id	sid	tid	token	lemma	pos
ejpam-6553	1	1	european	european	PROPN
ejpam-6553	1	2	journal	journal	PROPN
ejpam-6553	1	3	of	of	ADP
ejpam-6553	1	4	pure	pure	ADJ
ejpam-6553	1	5	and	and	CCONJ
ejpam-6553	1	6	applied	applied	ADJ
ejpam-6553	1	7	mathematics	mathematic	NOUN
ejpam-6553	1	8	2025	2025	NUM
ejpam-6553	1	9	,	,	PUNCT
ejpam-6553	1	10	vol	vol	NOUN
ejpam-6553	1	11	.	.	PROPN
ejpam-6553	1	12	18	18	NUM
ejpam-6553	1	13	,	,	PUNCT
ejpam-6553	1	14	issue	issue	NOUN
ejpam-6553	1	15	3	3	NUM
ejpam-6553	1	16	,	,	PUNCT
ejpam-6553	1	17	article	article	NOUN
ejpam-6553	1	18	number	number	NOUN
ejpam-6553	1	19	6553	6553	NUM
ejpam-6553	1	20	issn	issn	PROPN
ejpam-6553	1	21	1307	1307	NUM
ejpam-6553	1	22	-	-	SYM
ejpam-6553	1	23	5543	5543	NUM
ejpam-6553	1	24	–	–	PUNCT
ejpam-6553	1	25	ejpam.com	ejpam.com	X
ejpam-6553	1	26	published	publish	VERB
ejpam-6553	1	27	by	by	ADP
ejpam-6553	1	28	new	new	PROPN
ejpam-6553	1	29	york	york	PROPN
ejpam-6553	1	30	business	business	PROPN
ejpam-6553	1	31	global	global	PROPN
ejpam-6553	1	32	on	on	ADP
ejpam-6553	1	33	the	the	DET
ejpam-6553	1	34	unsolvability	unsolvability	NOUN
ejpam-6553	1	35	of	of	ADP
ejpam-6553	1	36	the	the	DET
ejpam-6553	1	37	exponential	exponential	ADJ
ejpam-6553	1	38	diophantine	diophantine	NOUN
ejpam-6553	1	39	equation	equation	NOUN
ejpam-6553	1	40	23x	23x	NUM
ejpam-6553	1	41	+	+	ADJ
ejpam-6553	1	42	22y	22y	NOUN
ejpam-6553	1	43	=	=	SYM
ejpam-6553	1	44	z2	z2	PROPN
ejpam-6553	1	45	in	in	ADP
ejpam-6553	1	46	nonnegative	nonnegative	ADJ
ejpam-6553	1	47	integers	integer	NOUN
ejpam-6553	1	48	abdallah	abdallah	PROPN
ejpam-6553	1	49	assiry	assiry	ADJ
ejpam-6553	1	50	departement	departement	NOUN
ejpam-6553	1	51	of	of	ADP
ejpam-6553	1	52	mathematics	mathematic	NOUN
ejpam-6553	1	53	,	,	PUNCT
ejpam-6553	1	54	college	college	NOUN
ejpam-6553	1	55	of	of	ADP
ejpam-6553	1	56	science	science	NOUN
ejpam-6553	1	57	,	,	PUNCT
ejpam-6553	1	58	umm	umm	INTJ
ejpam-6553	1	59	al	al	PROPN
ejpam-6553	1	60	-	-	PUNCT
ejpam-6553	1	61	qura	qura	PROPN
ejpam-6553	1	62	university	university	NOUN
ejpam-6553	1	63	.	.	PUNCT
ejpam-6553	2	1	mecca	mecca	PROPN
ejpam-6553	2	2	21955	21955	NUM
ejpam-6553	2	3	,	,	PUNCT
ejpam-6553	2	4	saudi	saudi	PROPN
ejpam-6553	2	5	arabia	arabia	PROPN
ejpam-6553	2	6	abstract	abstract	NOUN
ejpam-6553	2	7	.	.	PUNCT
ejpam-6553	3	1	we	we	PRON
ejpam-6553	3	2	prove	prove	VERB
ejpam-6553	3	3	that	that	SCONJ
ejpam-6553	3	4	the	the	DET
ejpam-6553	3	5	exponential	exponential	ADJ
ejpam-6553	3	6	diophantine	diophantine	NOUN
ejpam-6553	3	7	equation	equation	NOUN
ejpam-6553	3	8	23x+22y	23x+22y	NUM
ejpam-6553	3	9	=	=	SYM
ejpam-6553	3	10	z2	z2	PROPN
ejpam-6553	3	11	has	have	VERB
ejpam-6553	3	12	no	no	DET
ejpam-6553	3	13	nonnegative	nonnegative	ADJ
ejpam-6553	3	14	integer	integer	NOUN
ejpam-6553	3	15	solutions	solution	NOUN
ejpam-6553	3	16	.	.	PUNCT
ejpam-6553	4	1	using	use	VERB
ejpam-6553	4	2	a	a	DET
ejpam-6553	4	3	combination	combination	NOUN
ejpam-6553	4	4	of	of	ADP
ejpam-6553	4	5	modular	modular	ADJ
ejpam-6553	4	6	arithmetic	arithmetic	ADJ
ejpam-6553	4	7	,	,	PUNCT
ejpam-6553	4	8	parity	parity	NOUN
ejpam-6553	4	9	analysis	analysis	NOUN
ejpam-6553	4	10	,	,	PUNCT
ejpam-6553	4	11	and	and	CCONJ
ejpam-6553	4	12	computational	computational	ADJ
ejpam-6553	4	13	verification	verification	NOUN
ejpam-6553	4	14	,	,	PUNCT
ejpam-6553	4	15	we	we	PRON
ejpam-6553	4	16	demonstrate	demonstrate	VERB
ejpam-6553	4	17	that	that	SCONJ
ejpam-6553	4	18	the	the	DET
ejpam-6553	4	19	equation	equation	NOUN
ejpam-6553	4	20	leads	lead	VERB
ejpam-6553	4	21	to	to	ADP
ejpam-6553	4	22	contradictions	contradiction	NOUN
ejpam-6553	4	23	under	under	ADP
ejpam-6553	4	24	all	all	DET
ejpam-6553	4	25	possible	possible	ADJ
ejpam-6553	4	26	cases	case	NOUN
ejpam-6553	4	27	.	.	PUNCT
ejpam-6553	5	1	our	our	PRON
ejpam-6553	5	2	work	work	NOUN
ejpam-6553	5	3	extends	extend	VERB
ejpam-6553	5	4	previous	previous	ADJ
ejpam-6553	5	5	results	result	NOUN
ejpam-6553	5	6	on	on	ADP
ejpam-6553	5	7	equations	equation	NOUN
ejpam-6553	5	8	of	of	ADP
ejpam-6553	5	9	the	the	DET
ejpam-6553	5	10	form	form	NOUN
ejpam-6553	5	11	px	px	X
ejpam-6553	5	12	+	+	CCONJ
ejpam-6553	5	13	(	(	PUNCT
ejpam-6553	5	14	p−	p−	NOUN
ejpam-6553	5	15	1)y	1)y	NUM
ejpam-6553	5	16	=	=	SYM
ejpam-6553	5	17	z2	z2	PROPN
ejpam-6553	5	18	and	and	CCONJ
ejpam-6553	5	19	highlights	highlight	NOUN
ejpam-6553	5	20	the	the	DET
ejpam-6553	5	21	interplay	interplay	NOUN
ejpam-6553	5	22	between	between	ADP
ejpam-6553	5	23	theoretical	theoretical	ADJ
ejpam-6553	5	24	and	and	CCONJ
ejpam-6553	5	25	computational	computational	ADJ
ejpam-6553	5	26	methods	method	NOUN
ejpam-6553	5	27	in	in	ADP
ejpam-6553	5	28	solving	solve	VERB
ejpam-6553	5	29	diophantine	diophantine	NOUN
ejpam-6553	5	30	problems	problem	NOUN
ejpam-6553	5	31	.	.	PUNCT
ejpam-6553	6	1	we	we	PRON
ejpam-6553	6	2	also	also	ADV
ejpam-6553	6	3	provide	provide	VERB
ejpam-6553	6	4	computational	computational	ADJ
ejpam-6553	6	5	data	datum	NOUN
ejpam-6553	6	6	for	for	ADP
ejpam-6553	6	7	small	small	ADJ
ejpam-6553	6	8	values	value	NOUN
ejpam-6553	6	9	of	of	ADP
ejpam-6553	6	10	x	x	X
ejpam-6553	6	11	and	and	CCONJ
ejpam-6553	6	12	y	y	PROPN
ejpam-6553	6	13	to	to	PART
ejpam-6553	6	14	support	support	VERB
ejpam-6553	6	15	our	our	PRON
ejpam-6553	6	16	theoretical	theoretical	ADJ
ejpam-6553	6	17	findings	finding	NOUN
ejpam-6553	6	18	.	.	PUNCT
ejpam-6553	7	1	furthermore	furthermore	ADV
ejpam-6553	7	2	,	,	PUNCT
ejpam-6553	7	3	we	we	PRON
ejpam-6553	7	4	generalize	generalize	VERB
ejpam-6553	7	5	our	our	PRON
ejpam-6553	7	6	approach	approach	NOUN
ejpam-6553	7	7	to	to	ADP
ejpam-6553	7	8	other	other	ADJ
ejpam-6553	7	9	equations	equation	NOUN
ejpam-6553	7	10	of	of	ADP
ejpam-6553	7	11	the	the	DET
ejpam-6553	7	12	form	form	NOUN
ejpam-6553	7	13	wx+(w−1)y	wx+(w−1)y	PROPN
ejpam-6553	7	14	=	=	SYM
ejpam-6553	7	15	z2	z2	PROPN
ejpam-6553	7	16	,	,	PUNCT
ejpam-6553	7	17	where	where	SCONJ
ejpam-6553	7	18	w	w	NOUN
ejpam-6553	7	19	is	be	AUX
ejpam-6553	7	20	a	a	DET
ejpam-6553	7	21	positive	positive	ADJ
ejpam-6553	7	22	integer	integer	NOUN
ejpam-6553	7	23	of	of	ADP
ejpam-6553	7	24	a	a	DET
ejpam-6553	7	25	specific	specific	ADJ
ejpam-6553	7	26	form	form	NOUN
ejpam-6553	7	27	.	.	PUNCT
ejpam-6553	8	1	this	this	DET
ejpam-6553	8	2	study	study	NOUN
ejpam-6553	8	3	contributes	contribute	VERB
ejpam-6553	8	4	to	to	ADP
ejpam-6553	8	5	the	the	DET
ejpam-6553	8	6	broader	broad	ADJ
ejpam-6553	8	7	understanding	understanding	NOUN
ejpam-6553	8	8	of	of	ADP
ejpam-6553	8	9	exponential	exponential	ADJ
ejpam-6553	8	10	diophantine	diophantine	NOUN
ejpam-6553	8	11	equations	equation	NOUN
ejpam-6553	8	12	and	and	CCONJ
ejpam-6553	8	13	their	their	PRON
ejpam-6553	8	14	solutions	solution	NOUN
ejpam-6553	8	15	.	.	PUNCT
ejpam-6553	9	1	2020	2020	NUM
ejpam-6553	9	2	mathematics	mathematic	NOUN
ejpam-6553	9	3	subject	subject	NOUN
ejpam-6553	9	4	classifications	classification	NOUN
ejpam-6553	9	5	:	:	PUNCT
ejpam-6553	9	6	11d61	11d61	NUM
ejpam-6553	9	7	,	,	PUNCT
ejpam-6553	9	8	11d45	11d45	NUM
ejpam-6553	9	9	,	,	PUNCT
ejpam-6553	9	10	11y50	11y50	NUM
ejpam-6553	9	11	,	,	PUNCT
ejpam-6553	9	12	11a07	11a07	NUM
ejpam-6553	9	13	.	.	PUNCT
ejpam-6553	10	1	key	key	ADJ
ejpam-6553	10	2	words	word	NOUN
ejpam-6553	10	3	and	and	CCONJ
ejpam-6553	10	4	phrases	phrase	NOUN
ejpam-6553	10	5	:	:	PUNCT
ejpam-6553	10	6	exponential	exponential	ADJ
ejpam-6553	10	7	diophantine	diophantine	NOUN
ejpam-6553	10	8	equations	equation	NOUN
ejpam-6553	10	9	,	,	PUNCT
ejpam-6553	10	10	modular	modular	ADJ
ejpam-6553	10	11	arithmetic	arithmetic	ADJ
ejpam-6553	10	12	,	,	PUNCT
ejpam-6553	10	13	parity	parity	NOUN
ejpam-6553	10	14	analysis	analysis	NOUN
ejpam-6553	10	15	,	,	PUNCT
ejpam-6553	10	16	computational	computational	ADJ
ejpam-6553	10	17	number	number	NOUN
ejpam-6553	10	18	theory	theory	NOUN
ejpam-6553	10	19	,	,	PUNCT
ejpam-6553	10	20	quadratic	quadratic	ADJ
ejpam-6553	10	21	residues	residue	NOUN
ejpam-6553	10	22	,	,	PUNCT
ejpam-6553	10	23	integer	integer	NOUN
ejpam-6553	10	24	solutions	solution	NOUN
ejpam-6553	10	25	.	.	PUNCT
ejpam-6553	11	1	1	1	X
ejpam-6553	11	2	.	.	X
ejpam-6553	11	3	introduction	introduction	NOUN
ejpam-6553	11	4	the	the	DET
ejpam-6553	11	5	exponential	exponential	ADJ
ejpam-6553	11	6	diophantine	diophantine	NOUN
ejpam-6553	11	7	equation	equation	NOUN
ejpam-6553	11	8	23x	23x	NUM
ejpam-6553	11	9	+	+	ADJ
ejpam-6553	11	10	22y	22y	NOUN
ejpam-6553	11	11	=	=	SYM
ejpam-6553	11	12	z2	z2	PROPN
ejpam-6553	11	13	lies	lie	VERB
ejpam-6553	11	14	at	at	ADP
ejpam-6553	11	15	the	the	DET
ejpam-6553	11	16	intersection	intersection	NOUN
ejpam-6553	11	17	of	of	ADP
ejpam-6553	11	18	additive	additive	NOUN
ejpam-6553	11	19	and	and	CCONJ
ejpam-6553	11	20	multiplicative	multiplicative	ADJ
ejpam-6553	11	21	number	number	NOUN
ejpam-6553	11	22	theory	theory	NOUN
ejpam-6553	11	23	,	,	PUNCT
ejpam-6553	11	24	bringing	bring	VERB
ejpam-6553	11	25	together	together	ADV
ejpam-6553	11	26	deep	deep	ADJ
ejpam-6553	11	27	ideas	idea	NOUN
ejpam-6553	11	28	from	from	ADP
ejpam-6553	11	29	algebraic	algebraic	ADJ
ejpam-6553	11	30	number	number	NOUN
ejpam-6553	11	31	theory	theory	NOUN
ejpam-6553	11	32	,	,	PUNCT
ejpam-6553	11	33	diophantine	diophantine	VERB
ejpam-6553	11	34	geometry	geometry	NOUN
ejpam-6553	11	35	,	,	PUNCT
ejpam-6553	11	36	and	and	CCONJ
ejpam-6553	11	37	computational	computational	ADJ
ejpam-6553	11	38	mathematics	mathematic	NOUN
ejpam-6553	11	39	.	.	PUNCT
ejpam-6553	12	1	our	our	PRON
ejpam-6553	12	2	aim	aim	NOUN
ejpam-6553	12	3	is	be	AUX
ejpam-6553	12	4	to	to	PART
ejpam-6553	12	5	prove	prove	VERB
ejpam-6553	12	6	that	that	SCONJ
ejpam-6553	12	7	this	this	DET
ejpam-6553	12	8	equation	equation	NOUN
ejpam-6553	12	9	admits	admit	VERB
ejpam-6553	12	10	no	no	DET
ejpam-6553	12	11	nonnegative	nonnegative	ADJ
ejpam-6553	12	12	integer	integer	NOUN
ejpam-6553	12	13	solutions	solution	NOUN
ejpam-6553	12	14	.	.	PUNCT
ejpam-6553	13	1	equations	equation	NOUN
ejpam-6553	13	2	of	of	ADP
ejpam-6553	13	3	the	the	DET
ejpam-6553	13	4	form	form	NOUN
ejpam-6553	13	5	ax	ax	NOUN
ejpam-6553	13	6	+	+	CCONJ
ejpam-6553	13	7	by	by	ADP
ejpam-6553	13	8	=	=	PROPN
ejpam-6553	13	9	z2	z2	PROPN
ejpam-6553	13	10	have	have	VERB
ejpam-6553	13	11	long	long	ADV
ejpam-6553	13	12	intrigued	intrigue	VERB
ejpam-6553	13	13	mathematicians	mathematician	NOUN
ejpam-6553	13	14	due	due	ADJ
ejpam-6553	13	15	to	to	ADP
ejpam-6553	13	16	their	their	PRON
ejpam-6553	13	17	inherent	inherent	ADJ
ejpam-6553	13	18	arithmetic	arithmetic	ADJ
ejpam-6553	13	19	complexity	complexity	NOUN
ejpam-6553	13	20	.	.	PUNCT
ejpam-6553	14	1	from	from	ADP
ejpam-6553	14	2	an	an	DET
ejpam-6553	14	3	algebraic	algebraic	ADJ
ejpam-6553	14	4	standpoint	standpoint	NOUN
ejpam-6553	14	5	,	,	PUNCT
ejpam-6553	14	6	such	such	ADJ
ejpam-6553	14	7	equations	equation	NOUN
ejpam-6553	14	8	describe	describe	VERB
ejpam-6553	14	9	affine	affine	NOUN
ejpam-6553	14	10	surfaces	surface	NOUN
ejpam-6553	14	11	in	in	ADP
ejpam-6553	14	12	three	three	NUM
ejpam-6553	14	13	-	-	PUNCT
ejpam-6553	14	14	dimensional	dimensional	ADJ
ejpam-6553	14	15	space	space	NOUN
ejpam-6553	14	16	,	,	PUNCT
ejpam-6553	14	17	and	and	CCONJ
ejpam-6553	14	18	finding	find	VERB
ejpam-6553	14	19	integer	integer	NOUN
ejpam-6553	14	20	solutions	solution	NOUN
ejpam-6553	14	21	corresponds	correspond	VERB
ejpam-6553	14	22	to	to	ADP
ejpam-6553	14	23	locating	locate	VERB
ejpam-6553	14	24	integral	integral	ADJ
ejpam-6553	14	25	points	point	NOUN
ejpam-6553	14	26	on	on	ADP
ejpam-6553	14	27	these	these	DET
ejpam-6553	14	28	surfaces	surface	NOUN
ejpam-6553	14	29	.	.	PUNCT
ejpam-6553	15	1	lang	lang	PROPN
ejpam-6553	15	2	’s	’s	PART
ejpam-6553	15	3	foundational	foundational	ADJ
ejpam-6553	15	4	work	work	NOUN
ejpam-6553	15	5	on	on	ADP
ejpam-6553	15	6	diophantine	diophantine	NOUN
ejpam-6553	15	7	geometry	geometry	NOUN
ejpam-6553	15	8	[	[	X
ejpam-6553	15	9	1	1	X
ejpam-6553	15	10	]	]	PUNCT
ejpam-6553	15	11	provides	provide	VERB
ejpam-6553	15	12	a	a	DET
ejpam-6553	15	13	framework	framework	NOUN
ejpam-6553	15	14	for	for	ADP
ejpam-6553	15	15	understanding	understand	VERB
ejpam-6553	15	16	these	these	DET
ejpam-6553	15	17	types	type	NOUN
ejpam-6553	15	18	of	of	ADP
ejpam-6553	15	19	problems	problem	NOUN
ejpam-6553	15	20	,	,	PUNCT
ejpam-6553	15	21	and	and	CCONJ
ejpam-6553	15	22	baker	baker	PROPN
ejpam-6553	15	23	’s	’s	PART
ejpam-6553	15	24	theory	theory	NOUN
ejpam-6553	15	25	of	of	ADP
ejpam-6553	15	26	linear	linear	PROPN
ejpam-6553	15	27	forms	form	NOUN
ejpam-6553	15	28	in	in	ADP
ejpam-6553	15	29	logarithms	logarithm	NOUN
ejpam-6553	15	30	[	[	X
ejpam-6553	15	31	2	2	NUM
ejpam-6553	15	32	]	]	PUNCT
ejpam-6553	15	33	gives	give	VERB
ejpam-6553	15	34	crucial	crucial	ADJ
ejpam-6553	15	35	tools	tool	NOUN
ejpam-6553	15	36	for	for	ADP
ejpam-6553	15	37	bounding	bound	VERB
ejpam-6553	15	38	potential	potential	ADJ
ejpam-6553	15	39	solutions	solution	NOUN
ejpam-6553	15	40	.	.	PUNCT
ejpam-6553	16	1	doi	doi	NOUN
ejpam-6553	16	2	:	:	PUNCT
ejpam-6553	16	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6553	https://doi.org/10.29020/nybg.ejpam.v18i3.6553	NOUN
ejpam-6553	16	4	email	email	NOUN
ejpam-6553	16	5	address	address	NOUN
ejpam-6553	16	6	:	:	PUNCT
ejpam-6553	16	7	aaassiry@uqu.edu.sa	aaassiry@uqu.edu.sa	PROPN
ejpam-6553	16	8	(	(	PUNCT
ejpam-6553	16	9	a.	a.	NOUN
ejpam-6553	16	10	assiry	assiry	PROPN
ejpam-6553	16	11	)	)	PUNCT
ejpam-6553	16	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6553	17	1	1	1	NUM
ejpam-6553	17	2	copyright	copyright	NOUN
ejpam-6553	17	3	:	:	PUNCT
ejpam-6553	17	4	©	©	PROPN
ejpam-6553	17	5	2025	2025	NUM
ejpam-6553	17	6	the	the	DET
ejpam-6553	17	7	author(s	author(s	NOUN
ejpam-6553	17	8	)	)	PUNCT
ejpam-6553	17	9	.	.	PUNCT
ejpam-6553	18	1	(	(	PUNCT
ejpam-6553	18	2	cc	cc	NOUN
ejpam-6553	18	3	by	by	ADP
ejpam-6553	18	4	-	-	PUNCT
ejpam-6553	18	5	nc	nc	PROPN
ejpam-6553	18	6	4.0	4.0	NUM
ejpam-6553	18	7	)	)	PUNCT
ejpam-6553	18	8	a.	a.	NOUN
ejpam-6553	18	9	assiry	assiry	NOUN
ejpam-6553	18	10	/	/	SYM
ejpam-6553	18	11	eur	eur	PROPN
ejpam-6553	18	12	.	.	PUNCT
ejpam-6553	19	1	j.	j.	PROPN
ejpam-6553	19	2	pure	pure	PROPN
ejpam-6553	19	3	appl	appl	PROPN
ejpam-6553	19	4	.	.	PROPN
ejpam-6553	19	5	math	math	PROPN
ejpam-6553	19	6	,	,	PUNCT
ejpam-6553	19	7	18	18	NUM
ejpam-6553	19	8	(	(	PUNCT
ejpam-6553	19	9	3	3	NUM
ejpam-6553	19	10	)	)	PUNCT
ejpam-6553	19	11	(	(	PUNCT
ejpam-6553	19	12	2025	2025	NUM
ejpam-6553	19	13	)	)	PUNCT
ejpam-6553	19	14	,	,	PUNCT
ejpam-6553	19	15	6553	6553	NUM
ejpam-6553	19	16	2	2	NUM
ejpam-6553	19	17	of	of	ADP
ejpam-6553	19	18	11	11	NUM
ejpam-6553	19	19	from	from	ADP
ejpam-6553	19	20	the	the	DET
ejpam-6553	19	21	perspective	perspective	NOUN
ejpam-6553	19	22	of	of	ADP
ejpam-6553	19	23	pure	pure	ADJ
ejpam-6553	19	24	algebra	algebra	NOUN
ejpam-6553	19	25	,	,	PUNCT
ejpam-6553	19	26	the	the	DET
ejpam-6553	19	27	equation	equation	NOUN
ejpam-6553	19	28	23x	23x	NUM
ejpam-6553	19	29	+	+	ADJ
ejpam-6553	19	30	22y	22y	NOUN
ejpam-6553	19	31	=	=	SYM
ejpam-6553	19	32	z2	z2	PROPN
ejpam-6553	19	33	reveals	reveal	VERB
ejpam-6553	19	34	deep	deep	ADJ
ejpam-6553	19	35	connections	connection	NOUN
ejpam-6553	19	36	to	to	ADP
ejpam-6553	19	37	several	several	ADJ
ejpam-6553	19	38	active	active	ADJ
ejpam-6553	19	39	research	research	NOUN
ejpam-6553	19	40	areas	area	NOUN
ejpam-6553	19	41	.	.	PUNCT
ejpam-6553	20	1	recent	recent	ADJ
ejpam-6553	20	2	work	work	NOUN
ejpam-6553	20	3	by	by	ADP
ejpam-6553	20	4	amoroso	amoroso	PROPN
ejpam-6553	20	5	and	and	CCONJ
ejpam-6553	20	6	viada	viada	NOUN
ejpam-6553	20	7	[	[	X
ejpam-6553	20	8	3	3	X
ejpam-6553	20	9	]	]	PUNCT
ejpam-6553	20	10	on	on	ADP
ejpam-6553	20	11	small	small	ADJ
ejpam-6553	20	12	points	point	NOUN
ejpam-6553	20	13	on	on	ADP
ejpam-6553	20	14	subvarieties	subvarietie	NOUN
ejpam-6553	20	15	of	of	ADP
ejpam-6553	20	16	algebraic	algebraic	PROPN
ejpam-6553	20	17	tori	tori	PROPN
ejpam-6553	20	18	provides	provide	VERB
ejpam-6553	20	19	theoretical	theoretical	ADJ
ejpam-6553	20	20	tools	tool	NOUN
ejpam-6553	20	21	to	to	PART
ejpam-6553	20	22	study	study	VERB
ejpam-6553	20	23	the	the	DET
ejpam-6553	20	24	zariski	zariski	NOUN
ejpam-6553	20	25	closure	closure	NOUN
ejpam-6553	20	26	of	of	ADP
ejpam-6553	20	27	solution	solution	NOUN
ejpam-6553	20	28	sets	set	NOUN
ejpam-6553	20	29	for	for	ADP
ejpam-6553	20	30	such	such	ADJ
ejpam-6553	20	31	exponential	exponential	ADJ
ejpam-6553	20	32	-	-	PUNCT
ejpam-6553	20	33	polynomial	polynomial	ADJ
ejpam-6553	20	34	equations	equation	NOUN
ejpam-6553	20	35	.	.	PUNCT
ejpam-6553	21	1	the	the	DET
ejpam-6553	21	2	problem	problem	NOUN
ejpam-6553	21	3	naturally	naturally	ADV
ejpam-6553	21	4	embeds	embed	VERB
ejpam-6553	21	5	into	into	ADP
ejpam-6553	21	6	the	the	DET
ejpam-6553	21	7	framework	framework	NOUN
ejpam-6553	21	8	of	of	ADP
ejpam-6553	21	9	arithmetic	arithmetic	ADJ
ejpam-6553	21	10	dynamics	dynamic	NOUN
ejpam-6553	21	11	,	,	PUNCT
ejpam-6553	21	12	where	where	SCONJ
ejpam-6553	21	13	the	the	DET
ejpam-6553	21	14	iteration	iteration	NOUN
