve	NOUN
cana-3491	44	6	over	over	ADP
cana-3491	44	7	a	a	DET
cana-3491	44	8	field	field	NOUN
cana-3491	44	9	𝜿	𝜿	X
cana-3491	44	10	the	the	DET
cana-3491	44	11	hidden	hide	VERB
cana-3491	44	12	beauty	beauty	NOUN
cana-3491	44	13	of	of	ADP
cana-3491	44	14	ecc	ecc	PROPN
cana-3491	44	15	lies	lie	VERB
cana-3491	44	16	in	in	ADP
cana-3491	44	17	adding	add	VERB
cana-3491	44	18	two	two	NUM
cana-3491	44	19	points	point	NOUN
cana-3491	44	20	on	on	ADP
cana-3491	44	21	elliptic	elliptic	ADJ
cana-3491	44	22	curve	curve	NOUN
cana-3491	44	23	in	in	ADP
cana-3491	44	24	such	such	DET
cana-3491	44	25	a	a	DET
cana-3491	44	26	way	way	NOUN
cana-3491	44	27	that	that	PRON
cana-3491	44	28	it	it	PRON
cana-3491	44	29	is	be	AUX
cana-3491	44	30	completely	completely	ADV
cana-3491	44	31	different	different	ADJ
cana-3491	44	32	from	from	ADP
cana-3491	44	33	any	any	DET
cana-3491	44	34	other	other	ADJ
cana-3491	44	35	point	point	NOUN
cana-3491	44	36	additions	addition	NOUN
cana-3491	44	37	that	that	PRON
cana-3491	44	38	are	be	AUX
cana-3491	44	39	generally	generally	ADV
cana-3491	44	40	used	use	VERB
cana-3491	44	41	.	.	PUNCT
cana-3491	45	1	the	the	DET
cana-3491	45	2	addition	addition	NOUN
cana-3491	45	3	law	law	NOUN
cana-3491	45	4	on	on	ADP
cana-3491	45	5	elliptic	elliptic	ADJ
cana-3491	45	6	curves	curve	NOUN
cana-3491	45	7	is	be	AUX
cana-3491	45	8	explained	explain	VERB
cana-3491	45	9	geometrically	geometrically	ADV
cana-3491	45	10	below	below	ADV
cana-3491	45	11	.	.	PUNCT
cana-3491	46	1	let	let	VERB
cana-3491	46	2	us	we	PRON
cana-3491	46	3	suppose	suppose	VERB
cana-3491	46	4	an	an	DET
cana-3491	46	5	elliptic	elliptic	ADJ
cana-3491	46	6	curve	curve	NOUN
cana-3491	46	7	over	over	ADP
cana-3491	46	8	a	a	DET
cana-3491	46	9	field	field	NOUN
cana-3491	46	10	of	of	ADP
cana-3491	46	11	characteristic	characteristic	ADJ
cana-3491	46	12	𝜅	𝜅	PRON
cana-3491	46	13	≠	≠	PROPN
cana-3491	46	14	2,3	2,3	NUM
cana-3491	46	15	then	then	ADV
cana-3491	46	16	the	the	DET
cana-3491	46	17	curve	curve	NOUN
cana-3491	46	18	equation	equation	NOUN
cana-3491	46	19	over	over	ADP
cana-3491	46	20	the	the	DET
cana-3491	46	21	field	field	NOUN
cana-3491	46	22	𝜅	𝜅	PRON
cana-3491	46	23	is	be	AUX
cana-3491	46	24	given	give	VERB
cana-3491	46	25	as	as	ADP
cana-3491	46	26	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	46	27	):	):	PUNCT
cana-3491	46	28	𝛽2	𝛽2	NOUN
cana-3491	46	29	=	=	PRON
cana-3491	47	1	𝛼3	𝛼3	NOUN
cana-3491	47	2	+	+	CCONJ
cana-3491	48	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	48	2	+	+	CCONJ
cana-3491	48	3	𝐵	𝐵	NOUN
cana-3491	48	4	communications	communication	NOUN
cana-3491	48	5	on	on	ADP
cana-3491	48	6	applied	apply	VERB
cana-3491	48	7	nonlinear	nonlinear	ADJ
cana-3491	48	8	analysis	analysis	NOUN
cana-3491	48	9	issn	issn	NOUN
cana-3491	48	10	:	:	PUNCT
cana-3491	48	11	1074	1074	NUM
cana-3491	48	12	-	-	PUNCT
cana-3491	48	13	133x	133x	NUM
cana-3491	48	14	vol	vol	NOUN
cana-3491	48	15	32	32	NUM
cana-3491	48	16	no	no	NOUN
cana-3491	48	17	.	.	PUNCT
cana-3491	49	1	7s	7	NOUN
cana-3491	49	2	(	(	PUNCT
cana-3491	49	3	2025	2025	NUM
cana-3491	49	4	)	)	PUNCT
cana-3491	49	5	858	858	NUM
cana-3491	49	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	49	7	let	let	VERB
cana-3491	49	8	𝑃1	𝑃1	NOUN
cana-3491	49	9	=(	=(	NOUN
cana-3491	49	10	𝛼1	𝛼1	NOUN
cana-3491	49	11	,	,	PUNCT
cana-3491	49	12	𝛽1	𝛽1	NOUN
cana-3491	49	13	)	)	PUNCT
cana-3491	49	14	and	and	CCONJ
cana-3491	49	15	𝑃2	𝑃2	ADJ
cana-3491	49	16	=(	=(	PROPN
cana-3491	49	17	𝛼2	𝛼2	PROPN
cana-3491	49	18	,	,	PUNCT
cana-3491	49	19	𝛽2	𝛽2	NOUN
cana-3491	49	20	)	)	PUNCT
cana-3491	49	21	be	be	VERB
cana-3491	49	22	two	two	NUM
cana-3491	49	23	points	point	NOUN
cana-3491	49	24	on	on	ADP
cana-3491	49	25	the	the	DET
cana-3491	49	26	given	give	VERB
cana-3491	49	27	elliptic	elliptic	ADJ
cana-3491	49	28	curve	curve	NOUN
cana-3491	49	29	e.	e.	PROPN
cana-3491	49	30	draw	draw	VERB
cana-3491	49	31	a	a	DET
cana-3491	49	32	line	line	NOUN
cana-3491	49	33	𝐿	𝐿	NOUN
cana-3491	49	34	through	through	ADP
cana-3491	49	35	𝑃1	𝑃1	NOUN
cana-3491	49	36	and	and	CCONJ
cana-3491	49	37	𝑃2	𝑃2	NOUN
cana-3491	49	38	,	,	PUNCT
cana-3491	49	39	then	then	ADV
cana-3491	49	40	𝐿	𝐿	PROPN
cana-3491	49	41	intersects	intersect	VERB
cana-3491	49	42	𝐸	𝐸	PROPN
cana-3491	49	43	in	in	ADP
cana-3491	49	44	a	a	DET
cana-3491	49	45	third	third	ADJ
cana-3491	49	46	point	point	NOUN
cana-3491	49	47	𝑃3′	𝑃3′	NOUN
cana-3491	49	48	as	as	ADP
cana-3491	49	49	𝛥	𝛥	PROPN
cana-3491	49	50	≠	≠	PROPN
cana-3491	49	51	0	0	NUM
cana-3491	49	52	.	.	X
cana-3491	49	53	reflect	reflect	VERB
cana-3491	49	54	𝑃3′	𝑃3′	NOUN
cana-3491	49	55	across	across	ADP
cana-3491	49	56	𝑋-axis	𝑋-axis	PROPN
cana-3491	49	57	to	to	PART
cana-3491	49	58	obtain	obtain	VERB
cana-3491	49	59	𝑃3	𝑃3	NOUN
cana-3491	49	60	.	.	PUNCT
cana-3491	50	1	now	now	ADV
cana-3491	50	2	we	we	PRON
cana-3491	50	3	define	define	VERB
cana-3491	50	4	the	the	DET
cana-3491	50	5	sum	sum	NOUN
cana-3491	50	6	of	of	ADP
cana-3491	50	7	𝑃1	𝑃1	NOUN
cana-3491	50	8	and	and	CCONJ
cana-3491	50	9	𝑃2	𝑃2	NOUN
cana-3491	50	10	as	as	ADP
cana-3491	50	11	𝑃3	𝑃3	NOUN
cana-3491	50	12	and	and	CCONJ
cana-3491	50	13	is	be	AUX
cana-3491	50	14	denoted	denote	VERB
cana-3491	50	15	as	as	ADP
cana-3491	50	16	𝑃1	𝑃1	NOUN
cana-3491	50	17	+	+	CCONJ
cana-3491	50	18	𝑃2	𝑃2	ADJ
cana-3491	50	19	=	=	SYM
cana-3491	50	20	𝑃3	𝑃3	NOUN
cana-3491	50	21	given	give	VERB
cana-3491	50	22	as	as	ADP
cana-3491	50	23	𝑃3	𝑃3	NOUN
cana-3491	50	24	=	=	PUNCT
cana-3491	50	25	(	(	PUNCT
cana-3491	50	26	𝛼3	𝛼3	PROPN
cana-3491	50	27	,	,	PUNCT
cana-3491	50	28	𝛽3	𝛽3	PROPN
cana-3491	50	29	)	)	PUNCT
cana-3491	50	30	as	as	SCONJ
cana-3491	50	31	shown	show	VERB
cana-3491	50	32	in	in	ADP
cana-3491	50	33	fig	fig	NOUN
cana-3491	50	34	.	.	PUNCT
cana-3491	51	1	1	1	X
cana-3491	51	2	.	.	X
cana-3491	51	3	the	the	DET
cana-3491	51	4	set	set	NOUN
cana-3491	51	5	of	of	ADP
cana-3491	51	6	points	point	NOUN
cana-3491	51	7	on	on	ADP
cana-3491	51	8	an	an	DET
cana-3491	51	9	elliptic	elliptic	ADJ
cana-3491	51	10	curve	curve	NOUN
cana-3491	51	11	over	over	ADP
cana-3491	51	12	a	a	DET
cana-3491	51	13	field	field	NOUN
cana-3491	51	14	𝜅	𝜅	DET
cana-3491	51	15	forms	form	NOUN
cana-3491	51	16	a	a	DET
cana-3491	51	17	group	group	NOUN
cana-3491	51	18	which	which	PRON
cana-3491	51	19	is	be	AUX
cana-3491	51	20	also	also	ADV
cana-3491	51	21	abelian	abelian	ADJ
cana-3491	51	22	with	with	ADP
cana-3491	51	23	respect	respect	NOUN
cana-3491	51	24	to	to	PART
cana-3491	51	25	point	point	NOUN
cana-3491	51	26	addition	addition	NOUN
cana-3491	51	27	defined	define	VERB
cana-3491	51	28	above	above	ADV
cana-3491	51	29	.	.	PUNCT
cana-3491	52	1	figure	figure	VERB
cana-3491	52	2	1	1	NUM
cana-3491	52	3	:	:	PUNCT
cana-3491	52	4	geometric	geometric	ADJ
cana-3491	52	5	interpretation	interpretation	NOUN
cana-3491	52	6	of	of	ADP
cana-3491	52	7	point	point	NOUN
cana-3491	52	8	addition	addition	NOUN
cana-3491	52	9	on	on	ADP
cana-3491	52	10	curve	curve	NOUN
cana-3491	52	11	of	of	ADP
cana-3491	52	12	the	the	DET
cana-3491	52	13	form	form	NOUN
cana-3491	52	14	β2	β2	NOUN
cana-3491	52	15	=	=	PROPN
cana-3491	52	16	α3	α3	PROPN
cana-3491	53	1	+	+	NOUN
cana-3491	53	2	aα	aα	NOUN
cana-3491	54	1	+	+	NUM
cana-3491	54	2	b	b	NOUN
cana-3491	54	3	the	the	DET
cana-3491	54	4	point	point	NOUN
cana-3491	54	5	addition	addition	NOUN
cana-3491	54	6	and	and	CCONJ
cana-3491	54	7	doubling	double	VERB
cana-3491	54	8	formula	formula	NOUN
cana-3491	54	9	for	for	ADP
cana-3491	54	10	points	point	NOUN
cana-3491	54	11	on	on	ADP
cana-3491	54	12	an	an	DET
cana-3491	54	13	elliptic	elliptic	ADJ
cana-3491	54	14	curve	curve	NOUN
cana-3491	54	15	on	on	ADP
cana-3491	54	16	a	a	DET
cana-3491	54	17	field	field	NOUN
cana-3491	54	18	𝜅	𝜅	ADP
cana-3491	54	19	with	with	ADP
cana-3491	54	20	𝐶ℎ𝑎𝑟𝜅	𝐶ℎ𝑎𝑟𝜅	PROPN
cana-3491	54	21	=	=	SYM
cana-3491	54	22	2	2	NUM
cana-3491	54	23	and	and	CCONJ
cana-3491	54	24	𝐶ℎ𝑎𝑟𝜅	𝐶ℎ𝑎𝑟𝜅	PROPN
cana-3491	54	25	=	=	SYM
cana-3491	54	26	3	3	NUM
cana-3491	54	27	and	and	CCONJ
cana-3491	54	28	𝐶ℎ𝑎𝑟𝜅	𝐶ℎ𝑎𝑟𝜅	PROPN
cana-3491	54	29	≠	≠	PROPN
cana-3491	54	30	2,3	2,3	NUM
cana-3491	54	31	under	under	ADP
cana-3491	54	32	different	different	ADJ
cana-3491	54	33	conditions	condition	NOUN
cana-3491	54	34	at	at	ADP
cana-3491	54	35	points	point	NOUN
cana-3491	54	36	𝑃1	𝑃1	NOUN
cana-3491	54	37	and	and	CCONJ
cana-3491	54	38	𝑃2	𝑃2	NOUN
cana-3491	54	39	is	be	AUX
cana-3491	54	40	given	give	VERB
cana-3491	54	41	in	in	ADP
cana-3491	54	42	the	the	DET
cana-3491	54	43	table	table	NOUN
cana-3491	54	44	below	below	ADV
cana-3491	54	45	.	.	PUNCT
cana-3491	55	1	field	field	NOUN
cana-3491	55	2	κ	κ	ADP
cana-3491	55	3	elliptic	elliptic	ADJ
cana-3491	55	4	curve	curve	NOUN
cana-3491	55	5	slope	slope	NOUN
cana-3491	55	6	𝒎	𝒎	PRON
cana-3491	55	7	𝑷𝟏	𝑷𝟏	PROPN
cana-3491	56	1	+	+	CCONJ
cana-3491	56	2	𝑷𝟐	𝑷𝟐	NOUN
cana-3491	56	3	=	=	SYM
cana-3491	56	4	𝑷𝟑	𝑷𝟑	PROPN
cana-3491	56	5	p1	p1	NOUN
cana-3491	56	6	≠	≠	PROPN
cana-3491	56	7	p2	p2	NOUN
cana-3491	56	8	𝑃1	𝑃1	NOUN
cana-3491	56	9	=	=	SYM
cana-3491	56	10	𝑃2	𝑃2	PROPN
cana-3491	56	11	p1	p1	PROPN
cana-3491	56	12	≠	≠	PROPN
cana-3491	56	13	p2	p2	PROPN
cana-3491	56	14	∝1≠∝2	∝1≠∝2	PROPN
cana-3491	56	15	p1≠	p1≠	VERB
cana-3491	56	16	p2	p2	PROPN
cana-3491	56	17	∝1	∝1	NUM
cana-3491	56	18	=	=	SYM
cana-3491	56	19	∝2	∝2	NOUN
cana-3491	56	20	𝑃1	𝑃1	NOUN
cana-3491	56	21	=	=	PUNCT
cana-3491	56	22	𝑃2	𝑃2	PROPN
cana-3491	56	23	char	char	NOUN
cana-3491	56	24	κ	κ	PROPN
cana-3491	56	25	≠	≠	PROPN
cana-3491	56	26	2,3	2,3	NUM
cana-3491	56	27	𝛽2	𝛽2	NOUN
cana-3491	56	28	=	=	PRON
cana-3491	56	29	∝3+𝐴	∝3+𝐴	ADJ
cana-3491	56	30	∝	∝	PROPN
cana-3491	57	1	+	+	ADJ
cana-3491	57	2	𝐵	𝐵	PROPN
cana-3491	57	3	𝛽2−𝛽1	𝛽2−𝛽1	PUNCT
cana-3491	57	4	∝2−∝1	∝2−∝1	ADP
cana-3491	57	5	3	3	NUM
cana-3491	57	6	∝1	∝1	SYM
cana-3491	57	7	2	2	NUM
cana-3491	57	8	+	+	NUM
cana-3491	57	9	𝐴	𝐴	PROPN
cana-3491	57	10	2𝛽1	2𝛽1	NUM
cana-3491	57	11	∝3=𝑚	∝3=𝑚	PROPN
cana-3491	57	12	2	2	NUM
cana-3491	57	13	─	─	PROPN
cana-3491	57	14	∝1	∝1	NUM
cana-3491	57	15	─	─	PUNCT
cana-3491	57	16	∝2	∝2	PROPN
cana-3491	57	17	𝛽3	𝛽3	PROPN
cana-3491	57	18	=	=	PUNCT
cana-3491	57	19	𝑚(∝1−∝3	𝑚(∝1−∝3	ADV
cana-3491	57	20	)	)	PUNCT
cana-3491	57	21	─	─	X
cana-3491	57	22	𝛽1	𝛽1	NOUN
cana-3491	57	23	𝒪	𝒪	PROPN
cana-3491	57	24	∝3=	∝3=	PROPN
cana-3491	57	25	𝑚	𝑚	PROPN
cana-3491	57	26	2	2	NUM
cana-3491	57	27	─	─	NOUN
cana-3491	57	28	2	2	NUM
cana-3491	57	29	∝1	∝1	NOUN
cana-3491	57	30	𝛽3	𝛽3	NOUN
cana-3491	57	31	=	=	PUNCT
cana-3491	57	32	𝑚(∝1−∝3	𝑚(∝1−∝3	ADV
cana-3491	57	33	)	)	PUNCT
cana-3491	57	34	─	─	X
cana-3491	57	35	𝛽1	𝛽1	NOUN
cana-3491	57	36	char	char	NOUN
cana-3491	57	37	κ	κ	PROPN
cana-3491	57	38	=	=	SYM
cana-3491	57	39	3	3	NUM
cana-3491	57	40	𝛽2	𝛽2	NOUN
cana-3491	57	41	=	=	PRON
cana-3491	57	42	∝3+𝐴	∝3+𝐴	ADV
cana-3491	57	43	∝2+𝐵	∝2+𝐵	ADJ
cana-3491	57	44	∝	∝	PROPN
cana-3491	57	45	+	+	PROPN
cana-3491	57	46	𝐶	𝐶	PROPN
cana-3491	57	47	𝛽2−𝛽1	𝛽2−𝛽1	NOUN
cana-3491	57	48	∝2−∝1	∝2−∝1	ADP
cana-3491	57	49	3	3	NUM
cana-3491	57	50	∝1	∝1	NUM
cana-3491	57	51	2	2	NUM
cana-3491	57	52	+	+	NOUN
cana-3491	57	53	2𝐴	2𝐴	ADJ
cana-3491	58	1	∝	∝	PROPN
cana-3491	58	2	+	+	PROPN
cana-3491	58	3	𝐵	𝐵	PROPN
cana-3491	58	4	2𝛽1	2𝛽1	NUM
cana-3491	58	5	∝3=	∝3=	PROPN
cana-3491	58	6	𝑚	𝑚	PROPN
cana-3491	58	7	2	2	NUM
cana-3491	58	8	─	─	NOUN
cana-3491	58	9	𝐴	𝐴	PROPN
cana-3491	58	10	─	─	PROPN
cana-3491	58	11	∝1	∝1	NUM
cana-3491	58	12	─	─	PUNCT
cana-3491	58	13	∝2	∝2	PROPN
cana-3491	58	14	𝛽3	𝛽3	PROPN
cana-3491	58	15	=	=	PUNCT
cana-3491	58	16	𝑚(∝1−∝3	𝑚(∝1−∝3	ADV
cana-3491	58	17	)	)	PUNCT
cana-3491	58	18	─	─	X
cana-3491	58	19	𝛽1	𝛽1	NOUN
cana-3491	58	20	𝒪	𝒪	PROPN
cana-3491	58	21	∝3=	∝3=	PROPN
cana-3491	58	22	𝑚	𝑚	PROPN
cana-3491	58	23	2	2	NUM
cana-3491	58	24	─	─	ADJ
cana-3491	58	25	𝐴	𝐴	NOUN
cana-3491	58	26	─	─	PROPN
cana-3491	58	27	2	2	PROPN
cana-3491	58	28	∝1	∝1	NOUN
cana-3491	58	29	𝛽3	𝛽3	NOUN
cana-3491	58	30	=	=	PUNCT
cana-3491	58	31	𝑚(∝1−∝3	𝑚(∝1−∝3	ADV
cana-3491	58	32	)	)	PUNCT
cana-3491	58	33	─	─	X
cana-3491	58	34	𝛽1	𝛽1	NOUN
cana-3491	58	35	char	char	NOUN
cana-3491	58	36	κ	κ	NOUN
cana-3491	58	37	=	=	SYM
cana-3491	58	38	2	2	NUM
cana-3491	58	39	𝛽2+∝	𝛽2+∝	PROPN
cana-3491	58	40	𝛽	𝛽	NOUN
cana-3491	58	41	=	=	SYM
cana-3491	58	42	∝3+𝑎2	∝3+𝑎2	PROPN
cana-3491	58	43	∝	∝	PROPN
cana-3491	58	44	2	2	NUM
cana-3491	58	45	+	+	CCONJ
cana-3491	58	46	𝑎6	𝑎6	NOUN
cana-3491	58	47	or	or	CCONJ
cana-3491	58	48	𝛽2	𝛽2	PROPN
cana-3491	58	49	+	+	CCONJ
cana-3491	58	50	𝑎′3𝛽	𝑎′3𝛽	PROPN
cana-3491	58	51	=	=	SYM
cana-3491	58	52	∝3	∝3	PROPN
cana-3491	58	53	+	+	CCONJ
cana-3491	58	54	𝑎′2	𝑎′2	X
cana-3491	58	55	∝	∝	PROPN
cana-3491	58	56	2+𝑎′6	2+𝑎′6	NUM
cana-3491	58	57	𝛽2−𝛽1	𝛽2−𝛽1	NOUN
cana-3491	58	58	∝2−∝1	∝2−∝1	PROPN
cana-3491	58	59	𝛽2−𝛽1	𝛽2−𝛽1	NUM
cana-3491	58	60	∝2−∝1	∝2−∝1	PROPN
cana-3491	58	61	∝1	∝1	NOUN
cana-3491	58	62	2+𝛽1	2+𝛽1	NUM
cana-3491	58	63	∝1	∝1	NOUN
cana-3491	58	64	∝1	∝1	NUM
cana-3491	58	65	2	2	NUM
cana-3491	58	66	+	+	NUM
cana-3491	58	67	𝑎4	𝑎4	PROPN
cana-3491	58	68	𝑎3	𝑎3	PROPN
cana-3491	58	69	∝3=	∝3=	PROPN
cana-3491	58	70	𝑚	𝑚	PROPN
cana-3491	58	71	2	2	NUM
cana-3491	58	72	+	+	NOUN
cana-3491	58	73	𝑚	𝑚	X
cana-3491	58	74	+	+	NOUN
cana-3491	58	75	∝1	∝1	ADJ
cana-3491	58	76	+	+	ADJ
cana-3491	58	77	∝2	∝2	NOUN
cana-3491	58	78	+	+	CCONJ
cana-3491	58	79	𝑎2	𝑎2	PROPN
cana-3491	58	80	𝛽3	𝛽3	PROPN
cana-3491	58	81	=	=	SYM
cana-3491	58	82	𝑚(∝1+∝3	𝑚(∝1+∝3	PROPN
cana-3491	58	83	)	)	PUNCT
cana-3491	58	84	+	+	NOUN
cana-3491	58	85	∝3	∝3	NOUN
cana-3491	58	86	+	+	CCONJ
cana-3491	58	87	𝛽1	𝛽1	NOUN
cana-3491	58	88	𝒪	𝒪	PROPN
cana-3491	58	89	𝒪	𝒪	PROPN
cana-3491	58	90	∝3=	∝3=	X
cana-3491	58	91	∝1	∝1	NOUN
cana-3491	58	92	4+𝑎6	4+𝑎6	NUM
cana-3491	58	93	∝1	∝1	NUM
cana-3491	58	94	2	2	NUM
cana-3491	58	95	𝛽3=∝	𝛽3=∝	NOUN
cana-3491	58	96	+	+	ADP
cana-3491	58	97	𝛽	𝛽	NOUN
cana-3491	58	98	∝3=	∝3=	PROPN
cana-3491	58	99	∝1	∝1	NOUN
cana-3491	58	100	4+𝑎4	4+𝑎4	NOUN
cana-3491	58	101	2	2	NUM
cana-3491	58	102	𝑎3	𝑎3	NOUN
cana-3491	58	103	2	2	NUM
cana-3491	58	104	𝑦3=𝑎3	𝑦3=𝑎3	ADV
cana-3491	58	105	+	+	NOUN
cana-3491	58	106	𝛽	𝛽	NOUN
cana-3491	58	107	communications	communication	NOUN
cana-3491	58	108	on	on	ADP
cana-3491	58	109	applied	apply	VERB
cana-3491	58	110	nonlinear	nonlinear	ADJ
cana-3491	58	111	analysis	analysis	NOUN
cana-3491	58	112	issn	issn	NOUN
cana-3491	58	113	:	:	PUNCT
cana-3491	58	114	1074	1074	NUM
cana-3491	58	115	-	-	PUNCT
cana-3491	58	116	133x	133x	NUM
cana-3491	58	117	vol	vol	NOUN
cana-3491	58	118	32	32	NUM
cana-3491	58	119	no	no	NOUN
cana-3491	58	120	.	.	PUNCT
cana-3491	59	1	7s	7	NOUN
cana-3491	59	2	(	(	PUNCT
cana-3491	59	3	2025	2025	NUM
cana-3491	59	4	)	)	PUNCT
cana-3491	59	5	859	859	NUM
cana-3491	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	59	7	∝3=𝑚	∝3=𝑚	PROPN
cana-3491	59	8	2	2	NUM
cana-3491	59	9	+	+	PROPN
cana-3491	59	10	∝1+∝2	∝1+∝2	PROPN
cana-3491	59	11	𝛽3	𝛽3	PROPN
cana-3491	59	12	=	=	SYM
cana-3491	59	13	𝑚(∝1+∝3	𝑚(∝1+∝3	PROPN
cana-3491	59	14	)	)	PUNCT
cana-3491	60	1	+	+	NOUN
cana-3491	60	2	∝3	∝3	ADJ
cana-3491	60	3	+	+	CCONJ
cana-3491	60	4	𝑎3	𝑎3	ADJ
cana-3491	60	5	table	table	NOUN
cana-3491	60	6	1	1	NUM
cana-3491	60	7	:	:	PUNCT
cana-3491	60	8	table	table	NOUN
cana-3491	60	9	for	for	ADP
cana-3491	60	10	arithmetic	arithmetic	NOUN
cana-3491	60	11	of	of	ADP
cana-3491	60	12	points	point	NOUN
cana-3491	60	13	in	in	ADP
cana-3491	60	14	e(κ	e(κ	NOUN
cana-3491	60	15	)	)	PUNCT
cana-3491	60	16	2	2	NUM
cana-3491	60	17	.	.	X
cana-3491	60	18	𝒑-adic	𝒑-adic	ADJ
cana-3491	60	19	numbers	number	NOUN
cana-3491	60	20	definition	definition	NOUN
cana-3491	60	21	2.1	2.1	NUM
cana-3491	60	22	(	(	PUNCT
cana-3491	60	23	p	p	NOUN
cana-3491	60	24	-	-	PUNCT
cana-3491	60	25	adic	adic	ADJ
cana-3491	60	26	valuation	valuation	NOUN
cana-3491	60	27	)	)	PUNCT
cana-3491	60	28	.	.	PUNCT
cana-3491	61	1	the	the	DET
cana-3491	61	2	𝑝-adic	𝑝-adic	PROPN
cana-3491	61	3	valuation	valuation	PROPN
cana-3491	61	4	𝑣𝑝(𝛼	𝑣𝑝(𝛼	ADV
cana-3491	61	5	)	)	PUNCT
cana-3491	61	6	is	be	AUX
cana-3491	61	7	given	give	VERB
cana-3491	61	8	as	as	ADP
cana-3491	61	9	for	for	ADP
cana-3491	61	10	any	any	DET
cana-3491	61	11	𝛼	𝛼	PROPN
cana-3491	61	12	∈	∈	NOUN
cana-3491	61	13	𝑄∗	𝑄∗	NOUN
cana-3491	62	1	and	and	CCONJ
cana-3491	62	2	𝑥	𝑥	NOUN
cana-3491	62	3	=	=	SYM
cana-3491	62	4	𝑝𝜌	𝑝𝜌	NOUN
cana-3491	62	5	.	.	PUNCT
cana-3491	63	1	𝑚	𝑚	ADP
cana-3491	63	2	𝑛	𝑛	PROPN
cana-3491	63	3	with	with	ADP
cana-3491	63	4	𝑚,𝑛	𝑚,𝑛	PROPN
cana-3491	63	5	,	,	PUNCT
cana-3491	63	6	𝜌	𝜌	ADP
cana-3491	63	7	∈	∈	PROPN
cana-3491	63	8	𝑍	𝑍	NOUN
cana-3491	63	9	,	,	PUNCT
cana-3491	63	10	𝑝	𝑝	PROPN
cana-3491	63	11	is	be	AUX
cana-3491	63	12	a	a	DET
cana-3491	63	13	prime	prime	NOUN
cana-3491	63	14	,	,	PUNCT
cana-3491	63	15	and	and	CCONJ
cana-3491	63	16	𝑝	𝑝	PROPN
cana-3491	63	17	∤	∤	PROPN
cana-3491	63	18	𝑚𝑛.	𝑚𝑛.	PROPN
cana-3491	63	19	also	also	ADV
cana-3491	63	20	,	,	PUNCT
cana-3491	63	21	𝑣𝑝(0	𝑣𝑝(0	NUM
cana-3491	63	22	)	)	PUNCT
cana-3491	63	23	=	=	SYM
cana-3491	63	24	∞.	∞.	PROPN
cana-3491	63	25	remark	remark	VERB
cana-3491	63	26	2	2	NUM
cana-3491	63	27	.	.	PUNCT
cana-3491	64	1	the	the	DET
cana-3491	64	2	p	p	ADJ
cana-3491	64	3	-	-	PUNCT
cana-3491	64	4	adic	adic	ADJ
cana-3491	64	5	valuation	valuation	NOUN
cana-3491	64	6	satisfies	satisfy	VERB
cana-3491	64	7	the	the	DET
cana-3491	64	8	following	follow	VERB
cana-3491	64	9	properties	property	NOUN
cana-3491	64	10	:	:	PUNCT
cana-3491	64	11	for	for	ADP
cana-3491	64	12	any	any	DET
cana-3491	64	13	𝑎	𝑎	NOUN
cana-3491	64	14	,	,	PUNCT
cana-3491	64	15	𝑏	𝑏	PROPN
cana-3491	64	16	∈	∈	PROPN
cana-3491	64	17	𝑄	𝑄	PROPN
cana-3491	64	18	1	1	NUM
cana-3491	64	19	.	.	PUNCT
cana-3491	64	20	𝑣𝑝(𝑎𝑏	𝑣𝑝(𝑎𝑏	VERB
cana-3491	64	21	)	)	PUNCT
cana-3491	65	1	=	=	SYM
cana-3491	65	2	𝑣𝑝(𝑎	𝑣𝑝(𝑎	NOUN
cana-3491	65	3	)	)	PUNCT
cana-3491	66	1	+	+	CCONJ
cana-3491	66	2	𝑣𝑝(𝑏	𝑣𝑝(𝑏	X
cana-3491	66	3	)	)	PUNCT
cana-3491	66	4	2	2	NUM
cana-3491	66	5	.	.	NOUN
cana-3491	66	6	𝑣𝑝(𝑎	𝑣𝑝(𝑎	NOUN
cana-3491	67	1	+	+	CCONJ
cana-3491	67	2	𝑏	𝑏	X
cana-3491	67	3	)	)	PUNCT
cana-3491	67	4	≥	≥	NOUN
cana-3491	67	5	𝑚𝑖𝑛{𝑣𝑝(𝑎	𝑚𝑖𝑛{𝑣𝑝(𝑎	PROPN
cana-3491	67	6	)	)	PUNCT
cana-3491	67	7	,	,	PUNCT
cana-3491	67	8	𝑣𝑝(𝑏	𝑣𝑝(𝑏	NOUN
cana-3491	67	9	)	)	PUNCT
cana-3491	67	10	}	}	PUNCT
cana-3491	67	11	with	with	ADP
cana-3491	67	12	𝑣𝑝(𝑎	𝑣𝑝(𝑎	NOUN
cana-3491	67	13	)	)	PUNCT
cana-3491	67	14	≠	≠	PROPN
cana-3491	67	15	𝑣𝑝(𝑏	𝑣𝑝(𝑏	NUM
cana-3491	67	16	)	)	PUNCT
cana-3491	67	17	3	3	NUM
cana-3491	67	18	.	.	X
cana-3491	67	19	𝑣𝑝(𝑎	𝑣𝑝(𝑎	NOUN
cana-3491	67	20	)	)	PUNCT
cana-3491	68	1	=	=	SYM
cana-3491	68	2	∞	∞	NOUN
cana-3491	68	3	if	if	SCONJ
cana-3491	68	4	and	and	CCONJ
cana-3491	68	5	only	only	ADV
cana-3491	68	6	if	if	SCONJ
cana-3491	68	7	𝑥	𝑥	PRON
cana-3491	68	8	=	=	SYM
cana-3491	68	9	0	0	NUM
cana-3491	68	10	definition	definition	NOUN
cana-3491	68	11	2.2	2.2	NUM
cana-3491	68	12	(	(	PUNCT
cana-3491	68	13	𝑝-adic	𝑝-adic	ADJ
cana-3491	68	14	norm	norm	NOUN
cana-3491	68	15	)	)	PUNCT
cana-3491	68	16	.	.	PUNCT
cana-3491	69	1	let	let	VERB
cana-3491	69	2	𝑝	𝑝	PART
cana-3491	69	3	be	be	AUX
cana-3491	69	4	a	a	DET
cana-3491	69	5	prime	prime	NOUN
cana-3491	69	6	and	and	CCONJ
cana-3491	69	7	𝛼	𝛼	ADP
cana-3491	69	8	∈	∈	NOUN
cana-3491	69	9	𝑄	𝑄	PROPN
cana-3491	69	10	then	then	ADV
cana-3491	69	11	𝑝-adic	𝑝-adic	PROPN
cana-3491	69	12	norm	norm	NOUN
cana-3491	69	13	is	be	AUX
cana-3491	69	14	given	give	VERB
cana-3491	69	15	as	as	ADP
cana-3491	69	16	|𝛼|𝑝	|𝛼|𝑝	PROPN
cana-3491	69	17	=	=	PROPN
cana-3491	69	18	{	{	PUNCT
cana-3491	69	19	𝑝	𝑝	PROPN
cana-3491	69	20	−𝑣𝑝(𝛼	−𝑣𝑝(𝛼	PROPN
cana-3491	69	21	)	)	PUNCT
cana-3491	69	22	if	if	SCONJ
cana-3491	69	23	𝛼	𝛼	PRON
cana-3491	69	24	≠	≠	PROPN
cana-3491	69	25	0	0	NUM
cana-3491	69	26	0	0	NUM
cana-3491	70	1	if	if	SCONJ
cana-3491	70	2	𝛼	𝛼	X
cana-3491	70	3	=	=	SYM
cana-3491	70	4	0	0	NUM
cana-3491	70	5	𝑝-adic	𝑝-adic	ADJ
cana-3491	70	6	norm	norm	NOUN
cana-3491	71	1	|𝑥|𝑝	|𝑥|𝑝	PROPN
cana-3491	71	2	is	be	AUX
cana-3491	71	3	non	non	ADJ
cana-3491	71	4	-	-	ADJ
cana-3491	71	5	archimedean	archimedean	ADJ
cana-3491	71	6	norm	norm	NOUN
cana-3491	71	7	of	of	ADP
cana-3491	71	8	𝑥	𝑥	NOUN
cana-3491	71	9	on	on	ADP
cana-3491	71	10	𝑄.	𝑄.	PROPN
cana-3491	71	11	definition	definition	NOUN
cana-3491	71	12	2.3	2.3	NUM
cana-3491	71	13	(	(	PUNCT
cana-3491	71	14	𝑝-adic	𝑝-adic	ADJ
cana-3491	71	15	numbers	number	NOUN
cana-3491	71	16	)	)	PUNCT
cana-3491	71	17	.	.	PUNCT
cana-3491	72	1	for	for	ADP
cana-3491	72	2	any	any	DET
cana-3491	72	3	fixed	fix	VERB
cana-3491	72	4	prime	prime	NOUN
cana-3491	72	5	𝑝.	𝑝.	NOUN
cana-3491	72	6	the	the	DET
cana-3491	72	7	completion	completion	NOUN
cana-3491	72	8	of	of	ADP
cana-3491	72	9	𝑄	𝑄	PRON
cana-3491	72	10	with	with	ADP
cana-3491	72	11	respect	respect	NOUN
cana-3491	72	12	to	to	ADP
cana-3491	72	13	𝑝-adic	𝑝-adic	ADJ
cana-3491	72	14	norm	norm	NOUN
cana-3491	72	15	|	|	ADV
cana-3491	72	16	|𝑝	|𝑝	NOUN
cana-3491	72	17	is	be	AUX
cana-3491	72	18	denoted	denote	VERB
cana-3491	72	19	as	as	ADP
cana-3491	72	20	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	72	21	which	which	PRON
cana-3491	72	22	is	be	AUX
cana-3491	72	23	called	call	VERB
cana-3491	72	24	the	the	DET
cana-3491	72	25	field	field	NOUN
cana-3491	72	26	of	of	ADP
cana-3491	72	27	𝑝-adic	𝑝-adic	ADJ
cana-3491	72	28	numbers	number	NOUN
cana-3491	72	29	.	.	PUNCT
cana-3491	73	1	we	we	PRON
cana-3491	73	2	have	have	VERB
cana-3491	73	3	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	73	4	as	as	ADP
cana-3491	73	5	a	a	DET
cana-3491	73	6	field	field	NOUN
cana-3491	73	7	of	of	ADP
cana-3491	73	8	characteristic	characteristic	ADJ
cana-3491	73	9	0	0	NUM
cana-3491	73	10	.	.	PUNCT
cana-3491	74	1	proposition	proposition	NOUN
cana-3491	74	2	1	1	NUM
cana-3491	74	3	.	.	PUNCT
cana-3491	75	1	if	if	SCONJ
cana-3491	75	2	𝛼	𝛼	PRON
cana-3491	75	3	∈	∈	NOUN
cana-3491	76	1	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	76	2	then	then	ADV
cana-3491	76	3	there	there	PRON
cana-3491	76	4	exists	exist	VERB
cana-3491	76	5	a	a	DET
cana-3491	76	6	unique	unique	ADJ
cana-3491	76	7	sequence	sequence	NOUN
cana-3491	76	8	of	of	ADP
cana-3491	76	9	integers	integer	NOUN
cana-3491	76	10	𝛼𝑖	𝛼𝑖	NOUN
cana-3491	76	11	’s	’s	NOUN
cana-3491	76	12	with	with	ADP
cana-3491	76	13	0	0	NUM
cana-3491	76	14	≤	≤	NOUN
cana-3491	76	15	𝑥𝑖	𝑥𝑖	ADP
cana-3491	76	16	≤	≤	NUM
cana-3491	76	17	𝑝	𝑝	ADV
cana-3491	76	18	−	−	PROPN
cana-3491	76	19	1	1	NUM
cana-3491	76	20	and	and	CCONJ
cana-3491	76	21	𝑥𝑖	𝑥𝑖	X
cana-3491	76	22	=	=	NOUN
cana-3491	76	23	0	0	NUM
cana-3491	76	24	for	for	ADP
cana-3491	76	25	𝑖	𝑖	PRON
cana-3491	76	26	sufficiently	sufficiently	ADV
cana-3491	76	27	negative	negative	ADJ
cana-3491	76	28	such	such	ADJ
cana-3491	77	1	that	that	SCONJ
cana-3491	77	2	𝛼	𝛼	X
cana-3491	77	3	=	=	PUNCT
cana-3491	77	4	∑	∑	PUNCT
cana-3491	77	5	𝛼𝑖	𝛼𝑖	PROPN
cana-3491	77	6	∞	∞	PROPN
cana-3491	77	7	𝑖=−∞	𝑖=−∞	PUNCT
cana-3491	77	8	𝑝𝑖	𝑝𝑖	AUX
cana-3491	77	9	note	note	VERB
cana-3491	77	10	1	1	NUM
cana-3491	77	11	.	.	PUNCT
cana-3491	78	1	the	the	DET
cana-3491	78	2	partial	partial	ADJ
cana-3491	78	3	sums	sum	NOUN
cana-3491	78	4	of	of	ADP
cana-3491	78	5	the	the	DET
cana-3491	78	6	series	series	NOUN
cana-3491	78	7	𝛼	𝛼	PROPN
cana-3491	78	8	=	=	PUNCT
cana-3491	78	9	∑	∑	PUNCT
cana-3491	78	10	𝛼𝑖	𝛼𝑖	NUM
cana-3491	78	11	∞	∞	PROPN
cana-3491	78	12	𝑖=−∞	𝑖=−∞	PUNCT
cana-3491	78	13	𝑝𝑖	𝑝𝑖	NOUN
cana-3491	78	14	form	form	VERB
cana-3491	78	15	a	a	DET
cana-3491	78	16	cauchy	cauchy	ADJ
cana-3491	78	17	sequence	sequence	NOUN
cana-3491	78	18	and	and	CCONJ
cana-3491	78	19	𝑥	𝑥	NOUN
cana-3491	78	20	is	be	AUX
cana-3491	78	21	the	the	DET
cana-3491	78	22	limit	limit	NOUN
cana-3491	78	23	of	of	ADP
cana-3491	78	24	this	this	DET
cana-3491	78	25	sequence	sequence	NOUN
cana-3491	78	26	.	.	PUNCT
cana-3491	79	1	remark	remark	NOUN
cana-3491	79	2	3	3	NUM
cana-3491	79	3	.	.	PUNCT
cana-3491	80	1	every	every	DET
cana-3491	80	2	𝑥	𝑥	PRON
cana-3491	80	3	∈	∈	PROPN
cana-3491	80	4	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	80	5	has	have	VERB
cana-3491	80	6	unique	unique	ADJ
cana-3491	80	7	representation	representation	NOUN
cana-3491	80	8	depending	depend	VERB
cana-3491	80	9	on	on	ADP
cana-3491	80	10	its	its	PRON
cana-3491	80	11	𝑝-adic	𝑝-adic	ADJ
cana-3491	80	12	valuation	valuation	NOUN
cana-3491	80	13	|𝑥|𝑝	|𝑥|𝑝	ADP
cana-3491	80	14	i.e.	i.e.	ADV
cana-3491	80	15	,	,	PUNCT
cana-3491	80	16	either	either	CCONJ
cana-3491	80	17	|𝑥|𝑝	|𝑥|𝑝	PROPN
cana-3491	80	18	≥	≥	NOUN
cana-3491	80	19	1	1	NUM
cana-3491	80	20	or	or	CCONJ
cana-3491	80	21	|𝑥|𝑝	|𝑥|𝑝	PROPN
cana-3491	80	22	≤	≤	ADV
cana-3491	80	23	1	1	NUM
cana-3491	80	24	.	.	PUNCT
cana-3491	81	1	if	if	SCONJ
cana-3491	81	2	𝑥	𝑥	PRON
cana-3491	81	3	∈	∈	ADJ
cana-3491	81	4	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	81	5	with	with	ADP
cana-3491	81	6	|𝑥|𝑝	|𝑥|𝑝	PRON
cana-3491	81	7	≤	≤	NUM
cana-3491	81	8	1	1	NUM
cana-3491	81	9	,	,	PUNCT
cana-3491	81	10	we	we	PRON
cana-3491	81	11	can	can	AUX
cana-3491	81	12	represent	represent	VERB
cana-3491	81	13	𝑥	𝑥	PROPN
cana-3491	81	14	as	as	ADP
cana-3491	81	15	a	a	DET
cana-3491	81	16	sequence	sequence	NOUN
cana-3491	81	17	given	give	VERB
cana-3491	81	18	as	as	ADP
cana-3491	81	19	𝑥	𝑥	PRON
cana-3491	81	20	=	=	SYM
cana-3491	81	21	𝑥0	𝑥0	NOUN
cana-3491	81	22	+	+	CCONJ
cana-3491	81	23	𝑥1𝑝	𝑥1𝑝	PROPN
cana-3491	81	24	+	+	CCONJ
cana-3491	81	25	𝑥2𝑝	𝑥2𝑝	NUM
cana-3491	81	26	2	2	NUM
cana-3491	81	27	+	+	NOUN
cana-3491	81	28	.	.	PUNCT
cana-3491	81	29	.	.	PUNCT
cana-3491	81	30	.	.	PUNCT
cana-3491	82	1	+	+	VERB
cana-3491	82	2	𝑥𝑘−1𝑝	𝑥𝑘−1𝑝	PROPN
cana-3491	82	3	𝑘−1	𝑘−1	PROPN
cana-3491	82	4	=	=	PUNCT
cana-3491	82	5	∑𝑥𝑛	∑𝑥𝑛	NOUN
cana-3491	82	6	∞	∞	NUM
cana-3491	82	7	𝑛=0	𝑛=0	PROPN
cana-3491	82	8	𝑝𝑛	𝑝𝑛	ADP
cana-3491	82	9	where	where	SCONJ
cana-3491	82	10	𝑥𝑖	𝑥𝑖	X
cana-3491	82	11	∈	∈	PROPN
cana-3491	82	12	0,1,2	0,1,2	NOUN
cana-3491	82	13	,	,	PUNCT
cana-3491	82	14	.	.	PUNCT
cana-3491	82	15	.	.	PUNCT
cana-3491	82	16	.	.	PUNCT
cana-3491	83	1	𝑝	𝑝	X
cana-3491	83	2	−	−	NOUN
cana-3491	84	1	1	1	NUM
cana-3491	84	2	.	.	PUNCT
cana-3491	85	1	if	if	SCONJ
cana-3491	85	2	𝑥	𝑥	PRON
cana-3491	85	3	∈	∈	ADJ
cana-3491	86	1	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	86	2	with	with	ADP
cana-3491	86	3	|𝑥|𝑝	|𝑥|𝑝	PRON
cana-3491	86	4	≥	≥	NOUN
cana-3491	86	5	1	1	NUM
cana-3491	86	6	,	,	PUNCT
cana-3491	86	7	we	we	PRON
cana-3491	86	8	can	can	AUX
cana-3491	86	9	represent	represent	VERB
cana-3491	86	10	𝑥	𝑥	PROPN
cana-3491	86	11	as	as	ADP
cana-3491	86	12	a	a	DET
cana-3491	86	13	sequence	sequence	NOUN
cana-3491	86	14	given	give	VERB
cana-3491	86	15	as	as	ADP
cana-3491	86	16	𝑥	𝑥	PROPN
cana-3491	86	17	=	=	NOUN
cana-3491	86	18	.	.	PUNCT
cana-3491	86	19	.	.	PUNCT
cana-3491	86	20	.	.	PUNCT
cana-3491	87	1	+	+	ADV
cana-3491	87	2	𝑥−2𝑝	𝑥−2𝑝	NOUN
cana-3491	87	3	−2	−2	NOUN
cana-3491	87	4	+	+	CCONJ
cana-3491	87	5	𝑥−1𝑝	𝑥−1𝑝	NUM
cana-3491	87	6	−1	−1	NOUN
cana-3491	87	7	+	+	CCONJ
cana-3491	87	8	𝑥0	𝑥0	NOUN
cana-3491	87	9	+	+	CCONJ
cana-3491	87	10	𝑥1𝑝	𝑥1𝑝	PROPN
cana-3491	87	11	+	+	CCONJ
cana-3491	87	12	𝑥2𝑝	𝑥2𝑝	NUM
cana-3491	87	13	2	2	NUM
cana-3491	87	14	+	+	NOUN
cana-3491	87	15	.	.	PUNCT
cana-3491	87	16	.	.	PUNCT
cana-3491	87	17	.	.	PUNCT
cana-3491	88	1	+	+	VERB
cana-3491	88	2	𝑥𝑘−1𝑝	𝑥𝑘−1𝑝	PROPN
cana-3491	88	3	𝑘−1	𝑘−1	VERB
cana-3491	88	4	=	=	PUNCT
cana-3491	88	5	∑	∑	PUNCT
cana-3491	88	6	𝑥𝑛	𝑥𝑛	PROPN
cana-3491	88	7	∞	∞	NUM
cana-3491	88	8	𝑛=−∞	𝑛=−∞	NOUN
cana-3491	88	9	𝑝𝑛	𝑝𝑛	ADP
cana-3491	88	10	where	where	SCONJ
cana-3491	88	11	𝑥𝑖	𝑥𝑖	X
cana-3491	88	12	∈	∈	PROPN
cana-3491	88	13	0,1,2	0,1,2	NOUN
cana-3491	88	14	,	,	PUNCT
cana-3491	88	15	.	.	PUNCT
cana-3491	88	16	.	.	PUNCT
cana-3491	88	17	.	.	PUNCT
cana-3491	89	1	𝑝	𝑝	X
cana-3491	90	1	−	−	NOUN
cana-3491	90	2	1	1	NUM
cana-3491	90	3	.	.	PUNCT
cana-3491	90	4	communications	communication	NOUN
cana-3491	90	5	on	on	ADP
cana-3491	90	6	applied	apply	VERB
cana-3491	90	7	nonlinear	nonlinear	ADJ
cana-3491	90	8	analysis	analysis	NOUN
cana-3491	90	9	issn	issn	NOUN
cana-3491	90	10	:	:	PUNCT
cana-3491	90	11	1074	1074	NUM
cana-3491	90	12	-	-	PUNCT
cana-3491	90	13	133x	133x	NUM
cana-3491	90	14	vol	vol	NOUN
cana-3491	90	15	32	32	NUM
cana-3491	90	16	no	no	NOUN
cana-3491	90	17	.	.	PUNCT
cana-3491	91	1	7s	7	NOUN
cana-3491	91	2	(	(	PUNCT
cana-3491	91	3	2025	2025	NUM
cana-3491	91	4	)	)	PUNCT
cana-3491	91	5	860	860	NUM
cana-3491	91	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	91	7	remark	remark	NOUN
cana-3491	91	8	4	4	NUM
cana-3491	91	9	.	.	PUNCT
cana-3491	92	1	for	for	ADP
cana-3491	92	2	every	every	DET
cana-3491	92	3	𝑥	𝑥	PRON
cana-3491	92	4	∈	∈	NOUN
cana-3491	92	5	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	92	6	with	with	ADP
cana-3491	92	7	𝑥	𝑥	NOUN
cana-3491	92	8	=	=	PUNCT
cana-3491	92	9	∑	∑	PART
cana-3491	92	10	𝑥𝑛	𝑥𝑛	PROPN
cana-3491	92	11	∞	∞	PROPN
cana-3491	92	12	𝑛=−𝑚	𝑛=−𝑚	VERB
cana-3491	92	13	𝑝𝑛	𝑝𝑛	ADP
cana-3491	92	14	,	,	PUNCT
cana-3491	92	15	the	the	DET
cana-3491	92	16	canonical	canonical	ADJ
cana-3491	92	17	expression	expression	NOUN
cana-3491	92	18	of	of	ADP
cana-3491	92	19	𝑎	𝑎	NOUN
cana-3491	92	20	is	be	AUX
cana-3491	92	21	given	give	VERB
cana-3491	92	22	as	as	ADP
cana-3491	92	23	𝑥	𝑥	NOUN
cana-3491	92	24	=	=	NOUN
cana-3491	92	25	.	.	PUNCT
cana-3491	92	26	.	.	PUNCT
cana-3491	92	27	.	.	PUNCT
cana-3491	93	1	𝑥𝑛	𝑥𝑛	PROPN
cana-3491	93	2	.	.	PUNCT
cana-3491	93	3	.	.	PUNCT
cana-3491	93	4	.	.	PUNCT
cana-3491	94	1	𝑥2𝑥1𝑥0	𝑥2𝑥1𝑥0	AUX
cana-3491	94	2	.	.	PUNCT
cana-3491	94	3	𝑥−1𝑥−2𝑥−3	𝑥−1𝑥−2𝑥−3	VERB
cana-3491	94	4	.	.	PUNCT
cana-3491	94	5	.	.	PUNCT
cana-3491	94	6	.	.	PUNCT
cana-3491	95	1	𝑥−𝑚	𝑥−𝑚	NOUN
cana-3491	95	2	definition	definition	NOUN
cana-3491	95	3	2.4	2.4	NUM
cana-3491	95	4	(	(	PUNCT
cana-3491	95	5	𝑝-adic	𝑝-adic	ADJ
cana-3491	95	6	integers	integer	NOUN
cana-3491	95	7	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	95	8	)	)	PUNCT
cana-3491	95	9	.	.	PUNCT
cana-3491	96	1	a	a	DET
cana-3491	96	2	𝑝-adic	𝑝-adic	ADJ
cana-3491	96	3	number	number	NOUN
cana-3491	96	4	𝑥	𝑥	X
cana-3491	96	5	∈	∈	PROPN
cana-3491	97	1	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	97	2	is	be	AUX
cana-3491	97	3	said	say	VERB
cana-3491	97	4	to	to	PART
cana-3491	97	5	be	be	AUX
cana-3491	97	6	𝑝-adic	𝑝-adic	ADJ
cana-3491	97	7	integer	integer	NOUN
cana-3491	97	8	if	if	SCONJ
cana-3491	97	9	its	its	PRON
cana-3491	97	10	canonical	canonical	ADJ
cana-3491	97	11	expression	expression	NOUN
cana-3491	97	12	contains	contain	VERB
cana-3491	97	13	only	only	ADV
cana-3491	97	14	non	non	ADJ
cana-3491	97	15	-	-	ADJ
cana-3491	97	16	negative	negative	ADJ
cana-3491	97	17	powers	power	NOUN
cana-3491	97	18	of	of	ADP
cana-3491	97	19	𝑝.	𝑝.	PROPN
cana-3491	97	20	set	set	PROPN
cana-3491	97	21	of	of	ADP
cana-3491	97	22	𝑝-adic	𝑝-adic	ADJ
cana-3491	97	23	integers	integer	NOUN
cana-3491	97	24	is	be	AUX
cana-3491	97	25	denoted	denote	VERB
cana-3491	97	26	by	by	ADP
cana-3491	97	27	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	97	28	which	which	PRON
cana-3491	97	29	is	be	AUX
cana-3491	97	30	a	a	DET
cana-3491	97	31	sub	sub	NOUN
cana-3491	97	32	-	-	NOUN
cana-3491	97	33	ring	ring	NOUN
cana-3491	97	34	of	of	ADP
cana-3491	97	35	𝑄𝑝.	𝑄𝑝.	PROPN
cana-3491	97	36	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	97	37	=	=	PUNCT
cana-3491	97	38	{	{	PUNCT
cana-3491	97	39	𝑥	𝑥	X
cana-3491	97	40	∈	∈	PROPN
cana-3491	98	1	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	98	2	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-3491	98	3	|𝑥|	|𝑥|	ADJ
cana-3491	98	4	≤	≤	NUM
cana-3491	98	5	1	1	NUM
cana-3491	98	6	}	}	PUNCT
cana-3491	98	7	=	=	PRON
cana-3491	98	8	{	{	PUNCT
cana-3491	98	9	𝑥	𝑥	X
cana-3491	98	10	∈	∈	PROPN
cana-3491	99	1	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	99	2	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-3491	99	3	𝑣𝑝(𝑥	𝑣𝑝(𝑥	ADP
cana-3491	99	4	)	)	PUNCT
cana-3491	99	5	≥	≥	NOUN
cana-3491	99	6	0	0	NUM
cana-3491	99	7	}	}	PUNCT
cana-3491	99	8	one	one	NOUN
cana-3491	99	9	can	can	AUX
cana-3491	99	10	also	also	ADV
cana-3491	99	11	perform	perform	VERB
cana-3491	99	12	arithmetic	arithmetic	ADJ
cana-3491	99	13	operations	operation	NOUN
cana-3491	99	14	like	like	ADP
cana-3491	99	15	addition	addition	NOUN
cana-3491	99	16	,	,	PUNCT
cana-3491	99	17	subtraction	subtraction	NOUN
cana-3491	99	18	,	,	PUNCT
cana-3491	99	19	multiplication	multiplication	NOUN
cana-3491	99	20	and	and	CCONJ
cana-3491	99	21	division	division	NOUN
cana-3491	99	22	of	of	ADP
cana-3491	99	23	any	any	DET
cana-3491	99	24	two	two	NUM
cana-3491	99	25	𝑝-adic	𝑝-adic	ADJ
cana-3491	99	26	numbers	number	NOUN
cana-3491	99	27	.	.	PUNCT
cana-3491	100	1	2.1	2.1	NUM
cana-3491	100	2	arithmetic	arithmetic	ADJ
cana-3491	100	3	operations	operation	NOUN
cana-3491	100	4	on	on	ADP
cana-3491	100	5	𝒑-adic	𝒑-adic	ADJ
cana-3491	100	6	numbers	number	NOUN
cana-3491	100	7	.	.	PUNCT
cana-3491	101	1	every	every	DET
cana-3491	101	2	number	number	NOUN
cana-3491	101	3	𝛼	𝛼	NOUN
cana-3491	101	4	∈	∈	NOUN
cana-3491	101	5	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	101	6	for	for	ADP
cana-3491	101	7	𝑝	𝑝	PROPN
cana-3491	101	8	being	be	AUX
cana-3491	101	9	a	a	DET
cana-3491	101	10	prime	prime	NOUN
cana-3491	101	11	has	have	VERB
cana-3491	101	12	its	its	PRON
cana-3491	101	13	𝑝-adic	𝑝-adic	ADJ
cana-3491	101	14	expansion	expansion	NOUN
cana-3491	101	15	given	give	VERB
cana-3491	101	16	as	as	ADP
cana-3491	101	17	𝛼	𝛼	NOUN
cana-3491	101	18	=	=	NOUN
cana-3491	101	19	.	.	PUNCT
cana-3491	101	20	.	.	PUNCT
cana-3491	101	21	.	.	PUNCT
cana-3491	102	1	+	+	ADV
cana-3491	102	2	𝛼−2𝑝	𝛼−2𝑝	ADJ
cana-3491	102	3	−2	−2	NOUN
cana-3491	102	4	+	+	NUM
cana-3491	102	5	𝛼−1𝑝	𝛼−1𝑝	NOUN
cana-3491	102	6	−1	−1	NOUN
cana-3491	102	7	+	+	CCONJ
cana-3491	102	8	𝛼0	𝛼0	PROPN
cana-3491	102	9	+	+	CCONJ
cana-3491	102	10	𝛼1𝑝	𝛼1𝑝	PROPN
cana-3491	102	11	+	+	CCONJ
cana-3491	102	12	𝛼2𝑝	𝛼2𝑝	NUM
cana-3491	102	13	2	2	NUM
cana-3491	102	14	+	+	CCONJ
cana-3491	102	15	𝛼3𝑝	𝛼3𝑝	ADP
cana-3491	102	16	3	3	NUM
cana-3491	102	17	+	+	NOUN
cana-3491	102	18	.	.	PUNCT
cana-3491	102	19	..	..	PUNCT
cana-3491	103	1	while	while	SCONJ
cana-3491	103	2	some	some	PRON
cana-3491	103	3	are	be	AUX
cana-3491	103	4	finite	finite	ADJ
cana-3491	103	5	expansions	expansion	NOUN
cana-3491	103	6	including	include	VERB
cana-3491	103	7	only	only	ADV
cana-3491	103	8	positive	positive	ADJ
cana-3491	103	9	powers	power	NOUN
cana-3491	103	10	of	of	ADP
cana-3491	103	11	𝑝	𝑝	NOUN
cana-3491	103	12	in	in	ADP
cana-3491	103	13	its	its	PRON
cana-3491	103	14	expansion	expansion	NOUN
cana-3491	103	15	.	.	PUNCT
cana-3491	103	16	example	example	NOUN
cana-3491	104	1	2.1.1	2.1.1	NUM
cana-3491	104	2	.	.	NOUN
cana-3491	104	3	1	1	NUM
cana-3491	104	4	.	.	NOUN
cana-3491	104	5	320	320	NUM
cana-3491	104	6	=	=	SYM
cana-3491	104	7	5	5	NUM
cana-3491	104	8	+	+	CCONJ
cana-3491	104	9	3	3	NUM
cana-3491	104	10	×	×	NOUN
cana-3491	104	11	7	7	NUM
cana-3491	104	12	+	+	CCONJ
cana-3491	104	13	6	6	NUM
cana-3491	104	14	×	×	NOUN
cana-3491	104	15	72	72	NUM
cana-3491	104	16	=	=	SYM
cana-3491	104	17	635	635	NUM
cana-3491	104	18	in	in	ADP
cana-3491	104	19	𝑄7	𝑄7	NOUN
cana-3491	104	20	2	2	NUM
cana-3491	104	21	.	.	SYM
cana-3491	104	22	108	108	NUM
cana-3491	104	23	=	=	SYM
cana-3491	104	24	3	3	NUM
cana-3491	104	25	+	+	SYM
cana-3491	104	26	1	1	NUM
cana-3491	104	27	×	×	NOUN
cana-3491	104	28	7	7	NUM
cana-3491	104	29	+	+	CCONJ
cana-3491	104	30	2	2	NUM
cana-3491	104	31	×	×	NOUN
cana-3491	104	32	72	72	NUM
cana-3491	104	33	=	=	SYM
cana-3491	104	34	213	213	NUM
cana-3491	104	35	in	in	ADP
cana-3491	104	36	𝑄7	𝑄7	NOUN
cana-3491	104	37	3	3	NUM
cana-3491	104	38	.	.	NOUN
cana-3491	104	39	1	1	NUM
cana-3491	104	40	2	2	NUM
cana-3491	104	41	=	=	SYM
cana-3491	104	42	3	3	NUM
cana-3491	104	43	+	+	NUM
cana-3491	104	44	2	2	NUM
cana-3491	104	45	×	×	NOUN
cana-3491	104	46	5	5	NUM
cana-3491	104	47	+	+	CCONJ
cana-3491	104	48	2	2	NUM
cana-3491	104	49	×	×	NOUN
cana-3491	104	50	52	52	NUM
cana-3491	104	51	+	+	NOUN
cana-3491	104	52	.	.	PUNCT
cana-3491	104	53	.	.	PUNCT
cana-3491	104	54	.	.	PUNCT
cana-3491	105	1	=	=	PUNCT
cana-3491	105	2	...	...	PUNCT
cana-3491	105	3	2223	2223	NUM
cana-3491	105	4	in	in	ADP
cana-3491	105	5	𝑄5	𝑄5	PROPN
cana-3491	105	6	also	also	ADV
cana-3491	105	7	,	,	PUNCT
cana-3491	105	8	negative	negative	ADJ
cana-3491	105	9	of	of	ADP
cana-3491	105	10	𝑥	𝑥	PRON
cana-3491	105	11	∈	∈	PROPN
cana-3491	106	1	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	106	2	is	be	AUX
cana-3491	106	3	given	give	VERB
cana-3491	106	4	as	as	ADP
cana-3491	106	5	−𝑥	−𝑥	NOUN
cana-3491	106	6	=	=	SYM
cana-3491	106	7	𝑥	𝑥	PROPN
cana-3491	106	8	×	×	NOUN
cana-3491	106	9	(	(	PUNCT
cana-3491	106	10	−1	−1	NOUN
cana-3491	106	11	)	)	PUNCT
cana-3491	106	12	with	with	ADP
cana-3491	106	13	its	its	PRON
cana-3491	106	14	𝑝-adic	𝑝-adic	ADJ
cana-3491	106	15	expansion	expansion	NOUN
cana-3491	106	16	given	give	VERB
cana-3491	106	17	as	as	ADP
cana-3491	106	18	−1	−1	NOUN
cana-3491	106	19	=	=	PUNCT
cana-3491	106	20	(	(	PUNCT
cana-3491	106	21	𝑝	𝑝	NOUN
cana-3491	106	22	−	−	PROPN
cana-3491	106	23	1	1	NUM
cana-3491	106	24	)	)	PUNCT
cana-3491	107	1	+	+	CCONJ
cana-3491	107	2	(	(	PUNCT
cana-3491	107	3	𝑝	𝑝	NOUN
cana-3491	107	4	−	−	PROPN
cana-3491	107	5	1	1	NUM
cana-3491	107	6	)	)	PUNCT
cana-3491	107	7	×	×	PROPN
cana-3491	107	8	𝑝	𝑝	PROPN
cana-3491	107	9	+	+	CCONJ
cana-3491	107	10	(	(	PUNCT
cana-3491	107	11	𝑝	𝑝	NOUN
cana-3491	107	12	−	−	PROPN
cana-3491	107	13	1	1	NUM
cana-3491	107	14	)	)	PUNCT
cana-3491	107	15	×	×	NOUN
cana-3491	107	16	𝑝2	𝑝2	NOUN
cana-3491	107	17	+	+	PROPN
cana-3491	107	18	.	.	PROPN
cana-3491	107	19	..	..	PUNCT
cana-3491	108	1	i.e.	i.e.	X
cana-3491	108	2	,	,	PUNCT
cana-3491	108	3	−1	−1	NOUN
cana-3491	108	4	=	=	SYM
cana-3491	108	5	4	4	NUM
cana-3491	108	6	+	+	NUM
cana-3491	108	7	4.5	4.5	NUM
cana-3491	108	8	+	+	NUM
cana-3491	108	9	4.52	4.52	NUM
cana-3491	108	10	+	+	NOUN
cana-3491	108	11	.	.	PUNCT
cana-3491	108	12	..	..	PUNCT
cana-3491	109	1	in	in	ADP
cana-3491	109	2	𝑄5	𝑄5	PROPN
cana-3491	109	3	arithmetical	arithmetical	ADJ
cana-3491	109	4	operations	operation	NOUN
cana-3491	109	5	in	in	ADP
cana-3491	109	6	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	109	7	extend	extend	VERB
cana-3491	109	8	ordinary	ordinary	ADJ
cana-3491	109	9	arithmetic	arithmetic	ADJ
cana-3491	109	10	operations	operation	NOUN
cana-3491	109	11	on	on	ADP
cana-3491	109	12	natural	natural	ADJ
cana-3491	109	13	numbers	number	NOUN
cana-3491	109	14	𝑁.	𝑁.	PROPN
cana-3491	109	15	𝑝-adic	𝑝-adic	ADJ
cana-3491	109	16	addition	addition	NOUN
cana-3491	109	17	and	and	CCONJ
cana-3491	109	18	multiplication	multiplication	NOUN
cana-3491	109	19	are	be	AUX
cana-3491	109	20	performed	perform	VERB
cana-3491	109	21	from	from	ADP
cana-3491	109	22	right	right	ADJ
cana-3491	109	23	to	to	ADP
cana-3491	109	24	left	leave	VERB
cana-3491	109	25	with	with	ADP
cana-3491	109	26	a	a	DET
cana-3491	109	27	carry	carry	NOUN
cana-3491	109	28	.	.	PUNCT
cana-3491	110	1	also	also	ADV
cana-3491	110	2	,	,	PUNCT
cana-3491	110	3	𝑝-adic	𝑝-adic	ADJ
cana-3491	110	4	division	division	NOUN
cana-3491	110	5	will	will	AUX
cana-3491	110	6	be	be	AUX
cana-3491	110	7	performed	perform	VERB
cana-3491	110	8	from	from	ADP
cana-3491	110	9	right	right	ADJ
cana-3491	110	10	to	to	ADP
cana-3491	110	11	left	left	ADJ
cana-3491	110	12	but	but	CCONJ
cana-3491	110	13	different	different	ADJ
cana-3491	110	14	from	from	ADP
cana-3491	110	15	long	long	ADJ
cana-3491	110	16	division	division	NOUN
cana-3491	110	17	as	as	ADP
cana-3491	110	18	in	in	ADP
cana-3491	110	19	𝑁.	𝑁.	PROPN
cana-3491	110	20	example	example	NOUN
cana-3491	110	21	2.1.2	2.1.2	NUM
cana-3491	110	22	.	.	PUNCT
cana-3491	110	23	addition	addition	NOUN
cana-3491	110	24	,	,	PUNCT
cana-3491	110	25	multiplication	multiplication	NOUN
cana-3491	110	26	and	and	CCONJ
cana-3491	110	27	division	division	NOUN
cana-3491	110	28	in	in	ADP
cana-3491	110	29	7	7	NUM
cana-3491	110	30	-	-	PUNCT
cana-3491	110	31	adic	adic	ADJ
cana-3491	110	32	field	field	NOUN
cana-3491	110	33	𝑄7	𝑄7	NOUN
cana-3491	110	34	are	be	AUX
cana-3491	110	35	given	give	VERB
cana-3491	110	36	below	below	ADP
cana-3491	110	37	for	for	ADP
cana-3491	110	38	320	320	NUM
cana-3491	110	39	and	and	CCONJ
cana-3491	110	40	108	108	NUM
cana-3491	110	41	expanded	expand	VERB
cana-3491	110	42	in	in	ADP
cana-3491	110	43	𝑄7	𝑄7	NOUN
cana-3491	110	44	in	in	ADP
cana-3491	110	45	above	above	ADP
cana-3491	110	46	examples	example	NOUN
cana-3491	110	47	.	.	PUNCT
cana-3491	111	1	320	320	NUM
cana-3491	111	2	+	+	SYM
cana-3491	111	3	108	108	NUM
cana-3491	111	4	=	=	SYM
cana-3491	111	5	1151	1151	NUM
cana-3491	111	6	given	give	VERB
cana-3491	111	7	as	as	ADP
cana-3491	111	8	5	5	NUM
cana-3491	111	9	+	+	SYM
cana-3491	111	10	3	3	NUM
cana-3491	111	11	×	×	NOUN
cana-3491	111	12	7	7	NUM
cana-3491	111	13	+	+	CCONJ
cana-3491	111	14	6	6	NUM
cana-3491	111	15	×	×	NOUN
cana-3491	111	16	72	72	NUM
cana-3491	111	17	+	+	CCONJ
cana-3491	111	18	3	3	NUM
cana-3491	111	19	+	+	SYM
cana-3491	111	20	1	1	NUM
cana-3491	111	21	×	×	NOUN
cana-3491	111	22	7	7	NUM
cana-3491	111	23	+	+	CCONJ
cana-3491	111	24	2	2	NUM
cana-3491	111	25	×	×	NOUN
cana-3491	111	26	72	72	NUM
cana-3491	111	27	−−	−−	NOUN
cana-3491	111	28	−−	−−	PROPN
cana-3491	111	29	−−	−−	PROPN
cana-3491	111	30	−−	−−	NOUN
cana-3491	111	31	−−−	−−−	X
cana-3491	112	1	−−	−−	PROPN
cana-3491	112	2	−−	−−	PROPN
cana-3491	112	3	−−	−−	NOUN
cana-3491	112	4	1	1	NUM
cana-3491	112	5	+	+	SYM
cana-3491	112	6	5	5	NUM
cana-3491	112	7	×	×	NOUN
cana-3491	112	8	7	7	NUM
cana-3491	112	9	+	+	CCONJ
cana-3491	112	10	1	1	NUM
cana-3491	112	11	×	×	NOUN
cana-3491	112	12	72	72	NUM
cana-3491	112	13	+	+	CCONJ
cana-3491	112	14	1	1	NUM
cana-3491	112	15	×	×	NOUN
cana-3491	112	16	73	73	NUM
cana-3491	112	17	−−	−−	PROPN
cana-3491	112	18	−−	−−	PROPN
cana-3491	112	19	−−	−−	PROPN
cana-3491	112	20	−−	−−	NOUN
cana-3491	112	21	−−−	−−−	X
cana-3491	113	1	−−	−−	PROPN
cana-3491	113	2	−−	−−	PROPN
cana-3491	113	3	−−	−−	NOUN
cana-3491	113	4	320	320	NUM
cana-3491	113	5	−	−	NOUN
cana-3491	113	6	108	108	NUM
cana-3491	113	7	=	=	SYM
cana-3491	113	8	422	422	NUM
cana-3491	113	9	given	give	VERB
cana-3491	113	10	as	as	ADP
cana-3491	113	11	5	5	NUM
cana-3491	113	12	+	+	SYM
cana-3491	113	13	3	3	NUM
cana-3491	113	14	×	×	NOUN
cana-3491	113	15	7	7	NUM
cana-3491	113	16	+	+	CCONJ
cana-3491	113	17	6	6	NUM
cana-3491	113	18	×	×	NOUN
cana-3491	113	19	72	72	NUM
cana-3491	113	20	−	−	NOUN
cana-3491	113	21	3	3	NUM
cana-3491	113	22	+	+	SYM
cana-3491	113	23	1	1	NUM
cana-3491	113	24	×	×	NOUN
cana-3491	113	25	7	7	NUM
cana-3491	113	26	+	+	CCONJ
cana-3491	113	27	2	2	NUM
cana-3491	113	28	×	×	NOUN
cana-3491	113	29	72	72	NUM
cana-3491	113	30	−−	−−	NOUN
cana-3491	113	31	−−	−−	PROPN
cana-3491	113	32	−−	−−	PROPN
cana-3491	113	33	−−	−−	NOUN
cana-3491	113	34	−−−	−−−	X
cana-3491	114	1	−−	−−	PROPN
cana-3491	114	2	−−	−−	PROPN
cana-3491	114	3	−−	−−	PROPN
cana-3491	114	4	communications	communication	NOUN
cana-3491	114	5	on	on	ADP
cana-3491	114	6	applied	apply	VERB
cana-3491	114	7	nonlinear	nonlinear	ADJ
cana-3491	114	8	analysis	analysis	NOUN
cana-3491	114	9	issn	issn	NOUN
cana-3491	114	10	:	:	PUNCT
cana-3491	114	11	1074	1074	NUM
cana-3491	114	12	-	-	PUNCT
cana-3491	114	13	133x	133x	NUM
cana-3491	114	14	vol	vol	NOUN
cana-3491	114	15	32	32	NUM
cana-3491	114	16	no	no	NOUN
cana-3491	114	17	.	.	PUNCT
cana-3491	115	1	7s	7	NOUN
cana-3491	115	2	(	(	PUNCT
cana-3491	115	3	2025	2025	NUM
cana-3491	115	4	)	)	PUNCT
cana-3491	115	5	861	861	NUM
cana-3491	115	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	115	7	2	2	NUM
cana-3491	115	8	+	+	CCONJ
cana-3491	115	9	2	2	NUM
cana-3491	115	10	×	×	NOUN
cana-3491	115	11	7	7	NUM
cana-3491	115	12	+	+	CCONJ
cana-3491	115	13	4	4	NUM
cana-3491	115	14	×	×	NOUN
cana-3491	115	15	72	72	NUM
cana-3491	115	16	−−	−−	NOUN
cana-3491	115	17	−−	−−	PROPN
cana-3491	115	18	−−	−−	PROPN
cana-3491	115	19	−−	−−	NOUN
cana-3491	115	20	−−−	−−−	X
cana-3491	116	1	−−	−−	PROPN
cana-3491	116	2	−−	−−	PROPN
cana-3491	116	3	−−	−−	PROPN
cana-3491	116	4	also	also	ADV
cana-3491	116	5	,	,	PUNCT
cana-3491	116	6	320	320	NUM
cana-3491	116	7	×	×	NOUN
cana-3491	116	8	108	108	NUM
cana-3491	116	9	=	=	SYM
cana-3491	116	10	202521	202521	NUM
cana-3491	116	11	given	give	VERB
cana-3491	116	12	as	as	ADP
cana-3491	116	13	5	5	NUM
cana-3491	116	14	+	+	SYM
cana-3491	116	15	3	3	NUM
cana-3491	116	16	×	×	NOUN
cana-3491	116	17	7	7	NUM
cana-3491	116	18	+	+	CCONJ
cana-3491	116	19	6	6	NUM
cana-3491	116	20	×	×	NOUN
cana-3491	116	21	72	72	NUM
cana-3491	116	22	×	×	NOUN
cana-3491	116	23	3	3	NUM
cana-3491	116	24	+	+	SYM
cana-3491	116	25	1	1	NUM
cana-3491	116	26	×	×	NOUN
cana-3491	116	27	7	7	NUM
cana-3491	116	28	+	+	CCONJ
cana-3491	116	29	2	2	NUM
cana-3491	116	30	×	×	NOUN
cana-3491	116	31	72	72	NUM
cana-3491	116	32	−−	−−	NOUN
cana-3491	116	33	−−	−−	PROPN
cana-3491	116	34	−−	−−	PROPN
cana-3491	116	35	−−	−−	PROPN
cana-3491	116	36	−−	−−	NOUN
cana-3491	116	37	−−−	−−−	X
cana-3491	117	1	−−	−−	PROPN
cana-3491	117	2	−−	−−	PROPN
cana-3491	117	3	−−	−−	PROPN
cana-3491	117	4	−−	−−	NOUN
cana-3491	117	5	1	1	NUM
cana-3491	117	6	+	+	CCONJ
cana-3491	117	7	4	4	NUM
cana-3491	117	8	×	×	NOUN
cana-3491	117	9	7	7	NUM
cana-3491	117	10	+	+	CCONJ
cana-3491	117	11	5	5	NUM
cana-3491	117	12	×	×	NOUN
cana-3491	117	13	72	72	NUM
cana-3491	117	14	+	+	CCONJ
cana-3491	117	15	2	2	NUM
cana-3491	117	16	×	×	NOUN
cana-3491	117	17	73	73	NUM
cana-3491	117	18	5	5	NUM
cana-3491	117	19	×	×	NOUN
cana-3491	117	20	7	7	NUM
cana-3491	117	21	+	+	CCONJ
cana-3491	117	22	3	3	NUM
cana-3491	117	23	×	×	NOUN
cana-3491	117	24	72	72	NUM
cana-3491	117	25	+	+	CCONJ
cana-3491	117	26	6	6	NUM
cana-3491	117	27	×	×	NOUN
cana-3491	117	28	73	73	NUM
cana-3491	117	29	3	3	NUM
cana-3491	117	30	×	×	NOUN
cana-3491	117	31	72	72	NUM
cana-3491	117	32	+	+	CCONJ
cana-3491	117	33	0	0	NUM
cana-3491	117	34	×	×	NOUN
cana-3491	117	35	73	73	NUM
cana-3491	117	36	+	+	CCONJ
cana-3491	117	37	6	6	NUM
cana-3491	117	38	×	×	NOUN
cana-3491	117	39	74	74	NUM
cana-3491	117	40	+	+	CCONJ
