id	sid	tid	token	lemma	pos
ejpam-4702	1	1	european	european	PROPN
ejpam-4702	1	2	journal	journal	PROPN
ejpam-4702	1	3	of	of	ADP
ejpam-4702	1	4	pure	pure	ADJ
ejpam-4702	1	5	and	and	CCONJ
ejpam-4702	1	6	applied	apply	VERB
ejpam-4702	1	7	mathematics	mathematic	NOUN
ejpam-4702	1	8	vol	vol	NOUN
ejpam-4702	1	9	.	.	PUNCT
ejpam-4702	2	1	16	16	NUM
ejpam-4702	2	2	,	,	PUNCT
ejpam-4702	2	3	no	no	INTJ
ejpam-4702	2	4	.	.	NOUN
ejpam-4702	2	5	2	2	NUM
ejpam-4702	2	6	,	,	PUNCT
ejpam-4702	2	7	2023	2023	NUM
ejpam-4702	2	8	,	,	PUNCT
ejpam-4702	2	9	724	724	NUM
ejpam-4702	2	10	-	-	SYM
ejpam-4702	2	11	735	735	NUM
ejpam-4702	2	12	issn	issn	PROPN
ejpam-4702	2	13	1307	1307	NUM
ejpam-4702	2	14	-	-	SYM
ejpam-4702	2	15	5543	5543	NUM
ejpam-4702	2	16	–	–	PUNCT
ejpam-4702	3	1	ejpam.com	ejpam.com	X
ejpam-4702	3	2	published	publish	VERB
ejpam-4702	3	3	by	by	ADP
ejpam-4702	3	4	new	new	PROPN
ejpam-4702	3	5	york	york	PROPN
ejpam-4702	3	6	business	business	PROPN
ejpam-4702	3	7	global	global	ADJ
ejpam-4702	3	8	non	non	NOUN
ejpam-4702	3	9	-	-	NOUN
ejpam-4702	3	10	existence	existence	NOUN
ejpam-4702	3	11	of	of	ADP
ejpam-4702	3	12	positive	positive	ADJ
ejpam-4702	3	13	integer	integer	NOUN
ejpam-4702	3	14	solutions	solution	NOUN
ejpam-4702	3	15	of	of	ADP
ejpam-4702	3	16	the	the	DET
ejpam-4702	3	17	diophantine	diophantine	NOUN
ejpam-4702	3	18	equation	equation	NOUN
ejpam-4702	3	19	px	px	X
ejpam-4702	3	20	+	+	CCONJ
ejpam-4702	3	21	(	(	PUNCT
ejpam-4702	3	22	p+	p+	PROPN
ejpam-4702	3	23	2q)y	2q)y	NUM
ejpam-4702	3	24	=	=	SYM
ejpam-4702	3	25	z2	z2	PROPN
ejpam-4702	3	26	,	,	PUNCT
ejpam-4702	3	27	where	where	SCONJ
ejpam-4702	3	28	p	p	X
ejpam-4702	3	29	,	,	PUNCT
ejpam-4702	3	30	q	q	NOUN
ejpam-4702	3	31	and	and	CCONJ
ejpam-4702	3	32	p+	p+	PROPN
ejpam-4702	3	33	2q	2q	NUM
ejpam-4702	3	34	are	be	AUX
ejpam-4702	3	35	prime	prime	ADJ
ejpam-4702	3	36	numbers	number	NOUN
ejpam-4702	3	37	suton	suton	PROPN
ejpam-4702	3	38	tadee1	tadee1	PROPN
ejpam-4702	3	39	,	,	PUNCT
ejpam-4702	3	40	apirat	apirat	PROPN
ejpam-4702	3	41	siraworakun1,∗	siraworakun1,∗	PROPN
ejpam-4702	3	42	1	1	NUM
ejpam-4702	3	43	department	department	NOUN
ejpam-4702	3	44	of	of	ADP
ejpam-4702	3	45	mathematics	mathematic	NOUN
ejpam-4702	3	46	,	,	PUNCT
ejpam-4702	3	47	faculty	faculty	NOUN
ejpam-4702	3	48	of	of	ADP
ejpam-4702	3	49	science	science	NOUN
ejpam-4702	3	50	and	and	CCONJ
ejpam-4702	3	51	technology	technology	NOUN
ejpam-4702	3	52	,	,	PUNCT
ejpam-4702	3	53	thepsatri	thepsatri	VERB
ejpam-4702	3	54	rajabhat	rajabhat	ADJ
ejpam-4702	3	55	university	university	NOUN
ejpam-4702	3	56	,	,	PUNCT
ejpam-4702	3	57	lopburi	lopburi	PROPN
ejpam-4702	3	58	15000	15000	NUM
ejpam-4702	3	59	,	,	PUNCT
ejpam-4702	3	60	thailand	thailand	PROPN
ejpam-4702	3	61	abstract	abstract	PROPN
ejpam-4702	3	62	.	.	PUNCT
ejpam-4702	4	1	the	the	DET
ejpam-4702	4	2	diophantine	diophantine	NOUN
ejpam-4702	4	3	equation	equation	NOUN
ejpam-4702	4	4	px	px	X
ejpam-4702	4	5	+	+	CCONJ
ejpam-4702	4	6	(	(	PUNCT
ejpam-4702	4	7	p	p	X
ejpam-4702	4	8	+	+	CCONJ
ejpam-4702	4	9	2q)y	2q)y	NUM
ejpam-4702	4	10	=	=	SYM
ejpam-4702	4	11	z2	z2	PROPN
ejpam-4702	4	12	,	,	PUNCT
ejpam-4702	4	13	where	where	SCONJ
ejpam-4702	4	14	p	p	X
ejpam-4702	4	15	,	,	PUNCT
ejpam-4702	4	16	q	q	X
ejpam-4702	4	17	and	and	CCONJ
ejpam-4702	4	18	p	p	NOUN
ejpam-4702	4	19	+	+	CCONJ
ejpam-4702	4	20	2q	2q	NUM
ejpam-4702	4	21	are	be	AUX
ejpam-4702	4	22	prime	prime	ADJ
ejpam-4702	4	23	numbers	number	NOUN
ejpam-4702	4	24	,	,	PUNCT
ejpam-4702	4	25	is	be	AUX
ejpam-4702	4	26	studied	study	VERB
ejpam-4702	4	27	widely	widely	ADV
ejpam-4702	4	28	.	.	PUNCT
ejpam-4702	5	1	many	many	ADJ
ejpam-4702	5	2	authors	author	NOUN
ejpam-4702	5	3	give	give	VERB
ejpam-4702	5	4	q	q	PUNCT
ejpam-4702	5	5	as	as	ADP
ejpam-4702	5	6	an	an	DET
ejpam-4702	5	7	explicit	explicit	ADJ
ejpam-4702	5	8	prime	prime	ADJ
ejpam-4702	5	9	number	number	NOUN
ejpam-4702	5	10	and	and	CCONJ
ejpam-4702	5	11	investigate	investigate	VERB
ejpam-4702	5	12	the	the	DET
ejpam-4702	5	13	positive	positive	ADJ
ejpam-4702	5	14	integer	integer	NOUN
ejpam-4702	5	15	solutions	solution	NOUN
ejpam-4702	5	16	and	and	CCONJ
ejpam-4702	5	17	some	some	DET
ejpam-4702	5	18	conditions	condition	NOUN
ejpam-4702	5	19	for	for	ADP
ejpam-4702	5	20	non	non	ADJ
ejpam-4702	5	21	-	-	NOUN
ejpam-4702	5	22	existence	existence	NOUN
ejpam-4702	5	23	of	of	ADP
ejpam-4702	5	24	positive	positive	ADJ
ejpam-4702	5	25	integer	integer	NOUN
ejpam-4702	5	26	solutions	solution	NOUN
ejpam-4702	5	27	.	.	PUNCT
ejpam-4702	6	1	in	in	ADP
ejpam-4702	6	2	this	this	DET
ejpam-4702	6	3	work	work	NOUN
ejpam-4702	6	4	,	,	PUNCT
ejpam-4702	6	5	we	we	PRON
ejpam-4702	6	6	gather	gather	VERB
ejpam-4702	6	7	some	some	DET
ejpam-4702	6	8	conditions	condition	NOUN
ejpam-4702	6	9	for	for	ADP
ejpam-4702	6	10	odd	odd	ADJ
ejpam-4702	6	11	prime	prime	ADJ
ejpam-4702	6	12	numbers	number	NOUN
ejpam-4702	6	13	p	p	NOUN
ejpam-4702	6	14	and	and	CCONJ
ejpam-4702	6	15	q	q	NOUN
ejpam-4702	6	16	for	for	ADP
ejpam-4702	6	17	showing	show	VERB
ejpam-4702	6	18	that	that	SCONJ
ejpam-4702	6	19	the	the	DET
ejpam-4702	6	20	diophantine	diophantine	NOUN
ejpam-4702	6	21	equation	equation	NOUN
ejpam-4702	6	22	px	px	X
ejpam-4702	7	1	+	+	CCONJ
ejpam-4702	7	2	(	(	PUNCT
ejpam-4702	7	3	p	p	X
ejpam-4702	7	4	+	+	CCONJ
ejpam-4702	7	5	2q)y	2q)y	NUM
ejpam-4702	7	6	=	=	SYM
ejpam-4702	7	7	z2	z2	PROPN
ejpam-4702	7	8	has	have	VERB
ejpam-4702	7	9	no	no	DET
ejpam-4702	7	10	positive	positive	ADJ
ejpam-4702	7	11	integer	integer	NOUN
ejpam-4702	7	12	solution	solution	NOUN
ejpam-4702	7	13	.	.	PUNCT
ejpam-4702	8	1	moreover	moreover	ADV
ejpam-4702	8	2	,	,	PUNCT
ejpam-4702	8	3	many	many	ADJ
ejpam-4702	8	4	examples	example	NOUN
ejpam-4702	8	5	of	of	ADP
ejpam-4702	8	6	diophantine	diophantine	NOUN
ejpam-4702	8	7	equations	equation	NOUN
ejpam-4702	8	8	with	with	ADP
ejpam-4702	8	9	no	no	DET
ejpam-4702	8	10	positive	positive	ADJ
ejpam-4702	8	11	integer	integer	NOUN
ejpam-4702	8	12	solution	solution	NOUN
ejpam-4702	8	13	are	be	AUX
ejpam-4702	8	14	illustrated	illustrate	VERB
ejpam-4702	8	15	.	.	PUNCT
ejpam-4702	9	1	2020	2020	NUM
ejpam-4702	9	2	mathematics	mathematics	PROPN
ejpam-4702	9	3	subject	subject	NOUN
ejpam-4702	9	4	classifications	classification	NOUN
ejpam-4702	9	5	:	:	PUNCT
ejpam-4702	9	6	11d61	11d61	NUM
ejpam-4702	9	7	key	key	ADJ
ejpam-4702	9	8	words	word	NOUN
ejpam-4702	9	9	and	and	CCONJ
ejpam-4702	9	10	phrases	phrase	NOUN
ejpam-4702	9	11	:	:	PUNCT
ejpam-4702	9	12	diophantine	diophantine	VERB
ejpam-4702	9	13	equation	equation	NOUN
ejpam-4702	9	14	,	,	PUNCT
ejpam-4702	9	15	legendre	legendre	NOUN
ejpam-4702	9	16	symbol	symbol	NOUN
ejpam-4702	9	17	,	,	PUNCT
ejpam-4702	9	18	the	the	DET
ejpam-4702	9	19	chinese	chinese	ADJ
ejpam-4702	9	20	remainder	remainder	NOUN
ejpam-4702	9	21	theorem	theorem	NOUN
ejpam-4702	9	22	1	1	NUM
ejpam-4702	9	23	.	.	PUNCT
ejpam-4702	10	1	introduction	introduction	NOUN
ejpam-4702	10	2	studying	study	VERB
ejpam-4702	10	3	non	non	ADJ
ejpam-4702	10	4	-	-	ADJ
ejpam-4702	10	5	negative	negative	ADJ
ejpam-4702	10	6	integer	integer	NOUN
ejpam-4702	10	7	solutions	solution	NOUN
ejpam-4702	10	8	of	of	ADP
ejpam-4702	10	9	the	the	DET
ejpam-4702	10	10	diophantine	diophantine	NOUN
ejpam-4702	10	11	equation	equation	NOUN
ejpam-4702	10	12	px	px	X
ejpam-4702	10	13	+	+	CCONJ
ejpam-4702	10	14	qy	qy	NOUN
ejpam-4702	10	15	=	=	SYM
ejpam-4702	10	16	z2	z2	PROPN
ejpam-4702	10	17	,	,	PUNCT
ejpam-4702	10	18	where	where	SCONJ
ejpam-4702	10	19	p	p	NOUN
ejpam-4702	10	20	and	and	CCONJ
ejpam-4702	10	21	q	q	NOUN
ejpam-4702	10	22	are	be	AUX
ejpam-4702	10	23	prime	prime	ADJ
ejpam-4702	10	24	numbers	number	NOUN
ejpam-4702	10	25	,	,	PUNCT
ejpam-4702	10	26	has	have	AUX
ejpam-4702	10	27	been	be	AUX
ejpam-4702	10	28	done	do	VERB
ejpam-4702	10	29	in	in	ADP
ejpam-4702	10	30	numerous	numerous	ADJ
ejpam-4702	10	31	ways	way	NOUN
ejpam-4702	10	32	.	.	PUNCT
ejpam-4702	11	1	one	one	NUM
ejpam-4702	11	2	of	of	ADP
ejpam-4702	11	3	them	they	PRON
ejpam-4702	11	4	is	be	AUX
ejpam-4702	11	5	that	that	SCONJ
ejpam-4702	11	6	p	p	PROPN
ejpam-4702	11	7	and	and	CCONJ
ejpam-4702	11	8	q	q	NOUN
ejpam-4702	11	9	are	be	AUX
ejpam-4702	11	10	given	give	VERB
ejpam-4702	11	11	as	as	ADP
ejpam-4702	11	12	explicit	explicit	ADJ
ejpam-4702	11	13	prime	prime	ADJ
ejpam-4702	11	14	numbers	number	NOUN
ejpam-4702	11	15	.	.	PUNCT
ejpam-4702	12	1	for	for	ADP
ejpam-4702	12	2	example	example	NOUN
ejpam-4702	12	3	,	,	PUNCT
ejpam-4702	12	4	in	in	ADP
ejpam-4702	12	5	[	[	X
ejpam-4702	12	6	4	4	NUM
ejpam-4702	12	7	]	]	PUNCT
ejpam-4702	12	8	and	and	CCONJ
ejpam-4702	12	9	[	[	X
ejpam-4702	12	10	5	5	NUM
ejpam-4702	12	11	]	]	PUNCT
ejpam-4702	12	12	,	,	PUNCT
ejpam-4702	12	13	kumar	kumar	PROPN
ejpam-4702	12	14	,	,	PUNCT
ejpam-4702	12	15	gupta	gupta	PROPN
ejpam-4702	12	16	and	and	CCONJ
ejpam-4702	12	17	kishan	kishan	PROPN
ejpam-4702	12	18	showed	show	VERB
ejpam-4702	12	19	that	that	SCONJ
ejpam-4702	12	20	the	the	DET
ejpam-4702	12	21	diophantine	diophantine	NOUN
ejpam-4702	12	22	equations	equation	NOUN
ejpam-4702	12	23	61x+67y	61x+67y	NUM
ejpam-4702	12	24	=	=	SYM
ejpam-4702	12	25	z2	z2	PROPN
ejpam-4702	12	26	,	,	PUNCT
ejpam-4702	12	27	67x+73y	67x+73y	PUNCT
ejpam-4702	12	28	=	=	SYM
ejpam-4702	12	29	z2	z2	NOUN
ejpam-4702	12	30	,	,	PUNCT
ejpam-4702	12	31	31x	31x	PUNCT
ejpam-4702	12	32	+	+	CCONJ
ejpam-4702	12	33	41y	41y	X
ejpam-4702	12	34	=	=	SYM
ejpam-4702	12	35	z2	z2	PROPN
ejpam-4702	12	36	and	and	CCONJ
ejpam-4702	12	37	61x	61x	NOUN
ejpam-4702	12	38	+	+	SYM
ejpam-4702	12	39	71y	71y	X
ejpam-4702	12	40	=	=	SYM
ejpam-4702	12	41	z2	z2	PROPN
ejpam-4702	12	42	have	have	VERB
ejpam-4702	12	43	no	no	DET
ejpam-4702	12	44	non	non	ADJ
ejpam-4702	12	45	-	-	ADJ
ejpam-4702	12	46	negative	negative	ADJ
ejpam-4702	12	47	integer	integer	NOUN
ejpam-4702	12	48	solution	solution	NOUN
ejpam-4702	12	49	and	and	CCONJ
ejpam-4702	12	50	burshtein	burshtein	ADV
ejpam-4702	12	51	[	[	X
ejpam-4702	12	52	3	3	X
ejpam-4702	12	53	]	]	PUNCT
ejpam-4702	12	54	revealed	reveal	VERB
ejpam-4702	12	55	that	that	SCONJ
ejpam-4702	12	56	the	the	DET
ejpam-4702	12	57	diophantine	diophantine	NOUN
ejpam-4702	12	58	equations	equation	VERB
ejpam-4702	12	59	2x	2x	NUM
ejpam-4702	12	60	+	+	X
ejpam-4702	12	61	11y	11y	NOUN
ejpam-4702	12	62	=	=	SYM
ejpam-4702	12	63	z2	z2	PROPN
ejpam-4702	12	64	and	and	CCONJ
ejpam-4702	12	65	19x	19x	NOUN
ejpam-4702	12	66	+	+	CCONJ
ejpam-4702	12	67	29y	29y	NOUN
ejpam-4702	12	68	=	=	SYM
ejpam-4702	12	69	z2	z2	NOUN
ejpam-4702	12	70	have	have	VERB
ejpam-4702	12	71	no	no	DET
ejpam-4702	12	72	positive	positive	ADJ
ejpam-4702	12	73	integer	integer	NOUN
ejpam-4702	12	74	solutions	solution	NOUN
ejpam-4702	12	75	(	(	PUNCT
ejpam-4702	12	76	x	x	X
ejpam-4702	12	77	,	,	PUNCT
ejpam-4702	12	78	y	y	PROPN
ejpam-4702	12	79	,	,	PUNCT
ejpam-4702	12	80	z	z	NOUN
ejpam-4702	12	81	)	)	PUNCT
ejpam-4702	12	82	.	.	PUNCT
ejpam-4702	13	1	many	many	ADJ
ejpam-4702	13	2	researchers	researcher	NOUN
ejpam-4702	13	3	studied	study	VERB
ejpam-4702	13	4	the	the	DET
ejpam-4702	13	5	diophantine	diophantine	NOUN
ejpam-4702	13	6	equation	equation	NOUN
ejpam-4702	13	7	by	by	ADP
ejpam-4702	13	8	considering	consider	VERB
ejpam-4702	13	9	q	q	PROPN
ejpam-4702	13	10	=	=	SYM
ejpam-4702	13	11	p+k	p+k	PROPN
ejpam-4702	13	12	,	,	PUNCT
ejpam-4702	13	13	where	where	SCONJ
ejpam-4702	13	14	k	k	PROPN
ejpam-4702	13	15	is	be	AUX
ejpam-4702	13	16	an	an	DET
ejpam-4702	13	17	even	even	ADJ
ejpam-4702	13	18	number	number	NOUN
ejpam-4702	13	19	.	.	PUNCT
ejpam-4702	14	1	in	in	ADP
ejpam-4702	14	2	[	[	X
ejpam-4702	14	3	2	2	NUM
ejpam-4702	14	4	]	]	PUNCT
ejpam-4702	14	5	,	,	PUNCT
ejpam-4702	14	6	burshtein	burshtein	ADV
ejpam-4702	14	7	investigated	investigate	VERB
ejpam-4702	14	8	the	the	DET
ejpam-4702	14	9	solutions	solution	NOUN
ejpam-4702	14	10	of	of	ADP
ejpam-4702	14	11	the	the	DET
ejpam-4702	14	12	diophantine	diophantine	NOUN
ejpam-4702	14	13	equation	equation	NOUN
ejpam-4702	14	14	px	px	X
ejpam-4702	15	1	+	+	CCONJ
ejpam-4702	15	2	(	(	PUNCT
ejpam-4702	15	3	p	p	NOUN
ejpam-4702	15	4	+	+	NOUN
ejpam-4702	15	5	6)y	6)y	NOUN
ejpam-4702	15	6	=	=	SYM
ejpam-4702	15	7	z2	z2	PROPN
ejpam-4702	15	8	,	,	PUNCT
ejpam-4702	15	9	where	where	SCONJ
ejpam-4702	15	10	p	p	NOUN
ejpam-4702	15	11	and	and	CCONJ
ejpam-4702	15	12	p	p	NOUN
ejpam-4702	15	13	+	+	CCONJ
ejpam-4702	15	14	6	6	NUM
ejpam-4702	15	15	are	be	AUX
ejpam-4702	15	16	primes	prime	NOUN
ejpam-4702	15	17	and	and	CCONJ
ejpam-4702	15	18	x	x	PUNCT
ejpam-4702	16	1	+	+	CCONJ
ejpam-4702	16	2	y	y	NOUN
ejpam-4702	16	3	=	=	SYM
ejpam-4702	16	4	2	2	NUM
ejpam-4702	16	5	,	,	PUNCT
ejpam-4702	16	6	3	3	NUM
ejpam-4702	16	7	,	,	PUNCT
ejpam-4702	16	8	4	4	NUM
ejpam-4702	16	9	.	.	X
ejpam-4702	17	1	gupta	gupta	PROPN
ejpam-4702	17	2	,	,	PUNCT
ejpam-4702	17	3	kumar	kumar	PROPN
ejpam-4702	17	4	and	and	CCONJ
ejpam-4702	17	5	kishan	kishan	PROPN
ejpam-4702	18	1	[	[	X
ejpam-4702	18	2	6	6	NUM
ejpam-4702	18	3	]	]	PUNCT
ejpam-4702	18	4	studied	study	VERB
ejpam-4702	18	5	the	the	DET
ejpam-4702	18	6	diophantine	diophantine	NOUN
ejpam-4702	18	7	equation	equation	NOUN
ejpam-4702	18	8	px	px	X
ejpam-4702	19	1	+	+	CCONJ
ejpam-4702	19	2	(	(	PUNCT
ejpam-4702	19	3	p	p	NOUN
ejpam-4702	19	4	+	+	NOUN
ejpam-4702	19	5	6)y	6)y	NOUN
ejpam-4702	19	6	=	=	SYM
ejpam-4702	19	7	z2	z2	PROPN
ejpam-4702	19	8	,	,	PUNCT
ejpam-4702	19	9	where	where	SCONJ
ejpam-4702	19	10	p	p	NOUN
ejpam-4702	19	11	and	and	CCONJ
ejpam-4702	19	12	p	p	NOUN
ejpam-4702	19	13	+	+	NOUN
ejpam-4702	19	14	6	6	NUM
ejpam-4702	19	15	are	be	AUX
ejpam-4702	19	16	sexy	sexy	ADJ
ejpam-4702	19	17	primes	prime	NOUN
ejpam-4702	19	18	with	with	ADP
ejpam-4702	19	19	p	p	NOUN
ejpam-4702	19	20	=	=	PROPN
ejpam-4702	19	21	6n	6n	NOUN
ejpam-4702	20	1	+	+	CCONJ
ejpam-4702	20	2	1	1	NUM
ejpam-4702	20	3	and	and	CCONJ
ejpam-4702	20	4	n	n	PRON
ejpam-4702	20	5	is	be	AUX
ejpam-4702	20	6	a	a	DET
ejpam-4702	20	7	natural	natural	ADJ
ejpam-4702	20	8	number	number	NOUN
ejpam-4702	20	9	.	.	PUNCT
ejpam-4702	21	1	burshtein	burshtein	ADV
ejpam-4702	21	2	[	[	X
ejpam-4702	21	3	1	1	X
ejpam-4702	21	4	]	]	PUNCT
ejpam-4702	21	5	showed	show	VERB
ejpam-4702	21	6	that	that	SCONJ
ejpam-4702	21	7	∗corresponding	∗corresponde	VERB
ejpam-4702	21	8	author	author	NOUN
ejpam-4702	21	9	.	.	PUNCT
ejpam-4702	22	1	doi	doi	NOUN
ejpam-4702	22	2	:	:	PUNCT
ejpam-4702	22	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4702	https://doi.org/10.29020/nybg.ejpam.v16i2.4702	NOUN
ejpam-4702	22	4	email	email	NOUN
ejpam-4702	22	5	addresses	address	NOUN
ejpam-4702	22	6	:	:	PUNCT
ejpam-4702	22	7	suton.t@lawasri.tru.ac.th	suton.t@lawasri.tru.ac.th	PROPN
ejpam-4702	22	8	(	(	PUNCT
ejpam-4702	22	9	s.	s.	PROPN
ejpam-4702	22	10	tadee	tadee	PROPN
ejpam-4702	22	11	)	)	PUNCT
ejpam-4702	22	12	,	,	PUNCT
ejpam-4702	22	13	apirat.si@lawasri.tru.ac.th	apirat.si@lawasri.tru.ac.th	PROPN
ejpam-4702	22	14	(	(	PUNCT
ejpam-4702	22	15	a.	a.	NOUN
ejpam-4702	22	16	siraworakun	siraworakun	PROPN
ejpam-4702	22	17	)	)	PUNCT
ejpam-4702	22	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4702	22	19	724	724	NUM
ejpam-4702	23	1	©	©	PROPN
ejpam-4702	23	2	2023	2023	NUM
ejpam-4702	23	3	ejpam	ejpam	NOUN
ejpam-4702	23	4	all	all	DET
ejpam-4702	23	5	rights	right	NOUN
ejpam-4702	23	6	reserved	reserve	VERB
ejpam-4702	23	7	.	.	PUNCT
ejpam-4702	24	1	s.	s.	PROPN
ejpam-4702	24	2	tadee	tadee	PROPN
ejpam-4702	24	3	,	,	PUNCT
ejpam-4702	24	4	a.	a.	NOUN
ejpam-4702	24	5	siraworakun	siraworakun	PROPN
ejpam-4702	24	6	/	/	SYM
ejpam-4702	24	7	eur	eur	PROPN
ejpam-4702	24	8	.	.	PUNCT
ejpam-4702	25	1	j.	j.	PROPN
ejpam-4702	25	2	pure	pure	PROPN
ejpam-4702	25	3	appl	appl	PROPN
ejpam-4702	25	4	.	.	PROPN
ejpam-4702	25	5	math	math	PROPN
ejpam-4702	25	6	,	,	PUNCT
ejpam-4702	25	7	16	16	NUM
ejpam-4702	25	8	(	(	PUNCT
ejpam-4702	25	9	2	2	NUM
ejpam-4702	25	10	)	)	PUNCT
ejpam-4702	25	11	(	(	PUNCT
ejpam-4702	25	12	2023	2023	NUM
ejpam-4702	25	13	)	)	PUNCT
ejpam-4702	25	14	,	,	PUNCT
ejpam-4702	25	15	724	724	NUM
ejpam-4702	25	16	-	-	SYM
ejpam-4702	25	17	735	735	NUM
ejpam-4702	25	18	725	725	NUM
ejpam-4702	25	19	the	the	DET
ejpam-4702	25	20	diophantine	diophantine	NOUN
ejpam-4702	25	21	equation	equation	NOUN
ejpam-4702	25	22	px	px	X
ejpam-4702	25	23	+	+	CCONJ
ejpam-4702	25	24	(	(	PUNCT
ejpam-4702	25	25	p	p	X
ejpam-4702	25	26	+	+	NUM
ejpam-4702	25	27	4)y	4)y	PROPN
ejpam-4702	25	28	=	=	SYM
ejpam-4702	25	29	z2	z2	PROPN
ejpam-4702	25	30	,	,	PUNCT
ejpam-4702	25	31	where	where	SCONJ
ejpam-4702	25	32	p	p	PROPN
ejpam-4702	25	33	>	>	X
ejpam-4702	25	34	3	3	NUM
ejpam-4702	25	35	and	and	CCONJ
ejpam-4702	25	36	p	p	PRON
ejpam-4702	26	1	+	+	CCONJ
ejpam-4702	26	2	4	4	NUM
ejpam-4702	26	3	are	be	AUX
ejpam-4702	26	4	primes	prime	NOUN
ejpam-4702	26	5	,	,	PUNCT
ejpam-4702	26	6	has	have	VERB
ejpam-4702	26	7	no	no	DET
ejpam-4702	26	8	positive	positive	ADJ
ejpam-4702	26	9	integer	integer	NOUN
ejpam-4702	26	10	solutions	solution	NOUN
ejpam-4702	26	11	(	(	PUNCT
ejpam-4702	26	12	x	x	X
ejpam-4702	26	13	,	,	PUNCT
ejpam-4702	26	14	y	y	PROPN
ejpam-4702	26	15	,	,	PUNCT
ejpam-4702	26	16	z	z	NOUN
ejpam-4702	26	17	)	)	PUNCT
ejpam-4702	26	18	.	.	PUNCT
ejpam-4702	27	1	in	in	ADP
ejpam-4702	27	2	addition	addition	NOUN
ejpam-4702	27	3	,	,	PUNCT
ejpam-4702	27	4	rao	rao	NOUN
ejpam-4702	28	1	[	[	X
ejpam-4702	28	2	10	10	NUM
ejpam-4702	28	3	]	]	PUNCT
ejpam-4702	28	4	studied	study	VERB
ejpam-4702	28	5	the	the	DET
ejpam-4702	28	6	diophantine	diophantine	NOUN
ejpam-4702	28	7	equation	equation	NOUN
ejpam-4702	28	8	3x	3x	PRON
ejpam-4702	29	1	+	+	CCONJ
ejpam-4702	29	2	7y	7y	NOUN
ejpam-4702	29	3	=	=	SYM
ejpam-4702	29	4	z2	z2	PROPN
ejpam-4702	29	5	.	.	PUNCT
ejpam-4702	30	1	neres	nere	NOUN
ejpam-4702	31	1	[	[	X
ejpam-4702	31	2	9	9	NUM
ejpam-4702	31	3	]	]	PUNCT
ejpam-4702	31	4	investigated	investigate	VERB
ejpam-4702	31	5	the	the	DET
ejpam-4702	31	6	diophantine	diophantine	NOUN
ejpam-4702	31	7	equation	equation	NOUN
ejpam-4702	31	8	px	px	X
ejpam-4702	32	1	+	+	CCONJ
ejpam-4702	32	2	(	(	PUNCT
ejpam-4702	32	3	p	p	X
ejpam-4702	32	4	+	+	NUM
ejpam-4702	32	5	8)y	8)y	NOUN
ejpam-4702	32	6	=	=	SYM
ejpam-4702	32	7	z2	z2	PROPN
ejpam-4702	32	8	,	,	PUNCT
ejpam-4702	32	9	where	where	SCONJ
ejpam-4702	32	10	p	p	PROPN
ejpam-4702	32	11	>	>	X
ejpam-4702	32	12	3	3	NUM
ejpam-4702	32	13	and	and	CCONJ
ejpam-4702	32	14	p	p	PRON
ejpam-4702	32	15	+	+	NOUN
ejpam-4702	32	16	8	8	NUM
ejpam-4702	32	17	are	be	AUX
ejpam-4702	32	18	primes	prime	NOUN
ejpam-4702	32	19	.	.	PUNCT
ejpam-4702	33	1	moreover	moreover	ADV
ejpam-4702	33	2	,	,	PUNCT
ejpam-4702	33	3	tadee	tadee	X
ejpam-4702	33	4	(	(	PUNCT
ejpam-4702	33	5	[	[	X
ejpam-4702	33	6	13	13	NUM
ejpam-4702	33	7	]	]	PUNCT
ejpam-4702	33	8	,	,	PUNCT
ejpam-4702	33	9	[	[	X
ejpam-4702	33	10	14	14	NUM
ejpam-4702	33	11	]	]	PUNCT
ejpam-4702	33	12	)	)	PUNCT
ejpam-4702	33	13	has	have	AUX
ejpam-4702	33	14	given	give	VERB
ejpam-4702	33	15	the	the	DET
ejpam-4702	33	16	solutions	solution	NOUN
ejpam-4702	33	17	of	of	ADP
ejpam-4702	33	18	the	the	DET
ejpam-4702	33	19	diophantine	diophantine	NOUN
ejpam-4702	33	20	equations	equation	NOUN
ejpam-4702	33	21	px	px	X
ejpam-4702	34	1	+	+	CCONJ
ejpam-4702	34	2	(	(	PUNCT
ejpam-4702	34	3	p	p	X
ejpam-4702	34	4	+	+	NOUN
ejpam-4702	34	5	10)y	10)y	NUM
ejpam-4702	34	6	=	=	SYM
ejpam-4702	34	7	z2	z2	PROPN
ejpam-4702	34	8	and	and	CCONJ
ejpam-4702	34	9	px	px	PROPN
ejpam-4702	35	1	+	+	CCONJ
ejpam-4702	35	2	(	(	PUNCT
ejpam-4702	35	3	p	p	X
ejpam-4702	35	4	+	+	PROPN
ejpam-4702	35	5	14)y	14)y	NUM
ejpam-4702	35	6	=	=	SYM
ejpam-4702	35	7	z2	z2	PROPN
ejpam-4702	35	8	,	,	PUNCT
ejpam-4702	35	9	where	where	SCONJ
ejpam-4702	35	10	p	p	X
ejpam-4702	35	11	,	,	PUNCT
ejpam-4702	35	12	p	p	X
ejpam-4702	35	13	+	+	CCONJ
ejpam-4702	35	14	10	10	NUM
ejpam-4702	35	15	and	and	CCONJ
ejpam-4702	35	16	p+	p+	PROPN
ejpam-4702	35	17	14	14	NUM
ejpam-4702	35	18	are	be	AUX
ejpam-4702	35	19	primes	prime	NOUN
ejpam-4702	35	20	.	.	PUNCT
ejpam-4702	36	1	in	in	ADP
ejpam-4702	36	2	[	[	X
ejpam-4702	36	3	7	7	NUM
ejpam-4702	36	4	]	]	PUNCT
ejpam-4702	36	5	,	,	PUNCT
ejpam-4702	36	6	mina	mina	ADJ
ejpam-4702	36	7	and	and	CCONJ
ejpam-4702	36	8	bacani	bacani	PROPN
ejpam-4702	36	9	use	use	VERB
ejpam-4702	36	10	the	the	DET
ejpam-4702	36	11	concepts	concept	NOUN
ejpam-4702	36	12	of	of	ADP
ejpam-4702	36	13	legendre	legendre	PROPN
ejpam-4702	36	14	symbol	symbol	PROPN
ejpam-4702	36	15	and	and	CCONJ
ejpam-4702	36	16	jacobi	jacobi	PROPN
ejpam-4702	36	17	symbol	symbol	NOUN
ejpam-4702	36	18	to	to	PART
ejpam-4702	36	19	find	find	VERB
ejpam-4702	36	20	some	some	DET
ejpam-4702	36	21	condition	condition	NOUN
ejpam-4702	36	22	for	for	ADP
ejpam-4702	36	23	non	non	ADJ
ejpam-4702	36	24	-	-	NOUN
ejpam-4702	36	25	existence	existence	NOUN
ejpam-4702	36	26	of	of	ADP
ejpam-4702	36	27	solutions	solution	NOUN
ejpam-4702	36	28	of	of	ADP
ejpam-4702	36	29	the	the	DET
ejpam-4702	36	30	diophantine	diophantine	NOUN
ejpam-4702	36	31	equations	equation	NOUN
ejpam-4702	36	32	of	of	ADP
ejpam-4702	36	33	the	the	DET
ejpam-4702	36	34	form	form	NOUN
ejpam-4702	36	35	px	px	X
ejpam-4702	36	36	+	+	CCONJ
ejpam-4702	36	37	qy	qy	NOUN
ejpam-4702	36	38	=	=	SYM
ejpam-4702	36	39	z2n	z2n	PROPN
ejpam-4702	36	40	.	.	PUNCT
ejpam-4702	37	1	two	two	NUM
ejpam-4702	37	2	years	year	NOUN
ejpam-4702	37	3	later	later	ADV
ejpam-4702	37	4	,	,	PUNCT
ejpam-4702	37	5	the	the	DET
ejpam-4702	37	6	solutions	solution	NOUN
ejpam-4702	37	7	of	of	ADP
ejpam-4702	37	8	the	the	DET
ejpam-4702	37	9	diophantine	diophantine	NOUN
ejpam-4702	37	10	equation	equation	NOUN
ejpam-4702	37	11	px	px	X
ejpam-4702	37	12	+	+	CCONJ
ejpam-4702	37	13	(	(	PUNCT
ejpam-4702	37	14	p+	p+	VERB
ejpam-4702	37	15	4k)y	4k)y	X
ejpam-4702	37	16	=	=	SYM
ejpam-4702	37	17	z2	z2	PROPN
ejpam-4702	37	18	,	,	PUNCT
ejpam-4702	37	19	where	where	SCONJ
ejpam-4702	37	20	k	k	PROPN
ejpam-4702	37	21	is	be	AUX
ejpam-4702	37	22	a	a	DET
ejpam-4702	37	23	natural	natural	ADJ
ejpam-4702	37	24	number	number	NOUN
ejpam-4702	37	25	and	and	CCONJ
ejpam-4702	37	26	p	p	NOUN
ejpam-4702	37	27	,	,	PUNCT
ejpam-4702	37	28	p+	p+	VERB
ejpam-4702	37	29	4k	4k	PRON
ejpam-4702	37	30	are	be	AUX
ejpam-4702	37	31	prime	prime	ADJ
ejpam-4702	37	32	numbers	number	NOUN
ejpam-4702	37	33	,	,	PUNCT
ejpam-4702	37	34	were	be	AUX
ejpam-4702	37	35	investigated	investigate	VERB
ejpam-4702	37	36	[	[	PUNCT
ejpam-4702	37	37	8	8	NUM
ejpam-4702	37	38	]	]	PUNCT
ejpam-4702	37	39	.	.	PUNCT
ejpam-4702	38	1	the	the	DET
ejpam-4702	38	2	goal	goal	NOUN
ejpam-4702	38	3	of	of	ADP
ejpam-4702	38	4	this	this	DET
ejpam-4702	38	5	article	article	NOUN
ejpam-4702	38	6	is	be	AUX
ejpam-4702	38	7	to	to	PART
ejpam-4702	38	8	give	give	VERB
ejpam-4702	38	9	some	some	DET
ejpam-4702	38	10	conditions	condition	NOUN
ejpam-4702	38	11	on	on	ADP
ejpam-4702	38	12	primes	prime	NOUN
ejpam-4702	38	13	p	p	NOUN
ejpam-4702	38	14	and	and	CCONJ
ejpam-4702	38	15	q	q	NOUN
ejpam-4702	38	16	to	to	PART
ejpam-4702	38	17	show	show	VERB
ejpam-4702	38	18	that	that	SCONJ
ejpam-4702	38	19	the	the	DET
ejpam-4702	38	20	diophantine	diophantine	NOUN
ejpam-4702	38	21	equation	equation	NOUN
ejpam-4702	38	22	px+(p+2q)y	px+(p+2q)y	NOUN
ejpam-4702	38	23	=	=	SYM
ejpam-4702	38	24	z2	z2	PROPN
ejpam-4702	38	25	,	,	PUNCT
ejpam-4702	38	26	where	where	SCONJ
ejpam-4702	38	27	p	p	X
ejpam-4702	38	28	,	,	PUNCT
ejpam-4702	38	29	q	q	PUNCT
ejpam-4702	38	30	and	and	CCONJ
ejpam-4702	38	31	p+2q	p+2q	NOUN
ejpam-4702	38	32	are	be	AUX
ejpam-4702	38	33	prime	prime	ADJ
ejpam-4702	38	34	numbers	number	NOUN
ejpam-4702	38	35	,	,	PUNCT
ejpam-4702	38	36	has	have	VERB
ejpam-4702	38	37	no	no	DET
ejpam-4702	38	38	positive	positive	ADJ
ejpam-4702	38	39	integer	integer	NOUN
ejpam-4702	38	40	solution	solution	NOUN
ejpam-4702	38	41	.	.	PUNCT
ejpam-4702	39	1	moreover	moreover	ADV
ejpam-4702	39	2	,	,	PUNCT
ejpam-4702	39	3	the	the	DET
ejpam-4702	39	4	forms	form	NOUN
ejpam-4702	39	5	of	of	ADP
ejpam-4702	39	6	odd	odd	ADJ
ejpam-4702	39	7	prime	prime	ADJ
ejpam-4702	39	8	numbers	number	NOUN
ejpam-4702	39	9	p	p	X
ejpam-4702	39	10	,	,	PUNCT
ejpam-4702	39	11	when	when	SCONJ
ejpam-4702	39	12	q	q	NOUN
ejpam-4702	39	13	is	be	AUX
ejpam-4702	39	14	a	a	DET
ejpam-4702	39	15	prime	prime	ADJ
ejpam-4702	39	16	number	number	NOUN
ejpam-4702	39	17	,	,	PUNCT
ejpam-4702	39	18	are	be	AUX
ejpam-4702	39	19	investigated	investigate	VERB
ejpam-4702	39	20	and	and	CCONJ
ejpam-4702	39	21	many	many	ADJ
ejpam-4702	39	22	examples	example	NOUN
ejpam-4702	39	23	of	of	ADP
ejpam-4702	39	24	diophantine	diophantine	NOUN
ejpam-4702	39	25	equations	equation	NOUN
ejpam-4702	39	26	with	with	ADP
ejpam-4702	39	27	no	no	DET
ejpam-4702	39	28	positive	positive	ADJ
ejpam-4702	39	29	integer	integer	NOUN
ejpam-4702	39	30	solution	solution	NOUN
ejpam-4702	39	31	are	be	AUX
ejpam-4702	39	32	demonstrated	demonstrate	VERB
ejpam-4702	39	33	.	.	PUNCT
ejpam-4702	40	1	2	2	X
ejpam-4702	40	2	.	.	X
ejpam-4702	40	3	preliminaries	preliminary	NOUN
ejpam-4702	40	4	first	first	ADV
ejpam-4702	40	5	,	,	PUNCT
ejpam-4702	40	6	we	we	PRON
ejpam-4702	40	7	recall	recall	VERB
ejpam-4702	40	8	some	some	DET
ejpam-4702	40	9	elementary	elementary	ADJ
ejpam-4702	40	10	definitions	definition	NOUN
ejpam-4702	40	11	and	and	CCONJ
ejpam-4702	40	12	theorems	theorem	NOUN
ejpam-4702	40	13	in	in	ADP
ejpam-4702	40	14	number	number	NOUN
ejpam-4702	40	15	theory	theory	NOUN
ejpam-4702	40	16	.	.	PUNCT
ejpam-4702	41	1	see	see	VERB
ejpam-4702	41	2	[	[	X
ejpam-4702	41	3	11	11	NUM
ejpam-4702	41	4	]	]	PUNCT
ejpam-4702	41	5	for	for	ADP
ejpam-4702	41	6	instance	instance	NOUN
ejpam-4702	41	7	.	.	PUNCT
ejpam-4702	42	1	definition	definition	NOUN
ejpam-4702	42	2	1	1	NUM
ejpam-4702	42	3	.	.	PUNCT
ejpam-4702	43	1	let	let	VERB
ejpam-4702	43	2	n	n	PRON
ejpam-4702	43	3	be	be	AUX
ejpam-4702	43	4	a	a	DET
ejpam-4702	43	5	positive	positive	ADJ
ejpam-4702	43	6	integer	integer	NOUN
ejpam-4702	43	7	.	.	PUNCT
ejpam-4702	44	1	the	the	DET
ejpam-4702	44	2	euler	euler	NOUN
ejpam-4702	44	3	phi	phi	ADJ
ejpam-4702	44	4	-	-	PUNCT
ejpam-4702	44	5	function	function	NOUN
ejpam-4702	44	6	ϕ(n	ϕ(n	X
ejpam-4702	44	7	)	)	PUNCT
ejpam-4702	44	8	is	be	AUX
ejpam-4702	44	9	defined	define	VERB
ejpam-4702	44	10	to	to	PART
ejpam-4702	44	11	be	be	AUX
ejpam-4702	44	12	the	the	DET
ejpam-4702	44	13	number	number	NOUN
ejpam-4702	44	14	of	of	ADP
ejpam-4702	44	15	positive	positive	ADJ
ejpam-4702	44	16	integers	integer	NOUN
ejpam-4702	44	17	not	not	PART
ejpam-4702	44	18	exceeding	exceed	VERB
ejpam-4702	44	19	n	n	CCONJ
ejpam-4702	44	20	that	that	PRON
ejpam-4702	44	21	are	be	AUX
ejpam-4702	44	22	relatively	relatively	ADV
ejpam-4702	44	23	prime	prime	ADJ
ejpam-4702	44	24	to	to	ADP
ejpam-4702	44	25	n.	n.	VERB
ejpam-4702	44	26	definition	definition	NOUN
ejpam-4702	44	27	2	2	NUM
ejpam-4702	44	28	.	.	PUNCT
ejpam-4702	45	1	let	let	VERB
ejpam-4702	45	2	a	a	PRON
ejpam-4702	45	3	and	and	CCONJ
ejpam-4702	45	4	n	n	VERB
ejpam-4702	45	5	be	be	AUX
ejpam-4702	45	6	relatively	relatively	ADV
ejpam-4702	45	7	prime	prime	ADJ
ejpam-4702	45	8	integers	integer	NOUN
ejpam-4702	45	9	with	with	ADP
ejpam-4702	45	10	a	a	DET
ejpam-4702	45	11	̸=	̸=	PROPN
ejpam-4702	45	12	0	0	NUM
ejpam-4702	45	13	and	and	CCONJ
ejpam-4702	45	14	n	n	CCONJ
ejpam-4702	45	15	>	>	NOUN