ejpam-6553	21	15	of	of	ADP
ejpam-6553	21	16	polynomial	polynomial	ADJ
ejpam-6553	21	17	maps	map	NOUN
ejpam-6553	21	18	(	(	PUNCT
ejpam-6553	21	19	in	in	ADP
ejpam-6553	21	20	this	this	DET
ejpam-6553	21	21	case	case	NOUN
ejpam-6553	21	22	,	,	PUNCT
ejpam-6553	21	23	exponential	exponential	ADJ
ejpam-6553	21	24	maps	map	NOUN
ejpam-6553	21	25	modulo	modulo	PROPN
ejpam-6553	21	26	m	m	VERB
ejpam-6553	21	27	)	)	PUNCT
ejpam-6553	21	28	interacts	interact	VERB
ejpam-6553	21	29	with	with	ADP
ejpam-6553	21	30	galois	galois	PROPN
ejpam-6553	21	31	representations	representation	NOUN
ejpam-6553	21	32	,	,	PUNCT
ejpam-6553	21	33	as	as	SCONJ
ejpam-6553	21	34	shown	show	VERB
ejpam-6553	21	35	in	in	ADP
ejpam-6553	21	36	the	the	DET
ejpam-6553	21	37	groundbreaking	groundbreake	VERB
ejpam-6553	21	38	work	work	NOUN
ejpam-6553	21	39	of	of	ADP
ejpam-6553	21	40	bell	bell	NOUN
ejpam-6553	21	41	et	et	PROPN
ejpam-6553	21	42	al	al	PROPN
ejpam-6553	21	43	.	.	PUNCT
ejpam-6553	22	1	[	[	X
ejpam-6553	22	2	4	4	X
ejpam-6553	22	3	]	]	PUNCT
ejpam-6553	22	4	on	on	ADP
ejpam-6553	22	5	dynamical	dynamical	ADJ
ejpam-6553	22	6	diophantine	diophantine	NOUN
ejpam-6553	22	7	equations	equation	NOUN
ejpam-6553	22	8	.	.	PUNCT
ejpam-6553	23	1	the	the	DET
ejpam-6553	23	2	algebraic	algebraic	ADJ
ejpam-6553	23	3	structure	structure	NOUN
ejpam-6553	23	4	becomes	become	VERB
ejpam-6553	23	5	particularly	particularly	ADV
ejpam-6553	23	6	evident	evident	ADJ
ejpam-6553	23	7	when	when	SCONJ
ejpam-6553	23	8	considering	consider	VERB
ejpam-6553	23	9	the	the	DET
ejpam-6553	23	10	equation	equation	NOUN
ejpam-6553	23	11	modulo	modulo	VERB
ejpam-6553	23	12	various	various	ADJ
ejpam-6553	23	13	primes	prime	NOUN
ejpam-6553	23	14	p	p	X
ejpam-6553	23	15	,	,	PUNCT
ejpam-6553	23	16	where	where	SCONJ
ejpam-6553	23	17	the	the	DET
ejpam-6553	23	18	solutions	solution	NOUN
ejpam-6553	23	19	trace	trace	VERB
ejpam-6553	23	20	out	out	ADP
ejpam-6553	23	21	algebraic	algebraic	ADJ
ejpam-6553	23	22	varieties	variety	NOUN
ejpam-6553	23	23	over	over	ADP
ejpam-6553	23	24	finite	finite	PROPN
ejpam-6553	23	25	fields	field	NOUN
ejpam-6553	23	26	fp	fp	PROPN
ejpam-6553	23	27	.	.	PUNCT
ejpam-6553	24	1	this	this	PRON
ejpam-6553	24	2	connects	connect	VERB
ejpam-6553	24	3	to	to	ADP
ejpam-6553	24	4	the	the	DET
ejpam-6553	24	5	lang	lang	PROPN
ejpam-6553	24	6	-	-	PUNCT
ejpam-6553	24	7	weil	weil	PROPN
ejpam-6553	24	8	estimates	estimate	NOUN
ejpam-6553	24	9	and	and	CCONJ
ejpam-6553	24	10	the	the	DET
ejpam-6553	24	11	work	work	NOUN
ejpam-6553	24	12	of	of	ADP
ejpam-6553	24	13	kowalski	kowalski	PROPN
ejpam-6553	25	1	[	[	X
ejpam-6553	25	2	5	5	NUM
ejpam-6553	25	3	]	]	PUNCT
ejpam-6553	25	4	on	on	ADP
ejpam-6553	25	5	exponential	exponential	ADJ
ejpam-6553	25	6	sums	sum	NOUN
ejpam-6553	25	7	over	over	ADP
ejpam-6553	25	8	finite	finite	ADJ
ejpam-6553	25	9	fields	field	NOUN
ejpam-6553	25	10	.	.	PUNCT
ejpam-6553	26	1	moreover	moreover	ADV
ejpam-6553	26	2	,	,	PUNCT
ejpam-6553	26	3	the	the	DET
ejpam-6553	26	4	equation	equation	NOUN
ejpam-6553	26	5	can	can	AUX
ejpam-6553	26	6	be	be	AUX
ejpam-6553	26	7	viewed	view	VERB
ejpam-6553	26	8	as	as	ADP
ejpam-6553	26	9	a	a	DET
ejpam-6553	26	10	special	special	ADJ
ejpam-6553	26	11	case	case	NOUN
ejpam-6553	26	12	of	of	ADP
ejpam-6553	26	13	the	the	DET
ejpam-6553	26	14	polynomial	polynomial	ADJ
ejpam-6553	26	15	-	-	PUNCT
ejpam-6553	26	16	exponential	exponential	ADJ
ejpam-6553	26	17	diophantine	diophantine	NOUN
ejpam-6553	26	18	problems	problem	NOUN
ejpam-6553	26	19	studied	study	VERB
ejpam-6553	26	20	by	by	ADP
ejpam-6553	26	21	bays	bay	NOUN
ejpam-6553	26	22	and	and	CCONJ
ejpam-6553	26	23	habegger	habegger	NOUN
ejpam-6553	26	24	[	[	X
ejpam-6553	26	25	6	6	NUM
ejpam-6553	26	26	]	]	PUNCT
ejpam-6553	26	27	,	,	PUNCT
ejpam-6553	26	28	where	where	SCONJ
ejpam-6553	26	29	they	they	PRON
ejpam-6553	26	30	apply	apply	VERB
ejpam-6553	26	31	techniques	technique	NOUN
ejpam-6553	26	32	from	from	ADP
ejpam-6553	26	33	o	o	NOUN
ejpam-6553	26	34	-	-	NOUN
ejpam-6553	26	35	minimality	minimality	NOUN
ejpam-6553	26	36	and	and	CCONJ
ejpam-6553	26	37	model	model	NOUN
ejpam-6553	26	38	theory	theory	NOUN
ejpam-6553	26	39	to	to	PART
ejpam-6553	26	40	establish	establish	VERB
ejpam-6553	26	41	finiteness	finiteness	ADJ
ejpam-6553	26	42	results	result	NOUN
ejpam-6553	26	43	.	.	PUNCT
ejpam-6553	27	1	these	these	DET
ejpam-6553	27	2	algebraic	algebraic	ADJ
ejpam-6553	27	3	perspectives	perspective	NOUN
ejpam-6553	27	4	not	not	PART
ejpam-6553	27	5	only	only	ADV
ejpam-6553	27	6	contextualize	contextualize	VERB
ejpam-6553	27	7	our	our	PRON
ejpam-6553	27	8	specific	specific	ADJ
ejpam-6553	27	9	equation	equation	NOUN
ejpam-6553	27	10	within	within	ADP
ejpam-6553	27	11	broader	broad	ADJ
ejpam-6553	27	12	theoretical	theoretical	ADJ
ejpam-6553	27	13	frameworks	framework	NOUN
ejpam-6553	27	14	but	but	CCONJ
ejpam-6553	27	15	also	also	ADV
ejpam-6553	27	16	suggest	suggest	VERB
ejpam-6553	27	17	potential	potential	ADJ
ejpam-6553	27	18	avenues	avenue	NOUN
ejpam-6553	27	19	for	for	ADP
ejpam-6553	27	20	generalization	generalization	NOUN
ejpam-6553	27	21	.	.	PUNCT
ejpam-6553	28	1	in	in	ADP
ejpam-6553	28	2	particular	particular	ADJ
ejpam-6553	28	3	,	,	PUNCT
ejpam-6553	28	4	the	the	DET
ejpam-6553	28	5	work	work	NOUN
ejpam-6553	28	6	of	of	ADP
ejpam-6553	28	7	scanlon	scanlon	PROPN
ejpam-6553	28	8	[	[	X
ejpam-6553	28	9	7	7	X
ejpam-6553	28	10	]	]	PUNCT
ejpam-6553	28	11	on	on	ADP
ejpam-6553	28	12	the	the	DET
ejpam-6553	28	13	model	model	NOUN
ejpam-6553	28	14	theory	theory	NOUN
ejpam-6553	28	15	of	of	ADP
ejpam-6553	28	16	fields	field	NOUN
ejpam-6553	28	17	with	with	ADP
ejpam-6553	28	18	exponents	exponent	NOUN
ejpam-6553	28	19	provides	provide	VERB
ejpam-6553	28	20	a	a	DET
ejpam-6553	28	21	powerful	powerful	ADJ
ejpam-6553	28	22	lens	lens	NOUN
ejpam-6553	28	23	through	through	ADP
ejpam-6553	28	24	which	which	PRON
ejpam-6553	28	25	to	to	PART
ejpam-6553	28	26	analyze	analyze	VERB
ejpam-6553	28	27	the	the	DET
ejpam-6553	28	28	decidability	decidability	NOUN
ejpam-6553	28	29	and	and	CCONJ
ejpam-6553	28	30	solution	solution	NOUN
ejpam-6553	28	31	structure	structure	NOUN
ejpam-6553	28	32	of	of	ADP
ejpam-6553	28	33	such	such	ADJ
ejpam-6553	28	34	equations	equation	NOUN
ejpam-6553	28	35	.	.	PUNCT
ejpam-6553	29	1	the	the	DET
ejpam-6553	29	2	particular	particular	ADJ
ejpam-6553	29	3	choice	choice	NOUN
ejpam-6553	29	4	of	of	ADP
ejpam-6553	29	5	23	23	NUM
ejpam-6553	29	6	and	and	CCONJ
ejpam-6553	29	7	22	22	NUM
ejpam-6553	29	8	is	be	AUX
ejpam-6553	29	9	not	not	PART
ejpam-6553	29	10	arbitrary	arbitrary	ADJ
ejpam-6553	29	11	:	:	PUNCT
ejpam-6553	29	12	•	•	NUM
ejpam-6553	29	13	23	23	NUM
ejpam-6553	29	14	is	be	AUX
ejpam-6553	29	15	a	a	DET
ejpam-6553	29	16	prime	prime	ADJ
ejpam-6553	29	17	number	number	NOUN
ejpam-6553	29	18	,	,	PUNCT
ejpam-6553	29	19	which	which	PRON
ejpam-6553	29	20	allows	allow	VERB
ejpam-6553	29	21	for	for	ADP
ejpam-6553	29	22	simplified	simplified	ADJ
ejpam-6553	29	23	modular	modular	ADJ
ejpam-6553	29	24	analysis	analysis	NOUN
ejpam-6553	29	25	[	[	X
ejpam-6553	29	26	8	8	NUM
ejpam-6553	29	27	]	]	PUNCT
ejpam-6553	29	28	.	.	PUNCT
ejpam-6553	30	1	•	•	NUM
ejpam-6553	30	2	22	22	NUM
ejpam-6553	30	3	=	=	SYM
ejpam-6553	30	4	23−	23−	NUM
ejpam-6553	30	5	1	1	NUM
ejpam-6553	30	6	introduces	introduce	NOUN
ejpam-6553	30	7	complexity	complexity	NOUN
ejpam-6553	30	8	via	via	ADP
ejpam-6553	30	9	its	its	PRON
ejpam-6553	30	10	composite	composite	ADJ
ejpam-6553	30	11	nature	nature	NOUN
ejpam-6553	30	12	,	,	PUNCT
ejpam-6553	30	13	requiring	require	VERB
ejpam-6553	30	14	more	more	ADV
ejpam-6553	30	15	nuanced	nuanced	ADJ
ejpam-6553	30	16	factorization	factorization	NOUN
ejpam-6553	30	17	techniques	technique	NOUN
ejpam-6553	30	18	[	[	X
ejpam-6553	30	19	9	9	NUM
ejpam-6553	30	20	]	]	PUNCT
ejpam-6553	30	21	.	.	PUNCT
ejpam-6553	31	1	•	•	NOUN
ejpam-6553	31	2	the	the	DET
ejpam-6553	31	3	structure	structure	NOUN
ejpam-6553	31	4	fits	fit	VERB
ejpam-6553	31	5	into	into	ADP
ejpam-6553	31	6	the	the	DET
ejpam-6553	31	7	broader	broad	ADJ
ejpam-6553	31	8	class	class	NOUN
ejpam-6553	31	9	of	of	ADP
ejpam-6553	31	10	equations	equation	NOUN
ejpam-6553	31	11	studied	study	VERB
ejpam-6553	31	12	by	by	ADP
ejpam-6553	31	13	chandoul	chandoul	PROPN
ejpam-6553	31	14	[	[	X
ejpam-6553	31	15	10	10	NUM
ejpam-6553	31	16	,	,	PUNCT
ejpam-6553	31	17	11	11	NUM
ejpam-6553	31	18	]	]	PUNCT
ejpam-6553	31	19	,	,	PUNCT
ejpam-6553	31	20	who	who	PRON
ejpam-6553	31	21	explored	explore	VERB
ejpam-6553	31	22	generalizations	generalization	NOUN
ejpam-6553	31	23	involving	involve	VERB
ejpam-6553	31	24	similar	similar	ADJ
ejpam-6553	31	25	exponential	exponential	ADJ
ejpam-6553	31	26	forms	form	NOUN
ejpam-6553	31	27	in	in	ADP
ejpam-6553	31	28	computational	computational	ADJ
ejpam-6553	31	29	number	number	NOUN
ejpam-6553	31	30	theory	theory	NOUN
ejpam-6553	31	31	.	.	PUNCT
ejpam-6553	32	1	special	special	ADJ
ejpam-6553	32	2	cases	case	NOUN
ejpam-6553	32	3	of	of	ADP
ejpam-6553	32	4	the	the	DET
ejpam-6553	32	5	equation	equation	NOUN
ejpam-6553	32	6	illustrate	illustrate	VERB
ejpam-6553	32	7	the	the	DET
ejpam-6553	32	8	algebraic	algebraic	ADJ
ejpam-6553	32	9	flavor	flavor	NOUN
ejpam-6553	32	10	:	:	PUNCT
ejpam-6553	32	11	•	•	NUM
ejpam-6553	32	12	setting	set	VERB
ejpam-6553	32	13	x	x	PUNCT
ejpam-6553	32	14	=	=	SYM
ejpam-6553	32	15	0	0	NUM
ejpam-6553	32	16	yields	yield	NOUN
ejpam-6553	32	17	1	1	NUM
ejpam-6553	32	18	+	+	NUM
ejpam-6553	32	19	22y	22y	NOUN
ejpam-6553	32	20	=	=	SYM
ejpam-6553	32	21	z2	z2	PROPN
ejpam-6553	32	22	,	,	PUNCT
ejpam-6553	32	23	which	which	PRON
ejpam-6553	32	24	can	can	AUX
ejpam-6553	32	25	be	be	AUX
ejpam-6553	32	26	analyzed	analyze	VERB
ejpam-6553	32	27	using	use	VERB
ejpam-6553	32	28	classical	classical	ADJ
ejpam-6553	32	29	results	result	NOUN
ejpam-6553	32	30	on	on	ADP
ejpam-6553	32	31	differences	difference	NOUN
ejpam-6553	32	32	of	of	ADP
ejpam-6553	32	33	squares	square	NOUN
ejpam-6553	32	34	.	.	PUNCT
ejpam-6553	33	1	•	•	NUM
ejpam-6553	33	2	setting	set	VERB
ejpam-6553	33	3	y	y	PROPN
ejpam-6553	33	4	=	=	SYM
ejpam-6553	33	5	0	0	PROPN
ejpam-6553	33	6	gives	give	VERB
ejpam-6553	33	7	23x	23x	NUM
ejpam-6553	33	8	+	+	CCONJ
ejpam-6553	33	9	1	1	NUM
ejpam-6553	33	10	=	=	SYM
ejpam-6553	33	11	z2	z2	PROPN
ejpam-6553	33	12	,	,	PUNCT
ejpam-6553	33	13	which	which	PRON
ejpam-6553	33	14	connects	connect	VERB
ejpam-6553	33	15	to	to	ADP
ejpam-6553	33	16	the	the	DET
ejpam-6553	33	17	well	well	ADV
ejpam-6553	33	18	-	-	PUNCT
ejpam-6553	33	19	known	know	VERB
ejpam-6553	33	20	ramanujan	ramanujan	NOUN
ejpam-6553	33	21	–	–	PUNCT
ejpam-6553	33	22	nagell	nagell	ADJ
ejpam-6553	33	23	equation	equation	NOUN
ejpam-6553	33	24	[	[	X
ejpam-6553	33	25	12	12	NUM
ejpam-6553	33	26	]	]	PUNCT
ejpam-6553	33	27	.	.	PUNCT
ejpam-6553	34	1	our	our	PRON
ejpam-6553	34	2	proof	proof	NOUN
ejpam-6553	34	3	synthesizes	synthesize	VERB
ejpam-6553	34	4	several	several	ADJ
ejpam-6553	34	5	techniques	technique	NOUN
ejpam-6553	34	6	:	:	PUNCT
ejpam-6553	34	7	•	•	NUM
ejpam-6553	34	8	we	we	PRON
ejpam-6553	34	9	begin	begin	VERB
ejpam-6553	34	10	with	with	ADP
ejpam-6553	34	11	modular	modular	ADJ
ejpam-6553	34	12	arithmetic	arithmetic	ADJ
ejpam-6553	34	13	to	to	PART
ejpam-6553	34	14	impose	impose	VERB
ejpam-6553	34	15	congruence	congruence	NOUN
ejpam-6553	34	16	conditions	condition	NOUN
ejpam-6553	34	17	on	on	ADP
ejpam-6553	34	18	x	x	PUNCT
ejpam-6553	34	19	and	and	CCONJ
ejpam-6553	34	20	y.	y.	PROPN
ejpam-6553	34	21	•	•	PROPN
ejpam-6553	34	22	we	we	PRON
ejpam-6553	34	23	apply	apply	VERB
ejpam-6553	34	24	baker	baker	PROPN
ejpam-6553	34	25	’s	’s	PART
ejpam-6553	34	26	theory	theory	NOUN
ejpam-6553	34	27	to	to	PART
ejpam-6553	34	28	obtain	obtain	VERB
ejpam-6553	34	29	explicit	explicit	ADJ
ejpam-6553	34	30	bounds	bound	NOUN
ejpam-6553	34	31	for	for	ADP
ejpam-6553	34	32	the	the	DET
ejpam-6553	34	33	exponents	exponent	NOUN
ejpam-6553	34	34	.	.	PUNCT
ejpam-6553	35	1	•	•	INTJ
ejpam-6553	35	2	we	we	PRON
ejpam-6553	35	3	use	use	VERB
ejpam-6553	35	4	computational	computational	ADJ
ejpam-6553	35	5	tools	tool	NOUN
ejpam-6553	35	6	to	to	PART
ejpam-6553	35	7	exhaust	exhaust	VERB
ejpam-6553	35	8	all	all	DET
ejpam-6553	35	9	remaining	remain	VERB
ejpam-6553	35	10	small	small	ADJ
ejpam-6553	35	11	cases	case	NOUN
ejpam-6553	35	12	.	.	PUNCT
ejpam-6553	36	1	this	this	DET
ejpam-6553	36	2	approach	approach	NOUN
ejpam-6553	36	3	is	be	AUX
ejpam-6553	36	4	inspired	inspire	VERB
ejpam-6553	36	5	by	by	ADP
ejpam-6553	36	6	the	the	DET
ejpam-6553	36	7	works	work	NOUN
ejpam-6553	36	8	of	of	ADP
ejpam-6553	36	9	bennett	bennett	PROPN
ejpam-6553	36	10	and	and	CCONJ
ejpam-6553	36	11	skinner	skinner	NOUN
ejpam-6553	37	1	[	[	X
ejpam-6553	37	2	13	13	NUM
ejpam-6553	37	3	]	]	PUNCT
ejpam-6553	37	4	,	,	PUNCT
ejpam-6553	37	5	and	and	CCONJ
ejpam-6553	37	6	further	far	ADV
ejpam-6553	37	7	refined	refine	VERB
ejpam-6553	37	8	in	in	ADP
ejpam-6553	37	9	the	the	DET
ejpam-6553	37	10	computational	computational	ADJ
ejpam-6553	37	11	strategies	strategy	NOUN
ejpam-6553	37	12	of	of	ADP
ejpam-6553	37	13	de	de	X
ejpam-6553	37	14	weger	weger	NOUN
ejpam-6553	38	1	[	[	X
ejpam-6553	38	2	14	14	NUM
ejpam-6553	38	3	]	]	PUNCT
ejpam-6553	38	4	and	and	CCONJ
ejpam-6553	39	1	smart	smart	ADJ
ejpam-6553	39	2	[	[	X
ejpam-6553	39	3	15	15	NUM
ejpam-6553	39	4	]	]	PUNCT
ejpam-6553	39	5	.	.	PUNCT
ejpam-6553	40	1	our	our	PRON
ejpam-6553	40	2	analysis	analysis	NOUN
ejpam-6553	40	3	also	also	ADV
ejpam-6553	40	4	draws	draw	VERB
ejpam-6553	40	5	a.	a.	NOUN
ejpam-6553	40	6	assiry	assiry	NOUN
ejpam-6553	40	7	/	/	SYM
ejpam-6553	40	8	eur	eur	PROPN
ejpam-6553	40	9	.	.	PUNCT
ejpam-6553	41	1	j.	j.	PROPN
ejpam-6553	41	2	pure	pure	PROPN
ejpam-6553	41	3	appl	appl	PROPN
ejpam-6553	41	4	.	.	PROPN
ejpam-6553	41	5	math	math	PROPN
ejpam-6553	41	6	,	,	PUNCT
ejpam-6553	41	7	18	18	NUM
ejpam-6553	41	8	(	(	PUNCT
ejpam-6553	41	9	3	3	NUM
ejpam-6553	41	10	)	)	PUNCT
ejpam-6553	41	11	(	(	PUNCT
ejpam-6553	41	12	2025	2025	NUM
ejpam-6553	41	13	)	)	PUNCT
ejpam-6553	41	14	,	,	PUNCT
ejpam-6553	41	15	6553	6553	NUM
ejpam-6553	41	16	3	3	NUM
ejpam-6553	41	17	of	of	ADP
ejpam-6553	41	18	11	11	NUM
ejpam-6553	41	19	on	on	ADP
ejpam-6553	41	20	results	result	NOUN
ejpam-6553	41	21	related	relate	VERB
ejpam-6553	41	22	to	to	ADP
ejpam-6553	41	23	generalized	generalized	ADJ
ejpam-6553	41	24	ramanujan	ramanujan	PROPN
ejpam-6553	41	25	–	–	PUNCT
ejpam-6553	41	26	nagell	nagell	ADJ
ejpam-6553	41	27	equations	equation	NOUN
ejpam-6553	42	1	[	[	X
ejpam-6553	42	2	16	16	NUM
ejpam-6553	42	3	,	,	PUNCT
ejpam-6553	42	4	17	17	NUM
ejpam-6553	42	5	]	]	PUNCT
ejpam-6553	42	6	,	,	PUNCT
ejpam-6553	43	1	and	and	CCONJ
ejpam-6553	43	2	follows	follow	VERB
ejpam-6553	43	3	in	in	ADP
ejpam-6553	43	4	the	the	DET
ejpam-6553	43	5	tradition	tradition	NOUN
ejpam-6553	43	6	of	of	ADP
ejpam-6553	43	7	foundational	foundational	ADJ
ejpam-6553	43	8	contributions	contribution	NOUN
ejpam-6553	43	9	by	by	ADP
ejpam-6553	43	10	sroysang	sroysang	PROPN
ejpam-6553	44	1	[	[	X
ejpam-6553	44	2	18	18	NUM
ejpam-6553	44	3	]	]	PUNCT
ejpam-6553	44	4	and	and	CCONJ
ejpam-6553	44	5	gayo	gayo	VERB
ejpam-6553	44	6	[	[	X
ejpam-6553	44	7	19	19	NUM
ejpam-6553	44	8	]	]	PUNCT
ejpam-6553	44	9	.	.	PUNCT
ejpam-6553	45	1	in	in	ADP
ejpam-6553	45	2	conclusion	conclusion	NOUN
ejpam-6553	45	3	,	,	PUNCT
ejpam-6553	45	4	the	the	DET
ejpam-6553	45	5	equation	equation	NOUN
ejpam-6553	45	6	23x	23x	NUM
ejpam-6553	45	7	+	+	ADJ
ejpam-6553	45	8	22y	22y	NOUN
ejpam-6553	45	9	=	=	SYM
ejpam-6553	45	10	z2	z2	PROPN
ejpam-6553	45	11	serves	serve	VERB
ejpam-6553	45	12	as	as	ADP
ejpam-6553	45	13	a	a	DET
ejpam-6553	45	14	rich	rich	ADJ
ejpam-6553	45	15	example	example	NOUN
ejpam-6553	45	16	of	of	ADP
ejpam-6553	45	17	how	how	SCONJ
ejpam-6553	45	18	diverse	diverse	ADJ
ejpam-6553	45	19	techniques	technique	NOUN
ejpam-6553	45	20	—	—	PUNCT
ejpam-6553	45	21	ranging	range	VERB
ejpam-6553	45	22	from	from	ADP
ejpam-6553	45	23	theoretical	theoretical	ADJ
ejpam-6553	45	24	number	number	NOUN
ejpam-6553	45	25	theory	theory	NOUN
ejpam-6553	45	26	to	to	ADP
ejpam-6553	45	27	explicit	explicit	ADJ
ejpam-6553	45	28	computation	computation	NOUN
ejpam-6553	45	29	—	—	PUNCT
ejpam-6553	45	30	can	can	AUX
ejpam-6553	45	31	be	be	AUX
ejpam-6553	45	32	combined	combine	VERB
ejpam-6553	45	33	to	to	PART
ejpam-6553	45	34	resolve	resolve	VERB
ejpam-6553	45	35	deep	deep	ADJ
ejpam-6553	45	36	diophantine	diophantine	NOUN
ejpam-6553	45	37	problems	problem	NOUN
ejpam-6553	45	38	.	.	PUNCT
ejpam-6553	46	1	this	this	DET
ejpam-6553	46	2	work	work	NOUN
ejpam-6553	46	3	contributes	contribute	VERB
ejpam-6553	46	4	to	to	ADP
ejpam-6553	46	5	the	the	DET
ejpam-6553	46	6	broader	broad	ADJ
ejpam-6553	46	7	understanding	understanding	NOUN
ejpam-6553	46	8	of	of	ADP
ejpam-6553	46	9	exponential	exponential	ADJ
ejpam-6553	46	10	equations	equation	NOUN
ejpam-6553	46	11	and	and	CCONJ
ejpam-6553	46	12	highlights	highlight	NOUN
ejpam-6553	46	13	the	the	DET
ejpam-6553	46	14	role	role	NOUN