cana-3491	117	41	1	1	NUM
cana-3491	117	42	×	×	NOUN
cana-3491	117	43	75	75	NUM
cana-3491	117	44	−	−	NOUN
cana-3491	117	45	−−	−−	NOUN
cana-3491	117	46	−−	−−	NOUN
cana-3491	117	47	−−−	−−−	X
cana-3491	117	48	−−	−−	PROPN
cana-3491	117	49	−−	−−	PROPN
cana-3491	117	50	−−	−−	PROPN
cana-3491	117	51	−−	−−	PROPN
cana-3491	117	52	−−	−−	NOUN
cana-3491	117	53	−−−	−−−	X
cana-3491	118	1	−−	−−	NOUN
cana-3491	118	2	−	−	NOUN
cana-3491	118	3	1	1	NUM
cana-3491	119	1	+	+	SYM
cana-3491	119	2	2	2	NUM
cana-3491	119	3	×	×	NOUN
cana-3491	119	4	7	7	NUM
cana-3491	119	5	+	+	CCONJ
cana-3491	119	6	5	5	NUM
cana-3491	119	7	×	×	NOUN
cana-3491	119	8	72	72	NUM
cana-3491	119	9	+	+	CCONJ
cana-3491	119	10	2	2	NUM
cana-3491	119	11	×	×	NOUN
cana-3491	119	12	73	73	NUM
cana-3491	119	13	+	+	CCONJ
cana-3491	119	14	0	0	NUM
cana-3491	119	15	×	×	NOUN
cana-3491	119	16	74	74	NUM
cana-3491	119	17	+	+	CCONJ
cana-3491	119	18	2	2	NUM
cana-3491	119	19	×	×	NOUN
cana-3491	119	20	75	75	NUM
cana-3491	119	21	−	−	NOUN
cana-3491	119	22	−−	−−	NOUN
cana-3491	119	23	−−	−−	NOUN
cana-3491	119	24	−−−	−−−	X
cana-3491	120	1	−−	−−	PROPN
cana-3491	120	2	−−	−−	PROPN
cana-3491	120	3	−−	−−	PROPN
cana-3491	120	4	−−	−−	PROPN
cana-3491	120	5	−−	−−	NOUN
cana-3491	120	6	−−−	−−−	X
cana-3491	121	1	−−	−−	NOUN
cana-3491	121	2	−	−	PROPN
cana-3491	121	3	also	also	ADV
cana-3491	121	4	,	,	PUNCT
cana-3491	121	5	302	302	NUM
cana-3491	121	6	108	108	NUM
cana-3491	121	7	=	=	SYM
cana-3491	121	8	....	....	PUNCT
cana-3491	121	9	244	244	NUM
cana-3491	121	10	3	3	NUM
cana-3491	121	11	+	+	SYM
cana-3491	121	12	1	1	NUM
cana-3491	121	13	×	×	NOUN
cana-3491	121	14	7	7	NUM
cana-3491	121	15	+	+	CCONJ
cana-3491	121	16	2	2	NUM
cana-3491	121	17	×	×	NOUN
cana-3491	121	18	72	72	NUM
cana-3491	121	19	)	)	PUNCT
cana-3491	121	20	5	5	NUM
cana-3491	122	1	+	+	CCONJ
cana-3491	122	2	3	3	NUM
cana-3491	122	3	×	×	NOUN
cana-3491	122	4	7	7	NUM
cana-3491	122	5	+	+	CCONJ
cana-3491	122	6	6	6	NUM
cana-3491	122	7	×	×	NOUN
cana-3491	122	8	72	72	NUM
cana-3491	122	9	(	(	PUNCT
cana-3491	122	10	4	4	NUM
cana-3491	122	11	+	+	SYM
cana-3491	122	12	4	4	NUM
cana-3491	122	13	×	×	NOUN
cana-3491	122	14	7	7	NUM
cana-3491	122	15	+	+	CCONJ
cana-3491	122	16	2	2	NUM
cana-3491	122	17	×	×	NOUN
cana-3491	122	18	72	72	NUM
cana-3491	122	19	5	5	NUM
cana-3491	122	20	+	+	SYM
cana-3491	122	21	5	5	NUM
cana-3491	122	22	×	×	NOUN
cana-3491	122	23	7	7	NUM
cana-3491	122	24	+	+	CCONJ
cana-3491	122	25	1	1	NUM
cana-3491	122	26	×	×	NOUN
cana-3491	122	27	72	72	NUM
cana-3491	122	28	+	+	CCONJ
cana-3491	122	29	1	1	NUM
cana-3491	122	30	×	×	NOUN
cana-3491	122	31	73	73	NUM
cana-3491	122	32	−−	−−	PROPN
cana-3491	122	33	−−	−−	PROPN
cana-3491	122	34	−−	−−	PROPN
cana-3491	122	35	−−	−−	PROPN
cana-3491	122	36	−−	−−	PROPN
cana-3491	122	37	−−	−−	NOUN
cana-3491	122	38	−−−	−−−	X
cana-3491	123	1	−−	−−	PROPN
cana-3491	123	2	−−	−−	PROPN
cana-3491	123	3	−−	−−	PROPN
cana-3491	123	4	−−	−−	PROPN
cana-3491	123	5	−−	−−	NOUN
cana-3491	123	6	5	5	NUM
cana-3491	123	7	×	×	NOUN
cana-3491	123	8	7	7	NUM
cana-3491	123	9	+	+	CCONJ
cana-3491	123	10	4	4	NUM
cana-3491	123	11	×	×	NOUN
cana-3491	123	12	72	72	NUM
cana-3491	123	13	+	+	CCONJ
cana-3491	123	14	6	6	NUM
cana-3491	123	15	×	×	NOUN
cana-3491	123	16	73	73	NUM
cana-3491	123	17	+	+	CCONJ
cana-3491	123	18	6	6	NUM
cana-3491	123	19	×	×	NOUN
cana-3491	123	20	74	74	NUM
cana-3491	123	21	+	+	ADJ
cana-3491	123	22	⋯	⋯	ADP
cana-3491	123	23	5	5	NUM
cana-3491	123	24	×	×	NOUN
cana-3491	123	25	7	7	NUM
cana-3491	123	26	+	+	CCONJ
cana-3491	123	27	5	5	NUM
cana-3491	123	28	×	×	NOUN
cana-3491	123	29	72	72	NUM
cana-3491	123	30	+	+	CCONJ
cana-3491	123	31	1	1	NUM
cana-3491	123	32	×	×	NOUN
cana-3491	123	33	73	73	NUM
cana-3491	123	34	+	+	CCONJ
cana-3491	123	35	1	1	NUM
cana-3491	123	36	×	×	NOUN
cana-3491	123	37	74	74	NUM
cana-3491	123	38	−	−	NOUN
cana-3491	123	39	−	−	PROPN
cana-3491	123	40	−	−	PROPN
cana-3491	123	41	−−	−−	PROPN
cana-3491	123	42	−−	−−	PROPN
cana-3491	123	43	−−	−−	PROPN
cana-3491	123	44	−−	−−	PROPN
cana-3491	123	45	−−	−−	NOUN
cana-3491	123	46	−−−	−−−	X
cana-3491	124	1	−−	−−	PROPN
cana-3491	124	2	−−	−−	PROPN
cana-3491	124	3	−−	−−	PROPN
cana-3491	124	4	−−	−−	PROPN
cana-3491	124	5	−−	−−	NOUN
cana-3491	124	6	−	−	PROPN
cana-3491	124	7	6	6	NUM
cana-3491	124	8	×	×	NOUN
cana-3491	124	9	72	72	NUM
cana-3491	124	10	+	+	CCONJ
cana-3491	124	11	4	4	NUM
cana-3491	124	12	×	×	NOUN
cana-3491	124	13	73	73	NUM
cana-3491	124	14	+	+	CCONJ
cana-3491	124	15	6	6	NUM
cana-3491	124	16	×	×	NOUN
cana-3491	124	17	74	74	NUM
cana-3491	124	18	+	+	ADJ
cana-3491	124	19	⋯	⋯	VERB
cana-3491	124	20	6	6	NUM
cana-3491	124	21	×	×	NOUN
cana-3491	124	22	72	72	NUM
cana-3491	124	23	+	+	CCONJ
cana-3491	124	24	2	2	NUM
cana-3491	124	25	×	×	NOUN
cana-3491	124	26	73	73	NUM
cana-3491	124	27	+	+	CCONJ
cana-3491	124	28	4	4	NUM
cana-3491	124	29	×	×	NOUN
cana-3491	124	30	74	74	NUM
cana-3491	124	31	−−	−−	PROPN
cana-3491	124	32	−−	−−	PROPN
cana-3491	124	33	−−	−−	PROPN
cana-3491	124	34	−−	−−	PROPN
cana-3491	124	35	−−	−−	PROPN
cana-3491	124	36	−−	−−	PROPN
cana-3491	124	37	−−	−−	NOUN
cana-3491	124	38	−−−	−−−	X
cana-3491	125	1	−−	−−	PROPN
cana-3491	125	2	−−	−−	PROPN
cana-3491	125	3	−−	−−	PROPN
cana-3491	125	4	−−	−−	PROPN
cana-3491	125	5	−−	−−	NOUN
cana-3491	125	6	−−−	−−−	AUX
cana-3491	125	7	−−	−−	NOUN
cana-3491	125	8	−	−	PROPN
cana-3491	125	9	2	2	NUM
cana-3491	125	10	×	×	NOUN
cana-3491	125	11	73	73	NUM
cana-3491	126	1	+	+	CCONJ
cana-3491	126	2	0	0	NUM
cana-3491	126	3	×	×	NOUN
cana-3491	126	4	74	74	NUM
cana-3491	127	1	+	+	CCONJ
cana-3491	127	2	6	6	NUM
cana-3491	127	3	×	×	NOUN
cana-3491	127	4	75	75	NUM
cana-3491	128	1	+	+	ADJ
cana-3491	128	2	⋯	⋯	PROPN
cana-3491	128	3	−−	−−	PROPN
cana-3491	128	4	−−	−−	PROPN
cana-3491	128	5	−−	−−	PROPN
cana-3491	128	6	−−	−−	PROPN
cana-3491	128	7	−−	−−	NOUN
cana-3491	128	8	−−−	−−−	X
cana-3491	129	1	−−	−−	PROPN
cana-3491	129	2	−−	−−	PROPN
cana-3491	129	3	−−	−−	PROPN
cana-3491	129	4	−−	−−	PROPN
cana-3491	129	5	−−	−−	NOUN
cana-3491	129	6	−−−	−−−	X
cana-3491	130	1	−−	−−	PROPN
cana-3491	130	2	−−	−−	PROPN
cana-3491	130	3	−−	−−	PROPN
cana-3491	130	4	−−	−−	NOUN
cana-3491	130	5	3	3	X
cana-3491	130	6	.	.	PUNCT
cana-3491	130	7	elliptic	elliptic	ADJ
cana-3491	130	8	curves	curve	NOUN
cana-3491	130	9	𝑬(𝑸𝒑	𝑬(𝑸𝒑	NUM
cana-3491	130	10	)	)	PUNCT
cana-3491	130	11	over	over	ADP
cana-3491	130	12	𝒑-adic	𝒑-adic	ADJ
cana-3491	130	13	field	field	NOUN
cana-3491	130	14	𝑸𝒑	𝑸𝒑	PROPN
cana-3491	130	15	and	and	CCONJ
cana-3491	130	16	points	point	VERB
cana-3491	130	17	in	in	ADP
cana-3491	130	18	𝑬(𝑸𝒑)(𝒎𝒐𝒅	𝑬(𝑸𝒑)(𝒎𝒐𝒅	NOUN
cana-3491	130	19	𝒑𝒏	𝒑𝒏	NOUN
cana-3491	130	20	)	)	PUNCT
cana-3491	131	1	for	for	ADP
cana-3491	131	2	𝒏	𝒏	PROPN
cana-3491	131	3	>	>	X
cana-3491	131	4	𝟏	𝟏	PROPN
cana-3491	131	5	in	in	ADP
cana-3491	131	6	the	the	DET
cana-3491	131	7	context	context	NOUN
cana-3491	131	8	of	of	ADP
cana-3491	131	9	studying	study	VERB
cana-3491	131	10	point	point	NOUN
cana-3491	131	11	addition	addition	NOUN
cana-3491	131	12	on	on	ADP
cana-3491	131	13	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	131	14	)	)	PUNCT
cana-3491	131	15	,	,	PUNCT
cana-3491	131	16	we	we	PRON
cana-3491	131	17	have	have	VERB
cana-3491	131	18	to	to	PART
cana-3491	131	19	study	study	VERB
cana-3491	131	20	the	the	DET
cana-3491	131	21	relation	relation	NOUN
cana-3491	131	22	between	between	ADP
cana-3491	131	23	the	the	DET
cana-3491	131	24	points	point	NOUN
cana-3491	131	25	in	in	ADP
cana-3491	131	26	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	131	27	)	)	PUNCT
cana-3491	131	28	and	and	CCONJ
cana-3491	131	29	the	the	DET
cana-3491	131	30	points	point	NOUN
cana-3491	131	31	in	in	ADP
cana-3491	131	32	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	131	33	)	)	PUNCT
cana-3491	131	34	.	.	PUNCT
cana-3491	132	1	we	we	PRON
cana-3491	132	2	first	first	ADV
cana-3491	132	3	proceed	proceed	VERB
cana-3491	132	4	to	to	PART
cana-3491	132	5	study	study	VERB
cana-3491	132	6	the	the	DET
cana-3491	132	7	reduction	reduction	NOUN
cana-3491	132	8	of	of	ADP
cana-3491	132	9	points	point	NOUN
cana-3491	132	10	on	on	ADP
cana-3491	132	11	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	132	12	)	)	PUNCT
cana-3491	132	13	to	to	PART
cana-3491	132	14	𝑚𝑜𝑑𝑝.	𝑚𝑜𝑑𝑝.	VERB
cana-3491	132	15	for	for	ADP
cana-3491	132	16	any	any	DET
cana-3491	132	17	element	element	NOUN
cana-3491	132	18	�	�	PROPN
cana-3491	132	19	̃	̃	PROPN
cana-3491	132	20	�	�	PROPN
cana-3491	132	21	∈	∈	PROPN
cana-3491	132	22	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	132	23	,	,	PUNCT
cana-3491	132	24	there	there	PRON
cana-3491	132	25	is	be	VERB
cana-3491	132	26	a	a	DET
cana-3491	132	27	natural	natural	ADJ
cana-3491	132	28	reduction	reduction	NOUN
cana-3491	132	29	�	�	PROPN
cana-3491	132	30	̃	̃	PROPN
cana-3491	132	31	�	�	PROPN
cana-3491	132	32	→	→	SYM
cana-3491	132	33	𝑥	𝑥	PROPN
cana-3491	132	34	from	from	ADP
cana-3491	132	35	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	132	36	→	→	SYM
cana-3491	132	37	𝐹𝑝.	𝐹𝑝.	PROPN
cana-3491	132	38	but	but	CCONJ
cana-3491	132	39	,	,	PUNCT
cana-3491	132	40	such	such	ADJ
cana-3491	132	41	reduction	reduction	NOUN
cana-3491	132	42	can	can	AUX
cana-3491	132	43	not	not	PART
cana-3491	132	44	be	be	AUX
cana-3491	132	45	extended	extend	VERB
cana-3491	132	46	from	from	ADP
cana-3491	132	47	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	132	48	to	to	ADP
cana-3491	132	49	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	132	50	as	as	SCONJ
cana-3491	132	51	reduction	reduction	NOUN
cana-3491	132	52	is	be	AUX
cana-3491	132	53	not	not	PART
cana-3491	132	54	injection	injection	NOUN
cana-3491	132	55	whereas	whereas	SCONJ
cana-3491	132	56	any	any	DET
cana-3491	132	57	ring	ring	NOUN
cana-3491	132	58	homomorphism	homomorphism	NOUN
cana-3491	132	59	from	from	ADP
cana-3491	132	60	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	132	61	would	would	AUX
cana-3491	132	62	always	always	ADV
cana-3491	132	63	be	be	AUX
cana-3491	132	64	an	an	DET
cana-3491	132	65	injection	injection	NOUN
cana-3491	132	66	.	.	PUNCT
cana-3491	133	1	therefore	therefore	ADV
cana-3491	133	2	,	,	PUNCT
cana-3491	133	3	an	an	DET
cana-3491	133	4	elliptic	elliptic	ADJ
cana-3491	133	5	curve	curve	NOUN
cana-3491	133	6	over	over	ADP
cana-3491	133	7	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	133	8	in	in	ADP
cana-3491	133	9	general	general	ADJ
cana-3491	133	10	can	can	AUX
cana-3491	133	11	not	not	PART
cana-3491	133	12	be	be	AUX
cana-3491	133	13	reduced	reduce	VERB
cana-3491	133	14	to	to	ADP
cana-3491	133	15	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	133	16	.	.	PUNCT
cana-3491	134	1	but	but	CCONJ
cana-3491	134	2	,	,	PUNCT
cana-3491	134	3	note	note	VERB
cana-3491	134	4	as	as	ADP
cana-3491	134	5	for	for	ADP
cana-3491	134	6	any	any	DET
cana-3491	134	7	𝑥	𝑥	PRON
cana-3491	134	8	∈	∈	PROPN
cana-3491	134	9	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	134	10	,	,	PUNCT
cana-3491	134	11	as	as	SCONJ
cana-3491	134	12	𝑥	𝑥	PRON
cana-3491	134	13	can	can	AUX
cana-3491	134	14	be	be	AUX
cana-3491	134	15	reduced	reduce	VERB
cana-3491	134	16	to	to	ADP
cana-3491	134	17	an	an	DET
cana-3491	134	18	element	element	NOUN
cana-3491	134	19	in	in	ADP
cana-3491	134	20	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	134	21	naturally	naturally	ADV
cana-3491	134	22	,	,	PUNCT
cana-3491	134	23	to	to	PART
cana-3491	134	24	reduce	reduce	VERB
cana-3491	134	25	an	an	DET
cana-3491	134	26	elliptic	elliptic	ADJ
cana-3491	134	27	curve	curve	NOUN
cana-3491	134	28	over	over	ADP
cana-3491	134	29	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	134	30	,	,	PUNCT
cana-3491	134	31	we	we	PRON
cana-3491	134	32	need	need	VERB
cana-3491	134	33	to	to	PART
cana-3491	134	34	consider	consider	VERB
cana-3491	134	35	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	134	36	)	)	PUNCT
cana-3491	134	37	to	to	PART
cana-3491	134	38	be	be	AUX
cana-3491	134	39	defined	define	VERB
cana-3491	134	40	over	over	ADP
cana-3491	134	41	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	134	42	for	for	ADP
cana-3491	134	43	an	an	DET
cana-3491	134	44	elliptic	elliptic	ADJ
cana-3491	134	45	curve	curve	NOUN
cana-3491	134	46	defined	define	VERB
cana-3491	134	47	over	over	ADP
cana-3491	134	48	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	134	49	,	,	PUNCT
cana-3491	134	50	the	the	DET
cana-3491	134	51	point	point	NOUN
cana-3491	134	52	�	�	PROPN
cana-3491	134	53	̃	̃	NOUN
cana-3491	134	54	�	�	NOUN
cana-3491	134	55	=	=	SYM
cana-3491	134	56	(	(	PUNCT
cana-3491	134	57	�	�	PROPN
cana-3491	134	58	̃	̃	PROPN
cana-3491	134	59	�	�	PROPN
cana-3491	134	60	,	,	PUNCT
cana-3491	134	61	�	�	PROPN
cana-3491	134	62	̃	̃	PROPN
cana-3491	134	63	�	�	PROPN
cana-3491	134	64	)	)	PUNCT
cana-3491	134	65	∈	∈	PROPN
cana-3491	134	66	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	134	67	)	)	PUNCT
cana-3491	134	68	defined	define	VERB
cana-3491	134	69	over	over	ADP
cana-3491	134	70	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	134	71	is	be	AUX
cana-3491	134	72	such	such	ADJ
cana-3491	134	73	that	that	SCONJ
cana-3491	134	74	either	either	CCONJ
cana-3491	134	75	(	(	PUNCT
cana-3491	134	76	�	�	PROPN
cana-3491	134	77	̃	̃	PROPN
cana-3491	134	78	�	�	PROPN
cana-3491	134	79	,	,	PUNCT
cana-3491	134	80	�	�	PROPN
cana-3491	134	81	̃	̃	PROPN
cana-3491	134	82	�	�	PROPN
cana-3491	134	83	)	)	PUNCT
cana-3491	134	84	∈	∈	PROPN
cana-3491	135	1	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	135	2	×	×	NOUN
cana-3491	135	3	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	135	4	or	or	CCONJ
cana-3491	135	5	(	(	PUNCT
cana-3491	135	6	�	�	PROPN
cana-3491	135	7	̃	̃	PROPN
cana-3491	135	8	�	�	PROPN
cana-3491	135	9	,	,	PUNCT
cana-3491	135	10	�	�	PROPN
cana-3491	135	11	̃	̃	PROPN
cana-3491	135	12	�	�	PROPN
cana-3491	135	13	)	)	PUNCT
cana-3491	135	14	∉	∉	PROPN
cana-3491	136	1	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	136	2	×	×	NOUN
cana-3491	136	3	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	136	4	i.e.	i.e.	X
cana-3491	136	5	,	,	PUNCT
cana-3491	136	6	(	(	PUNCT
cana-3491	136	7	�	�	PROPN
cana-3491	136	8	̃	̃	NOUN
cana-3491	136	9	�	�	PROPN
cana-3491	136	10	,	,	PUNCT
cana-3491	136	11	�	�	PROPN
cana-3491	136	12	̃	̃	PROPN
cana-3491	136	13	�	�	PROPN
cana-3491	136	14	)	)	PUNCT
cana-3491	136	15	∈	∈	PROPN
cana-3491	137	1	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	137	2	×	×	NOUN
cana-3491	137	3	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	137	4	or	or	CCONJ
cana-3491	137	5	(	(	PUNCT
cana-3491	137	6	�	�	PROPN
cana-3491	137	7	̃	̃	PROPN
cana-3491	137	8	�	�	PROPN
cana-3491	137	9	,	,	PUNCT
cana-3491	137	10	�	�	PROPN
cana-3491	137	11	̃	̃	PROPN
cana-3491	137	12	�	�	PROPN
cana-3491	137	13	)	)	PUNCT
cana-3491	137	14	∈	∈	PROPN
cana-3491	138	1	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	138	2	×	×	NOUN
cana-3491	138	3	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	138	4	or	or	CCONJ
cana-3491	138	5	(	(	PUNCT
cana-3491	138	6	�	�	PROPN
cana-3491	138	7	̃	̃	PROPN
cana-3491	138	8	�	�	PROPN
cana-3491	138	9	,	,	PUNCT
cana-3491	138	10	�	�	PROPN
cana-3491	138	11	̃	̃	PROPN
cana-3491	138	12	�	�	PROPN
cana-3491	138	13	)	)	PUNCT
cana-3491	138	14	∈	∈	PROPN
cana-3491	139	1	𝑄𝑝	𝑄𝑝	NOUN
cana-3491	139	2	×	×	NOUN
cana-3491	139	3	𝑄𝑝.	𝑄𝑝.	PROPN
cana-3491	139	4	note	note	NOUN
cana-3491	139	5	for	for	ADP
cana-3491	139	6	(	(	PUNCT
cana-3491	139	7	�	�	PROPN
cana-3491	139	8	̃	̃	PROPN
cana-3491	139	9	�	�	PROPN
cana-3491	139	10	,	,	PUNCT
cana-3491	139	11	�	�	PROPN
cana-3491	139	12	̃	̃	PROPN
cana-3491	139	13	�	�	PROPN
cana-3491	139	14	)	)	PUNCT
cana-3491	139	15	∈	∈	PROPN
cana-3491	140	1	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	140	2	×	×	NOUN
cana-3491	140	3	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	140	4	,	,	PUNCT
cana-3491	140	5	we	we	PRON
cana-3491	140	6	have	have	VERB
cana-3491	140	7	�	�	PROPN
cana-3491	140	8	̃	̃	PROPN
cana-3491	140	9	�	�	PROPN
cana-3491	140	10	=	=	SYM
cana-3491	140	11	𝛼0	𝛼0	PROPN
cana-3491	140	12	+	+	CCONJ
cana-3491	140	13	𝛼1𝑝	𝛼1𝑝	PROPN
cana-3491	140	14	+	+	CCONJ
cana-3491	140	15	𝛼2𝑝	𝛼2𝑝	NUM
cana-3491	140	16	2	2	NUM
cana-3491	140	17	+	+	NOUN
cana-3491	140	18	.	.	PUNCT
cana-3491	140	19	..	..	PUNCT
cana-3491	140	20	and	and	CCONJ
cana-3491	140	21	�	�	PROPN
cana-3491	140	22	̃	̃	PROPN
cana-3491	140	23	�	�	NOUN
cana-3491	140	24	=	=	SYM
cana-3491	140	25	𝛽0	𝛽0	PROPN
cana-3491	140	26	+	+	CCONJ
cana-3491	140	27	𝛽1𝑝	𝛽1𝑝	PROPN
cana-3491	140	28	+	+	CCONJ
cana-3491	140	29	𝛽2𝑝	𝛽2𝑝	ADJ
cana-3491	140	30	2	2	NUM
cana-3491	140	31	+	+	NOUN
cana-3491	140	32	.	.	PUNCT
cana-3491	140	33	..	..	PUNCT
cana-3491	140	34	,	,	PUNCT
cana-3491	140	35	note	note	VERB
cana-3491	140	36	there	there	PRON
cana-3491	140	37	is	be	VERB
cana-3491	140	38	a	a	DET
cana-3491	140	39	natural	natural	ADJ
cana-3491	140	40	reduction	reduction	NOUN
cana-3491	140	41	to	to	ADP
cana-3491	140	42	𝐹𝑝(𝑚𝑜𝑑𝑝	𝐹𝑝(𝑚𝑜𝑑𝑝	NOUN
cana-3491	140	43	)	)	PUNCT
cana-3491	140	44	i.e.	i.e.	X
cana-3491	140	45	,	,	PUNCT
cana-3491	140	46	(	(	PUNCT
cana-3491	140	47	�	�	PROPN
cana-3491	140	48	̃	̃	NOUN
cana-3491	140	49	�	�	PROPN
cana-3491	140	50	,	,	PUNCT
cana-3491	140	51	�	�	PROPN
cana-3491	140	52	̃	̃	PROPN
cana-3491	140	53	�	�	PROPN
cana-3491	140	54	)	)	PUNCT
cana-3491	140	55	→	→	SYM
cana-3491	140	56	(	(	PUNCT
cana-3491	140	57	𝛼0	𝛼0	PROPN
cana-3491	140	58	,	,	PUNCT
cana-3491	140	59	𝛽0	𝛽0	PROPN
cana-3491	140	60	)	)	PUNCT
cana-3491	140	61	from	from	ADP
cana-3491	140	62	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	140	63	)	)	PUNCT
cana-3491	140	64	→	→	SYM
cana-3491	140	65	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	140	66	)	)	PUNCT
cana-3491	140	67	.	.	PUNCT
cana-3491	141	1	note	note	VERB
cana-3491	141	2	for	for	ADP
cana-3491	141	3	(	(	PUNCT
cana-3491	141	4	�	�	PROPN
cana-3491	141	5	̃	̃	PROPN
cana-3491	141	6	�	�	PROPN
cana-3491	141	7	,	,	PUNCT
cana-3491	141	8	�	�	PROPN
cana-3491	141	9	̃	̃	PROPN
cana-3491	141	10	�	�	PROPN
cana-3491	141	11	)	)	PUNCT
cana-3491	141	12	∉	∉	PROPN
cana-3491	142	1	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	142	2	×	×	NOUN
cana-3491	142	3	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	142	4	,	,	PUNCT
cana-3491	142	5	we	we	PRON
cana-3491	142	6	consider	consider	VERB
cana-3491	142	7	the	the	DET
cana-3491	142	8	corresponding	corresponding	ADJ
cana-3491	142	9	projective	projective	NOUN
cana-3491	142	10	co	co	NOUN
cana-3491	142	11	-	-	NOUN
cana-3491	142	12	ordinates	ordinate	NOUN
cana-3491	142	13	of	of	ADP
cana-3491	142	14	�	�	PROPN
cana-3491	142	15	̃	̃	PROPN
cana-3491	142	16	�	�	NOUN
cana-3491	142	17	=	=	PUNCT
cana-3491	143	1	[	[	X
cana-3491	143	2	𝐴:𝐵	𝐴:𝐵	NOUN
cana-3491	143	3	:	:	PUNCT
cana-3491	143	4	𝛤	𝛤	PROPN
cana-3491	143	5	]	]	PUNCT
cana-3491	143	6	∋	∋	NOUN
cana-3491	143	7	𝐴	𝐴	PROPN
cana-3491	143	8	,	,	PUNCT
cana-3491	143	9	𝐵	𝐵	PROPN
cana-3491	143	10	,	,	PUNCT
cana-3491	143	11	𝛤	𝛤	PROPN
cana-3491	143	12	∈	∈	PROPN
cana-3491	143	13	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	143	14	and	and	CCONJ
cana-3491	143	15	obtain	obtain	VERB
cana-3491	143	16	natural	natural	ADJ
cana-3491	143	17	reduction	reduction	NOUN
cana-3491	143	18	�	�	PROPN
cana-3491	143	19	̃	̃	PROPN
cana-3491	143	20	�	�	PROPN
cana-3491	143	21	→	→	SYM
cana-3491	143	22	𝑃	𝑃	NOUN
cana-3491	143	23	from	from	ADP
cana-3491	143	24	𝐸(𝑄𝑝	𝐸(𝑄𝑝	ADJ
cana-3491	143	25	)	)	PUNCT
cana-3491	143	26	→	→	SYM
cana-3491	143	27	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	143	28	)	)	PUNCT
cana-3491	143	29	.	.	PUNCT
cana-3491	144	1	communications	communication	NOUN
cana-3491	144	2	on	on	ADP
cana-3491	144	3	applied	apply	VERB
cana-3491	144	4	nonlinear	nonlinear	ADJ
cana-3491	144	5	analysis	analysis	NOUN
cana-3491	144	6	issn	issn	NOUN
cana-3491	144	7	:	:	PUNCT
cana-3491	144	8	1074	1074	NUM
cana-3491	144	9	-	-	PUNCT
cana-3491	144	10	133x	133x	NUM
cana-3491	144	11	vol	vol	NOUN
cana-3491	144	12	32	32	NUM
cana-3491	144	13	no	no	NOUN
cana-3491	144	14	.	.	PUNCT
cana-3491	145	1	7s	7	NOUN
cana-3491	145	2	(	(	PUNCT
cana-3491	145	3	2025	2025	NUM
cana-3491	145	4	)	)	PUNCT
cana-3491	145	5	862	862	NUM
cana-3491	145	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	145	7	hence	hence	ADV
cana-3491	145	8	,	,	PUNCT
cana-3491	145	9	for	for	ADP
cana-3491	145	10	an	an	DET
cana-3491	145	11	elliptic	elliptic	ADJ
cana-3491	145	12	curve	curve	NOUN
cana-3491	145	13	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	145	14	)	)	PUNCT
cana-3491	145	15	defined	define	VERB
cana-3491	145	16	over	over	ADP
cana-3491	145	17	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	145	18	,	,	PUNCT
cana-3491	145	19	its	its	PRON
cana-3491	145	20	points	point	NOUN
cana-3491	145	21	can	can	AUX
cana-3491	145	22	always	always	ADV
cana-3491	145	23	be	be	AUX
cana-3491	145	24	reduced	reduce	VERB
cana-3491	145	25	to	to	ADP
cana-3491	145	26	points	point	NOUN
cana-3491	145	27	in	in	ADP
cana-3491	145	28	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	145	29	)	)	PUNCT
cana-3491	145	30	,	,	PUNCT
cana-3491	145	31	the	the	DET
cana-3491	145	32	elliptic	elliptic	ADJ
cana-3491	145	33	curve	curve	NOUN
cana-3491	145	34	over	over	ADP
cana-3491	145	35	a	a	DET
cana-3491	145	36	finite	finite	ADJ
cana-3491	145	37	field	field	NOUN
cana-3491	145	38	𝐹𝑝.	𝐹𝑝.	PROPN
cana-3491	146	1	however	however	ADV
cana-3491	146	2	if	if	SCONJ
cana-3491	146	3	our	our	PRON
cana-3491	146	4	elliptic	elliptic	ADJ
cana-3491	146	5	curve	curve	NOUN
cana-3491	146	6	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	146	7	)	)	PUNCT
cana-3491	146	8	is	be	AUX
cana-3491	146	9	not	not	PART
cana-3491	146	10	defined	define	VERB
cana-3491	146	11	over	over	ADP
cana-3491	146	12	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	146	13	,	,	PUNCT
cana-3491	146	14	there	there	PRON
cana-3491	146	15	is	be	VERB
cana-3491	146	16	an	an	DET
cana-3491	146	17	elliptic	elliptic	ADJ
cana-3491	146	18	curve	curve	NOUN
cana-3491	146	19	𝐸′(𝑄𝑝	𝐸′(𝑄𝑝	NOUN
cana-3491	146	20	)	)	PUNCT
cana-3491	146	21	such	such	ADJ
cana-3491	146	22	that	that	SCONJ
cana-3491	146	23	𝐸′	𝐸′	PROPN
cana-3491	146	24	≅	≅	PROPN
cana-3491	146	25	𝐸	𝐸	PROPN
cana-3491	146	26	where	where	SCONJ
cana-3491	146	27	𝐸′	𝐸′	PROPN
cana-3491	146	28	is	be	AUX
cana-3491	146	29	the	the	DET
cana-3491	146	30	elliptic	elliptic	ADJ
cana-3491	146	31	curve	curve	NOUN
cana-3491	146	32	defined	define	VERB
cana-3491	146	33	over	over	ADP
cana-3491	146	34	𝑍𝑝	𝑍𝑝	PRON
cana-3491	146	35	i.e.	i.e.	X
cana-3491	146	36	,	,	PUNCT
cana-3491	146	37	the	the	DET
cana-3491	146	38	coefficients	coefficient	NOUN
cana-3491	146	39	of	of	ADP
cana-3491	146	40	curve	curve	NOUN
cana-3491	146	41	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	146	42	)	)	PUNCT
cana-3491	146	43	are	be	AUX
cana-3491	146	44	in	in	ADP
cana-3491	146	45	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	146	46	under	under	ADP
cana-3491	146	47	the	the	DET
cana-3491	146	48	mapping	mapping	NOUN
cana-3491	146	49	(	(	PUNCT
cana-3491	146	50	𝛼	𝛼	PROPN
cana-3491	146	51	,	,	PUNCT
cana-3491	146	52	𝛽	𝛽	NOUN
cana-3491	146	53	)	)	PUNCT
cana-3491	146	54	→	→	PUNCT
cana-3491	146	55	(	(	PUNCT
cana-3491	146	56	𝑣−2𝛼	𝑣−2𝛼	NOUN
cana-3491	146	57	,	,	PUNCT
cana-3491	146	58	𝑣−3𝛽	𝑣−3𝛽	ADV
cana-3491	146	59	)	)	PUNCT
cana-3491	146	60	.	.	PUNCT
cana-3491	147	1	hence	hence	ADV
cana-3491	147	2	for	for	ADP
cana-3491	147	3	further	further	ADJ
cana-3491	147	4	study	study	NOUN
cana-3491	147	5	,	,	PUNCT
cana-3491	147	6	without	without	ADP
cana-3491	147	7	loss	loss	NOUN
cana-3491	147	8	of	of	ADP
cana-3491	147	9	generality	generality	NOUN
cana-3491	147	10	,	,	PUNCT
cana-3491	147	11	we	we	PRON
cana-3491	147	12	consider	consider	VERB
cana-3491	147	13	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	147	14	)	)	PUNCT
cana-3491	147	15	is	be	AUX
cana-3491	147	16	always	always	ADV
cana-3491	147	17	defined	define	VERB
cana-3491	147	18	over	over	ADP
cana-3491	147	19	𝑍𝑝.	𝑍𝑝.	PROPN
cana-3491	147	20	definition	definition	NOUN
cana-3491	147	21	3.1	3.1	NUM
cana-3491	147	22	.	.	PUNCT
cana-3491	148	1	(	(	PUNCT
cana-3491	148	2	lift	lift	NOUN
cana-3491	148	3	of	of	ADP
cana-3491	148	4	a	a	DET
cana-3491	148	5	point	point	NOUN
cana-3491	148	6	)	)	PUNCT
cana-3491	148	7	.	.	PUNCT
cana-3491	149	1	the	the	DET
cana-3491	149	2	set	set	NOUN
cana-3491	149	3	of	of	ADP
cana-3491	149	4	all	all	DET
cana-3491	149	5	lifts	lift	NOUN
cana-3491	149	6	of	of	ADP
cana-3491	149	7	points	point	NOUN
cana-3491	149	8	from	from	ADP
cana-3491	149	9	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	149	10	to	to	ADP
cana-3491	149	11	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	149	12	is	be	AUX
cana-3491	149	13	denoted	denote	VERB
cana-3491	149	14	as	as	ADP
cana-3491	149	15	𝐿(𝐹𝑝	𝐿(𝐹𝑝	NOUN
cana-3491	149	16	)	)	PUNCT
cana-3491	149	17	.	.	PUNCT
cana-3491	150	1	for	for	ADP
cana-3491	150	2	any	any	DET
cana-3491	150	3	(	(	PUNCT
cana-3491	150	4	𝑥0	𝑥0	NOUN
cana-3491	150	5	,	,	PUNCT
cana-3491	150	6	𝑦0	𝑦0	NOUN
cana-3491	150	7	)	)	PUNCT
cana-3491	150	8	∈	∈	PROPN
cana-3491	150	9	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	150	10	×	×	PROPN
cana-3491	150	11	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	150	12	,	,	PUNCT
cana-3491	150	13	the	the	DET
cana-3491	150	14	point	point	NOUN
cana-3491	150	15	(	(	PUNCT
cana-3491	150	16	𝛼	𝛼	NOUN
cana-3491	150	17	,	,	PUNCT
cana-3491	150	18	𝛽	𝛽	NOUN
cana-3491	150	19	)	)	PUNCT
cana-3491	150	20	=	=	SYM
cana-3491	150	21	(	(	PUNCT
cana-3491	150	22	�	�	PROPN
cana-3491	150	23	̃	̃	PROPN
cana-3491	150	24	�	�	PROPN
cana-3491	150	25	,	,	PUNCT
cana-3491	150	26	�	�	PROPN
cana-3491	150	27	̃	̃	PROPN
cana-3491	150	28	�	�	PROPN
cana-3491	150	29	)	)	PUNCT
cana-3491	150	30	∈	∈	PROPN
cana-3491	151	1	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	151	2	×	×	NOUN
cana-3491	151	3	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	151	4	is	be	AUX
cana-3491	151	5	called	call	VERB
cana-3491	151	6	a	a	DET
cana-3491	151	7	lift	lift	NOUN
cana-3491	151	8	of	of	ADP
cana-3491	151	9	a	a	DET
cana-3491	151	10	point	point	NOUN
cana-3491	151	11	(	(	PUNCT
cana-3491	151	12	𝑥0	𝑥0	NOUN
cana-3491	151	13	,	,	PUNCT
cana-3491	151	14	𝑦0	𝑦0	NOUN
cana-3491	151	15	)	)	PUNCT
cana-3491	151	16	from	from	ADP
cana-3491	151	17	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	151	18	to	to	ADP
cana-3491	151	19	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	151	20	and	and	CCONJ
cana-3491	151	21	is	be	AUX
cana-3491	151	22	defined	define	VERB
cana-3491	151	23	as	as	ADP
cana-3491	151	24	𝐿(𝐹𝑝	𝐿(𝐹𝑝	NOUN
cana-3491	151	25	)	)	PUNCT
cana-3491	152	1	=	=	PRON
cana-3491	152	2	{	{	PUNCT
cana-3491	152	3	(	(	PUNCT
cana-3491	152	4	�	�	PROPN
cana-3491	152	5	̃	̃	PROPN
cana-3491	152	6	�	�	PROPN
cana-3491	152	7	,	,	PUNCT
cana-3491	152	8	�	�	PROPN
cana-3491	152	9	̃	̃	PROPN
cana-3491	152	10	�	�	PROPN
cana-3491	152	11	)	)	PUNCT
cana-3491	152	12	∈	∈	PROPN
cana-3491	153	1	𝑍𝑝	𝑍𝑝	PRON
cana-3491	153	2	×	×	NOUN
cana-3491	153	3	𝑍𝑝/	𝑍𝑝/	PROPN
cana-3491	153	4	�	�	PROPN
cana-3491	153	5	̃	̃	PROPN
cana-3491	153	6	�	�	NOUN
cana-3491	153	7	=	=	SYM
cana-3491	153	8	𝑥0	𝑥0	NOUN
cana-3491	153	9	+	+	CCONJ
cana-3491	153	10	𝑥1𝑝	𝑥1𝑝	PROPN
cana-3491	153	11	+	+	CCONJ
cana-3491	153	12	𝑥2𝑝	𝑥2𝑝	NUM
cana-3491	153	13	2	2	NUM
cana-3491	153	14	+	+	NOUN
cana-3491	153	15	.	.	PUNCT
cana-3491	153	16	.	.	PUNCT
cana-3491	153	17	.	.	PUNCT
cana-3491	154	1	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-3491	154	2	�	�	PROPN
cana-3491	154	3	̃	̃	PROPN
cana-3491	154	4	�	�	NOUN
cana-3491	154	5	=	=	SYM
cana-3491	154	6	𝑦0	𝑦0	PROPN
cana-3491	154	7	+	+	CCONJ
cana-3491	154	8	𝑦1𝑝	𝑦1𝑝	X
cana-3491	154	9	+	+	CCONJ
cana-3491	154	10	𝑦2𝑝	𝑦2𝑝	NOUN
cana-3491	154	11	2	2	NUM
cana-3491	154	12	+	+	PROPN
cana-3491	154	13	.	.	PUNCT
cana-3491	154	14	..	..	PUNCT
cana-3491	155	1	with	with	ADP
cana-3491	155	2	(	(	PUNCT
cana-3491	155	3	𝑥0	𝑥0	NOUN
cana-3491	155	4	,	,	PUNCT
cana-3491	155	5	𝑦0	𝑦0	NOUN
cana-3491	155	6	)	)	PUNCT
cana-3491	155	7	∈	∈	PROPN
cana-3491	156	1	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	156	2	×	×	NOUN
cana-3491	156	3	𝐹𝑝	𝐹𝑝	NOUN
cana-3491	156	4	}	}	PUNCT
cana-3491	156	5	definition	definition	NOUN
cana-3491	156	6	3.2	3.2	NUM
cana-3491	156	7	.	.	PUNCT
cana-3491	157	1	the	the	DET
cana-3491	157	2	set	set	NOUN
cana-3491	157	3	of	of	ADP
cana-3491	157	4	all	all	DET
cana-3491	157	5	lifts	lift	NOUN
cana-3491	157	6	of	of	ADP
cana-3491	157	7	points	point	NOUN
cana-3491	157	8	on	on	ADP
cana-3491	157	9	elliptic	elliptic	ADJ
cana-3491	157	10	curve	curve	NOUN
cana-3491	157	11	over	over	ADP
cana-3491	157	12	finite	finite	ADJ
cana-3491	157	13	field	field	NOUN
cana-3491	157	14	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	157	15	is	be	AUX
cana-3491	157	16	given	give	VERB
cana-3491	157	17	as	as	ADP
cana-3491	157	18	for	for	ADP
cana-3491	157	19	any	any	DET
cana-3491	157	20	(	(	PUNCT
cana-3491	157	21	𝛼	𝛼	NOUN
cana-3491	157	22	,	,	PUNCT
cana-3491	157	23	𝛽	𝛽	NOUN
cana-3491	157	24	)	)	PUNCT
cana-3491	157	25	=	=	SYM
cana-3491	157	26	(	(	PUNCT
cana-3491	157	27	𝑥0	𝑥0	NOUN
cana-3491	157	28	,	,	PUNCT
cana-3491	157	29	𝑦0	𝑦0	NOUN
cana-3491	157	30	)	)	PUNCT
cana-3491	157	31	∈	∈	PROPN
cana-3491	157	32	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	157	33	)	)	PUNCT
cana-3491	157	34	the	the	DET
cana-3491	157	35	point	point	NOUN
cana-3491	157	36	(	(	PUNCT
cana-3491	157	37	�	�	PROPN
cana-3491	157	38	̃	̃	NOUN
cana-3491	157	39	�	�	PROPN
cana-3491	157	40	,	,	PUNCT
cana-3491	157	41	�	�	PROPN
cana-3491	157	42	̃	̃	PROPN
cana-3491	157	43	�	�	PROPN
cana-3491	157	44	)	)	PUNCT
cana-3491	157	45	∈	∈	PROPN
cana-3491	157	46	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	157	47	)	)	PUNCT
cana-3491	157	48	is	be	AUX
cana-3491	157	49	called	call	VERB
cana-3491	157	50	a	a	DET
cana-3491	157	51	lift	lift	NOUN
cana-3491	157	52	of	of	ADP
cana-3491	157	53	a	a	DET
cana-3491	157	54	point	point	NOUN
cana-3491	157	55	(	(	PUNCT
cana-3491	157	56	𝑥0	𝑥0	NOUN
cana-3491	157	57	,	,	PUNCT
cana-3491	157	58	𝑦0	𝑦0	NOUN
cana-3491	157	59	)	)	PUNCT
cana-3491	157	60	from	from	ADP
cana-3491	157	61	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	157	62	)	)	PUNCT
cana-3491	157	63	to	to	ADP
cana-3491	157	64	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	157	65	)	)	PUNCT
cana-3491	157	66	and	and	CCONJ
cana-3491	157	67	is	be	AUX
cana-3491	157	68	defined	define	VERB
cana-3491	157	69	as	as	ADP
cana-3491	157	70	𝐿	𝐿	PROPN
cana-3491	157	71	(	(	PUNCT
cana-3491	157	72	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	157	73	)	)	PUNCT
cana-3491	157	74	)	)	PUNCT
cana-3491	158	1	=	=	PRON
cana-3491	158	2	{	{	PUNCT
cana-3491	158	3	(	(	PUNCT
cana-3491	158	4	�	�	PROPN
cana-3491	158	5	̃	̃	PROPN
cana-3491	158	6	�	�	PROPN
cana-3491	158	7	,	,	PUNCT
cana-3491	158	8	�	�	PROPN
cana-3491	158	9	̃	̃	PROPN
cana-3491	158	10	�	�	PROPN
cana-3491	158	11	)	)	PUNCT
cana-3491	158	12	∈	∈	PROPN
cana-3491	159	1	𝑍𝑝	𝑍𝑝	PRON
cana-3491	159	2	×	×	NOUN
cana-3491	159	3	𝑍𝑝/	𝑍𝑝/	PROPN
cana-3491	159	4	�	�	PROPN
cana-3491	159	5	̃	̃	PROPN
cana-3491	159	6	�	�	NOUN
cana-3491	159	7	=	=	SYM
cana-3491	159	8	𝑥0	𝑥0	NOUN
cana-3491	159	9	+	+	CCONJ
cana-3491	159	10	𝑥1𝑝	𝑥1𝑝	PUNCT
cana-3491	159	11	+	+	NOUN
cana-3491	159	12	𝑥2𝑝	𝑥2𝑝	NUM
cana-3491	159	13	2	2	NUM
cana-3491	159	14	+	+	ADJ
cana-3491	159	15	⋯	⋯	PROPN
cana-3491	159	16	and	and	CCONJ
cana-3491	159	17	�	�	PROPN
cana-3491	159	18	̃	̃	PROPN
cana-3491	159	19	�	�	NOUN
cana-3491	159	20	=	=	SYM
cana-3491	159	21	𝑦0	𝑦0	PROPN
cana-3491	159	22	+	+	CCONJ
cana-3491	159	23	𝑦1𝑝	𝑦1𝑝	X
cana-3491	159	24	+	+	CCONJ
cana-3491	159	25	𝑦2𝑝	𝑦2𝑝	PROPN
cana-3491	159	26	2	2	NUM
cana-3491	160	1	+	+	NOUN
cana-3491	160	2	⋯	⋯	VERB
cana-3491	160	3	with	with	ADP
cana-3491	160	4	(	(	PUNCT
cana-3491	160	5	𝑥0	𝑥0	NOUN
cana-3491	160	6	,	,	PUNCT
cana-3491	160	7	𝑦0	𝑦0	NOUN
cana-3491	160	8	)	)	PUNCT
cana-3491	160	9	∈	∈	PROPN
cana-3491	160	10	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	160	11	)	)	PUNCT
cana-3491	160	12	and	and	CCONJ
cana-3491	160	13	�	�	PROPN
cana-3491	160	14	̃	̃	PROPN
cana-3491	160	15	�	�	NOUN
cana-3491	160	16	2	2	NUM
cana-3491	160	17	=	=	SYM
cana-3491	160	18	�	�	PROPN
cana-3491	160	19	̃	̃	PROPN
cana-3491	160	20	�	�	NOUN
cana-3491	160	21	3	3	NUM
cana-3491	160	22	+	+	CCONJ
cana-3491	160	23	𝐴	𝐴	PROPN
cana-3491	160	24	�	�	PROPN
cana-3491	160	25	̃	̃	PROPN
cana-3491	160	26	�	�	PROPN
cana-3491	160	27	+	+	CCONJ
cana-3491	160	28	𝐵	𝐵	NOUN
cana-3491	160	29	}	}	PUNCT
cana-3491	160	30	now	now	ADV
cana-3491	160	31	in	in	ADP
cana-3491	160	32	the	the	DET
cana-3491	160	33	following	follow	VERB
cana-3491	160	34	theorems	theorem	NOUN
cana-3491	160	35	,	,	PUNCT
cana-3491	160	36	we	we	PRON
cana-3491	160	37	describe	describe	VERB
cana-3491	160	38	the	the	DET
cana-3491	160	39	points	point	NOUN
cana-3491	160	40	in	in	ADP
cana-3491	160	41	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	160	42	)	)	PUNCT
cana-3491	160	43	defined	define	VERB
cana-3491	160	44	over	over	ADP
cana-3491	160	45	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	160	46	as	as	ADP
cana-3491	160	47	as	as	ADP
cana-3491	160	48	lift	lift	NOUN
cana-3491	160	49	of	of	ADP
cana-3491	160	50	each	each	DET
cana-3491	160	51	point	point	NOUN
cana-3491	160	52	(	(	PUNCT
cana-3491	160	53	𝑥0	𝑥0	NOUN
cana-3491	160	54	,	,	PUNCT
cana-3491	160	55	𝑦0	𝑦0	NOUN
cana-3491	160	56	)	)	PUNCT
cana-3491	160	57	∈	∈	PROPN
cana-3491	160	58	𝐸(𝐹𝑝	𝐸(𝐹𝑝	PROPN
cana-3491	160	59	)	)	PUNCT
cana-3491	160	60	.	.	PUNCT
cana-3491	161	1	theorem	theorem	VERB
cana-3491	161	2	3.1	3.1	NUM
cana-3491	161	3	.	.	PUNCT
cana-3491	162	1	let	let	VERB
cana-3491	162	2	𝐸	𝐸	PRON
cana-3491	162	3	be	be	AUX
cana-3491	162	4	an	an	DET
cana-3491	162	5	elliptic	elliptic	ADJ
cana-3491	162	6	curve	curve	NOUN
cana-3491	162	7	over	over	ADP
cana-3491	162	8	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	162	9	defined	define	VERB
cana-3491	162	10	over	over	ADP
cana-3491	162	11	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	162	12	then	then	ADV
cana-3491	162	13	each	each	DET
cana-3491	162	14	point	point	NOUN
cana-3491	162	15	in	in	ADP
cana-3491	162	16	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	162	17	)	)	PUNCT
cana-3491	162	18	is	be	AUX
cana-3491	162	19	a	a	DET
cana-3491	162	20	reduction	reduction	NOUN
cana-3491	162	21	of	of	ADP
cana-3491	162	22	a	a	DET
cana-3491	162	23	point	point	NOUN
cana-3491	162	24	in	in	ADP
cana-3491	162	25	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	162	26	)	)	PUNCT
cana-3491	162	27	.	.	PUNCT
cana-3491	163	1	proof	proof	NOUN
cana-3491	163	2	.	.	PUNCT
cana-3491	164	1	by	by	ADP
cana-3491	164	2	definition	definition	NOUN
cana-3491	164	3	,	,	PUNCT
cana-3491	164	4	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	164	5	)	)	PUNCT
cana-3491	164	6	=	=	SYM
cana-3491	164	7	{	{	PUNCT
cana-3491	164	8	(	(	PUNCT
cana-3491	164	9	𝛼	𝛼	PROPN
cana-3491	164	10	,	,	PUNCT
cana-3491	164	11	𝛽	𝛽	NOUN
cana-3491	164	12	)	)	PUNCT
cana-3491	164	13	∈	∈	PROPN
cana-3491	165	1	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	165	2	×	×	NOUN
cana-3491	165	3	𝑄𝑝/𝛽	𝑄𝑝/𝛽	NUM
cana-3491	165	4	2	2	NUM
cana-3491	165	5	=	=	NOUN
cana-3491	165	6	𝛼3	𝛼3	NOUN
cana-3491	165	7	+	+	CCONJ
cana-3491	166	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	166	2	+	+	CCONJ
cana-3491	166	3	𝐵	𝐵	NOUN
cana-3491	166	4	}	}	PUNCT
cana-3491	166	5	∪	∪	VERB
cana-3491	166	6	{	{	PUNCT
cana-3491	166	7	�	�	PROPN
cana-3491	166	8	̃	̃	NOUN
cana-3491	166	9	�	�	PROPN
cana-3491	166	10	}	}	PUNCT
cana-3491	166	11	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3491	166	12	𝐸(𝐹𝑝	𝐸(𝐹𝑝	PROPN
cana-3491	166	13	)	)	PUNCT
cana-3491	167	1	=	=	PRON
cana-3491	167	2	{	{	PUNCT
cana-3491	167	3	(	(	PUNCT
cana-3491	167	4	𝛼	𝛼	PROPN
cana-3491	167	5	,	,	PUNCT
cana-3491	167	6	𝛽	𝛽	NOUN
cana-3491	167	7	)	)	PUNCT
cana-3491	167	8	∈	∈	PROPN
cana-3491	168	1	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	168	2	×	×	NOUN
cana-3491	168	3	𝐹𝑝/𝛽	𝐹𝑝/𝛽	NUM
cana-3491	168	4	2	2	NUM
cana-3491	168	5	=	=	NOUN
cana-3491	168	6	𝛼3	𝛼3	NOUN
cana-3491	168	7	+	+	CCONJ
cana-3491	169	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	169	2	+	+	CCONJ
cana-3491	169	3	𝐵	𝐵	NOUN
cana-3491	169	4	}	}	PUNCT
cana-3491	169	5	∪	∪	VERB
cana-3491	169	6	{	{	PUNCT
cana-3491	169	7	�	�	PROPN
cana-3491	169	8	̃	̃	NOUN
cana-3491	169	9	�	�	PROPN
cana-3491	169	10	}	}	PUNCT
cana-3491	169	11	let	let	VERB
cana-3491	169	12	𝑃	𝑃	NOUN
cana-3491	169	13	=	=	SYM
cana-3491	169	14	(	(	PUNCT
cana-3491	169	15	𝑥0	𝑥0	NOUN
cana-3491	169	16	,	,	PUNCT
cana-3491	169	17	𝑦0	𝑦0	NOUN
cana-3491	169	18	)	)	PUNCT
cana-3491	169	19	be	be	VERB
cana-3491	169	20	a	a	DET
cana-3491	169	21	point	point	NOUN
cana-3491	169	22	in	in	ADP
cana-3491	169	23	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	169	24	)	)	PUNCT
cana-3491	169	25	such	such	ADJ
cana-3491	169	26	that	that	SCONJ
cana-3491	169	27	𝑃	𝑃	PROPN
cana-3491	169	28	≠	≠	PROPN
cana-3491	169	29	𝒪	𝒪	PROPN
cana-3491	169	30	,	,	PUNCT
cana-3491	169	31	then	then	ADV
cana-3491	169	32	for	for	ADP
cana-3491	169	33	𝑃	𝑃	NOUN
cana-3491	169	34	=	=	SYM
cana-3491	169	35	(	(	PUNCT
cana-3491	169	36	𝑥0	𝑥0	NOUN
cana-3491	169	37	,	,	PUNCT
cana-3491	169	38	𝑦0	𝑦0	NOUN
cana-3491	169	39	)	)	PUNCT
cana-3491	169	40	we	we	PRON
cana-3491	169	41	have	have	VERB
cana-3491	169	42	𝑦0	𝑦0	NOUN
cana-3491	169	43	2	2	NUM
cana-3491	169	44	=	=	SYM
cana-3491	169	45	𝑥0	𝑥0	NOUN
cana-3491	169	46	3	3	NUM
cana-3491	169	47	+	+	NUM
cana-3491	169	48	𝐴𝑥0	𝐴𝑥0	NOUN
cana-3491	170	1	+	+	X
cana-3491	170	2	𝐵.	𝐵.	NOUN
cana-3491	170	3	now	now	ADV
cana-3491	170	4	to	to	PART
cana-3491	170	5	find	find	VERB
cana-3491	170	6	�	�	PROPN
cana-3491	170	7	̃	̃	PROPN
cana-3491	170	8	�	�	PROPN
cana-3491	170	9	lift	lift	NOUN
cana-3491	170	10	of	of	ADP
cana-3491	170	11	𝑃	𝑃	NOUN
cana-3491	170	12	such	such	ADJ
cana-3491	170	13	that	that	DET
cana-3491	170	14	�	�	PROPN
cana-3491	170	15	̃	̃	PROPN
cana-3491	170	16	�	�	PROPN
cana-3491	170	17	∈	∈	PROPN
cana-3491	170	18	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	170	19	)	)	PUNCT
cana-3491	170	20	.	.	PUNCT
cana-3491	171	1	let	let	VERB
cana-3491	171	2	�	�	PROPN
cana-3491	171	3	̃	̃	PROPN
cana-3491	171	4	�	�	PROPN
cana-3491	171	5	=	=	SYM
cana-3491	171	6	(	(	PUNCT
cana-3491	171	7	�	�	PROPN
cana-3491	171	8	̃	̃	PROPN
cana-3491	171	9	�	�	PROPN
cana-3491	171	10	,	,	PUNCT
cana-3491	171	11	�	�	PROPN
cana-3491	171	12	̃	̃	PROPN
cana-3491	171	13	�	�	PROPN
cana-3491	171	14	)	)	PUNCT
cana-3491	171	15	∈	∈	PROPN
cana-3491	172	1	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	172	2	×	×	NOUN
cana-3491	172	3	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	172	4	be	be	AUX
cana-3491	172	5	a	a	DET
cana-3491	172	6	lift	lift	NOUN
cana-3491	172	7	of	of	ADP
cana-3491	172	8	𝑃	𝑃	NOUN
cana-3491	172	9	=	=	SYM
cana-3491	172	10	(	(	PUNCT
cana-3491	172	11	𝑥0	𝑥0	NOUN
cana-3491	172	12	,	,	PUNCT
cana-3491	172	13	𝑦0	𝑦0	NOUN
cana-3491	172	14	)	)	PUNCT
cana-3491	172	15	,	,	PUNCT
cana-3491	172	16	then	then	ADV
cana-3491	172	17	we	we	PRON
cana-3491	172	18	have	have	VERB
cana-3491	172	19	�	�	PROPN
cana-3491	172	20	̃	̃	PROPN
cana-3491	172	21	�	�	PROPN
cana-3491	172	22	,	,	PUNCT
cana-3491	172	23	�	�	PROPN
cana-3491	172	24	̃	̃	PROPN
cana-3491	172	25	�	�	PROPN
cana-3491	172	26	are	be	AUX
cana-3491	172	27	𝑝-adic	𝑝-adic	ADJ
cana-3491	172	28	integers	integer	NOUN
cana-3491	172	29	having	have	VERB
cana-3491	172	30	the	the	DET
cana-3491	172	31	form	form	NOUN
cana-3491	172	32	�	�	NOUN
cana-3491	172	33	̃	̃	PROPN
cana-3491	172	34	�	�	NOUN
cana-3491	172	35	=	=	SYM
cana-3491	172	36	𝑥0	𝑥0	NOUN
cana-3491	172	37	+	+	CCONJ
cana-3491	172	38	𝑥1𝑝	𝑥1𝑝	PROPN
cana-3491	172	39	+	+	CCONJ
cana-3491	172	40	𝑥2𝑝	𝑥2𝑝	NUM
cana-3491	172	41	2	2	NUM
cana-3491	172	42	+	+	NOUN
cana-3491	172	43	.	.	PUNCT
cana-3491	172	44	..	..	PUNCT
cana-3491	172	45	�	�	PROPN
cana-3491	172	46	̃	̃	NOUN
cana-3491	172	47	�	�	NOUN
cana-3491	172	48	=	=	SYM
cana-3491	172	49	𝑦0	𝑦0	PROPN
cana-3491	172	50	+	+	CCONJ
cana-3491	172	51	𝑦1𝑝	𝑦1𝑝	X
cana-3491	172	52	+	+	CCONJ
cana-3491	172	53	𝑦2𝑝	𝑦2𝑝	NOUN
cana-3491	172	54	2	2	NUM
cana-3491	172	55	+	+	PROPN
cana-3491	172	56	.	.	PUNCT
cana-3491	172	57	..	..	PUNCT
cana-3491	173	1	communications	communication	NOUN
cana-3491	173	2	on	on	ADP
cana-3491	173	3	applied	apply	VERB
cana-3491	173	4	nonlinear	nonlinear	ADJ
cana-3491	173	5	analysis	analysis	NOUN
cana-3491	173	6	issn	issn	NOUN
cana-3491	173	7	:	:	PUNCT
cana-3491	173	8	1074	1074	NUM
cana-3491	173	9	-	-	PUNCT
cana-3491	173	10	133x	133x	NUM
cana-3491	173	11	vol	vol	NOUN
cana-3491	173	12	32	32	NUM
cana-3491	173	13	no	no	NOUN
cana-3491	173	14	.	.	PUNCT
cana-3491	174	1	7s	7	NOUN
cana-3491	174	2	(	(	PUNCT
cana-3491	174	3	2025	2025	NUM
cana-3491	174	4	)	)	PUNCT
cana-3491	174	5	863	863	NUM
cana-3491	174	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	174	7	with	with	ADP
cana-3491	174	8	𝑥0	𝑥0	NOUN
cana-3491	174	9	,	,	PUNCT
cana-3491	174	10	𝑦0	𝑦0	NOUN
cana-3491	174	11	,	,	PUNCT
cana-3491	174	12	𝑥1	𝑥1	NOUN
cana-3491	174	13	,	,	PUNCT
cana-3491	174	14	𝑦1	𝑦1	NOUN
cana-3491	174	15	,	,	PUNCT
cana-3491	174	16	𝑥2	𝑥2	NOUN
cana-3491	174	17	,	,	PUNCT
cana-3491	174	18	𝑦2	𝑦2	NOUN
cana-3491	174	19	,	,	PUNCT
cana-3491	174	20	.	.	PUNCT
cana-3491	174	21	.	.	PUNCT
cana-3491	174	22	.	.	PUNCT
cana-3491	175	1	∈	∈	PROPN
cana-3491	175	2	𝐹𝑝.	𝐹𝑝.	PROPN
cana-3491	176	1	now	now	ADV
cana-3491	176	2	if	if	SCONJ
cana-3491	176	3	�	�	PROPN
cana-3491	176	4	̃	̃	NOUN
cana-3491	176	5	�	�	NOUN
cana-3491	176	6	=	=	SYM
cana-3491	176	7	(	(	PUNCT
cana-3491	176	8	�	�	PROPN
cana-3491	176	9	̃	̃	PROPN
cana-3491	176	10	�	�	PROPN
cana-3491	176	11	,	,	PUNCT
cana-3491	176	12	�	�	PROPN
cana-3491	176	13	̃	̃	PROPN
cana-3491	176	14	�	�	PROPN
cana-3491	176	15	)	)	PUNCT
cana-3491	176	16	∈	∈	PROPN
cana-3491	176	17	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	176	18	)	)	PUNCT
cana-3491	176	19	then	then	ADV
cana-3491	176	20	�	�	PROPN
cana-3491	176	21	̃	̃	PROPN
cana-3491	176	22	�	�	PROPN
cana-3491	176	23	2	2	NUM
cana-3491	176	24	=	=	SYM
cana-3491	176	25	�	�	PROPN
cana-3491	176	26	̃	̃	PROPN
cana-3491	176	27	�	�	NOUN
cana-3491	176	28	3	3	NUM
cana-3491	176	29	+	+	CCONJ
cana-3491	176	30	𝐴	𝐴	PROPN
cana-3491	176	31	�	�	PROPN
cana-3491	176	32	̃	̃	PROPN
cana-3491	176	33	�	�	PROPN
cana-3491	176	34	+	+	CCONJ
cana-3491	176	35	𝐵	𝐵	PROPN
cana-3491	176	36	,	,	PUNCT
cana-3491	176	37	i.e.	i.e.	X
cana-3491	176	38	,	,	PUNCT
cana-3491	176	39	for	for	ADP
cana-3491	176	40	�	�	PROPN
cana-3491	176	41	̃	̃	PROPN
cana-3491	176	42	�	�	NOUN
cana-3491	176	43	=	=	SYM
cana-3491	176	44	(	(	PUNCT
cana-3491	176	45	𝑥0	𝑥0	NOUN
cana-3491	176	46	+	+	CCONJ
cana-3491	176	47	𝑝𝑥1	𝑝𝑥1	NOUN
cana-3491	176	48	+	+	CCONJ
cana-3491	176	49	𝑝	𝑝	NOUN
cana-3491	176	50	2𝑥2	2𝑥2	NUM
cana-3491	176	51	+	+	PROPN
cana-3491	176	52	.	.	PUNCT
cana-3491	176	53	.	.	PUNCT
cana-3491	176	54	.	.	PUNCT
cana-3491	177	1	,	,	PUNCT
cana-3491	177	2	𝑦0	𝑦0	NOUN
cana-3491	177	3	+	+	CCONJ
cana-3491	177	4	𝑝𝑦1	𝑝𝑦1	NOUN
cana-3491	177	5	+	+	X
cana-3491	177	6	𝑝	𝑝	NOUN
cana-3491	177	7	2𝑦2	2𝑦2	NUM
cana-3491	177	8	+	+	NOUN
cana-3491	177	9	.	.	PUNCT
cana-3491	177	10	.	.	PUNCT
cana-3491	177	11	.	.	PUNCT
cana-3491	177	12	)	)	PUNCT
cana-3491	178	1	we	we	PRON
cana-3491	178	2	have	have	VERB
cana-3491	178	3	(	(	PUNCT
cana-3491	178	4	𝑦0	𝑦0	NOUN
cana-3491	178	5	+	+	CCONJ
cana-3491	178	6	𝑦1𝑝	𝑦1𝑝	X
cana-3491	178	7	+	+	CCONJ
cana-3491	178	8	𝑦2𝑝	𝑦2𝑝	NOUN
cana-3491	178	9	2	2	NUM
cana-3491	178	10	+	+	PROPN
cana-3491	178	11	.	.	PUNCT
cana-3491	178	12	.	.	PUNCT
cana-3491	178	13	.	.	PUNCT
cana-3491	179	1	)	)	PUNCT
cana-3491	179	2	2	2	X
cana-3491	179	3	=	=	SYM
cana-3491	179	4	(	(	PUNCT
cana-3491	179	5	𝑥0	𝑥0	NOUN
cana-3491	179	6	+	+	CCONJ
cana-3491	179	7	𝑥1𝑝	𝑥1𝑝	PROPN
cana-3491	179	8	+	+	CCONJ
cana-3491	179	9	𝑥2𝑝	𝑥2𝑝	NUM
cana-3491	179	10	2	2	NUM
cana-3491	179	11	+	+	NOUN
cana-3491	179	12	.	.	PUNCT
cana-3491	179	13	.	.	PUNCT
cana-3491	179	14	.	.	PUNCT
cana-3491	180	1	)	)	PUNCT
cana-3491	180	2	3	3	X
cana-3491	181	1	+	+	CCONJ
cana-3491	181	2	𝐴(𝑥0	𝐴(𝑥0	VERB
cana-3491	181	3	+	+	CCONJ
cana-3491	181	4	𝑥1𝑝	𝑥1𝑝	PUNCT
cana-3491	181	5	+	+	CCONJ
cana-3491	181	6	𝑥2𝑝	𝑥2𝑝	NUM
cana-3491	181	7	2	2	NUM
cana-3491	181	8	+	+	NOUN
cana-3491	181	9	.	.	PUNCT
cana-3491	181	10	.	.	PUNCT
cana-3491	181	11	.	.	PUNCT
cana-3491	181	12	)	)	PUNCT
cana-3491	182	1	+	+	CCONJ
cana-3491	182	2	𝐵	𝐵	NOUN
cana-3491	182	3	𝑦0	𝑦0	NOUN
cana-3491	182	4	2	2	NUM
cana-3491	183	1	+	+	CCONJ
cana-3491	183	2	𝑦1	𝑦1	PROPN
cana-3491	183	3	2𝑝2	2𝑝2	NUM
cana-3491	184	1	+	+	CCONJ
cana-3491	184	2	2𝑦0𝑦1𝑝	2𝑦0𝑦1𝑝	NUM
cana-3491	184	3	+	+	CCONJ
cana-3491	184	4	𝑝	𝑝	NOUN
cana-3491	184	5	2𝑡	2𝑡	NOUN
cana-3491	184	6	=	=	SYM
cana-3491	184	7	𝑥0	𝑥0	NOUN
cana-3491	184	8	3	3	NUM
cana-3491	184	9	+	+	SYM
cana-3491	184	10	3𝑥0	3𝑥0	NUM
cana-3491	184	11	2𝑥1𝑝	2𝑥1𝑝	ADJ
cana-3491	185	1	+	+	NUM
cana-3491	185	2	3𝑥0𝑥1	3𝑥0𝑥1	NUM
cana-3491	185	3	2𝑝2	2𝑝2	NUM
cana-3491	186	1	+	+	CCONJ
cana-3491	186	2	𝑝3𝑥1	𝑝3𝑥1	X
cana-3491	186	3	3	3	NUM
cana-3491	186	4	+	+	NUM
cana-3491	186	5	𝐴𝑥0	𝐴𝑥0	NOUN
cana-3491	186	6	+	+	X
cana-3491	186	7	𝐴𝑝𝑥1	𝐴𝑝𝑥1	X
cana-3491	186	8	+	+	CCONJ
cana-3491	186	9	𝐵	𝐵	NOUN
cana-3491	186	10	+	+	CCONJ
cana-3491	186	11	𝑝	𝑝	ADJ
cana-3491	186	12	2𝑘	2𝑘	NUM
cana-3491	186	13	let	let	VERB
cana-3491	186	14	𝑃1̃	𝑃1̃	PROPN
cana-3491	186	15	be	be	AUX
cana-3491	186	16	the	the	DET
cana-3491	186	17	lift	lift	NOUN
cana-3491	186	18	of	of	ADP
cana-3491	186	19	𝑃	𝑃	NOUN
cana-3491	186	20	modulo	modulo	NOUN
cana-3491	186	21	𝑝2	𝑝2	NOUN
cana-3491	186	22	given	give	VERB
cana-3491	186	23	as	as	ADP
cana-3491	186	24	𝑃1̃	𝑃1̃	PROPN
cana-3491	186	25	=	=	SYM
cana-3491	186	26	(	(	PUNCT
cana-3491	186	27	�	�	PROPN
cana-3491	186	28	̃	̃	NOUN
cana-3491	186	29	�	�	NOUN
cana-3491	186	30	1	1	NUM