ejpam-4702	45	16	0	0	X
ejpam-4702	45	17	.	.	PUNCT
ejpam-4702	46	1	the	the	DET
ejpam-4702	46	2	least	least	ADV
ejpam-4702	46	3	positive	positive	ADJ
ejpam-4702	46	4	integer	integer	NOUN
ejpam-4702	46	5	x	x	X
ejpam-4702	46	6	such	such	ADJ
ejpam-4702	46	7	that	that	DET
ejpam-4702	46	8	ax	ax	NOUN
ejpam-4702	46	9	≡	≡	PROPN
ejpam-4702	46	10	1	1	NUM
ejpam-4702	46	11	(	(	PUNCT
ejpam-4702	46	12	mod	mod	NOUN
ejpam-4702	46	13	n	n	CCONJ
ejpam-4702	46	14	)	)	PUNCT
ejpam-4702	46	15	is	be	AUX
ejpam-4702	46	16	called	call	VERB
ejpam-4702	46	17	the	the	DET
ejpam-4702	46	18	order	order	NOUN
ejpam-4702	46	19	of	of	ADP
ejpam-4702	46	20	a	a	DET
ejpam-4702	46	21	modulo	modulo	NOUN
ejpam-4702	46	22	n	n	NOUN
ejpam-4702	46	23	and	and	CCONJ
ejpam-4702	46	24	is	be	AUX
ejpam-4702	46	25	denoted	denote	VERB
ejpam-4702	46	26	by	by	ADP
ejpam-4702	46	27	ordna	ordna	ADV
ejpam-4702	46	28	.	.	PUNCT
ejpam-4702	47	1	theorem	theorem	NOUN
ejpam-4702	47	2	1	1	NUM
ejpam-4702	47	3	.	.	PUNCT
ejpam-4702	48	1	(	(	PUNCT
ejpam-4702	48	2	fermat	fermat	PROPN
ejpam-4702	48	3	’s	’s	PART
ejpam-4702	48	4	little	little	ADJ
ejpam-4702	48	5	theorem	theorem	NOUN
ejpam-4702	48	6	)	)	PUNCT
ejpam-4702	48	7	.	.	PUNCT
ejpam-4702	49	1	if	if	SCONJ
ejpam-4702	49	2	p	p	NOUN
ejpam-4702	49	3	is	be	AUX
ejpam-4702	49	4	a	a	DET
ejpam-4702	49	5	prime	prime	ADJ
ejpam-4702	49	6	number	number	NOUN
ejpam-4702	49	7	and	and	CCONJ
ejpam-4702	49	8	a	a	PRON
ejpam-4702	49	9	is	be	AUX
ejpam-4702	49	10	an	an	DET
ejpam-4702	49	11	integer	integer	NOUN
ejpam-4702	49	12	with	with	ADP
ejpam-4702	49	13	p	p	PROPN
ejpam-4702	49	14	∤	∤	PROPN
ejpam-4702	49	15	a	a	PROPN
ejpam-4702	49	16	,	,	PUNCT
ejpam-4702	49	17	then	then	ADV
ejpam-4702	49	18	ap−1	ap−1	PROPN
ejpam-4702	49	19	≡	≡	PROPN
ejpam-4702	49	20	1	1	NUM
ejpam-4702	49	21	(	(	PUNCT
ejpam-4702	49	22	mod	mod	NOUN
ejpam-4702	49	23	p	p	NOUN
ejpam-4702	49	24	)	)	PUNCT
ejpam-4702	49	25	.	.	PUNCT
ejpam-4702	50	1	definition	definition	NOUN
ejpam-4702	50	2	3	3	X
ejpam-4702	50	3	.	.	PUNCT
ejpam-4702	51	1	let	let	VERB
ejpam-4702	51	2	r	r	NOUN
ejpam-4702	51	3	and	and	CCONJ
ejpam-4702	51	4	n	n	CCONJ
ejpam-4702	51	5	be	be	VERB
ejpam-4702	51	6	relatively	relatively	ADV
ejpam-4702	51	7	prime	prime	ADJ
ejpam-4702	51	8	integers	integer	NOUN
ejpam-4702	51	9	with	with	ADP
ejpam-4702	51	10	n	n	NOUN
ejpam-4702	51	11	>	>	X
ejpam-4702	51	12	0	0	NUM
ejpam-4702	51	13	.	.	PUNCT
ejpam-4702	52	1	the	the	DET
ejpam-4702	52	2	integer	integer	NOUN
ejpam-4702	52	3	r	r	NOUN
ejpam-4702	52	4	is	be	AUX
ejpam-4702	52	5	called	call	VERB
ejpam-4702	52	6	a	a	DET
ejpam-4702	52	7	primitive	primitive	ADJ
ejpam-4702	52	8	root	root	NOUN
ejpam-4702	52	9	modulo	modulo	NOUN
ejpam-4702	52	10	n	n	CCONJ
ejpam-4702	52	11	if	if	SCONJ
ejpam-4702	52	12	ordnr	ordnr	PROPN
ejpam-4702	52	13	=	=	SYM
ejpam-4702	52	14	ϕ(n	ϕ(n	PROPN
ejpam-4702	52	15	)	)	PUNCT
ejpam-4702	52	16	.	.	PUNCT
ejpam-4702	53	1	theorem	theorem	NOUN
ejpam-4702	53	2	2	2	NUM
ejpam-4702	53	3	.	.	PUNCT
ejpam-4702	54	1	every	every	DET
ejpam-4702	54	2	prime	prime	ADJ
ejpam-4702	54	3	number	number	NOUN
ejpam-4702	54	4	has	have	VERB
ejpam-4702	54	5	a	a	DET
ejpam-4702	54	6	primitive	primitive	ADJ
ejpam-4702	54	7	root	root	NOUN
ejpam-4702	54	8	.	.	PUNCT
ejpam-4702	55	1	the	the	DET
ejpam-4702	55	2	concepts	concept	NOUN
ejpam-4702	55	3	of	of	ADP
ejpam-4702	55	4	quadratic	quadratic	ADJ
ejpam-4702	55	5	residue	residue	NOUN
ejpam-4702	55	6	and	and	CCONJ
ejpam-4702	55	7	legendre	legendre	NOUN
ejpam-4702	55	8	symbol	symbol	PROPN
ejpam-4702	55	9	have	have	VERB
ejpam-4702	55	10	important	important	ADJ
ejpam-4702	55	11	roles	role	NOUN
ejpam-4702	55	12	in	in	ADP
ejpam-4702	55	13	this	this	DET
ejpam-4702	55	14	paper	paper	NOUN
ejpam-4702	55	15	.	.	PUNCT
ejpam-4702	56	1	definition	definition	NOUN
ejpam-4702	56	2	4	4	NUM
ejpam-4702	56	3	.	.	PUNCT
ejpam-4702	57	1	let	let	VERB
ejpam-4702	57	2	a	a	PRON
ejpam-4702	57	3	and	and	CCONJ
ejpam-4702	57	4	m	m	AUX
ejpam-4702	57	5	be	be	AUX
ejpam-4702	57	6	positive	positive	ADJ
ejpam-4702	57	7	integers	integer	NOUN
ejpam-4702	57	8	with	with	ADP
ejpam-4702	57	9	(	(	PUNCT
ejpam-4702	57	10	a	a	PRON
ejpam-4702	57	11	,	,	PUNCT
ejpam-4702	57	12	m	m	NOUN
ejpam-4702	57	13	)	)	PUNCT
ejpam-4702	57	14	=	=	SYM
ejpam-4702	58	1	1	1	X
ejpam-4702	58	2	.	.	X
ejpam-4702	59	1	we	we	PRON
ejpam-4702	59	2	say	say	VERB
ejpam-4702	59	3	that	that	SCONJ
ejpam-4702	59	4	a	a	PRON
ejpam-4702	59	5	is	be	AUX
ejpam-4702	59	6	a	a	DET
ejpam-4702	59	7	quadratic	quadratic	ADJ
ejpam-4702	59	8	residue	residue	NOUN
ejpam-4702	59	9	of	of	ADP
ejpam-4702	59	10	m	m	PRON
ejpam-4702	59	11	if	if	SCONJ
ejpam-4702	59	12	the	the	DET
ejpam-4702	59	13	congruence	congruence	NOUN
ejpam-4702	59	14	x2	x2	PROPN
ejpam-4702	59	15	≡	≡	PROPN
ejpam-4702	59	16	a	a	DET
ejpam-4702	59	17	(	(	PUNCT
ejpam-4702	59	18	mod	mod	PROPN
ejpam-4702	59	19	m	m	PROPN
ejpam-4702	59	20	)	)	PUNCT
ejpam-4702	59	21	has	have	VERB
ejpam-4702	59	22	a	a	DET
ejpam-4702	59	23	solution	solution	NOUN
ejpam-4702	59	24	.	.	PUNCT
ejpam-4702	60	1	otherwise	otherwise	ADV
ejpam-4702	60	2	,	,	PUNCT
ejpam-4702	60	3	a	a	PRON
ejpam-4702	60	4	is	be	AUX
ejpam-4702	60	5	a	a	DET
ejpam-4702	60	6	quadratic	quadratic	ADJ
ejpam-4702	60	7	nonresidue	nonresidue	NOUN
ejpam-4702	60	8	of	of	ADP
ejpam-4702	60	9	m.	m.	NOUN
ejpam-4702	60	10	s.	s.	PROPN
ejpam-4702	60	11	tadee	tadee	PROPN
ejpam-4702	60	12	,	,	PUNCT
ejpam-4702	60	13	a.	a.	NOUN
ejpam-4702	60	14	siraworakun	siraworakun	PROPN
ejpam-4702	60	15	/	/	SYM
ejpam-4702	60	16	eur	eur	PROPN
ejpam-4702	60	17	.	.	PUNCT
ejpam-4702	61	1	j.	j.	PROPN
ejpam-4702	61	2	pure	pure	PROPN
ejpam-4702	61	3	appl	appl	PROPN
ejpam-4702	61	4	.	.	PROPN
ejpam-4702	61	5	math	math	PROPN
ejpam-4702	61	6	,	,	PUNCT
ejpam-4702	61	7	16	16	NUM
ejpam-4702	61	8	(	(	PUNCT
ejpam-4702	61	9	2	2	NUM
ejpam-4702	61	10	)	)	PUNCT
ejpam-4702	61	11	(	(	PUNCT
ejpam-4702	61	12	2023	2023	NUM
ejpam-4702	61	13	)	)	PUNCT
ejpam-4702	61	14	,	,	PUNCT
ejpam-4702	61	15	724	724	NUM
ejpam-4702	61	16	-	-	SYM
ejpam-4702	61	17	735	735	NUM
ejpam-4702	61	18	726	726	NUM
ejpam-4702	61	19	definition	definition	NOUN
ejpam-4702	61	20	5	5	NUM
ejpam-4702	61	21	.	.	PUNCT
ejpam-4702	62	1	let	let	VERB
ejpam-4702	62	2	p	p	PRON
ejpam-4702	62	3	be	be	AUX
ejpam-4702	62	4	an	an	DET
ejpam-4702	62	5	odd	odd	ADJ
ejpam-4702	62	6	prime	prime	ADJ
ejpam-4702	62	7	number	number	NOUN
ejpam-4702	62	8	and	and	CCONJ
ejpam-4702	62	9	a	a	DET
ejpam-4702	62	10	be	be	AUX
ejpam-4702	62	11	an	an	DET
ejpam-4702	62	12	integer	integer	NOUN
ejpam-4702	62	13	with	with	ADP
ejpam-4702	62	14	p	p	PROPN
ejpam-4702	62	15	∤	∤	PROPN
ejpam-4702	62	16	a.	a.	NOUN
ejpam-4702	62	17	the	the	DET
ejpam-4702	62	18	legendre	legendre	PROPN
ejpam-4702	62	19	symbol	symbol	NOUN
ejpam-4702	62	20	(	(	PUNCT
ejpam-4702	62	21	ap	ap	PROPN
ejpam-4702	62	22	)	)	PUNCT
ejpam-4702	62	23	is	be	AUX
ejpam-4702	62	24	defined	define	VERB
ejpam-4702	62	25	by	by	ADP
ejpam-4702	62	26	(	(	PUNCT
ejpam-4702	62	27	a	a	DET
ejpam-4702	62	28	p	p	NOUN
ejpam-4702	62	29	)	)	PUNCT
ejpam-4702	63	1	=	=	PRON
ejpam-4702	63	2	{	{	PUNCT
ejpam-4702	63	3	1	1	NUM
ejpam-4702	63	4	if	if	SCONJ
ejpam-4702	63	5	a	a	PRON
ejpam-4702	63	6	is	be	AUX
ejpam-4702	63	7	a	a	DET
ejpam-4702	63	8	quadratic	quadratic	ADJ
ejpam-4702	63	9	residue	residue	NOUN
ejpam-4702	63	10	of	of	ADP
ejpam-4702	63	11	p	p	NOUN
ejpam-4702	63	12	−1	−1	NOUN
ejpam-4702	63	13	if	if	SCONJ
ejpam-4702	63	14	a	a	PRON
ejpam-4702	63	15	is	be	AUX
ejpam-4702	63	16	a	a	DET
ejpam-4702	63	17	quadratic	quadratic	ADJ
ejpam-4702	63	18	nonresidue	nonresidue	NOUN
ejpam-4702	63	19	of	of	ADP
ejpam-4702	63	20	p.	p.	NOUN
ejpam-4702	63	21	some	some	DET
ejpam-4702	63	22	properties	property	NOUN
ejpam-4702	63	23	of	of	ADP
ejpam-4702	63	24	legendre	legendre	PROPN
ejpam-4702	63	25	symbol	symbol	NOUN
ejpam-4702	63	26	are	be	AUX
ejpam-4702	63	27	given	give	VERB
ejpam-4702	63	28	in	in	ADP
ejpam-4702	63	29	theorems	theorem	NOUN
ejpam-4702	63	30	3	3	NUM
ejpam-4702	63	31	and	and	CCONJ
ejpam-4702	63	32	4	4	NUM
ejpam-4702	63	33	.	.	X
ejpam-4702	63	34	theorem	theorem	NOUN
ejpam-4702	63	35	3	3	X
ejpam-4702	63	36	.	.	PUNCT
ejpam-4702	64	1	let	let	VERB
ejpam-4702	64	2	p	p	PRON
ejpam-4702	64	3	be	be	AUX
ejpam-4702	64	4	an	an	DET
ejpam-4702	64	5	odd	odd	ADJ
ejpam-4702	64	6	prime	prime	ADJ
ejpam-4702	64	7	number	number	NOUN
ejpam-4702	64	8	and	and	CCONJ
ejpam-4702	64	9	a	a	DET
ejpam-4702	64	10	,	,	PUNCT
ejpam-4702	64	11	b	b	NOUN
ejpam-4702	64	12	be	be	AUX
ejpam-4702	64	13	integers	integer	NOUN
ejpam-4702	64	14	with	with	ADP
ejpam-4702	64	15	p	p	PROPN
ejpam-4702	64	16	∤	∤	PROPN
ejpam-4702	64	17	a	a	PROPN
ejpam-4702	64	18	and	and	CCONJ
ejpam-4702	64	19	p	p	PROPN
ejpam-4702	64	20	∤	∤	PROPN
ejpam-4702	64	21	b.	b.	PROPN
ejpam-4702	65	1	the	the	DET
ejpam-4702	65	2	following	follow	VERB
ejpam-4702	65	3	statements	statement	NOUN
ejpam-4702	65	4	hold	hold	VERB
ejpam-4702	65	5	.	.	PUNCT
ejpam-4702	66	1	(	(	PUNCT
ejpam-4702	66	2	i	i	NOUN
ejpam-4702	66	3	)	)	PUNCT
ejpam-4702	66	4	if	if	SCONJ
ejpam-4702	66	5	a	a	DET
ejpam-4702	66	6	≡	≡	PROPN
ejpam-4702	66	7	b	b	PROPN
ejpam-4702	66	8	(	(	PUNCT
ejpam-4702	66	9	mod	mod	PROPN
ejpam-4702	66	10	p	p	PROPN
ejpam-4702	66	11	)	)	PUNCT
ejpam-4702	66	12	,	,	PUNCT
ejpam-4702	66	13	then	then	ADV
ejpam-4702	66	14	(	(	PUNCT
ejpam-4702	66	15	a	a	DET
ejpam-4702	66	16	p	p	NOUN
ejpam-4702	66	17	)	)	PUNCT
ejpam-4702	66	18	=	=	PUNCT
ejpam-4702	66	19	(	(	PUNCT
ejpam-4702	66	20	b	b	NOUN
ejpam-4702	66	21	p	p	NOUN
ejpam-4702	66	22	)	)	PUNCT
ejpam-4702	66	23	.	.	PUNCT
ejpam-4702	67	1	(	(	PUNCT
ejpam-4702	67	2	ii	ii	NOUN
ejpam-4702	67	3	)	)	PUNCT
ejpam-4702	67	4	(	(	PUNCT
ejpam-4702	67	5	ab	ab	PROPN
ejpam-4702	67	6	p	p	NOUN
ejpam-4702	67	7	)	)	PUNCT
ejpam-4702	67	8	=	=	PRON
ejpam-4702	67	9	(	(	PUNCT
ejpam-4702	67	10	a	a	DET
ejpam-4702	67	11	p	p	NOUN
ejpam-4702	67	12	)	)	PUNCT
ejpam-4702	67	13	(	(	PUNCT
ejpam-4702	67	14	b	b	NOUN
ejpam-4702	67	15	p	p	NOUN
ejpam-4702	67	16	)	)	PUNCT
ejpam-4702	67	17	;	;	PUNCT
ejpam-4702	67	18	(	(	PUNCT
ejpam-4702	67	19	iii	iii	X
ejpam-4702	67	20	)	)	PUNCT
ejpam-4702	67	21	(	(	PUNCT
ejpam-4702	67	22	a2	a2	PROPN
ejpam-4702	67	23	p	p	NOUN
ejpam-4702	67	24	)	)	PUNCT
ejpam-4702	68	1	=	=	SYM
ejpam-4702	68	2	1	1	X
ejpam-4702	68	3	.	.	X
ejpam-4702	68	4	theorem	theorem	NOUN
ejpam-4702	68	5	4	4	NUM
ejpam-4702	68	6	.	.	PUNCT
ejpam-4702	69	1	(	(	PUNCT
ejpam-4702	69	2	the	the	DET
ejpam-4702	69	3	law	law	NOUN
ejpam-4702	69	4	of	of	ADP
ejpam-4702	69	5	quadratic	quadratic	ADJ
ejpam-4702	69	6	reciprocity	reciprocity	NOUN
ejpam-4702	69	7	)	)	PUNCT
ejpam-4702	69	8	.	.	PUNCT
ejpam-4702	70	1	let	let	VERB
ejpam-4702	70	2	p	p	NOUN
ejpam-4702	70	3	and	and	CCONJ
ejpam-4702	70	4	q	q	NOUN
ejpam-4702	70	5	be	be	AUX
ejpam-4702	70	6	distinct	distinct	ADJ
ejpam-4702	70	7	odd	odd	ADJ
ejpam-4702	70	8	prime	prime	ADJ
ejpam-4702	70	9	numbers	number	NOUN
ejpam-4702	70	10	.	.	PUNCT
ejpam-4702	71	1	then	then	ADV
ejpam-4702	71	2	(	(	PUNCT
ejpam-4702	71	3	p	p	X
ejpam-4702	71	4	q	q	NOUN
ejpam-4702	71	5	)	)	PUNCT
ejpam-4702	71	6	(	(	PUNCT
ejpam-4702	71	7	q	q	PROPN
ejpam-4702	71	8	p	p	NOUN
ejpam-4702	71	9	)	)	PUNCT
ejpam-4702	71	10	=	=	SYM
ejpam-4702	71	11	(	(	PUNCT
ejpam-4702	71	12	−1	−1	NOUN
ejpam-4702	71	13	)	)	PUNCT
ejpam-4702	71	14	(	(	PUNCT
ejpam-4702	71	15	p−1	p−1	PROPN
ejpam-4702	71	16	2	2	NUM
ejpam-4702	71	17	)	)	PUNCT
ejpam-4702	71	18	(	(	PUNCT
ejpam-4702	71	19	q−1	q−1	PROPN
ejpam-4702	71	20	2	2	NUM
ejpam-4702	71	21	)	)	PUNCT
ejpam-4702	71	22	.	.	PUNCT
ejpam-4702	72	1	the	the	DET
ejpam-4702	72	2	following	follow	VERB
ejpam-4702	72	3	theorem	theorem	NOUN
ejpam-4702	72	4	shows	show	VERB
ejpam-4702	72	5	the	the	DET
ejpam-4702	72	6	form	form	NOUN
ejpam-4702	72	7	of	of	ADP
ejpam-4702	72	8	odd	odd	ADJ
ejpam-4702	72	9	prime	prime	ADJ
ejpam-4702	72	10	number	number	NOUN
ejpam-4702	72	11	p	p	NOUN
ejpam-4702	72	12	in	in	ADP
ejpam-4702	72	13	the	the	DET
ejpam-4702	72	14	legendre	legendre	NOUN
ejpam-4702	72	15	symbol	symbol	NOUN
ejpam-4702	72	16	(	(	PUNCT
ejpam-4702	72	17	2	2	NUM
ejpam-4702	72	18	p	p	NOUN
ejpam-4702	72	19	)	)	PUNCT
ejpam-4702	72	20	.	.	PUNCT
ejpam-4702	73	1	theorem	theorem	ADJ
ejpam-4702	73	2	5	5	NUM
ejpam-4702	73	3	.	.	PUNCT
ejpam-4702	74	1	let	let	VERB
ejpam-4702	74	2	p	p	PRON
ejpam-4702	74	3	be	be	AUX
ejpam-4702	74	4	an	an	DET
ejpam-4702	74	5	odd	odd	ADJ
ejpam-4702	74	6	prime	prime	ADJ
ejpam-4702	74	7	number	number	NOUN
ejpam-4702	74	8	.	.	PUNCT
ejpam-4702	75	1	then	then	ADV
ejpam-4702	75	2	(	(	PUNCT
ejpam-4702	75	3	2	2	NUM
ejpam-4702	75	4	p	p	NOUN
ejpam-4702	75	5	)	)	PUNCT
ejpam-4702	75	6	=	=	PUNCT
ejpam-4702	75	7	{	{	PUNCT
ejpam-4702	75	8	1	1	NUM
ejpam-4702	75	9	if	if	SCONJ
ejpam-4702	75	10	p	p	DET
ejpam-4702	75	11	≡	≡	PROPN
ejpam-4702	75	12	±1	±1	VERB
ejpam-4702	75	13	(	(	PUNCT
ejpam-4702	75	14	mod	mod	PROPN
ejpam-4702	75	15	8)	8)	NUM
ejpam-4702	75	16	−1	−1	NOUN
ejpam-4702	75	17	if	if	SCONJ
ejpam-4702	75	18	p	p	PRON
ejpam-4702	75	19	≡	≡	PROPN
ejpam-4702	75	20	±3	±3	X
ejpam-4702	75	21	(	(	PUNCT
ejpam-4702	75	22	mod	mod	PROPN
ejpam-4702	75	23	8)	8)	NUM
ejpam-4702	75	24	.	.	PUNCT
ejpam-4702	75	25	theorem	theorem	NOUN
ejpam-4702	75	26	6	6	NUM
ejpam-4702	75	27	.	.	PUNCT
ejpam-4702	76	1	let	let	VERB
ejpam-4702	76	2	p	p	NOUN
ejpam-4702	76	3	and	and	CCONJ
ejpam-4702	76	4	q	q	NOUN
ejpam-4702	76	5	be	be	AUX
ejpam-4702	76	6	distinct	distinct	ADJ
ejpam-4702	76	7	odd	odd	ADJ
ejpam-4702	76	8	prime	prime	ADJ
ejpam-4702	76	9	numbers	number	NOUN
ejpam-4702	76	10	.	.	PUNCT
ejpam-4702	77	1	(	(	PUNCT
ejpam-4702	77	2	i	i	NOUN
ejpam-4702	77	3	)	)	PUNCT
ejpam-4702	77	4	if	if	SCONJ
ejpam-4702	77	5	p	p	PRON
ejpam-4702	77	6	≡	≡	PROPN
ejpam-4702	77	7	1	1	NUM
ejpam-4702	77	8	(	(	PUNCT
ejpam-4702	77	9	mod	mod	NOUN
ejpam-4702	77	10	4	4	NUM
ejpam-4702	77	11	)	)	PUNCT
ejpam-4702	77	12	or	or	CCONJ
ejpam-4702	77	13	q	q	PROPN
ejpam-4702	77	14	≡	≡	PROPN
ejpam-4702	77	15	1	1	NUM
ejpam-4702	77	16	(	(	PUNCT
ejpam-4702	77	17	mod	mod	NOUN
ejpam-4702	77	18	4	4	NUM
ejpam-4702	77	19	)	)	PUNCT
ejpam-4702	77	20	,	,	PUNCT
ejpam-4702	77	21	then	then	ADV
ejpam-4702	77	22	(	(	PUNCT
ejpam-4702	77	23	p	p	X
ejpam-4702	77	24	q	q	X
ejpam-4702	77	25	)	)	PUNCT
ejpam-4702	77	26	=	=	SYM
ejpam-4702	77	27	(	(	PUNCT
ejpam-4702	77	28	q	q	X
ejpam-4702	77	29	p	p	NOUN
ejpam-4702	77	30	)	)	PUNCT
ejpam-4702	77	31	.	.	PUNCT
ejpam-4702	78	1	(	(	PUNCT
ejpam-4702	78	2	ii	ii	X
ejpam-4702	78	3	)	)	PUNCT
ejpam-4702	78	4	if	if	SCONJ
ejpam-4702	78	5	p	p	PRON
ejpam-4702	78	6	≡	≡	PROPN
ejpam-4702	78	7	3	3	NUM
ejpam-4702	78	8	(	(	PUNCT
ejpam-4702	78	9	mod	mod	NOUN
ejpam-4702	78	10	4	4	NUM
ejpam-4702	78	11	)	)	PUNCT
ejpam-4702	78	12	and	and	CCONJ
ejpam-4702	78	13	q	q	PROPN
ejpam-4702	78	14	≡	≡	PROPN
ejpam-4702	78	15	3	3	NUM
ejpam-4702	78	16	(	(	PUNCT
ejpam-4702	78	17	mod	mod	NOUN
ejpam-4702	78	18	4	4	NUM
ejpam-4702	78	19	)	)	PUNCT
ejpam-4702	78	20	,	,	PUNCT
ejpam-4702	78	21	then	then	ADV
ejpam-4702	78	22	(	(	PUNCT
ejpam-4702	78	23	p	p	X
ejpam-4702	78	24	q	q	X
ejpam-4702	78	25	)	)	PUNCT
ejpam-4702	78	26	=	=	SYM
ejpam-4702	79	1	−	−	PROPN
ejpam-4702	79	2	(	(	PUNCT
ejpam-4702	79	3	q	q	NOUN
ejpam-4702	79	4	p	p	NOUN
ejpam-4702	79	5	)	)	PUNCT
ejpam-4702	79	6	.	.	PUNCT
ejpam-4702	80	1	theorem	theorem	ADJ
ejpam-4702	80	2	7	7	NUM
ejpam-4702	80	3	.	.	PUNCT
ejpam-4702	81	1	let	let	VERB
ejpam-4702	81	2	a1	a1	NOUN
ejpam-4702	81	3	,	,	PUNCT
ejpam-4702	81	4	a2	a2	PROPN
ejpam-4702	81	5	,	,	PUNCT
ejpam-4702	81	6	a3	a3	NOUN
ejpam-4702	81	7	,	,	PUNCT
ejpam-4702	81	8	.	.	PUNCT
ejpam-4702	81	9	.	.	PUNCT
ejpam-4702	82	1	.	.	PUNCT
ejpam-4702	83	1	,	,	PUNCT
ejpam-4702	83	2	an	an	DET
ejpam-4702	83	3	be	be	AUX
ejpam-4702	83	4	integers	integer	NOUN
ejpam-4702	83	5	and	and	CCONJ
ejpam-4702	83	6	m1,m2,m3	m1,m2,m3	NOUN
ejpam-4702	83	7	,	,	PUNCT
ejpam-4702	83	8	.	.	PUNCT
ejpam-4702	83	9	.	.	PUNCT
ejpam-4702	83	10	.	.	PUNCT
ejpam-4702	84	1	,	,	PUNCT
ejpam-4702	84	2	mn	mn	PROPN
ejpam-4702	84	3	be	be	VERB
ejpam-4702	84	4	positive	positive	ADJ
ejpam-4702	84	5	integers	integer	NOUN
ejpam-4702	84	6	.	.	PUNCT
ejpam-4702	85	1	then	then	ADV
ejpam-4702	85	2	,	,	PUNCT
ejpam-4702	85	3	the	the	DET
ejpam-4702	85	4	system	system	NOUN
ejpam-4702	85	5	of	of	ADP
ejpam-4702	85	6	congruences	congruence	NOUN
ejpam-4702	85	7	x	x	SYM
ejpam-4702	85	8	≡	≡	ADJ
ejpam-4702	85	9	a1	a1	NOUN
ejpam-4702	85	10	(	(	PUNCT
ejpam-4702	85	11	mod	mod	PROPN
ejpam-4702	85	12	m1	m1	PROPN
ejpam-4702	85	13	)	)	PUNCT
ejpam-4702	85	14	x	x	SYM
ejpam-4702	85	15	≡	≡	PROPN
ejpam-4702	85	16	a2	a2	PROPN
ejpam-4702	85	17	(	(	PUNCT
ejpam-4702	85	18	mod	mod	PROPN
ejpam-4702	85	19	m2	m2	PROPN
ejpam-4702	85	20	)	)	PUNCT
ejpam-4702	85	21	x	x	SYM
ejpam-4702	85	22	≡	≡	PROPN
ejpam-4702	85	23	a3	a3	NOUN
ejpam-4702	85	24	(	(	PUNCT
ejpam-4702	85	25	mod	mod	PROPN
ejpam-4702	85	26	m3	m3	PROPN
ejpam-4702	85	27	)	)	PUNCT
ejpam-4702	85	28	...	...	PUNCT
ejpam-4702	86	1	x	x	X
ejpam-4702	86	2	≡	≡	PROPN
ejpam-4702	86	3	an	an	DET
ejpam-4702	86	4	(	(	PUNCT
ejpam-4702	86	5	mod	mod	PROPN
ejpam-4702	86	6	mn	mn	PROPN
ejpam-4702	86	7	)	)	PUNCT
ejpam-4702	86	8	has	have	VERB
ejpam-4702	86	9	a	a	DET
ejpam-4702	86	10	solution	solution	NOUN
ejpam-4702	86	11	if	if	SCONJ
ejpam-4702	86	12	and	and	CCONJ
ejpam-4702	86	13	only	only	ADV
ejpam-4702	86	14	if	if	SCONJ
ejpam-4702	86	15	(	(	PUNCT
ejpam-4702	86	16	mi	mi	NOUN
ejpam-4702	86	17	,	,	PUNCT
ejpam-4702	86	18	mj)|(ai	mj)|(ai	AUX
ejpam-4702	86	19	−	−	PROPN
ejpam-4702	86	20	aj	aj	PROPN
ejpam-4702	86	21	)	)	PUNCT
ejpam-4702	86	22	for	for	ADP
ejpam-4702	86	23	all	all	DET
ejpam-4702	86	24	pairs	pair	NOUN
ejpam-4702	86	25	of	of	ADP
ejpam-4702	86	26	integer	integer	NOUN
ejpam-4702	86	27	(	(	PUNCT
ejpam-4702	86	28	i	i	PROPN
ejpam-4702	86	29	,	,	PUNCT
ejpam-4702	86	30	j	j	PROPN
ejpam-4702	86	31	)	)	PUNCT
ejpam-4702	86	32	.	.	PUNCT
ejpam-4702	87	1	s.	s.	PROPN
ejpam-4702	87	2	tadee	tadee	PROPN
ejpam-4702	87	3	,	,	PUNCT
ejpam-4702	87	4	a.	a.	NOUN
ejpam-4702	87	5	siraworakun	siraworakun	PROPN
ejpam-4702	87	6	/	/	SYM
ejpam-4702	87	7	eur	eur	PROPN
ejpam-4702	87	8	.	.	PUNCT
ejpam-4702	88	1	j.	j.	PROPN
ejpam-4702	88	2	pure	pure	PROPN
ejpam-4702	88	3	appl	appl	PROPN
ejpam-4702	88	4	.	.	PROPN
ejpam-4702	88	5	math	math	PROPN
ejpam-4702	88	6	,	,	PUNCT
ejpam-4702	88	7	16	16	NUM
ejpam-4702	88	8	(	(	PUNCT
ejpam-4702	88	9	2	2	NUM
ejpam-4702	88	10	)	)	PUNCT
ejpam-4702	88	11	(	(	PUNCT
ejpam-4702	88	12	2023	2023	NUM
ejpam-4702	88	13	)	)	PUNCT
ejpam-4702	88	14	,	,	PUNCT
ejpam-4702	88	15	724	724	NUM
ejpam-4702	88	16	-	-	SYM
ejpam-4702	88	17	735	735	NUM
ejpam-4702	88	18	727	727	NUM
ejpam-4702	88	19	in	in	ADP
ejpam-4702	88	20	2004	2004	NUM
ejpam-4702	88	21	,	,	PUNCT
ejpam-4702	88	22	siraworakun	siraworakun	NOUN
ejpam-4702	88	23	investigated	investigate	VERB
ejpam-4702	88	24	the	the	DET
ejpam-4702	88	25	forms	form	NOUN
ejpam-4702	88	26	of	of	ADP
ejpam-4702	88	27	odd	odd	ADJ
ejpam-4702	88	28	prime	prime	ADJ
ejpam-4702	88	29	numbers	number	NOUN
ejpam-4702	88	30	p	p	PROPN
ejpam-4702	88	31	in	in	ADP
ejpam-4702	88	32	legendre	legendre	PROPN
ejpam-4702	88	33	symbol	symbol	NOUN
ejpam-4702	88	34	(	(	PUNCT
ejpam-4702	88	35	q	q	PROPN
ejpam-4702	88	36	p	p	NOUN
ejpam-4702	88	37	)	)	PUNCT
ejpam-4702	88	38	,	,	PUNCT
ejpam-4702	88	39	where	where	SCONJ
ejpam-4702	88	40	q	q	NOUN
ejpam-4702	88	41	is	be	AUX
ejpam-4702	88	42	an	an	DET
ejpam-4702	88	43	odd	odd	ADJ
ejpam-4702	88	44	prime	prime	ADJ
ejpam-4702	88	45	number	number	NOUN
ejpam-4702	88	46	,	,	PUNCT
ejpam-4702	88	47	in	in	ADP
ejpam-4702	88	48	his	his	PRON
ejpam-4702	88	49	unplublished	unplublished	ADJ
ejpam-4702	88	50	senoir	senoir	NOUN
ejpam-4702	88	51	project	project	NOUN
ejpam-4702	88	52	.	.	PUNCT
ejpam-4702	89	1	we	we	PRON
ejpam-4702	89	2	review	review	VERB
ejpam-4702	89	3	some	some	DET
ejpam-4702	89	4	important	important	ADJ
ejpam-4702	89	5	results	result	NOUN
ejpam-4702	89	6	in	in	ADP
ejpam-4702	89	7	theorem	theorem	ADJ
ejpam-4702	89	8	8	8	NUM
ejpam-4702	89	9	11	11	NUM
ejpam-4702	89	10	.	.	PUNCT
ejpam-4702	90	1	theorem	theorem	VERB
ejpam-4702	90	2	8	8	NUM
ejpam-4702	90	3	.	.	PUNCT
ejpam-4702	91	1	let	let	VERB
ejpam-4702	91	2	p	p	PRON
ejpam-4702	91	3	be	be	AUX
ejpam-4702	91	4	an	an	DET
ejpam-4702	91	5	odd	odd	ADJ
ejpam-4702	91	6	prime	prime	ADJ
ejpam-4702	91	7	number	number	NOUN
ejpam-4702	91	8	.	.	PUNCT
ejpam-4702	92	1	then	then	ADV
ejpam-4702	92	2	there	there	PRON
ejpam-4702	92	3	is	be	VERB
ejpam-4702	92	4	no	no	DET
ejpam-4702	92	5	a	a	DET
ejpam-4702	92	6	primitive	primitive	ADJ
ejpam-4702	92	7	root	root	NOUN
ejpam-4702	92	8	modulo	modulo	NOUN
ejpam-4702	92	9	p	p	NOUN
ejpam-4702	92	10	in	in	ADP
ejpam-4702	92	11	the	the	DET
ejpam-4702	92	12	form	form	NOUN
ejpam-4702	92	13	n2	n2	NOUN
ejpam-4702	92	14	,	,	PUNCT
ejpam-4702	92	15	where	where	SCONJ
ejpam-4702	92	16	n	n	PRON
ejpam-4702	92	17	is	be	AUX
ejpam-4702	92	18	a	a	DET
ejpam-4702	92	19	natural	natural	ADJ
ejpam-4702	92	20	number	number	NOUN
ejpam-4702	92	21	with	with	ADP
ejpam-4702	92	22	n	n	NOUN
ejpam-4702	92	23	<	<	X
ejpam-4702	92	24	p.	p.	NOUN
ejpam-4702	92	25	proof	proof	NOUN
ejpam-4702	92	26	.	.	PUNCT
ejpam-4702	93	1	assume	assume	VERB
ejpam-4702	93	2	that	that	SCONJ
ejpam-4702	93	3	there	there	PRON
ejpam-4702	93	4	is	be	VERB
ejpam-4702	93	5	a	a	DET
ejpam-4702	93	6	primitive	primitive	ADJ
ejpam-4702	93	7	root	root	NOUN
ejpam-4702	93	8	n2	n2	NOUN
ejpam-4702	93	9	0	0	NUM
ejpam-4702	93	10	modulo	modulo	PROPN
ejpam-4702	93	11	p	p	X
ejpam-4702	93	12	,	,	PUNCT
ejpam-4702	93	13	where	where	SCONJ
ejpam-4702	93	14	n0	n0	PROPN
ejpam-4702	93	15	is	be	AUX
ejpam-4702	93	16	a	a	DET
ejpam-4702	93	17	natural	natural	ADJ
ejpam-4702	93	18	number	number	NOUN
ejpam-4702	93	19	with	with	ADP
ejpam-4702	93	20	n0	n0	X
ejpam-4702	93	21	<	<	X
ejpam-4702	93	22	p.	p.	NOUN
ejpam-4702	94	1	then	then	ADV
ejpam-4702	94	2	(	(	PUNCT
ejpam-4702	94	3	n2	n2	PROPN
ejpam-4702	94	4	0	0	NUM
ejpam-4702	94	5	)	)	PUNCT
ejpam-4702	94	6	p−1	p−1	PROPN
ejpam-4702	94	7	≡	≡	PROPN
ejpam-4702	94	8	1	1	NUM
ejpam-4702	94	9	(	(	PUNCT
ejpam-4702	94	10	mod	mod	NOUN
ejpam-4702	94	11	p	p	NOUN
ejpam-4702	94	12	)	)	PUNCT
ejpam-4702	94	13	.	.	PUNCT
ejpam-4702	95	1	so	so	ADV
ejpam-4702	95	2	(	(	PUNCT
ejpam-4702	95	3	np−1	np−1	NOUN
ejpam-4702	95	4	0	0	NUM
ejpam-4702	95	5	)	)	SYM
ejpam-4702	95	6	2	2	NUM
ejpam-4702	95	7	≡	≡	PROPN
ejpam-4702	95	8	1	1	NUM
ejpam-4702	95	9	(	(	PUNCT
ejpam-4702	95	10	mod	mod	NOUN
ejpam-4702	95	11	p	p	NOUN
ejpam-4702	95	12	)	)	PUNCT
ejpam-4702	95	13	.	.	PUNCT
ejpam-4702	96	1	thus	thus	ADV
ejpam-4702	96	2	,	,	PUNCT
ejpam-4702	96	3	np−1	np−1	PROPN
ejpam-4702	96	4	0	0	NUM
ejpam-4702	96	5	≡	≡	PROPN
ejpam-4702	96	6	1	1	NUM
ejpam-4702	96	7	(	(	PUNCT
ejpam-4702	96	8	mod	mod	PROPN
ejpam-4702	96	9	p	p	NOUN
ejpam-4702	96	10	)	)	PUNCT
ejpam-4702	96	11	or	or	CCONJ
ejpam-4702	96	12	np−1	np−1	PROPN
ejpam-4702	96	13	0	0	NUM
ejpam-4702	96	14	≡	≡	PROPN
ejpam-4702	96	15	−1	−1	NOUN
ejpam-4702	96	16	(	(	PUNCT
ejpam-4702	96	17	mod	mod	PROPN
ejpam-4702	96	18	p	p	X
ejpam-4702	96	19	)	)	PUNCT
ejpam-4702	96	20	.	.	PUNCT
ejpam-4702	97	1	if	if	SCONJ
ejpam-4702	97	2	np−1	np−1	PROPN
ejpam-4702	97	3	0	0	NUM
ejpam-4702	97	4	≡	≡	PROPN
ejpam-4702	97	5	1	1	NUM
ejpam-4702	97	6	(	(	PUNCT
ejpam-4702	97	7	mod	mod	PROPN
ejpam-4702	97	8	p	p	PROPN
ejpam-4702	97	9	)	)	PUNCT
ejpam-4702	97	10	,	,	PUNCT
ejpam-4702	97	11	then	then	ADV
ejpam-4702	97	12	(	(	PUNCT
ejpam-4702	97	13	n2	n2	NOUN
ejpam-4702	97	14	0	0	NUM
ejpam-4702	97	15	)	)	PUNCT
ejpam-4702	97	16	p−1	p−1	PROPN
ejpam-4702	97	17	2	2	NUM
ejpam-4702	97	18	≡	≡	PROPN
ejpam-4702	97	19	1	1	NUM
ejpam-4702	97	20	(	(	PUNCT
ejpam-4702	97	21	mod	mod	NOUN
ejpam-4702	97	22	p	p	NOUN
ejpam-4702	97	23	)	)	PUNCT
ejpam-4702	97	24	.	.	PUNCT
ejpam-4702	98	1	it	it	PRON
ejpam-4702	98	2	contradicts	contradict	VERB
ejpam-4702	98	3	to	to	ADP
ejpam-4702	98	4	the	the	DET
ejpam-4702	98	5	order	order	NOUN
ejpam-4702	98	6	of	of	ADP
ejpam-4702	98	7	n2	n2	ADJ
ejpam-4702	98	8	0	0	NUM
ejpam-4702	98	9	modulo	modulo	PROPN
ejpam-4702	98	10	p.	p.	NOUN
ejpam-4702	98	11	hence	hence	ADV
ejpam-4702	98	12	,	,	PUNCT
ejpam-4702	98	13	np−1	np−1	PROPN
ejpam-4702	98	14	0	0	NUM
ejpam-4702	98	15	≡	≡	PROPN
ejpam-4702	98	16	−1	−1	NOUN
ejpam-4702	98	17	(	(	PUNCT
ejpam-4702	98	18	mod	mod	PROPN
ejpam-4702	98	19	p	p	X
ejpam-4702	98	20	)	)	PUNCT
ejpam-4702	98	21	.	.	PUNCT
ejpam-4702	99	1	since	since	SCONJ
ejpam-4702	99	2	n0	n0	X
ejpam-4702	99	3	<	<	X
ejpam-4702	99	4	p	p	X
ejpam-4702	99	5	,	,	PUNCT
ejpam-4702	99	6	it	it	PRON
ejpam-4702	99	7	contradicts	contradict	VERB
ejpam-4702	99	8	to	to	ADP
ejpam-4702	99	9	fermat	fermat	PROPN
ejpam-4702	99	10	’s	’s	PART
ejpam-4702	99	11	little	little	ADJ
ejpam-4702	99	12	theorem	theorem	ADJ
ejpam-4702	99	13	.	.	PUNCT
ejpam-4702	100	1	therefore	therefore	ADV
ejpam-4702	100	2	,	,	PUNCT
ejpam-4702	100	3	there	there	PRON
ejpam-4702	100	4	is	be	VERB
ejpam-4702	100	5	no	no	DET
ejpam-4702	100	6	a	a	DET
ejpam-4702	100	7	primitive	primitive	ADJ
ejpam-4702	100	8	root	root	NOUN
ejpam-4702	100	9	modulo	modulo	NOUN
ejpam-4702	100	10	p	p	NOUN
ejpam-4702	100	11	in	in	ADP
ejpam-4702	100	12	the	the	DET
ejpam-4702	100	13	form	form	NOUN
ejpam-4702	100	14	n2	n2	NOUN
ejpam-4702	100	15	,	,	PUNCT
ejpam-4702	100	16	where	where	SCONJ
ejpam-4702	100	17	n	n	PRON
ejpam-4702	100	18	is	be	AUX
ejpam-4702	100	19	a	a	DET
ejpam-4702	100	20	natural	natural	ADJ
ejpam-4702	100	21	number	number	NOUN
ejpam-4702	100	22	with	with	ADP
ejpam-4702	100	23	n	n	CCONJ
ejpam-4702	100	24	<	<	X
ejpam-4702	100	25	p.	p.	NOUN
ejpam-4702	100	26	theorem	theorem	NOUN
ejpam-4702	100	27	9	9	NUM
ejpam-4702	100	28	.	.	PUNCT
ejpam-4702	101	1	let	let	VERB
ejpam-4702	101	2	p	p	PRON
ejpam-4702	101	3	be	be	AUX
ejpam-4702	101	4	an	an	DET
ejpam-4702	101	5	odd	odd	ADJ
ejpam-4702	101	6	prime	prime	ADJ
ejpam-4702	101	7	number	number	NOUN
ejpam-4702	101	8	and	and	CCONJ
ejpam-4702	101	9	r	r	NOUN
ejpam-4702	101	10	be	be	VERB
ejpam-4702	101	11	a	a	DET
ejpam-4702	101	12	primitive	primitive	ADJ
ejpam-4702	101	13	root	root	NOUN
ejpam-4702	101	14	modulo	modulo	NOUN
ejpam-4702	102	1	p.	p.	NOUN
ejpam-4702	102	2	then	then	ADV
ejpam-4702	102	3	r2	r2	PROPN
ejpam-4702	102	4	,	,	PUNCT
ejpam-4702	102	5	r4	r4	NOUN
ejpam-4702	102	6	,	,	PUNCT
ejpam-4702	102	7	r6	r6	NOUN
ejpam-4702	102	8	,	,	PUNCT
ejpam-4702	102	9	.	.	PUNCT
ejpam-4702	102	10	.	.	PUNCT
ejpam-4702	103	1	.	.	PUNCT
ejpam-4702	104	1	,	,	PUNCT
ejpam-4702	104	2	rp−1	rp−1	NOUN
ejpam-4702	104	3	are	be	AUX
ejpam-4702	104	4	quadratic	quadratic	ADJ
ejpam-4702	104	5	residues	residue	NOUN
ejpam-4702	104	6	of	of	ADP
ejpam-4702	104	7	p	p	NOUN
ejpam-4702	104	8	and	and	CCONJ
ejpam-4702	104	9	r1	r1	PROPN
ejpam-4702	104	10	,	,	PUNCT
ejpam-4702	104	11	r3	r3	PROPN
ejpam-4702	104	12	,	,	PUNCT
ejpam-4702	104	13	r5	r5	PROPN
ejpam-4702	104	14	,	,	PUNCT
ejpam-4702	104	15	.	.	PUNCT
ejpam-4702	104	16	.	.	PUNCT
ejpam-4702	105	1	.	.	PUNCT
ejpam-4702	106	1	,	,	PUNCT
ejpam-4702	106	2	rp−2	rp−2	NOUN
ejpam-4702	106	3	are	be	AUX
ejpam-4702	106	4	quadratic	quadratic	ADJ
ejpam-4702	106	5	nonresidues	nonresidue	NOUN
ejpam-4702	106	6	of	of	ADP
ejpam-4702	106	7	p.	p.	NOUN
ejpam-4702	106	8	proof	proof	NOUN
ejpam-4702	106	9	.	.	PUNCT
