id	sid	tid	token	lemma	pos
ejpam-6066	1	1	european	european	PROPN
ejpam-6066	1	2	journal	journal	PROPN
ejpam-6066	1	3	of	of	ADP
ejpam-6066	1	4	pure	pure	ADJ
ejpam-6066	1	5	and	and	CCONJ
ejpam-6066	1	6	applied	applied	ADJ
ejpam-6066	1	7	mathematics	mathematic	NOUN
ejpam-6066	1	8	2025	2025	NUM
ejpam-6066	1	9	,	,	PUNCT
ejpam-6066	1	10	vol	vol	NOUN
ejpam-6066	1	11	.	.	PROPN
ejpam-6066	1	12	18	18	NUM
ejpam-6066	1	13	,	,	PUNCT
ejpam-6066	1	14	issue	issue	NOUN
ejpam-6066	1	15	3	3	NUM
ejpam-6066	1	16	,	,	PUNCT
ejpam-6066	1	17	article	article	NOUN
ejpam-6066	1	18	number	number	NOUN
ejpam-6066	1	19	6066	6066	NUM
ejpam-6066	1	20	issn	issn	PROPN
ejpam-6066	1	21	1307	1307	NUM
ejpam-6066	1	22	-	-	SYM
ejpam-6066	1	23	5543	5543	NUM
ejpam-6066	1	24	–	–	PUNCT
ejpam-6066	1	25	ejpam.com	ejpam.com	X
ejpam-6066	1	26	published	publish	VERB
ejpam-6066	1	27	by	by	ADP
ejpam-6066	1	28	new	new	PROPN
ejpam-6066	1	29	york	york	PROPN
ejpam-6066	1	30	business	business	PROPN
ejpam-6066	1	31	global	global	PROPN
ejpam-6066	1	32	on	on	ADP
ejpam-6066	1	33	the	the	DET
ejpam-6066	1	34	diophantine	diophantine	NOUN
ejpam-6066	1	35	equation	equation	NOUN
ejpam-6066	1	36	4(7x)−	4(7x)−	PROPN
ejpam-6066	1	37	py	py	PROPN
ejpam-6066	1	38	=	=	PROPN
ejpam-6066	1	39	z2	z2	PROPN
ejpam-6066	1	40	kittipong	kittipong	PROPN
ejpam-6066	1	41	laipaporn1	laipaporn1	PROPN
ejpam-6066	1	42	,	,	PUNCT
ejpam-6066	1	43	ratcharut	ratcharut	PROPN
ejpam-6066	1	44	jankaew2	jankaew2	PROPN
ejpam-6066	1	45	,	,	PUNCT
ejpam-6066	1	46	poomipat	poomipat	PROPN
ejpam-6066	1	47	sae	sae	PROPN
ejpam-6066	1	48	-	-	NOUN
ejpam-6066	1	49	iu2	iu2	NOUN
ejpam-6066	1	50	,	,	PUNCT
ejpam-6066	1	51	adisak	adisak	PROPN
ejpam-6066	1	52	karnbanjong1,∗	karnbanjong1,∗	NOUN
ejpam-6066	1	53	1	1	NUM
ejpam-6066	1	54	department	department	NOUN
ejpam-6066	1	55	of	of	ADP
ejpam-6066	1	56	mathematics	mathematic	NOUN
ejpam-6066	1	57	and	and	CCONJ
ejpam-6066	1	58	statistics	statistic	NOUN
ejpam-6066	1	59	,	,	PUNCT
ejpam-6066	1	60	center	center	NOUN
ejpam-6066	1	61	of	of	ADP
ejpam-6066	1	62	excellence	excellence	NOUN
ejpam-6066	1	63	for	for	ADP
ejpam-6066	1	64	ecoinformatics	ecoinformatic	NOUN
ejpam-6066	1	65	,	,	PUNCT
ejpam-6066	1	66	school	school	NOUN
ejpam-6066	1	67	of	of	ADP
ejpam-6066	1	68	science	science	NOUN
ejpam-6066	1	69	,	,	PUNCT
ejpam-6066	1	70	walailak	walailak	ADJ
ejpam-6066	1	71	university	university	NOUN
ejpam-6066	1	72	,	,	PUNCT
ejpam-6066	1	73	nakhon	nakhon	PROPN
ejpam-6066	1	74	si	si	PROPN
ejpam-6066	1	75	thammarat	thammarat	PROPN
ejpam-6066	1	76	,	,	PUNCT
ejpam-6066	1	77	80160	80160	NUM
ejpam-6066	1	78	,	,	PUNCT
ejpam-6066	1	79	thailand	thailand	PROPN
ejpam-6066	1	80	2	2	NUM
ejpam-6066	1	81	princess	princess	NOUN
ejpam-6066	1	82	chulabhorn	chulabhorn	VERB
ejpam-6066	1	83	science	science	PROPN
ejpam-6066	1	84	high	high	ADJ
ejpam-6066	1	85	school	school	NOUN
ejpam-6066	1	86	,	,	PUNCT
ejpam-6066	1	87	nakhon	nakhon	PROPN
ejpam-6066	1	88	si	si	PROPN
ejpam-6066	1	89	thammarat	thammarat	PROPN
ejpam-6066	1	90	,	,	PUNCT
ejpam-6066	1	91	80330	80330	NUM
ejpam-6066	1	92	,	,	PUNCT
ejpam-6066	1	93	thailand	thailand	PROPN
ejpam-6066	1	94	abstract	abstract	NOUN
ejpam-6066	1	95	.	.	PUNCT
ejpam-6066	2	1	this	this	DET
ejpam-6066	2	2	paper	paper	NOUN
ejpam-6066	2	3	determines	determine	VERB
ejpam-6066	2	4	all	all	DET
ejpam-6066	2	5	non	non	ADJ
ejpam-6066	2	6	-	-	ADJ
ejpam-6066	2	7	negative	negative	ADJ
ejpam-6066	2	8	integer	integer	NOUN
ejpam-6066	2	9	solutions	solution	NOUN
ejpam-6066	2	10	to	to	ADP
ejpam-6066	2	11	the	the	DET
ejpam-6066	2	12	diophantine	diophantine	NOUN
ejpam-6066	2	13	equation	equation	NOUN
ejpam-6066	2	14	4(7x	4(7x	NUM
ejpam-6066	2	15	)	)	PUNCT
ejpam-6066	2	16	−	−	PROPN
ejpam-6066	2	17	py	py	PROPN
ejpam-6066	2	18	=	=	SYM
ejpam-6066	2	19	z2	z2	PROPN
ejpam-6066	2	20	,	,	PUNCT
ejpam-6066	2	21	where	where	SCONJ
ejpam-6066	2	22	p	p	NOUN
ejpam-6066	2	23	is	be	AUX
ejpam-6066	2	24	a	a	DET
ejpam-6066	2	25	prime	prime	NOUN
ejpam-6066	2	26	.	.	PUNCT
ejpam-6066	3	1	using	use	VERB
ejpam-6066	3	2	modular	modular	ADJ
ejpam-6066	3	3	arithmetic	arithmetic	ADJ
ejpam-6066	3	4	and	and	CCONJ
ejpam-6066	3	5	congruence	congruence	NOUN
ejpam-6066	3	6	arguments	argument	NOUN
ejpam-6066	3	7	,	,	PUNCT
ejpam-6066	3	8	we	we	PRON
ejpam-6066	3	9	classify	classify	VERB
ejpam-6066	3	10	all	all	DET
ejpam-6066	3	11	solutions	solution	NOUN
ejpam-6066	3	12	as	as	SCONJ
ejpam-6066	3	13	follows	follow	VERB
ejpam-6066	3	14	:	:	PUNCT
ejpam-6066	3	15	a	a	DET
ejpam-6066	3	16	unique	unique	ADJ
ejpam-6066	3	17	solution	solution	NOUN
ejpam-6066	3	18	for	for	ADP
ejpam-6066	3	19	p	p	NOUN
ejpam-6066	3	20	=	=	SYM
ejpam-6066	3	21	2	2	NUM
ejpam-6066	3	22	,	,	PUNCT
ejpam-6066	3	23	an	an	DET
ejpam-6066	3	24	infinite	infinite	ADJ
ejpam-6066	3	25	family	family	NOUN
ejpam-6066	3	26	of	of	ADP
ejpam-6066	3	27	solutions	solution	NOUN
ejpam-6066	3	28	for	for	ADP
ejpam-6066	3	29	p	p	NOUN
ejpam-6066	3	30	=	=	SYM
ejpam-6066	3	31	3	3	NUM
ejpam-6066	3	32	,	,	PUNCT
ejpam-6066	3	33	no	no	DET
ejpam-6066	3	34	solutions	solution	NOUN
ejpam-6066	3	35	for	for	ADP
ejpam-6066	3	36	5	5	NUM
ejpam-6066	3	37	≤	≤	NOUN
ejpam-6066	3	38	p	p	NOUN
ejpam-6066	3	39	≤	≤	NUM
ejpam-6066	3	40	17	17	NUM
ejpam-6066	3	41	,	,	PUNCT
ejpam-6066	3	42	and	and	CCONJ
ejpam-6066	3	43	–	–	PUNCT
ejpam-6066	3	44	for	for	ADP
ejpam-6066	3	45	p	p	PRON
ejpam-6066	3	46	≥	≥	NUM
ejpam-6066	3	47	19	19	NUM
ejpam-6066	3	48	–	–	PUNCT
ejpam-6066	3	49	the	the	DET
ejpam-6066	3	50	existence	existence	NOUN
ejpam-6066	3	51	of	of	ADP
ejpam-6066	3	52	solutions	solution	NOUN
ejpam-6066	3	53	requires	require	VERB
ejpam-6066	3	54	that	that	SCONJ
ejpam-6066	3	55	p	p	PROPN
ejpam-6066	3	56	≡	≡	PROPN
ejpam-6066	3	57	19	19	NUM
ejpam-6066	3	58	(	(	PUNCT
ejpam-6066	3	59	mod	mod	NOUN
ejpam-6066	3	60	24	24	NUM
ejpam-6066	3	61	)	)	PUNCT
ejpam-6066	3	62	subject	subject	NOUN
ejpam-6066	3	63	to	to	ADP
ejpam-6066	3	64	specific	specific	ADJ
ejpam-6066	3	65	modular	modular	ADJ
ejpam-6066	3	66	constraints	constraint	NOUN
ejpam-6066	3	67	.	.	PUNCT
ejpam-6066	4	1	computational	computational	ADJ
ejpam-6066	4	2	results	result	NOUN
ejpam-6066	4	3	support	support	VERB
ejpam-6066	4	4	the	the	DET
ejpam-6066	4	5	conjecture	conjecture	NOUN
ejpam-6066	4	6	that	that	SCONJ
ejpam-6066	4	7	no	no	DET
ejpam-6066	4	8	further	further	ADJ
ejpam-6066	4	9	solutions	solution	NOUN
ejpam-6066	4	10	exist	exist	VERB
ejpam-6066	4	11	beyond	beyond	ADP
ejpam-6066	4	12	those	those	PRON
ejpam-6066	4	13	identified	identify	VERB
ejpam-6066	4	14	.	.	PUNCT
ejpam-6066	5	1	this	this	DET
ejpam-6066	5	2	work	work	NOUN
ejpam-6066	5	3	illustrates	illustrate	VERB
ejpam-6066	5	4	how	how	SCONJ
ejpam-6066	5	5	modular	modular	ADJ
ejpam-6066	5	6	techniques	technique	NOUN
ejpam-6066	5	7	can	can	AUX
ejpam-6066	5	8	fully	fully	ADV
ejpam-6066	5	9	resolve	resolve	VERB
ejpam-6066	5	10	an	an	DET
ejpam-6066	5	11	exponential	exponential	ADJ
ejpam-6066	5	12	diophantine	diophantine	NOUN
ejpam-6066	5	13	equation	equation	NOUN
ejpam-6066	5	14	and	and	CCONJ
ejpam-6066	5	15	offers	offer	VERB
ejpam-6066	5	16	a	a	DET
ejpam-6066	5	17	framework	framework	NOUN
ejpam-6066	5	18	for	for	ADP
ejpam-6066	5	19	analyzing	analyze	VERB
ejpam-6066	5	20	similar	similar	ADJ
ejpam-6066	5	21	equations	equation	NOUN
ejpam-6066	5	22	involving	involve	VERB
ejpam-6066	5	23	mixed	mixed	ADJ
ejpam-6066	5	24	exponential	exponential	ADJ
ejpam-6066	5	25	and	and	CCONJ
ejpam-6066	5	26	polynomial	polynomial	ADJ
ejpam-6066	5	27	terms	term	NOUN
ejpam-6066	5	28	.	.	PUNCT
ejpam-6066	6	1	2020	2020	NUM
ejpam-6066	6	2	mathematics	mathematic	NOUN
ejpam-6066	6	3	subject	subject	NOUN
ejpam-6066	6	4	classifications	classification	NOUN
ejpam-6066	6	5	:	:	PUNCT
ejpam-6066	6	6	11a07	11a07	NUM
ejpam-6066	6	7	key	key	ADJ
ejpam-6066	6	8	words	word	NOUN
ejpam-6066	6	9	and	and	CCONJ
ejpam-6066	6	10	phrases	phrase	NOUN
ejpam-6066	6	11	:	:	PUNCT
ejpam-6066	6	12	exponential	exponential	ADJ
ejpam-6066	6	13	diophantine	diophantine	NOUN
ejpam-6066	6	14	equation	equation	NOUN
ejpam-6066	6	15	,	,	PUNCT
ejpam-6066	6	16	modulo	modulo	PROPN
ejpam-6066	6	17	1	1	NUM
ejpam-6066	6	18	.	.	X
ejpam-6066	7	1	introduction	introduction	NOUN
ejpam-6066	7	2	the	the	DET
ejpam-6066	7	3	study	study	NOUN
ejpam-6066	7	4	of	of	ADP
ejpam-6066	7	5	diophantine	diophantine	NOUN
ejpam-6066	7	6	equations	equation	NOUN
ejpam-6066	7	7	,	,	PUNCT
ejpam-6066	7	8	named	name	VERB
ejpam-6066	7	9	after	after	ADP
ejpam-6066	7	10	the	the	DET
ejpam-6066	7	11	ancient	ancient	ADJ
ejpam-6066	7	12	greek	greek	ADJ
ejpam-6066	7	13	mathematician	mathematician	ADJ
ejpam-6066	7	14	diophantus	diophantus	NOUN
ejpam-6066	7	15	of	of	ADP
ejpam-6066	7	16	alexandria	alexandria	PROPN
ejpam-6066	7	17	(	(	PUNCT
ejpam-6066	7	18	ca	ca	NOUN
ejpam-6066	7	19	.	.	PUNCT
ejpam-6066	8	1	250	250	NUM
ejpam-6066	8	2	ad	ad	NOUN
ejpam-6066	8	3	)	)	PUNCT
ejpam-6066	8	4	,	,	PUNCT
ejpam-6066	8	5	forms	form	VERB
ejpam-6066	8	6	a	a	DET
ejpam-6066	8	7	foundational	foundational	ADJ
ejpam-6066	8	8	aspect	aspect	NOUN
ejpam-6066	8	9	of	of	ADP
ejpam-6066	8	10	number	number	NOUN
ejpam-6066	8	11	theory	theory	NOUN
ejpam-6066	8	12	.	.	PUNCT
ejpam-6066	9	1	diophantus	diophantus	NOUN
ejpam-6066	9	2	is	be	AUX
ejpam-6066	9	3	often	often	ADV
ejpam-6066	9	4	regarded	regard	VERB
ejpam-6066	9	5	as	as	ADP
ejpam-6066	9	6	the	the	DET
ejpam-6066	9	7	“	"	PUNCT
ejpam-6066	9	8	father	father	NOUN
ejpam-6066	9	9	of	of	ADP
ejpam-6066	9	10	algebra	algebra	PROPN
ejpam-6066	9	11	”	"	PUNCT
ejpam-6066	9	12	due	due	ADP
ejpam-6066	9	13	to	to	ADP
ejpam-6066	9	14	his	his	PRON
ejpam-6066	9	15	pioneering	pioneer	VERB
ejpam-6066	9	16	work	work	NOUN
ejpam-6066	9	17	in	in	ADP
ejpam-6066	9	18	using	use	VERB
ejpam-6066	9	19	symbolic	symbolic	ADJ
ejpam-6066	9	20	methods	method	NOUN
ejpam-6066	9	21	to	to	PART
ejpam-6066	9	22	represent	represent	VERB
ejpam-6066	9	23	equations	equation	NOUN
ejpam-6066	9	24	in	in	ADP
ejpam-6066	9	25	his	his	PRON
ejpam-6066	9	26	famous	famous	ADJ
ejpam-6066	9	27	treatise	treatise	NOUN
ejpam-6066	9	28	arithmetica	arithmetica	ADJ
ejpam-6066	9	29	.	.	PUNCT
ejpam-6066	10	1	although	although	SCONJ
ejpam-6066	10	2	originally	originally	ADV
ejpam-6066	10	3	consisting	consist	VERB
ejpam-6066	10	4	of	of	ADP
ejpam-6066	10	5	thirteen	thirteen	NUM
ejpam-6066	10	6	books	book	NOUN
ejpam-6066	10	7	,	,	PUNCT
ejpam-6066	10	8	only	only	ADV
ejpam-6066	10	9	six	six	NUM
ejpam-6066	10	10	have	have	AUX
ejpam-6066	10	11	survived	survive	VERB
ejpam-6066	10	12	and	and	CCONJ
ejpam-6066	10	13	contain	contain	VERB
ejpam-6066	10	14	around	around	ADP
ejpam-6066	10	15	130	130	NUM
ejpam-6066	10	16	problems	problem	NOUN
ejpam-6066	10	17	with	with	ADP
ejpam-6066	10	18	solutions	solution	NOUN
ejpam-6066	10	19	.	.	PUNCT
ejpam-6066	11	1	these	these	DET
ejpam-6066	11	2	problems	problem	NOUN
ejpam-6066	11	3	demonstrate	demonstrate	VERB
ejpam-6066	11	4	early	early	ADV
ejpam-6066	11	5	algebraic	algebraic	ADJ
ejpam-6066	11	6	reasoning	reasoning	NOUN
ejpam-6066	11	7	and	and	CCONJ
ejpam-6066	11	8	have	have	AUX
ejpam-6066	11	9	inspired	inspire	VERB
ejpam-6066	11	10	generations	generation	NOUN
ejpam-6066	11	11	of	of	ADP
ejpam-6066	11	12	mathematicians	mathematician	NOUN
ejpam-6066	11	13	.	.	PUNCT
ejpam-6066	12	1	a	a	DET
ejpam-6066	12	2	well	well	ADV
ejpam-6066	12	3	-	-	PUNCT
ejpam-6066	12	4	known	know	VERB
ejpam-6066	12	5	anecdote	anecdote	NOUN
ejpam-6066	12	6	regarding	regard	VERB
ejpam-6066	12	7	diophantus	diophantus	NOUN
ejpam-6066	12	8	’	'	PUNCT
ejpam-6066	12	9	life	life	NOUN
ejpam-6066	12	10	is	be	AUX
ejpam-6066	12	11	his	his	PRON
ejpam-6066	12	12	purported	purport	VERB
ejpam-6066	12	13	age	age	NOUN
ejpam-6066	12	14	at	at	ADP
ejpam-6066	12	15	death	death	NOUN
ejpam-6066	12	16	,	,	PUNCT
ejpam-6066	12	17	deduced	deduce	VERB
ejpam-6066	12	18	from	from	ADP
ejpam-6066	12	19	a	a	DET
ejpam-6066	12	20	riddle	riddle	NOUN
ejpam-6066	12	21	composed	compose	VERB
ejpam-6066	12	22	by	by	ADP
ejpam-6066	12	23	the	the	DET
ejpam-6066	12	24	poet	poet	NOUN
ejpam-6066	12	25	metrodorus	metrodorus	PROPN
ejpam-6066	12	26	.	.	PUNCT
ejpam-6066	13	1	the	the	DET
ejpam-6066	13	2	riddle	riddle	NOUN
ejpam-6066	13	3	,	,	PUNCT
ejpam-6066	13	4	when	when	SCONJ
ejpam-6066	13	5	interpreted	interpret	VERB
ejpam-6066	13	6	algebraically	algebraically	ADV
ejpam-6066	13	7	,	,	PUNCT
ejpam-6066	13	8	leads	lead	VERB
ejpam-6066	13	9	to	to	ADP
ejpam-6066	13	10	the	the	DET
ejpam-6066	13	11	equation	equation	NOUN
ejpam-6066	13	12	x	x	PUNCT
ejpam-6066	14	1	=	=	SYM
ejpam-6066	14	2	1	1	NUM
ejpam-6066	14	3	6	6	NUM
ejpam-6066	14	4	x+	x+	SYM
ejpam-6066	14	5	1	1	NUM
ejpam-6066	14	6	12	12	NUM
ejpam-6066	14	7	x+	x+	SYM
ejpam-6066	14	8	1	1	NUM
ejpam-6066	14	9	7	7	NUM
ejpam-6066	14	10	x+	x+	SYM
ejpam-6066	14	11	5	5	NUM
ejpam-6066	14	12	+	+	CCONJ
ejpam-6066	14	13	1	1	NUM
ejpam-6066	14	14	2	2	NUM
ejpam-6066	14	15	x+	x+	SYM
ejpam-6066	14	16	4	4	NUM
ejpam-6066	14	17	,	,	PUNCT
ejpam-6066	14	18	∗corresponding	∗corresponde	VERB
ejpam-6066	14	19	author	author	NOUN
ejpam-6066	14	20	.	.	PUNCT
ejpam-6066	15	1	doi	doi	NOUN
ejpam-6066	15	2	:	:	PUNCT
ejpam-6066	15	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6066	https://doi.org/10.29020/nybg.ejpam.v18i3.6066	NUM
ejpam-6066	15	4	email	email	NOUN
ejpam-6066	15	5	addresses	address	NOUN
ejpam-6066	15	6	:	:	PUNCT
ejpam-6066	15	7	lkittipo@wu.ac.th	lkittipo@wu.ac.th	PROPN
ejpam-6066	15	8	(	(	PUNCT
ejpam-6066	15	9	k.	k.	PROPN
ejpam-6066	15	10	laipaporn	laipaporn	PROPN
ejpam-6066	15	11	)	)	PUNCT
ejpam-6066	15	12	,	,	PUNCT
ejpam-6066	15	13	6405825@pccnst.ac.th	6405825@pccnst.ac.th	NUM
ejpam-6066	15	14	(	(	PUNCT
ejpam-6066	15	15	r.	r.	PROPN
ejpam-6066	15	16	jankaew	jankaew	PROPN
ejpam-6066	15	17	)	)	PUNCT
ejpam-6066	15	18	,	,	PUNCT
ejpam-6066	16	1	6405826@pccnst.ac.th	6405826@pccnst.ac.th	PRON
ejpam-6066	16	2	(	(	PUNCT
ejpam-6066	16	3	p.	p.	PROPN
ejpam-6066	16	4	sae	sae	PROPN
ejpam-6066	16	5	-	-	PROPN
ejpam-6066	16	6	iu	iu	NOUN
ejpam-6066	16	7	)	)	PUNCT
ejpam-6066	16	8	,	,	PUNCT
ejpam-6066	16	9	kadisak@mail.wu.ac.th	kadisak@mail.wu.ac.th	PROPN
ejpam-6066	16	10	(	(	PUNCT
ejpam-6066	16	11	a.	a.	NOUN
ejpam-6066	16	12	karnbanjong	karnbanjong	PROPN
ejpam-6066	16	13	)	)	PUNCT
ejpam-6066	16	14	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6066	17	1	1	1	NUM
ejpam-6066	17	2	copyright	copyright	NOUN
ejpam-6066	17	3	:	:	PUNCT
ejpam-6066	17	4	©	©	PROPN
ejpam-6066	17	5	2025	2025	NUM
ejpam-6066	17	6	the	the	DET
ejpam-6066	17	7	author(s	author(s	NOUN
ejpam-6066	17	8	)	)	PUNCT
ejpam-6066	17	9	.	.	PUNCT
ejpam-6066	18	1	(	(	PUNCT
ejpam-6066	18	2	cc	cc	NOUN
ejpam-6066	18	3	by	by	ADP
ejpam-6066	18	4	-	-	PUNCT
ejpam-6066	18	5	nc	nc	PROPN
ejpam-6066	18	6	4.0	4.0	NUM
ejpam-6066	18	7	)	)	PUNCT
ejpam-6066	18	8	k.	k.	PROPN
ejpam-6066	19	1	laipaporn	laipaporn	VERB
ejpam-6066	19	2	et	et	PROPN
ejpam-6066	19	3	al	al	PROPN
ejpam-6066	19	4	.	.	PUNCT
ejpam-6066	19	5	/	/	SYM
ejpam-6066	19	6	eur	eur	PROPN
ejpam-6066	19	7	.	.	PUNCT
ejpam-6066	20	1	j.	j.	PROPN
ejpam-6066	20	2	pure	pure	PROPN
ejpam-6066	20	3	appl	appl	PROPN
ejpam-6066	20	4	.	.	PROPN
ejpam-6066	20	5	math	math	PROPN
ejpam-6066	20	6	,	,	PUNCT
ejpam-6066	20	7	18	18	NUM
ejpam-6066	20	8	(	(	PUNCT
ejpam-6066	20	9	3	3	NUM
ejpam-6066	20	10	)	)	PUNCT
ejpam-6066	20	11	(	(	PUNCT
ejpam-6066	20	12	2025	2025	NUM
ejpam-6066	20	13	)	)	PUNCT
ejpam-6066	20	14	,	,	PUNCT
ejpam-6066	20	15	6066	6066	NUM
ejpam-6066	20	16	2	2	NUM
ejpam-6066	20	17	of	of	ADP
ejpam-6066	20	18	13	13	NUM
ejpam-6066	20	19	whose	whose	DET
ejpam-6066	20	20	solution	solution	NOUN
ejpam-6066	20	21	is	be	AUX
ejpam-6066	20	22	x	x	X
ejpam-6066	20	23	=	=	SYM
ejpam-6066	20	24	84	84	NUM
ejpam-6066	20	25	,	,	PUNCT
ejpam-6066	20	26	suggesting	suggest	VERB
ejpam-6066	20	27	that	that	SCONJ
ejpam-6066	20	28	diophantus	diophantus	NOUN
ejpam-6066	20	29	lived	live	VERB
ejpam-6066	20	30	to	to	PART
ejpam-6066	20	31	be	be	AUX
ejpam-6066	20	32	84	84	NUM
ejpam-6066	20	33	years	year	NOUN
ejpam-6066	20	34	old	old	ADJ
ejpam-6066	20	35	.	.	PUNCT
ejpam-6066	21	1	this	this	DET
ejpam-6066	21	2	kind	kind	NOUN
ejpam-6066	21	3	of	of	ADP
ejpam-6066	21	4	problem	problem	NOUN
ejpam-6066	21	5	serves	serve	VERB
ejpam-6066	21	6	as	as	ADP
ejpam-6066	21	7	an	an	DET
ejpam-6066	21	8	early	early	ADJ
ejpam-6066	21	9	example	example	NOUN
ejpam-6066	21	10	of	of	ADP
ejpam-6066	21	11	a	a	DET
ejpam-6066	21	12	linear	linear	ADJ
ejpam-6066	21	13	diophantine	diophantine	NOUN
ejpam-6066	21	14	equation	equation	NOUN
ejpam-6066	21	15	,	,	PUNCT
ejpam-6066	21	16	which	which	PRON
ejpam-6066	21	17	is	be	AUX
ejpam-6066	21	18	an	an	DET
ejpam-6066	21	19	equation	equation	NOUN
ejpam-6066	21	20	of	of	ADP
ejpam-6066	21	21	the	the	DET
ejpam-6066	21	22	form	form	NOUN
ejpam-6066	21	23	ax+	ax+	NOUN
ejpam-6066	21	24	by	by	ADP
ejpam-6066	21	25	=	=	PROPN
ejpam-6066	21	26	c	c	PROPN
ejpam-6066	21	27	where	where	SCONJ
ejpam-6066	21	28	a	a	DET
ejpam-6066	21	29	,	,	PUNCT
ejpam-6066	21	30	b	b	NOUN
ejpam-6066	21	31	,	,	PUNCT
ejpam-6066	21	32	c	c	PROPN
ejpam-6066	21	33	∈	∈	PROPN
ejpam-6066	21	34	z	z	NOUN
ejpam-6066	21	35	,	,	PUNCT
ejpam-6066	21	36	and	and	CCONJ
ejpam-6066	21	37	the	the	DET
ejpam-6066	21	38	goal	goal	NOUN
ejpam-6066	21	39	is	be	AUX
ejpam-6066	21	40	to	to	PART
ejpam-6066	21	41	find	find	VERB
ejpam-6066	21	42	integer	integer	NOUN
ejpam-6066	21	43	solutions	solution	NOUN
ejpam-6066	21	44	for	for	ADP
ejpam-6066	21	45	x	x	PUNCT
ejpam-6066	21	46	and	and	CCONJ
ejpam-6066	21	47	y.	y.	PROPN
ejpam-6066	21	48	a	a	DET
ejpam-6066	21	49	necessary	necessary	ADJ
ejpam-6066	21	50	and	and	CCONJ
ejpam-6066	21	51	sufficient	sufficient	ADJ
ejpam-6066	21	52	condition	condition	NOUN
ejpam-6066	21	53	for	for	ADP
ejpam-6066	21	54	such	such	DET
ejpam-6066	21	55	an	an	DET
ejpam-6066	21	56	equation	equation	NOUN
ejpam-6066	21	57	to	to	PART
ejpam-6066	21	58	have	have	AUX
ejpam-6066	21	59	solutions	solution	NOUN
ejpam-6066	21	60	is	be	AUX
ejpam-6066	21	61	that	that	DET
ejpam-6066	21	62	gcd	gcd	NOUN
ejpam-6066	21	63	(	(	PUNCT
ejpam-6066	21	64	a	a	PRON
ejpam-6066	21	65	,	,	PUNCT
ejpam-6066	21	66	b)|c	b)|c	PROPN
ejpam-6066	21	67	.	.	PUNCT
ejpam-6066	22	1	over	over	ADP
ejpam-6066	22	2	the	the	DET
ejpam-6066	22	3	centuries	century	NOUN
ejpam-6066	22	4	,	,	PUNCT
ejpam-6066	22	5	linear	linear	ADJ
ejpam-6066	22	6	diophantine	diophantine	NOUN
ejpam-6066	22	7	equations	equation	NOUN
ejpam-6066	22	8	have	have	AUX
ejpam-6066	22	9	evolved	evolve	VERB
ejpam-6066	22	10	into	into	ADP
ejpam-6066	22	11	more	more	ADJ
ejpam-6066	22	12	intricate	intricate	ADJ
ejpam-6066	22	13	forms	form	NOUN
ejpam-6066	22	14	,	,	PUNCT
ejpam-6066	22	15	including	include	VERB
ejpam-6066	22	16	exponential	exponential	ADJ
ejpam-6066	22	17	and	and	CCONJ
ejpam-6066	22	18	nonlinear	nonlinear	ADJ
ejpam-6066	22	19	variants	variant	NOUN
ejpam-6066	22	20	.	.	PUNCT
ejpam-6066	23	1	table	table	NOUN
ejpam-6066	23	2	1	1	NUM
ejpam-6066	23	3	provides	provide	VERB
ejpam-6066	23	4	a	a	DET
ejpam-6066	23	5	brief	brief	ADJ
ejpam-6066	23	6	historical	historical	ADJ
ejpam-6066	23	7	overview	overview	NOUN
ejpam-6066	23	8	of	of	ADP
ejpam-6066	23	9	several	several	ADJ
ejpam-6066	23	10	notable	notable	ADJ
ejpam-6066	23	11	diophantine	diophantine	NOUN
ejpam-6066	23	12	equations	equation	NOUN
ejpam-6066	23	13	,	,	PUNCT
ejpam-6066	23	14	ranging	range	VERB
ejpam-6066	23	15	from	from	ADP
ejpam-6066	23	16	pell	pell	PROPN
ejpam-6066	23	17	’s	’s	PART
ejpam-6066	23	18	equation	equation	NOUN
ejpam-6066	23	19	to	to	ADP
ejpam-6066	23	20	fermat	fermat	PROPN
ejpam-6066	23	21	’s	’s	PART
ejpam-6066	23	22	last	last	ADJ
ejpam-6066	23	23	theorem	theorem	ADJ
ejpam-6066	23	24	.	.	PROPN
ejpam-6066	23	25	table	table	NOUN
ejpam-6066	23	26	1	1	NUM
ejpam-6066	23	27	:	:	PUNCT
ejpam-6066	23	28	historical	historical	ADJ
ejpam-6066	23	29	diophantine	diophantine	NOUN
ejpam-6066	23	30	equations	equation	NOUN
ejpam-6066	23	31	.	.	PUNCT
ejpam-6066	24	1	equation	equation	NOUN
ejpam-6066	24	2	reference	reference	NOUN
ejpam-6066	24	3	equations	equation	NOUN
ejpam-6066	24	4	equation	equation	NOUN
ejpam-6066	24	5	names	name	NOUN
ejpam-6066	24	6	discovered	discover	VERB
ejpam-6066	24	7	in	in	ADP
ejpam-6066	24	8	1768	1768	NUM
ejpam-6066	24	9	[	[	X
ejpam-6066	24	10	1	1	X
ejpam-6066	24	11	]	]	X
ejpam-6066	24	12	x2	x2	PRON
ejpam-6066	24	13	−	−	PROPN
ejpam-6066	24	14	ny2	ny2	PROPN
ejpam-6066	24	15	=	=	SYM
ejpam-6066	24	16	±1	±1	VERB
ejpam-6066	24	17	pell	pell	PROPN
ejpam-6066	24	18	’s	’s	PART
ejpam-6066	24	19	equation	equation	NOUN
ejpam-6066	24	20	1918	1918	NUM
ejpam-6066	25	1	[	[	X
ejpam-6066	25	2	2	2	NUM
ejpam-6066	25	3	]	]	X
ejpam-6066	25	4	w3	w3	NOUN
ejpam-6066	25	5	+	+	CCONJ
ejpam-6066	25	6	x3	x3	ADJ
ejpam-6066	25	7	=	=	SYM
ejpam-6066	25	8	y3	y3	NOUN
ejpam-6066	25	9	+	+	CCONJ
ejpam-6066	25	10	z3	z3	ADJ
ejpam-6066	25	11	hardy	hardy	ADJ
ejpam-6066	25	12	-	-	PUNCT
ejpam-6066	25	13	ramanujan	ramanujan	NOUN
ejpam-6066	25	14	number	number	NOUN
ejpam-6066	25	15	1948	1948	NUM
ejpam-6066	26	1	[	[	X
ejpam-6066	26	2	3	3	NUM
ejpam-6066	26	3	]	]	SYM
ejpam-6066	26	4	4	4	NUM
ejpam-6066	26	5	n	n	NOUN
ejpam-6066	26	6	=	=	SYM
ejpam-6066	26	7	1	1	NUM
ejpam-6066	26	8	x	x	SYM
ejpam-6066	26	9	+	+	NUM
ejpam-6066	26	10	1	1	NUM
ejpam-6066	26	11	y	y	NOUN
ejpam-6066	26	12	+	+	CCONJ
ejpam-6066	26	13	1	1	NUM
ejpam-6066	26	14	z	z	NUM
ejpam-6066	26	15	erdös	erdös	ADJ
ejpam-6066	26	16	-	-	PUNCT
ejpam-6066	26	17	straus	straus	NOUN
ejpam-6066	26	18	conjecture	conjecture	NOUN
ejpam-6066	26	19	1988	1988	NUM
ejpam-6066	26	20	[	[	X
ejpam-6066	26	21	4	4	NUM
ejpam-6066	26	22	]	]	X
ejpam-6066	26	23	x4	x4	PROPN
ejpam-6066	26	24	+	+	CCONJ
ejpam-6066	26	25	y4	y4	PROPN
ejpam-6066	26	26	+	+	CCONJ
ejpam-6066	26	27	z4	z4	PROPN
ejpam-6066	26	28	=	=	SYM
ejpam-6066	26	29	w4	w4	PROPN
ejpam-6066	26	30	euler	euler	PROPN
ejpam-6066	26	31	’s	’s	PART
ejpam-6066	26	32	conjecture	conjecture	NOUN
ejpam-6066	26	33	(	(	PUNCT
ejpam-6066	26	34	disproved	disprove	VERB
ejpam-6066	26	35	)	)	PUNCT
ejpam-6066	26	36	1995	1995	NUM
ejpam-6066	27	1	[	[	X
ejpam-6066	27	2	5	5	NUM
ejpam-6066	27	3	]	]	PUNCT
ejpam-6066	27	4	xn	xn	PUNCT
ejpam-6066	28	1	+	+	CCONJ
ejpam-6066	28	2	yn	yn	X
ejpam-6066	28	3	=	=	SYM
ejpam-6066	28	4	zn	zn	PROPN
ejpam-6066	28	5	,	,	PUNCT
ejpam-6066	28	6	n	n	PROPN
ejpam-6066	28	7	≥	≥	NOUN
ejpam-6066	28	8	3	3	NUM
ejpam-6066	28	9	fermat	fermat	PROPN
ejpam-6066	28	10	’s	’s	PART
ejpam-6066	28	11	last	last	ADJ
ejpam-6066	28	12	theorem	theorem	NOUN
ejpam-6066	28	13	among	among	ADP
ejpam-6066	28	14	these	these	DET
ejpam-6066	28	15	developments	development	NOUN
ejpam-6066	28	16	,	,	PUNCT
ejpam-6066	28	17	catalan	catalan	NOUN
ejpam-6066	28	18	’s	’s	PART
ejpam-6066	28	19	conjecture	conjecture	NOUN
ejpam-6066	28	20	—	—	PUNCT
ejpam-6066	28	21	proposed	propose	VERB
ejpam-6066	28	22	by	by	ADP
ejpam-6066	28	23	eugène	eugène	PROPN
ejpam-6066	28	24	catalan	catalan	NOUN
ejpam-6066	28	25	in	in	ADP
ejpam-6066	28	26	1844	1844	NUM
ejpam-6066	28	27	and	and	CCONJ
ejpam-6066	28	28	proved	prove	VERB
ejpam-6066	28	29	by	by	ADP
ejpam-6066	28	30	p.	p.	PROPN
ejpam-6066	28	31	mihăilescu	mihăilescu	PROPN
ejpam-6066	29	1	[	[	X
ejpam-6066	29	2	6	6	X
ejpam-6066	29	3	]	]	PUNCT
ejpam-6066	29	4	in	in	ADP
ejpam-6066	29	5	2004	2004	NUM
ejpam-6066	29	6	—	—	PUNCT
ejpam-6066	29	7	stands	stand	VERB
ejpam-6066	29	8	out	out	ADP
ejpam-6066	29	9	as	as	ADP
ejpam-6066	29	10	a	a	DET
ejpam-6066	29	11	landmark	landmark	NOUN
ejpam-6066	29	12	result	result	NOUN
ejpam-6066	29	13	.	.	PUNCT
ejpam-6066	30	1	it	it	PRON
ejpam-6066	30	2	asserts	assert	VERB
ejpam-6066	30	3	that	that	SCONJ
ejpam-6066	30	4	the	the	DET
ejpam-6066	30	5	equation	equation	NOUN
ejpam-6066	30	6	ax	ax	NOUN
ejpam-6066	30	7	−	−	PROPN
ejpam-6066	30	8	by	by	ADP
ejpam-6066	30	9	=	=	SYM
ejpam-6066	30	10	1	1	PROPN
ejpam-6066	30	11	has	have	VERB
ejpam-6066	30	12	a	a	DET
ejpam-6066	30	13	unique	unique	ADJ
ejpam-6066	30	14	solution	solution	NOUN
ejpam-6066	30	15	in	in	ADP
ejpam-6066	30	16	natural	natural	ADJ
ejpam-6066	30	17	numbers	number	NOUN
ejpam-6066	30	18	when	when	SCONJ
ejpam-6066	30	19	a	a	DET
ejpam-6066	30	20	,	,	PUNCT
ejpam-6066	30	21	b	b	NOUN
ejpam-6066	30	22	,	,	PUNCT
ejpam-6066	30	23	x	x	AUX
ejpam-6066	30	24	,	,	PUNCT
ejpam-6066	30	25	y	y	PROPN
ejpam-6066	30	26	≥	≥	NUM
ejpam-6066	30	27	2	2	NUM
ejpam-6066	30	28	,	,	PUNCT
ejpam-6066	30	29	namely	namely	ADV
ejpam-6066	30	30	(	(	PUNCT
ejpam-6066	30	31	a	a	PRON
ejpam-6066	30	32	,	,	PUNCT
ejpam-6066	30	33	b	b	NOUN
ejpam-6066	30	34	,	,	PUNCT
ejpam-6066	30	35	x	x	NOUN
ejpam-6066	30	36	,	,	PUNCT
ejpam-6066	30	37	y	y	NOUN
ejpam-6066	30	38	)	)	PUNCT
ejpam-6066	30	39	=	=	PUNCT
ejpam-6066	30	40	(	(	PUNCT
ejpam-6066	30	41	3	3	NUM
ejpam-6066	30	42	,	,	PUNCT
ejpam-6066	30	43	2	2	NUM
ejpam-6066	30	44	,	,	PUNCT
ejpam-6066	30	45	2	2	NUM
ejpam-6066	30	46	,	,	PUNCT
ejpam-6066	30	47	3	3	NUM
ejpam-6066	30	48	)	)	PUNCT
ejpam-6066	30	49	.	.	PUNCT
ejpam-6066	31	1	this	this	DET
ejpam-6066	31	2	result	result	NOUN
ejpam-6066	31	3	has	have	AUX
ejpam-6066	31	4	had	have	VERB
ejpam-6066	31	5	significant	significant	ADJ
ejpam-6066	31	6	implications	implication	NOUN
ejpam-6066	31	7	in	in	ADP
ejpam-6066	31	8	the	the	DET
ejpam-6066	31	9	field	field	NOUN
ejpam-6066	31	10	of	of	ADP
ejpam-6066	31	11	exponential	exponential	ADJ
ejpam-6066	31	12	diophantine	diophantine	NOUN
ejpam-6066	31	13	equations	equation	NOUN
ejpam-6066	31	14	.	.	PUNCT
ejpam-6066	32	1	in	in	ADP
ejpam-6066	32	2	2007	2007	NUM
ejpam-6066	32	3	,	,	PUNCT
ejpam-6066	32	4	acu	acu	PROPN
ejpam-6066	33	1	[	[	X
ejpam-6066	33	2	7	7	NUM
ejpam-6066	33	3	]	]	X
ejpam-6066	33	4	utilized	utilize	VERB
ejpam-6066	33	5	catalan	catalan	NOUN
ejpam-6066	33	6	’s	’s	PART
ejpam-6066	33	7	theorem	theorem	VERB
ejpam-6066	33	8	to	to	PART
ejpam-6066	33	9	analyze	analyze	VERB
ejpam-6066	33	10	the	the	DET
ejpam-6066	33	11	equation	equation	NOUN
ejpam-6066	33	12	2x	2x	NUM
ejpam-6066	34	1	+	+	CCONJ
ejpam-6066	34	2	5y	5y	NOUN
ejpam-6066	34	3	=	=	SYM
ejpam-6066	34	4	z2	z2	PROPN
ejpam-6066	34	5	,	,	PUNCT
ejpam-6066	34	6	which	which	PRON
ejpam-6066	34	7	has	have	AUX
ejpam-6066	34	8	inspired	inspire	VERB
ejpam-6066	34	9	the	the	DET
ejpam-6066	34	10	investigation	investigation	NOUN
ejpam-6066	34	11	of	of	ADP
ejpam-6066	34	12	equations	equation	NOUN
ejpam-6066	34	13	involving	involve	VERB
ejpam-6066	34	14	exponential	exponential	ADJ
ejpam-6066	34	15	terms	term	NOUN
ejpam-6066	34	16	equated	equate	VERB
ejpam-6066	34	17	to	to	ADP
ejpam-6066	34	18	perfect	perfect	ADJ
ejpam-6066	34	19	squares	square	NOUN
ejpam-6066	34	20	.	.	PUNCT
ejpam-6066	35	1	over	over	ADP
ejpam-6066	35	2	the	the	DET
ejpam-6066	35	3	past	past	ADJ
ejpam-6066	35	4	decade	decade	NOUN
ejpam-6066	35	5	,	,	PUNCT
ejpam-6066	35	6	many	many	ADJ
ejpam-6066	35	7	researchers	researcher	NOUN
ejpam-6066	35	8	have	have	AUX
ejpam-6066	35	9	investigated	investigate	VERB
ejpam-6066	35	10	equations	equation	NOUN
ejpam-6066	35	11	of	of	ADP
ejpam-6066	35	12	the	the	DET
ejpam-6066	35	13	general	general	ADJ
ejpam-6066	35	14	type	type	NOUN
ejpam-6066	35	15	a(px)±	a(px)±	NOUN
ejpam-6066	35	16	b(qy	b(qy	PROPN
ejpam-6066	35	17	)	)	PUNCT
ejpam-6066	35	18	=	=	SYM
ejpam-6066	35	19	z2	z2	PROPN
ejpam-6066	35	20	,	,	PUNCT
ejpam-6066	35	21	where	where	SCONJ
ejpam-6066	35	22	a	a	DET
ejpam-6066	35	23	,	,	PUNCT
ejpam-6066	35	24	b	b	PROPN
ejpam-6066	35	25	∈	∈	PROPN
ejpam-6066	35	26	z+	z+	NUM
ejpam-6066	35	27	,	,	PUNCT
ejpam-6066	35	28	and	and	CCONJ
ejpam-6066	35	29	p	p	X
ejpam-6066	35	30	,	,	PUNCT
ejpam-6066	35	31	q	q	X
ejpam-6066	35	32	are	be	AUX
ejpam-6066	35	33	primes	prime	NOUN
ejpam-6066	35	34	.	.	PUNCT
ejpam-6066	36	1	table	table	NOUN
ejpam-6066	36	2	2	2	NUM
ejpam-6066	36	3	summarizes	summarize	NOUN
ejpam-6066	36	4	several	several	ADJ
ejpam-6066	36	5	recent	recent	ADJ
ejpam-6066	36	6	contributions	contribution	NOUN
ejpam-6066	36	7	on	on	ADP
ejpam-6066	36	8	exponential	exponential	ADJ
ejpam-6066	36	9	diophantine	diophantine	NOUN
ejpam-6066	36	10	equations	equation	NOUN
ejpam-6066	36	11	of	of	ADP
ejpam-6066	36	12	the	the	DET
ejpam-6066	36	13	form	form	NOUN
ejpam-6066	36	14	a(px)±	a(px)±	NOUN
ejpam-6066	36	15	b(qy	b(qy	PROPN
ejpam-6066	36	16	)	)	PUNCT
ejpam-6066	36	17	=	=	SYM
ejpam-6066	36	18	z2	z2	PROPN
ejpam-6066	36	19	,	,	PUNCT
ejpam-6066	36	20	where	where	SCONJ
ejpam-6066	36	21	p	p	X
ejpam-6066	36	22	,	,	PUNCT
ejpam-6066	36	23	q	q	X
ejpam-6066	36	24	are	be	AUX
ejpam-6066	36	25	primes	prime	NOUN
ejpam-6066	36	26	.	.	PUNCT
ejpam-6066	37	1	some	some	PRON
ejpam-6066	37	2	of	of	ADP
ejpam-6066	37	3	these	these	DET