cana-3491	186	31	,	,	PUNCT
cana-3491	186	32	�	�	PROPN
cana-3491	186	33	̃	̃	NOUN
cana-3491	186	34	�	�	NOUN
cana-3491	186	35	1	1	NUM
cana-3491	186	36	)	)	PUNCT
cana-3491	186	37	with	with	ADP
cana-3491	186	38	�	�	PROPN
cana-3491	186	39	̃	̃	NOUN
cana-3491	186	40	�	�	NOUN
cana-3491	186	41	1	1	NUM
cana-3491	186	42	=	=	SYM
cana-3491	186	43	𝑥0	𝑥0	NOUN
cana-3491	186	44	+	+	CCONJ
cana-3491	186	45	𝑥1𝑝	𝑥1𝑝	PROPN
cana-3491	186	46	�	�	SYM
cana-3491	186	47	̃	̃	PROPN
cana-3491	186	48	�	�	NOUN
cana-3491	186	49	1	1	NUM
cana-3491	186	50	=	=	SYM
cana-3491	186	51	𝑦0	𝑦0	NOUN
cana-3491	186	52	+	+	CCONJ
cana-3491	186	53	𝑦1𝑝	𝑦1𝑝	PROPN
cana-3491	186	54	note	note	VERB
cana-3491	186	55	�	�	PROPN
cana-3491	186	56	̃	̃	PROPN
cana-3491	186	57	�	�	NOUN
cana-3491	186	58	1	1	NUM
cana-3491	186	59	∈	∈	PROPN
cana-3491	186	60	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	186	61	𝑝2	𝑝2	NOUN
cana-3491	186	62	)	)	PUNCT
cana-3491	186	63	𝑦0	𝑦0	NOUN
cana-3491	186	64	2	2	NUM
cana-3491	187	1	+	+	CCONJ
cana-3491	187	2	𝑦1	𝑦1	PROPN
cana-3491	187	3	2𝑝2	2𝑝2	NUM
cana-3491	187	4	+	+	NUM
cana-3491	187	5	2𝑦0𝑦1𝑝	2𝑦0𝑦1𝑝	NUM
cana-3491	187	6	=	=	SYM
cana-3491	187	7	𝑥0	𝑥0	NOUN
cana-3491	187	8	3	3	NUM
cana-3491	187	9	+	+	SYM
cana-3491	187	10	3𝑥0	3𝑥0	NUM
cana-3491	187	11	2𝑥1𝑝	2𝑥1𝑝	ADJ
cana-3491	187	12	+	+	NUM
cana-3491	187	13	3𝑥0𝑥1	3𝑥0𝑥1	NUM
cana-3491	187	14	2𝑝2	2𝑝2	NUM
cana-3491	187	15	+	+	CCONJ
cana-3491	187	16	𝑝3𝑥1	𝑝3𝑥1	X
cana-3491	187	17	3	3	NUM
cana-3491	187	18	+	+	NUM
cana-3491	187	19	𝐴𝑥0	𝐴𝑥0	NOUN
cana-3491	187	20	+	+	X
cana-3491	187	21	𝐴𝑝𝑥1	𝐴𝑝𝑥1	ADJ
cana-3491	187	22	+	+	CCONJ
cana-3491	187	23	𝐵	𝐵	NOUN
cana-3491	187	24	𝑦0	𝑦0	NOUN
cana-3491	187	25	2	2	NUM
cana-3491	187	26	+	+	NUM
cana-3491	187	27	2𝑝𝑦0𝑦1	2𝑝𝑦0𝑦1	NUM
cana-3491	187	28	≡	≡	PROPN
cana-3491	187	29	𝑥0	𝑥0	NOUN
cana-3491	187	30	3	3	NUM
cana-3491	187	31	+	+	SYM
cana-3491	187	32	3𝑥0	3𝑥0	NUM
cana-3491	187	33	2𝑥1𝑝	2𝑥1𝑝	ADJ
cana-3491	187	34	+	+	NUM
cana-3491	187	35	𝐴𝑥0	𝐴𝑥0	NOUN
cana-3491	187	36	+	+	X
cana-3491	187	37	𝐴𝑝𝑥1	𝐴𝑝𝑥1	ADJ
cana-3491	187	38	+	+	CCONJ
cana-3491	187	39	𝐵(𝑚𝑜𝑑	𝐵(𝑚𝑜𝑑	NUM
cana-3491	187	40	𝑝2	𝑝2	NOUN
cana-3491	187	41	)	)	PUNCT
cana-3491	187	42	now	now	ADV
cana-3491	187	43	as	as	ADP
cana-3491	187	44	(	(	PUNCT
cana-3491	187	45	𝑥0	𝑥0	NOUN
cana-3491	187	46	,	,	PUNCT
cana-3491	187	47	𝑦0	𝑦0	NOUN
cana-3491	187	48	)	)	PUNCT
cana-3491	187	49	∈	∈	PROPN
cana-3491	187	50	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	187	51	)	)	PUNCT
cana-3491	187	52	note	note	VERB
cana-3491	187	53	𝑦0	𝑦0	NOUN
cana-3491	187	54	2	2	NUM
cana-3491	187	55	=	=	SYM
cana-3491	187	56	𝑥0	𝑥0	NOUN
cana-3491	187	57	3	3	NUM
cana-3491	187	58	+	+	NUM
cana-3491	187	59	𝐴𝑥0	𝐴𝑥0	NOUN
cana-3491	188	1	+	+	X
cana-3491	188	2	𝐵	𝐵	NOUN
cana-3491	188	3	,	,	PUNCT
cana-3491	188	4	substituting	substitute	VERB
cana-3491	188	5	in	in	ADP
cana-3491	188	6	above	above	ADP
cana-3491	188	7	equation	equation	NOUN
cana-3491	188	8	,	,	PUNCT
cana-3491	188	9	we	we	PRON
cana-3491	188	10	have	have	VERB
cana-3491	188	11	2𝑝𝑦0𝑦1	2𝑝𝑦0𝑦1	NUM
cana-3491	188	12	≡	≡	PROPN
cana-3491	188	13	3𝑥0	3𝑥0	NUM
cana-3491	188	14	2𝑥1𝑝	2𝑥1𝑝	ADJ
cana-3491	188	15	+	+	CCONJ
cana-3491	188	16	𝐴𝑝𝑥1(𝑚𝑜𝑑	𝐴𝑝𝑥1(𝑚𝑜𝑑	X
cana-3491	188	17	𝑝2	𝑝2	NOUN
cana-3491	188	18	)	)	PUNCT
cana-3491	188	19	as	as	ADP
cana-3491	188	20	𝑥0	𝑥0	NOUN
cana-3491	188	21	,	,	PUNCT
cana-3491	188	22	𝑦0	𝑦0	NOUN
cana-3491	188	23	are	be	AUX
cana-3491	188	24	known	know	VERB
cana-3491	188	25	,	,	PUNCT
cana-3491	188	26	we	we	PRON
cana-3491	188	27	can	can	AUX
cana-3491	188	28	obtain	obtain	VERB
cana-3491	188	29	𝑦1	𝑦1	NOUN
cana-3491	188	30	in	in	ADP
cana-3491	188	31	terms	term	NOUN
cana-3491	188	32	of	of	ADP
cana-3491	188	33	𝑥1	𝑥1	NOUN
cana-3491	188	34	by	by	ADP
cana-3491	188	35	assigning	assign	VERB
cana-3491	188	36	values	value	NOUN
cana-3491	188	37	for	for	ADP
cana-3491	188	38	𝑥1	𝑥1	PROPN
cana-3491	188	39	in	in	ADP
cana-3491	188	40	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	188	41	,	,	PUNCT
cana-3491	188	42	therefore	therefore	ADV
cana-3491	188	43	𝑦1	𝑦1	PROPN
cana-3491	188	44	≡	≡	PROPN
cana-3491	188	45	(	(	PUNCT
cana-3491	188	46	2𝑝𝑦0	2𝑝𝑦0	NUM
cana-3491	188	47	)	)	PUNCT
cana-3491	188	48	−1(3𝑥0	−1(3𝑥0	NOUN
cana-3491	188	49	2𝑥1𝑝	2𝑥1𝑝	SYM
cana-3491	189	1	+	+	CCONJ
cana-3491	189	2	𝐴𝑝𝑥1)(𝑚𝑜𝑑	𝐴𝑝𝑥1)(𝑚𝑜𝑑	PROPN
cana-3491	189	3	𝑝2	𝑝2	NOUN
cana-3491	189	4	)	)	PUNCT
cana-3491	189	5	𝑃1̃	𝑃1̃	PROPN
cana-3491	189	6	=	=	SYM
cana-3491	189	7	(	(	PUNCT
cana-3491	189	8	�	�	PROPN
cana-3491	189	9	̃	̃	NOUN
cana-3491	189	10	�	�	NOUN
cana-3491	189	11	1	1	NUM
cana-3491	189	12	,	,	PUNCT
cana-3491	189	13	�	�	PROPN
cana-3491	189	14	̃	̃	NOUN
cana-3491	189	15	�	�	NOUN
cana-3491	189	16	1	1	NUM
cana-3491	189	17	)	)	PUNCT
cana-3491	189	18	=	=	PRON
cana-3491	189	19	(	(	PUNCT
cana-3491	189	20	𝑥0	𝑥0	NOUN
cana-3491	189	21	+	+	CCONJ
cana-3491	189	22	𝑥1𝑝	𝑥1𝑝	ADJ
cana-3491	189	23	,	,	PUNCT
cana-3491	189	24	𝑦0	𝑦0	NOUN
cana-3491	189	25	+	+	CCONJ
cana-3491	189	26	𝑦1𝑝	𝑦1𝑝	ADJ
cana-3491	189	27	)	)	PUNCT
cana-3491	189	28	=	=	SYM
cana-3491	189	29	(	(	PUNCT
cana-3491	189	30	𝑥0	𝑥0	NOUN
cana-3491	189	31	+	+	CCONJ
cana-3491	189	32	𝑝𝑥1	𝑝𝑥1	PROPN
cana-3491	189	33	,	,	PUNCT
cana-3491	189	34	𝑦0	𝑦0	NOUN
cana-3491	189	35	+	+	CCONJ
cana-3491	189	36	𝑝(2𝑝𝑦0	𝑝(2𝑝𝑦0	NOUN
cana-3491	189	37	)	)	PUNCT
cana-3491	189	38	−1(3𝑥0	−1(3𝑥0	NOUN
cana-3491	189	39	2𝑥1𝑝	2𝑥1𝑝	ADJ
cana-3491	190	1	+	+	CCONJ
cana-3491	190	2	𝐴𝑝𝑥1))(𝑚𝑜𝑑	𝐴𝑝𝑥1))(𝑚𝑜𝑑	NOUN
cana-3491	190	3	𝑝2	𝑝2	NOUN
cana-3491	190	4	)	)	PUNCT
cana-3491	190	5	note	note	PROPN
cana-3491	190	6	�	�	PROPN
cana-3491	190	7	̃	̃	PROPN
cana-3491	190	8	�	�	NOUN
cana-3491	190	9	1	1	NUM
cana-3491	190	10	2	2	NUM
cana-3491	190	11	=	=	SYM
cana-3491	190	12	�	�	PROPN
cana-3491	190	13	̃	̃	PROPN
cana-3491	190	14	�	�	NOUN
cana-3491	190	15	1	1	NUM
cana-3491	190	16	3	3	NUM
cana-3491	190	17	+	+	CCONJ
cana-3491	190	18	𝐴	𝐴	PROPN
cana-3491	190	19	�	�	PROPN
cana-3491	190	20	̃	̃	PROPN
cana-3491	190	21	�	�	NOUN
cana-3491	190	22	1	1	NUM
cana-3491	190	23	+	+	CCONJ
cana-3491	190	24	𝐵(𝑚𝑜𝑑	𝐵(𝑚𝑜𝑑	NUM
cana-3491	190	25	𝑝2	𝑝2	NOUN
cana-3491	190	26	)	)	PUNCT
cana-3491	190	27	consider	consider	VERB
cana-3491	190	28	(	(	PUNCT
cana-3491	190	29	�	�	NOUN
cana-3491	190	30	̃	̃	NOUN
cana-3491	190	31	�	�	NOUN
cana-3491	190	32	1	1	NUM
cana-3491	190	33	)	)	PUNCT
cana-3491	190	34	2	2	NUM
cana-3491	190	35	=	=	SYM
cana-3491	190	36	(	(	PUNCT
cana-3491	190	37	𝑦0	𝑦0	NOUN
cana-3491	190	38	+	+	CCONJ
cana-3491	190	39	𝑝(2𝑝𝑦0	𝑝(2𝑝𝑦0	NOUN
cana-3491	190	40	)	)	PUNCT
cana-3491	190	41	−1(3𝑥0	−1(3𝑥0	NOUN
cana-3491	190	42	2𝑥1𝑝	2𝑥1𝑝	NUM
cana-3491	190	43	+	+	CCONJ
cana-3491	190	44	𝐴𝑝𝑥1	𝐴𝑝𝑥1	NOUN
cana-3491	190	45	)	)	PUNCT
cana-3491	190	46	)	)	PUNCT
cana-3491	190	47	2	2	NUM
cana-3491	190	48	≡	≡	PROPN
cana-3491	190	49	(	(	PUNCT
cana-3491	190	50	�	�	PROPN
cana-3491	190	51	̃	̃	PROPN
cana-3491	190	52	�	�	PROPN
cana-3491	190	53	)2(𝑚𝑜𝑑	)2(𝑚𝑜𝑑	PROPN
cana-3491	190	54	𝑝2	𝑝2	NOUN
cana-3491	190	55	)	)	PUNCT
cana-3491	191	1	but	but	CCONJ
cana-3491	191	2	note	note	VERB
cana-3491	191	3	we	we	PRON
cana-3491	191	4	have	have	VERB
cana-3491	191	5	�	�	PROPN
cana-3491	191	6	̃	̃	NOUN
cana-3491	191	7	�	�	NOUN
cana-3491	191	8	1	1	NUM
cana-3491	191	9	3	3	NUM
cana-3491	191	10	+	+	CCONJ
cana-3491	191	11	𝐴	𝐴	PROPN
cana-3491	191	12	�	�	PROPN
cana-3491	191	13	̃	̃	PROPN
cana-3491	191	14	�	�	NOUN
cana-3491	191	15	1	1	NUM
cana-3491	191	16	+	+	NUM
cana-3491	191	17	𝐵	𝐵	NOUN
cana-3491	191	18	=	=	SYM
cana-3491	191	19	(	(	PUNCT
cana-3491	191	20	𝑥0	𝑥0	NOUN
cana-3491	191	21	+	+	CCONJ
cana-3491	191	22	𝑥1𝑝	𝑥1𝑝	NOUN
cana-3491	191	23	)	)	PUNCT
cana-3491	191	24	3	3	NUM
cana-3491	192	1	+	+	CCONJ
cana-3491	192	2	𝐴(𝑥0	𝐴(𝑥0	ADJ
cana-3491	192	3	+	+	CCONJ
cana-3491	192	4	𝑥1𝑝	𝑥1𝑝	ADJ
cana-3491	192	5	)	)	PUNCT
cana-3491	192	6	+	+	CCONJ
cana-3491	192	7	𝐵	𝐵	NOUN
cana-3491	192	8	=	=	SYM
cana-3491	192	9	𝑥0	𝑥0	NOUN
cana-3491	192	10	3	3	NUM
cana-3491	192	11	+	+	NUM
cana-3491	192	12	𝑥1	𝑥1	PROPN
cana-3491	192	13	3𝑝3	3𝑝3	PROPN
cana-3491	192	14	+	+	CCONJ
cana-3491	192	15	3𝑝2𝑥0𝑥1	3𝑝2𝑥0𝑥1	NUM
cana-3491	192	16	2	2	NUM
cana-3491	192	17	+	+	NUM
cana-3491	192	18	3𝑝𝑥0	3𝑝𝑥0	NUM
cana-3491	192	19	2𝑥1	2𝑥1	NUM
cana-3491	192	20	+	+	CCONJ
cana-3491	192	21	𝐴𝑥0	𝐴𝑥0	NOUN
cana-3491	192	22	+	+	X
cana-3491	192	23	𝐴𝑝𝑥1	𝐴𝑝𝑥1	X
cana-3491	192	24	+	+	CCONJ
cana-3491	192	25	𝐵	𝐵	PROPN
cana-3491	192	26	≡	≡	PROPN
cana-3491	192	27	𝑥0	𝑥0	PROPN
cana-3491	192	28	3	3	NUM
cana-3491	192	29	+	+	NUM
cana-3491	192	30	𝐴𝑥0	𝐴𝑥0	NOUN
cana-3491	192	31	+	+	CCONJ
cana-3491	192	32	𝐵(𝑚𝑜𝑑	𝐵(𝑚𝑜𝑑	NUM
cana-3491	192	33	𝑝2	𝑝2	NOUN
cana-3491	192	34	)	)	PUNCT
cana-3491	192	35	≡	≡	PROPN
cana-3491	192	36	𝑦0	𝑦0	PROPN
cana-3491	192	37	2(𝑚𝑜𝑑	2(𝑚𝑜𝑑	NOUN
cana-3491	192	38	𝑝2	𝑝2	NOUN
cana-3491	192	39	)	)	PUNCT
cana-3491	192	40	≡	≡	PROPN
cana-3491	192	41	(	(	PUNCT
cana-3491	192	42	�	�	PROPN
cana-3491	192	43	̃	̃	PROPN
cana-3491	192	44	�	�	PROPN
cana-3491	192	45	)2(𝑚𝑜𝑑	)2(𝑚𝑜𝑑	PROPN
cana-3491	192	46	𝑝2	𝑝2	PROPN
cana-3491	192	47	)	)	PUNCT
cana-3491	192	48	therefore	therefore	ADV
cana-3491	192	49	,	,	PUNCT
cana-3491	192	50	�	�	PROPN
cana-3491	192	51	̃	̃	PROPN
cana-3491	192	52	�	�	NOUN
cana-3491	192	53	1	1	NUM
cana-3491	192	54	2	2	NUM
cana-3491	192	55	=	=	SYM
cana-3491	192	56	�	�	PROPN
cana-3491	192	57	̃	̃	PROPN
cana-3491	192	58	�	�	NOUN
cana-3491	192	59	1	1	NUM
cana-3491	192	60	3	3	NUM
cana-3491	192	61	+	+	CCONJ
cana-3491	192	62	𝐴	𝐴	PROPN
cana-3491	192	63	�	�	PROPN
cana-3491	192	64	̃	̃	PROPN
cana-3491	192	65	�	�	NOUN
cana-3491	192	66	1	1	NUM
cana-3491	192	67	+	+	CCONJ
cana-3491	192	68	𝐵(𝑚𝑜𝑑	𝐵(𝑚𝑜𝑑	NUM
cana-3491	192	69	𝑝2	𝑝2	NOUN
cana-3491	192	70	)	)	PUNCT
cana-3491	192	71	.	.	PUNCT
cana-3491	193	1	hence	hence	ADV
cana-3491	193	2	,	,	PUNCT
cana-3491	193	3	(	(	PUNCT
cana-3491	193	4	�	�	PROPN
cana-3491	193	5	̃	̃	NOUN
cana-3491	193	6	�	�	NOUN
cana-3491	193	7	1	1	NUM
cana-3491	193	8	,	,	PUNCT
cana-3491	193	9	�	�	PROPN
cana-3491	193	10	̃	̃	NOUN
cana-3491	193	11	�	�	NOUN
cana-3491	193	12	1	1	NUM
cana-3491	193	13	)	)	PUNCT
cana-3491	193	14	satisfies	satisfy	VERB
cana-3491	193	15	the	the	DET
cana-3491	193	16	given	give	VERB
cana-3491	193	17	curve	curve	NOUN
cana-3491	193	18	𝛽2	𝛽2	NOUN
cana-3491	193	19	=	=	PRON
cana-3491	193	20	𝛼3	𝛼3	NOUN
cana-3491	193	21	+	+	CCONJ
cana-3491	193	22	𝐴𝛼	𝐴𝛼	VERB
cana-3491	193	23	+	+	NOUN
cana-3491	193	24	𝐵(𝑚𝑜𝑑	𝐵(𝑚𝑜𝑑	NUM
cana-3491	193	25	𝑝2	𝑝2	NOUN
cana-3491	193	26	)	)	PUNCT
cana-3491	193	27	,	,	PUNCT
cana-3491	193	28	now	now	ADV
cana-3491	193	29	repeating	repeat	VERB
cana-3491	193	30	the	the	DET
cana-3491	193	31	above	above	ADJ
cana-3491	193	32	argument	argument	NOUN
cana-3491	193	33	for	for	ADP
cana-3491	193	34	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3491	193	35	𝑝3	𝑝3	ADV
cana-3491	193	36	and	and	CCONJ
cana-3491	193	37	using	use	VERB
cana-3491	193	38	(	(	PUNCT
cana-3491	193	39	�	�	PROPN
cana-3491	193	40	̃	̃	NOUN
cana-3491	193	41	�	�	NOUN
cana-3491	193	42	1	1	NUM
cana-3491	193	43	,	,	PUNCT
cana-3491	193	44	�	�	PROPN
cana-3491	193	45	̃	̃	NOUN
cana-3491	193	46	�	�	NOUN
cana-3491	193	47	1	1	NUM
cana-3491	193	48	)	)	PUNCT
cana-3491	193	49	∈	∈	PROPN
cana-3491	193	50	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	193	51	𝑝2	𝑝2	NOUN
cana-3491	193	52	)	)	PUNCT
cana-3491	193	53	.	.	PUNCT
cana-3491	194	1	let	let	VERB
cana-3491	194	2	𝑃2̃	𝑃2̃	NOUN
cana-3491	194	3	be	be	AUX
cana-3491	194	4	the	the	DET
cana-3491	194	5	lift	lift	NOUN
cana-3491	194	6	of	of	ADP
cana-3491	194	7	𝑃	𝑃	NOUN
cana-3491	194	8	modulo	modulo	NOUN
cana-3491	194	9	𝑝3	𝑝3	ADV
cana-3491	194	10	given	give	VERB
cana-3491	194	11	as	as	ADP
cana-3491	194	12	𝑃2̃	𝑃2̃	NOUN
cana-3491	194	13	=	=	SYM
cana-3491	194	14	(	(	PUNCT
cana-3491	194	15	�	�	PROPN
cana-3491	194	16	̃	̃	NOUN
cana-3491	194	17	�	�	NOUN
cana-3491	194	18	2	2	NUM
cana-3491	194	19	,	,	PUNCT
cana-3491	194	20	�	�	PROPN
cana-3491	194	21	̃	̃	NOUN
cana-3491	194	22	�	�	NOUN
cana-3491	194	23	2	2	NUM
cana-3491	194	24	)	)	PUNCT
cana-3491	194	25	with	with	ADP
cana-3491	194	26	�	�	PROPN
cana-3491	194	27	̃	̃	NOUN
cana-3491	194	28	�	�	NOUN
cana-3491	194	29	2	2	NUM
cana-3491	194	30	=	=	SYM
cana-3491	194	31	𝑥0	𝑥0	NOUN
cana-3491	194	32	+	+	CCONJ
cana-3491	194	33	𝑥1𝑝	𝑥1𝑝	PUNCT
cana-3491	194	34	+	+	CCONJ
cana-3491	194	35	𝑥2𝑝	𝑥2𝑝	PROPN
cana-3491	194	36	2	2	NUM
cana-3491	194	37	�	�	PROPN
cana-3491	194	38	̃	̃	NOUN
cana-3491	194	39	�	�	NOUN
cana-3491	194	40	2	2	NUM
cana-3491	194	41	=	=	SYM
cana-3491	194	42	𝑦0	𝑦0	NOUN
cana-3491	194	43	+	+	CCONJ
cana-3491	194	44	𝑦1𝑝	𝑦1𝑝	X
cana-3491	194	45	+	+	CCONJ
cana-3491	194	46	𝑦2𝑝	𝑦2𝑝	PROPN
cana-3491	194	47	2	2	NUM
cana-3491	194	48	note	note	VERB
cana-3491	194	49	�	�	PROPN
cana-3491	194	50	̃	̃	PROPN
cana-3491	194	51	�	�	NOUN
cana-3491	194	52	2	2	NUM
cana-3491	194	53	∈	∈	NOUN
cana-3491	194	54	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	NOUN
cana-3491	194	55	𝑝3	𝑝3	NOUN
cana-3491	194	56	)	)	PUNCT
cana-3491	194	57	as	as	ADP
cana-3491	194	58	𝑥0	𝑥0	NOUN
cana-3491	194	59	,	,	PUNCT
cana-3491	194	60	𝑥1	𝑥1	NOUN
cana-3491	194	61	,	,	PUNCT
cana-3491	194	62	𝑦0	𝑦0	NOUN
cana-3491	194	63	,	,	PUNCT
cana-3491	194	64	𝑦1	𝑦1	PROPN
cana-3491	194	65	are	be	AUX
cana-3491	194	66	known	know	VERB
cana-3491	194	67	,	,	PUNCT
cana-3491	194	68	we	we	PRON
cana-3491	194	69	can	can	AUX
cana-3491	194	70	obtain	obtain	VERB
cana-3491	194	71	𝑦2	𝑦2	NOUN
cana-3491	194	72	in	in	ADP
cana-3491	194	73	terms	term	NOUN
cana-3491	194	74	of	of	ADP
cana-3491	194	75	𝑥2	𝑥2	NOUN
cana-3491	194	76	by	by	ADP
cana-3491	194	77	assigning	assign	VERB
cana-3491	194	78	values	value	NOUN
cana-3491	194	79	for	for	ADP
cana-3491	194	80	𝑥2	𝑥2	NOUN
cana-3491	194	81	in	in	ADP
cana-3491	194	82	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	194	83	.	.	PUNCT
cana-3491	195	1	also	also	ADV
cana-3491	195	2	communications	communication	NOUN
cana-3491	195	3	on	on	ADP
cana-3491	195	4	applied	apply	VERB
cana-3491	195	5	nonlinear	nonlinear	ADJ
cana-3491	195	6	analysis	analysis	NOUN
cana-3491	195	7	issn	issn	NOUN
cana-3491	195	8	:	:	PUNCT
cana-3491	195	9	1074	1074	NUM
cana-3491	195	10	-	-	PUNCT
cana-3491	195	11	133x	133x	NUM
cana-3491	195	12	vol	vol	NOUN
cana-3491	195	13	32	32	NUM
cana-3491	195	14	no	no	NOUN
cana-3491	195	15	.	.	PUNCT
cana-3491	196	1	7s	7	NOUN
cana-3491	196	2	(	(	PUNCT
cana-3491	196	3	2025	2025	NUM
cana-3491	196	4	)	)	PUNCT
cana-3491	196	5	864	864	NUM
cana-3491	197	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	197	2	we	we	PRON
cana-3491	197	3	can	can	AUX
cana-3491	197	4	see	see	VERB
cana-3491	197	5	that	that	SCONJ
cana-3491	197	6	(	(	PUNCT
cana-3491	197	7	�	�	PROPN
cana-3491	197	8	̃	̃	NOUN
cana-3491	197	9	�	�	NOUN
cana-3491	197	10	2	2	NUM
cana-3491	197	11	,	,	PUNCT
cana-3491	197	12	�	�	PROPN
cana-3491	197	13	̃	̃	NOUN
cana-3491	197	14	�	�	NOUN
cana-3491	197	15	2	2	NUM
cana-3491	197	16	)	)	PUNCT
cana-3491	197	17	satisfies	satisfy	VERB
cana-3491	197	18	the	the	DET
cana-3491	197	19	given	give	VERB
cana-3491	197	20	curve	curve	NOUN
cana-3491	197	21	𝛽2	𝛽2	NOUN
cana-3491	197	22	=	=	PRON
cana-3491	197	23	𝛼3	𝛼3	NOUN
cana-3491	197	24	+	+	CCONJ
cana-3491	198	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	198	2	+	+	CCONJ
cana-3491	198	3	𝐵	𝐵	NOUN
cana-3491	198	4	modulo	modulo	VERB
cana-3491	198	5	𝑝3	𝑝3	ADV
cana-3491	198	6	.	.	PUNCT
cana-3491	199	1	proceeding	proceed	VERB
cana-3491	199	2	so	so	ADV
cana-3491	199	3	on	on	ADV
cana-3491	199	4	,	,	PUNCT
cana-3491	199	5	let	let	VERB
cana-3491	199	6	𝑃	𝑃	PRON
cana-3491	199	7	�	�	NOUN
cana-3491	199	8	̃	̃	PROPN
cana-3491	199	9	�	�	PROPN
cana-3491	199	10	be	be	VERB
cana-3491	199	11	the	the	DET
cana-3491	199	12	lift	lift	NOUN
cana-3491	199	13	of	of	ADP
cana-3491	199	14	𝑃	𝑃	NOUN
cana-3491	199	15	modulo	modulo	NOUN
cana-3491	199	16	𝑝𝑖+1	𝑝𝑖+1	PUNCT
cana-3491	199	17	given	give	VERB
cana-3491	199	18	as	as	ADP
cana-3491	199	19	𝑃	𝑃	PROPN
cana-3491	199	20	�	�	NOUN
cana-3491	199	21	̃	̃	NOUN
cana-3491	199	22	�	�	NOUN
cana-3491	199	23	=	=	SYM
cana-3491	199	24	(	(	PUNCT
cana-3491	199	25	�	�	PROPN
cana-3491	199	26	̃	̃	PROPN
cana-3491	199	27	�	�	NOUN
cana-3491	199	28	𝑖	𝑖	SYM
cana-3491	199	29	,	,	PUNCT
cana-3491	199	30	�	�	PROPN
cana-3491	199	31	̃	̃	PROPN
cana-3491	199	32	�	�	NOUN
cana-3491	199	33	𝑖	𝑖	NUM
cana-3491	199	34	)	)	PUNCT
cana-3491	199	35	with	with	ADP
cana-3491	199	36	�	�	PROPN
cana-3491	199	37	̃	̃	NOUN
cana-3491	199	38	�	�	NOUN
cana-3491	199	39	𝑖	𝑖	SYM
cana-3491	199	40	=	=	NOUN
cana-3491	199	41	𝑥0	𝑥0	NOUN
cana-3491	199	42	+	+	CCONJ
cana-3491	199	43	𝑥1𝑝	𝑥1𝑝	PROPN
cana-3491	199	44	+	+	CCONJ
cana-3491	199	45	𝑥2𝑝	𝑥2𝑝	NUM
cana-3491	199	46	2	2	NUM
cana-3491	199	47	+	+	NOUN
cana-3491	199	48	.	.	PUNCT
cana-3491	199	49	.	.	PUNCT
cana-3491	199	50	.	.	PUNCT
cana-3491	200	1	+	+	CCONJ
cana-3491	200	2	𝑥𝑖𝑝	𝑥𝑖𝑝	NOUN
cana-3491	200	3	𝑖	𝑖	SYM
cana-3491	200	4	�	�	PROPN
cana-3491	200	5	̃	̃	NOUN
cana-3491	200	6	�	�	NOUN
cana-3491	200	7	𝑖	𝑖	DET
cana-3491	200	8	=	=	NOUN
cana-3491	200	9	𝑦0	𝑦0	NOUN
cana-3491	200	10	+	+	CCONJ
cana-3491	200	11	𝑦1𝑝	𝑦1𝑝	X
cana-3491	200	12	+	+	CCONJ
cana-3491	200	13	𝑦2𝑝	𝑦2𝑝	NOUN
cana-3491	200	14	2	2	NUM
cana-3491	200	15	+	+	PROPN
cana-3491	200	16	.	.	PUNCT
cana-3491	200	17	.	.	PUNCT
cana-3491	200	18	.	.	PUNCT
cana-3491	201	1	+	+	ADV
cana-3491	201	2	𝑦𝑖𝑝	𝑦𝑖𝑝	PROPN
cana-3491	201	3	𝑖	𝑖	SYM
cana-3491	201	4	note	note	VERB
cana-3491	201	5	�	�	PROPN
cana-3491	201	6	̃	̃	PROPN
cana-3491	201	7	�	�	NOUN
cana-3491	201	8	2	2	NUM
cana-3491	201	9	∈	∈	NOUN
cana-3491	201	10	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	NOUN
cana-3491	201	11	𝑝𝑖+1	𝑝𝑖+1	NOUN
cana-3491	201	12	)	)	PUNCT
cana-3491	201	13	as	as	ADP
cana-3491	201	14	𝑥0	𝑥0	NOUN
cana-3491	201	15	,	,	PUNCT
cana-3491	201	16	𝑥1	𝑥1	NOUN
cana-3491	201	17	,	,	PUNCT
cana-3491	201	18	.	.	PUNCT
cana-3491	201	19	.	.	PUNCT
cana-3491	201	20	.	.	PUNCT
cana-3491	202	1	𝑥𝑖−1	𝑥𝑖−1	NOUN
cana-3491	202	2	,	,	PUNCT
cana-3491	202	3	𝑦0	𝑦0	NOUN
cana-3491	202	4	,	,	PUNCT
cana-3491	202	5	𝑦1	𝑦1	NOUN
cana-3491	202	6	,	,	PUNCT
cana-3491	202	7	.	.	PUNCT
cana-3491	202	8	.	.	PUNCT
cana-3491	202	9	.	.	PUNCT
cana-3491	203	1	𝑦𝑖−1	𝑦𝑖−1	PROPN
cana-3491	203	2	are	be	AUX
cana-3491	203	3	known	know	VERB
cana-3491	203	4	,	,	PUNCT
cana-3491	203	5	we	we	PRON
cana-3491	203	6	can	can	AUX
cana-3491	203	7	obtain	obtain	VERB
cana-3491	203	8	𝑦𝑖	𝑦𝑖	ADP
cana-3491	203	9	in	in	ADP
cana-3491	203	10	terms	term	NOUN
cana-3491	203	11	of	of	ADP
cana-3491	203	12	𝑥𝑖	𝑥𝑖	NUM
cana-3491	203	13	by	by	ADP
cana-3491	203	14	assigning	assign	VERB
cana-3491	203	15	values	value	NOUN
cana-3491	203	16	for	for	ADP
cana-3491	203	17	𝑥𝑖	𝑥𝑖	PRON
cana-3491	203	18	in	in	ADP
cana-3491	203	19	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	203	20	.	.	PUNCT
cana-3491	204	1	also	also	ADV
cana-3491	204	2	we	we	PRON
cana-3491	204	3	can	can	AUX
cana-3491	204	4	see	see	VERB
cana-3491	204	5	that	that	SCONJ
cana-3491	204	6	(	(	PUNCT
cana-3491	204	7	�	�	PROPN
cana-3491	204	8	̃	̃	PROPN
cana-3491	204	9	�	�	NOUN
cana-3491	204	10	𝑖	𝑖	SYM
cana-3491	204	11	,	,	PUNCT
cana-3491	204	12	�	�	PROPN
cana-3491	204	13	̃	̃	PROPN
cana-3491	204	14	�	�	NOUN
cana-3491	204	15	𝑖	𝑖	SYM
cana-3491	204	16	)	)	PUNCT
cana-3491	204	17	satisfies	satisfy	VERB
cana-3491	204	18	the	the	DET
cana-3491	204	19	given	give	VERB
cana-3491	204	20	curve	curve	NOUN
cana-3491	204	21	𝛽2	𝛽2	NOUN
cana-3491	204	22	=	=	PRON
cana-3491	204	23	𝛼3	𝛼3	NOUN
cana-3491	204	24	+	+	CCONJ
cana-3491	205	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	205	2	+	+	CCONJ
cana-3491	205	3	𝐵	𝐵	NOUN
cana-3491	205	4	modulo	modulo	NOUN
cana-3491	205	5	𝑝𝑖+1	𝑝𝑖+1	PROPN
cana-3491	205	6	.	.	PROPN
cana-3491	205	7	hence	hence	ADV
cana-3491	205	8	,	,	PUNCT
cana-3491	205	9	(	(	PUNCT
cana-3491	205	10	�	�	PROPN
cana-3491	205	11	̃	̃	NOUN
cana-3491	205	12	�	�	PROPN
cana-3491	205	13	,	,	PUNCT
cana-3491	205	14	�	�	PROPN
cana-3491	205	15	̃	̃	PROPN
cana-3491	205	16	�	�	PROPN
cana-3491	205	17	)	)	PUNCT
cana-3491	205	18	=	=	PRON
cana-3491	205	19	(	(	PUNCT
cana-3491	205	20	𝑥0	𝑥0	NOUN
cana-3491	205	21	+	+	CCONJ
cana-3491	205	22	𝑝𝑥1	𝑝𝑥1	NOUN
cana-3491	205	23	+	+	CCONJ
cana-3491	205	24	𝑝	𝑝	NOUN
cana-3491	205	25	2𝑥2	2𝑥2	NUM
cana-3491	205	26	+	+	PROPN
cana-3491	205	27	.	.	PUNCT
cana-3491	205	28	.	.	PUNCT
cana-3491	205	29	.	.	PUNCT
cana-3491	206	1	,	,	PUNCT
cana-3491	206	2	𝑦0	𝑦0	NOUN
cana-3491	206	3	+	+	CCONJ
cana-3491	206	4	𝑝𝑦1	𝑝𝑦1	NOUN
cana-3491	206	5	+	+	X
cana-3491	206	6	𝑝	𝑝	NOUN
cana-3491	206	7	2𝑦2	2𝑦2	NUM
cana-3491	206	8	+	+	NOUN
cana-3491	206	9	.	.	PUNCT
cana-3491	206	10	.	.	PUNCT
cana-3491	206	11	.	.	PUNCT
cana-3491	206	12	)	)	PUNCT
cana-3491	206	13	satisfies	satisfy	VERB
cana-3491	206	14	the	the	DET
cana-3491	206	15	curve	curve	NOUN
cana-3491	206	16	𝛽2	𝛽2	NOUN
cana-3491	206	17	=	=	PRON
cana-3491	206	18	𝛼3	𝛼3	NOUN
cana-3491	206	19	+	+	CCONJ
cana-3491	207	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	207	2	+	+	CCONJ
cana-3491	207	3	𝐵.	𝐵.	PROPN
cana-3491	207	4	i.e.	i.e.	X
cana-3491	207	5	,	,	PUNCT
cana-3491	207	6	the	the	DET
cana-3491	207	7	lift	lift	NOUN
cana-3491	207	8	�	�	PROPN
cana-3491	207	9	̃	̃	PROPN
cana-3491	207	10	�	�	PROPN
cana-3491	207	11	of	of	ADP
cana-3491	207	12	𝑃	𝑃	PROPN
cana-3491	207	13	is	be	AUX
cana-3491	207	14	in	in	ADP
cana-3491	207	15	𝐸(𝑄𝑝	𝐸(𝑄𝑝	ADJ
cana-3491	207	16	)	)	PUNCT
cana-3491	207	17	.	.	PUNCT
cana-3491	208	1	therefore	therefore	ADV
cana-3491	208	2	,	,	PUNCT
cana-3491	208	3	for	for	ADP
cana-3491	208	4	𝑃	𝑃	PROPN
cana-3491	208	5	≠	≠	PROPN
cana-3491	208	6	𝒪	𝒪	PROPN
cana-3491	208	7	∈	∈	PROPN
cana-3491	208	8	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	208	9	)	)	PUNCT
cana-3491	208	10	there	there	PRON
cana-3491	208	11	lies	lie	VERB
cana-3491	208	12	a	a	DET
cana-3491	208	13	lift	lift	NOUN
cana-3491	208	14	�	�	NOUN
cana-3491	208	15	̃	̃	NOUN
cana-3491	208	16	�	�	PROPN
cana-3491	208	17	in	in	ADP
cana-3491	208	18	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	208	19	×	×	NOUN
cana-3491	208	20	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	208	21	such	such	ADJ
cana-3491	208	22	that	that	DET
cana-3491	208	23	�	�	PROPN
cana-3491	208	24	̃	̃	PROPN
cana-3491	208	25	�	�	PROPN
cana-3491	208	26	∈	∈	PROPN
cana-3491	208	27	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	208	28	)	)	PUNCT
cana-3491	208	29	,	,	PUNCT
cana-3491	208	30	with	with	ADP
cana-3491	208	31	reduction	reduction	NOUN
cana-3491	208	32	of	of	ADP
cana-3491	208	33	�	�	PROPN
cana-3491	208	34	̃	̃	PROPN
cana-3491	208	35	�	�	PROPN
cana-3491	208	36	is	be	AUX
cana-3491	208	37	𝑃	𝑃	VERB
cana-3491	208	38	itself	itself	PRON
cana-3491	208	39	.	.	PUNCT
cana-3491	209	1	hence	hence	ADV
cana-3491	209	2	,	,	PUNCT
cana-3491	209	3	for	for	ADP
cana-3491	209	4	all	all	DET
cana-3491	209	5	𝑃	𝑃	PROPN
cana-3491	209	6	∈	∈	NOUN
cana-3491	209	7	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	209	8	)	)	PUNCT
cana-3491	209	9	such	such	ADJ
cana-3491	209	10	that	that	SCONJ
cana-3491	209	11	𝑃	𝑃	PROPN
cana-3491	209	12	≠	≠	PROPN
cana-3491	209	13	𝒪	𝒪	PROPN
cana-3491	209	14	,	,	PUNCT
cana-3491	209	15	𝑃	𝑃	PROPN
cana-3491	209	16	is	be	AUX
cana-3491	209	17	the	the	DET
cana-3491	209	18	reduction	reduction	NOUN
cana-3491	209	19	of	of	ADP
cana-3491	209	20	some	some	DET
cana-3491	209	21	�	�	PROPN
cana-3491	209	22	̃	̃	PROPN
cana-3491	209	23	�	�	PROPN
cana-3491	209	24	∈	∈	PROPN
cana-3491	209	25	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	209	26	)	)	PUNCT
cana-3491	209	27	with	with	ADP
cana-3491	209	28	�	�	PROPN
cana-3491	209	29	̃	̃	NOUN
cana-3491	209	30	�	�	PROPN
cana-3491	209	31	in	in	ADP
cana-3491	209	32	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	209	33	×	×	NOUN
cana-3491	209	34	𝑍𝑝.	𝑍𝑝.	PROPN
cana-3491	209	35	now	now	ADV
cana-3491	209	36	for	for	ADP
cana-3491	209	37	𝑃	𝑃	PROPN
cana-3491	209	38	=	=	SYM
cana-3491	209	39	𝒪	𝒪	PROPN
cana-3491	209	40	,	,	PUNCT
cana-3491	209	41	the	the	DET
cana-3491	209	42	lift	lift	NOUN
cana-3491	209	43	of	of	ADP
cana-3491	209	44	point	point	NOUN
cana-3491	209	45	at	at	ADP
cana-3491	209	46	infinity	infinity	NOUN
cana-3491	209	47	𝒪	𝒪	PROPN
cana-3491	209	48	can	can	AUX
cana-3491	209	49	be	be	AUX
cana-3491	209	50	obtained	obtain	VERB
cana-3491	209	51	by	by	ADP
cana-3491	209	52	considering	consider	VERB
cana-3491	209	53	the	the	DET
cana-3491	209	54	corresponding	corresponding	ADJ
cana-3491	209	55	elliptic	elliptic	ADJ
cana-3491	209	56	curve	curve	NOUN
cana-3491	209	57	involving	involve	VERB
cana-3491	209	58	𝑍-coordinate	𝑍-coordinate	PROPN
cana-3491	209	59	is	be	AUX
cana-3491	209	60	as	as	ADP
cana-3491	209	61	𝑌2𝑍	𝑌2𝑍	PRON
cana-3491	209	62	=	=	PRON
cana-3491	209	63	𝑋3	𝑋3	NOUN
cana-3491	209	64	+	+	CCONJ
cana-3491	209	65	𝐴𝑋𝑍2	𝐴𝑋𝑍2	PROPN
cana-3491	209	66	+	+	X
cana-3491	209	67	𝐵𝑍3	𝐵𝑍3	PROPN
cana-3491	209	68	.	.	PUNCT
cana-3491	210	1	now	now	ADV
cana-3491	210	2	considering	consider	VERB
cana-3491	210	3	the	the	DET
cana-3491	210	4	point	point	NOUN
cana-3491	210	5	at	at	ADP
cana-3491	210	6	infinity	infinity	NOUN
cana-3491	210	7	𝒪	𝒪	PROPN
cana-3491	210	8	=	=	PUNCT
cana-3491	211	1	[	[	X
cana-3491	211	2	0:1	0:1	NUM
cana-3491	211	3	:	:	PUNCT
cana-3491	211	4	0	0	NUM
cana-3491	211	5	]	]	PUNCT
cana-3491	211	6	which	which	PRON
cana-3491	211	7	is	be	AUX
cana-3491	211	8	an	an	DET
cana-3491	211	9	equivalence	equivalence	NOUN
cana-3491	211	10	class	class	NOUN
cana-3491	211	11	of	of	ADP
cana-3491	211	12	points	point	NOUN
cana-3491	211	13	(	(	PUNCT
cana-3491	211	14	0	0	NUM
cana-3491	211	15	,	,	PUNCT
cana-3491	211	16	𝑘	𝑘	NOUN
cana-3491	211	17	,	,	PUNCT
cana-3491	211	18	0	0	NUM
cana-3491	211	19	)	)	PUNCT
cana-3491	211	20	.	.	PUNCT
cana-3491	212	1	we	we	PRON
cana-3491	212	2	have	have	VERB
cana-3491	212	3	for	for	ADP
cana-3491	212	4	𝒪	𝒪	PROPN
cana-3491	212	5	as	as	ADP
cana-3491	212	6	(	(	PUNCT
cana-3491	212	7	0	0	NUM
cana-3491	212	8	,	,	PUNCT
cana-3491	212	9	𝑘	𝑘	NOUN
cana-3491	212	10	,	,	PUNCT
cana-3491	212	11	0	0	NUM
cana-3491	212	12	)	)	PUNCT
cana-3491	212	13	,	,	PUNCT
cana-3491	212	14	the	the	DET
cana-3491	212	15	lift	lift	NOUN
cana-3491	212	16	of	of	ADP
cana-3491	212	17	𝑃	𝑃	NOUN
cana-3491	212	18	=	=	SYM
cana-3491	212	19	𝒪	𝒪	NOUN
cana-3491	212	20	=	=	SYM
cana-3491	212	21	(	(	PUNCT
cana-3491	212	22	0	0	NUM
cana-3491	212	23	,	,	PUNCT
cana-3491	212	24	𝑘	𝑘	NOUN
cana-3491	212	25	,	,	PUNCT
cana-3491	212	26	0	0	NUM
cana-3491	212	27	)	)	PUNCT
cana-3491	212	28	is	be	AUX
cana-3491	212	29	given	give	VERB
cana-3491	212	30	as	as	ADP
cana-3491	212	31	�	�	PROPN
cana-3491	212	32	̃	̃	PROPN
cana-3491	212	33	�	�	NOUN
cana-3491	212	34	=	=	PUNCT
cana-3491	212	35	[	[	X
cana-3491	212	36	�	�	PROPN
cana-3491	212	37	̃	̃	PROPN
cana-3491	212	38	�	�	PROPN
cana-3491	212	39	,	,	PUNCT
cana-3491	212	40	�	�	PROPN
cana-3491	212	41	̃	̃	PROPN
cana-3491	212	42	�	�	PROPN
cana-3491	212	43	,	,	PUNCT
cana-3491	212	44	𝑍	𝑍	PROPN
cana-3491	212	45	]	]	PUNCT
cana-3491	212	46	=	=	SYM
cana-3491	212	47	(	(	PUNCT
cana-3491	212	48	0	0	NUM
cana-3491	212	49	+	+	CCONJ
cana-3491	212	50	𝑋1𝑝	𝑋1𝑝	VERB
cana-3491	212	51	+	+	CCONJ
cana-3491	212	52	𝑋2𝑝	𝑋2𝑝	PROPN
cana-3491	212	53	2	2	NUM
cana-3491	212	54	+	+	NOUN
cana-3491	212	55	.	.	PUNCT
cana-3491	212	56	.	.	PUNCT
cana-3491	212	57	.	.	PUNCT
cana-3491	213	1	,	,	PUNCT
cana-3491	213	2	𝑘	𝑘	X
cana-3491	213	3	+	+	CCONJ
cana-3491	214	1	𝑌1𝑝	𝑌1𝑝	PROPN
cana-3491	214	2	+	+	CCONJ
cana-3491	214	3	𝑌2𝑝	𝑌2𝑝	PROPN
cana-3491	214	4	2	2	NUM
cana-3491	214	5	+	+	PROPN
cana-3491	214	6	.	.	PUNCT
cana-3491	214	7	.	.	PUNCT
cana-3491	214	8	.	.	PUNCT
cana-3491	215	1	,	,	PUNCT
cana-3491	215	2	0	0	NUM
cana-3491	216	1	+	+	CCONJ
cana-3491	216	2	𝑍1𝑝	𝑍1𝑝	PROPN
cana-3491	216	3	+	+	CCONJ
cana-3491	216	4	𝑍2𝑝	𝑍2𝑝	PROPN
cana-3491	216	5	2	2	NUM
cana-3491	216	6	+	+	NOUN
cana-3491	216	7	.	.	PUNCT
cana-3491	216	8	.	.	PUNCT
cana-3491	216	9	.	.	PUNCT
cana-3491	216	10	)	)	PUNCT
cana-3491	217	1	such	such	ADJ
cana-3491	217	2	that	that	DET
cana-3491	217	3	�	�	PROPN
cana-3491	217	4	̃	̃	PROPN
cana-3491	217	5	�	�	PROPN
cana-3491	217	6	satisfies	satisfy	VERB
cana-3491	217	7	the	the	DET
cana-3491	217	8	curve	curve	NOUN
cana-3491	217	9	𝑌2𝑍	𝑌2𝑍	PUNCT
cana-3491	217	10	=	=	PRON
cana-3491	218	1	𝑋3	𝑋3	NOUN
cana-3491	218	2	+	+	CCONJ
cana-3491	218	3	𝐴𝑋𝑍2	𝐴𝑋𝑍2	PROPN
cana-3491	218	4	+	+	X
cana-3491	218	5	𝐵𝑍3	𝐵𝑍3	PROPN
cana-3491	218	6	.	.	PUNCT
cana-3491	219	1	hence	hence	ADV
cana-3491	219	2	,	,	PUNCT
cana-3491	219	3	we	we	PRON
cana-3491	219	4	have	have	VERB
cana-3491	219	5	(	(	PUNCT
cana-3491	219	6	𝑘	𝑘	X
cana-3491	219	7	+	+	CCONJ
cana-3491	219	8	𝑌1𝑝	𝑌1𝑝	PROPN
cana-3491	220	1	+	+	CCONJ
cana-3491	220	2	𝑌2𝑝	𝑌2𝑝	PROPN
cana-3491	220	3	2	2	NUM
cana-3491	220	4	+	+	PROPN
cana-3491	220	5	.	.	PUNCT
cana-3491	220	6	.	.	PUNCT
cana-3491	220	7	.	.	PUNCT
cana-3491	221	1	)	)	PUNCT
cana-3491	222	1	2(0	2(0	NUM
cana-3491	223	1	+	+	NUM
cana-3491	223	2	𝑍1𝑝	𝑍1𝑝	PROPN
cana-3491	223	3	+	+	CCONJ
cana-3491	223	4	𝑍2𝑝	𝑍2𝑝	PROPN
cana-3491	223	5	2	2	NUM
cana-3491	223	6	+	+	NOUN
cana-3491	223	7	.	.	PUNCT
cana-3491	223	8	.	.	PUNCT
cana-3491	223	9	.	.	PUNCT
cana-3491	223	10	)	)	PUNCT
cana-3491	224	1	=	=	PUNCT
cana-3491	224	2	(	(	PUNCT
cana-3491	224	3	0	0	NUM
cana-3491	224	4	+	+	CCONJ
cana-3491	224	5	𝑋1𝑝	𝑋1𝑝	VERB
cana-3491	225	1	+	+	CCONJ
cana-3491	225	2	𝑋2𝑝	𝑋2𝑝	PROPN
cana-3491	225	3	2	2	NUM
cana-3491	225	4	+	+	NOUN
cana-3491	225	5	.	.	PUNCT
cana-3491	225	6	.	.	PUNCT
cana-3491	225	7	.	.	PUNCT
cana-3491	226	1	)	)	PUNCT
cana-3491	226	2	3	3	X
cana-3491	227	1	+	+	CCONJ
cana-3491	227	2	𝐴(0	𝐴(0	X
cana-3491	228	1	+	+	CCONJ
cana-3491	228	2	𝑋1𝑝	𝑋1𝑝	VERB
cana-3491	228	3	+	+	CCONJ
cana-3491	228	4	𝑋2𝑝	𝑋2𝑝	PROPN
cana-3491	228	5	2	2	NUM
cana-3491	228	6	+	+	NOUN
cana-3491	228	7	.	.	PUNCT
cana-3491	228	8	.	.	PUNCT
cana-3491	228	9	.	.	PUNCT
cana-3491	228	10	)	)	PUNCT
cana-3491	229	1	(	(	PUNCT
cana-3491	229	2	0	0	X
cana-3491	229	3	+	+	CCONJ
cana-3491	229	4	𝑍1𝑝	𝑍1𝑝	PROPN
cana-3491	229	5	+	+	CCONJ
cana-3491	229	6	𝑍2𝑝	𝑍2𝑝	PROPN
cana-3491	229	7	2	2	NUM
cana-3491	229	8	+	+	NOUN
cana-3491	229	9	.	.	PUNCT
cana-3491	229	10	.	.	PUNCT
cana-3491	229	11	.	.	PUNCT
cana-3491	230	1	)	)	PUNCT
cana-3491	230	2	2	2	NUM
cana-3491	231	1	+	+	NUM
cana-3491	231	2	𝐵(0	𝐵(0	X
cana-3491	231	3	+	+	CCONJ
cana-3491	231	4	𝑍1𝑝	𝑍1𝑝	PROPN
cana-3491	231	5	+	+	CCONJ
cana-3491	231	6	𝑍2𝑝	𝑍2𝑝	PROPN
cana-3491	231	7	2	2	NUM
cana-3491	231	8	+	+	NOUN
cana-3491	231	9	.	.	PUNCT
cana-3491	231	10	.	.	PUNCT
cana-3491	231	11	.	.	PUNCT
cana-3491	232	1	)	)	PUNCT
cana-3491	232	2	3	3	NUM
cana-3491	232	3	on	on	ADP
cana-3491	232	4	reducing	reduce	VERB
cana-3491	232	5	𝑚𝑜𝑑𝑝2	𝑚𝑜𝑑𝑝2	VERB
cana-3491	232	6	,	,	PUNCT
cana-3491	232	7	we	we	PRON
cana-3491	232	8	have	have	VERB
cana-3491	232	9	(	(	PUNCT
cana-3491	232	10	𝑘	𝑘	X
cana-3491	232	11	+	+	X
cana-3491	232	12	𝑌1𝑝	𝑌1𝑝	PROPN
cana-3491	232	13	)	)	PUNCT
cana-3491	232	14	2(0	2(0	NUM
cana-3491	233	1	+	+	PUNCT
cana-3491	233	2	𝑍1𝑝	𝑍1𝑝	PROPN
cana-3491	233	3	)	)	PUNCT
cana-3491	233	4	=	=	PUNCT
cana-3491	233	5	(	(	PUNCT
cana-3491	233	6	0	0	NUM
cana-3491	233	7	+	+	CCONJ
cana-3491	233	8	𝑋1𝑝	𝑋1𝑝	VERB
cana-3491	233	9	)	)	PUNCT
cana-3491	233	10	3	3	NUM
cana-3491	233	11	+	+	CCONJ
cana-3491	233	12	𝐴(0	𝐴(0	X
cana-3491	234	1	+	+	CCONJ
cana-3491	234	2	𝑋1𝑝)(0	𝑋1𝑝)(0	PROPN
cana-3491	234	3	+	+	SYM
cana-3491	234	4	𝑍1𝑝	𝑍1𝑝	PROPN
cana-3491	234	5	)	)	PUNCT
cana-3491	234	6	2	2	NUM
cana-3491	234	7	+	+	CCONJ
cana-3491	234	8	𝐵(0	𝐵(0	X
cana-3491	234	9	+	+	CCONJ
cana-3491	234	10	𝑍1𝑝	𝑍1𝑝	PROPN
cana-3491	234	11	)	)	PUNCT
cana-3491	234	12	3(𝑚𝑜𝑑	3(𝑚𝑜𝑑	NUM
cana-3491	234	13	𝑝2	𝑝2	NOUN
cana-3491	234	14	)	)	PUNCT
cana-3491	234	15	(	(	PUNCT
cana-3491	234	16	𝑘2	𝑘2	PROPN
cana-3491	234	17	+	+	CCONJ
cana-3491	234	18	𝑌1	𝑌1	PROPN
cana-3491	234	19	2𝑝2	2𝑝2	NUM
cana-3491	234	20	+	+	CCONJ
cana-3491	234	21	2𝑘𝑌1𝑝)𝑍1𝑝	2𝑘𝑌1𝑝)𝑍1𝑝	NUM
cana-3491	234	22	≡	≡	PROPN
cana-3491	234	23	𝑋1	𝑋1	NOUN
cana-3491	234	24	3	3	NUM
cana-3491	234	25	+	+	SYM
cana-3491	234	26	𝐴𝑋1𝑍1𝑝	𝐴𝑋1𝑍1𝑝	PROPN
cana-3491	234	27	2	2	NUM
cana-3491	234	28	+	+	NUM
cana-3491	234	29	𝐵𝑍1𝑝	𝐵𝑍1𝑝	ADJ
cana-3491	234	30	3(𝑚𝑜𝑑	3(𝑚𝑜𝑑	NUM
cana-3491	234	31	𝑝2	𝑝2	NOUN
cana-3491	234	32	)	)	PUNCT
cana-3491	234	33	𝑘2𝑍1𝑝	𝑘2𝑍1𝑝	ADJ
cana-3491	234	34	≡	≡	PROPN
cana-3491	234	35	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	PROPN
cana-3491	234	36	𝑝2	𝑝2	PROPN
cana-3491	234	37	)	)	PUNCT
cana-3491	234	38	𝑍1𝑝	𝑍1𝑝	PROPN
cana-3491	234	39	≡	≡	PROPN
cana-3491	234	40	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	PROPN
cana-3491	234	41	𝑝2	𝑝2	NOUN
cana-3491	234	42	)	)	PUNCT
cana-3491	234	43	𝑍1	𝑍1	PROPN
cana-3491	234	44	≡	≡	PROPN
cana-3491	234	45	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	PROPN
cana-3491	235	1	𝑝	𝑝	NOUN
cana-3491	235	2	)	)	PUNCT
cana-3491	235	3	also	also	ADV
cana-3491	235	4	,	,	PUNCT
cana-3491	235	5	note	note	VERB
cana-3491	235	6	𝑍1	𝑍1	PROPN
cana-3491	235	7	≡	≡	PROPN
cana-3491	235	8	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	PROPN
cana-3491	236	1	𝑝	𝑝	NOUN
cana-3491	236	2	)	)	PUNCT
cana-3491	236	3	⇒	⇒	VERB
cana-3491	236	4	𝑋1	𝑋1	PROPN
cana-3491	236	5	≡	≡	PROPN
cana-3491	236	6	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	PROPN
cana-3491	237	1	𝑝	𝑝	NOUN
cana-3491	237	2	)	)	PUNCT
cana-3491	237	3	therefore	therefore	ADV
cana-3491	237	4	,	,	PUNCT
cana-3491	237	5	substituting	substitute	VERB
cana-3491	237	6	in	in	ADP
cana-3491	237	7	�	�	PROPN
cana-3491	237	8	̃	̃	NOUN
cana-3491	237	9	�	�	PROPN
cana-3491	237	10	we	we	PRON
cana-3491	237	11	have	have	VERB
cana-3491	237	12	for	for	ADP
cana-3491	237	13	�	�	PROPN
cana-3491	237	14	̃	̃	PROPN
cana-3491	237	15	�	�	PROPN
cana-3491	237	16	(𝑚𝑜𝑑	(𝑚𝑜𝑑	PUNCT
cana-3491	237	17	𝑝2	𝑝2	NOUN
cana-3491	237	18	)	)	PUNCT
cana-3491	237	19	=	=	SYM
cana-3491	237	20	�	�	PROPN
cana-3491	237	21	̃	̃	PROPN
cana-3491	237	22	�	�	SYM
cana-3491	237	23	1	1	NUM
cana-3491	237	24	say	say	VERB
cana-3491	237	25	⇒	⇒	PROPN
cana-3491	237	26	�	�	PROPN
cana-3491	237	27	̃	̃	PROPN
cana-3491	237	28	�	�	NOUN
cana-3491	237	29	1	1	NUM
cana-3491	237	30	=	=	SYM
cana-3491	237	31	(	(	PUNCT
cana-3491	237	32	0	0	NUM
cana-3491	237	33	,	,	PUNCT
cana-3491	237	34	𝑘	𝑘	X
cana-3491	237	35	+	+	CCONJ
cana-3491	237	36	𝑝𝑌1	𝑝𝑌1	ADJ
cana-3491	237	37	,	,	PUNCT
cana-3491	237	38	0)(𝑚𝑜𝑑	0)(𝑚𝑜𝑑	ADJ
cana-3491	237	39	𝑝2	𝑝2	NOUN
cana-3491	237	40	)	)	PUNCT
cana-3491	237	41	proceeding	proceeding	NOUN
cana-3491	237	42	as	as	ADP
cana-3491	237	43	above	above	ADV
cana-3491	237	44	for	for	ADP
cana-3491	237	45	𝑚𝑜𝑑	𝑚𝑜𝑑	PROPN
cana-3491	237	46	𝑝3	𝑝3	NOUN
cana-3491	237	47	,	,	PUNCT
cana-3491	237	48	we	we	PRON
cana-3491	237	49	have	have	VERB
cana-3491	237	50	�	�	PROPN
cana-3491	237	51	̃	̃	NOUN
cana-3491	237	52	�	�	NOUN
cana-3491	237	53	2	2	NUM
cana-3491	237	54	=	=	SYM
cana-3491	237	55	(	(	PUNCT
cana-3491	237	56	0	0	NUM
cana-3491	237	57	,	,	PUNCT
cana-3491	237	58	𝑘	𝑘	X
cana-3491	237	59	+	+	CCONJ
cana-3491	237	60	𝑌1𝑝	𝑌1𝑝	PROPN
cana-3491	237	61	+	+	CCONJ
cana-3491	237	62	𝑌2𝑝	𝑌2𝑝	PROPN
cana-3491	237	63	2	2	NUM
cana-3491	237	64	,	,	PUNCT
cana-3491	237	65	0)(𝑚𝑜𝑑	0)(𝑚𝑜𝑑	NOUN
cana-3491	237	66	𝑝3	𝑝3	NOUN
cana-3491	237	67	)	)	PUNCT
cana-3491	237	68	on	on	ADP
cana-3491	237	69	continuing	continue	VERB
cana-3491	237	70	so	so	ADV
cana-3491	237	71	on	on	ADV
cana-3491	237	72	,	,	PUNCT
cana-3491	237	73	we	we	PRON
cana-3491	237	74	have	have	VERB
cana-3491	237	75	𝑍𝑛	𝑍𝑛	PROPN
cana-3491	237	76	≡	≡	PROPN
cana-3491	237	77	0(𝑚𝑜𝑑	0(𝑚𝑜𝑑	ADJ
cana-3491	238	1	𝑝	𝑝	NOUN
cana-3491	238	2	)	)	PUNCT
cana-3491	238	3	for	for	ADP
cana-3491	238	4	𝑛	𝑛	NOUN
cana-3491	238	5	=	=	SYM
cana-3491	238	6	0,1,2	0,1,2	NOUN
cana-3491	238	7	,	,	PUNCT
cana-3491	238	8	.	.	PUNCT
cana-3491	239	1	..	..	PUNCT
cana-3491	240	1	hence	hence	ADV
cana-3491	240	2	,	,	PUNCT
cana-3491	240	3	�	�	PROPN
cana-3491	240	4	̃	̃	NOUN
cana-3491	240	5	�	�	NOUN
cana-3491	240	6	=	=	SYM
cana-3491	240	7	(	(	PUNCT
cana-3491	240	8	0	0	NUM
cana-3491	240	9	,	,	PUNCT
cana-3491	240	10	𝑘	𝑘	X
cana-3491	240	11	+	+	CCONJ
cana-3491	241	1	𝑌1𝑝	𝑌1𝑝	PROPN
cana-3491	241	2	+	+	CCONJ
cana-3491	241	3	𝑌2𝑝	𝑌2𝑝	PROPN
cana-3491	241	4	2	2	NUM
cana-3491	241	5	+	+	PROPN
cana-3491	241	6	.	.	PUNCT
cana-3491	241	7	.	.	PUNCT
cana-3491	241	8	.	.	PUNCT
cana-3491	242	1	,	,	PUNCT
cana-3491	242	2	0	0	X
cana-3491	242	3	)	)	PUNCT
cana-3491	242	4	=	=	NOUN
cana-3491	242	5	(	(	PUNCT
cana-3491	242	6	0,1	0,1	NUM
cana-3491	243	1	+	+	CCONJ
cana-3491	243	2	𝑌1𝑝	𝑌1𝑝	PROPN
cana-3491	243	3	+	+	CCONJ
cana-3491	243	4	𝑌2𝑝	𝑌2𝑝	PROPN
cana-3491	243	5	2	2	NUM
cana-3491	243	6	+	+	PROPN
cana-3491	243	7	.	.	PUNCT
cana-3491	243	8	.	.	PUNCT
cana-3491	243	9	.	.	PUNCT
cana-3491	244	1	,	,	PUNCT
cana-3491	244	2	0	0	X
cana-3491	244	3	)	)	PUNCT
cana-3491	244	4	communications	communication	NOUN
cana-3491	244	5	on	on	ADP
cana-3491	244	6	applied	apply	VERB
cana-3491	244	7	nonlinear	nonlinear	ADJ
cana-3491	244	8	analysis	analysis	NOUN
cana-3491	244	9	issn	issn	NOUN
cana-3491	244	10	:	:	PUNCT
cana-3491	244	11	1074	1074	NUM
cana-3491	244	12	-	-	PUNCT
cana-3491	244	13	133x	133x	NUM
cana-3491	244	14	vol	vol	NOUN
cana-3491	244	15	32	32	NUM
cana-3491	244	16	no	no	NOUN
cana-3491	244	17	.	.	PUNCT
cana-3491	245	1	7s	7	NOUN
cana-3491	245	2	(	(	PUNCT
cana-3491	245	3	2025	2025	NUM
cana-3491	245	4	)	)	PUNCT
cana-3491	245	5	865	865	NUM
cana-3491	245	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	245	7	therefore	therefore	ADV
cana-3491	245	8	,	,	PUNCT
cana-3491	245	9	�	�	PROPN
cana-3491	245	10	̃	̃	NOUN
cana-3491	245	11	�	�	NOUN
cana-3491	245	12	=	=	PUNCT
cana-3491	246	1	[	[	X
cana-3491	246	2	�	�	PROPN
cana-3491	246	3	̃	̃	PROPN
cana-3491	246	4	�	�	PROPN
cana-3491	246	5	,	,	PUNCT
cana-3491	246	6	�	�	PROPN
cana-3491	246	7	̃	̃	PROPN
cana-3491	246	8	�	�	PROPN
cana-3491	246	9	,	,	PUNCT
cana-3491	246	10	�	�	PROPN
cana-3491	246	11	̃	̃	NOUN
cana-3491	246	12	�	�	NOUN
cana-3491	246	13	]	]	X
cana-3491	246	14	=	=	SYM
cana-3491	246	15	(	(	PUNCT
cana-3491	246	16	0,1	0,1	NUM
cana-3491	247	1	+	+	CCONJ
cana-3491	247	2	𝑌1𝑝	𝑌1𝑝	PROPN
cana-3491	247	3	+	+	CCONJ
cana-3491	247	4	𝑌2𝑝	𝑌2𝑝	PROPN
cana-3491	247	5	2	2	NUM
cana-3491	247	6	+	+	PROPN
cana-3491	247	7	.	.	PUNCT
cana-3491	247	8	.	.	PUNCT
cana-3491	247	9	.	.	PUNCT
cana-3491	248	1	,	,	PUNCT
cana-3491	248	2	0	0	NUM
cana-3491	248	3	)	)	PUNCT
cana-3491	248	4	for	for	ADP
cana-3491	248	5	all	all	DET
cana-3491	248	6	𝑌𝑖	𝑌𝑖	PROPN
cana-3491	248	7	∈	∈	PROPN
cana-3491	249	1	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	249	2	are	be	AUX
cana-3491	249	3	the	the	DET
cana-3491	249	4	lifts	lift	NOUN
cana-3491	249	5	of	of	ADP
cana-3491	249	6	point	point	NOUN
cana-3491	249	7	at	at	ADP
cana-3491	249	8	infinity	infinity	NOUN
cana-3491	249	9	𝒪	𝒪	PROPN
cana-3491	249	10	∈	∈	PROPN
cana-3491	249	11	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	249	12	)	)	PUNCT
cana-3491	249	13	and	and	CCONJ
cana-3491	249	14	is	be	AUX
cana-3491	249	15	denoted	denote	VERB
cana-3491	249	16	as	as	ADP
cana-3491	249	17	�	�	PROPN
cana-3491	249	18	̃	̃	PROPN
cana-3491	249	19	�	�	PROPN
cana-3491	249	20	.	.	PUNCT
cana-3491	250	1	therefore	therefore	ADV
cana-3491	250	2	,	,	PUNCT
cana-3491	250	3	for	for	ADP
cana-3491	250	4	𝑃	𝑃	NOUN
cana-3491	250	5	=	=	SYM
cana-3491	250	6	𝒪	𝒪	PROPN
cana-3491	250	7	∈	∈	PROPN
cana-3491	250	8	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	250	9	)	)	PUNCT
cana-3491	250	10	,	,	PUNCT
cana-3491	250	11	there	there	PRON
cana-3491	250	12	exists	exist	VERB
cana-3491	250	13	the	the	DET
cana-3491	250	14	lift	lift	NOUN
cana-3491	250	15	�	�	NOUN
cana-3491	250	16	̃	̃	PROPN
cana-3491	250	17	�	�	PROPN
cana-3491	250	18	∈	∈	PROPN
cana-3491	250	19	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	250	20	)	)	PUNCT
cana-3491	250	21	such	such	ADJ
cana-3491	250	22	that	that	SCONJ
cana-3491	250	23	𝒪	𝒪	PROPN
cana-3491	250	24	is	be	AUX
cana-3491	250	25	the	the	DET
cana-3491	250	26	reduction	reduction	NOUN
cana-3491	250	27	of	of	ADP
cana-3491	250	28	a	a	DET
cana-3491	250	29	point	point	NOUN
cana-3491	250	30	�	�	SYM
cana-3491	250	31	̃	̃	PROPN
cana-3491	250	32	�	�	PROPN
cana-3491	250	33	∈	∈	PROPN
cana-3491	250	34	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	250	35	)	)	PUNCT
cana-3491	250	36	.	.	PUNCT
cana-3491	251	1	therefore	therefore	ADV
cana-3491	251	2	,	,	PUNCT
cana-3491	251	3	each	each	DET
cana-3491	251	4	point	point	NOUN
cana-3491	251	5	in	in	ADP
cana-3491	251	6	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	251	7	)	)	PUNCT
cana-3491	251	8	is	be	AUX
cana-3491	251	9	a	a	DET
cana-3491	251	10	reduction	reduction	NOUN
cana-3491	251	11	of	of	ADP
cana-3491	251	12	a	a	DET
cana-3491	251	13	point	point	NOUN
cana-3491	251	14	in	in	ADP
cana-3491	251	15	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	251	16	)	)	PUNCT
cana-3491	251	17	.	.	PUNCT
cana-3491	252	1	theorem	theorem	ADJ
cana-3491	252	2	3.2	3.2	NUM
cana-3491	252	3	.	.	PUNCT
cana-3491	253	1	let	let	VERB
cana-3491	253	2	𝐸	𝐸	PRON
cana-3491	253	3	be	be	AUX
cana-3491	253	4	an	an	DET
cana-3491	253	5	elliptic	elliptic	ADJ
cana-3491	253	6	curve	curve	NOUN
cana-3491	253	7	over	over	ADP
cana-3491	253	8	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	253	9	defined	define	VERB
cana-3491	253	10	over	over	ADP
cana-3491	253	11	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	253	12	then	then	ADV
cana-3491	253	13	each	each	DET
cana-3491	253	14	point	point	NOUN
cana-3491	253	15	in	in	ADP
cana-3491	253	16	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	253	17	)	)	PUNCT
cana-3491	253	18	defined	define	VERB
cana-3491	253	19	over	over	ADP
cana-3491	253	20	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	253	21	is	be	AUX
cana-3491	253	22	a	a	DET
cana-3491	253	23	lift	lift	NOUN
cana-3491	253	24	of	of	ADP
cana-3491	253	25	a	a	DET
cana-3491	253	26	point	point	NOUN
cana-3491	253	27	in	in	ADP
cana-3491	253	28	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	253	29	)	)	PUNCT
cana-3491	253	30	.	.	PUNCT
cana-3491	254	1	proof	proof	NOUN
cana-3491	254	2	.	.	PUNCT
cana-3491	255	1	by	by	ADP
cana-3491	255	2	definition	definition	NOUN
cana-3491	255	3	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	255	4	)	)	PUNCT
cana-3491	255	5	=	=	PRON
cana-3491	255	6	{	{	PUNCT
cana-3491	255	7	(	(	PUNCT
cana-3491	255	8	𝛼	𝛼	PROPN
cana-3491	255	9	,	,	PUNCT
cana-3491	255	10	𝛽	𝛽	NOUN
cana-3491	255	11	)	)	PUNCT
cana-3491	255	12	∈	∈	PROPN
cana-3491	256	1	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	256	2	×	×	NOUN
cana-3491	256	3	𝑄𝑝/𝛽	𝑄𝑝/𝛽	NUM
cana-3491	256	4	2	2	NUM
cana-3491	256	5	=	=	NOUN
cana-3491	256	6	𝛼3	𝛼3	NOUN
cana-3491	256	7	+	+	CCONJ
cana-3491	257	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	257	2	+	+	CCONJ
cana-3491	257	3	𝐵	𝐵	NOUN
cana-3491	257	4	}	}	PUNCT
cana-3491	257	5	∪	∪	VERB
cana-3491	257	6	{	{	PUNCT
cana-3491	257	7	�	�	PROPN
cana-3491	257	8	̃	̃	NOUN
cana-3491	257	9	�	�	PROPN
cana-3491	257	10	}	}	PUNCT
cana-3491	257	11	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3491	257	12	𝐸(𝐹𝑝	𝐸(𝐹𝑝	PROPN
cana-3491	257	13	)	)	PUNCT
cana-3491	258	1	=	=	PRON
cana-3491	258	2	{	{	PUNCT
cana-3491	258	3	(	(	PUNCT
cana-3491	258	4	𝛼	𝛼	PROPN
cana-3491	258	5	,	,	PUNCT
cana-3491	258	6	𝛽	𝛽	NOUN
cana-3491	258	7	)	)	PUNCT
cana-3491	258	8	∈	∈	PROPN
cana-3491	259	1	𝐹𝑝	𝐹𝑝	PROPN
cana-3491	259	2	×	×	NOUN
cana-3491	259	3	𝐹𝑝/𝛽	𝐹𝑝/𝛽	NUM
cana-3491	259	4	2	2	NUM
cana-3491	259	5	=	=	NOUN
cana-3491	259	6	𝛼3	𝛼3	NOUN
cana-3491	259	7	+	+	CCONJ
cana-3491	260	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	260	2	+	+	CCONJ
cana-3491	260	3	𝐵	𝐵	NOUN
cana-3491	260	4	}	}	PUNCT
cana-3491	260	5	∪	∪	VERB
cana-3491	260	6	{	{	PUNCT
cana-3491	260	7	�	�	PROPN
cana-3491	260	8	̃	̃	NOUN
cana-3491	260	9	�	�	NOUN
cana-3491	260	10	}	}	PUNCT
cana-3491	260	11	now	now	ADV
cana-3491	260	12	,	,	PUNCT
cana-3491	260	13	note	note	VERB
cana-3491	260	14	for	for	ADP
cana-3491	260	15	any	any	DET
cana-3491	260	16	point	point	NOUN
cana-3491	260	17	�	�	SYM
cana-3491	260	18	̃	̃	NOUN
cana-3491	260	19	�	�	NOUN
cana-3491	260	20	=	=	SYM
cana-3491	260	21	(	(	PUNCT
cana-3491	260	22	�	�	PROPN
cana-3491	260	23	̃	̃	PROPN
cana-3491	260	24	�	�	PROPN
cana-3491	260	25	,	,	PUNCT
cana-3491	260	26	�	�	PROPN
cana-3491	260	27	̃	̃	PROPN
cana-3491	260	28	�	�	PROPN
cana-3491	260	29	)	)	PUNCT
cana-3491	260	30	∈	∈	PROPN
cana-3491	260	31	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	260	32	)	)	PUNCT
cana-3491	260	33	then	then	ADV
cana-3491	260	34	we	we	PRON
cana-3491	260	35	have	have	VERB
cana-3491	260	36	two	two	NUM
cana-3491	260	37	cases	case	NOUN
cana-3491	260	38	.	.	PUNCT
cana-3491	261	1	(	(	PUNCT
cana-3491	261	2	i	i	NOUN
cana-3491	261	3	)	)	PUNCT
cana-3491	261	4	�	�	PROPN
cana-3491	261	5	̃	̃	PROPN
cana-3491	261	6	�	�	PROPN
cana-3491	261	7	∈	∈	PROPN
cana-3491	261	8	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	261	9	×	×	NOUN
cana-3491	261	10	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	261	11	(	(	PUNCT
cana-3491	261	12	ii	ii	NOUN
cana-3491	261	13	)	)	PUNCT
cana-3491	261	14	�	�	PROPN
cana-3491	261	15	̃	̃	PROPN
cana-3491	261	16	�	�	PROPN
cana-3491	261	17	∉	∉	PROPN
cana-3491	262	1	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	262	2	×	×	PROPN
cana-3491	262	3	𝑍𝑝.	𝑍𝑝.	PROPN
cana-3491	262	4	case	case	NOUN
cana-3491	262	5	(	(	PUNCT
cana-3491	262	6	i	i	NOUN
cana-3491	262	7	):	):	PUNCT
cana-3491	262	8	for	for	ADP
cana-3491	262	9	�	�	PROPN
cana-3491	262	10	̃	̃	PROPN
cana-3491	262	11	�	�	PROPN
cana-3491	262	12	∈	∈	PROPN
cana-3491	263	1	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	263	2	×	×	NOUN
cana-3491	263	3	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	263	4	,	,	PUNCT
cana-3491	263	5	we	we	PRON
cana-3491	263	6	have	have	VERB
cana-3491	263	7	�	�	PROPN
cana-3491	263	8	̃	̃	NOUN
cana-3491	263	9	�	�	NOUN
cana-3491	263	10	=	=	SYM
cana-3491	263	11	(	(	PUNCT
cana-3491	263	12	𝑥0	𝑥0	NOUN
cana-3491	263	13	+	+	CCONJ
cana-3491	263	14	𝑝𝑥1	𝑝𝑥1	NOUN
cana-3491	263	15	+	+	CCONJ
cana-3491	263	16	𝑝	𝑝	NOUN
cana-3491	263	17	2𝑥2	2𝑥2	NUM
cana-3491	263	18	+	+	PROPN
cana-3491	263	19	.	.	PUNCT
cana-3491	263	20	.	.	PUNCT
cana-3491	263	21	.	.	PUNCT
cana-3491	264	1	,	,	PUNCT