ejpam-4702	107	1	it	it	PRON
ejpam-4702	107	2	is	be	AUX
ejpam-4702	107	3	obvious	obvious	ADJ
ejpam-4702	107	4	that	that	SCONJ
ejpam-4702	107	5	r2	r2	NOUN
ejpam-4702	107	6	,	,	PUNCT
ejpam-4702	107	7	r4	r4	NOUN
ejpam-4702	107	8	,	,	PUNCT
ejpam-4702	107	9	r6	r6	NOUN
ejpam-4702	107	10	,	,	PUNCT
ejpam-4702	107	11	.	.	PUNCT
ejpam-4702	107	12	.	.	PUNCT
ejpam-4702	108	1	.	.	PUNCT
ejpam-4702	109	1	,	,	PUNCT
ejpam-4702	109	2	rp−1	rp−1	NOUN
ejpam-4702	109	3	are	be	AUX
ejpam-4702	109	4	quadratic	quadratic	ADJ
ejpam-4702	109	5	residues	residue	NOUN
ejpam-4702	109	6	of	of	ADP
ejpam-4702	109	7	p.	p.	NOUN
ejpam-4702	109	8	since	since	SCONJ
ejpam-4702	109	9	r	r	NOUN
ejpam-4702	109	10	is	be	AUX
ejpam-4702	109	11	a	a	DET
ejpam-4702	109	12	primitive	primitive	ADJ
ejpam-4702	109	13	root	root	NOUN
ejpam-4702	109	14	modulo	modulo	NOUN
ejpam-4702	109	15	p	p	NOUN
ejpam-4702	109	16	and	and	CCONJ
ejpam-4702	109	17	theorem	theorem	ADJ
ejpam-4702	109	18	8	8	NUM
ejpam-4702	109	19	,	,	PUNCT
ejpam-4702	109	20	there	there	PRON
ejpam-4702	109	21	is	be	VERB
ejpam-4702	109	22	no	no	DET
ejpam-4702	109	23	a	a	DET
ejpam-4702	109	24	natural	natural	ADJ
ejpam-4702	109	25	number	number	NOUN
ejpam-4702	109	26	n0	n0	NUM
ejpam-4702	109	27	with	with	ADP
ejpam-4702	109	28	n0	n0	X
ejpam-4702	109	29	<	<	X
ejpam-4702	109	30	p	p	X
ejpam-4702	109	31	such	such	ADJ
ejpam-4702	109	32	that	that	DET
ejpam-4702	109	33	n2	n2	NOUN
ejpam-4702	109	34	0	0	NUM
ejpam-4702	110	1	≡	≡	PROPN
ejpam-4702	110	2	r	r	NOUN
ejpam-4702	110	3	(	(	PUNCT
ejpam-4702	110	4	mod	mod	PROPN
ejpam-4702	110	5	p	p	NOUN
ejpam-4702	110	6	)	)	PUNCT
ejpam-4702	110	7	.	.	PUNCT
ejpam-4702	111	1	then	then	ADV
ejpam-4702	111	2	(	(	PUNCT
ejpam-4702	111	3	r	r	NOUN
ejpam-4702	111	4	p	p	NOUN
ejpam-4702	111	5	)	)	PUNCT
ejpam-4702	111	6	=	=	SYM
ejpam-4702	111	7	−1	−1	NOUN
ejpam-4702	111	8	.	.	PUNCT
ejpam-4702	112	1	by	by	ADP
ejpam-4702	112	2	theorem	theorem	ADJ
ejpam-4702	112	3	3(ii	3(ii	NUM
ejpam-4702	112	4	)	)	PUNCT
ejpam-4702	112	5	,	,	PUNCT
ejpam-4702	112	6	(	(	PUNCT
ejpam-4702	112	7	iii	iii	NOUN
ejpam-4702	112	8	)	)	PUNCT
ejpam-4702	112	9	,	,	PUNCT
ejpam-4702	112	10	we	we	PRON
ejpam-4702	112	11	have	have	VERB
ejpam-4702	112	12	(	(	PUNCT
ejpam-4702	112	13	r3	r3	PROPN
ejpam-4702	112	14	p	p	NOUN
ejpam-4702	112	15	)	)	PUNCT
ejpam-4702	112	16	=	=	SYM
ejpam-4702	113	1	(	(	PUNCT
ejpam-4702	113	2	r5	r5	PROPN
ejpam-4702	113	3	p	p	NOUN
ejpam-4702	113	4	)	)	PUNCT
ejpam-4702	114	1	=	=	SYM
ejpam-4702	114	2	·	·	PUNCT
ejpam-4702	114	3	·	·	PUNCT
ejpam-4702	114	4	·	·	PUNCT
ejpam-4702	115	1	=	=	PUNCT
ejpam-4702	115	2	(	(	PUNCT
ejpam-4702	115	3	rp−2	rp−2	NOUN
ejpam-4702	115	4	p	p	NOUN
ejpam-4702	115	5	)	)	PUNCT
ejpam-4702	115	6	=	=	SYM
ejpam-4702	115	7	−1	−1	NOUN
ejpam-4702	115	8	.	.	PUNCT
ejpam-4702	116	1	hence	hence	ADV
ejpam-4702	116	2	,	,	PUNCT
ejpam-4702	116	3	r1	r1	PROPN
ejpam-4702	116	4	,	,	PUNCT
ejpam-4702	116	5	r3	r3	PROPN
ejpam-4702	116	6	,	,	PUNCT
ejpam-4702	116	7	r5	r5	PROPN
ejpam-4702	116	8	,	,	PUNCT
ejpam-4702	116	9	.	.	PUNCT
ejpam-4702	116	10	.	.	PUNCT
ejpam-4702	116	11	.	.	PUNCT
ejpam-4702	117	1	,	,	PUNCT
ejpam-4702	117	2	rp−2	rp−2	NOUN
ejpam-4702	117	3	are	be	AUX
ejpam-4702	117	4	quadratic	quadratic	ADJ
ejpam-4702	117	5	nonresidues	nonresidue	NOUN
ejpam-4702	117	6	of	of	ADP
ejpam-4702	117	7	p.	p.	NOUN
ejpam-4702	117	8	to	to	PART
ejpam-4702	117	9	find	find	VERB
ejpam-4702	117	10	the	the	DET
ejpam-4702	117	11	forms	form	NOUN
ejpam-4702	117	12	of	of	ADP
ejpam-4702	117	13	odd	odd	ADJ
ejpam-4702	117	14	prime	prime	ADJ
ejpam-4702	117	15	numbers	number	NOUN
ejpam-4702	117	16	p	p	PROPN
ejpam-4702	117	17	in	in	ADP
ejpam-4702	117	18	legendre	legendre	PROPN
ejpam-4702	117	19	symbol	symbol	NOUN
ejpam-4702	117	20	(	(	PUNCT
ejpam-4702	117	21	q	q	PROPN
ejpam-4702	117	22	p	p	NOUN
ejpam-4702	117	23	)	)	PUNCT
ejpam-4702	117	24	,	,	PUNCT
ejpam-4702	117	25	where	where	SCONJ
ejpam-4702	117	26	q	q	NOUN
ejpam-4702	117	27	is	be	AUX
ejpam-4702	117	28	an	an	DET
ejpam-4702	117	29	odd	odd	ADJ
ejpam-4702	117	30	prime	prime	ADJ
ejpam-4702	117	31	number	number	NOUN
ejpam-4702	117	32	,	,	PUNCT
ejpam-4702	117	33	we	we	PRON
ejpam-4702	117	34	use	use	VERB
ejpam-4702	117	35	the	the	DET
ejpam-4702	117	36	chinese	chinese	ADJ
ejpam-4702	117	37	remainder	remainder	NOUN
ejpam-4702	117	38	theorem	theorem	NOUN
ejpam-4702	117	39	for	for	ADP
ejpam-4702	117	40	solving	solve	VERB
ejpam-4702	117	41	the	the	DET
ejpam-4702	117	42	system	system	NOUN
ejpam-4702	117	43	of	of	ADP
ejpam-4702	117	44	congruences	congruence	NOUN
ejpam-4702	117	45	.	.	PUNCT
ejpam-4702	118	1	theorem	theorem	NOUN
ejpam-4702	118	2	10	10	NUM
ejpam-4702	118	3	.	.	PUNCT
ejpam-4702	119	1	let	let	VERB
ejpam-4702	119	2	p	p	NOUN
ejpam-4702	119	3	and	and	CCONJ
ejpam-4702	119	4	q	q	NOUN
ejpam-4702	119	5	be	be	AUX
ejpam-4702	119	6	distinct	distinct	ADJ
ejpam-4702	119	7	odd	odd	ADJ
ejpam-4702	119	8	prime	prime	ADJ
ejpam-4702	119	9	numbers	number	NOUN
ejpam-4702	119	10	with	with	ADP
ejpam-4702	119	11	q	q	PROPN
ejpam-4702	119	12	≡	≡	PROPN
ejpam-4702	119	13	1	1	NUM
ejpam-4702	119	14	(	(	PUNCT
ejpam-4702	119	15	mod	mod	NOUN
ejpam-4702	119	16	4	4	NUM
ejpam-4702	119	17	)	)	PUNCT
ejpam-4702	119	18	.	.	PUNCT
ejpam-4702	120	1	then	then	ADV
ejpam-4702	120	2	(	(	PUNCT
ejpam-4702	120	3	q	q	PROPN
ejpam-4702	120	4	p	p	NOUN
ejpam-4702	120	5	)	)	PUNCT
ejpam-4702	120	6	=	=	PUNCT
ejpam-4702	120	7	{	{	PUNCT
ejpam-4702	120	8	1	1	NUM
ejpam-4702	120	9	if	if	SCONJ
ejpam-4702	120	10	p	p	PRON
ejpam-4702	120	11	≡	≡	PROPN
ejpam-4702	120	12	q	q	PROPN
ejpam-4702	121	1	+	+	NUM
ejpam-4702	121	2	rs1q	rs1q	PROPN
ejpam-4702	121	3	+	+	CCONJ
ejpam-4702	121	4	rs1	rs1	PROPN
ejpam-4702	121	5	(	(	PUNCT
ejpam-4702	121	6	mod	mod	NOUN
ejpam-4702	121	7	2q	2q	NUM
ejpam-4702	121	8	)	)	PUNCT
ejpam-4702	121	9	−1	−1	NOUN
ejpam-4702	121	10	if	if	SCONJ
ejpam-4702	121	11	p	p	PRON
ejpam-4702	121	12	≡	≡	PROPN
ejpam-4702	121	13	q	q	PROPN
ejpam-4702	122	1	+	+	CCONJ
ejpam-4702	122	2	rs2q	rs2q	NUM
ejpam-4702	122	3	+	+	CCONJ
ejpam-4702	122	4	rs2	rs2	NOUN
ejpam-4702	122	5	(	(	PUNCT
ejpam-4702	122	6	mod	mod	ADJ
ejpam-4702	122	7	2q	2q	NUM
ejpam-4702	122	8	)	)	PUNCT
ejpam-4702	122	9	,	,	PUNCT
ejpam-4702	122	10	where	where	SCONJ
ejpam-4702	122	11	s1	s1	PROPN
ejpam-4702	122	12	∈	∈	PROPN
ejpam-4702	122	13	{	{	PUNCT
ejpam-4702	122	14	2	2	NUM
ejpam-4702	122	15	,	,	PUNCT
ejpam-4702	122	16	4	4	NUM
ejpam-4702	122	17	,	,	PUNCT
ejpam-4702	122	18	6	6	NUM
ejpam-4702	122	19	,	,	PUNCT
ejpam-4702	122	20	.	.	PUNCT
ejpam-4702	122	21	.	.	PUNCT
ejpam-4702	122	22	.	.	PUNCT
ejpam-4702	123	1	,	,	PUNCT
ejpam-4702	123	2	q−	q−	PROPN
ejpam-4702	123	3	1	1	NUM
ejpam-4702	123	4	}	}	PUNCT
ejpam-4702	123	5	,	,	PUNCT
ejpam-4702	123	6	s2	s2	PROPN
ejpam-4702	123	7	∈	∈	PROPN
ejpam-4702	123	8	{	{	PUNCT
ejpam-4702	123	9	1	1	NUM
ejpam-4702	123	10	,	,	PUNCT
ejpam-4702	123	11	3	3	NUM
ejpam-4702	123	12	,	,	PUNCT
ejpam-4702	123	13	5	5	NUM
ejpam-4702	123	14	,	,	PUNCT
ejpam-4702	123	15	.	.	PUNCT
ejpam-4702	123	16	.	.	PUNCT
ejpam-4702	124	1	.	.	PUNCT
ejpam-4702	125	1	,	,	PUNCT
ejpam-4702	125	2	q−	q−	PROPN
ejpam-4702	125	3	2	2	NUM
ejpam-4702	125	4	}	}	PUNCT
ejpam-4702	125	5	and	and	CCONJ
ejpam-4702	125	6	r	r	NOUN
ejpam-4702	125	7	is	be	AUX
ejpam-4702	125	8	a	a	DET
ejpam-4702	125	9	primitive	primitive	ADJ
ejpam-4702	125	10	root	root	NOUN
ejpam-4702	125	11	modulo	modulo	NOUN
ejpam-4702	125	12	q.	q.	NOUN
ejpam-4702	125	13	proof	proof	NOUN
ejpam-4702	125	14	.	.	PUNCT
ejpam-4702	126	1	since	since	SCONJ
ejpam-4702	126	2	q	q	PROPN
ejpam-4702	126	3	≡	≡	PROPN
ejpam-4702	126	4	1	1	NUM
ejpam-4702	126	5	(	(	PUNCT
ejpam-4702	126	6	mod	mod	NOUN
ejpam-4702	126	7	4	4	NUM
ejpam-4702	126	8	)	)	PUNCT
ejpam-4702	126	9	,	,	PUNCT
ejpam-4702	126	10	we	we	PRON
ejpam-4702	126	11	have	have	VERB
ejpam-4702	126	12	(	(	PUNCT
ejpam-4702	126	13	q	q	X
ejpam-4702	126	14	p	p	NOUN
ejpam-4702	126	15	)	)	PUNCT
ejpam-4702	126	16	=	=	PUNCT
ejpam-4702	127	1	(	(	PUNCT
ejpam-4702	127	2	p	p	X
ejpam-4702	127	3	q	q	X
ejpam-4702	127	4	)	)	PUNCT
ejpam-4702	127	5	by	by	ADP
ejpam-4702	127	6	theorem	theorem	NOUN
ejpam-4702	127	7	6(i	6(i	NUM
ejpam-4702	127	8	)	)	PUNCT
ejpam-4702	127	9	.	.	PUNCT
ejpam-4702	128	1	let	let	VERB
ejpam-4702	128	2	r	r	PRON
ejpam-4702	128	3	be	be	AUX
ejpam-4702	128	4	a	a	DET
ejpam-4702	128	5	primitive	primitive	ADJ
ejpam-4702	128	6	root	root	NOUN
ejpam-4702	128	7	modulo	modulo	NOUN
ejpam-4702	128	8	q.	q.	NOUN
ejpam-4702	128	9	by	by	ADP
ejpam-4702	128	10	theorem	theorem	NOUN
ejpam-4702	128	11	9	9	NUM
ejpam-4702	128	12	,	,	PUNCT
ejpam-4702	128	13	we	we	PRON
ejpam-4702	128	14	obtain	obtain	VERB
ejpam-4702	128	15	that	that	PRON
ejpam-4702	128	16	(	(	PUNCT
ejpam-4702	128	17	p	p	X
ejpam-4702	128	18	q	q	X
ejpam-4702	128	19	)	)	PUNCT
ejpam-4702	129	1	=	=	PRON
ejpam-4702	129	2	{	{	PUNCT
ejpam-4702	129	3	1	1	NUM
ejpam-4702	129	4	if	if	SCONJ
ejpam-4702	129	5	p	p	PRON
ejpam-4702	129	6	≡	≡	PROPN
ejpam-4702	129	7	rs1	rs1	PROPN
ejpam-4702	129	8	(	(	PUNCT
ejpam-4702	129	9	mod	mod	PROPN
ejpam-4702	129	10	q	q	NOUN
ejpam-4702	129	11	)	)	PUNCT
ejpam-4702	129	12	−1	−1	NOUN
ejpam-4702	129	13	if	if	SCONJ
ejpam-4702	129	14	p	p	PRON
ejpam-4702	129	15	≡	≡	PROPN
ejpam-4702	129	16	rs2	rs2	NOUN
ejpam-4702	129	17	(	(	PUNCT
ejpam-4702	129	18	mod	mod	PROPN
ejpam-4702	129	19	q	q	NOUN
ejpam-4702	129	20	)	)	PUNCT
ejpam-4702	129	21	,	,	PUNCT
ejpam-4702	129	22	where	where	SCONJ
ejpam-4702	129	23	s1	s1	PROPN
ejpam-4702	129	24	∈	∈	PROPN
ejpam-4702	129	25	{	{	PUNCT
ejpam-4702	129	26	2	2	NUM
ejpam-4702	129	27	,	,	PUNCT
ejpam-4702	129	28	4	4	NUM
ejpam-4702	129	29	,	,	PUNCT
ejpam-4702	129	30	6	6	NUM
ejpam-4702	129	31	,	,	PUNCT
ejpam-4702	129	32	.	.	PUNCT
ejpam-4702	129	33	.	.	PUNCT
ejpam-4702	129	34	.	.	PUNCT
ejpam-4702	130	1	,	,	PUNCT
ejpam-4702	130	2	q	q	NOUN
ejpam-4702	130	3	−	−	NOUN
ejpam-4702	130	4	1	1	NUM
ejpam-4702	130	5	}	}	PUNCT
ejpam-4702	130	6	and	and	CCONJ
ejpam-4702	130	7	s2	s2	PROPN
ejpam-4702	130	8	∈	∈	PROPN
ejpam-4702	130	9	{	{	PUNCT
ejpam-4702	130	10	1	1	NUM
ejpam-4702	130	11	,	,	PUNCT
ejpam-4702	130	12	3	3	NUM
ejpam-4702	130	13	,	,	PUNCT
ejpam-4702	130	14	5	5	NUM
ejpam-4702	130	15	,	,	PUNCT
ejpam-4702	130	16	.	.	PUNCT
ejpam-4702	130	17	.	.	PUNCT
ejpam-4702	131	1	.	.	PUNCT
ejpam-4702	132	1	,	,	PUNCT
ejpam-4702	132	2	q	q	X
ejpam-4702	132	3	−	−	PROPN
ejpam-4702	132	4	2	2	NUM
ejpam-4702	132	5	}	}	PUNCT
ejpam-4702	132	6	.	.	PUNCT
ejpam-4702	133	1	case	case	NOUN
ejpam-4702	133	2	1	1	X
ejpam-4702	133	3	.	.	PUNCT
ejpam-4702	134	1	(	(	PUNCT
ejpam-4702	134	2	q	q	X
ejpam-4702	134	3	p	p	NOUN
ejpam-4702	134	4	)	)	PUNCT
ejpam-4702	134	5	=	=	SYM
ejpam-4702	135	1	1	1	X
ejpam-4702	135	2	.	.	PUNCT
ejpam-4702	136	1	then	then	ADV
ejpam-4702	136	2	(	(	PUNCT
ejpam-4702	136	3	p	p	X
ejpam-4702	136	4	q	q	X
ejpam-4702	136	5	)	)	PUNCT
ejpam-4702	136	6	=	=	SYM
ejpam-4702	136	7	1	1	X
ejpam-4702	136	8	.	.	PUNCT
ejpam-4702	137	1	thus	thus	ADV
ejpam-4702	137	2	,	,	PUNCT
ejpam-4702	137	3	p	p	PROPN
ejpam-4702	137	4	≡	≡	PROPN
ejpam-4702	137	5	rs1	rs1	PROPN
ejpam-4702	137	6	(	(	PUNCT
ejpam-4702	137	7	mod	mod	PROPN
ejpam-4702	137	8	q	q	PROPN
ejpam-4702	137	9	)	)	PUNCT
ejpam-4702	137	10	.	.	PUNCT
ejpam-4702	138	1	since	since	SCONJ
ejpam-4702	138	2	p	p	PRON
ejpam-4702	138	3	≡	≡	PROPN
ejpam-4702	138	4	1	1	NUM
ejpam-4702	138	5	(	(	PUNCT
ejpam-4702	138	6	mod	mod	NOUN
ejpam-4702	138	7	2	2	NUM
ejpam-4702	138	8	)	)	PUNCT
ejpam-4702	138	9	and	and	CCONJ
ejpam-4702	138	10	by	by	ADP
ejpam-4702	138	11	the	the	DET
ejpam-4702	138	12	chinese	chinese	ADJ
ejpam-4702	138	13	remainder	remainder	NOUN
ejpam-4702	138	14	theorem	theorem	PROPN
ejpam-4702	138	15	,	,	PUNCT
ejpam-4702	138	16	we	we	PRON
ejpam-4702	138	17	obtain	obtain	VERB
ejpam-4702	138	18	that	that	SCONJ
ejpam-4702	138	19	p	p	PROPN
ejpam-4702	138	20	≡	≡	PROPN
ejpam-4702	138	21	q	q	PROPN
ejpam-4702	139	1	+	+	NUM
ejpam-4702	139	2	rs1q	rs1q	PROPN
ejpam-4702	139	3	+	+	CCONJ
ejpam-4702	139	4	rs1	rs1	PROPN
ejpam-4702	139	5	(	(	PUNCT
ejpam-4702	139	6	mod	mod	ADJ
ejpam-4702	139	7	2q	2q	NUM
ejpam-4702	139	8	)	)	PUNCT
ejpam-4702	139	9	.	.	PUNCT
ejpam-4702	140	1	s.	s.	PROPN
ejpam-4702	140	2	tadee	tadee	PROPN
ejpam-4702	140	3	,	,	PUNCT
ejpam-4702	140	4	a.	a.	NOUN
ejpam-4702	140	5	siraworakun	siraworakun	PROPN
ejpam-4702	140	6	/	/	SYM
ejpam-4702	140	7	eur	eur	PROPN
ejpam-4702	140	8	.	.	PUNCT
ejpam-4702	141	1	j.	j.	PROPN
ejpam-4702	141	2	pure	pure	PROPN
ejpam-4702	141	3	appl	appl	PROPN
ejpam-4702	141	4	.	.	PROPN
ejpam-4702	141	5	math	math	PROPN
ejpam-4702	141	6	,	,	PUNCT
ejpam-4702	141	7	16	16	NUM
ejpam-4702	141	8	(	(	PUNCT
ejpam-4702	141	9	2	2	NUM
ejpam-4702	141	10	)	)	PUNCT
ejpam-4702	141	11	(	(	PUNCT
ejpam-4702	141	12	2023	2023	NUM
ejpam-4702	141	13	)	)	PUNCT
ejpam-4702	141	14	,	,	PUNCT
ejpam-4702	141	15	724	724	NUM
ejpam-4702	141	16	-	-	SYM
ejpam-4702	141	17	735	735	NUM
ejpam-4702	141	18	728	728	NUM
ejpam-4702	141	19	case	case	NOUN
ejpam-4702	141	20	2	2	NUM
ejpam-4702	141	21	.	.	PUNCT
ejpam-4702	142	1	(	(	PUNCT
ejpam-4702	142	2	q	q	X
ejpam-4702	142	3	p	p	NOUN
ejpam-4702	142	4	)	)	PUNCT
ejpam-4702	142	5	=	=	SYM
ejpam-4702	142	6	−1	−1	NOUN
ejpam-4702	142	7	.	.	PUNCT
ejpam-4702	143	1	then	then	ADV
ejpam-4702	143	2	(	(	PUNCT
ejpam-4702	143	3	p	p	X
ejpam-4702	143	4	q	q	X
ejpam-4702	143	5	)	)	PUNCT
ejpam-4702	143	6	=	=	SYM
ejpam-4702	143	7	−1	−1	NOUN
ejpam-4702	143	8	.	.	PUNCT
ejpam-4702	144	1	thus	thus	ADV
ejpam-4702	144	2	,	,	PUNCT
ejpam-4702	144	3	p	p	PROPN
ejpam-4702	144	4	≡	≡	PROPN
ejpam-4702	144	5	rs2	rs2	NOUN
ejpam-4702	144	6	(	(	PUNCT
ejpam-4702	144	7	mod	mod	PROPN
ejpam-4702	144	8	q	q	NOUN
ejpam-4702	144	9	)	)	PUNCT
ejpam-4702	144	10	.	.	PUNCT
ejpam-4702	145	1	since	since	SCONJ
ejpam-4702	145	2	p	p	PRON
ejpam-4702	145	3	≡	≡	PROPN
ejpam-4702	145	4	1	1	NUM
ejpam-4702	145	5	(	(	PUNCT
ejpam-4702	145	6	mod	mod	NOUN
ejpam-4702	145	7	2	2	NUM
ejpam-4702	145	8	)	)	PUNCT
ejpam-4702	145	9	and	and	CCONJ
ejpam-4702	145	10	by	by	ADP
ejpam-4702	145	11	the	the	DET
ejpam-4702	145	12	chinese	chinese	ADJ
ejpam-4702	145	13	remainder	remainder	NOUN
ejpam-4702	145	14	theorem	theorem	PROPN
ejpam-4702	145	15	,	,	PUNCT
ejpam-4702	145	16	we	we	PRON
ejpam-4702	145	17	obtain	obtain	VERB
ejpam-4702	145	18	that	that	SCONJ
ejpam-4702	145	19	p	p	PROPN
ejpam-4702	145	20	≡	≡	PROPN
ejpam-4702	145	21	q	q	PROPN
ejpam-4702	146	1	+	+	CCONJ
ejpam-4702	146	2	rs2q	rs2q	NUM
ejpam-4702	146	3	+	+	CCONJ
ejpam-4702	146	4	rs2	rs2	NOUN
ejpam-4702	146	5	(	(	PUNCT
ejpam-4702	146	6	mod	mod	ADJ
ejpam-4702	146	7	2q	2q	NUM
ejpam-4702	146	8	)	)	PUNCT
ejpam-4702	146	9	.	.	PUNCT
ejpam-4702	147	1	theorem	theorem	VERB
ejpam-4702	147	2	11	11	NUM
ejpam-4702	147	3	.	.	PUNCT
ejpam-4702	148	1	let	let	VERB
ejpam-4702	148	2	p	p	NOUN
ejpam-4702	148	3	and	and	CCONJ
ejpam-4702	148	4	q	q	NOUN
ejpam-4702	148	5	be	be	AUX
ejpam-4702	148	6	distinct	distinct	ADJ
ejpam-4702	148	7	odd	odd	ADJ
ejpam-4702	148	8	prime	prime	ADJ
ejpam-4702	148	9	numbers	number	NOUN
ejpam-4702	148	10	with	with	ADP
ejpam-4702	148	11	q	q	PROPN
ejpam-4702	148	12	≡	≡	PROPN
ejpam-4702	148	13	3	3	NUM
ejpam-4702	148	14	(	(	PUNCT
ejpam-4702	148	15	mod	mod	NOUN
ejpam-4702	148	16	4	4	NUM
ejpam-4702	148	17	)	)	PUNCT
ejpam-4702	148	18	.	.	PUNCT
ejpam-4702	149	1	then	then	ADV
ejpam-4702	149	2	(	(	PUNCT
ejpam-4702	149	3	q	q	PROPN
ejpam-4702	149	4	p	p	NOUN
ejpam-4702	149	5	)	)	PUNCT
ejpam-4702	149	6	=	=	PUNCT
ejpam-4702	149	7	{	{	PUNCT
ejpam-4702	149	8	1	1	NUM
ejpam-4702	149	9	if	if	SCONJ
ejpam-4702	149	10	p	p	PRON
ejpam-4702	149	11	≡	≡	PROPN
ejpam-4702	149	12	3q	3q	NOUN
ejpam-4702	149	13	+	+	CCONJ
ejpam-4702	149	14	4n0r	4n0r	NOUN
ejpam-4702	149	15	s1	s1	NOUN
ejpam-4702	149	16	,	,	PUNCT
ejpam-4702	149	17	−	−	PROPN
ejpam-4702	149	18	3q	3q	NUM
ejpam-4702	149	19	+	+	CCONJ
ejpam-4702	149	20	4n0r	4n0r	NOUN
ejpam-4702	149	21	s2	s2	NOUN
ejpam-4702	149	22	(	(	PUNCT
ejpam-4702	149	23	mod	mod	PROPN
ejpam-4702	149	24	4q	4q	NOUN
ejpam-4702	149	25	)	)	PUNCT
ejpam-4702	149	26	−1	−1	NOUN
ejpam-4702	149	27	if	if	SCONJ
ejpam-4702	149	28	p	p	PRON
ejpam-4702	149	29	≡	≡	PROPN
ejpam-4702	149	30	3q	3q	NOUN
ejpam-4702	149	31	+	+	CCONJ
ejpam-4702	149	32	4n0r	4n0r	NOUN
ejpam-4702	149	33	s2	s2	NOUN
ejpam-4702	149	34	,	,	PUNCT
ejpam-4702	149	35	−	−	PROPN
ejpam-4702	149	36	3q	3q	NUM
ejpam-4702	149	37	+	+	CCONJ
ejpam-4702	149	38	4n0r	4n0r	NOUN
ejpam-4702	149	39	s1	s1	NOUN
ejpam-4702	149	40	(	(	PUNCT
ejpam-4702	149	41	mod	mod	PROPN
ejpam-4702	149	42	4q	4q	NOUN
ejpam-4702	149	43	)	)	PUNCT
ejpam-4702	149	44	,	,	PUNCT
ejpam-4702	149	45	where	where	SCONJ
ejpam-4702	149	46	s1	s1	PROPN
ejpam-4702	149	47	∈	∈	PROPN
ejpam-4702	149	48	{	{	PUNCT
ejpam-4702	149	49	2	2	NUM
ejpam-4702	149	50	,	,	PUNCT
ejpam-4702	149	51	4	4	NUM
ejpam-4702	149	52	,	,	PUNCT
ejpam-4702	149	53	6	6	NUM
ejpam-4702	149	54	,	,	PUNCT
ejpam-4702	149	55	.	.	PUNCT
ejpam-4702	149	56	.	.	PUNCT
ejpam-4702	149	57	.	.	PUNCT
ejpam-4702	150	1	,	,	PUNCT
ejpam-4702	150	2	q−	q−	PROPN
ejpam-4702	150	3	1	1	NUM
ejpam-4702	150	4	}	}	PUNCT
ejpam-4702	150	5	,	,	PUNCT
ejpam-4702	150	6	s2	s2	PROPN
ejpam-4702	150	7	∈	∈	PROPN
ejpam-4702	150	8	{	{	PUNCT
ejpam-4702	150	9	1	1	NUM
ejpam-4702	150	10	,	,	PUNCT
ejpam-4702	150	11	3	3	NUM
ejpam-4702	150	12	,	,	PUNCT
ejpam-4702	150	13	5	5	NUM
ejpam-4702	150	14	,	,	PUNCT
ejpam-4702	150	15	.	.	PUNCT
ejpam-4702	150	16	.	.	PUNCT
ejpam-4702	151	1	.	.	PUNCT
ejpam-4702	152	1	,	,	PUNCT
ejpam-4702	152	2	q−	q−	PROPN
ejpam-4702	152	3	2	2	NUM
ejpam-4702	152	4	}	}	PUNCT
ejpam-4702	152	5	,	,	PUNCT
ejpam-4702	152	6	r	r	NOUN
ejpam-4702	152	7	is	be	AUX
ejpam-4702	152	8	a	a	DET
ejpam-4702	152	9	primitive	primitive	ADJ
ejpam-4702	152	10	root	root	NOUN
ejpam-4702	152	11	modulo	modulo	NOUN
ejpam-4702	152	12	q	q	NOUN
ejpam-4702	152	13	and	and	CCONJ
ejpam-4702	152	14	n0	n0	NOUN
ejpam-4702	152	15	=	=	SYM
ejpam-4702	152	16	q	q	PROPN
ejpam-4702	153	1	+	+	NUM
ejpam-4702	153	2	1	1	NUM
ejpam-4702	153	3	4	4	NUM
ejpam-4702	153	4	.	.	PUNCT
ejpam-4702	154	1	proof	proof	NOUN
ejpam-4702	154	2	.	.	PUNCT
ejpam-4702	155	1	let	let	VERB
ejpam-4702	155	2	r	r	PRON
ejpam-4702	155	3	be	be	AUX
ejpam-4702	155	4	a	a	DET
ejpam-4702	155	5	primitive	primitive	ADJ
ejpam-4702	155	6	root	root	NOUN
ejpam-4702	155	7	modulo	modulo	NOUN
ejpam-4702	155	8	q.	q.	NOUN
ejpam-4702	155	9	by	by	ADP
ejpam-4702	155	10	theorems	theorem	NOUN
ejpam-4702	155	11	6	6	NUM
ejpam-4702	155	12	and	and	CCONJ
ejpam-4702	155	13	9	9	NUM
ejpam-4702	155	14	,	,	PUNCT
ejpam-4702	155	15	we	we	PRON
ejpam-4702	155	16	obtain	obtain	VERB
ejpam-4702	155	17	that	that	PRON
ejpam-4702	155	18	(	(	PUNCT
ejpam-4702	155	19	q	q	X
ejpam-4702	155	20	p	p	NOUN
ejpam-4702	155	21	)	)	PUNCT
ejpam-4702	155	22	=	=	PUNCT
ejpam-4702	156	1			PUNCT
ejpam-4702	156	2	(	(	PUNCT
ejpam-4702	156	3	p	p	X
ejpam-4702	156	4	q	q	X
ejpam-4702	156	5	)	)	PUNCT
ejpam-4702	156	6	if	if	SCONJ
ejpam-4702	156	7	p	p	PRON
ejpam-4702	156	8	≡	≡	PROPN
ejpam-4702	156	9	1	1	NUM
ejpam-4702	156	10	(	(	PUNCT
ejpam-4702	156	11	mod	mod	NOUN
ejpam-4702	156	12	4	4	NUM
ejpam-4702	156	13	)	)	PUNCT
ejpam-4702	156	14	−	−	PROPN
ejpam-4702	156	15	(	(	PUNCT
ejpam-4702	156	16	p	p	X
ejpam-4702	156	17	q	q	X
ejpam-4702	156	18	)	)	PUNCT
ejpam-4702	156	19	if	if	SCONJ
ejpam-4702	156	20	p	p	PRON
ejpam-4702	156	21	≡	≡	PROPN
ejpam-4702	156	22	3	3	NUM
ejpam-4702	156	23	(	(	PUNCT
ejpam-4702	156	24	mod	mod	NOUN
ejpam-4702	156	25	4	4	NUM
ejpam-4702	156	26	)	)	PUNCT
ejpam-4702	156	27	and	and	CCONJ
ejpam-4702	156	28	(	(	PUNCT
ejpam-4702	156	29	p	p	X
ejpam-4702	156	30	q	q	X
ejpam-4702	156	31	)	)	PUNCT
ejpam-4702	156	32	=	=	PRON
ejpam-4702	156	33	{	{	PUNCT
ejpam-4702	156	34	1	1	NUM
ejpam-4702	156	35	if	if	SCONJ
ejpam-4702	156	36	p	p	PRON
ejpam-4702	156	37	≡	≡	PROPN
ejpam-4702	156	38	rs1	rs1	PROPN
ejpam-4702	156	39	(	(	PUNCT
ejpam-4702	156	40	mod	mod	PROPN
ejpam-4702	156	41	q	q	NOUN
ejpam-4702	156	42	)	)	PUNCT
ejpam-4702	156	43	−1	−1	NOUN
ejpam-4702	156	44	if	if	SCONJ
ejpam-4702	156	45	p	p	PRON
ejpam-4702	156	46	≡	≡	PROPN
ejpam-4702	156	47	rs2	rs2	NOUN
ejpam-4702	156	48	(	(	PUNCT
ejpam-4702	156	49	mod	mod	PROPN
ejpam-4702	156	50	q	q	NOUN
ejpam-4702	156	51	)	)	PUNCT
ejpam-4702	156	52	,	,	PUNCT
ejpam-4702	156	53	where	where	SCONJ
ejpam-4702	156	54	s1	s1	PROPN
ejpam-4702	156	55	∈	∈	PROPN
ejpam-4702	156	56	{	{	PUNCT
ejpam-4702	156	57	2	2	NUM
ejpam-4702	156	58	,	,	PUNCT
ejpam-4702	156	59	4	4	NUM
ejpam-4702	156	60	,	,	PUNCT
ejpam-4702	156	61	6	6	NUM
ejpam-4702	156	62	,	,	PUNCT
ejpam-4702	156	63	.	.	PUNCT
ejpam-4702	156	64	.	.	PUNCT
ejpam-4702	156	65	.	.	PUNCT
ejpam-4702	157	1	,	,	PUNCT
ejpam-4702	157	2	q−1	q−1	PROPN
ejpam-4702	157	3	}	}	PUNCT
ejpam-4702	157	4	and	and	CCONJ
ejpam-4702	157	5	s2	s2	PROPN
ejpam-4702	157	6	∈	∈	PROPN
ejpam-4702	157	7	{	{	PUNCT
ejpam-4702	157	8	1	1	NUM
ejpam-4702	157	9	,	,	PUNCT
ejpam-4702	157	10	3	3	NUM
ejpam-4702	157	11	,	,	PUNCT
ejpam-4702	157	12	5	5	NUM
ejpam-4702	157	13	,	,	PUNCT
ejpam-4702	157	14	.	.	PUNCT
ejpam-4702	157	15	.	.	PUNCT
ejpam-4702	157	16	.	.	PUNCT
ejpam-4702	158	1	,	,	PUNCT
ejpam-4702	158	2	q−2	q−2	PROPN
ejpam-4702	158	3	}	}	PUNCT
ejpam-4702	158	4	.	.	PUNCT
ejpam-4702	159	1	since	since	SCONJ
ejpam-4702	159	2	q	q	PROPN
ejpam-4702	159	3	≡	≡	PROPN
ejpam-4702	159	4	3	3	NUM
ejpam-4702	159	5	(	(	PUNCT
ejpam-4702	159	6	mod	mod	NOUN
ejpam-4702	159	7	4	4	NUM
ejpam-4702	159	8	)	)	PUNCT
ejpam-4702	159	9	,	,	PUNCT
ejpam-4702	159	10	we	we	PRON
ejpam-4702	159	11	choose	choose	VERB
ejpam-4702	159	12	an	an	DET
ejpam-4702	159	13	integer	integer	NOUN
ejpam-4702	159	14	n0	n0	NOUN
ejpam-4702	159	15	=	=	PUNCT
ejpam-4702	159	16	q	q	PROPN
ejpam-4702	160	1	+	+	NUM
ejpam-4702	160	2	1	1	NUM
ejpam-4702	160	3	4	4	NUM
ejpam-4702	160	4	.	.	PUNCT
ejpam-4702	161	1	then	then	ADV
ejpam-4702	161	2	q	q	X
ejpam-4702	162	1	=	=	SYM
ejpam-4702	162	2	4n0	4n0	NUM
ejpam-4702	162	3	−	−	NOUN
ejpam-4702	162	4	1	1	NUM
ejpam-4702	162	5	.	.	PUNCT
ejpam-4702	162	6	thus	thus	ADV
ejpam-4702	162	7	3q	3q	NUM
ejpam-4702	162	8	≡	≡	PROPN
ejpam-4702	162	9	1	1	NUM
ejpam-4702	162	10	(	(	PUNCT
ejpam-4702	162	11	mod	mod	NOUN
ejpam-4702	162	12	4	4	NUM
ejpam-4702	162	13	)	)	PUNCT
ejpam-4702	162	14	and	and	CCONJ
ejpam-4702	162	15	4n0	4n0	NUM
ejpam-4702	162	16	≡	≡	PROPN
ejpam-4702	162	17	1	1	NUM
ejpam-4702	162	18	(	(	PUNCT
ejpam-4702	162	19	mod	mod	PROPN
ejpam-4702	162	20	q	q	NOUN
ejpam-4702	162	21	)	)	PUNCT
ejpam-4702	162	22	.	.	PUNCT
ejpam-4702	163	1	in	in	ADP
ejpam-4702	163	2	the	the	DET
ejpam-4702	163	3	following	following	ADJ
ejpam-4702	163	4	cases	case	NOUN
ejpam-4702	163	5	,	,	PUNCT
ejpam-4702	163	6	the	the	DET
ejpam-4702	163	7	systems	system	NOUN
ejpam-4702	163	8	of	of	ADP
ejpam-4702	163	9	congruences	congruence	NOUN
ejpam-4702	163	10	are	be	AUX
ejpam-4702	163	11	solved	solve	VERB
ejpam-4702	163	12	by	by	ADP
ejpam-4702	163	13	the	the	DET
ejpam-4702	163	14	chinese	chinese	ADJ
ejpam-4702	163	15	remainder	remainder	NOUN
ejpam-4702	163	16	theorem	theorem	PROPN
ejpam-4702	163	17	.	.	PUNCT
ejpam-4702	163	18	case	case	NOUN
ejpam-4702	163	19	1	1	NUM
ejpam-4702	163	20	.	.	PUNCT
ejpam-4702	164	1	(	(	PUNCT
ejpam-4702	164	2	q	q	X
ejpam-4702	164	3	p	p	NOUN
ejpam-4702	164	4	)	)	PUNCT
ejpam-4702	164	5	=	=	SYM
ejpam-4702	164	6	1	1	X
ejpam-4702	164	7	.	.	PUNCT
ejpam-4702	164	8	case	case	NOUN
ejpam-4702	164	9	1.1	1.1	NUM
ejpam-4702	164	10	(	(	PUNCT
ejpam-4702	164	11	q	q	NOUN
ejpam-4702	164	12	p	p	NOUN
ejpam-4702	164	13	)	)	PUNCT
ejpam-4702	164	14	=	=	PUNCT
ejpam-4702	164	15	(	(	PUNCT
ejpam-4702	164	16	p	p	X
ejpam-4702	164	17	q	q	X
ejpam-4702	164	18	)	)	PUNCT
ejpam-4702	164	19	and	and	CCONJ
ejpam-4702	164	20	(	(	PUNCT
ejpam-4702	164	21	p	p	X
ejpam-4702	164	22	q	q	X
ejpam-4702	164	23	)	)	PUNCT
ejpam-4702	164	24	=	=	SYM
ejpam-4702	165	1	1	1	X
ejpam-4702	165	2	.	.	PUNCT
ejpam-4702	165	3	then	then	ADV
ejpam-4702	165	4	p	p	PROPN
ejpam-4702	165	5	≡	≡	PROPN
ejpam-4702	165	6	1	1	NUM
ejpam-4702	165	7	(	(	PUNCT
ejpam-4702	165	8	mod	mod	NOUN
ejpam-4702	165	9	4	4	NUM
ejpam-4702	165	10	)	)	PUNCT
ejpam-4702	165	11	and	and	CCONJ
ejpam-4702	165	12	p	p	PROPN
ejpam-4702	165	13	≡	≡	PROPN
ejpam-4702	165	14	rs1	rs1	PROPN
ejpam-4702	165	15	(	(	PUNCT
ejpam-4702	165	16	mod	mod	PROPN
ejpam-4702	165	17	q	q	PROPN
ejpam-4702	165	18	)	)	PUNCT
ejpam-4702	165	19	.	.	PUNCT
ejpam-4702	166	1	thus	thus	ADV
ejpam-4702	166	2	,	,	PUNCT
ejpam-4702	166	3	p	p	PROPN
ejpam-4702	166	4	≡	≡	PROPN
ejpam-4702	166	5	3q	3q	NOUN
ejpam-4702	166	6	+	+	CCONJ
ejpam-4702	166	7	4n0r	4n0r	NOUN
ejpam-4702	166	8	s1	s1	NOUN
ejpam-4702	166	9	(	(	PUNCT
ejpam-4702	166	10	mod	mod	PROPN
ejpam-4702	166	11	4q	4q	NOUN
ejpam-4702	166	12	)	)	PUNCT
ejpam-4702	166	13	.	.	PUNCT
ejpam-4702	167	1	case	case	NOUN
ejpam-4702	167	2	1.2	1.2	NUM
ejpam-4702	167	3	(	(	PUNCT
ejpam-4702	167	4	q	q	NOUN
ejpam-4702	167	5	p	p	NOUN
ejpam-4702	167	6	)	)	PUNCT
ejpam-4702	168	1	=	=	SYM
ejpam-4702	168	2	−	−	PROPN
ejpam-4702	169	1	(	(	PUNCT
ejpam-4702	169	2	p	p	X
ejpam-4702	169	3	q	q	X
ejpam-4702	169	4	)	)	PUNCT
ejpam-4702	169	5	and	and	CCONJ
ejpam-4702	169	6	(	(	PUNCT
ejpam-4702	169	7	p	p	X
ejpam-4702	169	8	q	q	X
ejpam-4702	169	9	)	)	PUNCT
ejpam-4702	169	10	=	=	SYM
ejpam-4702	169	11	−1	−1	NOUN
ejpam-4702	169	12	.	.	PUNCT
ejpam-4702	170	1	then	then	ADV
ejpam-4702	170	2	p	p	PROPN
ejpam-4702	170	3	≡	≡	PROPN
ejpam-4702	170	4	3	3	NUM
ejpam-4702	170	5	(	(	PUNCT
ejpam-4702	170	6	mod	mod	NOUN
ejpam-4702	170	7	4	4	NUM
ejpam-4702	170	8	)	)	PUNCT
ejpam-4702	170	9	and	and	CCONJ
ejpam-4702	170	10	p	p	PROPN
ejpam-4702	170	11	≡	≡	PROPN
ejpam-4702	170	12	rs2	rs2	NOUN
ejpam-4702	170	13	(	(	PUNCT
ejpam-4702	170	14	mod	mod	PROPN
ejpam-4702	170	15	q	q	NOUN
ejpam-4702	170	16	)	)	PUNCT
ejpam-4702	170	17	.	.	PUNCT
ejpam-4702	171	1	thus	thus	ADV
ejpam-4702	171	2	,	,	PUNCT
ejpam-4702	171	3	p	p	PROPN
ejpam-4702	171	4	≡	≡	PROPN
ejpam-4702	171	5	−3q	−3q	PROPN
ejpam-4702	171	6	+	+	NUM
ejpam-4702	171	7	4n0r	4n0r	NOUN
ejpam-4702	171	8	s2	s2	NOUN
ejpam-4702	171	9	(	(	PUNCT
ejpam-4702	171	10	mod	mod	PROPN
ejpam-4702	171	11	4q	4q	NOUN
ejpam-4702	171	12	)	)	PUNCT
ejpam-4702	171	13	.	.	PUNCT
ejpam-4702	171	14	case	case	NOUN
ejpam-4702	171	15	2	2	X
ejpam-4702	171	16	.	.	PUNCT
ejpam-4702	172	1	(	(	PUNCT
ejpam-4702	172	2	q	q	X
ejpam-4702	172	3	p	p	NOUN
ejpam-4702	172	4	)	)	PUNCT
ejpam-4702	172	5	=	=	PUNCT
ejpam-4702	172	6	−1	−1	NOUN
ejpam-4702	172	7	.	.	PUNCT
ejpam-4702	173	1	case	case	NOUN
ejpam-4702	173	2	2.1	2.1	NUM
ejpam-4702	173	3	(	(	PUNCT
ejpam-4702	173	4	q	q	NOUN
ejpam-4702	173	5	p	p	NOUN
ejpam-4702	173	6	)	)	PUNCT
ejpam-4702	173	7	=	=	PUNCT
ejpam-4702	174	1	(	(	PUNCT
ejpam-4702	174	2	p	p	X
ejpam-4702	174	3	q	q	X
ejpam-4702	174	4	)	)	PUNCT
ejpam-4702	174	5	and	and	CCONJ
ejpam-4702	174	6	(	(	PUNCT
ejpam-4702	174	7	p	p	X
ejpam-4702	174	8	q	q	X
ejpam-4702	174	9	)	)	PUNCT
ejpam-4702	174	10	=	=	SYM
ejpam-4702	174	11	−1	−1	NOUN
ejpam-4702	174	12	.	.	PUNCT
ejpam-4702	175	1	then	then	ADV
ejpam-4702	175	2	p	p	PROPN
ejpam-4702	175	3	≡	≡	PROPN
ejpam-4702	175	4	1	1	NUM
ejpam-4702	175	5	(	(	PUNCT
ejpam-4702	175	6	mod	mod	NOUN
ejpam-4702	175	7	4	4	NUM
ejpam-4702	175	8	)	)	PUNCT
ejpam-4702	175	9	and	and	CCONJ
ejpam-4702	175	10	p	p	PROPN
ejpam-4702	175	11	≡	≡	PROPN
ejpam-4702	175	12	rs2	rs2	NOUN