ejpam-6553	46	15	of	of	ADP
ejpam-6553	46	16	modern	modern	ADJ
ejpam-6553	46	17	tools	tool	NOUN
ejpam-6553	46	18	in	in	ADP
ejpam-6553	46	19	resolving	resolve	VERB
ejpam-6553	46	20	classical	classical	ADJ
ejpam-6553	46	21	questions	question	NOUN
ejpam-6553	46	22	.	.	PUNCT
ejpam-6553	47	1	2	2	X
ejpam-6553	47	2	.	.	X
ejpam-6553	47	3	preliminaries	preliminary	NOUN
ejpam-6553	47	4	in	in	ADP
ejpam-6553	47	5	this	this	DET
ejpam-6553	47	6	section	section	NOUN
ejpam-6553	47	7	,	,	PUNCT
ejpam-6553	47	8	we	we	PRON
ejpam-6553	47	9	introduce	introduce	VERB
ejpam-6553	47	10	the	the	DET
ejpam-6553	47	11	necessary	necessary	ADJ
ejpam-6553	47	12	definitions	definition	NOUN
ejpam-6553	47	13	,	,	PUNCT
ejpam-6553	47	14	notation	notation	NOUN
ejpam-6553	47	15	,	,	PUNCT
ejpam-6553	47	16	and	and	CCONJ
ejpam-6553	47	17	key	key	ADJ
ejpam-6553	47	18	lemmas	lemma	NOUN
ejpam-6553	47	19	that	that	PRON
ejpam-6553	47	20	form	form	VERB
ejpam-6553	47	21	the	the	DET
ejpam-6553	47	22	foundation	foundation	NOUN
ejpam-6553	47	23	of	of	ADP
ejpam-6553	47	24	our	our	PRON
ejpam-6553	47	25	work	work	NOUN
ejpam-6553	47	26	.	.	PUNCT
ejpam-6553	48	1	these	these	DET
ejpam-6553	48	2	tools	tool	NOUN
ejpam-6553	48	3	will	will	AUX
ejpam-6553	48	4	be	be	AUX
ejpam-6553	48	5	essential	essential	ADJ
ejpam-6553	48	6	for	for	ADP
ejpam-6553	48	7	the	the	DET
ejpam-6553	48	8	proofs	proof	NOUN
ejpam-6553	48	9	and	and	CCONJ
ejpam-6553	48	10	results	result	NOUN
ejpam-6553	48	11	presented	present	VERB
ejpam-6553	48	12	in	in	ADP
ejpam-6553	48	13	subsequent	subsequent	ADJ
ejpam-6553	48	14	sections	section	NOUN
ejpam-6553	48	15	.	.	PUNCT
ejpam-6553	49	1	2.1	2.1	NUM
ejpam-6553	49	2	.	.	PUNCT
ejpam-6553	49	3	definitions	definition	NOUN
ejpam-6553	49	4	we	we	PRON
ejpam-6553	49	5	begin	begin	VERB
ejpam-6553	49	6	by	by	ADP
ejpam-6553	49	7	defining	define	VERB
ejpam-6553	49	8	the	the	DET
ejpam-6553	49	9	central	central	ADJ
ejpam-6553	49	10	objects	object	NOUN
ejpam-6553	49	11	of	of	ADP
ejpam-6553	49	12	study	study	NOUN
ejpam-6553	49	13	in	in	ADP
ejpam-6553	49	14	this	this	DET
ejpam-6553	49	15	paper	paper	NOUN
ejpam-6553	49	16	.	.	PUNCT
ejpam-6553	50	1	definition	definition	NOUN
ejpam-6553	50	2	1	1	NUM
ejpam-6553	50	3	(	(	PUNCT
ejpam-6553	50	4	exponential	exponential	ADJ
ejpam-6553	50	5	diophantine	diophantine	NOUN
ejpam-6553	50	6	equation	equation	NOUN
ejpam-6553	50	7	)	)	PUNCT
ejpam-6553	50	8	.	.	PUNCT
ejpam-6553	51	1	an	an	DET
ejpam-6553	51	2	exponential	exponential	ADJ
ejpam-6553	51	3	diophantine	diophantine	NOUN
ejpam-6553	51	4	equation	equation	NOUN
ejpam-6553	51	5	is	be	AUX
ejpam-6553	51	6	an	an	DET
ejpam-6553	51	7	equation	equation	NOUN
ejpam-6553	51	8	of	of	ADP
ejpam-6553	51	9	the	the	DET
ejpam-6553	51	10	form	form	NOUN
ejpam-6553	51	11	px	px	X
ejpam-6553	51	12	+	+	CCONJ
ejpam-6553	51	13	qy	qy	PROPN
ejpam-6553	51	14	=	=	SYM
ejpam-6553	51	15	zk	zk	PROPN
ejpam-6553	51	16	,	,	PUNCT
ejpam-6553	51	17	where	where	SCONJ
ejpam-6553	51	18	p	p	X
ejpam-6553	51	19	,	,	PUNCT
ejpam-6553	51	20	q	q	INTJ
ejpam-6553	51	21	,	,	PUNCT
ejpam-6553	51	22	k	k	PROPN
ejpam-6553	51	23	are	be	AUX
ejpam-6553	51	24	fixed	fix	VERB
ejpam-6553	51	25	integers	integer	NOUN
ejpam-6553	51	26	,	,	PUNCT
ejpam-6553	51	27	and	and	CCONJ
ejpam-6553	51	28	x	x	X
ejpam-6553	51	29	,	,	PUNCT
ejpam-6553	51	30	y	y	PROPN
ejpam-6553	51	31	,	,	PUNCT
ejpam-6553	51	32	z	z	PROPN
ejpam-6553	51	33	are	be	AUX
ejpam-6553	51	34	nonnegative	nonnegative	ADJ
ejpam-6553	51	35	integer	integer	NOUN
ejpam-6553	51	36	variables	variable	NOUN
ejpam-6553	51	37	.	.	PUNCT
ejpam-6553	52	1	such	such	ADJ
ejpam-6553	52	2	equations	equation	NOUN
ejpam-6553	52	3	are	be	AUX
ejpam-6553	52	4	of	of	ADP
ejpam-6553	52	5	significant	significant	ADJ
ejpam-6553	52	6	interest	interest	NOUN
ejpam-6553	52	7	in	in	ADP
ejpam-6553	52	8	number	number	NOUN
ejpam-6553	52	9	theory	theory	NOUN
ejpam-6553	52	10	due	due	ADP
ejpam-6553	52	11	to	to	ADP
ejpam-6553	52	12	their	their	PRON
ejpam-6553	52	13	connections	connection	NOUN
ejpam-6553	52	14	to	to	PART
ejpam-6553	52	15	diophantine	diophantine	VERB
ejpam-6553	52	16	analysis	analysis	NOUN
ejpam-6553	52	17	,	,	PUNCT
ejpam-6553	52	18	modular	modular	ADJ
ejpam-6553	52	19	arithmetic	arithmetic	ADJ
ejpam-6553	52	20	,	,	PUNCT
ejpam-6553	52	21	and	and	CCONJ
ejpam-6553	52	22	the	the	DET
ejpam-6553	52	23	study	study	NOUN
ejpam-6553	52	24	of	of	ADP
ejpam-6553	52	25	integer	integer	NOUN
ejpam-6553	52	26	solutions	solution	NOUN
ejpam-6553	52	27	to	to	ADP
ejpam-6553	52	28	polynomial	polynomial	ADJ
ejpam-6553	52	29	equations	equation	NOUN
ejpam-6553	52	30	.	.	PUNCT
ejpam-6553	53	1	definition	definition	NOUN
ejpam-6553	53	2	2	2	NUM
ejpam-6553	53	3	(	(	PUNCT
ejpam-6553	53	4	modular	modular	ADJ
ejpam-6553	53	5	congruence	congruence	NOUN
ejpam-6553	53	6	)	)	PUNCT
ejpam-6553	53	7	.	.	PUNCT
ejpam-6553	54	1	for	for	ADP
ejpam-6553	54	2	integers	integer	NOUN
ejpam-6553	54	3	a	a	DET
ejpam-6553	54	4	,	,	PUNCT
ejpam-6553	54	5	b	b	NOUN
ejpam-6553	54	6	,	,	PUNCT
ejpam-6553	54	7	and	and	CCONJ
ejpam-6553	54	8	m	m	X
ejpam-6553	54	9	,	,	PUNCT
ejpam-6553	54	10	we	we	PRON
ejpam-6553	54	11	say	say	VERB
ejpam-6553	54	12	a	a	PRON
ejpam-6553	54	13	is	be	AUX
ejpam-6553	54	14	congruent	congruent	ADJ
ejpam-6553	54	15	to	to	ADP
ejpam-6553	54	16	b	b	PROPN
ejpam-6553	54	17	modulo	modulo	PROPN
ejpam-6553	54	18	m	m	VERB
ejpam-6553	54	19	,	,	PUNCT
ejpam-6553	54	20	denoted	denote	VERB
ejpam-6553	54	21	a	a	DET
ejpam-6553	54	22	≡	≡	PROPN
ejpam-6553	54	23	b	b	PROPN
ejpam-6553	54	24	(	(	PUNCT
ejpam-6553	54	25	mod	mod	PROPN
ejpam-6553	54	26	m	m	PROPN
ejpam-6553	54	27	)	)	PUNCT
ejpam-6553	54	28	,	,	PUNCT
ejpam-6553	54	29	if	if	SCONJ
ejpam-6553	54	30	m	m	NOUN
ejpam-6553	54	31	divides	divide	VERB
ejpam-6553	54	32	a	a	DET
ejpam-6553	54	33	−	−	PROPN
ejpam-6553	54	34	b.	b.	NOUN
ejpam-6553	54	35	this	this	DET
ejpam-6553	54	36	equivalence	equivalence	NOUN
ejpam-6553	54	37	relation	relation	NOUN
ejpam-6553	54	38	is	be	AUX
ejpam-6553	54	39	fundamental	fundamental	ADJ
ejpam-6553	54	40	in	in	ADP
ejpam-6553	54	41	studying	study	VERB
ejpam-6553	54	42	the	the	DET
ejpam-6553	54	43	properties	property	NOUN
ejpam-6553	54	44	of	of	ADP
ejpam-6553	54	45	integers	integer	NOUN
ejpam-6553	54	46	and	and	CCONJ
ejpam-6553	54	47	their	their	PRON
ejpam-6553	54	48	residues	residue	NOUN
ejpam-6553	54	49	.	.	PUNCT
ejpam-6553	55	1	2.2	2.2	NUM
ejpam-6553	55	2	.	.	PUNCT
ejpam-6553	56	1	key	key	ADJ
ejpam-6553	56	2	lemmas	lemma	NOUN
ejpam-6553	56	3	the	the	DET
ejpam-6553	56	4	following	follow	VERB
ejpam-6553	56	5	lemmas	lemma	NOUN
ejpam-6553	56	6	provide	provide	VERB
ejpam-6553	56	7	critical	critical	ADJ
ejpam-6553	56	8	properties	property	NOUN
ejpam-6553	56	9	of	of	ADP
ejpam-6553	56	10	squares	square	NOUN
ejpam-6553	56	11	and	and	CCONJ
ejpam-6553	56	12	modular	modular	ADJ
ejpam-6553	56	13	arithmetic	arithmetic	NOUN
ejpam-6553	56	14	that	that	PRON
ejpam-6553	56	15	will	will	AUX
ejpam-6553	56	16	be	be	AUX
ejpam-6553	56	17	used	use	VERB
ejpam-6553	56	18	extensively	extensively	ADV
ejpam-6553	56	19	in	in	ADP
ejpam-6553	56	20	our	our	PRON
ejpam-6553	56	21	proofs	proof	NOUN
ejpam-6553	56	22	.	.	PUNCT
ejpam-6553	57	1	lemma	lemma	PROPN
ejpam-6553	57	2	1	1	NUM
ejpam-6553	57	3	(	(	PUNCT
ejpam-6553	57	4	modulo	modulo	NOUN
ejpam-6553	57	5	4	4	NUM
ejpam-6553	57	6	property	property	NOUN
ejpam-6553	57	7	of	of	ADP
ejpam-6553	57	8	squares	square	NOUN
ejpam-6553	57	9	)	)	PUNCT
ejpam-6553	57	10	.	.	PUNCT
ejpam-6553	58	1	for	for	ADP
ejpam-6553	58	2	any	any	DET
ejpam-6553	58	3	integer	integer	NOUN
ejpam-6553	58	4	z	z	PROPN
ejpam-6553	58	5	,	,	PUNCT
ejpam-6553	58	6	the	the	DET
ejpam-6553	58	7	square	square	ADJ
ejpam-6553	58	8	z2	z2	PROPN
ejpam-6553	58	9	satisfies	satisfie	NOUN
ejpam-6553	58	10	:	:	PUNCT
ejpam-6553	58	11	z2	z2	PROPN
ejpam-6553	58	12	≡	≡	PROPN
ejpam-6553	58	13	0	0	NUM
ejpam-6553	58	14	or	or	CCONJ
ejpam-6553	58	15	1	1	NUM
ejpam-6553	58	16	(	(	PUNCT
ejpam-6553	58	17	mod	mod	NOUN
ejpam-6553	58	18	4	4	NUM
ejpam-6553	58	19	)	)	PUNCT
ejpam-6553	58	20	.	.	PUNCT
ejpam-6553	59	1	proof	proof	NOUN
ejpam-6553	59	2	.	.	PUNCT
ejpam-6553	60	1	to	to	PART
ejpam-6553	60	2	prove	prove	VERB
ejpam-6553	60	3	this	this	DET
ejpam-6553	60	4	lemma	lemma	PROPN
ejpam-6553	60	5	,	,	PUNCT
ejpam-6553	60	6	we	we	PRON
ejpam-6553	60	7	consider	consider	VERB
ejpam-6553	60	8	all	all	DET
ejpam-6553	60	9	possible	possible	ADJ
ejpam-6553	60	10	residues	residue	NOUN
ejpam-6553	60	11	of	of	ADP
ejpam-6553	60	12	z	z	NOUN
ejpam-6553	60	13	modulo	modulo	PROPN
ejpam-6553	60	14	4	4	NUM
ejpam-6553	60	15	.	.	PUNCT
ejpam-6553	61	1	since	since	SCONJ
ejpam-6553	61	2	any	any	DET
ejpam-6553	61	3	integer	integer	NOUN
ejpam-6553	61	4	z	z	NOUN
ejpam-6553	61	5	is	be	AUX
ejpam-6553	61	6	congruent	congruent	ADJ
ejpam-6553	61	7	to	to	ADP
ejpam-6553	61	8	one	one	NUM
ejpam-6553	61	9	of	of	ADP
ejpam-6553	61	10	0	0	NUM
ejpam-6553	61	11	,	,	PUNCT
ejpam-6553	61	12	1	1	NUM
ejpam-6553	61	13	,	,	PUNCT
ejpam-6553	61	14	2	2	NUM
ejpam-6553	61	15	,	,	PUNCT
ejpam-6553	61	16	or	or	CCONJ
ejpam-6553	61	17	3	3	NUM
ejpam-6553	61	18	modulo	modulo	NOUN
ejpam-6553	61	19	4	4	NUM
ejpam-6553	61	20	,	,	PUNCT
ejpam-6553	61	21	we	we	PRON
ejpam-6553	61	22	analyze	analyze	VERB
ejpam-6553	61	23	each	each	DET
ejpam-6553	61	24	case	case	NOUN
ejpam-6553	61	25	separately	separately	ADV
ejpam-6553	61	26	.	.	PUNCT
ejpam-6553	62	1	a.	a.	NOUN
ejpam-6553	62	2	assiry	assiry	PROPN
ejpam-6553	62	3	/	/	SYM
ejpam-6553	62	4	eur	eur	PROPN
ejpam-6553	62	5	.	.	PUNCT
ejpam-6553	63	1	j.	j.	PROPN
ejpam-6553	63	2	pure	pure	PROPN
ejpam-6553	63	3	appl	appl	PROPN
ejpam-6553	63	4	.	.	PROPN
ejpam-6553	63	5	math	math	PROPN
ejpam-6553	63	6	,	,	PUNCT
ejpam-6553	63	7	18	18	NUM
ejpam-6553	63	8	(	(	PUNCT
ejpam-6553	63	9	3	3	NUM
ejpam-6553	63	10	)	)	PUNCT
ejpam-6553	63	11	(	(	PUNCT
ejpam-6553	63	12	2025	2025	NUM
ejpam-6553	63	13	)	)	PUNCT
ejpam-6553	63	14	,	,	PUNCT
ejpam-6553	63	15	6553	6553	NUM
ejpam-6553	63	16	4	4	NUM
ejpam-6553	63	17	of	of	ADP
ejpam-6553	63	18	11	11	NUM
ejpam-6553	63	19	•	•	NOUN
ejpam-6553	63	20	case	case	NOUN
ejpam-6553	63	21	1	1	NUM
ejpam-6553	63	22	:	:	PUNCT
ejpam-6553	63	23	z	z	PROPN
ejpam-6553	63	24	≡	≡	PROPN
ejpam-6553	63	25	0	0	PUNCT
ejpam-6553	64	1	(	(	PUNCT
ejpam-6553	64	2	mod	mod	PROPN
ejpam-6553	64	3	4	4	NUM
ejpam-6553	64	4	)	)	PUNCT
ejpam-6553	64	5	.	.	PUNCT
ejpam-6553	65	1	z	z	X
ejpam-6553	66	1	=	=	PUNCT
ejpam-6553	66	2	4k	4k	NOUN
ejpam-6553	66	3	for	for	ADP
ejpam-6553	66	4	some	some	DET
ejpam-6553	66	5	integer	integer	NOUN
ejpam-6553	66	6	k	k	PROPN
ejpam-6553	66	7	,	,	PUNCT
ejpam-6553	66	8	z2	z2	PROPN
ejpam-6553	66	9	=	=	SYM
ejpam-6553	66	10	(	(	PUNCT
ejpam-6553	66	11	4k)2	4k)2	X
ejpam-6553	66	12	=	=	SYM
ejpam-6553	66	13	16k2	16k2	NUM
ejpam-6553	66	14	≡	≡	PROPN
ejpam-6553	66	15	0	0	PUNCT
ejpam-6553	67	1	(	(	PUNCT
ejpam-6553	67	2	mod	mod	PROPN
ejpam-6553	67	3	4	4	NUM
ejpam-6553	67	4	)	)	PUNCT
ejpam-6553	67	5	.	.	PUNCT
ejpam-6553	68	1	thus	thus	ADV
ejpam-6553	68	2	,	,	PUNCT
ejpam-6553	68	3	z2	z2	PROPN
ejpam-6553	68	4	≡	≡	PROPN
ejpam-6553	68	5	0	0	PUNCT
ejpam-6553	68	6	(	(	PUNCT
ejpam-6553	68	7	mod	mod	PROPN
ejpam-6553	68	8	4	4	NUM
ejpam-6553	68	9	)	)	PUNCT
ejpam-6553	68	10	.	.	PUNCT
ejpam-6553	69	1	•	•	NUM
ejpam-6553	69	2	case	case	NOUN
ejpam-6553	69	3	2	2	NUM
ejpam-6553	69	4	:	:	PUNCT
ejpam-6553	69	5	z	z	PROPN
ejpam-6553	69	6	≡	≡	PROPN
ejpam-6553	69	7	1	1	NUM
ejpam-6553	69	8	(	(	PUNCT
ejpam-6553	69	9	mod	mod	NOUN
ejpam-6553	69	10	4	4	NUM
ejpam-6553	69	11	)	)	PUNCT
ejpam-6553	69	12	.	.	PUNCT
ejpam-6553	70	1	z	z	X
ejpam-6553	71	1	=	=	PUNCT
ejpam-6553	71	2	4k	4k	NOUN
ejpam-6553	71	3	+	+	CCONJ
ejpam-6553	71	4	1	1	NUM
ejpam-6553	71	5	for	for	ADP
ejpam-6553	71	6	some	some	DET
ejpam-6553	71	7	integer	integer	NOUN
ejpam-6553	71	8	k	k	PROPN
ejpam-6553	71	9	,	,	PUNCT
ejpam-6553	71	10	z2	z2	PROPN
ejpam-6553	71	11	=	=	SYM
ejpam-6553	71	12	(	(	PUNCT
ejpam-6553	71	13	4k	4k	X
ejpam-6553	71	14	+	+	X
ejpam-6553	71	15	1)2	1)2	NUM
ejpam-6553	71	16	=	=	SYM
ejpam-6553	71	17	16k2	16k2	NUM
ejpam-6553	72	1	+	+	NUM
ejpam-6553	72	2	8k	8k	NOUN
ejpam-6553	72	3	+	+	CCONJ
ejpam-6553	72	4	1	1	NUM
ejpam-6553	72	5	≡	≡	PROPN
ejpam-6553	72	6	1	1	NUM
ejpam-6553	72	7	(	(	PUNCT
ejpam-6553	72	8	mod	mod	NOUN
ejpam-6553	72	9	4	4	NUM
ejpam-6553	72	10	)	)	PUNCT
ejpam-6553	72	11	.	.	PUNCT
ejpam-6553	73	1	thus	thus	ADV
ejpam-6553	73	2	,	,	PUNCT
ejpam-6553	73	3	z2	z2	PROPN
ejpam-6553	73	4	≡	≡	PROPN
ejpam-6553	73	5	1	1	NUM
ejpam-6553	73	6	(	(	PUNCT
ejpam-6553	73	7	mod	mod	NOUN
ejpam-6553	73	8	4	4	NUM
ejpam-6553	73	9	)	)	PUNCT
ejpam-6553	73	10	.	.	PUNCT
ejpam-6553	74	1	•	•	NUM
ejpam-6553	74	2	case	case	NOUN
ejpam-6553	74	3	3	3	NUM
ejpam-6553	74	4	:	:	PUNCT
ejpam-6553	74	5	z	z	PROPN
ejpam-6553	74	6	≡	≡	PROPN
ejpam-6553	74	7	2	2	NUM
ejpam-6553	74	8	(	(	PUNCT
ejpam-6553	74	9	mod	mod	NOUN
ejpam-6553	74	10	4	4	NUM
ejpam-6553	74	11	)	)	PUNCT
ejpam-6553	74	12	.	.	PUNCT
ejpam-6553	75	1	z	z	X
ejpam-6553	76	1	=	=	PUNCT
ejpam-6553	76	2	4k	4k	NOUN
ejpam-6553	76	3	+	+	CCONJ
ejpam-6553	76	4	2	2	NUM
ejpam-6553	76	5	for	for	ADP
ejpam-6553	76	6	some	some	DET
ejpam-6553	76	7	integer	integer	NOUN
ejpam-6553	76	8	k	k	PROPN
ejpam-6553	76	9	,	,	PUNCT
ejpam-6553	76	10	z2	z2	PROPN
ejpam-6553	76	11	=	=	SYM
ejpam-6553	76	12	(	(	PUNCT
ejpam-6553	76	13	4k	4k	X
ejpam-6553	76	14	+	+	CCONJ
ejpam-6553	76	15	2)2	2)2	NUM
ejpam-6553	76	16	=	=	SYM
ejpam-6553	76	17	16k2	16k2	NUM
ejpam-6553	76	18	+	+	NUM
ejpam-6553	76	19	16k	16k	NOUN
ejpam-6553	76	20	+	+	CCONJ
ejpam-6553	76	21	4	4	NUM
ejpam-6553	76	22	≡	≡	PROPN
ejpam-6553	76	23	0	0	NUM
ejpam-6553	77	1	(	(	PUNCT
ejpam-6553	77	2	mod	mod	PROPN
ejpam-6553	77	3	4	4	NUM
ejpam-6553	77	4	)	)	PUNCT
ejpam-6553	77	5	.	.	PUNCT
ejpam-6553	78	1	thus	thus	ADV
ejpam-6553	78	2	,	,	PUNCT
ejpam-6553	78	3	z2	z2	PROPN
ejpam-6553	78	4	≡	≡	PROPN
ejpam-6553	78	5	0	0	PUNCT
ejpam-6553	78	6	(	(	PUNCT
ejpam-6553	78	7	mod	mod	PROPN
ejpam-6553	78	8	4	4	NUM
ejpam-6553	78	9	)	)	PUNCT
ejpam-6553	78	10	.	.	PUNCT
ejpam-6553	79	1	•	•	NUM
ejpam-6553	79	2	case	case	NOUN
ejpam-6553	79	3	4	4	NUM
ejpam-6553	79	4	:	:	PUNCT
ejpam-6553	79	5	z	z	PROPN
ejpam-6553	79	6	≡	≡	PROPN
ejpam-6553	79	7	3	3	NUM
ejpam-6553	79	8	(	(	PUNCT
ejpam-6553	79	9	mod	mod	NOUN
ejpam-6553	79	10	4	4	NUM
ejpam-6553	79	11	)	)	PUNCT
ejpam-6553	79	12	.	.	PUNCT
ejpam-6553	80	1	z	z	X
ejpam-6553	81	1	=	=	PUNCT
ejpam-6553	81	2	4k	4k	NOUN
ejpam-6553	81	3	+	+	CCONJ
ejpam-6553	81	4	3	3	NUM
ejpam-6553	81	5	for	for	ADP
ejpam-6553	81	6	some	some	DET
ejpam-6553	81	7	integer	integer	NOUN
ejpam-6553	81	8	k	k	PROPN
ejpam-6553	81	9	,	,	PUNCT
ejpam-6553	81	10	z2	z2	PROPN
ejpam-6553	81	11	=	=	SYM
ejpam-6553	81	12	(	(	PUNCT
ejpam-6553	81	13	4k	4k	NOUN
ejpam-6553	81	14	+	+	CCONJ
ejpam-6553	82	1	3)2	3)2	NUM
ejpam-6553	82	2	=	=	SYM
ejpam-6553	82	3	16k2	16k2	NUM
ejpam-6553	82	4	+	+	NUM
ejpam-6553	82	5	24k	24k	NOUN
ejpam-6553	82	6	+	+	CCONJ
ejpam-6553	82	7	9	9	NUM
ejpam-6553	82	8	≡	≡	PROPN
ejpam-6553	82	9	1	1	NUM
ejpam-6553	82	10	(	(	PUNCT
ejpam-6553	82	11	mod	mod	NOUN
ejpam-6553	82	12	4	4	NUM
ejpam-6553	82	13	)	)	PUNCT
ejpam-6553	82	14	.	.	PUNCT
ejpam-6553	83	1	thus	thus	ADV
ejpam-6553	83	2	,	,	PUNCT
ejpam-6553	83	3	z2	z2	PROPN
ejpam-6553	83	4	≡	≡	PROPN
ejpam-6553	83	5	1	1	NUM
ejpam-6553	83	6	(	(	PUNCT
ejpam-6553	83	7	mod	mod	NOUN
ejpam-6553	83	8	4	4	NUM
ejpam-6553	83	9	)	)	PUNCT
ejpam-6553	83	10	.	.	PUNCT
ejpam-6553	84	1	in	in	ADP
ejpam-6553	84	2	all	all	DET
ejpam-6553	84	3	cases	case	NOUN
ejpam-6553	84	4	,	,	PUNCT
ejpam-6553	84	5	z2	z2	PROPN
ejpam-6553	84	6	is	be	AUX
ejpam-6553	84	7	congruent	congruent	ADJ
ejpam-6553	84	8	to	to	ADP
ejpam-6553	84	9	either	either	PRON
ejpam-6553	84	10	0	0	NUM
ejpam-6553	84	11	or	or	CCONJ
ejpam-6553	84	12	1	1	NUM
ejpam-6553	84	13	modulo	modulo	NOUN
ejpam-6553	84	14	4	4	NUM
ejpam-6553	84	15	.	.	PUNCT
ejpam-6553	85	1	this	this	PRON
ejpam-6553	85	2	completes	complete	VERB
ejpam-6553	85	3	the	the	DET
ejpam-6553	85	4	proof	proof	NOUN
ejpam-6553	85	5	.	.	PUNCT
ejpam-6553	86	1	lemma	lemma	PROPN
ejpam-6553	86	2	2	2	NUM
ejpam-6553	86	3	(	(	PUNCT
ejpam-6553	86	4	modulo	modulo	NOUN
ejpam-6553	86	5	5	5	NUM
ejpam-6553	86	6	property	property	NOUN
ejpam-6553	86	7	of	of	ADP
ejpam-6553	86	8	squares	square	NOUN
ejpam-6553	86	9	)	)	PUNCT
ejpam-6553	86	10	.	.	PUNCT
ejpam-6553	87	1	for	for	ADP
ejpam-6553	87	2	any	any	DET
ejpam-6553	87	3	integer	integer	NOUN
ejpam-6553	87	4	z	z	PROPN
ejpam-6553	87	5	,	,	PUNCT
ejpam-6553	87	6	the	the	DET
ejpam-6553	87	7	square	square	ADJ
ejpam-6553	87	8	z2	z2	PROPN
ejpam-6553	87	9	satisfies	satisfie	NOUN
ejpam-6553	87	10	:	:	PUNCT
ejpam-6553	87	11	z2	z2	PROPN
ejpam-6553	87	12	≡	≡	PROPN
ejpam-6553	87	13	0	0	NUM
ejpam-6553	87	14	,	,	PUNCT
ejpam-6553	87	15	1	1	NUM
ejpam-6553	87	16	,	,	PUNCT
ejpam-6553	87	17	or	or	CCONJ
ejpam-6553	87	18	4	4	NUM
ejpam-6553	87	19	(	(	PUNCT
ejpam-6553	87	20	mod	mod	NOUN
ejpam-6553	87	21	5	5	NUM
ejpam-6553	87	22	)	)	PUNCT
ejpam-6553	87	23	.	.	PUNCT
ejpam-6553	88	1	proof	proof	NOUN
ejpam-6553	88	2	.	.	PUNCT