ejpam-6066	37	4	studies	study	NOUN
ejpam-6066	37	5	demonstrate	demonstrate	VERB
ejpam-6066	37	6	the	the	DET
ejpam-6066	37	7	effectiveness	effectiveness	NOUN
ejpam-6066	37	8	of	of	ADP
ejpam-6066	37	9	modular	modular	ADJ
ejpam-6066	37	10	arithmetic	arithmetic	ADJ
ejpam-6066	37	11	and	and	CCONJ
ejpam-6066	37	12	parametric	parametric	ADJ
ejpam-6066	37	13	forms	form	NOUN
ejpam-6066	37	14	in	in	ADP
ejpam-6066	37	15	analyzing	analyze	VERB
ejpam-6066	37	16	such	such	ADJ
ejpam-6066	37	17	equations	equation	NOUN
ejpam-6066	37	18	,	,	PUNCT
ejpam-6066	37	19	and	and	CCONJ
ejpam-6066	37	20	they	they	PRON
ejpam-6066	37	21	inform	inform	VERB
ejpam-6066	37	22	the	the	DET
ejpam-6066	37	23	approach	approach	NOUN
ejpam-6066	37	24	adopted	adopt	VERB
ejpam-6066	37	25	in	in	ADP
ejpam-6066	37	26	this	this	DET
ejpam-6066	37	27	article	article	NOUN
ejpam-6066	37	28	.	.	PUNCT
ejpam-6066	38	1	motivated	motivate	VERB
ejpam-6066	38	2	by	by	ADP
ejpam-6066	38	3	previous	previous	ADJ
ejpam-6066	38	4	studies	study	NOUN
ejpam-6066	38	5	on	on	ADP
ejpam-6066	38	6	exponential	exponential	ADJ
ejpam-6066	38	7	diophantine	diophantine	NOUN
ejpam-6066	38	8	equations	equation	NOUN
ejpam-6066	38	9	,	,	PUNCT
ejpam-6066	38	10	we	we	PRON
ejpam-6066	38	11	continue	continue	VERB
ejpam-6066	38	12	this	this	DET
ejpam-6066	38	13	line	line	NOUN
ejpam-6066	38	14	of	of	ADP
ejpam-6066	38	15	inquiry	inquiry	NOUN
ejpam-6066	38	16	by	by	ADP
ejpam-6066	38	17	examining	examine	VERB
ejpam-6066	38	18	the	the	DET
ejpam-6066	38	19	equation	equation	NOUN
ejpam-6066	38	20	4(7x)−	4(7x)−	PROPN
ejpam-6066	38	21	py	py	PROPN
ejpam-6066	38	22	=	=	SYM
ejpam-6066	38	23	z2	z2	PROPN
ejpam-6066	38	24	,	,	PUNCT
ejpam-6066	38	25	where	where	SCONJ
ejpam-6066	38	26	p	p	NOUN
ejpam-6066	38	27	is	be	AUX
ejpam-6066	38	28	a	a	DET
ejpam-6066	38	29	prime	prime	NOUN
ejpam-6066	38	30	and	and	CCONJ
ejpam-6066	38	31	x	x	NOUN
ejpam-6066	38	32	,	,	PUNCT
ejpam-6066	38	33	y	y	PROPN
ejpam-6066	38	34	,	,	PUNCT
ejpam-6066	38	35	z	z	PROPN
ejpam-6066	38	36	∈	∈	PROPN
ejpam-6066	38	37	z0	z0	PROPN
ejpam-6066	38	38	.	.	PUNCT
ejpam-6066	39	1	by	by	ADP
ejpam-6066	39	2	employing	employ	VERB
ejpam-6066	39	3	modular	modular	ADJ
ejpam-6066	39	4	arithmetic	arithmetic	ADJ
ejpam-6066	39	5	and	and	CCONJ
ejpam-6066	39	6	a	a	DET
ejpam-6066	39	7	detailed	detailed	ADJ
ejpam-6066	39	8	analysis	analysis	NOUN
ejpam-6066	39	9	of	of	ADP
ejpam-6066	39	10	congruence	congruence	NOUN
ejpam-6066	39	11	conditions	condition	NOUN
ejpam-6066	39	12	,	,	PUNCT
ejpam-6066	39	13	we	we	PRON
ejpam-6066	39	14	classify	classify	VERB
ejpam-6066	39	15	all	all	DET
ejpam-6066	39	16	possible	possible	ADJ
ejpam-6066	39	17	solutions	solution	NOUN
ejpam-6066	39	18	and	and	CCONJ
ejpam-6066	39	19	propose	propose	VERB
ejpam-6066	39	20	a	a	DET
ejpam-6066	39	21	conjecture	conjecture	NOUN
ejpam-6066	39	22	on	on	ADP
ejpam-6066	39	23	the	the	DET
ejpam-6066	39	24	nonexistence	nonexistence	NOUN
ejpam-6066	39	25	of	of	ADP
ejpam-6066	39	26	further	further	ADJ
ejpam-6066	39	27	solutions	solution	NOUN
ejpam-6066	39	28	beyond	beyond	ADP
ejpam-6066	39	29	those	those	PRON
ejpam-6066	39	30	explicitly	explicitly	ADV
ejpam-6066	39	31	identified	identify	VERB
ejpam-6066	39	32	.	.	PUNCT
ejpam-6066	40	1	k.	k.	PROPN
ejpam-6066	40	2	laipaporn	laipaporn	PROPN
ejpam-6066	40	3	et	et	PROPN
ejpam-6066	40	4	al	al	PROPN
ejpam-6066	40	5	.	.	PUNCT
ejpam-6066	40	6	/	/	SYM
ejpam-6066	40	7	eur	eur	PROPN
ejpam-6066	40	8	.	.	PUNCT
ejpam-6066	41	1	j.	j.	PROPN
ejpam-6066	41	2	pure	pure	PROPN
ejpam-6066	41	3	appl	appl	PROPN
ejpam-6066	41	4	.	.	PROPN
ejpam-6066	41	5	math	math	PROPN
ejpam-6066	41	6	,	,	PUNCT
ejpam-6066	41	7	18	18	NUM
ejpam-6066	41	8	(	(	PUNCT
ejpam-6066	41	9	3	3	NUM
ejpam-6066	41	10	)	)	PUNCT
ejpam-6066	41	11	(	(	PUNCT
ejpam-6066	41	12	2025	2025	NUM
ejpam-6066	41	13	)	)	PUNCT
ejpam-6066	41	14	,	,	PUNCT
ejpam-6066	41	15	6066	6066	NUM
ejpam-6066	41	16	3	3	NUM
ejpam-6066	41	17	of	of	ADP
ejpam-6066	41	18	13	13	NUM
ejpam-6066	41	19	table	table	NOUN
ejpam-6066	41	20	2	2	NUM
ejpam-6066	41	21	:	:	PUNCT
ejpam-6066	41	22	examples	example	NOUN
ejpam-6066	41	23	of	of	ADP
ejpam-6066	41	24	exponential	exponential	ADJ
ejpam-6066	41	25	diophantine	diophantine	NOUN
ejpam-6066	41	26	equations	equation	NOUN
ejpam-6066	41	27	of	of	ADP
ejpam-6066	41	28	the	the	DET
ejpam-6066	41	29	forms	form	NOUN
ejpam-6066	41	30	a(px)±	a(px)±	NOUN
ejpam-6066	41	31	b(qy	b(qy	PROPN
ejpam-6066	41	32	)	)	PUNCT
ejpam-6066	42	1	=	=	SYM
ejpam-6066	42	2	z2	z2	PROPN
ejpam-6066	42	3	where	where	SCONJ
ejpam-6066	42	4	a	a	DET
ejpam-6066	42	5	,	,	PUNCT
ejpam-6066	42	6	b	b	NOUN
ejpam-6066	42	7	are	be	AUX
ejpam-6066	42	8	positive	positive	ADJ
ejpam-6066	42	9	integers	integer	NOUN
ejpam-6066	42	10	and	and	CCONJ
ejpam-6066	42	11	p	p	X
ejpam-6066	42	12	,	,	PUNCT
ejpam-6066	42	13	q	q	X
ejpam-6066	42	14	are	be	AUX
ejpam-6066	42	15	primes	prime	NOUN
ejpam-6066	42	16	.	.	PUNCT
ejpam-6066	43	1	year	year	NOUN
ejpam-6066	43	2	authors	author	NOUN
ejpam-6066	43	3	equation	equation	NOUN
ejpam-6066	43	4	2018	2018	NUM
ejpam-6066	43	5	j.f.t	j.f.t	NOUN
ejpam-6066	43	6	.	.	PUNCT
ejpam-6066	44	1	rabago	rabago	PROPN
ejpam-6066	45	1	[	[	X
ejpam-6066	45	2	8	8	X
ejpam-6066	45	3	]	]	PUNCT
ejpam-6066	46	1	4x	4x	NUM
ejpam-6066	46	2	−	−	PROPN
ejpam-6066	46	3	py	py	PROPN
ejpam-6066	46	4	=	=	NOUN
ejpam-6066	46	5	3z2	3z2	NUM
ejpam-6066	46	6	2019	2019	NUM
ejpam-6066	46	7	k.	k.	PROPN
ejpam-6066	46	8	laipaporn	laipaporn	PROPN
ejpam-6066	46	9	,	,	PUNCT
ejpam-6066	46	10	et	et	PROPN
ejpam-6066	46	11	al	al	PROPN
ejpam-6066	46	12	.	.	PUNCT
ejpam-6066	47	1	[	[	X
ejpam-6066	47	2	9	9	NUM
ejpam-6066	47	3	]	]	SYM
ejpam-6066	47	4	3x	3x	NUM
ejpam-6066	47	5	+	+	CCONJ
ejpam-6066	47	6	p(5y	p(5y	NOUN
ejpam-6066	47	7	)	)	PUNCT
ejpam-6066	47	8	=	=	SYM
ejpam-6066	47	9	z2	z2	PROPN
ejpam-6066	47	10	2020	2020	NUM
ejpam-6066	47	11	a.	a.	NOUN
ejpam-6066	47	12	elshahed	elshahe	VERB
ejpam-6066	47	13	and	and	CCONJ
ejpam-6066	47	14	h.	h.	PROPN
ejpam-6066	47	15	kamarulhaili	kamarulhaili	NOUN
ejpam-6066	48	1	[	[	X
ejpam-6066	48	2	10	10	NUM
ejpam-6066	48	3	]	]	X
ejpam-6066	48	4	(	(	PUNCT
ejpam-6066	48	5	4n)x	4n)x	NUM
ejpam-6066	48	6	−	−	NOUN
ejpam-6066	49	1	py	py	PROPN
ejpam-6066	49	2	=	=	SYM
ejpam-6066	49	3	z2	z2	PROPN
ejpam-6066	49	4	2021	2021	NUM
ejpam-6066	49	5	s.	s.	PROPN
ejpam-6066	49	6	thongnak	thongnak	PROPN
ejpam-6066	49	7	,	,	PUNCT
ejpam-6066	49	8	et	et	PROPN
ejpam-6066	49	9	al	al	PROPN
ejpam-6066	49	10	.	.	PUNCT
ejpam-6066	50	1	[	[	X
ejpam-6066	50	2	11	11	NUM
ejpam-6066	50	3	]	]	SYM
ejpam-6066	50	4	7x	7x	NUM
ejpam-6066	50	5	−	−	NOUN
ejpam-6066	50	6	5y	5y	NOUN
ejpam-6066	50	7	=	=	SYM
ejpam-6066	50	8	z2	z2	PROPN
ejpam-6066	50	9	2022	2022	NUM
ejpam-6066	50	10	w.	w.	PROPN
ejpam-6066	50	11	tangjai	tangjai	PROPN
ejpam-6066	50	12	,	,	PUNCT
ejpam-6066	50	13	et	et	PROPN
ejpam-6066	50	14	al	al	PROPN
ejpam-6066	50	15	.	.	PUNCT
ejpam-6066	51	1	[	[	X
ejpam-6066	51	2	12	12	NUM
ejpam-6066	51	3	]	]	SYM
ejpam-6066	51	4	7x	7x	NUM
ejpam-6066	51	5	+	+	CCONJ
ejpam-6066	51	6	5(py	5(py	NOUN
ejpam-6066	51	7	)	)	PUNCT
ejpam-6066	51	8	=	=	SYM
ejpam-6066	51	9	z2	z2	PROPN
ejpam-6066	51	10	2022	2022	NUM
ejpam-6066	51	11	w.	w.	PROPN
ejpam-6066	51	12	orosram	orosram	PROPN
ejpam-6066	51	13	and	and	CCONJ
ejpam-6066	51	14	a.	a.	NOUN
ejpam-6066	51	15	unchai	unchai	PROPN
ejpam-6066	52	1	[	[	X
ejpam-6066	52	2	13	13	NUM
ejpam-6066	52	3	]	]	SYM
ejpam-6066	52	4	22nx	22nx	ADJ
ejpam-6066	52	5	−	−	PROPN
ejpam-6066	52	6	py	py	PROPN
ejpam-6066	52	7	=	=	SYM
ejpam-6066	52	8	z2	z2	PROPN
ejpam-6066	52	9	2022	2022	NUM
ejpam-6066	52	10	m.	m.	NOUN
ejpam-6066	52	11	buosi	buosi	NOUN
ejpam-6066	52	12	,	,	PUNCT
ejpam-6066	52	13	et	et	PROPN
ejpam-6066	52	14	al	al	PROPN
ejpam-6066	52	15	.	.	PUNCT
ejpam-6066	53	1	[	[	X
ejpam-6066	53	2	14	14	NUM
ejpam-6066	53	3	]	]	X
ejpam-6066	53	4	px	px	ADP
ejpam-6066	53	5	−	−	PROPN
ejpam-6066	53	6	2y	2y	PROPN
ejpam-6066	53	7	=	=	SYM
ejpam-6066	53	8	z2	z2	PROPN
ejpam-6066	53	9	2024	2024	NUM
ejpam-6066	53	10	s.	s.	PROPN
ejpam-6066	53	11	thongnak	thongnak	PROPN
ejpam-6066	53	12	,	,	PUNCT
ejpam-6066	53	13	et	et	PROPN
ejpam-6066	53	14	al	al	PROPN
ejpam-6066	53	15	.	.	PUNCT
ejpam-6066	54	1	[	[	X
ejpam-6066	54	2	15	15	NUM
ejpam-6066	54	3	]	]	PUNCT
ejpam-6066	54	4	11x	11x	NUM
ejpam-6066	54	5	−	−	PROPN
ejpam-6066	54	6	17y	17y	NOUN
ejpam-6066	54	7	=	=	SYM
ejpam-6066	54	8	z2	z2	PROPN
ejpam-6066	54	9	2024	2024	NUM
ejpam-6066	54	10	k.	k.	PROPN
ejpam-6066	54	11	laipaporn	laipaporn	PROPN
ejpam-6066	54	12	,	,	PUNCT
ejpam-6066	54	13	et	et	PROPN
ejpam-6066	54	14	al	al	PROPN
ejpam-6066	54	15	.	.	PUNCT
ejpam-6066	55	1	[	[	X
ejpam-6066	55	2	16	16	NUM
ejpam-6066	55	3	]	]	X
ejpam-6066	55	4	ax	ax	NOUN
ejpam-6066	55	5	±	±	NUM
ejpam-6066	55	6	ay	ay	PROPN
ejpam-6066	55	7	=	=	SYM
ejpam-6066	55	8	zn	zn	PROPN
ejpam-6066	55	9	2024	2024	NUM
ejpam-6066	55	10	y.	y.	PROPN
ejpam-6066	55	11	li	li	PROPN
ejpam-6066	55	12	,	,	PUNCT
ejpam-6066	55	13	et	et	PROPN
ejpam-6066	55	14	al	al	PROPN
ejpam-6066	55	15	.	.	PUNCT
ejpam-6066	56	1	[	[	X
ejpam-6066	56	2	17	17	NUM
ejpam-6066	56	3	]	]	SYM
ejpam-6066	56	4	2x	2x	NUM
ejpam-6066	56	5	±	±	NOUN
ejpam-6066	56	6	(	(	PUNCT
ejpam-6066	56	7	2kp)y	2kp)y	PROPN
ejpam-6066	56	8	=	=	SYM
ejpam-6066	56	9	z2	z2	PROPN
ejpam-6066	56	10	and	and	CCONJ
ejpam-6066	56	11	−2x	−2x	PROPN
ejpam-6066	56	12	+	+	CCONJ
ejpam-6066	56	13	(	(	PUNCT
ejpam-6066	56	14	2k3)y	2k3)y	NUM
ejpam-6066	56	15	=	=	SYM
ejpam-6066	56	16	z2	z2	PROPN
ejpam-6066	56	17	2024	2024	NUM
ejpam-6066	56	18	j.	j.	PROPN
ejpam-6066	56	19	zhang	zhang	PROPN
ejpam-6066	56	20	and	and	CCONJ
ejpam-6066	56	21	y.	y.	PROPN
ejpam-6066	56	22	li	li	PROPN
ejpam-6066	57	1	[	[	X
ejpam-6066	57	2	18	18	NUM
ejpam-6066	57	3	]	]	PUNCT
ejpam-6066	57	4	(	(	PUNCT
ejpam-6066	57	5	−1)αpx	−1)αpx	NOUN
ejpam-6066	57	6	+	+	CCONJ
ejpam-6066	57	7	(	(	PUNCT
ejpam-6066	57	8	−1)β(2k(2p−	−1)β(2k(2p−	NOUN
ejpam-6066	57	9	1))y	1))y	NUM
ejpam-6066	57	10	=	=	SYM
ejpam-6066	57	11	z2	z2	PROPN
ejpam-6066	57	12	2025	2025	NUM
ejpam-6066	57	13	k.	k.	PROPN
ejpam-6066	57	14	laipaporn	laipaporn	PROPN
ejpam-6066	57	15	,	,	PUNCT
ejpam-6066	57	16	et	et	PROPN
ejpam-6066	57	17	al	al	PROPN
ejpam-6066	57	18	.	.	PUNCT
ejpam-6066	58	1	[	[	X
ejpam-6066	58	2	19	19	NUM
ejpam-6066	58	3	]	]	X
ejpam-6066	58	4	px	px	X
ejpam-6066	58	5	+	+	CCONJ
ejpam-6066	58	6	q2y	q2y	NOUN
ejpam-6066	58	7	=	=	SYM
ejpam-6066	58	8	z2n	z2n	PROPN
ejpam-6066	58	9	2	2	NUM
ejpam-6066	58	10	.	.	PUNCT
ejpam-6066	58	11	main	main	ADJ
ejpam-6066	58	12	theorem	theorem	NOUN
ejpam-6066	58	13	we	we	PRON
ejpam-6066	58	14	begin	begin	VERB
ejpam-6066	58	15	by	by	ADP
ejpam-6066	58	16	analyzing	analyze	VERB
ejpam-6066	58	17	the	the	DET
ejpam-6066	58	18	structure	structure	NOUN
ejpam-6066	58	19	of	of	ADP
ejpam-6066	58	20	the	the	DET
ejpam-6066	58	21	equation	equation	NOUN
ejpam-6066	58	22	and	and	CCONJ
ejpam-6066	58	23	presenting	present	VERB
ejpam-6066	58	24	the	the	DET
ejpam-6066	58	25	main	main	ADJ
ejpam-6066	58	26	classification	classification	NOUN
ejpam-6066	58	27	result	result	NOUN
ejpam-6066	58	28	.	.	PUNCT
ejpam-6066	59	1	to	to	PART
ejpam-6066	59	2	support	support	VERB
ejpam-6066	59	3	the	the	DET
ejpam-6066	59	4	classification	classification	NOUN
ejpam-6066	59	5	,	,	PUNCT
ejpam-6066	59	6	we	we	PRON
ejpam-6066	59	7	first	first	ADV
ejpam-6066	59	8	introduce	introduce	VERB
ejpam-6066	59	9	an	an	DET
ejpam-6066	59	10	auxiliary	auxiliary	ADJ
ejpam-6066	59	11	lemma	lemma	PROPN
ejpam-6066	59	12	to	to	PART
ejpam-6066	59	13	understand	understand	VERB
ejpam-6066	59	14	the	the	DET
ejpam-6066	59	15	behavior	behavior	NOUN
ejpam-6066	59	16	of	of	ADP
ejpam-6066	59	17	power	power	NOUN
ejpam-6066	59	18	of	of	ADP
ejpam-6066	59	19	p	p	PRON
ejpam-6066	59	20	modulo	modulo	NOUN
ejpam-6066	59	21	9	9	NUM
ejpam-6066	59	22	.	.	PUNCT
ejpam-6066	60	1	lemma	lemma	PROPN
ejpam-6066	60	2	1	1	X
ejpam-6066	60	3	.	.	PUNCT
ejpam-6066	61	1	let	let	VERB
ejpam-6066	61	2	p	p	PRON
ejpam-6066	61	3	≡	≡	PROPN
ejpam-6066	61	4	6r+1	6r+1	PROPN
ejpam-6066	61	5	(	(	PUNCT
ejpam-6066	61	6	mod	mod	PROPN
ejpam-6066	61	7	9	9	NUM
ejpam-6066	61	8	)	)	PUNCT
ejpam-6066	61	9	for	for	ADP
ejpam-6066	61	10	some	some	DET
ejpam-6066	61	11	integer	integer	NOUN
ejpam-6066	61	12	r	r	NOUN
ejpam-6066	61	13	≥	≥	NOUN
ejpam-6066	61	14	0	0	NUM
ejpam-6066	61	15	.	.	PUNCT
ejpam-6066	62	1	then	then	ADV
ejpam-6066	62	2	for	for	ADP
ejpam-6066	62	3	all	all	DET
ejpam-6066	62	4	integers	integer	NOUN
ejpam-6066	62	5	x	x	X
ejpam-6066	62	6	,	,	PUNCT
ejpam-6066	62	7	y	y	PROPN
ejpam-6066	62	8	≥	≥	NUM
ejpam-6066	62	9	0	0	NUM
ejpam-6066	62	10	,	,	PUNCT
ejpam-6066	62	11	the	the	DET
ejpam-6066	62	12	congruence	congruence	ADJ
ejpam-6066	62	13	class	class	NOUN
ejpam-6066	62	14	of	of	ADP
ejpam-6066	62	15	4(7x	4(7x	NUM
ejpam-6066	62	16	)	)	PUNCT
ejpam-6066	62	17	−	−	PROPN
ejpam-6066	62	18	py	py	INTJ
ejpam-6066	62	19	(	(	PUNCT
ejpam-6066	62	20	mod	mod	PROPN
ejpam-6066	62	21	9	9	NUM
ejpam-6066	62	22	)	)	PUNCT
ejpam-6066	62	23	depends	depend	VERB
ejpam-6066	62	24	on	on	ADP
ejpam-6066	62	25	the	the	DET
ejpam-6066	62	26	values	value	NOUN
ejpam-6066	62	27	of	of	ADP
ejpam-6066	62	28	x	x	X
ejpam-6066	62	29	(	(	PUNCT
ejpam-6066	62	30	mod	mod	NOUN
ejpam-6066	62	31	3	3	NUM
ejpam-6066	62	32	)	)	PUNCT
ejpam-6066	62	33	and	and	CCONJ
ejpam-6066	62	34	y	y	PROPN
ejpam-6066	62	35	(	(	PUNCT
ejpam-6066	62	36	mod	mod	PROPN
ejpam-6066	62	37	3	3	NUM
ejpam-6066	62	38	)	)	PUNCT
ejpam-6066	62	39	as	as	SCONJ
ejpam-6066	62	40	follows	follow	VERB
ejpam-6066	62	41	:	:	PUNCT
ejpam-6066	62	42	4(7x)−	4(7x)−	PROPN
ejpam-6066	62	43	py	py	PROPN
ejpam-6066	62	44	≡	≡	PROPN
ejpam-6066	62	45			PRON
ejpam-6066	62	46	1	1	NUM
ejpam-6066	62	47	(	(	PUNCT
ejpam-6066	62	48	mod	mod	NOUN
ejpam-6066	62	49	9	9	NUM
ejpam-6066	62	50	)	)	PUNCT
ejpam-6066	62	51	if	if	SCONJ
ejpam-6066	62	52	x	x	SYM
ejpam-6066	62	53	≡	≡	PROPN
ejpam-6066	62	54	0	0	PUNCT
ejpam-6066	62	55	(	(	PUNCT
ejpam-6066	62	56	mod	mod	NOUN
ejpam-6066	62	57	3	3	NUM
ejpam-6066	62	58	)	)	PUNCT
ejpam-6066	62	59	and	and	CCONJ
ejpam-6066	62	60	y	y	PROPN
ejpam-6066	62	61	≡	≡	PROPN
ejpam-6066	62	62	0	0	PUNCT
ejpam-6066	63	1	(	(	PUNCT
ejpam-6066	63	2	mod	mod	PROPN
ejpam-6066	63	3	3	3	NUM
ejpam-6066	63	4	)	)	PUNCT
ejpam-6066	63	5	,	,	PUNCT
ejpam-6066	63	6	4−	4−	PROPN
ejpam-6066	64	1	p	p	NOUN
ejpam-6066	64	2	(	(	PUNCT
ejpam-6066	64	3	mod	mod	PROPN
ejpam-6066	64	4	9	9	NUM
ejpam-6066	64	5	)	)	PUNCT
ejpam-6066	64	6	if	if	SCONJ
ejpam-6066	64	7	x	x	SYM
ejpam-6066	64	8	≡	≡	PROPN
ejpam-6066	64	9	0	0	PUNCT
ejpam-6066	64	10	(	(	PUNCT
ejpam-6066	64	11	mod	mod	NOUN
ejpam-6066	64	12	3	3	NUM
ejpam-6066	64	13	)	)	PUNCT
ejpam-6066	64	14	and	and	CCONJ
ejpam-6066	64	15	y	y	PROPN
ejpam-6066	64	16	≡	≡	PROPN
ejpam-6066	64	17	1	1	NUM
ejpam-6066	64	18	(	(	PUNCT
ejpam-6066	64	19	mod	mod	NOUN
ejpam-6066	64	20	3	3	NUM
ejpam-6066	64	21	)	)	PUNCT
ejpam-6066	64	22	,	,	PUNCT
ejpam-6066	64	23	2	2	NUM
ejpam-6066	64	24	+	+	CCONJ
ejpam-6066	64	25	p	p	X
ejpam-6066	64	26	(	(	PUNCT
ejpam-6066	64	27	mod	mod	NOUN
ejpam-6066	64	28	9	9	NUM
ejpam-6066	64	29	)	)	PUNCT
ejpam-6066	64	30	if	if	SCONJ
ejpam-6066	64	31	x	x	SYM
ejpam-6066	64	32	≡	≡	PROPN
ejpam-6066	64	33	0	0	PUNCT
ejpam-6066	64	34	(	(	PUNCT
ejpam-6066	64	35	mod	mod	NOUN
ejpam-6066	64	36	3	3	NUM
ejpam-6066	64	37	)	)	PUNCT
ejpam-6066	64	38	and	and	CCONJ
ejpam-6066	64	39	y	y	PROPN
ejpam-6066	64	40	≡	≡	PROPN
ejpam-6066	64	41	2	2	NUM
ejpam-6066	64	42	(	(	PUNCT
ejpam-6066	64	43	mod	mod	NOUN
ejpam-6066	64	44	3	3	NUM
ejpam-6066	64	45	)	)	PUNCT
ejpam-6066	64	46	,	,	PUNCT
ejpam-6066	64	47	0	0	NUM
ejpam-6066	64	48	(	(	PUNCT
ejpam-6066	64	49	mod	mod	NOUN
ejpam-6066	64	50	9	9	NUM
ejpam-6066	64	51	)	)	PUNCT
ejpam-6066	64	52	if	if	SCONJ
ejpam-6066	64	53	x	x	SYM
ejpam-6066	64	54	≡	≡	PROPN
ejpam-6066	64	55	1	1	NUM
ejpam-6066	64	56	(	(	PUNCT
ejpam-6066	64	57	mod	mod	NOUN
ejpam-6066	64	58	3	3	NUM
ejpam-6066	64	59	)	)	PUNCT
ejpam-6066	64	60	and	and	CCONJ
ejpam-6066	64	61	y	y	PROPN
ejpam-6066	64	62	≡	≡	PROPN
ejpam-6066	64	63	0	0	PUNCT
ejpam-6066	64	64	(	(	PUNCT
ejpam-6066	64	65	mod	mod	PROPN
ejpam-6066	64	66	3	3	NUM
ejpam-6066	64	67	)	)	PUNCT
ejpam-6066	64	68	,	,	PUNCT
ejpam-6066	64	69	1−	1−	NUM
ejpam-6066	64	70	p	p	X
ejpam-6066	64	71	(	(	PUNCT
ejpam-6066	64	72	mod	mod	PROPN
ejpam-6066	64	73	9	9	NUM
ejpam-6066	64	74	)	)	PUNCT
ejpam-6066	65	1	if	if	SCONJ
ejpam-6066	65	2	x	x	SYM
ejpam-6066	65	3	≡	≡	PROPN
ejpam-6066	65	4	1	1	NUM
ejpam-6066	65	5	(	(	PUNCT
ejpam-6066	65	6	mod	mod	NOUN
ejpam-6066	65	7	3	3	NUM
ejpam-6066	65	8	)	)	PUNCT
ejpam-6066	65	9	and	and	CCONJ
ejpam-6066	65	10	y	y	PROPN
ejpam-6066	65	11	≡	≡	PROPN
ejpam-6066	65	12	1	1	NUM
ejpam-6066	65	13	(	(	PUNCT
ejpam-6066	65	14	mod	mod	NOUN
ejpam-6066	65	15	3	3	NUM
ejpam-6066	65	16	)	)	PUNCT
ejpam-6066	65	17	,	,	PUNCT
ejpam-6066	65	18	p−	p−	NOUN
ejpam-6066	65	19	1	1	NUM
ejpam-6066	65	20	(	(	PUNCT
ejpam-6066	65	21	mod	mod	NOUN
ejpam-6066	65	22	9	9	NUM
ejpam-6066	65	23	)	)	PUNCT
ejpam-6066	65	24	if	if	SCONJ
ejpam-6066	65	25	x	x	SYM
ejpam-6066	65	26	≡	≡	PROPN
ejpam-6066	65	27	1	1	NUM
ejpam-6066	65	28	(	(	PUNCT
ejpam-6066	65	29	mod	mod	NOUN
ejpam-6066	65	30	3	3	NUM
ejpam-6066	65	31	)	)	PUNCT
ejpam-6066	65	32	and	and	CCONJ
ejpam-6066	65	33	y	y	PROPN
ejpam-6066	65	34	≡	≡	PROPN
ejpam-6066	65	35	2	2	NUM
ejpam-6066	65	36	(	(	PUNCT
ejpam-6066	65	37	mod	mod	NOUN
ejpam-6066	65	38	3	3	NUM
ejpam-6066	65	39	)	)	PUNCT
ejpam-6066	65	40	,	,	PUNCT
ejpam-6066	65	41	6	6	NUM
ejpam-6066	65	42	(	(	PUNCT
ejpam-6066	65	43	mod	mod	NOUN
ejpam-6066	65	44	9	9	NUM
ejpam-6066	65	45	)	)	PUNCT
ejpam-6066	65	46	if	if	SCONJ
ejpam-6066	65	47	x	x	SYM
ejpam-6066	65	48	≡	≡	PROPN
ejpam-6066	65	49	2	2	NUM
ejpam-6066	65	50	(	(	PUNCT
ejpam-6066	65	51	mod	mod	NOUN
ejpam-6066	65	52	3	3	NUM
ejpam-6066	65	53	)	)	PUNCT
ejpam-6066	65	54	and	and	CCONJ
ejpam-6066	65	55	y	y	PROPN
ejpam-6066	65	56	≡	≡	PROPN
ejpam-6066	65	57	0	0	PUNCT
ejpam-6066	65	58	(	(	PUNCT
ejpam-6066	65	59	mod	mod	PROPN
ejpam-6066	65	60	3	3	NUM
ejpam-6066	65	61	)	)	PUNCT
ejpam-6066	65	62	,	,	PUNCT
ejpam-6066	65	63	7−	7−	NUM
ejpam-6066	65	64	p	p	NOUN
ejpam-6066	65	65	(	(	PUNCT
ejpam-6066	65	66	mod	mod	PROPN
ejpam-6066	65	67	9	9	NUM
ejpam-6066	65	68	)	)	PUNCT
ejpam-6066	65	69	if	if	SCONJ
ejpam-6066	65	70	x	x	SYM
ejpam-6066	65	71	≡	≡	PROPN
ejpam-6066	65	72	2	2	NUM
ejpam-6066	65	73	(	(	PUNCT
ejpam-6066	65	74	mod	mod	NOUN
ejpam-6066	65	75	3	3	NUM
ejpam-6066	65	76	)	)	PUNCT
ejpam-6066	65	77	and	and	CCONJ
ejpam-6066	65	78	y	y	PROPN
ejpam-6066	65	79	≡	≡	PROPN
ejpam-6066	65	80	1	1	NUM
ejpam-6066	65	81	(	(	PUNCT
ejpam-6066	65	82	mod	mod	NOUN
ejpam-6066	65	83	3	3	NUM
ejpam-6066	65	84	)	)	PUNCT
ejpam-6066	65	85	,	,	PUNCT
ejpam-6066	65	86	5	5	NUM
ejpam-6066	65	87	+	+	CCONJ
ejpam-6066	65	88	p	p	X
ejpam-6066	65	89	(	(	PUNCT
ejpam-6066	65	90	mod	mod	NOUN
ejpam-6066	65	91	9	9	NUM
ejpam-6066	65	92	)	)	PUNCT
ejpam-6066	65	93	if	if	SCONJ
ejpam-6066	65	94	x	x	SYM
ejpam-6066	65	95	≡	≡	PROPN
ejpam-6066	65	96	2	2	NUM
ejpam-6066	65	97	(	(	PUNCT
ejpam-6066	65	98	mod	mod	NOUN
ejpam-6066	65	99	3	3	NUM
ejpam-6066	65	100	)	)	PUNCT
ejpam-6066	65	101	and	and	CCONJ
ejpam-6066	65	102	y	y	PROPN
ejpam-6066	65	103	≡	≡	PROPN
ejpam-6066	65	104	2	2	NUM
ejpam-6066	65	105	(	(	PUNCT
ejpam-6066	65	106	mod	mod	NOUN
ejpam-6066	65	107	3	3	NUM
ejpam-6066	65	108	)	)	PUNCT
ejpam-6066	66	1	.	.	PUNCT
ejpam-6066	67	1	proof	proof	NOUN
ejpam-6066	67	2	.	.	PUNCT
ejpam-6066	68	1	since	since	SCONJ
ejpam-6066	68	2	p	p	PROPN
ejpam-6066	68	3	≡	≡	PROPN
ejpam-6066	68	4	6r+1	6r+1	PROPN
ejpam-6066	68	5	(	(	PUNCT
ejpam-6066	68	6	mod	mod	PROPN
ejpam-6066	68	7	9	9	NUM
ejpam-6066	68	8	)	)	PUNCT
ejpam-6066	68	9	,	,	PUNCT
ejpam-6066	68	10	we	we	PRON
ejpam-6066	68	11	derive	derive	VERB
ejpam-6066	68	12	p3	p3	PROPN
ejpam-6066	68	13	≡	≡	PROPN
ejpam-6066	68	14	216r3	216r3	NUM
ejpam-6066	68	15	+	+	NOUN
ejpam-6066	68	16	108r2	108r2	NUM
ejpam-6066	68	17	+	+	ADJ
ejpam-6066	68	18	18r+1	18r+1	ADJ
ejpam-6066	68	19	≡	≡	ADJ
ejpam-6066	68	20	1	1	NUM
ejpam-6066	68	21	(	(	PUNCT
ejpam-6066	68	22	mod	mod	NOUN
ejpam-6066	68	23	9	9	NUM
ejpam-6066	68	24	)	)	PUNCT
ejpam-6066	68	25	.	.	PUNCT
ejpam-6066	69	1	therefore	therefore	ADV
ejpam-6066	69	2	,	,	PUNCT
ejpam-6066	69	3	for	for	ADP
ejpam-6066	69	4	all	all	DET
ejpam-6066	69	5	integers	integer	NOUN
ejpam-6066	69	6	t	t	PROPN
ejpam-6066	69	7	,	,	PUNCT
ejpam-6066	69	8	y	y	PROPN
ejpam-6066	69	9	≥	≥	PROPN
ejpam-6066	69	10	0	0	NUM
ejpam-6066	69	11	,	,	PUNCT
ejpam-6066	69	12	py	py	PROPN
ejpam-6066	69	13	≡	≡	PROPN
ejpam-6066	69	14	pt	pt	PROPN
ejpam-6066	69	15	(	(	PUNCT
ejpam-6066	69	16	mod	mod	PROPN
ejpam-6066	69	17	9	9	NUM
ejpam-6066	69	18	)	)	PUNCT
ejpam-6066	69	19	where	where	SCONJ
ejpam-6066	69	20	y	y	PROPN
ejpam-6066	69	21	≡	≡	PROPN
ejpam-6066	69	22	t	t	PROPN
ejpam-6066	69	23	(	(	PUNCT
ejpam-6066	69	24	mod	mod	PROPN
ejpam-6066	69	25	3	3	NUM
ejpam-6066	69	26	)	)	PUNCT
ejpam-6066	69	27	.	.	PUNCT
ejpam-6066	70	1	it	it	PRON
ejpam-6066	70	2	follows	follow	VERB
ejpam-6066	70	3	that	that	SCONJ
ejpam-6066	70	4	py	py	PROPN
ejpam-6066	70	5	≡	≡	PROPN
ejpam-6066	71	1			ADV
ejpam-6066	71	2	1	1	NUM
ejpam-6066	71	3	(	(	PUNCT
ejpam-6066	71	4	mod	mod	NOUN
ejpam-6066	71	5	9	9	NUM
ejpam-6066	71	6	)	)	PUNCT
ejpam-6066	71	7	if	if	SCONJ
ejpam-6066	71	8	y	y	PROPN
ejpam-6066	71	9	≡	≡	PROPN
ejpam-6066	71	10	0	0	PUNCT
ejpam-6066	72	1	(	(	PUNCT
ejpam-6066	72	2	mod	mod	NOUN
ejpam-6066	72	3	3	3	NUM
ejpam-6066	72	4	)	)	PUNCT
ejpam-6066	72	5	,	,	PUNCT
ejpam-6066	72	6	p	p	X
ejpam-6066	72	7	(	(	PUNCT
ejpam-6066	72	8	mod	mod	PROPN
ejpam-6066	72	9	9	9	NUM
ejpam-6066	72	10	)	)	PUNCT
ejpam-6066	72	11	if	if	SCONJ
ejpam-6066	72	12	y	y	PROPN
ejpam-6066	72	13	≡	≡	PROPN
ejpam-6066	72	14	1	1	NUM
ejpam-6066	72	15	(	(	PUNCT
ejpam-6066	72	16	mod	mod	NOUN
ejpam-6066	72	17	3	3	NUM
ejpam-6066	72	18	)	)	PUNCT
ejpam-6066	72	19	,	,	PUNCT
ejpam-6066	72	20	p2	p2	PROPN
ejpam-6066	72	21	(	(	PUNCT
ejpam-6066	72	22	mod	mod	NOUN
ejpam-6066	72	23	9	9	NUM
ejpam-6066	72	24	)	)	PUNCT
ejpam-6066	72	25	if	if	SCONJ
ejpam-6066	72	26	y	y	PROPN
ejpam-6066	72	27	≡	≡	PROPN
ejpam-6066	72	28	2	2	NUM
ejpam-6066	72	29	(	(	PUNCT
ejpam-6066	72	30	mod	mod	NOUN
ejpam-6066	72	31	3	3	NUM
ejpam-6066	72	32	)	)	PUNCT
ejpam-6066	72	33	.	.	PUNCT
ejpam-6066	73	1	k.	k.	PROPN
ejpam-6066	73	2	laipaporn	laipaporn	PROPN
ejpam-6066	73	3	et	et	PROPN
ejpam-6066	73	4	al	al	PROPN
ejpam-6066	73	5	.	.	PUNCT
ejpam-6066	73	6	/	/	SYM
ejpam-6066	73	7	eur	eur	PROPN
ejpam-6066	73	8	.	.	PUNCT
ejpam-6066	74	1	j.	j.	PROPN
ejpam-6066	74	2	pure	pure	PROPN
ejpam-6066	74	3	appl	appl	PROPN
ejpam-6066	74	4	.	.	PROPN
ejpam-6066	74	5	math	math	PROPN
ejpam-6066	74	6	,	,	PUNCT
ejpam-6066	74	7	18	18	NUM
ejpam-6066	74	8	(	(	PUNCT
ejpam-6066	74	9	3	3	NUM
ejpam-6066	74	10	)	)	PUNCT
ejpam-6066	74	11	(	(	PUNCT
ejpam-6066	74	12	2025	2025	NUM
ejpam-6066	74	13	)	)	PUNCT
ejpam-6066	74	14	,	,	PUNCT
ejpam-6066	74	15	6066	6066	NUM
ejpam-6066	74	16	4	4	NUM
ejpam-6066	74	17	of	of	ADP
ejpam-6066	74	18	13	13	NUM
ejpam-6066	74	19	next	next	ADJ
ejpam-6066	75	1	,	,	PUNCT
ejpam-6066	75	2	we	we	PRON
ejpam-6066	75	3	compute	compute	VERB
ejpam-6066	75	4	p2	p2	PROPN
ejpam-6066	75	5	(	(	PUNCT
ejpam-6066	75	6	mod	mod	NOUN
ejpam-6066	75	7	9	9	NUM
ejpam-6066	75	8	)	)	PUNCT
ejpam-6066	75	9	using	use	VERB
ejpam-6066	75	10	the	the	DET
ejpam-6066	75	11	given	give	VERB
ejpam-6066	75	12	congruence	congruence	NOUN
ejpam-6066	75	13	for	for	ADP
ejpam-6066	75	14	p	p	X
ejpam-6066	75	15	:	:	PUNCT
ejpam-6066	75	16	p2	p2	PROPN
ejpam-6066	75	17	≡	≡	PROPN
ejpam-6066	75	18	36r2	36r2	NUM
ejpam-6066	76	1	+	+	NUM
ejpam-6066	76	2	12r	12r	NUM
ejpam-6066	76	3	+	+	CCONJ
ejpam-6066	76	4	1	1	NUM
ejpam-6066	76	5	≡	≡	PROPN
ejpam-6066	76	6	−6r	−6r	PROPN
ejpam-6066	77	1	+	+	CCONJ
ejpam-6066	77	2	1	1	NUM
ejpam-6066	77	3	≡	≡	PROPN
ejpam-6066	77	4	2−	2−	NUM
ejpam-6066	77	5	p	p	NOUN
ejpam-6066	77	6	(	(	PUNCT
ejpam-6066	77	7	mod	mod	PROPN
ejpam-6066	77	8	9	9	NUM
ejpam-6066	77	9	)	)	PUNCT
ejpam-6066	77	10	.	.	PUNCT
ejpam-6066	78	1	therefore	therefore	ADV
ejpam-6066	78	2	,	,	PUNCT
ejpam-6066	78	3	py	py	PROPN
ejpam-6066	78	4	≡	≡	PROPN
ejpam-6066	78	5			ADV
ejpam-6066	78	6	1	1	NUM
ejpam-6066	78	7	(	(	PUNCT
ejpam-6066	78	8	mod	mod	NOUN
ejpam-6066	78	9	9	9	NUM
ejpam-6066	78	10	)	)	PUNCT
ejpam-6066	78	11	if	if	SCONJ
ejpam-6066	78	12	y	y	PROPN
ejpam-6066	78	13	≡	≡	PROPN
ejpam-6066	78	14	0	0	PUNCT
ejpam-6066	79	1	(	(	PUNCT
ejpam-6066	79	2	mod	mod	NOUN
ejpam-6066	79	3	3	3	NUM
ejpam-6066	79	4	)	)	PUNCT
ejpam-6066	79	5	,	,	PUNCT
ejpam-6066	79	6	p	p	X
ejpam-6066	79	7	(	(	PUNCT
ejpam-6066	79	8	mod	mod	PROPN
ejpam-6066	79	9	9	9	NUM
ejpam-6066	79	10	)	)	PUNCT
ejpam-6066	79	11	if	if	SCONJ
ejpam-6066	79	12	y	y	PROPN
ejpam-6066	79	13	≡	≡	PROPN
ejpam-6066	79	14	1	1	NUM
ejpam-6066	79	15	(	(	PUNCT
ejpam-6066	79	16	mod	mod	NOUN
ejpam-6066	79	17	3	3	NUM
ejpam-6066	79	18	)	)	PUNCT
ejpam-6066	79	19	,	,	PUNCT
ejpam-6066	79	20	2−	2−	NUM
ejpam-6066	79	21	p	p	NOUN
ejpam-6066	79	22	(	(	PUNCT
ejpam-6066	79	23	mod	mod	PROPN
ejpam-6066	79	24	9	9	NUM
ejpam-6066	79	25	)	)	PUNCT
ejpam-6066	79	26	if	if	SCONJ
ejpam-6066	79	27	y	y	PROPN
ejpam-6066	79	28	≡	≡	PROPN
ejpam-6066	79	29	2	2	NUM
ejpam-6066	79	30	(	(	PUNCT
ejpam-6066	79	31	mod	mod	NOUN
ejpam-6066	79	32	3	3	NUM
ejpam-6066	79	33	)	)	PUNCT
ejpam-6066	79	34	.	.	PUNCT
ejpam-6066	80	1	hence	hence	ADV
ejpam-6066	80	2	,	,	PUNCT
ejpam-6066	80	3	4(7x	4(7x	PROPN
ejpam-6066	80	4	)	)	PUNCT
ejpam-6066	80	5	≡	≡	PROPN
ejpam-6066	81	1			ADP
ejpam-6066	81	2	4(1	4(1	NOUN
ejpam-6066	81	3	)	)	PUNCT
ejpam-6066	82	1	≡	≡	PROPN
ejpam-6066	82	2	4	4	NUM
ejpam-6066	82	3	(	(	PUNCT
ejpam-6066	82	4	mod	mod	NOUN
ejpam-6066	82	5	9	9	NUM
ejpam-6066	82	6	)	)	PUNCT
ejpam-6066	82	7	if	if	SCONJ
ejpam-6066	82	8	x	x	SYM
ejpam-6066	82	9	≡	≡	PROPN
ejpam-6066	82	10	0	0	PUNCT
ejpam-6066	82	11	(	(	PUNCT
ejpam-6066	82	12	mod	mod	NOUN
ejpam-6066	82	13	3	3	NUM
ejpam-6066	82	14	)	)	PUNCT
ejpam-6066	82	15	,	,	PUNCT
ejpam-6066	82	16	4(7	4(7	NUM
ejpam-6066	82	17	)	)	PUNCT
ejpam-6066	82	18	≡	≡	PROPN
ejpam-6066	82	19	1	1	NUM
ejpam-6066	82	20	(	(	PUNCT
ejpam-6066	82	21	mod	mod	NOUN
ejpam-6066	82	22	9	9	NUM
ejpam-6066	82	23	)	)	PUNCT
ejpam-6066	82	24	if	if	SCONJ
ejpam-6066	82	25	x	x	SYM
ejpam-6066	82	26	≡	≡	PROPN
ejpam-6066	82	27	1	1	NUM
ejpam-6066	82	28	(	(	PUNCT
ejpam-6066	82	29	mod	mod	NOUN
ejpam-6066	82	30	3	3	NUM
ejpam-6066	82	31	)	)	PUNCT
ejpam-6066	82	32	,	,	PUNCT
ejpam-6066	82	33	4(4	4(4	NUM
ejpam-6066	82	34	)	)	PUNCT
ejpam-6066	82	35	≡	≡	PROPN
ejpam-6066	82	36	7	7	NUM
ejpam-6066	82	37	(	(	PUNCT
ejpam-6066	82	38	mod	mod	NOUN
ejpam-6066	82	39	9	9	NUM
ejpam-6066	82	40	)	)	PUNCT
ejpam-6066	82	41	if	if	SCONJ
ejpam-6066	82	42	x	x	SYM
ejpam-6066	82	43	≡	≡	PROPN
ejpam-6066	82	44	2	2	NUM
ejpam-6066	82	45	(	(	PUNCT
ejpam-6066	82	46	mod	mod	NOUN
ejpam-6066	82	47	3	3	NUM
ejpam-6066	82	48	)	)	PUNCT
ejpam-6066	82	49	.	.	PUNCT
ejpam-6066	83	1	combining	combine	VERB
ejpam-6066	83	2	the	the	DET
ejpam-6066	83	3	congruences	congruence	NOUN
ejpam-6066	83	4	for	for	ADP
ejpam-6066	83	5	4(7x	4(7x	NUM
ejpam-6066	83	6	)	)	PUNCT
ejpam-6066	83	7	and	and	CCONJ
ejpam-6066	83	8	py	py	INTJ
ejpam-6066	83	9	,	,	PUNCT
ejpam-6066	83	10	we	we	PRON
ejpam-6066	83	11	obtain	obtain	VERB
ejpam-6066	83	12	the	the	DET
ejpam-6066	83	13	desired	desire	VERB
ejpam-6066	83	14	result	result	NOUN
ejpam-6066	83	15	.	.	PUNCT
ejpam-6066	84	1	we	we	PRON
ejpam-6066	84	2	now	now	ADV
ejpam-6066	84	3	proceed	proceed	VERB
ejpam-6066	84	4	to	to	ADP
ejpam-6066	84	5	a	a	DET
ejpam-6066	84	6	complete	complete	ADJ
ejpam-6066	84	7	classification	classification	NOUN
ejpam-6066	84	8	of	of	ADP
ejpam-6066	84	9	the	the	DET
ejpam-6066	84	10	solutions	solution	NOUN
ejpam-6066	84	11	based	base	VERB
ejpam-6066	84	12	on	on	ADP
ejpam-6066	84	13	the	the	DET
ejpam-6066	84	14	value	value	NOUN
ejpam-6066	84	15	of	of	ADP
ejpam-6066	84	16	p	p	X
ejpam-6066	84	17	:	:	PUNCT
ejpam-6066	84	18	•	•	ADJ
ejpam-6066	84	19	small	small	ADJ
ejpam-6066	84	20	primes	prime	NOUN
ejpam-6066	84	21	p	p	X
ejpam-6066	84	22	=	=	NOUN
ejpam-6066	84	23	2	2	NUM
ejpam-6066	84	24	,	,	PUNCT
ejpam-6066	84	25	3	3	NUM
ejpam-6066	84	26	,	,	PUNCT
ejpam-6066	84	27	where	where	SCONJ
ejpam-6066	84	28	elementary	elementary	ADJ
ejpam-6066	84	29	computations	computation	NOUN
ejpam-6066	84	30	can	can	AUX
ejpam-6066	84	31	be	be	AUX
ejpam-6066	84	32	used	use	VERB
ejpam-6066	84	33	.	.	PUNCT
ejpam-6066	85	1	•	•	NUM
ejpam-6066	85	2	intermediate	intermediate	ADJ
ejpam-6066	85	3	primes	prime	NOUN
ejpam-6066	85	4	5	5	NUM
ejpam-6066	85	5	≤	≤	NOUN
ejpam-6066	85	6	p	p	NOUN
ejpam-6066	85	7	≤	≤	NUM
ejpam-6066	85	8	17	17	NUM