cana-3491	264	2	𝑦0	𝑦0	NOUN
cana-3491	264	3	+	+	CCONJ
cana-3491	264	4	𝑝𝑦1	𝑝𝑦1	NOUN
cana-3491	264	5	+	+	X
cana-3491	264	6	𝑝	𝑝	NOUN
cana-3491	264	7	2𝑦2	2𝑦2	NUM
cana-3491	264	8	+	+	NOUN
cana-3491	264	9	.	.	PUNCT
cana-3491	264	10	.	.	PUNCT
cana-3491	264	11	.	.	PUNCT
cana-3491	264	12	)	)	PUNCT
cana-3491	265	1	reduction	reduction	NOUN
cana-3491	265	2	of	of	ADP
cana-3491	265	3	�	�	PROPN
cana-3491	265	4	̃	̃	PROPN
cana-3491	265	5	�	�	PROPN
cana-3491	265	6	is	be	AUX
cana-3491	265	7	�	�	PROPN
cana-3491	265	8	̃	̃	PROPN
cana-3491	265	9	�	�	PROPN
cana-3491	265	10	(𝑚𝑜𝑑	(𝑚𝑜𝑑	SYM
cana-3491	265	11	𝑝	𝑝	NOUN
cana-3491	265	12	)	)	PUNCT
cana-3491	265	13	=	=	SYM
cana-3491	265	14	(	(	PUNCT
cana-3491	265	15	𝑥0	𝑥0	NOUN
cana-3491	265	16	,	,	PUNCT
cana-3491	265	17	𝑦0	𝑦0	NOUN
cana-3491	265	18	)	)	PUNCT
cana-3491	265	19	∈	∈	PROPN
cana-3491	265	20	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	265	21	)	)	PUNCT
cana-3491	265	22	hence	hence	ADV
cana-3491	265	23	,	,	PUNCT
cana-3491	265	24	�	�	PROPN
cana-3491	265	25	̃	̃	PROPN
cana-3491	265	26	�	�	PROPN
cana-3491	265	27	is	be	AUX
cana-3491	265	28	a	a	DET
cana-3491	265	29	lift	lift	NOUN
cana-3491	265	30	of	of	ADP
cana-3491	265	31	(	(	PUNCT
cana-3491	265	32	𝑥0	𝑥0	NOUN
cana-3491	265	33	,	,	PUNCT
cana-3491	265	34	𝑦0	𝑦0	NOUN
cana-3491	265	35	)	)	PUNCT
cana-3491	265	36	∈	∈	PROPN
cana-3491	265	37	𝐸(𝐹𝑝	𝐸(𝐹𝑝	PROPN
cana-3491	265	38	)	)	PUNCT
cana-3491	265	39	.	.	PUNCT
cana-3491	266	1	case(ii	case(ii	VERB
cana-3491	266	2	):	):	PUNCT
cana-3491	266	3	if	if	SCONJ
cana-3491	266	4	�	�	PROPN
cana-3491	266	5	̃	̃	PROPN
cana-3491	266	6	�	�	PROPN
cana-3491	266	7	∉	∉	PROPN
cana-3491	267	1	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	267	2	×	×	NOUN
cana-3491	267	3	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	267	4	then	then	ADV
cana-3491	267	5	we	we	PRON
cana-3491	267	6	have	have	VERB
cana-3491	267	7	either	either	CCONJ
cana-3491	267	8	both	both	DET
cana-3491	267	9	�	�	PROPN
cana-3491	267	10	̃	̃	PROPN
cana-3491	267	11	�	�	PROPN
cana-3491	267	12	,	,	PUNCT
cana-3491	267	13	�	�	PROPN
cana-3491	267	14	̃	̃	PROPN
cana-3491	267	15	�	�	PROPN
cana-3491	267	16	∉	∉	PROPN
cana-3491	268	1	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	268	2	or	or	CCONJ
cana-3491	268	3	exactly	exactly	ADV
cana-3491	268	4	one	one	NUM
cana-3491	268	5	of	of	ADP
cana-3491	268	6	�	�	PROPN
cana-3491	268	7	̃	̃	PROPN
cana-3491	268	8	�	�	PROPN
cana-3491	268	9	,	,	PUNCT
cana-3491	268	10	�	�	PROPN
cana-3491	268	11	̃	̃	PROPN
cana-3491	268	12	�	�	PROPN
cana-3491	268	13	is	be	AUX
cana-3491	268	14	in	in	ADP
cana-3491	268	15	𝑍𝑝.	𝑍𝑝.	PROPN
cana-3491	268	16	now	now	ADV
cana-3491	268	17	for	for	ADP
cana-3491	268	18	�	�	PROPN
cana-3491	268	19	̃	̃	PROPN
cana-3491	268	20	�	�	NOUN
cana-3491	268	21	=	=	SYM
cana-3491	268	22	(	(	PUNCT
cana-3491	268	23	�	�	PROPN
cana-3491	268	24	̃	̃	PROPN
cana-3491	268	25	�	�	PROPN
cana-3491	268	26	,	,	PUNCT
cana-3491	268	27	�	�	PROPN
cana-3491	268	28	̃	̃	PROPN
cana-3491	268	29	�	�	PROPN
cana-3491	268	30	)	)	PUNCT
cana-3491	268	31	with	with	ADP
cana-3491	268	32	both	both	PRON
cana-3491	268	33	�	�	PROPN
cana-3491	268	34	̃	̃	PROPN
cana-3491	268	35	�	�	PROPN
cana-3491	268	36	,	,	PUNCT
cana-3491	268	37	�	�	PROPN
cana-3491	268	38	̃	̃	PROPN
cana-3491	268	39	�	�	PROPN
cana-3491	268	40	∉	∉	PROPN
cana-3491	268	41	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	268	42	,	,	PUNCT
cana-3491	268	43	�	�	PROPN
cana-3491	268	44	̃	̃	PROPN
cana-3491	268	45	�	�	PROPN
cana-3491	268	46	is	be	AUX
cana-3491	268	47	the	the	DET
cana-3491	268	48	lift	lift	NOUN
cana-3491	268	49	of	of	ADP
cana-3491	268	50	point	point	NOUN
cana-3491	268	51	at	at	ADP
cana-3491	268	52	infinity	infinity	NOUN
cana-3491	268	53	𝒪	𝒪	PROPN
cana-3491	268	54	∈	∈	PROPN
cana-3491	268	55	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	268	56	)	)	PUNCT
cana-3491	268	57	,	,	PUNCT
cana-3491	268	58	follows	follow	VERB
cana-3491	268	59	from	from	ADP
cana-3491	268	60	[	[	X
cana-3491	268	61	8	8	NUM
cana-3491	268	62	]	]	PUNCT
cana-3491	268	63	.	.	PUNCT
cana-3491	269	1	now	now	ADV
cana-3491	269	2	for	for	ADP
cana-3491	269	3	�	�	PROPN
cana-3491	269	4	̃	̃	PROPN
cana-3491	269	5	�	�	PROPN
cana-3491	269	6	∉	∉	PROPN
cana-3491	270	1	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	270	2	×	×	NOUN
cana-3491	270	3	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	270	4	with	with	ADP
cana-3491	270	5	exactly	exactly	ADV
cana-3491	270	6	one	one	NUM
cana-3491	270	7	of	of	ADP
cana-3491	270	8	�	�	PROPN
cana-3491	270	9	̃	̃	PROPN
cana-3491	270	10	�	�	PROPN
cana-3491	270	11	,	,	PUNCT
cana-3491	270	12	�	�	PROPN
cana-3491	270	13	̃	̃	PROPN
cana-3491	270	14	�	�	PROPN
cana-3491	270	15	is	be	AUX
cana-3491	270	16	in	in	ADP
cana-3491	270	17	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	270	18	then	then	ADV
cana-3491	270	19	we	we	PRON
cana-3491	270	20	have(	have(	VERB
cana-3491	270	21	�	�	PROPN
cana-3491	270	22	̃	̃	PROPN
cana-3491	270	23	�	�	PROPN
cana-3491	270	24	,	,	PUNCT
cana-3491	270	25	�	�	PROPN
cana-3491	270	26	̃	̃	PROPN
cana-3491	270	27	�	�	PROPN
cana-3491	270	28	)	)	PUNCT
cana-3491	270	29	such	such	ADJ
cana-3491	270	30	that	that	SCONJ
cana-3491	270	31	either	either	DET
cana-3491	270	32	�	�	PROPN
cana-3491	270	33	̃	̃	PROPN
cana-3491	270	34	�	�	PROPN
cana-3491	270	35	∈	∈	PROPN
cana-3491	270	36	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	270	37	,	,	PUNCT
cana-3491	270	38	�	�	PROPN
cana-3491	270	39	̃	̃	PROPN
cana-3491	270	40	�	�	PROPN
cana-3491	270	41	∉	∉	PROPN
cana-3491	270	42	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	270	43	or	or	CCONJ
cana-3491	270	44	�	�	PROPN
cana-3491	270	45	̃	̃	PROPN
cana-3491	270	46	�	�	PROPN
cana-3491	270	47	∉	∉	PROPN
cana-3491	270	48	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	270	49	,	,	PUNCT
cana-3491	270	50	�	�	PROPN
cana-3491	270	51	̃	̃	PROPN
cana-3491	270	52	�	�	PROPN
cana-3491	270	53	∈	∈	PROPN
cana-3491	270	54	𝑍𝑝.	𝑍𝑝.	PROPN
cana-3491	271	1	then	then	ADV
cana-3491	271	2	,	,	PUNCT
cana-3491	271	3	if	if	SCONJ
cana-3491	271	4	�	�	PROPN
cana-3491	271	5	̃	̃	PROPN
cana-3491	271	6	�	�	PROPN
cana-3491	271	7	is	be	AUX
cana-3491	271	8	not	not	PART
cana-3491	271	9	reduction	reduction	NOUN
cana-3491	271	10	of	of	ADP
cana-3491	271	11	point	point	NOUN
cana-3491	271	12	at	at	ADP
cana-3491	271	13	infinity	infinity	NOUN
cana-3491	271	14	𝒪	𝒪	PROPN
cana-3491	271	15	∈	∈	PROPN
cana-3491	271	16	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	271	17	)	)	PUNCT
cana-3491	271	18	then	then	ADV
cana-3491	271	19	�	�	PROPN
cana-3491	271	20	̃	̃	PROPN
cana-3491	271	21	�	�	PROPN
cana-3491	271	22	reduction	reduction	NOUN
cana-3491	271	23	is	be	AUX
cana-3491	271	24	of	of	ADP
cana-3491	271	25	the	the	DET
cana-3491	271	26	form	form	NOUN
cana-3491	271	27	[	[	X
cana-3491	271	28	𝑋	𝑋	PROPN
cana-3491	271	29	,	,	PUNCT
cana-3491	271	30	𝑌	𝑌	PROPN
cana-3491	271	31	,	,	PUNCT
cana-3491	271	32	𝑍	𝑍	PROPN
cana-3491	271	33	]	]	PUNCT
cana-3491	271	34	with	with	ADP
cana-3491	271	35	𝑍	𝑍	PROPN
cana-3491	271	36	≠	≠	PROPN
cana-3491	271	37	0	0	NUM
cana-3491	271	38	⇒	⇒	NOUN
cana-3491	271	39	(	(	PUNCT
cana-3491	271	40	𝑋	𝑋	PROPN
cana-3491	271	41	,	,	PUNCT
cana-3491	271	42	𝑌	𝑌	PROPN
cana-3491	271	43	,	,	PUNCT
cana-3491	271	44	𝑍	𝑍	PROPN
cana-3491	271	45	)	)	PUNCT
cana-3491	271	46	=	=	SYM
cana-3491	271	47	(	(	PUNCT
cana-3491	271	48	𝑥	𝑥	NOUN
cana-3491	271	49	,	,	PUNCT
cana-3491	271	50	𝑦	𝑦	NOUN
cana-3491	271	51	,	,	PUNCT
cana-3491	271	52	1	1	X
cana-3491	271	53	)	)	PUNCT
cana-3491	271	54	⇒	⇒	PROPN
cana-3491	271	55	�	�	PROPN
cana-3491	271	56	̃	̃	PROPN
cana-3491	271	57	�	�	PROPN
cana-3491	271	58	is	be	AUX
cana-3491	271	59	the	the	DET
cana-3491	271	60	lift	lift	NOUN
cana-3491	271	61	of	of	ADP
cana-3491	271	62	(	(	PUNCT
cana-3491	271	63	𝛼	𝛼	PROPN
cana-3491	271	64	,	,	PUNCT
cana-3491	271	65	𝛽	𝛽	NOUN
cana-3491	271	66	)	)	PUNCT
cana-3491	271	67	∈	∈	PROPN
cana-3491	271	68	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	271	69	)	)	PUNCT
cana-3491	271	70	,	,	PUNCT
cana-3491	271	71	then	then	ADV
cana-3491	271	72	note	note	VERB
cana-3491	271	73	(	(	PUNCT
cana-3491	271	74	�	�	PROPN
cana-3491	271	75	̃	̃	NOUN
cana-3491	271	76	�	�	PROPN
cana-3491	271	77	,	,	PUNCT
cana-3491	271	78	�	�	PROPN
cana-3491	271	79	̃	̃	PROPN
cana-3491	271	80	�	�	PROPN
cana-3491	271	81	)	)	PUNCT
cana-3491	271	82	is	be	AUX
cana-3491	271	83	a	a	DET
cana-3491	271	84	lift	lift	NOUN
cana-3491	271	85	of	of	ADP
cana-3491	271	86	(	(	PUNCT
cana-3491	271	87	𝛼	𝛼	PROPN
cana-3491	271	88	,	,	PUNCT
cana-3491	271	89	𝛽	𝛽	NOUN
cana-3491	271	90	)	)	PUNCT
cana-3491	271	91	∈	∈	PROPN
cana-3491	271	92	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	271	93	)	)	PUNCT
cana-3491	271	94	.	.	PUNCT
cana-3491	272	1	but	but	CCONJ
cana-3491	272	2	any	any	DET
cana-3491	272	3	lift	lift	NOUN
cana-3491	272	4	of	of	ADP
cana-3491	272	5	(	(	PUNCT
cana-3491	272	6	𝛼	𝛼	PROPN
cana-3491	272	7	,	,	PUNCT
cana-3491	272	8	𝛽	𝛽	NOUN
cana-3491	272	9	)	)	PUNCT
cana-3491	272	10	∈	∈	PROPN
cana-3491	272	11	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	272	12	)	)	PUNCT
cana-3491	272	13	are	be	AUX
cana-3491	272	14	in	in	ADP
cana-3491	272	15	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	272	16	×	×	NOUN
cana-3491	272	17	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	272	18	,	,	PUNCT
cana-3491	272	19	which	which	PRON
cana-3491	272	20	is	be	AUX
cana-3491	272	21	a	a	DET
cana-3491	272	22	contradiction	contradiction	NOUN
cana-3491	272	23	.	.	PUNCT
cana-3491	273	1	hence	hence	ADV
cana-3491	273	2	,	,	PUNCT
cana-3491	273	3	there	there	PRON
cana-3491	273	4	are	be	VERB
cana-3491	273	5	no	no	DET
cana-3491	273	6	�	�	PROPN
cana-3491	273	7	̃	̃	PROPN
cana-3491	273	8	�	�	PROPN
cana-3491	273	9	∈	∈	PROPN
cana-3491	273	10	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	273	11	)	)	PUNCT
cana-3491	273	12	such	such	ADJ
cana-3491	273	13	that	that	DET
cana-3491	273	14	�	�	PROPN
cana-3491	273	15	̃	̃	PROPN
cana-3491	273	16	�	�	PROPN
cana-3491	273	17	=	=	SYM
cana-3491	273	18	(	(	PUNCT
cana-3491	273	19	�	�	PROPN
cana-3491	273	20	̃	̃	PROPN
cana-3491	273	21	�	�	PROPN
cana-3491	273	22	,	,	PUNCT
cana-3491	273	23	�	�	PROPN
cana-3491	273	24	̃	̃	PROPN
cana-3491	273	25	�	�	PROPN
cana-3491	273	26	)	)	PUNCT
cana-3491	273	27	∉	∉	PROPN
cana-3491	274	1	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	274	2	×	×	NOUN
cana-3491	274	3	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	274	4	such	such	ADJ
cana-3491	274	5	that	that	SCONJ
cana-3491	274	6	exactly	exactly	ADV
cana-3491	274	7	one	one	NUM
cana-3491	274	8	of	of	ADP
cana-3491	274	9	�	�	PROPN
cana-3491	274	10	̃	̃	PROPN
cana-3491	274	11	�	�	PROPN
cana-3491	274	12	,	,	PUNCT
cana-3491	274	13	�	�	PROPN
cana-3491	274	14	̃	̃	PROPN
cana-3491	274	15	�	�	PROPN
cana-3491	274	16	∉	∉	PROPN
cana-3491	274	17	𝑍𝑝.	𝑍𝑝.	PROPN
cana-3491	274	18	hence	hence	ADV
cana-3491	274	19	,	,	PUNCT
cana-3491	274	20	for	for	ADP
cana-3491	274	21	all	all	DET
cana-3491	274	22	�	�	PROPN
cana-3491	274	23	̃	̃	PROPN
cana-3491	274	24	�	�	PROPN
cana-3491	274	25	∈	∈	PROPN
cana-3491	274	26	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	274	27	)	)	PUNCT
cana-3491	274	28	,	,	PUNCT
cana-3491	274	29	�	�	PROPN
cana-3491	274	30	̃	̃	PROPN
cana-3491	274	31	�	�	PROPN
cana-3491	274	32	is	be	AUX
cana-3491	274	33	a	a	DET
cana-3491	274	34	lift	lift	NOUN
cana-3491	274	35	of	of	ADP
cana-3491	274	36	a	a	DET
cana-3491	274	37	point	point	NOUN
cana-3491	274	38	𝑃	𝑃	NOUN
cana-3491	274	39	in	in	ADP
cana-3491	274	40	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	274	41	)	)	PUNCT
cana-3491	274	42	.	.	PUNCT
cana-3491	275	1	remark	remark	NOUN
cana-3491	275	2	5	5	NUM
cana-3491	275	3	.	.	PUNCT
cana-3491	276	1	we	we	PRON
cana-3491	276	2	can	can	AUX
cana-3491	276	3	obtain	obtain	VERB
cana-3491	276	4	all	all	DET
cana-3491	276	5	the	the	DET
cana-3491	276	6	points	point	NOUN
cana-3491	276	7	in	in	ADP
cana-3491	276	8	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	276	9	)	)	PUNCT
cana-3491	276	10	by	by	ADP
cana-3491	276	11	starting	start	VERB
cana-3491	276	12	with	with	ADP
cana-3491	276	13	points	point	NOUN
cana-3491	276	14	in	in	ADP
cana-3491	276	15	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	276	16	)	)	PUNCT
cana-3491	276	17	and	and	CCONJ
cana-3491	276	18	lifting	lift	VERB
cana-3491	276	19	each	each	DET
cana-3491	276	20	point	point	NOUN
cana-3491	276	21	in	in	ADP
cana-3491	276	22	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	276	23	)	)	PUNCT
cana-3491	276	24	to	to	ADP
cana-3491	276	25	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	276	26	𝑝2	𝑝2	NOUN
cana-3491	276	27	)	)	PUNCT
cana-3491	276	28	as	as	SCONJ
cana-3491	276	29	described	describe	VERB
cana-3491	276	30	in	in	ADP
cana-3491	276	31	theorem	theorem	ADJ
cana-3491	276	32	3.1	3.1	NUM
cana-3491	276	33	and	and	CCONJ
cana-3491	276	34	lifting	lifting	NOUN
cana-3491	276	35	of	of	ADP
cana-3491	276	36	each	each	DET
cana-3491	276	37	point	point	NOUN
cana-3491	276	38	in	in	ADP
cana-3491	276	39	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	276	40	𝑝2	𝑝2	NOUN
cana-3491	276	41	)	)	PUNCT
cana-3491	276	42	to	to	ADP
cana-3491	276	43	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADV
cana-3491	276	44	𝑝3	𝑝3	ADV
cana-3491	276	45	)	)	PUNCT
cana-3491	276	46	and	and	CCONJ
cana-3491	276	47	so	so	ADV
cana-3491	276	48	on	on	ADV
cana-3491	276	49	.	.	PUNCT
cana-3491	277	1	the	the	DET
cana-3491	277	2	lifting	lifting	NOUN
cana-3491	277	3	of	of	ADP
cana-3491	277	4	points	point	NOUN
cana-3491	277	5	from	from	ADP
cana-3491	277	6	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	277	7	)	)	PUNCT
cana-3491	277	8	to	to	PART
cana-3491	277	9	obtain	obtain	VERB
cana-3491	277	10	points	point	NOUN
cana-3491	277	11	in	in	ADP
cana-3491	277	12	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	277	13	)	)	PUNCT
cana-3491	277	14	is	be	AUX
cana-3491	277	15	depicted	depict	VERB
cana-3491	277	16	in	in	ADP
cana-3491	277	17	the	the	DET
cana-3491	277	18	following	follow	VERB
cana-3491	277	19	figure	figure	NOUN
cana-3491	277	20	2	2	NUM
cana-3491	277	21	.	.	PUNCT
cana-3491	277	22	communications	communication	NOUN
cana-3491	277	23	on	on	ADP
cana-3491	277	24	applied	apply	VERB
cana-3491	277	25	nonlinear	nonlinear	ADJ
cana-3491	277	26	analysis	analysis	NOUN
cana-3491	277	27	issn	issn	NOUN
cana-3491	277	28	:	:	PUNCT
cana-3491	277	29	1074	1074	NUM
cana-3491	277	30	-	-	PUNCT
cana-3491	277	31	133x	133x	NUM
cana-3491	277	32	vol	vol	NOUN
cana-3491	277	33	32	32	NUM
cana-3491	277	34	no	no	NOUN
cana-3491	277	35	.	.	PUNCT
cana-3491	278	1	7s	7	NOUN
cana-3491	278	2	(	(	PUNCT
cana-3491	278	3	2025	2025	NUM
cana-3491	278	4	)	)	PUNCT
cana-3491	278	5	866	866	NUM
cana-3491	278	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	278	7	figure	figure	NOUN
cana-3491	278	8	2	2	NUM
cana-3491	278	9	:	:	PUNCT
cana-3491	278	10	flowchart	flowchart	NOUN
cana-3491	278	11	for	for	ADP
cana-3491	278	12	points	point	NOUN
cana-3491	278	13	in	in	ADP
cana-3491	278	14	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	278	15	)	)	PUNCT
cana-3491	278	16	the	the	DET
cana-3491	278	17	points	point	NOUN
cana-3491	278	18	in	in	ADP
cana-3491	278	19	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	278	20	𝑝2	𝑝2	NOUN
cana-3491	278	21	)	)	PUNCT
cana-3491	278	22	which	which	PRON
cana-3491	278	23	are	be	AUX
cana-3491	278	24	obtained	obtain	VERB
cana-3491	278	25	by	by	ADP
cana-3491	278	26	lifting	lifting	NOUN
cana-3491	278	27	of	of	ADP
cana-3491	278	28	points	point	NOUN
cana-3491	278	29	in	in	ADP
cana-3491	278	30	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	278	31	)	)	PUNCT
cana-3491	278	32	are	be	AUX
cana-3491	278	33	given	give	VERB
cana-3491	278	34	in	in	ADP
cana-3491	278	35	the	the	DET
cana-3491	278	36	following	following	ADJ
cana-3491	278	37	example	example	NOUN
cana-3491	278	38	for	for	ADP
cana-3491	278	39	𝑝	𝑝	NOUN
cana-3491	278	40	=	=	SYM
cana-3491	278	41	5	5	NUM
cana-3491	278	42	.	.	PUNCT
cana-3491	278	43	example	example	NOUN
cana-3491	278	44	3.1	3.1	NUM
cana-3491	278	45	.	.	PUNCT
cana-3491	279	1	consider	consider	VERB
cana-3491	279	2	the	the	DET
cana-3491	279	3	elliptic	elliptic	ADJ
cana-3491	279	4	curve	curve	NOUN
cana-3491	279	5	𝑦2	𝑦2	PROPN
cana-3491	279	6	=	=	SYM
cana-3491	279	7	𝑥3	𝑥3	PROPN
cana-3491	279	8	+	+	CCONJ
cana-3491	279	9	𝑥	𝑥	X
cana-3491	280	1	+	+	NUM
cana-3491	280	2	1	1	NUM
cana-3491	280	3	,	,	PUNCT
cana-3491	280	4	then	then	ADV
cana-3491	280	5	we	we	PRON
cana-3491	280	6	have	have	VERB
cana-3491	280	7	𝐸(𝐹5	𝐸(𝐹5	NOUN
cana-3491	280	8	)	)	PUNCT
cana-3491	281	1	=	=	SYM
cana-3491	281	2	{	{	PUNCT
cana-3491	281	3	(	(	PUNCT
cana-3491	281	4	0,1	0,1	NUM
cana-3491	281	5	)	)	PUNCT
cana-3491	281	6	,	,	PUNCT
cana-3491	281	7	(	(	PUNCT
cana-3491	281	8	0,4	0,4	NOUN
cana-3491	281	9	)	)	PUNCT
cana-3491	281	10	,	,	PUNCT
cana-3491	281	11	(	(	PUNCT
cana-3491	281	12	2,1	2,1	NUM
cana-3491	281	13	)	)	PUNCT
cana-3491	281	14	,	,	PUNCT
cana-3491	281	15	(	(	PUNCT
cana-3491	281	16	2,4	2,4	NUM
cana-3491	281	17	)	)	PUNCT
cana-3491	281	18	,	,	PUNCT
cana-3491	281	19	(	(	PUNCT
cana-3491	281	20	3,1	3,1	NUM
cana-3491	281	21	)	)	PUNCT
cana-3491	281	22	,	,	PUNCT
cana-3491	281	23	(	(	PUNCT
cana-3491	281	24	3,4)(4,2	3,4)(4,2	NUM
cana-3491	281	25	)	)	PUNCT
cana-3491	281	26	,	,	PUNCT
cana-3491	281	27	(	(	PUNCT
cana-3491	281	28	4,3	4,3	NUM
cana-3491	281	29	)	)	PUNCT
cana-3491	281	30	}	}	PUNCT
cana-3491	281	31	∪	∪	X
cana-3491	281	32	{	{	PUNCT
cana-3491	281	33	𝒪	𝒪	PROPN
cana-3491	281	34	}	}	PUNCT
cana-3491	281	35	a	a	DET
cana-3491	281	36	lift	lift	NOUN
cana-3491	281	37	of	of	ADP
cana-3491	281	38	any	any	DET
cana-3491	281	39	point	point	NOUN
cana-3491	281	40	in	in	ADP
cana-3491	281	41	𝐸(𝐹5	𝐸(𝐹5	NOUN
cana-3491	281	42	)	)	PUNCT
cana-3491	281	43	to	to	ADP
cana-3491	281	44	a	a	DET
cana-3491	281	45	point	point	NOUN
cana-3491	281	46	in	in	ADP
cana-3491	281	47	𝑍5	𝑍5	NOUN
cana-3491	281	48	×	×	PROPN
cana-3491	281	49	𝑍5(𝑚𝑜𝑑5	𝑍5(𝑚𝑜𝑑5	PROPN
cana-3491	281	50	2	2	NUM
cana-3491	281	51	)	)	PUNCT
cana-3491	281	52	is	be	AUX
cana-3491	281	53	given	give	VERB
cana-3491	281	54	as	as	ADP
cana-3491	281	55	in	in	ADP
cana-3491	281	56	the	the	DET
cana-3491	281	57	following	follow	VERB
cana-3491	281	58	process	process	NOUN
cana-3491	281	59	:	:	PUNCT
cana-3491	281	60	let	let	VERB
cana-3491	281	61	𝑃	𝑃	VERB
cana-3491	281	62	=	=	SYM
cana-3491	281	63	(	(	PUNCT
cana-3491	281	64	2,1	2,1	NUM
cana-3491	281	65	)	)	PUNCT
cana-3491	281	66	∈	∈	PROPN
cana-3491	281	67	𝐸(𝐹5	𝐸(𝐹5	PROPN
cana-3491	281	68	)	)	PUNCT
cana-3491	281	69	.	.	PUNCT
cana-3491	282	1	lift	lift	PROPN
cana-3491	282	2	𝑃(2,1	𝑃(2,1	NOUN
cana-3491	282	3	)	)	PUNCT
cana-3491	283	1	to	to	ADP
cana-3491	283	2	�	�	PROPN
cana-3491	283	3	̃	̃	NOUN
cana-3491	283	4	�	�	NOUN
cana-3491	283	5	=	=	PUNCT
cana-3491	283	6	(	(	PUNCT
cana-3491	283	7	2	2	NUM
cana-3491	283	8	+	+	CCONJ
cana-3491	283	9	𝑝𝑥1	𝑝𝑥1	PROPN
cana-3491	283	10	,	,	PUNCT
cana-3491	283	11	1	1	NUM
cana-3491	283	12	+	+	CCONJ
cana-3491	283	13	𝑝𝑦1)(𝑚𝑜𝑑5	𝑝𝑦1)(𝑚𝑜𝑑5	PROPN
cana-3491	283	14	2	2	NUM
cana-3491	283	15	)	)	PUNCT
cana-3491	283	16	,	,	PUNCT
cana-3491	283	17	0	0	NUM
cana-3491	283	18	≤	≤	NUM
cana-3491	283	19	𝑥1	𝑥1	PROPN
cana-3491	283	20	,	,	PUNCT
cana-3491	283	21	𝑦1	𝑦1	PROPN
cana-3491	283	22	<	<	X
cana-3491	283	23	5	5	NUM
cana-3491	283	24	then	then	ADV
cana-3491	283	25	�	�	PROPN
cana-3491	283	26	̃	̃	PROPN
cana-3491	283	27	�	�	PROPN
cana-3491	283	28	∈	∈	PROPN
cana-3491	283	29	𝐸(𝑄5)(𝑚𝑜𝑑5	𝐸(𝑄5)(𝑚𝑜𝑑5	NOUN
cana-3491	283	30	2	2	NUM
cana-3491	283	31	)	)	PUNCT
cana-3491	283	32	then	then	ADV
cana-3491	283	33	as	as	SCONJ
cana-3491	283	34	𝑃	𝑃	NOUN
cana-3491	283	35	satisfies	satisfie	NOUN
cana-3491	283	36	𝑦2	𝑦2	NOUN
cana-3491	283	37	=	=	SYM
cana-3491	283	38	𝑥3	𝑥3	PROPN
cana-3491	283	39	+	+	CCONJ
cana-3491	283	40	𝑥	𝑥	X
cana-3491	284	1	+	+	NUM
cana-3491	284	2	1	1	NUM
cana-3491	284	3	,	,	PUNCT
cana-3491	284	4	note	note	NOUN
cana-3491	284	5	(	(	PUNCT
cana-3491	284	6	1	1	NUM
cana-3491	284	7	+	+	CCONJ
cana-3491	284	8	𝑝𝑦1	𝑝𝑦1	NOUN
cana-3491	284	9	)	)	PUNCT
cana-3491	284	10	2	2	NUM
cana-3491	284	11	=	=	SYM
cana-3491	284	12	(	(	PUNCT
cana-3491	284	13	2	2	NUM
cana-3491	284	14	+	+	CCONJ
cana-3491	284	15	𝑝𝑥1	𝑝𝑥1	PROPN
cana-3491	284	16	)	)	PUNCT
cana-3491	284	17	3	3	NUM
cana-3491	285	1	+	+	CCONJ
cana-3491	285	2	2	2	NUM
cana-3491	285	3	+	+	CCONJ
cana-3491	285	4	𝑝𝑥1	𝑝𝑥1	NOUN
cana-3491	285	5	+	+	CCONJ
cana-3491	285	6	1(𝑚𝑜𝑑5	1(𝑚𝑜𝑑5	NUM
cana-3491	285	7	2	2	NUM
cana-3491	285	8	)	)	PUNCT
cana-3491	285	9	now	now	ADV
cana-3491	285	10	expressing	express	VERB
cana-3491	285	11	𝑦1	𝑦1	NOUN
cana-3491	285	12	in	in	ADP
cana-3491	285	13	terms	term	NOUN
cana-3491	285	14	of	of	ADP
cana-3491	285	15	𝑥1	𝑥1	NOUN
cana-3491	285	16	,	,	PUNCT
cana-3491	285	17	we	we	PRON
cana-3491	285	18	have	have	VERB
cana-3491	285	19	𝑦1	𝑦1	PROPN
cana-3491	285	20	≡	≡	PROPN
cana-3491	285	21	𝑥1	𝑥1	PROPN
cana-3491	285	22	+	+	PROPN
cana-3491	285	23	4(𝑚𝑜𝑑5	4(𝑚𝑜𝑑5	NOUN
cana-3491	285	24	)	)	PUNCT
cana-3491	285	25	substituting	substitute	VERB
cana-3491	285	26	for	for	ADP
cana-3491	285	27	𝑦1	𝑦1	PROPN
cana-3491	285	28	in	in	ADP
cana-3491	285	29	�	�	PROPN
cana-3491	285	30	̃	̃	NOUN
cana-3491	285	31	�	�	NOUN
cana-3491	285	32	=	=	PUNCT
cana-3491	285	33	(	(	PUNCT
cana-3491	285	34	2	2	NUM
cana-3491	285	35	+	+	CCONJ
cana-3491	285	36	𝑝𝑥1	𝑝𝑥1	PROPN
cana-3491	285	37	,	,	PUNCT
cana-3491	285	38	1	1	NUM
cana-3491	286	1	+	+	CCONJ
cana-3491	286	2	𝑝𝑦1)(𝑚𝑜𝑑5	𝑝𝑦1)(𝑚𝑜𝑑5	PROPN
cana-3491	286	3	2	2	NUM
cana-3491	286	4	)	)	PUNCT
cana-3491	286	5	,	,	PUNCT
cana-3491	286	6	we	we	PRON
cana-3491	286	7	have	have	VERB
cana-3491	286	8	�	�	PROPN
cana-3491	286	9	̃	̃	NOUN
cana-3491	286	10	�	�	NOUN
cana-3491	286	11	=	=	PUNCT
cana-3491	286	12	(	(	PUNCT
cana-3491	286	13	2	2	NUM
cana-3491	286	14	+	+	SYM
cana-3491	286	15	5𝑥1	5𝑥1	NUM
cana-3491	286	16	,	,	PUNCT
cana-3491	286	17	21	21	NUM
cana-3491	286	18	+	+	CCONJ
cana-3491	286	19	5𝑥1)(𝑚𝑜𝑑5	5𝑥1)(𝑚𝑜𝑑5	NUM
cana-3491	286	20	2	2	NUM
cana-3491	286	21	)	)	PUNCT
cana-3491	286	22	each	each	DET
cana-3491	286	23	point	point	NOUN
cana-3491	286	24	in	in	ADP
cana-3491	286	25	𝐸(𝐹5	𝐸(𝐹5	PROPN
cana-3491	286	26	)	)	PUNCT
cana-3491	286	27	may	may	AUX
cana-3491	286	28	be	be	AUX
cana-3491	286	29	lifted	lift	VERB
cana-3491	286	30	in	in	ADP
cana-3491	286	31	a	a	DET
cana-3491	286	32	similar	similar	ADJ
cana-3491	286	33	manner	manner	NOUN
cana-3491	286	34	to	to	PART
cana-3491	286	35	obtain	obtain	VERB
cana-3491	286	36	points	point	NOUN
cana-3491	286	37	in	in	ADP
cana-3491	286	38	𝐸(𝑄5)(𝑚𝑜𝑑5	𝐸(𝑄5)(𝑚𝑜𝑑5	NOUN
cana-3491	286	39	2	2	NUM
cana-3491	286	40	)	)	PUNCT
cana-3491	286	41	as	as	SCONJ
cana-3491	286	42	given	give	VERB
cana-3491	286	43	in	in	ADP
cana-3491	286	44	the	the	DET
cana-3491	286	45	table	table	NOUN
cana-3491	286	46	below	below	ADV
cana-3491	286	47	.	.	PUNCT
cana-3491	287	1	communications	communication	NOUN
cana-3491	287	2	on	on	ADP
cana-3491	287	3	applied	apply	VERB
cana-3491	287	4	nonlinear	nonlinear	ADJ
cana-3491	287	5	analysis	analysis	NOUN
cana-3491	287	6	issn	issn	NOUN
cana-3491	287	7	:	:	PUNCT
cana-3491	287	8	1074	1074	NUM
cana-3491	287	9	-	-	PUNCT
cana-3491	287	10	133x	133x	NUM
cana-3491	287	11	vol	vol	NOUN
cana-3491	287	12	32	32	NUM
cana-3491	287	13	no	no	NOUN
cana-3491	287	14	.	.	PUNCT
cana-3491	288	1	7s	7	NOUN
cana-3491	288	2	(	(	PUNCT
cana-3491	288	3	2025	2025	NUM
cana-3491	288	4	)	)	PUNCT
cana-3491	288	5	867	867	NUM
cana-3491	288	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	288	7	points	point	NOUN
cana-3491	288	8	in	in	ADP
cana-3491	288	9	𝑬(𝑭𝟓	𝑬(𝑭𝟓	NOUN
cana-3491	288	10	)	)	PUNCT
cana-3491	288	11	lifting	lifting	NOUN
cana-3491	288	12	point	point	NOUN
cana-3491	288	13	in	in	ADP
cana-3491	288	14	𝑬(𝑸𝟓)(𝒎𝒐𝒅𝟓	𝑬(𝑸𝟓)(𝒎𝒐𝒅𝟓	PROPN
cana-3491	288	15	𝟐	𝟐	NUM
cana-3491	288	16	)	)	PUNCT
cana-3491	288	17	𝑃1	𝑃1	NOUN
cana-3491	288	18	=	=	SYM
cana-3491	288	19	(	(	PUNCT
cana-3491	288	20	0,1	0,1	NUM
cana-3491	288	21	)	)	PUNCT
cana-3491	288	22	(	(	PUNCT
cana-3491	288	23	5𝑥1	5𝑥1	NUM
cana-3491	288	24	,	,	PUNCT
cana-3491	288	25	1	1	NUM
cana-3491	288	26	+	+	SYM
cana-3491	288	27	3𝑥1	3𝑥1	NUM
cana-3491	288	28	.	.	PUNCT
cana-3491	289	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	289	2	2	2	NUM
cana-3491	289	3	)	)	PUNCT
cana-3491	289	4	𝑃2	𝑃2	NOUN
cana-3491	289	5	=	=	SYM
cana-3491	289	6	(	(	PUNCT
cana-3491	289	7	0,4	0,4	NOUN
cana-3491	289	8	)	)	PUNCT
cana-3491	289	9	(	(	PUNCT
cana-3491	289	10	5𝑥1	5𝑥1	NUM
cana-3491	289	11	,	,	PUNCT
cana-3491	289	12	4	4	NUM
cana-3491	289	13	+	+	CCONJ
cana-3491	289	14	(	(	PUNCT
cana-3491	289	15	2𝑥1	2𝑥1	NUM
cana-3491	289	16	+	+	CCONJ
cana-3491	289	17	4	4	NUM
cana-3491	289	18	)	)	PUNCT
cana-3491	289	19	.	.	PUNCT
cana-3491	290	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	290	2	2	2	NUM
cana-3491	290	3	)	)	PUNCT
cana-3491	290	4	𝑃3	𝑃3	NOUN
cana-3491	290	5	=	=	SYM
cana-3491	290	6	(	(	PUNCT
cana-3491	290	7	2,1	2,1	NUM
cana-3491	290	8	)	)	PUNCT
cana-3491	290	9	(	(	PUNCT
cana-3491	290	10	2	2	NUM
cana-3491	290	11	+	+	SYM
cana-3491	290	12	5𝑥1	5𝑥1	NUM
cana-3491	290	13	,	,	PUNCT
cana-3491	290	14	1	1	NUM
cana-3491	290	15	+	+	CCONJ
cana-3491	290	16	(	(	PUNCT
cana-3491	290	17	4𝑥1	4𝑥1	NUM
cana-3491	290	18	+	+	NOUN
cana-3491	290	19	1	1	NUM
cana-3491	290	20	)	)	PUNCT
cana-3491	290	21	.	.	PUNCT
cana-3491	291	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	291	2	2	2	NUM
cana-3491	291	3	)	)	PUNCT
cana-3491	291	4	𝑃4	𝑃4	NOUN
cana-3491	291	5	=	=	SYM
cana-3491	291	6	(	(	PUNCT
cana-3491	291	7	2,4	2,4	NUM
cana-3491	291	8	)	)	PUNCT
cana-3491	291	9	(	(	PUNCT
cana-3491	291	10	2	2	NUM
cana-3491	291	11	+	+	SYM
cana-3491	291	12	5𝑥1	5𝑥1	NUM
cana-3491	291	13	,	,	PUNCT
cana-3491	291	14	4	4	NUM
cana-3491	291	15	+	+	CCONJ
cana-3491	291	16	(	(	PUNCT
cana-3491	291	17	𝑥1	𝑥1	NOUN
cana-3491	291	18	+	+	NOUN
cana-3491	291	19	3	3	NUM
cana-3491	291	20	)	)	PUNCT
cana-3491	291	21	.	.	PUNCT
cana-3491	292	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	292	2	2	2	NUM
cana-3491	292	3	)	)	PUNCT
cana-3491	292	4	𝑃5	𝑃5	NOUN
cana-3491	292	5	=	=	SYM
cana-3491	292	6	(	(	PUNCT
cana-3491	292	7	3,1	3,1	NUM
cana-3491	292	8	)	)	PUNCT
cana-3491	292	9	(	(	PUNCT
cana-3491	292	10	3	3	NUM
cana-3491	292	11	+	+	SYM
cana-3491	292	12	5𝑥1	5𝑥1	NUM
cana-3491	292	13	,	,	PUNCT
cana-3491	292	14	1	1	NUM
cana-3491	292	15	+	+	CCONJ
cana-3491	292	16	(	(	PUNCT
cana-3491	292	17	3	3	NUM
cana-3491	292	18	+	+	NUM
cana-3491	292	19	4𝑥1	4𝑥1	NUM
cana-3491	292	20	)	)	PUNCT
cana-3491	292	21	.	.	PUNCT
cana-3491	293	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	293	2	2	2	NUM
cana-3491	293	3	)	)	PUNCT
cana-3491	293	4	𝑃6	𝑃6	NOUN
cana-3491	293	5	=	=	X
cana-3491	293	6	(	(	PUNCT
cana-3491	293	7	3,4	3,4	NUM
cana-3491	293	8	)	)	PUNCT
cana-3491	293	9	(	(	PUNCT
cana-3491	293	10	3	3	NUM
cana-3491	293	11	+	+	SYM
cana-3491	293	12	5𝑥1	5𝑥1	NUM
cana-3491	293	13	,	,	PUNCT
cana-3491	293	14	4	4	NUM
cana-3491	293	15	+	+	CCONJ
cana-3491	293	16	(	(	PUNCT
cana-3491	293	17	𝑥1	𝑥1	NOUN
cana-3491	293	18	+	+	NOUN
cana-3491	293	19	1	1	NUM
cana-3491	293	20	)	)	PUNCT
cana-3491	293	21	.	.	PUNCT
cana-3491	294	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	294	2	2	2	NUM
cana-3491	294	3	)	)	PUNCT
cana-3491	294	4	𝑃7	𝑃7	NOUN
cana-3491	294	5	=	=	SYM
cana-3491	294	6	(	(	PUNCT
cana-3491	294	7	4,2	4,2	NUM
cana-3491	294	8	)	)	PUNCT
cana-3491	294	9	(	(	PUNCT
cana-3491	294	10	4	4	NUM
cana-3491	294	11	+	+	SYM
cana-3491	294	12	5𝑥1	5𝑥1	NUM
cana-3491	294	13	,	,	PUNCT
cana-3491	294	14	2	2	NUM
cana-3491	294	15	+	+	NUM
cana-3491	294	16	5(2	5(2	NUM
cana-3491	294	17	+	+	CCONJ
cana-3491	294	18	𝑥1))(𝑚𝑜𝑑5	𝑥1))(𝑚𝑜𝑑5	ADJ
cana-3491	294	19	2	2	NUM
cana-3491	294	20	)	)	PUNCT
cana-3491	294	21	𝑃8	𝑃8	NOUN
cana-3491	294	22	=	=	SYM
cana-3491	294	23	(	(	PUNCT
cana-3491	294	24	4,3	4,3	NUM
cana-3491	294	25	)	)	PUNCT
cana-3491	294	26	(	(	PUNCT
cana-3491	294	27	4	4	NUM
cana-3491	294	28	+	+	SYM
cana-3491	294	29	5𝑥1	5𝑥1	NUM
cana-3491	294	30	,	,	PUNCT
cana-3491	294	31	3	3	NUM
cana-3491	294	32	+	+	CCONJ
cana-3491	294	33	(	(	PUNCT
cana-3491	294	34	2	2	NUM
cana-3491	294	35	+	+	NUM
cana-3491	294	36	4𝑥1	4𝑥1	NUM
cana-3491	294	37	)	)	PUNCT
cana-3491	294	38	.	.	PUNCT
cana-3491	295	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	295	2	2	2	NUM
cana-3491	295	3	)	)	PUNCT
cana-3491	295	4	𝒪	𝒪	PROPN
cana-3491	295	5	�	�	PROPN
cana-3491	295	6	̃	̃	PROPN
cana-3491	295	7	�	�	PROPN
cana-3491	295	8	figure	figure	NOUN
cana-3491	295	9	3	3	NUM
cana-3491	295	10	:	:	PUNCT
cana-3491	295	11	lifting	lifting	NOUN
cana-3491	295	12	of	of	ADP
cana-3491	295	13	points	point	NOUN
cana-3491	295	14	in	in	ADP
cana-3491	295	15	𝐸(𝐹5	𝐸(𝐹5	NOUN
cana-3491	295	16	)	)	PUNCT
cana-3491	295	17	to	to	ADP
cana-3491	295	18	𝐸(𝑄5)(𝑚𝑜𝑑	𝐸(𝑄5)(𝑚𝑜𝑑	ADJ
cana-3491	295	19	52	52	NUM
cana-3491	295	20	)	)	PUNCT
cana-3491	295	21	assigning	assign	VERB
cana-3491	295	22	𝑥1	𝑥1	NOUN
cana-3491	295	23	=	=	NOUN
cana-3491	295	24	0,1,2,3,4	0,1,2,3,4	NOUN
cana-3491	295	25	in	in	ADP
cana-3491	295	26	the	the	DET
cana-3491	295	27	above	above	ADJ
cana-3491	295	28	table	table	NOUN
cana-3491	295	29	,	,	PUNCT
cana-3491	295	30	we	we	PRON
cana-3491	295	31	have	have	AUX
cana-3491	295	32	all	all	PRON
cana-3491	295	33	lifting	lift	VERB
cana-3491	295	34	points	point	NOUN
cana-3491	295	35	in	in	ADP
cana-3491	295	36	𝐸(𝑄5)(𝑚𝑜𝑑5	𝐸(𝑄5)(𝑚𝑜𝑑5	NOUN
cana-3491	295	37	2	2	NUM
cana-3491	295	38	)	)	PUNCT
cana-3491	295	39	in	in	ADP
cana-3491	295	40	the	the	DET
cana-3491	295	41	following	follow	VERB
cana-3491	295	42	table	table	NOUN
cana-3491	295	43	.	.	PUNCT
cana-3491	296	1	lifting	lift	VERB
cana-3491	296	2	point	point	NOUN
cana-3491	296	3	in	in	ADP
cana-3491	296	4	𝑬(𝑸𝟓)(𝒎𝒐𝒅𝟓	𝑬(𝑸𝟓)(𝒎𝒐𝒅𝟓	PROPN
cana-3491	296	5	𝟐	𝟐	NUM
cana-3491	296	6	)	)	PUNCT
cana-3491	296	7	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	296	8	=	=	SYM
cana-3491	296	9	𝟎	𝟎	NUM
cana-3491	296	10	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	296	11	=	=	SYM
cana-3491	296	12	𝟏	𝟏	NUM
cana-3491	296	13	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	296	14	=	=	SYM
cana-3491	296	15	𝟐	𝟐	NUM
cana-3491	296	16	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	296	17	=	=	SYM
cana-3491	296	18	𝟑	𝟑	NUM
cana-3491	296	19	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	296	20	=	=	SYM
cana-3491	296	21	𝟒	𝟒	PROPN
cana-3491	296	22	(	(	PUNCT
cana-3491	296	23	5𝑥1	5𝑥1	NUM
cana-3491	296	24	,	,	PUNCT
cana-3491	296	25	1	1	NUM
cana-3491	296	26	+	+	SYM
cana-3491	296	27	3𝑥1	3𝑥1	NUM
cana-3491	296	28	.	.	PUNCT
cana-3491	297	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	297	2	2	2	NUM
cana-3491	297	3	)	)	PUNCT
cana-3491	297	4	(	(	PUNCT
cana-3491	297	5	0,1	0,1	NUM
cana-3491	297	6	)	)	PUNCT
cana-3491	297	7	(	(	PUNCT
cana-3491	297	8	1.5,1	1.5,1	NUM
cana-3491	297	9	+	+	NOUN
cana-3491	297	10	3.5	3.5	NUM
cana-3491	297	11	)	)	PUNCT
cana-3491	297	12	(	(	PUNCT
cana-3491	297	13	2.5,1	2.5,1	NUM
cana-3491	297	14	+	+	NUM
cana-3491	297	15	1.5	1.5	NUM
cana-3491	297	16	)	)	PUNCT
cana-3491	297	17	(	(	PUNCT
cana-3491	297	18	3.5,1	3.5,1	NUM
cana-3491	297	19	+	+	NOUN
cana-3491	297	20	4.5	4.5	NUM
cana-3491	297	21	)	)	PUNCT
cana-3491	297	22	(	(	PUNCT
cana-3491	297	23	4.5,1	4.5,1	X
cana-3491	297	24	+	+	NOUN
cana-3491	297	25	2.5	2.5	NUM
cana-3491	297	26	)	)	PUNCT
cana-3491	297	27	(	(	PUNCT
cana-3491	297	28	5𝑥1	5𝑥1	NUM
cana-3491	297	29	,	,	PUNCT
cana-3491	297	30	4	4	NUM
cana-3491	297	31	+	+	CCONJ
cana-3491	297	32	(	(	PUNCT
cana-3491	297	33	2𝑥1	2𝑥1	NUM
cana-3491	297	34	+	+	CCONJ
cana-3491	297	35	4	4	NUM
cana-3491	297	36	)	)	PUNCT
cana-3491	297	37	.	.	PUNCT
cana-3491	298	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	298	2	2	2	NUM
cana-3491	298	3	)	)	PUNCT
cana-3491	298	4	(	(	PUNCT
cana-3491	298	5	0,4	0,4	NUM
cana-3491	298	6	+	+	SYM
cana-3491	298	7	4.5	4.5	NUM
cana-3491	298	8	)	)	PUNCT
cana-3491	298	9	(	(	PUNCT
cana-3491	298	10	1.5,4	1.5,4	NOUN
cana-3491	298	11	+	+	CCONJ
cana-3491	298	12	1.5	1.5	NUM
cana-3491	298	13	)	)	PUNCT
cana-3491	298	14	(	(	PUNCT
cana-3491	298	15	2.5,4	2.5,4	NUM
cana-3491	298	16	+	+	NOUN
cana-3491	298	17	3.5	3.5	NUM
cana-3491	298	18	)	)	PUNCT
cana-3491	298	19	(	(	PUNCT
cana-3491	298	20	3.5,4	3.5,4	NUM
cana-3491	298	21	)	)	PUNCT
cana-3491	298	22	(	(	PUNCT
cana-3491	298	23	4.5,4	4.5,4	NOUN
cana-3491	298	24	+	+	NOUN
cana-3491	298	25	2.5	2.5	NUM
cana-3491	298	26	)	)	PUNCT
cana-3491	298	27	(	(	PUNCT
cana-3491	298	28	2	2	NUM
cana-3491	298	29	+	+	SYM
cana-3491	298	30	5𝑥1	5𝑥1	NUM
cana-3491	298	31	,	,	PUNCT
cana-3491	298	32	1	1	NUM
cana-3491	298	33	+	+	CCONJ
cana-3491	298	34	(	(	PUNCT
cana-3491	298	35	4𝑥1	4𝑥1	NUM
cana-3491	298	36	+	+	NOUN
cana-3491	298	37	1	1	NUM
cana-3491	298	38	)	)	PUNCT
cana-3491	298	39	.	.	PUNCT
cana-3491	299	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	299	2	2	2	NUM
cana-3491	299	3	)	)	PUNCT
cana-3491	299	4	(	(	PUNCT
cana-3491	299	5	2,1	2,1	NUM
cana-3491	299	6	+	+	NUM
cana-3491	299	7	1.5	1.5	NUM
cana-3491	299	8	)	)	PUNCT
cana-3491	299	9	(	(	PUNCT
cana-3491	299	10	2	2	NUM
cana-3491	299	11	+	+	SYM
cana-3491	299	12	1.5,1	1.5,1	NUM
cana-3491	299	13	)	)	PUNCT
cana-3491	299	14	(	(	PUNCT
cana-3491	299	15	2	2	NUM
cana-3491	299	16	+	+	NUM
cana-3491	299	17	2.5,1	2.5,1	NUM
cana-3491	299	18	+	+	NUM
cana-3491	299	19	4.5	4.5	NUM
cana-3491	299	20	)	)	PUNCT
cana-3491	299	21	(	(	PUNCT
cana-3491	299	22	2	2	NUM
cana-3491	299	23	+	+	NUM
cana-3491	299	24	3.5,1	3.5,1	NUM
cana-3491	299	25	+	+	NOUN
cana-3491	299	26	3.5	3.5	NUM
cana-3491	299	27	)	)	PUNCT
cana-3491	299	28	(	(	PUNCT
cana-3491	299	29	2	2	NUM
cana-3491	299	30	+	+	NUM
cana-3491	299	31	4.5,1	4.5,1	NOUN
cana-3491	299	32	+	+	NOUN
cana-3491	299	33	2.5	2.5	NUM
cana-3491	299	34	)	)	PUNCT
cana-3491	299	35	(	(	PUNCT
cana-3491	299	36	2	2	NUM
cana-3491	299	37	+	+	SYM
cana-3491	299	38	5𝑥1	5𝑥1	NUM
cana-3491	299	39	,	,	PUNCT
cana-3491	299	40	4	4	NUM
cana-3491	299	41	+	+	CCONJ
cana-3491	299	42	(	(	PUNCT
cana-3491	299	43	𝑥1	𝑥1	NOUN
cana-3491	299	44	+	+	NOUN
cana-3491	299	45	3	3	NUM
cana-3491	299	46	)	)	PUNCT
cana-3491	299	47	.	.	PUNCT
cana-3491	300	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	300	2	2	2	NUM
cana-3491	300	3	)	)	PUNCT
cana-3491	300	4	(	(	PUNCT
cana-3491	300	5	2,4	2,4	NUM
cana-3491	300	6	+	+	NUM
cana-3491	300	7	3.5	3.5	NUM
cana-3491	300	8	)	)	PUNCT
cana-3491	300	9	(	(	PUNCT
cana-3491	300	10	2	2	NUM
cana-3491	300	11	+	+	NUM
cana-3491	300	12	1.5,4	1.5,4	NOUN
cana-3491	300	13	+	+	NUM
cana-3491	300	14	4.5	4.5	NUM
cana-3491	300	15	)	)	PUNCT
cana-3491	300	16	(	(	PUNCT
cana-3491	300	17	2	2	NUM
cana-3491	300	18	+	+	NUM
cana-3491	300	19	2.5,4	2.5,4	NUM
cana-3491	300	20	)	)	PUNCT
cana-3491	300	21	(	(	PUNCT
cana-3491	300	22	2	2	NUM
cana-3491	300	23	+	+	NUM
cana-3491	300	24	3.5,4	3.5,4	NUM
cana-3491	300	25	+	+	CCONJ
cana-3491	300	26	1.5	1.5	NUM
cana-3491	300	27	)	)	PUNCT
cana-3491	300	28	(	(	PUNCT
cana-3491	300	29	2	2	NUM
cana-3491	300	30	+	+	NUM
cana-3491	300	31	4.5,4	4.5,4	NOUN
cana-3491	300	32	+	+	CCONJ
cana-3491	300	33	2.5	2.5	NUM
cana-3491	300	34	)	)	PUNCT
cana-3491	300	35	communications	communication	NOUN
cana-3491	300	36	on	on	ADP
cana-3491	300	37	applied	apply	VERB
cana-3491	300	38	nonlinear	nonlinear	ADJ
cana-3491	300	39	analysis	analysis	NOUN
cana-3491	300	40	issn	issn	NOUN
cana-3491	300	41	:	:	PUNCT
cana-3491	300	42	1074	1074	NUM
cana-3491	300	43	-	-	PUNCT
cana-3491	300	44	133x	133x	NUM
cana-3491	300	45	vol	vol	NOUN
cana-3491	300	46	32	32	NUM
cana-3491	300	47	no	no	NOUN
cana-3491	300	48	.	.	PUNCT
cana-3491	301	1	7s	7	NOUN
cana-3491	301	2	(	(	PUNCT
cana-3491	301	3	2025	2025	NUM
cana-3491	301	4	)	)	PUNCT
cana-3491	301	5	868	868	NUM
cana-3491	301	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	301	7	lifting	lifting	NOUN
cana-3491	301	8	point	point	NOUN
cana-3491	301	9	in	in	ADP
cana-3491	301	10	𝑬(𝑸𝟓)(𝒎𝒐𝒅𝟓	𝑬(𝑸𝟓)(𝒎𝒐𝒅𝟓	PROPN
cana-3491	301	11	𝟐	𝟐	NUM
cana-3491	301	12	)	)	PUNCT
cana-3491	302	1	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	302	2	=	=	SYM
cana-3491	302	3	𝟎	𝟎	NUM
cana-3491	302	4	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	302	5	=	=	SYM
cana-3491	302	6	𝟏	𝟏	NUM
cana-3491	302	7	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	302	8	=	=	SYM
cana-3491	302	9	𝟐	𝟐	NUM
cana-3491	302	10	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	302	11	=	=	SYM
cana-3491	302	12	𝟑	𝟑	NUM
cana-3491	302	13	𝒙𝟏	𝒙𝟏	NOUN
cana-3491	302	14	=	=	SYM
cana-3491	302	15	𝟒	𝟒	PROPN
cana-3491	302	16	(	(	PUNCT
cana-3491	302	17	3	3	NUM
cana-3491	302	18	+	+	SYM
cana-3491	302	19	5𝑥1	5𝑥1	NUM
cana-3491	302	20	,	,	PUNCT
cana-3491	302	21	1	1	NUM
cana-3491	302	22	+	+	CCONJ
cana-3491	302	23	(	(	PUNCT
cana-3491	302	24	3	3	NUM
cana-3491	302	25	+	+	NUM
cana-3491	302	26	4𝑥1	4𝑥1	NUM
cana-3491	302	27	)	)	PUNCT
cana-3491	302	28	.	.	PUNCT
cana-3491	303	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	303	2	2	2	NUM
cana-3491	303	3	)	)	PUNCT
cana-3491	303	4	(	(	PUNCT
cana-3491	303	5	3,1	3,1	NUM
cana-3491	303	6	+	+	CCONJ
cana-3491	303	7	3.5	3.5	NUM
cana-3491	303	8	)	)	PUNCT
cana-3491	303	9	(	(	PUNCT
cana-3491	303	10	3	3	NUM
cana-3491	303	11	+	+	NUM
cana-3491	303	12	1.5,1	1.5,1	NUM
cana-3491	303	13	+	+	NUM
cana-3491	303	14	2.5	2.5	NUM
cana-3491	303	15	)	)	PUNCT
cana-3491	303	16	(	(	PUNCT
cana-3491	303	17	3	3	NUM
cana-3491	303	18	+	+	NUM
cana-3491	303	19	2.5,1	2.5,1	NUM
cana-3491	303	20	+	+	NUM
cana-3491	303	21	1.5	1.5	NUM
cana-3491	303	22	)	)	PUNCT
cana-3491	303	23	(	(	PUNCT
cana-3491	303	24	3	3	NUM
cana-3491	303	25	+	+	SYM
cana-3491	303	26	3.5,1	3.5,1	NUM
cana-3491	303	27	)	)	PUNCT
cana-3491	303	28	(	(	PUNCT
cana-3491	303	29	3	3	NUM
cana-3491	303	30	+	+	NUM
cana-3491	303	31	4.5,1	4.5,1	NOUN
cana-3491	303	32	+	+	NOUN
cana-3491	303	33	4.5	4.5	NUM
cana-3491	303	34	)	)	PUNCT
cana-3491	303	35	(	(	PUNCT
cana-3491	303	36	3	3	NUM
cana-3491	303	37	+	+	SYM
cana-3491	303	38	5𝑥1	5𝑥1	NUM
cana-3491	303	39	,	,	PUNCT
cana-3491	303	40	4	4	NUM
cana-3491	303	41	+	+	CCONJ
cana-3491	303	42	(	(	PUNCT
cana-3491	303	43	𝑥1	𝑥1	NOUN
cana-3491	303	44	+	+	NOUN
cana-3491	303	45	1	1	NUM
cana-3491	303	46	)	)	PUNCT
cana-3491	303	47	.	.	PUNCT
cana-3491	304	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	304	2	2	2	NUM
cana-3491	304	3	)	)	PUNCT
cana-3491	304	4	(	(	PUNCT
cana-3491	304	5	3,4	3,4	NUM
cana-3491	304	6	+	+	NUM
cana-3491	304	7	1.5	1.5	NUM
cana-3491	304	8	)	)	PUNCT
cana-3491	304	9	(	(	PUNCT
cana-3491	304	10	3	3	NUM
cana-3491	304	11	+	+	NUM
cana-3491	304	12	1.5,4	1.5,4	NOUN
cana-3491	304	13	+	+	NUM
cana-3491	304	14	2.5	2.5	NUM
cana-3491	304	15	)	)	PUNCT
cana-3491	304	16	(	(	PUNCT
cana-3491	304	17	3	3	NUM
cana-3491	304	18	+	+	NUM
cana-3491	304	19	2.5,4	2.5,4	NUM
cana-3491	304	20	+	+	NOUN
cana-3491	304	21	3.5	3.5	NUM
cana-3491	304	22	)	)	PUNCT
cana-3491	304	23	(	(	PUNCT
cana-3491	304	24	3	3	NUM
cana-3491	304	25	+	+	NUM
cana-3491	304	26	3.5,4	3.5,4	NUM
cana-3491	304	27	+	+	CCONJ
cana-3491	304	28	4.5	4.5	NUM
cana-3491	304	29	)	)	PUNCT
cana-3491	304	30	(	(	PUNCT
cana-3491	304	31	3	3	NUM
cana-3491	304	32	+	+	SYM
cana-3491	304	33	4.5,4	4.5,4	NOUN
cana-3491	304	34	)	)	PUNCT
cana-3491	304	35	(	(	PUNCT
cana-3491	304	36	4	4	NUM
cana-3491	304	37	+	+	SYM
cana-3491	304	38	5𝑥1	5𝑥1	NUM
cana-3491	304	39	,	,	PUNCT
cana-3491	304	40	2	2	NUM
cana-3491	304	41	+	+	NUM
cana-3491	304	42	5(2	5(2	NUM
cana-3491	304	43	+	+	CCONJ
cana-3491	304	44	𝑥1))(𝑚𝑜𝑑5	𝑥1))(𝑚𝑜𝑑5	ADJ
cana-3491	304	45	2	2	NUM
cana-3491	304	46	)	)	PUNCT
cana-3491	304	47	(	(	PUNCT
cana-3491	304	48	4,2	4,2	NUM
cana-3491	304	49	+	+	X
cana-3491	304	50	2.5	2.5	NUM
cana-3491	304	51	)	)	PUNCT
cana-3491	304	52	(	(	PUNCT
cana-3491	304	53	4	4	NUM
cana-3491	304	54	+	+	NUM
cana-3491	304	55	1.5,2	1.5,2	NOUN
cana-3491	304	56	+	+	CCONJ
cana-3491	304	57	3.5	3.5	NUM
cana-3491	304	58	)	)	PUNCT
cana-3491	304	59	(	(	PUNCT
cana-3491	304	60	4	4	NUM
cana-3491	304	61	+	+	NUM
cana-3491	304	62	2.5,2	2.5,2	NOUN
cana-3491	304	63	+	+	CCONJ
cana-3491	304	64	4.5	4.5	NUM
cana-3491	304	65	)	)	PUNCT
cana-3491	304	66	(	(	PUNCT
cana-3491	304	67	4	4	NUM
cana-3491	304	68	+	+	NUM
cana-3491	304	69	3.5,2	3.5,2	NUM
cana-3491	304	70	)	)	PUNCT
cana-3491	304	71	(	(	PUNCT
cana-3491	304	72	4	4	NUM
cana-3491	304	73	+	+	NUM
cana-3491	304	74	4.5,2	4.5,2	NOUN
cana-3491	304	75	+	+	NOUN
cana-3491	304	76	1.5	1.5	NUM
cana-3491	304	77	)	)	PUNCT
cana-3491	304	78	(	(	PUNCT
cana-3491	304	79	4	4	NUM
cana-3491	304	80	+	+	SYM
cana-3491	304	81	5𝑥1	5𝑥1	NUM
cana-3491	304	82	,	,	PUNCT
cana-3491	304	83	3	3	NUM
cana-3491	304	84	+	+	CCONJ
cana-3491	304	85	(	(	PUNCT
cana-3491	304	86	2	2	NUM
cana-3491	304	87	+	+	NUM
cana-3491	304	88	4𝑥1	4𝑥1	NUM
cana-3491	304	89	)	)	PUNCT
cana-3491	304	90	.	.	PUNCT
cana-3491	305	1	5)(𝑚𝑜𝑑5	5)(𝑚𝑜𝑑5	NUM
cana-3491	305	2	2	2	NUM
cana-3491	305	3	)	)	PUNCT
cana-3491	305	4	(	(	PUNCT
cana-3491	305	5	4,3	4,3	NUM
cana-3491	305	6	+	+	NOUN
cana-3491	305	7	2.5	2.5	NUM
cana-3491	305	8	)	)	PUNCT
cana-3491	305	9	(	(	PUNCT
cana-3491	305	10	4	4	NUM
cana-3491	305	11	+	+	NUM
cana-3491	305	12	1.5,3	1.5,3	NUM
cana-3491	305	13	+	+	CCONJ
cana-3491	305	14	1.5	1.5	NUM
cana-3491	305	15	)	)	PUNCT
cana-3491	305	16	(	(	PUNCT
cana-3491	305	17	4	4	NUM
cana-3491	305	18	+	+	SYM
cana-3491	305	19	2.5,3	2.5,3	NUM
cana-3491	305	20	)	)	PUNCT
cana-3491	305	21	(	(	PUNCT
cana-3491	305	22	4	4	NUM
cana-3491	305	23	+	+	NUM
cana-3491	305	24	3.5,3	3.5,3	NUM
cana-3491	305	25	+	+	NOUN
cana-3491	305	26	4.5	4.5	NUM
cana-3491	305	27	)	)	PUNCT
cana-3491	305	28	(	(	PUNCT
cana-3491	305	29	4	4	NUM
cana-3491	305	30	+	+	NUM
cana-3491	305	31	4.5,3	4.5,3	NUM
cana-3491	305	32	+	+	CCONJ
cana-3491	305	33	3.5	3.5	NUM
cana-3491	305	34	)	)	PUNCT
cana-3491	305	35	�	�	PROPN
cana-3491	305	36	̃	̃	PROPN
cana-3491	305	37	�	�	PROPN
cana-3491	305	38	�	�	PROPN
cana-3491	305	39	̃	̃	PROPN
cana-3491	305	40	�	�	PROPN
cana-3491	305	41	�	�	PROPN
cana-3491	305	42	̃	̃	PROPN
cana-3491	305	43	�	�	PROPN
cana-3491	305	44	�	�	PROPN
cana-3491	305	45	̃	̃	PROPN
cana-3491	305	46	�	�	PROPN
cana-3491	305	47	�	�	PROPN
cana-3491	305	48	̃	̃	PROPN
cana-3491	305	49	�	�	PROPN
cana-3491	305	50	�	�	PROPN
cana-3491	305	51	̃	̃	PROPN
cana-3491	305	52	�	�	PROPN
cana-3491	305	53	in	in	ADP
cana-3491	305	54	order	order	NOUN
cana-3491	305	55	to	to	PART
cana-3491	305	56	obtain	obtain	VERB
cana-3491	305	57	the	the	DET
cana-3491	305	58	lift	lift	NOUN
cana-3491	305	59	points	point	NOUN
cana-3491	305	60	e(q5)(mod5	e(q5)(mod5	NOUN
cana-3491	305	61	3	3	NUM
cana-3491	305	62	)	)	PUNCT
cana-3491	305	63	,	,	PUNCT
cana-3491	305	64	we	we	PRON
cana-3491	305	65	repeat	repeat	VERB
cana-3491	305	66	the	the	DET
cana-3491	305	67	above	above	ADJ
cana-3491	305	68	process	process	NOUN
cana-3491	305	69	for	for	ADP
cana-3491	305	70	considered	considered	ADJ
cana-3491	305	71	point	point	NOUN
cana-3491	305	72	in	in	ADP
cana-3491	305	73	e(q5)(mod5	e(q5)(mod5	PROPN
cana-3491	305	74	2	2	NUM
cana-3491	305	75	)	)	PUNCT
cana-3491	305	76	and	and	CCONJ
cana-3491	305	77	obtain	obtain	VERB
cana-3491	305	78	all	all	DET
cana-3491	305	79	lifts	lift	NOUN
cana-3491	305	80	in	in	ADP
cana-3491	305	81	e(q5)(mod5	e(q5)(mod5	PROPN
cana-3491	305	82	3	3	NUM
cana-3491	305	83	)	)	PUNCT
cana-3491	305	84	.	.	PUNCT
cana-3491	306	1	on	on	ADP
cana-3491	306	2	repeating	repeat	VERB
cana-3491	306	3	in	in	ADP
cana-3491	306	4	such	such	DET
cana-3491	306	5	a	a	DET
cana-3491	306	6	manner	manner	NOUN
cana-3491	306	7	,	,	PUNCT
cana-3491	306	8	we	we	PRON
cana-3491	306	9	can	can	AUX
cana-3491	306	10	obtain	obtain	VERB
cana-3491	306	11	lift	lift	NOUN
cana-3491	306	12	points	point	NOUN
cana-3491	306	13	in	in	ADP
cana-3491	306	14	e(q5)(mod5	e(q5)(mod5	NOUN
cana-3491	306	15	n	n	CCONJ
cana-3491	306	16	)	)	PUNCT
cana-3491	306	17	for	for	ADP
cana-3491	306	18	any	any	DET
cana-3491	306	19	positive	positive	ADJ
cana-3491	306	20	integer	integer	NOUN
cana-3491	306	21	n.	n.	NOUN
cana-3491	306	22	the	the	DET
cana-3491	306	23	lifting	lifting	NOUN
cana-3491	306	24	of	of	ADP
cana-3491	306	25	points	point	NOUN
cana-3491	306	26	from	from	ADP
cana-3491	306	27	e(f5	e(f5	NOUN
cana-3491	306	28	)	)	PUNCT
cana-3491	306	29	to	to	PART
cana-3491	306	30	e(q5)(mod5	e(q5)(mod5	VERB
cana-3491	306	31	3	3	NUM
cana-3491	306	32	)	)	PUNCT
cana-3491	306	33	are	be	AUX
cana-3491	306	34	represented	represent	VERB
cana-3491	306	35	pictorially	pictorially	ADV
cana-3491	306	36	in	in	ADP
cana-3491	306	37	fig	fig	NOUN
cana-3491	306	38	4	4	NUM
cana-3491	306	39	.	.	PUNCT
cana-3491	306	40	figure	figure	VERB
cana-3491	306	41	4	4	NUM
cana-3491	306	42	:	:	PUNCT
cana-3491	306	43	lifting	lifting	NOUN
cana-3491	306	44	of	of	ADP
cana-3491	306	45	points	point	NOUN
cana-3491	306	46	in	in	ADP
cana-3491	306	47	e(f5	e(f5	NOUN
cana-3491	306	48	)	)	PUNCT
cana-3491	306	49	to	to	ADP
cana-3491	306	50	e(q5)(mod	e(q5)(mod	DET
cana-3491	306	51	52	52	NUM
cana-3491	306	52	)	)	PUNCT
cana-3491	306	53	the	the	DET
cana-3491	306	54	table	table	NOUN
cana-3491	306	55	below	below	ADV
cana-3491	306	56	shows	show	VERB
cana-3491	306	57	the	the	DET
cana-3491	306	58	difference	difference	NOUN
cana-3491	306	59	in	in	ADP
cana-3491	306	60	number	number	NOUN
cana-3491	306	61	of	of	ADP
cana-3491	306	62	points	point	NOUN
cana-3491	306	63	from	from	ADP
cana-3491	306	64	e(f5	e(f5	NOUN
cana-3491	306	65	)	)	PUNCT
cana-3491	306	66	to	to	ADP
cana-3491	306	67	e(q5)(mod	e(q5)(mod	DET
cana-3491	306	68	52	52	NUM
cana-3491	306	69	)	)	PUNCT
cana-3491	306	70	which	which	PRON
cana-3491	306	71	provides	provide	VERB
cana-3491	306	72	large	large	ADJ
cana-3491	306	73	key	key	ADJ
cana-3491	306	74	space	space	NOUN
cana-3491	306	75	for	for	ADP
cana-3491	306	76	cryptographic	cryptographic	ADJ
cana-3491	306	77	purpose	purpose	NOUN
cana-3491	306	78	.	.	PUNCT
cana-3491	307	1	points	point	NOUN
cana-3491	307	2	in	in	ADP
cana-3491	307	3	𝐸(𝐹5	𝐸(𝐹5	NOUN