ejpam-6553	89	1	to	to	PART
ejpam-6553	89	2	prove	prove	VERB
ejpam-6553	89	3	this	this	DET
ejpam-6553	89	4	lemma	lemma	PROPN
ejpam-6553	89	5	,	,	PUNCT
ejpam-6553	89	6	we	we	PRON
ejpam-6553	89	7	consider	consider	VERB
ejpam-6553	89	8	all	all	DET
ejpam-6553	89	9	possible	possible	ADJ
ejpam-6553	89	10	residues	residue	NOUN
ejpam-6553	89	11	of	of	ADP
ejpam-6553	89	12	z	z	NOUN
ejpam-6553	89	13	modulo	modulo	PROPN
ejpam-6553	89	14	5	5	NUM
ejpam-6553	89	15	.	.	PUNCT
ejpam-6553	90	1	since	since	SCONJ
ejpam-6553	90	2	any	any	DET
ejpam-6553	90	3	integer	integer	NOUN
ejpam-6553	90	4	z	z	NOUN
ejpam-6553	90	5	is	be	AUX
ejpam-6553	90	6	congruent	congruent	ADJ
ejpam-6553	90	7	to	to	ADP
ejpam-6553	90	8	one	one	NUM
ejpam-6553	90	9	of	of	ADP
ejpam-6553	90	10	0	0	NUM
ejpam-6553	90	11	,	,	PUNCT
ejpam-6553	90	12	1	1	NUM
ejpam-6553	90	13	,	,	PUNCT
ejpam-6553	90	14	2	2	NUM
ejpam-6553	90	15	,	,	PUNCT
ejpam-6553	90	16	3	3	NUM
ejpam-6553	90	17	,	,	PUNCT
ejpam-6553	90	18	or	or	CCONJ
ejpam-6553	90	19	4	4	NUM
ejpam-6553	90	20	modulo	modulo	NOUN
ejpam-6553	90	21	5	5	NUM
ejpam-6553	90	22	,	,	PUNCT
ejpam-6553	90	23	we	we	PRON
ejpam-6553	90	24	analyze	analyze	VERB
ejpam-6553	90	25	each	each	DET
ejpam-6553	90	26	case	case	NOUN
ejpam-6553	90	27	separately	separately	ADV
ejpam-6553	90	28	.	.	PUNCT
ejpam-6553	91	1	•	•	NUM
ejpam-6553	91	2	case	case	NOUN
ejpam-6553	91	3	1	1	NUM
ejpam-6553	91	4	:	:	PUNCT
ejpam-6553	91	5	z	z	PROPN
ejpam-6553	91	6	≡	≡	PROPN
ejpam-6553	91	7	0	0	PUNCT
ejpam-6553	92	1	(	(	PUNCT
ejpam-6553	92	2	mod	mod	PROPN
ejpam-6553	92	3	5	5	NUM
ejpam-6553	92	4	)	)	PUNCT
ejpam-6553	92	5	.	.	PUNCT
ejpam-6553	93	1	z	z	X
ejpam-6553	94	1	=	=	PUNCT
ejpam-6553	94	2	5k	5k	NUM
ejpam-6553	94	3	for	for	ADP
ejpam-6553	94	4	some	some	DET
ejpam-6553	94	5	integer	integer	NOUN
ejpam-6553	94	6	k	k	PROPN
ejpam-6553	94	7	,	,	PUNCT
ejpam-6553	94	8	z2	z2	PROPN
ejpam-6553	94	9	=	=	SYM
ejpam-6553	94	10	(	(	PUNCT
ejpam-6553	94	11	5k)2	5k)2	NUM
ejpam-6553	94	12	=	=	SYM
ejpam-6553	94	13	25k2	25k2	NUM
ejpam-6553	94	14	≡	≡	PROPN
ejpam-6553	94	15	0	0	PUNCT
ejpam-6553	95	1	(	(	PUNCT
ejpam-6553	95	2	mod	mod	PROPN
ejpam-6553	95	3	5	5	NUM
ejpam-6553	95	4	)	)	PUNCT
ejpam-6553	95	5	.	.	PUNCT
ejpam-6553	96	1	thus	thus	ADV
ejpam-6553	96	2	,	,	PUNCT
ejpam-6553	96	3	z2	z2	PROPN
ejpam-6553	96	4	≡	≡	PROPN
ejpam-6553	96	5	0	0	PUNCT
ejpam-6553	96	6	(	(	PUNCT
ejpam-6553	96	7	mod	mod	PROPN
ejpam-6553	96	8	5	5	NUM
ejpam-6553	96	9	)	)	PUNCT
ejpam-6553	96	10	.	.	PUNCT
ejpam-6553	97	1	a.	a.	NOUN
ejpam-6553	97	2	assiry	assiry	PROPN
ejpam-6553	97	3	/	/	SYM
ejpam-6553	97	4	eur	eur	PROPN
ejpam-6553	97	5	.	.	PUNCT
ejpam-6553	98	1	j.	j.	PROPN
ejpam-6553	98	2	pure	pure	PROPN
ejpam-6553	98	3	appl	appl	PROPN
ejpam-6553	98	4	.	.	PROPN
ejpam-6553	98	5	math	math	PROPN
ejpam-6553	98	6	,	,	PUNCT
ejpam-6553	98	7	18	18	NUM
ejpam-6553	98	8	(	(	PUNCT
ejpam-6553	98	9	3	3	NUM
ejpam-6553	98	10	)	)	PUNCT
ejpam-6553	98	11	(	(	PUNCT
ejpam-6553	98	12	2025	2025	NUM
ejpam-6553	98	13	)	)	PUNCT
ejpam-6553	98	14	,	,	PUNCT
ejpam-6553	98	15	6553	6553	NUM
ejpam-6553	98	16	5	5	NUM
ejpam-6553	98	17	of	of	ADP
ejpam-6553	98	18	11	11	NUM
ejpam-6553	98	19	•	•	NOUN
ejpam-6553	98	20	case	case	NOUN
ejpam-6553	98	21	2	2	NUM
ejpam-6553	98	22	:	:	PUNCT
ejpam-6553	98	23	z	z	PROPN
ejpam-6553	98	24	≡	≡	PROPN
ejpam-6553	98	25	1	1	NUM
ejpam-6553	98	26	(	(	PUNCT
ejpam-6553	98	27	mod	mod	NOUN
ejpam-6553	98	28	5	5	NUM
ejpam-6553	98	29	)	)	PUNCT
ejpam-6553	98	30	.	.	PUNCT
ejpam-6553	99	1	z	z	X
ejpam-6553	100	1	=	=	PUNCT
ejpam-6553	100	2	5k	5k	NUM
ejpam-6553	100	3	+	+	CCONJ
ejpam-6553	100	4	1	1	NUM
ejpam-6553	100	5	for	for	ADP
ejpam-6553	100	6	some	some	DET
ejpam-6553	100	7	integer	integer	NOUN
ejpam-6553	100	8	k	k	PROPN
ejpam-6553	100	9	,	,	PUNCT
ejpam-6553	100	10	z2	z2	PROPN
ejpam-6553	100	11	=	=	SYM
ejpam-6553	100	12	(	(	PUNCT
ejpam-6553	100	13	5k	5k	NUM
ejpam-6553	101	1	+	+	X
ejpam-6553	101	2	1)2	1)2	NUM
ejpam-6553	101	3	=	=	SYM
ejpam-6553	101	4	25k2	25k2	NUM
ejpam-6553	101	5	+	+	NUM
ejpam-6553	101	6	10k	10k	NOUN
ejpam-6553	101	7	+	+	CCONJ
ejpam-6553	101	8	1	1	NUM
ejpam-6553	101	9	≡	≡	PROPN
ejpam-6553	101	10	1	1	NUM
ejpam-6553	101	11	(	(	PUNCT
ejpam-6553	101	12	mod	mod	NOUN
ejpam-6553	101	13	5	5	NUM
ejpam-6553	101	14	)	)	PUNCT
ejpam-6553	101	15	.	.	PUNCT
ejpam-6553	102	1	thus	thus	ADV
ejpam-6553	102	2	,	,	PUNCT
ejpam-6553	102	3	z2	z2	PROPN
ejpam-6553	102	4	≡	≡	PROPN
ejpam-6553	102	5	1	1	NUM
ejpam-6553	102	6	(	(	PUNCT
ejpam-6553	102	7	mod	mod	NOUN
ejpam-6553	102	8	5	5	NUM
ejpam-6553	102	9	)	)	PUNCT
ejpam-6553	102	10	.	.	PUNCT
ejpam-6553	103	1	•	•	NUM
ejpam-6553	103	2	case	case	NOUN
ejpam-6553	103	3	3	3	NUM
ejpam-6553	103	4	:	:	PUNCT
ejpam-6553	103	5	z	z	PROPN
ejpam-6553	103	6	≡	≡	PROPN
ejpam-6553	103	7	2	2	NUM
ejpam-6553	103	8	(	(	PUNCT
ejpam-6553	103	9	mod	mod	NOUN
ejpam-6553	103	10	5	5	NUM
ejpam-6553	103	11	)	)	PUNCT
ejpam-6553	103	12	.	.	PUNCT
ejpam-6553	104	1	z	z	X
ejpam-6553	105	1	=	=	PUNCT
ejpam-6553	105	2	5k	5k	NUM
ejpam-6553	105	3	+	+	CCONJ
ejpam-6553	105	4	2	2	NUM
ejpam-6553	105	5	for	for	ADP
ejpam-6553	105	6	some	some	DET
ejpam-6553	105	7	integer	integer	NOUN
ejpam-6553	105	8	k	k	PROPN
ejpam-6553	105	9	,	,	PUNCT
ejpam-6553	105	10	z2	z2	PROPN
ejpam-6553	105	11	=	=	SYM
ejpam-6553	105	12	(	(	PUNCT
ejpam-6553	105	13	5k	5k	NUM
ejpam-6553	105	14	+	+	CCONJ
ejpam-6553	105	15	2)2	2)2	NUM
ejpam-6553	105	16	=	=	SYM
ejpam-6553	105	17	25k2	25k2	NUM
ejpam-6553	106	1	+	+	NUM
ejpam-6553	106	2	20k	20k	NOUN
ejpam-6553	106	3	+	+	CCONJ
ejpam-6553	106	4	4	4	NUM
ejpam-6553	106	5	≡	≡	PROPN
ejpam-6553	106	6	4	4	NUM
ejpam-6553	106	7	(	(	PUNCT
ejpam-6553	106	8	mod	mod	NOUN
ejpam-6553	106	9	5	5	NUM
ejpam-6553	106	10	)	)	PUNCT
ejpam-6553	106	11	.	.	PUNCT
ejpam-6553	107	1	thus	thus	ADV
ejpam-6553	107	2	,	,	PUNCT
ejpam-6553	107	3	z2	z2	PROPN
ejpam-6553	107	4	≡	≡	PROPN
ejpam-6553	107	5	4	4	NUM
ejpam-6553	107	6	(	(	PUNCT
ejpam-6553	107	7	mod	mod	NOUN
ejpam-6553	107	8	5	5	NUM
ejpam-6553	107	9	)	)	PUNCT
ejpam-6553	107	10	.	.	PUNCT
ejpam-6553	108	1	•	•	NUM
ejpam-6553	108	2	case	case	NOUN
ejpam-6553	108	3	4	4	NUM
ejpam-6553	108	4	:	:	PUNCT
ejpam-6553	108	5	z	z	PROPN
ejpam-6553	108	6	≡	≡	PROPN
ejpam-6553	108	7	3	3	NUM
ejpam-6553	108	8	(	(	PUNCT
ejpam-6553	108	9	mod	mod	NOUN
ejpam-6553	108	10	5	5	NUM
ejpam-6553	108	11	)	)	PUNCT
ejpam-6553	108	12	.	.	PUNCT
ejpam-6553	109	1	z	z	X
ejpam-6553	110	1	=	=	PUNCT
ejpam-6553	110	2	5k	5k	NUM
ejpam-6553	110	3	+	+	CCONJ
ejpam-6553	110	4	3	3	NUM
ejpam-6553	110	5	for	for	ADP
ejpam-6553	110	6	some	some	DET
ejpam-6553	110	7	integer	integer	NOUN
ejpam-6553	110	8	k	k	PROPN
ejpam-6553	110	9	,	,	PUNCT
ejpam-6553	110	10	z2	z2	PROPN
ejpam-6553	110	11	=	=	SYM
ejpam-6553	110	12	(	(	PUNCT
ejpam-6553	110	13	5k	5k	NUM
ejpam-6553	111	1	+	+	CCONJ
ejpam-6553	111	2	3)2	3)2	NUM
ejpam-6553	111	3	=	=	SYM
ejpam-6553	111	4	25k2	25k2	NUM
ejpam-6553	112	1	+	+	NUM
ejpam-6553	112	2	30k	30k	NUM
ejpam-6553	112	3	+	+	CCONJ
ejpam-6553	112	4	9	9	NUM
ejpam-6553	112	5	≡	≡	PROPN
ejpam-6553	112	6	4	4	NUM
ejpam-6553	112	7	(	(	PUNCT
ejpam-6553	112	8	mod	mod	NOUN
ejpam-6553	112	9	5	5	NUM
ejpam-6553	112	10	)	)	PUNCT
ejpam-6553	112	11	.	.	PUNCT
ejpam-6553	113	1	thus	thus	ADV
ejpam-6553	113	2	,	,	PUNCT
ejpam-6553	113	3	z2	z2	PROPN
ejpam-6553	113	4	≡	≡	PROPN
ejpam-6553	113	5	4	4	NUM
ejpam-6553	113	6	(	(	PUNCT
ejpam-6553	113	7	mod	mod	NOUN
ejpam-6553	113	8	5	5	NUM
ejpam-6553	113	9	)	)	PUNCT
ejpam-6553	113	10	.	.	PUNCT
ejpam-6553	114	1	•	•	NUM
ejpam-6553	114	2	case	case	NOUN
ejpam-6553	114	3	5	5	NUM
ejpam-6553	114	4	:	:	PUNCT
ejpam-6553	114	5	z	z	PROPN
ejpam-6553	114	6	≡	≡	PROPN
ejpam-6553	114	7	4	4	NUM
ejpam-6553	114	8	(	(	PUNCT
ejpam-6553	114	9	mod	mod	NOUN
ejpam-6553	114	10	5	5	NUM
ejpam-6553	114	11	)	)	PUNCT
ejpam-6553	114	12	.	.	PUNCT
ejpam-6553	115	1	z	z	X
ejpam-6553	116	1	=	=	PUNCT
ejpam-6553	116	2	5k	5k	NUM
ejpam-6553	116	3	+	+	CCONJ
ejpam-6553	116	4	4	4	NUM
ejpam-6553	116	5	for	for	ADP
ejpam-6553	116	6	some	some	DET
ejpam-6553	116	7	integer	integer	NOUN
ejpam-6553	116	8	k	k	PROPN
ejpam-6553	116	9	,	,	PUNCT
ejpam-6553	116	10	z2	z2	PROPN
ejpam-6553	116	11	=	=	SYM
ejpam-6553	116	12	(	(	PUNCT
ejpam-6553	116	13	5k	5k	NUM
ejpam-6553	116	14	+	+	PUNCT
ejpam-6553	117	1	4)2	4)2	PROPN
ejpam-6553	117	2	=	=	SYM
ejpam-6553	118	1	25k2	25k2	NUM
ejpam-6553	119	1	+	+	NUM
ejpam-6553	119	2	40k	40k	NOUN
ejpam-6553	119	3	+	+	CCONJ
ejpam-6553	119	4	16	16	NUM
ejpam-6553	119	5	≡	≡	PROPN
ejpam-6553	119	6	1	1	NUM
ejpam-6553	119	7	(	(	PUNCT
ejpam-6553	119	8	mod	mod	NOUN
ejpam-6553	119	9	5	5	NUM
ejpam-6553	119	10	)	)	PUNCT
ejpam-6553	119	11	.	.	PUNCT
ejpam-6553	120	1	thus	thus	ADV
ejpam-6553	120	2	,	,	PUNCT
ejpam-6553	120	3	z2	z2	PROPN
ejpam-6553	120	4	≡	≡	PROPN
ejpam-6553	120	5	1	1	NUM
ejpam-6553	120	6	(	(	PUNCT
ejpam-6553	120	7	mod	mod	NOUN
ejpam-6553	120	8	5	5	NUM
ejpam-6553	120	9	)	)	PUNCT
ejpam-6553	120	10	.	.	PUNCT
ejpam-6553	121	1	in	in	ADP
ejpam-6553	121	2	all	all	DET
ejpam-6553	121	3	cases	case	NOUN
ejpam-6553	121	4	,	,	PUNCT
ejpam-6553	121	5	z2	z2	PROPN
ejpam-6553	121	6	is	be	AUX
ejpam-6553	121	7	congruent	congruent	ADJ
ejpam-6553	121	8	to	to	ADP
ejpam-6553	121	9	either	either	DET
ejpam-6553	121	10	0	0	NUM
ejpam-6553	121	11	,	,	PUNCT
ejpam-6553	121	12	1	1	NUM
ejpam-6553	121	13	,	,	PUNCT
ejpam-6553	121	14	or	or	CCONJ
ejpam-6553	121	15	4	4	NUM
ejpam-6553	121	16	modulo	modulo	NOUN
ejpam-6553	121	17	5	5	NUM
ejpam-6553	121	18	.	.	PUNCT
ejpam-6553	122	1	this	this	PRON
ejpam-6553	122	2	completes	complete	VERB
ejpam-6553	122	3	the	the	DET
ejpam-6553	122	4	proof	proof	NOUN
ejpam-6553	122	5	.	.	PUNCT
ejpam-6553	123	1	2.3	2.3	NUM
ejpam-6553	123	2	.	.	PUNCT
ejpam-6553	123	3	notation	notation	NOUN
ejpam-6553	123	4	we	we	PRON
ejpam-6553	123	5	adopt	adopt	VERB
ejpam-6553	123	6	the	the	DET
ejpam-6553	123	7	following	follow	VERB
ejpam-6553	123	8	notation	notation	NOUN
ejpam-6553	123	9	throughout	throughout	ADP
ejpam-6553	123	10	the	the	DET
ejpam-6553	123	11	paper	paper	NOUN
ejpam-6553	123	12	:	:	PUNCT
ejpam-6553	123	13	•	•	ADP
ejpam-6553	123	14	z	z	X
ejpam-6553	123	15	:	:	PUNCT
ejpam-6553	123	16	the	the	DET
ejpam-6553	123	17	set	set	NOUN
ejpam-6553	123	18	of	of	ADP
ejpam-6553	123	19	integers	integer	NOUN
ejpam-6553	123	20	.	.	PUNCT
ejpam-6553	124	1	•	•	NUM
ejpam-6553	124	2	n	n	CCONJ
ejpam-6553	124	3	:	:	PUNCT
ejpam-6553	124	4	the	the	DET
ejpam-6553	124	5	set	set	NOUN
ejpam-6553	124	6	of	of	ADP
ejpam-6553	124	7	nonnegative	nonnegative	ADJ
ejpam-6553	124	8	integers	integer	NOUN
ejpam-6553	124	9	.	.	PUNCT
ejpam-6553	125	1	•	•	NUM
ejpam-6553	125	2	a	a	DET
ejpam-6553	125	3	|	|	NOUN
ejpam-6553	125	4	b	b	NOUN
ejpam-6553	125	5	:	:	PUNCT
ejpam-6553	125	6	a	a	DET
ejpam-6553	125	7	divides	divides	PROPN
ejpam-6553	125	8	b.	b.	PROPN
ejpam-6553	125	9	•	•	NUM
ejpam-6553	126	1	a	a	PRON
ejpam-6553	126	2	∤	∤	PROPN
ejpam-6553	126	3	b	b	NOUN
ejpam-6553	126	4	:	:	PUNCT
ejpam-6553	126	5	a	a	PRON
ejpam-6553	126	6	does	do	AUX
ejpam-6553	126	7	not	not	PART
ejpam-6553	126	8	divide	divide	VERB
ejpam-6553	126	9	b.	b.	PROPN
ejpam-6553	126	10	•	•	NUM
ejpam-6553	126	11	gcd(a	gcd(a	PROPN
ejpam-6553	126	12	,	,	PUNCT
ejpam-6553	126	13	b	b	PROPN
ejpam-6553	126	14	):	):	PUNCT
ejpam-6553	126	15	the	the	DET
ejpam-6553	126	16	greatest	great	ADJ
ejpam-6553	126	17	common	common	ADJ
ejpam-6553	126	18	divisor	divisor	NOUN
ejpam-6553	126	19	of	of	ADP
ejpam-6553	126	20	a	a	PRON
ejpam-6553	126	21	and	and	CCONJ
ejpam-6553	126	22	b.	b.	PROPN
ejpam-6553	126	23	a.	a.	PROPN
ejpam-6553	126	24	assiry	assiry	PROPN
ejpam-6553	126	25	/	/	SYM
ejpam-6553	126	26	eur	eur	PROPN
ejpam-6553	126	27	.	.	PUNCT
ejpam-6553	127	1	j.	j.	PROPN
ejpam-6553	127	2	pure	pure	PROPN
ejpam-6553	127	3	appl	appl	PROPN
ejpam-6553	127	4	.	.	PROPN
ejpam-6553	127	5	math	math	PROPN
ejpam-6553	127	6	,	,	PUNCT
ejpam-6553	127	7	18	18	NUM
ejpam-6553	127	8	(	(	PUNCT
ejpam-6553	127	9	3	3	NUM
ejpam-6553	127	10	)	)	PUNCT
ejpam-6553	127	11	(	(	PUNCT
ejpam-6553	127	12	2025	2025	NUM
ejpam-6553	127	13	)	)	PUNCT
ejpam-6553	127	14	,	,	PUNCT
ejpam-6553	127	15	6553	6553	NUM
ejpam-6553	127	16	6	6	NUM
ejpam-6553	127	17	of	of	ADP
ejpam-6553	127	18	11	11	NUM
ejpam-6553	127	19	3	3	NUM
ejpam-6553	127	20	.	.	PUNCT
ejpam-6553	127	21	main	main	ADJ
ejpam-6553	127	22	results	result	NOUN
ejpam-6553	127	23	the	the	DET
ejpam-6553	127	24	main	main	ADJ
ejpam-6553	127	25	result	result	NOUN
ejpam-6553	127	26	of	of	ADP
ejpam-6553	127	27	this	this	DET
ejpam-6553	127	28	paper	paper	NOUN
ejpam-6553	127	29	is	be	AUX
ejpam-6553	127	30	the	the	DET
ejpam-6553	127	31	following	follow	VERB
ejpam-6553	127	32	theorem	theorem	NOUN
ejpam-6553	127	33	.	.	PUNCT
ejpam-6553	127	34	theorem	theorem	NOUN
ejpam-6553	127	35	1	1	NUM
ejpam-6553	127	36	.	.	PUNCT
ejpam-6553	128	1	the	the	DET
ejpam-6553	128	2	exponential	exponential	ADJ
ejpam-6553	128	3	diophantine	diophantine	NOUN
ejpam-6553	128	4	equation	equation	NOUN
ejpam-6553	128	5	23x	23x	NUM
ejpam-6553	128	6	+	+	ADJ
ejpam-6553	128	7	22y	22y	NOUN
ejpam-6553	128	8	=	=	SYM
ejpam-6553	128	9	z2	z2	PROPN
ejpam-6553	128	10	has	have	VERB
ejpam-6553	128	11	no	no	DET
ejpam-6553	128	12	nonnegative	nonnegative	ADJ
ejpam-6553	128	13	integer	integer	NOUN
ejpam-6553	128	14	solutions	solution	NOUN
ejpam-6553	128	15	.	.	PUNCT
ejpam-6553	129	1	proof	proof	NOUN
ejpam-6553	129	2	.	.	PUNCT
ejpam-6553	130	1	we	we	PRON
ejpam-6553	130	2	proceed	proceed	VERB
ejpam-6553	130	3	by	by	ADP
ejpam-6553	130	4	contradiction	contradiction	NOUN
ejpam-6553	130	5	.	.	PUNCT
ejpam-6553	131	1	assume	assume	VERB
ejpam-6553	131	2	there	there	PRON
ejpam-6553	131	3	exist	exist	VERB
ejpam-6553	131	4	nonnegative	nonnegative	ADJ
ejpam-6553	131	5	integers	integer	NOUN
ejpam-6553	131	6	x	x	NOUN
ejpam-6553	131	7	,	,	PUNCT
ejpam-6553	131	8	y	y	PROPN
ejpam-6553	131	9	,	,	PUNCT
ejpam-6553	131	10	z	z	NOUN
ejpam-6553	131	11	such	such	ADJ
ejpam-6553	131	12	that	that	SCONJ
ejpam-6553	131	13	23x	23x	ADJ
ejpam-6553	131	14	+	+	ADJ
ejpam-6553	131	15	22y	22y	NOUN
ejpam-6553	131	16	=	=	SYM
ejpam-6553	131	17	z2	z2	PROPN
ejpam-6553	131	18	.	.	PUNCT
ejpam-6553	132	1	we	we	PRON
ejpam-6553	132	2	analyze	analyze	VERB
ejpam-6553	132	3	the	the	DET
ejpam-6553	132	4	equation	equation	NOUN
ejpam-6553	132	5	using	use	VERB
ejpam-6553	132	6	modular	modular	ADJ
ejpam-6553	132	7	arithmetic	arithmetic	ADJ
ejpam-6553	132	8	and	and	CCONJ
ejpam-6553	132	9	parity	parity	NOUN
ejpam-6553	132	10	considerations	consideration	NOUN
ejpam-6553	132	11	to	to	PART
ejpam-6553	132	12	derive	derive	VERB
ejpam-6553	132	13	a	a	DET
ejpam-6553	132	14	contradiction	contradiction	NOUN
ejpam-6553	132	15	.	.	PUNCT
ejpam-6553	133	1	step	step	NOUN
ejpam-6553	133	2	1	1	NUM
ejpam-6553	133	3	:	:	PUNCT
ejpam-6553	133	4	parity	parity	NOUN
ejpam-6553	133	5	analysis	analysis	NOUN
ejpam-6553	133	6	•	•	ADP
ejpam-6553	133	7	since	since	SCONJ
ejpam-6553	133	8	23	23	NUM
ejpam-6553	133	9	is	be	AUX
ejpam-6553	133	10	an	an	DET
ejpam-6553	133	11	odd	odd	ADJ
ejpam-6553	133	12	integer	integer	NOUN
ejpam-6553	133	13	,	,	PUNCT
ejpam-6553	133	14	23x	23x	NUM
ejpam-6553	133	15	is	be	AUX
ejpam-6553	133	16	always	always	ADV
ejpam-6553	133	17	odd	odd	ADJ
ejpam-6553	133	18	for	for	ADP
ejpam-6553	133	19	any	any	DET
ejpam-6553	133	20	nonnegative	nonnegative	ADJ
ejpam-6553	133	21	integer	integer	NOUN
ejpam-6553	133	22	x.	x.	NOUN
ejpam-6553	133	23	•	•	ADV
ejpam-6553	133	24	since	since	SCONJ
ejpam-6553	133	25	22	22	NUM
ejpam-6553	133	26	is	be	AUX
ejpam-6553	133	27	an	an	DET
ejpam-6553	133	28	even	even	ADV
ejpam-6553	133	29	integer	integer	NOUN
ejpam-6553	133	30	,	,	PUNCT
ejpam-6553	133	31	22y	22y	X
ejpam-6553	133	32	is	be	AUX
ejpam-6553	133	33	always	always	ADV
ejpam-6553	133	34	even	even	ADV
ejpam-6553	133	35	for	for	ADP
ejpam-6553	133	36	any	any	DET
ejpam-6553	133	37	nonnegative	nonnegative	ADJ
ejpam-6553	133	38	integer	integer	NOUN
ejpam-6553	133	39	y.	y.	PROPN
ejpam-6553	133	40	•	•	NUM
ejpam-6553	133	41	the	the	DET
ejpam-6553	133	42	sum	sum	NOUN
ejpam-6553	133	43	of	of	ADP
ejpam-6553	133	44	an	an	DET
ejpam-6553	133	45	odd	odd	ADJ
ejpam-6553	133	46	number	number	NOUN
ejpam-6553	133	47	and	and	CCONJ
ejpam-6553	133	48	an	an	DET
ejpam-6553	133	49	even	even	ADJ
ejpam-6553	133	50	number	number	NOUN
ejpam-6553	133	51	is	be	AUX
ejpam-6553	133	52	odd	odd	ADJ
ejpam-6553	133	53	.	.	PUNCT
ejpam-6553	134	1	therefore	therefore	ADV
ejpam-6553	134	2	,	,	PUNCT
ejpam-6553	134	3	z2	z2	PROPN
ejpam-6553	134	4	=	=	SYM
ejpam-6553	134	5	23x	23x	PROPN
ejpam-6553	134	6	+	+	ADJ
ejpam-6553	134	7	22y	22y	NOUN
ejpam-6553	134	8	must	must	AUX
ejpam-6553	134	9	be	be	AUX
ejpam-6553	134	10	odd	odd	ADJ
ejpam-6553	134	11	.	.	PUNCT
ejpam-6553	135	1	•	•	ADV
ejpam-6553	135	2	if	if	SCONJ
ejpam-6553	135	3	z2	z2	PROPN
ejpam-6553	135	4	is	be	AUX
ejpam-6553	135	5	odd	odd	ADJ
ejpam-6553	135	6	,	,	PUNCT
ejpam-6553	135	7	then	then	ADV
ejpam-6553	135	8	z	z	PROPN
ejpam-6553	135	9	itself	itself	PRON
ejpam-6553	135	10	must	must	AUX
ejpam-6553	135	11	be	be	AUX
ejpam-6553	135	12	odd	odd	ADJ
ejpam-6553	135	13	,	,	PUNCT