ejpam-6066	85	9	,	,	PUNCT
ejpam-6066	85	10	where	where	SCONJ
ejpam-6066	85	11	contradiction	contradiction	NOUN
ejpam-6066	85	12	via	via	ADP
ejpam-6066	85	13	modular	modular	ADJ
ejpam-6066	85	14	arguments	argument	NOUN
ejpam-6066	85	15	can	can	AUX
ejpam-6066	85	16	be	be	AUX
ejpam-6066	85	17	established	establish	VERB
ejpam-6066	85	18	,	,	PUNCT
ejpam-6066	85	19	•	•	ADP
ejpam-6066	85	20	large	large	ADJ
ejpam-6066	85	21	prime	prime	ADJ
ejpam-6066	85	22	p	p	X
ejpam-6066	85	23	≥	≥	NUM
ejpam-6066	85	24	19	19	NUM
ejpam-6066	85	25	,	,	PUNCT
ejpam-6066	85	26	where	where	SCONJ
ejpam-6066	85	27	structural	structural	ADJ
ejpam-6066	85	28	congruence	congruence	NOUN
ejpam-6066	85	29	restrictions	restriction	NOUN
ejpam-6066	85	30	guide	guide	VERB
ejpam-6066	85	31	the	the	DET
ejpam-6066	85	32	solution	solution	NOUN
ejpam-6066	85	33	forms	form	NOUN
ejpam-6066	85	34	.	.	PUNCT
ejpam-6066	86	1	theorem	theorem	NOUN
ejpam-6066	86	2	1	1	NUM
ejpam-6066	86	3	.	.	PUNCT
ejpam-6066	87	1	let	let	VERB
ejpam-6066	87	2	p	p	PRON
ejpam-6066	87	3	be	be	AUX
ejpam-6066	87	4	any	any	DET
ejpam-6066	87	5	prime	prime	ADJ
ejpam-6066	87	6	number	number	NOUN
ejpam-6066	87	7	.	.	PUNCT
ejpam-6066	88	1	then	then	ADV
ejpam-6066	88	2	the	the	DET
ejpam-6066	88	3	solutions	solution	NOUN
ejpam-6066	88	4	to	to	ADP
ejpam-6066	88	5	the	the	DET
ejpam-6066	88	6	diophantine	diophantine	NOUN
ejpam-6066	88	7	equation	equation	NOUN
ejpam-6066	88	8	4(7x)−	4(7x)−	PROPN
ejpam-6066	88	9	py	py	PROPN
ejpam-6066	88	10	=	=	PROPN
ejpam-6066	88	11	z2	z2	PROPN
ejpam-6066	88	12	where	where	SCONJ
ejpam-6066	88	13	x	x	X
ejpam-6066	88	14	,	,	PUNCT
ejpam-6066	88	15	y	y	PROPN
ejpam-6066	88	16	and	and	CCONJ
ejpam-6066	88	17	z	z	PROPN
ejpam-6066	88	18	are	be	AUX
ejpam-6066	88	19	non	non	ADJ
ejpam-6066	88	20	-	-	ADJ
ejpam-6066	88	21	negative	negative	ADJ
ejpam-6066	88	22	integers	integer	NOUN
ejpam-6066	88	23	satisfy	satisfy	VERB
ejpam-6066	88	24	the	the	DET
ejpam-6066	88	25	following	following	NOUN
ejpam-6066	88	26	:	:	PUNCT
ejpam-6066	88	27	(	(	PUNCT
ejpam-6066	88	28	i	i	NOUN
ejpam-6066	88	29	)	)	PUNCT
ejpam-6066	88	30	for	for	ADP
ejpam-6066	88	31	the	the	DET
ejpam-6066	88	32	prime	prime	NOUN
ejpam-6066	88	33	p	p	NOUN
ejpam-6066	88	34	=	=	NOUN
ejpam-6066	88	35	2	2	NUM
ejpam-6066	88	36	,	,	PUNCT
ejpam-6066	88	37	the	the	DET
ejpam-6066	88	38	unique	unique	ADJ
ejpam-6066	88	39	solution	solution	NOUN
ejpam-6066	88	40	is	be	AUX
ejpam-6066	88	41	(	(	PUNCT
ejpam-6066	88	42	x	x	NOUN
ejpam-6066	88	43	,	,	PUNCT
ejpam-6066	88	44	y	y	PROPN
ejpam-6066	88	45	,	,	PUNCT
ejpam-6066	88	46	z	z	PROPN
ejpam-6066	88	47	,	,	PUNCT
ejpam-6066	88	48	p	p	NOUN
ejpam-6066	88	49	)	)	PUNCT
ejpam-6066	88	50	=	=	SYM
ejpam-6066	88	51	(	(	PUNCT
ejpam-6066	88	52	0	0	NUM
ejpam-6066	88	53	,	,	PUNCT
ejpam-6066	88	54	2	2	NUM
ejpam-6066	88	55	,	,	PUNCT
ejpam-6066	88	56	0	0	NUM
ejpam-6066	88	57	,	,	PUNCT
ejpam-6066	88	58	2	2	NUM
ejpam-6066	88	59	)	)	PUNCT
ejpam-6066	88	60	.	.	PUNCT
ejpam-6066	89	1	(	(	PUNCT
ejpam-6066	89	2	ii	ii	NOUN
ejpam-6066	89	3	)	)	PUNCT
ejpam-6066	89	4	for	for	ADP
ejpam-6066	89	5	the	the	DET
ejpam-6066	89	6	prime	prime	NOUN
ejpam-6066	89	7	p	p	NOUN
ejpam-6066	89	8	=	=	NOUN
ejpam-6066	89	9	3	3	NUM
ejpam-6066	89	10	,	,	PUNCT
ejpam-6066	89	11	if	if	SCONJ
ejpam-6066	89	12	a	a	DET
ejpam-6066	89	13	solution	solution	NOUN
ejpam-6066	89	14	exists	exist	VERB
ejpam-6066	89	15	,	,	PUNCT
ejpam-6066	89	16	then	then	ADV
ejpam-6066	89	17	it	it	PRON
ejpam-6066	89	18	must	must	AUX
ejpam-6066	89	19	be	be	AUX
ejpam-6066	89	20	of	of	ADP
ejpam-6066	89	21	the	the	DET
ejpam-6066	89	22	following	follow	VERB
ejpam-6066	89	23	set	set	NOUN
ejpam-6066	89	24	(	(	PUNCT
ejpam-6066	89	25	x	x	NOUN
ejpam-6066	89	26	,	,	PUNCT
ejpam-6066	89	27	y	y	PROPN
ejpam-6066	89	28	,	,	PUNCT
ejpam-6066	89	29	z	z	PROPN
ejpam-6066	89	30	,	,	PUNCT
ejpam-6066	89	31	p	p	NOUN
ejpam-6066	89	32	)	)	PUNCT
ejpam-6066	89	33	∈	∈	PROPN
ejpam-6066	89	34	{	{	PUNCT
ejpam-6066	89	35	(	(	PUNCT
ejpam-6066	89	36	0	0	NUM
ejpam-6066	89	37	,	,	PUNCT
ejpam-6066	89	38	1	1	NUM
ejpam-6066	89	39	,	,	PUNCT
ejpam-6066	89	40	1	1	NUM
ejpam-6066	89	41	,	,	PUNCT
ejpam-6066	89	42	3	3	NUM
ejpam-6066	89	43	)	)	PUNCT
ejpam-6066	89	44	,	,	PUNCT
ejpam-6066	89	45	(	(	PUNCT
ejpam-6066	89	46	1	1	NUM
ejpam-6066	89	47	,	,	PUNCT
ejpam-6066	89	48	1	1	NUM
ejpam-6066	89	49	,	,	PUNCT
ejpam-6066	89	50	5	5	NUM
ejpam-6066	89	51	,	,	PUNCT
ejpam-6066	89	52	3	3	NUM
ejpam-6066	89	53	)	)	PUNCT
ejpam-6066	89	54	,	,	PUNCT
ejpam-6066	89	55	(	(	PUNCT
ejpam-6066	89	56	1	1	NUM
ejpam-6066	89	57	,	,	PUNCT
ejpam-6066	89	58	3	3	NUM
ejpam-6066	89	59	,	,	PUNCT
ejpam-6066	89	60	1	1	NUM
ejpam-6066	89	61	,	,	PUNCT
ejpam-6066	89	62	3	3	NUM
ejpam-6066	89	63	)	)	PUNCT
ejpam-6066	89	64	,	,	PUNCT
ejpam-6066	89	65	(	(	PUNCT
ejpam-6066	89	66	2	2	NUM
ejpam-6066	89	67	,	,	PUNCT
ejpam-6066	89	68	3	3	NUM
ejpam-6066	89	69	,	,	PUNCT
ejpam-6066	89	70	13	13	NUM
ejpam-6066	89	71	,	,	PUNCT
ejpam-6066	89	72	3	3	NUM
ejpam-6066	89	73	)	)	PUNCT
ejpam-6066	89	74	,	,	PUNCT
ejpam-6066	89	75	(	(	PUNCT
ejpam-6066	89	76	3	3	NUM
ejpam-6066	89	77	,	,	PUNCT
ejpam-6066	89	78	1	1	NUM
ejpam-6066	89	79	,	,	PUNCT
ejpam-6066	89	80	37	37	NUM
ejpam-6066	89	81	,	,	PUNCT
ejpam-6066	89	82	3	3	NUM
ejpam-6066	89	83	)	)	PUNCT
ejpam-6066	89	84	}	}	PUNCT
ejpam-6066	89	85	∪	∪	X
ejpam-6066	89	86	{	{	PUNCT
ejpam-6066	89	87	(	(	PUNCT
ejpam-6066	89	88	2k	2k	NUM
ejpam-6066	89	89	+	+	CCONJ
ejpam-6066	89	90	1	1	NUM
ejpam-6066	89	91	,	,	PUNCT
ejpam-6066	89	92	4l	4l	NOUN
ejpam-6066	89	93	+	+	X
ejpam-6066	89	94	1	1	NUM
ejpam-6066	89	95	,	,	PUNCT
ejpam-6066	89	96	16m+	16m+	NUM
ejpam-6066	89	97	n	n	CCONJ
ejpam-6066	89	98	,	,	PUNCT
ejpam-6066	89	99	3)|	3)|	NUM
ejpam-6066	89	100	for	for	ADP
ejpam-6066	89	101	any	any	DET
ejpam-6066	89	102	integers	integer	NOUN
ejpam-6066	89	103	k	k	X
ejpam-6066	89	104	≥	≥	NUM
ejpam-6066	89	105	2	2	NUM
ejpam-6066	89	106	,	,	PUNCT
ejpam-6066	89	107	l	l	NOUN
ejpam-6066	89	108	,	,	PUNCT
ejpam-6066	89	109	m	m	PROPN
ejpam-6066	89	110	≥	≥	NOUN
ejpam-6066	89	111	0	0	NUM
ejpam-6066	89	112	and	and	CCONJ
ejpam-6066	89	113	n	n	CCONJ
ejpam-6066	89	114	=	=	SYM
ejpam-6066	89	115	3	3	NUM
ejpam-6066	89	116	,	,	PUNCT
ejpam-6066	89	117	5	5	NUM
ejpam-6066	89	118	,	,	PUNCT
ejpam-6066	89	119	11	11	NUM
ejpam-6066	89	120	,	,	PUNCT
ejpam-6066	89	121	13	13	NUM
ejpam-6066	89	122	}	}	PUNCT
ejpam-6066	89	123	∪	∪	X
ejpam-6066	89	124	{	{	PUNCT
ejpam-6066	89	125	(	(	PUNCT
ejpam-6066	89	126	2k	2k	NUM
ejpam-6066	89	127	+	+	CCONJ
ejpam-6066	89	128	1	1	NUM
ejpam-6066	89	129	,	,	PUNCT
ejpam-6066	89	130	4l	4l	NOUN
ejpam-6066	89	131	+	+	X
ejpam-6066	89	132	3	3	NUM
ejpam-6066	89	133	,	,	PUNCT
ejpam-6066	89	134	16m+	16m+	NUM
ejpam-6066	89	135	n	n	CCONJ
ejpam-6066	89	136	,	,	PUNCT
ejpam-6066	89	137	3)|	3)|	NUM
ejpam-6066	89	138	for	for	ADP
ejpam-6066	89	139	any	any	DET
ejpam-6066	89	140	integers	integer	NOUN
ejpam-6066	89	141	k	k	X
ejpam-6066	89	142	≥	≥	NUM
ejpam-6066	89	143	2	2	NUM
ejpam-6066	89	144	,	,	PUNCT
ejpam-6066	89	145	l	l	NOUN
ejpam-6066	89	146	,	,	PUNCT
ejpam-6066	89	147	m	m	PROPN
ejpam-6066	89	148	≥	≥	NOUN
ejpam-6066	89	149	0	0	NUM
ejpam-6066	89	150	and	and	CCONJ
ejpam-6066	89	151	n	n	CCONJ
ejpam-6066	89	152	=	=	SYM
ejpam-6066	89	153	1	1	NUM
ejpam-6066	89	154	,	,	PUNCT
ejpam-6066	89	155	7	7	NUM
ejpam-6066	89	156	,	,	PUNCT
ejpam-6066	89	157	9	9	NUM
ejpam-6066	89	158	,	,	PUNCT
ejpam-6066	89	159	15	15	NUM
ejpam-6066	89	160	}	}	PUNCT
ejpam-6066	89	161	.	.	PUNCT
ejpam-6066	90	1	(	(	PUNCT
ejpam-6066	90	2	iii	iii	X
ejpam-6066	90	3	)	)	PUNCT
ejpam-6066	90	4	there	there	PRON
ejpam-6066	90	5	is	be	VERB
ejpam-6066	90	6	no	no	DET
ejpam-6066	90	7	solution	solution	NOUN
ejpam-6066	90	8	for	for	ADP
ejpam-6066	90	9	any	any	DET
ejpam-6066	90	10	prime	prime	NOUN
ejpam-6066	90	11	p	p	NOUN
ejpam-6066	90	12	with	with	ADP
ejpam-6066	90	13	5	5	NUM
ejpam-6066	90	14	≤	≤	NOUN
ejpam-6066	90	15	p	p	NOUN
ejpam-6066	90	16	≤	≤	NUM
ejpam-6066	90	17	17	17	NUM
ejpam-6066	90	18	.	.	PUNCT
ejpam-6066	91	1	(	(	PUNCT
ejpam-6066	91	2	iv	iv	X
ejpam-6066	91	3	)	)	PUNCT
ejpam-6066	91	4	for	for	ADP
ejpam-6066	91	5	the	the	DET
ejpam-6066	91	6	prime	prime	NOUN
ejpam-6066	91	7	p	p	X
ejpam-6066	91	8	≥	≥	NUM
ejpam-6066	91	9	19	19	NUM
ejpam-6066	91	10	,	,	PUNCT
ejpam-6066	91	11	a	a	DET
ejpam-6066	91	12	necessary	necessary	ADJ
ejpam-6066	91	13	condition	condition	NOUN
ejpam-6066	91	14	for	for	ADP
ejpam-6066	91	15	the	the	DET
ejpam-6066	91	16	existence	existence	NOUN
ejpam-6066	91	17	of	of	ADP
ejpam-6066	91	18	solutions	solution	NOUN
ejpam-6066	91	19	is	be	AUX
ejpam-6066	91	20	that	that	SCONJ
ejpam-6066	91	21	p	p	PROPN
ejpam-6066	91	22	≡	≡	PROPN
ejpam-6066	91	23	19	19	NUM
ejpam-6066	91	24	(	(	PUNCT
ejpam-6066	91	25	mod	mod	NOUN
ejpam-6066	91	26	24	24	NUM
ejpam-6066	91	27	)	)	PUNCT
ejpam-6066	91	28	.	.	PUNCT
ejpam-6066	92	1	in	in	ADP
ejpam-6066	92	2	such	such	ADJ
ejpam-6066	92	3	cases	case	NOUN
ejpam-6066	92	4	,	,	PUNCT
ejpam-6066	92	5	all	all	DET
ejpam-6066	92	6	solutions	solution	NOUN
ejpam-6066	92	7	must	must	AUX
ejpam-6066	92	8	satisfy	satisfy	VERB
ejpam-6066	92	9	:	:	PUNCT
ejpam-6066	92	10	(	(	PUNCT
ejpam-6066	92	11	x	x	X
ejpam-6066	92	12	,	,	PUNCT
ejpam-6066	92	13	y	y	PROPN
ejpam-6066	92	14	,	,	PUNCT
ejpam-6066	92	15	z	z	PROPN
ejpam-6066	92	16	,	,	PUNCT
ejpam-6066	92	17	p	p	NOUN
ejpam-6066	92	18	)	)	PUNCT
ejpam-6066	92	19	∈	∈	PROPN
ejpam-6066	92	20	{	{	PUNCT
ejpam-6066	92	21	(	(	PUNCT
ejpam-6066	92	22	2k+1	2k+1	PROPN
ejpam-6066	92	23	,	,	PUNCT
ejpam-6066	92	24	2l+1	2l+1	PROPN
ejpam-6066	92	25	,	,	PUNCT
ejpam-6066	92	26	24m+n	24m+n	NUM
ejpam-6066	92	27	,	,	PUNCT
ejpam-6066	92	28	24r+19)|	24r+19)|	NUM
ejpam-6066	92	29	for	for	ADP
ejpam-6066	92	30	any	any	DET
ejpam-6066	92	31	integers	integer	NOUN
ejpam-6066	92	32	k	k	NOUN
ejpam-6066	92	33	,	,	PUNCT
ejpam-6066	92	34	l	l	NOUN
ejpam-6066	92	35	,	,	PUNCT
ejpam-6066	92	36	m	m	PROPN
ejpam-6066	92	37	,	,	PUNCT
ejpam-6066	92	38	r	r	NOUN
ejpam-6066	92	39	≥	≥	NOUN
ejpam-6066	92	40	0	0	NUM
ejpam-6066	92	41	and	and	CCONJ
ejpam-6066	92	42	n	n	CCONJ
ejpam-6066	92	43	=	=	SYM
ejpam-6066	92	44	3	3	NUM
ejpam-6066	92	45	,	,	PUNCT
ejpam-6066	92	46	9	9	NUM
ejpam-6066	92	47	,	,	PUNCT
ejpam-6066	92	48	15	15	NUM
ejpam-6066	92	49	,	,	PUNCT
ejpam-6066	92	50	21	21	NUM
ejpam-6066	92	51	}	}	PUNCT
ejpam-6066	92	52	.	.	PUNCT
ejpam-6066	93	1	proof	proof	NOUN
ejpam-6066	93	2	.	.	PUNCT
ejpam-6066	94	1	we	we	PRON
ejpam-6066	94	2	begin	begin	VERB
ejpam-6066	94	3	by	by	ADP
ejpam-6066	94	4	examining	examine	VERB
ejpam-6066	94	5	the	the	DET
ejpam-6066	94	6	degenerate	degenerate	ADJ
ejpam-6066	94	7	cases	case	NOUN
ejpam-6066	94	8	x	x	PUNCT
ejpam-6066	94	9	=	=	SYM
ejpam-6066	94	10	0	0	NUM
ejpam-6066	94	11	and	and	CCONJ
ejpam-6066	94	12	y	y	PROPN
ejpam-6066	94	13	≥	≥	PROPN
ejpam-6066	94	14	0	0	NUM
ejpam-6066	94	15	.	.	PUNCT
ejpam-6066	95	1	in	in	ADP
ejpam-6066	95	2	this	this	DET
ejpam-6066	95	3	case	case	NOUN
ejpam-6066	95	4	,	,	PUNCT
ejpam-6066	95	5	direct	direct	ADJ
ejpam-6066	95	6	computation	computation	NOUN
ejpam-6066	95	7	shows	show	VERB
ejpam-6066	95	8	that	that	SCONJ
ejpam-6066	95	9	(	(	PUNCT
ejpam-6066	95	10	x	x	X
ejpam-6066	95	11	,	,	PUNCT
ejpam-6066	95	12	y	y	PROPN
ejpam-6066	95	13	,	,	PUNCT
ejpam-6066	95	14	z	z	PROPN
ejpam-6066	95	15	,	,	PUNCT
ejpam-6066	95	16	p	p	NOUN
ejpam-6066	95	17	)	)	PUNCT
ejpam-6066	95	18	=	=	SYM
ejpam-6066	95	19	(	(	PUNCT
ejpam-6066	95	20	0	0	NUM
ejpam-6066	95	21	,	,	PUNCT
ejpam-6066	95	22	1	1	NUM
ejpam-6066	95	23	,	,	PUNCT
ejpam-6066	95	24	1	1	NUM
ejpam-6066	95	25	,	,	PUNCT
ejpam-6066	95	26	3	3	NUM
ejpam-6066	95	27	)	)	PUNCT
ejpam-6066	95	28	and	and	CCONJ
ejpam-6066	95	29	(	(	PUNCT
ejpam-6066	95	30	0	0	NUM
ejpam-6066	95	31	,	,	PUNCT
ejpam-6066	95	32	2	2	NUM
ejpam-6066	95	33	,	,	PUNCT
ejpam-6066	95	34	0	0	NUM
ejpam-6066	95	35	,	,	PUNCT
ejpam-6066	95	36	2	2	NUM
ejpam-6066	95	37	)	)	PUNCT
ejpam-6066	95	38	are	be	AUX
ejpam-6066	95	39	valid	valid	ADJ
ejpam-6066	95	40	solutions	solution	NOUN
ejpam-6066	95	41	,	,	PUNCT
ejpam-6066	95	42	as	as	ADP
ejpam-6066	95	43	the	the	DET
ejpam-6066	95	44	expression	expression	NOUN
ejpam-6066	95	45	4(1)−py	4(1)−py	NUM
ejpam-6066	96	1	=	=	SYM
ejpam-6066	96	2	z2	z2	NOUN
ejpam-6066	96	3	yields	yield	VERB
ejpam-6066	96	4	a	a	DET
ejpam-6066	96	5	non	non	ADJ
ejpam-6066	96	6	-	-	ADJ
ejpam-6066	96	7	negative	negative	ADJ
ejpam-6066	96	8	perfect	perfect	ADJ
ejpam-6066	96	9	square	square	NOUN
ejpam-6066	96	10	.	.	PUNCT
ejpam-6066	97	1	next	next	ADV
ejpam-6066	97	2	,	,	PUNCT
ejpam-6066	97	3	suppose	suppose	VERB
ejpam-6066	97	4	x	x	X
ejpam-6066	97	5	≥	≥	NUM
ejpam-6066	97	6	1	1	NUM
ejpam-6066	97	7	and	and	CCONJ
ejpam-6066	97	8	y	y	PROPN
ejpam-6066	97	9	=	=	SYM
ejpam-6066	97	10	0	0	PROPN
ejpam-6066	97	11	.	.	PUNCT
ejpam-6066	98	1	then	then	ADV
ejpam-6066	98	2	the	the	DET
ejpam-6066	98	3	equation	equation	NOUN
ejpam-6066	98	4	becomes	become	VERB
ejpam-6066	98	5	4(7x)−1	4(7x)−1	NOUN
ejpam-6066	98	6	=	=	SYM
ejpam-6066	98	7	z2	z2	PROPN
ejpam-6066	98	8	,	,	PUNCT
ejpam-6066	98	9	which	which	PRON
ejpam-6066	98	10	is	be	AUX
ejpam-6066	98	11	impossible	impossible	ADJ
ejpam-6066	98	12	modulo	modulo	NOUN
ejpam-6066	98	13	4	4	NUM
ejpam-6066	98	14	since	since	SCONJ
ejpam-6066	98	15	z2	z2	PROPN
ejpam-6066	98	16	≡	≡	PROPN
ejpam-6066	98	17	0	0	NUM
ejpam-6066	98	18	or	or	CCONJ
ejpam-6066	98	19	1	1	NUM
ejpam-6066	98	20	(	(	PUNCT
ejpam-6066	98	21	mod	mod	NOUN
ejpam-6066	98	22	4	4	NUM
ejpam-6066	98	23	)	)	PUNCT
ejpam-6066	98	24	,	,	PUNCT
ejpam-6066	98	25	but	but	CCONJ
ejpam-6066	98	26	4(7x)−	4(7x)−	PROPN
ejpam-6066	98	27	1	1	NUM
ejpam-6066	98	28	≡	≡	PROPN
ejpam-6066	98	29	0−	0−	NUM
ejpam-6066	98	30	1	1	NUM
ejpam-6066	98	31	≡	≡	PROPN
ejpam-6066	98	32	3	3	NUM
ejpam-6066	98	33	(	(	PUNCT
ejpam-6066	98	34	mod	mod	NOUN
ejpam-6066	98	35	4	4	NUM
ejpam-6066	98	36	)	)	PUNCT
ejpam-6066	98	37	.	.	PUNCT
ejpam-6066	99	1	hence	hence	ADV
ejpam-6066	99	2	,	,	PUNCT
ejpam-6066	99	3	no	no	DET
ejpam-6066	99	4	solution	solution	NOUN
ejpam-6066	99	5	exists	exist	VERB
ejpam-6066	99	6	in	in	ADP
ejpam-6066	99	7	this	this	DET
ejpam-6066	99	8	case	case	NOUN
ejpam-6066	99	9	.	.	PUNCT
ejpam-6066	100	1	we	we	PRON
ejpam-6066	100	2	now	now	ADV
ejpam-6066	100	3	focus	focus	VERB
ejpam-6066	100	4	on	on	ADP
ejpam-6066	100	5	the	the	DET
ejpam-6066	100	6	remaining	remain	VERB
ejpam-6066	100	7	case	case	NOUN
ejpam-6066	100	8	where	where	SCONJ
ejpam-6066	100	9	both	both	DET
ejpam-6066	100	10	x	x	X
ejpam-6066	100	11	and	and	CCONJ
ejpam-6066	100	12	y	y	PROPN
ejpam-6066	100	13	are	be	AUX
ejpam-6066	100	14	positive	positive	ADJ
ejpam-6066	100	15	.	.	PUNCT
ejpam-6066	101	1	to	to	PART
ejpam-6066	101	2	proceed	proceed	VERB
ejpam-6066	101	3	,	,	PUNCT
ejpam-6066	101	4	we	we	PRON
ejpam-6066	101	5	consider	consider	VERB
ejpam-6066	101	6	three	three	NUM
ejpam-6066	101	7	cases	case	NOUN
ejpam-6066	101	8	based	base	VERB
ejpam-6066	101	9	on	on	ADP
ejpam-6066	101	10	the	the	DET
ejpam-6066	101	11	value	value	NOUN
ejpam-6066	101	12	of	of	ADP
ejpam-6066	101	13	the	the	DET
ejpam-6066	101	14	prime	prime	ADJ
ejpam-6066	101	15	p	p	X
ejpam-6066	101	16	:	:	PUNCT
ejpam-6066	101	17	k.	k.	PROPN
ejpam-6066	101	18	laipaporn	laipaporn	PROPN
ejpam-6066	101	19	et	et	PROPN
ejpam-6066	101	20	al	al	PROPN
ejpam-6066	101	21	.	.	PUNCT
ejpam-6066	101	22	/	/	SYM
ejpam-6066	101	23	eur	eur	PROPN
ejpam-6066	101	24	.	.	PUNCT
ejpam-6066	102	1	j.	j.	PROPN
ejpam-6066	102	2	pure	pure	PROPN
ejpam-6066	102	3	appl	appl	PROPN
ejpam-6066	102	4	.	.	PROPN
ejpam-6066	102	5	math	math	PROPN
ejpam-6066	102	6	,	,	PUNCT
ejpam-6066	102	7	18	18	NUM
ejpam-6066	102	8	(	(	PUNCT
ejpam-6066	102	9	3	3	NUM
ejpam-6066	102	10	)	)	PUNCT
ejpam-6066	102	11	(	(	PUNCT
ejpam-6066	102	12	2025	2025	NUM
ejpam-6066	102	13	)	)	PUNCT
ejpam-6066	102	14	,	,	PUNCT
ejpam-6066	102	15	6066	6066	NUM
ejpam-6066	102	16	5	5	NUM
ejpam-6066	102	17	of	of	ADP
ejpam-6066	102	18	13	13	NUM
ejpam-6066	102	19	case	case	NOUN
ejpam-6066	102	20	1	1	NUM
ejpam-6066	102	21	p	p	NOUN
ejpam-6066	102	22	=	=	NOUN
ejpam-6066	102	23	2	2	X
ejpam-6066	102	24	.	.	PUNCT
ejpam-6066	103	1	then	then	ADV
ejpam-6066	103	2	4(7x)−	4(7x)−	PROPN
ejpam-6066	103	3	2y	2y	PROPN
ejpam-6066	103	4	≡	≡	PROPN
ejpam-6066	103	5			PROPN
ejpam-6066	103	6	0−	0−	NUM
ejpam-6066	103	7	1	1	NUM
ejpam-6066	103	8	≡	≡	PROPN
ejpam-6066	103	9	6	6	NUM
ejpam-6066	103	10	(	(	PUNCT
ejpam-6066	103	11	mod	mod	PROPN
ejpam-6066	103	12	7	7	NUM
ejpam-6066	103	13	)	)	PUNCT
ejpam-6066	103	14	if	if	SCONJ
ejpam-6066	103	15	y	y	PROPN
ejpam-6066	103	16	≡	≡	PROPN
ejpam-6066	103	17	0	0	PUNCT
ejpam-6066	104	1	(	(	PUNCT
ejpam-6066	104	2	mod	mod	NOUN
ejpam-6066	104	3	3	3	NUM
ejpam-6066	104	4	)	)	PUNCT
ejpam-6066	104	5	,	,	PUNCT
ejpam-6066	104	6	0−	0−	NUM
ejpam-6066	104	7	2	2	NUM
ejpam-6066	104	8	≡	≡	PROPN
ejpam-6066	104	9	5	5	NUM
ejpam-6066	104	10	(	(	PUNCT
ejpam-6066	104	11	mod	mod	NOUN
ejpam-6066	104	12	7	7	NUM
ejpam-6066	104	13	)	)	PUNCT
ejpam-6066	104	14	if	if	SCONJ
ejpam-6066	104	15	y	y	PROPN
ejpam-6066	104	16	≡	≡	PROPN
ejpam-6066	104	17	1	1	NUM
ejpam-6066	104	18	(	(	PUNCT
ejpam-6066	104	19	mod	mod	NOUN
ejpam-6066	104	20	3	3	NUM
ejpam-6066	104	21	)	)	PUNCT
ejpam-6066	104	22	,	,	PUNCT
ejpam-6066	104	23	0−	0−	NUM
ejpam-6066	104	24	4	4	NUM
ejpam-6066	104	25	≡	≡	PROPN
ejpam-6066	104	26	3	3	NUM
ejpam-6066	104	27	(	(	PUNCT
ejpam-6066	104	28	mod	mod	PROPN
ejpam-6066	104	29	7	7	NUM
ejpam-6066	104	30	)	)	PUNCT
ejpam-6066	104	31	if	if	SCONJ
ejpam-6066	104	32	y	y	PROPN
ejpam-6066	104	33	≡	≡	PROPN
ejpam-6066	104	34	2	2	NUM
ejpam-6066	104	35	(	(	PUNCT
ejpam-6066	104	36	mod	mod	NOUN
ejpam-6066	104	37	3	3	NUM
ejpam-6066	104	38	)	)	PUNCT
ejpam-6066	104	39	.	.	PUNCT
ejpam-6066	105	1	however	however	ADV
ejpam-6066	105	2	,	,	PUNCT
ejpam-6066	105	3	for	for	ADP
ejpam-6066	105	4	any	any	DET
ejpam-6066	105	5	integer	integer	NOUN
ejpam-6066	105	6	z	z	PROPN
ejpam-6066	105	7	,	,	PUNCT
ejpam-6066	105	8	z2	z2	PROPN
ejpam-6066	105	9	≡	≡	PROPN
ejpam-6066	105	10	0	0	NUM
ejpam-6066	105	11	,	,	PUNCT
ejpam-6066	105	12	1	1	NUM
ejpam-6066	105	13	,	,	PUNCT
ejpam-6066	105	14	2	2	NUM
ejpam-6066	105	15	,	,	PUNCT
ejpam-6066	105	16	4	4	NUM
ejpam-6066	105	17	(	(	PUNCT
ejpam-6066	105	18	mod	mod	PROPN
ejpam-6066	105	19	7	7	NUM
ejpam-6066	105	20	)	)	PUNCT
ejpam-6066	105	21	so	so	ADV
ejpam-6066	105	22	no	no	DET
ejpam-6066	105	23	solution	solution	NOUN
ejpam-6066	105	24	is	be	AUX
ejpam-6066	105	25	possible	possible	ADJ
ejpam-6066	105	26	in	in	ADP
ejpam-6066	105	27	this	this	DET
ejpam-6066	105	28	case	case	NOUN
ejpam-6066	105	29	.	.	PUNCT
ejpam-6066	106	1	case	case	NOUN
ejpam-6066	106	2	2	2	NUM
ejpam-6066	106	3	p	p	NOUN
ejpam-6066	106	4	=	=	NOUN
ejpam-6066	106	5	3	3	X
ejpam-6066	106	6	.	.	PUNCT
ejpam-6066	107	1	since	since	SCONJ
ejpam-6066	107	2	z	z	PROPN
ejpam-6066	107	3	is	be	AUX
ejpam-6066	107	4	odd	odd	ADJ
ejpam-6066	107	5	and	and	CCONJ
ejpam-6066	107	6	not	not	PART
ejpam-6066	107	7	divisible	divisible	ADJ
ejpam-6066	107	8	by	by	ADP
ejpam-6066	107	9	3	3	NUM
ejpam-6066	107	10	,	,	PUNCT
ejpam-6066	107	11	it	it	PRON
ejpam-6066	107	12	follows	follow	VERB
ejpam-6066	107	13	that	that	SCONJ
ejpam-6066	107	14	z2	z2	PROPN
ejpam-6066	107	15	≡	≡	PROPN
ejpam-6066	107	16	1	1	NUM
ejpam-6066	107	17	(	(	PUNCT
ejpam-6066	107	18	mod	mod	NOUN
ejpam-6066	107	19	24	24	NUM
ejpam-6066	107	20	)	)	PUNCT
ejpam-6066	107	21	.	.	PUNCT
ejpam-6066	108	1	observe	observe	VERB
ejpam-6066	108	2	that	that	SCONJ
ejpam-6066	108	3	for	for	ADP
ejpam-6066	108	4	any	any	DET
ejpam-6066	108	5	x	x	SYM
ejpam-6066	108	6	>	>	X
ejpam-6066	108	7	0	0	NUM
ejpam-6066	108	8	,	,	PUNCT
ejpam-6066	108	9	we	we	PRON
ejpam-6066	108	10	have	have	VERB
ejpam-6066	108	11	4(7x)−	4(7x)−	PROPN
ejpam-6066	108	12	3y	3y	NUM
ejpam-6066	108	13	≡	≡	PROPN
ejpam-6066	108	14	{	{	PUNCT
ejpam-6066	108	15	4−	4−	NOUN
ejpam-6066	108	16	3	3	NUM
ejpam-6066	108	17	≡	≡	PROPN
ejpam-6066	108	18	1	1	NUM
ejpam-6066	108	19	(	(	PUNCT
ejpam-6066	108	20	mod	mod	NOUN
ejpam-6066	108	21	24	24	NUM
ejpam-6066	108	22	)	)	PUNCT
ejpam-6066	108	23	if	if	SCONJ
ejpam-6066	108	24	y	y	PROPN
ejpam-6066	108	25	is	be	AUX
ejpam-6066	108	26	odd	odd	ADJ
ejpam-6066	108	27	,	,	PUNCT
ejpam-6066	108	28	4−	4−	PROPN
ejpam-6066	108	29	9	9	NUM
ejpam-6066	108	30	≡	≡	PROPN
ejpam-6066	108	31	19	19	NUM
ejpam-6066	108	32	(	(	PUNCT
ejpam-6066	108	33	mod	mod	NOUN
ejpam-6066	108	34	24	24	NUM
ejpam-6066	108	35	)	)	PUNCT
ejpam-6066	108	36	if	if	SCONJ
ejpam-6066	108	37	y	y	PROPN
ejpam-6066	108	38	is	be	AUX
ejpam-6066	108	39	even	even	ADV
ejpam-6066	108	40	.	.	PUNCT
ejpam-6066	109	1	this	this	PRON
ejpam-6066	109	2	contradicts	contradict	VERB
ejpam-6066	109	3	the	the	DET
ejpam-6066	109	4	fact	fact	NOUN
ejpam-6066	109	5	z2	z2	PROPN
ejpam-6066	109	6	≡	≡	PROPN
ejpam-6066	109	7	1	1	NUM
ejpam-6066	109	8	(	(	PUNCT
ejpam-6066	109	9	mod	mod	NOUN
ejpam-6066	109	10	24	24	NUM
ejpam-6066	109	11	)	)	PUNCT
ejpam-6066	109	12	when	when	SCONJ
ejpam-6066	109	13	y	y	PROPN
ejpam-6066	109	14	is	be	AUX
ejpam-6066	109	15	even	even	ADV
ejpam-6066	109	16	,	,	PUNCT
ejpam-6066	109	17	so	so	ADV
ejpam-6066	109	18	we	we	PRON
ejpam-6066	109	19	only	only	ADV
ejpam-6066	109	20	consider	consider	VERB
ejpam-6066	109	21	the	the	DET
ejpam-6066	109	22	case	case	NOUN
ejpam-6066	109	23	when	when	SCONJ
ejpam-6066	109	24	y	y	PROPN
ejpam-6066	109	25	is	be	AUX
ejpam-6066	109	26	an	an	DET
ejpam-6066	109	27	odd	odd	ADJ
ejpam-6066	109	28	number	number	NOUN
ejpam-6066	109	29	.	.	PUNCT
ejpam-6066	110	1	subcase	subcase	NOUN
ejpam-6066	110	2	2.1	2.1	NUM
ejpam-6066	110	3	x	x	SYM
ejpam-6066	110	4	=	=	PUNCT
ejpam-6066	110	5	2k	2k	NUM
ejpam-6066	110	6	for	for	ADP
ejpam-6066	110	7	some	some	DET
ejpam-6066	110	8	k	k	PROPN
ejpam-6066	110	9	≥	≥	NUM
ejpam-6066	110	10	1	1	NUM
ejpam-6066	110	11	.	.	PUNCT
ejpam-6066	111	1	in	in	ADP
ejpam-6066	111	2	this	this	DET
ejpam-6066	111	3	case	case	NOUN
ejpam-6066	111	4	,	,	PUNCT
ejpam-6066	111	5	the	the	DET
ejpam-6066	111	6	equation	equation	NOUN
ejpam-6066	111	7	4(7x)−	4(7x)−	PROPN
ejpam-6066	111	8	py	py	PROPN
ejpam-6066	111	9	=	=	PROPN
ejpam-6066	111	10	z2	z2	PROPN
ejpam-6066	111	11	becomes	become	VERB
ejpam-6066	111	12	3y	3y	NUM
ejpam-6066	111	13	=	=	SYM
ejpam-6066	111	14	(	(	PUNCT
ejpam-6066	111	15	2(7k)−	2(7k)−	PROPN
ejpam-6066	111	16	z)(2(7k	z)(2(7k	NOUN
ejpam-6066	111	17	)	)	PUNCT
ejpam-6066	111	18	+	+	SYM
ejpam-6066	112	1	z	z	X
ejpam-6066	112	2	)	)	PUNCT
ejpam-6066	112	3	,	,	PUNCT
ejpam-6066	112	4	which	which	PRON
ejpam-6066	112	5	is	be	AUX
ejpam-6066	112	6	factorization	factorization	NOUN
ejpam-6066	112	7	of	of	ADP
ejpam-6066	112	8	3y	3y	NUM
ejpam-6066	112	9	into	into	ADP
ejpam-6066	112	10	two	two	NUM
ejpam-6066	112	11	positive	positive	ADJ
ejpam-6066	112	12	integers	integer	NOUN
ejpam-6066	112	13	.	.	PUNCT
ejpam-6066	113	1	let	let	VERB
ejpam-6066	113	2	3u	3u	NUM
ejpam-6066	113	3	=	=	SYM
ejpam-6066	113	4	2(7k	2(7k	NUM
ejpam-6066	113	5	)	)	PUNCT
ejpam-6066	113	6	−	−	PROPN
ejpam-6066	113	7	z	z	NOUN
ejpam-6066	113	8	and	and	CCONJ
ejpam-6066	113	9	3y−u	3y−u	NUM
ejpam-6066	113	10	=	=	SYM
ejpam-6066	113	11	2(7k	2(7k	NUM
ejpam-6066	113	12	)	)	PUNCT
ejpam-6066	114	1	+	+	CCONJ
ejpam-6066	114	2	z	z	NOUN
ejpam-6066	114	3	for	for	ADP
ejpam-6066	114	4	some	some	DET
ejpam-6066	114	5	0	0	NUM
ejpam-6066	114	6	≤	≤	NUM
ejpam-6066	114	7	u	u	NOUN
ejpam-6066	114	8	<	<	X
ejpam-6066	114	9	y.	y.	NOUN
ejpam-6066	114	10	adding	add	VERB
ejpam-6066	114	11	these	these	DET
ejpam-6066	114	12	two	two	NUM
ejpam-6066	114	13	expressions	expression	NOUN
ejpam-6066	114	14	yields	yield	NOUN
ejpam-6066	114	15	4(7k	4(7k	NOUN
ejpam-6066	114	16	)	)	PUNCT
ejpam-6066	114	17	=	=	NOUN
ejpam-6066	115	1	3(3u−1	3(3u−1	NUM
ejpam-6066	115	2	+	+	NUM
ejpam-6066	115	3	3y−u−1	3y−u−1	NUM
ejpam-6066	115	4	)	)	PUNCT
ejpam-6066	115	5	.	.	PUNCT
ejpam-6066	116	1	if	if	SCONJ
ejpam-6066	116	2	0	0	NUM
ejpam-6066	116	3	<	<	X
ejpam-6066	116	4	u	u	X
ejpam-6066	116	5	<	<	X
ejpam-6066	116	6	y	y	PROPN
ejpam-6066	116	7	,	,	PUNCT
ejpam-6066	116	8	then	then	ADV
ejpam-6066	116	9	the	the	DET
ejpam-6066	116	10	right	right	ADJ
ejpam-6066	116	11	-	-	PUNCT
ejpam-6066	116	12	hand	hand	NOUN
ejpam-6066	116	13	side	side	NOUN
ejpam-6066	116	14	is	be	AUX
ejpam-6066	116	15	divisible	divisible	ADJ
ejpam-6066	116	16	by	by	ADP
ejpam-6066	116	17	3	3	NUM
ejpam-6066	116	18	while	while	SCONJ
ejpam-6066	116	19	the	the	DET
ejpam-6066	116	20	left	leave	VERB
ejpam-6066	116	21	-	-	PUNCT
ejpam-6066	116	22	hand	hand	NOUN
ejpam-6066	116	23	side	side	NOUN
ejpam-6066	116	24	is	be	AUX
ejpam-6066	116	25	not	not	PART
ejpam-6066	116	26	,	,	PUNCT
ejpam-6066	116	27	giving	give	VERB
ejpam-6066	116	28	a	a	DET
ejpam-6066	116	29	contradiction	contradiction	NOUN
ejpam-6066	116	30	.	.	PUNCT
ejpam-6066	117	1	hence	hence	ADV
ejpam-6066	117	2	,	,	PUNCT
ejpam-6066	117	3	the	the	DET
ejpam-6066	117	4	only	only	ADJ
ejpam-6066	117	5	possibility	possibility	NOUN
ejpam-6066	117	6	is	be	AUX
ejpam-6066	117	7	u	u	NOUN
ejpam-6066	117	8	=	=	PROPN
ejpam-6066	117	9	0	0	NUM
ejpam-6066	117	10	,	,	PUNCT
ejpam-6066	117	11	leading	lead	VERB
ejpam-6066	117	12	to	to	ADP
ejpam-6066	117	13	1	1	NUM
ejpam-6066	117	14	=	=	SYM
ejpam-6066	117	15	2(7k)−	2(7k)−	PROPN
ejpam-6066	117	16	z	z	NOUN
ejpam-6066	117	17	and	and	CCONJ
ejpam-6066	117	18	3y	3y	NUM
ejpam-6066	117	19	=	=	SYM
ejpam-6066	117	20	2(7k	2(7k	NUM
ejpam-6066	117	21	)	)	PUNCT
ejpam-6066	117	22	+	+	CCONJ
ejpam-6066	118	1	z.	z.	PROPN
ejpam-6066	118	2	substituting	substitute	VERB
ejpam-6066	118	3	the	the	DET
ejpam-6066	118	4	first	first	ADJ
ejpam-6066	118	5	equation	equation	NOUN
ejpam-6066	118	6	into	into	ADP
ejpam-6066	118	7	the	the	DET
ejpam-6066	118	8	second	second	NOUN
ejpam-6066	118	9	gives	give	VERB
ejpam-6066	118	10	3y	3y	NUM
ejpam-6066	118	11	=	=	SYM
ejpam-6066	118	12	4(7k)−1	4(7k)−1	NOUN
ejpam-6066	118	13	.	.	PUNCT
ejpam-6066	119	1	we	we	PRON
ejpam-6066	119	2	now	now	ADV
ejpam-6066	119	3	consider	consider	VERB
ejpam-6066	119	4	whether	whether	SCONJ
ejpam-6066	119	5	k	k	PROPN
ejpam-6066	119	6	is	be	AUX
ejpam-6066	119	7	even	even	ADV
ejpam-6066	119	8	or	or	CCONJ
ejpam-6066	119	9	odd	odd	ADJ
ejpam-6066	119	10	.	.	PUNCT
ejpam-6066	120	1	if	if	SCONJ
ejpam-6066	120	2	k	k	PROPN
ejpam-6066	120	3	=	=	PUNCT
ejpam-6066	120	4	2l	2l	PROPN
ejpam-6066	120	5	for	for	ADP
ejpam-6066	120	6	some	some	DET
ejpam-6066	120	7	l	l	NOUN
ejpam-6066	120	8	≥	≥	NOUN
ejpam-6066	120	9	1	1	NUM
ejpam-6066	120	10	.	.	PUNCT
ejpam-6066	121	1	then	then	ADV
ejpam-6066	121	2	we	we	PRON
ejpam-6066	121	3	obtain	obtain	VERB
ejpam-6066	121	4	3y	3y	NOUN
ejpam-6066	121	5	=	=	SYM
ejpam-6066	121	6	4(72l	4(72l	NUM
ejpam-6066	121	7	)	)	PUNCT
ejpam-6066	121	8	−	−	PROPN
ejpam-6066	122	1	1	1	NUM
ejpam-6066	122	2	=	=	SYM
ejpam-6066	122	3	(	(	PUNCT
ejpam-6066	122	4	2(7l)−	2(7l)−	NUM
ejpam-6066	122	5	1)(2(7l	1)(2(7l	NUM
ejpam-6066	122	6	)	)	PUNCT
ejpam-6066	123	1	+	+	CCONJ
ejpam-6066	123	2	1	1	NUM
ejpam-6066	123	3	)	)	PUNCT
ejpam-6066	123	4	.	.	PUNCT
ejpam-6066	124	1	since	since	SCONJ
ejpam-6066	124	2	l	l	PROPN
ejpam-6066	124	3	̸=	̸=	PROPN
ejpam-6066	124	4	0	0	NUM
ejpam-6066	124	5	,	,	PUNCT
ejpam-6066	124	6	both	both	DET
ejpam-6066	124	7	factors	factor	NOUN
ejpam-6066	124	8	are	be	AUX
ejpam-6066	124	9	greater	great	ADJ
ejpam-6066	124	10	than	than	ADP
ejpam-6066	124	11	1	1	NUM
ejpam-6066	124	12	and	and	CCONJ
ejpam-6066	124	13	differ	differ	VERB
ejpam-6066	124	14	by	by	ADP
ejpam-6066	124	15	2	2	NUM
ejpam-6066	124	16	,	,	PUNCT
ejpam-6066	124	17	so	so	ADV
ejpam-6066	124	18	both	both	PRON
ejpam-6066	124	19	must	must	AUX
ejpam-6066	124	20	be	be	AUX
ejpam-6066	124	21	divisible	divisible	ADJ
ejpam-6066	124	22	by	by	ADP
ejpam-6066	124	23	3	3	NUM
ejpam-6066	124	24	,	,	PUNCT
ejpam-6066	124	25	which	which	PRON
ejpam-6066	124	26	is	be	AUX
ejpam-6066	124	27	impossible	impossible	ADJ
ejpam-6066	124	28	since	since	SCONJ
ejpam-6066	124	29	3	3	NUM
ejpam-6066	124	30	does	do	AUX
ejpam-6066	124	31	not	not	PART
ejpam-6066	124	32	divide	divide	VERB
ejpam-6066	124	33	2	2	NUM
ejpam-6066	124	34	.	.	PUNCT
ejpam-6066	125	1	therefore	therefore	ADV
ejpam-6066	125	2	,	,	PUNCT
ejpam-6066	125	3	no	no	DET
ejpam-6066	125	4	solution	solution	NOUN
ejpam-6066	125	5	exists	exist	VERB
ejpam-6066	125	6	in	in	ADP
ejpam-6066	125	7	the	the	DET
ejpam-6066	125	8	case	case	NOUN
ejpam-6066	125	9	k	k	NOUN
ejpam-6066	125	10	is	be	AUX
ejpam-6066	125	11	even	even	ADV
ejpam-6066	125	12	.	.	PUNCT
ejpam-6066	126	1	next	next	ADV
ejpam-6066	126	2	,	,	PUNCT
ejpam-6066	126	3	we	we	PRON
ejpam-6066	126	4	suppose	suppose	VERB
ejpam-6066	126	5	k	k	X
ejpam-6066	126	6	=	=	SYM
ejpam-6066	126	7	2h+1	2h+1	PROPN
ejpam-6066	126	8	for	for	ADP