cana-3491	307	4	)	)	PUNCT
cana-3491	307	5	points	point	NOUN
cana-3491	307	6	in	in	ADP
cana-3491	307	7	𝐸(𝑄5)(mod	𝐸(𝑄5)(mod	NUM
cana-3491	307	8	52	52	NUM
cana-3491	307	9	)	)	PUNCT
cana-3491	307	10	(	(	PUNCT
cana-3491	307	11	0,1	0,1	NUM
cana-3491	307	12	)	)	PUNCT
cana-3491	307	13	(	(	PUNCT
cana-3491	307	14	0,1	0,1	NUM
cana-3491	307	15	)	)	PUNCT
cana-3491	307	16	(	(	PUNCT
cana-3491	307	17	1.5,1	1.5,1	NUM
cana-3491	307	18	+	+	NOUN
cana-3491	307	19	3.5	3.5	NUM
cana-3491	307	20	)	)	PUNCT
cana-3491	307	21	(	(	PUNCT
cana-3491	307	22	2.5,1	2.5,1	NUM
cana-3491	307	23	+	+	NUM
cana-3491	307	24	1.5	1.5	NUM
cana-3491	307	25	)	)	PUNCT
cana-3491	307	26	(	(	PUNCT
cana-3491	307	27	3.5,1	3.5,1	NUM
cana-3491	307	28	+	+	NOUN
cana-3491	307	29	4.5	4.5	NUM
cana-3491	307	30	)	)	PUNCT
cana-3491	307	31	(	(	PUNCT
cana-3491	307	32	4.5,1	4.5,1	X
cana-3491	307	33	+	+	NOUN
cana-3491	307	34	2.5	2.5	NUM
cana-3491	307	35	)	)	PUNCT
cana-3491	307	36	(	(	PUNCT
cana-3491	307	37	0,4	0,4	NOUN
cana-3491	307	38	)	)	PUNCT
cana-3491	307	39	(	(	PUNCT
cana-3491	307	40	0,4	0,4	NUM
cana-3491	307	41	+	+	SYM
cana-3491	307	42	4.5	4.5	NUM
cana-3491	307	43	)	)	PUNCT
cana-3491	307	44	(	(	PUNCT
cana-3491	307	45	1.5,4	1.5,4	NOUN
cana-3491	307	46	+	+	CCONJ
cana-3491	307	47	1.5	1.5	NUM
cana-3491	307	48	)	)	PUNCT
cana-3491	307	49	(	(	PUNCT
cana-3491	307	50	2.5,4	2.5,4	NUM
cana-3491	307	51	+	+	NOUN
cana-3491	307	52	3.5	3.5	NUM
cana-3491	307	53	)	)	PUNCT
cana-3491	307	54	(	(	PUNCT
cana-3491	307	55	3.5,4	3.5,4	NUM
cana-3491	307	56	)	)	PUNCT
cana-3491	307	57	(	(	PUNCT
cana-3491	307	58	4.5,4	4.5,4	NOUN
cana-3491	307	59	+	+	NOUN
cana-3491	307	60	2.5	2.5	NUM
cana-3491	307	61	)	)	PUNCT
cana-3491	307	62	(	(	PUNCT
cana-3491	307	63	2,1	2,1	NUM
cana-3491	307	64	)	)	PUNCT
cana-3491	307	65	(	(	PUNCT
cana-3491	307	66	2,1	2,1	NUM
cana-3491	307	67	+	+	NUM
cana-3491	307	68	1.5	1.5	NUM
cana-3491	307	69	)	)	PUNCT
cana-3491	307	70	(	(	PUNCT
cana-3491	307	71	2	2	NUM
cana-3491	307	72	+	+	SYM
cana-3491	307	73	1.5,1	1.5,1	NUM
cana-3491	307	74	)	)	PUNCT
cana-3491	307	75	(	(	PUNCT
cana-3491	307	76	2	2	NUM
cana-3491	307	77	+	+	NUM
cana-3491	307	78	2.5	2.5	NUM
cana-3491	307	79	,	,	PUNCT
cana-3491	307	80	1	1	NUM
cana-3491	307	81	+	+	NUM
cana-3491	307	82	4.5	4.5	NUM
cana-3491	307	83	)	)	PUNCT
cana-3491	307	84	(	(	PUNCT
cana-3491	307	85	2	2	NUM
cana-3491	308	1	+	+	NUM
cana-3491	308	2	3.5,1	3.5,1	NUM
cana-3491	308	3	+	+	NOUN
cana-3491	308	4	3.5	3.5	NUM
cana-3491	308	5	)	)	PUNCT
cana-3491	308	6	(	(	PUNCT
cana-3491	308	7	2	2	NUM
cana-3491	308	8	+	+	NUM
cana-3491	308	9	4.5,1	4.5,1	NOUN
cana-3491	308	10	+	+	NOUN
cana-3491	308	11	2.5	2.5	NUM
cana-3491	308	12	)	)	PUNCT
cana-3491	308	13	communications	communication	NOUN
cana-3491	308	14	on	on	ADP
cana-3491	308	15	applied	apply	VERB
cana-3491	308	16	nonlinear	nonlinear	ADJ
cana-3491	308	17	analysis	analysis	NOUN
cana-3491	308	18	issn	issn	NOUN
cana-3491	308	19	:	:	PUNCT
cana-3491	308	20	1074	1074	NUM
cana-3491	308	21	-	-	PUNCT
cana-3491	308	22	133x	133x	NUM
cana-3491	308	23	vol	vol	NOUN
cana-3491	308	24	32	32	NUM
cana-3491	308	25	no	no	NOUN
cana-3491	308	26	.	.	PUNCT
cana-3491	309	1	7s	7	NOUN
cana-3491	309	2	(	(	PUNCT
cana-3491	309	3	2025	2025	NUM
cana-3491	309	4	)	)	PUNCT
cana-3491	309	5	869	869	NUM
cana-3491	309	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	309	7	points	point	NOUN
cana-3491	309	8	in	in	ADP
cana-3491	309	9	𝐸(𝐹5	𝐸(𝐹5	NOUN
cana-3491	309	10	)	)	PUNCT
cana-3491	309	11	points	point	NOUN
cana-3491	309	12	in	in	ADP
cana-3491	309	13	𝐸(𝑄5)(mod	𝐸(𝑄5)(mod	NUM
cana-3491	309	14	52	52	NUM
cana-3491	309	15	)	)	PUNCT
cana-3491	309	16	(	(	PUNCT
cana-3491	309	17	2,4	2,4	NUM
cana-3491	309	18	)	)	PUNCT
cana-3491	309	19	(	(	PUNCT
cana-3491	309	20	2,4	2,4	NUM
cana-3491	309	21	+	+	NUM
cana-3491	309	22	3.5	3.5	NUM
cana-3491	309	23	)	)	PUNCT
cana-3491	309	24	(	(	PUNCT
cana-3491	309	25	2	2	NUM
cana-3491	309	26	+	+	NUM
cana-3491	309	27	1.5,4	1.5,4	NOUN
cana-3491	309	28	+	+	NUM
cana-3491	309	29	4.5	4.5	NUM
cana-3491	309	30	)	)	PUNCT
cana-3491	309	31	(	(	PUNCT
cana-3491	309	32	2	2	NUM
cana-3491	309	33	+	+	NUM
cana-3491	309	34	2.5,4	2.5,4	NUM
cana-3491	309	35	)	)	PUNCT
cana-3491	309	36	(	(	PUNCT
cana-3491	309	37	2	2	NUM
cana-3491	309	38	+	+	NUM
cana-3491	309	39	3.5,4	3.5,4	NUM
cana-3491	309	40	+	+	CCONJ
cana-3491	309	41	1.5	1.5	NUM
cana-3491	309	42	)	)	PUNCT
cana-3491	309	43	(	(	PUNCT
cana-3491	309	44	2	2	NUM
cana-3491	309	45	+	+	NUM
cana-3491	309	46	4.5,4	4.5,4	NOUN
cana-3491	309	47	+	+	NOUN
cana-3491	309	48	2.5	2.5	NUM
cana-3491	309	49	)	)	PUNCT
cana-3491	309	50	(	(	PUNCT
cana-3491	309	51	3,1	3,1	NUM
cana-3491	309	52	)	)	PUNCT
cana-3491	309	53	(	(	PUNCT
cana-3491	309	54	3,1	3,1	NUM
cana-3491	309	55	+	+	CCONJ
cana-3491	309	56	3.5	3.5	NUM
cana-3491	309	57	)	)	PUNCT
cana-3491	309	58	(	(	PUNCT
cana-3491	309	59	3	3	NUM
cana-3491	309	60	+	+	NUM
cana-3491	309	61	1.5,1	1.5,1	NUM
cana-3491	309	62	+	+	NUM
cana-3491	309	63	2.5	2.5	NUM
cana-3491	309	64	)	)	PUNCT
cana-3491	309	65	(	(	PUNCT
cana-3491	309	66	3	3	NUM
cana-3491	309	67	+	+	NUM
cana-3491	309	68	2.5,1	2.5,1	NUM
cana-3491	309	69	+	+	NUM
cana-3491	309	70	1.5	1.5	NUM
cana-3491	309	71	)	)	PUNCT
cana-3491	309	72	(	(	PUNCT
cana-3491	309	73	3	3	NUM
cana-3491	309	74	+	+	SYM
cana-3491	309	75	3.5,1	3.5,1	NUM
cana-3491	309	76	)	)	PUNCT
cana-3491	309	77	(	(	PUNCT
cana-3491	309	78	3	3	NUM
cana-3491	309	79	+	+	NUM
cana-3491	309	80	4.5,1	4.5,1	NOUN
cana-3491	309	81	+	+	NOUN
cana-3491	309	82	4.5	4.5	NUM
cana-3491	309	83	)	)	PUNCT
cana-3491	309	84	(	(	PUNCT
cana-3491	309	85	3,4	3,4	NUM
cana-3491	309	86	)	)	PUNCT
cana-3491	309	87	(	(	PUNCT
cana-3491	309	88	3,4	3,4	NUM
cana-3491	309	89	+	+	NUM
cana-3491	309	90	1.5	1.5	NUM
cana-3491	309	91	)	)	PUNCT
cana-3491	309	92	(	(	PUNCT
cana-3491	309	93	3	3	NUM
cana-3491	309	94	+	+	NUM
cana-3491	309	95	1.5,4	1.5,4	NOUN
cana-3491	309	96	+	+	NUM
cana-3491	309	97	2.5	2.5	NUM
cana-3491	309	98	)	)	PUNCT
cana-3491	309	99	(	(	PUNCT
cana-3491	309	100	3	3	NUM
cana-3491	309	101	+	+	NUM
cana-3491	309	102	2.5,4	2.5,4	NUM
cana-3491	309	103	+	+	NOUN
cana-3491	309	104	3.5	3.5	NUM
cana-3491	309	105	)	)	PUNCT
cana-3491	309	106	(	(	PUNCT
cana-3491	309	107	3	3	NUM
cana-3491	309	108	+	+	NUM
cana-3491	309	109	3.5,4	3.5,4	NUM
cana-3491	309	110	+	+	CCONJ
cana-3491	309	111	4.5	4.5	NUM
cana-3491	309	112	)	)	PUNCT
cana-3491	309	113	(	(	PUNCT
cana-3491	309	114	3	3	NUM
cana-3491	309	115	+	+	SYM
cana-3491	309	116	4.5,4	4.5,4	NOUN
cana-3491	309	117	)	)	PUNCT
cana-3491	309	118	(	(	PUNCT
cana-3491	309	119	4,2	4,2	NUM
cana-3491	309	120	)	)	PUNCT
cana-3491	309	121	(	(	PUNCT
cana-3491	309	122	4,2	4,2	NUM
cana-3491	309	123	+	+	X
cana-3491	309	124	2.5	2.5	NUM
cana-3491	309	125	)	)	PUNCT
cana-3491	309	126	(	(	PUNCT
cana-3491	309	127	4	4	NUM
cana-3491	310	1	+	+	NUM
cana-3491	310	2	1.5,2	1.5,2	NOUN
cana-3491	310	3	+	+	CCONJ
cana-3491	310	4	3.5	3.5	NUM
cana-3491	310	5	)	)	PUNCT
cana-3491	310	6	(	(	PUNCT
cana-3491	310	7	4	4	NUM
cana-3491	310	8	+	+	NUM
cana-3491	310	9	2.5,2	2.5,2	NOUN
cana-3491	310	10	+	+	CCONJ
cana-3491	310	11	4.5	4.5	NUM
cana-3491	310	12	)	)	PUNCT
cana-3491	310	13	(	(	PUNCT
cana-3491	310	14	4	4	NUM
cana-3491	310	15	+	+	NUM
cana-3491	310	16	3.5,2	3.5,2	NUM
cana-3491	310	17	)	)	PUNCT
cana-3491	310	18	(	(	PUNCT
cana-3491	310	19	4	4	NUM
cana-3491	310	20	+	+	NUM
cana-3491	310	21	4.5,2	4.5,2	NOUN
cana-3491	310	22	+	+	NOUN
cana-3491	310	23	1.5	1.5	NUM
cana-3491	310	24	)	)	PUNCT
cana-3491	310	25	(	(	PUNCT
cana-3491	310	26	4,3	4,3	NUM
cana-3491	310	27	)	)	PUNCT
cana-3491	310	28	(	(	PUNCT
cana-3491	310	29	4,3	4,3	NUM
cana-3491	310	30	+	+	NOUN
cana-3491	310	31	2.5	2.5	NUM
cana-3491	310	32	)	)	PUNCT
cana-3491	310	33	(	(	PUNCT
cana-3491	310	34	4	4	NUM
cana-3491	310	35	+	+	NUM
cana-3491	310	36	1.5,3	1.5,3	NUM
cana-3491	310	37	+	+	CCONJ
cana-3491	310	38	1.5	1.5	NUM
cana-3491	310	39	)	)	PUNCT
cana-3491	310	40	(	(	PUNCT
cana-3491	310	41	4	4	NUM
cana-3491	310	42	+	+	SYM
cana-3491	310	43	2.5,3	2.5,3	NUM
cana-3491	310	44	)	)	PUNCT
cana-3491	310	45	(	(	PUNCT
cana-3491	310	46	4	4	NUM
cana-3491	310	47	+	+	NUM
cana-3491	310	48	3.5,3	3.5,3	NUM
cana-3491	310	49	+	+	NOUN
cana-3491	310	50	4.5	4.5	NUM
cana-3491	310	51	)	)	PUNCT
cana-3491	310	52	(	(	PUNCT
cana-3491	310	53	4	4	NUM
cana-3491	310	54	+	+	NUM
cana-3491	310	55	4.5,3	4.5,3	NUM
cana-3491	310	56	+	+	CCONJ
cana-3491	310	57	3.5	3.5	NUM
cana-3491	310	58	)	)	PUNCT
cana-3491	310	59	𝒪	𝒪	PROPN
cana-3491	310	60	�	�	PROPN
cana-3491	310	61	̃	̃	PROPN
cana-3491	310	62	�	�	PROPN
cana-3491	310	63	�	�	PROPN
cana-3491	310	64	̃	̃	PROPN
cana-3491	310	65	�	�	PROPN
cana-3491	310	66	�	�	PROPN
cana-3491	310	67	̃	̃	PROPN
cana-3491	310	68	�	�	PROPN
cana-3491	310	69	�	�	PROPN
cana-3491	310	70	̃	̃	PROPN
cana-3491	310	71	�	�	PROPN
cana-3491	310	72	�	�	PROPN
cana-3491	310	73	̃	̃	PROPN
cana-3491	310	74	�	�	PROPN
cana-3491	310	75	4	4	NUM
cana-3491	310	76	.	.	PUNCT
cana-3491	310	77	implementing	implement	VERB
cana-3491	310	78	arithmetic	arithmetic	NOUN
cana-3491	310	79	of	of	ADP
cana-3491	310	80	points	point	NOUN
cana-3491	310	81	on	on	ADP
cana-3491	310	82	elliptic	elliptic	ADJ
cana-3491	310	83	curve	curve	NOUN
cana-3491	310	84	𝑬(𝑸𝒑	𝑬(𝑸𝒑	NUM
cana-3491	310	85	)	)	PUNCT
cana-3491	310	86	over	over	ADP
cana-3491	310	87	𝒑-adic	𝒑-adic	ADJ
cana-3491	310	88	field	field	NOUN
cana-3491	310	89	𝑸𝒑	𝑸𝒑	PROPN
cana-3491	310	90	to	to	PART
cana-3491	310	91	𝑬(𝑸𝒑)(𝒎𝒐𝒅	𝑬(𝑸𝒑)(𝒎𝒐𝒅	VERB
cana-3491	310	92	𝒑𝟐	𝒑𝟐	NOUN
cana-3491	310	93	)	)	PUNCT
cana-3491	310	94	in	in	ADP
cana-3491	310	95	the	the	DET
cana-3491	310	96	context	context	NOUN
cana-3491	310	97	of	of	ADP
cana-3491	310	98	improvising	improvise	VERB
cana-3491	310	99	the	the	DET
cana-3491	310	100	efficiency	efficiency	NOUN
cana-3491	310	101	of	of	ADP
cana-3491	310	102	cryptosystems	cryptosystem	NOUN
cana-3491	310	103	with	with	ADP
cana-3491	310	104	elliptic	elliptic	ADJ
cana-3491	310	105	curves	curve	NOUN
cana-3491	310	106	,	,	PUNCT
cana-3491	310	107	the	the	DET
cana-3491	310	108	purpose	purpose	NOUN
cana-3491	310	109	of	of	ADP
cana-3491	310	110	studying	study	VERB
cana-3491	310	111	arithmetic	arithmetic	NOUN
cana-3491	310	112	of	of	ADP
cana-3491	310	113	points	point	NOUN
cana-3491	310	114	in	in	ADP
cana-3491	310	115	elliptic	elliptic	ADJ
cana-3491	310	116	curves	curve	NOUN
cana-3491	310	117	𝐸(𝑄𝑝	𝐸(𝑄𝑝	ADJ
cana-3491	310	118	)	)	PUNCT
cana-3491	310	119	over	over	ADP
cana-3491	310	120	𝑝-adic	𝑝-adic	ADJ
cana-3491	310	121	field	field	NOUN
cana-3491	310	122	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	310	123	is	be	AUX
cana-3491	310	124	necessary	necessary	ADJ
cana-3491	310	125	.	.	PUNCT
cana-3491	311	1	to	to	PART
cana-3491	311	2	study	study	VERB
cana-3491	311	3	arithmetic	arithmetic	ADJ
cana-3491	311	4	of	of	ADP
cana-3491	311	5	points	point	NOUN
cana-3491	311	6	in	in	ADP
cana-3491	311	7	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	311	8	)	)	PUNCT
cana-3491	311	9	,	,	PUNCT
cana-3491	311	10	we	we	PRON
cana-3491	311	11	first	first	ADV
cana-3491	311	12	study	study	VERB
cana-3491	311	13	arithmetic	arithmetic	ADJ
cana-3491	311	14	of	of	ADP
cana-3491	311	15	points	point	NOUN
cana-3491	311	16	in	in	ADP
cana-3491	311	17	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	311	18	)	)	PUNCT
cana-3491	311	19	(	(	PUNCT
cana-3491	311	20	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3491	311	21	𝑝2	𝑝2	NOUN
cana-3491	311	22	)	)	PUNCT
cana-3491	311	23	by	by	ADP
cana-3491	311	24	extending	extend	VERB
cana-3491	311	25	arithmetic	arithmetic	NOUN
cana-3491	311	26	of	of	ADP
cana-3491	311	27	points	point	NOUN
cana-3491	311	28	in	in	ADP
cana-3491	311	29	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	311	30	)	)	PUNCT
cana-3491	311	31	and	and	CCONJ
cana-3491	311	32	then	then	ADV
cana-3491	311	33	the	the	DET
cana-3491	311	34	arithmetic	arithmetic	NOUN
cana-3491	311	35	of	of	ADP
cana-3491	311	36	points	point	NOUN
cana-3491	311	37	in	in	ADP
cana-3491	311	38	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	311	39	)	)	PUNCT
cana-3491	311	40	(	(	PUNCT
cana-3491	311	41	𝑚𝑜𝑑	𝑚𝑜𝑑	PROPN
cana-3491	311	42	𝑝3	𝑝3	NOUN
cana-3491	311	43	)	)	PUNCT
cana-3491	311	44	by	by	ADP
cana-3491	311	45	extending	extend	VERB
cana-3491	311	46	arithmetic	arithmetic	NOUN
cana-3491	311	47	of	of	ADP
cana-3491	311	48	points	point	NOUN
cana-3491	311	49	in	in	ADP
cana-3491	311	50	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	311	51	)	)	PUNCT
cana-3491	311	52	(	(	PUNCT
cana-3491	311	53	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3491	311	54	𝑝2	𝑝2	NOUN
cana-3491	311	55	)	)	PUNCT
cana-3491	311	56	and	and	CCONJ
cana-3491	311	57	so	so	ADV
cana-3491	311	58	on	on	ADV
cana-3491	311	59	.	.	PUNCT
cana-3491	312	1	in	in	ADP
cana-3491	312	2	this	this	DET
cana-3491	312	3	section	section	NOUN
cana-3491	312	4	,	,	PUNCT
cana-3491	312	5	we	we	PRON
cana-3491	312	6	have	have	AUX
cana-3491	312	7	derived	derive	VERB
cana-3491	312	8	formula	formula	NOUN
cana-3491	312	9	for	for	ADP
cana-3491	312	10	arithmetic	arithmetic	NOUN
cana-3491	312	11	of	of	ADP
cana-3491	312	12	points	point	NOUN
cana-3491	312	13	in	in	ADP
cana-3491	312	14	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	312	15	𝑝2	𝑝2	NOUN
cana-3491	312	16	)	)	PUNCT
cana-3491	312	17	and	and	CCONJ
cana-3491	312	18	had	have	AUX
cana-3491	312	19	given	give	VERB
cana-3491	312	20	an	an	DET
cana-3491	312	21	algorithm	algorithm	NOUN
cana-3491	312	22	along	along	ADP
cana-3491	312	23	with	with	ADP
cana-3491	312	24	code	code	NOUN
cana-3491	312	25	in	in	ADP
cana-3491	312	26	python	python	NOUN
cana-3491	312	27	language	language	NOUN
cana-3491	312	28	.	.	PUNCT
cana-3491	313	1	now	now	ADV
cana-3491	313	2	,	,	PUNCT
cana-3491	313	3	to	to	PART
cana-3491	313	4	implement	implement	VERB
cana-3491	313	5	the	the	DET
cana-3491	313	6	arithmetic	arithmetic	NOUN
cana-3491	313	7	of	of	ADP
cana-3491	313	8	points	point	NOUN
cana-3491	313	9	on	on	ADP
cana-3491	313	10	elliptic	elliptic	ADJ
cana-3491	313	11	curve	curve	NOUN
cana-3491	313	12	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	313	13	)	)	PUNCT
cana-3491	313	14	over	over	ADP
cana-3491	313	15	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	313	16	to	to	ADP
cana-3491	313	17	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	313	18	)	)	PUNCT
cana-3491	313	19	(	(	PUNCT
cana-3491	313	20	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3491	313	21	𝑝2	𝑝2	PROPN
cana-3491	313	22	)	)	PUNCT
cana-3491	313	23	,	,	PUNCT
cana-3491	313	24	if𝐸(𝜅	if𝐸(𝜅	PROPN
cana-3491	313	25	)	)	PUNCT
cana-3491	313	26	is	be	AUX
cana-3491	313	27	an	an	DET
cana-3491	313	28	elliptic	elliptic	ADJ
cana-3491	313	29	curve	curve	NOUN
cana-3491	313	30	defined	define	VERB
cana-3491	313	31	over	over	ADP
cana-3491	313	32	a	a	DET
cana-3491	313	33	field	field	NOUN
cana-3491	313	34	𝜅	𝜅	ADP
cana-3491	313	35	with	with	ADP
cana-3491	313	36	𝐶ℎ𝑎𝑟𝜅	𝐶ℎ𝑎𝑟𝜅	PROPN
cana-3491	313	37	≠	≠	PROPN
cana-3491	313	38	2,3	2,3	NUM
cana-3491	313	39	,	,	PUNCT
cana-3491	313	40	given	give	VERB
cana-3491	313	41	as	as	ADP
cana-3491	313	42	𝐸	𝐸	PROPN
cana-3491	313	43	:	:	PUNCT
cana-3491	313	44	𝛽2	𝛽2	NOUN
cana-3491	313	45	=	=	PRON
cana-3491	313	46	𝛼3	𝛼3	NOUN
cana-3491	313	47	+	+	CCONJ
cana-3491	313	48	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	313	49	+	+	CCONJ
cana-3491	313	50	𝐵	𝐵	NOUN
cana-3491	313	51	then	then	ADV
cana-3491	313	52	for	for	ADP
cana-3491	313	53	𝑃1	𝑃1	NOUN
cana-3491	313	54	=(	=(	NOUN
cana-3491	313	55	𝛼1	𝛼1	NOUN
cana-3491	313	56	,	,	PUNCT
cana-3491	313	57	𝛽1	𝛽1	NOUN
cana-3491	313	58	)	)	PUNCT
cana-3491	313	59	and	and	CCONJ
cana-3491	313	60	𝑃2	𝑃2	ADJ
cana-3491	313	61	=(	=(	PROPN
cana-3491	313	62	𝛼2	𝛼2	PROPN
cana-3491	313	63	,	,	PUNCT
cana-3491	313	64	𝛽2	𝛽2	NOUN
cana-3491	313	65	)	)	PUNCT
cana-3491	313	66	in	in	ADP
cana-3491	313	67	𝐸(𝜅	𝐸(𝜅	NOUN
cana-3491	313	68	)	)	PUNCT
cana-3491	313	69	,	,	PUNCT
cana-3491	313	70	the	the	DET
cana-3491	313	71	point	point	NOUN
cana-3491	313	72	addition	addition	NOUN
cana-3491	313	73	of	of	ADP
cana-3491	313	74	𝑃1	𝑃1	NOUN
cana-3491	313	75	and	and	CCONJ
cana-3491	313	76	𝑃2	𝑃2	NOUN
cana-3491	313	77	denoted	denote	VERB
cana-3491	313	78	as	as	ADP
cana-3491	313	79	𝑃1	𝑃1	NOUN
cana-3491	313	80	+	+	CCONJ
cana-3491	313	81	𝑃2	𝑃2	NOUN
cana-3491	313	82	and	and	CCONJ
cana-3491	313	83	is	be	AUX
cana-3491	313	84	given	give	VERB
cana-3491	313	85	as	as	ADP
cana-3491	313	86	𝑃1	𝑃1	NOUN
cana-3491	313	87	+	+	CCONJ
cana-3491	313	88	𝑃2	𝑃2	NOUN
cana-3491	313	89	=	=	SYM
cana-3491	313	90	{	{	PUNCT
cana-3491	313	91	(	(	PUNCT
cana-3491	313	92	𝑚2	𝑚2	NOUN
cana-3491	313	93	−	−	NOUN
cana-3491	313	94	𝛼1	𝛼1	NOUN
cana-3491	313	95	−	−	PROPN
cana-3491	313	96	𝛼2	𝛼2	PROPN
cana-3491	313	97	,	,	PUNCT
cana-3491	313	98	𝑚(𝛼1	𝑚(𝛼1	VERB
cana-3491	313	99	−	−	NOUN
cana-3491	313	100	𝛼3	𝛼3	NOUN
cana-3491	313	101	)	)	PUNCT
cana-3491	313	102	−	−	PROPN
cana-3491	313	103	𝛽1	𝛽1	NOUN
cana-3491	313	104	)	)	PUNCT
cana-3491	313	105	with	with	ADP
cana-3491	313	106	𝑚	𝑚	NOUN
cana-3491	313	107	=	=	SYM
cana-3491	313	108	𝛽2−𝛽1	𝛽2−𝛽1	NUM
cana-3491	313	109	𝛼2−𝛼1	𝛼2−𝛼1	NUM
cana-3491	313	110	if	if	SCONJ
cana-3491	313	111	𝑃1	𝑃1	NOUN
cana-3491	313	112	≠	≠	PROPN
cana-3491	313	113	𝑃2	𝑃2	NOUN
cana-3491	313	114	(	(	PUNCT
cana-3491	313	115	𝑚2	𝑚2	NOUN
cana-3491	313	116	−	−	PROPN
cana-3491	313	117	2𝛼1	2𝛼1	NUM
cana-3491	313	118	,	,	PUNCT
cana-3491	313	119	𝑚(𝛼1	𝑚(𝛼1	VERB
cana-3491	313	120	−	−	NOUN
cana-3491	313	121	𝛼3	𝛼3	NOUN
cana-3491	313	122	)	)	PUNCT
cana-3491	313	123	−	−	PROPN
cana-3491	313	124	𝛽1	𝛽1	NOUN
cana-3491	313	125	)	)	PUNCT
cana-3491	313	126	with	with	ADP
cana-3491	313	127	𝑚	𝑚	NOUN
cana-3491	313	128	=	=	SYM
cana-3491	313	129	3𝛼1	3𝛼1	NUM
cana-3491	313	130	2+𝐴	2+𝐴	NUM
cana-3491	313	131	2𝛽1	2𝛽1	NUM
cana-3491	313	132	if	if	SCONJ
cana-3491	313	133	𝑃1	𝑃1	NOUN
cana-3491	313	134	=	=	SYM
cana-3491	313	135	𝑃2	𝑃2	PROPN
cana-3491	313	136	------------(1	------------(1	PROPN
cana-3491	313	137	)	)	PUNCT
cana-3491	313	138	now	now	ADV
cana-3491	313	139	,	,	PUNCT
cana-3491	313	140	in	in	ADP
cana-3491	313	141	particular	particular	ADJ
cana-3491	313	142	for	for	ADP
cana-3491	313	143	𝐾	𝐾	PROPN
cana-3491	313	144	=	=	SYM
cana-3491	313	145	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	313	146	,	,	PUNCT
cana-3491	313	147	the	the	DET
cana-3491	313	148	𝑝-adic	𝑝-adic	ADJ
cana-3491	313	149	field	field	NOUN
cana-3491	313	150	for	for	ADP
cana-3491	313	151	an	an	DET
cana-3491	313	152	elliptic	elliptic	ADJ
cana-3491	313	153	curve	curve	NOUN
cana-3491	313	154	𝐸	𝐸	PROPN
cana-3491	313	155	over	over	ADP
cana-3491	313	156	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	313	157	defined	define	VERB
cana-3491	313	158	over	over	ADP
cana-3491	313	159	𝑍𝑝	𝑍𝑝	PROPN
cana-3491	313	160	,	,	PUNCT
cana-3491	313	161	the	the	DET
cana-3491	313	162	points	point	NOUN
cana-3491	313	163	�	�	PROPN
cana-3491	313	164	̃	̃	PROPN
cana-3491	313	165	�	�	NOUN
cana-3491	313	166	1	1	NUM
cana-3491	313	167	and	and	CCONJ
cana-3491	313	168	�	�	PROPN
cana-3491	313	169	̃	̃	PROPN
cana-3491	313	170	�	�	NOUN
cana-3491	313	171	2	2	NUM
cana-3491	313	172	are	be	AUX
cana-3491	313	173	given	give	VERB
cana-3491	313	174	as	as	SCONJ
cana-3491	313	175	communications	communication	NOUN
cana-3491	313	176	on	on	ADP
cana-3491	313	177	applied	apply	VERB
cana-3491	313	178	nonlinear	nonlinear	ADJ
cana-3491	313	179	analysis	analysis	NOUN
cana-3491	313	180	issn	issn	NOUN
cana-3491	313	181	:	:	PUNCT
cana-3491	313	182	1074	1074	NUM
cana-3491	313	183	-	-	PUNCT
cana-3491	313	184	133x	133x	NUM
cana-3491	313	185	vol	vol	NOUN
cana-3491	313	186	32	32	NUM
cana-3491	313	187	no	no	NOUN
cana-3491	313	188	.	.	PUNCT
cana-3491	314	1	7s	7	NOUN
cana-3491	314	2	(	(	PUNCT
cana-3491	314	3	2025	2025	NUM
cana-3491	314	4	)	)	PUNCT
cana-3491	314	5	870	870	NUM
cana-3491	314	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	314	7	�	�	PROPN
cana-3491	314	8	̃	̃	PROPN
cana-3491	314	9	�	�	NOUN
cana-3491	314	10	1	1	NUM
cana-3491	314	11	=	=	SYM
cana-3491	314	12	(	(	PUNCT
cana-3491	314	13	�	�	PROPN
cana-3491	314	14	̃	̃	NOUN
cana-3491	314	15	�	�	NOUN
cana-3491	314	16	1	1	NUM
cana-3491	314	17	,	,	PUNCT
cana-3491	314	18	�	�	PROPN
cana-3491	314	19	̃	̃	NOUN
cana-3491	314	20	�	�	NOUN
cana-3491	314	21	1	1	NUM
cana-3491	314	22	)	)	PUNCT
cana-3491	314	23	=	=	NOUN
cana-3491	314	24	(	(	PUNCT
cana-3491	314	25	𝑥10	𝑥10	NOUN
cana-3491	314	26	+	+	SYM
cana-3491	314	27	𝑥11𝑝	𝑥11𝑝	NUM
cana-3491	314	28	+	+	NUM
cana-3491	314	29	𝑥12𝑝	𝑥12𝑝	NUM
cana-3491	314	30	2	2	NUM
cana-3491	314	31	+	+	NOUN
cana-3491	314	32	.	.	PUNCT
cana-3491	314	33	.	.	PUNCT
cana-3491	314	34	.	.	PUNCT
cana-3491	315	1	,	,	PUNCT
cana-3491	315	2	𝑦10	𝑦10	ADJ
cana-3491	315	3	+	+	CCONJ
cana-3491	315	4	𝑦11𝑝	𝑦11𝑝	NOUN
cana-3491	315	5	+	+	SYM
cana-3491	315	6	𝑦12𝑝	𝑦12𝑝	NUM
cana-3491	315	7	2	2	NUM
cana-3491	315	8	+	+	NOUN
cana-3491	315	9	.	.	PUNCT
cana-3491	315	10	.	.	PUNCT
cana-3491	315	11	.	.	PUNCT
cana-3491	315	12	)	)	PUNCT
cana-3491	316	1	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-3491	316	2	�	�	PROPN
cana-3491	316	3	̃	̃	PROPN
cana-3491	316	4	�	�	NOUN
cana-3491	316	5	2	2	NUM
cana-3491	316	6	=	=	SYM
cana-3491	316	7	(	(	PUNCT
cana-3491	316	8	�	�	PROPN
cana-3491	316	9	̃	̃	NOUN
cana-3491	316	10	�	�	NOUN
cana-3491	316	11	2	2	NUM
cana-3491	316	12	,	,	PUNCT
cana-3491	316	13	�	�	PROPN
cana-3491	316	14	̃	̃	NOUN
cana-3491	316	15	�	�	NOUN
cana-3491	316	16	2	2	NUM
cana-3491	316	17	)	)	PUNCT
cana-3491	316	18	=	=	NOUN
cana-3491	317	1	(	(	PUNCT
cana-3491	317	2	𝑥20	𝑥20	NOUN
cana-3491	317	3	+	+	CCONJ
cana-3491	317	4	𝑥21𝑝	𝑥21𝑝	PRON
cana-3491	317	5	+	+	NUM
cana-3491	317	6	𝑥22𝑝	𝑥22𝑝	NUM
cana-3491	317	7	2	2	NUM
cana-3491	317	8	+	+	NOUN
cana-3491	317	9	.	.	PUNCT
cana-3491	317	10	.	.	PUNCT
cana-3491	317	11	.	.	PUNCT
cana-3491	318	1	,	,	PUNCT
cana-3491	318	2	𝑦20	𝑦20	NOUN
cana-3491	318	3	+	+	CCONJ
cana-3491	318	4	𝑦21𝑝	𝑦21𝑝	PROPN
cana-3491	318	5	+	+	CCONJ
cana-3491	318	6	𝑦22𝑝	𝑦22𝑝	PROPN
cana-3491	318	7	2	2	NUM
cana-3491	318	8	+	+	NOUN
cana-3491	318	9	.	.	PUNCT
cana-3491	318	10	.	.	PUNCT
cana-3491	318	11	.	.	PUNCT
cana-3491	318	12	)	)	PUNCT
cana-3491	319	1	with	with	ADP
cana-3491	319	2	each	each	DET
cana-3491	319	3	coordinate	coordinate	NOUN
cana-3491	319	4	in	in	ADP
cana-3491	319	5	its	its	PRON
cana-3491	319	6	𝑝-adic	𝑝-adic	ADJ
cana-3491	319	7	expansion	expansion	NOUN
cana-3491	319	8	and	and	CCONJ
cana-3491	319	9	by	by	ADP
cana-3491	319	10	the	the	DET
cana-3491	319	11	point	point	NOUN
cana-3491	319	12	addition	addition	NOUN
cana-3491	319	13	in	in	ADP
cana-3491	319	14	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	319	15	)	)	PUNCT
cana-3491	319	16	,	,	PUNCT
cana-3491	319	17	we	we	PRON
cana-3491	319	18	have	have	AUX
cana-3491	319	19	�	�	PROPN
cana-3491	319	20	̃	̃	NOUN
cana-3491	319	21	�	�	PROPN
cana-3491	319	22	1	1	NUM
cana-3491	319	23	+	+	NUM
cana-3491	319	24	�	�	PROPN
cana-3491	319	25	̃	̃	NOUN
cana-3491	319	26	�	�	NOUN
cana-3491	319	27	2	2	NUM
cana-3491	319	28	=	=	SYM
cana-3491	319	29	�	�	PROPN
cana-3491	319	30	̃	̃	PROPN
cana-3491	319	31	�	�	NOUN
cana-3491	319	32	3	3	NUM
cana-3491	319	33	say	say	VERB
cana-3491	319	34	with	with	ADP
cana-3491	319	35	�	�	PROPN
cana-3491	319	36	̃	̃	NOUN
cana-3491	319	37	�	�	NOUN
cana-3491	319	38	3	3	NUM
cana-3491	319	39	=	=	SYM
cana-3491	319	40	(	(	PUNCT
cana-3491	319	41	�	�	PROPN
cana-3491	319	42	̃	̃	NOUN
cana-3491	319	43	�	�	NOUN
cana-3491	319	44	3	3	NUM
cana-3491	319	45	,	,	PUNCT
cana-3491	319	46	�	�	PROPN
cana-3491	319	47	̃	̃	PROPN
cana-3491	319	48	�	�	NOUN
cana-3491	319	49	3	3	NUM
cana-3491	319	50	)	)	PUNCT
cana-3491	319	51	given	give	VERB
cana-3491	319	52	as	as	ADP
cana-3491	319	53	�	�	PROPN
cana-3491	319	54	̃	̃	PROPN
cana-3491	319	55	�	�	NOUN
cana-3491	319	56	3	3	NUM
cana-3491	319	57	=	=	SYM
cana-3491	319	58	(	(	PUNCT
cana-3491	319	59	�	�	PROPN
cana-3491	319	60	̃	̃	NOUN
cana-3491	319	61	�	�	NOUN
cana-3491	319	62	3	3	NUM
cana-3491	319	63	,	,	PUNCT
cana-3491	319	64	�	�	PROPN
cana-3491	319	65	̃	̃	NOUN
cana-3491	319	66	�	�	NOUN
cana-3491	319	67	3	3	NUM
cana-3491	319	68	)	)	PUNCT
cana-3491	319	69	=	=	SYM
cana-3491	319	70	(	(	PUNCT
cana-3491	319	71	𝑥30	𝑥30	NOUN
cana-3491	319	72	+	+	CCONJ
cana-3491	319	73	𝑥31𝑝	𝑥31𝑝	PROPN
cana-3491	320	1	+	+	CCONJ
cana-3491	320	2	𝑥32𝑝	𝑥32𝑝	X
cana-3491	320	3	2	2	NUM
cana-3491	320	4	+	+	NOUN
cana-3491	320	5	.	.	PUNCT
cana-3491	320	6	.	.	PUNCT
cana-3491	320	7	.	.	PUNCT
cana-3491	321	1	,	,	PUNCT
cana-3491	321	2	𝑦30	𝑦30	VERB
cana-3491	321	3	+	+	CCONJ
cana-3491	321	4	𝑦31𝑝	𝑦31𝑝	NOUN
cana-3491	321	5	+	+	CCONJ
cana-3491	321	6	𝑦32𝑝	𝑦32𝑝	NUM
cana-3491	321	7	2	2	NUM
cana-3491	321	8	+	+	NOUN
cana-3491	321	9	.	.	PUNCT
cana-3491	321	10	.	.	PUNCT
cana-3491	321	11	.	.	PUNCT
cana-3491	321	12	)	)	PUNCT
cana-3491	322	1	the	the	DET
cana-3491	322	2	point	point	NOUN
cana-3491	322	3	�	�	PROPN
cana-3491	322	4	̃	̃	PROPN
cana-3491	322	5	�	�	NOUN
cana-3491	322	6	3	3	NUM
cana-3491	322	7	could	could	AUX
cana-3491	322	8	be	be	AUX
cana-3491	322	9	known	know	VERB
cana-3491	322	10	with	with	ADP
cana-3491	322	11	the	the	DET
cana-3491	322	12	evaluation	evaluation	NOUN
cana-3491	322	13	of	of	ADP
cana-3491	322	14	𝑥3𝑖	𝑥3𝑖	PROPN
cana-3491	322	15	’s	’s	PART
cana-3491	322	16	and	and	CCONJ
cana-3491	322	17	𝑦3𝑖	𝑦3𝑖	PROPN
cana-3491	322	18	’s	’s	PART
cana-3491	322	19	for	for	ADP
cana-3491	322	20	all	all	PRON
cana-3491	322	21	𝑖	𝑖	NOUN
cana-3491	323	1	=	=	NOUN
cana-3491	323	2	0,1,2	0,1,2	NUM
cana-3491	323	3	,	,	PUNCT
cana-3491	323	4	.	.	PUNCT
cana-3491	324	1	..	..	PUNCT
cana-3491	324	2	.	.	PUNCT
cana-3491	325	1	now	now	ADV
cana-3491	325	2	to	to	PART
cana-3491	325	3	obtain	obtain	VERB
cana-3491	325	4	𝑥3𝑖	𝑥3𝑖	PRON
cana-3491	325	5	’s	’s	PART
cana-3491	325	6	and	and	CCONJ
cana-3491	325	7	𝑦3𝑖	𝑦3𝑖	PROPN
cana-3491	325	8	’s	’s	PART
cana-3491	325	9	for	for	ADP
cana-3491	325	10	all	all	PRON
cana-3491	325	11	𝑖	𝑖	NOUN
cana-3491	326	1	=	=	NOUN
cana-3491	326	2	0,1,2	0,1,2	NUM
cana-3491	326	3	,	,	PUNCT
cana-3491	326	4	.	.	PUNCT
cana-3491	327	1	..	..	PUNCT
cana-3491	327	2	,	,	PUNCT
cana-3491	327	3	we	we	PRON
cana-3491	327	4	implement	implement	VERB
cana-3491	327	5	the	the	DET
cana-3491	327	6	arithmetic	arithmetic	NOUN
cana-3491	327	7	of	of	ADP
cana-3491	327	8	points	point	NOUN
cana-3491	327	9	�	�	PROPN
cana-3491	327	10	̃	̃	PROPN
cana-3491	327	11	�	�	NOUN
cana-3491	327	12	1	1	NUM
cana-3491	327	13	and	and	CCONJ
cana-3491	327	14	�	�	PROPN
cana-3491	327	15	̃	̃	PROPN
cana-3491	327	16	�	�	NOUN
cana-3491	327	17	2	2	NUM
cana-3491	327	18	in	in	ADP
cana-3491	327	19	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	327	20	)	)	PUNCT
cana-3491	327	21	to	to	ADP
cana-3491	327	22	the	the	DET
cana-3491	327	23	points	point	NOUN
cana-3491	327	24	�	�	PROPN
cana-3491	327	25	̃	̃	PROPN
cana-3491	327	26	�	�	NOUN
cana-3491	327	27	1𝑛	1𝑛	NOUN
cana-3491	327	28	and	and	CCONJ
cana-3491	327	29	�	�	PROPN
cana-3491	327	30	̃	̃	PROPN
cana-3491	327	31	�	�	PROPN
cana-3491	327	32	2𝑛	2𝑛	PROPN
cana-3491	327	33	in	in	ADP
cana-3491	327	34	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	327	35	𝑝𝑛+1	𝑝𝑛+1	NUM
cana-3491	327	36	)	)	PUNCT
cana-3491	327	37	and	and	CCONJ
cana-3491	327	38	obtain	obtain	VERB
cana-3491	327	39	𝑥3𝑖	𝑥3𝑖	PRON
cana-3491	327	40	’s	’s	PART
cana-3491	327	41	and	and	CCONJ
cana-3491	327	42	𝑦3𝑖	𝑦3𝑖	PROPN
cana-3491	327	43	’s	’s	PART
cana-3491	327	44	for	for	ADP
cana-3491	327	45	all	all	DET
cana-3491	327	46	𝑖	𝑖	NOUN
cana-3491	327	47	=	=	NOUN
cana-3491	327	48	0,1,2	0,1,2	NUM
cana-3491	327	49	,	,	PUNCT
cana-3491	327	50	.	.	PUNCT
cana-3491	327	51	.	.	PUNCT
cana-3491	327	52	.	.	PUNCT
cana-3491	328	1	𝑛	𝑛	X
cana-3491	328	2	,	,	PUNCT
cana-3491	328	3	where	where	SCONJ
cana-3491	328	4	the	the	DET
cana-3491	328	5	points	point	NOUN
cana-3491	328	6	�	�	PROPN
cana-3491	328	7	̃	̃	PROPN
cana-3491	328	8	�	�	NOUN
cana-3491	328	9	1𝑛	1𝑛	NOUN
cana-3491	328	10	and	and	CCONJ
cana-3491	328	11	�	�	PROPN
cana-3491	328	12	̃	̃	PROPN
cana-3491	328	13	�	�	PROPN
cana-3491	328	14	2𝑛	2𝑛	PROPN
cana-3491	328	15	are	be	AUX
cana-3491	328	16	the	the	DET
cana-3491	328	17	points	point	NOUN
cana-3491	328	18	obtained	obtain	VERB
cana-3491	328	19	by	by	ADP
cana-3491	328	20	considering	consider	VERB
cana-3491	328	21	�	�	PROPN
cana-3491	328	22	̃	̃	PROPN
cana-3491	328	23	�	�	PROPN
cana-3491	328	24	1	1	NUM
cana-3491	328	25	and	and	CCONJ
cana-3491	328	26	�	�	PROPN
cana-3491	328	27	̃	̃	PROPN
cana-3491	328	28	�	�	NOUN
cana-3491	328	29	2	2	NUM
cana-3491	328	30	modulo	modulo	NOUN
cana-3491	328	31	𝑝𝑛+1	𝑝𝑛+1	NOUN
cana-3491	328	32	.	.	PUNCT
cana-3491	329	1	in	in	ADP
cana-3491	329	2	particular	particular	ADJ
cana-3491	329	3	,	,	PUNCT
cana-3491	329	4	on	on	ADP
cana-3491	329	5	considering	consider	VERB
cana-3491	329	6	�	�	PROPN
cana-3491	329	7	̃	̃	NOUN
cana-3491	329	8	�	�	NOUN
cana-3491	329	9	1	1	NUM
cana-3491	329	10	,	,	PUNCT
cana-3491	329	11	�	�	PROPN
cana-3491	329	12	̃	̃	NOUN
cana-3491	329	13	�	�	NOUN
cana-3491	329	14	2	2	NUM
cana-3491	329	15	to	to	ADP
cana-3491	329	16	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3491	329	17	𝑝2	𝑝2	NOUN
cana-3491	329	18	,	,	PUNCT
cana-3491	329	19	we	we	PRON
cana-3491	329	20	have	have	VERB
cana-3491	329	21	�	�	PROPN
cana-3491	329	22	̃	̃	NOUN
cana-3491	329	23	�	�	NOUN
cana-3491	329	24	11	11	NUM
cana-3491	329	25	=	=	SYM
cana-3491	329	26	(	(	PUNCT
cana-3491	329	27	�	�	PROPN
cana-3491	329	28	̃	̃	NOUN
cana-3491	329	29	�	�	NOUN
cana-3491	329	30	11	11	NUM
cana-3491	329	31	,	,	PUNCT
cana-3491	329	32	�	�	PROPN
cana-3491	329	33	̃	̃	PROPN
cana-3491	329	34	�	�	NOUN
cana-3491	329	35	11	11	NUM
cana-3491	329	36	)	)	PUNCT
cana-3491	329	37	=	=	PUNCT
cana-3491	329	38	(	(	PUNCT
cana-3491	329	39	𝑥10	𝑥10	X
cana-3491	329	40	+	+	SYM
cana-3491	329	41	𝑥11𝑝	𝑥11𝑝	ADJ
cana-3491	329	42	,	,	PUNCT
cana-3491	329	43	𝑦10	𝑦10	ADJ
cana-3491	329	44	+	+	SYM
cana-3491	329	45	𝑦11𝑝)𝑎𝑛𝑑	𝑦11𝑝)𝑎𝑛𝑑	PROPN
cana-3491	329	46	�	�	PROPN
cana-3491	329	47	̃	̃	PROPN
cana-3491	329	48	�	�	NOUN
cana-3491	329	49	21	21	NUM
cana-3491	329	50	=	=	SYM
cana-3491	329	51	(	(	PUNCT
cana-3491	329	52	�	�	PROPN
cana-3491	329	53	̃	̃	NOUN
cana-3491	329	54	�	�	NOUN
cana-3491	329	55	21	21	NUM
cana-3491	329	56	,	,	PUNCT
cana-3491	329	57	�	�	PROPN
cana-3491	329	58	̃	̃	PROPN
cana-3491	329	59	�	�	NOUN
cana-3491	329	60	21	21	NUM
cana-3491	329	61	)	)	PUNCT
cana-3491	329	62	=	=	PUNCT
cana-3491	330	1	(	(	PUNCT
cana-3491	330	2	𝑥20	𝑥20	NOUN
cana-3491	330	3	+	+	CCONJ
cana-3491	330	4	𝑥21𝑝	𝑥21𝑝	NOUN
cana-3491	330	5	,	,	PUNCT
cana-3491	330	6	𝑦20	𝑦20	NOUN
cana-3491	330	7	+	+	CCONJ
cana-3491	330	8	𝑦21𝑝	𝑦21𝑝	NOUN
cana-3491	330	9	)	)	PUNCT
cana-3491	330	10	now	now	ADV
cana-3491	330	11	we	we	PRON
cana-3491	330	12	obtain	obtain	VERB
cana-3491	330	13	the	the	DET
cana-3491	330	14	arithmetic	arithmetic	NOUN
cana-3491	330	15	of	of	ADP
cana-3491	330	16	points	point	NOUN
cana-3491	330	17	�	�	PROPN
cana-3491	330	18	̃	̃	PROPN
cana-3491	330	19	�	�	PROPN
cana-3491	330	20	11	11	NUM
cana-3491	330	21	and	and	CCONJ
cana-3491	330	22	�	�	PROPN
cana-3491	330	23	̃	̃	PROPN
cana-3491	330	24	�	�	NOUN
cana-3491	330	25	21	21	NUM
cana-3491	330	26	in	in	ADP
cana-3491	330	27	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	330	28	𝑝2	𝑝2	NOUN
cana-3491	330	29	)	)	PUNCT
cana-3491	330	30	by	by	ADP
cana-3491	330	31	implementing	implement	VERB
cana-3491	330	32	the	the	DET
cana-3491	330	33	point	point	NOUN
cana-3491	330	34	addition	addition	NOUN
cana-3491	330	35	in	in	ADP
cana-3491	330	36	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	330	37	)	)	PUNCT
cana-3491	330	38	and	and	CCONJ
cana-3491	330	39	obtain	obtain	VERB
cana-3491	330	40	𝑥30	𝑥30	NOUN
cana-3491	330	41	,	,	PUNCT
cana-3491	330	42	𝑥31	𝑥31	PROPN
cana-3491	330	43	,	,	PUNCT
cana-3491	330	44	𝑦30	𝑦30	VERB
cana-3491	330	45	,	,	PUNCT
cana-3491	330	46	𝑦31	𝑦31	NOUN
cana-3491	330	47	.	.	PUNCT
cana-3491	331	1	in	in	ADP
cana-3491	331	2	the	the	DET
cana-3491	331	3	following	follow	VERB
cana-3491	331	4	theorem	theorem	NOUN
cana-3491	331	5	we	we	PRON
cana-3491	331	6	describe	describe	VERB
cana-3491	331	7	the	the	DET
cana-3491	331	8	implementation	implementation	NOUN
cana-3491	331	9	of	of	ADP
cana-3491	331	10	the	the	DET
cana-3491	331	11	arithmetic	arithmetic	NOUN
cana-3491	331	12	of	of	ADP
cana-3491	331	13	points	point	NOUN
cana-3491	331	14	on	on	ADP
cana-3491	331	15	elliptic	elliptic	ADJ
cana-3491	331	16	curve	curve	NOUN
cana-3491	331	17	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	331	18	)	)	PUNCT
cana-3491	331	19	defined	define	VERB
cana-3491	331	20	over	over	ADP
cana-3491	331	21	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	331	22	to	to	ADP
cana-3491	331	23	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	331	24	)	)	PUNCT
cana-3491	331	25	(	(	PUNCT
cana-3491	331	26	𝑚𝑜𝑑	𝑚𝑜𝑑	NOUN
cana-3491	331	27	𝑝2	𝑝2	NOUN
cana-3491	331	28	)	)	PUNCT
cana-3491	331	29	.	.	PUNCT
cana-3491	332	1	theorem	theorem	VERB
cana-3491	332	2	4.1	4.1	NUM
cana-3491	332	3	.	.	PUNCT
cana-3491	333	1	consider	consider	VERB
cana-3491	333	2	an	an	DET
cana-3491	333	3	elliptic	elliptic	ADJ
cana-3491	333	4	curve	curve	NOUN
cana-3491	333	5	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	333	6	)	)	PUNCT
cana-3491	333	7	defined	define	VERB
cana-3491	333	8	over	over	ADP
cana-3491	333	9	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	333	10	given	give	VERB
cana-3491	333	11	as	as	ADP
cana-3491	333	12	𝐸(𝑄𝑝	𝐸(𝑄𝑝	ADJ
cana-3491	333	13	):	):	PUNCT
cana-3491	333	14	𝑦	𝑦	NOUN
cana-3491	333	15	2	2	NUM
cana-3491	333	16	=	=	SYM
cana-3491	333	17	𝑥3	𝑥3	NOUN
cana-3491	333	18	+	+	CCONJ
cana-3491	334	1	𝐴𝑥	𝐴𝑥	X
cana-3491	334	2	+	+	CCONJ
cana-3491	334	3	𝐵	𝐵	NOUN
cana-3491	334	4	over	over	ADP
cana-3491	334	5	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	334	6	defined	define	VERB
cana-3491	334	7	over	over	ADP
cana-3491	334	8	𝑍𝑝.	𝑍𝑝.	PROPN
cana-3491	334	9	for	for	ADP
cana-3491	334	10	any	any	DET
cana-3491	334	11	points	point	NOUN
cana-3491	334	12	�	�	PROPN
cana-3491	334	13	̃	̃	NOUN
cana-3491	334	14	�	�	NOUN
cana-3491	334	15	1	1	NUM
cana-3491	334	16	,	,	PUNCT
cana-3491	334	17	�	�	PROPN
cana-3491	334	18	̃	̃	NOUN
cana-3491	334	19	�	�	NOUN
cana-3491	334	20	2	2	NUM
cana-3491	334	21	in	in	ADP
cana-3491	334	22	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	334	23	)	)	PUNCT
cana-3491	334	24	,	,	PUNCT
cana-3491	334	25	the	the	DET
cana-3491	334	26	arithmetic	arithmetic	NOUN
cana-3491	334	27	of	of	ADP
cana-3491	334	28	points	point	NOUN
cana-3491	334	29	�	�	PROPN
cana-3491	334	30	̃	̃	NOUN
cana-3491	334	31	�	�	NOUN
cana-3491	334	32	11	11	NUM
cana-3491	334	33	=	=	SYM
cana-3491	334	34	(	(	PUNCT
cana-3491	334	35	�	�	PROPN
cana-3491	334	36	̃	̃	NOUN
cana-3491	334	37	�	�	NOUN
cana-3491	334	38	11	11	NUM
cana-3491	334	39	,	,	PUNCT
cana-3491	334	40	�	�	PROPN
cana-3491	334	41	̃	̃	PROPN
cana-3491	334	42	�	�	NOUN
cana-3491	334	43	11	11	NUM
cana-3491	334	44	)	)	PUNCT
cana-3491	334	45	=	=	PUNCT
cana-3491	334	46	(	(	PUNCT
cana-3491	334	47	𝑥10	𝑥10	X
cana-3491	334	48	+	+	SYM
cana-3491	334	49	𝑥11𝑝	𝑥11𝑝	ADJ
cana-3491	334	50	,	,	PUNCT
cana-3491	334	51	𝑦10	𝑦10	ADJ
cana-3491	334	52	+	+	CCONJ
cana-3491	334	53	𝑦11𝑝	𝑦11𝑝	NOUN
cana-3491	334	54	)	)	PUNCT
cana-3491	334	55	and	and	CCONJ
cana-3491	334	56	�	�	PROPN
cana-3491	334	57	̃	̃	PROPN
cana-3491	334	58	�	�	NOUN
cana-3491	334	59	21	21	NUM
cana-3491	334	60	=	=	SYM
cana-3491	334	61	(	(	PUNCT
cana-3491	334	62	�	�	PROPN
cana-3491	334	63	̃	̃	NOUN
cana-3491	334	64	�	�	NOUN
cana-3491	334	65	21	21	NUM
cana-3491	334	66	,	,	PUNCT
cana-3491	334	67	�	�	PROPN
cana-3491	334	68	̃	̃	PROPN
cana-3491	334	69	�	�	NOUN
cana-3491	334	70	21	21	NUM
cana-3491	334	71	)	)	PUNCT
cana-3491	334	72	=	=	PUNCT
cana-3491	335	1	(	(	PUNCT
cana-3491	335	2	𝑥20	𝑥20	NOUN
cana-3491	335	3	+	+	CCONJ
cana-3491	335	4	𝑥21𝑝	𝑥21𝑝	NOUN
cana-3491	335	5	,	,	PUNCT
cana-3491	335	6	𝑦20	𝑦20	NOUN
cana-3491	335	7	+	+	CCONJ
cana-3491	335	8	𝑦21𝑝	𝑦21𝑝	NOUN
cana-3491	335	9	)	)	PUNCT
cana-3491	335	10	in	in	ADP
cana-3491	335	11	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	335	12	𝑝2	𝑝2	NOUN
cana-3491	335	13	)	)	PUNCT
cana-3491	335	14	may	may	AUX
cana-3491	335	15	be	be	AUX
cana-3491	335	16	obtained	obtain	VERB
cana-3491	335	17	by	by	ADP
cana-3491	335	18	implementing	implement	VERB
cana-3491	335	19	the	the	DET
cana-3491	335	20	point	point	NOUN
cana-3491	335	21	addition	addition	NOUN
cana-3491	335	22	of	of	ADP
cana-3491	335	23	points	point	NOUN
cana-3491	335	24	�	�	PROPN
cana-3491	335	25	̃	̃	NOUN
cana-3491	335	26	�	�	NOUN
cana-3491	335	27	1	1	NUM
cana-3491	335	28	,	,	PUNCT
cana-3491	335	29	�	�	PROPN
cana-3491	335	30	̃	̃	NOUN
cana-3491	335	31	�	�	NOUN
cana-3491	335	32	2	2	NUM
cana-3491	335	33	in	in	ADP
cana-3491	335	34	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	335	35	)	)	PUNCT
cana-3491	335	36	to	to	ADP
cana-3491	335	37	the	the	DET
cana-3491	335	38	points	point	NOUN
cana-3491	335	39	�	�	PROPN
cana-3491	335	40	̃	̃	PROPN
cana-3491	335	41	�	�	NOUN
cana-3491	335	42	11	11	NUM
cana-3491	335	43	,	,	PUNCT
cana-3491	335	44	�	�	PROPN
cana-3491	335	45	̃	̃	NOUN
cana-3491	335	46	�	�	NOUN
cana-3491	335	47	21	21	NUM
cana-3491	335	48	in	in	ADP
cana-3491	335	49	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	335	50	𝑝2	𝑝2	NOUN
cana-3491	335	51	)	)	PUNCT
cana-3491	335	52	and	and	CCONJ
cana-3491	335	53	�	�	PROPN
cana-3491	335	54	̃	̃	PROPN
cana-3491	335	55	�	�	NOUN
cana-3491	335	56	31is	31is	NOUN
cana-3491	335	57	given	give	VERB
cana-3491	335	58	as	as	ADP
cana-3491	335	59	�	�	PROPN
cana-3491	335	60	̃	̃	PROPN
cana-3491	335	61	�	�	PROPN
cana-3491	335	62	31	31	NUM
cana-3491	335	63	=	=	SYM
cana-3491	335	64	�	�	PROPN
cana-3491	335	65	̃	̃	PROPN
cana-3491	335	66	�	�	PROPN
cana-3491	335	67	11	11	NUM
cana-3491	335	68	+	+	SYM
cana-3491	335	69	�	�	PROPN
cana-3491	335	70	̃	̃	PROPN
cana-3491	335	71	�	�	NOUN
cana-3491	335	72	21	21	NUM
cana-3491	335	73	=	=	SYM
cana-3491	335	74	(	(	PUNCT
cana-3491	335	75	�	�	PROPN
cana-3491	335	76	̃	̃	PROPN
cana-3491	335	77	�	�	NOUN
cana-3491	335	78	31	31	NUM
cana-3491	335	79	,	,	PUNCT
cana-3491	335	80	�	�	PROPN
cana-3491	335	81	̃	̃	PROPN
cana-3491	335	82	�	�	NOUN
cana-3491	335	83	31	31	NUM
cana-3491	335	84	)	)	PUNCT
cana-3491	336	1	=	=	PUNCT
cana-3491	336	2	(	(	PUNCT
cana-3491	336	3	𝑥30	𝑥30	PROPN
cana-3491	336	4	+	+	CCONJ
cana-3491	336	5	𝑥31𝑝	𝑥31𝑝	NOUN
cana-3491	336	6	,	,	PUNCT
cana-3491	336	7	𝑦30	𝑦30	VERB
cana-3491	336	8	+	+	CCONJ
cana-3491	336	9	𝑦31𝑝	𝑦31𝑝	NOUN
cana-3491	336	10	)	)	PUNCT
cana-3491	336	11	such	such	ADJ
cana-3491	336	12	that	that	SCONJ
cana-3491	336	13	{	{	PUNCT
cana-3491	336	14	𝑥30	𝑥30	NOUN
cana-3491	336	15	+	+	CCONJ
cana-3491	336	16	𝑥31𝑝	𝑥31𝑝	PRON
cana-3491	336	17	is	be	AUX
cana-3491	336	18	the	the	DET
cana-3491	336	19	𝑝-adic	𝑝-adic	ADJ
cana-3491	336	20	expansion	expansion	NOUN
cana-3491	336	21	of	of	ADP
cana-3491	336	22	𝑥30′	𝑥30′	PROPN
cana-3491	336	23	+	+	CCONJ
cana-3491	337	1	𝑥31′𝑝	𝑥31′𝑝	DET
cana-3491	337	2	modulo	modulo	NOUN
cana-3491	337	3	𝑝2	𝑝2	NOUN
cana-3491	337	4	𝑦30	𝑦30	NOUN
cana-3491	337	5	+	+	CCONJ
cana-3491	337	6	𝑦31𝑝	𝑦31𝑝	NOUN
cana-3491	337	7	is	be	AUX
cana-3491	337	8	the	the	DET
cana-3491	337	9	𝑝-adic	𝑝-adic	ADJ
cana-3491	337	10	expansion	expansion	NOUN
cana-3491	337	11	of	of	ADP
cana-3491	337	12	𝑦30′	𝑦30′	NUM
cana-3491	337	13	+	+	CCONJ
cana-3491	337	14	𝑦31′𝑝	𝑦31′𝑝	PROPN
cana-3491	337	15	modulo	modulo	NOUN
cana-3491	337	16	𝑝2	𝑝2	NOUN
cana-3491	337	17	where	where	SCONJ
cana-3491	337	18	𝑥30′	𝑥30′	NOUN
cana-3491	337	19	,	,	PUNCT
cana-3491	337	20	𝑥31′	𝑥31′	ADJ
cana-3491	337	21	,	,	PUNCT
cana-3491	337	22	𝑦30′	𝑦30′	NUM
cana-3491	337	23	,	,	PUNCT
cana-3491	337	24	𝑦31′	𝑦31′	NOUN
cana-3491	337	25	given	give	VERB
cana-3491	337	26	as	as	SCONJ
cana-3491	337	27	follows	follow	VERB
cana-3491	337	28	for	for	ADP
cana-3491	337	29	�	�	PROPN
cana-3491	337	30	̃	̃	PROPN
cana-3491	337	31	�	�	PROPN
cana-3491	337	32	11	11	NUM
cana-3491	337	33	≠	≠	PROPN
cana-3491	337	34	�	�	PROPN
cana-3491	337	35	̃	̃	PROPN
cana-3491	337	36	�	�	NOUN
cana-3491	337	37	21	21	NUM
cana-3491	337	38	{	{	PUNCT
cana-3491	337	39	𝑥30′	𝑥30′	NOUN
cana-3491	337	40	=	=	PUNCT
cana-3491	337	41	𝑚0	𝑚0	NOUN
cana-3491	337	42	2	2	NUM
cana-3491	337	43	+	+	CCONJ
cana-3491	337	44	(	(	PUNCT
cana-3491	337	45	𝑝	𝑝	PROPN
cana-3491	337	46	−	−	PROPN
cana-3491	337	47	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	337	48	+	+	CCONJ
cana-3491	337	49	𝑥20	𝑥20	X
cana-3491	337	50	)	)	PUNCT
cana-3491	337	51	𝑥31′	𝑥31′	PROPN
cana-3491	337	52	=	=	SYM
cana-3491	337	53	2𝑚0𝑚1	2𝑚0𝑚1	NOUN
cana-3491	337	54	+	+	CCONJ
cana-3491	338	1	(	(	PUNCT
cana-3491	338	2	𝑝	𝑝	PROPN
cana-3491	338	3	−	−	PROPN
cana-3491	338	4	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	338	5	+	+	CCONJ
cana-3491	338	6	𝑥11	𝑥11	ADV
cana-3491	339	1	+	+	CCONJ
cana-3491	339	2	𝑥20	𝑥20	ADJ
cana-3491	339	3	+	+	NUM
cana-3491	339	4	𝑥21	𝑥21	NOUN
cana-3491	339	5	)	)	PUNCT
cana-3491	339	6	𝑦30′	𝑦30′	NOUN
cana-3491	339	7	=	=	PUNCT
cana-3491	339	8	𝑚0𝑥10	𝑚0𝑥10	VERB
cana-3491	339	9	+	+	CCONJ
cana-3491	339	10	(	(	PUNCT
cana-3491	339	11	𝑝	𝑝	PROPN
cana-3491	339	12	−	−	ADP
cana-3491	339	13	1)(𝑚0𝑥30	1)(𝑚0𝑥30	NUM
cana-3491	339	14	+	+	CCONJ
cana-3491	339	15	𝑦10	𝑦10	ADJ
cana-3491	339	16	)	)	PUNCT
cana-3491	339	17	𝑦31′	𝑦31′	NOUN
cana-3491	339	18	=	=	PUNCT
cana-3491	339	19	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	339	20	+	+	ADV
cana-3491	339	21	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	339	22	+	+	NUM
cana-3491	339	23	(	(	PUNCT
cana-3491	339	24	𝑝	𝑝	PROPN
cana-3491	339	25	−	−	PROPN
cana-3491	339	26	1)(𝑚1𝑥30	1)(𝑚1𝑥30	NUM
cana-3491	339	27	+	+	NOUN
cana-3491	339	28	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	339	29	+	+	ADJ
cana-3491	339	30	𝑚0𝑥31	𝑚0𝑥31	ADJ
cana-3491	339	31	+	+	CCONJ
cana-3491	339	32	𝑦10	𝑦10	ADJ
cana-3491	339	33	+	+	CCONJ
cana-3491	339	34	𝑦11	𝑦11	ADJ
cana-3491	339	35	)	)	PUNCT
cana-3491	339	36	communications	communication	NOUN
cana-3491	339	37	on	on	ADP
cana-3491	339	38	applied	apply	VERB
cana-3491	339	39	nonlinear	nonlinear	ADJ
cana-3491	339	40	analysis	analysis	NOUN
cana-3491	339	41	issn	issn	NOUN
cana-3491	339	42	:	:	PUNCT
cana-3491	339	43	1074	1074	NUM
cana-3491	339	44	-	-	PUNCT
cana-3491	339	45	133x	133x	NUM
cana-3491	339	46	vol	vol	NOUN
cana-3491	339	47	32	32	NUM
cana-3491	339	48	no	no	NOUN
cana-3491	339	49	.	.	PUNCT
cana-3491	340	1	7s	7	NOUN
cana-3491	340	2	(	(	PUNCT
cana-3491	340	3	2025	2025	NUM
cana-3491	340	4	)	)	PUNCT
cana-3491	340	5	871	871	NUM
cana-3491	340	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	341	1	where	where	SCONJ
cana-3491	341	2	�	�	PROPN
cana-3491	341	3	̃	̃	PROPN
cana-3491	341	4	�	�	NOUN
cana-3491	341	5	1	1	NUM
cana-3491	341	6	=	=	SYM
cana-3491	341	7	�	�	PROPN
cana-3491	341	8	̃	̃	PROPN
cana-3491	341	9	�	�	PROPN
cana-3491	341	10	21−	21−	NUM
cana-3491	341	11	�	�	PROPN
cana-3491	341	12	̃	̃	PROPN
cana-3491	341	13	�	�	PROPN
cana-3491	341	14	11	11	NUM
cana-3491	341	15	�	�	PROPN
cana-3491	341	16	̃	̃	PROPN
cana-3491	341	17	�	�	PROPN
cana-3491	341	18	21−	21−	NUM
cana-3491	341	19	�	�	PROPN
cana-3491	341	20	̃	̃	PROPN
cana-3491	341	21	�	�	PROPN
cana-3491	341	22	11	11	NUM
cana-3491	341	23	=	=	SYM
cana-3491	341	24	𝑚0	𝑚0	NOUN
cana-3491	341	25	+	+	ADP
cana-3491	341	26	𝑚1𝑝.	𝑚1𝑝.	PROPN
cana-3491	341	27	for	for	ADP
cana-3491	341	28	�	�	PROPN
cana-3491	341	29	̃	̃	PROPN
cana-3491	341	30	�	�	PROPN
cana-3491	341	31	11	11	NUM
cana-3491	341	32	=	=	SYM
cana-3491	341	33	�	�	PROPN
cana-3491	341	34	̃	̃	PROPN
cana-3491	341	35	�	�	NOUN
cana-3491	341	36	21	21	NUM
cana-3491	341	37	{	{	PUNCT
cana-3491	341	38	𝑥30′	𝑥30′	NOUN
cana-3491	341	39	=	=	PUNCT
cana-3491	341	40	𝑚0	𝑚0	NOUN
cana-3491	341	41	2	2	NUM
cana-3491	341	42	+	+	CCONJ
cana-3491	341	43	(	(	PUNCT
cana-3491	341	44	𝑝	𝑝	PROPN
cana-3491	341	45	−	−	PROPN
cana-3491	341	46	1)2𝑥10	1)2𝑥10	NUM
cana-3491	341	47	𝑥31′	𝑥31′	NOUN
cana-3491	341	48	=	=	SYM
cana-3491	341	49	2(𝑚0𝑚1	2(𝑚0𝑚1	NUM
cana-3491	341	50	+	+	CCONJ
cana-3491	341	51	(	(	PUNCT
cana-3491	341	52	𝑝	𝑝	PROPN
cana-3491	341	53	−	−	PROPN
cana-3491	341	54	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	341	55	+	+	CCONJ
cana-3491	341	56	𝑥11	𝑥11	ADV
cana-3491	341	57	)	)	PUNCT
cana-3491	341	58	)	)	PUNCT
cana-3491	341	59	𝑦30′	𝑦30′	NOUN
cana-3491	341	60	=	=	PUNCT
cana-3491	341	61	𝑚0𝑥10	𝑚0𝑥10	VERB
cana-3491	341	62	+	+	CCONJ
cana-3491	341	63	(	(	PUNCT
cana-3491	341	64	𝑝	𝑝	PROPN
cana-3491	341	65	−	−	ADP
cana-3491	341	66	1)(𝑚0𝑥30	1)(𝑚0𝑥30	NUM
cana-3491	341	67	+	+	CCONJ
cana-3491	341	68	𝑦10	𝑦10	ADJ
cana-3491	341	69	)	)	PUNCT
cana-3491	341	70	𝑦31′	𝑦31′	NOUN
cana-3491	341	71	=	=	PUNCT
cana-3491	341	72	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	341	73	+	+	ADV
cana-3491	341	74	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	341	75	+	+	NUM
cana-3491	341	76	(	(	PUNCT
cana-3491	341	77	𝑝	𝑝	PROPN
cana-3491	341	78	−	−	PROPN
cana-3491	341	79	1)(𝑚1𝑥30	1)(𝑚1𝑥30	NUM
cana-3491	341	80	+	+	NOUN
cana-3491	341	81	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	341	82	+	+	ADJ
cana-3491	341	83	𝑚0𝑥31	𝑚0𝑥31	ADJ
cana-3491	341	84	+	+	CCONJ
cana-3491	341	85	𝑦10	𝑦10	ADJ
cana-3491	341	86	+	+	CCONJ
cana-3491	341	87	𝑦11	𝑦11	NOUN
cana-3491	341	88	)	)	PUNCT
cana-3491	341	89	where	where	SCONJ
cana-3491	341	90	�	�	PROPN
cana-3491	341	91	̃	̃	PROPN
cana-3491	341	92	�	�	NOUN
cana-3491	341	93	1	1	NUM
cana-3491	341	94	=	=	SYM
cana-3491	341	95	3	3	NUM
cana-3491	341	96	�	�	PROPN
cana-3491	341	97	̃	̃	NOUN
cana-3491	341	98	�	�	NOUN
cana-3491	341	99	11	11	NUM
cana-3491	341	100	2	2	NUM
cana-3491	341	101	+	+	NOUN
cana-3491	341	102	𝐴	𝐴	PROPN
cana-3491	341	103	2	2	NUM
cana-3491	341	104	�	�	PROPN
cana-3491	341	105	̃	̃	PROPN
cana-3491	341	106	�	�	NOUN
cana-3491	341	107	11	11	NUM
cana-3491	341	108	=	=	SYM
cana-3491	341	109	𝑚0	𝑚0	NOUN
cana-3491	341	110	+	+	ADP
cana-3491	341	111	𝑚1𝑝.	𝑚1𝑝.	ADJ