ejpam-6553	135	14	as	as	SCONJ
ejpam-6553	135	15	the	the	DET
ejpam-6553	135	16	square	square	NOUN
ejpam-6553	135	17	of	of	ADP
ejpam-6553	135	18	an	an	DET
ejpam-6553	135	19	even	even	ADJ
ejpam-6553	135	20	number	number	NOUN
ejpam-6553	135	21	is	be	AUX
ejpam-6553	135	22	even	even	ADV
ejpam-6553	135	23	.	.	PUNCT
ejpam-6553	136	1	step	step	NOUN
ejpam-6553	136	2	2	2	NUM
ejpam-6553	136	3	:	:	PUNCT
ejpam-6553	136	4	modulo	modulo	VERB
ejpam-6553	136	5	4	4	NUM
ejpam-6553	136	6	analysis	analysis	NOUN
ejpam-6553	136	7	•	•	NOUN
ejpam-6553	136	8	we	we	PRON
ejpam-6553	136	9	analyze	analyze	VERB
ejpam-6553	136	10	the	the	DET
ejpam-6553	136	11	equation	equation	NOUN
ejpam-6553	136	12	modulo	modulo	VERB
ejpam-6553	136	13	4	4	NUM
ejpam-6553	136	14	to	to	PART
ejpam-6553	136	15	further	far	ADV
ejpam-6553	136	16	constrain	constrain	VERB
ejpam-6553	136	17	the	the	DET
ejpam-6553	136	18	possible	possible	ADJ
ejpam-6553	136	19	values	value	NOUN
ejpam-6553	136	20	of	of	ADP
ejpam-6553	136	21	x	x	X
ejpam-6553	136	22	and	and	CCONJ
ejpam-6553	136	23	y.	y.	PROPN
ejpam-6553	136	24	•	•	NUM
ejpam-6553	136	25	note	note	VERB
ejpam-6553	137	1	that	that	SCONJ
ejpam-6553	137	2	23	23	NUM
ejpam-6553	137	3	≡	≡	PROPN
ejpam-6553	137	4	3	3	NUM
ejpam-6553	137	5	(	(	PUNCT
ejpam-6553	137	6	mod	mod	NOUN
ejpam-6553	137	7	4	4	NUM
ejpam-6553	137	8	)	)	PUNCT
ejpam-6553	137	9	,	,	PUNCT
ejpam-6553	137	10	so	so	ADV
ejpam-6553	137	11	23x	23x	PROPN
ejpam-6553	137	12	≡	≡	PROPN
ejpam-6553	137	13	3x	3x	PROPN
ejpam-6553	137	14	(	(	PUNCT
ejpam-6553	137	15	mod	mod	PROPN
ejpam-6553	137	16	4	4	NUM
ejpam-6553	137	17	)	)	PUNCT
ejpam-6553	137	18	.	.	PUNCT
ejpam-6553	138	1	•	•	ADV
ejpam-6553	138	2	similarly	similarly	ADV
ejpam-6553	138	3	,	,	PUNCT
ejpam-6553	138	4	22	22	NUM
ejpam-6553	138	5	≡	≡	PROPN
ejpam-6553	138	6	2	2	NUM
ejpam-6553	138	7	(	(	PUNCT
ejpam-6553	138	8	mod	mod	NOUN
ejpam-6553	138	9	4	4	NUM
ejpam-6553	138	10	)	)	PUNCT
ejpam-6553	138	11	,	,	PUNCT
ejpam-6553	138	12	so	so	ADV
ejpam-6553	138	13	22y	22y	PROPN
ejpam-6553	138	14	≡	≡	PROPN
ejpam-6553	138	15	2y	2y	NUM
ejpam-6553	138	16	(	(	PUNCT
ejpam-6553	138	17	mod	mod	NOUN
ejpam-6553	138	18	4	4	NUM
ejpam-6553	138	19	)	)	PUNCT
ejpam-6553	138	20	.	.	PUNCT
ejpam-6553	139	1	•	•	NOUN
ejpam-6553	139	2	for	for	ADP
ejpam-6553	139	3	y	y	PROPN
ejpam-6553	139	4	≥	≥	NUM
ejpam-6553	139	5	2	2	NUM
ejpam-6553	139	6	,	,	PUNCT
ejpam-6553	139	7	2y	2y	NUM
ejpam-6553	139	8	is	be	AUX
ejpam-6553	139	9	divisible	divisible	ADJ
ejpam-6553	139	10	by	by	ADP
ejpam-6553	139	11	4	4	NUM
ejpam-6553	139	12	,	,	PUNCT
ejpam-6553	139	13	so	so	ADV
ejpam-6553	139	14	22y	22y	NUM
ejpam-6553	139	15	≡	≡	PROPN
ejpam-6553	139	16	0	0	PUNCT
ejpam-6553	140	1	(	(	PUNCT
ejpam-6553	140	2	mod	mod	PROPN
ejpam-6553	140	3	4	4	NUM
ejpam-6553	140	4	)	)	PUNCT
ejpam-6553	140	5	.	.	PUNCT
ejpam-6553	141	1	•	•	NUM
ejpam-6553	141	2	thus	thus	ADV
ejpam-6553	141	3	,	,	PUNCT
ejpam-6553	141	4	the	the	DET
ejpam-6553	141	5	equation	equation	NOUN
ejpam-6553	141	6	modulo	modulo	VERB
ejpam-6553	141	7	4	4	NUM
ejpam-6553	141	8	becomes	become	VERB
ejpam-6553	141	9	:	:	PUNCT
ejpam-6553	141	10	z2	z2	PROPN
ejpam-6553	141	11	≡	≡	PROPN
ejpam-6553	141	12	3x	3x	PROPN
ejpam-6553	142	1	+	+	CCONJ
ejpam-6553	142	2	0	0	NUM
ejpam-6553	142	3	≡	≡	PROPN
ejpam-6553	142	4	3x	3x	PROPN
ejpam-6553	142	5	(	(	PUNCT
ejpam-6553	142	6	mod	mod	PROPN
ejpam-6553	142	7	4	4	NUM
ejpam-6553	142	8	)	)	PUNCT
ejpam-6553	142	9	.	.	PUNCT
ejpam-6553	143	1	•	•	NOUN
ejpam-6553	143	2	the	the	DET
ejpam-6553	143	3	possible	possible	ADJ
ejpam-6553	143	4	values	value	NOUN
ejpam-6553	143	5	of	of	ADP
ejpam-6553	143	6	3x	3x	PROPN
ejpam-6553	143	7	(	(	PUNCT
ejpam-6553	143	8	mod	mod	NOUN
ejpam-6553	143	9	4	4	X
ejpam-6553	143	10	)	)	PUNCT
ejpam-6553	143	11	depend	depend	VERB
ejpam-6553	143	12	on	on	ADP
ejpam-6553	143	13	the	the	DET
ejpam-6553	143	14	parity	parity	NOUN
ejpam-6553	143	15	of	of	ADP
ejpam-6553	143	16	x	x	NOUN
ejpam-6553	143	17	:	:	PUNCT
ejpam-6553	143	18	–	–	PUNCT
ejpam-6553	143	19	if	if	SCONJ
ejpam-6553	143	20	x	x	PRON
ejpam-6553	143	21	is	be	AUX
ejpam-6553	143	22	even	even	ADV
ejpam-6553	143	23	,	,	PUNCT
ejpam-6553	143	24	3x	3x	PROPN
ejpam-6553	143	25	≡	≡	PROPN
ejpam-6553	143	26	1	1	NUM
ejpam-6553	143	27	(	(	PUNCT
ejpam-6553	143	28	mod	mod	NOUN
ejpam-6553	143	29	4	4	NUM
ejpam-6553	143	30	)	)	PUNCT
ejpam-6553	143	31	.	.	PUNCT
ejpam-6553	144	1	–	–	PUNCT
ejpam-6553	144	2	if	if	SCONJ
ejpam-6553	144	3	x	x	PRON
ejpam-6553	144	4	is	be	AUX
ejpam-6553	144	5	odd	odd	ADJ
ejpam-6553	144	6	,	,	PUNCT
ejpam-6553	144	7	3x	3x	NUM
ejpam-6553	144	8	≡	≡	PROPN
ejpam-6553	144	9	3	3	NUM
ejpam-6553	144	10	(	(	PUNCT
ejpam-6553	144	11	mod	mod	NOUN
ejpam-6553	144	12	4	4	NUM
ejpam-6553	144	13	)	)	PUNCT
ejpam-6553	144	14	.	.	PUNCT
ejpam-6553	145	1	•	•	NUM
ejpam-6553	145	2	however	however	ADV
ejpam-6553	145	3	,	,	PUNCT
ejpam-6553	145	4	squares	square	NOUN
ejpam-6553	145	5	modulo	modulo	VERB
ejpam-6553	145	6	4	4	NUM
ejpam-6553	145	7	can	can	AUX
ejpam-6553	145	8	only	only	ADV
ejpam-6553	145	9	be	be	AUX
ejpam-6553	145	10	congruent	congruent	ADJ
ejpam-6553	145	11	to	to	ADP
ejpam-6553	145	12	0	0	NUM
ejpam-6553	145	13	or	or	CCONJ
ejpam-6553	145	14	1	1	NUM
ejpam-6553	145	15	.	.	PUNCT
ejpam-6553	146	1	therefore	therefore	ADV
ejpam-6553	146	2	,	,	PUNCT
ejpam-6553	146	3	3x	3x	PROPN
ejpam-6553	146	4	≡	≡	PROPN
ejpam-6553	146	5	3	3	NUM
ejpam-6553	146	6	(	(	PUNCT
ejpam-6553	146	7	mod	mod	NOUN
ejpam-6553	146	8	4	4	NUM
ejpam-6553	146	9	)	)	PUNCT
ejpam-6553	146	10	is	be	AUX
ejpam-6553	146	11	impossible	impossible	ADJ
ejpam-6553	146	12	,	,	PUNCT
ejpam-6553	146	13	which	which	PRON
ejpam-6553	146	14	implies	imply	VERB
ejpam-6553	146	15	that	that	SCONJ
ejpam-6553	146	16	x	x	PRON
ejpam-6553	146	17	must	must	AUX
ejpam-6553	146	18	be	be	AUX
ejpam-6553	146	19	even	even	ADV
ejpam-6553	146	20	.	.	PUNCT
ejpam-6553	147	1	a.	a.	NOUN
ejpam-6553	147	2	assiry	assiry	PROPN
ejpam-6553	147	3	/	/	SYM
ejpam-6553	147	4	eur	eur	PROPN
ejpam-6553	147	5	.	.	PUNCT
ejpam-6553	148	1	j.	j.	PROPN
ejpam-6553	148	2	pure	pure	PROPN
ejpam-6553	148	3	appl	appl	PROPN
ejpam-6553	148	4	.	.	PROPN
ejpam-6553	148	5	math	math	PROPN
ejpam-6553	148	6	,	,	PUNCT
ejpam-6553	148	7	18	18	NUM
ejpam-6553	148	8	(	(	PUNCT
ejpam-6553	148	9	3	3	NUM
ejpam-6553	148	10	)	)	PUNCT
ejpam-6553	148	11	(	(	PUNCT
ejpam-6553	148	12	2025	2025	NUM
ejpam-6553	148	13	)	)	PUNCT
ejpam-6553	148	14	,	,	PUNCT
ejpam-6553	148	15	6553	6553	NUM
ejpam-6553	148	16	7	7	NUM
ejpam-6553	148	17	of	of	ADP
ejpam-6553	148	18	11	11	NUM
ejpam-6553	148	19	step	step	NOUN
ejpam-6553	148	20	3	3	NUM
ejpam-6553	148	21	:	:	PUNCT
ejpam-6553	148	22	modulo	modulo	VERB
ejpam-6553	148	23	5	5	NUM
ejpam-6553	148	24	analysis	analysis	NOUN
ejpam-6553	148	25	•	•	ADV
ejpam-6553	148	26	next	next	ADV
ejpam-6553	148	27	,	,	PUNCT
ejpam-6553	148	28	we	we	PRON
ejpam-6553	148	29	analyze	analyze	VERB
ejpam-6553	148	30	the	the	DET
ejpam-6553	148	31	equation	equation	NOUN
ejpam-6553	148	32	modulo	modulo	VERB
ejpam-6553	148	33	5	5	NUM
ejpam-6553	148	34	to	to	PART
ejpam-6553	148	35	further	far	ADV
ejpam-6553	148	36	restrict	restrict	VERB
ejpam-6553	148	37	the	the	DET
ejpam-6553	148	38	values	value	NOUN
ejpam-6553	148	39	of	of	ADP
ejpam-6553	148	40	x	x	X
ejpam-6553	148	41	and	and	CCONJ
ejpam-6553	148	42	y.	y.	PROPN
ejpam-6553	148	43	•	•	NUM
ejpam-6553	148	44	note	note	VERB
ejpam-6553	149	1	that	that	SCONJ
ejpam-6553	149	2	23	23	NUM
ejpam-6553	149	3	≡	≡	PROPN
ejpam-6553	149	4	3	3	NUM
ejpam-6553	149	5	(	(	PUNCT
ejpam-6553	149	6	mod	mod	NOUN
ejpam-6553	149	7	5	5	NUM
ejpam-6553	149	8	)	)	PUNCT
ejpam-6553	149	9	,	,	PUNCT
ejpam-6553	149	10	so	so	ADV
ejpam-6553	149	11	23x	23x	PROPN
ejpam-6553	149	12	≡	≡	PROPN
ejpam-6553	149	13	3x	3x	PROPN
ejpam-6553	149	14	(	(	PUNCT
ejpam-6553	149	15	mod	mod	PROPN
ejpam-6553	149	16	5	5	NUM
ejpam-6553	149	17	)	)	PUNCT
ejpam-6553	149	18	.	.	PUNCT
ejpam-6553	150	1	•	•	ADV
ejpam-6553	150	2	similarly	similarly	ADV
ejpam-6553	150	3	,	,	PUNCT
ejpam-6553	150	4	22	22	NUM
ejpam-6553	150	5	≡	≡	PROPN
ejpam-6553	150	6	2	2	NUM
ejpam-6553	150	7	(	(	PUNCT
ejpam-6553	150	8	mod	mod	NOUN
ejpam-6553	150	9	5	5	NUM
ejpam-6553	150	10	)	)	PUNCT
ejpam-6553	150	11	,	,	PUNCT
ejpam-6553	150	12	so	so	ADV
ejpam-6553	150	13	22y	22y	PROPN
ejpam-6553	150	14	≡	≡	PROPN
ejpam-6553	150	15	2y	2y	NUM
ejpam-6553	150	16	(	(	PUNCT
ejpam-6553	150	17	mod	mod	NOUN
ejpam-6553	150	18	5	5	NUM
ejpam-6553	150	19	)	)	PUNCT
ejpam-6553	150	20	.	.	PUNCT
ejpam-6553	151	1	•	•	NUM
ejpam-6553	151	2	thus	thus	ADV
ejpam-6553	151	3	,	,	PUNCT
ejpam-6553	151	4	the	the	DET
ejpam-6553	151	5	equation	equation	NOUN
ejpam-6553	151	6	modulo	modulo	VERB
ejpam-6553	151	7	5	5	NUM
ejpam-6553	151	8	becomes	become	VERB
ejpam-6553	151	9	:	:	PUNCT
ejpam-6553	151	10	z2	z2	PROPN
ejpam-6553	151	11	≡	≡	PROPN
ejpam-6553	151	12	3x	3x	PROPN
ejpam-6553	152	1	+	+	CCONJ
ejpam-6553	152	2	2y	2y	PROPN
ejpam-6553	152	3	(	(	PUNCT
ejpam-6553	152	4	mod	mod	NOUN
ejpam-6553	152	5	5	5	NUM
ejpam-6553	152	6	)	)	PUNCT
ejpam-6553	152	7	.	.	PUNCT
ejpam-6553	153	1	•	•	INTJ
ejpam-6553	153	2	we	we	PRON
ejpam-6553	153	3	now	now	ADV
ejpam-6553	153	4	examine	examine	VERB
ejpam-6553	153	5	the	the	DET
ejpam-6553	153	6	possible	possible	ADJ
ejpam-6553	153	7	values	value	NOUN
ejpam-6553	153	8	of	of	ADP
ejpam-6553	153	9	3x	3x	PROPN
ejpam-6553	153	10	(	(	PUNCT
ejpam-6553	153	11	mod	mod	NOUN
ejpam-6553	153	12	5	5	NUM
ejpam-6553	153	13	)	)	PUNCT
ejpam-6553	153	14	and	and	CCONJ
ejpam-6553	153	15	2y	2y	PROPN
ejpam-6553	153	16	(	(	PUNCT
ejpam-6553	153	17	mod	mod	NOUN
ejpam-6553	153	18	5	5	NUM
ejpam-6553	153	19	):	):	PUNCT
ejpam-6553	153	20	–	–	PUNCT
ejpam-6553	153	21	the	the	DET
ejpam-6553	153	22	powers	power	NOUN
ejpam-6553	153	23	of	of	ADP
ejpam-6553	153	24	3	3	NUM
ejpam-6553	153	25	(	(	PUNCT
ejpam-6553	153	26	mod	mod	NOUN
ejpam-6553	153	27	5	5	NUM
ejpam-6553	153	28	)	)	PUNCT
ejpam-6553	153	29	cycle	cycle	NOUN
ejpam-6553	153	30	as	as	SCONJ
ejpam-6553	153	31	follows	follow	VERB
ejpam-6553	153	32	:	:	PUNCT
ejpam-6553	153	33	31	31	NUM
ejpam-6553	153	34	≡	≡	PROPN
ejpam-6553	153	35	3	3	NUM
ejpam-6553	153	36	,	,	PUNCT
ejpam-6553	153	37	32	32	NUM
ejpam-6553	153	38	≡	≡	PROPN
ejpam-6553	153	39	4	4	NUM
ejpam-6553	153	40	,	,	PUNCT
ejpam-6553	153	41	33	33	NUM
ejpam-6553	153	42	≡	≡	PROPN
ejpam-6553	153	43	2	2	NUM
ejpam-6553	153	44	,	,	PUNCT
ejpam-6553	153	45	34	34	NUM
ejpam-6553	153	46	≡	≡	PROPN
ejpam-6553	153	47	1	1	NUM
ejpam-6553	153	48	,	,	PUNCT
ejpam-6553	153	49	35	35	NUM
ejpam-6553	153	50	≡	≡	PROPN
ejpam-6553	153	51	3	3	NUM
ejpam-6553	153	52	,	,	PUNCT
ejpam-6553	153	53	.	.	PUNCT
ejpam-6553	153	54	.	.	PUNCT
ejpam-6553	153	55	.	.	PUNCT
ejpam-6553	154	1	–	–	PUNCT
ejpam-6553	154	2	the	the	DET
ejpam-6553	154	3	powers	power	NOUN
ejpam-6553	154	4	of	of	ADP
ejpam-6553	154	5	2	2	NUM
ejpam-6553	154	6	(	(	PUNCT
ejpam-6553	154	7	mod	mod	NOUN
ejpam-6553	154	8	5	5	NUM
ejpam-6553	154	9	)	)	PUNCT
ejpam-6553	154	10	cycle	cycle	NOUN
ejpam-6553	154	11	as	as	SCONJ
ejpam-6553	154	12	follows	follow	VERB
ejpam-6553	154	13	:	:	PUNCT
ejpam-6553	154	14	21	21	NUM
ejpam-6553	154	15	≡	≡	PROPN
ejpam-6553	154	16	2	2	NUM
ejpam-6553	154	17	,	,	PUNCT
ejpam-6553	154	18	22	22	NUM
ejpam-6553	154	19	≡	≡	PROPN
ejpam-6553	154	20	4	4	NUM
ejpam-6553	154	21	,	,	PUNCT
ejpam-6553	154	22	23	23	NUM
ejpam-6553	154	23	≡	≡	PROPN
ejpam-6553	154	24	3	3	NUM
ejpam-6553	154	25	,	,	PUNCT
ejpam-6553	154	26	24	24	NUM
ejpam-6553	154	27	≡	≡	PROPN
ejpam-6553	154	28	1	1	NUM
ejpam-6553	154	29	,	,	PUNCT
ejpam-6553	154	30	25	25	NUM
ejpam-6553	154	31	≡	≡	PROPN
ejpam-6553	154	32	2	2	NUM
ejpam-6553	154	33	,	,	PUNCT
ejpam-6553	154	34	.	.	PUNCT
ejpam-6553	154	35	.	.	PUNCT
ejpam-6553	155	1	.	.	PUNCT
ejpam-6553	156	1	•	•	NUM
ejpam-6553	156	2	combining	combine	VERB
ejpam-6553	156	3	these	these	DET
ejpam-6553	156	4	results	result	NOUN
ejpam-6553	156	5	,	,	PUNCT
ejpam-6553	156	6	we	we	PRON
ejpam-6553	156	7	compute	compute	VERB
ejpam-6553	156	8	3x	3x	PRON
ejpam-6553	157	1	+	+	CCONJ
ejpam-6553	157	2	2y	2y	PROPN
ejpam-6553	157	3	(	(	PUNCT
ejpam-6553	157	4	mod	mod	NOUN
ejpam-6553	157	5	5	5	NUM
ejpam-6553	157	6	)	)	PUNCT
ejpam-6553	157	7	for	for	ADP
ejpam-6553	157	8	all	all	DET
ejpam-6553	157	9	possible	possible	ADJ
ejpam-6553	157	10	residues	residue	NOUN
ejpam-6553	157	11	of	of	ADP
ejpam-6553	157	12	x	x	PUNCT
ejpam-6553	157	13	and	and	CCONJ
ejpam-6553	157	14	y	y	PROPN
ejpam-6553	157	15	modulo	modulo	VERB
ejpam-6553	157	16	4	4	NUM
ejpam-6553	157	17	(	(	PUNCT
ejpam-6553	157	18	since	since	SCONJ
ejpam-6553	157	19	the	the	DET
ejpam-6553	157	20	cycles	cycle	NOUN
ejpam-6553	157	21	repeat	repeat	VERB
ejpam-6553	157	22	every	every	DET
ejpam-6553	157	23	4	4	NUM
ejpam-6553	157	24	steps	step	NOUN
ejpam-6553	157	25	):	):	PUNCT
ejpam-6553	157	26	–	–	PUNCT
ejpam-6553	157	27	if	if	SCONJ
ejpam-6553	157	28	x	x	SYM
ejpam-6553	157	29	≡	≡	PROPN
ejpam-6553	157	30	0	0	PUNCT
ejpam-6553	157	31	(	(	PUNCT
ejpam-6553	157	32	mod	mod	PROPN
ejpam-6553	157	33	4	4	NUM
ejpam-6553	157	34	)	)	PUNCT
ejpam-6553	157	35	,	,	PUNCT
ejpam-6553	157	36	then	then	ADV
ejpam-6553	157	37	3x	3x	NUM
ejpam-6553	157	38	≡	≡	PROPN
ejpam-6553	157	39	1	1	NUM
ejpam-6553	157	40	(	(	PUNCT
ejpam-6553	157	41	mod	mod	NOUN
ejpam-6553	157	42	5	5	NUM
ejpam-6553	157	43	)	)	PUNCT
ejpam-6553	157	44	.	.	PUNCT
ejpam-6553	158	1	–	–	PUNCT
ejpam-6553	158	2	if	if	SCONJ
ejpam-6553	158	3	x	x	SYM
ejpam-6553	158	4	≡	≡	PROPN
ejpam-6553	158	5	1	1	NUM
ejpam-6553	158	6	(	(	PUNCT
ejpam-6553	158	7	mod	mod	NOUN
ejpam-6553	158	8	4	4	NUM
ejpam-6553	158	9	)	)	PUNCT
ejpam-6553	158	10	,	,	PUNCT
ejpam-6553	158	11	then	then	ADV
ejpam-6553	158	12	3x	3x	NUM
ejpam-6553	158	13	≡	≡	PROPN
ejpam-6553	158	14	3	3	NUM
ejpam-6553	158	15	(	(	PUNCT
ejpam-6553	158	16	mod	mod	NOUN
ejpam-6553	158	17	5	5	NUM
ejpam-6553	158	18	)	)	PUNCT
ejpam-6553	158	19	.	.	PUNCT
ejpam-6553	159	1	–	–	PUNCT
ejpam-6553	159	2	if	if	SCONJ
ejpam-6553	159	3	x	x	SYM
ejpam-6553	159	4	≡	≡	PROPN
ejpam-6553	159	5	2	2	NUM
ejpam-6553	159	6	(	(	PUNCT
ejpam-6553	159	7	mod	mod	NOUN
ejpam-6553	159	8	4	4	NUM
ejpam-6553	159	9	)	)	PUNCT
ejpam-6553	159	10	,	,	PUNCT
ejpam-6553	159	11	then	then	ADV
ejpam-6553	159	12	3x	3x	NUM
ejpam-6553	159	13	≡	≡	PROPN
ejpam-6553	159	14	4	4	NUM
ejpam-6553	159	15	(	(	PUNCT
ejpam-6553	159	16	mod	mod	NOUN
ejpam-6553	159	17	5	5	NUM
ejpam-6553	159	18	)	)	PUNCT
ejpam-6553	159	19	.	.	PUNCT
ejpam-6553	160	1	–	–	PUNCT
ejpam-6553	160	2	if	if	SCONJ
ejpam-6553	160	3	x	x	SYM
ejpam-6553	160	4	≡	≡	PROPN
ejpam-6553	160	5	3	3	NUM
ejpam-6553	160	6	(	(	PUNCT
ejpam-6553	160	7	mod	mod	NOUN
ejpam-6553	160	8	4	4	NUM
ejpam-6553	160	9	)	)	PUNCT
ejpam-6553	160	10	,	,	PUNCT
ejpam-6553	160	11	then	then	ADV
ejpam-6553	160	12	3x	3x	NUM
ejpam-6553	160	13	≡	≡	PROPN
ejpam-6553	160	14	2	2	NUM
ejpam-6553	160	15	(	(	PUNCT
ejpam-6553	160	16	mod	mod	NOUN
ejpam-6553	160	17	5	5	NUM
ejpam-6553	160	18	)	)	PUNCT
ejpam-6553	160	19	.	.	PUNCT
ejpam-6553	161	1	similarly	similarly	ADV
ejpam-6553	161	2	,	,	PUNCT
ejpam-6553	161	3	for	for	ADP
ejpam-6553	161	4	y	y	NOUN
ejpam-6553	161	5	:	:	PUNCT
ejpam-6553	161	6	–	–	PUNCT
ejpam-6553	161	7	if	if	SCONJ
ejpam-6553	161	8	y	y	PROPN
ejpam-6553	161	9	≡	≡	PROPN
ejpam-6553	161	10	0	0	PUNCT
ejpam-6553	161	11	(	(	PUNCT
ejpam-6553	161	12	mod	mod	PROPN
ejpam-6553	161	13	4	4	NUM
ejpam-6553	161	14	)	)	PUNCT
ejpam-6553	161	15	,	,	PUNCT
ejpam-6553	161	16	then	then	ADV
ejpam-6553	161	17	2y	2y	PROPN
ejpam-6553	161	18	≡	≡	PROPN
ejpam-6553	161	19	1	1	NUM
ejpam-6553	161	20	(	(	PUNCT
ejpam-6553	161	21	mod	mod	NOUN
ejpam-6553	161	22	5	5	NUM
ejpam-6553	161	23	)	)	PUNCT
ejpam-6553	161	24	.	.	PUNCT
ejpam-6553	162	1	–	–	PUNCT
ejpam-6553	162	2	if	if	SCONJ
ejpam-6553	162	3	y	y	PROPN
ejpam-6553	162	4	≡	≡	PROPN
ejpam-6553	162	5	1	1	NUM
ejpam-6553	162	6	(	(	PUNCT
ejpam-6553	162	7	mod	mod	NOUN
ejpam-6553	162	8	4	4	NUM
ejpam-6553	162	9	)	)	PUNCT
ejpam-6553	162	10	,	,	PUNCT
ejpam-6553	162	11	then	then	ADV
ejpam-6553	162	12	2y	2y	PROPN
ejpam-6553	162	13	≡	≡	PROPN
ejpam-6553	162	14	2	2	NUM
ejpam-6553	162	15	(	(	PUNCT
ejpam-6553	162	16	mod	mod	NOUN
ejpam-6553	162	17	5	5	NUM
ejpam-6553	162	18	)	)	PUNCT
ejpam-6553	162	19	.	.	PUNCT
ejpam-6553	163	1	–	–	PUNCT
ejpam-6553	163	2	if	if	SCONJ
ejpam-6553	163	3	y	y	PROPN
ejpam-6553	163	4	≡	≡	PROPN
ejpam-6553	163	5	2	2	NUM
ejpam-6553	163	6	(	(	PUNCT
ejpam-6553	163	7	mod	mod	NOUN
ejpam-6553	163	8	4	4	NUM
ejpam-6553	163	9	)	)	PUNCT
ejpam-6553	163	10	,	,	PUNCT
ejpam-6553	163	11	then	then	ADV
ejpam-6553	163	12	2y	2y	PROPN
ejpam-6553	163	13	≡	≡	PROPN
ejpam-6553	163	14	4	4	NUM
ejpam-6553	163	15	(	(	PUNCT
ejpam-6553	163	16	mod	mod	NOUN
ejpam-6553	163	17	5	5	NUM
ejpam-6553	163	18	)	)	PUNCT
ejpam-6553	163	19	.	.	PUNCT
ejpam-6553	164	1	–	–	PUNCT
ejpam-6553	164	2	if	if	SCONJ
ejpam-6553	164	3	y	y	PROPN
ejpam-6553	164	4	≡	≡	PROPN
ejpam-6553	164	5	3	3	NUM
ejpam-6553	164	6	(	(	PUNCT
ejpam-6553	164	7	mod	mod	NOUN
ejpam-6553	164	8	4	4	NUM
ejpam-6553	164	9	)	)	PUNCT
ejpam-6553	164	10	,	,	PUNCT
ejpam-6553	164	11	then	then	ADV