ejpam-6066	126	9	some	some	DET
ejpam-6066	126	10	h	h	NOUN
ejpam-6066	126	11	≥	≥	NOUN
ejpam-6066	126	12	0	0	NUM
ejpam-6066	126	13	.	.	PUNCT
ejpam-6066	127	1	then	then	ADV
ejpam-6066	127	2	we	we	PRON
ejpam-6066	127	3	have	have	VERB
ejpam-6066	127	4	3y	3y	NUM
ejpam-6066	127	5	=	=	SYM
ejpam-6066	127	6	4(74h+2)−1	4(74h+2)−1	X
ejpam-6066	127	7	=	=	SYM
ejpam-6066	127	8	(	(	PUNCT
ejpam-6066	127	9	2(72h+1)−1)(2(72h+1)+1	2(72h+1)−1)(2(72h+1)+1	NOUN
ejpam-6066	127	10	)	)	PUNCT
ejpam-6066	127	11	,	,	PUNCT
ejpam-6066	127	12	which	which	PRON
ejpam-6066	127	13	again	again	ADV
ejpam-6066	127	14	leads	lead	VERB
ejpam-6066	127	15	to	to	ADP
ejpam-6066	127	16	a	a	DET
ejpam-6066	127	17	contradiction	contradiction	NOUN
ejpam-6066	127	18	unless	unless	SCONJ
ejpam-6066	127	19	h	h	NOUN
ejpam-6066	127	20	=	=	NOUN
ejpam-6066	127	21	0	0	NUM
ejpam-6066	127	22	,	,	PUNCT
ejpam-6066	127	23	i.e.	i.e.	X
ejpam-6066	127	24	,	,	PUNCT
ejpam-6066	127	25	x	x	SYM
ejpam-6066	127	26	=	=	SYM
ejpam-6066	127	27	2	2	NUM
ejpam-6066	127	28	.	.	PUNCT
ejpam-6066	128	1	hence	hence	ADV
ejpam-6066	128	2	,	,	PUNCT
ejpam-6066	128	3	the	the	DET
ejpam-6066	128	4	only	only	ADJ
ejpam-6066	128	5	solution	solution	NOUN
ejpam-6066	128	6	in	in	ADP
ejpam-6066	128	7	this	this	DET
ejpam-6066	128	8	case	case	NOUN
ejpam-6066	128	9	(	(	PUNCT
ejpam-6066	128	10	x	x	X
ejpam-6066	128	11	,	,	PUNCT
ejpam-6066	128	12	y	y	PROPN
ejpam-6066	128	13	,	,	PUNCT
ejpam-6066	128	14	z	z	NOUN
ejpam-6066	128	15	)	)	PUNCT
ejpam-6066	128	16	=	=	SYM
ejpam-6066	128	17	(	(	PUNCT
ejpam-6066	128	18	2	2	NUM
ejpam-6066	128	19	,	,	PUNCT
ejpam-6066	128	20	3	3	NUM
ejpam-6066	128	21	,	,	PUNCT
ejpam-6066	128	22	13	13	NUM
ejpam-6066	128	23	)	)	PUNCT
ejpam-6066	128	24	.	.	PUNCT
ejpam-6066	129	1	subcase	subcase	PROPN
ejpam-6066	129	2	2.2	2.2	NUM
ejpam-6066	129	3	x	x	SYM
ejpam-6066	129	4	=	=	PUNCT
ejpam-6066	129	5	2k	2k	NUM
ejpam-6066	129	6	+	+	CCONJ
ejpam-6066	129	7	1	1	NUM
ejpam-6066	129	8	for	for	ADP
ejpam-6066	129	9	some	some	DET
ejpam-6066	129	10	k	k	PROPN
ejpam-6066	129	11	≥	≥	NOUN
ejpam-6066	129	12	0	0	NUM
ejpam-6066	129	13	.	.	PUNCT
ejpam-6066	130	1	for	for	ADP
ejpam-6066	130	2	k	k	PROPN
ejpam-6066	130	3	=	=	SYM
ejpam-6066	130	4	0	0	NUM
ejpam-6066	130	5	or	or	CCONJ
ejpam-6066	130	6	1	1	NUM
ejpam-6066	130	7	,	,	PUNCT
ejpam-6066	130	8	direct	direct	ADJ
ejpam-6066	130	9	computation	computation	NOUN
ejpam-6066	130	10	shows	show	VERB
ejpam-6066	130	11	that	that	SCONJ
ejpam-6066	130	12	(	(	PUNCT
ejpam-6066	130	13	x	x	X
ejpam-6066	130	14	,	,	PUNCT
ejpam-6066	130	15	y	y	PROPN
ejpam-6066	130	16	,	,	PUNCT
ejpam-6066	130	17	z	z	NOUN
ejpam-6066	130	18	)	)	PUNCT
ejpam-6066	130	19	=	=	SYM
ejpam-6066	130	20	(	(	PUNCT
ejpam-6066	130	21	1	1	NUM
ejpam-6066	130	22	,	,	PUNCT
ejpam-6066	130	23	1	1	NUM
ejpam-6066	130	24	,	,	PUNCT
ejpam-6066	130	25	5	5	NUM
ejpam-6066	130	26	)	)	PUNCT
ejpam-6066	130	27	,	,	PUNCT
ejpam-6066	130	28	(	(	PUNCT
ejpam-6066	130	29	1	1	NUM
ejpam-6066	130	30	,	,	PUNCT
ejpam-6066	130	31	3	3	NUM
ejpam-6066	130	32	,	,	PUNCT
ejpam-6066	130	33	1	1	NUM
ejpam-6066	130	34	)	)	PUNCT
ejpam-6066	130	35	and	and	CCONJ
ejpam-6066	130	36	(	(	PUNCT
ejpam-6066	130	37	3	3	NUM
ejpam-6066	130	38	,	,	PUNCT
ejpam-6066	130	39	1	1	NUM
ejpam-6066	130	40	,	,	PUNCT
ejpam-6066	130	41	37	37	NUM
ejpam-6066	130	42	)	)	PUNCT
ejpam-6066	130	43	are	be	AUX
ejpam-6066	130	44	valid	valid	ADJ
ejpam-6066	130	45	solutions	solution	NOUN
ejpam-6066	130	46	,	,	PUNCT
ejpam-6066	130	47	as	as	ADP
ejpam-6066	130	48	the	the	DET
ejpam-6066	130	49	expression	expression	NOUN
ejpam-6066	130	50	4(7x	4(7x	NUM
ejpam-6066	130	51	)	)	PUNCT
ejpam-6066	131	1	−	−	NOUN
ejpam-6066	132	1	3y	3y	NOUN
ejpam-6066	132	2	=	=	SYM
ejpam-6066	132	3	z2	z2	NOUN
ejpam-6066	132	4	yields	yield	VERB
ejpam-6066	132	5	a	a	DET
ejpam-6066	132	6	nonnegative	nonnegative	ADJ
ejpam-6066	132	7	perfect	perfect	ADJ
ejpam-6066	132	8	square	square	NOUN
ejpam-6066	132	9	.	.	PUNCT
ejpam-6066	133	1	for	for	ADP
ejpam-6066	133	2	k	k	PROPN
ejpam-6066	133	3	≥	≥	NUM
ejpam-6066	133	4	2	2	NUM
ejpam-6066	133	5	,	,	PUNCT
ejpam-6066	133	6	consider	consider	VERB
ejpam-6066	133	7	the	the	DET
ejpam-6066	133	8	equation	equation	NOUN
ejpam-6066	133	9	with	with	ADP
ejpam-6066	133	10	modulo	modulo	PROPN
ejpam-6066	133	11	16	16	NUM
ejpam-6066	133	12	,	,	PUNCT
ejpam-6066	133	13	we	we	PRON
ejpam-6066	133	14	have	have	VERB
ejpam-6066	133	15	z2	z2	PROPN
ejpam-6066	133	16	≡	≡	PROPN
ejpam-6066	133	17	4(7x)−	4(7x)−	PROPN
ejpam-6066	133	18	3y	3y	PROPN
ejpam-6066	133	19	≡	≡	PROPN
ejpam-6066	133	20	{	{	PUNCT
ejpam-6066	133	21	4(7)−	4(7)−	PROPN
ejpam-6066	133	22	3	3	NUM
ejpam-6066	133	23	≡	≡	PROPN
ejpam-6066	133	24	9	9	NUM
ejpam-6066	133	25	(	(	PUNCT
ejpam-6066	133	26	mod	mod	NOUN
ejpam-6066	133	27	16	16	NUM
ejpam-6066	133	28	)	)	PUNCT
ejpam-6066	133	29	if	if	SCONJ
ejpam-6066	133	30	y	y	PROPN
ejpam-6066	133	31	≡	≡	PROPN
ejpam-6066	133	32	1	1	NUM
ejpam-6066	133	33	(	(	PUNCT
ejpam-6066	133	34	mod	mod	NOUN
ejpam-6066	133	35	4	4	NUM
ejpam-6066	133	36	)	)	PUNCT
ejpam-6066	133	37	,	,	PUNCT
ejpam-6066	133	38	4(7)−	4(7)−	PROPN
ejpam-6066	133	39	11	11	NUM
ejpam-6066	133	40	≡	≡	PROPN
ejpam-6066	133	41	1	1	NUM
ejpam-6066	133	42	(	(	PUNCT
ejpam-6066	133	43	mod	mod	NOUN
ejpam-6066	133	44	16	16	NUM
ejpam-6066	133	45	)	)	PUNCT
ejpam-6066	133	46	if	if	SCONJ
ejpam-6066	133	47	y	y	PROPN
ejpam-6066	133	48	≡	≡	PROPN
ejpam-6066	133	49	3	3	NUM
ejpam-6066	133	50	(	(	PUNCT
ejpam-6066	133	51	mod	mod	NOUN
ejpam-6066	133	52	4	4	NUM
ejpam-6066	133	53	)	)	PUNCT
ejpam-6066	133	54	.	.	PUNCT
ejpam-6066	134	1	since	since	SCONJ
ejpam-6066	134	2	z	z	PROPN
ejpam-6066	134	3	≡	≡	PROPN
ejpam-6066	134	4	{	{	PUNCT
ejpam-6066	134	5	3	3	NUM
ejpam-6066	134	6	,	,	PUNCT
ejpam-6066	134	7	5	5	NUM
ejpam-6066	134	8	,	,	PUNCT
ejpam-6066	134	9	11	11	NUM
ejpam-6066	134	10	,	,	PUNCT
ejpam-6066	134	11	13	13	NUM
ejpam-6066	134	12	(	(	PUNCT
ejpam-6066	134	13	mod	mod	PROPN
ejpam-6066	134	14	16	16	NUM
ejpam-6066	134	15	)	)	PUNCT
ejpam-6066	134	16	if	if	SCONJ
ejpam-6066	134	17	z2	z2	PROPN
ejpam-6066	134	18	≡	≡	PROPN
ejpam-6066	134	19	9	9	NUM
ejpam-6066	134	20	(	(	PUNCT
ejpam-6066	134	21	mod	mod	PROPN
ejpam-6066	134	22	16	16	NUM
ejpam-6066	134	23	)	)	PUNCT
ejpam-6066	134	24	,	,	PUNCT
ejpam-6066	134	25	1	1	NUM
ejpam-6066	134	26	,	,	PUNCT
ejpam-6066	134	27	7	7	NUM
ejpam-6066	134	28	,	,	PUNCT
ejpam-6066	134	29	9	9	NUM
ejpam-6066	134	30	,	,	PUNCT
ejpam-6066	134	31	15	15	NUM
ejpam-6066	134	32	(	(	PUNCT
ejpam-6066	134	33	mod	mod	PROPN
ejpam-6066	134	34	16	16	NUM
ejpam-6066	134	35	)	)	PUNCT
ejpam-6066	134	36	if	if	SCONJ
ejpam-6066	134	37	z2	z2	PROPN
ejpam-6066	134	38	≡	≡	PROPN
ejpam-6066	134	39	1	1	NUM
ejpam-6066	134	40	(	(	PUNCT
ejpam-6066	134	41	mod	mod	PROPN
ejpam-6066	134	42	16	16	NUM
ejpam-6066	134	43	)	)	PUNCT
ejpam-6066	134	44	,	,	PUNCT
ejpam-6066	134	45	k.	k.	PROPN
ejpam-6066	134	46	laipaporn	laipaporn	VERB
ejpam-6066	134	47	et	et	PROPN
ejpam-6066	134	48	al	al	PROPN
ejpam-6066	134	49	.	.	PUNCT
ejpam-6066	134	50	/	/	SYM
ejpam-6066	134	51	eur	eur	PROPN
ejpam-6066	134	52	.	.	PUNCT
ejpam-6066	135	1	j.	j.	PROPN
ejpam-6066	135	2	pure	pure	PROPN
ejpam-6066	135	3	appl	appl	PROPN
ejpam-6066	135	4	.	.	PROPN
ejpam-6066	135	5	math	math	PROPN
ejpam-6066	135	6	,	,	PUNCT
ejpam-6066	135	7	18	18	NUM
ejpam-6066	135	8	(	(	PUNCT
ejpam-6066	135	9	3	3	NUM
ejpam-6066	135	10	)	)	PUNCT
ejpam-6066	135	11	(	(	PUNCT
ejpam-6066	135	12	2025	2025	NUM
ejpam-6066	135	13	)	)	PUNCT
ejpam-6066	135	14	,	,	PUNCT
ejpam-6066	135	15	6066	6066	NUM
ejpam-6066	135	16	6	6	NUM
ejpam-6066	135	17	of	of	ADP
ejpam-6066	135	18	13	13	NUM
ejpam-6066	135	19	it	it	PRON
ejpam-6066	135	20	follows	follow	VERB
ejpam-6066	135	21	that	that	SCONJ
ejpam-6066	135	22	any	any	DET
ejpam-6066	135	23	solution	solution	NOUN
ejpam-6066	135	24	(	(	PUNCT
ejpam-6066	135	25	x	x	X
ejpam-6066	135	26	,	,	PUNCT
ejpam-6066	135	27	y	y	PROPN
ejpam-6066	135	28	,	,	PUNCT
ejpam-6066	135	29	z	z	NOUN
ejpam-6066	135	30	)	)	PUNCT
ejpam-6066	135	31	with	with	ADP
ejpam-6066	135	32	x	x	X
ejpam-6066	135	33	=	=	PUNCT
ejpam-6066	135	34	2k	2k	NUM
ejpam-6066	135	35	+	+	CCONJ
ejpam-6066	135	36	1	1	NUM
ejpam-6066	135	37	,	,	PUNCT
ejpam-6066	135	38	k	k	PROPN
ejpam-6066	135	39	≥	≥	NUM
ejpam-6066	135	40	2	2	NUM
ejpam-6066	135	41	and	and	CCONJ
ejpam-6066	135	42	y	y	PROPN
ejpam-6066	135	43	≡	≡	PROPN
ejpam-6066	135	44	1	1	NUM
ejpam-6066	135	45	or	or	CCONJ
ejpam-6066	135	46	3	3	NUM
ejpam-6066	135	47	(	(	PUNCT
ejpam-6066	135	48	mod	mod	NOUN
ejpam-6066	135	49	4	4	NUM
ejpam-6066	135	50	)	)	PUNCT
ejpam-6066	135	51	must	must	AUX
ejpam-6066	135	52	satisfy	satisfy	VERB
ejpam-6066	135	53	:	:	PUNCT
ejpam-6066	135	54	(	(	PUNCT
ejpam-6066	135	55	x	x	X
ejpam-6066	135	56	,	,	PUNCT
ejpam-6066	135	57	y	y	PROPN
ejpam-6066	135	58	,	,	PUNCT
ejpam-6066	135	59	z	z	NOUN
ejpam-6066	135	60	)	)	PUNCT
ejpam-6066	135	61	∈	∈	NOUN
ejpam-6066	135	62	{	{	PUNCT
ejpam-6066	135	63	(	(	PUNCT
ejpam-6066	135	64	2k+1	2k+1	PROPN
ejpam-6066	135	65	,	,	PUNCT
ejpam-6066	135	66	4l+1	4l+1	PROPN
ejpam-6066	135	67	,	,	PUNCT
ejpam-6066	135	68	16m+n)|	16m+n)|	NUM
ejpam-6066	135	69	for	for	ADP
ejpam-6066	135	70	any	any	DET
ejpam-6066	135	71	integers	integer	NOUN
ejpam-6066	135	72	k	k	X
ejpam-6066	135	73	≥	≥	NUM
ejpam-6066	135	74	2	2	NUM
ejpam-6066	135	75	,	,	PUNCT
ejpam-6066	135	76	l	l	NOUN
ejpam-6066	135	77	,	,	PUNCT
ejpam-6066	135	78	m	m	PROPN
ejpam-6066	135	79	≥	≥	NOUN
ejpam-6066	135	80	0	0	NUM
ejpam-6066	135	81	and	and	CCONJ
ejpam-6066	135	82	n	n	CCONJ
ejpam-6066	135	83	=	=	SYM
ejpam-6066	135	84	3	3	NUM
ejpam-6066	135	85	,	,	PUNCT
ejpam-6066	135	86	5	5	NUM
ejpam-6066	135	87	,	,	PUNCT
ejpam-6066	135	88	11	11	NUM
ejpam-6066	135	89	,	,	PUNCT
ejpam-6066	135	90	13}∪{(2k+1	13}∪{(2k+1	NUM
ejpam-6066	135	91	,	,	PUNCT
ejpam-6066	135	92	4l+3	4l+3	PROPN
ejpam-6066	135	93	,	,	PUNCT
ejpam-6066	135	94	16m+n)|	16m+n)|	NUM
ejpam-6066	135	95	for	for	ADP
ejpam-6066	135	96	any	any	DET
ejpam-6066	135	97	integers	integer	NOUN
ejpam-6066	135	98	k	k	X
ejpam-6066	135	99	≥	≥	NUM
ejpam-6066	135	100	2	2	NUM
ejpam-6066	135	101	,	,	PUNCT
ejpam-6066	135	102	l	l	NOUN
ejpam-6066	135	103	,	,	PUNCT
ejpam-6066	135	104	m	m	PROPN
ejpam-6066	135	105	≥	≥	NOUN
ejpam-6066	135	106	0	0	NUM
ejpam-6066	135	107	and	and	CCONJ
ejpam-6066	135	108	n	n	CCONJ
ejpam-6066	135	109	=	=	SYM
ejpam-6066	135	110	1	1	NUM
ejpam-6066	135	111	,	,	PUNCT
ejpam-6066	135	112	7	7	NUM
ejpam-6066	135	113	,	,	PUNCT
ejpam-6066	135	114	9	9	NUM
ejpam-6066	135	115	,	,	PUNCT
ejpam-6066	135	116	15	15	NUM
ejpam-6066	135	117	}	}	PUNCT
ejpam-6066	135	118	.	.	PUNCT
ejpam-6066	136	1	case	case	NOUN
ejpam-6066	136	2	3	3	NUM
ejpam-6066	136	3	p	p	NOUN
ejpam-6066	136	4	≥	≥	NUM
ejpam-6066	136	5	5	5	NUM
ejpam-6066	136	6	.	.	PUNCT
ejpam-6066	136	7	subcase	subcase	PROPN
ejpam-6066	136	8	3.1	3.1	NUM
ejpam-6066	136	9	x	x	SYM
ejpam-6066	136	10	=	=	PUNCT
ejpam-6066	136	11	2k	2k	NUM
ejpam-6066	136	12	for	for	ADP
ejpam-6066	136	13	some	some	DET
ejpam-6066	136	14	k	k	PROPN
ejpam-6066	136	15	≥	≥	NUM
ejpam-6066	136	16	1	1	NUM
ejpam-6066	136	17	.	.	PUNCT
ejpam-6066	137	1	since	since	SCONJ
ejpam-6066	137	2	z	z	PROPN
ejpam-6066	137	3	is	be	AUX
ejpam-6066	137	4	odd	odd	ADJ
ejpam-6066	137	5	,	,	PUNCT
ejpam-6066	137	6	the	the	DET
ejpam-6066	137	7	quantity	quantity	NOUN
ejpam-6066	138	1	d	d	PROPN
ejpam-6066	138	2	=	=	SYM
ejpam-6066	138	3	gcd	gcd	PROPN
ejpam-6066	138	4	(	(	PUNCT
ejpam-6066	138	5	2(7k)−	2(7k)−	PROPN
ejpam-6066	138	6	z	z	PROPN
ejpam-6066	138	7	,	,	PUNCT
ejpam-6066	138	8	2(7k	2(7k	NUM
ejpam-6066	138	9	)	)	PUNCT
ejpam-6066	139	1	+	+	SYM
ejpam-6066	139	2	z	z	X
ejpam-6066	139	3	)	)	PUNCT
ejpam-6066	139	4	is	be	AUX
ejpam-6066	139	5	also	also	ADV
ejpam-6066	139	6	odd	odd	ADJ
ejpam-6066	139	7	.	.	PUNCT
ejpam-6066	140	1	note	note	VERB
ejpam-6066	140	2	that	that	SCONJ
ejpam-6066	140	3	the	the	DET
ejpam-6066	140	4	product	product	NOUN
ejpam-6066	140	5	py	py	INTJ
ejpam-6066	140	6	=	=	SYM
ejpam-6066	140	7	(	(	PUNCT
ejpam-6066	140	8	2(7k)−z)(2(7k)+z	2(7k)−z)(2(7k)+z	NUM
ejpam-6066	140	9	)	)	PUNCT
ejpam-6066	140	10	consists	consist	VERB
ejpam-6066	140	11	of	of	ADP
ejpam-6066	140	12	two	two	NUM
ejpam-6066	140	13	factors	factor	NOUN
ejpam-6066	140	14	whose	whose	DET
ejpam-6066	140	15	sum	sum	NOUN
ejpam-6066	140	16	is	be	AUX
ejpam-6066	140	17	4(7k	4(7k	NOUN
ejpam-6066	140	18	)	)	PUNCT
ejpam-6066	140	19	,	,	PUNCT
ejpam-6066	140	20	implying	imply	VERB
ejpam-6066	140	21	d|4(7k	d|4(7k	PROPN
ejpam-6066	140	22	)	)	PUNCT
ejpam-6066	140	23	.	.	PUNCT
ejpam-6066	141	1	therefore	therefore	ADV
ejpam-6066	141	2	,	,	PUNCT
ejpam-6066	141	3	d	d	PROPN
ejpam-6066	141	4	=	=	SYM
ejpam-6066	141	5	1	1	NUM
ejpam-6066	141	6	or	or	CCONJ
ejpam-6066	141	7	7|d	7|d	NOUN
ejpam-6066	141	8	.	.	PUNCT
ejpam-6066	142	1	if	if	SCONJ
ejpam-6066	142	2	d	d	NOUN
ejpam-6066	142	3	=	=	SYM
ejpam-6066	142	4	1	1	NUM
ejpam-6066	142	5	then	then	ADV
ejpam-6066	142	6	2(7k	2(7k	NUM
ejpam-6066	142	7	)	)	PUNCT
ejpam-6066	143	1	−	−	PROPN
ejpam-6066	143	2	z	z	NOUN
ejpam-6066	143	3	=	=	SYM
ejpam-6066	143	4	1	1	NUM
ejpam-6066	143	5	,	,	PUNCT
ejpam-6066	143	6	which	which	PRON
ejpam-6066	143	7	gives	give	VERB
ejpam-6066	143	8	py	py	NOUN
ejpam-6066	143	9	=	=	SYM
ejpam-6066	144	1	4(7k)−	4(7k)−	PROPN
ejpam-6066	144	2	1	1	NUM
ejpam-6066	144	3	.	.	PUNCT
ejpam-6066	145	1	reducing	reduce	VERB
ejpam-6066	145	2	both	both	DET
ejpam-6066	145	3	sides	side	NOUN
ejpam-6066	145	4	modulo	modulo	VERB
ejpam-6066	145	5	3	3	NUM
ejpam-6066	145	6	yields	yield	NOUN
ejpam-6066	145	7	py	py	PROPN
ejpam-6066	145	8	≡	≡	PROPN
ejpam-6066	145	9	0	0	PUNCT
ejpam-6066	146	1	(	(	PUNCT
ejpam-6066	146	2	mod	mod	NOUN
ejpam-6066	146	3	3	3	NUM
ejpam-6066	146	4	)	)	PUNCT
ejpam-6066	146	5	,	,	PUNCT
ejpam-6066	146	6	so	so	SCONJ
ejpam-6066	146	7	p	p	X
ejpam-6066	146	8	=	=	NOUN
ejpam-6066	146	9	3	3	NUM
ejpam-6066	146	10	,	,	PUNCT
ejpam-6066	146	11	contradicting	contradict	VERB
ejpam-6066	146	12	the	the	DET
ejpam-6066	146	13	assumption	assumption	NOUN
ejpam-6066	146	14	p	p	X
ejpam-6066	146	15	≥	≥	NUM
ejpam-6066	146	16	5	5	NUM
ejpam-6066	146	17	.	.	PUNCT
ejpam-6066	147	1	hence	hence	ADV
ejpam-6066	147	2	,	,	PUNCT
ejpam-6066	147	3	we	we	PRON
ejpam-6066	147	4	must	must	AUX
ejpam-6066	147	5	have	have	VERB
ejpam-6066	147	6	7|d	7|d	PROPN
ejpam-6066	147	7	,	,	PUNCT
ejpam-6066	147	8	which	which	PRON
ejpam-6066	147	9	implies	imply	VERB
ejpam-6066	147	10	p	p	X
ejpam-6066	147	11	=	=	NOUN
ejpam-6066	147	12	7	7	X
ejpam-6066	147	13	.	.	PUNCT
ejpam-6066	148	1	in	in	ADP
ejpam-6066	148	2	this	this	DET
ejpam-6066	148	3	case	case	NOUN
ejpam-6066	148	4	,	,	PUNCT
ejpam-6066	148	5	the	the	DET
ejpam-6066	148	6	equation	equation	NOUN
ejpam-6066	148	7	becomes	become	VERB
ejpam-6066	148	8	7y	7y	NOUN
ejpam-6066	148	9	=	=	PUNCT
ejpam-6066	148	10	(	(	PUNCT
ejpam-6066	148	11	2(7k	2(7k	NUM
ejpam-6066	148	12	)	)	PUNCT
ejpam-6066	148	13	−	−	PROPN
ejpam-6066	148	14	z)(2(7k	z)(2(7k	NOUN
ejpam-6066	148	15	)	)	PUNCT
ejpam-6066	148	16	+	+	SYM
ejpam-6066	149	1	z	z	X
ejpam-6066	149	2	)	)	PUNCT
ejpam-6066	149	3	,	,	PUNCT
ejpam-6066	149	4	where	where	SCONJ
ejpam-6066	149	5	the	the	DET
ejpam-6066	149	6	two	two	NUM
ejpam-6066	149	7	factors	factor	NOUN
ejpam-6066	149	8	are	be	AUX
ejpam-6066	149	9	coprime	coprime	ADJ
ejpam-6066	149	10	and	and	CCONJ
ejpam-6066	149	11	their	their	PRON
ejpam-6066	149	12	product	product	NOUN
ejpam-6066	149	13	is	be	AUX
ejpam-6066	149	14	a	a	DET
ejpam-6066	149	15	power	power	NOUN
ejpam-6066	149	16	of	of	ADP
ejpam-6066	149	17	7	7	NUM
ejpam-6066	149	18	.	.	PUNCT
ejpam-6066	150	1	therefore	therefore	ADV
ejpam-6066	150	2	,	,	PUNCT
ejpam-6066	150	3	we	we	PRON
ejpam-6066	150	4	can	can	AUX
ejpam-6066	150	5	write	write	VERB
ejpam-6066	150	6	7u	7u	NOUN
ejpam-6066	150	7	=	=	SYM
ejpam-6066	150	8	2(7k	2(7k	NUM
ejpam-6066	150	9	)	)	PUNCT
ejpam-6066	150	10	−	−	PROPN
ejpam-6066	151	1	z	z	X
ejpam-6066	151	2	,	,	PUNCT
ejpam-6066	151	3	7y−u	7y−u	NOUN
ejpam-6066	151	4	=	=	SYM
ejpam-6066	151	5	2(7k	2(7k	NUM
ejpam-6066	151	6	)	)	PUNCT
ejpam-6066	152	1	+	+	CCONJ
ejpam-6066	152	2	z	z	NOUN
ejpam-6066	152	3	for	for	ADP
ejpam-6066	152	4	some	some	DET
ejpam-6066	152	5	1	1	NUM
ejpam-6066	152	6	≤	≤	NOUN
ejpam-6066	152	7	u	u	NOUN
ejpam-6066	152	8	<	<	X
ejpam-6066	152	9	y.	y.	NOUN
ejpam-6066	152	10	adding	add	VERB
ejpam-6066	152	11	these	these	DET
ejpam-6066	152	12	two	two	NUM
ejpam-6066	152	13	equations	equation	NOUN
ejpam-6066	152	14	yields	yield	VERB
ejpam-6066	152	15	4(7k	4(7k	NOUN
ejpam-6066	152	16	)	)	PUNCT
ejpam-6066	152	17	=	=	SYM
ejpam-6066	152	18	7u	7u	NOUN
ejpam-6066	152	19	+	+	CCONJ
ejpam-6066	152	20	7y−u	7y−u	NOUN
ejpam-6066	152	21	.	.	PUNCT
ejpam-6066	153	1	reducing	reduce	VERB
ejpam-6066	153	2	modulo	modulo	NOUN
ejpam-6066	153	3	6	6	NUM
ejpam-6066	153	4	,	,	PUNCT
ejpam-6066	153	5	we	we	PRON
ejpam-6066	153	6	find	find	VERB
ejpam-6066	153	7	7u	7u	NOUN
ejpam-6066	153	8	+	+	CCONJ
ejpam-6066	153	9	7y−u	7y−u	NUM
ejpam-6066	153	10	≡	≡	PROPN
ejpam-6066	153	11	2	2	NUM
ejpam-6066	153	12	(	(	PUNCT
ejpam-6066	153	13	mod	mod	PROPN
ejpam-6066	153	14	6	6	NUM
ejpam-6066	153	15	)	)	PUNCT
ejpam-6066	153	16	,	,	PUNCT
ejpam-6066	153	17	while	while	SCONJ
ejpam-6066	153	18	4(7k	4(7k	NOUN
ejpam-6066	153	19	)	)	PUNCT
ejpam-6066	153	20	≡	≡	PROPN
ejpam-6066	153	21	4	4	NUM
ejpam-6066	153	22	(	(	PUNCT
ejpam-6066	153	23	mod	mod	PROPN
ejpam-6066	153	24	6	6	NUM
ejpam-6066	153	25	)	)	PUNCT
ejpam-6066	153	26	.	.	PUNCT
ejpam-6066	154	1	since	since	SCONJ
ejpam-6066	154	2	both	both	DET
ejpam-6066	154	3	sides	side	NOUN
ejpam-6066	154	4	are	be	AUX
ejpam-6066	154	5	not	not	PART
ejpam-6066	154	6	congruent	congruent	ADJ
ejpam-6066	154	7	modulo	modulo	NOUN
ejpam-6066	154	8	6	6	NUM
ejpam-6066	154	9	,	,	PUNCT
ejpam-6066	154	10	the	the	DET
ejpam-6066	154	11	equality	equality	NOUN
ejpam-6066	154	12	can	can	AUX
ejpam-6066	154	13	not	not	PART
ejpam-6066	154	14	hold	hold	VERB
ejpam-6066	154	15	.	.	PUNCT
ejpam-6066	155	1	hence	hence	ADV
ejpam-6066	155	2	there	there	PRON
ejpam-6066	155	3	is	be	VERB
ejpam-6066	155	4	no	no	DET
ejpam-6066	155	5	solution	solution	NOUN
ejpam-6066	155	6	when	when	SCONJ
ejpam-6066	155	7	x	x	PRON
ejpam-6066	155	8	is	be	AUX
ejpam-6066	155	9	even	even	ADV
ejpam-6066	155	10	and	and	CCONJ
ejpam-6066	155	11	p	p	X
ejpam-6066	155	12	≥	≥	NUM
ejpam-6066	155	13	5	5	NUM
ejpam-6066	155	14	.	.	PUNCT
ejpam-6066	155	15	subcase	subcase	PROPN
ejpam-6066	155	16	3.2	3.2	NUM
ejpam-6066	155	17	x	x	SYM
ejpam-6066	155	18	=	=	PUNCT
ejpam-6066	155	19	2k	2k	NUM
ejpam-6066	155	20	+	+	CCONJ
ejpam-6066	155	21	1	1	NUM
ejpam-6066	155	22	for	for	ADP
ejpam-6066	155	23	some	some	PRON
ejpam-6066	155	24	k	k	PROPN
ejpam-6066	155	25	≥	≥	PROPN
ejpam-6066	155	26	0	0	NUM
ejpam-6066	155	27	.	.	PUNCT
ejpam-6066	156	1	note	note	VERB
ejpam-6066	156	2	that	that	SCONJ
ejpam-6066	156	3	72k+1	72k+1	NUM
ejpam-6066	156	4	≡	≡	PROPN
ejpam-6066	156	5	(	(	PUNCT
ejpam-6066	156	6	49k)7	49k)7	NUM
ejpam-6066	156	7	≡	≡	PROPN
ejpam-6066	156	8	7	7	NUM
ejpam-6066	156	9	(	(	PUNCT
ejpam-6066	156	10	mod	mod	NOUN
ejpam-6066	156	11	24	24	NUM
ejpam-6066	156	12	)	)	PUNCT
ejpam-6066	156	13	.	.	PUNCT
ejpam-6066	157	1	since	since	SCONJ
ejpam-6066	157	2	p	p	NOUN
ejpam-6066	157	3	is	be	AUX
ejpam-6066	157	4	an	an	DET
ejpam-6066	157	5	odd	odd	ADJ
ejpam-6066	157	6	prime	prime	NOUN
ejpam-6066	157	7	,	,	PUNCT
ejpam-6066	157	8	we	we	PRON
ejpam-6066	157	9	have	have	VERB
ejpam-6066	157	10	p	p	PRON
ejpam-6066	157	11	≡	≡	PROPN
ejpam-6066	157	12	±1,±5,±7,±11	±1,±5,±7,±11	PROPN
ejpam-6066	157	13	or	or	CCONJ
ejpam-6066	157	14	13	13	NUM
ejpam-6066	157	15	(	(	PUNCT
ejpam-6066	157	16	mod	mod	NOUN
ejpam-6066	157	17	24	24	NUM
ejpam-6066	157	18	)	)	PUNCT
ejpam-6066	157	19	,	,	PUNCT
ejpam-6066	157	20	so	so	ADV
ejpam-6066	157	21	py	py	PROPN
ejpam-6066	157	22	≡	≡	PROPN
ejpam-6066	157	23	{	{	PUNCT
ejpam-6066	157	24	1	1	NUM
ejpam-6066	157	25	(	(	PUNCT
ejpam-6066	157	26	mod	mod	NOUN
ejpam-6066	157	27	24	24	NUM
ejpam-6066	157	28	)	)	PUNCT
ejpam-6066	157	29	if	if	SCONJ
ejpam-6066	157	30	y	y	PROPN
ejpam-6066	157	31	is	be	AUX
ejpam-6066	157	32	even	even	ADV
ejpam-6066	157	33	,	,	PUNCT
ejpam-6066	157	34	p	p	X
ejpam-6066	157	35	(	(	PUNCT
ejpam-6066	157	36	mod	mod	PROPN
ejpam-6066	157	37	24	24	NUM
ejpam-6066	157	38	)	)	PUNCT
ejpam-6066	157	39	if	if	SCONJ
ejpam-6066	157	40	y	y	PROPN
ejpam-6066	157	41	is	be	AUX
ejpam-6066	157	42	odd	odd	ADJ
ejpam-6066	157	43	.	.	PUNCT
ejpam-6066	158	1	therefore	therefore	ADV
ejpam-6066	158	2	,	,	PUNCT
ejpam-6066	158	3	4(7x)−	4(7x)−	PROPN
ejpam-6066	158	4	py	py	PROPN
ejpam-6066	158	5	≡	≡	PROPN
ejpam-6066	158	6	{	{	PUNCT
ejpam-6066	158	7	4(7)−	4(7)−	PROPN
ejpam-6066	158	8	1	1	NUM
ejpam-6066	158	9	≡	≡	PROPN
ejpam-6066	158	10	3	3	NUM
ejpam-6066	158	11	(	(	PUNCT
ejpam-6066	158	12	mod	mod	NOUN
ejpam-6066	158	13	24	24	NUM
ejpam-6066	158	14	)	)	PUNCT
ejpam-6066	158	15	if	if	SCONJ
ejpam-6066	158	16	y	y	PROPN
ejpam-6066	158	17	is	be	AUX
ejpam-6066	158	18	even	even	ADV
ejpam-6066	158	19	,	,	PUNCT
ejpam-6066	158	20	4(7)−	4(7)−	PROPN
ejpam-6066	158	21	p	p	PROPN
ejpam-6066	158	22	≡	≡	PROPN
ejpam-6066	159	1	4−	4−	PROPN
ejpam-6066	160	1	p	p	NOUN
ejpam-6066	160	2	(	(	PUNCT
ejpam-6066	160	3	mod	mod	PROPN
ejpam-6066	160	4	24	24	NUM
ejpam-6066	160	5	)	)	PUNCT
ejpam-6066	160	6	if	if	SCONJ
ejpam-6066	160	7	y	y	PROPN
ejpam-6066	160	8	is	be	AUX
ejpam-6066	160	9	odd	odd	ADJ
ejpam-6066	160	10	.	.	PUNCT
ejpam-6066	161	1	since	since	SCONJ
ejpam-6066	161	2	z	z	PROPN
ejpam-6066	161	3	is	be	AUX
ejpam-6066	161	4	odd	odd	ADJ
ejpam-6066	161	5	,	,	PUNCT
ejpam-6066	161	6	it	it	PRON
ejpam-6066	161	7	square	square	ADJ
ejpam-6066	161	8	must	must	AUX
ejpam-6066	161	9	satisfy	satisfy	VERB
ejpam-6066	161	10	z2	z2	PROPN
ejpam-6066	161	11	≡	≡	PROPN
ejpam-6066	161	12	1	1	NUM
ejpam-6066	161	13	or	or	CCONJ
ejpam-6066	161	14	9	9	NUM
ejpam-6066	161	15	(	(	PUNCT
ejpam-6066	161	16	mod	mod	NOUN
ejpam-6066	161	17	24	24	NUM
ejpam-6066	161	18	)	)	PUNCT
ejpam-6066	161	19	.	.	PUNCT
ejpam-6066	162	1	but	but	CCONJ
ejpam-6066	162	2	4(7x)−	4(7x)−	PROPN
ejpam-6066	162	3	py	py	PROPN
ejpam-6066	162	4	≡	≡	PROPN
ejpam-6066	162	5	3	3	NUM
ejpam-6066	162	6	(	(	PUNCT
ejpam-6066	162	7	mod	mod	NOUN
ejpam-6066	162	8	24	24	NUM
ejpam-6066	162	9	)	)	PUNCT
ejpam-6066	162	10	where	where	SCONJ
ejpam-6066	162	11	y	y	PROPN
ejpam-6066	162	12	is	be	AUX
ejpam-6066	162	13	even	even	ADV
ejpam-6066	162	14	,	,	PUNCT
ejpam-6066	162	15	which	which	PRON
ejpam-6066	162	16	is	be	AUX
ejpam-6066	162	17	not	not	PART
ejpam-6066	162	18	a	a	DET
ejpam-6066	162	19	quadratic	quadratic	ADJ
ejpam-6066	162	20	residue	residue	NOUN
ejpam-6066	162	21	modulo	modulo	NOUN
ejpam-6066	162	22	24	24	NUM
ejpam-6066	162	23	.	.	PUNCT
ejpam-6066	163	1	now	now	ADV
ejpam-6066	163	2	,	,	PUNCT
ejpam-6066	163	3	suppose	suppose	VERB
ejpam-6066	163	4	y	y	PROPN
ejpam-6066	163	5	is	be	AUX
ejpam-6066	163	6	odd	odd	ADJ
ejpam-6066	163	7	.	.	PUNCT
ejpam-6066	164	1	then	then	ADV
ejpam-6066	164	2	4	4	NUM
ejpam-6066	164	3	−	−	PROPN
ejpam-6066	164	4	p	p	PROPN
ejpam-6066	164	5	≡	≡	PROPN
ejpam-6066	164	6	4(7x	4(7x	PROPN
ejpam-6066	164	7	)	)	PUNCT
ejpam-6066	165	1	−	−	PROPN
ejpam-6066	165	2	py	py	PROPN
ejpam-6066	165	3	≡	≡	PROPN
ejpam-6066	165	4	z2	z2	PROPN
ejpam-6066	165	5	≡	≡	PROPN
ejpam-6066	165	6	1	1	NUM
ejpam-6066	165	7	or	or	CCONJ
ejpam-6066	165	8	9	9	NUM
ejpam-6066	165	9	(	(	PUNCT
ejpam-6066	165	10	mod	mod	NOUN
ejpam-6066	165	11	24	24	NUM
ejpam-6066	165	12	)	)	PUNCT
ejpam-6066	165	13	.	.	PUNCT
ejpam-6066	166	1	this	this	PRON
ejpam-6066	166	2	implies	imply	VERB
ejpam-6066	166	3	p	p	PROPN
ejpam-6066	166	4	≡	≡	PROPN
ejpam-6066	166	5	3	3	NUM
ejpam-6066	166	6	or	or	CCONJ
ejpam-6066	166	7	19	19	NUM
ejpam-6066	166	8	(	(	PUNCT
ejpam-6066	166	9	mod	mod	NOUN
ejpam-6066	166	10	24	24	NUM
ejpam-6066	166	11	)	)	PUNCT
ejpam-6066	166	12	.	.	PUNCT
ejpam-6066	167	1	however	however	ADV
ejpam-6066	167	2	,	,	PUNCT
ejpam-6066	167	3	gcd	gcd	X
ejpam-6066	167	4	(	(	PUNCT
ejpam-6066	167	5	p	p	X
ejpam-6066	167	6	,	,	PUNCT
ejpam-6066	167	7	3	3	NUM
ejpam-6066	167	8	)	)	PUNCT
ejpam-6066	167	9	=	=	SYM
ejpam-6066	167	10	1	1	NUM
ejpam-6066	167	11	so	so	ADV
ejpam-6066	167	12	p	p	X
ejpam-6066	167	13	̸≡	̸≡	PROPN
ejpam-6066	167	14	3	3	NUM
ejpam-6066	167	15	(	(	PUNCT
ejpam-6066	167	16	mod	mod	NOUN
ejpam-6066	167	17	24	24	NUM
ejpam-6066	167	18	)	)	PUNCT
ejpam-6066	167	19	.	.	PUNCT
ejpam-6066	168	1	therefore	therefore	ADV
ejpam-6066	168	2	,	,	PUNCT
ejpam-6066	168	3	we	we	PRON
ejpam-6066	168	4	must	must	AUX
ejpam-6066	168	5	have	have	VERB
ejpam-6066	168	6	p	p	X
ejpam-6066	168	7	≡	≡	PROPN
ejpam-6066	168	8	19	19	NUM
ejpam-6066	168	9	(	(	PUNCT
ejpam-6066	168	10	mod	mod	NOUN
ejpam-6066	168	11	24	24	NUM
ejpam-6066	168	12	)	)	PUNCT
ejpam-6066	168	13	,	,	PUNCT
ejpam-6066	168	14	and	and	CCONJ
ejpam-6066	168	15	hence	hence	ADV
ejpam-6066	168	16	z2	z2	PROPN
ejpam-6066	168	17	≡	≡	PROPN
ejpam-6066	168	18	9	9	NUM
ejpam-6066	168	19	(	(	PUNCT
ejpam-6066	168	20	mod	mod	NOUN
ejpam-6066	168	21	24	24	NUM
ejpam-6066	168	22	)	)	PUNCT
ejpam-6066	168	23	.	.	PUNCT
ejpam-6066	169	1	finally	finally	ADV
ejpam-6066	169	2	,	,	PUNCT
ejpam-6066	169	3	we	we	PRON
ejpam-6066	169	4	can	can	AUX
ejpam-6066	169	5	conclude	conclude	VERB
ejpam-6066	169	6	that	that	SCONJ
ejpam-6066	169	7	if	if	SCONJ
ejpam-6066	169	8	4(7x	4(7x	NUM
ejpam-6066	169	9	)	)	PUNCT
ejpam-6066	170	1	−	−	PROPN
ejpam-6066	171	1	py	py	NOUN
ejpam-6066	171	2	=	=	PROPN
ejpam-6066	171	3	z2	z2	PROPN
ejpam-6066	171	4	has	have	VERB
ejpam-6066	171	5	a	a	DET
ejpam-6066	171	6	solution	solution	NOUN
ejpam-6066	171	7	with	with	ADP
ejpam-6066	171	8	prime	prime	ADJ
ejpam-6066	171	9	p	p	X
ejpam-6066	171	10	≥	≥	NUM
ejpam-6066	171	11	19	19	NUM
ejpam-6066	171	12	then	then	ADV
ejpam-6066	171	13	both	both	CCONJ
ejpam-6066	171	14	x	x	PUNCT
ejpam-6066	171	15	and	and	CCONJ
ejpam-6066	171	16	y	y	PROPN
ejpam-6066	171	17	must	must	AUX
ejpam-6066	171	18	be	be	AUX
ejpam-6066	171	19	odd	odd	ADJ
ejpam-6066	171	20	,	,	PUNCT
ejpam-6066	171	21	z	z	PROPN
ejpam-6066	171	22	≡	≡	PROPN
ejpam-6066	171	23	3	3	NUM
ejpam-6066	171	24	,	,	PUNCT
ejpam-6066	171	25	9	9	NUM
ejpam-6066	171	26	,	,	PUNCT
ejpam-6066	171	27	15	15	NUM
ejpam-6066	171	28	or	or	CCONJ
ejpam-6066	171	29	21	21	NUM
ejpam-6066	171	30	(	(	PUNCT
ejpam-6066	171	31	mod	mod	PROPN
ejpam-6066	171	32	24	24	NUM
ejpam-6066	171	33	)	)	PUNCT
ejpam-6066	171	34	,	,	PUNCT
ejpam-6066	171	35	and	and	CCONJ
ejpam-6066	171	36	p	p	PROPN
ejpam-6066	171	37	≡	≡	PROPN
ejpam-6066	171	38	19	19	NUM
ejpam-6066	171	39	(	(	PUNCT
ejpam-6066	171	40	mod	mod	NOUN
ejpam-6066	171	41	24	24	NUM
ejpam-6066	171	42	)	)	PUNCT
ejpam-6066	171	43	.	.	PUNCT
ejpam-6066	172	1	based	base	VERB
ejpam-6066	172	2	on	on	ADP
ejpam-6066	172	3	theorem	theorem	NOUN
ejpam-6066	172	4	1(iv	1(iv	NUM
ejpam-6066	172	5	)	)	PUNCT
ejpam-6066	172	6	,	,	PUNCT
ejpam-6066	172	7	we	we	PRON
ejpam-6066	172	8	refine	refine	VERB
ejpam-6066	172	9	the	the	DET
ejpam-6066	172	10	possible	possible	ADJ
ejpam-6066	172	11	values	value	NOUN
ejpam-6066	172	12	of	of	ADP
ejpam-6066	172	13	the	the	DET
ejpam-6066	172	14	prime	prime	NOUN
ejpam-6066	172	15	p	p	NOUN
ejpam-6066	172	16	by	by	ADP
ejpam-6066	172	17	analyzing	analyze	VERB
ejpam-6066	172	18	the	the	DET
ejpam-6066	172	19	divisibility	divisibility	NOUN
ejpam-6066	172	20	properties	property	NOUN
ejpam-6066	172	21	of	of	ADP
ejpam-6066	172	22	z	z	NOUN
ejpam-6066	172	23	,	,	PUNCT
ejpam-6066	172	24	particularly	particularly	ADV
ejpam-6066	172	25	with	with	ADP
ejpam-6066	172	26	respect	respect	NOUN
ejpam-6066	172	27	to	to	ADP
ejpam-6066	172	28	modulo	modulo	NOUN
ejpam-6066	172	29	3	3	NUM
ejpam-6066	172	30	.	.	PUNCT
ejpam-6066	173	1	this	this	PRON
ejpam-6066	173	2	leads	lead	VERB
ejpam-6066	173	3	to	to	ADP
ejpam-6066	173	4	the	the	DET
ejpam-6066	173	5	following	follow	VERB
ejpam-6066	173	6	corollary	corollary	NOUN
ejpam-6066	173	7	.	.	PUNCT
ejpam-6066	174	1	corollary	corollary	ADJ
ejpam-6066	174	2	1	1	NUM
ejpam-6066	174	3	.	.	PUNCT
ejpam-6066	175	1	let	let	VERB
ejpam-6066	175	2	p	p	PRON
ejpam-6066	175	3	≥	≥	NUM
ejpam-6066	175	4	19	19	NUM
ejpam-6066	175	5	be	be	AUX
ejpam-6066	175	6	a	a	DET
ejpam-6066	175	7	prime	prime	NOUN
ejpam-6066	175	8	.	.	PUNCT
ejpam-6066	176	1	then	then	ADV
ejpam-6066	176	2	all	all	DET
ejpam-6066	176	3	non	non	ADJ
ejpam-6066	176	4	-	-	ADJ
ejpam-6066	176	5	negative	negative	ADJ
ejpam-6066	176	6	integer	integer	NOUN
ejpam-6066	176	7	solutions	solution	NOUN
ejpam-6066	176	8	(	(	PUNCT
ejpam-6066	176	9	x	x	X
ejpam-6066	176	10	,	,	PUNCT
ejpam-6066	176	11	y	y	PROPN
ejpam-6066	176	12	,	,	PUNCT
ejpam-6066	176	13	z	z	PROPN
ejpam-6066	176	14	,	,	PUNCT