cana-3491	341	112	proof	proof	NOUN
cana-3491	341	113	.	.	PUNCT
cana-3491	342	1	consider	consider	VERB
cana-3491	342	2	�	�	PROPN
cana-3491	342	3	̃	̃	NOUN
cana-3491	342	4	�	�	NOUN
cana-3491	342	5	11	11	NUM
cana-3491	342	6	=	=	SYM
cana-3491	342	7	(	(	PUNCT
cana-3491	342	8	�	�	PROPN
cana-3491	342	9	̃	̃	NOUN
cana-3491	342	10	�	�	NOUN
cana-3491	342	11	11	11	NUM
cana-3491	342	12	,	,	PUNCT
cana-3491	342	13	�	�	PROPN
cana-3491	342	14	̃	̃	PROPN
cana-3491	342	15	�	�	NOUN
cana-3491	342	16	11	11	NUM
cana-3491	342	17	)	)	PUNCT
cana-3491	342	18	=	=	PUNCT
cana-3491	342	19	(	(	PUNCT
cana-3491	342	20	𝑥10	𝑥10	X
cana-3491	342	21	+	+	SYM
cana-3491	342	22	𝑥11𝑝	𝑥11𝑝	ADJ
cana-3491	342	23	,	,	PUNCT
cana-3491	342	24	𝑦10	𝑦10	ADJ
cana-3491	342	25	+	+	CCONJ
cana-3491	342	26	𝑦11𝑝	𝑦11𝑝	NOUN
cana-3491	342	27	)	)	PUNCT
cana-3491	342	28	and	and	CCONJ
cana-3491	342	29	�	�	PROPN
cana-3491	342	30	̃	̃	PROPN
cana-3491	342	31	�	�	NOUN
cana-3491	342	32	21	21	NUM
cana-3491	342	33	=	=	SYM
cana-3491	342	34	(	(	PUNCT
cana-3491	342	35	�	�	PROPN
cana-3491	342	36	̃	̃	NOUN
cana-3491	342	37	�	�	NOUN
cana-3491	342	38	21	21	NUM
cana-3491	342	39	,	,	PUNCT
cana-3491	342	40	�	�	PROPN
cana-3491	342	41	̃	̃	PROPN
cana-3491	342	42	�	�	NOUN
cana-3491	342	43	21	21	NUM
cana-3491	342	44	)	)	PUNCT
cana-3491	342	45	=	=	PUNCT
cana-3491	343	1	(	(	PUNCT
cana-3491	343	2	𝑥20	𝑥20	NOUN
cana-3491	343	3	+	+	CCONJ
cana-3491	343	4	𝑎21𝑝	𝑎21𝑝	NOUN
cana-3491	343	5	,	,	PUNCT
cana-3491	343	6	𝑦20	𝑦20	NOUN
cana-3491	343	7	+	+	CCONJ
cana-3491	343	8	𝑦21𝑝	𝑦21𝑝	NOUN
cana-3491	343	9	)	)	PUNCT
cana-3491	343	10	for	for	ADP
cana-3491	343	11	�	�	PROPN
cana-3491	343	12	̃	̃	PROPN
cana-3491	343	13	�	�	PROPN
cana-3491	343	14	11	11	NUM
cana-3491	343	15	≠	≠	PROPN
cana-3491	343	16	�	�	PROPN
cana-3491	343	17	̃	̃	PROPN
cana-3491	343	18	�	�	NOUN
cana-3491	343	19	21	21	NUM
cana-3491	343	20	:	:	PUNCT
cana-3491	343	21	by	by	ADP
cana-3491	343	22	the	the	DET
cana-3491	343	23	implementation	implementation	NOUN
cana-3491	343	24	of	of	ADP
cana-3491	343	25	arithmetic	arithmetic	NOUN
cana-3491	343	26	of	of	ADP
cana-3491	343	27	points	point	NOUN
cana-3491	343	28	�	�	PROPN
cana-3491	343	29	̃	̃	NOUN
cana-3491	343	30	�	�	NOUN
cana-3491	343	31	1	1	NUM
cana-3491	343	32	,	,	PUNCT
cana-3491	343	33	�	�	PROPN
cana-3491	343	34	̃	̃	NOUN
cana-3491	343	35	�	�	NOUN
cana-3491	343	36	2	2	NUM
cana-3491	343	37	in	in	ADP
cana-3491	343	38	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	343	39	)	)	PUNCT
cana-3491	343	40	as	as	ADP
cana-3491	343	41	in	in	ADP
cana-3491	343	42	(	(	PUNCT
cana-3491	343	43	1	1	NUM
cana-3491	343	44	)	)	PUNCT
cana-3491	343	45	to	to	ADP
cana-3491	343	46	the	the	DET
cana-3491	343	47	points	point	NOUN
cana-3491	343	48	�	�	PROPN
cana-3491	343	49	̃	̃	PROPN
cana-3491	343	50	�	�	NOUN
cana-3491	343	51	11	11	NUM
cana-3491	343	52	,	,	PUNCT
cana-3491	343	53	�	�	PROPN
cana-3491	343	54	̃	̃	NOUN
cana-3491	343	55	�	�	NOUN
cana-3491	343	56	21	21	NUM
cana-3491	343	57	in	in	ADP
cana-3491	343	58	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	343	59	𝑝2	𝑝2	NOUN
cana-3491	343	60	)	)	PUNCT
cana-3491	343	61	,	,	PUNCT
cana-3491	343	62	we	we	PRON
cana-3491	343	63	have	have	VERB
cana-3491	343	64	�	�	PROPN
cana-3491	343	65	̃	̃	PROPN
cana-3491	343	66	�	�	PROPN
cana-3491	343	67	31	31	NUM
cana-3491	343	68	=	=	SYM
cana-3491	343	69	�	�	PROPN
cana-3491	343	70	̃	̃	PROPN
cana-3491	343	71	�	�	PROPN
cana-3491	343	72	11	11	NUM
cana-3491	343	73	+	+	SYM
cana-3491	343	74	�	�	PROPN
cana-3491	343	75	̃	̃	PROPN
cana-3491	343	76	�	�	NOUN
cana-3491	343	77	21	21	NUM
cana-3491	343	78	=	=	SYM
cana-3491	343	79	(	(	PUNCT
cana-3491	343	80	�	�	PROPN
cana-3491	343	81	̃	̃	NOUN
cana-3491	343	82	�	�	NOUN
cana-3491	343	83	1	1	NUM
cana-3491	343	84	2	2	NUM
cana-3491	343	85	−	−	PROPN
cana-3491	343	86	�	�	PROPN
cana-3491	343	87	̃	̃	PROPN
cana-3491	343	88	�	�	NOUN
cana-3491	343	89	11	11	NUM
cana-3491	343	90	2	2	NUM
cana-3491	343	91	−	−	PROPN
cana-3491	343	92	�	�	PROPN
cana-3491	343	93	̃	̃	PROPN
cana-3491	343	94	�	�	NOUN
cana-3491	343	95	21	21	NUM
cana-3491	343	96	2	2	NUM
cana-3491	343	97	,	,	PUNCT
cana-3491	343	98	�	�	PROPN
cana-3491	343	99	̃	̃	PROPN
cana-3491	343	100	�	�	PROPN
cana-3491	343	101	1(	1(	NUM
cana-3491	343	102	�	�	NOUN
cana-3491	343	103	̃	̃	NOUN
cana-3491	343	104	�	�	PROPN
cana-3491	343	105	11	11	NUM
cana-3491	343	106	−	−	PROPN
cana-3491	343	107	�	�	PROPN
cana-3491	343	108	̃	̃	PROPN
cana-3491	343	109	�	�	NOUN
cana-3491	343	110	31	31	NUM
cana-3491	343	111	)	)	PUNCT
cana-3491	343	112	−	−	PROPN
cana-3491	343	113	�	�	PROPN
cana-3491	343	114	̃	̃	PROPN
cana-3491	343	115	�	�	NOUN
cana-3491	343	116	11	11	NUM
cana-3491	343	117	)	)	PUNCT
cana-3491	343	118	the	the	DET
cana-3491	343	119	𝑥-coordinate	𝑥-coordinate	NOUN
cana-3491	343	120	of	of	ADP
cana-3491	343	121	�	�	PROPN
cana-3491	343	122	̃	̃	PROPN
cana-3491	343	123	�	�	PROPN
cana-3491	343	124	31	31	NUM
cana-3491	343	125	is	be	AUX
cana-3491	343	126	given	give	VERB
cana-3491	343	127	as	as	ADP
cana-3491	343	128	𝑥(	𝑥(	PROPN
cana-3491	343	129	�	�	PROPN
cana-3491	343	130	̃	̃	PROPN
cana-3491	343	131	�	�	NOUN
cana-3491	343	132	31	31	NUM
cana-3491	343	133	)	)	PUNCT
cana-3491	343	134	=	=	SYM
cana-3491	343	135	�	�	PROPN
cana-3491	343	136	̃	̃	PROPN
cana-3491	343	137	�	�	NOUN
cana-3491	343	138	1	1	NUM
cana-3491	343	139	2	2	NUM
cana-3491	343	140	−	−	PROPN
cana-3491	343	141	�	�	PROPN
cana-3491	343	142	̃	̃	PROPN
cana-3491	343	143	�	�	NOUN
cana-3491	343	144	11	11	NUM
cana-3491	343	145	2	2	NUM
cana-3491	343	146	−	−	PROPN
cana-3491	343	147	�	�	PROPN
cana-3491	343	148	̃	̃	PROPN
cana-3491	343	149	�	�	NOUN
cana-3491	343	150	21	21	NUM
cana-3491	343	151	2	2	NUM
cana-3491	343	152	=	=	SYM
cana-3491	343	153	(	(	PUNCT
cana-3491	343	154	𝑚0	𝑚0	NOUN
cana-3491	343	155	2	2	NUM
cana-3491	343	156	+	+	NUM
cana-3491	343	157	2𝑚0𝑚1𝑝	2𝑚0𝑚1𝑝	NUM
cana-3491	343	158	)	)	PUNCT
cana-3491	344	1	+	+	CCONJ
cana-3491	344	2	(	(	PUNCT
cana-3491	344	3	(	(	PUNCT
cana-3491	344	4	𝑝	𝑝	NOUN
cana-3491	344	5	−	−	NOUN
cana-3491	344	6	1	1	NUM
cana-3491	344	7	)	)	PUNCT
cana-3491	344	8	+	+	CCONJ
cana-3491	344	9	(	(	PUNCT
cana-3491	344	10	𝑝	𝑝	PROPN
cana-3491	344	11	−	−	NOUN
cana-3491	344	12	1)𝑝)(𝑥10	1)𝑝)(𝑥10	NUM
cana-3491	344	13	+	+	CCONJ
cana-3491	344	14	𝑥11𝑝	𝑥11𝑝	PUNCT
cana-3491	344	15	)	)	PUNCT
cana-3491	344	16	+	+	CCONJ
cana-3491	344	17	(	(	PUNCT
cana-3491	344	18	(	(	PUNCT
cana-3491	344	19	𝑝	𝑝	NOUN
cana-3491	344	20	−	−	NOUN
cana-3491	344	21	1	1	NUM
cana-3491	344	22	)	)	PUNCT
cana-3491	344	23	+	+	CCONJ
cana-3491	344	24	(	(	PUNCT
cana-3491	344	25	𝑝	𝑝	PROPN
cana-3491	344	26	−	−	PROPN
cana-3491	344	27	1)𝑝)(𝑥20	1)𝑝)(𝑥20	NUM
cana-3491	344	28	+	+	NUM
cana-3491	344	29	𝑥21𝑝	𝑥21𝑝	NOUN
cana-3491	344	30	)	)	PUNCT
cana-3491	344	31	=	=	SYM
cana-3491	344	32	𝑚0	𝑚0	NOUN
cana-3491	344	33	2	2	NUM
cana-3491	344	34	+	+	NUM
cana-3491	344	35	2𝑚0𝑚1𝑝	2𝑚0𝑚1𝑝	NUM
cana-3491	344	36	+	+	CCONJ
cana-3491	344	37	(	(	PUNCT
cana-3491	344	38	𝑝	𝑝	NOUN
cana-3491	344	39	−	−	NOUN
cana-3491	344	40	1)𝑥10	1)𝑥10	NOUN
cana-3491	344	41	+	+	CCONJ
cana-3491	344	42	(	(	PUNCT
cana-3491	344	43	(	(	PUNCT
cana-3491	344	44	𝑝	𝑝	PROPN
cana-3491	344	45	−	−	PROPN
cana-3491	344	46	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	344	47	+	+	CCONJ
cana-3491	344	48	𝑥11))𝑝	𝑥11))𝑝	ADJ
cana-3491	344	49	+	+	CCONJ
cana-3491	344	50	(	(	PUNCT
cana-3491	344	51	𝑝	𝑝	PROPN
cana-3491	344	52	−	−	PROPN
cana-3491	344	53	1)𝑥20	1)𝑥20	NUM
cana-3491	344	54	+	+	CCONJ
cana-3491	344	55	(	(	PUNCT
cana-3491	344	56	(	(	PUNCT
cana-3491	344	57	𝑝	𝑝	NOUN
cana-3491	344	58	−	−	PROPN
cana-3491	344	59	1)(𝑥20	1)(𝑥20	PROPN
cana-3491	344	60	+	+	NUM
cana-3491	344	61	𝑥21))𝑝	𝑥21))𝑝	NOUN
cana-3491	344	62	=	=	SYM
cana-3491	344	63	𝑚0	𝑚0	NOUN
cana-3491	344	64	2	2	NUM
cana-3491	344	65	+	+	CCONJ
cana-3491	344	66	(	(	PUNCT
cana-3491	344	67	𝑝	𝑝	NOUN
cana-3491	344	68	−	−	NOUN
cana-3491	344	69	1)𝑥10	1)𝑥10	NOUN
cana-3491	344	70	+	+	CCONJ
cana-3491	344	71	(	(	PUNCT
cana-3491	344	72	𝑝	𝑝	NOUN
cana-3491	344	73	−	−	PROPN
cana-3491	344	74	1)𝑥20	1)𝑥20	NUM
cana-3491	344	75	+	+	CCONJ
cana-3491	344	76	2𝑚0𝑚1𝑝	2𝑚0𝑚1𝑝	NUM
cana-3491	344	77	+	+	CCONJ
cana-3491	344	78	(	(	PUNCT
cana-3491	344	79	(	(	PUNCT
cana-3491	344	80	𝑝	𝑝	PROPN
cana-3491	344	81	−	−	PROPN
cana-3491	344	82	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	344	83	+	+	CCONJ
cana-3491	344	84	𝑥11))𝑝	𝑥11))𝑝	ADJ
cana-3491	344	85	+	+	CCONJ
cana-3491	344	86	(	(	PUNCT
cana-3491	344	87	(	(	PUNCT
cana-3491	344	88	𝑝	𝑝	NOUN
cana-3491	344	89	−	−	PROPN
cana-3491	344	90	1)(𝑥20	1)(𝑥20	PROPN
cana-3491	344	91	+	+	NUM
cana-3491	344	92	𝑥21))𝑝	𝑥21))𝑝	NOUN
cana-3491	344	93	=	=	SYM
cana-3491	344	94	𝑚0	𝑚0	NOUN
cana-3491	344	95	2	2	NUM
cana-3491	344	96	+	+	CCONJ
cana-3491	344	97	(	(	PUNCT
cana-3491	344	98	𝑝	𝑝	PROPN
cana-3491	344	99	−	−	PROPN
cana-3491	344	100	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	344	101	+	+	CCONJ
cana-3491	344	102	𝑥20	𝑥20	NUM
cana-3491	344	103	)	)	PUNCT
cana-3491	345	1	+	+	CCONJ
cana-3491	345	2	(	(	PUNCT
cana-3491	345	3	2𝑚0𝑚1	2𝑚0𝑚1	NUM
cana-3491	345	4	+	+	CCONJ
cana-3491	345	5	(	(	PUNCT
cana-3491	345	6	𝑝	𝑝	PROPN
cana-3491	345	7	−	−	PROPN
cana-3491	345	8	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	345	9	+	+	CCONJ
cana-3491	345	10	𝑥11	𝑥11	ADV
cana-3491	345	11	)	)	PUNCT
cana-3491	346	1	+	+	CCONJ
cana-3491	346	2	(	(	PUNCT
cana-3491	346	3	𝑝	𝑝	PROPN
cana-3491	346	4	−	−	PROPN
cana-3491	346	5	1)(𝑥20	1)(𝑥20	PROPN
cana-3491	346	6	+	+	NUM
cana-3491	346	7	𝑥21))𝑝	𝑥21))𝑝	NOUN
cana-3491	346	8	therefore	therefore	ADV
cana-3491	346	9	,	,	PUNCT
cana-3491	346	10	𝑥30′	𝑥30′	X
cana-3491	346	11	=	=	PUNCT
cana-3491	346	12	𝑚0	𝑚0	NOUN
cana-3491	346	13	2	2	NUM
cana-3491	346	14	+	+	CCONJ
cana-3491	346	15	(	(	PUNCT
cana-3491	346	16	𝑝	𝑝	PROPN
cana-3491	346	17	−	−	PROPN
cana-3491	346	18	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	346	19	+	+	CCONJ
cana-3491	346	20	𝑥20	𝑥20	X
cana-3491	346	21	)	)	PUNCT
cana-3491	346	22	𝑥31′	𝑥31′	PROPN
cana-3491	346	23	=	=	SYM
cana-3491	346	24	2𝑚0𝑚1	2𝑚0𝑚1	NOUN
cana-3491	346	25	+	+	CCONJ
cana-3491	346	26	(	(	PUNCT
cana-3491	346	27	𝑝	𝑝	PROPN
cana-3491	346	28	−	−	PROPN
cana-3491	346	29	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	346	30	+	+	CCONJ
cana-3491	346	31	𝑥11	𝑥11	ADV
cana-3491	347	1	+	+	CCONJ
cana-3491	347	2	𝑥20	𝑥20	ADJ
cana-3491	347	3	+	+	NUM
cana-3491	347	4	𝑥21	𝑥21	NOUN
cana-3491	347	5	)	)	PUNCT
cana-3491	347	6	now	now	ADV
cana-3491	347	7	,	,	PUNCT
cana-3491	347	8	by	by	ADP
cana-3491	347	9	considering	consider	VERB
cana-3491	347	10	𝑝-adic	𝑝-adic	ADJ
cana-3491	347	11	expansion	expansion	NOUN
cana-3491	347	12	of	of	ADP
cana-3491	347	13	𝑥30′	𝑥30′	PROPN
cana-3491	347	14	+	+	CCONJ
cana-3491	347	15	𝑥31′𝑝	𝑥31′𝑝	DET
cana-3491	347	16	modulo	modulo	NOUN
cana-3491	347	17	𝑝2	𝑝2	NOUN
cana-3491	347	18	,	,	PUNCT
cana-3491	347	19	we	we	PRON
cana-3491	347	20	have	have	VERB
cana-3491	347	21	𝑥(	𝑥(	PROPN
cana-3491	347	22	�	�	PROPN
cana-3491	347	23	̃	̃	PROPN
cana-3491	347	24	�	�	NOUN
cana-3491	347	25	31	31	NUM
cana-3491	347	26	)	)	PUNCT
cana-3491	347	27	=	=	SYM
cana-3491	347	28	𝑥30	𝑥30	NOUN
cana-3491	347	29	+	+	CCONJ
cana-3491	347	30	𝑥31𝑝	𝑥31𝑝	VERB
cana-3491	347	31	the	the	DET
cana-3491	347	32	𝑦-coordinate	𝑦-coordinate	NOUN
cana-3491	347	33	of	of	ADP
cana-3491	347	34	�	�	PROPN
cana-3491	347	35	̃	̃	PROPN
cana-3491	347	36	�	�	PROPN
cana-3491	347	37	31	31	NUM
cana-3491	347	38	is	be	AUX
cana-3491	347	39	given	give	VERB
cana-3491	347	40	as	as	ADP
cana-3491	347	41	𝑦(	𝑦(	NUM
cana-3491	347	42	�	�	PROPN
cana-3491	347	43	̃	̃	PROPN
cana-3491	347	44	�	�	NOUN
cana-3491	347	45	31	31	NUM
cana-3491	347	46	)	)	PUNCT
cana-3491	347	47	=	=	SYM
cana-3491	347	48	�	�	PROPN
cana-3491	347	49	̃	̃	PROPN
cana-3491	347	50	�	�	PROPN
cana-3491	347	51	1(	1(	NUM
cana-3491	347	52	�	�	NOUN
cana-3491	347	53	̃	̃	NOUN
cana-3491	347	54	�	�	PROPN
cana-3491	347	55	11	11	NUM
cana-3491	347	56	−	−	PROPN
cana-3491	347	57	�	�	PROPN
cana-3491	347	58	̃	̃	PROPN
cana-3491	347	59	�	�	NOUN
cana-3491	347	60	31	31	NUM
cana-3491	347	61	)	)	PUNCT
cana-3491	347	62	−	−	PROPN
cana-3491	347	63	�	�	PROPN
cana-3491	347	64	̃	̃	PROPN
cana-3491	347	65	�	�	NOUN
cana-3491	347	66	11	11	NUM
cana-3491	347	67	=	=	SYM
cana-3491	347	68	(	(	PUNCT
cana-3491	347	69	𝑚0	𝑚0	NOUN
cana-3491	347	70	+	+	PROPN
cana-3491	347	71	𝑚1𝑝	𝑚1𝑝	ADV
cana-3491	347	72	)	)	PUNCT
cana-3491	347	73	(	(	PUNCT
cana-3491	347	74	𝑥10	𝑥10	NOUN
cana-3491	347	75	+	+	X
cana-3491	348	1	𝑥11𝑝	𝑥11𝑝	NUM
cana-3491	349	1	+	+	PUNCT
cana-3491	349	2	(	(	PUNCT
cana-3491	349	3	(	(	PUNCT
cana-3491	349	4	𝑝	𝑝	NOUN
cana-3491	349	5	−	−	NOUN
cana-3491	349	6	1	1	NUM
cana-3491	349	7	)	)	PUNCT
cana-3491	349	8	+	+	CCONJ
cana-3491	349	9	(	(	PUNCT
cana-3491	349	10	𝑝	𝑝	ADP
cana-3491	349	11	−	−	PROPN
cana-3491	349	12	1)𝑝)(𝑥30	1)𝑝)(𝑥30	PROPN
cana-3491	349	13	+	+	NUM
cana-3491	349	14	𝑥31𝑝	𝑥31𝑝	NOUN
cana-3491	349	15	)	)	PUNCT
cana-3491	349	16	)	)	PUNCT
cana-3491	350	1	+	+	CCONJ
cana-3491	350	2	(	(	PUNCT
cana-3491	350	3	(	(	PUNCT
cana-3491	350	4	(	(	PUNCT
cana-3491	350	5	𝑝	𝑝	NOUN
cana-3491	350	6	−	−	NOUN
cana-3491	350	7	1	1	NUM
cana-3491	350	8	)	)	PUNCT
cana-3491	350	9	+	+	CCONJ
cana-3491	350	10	(	(	PUNCT
cana-3491	350	11	𝑝	𝑝	ADP
cana-3491	350	12	−	−	NUM
cana-3491	350	13	1)𝑝)(𝑦10	1)𝑝)(𝑦10	NUM
cana-3491	350	14	+	+	CCONJ
cana-3491	350	15	𝑦11𝑝	𝑦11𝑝	NOUN
cana-3491	350	16	)	)	PUNCT
cana-3491	350	17	)	)	PUNCT
cana-3491	350	18	communications	communication	NOUN
cana-3491	350	19	on	on	ADP
cana-3491	350	20	applied	apply	VERB
cana-3491	350	21	nonlinear	nonlinear	ADJ
cana-3491	350	22	analysis	analysis	NOUN
cana-3491	350	23	issn	issn	NOUN
cana-3491	350	24	:	:	PUNCT
cana-3491	350	25	1074	1074	NUM
cana-3491	350	26	-	-	PUNCT
cana-3491	350	27	133x	133x	NUM
cana-3491	350	28	vol	vol	NOUN
cana-3491	350	29	32	32	NUM
cana-3491	350	30	no	no	NOUN
cana-3491	350	31	.	.	PUNCT
cana-3491	351	1	7s	7	NOUN
cana-3491	351	2	(	(	PUNCT
cana-3491	351	3	2025	2025	NUM
cana-3491	351	4	)	)	PUNCT
cana-3491	351	5	872	872	NUM
cana-3491	351	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	351	7	=	=	PUNCT
cana-3491	351	8	(	(	PUNCT
cana-3491	351	9	𝑚0	𝑚0	NOUN
cana-3491	351	10	+	+	VERB
cana-3491	351	11	𝑚1𝑝)(𝑥10	𝑚1𝑝)(𝑥10	NOUN
cana-3491	351	12	+	+	CCONJ
cana-3491	351	13	(	(	PUNCT
cana-3491	351	14	𝑝	𝑝	NOUN
cana-3491	351	15	−	−	PROPN
cana-3491	351	16	1)𝑥30	1)𝑥30	PROPN
cana-3491	351	17	+	+	CCONJ
cana-3491	351	18	(	(	PUNCT
cana-3491	351	19	𝑥11	𝑥11	ADV
cana-3491	351	20	+	+	CCONJ
cana-3491	351	21	(	(	PUNCT
cana-3491	351	22	𝑝	𝑝	PROPN
cana-3491	351	23	−	−	PROPN
cana-3491	351	24	1)(𝑥30	1)(𝑥30	PROPN
cana-3491	351	25	+	+	NUM
cana-3491	351	26	𝑥31))𝑝	𝑥31))𝑝	NUM
cana-3491	351	27	)	)	PUNCT
cana-3491	351	28	+	+	CCONJ
cana-3491	351	29	(	(	PUNCT
cana-3491	351	30	𝑝	𝑝	NOUN
cana-3491	351	31	−	−	PROPN
cana-3491	351	32	1)𝑦10	1)𝑦10	PROPN
cana-3491	351	33	+	+	CCONJ
cana-3491	351	34	(	(	PUNCT
cana-3491	351	35	𝑝	𝑝	PROPN
cana-3491	351	36	−	−	PROPN
cana-3491	351	37	1)(𝑦10	1)(𝑦10	PROPN
cana-3491	351	38	+	+	NUM
cana-3491	351	39	𝑦11))𝑝	𝑦11))𝑝	NOUN
cana-3491	351	40	=	=	VERB
cana-3491	351	41	𝑚0𝑥10	𝑚0𝑥10	X
cana-3491	351	42	+	+	CCONJ
cana-3491	351	43	(	(	PUNCT
cana-3491	351	44	𝑝	𝑝	NOUN
cana-3491	351	45	−	−	ADP
cana-3491	351	46	1)𝑚0𝑥30	1)𝑚0𝑥30	NUM
cana-3491	351	47	+	+	CCONJ
cana-3491	351	48	(	(	PUNCT
cana-3491	351	49	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	351	50	+	+	NOUN
cana-3491	351	51	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	351	52	+	+	NUM
cana-3491	351	53	(	(	PUNCT
cana-3491	351	54	𝑝	𝑝	PROPN
cana-3491	351	55	−	−	PROPN
cana-3491	351	56	1)(𝑚1𝑥30	1)(𝑚1𝑥30	NUM
cana-3491	351	57	+	+	NOUN
cana-3491	351	58	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	351	59	+	+	ADJ
cana-3491	351	60	𝑚0𝑥31)𝑝	𝑚0𝑥31)𝑝	PROPN
cana-3491	351	61	)	)	PUNCT
cana-3491	352	1	+	+	CCONJ
cana-3491	352	2	(	(	PUNCT
cana-3491	352	3	𝑝	𝑝	NOUN
cana-3491	352	4	−	−	PROPN
cana-3491	352	5	1)𝑦10	1)𝑦10	PROPN
cana-3491	352	6	+	+	CCONJ
cana-3491	352	7	(	(	PUNCT
cana-3491	352	8	𝑝	𝑝	PROPN
cana-3491	352	9	−	−	PROPN
cana-3491	352	10	1)(𝑦10	1)(𝑦10	PROPN
cana-3491	352	11	+	+	NUM
cana-3491	352	12	𝑦11	𝑦11	NOUN
cana-3491	352	13	)	)	PUNCT
cana-3491	352	14	=	=	PUNCT
cana-3491	352	15	𝑚0𝑥10	𝑚0𝑥10	VERB
cana-3491	352	16	+	+	PUNCT
cana-3491	352	17	(	(	PUNCT
cana-3491	352	18	𝑝	𝑝	PROPN
cana-3491	352	19	−	−	ADP
cana-3491	352	20	1)(𝑚0𝑥30	1)(𝑚0𝑥30	NUM
cana-3491	352	21	+	+	CCONJ
cana-3491	352	22	𝑦10	𝑦10	ADJ
cana-3491	352	23	)	)	PUNCT
cana-3491	352	24	+	+	CCONJ
cana-3491	352	25	(	(	PUNCT
cana-3491	352	26	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	352	27	+	+	NOUN
cana-3491	352	28	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	352	29	+	+	NUM
cana-3491	352	30	(	(	PUNCT
cana-3491	352	31	𝑝	𝑝	PROPN
cana-3491	352	32	−	−	PROPN
cana-3491	352	33	1)(𝑚1𝑥30	1)(𝑚1𝑥30	NUM
cana-3491	352	34	+	+	NOUN
cana-3491	352	35	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	352	36	+	+	ADJ
cana-3491	352	37	𝑚0𝑥31	𝑚0𝑥31	ADJ
cana-3491	352	38	+	+	CCONJ
cana-3491	352	39	𝑦10	𝑦10	ADJ
cana-3491	352	40	+	+	CCONJ
cana-3491	352	41	𝑦11))𝑝	𝑦11))𝑝	NOUN
cana-3491	352	42	therefore	therefore	ADV
cana-3491	352	43	,	,	PUNCT
cana-3491	352	44	𝑦30′	𝑦30′	NOUN
cana-3491	352	45	=	=	PUNCT
cana-3491	352	46	𝑚0𝑥10	𝑚0𝑥10	VERB
cana-3491	352	47	+	+	CCONJ
cana-3491	352	48	(	(	PUNCT
cana-3491	352	49	𝑝	𝑝	PROPN
cana-3491	352	50	−	−	ADP
cana-3491	352	51	1)(𝑚0𝑥30	1)(𝑚0𝑥30	NUM
cana-3491	352	52	+	+	CCONJ
cana-3491	352	53	𝑦10	𝑦10	ADJ
cana-3491	352	54	)	)	PUNCT
cana-3491	352	55	𝑦31′	𝑦31′	NOUN
cana-3491	352	56	=	=	PUNCT
cana-3491	352	57	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	352	58	+	+	ADV
cana-3491	352	59	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	352	60	+	+	NUM
cana-3491	352	61	(	(	PUNCT
cana-3491	352	62	𝑝	𝑝	PROPN
cana-3491	352	63	−	−	PROPN
cana-3491	352	64	1)(𝑚1𝑥30	1)(𝑚1𝑥30	NUM
cana-3491	353	1	+	+	NOUN
cana-3491	353	2	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	353	3	+	+	ADJ
cana-3491	353	4	𝑚0𝑥31	𝑚0𝑥31	ADJ
cana-3491	353	5	+	+	CCONJ
cana-3491	353	6	𝑦10	𝑦10	ADJ
cana-3491	353	7	+	+	CCONJ
cana-3491	353	8	𝑦11	𝑦11	NOUN
cana-3491	353	9	)	)	PUNCT
cana-3491	353	10	now	now	ADV
cana-3491	353	11	,	,	PUNCT
cana-3491	353	12	by	by	ADP
cana-3491	353	13	considering	consider	VERB
cana-3491	353	14	𝑝-adic	𝑝-adic	ADJ
cana-3491	353	15	expansion	expansion	NOUN
cana-3491	353	16	of	of	ADP
cana-3491	353	17	𝑦30′	𝑦30′	NUM
cana-3491	353	18	+	+	CCONJ
cana-3491	353	19	𝑦31′𝑝	𝑦31′𝑝	PROPN
cana-3491	353	20	modulo	modulo	NOUN
cana-3491	353	21	𝑝2	𝑝2	NOUN
cana-3491	353	22	,	,	PUNCT
cana-3491	353	23	we	we	PRON
cana-3491	353	24	have	have	VERB
cana-3491	353	25	𝑦(	𝑦(	NUM
cana-3491	353	26	�	�	PROPN
cana-3491	353	27	̃	̃	NOUN
cana-3491	353	28	�	�	NOUN
cana-3491	353	29	31	31	NUM
cana-3491	353	30	)	)	PUNCT
cana-3491	354	1	=	=	VERB
cana-3491	354	2	𝑦30	𝑦30	VERB
cana-3491	354	3	+	+	CCONJ
cana-3491	354	4	𝑦31𝑝	𝑦31𝑝	NOUN
cana-3491	354	5	for	for	ADP
cana-3491	354	6	�	�	PROPN
cana-3491	354	7	̃	̃	PROPN
cana-3491	354	8	�	�	PROPN
cana-3491	354	9	11	11	NUM
cana-3491	354	10	=	=	SYM
cana-3491	354	11	�	�	PROPN
cana-3491	354	12	̃	̃	PROPN
cana-3491	354	13	�	�	NOUN
cana-3491	354	14	21	21	NUM
cana-3491	354	15	:	:	PUNCT
cana-3491	354	16	by	by	ADP
cana-3491	354	17	the	the	DET
cana-3491	354	18	implementation	implementation	NOUN
cana-3491	354	19	of	of	ADP
cana-3491	354	20	arithmetic	arithmetic	NOUN
cana-3491	354	21	of	of	ADP
cana-3491	354	22	points	point	NOUN
cana-3491	354	23	�	�	PROPN
cana-3491	354	24	̃	̃	NOUN
cana-3491	354	25	�	�	NOUN
cana-3491	354	26	1	1	NUM
cana-3491	354	27	,	,	PUNCT
cana-3491	354	28	�	�	PROPN
cana-3491	354	29	̃	̃	NOUN
cana-3491	354	30	�	�	NOUN
cana-3491	354	31	2	2	NUM
cana-3491	354	32	in	in	ADP
cana-3491	354	33	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NUM
cana-3491	354	34	)	)	PUNCT
cana-3491	354	35	as	as	ADP
cana-3491	354	36	in	in	ADP
cana-3491	354	37	(	(	PUNCT
cana-3491	354	38	1	1	NUM
cana-3491	354	39	)	)	PUNCT
cana-3491	354	40	to	to	ADP
cana-3491	354	41	the	the	DET
cana-3491	354	42	points	point	NOUN
cana-3491	354	43	�	�	PROPN
cana-3491	354	44	̃	̃	PROPN
cana-3491	354	45	�	�	NOUN
cana-3491	354	46	11	11	NUM
cana-3491	354	47	,	,	PUNCT
cana-3491	354	48	�	�	PROPN
cana-3491	354	49	̃	̃	NOUN
cana-3491	354	50	�	�	NOUN
cana-3491	354	51	21	21	NUM
cana-3491	354	52	in	in	ADP
cana-3491	354	53	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	354	54	𝑝2	𝑝2	NOUN
cana-3491	354	55	)	)	PUNCT
cana-3491	354	56	,	,	PUNCT
cana-3491	354	57	we	we	PRON
cana-3491	354	58	have	have	VERB
cana-3491	354	59	�	�	PROPN
cana-3491	354	60	̃	̃	PROPN
cana-3491	354	61	�	�	PROPN
cana-3491	354	62	31	31	NUM
cana-3491	354	63	=	=	SYM
cana-3491	354	64	�	�	PROPN
cana-3491	354	65	̃	̃	PROPN
cana-3491	354	66	�	�	PROPN
cana-3491	354	67	11	11	NUM
cana-3491	354	68	+	+	SYM
cana-3491	354	69	�	�	PROPN
cana-3491	354	70	̃	̃	PROPN
cana-3491	354	71	�	�	NOUN
cana-3491	354	72	21	21	NUM
cana-3491	354	73	=	=	SYM
cana-3491	354	74	(	(	PUNCT
cana-3491	354	75	�	�	PROPN
cana-3491	354	76	̃	̃	NOUN
cana-3491	354	77	�	�	NOUN
cana-3491	354	78	1	1	NUM
cana-3491	354	79	2	2	NUM
cana-3491	354	80	−	−	NUM
cana-3491	354	81	2	2	NUM
cana-3491	354	82	�	�	PROPN
cana-3491	354	83	̃	̃	PROPN
cana-3491	354	84	�	�	NOUN
cana-3491	354	85	11	11	NUM
cana-3491	354	86	2	2	NUM
cana-3491	354	87	−	−	PROPN
cana-3491	354	88	�	�	PROPN
cana-3491	354	89	̃	̃	PROPN
cana-3491	354	90	�	�	NOUN
cana-3491	354	91	21	21	NUM
cana-3491	354	92	2	2	NUM
cana-3491	354	93	,	,	PUNCT
cana-3491	354	94	�	�	PROPN
cana-3491	354	95	̃	̃	PROPN
cana-3491	354	96	�	�	PROPN
cana-3491	354	97	1(	1(	NUM
cana-3491	354	98	�	�	NOUN
cana-3491	354	99	̃	̃	NOUN
cana-3491	354	100	�	�	PROPN
cana-3491	354	101	11	11	NUM
cana-3491	354	102	−	−	PROPN
cana-3491	354	103	�	�	PROPN
cana-3491	354	104	̃	̃	PROPN
cana-3491	354	105	�	�	NOUN
cana-3491	354	106	31	31	NUM
cana-3491	354	107	)	)	PUNCT
cana-3491	354	108	−	−	PROPN
cana-3491	354	109	�	�	PROPN
cana-3491	354	110	̃	̃	PROPN
cana-3491	354	111	�	�	NOUN
cana-3491	354	112	11	11	NUM
cana-3491	354	113	)	)	PUNCT
cana-3491	354	114	the	the	DET
cana-3491	354	115	𝑥-coordinate	𝑥-coordinate	NOUN
cana-3491	354	116	of	of	ADP
cana-3491	354	117	�	�	PROPN
cana-3491	354	118	̃	̃	PROPN
cana-3491	354	119	�	�	PROPN
cana-3491	354	120	31	31	NUM
cana-3491	354	121	is	be	AUX
cana-3491	354	122	given	give	VERB
cana-3491	354	123	as	as	ADP
cana-3491	354	124	𝑥(	𝑥(	PROPN
cana-3491	354	125	�	�	PROPN
cana-3491	354	126	̃	̃	PROPN
cana-3491	354	127	�	�	NOUN
cana-3491	354	128	31	31	NUM
cana-3491	354	129	)	)	PUNCT
cana-3491	354	130	=	=	SYM
cana-3491	354	131	�	�	PROPN
cana-3491	354	132	̃	̃	PROPN
cana-3491	354	133	�	�	NOUN
cana-3491	354	134	1	1	NUM
cana-3491	354	135	2	2	NUM
cana-3491	354	136	−	−	NUM
cana-3491	354	137	2	2	NUM
cana-3491	354	138	�	�	PROPN
cana-3491	354	139	̃	̃	PROPN
cana-3491	354	140	�	�	NOUN
cana-3491	354	141	11	11	NUM
cana-3491	354	142	2	2	NUM
cana-3491	354	143	=	=	SYM
cana-3491	354	144	(	(	PUNCT
cana-3491	354	145	𝑚0	𝑚0	NOUN
cana-3491	354	146	2	2	NUM
cana-3491	354	147	+	+	NUM
cana-3491	354	148	2𝑚1𝑝	2𝑚1𝑝	NOUN
cana-3491	354	149	)	)	PUNCT
cana-3491	355	1	+	+	CCONJ
cana-3491	355	2	2((𝑝	2((𝑝	NUM
cana-3491	356	1	−	−	NOUN
cana-3491	356	2	1	1	NUM
cana-3491	356	3	)	)	PUNCT
cana-3491	356	4	+	+	CCONJ
cana-3491	356	5	(	(	PUNCT
cana-3491	356	6	𝑝	𝑝	PROPN
cana-3491	356	7	−	−	NOUN
cana-3491	356	8	1)𝑝)(𝑥10	1)𝑝)(𝑥10	NUM
cana-3491	356	9	+	+	SYM
cana-3491	356	10	𝑥11𝑝	𝑥11𝑝	X
cana-3491	356	11	)	)	PUNCT
cana-3491	356	12	=	=	SYM
cana-3491	356	13	𝑚0	𝑚0	NOUN
cana-3491	356	14	2	2	NUM
cana-3491	356	15	+	+	NUM
cana-3491	356	16	2𝑚0𝑚1𝑝	2𝑚0𝑚1𝑝	NUM
cana-3491	356	17	+	+	CCONJ
cana-3491	356	18	2(𝑝	2(𝑝	NUM
cana-3491	356	19	−	−	NOUN
cana-3491	356	20	1)𝑥10	1)𝑥10	NOUN
cana-3491	356	21	+	+	CCONJ
cana-3491	356	22	2((𝑝	2((𝑝	NUM
cana-3491	356	23	−	−	PROPN
cana-3491	356	24	1)(𝑥10	1)(𝑥10	NUM
cana-3491	356	25	+	+	CCONJ
cana-3491	356	26	𝑥11))𝑝	𝑥11))𝑝	ADJ
cana-3491	356	27	=	=	SYM
cana-3491	356	28	𝑚0	𝑚0	NOUN
cana-3491	356	29	2	2	NUM
cana-3491	356	30	+	+	CCONJ
cana-3491	356	31	2(𝑝	2(𝑝	NUM
cana-3491	356	32	−	−	NOUN
cana-3491	356	33	1)𝑥10	1)𝑥10	NOUN
cana-3491	356	34	+	+	CCONJ
cana-3491	356	35	2𝑚0𝑚1𝑝	2𝑚0𝑚1𝑝	NUM
cana-3491	356	36	+	+	CCONJ
cana-3491	356	37	2((𝑝	2((𝑝	NUM
cana-3491	356	38	−	−	PROPN
cana-3491	356	39	1)(𝑥10	1)(𝑥10	NUM
cana-3491	356	40	+	+	CCONJ
cana-3491	356	41	𝑥11))𝑝	𝑥11))𝑝	ADJ
cana-3491	356	42	=	=	SYM
cana-3491	356	43	𝑚0	𝑚0	NOUN
cana-3491	356	44	2	2	NUM
cana-3491	356	45	+	+	CCONJ
cana-3491	356	46	2(𝑝	2(𝑝	NUM
cana-3491	356	47	−	−	ADP
cana-3491	356	48	1)𝑥10	1)𝑥10	NOUN
cana-3491	356	49	+	+	CCONJ
cana-3491	356	50	(	(	PUNCT
cana-3491	356	51	2𝑚0𝑚1	2𝑚0𝑚1	NUM
cana-3491	356	52	+	+	CCONJ
cana-3491	356	53	2(𝑝	2(𝑝	NUM
cana-3491	356	54	−	−	NUM
cana-3491	356	55	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	356	56	+	+	CCONJ
cana-3491	356	57	𝑥11))𝑝	𝑥11))𝑝	ADJ
cana-3491	356	58	therefore	therefore	ADV
cana-3491	356	59	,	,	PUNCT
cana-3491	356	60	𝑥30′	𝑥30′	PROPN
cana-3491	356	61	=	=	PUNCT
cana-3491	356	62	𝑚0	𝑚0	NOUN
cana-3491	356	63	2	2	NUM
cana-3491	356	64	+	+	CCONJ
cana-3491	356	65	2(𝑝	2(𝑝	NUM
cana-3491	356	66	−	−	NOUN
cana-3491	356	67	1)𝑥10	1)𝑥10	NOUN
cana-3491	356	68	𝑥31′	𝑥31′	ADJ
cana-3491	356	69	=	=	SYM
cana-3491	357	1	2(𝑚0𝑚1	2(𝑚0𝑚1	NUM
cana-3491	358	1	+	+	CCONJ
cana-3491	358	2	(	(	PUNCT
cana-3491	358	3	𝑝	𝑝	PROPN
cana-3491	358	4	−	−	PROPN
cana-3491	358	5	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	358	6	+	+	CCONJ
cana-3491	358	7	𝑥11	𝑥11	ADV
cana-3491	358	8	)	)	PUNCT
cana-3491	358	9	)	)	PUNCT
cana-3491	359	1	now	now	ADV
cana-3491	359	2	,	,	PUNCT
cana-3491	359	3	by	by	ADP
cana-3491	359	4	considering	consider	VERB
cana-3491	359	5	𝑝-adic	𝑝-adic	ADJ
cana-3491	359	6	expansion	expansion	NOUN
cana-3491	359	7	of	of	ADP
cana-3491	359	8	𝑥30′	𝑥30′	PROPN
cana-3491	359	9	+	+	CCONJ
cana-3491	359	10	𝑥31′𝑝	𝑥31′𝑝	DET
cana-3491	359	11	modulo	modulo	NOUN
cana-3491	359	12	𝑝2	𝑝2	NOUN
cana-3491	359	13	,	,	PUNCT
cana-3491	359	14	we	we	PRON
cana-3491	359	15	have	have	VERB
cana-3491	359	16	𝑥(	𝑥(	PROPN
cana-3491	359	17	�	�	PROPN
cana-3491	359	18	̃	̃	PROPN
cana-3491	359	19	�	�	NOUN
cana-3491	359	20	31	31	NUM
cana-3491	359	21	)	)	PUNCT
cana-3491	359	22	=	=	SYM
cana-3491	359	23	𝑥30	𝑥30	NOUN
cana-3491	359	24	+	+	CCONJ
cana-3491	359	25	𝑥31𝑝	𝑥31𝑝	VERB
cana-3491	359	26	the	the	DET
cana-3491	359	27	𝑦-coordinate	𝑦-coordinate	NOUN
cana-3491	359	28	of	of	ADP
cana-3491	359	29	�	�	PROPN
cana-3491	359	30	̃	̃	PROPN
cana-3491	359	31	�	�	PROPN
cana-3491	359	32	31	31	NUM
cana-3491	359	33	is	be	AUX
cana-3491	359	34	given	give	VERB
cana-3491	359	35	as	as	ADP
cana-3491	359	36	𝑦(	𝑦(	NUM
cana-3491	359	37	�	�	PROPN
cana-3491	359	38	̃	̃	PROPN
cana-3491	359	39	�	�	NOUN
cana-3491	359	40	31	31	NUM
cana-3491	359	41	)	)	PUNCT
cana-3491	360	1	=	=	SYM
cana-3491	360	2	�	�	PROPN
cana-3491	360	3	̃	̃	PROPN
cana-3491	360	4	�	�	PROPN
cana-3491	360	5	1(	1(	NUM
cana-3491	360	6	�	�	NOUN
cana-3491	360	7	̃	̃	NOUN
cana-3491	360	8	�	�	PROPN
cana-3491	360	9	11	11	NUM
cana-3491	360	10	−	−	PROPN
cana-3491	360	11	�	�	PROPN
cana-3491	360	12	̃	̃	PROPN
cana-3491	360	13	�	�	NOUN
cana-3491	360	14	31	31	NUM
cana-3491	360	15	)	)	PUNCT
cana-3491	360	16	−	−	PROPN
cana-3491	360	17	�	�	PROPN
cana-3491	360	18	̃	̃	PROPN
cana-3491	360	19	�	�	NOUN
cana-3491	360	20	11	11	NUM
cana-3491	360	21	which	which	PRON
cana-3491	360	22	on	on	ADP
cana-3491	360	23	repeating	repeat	VERB
cana-3491	360	24	above	above	ADP
cana-3491	360	25	process	process	NOUN
cana-3491	360	26	,	,	PUNCT
cana-3491	360	27	we	we	PRON
cana-3491	360	28	have	have	VERB
cana-3491	360	29	communications	communication	NOUN
cana-3491	360	30	on	on	ADP
cana-3491	360	31	applied	apply	VERB
cana-3491	360	32	nonlinear	nonlinear	ADJ
cana-3491	360	33	analysis	analysis	NOUN
cana-3491	360	34	issn	issn	NOUN
cana-3491	360	35	:	:	PUNCT
cana-3491	360	36	1074	1074	NUM
cana-3491	360	37	-	-	PUNCT
cana-3491	360	38	133x	133x	NUM
cana-3491	360	39	vol	vol	NOUN
cana-3491	360	40	32	32	NUM
cana-3491	360	41	no	no	NOUN
cana-3491	360	42	.	.	PUNCT
cana-3491	361	1	7s	7	NOUN
cana-3491	361	2	(	(	PUNCT
cana-3491	361	3	2025	2025	NUM
cana-3491	361	4	)	)	PUNCT
cana-3491	361	5	873	873	NUM
cana-3491	361	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	361	7	𝑦30	𝑦30	NOUN
cana-3491	361	8	=	=	PUNCT
cana-3491	361	9	𝑚0𝑥10	𝑚0𝑥10	X
cana-3491	361	10	+	+	CCONJ
cana-3491	361	11	(	(	PUNCT
cana-3491	361	12	𝑝	𝑝	PROPN
cana-3491	361	13	−	−	ADP
cana-3491	361	14	1)(𝑚0𝑥30	1)(𝑚0𝑥30	NUM
cana-3491	361	15	+	+	CCONJ
cana-3491	361	16	𝑦10	𝑦10	ADJ
cana-3491	361	17	)	)	PUNCT
cana-3491	361	18	𝑦31	𝑦31	NOUN
cana-3491	361	19	=	=	PUNCT
cana-3491	361	20	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	361	21	+	+	ADV
cana-3491	361	22	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	361	23	+	+	NUM
cana-3491	361	24	(	(	PUNCT
cana-3491	361	25	𝑝	𝑝	PROPN
cana-3491	361	26	−	−	PROPN
cana-3491	361	27	1)(𝑚1𝑥30	1)(𝑚1𝑥30	NUM
cana-3491	362	1	+	+	NOUN
cana-3491	362	2	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	362	3	+	+	ADJ
cana-3491	362	4	𝑚0𝑥31	𝑚0𝑥31	ADJ
cana-3491	362	5	+	+	CCONJ
cana-3491	362	6	𝑦10	𝑦10	ADJ
cana-3491	362	7	+	+	CCONJ
cana-3491	362	8	𝑦11	𝑦11	NOUN
cana-3491	362	9	)	)	PUNCT
cana-3491	362	10	now	now	ADV
cana-3491	362	11	,	,	PUNCT
cana-3491	362	12	by	by	ADP
cana-3491	362	13	considering	consider	VERB
cana-3491	362	14	𝑝-adic	𝑝-adic	ADJ
cana-3491	362	15	expansion	expansion	NOUN
cana-3491	362	16	of	of	ADP
cana-3491	362	17	𝑦30′	𝑦30′	NUM
cana-3491	362	18	+	+	CCONJ
cana-3491	362	19	𝑦31′𝑝	𝑦31′𝑝	PROPN
cana-3491	362	20	modulo	modulo	NOUN
cana-3491	362	21	𝑝2	𝑝2	NOUN
cana-3491	362	22	,	,	PUNCT
cana-3491	362	23	we	we	PRON
cana-3491	362	24	have	have	VERB
cana-3491	362	25	𝑦(	𝑦(	NUM
cana-3491	362	26	�	�	PROPN
cana-3491	362	27	̃	̃	NOUN
cana-3491	362	28	�	�	NOUN
cana-3491	362	29	31	31	NUM
cana-3491	362	30	)	)	PUNCT
cana-3491	363	1	=	=	VERB
cana-3491	363	2	𝑦30	𝑦30	VERB
cana-3491	363	3	+	+	CCONJ
cana-3491	363	4	𝑦31𝑝	𝑦31𝑝	NOUN
cana-3491	363	5	fi	fi	NOUN
cana-3491	363	6	el	el	PROPN
cana-3491	363	7	d	d	PROPN
cana-3491	363	8	κ	κ	PROPN
cana-3491	363	9	elli	elli	PROPN
cana-3491	363	10	pti	pti	PROPN
cana-3491	363	11	c	c	PROPN
cana-3491	363	12	cur	cur	AUX
cana-3491	363	13	ve	ve	AUX
cana-3491	363	14	slope	slope	VERB
cana-3491	363	15	𝒎	𝒎	PROPN
cana-3491	364	1	𝑷𝟏𝟏	𝑷𝟏𝟏	NUM
cana-3491	364	2	+	+	CCONJ
cana-3491	364	3	𝑷𝟐𝟏	𝑷𝟐𝟏	PROPN
cana-3491	364	4	=	=	SYM
cana-3491	364	5	𝑷𝟑𝟏	𝑷𝟑𝟏	PROPN
cana-3491	364	6	p11	p11	NOUN
cana-3491	364	7	≠	≠	PROPN
cana-3491	365	1	p21	p21	NOUN
cana-3491	365	2	𝑃11	𝑃11	PROPN
cana-3491	365	3	=	=	SYM
cana-3491	365	4	𝑃21	𝑃21	PROPN
cana-3491	365	5	p11	p11	NOUN
cana-3491	365	6	≠	≠	PROPN
cana-3491	365	7	p21	p21	NOUN
cana-3491	365	8	with	with	ADP
cana-3491	365	9	�	�	PROPN
cana-3491	365	10	̃	̃	PROPN
cana-3491	365	11	�	�	PROPN
cana-3491	365	12	11	11	NUM
cana-3491	365	13	=	=	SYM
cana-3491	365	14	�	�	PROPN
cana-3491	365	15	̃	̃	PROPN
cana-3491	365	16	�	�	PROPN
cana-3491	365	17	21	21	NUM
cana-3491	365	18	p11	p11	NOUN
cana-3491	365	19	≠	≠	PROPN
cana-3491	365	20	p21	p21	NOUN
cana-3491	365	21	wi	wi	PROPN
cana-3491	365	22	th	th	PROPN
cana-3491	365	23	�	�	PROPN
cana-3491	365	24	̃	̃	PROPN
cana-3491	365	25	�	�	PROPN
cana-3491	365	26	11	11	NUM
cana-3491	365	27	≠	≠	PROPN
cana-3491	365	28	�	�	PROPN
cana-3491	365	29	̃	̃	PROPN
cana-3491	365	30	�	�	PROPN
cana-3491	365	31	21	21	NUM
cana-3491	365	32	𝑃11	𝑃11	PROPN
cana-3491	365	33	=	=	PUNCT
cana-3491	366	1	𝑃21	𝑃21	PROPN
cana-3491	366	2	c	c	X
cana-3491	367	1	ha	ha	INTJ
cana-3491	367	2	r	r	NOUN
cana-3491	367	3	κ	κ	X
cana-3491	367	4	≠	≠	PROPN
cana-3491	367	5	2	2	NUM
cana-3491	367	6	,	,	PUNCT
cana-3491	367	7	3	3	NUM
cana-3491	367	8	𝛽2	𝛽2	NOUN
cana-3491	367	9	=	=	PUNCT
cana-3491	367	10	∝3	∝3	NUM
cana-3491	368	1	+	+	CCONJ
cana-3491	368	2	𝐴	𝐴	PROPN
cana-3491	368	3	∝	∝	PROPN
cana-3491	368	4	+	+	PROPN
cana-3491	368	5	𝐵	𝐵	PROPN
cana-3491	368	6	�	�	PROPN
cana-3491	368	7	̃	̃	PROPN
cana-3491	368	8	�	�	PROPN
cana-3491	368	9	21	21	NUM
cana-3491	368	10	−	−	PROPN
cana-3491	368	11	�	�	PROPN
cana-3491	368	12	̃	̃	PROPN
cana-3491	368	13	�	�	PROPN
cana-3491	368	14	11	11	NUM
cana-3491	368	15	�	�	PROPN
cana-3491	368	16	̃	̃	PROPN
cana-3491	368	17	�	�	PROPN
cana-3491	368	18	21	21	NUM
cana-3491	368	19	−	−	PROPN
cana-3491	368	20	�	�	PROPN
cana-3491	368	21	̃	̃	PROPN
cana-3491	368	22	�	�	PROPN
cana-3491	368	23	11	11	NUM
cana-3491	368	24	3	3	NUM
cana-3491	368	25	�	�	PROPN
cana-3491	368	26	̃	̃	PROPN
cana-3491	368	27	�	�	NOUN
cana-3491	368	28	11	11	NUM
cana-3491	368	29	2	2	NUM
cana-3491	368	30	+	+	CCONJ
cana-3491	368	31	𝐴	𝐴	PROPN
cana-3491	368	32	2	2	NUM
cana-3491	368	33	�	�	PROPN
cana-3491	368	34	̃	̃	PROPN
cana-3491	368	35	�	�	NOUN
cana-3491	368	36	11	11	NUM
cana-3491	368	37	{	{	PUNCT
cana-3491	368	38	𝑥30′	𝑥30′	NOUN
cana-3491	368	39	=	=	PUNCT
cana-3491	368	40	𝑚0	𝑚0	NOUN
cana-3491	368	41	2	2	NUM
cana-3491	368	42	+	+	CCONJ
cana-3491	368	43	(	(	PUNCT
cana-3491	368	44	𝑝	𝑝	PROPN
cana-3491	368	45	−	−	PROPN
cana-3491	368	46	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	368	47	+	+	CCONJ
cana-3491	368	48	𝑥20	𝑥20	X
cana-3491	368	49	)	)	PUNCT
cana-3491	368	50	𝑥31′	𝑥31′	PROPN
cana-3491	368	51	=	=	SYM
cana-3491	368	52	2𝑚0𝑚1	2𝑚0𝑚1	NOUN
cana-3491	368	53	+	+	CCONJ
cana-3491	368	54	(	(	PUNCT
cana-3491	368	55	𝑝	𝑝	PROPN
cana-3491	368	56	−	−	PROPN
cana-3491	368	57	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	368	58	+	+	CCONJ
cana-3491	368	59	𝑥11	𝑥11	ADV
cana-3491	368	60	+	+	CCONJ
cana-3491	368	61	𝑥20	𝑥20	ADJ
cana-3491	368	62	+	+	NUM
cana-3491	368	63	𝑥21	𝑥21	NOUN
cana-3491	368	64	)	)	PUNCT
cana-3491	368	65	𝑦30′	𝑦30′	NOUN
cana-3491	368	66	=	=	PUNCT
cana-3491	368	67	𝑚0𝑥10	𝑚0𝑥10	VERB
cana-3491	368	68	+	+	CCONJ
cana-3491	368	69	(	(	PUNCT
cana-3491	368	70	𝑝	𝑝	PROPN
cana-3491	368	71	−	−	ADP
cana-3491	368	72	1)(𝑚0𝑥30	1)(𝑚0𝑥30	NUM
cana-3491	368	73	+	+	CCONJ
cana-3491	368	74	𝑦10	𝑦10	ADJ
cana-3491	368	75	)	)	PUNCT
cana-3491	368	76	𝑦31′	𝑦31′	NOUN
cana-3491	368	77	=	=	PUNCT
cana-3491	368	78	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	368	79	+	+	ADV
cana-3491	368	80	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	368	81	+	+	NUM
cana-3491	368	82	(	(	PUNCT
cana-3491	368	83	𝑝	𝑝	NOUN
cana-3491	368	84	−	−	PROPN
cana-3491	368	85	1	1	NUM
cana-3491	368	86	)	)	PUNCT
cana-3491	368	87	(	(	PUNCT
cana-3491	368	88	𝑚1𝑥30	𝑚1𝑥30	X
cana-3491	368	89	+	+	ADV
cana-3491	368	90	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	368	91	+	+	CCONJ
cana-3491	368	92	𝑚0𝑥31	𝑚0𝑥31	NOUN
cana-3491	368	93	+	+	CCONJ
cana-3491	368	94	𝑦10	𝑦10	ADJ
cana-3491	368	95	+	+	CCONJ
cana-3491	368	96	𝑦11	𝑦11	ADJ
cana-3491	368	97	)	)	PUNCT
cana-3491	368	98	𝒪	𝒪	PROPN
cana-3491	368	99	{	{	PUNCT
cana-3491	368	100	𝑥30′	𝑥30′	NOUN
cana-3491	368	101	=	=	PUNCT
cana-3491	368	102	𝑚0	𝑚0	NOUN
cana-3491	368	103	2	2	NUM
cana-3491	368	104	+	+	CCONJ
cana-3491	368	105	(	(	PUNCT
cana-3491	368	106	𝑝	𝑝	PROPN
cana-3491	368	107	−	−	PROPN
cana-3491	368	108	1)2𝑥10	1)2𝑥10	NUM
cana-3491	368	109	𝑥31′	𝑥31′	NOUN
cana-3491	368	110	=	=	SYM
cana-3491	368	111	2(𝑚0𝑚1	2(𝑚0𝑚1	NUM
cana-3491	369	1	+	+	CCONJ
cana-3491	369	2	(	(	PUNCT
cana-3491	369	3	𝑝	𝑝	PROPN
cana-3491	369	4	−	−	PROPN
cana-3491	369	5	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	369	6	+	+	CCONJ
cana-3491	369	7	𝑥11	𝑥11	ADV
cana-3491	369	8	)	)	PUNCT
cana-3491	369	9	)	)	PUNCT
cana-3491	370	1	𝑦30′	𝑦30′	NOUN
cana-3491	370	2	=	=	PUNCT
cana-3491	370	3	𝑚0𝑥10	𝑚0𝑥10	VERB
cana-3491	370	4	+	+	CCONJ
cana-3491	370	5	(	(	PUNCT
cana-3491	370	6	𝑝	𝑝	PROPN
cana-3491	370	7	−	−	ADP
cana-3491	370	8	1)(𝑚0𝑥30	1)(𝑚0𝑥30	NUM
cana-3491	370	9	+	+	CCONJ
cana-3491	370	10	𝑦10	𝑦10	ADJ
cana-3491	370	11	)	)	PUNCT
cana-3491	370	12	𝑦31′	𝑦31′	NOUN
cana-3491	370	13	=	=	PUNCT
cana-3491	370	14	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	370	15	+	+	ADV
cana-3491	370	16	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	370	17	+	+	NUM
cana-3491	370	18	(	(	PUNCT
cana-3491	370	19	𝑝	𝑝	PROPN
cana-3491	370	20	−	−	PROPN
cana-3491	370	21	1)(𝑚1𝑥30	1)(𝑚1𝑥30	NUM
cana-3491	370	22	+	+	NOUN
cana-3491	370	23	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	370	24	+	+	ADJ
cana-3491	370	25	𝑚0𝑥31	𝑚0𝑥31	ADJ
cana-3491	370	26	+	+	CCONJ
cana-3491	370	27	𝑦10	𝑦10	ADJ
cana-3491	370	28	+	+	CCONJ
cana-3491	370	29	𝑦11	𝑦11	ADJ
cana-3491	370	30	)	)	PUNCT
cana-3491	370	31	table	table	NOUN
cana-3491	370	32	2	2	NUM
cana-3491	370	33	:	:	PUNCT
cana-3491	370	34	arithmetic	arithmetic	ADJ
cana-3491	370	35	of	of	ADP
cana-3491	370	36	points	point	NOUN
cana-3491	370	37	in	in	ADP
cana-3491	370	38	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	370	39	𝑝2	𝑝2	NOUN
cana-3491	370	40	example	example	NOUN
cana-3491	370	41	4.1	4.1	NUM
cana-3491	370	42	.	.	PUNCT
cana-3491	371	1	for	for	ADP
cana-3491	371	2	�	�	PROPN
cana-3491	371	3	̃	̃	PROPN
cana-3491	371	4	�	�	NOUN
cana-3491	371	5	11	11	NUM
cana-3491	371	6	=	=	SYM
cana-3491	371	7	(	(	PUNCT
cana-3491	371	8	4	4	NUM
cana-3491	371	9	+	+	NUM
cana-3491	371	10	2.5,2	2.5,2	NOUN
cana-3491	371	11	+	+	CCONJ
cana-3491	371	12	4.5	4.5	NUM
cana-3491	371	13	)	)	PUNCT
cana-3491	371	14	and	and	CCONJ
cana-3491	371	15	�	�	PROPN
cana-3491	371	16	̃	̃	PROPN
cana-3491	371	17	�	�	NOUN
cana-3491	371	18	21	21	NUM
cana-3491	371	19	=	=	SYM
cana-3491	371	20	(	(	PUNCT
cana-3491	371	21	2	2	NUM
cana-3491	371	22	+	+	NUM
cana-3491	371	23	1.5,4	1.5,4	NOUN
cana-3491	371	24	+	+	NUM
cana-3491	371	25	4.5	4.5	NUM
cana-3491	371	26	)	)	PUNCT
cana-3491	371	27	then	then	ADV
cana-3491	371	28	by	by	ADP
cana-3491	371	29	above	above	ADP
cana-3491	371	30	formulas	formula	NOUN
cana-3491	371	31	,	,	PUNCT
cana-3491	371	32	we	we	PRON
cana-3491	371	33	have	have	VERB
cana-3491	371	34	�	�	PROPN
cana-3491	371	35	̃	̃	PROPN
cana-3491	371	36	�	�	PROPN
cana-3491	371	37	11	11	NUM
cana-3491	371	38	+	+	SYM
cana-3491	371	39	�	�	PROPN
cana-3491	371	40	̃	̃	PROPN
cana-3491	371	41	�	�	NOUN
cana-3491	371	42	21	21	NUM
cana-3491	371	43	=	=	SYM
cana-3491	371	44	(	(	PUNCT
cana-3491	371	45	3.5	3.5	NUM
cana-3491	371	46	+	+	CCONJ
cana-3491	371	47	(	(	PUNCT
cana-3491	371	48	2	2	NUM
cana-3491	371	49	+	+	NUM
cana-3491	371	50	2.5)5,4	2.5)5,4	NUM
cana-3491	371	51	+	+	NUM
cana-3491	371	52	4.5	4.5	NUM
cana-3491	371	53	+	+	CCONJ
cana-3491	371	54	(	(	PUNCT
cana-3491	371	55	3.5	3.5	NUM
cana-3491	371	56	+	+	NUM
cana-3491	371	57	1.5	1.5	NUM
cana-3491	371	58	2)5	2)5	NUM
cana-3491	371	59	)	)	PUNCT
cana-3491	371	60	=	=	NOUN
cana-3491	372	1	(	(	PUNCT
cana-3491	372	2	3.5	3.5	NUM
cana-3491	372	3	+	+	NUM
cana-3491	372	4	2.5	2.5	NUM
cana-3491	372	5	+	+	CCONJ
cana-3491	372	6	2.52	2.52	NUM
cana-3491	372	7	,	,	PUNCT
cana-3491	372	8	4	4	NUM
cana-3491	372	9	+	+	SYM
cana-3491	372	10	4.5	4.5	NUM
cana-3491	372	11	+	+	CCONJ
cana-3491	372	12	3.52	3.52	NUM
cana-3491	372	13	+	+	NUM
cana-3491	372	14	1.53	1.53	NUM
cana-3491	372	15	)	)	PUNCT
cana-3491	372	16	=	=	PUNCT
cana-3491	372	17	(	(	PUNCT
cana-3491	372	18	3.53	3.53	NUM
cana-3491	372	19	,	,	PUNCT
cana-3491	372	20	4	4	NUM
cana-3491	372	21	+	+	SYM
cana-3491	372	22	4.5	4.5	NUM
cana-3491	372	23	+	+	CCONJ
cana-3491	372	24	3.52	3.52	NUM
cana-3491	372	25	+	+	NUM
cana-3491	372	26	1.53	1.53	NUM
cana-3491	372	27	)	)	PUNCT
cana-3491	372	28	=	=	NOUN
cana-3491	372	29	(	(	PUNCT
cana-3491	372	30	0,4	0,4	NUM
cana-3491	372	31	+	+	CCONJ
cana-3491	372	32	4.5)(𝑚𝑜𝑑52	4.5)(𝑚𝑜𝑑52	NUM
cana-3491	372	33	)	)	PUNCT
cana-3491	372	34	the	the	DET
cana-3491	372	35	step	step	NOUN
cana-3491	372	36	-	-	PUNCT
cana-3491	372	37	by	by	ADP
cana-3491	372	38	-	-	PUNCT
cana-3491	372	39	step	step	NOUN
cana-3491	372	40	procedure	procedure	NOUN
cana-3491	372	41	for	for	ADP
cana-3491	372	42	point	point	NOUN
cana-3491	372	43	addition	addition	NOUN
cana-3491	372	44	on	on	ADP
cana-3491	372	45	elliptic	elliptic	ADJ
cana-3491	372	46	curve	curve	NOUN
cana-3491	372	47	𝐸(𝑄𝑝)𝑚𝑜𝑑	𝐸(𝑄𝑝)𝑚𝑜𝑑	NOUN
cana-3491	372	48	𝑝2	𝑝2	NOUN
cana-3491	372	49	over	over	ADP
cana-3491	372	50	p	p	ADJ
cana-3491	372	51	-	-	PUNCT
cana-3491	372	52	adic	adic	ADJ
cana-3491	372	53	field	field	NOUN
cana-3491	372	54	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	372	55	defined	define	VERB
cana-3491	372	56	over	over	ADP
cana-3491	372	57	𝑍𝑝	𝑍𝑝	NOUN
cana-3491	372	58	using	use	VERB
cana-3491	372	59	addition	addition	NOUN
cana-3491	372	60	and	and	CCONJ
cana-3491	372	61	multiplication	multiplication	NOUN
cana-3491	372	62	process	process	NOUN
cana-3491	372	63	as	as	ADP
cana-3491	372	64	in	in	ADP
cana-3491	372	65	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	372	66	was	be	AUX
cana-3491	372	67	discussed	discuss	VERB
cana-3491	372	68	below	below	ADV
cana-3491	372	69	.	.	PUNCT
cana-3491	373	1	the	the	DET
cana-3491	373	2	code	code	NOUN
cana-3491	373	3	for	for	ADP
cana-3491	373	4	arithmetic	arithmetic	NOUN
cana-3491	373	5	of	of	ADP
cana-3491	373	6	points	point	NOUN
cana-3491	373	7	in	in	ADP
cana-3491	373	8	𝐸(𝑄𝑝)𝑚𝑜𝑑	𝐸(𝑄𝑝)𝑚𝑜𝑑	NOUN
cana-3491	373	9	𝑝2	𝑝2	NOUN
cana-3491	373	10	and	and	CCONJ
cana-3491	373	11	arithmetic	arithmetic	ADJ
cana-3491	373	12	operations	operation	NOUN
cana-3491	373	13	of	of	ADP
cana-3491	373	14	numbers	number	NOUN
cana-3491	373	15	in	in	ADP
cana-3491	373	16	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	373	17	are	be	AUX
cana-3491	373	18	included	include	VERB
cana-3491	373	19	below	below	ADV
cana-3491	373	20	.	.	PUNCT
cana-3491	374	1	algorithm	algorithm	NOUN
cana-3491	374	2	:	:	PUNCT
cana-3491	374	3	the	the	DET
cana-3491	374	4	step	step	NOUN
cana-3491	374	5	-	-	PUNCT
cana-3491	374	6	by	by	ADP
cana-3491	374	7	-	-	PUNCT
cana-3491	374	8	step	step	NOUN
cana-3491	374	9	procedure	procedure	NOUN
cana-3491	374	10	is	be	AUX
cana-3491	374	11	termed	term	VERB
cana-3491	374	12	as	as	ADP
cana-3491	374	13	algorithm	algorithm	NOUN
cana-3491	374	14	.	.	PUNCT
cana-3491	375	1	communications	communication	NOUN
cana-3491	375	2	on	on	ADP
cana-3491	375	3	applied	apply	VERB
cana-3491	375	4	nonlinear	nonlinear	ADJ
cana-3491	375	5	analysis	analysis	NOUN
cana-3491	375	6	issn	issn	NOUN
cana-3491	375	7	:	:	PUNCT
cana-3491	375	8	1074	1074	NUM
cana-3491	375	9	-	-	PUNCT
cana-3491	375	10	133x	133x	NUM
cana-3491	375	11	vol	vol	NOUN
cana-3491	375	12	32	32	NUM
cana-3491	375	13	no	no	NOUN
cana-3491	375	14	.	.	PUNCT
cana-3491	376	1	7s	7	NOUN
cana-3491	376	2	(	(	PUNCT
cana-3491	376	3	2025	2025	NUM
cana-3491	376	4	)	)	PUNCT
cana-3491	376	5	874	874	NUM
cana-3491	377	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	377	2	algorithm	algorithm	NOUN
cana-3491	377	3	for	for	ADP
cana-3491	377	4	point	point	NOUN
cana-3491	377	5	addition	addition	NOUN
cana-3491	377	6	in	in	ADP
cana-3491	377	7	(	(	PUNCT
cana-3491	377	8	𝑸𝒑)(𝒎𝒐𝒅	𝑸𝒑)(𝒎𝒐𝒅	ADJ
cana-3491	377	9	𝒑𝟐	𝒑𝟐	NOUN
cana-3491	377	10	)	)	PUNCT
cana-3491	377	11	:	:	PUNCT
cana-3491	377	12	consider	consider	VERB
cana-3491	377	13	an	an	DET
cana-3491	377	14	elliptic	elliptic	ADJ
cana-3491	377	15	curve𝛽2	curve𝛽2	NOUN
cana-3491	378	1	=	=	PUNCT
cana-3491	378	2	𝛼3	𝛼3	NOUN
cana-3491	378	3	+	+	CCONJ
cana-3491	379	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3491	379	2	+	+	CCONJ
cana-3491	379	3	𝐵	𝐵	NOUN
cana-3491	379	4	over	over	ADP
cana-3491	379	5	𝑄𝑝.	𝑄𝑝.	PROPN
cana-3491	379	6	let	let	VERB
cana-3491	379	7	�	�	PROPN
cana-3491	379	8	̃	̃	NOUN
cana-3491	379	9	�	�	NOUN
cana-3491	379	10	1	1	NUM
cana-3491	379	11	,	,	PUNCT
cana-3491	379	12	�	�	PROPN
cana-3491	379	13	̃	̃	PROPN
cana-3491	379	14	�	�	NOUN
cana-3491	379	15	2	2	NUM
cana-3491	379	16	be	be	VERB
cana-3491	379	17	two	two	NUM
cana-3491	379	18	points	point	NOUN
cana-3491	379	19	in	in	ADP
cana-3491	379	20	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	379	21	)	)	PUNCT
cana-3491	379	22	then	then	ADV
cana-3491	379	23	for	for	ADP
cana-3491	379	24	�	�	PROPN
cana-3491	379	25	̃	̃	PROPN
cana-3491	379	26	�	�	PROPN
cana-3491	379	27	1	1	NUM
cana-3491	379	28	and	and	CCONJ
cana-3491	379	29	�	�	PROPN
cana-3491	379	30	̃	̃	PROPN
cana-3491	379	31	�	�	NOUN
cana-3491	379	32	2	2	NUM
cana-3491	379	33	considered	consider	VERB
cana-3491	379	34	in	in	ADP
cana-3491	379	35	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	379	36	𝑝2	𝑝2	NOUN
cana-3491	379	37	)	)	PUNCT
cana-3491	379	38	given	give	VERB
cana-3491	379	39	as	as	ADP
cana-3491	379	40	�	�	PROPN
cana-3491	379	41	̃	̃	PROPN
cana-3491	379	42	�	�	NOUN
cana-3491	379	43	11	11	NUM
cana-3491	379	44	=	=	SYM
cana-3491	379	45	(	(	PUNCT
cana-3491	379	46	�	�	PROPN
cana-3491	379	47	̃	̃	NOUN
cana-3491	379	48	�	�	NOUN
cana-3491	379	49	11	11	NUM
cana-3491	379	50	,	,	PUNCT
cana-3491	379	51	�	�	PROPN
cana-3491	379	52	̃	̃	PROPN
cana-3491	379	53	�	�	NOUN
cana-3491	379	54	11	11	NUM
cana-3491	379	55	)	)	PUNCT
cana-3491	379	56	=	=	PUNCT
cana-3491	379	57	(	(	PUNCT
cana-3491	379	58	𝑥10	𝑥10	X
cana-3491	379	59	+	+	SYM
cana-3491	379	60	𝑥11𝑝	𝑥11𝑝	ADJ
cana-3491	379	61	,	,	PUNCT
cana-3491	379	62	𝑦10	𝑦10	ADJ
cana-3491	379	63	+	+	CCONJ
cana-3491	379	64	𝑦11𝑝	𝑦11𝑝	NOUN
cana-3491	379	65	)	)	PUNCT
cana-3491	379	66	and	and	CCONJ
cana-3491	379	67	�	�	PROPN
cana-3491	379	68	̃	̃	PROPN
cana-3491	379	69	�	�	NOUN
cana-3491	379	70	21	21	NUM
cana-3491	379	71	=	=	SYM
cana-3491	379	72	(	(	PUNCT
cana-3491	379	73	𝑥20	𝑥20	NOUN
cana-3491	379	74	+	+	CCONJ
cana-3491	379	75	𝑥21𝑝	𝑥21𝑝	NOUN
cana-3491	379	76	,	,	PUNCT
cana-3491	379	77	𝑦20	𝑦20	NOUN
cana-3491	379	78	+	+	CCONJ
cana-3491	379	79	𝑦21𝑝	𝑦21𝑝	NOUN
cana-3491	379	80	)	)	PUNCT
cana-3491	379	81	respectively	respectively	ADV
cana-3491	379	82	.	.	PUNCT
cana-3491	380	1	the	the	DET
cana-3491	380	2	point	point	NOUN
cana-3491	380	3	addition	addition	NOUN
cana-3491	380	4	�	�	PROPN
cana-3491	380	5	̃	̃	NOUN
cana-3491	380	6	�	�	PROPN
cana-3491	380	7	11	11	NUM
cana-3491	380	8	+	+	SYM
cana-3491	380	9	�	�	PROPN
cana-3491	380	10	̃	̃	PROPN
cana-3491	380	11	�	�	NOUN
cana-3491	380	12	21	21	NUM
cana-3491	380	13	=	=	SYM
cana-3491	380	14	�	�	PROPN
cana-3491	380	15	̃	̃	PROPN
cana-3491	380	16	�	�	PROPN
