id	sid	tid	token	lemma	pos
ejpam-6004	1	1	european	european	PROPN
ejpam-6004	1	2	journal	journal	PROPN
ejpam-6004	1	3	of	of	ADP
ejpam-6004	1	4	pure	pure	ADJ
ejpam-6004	1	5	and	and	CCONJ
ejpam-6004	1	6	applied	applied	ADJ
ejpam-6004	1	7	mathematics	mathematic	NOUN
ejpam-6004	1	8	2025	2025	NUM
ejpam-6004	1	9	,	,	PUNCT
ejpam-6004	1	10	vol	vol	NOUN
ejpam-6004	1	11	.	.	PROPN
ejpam-6004	1	12	18	18	NUM
ejpam-6004	1	13	,	,	PUNCT
ejpam-6004	1	14	issue	issue	NOUN
ejpam-6004	1	15	2	2	NUM
ejpam-6004	1	16	,	,	PUNCT
ejpam-6004	1	17	article	article	NOUN
ejpam-6004	1	18	number	number	NOUN
ejpam-6004	1	19	6004	6004	NUM
ejpam-6004	1	20	issn	issn	PROPN
ejpam-6004	1	21	1307	1307	NUM
ejpam-6004	1	22	-	-	SYM
ejpam-6004	1	23	5543	5543	NUM
ejpam-6004	1	24	–	–	PUNCT
ejpam-6004	1	25	ejpam.com	ejpam.com	X
ejpam-6004	1	26	published	publish	VERB
ejpam-6004	1	27	by	by	ADP
ejpam-6004	1	28	new	new	PROPN
ejpam-6004	1	29	york	york	PROPN
ejpam-6004	1	30	business	business	PROPN
ejpam-6004	1	31	global	global	PROPN
ejpam-6004	1	32	on	on	ADP
ejpam-6004	1	33	the	the	DET
ejpam-6004	1	34	diophantine	diophantine	NOUN
ejpam-6004	1	35	equations	equation	NOUN
ejpam-6004	1	36	of	of	ADP
ejpam-6004	1	37	the	the	DET
ejpam-6004	1	38	form	form	NOUN
ejpam-6004	2	1	x2	x2	CCONJ
ejpam-6004	2	2	−	−	PROPN
ejpam-6004	2	3	kxy	kxy	NOUN
ejpam-6004	2	4	+	+	CCONJ
ejpam-6004	2	5	ky2	ky2	NOUN
ejpam-6004	2	6	+	+	CCONJ
ejpam-6004	2	7	2ny	2ny	ADJ
ejpam-6004	2	8	=	=	SYM
ejpam-6004	2	9	0	0	NUM
ejpam-6004	2	10	supawadee	supawadee	PROPN
ejpam-6004	2	11	prugsapitak1,∗	prugsapitak1,∗	NOUN
ejpam-6004	2	12	,	,	PUNCT
ejpam-6004	2	13	nattaporn	nattaporn	ADJ
ejpam-6004	2	14	thongngam2	thongngam2	NOUN
ejpam-6004	2	15	1	1	NUM
ejpam-6004	2	16	division	division	NOUN
ejpam-6004	2	17	of	of	ADP
ejpam-6004	2	18	computational	computational	ADJ
ejpam-6004	2	19	science	science	NOUN
ejpam-6004	2	20	,	,	PUNCT
ejpam-6004	2	21	faculty	faculty	NOUN
ejpam-6004	2	22	of	of	ADP
ejpam-6004	2	23	science	science	NOUN
ejpam-6004	2	24	,	,	PUNCT
ejpam-6004	2	25	prince	prince	NOUN
ejpam-6004	2	26	of	of	ADP
ejpam-6004	2	27	songkla	songkla	PROPN
ejpam-6004	2	28	university	university	NOUN
ejpam-6004	2	29	abstract	abstract	NOUN
ejpam-6004	2	30	.	.	PUNCT
ejpam-6004	3	1	in	in	ADP
ejpam-6004	3	2	this	this	DET
ejpam-6004	3	3	article	article	NOUN
ejpam-6004	3	4	,	,	PUNCT
ejpam-6004	3	5	we	we	PRON
ejpam-6004	3	6	determine	determine	VERB
ejpam-6004	3	7	all	all	DET
ejpam-6004	3	8	values	value	NOUN
ejpam-6004	3	9	of	of	ADP
ejpam-6004	3	10	k	k	PROPN
ejpam-6004	3	11	for	for	ADP
ejpam-6004	3	12	which	which	PRON
ejpam-6004	3	13	the	the	DET
ejpam-6004	3	14	equation	equation	NOUN
ejpam-6004	3	15	x2−kxy+ky2	x2−kxy+ky2	PUNCT
ejpam-6004	4	1	+	+	ADJ
ejpam-6004	4	2	2ny	2ny	ADJ
ejpam-6004	4	3	=	=	X
ejpam-6004	4	4	0	0	NUM
ejpam-6004	4	5	where	where	SCONJ
ejpam-6004	4	6	n	n	ADV
ejpam-6004	4	7	=	=	SYM
ejpam-6004	4	8	3	3	NUM
ejpam-6004	4	9	,	,	PUNCT
ejpam-6004	4	10	4	4	NUM
ejpam-6004	4	11	,	,	PUNCT
ejpam-6004	4	12	5	5	NUM
ejpam-6004	4	13	,	,	PUNCT
ejpam-6004	4	14	6	6	NUM
ejpam-6004	4	15	,	,	PUNCT
ejpam-6004	4	16	7	7	NUM
ejpam-6004	4	17	has	have	VERB
ejpam-6004	4	18	infinitely	infinitely	ADV
ejpam-6004	4	19	many	many	ADJ
ejpam-6004	4	20	positive	positive	ADJ
ejpam-6004	4	21	solutions	solution	NOUN
ejpam-6004	4	22	.	.	PUNCT
ejpam-6004	5	1	2020	2020	NUM
ejpam-6004	5	2	mathematics	mathematic	NOUN
ejpam-6004	5	3	subject	subject	NOUN
ejpam-6004	5	4	classifications	classification	NOUN
ejpam-6004	5	5	:	:	PUNCT
ejpam-6004	5	6	11d09	11d09	NUM
ejpam-6004	5	7	,	,	PUNCT
ejpam-6004	5	8	11d72	11d72	NUM
ejpam-6004	5	9	key	key	ADJ
ejpam-6004	5	10	words	word	NOUN
ejpam-6004	5	11	and	and	CCONJ
ejpam-6004	5	12	phrases	phrase	NOUN
ejpam-6004	5	13	:	:	PUNCT
ejpam-6004	5	14	diophantine	diophantine	VERB
ejpam-6004	5	15	equation	equation	NOUN
ejpam-6004	5	16	,	,	PUNCT
ejpam-6004	5	17	pell	pell	NOUN
ejpam-6004	5	18	equation	equation	NOUN
ejpam-6004	5	19	,	,	PUNCT
ejpam-6004	5	20	quadratic	quadratic	ADJ
ejpam-6004	5	21	diophantine	diophantine	NOUN
ejpam-6004	5	22	equation	equation	NOUN
ejpam-6004	5	23	1	1	NUM
ejpam-6004	5	24	.	.	PUNCT
ejpam-6004	5	25	introduction	introduction	NOUN
ejpam-6004	5	26	the	the	DET
ejpam-6004	5	27	diophantine	diophantine	NOUN
ejpam-6004	5	28	equation	equation	NOUN
ejpam-6004	5	29	x2	x2	PROPN
ejpam-6004	6	1	+	+	CCONJ
ejpam-6004	6	2	axy	axy	PROPN
ejpam-6004	6	3	+	+	CCONJ
ejpam-6004	6	4	by2	by2	PROPN
ejpam-6004	6	5	+	+	CCONJ
ejpam-6004	6	6	cx+	cx+	NOUN
ejpam-6004	6	7	dy	dy	NOUN
ejpam-6004	6	8	+	+	NOUN
ejpam-6004	6	9	e	e	NOUN
ejpam-6004	6	10	=	=	SYM
ejpam-6004	6	11	0	0	PROPN
ejpam-6004	6	12	has	have	AUX
ejpam-6004	6	13	been	be	AUX
ejpam-6004	6	14	analyzed	analyze	VERB
ejpam-6004	6	15	for	for	ADP
ejpam-6004	6	16	various	various	ADJ
ejpam-6004	6	17	integer	integer	NOUN
ejpam-6004	6	18	values	value	NOUN
ejpam-6004	6	19	of	of	ADP
ejpam-6004	6	20	a	a	DET
ejpam-6004	6	21	,	,	PUNCT
ejpam-6004	6	22	b	b	NOUN
ejpam-6004	6	23	,	,	PUNCT
ejpam-6004	6	24	c	c	NOUN
ejpam-6004	6	25	,	,	PUNCT
ejpam-6004	6	26	d	d	PROPN
ejpam-6004	6	27	and	and	CCONJ
ejpam-6004	6	28	e.	e.	PROPN
ejpam-6004	6	29	specifically	specifically	ADV
ejpam-6004	6	30	,	,	PUNCT
ejpam-6004	6	31	keskin	keskin	PROPN
ejpam-6004	6	32	,	,	PUNCT
ejpam-6004	6	33	o.	o.	PROPN
ejpam-6004	6	34	karaatli	karaatli	PROPN
ejpam-6004	6	35	,	,	PUNCT
ejpam-6004	6	36	and	and	CCONJ
ejpam-6004	6	37	z.	z.	PROPN
ejpam-6004	6	38	siar	siar	PROPN
ejpam-6004	7	1	[	[	X
ejpam-6004	7	2	1	1	NUM
ejpam-6004	7	3	,	,	PUNCT
ejpam-6004	7	4	2	2	NUM
ejpam-6004	7	5	]	]	PUNCT
ejpam-6004	7	6	identified	identify	VERB
ejpam-6004	7	7	conditions	condition	NOUN
ejpam-6004	7	8	under	under	ADP
ejpam-6004	7	9	which	which	PRON
ejpam-6004	7	10	the	the	DET
ejpam-6004	7	11	equation	equation	NOUN
ejpam-6004	7	12	x2	x2	NOUN
ejpam-6004	7	13	−	−	PROPN
ejpam-6004	7	14	kxy	kxy	NOUN
ejpam-6004	7	15	+	+	CCONJ
ejpam-6004	7	16	y2	y2	PROPN
ejpam-6004	7	17	±	±	NOUN
ejpam-6004	7	18	2n	2n	NUM
ejpam-6004	7	19	=	=	SYM
ejpam-6004	7	20	0	0	NUM
ejpam-6004	7	21	possesses	possess	VERB
ejpam-6004	7	22	an	an	DET
ejpam-6004	7	23	infinite	infinite	ADJ
ejpam-6004	7	24	set	set	NOUN
ejpam-6004	7	25	of	of	ADP
ejpam-6004	7	26	positive	positive	ADJ
ejpam-6004	7	27	integer	integer	NOUN
ejpam-6004	7	28	solutions	solution	NOUN
ejpam-6004	7	29	(	(	PUNCT
ejpam-6004	7	30	x	x	X
ejpam-6004	7	31	,	,	PUNCT
ejpam-6004	7	32	y	y	NOUN
ejpam-6004	7	33	)	)	PUNCT
ejpam-6004	7	34	for	for	ADP
ejpam-6004	7	35	0	0	NUM
ejpam-6004	7	36	≤	≤	NUM
ejpam-6004	7	37	n	n	DET
ejpam-6004	7	38	≤	≤	NOUN
ejpam-6004	7	39	10	10	NUM
ejpam-6004	7	40	and	and	CCONJ
ejpam-6004	7	41	provided	provide	VERB
ejpam-6004	7	42	all	all	DET
ejpam-6004	7	43	such	such	ADJ
ejpam-6004	7	44	solutions	solution	NOUN
ejpam-6004	7	45	for	for	ADP
ejpam-6004	7	46	this	this	DET
ejpam-6004	7	47	range	range	NOUN
ejpam-6004	7	48	of	of	ADP
ejpam-6004	7	49	n.	n.	NOUN
ejpam-6004	7	50	moreover	moreover	ADV
ejpam-6004	7	51	,	,	PUNCT
ejpam-6004	7	52	they	they	PRON
ejpam-6004	7	53	proposed	propose	VERB
ejpam-6004	7	54	hypotheses	hypothesis	NOUN
ejpam-6004	7	55	based	base	VERB
ejpam-6004	7	56	on	on	ADP
ejpam-6004	7	57	the	the	DET
ejpam-6004	7	58	parity	parity	NOUN
ejpam-6004	7	59	of	of	ADP
ejpam-6004	7	60	r	r	NOUN
ejpam-6004	7	61	regarding	regard	VERB
ejpam-6004	7	62	integer	integer	NOUN
ejpam-6004	7	63	solutions	solution	NOUN
ejpam-6004	7	64	of	of	ADP
ejpam-6004	7	65	the	the	DET
ejpam-6004	7	66	equation	equation	NOUN
ejpam-6004	7	67	x2	x2	NOUN
ejpam-6004	8	1	−	−	PROPN