ejpam-6066	176	15	p	p	NOUN
ejpam-6066	176	16	)	)	PUNCT
ejpam-6066	176	17	to	to	ADP
ejpam-6066	176	18	the	the	DET
ejpam-6066	176	19	equation	equation	NOUN
ejpam-6066	176	20	4(7x)−	4(7x)−	PROPN
ejpam-6066	176	21	py	py	PROPN
ejpam-6066	176	22	=	=	PROPN
ejpam-6066	176	23	z2	z2	PROPN
ejpam-6066	176	24	are	be	AUX
ejpam-6066	176	25	of	of	ADP
ejpam-6066	176	26	the	the	DET
ejpam-6066	176	27	following	follow	VERB
ejpam-6066	176	28	form	form	NOUN
ejpam-6066	176	29	:	:	PUNCT
ejpam-6066	176	30	(	(	PUNCT
ejpam-6066	176	31	x	x	X
ejpam-6066	176	32	,	,	PUNCT
ejpam-6066	176	33	y	y	PROPN
ejpam-6066	176	34	,	,	PUNCT
ejpam-6066	176	35	z	z	PROPN
ejpam-6066	176	36	,	,	PUNCT
ejpam-6066	176	37	p	p	NOUN
ejpam-6066	176	38	)	)	PUNCT
ejpam-6066	176	39	∈	∈	PROPN
ejpam-6066	176	40	{	{	PUNCT
ejpam-6066	176	41	(	(	PUNCT
ejpam-6066	176	42	6k	6k	NOUN
ejpam-6066	176	43	+	+	NOUN
ejpam-6066	176	44	1	1	NUM
ejpam-6066	176	45	,	,	PUNCT
ejpam-6066	176	46	6l	6l	NUM
ejpam-6066	176	47	+	+	CCONJ
ejpam-6066	176	48	3	3	NUM
ejpam-6066	176	49	,	,	PUNCT
ejpam-6066	176	50	24m+	24m+	NUM
ejpam-6066	176	51	n	n	CCONJ
ejpam-6066	176	52	,	,	PUNCT
ejpam-6066	176	53	24r	24r	PUNCT
ejpam-6066	177	1	+	+	X
ejpam-6066	177	2	19)|k	19)|k	NUM
ejpam-6066	177	3	,	,	PUNCT
ejpam-6066	177	4	l	l	NOUN
ejpam-6066	177	5	,	,	PUNCT
ejpam-6066	177	6	m	m	PROPN
ejpam-6066	177	7	,	,	PUNCT
ejpam-6066	177	8	r	r	NOUN
ejpam-6066	177	9	∈	∈	PROPN
ejpam-6066	177	10	z+	z+	NUM
ejpam-6066	177	11	0	0	NUM
ejpam-6066	177	12	and	and	CCONJ
ejpam-6066	177	13	n	n	CCONJ
ejpam-6066	177	14	=	=	SYM
ejpam-6066	177	15	3	3	NUM
ejpam-6066	177	16	,	,	PUNCT
ejpam-6066	177	17	9	9	NUM
ejpam-6066	177	18	,	,	PUNCT
ejpam-6066	177	19	15	15	NUM
ejpam-6066	177	20	,	,	PUNCT
ejpam-6066	177	21	21	21	NUM
ejpam-6066	177	22	}	}	PUNCT
ejpam-6066	177	23	k.	k.	PROPN
ejpam-6066	177	24	laipaporn	laipaporn	VERB
ejpam-6066	177	25	et	et	PROPN
ejpam-6066	177	26	al	al	PROPN
ejpam-6066	177	27	.	.	PUNCT
ejpam-6066	177	28	/	/	SYM
ejpam-6066	177	29	eur	eur	PROPN
ejpam-6066	177	30	.	.	PUNCT
ejpam-6066	178	1	j.	j.	PROPN
ejpam-6066	178	2	pure	pure	PROPN
ejpam-6066	178	3	appl	appl	PROPN
ejpam-6066	178	4	.	.	PROPN
ejpam-6066	178	5	math	math	PROPN
ejpam-6066	178	6	,	,	PUNCT
ejpam-6066	178	7	18	18	NUM
ejpam-6066	178	8	(	(	PUNCT
ejpam-6066	178	9	3	3	NUM
ejpam-6066	178	10	)	)	PUNCT
ejpam-6066	178	11	(	(	PUNCT
ejpam-6066	178	12	2025	2025	NUM
ejpam-6066	178	13	)	)	PUNCT
ejpam-6066	178	14	,	,	PUNCT
ejpam-6066	178	15	6066	6066	NUM
ejpam-6066	178	16	7	7	NUM
ejpam-6066	178	17	of	of	ADP
ejpam-6066	178	18	13	13	NUM
ejpam-6066	178	19	∪	∪	X
ejpam-6066	178	20	{	{	PUNCT
ejpam-6066	178	21	(	(	PUNCT
ejpam-6066	178	22	6k	6k	NOUN
ejpam-6066	178	23	+	+	NOUN
ejpam-6066	178	24	1	1	NUM
ejpam-6066	178	25	,	,	PUNCT
ejpam-6066	178	26	6l	6l	NUM
ejpam-6066	178	27	+	+	CCONJ
ejpam-6066	178	28	1	1	NUM
ejpam-6066	178	29	,	,	PUNCT
ejpam-6066	178	30	24m+	24m+	NUM
ejpam-6066	178	31	n	n	CCONJ
ejpam-6066	178	32	,	,	PUNCT
ejpam-6066	178	33	72r	72r	NUM
ejpam-6066	178	34	+	+	X
ejpam-6066	178	35	19)|k	19)|k	NUM
ejpam-6066	178	36	,	,	PUNCT
ejpam-6066	178	37	l	l	NOUN
ejpam-6066	178	38	,	,	PUNCT
ejpam-6066	178	39	m	m	PROPN
ejpam-6066	178	40	,	,	PUNCT
ejpam-6066	178	41	r	r	NOUN
ejpam-6066	178	42	∈	∈	PROPN
ejpam-6066	178	43	z+	z+	NUM
ejpam-6066	178	44	0	0	NUM
ejpam-6066	178	45	and	and	CCONJ
ejpam-6066	178	46	n	n	CCONJ
ejpam-6066	178	47	=	=	SYM
ejpam-6066	178	48	3	3	NUM
ejpam-6066	178	49	,	,	PUNCT
ejpam-6066	178	50	9	9	NUM
ejpam-6066	178	51	,	,	PUNCT
ejpam-6066	178	52	15	15	NUM
ejpam-6066	178	53	,	,	PUNCT
ejpam-6066	178	54	21	21	NUM
ejpam-6066	178	55	}	}	PUNCT
ejpam-6066	178	56	∪	∪	X
ejpam-6066	178	57	{	{	PUNCT
ejpam-6066	178	58	(	(	PUNCT
ejpam-6066	178	59	6k	6k	NOUN
ejpam-6066	178	60	+	+	NOUN
ejpam-6066	178	61	1	1	NUM
ejpam-6066	178	62	,	,	PUNCT
ejpam-6066	178	63	6l	6l	NUM
ejpam-6066	178	64	+	+	CCONJ
ejpam-6066	178	65	5	5	NUM
ejpam-6066	178	66	,	,	PUNCT
ejpam-6066	178	67	24m+	24m+	NUM
ejpam-6066	178	68	n	n	CCONJ
ejpam-6066	178	69	,	,	PUNCT
ejpam-6066	178	70	72r	72r	NUM
ejpam-6066	178	71	+	+	X
ejpam-6066	178	72	19)|k	19)|k	NUM
ejpam-6066	178	73	,	,	PUNCT
ejpam-6066	178	74	l	l	NOUN
ejpam-6066	178	75	,	,	PUNCT
ejpam-6066	178	76	m	m	PROPN
ejpam-6066	178	77	,	,	PUNCT
ejpam-6066	178	78	r	r	NOUN
ejpam-6066	178	79	∈	∈	PROPN
ejpam-6066	178	80	z+	z+	NUM
ejpam-6066	178	81	0	0	NUM
ejpam-6066	178	82	and	and	CCONJ
ejpam-6066	178	83	n	n	CCONJ
ejpam-6066	178	84	=	=	SYM
ejpam-6066	178	85	3	3	NUM
ejpam-6066	178	86	,	,	PUNCT
ejpam-6066	178	87	9	9	NUM
ejpam-6066	178	88	,	,	PUNCT
ejpam-6066	178	89	15	15	NUM
ejpam-6066	178	90	,	,	PUNCT
ejpam-6066	178	91	21	21	NUM
ejpam-6066	178	92	}	}	PUNCT
ejpam-6066	178	93	∪	∪	X
ejpam-6066	178	94	{	{	PUNCT
ejpam-6066	178	95	(	(	PUNCT
ejpam-6066	178	96	6k	6k	NOUN
ejpam-6066	178	97	+	+	NOUN
ejpam-6066	178	98	3	3	NUM
ejpam-6066	178	99	,	,	PUNCT
ejpam-6066	178	100	6l	6l	NUM
ejpam-6066	178	101	+	+	CCONJ
ejpam-6066	178	102	5	5	NUM
ejpam-6066	178	103	,	,	PUNCT
ejpam-6066	178	104	24m+	24m+	NUM
ejpam-6066	178	105	n	n	CCONJ
ejpam-6066	178	106	,	,	PUNCT
ejpam-6066	178	107	72r	72r	NUM
ejpam-6066	179	1	+	+	CCONJ
ejpam-6066	179	2	43)|k	43)|k	NUM
ejpam-6066	179	3	,	,	PUNCT
ejpam-6066	179	4	l	l	NOUN
ejpam-6066	179	5	,	,	PUNCT
ejpam-6066	179	6	m	m	PROPN
ejpam-6066	179	7	,	,	PUNCT
ejpam-6066	179	8	r	r	NOUN
ejpam-6066	179	9	∈	∈	PROPN
ejpam-6066	179	10	z+	z+	NUM
ejpam-6066	179	11	0	0	NUM
ejpam-6066	179	12	and	and	CCONJ
ejpam-6066	179	13	n	n	CCONJ
ejpam-6066	179	14	=	=	SYM
ejpam-6066	179	15	3	3	NUM
ejpam-6066	179	16	,	,	PUNCT
ejpam-6066	179	17	9	9	NUM
ejpam-6066	179	18	,	,	PUNCT
ejpam-6066	179	19	15	15	NUM
ejpam-6066	179	20	,	,	PUNCT
ejpam-6066	179	21	21	21	NUM
ejpam-6066	179	22	}	}	PUNCT
ejpam-6066	179	23	∪	∪	X
ejpam-6066	179	24	{	{	PUNCT
ejpam-6066	179	25	(	(	PUNCT
ejpam-6066	179	26	6k	6k	NOUN
ejpam-6066	179	27	+	+	X
ejpam-6066	179	28	5	5	NUM
ejpam-6066	179	29	,	,	PUNCT
ejpam-6066	179	30	6l	6l	NUM
ejpam-6066	179	31	+	+	CCONJ
ejpam-6066	179	32	1	1	NUM
ejpam-6066	179	33	,	,	PUNCT
ejpam-6066	179	34	24m+	24m+	NUM
ejpam-6066	179	35	n	n	CCONJ
ejpam-6066	179	36	,	,	PUNCT
ejpam-6066	179	37	72r	72r	NUM
ejpam-6066	180	1	+	+	CCONJ
ejpam-6066	180	2	43)|k	43)|k	NUM
ejpam-6066	180	3	,	,	PUNCT
ejpam-6066	180	4	l	l	NOUN
ejpam-6066	180	5	,	,	PUNCT
ejpam-6066	180	6	m	m	PROPN
ejpam-6066	180	7	,	,	PUNCT
ejpam-6066	180	8	r	r	NOUN
ejpam-6066	180	9	∈	∈	PROPN
ejpam-6066	180	10	z+	z+	NUM
ejpam-6066	180	11	0	0	NUM
ejpam-6066	180	12	and	and	CCONJ
ejpam-6066	180	13	n	n	CCONJ
ejpam-6066	180	14	=	=	SYM
ejpam-6066	180	15	3	3	NUM
ejpam-6066	180	16	,	,	PUNCT
ejpam-6066	180	17	9	9	NUM
ejpam-6066	180	18	,	,	PUNCT
ejpam-6066	180	19	15	15	NUM
ejpam-6066	180	20	,	,	PUNCT
ejpam-6066	180	21	21	21	NUM
ejpam-6066	180	22	}	}	PUNCT
ejpam-6066	180	23	∪	∪	X
ejpam-6066	180	24	{	{	PUNCT
ejpam-6066	180	25	(	(	PUNCT
ejpam-6066	180	26	6k	6k	NOUN
ejpam-6066	180	27	+	+	NOUN
ejpam-6066	180	28	3	3	NUM
ejpam-6066	180	29	,	,	PUNCT
ejpam-6066	180	30	6l	6l	NUM
ejpam-6066	180	31	+	+	CCONJ
ejpam-6066	180	32	1	1	NUM
ejpam-6066	180	33	,	,	PUNCT
ejpam-6066	180	34	24m+	24m+	NUM
ejpam-6066	180	35	n	n	CCONJ
ejpam-6066	180	36	,	,	PUNCT
ejpam-6066	180	37	72r	72r	NUM
ejpam-6066	181	1	+	+	CCONJ
ejpam-6066	181	2	67)|k	67)|k	NUM
ejpam-6066	181	3	,	,	PUNCT
ejpam-6066	181	4	l	l	NOUN
ejpam-6066	181	5	,	,	PUNCT
ejpam-6066	181	6	m	m	PROPN
ejpam-6066	181	7	,	,	PUNCT
ejpam-6066	181	8	r	r	NOUN
ejpam-6066	181	9	∈	∈	PROPN
ejpam-6066	181	10	z+	z+	NUM
ejpam-6066	181	11	0	0	NUM
ejpam-6066	181	12	and	and	CCONJ
ejpam-6066	181	13	n	n	CCONJ
ejpam-6066	181	14	=	=	SYM
ejpam-6066	181	15	3	3	NUM
ejpam-6066	181	16	,	,	PUNCT
ejpam-6066	181	17	9	9	NUM
ejpam-6066	181	18	,	,	PUNCT
ejpam-6066	181	19	15	15	NUM
ejpam-6066	181	20	,	,	PUNCT
ejpam-6066	181	21	21	21	NUM
ejpam-6066	181	22	}	}	PUNCT
ejpam-6066	181	23	∪	∪	X
ejpam-6066	181	24	{	{	PUNCT
ejpam-6066	181	25	(	(	PUNCT
ejpam-6066	181	26	6k	6k	NOUN
ejpam-6066	181	27	+	+	X
ejpam-6066	181	28	5	5	NUM
ejpam-6066	181	29	,	,	PUNCT
ejpam-6066	181	30	6l	6l	NUM
ejpam-6066	181	31	+	+	CCONJ
ejpam-6066	181	32	5	5	NUM
ejpam-6066	181	33	,	,	PUNCT
ejpam-6066	181	34	24m+	24m+	NUM
ejpam-6066	181	35	n	n	CCONJ
ejpam-6066	181	36	,	,	PUNCT
ejpam-6066	181	37	72r	72r	NUM
ejpam-6066	181	38	+	+	CCONJ
ejpam-6066	181	39	67)|k	67)|k	NUM
ejpam-6066	181	40	,	,	PUNCT
ejpam-6066	181	41	l	l	NOUN
ejpam-6066	181	42	,	,	PUNCT
ejpam-6066	181	43	m	m	PROPN
ejpam-6066	181	44	,	,	PUNCT
ejpam-6066	181	45	r	r	NOUN
ejpam-6066	181	46	∈	∈	PROPN
ejpam-6066	181	47	z+	z+	NUM
ejpam-6066	181	48	0	0	NUM
ejpam-6066	181	49	and	and	CCONJ
ejpam-6066	181	50	n	n	CCONJ
ejpam-6066	181	51	=	=	SYM
ejpam-6066	181	52	3	3	NUM
ejpam-6066	181	53	,	,	PUNCT
ejpam-6066	181	54	9	9	NUM
ejpam-6066	181	55	,	,	PUNCT
ejpam-6066	181	56	15	15	NUM
ejpam-6066	181	57	,	,	PUNCT
ejpam-6066	181	58	21	21	NUM
ejpam-6066	181	59	}	}	PUNCT
ejpam-6066	181	60	.	.	PUNCT
ejpam-6066	182	1	remark	remark	NOUN
ejpam-6066	182	2	1	1	NUM
ejpam-6066	182	3	.	.	PUNCT
ejpam-6066	183	1	this	this	DET
ejpam-6066	183	2	corollary	corollary	NOUN
ejpam-6066	183	3	implies	imply	VERB
ejpam-6066	183	4	that	that	SCONJ
ejpam-6066	183	5	for	for	ADP
ejpam-6066	183	6	any	any	DET
ejpam-6066	183	7	prime	prime	NOUN
ejpam-6066	183	8	p	p	NOUN
ejpam-6066	183	9	≥	≥	NUM
ejpam-6066	183	10	19	19	NUM
ejpam-6066	183	11	,	,	PUNCT
ejpam-6066	183	12	the	the	DET
ejpam-6066	183	13	equation	equation	NOUN
ejpam-6066	183	14	4(7x)−py	4(7x)−py	NOUN
ejpam-6066	183	15	=	=	SYM
ejpam-6066	183	16	z2	z2	PROPN
ejpam-6066	183	17	has	have	VERB
ejpam-6066	183	18	no	no	DET
ejpam-6066	183	19	solution	solution	NOUN
ejpam-6066	183	20	if	if	SCONJ
ejpam-6066	183	21	x	x	SYM
ejpam-6066	183	22	≡	≡	PROPN
ejpam-6066	183	23	3	3	NUM
ejpam-6066	183	24	or	or	CCONJ
ejpam-6066	183	25	5	5	NUM
ejpam-6066	183	26	(	(	PUNCT
ejpam-6066	183	27	mod	mod	NOUN
ejpam-6066	183	28	6	6	NUM
ejpam-6066	183	29	)	)	PUNCT
ejpam-6066	183	30	and	and	CCONJ
ejpam-6066	183	31	y	y	PROPN
ejpam-6066	183	32	≡	≡	PROPN
ejpam-6066	183	33	3	3	NUM
ejpam-6066	183	34	(	(	PUNCT
ejpam-6066	183	35	mod	mod	PROPN
ejpam-6066	183	36	6	6	NUM
ejpam-6066	183	37	)	)	PUNCT
ejpam-6066	183	38	.	.	PUNCT
ejpam-6066	184	1	proof	proof	NOUN
ejpam-6066	184	2	.	.	PUNCT
ejpam-6066	185	1	from	from	ADP
ejpam-6066	185	2	theorem	theorem	ADJ
ejpam-6066	185	3	1(iv	1(iv	NUM
ejpam-6066	185	4	)	)	PUNCT
ejpam-6066	185	5	,	,	PUNCT
ejpam-6066	185	6	we	we	PRON
ejpam-6066	185	7	know	know	VERB
ejpam-6066	185	8	that	that	SCONJ
ejpam-6066	185	9	all	all	DET
ejpam-6066	185	10	solutions	solution	NOUN
ejpam-6066	185	11	are	be	AUX
ejpam-6066	185	12	of	of	ADP
ejpam-6066	185	13	the	the	DET
ejpam-6066	185	14	form	form	NOUN
ejpam-6066	185	15	(	(	PUNCT
ejpam-6066	185	16	x	x	X
ejpam-6066	185	17	,	,	PUNCT
ejpam-6066	185	18	y	y	PROPN
ejpam-6066	185	19	,	,	PUNCT
ejpam-6066	185	20	z	z	PROPN
ejpam-6066	185	21	,	,	PUNCT
ejpam-6066	185	22	p	p	NOUN
ejpam-6066	185	23	)	)	PUNCT
ejpam-6066	185	24	∈	∈	PROPN
ejpam-6066	185	25	{	{	PUNCT
ejpam-6066	185	26	(	(	PUNCT
ejpam-6066	185	27	2k	2k	NUM
ejpam-6066	185	28	+	+	CCONJ
ejpam-6066	185	29	1	1	NUM
ejpam-6066	185	30	,	,	PUNCT
ejpam-6066	185	31	2l	2l	NOUN
ejpam-6066	185	32	+	+	CCONJ
ejpam-6066	185	33	1	1	NUM
ejpam-6066	185	34	,	,	PUNCT
ejpam-6066	185	35	24m+	24m+	NUM
ejpam-6066	185	36	n	n	CCONJ
ejpam-6066	185	37	,	,	PUNCT
ejpam-6066	185	38	24r	24r	PUNCT
ejpam-6066	186	1	+	+	X
ejpam-6066	186	2	19)|k	19)|k	NUM
ejpam-6066	186	3	,	,	PUNCT
ejpam-6066	186	4	l	l	NOUN
ejpam-6066	186	5	,	,	PUNCT
ejpam-6066	186	6	m	m	PROPN
ejpam-6066	186	7	,	,	PUNCT
ejpam-6066	186	8	r	r	NOUN
ejpam-6066	186	9	∈	∈	PROPN
ejpam-6066	186	10	z+	z+	NUM
ejpam-6066	186	11	0	0	NUM
ejpam-6066	186	12	and	and	CCONJ
ejpam-6066	186	13	n	n	CCONJ
ejpam-6066	186	14	=	=	SYM
ejpam-6066	186	15	3	3	NUM
ejpam-6066	186	16	,	,	PUNCT
ejpam-6066	186	17	9	9	NUM
ejpam-6066	186	18	,	,	PUNCT
ejpam-6066	186	19	15	15	NUM
ejpam-6066	186	20	,	,	PUNCT
ejpam-6066	186	21	21	21	NUM
ejpam-6066	186	22	}	}	PUNCT
ejpam-6066	186	23	.	.	PUNCT
ejpam-6066	187	1	we	we	PRON
ejpam-6066	187	2	first	first	ADV
ejpam-6066	187	3	consider	consider	VERB
ejpam-6066	187	4	z	z	NOUN
ejpam-6066	187	5	=	=	SYM
ejpam-6066	187	6	24m+	24m+	NOUN
ejpam-6066	187	7	n	n	CCONJ
ejpam-6066	187	8	where	where	SCONJ
ejpam-6066	187	9	m	m	VERB
ejpam-6066	187	10	≥	≥	NOUN
ejpam-6066	187	11	0	0	NUM
ejpam-6066	187	12	and	and	CCONJ
ejpam-6066	187	13	n	n	PRON
ejpam-6066	187	14	∈	∈	PROPN
ejpam-6066	187	15	{	{	PUNCT
ejpam-6066	187	16	3	3	NUM
ejpam-6066	187	17	,	,	PUNCT
ejpam-6066	187	18	9	9	NUM
ejpam-6066	187	19	,	,	PUNCT
ejpam-6066	187	20	15	15	NUM
ejpam-6066	187	21	,	,	PUNCT
ejpam-6066	187	22	21	21	NUM
ejpam-6066	187	23	}	}	PUNCT
ejpam-6066	187	24	.	.	PUNCT
ejpam-6066	188	1	since	since	SCONJ
ejpam-6066	188	2	every	every	DET
ejpam-6066	188	3	n	n	NOUN
ejpam-6066	188	4	is	be	AUX
ejpam-6066	188	5	divisible	divisible	ADJ
ejpam-6066	188	6	by	by	ADP
ejpam-6066	188	7	3	3	NUM
ejpam-6066	188	8	,	,	PUNCT
ejpam-6066	188	9	it	it	PRON
ejpam-6066	188	10	follows	follow	VERB
ejpam-6066	188	11	that	that	SCONJ
ejpam-6066	188	12	z2	z2	PROPN
ejpam-6066	188	13	≡	≡	PROPN
ejpam-6066	188	14	0	0	PUNCT
ejpam-6066	189	1	(	(	PUNCT
ejpam-6066	189	2	mod	mod	NOUN
ejpam-6066	189	3	9	9	NUM
ejpam-6066	189	4	)	)	PUNCT
ejpam-6066	189	5	.	.	PUNCT
ejpam-6066	190	1	next	next	ADV
ejpam-6066	190	2	,	,	PUNCT
ejpam-6066	190	3	we	we	PRON
ejpam-6066	190	4	observe	observe	VERB
ejpam-6066	190	5	that	that	SCONJ
ejpam-6066	190	6	p	p	NOUN
ejpam-6066	190	7	=	=	X
ejpam-6066	190	8	24r	24r	NOUN
ejpam-6066	191	1	+	+	CCONJ
ejpam-6066	191	2	19	19	NUM
ejpam-6066	191	3	for	for	ADP
ejpam-6066	191	4	some	some	DET
ejpam-6066	191	5	non	non	ADJ
ejpam-6066	191	6	-	-	ADJ
ejpam-6066	191	7	negative	negative	ADJ
ejpam-6066	191	8	integer	integer	NOUN
ejpam-6066	191	9	r	r	NOUN
ejpam-6066	191	10	,	,	PUNCT
ejpam-6066	191	11	which	which	PRON
ejpam-6066	191	12	implies	imply	VERB
ejpam-6066	191	13	p	p	X
ejpam-6066	191	14	≡	≡	PROPN
ejpam-6066	191	15	6r	6r	NUM
ejpam-6066	192	1	+	+	CCONJ
ejpam-6066	192	2	1	1	NUM
ejpam-6066	192	3	(	(	PUNCT
ejpam-6066	192	4	mod	mod	NOUN
ejpam-6066	192	5	9	9	NUM
ejpam-6066	192	6	)	)	PUNCT
ejpam-6066	192	7	.	.	PUNCT
ejpam-6066	193	1	by	by	ADP
ejpam-6066	193	2	lemma	lemma	PROPN
ejpam-6066	193	3	1	1	NUM
ejpam-6066	193	4	we	we	PRON
ejpam-6066	193	5	have	have	VERB
ejpam-6066	193	6	that	that	DET
ejpam-6066	193	7	py	py	PROPN
ejpam-6066	193	8	≡	≡	PROPN
ejpam-6066	194	1			ADV
ejpam-6066	194	2	1	1	NUM
ejpam-6066	194	3	(	(	PUNCT
ejpam-6066	194	4	mod	mod	NOUN
ejpam-6066	194	5	9	9	NUM
ejpam-6066	194	6	)	)	PUNCT
ejpam-6066	194	7	if	if	SCONJ
ejpam-6066	194	8	y	y	PROPN
ejpam-6066	194	9	≡	≡	PROPN
ejpam-6066	194	10	0	0	PUNCT
ejpam-6066	195	1	(	(	PUNCT
ejpam-6066	195	2	mod	mod	NOUN
ejpam-6066	195	3	3	3	NUM
ejpam-6066	195	4	)	)	PUNCT
ejpam-6066	195	5	,	,	PUNCT
ejpam-6066	195	6	p	p	X
ejpam-6066	195	7	(	(	PUNCT
ejpam-6066	195	8	mod	mod	PROPN
ejpam-6066	195	9	9	9	NUM
ejpam-6066	195	10	)	)	PUNCT
ejpam-6066	195	11	if	if	SCONJ
ejpam-6066	195	12	y	y	PROPN
ejpam-6066	195	13	≡	≡	PROPN
ejpam-6066	195	14	1	1	NUM
ejpam-6066	195	15	(	(	PUNCT
ejpam-6066	195	16	mod	mod	NOUN
ejpam-6066	195	17	3	3	NUM
ejpam-6066	195	18	)	)	PUNCT
ejpam-6066	195	19	,	,	PUNCT
ejpam-6066	195	20	2−	2−	NUM
ejpam-6066	195	21	p	p	NOUN
ejpam-6066	195	22	(	(	PUNCT
ejpam-6066	195	23	mod	mod	PROPN
ejpam-6066	195	24	9	9	NUM
ejpam-6066	195	25	)	)	PUNCT
ejpam-6066	195	26	if	if	SCONJ
ejpam-6066	195	27	y	y	PROPN
ejpam-6066	195	28	≡	≡	PROPN
ejpam-6066	195	29	2	2	NUM
ejpam-6066	195	30	(	(	PUNCT
ejpam-6066	195	31	mod	mod	NOUN
ejpam-6066	195	32	3	3	NUM
ejpam-6066	195	33	)	)	PUNCT
ejpam-6066	195	34	.	.	PUNCT
ejpam-6066	196	1	next	next	ADV
ejpam-6066	196	2	,	,	PUNCT
ejpam-6066	196	3	we	we	PRON
ejpam-6066	196	4	examine	examine	VERB
ejpam-6066	196	5	the	the	DET
ejpam-6066	196	6	behavior	behavior	NOUN
ejpam-6066	196	7	4(7x	4(7x	NUM
ejpam-6066	196	8	)	)	PUNCT
ejpam-6066	196	9	with	with	ADP
ejpam-6066	196	10	mod	mod	PROPN
ejpam-6066	196	11	9	9	NUM
ejpam-6066	196	12	and	and	CCONJ
ejpam-6066	196	13	we	we	PRON
ejpam-6066	196	14	have	have	VERB
ejpam-6066	196	15	4(7x	4(7x	NUM
ejpam-6066	196	16	)	)	PUNCT
ejpam-6066	196	17	≡	≡	PROPN
ejpam-6066	197	1			ADP
ejpam-6066	197	2	4(1	4(1	NOUN
ejpam-6066	197	3	)	)	PUNCT
ejpam-6066	198	1	≡	≡	PROPN
ejpam-6066	198	2	4	4	NUM
ejpam-6066	198	3	(	(	PUNCT
ejpam-6066	198	4	mod	mod	NOUN
ejpam-6066	198	5	9	9	NUM
ejpam-6066	198	6	)	)	PUNCT
ejpam-6066	198	7	if	if	SCONJ
ejpam-6066	198	8	x	x	SYM
ejpam-6066	198	9	≡	≡	PROPN
ejpam-6066	198	10	0	0	PUNCT
ejpam-6066	198	11	(	(	PUNCT
ejpam-6066	198	12	mod	mod	NOUN
ejpam-6066	198	13	3	3	NUM
ejpam-6066	198	14	)	)	PUNCT
ejpam-6066	198	15	,	,	PUNCT
ejpam-6066	198	16	4(7	4(7	NUM
ejpam-6066	198	17	)	)	PUNCT
ejpam-6066	198	18	≡	≡	PROPN
ejpam-6066	198	19	1	1	NUM
ejpam-6066	198	20	(	(	PUNCT
ejpam-6066	198	21	mod	mod	NOUN
ejpam-6066	198	22	9	9	NUM
ejpam-6066	198	23	)	)	PUNCT
ejpam-6066	198	24	if	if	SCONJ
ejpam-6066	198	25	x	x	SYM
ejpam-6066	198	26	≡	≡	PROPN
ejpam-6066	198	27	1	1	NUM
ejpam-6066	198	28	(	(	PUNCT
ejpam-6066	198	29	mod	mod	NOUN
ejpam-6066	198	30	3	3	NUM
ejpam-6066	198	31	)	)	PUNCT
ejpam-6066	198	32	,	,	PUNCT
ejpam-6066	198	33	4(7	4(7	NUM
ejpam-6066	198	34	)	)	PUNCT
ejpam-6066	198	35	≡	≡	PROPN
ejpam-6066	198	36	7	7	NUM
ejpam-6066	198	37	(	(	PUNCT
ejpam-6066	198	38	mod	mod	NOUN
ejpam-6066	198	39	9	9	NUM
ejpam-6066	198	40	)	)	PUNCT
ejpam-6066	198	41	if	if	SCONJ
ejpam-6066	198	42	x	x	SYM
ejpam-6066	198	43	≡	≡	PROPN
ejpam-6066	198	44	2	2	NUM
ejpam-6066	198	45	(	(	PUNCT
ejpam-6066	198	46	mod	mod	NOUN
ejpam-6066	198	47	3	3	NUM
ejpam-6066	198	48	)	)	PUNCT
ejpam-6066	198	49	.	.	PUNCT
ejpam-6066	199	1	then	then	ADV
ejpam-6066	199	2	we	we	PRON
ejpam-6066	199	3	analyze	analyze	VERB
ejpam-6066	199	4	the	the	DET
ejpam-6066	199	5	equation	equation	NOUN
ejpam-6066	199	6	4(7x	4(7x	NUM
ejpam-6066	199	7	)	)	PUNCT
ejpam-6066	200	1	−	−	PROPN
ejpam-6066	200	2	py	py	PROPN
ejpam-6066	200	3	≡	≡	PROPN
ejpam-6066	200	4	z2	z2	PROPN
ejpam-6066	200	5	≡	≡	PROPN
ejpam-6066	200	6	0	0	PUNCT
ejpam-6066	201	1	(	(	PUNCT
ejpam-6066	201	2	mod	mod	NOUN
ejpam-6066	201	3	9	9	NUM
ejpam-6066	201	4	)	)	PUNCT
ejpam-6066	201	5	.	.	PUNCT
ejpam-6066	202	1	the	the	DET
ejpam-6066	202	2	resulting	result	VERB
ejpam-6066	202	3	congruence	congruence	NOUN
ejpam-6066	202	4	modulo	modulo	NOUN
ejpam-6066	202	5	9	9	NUM
ejpam-6066	202	6	for	for	ADP
ejpam-6066	202	7	each	each	DET
ejpam-6066	202	8	pair	pair	NOUN
ejpam-6066	202	9	(	(	PUNCT
ejpam-6066	202	10	x	x	SYM
ejpam-6066	202	11	mod	mod	PROPN
ejpam-6066	202	12	3	3	NUM
ejpam-6066	202	13	,	,	PUNCT
ejpam-6066	202	14	y	y	PROPN
ejpam-6066	202	15	mod	mod	PROPN
ejpam-6066	202	16	3	3	X
ejpam-6066	202	17	)	)	PUNCT
ejpam-6066	202	18	yields	yield	VERB
ejpam-6066	202	19	the	the	DET
ejpam-6066	202	20	following	following	NOUN
ejpam-6066	202	21	:	:	PUNCT
ejpam-6066	202	22	no	no	DET
ejpam-6066	202	23	solution	solution	NOUN
ejpam-6066	202	24	if	if	SCONJ
ejpam-6066	202	25	x	x	SYM
ejpam-6066	202	26	≡	≡	PROPN
ejpam-6066	202	27	0	0	PUNCT
ejpam-6066	202	28	(	(	PUNCT
ejpam-6066	202	29	mod	mod	NOUN
ejpam-6066	202	30	3	3	NUM
ejpam-6066	202	31	)	)	PUNCT
ejpam-6066	202	32	and	and	CCONJ
ejpam-6066	202	33	y	y	PROPN
ejpam-6066	202	34	≡	≡	PROPN
ejpam-6066	202	35	0	0	PUNCT
ejpam-6066	203	1	(	(	PUNCT
ejpam-6066	203	2	mod	mod	NOUN
ejpam-6066	203	3	3	3	NUM
ejpam-6066	203	4	)	)	PUNCT
ejpam-6066	203	5	,	,	PUNCT
ejpam-6066	203	6	p	p	PROPN
ejpam-6066	203	7	≡	≡	PROPN
ejpam-6066	203	8	4	4	NUM
ejpam-6066	203	9	(	(	PUNCT
ejpam-6066	203	10	mod	mod	NOUN
ejpam-6066	203	11	9	9	NUM
ejpam-6066	203	12	)	)	PUNCT
ejpam-6066	203	13	if	if	SCONJ
ejpam-6066	203	14	x	x	SYM
ejpam-6066	203	15	≡	≡	PROPN
ejpam-6066	203	16	0	0	PUNCT
ejpam-6066	203	17	(	(	PUNCT
ejpam-6066	203	18	mod	mod	NOUN
ejpam-6066	203	19	3	3	NUM
ejpam-6066	203	20	)	)	PUNCT
ejpam-6066	203	21	and	and	CCONJ
ejpam-6066	203	22	y	y	PROPN
ejpam-6066	203	23	≡	≡	PROPN
ejpam-6066	203	24	1	1	NUM
ejpam-6066	203	25	(	(	PUNCT
ejpam-6066	203	26	mod	mod	NOUN
ejpam-6066	203	27	3	3	NUM
ejpam-6066	203	28	)	)	PUNCT
ejpam-6066	203	29	,	,	PUNCT
ejpam-6066	203	30	p	p	PROPN
ejpam-6066	203	31	≡	≡	PROPN
ejpam-6066	203	32	7	7	NUM
ejpam-6066	203	33	(	(	PUNCT
ejpam-6066	203	34	mod	mod	NOUN
ejpam-6066	203	35	9	9	NUM
ejpam-6066	203	36	)	)	PUNCT
ejpam-6066	203	37	if	if	SCONJ
ejpam-6066	203	38	x	x	SYM
ejpam-6066	203	39	≡	≡	PROPN
ejpam-6066	203	40	0	0	PUNCT
ejpam-6066	203	41	(	(	PUNCT
ejpam-6066	203	42	mod	mod	NOUN
ejpam-6066	203	43	3	3	NUM
ejpam-6066	203	44	)	)	PUNCT
ejpam-6066	203	45	and	and	CCONJ
ejpam-6066	203	46	y	y	PROPN
ejpam-6066	203	47	≡	≡	PROPN
ejpam-6066	203	48	2	2	NUM
ejpam-6066	203	49	(	(	PUNCT
ejpam-6066	203	50	mod	mod	NOUN
ejpam-6066	203	51	3	3	NUM
ejpam-6066	203	52	)	)	PUNCT
ejpam-6066	203	53	,	,	PUNCT
ejpam-6066	203	54	no	no	DET
ejpam-6066	203	55	additional	additional	ADJ
ejpam-6066	203	56	information	information	NOUN
ejpam-6066	203	57	if	if	SCONJ
ejpam-6066	203	58	x	x	SYM
ejpam-6066	203	59	≡	≡	PROPN
ejpam-6066	203	60	1	1	NUM
ejpam-6066	203	61	(	(	PUNCT
ejpam-6066	203	62	mod	mod	NOUN
ejpam-6066	203	63	3	3	NUM
ejpam-6066	203	64	)	)	PUNCT
ejpam-6066	203	65	and	and	CCONJ
ejpam-6066	203	66	y	y	PROPN
ejpam-6066	203	67	≡	≡	PROPN
ejpam-6066	203	68	0	0	PUNCT
ejpam-6066	204	1	(	(	PUNCT
ejpam-6066	204	2	mod	mod	NOUN
ejpam-6066	204	3	3	3	NUM
ejpam-6066	204	4	)	)	PUNCT
ejpam-6066	204	5	,	,	PUNCT
ejpam-6066	204	6	p	p	PROPN
ejpam-6066	204	7	≡	≡	PROPN
ejpam-6066	204	8	1	1	NUM
ejpam-6066	204	9	(	(	PUNCT
ejpam-6066	204	10	mod	mod	NOUN
ejpam-6066	204	11	9	9	NUM
ejpam-6066	204	12	)	)	PUNCT
ejpam-6066	204	13	if	if	SCONJ
ejpam-6066	204	14	x	x	SYM
ejpam-6066	204	15	≡	≡	PROPN
ejpam-6066	204	16	1	1	NUM
ejpam-6066	204	17	(	(	PUNCT
ejpam-6066	204	18	mod	mod	NOUN
ejpam-6066	204	19	3	3	NUM
ejpam-6066	204	20	)	)	PUNCT
ejpam-6066	204	21	and	and	CCONJ
ejpam-6066	204	22	y	y	PROPN
ejpam-6066	204	23	≡	≡	PROPN
ejpam-6066	204	24	1	1	NUM
ejpam-6066	204	25	(	(	PUNCT
ejpam-6066	204	26	mod	mod	NOUN
ejpam-6066	204	27	3	3	NUM
ejpam-6066	204	28	)	)	PUNCT
ejpam-6066	204	29	,	,	PUNCT
ejpam-6066	204	30	p	p	PROPN
ejpam-6066	204	31	≡	≡	PROPN
ejpam-6066	204	32	1	1	NUM
ejpam-6066	204	33	(	(	PUNCT
ejpam-6066	204	34	mod	mod	NOUN
ejpam-6066	204	35	9	9	NUM
ejpam-6066	204	36	)	)	PUNCT
ejpam-6066	204	37	if	if	SCONJ
ejpam-6066	204	38	x	x	SYM
ejpam-6066	204	39	≡	≡	PROPN
ejpam-6066	204	40	1	1	NUM
ejpam-6066	204	41	(	(	PUNCT
ejpam-6066	204	42	mod	mod	NOUN
ejpam-6066	204	43	3	3	NUM
ejpam-6066	204	44	)	)	PUNCT
ejpam-6066	204	45	and	and	CCONJ
ejpam-6066	204	46	y	y	PROPN
ejpam-6066	204	47	≡	≡	PROPN
ejpam-6066	204	48	2	2	NUM
ejpam-6066	204	49	(	(	PUNCT
ejpam-6066	204	50	mod	mod	NOUN
ejpam-6066	204	51	3	3	NUM
ejpam-6066	204	52	)	)	PUNCT
ejpam-6066	204	53	,	,	PUNCT
ejpam-6066	204	54	no	no	DET
ejpam-6066	204	55	solution	solution	NOUN
ejpam-6066	204	56	if	if	SCONJ
ejpam-6066	204	57	x	x	SYM
ejpam-6066	204	58	≡	≡	PROPN
ejpam-6066	204	59	2	2	NUM
ejpam-6066	204	60	(	(	PUNCT
ejpam-6066	204	61	mod	mod	NOUN
ejpam-6066	204	62	3	3	NUM
ejpam-6066	204	63	)	)	PUNCT
ejpam-6066	204	64	and	and	CCONJ
ejpam-6066	204	65	y	y	PROPN
ejpam-6066	204	66	≡	≡	PROPN
ejpam-6066	204	67	0	0	PUNCT
ejpam-6066	205	1	(	(	PUNCT
ejpam-6066	205	2	mod	mod	NOUN
ejpam-6066	205	3	3	3	NUM
ejpam-6066	205	4	)	)	PUNCT
ejpam-6066	205	5	,	,	PUNCT
ejpam-6066	205	6	p	p	PROPN
ejpam-6066	205	7	≡	≡	PROPN
ejpam-6066	205	8	7	7	NUM
ejpam-6066	205	9	(	(	PUNCT
ejpam-6066	205	10	mod	mod	NOUN
ejpam-6066	205	11	9	9	NUM
ejpam-6066	205	12	)	)	PUNCT
ejpam-6066	205	13	if	if	SCONJ
ejpam-6066	205	14	x	x	SYM
ejpam-6066	205	15	≡	≡	PROPN
ejpam-6066	205	16	2	2	NUM
ejpam-6066	205	17	(	(	PUNCT
ejpam-6066	205	18	mod	mod	NOUN
ejpam-6066	205	19	3	3	NUM
ejpam-6066	205	20	)	)	PUNCT
ejpam-6066	205	21	and	and	CCONJ
ejpam-6066	205	22	y	y	PROPN
ejpam-6066	205	23	≡	≡	PROPN
ejpam-6066	205	24	1	1	NUM
ejpam-6066	205	25	(	(	PUNCT
ejpam-6066	205	26	mod	mod	NOUN
ejpam-6066	205	27	3	3	NUM
ejpam-6066	205	28	)	)	PUNCT
ejpam-6066	205	29	,	,	PUNCT
ejpam-6066	205	30	p	p	PROPN
ejpam-6066	205	31	≡	≡	PROPN
ejpam-6066	205	32	4	4	NUM
ejpam-6066	205	33	(	(	PUNCT
ejpam-6066	205	34	mod	mod	NOUN
ejpam-6066	205	35	9	9	NUM
ejpam-6066	205	36	)	)	PUNCT
ejpam-6066	205	37	if	if	SCONJ
ejpam-6066	205	38	x	x	SYM
ejpam-6066	205	39	≡	≡	PROPN
ejpam-6066	205	40	2	2	NUM
ejpam-6066	205	41	(	(	PUNCT
ejpam-6066	205	42	mod	mod	NOUN
ejpam-6066	205	43	3	3	NUM
ejpam-6066	205	44	)	)	PUNCT
ejpam-6066	205	45	and	and	CCONJ
ejpam-6066	205	46	y	y	PROPN
ejpam-6066	205	47	≡	≡	PROPN
ejpam-6066	205	48	2	2	NUM
ejpam-6066	205	49	(	(	PUNCT
ejpam-6066	205	50	mod	mod	NOUN
ejpam-6066	205	51	3	3	NUM
ejpam-6066	205	52	)	)	PUNCT
ejpam-6066	205	53	.	.	PUNCT
ejpam-6066	206	1	combining	combine	VERB
ejpam-6066	206	2	these	these	DET
ejpam-6066	206	3	observations	observation	NOUN
ejpam-6066	206	4	with	with	ADP
ejpam-6066	206	5	theorem	theorem	ADJ
ejpam-6066	206	6	1(iv	1(iv	NUM
ejpam-6066	206	7	)	)	PUNCT
ejpam-6066	206	8	and	and	CCONJ
ejpam-6066	206	9	applying	apply	VERB
ejpam-6066	206	10	the	the	DET
ejpam-6066	206	11	chinese	chinese	ADJ
ejpam-6066	206	12	remainder	remainder	NOUN
ejpam-6066	206	13	theorem	theorem	NOUN
ejpam-6066	206	14	to	to	ADP
ejpam-6066	206	15	parameters	parameter	NOUN
ejpam-6066	206	16	x	x	SYM
ejpam-6066	206	17	,	,	PUNCT
ejpam-6066	206	18	y	y	PROPN
ejpam-6066	206	19	and	and	CCONJ
ejpam-6066	206	20	p	p	X
ejpam-6066	206	21	,	,	PUNCT
ejpam-6066	206	22	we	we	PRON
ejpam-6066	206	23	derive	derive	VERB
ejpam-6066	206	24	the	the	DET
ejpam-6066	206	25	following	following	ADJ
ejpam-6066	206	26	refined	refined	ADJ
ejpam-6066	206	27	conditions	condition	NOUN
ejpam-6066	206	28	for	for	ADP
ejpam-6066	206	29	a	a	DET
ejpam-6066	206	30	prime	prime	NOUN
ejpam-6066	206	31	p	p	X
ejpam-6066	206	32	≥	≥	NUM
ejpam-6066	206	33	19	19	NUM
ejpam-6066	206	34	:	:	PUNCT
ejpam-6066	206	35	k.	k.	PROPN
ejpam-6066	207	1	laipaporn	laipaporn	PROPN
ejpam-6066	207	2	et	et	PROPN
ejpam-6066	207	3	al	al	PROPN
ejpam-6066	207	4	.	.	PUNCT
ejpam-6066	207	5	/	/	SYM
ejpam-6066	207	6	eur	eur	PROPN
ejpam-6066	207	7	.	.	PUNCT
ejpam-6066	208	1	j.	j.	PROPN
ejpam-6066	208	2	pure	pure	PROPN
ejpam-6066	208	3	appl	appl	PROPN
ejpam-6066	208	4	.	.	PROPN
ejpam-6066	208	5	math	math	PROPN
ejpam-6066	208	6	,	,	PUNCT
ejpam-6066	208	7	18	18	NUM
ejpam-6066	208	8	(	(	PUNCT
ejpam-6066	208	9	3	3	NUM
ejpam-6066	208	10	)	)	PUNCT
ejpam-6066	208	11	(	(	PUNCT
ejpam-6066	208	12	2025	2025	NUM
ejpam-6066	208	13	)	)	PUNCT
ejpam-6066	208	14	,	,	PUNCT
ejpam-6066	208	15	6066	6066	NUM
ejpam-6066	208	16	8	8	NUM
ejpam-6066	208	17	of	of	ADP
ejpam-6066	208	18	13	13	NUM
ejpam-6066	208	19	(	(	PUNCT
ejpam-6066	208	20	i	i	NOUN
ejpam-6066	208	21	)	)	PUNCT
ejpam-6066	208	22	if	if	SCONJ
ejpam-6066	208	23	x	x	SYM
ejpam-6066	208	24	≡	≡	PROPN
ejpam-6066	208	25	3	3	NUM
ejpam-6066	208	26	or	or	CCONJ
ejpam-6066	208	27	5	5	NUM
ejpam-6066	208	28	(	(	PUNCT
ejpam-6066	208	29	mod	mod	NOUN
ejpam-6066	208	30	6	6	NUM
ejpam-6066	208	31	)	)	PUNCT
ejpam-6066	208	32	and	and	CCONJ
ejpam-6066	208	33	y	y	PROPN
ejpam-6066	208	34	≡	≡	PROPN
ejpam-6066	208	35	3	3	NUM
ejpam-6066	208	36	(	(	PUNCT
ejpam-6066	208	37	mod	mod	PROPN
ejpam-6066	208	38	6	6	NUM
ejpam-6066	208	39	)	)	PUNCT
ejpam-6066	208	40	then	then	ADV
ejpam-6066	208	41	the	the	DET
ejpam-6066	208	42	equation	equation	NOUN
ejpam-6066	208	43	has	have	VERB
ejpam-6066	208	44	no	no	DET
ejpam-6066	208	45	solution	solution	NOUN
ejpam-6066	208	46	.	.	PUNCT
ejpam-6066	209	1	(	(	PUNCT
ejpam-6066	209	2	ii	ii	NOUN
ejpam-6066	209	3	)	)	PUNCT
ejpam-6066	209	4	under	under	ADP
ejpam-6066	209	5	the	the	DET
ejpam-6066	209	6	assumption	assumption	NOUN
ejpam-6066	209	7	that	that	SCONJ
ejpam-6066	209	8	a	a	DET
ejpam-6066	209	9	solution	solution	NOUN
ejpam-6066	209	10	exists	exist	VERB
ejpam-6066	209	11	:	:	PUNCT
ejpam-6066	209	12	(	(	PUNCT
ejpam-6066	209	13	a	a	X
ejpam-6066	209	14	)	)	PUNCT
ejpam-6066	209	15	if	if	SCONJ
ejpam-6066	209	16	(	(	PUNCT
ejpam-6066	209	17	x	x	SYM
ejpam-6066	209	18	≡	≡	PROPN
ejpam-6066	209	19	3	3	NUM
ejpam-6066	209	20	(	(	PUNCT
ejpam-6066	209	21	mod	mod	NOUN
ejpam-6066	209	22	6	6	NUM
ejpam-6066	209	23	)	)	PUNCT
ejpam-6066	209	24	and	and	CCONJ
ejpam-6066	209	25	y	y	PROPN
ejpam-6066	209	26	≡	≡	PROPN
ejpam-6066	209	27	1	1	NUM
ejpam-6066	209	28	(	(	PUNCT
ejpam-6066	209	29	mod	mod	PROPN
ejpam-6066	209	30	6	6	NUM
ejpam-6066	209	31	)	)	PUNCT
ejpam-6066	209	32	)	)	PUNCT
ejpam-6066	209	33	or	or	CCONJ
ejpam-6066	209	34	(	(	PUNCT
ejpam-6066	209	35	x	x	SYM
ejpam-6066	209	36	≡	≡	PROPN
ejpam-6066	209	37	5	5	NUM
ejpam-6066	209	38	(	(	PUNCT
ejpam-6066	209	39	mod	mod	NOUN