ejpam-4702	175	13	(	(	PUNCT
ejpam-4702	175	14	mod	mod	PROPN
ejpam-4702	175	15	q	q	NOUN
ejpam-4702	175	16	)	)	PUNCT
ejpam-4702	175	17	.	.	PUNCT
ejpam-4702	176	1	thus	thus	ADV
ejpam-4702	176	2	,	,	PUNCT
ejpam-4702	176	3	p	p	PROPN
ejpam-4702	176	4	≡	≡	PROPN
ejpam-4702	176	5	3q	3q	NOUN
ejpam-4702	176	6	+	+	CCONJ
ejpam-4702	176	7	4n0r	4n0r	NOUN
ejpam-4702	176	8	s2	s2	NOUN
ejpam-4702	176	9	(	(	PUNCT
ejpam-4702	176	10	mod	mod	PROPN
ejpam-4702	176	11	4q	4q	NOUN
ejpam-4702	176	12	)	)	PUNCT
ejpam-4702	176	13	.	.	PUNCT
ejpam-4702	177	1	case	case	NOUN
ejpam-4702	177	2	2.2	2.2	NUM
ejpam-4702	177	3	(	(	PUNCT
ejpam-4702	177	4	q	q	NOUN
ejpam-4702	177	5	p	p	NOUN
ejpam-4702	177	6	)	)	PUNCT
ejpam-4702	178	1	=	=	SYM
ejpam-4702	178	2	−	−	PROPN
ejpam-4702	179	1	(	(	PUNCT
ejpam-4702	179	2	p	p	X
ejpam-4702	179	3	q	q	X
ejpam-4702	179	4	)	)	PUNCT
ejpam-4702	179	5	and	and	CCONJ
ejpam-4702	179	6	(	(	PUNCT
ejpam-4702	179	7	p	p	X
ejpam-4702	179	8	q	q	X
ejpam-4702	179	9	)	)	PUNCT
ejpam-4702	179	10	=	=	SYM
ejpam-4702	180	1	1	1	X
ejpam-4702	180	2	.	.	PUNCT
ejpam-4702	180	3	then	then	ADV
ejpam-4702	180	4	p	p	PROPN
ejpam-4702	180	5	≡	≡	PROPN
ejpam-4702	180	6	3	3	NUM
ejpam-4702	180	7	(	(	PUNCT
ejpam-4702	180	8	mod	mod	NOUN
ejpam-4702	180	9	4	4	NUM
ejpam-4702	180	10	)	)	PUNCT
ejpam-4702	180	11	and	and	CCONJ
ejpam-4702	180	12	p	p	PROPN
ejpam-4702	180	13	≡	≡	PROPN
ejpam-4702	180	14	rs1	rs1	PROPN
ejpam-4702	180	15	(	(	PUNCT
ejpam-4702	180	16	mod	mod	PROPN
ejpam-4702	180	17	q	q	X
ejpam-4702	180	18	)	)	PUNCT
ejpam-4702	180	19	.	.	PUNCT
ejpam-4702	181	1	thus	thus	ADV
ejpam-4702	181	2	,	,	PUNCT
ejpam-4702	181	3	p	p	PROPN
ejpam-4702	181	4	≡	≡	PROPN
ejpam-4702	181	5	−3q	−3q	PROPN
ejpam-4702	181	6	+	+	NUM
ejpam-4702	181	7	4n0r	4n0r	NOUN
ejpam-4702	181	8	s1	s1	NOUN
ejpam-4702	181	9	(	(	PUNCT
ejpam-4702	181	10	mod	mod	PROPN
ejpam-4702	181	11	4q	4q	NOUN
ejpam-4702	181	12	)	)	PUNCT
ejpam-4702	181	13	.	.	PUNCT
ejpam-4702	182	1	moreover	moreover	ADV
ejpam-4702	182	2	,	,	PUNCT
ejpam-4702	182	3	siraworakun	siraworakun	NOUN
ejpam-4702	182	4	has	have	AUX
ejpam-4702	182	5	given	give	VERB
ejpam-4702	182	6	the	the	DET
ejpam-4702	182	7	forms	form	NOUN
ejpam-4702	182	8	of	of	ADP
ejpam-4702	182	9	prime	prime	ADJ
ejpam-4702	182	10	numbers	number	NOUN
ejpam-4702	182	11	p	p	PROPN
ejpam-4702	182	12	in	in	ADP
ejpam-4702	182	13	legendre	legendre	PROPN
ejpam-4702	182	14	symbol	symbol	NOUN
ejpam-4702	182	15	(	(	PUNCT
ejpam-4702	182	16	2q	2q	NUM
ejpam-4702	182	17	p	p	NOUN
ejpam-4702	182	18	)	)	PUNCT
ejpam-4702	182	19	,	,	PUNCT
ejpam-4702	182	20	where	where	SCONJ
ejpam-4702	182	21	q	q	NOUN
ejpam-4702	182	22	is	be	AUX
ejpam-4702	182	23	a	a	DET
ejpam-4702	182	24	prime	prime	ADJ
ejpam-4702	182	25	number	number	NOUN
ejpam-4702	182	26	.	.	PUNCT
ejpam-4702	183	1	s.	s.	PROPN
ejpam-4702	183	2	tadee	tadee	PROPN
ejpam-4702	183	3	,	,	PUNCT
ejpam-4702	183	4	a.	a.	NOUN
ejpam-4702	183	5	siraworakun	siraworakun	PROPN
ejpam-4702	183	6	/	/	SYM
ejpam-4702	183	7	eur	eur	PROPN
ejpam-4702	183	8	.	.	PUNCT
ejpam-4702	184	1	j.	j.	PROPN
ejpam-4702	184	2	pure	pure	PROPN
ejpam-4702	184	3	appl	appl	PROPN
ejpam-4702	184	4	.	.	PROPN
ejpam-4702	184	5	math	math	PROPN
ejpam-4702	184	6	,	,	PUNCT
ejpam-4702	184	7	16	16	NUM
ejpam-4702	184	8	(	(	PUNCT
ejpam-4702	184	9	2	2	NUM
ejpam-4702	184	10	)	)	PUNCT
ejpam-4702	184	11	(	(	PUNCT
ejpam-4702	184	12	2023	2023	NUM
ejpam-4702	184	13	)	)	PUNCT
ejpam-4702	184	14	,	,	PUNCT
ejpam-4702	184	15	724	724	NUM
ejpam-4702	184	16	-	-	SYM
ejpam-4702	184	17	735	735	NUM
ejpam-4702	184	18	729	729	NUM
ejpam-4702	184	19	theorem	theorem	NOUN
ejpam-4702	184	20	12	12	NUM
ejpam-4702	184	21	.	.	PUNCT
ejpam-4702	185	1	let	let	VERB
ejpam-4702	185	2	p	p	NOUN
ejpam-4702	185	3	and	and	CCONJ
ejpam-4702	185	4	q	q	NOUN
ejpam-4702	185	5	be	be	AUX
ejpam-4702	185	6	distinct	distinct	ADJ
ejpam-4702	185	7	odd	odd	ADJ
ejpam-4702	185	8	prime	prime	ADJ
ejpam-4702	185	9	numbers	number	NOUN
ejpam-4702	185	10	with	with	ADP
ejpam-4702	185	11	q	q	PROPN
ejpam-4702	185	12	≡	≡	PROPN
ejpam-4702	185	13	1	1	NUM
ejpam-4702	185	14	(	(	PUNCT
ejpam-4702	185	15	mod	mod	NOUN
ejpam-4702	185	16	4	4	NUM
ejpam-4702	185	17	)	)	PUNCT
ejpam-4702	185	18	.	.	PUNCT
ejpam-4702	186	1	then	then	ADV
ejpam-4702	186	2	(	(	PUNCT
ejpam-4702	186	3	2q	2q	NUM
ejpam-4702	186	4	p	p	NOUN
ejpam-4702	186	5	)	)	PUNCT
ejpam-4702	186	6	=	=	SYM
ejpam-4702	186	7	1	1	NUM
ejpam-4702	186	8	if	if	SCONJ
ejpam-4702	186	9	p	p	DET
ejpam-4702	186	10	≡	≡	PROPN
ejpam-4702	186	11	q2	q2	PROPN
ejpam-4702	186	12	+	+	CCONJ
ejpam-4702	186	13	8n1r	8n1r	ADJ
ejpam-4702	186	14	s1	s1	NOUN
ejpam-4702	186	15	,	,	PUNCT
ejpam-4702	186	16	−	−	PROPN
ejpam-4702	186	17	q2	q2	NOUN
ejpam-4702	186	18	+	+	CCONJ
ejpam-4702	186	19	8n1r	8n1r	ADJ
ejpam-4702	186	20	s1	s1	NOUN
ejpam-4702	186	21	,	,	PUNCT
ejpam-4702	186	22	3q2	3q2	NUM
ejpam-4702	186	23	+	+	CCONJ
ejpam-4702	186	24	8n1r	8n1r	ADJ
ejpam-4702	186	25	s2	s2	NOUN
ejpam-4702	186	26	,	,	PUNCT
ejpam-4702	186	27	−3q2	−3q2	X
ejpam-4702	186	28	+	+	CCONJ
ejpam-4702	186	29	8n1r	8n1r	ADJ
ejpam-4702	186	30	s2	s2	NOUN
ejpam-4702	186	31	(	(	PUNCT
ejpam-4702	186	32	mod	mod	PROPN
ejpam-4702	186	33	8q	8q	PROPN
ejpam-4702	186	34	)	)	PUNCT
ejpam-4702	186	35	,	,	PUNCT
ejpam-4702	186	36	(	(	PUNCT
ejpam-4702	186	37	2q	2q	NUM
ejpam-4702	186	38	p	p	NOUN
ejpam-4702	186	39	)	)	PUNCT
ejpam-4702	186	40	=	=	PUNCT
ejpam-4702	186	41	−1	−1	NOUN
ejpam-4702	186	42	if	if	SCONJ
ejpam-4702	186	43	p	p	PRON
ejpam-4702	186	44	≡	≡	PROPN
ejpam-4702	186	45	q2	q2	PROPN
ejpam-4702	186	46	+	+	CCONJ
ejpam-4702	186	47	8n1r	8n1r	ADJ
ejpam-4702	186	48	s2	s2	NOUN
ejpam-4702	186	49	,	,	PUNCT
ejpam-4702	186	50	−q2	−q2	PROPN
ejpam-4702	186	51	+	+	CCONJ
ejpam-4702	186	52	8n1r	8n1r	ADJ
ejpam-4702	186	53	s2	s2	NOUN
ejpam-4702	186	54	,	,	PUNCT
ejpam-4702	186	55	3q2	3q2	NUM
ejpam-4702	186	56	+	+	CCONJ
ejpam-4702	186	57	8n1r	8n1r	ADJ
ejpam-4702	186	58	s1	s1	NOUN
ejpam-4702	186	59	,	,	PUNCT
ejpam-4702	186	60	−3q2	−3q2	X
ejpam-4702	186	61	+	+	CCONJ
ejpam-4702	186	62	8n1r	8n1r	ADJ
ejpam-4702	186	63	s1	s1	NOUN
ejpam-4702	186	64	(	(	PUNCT
ejpam-4702	186	65	mod	mod	PROPN
ejpam-4702	186	66	8q	8q	PROPN
ejpam-4702	186	67	)	)	PUNCT
ejpam-4702	186	68	,	,	PUNCT
ejpam-4702	186	69	where	where	SCONJ
ejpam-4702	186	70	s1	s1	PROPN
ejpam-4702	186	71	∈	∈	PROPN
ejpam-4702	186	72	{	{	PUNCT
ejpam-4702	186	73	2	2	NUM
ejpam-4702	186	74	,	,	PUNCT
ejpam-4702	186	75	4	4	NUM
ejpam-4702	186	76	,	,	PUNCT
ejpam-4702	186	77	6	6	NUM
ejpam-4702	186	78	,	,	PUNCT
ejpam-4702	186	79	.	.	PUNCT
ejpam-4702	186	80	.	.	PUNCT
ejpam-4702	186	81	.	.	PUNCT
ejpam-4702	187	1	,	,	PUNCT
ejpam-4702	187	2	q−	q−	PROPN
ejpam-4702	187	3	1	1	NUM
ejpam-4702	187	4	}	}	PUNCT
ejpam-4702	187	5	,	,	PUNCT
ejpam-4702	187	6	s2	s2	PROPN
ejpam-4702	187	7	∈	∈	PROPN
ejpam-4702	187	8	{	{	PUNCT
ejpam-4702	187	9	1	1	NUM
ejpam-4702	187	10	,	,	PUNCT
ejpam-4702	187	11	3	3	NUM
ejpam-4702	187	12	,	,	PUNCT
ejpam-4702	187	13	5	5	NUM
ejpam-4702	187	14	,	,	PUNCT
ejpam-4702	187	15	.	.	PUNCT
ejpam-4702	187	16	.	.	PUNCT
ejpam-4702	188	1	.	.	PUNCT
ejpam-4702	189	1	,	,	PUNCT
ejpam-4702	189	2	q−	q−	PROPN
ejpam-4702	189	3	2	2	NUM
ejpam-4702	189	4	}	}	PUNCT
ejpam-4702	189	5	,	,	PUNCT
ejpam-4702	189	6	r	r	NOUN
ejpam-4702	189	7	is	be	AUX
ejpam-4702	189	8	a	a	DET
ejpam-4702	189	9	primitive	primitive	ADJ
ejpam-4702	189	10	root	root	NOUN
ejpam-4702	189	11	modulo	modulo	NOUN
ejpam-4702	189	12	q	q	NOUN
ejpam-4702	190	1	and	and	CCONJ
ejpam-4702	190	2	if	if	SCONJ
ejpam-4702	190	3	q	q	PUNCT
ejpam-4702	190	4	−	−	NOUN
ejpam-4702	190	5	1	1	NUM
ejpam-4702	190	6	4	4	NUM
ejpam-4702	190	7	is	be	AUX
ejpam-4702	190	8	an	an	DET
ejpam-4702	190	9	even	even	ADJ
ejpam-4702	190	10	number	number	NOUN
ejpam-4702	190	11	,	,	PUNCT
ejpam-4702	190	12	then	then	ADV
ejpam-4702	190	13	n1	n1	ADJ
ejpam-4702	190	14	=	=	SYM
ejpam-4702	190	15	−q	−q	ADJ
ejpam-4702	190	16	+	+	CCONJ
ejpam-4702	190	17	1	1	NUM
ejpam-4702	190	18	8	8	NUM
ejpam-4702	190	19	,	,	PUNCT
ejpam-4702	190	20	and	and	CCONJ
ejpam-4702	190	21	if	if	SCONJ
ejpam-4702	190	22	otherwise	otherwise	ADV
ejpam-4702	190	23	,	,	PUNCT
ejpam-4702	190	24	then	then	ADV
ejpam-4702	190	25	n1	n1	PROPN
ejpam-4702	190	26	=	=	SYM
ejpam-4702	190	27	3q	3q	NUM
ejpam-4702	190	28	+	+	CCONJ
ejpam-4702	190	29	1	1	NUM
ejpam-4702	190	30	8	8	NUM
ejpam-4702	190	31	.	.	PUNCT
ejpam-4702	191	1	proof	proof	NOUN
ejpam-4702	191	2	.	.	PUNCT
ejpam-4702	192	1	by	by	ADP
ejpam-4702	192	2	theorem	theorem	ADJ
ejpam-4702	192	3	3(ii	3(ii	NUM
ejpam-4702	192	4	)	)	PUNCT
ejpam-4702	192	5	,	,	PUNCT
ejpam-4702	192	6	we	we	PRON
ejpam-4702	192	7	have	have	VERB
ejpam-4702	192	8	(	(	PUNCT
ejpam-4702	192	9	2q	2q	NUM
ejpam-4702	192	10	p	p	NOUN
ejpam-4702	192	11	)	)	PUNCT
ejpam-4702	192	12	=	=	PUNCT
ejpam-4702	192	13	(	(	PUNCT
ejpam-4702	192	14	2	2	NUM
ejpam-4702	192	15	p	p	NOUN
ejpam-4702	192	16	)	)	PUNCT
ejpam-4702	192	17	(	(	PUNCT
ejpam-4702	192	18	q	q	PROPN
ejpam-4702	192	19	p	p	NOUN
ejpam-4702	192	20	)	)	PUNCT
ejpam-4702	192	21	.	.	PUNCT
ejpam-4702	193	1	let	let	VERB
ejpam-4702	193	2	r	r	PRON
ejpam-4702	193	3	be	be	AUX
ejpam-4702	193	4	a	a	DET
ejpam-4702	193	5	primitive	primitive	ADJ
ejpam-4702	193	6	root	root	NOUN
ejpam-4702	193	7	modulo	modulo	NOUN
ejpam-4702	193	8	q.	q.	NOUN
ejpam-4702	193	9	by	by	ADP
ejpam-4702	193	10	theorems	theorem	NOUN
ejpam-4702	193	11	5	5	NUM
ejpam-4702	193	12	and	and	CCONJ
ejpam-4702	193	13	10	10	NUM
ejpam-4702	193	14	,	,	PUNCT
ejpam-4702	193	15	we	we	PRON
ejpam-4702	193	16	know	know	VERB
ejpam-4702	193	17	that	that	SCONJ
ejpam-4702	193	18	(	(	PUNCT
ejpam-4702	193	19	2	2	NUM
ejpam-4702	193	20	p	p	NOUN
ejpam-4702	193	21	)	)	PUNCT
ejpam-4702	193	22	=	=	PUNCT
ejpam-4702	193	23	{	{	PUNCT
ejpam-4702	193	24	1	1	NUM
ejpam-4702	193	25	if	if	SCONJ
ejpam-4702	193	26	p	p	DET
ejpam-4702	193	27	≡	≡	PROPN
ejpam-4702	193	28	±1	±1	VERB
ejpam-4702	193	29	(	(	PUNCT
ejpam-4702	193	30	mod	mod	PROPN
ejpam-4702	193	31	8)	8)	NUM
ejpam-4702	193	32	−1	−1	NOUN
ejpam-4702	193	33	if	if	SCONJ
ejpam-4702	193	34	p	p	PRON
ejpam-4702	193	35	≡	≡	PROPN
ejpam-4702	193	36	±3	±3	X
ejpam-4702	193	37	(	(	PUNCT
ejpam-4702	193	38	mod	mod	PROPN
ejpam-4702	193	39	8)	8)	NUM
ejpam-4702	193	40	and	and	CCONJ
ejpam-4702	193	41	(	(	PUNCT
ejpam-4702	193	42	q	q	X
ejpam-4702	193	43	p	p	NOUN
ejpam-4702	193	44	)	)	PUNCT
ejpam-4702	193	45	=	=	PUNCT
ejpam-4702	193	46	{	{	PUNCT
ejpam-4702	193	47	1	1	NUM
ejpam-4702	193	48	if	if	SCONJ
ejpam-4702	193	49	p	p	PRON
ejpam-4702	193	50	≡	≡	PROPN
ejpam-4702	193	51	q	q	PROPN
ejpam-4702	194	1	+	+	NUM
ejpam-4702	194	2	rs1q	rs1q	PROPN
ejpam-4702	194	3	+	+	CCONJ
ejpam-4702	194	4	rs1	rs1	PROPN
ejpam-4702	194	5	(	(	PUNCT
ejpam-4702	194	6	mod	mod	NOUN
ejpam-4702	194	7	2q	2q	NUM
ejpam-4702	194	8	)	)	PUNCT
ejpam-4702	194	9	−1	−1	NOUN
ejpam-4702	194	10	if	if	SCONJ
ejpam-4702	194	11	p	p	PRON
ejpam-4702	194	12	≡	≡	PROPN
ejpam-4702	194	13	q	q	PROPN
ejpam-4702	195	1	+	+	CCONJ
ejpam-4702	195	2	rs2q	rs2q	NUM
ejpam-4702	195	3	+	+	CCONJ
ejpam-4702	195	4	rs2	rs2	NOUN
ejpam-4702	195	5	(	(	PUNCT
ejpam-4702	195	6	mod	mod	ADJ
ejpam-4702	195	7	2q	2q	NUM
ejpam-4702	195	8	)	)	PUNCT
ejpam-4702	195	9	,	,	PUNCT
ejpam-4702	195	10	where	where	SCONJ
ejpam-4702	195	11	s1	s1	PROPN
ejpam-4702	195	12	∈	∈	PROPN
ejpam-4702	195	13	{	{	PUNCT
ejpam-4702	195	14	2	2	NUM
ejpam-4702	195	15	,	,	PUNCT
ejpam-4702	195	16	4	4	NUM
ejpam-4702	195	17	,	,	PUNCT
ejpam-4702	195	18	6	6	NUM
ejpam-4702	195	19	,	,	PUNCT
ejpam-4702	195	20	.	.	PUNCT
ejpam-4702	195	21	.	.	PUNCT
ejpam-4702	195	22	.	.	PUNCT
ejpam-4702	196	1	,	,	PUNCT
ejpam-4702	196	2	q	q	NOUN
ejpam-4702	196	3	−	−	NOUN
ejpam-4702	196	4	1	1	NUM
ejpam-4702	196	5	}	}	PUNCT
ejpam-4702	196	6	and	and	CCONJ
ejpam-4702	196	7	s2	s2	PROPN
ejpam-4702	196	8	∈	∈	PROPN
ejpam-4702	196	9	{	{	PUNCT
ejpam-4702	196	10	1	1	NUM
ejpam-4702	196	11	,	,	PUNCT
ejpam-4702	196	12	3	3	NUM
ejpam-4702	196	13	,	,	PUNCT
ejpam-4702	196	14	5	5	NUM
ejpam-4702	196	15	,	,	PUNCT
ejpam-4702	196	16	.	.	PUNCT
ejpam-4702	196	17	.	.	PUNCT
ejpam-4702	197	1	.	.	PUNCT
ejpam-4702	198	1	,	,	PUNCT
ejpam-4702	198	2	q	q	X
ejpam-4702	198	3	−	−	PROPN
ejpam-4702	198	4	2	2	NUM
ejpam-4702	198	5	}	}	PUNCT
ejpam-4702	198	6	.	.	PUNCT
ejpam-4702	199	1	since	since	SCONJ
ejpam-4702	199	2	q	q	PROPN
ejpam-4702	199	3	≡	≡	PROPN
ejpam-4702	199	4	1	1	NUM
ejpam-4702	199	5	(	(	PUNCT
ejpam-4702	199	6	mod	mod	NOUN
ejpam-4702	199	7	4	4	NUM
ejpam-4702	199	8	)	)	PUNCT
ejpam-4702	199	9	,	,	PUNCT
ejpam-4702	199	10	we	we	PRON
ejpam-4702	199	11	have	have	VERB
ejpam-4702	199	12	q	q	NOUN
ejpam-4702	199	13	=	=	PUNCT
ejpam-4702	199	14	4k	4k	NOUN
ejpam-4702	199	15	+	+	NOUN
ejpam-4702	199	16	1	1	NUM
ejpam-4702	199	17	for	for	ADP
ejpam-4702	199	18	some	some	DET
ejpam-4702	199	19	integer	integer	NOUN
ejpam-4702	199	20	k.	k.	PROPN
ejpam-4702	199	21	then	then	ADV
ejpam-4702	199	22	q2	q2	PROPN
ejpam-4702	199	23	≡	≡	PROPN
ejpam-4702	199	24	1	1	NUM
ejpam-4702	199	25	(	(	PUNCT
ejpam-4702	199	26	mod	mod	PROPN
ejpam-4702	199	27	8)	8)	NUM
ejpam-4702	199	28	.	.	PUNCT
ejpam-4702	200	1	if	if	SCONJ
ejpam-4702	200	2	k	k	PROPN
ejpam-4702	200	3	is	be	AUX
ejpam-4702	200	4	even	even	ADV
ejpam-4702	200	5	,	,	PUNCT
ejpam-4702	200	6	then	then	ADV
ejpam-4702	200	7	k	k	PROPN
ejpam-4702	200	8	=	=	PUNCT
ejpam-4702	200	9	−2l	−2l	PROPN
ejpam-4702	200	10	for	for	ADP
ejpam-4702	200	11	some	some	DET
ejpam-4702	200	12	integer	integer	NOUN
ejpam-4702	200	13	l.	l.	PROPN
ejpam-4702	200	14	it	it	PRON
ejpam-4702	200	15	leads	lead	VERB
ejpam-4702	200	16	to	to	ADP
ejpam-4702	200	17	8l	8l	NUM
ejpam-4702	200	18	−	−	PROPN
ejpam-4702	200	19	1	1	NUM
ejpam-4702	200	20	=	=	NOUN
ejpam-4702	200	21	−q	−q	NOUN
ejpam-4702	200	22	.	.	PUNCT
ejpam-4702	201	1	otherwise	otherwise	ADV
ejpam-4702	201	2	,	,	PUNCT
ejpam-4702	201	3	q	q	X
ejpam-4702	201	4	=	=	SYM
ejpam-4702	201	5	8s	8	NOUN
ejpam-4702	202	1	+	+	CCONJ
ejpam-4702	202	2	5	5	NUM
ejpam-4702	202	3	for	for	ADP
ejpam-4702	202	4	some	some	DET
ejpam-4702	202	5	integer	integer	NOUN
ejpam-4702	202	6	s	s	NOUN
ejpam-4702	202	7	,	,	PUNCT
ejpam-4702	202	8	so	so	ADV
ejpam-4702	202	9	8(3s	8(3s	NUM
ejpam-4702	202	10	+	+	CCONJ
ejpam-4702	202	11	2	2	X
ejpam-4702	202	12	)	)	PUNCT
ejpam-4702	202	13	−	−	PROPN
ejpam-4702	202	14	1	1	NUM
ejpam-4702	202	15	=	=	SYM
ejpam-4702	202	16	3q	3q	NUM
ejpam-4702	202	17	.	.	PUNCT
ejpam-4702	203	1	choose	choose	VERB
ejpam-4702	203	2	n1	n1	NOUN
ejpam-4702	203	3	=	=	SYM
ejpam-4702	203	4	−q	−q	NOUN
ejpam-4702	203	5	+	+	CCONJ
ejpam-4702	203	6	1	1	NUM
ejpam-4702	203	7	8	8	NUM
ejpam-4702	203	8	,	,	PUNCT
ejpam-4702	203	9	when	when	SCONJ
ejpam-4702	203	10	k	k	PROPN
ejpam-4702	203	11	is	be	AUX
ejpam-4702	203	12	even	even	ADV
ejpam-4702	203	13	and	and	CCONJ
ejpam-4702	203	14	otherwise	otherwise	ADV
ejpam-4702	203	15	,	,	PUNCT
ejpam-4702	203	16	n1	n1	NOUN
ejpam-4702	203	17	=	=	SYM
ejpam-4702	203	18	3q	3q	NUM
ejpam-4702	203	19	+	+	CCONJ
ejpam-4702	203	20	1	1	NUM
ejpam-4702	203	21	8	8	NUM
ejpam-4702	203	22	.	.	PUNCT
ejpam-4702	204	1	thus	thus	ADV
ejpam-4702	204	2	,	,	PUNCT
ejpam-4702	204	3	8n1	8n1	NUM
ejpam-4702	204	4	≡	≡	PROPN
ejpam-4702	204	5	1	1	NUM
ejpam-4702	204	6	(	(	PUNCT
ejpam-4702	204	7	mod	mod	PROPN
ejpam-4702	204	8	q	q	NOUN
ejpam-4702	204	9	)	)	PUNCT
ejpam-4702	204	10	.	.	PUNCT
ejpam-4702	205	1	in	in	ADP
ejpam-4702	205	2	the	the	DET
ejpam-4702	205	3	following	following	ADJ
ejpam-4702	205	4	cases	case	NOUN
ejpam-4702	205	5	,	,	PUNCT
ejpam-4702	205	6	we	we	PRON
ejpam-4702	205	7	use	use	VERB
ejpam-4702	205	8	the	the	DET
ejpam-4702	205	9	chinese	chinese	ADJ
ejpam-4702	205	10	remainder	remainder	NOUN
ejpam-4702	205	11	theorem	theorem	NOUN
ejpam-4702	205	12	for	for	ADP
ejpam-4702	205	13	solving	solve	VERB
ejpam-4702	205	14	the	the	DET
ejpam-4702	205	15	systems	system	NOUN
ejpam-4702	205	16	of	of	ADP
ejpam-4702	205	17	congruences	congruence	NOUN
ejpam-4702	205	18	.	.	PUNCT
ejpam-4702	206	1	case	case	NOUN
ejpam-4702	206	2	1	1	NUM
ejpam-4702	206	3	.	.	PUNCT
ejpam-4702	207	1	(	(	PUNCT
ejpam-4702	207	2	2q	2q	NUM
ejpam-4702	207	3	p	p	NOUN
ejpam-4702	207	4	)	)	PUNCT
ejpam-4702	207	5	=	=	SYM
ejpam-4702	207	6	1	1	X
ejpam-4702	207	7	.	.	PUNCT
ejpam-4702	207	8	case	case	NOUN
ejpam-4702	207	9	1.1	1.1	NUM
ejpam-4702	207	10	(	(	PUNCT
ejpam-4702	207	11	2	2	NUM
ejpam-4702	207	12	p	p	NOUN
ejpam-4702	207	13	)	)	PUNCT
ejpam-4702	207	14	=	=	SYM
ejpam-4702	207	15	1	1	NUM
ejpam-4702	207	16	and	and	CCONJ
ejpam-4702	207	17	(	(	PUNCT
ejpam-4702	207	18	q	q	X
ejpam-4702	207	19	p	p	NOUN
ejpam-4702	207	20	)	)	PUNCT
ejpam-4702	207	21	=	=	SYM
ejpam-4702	208	1	1	1	X
ejpam-4702	208	2	.	.	PUNCT
ejpam-4702	208	3	then	then	ADV
ejpam-4702	208	4	p	p	PROPN
ejpam-4702	208	5	≡	≡	PROPN
ejpam-4702	208	6	±1	±1	PROPN
ejpam-4702	208	7	(	(	PUNCT
ejpam-4702	208	8	mod	mod	PROPN
ejpam-4702	208	9	8)	8)	NUM
ejpam-4702	208	10	and	and	CCONJ
ejpam-4702	208	11	p	p	PROPN
ejpam-4702	208	12	≡	≡	PROPN
ejpam-4702	208	13	q	q	PROPN
ejpam-4702	209	1	+	+	NUM
ejpam-4702	209	2	rs1q	rs1q	PROPN
ejpam-4702	209	3	+	+	CCONJ
ejpam-4702	209	4	rs1	rs1	PROPN
ejpam-4702	209	5	(	(	PUNCT
ejpam-4702	209	6	mod	mod	ADJ
ejpam-4702	209	7	2q	2q	NUM
ejpam-4702	209	8	)	)	PUNCT
ejpam-4702	209	9	.	.	PUNCT
ejpam-4702	210	1	so	so	ADV
ejpam-4702	210	2	p	p	PROPN
ejpam-4702	210	3	≡	≡	PROPN
ejpam-4702	210	4	rs1	rs1	PROPN
ejpam-4702	210	5	(	(	PUNCT
ejpam-4702	210	6	mod	mod	PROPN
ejpam-4702	210	7	q	q	PROPN
ejpam-4702	210	8	)	)	PUNCT
ejpam-4702	210	9	.	.	PUNCT
ejpam-4702	211	1	it	it	PRON
ejpam-4702	211	2	implies	imply	VERB
ejpam-4702	211	3	that	that	SCONJ
ejpam-4702	211	4	p	p	PROPN
ejpam-4702	211	5	≡	≡	PROPN
ejpam-4702	211	6	q2	q2	PROPN
ejpam-4702	211	7	+	+	CCONJ
ejpam-4702	211	8	8n1r	8n1r	ADJ
ejpam-4702	211	9	s1	s1	NOUN
ejpam-4702	211	10	(	(	PUNCT
ejpam-4702	211	11	mod	mod	PROPN
ejpam-4702	211	12	8q	8q	NUM
ejpam-4702	211	13	)	)	PUNCT
ejpam-4702	211	14	or	or	CCONJ
ejpam-4702	211	15	p	p	PROPN
ejpam-4702	211	16	≡	≡	PROPN
ejpam-4702	211	17	−q2	−q2	PROPN
ejpam-4702	211	18	+	+	CCONJ
ejpam-4702	211	19	8n1r	8n1r	ADJ
ejpam-4702	211	20	s1	s1	NOUN
ejpam-4702	211	21	(	(	PUNCT
ejpam-4702	211	22	mod	mod	PROPN
ejpam-4702	211	23	8q	8q	NUM
ejpam-4702	211	24	)	)	PUNCT
ejpam-4702	211	25	.	.	PUNCT
ejpam-4702	212	1	case	case	NOUN
ejpam-4702	212	2	1.2	1.2	NUM
ejpam-4702	212	3	(	(	PUNCT
ejpam-4702	212	4	2	2	NUM
ejpam-4702	212	5	p	p	NOUN
ejpam-4702	212	6	)	)	PUNCT
ejpam-4702	212	7	=	=	SYM
ejpam-4702	212	8	−1	−1	NOUN
ejpam-4702	212	9	and	and	CCONJ
ejpam-4702	212	10	(	(	PUNCT
ejpam-4702	212	11	q	q	X
ejpam-4702	212	12	p	p	NOUN
ejpam-4702	212	13	)	)	PUNCT
ejpam-4702	212	14	=	=	SYM
ejpam-4702	212	15	−1	−1	NOUN
ejpam-4702	212	16	.	.	PUNCT
ejpam-4702	213	1	then	then	ADV
ejpam-4702	213	2	p	p	PROPN
ejpam-4702	213	3	≡	≡	PROPN
ejpam-4702	213	4	±3	±3	X
ejpam-4702	213	5	(	(	PUNCT
ejpam-4702	213	6	mod	mod	PROPN
ejpam-4702	213	7	8)	8)	NUM
ejpam-4702	213	8	and	and	CCONJ
ejpam-4702	213	9	p	p	PROPN
ejpam-4702	213	10	≡	≡	PROPN
ejpam-4702	213	11	q	q	PROPN
ejpam-4702	214	1	+	+	CCONJ
ejpam-4702	214	2	rs2q	rs2q	NUM
ejpam-4702	214	3	+	+	CCONJ
ejpam-4702	214	4	rs2	rs2	NOUN
ejpam-4702	214	5	(	(	PUNCT
ejpam-4702	214	6	mod	mod	ADJ
ejpam-4702	214	7	2q	2q	NUM
ejpam-4702	214	8	)	)	PUNCT
ejpam-4702	214	9	.	.	PUNCT
ejpam-4702	215	1	so	so	ADV
ejpam-4702	215	2	p	p	PRON
ejpam-4702	215	3	≡	≡	PROPN
ejpam-4702	215	4	rs2	rs2	NOUN
ejpam-4702	215	5	(	(	PUNCT
ejpam-4702	215	6	mod	mod	PROPN
ejpam-4702	215	7	q	q	NOUN
ejpam-4702	215	8	)	)	PUNCT
ejpam-4702	215	9	.	.	PUNCT
ejpam-4702	216	1	it	it	PRON
ejpam-4702	216	2	implies	imply	VERB
ejpam-4702	216	3	that	that	SCONJ
ejpam-4702	216	4	p	p	PROPN
ejpam-4702	216	5	≡	≡	PROPN
ejpam-4702	216	6	3q2	3q2	PROPN
ejpam-4702	216	7	+	+	CCONJ
ejpam-4702	216	8	8n1r	8n1r	ADJ
ejpam-4702	216	9	s2	s2	NOUN
ejpam-4702	216	10	(	(	PUNCT
ejpam-4702	216	11	mod	mod	PROPN
ejpam-4702	216	12	8q	8q	NUM
ejpam-4702	216	13	)	)	PUNCT
ejpam-4702	216	14	or	or	CCONJ
ejpam-4702	216	15	p	p	PRON
ejpam-4702	216	16	≡	≡	PROPN
ejpam-4702	216	17	−3q2	−3q2	NUM
ejpam-4702	216	18	+	+	CCONJ
ejpam-4702	216	19	8n1r	8n1r	ADJ
ejpam-4702	216	20	s2	s2	NOUN
ejpam-4702	216	21	(	(	PUNCT
ejpam-4702	216	22	mod	mod	PROPN
ejpam-4702	216	23	8q	8q	NUM
ejpam-4702	216	24	)	)	PUNCT
ejpam-4702	216	25	.	.	PUNCT
ejpam-4702	217	1	case	case	NOUN
ejpam-4702	217	2	2	2	X
ejpam-4702	217	3	.	.	PUNCT
ejpam-4702	218	1	(	(	PUNCT
ejpam-4702	218	2	2q	2q	NUM
ejpam-4702	218	3	p	p	NOUN
ejpam-4702	218	4	)	)	PUNCT
ejpam-4702	218	5	=	=	PUNCT
ejpam-4702	218	6	−1	−1	NOUN
ejpam-4702	218	7	.	.	PUNCT
ejpam-4702	219	1	case	case	NOUN
ejpam-4702	219	2	2.1	2.1	NUM
ejpam-4702	219	3	(	(	PUNCT
ejpam-4702	219	4	2	2	NUM
ejpam-4702	219	5	p	p	NOUN
ejpam-4702	219	6	)	)	PUNCT
ejpam-4702	219	7	=	=	SYM
ejpam-4702	219	8	1	1	NUM
ejpam-4702	219	9	and	and	CCONJ
ejpam-4702	219	10	(	(	PUNCT
ejpam-4702	219	11	q	q	X
ejpam-4702	219	12	p	p	NOUN
ejpam-4702	219	13	)	)	PUNCT
ejpam-4702	219	14	=	=	SYM
ejpam-4702	219	15	−1	−1	NOUN
ejpam-4702	219	16	.	.	PUNCT
ejpam-4702	220	1	then	then	ADV
ejpam-4702	220	2	p	p	PROPN
ejpam-4702	220	3	≡	≡	PROPN
ejpam-4702	220	4	±1	±1	PROPN
ejpam-4702	220	5	(	(	PUNCT
ejpam-4702	220	6	mod	mod	PROPN
ejpam-4702	220	7	8)	8)	NUM
ejpam-4702	220	8	and	and	CCONJ
ejpam-4702	220	9	p	p	PROPN
ejpam-4702	220	10	≡	≡	PROPN
ejpam-4702	220	11	q	q	PROPN
ejpam-4702	221	1	+	+	CCONJ
ejpam-4702	221	2	rs2q	rs2q	NUM
ejpam-4702	221	3	+	+	CCONJ
ejpam-4702	221	4	rs2	rs2	NOUN
ejpam-4702	221	5	(	(	PUNCT
ejpam-4702	221	6	mod	mod	ADJ
ejpam-4702	221	7	2q	2q	NUM
ejpam-4702	221	8	)	)	PUNCT
ejpam-4702	221	9	.	.	PUNCT
ejpam-4702	222	1	so	so	ADV
ejpam-4702	222	2	p	p	PRON
ejpam-4702	222	3	≡	≡	PROPN
ejpam-4702	222	4	rs2	rs2	NOUN
ejpam-4702	222	5	(	(	PUNCT
ejpam-4702	222	6	mod	mod	PROPN
ejpam-4702	222	7	q	q	NOUN
ejpam-4702	222	8	)	)	PUNCT
ejpam-4702	222	9	.	.	PUNCT
ejpam-4702	223	1	it	it	PRON
ejpam-4702	223	2	implies	imply	VERB
ejpam-4702	223	3	that	that	SCONJ
ejpam-4702	223	4	p	p	PROPN
ejpam-4702	223	5	≡	≡	PROPN
ejpam-4702	223	6	q2	q2	PROPN
ejpam-4702	223	7	+	+	CCONJ
ejpam-4702	223	8	8n1r	8n1r	ADJ
ejpam-4702	223	9	s2	s2	NOUN
ejpam-4702	223	10	(	(	PUNCT
ejpam-4702	223	11	mod	mod	PROPN
ejpam-4702	223	12	8q	8q	NUM
ejpam-4702	223	13	)	)	PUNCT
ejpam-4702	223	14	or	or	CCONJ
ejpam-4702	223	15	p	p	PROPN
ejpam-4702	223	16	≡	≡	PROPN
ejpam-4702	223	17	−q2	−q2	PROPN
ejpam-4702	223	18	+	+	CCONJ
ejpam-4702	223	19	8n1r	8n1r	ADJ
ejpam-4702	223	20	s2	s2	NOUN
ejpam-4702	223	21	(	(	PUNCT
ejpam-4702	223	22	mod	mod	PROPN
ejpam-4702	223	23	8q	8q	NUM
ejpam-4702	223	24	)	)	PUNCT
ejpam-4702	223	25	.	.	PUNCT
ejpam-4702	224	1	case	case	NOUN
ejpam-4702	224	2	2.2	2.2	NUM
ejpam-4702	224	3	(	(	PUNCT
ejpam-4702	224	4	2	2	NUM
ejpam-4702	224	5	p	p	NOUN
ejpam-4702	224	6	)	)	PUNCT
ejpam-4702	224	7	=	=	SYM
ejpam-4702	224	8	−1	−1	NOUN
ejpam-4702	224	9	and	and	CCONJ
ejpam-4702	224	10	(	(	PUNCT
ejpam-4702	224	11	q	q	X
ejpam-4702	224	12	p	p	NOUN
ejpam-4702	224	13	)	)	PUNCT
ejpam-4702	225	1	=	=	SYM
ejpam-4702	225	2	1	1	X
ejpam-4702	225	3	.	.	PUNCT
ejpam-4702	226	1	then	then	ADV
ejpam-4702	226	2	p	p	PROPN
ejpam-4702	226	3	≡	≡	PROPN
ejpam-4702	226	4	±3	±3	X
ejpam-4702	226	5	(	(	PUNCT
ejpam-4702	226	6	mod	mod	PROPN
ejpam-4702	226	7	8)	8)	NUM
ejpam-4702	226	8	and	and	CCONJ
ejpam-4702	226	9	p	p	PROPN
ejpam-4702	226	10	≡	≡	PROPN
ejpam-4702	226	11	q	q	PROPN
ejpam-4702	227	1	+	+	NUM
ejpam-4702	227	2	rs1q	rs1q	PROPN
ejpam-4702	227	3	+	+	CCONJ
ejpam-4702	227	4	rs1	rs1	PROPN
ejpam-4702	227	5	(	(	PUNCT
ejpam-4702	227	6	mod	mod	ADJ
ejpam-4702	227	7	2q	2q	NUM
ejpam-4702	227	8	)	)	PUNCT
ejpam-4702	227	9	.	.	PUNCT
ejpam-4702	228	1	so	so	ADV
ejpam-4702	228	2	p	p	PROPN
ejpam-4702	228	3	≡	≡	PROPN
ejpam-4702	228	4	rs1	rs1	PROPN
ejpam-4702	228	5	(	(	PUNCT
ejpam-4702	228	6	mod	mod	PROPN
ejpam-4702	228	7	q	q	PROPN
ejpam-4702	228	8	)	)	PUNCT
ejpam-4702	228	9	.	.	PUNCT
ejpam-4702	229	1	it	it	PRON
ejpam-4702	229	2	implies	imply	VERB
ejpam-4702	229	3	that	that	SCONJ
ejpam-4702	229	4	p	p	PROPN
ejpam-4702	229	5	≡	≡	PROPN
ejpam-4702	229	6	3q2	3q2	NUM
ejpam-4702	229	7	+	+	CCONJ
ejpam-4702	229	8	8n1r	8n1r	ADJ
ejpam-4702	229	9	s1	s1	NOUN
ejpam-4702	229	10	(	(	PUNCT
ejpam-4702	229	11	mod	mod	PROPN
ejpam-4702	229	12	8q	8q	NUM
ejpam-4702	229	13	)	)	PUNCT
ejpam-4702	229	14	or	or	CCONJ
ejpam-4702	229	15	p	p	PRON
ejpam-4702	229	16	≡	≡	PROPN
ejpam-4702	229	17	−3q2	−3q2	NUM
ejpam-4702	229	18	+	+	CCONJ
ejpam-4702	229	19	8n1r	8n1r	ADJ
ejpam-4702	229	20	s1	s1	NOUN
ejpam-4702	229	21	(	(	PUNCT
ejpam-4702	229	22	mod	mod	PROPN
ejpam-4702	229	23	8q	8q	NUM
ejpam-4702	229	24	)	)	PUNCT
ejpam-4702	229	25	.	.	PUNCT
ejpam-4702	230	1	s.	s.	PROPN
ejpam-4702	230	2	tadee	tadee	PROPN
ejpam-4702	230	3	,	,	PUNCT
ejpam-4702	230	4	a.	a.	NOUN
ejpam-4702	230	5	siraworakun	siraworakun	PROPN
ejpam-4702	230	6	/	/	SYM
ejpam-4702	230	7	eur	eur	PROPN
ejpam-4702	230	8	.	.	PUNCT
ejpam-4702	231	1	j.	j.	PROPN
ejpam-4702	231	2	pure	pure	PROPN
ejpam-4702	231	3	appl	appl	PROPN
ejpam-4702	231	4	.	.	PROPN
ejpam-4702	231	5	math	math	PROPN
ejpam-4702	231	6	,	,	PUNCT
ejpam-4702	231	7	16	16	NUM
ejpam-4702	231	8	(	(	PUNCT
ejpam-4702	231	9	2	2	NUM
ejpam-4702	231	10	)	)	PUNCT
ejpam-4702	231	11	(	(	PUNCT
ejpam-4702	231	12	2023	2023	NUM
ejpam-4702	231	13	)	)	PUNCT
ejpam-4702	231	14	,	,	PUNCT
ejpam-4702	231	15	724	724	NUM
ejpam-4702	231	16	-	-	SYM
ejpam-4702	231	17	735	735	NUM
ejpam-4702	231	18	730	730	NUM
ejpam-4702	231	19	theorem	theorem	NOUN
ejpam-4702	231	20	13	13	NUM
ejpam-4702	231	21	.	.	PUNCT
ejpam-4702	232	1	let	let	VERB
ejpam-4702	232	2	p	p	NOUN
ejpam-4702	232	3	and	and	CCONJ
ejpam-4702	232	4	q	q	NOUN
ejpam-4702	232	5	be	be	AUX
ejpam-4702	232	6	distinct	distinct	ADJ
ejpam-4702	232	7	odd	odd	ADJ
ejpam-4702	232	8	prime	prime	ADJ
ejpam-4702	232	9	numbers	number	NOUN
ejpam-4702	232	10	with	with	ADP
ejpam-4702	232	11	q	q	PROPN
ejpam-4702	232	12	≡	≡	PROPN
ejpam-4702	232	13	3	3	NUM
ejpam-4702	232	14	(	(	PUNCT
ejpam-4702	232	15	mod	mod	NOUN
ejpam-4702	232	16	4	4	NUM
ejpam-4702	232	17	)	)	PUNCT
ejpam-4702	232	18	.	.	PUNCT
ejpam-4702	233	1	then	then	ADV
ejpam-4702	233	2	(	(	PUNCT
ejpam-4702	233	3	2q	2q	NUM
ejpam-4702	233	4	p	p	NOUN
ejpam-4702	233	5	)	)	PUNCT
ejpam-4702	233	6	=	=	SYM
ejpam-4702	233	7	1	1	NUM
ejpam-4702	233	8	if	if	SCONJ
ejpam-4702	233	9	p	p	DET
ejpam-4702	233	10	≡	≡	PROPN
ejpam-4702	233	11	q2	q2	PROPN
ejpam-4702	233	12	+	+	CCONJ
ejpam-4702	233	13	32n0n1r	32n0n1r	CCONJ
ejpam-4702	233	14	s1	s1	NOUN
ejpam-4702	233	15	,	,	PUNCT
ejpam-4702	233	16	−q2	−q2	PROPN
ejpam-4702	233	17	+	+	CCONJ
ejpam-4702	233	18	32n0n1r	32n0n1r	NUM
ejpam-4702	233	19	s2	s2	NOUN
ejpam-4702	233	20	,	,	PUNCT
ejpam-4702	233	21	3q2	3q2	NUM
ejpam-4702	233	22	+	+	CCONJ
ejpam-4702	233	23	32n0n1r	32n0n1r	CCONJ
ejpam-4702	233	24	s1	s1	NOUN
ejpam-4702	233	25	,	,	PUNCT
ejpam-4702	233	26	−3q2	−3q2	X
ejpam-4702	233	27	+	+	CCONJ
ejpam-4702	233	28	32n0n1r	32n0n1r	CCONJ
ejpam-4702	233	29	s2	s2	PROPN
ejpam-4702	233	30	(	(	PUNCT
ejpam-4702	233	31	mod	mod	PROPN
ejpam-4702	233	32	8q	8q	PROPN
ejpam-4702	233	33	)	)	PUNCT
ejpam-4702	233	34	,	,	PUNCT
ejpam-4702	233	35	(	(	PUNCT
ejpam-4702	233	36	2q	2q	NUM
ejpam-4702	233	37	p	p	NOUN
ejpam-4702	233	38	)	)	PUNCT
ejpam-4702	233	39	=	=	PUNCT
ejpam-4702	233	40	−1	−1	NOUN