ejpam-6004	8	2	kxy	kxy	NOUN
ejpam-6004	8	3	+	+	CCONJ
ejpam-6004	8	4	y2	y2	NOUN
ejpam-6004	8	5	=	=	SYM
ejpam-6004	8	6	2r	2r	NUM
ejpam-6004	8	7	.	.	PUNCT
ejpam-6004	9	1	r.	r.	PROPN
ejpam-6004	9	2	boumahdi	boumahdi	PROPN
ejpam-6004	9	3	,	,	PUNCT
ejpam-6004	9	4	o.	o.	PROPN
ejpam-6004	9	5	kihel	kihel	PROPN
ejpam-6004	9	6	,	,	PUNCT
ejpam-6004	9	7	and	and	CCONJ
ejpam-6004	9	8	s.	s.	PROPN
ejpam-6004	9	9	mavecha	mavecha	PROPN
ejpam-6004	10	1	[	[	X
ejpam-6004	10	2	3	3	X
ejpam-6004	10	3	]	]	PUNCT
ejpam-6004	10	4	gave	give	VERB
ejpam-6004	10	5	a	a	DET
ejpam-6004	10	6	proof	proof	NOUN
ejpam-6004	10	7	of	of	ADP
ejpam-6004	10	8	this	this	DET
ejpam-6004	10	9	conjecture	conjecture	NOUN
ejpam-6004	10	10	.	.	PUNCT
ejpam-6004	11	1	for	for	ADP
ejpam-6004	11	2	any	any	DET
ejpam-6004	11	3	integer	integer	NOUN
ejpam-6004	11	4	l	l	NOUN
ejpam-6004	11	5	,	,	PUNCT
ejpam-6004	11	6	let	let	VERB
ejpam-6004	11	7	t	t	PROPN
ejpam-6004	11	8	(	(	PUNCT
ejpam-6004	11	9	l	l	NOUN
ejpam-6004	11	10	)	)	PUNCT
ejpam-6004	11	11	be	be	VERB
ejpam-6004	11	12	the	the	DET
ejpam-6004	11	13	set	set	NOUN
ejpam-6004	11	14	of	of	ADP
ejpam-6004	11	15	positive	positive	ADJ
ejpam-6004	11	16	integers	integer	NOUN
ejpam-6004	11	17	k	k	X
ejpam-6004	11	18	for	for	ADP
ejpam-6004	11	19	which	which	PRON
ejpam-6004	11	20	the	the	DET
ejpam-6004	11	21	equation	equation	NOUN
ejpam-6004	11	22	x2	x2	NOUN
ejpam-6004	12	1	−	−	PROPN
ejpam-6004	12	2	kxy	kxy	NOUN
ejpam-6004	12	3	+	+	CCONJ
ejpam-6004	12	4	ky2	ky2	NOUN
ejpam-6004	12	5	+	+	CCONJ
ejpam-6004	12	6	ly	ly	X
ejpam-6004	12	7	=	=	SYM
ejpam-6004	12	8	0	0	PROPN
ejpam-6004	12	9	has	have	VERB
ejpam-6004	12	10	infinitely	infinitely	ADV
ejpam-6004	12	11	many	many	ADJ
ejpam-6004	12	12	positive	positive	ADJ
ejpam-6004	12	13	integer	integer	NOUN
ejpam-6004	12	14	solutions	solution	NOUN
ejpam-6004	12	15	and	and	CCONJ
ejpam-6004	12	16	let	let	VERB
ejpam-6004	12	17	t	t	PROPN
ejpam-6004	12	18	′(l	′(l	ADV
ejpam-6004	12	19	)	)	PUNCT
ejpam-6004	12	20	be	be	AUX
ejpam-6004	12	21	the	the	DET
ejpam-6004	12	22	set	set	NOUN
ejpam-6004	12	23	of	of	ADP
ejpam-6004	12	24	positive	positive	ADJ
ejpam-6004	12	25	integers	integer	NOUN
ejpam-6004	12	26	k	k	X
ejpam-6004	12	27	for	for	ADP
ejpam-6004	12	28	which	which	PRON
ejpam-6004	12	29	the	the	DET
ejpam-6004	12	30	equation	equation	NOUN
ejpam-6004	12	31	x2	x2	NOUN
ejpam-6004	13	1	−	−	PROPN
ejpam-6004	13	2	kxy	kxy	NOUN
ejpam-6004	13	3	+	+	CCONJ
ejpam-6004	13	4	ky2	ky2	NOUN
ejpam-6004	13	5	+	+	CCONJ
ejpam-6004	13	6	l	l	NOUN
ejpam-6004	13	7	=	=	SYM
ejpam-6004	13	8	0	0	NUM
ejpam-6004	13	9	,	,	PUNCT
ejpam-6004	13	10	where	where	SCONJ
ejpam-6004	13	11	(	(	PUNCT
ejpam-6004	13	12	x	x	NOUN
ejpam-6004	13	13	,	,	PUNCT
ejpam-6004	13	14	y	y	NOUN
ejpam-6004	13	15	)	)	PUNCT
ejpam-6004	13	16	=	=	SYM
ejpam-6004	13	17	1	1	NUM
ejpam-6004	13	18	has	have	VERB
ejpam-6004	13	19	infinitely	infinitely	ADV
ejpam-6004	13	20	many	many	ADJ
ejpam-6004	13	21	positive	positive	ADJ
ejpam-6004	13	22	integer	integer	NOUN
ejpam-6004	13	23	solutions	solution	NOUN
ejpam-6004	13	24	(	(	PUNCT
ejpam-6004	13	25	x	x	X
ejpam-6004	13	26	,	,	PUNCT
ejpam-6004	13	27	y	y	PROPN
ejpam-6004	13	28	)	)	PUNCT
ejpam-6004	13	29	.	.	PUNCT
ejpam-6004	14	1	∗corresponding	∗corresponde	VERB
ejpam-6004	14	2	author	author	NOUN
ejpam-6004	14	3	.	.	PUNCT
ejpam-6004	15	1	doi	doi	NOUN
ejpam-6004	15	2	:	:	PUNCT
ejpam-6004	15	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6004	https://doi.org/10.29020/nybg.ejpam.v18i2.6004	VERB
ejpam-6004	15	4	email	email	NOUN
ejpam-6004	15	5	addresses	address	NOUN
ejpam-6004	15	6	:	:	PUNCT
ejpam-6004	15	7	supawadee.p@psu.ac.th	supawadee.p@psu.ac.th	PROPN
ejpam-6004	15	8	(	(	PUNCT
ejpam-6004	15	9	s.	s.	PROPN
ejpam-6004	15	10	prugsapitak	prugsapitak	PROPN
ejpam-6004	15	11	)	)	PUNCT
ejpam-6004	15	12	,	,	PUNCT
ejpam-6004	15	13	6510230021@psu.ac.th	6510230021@psu.ac.th	NUM
ejpam-6004	15	14	(	(	PUNCT
ejpam-6004	15	15	n.	n.	PROPN
ejpam-6004	15	16	thongngam	thongngam	NOUN
ejpam-6004	15	17	)	)	PUNCT
ejpam-6004	15	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6004	16	1	1	1	NUM
ejpam-6004	16	2	copyright	copyright	NOUN
ejpam-6004	16	3	:	:	PUNCT
ejpam-6004	16	4	©	©	PROPN
ejpam-6004	16	5	2025	2025	NUM
ejpam-6004	16	6	the	the	DET
ejpam-6004	16	7	author(s	author(s	NOUN
ejpam-6004	16	8	)	)	PUNCT
ejpam-6004	16	9	.	.	PUNCT
ejpam-6004	17	1	(	(	PUNCT
ejpam-6004	17	2	cc	cc	NOUN
ejpam-6004	17	3	by	by	ADP
ejpam-6004	17	4	-	-	PUNCT
ejpam-6004	17	5	nc	nc	PROPN
ejpam-6004	17	6	4.0	4.0	NUM
ejpam-6004	17	7	)	)	PUNCT
ejpam-6004	17	8	s.	s.	PROPN
ejpam-6004	17	9	prugsapitak	prugsapitak	PROPN
ejpam-6004	17	10	,	,	PUNCT
ejpam-6004	17	11	n.	n.	PROPN
ejpam-6004	17	12	thongngam	thongngam	PROPN
ejpam-6004	17	13	/	/	SYM
ejpam-6004	17	14	eur	eur	NOUN
ejpam-6004	17	15	.	.	PUNCT
ejpam-6004	18	1	j.	j.	PROPN
ejpam-6004	18	2	pure	pure	PROPN
ejpam-6004	18	3	appl	appl	PROPN
ejpam-6004	18	4	.	.	PROPN
ejpam-6004	18	5	math	math	PROPN
ejpam-6004	18	6	,	,	PUNCT
ejpam-6004	18	7	18	18	NUM
ejpam-6004	18	8	(	(	PUNCT
ejpam-6004	18	9	2	2	NUM
ejpam-6004	18	10	)	)	PUNCT
ejpam-6004	18	11	(	(	PUNCT
ejpam-6004	18	12	2025	2025	NUM
ejpam-6004	18	13	)	)	PUNCT
ejpam-6004	18	14	,	,	PUNCT
ejpam-6004	18	15	6004	6004	NUM
ejpam-6004	18	16	2	2	NUM
ejpam-6004	18	17	of	of	ADP
ejpam-6004	18	18	8	8	NUM
ejpam-6004	18	19	in	in	ADP
ejpam-6004	18	20	2012	2012	NUM
ejpam-6004	18	21	,	,	PUNCT
ejpam-6004	18	22	o.	o.	PROPN
ejpam-6004	18	23	karaatli	karaatli	PROPN
ejpam-6004	18	24	and	and	CCONJ
ejpam-6004	18	25	z.	z.	PROPN
ejpam-6004	18	26	siar	siar	PROPN
ejpam-6004	19	1	[	[	X
ejpam-6004	19	2	4	4	X
ejpam-6004	19	3	]	]	PUNCT
ejpam-6004	19	4	showed	show	VERB
ejpam-6004	19	5	that	that	SCONJ
ejpam-6004	19	6	t	t	NOUN
ejpam-6004	19	7	(	(	PUNCT
ejpam-6004	19	8	1	1	NUM
ejpam-6004	19	9	)	)	PUNCT
ejpam-6004	19	10	=	=	NOUN
ejpam-6004	19	11	{	{	PUNCT
ejpam-6004	19	12	5	5	NUM
ejpam-6004	19	13	}	}	PUNCT
ejpam-6004	19	14	,	,	PUNCT
ejpam-6004	19	15	t	t	PROPN
ejpam-6004	19	16	(	(	PUNCT
ejpam-6004	19	17	2	2	NUM
ejpam-6004	19	18	)	)	PUNCT
ejpam-6004	19	19	=	=	NOUN
ejpam-6004	19	20	{	{	PUNCT
ejpam-6004	19	21	5	5	NUM
ejpam-6004	19	22	,	,	PUNCT
ejpam-6004	19	23	6	6	NUM
ejpam-6004	19	24	}	}	PUNCT
ejpam-6004	19	25	,	,	PUNCT
ejpam-6004	19	26	t	t	PROPN
ejpam-6004	19	27	(	(	PUNCT
ejpam-6004	19	28	4	4	NUM
ejpam-6004	19	29	)	)	PUNCT
ejpam-6004	19	30	=	=	NOUN
ejpam-6004	19	31	{	{	PUNCT
ejpam-6004	19	32	5	5	NUM
ejpam-6004	19	33	,	,	PUNCT
ejpam-6004	19	34	6	6	NUM
ejpam-6004	19	35	,	,	PUNCT
ejpam-6004	19	36	8	8	NUM
ejpam-6004	19	37	}	}	PUNCT
ejpam-6004	19	38	,	,	PUNCT
ejpam-6004	19	39	and	and	CCONJ
ejpam-6004	19	40	t	t	PROPN
ejpam-6004	19	41	(	(	PUNCT
ejpam-6004	19	42	8)	8)	NUM
ejpam-6004	19	43	=	=	SYM
ejpam-6004	19	44	{	{	PUNCT
ejpam-6004	19	45	5	5	NUM
ejpam-6004	19	46	,	,	PUNCT
ejpam-6004	19	47	6	6	NUM
ejpam-6004	19	48	,	,	PUNCT
ejpam-6004	19	49	8	8	NUM
ejpam-6004	19	50	,	,	PUNCT
ejpam-6004	19	51	12	12	NUM
ejpam-6004	19	52	}	}	PUNCT
ejpam-6004	19	53	.	.	PUNCT
ejpam-6004	20	1	in	in	ADP
ejpam-6004	20	2	2017	2017	NUM
ejpam-6004	20	3	,	,	PUNCT
ejpam-6004	20	4	mavecha	mavecha	ADJ
ejpam-6004	20	5	[	[	X
ejpam-6004	20	6	5	5	NUM
ejpam-6004	20	7	]	]	PUNCT
ejpam-6004	20	8	studied	study	VERB
ejpam-6004	20	9	t	t	PROPN
ejpam-6004	20	10	(	(	PUNCT
ejpam-6004	20	11	2n	2n	NUM
ejpam-6004	20	12	)	)	PUNCT
ejpam-6004	20	13	for	for	ADP
ejpam-6004	20	14	a	a	DET
ejpam-6004	20	15	non	non	ADJ
ejpam-6004	20	16	-	-	ADJ
ejpam-6004	20	17	negative	negative	ADJ
ejpam-6004	20	18	integer	integer	NOUN