ejpam-6066	209	40	6	6	NUM
ejpam-6066	209	41	)	)	PUNCT
ejpam-6066	209	42	and	and	CCONJ
ejpam-6066	209	43	y	y	PROPN
ejpam-6066	209	44	≡	≡	PROPN
ejpam-6066	209	45	5	5	NUM
ejpam-6066	209	46	(	(	PUNCT
ejpam-6066	209	47	mod	mod	PROPN
ejpam-6066	209	48	6	6	NUM
ejpam-6066	209	49	)	)	PUNCT
ejpam-6066	209	50	)	)	PUNCT
ejpam-6066	209	51	then	then	ADV
ejpam-6066	209	52	p	p	PROPN
ejpam-6066	209	53	≡	≡	PROPN
ejpam-6066	209	54	67	67	NUM
ejpam-6066	209	55	(	(	PUNCT
ejpam-6066	209	56	mod	mod	PROPN
ejpam-6066	209	57	72	72	NUM
ejpam-6066	209	58	)	)	PUNCT
ejpam-6066	209	59	.	.	PUNCT
ejpam-6066	210	1	(	(	PUNCT
ejpam-6066	210	2	b	b	X
ejpam-6066	210	3	)	)	PUNCT
ejpam-6066	210	4	if	if	SCONJ
ejpam-6066	210	5	(	(	PUNCT
ejpam-6066	210	6	x	x	SYM
ejpam-6066	210	7	≡	≡	PROPN
ejpam-6066	210	8	3	3	NUM
ejpam-6066	210	9	(	(	PUNCT
ejpam-6066	210	10	mod	mod	NOUN
ejpam-6066	210	11	6	6	NUM
ejpam-6066	210	12	)	)	PUNCT
ejpam-6066	210	13	and	and	CCONJ
ejpam-6066	210	14	y	y	PROPN
ejpam-6066	210	15	≡	≡	PROPN
ejpam-6066	210	16	5	5	NUM
ejpam-6066	210	17	(	(	PUNCT
ejpam-6066	210	18	mod	mod	PROPN
ejpam-6066	210	19	6	6	NUM
ejpam-6066	210	20	)	)	PUNCT
ejpam-6066	210	21	)	)	PUNCT
ejpam-6066	210	22	or	or	CCONJ
ejpam-6066	210	23	(	(	PUNCT
ejpam-6066	210	24	x	x	SYM
ejpam-6066	210	25	≡	≡	PROPN
ejpam-6066	210	26	5	5	NUM
ejpam-6066	210	27	(	(	PUNCT
ejpam-6066	210	28	mod	mod	NOUN
ejpam-6066	210	29	6	6	NUM
ejpam-6066	210	30	)	)	PUNCT
ejpam-6066	210	31	and	and	CCONJ
ejpam-6066	210	32	y	y	PROPN
ejpam-6066	210	33	≡	≡	PROPN
ejpam-6066	210	34	1	1	NUM
ejpam-6066	210	35	(	(	PUNCT
ejpam-6066	210	36	mod	mod	PROPN
ejpam-6066	210	37	6	6	NUM
ejpam-6066	210	38	)	)	PUNCT
ejpam-6066	210	39	)	)	PUNCT
ejpam-6066	210	40	then	then	ADV
ejpam-6066	210	41	p	p	PROPN
ejpam-6066	210	42	≡	≡	PROPN
ejpam-6066	210	43	43	43	NUM
ejpam-6066	210	44	(	(	PUNCT
ejpam-6066	210	45	mod	mod	PROPN
ejpam-6066	210	46	72	72	NUM
ejpam-6066	210	47	)	)	PUNCT
ejpam-6066	210	48	.	.	PUNCT
ejpam-6066	211	1	(	(	PUNCT
ejpam-6066	211	2	c	c	X
ejpam-6066	211	3	)	)	PUNCT
ejpam-6066	211	4	if	if	SCONJ
ejpam-6066	211	5	x	x	SYM
ejpam-6066	211	6	≡	≡	PROPN
ejpam-6066	211	7	1	1	NUM
ejpam-6066	211	8	(	(	PUNCT
ejpam-6066	211	9	mod	mod	NOUN
ejpam-6066	211	10	6	6	NUM
ejpam-6066	211	11	)	)	PUNCT
ejpam-6066	211	12	and	and	CCONJ
ejpam-6066	211	13	y	y	PROPN
ejpam-6066	211	14	≡	≡	PROPN
ejpam-6066	211	15	1	1	NUM
ejpam-6066	211	16	or	or	CCONJ
ejpam-6066	211	17	5	5	NUM
ejpam-6066	211	18	(	(	PUNCT
ejpam-6066	211	19	mod	mod	NOUN
ejpam-6066	211	20	6	6	NUM
ejpam-6066	211	21	)	)	PUNCT
ejpam-6066	211	22	then	then	ADV
ejpam-6066	211	23	p	p	PROPN
ejpam-6066	211	24	≡	≡	PROPN
ejpam-6066	211	25	19	19	NUM
ejpam-6066	211	26	(	(	PUNCT
ejpam-6066	211	27	mod	mod	PROPN
ejpam-6066	211	28	72	72	NUM
ejpam-6066	211	29	)	)	PUNCT
ejpam-6066	211	30	.	.	PUNCT
ejpam-6066	212	1	(	(	PUNCT
ejpam-6066	212	2	d	d	X
ejpam-6066	212	3	)	)	PUNCT
ejpam-6066	212	4	if	if	SCONJ
ejpam-6066	212	5	x	x	SYM
ejpam-6066	212	6	≡	≡	PROPN
ejpam-6066	212	7	1	1	NUM
ejpam-6066	212	8	(	(	PUNCT
ejpam-6066	212	9	mod	mod	NOUN
ejpam-6066	212	10	6	6	NUM
ejpam-6066	212	11	)	)	PUNCT
ejpam-6066	212	12	and	and	CCONJ
ejpam-6066	212	13	y	y	PROPN
ejpam-6066	212	14	≡	≡	PROPN
ejpam-6066	212	15	3	3	NUM
ejpam-6066	212	16	(	(	PUNCT
ejpam-6066	212	17	mod	mod	PROPN
ejpam-6066	212	18	6	6	NUM
ejpam-6066	212	19	)	)	PUNCT
ejpam-6066	212	20	then	then	ADV
ejpam-6066	212	21	p	p	PROPN
ejpam-6066	212	22	≡	≡	PROPN
ejpam-6066	212	23	19	19	NUM
ejpam-6066	212	24	(	(	PUNCT
ejpam-6066	212	25	mod	mod	NOUN
ejpam-6066	212	26	24	24	NUM
ejpam-6066	212	27	)	)	PUNCT
ejpam-6066	212	28	.	.	PUNCT
ejpam-6066	213	1	3	3	X
ejpam-6066	213	2	.	.	X
ejpam-6066	213	3	conclusion	conclusion	NOUN
ejpam-6066	213	4	in	in	ADP
ejpam-6066	213	5	this	this	DET
ejpam-6066	213	6	article	article	NOUN
ejpam-6066	213	7	,	,	PUNCT
ejpam-6066	213	8	we	we	PRON
ejpam-6066	213	9	have	have	AUX
ejpam-6066	213	10	determined	determine	VERB
ejpam-6066	213	11	all	all	DET
ejpam-6066	213	12	non	non	ADJ
ejpam-6066	213	13	-	-	ADJ
ejpam-6066	213	14	negative	negative	ADJ
ejpam-6066	213	15	integer	integer	NOUN
ejpam-6066	213	16	solutions	solution	NOUN
ejpam-6066	213	17	to	to	ADP
ejpam-6066	213	18	the	the	DET
ejpam-6066	213	19	exponential	exponential	ADJ
ejpam-6066	213	20	diophantine	diophantine	NOUN
ejpam-6066	213	21	equation	equation	NOUN
ejpam-6066	213	22	4(7x)−py	4(7x)−py	NOUN
ejpam-6066	213	23	=	=	SYM
ejpam-6066	213	24	z2	z2	PROPN
ejpam-6066	213	25	,	,	PUNCT
ejpam-6066	213	26	where	where	SCONJ
ejpam-6066	213	27	p	p	NOUN
ejpam-6066	213	28	is	be	AUX
ejpam-6066	213	29	a	a	DET
ejpam-6066	213	30	prime	prime	ADJ
ejpam-6066	213	31	number	number	NOUN
ejpam-6066	213	32	.	.	PUNCT
ejpam-6066	214	1	through	through	ADP
ejpam-6066	214	2	a	a	DET
ejpam-6066	214	3	careful	careful	ADJ
ejpam-6066	214	4	caseby	caseby	NOUN
ejpam-6066	214	5	-	-	PUNCT
ejpam-6066	214	6	case	case	NOUN
ejpam-6066	214	7	analysis	analysis	NOUN
ejpam-6066	214	8	based	base	VERB
ejpam-6066	214	9	on	on	ADP
ejpam-6066	214	10	the	the	DET
ejpam-6066	214	11	value	value	NOUN
ejpam-6066	214	12	of	of	ADP
ejpam-6066	214	13	the	the	DET
ejpam-6066	214	14	prime	prime	ADJ
ejpam-6066	214	15	p	p	NOUN
ejpam-6066	214	16	,	,	PUNCT
ejpam-6066	214	17	we	we	PRON
ejpam-6066	214	18	find	find	VERB
ejpam-6066	214	19	:	:	PUNCT
ejpam-6066	214	20	•	•	ADP
ejpam-6066	214	21	a	a	DET
ejpam-6066	214	22	unique	unique	ADJ
ejpam-6066	214	23	solution	solution	NOUN
ejpam-6066	214	24	for	for	ADP
ejpam-6066	214	25	p	p	NOUN
ejpam-6066	214	26	=	=	SYM
ejpam-6066	214	27	2	2	NUM
ejpam-6066	214	28	,	,	PUNCT
ejpam-6066	214	29	namely	namely	ADV
ejpam-6066	214	30	(	(	PUNCT
ejpam-6066	214	31	x	x	X
ejpam-6066	214	32	,	,	PUNCT
ejpam-6066	214	33	y	y	PROPN
ejpam-6066	214	34	,	,	PUNCT
ejpam-6066	214	35	z	z	PROPN
ejpam-6066	214	36	,	,	PUNCT
ejpam-6066	214	37	p	p	NOUN
ejpam-6066	214	38	)	)	PUNCT
ejpam-6066	214	39	=	=	SYM
ejpam-6066	214	40	(	(	PUNCT
ejpam-6066	214	41	0	0	NUM
ejpam-6066	214	42	,	,	PUNCT
ejpam-6066	214	43	2	2	NUM
ejpam-6066	214	44	,	,	PUNCT
ejpam-6066	214	45	0	0	NUM
ejpam-6066	214	46	,	,	PUNCT
ejpam-6066	214	47	2	2	NUM
ejpam-6066	214	48	)	)	PUNCT
ejpam-6066	214	49	.	.	PUNCT
ejpam-6066	215	1	•	•	NUM
ejpam-6066	215	2	a	a	DET
ejpam-6066	215	3	parametrized	parametrized	ADJ
ejpam-6066	215	4	family	family	NOUN
ejpam-6066	215	5	of	of	ADP
ejpam-6066	215	6	possible	possible	ADJ
ejpam-6066	215	7	solutions	solution	NOUN
ejpam-6066	215	8	satisfying	satisfy	VERB
ejpam-6066	215	9	specific	specific	ADJ
ejpam-6066	215	10	congruence	congruence	NOUN
ejpam-6066	215	11	conditions	condition	NOUN
ejpam-6066	215	12	for	for	ADP
ejpam-6066	215	13	p	p	NOUN
ejpam-6066	215	14	=	=	SYM
ejpam-6066	215	15	3	3	NUM
ejpam-6066	215	16	,	,	PUNCT
ejpam-6066	215	17	where	where	SCONJ
ejpam-6066	215	18	all	all	DET
ejpam-6066	215	19	solutions	solution	NOUN
ejpam-6066	215	20	require	require	VERB
ejpam-6066	215	21	x	x	X
ejpam-6066	215	22	and	and	CCONJ
ejpam-6066	215	23	y	y	PROPN
ejpam-6066	215	24	to	to	PART
ejpam-6066	215	25	be	be	AUX
ejpam-6066	215	26	odd	odd	ADJ
ejpam-6066	215	27	,	,	PUNCT
ejpam-6066	215	28	except	except	SCONJ
ejpam-6066	215	29	for	for	ADP
ejpam-6066	215	30	the	the	DET
ejpam-6066	215	31	cases	case	NOUN
ejpam-6066	215	32	(	(	PUNCT
ejpam-6066	215	33	x	x	X
ejpam-6066	215	34	,	,	PUNCT
ejpam-6066	215	35	y	y	PROPN
ejpam-6066	215	36	,	,	PUNCT
ejpam-6066	215	37	z	z	PROPN
ejpam-6066	215	38	,	,	PUNCT
ejpam-6066	215	39	p	p	NOUN
ejpam-6066	215	40	)	)	PUNCT
ejpam-6066	215	41	=	=	SYM
ejpam-6066	215	42	(	(	PUNCT
ejpam-6066	215	43	0	0	NUM
ejpam-6066	215	44	,	,	PUNCT
ejpam-6066	215	45	1	1	NUM
ejpam-6066	215	46	,	,	PUNCT
ejpam-6066	215	47	1	1	NUM
ejpam-6066	215	48	,	,	PUNCT
ejpam-6066	215	49	3	3	NUM
ejpam-6066	215	50	)	)	PUNCT
ejpam-6066	215	51	and	and	CCONJ
ejpam-6066	215	52	(	(	PUNCT
ejpam-6066	215	53	2	2	NUM
ejpam-6066	215	54	,	,	PUNCT
ejpam-6066	215	55	3	3	NUM
ejpam-6066	215	56	,	,	PUNCT
ejpam-6066	215	57	13	13	NUM
ejpam-6066	215	58	,	,	PUNCT
ejpam-6066	215	59	3	3	NUM
ejpam-6066	215	60	)	)	PUNCT
ejpam-6066	215	61	,	,	PUNCT
ejpam-6066	215	62	in	in	ADP
ejpam-6066	215	63	which	which	PRON
ejpam-6066	215	64	x	x	PRON
ejpam-6066	215	65	is	be	AUX
ejpam-6066	215	66	even	even	ADV
ejpam-6066	215	67	.	.	PUNCT
ejpam-6066	216	1	•	•	NUM
ejpam-6066	216	2	no	no	DET
ejpam-6066	216	3	solutions	solution	NOUN
ejpam-6066	216	4	exist	exist	VERB
ejpam-6066	216	5	for	for	ADP
ejpam-6066	216	6	5	5	NUM
ejpam-6066	216	7	≤	≤	NOUN
ejpam-6066	216	8	p	p	NOUN
ejpam-6066	216	9	≤	≤	NUM
ejpam-6066	216	10	17	17	NUM
ejpam-6066	216	11	.	.	NOUN
ejpam-6066	216	12	•	•	NOUN
ejpam-6066	216	13	for	for	ADP
ejpam-6066	216	14	p	p	PRON
ejpam-6066	216	15	≥	≥	NUM
ejpam-6066	216	16	19	19	NUM
ejpam-6066	216	17	,	,	PUNCT
ejpam-6066	216	18	a	a	DET
ejpam-6066	216	19	necessary	necessary	ADJ
ejpam-6066	216	20	condition	condition	NOUN
ejpam-6066	216	21	for	for	ADP
ejpam-6066	216	22	the	the	DET
ejpam-6066	216	23	existence	existence	NOUN
ejpam-6066	216	24	of	of	ADP
ejpam-6066	216	25	the	the	DET
ejpam-6066	216	26	solutions	solution	NOUN
ejpam-6066	216	27	is	be	AUX
ejpam-6066	216	28	that	that	SCONJ
ejpam-6066	216	29	p	p	PROPN
ejpam-6066	216	30	≡	≡	PROPN
ejpam-6066	216	31	19	19	NUM
ejpam-6066	216	32	(	(	PUNCT
ejpam-6066	216	33	mod	mod	NOUN
ejpam-6066	216	34	24	24	NUM
ejpam-6066	216	35	)	)	PUNCT
ejpam-6066	216	36	,	,	PUNCT
ejpam-6066	216	37	and	and	CCONJ
ejpam-6066	216	38	the	the	DET
ejpam-6066	216	39	values	value	NOUN
ejpam-6066	216	40	of	of	ADP
ejpam-6066	216	41	x	x	PROPN
ejpam-6066	216	42	,	,	PUNCT
ejpam-6066	216	43	y	y	PROPN
ejpam-6066	216	44	,	,	PUNCT
ejpam-6066	216	45	z	z	PROPN
ejpam-6066	216	46	and	and	CCONJ
ejpam-6066	216	47	p	p	PROPN
ejpam-6066	216	48	must	must	AUX
ejpam-6066	216	49	satisfy	satisfy	VERB
ejpam-6066	216	50	precise	precise	ADJ
ejpam-6066	216	51	congruence	congruence	NOUN
ejpam-6066	216	52	conditions	condition	NOUN
ejpam-6066	216	53	derived	derive	VERB
ejpam-6066	216	54	using	use	VERB
ejpam-6066	216	55	quadratic	quadratic	ADJ
ejpam-6066	216	56	residues	residue	NOUN
ejpam-6066	216	57	and	and	CCONJ
ejpam-6066	216	58	the	the	DET
ejpam-6066	216	59	chinese	chinese	ADJ
ejpam-6066	216	60	remainder	remainder	NOUN
ejpam-6066	216	61	theorem	theorem	PROPN
ejpam-6066	216	62	.	.	PUNCT
ejpam-6066	217	1	moreover	moreover	ADV
ejpam-6066	217	2	,	,	PUNCT
ejpam-6066	217	3	there	there	PRON
ejpam-6066	217	4	is	be	VERB
ejpam-6066	217	5	no	no	DET
ejpam-6066	217	6	solution	solution	NOUN
ejpam-6066	217	7	when	when	SCONJ
ejpam-6066	217	8	x	x	PRON
ejpam-6066	217	9	≡	≡	PROPN
ejpam-6066	217	10	3	3	NUM
ejpam-6066	217	11	or	or	CCONJ
ejpam-6066	217	12	5	5	NUM
ejpam-6066	217	13	(	(	PUNCT
ejpam-6066	217	14	mod	mod	NOUN
ejpam-6066	217	15	6	6	NUM
ejpam-6066	217	16	)	)	PUNCT
ejpam-6066	217	17	and	and	CCONJ
ejpam-6066	217	18	y	y	PROPN
ejpam-6066	217	19	≡	≡	PROPN
ejpam-6066	217	20	3	3	NUM
ejpam-6066	217	21	(	(	PUNCT
ejpam-6066	217	22	mod	mod	PROPN
ejpam-6066	217	23	6	6	NUM
ejpam-6066	217	24	)	)	PUNCT
ejpam-6066	217	25	.	.	PUNCT
ejpam-6066	218	1	in	in	ADP
ejpam-6066	218	2	contrast	contrast	NOUN
ejpam-6066	218	3	to	to	ADP
ejpam-6066	218	4	the	the	DET
ejpam-6066	218	5	previous	previous	ADJ
ejpam-6066	218	6	studies	study	NOUN
ejpam-6066	218	7	summarized	summarize	VERB
ejpam-6066	218	8	in	in	ADP
ejpam-6066	218	9	table	table	NOUN
ejpam-6066	218	10	2	2	NUM
ejpam-6066	218	11	,	,	PUNCT
ejpam-6066	218	12	our	our	PRON
ejpam-6066	218	13	analysis	analysis	NOUN
ejpam-6066	218	14	addresses	address	VERB
ejpam-6066	218	15	a	a	DET
ejpam-6066	218	16	distinct	distinct	ADJ
ejpam-6066	218	17	class	class	NOUN
ejpam-6066	218	18	of	of	ADP
ejpam-6066	218	19	exponential	exponential	ADJ
ejpam-6066	218	20	diophantine	diophantine	NOUN
ejpam-6066	218	21	equations	equation	NOUN
ejpam-6066	218	22	in	in	ADP
ejpam-6066	218	23	which	which	PRON
ejpam-6066	218	24	the	the	DET
ejpam-6066	218	25	term	term	NOUN
ejpam-6066	218	26	py	py	PROPN
ejpam-6066	218	27	is	be	AUX
ejpam-6066	218	28	subtracted	subtract	VERB
ejpam-6066	218	29	from	from	ADP
ejpam-6066	218	30	a	a	DET
ejpam-6066	218	31	fixed	fix	VERB
ejpam-6066	218	32	exponential	exponential	ADJ
ejpam-6066	218	33	base	base	NOUN
ejpam-6066	218	34	7x	7x	NUM
ejpam-6066	218	35	,	,	PUNCT
ejpam-6066	218	36	rather	rather	ADV
ejpam-6066	218	37	than	than	ADP
ejpam-6066	218	38	involving	involve	VERB
ejpam-6066	218	39	variable	variable	ADJ
ejpam-6066	218	40	coefficients	coefficient	NOUN
ejpam-6066	218	41	or	or	CCONJ
ejpam-6066	218	42	dual	dual	ADJ
ejpam-6066	218	43	exponential	exponential	ADJ
ejpam-6066	218	44	terms	term	NOUN
ejpam-6066	218	45	.	.	PUNCT
ejpam-6066	219	1	for	for	ADP
ejpam-6066	219	2	instance	instance	NOUN
ejpam-6066	219	3	,	,	PUNCT
ejpam-6066	219	4	rabago	rabago	VERB
ejpam-6066	219	5	[	[	X
ejpam-6066	219	6	8	8	NUM
ejpam-6066	219	7	]	]	PUNCT
ejpam-6066	219	8	analyzed	analyze	VERB
ejpam-6066	219	9	4x	4x	NOUN
ejpam-6066	219	10	−	−	PROPN
ejpam-6066	220	1	py	py	PROPN
ejpam-6066	220	2	=	=	SYM
ejpam-6066	220	3	3z2	3z2	NUM
ejpam-6066	220	4	,	,	PUNCT
ejpam-6066	220	5	while	while	SCONJ
ejpam-6066	220	6	laipaporn	laipaporn	VERB
ejpam-6066	220	7	et	et	PROPN
ejpam-6066	220	8	al	al	PROPN
ejpam-6066	220	9	.	.	PUNCT
ejpam-6066	221	1	[	[	X
ejpam-6066	221	2	9	9	NUM
ejpam-6066	221	3	]	]	PUNCT
ejpam-6066	221	4	examined	examine	VERB
ejpam-6066	221	5	3x	3x	NUM
ejpam-6066	221	6	+	+	CCONJ
ejpam-6066	221	7	p(5y	p(5y	NOUN
ejpam-6066	221	8	)	)	PUNCT
ejpam-6066	221	9	=	=	SYM
ejpam-6066	221	10	z2	z2	PROPN
ejpam-6066	221	11	,	,	PUNCT
ejpam-6066	221	12	which	which	PRON
ejpam-6066	221	13	is	be	AUX
ejpam-6066	221	14	additive	additive	ADJ
ejpam-6066	221	15	in	in	ADP
ejpam-6066	221	16	nature	nature	NOUN
ejpam-6066	221	17	.	.	PUNCT
ejpam-6066	222	1	in	in	ADP
ejpam-6066	222	2	contrast	contrast	NOUN
ejpam-6066	222	3	,	,	PUNCT
ejpam-6066	222	4	our	our	PRON
ejpam-6066	222	5	equation	equation	NOUN
ejpam-6066	222	6	exhibits	exhibit	VERB
ejpam-6066	222	7	a	a	DET
ejpam-6066	222	8	more	more	ADV
ejpam-6066	222	9	intricate	intricate	ADJ
ejpam-6066	222	10	balance	balance	NOUN
ejpam-6066	222	11	between	between	ADP
ejpam-6066	222	12	exponential	exponential	ADJ
ejpam-6066	222	13	growth	growth	NOUN
ejpam-6066	222	14	and	and	CCONJ
ejpam-6066	222	15	quadratic	quadratic	ADJ
ejpam-6066	222	16	structure	structure	NOUN
ejpam-6066	222	17	,	,	PUNCT
ejpam-6066	222	18	requiring	require	VERB
ejpam-6066	222	19	careful	careful	ADJ
ejpam-6066	222	20	refinement	refinement	NOUN
ejpam-6066	222	21	through	through	ADP
ejpam-6066	222	22	modular	modular	ADJ
ejpam-6066	222	23	analysis	analysis	NOUN
ejpam-6066	222	24	.	.	PUNCT
ejpam-6066	223	1	interestingly	interestingly	ADV
ejpam-6066	223	2	,	,	PUNCT
ejpam-6066	223	3	the	the	DET
ejpam-6066	223	4	diophantine	diophantine	NOUN
ejpam-6066	223	5	equation	equation	NOUN
ejpam-6066	223	6	4(7x	4(7x	NUM
ejpam-6066	223	7	)	)	PUNCT
ejpam-6066	224	1	−	−	PROPN
ejpam-6066	224	2	py	py	NOUN
ejpam-6066	224	3	=	=	PROPN
ejpam-6066	224	4	z2	z2	PROPN
ejpam-6066	224	5	was	be	AUX
ejpam-6066	224	6	also	also	ADV
ejpam-6066	224	7	examined	examine	VERB
ejpam-6066	224	8	computationally	computationally	ADV
ejpam-6066	224	9	using	use	VERB
ejpam-6066	224	10	algorithm	algorithm	NOUN
ejpam-6066	224	11	1	1	NUM
ejpam-6066	224	12	,	,	PUNCT
ejpam-6066	224	13	which	which	PRON
ejpam-6066	224	14	internally	internally	ADV
ejpam-6066	224	15	calls	call	VERB
ejpam-6066	224	16	two	two	NUM
ejpam-6066	224	17	functions	function	NOUN
ejpam-6066	224	18	based	base	VERB
ejpam-6066	224	19	on	on	ADP
ejpam-6066	224	20	algorithm	algorithm	NOUN
ejpam-6066	224	21	2	2	NUM
ejpam-6066	224	22	,	,	PUNCT
ejpam-6066	224	23	and	and	CCONJ
ejpam-6066	224	24	algorithm	algorithm	NOUN
ejpam-6066	224	25	3	3	NUM
ejpam-6066	224	26	.	.	PUNCT
ejpam-6066	225	1	the	the	DET
ejpam-6066	225	2	complete	complete	ADJ
ejpam-6066	225	3	procedure	procedure	NOUN
ejpam-6066	225	4	was	be	AUX
ejpam-6066	225	5	implemented	implement	VERB
ejpam-6066	225	6	in	in	ADP
ejpam-6066	225	7	python	python	PROPN
ejpam-6066	225	8	code	code	NOUN
ejpam-6066	225	9	(	(	PUNCT
ejpam-6066	225	10	accessible	accessible	ADJ
ejpam-6066	225	11	at	at	ADP
ejpam-6066	225	12	https://github.com/kadisak/ejpam6066.git	https://github.com/kadisak/ejpam6066.git	NOUN
ejpam-6066	225	13	)	)	PUNCT
ejpam-6066	225	14	.	.	PUNCT
ejpam-6066	226	1	for	for	ADP
ejpam-6066	226	2	the	the	DET
ejpam-6066	226	3	case	case	NOUN
ejpam-6066	226	4	p	p	X
ejpam-6066	226	5	=	=	NOUN
ejpam-6066	226	6	3	3	NUM
ejpam-6066	226	7	,	,	PUNCT
ejpam-6066	226	8	the	the	DET
ejpam-6066	226	9	computational	computational	ADJ
ejpam-6066	226	10	search	search	NOUN
ejpam-6066	226	11	was	be	AUX
ejpam-6066	226	12	performed	perform	VERB
ejpam-6066	226	13	over	over	ADP
ejpam-6066	226	14	the	the	DET
ejpam-6066	226	15	range	range	NOUN
ejpam-6066	226	16	0	0	NUM
ejpam-6066	226	17	≤	≤	NUM
ejpam-6066	226	18	x	x	SYM
ejpam-6066	226	19	≤	≤	NUM
ejpam-6066	226	20	100	100	NUM
ejpam-6066	226	21	,	,	PUNCT
ejpam-6066	226	22	000	000	NUM
ejpam-6066	226	23	and	and	CCONJ
ejpam-6066	226	24	1	1	NUM
ejpam-6066	226	25	≤	≤	NUM
ejpam-6066	226	26	y	y	PROPN
ejpam-6066	226	27	≤	≤	ADJ
ejpam-6066	226	28	⌊logp	⌊logp	ADV
ejpam-6066	226	29	(	(	PUNCT
ejpam-6066	226	30	4(7x))⌋.	4(7x))⌋.	NOUN
ejpam-6066	226	31	within	within	ADP
ejpam-6066	226	32	this	this	DET
ejpam-6066	226	33	domain	domain	NOUN
ejpam-6066	226	34	,	,	PUNCT
ejpam-6066	226	35	exactly	exactly	ADV
ejpam-6066	226	36	five	five	NUM
ejpam-6066	226	37	solutions	solution	NOUN
ejpam-6066	226	38	were	be	AUX
ejpam-6066	226	39	obtained	obtain	VERB
ejpam-6066	226	40	,	,	PUNCT
ejpam-6066	226	41	namely	namely	ADV
ejpam-6066	226	42	(	(	PUNCT
ejpam-6066	226	43	x	x	X
ejpam-6066	226	44	,	,	PUNCT
ejpam-6066	226	45	y	y	PROPN
ejpam-6066	226	46	,	,	PUNCT
ejpam-6066	226	47	z	z	NOUN
ejpam-6066	226	48	)	)	PUNCT
ejpam-6066	226	49	=	=	SYM
ejpam-6066	227	1	(	(	PUNCT
ejpam-6066	227	2	0	0	NUM
ejpam-6066	227	3	,	,	PUNCT
ejpam-6066	227	4	1	1	NUM
ejpam-6066	227	5	,	,	PUNCT
ejpam-6066	227	6	1	1	NUM
ejpam-6066	227	7	)	)	PUNCT
ejpam-6066	227	8	,	,	PUNCT
ejpam-6066	227	9	(	(	PUNCT
ejpam-6066	227	10	1	1	NUM
ejpam-6066	227	11	,	,	PUNCT
ejpam-6066	227	12	1	1	NUM
ejpam-6066	227	13	,	,	PUNCT
ejpam-6066	227	14	5	5	NUM
ejpam-6066	227	15	)	)	PUNCT
ejpam-6066	227	16	,	,	PUNCT
ejpam-6066	227	17	(	(	PUNCT
ejpam-6066	227	18	1	1	NUM
ejpam-6066	227	19	,	,	PUNCT
ejpam-6066	227	20	3	3	NUM
ejpam-6066	227	21	,	,	PUNCT
ejpam-6066	227	22	1	1	NUM
ejpam-6066	227	23	)	)	PUNCT
ejpam-6066	227	24	,	,	PUNCT
ejpam-6066	227	25	(	(	PUNCT
ejpam-6066	227	26	2	2	NUM
ejpam-6066	227	27	,	,	PUNCT
ejpam-6066	227	28	3	3	NUM
ejpam-6066	227	29	,	,	PUNCT
ejpam-6066	227	30	13	13	NUM
ejpam-6066	227	31	)	)	PUNCT
ejpam-6066	227	32	,	,	PUNCT
ejpam-6066	227	33	and	and	CCONJ
ejpam-6066	227	34	(	(	PUNCT
ejpam-6066	227	35	3	3	NUM
ejpam-6066	227	36	,	,	PUNCT
ejpam-6066	227	37	1	1	NUM
ejpam-6066	227	38	,	,	PUNCT
ejpam-6066	227	39	37	37	NUM
ejpam-6066	227	40	)	)	PUNCT
ejpam-6066	227	41	,	,	PUNCT
ejpam-6066	227	42	see	see	VERB
ejpam-6066	227	43	table	table	NOUN
ejpam-6066	227	44	3	3	NUM
ejpam-6066	227	45	.	.	PUNCT
ejpam-6066	228	1	k.	k.	PROPN
ejpam-6066	228	2	laipaporn	laipaporn	PROPN
ejpam-6066	228	3	et	et	PROPN
ejpam-6066	228	4	al	al	PROPN
ejpam-6066	228	5	.	.	PUNCT
ejpam-6066	228	6	/	/	SYM
ejpam-6066	228	7	eur	eur	PROPN
ejpam-6066	228	8	.	.	PUNCT
ejpam-6066	229	1	j.	j.	PROPN
ejpam-6066	229	2	pure	pure	PROPN
ejpam-6066	229	3	appl	appl	PROPN
ejpam-6066	229	4	.	.	PROPN
ejpam-6066	229	5	math	math	PROPN
ejpam-6066	229	6	,	,	PUNCT
ejpam-6066	229	7	18	18	NUM
ejpam-6066	229	8	(	(	PUNCT
ejpam-6066	229	9	3	3	NUM
ejpam-6066	229	10	)	)	PUNCT
ejpam-6066	229	11	(	(	PUNCT
ejpam-6066	229	12	2025	2025	NUM
ejpam-6066	229	13	)	)	PUNCT
ejpam-6066	229	14	,	,	PUNCT
ejpam-6066	229	15	6066	6066	NUM
ejpam-6066	229	16	9	9	NUM
ejpam-6066	229	17	of	of	ADP
ejpam-6066	229	18	13	13	NUM
ejpam-6066	229	19	table	table	NOUN
ejpam-6066	230	1	3	3	NUM
ejpam-6066	230	2	:	:	PUNCT
ejpam-6066	230	3	valid	valid	ADJ
ejpam-6066	230	4	solutions	solution	NOUN
ejpam-6066	230	5	to	to	ADP
ejpam-6066	230	6	the	the	DET
ejpam-6066	230	7	equation	equation	NOUN
ejpam-6066	230	8	4(7x)−	4(7x)−	PROPN
ejpam-6066	230	9	py	py	PROPN
ejpam-6066	230	10	=	=	PROPN
ejpam-6066	230	11	z2	z2	PROPN
ejpam-6066	230	12	for	for	ADP
ejpam-6066	230	13	p	p	NOUN
ejpam-6066	230	14	=	=	PROPN
ejpam-6066	230	15	3	3	NUM
ejpam-6066	230	16	,	,	PUNCT
ejpam-6066	230	17	0	0	NUM
ejpam-6066	230	18	≤	≤	NUM
ejpam-6066	230	19	x	x	SYM
ejpam-6066	230	20	≤	≤	NUM
ejpam-6066	230	21	100	100	NUM
ejpam-6066	230	22	,	,	PUNCT
ejpam-6066	230	23	000	000	NUM
ejpam-6066	230	24	,	,	PUNCT
ejpam-6066	230	25	and	and	CCONJ
ejpam-6066	230	26	1	1	NUM
ejpam-6066	230	27	≤	≤	NUM
ejpam-6066	230	28	y	y	PROPN
ejpam-6066	230	29	≤	≤	ADJ
ejpam-6066	230	30	⌊logp	⌊logp	ADV
ejpam-6066	230	31	(	(	PUNCT
ejpam-6066	230	32	4(7x))⌋.	4(7x))⌋.	PROPN
ejpam-6066	230	33	p	p	X
ejpam-6066	230	34	x	x	X
ejpam-6066	230	35	y	y	PROPN
ejpam-6066	230	36	z	z	PROPN
ejpam-6066	230	37	3	3	NUM
ejpam-6066	230	38	0	0	NUM
ejpam-6066	230	39	1	1	NUM
ejpam-6066	230	40	1	1	NUM
ejpam-6066	230	41	3	3	NUM
ejpam-6066	230	42	1	1	NUM
ejpam-6066	230	43	1	1	NUM
ejpam-6066	230	44	5	5	NUM
ejpam-6066	230	45	3	3	NUM
ejpam-6066	230	46	1	1	NUM
ejpam-6066	230	47	3	3	NUM
ejpam-6066	230	48	1	1	NUM
ejpam-6066	230	49	3	3	NUM
ejpam-6066	230	50	2	2	NUM
ejpam-6066	230	51	3	3	NUM
ejpam-6066	230	52	13	13	NUM
ejpam-6066	230	53	3	3	NUM
ejpam-6066	230	54	3	3	NUM
ejpam-6066	230	55	1	1	NUM
ejpam-6066	230	56	37	37	NUM
ejpam-6066	230	57	for	for	ADP
ejpam-6066	230	58	prime	prime	ADJ
ejpam-6066	230	59	numbers	number	NOUN
ejpam-6066	230	60	p	p	NOUN
ejpam-6066	230	61	satisfying	satisfy	VERB
ejpam-6066	230	62	2	2	NUM
ejpam-6066	230	63	≤	≤	NOUN
ejpam-6066	230	64	p	p	NOUN
ejpam-6066	230	65	≤	≤	NUM
ejpam-6066	230	66	100	100	NUM
ejpam-6066	230	67	,	,	PUNCT
ejpam-6066	230	68	000	000	NUM
ejpam-6066	230	69	with	with	ADP
ejpam-6066	230	70	p	p	NOUN
ejpam-6066	230	71	̸=	̸=	PROPN
ejpam-6066	230	72	3	3	NUM
ejpam-6066	230	73	,	,	PUNCT
ejpam-6066	230	74	and	and	CCONJ
ejpam-6066	230	75	under	under	ADP
ejpam-6066	230	76	the	the	DET
ejpam-6066	230	77	parameter	parameter	NOUN
ejpam-6066	230	78	constraints	constraint	VERB
ejpam-6066	230	79	1	1	NUM
ejpam-6066	230	80	≤	≤	NUM
ejpam-6066	230	81	x	x	SYM
ejpam-6066	230	82	≤	≤	NUM
ejpam-6066	230	83	10	10	NUM
ejpam-6066	230	84	,	,	PUNCT
ejpam-6066	230	85	000	000	NUM
ejpam-6066	230	86	and	and	CCONJ
ejpam-6066	230	87	1	1	NUM
ejpam-6066	230	88	≤	≤	NUM
ejpam-6066	230	89	y	y	PROPN
ejpam-6066	230	90	≤	≤	NOUN
ejpam-6066	230	91	⌊logp	⌊logp	ADV
ejpam-6066	230	92	(	(	PUNCT
ejpam-6066	230	93	4(7x))⌋	4(7x))⌋	NUM
ejpam-6066	230	94	,	,	PUNCT
ejpam-6066	230	95	a	a	DET
ejpam-6066	230	96	total	total	NOUN
ejpam-6066	230	97	of	of	ADP
ejpam-6066	230	98	31	31	NUM
ejpam-6066	230	99	solutions	solution	NOUN
ejpam-6066	230	100	were	be	AUX
ejpam-6066	230	101	found	find	VERB
ejpam-6066	230	102	when	when	SCONJ
ejpam-6066	230	103	p	p	PROPN
ejpam-6066	230	104	≡	≡	PROPN
ejpam-6066	230	105	19	19	NUM
ejpam-6066	230	106	(	(	PUNCT
ejpam-6066	230	107	mod	mod	NOUN
ejpam-6066	230	108	24	24	NUM
ejpam-6066	230	109	)	)	PUNCT
ejpam-6066	230	110	,	,	PUNCT
ejpam-6066	230	111	summarized	summarize	VERB
ejpam-6066	230	112	in	in	ADP
ejpam-6066	230	113	table	table	NOUN
ejpam-6066	230	114	4	4	NUM
ejpam-6066	230	115	.	.	PUNCT
ejpam-6066	230	116	table	table	NOUN
ejpam-6066	230	117	4	4	NUM
ejpam-6066	230	118	:	:	PUNCT
ejpam-6066	230	119	all	all	PRON
ejpam-6066	230	120	computed	compute	VERB
ejpam-6066	230	121	solutions	solution	NOUN
ejpam-6066	230	122	to	to	ADP
ejpam-6066	230	123	the	the	DET
ejpam-6066	230	124	equation	equation	NOUN
ejpam-6066	230	125	4(7x)−py	4(7x)−py	NOUN
ejpam-6066	230	126	=	=	SYM
ejpam-6066	230	127	z2	z2	PROPN
ejpam-6066	230	128	.	.	PUNCT
ejpam-6066	231	1	the	the	DET
ejpam-6066	231	2	search	search	NOUN
ejpam-6066	231	3	was	be	AUX
ejpam-6066	231	4	conducted	conduct	VERB
ejpam-6066	231	5	over	over	ADP
ejpam-6066	231	6	the	the	DET
ejpam-6066	231	7	range	range	NOUN
ejpam-6066	231	8	2	2	NUM
ejpam-6066	231	9	≤	≤	NOUN
ejpam-6066	231	10	p	p	NOUN
ejpam-6066	231	11	≤	≤	NUM
ejpam-6066	231	12	100	100	NUM
ejpam-6066	231	13	,	,	PUNCT
ejpam-6066	231	14	000	000	NUM
ejpam-6066	231	15	,	,	PUNCT
ejpam-6066	231	16	1	1	NUM
ejpam-6066	231	17	≤	≤	NUM
ejpam-6066	231	18	x	x	SYM
ejpam-6066	231	19	≤	≤	NUM
ejpam-6066	231	20	10	10	NUM
ejpam-6066	231	21	,	,	PUNCT
ejpam-6066	231	22	000	000	NUM
ejpam-6066	231	23	,	,	PUNCT
ejpam-6066	231	24	and	and	CCONJ
ejpam-6066	231	25	1	1	NUM
ejpam-6066	231	26	≤	≤	NUM
ejpam-6066	231	27	y	y	PROPN
ejpam-6066	231	28	≤	≤	NOUN
ejpam-6066	231	29	⌊logp	⌊logp	ADV
ejpam-6066	231	30	(	(	PUNCT
ejpam-6066	231	31	4(7x))⌋	4(7x))⌋	NUM
ejpam-6066	231	32	,	,	PUNCT
ejpam-6066	231	33	for	for	ADP
ejpam-6066	231	34	primes	prime	NOUN
ejpam-6066	231	35	p	p	PROPN
ejpam-6066	231	36	≡	≡	PROPN
ejpam-6066	231	37	19	19	NUM
ejpam-6066	231	38	(	(	PUNCT
ejpam-6066	231	39	mod	mod	NOUN
ejpam-6066	231	40	24	24	NUM
ejpam-6066	231	41	)	)	PUNCT
ejpam-6066	231	42	,	,	PUNCT
ejpam-6066	231	43	as	as	SCONJ
ejpam-6066	231	44	related	relate	VERB
ejpam-6066	231	45	to	to	ADP
ejpam-6066	231	46	corollary	corollary	ADJ
ejpam-6066	231	47	1	1	NUM
ejpam-6066	231	48	.	.	PUNCT
ejpam-6066	232	1	congruence	congruence	PROPN
ejpam-6066	232	2	class	class	PROPN
ejpam-6066	232	3	p	p	PROPN
ejpam-6066	232	4	x	x	X
ejpam-6066	232	5	y	y	PROPN
ejpam-6066	232	6	z	z	PROPN
ejpam-6066	232	7	p	p	PROPN
ejpam-6066	232	8	≡	≡	PROPN
ejpam-6066	232	9	19	19	NUM
ejpam-6066	232	10	(	(	PUNCT
ejpam-6066	232	11	mod	mod	PROPN
ejpam-6066	232	12	72	72	NUM
ejpam-6066	232	13	)	)	PUNCT
ejpam-6066	232	14	19	19	NUM
ejpam-6066	232	15	1	1	NUM
ejpam-6066	232	16	1	1	NUM
ejpam-6066	232	17	3	3	NUM
ejpam-6066	232	18	86491	86491	NUM
ejpam-6066	232	19	7	7	NUM
ejpam-6066	232	20	1	1	NUM
ejpam-6066	232	21	1791	1791	NUM
ejpam-6066	232	22	p	p	PROPN
ejpam-6066	232	23	≡	≡	PROPN
ejpam-6066	232	24	67	67	NUM
ejpam-6066	232	25	(	(	PUNCT
ejpam-6066	232	26	mod	mod	PROPN
ejpam-6066	232	27	72	72	NUM
ejpam-6066	232	28	)	)	PUNCT
ejpam-6066	232	29	283	283	NUM
ejpam-6066	232	30	3	3	NUM
ejpam-6066	232	31	1	1	NUM
ejpam-6066	232	32	33	33	NUM
ejpam-6066	232	33	643	643	NUM
ejpam-6066	232	34	3	3	NUM
ejpam-6066	232	35	1	1	NUM
ejpam-6066	232	36	27	27	NUM
ejpam-6066	232	37	1291	1291	NUM
ejpam-6066	232	38	3	3	NUM
ejpam-6066	232	39	1	1	NUM
ejpam-6066	232	40	9	9	NUM
ejpam-6066	232	41	p	p	PRON
ejpam-6066	232	42	≡	≡	PROPN
ejpam-6066	232	43	43	43	NUM
ejpam-6066	232	44	(	(	PUNCT
ejpam-6066	232	45	mod	mod	PROPN
ejpam-6066	232	46	72	72	NUM
ejpam-6066	232	47	)	)	PUNCT
ejpam-6066	232	48	2203	2203	NUM
ejpam-6066	232	49	5	5	NUM
ejpam-6066	232	50	1	1	NUM
ejpam-6066	232	51	255	255	NUM
ejpam-6066	232	52	5227	5227	NUM
ejpam-6066	232	53	5	5	NUM
ejpam-6066	232	54	1	1	NUM
ejpam-6066	232	55	249	249	NUM
ejpam-6066	232	56	8179	8179	NUM
ejpam-6066	232	57	5	5	NUM
ejpam-6066	232	58	1	1	NUM
ejpam-6066	232	59	243	243	NUM
ejpam-6066	232	60	11059	11059	NUM
ejpam-6066	232	61	5	5	NUM
ejpam-6066	232	62	1	1	NUM
ejpam-6066	232	63	237	237	NUM
ejpam-6066	232	64	16603	16603	NUM
ejpam-6066	232	65	5	5	NUM
ejpam-6066	232	66	1	1	NUM
ejpam-6066	232	67	225	225	NUM
ejpam-6066	232	68	19267	19267	NUM
ejpam-6066	232	69	5	5	NUM
ejpam-6066	232	70	1	1	NUM
ejpam-6066	232	71	219	219	NUM
ejpam-6066	232	72	21859	21859	NUM
ejpam-6066	232	73	5	5	NUM
ejpam-6066	232	74	1	1	NUM
ejpam-6066	232	75	213	213	NUM
ejpam-6066	232	76	24379	24379	NUM
ejpam-6066	232	77	5	5	NUM
ejpam-6066	232	78	1	1	NUM
ejpam-6066	232	79	207	207	NUM
ejpam-6066	232	80	33739	33739	NUM
ejpam-6066	232	81	5	5	NUM
ejpam-6066	232	82	1	1	NUM
ejpam-6066	232	83	183	183	NUM
ejpam-6066	232	84	35899	35899	NUM
ejpam-6066	232	85	5	5	NUM
ejpam-6066	232	86	1	1	NUM
ejpam-6066	232	87	177	177	NUM
ejpam-6066	232	88	37987	37987	NUM
ejpam-6066	232	89	5	5	NUM
ejpam-6066	232	90	1	1	NUM
ejpam-6066	232	91	171	171	NUM
ejpam-6066	232	92	congruence	congruence	NOUN
ejpam-6066	232	93	class	class	NOUN
ejpam-6066	232	94	p	p	NOUN
ejpam-6066	232	95	x	x	X
ejpam-6066	232	96	y	y	PROPN
ejpam-6066	232	97	z	z	PROPN
ejpam-6066	232	98	p	p	PROPN
ejpam-6066	232	99	≡	≡	PROPN
ejpam-6066	232	100	43	43	NUM
ejpam-6066	232	101	(	(	PUNCT
ejpam-6066	232	102	mod	mod	PROPN
ejpam-6066	232	103	72	72	NUM
ejpam-6066	232	104	)	)	PUNCT
ejpam-6066	232	105	41947	41947	NUM
ejpam-6066	232	106	5	5	NUM
ejpam-6066	232	107	1	1	NUM
ejpam-6066	232	108	159	159	NUM
ejpam-6066	232	109	49003	49003	NUM
ejpam-6066	232	110	5	5	NUM
ejpam-6066	232	111	1	1	NUM
ejpam-6066	232	112	135	135	NUM
ejpam-6066	232	113	50587	50587	NUM
ejpam-6066	232	114	5	5	NUM