ejpam-4702	233	41	if	if	SCONJ
ejpam-4702	233	42	p	p	PRON
ejpam-4702	233	43	≡	≡	PROPN
ejpam-4702	233	44	q2	q2	PROPN
ejpam-4702	233	45	+	+	CCONJ
ejpam-4702	233	46	32n0n1r	32n0n1r	CCONJ
ejpam-4702	233	47	s2	s2	NOUN
ejpam-4702	233	48	,	,	PUNCT
ejpam-4702	233	49	−q2	−q2	PROPN
ejpam-4702	233	50	+	+	CCONJ
ejpam-4702	233	51	32n0n1r	32n0n1r	NUM
ejpam-4702	233	52	s1	s1	NOUN
ejpam-4702	233	53	,	,	PUNCT
ejpam-4702	233	54	3q2	3q2	NUM
ejpam-4702	233	55	+	+	CCONJ
ejpam-4702	233	56	32n0n1r	32n0n1r	CCONJ
ejpam-4702	233	57	s2	s2	NOUN
ejpam-4702	233	58	,	,	PUNCT
ejpam-4702	233	59	−3q2	−3q2	X
ejpam-4702	233	60	+	+	CCONJ
ejpam-4702	233	61	32n0n1r	32n0n1r	CCONJ
ejpam-4702	233	62	s1	s1	NOUN
ejpam-4702	233	63	(	(	PUNCT
ejpam-4702	233	64	mod	mod	PROPN
ejpam-4702	233	65	8q	8q	PROPN
ejpam-4702	233	66	)	)	PUNCT
ejpam-4702	233	67	,	,	PUNCT
ejpam-4702	233	68	where	where	SCONJ
ejpam-4702	233	69	s1	s1	PROPN
ejpam-4702	233	70	∈	∈	PROPN
ejpam-4702	233	71	{	{	PUNCT
ejpam-4702	233	72	2	2	NUM
ejpam-4702	233	73	,	,	PUNCT
ejpam-4702	233	74	4	4	NUM
ejpam-4702	233	75	,	,	PUNCT
ejpam-4702	233	76	6	6	NUM
ejpam-4702	233	77	,	,	PUNCT
ejpam-4702	233	78	.	.	PUNCT
ejpam-4702	233	79	.	.	PUNCT
ejpam-4702	233	80	.	.	PUNCT
ejpam-4702	234	1	,	,	PUNCT
ejpam-4702	234	2	q	q	NOUN
ejpam-4702	234	3	−	−	NOUN
ejpam-4702	234	4	1	1	NUM
ejpam-4702	234	5	}	}	PUNCT
ejpam-4702	234	6	,	,	PUNCT
ejpam-4702	234	7	s2	s2	PROPN
ejpam-4702	234	8	∈	∈	PROPN
ejpam-4702	234	9	{	{	PUNCT
ejpam-4702	234	10	1	1	NUM
ejpam-4702	234	11	,	,	PUNCT
ejpam-4702	234	12	3	3	NUM
ejpam-4702	234	13	,	,	PUNCT
ejpam-4702	234	14	5	5	NUM
ejpam-4702	234	15	,	,	PUNCT
ejpam-4702	234	16	.	.	PUNCT
ejpam-4702	234	17	.	.	PUNCT
ejpam-4702	235	1	.	.	PUNCT
ejpam-4702	236	1	,	,	PUNCT
ejpam-4702	236	2	q	q	X
ejpam-4702	237	1	−	−	NOUN
ejpam-4702	237	2	2	2	NUM
ejpam-4702	237	3	}	}	PUNCT
ejpam-4702	237	4	,	,	PUNCT
ejpam-4702	237	5	r	r	NOUN
ejpam-4702	237	6	is	be	AUX
ejpam-4702	237	7	a	a	DET
ejpam-4702	237	8	primitive	primitive	ADJ
ejpam-4702	237	9	root	root	NOUN
ejpam-4702	237	10	modulo	modulo	NOUN
ejpam-4702	237	11	q	q	X
ejpam-4702	237	12	,	,	PUNCT
ejpam-4702	237	13	n0	n0	X
ejpam-4702	237	14	=	=	PUNCT
ejpam-4702	237	15	q	q	PROPN
ejpam-4702	238	1	+	+	NUM
ejpam-4702	238	2	1	1	NUM
ejpam-4702	238	3	4	4	NUM
ejpam-4702	238	4	and	and	CCONJ
ejpam-4702	238	5	if	if	SCONJ
ejpam-4702	238	6	q	q	PUNCT
ejpam-4702	238	7	−	−	NOUN
ejpam-4702	238	8	3	3	NUM
ejpam-4702	238	9	4	4	NUM
ejpam-4702	238	10	is	be	AUX
ejpam-4702	238	11	an	an	DET
ejpam-4702	238	12	even	even	ADJ
ejpam-4702	238	13	number	number	NOUN
ejpam-4702	238	14	,	,	PUNCT
ejpam-4702	238	15	then	then	ADV
ejpam-4702	238	16	n1	n1	PROPN
ejpam-4702	238	17	=	=	SYM
ejpam-4702	238	18	5q	5q	NOUN
ejpam-4702	238	19	+	+	CCONJ
ejpam-4702	238	20	1	1	NUM
ejpam-4702	238	21	8	8	NUM
ejpam-4702	238	22	,	,	PUNCT
ejpam-4702	238	23	and	and	CCONJ
ejpam-4702	238	24	if	if	SCONJ
ejpam-4702	238	25	otherwise	otherwise	ADV
ejpam-4702	238	26	,	,	PUNCT
ejpam-4702	238	27	then	then	ADV
ejpam-4702	238	28	n1	n1	PROPN
ejpam-4702	238	29	=	=	SYM
ejpam-4702	238	30	q	q	PROPN
ejpam-4702	239	1	+	+	NUM
ejpam-4702	239	2	1	1	NUM
ejpam-4702	239	3	8	8	NUM
ejpam-4702	239	4	.	.	PUNCT
ejpam-4702	240	1	proof	proof	NOUN
ejpam-4702	240	2	.	.	PUNCT
ejpam-4702	241	1	by	by	ADP
ejpam-4702	241	2	theorem	theorem	ADJ
ejpam-4702	241	3	3(ii	3(ii	NUM
ejpam-4702	241	4	)	)	PUNCT
ejpam-4702	241	5	,	,	PUNCT
ejpam-4702	241	6	we	we	PRON
ejpam-4702	241	7	have	have	VERB
ejpam-4702	241	8	(	(	PUNCT
ejpam-4702	241	9	2q	2q	NUM
ejpam-4702	241	10	p	p	NOUN
ejpam-4702	241	11	)	)	PUNCT
ejpam-4702	241	12	=	=	PUNCT
ejpam-4702	241	13	(	(	PUNCT
ejpam-4702	241	14	2	2	NUM
ejpam-4702	241	15	p	p	NOUN
ejpam-4702	241	16	)	)	PUNCT
ejpam-4702	241	17	(	(	PUNCT
ejpam-4702	241	18	q	q	PROPN
ejpam-4702	241	19	p	p	NOUN
ejpam-4702	241	20	)	)	PUNCT
ejpam-4702	241	21	.	.	PUNCT
ejpam-4702	242	1	let	let	VERB
ejpam-4702	242	2	r	r	PRON
ejpam-4702	242	3	be	be	AUX
ejpam-4702	242	4	a	a	DET
ejpam-4702	242	5	primitive	primitive	ADJ
ejpam-4702	242	6	root	root	NOUN
ejpam-4702	242	7	modulo	modulo	NOUN
ejpam-4702	242	8	q.	q.	NOUN
ejpam-4702	242	9	by	by	ADP
ejpam-4702	242	10	theorems	theorem	NOUN
ejpam-4702	242	11	5	5	NUM
ejpam-4702	242	12	and	and	CCONJ
ejpam-4702	242	13	11	11	NUM
ejpam-4702	242	14	,	,	PUNCT
ejpam-4702	242	15	we	we	PRON
ejpam-4702	242	16	know	know	VERB
ejpam-4702	242	17	that	that	SCONJ
ejpam-4702	242	18	(	(	PUNCT
ejpam-4702	242	19	2	2	NUM
ejpam-4702	242	20	p	p	NOUN
ejpam-4702	242	21	)	)	PUNCT
ejpam-4702	243	1	=	=	PUNCT
ejpam-4702	243	2	{	{	PUNCT
ejpam-4702	243	3	1	1	NUM
ejpam-4702	243	4	if	if	SCONJ
ejpam-4702	243	5	p	p	DET
ejpam-4702	243	6	≡	≡	PROPN
ejpam-4702	243	7	±1	±1	VERB
ejpam-4702	243	8	(	(	PUNCT
ejpam-4702	243	9	mod	mod	PROPN
ejpam-4702	243	10	8)	8)	NUM
ejpam-4702	243	11	−1	−1	NOUN
ejpam-4702	243	12	if	if	SCONJ
ejpam-4702	243	13	p	p	PRON
ejpam-4702	243	14	≡	≡	PROPN
ejpam-4702	243	15	±3	±3	X
ejpam-4702	243	16	(	(	PUNCT
ejpam-4702	243	17	mod	mod	PROPN
ejpam-4702	243	18	8)	8)	NUM
ejpam-4702	243	19	and	and	CCONJ
ejpam-4702	243	20	(	(	PUNCT
ejpam-4702	243	21	q	q	X
ejpam-4702	243	22	p	p	NOUN
ejpam-4702	243	23	)	)	PUNCT
ejpam-4702	243	24	=	=	PUNCT
ejpam-4702	243	25	{	{	PUNCT
ejpam-4702	243	26	1	1	NUM
ejpam-4702	243	27	if	if	SCONJ
ejpam-4702	243	28	p	p	PRON
ejpam-4702	243	29	≡	≡	PROPN
ejpam-4702	243	30	3q	3q	NOUN
ejpam-4702	243	31	+	+	CCONJ
ejpam-4702	243	32	4n0r	4n0r	NOUN
ejpam-4702	243	33	s1	s1	NOUN
ejpam-4702	243	34	,	,	PUNCT
ejpam-4702	243	35	−	−	PROPN
ejpam-4702	243	36	3q	3q	NUM
ejpam-4702	243	37	+	+	CCONJ
ejpam-4702	243	38	4n0r	4n0r	NOUN
ejpam-4702	243	39	s2	s2	NOUN
ejpam-4702	243	40	(	(	PUNCT
ejpam-4702	243	41	mod	mod	PROPN
ejpam-4702	243	42	4q	4q	NOUN
ejpam-4702	243	43	)	)	PUNCT
ejpam-4702	243	44	−1	−1	NOUN
ejpam-4702	243	45	if	if	SCONJ
ejpam-4702	243	46	p	p	PRON
ejpam-4702	243	47	≡	≡	PROPN
ejpam-4702	243	48	3q	3q	NOUN
ejpam-4702	243	49	+	+	CCONJ
ejpam-4702	243	50	4n0r	4n0r	NOUN
ejpam-4702	243	51	s2	s2	NOUN
ejpam-4702	243	52	,	,	PUNCT
ejpam-4702	243	53	−	−	PROPN
ejpam-4702	243	54	3q	3q	NUM
ejpam-4702	243	55	+	+	CCONJ
ejpam-4702	243	56	4n0r	4n0r	NOUN
ejpam-4702	243	57	s1	s1	NOUN
ejpam-4702	243	58	(	(	PUNCT
ejpam-4702	243	59	mod	mod	PROPN
ejpam-4702	243	60	4q	4q	NOUN
ejpam-4702	243	61	)	)	PUNCT
ejpam-4702	243	62	,	,	PUNCT
ejpam-4702	243	63	where	where	SCONJ
ejpam-4702	243	64	s1	s1	PROPN
ejpam-4702	243	65	∈	∈	PROPN
ejpam-4702	243	66	{	{	PUNCT
ejpam-4702	243	67	2	2	NUM
ejpam-4702	243	68	,	,	PUNCT
ejpam-4702	243	69	4	4	NUM
ejpam-4702	243	70	,	,	PUNCT
ejpam-4702	243	71	6	6	NUM
ejpam-4702	243	72	,	,	PUNCT
ejpam-4702	243	73	.	.	PUNCT
ejpam-4702	243	74	.	.	PUNCT
ejpam-4702	243	75	.	.	PUNCT
ejpam-4702	244	1	,	,	PUNCT
ejpam-4702	244	2	q−	q−	PROPN
ejpam-4702	244	3	1	1	NUM
ejpam-4702	244	4	}	}	PUNCT
ejpam-4702	244	5	,	,	PUNCT
ejpam-4702	244	6	s2	s2	PROPN
ejpam-4702	244	7	∈	∈	PROPN
ejpam-4702	244	8	{	{	PUNCT
ejpam-4702	244	9	1	1	NUM
ejpam-4702	244	10	,	,	PUNCT
ejpam-4702	244	11	3	3	NUM
ejpam-4702	244	12	,	,	PUNCT
ejpam-4702	244	13	5	5	NUM
ejpam-4702	244	14	,	,	PUNCT
ejpam-4702	244	15	.	.	PUNCT
ejpam-4702	244	16	.	.	PUNCT
ejpam-4702	245	1	.	.	PUNCT
ejpam-4702	246	1	,	,	PUNCT
ejpam-4702	246	2	q−	q−	PROPN
ejpam-4702	246	3	2	2	NUM
ejpam-4702	246	4	}	}	PUNCT
ejpam-4702	246	5	,	,	PUNCT
ejpam-4702	246	6	r	r	NOUN
ejpam-4702	246	7	is	be	AUX
ejpam-4702	246	8	a	a	DET
ejpam-4702	246	9	primitive	primitive	ADJ
ejpam-4702	246	10	root	root	NOUN
ejpam-4702	246	11	modulo	modulo	NOUN
ejpam-4702	246	12	q	q	NOUN
ejpam-4702	246	13	and	and	CCONJ
ejpam-4702	246	14	n0	n0	NOUN
ejpam-4702	246	15	=	=	SYM
ejpam-4702	246	16	q	q	PROPN
ejpam-4702	247	1	+	+	NUM
ejpam-4702	247	2	1	1	NUM
ejpam-4702	247	3	4	4	NUM
ejpam-4702	247	4	.	.	PUNCT
ejpam-4702	248	1	since	since	SCONJ
ejpam-4702	248	2	q	q	PROPN
ejpam-4702	248	3	≡	≡	PROPN
ejpam-4702	248	4	3	3	NUM
ejpam-4702	248	5	(	(	PUNCT
ejpam-4702	248	6	mod	mod	NOUN
ejpam-4702	248	7	4	4	NUM
ejpam-4702	248	8	)	)	PUNCT
ejpam-4702	248	9	,	,	PUNCT
ejpam-4702	248	10	we	we	PRON
ejpam-4702	248	11	have	have	VERB
ejpam-4702	248	12	q	q	NOUN
ejpam-4702	248	13	=	=	SYM
ejpam-4702	248	14	4k	4k	NUM
ejpam-4702	248	15	+	+	CCONJ
ejpam-4702	248	16	3	3	NUM
ejpam-4702	248	17	for	for	ADP
ejpam-4702	248	18	some	some	DET
ejpam-4702	248	19	integer	integer	NOUN
ejpam-4702	248	20	k.	k.	PROPN
ejpam-4702	248	21	then	then	ADV
ejpam-4702	248	22	q2	q2	PROPN
ejpam-4702	248	23	≡	≡	PROPN
ejpam-4702	248	24	1	1	NUM
ejpam-4702	248	25	(	(	PUNCT
ejpam-4702	248	26	mod	mod	PROPN
ejpam-4702	248	27	8)	8)	NUM
ejpam-4702	248	28	.	.	PUNCT
ejpam-4702	249	1	if	if	SCONJ
ejpam-4702	249	2	k	k	PROPN
ejpam-4702	249	3	is	be	AUX
ejpam-4702	249	4	even	even	ADV
ejpam-4702	249	5	,	,	PUNCT
ejpam-4702	249	6	then	then	ADV
ejpam-4702	249	7	k	k	PROPN
ejpam-4702	249	8	=	=	PUNCT
ejpam-4702	249	9	2l	2l	PROPN
ejpam-4702	249	10	for	for	ADP
ejpam-4702	249	11	some	some	DET
ejpam-4702	249	12	integer	integer	NOUN
ejpam-4702	249	13	l.	l.	PROPN
ejpam-4702	249	14	it	it	PRON
ejpam-4702	249	15	leads	lead	VERB
ejpam-4702	249	16	to	to	ADP
ejpam-4702	249	17	8(5l	8(5l	NUM
ejpam-4702	249	18	+	+	CCONJ
ejpam-4702	249	19	2	2	NUM
ejpam-4702	249	20	)	)	PUNCT
ejpam-4702	249	21	−	−	PROPN
ejpam-4702	249	22	1	1	NUM
ejpam-4702	250	1	=	=	SYM
ejpam-4702	250	2	5q	5q	X
ejpam-4702	250	3	.	.	PUNCT
ejpam-4702	251	1	otherwise	otherwise	ADV
ejpam-4702	251	2	,	,	PUNCT
ejpam-4702	251	3	q	q	X
ejpam-4702	251	4	=	=	PUNCT
ejpam-4702	251	5	8s+	8s+	NUM
ejpam-4702	251	6	7	7	NUM
ejpam-4702	251	7	for	for	ADP
ejpam-4702	251	8	some	some	DET
ejpam-4702	251	9	integer	integer	NOUN
ejpam-4702	251	10	s	s	NOUN
ejpam-4702	251	11	,	,	PUNCT
ejpam-4702	251	12	so	so	ADV
ejpam-4702	251	13	8(s+	8(s+	PROPN
ejpam-4702	252	1	1)−	1)−	NUM
ejpam-4702	252	2	1	1	NUM
ejpam-4702	252	3	=	=	SYM
ejpam-4702	252	4	q.	q.	NOUN
ejpam-4702	252	5	choose	choose	VERB
ejpam-4702	252	6	n1	n1	PROPN
ejpam-4702	252	7	=	=	SYM
ejpam-4702	252	8	5q	5q	NOUN
ejpam-4702	253	1	+	+	CCONJ
ejpam-4702	253	2	1	1	NUM
ejpam-4702	253	3	8	8	NUM
ejpam-4702	253	4	,	,	PUNCT
ejpam-4702	253	5	when	when	SCONJ
ejpam-4702	253	6	k	k	PROPN
ejpam-4702	253	7	is	be	AUX
ejpam-4702	253	8	even	even	ADV
ejpam-4702	253	9	and	and	CCONJ
ejpam-4702	253	10	otherwise	otherwise	ADV
ejpam-4702	253	11	,	,	PUNCT
ejpam-4702	253	12	n1	n1	PROPN
ejpam-4702	253	13	=	=	SYM
ejpam-4702	253	14	q	q	NOUN
ejpam-4702	254	1	+	+	NUM
ejpam-4702	254	2	1	1	NUM
ejpam-4702	254	3	8	8	NUM
ejpam-4702	254	4	.	.	PUNCT
ejpam-4702	255	1	thus	thus	ADV
ejpam-4702	255	2	,	,	PUNCT
ejpam-4702	255	3	8n1	8n1	NUM
ejpam-4702	255	4	≡	≡	PROPN
ejpam-4702	255	5	1	1	NUM
ejpam-4702	255	6	(	(	PUNCT
ejpam-4702	255	7	mod	mod	PROPN
ejpam-4702	255	8	q	q	NOUN
ejpam-4702	255	9	)	)	PUNCT
ejpam-4702	255	10	.	.	PUNCT
ejpam-4702	256	1	in	in	ADP
ejpam-4702	256	2	the	the	DET
ejpam-4702	256	3	following	following	ADJ
ejpam-4702	256	4	cases	case	NOUN
ejpam-4702	256	5	,	,	PUNCT
ejpam-4702	256	6	the	the	DET
ejpam-4702	256	7	systems	system	NOUN
ejpam-4702	256	8	of	of	ADP
ejpam-4702	256	9	congruences	congruence	NOUN
ejpam-4702	256	10	are	be	AUX
ejpam-4702	256	11	solved	solve	VERB
ejpam-4702	256	12	by	by	ADP
ejpam-4702	256	13	the	the	DET
ejpam-4702	256	14	chinese	chinese	ADJ
ejpam-4702	256	15	remainder	remainder	NOUN
ejpam-4702	256	16	theorem	theorem	NOUN
ejpam-4702	256	17	and	and	CCONJ
ejpam-4702	256	18	theorem	theorem	VERB
ejpam-4702	256	19	7	7	NUM
ejpam-4702	256	20	.	.	NOUN
ejpam-4702	256	21	case	case	NOUN
ejpam-4702	256	22	1	1	NUM
ejpam-4702	256	23	.	.	PUNCT
ejpam-4702	257	1	(	(	PUNCT
ejpam-4702	257	2	2q	2q	NUM
ejpam-4702	257	3	p	p	NOUN
ejpam-4702	257	4	)	)	PUNCT
ejpam-4702	257	5	=	=	SYM
ejpam-4702	257	6	1	1	X
ejpam-4702	257	7	.	.	PUNCT
ejpam-4702	257	8	case	case	NOUN
ejpam-4702	257	9	1.1	1.1	NUM
ejpam-4702	257	10	(	(	PUNCT
ejpam-4702	257	11	2	2	NUM
ejpam-4702	257	12	p	p	NOUN
ejpam-4702	257	13	)	)	PUNCT
ejpam-4702	257	14	=	=	SYM
ejpam-4702	257	15	1	1	NUM
ejpam-4702	257	16	and	and	CCONJ
ejpam-4702	257	17	(	(	PUNCT
ejpam-4702	257	18	q	q	X
ejpam-4702	257	19	p	p	NOUN
ejpam-4702	257	20	)	)	PUNCT
ejpam-4702	257	21	=	=	SYM
ejpam-4702	258	1	1	1	X
ejpam-4702	258	2	.	.	PUNCT
ejpam-4702	258	3	then	then	ADV
ejpam-4702	258	4	p	p	PROPN
ejpam-4702	258	5	≡	≡	PROPN
ejpam-4702	258	6	±1	±1	PROPN
ejpam-4702	258	7	(	(	PUNCT
ejpam-4702	258	8	mod	mod	PROPN
ejpam-4702	258	9	8)	8)	NUM
ejpam-4702	258	10	and	and	CCONJ
ejpam-4702	258	11	p	p	PROPN
ejpam-4702	258	12	≡	≡	PROPN
ejpam-4702	258	13	3q	3q	NOUN
ejpam-4702	258	14	+	+	CCONJ
ejpam-4702	258	15	4n0r	4n0r	NOUN
ejpam-4702	258	16	s1	s1	NOUN
ejpam-4702	258	17	,	,	PUNCT
ejpam-4702	258	18	−3q	−3q	PROPN
ejpam-4702	258	19	+	+	NUM
ejpam-4702	258	20	4n0r	4n0r	NOUN
ejpam-4702	258	21	s2	s2	NOUN
ejpam-4702	258	22	(	(	PUNCT
ejpam-4702	258	23	mod	mod	PROPN
ejpam-4702	258	24	4q	4q	NOUN
ejpam-4702	258	25	)	)	PUNCT
ejpam-4702	258	26	.	.	PUNCT
ejpam-4702	259	1	so	so	ADV
ejpam-4702	259	2	p	p	PROPN
ejpam-4702	259	3	≡	≡	PROPN
ejpam-4702	259	4	4n0r	4n0r	PROPN
ejpam-4702	259	5	s1	s1	PROPN
ejpam-4702	259	6	,	,	PUNCT
ejpam-4702	259	7	4n0r	4n0r	NOUN
ejpam-4702	259	8	s2	s2	NOUN
ejpam-4702	259	9	(	(	PUNCT
ejpam-4702	259	10	mod	mod	PROPN
ejpam-4702	259	11	q	q	NOUN
ejpam-4702	259	12	)	)	PUNCT
ejpam-4702	259	13	.	.	PUNCT
ejpam-4702	260	1	since	since	SCONJ
ejpam-4702	260	2	(	(	PUNCT
ejpam-4702	260	3	8	8	NUM
ejpam-4702	260	4	,	,	PUNCT
ejpam-4702	260	5	4q	4q	NOUN
ejpam-4702	260	6	)	)	PUNCT
ejpam-4702	260	7	=	=	SYM
ejpam-4702	260	8	4	4	NUM
ejpam-4702	260	9	and	and	CCONJ
ejpam-4702	260	10	q	q	PROPN
ejpam-4702	260	11	≡	≡	PROPN
ejpam-4702	260	12	3	3	NUM
ejpam-4702	260	13	(	(	PUNCT
ejpam-4702	260	14	mod	mod	NOUN
ejpam-4702	260	15	4	4	NUM
ejpam-4702	260	16	)	)	PUNCT
ejpam-4702	260	17	,	,	PUNCT
ejpam-4702	260	18	we	we	PRON
ejpam-4702	260	19	get	get	VERB
ejpam-4702	260	20	that	that	PRON
ejpam-4702	260	21	4	4	NUM
ejpam-4702	260	22	∤	∤	SYM
ejpam-4702	260	23	(	(	PUNCT
ejpam-4702	260	24	−3q+4n0r	−3q+4n0r	NOUN
ejpam-4702	260	25	s2)−1	s2)−1	NOUN
ejpam-4702	260	26	and	and	CCONJ
ejpam-4702	260	27	4	4	NUM
ejpam-4702	260	28	∤	∤	NUM
ejpam-4702	260	29	(	(	PUNCT
ejpam-4702	260	30	3q+4n0r	3q+4n0r	PROPN
ejpam-4702	260	31	s1)+1	s1)+1	PROPN
ejpam-4702	260	32	.	.	PROPN
ejpam-4702	260	33	hence	hence	ADV
ejpam-4702	260	34	,	,	PUNCT
ejpam-4702	260	35	the	the	DET
ejpam-4702	260	36	system	system	NOUN
ejpam-4702	260	37	of	of	ADP
ejpam-4702	260	38	congruences	congruence	NOUN
ejpam-4702	260	39	p	p	PROPN
ejpam-4702	260	40	≡	≡	PROPN
ejpam-4702	260	41	1	1	NUM
ejpam-4702	260	42	(	(	PUNCT
ejpam-4702	260	43	mod	mod	PROPN
ejpam-4702	260	44	8)	8)	NUM
ejpam-4702	260	45	and	and	CCONJ
ejpam-4702	260	46	p	p	PROPN
ejpam-4702	260	47	≡	≡	PROPN
ejpam-4702	260	48	−3q	−3q	PROPN
ejpam-4702	260	49	+	+	NUM
ejpam-4702	260	50	4n0r	4n0r	NOUN
ejpam-4702	260	51	s1	s1	NOUN
ejpam-4702	260	52	(	(	PUNCT
ejpam-4702	260	53	mod	mod	ADJ
ejpam-4702	260	54	4q	4q	NOUN
ejpam-4702	260	55	)	)	PUNCT
ejpam-4702	260	56	and	and	CCONJ
ejpam-4702	260	57	the	the	DET
ejpam-4702	260	58	system	system	NOUN
ejpam-4702	260	59	of	of	ADP
ejpam-4702	260	60	congruences	congruence	NOUN
ejpam-4702	260	61	p	p	PROPN
ejpam-4702	260	62	≡	≡	PROPN
ejpam-4702	260	63	−1	−1	NOUN
ejpam-4702	260	64	(	(	PUNCT
ejpam-4702	260	65	mod	mod	PROPN
ejpam-4702	260	66	8)	8)	NUM
ejpam-4702	260	67	and	and	CCONJ
ejpam-4702	260	68	p	p	PROPN
ejpam-4702	260	69	≡	≡	PROPN
ejpam-4702	260	70	3q	3q	NOUN
ejpam-4702	260	71	+	+	CCONJ
ejpam-4702	260	72	4n0r	4n0r	NOUN
ejpam-4702	260	73	s1	s1	NOUN
ejpam-4702	260	74	(	(	PUNCT
ejpam-4702	260	75	mod	mod	PROPN
ejpam-4702	260	76	4q	4q	NOUN
ejpam-4702	260	77	)	)	PUNCT
ejpam-4702	260	78	have	have	VERB
ejpam-4702	260	79	no	no	DET
ejpam-4702	260	80	solution	solution	NOUN
ejpam-4702	260	81	.	.	PUNCT
ejpam-4702	261	1	thus	thus	ADV
ejpam-4702	261	2	,	,	PUNCT
ejpam-4702	261	3	p	p	PROPN
ejpam-4702	261	4	≡	≡	PROPN
ejpam-4702	261	5	q2	q2	NOUN
ejpam-4702	261	6	+	+	CCONJ
ejpam-4702	261	7	32n0n1r	32n0n1r	CCONJ
ejpam-4702	261	8	s1	s1	NOUN
ejpam-4702	261	9	(	(	PUNCT
ejpam-4702	261	10	mod	mod	PROPN
ejpam-4702	261	11	8q	8q	NUM
ejpam-4702	261	12	)	)	PUNCT
ejpam-4702	261	13	or	or	CCONJ
ejpam-4702	261	14	p	p	PROPN
ejpam-4702	261	15	≡	≡	PROPN
ejpam-4702	261	16	−q2	−q2	PROPN
ejpam-4702	262	1	+	+	CCONJ
ejpam-4702	262	2	32n0n1r	32n0n1r	NUM
ejpam-4702	262	3	s2	s2	PROPN
ejpam-4702	262	4	(	(	PUNCT
ejpam-4702	262	5	mod	mod	PROPN
ejpam-4702	262	6	8q	8q	NUM
ejpam-4702	262	7	)	)	PUNCT
ejpam-4702	262	8	.	.	PUNCT
ejpam-4702	263	1	case	case	NOUN
ejpam-4702	263	2	1.2	1.2	NUM
ejpam-4702	263	3	(	(	PUNCT
ejpam-4702	263	4	2	2	NUM
ejpam-4702	263	5	p	p	NOUN
ejpam-4702	263	6	)	)	PUNCT
ejpam-4702	263	7	=	=	SYM
ejpam-4702	263	8	−1	−1	NOUN
ejpam-4702	263	9	and	and	CCONJ
ejpam-4702	263	10	(	(	PUNCT
ejpam-4702	263	11	q	q	X
ejpam-4702	263	12	p	p	NOUN
ejpam-4702	263	13	)	)	PUNCT
ejpam-4702	263	14	=	=	SYM
ejpam-4702	263	15	−1	−1	NOUN
ejpam-4702	263	16	.	.	PUNCT
ejpam-4702	264	1	then	then	ADV
ejpam-4702	264	2	p	p	PROPN
ejpam-4702	264	3	≡	≡	PROPN
ejpam-4702	264	4	±3	±3	X
ejpam-4702	264	5	(	(	PUNCT
ejpam-4702	264	6	mod	mod	PROPN
ejpam-4702	264	7	8)	8)	NUM
ejpam-4702	264	8	and	and	CCONJ
ejpam-4702	264	9	p	p	PROPN
ejpam-4702	264	10	≡	≡	PROPN
ejpam-4702	264	11	3q+4n0r	3q+4n0r	PROPN
ejpam-4702	264	12	s2	s2	NOUN
ejpam-4702	264	13	,	,	PUNCT
ejpam-4702	264	14	−3q+	−3q+	PROPN
ejpam-4702	264	15	4n0r	4n0r	NUM
ejpam-4702	264	16	s1	s1	NOUN
ejpam-4702	264	17	(	(	PUNCT
ejpam-4702	264	18	mod	mod	PROPN
ejpam-4702	264	19	4q	4q	NOUN
ejpam-4702	264	20	)	)	PUNCT
ejpam-4702	264	21	.	.	PUNCT
ejpam-4702	265	1	so	so	ADV
ejpam-4702	265	2	p	p	PROPN
ejpam-4702	265	3	≡	≡	PROPN
ejpam-4702	265	4	4n0r	4n0r	PROPN
ejpam-4702	265	5	s2	s2	PROPN
ejpam-4702	265	6	,	,	PUNCT
ejpam-4702	265	7	4n0r	4n0r	NOUN
ejpam-4702	265	8	s1	s1	NOUN
ejpam-4702	265	9	(	(	PUNCT
ejpam-4702	265	10	mod	mod	PROPN
ejpam-4702	265	11	q	q	NOUN
ejpam-4702	265	12	)	)	PUNCT
ejpam-4702	265	13	.	.	PUNCT
ejpam-4702	266	1	since	since	SCONJ
ejpam-4702	266	2	(	(	PUNCT
ejpam-4702	266	3	8	8	NUM
ejpam-4702	266	4	,	,	PUNCT
ejpam-4702	266	5	4q	4q	NOUN
ejpam-4702	266	6	)	)	PUNCT
ejpam-4702	266	7	=	=	SYM
ejpam-4702	266	8	4	4	NUM
ejpam-4702	266	9	and	and	CCONJ
ejpam-4702	266	10	q	q	PROPN
ejpam-4702	266	11	≡	≡	PROPN
ejpam-4702	266	12	3	3	NUM
ejpam-4702	266	13	(	(	PUNCT
ejpam-4702	266	14	mod	mod	NOUN
ejpam-4702	266	15	4	4	NUM
ejpam-4702	266	16	)	)	PUNCT
ejpam-4702	266	17	,	,	PUNCT
ejpam-4702	266	18	we	we	PRON
ejpam-4702	266	19	get	get	VERB
ejpam-4702	266	20	that	that	PRON
ejpam-4702	266	21	4	4	NUM
ejpam-4702	266	22	∤	∤	SYM
ejpam-4702	266	23	(	(	PUNCT
ejpam-4702	266	24	3q+4n0r	3q+4n0r	NOUN
ejpam-4702	266	25	s2)−3	s2)−3	NOUN
ejpam-4702	266	26	and	and	CCONJ
ejpam-4702	266	27	4	4	NUM
ejpam-4702	266	28	∤	∤	NUM
ejpam-4702	266	29	(	(	PUNCT
ejpam-4702	266	30	−3q+4n0r	−3q+4n0r	NOUN
ejpam-4702	266	31	s1)+3	s1)+3	PROPN
ejpam-4702	266	32	.	.	PUNCT
ejpam-4702	267	1	hence	hence	ADV
ejpam-4702	267	2	,	,	PUNCT
ejpam-4702	267	3	the	the	DET
ejpam-4702	267	4	system	system	NOUN
ejpam-4702	267	5	of	of	ADP
ejpam-4702	267	6	congruences	congruence	NOUN
ejpam-4702	267	7	p	p	PROPN
ejpam-4702	267	8	≡	≡	PROPN
ejpam-4702	267	9	3	3	NUM
ejpam-4702	267	10	(	(	PUNCT
ejpam-4702	267	11	mod	mod	PROPN
ejpam-4702	267	12	8)	8)	NUM
ejpam-4702	267	13	and	and	CCONJ
ejpam-4702	267	14	p	p	PROPN
ejpam-4702	267	15	≡	≡	PROPN
ejpam-4702	267	16	3q	3q	NOUN
ejpam-4702	267	17	+	+	CCONJ
ejpam-4702	267	18	4n0r	4n0r	NOUN
ejpam-4702	267	19	s2	s2	NOUN
ejpam-4702	267	20	(	(	PUNCT
ejpam-4702	267	21	mod	mod	ADJ
ejpam-4702	267	22	4q	4q	NOUN
ejpam-4702	267	23	)	)	PUNCT
ejpam-4702	267	24	and	and	CCONJ
ejpam-4702	267	25	the	the	DET
ejpam-4702	267	26	system	system	NOUN
ejpam-4702	267	27	of	of	ADP
ejpam-4702	267	28	congruences	congruence	NOUN
ejpam-4702	267	29	p	p	PROPN
ejpam-4702	267	30	≡	≡	PROPN
ejpam-4702	267	31	−3	−3	PROPN
ejpam-4702	267	32	(	(	PUNCT
ejpam-4702	267	33	mod	mod	PROPN
ejpam-4702	267	34	8)	8)	NUM
ejpam-4702	267	35	and	and	CCONJ
ejpam-4702	267	36	p	p	PROPN
ejpam-4702	267	37	≡	≡	PROPN
ejpam-4702	267	38	−3q	−3q	PROPN
ejpam-4702	267	39	+	+	NUM
ejpam-4702	267	40	4n0r	4n0r	NOUN
ejpam-4702	267	41	s1	s1	NOUN
ejpam-4702	267	42	(	(	PUNCT
ejpam-4702	267	43	mod	mod	PROPN
ejpam-4702	267	44	4q	4q	NOUN
ejpam-4702	267	45	)	)	PUNCT
ejpam-4702	267	46	have	have	VERB
ejpam-4702	267	47	no	no	DET
ejpam-4702	267	48	solution	solution	NOUN
ejpam-4702	267	49	.	.	PUNCT
ejpam-4702	268	1	thus	thus	ADV
ejpam-4702	268	2	,	,	PUNCT
ejpam-4702	268	3	p	p	PROPN
ejpam-4702	268	4	≡	≡	PROPN
ejpam-4702	268	5	3q2	3q2	NUM
ejpam-4702	269	1	+	+	CCONJ
ejpam-4702	269	2	32n0n1r	32n0n1r	CCONJ
ejpam-4702	269	3	s1	s1	NOUN
ejpam-4702	269	4	(	(	PUNCT
ejpam-4702	269	5	mod	mod	PROPN
ejpam-4702	269	6	8q	8q	NUM
ejpam-4702	269	7	)	)	PUNCT
ejpam-4702	269	8	or	or	CCONJ
ejpam-4702	269	9	p	p	PRON
ejpam-4702	269	10	≡	≡	PROPN
ejpam-4702	269	11	−3q2	−3q2	NUM
ejpam-4702	269	12	+	+	CCONJ
ejpam-4702	269	13	32n0n1r	32n0n1r	NUM
ejpam-4702	269	14	s2	s2	PROPN
ejpam-4702	269	15	(	(	PUNCT
ejpam-4702	269	16	mod	mod	PROPN
ejpam-4702	269	17	8q	8q	NUM
ejpam-4702	269	18	)	)	PUNCT
ejpam-4702	269	19	.	.	PUNCT
ejpam-4702	270	1	s.	s.	PROPN
ejpam-4702	270	2	tadee	tadee	PROPN
ejpam-4702	270	3	,	,	PUNCT
ejpam-4702	270	4	a.	a.	NOUN
ejpam-4702	270	5	siraworakun	siraworakun	PROPN
ejpam-4702	270	6	/	/	SYM
ejpam-4702	270	7	eur	eur	PROPN
ejpam-4702	270	8	.	.	PUNCT
ejpam-4702	271	1	j.	j.	PROPN
ejpam-4702	271	2	pure	pure	PROPN
ejpam-4702	271	3	appl	appl	PROPN
ejpam-4702	271	4	.	.	PROPN
ejpam-4702	271	5	math	math	PROPN
ejpam-4702	271	6	,	,	PUNCT
ejpam-4702	271	7	16	16	NUM
ejpam-4702	271	8	(	(	PUNCT
ejpam-4702	271	9	2	2	NUM
ejpam-4702	271	10	)	)	PUNCT
ejpam-4702	271	11	(	(	PUNCT
ejpam-4702	271	12	2023	2023	NUM
ejpam-4702	271	13	)	)	PUNCT
ejpam-4702	271	14	,	,	PUNCT
ejpam-4702	271	15	724	724	NUM
ejpam-4702	271	16	-	-	SYM
ejpam-4702	271	17	735	735	NUM
ejpam-4702	271	18	731	731	NUM
ejpam-4702	271	19	case	case	NOUN
ejpam-4702	271	20	2	2	NUM
ejpam-4702	271	21	.	.	PUNCT
ejpam-4702	272	1	(	(	PUNCT
ejpam-4702	272	2	2q	2q	NUM
ejpam-4702	272	3	p	p	NOUN
ejpam-4702	272	4	)	)	PUNCT
ejpam-4702	272	5	=	=	PUNCT
ejpam-4702	272	6	−1	−1	NOUN
ejpam-4702	272	7	.	.	PUNCT
ejpam-4702	273	1	case	case	NOUN
ejpam-4702	273	2	2.1	2.1	NUM
ejpam-4702	273	3	(	(	PUNCT
ejpam-4702	273	4	2	2	NUM
ejpam-4702	273	5	p	p	NOUN
ejpam-4702	273	6	)	)	PUNCT
ejpam-4702	273	7	=	=	SYM
ejpam-4702	273	8	1	1	NUM
ejpam-4702	273	9	and	and	CCONJ
ejpam-4702	273	10	(	(	PUNCT
ejpam-4702	273	11	q	q	X
ejpam-4702	273	12	p	p	NOUN
ejpam-4702	273	13	)	)	PUNCT
ejpam-4702	273	14	=	=	SYM
ejpam-4702	273	15	−1	−1	NOUN
ejpam-4702	273	16	.	.	PUNCT
ejpam-4702	274	1	then	then	ADV
ejpam-4702	274	2	p	p	PROPN
ejpam-4702	274	3	≡	≡	PROPN
ejpam-4702	274	4	±1	±1	PROPN
ejpam-4702	274	5	(	(	PUNCT
ejpam-4702	274	6	mod	mod	PROPN
ejpam-4702	274	7	8)	8)	NUM
ejpam-4702	274	8	and	and	CCONJ
ejpam-4702	274	9	p	p	PROPN
ejpam-4702	274	10	≡	≡	PROPN
ejpam-4702	274	11	3q+4n0r	3q+4n0r	PROPN
ejpam-4702	274	12	s2	s2	NOUN
ejpam-4702	274	13	,	,	PUNCT
ejpam-4702	274	14	−3q+	−3q+	PROPN
ejpam-4702	274	15	4n0r	4n0r	NUM
ejpam-4702	274	16	s1	s1	NOUN
ejpam-4702	274	17	(	(	PUNCT
ejpam-4702	274	18	mod	mod	PROPN
ejpam-4702	274	19	4q	4q	NOUN
ejpam-4702	274	20	)	)	PUNCT
ejpam-4702	274	21	.	.	PUNCT
ejpam-4702	275	1	so	so	ADV
ejpam-4702	275	2	p	p	PROPN
ejpam-4702	275	3	≡	≡	PROPN
ejpam-4702	275	4	4n0r	4n0r	PROPN
ejpam-4702	275	5	s2	s2	PROPN
ejpam-4702	275	6	,	,	PUNCT
ejpam-4702	275	7	4n0r	4n0r	NOUN
ejpam-4702	275	8	s1	s1	NOUN
ejpam-4702	275	9	(	(	PUNCT
ejpam-4702	275	10	mod	mod	PROPN
ejpam-4702	275	11	q	q	NOUN
ejpam-4702	275	12	)	)	PUNCT
ejpam-4702	275	13	.	.	PUNCT
ejpam-4702	276	1	since	since	SCONJ
ejpam-4702	276	2	(	(	PUNCT
ejpam-4702	276	3	8	8	NUM
ejpam-4702	276	4	,	,	PUNCT
ejpam-4702	276	5	4q	4q	NOUN
ejpam-4702	276	6	)	)	PUNCT
ejpam-4702	276	7	=	=	SYM
ejpam-4702	276	8	4	4	NUM
ejpam-4702	276	9	and	and	CCONJ
ejpam-4702	276	10	q	q	PROPN
ejpam-4702	276	11	≡	≡	PROPN
ejpam-4702	276	12	3	3	NUM
ejpam-4702	276	13	(	(	PUNCT
ejpam-4702	276	14	mod	mod	NOUN
ejpam-4702	276	15	4	4	NUM
ejpam-4702	276	16	)	)	PUNCT
ejpam-4702	276	17	,	,	PUNCT
ejpam-4702	276	18	we	we	PRON
ejpam-4702	276	19	get	get	VERB
ejpam-4702	276	20	that	that	PRON
ejpam-4702	276	21	4	4	NUM
ejpam-4702	276	22	∤	∤	SYM
ejpam-4702	276	23	(	(	PUNCT
ejpam-4702	276	24	−3q+4n0r	−3q+4n0r	NOUN
ejpam-4702	276	25	s1)−1	s1)−1	NOUN
ejpam-4702	276	26	and	and	CCONJ
ejpam-4702	276	27	4	4	NUM
ejpam-4702	276	28	∤	∤	NUM
ejpam-4702	276	29	(	(	PUNCT
ejpam-4702	276	30	3q+4n0r	3q+4n0r	PROPN
ejpam-4702	276	31	s2)+1	s2)+1	ADJ
ejpam-4702	276	32	.	.	PUNCT
ejpam-4702	277	1	hence	hence	ADV
ejpam-4702	277	2	,	,	PUNCT
ejpam-4702	277	3	the	the	DET
ejpam-4702	277	4	system	system	NOUN
ejpam-4702	277	5	of	of	ADP
ejpam-4702	277	6	congruences	congruence	NOUN
ejpam-4702	277	7	p	p	PROPN
ejpam-4702	277	8	≡	≡	PROPN
ejpam-4702	277	9	1	1	NUM
ejpam-4702	277	10	(	(	PUNCT
ejpam-4702	277	11	mod	mod	PROPN
ejpam-4702	277	12	8)	8)	NUM
ejpam-4702	277	13	and	and	CCONJ
ejpam-4702	277	14	p	p	PROPN
ejpam-4702	277	15	≡	≡	PROPN
ejpam-4702	277	16	−3q	−3q	PROPN
ejpam-4702	277	17	+	+	NUM
ejpam-4702	277	18	4n0r	4n0r	NOUN
ejpam-4702	277	19	s1	s1	NOUN
ejpam-4702	277	20	(	(	PUNCT
ejpam-4702	277	21	mod	mod	ADJ
ejpam-4702	277	22	4q	4q	NOUN
ejpam-4702	277	23	)	)	PUNCT
ejpam-4702	277	24	and	and	CCONJ
ejpam-4702	277	25	the	the	DET
ejpam-4702	277	26	system	system	NOUN
ejpam-4702	277	27	of	of	ADP
ejpam-4702	277	28	congruences	congruence	NOUN
ejpam-4702	277	29	p	p	PROPN
ejpam-4702	277	30	≡	≡	PROPN
ejpam-4702	277	31	−1	−1	NOUN
ejpam-4702	277	32	(	(	PUNCT
ejpam-4702	277	33	mod	mod	PROPN
ejpam-4702	277	34	8)	8)	NUM
ejpam-4702	277	35	and	and	CCONJ
ejpam-4702	277	36	p	p	PROPN
ejpam-4702	277	37	≡	≡	PROPN
ejpam-4702	277	38	3q	3q	NOUN
ejpam-4702	277	39	+	+	CCONJ
ejpam-4702	277	40	4n0r	4n0r	NOUN
ejpam-4702	277	41	s1	s1	NOUN
ejpam-4702	277	42	(	(	PUNCT
ejpam-4702	277	43	mod	mod	PROPN
ejpam-4702	277	44	4q	4q	NOUN
ejpam-4702	277	45	)	)	PUNCT
ejpam-4702	277	46	have	have	VERB
ejpam-4702	277	47	no	no	DET
ejpam-4702	277	48	solution	solution	NOUN
ejpam-4702	277	49	.	.	PUNCT
ejpam-4702	278	1	thus	thus	ADV
ejpam-4702	278	2	,	,	PUNCT
ejpam-4702	278	3	p	p	PROPN
ejpam-4702	278	4	≡	≡	PROPN
ejpam-4702	278	5	q2	q2	NOUN
ejpam-4702	278	6	+	+	CCONJ
ejpam-4702	278	7	32n0n1r	32n0n1r	CCONJ
ejpam-4702	278	8	s2	s2	PROPN
ejpam-4702	278	9	(	(	PUNCT
ejpam-4702	278	10	mod	mod	PROPN
ejpam-4702	278	11	8q	8q	NUM
ejpam-4702	278	12	)	)	PUNCT
ejpam-4702	278	13	or	or	CCONJ
ejpam-4702	278	14	p	p	PROPN
ejpam-4702	278	15	≡	≡	PROPN
ejpam-4702	278	16	−q2	−q2	PROPN
ejpam-4702	278	17	+	+	CCONJ
ejpam-4702	278	18	32n0n1r	32n0n1r	PROPN
ejpam-4702	278	19	s1	s1	NOUN
ejpam-4702	278	20	(	(	PUNCT
ejpam-4702	278	21	mod	mod	PROPN
ejpam-4702	278	22	8q	8q	NUM
ejpam-4702	278	23	)	)	PUNCT
ejpam-4702	278	24	.	.	PUNCT
ejpam-4702	279	1	case	case	NOUN
ejpam-4702	279	2	2.2	2.2	NUM
ejpam-4702	279	3	(	(	PUNCT
ejpam-4702	279	4	2	2	NUM
ejpam-4702	279	5	p	p	NOUN
ejpam-4702	279	6	)	)	PUNCT
ejpam-4702	279	7	=	=	SYM
ejpam-4702	279	8	−1	−1	NOUN
ejpam-4702	279	9	and	and	CCONJ
ejpam-4702	279	10	(	(	PUNCT
ejpam-4702	279	11	q	q	X