ejpam-6004	20	19	n	n	NOUN
ejpam-6004	20	20	and	and	CCONJ
ejpam-6004	20	21	showed	show	VERB
ejpam-6004	20	22	that	that	SCONJ
ejpam-6004	20	23	5	5	NUM
ejpam-6004	20	24	is	be	AUX
ejpam-6004	20	25	the	the	DET
ejpam-6004	20	26	only	only	ADJ
ejpam-6004	20	27	odd	odd	ADJ
ejpam-6004	20	28	integer	integer	NOUN
ejpam-6004	20	29	in	in	ADP
ejpam-6004	20	30	the	the	DET
ejpam-6004	20	31	set	set	NOUN
ejpam-6004	20	32	t	t	PROPN
ejpam-6004	20	33	(	(	PUNCT
ejpam-6004	20	34	2n	2n	NUM
ejpam-6004	20	35	)	)	PUNCT
ejpam-6004	20	36	for	for	ADP
ejpam-6004	20	37	all	all	DET
ejpam-6004	20	38	non	non	ADJ
ejpam-6004	20	39	-	-	ADJ
ejpam-6004	20	40	negative	negative	ADJ
ejpam-6004	20	41	integers	integer	NOUN
ejpam-6004	20	42	n.	n.	VERB
ejpam-6004	20	43	in	in	ADP
ejpam-6004	20	44	2021	2021	NUM
ejpam-6004	20	45	,	,	PUNCT
ejpam-6004	20	46	alkabouss	alkabouss	ADV
ejpam-6004	20	47	et	et	NOUN
ejpam-6004	20	48	al	al	PROPN
ejpam-6004	20	49	.	.	PUNCT
ejpam-6004	21	1	[	[	X
ejpam-6004	21	2	6	6	NUM
ejpam-6004	21	3	]	]	PUNCT
ejpam-6004	21	4	determined	determine	VERB
ejpam-6004	21	5	conditions	condition	NOUN
ejpam-6004	21	6	for	for	ADP
ejpam-6004	21	7	a	a	DET
ejpam-6004	21	8	positive	positive	ADJ
ejpam-6004	21	9	integer	integer	NOUN
ejpam-6004	21	10	k	k	PROPN
ejpam-6004	21	11	in	in	ADP
ejpam-6004	21	12	t	t	PROPN
ejpam-6004	21	13	(	(	PUNCT
ejpam-6004	21	14	l	l	NOUN
ejpam-6004	21	15	)	)	PUNCT
ejpam-6004	21	16	.	.	PUNCT
ejpam-6004	22	1	they	they	PRON
ejpam-6004	22	2	showed	show	VERB
ejpam-6004	22	3	that	that	SCONJ
ejpam-6004	22	4	if	if	SCONJ
ejpam-6004	22	5	l2	l2	NOUN
ejpam-6004	22	6	<	<	X
ejpam-6004	22	7	k	k	X
ejpam-6004	22	8	then	then	ADV
ejpam-6004	22	9	(	(	PUNCT
ejpam-6004	22	10	l	l	NOUN
ejpam-6004	22	11	,	,	PUNCT
ejpam-6004	22	12	k	k	NOUN
ejpam-6004	22	13	)	)	PUNCT
ejpam-6004	22	14	is	be	AUX
ejpam-6004	22	15	one	one	NUM
ejpam-6004	22	16	of	of	ADP
ejpam-6004	22	17	(	(	PUNCT
ejpam-6004	22	18	1	1	NUM
ejpam-6004	22	19	,	,	PUNCT
ejpam-6004	22	20	5	5	NUM
ejpam-6004	22	21	)	)	PUNCT
ejpam-6004	22	22	,	,	PUNCT
ejpam-6004	22	23	(	(	PUNCT
ejpam-6004	22	24	2	2	NUM
ejpam-6004	22	25	,	,	PUNCT
ejpam-6004	22	26	5	5	NUM
ejpam-6004	22	27	)	)	PUNCT
ejpam-6004	22	28	and	and	CCONJ
ejpam-6004	22	29	(	(	PUNCT
ejpam-6004	22	30	2	2	NUM
ejpam-6004	22	31	,	,	PUNCT
ejpam-6004	22	32	6	6	NUM
ejpam-6004	22	33	)	)	PUNCT
ejpam-6004	22	34	.	.	PUNCT
ejpam-6004	23	1	later	later	ADV
ejpam-6004	23	2	in	in	ADP
ejpam-6004	23	3	2024	2024	NUM
ejpam-6004	23	4	,	,	PUNCT
ejpam-6004	23	5	s.	s.	PROPN
ejpam-6004	23	6	prugsapitak	prugsapitak	PROPN
ejpam-6004	23	7	and	and	CCONJ
ejpam-6004	23	8	n.	n.	NOUN
ejpam-6004	23	9	thongngam	thongngam	NOUN
ejpam-6004	24	1	[	[	X
ejpam-6004	24	2	7	7	X
ejpam-6004	24	3	]	]	PUNCT
ejpam-6004	24	4	showed	show	VERB
ejpam-6004	24	5	that	that	SCONJ
ejpam-6004	24	6	for	for	ADP
ejpam-6004	24	7	a	a	DET
ejpam-6004	24	8	prime	prime	ADJ
ejpam-6004	24	9	p	p	NOUN
ejpam-6004	24	10	and	and	CCONJ
ejpam-6004	24	11	a	a	DET
ejpam-6004	24	12	positive	positive	ADJ
ejpam-6004	24	13	integer	integer	NOUN
ejpam-6004	24	14	n	n	CCONJ
ejpam-6004	24	15	,	,	PUNCT
ejpam-6004	24	16	t	t	PROPN
ejpam-6004	24	17	(	(	PUNCT
ejpam-6004	24	18	pn	pn	PROPN
ejpam-6004	24	19	)	)	PUNCT
ejpam-6004	24	20	=	=	NOUN
ejpam-6004	24	21	n⋃	n⋃	DET
ejpam-6004	24	22	k=0	k=0	PROPN
ejpam-6004	24	23	t	t	NOUN
ejpam-6004	24	24	′(pk	′(pk	PROPN
ejpam-6004	24	25	)	)	PUNCT
ejpam-6004	24	26	.	.	PUNCT
ejpam-6004	25	1	their	their	PRON
ejpam-6004	25	2	findings	finding	NOUN
ejpam-6004	25	3	offer	offer	VERB
ejpam-6004	25	4	a	a	DET
ejpam-6004	25	5	practical	practical	ADJ
ejpam-6004	25	6	method	method	NOUN
ejpam-6004	25	7	for	for	ADP
ejpam-6004	25	8	determining	determine	VERB
ejpam-6004	25	9	t	t	PROPN
ejpam-6004	25	10	(	(	PUNCT
ejpam-6004	25	11	3n	3n	NUM
ejpam-6004	25	12	)	)	PUNCT
ejpam-6004	25	13	for	for	ADP
ejpam-6004	25	14	n	n	NOUN
ejpam-6004	25	15	=	=	SYM
ejpam-6004	25	16	1	1	NUM
ejpam-6004	25	17	,	,	PUNCT
ejpam-6004	25	18	2	2	NUM
ejpam-6004	25	19	,	,	PUNCT
ejpam-6004	25	20	3	3	NUM
ejpam-6004	25	21	.	.	PUNCT
ejpam-6004	26	1	in	in	ADP
ejpam-6004	26	2	this	this	DET
ejpam-6004	26	3	article	article	NOUN
ejpam-6004	26	4	,	,	PUNCT
ejpam-6004	26	5	we	we	PRON
ejpam-6004	26	6	will	will	AUX
ejpam-6004	26	7	find	find	VERB
ejpam-6004	26	8	sets	set	NOUN
ejpam-6004	26	9	t	t	PROPN
ejpam-6004	26	10	′(l	′(l	NOUN
ejpam-6004	26	11	)	)	PUNCT
ejpam-6004	26	12	for	for	ADP
ejpam-6004	26	13	l	l	NOUN
ejpam-6004	26	14	=	=	SYM
ejpam-6004	26	15	8	8	NUM
ejpam-6004	26	16	,	,	PUNCT
ejpam-6004	26	17	16	16	NUM
ejpam-6004	26	18	,	,	PUNCT
ejpam-6004	26	19	64	64	NUM
ejpam-6004	26	20	and	and	CCONJ
ejpam-6004	26	21	128	128	NUM
ejpam-6004	26	22	in	in	ADP
ejpam-6004	26	23	order	order	NOUN
ejpam-6004	26	24	to	to	PART
ejpam-6004	26	25	find	find	VERB
ejpam-6004	26	26	t	t	PROPN
ejpam-6004	26	27	(	(	PUNCT
ejpam-6004	26	28	8)	8)	NUM
ejpam-6004	26	29	,	,	PUNCT
ejpam-6004	26	30	t	t	PROPN
ejpam-6004	26	31	(	(	PUNCT
ejpam-6004	26	32	16	16	NUM
ejpam-6004	26	33	)	)	PUNCT
ejpam-6004	26	34	,	,	PUNCT
ejpam-6004	26	35	t	t	PROPN
ejpam-6004	26	36	(	(	PUNCT
ejpam-6004	26	37	64	64	NUM
ejpam-6004	26	38	)	)	PUNCT
ejpam-6004	26	39	and	and	CCONJ
ejpam-6004	26	40	t	t	PROPN
ejpam-6004	26	41	(	(	PUNCT
ejpam-6004	26	42	128	128	NUM
ejpam-6004	26	43	)	)	PUNCT
ejpam-6004	26	44	.	.	PUNCT
ejpam-6004	27	1	our	our	PRON
ejpam-6004	27	2	approach	approach	NOUN
ejpam-6004	27	3	differs	differ	VERB
ejpam-6004	27	4	from	from	ADP
ejpam-6004	27	5	o.	o.	PROPN
ejpam-6004	27	6	karaatli	karaatli	PROPN
ejpam-6004	27	7	and	and	CCONJ
ejpam-6004	27	8	z.	z.	PROPN
ejpam-6004	27	9	siar	siar	PROPN
ejpam-6004	27	10	’s	’s	PART
ejpam-6004	27	11	method	method	NOUN
ejpam-6004	27	12	[	[	X
ejpam-6004	27	13	4	4	NUM
ejpam-6004	27	14	]	]	PUNCT
ejpam-6004	27	15	,	,	PUNCT
ejpam-6004	27	16	as	as	ADV
ejpam-6004	27	17	well	well	ADV
ejpam-6004	27	18	as	as	ADP
ejpam-6004	27	19	the	the	DET
ejpam-6004	27	20	method	method	NOUN
ejpam-6004	27	21	presented	present	VERB
ejpam-6004	27	22	in	in	ADP
ejpam-6004	27	23	[	[	X
ejpam-6004	27	24	7	7	NUM
ejpam-6004	27	25	]	]	PUNCT
ejpam-6004	27	26	.	.	PUNCT
ejpam-6004	28	1	2	2	X
ejpam-6004	28	2	.	.	X
ejpam-6004	28	3	preliminaries	preliminary	NOUN
ejpam-6004	28	4	to	to	PART
ejpam-6004	28	5	lay	lay	VERB
ejpam-6004	28	6	the	the	DET
ejpam-6004	28	7	groundwork	groundwork	NOUN
ejpam-6004	28	8	for	for	ADP
ejpam-6004	28	9	proving	prove	VERB
ejpam-6004	28	10	our	our	PRON
ejpam-6004	28	11	main	main	ADJ
ejpam-6004	28	12	theorems	theorem	NOUN
ejpam-6004	28	13	,	,	PUNCT
ejpam-6004	28	14	this	this	DET
ejpam-6004	28	15	section	section	NOUN
ejpam-6004	28	16	establishes	establish	VERB
ejpam-6004	28	17	some	some	DET
ejpam-6004	28	18	essential	essential	ADJ
ejpam-6004	28	19	results	result	NOUN
ejpam-6004	28	20	.	.	PUNCT
ejpam-6004	29	1	definition	definition	NOUN
ejpam-6004	29	2	1	1	NUM
ejpam-6004	29	3	.	.	PUNCT
ejpam-6004	30	1	[	[	X
ejpam-6004	30	2	7	7	X
ejpam-6004	30	3	]	]	PUNCT
ejpam-6004	30	4	for	for	ADP
ejpam-6004	30	5	a	a	DET
ejpam-6004	30	6	positive	positive	ADJ
ejpam-6004	30	7	integer	integer	NOUN
ejpam-6004	30	8	l	l	NOUN
ejpam-6004	30	9	,	,	PUNCT
ejpam-6004	30	10	let	let	VERB
ejpam-6004	30	11	t	t	PROPN
ejpam-6004	30	12	(	(	PUNCT
ejpam-6004	30	13	l	l	NOUN
ejpam-6004	30	14	)	)	PUNCT
ejpam-6004	30	15	be	be	VERB
ejpam-6004	30	16	the	the	DET
ejpam-6004	30	17	set	set	NOUN
ejpam-6004	30	18	of	of	ADP
ejpam-6004	30	19	integers	integer	NOUN
ejpam-6004	30	20	k	k	X
ejpam-6004	30	21	for	for	ADP
ejpam-6004	30	22	which	which	PRON
ejpam-6004	30	23	the	the	DET