ejpam-6066	232	115	1	1	NUM
ejpam-6066	232	116	129	129	NUM
ejpam-6066	232	117	54907	54907	NUM
ejpam-6066	232	118	5	5	NUM
ejpam-6066	232	119	1	1	NUM
ejpam-6066	232	120	111	111	NUM
ejpam-6066	232	121	57427	57427	NUM
ejpam-6066	232	122	5	5	NUM
ejpam-6066	232	123	1	1	NUM
ejpam-6066	232	124	99	99	NUM
ejpam-6066	232	125	58579	58579	NUM
ejpam-6066	232	126	5	5	NUM
ejpam-6066	232	127	1	1	NUM
ejpam-6066	232	128	93	93	NUM
ejpam-6066	232	129	59659	59659	NUM
ejpam-6066	232	130	5	5	NUM
ejpam-6066	232	131	1	1	NUM
ejpam-6066	232	132	87	87	NUM
ejpam-6066	232	133	61603	61603	NUM
ejpam-6066	232	134	5	5	NUM
ejpam-6066	232	135	1	1	NUM
ejpam-6066	232	136	75	75	NUM
ejpam-6066	232	137	62467	62467	NUM
ejpam-6066	232	138	5	5	NUM
ejpam-6066	232	139	1	1	NUM
ejpam-6066	232	140	69	69	NUM
ejpam-6066	232	141	64627	64627	NUM
ejpam-6066	232	142	5	5	NUM
ejpam-6066	232	143	1	1	NUM
ejpam-6066	232	144	51	51	NUM
ejpam-6066	232	145	65203	65203	NUM
ejpam-6066	232	146	5	5	NUM
ejpam-6066	232	147	1	1	NUM
ejpam-6066	232	148	45	45	NUM
ejpam-6066	232	149	65707	65707	NUM
ejpam-6066	232	150	5	5	NUM
ejpam-6066	232	151	1	1	NUM
ejpam-6066	232	152	39	39	NUM
ejpam-6066	232	153	66499	66499	NUM
ejpam-6066	232	154	5	5	NUM
ejpam-6066	232	155	1	1	NUM
ejpam-6066	232	156	27	27	NUM
ejpam-6066	232	157	67003	67003	NUM
ejpam-6066	232	158	5	5	NUM
ejpam-6066	232	159	1	1	NUM
ejpam-6066	232	160	15	15	NUM
ejpam-6066	232	161	67219	67219	NUM
ejpam-6066	232	162	5	5	NUM
ejpam-6066	232	163	1	1	NUM
ejpam-6066	232	164	3	3	NUM
ejpam-6066	232	165	based	base	VERB
ejpam-6066	232	166	on	on	ADP
ejpam-6066	232	167	both	both	CCONJ
ejpam-6066	232	168	our	our	PRON
ejpam-6066	232	169	computational	computational	ADJ
ejpam-6066	232	170	findings	finding	NOUN
ejpam-6066	232	171	and	and	CCONJ
ejpam-6066	232	172	theorem	theorem	VERB
ejpam-6066	232	173	1	1	NUM
ejpam-6066	232	174	,	,	PUNCT
ejpam-6066	232	175	we	we	PRON
ejpam-6066	232	176	propose	propose	VERB
ejpam-6066	232	177	the	the	DET
ejpam-6066	232	178	following	following	NOUN
ejpam-6066	232	179	conjectures	conjecture	NOUN
ejpam-6066	232	180	:	:	PUNCT
ejpam-6066	232	181	•	•	ADP
ejpam-6066	232	182	if	if	SCONJ
ejpam-6066	232	183	z	z	NOUN
ejpam-6066	232	184	̸≡	̸≡	VERB
ejpam-6066	232	185	1	1	NUM
ejpam-6066	232	186	,	,	PUNCT
ejpam-6066	232	187	5	5	NUM
ejpam-6066	232	188	and	and	CCONJ
ejpam-6066	232	189	13	13	NUM
ejpam-6066	232	190	(	(	PUNCT
ejpam-6066	232	191	mod	mod	PROPN
ejpam-6066	232	192	16	16	NUM
ejpam-6066	232	193	)	)	PUNCT
ejpam-6066	232	194	,	,	PUNCT
ejpam-6066	232	195	then	then	ADV
ejpam-6066	232	196	the	the	DET
ejpam-6066	232	197	equation	equation	NOUN
ejpam-6066	232	198	4(7x)−	4(7x)−	PROPN
ejpam-6066	232	199	3y	3y	NUM
ejpam-6066	232	200	=	=	SYM
ejpam-6066	232	201	z2	z2	PROPN
ejpam-6066	232	202	has	have	VERB
ejpam-6066	232	203	no	no	DET
ejpam-6066	232	204	solution	solution	NOUN
ejpam-6066	232	205	.	.	PUNCT
ejpam-6066	233	1	•	•	NOUN
ejpam-6066	233	2	for	for	ADP
ejpam-6066	233	3	any	any	DET
ejpam-6066	233	4	prime	prime	NOUN
ejpam-6066	233	5	p	p	PROPN
ejpam-6066	233	6	≡	≡	PROPN
ejpam-6066	233	7	19	19	NUM
ejpam-6066	233	8	(	(	PUNCT
ejpam-6066	233	9	mod	mod	NOUN
ejpam-6066	233	10	24	24	NUM
ejpam-6066	233	11	)	)	PUNCT
ejpam-6066	233	12	and	and	CCONJ
ejpam-6066	233	13	y	y	PROPN
ejpam-6066	233	14	≥	≥	PROPN
ejpam-6066	233	15	3	3	NUM
ejpam-6066	233	16	,	,	PUNCT
ejpam-6066	233	17	our	our	PRON
ejpam-6066	233	18	computational	computational	ADJ
ejpam-6066	233	19	results	result	NOUN
ejpam-6066	233	20	suggest	suggest	VERB
ejpam-6066	233	21	that	that	SCONJ
ejpam-6066	233	22	no	no	DET
ejpam-6066	233	23	solutions	solution	NOUN
ejpam-6066	233	24	exist	exist	VERB
ejpam-6066	233	25	within	within	ADP
ejpam-6066	233	26	the	the	DET
ejpam-6066	233	27	tested	test	VERB
ejpam-6066	233	28	parameter	parameter	NOUN
ejpam-6066	233	29	range	range	NOUN
ejpam-6066	233	30	.	.	PUNCT
ejpam-6066	234	1	this	this	PRON
ejpam-6066	234	2	leads	lead	VERB
ejpam-6066	234	3	us	we	PRON
ejpam-6066	234	4	to	to	PART
ejpam-6066	234	5	conjecture	conjecture	VERB
ejpam-6066	234	6	that	that	SCONJ
ejpam-6066	234	7	such	such	ADJ
ejpam-6066	234	8	solutions	solution	NOUN
ejpam-6066	234	9	are	be	AUX
ejpam-6066	234	10	either	either	CCONJ
ejpam-6066	234	11	extremely	extremely	ADV
ejpam-6066	234	12	rare	rare	ADJ
ejpam-6066	234	13	or	or	CCONJ
ejpam-6066	234	14	do	do	AUX
ejpam-6066	234	15	not	not	PART
ejpam-6066	234	16	exist	exist	VERB
ejpam-6066	234	17	at	at	ADV
ejpam-6066	234	18	all	all	ADV
ejpam-6066	234	19	.	.	PUNCT
ejpam-6066	235	1	k.	k.	PROPN
ejpam-6066	235	2	laipaporn	laipaporn	PROPN
ejpam-6066	235	3	et	et	PROPN
ejpam-6066	235	4	al	al	PROPN
ejpam-6066	235	5	.	.	PUNCT
ejpam-6066	235	6	/	/	SYM
ejpam-6066	235	7	eur	eur	PROPN
ejpam-6066	235	8	.	.	PUNCT
ejpam-6066	236	1	j.	j.	PROPN
ejpam-6066	236	2	pure	pure	PROPN
ejpam-6066	236	3	appl	appl	PROPN
ejpam-6066	236	4	.	.	PROPN
ejpam-6066	236	5	math	math	PROPN
ejpam-6066	236	6	,	,	PUNCT
ejpam-6066	236	7	18	18	NUM
ejpam-6066	236	8	(	(	PUNCT
ejpam-6066	236	9	3	3	NUM
ejpam-6066	236	10	)	)	PUNCT
ejpam-6066	236	11	(	(	PUNCT
ejpam-6066	236	12	2025	2025	NUM
ejpam-6066	236	13	)	)	PUNCT
ejpam-6066	236	14	,	,	PUNCT
ejpam-6066	236	15	6066	6066	NUM
ejpam-6066	236	16	10	10	NUM
ejpam-6066	236	17	of	of	ADP
ejpam-6066	236	18	13	13	NUM
ejpam-6066	236	19	overall	overall	NOUN
ejpam-6066	236	20	,	,	PUNCT
ejpam-6066	236	21	this	this	DET
ejpam-6066	236	22	work	work	NOUN
ejpam-6066	236	23	contributes	contribute	VERB
ejpam-6066	236	24	to	to	ADP
ejpam-6066	236	25	the	the	DET
ejpam-6066	236	26	classification	classification	NOUN
ejpam-6066	236	27	program	program	NOUN
ejpam-6066	236	28	of	of	ADP
ejpam-6066	236	29	exponential	exponential	ADJ
ejpam-6066	236	30	diophantine	diophantine	NOUN
ejpam-6066	236	31	equations	equation	NOUN
ejpam-6066	236	32	of	of	ADP
ejpam-6066	236	33	the	the	DET
ejpam-6066	236	34	form	form	NOUN
ejpam-6066	236	35	a(px)−b(qy	a(px)−b(qy	NOUN
ejpam-6066	236	36	)	)	PUNCT
ejpam-6066	237	1	=	=	SYM
ejpam-6066	237	2	z2	z2	PROPN
ejpam-6066	237	3	,	,	PUNCT
ejpam-6066	237	4	especially	especially	ADV
ejpam-6066	237	5	when	when	SCONJ
ejpam-6066	237	6	one	one	NUM
ejpam-6066	237	7	exponential	exponential	ADJ
ejpam-6066	237	8	base	base	NOUN
ejpam-6066	237	9	is	be	AUX
ejpam-6066	237	10	fixed	fix	VERB
ejpam-6066	237	11	.	.	PUNCT
ejpam-6066	238	1	by	by	ADP
ejpam-6066	238	2	combining	combine	VERB
ejpam-6066	238	3	classical	classical	ADJ
ejpam-6066	238	4	number	number	NOUN
ejpam-6066	238	5	-	-	PUNCT
ejpam-6066	238	6	theoretic	theoretic	NOUN
ejpam-6066	238	7	tools	tool	NOUN
ejpam-6066	238	8	with	with	ADP
ejpam-6066	238	9	modern	modern	ADJ
ejpam-6066	238	10	computation	computation	NOUN
ejpam-6066	238	11	,	,	PUNCT
ejpam-6066	238	12	we	we	PRON
ejpam-6066	238	13	not	not	PART
ejpam-6066	238	14	only	only	ADV
ejpam-6066	238	15	resolve	resolve	VERB
ejpam-6066	238	16	the	the	DET
ejpam-6066	238	17	given	give	VERB
ejpam-6066	238	18	equation	equation	NOUN
ejpam-6066	238	19	but	but	CCONJ
ejpam-6066	238	20	also	also	ADV
ejpam-6066	238	21	establish	establish	VERB
ejpam-6066	238	22	a	a	DET
ejpam-6066	238	23	pathway	pathway	NOUN
ejpam-6066	238	24	for	for	ADP
ejpam-6066	238	25	studying	study	VERB
ejpam-6066	238	26	similar	similar	ADJ
ejpam-6066	238	27	equations	equation	NOUN
ejpam-6066	238	28	involving	involve	VERB
ejpam-6066	238	29	asymmetrical	asymmetrical	ADJ
ejpam-6066	238	30	exponential	exponential	ADJ
ejpam-6066	238	31	and	and	CCONJ
ejpam-6066	238	32	quadratic	quadratic	ADJ
ejpam-6066	238	33	forms	form	NOUN
ejpam-6066	238	34	.	.	PUNCT
ejpam-6066	239	1	we	we	PRON
ejpam-6066	239	2	hope	hope	VERB
ejpam-6066	239	3	these	these	DET
ejpam-6066	239	4	results	result	NOUN
ejpam-6066	239	5	will	will	AUX
ejpam-6066	239	6	inspire	inspire	VERB
ejpam-6066	239	7	further	further	ADJ
ejpam-6066	239	8	theoretical	theoretical	ADJ
ejpam-6066	239	9	generalizations	generalization	NOUN
ejpam-6066	239	10	and	and	CCONJ
ejpam-6066	239	11	computational	computational	ADJ
ejpam-6066	239	12	techniques	technique	NOUN
ejpam-6066	239	13	for	for	ADP
ejpam-6066	239	14	analyzing	analyze	VERB
ejpam-6066	239	15	diophantine	diophantine	NOUN
ejpam-6066	239	16	equations	equation	NOUN
ejpam-6066	239	17	of	of	ADP
ejpam-6066	239	18	this	this	DET
ejpam-6066	239	19	type	type	NOUN
ejpam-6066	239	20	.	.	PUNCT
ejpam-6066	240	1	algorithm	algorithm	NOUN
ejpam-6066	240	2	1	1	NUM
ejpam-6066	240	3	pseudocode	pseudocode	NOUN
ejpam-6066	240	4	for	for	ADP
ejpam-6066	240	5	solution	solution	NOUN
ejpam-6066	240	6	search	search	NOUN
ejpam-6066	240	7	and	and	CCONJ
ejpam-6066	240	8	verification	verification	NOUN
ejpam-6066	240	9	.	.	PUNCT
ejpam-6066	241	1	input	input	NOUN
ejpam-6066	241	2	:	:	PUNCT
ejpam-6066	241	3	pmin	pmin	NOUN
ejpam-6066	241	4	:	:	PUNCT
ejpam-6066	241	5	minimum	minimum	ADJ
ejpam-6066	241	6	prime	prime	NOUN
ejpam-6066	241	7	considered	consider	VERB
ejpam-6066	241	8	;	;	PUNCT
ejpam-6066	241	9	pmax	pmax	ADJ
ejpam-6066	241	10	:	:	PUNCT
ejpam-6066	241	11	maximum	maximum	ADJ
ejpam-6066	241	12	prime	prime	NOUN
ejpam-6066	241	13	considered	consider	VERB
ejpam-6066	241	14	;	;	PUNCT
ejpam-6066	241	15	xmax	xmax	PROPN
ejpam-6066	241	16	:	:	PUNCT
ejpam-6066	241	17	maximum	maximum	ADJ
ejpam-6066	241	18	value	value	NOUN
ejpam-6066	241	19	investigated	investigate	VERB
ejpam-6066	241	20	for	for	ADP
ejpam-6066	241	21	x.	x.	NOUN
ejpam-6066	241	22	output	output	NOUN
ejpam-6066	241	23	:	:	PUNCT
ejpam-6066	241	24	verification	verification	NOUN
ejpam-6066	241	25	results	result	NOUN
ejpam-6066	241	26	:	:	PUNCT
ejpam-6066	241	27	diophantine	diophantine	VERB
ejpam-6066	241	28	equation	equation	NOUN
ejpam-6066	241	29	solutions	solution	NOUN
ejpam-6066	241	30	.	.	PUNCT
ejpam-6066	242	1	for	for	ADP
ejpam-6066	242	2	p←	p←	DET
ejpam-6066	242	3	pmin	pmin	NOUN
ejpam-6066	242	4	to	to	PART
ejpam-6066	242	5	pmax	pmax	VERB
ejpam-6066	242	6	do	do	AUX
ejpam-6066	242	7	for	for	ADP
ejpam-6066	242	8	x←	x←	PROPN
ejpam-6066	242	9	1	1	NUM
ejpam-6066	242	10	to	to	AUX
ejpam-6066	242	11	xmax	xmax	PROPN
ejpam-6066	242	12	do	do	AUX
ejpam-6066	242	13	ymax	ymax	PROPN
ejpam-6066	242	14	←	←	PROPN
ejpam-6066	242	15	⌊logp	⌊logp	ADV
ejpam-6066	242	16	4(7x)⌋	4(7x)⌋	NUM
ejpam-6066	242	17	;	;	PUNCT
ejpam-6066	242	18	//	//	NUM
ejpam-6066	242	19	the	the	DET
ejpam-6066	242	20	maximum	maximum	ADJ
ejpam-6066	242	21	value	value	NOUN
ejpam-6066	242	22	of	of	ADP
ejpam-6066	242	23	y	y	PROPN
ejpam-6066	242	24	for	for	ADP
ejpam-6066	242	25	each	each	DET
ejpam-6066	242	26	pair	pair	NOUN
ejpam-6066	242	27	(	(	PUNCT
ejpam-6066	242	28	p	p	X
ejpam-6066	242	29	,	,	PUNCT
ejpam-6066	242	30	x	x	NOUN
ejpam-6066	242	31	)	)	PUNCT
ejpam-6066	242	32	for	for	ADP
ejpam-6066	242	33	y	y	PROPN
ejpam-6066	242	34	←	←	PROPN
ejpam-6066	242	35	1	1	NUM
ejpam-6066	242	36	to	to	PART
ejpam-6066	242	37	ymax	ymax	ADV
ejpam-6066	242	38	do	do	VERB
ejpam-6066	242	39	if	if	SCONJ
ejpam-6066	242	40	4(7x)−	4(7x)−	PROPN
ejpam-6066	242	41	py	py	PROPN
ejpam-6066	242	42	is	be	AUX
ejpam-6066	242	43	perfect	perfect	ADJ
ejpam-6066	242	44	square	square	ADJ
ejpam-6066	243	1	then	then	ADV
ejpam-6066	243	2	z	z	PROPN
ejpam-6066	243	3	←	←	PROPN
ejpam-6066	243	4	√	√	INTJ
ejpam-6066	243	5	4(7x)−	4(7x)−	PROPN
ejpam-6066	244	1	py	py	INTJ
ejpam-6066	245	1	if	if	SCONJ
ejpam-6066	245	2	(	(	PUNCT
ejpam-6066	245	3	x	x	X
ejpam-6066	245	4	,	,	PUNCT
ejpam-6066	245	5	y	y	PROPN
ejpam-6066	245	6	,	,	PUNCT
ejpam-6066	245	7	z	z	PROPN
ejpam-6066	245	8	,	,	PUNCT
ejpam-6066	245	9	p	p	NOUN
ejpam-6066	245	10	)	)	PUNCT
ejpam-6066	245	11	satisfies	satisfie	NOUN
ejpam-6066	245	12	theorem	theorem	VERB
ejpam-6066	245	13	1	1	NUM
ejpam-6066	245	14	(	(	PUNCT
ejpam-6066	245	15	verified	verify	VERB
ejpam-6066	245	16	by	by	ADP
ejpam-6066	245	17	algorithm	algorithm	NOUN
ejpam-6066	245	18	2	2	NUM
ejpam-6066	245	19	)	)	PUNCT
ejpam-6066	245	20	then	then	ADV
ejpam-6066	245	21	output	output	VERB
ejpam-6066	245	22	:	:	PUNCT
ejpam-6066	245	23	”	"	PUNCT
ejpam-6066	245	24	the	the	DET
ejpam-6066	245	25	solution	solution	NOUN
ejpam-6066	245	26	(	(	PUNCT
ejpam-6066	245	27	x	x	X
ejpam-6066	245	28	,	,	PUNCT
ejpam-6066	245	29	y	y	PROPN
ejpam-6066	245	30	,	,	PUNCT
ejpam-6066	245	31	z	z	PROPN
ejpam-6066	245	32	,	,	PUNCT
ejpam-6066	245	33	p	p	NOUN
ejpam-6066	245	34	)	)	PUNCT
ejpam-6066	245	35	satisfies	satisfy	VERB
ejpam-6066	245	36	the	the	DET
ejpam-6066	245	37	theorem	theorem	NOUN
ejpam-6066	245	38	1	1	NUM
ejpam-6066	245	39	”	"	PUNCT
ejpam-6066	245	40	else	else	ADV
ejpam-6066	245	41	output	output	NOUN
ejpam-6066	245	42	:	:	PUNCT
ejpam-6066	245	43	”	"	PUNCT
ejpam-6066	245	44	the	the	DET
ejpam-6066	245	45	solution	solution	NOUN
ejpam-6066	245	46	(	(	PUNCT
ejpam-6066	245	47	x	x	X
ejpam-6066	245	48	,	,	PUNCT
ejpam-6066	245	49	y	y	PROPN
ejpam-6066	245	50	,	,	PUNCT
ejpam-6066	245	51	z	z	PROPN
ejpam-6066	245	52	,	,	PUNCT
ejpam-6066	245	53	p	p	NOUN
ejpam-6066	245	54	)	)	PUNCT
ejpam-6066	245	55	does	do	AUX
ejpam-6066	245	56	not	not	PART
ejpam-6066	245	57	satisfy	satisfy	VERB
ejpam-6066	245	58	the	the	DET
ejpam-6066	245	59	theorem	theorem	ADJ
ejpam-6066	245	60	1	1	NUM
ejpam-6066	245	61	”	"	PUNCT
ejpam-6066	245	62	end	end	NOUN
ejpam-6066	245	63	if	if	SCONJ
ejpam-6066	245	64	p	p	NOUN
ejpam-6066	245	65	≥	≥	NOUN
ejpam-6066	245	66	19	19	NUM
ejpam-6066	245	67	then	then	ADV
ejpam-6066	245	68	if	if	SCONJ
ejpam-6066	245	69	(	(	PUNCT
ejpam-6066	245	70	x	x	X
ejpam-6066	245	71	,	,	PUNCT
ejpam-6066	245	72	y	y	PROPN
ejpam-6066	245	73	,	,	PUNCT
ejpam-6066	245	74	z	z	PROPN
ejpam-6066	245	75	,	,	PUNCT
ejpam-6066	245	76	p	p	NOUN
ejpam-6066	245	77	)	)	PUNCT
ejpam-6066	245	78	satisfies	satisfie	NOUN
ejpam-6066	245	79	corollary	corollary	ADJ
ejpam-6066	245	80	1	1	NUM
ejpam-6066	245	81	(	(	PUNCT
ejpam-6066	245	82	verified	verify	VERB
ejpam-6066	245	83	by	by	ADP
ejpam-6066	245	84	algorithm	algorithm	NOUN
ejpam-6066	245	85	3	3	NUM
ejpam-6066	245	86	)	)	PUNCT
ejpam-6066	245	87	then	then	ADV
ejpam-6066	245	88	output	output	VERB
ejpam-6066	245	89	:	:	PUNCT
ejpam-6066	245	90	”	"	PUNCT
ejpam-6066	245	91	the	the	DET
ejpam-6066	245	92	solution	solution	NOUN
ejpam-6066	245	93	(	(	PUNCT
ejpam-6066	245	94	x	x	X
ejpam-6066	245	95	,	,	PUNCT
ejpam-6066	245	96	y	y	PROPN
ejpam-6066	245	97	,	,	PUNCT
ejpam-6066	245	98	z	z	PROPN
ejpam-6066	245	99	,	,	PUNCT
ejpam-6066	245	100	p	p	NOUN
ejpam-6066	245	101	)	)	PUNCT
ejpam-6066	245	102	satisfies	satisfy	VERB
ejpam-6066	245	103	the	the	DET
ejpam-6066	245	104	corollary	corollary	ADJ
ejpam-6066	245	105	1	1	NUM
ejpam-6066	245	106	”	"	PUNCT
ejpam-6066	245	107	else	else	ADV
ejpam-6066	245	108	output	output	NOUN
ejpam-6066	245	109	:	:	PUNCT
ejpam-6066	245	110	”	"	PUNCT
ejpam-6066	245	111	the	the	DET
ejpam-6066	245	112	solution	solution	NOUN
ejpam-6066	245	113	(	(	PUNCT
ejpam-6066	245	114	x	x	X
ejpam-6066	245	115	,	,	PUNCT
ejpam-6066	245	116	y	y	PROPN
ejpam-6066	245	117	,	,	PUNCT
ejpam-6066	245	118	z	z	PROPN
ejpam-6066	245	119	,	,	PUNCT
ejpam-6066	245	120	p	p	NOUN
ejpam-6066	245	121	)	)	PUNCT
ejpam-6066	245	122	does	do	AUX
ejpam-6066	245	123	not	not	PART
ejpam-6066	245	124	satisfy	satisfy	VERB
ejpam-6066	245	125	the	the	DET
ejpam-6066	245	126	corollary	corollary	ADJ
ejpam-6066	245	127	1	1	NUM
ejpam-6066	245	128	”	"	PUNCT
ejpam-6066	245	129	end	end	NOUN
ejpam-6066	245	130	end	end	NOUN
ejpam-6066	245	131	end	end	NOUN
ejpam-6066	245	132	output	output	NOUN
ejpam-6066	245	133	:	:	PUNCT
ejpam-6066	245	134	”	"	PUNCT
ejpam-6066	245	135	(	(	PUNCT
ejpam-6066	245	136	x	x	X
ejpam-6066	245	137	,	,	PUNCT
ejpam-6066	245	138	y	y	PROPN
ejpam-6066	245	139	,	,	PUNCT
ejpam-6066	245	140	z	z	PROPN
ejpam-6066	245	141	,	,	PUNCT
ejpam-6066	245	142	p	p	NOUN
ejpam-6066	245	143	)	)	PUNCT
ejpam-6066	245	144	is	be	AUX
ejpam-6066	245	145	not	not	PART
ejpam-6066	245	146	a	a	DET
ejpam-6066	245	147	solution	solution	NOUN
ejpam-6066	245	148	to	to	ADP
ejpam-6066	245	149	the	the	DET
ejpam-6066	245	150	equation	equation	NOUN
ejpam-6066	245	151	”	"	PUNCT
ejpam-6066	245	152	end	end	NOUN
ejpam-6066	245	153	end	end	NOUN
ejpam-6066	245	154	end	end	NOUN
ejpam-6066	246	1	k.	k.	PROPN
ejpam-6066	246	2	laipaporn	laipaporn	PROPN
ejpam-6066	246	3	et	et	PROPN
ejpam-6066	246	4	al	al	PROPN
ejpam-6066	246	5	.	.	PUNCT
ejpam-6066	246	6	/	/	SYM
ejpam-6066	246	7	eur	eur	PROPN
ejpam-6066	246	8	.	.	PUNCT
ejpam-6066	247	1	j.	j.	PROPN
ejpam-6066	247	2	pure	pure	PROPN
ejpam-6066	247	3	appl	appl	PROPN
ejpam-6066	247	4	.	.	PROPN
ejpam-6066	247	5	math	math	PROPN
ejpam-6066	247	6	,	,	PUNCT
ejpam-6066	247	7	18	18	NUM
ejpam-6066	247	8	(	(	PUNCT
ejpam-6066	247	9	3	3	NUM
ejpam-6066	247	10	)	)	PUNCT
ejpam-6066	247	11	(	(	PUNCT
ejpam-6066	247	12	2025	2025	NUM
ejpam-6066	247	13	)	)	PUNCT
ejpam-6066	247	14	,	,	PUNCT
ejpam-6066	247	15	6066	6066	NUM
ejpam-6066	247	16	11	11	NUM
ejpam-6066	247	17	of	of	ADP
ejpam-6066	247	18	13	13	NUM
ejpam-6066	247	19	algorithm	algorithm	NOUN
ejpam-6066	247	20	2	2	NUM
ejpam-6066	247	21	pseudocode	pseudocode	NOUN
ejpam-6066	247	22	for	for	ADP
ejpam-6066	247	23	verifying	verify	VERB
ejpam-6066	247	24	a	a	DET
ejpam-6066	247	25	solution	solution	NOUN
ejpam-6066	247	26	(	(	PUNCT
ejpam-6066	247	27	x	x	X
ejpam-6066	247	28	,	,	PUNCT
ejpam-6066	247	29	y	y	PROPN
ejpam-6066	247	30	,	,	PUNCT
ejpam-6066	247	31	z	z	PROPN
ejpam-6066	247	32	,	,	PUNCT
ejpam-6066	247	33	p	p	NOUN
ejpam-6066	247	34	)	)	PUNCT
ejpam-6066	247	35	to	to	ADP
ejpam-6066	247	36	the	the	DET
ejpam-6066	247	37	theorem	theorem	ADJ
ejpam-6066	247	38	1	1	X
ejpam-6066	247	39	.	.	PUNCT
ejpam-6066	247	40	input	input	NOUN
ejpam-6066	247	41	:	:	PUNCT
ejpam-6066	247	42	(	(	PUNCT
ejpam-6066	247	43	x	x	X
ejpam-6066	247	44	,	,	PUNCT
ejpam-6066	247	45	y	y	PROPN
ejpam-6066	247	46	,	,	PUNCT
ejpam-6066	247	47	z	z	PROPN
ejpam-6066	247	48	,	,	PUNCT
ejpam-6066	247	49	p	p	NOUN
ejpam-6066	247	50	):	):	PUNCT
ejpam-6066	247	51	solution	solution	NOUN
ejpam-6066	247	52	of	of	ADP
ejpam-6066	247	53	the	the	DET
ejpam-6066	247	54	diophantine	diophantine	NOUN
ejpam-6066	247	55	equation	equation	NOUN
ejpam-6066	247	56	output	output	NOUN
ejpam-6066	247	57	:	:	PUNCT
ejpam-6066	247	58	1	1	NUM
ejpam-6066	247	59	:	:	PUNCT
ejpam-6066	247	60	the	the	DET
ejpam-6066	247	61	solution	solution	NOUN
ejpam-6066	247	62	satisfies	satisfy	VERB
ejpam-6066	247	63	the	the	DET
ejpam-6066	247	64	theorem	theorem	NOUN
ejpam-6066	247	65	,	,	PUNCT
ejpam-6066	247	66	0	0	NUM
ejpam-6066	247	67	:	:	PUNCT
ejpam-6066	247	68	the	the	DET
ejpam-6066	247	69	solution	solution	NOUN
ejpam-6066	247	70	does	do	AUX
ejpam-6066	247	71	not	not	PART
ejpam-6066	247	72	satisfy	satisfy	VERB
ejpam-6066	247	73	theorem	theorem	ADJ
ejpam-6066	247	74	function	function	NOUN
ejpam-6066	247	75	satisfymaintheorem(x	satisfymaintheorem(x	PROPN
ejpam-6066	247	76	,	,	PUNCT
ejpam-6066	247	77	y	y	PROPN
ejpam-6066	247	78	,	,	PUNCT
ejpam-6066	247	79	z	z	PROPN
ejpam-6066	247	80	,	,	PUNCT
ejpam-6066	247	81	p	p	NOUN
ejpam-6066	247	82	)	)	PUNCT
ejpam-6066	247	83	if	if	SCONJ
ejpam-6066	247	84	(	(	PUNCT
ejpam-6066	247	85	x	x	NOUN
ejpam-6066	247	86	,	,	PUNCT
ejpam-6066	247	87	y	y	PROPN
ejpam-6066	247	88	,	,	PUNCT
ejpam-6066	247	89	z	z	PROPN
ejpam-6066	247	90	,	,	PUNCT
ejpam-6066	247	91	p	p	NOUN
ejpam-6066	247	92	)	)	PUNCT
ejpam-6066	247	93	∈	∈	PROPN
ejpam-6066	247	94	{	{	PUNCT
ejpam-6066	247	95	(	(	PUNCT
ejpam-6066	247	96	0	0	NUM
ejpam-6066	247	97	,	,	PUNCT
ejpam-6066	247	98	2	2	NUM
ejpam-6066	247	99	,	,	PUNCT
ejpam-6066	247	100	0	0	NUM
ejpam-6066	247	101	,	,	PUNCT
ejpam-6066	247	102	2	2	NUM
ejpam-6066	247	103	)	)	PUNCT
ejpam-6066	247	104	}	}	PUNCT
ejpam-6066	247	105	then	then	ADV
ejpam-6066	247	106	return	return	VERB
ejpam-6066	247	107	1	1	NUM
ejpam-6066	247	108	else	else	ADV
ejpam-6066	247	109	if	if	SCONJ
ejpam-6066	247	110	p	p	NOUN
ejpam-6066	247	111	=	=	NOUN
ejpam-6066	247	112	3	3	NUM
ejpam-6066	247	113	then	then	ADV
ejpam-6066	247	114	if	if	SCONJ
ejpam-6066	247	115	(	(	PUNCT
ejpam-6066	247	116	x	x	X
ejpam-6066	247	117	≥	≥	NUM
ejpam-6066	247	118	5	5	NUM
ejpam-6066	247	119	and	and	CCONJ
ejpam-6066	247	120	x	x	NOUN
ejpam-6066	247	121	mod	mod	PROPN
ejpam-6066	247	122	2	2	NUM
ejpam-6066	247	123	=	=	SYM
ejpam-6066	247	124	1	1	NUM
ejpam-6066	247	125	and	and	CCONJ
ejpam-6066	247	126	y	y	PROPN
ejpam-6066	247	127	mod	mod	NOUN
ejpam-6066	247	128	4	4	NUM
ejpam-6066	247	129	=	=	SYM
ejpam-6066	247	130	1	1	NUM
ejpam-6066	247	131	and	and	CCONJ
ejpam-6066	247	132	z	z	NOUN
ejpam-6066	247	133	mod	mod	NOUN
ejpam-6066	248	1	16	16	NUM
ejpam-6066	248	2	∈	∈	PROPN
ejpam-6066	248	3	{	{	PUNCT
ejpam-6066	248	4	3	3	NUM
ejpam-6066	248	5	,	,	PUNCT
ejpam-6066	248	6	5	5	NUM
ejpam-6066	248	7	,	,	PUNCT
ejpam-6066	248	8	11	11	NUM
ejpam-6066	248	9	,	,	PUNCT
ejpam-6066	248	10	13	13	NUM
ejpam-6066	248	11	}	}	PUNCT
ejpam-6066	248	12	)	)	PUNCT
ejpam-6066	248	13	or	or	CCONJ
ejpam-6066	248	14	(	(	PUNCT
ejpam-6066	248	15	x	x	X
ejpam-6066	248	16	≥	≥	NUM
ejpam-6066	248	17	5	5	NUM
ejpam-6066	248	18	and	and	CCONJ
ejpam-6066	248	19	x	x	NOUN
ejpam-6066	248	20	mod	mod	PROPN
ejpam-6066	248	21	2	2	NUM
ejpam-6066	248	22	=	=	SYM
ejpam-6066	248	23	1	1	NUM
ejpam-6066	248	24	and	and	CCONJ
ejpam-6066	248	25	y	y	PROPN
ejpam-6066	248	26	mod	mod	NOUN
ejpam-6066	248	27	4	4	NUM
ejpam-6066	248	28	=	=	SYM
ejpam-6066	248	29	3	3	NUM
ejpam-6066	248	30	and	and	CCONJ
ejpam-6066	248	31	z	z	PROPN
ejpam-6066	248	32	mod	mod	NOUN
ejpam-6066	249	1	16	16	NUM
ejpam-6066	249	2	∈	∈	PROPN
ejpam-6066	249	3	{	{	PUNCT
ejpam-6066	249	4	1	1	NUM
ejpam-6066	249	5	,	,	PUNCT
ejpam-6066	249	6	7	7	NUM
ejpam-6066	249	7	,	,	PUNCT
ejpam-6066	249	8	9	9	NUM
ejpam-6066	249	9	,	,	PUNCT
ejpam-6066	249	10	15	15	NUM
ejpam-6066	249	11	}	}	PUNCT
ejpam-6066	249	12	)	)	PUNCT
ejpam-6066	249	13	or	or	CCONJ
ejpam-6066	249	14	(	(	PUNCT
ejpam-6066	249	15	(	(	PUNCT
ejpam-6066	249	16	x	x	NOUN
ejpam-6066	249	17	,	,	PUNCT
ejpam-6066	249	18	y	y	PROPN
ejpam-6066	249	19	,	,	PUNCT
ejpam-6066	249	20	z	z	NOUN
ejpam-6066	249	21	)	)	PUNCT
ejpam-6066	249	22	∈	∈	NOUN
ejpam-6066	249	23	{	{	PUNCT
ejpam-6066	249	24	(	(	PUNCT
ejpam-6066	249	25	0	0	NUM
ejpam-6066	249	26	,	,	PUNCT
ejpam-6066	249	27	1	1	NUM
ejpam-6066	249	28	,	,	PUNCT
ejpam-6066	249	29	1	1	NUM
ejpam-6066	249	30	)	)	PUNCT
ejpam-6066	249	31	,	,	PUNCT
ejpam-6066	249	32	(	(	PUNCT
ejpam-6066	249	33	1	1	NUM
ejpam-6066	249	34	,	,	PUNCT
ejpam-6066	249	35	1	1	NUM
ejpam-6066	249	36	,	,	PUNCT
ejpam-6066	249	37	5	5	NUM
ejpam-6066	249	38	)	)	PUNCT
ejpam-6066	249	39	,	,	PUNCT
ejpam-6066	249	40	(	(	PUNCT
ejpam-6066	249	41	1	1	NUM
ejpam-6066	249	42	,	,	PUNCT
ejpam-6066	249	43	3	3	NUM
ejpam-6066	249	44	,	,	PUNCT
ejpam-6066	249	45	1	1	NUM
ejpam-6066	249	46	)	)	PUNCT
ejpam-6066	249	47	,	,	PUNCT
ejpam-6066	249	48	(	(	PUNCT
ejpam-6066	249	49	2	2	NUM
ejpam-6066	249	50	,	,	PUNCT
ejpam-6066	249	51	3	3	NUM
ejpam-6066	249	52	,	,	PUNCT
ejpam-6066	249	53	13	13	NUM
ejpam-6066	249	54	)	)	PUNCT
ejpam-6066	249	55	,	,	PUNCT
ejpam-6066	249	56	(	(	PUNCT
ejpam-6066	249	57	3	3	NUM
ejpam-6066	249	58	,	,	PUNCT
ejpam-6066	249	59	1	1	NUM
ejpam-6066	249	60	,	,	PUNCT
ejpam-6066	249	61	37	37	NUM
ejpam-6066	249	62	)	)	PUNCT
ejpam-6066	249	63	}	}	PUNCT
ejpam-6066	249	64	)	)	PUNCT
ejpam-6066	249	65	then	then	ADV
ejpam-6066	249	66	return	return	VERB
ejpam-6066	249	67	1	1	NUM
ejpam-6066	249	68	end	end	NOUN
ejpam-6066	249	69	else	else	ADV
ejpam-6066	249	70	if	if	SCONJ
ejpam-6066	249	71	p	p	PROPN
ejpam-6066	249	72	≥	≥	PUNCT
ejpam-6066	249	73	19	19	NUM
ejpam-6066	250	1	and	and	CCONJ
ejpam-6066	250	2	x	x	SYM
ejpam-6066	250	3	mod	mod	PROPN
ejpam-6066	250	4	2	2	NUM
ejpam-6066	250	5	=	=	SYM
ejpam-6066	250	6	1	1	NUM
ejpam-6066	250	7	and	and	CCONJ
ejpam-6066	250	8	y	y	PROPN
ejpam-6066	250	9	mod	mod	NOUN
ejpam-6066	250	10	2	2	NUM
ejpam-6066	250	11	=	=	SYM
ejpam-6066	250	12	1	1	NUM
ejpam-6066	250	13	and	and	CCONJ
ejpam-6066	250	14	z	z	NOUN
ejpam-6066	250	15	mod	mod	NOUN
ejpam-6066	250	16	24	24	NUM
ejpam-6066	250	17	∈	∈	PROPN
ejpam-6066	250	18	{	{	PUNCT
ejpam-6066	250	19	3	3	NUM
ejpam-6066	250	20	,	,	PUNCT
ejpam-6066	250	21	9	9	NUM
ejpam-6066	250	22	,	,	PUNCT
ejpam-6066	250	23	15	15	NUM
ejpam-6066	250	24	,	,	PUNCT
ejpam-6066	250	25	21	21	NUM
ejpam-6066	250	26	}	}	PUNCT
ejpam-6066	250	27	then	then	ADV
ejpam-6066	250	28	return	return	VERB
ejpam-6066	250	29	1	1	NUM
ejpam-6066	250	30	return	return	NOUN
ejpam-6066	250	31	0	0	NUM
ejpam-6066	250	32	algorithm	algorithm	NOUN
ejpam-6066	250	33	3	3	NUM
ejpam-6066	250	34	pseudocode	pseudocode	NOUN
ejpam-6066	250	35	for	for	ADP
ejpam-6066	250	36	verifying	verify	VERB
ejpam-6066	250	37	a	a	DET
ejpam-6066	250	38	solution	solution	NOUN
ejpam-6066	250	39	(	(	PUNCT
ejpam-6066	250	40	x	x	X
ejpam-6066	250	41	,	,	PUNCT
ejpam-6066	250	42	y	y	PROPN
ejpam-6066	250	43	,	,	PUNCT
ejpam-6066	250	44	z	z	PROPN
ejpam-6066	250	45	,	,	PUNCT
ejpam-6066	250	46	p	p	NOUN
ejpam-6066	250	47	)	)	PUNCT
ejpam-6066	250	48	to	to	ADP
ejpam-6066	250	49	the	the	DET
ejpam-6066	250	50	corollary	corollary	ADJ
ejpam-6066	250	51	1	1	NUM
ejpam-6066	250	52	.	.	PUNCT
ejpam-6066	250	53	input	input	NOUN
ejpam-6066	250	54	:	:	PUNCT
ejpam-6066	250	55	(	(	PUNCT
ejpam-6066	250	56	x	x	X
ejpam-6066	250	57	,	,	PUNCT
ejpam-6066	250	58	y	y	PROPN
ejpam-6066	250	59	,	,	PUNCT
ejpam-6066	250	60	z	z	PROPN
ejpam-6066	250	61	,	,	PUNCT
ejpam-6066	250	62	p	p	NOUN
ejpam-6066	250	63	):	):	PUNCT
ejpam-6066	250	64	solution	solution	NOUN
ejpam-6066	250	65	of	of	ADP
ejpam-6066	250	66	the	the	DET
ejpam-6066	250	67	diophantine	diophantine	NOUN
ejpam-6066	250	68	equation	equation	NOUN
ejpam-6066	250	69	output	output	NOUN
ejpam-6066	250	70	:	:	PUNCT
ejpam-6066	251	1	1	1	NUM
ejpam-6066	251	2	:	:	PUNCT
ejpam-6066	251	3	the	the	DET
ejpam-6066	251	4	solution	solution	NOUN
ejpam-6066	251	5	satisfies	satisfy	VERB
ejpam-6066	251	6	the	the	DET
ejpam-6066	251	7	corollary	corollary	ADJ
ejpam-6066	251	8	,	,	PUNCT
ejpam-6066	251	9	0	0	NUM
ejpam-6066	251	10	:	:	PUNCT
ejpam-6066	251	11	the	the	DET
ejpam-6066	251	12	solution	solution	NOUN
ejpam-6066	251	13	does	do	AUX
ejpam-6066	251	14	not	not	PART
ejpam-6066	251	15	satisfy	satisfy	VERB
ejpam-6066	251	16	corollary	corollary	ADJ
ejpam-6066	251	17	function	function	NOUN
ejpam-6066	251	18	satisfycorollary(x	satisfycorollary(x	PROPN
ejpam-6066	251	19	,	,	PUNCT
ejpam-6066	251	20	y	y	PROPN
ejpam-6066	251	21	,	,	PUNCT
ejpam-6066	251	22	z	z	PROPN
ejpam-6066	251	23	,	,	PUNCT
ejpam-6066	251	24	p	p	NOUN
ejpam-6066	251	25	)	)	PUNCT
ejpam-6066	252	1	if	if	SCONJ
ejpam-6066	252	2	z	z	NOUN
ejpam-6066	252	3	mod	mod	NOUN
ejpam-6066	252	4	24	24	NUM
ejpam-6066	252	5	∈	∈	PROPN
ejpam-6066	252	6	{	{	PUNCT
ejpam-6066	252	7	3	3	NUM
ejpam-6066	252	8	,	,	PUNCT
ejpam-6066	252	9	9	9	NUM
ejpam-6066	252	10	,	,	PUNCT
ejpam-6066	252	11	15	15	NUM
ejpam-6066	252	12	,	,	PUNCT
ejpam-6066	252	13	21	21	NUM
ejpam-6066	252	14	}	}	PUNCT
ejpam-6066	252	15	then	then	ADV
ejpam-6066	252	16	if	if	SCONJ
ejpam-6066	252	17	x	x	X
ejpam-6066	252	18	mod	mod	NOUN
ejpam-6066	252	19	6	6	NUM
ejpam-6066	252	20	=	=	SYM
ejpam-6066	252	21	1	1	NUM
ejpam-6066	252	22	then	then	ADV
ejpam-6066	252	23	if	if	SCONJ
ejpam-6066	252	24	y	y	PROPN
ejpam-6066	252	25	mod	mod	PROPN
ejpam-6066	252	26	6	6	NUM
ejpam-6066	252	27	=	=	SYM
ejpam-6066	252	28	3	3	NUM
ejpam-6066	252	29	and	and	CCONJ
ejpam-6066	252	30	p	p	NOUN
ejpam-6066	252	31	mod	mod	ADJ
ejpam-6066	252	32	24	24	NUM
ejpam-6066	252	33	=	=	SYM
ejpam-6066	252	34	19	19	NUM
ejpam-6066	252	35	then	then	ADV
ejpam-6066	252	36	return	return	VERB
ejpam-6066	252	37	1	1	NUM
ejpam-6066	252	38	;	;	PUNCT
ejpam-6066	252	39	//	//	NUM
ejpam-6066	252	40	congruence	congruence	PROPN
ejpam-6066	252	41	class	class	NOUN
ejpam-6066	252	42	:	:	PUNCT
ejpam-6066	252	43	p	p	PROPN
ejpam-6066	252	44	≡	≡	PROPN
ejpam-6066	252	45	19	19	NUM
ejpam-6066	252	46	(	(	PUNCT
ejpam-6066	252	47	mod	mod	NOUN
ejpam-6066	252	48	24	24	NUM
ejpam-6066	252	49	)	)	PUNCT
ejpam-6066	252	50	else	else	ADV
ejpam-6066	252	51	if	if	SCONJ
ejpam-6066	252	52	y	y	PROPN
ejpam-6066	252	53	mod	mod	PROPN
ejpam-6066	252	54	6	6	NUM
ejpam-6066	252	55	=	=	SYM
ejpam-6066	252	56	1	1	NUM
ejpam-6066	252	57	and	and	CCONJ
ejpam-6066	252	58	p	p	ADJ
ejpam-6066	252	59	mod	mod	NOUN
ejpam-6066	252	60	72	72	NUM
ejpam-6066	252	61	=	=	SYM
ejpam-6066	252	62	19	19	NUM
ejpam-6066	252	63	then	then	ADV
ejpam-6066	252	64	return	return	VERB
ejpam-6066	252	65	1	1	NUM
ejpam-6066	252	66	;	;	PUNCT
ejpam-6066	252	67	//	//	NUM
ejpam-6066	252	68	congruence	congruence	PROPN
ejpam-6066	252	69	class	class	NOUN
ejpam-6066	252	70	:	:	PUNCT
ejpam-6066	252	71	p	p	PROPN
ejpam-6066	252	72	≡	≡	PROPN
ejpam-6066	252	73	19	19	NUM
ejpam-6066	252	74	(	(	PUNCT
ejpam-6066	252	75	mod	mod	PROPN