ejpam-4702	279	12	p	p	NOUN
ejpam-4702	279	13	)	)	PUNCT
ejpam-4702	280	1	=	=	SYM
ejpam-4702	280	2	1	1	X
ejpam-4702	280	3	.	.	PUNCT
ejpam-4702	281	1	then	then	ADV
ejpam-4702	281	2	p	p	PROPN
ejpam-4702	281	3	≡	≡	PROPN
ejpam-4702	281	4	±3	±3	X
ejpam-4702	281	5	(	(	PUNCT
ejpam-4702	281	6	mod	mod	PROPN
ejpam-4702	281	7	8)	8)	NUM
ejpam-4702	281	8	and	and	CCONJ
ejpam-4702	281	9	p	p	PROPN
ejpam-4702	281	10	≡	≡	PROPN
ejpam-4702	281	11	3q+4n0r	3q+4n0r	PROPN
ejpam-4702	281	12	s1	s1	PROPN
ejpam-4702	281	13	,	,	PUNCT
ejpam-4702	281	14	−3q+	−3q+	PROPN
ejpam-4702	281	15	4n0r	4n0r	PROPN
ejpam-4702	281	16	s2	s2	PROPN
ejpam-4702	281	17	(	(	PUNCT
ejpam-4702	281	18	mod	mod	PROPN
ejpam-4702	281	19	4q	4q	NOUN
ejpam-4702	281	20	)	)	PUNCT
ejpam-4702	281	21	.	.	PUNCT
ejpam-4702	282	1	so	so	ADV
ejpam-4702	282	2	p	p	PROPN
ejpam-4702	282	3	≡	≡	PROPN
ejpam-4702	282	4	4n0r	4n0r	PROPN
ejpam-4702	282	5	s1	s1	PROPN
ejpam-4702	282	6	,	,	PUNCT
ejpam-4702	282	7	4n0r	4n0r	NOUN
ejpam-4702	282	8	s2	s2	NOUN
ejpam-4702	282	9	(	(	PUNCT
ejpam-4702	282	10	mod	mod	PROPN
ejpam-4702	282	11	q	q	NOUN
ejpam-4702	282	12	)	)	PUNCT
ejpam-4702	282	13	.	.	PUNCT
ejpam-4702	283	1	since	since	SCONJ
ejpam-4702	283	2	(	(	PUNCT
ejpam-4702	283	3	8	8	NUM
ejpam-4702	283	4	,	,	PUNCT
ejpam-4702	283	5	4q	4q	NOUN
ejpam-4702	283	6	)	)	PUNCT
ejpam-4702	283	7	=	=	SYM
ejpam-4702	283	8	4	4	NUM
ejpam-4702	283	9	and	and	CCONJ
ejpam-4702	283	10	q	q	PROPN
ejpam-4702	283	11	≡	≡	PROPN
ejpam-4702	283	12	3	3	NUM
ejpam-4702	283	13	(	(	PUNCT
ejpam-4702	283	14	mod	mod	NOUN
ejpam-4702	283	15	4	4	NUM
ejpam-4702	283	16	)	)	PUNCT
ejpam-4702	283	17	,	,	PUNCT
ejpam-4702	283	18	we	we	PRON
ejpam-4702	283	19	get	get	VERB
ejpam-4702	283	20	that	that	PRON
ejpam-4702	283	21	4	4	NUM
ejpam-4702	283	22	∤	∤	SYM
ejpam-4702	283	23	(	(	PUNCT
ejpam-4702	283	24	3q+4n0r	3q+4n0r	NOUN
ejpam-4702	283	25	s1)−3	s1)−3	NOUN
ejpam-4702	283	26	and	and	CCONJ
ejpam-4702	283	27	4	4	NUM
ejpam-4702	283	28	∤	∤	NUM
ejpam-4702	283	29	(	(	PUNCT
ejpam-4702	283	30	−3q+4n0r	−3q+4n0r	NOUN
ejpam-4702	283	31	s1)+3	s1)+3	PROPN
ejpam-4702	283	32	.	.	PUNCT
ejpam-4702	284	1	hence	hence	ADV
ejpam-4702	284	2	,	,	PUNCT
ejpam-4702	284	3	the	the	DET
ejpam-4702	284	4	system	system	NOUN
ejpam-4702	284	5	of	of	ADP
ejpam-4702	284	6	congruences	congruence	NOUN
ejpam-4702	284	7	p	p	PROPN
ejpam-4702	284	8	≡	≡	PROPN
ejpam-4702	284	9	3	3	NUM
ejpam-4702	284	10	(	(	PUNCT
ejpam-4702	284	11	mod	mod	PROPN
ejpam-4702	284	12	8)	8)	NUM
ejpam-4702	284	13	and	and	CCONJ
ejpam-4702	284	14	p	p	PROPN
ejpam-4702	284	15	≡	≡	PROPN
ejpam-4702	284	16	3q	3q	NOUN
ejpam-4702	284	17	+	+	CCONJ
ejpam-4702	284	18	4n0r	4n0r	NOUN
ejpam-4702	284	19	s1	s1	NOUN
ejpam-4702	284	20	(	(	PUNCT
ejpam-4702	284	21	mod	mod	ADJ
ejpam-4702	284	22	4q	4q	NOUN
ejpam-4702	284	23	)	)	PUNCT
ejpam-4702	284	24	and	and	CCONJ
ejpam-4702	284	25	the	the	DET
ejpam-4702	284	26	system	system	NOUN
ejpam-4702	284	27	of	of	ADP
ejpam-4702	284	28	congruences	congruence	NOUN
ejpam-4702	284	29	p	p	PROPN
ejpam-4702	284	30	≡	≡	PROPN
ejpam-4702	284	31	−3	−3	PROPN
ejpam-4702	284	32	(	(	PUNCT
ejpam-4702	284	33	mod	mod	PROPN
ejpam-4702	284	34	8)	8)	NUM
ejpam-4702	284	35	and	and	CCONJ
ejpam-4702	284	36	p	p	PROPN
ejpam-4702	284	37	≡	≡	PROPN
ejpam-4702	284	38	−3q	−3q	PROPN
ejpam-4702	284	39	+	+	NUM
ejpam-4702	284	40	4n0r	4n0r	NOUN
ejpam-4702	284	41	s2	s2	NOUN
ejpam-4702	284	42	(	(	PUNCT
ejpam-4702	284	43	mod	mod	PROPN
ejpam-4702	284	44	4q	4q	NOUN
ejpam-4702	284	45	)	)	PUNCT
ejpam-4702	284	46	have	have	VERB
ejpam-4702	284	47	no	no	DET
ejpam-4702	284	48	solution	solution	NOUN
ejpam-4702	284	49	.	.	PUNCT
ejpam-4702	285	1	thus	thus	ADV
ejpam-4702	285	2	,	,	PUNCT
ejpam-4702	285	3	p	p	PROPN
ejpam-4702	285	4	≡	≡	PROPN
ejpam-4702	285	5	3q2	3q2	NUM
ejpam-4702	286	1	+	+	CCONJ
ejpam-4702	286	2	32n0n1r	32n0n1r	CCONJ
ejpam-4702	286	3	s2	s2	PROPN
ejpam-4702	286	4	(	(	PUNCT
ejpam-4702	286	5	mod	mod	PROPN
ejpam-4702	286	6	8q	8q	NUM
ejpam-4702	286	7	)	)	PUNCT
ejpam-4702	286	8	or	or	CCONJ
ejpam-4702	286	9	p	p	PRON
ejpam-4702	286	10	≡	≡	PROPN
ejpam-4702	286	11	−3q2	−3q2	NUM
ejpam-4702	286	12	+	+	CCONJ
ejpam-4702	286	13	32n0n1r	32n0n1r	NUM
ejpam-4702	286	14	s1	s1	NOUN
ejpam-4702	286	15	(	(	PUNCT
ejpam-4702	286	16	mod	mod	PROPN
ejpam-4702	286	17	8q	8q	NUM
ejpam-4702	286	18	)	)	PUNCT
ejpam-4702	286	19	.	.	PUNCT
ejpam-4702	287	1	3	3	X
ejpam-4702	287	2	.	.	X
ejpam-4702	287	3	main	main	ADJ
ejpam-4702	287	4	results	result	NOUN
ejpam-4702	287	5	in	in	ADP
ejpam-4702	287	6	this	this	DET
ejpam-4702	287	7	section	section	NOUN
ejpam-4702	287	8	,	,	PUNCT
ejpam-4702	287	9	we	we	PRON
ejpam-4702	287	10	study	study	VERB
ejpam-4702	287	11	the	the	DET
ejpam-4702	287	12	diophantine	diophantine	NOUN
ejpam-4702	287	13	equation	equation	NOUN
ejpam-4702	287	14	px+(p+2q)y	px+(p+2q)y	NOUN
ejpam-4702	287	15	=	=	SYM
ejpam-4702	287	16	z2	z2	PROPN
ejpam-4702	287	17	,	,	PUNCT
ejpam-4702	287	18	where	where	SCONJ
ejpam-4702	287	19	p	p	X
ejpam-4702	287	20	,	,	PUNCT
ejpam-4702	287	21	q	q	NOUN
ejpam-4702	287	22	and	and	CCONJ
ejpam-4702	287	23	p+	p+	PROPN
ejpam-4702	287	24	2q	2q	NUM
ejpam-4702	287	25	are	be	AUX
ejpam-4702	287	26	prime	prime	ADJ
ejpam-4702	287	27	numbers	number	NOUN
ejpam-4702	287	28	.	.	PUNCT
ejpam-4702	288	1	thus	thus	ADV
ejpam-4702	288	2	,	,	PUNCT
ejpam-4702	288	3	p	p	PROPN
ejpam-4702	288	4	is	be	AUX
ejpam-4702	288	5	an	an	DET
ejpam-4702	288	6	odd	odd	ADJ
ejpam-4702	288	7	prime	prime	ADJ
ejpam-4702	288	8	number	number	NOUN
ejpam-4702	288	9	with	with	ADP
ejpam-4702	288	10	(	(	PUNCT
ejpam-4702	288	11	p	p	X
ejpam-4702	288	12	,	,	PUNCT
ejpam-4702	288	13	q	q	NOUN
ejpam-4702	288	14	)	)	PUNCT
ejpam-4702	288	15	=	=	SYM
ejpam-4702	288	16	1	1	X
ejpam-4702	288	17	.	.	X
ejpam-4702	289	1	for	for	ADP
ejpam-4702	289	2	the	the	DET
ejpam-4702	289	3	case	case	NOUN
ejpam-4702	289	4	q	q	X
ejpam-4702	289	5	=	=	SYM
ejpam-4702	289	6	2	2	NUM
ejpam-4702	289	7	,	,	PUNCT
ejpam-4702	289	8	burshtein	burshtein	ADV
ejpam-4702	289	9	[	[	X
ejpam-4702	289	10	1	1	X
ejpam-4702	289	11	]	]	PUNCT
ejpam-4702	289	12	showed	show	VERB
ejpam-4702	289	13	that	that	SCONJ
ejpam-4702	289	14	the	the	DET
ejpam-4702	289	15	diophantine	diophantine	NOUN
ejpam-4702	289	16	equation	equation	NOUN
ejpam-4702	289	17	px+(p+4)y	px+(p+4)y	NOUN
ejpam-4702	289	18	=	=	SYM
ejpam-4702	289	19	z2	z2	PROPN
ejpam-4702	289	20	,	,	PUNCT
ejpam-4702	289	21	where	where	SCONJ
ejpam-4702	289	22	p	p	PROPN
ejpam-4702	289	23	>	>	X
ejpam-4702	289	24	3	3	NUM
ejpam-4702	289	25	and	and	CCONJ
ejpam-4702	289	26	p	p	PRON
ejpam-4702	289	27	+	+	CCONJ
ejpam-4702	289	28	4	4	NUM
ejpam-4702	289	29	are	be	AUX
ejpam-4702	289	30	primes	prime	NOUN
ejpam-4702	289	31	,	,	PUNCT
ejpam-4702	289	32	has	have	VERB
ejpam-4702	289	33	no	no	DET
ejpam-4702	289	34	non	non	ADJ
ejpam-4702	289	35	-	-	ADJ
ejpam-4702	289	36	negative	negative	ADJ
ejpam-4702	289	37	solution	solution	NOUN
ejpam-4702	289	38	.	.	PUNCT
ejpam-4702	290	1	moreover	moreover	ADV
ejpam-4702	290	2	,	,	PUNCT
ejpam-4702	290	3	rao	rao	PROPN
ejpam-4702	291	1	[	[	X
ejpam-4702	291	2	10	10	NUM
ejpam-4702	291	3	]	]	PUNCT
ejpam-4702	291	4	investigated	investigate	VERB
ejpam-4702	291	5	the	the	DET
ejpam-4702	291	6	same	same	ADJ
ejpam-4702	291	7	equation	equation	NOUN
ejpam-4702	291	8	,	,	PUNCT
ejpam-4702	291	9	when	when	SCONJ
ejpam-4702	291	10	q	q	X
ejpam-4702	291	11	=	=	SYM
ejpam-4702	291	12	2	2	NUM
ejpam-4702	291	13	and	and	CCONJ
ejpam-4702	291	14	p	p	NOUN
ejpam-4702	291	15	=	=	NOUN
ejpam-4702	291	16	3	3	X
ejpam-4702	291	17	.	.	NOUN
ejpam-4702	291	18	from	from	ADP
ejpam-4702	291	19	now	now	ADV
ejpam-4702	291	20	on	on	ADV
ejpam-4702	291	21	,	,	PUNCT
ejpam-4702	291	22	x	x	PRON
ejpam-4702	291	23	,	,	PUNCT
ejpam-4702	291	24	y	y	PROPN
ejpam-4702	291	25	are	be	AUX
ejpam-4702	291	26	positive	positive	ADJ
ejpam-4702	291	27	integers	integer	NOUN
ejpam-4702	291	28	and	and	CCONJ
ejpam-4702	291	29	p	p	X
ejpam-4702	291	30	,	,	PUNCT
ejpam-4702	291	31	q	q	X
ejpam-4702	291	32	are	be	AUX
ejpam-4702	291	33	distinct	distinct	ADJ
ejpam-4702	291	34	odd	odd	ADJ
ejpam-4702	291	35	prime	prime	ADJ
ejpam-4702	291	36	numbers	number	NOUN
ejpam-4702	291	37	.	.	PUNCT
ejpam-4702	292	1	lemma	lemma	PROPN
ejpam-4702	292	2	1	1	X
ejpam-4702	292	3	.	.	PUNCT
ejpam-4702	293	1	let	let	VERB
ejpam-4702	293	2	x	x	PRON
ejpam-4702	293	3	be	be	AUX
ejpam-4702	293	4	an	an	DET
ejpam-4702	293	5	even	even	ADJ
ejpam-4702	293	6	number	number	NOUN
ejpam-4702	293	7	.	.	PUNCT
ejpam-4702	294	1	if	if	SCONJ
ejpam-4702	294	2	the	the	DET
ejpam-4702	294	3	diophantine	diophantine	NOUN
ejpam-4702	294	4	equation	equation	NOUN
ejpam-4702	294	5	px+(p+2q)y	px+(p+2q)y	NOUN
ejpam-4702	294	6	=	=	SYM
ejpam-4702	294	7	z2	z2	PROPN
ejpam-4702	294	8	has	have	VERB
ejpam-4702	294	9	a	a	DET
ejpam-4702	294	10	positive	positive	ADJ
ejpam-4702	294	11	integer	integer	NOUN
ejpam-4702	294	12	solution	solution	NOUN
ejpam-4702	294	13	,	,	PUNCT
ejpam-4702	294	14	then	then	ADV
ejpam-4702	294	15	2q	2q	NUM
ejpam-4702	294	16	≡	≡	PROPN
ejpam-4702	294	17	1	1	NUM
ejpam-4702	294	18	(	(	PUNCT
ejpam-4702	294	19	mod	mod	NOUN
ejpam-4702	294	20	p	p	NOUN
ejpam-4702	294	21	)	)	PUNCT
ejpam-4702	294	22	.	.	PUNCT
ejpam-4702	295	1	proof	proof	NOUN
ejpam-4702	295	2	.	.	PUNCT
ejpam-4702	296	1	assume	assume	VERB
ejpam-4702	296	2	that	that	SCONJ
ejpam-4702	296	3	the	the	DET
ejpam-4702	296	4	diophantine	diophantine	NOUN
ejpam-4702	296	5	equation	equation	NOUN
ejpam-4702	296	6	px	px	X
ejpam-4702	296	7	+	+	CCONJ
ejpam-4702	296	8	(	(	PUNCT
ejpam-4702	296	9	p	p	X
ejpam-4702	297	1	+	+	CCONJ
ejpam-4702	297	2	2q)y	2q)y	NUM
ejpam-4702	297	3	=	=	SYM
ejpam-4702	297	4	z2	z2	PROPN
ejpam-4702	297	5	has	have	VERB
ejpam-4702	297	6	a	a	DET
ejpam-4702	297	7	positive	positive	ADJ
ejpam-4702	297	8	integer	integer	NOUN
ejpam-4702	297	9	solution	solution	NOUN
ejpam-4702	297	10	.	.	PUNCT
ejpam-4702	298	1	since	since	SCONJ
ejpam-4702	298	2	x	x	PRON
ejpam-4702	298	3	is	be	AUX
ejpam-4702	298	4	even	even	ADV
ejpam-4702	298	5	,	,	PUNCT
ejpam-4702	298	6	there	there	PRON
ejpam-4702	298	7	exists	exist	VERB
ejpam-4702	298	8	a	a	DET
ejpam-4702	298	9	positive	positive	ADJ
ejpam-4702	298	10	integer	integer	NOUN
ejpam-4702	298	11	k	k	PROPN
ejpam-4702	298	12	such	such	ADJ
ejpam-4702	298	13	that	that	SCONJ
ejpam-4702	298	14	x	x	PROPN
ejpam-4702	298	15	=	=	PUNCT
ejpam-4702	298	16	2k	2k	NUM
ejpam-4702	298	17	.	.	PUNCT
ejpam-4702	299	1	thus	thus	ADV
ejpam-4702	299	2	,	,	PUNCT
ejpam-4702	299	3	(	(	PUNCT
ejpam-4702	299	4	p	p	X
ejpam-4702	299	5	+	+	NOUN
ejpam-4702	299	6	2q)y	2q)y	NUM
ejpam-4702	299	7	=	=	SYM
ejpam-4702	299	8	z2	z2	PROPN
ejpam-4702	299	9	−	−	PROPN
ejpam-4702	299	10	p2k	p2k	PROPN
ejpam-4702	299	11	=	=	PUNCT
ejpam-4702	299	12	(	(	PUNCT
ejpam-4702	299	13	z	z	NOUN
ejpam-4702	299	14	−	−	PROPN
ejpam-4702	299	15	pk)(z	pk)(z	PROPN
ejpam-4702	299	16	+	+	CCONJ
ejpam-4702	299	17	pk	pk	NOUN
ejpam-4702	299	18	)	)	PUNCT
ejpam-4702	299	19	.	.	PUNCT
ejpam-4702	300	1	since	since	SCONJ
ejpam-4702	300	2	p	p	NOUN
ejpam-4702	300	3	+	+	CCONJ
ejpam-4702	300	4	2q	2q	NUM
ejpam-4702	300	5	is	be	AUX
ejpam-4702	300	6	a	a	DET
ejpam-4702	300	7	prime	prime	ADJ
ejpam-4702	300	8	number	number	NOUN
ejpam-4702	300	9	,	,	PUNCT
ejpam-4702	300	10	we	we	PRON
ejpam-4702	300	11	have	have	VERB
ejpam-4702	300	12	z−	z−	PROPN
ejpam-4702	300	13	pk	pk	NOUN
ejpam-4702	300	14	=	=	SYM
ejpam-4702	300	15	(	(	PUNCT
ejpam-4702	300	16	p+2q)u	p+2q)u	NOUN
ejpam-4702	300	17	and	and	CCONJ
ejpam-4702	300	18	z+	z+	NUM
ejpam-4702	300	19	pk	pk	NOUN
ejpam-4702	300	20	=	=	SYM
ejpam-4702	300	21	(	(	PUNCT
ejpam-4702	300	22	p+2q)y−u	p+2q)y−u	NOUN
ejpam-4702	300	23	,	,	PUNCT
ejpam-4702	300	24	where	where	SCONJ
ejpam-4702	300	25	u	u	NOUN
ejpam-4702	300	26	is	be	AUX
ejpam-4702	300	27	a	a	DET
ejpam-4702	300	28	non	non	ADJ
ejpam-4702	300	29	-	-	ADJ
ejpam-4702	300	30	negative	negative	ADJ
ejpam-4702	300	31	integer	integer	NOUN
ejpam-4702	300	32	.	.	PUNCT
ejpam-4702	301	1	so	so	ADV
ejpam-4702	301	2	y	y	PROPN
ejpam-4702	301	3	>	>	X
ejpam-4702	301	4	2u	2u	PROPN
ejpam-4702	301	5	and	and	CCONJ
ejpam-4702	301	6	2pk	2pk	NOUN
ejpam-4702	301	7	=	=	SYM
ejpam-4702	301	8	(	(	PUNCT
ejpam-4702	301	9	p+	p+	NOUN
ejpam-4702	301	10	2q)u((p+	2q)u((p+	NUM
ejpam-4702	301	11	2q)y−2u	2q)y−2u	NUM
ejpam-4702	301	12	−	−	NUM
ejpam-4702	301	13	1	1	NUM
ejpam-4702	301	14	)	)	PUNCT
ejpam-4702	301	15	.	.	PUNCT
ejpam-4702	302	1	since	since	SCONJ
ejpam-4702	302	2	p	p	PROPN
ejpam-4702	302	3	and	and	CCONJ
ejpam-4702	302	4	p+	p+	PROPN
ejpam-4702	302	5	2q	2q	NUM
ejpam-4702	302	6	are	be	AUX
ejpam-4702	302	7	prime	prime	ADJ
ejpam-4702	302	8	numbers	number	NOUN
ejpam-4702	302	9	,	,	PUNCT
ejpam-4702	302	10	we	we	PRON
ejpam-4702	302	11	obtain	obtain	VERB
ejpam-4702	302	12	that	that	DET
ejpam-4702	302	13	u	u	NOUN
ejpam-4702	302	14	=	=	NOUN
ejpam-4702	302	15	0	0	NUM
ejpam-4702	302	16	and	and	CCONJ
ejpam-4702	302	17	so	so	ADV
ejpam-4702	302	18	2pk	2pk	ADJ
ejpam-4702	302	19	=	=	SYM
ejpam-4702	302	20	(	(	PUNCT
ejpam-4702	302	21	p+	p+	PROPN
ejpam-4702	302	22	2q)y	2q)y	NUM
ejpam-4702	302	23	−	−	NOUN
ejpam-4702	302	24	1	1	NUM
ejpam-4702	302	25	=	=	SYM
ejpam-4702	302	26	(	(	PUNCT
ejpam-4702	302	27	p+	p+	NOUN
ejpam-4702	302	28	2q	2q	NUM
ejpam-4702	302	29	−	−	PROPN
ejpam-4702	302	30	1)((p+	1)((p+	NUM
ejpam-4702	302	31	2q)y−1	2q)y−1	PROPN
ejpam-4702	302	32	+	+	CCONJ
ejpam-4702	302	33	(	(	PUNCT
ejpam-4702	302	34	p+	p+	PROPN
ejpam-4702	302	35	2q)y−2	2q)y−2	NUM
ejpam-4702	302	36	+	+	CCONJ
ejpam-4702	302	37	·	·	PUNCT
ejpam-4702	302	38	·	·	PUNCT
ejpam-4702	302	39	·	·	PUNCT
ejpam-4702	303	1	+	+	NUM
ejpam-4702	303	2	1	1	NUM
ejpam-4702	303	3	)	)	PUNCT
ejpam-4702	303	4	.	.	PUNCT
ejpam-4702	304	1	hence	hence	ADV
ejpam-4702	304	2	p|(p+	p|(p+	VERB
ejpam-4702	304	3	2q	2q	NOUN
ejpam-4702	304	4	−	−	NOUN
ejpam-4702	304	5	1	1	NUM
ejpam-4702	304	6	)	)	PUNCT
ejpam-4702	304	7	.	.	PUNCT
ejpam-4702	305	1	therefore	therefore	ADV
ejpam-4702	305	2	2q	2q	NUM
ejpam-4702	305	3	≡	≡	PROPN
ejpam-4702	305	4	1	1	NUM
ejpam-4702	305	5	(	(	PUNCT
ejpam-4702	305	6	mod	mod	NOUN
ejpam-4702	305	7	p	p	NOUN
ejpam-4702	305	8	)	)	PUNCT
ejpam-4702	305	9	.	.	PUNCT
ejpam-4702	306	1	lemma	lemma	PROPN
ejpam-4702	306	2	2	2	X
ejpam-4702	306	3	.	.	PUNCT
ejpam-4702	307	1	let	let	VERB
ejpam-4702	307	2	x	x	PRON
ejpam-4702	307	3	be	be	AUX
ejpam-4702	307	4	an	an	DET
ejpam-4702	307	5	odd	odd	ADJ
ejpam-4702	307	6	number	number	NOUN
ejpam-4702	307	7	.	.	PUNCT
ejpam-4702	308	1	if	if	SCONJ
ejpam-4702	308	2	the	the	DET
ejpam-4702	308	3	diophantine	diophantine	NOUN
ejpam-4702	308	4	equation	equation	NOUN
ejpam-4702	308	5	px	px	X
ejpam-4702	308	6	+	+	CCONJ
ejpam-4702	308	7	(	(	PUNCT
ejpam-4702	308	8	p+	p+	PROPN
ejpam-4702	308	9	2q)y	2q)y	PROPN
ejpam-4702	308	10	=	=	SYM
ejpam-4702	308	11	z2	z2	PROPN
ejpam-4702	308	12	has	have	VERB
ejpam-4702	308	13	a	a	DET
ejpam-4702	308	14	positive	positive	ADJ
ejpam-4702	308	15	integer	integer	NOUN
ejpam-4702	308	16	solution	solution	NOUN
ejpam-4702	308	17	,	,	PUNCT
ejpam-4702	308	18	then	then	ADV
ejpam-4702	308	19	(	(	PUNCT
ejpam-4702	308	20	2q	2q	NUM
ejpam-4702	308	21	p	p	NOUN
ejpam-4702	308	22	)	)	PUNCT
ejpam-4702	308	23	=	=	SYM
ejpam-4702	308	24	1	1	X
ejpam-4702	308	25	.	.	PUNCT
ejpam-4702	308	26	proof	proof	NOUN
ejpam-4702	308	27	.	.	PUNCT
ejpam-4702	309	1	assume	assume	VERB
ejpam-4702	309	2	that	that	SCONJ
ejpam-4702	309	3	the	the	DET
ejpam-4702	309	4	diophantine	diophantine	NOUN
ejpam-4702	309	5	equation	equation	NOUN
ejpam-4702	309	6	px	px	X
ejpam-4702	309	7	+	+	CCONJ
ejpam-4702	309	8	(	(	PUNCT
ejpam-4702	309	9	p	p	X
ejpam-4702	310	1	+	+	CCONJ
ejpam-4702	310	2	2q)y	2q)y	NUM
ejpam-4702	310	3	=	=	SYM
ejpam-4702	310	4	z2	z2	PROPN
ejpam-4702	310	5	has	have	VERB
ejpam-4702	310	6	a	a	DET
ejpam-4702	310	7	positive	positive	ADJ
ejpam-4702	310	8	integer	integer	NOUN
ejpam-4702	310	9	solution	solution	NOUN
ejpam-4702	310	10	.	.	PUNCT
ejpam-4702	311	1	then	then	ADV
ejpam-4702	311	2	px	px	PROPN
ejpam-4702	311	3	≡	≡	PROPN
ejpam-4702	311	4	z2	z2	PROPN
ejpam-4702	311	5	(	(	PUNCT
ejpam-4702	311	6	mod	mod	PROPN
ejpam-4702	311	7	p	p	NOUN
ejpam-4702	311	8	+	+	NOUN
ejpam-4702	311	9	2q	2q	NUM
ejpam-4702	311	10	)	)	PUNCT
ejpam-4702	311	11	.	.	PUNCT
ejpam-4702	312	1	by	by	ADP
ejpam-4702	312	2	division	division	NOUN
ejpam-4702	312	3	algorithm	algorithm	NOUN
ejpam-4702	312	4	,	,	PUNCT
ejpam-4702	312	5	we	we	PRON
ejpam-4702	312	6	can	can	AUX
ejpam-4702	312	7	write	write	VERB
ejpam-4702	312	8	x	x	PUNCT
ejpam-4702	312	9	=	=	PUNCT
ejpam-4702	312	10	(	(	PUNCT
ejpam-4702	312	11	p	p	X
ejpam-4702	313	1	+	+	NOUN
ejpam-4702	313	2	2q	2q	NUM
ejpam-4702	313	3	−	−	NUM
ejpam-4702	313	4	1)m	1)m	NUM
ejpam-4702	314	1	+	+	NUM
ejpam-4702	314	2	l	l	NOUN
ejpam-4702	314	3	,	,	PUNCT
ejpam-4702	314	4	where	where	SCONJ
ejpam-4702	314	5	m	m	VERB
ejpam-4702	314	6	and	and	CCONJ
ejpam-4702	314	7	l	l	NOUN
ejpam-4702	314	8	are	be	AUX
ejpam-4702	314	9	integers	integer	NOUN
ejpam-4702	314	10	with	with	ADP
ejpam-4702	314	11	0	0	NUM
ejpam-4702	314	12	≤	≤	NUM
ejpam-4702	314	13	l	l	NOUN
ejpam-4702	315	1	<	<	X
ejpam-4702	315	2	p	p	X
ejpam-4702	315	3	+	+	X
ejpam-4702	315	4	2q	2q	NUM
ejpam-4702	315	5	−	−	NOUN
ejpam-4702	315	6	1	1	X
ejpam-4702	315	7	.	.	PUNCT
ejpam-4702	316	1	since	since	SCONJ
ejpam-4702	316	2	x	x	PRON
ejpam-4702	316	3	is	be	AUX
ejpam-4702	316	4	odd	odd	ADJ
ejpam-4702	316	5	,	,	PUNCT
ejpam-4702	316	6	we	we	PRON
ejpam-4702	316	7	obtain	obtain	VERB
ejpam-4702	316	8	l	l	NOUN
ejpam-4702	316	9	is	be	AUX
ejpam-4702	316	10	odd	odd	ADJ
ejpam-4702	316	11	.	.	PUNCT
ejpam-4702	317	1	by	by	ADP
ejpam-4702	317	2	theorem	theorem	NOUN
ejpam-4702	317	3	1	1	NUM
ejpam-4702	317	4	,	,	PUNCT
ejpam-4702	317	5	we	we	PRON
ejpam-4702	317	6	obtain	obtain	VERB
ejpam-4702	317	7	that	that	SCONJ
ejpam-4702	317	8	pp+2q−1	pp+2q−1	PRON
ejpam-4702	317	9	≡	≡	PROPN
ejpam-4702	317	10	1	1	NUM
ejpam-4702	317	11	(	(	PUNCT
ejpam-4702	317	12	mod	mod	NOUN
ejpam-4702	317	13	p	p	NOUN
ejpam-4702	317	14	+	+	NOUN
ejpam-4702	317	15	2q	2q	NUM
ejpam-4702	317	16	)	)	PUNCT
ejpam-4702	317	17	.	.	PUNCT
ejpam-4702	318	1	so	so	ADV
ejpam-4702	318	2	p(p+2q−1)m+l	p(p+2q−1)m+l	PROPN
ejpam-4702	318	3	≡	≡	PROPN
ejpam-4702	318	4	pl	pl	PROPN
ejpam-4702	318	5	(	(	PUNCT
ejpam-4702	318	6	mod	mod	PROPN
ejpam-4702	318	7	p+2q	p+2q	PROPN
ejpam-4702	318	8	)	)	PUNCT
ejpam-4702	318	9	.	.	PUNCT
ejpam-4702	319	1	then	then	ADV
ejpam-4702	319	2	px	px	PROPN
ejpam-4702	319	3	≡	≡	PROPN
ejpam-4702	319	4	pl	pl	PROPN
ejpam-4702	319	5	(	(	PUNCT
ejpam-4702	319	6	mod	mod	PROPN
ejpam-4702	319	7	p+2q	p+2q	PROPN
ejpam-4702	319	8	)	)	PUNCT
ejpam-4702	319	9	.	.	PUNCT
ejpam-4702	320	1	thus	thus	ADV
ejpam-4702	320	2	,	,	PUNCT
ejpam-4702	320	3	z2	z2	PROPN
ejpam-4702	320	4	≡	≡	PROPN
ejpam-4702	320	5	pl	pl	PROPN
ejpam-4702	320	6	(	(	PUNCT
ejpam-4702	320	7	mod	mod	PROPN
ejpam-4702	320	8	p+2q	p+2q	PROPN
ejpam-4702	320	9	)	)	PUNCT
ejpam-4702	320	10	.	.	PUNCT
ejpam-4702	321	1	s.	s.	PROPN
ejpam-4702	321	2	tadee	tadee	PROPN
ejpam-4702	321	3	,	,	PUNCT
ejpam-4702	321	4	a.	a.	NOUN
ejpam-4702	321	5	siraworakun	siraworakun	PROPN
ejpam-4702	321	6	/	/	SYM
ejpam-4702	321	7	eur	eur	PROPN
ejpam-4702	321	8	.	.	PUNCT
ejpam-4702	322	1	j.	j.	PROPN
ejpam-4702	322	2	pure	pure	PROPN
ejpam-4702	322	3	appl	appl	PROPN
ejpam-4702	322	4	.	.	PROPN
ejpam-4702	322	5	math	math	PROPN
ejpam-4702	322	6	,	,	PUNCT
ejpam-4702	322	7	16	16	NUM
ejpam-4702	322	8	(	(	PUNCT
ejpam-4702	322	9	2	2	NUM
ejpam-4702	322	10	)	)	PUNCT
ejpam-4702	322	11	(	(	PUNCT
ejpam-4702	322	12	2023	2023	NUM
ejpam-4702	322	13	)	)	PUNCT
ejpam-4702	322	14	,	,	PUNCT
ejpam-4702	322	15	724	724	NUM
ejpam-4702	322	16	-	-	SYM
ejpam-4702	322	17	735	735	NUM
ejpam-4702	322	18	732	732	NUM
ejpam-4702	322	19	hence	hence	ADV
ejpam-4702	322	20	,	,	PUNCT
ejpam-4702	322	21	(	(	PUNCT
ejpam-4702	322	22	pl	pl	X
ejpam-4702	322	23	p+2q	p+2q	PROPN
ejpam-4702	322	24	)	)	PUNCT
ejpam-4702	322	25	=	=	PUNCT
ejpam-4702	323	1	1	1	X
ejpam-4702	323	2	.	.	PUNCT
ejpam-4702	323	3	since	since	SCONJ
ejpam-4702	323	4	l	l	NOUN
ejpam-4702	323	5	is	be	AUX
ejpam-4702	323	6	an	an	DET
ejpam-4702	323	7	odd	odd	ADJ
ejpam-4702	323	8	number	number	NOUN
ejpam-4702	323	9	,	,	PUNCT
ejpam-4702	323	10	we	we	PRON
ejpam-4702	323	11	obtain	obtain	VERB
ejpam-4702	323	12	(	(	PUNCT
ejpam-4702	323	13	p	p	NOUN
ejpam-4702	323	14	p+2q	p+2q	PROPN
ejpam-4702	323	15	)	)	PUNCT
ejpam-4702	323	16	=	=	SYM
ejpam-4702	323	17	1	1	NUM
ejpam-4702	323	18	by	by	ADP
ejpam-4702	323	19	theorem	theorem	ADJ
ejpam-4702	323	20	3(ii	3(ii	NUM
ejpam-4702	323	21	)	)	PUNCT
ejpam-4702	323	22	.	.	PUNCT
ejpam-4702	324	1	by	by	ADP
ejpam-4702	324	2	theorem	theorem	NOUN
ejpam-4702	324	3	4	4	NUM
ejpam-4702	324	4	,	,	PUNCT
ejpam-4702	324	5	we	we	PRON
ejpam-4702	324	6	obtain	obtain	VERB
ejpam-4702	324	7	(	(	PUNCT
ejpam-4702	324	8	p	p	X
ejpam-4702	324	9	p+2q	p+2q	NOUN
ejpam-4702	324	10	)	)	PUNCT
ejpam-4702	324	11	(	(	PUNCT
ejpam-4702	324	12	p+2q	p+2q	NOUN
ejpam-4702	324	13	p	p	NOUN
ejpam-4702	324	14	)	)	PUNCT
ejpam-4702	324	15	=	=	SYM
ejpam-4702	324	16	(	(	PUNCT
ejpam-4702	324	17	−1	−1	NOUN
ejpam-4702	324	18	)	)	PUNCT
ejpam-4702	324	19	(	(	PUNCT
ejpam-4702	324	20	p−1	p−1	PROPN
ejpam-4702	324	21	2	2	NUM
ejpam-4702	324	22	)	)	PUNCT
ejpam-4702	324	23	(	(	PUNCT
ejpam-4702	324	24	p+2q−1	p+2q−1	NOUN
ejpam-4702	324	25	2	2	X
ejpam-4702	324	26	)	)	PUNCT
ejpam-4702	324	27	=	=	SYM
ejpam-4702	325	1	1	1	X
ejpam-4702	325	2	.	.	PUNCT
ejpam-4702	325	3	thus	thus	ADV
ejpam-4702	325	4	,	,	PUNCT
ejpam-4702	325	5	(	(	PUNCT
ejpam-4702	325	6	p+2q	p+2q	NOUN
ejpam-4702	325	7	p	p	NOUN
ejpam-4702	325	8	)	)	PUNCT
ejpam-4702	325	9	=	=	SYM
ejpam-4702	326	1	1	1	X
ejpam-4702	326	2	.	.	PUNCT
ejpam-4702	326	3	by	by	ADP
ejpam-4702	326	4	theorem	theorem	NOUN
ejpam-4702	326	5	3(i	3(i	NUM
ejpam-4702	326	6	)	)	PUNCT
ejpam-4702	326	7	,	,	PUNCT
ejpam-4702	326	8	we	we	PRON
ejpam-4702	326	9	have	have	VERB
ejpam-4702	326	10	(	(	PUNCT
ejpam-4702	326	11	2q	2q	NUM
ejpam-4702	326	12	p	p	NOUN
ejpam-4702	326	13	)	)	PUNCT
ejpam-4702	326	14	=	=	PUNCT
ejpam-4702	326	15	(	(	PUNCT
ejpam-4702	326	16	p+2q	p+2q	NOUN
ejpam-4702	326	17	p	p	NOUN
ejpam-4702	326	18	)	)	PUNCT
ejpam-4702	326	19	.	.	PUNCT
ejpam-4702	327	1	then	then	ADV
ejpam-4702	327	2	(	(	PUNCT
ejpam-4702	327	3	2q	2q	X
ejpam-4702	327	4	p	p	NOUN
ejpam-4702	327	5	)	)	PUNCT
ejpam-4702	327	6	=	=	PUNCT
ejpam-4702	328	1	1	1	X
ejpam-4702	328	2	.	.	X
ejpam-4702	328	3	from	from	ADP
ejpam-4702	328	4	above	above	ADP
ejpam-4702	328	5	lemmas	lemmas	PROPN
ejpam-4702	328	6	,	,	PUNCT
ejpam-4702	328	7	we	we	PRON
ejpam-4702	328	8	have	have	VERB
ejpam-4702	328	9	the	the	DET
ejpam-4702	328	10	following	follow	VERB
ejpam-4702	328	11	result	result	NOUN
ejpam-4702	328	12	.	.	PUNCT
ejpam-4702	329	1	theorem	theorem	ADJ
ejpam-4702	329	2	14	14	NUM
ejpam-4702	329	3	.	.	PUNCT
ejpam-4702	330	1	let	let	VERB
ejpam-4702	330	2	p	p	NOUN
ejpam-4702	330	3	and	and	CCONJ
ejpam-4702	330	4	q	q	NOUN
ejpam-4702	330	5	be	be	AUX
ejpam-4702	330	6	distinct	distinct	ADJ
ejpam-4702	330	7	prime	prime	ADJ
ejpam-4702	330	8	numbers	number	NOUN
ejpam-4702	330	9	with	with	ADP
ejpam-4702	330	10	2q	2q	NUM
ejpam-4702	330	11	̸≡	̸≡	NOUN
ejpam-4702	330	12	1	1	NUM
ejpam-4702	330	13	(	(	PUNCT
ejpam-4702	330	14	mod	mod	PROPN
ejpam-4702	330	15	p	p	X
ejpam-4702	330	16	)	)	PUNCT
ejpam-4702	330	17	and	and	CCONJ
ejpam-4702	331	1	(	(	PUNCT
ejpam-4702	331	2	2q	2q	NUM
ejpam-4702	331	3	p	p	NOUN
ejpam-4702	331	4	)	)	PUNCT
ejpam-4702	331	5	=	=	PUNCT
ejpam-4702	331	6	−1	−1	NOUN
ejpam-4702	331	7	.	.	PUNCT
ejpam-4702	332	1	then	then	ADV
ejpam-4702	332	2	the	the	DET
ejpam-4702	332	3	diophantine	diophantine	NOUN
ejpam-4702	332	4	equation	equation	NOUN
ejpam-4702	332	5	px	px	X
ejpam-4702	332	6	+	+	CCONJ
ejpam-4702	332	7	(	(	PUNCT
ejpam-4702	332	8	p+	p+	PROPN
ejpam-4702	332	9	2q)y	2q)y	PROPN
ejpam-4702	332	10	=	=	SYM
ejpam-4702	332	11	z2	z2	PROPN
ejpam-4702	332	12	has	have	VERB
ejpam-4702	332	13	no	no	DET
ejpam-4702	332	14	positive	positive	ADJ
ejpam-4702	332	15	integer	integer	NOUN
ejpam-4702	332	16	solution	solution	NOUN
ejpam-4702	332	17	.	.	PUNCT
ejpam-4702	333	1	by	by	ADP
ejpam-4702	333	2	applying	apply	VERB
ejpam-4702	333	3	theorems	theorem	NOUN
ejpam-4702	333	4	12	12	NUM
ejpam-4702	333	5	and	and	CCONJ
ejpam-4702	333	6	14	14	NUM
ejpam-4702	333	7	,	,	PUNCT
ejpam-4702	333	8	the	the	DET
ejpam-4702	333	9	forms	form	NOUN
ejpam-4702	333	10	of	of	ADP
ejpam-4702	333	11	odd	odd	ADJ
ejpam-4702	333	12	prime	prime	ADJ
ejpam-4702	333	13	number	number	NOUN
ejpam-4702	333	14	p	p	NOUN
ejpam-4702	333	15	are	be	AUX
ejpam-4702	333	16	identified	identify	VERB
ejpam-4702	333	17	,	,	PUNCT
ejpam-4702	333	18	when	when	SCONJ
ejpam-4702	333	19	q	q	PROPN
ejpam-4702	333	20	≡	≡	PROPN
ejpam-4702	333	21	1	1	NUM
ejpam-4702	333	22	(	(	PUNCT
ejpam-4702	333	23	mod	mod	NOUN
ejpam-4702	333	24	4	4	NUM
ejpam-4702	333	25	)	)	PUNCT
ejpam-4702	333	26	.	.	PUNCT
ejpam-4702	334	1	theorem	theorem	NOUN
ejpam-4702	334	2	15	15	NUM
ejpam-4702	334	3	.	.	PUNCT
ejpam-4702	335	1	let	let	VERB
ejpam-4702	335	2	q	q	PART
ejpam-4702	335	3	be	be	AUX
ejpam-4702	335	4	a	a	DET
ejpam-4702	335	5	prime	prime	ADJ
ejpam-4702	335	6	number	number	NOUN
ejpam-4702	335	7	such	such	ADJ
ejpam-4702	335	8	that	that	DET
ejpam-4702	335	9	q	q	PROPN
ejpam-4702	335	10	≡	≡	PROPN
ejpam-4702	335	11	1	1	NUM
ejpam-4702	335	12	(	(	PUNCT
ejpam-4702	335	13	mod	mod	NOUN
ejpam-4702	335	14	4	4	NUM
ejpam-4702	335	15	)	)	PUNCT
ejpam-4702	335	16	.	.	PUNCT
ejpam-4702	336	1	if	if	SCONJ
ejpam-4702	336	2	p	p	NOUN
ejpam-4702	336	3	is	be	AUX
ejpam-4702	336	4	a	a	DET
ejpam-4702	336	5	prime	prime	ADJ
ejpam-4702	336	6	number	number	NOUN
ejpam-4702	336	7	with	with	ADP
ejpam-4702	336	8	2q	2q	NUM
ejpam-4702	336	9	̸≡	̸≡	NOUN
ejpam-4702	336	10	1	1	NUM
ejpam-4702	336	11	(	(	PUNCT
ejpam-4702	336	12	mod	mod	PROPN
ejpam-4702	336	13	p	p	X
ejpam-4702	336	14	)	)	PUNCT
ejpam-4702	336	15	and	and	CCONJ
ejpam-4702	336	16	satisfies	satisfy	VERB
ejpam-4702	336	17	any	any	PRON
ejpam-4702	336	18	of	of	ADP
ejpam-4702	336	19	the	the	DET
ejpam-4702	336	20	following	follow	VERB
ejpam-4702	336	21	conditions	condition	NOUN
ejpam-4702	336	22	:	:	PUNCT
ejpam-4702	336	23	(	(	PUNCT
ejpam-4702	336	24	i	i	NOUN
ejpam-4702	336	25	)	)	PUNCT
ejpam-4702	336	26	p	p	PROPN
ejpam-4702	336	27	≡	≡	PROPN
ejpam-4702	336	28	q2	q2	PROPN
ejpam-4702	336	29	+	+	CCONJ
ejpam-4702	336	30	8n1r	8n1r	ADJ
ejpam-4702	336	31	s2	s2	NOUN
ejpam-4702	336	32	(	(	PUNCT
ejpam-4702	336	33	mod	mod	PROPN
ejpam-4702	336	34	8q	8q	PROPN
ejpam-4702	336	35	)	)	PUNCT
ejpam-4702	336	36	,	,	PUNCT
ejpam-4702	336	37	(	(	PUNCT
ejpam-4702	336	38	ii	ii	NOUN
ejpam-4702	336	39	)	)	PUNCT
ejpam-4702	336	40	p	p	PROPN
ejpam-4702	336	41	≡	≡	PROPN
ejpam-4702	336	42	−	−	PROPN
ejpam-4702	336	43	q2	q2	PROPN
ejpam-4702	336	44	+	+	CCONJ
ejpam-4702	336	45	8n1r	8n1r	ADJ
ejpam-4702	336	46	s2	s2	NOUN
ejpam-4702	336	47	(	(	PUNCT
ejpam-4702	336	48	mod	mod	PROPN
ejpam-4702	336	49	8q	8q	PROPN
ejpam-4702	336	50	)	)	PUNCT
ejpam-4702	336	51	,	,	PUNCT
ejpam-4702	336	52	(	(	PUNCT
ejpam-4702	336	53	iii	iii	X
ejpam-4702	336	54	)	)	PUNCT
ejpam-4702	336	55	p	p	PROPN
ejpam-4702	336	56	≡	≡	PROPN
ejpam-4702	336	57	3q2	3q2	NUM