ejpam-6004	30	24	equation	equation	NOUN
ejpam-6004	30	25	x2	x2	NOUN
ejpam-6004	30	26	−	−	PROPN
ejpam-6004	30	27	kxy	kxy	NOUN
ejpam-6004	30	28	+	+	CCONJ
ejpam-6004	30	29	ky2	ky2	NOUN
ejpam-6004	30	30	+	+	CCONJ
ejpam-6004	30	31	ly	ly	X
ejpam-6004	30	32	=	=	SYM
ejpam-6004	30	33	0	0	NUM
ejpam-6004	30	34	(	(	PUNCT
ejpam-6004	30	35	1	1	X
ejpam-6004	30	36	)	)	PUNCT
ejpam-6004	30	37	has	have	VERB
ejpam-6004	30	38	infinitely	infinitely	ADV
ejpam-6004	30	39	many	many	ADJ
ejpam-6004	30	40	positive	positive	ADJ
ejpam-6004	30	41	integer	integer	NOUN
ejpam-6004	30	42	solutions	solution	NOUN
ejpam-6004	30	43	and	and	CCONJ
ejpam-6004	30	44	let	let	VERB
ejpam-6004	30	45	t	t	PROPN
ejpam-6004	30	46	′(l	′(l	ADV
ejpam-6004	30	47	)	)	PUNCT
ejpam-6004	30	48	be	be	AUX
ejpam-6004	30	49	the	the	DET
ejpam-6004	30	50	set	set	NOUN
ejpam-6004	30	51	of	of	ADP
ejpam-6004	30	52	integers	integer	NOUN
ejpam-6004	30	53	k	k	X
ejpam-6004	31	1	for	for	ADP
ejpam-6004	31	2	which	which	PRON
ejpam-6004	31	3	the	the	DET
ejpam-6004	31	4	equation	equation	NOUN
ejpam-6004	31	5	x2	x2	NOUN
ejpam-6004	31	6	−	−	PROPN
ejpam-6004	31	7	kxy	kxy	NOUN
ejpam-6004	31	8	+	+	CCONJ
ejpam-6004	31	9	ky2	ky2	NOUN
ejpam-6004	31	10	+	+	CCONJ
ejpam-6004	31	11	l	l	NOUN
ejpam-6004	31	12	=	=	SYM
ejpam-6004	31	13	0	0	PUNCT
ejpam-6004	32	1	(	(	PUNCT
ejpam-6004	32	2	2	2	NUM
ejpam-6004	32	3	)	)	PUNCT
ejpam-6004	32	4	has	have	VERB
ejpam-6004	32	5	infinitely	infinitely	ADV
ejpam-6004	32	6	many	many	ADJ
ejpam-6004	32	7	positive	positive	ADJ
ejpam-6004	32	8	integer	integer	NOUN
ejpam-6004	32	9	solutions	solution	NOUN
ejpam-6004	32	10	(	(	PUNCT
ejpam-6004	32	11	x	x	X
ejpam-6004	32	12	,	,	PUNCT
ejpam-6004	32	13	y	y	PROPN
ejpam-6004	32	14	)	)	PUNCT
ejpam-6004	32	15	where	where	SCONJ
ejpam-6004	32	16	gcd(x	gcd(x	PROPN
ejpam-6004	32	17	,	,	PUNCT
ejpam-6004	32	18	y	y	NOUN
ejpam-6004	32	19	)	)	PUNCT
ejpam-6004	33	1	=	=	SYM
ejpam-6004	33	2	1	1	X
ejpam-6004	33	3	.	.	X
ejpam-6004	33	4	theorem	theorem	NOUN
ejpam-6004	33	5	1	1	NUM
ejpam-6004	33	6	.	.	PUNCT
ejpam-6004	34	1	[	[	X
ejpam-6004	34	2	7	7	X
ejpam-6004	34	3	]	]	PUNCT
ejpam-6004	34	4	let	let	VERB
ejpam-6004	34	5	p	p	PRON
ejpam-6004	34	6	be	be	AUX
ejpam-6004	34	7	a	a	DET
ejpam-6004	34	8	prime	prime	NOUN
ejpam-6004	34	9	and	and	CCONJ
ejpam-6004	34	10	n	n	CCONJ
ejpam-6004	34	11	be	be	VERB
ejpam-6004	34	12	a	a	DET
ejpam-6004	34	13	positive	positive	ADJ
ejpam-6004	34	14	integer	integer	NOUN
ejpam-6004	34	15	.	.	PUNCT
ejpam-6004	35	1	then	then	ADV
ejpam-6004	35	2	t	t	PROPN
ejpam-6004	35	3	(	(	PUNCT
ejpam-6004	35	4	pn	pn	PROPN
ejpam-6004	35	5	)	)	PUNCT
ejpam-6004	35	6	=	=	NOUN
ejpam-6004	35	7	n⋃	n⋃	DET
ejpam-6004	35	8	k=0	k=0	PROPN
ejpam-6004	35	9	t	t	NOUN
ejpam-6004	35	10	′(pk	′(pk	PROPN
ejpam-6004	35	11	)	)	PUNCT
ejpam-6004	35	12	.	.	PUNCT
ejpam-6004	36	1	moreover	moreover	ADV
ejpam-6004	36	2	,	,	PUNCT
ejpam-6004	36	3	t	t	PROPN
ejpam-6004	36	4	(	(	PUNCT
ejpam-6004	36	5	pn	pn	PROPN
ejpam-6004	36	6	)	)	PUNCT
ejpam-6004	36	7	=	=	SYM
ejpam-6004	36	8	t	t	PROPN
ejpam-6004	36	9	(	(	PUNCT
ejpam-6004	36	10	pn−1	pn−1	PROPN
ejpam-6004	36	11	)	)	PUNCT
ejpam-6004	36	12	∪	∪	ADP
ejpam-6004	36	13	t	t	NOUN
ejpam-6004	36	14	′(pn	′(pn	NOUN
ejpam-6004	36	15	)	)	PUNCT
ejpam-6004	36	16	.	.	PUNCT
ejpam-6004	36	17	definition	definition	NOUN
ejpam-6004	36	18	2	2	NUM
ejpam-6004	36	19	.	.	PUNCT
ejpam-6004	37	1	[	[	X
ejpam-6004	37	2	8	8	NUM
ejpam-6004	37	3	]	]	PUNCT
ejpam-6004	37	4	let	let	VERB
ejpam-6004	37	5	p	p	PRON
ejpam-6004	37	6	be	be	AUX
ejpam-6004	37	7	an	an	DET
ejpam-6004	37	8	odd	odd	ADJ
ejpam-6004	37	9	prime	prime	NOUN
ejpam-6004	37	10	and	and	CCONJ
ejpam-6004	37	11	a	a	PRON
ejpam-6004	37	12	be	be	AUX
ejpam-6004	37	13	an	an	DET
ejpam-6004	37	14	integer	integer	NOUN
ejpam-6004	37	15	.	.	PUNCT
ejpam-6004	38	1	the	the	DET
ejpam-6004	38	2	legendre	legendre	PROPN
ejpam-6004	38	3	symbol	symbol	NOUN
ejpam-6004	38	4	defined	define	VERB
ejpam-6004	38	5	as	as	ADP
ejpam-6004	38	6	(	(	PUNCT
ejpam-6004	38	7	a	a	DET
ejpam-6004	38	8	p	p	NOUN
ejpam-6004	38	9	)	)	PUNCT
ejpam-6004	39	1	=	=	SYM
ejpam-6004	39	2			NOUN
ejpam-6004	39	3	1	1	NUM
ejpam-6004	39	4	,	,	PUNCT
ejpam-6004	39	5	if	if	SCONJ
ejpam-6004	39	6	a	a	PRON
ejpam-6004	39	7	is	be	AUX
ejpam-6004	39	8	a	a	DET
ejpam-6004	39	9	quadratic	quadratic	ADJ
ejpam-6004	39	10	residue	residue	NOUN
ejpam-6004	39	11	modulo	modulo	NOUN
ejpam-6004	39	12	p	p	NOUN
ejpam-6004	39	13	,	,	PUNCT
ejpam-6004	39	14	−1	−1	ADP
ejpam-6004	39	15	,	,	PUNCT
ejpam-6004	39	16	if	if	SCONJ
ejpam-6004	39	17	a	a	PRON
ejpam-6004	39	18	is	be	AUX
ejpam-6004	39	19	a	a	DET
ejpam-6004	39	20	quadratic	quadratic	ADJ
ejpam-6004	39	21	non	non	ADJ
ejpam-6004	39	22	-	-	ADJ
ejpam-6004	39	23	residue	residue	ADJ
ejpam-6004	39	24	modulo	modulo	NOUN
ejpam-6004	39	25	p	p	X
ejpam-6004	39	26	,	,	PUNCT
ejpam-6004	39	27	0	0	NUM
ejpam-6004	39	28	,	,	PUNCT
ejpam-6004	39	29	if	if	SCONJ
ejpam-6004	39	30	p	p	PROPN
ejpam-6004	39	31	|	|	ADV
ejpam-6004	39	32	a.	a.	PROPN
ejpam-6004	39	33	s.	s.	PROPN
ejpam-6004	39	34	prugsapitak	prugsapitak	PROPN
ejpam-6004	39	35	,	,	PUNCT
ejpam-6004	39	36	n.	n.	PROPN
ejpam-6004	39	37	thongngam	thongngam	PROPN
ejpam-6004	39	38	/	/	SYM
ejpam-6004	39	39	eur	eur	NOUN
ejpam-6004	39	40	.	.	PUNCT
ejpam-6004	40	1	j.	j.	PROPN
ejpam-6004	40	2	pure	pure	PROPN
ejpam-6004	40	3	appl	appl	PROPN
ejpam-6004	40	4	.	.	PROPN
ejpam-6004	40	5	math	math	PROPN
ejpam-6004	40	6	,	,	PUNCT
ejpam-6004	40	7	18	18	NUM
ejpam-6004	40	8	(	(	PUNCT
ejpam-6004	40	9	2	2	NUM
ejpam-6004	40	10	)	)	PUNCT
ejpam-6004	40	11	(	(	PUNCT
ejpam-6004	40	12	2025	2025	NUM
ejpam-6004	40	13	)	)	PUNCT
ejpam-6004	40	14	,	,	PUNCT
ejpam-6004	40	15	6004	6004	NUM
ejpam-6004	40	16	3	3	NUM
ejpam-6004	40	17	of	of	ADP
ejpam-6004	40	18	8	8	NUM
ejpam-6004	40	19	lemma	lemma	PROPN
ejpam-6004	40	20	1	1	NUM
ejpam-6004	40	21	.	.	PUNCT
ejpam-6004	41	1	[	[	X
ejpam-6004	41	2	8	8	NUM
ejpam-6004	41	3	]	]	PUNCT
ejpam-6004	41	4	let	let	VERB
ejpam-6004	41	5	p	p	PRON
ejpam-6004	41	6	be	be	AUX
ejpam-6004	41	7	an	an	DET
ejpam-6004	41	8	odd	odd	ADJ
ejpam-6004	41	9	prime	prime	NOUN
ejpam-6004	41	10	.	.	PUNCT
ejpam-6004	42	1	then	then	ADV
ejpam-6004	42	2	(	(	PUNCT
ejpam-6004	42	3	−1	−1	NOUN
ejpam-6004	42	4	p	p	NOUN
ejpam-6004	42	5	)	)	PUNCT
ejpam-6004	42	6	=	=	SYM
ejpam-6004	42	7	(	(	PUNCT
ejpam-6004	42	8	−1	−1	NOUN
ejpam-6004	42	9	)	)	PUNCT
ejpam-6004	43	1	p−1	p−1	PROPN
ejpam-6004	43	2	2	2	NUM
ejpam-6004	43	3	=	=	SYM
ejpam-6004	43	4	{	{	PUNCT
ejpam-6004	43	5	1	1	NUM
ejpam-6004	43	6	,	,	PUNCT
ejpam-6004	43	7	if	if	SCONJ
ejpam-6004	43	8	p	p	PRON
ejpam-6004	43	9	≡	≡	PROPN
ejpam-6004	43	10	1	1	NUM
ejpam-6004	43	11	(	(	PUNCT
ejpam-6004	43	12	mod	mod	NOUN
ejpam-6004	43	13	4	4	NUM
ejpam-6004	43	14	)	)	PUNCT
ejpam-6004	43	15	,	,	PUNCT
ejpam-6004	43	16	−1	−1	ADP
ejpam-6004	43	17	,	,	PUNCT
ejpam-6004	43	18	if	if	SCONJ
ejpam-6004	43	19	p	p	PRON
ejpam-6004	43	20	≡	≡	PROPN
ejpam-6004	43	21	3	3	NUM
ejpam-6004	43	22	(	(	PUNCT
ejpam-6004	43	23	mod	mod	NOUN
ejpam-6004	43	24	4	4	NUM
ejpam-6004	43	25	)	)	PUNCT
ejpam-6004	43	26	.	.	PUNCT
ejpam-6004	44	1	lemma	lemma	PROPN
ejpam-6004	44	2	2	2	NUM
ejpam-6004	44	3	.	.	PUNCT
ejpam-6004	45	1	[	[	X
ejpam-6004	45	2	8	8	NUM
ejpam-6004	45	3	]	]	PUNCT
ejpam-6004	45	4	let	let	VERB
ejpam-6004	45	5	p	p	PRON
ejpam-6004	45	6	be	be	AUX
ejpam-6004	45	7	an	an	DET
ejpam-6004	45	8	odd	odd	ADJ
ejpam-6004	45	9	prime	prime	NOUN
ejpam-6004	45	10	.	.	PUNCT
ejpam-6004	46	1	then	then	ADV
ejpam-6004	46	2	(	(	PUNCT
ejpam-6004	46	3	2	2	NUM
ejpam-6004	46	4	p	p	NOUN