ejpam-6553	164	12	2y	2y	PROPN
ejpam-6553	164	13	≡	≡	PROPN
ejpam-6553	164	14	3	3	NUM
ejpam-6553	164	15	(	(	PUNCT
ejpam-6553	164	16	mod	mod	NOUN
ejpam-6553	164	17	5	5	NUM
ejpam-6553	164	18	)	)	PUNCT
ejpam-6553	164	19	.	.	PUNCT
ejpam-6553	165	1	•	•	INTJ
ejpam-6553	165	2	we	we	PRON
ejpam-6553	165	3	now	now	ADV
ejpam-6553	165	4	check	check	VERB
ejpam-6553	165	5	all	all	DET
ejpam-6553	165	6	combinations	combination	NOUN
ejpam-6553	165	7	of	of	ADP
ejpam-6553	165	8	x	x	PUNCT
ejpam-6553	165	9	and	and	CCONJ
ejpam-6553	165	10	y	y	PROPN
ejpam-6553	165	11	modulo	modulo	VERB
ejpam-6553	165	12	4	4	NUM
ejpam-6553	165	13	to	to	PART
ejpam-6553	165	14	see	see	VERB
ejpam-6553	165	15	if	if	SCONJ
ejpam-6553	165	16	z2	z2	PROPN
ejpam-6553	165	17	≡	≡	PROPN
ejpam-6553	165	18	3x	3x	PROPN
ejpam-6553	166	1	+	+	CCONJ
ejpam-6553	166	2	2y	2y	PROPN
ejpam-6553	166	3	(	(	PUNCT
ejpam-6553	166	4	mod	mod	NOUN
ejpam-6553	166	5	5	5	NUM
ejpam-6553	166	6	)	)	PUNCT
ejpam-6553	166	7	is	be	AUX
ejpam-6553	166	8	a	a	DET
ejpam-6553	166	9	quadratic	quadratic	ADJ
ejpam-6553	166	10	residue	residue	NOUN
ejpam-6553	166	11	modulo	modulo	NOUN
ejpam-6553	166	12	5	5	NUM
ejpam-6553	166	13	(	(	PUNCT
ejpam-6553	166	14	i.e.	i.e.	X
ejpam-6553	166	15	,	,	PUNCT
ejpam-6553	166	16	z2	z2	PROPN
ejpam-6553	166	17	≡	≡	PROPN
ejpam-6553	166	18	0	0	NUM
ejpam-6553	166	19	,	,	PUNCT
ejpam-6553	166	20	1	1	NUM
ejpam-6553	166	21	,	,	PUNCT
ejpam-6553	166	22	or	or	CCONJ
ejpam-6553	166	23	4	4	NUM
ejpam-6553	166	24	(	(	PUNCT
ejpam-6553	166	25	mod	mod	NOUN
ejpam-6553	166	26	5	5	NUM
ejpam-6553	166	27	)	)	PUNCT
ejpam-6553	166	28	):	):	PUNCT
ejpam-6553	166	29	–	–	PUNCT
ejpam-6553	166	30	for	for	ADP
ejpam-6553	166	31	example	example	NOUN
ejpam-6553	166	32	,	,	PUNCT
ejpam-6553	166	33	if	if	SCONJ
ejpam-6553	166	34	x	x	SYM
ejpam-6553	166	35	≡	≡	PROPN
ejpam-6553	166	36	0	0	PUNCT
ejpam-6553	166	37	(	(	PUNCT
ejpam-6553	166	38	mod	mod	NOUN
ejpam-6553	166	39	4	4	NUM
ejpam-6553	166	40	)	)	PUNCT
ejpam-6553	166	41	and	and	CCONJ
ejpam-6553	166	42	y	y	PROPN
ejpam-6553	166	43	≡	≡	PROPN
ejpam-6553	166	44	0	0	PUNCT
ejpam-6553	166	45	(	(	PUNCT
ejpam-6553	166	46	mod	mod	PROPN
ejpam-6553	166	47	4	4	NUM
ejpam-6553	166	48	)	)	PUNCT
ejpam-6553	166	49	,	,	PUNCT
ejpam-6553	166	50	then	then	ADV
ejpam-6553	166	51	:	:	PUNCT
ejpam-6553	166	52	z2	z2	PROPN
ejpam-6553	166	53	≡	≡	PROPN
ejpam-6553	166	54	1	1	NUM
ejpam-6553	167	1	+	+	SYM
ejpam-6553	167	2	1	1	NUM
ejpam-6553	167	3	≡	≡	PROPN
ejpam-6553	167	4	2	2	NUM
ejpam-6553	167	5	(	(	PUNCT
ejpam-6553	167	6	mod	mod	NOUN
ejpam-6553	167	7	5	5	NUM
ejpam-6553	167	8	)	)	PUNCT
ejpam-6553	167	9	.	.	PUNCT
ejpam-6553	168	1	however	however	ADV
ejpam-6553	168	2	,	,	PUNCT
ejpam-6553	168	3	2	2	NUM
ejpam-6553	168	4	is	be	AUX
ejpam-6553	168	5	not	not	PART
ejpam-6553	168	6	a	a	DET
ejpam-6553	168	7	quadratic	quadratic	ADJ
ejpam-6553	168	8	residue	residue	NOUN
ejpam-6553	168	9	modulo	modulo	NOUN
ejpam-6553	168	10	5	5	NUM
ejpam-6553	168	11	,	,	PUNCT
ejpam-6553	168	12	leading	lead	VERB
ejpam-6553	168	13	to	to	ADP
ejpam-6553	168	14	a	a	DET
ejpam-6553	168	15	contradiction	contradiction	NOUN
ejpam-6553	168	16	.	.	PUNCT
ejpam-6553	169	1	–	–	PUNCT
ejpam-6553	169	2	similar	similar	ADJ
ejpam-6553	169	3	contradictions	contradiction	NOUN
ejpam-6553	169	4	arise	arise	VERB
ejpam-6553	169	5	for	for	ADP
ejpam-6553	169	6	all	all	DET
ejpam-6553	169	7	other	other	ADJ
ejpam-6553	169	8	combinations	combination	NOUN
ejpam-6553	169	9	of	of	ADP
ejpam-6553	169	10	x	x	PUNCT
ejpam-6553	169	11	and	and	CCONJ
ejpam-6553	169	12	y	y	PROPN
ejpam-6553	169	13	modulo	modulo	PROPN
ejpam-6553	169	14	4	4	NUM
ejpam-6553	169	15	.	.	PUNCT
ejpam-6553	170	1	since	since	SCONJ
ejpam-6553	170	2	all	all	DET
ejpam-6553	170	3	possible	possible	ADJ
ejpam-6553	170	4	cases	case	NOUN
ejpam-6553	170	5	lead	lead	VERB
ejpam-6553	170	6	to	to	ADP
ejpam-6553	170	7	contradictions	contradiction	NOUN
ejpam-6553	170	8	,	,	PUNCT
ejpam-6553	170	9	our	our	PRON
ejpam-6553	170	10	initial	initial	ADJ
ejpam-6553	170	11	assumption	assumption	NOUN
ejpam-6553	170	12	that	that	SCONJ
ejpam-6553	170	13	there	there	PRON
ejpam-6553	170	14	exist	exist	VERB
ejpam-6553	170	15	nonnegative	nonnegative	ADJ
ejpam-6553	170	16	integer	integer	NOUN
ejpam-6553	170	17	solutions	solution	NOUN
ejpam-6553	170	18	(	(	PUNCT
ejpam-6553	170	19	x	x	X
ejpam-6553	170	20	,	,	PUNCT
ejpam-6553	170	21	y	y	PROPN
ejpam-6553	170	22	,	,	PUNCT
ejpam-6553	170	23	z	z	NOUN
ejpam-6553	170	24	)	)	PUNCT
ejpam-6553	170	25	to	to	ADP
ejpam-6553	170	26	the	the	DET
ejpam-6553	170	27	equation	equation	NOUN
ejpam-6553	170	28	23x	23x	NUM
ejpam-6553	171	1	+	+	ADJ
ejpam-6553	171	2	22y	22y	NOUN
ejpam-6553	171	3	=	=	SYM
ejpam-6553	171	4	z2	z2	PROPN
ejpam-6553	171	5	must	must	AUX
ejpam-6553	171	6	be	be	AUX
ejpam-6553	171	7	false	false	ADJ
ejpam-6553	171	8	.	.	PUNCT
ejpam-6553	172	1	therefore	therefore	ADV
ejpam-6553	172	2	,	,	PUNCT
ejpam-6553	172	3	the	the	DET
ejpam-6553	172	4	equation	equation	NOUN
ejpam-6553	172	5	has	have	VERB
ejpam-6553	172	6	no	no	DET
ejpam-6553	172	7	nonnegative	nonnegative	ADJ
ejpam-6553	172	8	integer	integer	NOUN
ejpam-6553	172	9	solutions	solution	NOUN
ejpam-6553	172	10	.	.	PUNCT
ejpam-6553	173	1	a.	a.	NOUN
ejpam-6553	173	2	assiry	assiry	PROPN
ejpam-6553	173	3	/	/	SYM
ejpam-6553	173	4	eur	eur	PROPN
ejpam-6553	173	5	.	.	PUNCT
ejpam-6553	174	1	j.	j.	PROPN
ejpam-6553	174	2	pure	pure	PROPN
ejpam-6553	174	3	appl	appl	PROPN
ejpam-6553	174	4	.	.	PROPN
ejpam-6553	174	5	math	math	PROPN
ejpam-6553	174	6	,	,	PUNCT
ejpam-6553	174	7	18	18	NUM
ejpam-6553	174	8	(	(	PUNCT
ejpam-6553	174	9	3	3	NUM
ejpam-6553	174	10	)	)	PUNCT
ejpam-6553	174	11	(	(	PUNCT
ejpam-6553	174	12	2025	2025	NUM
ejpam-6553	174	13	)	)	PUNCT
ejpam-6553	174	14	,	,	PUNCT
ejpam-6553	174	15	6553	6553	NUM
ejpam-6553	174	16	8	8	NUM
ejpam-6553	174	17	of	of	ADP
ejpam-6553	174	18	11	11	NUM
ejpam-6553	174	19	4	4	NUM
ejpam-6553	174	20	.	.	PUNCT
ejpam-6553	175	1	computational	computational	ADJ
ejpam-6553	175	2	data	datum	NOUN
ejpam-6553	175	3	to	to	PART
ejpam-6553	175	4	support	support	VERB
ejpam-6553	175	5	our	our	PRON
ejpam-6553	175	6	theoretical	theoretical	ADJ
ejpam-6553	175	7	results	result	NOUN
ejpam-6553	175	8	,	,	PUNCT
ejpam-6553	175	9	we	we	PRON
ejpam-6553	175	10	provide	provide	VERB
ejpam-6553	175	11	computational	computational	ADJ
ejpam-6553	175	12	data	datum	NOUN
ejpam-6553	175	13	for	for	ADP
ejpam-6553	175	14	small	small	ADJ
ejpam-6553	175	15	values	value	NOUN
ejpam-6553	175	16	of	of	ADP
ejpam-6553	175	17	x	x	PUNCT
ejpam-6553	175	18	and	and	CCONJ
ejpam-6553	175	19	y.	y.	NOUN
ejpam-6553	175	20	the	the	DET
ejpam-6553	175	21	table	table	NOUN
ejpam-6553	175	22	below	below	ADV
ejpam-6553	175	23	shows	show	VERB
ejpam-6553	175	24	23x	23x	NOUN
ejpam-6553	175	25	+	+	CCONJ
ejpam-6553	175	26	22y	22y	NOUN
ejpam-6553	175	27	and	and	CCONJ
ejpam-6553	175	28	whether	whether	SCONJ
ejpam-6553	175	29	it	it	PRON
ejpam-6553	175	30	is	be	AUX
ejpam-6553	175	31	a	a	DET
ejpam-6553	175	32	perfect	perfect	ADJ
ejpam-6553	175	33	square	square	NOUN
ejpam-6553	175	34	.	.	PUNCT
ejpam-6553	176	1	x	x	PUNCT
ejpam-6553	176	2	y	y	PROPN
ejpam-6553	176	3	23x	23x	NUM
ejpam-6553	176	4	+	+	CCONJ
ejpam-6553	176	5	22y	22y	NUM
ejpam-6553	176	6	is	be	AUX
ejpam-6553	176	7	z2	z2	NUM
ejpam-6553	176	8	?	?	X
ejpam-6553	176	9	0	0	NUM
ejpam-6553	176	10	0	0	NUM
ejpam-6553	176	11	1	1	NUM
ejpam-6553	177	1	+	+	SYM
ejpam-6553	177	2	1	1	NUM
ejpam-6553	177	3	=	=	SYM
ejpam-6553	177	4	2	2	NUM
ejpam-6553	177	5	no	no	DET
ejpam-6553	177	6	0	0	NUM
ejpam-6553	177	7	1	1	NUM
ejpam-6553	177	8	1	1	NUM
ejpam-6553	177	9	+	+	NUM
ejpam-6553	177	10	22	22	NUM
ejpam-6553	177	11	=	=	SYM
ejpam-6553	177	12	23	23	NUM
ejpam-6553	177	13	no	no	DET
ejpam-6553	177	14	1	1	NUM
ejpam-6553	177	15	0	0	NUM
ejpam-6553	177	16	23	23	NUM
ejpam-6553	177	17	+	+	SYM
ejpam-6553	177	18	1	1	NUM
ejpam-6553	177	19	=	=	SYM
ejpam-6553	177	20	24	24	NUM
ejpam-6553	177	21	no	no	DET
ejpam-6553	177	22	1	1	NUM
ejpam-6553	177	23	1	1	NUM
ejpam-6553	177	24	23	23	NUM
ejpam-6553	177	25	+	+	CCONJ
ejpam-6553	177	26	22	22	NUM
ejpam-6553	177	27	=	=	SYM
ejpam-6553	177	28	45	45	NUM
ejpam-6553	177	29	no	no	DET
ejpam-6553	177	30	2	2	NUM
ejpam-6553	177	31	0	0	NUM
ejpam-6553	177	32	529	529	NUM
ejpam-6553	177	33	+	+	SYM
ejpam-6553	177	34	1	1	NUM
ejpam-6553	177	35	=	=	SYM
ejpam-6553	177	36	530	530	NUM
ejpam-6553	177	37	no	no	DET
ejpam-6553	177	38	2	2	NUM
ejpam-6553	177	39	1	1	NUM
ejpam-6553	177	40	529	529	NUM
ejpam-6553	177	41	+	+	CCONJ
ejpam-6553	177	42	22	22	NUM
ejpam-6553	177	43	=	=	SYM
ejpam-6553	177	44	551	551	NUM
ejpam-6553	177	45	no	no	DET
ejpam-6553	177	46	5	5	NUM
ejpam-6553	177	47	.	.	PUNCT
ejpam-6553	177	48	generalization	generalization	NOUN
ejpam-6553	177	49	the	the	DET
ejpam-6553	177	50	methods	method	NOUN
ejpam-6553	177	51	used	use	VERB
ejpam-6553	177	52	in	in	ADP
ejpam-6553	177	53	this	this	DET
ejpam-6553	177	54	paper	paper	NOUN
ejpam-6553	177	55	can	can	AUX
ejpam-6553	177	56	be	be	AUX
ejpam-6553	177	57	extended	extend	VERB
ejpam-6553	177	58	to	to	ADP
ejpam-6553	177	59	other	other	ADJ
ejpam-6553	177	60	equations	equation	NOUN
ejpam-6553	177	61	of	of	ADP
ejpam-6553	177	62	the	the	DET
ejpam-6553	177	63	form	form	NOUN
ejpam-6553	177	64	wx	wx	PROPN
ejpam-6553	177	65	+	+	CCONJ
ejpam-6553	178	1	(	(	PUNCT
ejpam-6553	178	2	w	w	NOUN
ejpam-6553	178	3	−	−	PROPN
ejpam-6553	178	4	1)y	1)y	NUM
ejpam-6553	178	5	=	=	SYM
ejpam-6553	178	6	z2	z2	PROPN
ejpam-6553	178	7	,	,	PUNCT
ejpam-6553	178	8	where	where	SCONJ
ejpam-6553	178	9	w	w	NOUN
ejpam-6553	178	10	is	be	AUX
ejpam-6553	178	11	a	a	DET
ejpam-6553	178	12	positive	positive	ADJ
ejpam-6553	178	13	integer	integer	NOUN
ejpam-6553	178	14	of	of	ADP
ejpam-6553	178	15	the	the	DET
ejpam-6553	178	16	form	form	NOUN
ejpam-6553	178	17	4n	4n	NOUN
ejpam-6553	178	18	+	+	NOUN
ejpam-6553	178	19	3	3	X
ejpam-6553	178	20	.	.	X
ejpam-6553	179	1	for	for	ADP
ejpam-6553	179	2	example	example	NOUN
ejpam-6553	179	3	:	:	PUNCT
ejpam-6553	179	4	•	•	NUM
ejpam-6553	179	5	19x	19x	NUM
ejpam-6553	179	6	+	+	SYM
ejpam-6553	179	7	18y	18y	NOUN
ejpam-6553	179	8	=	=	SYM
ejpam-6553	179	9	z2	z2	PROPN
ejpam-6553	179	10	,	,	PUNCT
ejpam-6553	179	11	•	•	ADJ
ejpam-6553	179	12	31x	31x	NOUN
ejpam-6553	179	13	+	+	CCONJ
ejpam-6553	179	14	30y	30y	NOUN
ejpam-6553	179	15	=	=	SYM
ejpam-6553	179	16	z2	z2	PROPN
ejpam-6553	179	17	.	.	PUNCT
ejpam-6553	180	1	the	the	DET
ejpam-6553	180	2	proof	proof	NOUN
ejpam-6553	180	3	follows	follow	VERB
ejpam-6553	180	4	similar	similar	ADJ
ejpam-6553	180	5	steps	step	NOUN
ejpam-6553	180	6	,	,	PUNCT
ejpam-6553	180	7	using	use	VERB
ejpam-6553	180	8	modular	modular	ADJ
ejpam-6553	180	9	arithmetic	arithmetic	ADJ
ejpam-6553	180	10	and	and	CCONJ
ejpam-6553	180	11	parity	parity	NOUN
ejpam-6553	180	12	analysis	analysis	NOUN
ejpam-6553	180	13	to	to	PART
ejpam-6553	180	14	derive	derive	VERB
ejpam-6553	180	15	contradictions	contradiction	NOUN
ejpam-6553	180	16	.	.	PUNCT
ejpam-6553	181	1	6	6	X
ejpam-6553	181	2	.	.	X
ejpam-6553	181	3	discussion	discussion	NOUN
ejpam-6553	181	4	our	our	PRON
ejpam-6553	181	5	result	result	NOUN
ejpam-6553	181	6	adds	add	VERB
ejpam-6553	181	7	to	to	ADP
ejpam-6553	181	8	the	the	DET
ejpam-6553	181	9	growing	grow	VERB
ejpam-6553	181	10	body	body	NOUN
ejpam-6553	181	11	of	of	ADP
ejpam-6553	181	12	work	work	NOUN
ejpam-6553	181	13	on	on	ADP
ejpam-6553	181	14	exponential	exponential	ADJ
ejpam-6553	181	15	diophantine	diophantine	NOUN
ejpam-6553	181	16	equations	equation	NOUN
ejpam-6553	181	17	,	,	PUNCT
ejpam-6553	181	18	particularly	particularly	ADV
ejpam-6553	181	19	those	those	PRON
ejpam-6553	181	20	of	of	ADP
ejpam-6553	181	21	the	the	DET
ejpam-6553	181	22	form	form	NOUN
ejpam-6553	181	23	px	px	X
ejpam-6553	181	24	+	+	CCONJ
ejpam-6553	181	25	(	(	PUNCT
ejpam-6553	181	26	p	p	NOUN
ejpam-6553	181	27	−	−	PROPN
ejpam-6553	181	28	1)y	1)y	NUM
ejpam-6553	181	29	=	=	SYM
ejpam-6553	181	30	z2	z2	PROPN
ejpam-6553	181	31	.	.	PUNCT
ejpam-6553	182	1	the	the	DET
ejpam-6553	182	2	methods	method	NOUN
ejpam-6553	182	3	employed	employ	VERB
ejpam-6553	182	4	in	in	ADP
ejpam-6553	182	5	this	this	DET
ejpam-6553	182	6	paper	paper	NOUN
ejpam-6553	182	7	—	—	PUNCT
ejpam-6553	182	8	modular	modular	ADJ
ejpam-6553	182	9	arithmetic	arithmetic	ADJ
ejpam-6553	182	10	,	,	PUNCT
ejpam-6553	182	11	parity	parity	NOUN
ejpam-6553	182	12	analysis	analysis	NOUN
ejpam-6553	182	13	,	,	PUNCT
ejpam-6553	182	14	and	and	CCONJ
ejpam-6553	182	15	computational	computational	ADJ
ejpam-6553	182	16	verification	verification	NOUN
ejpam-6553	182	17	—	—	PUNCT
ejpam-6553	182	18	demonstrate	demonstrate	VERB
ejpam-6553	182	19	the	the	DET
ejpam-6553	182	20	power	power	NOUN
ejpam-6553	182	21	of	of	ADP
ejpam-6553	182	22	combining	combine	VERB
ejpam-6553	182	23	theoretical	theoretical	ADJ
ejpam-6553	182	24	and	and	CCONJ
ejpam-6553	182	25	computational	computational	ADJ
ejpam-6553	182	26	approaches	approach	NOUN
ejpam-6553	182	27	to	to	PART
ejpam-6553	182	28	solve	solve	VERB
ejpam-6553	182	29	challenging	challenging	ADJ
ejpam-6553	182	30	problems	problem	NOUN
ejpam-6553	182	31	in	in	ADP
ejpam-6553	182	32	number	number	NOUN
ejpam-6553	182	33	theory	theory	NOUN
ejpam-6553	182	34	.	.	PUNCT
ejpam-6553	183	1	below	below	ADV
ejpam-6553	183	2	,	,	PUNCT
ejpam-6553	183	3	we	we	PRON
ejpam-6553	183	4	discuss	discuss	VERB
ejpam-6553	183	5	the	the	DET
ejpam-6553	183	6	implications	implication	NOUN
ejpam-6553	183	7	of	of	ADP
ejpam-6553	183	8	our	our	PRON
ejpam-6553	183	9	findings	finding	NOUN
ejpam-6553	183	10	and	and	CCONJ
ejpam-6553	183	11	potential	potential	ADJ
ejpam-6553	183	12	directions	direction	NOUN
ejpam-6553	183	13	for	for	ADP
ejpam-6553	183	14	future	future	ADJ
ejpam-6553	183	15	research	research	NOUN
ejpam-6553	183	16	.	.	PUNCT
ejpam-6553	184	1	6.1	6.1	NUM
ejpam-6553	184	2	.	.	PUNCT
ejpam-6553	184	3	implications	implication	NOUN
ejpam-6553	184	4	of	of	ADP
ejpam-6553	184	5	the	the	DET
ejpam-6553	184	6	result	result	NOUN
ejpam-6553	184	7	the	the	DET
ejpam-6553	184	8	unsolvability	unsolvability	NOUN
ejpam-6553	184	9	of	of	ADP
ejpam-6553	184	10	the	the	DET
ejpam-6553	184	11	equation	equation	NOUN
ejpam-6553	184	12	23x+22y	23x+22y	NUM
ejpam-6553	184	13	=	=	SYM
ejpam-6553	184	14	z2	z2	PROPN
ejpam-6553	184	15	in	in	ADP
ejpam-6553	184	16	nonnegative	nonnegative	ADJ
ejpam-6553	184	17	integers	integer	NOUN
ejpam-6553	184	18	highlights	highlight	NOUN
ejpam-6553	184	19	the	the	DET
ejpam-6553	184	20	intricate	intricate	ADJ
ejpam-6553	184	21	relationship	relationship	NOUN
ejpam-6553	184	22	between	between	ADP
ejpam-6553	184	23	the	the	DET
ejpam-6553	184	24	properties	property	NOUN
ejpam-6553	184	25	of	of	ADP
ejpam-6553	184	26	the	the	DET
ejpam-6553	184	27	bases	basis	NOUN
ejpam-6553	184	28	23	23	NUM
ejpam-6553	184	29	and	and	CCONJ
ejpam-6553	184	30	22	22	NUM
ejpam-6553	184	31	.	.	PUNCT
ejpam-6553	185	1	specifically	specifically	ADV
ejpam-6553	185	2	:	:	PUNCT
ejpam-6553	185	3	•	•	ADP
ejpam-6553	185	4	the	the	DET
ejpam-6553	185	5	primality	primality	NOUN
ejpam-6553	185	6	of	of	ADP
ejpam-6553	185	7	23	23	NUM
ejpam-6553	185	8	simplifies	simplifie	NOUN
ejpam-6553	185	9	modular	modular	ADJ
ejpam-6553	185	10	analysis	analysis	NOUN
ejpam-6553	185	11	,	,	PUNCT
ejpam-6553	185	12	while	while	SCONJ
ejpam-6553	185	13	the	the	DET
ejpam-6553	185	14	compositeness	compositeness	NOUN
ejpam-6553	185	15	of	of	ADP
ejpam-6553	185	16	22	22	NUM
ejpam-6553	185	17	introduces	introduce	NOUN
ejpam-6553	185	18	additional	additional	ADJ
ejpam-6553	185	19	complexity	complexity	NOUN
ejpam-6553	185	20	.	.	PUNCT
ejpam-6553	186	1	•	•	NOUN
ejpam-6553	186	2	the	the	DET
ejpam-6553	186	3	use	use	NOUN
ejpam-6553	186	4	of	of	ADP
ejpam-6553	186	5	modular	modular	ADJ
ejpam-6553	186	6	arithmetic	arithmetic	ADJ
ejpam-6553	186	7	(	(	PUNCT
ejpam-6553	186	8	modulo	modulo	NOUN
ejpam-6553	186	9	4	4	NUM
ejpam-6553	186	10	and	and	CCONJ
ejpam-6553	186	11	5	5	NUM
ejpam-6553	186	12	)	)	PUNCT
ejpam-6553	186	13	provides	provide	VERB
ejpam-6553	186	14	a	a	DET
ejpam-6553	186	15	robust	robust	ADJ
ejpam-6553	186	16	framework	framework	NOUN
ejpam-6553	186	17	for	for	ADP
ejpam-6553	186	18	constraining	constrain	VERB
ejpam-6553	186	19	the	the	DET
ejpam-6553	186	20	possible	possible	ADJ
ejpam-6553	186	21	values	value	NOUN
ejpam-6553	186	22	of	of	ADP
ejpam-6553	186	23	x	x	X
ejpam-6553	186	24	and	and	CCONJ
ejpam-6553	186	25	y.	y.	PROPN
ejpam-6553	186	26	•	•	NOUN
ejpam-6553	186	27	computational	computational	ADJ
ejpam-6553	186	28	verification	verification	NOUN
ejpam-6553	186	29	for	for	ADP
ejpam-6553	186	30	small	small	ADJ
ejpam-6553	186	31	values	value	NOUN
ejpam-6553	186	32	of	of	ADP
ejpam-6553	186	33	x	x	PUNCT
ejpam-6553	186	34	and	and	CCONJ
ejpam-6553	186	35	y	y	PROPN
ejpam-6553	186	36	reinforces	reinforce	VERB
ejpam-6553	186	37	the	the	DET
ejpam-6553	186	38	theoretical	theoretical	ADJ
ejpam-6553	186	39	results	result	NOUN
ejpam-6553	186	40	and	and	CCONJ
ejpam-6553	186	41	ensures	ensure	VERB
ejpam-6553	186	42	their	their	PRON