ejpam-4702	337	1	+	+	CCONJ
ejpam-4702	337	2	8n1r	8n1r	ADJ
ejpam-4702	337	3	s1	s1	NOUN
ejpam-4702	337	4	(	(	PUNCT
ejpam-4702	337	5	mod	mod	PROPN
ejpam-4702	337	6	8q	8q	NUM
ejpam-4702	337	7	)	)	PUNCT
ejpam-4702	337	8	,	,	PUNCT
ejpam-4702	337	9	or	or	CCONJ
ejpam-4702	337	10	(	(	PUNCT
ejpam-4702	337	11	iv	iv	X
ejpam-4702	337	12	)	)	PUNCT
ejpam-4702	337	13	p	p	PROPN
ejpam-4702	337	14	≡	≡	PROPN
ejpam-4702	337	15	−	−	PROPN
ejpam-4702	337	16	3q2	3q2	NUM
ejpam-4702	337	17	+	+	CCONJ
ejpam-4702	337	18	8n1r	8n1r	ADJ
ejpam-4702	337	19	s1	s1	NOUN
ejpam-4702	337	20	(	(	PUNCT
ejpam-4702	337	21	mod	mod	PROPN
ejpam-4702	337	22	8q	8q	PROPN
ejpam-4702	337	23	)	)	PUNCT
ejpam-4702	337	24	,	,	PUNCT
ejpam-4702	337	25	where	where	SCONJ
ejpam-4702	337	26	s1	s1	PROPN
ejpam-4702	337	27	∈	∈	PROPN
ejpam-4702	337	28	{	{	PUNCT
ejpam-4702	337	29	2	2	NUM
ejpam-4702	337	30	,	,	PUNCT
ejpam-4702	337	31	4	4	NUM
ejpam-4702	337	32	,	,	PUNCT
ejpam-4702	337	33	6	6	NUM
ejpam-4702	337	34	,	,	PUNCT
ejpam-4702	337	35	.	.	PUNCT
ejpam-4702	337	36	.	.	PUNCT
ejpam-4702	337	37	.	.	PUNCT
ejpam-4702	337	38	,	,	PUNCT
ejpam-4702	337	39	q−1	q−1	PROPN
ejpam-4702	337	40	}	}	PUNCT
ejpam-4702	337	41	,	,	PUNCT
ejpam-4702	337	42	s2	s2	PROPN
ejpam-4702	337	43	∈	∈	PROPN
ejpam-4702	337	44	{	{	PUNCT
ejpam-4702	337	45	1	1	NUM
ejpam-4702	337	46	,	,	PUNCT
ejpam-4702	337	47	3	3	NUM
ejpam-4702	337	48	,	,	PUNCT
ejpam-4702	337	49	5	5	NUM
ejpam-4702	337	50	,	,	PUNCT
ejpam-4702	337	51	.	.	PUNCT
ejpam-4702	337	52	.	.	PUNCT
ejpam-4702	337	53	.	.	PUNCT
ejpam-4702	337	54	,	,	PUNCT
ejpam-4702	337	55	q−2	q−2	PROPN
ejpam-4702	337	56	}	}	PUNCT
ejpam-4702	337	57	,	,	PUNCT
ejpam-4702	337	58	r	r	NOUN
ejpam-4702	337	59	is	be	AUX
ejpam-4702	337	60	a	a	DET
ejpam-4702	337	61	primitive	primitive	ADJ
ejpam-4702	337	62	root	root	NOUN
ejpam-4702	337	63	modulo	modulo	NOUN
ejpam-4702	337	64	q	q	NOUN
ejpam-4702	337	65	,	,	PUNCT
ejpam-4702	337	66	and	and	CCONJ
ejpam-4702	337	67	if	if	SCONJ
ejpam-4702	337	68	q	q	PUNCT
ejpam-4702	337	69	−	−	NOUN
ejpam-4702	337	70	1	1	NUM
ejpam-4702	337	71	4	4	NUM
ejpam-4702	337	72	is	be	AUX
ejpam-4702	337	73	an	an	DET
ejpam-4702	337	74	even	even	ADJ
ejpam-4702	337	75	number	number	NOUN
ejpam-4702	337	76	,	,	PUNCT
ejpam-4702	337	77	then	then	ADV
ejpam-4702	337	78	n1	n1	ADJ
ejpam-4702	337	79	=	=	SYM
ejpam-4702	337	80	−q	−q	ADJ
ejpam-4702	337	81	+	+	CCONJ
ejpam-4702	337	82	1	1	NUM
ejpam-4702	337	83	8	8	NUM
ejpam-4702	337	84	,	,	PUNCT
ejpam-4702	337	85	and	and	CCONJ
ejpam-4702	337	86	if	if	SCONJ
ejpam-4702	337	87	otherwise	otherwise	ADV
ejpam-4702	337	88	,	,	PUNCT
ejpam-4702	337	89	then	then	ADV
ejpam-4702	337	90	n1	n1	PROPN
ejpam-4702	337	91	=	=	SYM
ejpam-4702	337	92	3q	3q	NUM
ejpam-4702	337	93	+	+	CCONJ
ejpam-4702	337	94	1	1	NUM
ejpam-4702	337	95	8	8	NUM
ejpam-4702	337	96	.	.	PUNCT
ejpam-4702	338	1	then	then	ADV
ejpam-4702	338	2	,	,	PUNCT
ejpam-4702	338	3	the	the	DET
ejpam-4702	338	4	diophantine	diophantine	NOUN
ejpam-4702	338	5	equation	equation	NOUN
ejpam-4702	338	6	px	px	X
ejpam-4702	338	7	+	+	CCONJ
ejpam-4702	338	8	(	(	PUNCT
ejpam-4702	338	9	p+	p+	PROPN
ejpam-4702	338	10	2q)y	2q)y	PROPN
ejpam-4702	338	11	=	=	SYM
ejpam-4702	338	12	z2	z2	PROPN
ejpam-4702	338	13	has	have	VERB
ejpam-4702	338	14	no	no	DET
ejpam-4702	338	15	positive	positive	ADJ
ejpam-4702	338	16	integer	integer	NOUN
ejpam-4702	338	17	solution	solution	NOUN
ejpam-4702	338	18	.	.	PUNCT
ejpam-4702	339	1	example	example	NOUN
ejpam-4702	340	1	1	1	NUM
ejpam-4702	340	2	.	.	PUNCT
ejpam-4702	340	3	let	let	VERB
ejpam-4702	340	4	q	q	NOUN
ejpam-4702	341	1	=	=	SYM
ejpam-4702	341	2	17	17	NUM
ejpam-4702	341	3	and	and	CCONJ
ejpam-4702	341	4	r	r	NOUN
ejpam-4702	341	5	=	=	SYM
ejpam-4702	341	6	3	3	X
ejpam-4702	341	7	.	.	PUNCT
ejpam-4702	342	1	then	then	ADV
ejpam-4702	342	2	r	r	NOUN
ejpam-4702	342	3	is	be	AUX
ejpam-4702	342	4	a	a	DET
ejpam-4702	342	5	primitive	primitive	ADJ
ejpam-4702	342	6	root	root	NOUN
ejpam-4702	342	7	of	of	ADP
ejpam-4702	342	8	q	q	NOUN
ejpam-4702	342	9	and	and	CCONJ
ejpam-4702	342	10	n1	n1	NOUN
ejpam-4702	342	11	=	=	SYM
ejpam-4702	342	12	−17	−17	NOUN
ejpam-4702	342	13	+	+	CCONJ
ejpam-4702	342	14	1	1	NUM
ejpam-4702	342	15	8	8	NUM
ejpam-4702	342	16	=	=	NOUN
ejpam-4702	342	17	−2	−2	NOUN
ejpam-4702	342	18	since	since	SCONJ
ejpam-4702	342	19	17−	17−	NUM
ejpam-4702	342	20	1	1	NUM
ejpam-4702	342	21	4	4	NUM
ejpam-4702	342	22	is	be	AUX
ejpam-4702	342	23	an	an	DET
ejpam-4702	342	24	even	even	ADJ
ejpam-4702	342	25	number	number	NOUN
ejpam-4702	342	26	.	.	PUNCT
ejpam-4702	343	1	consider	consider	VERB
ejpam-4702	343	2	a	a	DET
ejpam-4702	343	3	prime	prime	ADJ
ejpam-4702	343	4	number	number	NOUN
ejpam-4702	343	5	p	p	NOUN
ejpam-4702	343	6	that	that	PRON
ejpam-4702	343	7	satisfies	satisfy	VERB
ejpam-4702	343	8	any	any	PRON
ejpam-4702	343	9	of	of	ADP
ejpam-4702	343	10	the	the	DET
ejpam-4702	343	11	following	follow	VERB
ejpam-4702	343	12	congruences	congruence	NOUN
ejpam-4702	343	13	:	:	PUNCT
ejpam-4702	343	14	(	(	PUNCT
ejpam-4702	343	15	i	i	NOUN
ejpam-4702	343	16	)	)	PUNCT
ejpam-4702	343	17	p	p	PROPN
ejpam-4702	343	18	≡	≡	PROPN
ejpam-4702	343	19	289	289	NUM
ejpam-4702	344	1	+	+	CCONJ
ejpam-4702	344	2	(	(	PUNCT
ejpam-4702	344	3	−16	−16	PROPN
ejpam-4702	344	4	)	)	PUNCT
ejpam-4702	344	5	·	·	PUNCT
ejpam-4702	345	1	(	(	PUNCT
ejpam-4702	345	2	3)s2	3)s2	NUM
ejpam-4702	345	3	(	(	PUNCT
ejpam-4702	345	4	mod	mod	PROPN
ejpam-4702	345	5	136	136	NUM
ejpam-4702	345	6	)	)	PUNCT
ejpam-4702	345	7	,	,	PUNCT
ejpam-4702	345	8	(	(	PUNCT
ejpam-4702	345	9	ii	ii	NOUN
ejpam-4702	345	10	)	)	PUNCT
ejpam-4702	345	11	p	p	PROPN
ejpam-4702	345	12	≡	≡	PROPN
ejpam-4702	345	13	−	−	PROPN
ejpam-4702	345	14	289	289	NUM
ejpam-4702	346	1	+	+	CCONJ
ejpam-4702	346	2	(	(	PUNCT
ejpam-4702	346	3	−16	−16	PROPN
ejpam-4702	346	4	)	)	PUNCT
ejpam-4702	346	5	·	·	PUNCT
ejpam-4702	347	1	(	(	PUNCT
ejpam-4702	347	2	3)s2	3)s2	NUM
ejpam-4702	347	3	(	(	PUNCT
ejpam-4702	347	4	mod	mod	PROPN
ejpam-4702	347	5	136	136	NUM
ejpam-4702	347	6	)	)	PUNCT
ejpam-4702	347	7	,	,	PUNCT
ejpam-4702	347	8	(	(	PUNCT
ejpam-4702	347	9	iii	iii	X
ejpam-4702	347	10	)	)	PUNCT
ejpam-4702	347	11	p	p	PROPN
ejpam-4702	347	12	≡	≡	PROPN
ejpam-4702	347	13	867	867	NUM
ejpam-4702	347	14	+	+	CCONJ
ejpam-4702	347	15	(	(	PUNCT
ejpam-4702	347	16	−16	−16	PROPN
ejpam-4702	347	17	)	)	PUNCT
ejpam-4702	347	18	·	·	PUNCT
ejpam-4702	347	19	(	(	PUNCT
ejpam-4702	347	20	3)s1	3)s1	NUM
ejpam-4702	347	21	(	(	PUNCT
ejpam-4702	347	22	mod	mod	PROPN
ejpam-4702	347	23	136	136	NUM
ejpam-4702	347	24	)	)	PUNCT
ejpam-4702	347	25	,	,	PUNCT
ejpam-4702	347	26	or	or	CCONJ
ejpam-4702	347	27	(	(	PUNCT
ejpam-4702	347	28	iv	iv	X
ejpam-4702	347	29	)	)	PUNCT
ejpam-4702	347	30	p	p	PROPN
ejpam-4702	347	31	≡	≡	PROPN
ejpam-4702	347	32	−	−	PROPN
ejpam-4702	347	33	867	867	NUM
ejpam-4702	347	34	+	+	CCONJ
ejpam-4702	347	35	(	(	PUNCT
ejpam-4702	347	36	−16	−16	PROPN
ejpam-4702	347	37	)	)	PUNCT
ejpam-4702	347	38	·	·	PUNCT
ejpam-4702	347	39	(	(	PUNCT
ejpam-4702	347	40	3)s1	3)s1	NUM
ejpam-4702	347	41	(	(	PUNCT
ejpam-4702	347	42	mod	mod	PROPN
ejpam-4702	347	43	136	136	NUM
ejpam-4702	347	44	)	)	PUNCT
ejpam-4702	347	45	,	,	PUNCT
ejpam-4702	347	46	where	where	SCONJ
ejpam-4702	347	47	s1	s1	PROPN
ejpam-4702	347	48	∈	∈	PROPN
ejpam-4702	347	49	{	{	PUNCT
ejpam-4702	347	50	2	2	NUM
ejpam-4702	347	51	,	,	PUNCT
ejpam-4702	347	52	4	4	NUM
ejpam-4702	347	53	,	,	PUNCT
ejpam-4702	347	54	6	6	NUM
ejpam-4702	347	55	,	,	PUNCT
ejpam-4702	347	56	8	8	NUM
ejpam-4702	347	57	,	,	PUNCT
ejpam-4702	347	58	10	10	NUM
ejpam-4702	347	59	,	,	PUNCT
ejpam-4702	347	60	12	12	NUM
ejpam-4702	347	61	,	,	PUNCT
ejpam-4702	347	62	14	14	NUM
ejpam-4702	347	63	,	,	PUNCT
ejpam-4702	347	64	16	16	NUM
ejpam-4702	347	65	}	}	PUNCT
ejpam-4702	347	66	and	and	CCONJ
ejpam-4702	347	67	s2	s2	PROPN
ejpam-4702	347	68	∈	∈	PROPN
ejpam-4702	347	69	{	{	PUNCT
ejpam-4702	347	70	1	1	NUM
ejpam-4702	347	71	,	,	PUNCT
ejpam-4702	347	72	3	3	NUM
ejpam-4702	347	73	,	,	PUNCT
ejpam-4702	347	74	5	5	NUM
ejpam-4702	347	75	,	,	PUNCT
ejpam-4702	347	76	7	7	NUM
ejpam-4702	347	77	,	,	PUNCT
ejpam-4702	347	78	9	9	NUM
ejpam-4702	347	79	,	,	PUNCT
ejpam-4702	347	80	11	11	NUM
ejpam-4702	347	81	,	,	PUNCT
ejpam-4702	347	82	13	13	NUM
ejpam-4702	347	83	,	,	PUNCT
ejpam-4702	347	84	15	15	NUM
ejpam-4702	347	85	}	}	PUNCT
ejpam-4702	347	86	.	.	PUNCT
ejpam-4702	348	1	thus	thus	ADV
ejpam-4702	348	2	,	,	PUNCT
ejpam-4702	348	3	p	p	PROPN
ejpam-4702	348	4	≡	≡	PROPN
ejpam-4702	348	5	±7,±13,±19,±21,±23,±31,±35,±39,±41,±43,±53,±57,±59,±63,±65,±67	±7,±13,±19,±21,±23,±31,±35,±39,±41,±43,±53,±57,±59,±63,±65,±67	PROPN
ejpam-4702	348	6	(	(	PUNCT
ejpam-4702	348	7	mod	mod	PROPN
ejpam-4702	348	8	136	136	NUM
ejpam-4702	348	9	)	)	PUNCT
ejpam-4702	348	10	.	.	PUNCT
ejpam-4702	349	1	by	by	ADP
ejpam-4702	349	2	theorem	theorem	NOUN
ejpam-4702	349	3	15	15	NUM
ejpam-4702	349	4	,	,	PUNCT
ejpam-4702	349	5	we	we	PRON
ejpam-4702	349	6	obtain	obtain	VERB
ejpam-4702	349	7	that	that	SCONJ
ejpam-4702	349	8	the	the	DET
ejpam-4702	349	9	diophantine	diophantine	NOUN
ejpam-4702	349	10	equation	equation	NOUN
ejpam-4702	349	11	px	px	X
ejpam-4702	350	1	+	+	CCONJ
ejpam-4702	350	2	(	(	PUNCT
ejpam-4702	350	3	p	p	X
ejpam-4702	350	4	+	+	NOUN
ejpam-4702	350	5	34)y	34)y	NUM
ejpam-4702	350	6	=	=	SYM
ejpam-4702	350	7	z2	z2	PROPN
ejpam-4702	350	8	has	have	VERB
ejpam-4702	350	9	no	no	DET
ejpam-4702	350	10	positive	positive	ADJ
ejpam-4702	350	11	integer	integer	NOUN
ejpam-4702	350	12	solution	solution	NOUN
ejpam-4702	350	13	.	.	PUNCT
ejpam-4702	351	1	for	for	ADP
ejpam-4702	351	2	example	example	NOUN
ejpam-4702	351	3	,	,	PUNCT
ejpam-4702	351	4	7x	7x	NUM
ejpam-4702	351	5	+	+	CCONJ
ejpam-4702	351	6	41y	41y	NOUN
ejpam-4702	351	7	=	=	SYM
ejpam-4702	351	8	z2	z2	PROPN
ejpam-4702	351	9	,	,	PUNCT
ejpam-4702	351	10	13x	13x	NUM
ejpam-4702	352	1	+	+	SYM
ejpam-4702	352	2	47y	47y	NOUN
ejpam-4702	352	3	=	=	SYM
ejpam-4702	352	4	z2	z2	PROPN
ejpam-4702	352	5	,	,	PUNCT
ejpam-4702	352	6	19x	19x	NUM
ejpam-4702	352	7	+	+	CCONJ
ejpam-4702	352	8	47y	47y	NOUN
ejpam-4702	352	9	=	=	SYM
ejpam-4702	352	10	z2	z2	PROPN
ejpam-4702	352	11	,	,	PUNCT
ejpam-4702	352	12	53x	53x	NUM
ejpam-4702	352	13	+	+	CCONJ
ejpam-4702	352	14	87y	87y	NUM
ejpam-4702	352	15	=	=	SYM
ejpam-4702	352	16	z2	z2	PROPN
ejpam-4702	352	17	and	and	CCONJ
ejpam-4702	352	18	67x	67x	NUM
ejpam-4702	352	19	+	+	SYM
ejpam-4702	352	20	101y	101y	NUM
ejpam-4702	352	21	=	=	SYM
ejpam-4702	352	22	z2	z2	PROPN
ejpam-4702	352	23	.	.	PUNCT
ejpam-4702	353	1	s.	s.	PROPN
ejpam-4702	353	2	tadee	tadee	PROPN
ejpam-4702	353	3	,	,	PUNCT
ejpam-4702	353	4	a.	a.	NOUN
ejpam-4702	353	5	siraworakun	siraworakun	PROPN
ejpam-4702	353	6	/	/	SYM
ejpam-4702	353	7	eur	eur	PROPN
ejpam-4702	353	8	.	.	PUNCT
ejpam-4702	354	1	j.	j.	PROPN
ejpam-4702	354	2	pure	pure	PROPN
ejpam-4702	354	3	appl	appl	PROPN
ejpam-4702	354	4	.	.	PROPN
ejpam-4702	354	5	math	math	PROPN
ejpam-4702	354	6	,	,	PUNCT
ejpam-4702	354	7	16	16	NUM
ejpam-4702	354	8	(	(	PUNCT
ejpam-4702	354	9	2	2	NUM
ejpam-4702	354	10	)	)	PUNCT
ejpam-4702	354	11	(	(	PUNCT
ejpam-4702	354	12	2023	2023	NUM
ejpam-4702	354	13	)	)	PUNCT
ejpam-4702	354	14	,	,	PUNCT
ejpam-4702	354	15	724	724	NUM
ejpam-4702	354	16	-	-	SYM
ejpam-4702	354	17	735	735	NUM
ejpam-4702	354	18	733	733	NUM
ejpam-4702	354	19	example	example	NOUN
ejpam-4702	355	1	2	2	NUM
ejpam-4702	355	2	.	.	PUNCT
ejpam-4702	355	3	let	let	VERB
ejpam-4702	355	4	q	q	NOUN
ejpam-4702	355	5	=	=	SYM
ejpam-4702	355	6	5	5	NUM
ejpam-4702	355	7	and	and	CCONJ
ejpam-4702	355	8	r	r	NOUN
ejpam-4702	355	9	=	=	SYM
ejpam-4702	355	10	2	2	NUM
ejpam-4702	355	11	.	.	PUNCT
ejpam-4702	356	1	then	then	ADV
ejpam-4702	356	2	r	r	NOUN
ejpam-4702	356	3	is	be	AUX
ejpam-4702	356	4	a	a	DET
ejpam-4702	356	5	primitive	primitive	ADJ
ejpam-4702	356	6	root	root	NOUN
ejpam-4702	356	7	of	of	ADP
ejpam-4702	356	8	q	q	NOUN
ejpam-4702	356	9	and	and	CCONJ
ejpam-4702	356	10	n1	n1	NOUN
ejpam-4702	356	11	=	=	SYM
ejpam-4702	356	12	3(5	3(5	NUM
ejpam-4702	356	13	)	)	PUNCT
ejpam-4702	356	14	+	+	CCONJ
ejpam-4702	356	15	1	1	NUM
ejpam-4702	356	16	8	8	NUM
ejpam-4702	356	17	=	=	SYM
ejpam-4702	356	18	2	2	NUM
ejpam-4702	356	19	since	since	SCONJ
ejpam-4702	356	20	5−	5−	NUM
ejpam-4702	356	21	1	1	NUM
ejpam-4702	356	22	4	4	NUM
ejpam-4702	356	23	is	be	AUX
ejpam-4702	356	24	an	an	DET
ejpam-4702	356	25	odd	odd	ADJ
ejpam-4702	356	26	number	number	NOUN
ejpam-4702	356	27	.	.	PUNCT
ejpam-4702	357	1	consider	consider	VERB
ejpam-4702	357	2	a	a	DET
ejpam-4702	357	3	prime	prime	ADJ
ejpam-4702	357	4	number	number	NOUN
ejpam-4702	357	5	p	p	NOUN
ejpam-4702	357	6	that	that	PRON
ejpam-4702	357	7	satisfies	satisfy	VERB
ejpam-4702	357	8	any	any	PRON
ejpam-4702	357	9	of	of	ADP
ejpam-4702	357	10	the	the	DET
ejpam-4702	357	11	following	follow	VERB
ejpam-4702	357	12	congruences	congruence	NOUN
ejpam-4702	357	13	:	:	PUNCT
ejpam-4702	357	14	(	(	PUNCT
ejpam-4702	357	15	i	i	NOUN
ejpam-4702	357	16	)	)	PUNCT
ejpam-4702	357	17	p	p	PROPN
ejpam-4702	357	18	≡	≡	PROPN
ejpam-4702	357	19	25	25	NUM
ejpam-4702	358	1	+	+	CCONJ
ejpam-4702	358	2	(	(	PUNCT
ejpam-4702	358	3	16	16	NUM
ejpam-4702	358	4	)	)	PUNCT
ejpam-4702	358	5	·	·	PUNCT
ejpam-4702	358	6	(	(	PUNCT
ejpam-4702	358	7	2)s2	2)s2	PROPN
ejpam-4702	358	8	(	(	PUNCT
ejpam-4702	358	9	mod	mod	PROPN
ejpam-4702	358	10	40	40	NUM
ejpam-4702	358	11	)	)	PUNCT
ejpam-4702	358	12	,	,	PUNCT
ejpam-4702	358	13	(	(	PUNCT
ejpam-4702	358	14	ii	ii	NOUN
ejpam-4702	358	15	)	)	PUNCT
ejpam-4702	358	16	p	p	PROPN
ejpam-4702	358	17	≡	≡	PROPN
ejpam-4702	358	18	−	−	PROPN
ejpam-4702	359	1	25	25	NUM
ejpam-4702	359	2	+	+	CCONJ
ejpam-4702	359	3	(	(	PUNCT
ejpam-4702	359	4	16	16	NUM
ejpam-4702	359	5	)	)	PUNCT
ejpam-4702	359	6	·	·	PUNCT
ejpam-4702	360	1	(	(	PUNCT
ejpam-4702	360	2	2)s2	2)s2	PROPN
ejpam-4702	360	3	(	(	PUNCT
ejpam-4702	360	4	mod	mod	PROPN
ejpam-4702	360	5	40	40	NUM
ejpam-4702	360	6	)	)	PUNCT
ejpam-4702	360	7	,	,	PUNCT
ejpam-4702	360	8	(	(	PUNCT
ejpam-4702	360	9	iii	iii	X
ejpam-4702	360	10	)	)	PUNCT
ejpam-4702	360	11	p	p	PROPN
ejpam-4702	360	12	≡	≡	PROPN
ejpam-4702	360	13	75	75	NUM
ejpam-4702	360	14	+	+	CCONJ
ejpam-4702	360	15	(	(	PUNCT
ejpam-4702	360	16	16	16	NUM
ejpam-4702	360	17	)	)	PUNCT
ejpam-4702	360	18	·	·	PUNCT
ejpam-4702	360	19	(	(	PUNCT
ejpam-4702	360	20	2)s1	2)s1	NUM
ejpam-4702	360	21	(	(	PUNCT
ejpam-4702	360	22	mod	mod	PROPN
ejpam-4702	360	23	40	40	NUM
ejpam-4702	360	24	)	)	PUNCT
ejpam-4702	360	25	,	,	PUNCT
ejpam-4702	360	26	or	or	CCONJ
ejpam-4702	360	27	(	(	PUNCT
ejpam-4702	360	28	iv	iv	X
ejpam-4702	360	29	)	)	PUNCT
ejpam-4702	360	30	p	p	PROPN
ejpam-4702	360	31	≡	≡	PROPN
ejpam-4702	360	32	−	−	PROPN
ejpam-4702	361	1	75	75	NUM
ejpam-4702	361	2	+	+	CCONJ
ejpam-4702	361	3	(	(	PUNCT
ejpam-4702	361	4	16	16	NUM
ejpam-4702	361	5	)	)	PUNCT
ejpam-4702	361	6	·	·	PUNCT
ejpam-4702	362	1	(	(	PUNCT
ejpam-4702	362	2	2)s1	2)s1	NUM
ejpam-4702	362	3	(	(	PUNCT
ejpam-4702	362	4	mod	mod	PROPN
ejpam-4702	362	5	40	40	NUM
ejpam-4702	362	6	)	)	PUNCT
ejpam-4702	362	7	,	,	PUNCT
ejpam-4702	362	8	where	where	SCONJ
ejpam-4702	362	9	s1	s1	PROPN
ejpam-4702	362	10	∈	∈	PROPN
ejpam-4702	362	11	{	{	PUNCT
ejpam-4702	362	12	2	2	NUM
ejpam-4702	362	13	,	,	PUNCT
ejpam-4702	362	14	4	4	NUM
ejpam-4702	362	15	}	}	PUNCT
ejpam-4702	362	16	and	and	CCONJ
ejpam-4702	362	17	s2	s2	PROPN
ejpam-4702	362	18	∈	∈	PROPN
ejpam-4702	362	19	{	{	PUNCT
ejpam-4702	362	20	1	1	NUM
ejpam-4702	362	21	,	,	PUNCT
ejpam-4702	362	22	3	3	NUM
ejpam-4702	362	23	}	}	PUNCT
ejpam-4702	362	24	.	.	PUNCT
ejpam-4702	363	1	therefore	therefore	ADV
ejpam-4702	363	2	,	,	PUNCT
ejpam-4702	363	3	p	p	PROPN
ejpam-4702	363	4	≡	≡	PROPN
ejpam-4702	363	5	±7,±11,±17,±19	±7,±11,±17,±19	PROPN
ejpam-4702	363	6	(	(	PUNCT
ejpam-4702	363	7	mod	mod	NOUN
ejpam-4702	363	8	40	40	NUM
ejpam-4702	363	9	)	)	PUNCT
ejpam-4702	363	10	.	.	PUNCT
ejpam-4702	364	1	by	by	ADP
ejpam-4702	364	2	theorem	theorem	NOUN
ejpam-4702	364	3	15	15	NUM
ejpam-4702	364	4	,	,	PUNCT
ejpam-4702	364	5	we	we	PRON
ejpam-4702	364	6	obtain	obtain	VERB
ejpam-4702	364	7	that	that	SCONJ
ejpam-4702	364	8	the	the	DET
ejpam-4702	364	9	diophantine	diophantine	NOUN
ejpam-4702	364	10	equation	equation	NOUN
ejpam-4702	364	11	px	px	X
ejpam-4702	364	12	+	+	CCONJ
ejpam-4702	364	13	(	(	PUNCT
ejpam-4702	364	14	p+	p+	NOUN
ejpam-4702	364	15	10)y	10)y	NUM
ejpam-4702	364	16	=	=	SYM
ejpam-4702	364	17	z2	z2	PROPN
ejpam-4702	364	18	has	have	VERB
ejpam-4702	364	19	no	no	DET
ejpam-4702	364	20	positive	positive	ADJ
ejpam-4702	364	21	integer	integer	NOUN
ejpam-4702	364	22	solution	solution	NOUN
ejpam-4702	364	23	.	.	PUNCT
ejpam-4702	365	1	for	for	ADP
ejpam-4702	365	2	example	example	NOUN
ejpam-4702	365	3	,	,	PUNCT
ejpam-4702	365	4	7x+17y	7x+17y	NUM
ejpam-4702	365	5	=	=	SYM
ejpam-4702	365	6	z2	z2	PROPN
ejpam-4702	365	7	,	,	PUNCT
ejpam-4702	365	8	19x+29y	19x+29y	NUM
ejpam-4702	365	9	=	=	SYM
ejpam-4702	365	10	z2	z2	PROPN
ejpam-4702	365	11	(	(	PUNCT
ejpam-4702	365	12	burshtein	burshtein	ADV
ejpam-4702	365	13	[	[	X
ejpam-4702	365	14	3	3	NUM
ejpam-4702	365	15	]	]	NUM
ejpam-4702	365	16	)	)	PUNCT
ejpam-4702	365	17	,	,	PUNCT
ejpam-4702	365	18	61x+71y	61x+71y	NUM
ejpam-4702	365	19	=	=	SYM
ejpam-4702	365	20	z2	z2	PROPN
ejpam-4702	365	21	(	(	PUNCT
ejpam-4702	365	22	kumar	kumar	PROPN
ejpam-4702	365	23	[	[	X
ejpam-4702	365	24	5	5	NUM
ejpam-4702	365	25	]	]	NUM
ejpam-4702	365	26	)	)	PUNCT
ejpam-4702	365	27	,	,	PUNCT
ejpam-4702	365	28	73x	73x	X
ejpam-4702	365	29	+	+	CCONJ
ejpam-4702	365	30	83y	83y	NOUN
ejpam-4702	365	31	=	=	SYM
ejpam-4702	365	32	z2	z2	NOUN
ejpam-4702	365	33	and	and	CCONJ
ejpam-4702	365	34	97x	97x	NOUN
ejpam-4702	365	35	+	+	CCONJ
ejpam-4702	365	36	107y	107y	PROPN
ejpam-4702	365	37	=	=	SYM
ejpam-4702	365	38	z2	z2	PROPN
ejpam-4702	365	39	.	.	PUNCT
ejpam-4702	366	1	from	from	ADP
ejpam-4702	366	2	theorems	theorems	PROPN
ejpam-4702	366	3	13	13	NUM
ejpam-4702	366	4	and	and	CCONJ
ejpam-4702	366	5	14	14	NUM
ejpam-4702	366	6	,	,	PUNCT
ejpam-4702	366	7	the	the	DET
ejpam-4702	366	8	forms	form	NOUN
ejpam-4702	366	9	of	of	ADP
ejpam-4702	366	10	odd	odd	ADJ
ejpam-4702	366	11	prime	prime	ADJ
ejpam-4702	366	12	number	number	NOUN
ejpam-4702	366	13	p	p	NOUN
ejpam-4702	366	14	can	can	AUX
ejpam-4702	366	15	be	be	AUX
ejpam-4702	366	16	obtained	obtain	VERB
ejpam-4702	366	17	,	,	PUNCT
ejpam-4702	366	18	when	when	SCONJ
ejpam-4702	366	19	q	q	PROPN
ejpam-4702	366	20	≡	≡	PROPN
ejpam-4702	366	21	3	3	NUM
ejpam-4702	366	22	(	(	PUNCT
ejpam-4702	366	23	mod	mod	NOUN
ejpam-4702	366	24	4	4	NUM
ejpam-4702	366	25	)	)	PUNCT
ejpam-4702	366	26	.	.	PUNCT
ejpam-4702	367	1	theorem	theorem	VERB
ejpam-4702	367	2	16	16	NUM
ejpam-4702	367	3	.	.	PUNCT
ejpam-4702	368	1	let	let	VERB
ejpam-4702	368	2	p	p	NOUN
ejpam-4702	368	3	and	and	CCONJ
ejpam-4702	368	4	q	q	NOUN
ejpam-4702	368	5	be	be	AUX
ejpam-4702	368	6	distinct	distinct	ADJ
ejpam-4702	368	7	prime	prime	ADJ
ejpam-4702	368	8	numbers	number	NOUN
ejpam-4702	368	9	such	such	ADJ
ejpam-4702	368	10	that	that	PRON
ejpam-4702	368	11	q	q	PROPN
ejpam-4702	368	12	≡	≡	PROPN
ejpam-4702	368	13	3	3	NUM
ejpam-4702	368	14	(	(	PUNCT
ejpam-4702	368	15	mod	mod	NOUN
ejpam-4702	368	16	4	4	NUM
ejpam-4702	368	17	)	)	PUNCT
ejpam-4702	368	18	and	and	CCONJ
ejpam-4702	368	19	2q	2q	NUM
ejpam-4702	368	20	̸≡	̸≡	NOUN
ejpam-4702	368	21	1	1	NUM
ejpam-4702	368	22	(	(	PUNCT
ejpam-4702	368	23	mod	mod	PROPN
ejpam-4702	368	24	p	p	NOUN
ejpam-4702	368	25	)	)	PUNCT
ejpam-4702	368	26	.	.	PUNCT
ejpam-4702	369	1	if	if	SCONJ
ejpam-4702	369	2	p	p	NOUN
ejpam-4702	369	3	satisfies	satisfy	VERB
ejpam-4702	369	4	any	any	PRON
ejpam-4702	369	5	of	of	ADP
ejpam-4702	369	6	the	the	DET
ejpam-4702	369	7	following	follow	VERB
ejpam-4702	369	8	conditions	condition	NOUN
ejpam-4702	369	9	:	:	PUNCT
ejpam-4702	369	10	(	(	PUNCT
ejpam-4702	369	11	i	i	NOUN
ejpam-4702	369	12	)	)	PUNCT
ejpam-4702	369	13	p	p	PROPN
ejpam-4702	369	14	≡	≡	PROPN
ejpam-4702	369	15	q2	q2	NOUN
ejpam-4702	369	16	+	+	CCONJ
ejpam-4702	370	1	32n0n1r	32n0n1r	CCONJ
ejpam-4702	370	2	s2	s2	PROPN
ejpam-4702	370	3	(	(	PUNCT
ejpam-4702	370	4	mod	mod	PROPN
ejpam-4702	370	5	8q	8q	PROPN
ejpam-4702	370	6	)	)	PUNCT
ejpam-4702	370	7	,	,	PUNCT
ejpam-4702	370	8	(	(	PUNCT
ejpam-4702	370	9	ii	ii	NOUN
ejpam-4702	370	10	)	)	PUNCT
ejpam-4702	370	11	p	p	PROPN
ejpam-4702	370	12	≡	≡	PROPN
ejpam-4702	370	13	−	−	PROPN
ejpam-4702	370	14	q2	q2	PROPN
ejpam-4702	370	15	+	+	CCONJ
ejpam-4702	370	16	32n0n1r	32n0n1r	CCONJ
ejpam-4702	370	17	s1	s1	NOUN
ejpam-4702	370	18	(	(	PUNCT
ejpam-4702	370	19	mod	mod	PROPN
ejpam-4702	370	20	8q	8q	PROPN
ejpam-4702	370	21	)	)	PUNCT
ejpam-4702	370	22	,	,	PUNCT
ejpam-4702	370	23	(	(	PUNCT
ejpam-4702	370	24	iii	iii	X
ejpam-4702	370	25	)	)	PUNCT
ejpam-4702	370	26	p	p	PROPN
ejpam-4702	370	27	≡	≡	PROPN
ejpam-4702	370	28	3q2	3q2	NUM
ejpam-4702	370	29	+	+	CCONJ
ejpam-4702	370	30	32n0n1r	32n0n1r	CCONJ
ejpam-4702	370	31	s2	s2	PROPN
ejpam-4702	370	32	(	(	PUNCT
ejpam-4702	370	33	mod	mod	PROPN
ejpam-4702	370	34	8q	8q	NUM
ejpam-4702	370	35	)	)	PUNCT
ejpam-4702	370	36	,	,	PUNCT
ejpam-4702	370	37	or	or	CCONJ
ejpam-4702	370	38	(	(	PUNCT
ejpam-4702	370	39	iv	iv	X
ejpam-4702	370	40	)	)	PUNCT
ejpam-4702	370	41	p	p	PROPN
ejpam-4702	370	42	≡	≡	PROPN
ejpam-4702	370	43	−	−	PROPN
ejpam-4702	370	44	3q2	3q2	NUM
ejpam-4702	370	45	+	+	CCONJ
ejpam-4702	370	46	32n0n1r	32n0n1r	CCONJ
ejpam-4702	370	47	s1	s1	NOUN
ejpam-4702	370	48	(	(	PUNCT
ejpam-4702	370	49	mod	mod	PROPN
ejpam-4702	370	50	8q	8q	PROPN
ejpam-4702	370	51	)	)	PUNCT
ejpam-4702	370	52	,	,	PUNCT
ejpam-4702	370	53	where	where	SCONJ
ejpam-4702	370	54	s1	s1	PROPN
ejpam-4702	370	55	∈	∈	PROPN
ejpam-4702	370	56	{	{	PUNCT
ejpam-4702	370	57	2	2	NUM
ejpam-4702	370	58	,	,	PUNCT
ejpam-4702	370	59	4	4	NUM
ejpam-4702	370	60	,	,	PUNCT
ejpam-4702	370	61	6	6	NUM
ejpam-4702	370	62	,	,	PUNCT
ejpam-4702	370	63	.	.	PUNCT
ejpam-4702	370	64	.	.	PUNCT
ejpam-4702	370	65	.	.	PUNCT
ejpam-4702	371	1	,	,	PUNCT
ejpam-4702	371	2	q	q	NOUN
ejpam-4702	371	3	−	−	NOUN
ejpam-4702	371	4	1	1	NUM
ejpam-4702	371	5	}	}	PUNCT
ejpam-4702	371	6	,	,	PUNCT
ejpam-4702	371	7	s2	s2	PROPN
ejpam-4702	371	8	∈	∈	PROPN
ejpam-4702	371	9	{	{	PUNCT
ejpam-4702	371	10	1	1	NUM
ejpam-4702	371	11	,	,	PUNCT
ejpam-4702	371	12	3	3	NUM
ejpam-4702	371	13	,	,	PUNCT
ejpam-4702	371	14	5	5	NUM
ejpam-4702	371	15	,	,	PUNCT
ejpam-4702	371	16	.	.	PUNCT
ejpam-4702	371	17	.	.	PUNCT
ejpam-4702	372	1	.	.	PUNCT
ejpam-4702	373	1	,	,	PUNCT
ejpam-4702	373	2	q	q	X
ejpam-4702	374	1	−	−	NOUN
ejpam-4702	374	2	2	2	NUM
ejpam-4702	374	3	}	}	PUNCT
ejpam-4702	374	4	,	,	PUNCT
ejpam-4702	374	5	r	r	NOUN
ejpam-4702	374	6	is	be	AUX
ejpam-4702	374	7	a	a	DET
ejpam-4702	374	8	primitive	primitive	ADJ
ejpam-4702	374	9	root	root	NOUN
ejpam-4702	374	10	modulo	modulo	NOUN
ejpam-4702	374	11	q	q	X
ejpam-4702	374	12	,	,	PUNCT
ejpam-4702	374	13	n0	n0	X
ejpam-4702	374	14	=	=	PUNCT
ejpam-4702	374	15	q	q	PROPN
ejpam-4702	375	1	+	+	NUM
ejpam-4702	375	2	1	1	NUM
ejpam-4702	375	3	4	4	NUM
ejpam-4702	375	4	and	and	CCONJ
ejpam-4702	375	5	if	if	SCONJ
ejpam-4702	375	6	q	q	PUNCT
ejpam-4702	375	7	−	−	NOUN
ejpam-4702	375	8	3	3	NUM
ejpam-4702	375	9	4	4	NUM
ejpam-4702	375	10	is	be	AUX
ejpam-4702	375	11	an	an	DET
ejpam-4702	375	12	even	even	ADJ
ejpam-4702	375	13	number	number	NOUN
ejpam-4702	375	14	,	,	PUNCT
ejpam-4702	375	15	then	then	ADV
ejpam-4702	375	16	n1	n1	PROPN
ejpam-4702	375	17	=	=	SYM
ejpam-4702	375	18	5q	5q	NOUN
ejpam-4702	375	19	+	+	CCONJ
ejpam-4702	375	20	1	1	NUM
ejpam-4702	375	21	8	8	NUM
ejpam-4702	375	22	,	,	PUNCT
ejpam-4702	375	23	and	and	CCONJ
ejpam-4702	375	24	if	if	SCONJ
ejpam-4702	375	25	otherwise	otherwise	ADV
ejpam-4702	375	26	,	,	PUNCT
ejpam-4702	375	27	then	then	ADV
ejpam-4702	375	28	n1	n1	PROPN
ejpam-4702	375	29	=	=	SYM
ejpam-4702	375	30	q	q	PROPN
ejpam-4702	376	1	+	+	NUM
ejpam-4702	376	2	1	1	NUM
ejpam-4702	376	3	8	8	NUM
ejpam-4702	376	4	.	.	PUNCT
ejpam-4702	377	1	then	then	ADV
ejpam-4702	377	2	,	,	PUNCT
ejpam-4702	377	3	the	the	DET
ejpam-4702	377	4	diophantine	diophantine	NOUN
ejpam-4702	377	5	equation	equation	NOUN
ejpam-4702	377	6	px	px	X
ejpam-4702	377	7	+	+	CCONJ
ejpam-4702	377	8	(	(	PUNCT
ejpam-4702	377	9	p	p	X
ejpam-4702	377	10	+	+	CCONJ
ejpam-4702	377	11	2q)y	2q)y	NUM
ejpam-4702	377	12	=	=	SYM
ejpam-4702	377	13	z2	z2	PROPN
ejpam-4702	377	14	has	have	VERB
ejpam-4702	377	15	no	no	DET
ejpam-4702	377	16	positive	positive	ADJ
ejpam-4702	377	17	integer	integer	NOUN
ejpam-4702	377	18	solution	solution	NOUN
ejpam-4702	377	19	.	.	PUNCT
ejpam-4702	378	1	example	example	NOUN
ejpam-4702	379	1	3	3	X
ejpam-4702	379	2	.	.	PUNCT
ejpam-4702	379	3	let	let	VERB
ejpam-4702	379	4	q	q	NOUN
ejpam-4702	380	1	=	=	SYM
ejpam-4702	380	2	3	3	NUM
ejpam-4702	380	3	and	and	CCONJ
ejpam-4702	380	4	r	r	NOUN
ejpam-4702	380	5	=	=	SYM
ejpam-4702	380	6	2	2	NUM
ejpam-4702	380	7	.	.	PUNCT
ejpam-4702	381	1	then	then	ADV
ejpam-4702	381	2	r	r	NOUN
ejpam-4702	381	3	is	be	AUX
ejpam-4702	381	4	a	a	DET
ejpam-4702	381	5	primitive	primitive	ADJ
ejpam-4702	381	6	root	root	NOUN
ejpam-4702	381	7	of	of	ADP
ejpam-4702	381	8	q	q	NOUN
ejpam-4702	381	9	,	,	PUNCT
ejpam-4702	381	10	s1	s1	NOUN
ejpam-4702	381	11	=	=	SYM
ejpam-4702	381	12	2	2	NUM
ejpam-4702	381	13	,	,	PUNCT
ejpam-4702	381	14	s2	s2	X
ejpam-4702	381	15	=	=	SYM
ejpam-4702	381	16	1	1	NUM
ejpam-4702	381	17	,	,	PUNCT
ejpam-4702	381	18	n0	n0	X
ejpam-4702	381	19	=	=	SYM
ejpam-4702	381	20	3	3	NUM
ejpam-4702	381	21	+	+	CCONJ
ejpam-4702	381	22	1	1	NUM
ejpam-4702	381	23	4	4	NUM
ejpam-4702	381	24	=	=	SYM
ejpam-4702	381	25	1	1	NUM
ejpam-4702	381	26	and	and	CCONJ
ejpam-4702	381	27	n1	n1	NOUN
ejpam-4702	381	28	=	=	SYM
ejpam-4702	381	29	5(3	5(3	NUM
ejpam-4702	381	30	)	)	PUNCT
ejpam-4702	381	31	+	+	CCONJ
ejpam-4702	381	32	1	1	NUM
ejpam-4702	381	33	8	8	NUM
ejpam-4702	381	34	=	=	SYM
ejpam-4702	381	35	2	2	NUM
ejpam-4702	381	36	since	since	SCONJ
ejpam-4702	381	37	3−	3−	NUM
ejpam-4702	381	38	3	3	NUM
ejpam-4702	381	39	4	4	NUM
ejpam-4702	381	40	is	be	AUX
ejpam-4702	381	41	an	an	DET
ejpam-4702	381	42	even	even	ADJ
ejpam-4702	381	43	number	number	NOUN
ejpam-4702	381	44	.	.	PUNCT
ejpam-4702	382	1	consider	consider	VERB
ejpam-4702	382	2	a	a	DET
ejpam-4702	382	3	prime	prime	ADJ
ejpam-4702	382	4	number	number	NOUN
ejpam-4702	382	5	p	p	NOUN
ejpam-4702	382	6	that	that	PRON
ejpam-4702	382	7	satisfies	satisfy	VERB
ejpam-4702	382	8	any	any	PRON
ejpam-4702	382	9	of	of	ADP
ejpam-4702	382	10	the	the	DET
ejpam-4702	382	11	following	follow	VERB
ejpam-4702	382	12	congruences	congruence	NOUN
ejpam-4702	382	13	:	:	PUNCT
ejpam-4702	382	14	(	(	PUNCT
ejpam-4702	382	15	i	i	NOUN
ejpam-4702	382	16	)	)	PUNCT
ejpam-4702	382	17	p	p	PROPN
ejpam-4702	382	18	≡	≡	PROPN
ejpam-4702	382	19	9	9	NUM
ejpam-4702	382	20	+	+	CCONJ
ejpam-4702	382	21	(	(	PUNCT
ejpam-4702	382	22	64	64	NUM
ejpam-4702	382	23	)	)	PUNCT
ejpam-4702	382	24	·	·	PUNCT
ejpam-4702	383	1	(	(	PUNCT
ejpam-4702	383	2	2)1	2)1	NUM
ejpam-4702	383	3	(	(	PUNCT
ejpam-4702	383	4	mod	mod	PROPN
ejpam-4702	383	5	24	24	NUM
ejpam-4702	383	6	)	)	PUNCT
ejpam-4702	383	7	,	,	PUNCT
ejpam-4702	383	8	(	(	PUNCT
ejpam-4702	383	9	ii	ii	NOUN
ejpam-4702	383	10	)	)	PUNCT
ejpam-4702	383	11	p	p	PROPN
ejpam-4702	383	12	≡	≡	PROPN
ejpam-4702	383	13	−	−	ADP