ejpam-6004	46	5	)	)	PUNCT
ejpam-6004	46	6	=	=	SYM
ejpam-6004	46	7	(	(	PUNCT
ejpam-6004	46	8	−1	−1	NOUN
ejpam-6004	46	9	)	)	PUNCT
ejpam-6004	46	10	p2−1	p2−1	ADP
ejpam-6004	46	11	8	8	NUM
ejpam-6004	46	12	=	=	SYM
ejpam-6004	46	13	{	{	PUNCT
ejpam-6004	46	14	1	1	NUM
ejpam-6004	46	15	,	,	PUNCT
ejpam-6004	46	16	if	if	SCONJ
ejpam-6004	46	17	p	p	PRON
ejpam-6004	46	18	≡	≡	PROPN
ejpam-6004	46	19	1	1	NUM
ejpam-6004	46	20	,	,	PUNCT
ejpam-6004	46	21	7	7	NUM
ejpam-6004	46	22	(	(	PUNCT
ejpam-6004	46	23	mod	mod	NOUN
ejpam-6004	46	24	8)	8)	NUM
ejpam-6004	46	25	,	,	PUNCT
ejpam-6004	46	26	−1	−1	NOUN
ejpam-6004	46	27	,	,	PUNCT
ejpam-6004	46	28	if	if	SCONJ
ejpam-6004	46	29	p	p	PRON
ejpam-6004	46	30	≡	≡	PROPN
ejpam-6004	46	31	3	3	NUM
ejpam-6004	46	32	,	,	PUNCT
ejpam-6004	46	33	5	5	NUM
ejpam-6004	46	34	(	(	PUNCT
ejpam-6004	46	35	mod	mod	NOUN
ejpam-6004	46	36	8)	8)	NUM
ejpam-6004	46	37	.	.	PUNCT
ejpam-6004	47	1	definition	definition	NOUN
ejpam-6004	47	2	3	3	NUM
ejpam-6004	47	3	.	.	PUNCT
ejpam-6004	48	1	[	[	X
ejpam-6004	48	2	8	8	NUM
ejpam-6004	48	3	]	]	PUNCT
ejpam-6004	48	4	let	let	VERB
ejpam-6004	48	5	n	n	PRON
ejpam-6004	48	6	be	be	AUX
ejpam-6004	48	7	a	a	DET
ejpam-6004	48	8	nonzero	nonzero	NOUN
ejpam-6004	48	9	integer	integer	NOUN
ejpam-6004	48	10	and	and	CCONJ
ejpam-6004	48	11	d	d	NOUN
ejpam-6004	48	12	be	be	AUX
ejpam-6004	48	13	a	a	DET
ejpam-6004	48	14	positive	positive	ADJ
ejpam-6004	48	15	integer	integer	NOUN
ejpam-6004	48	16	which	which	PRON
ejpam-6004	48	17	is	be	AUX
ejpam-6004	48	18	not	not	PART
ejpam-6004	48	19	a	a	DET
ejpam-6004	48	20	perfect	perfect	ADJ
ejpam-6004	48	21	square	square	NOUN
ejpam-6004	48	22	.	.	PUNCT
ejpam-6004	49	1	the	the	DET
ejpam-6004	49	2	least	least	ADV
ejpam-6004	49	3	positive	positive	ADJ
ejpam-6004	49	4	integer	integer	NOUN
ejpam-6004	49	5	solution	solution	NOUN
ejpam-6004	49	6	(	(	PUNCT
ejpam-6004	49	7	x1	x1	PROPN
ejpam-6004	49	8	,	,	PUNCT
ejpam-6004	49	9	y1	y1	PROPN
ejpam-6004	49	10	)	)	PUNCT
ejpam-6004	49	11	of	of	ADP
ejpam-6004	49	12	the	the	DET
ejpam-6004	49	13	equation	equation	NOUN
ejpam-6004	49	14	x2	x2	PROPN
ejpam-6004	49	15	−dy2	−dy2	PROPN
ejpam-6004	49	16	=	=	PUNCT
ejpam-6004	50	1	n	n	PROPN
ejpam-6004	50	2	is	be	AUX
ejpam-6004	50	3	called	call	VERB
ejpam-6004	50	4	the	the	DET
ejpam-6004	50	5	fundamental	fundamental	ADJ
ejpam-6004	50	6	solution	solution	NOUN
ejpam-6004	50	7	.	.	PUNCT
ejpam-6004	51	1	lemma	lemma	PROPN
ejpam-6004	51	2	3	3	X
ejpam-6004	51	3	.	.	PUNCT
ejpam-6004	52	1	[	[	X
ejpam-6004	52	2	8	8	NUM
ejpam-6004	52	3	]	]	X
ejpam-6004	52	4	let	let	NOUN
ejpam-6004	52	5	n	n	PRON
ejpam-6004	52	6	,	,	PUNCT
ejpam-6004	52	7	d	d	X
ejpam-6004	52	8	be	be	AUX
ejpam-6004	52	9	odd	odd	ADJ
ejpam-6004	52	10	positive	positive	ADJ
ejpam-6004	52	11	integers	integer	NOUN
ejpam-6004	52	12	with	with	ADP
ejpam-6004	52	13	d	d	PROPN
ejpam-6004	52	14	non	non	ADJ
ejpam-6004	52	15	-	-	ADJ
ejpam-6004	52	16	square	square	ADJ
ejpam-6004	52	17	.	.	PUNCT
ejpam-6004	53	1	suppose	suppose	VERB
ejpam-6004	53	2	that	that	SCONJ
ejpam-6004	53	3	the	the	DET
ejpam-6004	53	4	equation	equation	NOUN
ejpam-6004	53	5	x2	x2	PROPN
ejpam-6004	53	6	−dy2	−dy2	X
ejpam-6004	53	7	=	=	PUNCT
ejpam-6004	53	8	4	4	NUM
ejpam-6004	53	9	,	,	PUNCT
ejpam-6004	53	10	gcd(x	gcd(x	PROPN
ejpam-6004	53	11	,	,	PUNCT
ejpam-6004	53	12	y	y	NOUN
ejpam-6004	53	13	)	)	PUNCT
ejpam-6004	53	14	=	=	SYM
ejpam-6004	53	15	1	1	NUM
ejpam-6004	53	16	is	be	AUX
ejpam-6004	53	17	solvable	solvable	ADJ
ejpam-6004	53	18	and	and	CCONJ
ejpam-6004	53	19	let	let	VERB
ejpam-6004	53	20	x0	x0	PROPN
ejpam-6004	54	1	+	+	CCONJ
ejpam-6004	55	1	y0	y0	NOUN
ejpam-6004	55	2	√	√	NOUN
ejpam-6004	55	3	d	d	NOUN
ejpam-6004	55	4	be	be	VERB
ejpam-6004	55	5	the	the	DET
ejpam-6004	55	6	fundamental	fundamental	ADJ
ejpam-6004	55	7	solution	solution	NOUN
ejpam-6004	55	8	.	.	PUNCT
ejpam-6004	56	1	if	if	SCONJ
ejpam-6004	56	2	the	the	DET
ejpam-6004	56	3	equation	equation	NOUN
ejpam-6004	56	4	u2	u2	PROPN
ejpam-6004	56	5	−dv2	−dv2	PROPN
ejpam-6004	56	6	=	=	SYM
ejpam-6004	56	7	−4n	−4n	PROPN
ejpam-6004	56	8	,	,	PUNCT
ejpam-6004	56	9	where	where	SCONJ
ejpam-6004	56	10	u	u	NOUN
ejpam-6004	56	11	,	,	PUNCT
ejpam-6004	56	12	v	v	ADP
ejpam-6004	56	13	∈	∈	PROPN
ejpam-6004	56	14	z	z	PROPN
ejpam-6004	56	15	,	,	PUNCT
ejpam-6004	56	16	gcd(u	gcd(u	PROPN
ejpam-6004	56	17	,	,	PUNCT
ejpam-6004	56	18	v	v	NOUN
ejpam-6004	56	19	)	)	PUNCT
ejpam-6004	56	20	|	|	ADV
ejpam-6004	56	21	2	2	NUM
ejpam-6004	56	22	,	,	PUNCT
ejpam-6004	56	23	is	be	AUX
ejpam-6004	56	24	solvable	solvable	ADJ
ejpam-6004	56	25	,	,	PUNCT
ejpam-6004	56	26	then	then	ADV
ejpam-6004	56	27	u2	u2	PROPN
ejpam-6004	56	28	−dv2	−dv2	PROPN
ejpam-6004	56	29	=	=	PRON
ejpam-6004	56	30	−4n	−4n	PROPN
ejpam-6004	56	31	has	have	VERB
ejpam-6004	56	32	a	a	DET
ejpam-6004	56	33	solution	solution	NOUN
ejpam-6004	56	34	u0	u0	ADJ
ejpam-6004	56	35	+	+	X
ejpam-6004	56	36	v0	v0	NOUN
ejpam-6004	56	37	√	√	NOUN
ejpam-6004	56	38	d	d	NOUN
ejpam-6004	56	39	with	with	ADP
ejpam-6004	56	40	the	the	DET
ejpam-6004	56	41	following	follow	VERB
ejpam-6004	56	42	property	property	NOUN
ejpam-6004	56	43	:	:	PUNCT
ejpam-6004	56	44	0	0	PUNCT
ejpam-6004	56	45	<	<	X
ejpam-6004	56	46	v0	v0	NOUN
ejpam-6004	56	47	≤	≤	NOUN
ejpam-6004	56	48	y0	y0	NOUN
ejpam-6004	56	49	√	√	NUM
ejpam-6004	56	50	n√	n√	PROPN
ejpam-6004	56	51	(	(	PUNCT
ejpam-6004	56	52	x0	x0	PROPN
ejpam-6004	56	53	−	−	PROPN
ejpam-6004	56	54	2	2	NUM
ejpam-6004	56	55	)	)	PUNCT
ejpam-6004	56	56	,	,	PUNCT
ejpam-6004	56	57	0	0	NUM
ejpam-6004	56	58	≤	≤	NUM
ejpam-6004	56	59	u0	u0	ADJ
ejpam-6004	56	60	≤	≤	NOUN
ejpam-6004	56	61	√	√	NUM
ejpam-6004	56	62	(	(	PUNCT
ejpam-6004	56	63	x0	x0	PROPN
ejpam-6004	56	64	−	−	PROPN
ejpam-6004	56	65	2)n	2)n	NOUN
ejpam-6004	56	66	.	.	PUNCT
ejpam-6004	57	1	lemma	lemma	PROPN
ejpam-6004	57	2	4	4	NUM
ejpam-6004	57	3	.	.	PUNCT
ejpam-6004	58	1	[	[	X
ejpam-6004	58	2	8	8	NUM
ejpam-6004	58	3	]	]	X
ejpam-6004	58	4	let	let	NOUN
ejpam-6004	58	5	n	n	PRON
ejpam-6004	58	6	,	,	PUNCT
ejpam-6004	58	7	d	d	X
ejpam-6004	58	8	be	be	AUX
ejpam-6004	58	9	positive	positive	ADJ
ejpam-6004	58	10	integers	integer	NOUN
ejpam-6004	58	11	with	with	ADP
ejpam-6004	58	12	d	d	PROPN
ejpam-6004	58	13	non	non	ADJ
ejpam-6004	58	14	-	-	ADJ
ejpam-6004	58	15	square	square	ADJ
ejpam-6004	58	16	.	.	PUNCT
ejpam-6004	59	1	suppose	suppose	VERB
ejpam-6004	59	2	that	that	SCONJ
ejpam-6004	59	3	x0	x0	PROPN
ejpam-6004	59	4	+	+	CCONJ
ejpam-6004	59	5	y0	y0	NOUN
ejpam-6004	59	6	√	√	ADJ
ejpam-6004	59	7	d	d	NOUN
ejpam-6004	59	8	is	be	AUX
ejpam-6004	59	9	the	the	DET
ejpam-6004	59	10	fundamental	fundamental	ADJ
ejpam-6004	59	11	solution	solution	NOUN
ejpam-6004	59	12	of	of	ADP
ejpam-6004	59	13	the	the	DET
ejpam-6004	59	14	pell	pell	NOUN
ejpam-6004	59	15	equation	equation	NOUN
ejpam-6004	59	16	x2	x2	PROPN
ejpam-6004	59	17	−dy2	−dy2	X
ejpam-6004	60	1	=	=	PUNCT
ejpam-6004	60	2	1	1	NUM
ejpam-6004	60	3	and	and	CCONJ
ejpam-6004	60	4	the	the	DET
ejpam-6004	60	5	equation	equation	NOUN
ejpam-6004	60	6	u2	u2	PROPN
ejpam-6004	60	7	−dv2	−dv2	PROPN
ejpam-6004	60	8	=	=	SYM
ejpam-6004	60	9	−n	−n	NOUN
ejpam-6004	60	10	,	,	PUNCT
ejpam-6004	60	11	gcd(u	gcd(u	PROPN
ejpam-6004	60	12	,	,	PUNCT
ejpam-6004	60	13	v	v	NOUN
ejpam-6004	60	14	)	)	PUNCT
ejpam-6004	60	15	=	=	SYM
ejpam-6004	60	16	1	1	NUM
ejpam-6004	60	17	is	be	AUX
ejpam-6004	60	18	solvable	solvable	ADJ
ejpam-6004	60	19	.	.	PUNCT
ejpam-6004	61	1	then	then	ADV
ejpam-6004	61	2	u2	u2	PROPN
ejpam-6004	61	3	−dv2	−dv2	PROPN
ejpam-6004	61	4	=	=	PUNCT
ejpam-6004	61	5	−n	−n	NOUN