ejpam-6066	252	76	72	72	NUM
ejpam-6066	252	77	)	)	PUNCT
ejpam-6066	252	78	else	else	ADV
ejpam-6066	252	79	if	if	SCONJ
ejpam-6066	252	80	y	y	PROPN
ejpam-6066	252	81	mod	mod	PROPN
ejpam-6066	252	82	6	6	NUM
ejpam-6066	252	83	=	=	SYM
ejpam-6066	252	84	5	5	NUM
ejpam-6066	252	85	and	and	CCONJ
ejpam-6066	252	86	p	p	ADJ
ejpam-6066	252	87	mod	mod	NOUN
ejpam-6066	252	88	72	72	NUM
ejpam-6066	252	89	=	=	SYM
ejpam-6066	252	90	19	19	NUM
ejpam-6066	252	91	then	then	ADV
ejpam-6066	252	92	return	return	VERB
ejpam-6066	252	93	1	1	NUM
ejpam-6066	252	94	;	;	PUNCT
ejpam-6066	252	95	//	//	NUM
ejpam-6066	252	96	congruence	congruence	PROPN
ejpam-6066	252	97	class	class	NOUN
ejpam-6066	252	98	:	:	PUNCT
ejpam-6066	252	99	p	p	PROPN
ejpam-6066	252	100	≡	≡	PROPN
ejpam-6066	252	101	19	19	NUM
ejpam-6066	252	102	(	(	PUNCT
ejpam-6066	252	103	mod	mod	PROPN
ejpam-6066	252	104	72	72	NUM
ejpam-6066	252	105	)	)	PUNCT
ejpam-6066	252	106	else	else	ADV
ejpam-6066	252	107	if	if	SCONJ
ejpam-6066	252	108	x	x	X
ejpam-6066	252	109	mod	mod	NOUN
ejpam-6066	252	110	6	6	NUM
ejpam-6066	252	111	=	=	SYM
ejpam-6066	252	112	3	3	NUM
ejpam-6066	252	113	then	then	ADV
ejpam-6066	252	114	if	if	SCONJ
ejpam-6066	252	115	y	y	PROPN
ejpam-6066	252	116	mod	mod	PROPN
ejpam-6066	252	117	6	6	NUM
ejpam-6066	252	118	=	=	SYM
ejpam-6066	252	119	5	5	NUM
ejpam-6066	252	120	and	and	CCONJ
ejpam-6066	252	121	p	p	ADJ
ejpam-6066	252	122	mod	mod	NOUN
ejpam-6066	252	123	72	72	NUM
ejpam-6066	252	124	=	=	SYM
ejpam-6066	252	125	43	43	NUM
ejpam-6066	252	126	then	then	ADV
ejpam-6066	252	127	return	return	VERB
ejpam-6066	252	128	1	1	NUM
ejpam-6066	252	129	;	;	PUNCT
ejpam-6066	252	130	//	//	NUM
ejpam-6066	252	131	congruence	congruence	PROPN
ejpam-6066	252	132	class	class	NOUN
ejpam-6066	252	133	:	:	PUNCT
ejpam-6066	252	134	p	p	PROPN
ejpam-6066	252	135	≡	≡	PROPN
ejpam-6066	252	136	43	43	NUM
ejpam-6066	252	137	(	(	PUNCT
ejpam-6066	252	138	mod	mod	PROPN
ejpam-6066	252	139	72	72	NUM
ejpam-6066	252	140	)	)	PUNCT
ejpam-6066	252	141	else	else	ADV
ejpam-6066	253	1	if	if	SCONJ
ejpam-6066	253	2	y	y	PROPN
ejpam-6066	253	3	mod	mod	PROPN
ejpam-6066	253	4	6	6	NUM
ejpam-6066	253	5	=	=	SYM
ejpam-6066	253	6	1	1	NUM
ejpam-6066	253	7	and	and	CCONJ
ejpam-6066	253	8	p	p	ADJ
ejpam-6066	253	9	mod	mod	NOUN
ejpam-6066	253	10	72	72	NUM
ejpam-6066	253	11	=	=	SYM
ejpam-6066	253	12	67	67	NUM
ejpam-6066	253	13	then	then	ADV
ejpam-6066	253	14	return	return	VERB
ejpam-6066	253	15	1	1	NUM
ejpam-6066	253	16	;	;	PUNCT
ejpam-6066	253	17	//	//	NUM
ejpam-6066	253	18	congruence	congruence	PROPN
ejpam-6066	253	19	class	class	NOUN
ejpam-6066	253	20	:	:	PUNCT
ejpam-6066	253	21	p	p	PROPN
ejpam-6066	253	22	≡	≡	PROPN
ejpam-6066	253	23	67	67	NUM
ejpam-6066	253	24	(	(	PUNCT
ejpam-6066	253	25	mod	mod	PROPN
ejpam-6066	253	26	72	72	NUM
ejpam-6066	253	27	)	)	PUNCT
ejpam-6066	253	28	else	else	ADV
ejpam-6066	253	29	if	if	SCONJ
ejpam-6066	253	30	x	x	X
ejpam-6066	253	31	mod	mod	NOUN
ejpam-6066	253	32	6	6	NUM
ejpam-6066	253	33	=	=	SYM
ejpam-6066	253	34	5	5	NUM
ejpam-6066	253	35	then	then	ADV
ejpam-6066	253	36	if	if	SCONJ
ejpam-6066	253	37	y	y	PROPN
ejpam-6066	253	38	mod	mod	PROPN
ejpam-6066	253	39	6	6	NUM
ejpam-6066	253	40	=	=	SYM
ejpam-6066	253	41	5	5	NUM
ejpam-6066	253	42	and	and	CCONJ
ejpam-6066	253	43	p	p	ADJ
ejpam-6066	253	44	mod	mod	NOUN
ejpam-6066	253	45	72	72	NUM
ejpam-6066	253	46	=	=	SYM
ejpam-6066	253	47	67	67	NUM
ejpam-6066	253	48	then	then	ADV
ejpam-6066	253	49	return	return	VERB
ejpam-6066	253	50	1	1	NUM
ejpam-6066	253	51	;	;	PUNCT
ejpam-6066	253	52	//	//	NUM
ejpam-6066	253	53	congruence	congruence	PROPN
ejpam-6066	253	54	class	class	NOUN
ejpam-6066	253	55	:	:	PUNCT
ejpam-6066	253	56	p	p	PROPN
ejpam-6066	253	57	≡	≡	PROPN
ejpam-6066	253	58	67	67	NUM
ejpam-6066	253	59	(	(	PUNCT
ejpam-6066	253	60	mod	mod	PROPN
ejpam-6066	253	61	72	72	NUM
ejpam-6066	253	62	)	)	PUNCT
ejpam-6066	253	63	else	else	ADV
ejpam-6066	254	1	if	if	SCONJ
ejpam-6066	254	2	y	y	PROPN
ejpam-6066	254	3	mod	mod	PROPN
ejpam-6066	254	4	6	6	NUM
ejpam-6066	254	5	=	=	SYM
ejpam-6066	254	6	1	1	NUM
ejpam-6066	254	7	and	and	CCONJ
ejpam-6066	254	8	p	p	ADJ
ejpam-6066	254	9	mod	mod	NOUN
ejpam-6066	254	10	72	72	NUM
ejpam-6066	254	11	=	=	SYM
ejpam-6066	254	12	43	43	NUM
ejpam-6066	254	13	then	then	ADV
ejpam-6066	254	14	return	return	VERB
ejpam-6066	254	15	1	1	NUM
ejpam-6066	254	16	;	;	PUNCT
ejpam-6066	254	17	//	//	NUM
ejpam-6066	254	18	congruence	congruence	PROPN
ejpam-6066	254	19	class	class	NOUN
ejpam-6066	254	20	:	:	PUNCT
ejpam-6066	254	21	p	p	PROPN
ejpam-6066	254	22	≡	≡	PROPN
ejpam-6066	254	23	43	43	NUM
ejpam-6066	254	24	(	(	PUNCT
ejpam-6066	254	25	mod	mod	PROPN
ejpam-6066	254	26	72	72	NUM
ejpam-6066	254	27	)	)	PUNCT
ejpam-6066	254	28	end	end	NOUN
ejpam-6066	254	29	return	return	NOUN
ejpam-6066	254	30	0	0	PUNCT
ejpam-6066	255	1	k.	k.	PROPN
ejpam-6066	255	2	laipaporn	laipaporn	VERB
ejpam-6066	255	3	et	et	PROPN
ejpam-6066	255	4	al	al	PROPN
ejpam-6066	255	5	.	.	PUNCT
ejpam-6066	255	6	/	/	SYM
ejpam-6066	255	7	eur	eur	PROPN
ejpam-6066	255	8	.	.	PUNCT
ejpam-6066	256	1	j.	j.	PROPN
ejpam-6066	256	2	pure	pure	PROPN
ejpam-6066	256	3	appl	appl	PROPN
ejpam-6066	256	4	.	.	PROPN
ejpam-6066	256	5	math	math	PROPN
ejpam-6066	256	6	,	,	PUNCT
ejpam-6066	256	7	18	18	NUM
ejpam-6066	256	8	(	(	PUNCT
ejpam-6066	256	9	3	3	NUM
ejpam-6066	256	10	)	)	PUNCT
ejpam-6066	256	11	(	(	PUNCT
ejpam-6066	256	12	2025	2025	NUM
ejpam-6066	256	13	)	)	PUNCT
ejpam-6066	256	14	,	,	PUNCT
ejpam-6066	256	15	6066	6066	NUM
ejpam-6066	256	16	12	12	NUM
ejpam-6066	256	17	of	of	ADP
ejpam-6066	256	18	13	13	NUM
ejpam-6066	256	19	acknowledgements	acknowledgement	NOUN
ejpam-6066	256	20	the	the	DET
ejpam-6066	256	21	authors	author	NOUN
ejpam-6066	256	22	gratefully	gratefully	ADV
ejpam-6066	256	23	acknowledge	acknowledge	VERB
ejpam-6066	256	24	the	the	DET
ejpam-6066	256	25	valuable	valuable	ADJ
ejpam-6066	256	26	and	and	CCONJ
ejpam-6066	256	27	constructive	constructive	ADJ
ejpam-6066	256	28	suggestions	suggestion	NOUN
ejpam-6066	256	29	and	and	CCONJ
ejpam-6066	256	30	comments	comment	NOUN
ejpam-6066	256	31	provided	provide	VERB
ejpam-6066	256	32	by	by	ADP
ejpam-6066	256	33	the	the	DET
ejpam-6066	256	34	reviewers	reviewer	NOUN
ejpam-6066	256	35	,	,	PUNCT
ejpam-6066	256	36	which	which	PRON
ejpam-6066	256	37	have	have	AUX
ejpam-6066	256	38	substantially	substantially	ADV
ejpam-6066	256	39	improved	improve	VERB
ejpam-6066	256	40	the	the	DET
ejpam-6066	256	41	presentation	presentation	NOUN
ejpam-6066	256	42	and	and	CCONJ
ejpam-6066	256	43	clarity	clarity	NOUN
ejpam-6066	256	44	of	of	ADP
ejpam-6066	256	45	this	this	DET
ejpam-6066	256	46	article	article	NOUN
ejpam-6066	256	47	.	.	PUNCT
ejpam-6066	257	1	the	the	DET
ejpam-6066	257	2	authors	author	NOUN
ejpam-6066	257	3	also	also	ADV
ejpam-6066	257	4	thank	thank	VERB
ejpam-6066	257	5	the	the	DET
ejpam-6066	257	6	center	center	NOUN
ejpam-6066	257	7	of	of	ADP
ejpam-6066	257	8	excellence	excellence	PROPN
ejpam-6066	257	9	for	for	ADP
ejpam-6066	257	10	ecoinformatics	ecoinformatic	NOUN
ejpam-6066	257	11	,	,	PUNCT
ejpam-6066	257	12	walailak	walailak	ADJ
ejpam-6066	257	13	university	university	NOUN
ejpam-6066	257	14	,	,	PUNCT
ejpam-6066	257	15	for	for	ADP
ejpam-6066	257	16	their	their	PRON
ejpam-6066	257	17	support	support	NOUN
ejpam-6066	257	18	.	.	PUNCT
ejpam-6066	258	1	references	reference	NOUN
ejpam-6066	258	2	[	[	X
ejpam-6066	258	3	1	1	NUM
ejpam-6066	258	4	]	]	X
ejpam-6066	258	5	h	h	PROPN
ejpam-6066	258	6	w	w	PROPN
ejpam-6066	258	7	lenstra	lenstra	PROPN
ejpam-6066	258	8	jr	jr	PROPN
ejpam-6066	258	9	.	.	PUNCT
ejpam-6066	258	10	solving	solve	VERB
ejpam-6066	258	11	the	the	DET
ejpam-6066	258	12	pell	pell	NOUN
ejpam-6066	258	13	equation	equation	NOUN
ejpam-6066	258	14	.	.	PUNCT
ejpam-6066	259	1	notices	notice	NOUN
ejpam-6066	259	2	of	of	ADP
ejpam-6066	259	3	the	the	DET
ejpam-6066	259	4	american	american	PROPN
ejpam-6066	259	5	mathematical	mathematical	PROPN
ejpam-6066	259	6	society	society	NOUN
ejpam-6066	259	7	,	,	PUNCT
ejpam-6066	259	8	49(2):182–192	49(2):182–192	PROPN
ejpam-6066	259	9	,	,	PUNCT
ejpam-6066	259	10	2002	2002	NUM
ejpam-6066	259	11	.	.	PUNCT
ejpam-6066	260	1	[	[	X
ejpam-6066	260	2	2	2	NUM
ejpam-6066	260	3	]	]	X
ejpam-6066	260	4	r	r	NOUN
ejpam-6066	260	5	guy	guy	NOUN
ejpam-6066	260	6	.	.	PUNCT
ejpam-6066	261	1	unsolved	unsolved	ADJ
ejpam-6066	261	2	problems	problem	NOUN
ejpam-6066	261	3	in	in	ADP
ejpam-6066	261	4	number	number	NOUN
ejpam-6066	261	5	theory	theory	NOUN
ejpam-6066	261	6	.	.	PUNCT
ejpam-6066	262	1	springer	springer	NOUN
ejpam-6066	262	2	,	,	PUNCT
ejpam-6066	262	3	new	new	PROPN
ejpam-6066	262	4	york	york	PROPN
ejpam-6066	262	5	,	,	PUNCT
ejpam-6066	262	6	ny	ny	PROPN
ejpam-6066	262	7	,	,	PUNCT
ejpam-6066	262	8	3rd	3rd	ADJ
ejpam-6066	262	9	edition	edition	NOUN
ejpam-6066	262	10	,	,	PUNCT
ejpam-6066	262	11	2004	2004	NUM
ejpam-6066	262	12	.	.	PUNCT
ejpam-6066	263	1	[	[	X
ejpam-6066	263	2	3	3	NUM
ejpam-6066	263	3	]	]	X
ejpam-6066	263	4	e	e	X
ejpam-6066	263	5	j	j	PROPN
ejpam-6066	263	6	ionascu	ionascu	PROPN
ejpam-6066	263	7	and	and	CCONJ
ejpam-6066	263	8	a	a	DET
ejpam-6066	263	9	wilson	wilson	PROPN
ejpam-6066	263	10	.	.	PUNCT
ejpam-6066	264	1	on	on	ADP
ejpam-6066	264	2	the	the	DET
ejpam-6066	264	3	erdos	erdo	NOUN
ejpam-6066	264	4	-	-	PUNCT
ejpam-6066	264	5	straus	straus	NOUN
ejpam-6066	264	6	conjecture	conjecture	NOUN
ejpam-6066	264	7	.	.	PUNCT
ejpam-6066	265	1	rev	rev	PROPN
ejpam-6066	265	2	.	.	PROPN
ejpam-6066	265	3	roumaine	roumaine	PROPN
ejpam-6066	265	4	math	math	NOUN
ejpam-6066	265	5	.	.	PUNCT
ejpam-6066	266	1	pures	pure	NOUN
ejpam-6066	266	2	appl	appl	PROPN
ejpam-6066	266	3	.	.	PROPN
ejpam-6066	266	4	,	,	PUNCT
ejpam-6066	266	5	56(1):21–30	56(1):21–30	NUM
ejpam-6066	266	6	,	,	PUNCT
ejpam-6066	266	7	2011	2011	NUM
ejpam-6066	266	8	.	.	PUNCT
ejpam-6066	267	1	[	[	X
ejpam-6066	267	2	4	4	X
ejpam-6066	267	3	]	]	SYM
ejpam-6066	267	4	n	n	NOUN
ejpam-6066	267	5	d	d	NOUN
ejpam-6066	267	6	elkies	elkie	NOUN
ejpam-6066	267	7	.	.	PUNCT
ejpam-6066	268	1	on	on	ADP
ejpam-6066	268	2	a4	a4	NOUN
ejpam-6066	268	3	+	+	CCONJ
ejpam-6066	268	4	b4	b4	NOUN
ejpam-6066	268	5	+	+	CCONJ
ejpam-6066	268	6	c4	c4	NOUN
ejpam-6066	268	7	=	=	SYM
ejpam-6066	268	8	d4	d4	PROPN
ejpam-6066	268	9	.	.	PUNCT
ejpam-6066	269	1	mathematics	mathematic	NOUN
ejpam-6066	269	2	of	of	ADP
ejpam-6066	269	3	computation	computation	NOUN
ejpam-6066	269	4	,	,	PUNCT
ejpam-6066	269	5	51(184):825–835	51(184):825–835	PROPN
ejpam-6066	269	6	,	,	PUNCT
ejpam-6066	269	7	1988	1988	NUM
ejpam-6066	269	8	.	.	PUNCT
ejpam-6066	270	1	[	[	X
ejpam-6066	270	2	5	5	NUM
ejpam-6066	270	3	]	]	PUNCT
ejpam-6066	270	4	a	a	DET
ejpam-6066	270	5	wiles	wile	NOUN
ejpam-6066	270	6	.	.	PUNCT
ejpam-6066	271	1	modular	modular	ADJ
ejpam-6066	271	2	elliptic	elliptic	ADJ
ejpam-6066	271	3	curves	curve	NOUN
ejpam-6066	271	4	and	and	CCONJ
ejpam-6066	271	5	fermat	fermat	PROPN
ejpam-6066	271	6	’s	’s	PART
ejpam-6066	271	7	last	last	ADJ
ejpam-6066	271	8	theorem	theorem	NOUN
ejpam-6066	271	9	.	.	PUNCT
ejpam-6066	271	10	annals	annal	NOUN
ejpam-6066	271	11	of	of	ADP
ejpam-6066	271	12	mathematics	mathematic	NOUN
ejpam-6066	271	13	,	,	PUNCT
ejpam-6066	271	14	141(3):443–551	141(3):443–551	NUM
ejpam-6066	271	15	,	,	PUNCT
ejpam-6066	271	16	1995	1995	NUM
ejpam-6066	271	17	.	.	PUNCT
ejpam-6066	272	1	[	[	X
ejpam-6066	272	2	6	6	NUM
ejpam-6066	272	3	]	]	X
ejpam-6066	272	4	p	p	X
ejpam-6066	272	5	mihăilescu	mihăilescu	PROPN
ejpam-6066	272	6	.	.	PUNCT
ejpam-6066	273	1	primary	primary	ADJ
ejpam-6066	273	2	cyclotomic	cyclotomic	ADJ
ejpam-6066	273	3	units	unit	NOUN
ejpam-6066	273	4	and	and	CCONJ
ejpam-6066	273	5	a	a	DET
ejpam-6066	273	6	proof	proof	NOUN
ejpam-6066	273	7	of	of	ADP
ejpam-6066	273	8	catalan	catalan	NOUN
ejpam-6066	273	9	’s	’s	PART
ejpam-6066	273	10	conjecture	conjecture	NOUN
ejpam-6066	273	11	.	.	PUNCT
ejpam-6066	274	1	journal	journal	PROPN
ejpam-6066	274	2	für	für	AUX
ejpam-6066	274	3	die	die	VERB
ejpam-6066	274	4	reine	reine	PROPN
ejpam-6066	274	5	und	und	PROPN
ejpam-6066	274	6	angewandte	angewandte	PROPN
ejpam-6066	274	7	mathematik	mathematik	PROPN
ejpam-6066	274	8	,	,	PUNCT
ejpam-6066	274	9	2004(572):167–195	2004(572):167–195	PROPN
ejpam-6066	274	10	,	,	PUNCT
ejpam-6066	274	11	2004	2004	NUM
ejpam-6066	274	12	.	.	PUNCT
ejpam-6066	275	1	[	[	X
ejpam-6066	275	2	7	7	NUM
ejpam-6066	275	3	]	]	X
ejpam-6066	275	4	d	d	X
ejpam-6066	275	5	acu	acu	PROPN
ejpam-6066	275	6	.	.	PUNCT
ejpam-6066	276	1	on	on	ADP
ejpam-6066	276	2	the	the	DET
ejpam-6066	276	3	diophantine	diophantine	NOUN
ejpam-6066	276	4	equation	equation	NOUN
ejpam-6066	276	5	2x+5y	2x+5y	NUM
ejpam-6066	276	6	=	=	SYM
ejpam-6066	276	7	z2	z2	PROPN
ejpam-6066	276	8	.	.	PUNCT
ejpam-6066	276	9	general	general	ADJ
ejpam-6066	276	10	mathematics	mathematic	NOUN
ejpam-6066	276	11	,	,	PUNCT
ejpam-6066	276	12	15(4):145	15(4):145	NUM
ejpam-6066	276	13	–	–	PUNCT
ejpam-6066	276	14	148	148	NUM
ejpam-6066	276	15	,	,	PUNCT
ejpam-6066	276	16	2007	2007	NUM
ejpam-6066	276	17	.	.	PUNCT
ejpam-6066	277	1	[	[	X
ejpam-6066	277	2	8	8	NUM
ejpam-6066	277	3	]	]	X
ejpam-6066	277	4	j	j	PROPN
ejpam-6066	277	5	f	f	PROPN
ejpam-6066	277	6	t	t	PROPN
ejpam-6066	277	7	rabago	rabago	PROPN
ejpam-6066	277	8	.	.	PUNCT
ejpam-6066	278	1	on	on	ADP
ejpam-6066	278	2	the	the	DET
ejpam-6066	278	3	diophantine	diophantine	NOUN
ejpam-6066	278	4	equation	equation	NOUN
ejpam-6066	278	5	4x	4x	NUM
ejpam-6066	278	6	−	−	PROPN
ejpam-6066	278	7	py	py	PROPN
ejpam-6066	278	8	=	=	NOUN
ejpam-6066	278	9	3z2	3z2	NUM
ejpam-6066	278	10	where	where	SCONJ
ejpam-6066	278	11	p	p	NOUN
ejpam-6066	278	12	is	be	AUX
ejpam-6066	278	13	a	a	DET
ejpam-6066	278	14	prime	prime	NOUN
ejpam-6066	278	15	.	.	PUNCT
ejpam-6066	279	1	thai	thai	PROPN
ejpam-6066	279	2	journal	journal	PROPN
ejpam-6066	279	3	of	of	ADP
ejpam-6066	279	4	mathematics	mathematic	NOUN
ejpam-6066	279	5	,	,	PUNCT
ejpam-6066	279	6	16(3):643–650	16(3):643–650	NUM
ejpam-6066	279	7	,	,	PUNCT
ejpam-6066	279	8	2018	2018	NUM
ejpam-6066	279	9	.	.	PUNCT
ejpam-6066	280	1	[	[	X
ejpam-6066	280	2	9	9	NUM
ejpam-6066	280	3	]	]	X
ejpam-6066	280	4	k	k	X
ejpam-6066	280	5	laipaporn	laipaporn	VERB
ejpam-6066	280	6	,	,	PUNCT
ejpam-6066	280	7	s	s	PART
ejpam-6066	280	8	wananiyakul	wananiyakul	NOUN
ejpam-6066	280	9	,	,	PUNCT
ejpam-6066	280	10	and	and	CCONJ
ejpam-6066	280	11	p	p	NOUN
ejpam-6066	280	12	khachorncharoenkul	khachorncharoenkul	NOUN
ejpam-6066	280	13	.	.	PUNCT
ejpam-6066	281	1	on	on	ADP
ejpam-6066	281	2	the	the	DET
ejpam-6066	281	3	diophantine	diophantine	NOUN
ejpam-6066	281	4	equation	equation	NOUN
ejpam-6066	281	5	3x	3x	PRON
ejpam-6066	281	6	+	+	CCONJ
ejpam-6066	281	7	p(5y	p(5y	NOUN
ejpam-6066	281	8	)	)	PUNCT
ejpam-6066	281	9	=	=	SYM
ejpam-6066	281	10	z2	z2	PROPN
ejpam-6066	281	11	.	.	PROPN
ejpam-6066	281	12	walailak	walailak	ADJ
ejpam-6066	281	13	journal	journal	NOUN
ejpam-6066	281	14	of	of	ADP
ejpam-6066	281	15	science	science	NOUN
ejpam-6066	281	16	and	and	CCONJ
ejpam-6066	281	17	technology	technology	NOUN
ejpam-6066	281	18	,	,	PUNCT
ejpam-6066	281	19	16(9):647–653	16(9):647–653	NUM
ejpam-6066	281	20	,	,	PUNCT
ejpam-6066	281	21	2019	2019	NUM
ejpam-6066	281	22	.	.	PUNCT
ejpam-6066	282	1	[	[	X
ejpam-6066	282	2	10	10	NUM
ejpam-6066	282	3	]	]	X
ejpam-6066	282	4	a	a	DET
ejpam-6066	282	5	elshahed	elshahed	ADJ
ejpam-6066	282	6	and	and	CCONJ
ejpam-6066	282	7	h	h	NOUN
ejpam-6066	282	8	kamarulhaili	kamarulhaili	NOUN
ejpam-6066	282	9	.	.	PUNCT
ejpam-6066	283	1	on	on	ADP
ejpam-6066	283	2	the	the	DET
ejpam-6066	283	3	diophantine	diophantine	NOUN
ejpam-6066	283	4	equation	equation	NOUN
ejpam-6066	283	5	(	(	PUNCT
ejpam-6066	283	6	4n)x	4n)x	NUM
ejpam-6066	283	7	−	−	NOUN
ejpam-6066	284	1	py	py	PROPN
ejpam-6066	284	2	=	=	SYM
ejpam-6066	284	3	z2	z2	PROPN
ejpam-6066	284	4	.	.	PUNCT
ejpam-6066	285	1	wseas	wseas	PROPN
ejpam-6066	285	2	transactions	transaction	NOUN
ejpam-6066	285	3	on	on	ADP
ejpam-6066	285	4	mathematics	mathematic	NOUN
ejpam-6066	285	5	,	,	PUNCT
ejpam-6066	285	6	19:349–352	19:349–352	NUM
ejpam-6066	285	7	,	,	PUNCT
ejpam-6066	285	8	2020	2020	NUM
ejpam-6066	285	9	.	.	PUNCT
ejpam-6066	286	1	[	[	X
ejpam-6066	286	2	11	11	NUM
ejpam-6066	286	3	]	]	X
ejpam-6066	286	4	s	s	VERB
ejpam-6066	286	5	thongnak	thongnak	NOUN
ejpam-6066	286	6	,	,	PUNCT
ejpam-6066	286	7	w	w	PROPN
ejpam-6066	286	8	chuayjan	chuayjan	PROPN
ejpam-6066	286	9	,	,	PUNCT
ejpam-6066	286	10	and	and	CCONJ
ejpam-6066	286	11	t	t	PROPN
ejpam-6066	286	12	kaewong	kaewong	PROPN
ejpam-6066	286	13	.	.	PUNCT
ejpam-6066	287	1	the	the	DET
ejpam-6066	287	2	solution	solution	NOUN
ejpam-6066	287	3	of	of	ADP
ejpam-6066	287	4	the	the	DET
ejpam-6066	287	5	exponential	exponential	ADJ
ejpam-6066	287	6	diophantine	diophantine	NOUN
ejpam-6066	287	7	equation	equation	NOUN
ejpam-6066	287	8	7x	7x	NOUN
ejpam-6066	287	9	−	−	PROPN
ejpam-6066	287	10	5y	5y	NOUN
ejpam-6066	287	11	=	=	SYM
ejpam-6066	287	12	z2	z2	PROPN
ejpam-6066	287	13	.	.	PUNCT
ejpam-6066	287	14	math	math	PROPN
ejpam-6066	287	15	.	.	PUNCT
ejpam-6066	288	1	j.	j.	PROPN
ejpam-6066	288	2	of	of	ADP
ejpam-6066	288	3	the	the	DET
ejpam-6066	288	4	mathematical	mathematical	ADJ
ejpam-6066	288	5	association	association	NOUN
ejpam-6066	288	6	of	of	ADP
ejpam-6066	288	7	thailand	thailand	PROPN
ejpam-6066	288	8	,	,	PUNCT
ejpam-6066	288	9	66(703):62–67	66(703):62–67	NOUN
ejpam-6066	288	10	,	,	PUNCT
ejpam-6066	288	11	2021	2021	NUM
ejpam-6066	288	12	.	.	PUNCT
ejpam-6066	289	1	[	[	X
ejpam-6066	289	2	12	12	NUM
ejpam-6066	289	3	]	]	X
ejpam-6066	289	4	w	w	NOUN
ejpam-6066	289	5	tangjai	tangjai	PROPN
ejpam-6066	289	6	,	,	PUNCT
ejpam-6066	289	7	s	s	VERB
ejpam-6066	289	8	chaeoueng	chaeoueng	NOUN
ejpam-6066	289	9	,	,	PUNCT
ejpam-6066	289	10	and	and	CCONJ
ejpam-6066	289	11	n	n	PRON
ejpam-6066	289	12	phumchaichot	phumchaichot	ADJ
ejpam-6066	289	13	.	.	PUNCT
ejpam-6066	290	1	on	on	ADP
ejpam-6066	290	2	the	the	DET
ejpam-6066	290	3	diophantine	diophantine	NOUN
ejpam-6066	290	4	equation	equation	NOUN
ejpam-6066	290	5	7x	7x	NOUN
ejpam-6066	290	6	+	+	CCONJ
ejpam-6066	290	7	5(py	5(py	NOUN
ejpam-6066	290	8	)	)	PUNCT
ejpam-6066	290	9	=	=	SYM
ejpam-6066	290	10	z2	z2	NOUN
ejpam-6066	290	11	where	where	SCONJ
ejpam-6066	290	12	p	p	PROPN
ejpam-6066	290	13	≡	≡	PROPN
ejpam-6066	290	14	1	1	NUM
ejpam-6066	290	15	,	,	PUNCT
ejpam-6066	290	16	2	2	NUM
ejpam-6066	290	17	,	,	PUNCT
ejpam-6066	290	18	4	4	NUM
ejpam-6066	290	19	(	(	PUNCT
ejpam-6066	290	20	mod	mod	PROPN
ejpam-6066	290	21	7	7	NUM
ejpam-6066	290	22	)	)	PUNCT
ejpam-6066	290	23	.	.	PUNCT
ejpam-6066	291	1	international	international	ADJ
ejpam-6066	291	2	journal	journal	PROPN
ejpam-6066	291	3	of	of	ADP
ejpam-6066	291	4	mathematics	mathematic	NOUN
ejpam-6066	291	5	and	and	CCONJ
ejpam-6066	291	6	computer	computer	NOUN
ejpam-6066	291	7	science	science	NOUN
ejpam-6066	291	8	,	,	PUNCT
ejpam-6066	291	9	17(4):1483–1489	17(4):1483–1489	NUM
ejpam-6066	291	10	,	,	PUNCT
ejpam-6066	291	11	2022	2022	NUM
ejpam-6066	291	12	.	.	PUNCT
ejpam-6066	292	1	[	[	X
ejpam-6066	292	2	13	13	NUM
ejpam-6066	292	3	]	]	SYM
ejpam-6066	292	4	w	w	NOUN
ejpam-6066	292	5	orosram	orosram	NOUN
ejpam-6066	292	6	and	and	CCONJ
ejpam-6066	292	7	a	a	DET
ejpam-6066	292	8	unchai	unchai	PROPN
ejpam-6066	292	9	.	.	PUNCT
ejpam-6066	293	1	on	on	ADP
ejpam-6066	293	2	the	the	DET
ejpam-6066	293	3	diophantine	diophantine	NOUN
ejpam-6066	293	4	equation	equation	NOUN
ejpam-6066	293	5	22nx−	22nx−	NUM
ejpam-6066	293	6	py	py	PROPN
ejpam-6066	293	7	=	=	PROPN
ejpam-6066	293	8	z2	z2	PROPN
ejpam-6066	293	9	where	where	SCONJ
ejpam-6066	293	10	p	p	NOUN
ejpam-6066	293	11	is	be	AUX
ejpam-6066	293	12	a	a	DET
ejpam-6066	293	13	prime	prime	NOUN
ejpam-6066	293	14	.	.	PUNCT
ejpam-6066	294	1	international	international	ADJ
ejpam-6066	294	2	journal	journal	PROPN
ejpam-6066	294	3	of	of	ADP
ejpam-6066	294	4	mathematics	mathematic	NOUN
ejpam-6066	294	5	and	and	CCONJ
ejpam-6066	294	6	computer	computer	NOUN
ejpam-6066	294	7	science	science	NOUN
ejpam-6066	294	8	,	,	PUNCT
ejpam-6066	294	9	17(1):447–451	17(1):447–451	PROPN
ejpam-6066	294	10	,	,	PUNCT
ejpam-6066	294	11	2022	2022	NUM
ejpam-6066	294	12	.	.	PUNCT
ejpam-6066	295	1	[	[	X
ejpam-6066	295	2	14	14	NUM
ejpam-6066	295	3	]	]	X
ejpam-6066	295	4	m	m	NOUN
ejpam-6066	295	5	buosi	buosi	NOUN
ejpam-6066	295	6	,	,	PUNCT
ejpam-6066	295	7	a	a	DET
ejpam-6066	295	8	lemos	lemos	PROPN
ejpam-6066	295	9	,	,	PUNCT
ejpam-6066	295	10	a	a	DET
ejpam-6066	295	11	l	l	NOUN
ejpam-6066	295	12	p	p	X
ejpam-6066	295	13	porto	porto	PROPN
ejpam-6066	295	14	,	,	PUNCT
ejpam-6066	295	15	and	and	CCONJ
ejpam-6066	295	16	d	d	X
ejpam-6066	295	17	f	f	PROPN
ejpam-6066	295	18	g	g	PROPN
ejpam-6066	295	19	santiago	santiago	PROPN
ejpam-6066	295	20	.	.	PUNCT
ejpam-6066	296	1	on	on	ADP
ejpam-6066	296	2	the	the	DET
ejpam-6066	296	3	exponential	exponential	ADJ
ejpam-6066	296	4	diophantine	diophantine	NOUN
ejpam-6066	296	5	equation	equation	NOUN
ejpam-6066	296	6	px	px	ADP
ejpam-6066	296	7	−	−	PROPN
ejpam-6066	296	8	2y	2y	PROPN
ejpam-6066	296	9	=	=	SYM
ejpam-6066	296	10	z2	z2	NOUN
ejpam-6066	296	11	with	with	ADP
ejpam-6066	296	12	p	p	PROPN
ejpam-6066	296	13	=	=	PROPN
ejpam-6066	296	14	k2	k2	PROPN
ejpam-6066	296	15	+	+	CCONJ
ejpam-6066	296	16	4	4	NUM
ejpam-6066	296	17	a	a	DET
ejpam-6066	296	18	prime	prime	ADJ
ejpam-6066	296	19	number	number	NOUN
ejpam-6066	296	20	.	.	PUNCT
ejpam-6066	297	1	palestine	palestine	PROPN
ejpam-6066	297	2	journal	journal	PROPN
ejpam-6066	297	3	of	of	ADP
ejpam-6066	297	4	mathematics	mathematic	NOUN
ejpam-6066	297	5	,	,	PUNCT
ejpam-6066	297	6	11(4):130–135	11(4):130–135	NUM
ejpam-6066	297	7	,	,	PUNCT
ejpam-6066	297	8	2022	2022	NUM
ejpam-6066	297	9	.	.	PUNCT
ejpam-6066	298	1	k.	k.	PROPN
ejpam-6066	298	2	laipaporn	laipaporn	PROPN
ejpam-6066	298	3	et	et	PROPN
ejpam-6066	298	4	al	al	PROPN
ejpam-6066	298	5	.	.	PUNCT
ejpam-6066	298	6	/	/	SYM
ejpam-6066	298	7	eur	eur	PROPN
ejpam-6066	298	8	.	.	PUNCT
ejpam-6066	299	1	j.	j.	PROPN
ejpam-6066	299	2	pure	pure	PROPN
ejpam-6066	299	3	appl	appl	PROPN
ejpam-6066	299	4	.	.	PROPN
ejpam-6066	299	5	math	math	PROPN
ejpam-6066	299	6	,	,	PUNCT
ejpam-6066	299	7	18	18	NUM
ejpam-6066	299	8	(	(	PUNCT
ejpam-6066	299	9	3	3	NUM
ejpam-6066	299	10	)	)	PUNCT
ejpam-6066	299	11	(	(	PUNCT
ejpam-6066	299	12	2025	2025	NUM
ejpam-6066	299	13	)	)	PUNCT
ejpam-6066	299	14	,	,	PUNCT
ejpam-6066	299	15	6066	6066	NUM
ejpam-6066	299	16	13	13	NUM
ejpam-6066	299	17	of	of	ADP
ejpam-6066	299	18	13	13	NUM
ejpam-6066	300	1	[	[	SYM
ejpam-6066	300	2	15	15	NUM
ejpam-6066	300	3	]	]	X
ejpam-6066	300	4	s	s	VERB
ejpam-6066	300	5	thongnak	thongnak	NOUN
ejpam-6066	300	6	,	,	PUNCT
ejpam-6066	300	7	t	t	PROPN
ejpam-6066	300	8	kaewong	kaewong	PROPN
ejpam-6066	300	9	,	,	PUNCT
ejpam-6066	300	10	and	and	CCONJ
ejpam-6066	300	11	w	w	ADP
ejpam-6066	300	12	chuayjan	chuayjan	PROPN
ejpam-6066	300	13	.	.	PUNCT
ejpam-6066	301	1	on	on	ADP
ejpam-6066	301	2	the	the	DET
ejpam-6066	301	3	diophantine	diophantine	NOUN
ejpam-6066	301	4	equation	equation	NOUN
ejpam-6066	301	5	11x−17y	11x−17y	PROPN
ejpam-6066	301	6	=	=	SYM
ejpam-6066	301	7	z2	z2	PROPN
ejpam-6066	301	8	.	.	PUNCT
ejpam-6066	302	1	international	international	ADJ
ejpam-6066	302	2	journal	journal	PROPN
ejpam-6066	302	3	of	of	ADP
ejpam-6066	302	4	mathematics	mathematic	NOUN
ejpam-6066	302	5	and	and	CCONJ
ejpam-6066	302	6	computer	computer	NOUN
ejpam-6066	302	7	science	science	NOUN
ejpam-6066	302	8	,	,	PUNCT
ejpam-6066	302	9	19(1):181–184	19(1):181–184	PROPN
ejpam-6066	302	10	,	,	PUNCT
ejpam-6066	302	11	2024	2024	NUM
ejpam-6066	302	12	.	.	PUNCT
ejpam-6066	303	1	[	[	X
ejpam-6066	303	2	16	16	NUM
ejpam-6066	303	3	]	]	X
ejpam-6066	303	4	k	k	X
ejpam-6066	303	5	laipaporn	laipaporn	PROPN
ejpam-6066	303	6	,	,	PUNCT
ejpam-6066	303	7	s	s	PART
ejpam-6066	303	8	wananiyakul	wananiyakul	NOUN
ejpam-6066	303	9	,	,	PUNCT
ejpam-6066	303	10	and	and	CCONJ
ejpam-6066	303	11	p	p	NOUN
ejpam-6066	303	12	khachorncharoenkul	khachorncharoenkul	NOUN
ejpam-6066	303	13	.	.	PUNCT
ejpam-6066	304	1	the	the	DET
ejpam-6066	304	2	diophantine	diophantine	NOUN
ejpam-6066	304	3	equation	equation	NOUN
ejpam-6066	304	4	ax±ay	ax±ay	PROPN
ejpam-6066	304	5	=	=	SYM
ejpam-6066	304	6	zn	zn	PROPN
ejpam-6066	304	7	when	when	SCONJ
ejpam-6066	304	8	a	a	PRON
ejpam-6066	304	9	is	be	AUX
ejpam-6066	304	10	any	any	DET
ejpam-6066	304	11	nonnegative	nonnegative	ADJ
ejpam-6066	304	12	integer	integer	NOUN
ejpam-6066	304	13	.	.	PUNCT
ejpam-6066	305	1	journal	journal	NOUN
ejpam-6066	305	2	of	of	ADP
ejpam-6066	305	3	mathematics	mathematic	NOUN
ejpam-6066	305	4	and	and	CCONJ
ejpam-6066	305	5	computer	computer	NOUN
ejpam-6066	305	6	science	science	NOUN
ejpam-6066	305	7	,	,	PUNCT
ejpam-6066	305	8	32(3):213–221	32(3):213–221	NUM
ejpam-6066	305	9	,	,	PUNCT
ejpam-6066	305	10	2024	2024	NUM
ejpam-6066	305	11	.	.	PUNCT
ejpam-6066	306	1	[	[	X
ejpam-6066	306	2	17	17	NUM
ejpam-6066	306	3	]	]	X
ejpam-6066	306	4	y	y	PROPN
ejpam-6066	306	5	li	li	PROPN
ejpam-6066	306	6	,	,	PUNCT
ejpam-6066	306	7	t	t	PROPN
ejpam-6066	306	8	lloyd	lloyd	PROPN
ejpam-6066	306	9	,	,	PUNCT
ejpam-6066	306	10	and	and	CCONJ
ejpam-6066	306	11	a	a	DET
ejpam-6066	306	12	clinton	clinton	PROPN
ejpam-6066	306	13	.	.	PUNCT
ejpam-6066	307	1	on	on	ADP
ejpam-6066	307	2	diophantine	diophantine	NOUN
ejpam-6066	307	3	equations	equation	NOUN
ejpam-6066	307	4	2x	2x	NUM
ejpam-6066	307	5	±	±	NOUN
ejpam-6066	307	6	(	(	PUNCT
ejpam-6066	307	7	2kp)y	2kp)y	PROPN
ejpam-6066	307	8	=	=	SYM
ejpam-6066	307	9	z2	z2	PROPN
ejpam-6066	307	10	and	and	CCONJ
ejpam-6066	307	11	−2x	−2x	PROPN
ejpam-6066	307	12	+	+	CCONJ
ejpam-6066	307	13	(	(	PUNCT
ejpam-6066	307	14	2k3)y	2k3)y	NUM
ejpam-6066	307	15	=	=	SYM
ejpam-6066	307	16	z2	z2	PROPN
ejpam-6066	307	17	.	.	PUNCT
ejpam-6066	308	1	mathematics	mathematic	NOUN
ejpam-6066	308	2	,	,	PUNCT
ejpam-6066	308	3	12(24):4027	12(24):4027	NUM
ejpam-6066	308	4	,	,	PUNCT
ejpam-6066	308	5	2024	2024	NUM
ejpam-6066	308	6	.	.	PUNCT
ejpam-6066	309	1	[	[	X
ejpam-6066	309	2	18	18	NUM
ejpam-6066	309	3	]	]	X
ejpam-6066	309	4	j	j	PROPN
ejpam-6066	309	5	zhang	zhang	PROPN
ejpam-6066	309	6	and	and	CCONJ
ejpam-6066	309	7	y	y	PROPN
ejpam-6066	309	8	li	li	PROPN
ejpam-6066	309	9	.	.	PROPN
ejpam-6066	310	1	on	on	ADP
ejpam-6066	310	2	the	the	DET
ejpam-6066	310	3	equation	equation	NOUN
ejpam-6066	310	4	(	(	PUNCT
ejpam-6066	310	5	−1)αpx	−1)αpx	NOUN
ejpam-6066	310	6	+	+	CCONJ
ejpam-6066	310	7	(	(	PUNCT
ejpam-6066	310	8	−1)β(2k(2p	−1)β(2k(2p	NOUN
ejpam-6066	310	9	−	−	PROPN
ejpam-6066	310	10	1))y	1))y	NUM
ejpam-6066	310	11	=	=	SYM
ejpam-6066	310	12	z2	z2	PROPN
ejpam-6066	310	13	for	for	ADP
ejpam-6066	310	14	prime	prime	ADJ
ejpam-6066	310	15	pairs	pair	NOUN
ejpam-6066	310	16	(	(	PUNCT
ejpam-6066	310	17	p	p	X
ejpam-6066	310	18	,	,	PUNCT
ejpam-6066	310	19	2p−	2p−	NOUN
ejpam-6066	310	20	1	1	NUM
ejpam-6066	310	21	)	)	PUNCT
ejpam-6066	310	22	.	.	PUNCT
ejpam-6066	311	1	integers	integer	NOUN
ejpam-6066	311	2	,	,	PUNCT
ejpam-6066	311	3	24	24	NUM
ejpam-6066	311	4	:	:	PUNCT
ejpam-6066	311	5	a64	a64	NOUN
ejpam-6066	311	6	,	,	PUNCT
ejpam-6066	311	7	2024	2024	NUM
ejpam-6066	311	8	.	.	PUNCT
ejpam-6066	312	1	[	[	X
ejpam-6066	312	2	19	19	NUM
ejpam-6066	312	3	]	]	X
ejpam-6066	312	4	k	k	X
ejpam-6066	312	5	laipaporn	laipaporn	PROPN
ejpam-6066	312	6	,	,	PUNCT
ejpam-6066	312	7	s	s	PART
ejpam-6066	312	8	wananiyakul	wananiyakul	NOUN
ejpam-6066	312	9	,	,	PUNCT
ejpam-6066	312	10	and	and	CCONJ
ejpam-6066	312	11	p	p	PRON
ejpam-6066	312	12	khachorncharoenkul	khachorncharoenkul	PROPN
ejpam-6066	312	13	.	.	PUNCT
ejpam-6066	313	1	explicit	explicit	ADJ
ejpam-6066	313	2	solutions	solution	NOUN
ejpam-6066	313	3	and	and	CCONJ
ejpam-6066	313	4	non	non	NOUN
ejpam-6066	313	5	-	-	NOUN
ejpam-6066	313	6	solutions	solution	NOUN
ejpam-6066	313	7	for	for	ADP
ejpam-6066	313	8	the	the	DET
ejpam-6066	313	9	diophantine	diophantine	NOUN
ejpam-6066	313	10	equation	equation	NOUN
ejpam-6066	313	11	px	px	X
ejpam-6066	313	12	+	+	CCONJ
ejpam-6066	313	13	q2y	q2y	NOUN
ejpam-6066	313	14	=	=	SYM
ejpam-6066	313	15	z2n	z2n	NOUN
ejpam-6066	313	16	involving	involve	VERB
ejpam-6066	313	17	primes	prime	NOUN
ejpam-6066	313	18	p	p	X
ejpam-6066	313	19	̸≡	̸≡	PROPN
ejpam-6066	313	20	q	q	X
ejpam-6066	313	21	(	(	PUNCT
ejpam-6066	313	22	mod	mod	PROPN
ejpam-6066	313	23	4	4	NUM
ejpam-6066	313	24	)	)	PUNCT
ejpam-6066	313	25	.	.	PUNCT
ejpam-6066	314	1	aims	aim	VERB
ejpam-6066	314	2	mathematics	mathematic	NOUN
ejpam-6066	314	3	,	,	PUNCT
ejpam-6066	314	4	10(7):15720–15736	10(7):15720–15736	NUM
ejpam-6066	314	5	,	,	PUNCT
ejpam-6066	314	6	2025	2025	NUM
ejpam-6066	314	7	.	.	PUNCT