ejpam-4702	383	14	9	9	NUM
ejpam-4702	383	15	+	+	CCONJ
ejpam-4702	383	16	(	(	PUNCT
ejpam-4702	383	17	64	64	NUM
ejpam-4702	383	18	)	)	PUNCT
ejpam-4702	383	19	·	·	PUNCT
ejpam-4702	383	20	(	(	PUNCT
ejpam-4702	383	21	2)2	2)2	NUM
ejpam-4702	383	22	(	(	PUNCT
ejpam-4702	383	23	mod	mod	NOUN
ejpam-4702	383	24	24	24	NUM
ejpam-4702	383	25	)	)	PUNCT
ejpam-4702	383	26	,	,	PUNCT
ejpam-4702	383	27	(	(	PUNCT
ejpam-4702	383	28	iii	iii	X
ejpam-4702	383	29	)	)	PUNCT
ejpam-4702	383	30	p	p	PROPN
ejpam-4702	383	31	≡	≡	PROPN
ejpam-4702	383	32	27	27	NUM
ejpam-4702	383	33	+	+	CCONJ
ejpam-4702	383	34	(	(	PUNCT
ejpam-4702	383	35	64	64	NUM
ejpam-4702	383	36	)	)	PUNCT
ejpam-4702	383	37	·	·	PUNCT
ejpam-4702	384	1	(	(	PUNCT
ejpam-4702	384	2	2)1	2)1	NUM
ejpam-4702	384	3	(	(	PUNCT
ejpam-4702	384	4	mod	mod	PROPN
ejpam-4702	384	5	24	24	NUM
ejpam-4702	384	6	)	)	PUNCT
ejpam-4702	384	7	,	,	PUNCT
ejpam-4702	384	8	or	or	CCONJ
ejpam-4702	384	9	references	reference	NOUN
ejpam-4702	384	10	734	734	NUM
ejpam-4702	384	11	(	(	PUNCT
ejpam-4702	384	12	iv	iv	X
ejpam-4702	384	13	)	)	PUNCT
ejpam-4702	384	14	p	p	PROPN
ejpam-4702	384	15	≡	≡	PROPN
ejpam-4702	384	16	−	−	PROPN
ejpam-4702	384	17	27	27	NUM
ejpam-4702	384	18	+	+	CCONJ
ejpam-4702	384	19	(	(	PUNCT
ejpam-4702	384	20	64	64	NUM
ejpam-4702	384	21	)	)	PUNCT
ejpam-4702	384	22	·	·	PUNCT
ejpam-4702	384	23	(	(	PUNCT
ejpam-4702	384	24	2)2	2)2	NUM
ejpam-4702	384	25	(	(	PUNCT
ejpam-4702	384	26	mod	mod	NOUN
ejpam-4702	384	27	24	24	NUM
ejpam-4702	384	28	)	)	PUNCT
ejpam-4702	384	29	.	.	PUNCT
ejpam-4702	385	1	thus	thus	ADV
ejpam-4702	385	2	,	,	PUNCT
ejpam-4702	385	3	p	p	PROPN
ejpam-4702	385	4	≡	≡	PROPN
ejpam-4702	385	5	±7,±11	±7,±11	PROPN
ejpam-4702	385	6	(	(	PUNCT
ejpam-4702	385	7	mod	mod	PROPN
ejpam-4702	385	8	24	24	NUM
ejpam-4702	385	9	)	)	PUNCT
ejpam-4702	385	10	.	.	PUNCT
ejpam-4702	386	1	by	by	ADP
ejpam-4702	386	2	theorem	theorem	NOUN
ejpam-4702	386	3	16	16	NUM
ejpam-4702	386	4	,	,	PUNCT
ejpam-4702	386	5	we	we	PRON
ejpam-4702	386	6	obtain	obtain	VERB
ejpam-4702	386	7	that	that	SCONJ
ejpam-4702	386	8	the	the	DET
ejpam-4702	386	9	diophantine	diophantine	NOUN
ejpam-4702	386	10	equation	equation	NOUN
ejpam-4702	386	11	px	px	X
ejpam-4702	387	1	+	+	CCONJ
ejpam-4702	387	2	(	(	PUNCT
ejpam-4702	387	3	p	p	NOUN
ejpam-4702	387	4	+	+	NOUN
ejpam-4702	387	5	6)y	6)y	NOUN
ejpam-4702	387	6	=	=	SYM
ejpam-4702	387	7	z2	z2	PROPN
ejpam-4702	387	8	has	have	VERB
ejpam-4702	387	9	no	no	DET
ejpam-4702	387	10	positive	positive	ADJ
ejpam-4702	387	11	integer	integer	NOUN
ejpam-4702	387	12	solution	solution	NOUN
ejpam-4702	387	13	.	.	PUNCT
ejpam-4702	388	1	for	for	ADP
ejpam-4702	388	2	example	example	NOUN
ejpam-4702	388	3	,	,	PUNCT
ejpam-4702	388	4	7x	7x	NUM
ejpam-4702	388	5	+	+	CCONJ
ejpam-4702	388	6	13y	13y	NOUN
ejpam-4702	388	7	=	=	SYM
ejpam-4702	388	8	z2	z2	PROPN
ejpam-4702	388	9	,	,	PUNCT
ejpam-4702	388	10	11x	11x	NOUN
ejpam-4702	388	11	+	+	ADJ
ejpam-4702	388	12	17y	17y	NOUN
ejpam-4702	388	13	=	=	SYM
ejpam-4702	388	14	z2	z2	PROPN
ejpam-4702	388	15	,	,	PUNCT
ejpam-4702	388	16	13x	13x	NUM
ejpam-4702	389	1	+	+	CCONJ
ejpam-4702	389	2	19y	19y	NOUN
ejpam-4702	389	3	=	=	SYM
ejpam-4702	389	4	z2	z2	PROPN
ejpam-4702	389	5	,	,	PUNCT
ejpam-4702	389	6	17x	17x	NOUN
ejpam-4702	389	7	+	+	SYM
ejpam-4702	389	8	23y	23y	NUM
ejpam-4702	389	9	=	=	SYM
ejpam-4702	389	10	z2	z2	PROPN
ejpam-4702	389	11	and	and	CCONJ
ejpam-4702	389	12	61x	61x	NUM
ejpam-4702	389	13	+	+	CCONJ
ejpam-4702	389	14	67y	67y	NOUN
ejpam-4702	389	15	=	=	SYM
ejpam-4702	389	16	z2(kumar	z2(kumar	NUM
ejpam-4702	390	1	[	[	X
ejpam-4702	390	2	4	4	NUM
ejpam-4702	390	3	]	]	NUM
ejpam-4702	390	4	)	)	PUNCT
ejpam-4702	390	5	.	.	PUNCT
ejpam-4702	391	1	example	example	NOUN
ejpam-4702	392	1	4	4	X
ejpam-4702	392	2	.	.	PUNCT
ejpam-4702	392	3	let	let	VERB
ejpam-4702	392	4	q	q	NOUN
ejpam-4702	393	1	=	=	SYM
ejpam-4702	393	2	7	7	NUM
ejpam-4702	393	3	and	and	CCONJ
ejpam-4702	393	4	r	r	NOUN
ejpam-4702	393	5	=	=	SYM
ejpam-4702	393	6	3	3	X
ejpam-4702	393	7	.	.	PUNCT
ejpam-4702	394	1	then	then	ADV
ejpam-4702	394	2	r	r	NOUN
ejpam-4702	394	3	is	be	AUX
ejpam-4702	394	4	a	a	DET
ejpam-4702	394	5	primitive	primitive	ADJ
ejpam-4702	394	6	root	root	NOUN
ejpam-4702	394	7	of	of	ADP
ejpam-4702	394	8	q	q	NOUN
ejpam-4702	394	9	,	,	PUNCT
ejpam-4702	394	10	n0	n0	X
ejpam-4702	394	11	=	=	SYM
ejpam-4702	394	12	7	7	NUM
ejpam-4702	394	13	+	+	SYM
ejpam-4702	394	14	1	1	NUM
ejpam-4702	394	15	4	4	NUM
ejpam-4702	394	16	=	=	SYM
ejpam-4702	394	17	2	2	NUM
ejpam-4702	394	18	and	and	CCONJ
ejpam-4702	394	19	n1	n1	NOUN
ejpam-4702	394	20	=	=	SYM
ejpam-4702	394	21	7	7	NUM
ejpam-4702	394	22	+	+	CCONJ
ejpam-4702	394	23	1	1	NUM
ejpam-4702	394	24	8	8	NUM
ejpam-4702	394	25	=	=	SYM
ejpam-4702	394	26	1	1	NUM
ejpam-4702	394	27	since	since	SCONJ
ejpam-4702	394	28	7−	7−	NUM
ejpam-4702	394	29	3	3	NUM
ejpam-4702	394	30	4	4	NUM
ejpam-4702	394	31	is	be	AUX
ejpam-4702	394	32	an	an	DET
ejpam-4702	394	33	odd	odd	ADJ
ejpam-4702	394	34	number	number	NOUN
ejpam-4702	394	35	.	.	PUNCT
ejpam-4702	395	1	consider	consider	VERB
ejpam-4702	395	2	a	a	DET
ejpam-4702	395	3	prime	prime	ADJ
ejpam-4702	395	4	number	number	NOUN
ejpam-4702	395	5	p	p	NOUN
ejpam-4702	395	6	that	that	PRON
ejpam-4702	395	7	satisfies	satisfy	VERB
ejpam-4702	395	8	any	any	PRON
ejpam-4702	395	9	of	of	ADP
ejpam-4702	395	10	the	the	DET
ejpam-4702	395	11	following	follow	VERB
ejpam-4702	395	12	congruences	congruence	NOUN
ejpam-4702	395	13	:	:	PUNCT
ejpam-4702	395	14	(	(	PUNCT
ejpam-4702	395	15	i	i	NOUN
ejpam-4702	395	16	)	)	PUNCT
ejpam-4702	395	17	p	p	PROPN
ejpam-4702	395	18	≡	≡	PROPN
ejpam-4702	395	19	49	49	NUM
ejpam-4702	395	20	+	+	CCONJ
ejpam-4702	395	21	64(3s2	64(3s2	NUM
ejpam-4702	395	22	)	)	PUNCT
ejpam-4702	395	23	(	(	PUNCT
ejpam-4702	395	24	mod	mod	PROPN
ejpam-4702	395	25	56	56	NUM
ejpam-4702	395	26	)	)	PUNCT
ejpam-4702	395	27	,	,	PUNCT
ejpam-4702	395	28	(	(	PUNCT
ejpam-4702	395	29	ii	ii	NOUN
ejpam-4702	395	30	)	)	PUNCT
ejpam-4702	395	31	p	p	PROPN
ejpam-4702	395	32	≡	≡	PROPN
ejpam-4702	395	33	−	−	PROPN
ejpam-4702	395	34	49	49	NUM
ejpam-4702	396	1	+	+	CCONJ
ejpam-4702	396	2	64(3s1	64(3s1	NUM
ejpam-4702	396	3	)	)	PUNCT
ejpam-4702	396	4	(	(	PUNCT
ejpam-4702	396	5	mod	mod	PROPN
ejpam-4702	396	6	56	56	NUM
ejpam-4702	396	7	)	)	PUNCT
ejpam-4702	396	8	,	,	PUNCT
ejpam-4702	396	9	(	(	PUNCT
ejpam-4702	396	10	iii	iii	X
ejpam-4702	396	11	)	)	PUNCT
ejpam-4702	396	12	p	p	PROPN
ejpam-4702	396	13	≡	≡	PROPN
ejpam-4702	396	14	147	147	NUM
ejpam-4702	396	15	+	+	CCONJ
ejpam-4702	396	16	64(3s2	64(3s2	NUM
ejpam-4702	396	17	)	)	PUNCT
ejpam-4702	396	18	(	(	PUNCT
ejpam-4702	396	19	mod	mod	NOUN
ejpam-4702	396	20	56	56	NUM
ejpam-4702	396	21	)	)	PUNCT
ejpam-4702	396	22	,	,	PUNCT
ejpam-4702	396	23	or	or	CCONJ
ejpam-4702	396	24	(	(	PUNCT
ejpam-4702	396	25	iv	iv	X
ejpam-4702	396	26	)	)	PUNCT
ejpam-4702	396	27	p	p	PROPN
ejpam-4702	396	28	≡	≡	PROPN
ejpam-4702	396	29	−	−	PROPN
ejpam-4702	396	30	147	147	NUM
ejpam-4702	397	1	+	+	CCONJ
ejpam-4702	397	2	64(3s1	64(3s1	NUM
ejpam-4702	397	3	)	)	PUNCT
ejpam-4702	397	4	(	(	PUNCT
ejpam-4702	397	5	mod	mod	PROPN
ejpam-4702	397	6	56	56	NUM
ejpam-4702	397	7	)	)	PUNCT
ejpam-4702	397	8	,	,	PUNCT
ejpam-4702	397	9	where	where	SCONJ
ejpam-4702	397	10	s1	s1	PROPN
ejpam-4702	397	11	∈	∈	PROPN
ejpam-4702	397	12	{	{	PUNCT
ejpam-4702	397	13	2	2	NUM
ejpam-4702	397	14	,	,	PUNCT
ejpam-4702	397	15	4	4	NUM
ejpam-4702	397	16	,	,	PUNCT
ejpam-4702	397	17	6	6	NUM
ejpam-4702	397	18	}	}	PUNCT
ejpam-4702	397	19	and	and	CCONJ
ejpam-4702	397	20	s2	s2	PROPN
ejpam-4702	397	21	∈	∈	PROPN
ejpam-4702	397	22	{	{	PUNCT
ejpam-4702	397	23	1	1	NUM
ejpam-4702	397	24	,	,	PUNCT
ejpam-4702	397	25	3	3	NUM
ejpam-4702	397	26	,	,	PUNCT
ejpam-4702	397	27	5	5	NUM
ejpam-4702	397	28	}	}	PUNCT
ejpam-4702	397	29	.	.	PUNCT
ejpam-4702	398	1	thus	thus	ADV
ejpam-4702	398	2	,	,	PUNCT
ejpam-4702	398	3	p	p	PROPN
ejpam-4702	398	4	≡	≡	PROPN
ejpam-4702	398	5	±3,±15,±17,±19,±23,±27	±3,±15,±17,±19,±23,±27	PROPN
ejpam-4702	398	6	(	(	PUNCT
ejpam-4702	398	7	mod	mod	PROPN
ejpam-4702	398	8	56	56	NUM
ejpam-4702	398	9	)	)	PUNCT
ejpam-4702	398	10	.	.	PUNCT
ejpam-4702	399	1	by	by	ADP
ejpam-4702	399	2	theorem	theorem	NOUN
ejpam-4702	399	3	16	16	NUM
ejpam-4702	399	4	,	,	PUNCT
ejpam-4702	399	5	we	we	PRON
ejpam-4702	399	6	obtain	obtain	VERB
ejpam-4702	399	7	that	that	SCONJ
ejpam-4702	399	8	the	the	DET
ejpam-4702	399	9	diophantine	diophantine	NOUN
ejpam-4702	399	10	equation	equation	NOUN
ejpam-4702	399	11	px	px	X
ejpam-4702	400	1	+	+	CCONJ
ejpam-4702	400	2	(	(	PUNCT
ejpam-4702	400	3	p	p	X
ejpam-4702	400	4	+	+	NOUN
ejpam-4702	400	5	14)y	14)y	NUM
ejpam-4702	400	6	=	=	SYM
ejpam-4702	400	7	z2	z2	PROPN
ejpam-4702	400	8	has	have	VERB
ejpam-4702	400	9	no	no	DET
ejpam-4702	400	10	positive	positive	ADJ
ejpam-4702	400	11	integer	integer	NOUN
ejpam-4702	400	12	solution	solution	NOUN
ejpam-4702	400	13	.	.	PUNCT
ejpam-4702	401	1	for	for	ADP
ejpam-4702	401	2	example	example	NOUN
ejpam-4702	401	3	,	,	PUNCT
ejpam-4702	401	4	3x	3x	NUM
ejpam-4702	401	5	+	+	SYM
ejpam-4702	401	6	17y	17y	NOUN
ejpam-4702	401	7	=	=	SYM
ejpam-4702	401	8	z2(sroysang	z2(sroysang	X
ejpam-4702	402	1	[	[	X
ejpam-4702	402	2	12	12	NUM
ejpam-4702	402	3	]	]	NUM
ejpam-4702	402	4	)	)	PUNCT
ejpam-4702	402	5	,	,	PUNCT
ejpam-4702	402	6	17x	17x	NUM
ejpam-4702	402	7	+	+	CCONJ
ejpam-4702	402	8	31y	31y	NUM
ejpam-4702	402	9	=	=	SYM
ejpam-4702	402	10	z2	z2	PROPN
ejpam-4702	402	11	,	,	PUNCT
ejpam-4702	402	12	23x	23x	X
ejpam-4702	402	13	+	+	SYM
ejpam-4702	402	14	37y	37y	NUM
ejpam-4702	402	15	=	=	SYM
ejpam-4702	402	16	z2	z2	PROPN
ejpam-4702	402	17	,	,	PUNCT
ejpam-4702	402	18	29x	29x	NUM
ejpam-4702	402	19	+	+	NUM
ejpam-4702	402	20	43y	43y	NOUN
ejpam-4702	402	21	=	=	SYM
ejpam-4702	402	22	z2	z2	NOUN
ejpam-4702	402	23	,	,	PUNCT
ejpam-4702	402	24	and	and	CCONJ
ejpam-4702	402	25	53x	53x	NUM
ejpam-4702	402	26	+	+	CCONJ
ejpam-4702	402	27	67y	67y	NOUN
ejpam-4702	402	28	=	=	SYM
ejpam-4702	402	29	z2	z2	PROPN
ejpam-4702	402	30	.	.	PUNCT
ejpam-4702	403	1	acknowledgements	acknowledgement	NOUN
ejpam-4702	403	2	this	this	DET
ejpam-4702	403	3	work	work	NOUN
ejpam-4702	403	4	was	be	AUX
ejpam-4702	403	5	supported	support	VERB
ejpam-4702	403	6	by	by	ADP
ejpam-4702	403	7	research	research	NOUN
ejpam-4702	403	8	and	and	CCONJ
ejpam-4702	403	9	development	development	NOUN
ejpam-4702	403	10	institute	institute	PROPN
ejpam-4702	403	11	and	and	CCONJ
ejpam-4702	403	12	faculty	faculty	NOUN
ejpam-4702	403	13	of	of	ADP
ejpam-4702	403	14	science	science	NOUN
ejpam-4702	403	15	and	and	CCONJ
ejpam-4702	403	16	technology	technology	NOUN
ejpam-4702	403	17	,	,	PUNCT
ejpam-4702	403	18	thepsatri	thepsatri	VERB
ejpam-4702	403	19	rajabhat	rajabhat	ADJ
ejpam-4702	403	20	university	university	NOUN
ejpam-4702	403	21	,	,	PUNCT
ejpam-4702	403	22	thailand	thailand	PROPN
ejpam-4702	403	23	.	.	PUNCT
ejpam-4702	404	1	references	reference	NOUN
ejpam-4702	404	2	[	[	X
ejpam-4702	404	3	1	1	NUM
ejpam-4702	404	4	]	]	PUNCT
ejpam-4702	404	5	n	n	CCONJ
ejpam-4702	404	6	burshtein	burshtein	ADV
ejpam-4702	404	7	.	.	PUNCT
ejpam-4702	405	1	the	the	DET
ejpam-4702	405	2	diophantine	diophantine	NOUN
ejpam-4702	405	3	equation	equation	NOUN
ejpam-4702	405	4	px	px	X
ejpam-4702	405	5	+	+	CCONJ
ejpam-4702	405	6	(	(	PUNCT
ejpam-4702	405	7	p	p	X
ejpam-4702	405	8	+	+	NUM
ejpam-4702	405	9	4)y	4)y	X
ejpam-4702	405	10	=	=	SYM
ejpam-4702	405	11	z2	z2	PROPN
ejpam-4702	405	12	when	when	SCONJ
ejpam-4702	405	13	p	p	PROPN
ejpam-4702	405	14	>	>	X
ejpam-4702	405	15	3	3	NUM
ejpam-4702	405	16	and	and	CCONJ
ejpam-4702	405	17	p	p	PRON
ejpam-4702	405	18	+	+	CCONJ
ejpam-4702	405	19	4	4	NUM
ejpam-4702	405	20	are	be	AUX
ejpam-4702	405	21	primes	prime	NOUN
ejpam-4702	405	22	is	be	AUX
ejpam-4702	405	23	insolvable	insolvable	ADJ
ejpam-4702	405	24	in	in	ADP
ejpam-4702	405	25	positive	positive	ADJ
ejpam-4702	405	26	integers	integer	NOUN
ejpam-4702	405	27	x	x	NOUN
ejpam-4702	405	28	,	,	PUNCT
ejpam-4702	405	29	y	y	PROPN
ejpam-4702	405	30	,	,	PUNCT
ejpam-4702	405	31	z.	z.	PROPN
ejpam-4702	405	32	annals	annal	NOUN
ejpam-4702	405	33	of	of	ADP
ejpam-4702	405	34	pure	pure	ADJ
ejpam-4702	405	35	and	and	CCONJ
ejpam-4702	405	36	applied	applied	ADJ
ejpam-4702	405	37	mathematics	mathematic	NOUN
ejpam-4702	405	38	,	,	PUNCT
ejpam-4702	405	39	16(2):283–286	16(2):283–286	NUM
ejpam-4702	405	40	,	,	PUNCT
ejpam-4702	405	41	2018	2018	NUM
ejpam-4702	405	42	.	.	PUNCT
ejpam-4702	406	1	[	[	X
ejpam-4702	406	2	2	2	NUM
ejpam-4702	406	3	]	]	PUNCT
ejpam-4702	406	4	n	n	PRON
ejpam-4702	406	5	burshtein	burshtein	ADV
ejpam-4702	406	6	.	.	PUNCT
ejpam-4702	407	1	solutions	solution	NOUN
ejpam-4702	407	2	of	of	ADP
ejpam-4702	407	3	the	the	DET
ejpam-4702	407	4	diophantine	diophantine	NOUN
ejpam-4702	407	5	equation	equation	NOUN
ejpam-4702	407	6	px+(p+6)y	px+(p+6)y	NOUN
ejpam-4702	407	7	=	=	SYM
ejpam-4702	407	8	z2	z2	NOUN
ejpam-4702	407	9	when	when	SCONJ
ejpam-4702	407	10	p	p	X
ejpam-4702	407	11	,	,	PUNCT
ejpam-4702	407	12	p+6	p+6	NUM
ejpam-4702	407	13	are	be	AUX
ejpam-4702	407	14	primes	prime	NOUN
ejpam-4702	407	15	and	and	CCONJ
ejpam-4702	407	16	x+	x+	ADJ
ejpam-4702	407	17	y	y	NOUN
ejpam-4702	407	18	=	=	SYM
ejpam-4702	407	19	2	2	NUM
ejpam-4702	407	20	,	,	PUNCT
ejpam-4702	407	21	3	3	NUM
ejpam-4702	407	22	,	,	PUNCT
ejpam-4702	407	23	4	4	NUM
ejpam-4702	407	24	.	.	PUNCT
ejpam-4702	407	25	annals	annal	NOUN
ejpam-4702	407	26	of	of	ADP
ejpam-4702	407	27	pure	pure	ADJ
ejpam-4702	407	28	and	and	CCONJ
ejpam-4702	407	29	applied	applied	ADJ
ejpam-4702	407	30	mathematics	mathematic	NOUN
ejpam-4702	407	31	,	,	PUNCT
ejpam-4702	407	32	17(1):101–106	17(1):101–106	NUM
ejpam-4702	407	33	,	,	PUNCT
ejpam-4702	407	34	2018	2018	NUM
ejpam-4702	407	35	.	.	PUNCT
ejpam-4702	408	1	[	[	X
ejpam-4702	408	2	3	3	X
ejpam-4702	408	3	]	]	PUNCT
ejpam-4702	408	4	n	n	PRON
ejpam-4702	408	5	burshtein	burshtein	ADV
ejpam-4702	408	6	.	.	PUNCT
ejpam-4702	409	1	the	the	DET
ejpam-4702	409	2	diophantine	diophantine	NOUN
ejpam-4702	409	3	equations	equation	VERB
ejpam-4702	409	4	2x+11y	2x+11y	NUM
ejpam-4702	409	5	=	=	SYM
ejpam-4702	409	6	z2	z2	PROPN
ejpam-4702	409	7	and	and	CCONJ
ejpam-4702	409	8	19x+29y	19x+29y	NUM
ejpam-4702	409	9	=	=	SYM
ejpam-4702	409	10	z2	z2	PROPN
ejpam-4702	409	11	are	be	AUX
ejpam-4702	409	12	insolvable	insolvable	ADJ
ejpam-4702	409	13	in	in	ADP
ejpam-4702	409	14	positive	positive	ADJ
ejpam-4702	409	15	integers	integer	NOUN
ejpam-4702	409	16	x	x	NOUN
ejpam-4702	409	17	,	,	PUNCT
ejpam-4702	409	18	y	y	PROPN
ejpam-4702	409	19	,	,	PUNCT
ejpam-4702	409	20	z.	z.	PROPN
ejpam-4702	409	21	annals	annal	NOUN
ejpam-4702	409	22	of	of	ADP
ejpam-4702	409	23	pure	pure	ADJ
ejpam-4702	409	24	and	and	CCONJ
ejpam-4702	409	25	applied	applied	ADJ
ejpam-4702	409	26	mathematics	mathematic	NOUN
ejpam-4702	409	27	,	,	PUNCT
ejpam-4702	409	28	22(2):119	22(2):119	NUM
ejpam-4702	409	29	–	–	PUNCT
ejpam-4702	409	30	123	123	NUM
ejpam-4702	409	31	,	,	PUNCT
ejpam-4702	409	32	2020	2020	NUM
ejpam-4702	409	33	.	.	PUNCT
ejpam-4702	410	1	[	[	X
ejpam-4702	410	2	4	4	NUM
ejpam-4702	410	3	]	]	X
ejpam-4702	410	4	s	s	VERB
ejpam-4702	410	5	kumar	kumar	PROPN
ejpam-4702	410	6	s	s	PROPN
ejpam-4702	410	7	gupta	gupta	PROPN
ejpam-4702	410	8	and	and	CCONJ
ejpam-4702	410	9	h	h	PROPN
ejpam-4702	410	10	kishan	kishan	PROPN
ejpam-4702	410	11	.	.	PUNCT
ejpam-4702	411	1	on	on	ADP
ejpam-4702	411	2	the	the	DET
ejpam-4702	411	3	non	non	ADJ
ejpam-4702	411	4	-	-	ADJ
ejpam-4702	411	5	linear	linear	ADJ
ejpam-4702	411	6	diophantine	diophantine	NOUN
ejpam-4702	411	7	equation	equation	NOUN
ejpam-4702	411	8	61x+67y	61x+67y	NUM
ejpam-4702	411	9	=	=	SYM
ejpam-4702	411	10	z2	z2	PROPN
ejpam-4702	411	11	and	and	CCONJ
ejpam-4702	411	12	67x	67x	NUM
ejpam-4702	411	13	+	+	NUM
ejpam-4702	411	14	73y	73y	NOUN
ejpam-4702	411	15	=	=	SYM
ejpam-4702	411	16	z2	z2	PROPN
ejpam-4702	411	17	.	.	PUNCT
ejpam-4702	411	18	annals	annal	NOUN
ejpam-4702	411	19	of	of	ADP
ejpam-4702	411	20	pure	pure	ADJ
ejpam-4702	411	21	and	and	CCONJ
ejpam-4702	411	22	applied	applied	ADJ
ejpam-4702	411	23	mathematics	mathematic	NOUN
ejpam-4702	411	24	,	,	PUNCT
ejpam-4702	411	25	18(1):91–94	18(1):91–94	NUM
ejpam-4702	411	26	,	,	PUNCT
ejpam-4702	411	27	2018	2018	NUM
ejpam-4702	411	28	.	.	PUNCT
ejpam-4702	412	1	[	[	X
ejpam-4702	412	2	5	5	NUM
ejpam-4702	412	3	]	]	X
ejpam-4702	412	4	s	s	PART
ejpam-4702	412	5	kumar	kumar	PROPN
ejpam-4702	412	6	s	s	PROPN
ejpam-4702	412	7	gupta	gupta	PROPN
ejpam-4702	412	8	and	and	CCONJ
ejpam-4702	412	9	h	h	PROPN
ejpam-4702	412	10	kishan	kishan	PROPN
ejpam-4702	412	11	.	.	PUNCT
ejpam-4702	413	1	on	on	ADP
ejpam-4702	413	2	the	the	DET
ejpam-4702	413	3	non	non	ADJ
ejpam-4702	413	4	-	-	ADJ
ejpam-4702	413	5	linear	linear	ADJ
ejpam-4702	413	6	diophantine	diophantine	NOUN
ejpam-4702	413	7	equations	equation	NOUN
ejpam-4702	413	8	31x+41y	31x+41y	NUM
ejpam-4702	413	9	=	=	SYM
ejpam-4702	413	10	z2	z2	PROPN
ejpam-4702	413	11	and	and	CCONJ
ejpam-4702	413	12	61x+71y	61x+71y	NUM
ejpam-4702	413	13	=	=	SYM
ejpam-4702	413	14	z2	z2	PROPN
ejpam-4702	413	15	.	.	PUNCT
ejpam-4702	413	16	annals	annal	NOUN
ejpam-4702	413	17	of	of	ADP
ejpam-4702	413	18	pure	pure	ADJ
ejpam-4702	413	19	and	and	CCONJ
ejpam-4702	413	20	applied	applied	ADJ
ejpam-4702	413	21	mathematics	mathematic	NOUN
ejpam-4702	413	22	,	,	PUNCT
ejpam-4702	413	23	18(2):185–188	18(2):185–188	PROPN
ejpam-4702	413	24	,	,	PUNCT
ejpam-4702	413	25	2018	2018	NUM
ejpam-4702	413	26	.	.	PUNCT
ejpam-4702	414	1	references	reference	NOUN
ejpam-4702	414	2	735	735	NUM
ejpam-4702	414	3	[	[	X
ejpam-4702	414	4	6	6	NUM
ejpam-4702	414	5	]	]	SYM
ejpam-4702	414	6	s	s	PART
ejpam-4702	414	7	gupta	gupta	PROPN
ejpam-4702	414	8	s	s	PART
ejpam-4702	414	9	kumar	kumar	PROPN
ejpam-4702	414	10	and	and	CCONJ
ejpam-4702	414	11	h	h	PROPN
ejpam-4702	414	12	kishan	kishan	PROPN
ejpam-4702	414	13	.	.	PUNCT
ejpam-4702	415	1	on	on	ADP
ejpam-4702	415	2	the	the	DET
ejpam-4702	415	3	non	non	ADJ
ejpam-4702	415	4	-	-	ADJ
ejpam-4702	415	5	linear	linear	ADJ
ejpam-4702	415	6	diophantine	diophantine	NOUN
ejpam-4702	415	7	equation	equation	NOUN
ejpam-4702	415	8	px	px	X
ejpam-4702	415	9	+	+	CCONJ
ejpam-4702	415	10	(	(	PUNCT
ejpam-4702	415	11	p+	p+	NOUN
ejpam-4702	415	12	6)y	6)y	X
ejpam-4702	415	13	=	=	SYM
ejpam-4702	415	14	z2	z2	PROPN
ejpam-4702	415	15	.	.	PUNCT
ejpam-4702	415	16	annals	annal	NOUN
ejpam-4702	415	17	of	of	ADP
ejpam-4702	415	18	pure	pure	ADJ
ejpam-4702	415	19	and	and	CCONJ
ejpam-4702	415	20	applied	applied	ADJ
ejpam-4702	415	21	mathematics	mathematic	NOUN
ejpam-4702	415	22	,	,	PUNCT
ejpam-4702	415	23	18(1):125–128	18(1):125–128	NUM
ejpam-4702	415	24	,	,	PUNCT
ejpam-4702	415	25	2018	2018	NUM
ejpam-4702	415	26	.	.	PUNCT
ejpam-4702	416	1	[	[	X
ejpam-4702	416	2	7	7	X
ejpam-4702	416	3	]	]	X
ejpam-4702	416	4	r	r	NOUN
ejpam-4702	416	5	j	j	PROPN
ejpam-4702	416	6	s	s	X
ejpam-4702	416	7	mina	mina	PROPN
ejpam-4702	416	8	and	and	CCONJ
ejpam-4702	416	9	j	j	PROPN
ejpam-4702	416	10	b	b	PROPN
ejpam-4702	416	11	bacani	bacani	PROPN
ejpam-4702	416	12	.	.	PUNCT
ejpam-4702	417	1	non	non	ADJ
ejpam-4702	417	2	-	-	NOUN
ejpam-4702	417	3	existence	existence	NOUN
ejpam-4702	417	4	of	of	ADP
ejpam-4702	417	5	solutions	solution	NOUN
ejpam-4702	417	6	of	of	ADP
ejpam-4702	417	7	diophantine	diophantine	NOUN
ejpam-4702	417	8	equations	equation	NOUN
ejpam-4702	417	9	of	of	ADP
ejpam-4702	417	10	the	the	DET
ejpam-4702	417	11	form	form	NOUN
ejpam-4702	417	12	px	px	X
ejpam-4702	417	13	+	+	CCONJ
ejpam-4702	417	14	qy	qy	NOUN
ejpam-4702	417	15	=	=	SYM
ejpam-4702	417	16	z2n	z2n	PROPN
ejpam-4702	417	17	.	.	PUNCT
ejpam-4702	418	1	mathematics	mathematic	NOUN
ejpam-4702	418	2	and	and	CCONJ
ejpam-4702	418	3	statistics	statistic	NOUN
ejpam-4702	418	4	,	,	PUNCT
ejpam-4702	418	5	7(3):78–81	7(3):78–81	NUM
ejpam-4702	418	6	,	,	PUNCT
ejpam-4702	418	7	2019	2019	NUM
ejpam-4702	418	8	.	.	PUNCT
ejpam-4702	419	1	[	[	X
ejpam-4702	419	2	8	8	NUM
ejpam-4702	419	3	]	]	X
ejpam-4702	419	4	r	r	NOUN
ejpam-4702	419	5	j	j	PROPN
ejpam-4702	419	6	s	s	X
ejpam-4702	419	7	mina	mina	PROPN
ejpam-4702	419	8	and	and	CCONJ
ejpam-4702	419	9	j	j	PROPN
ejpam-4702	419	10	b	b	PROPN
ejpam-4702	419	11	bacani	bacani	PROPN
ejpam-4702	419	12	.	.	PUNCT
ejpam-4702	420	1	on	on	ADP
ejpam-4702	420	2	the	the	DET
ejpam-4702	420	3	solutions	solution	NOUN
ejpam-4702	420	4	of	of	ADP
ejpam-4702	420	5	the	the	DET
ejpam-4702	420	6	diophantine	diophantine	NOUN
ejpam-4702	420	7	equation	equation	NOUN
ejpam-4702	420	8	px	px	X
ejpam-4702	420	9	+	+	CCONJ
ejpam-4702	420	10	(	(	PUNCT
ejpam-4702	420	11	p	p	X
ejpam-4702	420	12	+	+	NOUN
ejpam-4702	420	13	4k)y	4k)y	NOUN
ejpam-4702	420	14	=	=	SYM
ejpam-4702	420	15	z2	z2	PROPN
ejpam-4702	420	16	for	for	ADP
ejpam-4702	420	17	prime	prime	ADJ
ejpam-4702	420	18	pairs	pair	NOUN
ejpam-4702	420	19	p	p	NOUN
ejpam-4702	420	20	and	and	CCONJ
ejpam-4702	420	21	p	p	NOUN
ejpam-4702	420	22	+	+	X
ejpam-4702	420	23	4k	4k	NOUN
ejpam-4702	420	24	.	.	PUNCT
ejpam-4702	421	1	european	european	ADJ
ejpam-4702	421	2	journal	journal	PROPN
ejpam-4702	421	3	of	of	ADP
ejpam-4702	421	4	pure	pure	ADJ
ejpam-4702	421	5	and	and	CCONJ
ejpam-4702	421	6	applied	applied	ADJ
ejpam-4702	421	7	mathematics	mathematic	NOUN
ejpam-4702	421	8	,	,	PUNCT
ejpam-4702	421	9	14(2):471–479	14(2):471–479	NUM
ejpam-4702	421	10	,	,	PUNCT
ejpam-4702	421	11	2021	2021	NUM
ejpam-4702	421	12	.	.	PUNCT
ejpam-4702	422	1	[	[	X
ejpam-4702	422	2	9	9	NUM
ejpam-4702	422	3	]	]	SYM
ejpam-4702	422	4	f	f	PROPN
ejpam-4702	422	5	neres	nere	NOUN
ejpam-4702	422	6	.	.	PUNCT
ejpam-4702	423	1	on	on	ADP
ejpam-4702	423	2	the	the	DET
ejpam-4702	423	3	solvability	solvability	NOUN
ejpam-4702	423	4	of	of	ADP
ejpam-4702	423	5	the	the	DET
ejpam-4702	423	6	diophantine	diophantine	NOUN
ejpam-4702	423	7	equation	equation	NOUN
ejpam-4702	423	8	px+(p+8)y	px+(p+8)y	NOUN
ejpam-4702	423	9	=	=	SYM
ejpam-4702	423	10	z2	z2	PROPN
ejpam-4702	423	11	when	when	SCONJ
ejpam-4702	423	12	p	p	PROPN
ejpam-4702	423	13	>	>	X
ejpam-4702	423	14	3	3	NUM
ejpam-4702	423	15	and	and	CCONJ
ejpam-4702	423	16	p+	p+	NOUN
ejpam-4702	423	17	8	8	NUM
ejpam-4702	423	18	are	be	AUX
ejpam-4702	423	19	primes	prime	NOUN
ejpam-4702	423	20	.	.	PUNCT
ejpam-4702	423	21	annals	annal	NOUN
ejpam-4702	423	22	of	of	ADP
ejpam-4702	423	23	pure	pure	ADJ
ejpam-4702	423	24	and	and	CCONJ
ejpam-4702	423	25	applied	applied	ADJ
ejpam-4702	423	26	mathematics	mathematic	NOUN
ejpam-4702	423	27	,	,	PUNCT
ejpam-4702	423	28	18(1):9–13	18(1):9–13	NUM
ejpam-4702	423	29	,	,	PUNCT
ejpam-4702	423	30	2018	2018	NUM
ejpam-4702	423	31	.	.	PUNCT
ejpam-4702	424	1	[	[	X
ejpam-4702	424	2	10	10	NUM
ejpam-4702	424	3	]	]	X
ejpam-4702	424	4	c	c	PROPN
ejpam-4702	424	5	g	g	PROPN
ejpam-4702	424	6	rao	rao	PROPN
ejpam-4702	424	7	.	.	PUNCT
ejpam-4702	425	1	on	on	ADP
ejpam-4702	425	2	the	the	DET
ejpam-4702	425	3	diophantine	diophantine	NOUN
ejpam-4702	425	4	equation	equation	NOUN
ejpam-4702	425	5	3x	3x	PRON
ejpam-4702	425	6	+	+	CCONJ
ejpam-4702	425	7	7y	7y	NOUN
ejpam-4702	425	8	=	=	SYM
ejpam-4702	425	9	z2	z2	PROPN
ejpam-4702	425	10	.	.	PUNCT
ejpam-4702	426	1	epra	epra	PROPN
ejpam-4702	426	2	international	international	PROPN
ejpam-4702	426	3	journal	journal	PROPN
ejpam-4702	426	4	of	of	ADP
ejpam-4702	426	5	research	research	NOUN
ejpam-4702	426	6	and	and	CCONJ
ejpam-4702	426	7	development	development	NOUN
ejpam-4702	426	8	,	,	PUNCT
ejpam-4702	426	9	3(6):93–95	3(6):93–95	NUM
ejpam-4702	426	10	,	,	PUNCT
ejpam-4702	426	11	2018	2018	NUM
ejpam-4702	426	12	.	.	PUNCT
ejpam-4702	427	1	[	[	X
ejpam-4702	427	2	11	11	NUM
ejpam-4702	427	3	]	]	X
ejpam-4702	427	4	k	k	PROPN
ejpam-4702	427	5	h	h	PROPN
ejpam-4702	427	6	rosen	rosen	PROPN
ejpam-4702	427	7	.	.	PUNCT
ejpam-4702	428	1	elementary	elementary	ADJ
ejpam-4702	428	2	number	number	NOUN
ejpam-4702	428	3	theory	theory	NOUN
ejpam-4702	428	4	.	.	PUNCT
ejpam-4702	429	1	clays	clays	PROPN
ejpam-4702	429	2	ltd	ltd	PROPN
ejpam-4702	429	3	,	,	PUNCT
ejpam-4702	429	4	great	great	ADJ
ejpam-4702	429	5	britain	britain	PROPN
ejpam-4702	429	6	,	,	PUNCT
ejpam-4702	429	7	2014	2014	NUM
ejpam-4702	429	8	.	.	PUNCT
ejpam-4702	430	1	[	[	X
ejpam-4702	430	2	12	12	NUM
ejpam-4702	430	3	]	]	X
ejpam-4702	430	4	b	b	X
ejpam-4702	430	5	sroysang	sroysang	PROPN
ejpam-4702	430	6	.	.	PUNCT
ejpam-4702	431	1	on	on	ADP
ejpam-4702	431	2	the	the	DET
ejpam-4702	431	3	diphantine	diphantine	ADJ
ejpam-4702	431	4	equation	equation	NOUN
ejpam-4702	431	5	3x+17y	3x+17y	NUM
ejpam-4702	431	6	=	=	SYM
ejpam-4702	431	7	z2	z2	PROPN
ejpam-4702	431	8	.	.	PUNCT
ejpam-4702	432	1	international	international	ADJ
ejpam-4702	432	2	journal	journal	NOUN
ejpam-4702	432	3	of	of	ADP
ejpam-4702	432	4	pure	pure	ADJ
ejpam-4702	432	5	and	and	CCONJ
ejpam-4702	432	6	applied	applied	ADJ
ejpam-4702	432	7	mathematics	mathematic	NOUN
ejpam-4702	432	8	,	,	PUNCT
ejpam-4702	432	9	89(1):111–114	89(1):111–114	NOUN
ejpam-4702	432	10	,	,	PUNCT
ejpam-4702	432	11	2013	2013	NUM
ejpam-4702	432	12	.	.	PUNCT
ejpam-4702	433	1	[	[	X
ejpam-4702	433	2	13	13	NUM
ejpam-4702	433	3	]	]	X
ejpam-4702	433	4	s	s	AUX
ejpam-4702	433	5	tadee	tadee	NOUN
ejpam-4702	433	6	.	.	PUNCT
ejpam-4702	434	1	on	on	ADP
ejpam-4702	434	2	the	the	DET
ejpam-4702	434	3	diophantine	diophantine	NOUN
ejpam-4702	434	4	equation	equation	NOUN
ejpam-4702	434	5	px+(p+10)y	px+(p+10)y	NOUN
ejpam-4702	434	6	=	=	SYM
ejpam-4702	434	7	z2	z2	PROPN
ejpam-4702	434	8	when	when	SCONJ
ejpam-4702	434	9	p	p	NOUN
ejpam-4702	434	10	and	and	CCONJ
ejpam-4702	434	11	p+10	p+10	NOUN
ejpam-4702	434	12	are	be	AUX
ejpam-4702	434	13	primes	prime	NOUN
ejpam-4702	434	14	.	.	PUNCT
ejpam-4702	435	1	udon	udon	PROPN
ejpam-4702	435	2	thani	thani	PROPN
ejpam-4702	435	3	rajabhat	rajabhat	PROPN
ejpam-4702	435	4	university	university	PROPN
ejpam-4702	435	5	journal	journal	NOUN
ejpam-4702	435	6	of	of	ADP
ejpam-4702	435	7	sciences	science	NOUN
ejpam-4702	435	8	and	and	CCONJ
ejpam-4702	435	9	technology	technology	NOUN
ejpam-4702	435	10	,	,	PUNCT
ejpam-4702	435	11	10(2):155–162	10(2):155–162	PROPN
ejpam-4702	435	12	,	,	PUNCT
ejpam-4702	435	13	2022	2022	NUM
ejpam-4702	435	14	.	.	PUNCT
ejpam-4702	436	1	[	[	X
ejpam-4702	436	2	14	14	NUM
ejpam-4702	436	3	]	]	X
ejpam-4702	436	4	s	s	VERB
ejpam-4702	436	5	tadee	tadee	VERB
ejpam-4702	436	6	.	.	PUNCT
ejpam-4702	437	1	on	on	ADP
ejpam-4702	437	2	the	the	DET
ejpam-4702	437	3	diophantine	diophantine	NOUN
ejpam-4702	437	4	equation	equation	NOUN
ejpam-4702	437	5	px+(p+14)y	px+(p+14)y	VERB
ejpam-4702	437	6	=	=	SYM
ejpam-4702	437	7	z2	z2	PROPN
ejpam-4702	437	8	,	,	PUNCT
ejpam-4702	437	9	where	where	SCONJ
ejpam-4702	437	10	p	p	X
ejpam-4702	437	11	,	,	PUNCT
ejpam-4702	437	12	p+14	p+14	NOUN
ejpam-4702	437	13	are	be	AUX
ejpam-4702	437	14	primes	prime	NOUN
ejpam-4702	437	15	.	.	PUNCT
ejpam-4702	437	16	annals	annal	NOUN
ejpam-4702	437	17	of	of	ADP
ejpam-4702	437	18	pure	pure	ADJ
ejpam-4702	437	19	and	and	CCONJ
ejpam-4702	437	20	applied	applied	ADJ
ejpam-4702	437	21	mathematics	mathematic	NOUN
ejpam-4702	437	22	,	,	PUNCT
ejpam-4702	437	23	26(2):125–130	26(2):125–130	PROPN
ejpam-4702	437	24	,	,	PUNCT
ejpam-4702	437	25	2022	2022	NUM
ejpam-4702	437	26	.	.	PUNCT