ejpam-6004	61	6	has	have	VERB
ejpam-6004	61	7	a	a	DET
ejpam-6004	61	8	solution	solution	NOUN
ejpam-6004	61	9	u0	u0	ADJ
ejpam-6004	61	10	+	+	X
ejpam-6004	61	11	v0	v0	NOUN
ejpam-6004	61	12	√	√	NOUN
ejpam-6004	61	13	d	d	NOUN
ejpam-6004	61	14	with	with	ADP
ejpam-6004	61	15	the	the	DET
ejpam-6004	61	16	following	follow	VERB
ejpam-6004	61	17	property	property	NOUN
ejpam-6004	61	18	:	:	PUNCT
ejpam-6004	61	19	0	0	PUNCT
ejpam-6004	62	1	<	<	X
ejpam-6004	62	2	v0	v0	NOUN
ejpam-6004	62	3	≤	≤	NOUN
ejpam-6004	62	4	y0	y0	NOUN
ejpam-6004	62	5	√	√	NUM
ejpam-6004	63	1	n√	n√	NUM
ejpam-6004	64	1	2(x0	2(x0	NUM
ejpam-6004	65	1	−	−	NOUN
ejpam-6004	65	2	1	1	NUM
ejpam-6004	65	3	)	)	PUNCT
ejpam-6004	65	4	,	,	PUNCT
ejpam-6004	65	5	0	0	NUM
ejpam-6004	65	6	≤	≤	NUM
ejpam-6004	65	7	u0	u0	ADJ
ejpam-6004	65	8	≤	≤	NOUN
ejpam-6004	65	9	√	√	ADV
ejpam-6004	65	10	1	1	NUM
ejpam-6004	65	11	2	2	NUM
ejpam-6004	65	12	(	(	PUNCT
ejpam-6004	65	13	x0	x0	PROPN
ejpam-6004	65	14	−	−	PROPN
ejpam-6004	65	15	2)n	2)n	NOUN
ejpam-6004	65	16	.	.	PUNCT
ejpam-6004	66	1	lemma	lemma	PROPN
ejpam-6004	66	2	5	5	NUM
ejpam-6004	66	3	.	.	PUNCT
ejpam-6004	67	1	[	[	X
ejpam-6004	67	2	9	9	NUM
ejpam-6004	67	3	]	]	PUNCT
ejpam-6004	67	4	let	let	VERB
ejpam-6004	67	5	k	k	PRON
ejpam-6004	67	6	>	>	X
ejpam-6004	67	7	3	3	X
ejpam-6004	67	8	.	.	PUNCT
ejpam-6004	68	1	then	then	ADV
ejpam-6004	68	2	the	the	DET
ejpam-6004	68	3	equation	equation	NOUN
ejpam-6004	68	4	u2−(k2−4)v2	u2−(k2−4)v2	NOUN
ejpam-6004	68	5	=	=	SYM
ejpam-6004	68	6	−4	−4	X
ejpam-6004	68	7	has	have	VERB
ejpam-6004	68	8	no	no	DET
ejpam-6004	68	9	integer	integer	NOUN
ejpam-6004	68	10	solutions	solution	NOUN
ejpam-6004	68	11	.	.	PUNCT
ejpam-6004	69	1	s.	s.	PROPN
ejpam-6004	69	2	prugsapitak	prugsapitak	PROPN
ejpam-6004	69	3	,	,	PUNCT
ejpam-6004	69	4	n.	n.	PROPN
ejpam-6004	69	5	thongngam	thongngam	PROPN
ejpam-6004	69	6	/	/	SYM
ejpam-6004	69	7	eur	eur	NOUN
ejpam-6004	69	8	.	.	PUNCT
ejpam-6004	70	1	j.	j.	PROPN
ejpam-6004	70	2	pure	pure	PROPN
ejpam-6004	70	3	appl	appl	PROPN
ejpam-6004	70	4	.	.	PROPN
ejpam-6004	70	5	math	math	PROPN
ejpam-6004	70	6	,	,	PUNCT
ejpam-6004	70	7	18	18	NUM
ejpam-6004	70	8	(	(	PUNCT
ejpam-6004	70	9	2	2	NUM
ejpam-6004	70	10	)	)	PUNCT
ejpam-6004	70	11	(	(	PUNCT
ejpam-6004	70	12	2025	2025	NUM
ejpam-6004	70	13	)	)	PUNCT
ejpam-6004	70	14	,	,	PUNCT
ejpam-6004	70	15	6004	6004	NUM
ejpam-6004	70	16	4	4	NUM
ejpam-6004	70	17	of	of	ADP
ejpam-6004	70	18	8	8	NUM
ejpam-6004	70	19	3	3	NUM
ejpam-6004	70	20	.	.	PUNCT
ejpam-6004	70	21	main	main	ADJ
ejpam-6004	70	22	results	result	NOUN
ejpam-6004	70	23	to	to	PART
ejpam-6004	70	24	begin	begin	VERB
ejpam-6004	70	25	our	our	PRON
ejpam-6004	70	26	analysis	analysis	NOUN
ejpam-6004	70	27	,	,	PUNCT
ejpam-6004	70	28	we	we	PRON
ejpam-6004	70	29	will	will	AUX
ejpam-6004	70	30	consider	consider	VERB
ejpam-6004	70	31	solutions	solution	NOUN
ejpam-6004	70	32	to	to	ADP
ejpam-6004	70	33	certain	certain	ADJ
ejpam-6004	70	34	pell	pell	NOUN
ejpam-6004	70	35	’s	’s	PART
ejpam-6004	70	36	equations	equation	NOUN
ejpam-6004	70	37	as	as	SCONJ
ejpam-6004	70	38	follows	follow	VERB
ejpam-6004	70	39	:	:	PUNCT
ejpam-6004	70	40	lemma	lemma	PROPN
ejpam-6004	70	41	6	6	X
ejpam-6004	70	42	.	.	PUNCT
ejpam-6004	71	1	let	let	VERB
ejpam-6004	71	2	k	k	PRON
ejpam-6004	71	3	be	be	AUX
ejpam-6004	71	4	a	a	DET
ejpam-6004	71	5	positive	positive	ADJ
ejpam-6004	71	6	integer	integer	NOUN
ejpam-6004	71	7	.	.	PUNCT
ejpam-6004	72	1	the	the	DET
ejpam-6004	72	2	equation	equation	NOUN
ejpam-6004	72	3	u2	u2	PROPN
ejpam-6004	72	4	−	−	PROPN
ejpam-6004	72	5	(	(	PUNCT
ejpam-6004	72	6	k2	k2	PROPN
ejpam-6004	72	7	−	−	PROPN
ejpam-6004	72	8	4)v2	4)v2	PROPN
ejpam-6004	72	9	=	=	SYM
ejpam-6004	72	10	−4	−4	X
ejpam-6004	72	11	has	have	VERB
ejpam-6004	72	12	positive	positive	ADJ
ejpam-6004	72	13	integer	integer	NOUN
ejpam-6004	72	14	solutions	solution	NOUN
ejpam-6004	72	15	u	u	NOUN
ejpam-6004	72	16	and	and	CCONJ
ejpam-6004	72	17	v	v	NOUN
ejpam-6004	72	18	if	if	SCONJ
ejpam-6004	73	1	and	and	CCONJ
ejpam-6004	73	2	only	only	ADV
ejpam-6004	73	3	if	if	SCONJ
ejpam-6004	73	4	k	k	PROPN
ejpam-6004	73	5	=	=	SYM
ejpam-6004	73	6	3	3	X
ejpam-6004	73	7	.	.	PUNCT
ejpam-6004	74	1	proof	proof	NOUN
ejpam-6004	74	2	.	.	PUNCT
ejpam-6004	75	1	suppose	suppose	VERB
ejpam-6004	75	2	that	that	SCONJ
ejpam-6004	75	3	u2	u2	PROPN
ejpam-6004	75	4	−	−	PROPN
ejpam-6004	75	5	(	(	PUNCT
ejpam-6004	75	6	k2	k2	PROPN
ejpam-6004	75	7	−	−	PROPN
ejpam-6004	75	8	4)v2	4)v2	PROPN
ejpam-6004	75	9	=	=	SYM
ejpam-6004	75	10	−4	−4	X
ejpam-6004	75	11	has	have	VERB
ejpam-6004	75	12	positive	positive	ADJ
ejpam-6004	75	13	integer	integer	NOUN
ejpam-6004	75	14	solutions	solution	NOUN
ejpam-6004	75	15	u	u	NOUN
ejpam-6004	75	16	and	and	CCONJ
ejpam-6004	75	17	v.	v.	X
ejpam-6004	75	18	by	by	ADP
ejpam-6004	75	19	lemma	lemma	PROPN
ejpam-6004	75	20	5	5	NUM
ejpam-6004	75	21	,	,	PUNCT
ejpam-6004	75	22	the	the	DET
ejpam-6004	75	23	equation	equation	NOUN
ejpam-6004	75	24	u2	u2	PROPN
ejpam-6004	75	25	−	−	PROPN
ejpam-6004	75	26	(	(	PUNCT
ejpam-6004	75	27	k2	k2	PROPN
ejpam-6004	75	28	−	−	PROPN
ejpam-6004	75	29	4)v2	4)v2	PROPN
ejpam-6004	75	30	=	=	SYM
ejpam-6004	75	31	−4	−4	X
ejpam-6004	75	32	has	have	VERB
ejpam-6004	75	33	no	no	DET
ejpam-6004	75	34	integer	integer	NOUN
ejpam-6004	75	35	solutions	solution	NOUN
ejpam-6004	75	36	u	u	NOUN
ejpam-6004	75	37	and	and	CCONJ
ejpam-6004	75	38	v	v	NOUN
ejpam-6004	75	39	for	for	ADP
ejpam-6004	75	40	k	k	PROPN
ejpam-6004	75	41	>	>	X
ejpam-6004	76	1	3	3	X
ejpam-6004	76	2	.	.	PUNCT
ejpam-6004	77	1	if	if	SCONJ
ejpam-6004	77	2	k	k	PROPN
ejpam-6004	77	3	=	=	SYM
ejpam-6004	77	4	1	1	NUM
ejpam-6004	77	5	,	,	PUNCT
ejpam-6004	77	6	then	then	ADV
ejpam-6004	77	7	u2	u2	PROPN
ejpam-6004	77	8	+	+	CCONJ
ejpam-6004	77	9	3v2	3v2	NUM
ejpam-6004	77	10	=	=	SYM
ejpam-6004	77	11	−4	−4	PROPN
ejpam-6004	77	12	,	,	PUNCT
ejpam-6004	77	13	which	which	PRON
ejpam-6004	77	14	is	be	AUX
ejpam-6004	77	15	impossible	impossible	ADJ
ejpam-6004	77	16	.	.	PUNCT
ejpam-6004	78	1	if	if	SCONJ
ejpam-6004	78	2	k	k	PROPN
ejpam-6004	78	3	=	=	SYM
ejpam-6004	78	4	2	2	NUM
ejpam-6004	78	5	,	,	PUNCT
ejpam-6004	78	6	then	then	ADV
ejpam-6004	78	7	u2	u2	PROPN
ejpam-6004	78	8	=	=	SYM
ejpam-6004	78	9	−4	−4	PROPN
ejpam-6004	78	10	,	,	PUNCT
ejpam-6004	78	11	which	which	PRON
ejpam-6004	78	12	is	be	AUX
ejpam-6004	78	13	impossible	impossible	ADJ
ejpam-6004	78	14	.	.	PUNCT
ejpam-6004	79	1	this	this	PRON
ejpam-6004	79	2	implies	imply	VERB
ejpam-6004	79	3	that	that	SCONJ
ejpam-6004	79	4	k	k	PROPN
ejpam-6004	79	5	=	=	SYM
ejpam-6004	79	6	3	3	X
ejpam-6004	79	7	.	.	PUNCT
ejpam-6004	79	8	conversely	conversely	ADV
ejpam-6004	79	9	,	,	PUNCT
ejpam-6004	79	10	let	let	VERB
ejpam-6004	79	11	k	k	NOUN
ejpam-6004	79	12	=	=	SYM
ejpam-6004	79	13	3	3	X
ejpam-6004	79	14	.	.	PUNCT
ejpam-6004	79	15	then	then	ADV
ejpam-6004	79	16	u2	u2	PROPN
ejpam-6004	79	17	−	−	PROPN
ejpam-6004	79	18	5v2	5v2	NUM
ejpam-6004	79	19	=	=	SYM
ejpam-6004	79	20	−4	−4	X
ejpam-6004	79	21	.	.	PUNCT
ejpam-6004	80	1	it	it	PRON
ejpam-6004	80	2	is	be	AUX
ejpam-6004	80	3	easy	easy	ADJ
ejpam-6004	80	4	to	to	PART
ejpam-6004	80	5	see	see	VERB
ejpam-6004	80	6	that	that	PRON
ejpam-6004	80	7	(	(	PUNCT
ejpam-6004	80	8	u	u	NOUN
ejpam-6004	80	9	,	,	PUNCT
ejpam-6004	80	10	v	v	NOUN
ejpam-6004	80	11	)	)	PUNCT
ejpam-6004	80	12	=	=	PUNCT
ejpam-6004	80	13	(	(	PUNCT
ejpam-6004	80	14	1	1	NUM
ejpam-6004	80	15	,	,	PUNCT
ejpam-6004	80	16	1	1	NUM
ejpam-6004	80	17	)	)	PUNCT
ejpam-6004	80	18	is	be	AUX
ejpam-6004	80	19	a	a	DET