cana-3491	380	17	31	31	NUM
cana-3491	380	18	=	=	SYM
cana-3491	380	19	(	(	PUNCT
cana-3491	380	20	𝑥30	𝑥30	PROPN
cana-3491	380	21	+	+	CCONJ
cana-3491	380	22	𝑥31𝑝	𝑥31𝑝	NOUN
cana-3491	380	23	,	,	PUNCT
cana-3491	380	24	𝑦30	𝑦30	VERB
cana-3491	380	25	+	+	CCONJ
cana-3491	380	26	𝑦31𝑝	𝑦31𝑝	NOUN
cana-3491	380	27	)	)	PUNCT
cana-3491	380	28	in	in	ADP
cana-3491	380	29	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	380	30	𝑝2	𝑝2	NOUN
cana-3491	380	31	)	)	PUNCT
cana-3491	380	32	is	be	AUX
cana-3491	380	33	obtained	obtain	VERB
cana-3491	380	34	by	by	ADP
cana-3491	380	35	the	the	DET
cana-3491	380	36	steps	step	NOUN
cana-3491	380	37	in	in	ADP
cana-3491	380	38	the	the	DET
cana-3491	380	39	following	follow	VERB
cana-3491	380	40	algorithm	algorithm	NOUN
cana-3491	380	41	.	.	PUNCT
cana-3491	381	1	step	step	NOUN
cana-3491	381	2	-	-	PUNCT
cana-3491	381	3	i	i	PRON
cana-3491	381	4	:	:	PUNCT
cana-3491	381	5	compute	compute	VERB
cana-3491	381	6	the	the	DET
cana-3491	381	7	slope	slope	PROPN
cana-3491	381	8	�	�	PROPN
cana-3491	381	9	̃	̃	PROPN
cana-3491	381	10	�	�	NOUN
cana-3491	381	11	1	1	NUM
cana-3491	381	12	=	=	SYM
cana-3491	381	13	𝑚0	𝑚0	NOUN
cana-3491	382	1	+	+	NOUN
cana-3491	382	2	𝑚1𝑝	𝑚1𝑝	ADP
cana-3491	382	3	of	of	ADP
cana-3491	382	4	�	�	PROPN
cana-3491	382	5	̃	̃	PROPN
cana-3491	382	6	�	�	PROPN
cana-3491	382	7	11	11	NUM
cana-3491	382	8	and	and	CCONJ
cana-3491	382	9	�	�	PROPN
cana-3491	382	10	̃	̃	PROPN
cana-3491	382	11	�	�	PROPN
cana-3491	382	12	21	21	NUM
cana-3491	382	13	given	give	VERB
cana-3491	382	14	as	as	ADP
cana-3491	382	15	�	�	PROPN
cana-3491	382	16	̃	̃	PROPN
cana-3491	382	17	�	�	NOUN
cana-3491	382	18	1	1	NUM
cana-3491	382	19	=	=	SYM
cana-3491	382	20	{	{	PUNCT
cana-3491	382	21	�	�	PROPN
cana-3491	382	22	̃	̃	PROPN
cana-3491	382	23	�	�	PROPN
cana-3491	382	24	21	21	NUM
cana-3491	382	25	−	−	PROPN
cana-3491	382	26	�	�	PROPN
cana-3491	382	27	̃	̃	PROPN
cana-3491	382	28	�	�	PROPN
cana-3491	382	29	11	11	NUM
cana-3491	382	30	�	�	PROPN
cana-3491	382	31	̃	̃	PROPN
cana-3491	382	32	�	�	PROPN
cana-3491	382	33	21	21	NUM
cana-3491	382	34	−	−	PROPN
cana-3491	382	35	�	�	PROPN
cana-3491	382	36	̃	̃	PROPN
cana-3491	382	37	�	�	NOUN
cana-3491	382	38	11	11	NUM
cana-3491	382	39	for	for	ADP
cana-3491	382	40	�	�	PROPN
cana-3491	382	41	̃	̃	PROPN
cana-3491	382	42	�	�	PROPN
cana-3491	382	43	11	11	NUM
cana-3491	382	44	≠	≠	PROPN
cana-3491	382	45	�	�	PROPN
cana-3491	382	46	̃	̃	PROPN
cana-3491	382	47	�	�	PROPN
cana-3491	382	48	21	21	NUM
cana-3491	382	49	3	3	NUM
cana-3491	382	50	�	�	PROPN
cana-3491	382	51	̃	̃	PROPN
cana-3491	382	52	�	�	NOUN
cana-3491	382	53	11	11	NUM
cana-3491	382	54	2	2	NUM
cana-3491	382	55	+	+	CCONJ
cana-3491	382	56	𝐴	𝐴	PROPN
cana-3491	382	57	2	2	NUM
cana-3491	382	58	�	�	PROPN
cana-3491	382	59	̃	̃	PROPN
cana-3491	382	60	�	�	NOUN
cana-3491	382	61	11	11	NUM
cana-3491	382	62	for	for	ADP
cana-3491	382	63	�	�	PROPN
cana-3491	382	64	̃	̃	PROPN
cana-3491	382	65	�	�	PROPN
cana-3491	382	66	11	11	NUM
cana-3491	382	67	=	=	SYM
cana-3491	382	68	�	�	PROPN
cana-3491	382	69	̃	̃	PROPN
cana-3491	382	70	�	�	NOUN
cana-3491	382	71	21	21	NUM
cana-3491	382	72	step	step	NOUN
cana-3491	382	73	-	-	PUNCT
cana-3491	382	74	ii	ii	NOUN
cana-3491	382	75	:	:	PUNCT
cana-3491	382	76	compute	compute	PROPN
cana-3491	382	77	𝑥30′	𝑥30′	PROPN
cana-3491	382	78	,	,	PUNCT
cana-3491	382	79	𝑥31′	𝑥31′	ADJ
cana-3491	382	80	,	,	PUNCT
cana-3491	382	81	𝑦30′	𝑦30′	NUM
cana-3491	382	82	,	,	PUNCT
cana-3491	382	83	𝑦31′	𝑦31′	NOUN
cana-3491	382	84	using	use	VERB
cana-3491	382	85	formulas	formula	NOUN
cana-3491	382	86	for	for	ADP
cana-3491	382	87	�	�	PROPN
cana-3491	382	88	̃	̃	PROPN
cana-3491	382	89	�	�	PROPN
cana-3491	382	90	11	11	NUM
cana-3491	382	91	≠	≠	PROPN
cana-3491	382	92	�	�	PROPN
cana-3491	382	93	̃	̃	PROPN
cana-3491	382	94	�	�	NOUN
cana-3491	382	95	21	21	NUM
cana-3491	382	96	:	:	PUNCT
cana-3491	382	97	{	{	PUNCT
cana-3491	382	98	𝑥30′	𝑥30′	X
cana-3491	382	99	=	=	PUNCT
cana-3491	382	100	𝑚0	𝑚0	NOUN
cana-3491	382	101	2	2	NUM
cana-3491	382	102	+	+	CCONJ
cana-3491	382	103	(	(	PUNCT
cana-3491	382	104	𝑝	𝑝	PROPN
cana-3491	382	105	−	−	PROPN
cana-3491	382	106	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	382	107	+	+	CCONJ
cana-3491	382	108	𝑥20	𝑥20	NUM
cana-3491	382	109	)	)	PUNCT
cana-3491	382	110	,	,	PUNCT
cana-3491	382	111	𝑥31′	𝑥31′	PROPN
cana-3491	382	112	=	=	SYM
cana-3491	382	113	2𝑚0𝑚1	2𝑚0𝑚1	NOUN
cana-3491	382	114	+	+	CCONJ
cana-3491	382	115	(	(	PUNCT
cana-3491	382	116	𝑝	𝑝	PROPN
cana-3491	382	117	−	−	PROPN
cana-3491	382	118	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	382	119	+	+	CCONJ
cana-3491	382	120	𝑥11	𝑥11	ADV
cana-3491	383	1	+	+	CCONJ
cana-3491	383	2	𝑥20	𝑥20	ADJ
cana-3491	383	3	+	+	NUM
cana-3491	383	4	𝑥21	𝑥21	NOUN
cana-3491	383	5	)	)	PUNCT
cana-3491	383	6	,	,	PUNCT
cana-3491	383	7	𝑦30′	𝑦30′	NOUN
cana-3491	383	8	=	=	PUNCT
cana-3491	383	9	𝑚0𝑥10	𝑚0𝑥10	VERB
cana-3491	383	10	+	+	CCONJ
cana-3491	383	11	(	(	PUNCT
cana-3491	383	12	𝑝	𝑝	PROPN
cana-3491	383	13	−	−	ADP
cana-3491	383	14	1)(𝑚0𝑥30	1)(𝑚0𝑥30	NUM
cana-3491	383	15	+	+	CCONJ
cana-3491	383	16	𝑦10	𝑦10	ADJ
cana-3491	383	17	)	)	PUNCT
cana-3491	383	18	,	,	PUNCT
cana-3491	383	19	𝑦31′	𝑦31′	NOUN
cana-3491	383	20	=	=	PUNCT
cana-3491	383	21	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	383	22	+	+	ADV
cana-3491	383	23	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	383	24	+	+	NUM
cana-3491	383	25	(	(	PUNCT
cana-3491	383	26	𝑝	𝑝	PROPN
cana-3491	383	27	−	−	PROPN
cana-3491	383	28	1)(𝑚1𝑥30	1)(𝑚1𝑥30	NUM
cana-3491	384	1	+	+	NOUN
cana-3491	384	2	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	384	3	+	+	ADJ
cana-3491	384	4	𝑚0𝑥31	𝑚0𝑥31	ADJ
cana-3491	384	5	+	+	CCONJ
cana-3491	384	6	𝑦10	𝑦10	ADJ
cana-3491	384	7	+	+	CCONJ
cana-3491	384	8	𝑦11	𝑦11	NOUN
cana-3491	384	9	)	)	PUNCT
cana-3491	384	10	for	for	ADP
cana-3491	384	11	�	�	PROPN
cana-3491	384	12	̃	̃	PROPN
cana-3491	384	13	�	�	PROPN
cana-3491	384	14	11	11	NUM
cana-3491	384	15	=	=	SYM
cana-3491	384	16	�	�	PROPN
cana-3491	384	17	̃	̃	PROPN
cana-3491	384	18	�	�	NOUN
cana-3491	384	19	21	21	NUM
cana-3491	384	20	:	:	PUNCT
cana-3491	384	21	{	{	PUNCT
cana-3491	384	22	𝑥30′	𝑥30′	X
cana-3491	384	23	=	=	PUNCT
cana-3491	384	24	𝑚0	𝑚0	NOUN
cana-3491	384	25	2	2	NUM
cana-3491	384	26	+	+	CCONJ
cana-3491	384	27	(	(	PUNCT
cana-3491	384	28	𝑝	𝑝	PROPN
cana-3491	384	29	−	−	PROPN
cana-3491	384	30	1)2𝑥10	1)2𝑥10	NUM
cana-3491	384	31	,	,	PUNCT
cana-3491	384	32	𝑥31′	𝑥31′	PROPN
cana-3491	384	33	=	=	PROPN
cana-3491	385	1	2(𝑚0𝑚1	2(𝑚0𝑚1	NUM
cana-3491	386	1	+	+	CCONJ
cana-3491	386	2	(	(	PUNCT
cana-3491	386	3	𝑝	𝑝	PROPN
cana-3491	386	4	−	−	PROPN
cana-3491	386	5	1)(𝑥10	1)(𝑥10	PROPN
cana-3491	386	6	+	+	CCONJ
cana-3491	386	7	𝑥11	𝑥11	ADV
cana-3491	386	8	)	)	PUNCT
cana-3491	386	9	)	)	PUNCT
cana-3491	386	10	,	,	PUNCT
cana-3491	386	11	𝑦30′	𝑦30′	NOUN
cana-3491	386	12	=	=	PUNCT
cana-3491	386	13	𝑚0𝑥10	𝑚0𝑥10	VERB
cana-3491	386	14	+	+	CCONJ
cana-3491	386	15	(	(	PUNCT
cana-3491	386	16	𝑝	𝑝	PROPN
cana-3491	386	17	−	−	ADP
cana-3491	386	18	1)(𝑚0𝑥30	1)(𝑚0𝑥30	NUM
cana-3491	386	19	+	+	CCONJ
cana-3491	386	20	𝑦10	𝑦10	ADJ
cana-3491	386	21	)	)	PUNCT
cana-3491	386	22	,	,	PUNCT
cana-3491	386	23	𝑦31′	𝑦31′	NOUN
cana-3491	386	24	=	=	PUNCT
cana-3491	386	25	𝑚1𝑥10	𝑚1𝑥10	ADP
cana-3491	386	26	+	+	ADV
cana-3491	386	27	𝑚0𝑥11	𝑚0𝑥11	PROPN
cana-3491	386	28	+	+	NUM
cana-3491	386	29	(	(	PUNCT
cana-3491	386	30	𝑝	𝑝	PROPN
cana-3491	386	31	−	−	PROPN
cana-3491	386	32	1)(𝑚1𝑥30	1)(𝑚1𝑥30	NUM
cana-3491	386	33	+	+	NOUN
cana-3491	386	34	𝑚0𝑥30	𝑚0𝑥30	PRON
cana-3491	386	35	+	+	ADJ
cana-3491	386	36	𝑚0𝑥31	𝑚0𝑥31	ADJ
cana-3491	386	37	+	+	CCONJ
cana-3491	386	38	𝑦10	𝑦10	ADJ
cana-3491	386	39	+	+	CCONJ
cana-3491	386	40	𝑦11	𝑦11	ADJ
cana-3491	386	41	)	)	PUNCT
cana-3491	386	42	step	step	NOUN
cana-3491	386	43	-	-	PUNCT
cana-3491	386	44	iii	iii	NOUN
cana-3491	386	45	:	:	PUNCT
cana-3491	386	46	to	to	PART
cana-3491	386	47	evaluate	evaluate	VERB
cana-3491	386	48	𝑥30	𝑥30	PROPN
cana-3491	386	49	and	and	CCONJ
cana-3491	386	50	𝑥31	𝑥31	NOUN
cana-3491	386	51	,	,	PUNCT
cana-3491	386	52	consider	consider	VERB
cana-3491	386	53	the	the	DET
cana-3491	386	54	value	value	NOUN
cana-3491	386	55	𝑁	𝑁	PROPN
cana-3491	386	56	=	=	PUNCT
cana-3491	386	57	𝑥30′	𝑥30′	PROPN
cana-3491	386	58	+	+	CCONJ
cana-3491	386	59	𝑥31′𝑝	𝑥31′𝑝	PROPN
cana-3491	386	60	and	and	CCONJ
cana-3491	386	61	write	write	VERB
cana-3491	386	62	the	the	DET
cana-3491	386	63	𝑝-adic	𝑝-adic	ADJ
cana-3491	386	64	expansion	expansion	NOUN
cana-3491	386	65	of	of	ADP
cana-3491	386	66	𝑁	𝑁	PROPN
cana-3491	386	67	and	and	CCONJ
cana-3491	386	68	consider	consider	VERB
cana-3491	386	69	𝑁(𝑚𝑜𝑑	𝑁(𝑚𝑜𝑑	ADJ
cana-3491	386	70	𝑝2	𝑝2	NOUN
cana-3491	386	71	)	)	PUNCT
cana-3491	386	72	to	to	PART
cana-3491	386	73	obtain	obtain	VERB
cana-3491	386	74	𝑥30	𝑥30	PROPN
cana-3491	386	75	and	and	CCONJ
cana-3491	386	76	𝑥31	𝑥31	NOUN
cana-3491	386	77	.	.	PUNCT
cana-3491	387	1	step	step	NOUN
cana-3491	387	2	-	-	PUNCT
cana-3491	387	3	iv	iv	NUM
cana-3491	387	4	:	:	PUNCT
cana-3491	387	5	to	to	PART
cana-3491	387	6	evaluate	evaluate	VERB
cana-3491	387	7	𝑦30	𝑦30	NOUN
cana-3491	387	8	and	and	CCONJ
cana-3491	387	9	𝑦31	𝑦31	NOUN
cana-3491	387	10	,	,	PUNCT
cana-3491	387	11	consider	consider	VERB
cana-3491	387	12	the	the	DET
cana-3491	387	13	value	value	NOUN
cana-3491	387	14	𝑀	𝑀	PROPN
cana-3491	387	15	=	=	PUNCT
cana-3491	388	1	𝑦30′	𝑦30′	PROPN
cana-3491	388	2	+	+	CCONJ
cana-3491	388	3	𝑦31′𝑝	𝑦31′𝑝	PROPN
cana-3491	388	4	and	and	CCONJ
cana-3491	388	5	write	write	VERB
cana-3491	388	6	the	the	DET
cana-3491	388	7	𝑝-adic	𝑝-adic	ADJ
cana-3491	388	8	expansion	expansion	NOUN
cana-3491	388	9	of	of	ADP
cana-3491	388	10	𝑀	𝑀	PROPN
cana-3491	388	11	and	and	CCONJ
cana-3491	388	12	consider	consider	VERB
cana-3491	388	13	𝑀(𝑚𝑜𝑑	𝑀(𝑚𝑜𝑑	PROPN
cana-3491	388	14	𝑝2	𝑝2	NOUN
cana-3491	388	15	)	)	PUNCT
cana-3491	388	16	to	to	PART
cana-3491	388	17	obtain	obtain	VERB
cana-3491	388	18	𝑦30	𝑦30	ADJ
cana-3491	388	19	and	and	CCONJ
cana-3491	388	20	𝑦31	𝑦31	NOUN
cana-3491	388	21	.	.	PUNCT
cana-3491	389	1	step	step	NOUN
cana-3491	389	2	-	-	PUNCT
cana-3491	389	3	v	v	NOUN
cana-3491	389	4	:	:	PUNCT
cana-3491	389	5	from	from	ADP
cana-3491	389	6	the	the	DET
cana-3491	389	7	values	value	NOUN
cana-3491	389	8	of	of	ADP
cana-3491	389	9	𝑥30	𝑥30	NOUN
cana-3491	389	10	and	and	CCONJ
cana-3491	389	11	𝑥31	𝑥31	VERB
cana-3491	389	12	in	in	ADP
cana-3491	389	13	step	step	NOUN
cana-3491	389	14	-	-	PUNCT
cana-3491	389	15	iii	iii	NOUN
cana-3491	389	16	and	and	CCONJ
cana-3491	389	17	𝑦30	𝑦30	VERB
cana-3491	389	18	and	and	CCONJ
cana-3491	389	19	𝑦31	𝑦31	NOUN
cana-3491	389	20	in	in	ADP
cana-3491	389	21	step	step	NOUN
cana-3491	389	22	-	-	PUNCT
cana-3491	389	23	iv	iv	NUM
cana-3491	389	24	,	,	PUNCT
cana-3491	389	25	the	the	DET
cana-3491	389	26	point	point	NOUN
cana-3491	389	27	addition	addition	NOUN
cana-3491	389	28	�	�	PROPN
cana-3491	389	29	̃	̃	NOUN
cana-3491	389	30	�	�	PROPN
cana-3491	389	31	11	11	NUM
cana-3491	389	32	+	+	SYM
cana-3491	389	33	�	�	PROPN
cana-3491	389	34	̃	̃	PROPN
cana-3491	389	35	�	�	NOUN
cana-3491	389	36	21	21	NUM
cana-3491	389	37	is	be	AUX
cana-3491	389	38	given	give	VERB
cana-3491	389	39	as	as	ADP
cana-3491	389	40	:	:	PUNCT
cana-3491	389	41	�	�	PROPN
cana-3491	389	42	̃	̃	PROPN
cana-3491	389	43	�	�	PROPN
cana-3491	389	44	31	31	NUM
cana-3491	389	45	=	=	SYM
cana-3491	389	46	(	(	PUNCT
cana-3491	389	47	𝑥30	𝑥30	PROPN
cana-3491	389	48	+	+	CCONJ
cana-3491	389	49	𝑥31𝑝	𝑥31𝑝	NOUN
cana-3491	389	50	,	,	PUNCT
cana-3491	389	51	𝑦30	𝑦30	VERB
cana-3491	389	52	+	+	CCONJ
cana-3491	389	53	𝑦31𝑝	𝑦31𝑝	NOUN
cana-3491	389	54	)	)	PUNCT
cana-3491	389	55	for	for	ADP
cana-3491	389	56	an	an	DET
cana-3491	389	57	elliptic	elliptic	ADJ
cana-3491	389	58	curve	curve	NOUN
cana-3491	389	59	𝑦2	𝑦2	PROPN
cana-3491	389	60	=	=	SYM
cana-3491	389	61	𝑥3	𝑥3	PROPN
cana-3491	389	62	+	+	CCONJ
cana-3491	389	63	𝑥	𝑥	X
cana-3491	390	1	+	+	NUM
cana-3491	390	2	1	1	NUM
cana-3491	390	3	over	over	ADP
cana-3491	390	4	𝑄999983(𝑚𝑜𝑑	𝑄999983(𝑚𝑜𝑑	PROPN
cana-3491	390	5	9999832	9999832	NUM
cana-3491	390	6	)	)	PUNCT
cana-3491	390	7	,	,	PUNCT
cana-3491	390	8	consider	consider	VERB
cana-3491	390	9	two	two	NUM
cana-3491	390	10	points	point	NOUN
cana-3491	390	11	�	�	PROPN
cana-3491	390	12	̃	̃	NOUN
cana-3491	390	13	�	�	NOUN
cana-3491	390	14	11	11	NUM
cana-3491	390	15	=	=	SYM
cana-3491	390	16	(	(	PUNCT
cana-3491	390	17	371181	371181	NUM
cana-3491	390	18	+	+	SYM
cana-3491	390	19	9738	9738	NUM
cana-3491	390	20	×	×	NOUN
cana-3491	390	21	999983	999983	NUM
cana-3491	390	22	,	,	PUNCT
cana-3491	390	23	209555	209555	NUM
cana-3491	390	24	+	+	SYM
cana-3491	390	25	151202	151202	NUM
cana-3491	390	26	×	×	NOUN
cana-3491	390	27	999983	999983	NUM
cana-3491	390	28	)	)	PUNCT
cana-3491	390	29	and	and	CCONJ
cana-3491	390	30	�	�	PROPN
cana-3491	390	31	̃	̃	PROPN
cana-3491	390	32	�	�	NOUN
cana-3491	390	33	21	21	NUM
cana-3491	390	34	=	=	SYM
cana-3491	390	35	(	(	PUNCT
cana-3491	390	36	540108	540108	NUM
cana-3491	390	37	+	+	SYM
cana-3491	390	38	4976	4976	NUM
cana-3491	390	39	×	×	NOUN
cana-3491	390	40	999983	999983	NUM
cana-3491	390	41	,	,	PUNCT
cana-3491	390	42	254286	254286	NUM
cana-3491	390	43	+	+	NUM
cana-3491	390	44	183355	183355	NUM
cana-3491	390	45	×	×	NOUN
cana-3491	390	46	999983	999983	NUM
cana-3491	390	47	)	)	PUNCT
cana-3491	390	48	with	with	ADP
cana-3491	390	49	slope	slope	NOUN
cana-3491	390	50	of	of	ADP
cana-3491	390	51	𝑃1	𝑃1	NOUN
cana-3491	390	52	and	and	CCONJ
cana-3491	390	53	𝑃2	𝑃2	NOUN
cana-3491	390	54	given	give	VERB
cana-3491	390	55	as	as	ADP
cana-3491	390	56	�	�	PROPN
cana-3491	390	57	̃	̃	NOUN
cana-3491	390	58	�	�	NOUN
cana-3491	390	59	′	′	NUM
cana-3491	390	60	=	=	SYM
cana-3491	390	61	383473	383473	NUM
cana-3491	390	62	+	+	SYM
cana-3491	390	63	214267	214267	NUM
cana-3491	390	64	×	×	NOUN
cana-3491	390	65	999983	999983	NUM
cana-3491	390	66	and	and	CCONJ
cana-3491	390	67	�	�	PROPN
cana-3491	390	68	̃	̃	PROPN
cana-3491	390	69	�	�	PROPN
cana-3491	390	70	11	11	NUM
cana-3491	390	71	+	+	SYM
cana-3491	390	72	�	�	PROPN
cana-3491	390	73	̃	̃	PROPN
cana-3491	390	74	�	�	NOUN
cana-3491	390	75	21	21	NUM
cana-3491	390	76	=	=	SYM
cana-3491	390	77	�	�	PROPN
cana-3491	390	78	̃	̃	PROPN
cana-3491	390	79	�	�	PROPN
cana-3491	390	80	31	31	NUM
cana-3491	390	81	is	be	AUX
cana-3491	390	82	given	give	VERB
cana-3491	390	83	as	as	ADP
cana-3491	390	84	�	�	PROPN
cana-3491	390	85	̃	̃	PROPN
cana-3491	390	86	�	�	PROPN
cana-3491	390	87	31	31	NUM
cana-3491	390	88	=	=	SYM
cana-3491	390	89	(	(	PUNCT
cana-3491	390	90	130341	130341	NUM
cana-3491	390	91	+	+	SYM
cana-3491	390	92	144599×	144599×	NUM
cana-3491	390	93	999983	999983	NUM
cana-3491	390	94	,	,	PUNCT
cana-3491	390	95	997817	997817	NUM
cana-3491	390	96	+	+	CCONJ
cana-3491	390	97	451277	451277	NUM
cana-3491	390	98	×	×	NOUN
cana-3491	390	99	999983	999983	NUM
cana-3491	390	100	)	)	PUNCT
cana-3491	390	101	which	which	PRON
cana-3491	390	102	is	be	AUX
cana-3491	390	103	a	a	DET
cana-3491	390	104	lift	lift	NOUN
cana-3491	390	105	of	of	ADP
cana-3491	390	106	a	a	DET
cana-3491	390	107	point	point	NOUN
cana-3491	390	108	(	(	PUNCT
cana-3491	390	109	130341,997817	130341,997817	X
cana-3491	390	110	)	)	PUNCT
cana-3491	390	111	∈	∈	PROPN
cana-3491	390	112	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	390	113	)	)	PUNCT
cana-3491	390	114	.	.	PUNCT
cana-3491	391	1	the	the	DET
cana-3491	391	2	code	code	NOUN
cana-3491	391	3	for	for	ADP
cana-3491	391	4	addition	addition	NOUN
cana-3491	391	5	,	,	PUNCT
cana-3491	391	6	subtraction	subtraction	NOUN
cana-3491	391	7	,	,	PUNCT
cana-3491	391	8	multiplication	multiplication	NOUN
cana-3491	391	9	and	and	CCONJ
cana-3491	391	10	division	division	NOUN
cana-3491	391	11	of	of	ADP
cana-3491	391	12	𝑝-adic	𝑝-adic	ADJ
cana-3491	391	13	numbers	number	NOUN
cana-3491	391	14	and	and	CCONJ
cana-3491	391	15	finding	find	VERB
cana-3491	391	16	slope	slope	NOUN
cana-3491	391	17	points	point	NOUN
cana-3491	391	18	in	in	ADP
cana-3491	391	19	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	391	20	𝑝2	𝑝2	NOUN
cana-3491	391	21	)	)	PUNCT
cana-3491	391	22	and	and	CCONJ
cana-3491	391	23	arithmetic	arithmetic	NOUN
cana-3491	391	24	of	of	ADP
cana-3491	391	25	points	point	NOUN
cana-3491	391	26	in	in	ADP
cana-3491	391	27	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	391	28	𝑝2	𝑝2	NOUN
cana-3491	391	29	)	)	PUNCT
cana-3491	391	30	were	be	AUX
cana-3491	391	31	given	give	VERB
cana-3491	391	32	below	below	ADV
cana-3491	391	33	in	in	ADP
cana-3491	391	34	python	python	PROPN
cana-3491	391	35	.	.	PUNCT
cana-3491	392	1	communications	communication	NOUN
cana-3491	392	2	on	on	ADP
cana-3491	392	3	applied	apply	VERB
cana-3491	392	4	nonlinear	nonlinear	ADJ
cana-3491	392	5	analysis	analysis	NOUN
cana-3491	392	6	issn	issn	NOUN
cana-3491	392	7	:	:	PUNCT
cana-3491	392	8	1074	1074	NUM
cana-3491	392	9	-	-	PUNCT
cana-3491	392	10	133x	133x	NUM
cana-3491	392	11	vol	vol	NOUN
cana-3491	392	12	32	32	NUM
cana-3491	392	13	no	no	NOUN
cana-3491	392	14	.	.	PUNCT
cana-3491	393	1	7s	7	NOUN
cana-3491	393	2	(	(	PUNCT
cana-3491	393	3	2025	2025	NUM
cana-3491	393	4	)	)	PUNCT
cana-3491	393	5	875	875	NUM
cana-3491	393	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	393	7	#	#	NOUN
cana-3491	393	8	to	to	PART
cana-3491	393	9	initialize	initialize	VERB
cana-3491	393	10	run	run	VERB
cana-3491	393	11	this	this	DET
cana-3491	393	12	cell	cell	NOUN
cana-3491	393	13	from	from	ADP
cana-3491	393	14	sage.all	sage.all	DET
cana-3491	393	15	import	import	NOUN
cana-3491	393	16	*	*	PUNCT
cana-3491	393	17	#	#	NOUN
cana-3491	393	18	make	make	VERB
cana-3491	393	19	same	same	ADJ
cana-3491	393	20	length	length	NOUN
cana-3491	393	21	def	def	PROPN
cana-3491	393	22	pad_with_zeros(expansion	pad_with_zeros(expansion	NOUN
cana-3491	393	23	,	,	PUNCT
cana-3491	393	24	length	length	NOUN
cana-3491	393	25	):	):	PUNCT
cana-3491	393	26	return	return	NOUN
cana-3491	393	27	expansion	expansion	NOUN
cana-3491	393	28	+	+	CCONJ
cana-3491	394	1	[	[	X
cana-3491	394	2	0	0	X
cana-3491	394	3	]	]	X
cana-3491	394	4	*	*	PUNCT
cana-3491	394	5	(	(	PUNCT
cana-3491	394	6	length	length	NOUN
cana-3491	394	7	len(expansion	len(expansion	NOUN
cana-3491	394	8	)	)	PUNCT
cana-3491	394	9	)	)	PUNCT
cana-3491	394	10	#	#	NOUN
cana-3491	394	11	function	function	NOUN
cana-3491	394	12	to	to	PART
cana-3491	394	13	add	add	VERB
cana-3491	394	14	def	def	ADJ
cana-3491	394	15	add_p_adic(expansion1	add_p_adic(expansion1	NOUN
cana-3491	394	16	,	,	PUNCT
cana-3491	394	17	expansion2	expansion2	PROPN
cana-3491	394	18	,	,	PUNCT
cana-3491	394	19	p	p	X
cana-3491	394	20	):	):	PUNCT
cana-3491	394	21	#	#	PUNCT
cana-3491	394	22	ensure	ensure	VERB
cana-3491	394	23	both	both	DET
cana-3491	394	24	expansions	expansion	NOUN
cana-3491	394	25	are	be	AUX
cana-3491	394	26	of	of	ADP
cana-3491	394	27	same	same	ADJ
cana-3491	394	28	lengh	lengh	PROPN
cana-3491	394	29	max_length	max_length	NOUN
cana-3491	394	30	=	=	SYM
cana-3491	394	31	max(len(expansion1	max(len(expansion1	ADJ
cana-3491	394	32	)	)	PUNCT
cana-3491	394	33	,	,	PUNCT
cana-3491	394	34	len(expansion2	len(expansion2	NOUN
cana-3491	394	35	)	)	PUNCT
cana-3491	394	36	)	)	PUNCT
cana-3491	394	37	expansion1	expansion1	NOUN
cana-3491	395	1	=	=	SYM
cana-3491	395	2	pad_with_zeros(expansion1	pad_with_zeros(expansion1	ADJ
cana-3491	395	3	,	,	PUNCT
cana-3491	395	4	max_length	max_length	NOUN
cana-3491	395	5	)	)	PUNCT
cana-3491	395	6	expansion2	expansion2	NOUN
cana-3491	396	1	=	=	SYM
cana-3491	396	2	pad_with_zeros(expansion2	pad_with_zeros(expansion2	NOUN
cana-3491	396	3	,	,	PUNCT
cana-3491	396	4	max_length	max_length	NOUN
cana-3491	396	5	)	)	PUNCT
cana-3491	396	6	result	result	NOUN
cana-3491	397	1	=	=	X
cana-3491	398	1	[	[	X
cana-3491	398	2	]	]	X
cana-3491	398	3	carry	carry	VERB
cana-3491	398	4	=	=	SYM
cana-3491	398	5	0	0	NUM
cana-3491	398	6	for	for	ADP
cana-3491	398	7	digit1	digit1	NOUN
cana-3491	398	8	,	,	PUNCT
cana-3491	398	9	digit2	digit2	NOUN
cana-3491	398	10	in	in	ADP
cana-3491	398	11	zip(expansion1	zip(expansion1	PROPN
cana-3491	398	12	,	,	PUNCT
cana-3491	398	13	expansion2	expansion2	PROPN
cana-3491	398	14	):	):	PUNCT
cana-3491	398	15	total	total	ADJ
cana-3491	398	16	=	=	PUNCT
cana-3491	398	17	digit1	digit1	NOUN
cana-3491	399	1	+	+	CCONJ
cana-3491	399	2	digit2	digit2	NOUN
cana-3491	400	1	+	+	CCONJ
cana-3491	400	2	carry	carry	VERB
cana-3491	400	3	result.append(total	result.append(total	ADJ
cana-3491	400	4	%	%	NOUN
cana-3491	400	5	p	p	NOUN
cana-3491	400	6	)	)	PUNCT
cana-3491	400	7	carry	carry	NOUN
cana-3491	400	8	=	=	PUNCT
cana-3491	400	9	total	total	NOUN
cana-3491	400	10	//	//	NOUN
cana-3491	400	11	p	p	NOUN
cana-3491	400	12	if	if	SCONJ
cana-3491	400	13	carry	carry	VERB
cana-3491	400	14	>	>	X
cana-3491	400	15	0	0	NUM
cana-3491	400	16	:	:	PUNCT
cana-3491	400	17	result.append(carry	result.append(carry	NOUN
cana-3491	400	18	)	)	PUNCT
cana-3491	400	19	return	return	NOUN
cana-3491	400	20	result	result	NOUN
cana-3491	400	21	#	#	NOUN
cana-3491	400	22	function	function	NOUN
cana-3491	400	23	to	to	PART
cana-3491	400	24	subtract	subtract	VERB
cana-3491	400	25	def	def	PROPN
cana-3491	400	26	subtract_p_adic(expansion1	subtract_p_adic(expansion1	PROPN
cana-3491	400	27	,	,	PUNCT
cana-3491	400	28	expansion2	expansion2	PROPN
cana-3491	400	29	,	,	PUNCT
cana-3491	400	30	p	p	X
cana-3491	400	31	):	):	PUNCT
cana-3491	400	32	#	#	NOUN
cana-3491	400	33	ensure	ensure	VERB
cana-3491	400	34	both	both	DET
cana-3491	400	35	expansions	expansion	NOUN
cana-3491	400	36	are	be	AUX
cana-3491	400	37	of	of	ADP
cana-3491	400	38	same	same	ADJ
cana-3491	400	39	length	length	NOUN
cana-3491	400	40	max_length	max_length	NOUN
cana-3491	400	41	=	=	SYM
cana-3491	400	42	max(len(expansion1	max(len(expansion1	ADJ
cana-3491	400	43	)	)	PUNCT
cana-3491	400	44	,	,	PUNCT
cana-3491	400	45	len(expansion2	len(expansion2	NOUN
cana-3491	400	46	)	)	PUNCT
cana-3491	400	47	)	)	PUNCT
cana-3491	400	48	expansion1	expansion1	NOUN
cana-3491	401	1	=	=	SYM
cana-3491	401	2	pad_with_zeros(expansion1	pad_with_zeros(expansion1	ADJ
cana-3491	401	3	,	,	PUNCT
cana-3491	401	4	max_length	max_length	NOUN
cana-3491	401	5	)	)	PUNCT
cana-3491	401	6	expansion2	expansion2	NOUN
cana-3491	402	1	=	=	SYM
cana-3491	402	2	pad_with_zeros(expansion2	pad_with_zeros(expansion2	NOUN
cana-3491	402	3	,	,	PUNCT
cana-3491	402	4	max_length	max_length	NOUN
cana-3491	402	5	)	)	PUNCT
cana-3491	402	6	result	result	NOUN
cana-3491	403	1	=	=	X
cana-3491	404	1	[	[	X
cana-3491	404	2	]	]	X
cana-3491	404	3	borrow	borrow	VERB
cana-3491	404	4	=	=	NOUN
cana-3491	404	5	0	0	NUM
cana-3491	404	6	for	for	ADP
cana-3491	404	7	digit1	digit1	NOUN
cana-3491	404	8	,	,	PUNCT
cana-3491	404	9	digit2	digit2	NOUN
cana-3491	404	10	in	in	ADP
cana-3491	404	11	zip(expansion1	zip(expansion1	PROPN
cana-3491	404	12	,	,	PUNCT
cana-3491	404	13	expansion2	expansion2	PROPN
cana-3491	404	14	):	):	PUNCT
cana-3491	404	15	total	total	ADJ
cana-3491	404	16	=	=	PUNCT
cana-3491	404	17	digit1	digit1	NOUN
cana-3491	404	18	digit2	digit2	NOUN
cana-3491	404	19	borrow	borrow	VERB
cana-3491	404	20	if	if	SCONJ
cana-3491	404	21	total	total	ADJ
cana-3491	404	22	<	<	X
cana-3491	404	23	0	0	NUM
cana-3491	404	24	:	:	PUNCT
cana-3491	404	25	total	total	NOUN
cana-3491	404	26	+	+	NOUN
cana-3491	404	27	=	=	X
cana-3491	404	28	p	p	ADJ
cana-3491	404	29	communications	communication	NOUN
cana-3491	404	30	on	on	ADP
cana-3491	404	31	applied	apply	VERB
cana-3491	404	32	nonlinear	nonlinear	ADJ
cana-3491	404	33	analysis	analysis	NOUN
cana-3491	404	34	issn	issn	NOUN
cana-3491	404	35	:	:	PUNCT
cana-3491	404	36	1074	1074	NUM
cana-3491	404	37	-	-	PUNCT
cana-3491	404	38	133x	133x	NUM
cana-3491	404	39	vol	vol	NOUN
cana-3491	404	40	32	32	NUM
cana-3491	404	41	no	no	NOUN
cana-3491	404	42	.	.	PUNCT
cana-3491	405	1	7s	7	NOUN
cana-3491	405	2	(	(	PUNCT
cana-3491	405	3	2025	2025	NUM
cana-3491	405	4	)	)	PUNCT
cana-3491	405	5	876	876	NUM
cana-3491	405	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	405	7	borrow	borrow	VERB
cana-3491	405	8	=	=	NOUN
cana-3491	405	9	1	1	NUM
cana-3491	405	10	else	else	ADV
cana-3491	405	11	:	:	PUNCT
cana-3491	405	12	borrow	borrow	VERB
cana-3491	405	13	=	=	NOUN
cana-3491	405	14	0	0	NUM
cana-3491	405	15	result.append(total	result.append(total	ADJ
cana-3491	405	16	)	)	PUNCT
cana-3491	405	17	#	#	NOUN
cana-3491	405	18	remove	remove	VERB
cana-3491	405	19	trailing	trail	VERB
cana-3491	405	20	zeros	zero	NOUN
cana-3491	405	21	while	while	SCONJ
cana-3491	405	22	result	result	NOUN
cana-3491	405	23	and	and	CCONJ
cana-3491	405	24	result[-1	result[-1	PRON
cana-3491	405	25	]	]	X
cana-3491	406	1	=	=	X
cana-3491	406	2	=	=	SYM
cana-3491	406	3	0	0	NUM
cana-3491	406	4	:	:	PUNCT
cana-3491	406	5	result.pop	result.pop	NOUN
cana-3491	406	6	(	(	PUNCT
cana-3491	406	7	)	)	PUNCT
cana-3491	406	8	return	return	NOUN
cana-3491	406	9	result	result	NOUN
cana-3491	406	10	#	#	NOUN
cana-3491	406	11	function	function	NOUN
cana-3491	406	12	to	to	PART
cana-3491	406	13	multiply	multiply	VERB
cana-3491	406	14	def	def	ADJ
cana-3491	406	15	multiply_p_adic(expansion1	multiply_p_adic(expansion1	ADJ
cana-3491	406	16	,	,	PUNCT
cana-3491	406	17	expansion2	expansion2	PROPN
cana-3491	406	18	,	,	PUNCT
cana-3491	406	19	p	p	X
cana-3491	406	20	):	):	PUNCT
cana-3491	406	21	#	#	PUNCT
cana-3491	406	22	ensure	ensure	VERB
cana-3491	406	23	both	both	DET
cana-3491	406	24	expansions	expansion	NOUN
cana-3491	406	25	are	be	AUX
cana-3491	406	26	of	of	ADP
cana-3491	406	27	same	same	ADJ
cana-3491	406	28	length	length	NOUN
cana-3491	406	29	max_length	max_length	NOUN
cana-3491	406	30	=	=	SYM
cana-3491	406	31	max(len(expansion1	max(len(expansion1	ADJ
cana-3491	406	32	)	)	PUNCT
cana-3491	406	33	,	,	PUNCT
cana-3491	406	34	len(expansion2	len(expansion2	NOUN
cana-3491	406	35	)	)	PUNCT
cana-3491	406	36	)	)	PUNCT
cana-3491	406	37	expansion1	expansion1	NOUN
cana-3491	407	1	=	=	SYM
cana-3491	407	2	pad_with_zeros(expansion1	pad_with_zeros(expansion1	ADJ
cana-3491	407	3	,	,	PUNCT
cana-3491	407	4	max_length	max_length	NOUN
cana-3491	407	5	)	)	PUNCT
cana-3491	407	6	expansion2	expansion2	NOUN
cana-3491	408	1	=	=	SYM
cana-3491	408	2	pad_with_zeros(expansion2	pad_with_zeros(expansion2	NOUN
cana-3491	408	3	,	,	PUNCT
cana-3491	408	4	max_length	max_length	NOUN
cana-3491	408	5	)	)	PUNCT
cana-3491	408	6	result	result	NOUN
cana-3491	408	7	=	=	PUNCT
cana-3491	409	1	[	[	X
cana-3491	409	2	0	0	X
cana-3491	409	3	]	]	X
cana-3491	409	4	*	*	X
cana-3491	409	5	(	(	PUNCT
cana-3491	409	6	2	2	NUM
cana-3491	409	7	*	*	NOUN
cana-3491	409	8	max_length	max_length	NOUN
cana-3491	409	9	)	)	PUNCT
cana-3491	409	10	for	for	ADP
cana-3491	409	11	i	i	PRON
cana-3491	409	12	in	in	ADP
cana-3491	409	13	range(max_length	range(max_length	NOUN
cana-3491	409	14	):	):	PUNCT
cana-3491	409	15	carry	carry	NOUN
cana-3491	409	16	=	=	SYM
cana-3491	409	17	0	0	NUM
cana-3491	409	18	for	for	ADP
cana-3491	409	19	j	j	PROPN
cana-3491	409	20	in	in	ADP
cana-3491	409	21	range(max_length	range(max_length	NOUN
cana-3491	409	22	):	):	PUNCT
cana-3491	409	23	total	total	ADJ
cana-3491	409	24	=	=	SYM
cana-3491	409	25	expansion1[i	expansion1[i	NOUN
cana-3491	409	26	]	]	X
cana-3491	409	27	*	*	PUNCT
cana-3491	410	1	expansion2[j	expansion2[j	PROPN
cana-3491	410	2	]	]	X
cana-3491	410	3	+	+	PUNCT
cana-3491	410	4	result[i+j	result[i+j	NUM
cana-3491	410	5	]	]	PUNCT
cana-3491	410	6	+	+	CCONJ
cana-3491	410	7	carry	carry	VERB
cana-3491	410	8	result[i	result[i	ADJ
cana-3491	410	9	+	+	CCONJ
cana-3491	410	10	j	j	X
cana-3491	410	11	]	]	X
cana-3491	410	12	=	=	PUNCT
cana-3491	410	13	total	total	ADJ
cana-3491	410	14	%	%	NOUN
cana-3491	410	15	p	p	NOUN
cana-3491	410	16	carry	carry	NOUN
cana-3491	410	17	=	=	SYM
cana-3491	410	18	total	total	ADJ
cana-3491	410	19	//	//	NOUN
cana-3491	410	20	p	p	NOUN
cana-3491	410	21	result[i	result[i	VERB
cana-3491	410	22	+	+	SYM
cana-3491	410	23	max_length	max_length	NOUN
cana-3491	410	24	]	]	PUNCT
cana-3491	411	1	+	+	PUNCT
cana-3491	411	2	=	=	SYM
cana-3491	411	3	carry	carry	VERB
cana-3491	411	4	#	#	NOUN
cana-3491	411	5	removing	remove	VERB
cana-3491	411	6	trailing	trail	VERB
cana-3491	411	7	zeros	zero	NOUN
cana-3491	411	8	while	while	SCONJ
cana-3491	411	9	len(result	len(result	PROPN
cana-3491	411	10	)	)	PUNCT
cana-3491	411	11	>	>	X
cana-3491	411	12	1	1	NUM
cana-3491	411	13	and	and	CCONJ
cana-3491	411	14	result[-1	result[-1	NOUN
cana-3491	411	15	]	]	X
cana-3491	412	1	=	=	X
cana-3491	412	2	=	=	SYM
cana-3491	412	3	0	0	NUM
cana-3491	412	4	:	:	PUNCT
cana-3491	412	5	result.pop	result.pop	NOUN
cana-3491	412	6	(	(	PUNCT
cana-3491	412	7	)	)	PUNCT
cana-3491	412	8	return	return	NOUN
cana-3491	412	9	result	result	NOUN
cana-3491	412	10	def	def	PROPN
cana-3491	412	11	divide_p_adic(expansion1	divide_p_adic(expansion1	PROPN
cana-3491	412	12	,	,	PUNCT
cana-3491	412	13	expansion2	expansion2	PROPN
cana-3491	412	14	,	,	PUNCT
cana-3491	412	15	p	p	X
cana-3491	412	16	):	):	PUNCT
cana-3491	412	17	#	#	PUNCT
cana-3491	412	18	ensure	ensure	VERB
cana-3491	412	19	both	both	DET
cana-3491	412	20	expansions	expansion	NOUN
cana-3491	412	21	are	be	AUX
cana-3491	412	22	of	of	ADP
cana-3491	412	23	same	same	ADJ
cana-3491	412	24	length	length	NOUN
cana-3491	412	25	max_length	max_length	NOUN
cana-3491	412	26	=	=	SYM
cana-3491	412	27	max(len(expansion1	max(len(expansion1	ADJ
cana-3491	412	28	)	)	PUNCT
cana-3491	412	29	,	,	PUNCT
cana-3491	412	30	len(expansion2	len(expansion2	NOUN
cana-3491	412	31	)	)	PUNCT
cana-3491	412	32	)	)	PUNCT
cana-3491	412	33	expansion1	expansion1	NOUN
cana-3491	413	1	=	=	SYM
cana-3491	413	2	pad_with_zeros(expansion1	pad_with_zeros(expansion1	ADJ
cana-3491	413	3	,	,	PUNCT
cana-3491	413	4	max_length	max_length	NOUN
cana-3491	413	5	)	)	PUNCT
cana-3491	413	6	expansion2	expansion2	NOUN
cana-3491	414	1	=	=	SYM
cana-3491	414	2	pad_with_zeros(expansion2	pad_with_zeros(expansion2	NOUN
cana-3491	414	3	,	,	PUNCT
cana-3491	414	4	max_length	max_length	NOUN
cana-3491	414	5	)	)	PUNCT
cana-3491	414	6	communications	communication	NOUN
cana-3491	414	7	on	on	ADP
cana-3491	414	8	applied	apply	VERB
cana-3491	414	9	nonlinear	nonlinear	ADJ
cana-3491	414	10	analysis	analysis	NOUN
cana-3491	414	11	issn	issn	NOUN
cana-3491	414	12	:	:	PUNCT
cana-3491	414	13	1074	1074	NUM
cana-3491	414	14	-	-	PUNCT
cana-3491	414	15	133x	133x	NUM
cana-3491	414	16	vol	vol	NOUN
cana-3491	414	17	32	32	NUM
cana-3491	414	18	no	no	NOUN
cana-3491	414	19	.	.	PUNCT
cana-3491	415	1	7s	7	NOUN
cana-3491	415	2	(	(	PUNCT
cana-3491	415	3	2025	2025	NUM
cana-3491	415	4	)	)	PUNCT
cana-3491	415	5	877	877	NUM
cana-3491	415	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	415	7	k	k	X
cana-3491	415	8	=	=	SYM
cana-3491	415	9	padicfield(p	padicfield(p	PROPN
cana-3491	415	10	,	,	PUNCT
cana-3491	415	11	max_length	max_length	NOUN
cana-3491	415	12	)	)	PUNCT
cana-3491	415	13	padic_number_a	padic_number_a	VERB
cana-3491	416	1	=	=	PUNCT
cana-3491	416	2	sum(c	sum(c	PROPN
cana-3491	416	3	*	*	PUNCT
cana-3491	416	4	k(p**i	k(p**i	NOUN
cana-3491	416	5	)	)	PUNCT
cana-3491	416	6	for	for	ADP
cana-3491	416	7	i	i	PRON
cana-3491	416	8	,	,	PUNCT
cana-3491	416	9	c	c	PROPN
cana-3491	416	10	in	in	ADP
cana-3491	416	11	enumerate(expansion1	enumerate(expansion1	NOUN
cana-3491	416	12	)	)	PUNCT
cana-3491	416	13	)	)	PUNCT
cana-3491	417	1	padic_number_b	padic_number_b	NOUN
cana-3491	418	1	=	=	PUNCT
cana-3491	418	2	sum(c	sum(c	PROPN
cana-3491	418	3	*	*	PUNCT
cana-3491	418	4	k(p**i	k(p**i	NOUN
cana-3491	418	5	)	)	PUNCT
cana-3491	418	6	for	for	ADP
cana-3491	418	7	i	i	PRON
cana-3491	418	8	,	,	PUNCT
cana-3491	418	9	c	c	PROPN
cana-3491	418	10	in	in	ADP
cana-3491	418	11	enumerate(expansion2	enumerate(expansion2	NOUN
cana-3491	418	12	)	)	PUNCT
cana-3491	418	13	)	)	PUNCT
cana-3491	419	1	result	result	NOUN
cana-3491	419	2	=	=	SYM
cana-3491	419	3	padic_number_a	padic_number_a	ADJ
cana-3491	419	4	/	/	SYM
cana-3491	419	5	padic_number_b	padic_number_b	ADJ
cana-3491	419	6	expansion	expansion	NOUN
cana-3491	419	7	=	=	SYM
cana-3491	419	8	result.expansion	result.expansion	PROPN
cana-3491	419	9	(	(	PUNCT
cana-3491	419	10	)	)	PUNCT
cana-3491	419	11	coefficients	coefficient	NOUN
cana-3491	419	12	=	=	PUNCT
cana-3491	420	1	[	[	X
cana-3491	420	2	int(coef	int(coef	X
cana-3491	420	3	)	)	PUNCT
cana-3491	420	4	for	for	ADP
cana-3491	420	5	coef	coef	PROPN
cana-3491	420	6	in	in	ADP
cana-3491	420	7	expansion	expansion	NOUN
cana-3491	420	8	]	]	PUNCT
cana-3491	420	9	return	return	NOUN
cana-3491	420	10	coefficients	coefficient	NOUN
cana-3491	420	11	def	def	PROPN
cana-3491	420	12	to_p_adic(n	to_p_adic(n	PROPN
cana-3491	420	13	,	,	PUNCT
cana-3491	420	14	p	p	NOUN
cana-3491	420	15	,	,	PUNCT
cana-3491	420	16	precision	precision	NOUN
cana-3491	420	17	):	):	PUNCT
cana-3491	420	18	if	if	SCONJ
cana-3491	420	19	p<=1	p<=1	PROPN
cana-3491	420	20	:	:	PUNCT
cana-3491	420	21	raise	raise	VERB
cana-3491	420	22	valueerror("base	valueerror("base	NOUN
cana-3491	420	23	p	p	NOUN
cana-3491	420	24	must	must	AUX
cana-3491	420	25	be	be	AUX
cana-3491	420	26	a	a	DET
cana-3491	420	27	prime	prime	ADJ
cana-3491	420	28	number	number	NOUN
cana-3491	420	29	greater	great	ADJ
cana-3491	420	30	then	then	ADP
cana-3491	420	31	1	1	NUM
cana-3491	420	32	.	.	PUNCT
cana-3491	420	33	"	"	PUNCT
cana-3491	420	34	)	)	PUNCT
cana-3491	420	35	if	if	SCONJ
cana-3491	420	36	n	n	PRON
cana-3491	420	37	=	=	NOUN
cana-3491	420	38	=	=	SYM
cana-3491	420	39	0	0	NUM
cana-3491	420	40	:	:	PUNCT
cana-3491	420	41	return	return	VERB
cana-3491	420	42	[	[	X
cana-3491	420	43	0	0	NUM
cana-3491	420	44	]	]	PUNCT
cana-3491	420	45	digits	digit	NOUN
cana-3491	420	46	=	=	PUNCT
cana-3491	421	1	[	[	X
cana-3491	421	2	]	]	X
cana-3491	421	3	while	while	SCONJ
cana-3491	421	4	n	n	PRON
cana-3491	421	5	!	!	PUNCT
cana-3491	421	6	=	=	SYM
cana-3491	422	1	0	0	NUM
cana-3491	422	2	:	:	PUNCT
cana-3491	422	3	digits.append(n	digits.append(n	NOUN
cana-3491	422	4	%	%	NOUN
cana-3491	422	5	p	p	NOUN
cana-3491	422	6	)	)	PUNCT
cana-3491	422	7	n	n	NOUN
cana-3491	422	8	//=	//=	NUM
cana-3491	423	1	p	p	NOUN
cana-3491	423	2	return	return	NOUN
cana-3491	423	3	digits[0	digits[0	ADJ
cana-3491	423	4	:	:	PUNCT
cana-3491	423	5	precision	precision	NOUN
cana-3491	423	6	]	]	PUNCT
cana-3491	423	7	def	def	PROPN
cana-3491	423	8	print_expansion(value	print_expansion(value	PROPN
cana-3491	423	9	,	,	PUNCT
cana-3491	423	10	base	base	NOUN
cana-3491	423	11	,	,	PUNCT
cana-3491	423	12	coefficients	coefficient	NOUN
cana-3491	423	13	,	,	PUNCT
cana-3491	423	14	precision	precision	NOUN
cana-3491	423	15	):	):	PUNCT
cana-3491	423	16	print(f"the	print(f"the	DET
cana-3491	423	17	{	{	PUNCT
cana-3491	423	18	base}-adic	base}-adic	ADJ
cana-3491	423	19	expansion	expansion	NOUN
cana-3491	423	20	of	of	ADP
cana-3491	423	21	{	{	PUNCT
cana-3491	423	22	value	value	NOUN
cana-3491	423	23	}	}	PUNCT
cana-3491	423	24	is	be	AUX
cana-3491	423	25	:	:	PUNCT
cana-3491	423	26	"	"	PUNCT
cana-3491	423	27	,	,	PUNCT
cana-3491	423	28	end=	end=	PROPN
cana-3491	423	29	'	'	PART
cana-3491	423	30	'	'	PUNCT
cana-3491	423	31	)	)	PUNCT
cana-3491	423	32	for	for	ADP
cana-3491	423	33	i	i	PRON
cana-3491	423	34	in	in	ADP
cana-3491	423	35	range(min(precision	range(min(precision	NOUN
cana-3491	423	36	,	,	PUNCT
cana-3491	423	37	len(coefficients	len(coefficient	NOUN
cana-3491	423	38	)	)	PUNCT
cana-3491	423	39	)	)	PUNCT
cana-3491	423	40	):	):	PUNCT
cana-3491	424	1	print(f"{coefficients[i]}*{base}^{i	print(f"{coefficients[i]}*{base}^{i	VERB
cana-3491	424	2	}	}	PUNCT
cana-3491	424	3	"	"	PUNCT
cana-3491	424	4	,	,	PUNCT
cana-3491	424	5	end=	end=	PROPN
cana-3491	424	6	'	'	PART
cana-3491	424	7	'	'	PUNCT
cana-3491	424	8	)	)	PUNCT
cana-3491	424	9	if	if	SCONJ
cana-3491	424	10	(	(	PUNCT
cana-3491	424	11	i	i	PRON
cana-3491	424	12	<	<	X
cana-3491	424	13	min(precision	min(precision	NOUN
cana-3491	424	14	,	,	PUNCT
cana-3491	424	15	len(coefficients	len(coefficient	NOUN
cana-3491	424	16	)	)	PUNCT
cana-3491	424	17	)	)	PUNCT
cana-3491	424	18	1	1	NUM
cana-3491	424	19	):	):	PUNCT
cana-3491	424	20	print(f"+	print(f"+	X
cana-3491	424	21	"	"	PUNCT
cana-3491	424	22	,	,	PUNCT
cana-3491	424	23	end=	end=	PROPN
cana-3491	424	24	'	'	PART
cana-3491	424	25	'	'	PUNCT
cana-3491	424	26	)	)	PUNCT
cana-3491	424	27	else	else	ADV
cana-3491	424	28	:	:	PUNCT
cana-3491	424	29	print	print	NOUN
cana-3491	424	30	(	(	PUNCT
cana-3491	424	31	"	"	PUNCT
cana-3491	424	32	"	"	PUNCT
cana-3491	424	33	)	)	PUNCT
cana-3491	424	34	def	def	PROPN
cana-3491	424	35	calculate_p3(x1	calculate_p3(x1	PROPN
cana-3491	424	36	,	,	PUNCT
cana-3491	424	37	x2	x2	PROPN
cana-3491	424	38	,	,	PUNCT
cana-3491	424	39	y1	y1	NOUN
cana-3491	424	40	,	,	PUNCT
cana-3491	424	41	y2	y2	PROPN
cana-3491	424	42	,	,	PUNCT
cana-3491	424	43	p	p	X
cana-3491	424	44	,	,	PUNCT
cana-3491	424	45	a	a	DET
cana-3491	424	46	,	,	PUNCT
cana-3491	424	47	b	b	NOUN
cana-3491	424	48	):	):	PUNCT
cana-3491	424	49	max_length	max_length	NOUN
cana-3491	424	50	=	=	SYM
cana-3491	424	51	max(len(x1	max(len(x1	PROPN
cana-3491	424	52	)	)	PUNCT
cana-3491	424	53	,	,	PUNCT
cana-3491	424	54	len(x2	len(x2	PROPN
cana-3491	424	55	)	)	PUNCT
cana-3491	424	56	,	,	PUNCT
cana-3491	424	57	len(y1	len(y1	PROPN
cana-3491	424	58	)	)	PUNCT
cana-3491	424	59	,	,	PUNCT
cana-3491	424	60	len(y2	len(y2	NOUN
cana-3491	424	61	)	)	PUNCT
cana-3491	424	62	)	)	PUNCT
cana-3491	425	1	x1	x1	PROPN
cana-3491	425	2	=	=	PUNCT
cana-3491	426	1	pad_with_zeros(x1	pad_with_zeros(x1	ADJ
cana-3491	426	2	,	,	PUNCT
cana-3491	426	3	max_length	max_length	NOUN
cana-3491	426	4	)	)	PUNCT
cana-3491	426	5	x2	x2	NOUN
cana-3491	427	1	=	=	PUNCT
cana-3491	427	2	pad_with_zeros(x2	pad_with_zeros(x2	NOUN
cana-3491	427	3	,	,	PUNCT
cana-3491	427	4	max_length	max_length	NOUN
cana-3491	427	5	)	)	PUNCT
cana-3491	427	6	y1	y1	NOUN
cana-3491	427	7	=	=	SYM
cana-3491	427	8	pad_with_zeros(y1	pad_with_zeros(y1	NOUN
cana-3491	427	9	,	,	PUNCT
cana-3491	427	10	max_length	max_length	NOUN
cana-3491	427	11	)	)	PUNCT
cana-3491	427	12	communications	communication	NOUN
cana-3491	427	13	on	on	ADP
cana-3491	427	14	applied	apply	VERB
cana-3491	427	15	nonlinear	nonlinear	ADJ
cana-3491	427	16	analysis	analysis	NOUN
cana-3491	427	17	issn	issn	NOUN
cana-3491	427	18	:	:	PUNCT
cana-3491	427	19	1074	1074	NUM
cana-3491	427	20	-	-	PUNCT
cana-3491	427	21	133x	133x	NUM
cana-3491	427	22	vol	vol	NOUN
cana-3491	427	23	32	32	NUM
cana-3491	427	24	no	no	NOUN
cana-3491	427	25	.	.	PUNCT
cana-3491	428	1	7s	7	NOUN
cana-3491	428	2	(	(	PUNCT
cana-3491	428	3	2025	2025	NUM
cana-3491	428	4	)	)	PUNCT
cana-3491	428	5	878	878	NUM
cana-3491	428	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	428	7	y2	y2	NOUN
cana-3491	428	8	=	=	SYM
cana-3491	428	9	pad_with_zeros(y2	pad_with_zeros(y2	NOUN
cana-3491	428	10	,	,	PUNCT
cana-3491	428	11	max_length	max_length	NOUN
cana-3491	428	12	)	)	PUNCT
cana-3491	428	13	precision	precision	NOUN
cana-3491	428	14	=	=	SYM
cana-3491	428	15	max_length	max_length	NOUN
cana-3491	428	16	x10	x10	NOUN
cana-3491	428	17	=	=	SYM
cana-3491	428	18	x1[0	x1[0	PROPN
cana-3491	428	19	]	]	X
cana-3491	428	20	y10	y10	NOUN
cana-3491	429	1	=	=	PUNCT
cana-3491	429	2	y1[0	y1[0	ADJ
cana-3491	429	3	]	]	X
cana-3491	429	4	x11	x11	NOUN
cana-3491	429	5	=	=	PUNCT
cana-3491	429	6	x1[1	x1[1	X
cana-3491	429	7	]	]	X
cana-3491	429	8	y11	y11	NOUN
cana-3491	429	9	=	=	SYM
cana-3491	429	10	y1[1	y1[1	X
cana-3491	429	11	]	]	X
cana-3491	429	12	x20	x20	NOUN
cana-3491	430	1	=	=	SYM
cana-3491	430	2	x2[0	x2[0	PROPN
cana-3491	430	3	]	]	X
cana-3491	430	4	y20	y20	NOUN
cana-3491	430	5	=	=	SYM
cana-3491	430	6	y2[0	y2[0	PROPN
cana-3491	430	7	]	]	X
cana-3491	430	8	x21	x21	PROPN
cana-3491	430	9	=	=	SYM
cana-3491	431	1	x2[1	x2[1	PROPN
cana-3491	431	2	]	]	X
cana-3491	431	3	y21	y21	NOUN
cana-3491	431	4	=	=	SYM
cana-3491	431	5	y2[1	y2[1	PROPN
cana-3491	431	6	]	]	X
cana-3491	431	7	#	#	NOUN
cana-3491	431	8	cheking	cheke	VERB
cana-3491	431	9	p1	p1	PROPN
cana-3491	431	10	=	=	NOUN
cana-3491	431	11	p2	p2	PROPN
cana-3491	431	12	pointequality	pointequality	NOUN
cana-3491	431	13	=	=	PUNCT
cana-3491	431	14	x10	x10	NOUN
cana-3491	431	15	=	=	PROPN
cana-3491	431	16	=	=	SYM
cana-3491	431	17	x20	x20	PROPN
cana-3491	431	18	and	and	CCONJ
cana-3491	431	19	x11	x11	NOUN
cana-3491	432	1	=	=	NOUN
cana-3491	432	2	=	=	SYM
cana-3491	432	3	x21	x21	PROPN
cana-3491	432	4	and	and	CCONJ
cana-3491	432	5	y10	y10	NOUN
cana-3491	432	6	=	=	NOUN
cana-3491	432	7	=	=	NOUN
cana-3491	432	8	y20	y20	NOUN
cana-3491	432	9	and	and	CCONJ
cana-3491	432	10	y11	y11	NOUN
cana-3491	432	11	=	=	NOUN
cana-3491	433	1	=	=	NOUN
cana-3491	433	2	y21	y21	NOUN
cana-3491	433	3	if	if	SCONJ
cana-3491	433	4	pointequality	pointequality	NOUN
cana-3491	433	5	:	:	PUNCT
cana-3491	433	6	print("p1	print("p1	NOUN
cana-3491	433	7	=	=	NOUN
cana-3491	433	8	=	=	NOUN
cana-3491	433	9	p2	p2	NOUN
cana-3491	433	10	"	"	PUNCT
cana-3491	433	11	)	)	PUNCT
cana-3491	434	1	kx	kx	PROPN
cana-3491	434	2	=	=	SYM
cana-3491	434	3	padicfield(p	padicfield(p	PROPN
cana-3491	434	4	,	,	PUNCT
cana-3491	434	5	max_length	max_length	NOUN
cana-3491	434	6	)	)	PUNCT
cana-3491	434	7	x_for_solpe	x_for_solpe	X
cana-3491	435	1	=	=	PUNCT
cana-3491	436	1	sum(c	sum(c	PROPN
cana-3491	436	2	*	*	PUNCT
cana-3491	436	3	kx(p**i	kx(p**i	PROPN
cana-3491	436	4	)	)	PUNCT
cana-3491	436	5	for	for	ADP
cana-3491	436	6	i	i	PRON
cana-3491	436	7	,	,	PUNCT
cana-3491	436	8	c	c	PROPN
cana-3491	436	9	in	in	ADP
cana-3491	436	10	enumerate(x1	enumerate(x1	NOUN
cana-3491	436	11	)	)	PUNCT
cana-3491	436	12	)	)	PUNCT
cana-3491	436	13	y_for_solpe	y_for_solpe	X
cana-3491	437	1	=	=	PUNCT
cana-3491	437	2	sum(c	sum(c	PROPN
cana-3491	437	3	*	*	PUNCT
cana-3491	437	4	kx(p**i	kx(p**i	PROPN
cana-3491	437	5	)	)	PUNCT
cana-3491	437	6	for	for	ADP
cana-3491	437	7	i	i	PRON
cana-3491	437	8	,	,	PUNCT
cana-3491	437	9	c	c	PROPN
cana-3491	437	10	in	in	ADP
cana-3491	437	11	enumerate(y1	enumerate(y1	NOUN
cana-3491	437	12	)	)	PUNCT
cana-3491	437	13	)	)	PUNCT
cana-3491	437	14	m_out	m_out	NOUN
cana-3491	438	1	=	=	SYM
cana-3491	438	2	(	(	PUNCT
cana-3491	438	3	(	(	PUNCT
cana-3491	438	4	3	3	NUM
cana-3491	438	5	*	*	PUNCT
cana-3491	438	6	x_for_solpe^2	x_for_solpe^2	PROPN
cana-3491	438	7	)	)	PUNCT
cana-3491	439	1	+	+	CCONJ
cana-3491	439	2	a	a	X
cana-3491	439	3	)	)	PUNCT
cana-3491	439	4	/	/	SYM
cana-3491	439	5	(	(	PUNCT
cana-3491	439	6	2	2	NUM
cana-3491	439	7	*	*	SYM
cana-3491	439	8	y_for_solpe	y_for_solpe	NUM
cana-3491	439	9	)	)	PUNCT
cana-3491	440	1	m_exp	m_exp	PROPN
cana-3491	440	2	=	=	PROPN
cana-3491	440	3	m_out.expansion	m_out.expansion	NOUN
cana-3491	440	4	(	(	PUNCT
cana-3491	440	5	)	)	PUNCT
cana-3491	440	6	m	m	VERB
cana-3491	440	7	=	=	X
cana-3491	441	1	[	[	X
cana-3491	441	2	int(coef	int(coef	X
cana-3491	441	3	)	)	PUNCT
cana-3491	441	4	for	for	ADP
cana-3491	441	5	coef	coef	PROPN
cana-3491	441	6	in	in	ADP
cana-3491	441	7	m_exp	m_exp	NOUN
cana-3491	441	8	]	]	X
cana-3491	441	9	else	else	ADV
cana-3491	441	10	:	:	PUNCT
cana-3491	441	11	print("p1	print("p1	VERB
cana-3491	441	12	!	!	PUNCT
cana-3491	441	13	=	=	SYM
cana-3491	441	14	p2	p2	PROPN
cana-3491	441	15	"	"	PUNCT
cana-3491	441	16	)	)	PUNCT
cana-3491	441	17	m	m	PROPN
cana-3491	441	18	=	=	PUNCT
cana-3491	441	19	divide_p_adic(subtract_p_adic(y2	divide_p_adic(subtract_p_adic(y2	NOUN
cana-3491	441	20	,	,	PUNCT
cana-3491	441	21	y1	y1	X
cana-3491	441	22	,	,	PUNCT
cana-3491	441	23	p	p	NOUN
cana-3491	441	24	)	)	PUNCT
cana-3491	441	25	,	,	PUNCT
cana-3491	441	26	subtract_p_adic(x2	subtract_p_adic(x2	PROPN
cana-3491	441	27	,	,	PUNCT
cana-3491	441	28	x1	x1	PROPN
cana-3491	441	29	,	,	PUNCT
cana-3491	441	30	p	p	NOUN
cana-3491	441	31	)	)	PUNCT
cana-3491	441	32	,	,	PUNCT
cana-3491	441	33	p	p	X
cana-3491	441	34	)	)	PUNCT
cana-3491	441	35	m0	m0	NOUN
cana-3491	441	36	=	=	SYM
cana-3491	441	37	m[0	m[0	PROPN
cana-3491	441	38	]	]	X
cana-3491	441	39	m1	m1	PROPN
cana-3491	441	40	=	=	SYM
cana-3491	441	41	m[1	m[1	PROPN
cana-3491	441	42	]	]	X
cana-3491	441	43	print(f"m0	print(f"m0	NOUN
cana-3491	441	44	:	:	PUNCT
cana-3491	441	45	{	{	PUNCT
cana-3491	441	46	m0	m0	NOUN
cana-3491	441	47	}	}	PUNCT
cana-3491	441	48	"	"	PUNCT
cana-3491	441	49	)	)	PUNCT
cana-3491	441	50	print(f"m1	print(f"m1	NOUN
cana-3491	441	51	:	:	PUNCT
cana-3491	441	52	{	{	PUNCT
cana-3491	441	53	m1	m1	NOUN
cana-3491	441	54	}	}	PUNCT
cana-3491	441	55	"	"	PUNCT
cana-3491	441	56	)	)	PUNCT
cana-3491	441	57	#	#	NOUN
cana-3491	441	58	p1	p1	NOUN
cana-3491	441	59	+	+	CCONJ
cana-3491	441	60	p2	p2	PROPN
cana-3491	441	61	=	=	SYM
cana-3491	441	62	p3	p3	NOUN
cana-3491	441	63	if	if	SCONJ
cana-3491	441	64	not	not	PART
cana-3491	441	65	pointequality	pointequality	NOUN
cana-3491	441	66	:	:	PUNCT
cana-3491	441	67	x3	x3	PROPN
cana-3491	441	68	=	=	SYM
cana-3491	441	69	(	(	PUNCT
cana-3491	441	70	m0	m0	PROPN
cana-3491	441	71	*	*	PUNCT
cana-3491	441	72	*	*	PROPN
cana-3491	441	73	2	2	NUM
cana-3491	441	74	)	)	PUNCT
cana-3491	441	75	+	+	CCONJ
cana-3491	441	76	(	(	PUNCT
cana-3491	441	77	p	p	NOUN
cana-3491	441	78	1	1	NUM
cana-3491	441	79	)	)	PUNCT
cana-3491	441	80	*	*	PUNCT
cana-3491	442	1	(	(	PUNCT
cana-3491	442	2	x10	x10	NOUN
cana-3491	442	3	+	+	CCONJ
cana-3491	442	4	x20	x20	NOUN
cana-3491	442	5	)	)	PUNCT
cana-3491	443	1	+	+	CCONJ
cana-3491	443	2	2	2	X
cana-3491	443	3	*	*	PUNCT
cana-3491	443	4	m0	m0	NOUN
cana-3491	443	5	*	*	PUNCT
cana-3491	443	6	m1	m1	NOUN
cana-3491	443	7	*	*	PUNCT
cana-3491	443	8	communications	communication	NOUN
cana-3491	443	9	on	on	ADP
cana-3491	443	10	applied	apply	VERB
cana-3491	443	11	nonlinear	nonlinear	ADJ
cana-3491	443	12	analysis	analysis	NOUN
cana-3491	443	13	issn	issn	NOUN
cana-3491	443	14	:	:	PUNCT
cana-3491	443	15	1074	1074	NUM
cana-3491	443	16	-	-	PUNCT
cana-3491	443	17	133x	133x	NUM