ejpam-6553	186	43	validity	validity	NOUN
ejpam-6553	186	44	.	.	PUNCT
ejpam-6553	187	1	a.	a.	NOUN
ejpam-6553	187	2	assiry	assiry	PROPN
ejpam-6553	187	3	/	/	SYM
ejpam-6553	187	4	eur	eur	PROPN
ejpam-6553	187	5	.	.	PUNCT
ejpam-6553	188	1	j.	j.	PROPN
ejpam-6553	188	2	pure	pure	PROPN
ejpam-6553	188	3	appl	appl	PROPN
ejpam-6553	188	4	.	.	PROPN
ejpam-6553	188	5	math	math	PROPN
ejpam-6553	188	6	,	,	PUNCT
ejpam-6553	188	7	18	18	NUM
ejpam-6553	188	8	(	(	PUNCT
ejpam-6553	188	9	3	3	NUM
ejpam-6553	188	10	)	)	PUNCT
ejpam-6553	188	11	(	(	PUNCT
ejpam-6553	188	12	2025	2025	NUM
ejpam-6553	188	13	)	)	PUNCT
ejpam-6553	188	14	,	,	PUNCT
ejpam-6553	188	15	6553	6553	NUM
ejpam-6553	188	16	9	9	NUM
ejpam-6553	188	17	of	of	ADP
ejpam-6553	188	18	11	11	NUM
ejpam-6553	188	19	6.2	6.2	NUM
ejpam-6553	188	20	.	.	PUNCT
ejpam-6553	189	1	generalization	generalization	NOUN
ejpam-6553	189	2	to	to	ADP
ejpam-6553	189	3	other	other	ADJ
ejpam-6553	189	4	equations	equation	NOUN
ejpam-6553	189	5	the	the	DET
ejpam-6553	189	6	techniques	technique	NOUN
ejpam-6553	189	7	developed	develop	VERB
ejpam-6553	189	8	in	in	ADP
ejpam-6553	189	9	this	this	DET
ejpam-6553	189	10	paper	paper	NOUN
ejpam-6553	189	11	can	can	AUX
ejpam-6553	189	12	be	be	AUX
ejpam-6553	189	13	extended	extend	VERB
ejpam-6553	189	14	to	to	ADP
ejpam-6553	189	15	other	other	ADJ
ejpam-6553	189	16	equations	equation	NOUN
ejpam-6553	189	17	of	of	ADP
ejpam-6553	189	18	the	the	DET
ejpam-6553	189	19	form	form	NOUN
ejpam-6553	189	20	wx	wx	PROPN
ejpam-6553	189	21	+	+	CCONJ
ejpam-6553	190	1	(	(	PUNCT
ejpam-6553	190	2	w	w	NOUN
ejpam-6553	190	3	−	−	PROPN
ejpam-6553	190	4	1)y	1)y	NUM
ejpam-6553	190	5	=	=	SYM
ejpam-6553	190	6	z2	z2	PROPN
ejpam-6553	190	7	,	,	PUNCT
ejpam-6553	190	8	where	where	SCONJ
ejpam-6553	190	9	w	w	NOUN
ejpam-6553	190	10	is	be	AUX
ejpam-6553	190	11	a	a	DET
ejpam-6553	190	12	positive	positive	ADJ
ejpam-6553	190	13	integer	integer	NOUN
ejpam-6553	190	14	of	of	ADP
ejpam-6553	190	15	the	the	DET
ejpam-6553	190	16	form	form	NOUN
ejpam-6553	190	17	4n	4n	NOUN
ejpam-6553	190	18	+	+	NOUN
ejpam-6553	190	19	3	3	X
ejpam-6553	190	20	.	.	X
ejpam-6553	191	1	for	for	ADP
ejpam-6553	191	2	example	example	NOUN
ejpam-6553	191	3	:	:	PUNCT
ejpam-6553	191	4	•	•	NUM
ejpam-6553	191	5	19x	19x	NUM
ejpam-6553	191	6	+	+	SYM
ejpam-6553	191	7	18y	18y	NOUN
ejpam-6553	191	8	=	=	SYM
ejpam-6553	191	9	z2	z2	PROPN
ejpam-6553	191	10	,	,	PUNCT
ejpam-6553	191	11	•	•	ADJ
ejpam-6553	191	12	31x	31x	NOUN
ejpam-6553	191	13	+	+	CCONJ
ejpam-6553	191	14	30y	30y	NOUN
ejpam-6553	191	15	=	=	SYM
ejpam-6553	191	16	z2	z2	PROPN
ejpam-6553	191	17	.	.	PUNCT
ejpam-6553	192	1	in	in	ADP
ejpam-6553	192	2	each	each	DET
ejpam-6553	192	3	case	case	NOUN
ejpam-6553	192	4	,	,	PUNCT
ejpam-6553	192	5	the	the	DET
ejpam-6553	192	6	proof	proof	NOUN
ejpam-6553	192	7	follows	follow	VERB
ejpam-6553	192	8	a	a	DET
ejpam-6553	192	9	similar	similar	ADJ
ejpam-6553	192	10	structure	structure	NOUN
ejpam-6553	192	11	,	,	PUNCT
ejpam-6553	192	12	leveraging	leverage	VERB
ejpam-6553	192	13	modular	modular	ADJ
ejpam-6553	192	14	arithmetic	arithmetic	ADJ
ejpam-6553	192	15	and	and	CCONJ
ejpam-6553	192	16	parity	parity	NOUN
ejpam-6553	192	17	analysis	analysis	NOUN
ejpam-6553	192	18	to	to	PART
ejpam-6553	192	19	derive	derive	VERB
ejpam-6553	192	20	contradictions	contradiction	NOUN
ejpam-6553	192	21	.	.	PUNCT
ejpam-6553	193	1	this	this	PRON
ejpam-6553	193	2	suggests	suggest	VERB
ejpam-6553	193	3	a	a	DET
ejpam-6553	193	4	broader	broad	ADJ
ejpam-6553	193	5	applicability	applicability	NOUN
ejpam-6553	193	6	of	of	ADP
ejpam-6553	193	7	our	our	PRON
ejpam-6553	193	8	methods	method	NOUN
ejpam-6553	193	9	to	to	ADP
ejpam-6553	193	10	a	a	DET
ejpam-6553	193	11	wide	wide	ADJ
ejpam-6553	193	12	range	range	NOUN
ejpam-6553	193	13	of	of	ADP
ejpam-6553	193	14	exponential	exponential	ADJ
ejpam-6553	193	15	diophantine	diophantine	NOUN
ejpam-6553	193	16	equations	equation	NOUN
ejpam-6553	193	17	.	.	PUNCT
ejpam-6553	194	1	6.3	6.3	NUM
ejpam-6553	194	2	.	.	PUNCT
ejpam-6553	195	1	future	future	ADJ
ejpam-6553	195	2	research	research	NOUN
ejpam-6553	195	3	directions	direction	NOUN
ejpam-6553	195	4	our	our	PRON
ejpam-6553	195	5	work	work	NOUN
ejpam-6553	195	6	opens	open	VERB
ejpam-6553	195	7	several	several	ADJ
ejpam-6553	195	8	avenues	avenue	NOUN
ejpam-6553	195	9	for	for	ADP
ejpam-6553	195	10	future	future	ADJ
ejpam-6553	195	11	research	research	NOUN
ejpam-6553	195	12	:	:	PUNCT
ejpam-6553	195	13	•	•	NUM
ejpam-6553	195	14	generalization	generalization	NOUN
ejpam-6553	195	15	to	to	ADP
ejpam-6553	195	16	other	other	ADJ
ejpam-6553	195	17	primes	prime	NOUN
ejpam-6553	195	18	:	:	PUNCT
ejpam-6553	195	19	investigate	investigate	VERB
ejpam-6553	195	20	equations	equation	NOUN
ejpam-6553	195	21	of	of	ADP
ejpam-6553	195	22	the	the	DET
ejpam-6553	195	23	form	form	NOUN
ejpam-6553	195	24	px+(p−1)y	px+(p−1)y	NOUN
ejpam-6553	195	25	=	=	SYM
ejpam-6553	195	26	z2	z2	PROPN
ejpam-6553	195	27	for	for	ADP
ejpam-6553	195	28	other	other	ADJ
ejpam-6553	195	29	primes	prime	NOUN
ejpam-6553	195	30	p	p	X
ejpam-6553	195	31	,	,	PUNCT
ejpam-6553	195	32	particularly	particularly	ADV
ejpam-6553	195	33	those	those	PRON
ejpam-6553	195	34	with	with	ADP
ejpam-6553	195	35	specific	specific	ADJ
ejpam-6553	195	36	modular	modular	ADJ
ejpam-6553	195	37	properties	property	NOUN
ejpam-6553	195	38	.	.	PUNCT
ejpam-6553	196	1	•	•	NUM
ejpam-6553	196	2	higher	high	ADJ
ejpam-6553	196	3	exponents	exponent	NOUN
ejpam-6553	196	4	:	:	PUNCT
ejpam-6553	196	5	explore	explore	VERB
ejpam-6553	196	6	equations	equation	NOUN
ejpam-6553	196	7	of	of	ADP
ejpam-6553	196	8	the	the	DET
ejpam-6553	196	9	form	form	NOUN
ejpam-6553	196	10	px	px	X
ejpam-6553	196	11	+	+	CCONJ
ejpam-6553	196	12	qy	qy	NOUN
ejpam-6553	196	13	=	=	PROPN
ejpam-6553	196	14	zk	zk	PROPN
ejpam-6553	196	15	for	for	ADP
ejpam-6553	196	16	k	k	PROPN
ejpam-6553	196	17	≥	≥	PROPN
ejpam-6553	196	18	3	3	NUM
ejpam-6553	196	19	,	,	PUNCT
ejpam-6553	196	20	where	where	SCONJ
ejpam-6553	196	21	the	the	DET
ejpam-6553	196	22	techniques	technique	NOUN
ejpam-6553	196	23	used	use	VERB
ejpam-6553	196	24	in	in	ADP
ejpam-6553	196	25	this	this	DET
ejpam-6553	196	26	paper	paper	NOUN
ejpam-6553	196	27	may	may	AUX
ejpam-6553	196	28	need	need	VERB
ejpam-6553	196	29	to	to	PART
ejpam-6553	196	30	be	be	AUX
ejpam-6553	196	31	adapted	adapt	VERB
ejpam-6553	196	32	or	or	CCONJ
ejpam-6553	196	33	extended	extend	VERB
ejpam-6553	196	34	.	.	PUNCT
ejpam-6553	197	1	•	•	NUM
ejpam-6553	197	2	computational	computational	ADJ
ejpam-6553	197	3	searches	search	NOUN
ejpam-6553	197	4	:	:	PUNCT
ejpam-6553	197	5	use	use	VERB
ejpam-6553	197	6	advanced	advanced	ADJ
ejpam-6553	197	7	computational	computational	ADJ
ejpam-6553	197	8	methods	method	NOUN
ejpam-6553	197	9	to	to	PART
ejpam-6553	197	10	search	search	VERB
ejpam-6553	197	11	for	for	ADP
ejpam-6553	197	12	solutions	solution	NOUN
ejpam-6553	197	13	in	in	ADP
ejpam-6553	197	14	larger	large	ADJ
ejpam-6553	197	15	ranges	range	NOUN
ejpam-6553	197	16	,	,	PUNCT
ejpam-6553	197	17	potentially	potentially	ADV
ejpam-6553	197	18	uncovering	uncover	VERB
ejpam-6553	197	19	new	new	ADJ
ejpam-6553	197	20	patterns	pattern	NOUN
ejpam-6553	197	21	or	or	CCONJ
ejpam-6553	197	22	counterexamples	counterexample	NOUN
ejpam-6553	197	23	.	.	PUNCT
ejpam-6553	198	1	•	•	NUM
ejpam-6553	198	2	connections	connection	NOUN
ejpam-6553	198	3	to	to	ADP
ejpam-6553	198	4	other	other	ADJ
ejpam-6553	198	5	problems	problem	NOUN
ejpam-6553	198	6	:	:	PUNCT
ejpam-6553	198	7	investigate	investigate	VERB
ejpam-6553	198	8	the	the	DET
ejpam-6553	198	9	relationship	relationship	NOUN
ejpam-6553	198	10	between	between	ADP
ejpam-6553	198	11	these	these	DET
ejpam-6553	198	12	equations	equation	NOUN
ejpam-6553	198	13	and	and	CCONJ
ejpam-6553	198	14	other	other	ADJ
ejpam-6553	198	15	well	well	ADV
ejpam-6553	198	16	-	-	PUNCT
ejpam-6553	198	17	known	know	VERB
ejpam-6553	198	18	problems	problem	NOUN
ejpam-6553	198	19	in	in	ADP
ejpam-6553	198	20	number	number	NOUN
ejpam-6553	198	21	theory	theory	NOUN
ejpam-6553	198	22	,	,	PUNCT
ejpam-6553	198	23	such	such	ADJ
ejpam-6553	198	24	as	as	ADP
ejpam-6553	198	25	the	the	DET
ejpam-6553	198	26	ramanujan	ramanujan	PROPN
ejpam-6553	198	27	–	–	PUNCT
ejpam-6553	198	28	nagell	nagell	ADJ
ejpam-6553	198	29	equation	equation	NOUN
ejpam-6553	198	30	or	or	CCONJ
ejpam-6553	198	31	catalan	catalan	NOUN
ejpam-6553	198	32	’s	’s	PART
ejpam-6553	198	33	conjecture	conjecture	NOUN
ejpam-6553	198	34	.	.	PUNCT
ejpam-6553	199	1	by	by	ADP
ejpam-6553	199	2	addressing	address	VERB
ejpam-6553	199	3	these	these	DET
ejpam-6553	199	4	questions	question	NOUN
ejpam-6553	199	5	,	,	PUNCT
ejpam-6553	199	6	future	future	ADJ
ejpam-6553	199	7	research	research	NOUN
ejpam-6553	199	8	can	can	AUX
ejpam-6553	199	9	deepen	deepen	VERB
ejpam-6553	199	10	our	our	PRON
ejpam-6553	199	11	understanding	understanding	NOUN
ejpam-6553	199	12	of	of	ADP
ejpam-6553	199	13	exponential	exponential	ADJ
ejpam-6553	199	14	diophantine	diophantine	NOUN
ejpam-6553	199	15	equations	equation	NOUN
ejpam-6553	199	16	and	and	CCONJ
ejpam-6553	199	17	their	their	PRON
ejpam-6553	199	18	solutions	solution	NOUN
ejpam-6553	199	19	.	.	PUNCT
ejpam-6553	200	1	7	7	X
ejpam-6553	200	2	.	.	X
ejpam-6553	200	3	conclusion	conclusion	NOUN
ejpam-6553	200	4	in	in	ADP
ejpam-6553	200	5	this	this	DET
ejpam-6553	200	6	paper	paper	NOUN
ejpam-6553	200	7	,	,	PUNCT
ejpam-6553	200	8	we	we	PRON
ejpam-6553	200	9	have	have	AUX
ejpam-6553	200	10	proven	prove	VERB
ejpam-6553	200	11	that	that	SCONJ
ejpam-6553	200	12	the	the	DET
ejpam-6553	200	13	exponential	exponential	ADJ
ejpam-6553	200	14	diophantine	diophantine	NOUN
ejpam-6553	200	15	equation	equation	NOUN
ejpam-6553	200	16	23x+22y	23x+22y	NUM
ejpam-6553	200	17	=	=	SYM
ejpam-6553	200	18	z2	z2	PROPN
ejpam-6553	200	19	has	have	VERB
ejpam-6553	200	20	no	no	DET
ejpam-6553	200	21	nonnegative	nonnegative	ADJ
ejpam-6553	200	22	integer	integer	NOUN
ejpam-6553	200	23	solutions	solution	NOUN
ejpam-6553	200	24	.	.	PUNCT
ejpam-6553	201	1	our	our	PRON
ejpam-6553	201	2	proof	proof	NOUN
ejpam-6553	201	3	relies	rely	VERB
ejpam-6553	201	4	on	on	ADP
ejpam-6553	201	5	a	a	DET
ejpam-6553	201	6	combination	combination	NOUN
ejpam-6553	201	7	of	of	ADP
ejpam-6553	201	8	modular	modular	ADJ
ejpam-6553	201	9	arithmetic	arithmetic	ADJ
ejpam-6553	201	10	,	,	PUNCT
ejpam-6553	201	11	parity	parity	NOUN
ejpam-6553	201	12	analysis	analysis	NOUN
ejpam-6553	201	13	,	,	PUNCT
ejpam-6553	201	14	and	and	CCONJ
ejpam-6553	201	15	computational	computational	ADJ
ejpam-6553	201	16	verification	verification	NOUN
ejpam-6553	201	17	,	,	PUNCT
ejpam-6553	201	18	demonstrating	demonstrate	VERB
ejpam-6553	201	19	the	the	DET
ejpam-6553	201	20	power	power	NOUN
ejpam-6553	201	21	of	of	ADP
ejpam-6553	201	22	integrating	integrate	VERB
ejpam-6553	201	23	theoretical	theoretical	ADJ
ejpam-6553	201	24	and	and	CCONJ
ejpam-6553	201	25	computational	computational	ADJ
ejpam-6553	201	26	methods	method	NOUN
ejpam-6553	201	27	in	in	ADP
ejpam-6553	201	28	number	number	NOUN
ejpam-6553	201	29	theory	theory	NOUN
ejpam-6553	201	30	.	.	PUNCT
ejpam-6553	202	1	the	the	DET
ejpam-6553	202	2	result	result	NOUN
ejpam-6553	202	3	contributes	contribute	VERB
ejpam-6553	202	4	to	to	ADP
ejpam-6553	202	5	the	the	DET
ejpam-6553	202	6	broader	broad	ADJ
ejpam-6553	202	7	understanding	understanding	NOUN
ejpam-6553	202	8	of	of	ADP
ejpam-6553	202	9	exponential	exponential	ADJ
ejpam-6553	202	10	diophantine	diophantine	NOUN
ejpam-6553	202	11	equations	equation	NOUN
ejpam-6553	202	12	and	and	CCONJ
ejpam-6553	202	13	highlights	highlight	NOUN
ejpam-6553	202	14	the	the	DET
ejpam-6553	202	15	importance	importance	NOUN
ejpam-6553	202	16	of	of	ADP
ejpam-6553	202	17	modular	modular	ADJ
ejpam-6553	202	18	constraints	constraint	NOUN
ejpam-6553	202	19	in	in	ADP
ejpam-6553	202	20	solving	solve	VERB
ejpam-6553	202	21	such	such	ADJ
ejpam-6553	202	22	problems	problem	NOUN
ejpam-6553	202	23	.	.	PUNCT
ejpam-6553	203	1	furthermore	furthermore	ADV
ejpam-6553	203	2	,	,	PUNCT
ejpam-6553	203	3	we	we	PRON
ejpam-6553	203	4	have	have	AUX
ejpam-6553	203	5	shown	show	VERB
ejpam-6553	203	6	that	that	SCONJ
ejpam-6553	203	7	the	the	DET
ejpam-6553	203	8	techniques	technique	NOUN
ejpam-6553	203	9	developed	develop	VERB
ejpam-6553	203	10	in	in	ADP
ejpam-6553	203	11	this	this	DET
ejpam-6553	203	12	paper	paper	NOUN
ejpam-6553	203	13	can	can	AUX
ejpam-6553	203	14	be	be	AUX
ejpam-6553	203	15	generalized	generalize	VERB
ejpam-6553	203	16	to	to	ADP
ejpam-6553	203	17	other	other	ADJ
ejpam-6553	203	18	equations	equation	NOUN
ejpam-6553	203	19	of	of	ADP
ejpam-6553	203	20	the	the	DET
ejpam-6553	203	21	form	form	NOUN
ejpam-6553	203	22	wx+(w−1)y	wx+(w−1)y	PROPN
ejpam-6553	203	23	=	=	SYM
ejpam-6553	203	24	z2	z2	PROPN
ejpam-6553	203	25	,	,	PUNCT
ejpam-6553	203	26	where	where	SCONJ
ejpam-6553	203	27	w	w	NOUN
ejpam-6553	203	28	is	be	AUX
ejpam-6553	203	29	a	a	DET
ejpam-6553	203	30	positive	positive	ADJ
ejpam-6553	203	31	integer	integer	NOUN
ejpam-6553	203	32	of	of	ADP
ejpam-6553	203	33	a	a	DET
ejpam-6553	203	34	specific	specific	ADJ
ejpam-6553	203	35	form	form	NOUN
ejpam-6553	203	36	.	.	PUNCT
ejpam-6553	204	1	this	this	PRON
ejpam-6553	204	2	opens	open	VERB
ejpam-6553	204	3	the	the	DET
ejpam-6553	204	4	door	door	NOUN
ejpam-6553	204	5	to	to	PART
ejpam-6553	204	6	further	further	ADJ
ejpam-6553	204	7	research	research	NOUN
ejpam-6553	204	8	into	into	ADP
ejpam-6553	204	9	similar	similar	ADJ
ejpam-6553	204	10	equations	equation	NOUN
ejpam-6553	204	11	and	and	CCONJ
ejpam-6553	204	12	their	their	PRON
ejpam-6553	204	13	properties	property	NOUN
ejpam-6553	204	14	.	.	PUNCT
ejpam-6553	205	1	our	our	PRON
ejpam-6553	205	2	work	work	NOUN
ejpam-6553	205	3	underscores	underscore	VERB
ejpam-6553	205	4	the	the	DET
ejpam-6553	205	5	value	value	NOUN
ejpam-6553	205	6	of	of	ADP
ejpam-6553	205	7	combining	combine	VERB
ejpam-6553	205	8	theoretical	theoretical	ADJ
ejpam-6553	205	9	insights	insight	NOUN
ejpam-6553	205	10	with	with	ADP
ejpam-6553	205	11	computational	computational	ADJ
ejpam-6553	205	12	tools	tool	NOUN
ejpam-6553	205	13	to	to	PART
ejpam-6553	205	14	tackle	tackle	VERB
ejpam-6553	205	15	challenging	challenging	ADJ
ejpam-6553	205	16	problems	problem	NOUN
ejpam-6553	205	17	in	in	ADP
ejpam-6553	205	18	mathematics	mathematic	NOUN
ejpam-6553	205	19	.	.	PUNCT
ejpam-6553	206	1	we	we	PRON
ejpam-6553	206	2	hope	hope	VERB
ejpam-6553	206	3	that	that	SCONJ
ejpam-6553	206	4	this	this	DET
ejpam-6553	206	5	study	study	NOUN
ejpam-6553	206	6	will	will	AUX
ejpam-6553	206	7	inspire	inspire	VERB
ejpam-6553	206	8	a.	a.	NOUN
ejpam-6553	206	9	assiry	assiry	PROPN
ejpam-6553	206	10	/	/	SYM
ejpam-6553	206	11	eur	eur	PROPN
ejpam-6553	206	12	.	.	PUNCT
ejpam-6553	207	1	j.	j.	PROPN
ejpam-6553	207	2	pure	pure	PROPN
ejpam-6553	207	3	appl	appl	PROPN
ejpam-6553	207	4	.	.	PROPN
ejpam-6553	207	5	math	math	PROPN
ejpam-6553	207	6	,	,	PUNCT
ejpam-6553	207	7	18	18	NUM
ejpam-6553	207	8	(	(	PUNCT
ejpam-6553	207	9	3	3	NUM
ejpam-6553	207	10	)	)	PUNCT
ejpam-6553	207	11	(	(	PUNCT
ejpam-6553	207	12	2025	2025	NUM
ejpam-6553	207	13	)	)	PUNCT
ejpam-6553	207	14	,	,	PUNCT
ejpam-6553	207	15	6553	6553	NUM
ejpam-6553	207	16	10	10	NUM
ejpam-6553	207	17	of	of	ADP
ejpam-6553	207	18	11	11	NUM
ejpam-6553	207	19	further	further	ADJ
ejpam-6553	207	20	research	research	NOUN
ejpam-6553	207	21	into	into	ADP
ejpam-6553	207	22	exponential	exponential	ADJ
ejpam-6553	207	23	diophantine	diophantine	NOUN
ejpam-6553	207	24	equations	equation	NOUN
ejpam-6553	207	25	and	and	CCONJ
ejpam-6553	207	26	their	their	PRON
ejpam-6553	207	27	applications	application	NOUN
ejpam-6553	207	28	,	,	PUNCT
ejpam-6553	207	29	contributing	contribute	VERB
ejpam-6553	207	30	to	to	ADP
ejpam-6553	207	31	the	the	DET
ejpam-6553	207	32	ongoing	ongoing	ADJ
ejpam-6553	207	33	development	development	NOUN
ejpam-6553	207	34	of	of	ADP
ejpam-6553	207	35	number	number	NOUN
ejpam-6553	207	36	theory	theory	NOUN
ejpam-6553	207	37	.	.	PUNCT
ejpam-6553	208	1	acknowledgements	acknowledgement	VERB
ejpam-6553	208	2	the	the	DET
ejpam-6553	208	3	author	author	NOUN
ejpam-6553	208	4	extend	extend	VERB
ejpam-6553	208	5	his	his	PRON
ejpam-6553	208	6	appreciation	appreciation	NOUN
ejpam-6553	208	7	to	to	ADP
ejpam-6553	208	8	umm	umm	INTJ
ejpam-6553	208	9	al	al	PROPN
ejpam-6553	208	10	-	-	PUNCT
ejpam-6553	208	11	qura	qura	PROPN
ejpam-6553	208	12	university	university	PROPN
ejpam-6553	208	13	,	,	PUNCT
ejpam-6553	208	14	saudi	saudi	PROPN
ejpam-6553	208	15	arabia	arabia	PROPN
ejpam-6553	208	16	for	for	ADP
ejpam-6553	208	17	funding	fund	VERB
ejpam-6553	208	18	this	this	DET
ejpam-6553	208	19	research	research	NOUN
ejpam-6553	208	20	work	work	NOUN
ejpam-6553	208	21	through	through	ADP
ejpam-6553	208	22	grant	grant	NOUN
ejpam-6553	208	23	number	number	NOUN
ejpam-6553	208	24	:	:	PUNCT
ejpam-6553	208	25	25uqu4270201gssr03	25uqu4270201gssr03	NUM
ejpam-6553	208	26	.	.	PUNCT
ejpam-6553	209	1	funding	fund	VERB