ejpam-6004	80	20	solution	solution	NOUN
ejpam-6004	80	21	of	of	ADP
ejpam-6004	80	22	the	the	DET
ejpam-6004	80	23	equation	equation	NOUN
ejpam-6004	80	24	u2	u2	NOUN
ejpam-6004	81	1	−	−	PROPN
ejpam-6004	81	2	5v2	5v2	NUM
ejpam-6004	81	3	=	=	SYM
ejpam-6004	81	4	−4	−4	X
ejpam-6004	81	5	.	.	PUNCT
ejpam-6004	82	1	lemma	lemma	PROPN
ejpam-6004	82	2	7	7	X
ejpam-6004	82	3	.	.	PUNCT
ejpam-6004	83	1	let	let	VERB
ejpam-6004	83	2	l	l	NOUN
ejpam-6004	83	3	≥	≥	NUM
ejpam-6004	83	4	1	1	NUM
ejpam-6004	83	5	and	and	CCONJ
ejpam-6004	83	6	s	s	PRON
ejpam-6004	83	7	≥	≥	NUM
ejpam-6004	83	8	2	2	NUM
ejpam-6004	83	9	be	be	AUX
ejpam-6004	83	10	positive	positive	ADJ
ejpam-6004	83	11	integers	integer	NOUN
ejpam-6004	83	12	.	.	PUNCT
ejpam-6004	84	1	if	if	SCONJ
ejpam-6004	84	2	u2	u2	PROPN
ejpam-6004	84	3	−	−	PROPN
ejpam-6004	84	4	s(s	s(s	PROPN
ejpam-6004	84	5	−	−	PROPN
ejpam-6004	84	6	1)v2	1)v2	PROPN
ejpam-6004	84	7	=	=	SYM
ejpam-6004	84	8	−2l	−2l	PROPN
ejpam-6004	84	9	has	have	VERB
ejpam-6004	84	10	a	a	DET
ejpam-6004	84	11	solution	solution	NOUN
ejpam-6004	84	12	,	,	PUNCT
ejpam-6004	84	13	then	then	ADV
ejpam-6004	84	14	s	s	VERB
ejpam-6004	84	15	≤	≤	X
ejpam-6004	84	16	2l	2l	NOUN
ejpam-6004	84	17	+	+	CCONJ
ejpam-6004	84	18	1	1	X
ejpam-6004	84	19	.	.	X
ejpam-6004	84	20	proof	proof	NOUN
ejpam-6004	84	21	.	.	PUNCT
ejpam-6004	85	1	suppose	suppose	VERB
ejpam-6004	85	2	u2−	u2−	NOUN
ejpam-6004	85	3	s(s−1)v2	s(s−1)v2	NOUN
ejpam-6004	85	4	=	=	SYM
ejpam-6004	85	5	−2l	−2l	PROPN
ejpam-6004	85	6	has	have	VERB
ejpam-6004	85	7	a	a	DET
ejpam-6004	85	8	solution	solution	NOUN
ejpam-6004	85	9	.	.	PUNCT
ejpam-6004	86	1	we	we	PRON
ejpam-6004	86	2	consider	consider	VERB
ejpam-6004	86	3	two	two	NUM
ejpam-6004	86	4	cases	case	NOUN
ejpam-6004	86	5	as	as	SCONJ
ejpam-6004	86	6	follows	follow	VERB
ejpam-6004	86	7	.	.	PUNCT
ejpam-6004	87	1	case	case	NOUN
ejpam-6004	87	2	1	1	NUM
ejpam-6004	87	3	.	.	PUNCT
ejpam-6004	88	1	gcd(u	gcd(u	PROPN
ejpam-6004	88	2	,	,	PUNCT
ejpam-6004	88	3	v	v	NOUN
ejpam-6004	88	4	)	)	PUNCT
ejpam-6004	88	5	=	=	SYM
ejpam-6004	88	6	1	1	X
ejpam-6004	88	7	.	.	PUNCT
ejpam-6004	89	1	since	since	SCONJ
ejpam-6004	89	2	(	(	PUNCT
ejpam-6004	89	3	2s−1	2s−1	NUM
ejpam-6004	89	4	,	,	PUNCT
ejpam-6004	89	5	2	2	NUM
ejpam-6004	89	6	)	)	PUNCT
ejpam-6004	89	7	is	be	AUX
ejpam-6004	89	8	the	the	DET
ejpam-6004	89	9	fundamental	fundamental	ADJ
ejpam-6004	89	10	solution	solution	NOUN
ejpam-6004	89	11	of	of	ADP
ejpam-6004	89	12	u2−s(s−1)v2	u2−s(s−1)v2	NOUN
ejpam-6004	89	13	=	=	SYM
ejpam-6004	89	14	1	1	NUM
ejpam-6004	89	15	,	,	PUNCT
ejpam-6004	89	16	it	it	PRON
ejpam-6004	89	17	follows	follow	VERB
ejpam-6004	89	18	from	from	ADP
ejpam-6004	89	19	lemma	lemma	PROPN
ejpam-6004	89	20	4	4	NUM
ejpam-6004	89	21	that	that	PRON
ejpam-6004	89	22	4(s−	4(s−	PROPN
ejpam-6004	89	23	1	1	NUM
ejpam-6004	89	24	)	)	PUNCT
ejpam-6004	89	25	≤	≤	NOUN
ejpam-6004	89	26	2l+2	2l+2	NUM
ejpam-6004	89	27	.	.	PUNCT
ejpam-6004	90	1	thus	thus	ADV
ejpam-6004	90	2	s	s	VERB
ejpam-6004	90	3	≤	≤	X
ejpam-6004	90	4	2l	2l	NOUN
ejpam-6004	90	5	+	+	CCONJ
ejpam-6004	90	6	1	1	X
ejpam-6004	90	7	.	.	X
ejpam-6004	90	8	case	case	NOUN
ejpam-6004	90	9	2	2	NUM
ejpam-6004	90	10	.	.	X
ejpam-6004	91	1	gcd(u	gcd(u	PROPN
ejpam-6004	91	2	,	,	PUNCT
ejpam-6004	91	3	v	v	NOUN
ejpam-6004	91	4	)	)	PUNCT
ejpam-6004	91	5	̸=	̸=	PROPN
ejpam-6004	91	6	1	1	NUM
ejpam-6004	91	7	.	.	PUNCT
ejpam-6004	91	8	assume	assume	VERB
ejpam-6004	91	9	that	that	SCONJ
ejpam-6004	91	10	gcd(u	gcd(u	PROPN
ejpam-6004	91	11	,	,	PUNCT
ejpam-6004	91	12	v	v	NOUN
ejpam-6004	91	13	)	)	PUNCT
ejpam-6004	91	14	=	=	PUNCT
ejpam-6004	92	1	d.	d.	NOUN
ejpam-6004	92	2	thus	thus	ADV
ejpam-6004	92	3	d|u	d|u	NOUN
ejpam-6004	92	4	and	and	CCONJ
ejpam-6004	92	5	d|v	d|v	NOUN
ejpam-6004	92	6	.	.	PUNCT
ejpam-6004	93	1	then	then	ADV
ejpam-6004	93	2	there	there	PRON
ejpam-6004	93	3	exist	exist	VERB
ejpam-6004	93	4	integers	integer	NOUN
ejpam-6004	93	5	m	m	VERB
ejpam-6004	93	6	and	and	CCONJ
ejpam-6004	93	7	n	n	CCONJ
ejpam-6004	93	8	such	such	ADJ
ejpam-6004	93	9	that	that	DET
ejpam-6004	93	10	u	u	NOUN
ejpam-6004	93	11	=	=	X
ejpam-6004	93	12	dm	dm	PROPN
ejpam-6004	93	13	and	and	CCONJ
ejpam-6004	93	14	v	v	NOUN
ejpam-6004	93	15	=	=	PUNCT
ejpam-6004	93	16	dn	dn	ADP
ejpam-6004	93	17	where	where	SCONJ
ejpam-6004	93	18	gcd(m	gcd(m	NOUN
ejpam-6004	93	19	,	,	PUNCT
ejpam-6004	93	20	n	n	CCONJ
ejpam-6004	93	21	)	)	PUNCT
ejpam-6004	93	22	=	=	SYM
ejpam-6004	93	23	1	1	X
ejpam-6004	93	24	.	.	X
ejpam-6004	93	25	substituting	substitute	VERB
ejpam-6004	93	26	the	the	DET
ejpam-6004	93	27	values	value	NOUN
ejpam-6004	93	28	of	of	ADP
ejpam-6004	93	29	u	u	NOUN
ejpam-6004	93	30	and	and	CCONJ
ejpam-6004	93	31	v	v	NOUN
ejpam-6004	93	32	into	into	ADP
ejpam-6004	93	33	the	the	DET
ejpam-6004	93	34	equation	equation	NOUN
ejpam-6004	93	35	u2	u2	NOUN
ejpam-6004	93	36	−	−	PROPN
ejpam-6004	93	37	s(s−	s(s−	PROPN
ejpam-6004	93	38	1)v2	1)v2	PROPN
ejpam-6004	93	39	=	=	SYM
ejpam-6004	93	40	−2l	−2l	PROPN
ejpam-6004	93	41	,	,	PUNCT
ejpam-6004	93	42	we	we	PRON
ejpam-6004	93	43	have	have	VERB
ejpam-6004	93	44	(	(	PUNCT
ejpam-6004	93	45	dm)2	dm)2	ADJ
ejpam-6004	93	46	−	−	PROPN
ejpam-6004	93	47	s(s−	s(s−	NOUN
ejpam-6004	93	48	1)(dn)2	1)(dn)2	NUM
ejpam-6004	93	49	=	=	SYM
ejpam-6004	93	50	−2l	−2l	PROPN
ejpam-6004	93	51	d2(m2	d2(m2	PRON
ejpam-6004	93	52	−	−	NOUN
ejpam-6004	93	53	s(s−	s(s−	PROPN
ejpam-6004	93	54	1)n2	1)n2	NUM
ejpam-6004	93	55	)	)	PUNCT
ejpam-6004	93	56	=	=	SYM
ejpam-6004	93	57	−2l	−2l	PROPN
ejpam-6004	93	58	.	.	PUNCT
ejpam-6004	94	1	then	then	ADV
ejpam-6004	94	2	d2|2l	d2|2l	NOUN
ejpam-6004	94	3	.	.	PUNCT
ejpam-6004	95	1	we	we	PRON
ejpam-6004	95	2	see	see	VERB
ejpam-6004	95	3	that	that	DET
ejpam-6004	95	4	d	d	NOUN
ejpam-6004	95	5	=	=	SYM
ejpam-6004	95	6	2k	2k	NUM
ejpam-6004	95	7	for	for	ADP
ejpam-6004	95	8	some	some	PRON
ejpam-6004	95	9	0	0	NUM
ejpam-6004	95	10	<	<	X
ejpam-6004	95	11	k	k	PROPN
ejpam-6004	95	12	≤	≤	PROPN
ejpam-6004	95	13	l/2	l/2	NUM
ejpam-6004	95	14	.	.	PUNCT
ejpam-6004	96	1	then	then	ADV
ejpam-6004	96	2	m2	m2	PROPN
ejpam-6004	96	3	−	−	PROPN
ejpam-6004	96	4	s(s	s(s	PROPN
ejpam-6004	96	5	−	−	PROPN
ejpam-6004	97	1	1)n2	1)n2	NUM
ejpam-6004	97	2	=	=	PUNCT
ejpam-6004	97	3	−2l−2k	−2l−2k	NOUN
ejpam-6004	97	4	where	where	SCONJ
ejpam-6004	97	5	gcd(m	gcd(m	NOUN
ejpam-6004	97	6	,	,	PUNCT
ejpam-6004	97	7	n	n	CCONJ
ejpam-6004	97	8	)	)	PUNCT
ejpam-6004	97	9	=	=	SYM
ejpam-6004	97	10	1	1	X
ejpam-6004	97	11	.	.	PUNCT
ejpam-6004	97	12	by	by	ADP
ejpam-6004	97	13	lemma	lemma	PROPN
ejpam-6004	97	14	4	4	NUM
ejpam-6004	97	15	,	,	PUNCT
ejpam-6004	97	16	0	0	PUNCT
ejpam-6004	97	17	<	<	X
ejpam-6004	97	18	n	n	X
ejpam-6004	97	19	≤	≤	NOUN
ejpam-6004	97	20	√	√	CCONJ
ejpam-6004	97	21	2l−2k√	2l−2k√	NUM
ejpam-6004	97	22	s−1	s−1	PROPN
ejpam-6004	97	23	.	.	PUNCT
ejpam-6004	98	1	thus	thus	ADV
ejpam-6004	98	2	s	s	VERB
ejpam-6004	98	3	≤	≤	NUM
ejpam-6004	98	4	2l−2k	2l−2k	NUM
ejpam-6004	99	1	+	+	CCONJ
ejpam-6004	99	2	1	1	X
ejpam-6004	99	3	.	.	X
ejpam-6004	100	1	lemma	lemma	PROPN
ejpam-6004	100	2	8	8	NUM
ejpam-6004	100	3	.	.	PUNCT
ejpam-6004	101	1	let	let	VERB
ejpam-6004	101	2	k	k	NOUN
ejpam-6004	101	3	and	and	CCONJ
ejpam-6004	101	4	l	l	PROPN
ejpam-6004	101	5	≥	≥	NUM
ejpam-6004	101	6	2	2	NUM
ejpam-6004	101	7	be	be	AUX
ejpam-6004	101	8	positive	positive	ADJ
ejpam-6004	101	9	integers	integer	NOUN
ejpam-6004	101	10	.	.	PUNCT
ejpam-6004	102	1	if	if	SCONJ
ejpam-6004	102	2	x2	x2	PRON
ejpam-6004	102	3	−	−	PROPN
ejpam-6004	102	4	kxy	kxy	NOUN
ejpam-6004	102	5	+	+	CCONJ