cana-3491	443	18	vol	vol	NOUN
cana-3491	443	19	32	32	NUM
cana-3491	443	20	no	no	NOUN
cana-3491	443	21	.	.	PUNCT
cana-3491	444	1	7s	7	NOUN
cana-3491	444	2	(	(	PUNCT
cana-3491	444	3	2025	2025	NUM
cana-3491	444	4	)	)	PUNCT
cana-3491	444	5	879	879	NUM
cana-3491	444	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	445	1	p	p	X
cana-3491	445	2	+	+	X
cana-3491	445	3	(	(	PUNCT
cana-3491	445	4	p	p	X
cana-3491	445	5	*	*	X
cana-3491	445	6	(	(	PUNCT
cana-3491	445	7	p	p	NOUN
cana-3491	445	8	1	1	NUM
cana-3491	445	9	)	)	PUNCT
cana-3491	445	10	*	*	PUNCT
cana-3491	446	1	(	(	PUNCT
cana-3491	446	2	x10	x10	NOUN
cana-3491	446	3	+	+	NUM
cana-3491	446	4	x11	x11	NOUN
cana-3491	446	5	+	+	CCONJ
cana-3491	446	6	x20	x20	NOUN
cana-3491	446	7	+	+	CCONJ
cana-3491	446	8	x21	x21	NUM
cana-3491	446	9	)	)	PUNCT
cana-3491	446	10	)	)	PUNCT
cana-3491	446	11	print(f"x3	print(f"x3	NOUN
cana-3491	446	12	=	=	SYM
cana-3491	446	13	{	{	PUNCT
cana-3491	446	14	x3	x3	ADJ
cana-3491	446	15	}	}	PUNCT
cana-3491	446	16	"	"	PUNCT
cana-3491	446	17	)	)	PUNCT
cana-3491	446	18	try	try	VERB
cana-3491	446	19	:	:	PUNCT
cana-3491	446	20	p_adic_expansion_x	p_adic_expansion_x	X
cana-3491	446	21	=	=	PUNCT
cana-3491	446	22	to_p_adic(x3	to_p_adic(x3	PROPN
cana-3491	446	23	,	,	PUNCT
cana-3491	446	24	p	p	NOUN
cana-3491	446	25	,	,	PUNCT
cana-3491	446	26	precision	precision	NOUN
cana-3491	446	27	)	)	PUNCT
cana-3491	446	28	print_expansion(x3	print_expansion(x3	ADV
cana-3491	446	29	,	,	PUNCT
cana-3491	446	30	p	p	X
cana-3491	446	31	,	,	PUNCT
cana-3491	446	32	p_adic_expansion_x	p_adic_expansion_x	ADJ
cana-3491	446	33	,	,	PUNCT
cana-3491	446	34	precision	precision	NOUN
cana-3491	446	35	)	)	PUNCT
cana-3491	446	36	except	except	SCONJ
cana-3491	446	37	valueerror	valueerror	NOUN
cana-3491	446	38	as	as	ADP
cana-3491	446	39	e	e	NOUN
cana-3491	446	40	:	:	PUNCT
cana-3491	446	41	print(e	print(e	ADJ
cana-3491	446	42	)	)	PUNCT
cana-3491	446	43	y3	y3	NOUN
cana-3491	446	44	=	=	SYM
cana-3491	446	45	m0	m0	PROPN
cana-3491	446	46	*	*	PUNCT
cana-3491	447	1	x10	x10	X
cana-3491	447	2	+	+	CCONJ
cana-3491	447	3	(	(	PUNCT
cana-3491	447	4	p	p	NOUN
cana-3491	447	5	1	1	NUM
cana-3491	447	6	)	)	PUNCT
cana-3491	447	7	*	*	PUNCT
cana-3491	448	1	(	(	PUNCT
cana-3491	448	2	m0	m0	NOUN
cana-3491	448	3	*	*	PUNCT
cana-3491	448	4	p_adic_expansion_x[0	p_adic_expansion_x[0	PROPN
cana-3491	448	5	]	]	X
cana-3491	448	6	+	+	CCONJ
cana-3491	448	7	y10	y10	NUM
cana-3491	448	8	)	)	PUNCT
cana-3491	449	1	+	+	NOUN
cana-3491	450	1	p	p	X
cana-3491	450	2	*	*	PUNCT
cana-3491	450	3	(	(	PUNCT
cana-3491	450	4	m1	m1	PROPN
cana-3491	450	5	*	*	PUNCT
cana-3491	450	6	x10	x10	PROPN
cana-3491	450	7	+	+	CCONJ
cana-3491	450	8	m0	m0	PROPN
cana-3491	450	9	*	*	PUNCT
cana-3491	450	10	x11	x11	PROPN
cana-3491	450	11	)	)	PUNCT
cana-3491	451	1	+	+	CCONJ
cana-3491	452	1	p	p	X
cana-3491	452	2	*	*	PUNCT
cana-3491	452	3	(	(	PUNCT
cana-3491	452	4	p	p	NOUN
cana-3491	452	5	1	1	NUM
cana-3491	452	6	)	)	PUNCT
cana-3491	452	7	*	*	PUNCT
cana-3491	453	1	(	(	PUNCT
cana-3491	453	2	m1	m1	PROPN
cana-3491	453	3	*	*	PUNCT
cana-3491	453	4	p_adic_expansion_x[0	p_adic_expansion_x[0	PROPN
cana-3491	453	5	]	]	X
cana-3491	453	6	+	+	CCONJ
cana-3491	453	7	m0	m0	PROPN
cana-3491	453	8	*	*	PUNCT
cana-3491	453	9	p_adic_expansion_x[0	p_adic_expansion_x[0	PROPN
cana-3491	453	10	]	]	X
cana-3491	453	11	+	+	CCONJ
cana-3491	453	12	m0	m0	PROPN
cana-3491	453	13	*	*	PUNCT
cana-3491	453	14	p_adic_expansion_x[1	p_adic_expansion_x[1	PROPN
cana-3491	453	15	]	]	X
cana-3491	453	16	+	+	CCONJ
cana-3491	453	17	y10	y10	ADJ
cana-3491	453	18	+	+	NUM
cana-3491	453	19	y11	y11	NOUN
cana-3491	453	20	)	)	PUNCT
cana-3491	453	21	print(f"y3	print(f"y3	NOUN
cana-3491	453	22	=	=	SYM
cana-3491	453	23	{	{	PUNCT
cana-3491	453	24	y3	y3	NOUN
cana-3491	453	25	}	}	PUNCT
cana-3491	453	26	"	"	PUNCT
cana-3491	453	27	)	)	PUNCT
cana-3491	453	28	try	try	VERB
cana-3491	453	29	:	:	PUNCT
cana-3491	453	30	p_adic_expansion_y	p_adic_expansion_y	NOUN
cana-3491	453	31	=	=	SYM
cana-3491	453	32	to_p_adic(y3	to_p_adic(y3	PROPN
cana-3491	453	33	,	,	PUNCT
cana-3491	453	34	p	p	NOUN
cana-3491	453	35	,	,	PUNCT
cana-3491	453	36	precision	precision	NOUN
cana-3491	453	37	)	)	PUNCT
cana-3491	453	38	print_expansion(y3	print_expansion(y3	NOUN
cana-3491	453	39	,	,	PUNCT
cana-3491	453	40	p	p	NOUN
cana-3491	453	41	,	,	PUNCT
cana-3491	453	42	p_adic_expansion_y	p_adic_expansion_y	NOUN
cana-3491	453	43	,	,	PUNCT
cana-3491	453	44	precision	precision	NOUN
cana-3491	453	45	)	)	PUNCT
cana-3491	453	46	except	except	SCONJ
cana-3491	453	47	valueerror	valueerror	NOUN
cana-3491	453	48	as	as	ADP
cana-3491	453	49	e	e	NOUN
cana-3491	453	50	:	:	PUNCT
cana-3491	453	51	print(e	print(e	ADJ
cana-3491	453	52	)	)	PUNCT
cana-3491	453	53	return	return	VERB
cana-3491	453	54	p_adic_expansion_x	p_adic_expansion_x	NOUN
cana-3491	453	55	,	,	PUNCT
cana-3491	453	56	p_adic_expansion_y	p_adic_expansion_y	NOUN
cana-3491	453	57	else	else	ADV
cana-3491	453	58	:	:	PUNCT
cana-3491	453	59	x3	x3	ADJ
cana-3491	453	60	=	=	SYM
cana-3491	453	61	m0	m0	PROPN
cana-3491	454	1	*	*	PUNCT
cana-3491	455	1	*	*	PUNCT
cana-3491	455	2	2	2	NUM
cana-3491	455	3	+	+	CCONJ
cana-3491	455	4	(	(	PUNCT
cana-3491	455	5	p	p	NOUN
cana-3491	455	6	1	1	NUM
cana-3491	455	7	)	)	PUNCT
cana-3491	455	8	*	*	PUNCT
cana-3491	455	9	2	2	NUM
cana-3491	455	10	*	*	PUNCT
cana-3491	455	11	x10	x10	NOUN
cana-3491	455	12	+	+	NOUN
cana-3491	455	13	2	2	NUM
cana-3491	455	14	*	*	PUNCT
cana-3491	455	15	p	p	X
cana-3491	455	16	*	*	X
cana-3491	455	17	(	(	PUNCT
cana-3491	455	18	m1	m1	PROPN
cana-3491	455	19	+	+	CCONJ
cana-3491	455	20	(	(	PUNCT
cana-3491	455	21	p	p	NOUN
cana-3491	455	22	1	1	NUM
cana-3491	455	23	)	)	PUNCT
cana-3491	455	24	*	*	PUNCT
cana-3491	456	1	(	(	PUNCT
cana-3491	456	2	x10	x10	NOUN
cana-3491	456	3	+	+	CCONJ
cana-3491	456	4	x11	x11	NOUN
cana-3491	456	5	)	)	PUNCT
cana-3491	456	6	)	)	PUNCT
cana-3491	456	7	print(f"x3	print(f"x3	NOUN
cana-3491	456	8	=	=	SYM
cana-3491	456	9	{	{	PUNCT
cana-3491	456	10	x3	x3	ADJ
cana-3491	456	11	}	}	PUNCT
cana-3491	456	12	"	"	PUNCT
cana-3491	456	13	)	)	PUNCT
cana-3491	456	14	try	try	VERB
cana-3491	456	15	:	:	PUNCT
cana-3491	456	16	p_adic_expansion_x	p_adic_expansion_x	X
cana-3491	456	17	=	=	PUNCT
cana-3491	456	18	to_p_adic(x3	to_p_adic(x3	PROPN
cana-3491	456	19	,	,	PUNCT
cana-3491	456	20	p	p	NOUN
cana-3491	456	21	,	,	PUNCT
cana-3491	456	22	precision	precision	NOUN
cana-3491	456	23	)	)	PUNCT
cana-3491	456	24	print_expansion(x3	print_expansion(x3	ADV
cana-3491	456	25	,	,	PUNCT
cana-3491	456	26	p	p	X
cana-3491	456	27	,	,	PUNCT
cana-3491	456	28	p_adic_expansion_x	p_adic_expansion_x	ADJ
cana-3491	456	29	,	,	PUNCT
cana-3491	456	30	precision	precision	NOUN
cana-3491	456	31	)	)	PUNCT
cana-3491	456	32	except	except	SCONJ
cana-3491	456	33	valueerror	valueerror	NOUN
cana-3491	456	34	as	as	ADP
cana-3491	456	35	e	e	NOUN
cana-3491	456	36	:	:	PUNCT
cana-3491	456	37	print(e	print(e	ADJ
cana-3491	456	38	)	)	PUNCT
cana-3491	456	39	y3	y3	NOUN
cana-3491	456	40	=	=	SYM
cana-3491	456	41	m0	m0	PROPN
cana-3491	456	42	*	*	PUNCT
cana-3491	457	1	x10	x10	X
cana-3491	457	2	+	+	CCONJ
cana-3491	457	3	(	(	PUNCT
cana-3491	457	4	p	p	NOUN
cana-3491	457	5	1	1	NUM
cana-3491	457	6	)	)	PUNCT
cana-3491	457	7	*	*	PUNCT
cana-3491	458	1	(	(	PUNCT
cana-3491	458	2	m0	m0	NOUN
cana-3491	458	3	*	*	PUNCT
cana-3491	458	4	p_adic_expansion_x[0	p_adic_expansion_x[0	PROPN
cana-3491	458	5	]	]	X
cana-3491	458	6	+	+	CCONJ
cana-3491	458	7	y10	y10	NUM
cana-3491	458	8	)	)	PUNCT
cana-3491	459	1	+	+	NOUN
cana-3491	460	1	p	p	X
cana-3491	460	2	*	*	PUNCT
cana-3491	460	3	(	(	PUNCT
cana-3491	460	4	m1	m1	PROPN
cana-3491	460	5	*	*	PUNCT
cana-3491	460	6	x10	x10	PROPN
cana-3491	460	7	+	+	CCONJ
cana-3491	460	8	m0	m0	PROPN
cana-3491	460	9	*	*	PUNCT
cana-3491	460	10	x11	x11	PROPN
cana-3491	460	11	)	)	PUNCT
cana-3491	461	1	+	+	CCONJ
cana-3491	462	1	p	p	X
cana-3491	462	2	*	*	PUNCT
cana-3491	462	3	(	(	PUNCT
cana-3491	462	4	p	p	NOUN
cana-3491	462	5	1	1	NUM
cana-3491	462	6	)	)	PUNCT
cana-3491	462	7	*	*	PUNCT
cana-3491	463	1	(	(	PUNCT
cana-3491	463	2	m1	m1	PROPN
cana-3491	463	3	*	*	PUNCT
cana-3491	463	4	p_adic_expansion_x[0	p_adic_expansion_x[0	PROPN
cana-3491	463	5	]	]	X
cana-3491	463	6	+	+	CCONJ
cana-3491	463	7	m0	m0	PROPN
cana-3491	463	8	*	*	PUNCT
cana-3491	463	9	p_adic_expansion_x[0	p_adic_expansion_x[0	PROPN
cana-3491	463	10	]	]	X
cana-3491	463	11	+	+	CCONJ
cana-3491	463	12	m0	m0	PROPN
cana-3491	463	13	*	*	PUNCT
cana-3491	463	14	p_adic_expansion_x[1	p_adic_expansion_x[1	PROPN
cana-3491	463	15	]	]	X
cana-3491	463	16	+	+	CCONJ
cana-3491	463	17	y10	y10	ADJ
cana-3491	463	18	+	+	NUM
cana-3491	463	19	y11	y11	NOUN
cana-3491	463	20	)	)	PUNCT
cana-3491	463	21	communications	communication	NOUN
cana-3491	463	22	on	on	ADP
cana-3491	463	23	applied	apply	VERB
cana-3491	463	24	nonlinear	nonlinear	ADJ
cana-3491	463	25	analysis	analysis	NOUN
cana-3491	463	26	issn	issn	NOUN
cana-3491	463	27	:	:	PUNCT
cana-3491	463	28	1074	1074	NUM
cana-3491	463	29	-	-	PUNCT
cana-3491	463	30	133x	133x	NUM
cana-3491	463	31	vol	vol	NOUN
cana-3491	463	32	32	32	NUM
cana-3491	463	33	no	no	NOUN
cana-3491	463	34	.	.	PUNCT
cana-3491	464	1	7s	7	NOUN
cana-3491	464	2	(	(	PUNCT
cana-3491	464	3	2025	2025	NUM
cana-3491	464	4	)	)	PUNCT
cana-3491	464	5	880	880	NUM
cana-3491	464	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	464	7	print(f"y3	print(f"y3	NOUN
cana-3491	464	8	=	=	SYM
cana-3491	464	9	{	{	PUNCT
cana-3491	464	10	y3	y3	NOUN
cana-3491	464	11	}	}	PUNCT
cana-3491	464	12	"	"	PUNCT
cana-3491	464	13	)	)	PUNCT
cana-3491	464	14	try	try	VERB
cana-3491	464	15	:	:	PUNCT
cana-3491	464	16	p_adic_expansion_y	p_adic_expansion_y	NOUN
cana-3491	465	1	=	=	SYM
cana-3491	466	1	to_p_adic(y3	to_p_adic(y3	PROPN
cana-3491	466	2	,	,	PUNCT
cana-3491	466	3	p	p	NOUN
cana-3491	466	4	,	,	PUNCT
cana-3491	466	5	precision	precision	NOUN
cana-3491	466	6	)	)	PUNCT
cana-3491	466	7	print_expansion(y3	print_expansion(y3	NOUN
cana-3491	466	8	,	,	PUNCT
cana-3491	466	9	p	p	NOUN
cana-3491	466	10	,	,	PUNCT
cana-3491	466	11	p_adic_expansion_y	p_adic_expansion_y	NOUN
cana-3491	466	12	,	,	PUNCT
cana-3491	466	13	precision	precision	NOUN
cana-3491	466	14	)	)	PUNCT
cana-3491	466	15	except	except	SCONJ
cana-3491	466	16	valueerror	valueerror	NOUN
cana-3491	466	17	as	as	ADP
cana-3491	466	18	e	e	NOUN
cana-3491	466	19	:	:	PUNCT
cana-3491	466	20	print(e	print(e	ADJ
cana-3491	466	21	)	)	PUNCT
cana-3491	466	22	return	return	VERB
cana-3491	466	23	p_adic_expansion_x	p_adic_expansion_x	NOUN
cana-3491	466	24	,	,	PUNCT
cana-3491	466	25	p_adic_expansion_y	p_adic_expansion_y	NOUN
cana-3491	466	26	def	def	PROPN
cana-3491	466	27	take_input	take_input	PROPN
cana-3491	466	28	(	(	PUNCT
cana-3491	466	29	):	):	PUNCT
cana-3491	466	30	expansion1	expansion1	NOUN
cana-3491	466	31	=	=	PUNCT
cana-3491	467	1	[	[	X
cana-3491	467	2	int(x	int(x	NOUN
cana-3491	467	3	)	)	PUNCT
cana-3491	467	4	for	for	ADP
cana-3491	467	5	x	x	PUNCT
cana-3491	467	6	in	in	ADP
cana-3491	467	7	input("enter	input("enter	X
cana-3491	467	8	the	the	DET
cana-3491	467	9	first	first	ADJ
cana-3491	467	10	p	p	ADJ
cana-3491	467	11	-	-	PUNCT
cana-3491	467	12	adic	adic	ADJ
cana-3491	467	13	expansion	expansion	NOUN
cana-3491	467	14	(	(	PUNCT
cana-3491	467	15	comma	comma	NOUN
cana-3491	467	16	-	-	PUNCT
cana-3491	467	17	separated	separate	VERB
cana-3491	467	18	):	):	PUNCT
cana-3491	467	19	"	"	PUNCT
cana-3491	467	20	)	)	PUNCT
cana-3491	467	21	.split	.split	NOUN
cana-3491	467	22	(	(	PUNCT
cana-3491	467	23	'	'	PUNCT
cana-3491	467	24	,	,	PUNCT
cana-3491	467	25	'	'	PUNCT
cana-3491	467	26	)	)	PUNCT
cana-3491	467	27	]	]	PUNCT
cana-3491	468	1	expansion2	expansion2	NOUN
cana-3491	469	1	=	=	PUNCT
cana-3491	470	1	[	[	X
cana-3491	470	2	int(x	int(x	NOUN
cana-3491	470	3	)	)	PUNCT
cana-3491	470	4	for	for	ADP
cana-3491	470	5	x	x	PUNCT
cana-3491	470	6	in	in	ADP
cana-3491	470	7	input("enter	input("enter	X
cana-3491	470	8	the	the	DET
cana-3491	470	9	second	second	ADJ
cana-3491	470	10	p	p	ADJ
cana-3491	470	11	-	-	PUNCT
cana-3491	470	12	adic	adic	ADJ
cana-3491	470	13	expansion	expansion	NOUN
cana-3491	470	14	(	(	PUNCT
cana-3491	470	15	comma	comma	NOUN
cana-3491	470	16	-	-	PUNCT
cana-3491	470	17	separated	separate	VERB
cana-3491	470	18	):	):	PUNCT
cana-3491	470	19	"	"	PUNCT
cana-3491	470	20	)	)	PUNCT
cana-3491	470	21	.split	.split	NOUN
cana-3491	470	22	(	(	PUNCT
cana-3491	470	23	'	'	PUNCT
cana-3491	470	24	,	,	PUNCT
cana-3491	470	25	'	'	PUNCT
cana-3491	470	26	)	)	PUNCT
cana-3491	470	27	]	]	PUNCT
cana-3491	471	1	p	p	X
cana-3491	471	2	=	=	PUNCT
cana-3491	471	3	int(input("enter	int(input("enter	VERB
cana-3491	471	4	the	the	DET
cana-3491	471	5	base	base	NOUN
cana-3491	471	6	(	(	PUNCT
cana-3491	471	7	prime	prime	ADJ
cana-3491	471	8	number	number	NOUN
cana-3491	471	9	):	):	PUNCT
cana-3491	471	10	"	"	PUNCT
cana-3491	471	11	)	)	PUNCT
cana-3491	471	12	)	)	PUNCT
cana-3491	471	13	return	return	VERB
cana-3491	471	14	expansion1	expansion1	PROPN
cana-3491	471	15	,	,	PUNCT
cana-3491	471	16	expansion2	expansion2	PROPN
cana-3491	471	17	,	,	PUNCT
cana-3491	471	18	p	p	PROPN
cana-3491	471	19	#	#	SYM
cana-3491	471	20	%	%	NOUN
cana-3491	471	21	%	%	NOUN
cana-3491	471	22	#	#	NOUN
cana-3491	471	23	for	for	ADP
cana-3491	471	24	addition	addition	NOUN
cana-3491	471	25	run	run	VERB
cana-3491	471	26	this	this	DET
cana-3491	471	27	cell	cell	NOUN
cana-3491	471	28	exp1	exp1	NOUN
cana-3491	471	29	,	,	PUNCT
cana-3491	471	30	exp2	exp2	PROPN
cana-3491	471	31	,	,	PUNCT
cana-3491	471	32	p	p	X
cana-3491	471	33	=	=	SYM
cana-3491	471	34	take_input	take_input	PROPN
cana-3491	471	35	(	(	PUNCT
cana-3491	471	36	)	)	PUNCT
cana-3491	471	37	result	result	NOUN
cana-3491	471	38	=	=	SYM
cana-3491	471	39	add_p_adic(exp1	add_p_adic(exp1	ADJ
cana-3491	471	40	,	,	PUNCT
cana-3491	471	41	exp2	exp2	NOUN
cana-3491	471	42	,	,	PUNCT
cana-3491	471	43	p	p	X
cana-3491	471	44	)	)	PUNCT
cana-3491	471	45	print(f"the	print(f"the	DET
cana-3491	471	46	sum	sum	NOUN
cana-3491	471	47	of	of	ADP
cana-3491	471	48	the	the	DET
cana-3491	471	49	p	p	NOUN
cana-3491	471	50	-	-	PUNCT
cana-3491	471	51	adic	adic	ADJ
cana-3491	471	52	expansions	expansion	NOUN
cana-3491	471	53	is	be	AUX
cana-3491	471	54	:	:	PUNCT
cana-3491	471	55	{	{	PUNCT
cana-3491	471	56	result	result	NOUN
cana-3491	471	57	}	}	PUNCT
cana-3491	471	58	"	"	PUNCT
cana-3491	471	59	)	)	PUNCT
cana-3491	471	60	#	#	NUM
cana-3491	471	61	%	%	NOUN
cana-3491	472	1	%	%	NOUN
cana-3491	472	2	#	#	NOUN
cana-3491	472	3	for	for	ADP
cana-3491	472	4	subtraction	subtraction	NOUN
cana-3491	472	5	run	run	NOUN
cana-3491	472	6	this	this	DET
cana-3491	472	7	cell	cell	NOUN
cana-3491	472	8	exp1	exp1	NOUN
cana-3491	472	9	,	,	PUNCT
cana-3491	472	10	exp2	exp2	PROPN
cana-3491	472	11	,	,	PUNCT
cana-3491	472	12	p	p	X
cana-3491	472	13	=	=	SYM
cana-3491	472	14	take_input	take_input	PROPN
cana-3491	472	15	(	(	PUNCT
cana-3491	472	16	)	)	PUNCT
cana-3491	472	17	result	result	NOUN
cana-3491	472	18	=	=	SYM
cana-3491	472	19	subtract_p_adic(exp1	subtract_p_adic(exp1	ADJ
cana-3491	472	20	,	,	PUNCT
cana-3491	473	1	exp2	exp2	NOUN
cana-3491	473	2	,	,	PUNCT
cana-3491	473	3	p	p	X
cana-3491	473	4	)	)	PUNCT
cana-3491	473	5	print(f"the	print(f"the	DET
cana-3491	473	6	sum	sum	NOUN
cana-3491	473	7	of	of	ADP
cana-3491	473	8	the	the	DET
cana-3491	473	9	p	p	NOUN
cana-3491	473	10	-	-	PUNCT
cana-3491	473	11	adic	adic	ADJ
cana-3491	473	12	expansions	expansion	NOUN
cana-3491	473	13	is	be	AUX
cana-3491	473	14	:	:	PUNCT
cana-3491	473	15	{	{	PUNCT
cana-3491	473	16	result	result	NOUN
cana-3491	473	17	}	}	PUNCT
cana-3491	473	18	"	"	PUNCT
cana-3491	473	19	)	)	PUNCT
cana-3491	473	20	#	#	NUM
cana-3491	473	21	%	%	NOUN
cana-3491	474	1	%	%	NOUN
cana-3491	474	2	#	#	NOUN
cana-3491	474	3	for	for	ADP
cana-3491	474	4	multiplication	multiplication	NOUN
cana-3491	474	5	run	run	VERB
cana-3491	474	6	this	this	DET
cana-3491	474	7	cell	cell	NOUN
cana-3491	474	8	exp1	exp1	NOUN
cana-3491	474	9	,	,	PUNCT
cana-3491	474	10	exp2	exp2	PROPN
cana-3491	474	11	,	,	PUNCT
cana-3491	474	12	p	p	X
cana-3491	474	13	=	=	SYM
cana-3491	474	14	take_input	take_input	PROPN
cana-3491	474	15	(	(	PUNCT
cana-3491	474	16	)	)	PUNCT
cana-3491	474	17	result	result	NOUN
cana-3491	474	18	=	=	SYM
cana-3491	475	1	multiply_p_adic(exp1	multiply_p_adic(exp1	ADJ
cana-3491	475	2	,	,	PUNCT
cana-3491	475	3	exp2	exp2	NOUN
cana-3491	475	4	,	,	PUNCT
cana-3491	475	5	p	p	X
cana-3491	475	6	)	)	PUNCT
cana-3491	475	7	print(f"the	print(f"the	DET
cana-3491	475	8	sum	sum	NOUN
cana-3491	475	9	of	of	ADP
cana-3491	475	10	the	the	DET
cana-3491	475	11	p	p	NOUN
cana-3491	475	12	-	-	PUNCT
cana-3491	475	13	adic	adic	ADJ
cana-3491	475	14	expansions	expansion	NOUN
cana-3491	475	15	is	be	AUX
cana-3491	475	16	:	:	PUNCT
cana-3491	475	17	{	{	PUNCT
cana-3491	475	18	result	result	NOUN
cana-3491	475	19	}	}	PUNCT
cana-3491	475	20	"	"	PUNCT
cana-3491	475	21	)	)	PUNCT
cana-3491	475	22	#	#	NUM
cana-3491	475	23	%	%	NOUN
cana-3491	476	1	%	%	NOUN
cana-3491	476	2	#	#	NOUN
cana-3491	476	3	for	for	ADP
cana-3491	476	4	division	division	NOUN
cana-3491	476	5	run	run	VERB
cana-3491	476	6	this	this	DET
cana-3491	476	7	cell	cell	NOUN
cana-3491	476	8	exp1	exp1	NOUN
cana-3491	476	9	,	,	PUNCT
cana-3491	476	10	exp2	exp2	PROPN
cana-3491	476	11	,	,	PUNCT
cana-3491	476	12	p	p	X
cana-3491	476	13	=	=	SYM
cana-3491	476	14	take_input	take_input	PROPN
cana-3491	476	15	(	(	PUNCT
cana-3491	476	16	)	)	PUNCT
cana-3491	476	17	result	result	NOUN
cana-3491	476	18	=	=	SYM
cana-3491	476	19	divide_p_adic(exp1	divide_p_adic(exp1	ADJ
cana-3491	476	20	,	,	PUNCT
cana-3491	476	21	exp2	exp2	NOUN
cana-3491	476	22	,	,	PUNCT
cana-3491	476	23	p	p	X
cana-3491	476	24	)	)	PUNCT
cana-3491	476	25	print(f"the	print(f"the	DET
cana-3491	476	26	sum	sum	NOUN
cana-3491	476	27	of	of	ADP
cana-3491	476	28	the	the	DET
cana-3491	476	29	p	p	NOUN
cana-3491	476	30	-	-	PUNCT
cana-3491	476	31	adic	adic	ADJ
cana-3491	476	32	expansions	expansion	NOUN
cana-3491	476	33	is	be	AUX
cana-3491	476	34	:	:	PUNCT
cana-3491	476	35	{	{	PUNCT
cana-3491	476	36	result	result	NOUN
cana-3491	476	37	}	}	PUNCT
cana-3491	476	38	"	"	PUNCT
cana-3491	476	39	)	)	PUNCT
cana-3491	476	40	#	#	NUM
cana-3491	476	41	%	%	NOUN
cana-3491	477	1	%	%	NOUN
cana-3491	477	2	#	#	NOUN
cana-3491	477	3	for	for	ADP
cana-3491	477	4	calculation	calculation	NOUN
cana-3491	477	5	of	of	ADP
cana-3491	477	6	slope	slope	NOUN
cana-3491	477	7	run	run	VERB
cana-3491	477	8	this	this	DET
cana-3491	477	9	cell	cell	NOUN
cana-3491	477	10	x1	x1	NOUN
cana-3491	478	1	=	=	PUNCT
cana-3491	479	1	[	[	X
cana-3491	479	2	int(x	int(x	NOUN
cana-3491	479	3	)	)	PUNCT
cana-3491	479	4	for	for	ADP
cana-3491	479	5	x	x	PUNCT
cana-3491	479	6	in	in	ADP
cana-3491	479	7	input("enter	input("enter	X
cana-3491	479	8	the	the	DET
cana-3491	479	9	x1	x1	PROPN
cana-3491	479	10	p	p	ADJ
cana-3491	479	11	-	-	PUNCT
cana-3491	479	12	adic	adic	ADJ
cana-3491	479	13	expansion	expansion	NOUN
cana-3491	479	14	(	(	PUNCT
cana-3491	479	15	comma	comma	NOUN
cana-3491	479	16	-	-	PUNCT
cana-3491	479	17	separated	separate	VERB
cana-3491	479	18	):	):	PUNCT
cana-3491	479	19	"	"	PUNCT
cana-3491	479	20	)	)	PUNCT
cana-3491	479	21	.split	.split	NOUN
cana-3491	479	22	(	(	PUNCT
cana-3491	479	23	'	'	PUNCT
cana-3491	479	24	,	,	PUNCT
cana-3491	479	25	'	'	PUNCT
cana-3491	479	26	)	)	PUNCT
cana-3491	479	27	]	]	PUNCT
cana-3491	480	1	communications	communication	NOUN
cana-3491	480	2	on	on	ADP
cana-3491	480	3	applied	apply	VERB
cana-3491	480	4	nonlinear	nonlinear	ADJ
cana-3491	480	5	analysis	analysis	NOUN
cana-3491	480	6	issn	issn	NOUN
cana-3491	480	7	:	:	PUNCT
cana-3491	480	8	1074	1074	NUM
cana-3491	480	9	-	-	PUNCT
cana-3491	480	10	133x	133x	NUM
cana-3491	480	11	vol	vol	NOUN
cana-3491	480	12	32	32	NUM
cana-3491	480	13	no	no	NOUN
cana-3491	480	14	.	.	PUNCT
cana-3491	481	1	7s	7	NOUN
cana-3491	481	2	(	(	PUNCT
cana-3491	481	3	2025	2025	NUM
cana-3491	481	4	)	)	PUNCT
cana-3491	481	5	881	881	NUM
cana-3491	481	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	482	1	x2	x2	NOUN
cana-3491	482	2	=	=	PUNCT
cana-3491	483	1	[	[	X
cana-3491	483	2	int(x	int(x	NOUN
cana-3491	483	3	)	)	PUNCT
cana-3491	483	4	for	for	ADP
cana-3491	483	5	x	x	PUNCT
cana-3491	483	6	in	in	ADP
cana-3491	483	7	input("enter	input("enter	X
cana-3491	483	8	the	the	DET
cana-3491	483	9	x2	x2	PROPN
cana-3491	483	10	p	p	ADJ
cana-3491	483	11	-	-	PUNCT
cana-3491	483	12	adic	adic	ADJ
cana-3491	483	13	expansion	expansion	NOUN
cana-3491	483	14	(	(	PUNCT
cana-3491	483	15	comma	comma	NOUN
cana-3491	483	16	-	-	PUNCT
cana-3491	483	17	separated	separate	VERB
cana-3491	483	18	):	):	PUNCT
cana-3491	483	19	"	"	PUNCT
cana-3491	483	20	)	)	PUNCT
cana-3491	483	21	.split	.split	NOUN
cana-3491	483	22	(	(	PUNCT
cana-3491	483	23	'	'	PUNCT
cana-3491	483	24	,	,	PUNCT
cana-3491	483	25	'	'	PUNCT
cana-3491	483	26	)	)	PUNCT
cana-3491	483	27	]	]	PUNCT
cana-3491	484	1	y1	y1	NOUN
cana-3491	484	2	=	=	PUNCT
cana-3491	485	1	[	[	X
cana-3491	485	2	int(x	int(x	NOUN
cana-3491	485	3	)	)	PUNCT
cana-3491	485	4	for	for	ADP
cana-3491	485	5	x	x	PUNCT
cana-3491	485	6	in	in	ADP
cana-3491	485	7	input("enter	input("enter	X
cana-3491	485	8	the	the	DET
cana-3491	485	9	y1	y1	ADJ
cana-3491	485	10	p	p	NOUN
cana-3491	485	11	-	-	PUNCT
cana-3491	485	12	adic	adic	ADJ
cana-3491	485	13	expansion	expansion	NOUN
cana-3491	485	14	(	(	PUNCT
cana-3491	485	15	comma	comma	NOUN
cana-3491	485	16	-	-	PUNCT
cana-3491	485	17	separated	separate	VERB
cana-3491	485	18	):	):	PUNCT
cana-3491	485	19	"	"	PUNCT
cana-3491	485	20	)	)	PUNCT
cana-3491	485	21	.split	.split	NOUN
cana-3491	485	22	(	(	PUNCT
cana-3491	485	23	'	'	PUNCT
cana-3491	485	24	,	,	PUNCT
cana-3491	485	25	'	'	PUNCT
cana-3491	485	26	)	)	PUNCT
cana-3491	485	27	]	]	PUNCT
cana-3491	486	1	y2	y2	NOUN
cana-3491	486	2	=	=	PUNCT
cana-3491	487	1	[	[	X
cana-3491	487	2	int(x	int(x	NOUN
cana-3491	487	3	)	)	PUNCT
cana-3491	487	4	for	for	ADP
cana-3491	487	5	x	x	PUNCT
cana-3491	487	6	in	in	ADP
cana-3491	487	7	input("enter	input("enter	X
cana-3491	487	8	the	the	DET
cana-3491	487	9	y2	y2	PROPN
cana-3491	487	10	p	p	NOUN
cana-3491	487	11	-	-	PUNCT
cana-3491	487	12	adic	adic	ADJ
cana-3491	487	13	expansion	expansion	NOUN
cana-3491	487	14	(	(	PUNCT
cana-3491	487	15	comma	comma	NOUN
cana-3491	487	16	-	-	PUNCT
cana-3491	487	17	separated	separate	VERB
cana-3491	487	18	):	):	PUNCT
cana-3491	487	19	"	"	PUNCT
cana-3491	487	20	)	)	PUNCT
cana-3491	487	21	.split	.split	NOUN
cana-3491	487	22	(	(	PUNCT
cana-3491	487	23	'	'	PUNCT
cana-3491	487	24	,	,	PUNCT
cana-3491	487	25	'	'	PUNCT
cana-3491	487	26	)	)	PUNCT
cana-3491	487	27	]	]	PUNCT
cana-3491	488	1	p	p	X
cana-3491	488	2	=	=	PUNCT
cana-3491	488	3	int(input("enter	int(input("enter	VERB
cana-3491	488	4	the	the	DET
cana-3491	488	5	base	base	NOUN
cana-3491	488	6	(	(	PUNCT
cana-3491	488	7	prime	prime	ADJ
cana-3491	488	8	number	number	NOUN
cana-3491	488	9	):	):	PUNCT
cana-3491	488	10	"	"	PUNCT
cana-3491	488	11	)	)	PUNCT
cana-3491	488	12	)	)	PUNCT
cana-3491	488	13	max_length	max_length	NOUN
cana-3491	488	14	=	=	SYM
cana-3491	488	15	max(len(x1	max(len(x1	PROPN
cana-3491	488	16	)	)	PUNCT
cana-3491	488	17	,	,	PUNCT
cana-3491	488	18	len(x2	len(x2	PROPN
cana-3491	488	19	)	)	PUNCT
cana-3491	488	20	,	,	PUNCT
cana-3491	488	21	len(y1	len(y1	PROPN
cana-3491	488	22	)	)	PUNCT
cana-3491	488	23	,	,	PUNCT
cana-3491	488	24	len(y2	len(y2	NOUN
cana-3491	488	25	)	)	PUNCT
cana-3491	488	26	)	)	PUNCT
cana-3491	489	1	x1	x1	PROPN
cana-3491	489	2	=	=	PUNCT
cana-3491	490	1	pad_with_zeros(x1	pad_with_zeros(x1	ADJ
cana-3491	490	2	,	,	PUNCT
cana-3491	490	3	max_length	max_length	NOUN
cana-3491	490	4	)	)	PUNCT
cana-3491	490	5	x2	x2	NOUN
cana-3491	491	1	=	=	PUNCT
cana-3491	491	2	pad_with_zeros(x2	pad_with_zeros(x2	NOUN
cana-3491	491	3	,	,	PUNCT
cana-3491	491	4	max_length	max_length	NOUN
cana-3491	491	5	)	)	PUNCT
cana-3491	491	6	y1	y1	NOUN
cana-3491	491	7	=	=	SYM
cana-3491	491	8	pad_with_zeros(y1	pad_with_zeros(y1	NOUN
cana-3491	491	9	,	,	PUNCT
cana-3491	491	10	max_length	max_length	NOUN
cana-3491	491	11	)	)	PUNCT
cana-3491	491	12	y2	y2	NOUN
cana-3491	491	13	=	=	SYM
cana-3491	491	14	pad_with_zeros(y2	pad_with_zeros(y2	NOUN
cana-3491	491	15	,	,	PUNCT
cana-3491	491	16	max_length	max_length	NOUN
cana-3491	491	17	)	)	PUNCT
cana-3491	491	18	slope_result	slope_result	PROPN
cana-3491	491	19	=	=	SYM
cana-3491	492	1	divide_p_adic(subtract_p_adic(y2	divide_p_adic(subtract_p_adic(y2	PROPN
cana-3491	492	2	,	,	PUNCT
cana-3491	492	3	y1,p	y1,p	PROPN
cana-3491	492	4	)	)	PUNCT
cana-3491	492	5	,	,	PUNCT
cana-3491	492	6	subtract_p_adic(x2	subtract_p_adic(x2	PROPN
cana-3491	492	7	,	,	PUNCT
cana-3491	492	8	x1	x1	PROPN
cana-3491	492	9	,	,	PUNCT
cana-3491	492	10	p	p	NOUN
cana-3491	492	11	)	)	PUNCT
cana-3491	492	12	,	,	PUNCT
cana-3491	492	13	p	p	X
cana-3491	492	14	)	)	PUNCT
cana-3491	492	15	print(slope_result	print(slope_result	NOUN
cana-3491	492	16	)	)	PUNCT
cana-3491	492	17	#	#	SYM
cana-3491	492	18	%	%	NOUN
cana-3491	492	19	%	%	NOUN
cana-3491	492	20	#	#	NOUN
cana-3491	492	21	to	to	PART
cana-3491	492	22	calculate	calculate	VERB
cana-3491	492	23	p3	p3	PROPN
cana-3491	492	24	run	run	VERB
cana-3491	492	25	this	this	DET
cana-3491	492	26	cell	cell	NOUN
cana-3491	492	27	print("calculation	print("calculation	NOUN
cana-3491	492	28	of	of	ADP
cana-3491	492	29	point	point	NOUN
cana-3491	492	30	on	on	ADP
cana-3491	492	31	curve	curve	NOUN
cana-3491	492	32	y^2	y^2	PROPN
cana-3491	492	33	=	=	PROPN
cana-3491	492	34	x^3	x^3	PROPN
cana-3491	492	35	+	+	PUNCT
cana-3491	492	36	ax	ax	NOUN
cana-3491	492	37	+	+	CCONJ
cana-3491	492	38	b	b	NOUN
cana-3491	492	39	"	"	PUNCT
cana-3491	492	40	)	)	PUNCT
cana-3491	492	41	x1	x1	PROPN
cana-3491	493	1	=	=	PUNCT
cana-3491	494	1	[	[	X
cana-3491	494	2	int(x	int(x	NOUN
cana-3491	494	3	)	)	PUNCT
cana-3491	494	4	for	for	ADP
cana-3491	494	5	x	x	PUNCT
cana-3491	494	6	in	in	ADP
cana-3491	494	7	input("enter	input("enter	X
cana-3491	494	8	the	the	DET
cana-3491	494	9	x1	x1	PROPN
cana-3491	494	10	p	p	ADJ
cana-3491	494	11	-	-	PUNCT
cana-3491	494	12	adic	adic	ADJ
cana-3491	494	13	expansion	expansion	NOUN
cana-3491	494	14	(	(	PUNCT
cana-3491	494	15	comma	comma	NOUN
cana-3491	494	16	-	-	PUNCT
cana-3491	494	17	separated	separate	VERB
cana-3491	494	18	):	):	PUNCT
cana-3491	494	19	"	"	PUNCT
cana-3491	494	20	)	)	PUNCT
cana-3491	494	21	.split	.split	NOUN
cana-3491	494	22	(	(	PUNCT
cana-3491	494	23	'	'	PUNCT
cana-3491	494	24	,	,	PUNCT
cana-3491	494	25	'	'	PUNCT
cana-3491	494	26	)	)	PUNCT
cana-3491	494	27	]	]	PUNCT
cana-3491	495	1	x2	x2	NOUN
cana-3491	495	2	=	=	PUNCT
cana-3491	496	1	[	[	X
cana-3491	496	2	int(x	int(x	NOUN
cana-3491	496	3	)	)	PUNCT
cana-3491	496	4	for	for	ADP
cana-3491	496	5	x	x	PUNCT
cana-3491	496	6	in	in	ADP
cana-3491	496	7	input("enter	input("enter	X
cana-3491	496	8	the	the	DET
cana-3491	496	9	x2	x2	PROPN
cana-3491	496	10	p	p	ADJ
cana-3491	496	11	-	-	PUNCT
cana-3491	496	12	adic	adic	ADJ
cana-3491	496	13	expansion	expansion	NOUN
cana-3491	496	14	(	(	PUNCT
cana-3491	496	15	comma	comma	NOUN
cana-3491	496	16	-	-	PUNCT
cana-3491	496	17	separated	separate	VERB
cana-3491	496	18	):	):	PUNCT
cana-3491	496	19	"	"	PUNCT
cana-3491	496	20	)	)	PUNCT
cana-3491	496	21	.split	.split	NOUN
cana-3491	496	22	(	(	PUNCT
cana-3491	496	23	'	'	PUNCT
cana-3491	496	24	,	,	PUNCT
cana-3491	496	25	'	'	PUNCT
cana-3491	496	26	)	)	PUNCT
cana-3491	496	27	]	]	PUNCT
cana-3491	497	1	y1	y1	NOUN
cana-3491	497	2	=	=	PUNCT
cana-3491	498	1	[	[	X
cana-3491	498	2	int(x	int(x	NOUN
cana-3491	498	3	)	)	PUNCT
cana-3491	498	4	for	for	ADP
cana-3491	498	5	x	x	PUNCT
cana-3491	498	6	in	in	ADP
cana-3491	498	7	input("enter	input("enter	X
cana-3491	498	8	the	the	DET
cana-3491	498	9	y1	y1	ADJ
cana-3491	498	10	p	p	NOUN
cana-3491	498	11	-	-	PUNCT
cana-3491	498	12	adic	adic	ADJ
cana-3491	498	13	expansion	expansion	NOUN
cana-3491	498	14	(	(	PUNCT
cana-3491	498	15	comma	comma	NOUN
cana-3491	498	16	-	-	PUNCT
cana-3491	498	17	separated	separate	VERB
cana-3491	498	18	):	):	PUNCT
cana-3491	498	19	"	"	PUNCT
cana-3491	498	20	)	)	PUNCT
cana-3491	498	21	.split	.split	NOUN
cana-3491	498	22	(	(	PUNCT
cana-3491	498	23	'	'	PUNCT
cana-3491	498	24	,	,	PUNCT
cana-3491	498	25	'	'	PUNCT
cana-3491	498	26	)	)	PUNCT
cana-3491	498	27	]	]	PUNCT
cana-3491	499	1	y2	y2	NOUN
cana-3491	499	2	=	=	PUNCT
cana-3491	500	1	[	[	X
cana-3491	500	2	int(x	int(x	NOUN
cana-3491	500	3	)	)	PUNCT
cana-3491	500	4	for	for	ADP
cana-3491	500	5	x	x	PUNCT
cana-3491	500	6	in	in	ADP
cana-3491	500	7	input("enter	input("enter	X
cana-3491	500	8	the	the	DET
cana-3491	500	9	y2	y2	PROPN
cana-3491	500	10	p	p	NOUN
cana-3491	500	11	-	-	PUNCT
cana-3491	500	12	adic	adic	ADJ
cana-3491	500	13	expansion	expansion	NOUN
cana-3491	500	14	(	(	PUNCT
cana-3491	500	15	comma	comma	NOUN
cana-3491	500	16	-	-	PUNCT
cana-3491	500	17	separated	separate	VERB
cana-3491	500	18	):	):	PUNCT
cana-3491	500	19	"	"	PUNCT
cana-3491	500	20	)	)	PUNCT
cana-3491	500	21	.split	.split	NOUN
cana-3491	500	22	(	(	PUNCT
cana-3491	500	23	'	'	PUNCT
cana-3491	500	24	,	,	PUNCT
cana-3491	500	25	'	'	PUNCT
cana-3491	500	26	)	)	PUNCT
cana-3491	500	27	]	]	PUNCT
cana-3491	501	1	p	p	X
cana-3491	501	2	=	=	PUNCT
cana-3491	501	3	int(input("enter	int(input("enter	VERB
cana-3491	501	4	the	the	DET
cana-3491	501	5	base	base	NOUN
cana-3491	501	6	(	(	PUNCT
cana-3491	501	7	prime	prime	ADJ
cana-3491	501	8	number	number	NOUN
cana-3491	501	9	):	):	PUNCT
cana-3491	501	10	"	"	PUNCT
cana-3491	501	11	)	)	PUNCT
cana-3491	501	12	)	)	PUNCT
cana-3491	502	1	a	a	DET
cana-3491	502	2	=	=	PUNCT
cana-3491	502	3	int(input("enter	int(input("enter	NOUN
cana-3491	502	4	the	the	DET
cana-3491	502	5	value	value	NOUN
cana-3491	502	6	of	of	ADP
cana-3491	502	7	a	a	PRON
cana-3491	502	8	:	:	PUNCT
cana-3491	502	9	"	"	PUNCT
cana-3491	502	10	)	)	PUNCT
cana-3491	502	11	)	)	PUNCT
cana-3491	503	1	b	b	X
cana-3491	503	2	=	=	SYM
cana-3491	503	3	int(input("enter	int(input("enter	VERB
cana-3491	503	4	the	the	DET
cana-3491	503	5	value	value	NOUN
cana-3491	503	6	of	of	ADP
cana-3491	503	7	b	b	NOUN
cana-3491	503	8	:	:	PUNCT
cana-3491	503	9	"	"	PUNCT
cana-3491	503	10	)	)	PUNCT
cana-3491	503	11	)	)	PUNCT
cana-3491	503	12	p3_x	p3_x	NOUN
cana-3491	503	13	,	,	PUNCT
cana-3491	503	14	p3_y	p3_y	NOUN
cana-3491	503	15	=	=	SYM
cana-3491	504	1	calculate_p3(x1	calculate_p3(x1	NOUN
cana-3491	504	2	,	,	PUNCT
cana-3491	504	3	x2	x2	PROPN
cana-3491	504	4	,	,	PUNCT
cana-3491	504	5	y1	y1	NOUN
cana-3491	504	6	,	,	PUNCT
cana-3491	504	7	y2	y2	PROPN
cana-3491	504	8	,	,	PUNCT
cana-3491	504	9	p	p	X
cana-3491	504	10	,	,	PUNCT
cana-3491	504	11	a	a	PRON
cana-3491	504	12	,	,	PUNCT
cana-3491	504	13	b	b	NOUN
cana-3491	504	14	)	)	PUNCT
cana-3491	504	15	print(f"p3	print(f"p3	NOUN
cana-3491	504	16	:	:	PUNCT
cana-3491	504	17	(	(	PUNCT
cana-3491	504	18	x	x	X
cana-3491	504	19	:	:	PUNCT
cana-3491	504	20	{	{	PUNCT
cana-3491	504	21	p3_x	p3_x	NOUN
cana-3491	504	22	}	}	PUNCT
cana-3491	504	23	,	,	PUNCT
cana-3491	504	24	y:{p3_y	y:{p3_y	NUM
cana-3491	504	25	}	}	PUNCT
cana-3491	504	26	)	)	PUNCT
cana-3491	504	27	"	"	PUNCT
cana-3491	504	28	)	)	PUNCT
cana-3491	505	1	#	#	NUM
cana-3491	505	2	%	%	NOUN
cana-3491	505	3	%	%	NOUN
cana-3491	505	4	5	5	NUM
cana-3491	505	5	.	.	PUNCT
cana-3491	506	1	conclusions	conclusion	NOUN
cana-3491	506	2	communications	communication	NOUN
cana-3491	506	3	on	on	ADP
cana-3491	506	4	applied	apply	VERB
cana-3491	506	5	nonlinear	nonlinear	ADJ
cana-3491	506	6	analysis	analysis	NOUN
cana-3491	506	7	issn	issn	NOUN
cana-3491	506	8	:	:	PUNCT
cana-3491	506	9	1074	1074	NUM
cana-3491	506	10	-	-	PUNCT
cana-3491	506	11	133x	133x	NUM
cana-3491	506	12	vol	vol	NOUN
cana-3491	506	13	32	32	NUM
cana-3491	506	14	no	no	NOUN
cana-3491	506	15	.	.	PUNCT
cana-3491	507	1	7s	7	NOUN
cana-3491	507	2	(	(	PUNCT
cana-3491	507	3	2025	2025	NUM
cana-3491	507	4	)	)	PUNCT
cana-3491	507	5	882	882	NUM
cana-3491	507	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3491	507	7	in	in	ADP
cana-3491	507	8	this	this	DET
cana-3491	507	9	paper	paper	NOUN
cana-3491	507	10	,	,	PUNCT
cana-3491	507	11	the	the	DET
cana-3491	507	12	focus	focus	NOUN
cana-3491	507	13	is	be	AUX
cana-3491	507	14	on	on	ADP
cana-3491	507	15	the	the	DET
cana-3491	507	16	points	point	NOUN
cana-3491	507	17	on	on	ADP
cana-3491	507	18	elliptic	elliptic	ADJ
cana-3491	507	19	curve	curve	NOUN
cana-3491	507	20	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	507	21	)	)	PUNCT
cana-3491	507	22	over	over	ADP
cana-3491	507	23	𝑝-adic	𝑝-adic	ADJ
cana-3491	507	24	field	field	NOUN
cana-3491	507	25	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	507	26	and	and	CCONJ
cana-3491	507	27	its	its	PRON
cana-3491	507	28	arithmetic	arithmetic	NOUN
cana-3491	507	29	.	.	PUNCT
cana-3491	508	1	all	all	DET
cana-3491	508	2	the	the	DET
cana-3491	508	3	points	point	NOUN
cana-3491	508	4	in	in	ADP
cana-3491	508	5	elliptic	elliptic	ADJ
cana-3491	508	6	curve	curve	NOUN
cana-3491	508	7	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	508	8	)	)	PUNCT
cana-3491	508	9	are	be	AUX
cana-3491	508	10	obtained	obtain	VERB
cana-3491	508	11	by	by	ADP
cana-3491	508	12	starting	start	VERB
cana-3491	508	13	with	with	ADP
cana-3491	508	14	points	point	NOUN
cana-3491	508	15	in	in	ADP
cana-3491	508	16	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	508	17	)	)	PUNCT
cana-3491	508	18	and	and	CCONJ
cana-3491	508	19	lifting	lift	VERB
cana-3491	508	20	to	to	ADP
cana-3491	508	21	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	508	22	𝑝2	𝑝2	NOUN
cana-3491	508	23	)	)	PUNCT
cana-3491	508	24	and	and	CCONJ
cana-3491	508	25	then	then	ADV
cana-3491	508	26	lifting	lift	VERB
cana-3491	508	27	each	each	DET
cana-3491	508	28	point	point	NOUN
cana-3491	508	29	in	in	ADP
cana-3491	508	30	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	508	31	𝑝2	𝑝2	NOUN
cana-3491	508	32	)	)	PUNCT
cana-3491	508	33	to	to	ADP
cana-3491	508	34	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADV
cana-3491	508	35	𝑝3	𝑝3	ADV
cana-3491	508	36	)	)	PUNCT
cana-3491	508	37	and	and	CCONJ
cana-3491	508	38	so	so	ADV
cana-3491	508	39	on	on	ADV
cana-3491	508	40	.	.	PUNCT
cana-3491	509	1	the	the	DET
cana-3491	509	2	arithmetic	arithmetic	NOUN
cana-3491	509	3	of	of	ADP
cana-3491	509	4	points	point	NOUN
cana-3491	509	5	on	on	ADP
cana-3491	509	6	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	509	7	)	)	PUNCT
cana-3491	509	8	may	may	AUX
cana-3491	509	9	be	be	AUX
cana-3491	509	10	evaluated	evaluate	VERB
cana-3491	509	11	by	by	ADP
cana-3491	509	12	implementing	implement	VERB
cana-3491	509	13	arithmetic	arithmetic	NOUN
cana-3491	509	14	of	of	ADP
cana-3491	509	15	points	point	NOUN
cana-3491	509	16	in	in	ADP
cana-3491	509	17	𝐸(𝑄𝑝	𝐸(𝑄𝑝	NOUN
cana-3491	509	18	)	)	PUNCT
cana-3491	509	19	to	to	ADP
cana-3491	509	20	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	509	21	𝑝2	𝑝2	NOUN
cana-3491	509	22	)	)	PUNCT
cana-3491	509	23	.	.	PUNCT
cana-3491	510	1	this	this	DET
cana-3491	510	2	study	study	NOUN
cana-3491	510	3	of	of	ADP
cana-3491	510	4	arithmetic	arithmetic	NOUN
cana-3491	510	5	in	in	ADP
cana-3491	510	6	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	510	7	𝑝2	𝑝2	NOUN
cana-3491	510	8	)	)	PUNCT
cana-3491	510	9	provides	provide	VERB
cana-3491	510	10	a	a	DET
cana-3491	510	11	key	key	ADJ
cana-3491	510	12	space	space	NOUN
cana-3491	510	13	for	for	ADP
cana-3491	510	14	cryptosystems	cryptosystem	NOUN
cana-3491	510	15	with	with	ADP
cana-3491	510	16	𝐸(𝑄𝑝)(𝑚𝑜𝑑	𝐸(𝑄𝑝)(𝑚𝑜𝑑	ADJ
cana-3491	510	17	𝑝2	𝑝2	NOUN
cana-3491	510	18	)	)	PUNCT
cana-3491	510	19	bigger	big	ADJ
cana-3491	510	20	than	than	ADP
cana-3491	510	21	the	the	DET
cana-3491	510	22	key	key	ADJ
cana-3491	510	23	space	space	NOUN
cana-3491	510	24	of	of	ADP
cana-3491	510	25	cryptosystems	cryptosystem	NOUN
cana-3491	510	26	with	with	ADP
cana-3491	510	27	𝐸(𝐹𝑝	𝐸(𝐹𝑝	NOUN
cana-3491	510	28	)	)	PUNCT
cana-3491	510	29	with	with	ADP
cana-3491	510	30	an	an	DET
cana-3491	510	31	increase	increase	NOUN
cana-3491	510	32	in	in	ADP
cana-3491	510	33	level	level	NOUN
cana-3491	510	34	of	of	ADP
cana-3491	510	35	security	security	NOUN
cana-3491	510	36	.	.	PUNCT
cana-3491	511	1	references	reference	NOUN
cana-3491	511	2	[	[	X
cana-3491	511	3	1	1	X
cana-3491	511	4	]	]	PUNCT
cana-3491	511	5	j.	j.	PROPN
cana-3491	511	6	h.	h.	PROPN
cana-3491	511	7	silverman	silverman	PROPN
cana-3491	511	8	and	and	CCONJ
cana-3491	511	9	j.	j.	PROPN
cana-3491	511	10	tate	tate	PROPN
cana-3491	511	11	.	.	PUNCT
cana-3491	512	1	“	"	PUNCT
cana-3491	512	2	rational	rational	ADJ
cana-3491	512	3	points	point	NOUN
cana-3491	512	4	on	on	ADP
cana-3491	512	5	elliptic	elliptic	ADJ
cana-3491	512	6	curves	curve	NOUN
cana-3491	512	7	”	"	PUNCT
cana-3491	512	8	undergraduate	undergraduate	NOUN
cana-3491	512	9	texts	text	NOUN
cana-3491	512	10	in	in	ADP
cana-3491	512	11	mathematics	mathematic	NOUN
cana-3491	512	12	.	.	PUNCT
cana-3491	513	1	springer	springer	NOUN
cana-3491	513	2	-	-	PUNCT
cana-3491	513	3	verlag	verlag	PROPN
cana-3491	513	4	,	,	PUNCT
cana-3491	513	5	new	new	PROPN
cana-3491	513	6	york	york	PROPN
cana-3491	513	7	,	,	PUNCT
cana-3491	513	8	1992	1992	NUM
cana-3491	513	9	.	.	PUNCT
cana-3491	514	1	[	[	X
cana-3491	514	2	2	2	X
cana-3491	514	3	]	]	PUNCT
cana-3491	514	4	j.	j.	PROPN
cana-3491	514	5	h.	h.	PROPN
cana-3491	514	6	silverman	silverman	PROPN
cana-3491	514	7	“	"	PUNCT
cana-3491	514	8	the	the	DET
cana-3491	514	9	arithmetic	arithmetic	NOUN
cana-3491	514	10	of	of	ADP
cana-3491	514	11	elliptic	elliptic	ADJ
cana-3491	514	12	curves	curve	NOUN
cana-3491	514	13	”	"	PUNCT
cana-3491	514	14	volume	volume	NOUN
cana-3491	514	15	106	106	NUM
cana-3491	514	16	of	of	ADP
cana-3491	514	17	graduate	graduate	NOUN
cana-3491	514	18	texts	text	NOUN
cana-3491	514	19	in	in	ADP
cana-3491	514	20	mathematics	mathematic	NOUN
cana-3491	514	21	.	.	PUNCT
cana-3491	515	1	springerverlag	springerverlag	PROPN
cana-3491	515	2	,	,	PUNCT
cana-3491	515	3	new	new	PROPN
cana-3491	515	4	york	york	PROPN
cana-3491	515	5	,	,	PUNCT
cana-3491	515	6	1996	1996	NUM
cana-3491	515	7	.	.	PUNCT
cana-3491	516	1	[	[	X
cana-3491	516	2	3	3	NUM
cana-3491	516	3	]	]	PUNCT
cana-3491	516	4	fernando	fernando	PROPN
cana-3491	516	5	gouvea	gouvea	PROPN
cana-3491	516	6	p	p	NOUN
cana-3491	516	7	-	-	PUNCT
cana-3491	516	8	adic	adic	ADJ
cana-3491	516	9	numbers	number	NOUN
cana-3491	516	10	:	:	PUNCT
cana-3491	516	11	an	an	DET
cana-3491	516	12	introduction	introduction	NOUN
cana-3491	516	13	(	(	PUNCT
cana-3491	516	14	second	second	ADJ
cana-3491	516	15	editon	editon	NOUN
cana-3491	516	16	)	)	PUNCT
cana-3491	516	17	.	.	PUNCT
cana-3491	517	1	springer	springer	NOUN
cana-3491	517	2	,	,	PUNCT
cana-3491	517	3	newyork	newyork	PROPN
cana-3491	517	4	,	,	PUNCT
cana-3491	517	5	1997	1997	NUM
cana-3491	517	6	.	.	PUNCT
cana-3491	518	1	[	[	X
cana-3491	518	2	4	4	NUM
cana-3491	518	3	]	]	X
cana-3491	518	4	neal	neal	PROPN
cana-3491	518	5	koblitz	koblitz	PROPN
cana-3491	518	6	“	"	PUNCT
cana-3491	518	7	a	a	DET
cana-3491	518	8	course	course	NOUN
cana-3491	518	9	in	in	ADP
cana-3491	518	10	number	number	NOUN
cana-3491	518	11	theory	theory	NOUN
cana-3491	518	12	and	and	CCONJ
cana-3491	518	13	cryptography	cryptography	NOUN
cana-3491	518	14	”	"	PUNCT
cana-3491	518	15	isbn	isbn	ADJ
cana-3491	518	16	3	3	NUM
cana-3491	518	17	-	-	SYM
cana-3491	518	18	5780718	5780718	NUM
cana-3491	518	19	spin	spin	NOUN
cana-3491	518	20	10893308	10893308	NUM
cana-3491	518	21	.	.	PUNCT
cana-3491	519	1	[	[	X
cana-3491	519	2	5	5	X
cana-3491	519	3	]	]	PUNCT
cana-3491	519	4	j.	j.	PROPN
cana-3491	519	5	buchmann	buchmann	PROPN
cana-3491	519	6	“	"	PUNCT
cana-3491	519	7	introduction	introduction	NOUN
cana-3491	519	8	to	to	ADP
cana-3491	519	9	cryptography	cryptography	NOUN
cana-3491	519	10	”	"	PUNCT
cana-3491	519	11	,	,	PUNCT
cana-3491	519	12	springer	springer	NOUN
cana-3491	519	13	-	-	PUNCT
cana-3491	519	14	verlag	verlag	PROPN
cana-3491	519	15	2001	2001	NUM
cana-3491	519	16	.	.	PUNCT
cana-3491	520	1	[	[	X
cana-3491	520	2	6	6	NUM
cana-3491	520	3	]	]	PUNCT
cana-3491	520	4	lawrence	lawrence	PROPN
cana-3491	520	5	c.	c.	PROPN
cana-3491	520	6	washington	washington	PROPN
cana-3491	520	7	.	.	PUNCT
cana-3491	521	1	“elliptic	“elliptic	ADJ
cana-3491	521	2	curves	curve	NOUN
cana-3491	521	3	number	number	NOUN
cana-3491	521	4	theory	theory	NOUN
cana-3491	521	5	and	and	CCONJ
cana-3491	521	6	cryptography	cryptography	NOUN
cana-3491	521	7	”	"	PUNCT
cana-3491	521	8	crc	crc	NOUN
cana-3491	521	9	press	press	NOUN
cana-3491	521	10	,	,	PUNCT
cana-3491	521	11	second	second	ADJ
cana-3491	521	12	edition	edition	NOUN
cana-3491	522	1	[	[	X
cana-3491	522	2	7	7	NUM
cana-3491	522	3	]	]	PUNCT
cana-3491	522	4	“	"	PUNCT
cana-3491	522	5	𝑝-adic	𝑝-adic	ADJ
cana-3491	522	6	numbers	number	NOUN
cana-3491	522	7	applied	apply	VERB
cana-3491	522	8	on	on	ADP
cana-3491	522	9	elliptic	elliptic	ADJ
cana-3491	522	10	curve	curve	NOUN
cana-3491	522	11	cryptography	cryptography	NOUN
cana-3491	522	12	”	"	PUNCT
cana-3491	522	13	maherindrainibelahasa	maherindrainibelahasa	PROPN
cana-3491	522	14	,	,	PUNCT
cana-3491	522	15	ravaliminoarimalalason	ravaliminoarimalalason	PROPN
cana-3491	522	16	,	,	PUNCT
cana-3491	522	17	randimbindrainibe	randimbindrainibe	PROPN
cana-3491	522	18	.	.	PUNCT
cana-3491	523	1	vol-5	vol-5	X
cana-3491	523	2	issue-2	issue-2	PROPN
cana-3491	523	3	2019	2019	NUM
cana-3491	523	4	,	,	PUNCT
cana-3491	523	5	ijariie	ijariie	NOUN
cana-3491	523	6	-	-	PUNCT
cana-3491	523	7	issn(o)-2395	issn(o)-2395	NOUN
cana-3491	523	8	-	-	PUNCT
cana-3491	523	9	4396	4396	NUM
cana-3491	523	10	.	.	PUNCT
cana-3491	524	1	[	[	X
cana-3491	524	2	8	8	X
cana-3491	524	3	]	]	X
cana-3491	524	4	rosa	rosa	PROPN
cana-3491	524	5	winter	winter	PROPN
cana-3491	524	6	“	"	PUNCT
cana-3491	524	7	elliptic	elliptic	ADJ
cana-3491	524	8	curves	curve	NOUN
cana-3491	524	9	over	over	ADP
cana-3491	524	10	𝑄𝑝	𝑄𝑝	PROPN
cana-3491	524	11	”	"	PUNCT
cana-3491	524	12	[	[	X
cana-3491	524	13	9	9	NUM
cana-3491	524	14	]	]	PUNCT
cana-3491	524	15	s.	s.	PROPN
cana-3491	524	16	katok	katok	PROPN
cana-3491	524	17	.	.	PUNCT
cana-3491	525	1	"	"	PUNCT
cana-3491	525	2	𝑝-adic	𝑝-adic	ADJ
cana-3491	525	3	analysis	analysis	NOUN
cana-3491	525	4	compared	compare	VERB
cana-3491	525	5	with	with	ADP
cana-3491	525	6	real	real	ADJ
cana-3491	525	7	"	"	PUNCT
cana-3491	525	8	volume	volume	NOUN
cana-3491	525	9	37	37	NUM
cana-3491	525	10	of	of	ADP
cana-3491	525	11	student	student	NOUN
cana-3491	525	12	mathematical	mathematical	ADJ
cana-3491	525	13	library	library	NOUN
cana-3491	525	14	.	.	PUNCT
cana-3491	526	1	american	american	PROPN
cana-3491	526	2	mathematical	mathematical	PROPN
cana-3491	526	3	society	society	NOUN
cana-3491	526	4	,	,	PUNCT
cana-3491	526	5	providence	providence	NOUN
cana-3491	526	6	,	,	PUNCT
cana-3491	526	7	ri	ri	NOUN
cana-3491	526	8	,	,	PUNCT
cana-3491	526	9	2007	2007	NUM
cana-3491	527	1	[	[	X
cana-3491	527	2	10	10	NUM
cana-3491	527	3	]	]	PUNCT
cana-3491	527	4	l.	l.	PROPN
cana-3491	527	5	praveen	praveen	PROPN
cana-3491	527	6	kumar	kumar	PROPN
cana-3491	527	7	"	"	PUNCT
cana-3491	527	8	arithmetic	arithmetic	NOUN
cana-3491	527	9	of	of	ADP
cana-3491	527	10	elliptic	elliptic	ADJ
cana-3491	527	11	curves	curve	NOUN
cana-3491	527	12	with	with	ADP
cana-3491	527	13	affine	affine	NOUN
cana-3491	527	14	and	and	CCONJ
cana-3491	527	15	projective	projective	ADJ
cana-3491	527	16	coordinates	coordinate	NOUN
cana-3491	527	17	and	and	CCONJ
cana-3491	527	18	some	some	DET
cana-3491	527	19	cryptographic	cryptographic	ADJ
cana-3491	527	20	aspects	aspect	NOUN
cana-3491	527	21	"	"	PUNCT
cana-3491	527	22	2015	2015	NUM
cana-3491	527	23	.	.	PUNCT
cana-3491	528	1	[	[	X
cana-3491	528	2	11	11	NUM
cana-3491	528	3	]	]	X
cana-3491	528	4	n.	n.	NOUN
cana-3491	528	5	koblitz	koblitz	PROPN
cana-3491	528	6	.	.	PUNCT
cana-3491	529	1	"	"	PUNCT
cana-3491	529	2	𝑝-adic	𝑝-adic	ADJ
cana-3491	529	3	numbers	number	NOUN
cana-3491	529	4	,	,	PUNCT
cana-3491	529	5	𝑝-adic	𝑝-adic	ADJ
cana-3491	529	6	analysis	analysis	NOUN
cana-3491	529	7	and	and	CCONJ
cana-3491	529	8	zeta	zeta	NOUN
cana-3491	529	9	-	-	PUNCT
cana-3491	529	10	functions	function	NOUN
cana-3491	529	11	"	"	PUNCT
cana-3491	529	12	volume	volume	NOUN
cana-3491	529	13	58	58	NUM
cana-3491	529	14	of	of	ADP
cana-3491	529	15	graduate	graduate	ADJ
cana-3491	529	16	texts	text	NOUN
cana-3491	529	17	in	in	ADP
cana-3491	529	18	mathematics	mathematic	NOUN
cana-3491	529	19	.	.	PUNCT
cana-3491	530	1	springer	springer	NOUN
cana-3491	530	2	-	-	PUNCT
cana-3491	530	3	verlag	verlag	PROPN
cana-3491	530	4	,	,	PUNCT
cana-3491	530	5	new	new	PROPN
cana-3491	530	6	york	york	PROPN
cana-3491	530	7	,	,	PUNCT
cana-3491	530	8	second	second	ADJ
cana-3491	530	9	edition	edition	NOUN
cana-3491	530	10	,	,	PUNCT
cana-3491	530	11	1984	1984	NUM
cana-3491	530	12	.	.	PUNCT
cana-3491	531	1	[	[	X
cana-3491	531	2	12	12	NUM
cana-3491	531	3	]	]	PUNCT
cana-3491	531	4	alain	alain	PROPN
cana-3491	531	5	m.	m.	PROPN
cana-3491	531	6	robert	robert	PROPN
cana-3491	531	7	"	"	PUNCT
cana-3491	531	8	a	a	DET
cana-3491	531	9	course	course	NOUN
cana-3491	531	10	in	in	ADP
cana-3491	531	11	𝑝-adic	𝑝-adic	ADJ
cana-3491	531	12	analysis	analysis	NOUN
cana-3491	531	13	"	"	PUNCT
cana-3491	531	14	volume	volume	NOUN
cana-3491	531	15	198	198	NUM
cana-3491	531	16	of	of	ADP
cana-3491	531	17	graduate	graduate	ADJ
cana-3491	531	18	texts	text	NOUN
cana-3491	531	19	in	in	ADP
cana-3491	531	20	mathematics	mathematics	PROPN
cana-3491	531	21	springer	springer	NOUN
cana-3491	531	22	-	-	PUNCT
cana-3491	531	23	verlag	verlag	PROPN
cana-3491	531	24	2000	2000	NUM
cana-3491	531	25	.	.	PUNCT