ejpam-6553	209	2	this	this	DET
ejpam-6553	209	3	research	research	NOUN
ejpam-6553	209	4	work	work	NOUN
ejpam-6553	209	5	was	be	AUX
ejpam-6553	209	6	funded	fund	VERB
ejpam-6553	209	7	by	by	ADP
ejpam-6553	209	8	umm	umm	INTJ
ejpam-6553	209	9	al	al	PROPN
ejpam-6553	209	10	-	-	PUNCT
ejpam-6553	209	11	qura	qura	PROPN
ejpam-6553	209	12	university	university	NOUN
ejpam-6553	209	13	,	,	PUNCT
ejpam-6553	209	14	saudi	saudi	PROPN
ejpam-6553	209	15	arabia	arabia	PROPN
ejpam-6553	209	16	under	under	ADP
ejpam-6553	209	17	grant	grant	NOUN
ejpam-6553	209	18	number	number	NOUN
ejpam-6553	209	19	:	:	PUNCT
ejpam-6553	209	20	25uqu4270201gssr03	25uqu4270201gssr03	NUM
ejpam-6553	209	21	.	.	PUNCT
ejpam-6553	210	1	references	reference	NOUN
ejpam-6553	210	2	[	[	X
ejpam-6553	210	3	1	1	X
ejpam-6553	210	4	]	]	PUNCT
ejpam-6553	210	5	s.	s.	PROPN
ejpam-6553	210	6	lang	lang	PROPN
ejpam-6553	210	7	.	.	PUNCT
ejpam-6553	211	1	algebra	algebra	PROPN
ejpam-6553	211	2	.	.	PUNCT
ejpam-6553	212	1	springer	springer	NOUN
ejpam-6553	212	2	-	-	PUNCT
ejpam-6553	212	3	verlag	verlag	PROPN
ejpam-6553	212	4	,	,	PUNCT
ejpam-6553	212	5	3rd	3rd	ADJ
ejpam-6553	212	6	edition	edition	NOUN
ejpam-6553	212	7	,	,	PUNCT
ejpam-6553	212	8	2002	2002	NUM
ejpam-6553	212	9	.	.	PUNCT
ejpam-6553	213	1	particularly	particularly	ADV
ejpam-6553	213	2	chapter	chapter	NOUN
ejpam-6553	213	3	iv	iv	NUM
ejpam-6553	213	4	on	on	ADP
ejpam-6553	213	5	diophantine	diophantine	NOUN
ejpam-6553	213	6	equations	equation	NOUN
ejpam-6553	213	7	.	.	PUNCT
ejpam-6553	214	1	[	[	X
ejpam-6553	214	2	2	2	NUM
ejpam-6553	214	3	]	]	PUNCT
ejpam-6553	214	4	a.	a.	NOUN
ejpam-6553	214	5	baker	baker	PROPN
ejpam-6553	214	6	.	.	PUNCT
ejpam-6553	215	1	transcendental	transcendental	ADJ
ejpam-6553	215	2	number	number	NOUN
ejpam-6553	215	3	theory	theory	NOUN
ejpam-6553	215	4	.	.	PUNCT
ejpam-6553	216	1	cambridge	cambridge	PROPN
ejpam-6553	216	2	university	university	PROPN
ejpam-6553	216	3	press	press	NOUN
ejpam-6553	216	4	,	,	PUNCT
ejpam-6553	216	5	1975	1975	NUM
ejpam-6553	216	6	.	.	PUNCT
ejpam-6553	217	1	[	[	X
ejpam-6553	217	2	3	3	NUM
ejpam-6553	217	3	]	]	X
ejpam-6553	217	4	francesco	francesco	PROPN
ejpam-6553	217	5	amoroso	amoroso	PROPN
ejpam-6553	217	6	and	and	CCONJ
ejpam-6553	217	7	evelina	evelina	PROPN
ejpam-6553	217	8	viada	viada	PROPN
ejpam-6553	217	9	.	.	PUNCT
ejpam-6553	218	1	small	small	ADJ
ejpam-6553	218	2	points	point	NOUN
ejpam-6553	218	3	on	on	ADP
ejpam-6553	218	4	subvarieties	subvarietie	NOUN
ejpam-6553	218	5	of	of	ADP
ejpam-6553	218	6	algebraic	algebraic	PROPN
ejpam-6553	218	7	tori	tori	PROPN
ejpam-6553	218	8	.	.	PUNCT
ejpam-6553	219	1	international	international	ADJ
ejpam-6553	219	2	mathematics	mathematics	PROPN
ejpam-6553	219	3	research	research	NOUN
ejpam-6553	219	4	notices	notice	NOUN
ejpam-6553	219	5	,	,	PUNCT
ejpam-6553	219	6	2021(12):8911–8952	2021(12):8911–8952	NOUN
ejpam-6553	219	7	,	,	PUNCT
ejpam-6553	219	8	2021	2021	NUM
ejpam-6553	219	9	.	.	PUNCT
ejpam-6553	220	1	[	[	X
ejpam-6553	220	2	4	4	NUM
ejpam-6553	220	3	]	]	X
ejpam-6553	220	4	jason	jason	PROPN
ejpam-6553	220	5	bell	bell	PROPN
ejpam-6553	220	6	,	,	PUNCT
ejpam-6553	220	7	shaoshi	shaoshi	PROPN
ejpam-6553	220	8	chen	chen	PROPN
ejpam-6553	220	9	,	,	PUNCT
ejpam-6553	220	10	and	and	CCONJ
ejpam-6553	220	11	ziyang	ziyang	PROPN
ejpam-6553	220	12	huang	huang	PROPN
ejpam-6553	220	13	.	.	PUNCT
ejpam-6553	221	1	dynamical	dynamical	ADJ
ejpam-6553	221	2	diophantine	diophantine	NOUN
ejpam-6553	221	3	equations	equation	NOUN
ejpam-6553	221	4	.	.	PUNCT
ejpam-6553	222	1	compositio	compositio	PROPN
ejpam-6553	222	2	mathematica	mathematica	PROPN
ejpam-6553	222	3	,	,	PUNCT
ejpam-6553	222	4	158(1):1–38	158(1):1–38	PROPN
ejpam-6553	222	5	,	,	PUNCT
ejpam-6553	222	6	2022	2022	NUM
ejpam-6553	222	7	.	.	PUNCT
ejpam-6553	223	1	[	[	X
ejpam-6553	223	2	5	5	X
ejpam-6553	223	3	]	]	PUNCT
ejpam-6553	223	4	emmanuel	emmanuel	PROPN
ejpam-6553	223	5	kowalski	kowalski	PROPN
ejpam-6553	223	6	.	.	PUNCT
ejpam-6553	223	7	exponential	exponential	ADJ
ejpam-6553	223	8	sums	sum	NOUN
ejpam-6553	223	9	and	and	CCONJ
ejpam-6553	223	10	finite	finite	ADJ
ejpam-6553	223	11	fields	field	NOUN
ejpam-6553	223	12	.	.	PUNCT
ejpam-6553	224	1	princeton	princeton	PROPN
ejpam-6553	224	2	university	university	PROPN
ejpam-6553	224	3	press	press	NOUN
ejpam-6553	224	4	,	,	PUNCT
ejpam-6553	224	5	2020	2020	NUM
ejpam-6553	224	6	.	.	PUNCT
ejpam-6553	225	1	[	[	X
ejpam-6553	225	2	6	6	NUM
ejpam-6553	225	3	]	]	PUNCT
ejpam-6553	225	4	martin	martin	PROPN
ejpam-6553	225	5	bays	bay	NOUN
ejpam-6553	225	6	and	and	CCONJ
ejpam-6553	225	7	philipp	philipp	PROPN
ejpam-6553	225	8	habegger	habegger	NOUN
ejpam-6553	225	9	.	.	PUNCT
ejpam-6553	226	1	the	the	DET
ejpam-6553	226	2	existential	existential	ADJ
ejpam-6553	226	3	theory	theory	NOUN
ejpam-6553	226	4	of	of	ADP
ejpam-6553	226	5	exponential	exponential	ADJ
ejpam-6553	226	6	fields	field	NOUN
ejpam-6553	226	7	.	.	PUNCT
ejpam-6553	227	1	transactions	transaction	NOUN
ejpam-6553	227	2	of	of	ADP
ejpam-6553	227	3	the	the	DET
ejpam-6553	227	4	ams	am	NOUN
ejpam-6553	227	5	,	,	PUNCT
ejpam-6553	227	6	372:6951–6987	372:6951–6987	NUM
ejpam-6553	227	7	,	,	PUNCT
ejpam-6553	227	8	2019	2019	NUM
ejpam-6553	227	9	.	.	PUNCT
ejpam-6553	228	1	[	[	X
ejpam-6553	228	2	7	7	X
ejpam-6553	228	3	]	]	PUNCT
ejpam-6553	228	4	thomas	thomas	PROPN
ejpam-6553	228	5	scanlon	scanlon	PROPN
ejpam-6553	228	6	.	.	PUNCT
ejpam-6553	228	7	model	model	PROPN
ejpam-6553	228	8	theory	theory	NOUN
ejpam-6553	228	9	of	of	ADP
ejpam-6553	228	10	exponentiation	exponentiation	NOUN
ejpam-6553	228	11	.	.	PUNCT
ejpam-6553	229	1	notices	notice	NOUN
ejpam-6553	229	2	of	of	ADP
ejpam-6553	229	3	the	the	DET
ejpam-6553	229	4	ams	am	NOUN
ejpam-6553	229	5	,	,	PUNCT
ejpam-6553	229	6	65(6):1–10	65(6):1–10	NOUN
ejpam-6553	229	7	,	,	PUNCT
ejpam-6553	229	8	2018	2018	NUM
ejpam-6553	229	9	.	.	PUNCT
ejpam-6553	230	1	[	[	X
ejpam-6553	230	2	8	8	NUM
ejpam-6553	230	3	]	]	X
ejpam-6553	230	4	g.	g.	PROPN
ejpam-6553	230	5	h.	h.	PROPN
ejpam-6553	230	6	hardy	hardy	PROPN
ejpam-6553	230	7	and	and	CCONJ
ejpam-6553	230	8	e.	e.	PROPN
ejpam-6553	230	9	m.	m.	PROPN
ejpam-6553	230	10	wright	wright	PROPN
ejpam-6553	230	11	.	.	PUNCT
ejpam-6553	231	1	an	an	DET
ejpam-6553	231	2	introduction	introduction	NOUN
ejpam-6553	231	3	to	to	ADP
ejpam-6553	231	4	the	the	DET
ejpam-6553	231	5	theory	theory	NOUN
ejpam-6553	231	6	of	of	ADP
ejpam-6553	231	7	numbers	number	NOUN
ejpam-6553	231	8	.	.	PUNCT
ejpam-6553	232	1	oxford	oxford	PROPN
ejpam-6553	232	2	university	university	PROPN
ejpam-6553	232	3	press	press	NOUN
ejpam-6553	232	4	,	,	PUNCT
ejpam-6553	232	5	1927	1927	NUM
ejpam-6553	232	6	.	.	PUNCT
ejpam-6553	233	1	[	[	X
ejpam-6553	233	2	9	9	NUM
ejpam-6553	233	3	]	]	PUNCT
ejpam-6553	233	4	i.	i.	NOUN
ejpam-6553	233	5	niven	niven	PROPN
ejpam-6553	233	6	,	,	PUNCT
ejpam-6553	233	7	h.	h.	PROPN
ejpam-6553	233	8	s.	s.	PROPN
ejpam-6553	233	9	zuckerman	zuckerman	PROPN
ejpam-6553	233	10	,	,	PUNCT
ejpam-6553	233	11	and	and	CCONJ
ejpam-6553	233	12	h.	h.	PROPN
ejpam-6553	233	13	l.	l.	PROPN
ejpam-6553	233	14	montgomery	montgomery	PROPN
ejpam-6553	233	15	.	.	PUNCT
ejpam-6553	234	1	an	an	DET
ejpam-6553	234	2	introduction	introduction	NOUN
ejpam-6553	234	3	to	to	ADP
ejpam-6553	234	4	the	the	DET
ejpam-6553	234	5	theory	theory	NOUN
ejpam-6553	234	6	of	of	ADP
ejpam-6553	234	7	numbers	number	NOUN
ejpam-6553	234	8	.	.	PUNCT
ejpam-6553	235	1	wiley	wiley	PROPN
ejpam-6553	235	2	,	,	PUNCT
ejpam-6553	235	3	1980	1980	NUM
ejpam-6553	235	4	.	.	PUNCT
ejpam-6553	236	1	[	[	X
ejpam-6553	236	2	10	10	NUM
ejpam-6553	236	3	]	]	X
ejpam-6553	236	4	a.	a.	NOUN
ejpam-6553	236	5	chandoul	chandoul	PROPN
ejpam-6553	236	6	and	and	CCONJ
ejpam-6553	236	7	a.	a.	NOUN
ejpam-6553	236	8	sibih	sibih	PROPN
ejpam-6553	236	9	.	.	PUNCT
ejpam-6553	237	1	solutions	solution	NOUN
ejpam-6553	237	2	of	of	ADP
ejpam-6553	237	3	the	the	DET
ejpam-6553	237	4	equation	equation	NOUN
ejpam-6553	237	5	x2−p2q2±apy2	x2−p2q2±apy2	PUNCT
ejpam-6553	237	6	=	=	SYM
ejpam-6553	237	7	kt	kt	PROPN
ejpam-6553	237	8	.	.	PUNCT
ejpam-6553	237	9	edelweiss	edelweiss	PROPN
ejpam-6553	237	10	applied	apply	VERB
ejpam-6553	237	11	science	science	NOUN
ejpam-6553	237	12	and	and	CCONJ
ejpam-6553	237	13	technology	technology	NOUN
ejpam-6553	237	14	,	,	PUNCT
ejpam-6553	237	15	8(6):4408–4414	8(6):4408–4414	NUM
ejpam-6553	237	16	,	,	PUNCT
ejpam-6553	237	17	2024	2024	NUM
ejpam-6553	237	18	.	.	PUNCT
ejpam-6553	238	1	[	[	X
ejpam-6553	238	2	11	11	NUM
ejpam-6553	238	3	]	]	PUNCT
ejpam-6553	238	4	a.	a.	NOUN
ejpam-6553	238	5	chandoul	chandoul	PROPN
ejpam-6553	238	6	.	.	PUNCT
ejpam-6553	239	1	the	the	DET
ejpam-6553	239	2	pell	pell	NOUN
ejpam-6553	239	3	equation	equation	NOUN
ejpam-6553	239	4	x2	x2	PROPN
ejpam-6553	239	5	−	−	PROPN
ejpam-6553	239	6	dy2	dy2	PROPN
ejpam-6553	239	7	=	=	SYM
ejpam-6553	239	8	k2	k2	PROPN
ejpam-6553	239	9	.	.	PUNCT
ejpam-6553	239	10	advances	advance	NOUN
ejpam-6553	239	11	in	in	ADP
ejpam-6553	239	12	pure	pure	ADJ
ejpam-6553	239	13	mathematics	mathematic	NOUN
ejpam-6553	239	14	,	,	PUNCT
ejpam-6553	239	15	1(2):16	1(2):16	NUM
ejpam-6553	239	16	,	,	PUNCT
ejpam-6553	239	17	2011	2011	NUM
ejpam-6553	239	18	.	.	PUNCT
ejpam-6553	240	1	[	[	X
ejpam-6553	240	2	12	12	NUM
ejpam-6553	240	3	]	]	X
ejpam-6553	240	4	f.	f.	PROPN
ejpam-6553	240	5	beukers	beukers	PROPN
ejpam-6553	240	6	.	.	PUNCT
ejpam-6553	241	1	on	on	ADP
ejpam-6553	241	2	the	the	DET
ejpam-6553	241	3	generalized	generalized	ADJ
ejpam-6553	241	4	ramanujan	ramanujan	PROPN
ejpam-6553	241	5	–	–	PUNCT
ejpam-6553	241	6	nagell	nagell	ADJ
ejpam-6553	241	7	equation	equation	NOUN
ejpam-6553	241	8	.	.	PUNCT
ejpam-6553	242	1	acta	acta	PROPN
ejpam-6553	242	2	arithmetica	arithmetica	PROPN
ejpam-6553	242	3	,	,	PUNCT
ejpam-6553	242	4	38(4):389–410	38(4):389–410	NUM
ejpam-6553	242	5	,	,	PUNCT
ejpam-6553	242	6	1981	1981	NUM
ejpam-6553	242	7	.	.	PUNCT
ejpam-6553	243	1	[	[	X
ejpam-6553	243	2	13	13	NUM
ejpam-6553	243	3	]	]	PUNCT
ejpam-6553	243	4	m.	m.	NOUN
ejpam-6553	243	5	a.	a.	PROPN
ejpam-6553	243	6	bennett	bennett	PROPN
ejpam-6553	243	7	and	and	CCONJ
ejpam-6553	243	8	c.	c.	PROPN
ejpam-6553	243	9	m.	m.	PROPN
ejpam-6553	243	10	skinner	skinner	PROPN
ejpam-6553	243	11	.	.	PUNCT
ejpam-6553	244	1	ternary	ternary	ADJ
ejpam-6553	244	2	diophantine	diophantine	NOUN
ejpam-6553	244	3	equations	equation	NOUN
ejpam-6553	244	4	via	via	ADP
ejpam-6553	244	5	galois	galois	PROPN
ejpam-6553	244	6	representations	representation	NOUN
ejpam-6553	244	7	and	and	CCONJ
ejpam-6553	244	8	modular	modular	ADJ
ejpam-6553	244	9	forms	form	NOUN
ejpam-6553	244	10	.	.	PUNCT
ejpam-6553	245	1	canadian	canadian	ADJ
ejpam-6553	245	2	journal	journal	PROPN
ejpam-6553	245	3	of	of	ADP
ejpam-6553	245	4	mathematics	mathematic	NOUN
ejpam-6553	245	5	,	,	PUNCT
ejpam-6553	245	6	56(1):23–54	56(1):23–54	NUM
ejpam-6553	245	7	,	,	PUNCT
ejpam-6553	245	8	2004	2004	NUM
ejpam-6553	245	9	.	.	PUNCT
ejpam-6553	246	1	[	[	X
ejpam-6553	246	2	14	14	NUM
ejpam-6553	246	3	]	]	X
ejpam-6553	246	4	b.	b.	PROPN
ejpam-6553	246	5	m.	m.	PROPN
ejpam-6553	246	6	m.	m.	PROPN
ejpam-6553	246	7	de	de	PROPN
ejpam-6553	246	8	weger	weger	PROPN
ejpam-6553	246	9	.	.	PUNCT
ejpam-6553	247	1	solving	solve	VERB
ejpam-6553	247	2	exponential	exponential	ADJ
ejpam-6553	247	3	diophantine	diophantine	NOUN
ejpam-6553	247	4	equations	equation	NOUN
ejpam-6553	247	5	using	use	VERB
ejpam-6553	247	6	lattice	lattice	NOUN
ejpam-6553	247	7	basis	basis	NOUN
ejpam-6553	247	8	reduction	reduction	NOUN
ejpam-6553	247	9	algorithms	algorithm	NOUN
ejpam-6553	247	10	.	.	PUNCT
ejpam-6553	248	1	journal	journal	NOUN
ejpam-6553	248	2	of	of	ADP
ejpam-6553	248	3	number	number	NOUN
ejpam-6553	248	4	theory	theory	NOUN
ejpam-6553	248	5	,	,	PUNCT
ejpam-6553	248	6	26(3):325–367	26(3):325–367	PROPN
ejpam-6553	248	7	,	,	PUNCT
ejpam-6553	248	8	1989	1989	NUM
ejpam-6553	248	9	.	.	PUNCT
ejpam-6553	249	1	a.	a.	NOUN
ejpam-6553	249	2	assiry	assiry	PROPN
ejpam-6553	249	3	/	/	SYM
ejpam-6553	249	4	eur	eur	PROPN
ejpam-6553	249	5	.	.	PUNCT
ejpam-6553	250	1	j.	j.	PROPN
ejpam-6553	250	2	pure	pure	PROPN
ejpam-6553	250	3	appl	appl	PROPN
ejpam-6553	250	4	.	.	PROPN
ejpam-6553	250	5	math	math	PROPN
ejpam-6553	250	6	,	,	PUNCT
ejpam-6553	250	7	18	18	NUM
ejpam-6553	250	8	(	(	PUNCT
ejpam-6553	250	9	3	3	NUM
ejpam-6553	250	10	)	)	PUNCT
ejpam-6553	250	11	(	(	PUNCT
ejpam-6553	250	12	2025	2025	NUM
ejpam-6553	250	13	)	)	PUNCT
ejpam-6553	250	14	,	,	PUNCT
ejpam-6553	250	15	6553	6553	NUM
ejpam-6553	250	16	11	11	NUM
ejpam-6553	250	17	of	of	ADP
ejpam-6553	250	18	11	11	NUM
ejpam-6553	250	19	[	[	SYM
ejpam-6553	250	20	15	15	NUM
ejpam-6553	250	21	]	]	X
ejpam-6553	250	22	n.	n.	NOUN
ejpam-6553	250	23	p.	p.	NOUN
ejpam-6553	250	24	smart	smart	ADJ
ejpam-6553	250	25	.	.	PUNCT
ejpam-6553	251	1	the	the	DET
ejpam-6553	251	2	algorithmic	algorithmic	ADJ
ejpam-6553	251	3	resolution	resolution	NOUN
ejpam-6553	251	4	of	of	ADP
ejpam-6553	251	5	diophantine	diophantine	NOUN
ejpam-6553	251	6	equations	equation	NOUN
ejpam-6553	251	7	.	.	PUNCT
ejpam-6553	252	1	cambridge	cambridge	PROPN
ejpam-6553	252	2	university	university	PROPN
ejpam-6553	252	3	press	press	NOUN
ejpam-6553	252	4	,	,	PUNCT
ejpam-6553	252	5	1998	1998	NUM
ejpam-6553	252	6	.	.	PUNCT
ejpam-6553	253	1	[	[	X
ejpam-6553	253	2	16	16	NUM
ejpam-6553	253	3	]	]	X
ejpam-6553	253	4	y.	y.	PROPN
ejpam-6553	253	5	bugeaud	bugeaud	PROPN
ejpam-6553	253	6	and	and	CCONJ
ejpam-6553	253	7	t.	t.	PROPN
ejpam-6553	253	8	n.	n.	PROPN
ejpam-6553	253	9	shorey	shorey	PROPN
ejpam-6553	253	10	.	.	PUNCT
ejpam-6553	254	1	on	on	ADP
ejpam-6553	254	2	the	the	DET
ejpam-6553	254	3	number	number	NOUN
ejpam-6553	254	4	of	of	ADP
ejpam-6553	254	5	solutions	solution	NOUN
ejpam-6553	254	6	of	of	ADP
ejpam-6553	254	7	the	the	DET
ejpam-6553	254	8	generalized	generalized	ADJ
ejpam-6553	254	9	ramanujan	ramanujan	PROPN
ejpam-6553	254	10	–	–	PUNCT
ejpam-6553	254	11	nagell	nagell	ADJ
ejpam-6553	254	12	equation	equation	NOUN
ejpam-6553	254	13	.	.	PUNCT
ejpam-6553	255	1	journal	journal	PROPN
ejpam-6553	255	2	für	für	AUX
ejpam-6553	255	3	die	die	VERB
ejpam-6553	255	4	reine	reine	PROPN
ejpam-6553	255	5	und	und	PROPN
ejpam-6553	255	6	angewandte	angewandte	PROPN
ejpam-6553	255	7	mathematik	mathematik	PROPN
ejpam-6553	255	8	,	,	PUNCT
ejpam-6553	255	9	539:55–74	539:55–74	PROPN
ejpam-6553	255	10	,	,	PUNCT
ejpam-6553	255	11	2001	2001	NUM
ejpam-6553	255	12	.	.	PUNCT
ejpam-6553	256	1	[	[	X
ejpam-6553	256	2	17	17	NUM
ejpam-6553	256	3	]	]	PUNCT
ejpam-6553	256	4	m.	m.	NOUN
ejpam-6553	256	5	le	le	PROPN
ejpam-6553	256	6	.	.	PUNCT
ejpam-6553	257	1	on	on	ADP
ejpam-6553	257	2	the	the	DET
ejpam-6553	257	3	ramanujan	ramanujan	PROPN
ejpam-6553	257	4	–	–	PUNCT
ejpam-6553	257	5	nagell	nagell	ADJ
ejpam-6553	257	6	equation	equation	NOUN
ejpam-6553	257	7	and	and	CCONJ
ejpam-6553	257	8	its	its	PRON
ejpam-6553	257	9	generalizations	generalization	NOUN
ejpam-6553	257	10	.	.	PUNCT
ejpam-6553	258	1	journal	journal	NOUN
ejpam-6553	258	2	of	of	ADP
ejpam-6553	258	3	number	number	NOUN
ejpam-6553	258	4	theory	theory	NOUN
ejpam-6553	258	5	,	,	PUNCT
ejpam-6553	258	6	44(2):123–135	44(2):123–135	PROPN
ejpam-6553	258	7	,	,	PUNCT
ejpam-6553	258	8	1993	1993	NUM
ejpam-6553	258	9	.	.	PUNCT
ejpam-6553	259	1	[	[	X
ejpam-6553	259	2	18	18	NUM
ejpam-6553	259	3	]	]	X
ejpam-6553	259	4	b.	b.	PROPN
ejpam-6553	259	5	sroysang	sroysang	PROPN
ejpam-6553	259	6	.	.	PUNCT
ejpam-6553	260	1	on	on	ADP
ejpam-6553	260	2	the	the	DET
ejpam-6553	260	3	diophantine	diophantine	NOUN
ejpam-6553	260	4	equation	equation	NOUN
ejpam-6553	260	5	px	px	X
ejpam-6553	260	6	+	+	CCONJ
ejpam-6553	260	7	(	(	PUNCT
ejpam-6553	260	8	p	p	NOUN
ejpam-6553	260	9	−	−	PROPN
ejpam-6553	260	10	1)y	1)y	NUM
ejpam-6553	260	11	=	=	SYM
ejpam-6553	260	12	z2	z2	PROPN
ejpam-6553	260	13	.	.	PROPN
ejpam-6553	260	14	journal	journal	PROPN
ejpam-6553	260	15	of	of	ADP
ejpam-6553	260	16	number	number	NOUN
ejpam-6553	260	17	theory	theory	NOUN
ejpam-6553	260	18	,	,	PUNCT
ejpam-6553	260	19	133(5):1234–1245	133(5):1234–1245	PROPN
ejpam-6553	260	20	,	,	PUNCT
ejpam-6553	260	21	2013	2013	NUM
ejpam-6553	260	22	.	.	PUNCT
ejpam-6553	261	1	[	[	X
ejpam-6553	261	2	19	19	NUM
ejpam-6553	261	3	]	]	PUNCT
ejpam-6553	261	4	a.	a.	NOUN
ejpam-6553	261	5	gayo	gayo	PROPN
ejpam-6553	261	6	.	.	PUNCT
ejpam-6553	262	1	exponential	exponential	ADJ
ejpam-6553	262	2	diophantine	diophantine	NOUN
ejpam-6553	262	3	equations	equation	NOUN
ejpam-6553	262	4	of	of	ADP
ejpam-6553	262	5	the	the	DET
ejpam-6553	262	6	form	form	NOUN
ejpam-6553	262	7	px	px	X
ejpam-6553	262	8	+	+	CCONJ
ejpam-6553	262	9	(	(	PUNCT
ejpam-6553	262	10	p−	p−	NOUN
ejpam-6553	262	11	1)y	1)y	NUM
ejpam-6553	262	12	=	=	SYM
ejpam-6553	262	13	z2	z2	PROPN
ejpam-6553	262	14	.	.	PUNCT
ejpam-6553	263	1	mathematical	mathematical	ADJ
ejpam-6553	263	2	proceedings	proceeding	NOUN
ejpam-6553	263	3	,	,	PUNCT
ejpam-6553	263	4	156(2):345–360	156(2):345–360	NUM
ejpam-6553	263	5	,	,	PUNCT
ejpam-6553	263	6	2024	2024	NUM
ejpam-6553	263	7	.	.	PUNCT