ejpam-6004	102	6	ky2	ky2	NOUN
ejpam-6004	102	7	+	+	CCONJ
ejpam-6004	102	8	2l	2l	NUM
ejpam-6004	102	9	=	=	SYM
ejpam-6004	102	10	0	0	NUM
ejpam-6004	102	11	,	,	PUNCT
ejpam-6004	102	12	where	where	SCONJ
ejpam-6004	102	13	gcd(x	gcd(x	PROPN
ejpam-6004	102	14	,	,	PUNCT
ejpam-6004	102	15	y	y	NOUN
ejpam-6004	102	16	)	)	PUNCT
ejpam-6004	102	17	=	=	SYM
ejpam-6004	102	18	1	1	NUM
ejpam-6004	102	19	,	,	PUNCT
ejpam-6004	102	20	then	then	ADV
ejpam-6004	102	21	x	x	SYM
ejpam-6004	102	22	≡	≡	PROPN
ejpam-6004	102	23	0	0	PUNCT
ejpam-6004	103	1	(	(	PUNCT
ejpam-6004	103	2	mod	mod	NOUN
ejpam-6004	103	3	2	2	NUM
ejpam-6004	103	4	)	)	PUNCT
ejpam-6004	103	5	,	,	PUNCT
ejpam-6004	103	6	y	y	PROPN
ejpam-6004	103	7	≡	≡	PROPN
ejpam-6004	103	8	1	1	NUM
ejpam-6004	103	9	(	(	PUNCT
ejpam-6004	103	10	mod	mod	NOUN
ejpam-6004	103	11	2	2	NUM
ejpam-6004	103	12	)	)	PUNCT
ejpam-6004	103	13	,	,	PUNCT
ejpam-6004	103	14	and	and	CCONJ
ejpam-6004	103	15	k	k	PROPN
ejpam-6004	103	16	≡	≡	PROPN
ejpam-6004	103	17	0	0	PUNCT
ejpam-6004	103	18	(	(	PUNCT
ejpam-6004	103	19	mod	mod	PROPN
ejpam-6004	103	20	4	4	NUM
ejpam-6004	103	21	)	)	PUNCT
ejpam-6004	103	22	.	.	PUNCT
ejpam-6004	104	1	proof	proof	NOUN
ejpam-6004	104	2	.	.	PUNCT
ejpam-6004	105	1	suppose	suppose	VERB
ejpam-6004	105	2	x	x	PRON
ejpam-6004	105	3	and	and	CCONJ
ejpam-6004	105	4	y	y	PROPN
ejpam-6004	105	5	satisfy	satisfy	VERB
ejpam-6004	105	6	the	the	DET
ejpam-6004	105	7	equation	equation	NOUN
ejpam-6004	105	8	x2−kxy+ky2	x2−kxy+ky2	PUNCT
ejpam-6004	106	1	+	+	ADJ
ejpam-6004	106	2	2l	2l	X
ejpam-6004	106	3	=	=	SYM
ejpam-6004	106	4	0	0	NUM
ejpam-6004	106	5	,	,	PUNCT
ejpam-6004	106	6	where	where	SCONJ
ejpam-6004	106	7	gcd(x	gcd(x	PROPN
ejpam-6004	106	8	,	,	PUNCT
ejpam-6004	106	9	y	y	NOUN
ejpam-6004	106	10	)	)	PUNCT
ejpam-6004	106	11	=	=	SYM
ejpam-6004	107	1	1	1	X
ejpam-6004	107	2	.	.	PUNCT
ejpam-6004	108	1	this	this	PRON
ejpam-6004	108	2	implies	imply	VERB
ejpam-6004	108	3	that	that	SCONJ
ejpam-6004	108	4	y	y	PROPN
ejpam-6004	108	5	is	be	AUX
ejpam-6004	108	6	odd	odd	ADJ
ejpam-6004	108	7	.	.	PUNCT
ejpam-6004	109	1	otherwise	otherwise	ADV
ejpam-6004	109	2	x	x	X
ejpam-6004	109	3	and	and	CCONJ
ejpam-6004	109	4	y	y	PROPN
ejpam-6004	109	5	are	be	AUX
ejpam-6004	109	6	both	both	PRON
ejpam-6004	109	7	even	even	ADV
ejpam-6004	109	8	,	,	PUNCT
ejpam-6004	109	9	which	which	PRON
ejpam-6004	109	10	is	be	AUX
ejpam-6004	109	11	a	a	DET
ejpam-6004	109	12	contradiction	contradiction	NOUN
ejpam-6004	109	13	.	.	PUNCT
ejpam-6004	110	1	hence	hence	ADV
ejpam-6004	110	2	x2−kx+k	x2−kx+k	PROPN
ejpam-6004	110	3	is	be	AUX
ejpam-6004	110	4	even	even	ADV
ejpam-6004	110	5	.	.	PUNCT
ejpam-6004	111	1	it	it	PRON
ejpam-6004	111	2	is	be	AUX
ejpam-6004	111	3	easy	easy	ADJ
ejpam-6004	111	4	to	to	PART
ejpam-6004	111	5	see	see	VERB
ejpam-6004	111	6	that	that	PRON
ejpam-6004	111	7	x	x	PROPN
ejpam-6004	111	8	and	and	CCONJ
ejpam-6004	111	9	k	k	PROPN
ejpam-6004	111	10	are	be	AUX
ejpam-6004	111	11	both	both	PRON
ejpam-6004	111	12	even	even	ADV
ejpam-6004	111	13	.	.	PUNCT
ejpam-6004	112	1	since	since	SCONJ
ejpam-6004	112	2	x2−kxy	x2−kxy	PROPN
ejpam-6004	112	3	≡	≡	PROPN
ejpam-6004	112	4	0	0	PUNCT
ejpam-6004	113	1	(	(	PUNCT
ejpam-6004	113	2	mod	mod	NOUN
ejpam-6004	113	3	4	4	NUM
ejpam-6004	113	4	)	)	PUNCT
ejpam-6004	113	5	and	and	CCONJ
ejpam-6004	113	6	x2	x2	NUM
ejpam-6004	113	7	−	−	PROPN
ejpam-6004	113	8	kxy	kxy	NOUN
ejpam-6004	113	9	+	+	CCONJ
ejpam-6004	113	10	ky2	ky2	PROPN
ejpam-6004	113	11	≡	≡	PROPN
ejpam-6004	113	12	0	0	PUNCT
ejpam-6004	114	1	(	(	PUNCT
ejpam-6004	114	2	mod	mod	PROPN
ejpam-6004	114	3	4	4	NUM
ejpam-6004	114	4	)	)	PUNCT
ejpam-6004	114	5	,	,	PUNCT
ejpam-6004	114	6	we	we	PRON
ejpam-6004	114	7	have	have	VERB
ejpam-6004	114	8	k	k	PROPN
ejpam-6004	114	9	≡	≡	PROPN
ejpam-6004	114	10	0	0	PUNCT
ejpam-6004	115	1	(	(	PUNCT
ejpam-6004	115	2	mod	mod	PROPN
ejpam-6004	115	3	4	4	NUM
ejpam-6004	115	4	)	)	PUNCT
ejpam-6004	115	5	.	.	PUNCT
ejpam-6004	116	1	the	the	DET
ejpam-6004	116	2	diophantine	diophantine	NOUN
ejpam-6004	116	3	equations	equation	NOUN
ejpam-6004	116	4	x2−kxy+ky2	x2−kxy+ky2	PUNCT
ejpam-6004	117	1	+	+	ADJ
ejpam-6004	117	2	2l	2l	X
ejpam-6004	117	3	=	=	SYM
ejpam-6004	117	4	0	0	NUM
ejpam-6004	117	5	and	and	CCONJ
ejpam-6004	117	6	u2−s(s−1)v2	u2−s(s−1)v2	PROPN
ejpam-6004	117	7	=	=	SYM
ejpam-6004	117	8	−2l−2	−2l−2	PROPN
ejpam-6004	117	9	,	,	PUNCT
ejpam-6004	117	10	where	where	SCONJ
ejpam-6004	117	11	k	k	PROPN
ejpam-6004	117	12	=	=	SYM
ejpam-6004	117	13	4s	4s	NUM
ejpam-6004	117	14	,	,	PUNCT
ejpam-6004	117	15	are	be	AUX
ejpam-6004	117	16	now	now	ADV
ejpam-6004	117	17	related	relate	VERB
ejpam-6004	117	18	.	.	PUNCT
ejpam-6004	118	1	there	there	PRON
ejpam-6004	118	2	are	be	VERB
ejpam-6004	118	3	several	several	ADJ
ejpam-6004	118	4	details	detail	NOUN
ejpam-6004	118	5	on	on	ADP
ejpam-6004	118	6	both	both	DET
ejpam-6004	118	7	equations	equation	NOUN
ejpam-6004	118	8	given	give	VERB
ejpam-6004	118	9	.	.	PUNCT
ejpam-6004	119	1	lemma	lemma	PROPN
ejpam-6004	119	2	9	9	NUM
ejpam-6004	119	3	.	.	PUNCT
ejpam-6004	120	1	for	for	ADP
ejpam-6004	120	2	any	any	DET
ejpam-6004	120	3	positive	positive	ADJ
ejpam-6004	120	4	integers	integer	NOUN
ejpam-6004	120	5	k	k	NOUN
ejpam-6004	120	6	and	and	CCONJ
ejpam-6004	120	7	l	l	PROPN
ejpam-6004	120	8	≥	≥	NOUN
ejpam-6004	120	9	3	3	NUM
ejpam-6004	120	10	,	,	PUNCT
ejpam-6004	120	11	the	the	DET
ejpam-6004	120	12	diophantine	diophantine	NOUN
ejpam-6004	120	13	equation	equation	NOUN
ejpam-6004	120	14	x2	x2	NOUN
ejpam-6004	121	1	−	−	PROPN
ejpam-6004	121	2	kxy	kxy	NOUN
ejpam-6004	121	3	+	+	CCONJ
ejpam-6004	121	4	ky2	ky2	NOUN
ejpam-6004	121	5	+	+	CCONJ
ejpam-6004	121	6	2l	2l	X
ejpam-6004	121	7	=	=	SYM
ejpam-6004	121	8	0	0	NUM
ejpam-6004	121	9	has	have	VERB
ejpam-6004	121	10	a	a	DET
ejpam-6004	121	11	solution	solution	NOUN
ejpam-6004	121	12	(	(	PUNCT
ejpam-6004	121	13	x	x	X
ejpam-6004	121	14	,	,	PUNCT
ejpam-6004	121	15	y	y	PROPN
ejpam-6004	121	16	)	)	PUNCT
ejpam-6004	121	17	where	where	SCONJ
ejpam-6004	121	18	k	k	NOUN
ejpam-6004	122	1	=	=	PUNCT
ejpam-6004	122	2	4s	4s	NUM
ejpam-6004	122	3	and	and	CCONJ
ejpam-6004	122	4	gcd(x	gcd(x	PROPN
ejpam-6004	122	5	,	,	PUNCT
ejpam-6004	122	6	y	y	NOUN
ejpam-6004	122	7	)	)	PUNCT
ejpam-6004	122	8	=	=	SYM
ejpam-6004	122	9	1	1	NUM
ejpam-6004	123	1	if	if	SCONJ
ejpam-6004	123	2	and	and	CCONJ
ejpam-6004	123	3	only	only	ADV
ejpam-6004	123	4	if	if	SCONJ
ejpam-6004	123	5	u2	u2	PROPN
ejpam-6004	123	6	−	−	PROPN
ejpam-6004	123	7	s(s−	s(s−	PROPN
ejpam-6004	123	8	1)v2	1)v2	PROPN
ejpam-6004	123	9	=	=	SYM
ejpam-6004	123	10	−2l−2	−2l−2	PROPN
ejpam-6004	123	11	has	have	VERB
ejpam-6004	123	12	a	a	DET
ejpam-6004	123	13	solution	solution	NOUN
ejpam-6004	123	14	(	(	PUNCT
ejpam-6004	123	15	u	u	NOUN
ejpam-6004	123	16	,	,	PUNCT
ejpam-6004	123	17	v	v	NOUN
ejpam-6004	123	18	)	)	PUNCT
ejpam-6004	123	19	where	where	SCONJ
ejpam-6004	123	20	gcd(u	gcd(u	PROPN
ejpam-6004	123	21	,	,	PUNCT
ejpam-6004	123	22	v	v	NOUN
ejpam-6004	123	23	)	)	PUNCT
ejpam-6004	123	24	=	=	SYM
ejpam-6004	124	1	1	1	X
ejpam-6004	124	2	.	.	PUNCT
ejpam-6004	124	3	moreover	moreover	ADV
ejpam-6004	124	4	,	,	PUNCT
ejpam-6004	124	5	both	both	DET
ejpam-6004	124	6	equations	equation	NOUN
ejpam-6004	124	7	have	have	VERB
ejpam-6004	124	8	infinitely	infinitely	ADV
ejpam-6004	124	9	many	many	ADJ
ejpam-6004	124	10	solutions	solution	NOUN
ejpam-6004	124	11	.	.	PUNCT
ejpam-6004	125	1	s.	s.	PROPN
ejpam-6004	125	2	prugsapitak	prugsapitak	PROPN
ejpam-6004	125	3	,	,	PUNCT
ejpam-6004	125	4	n.	n.	PROPN
ejpam-6004	125	5	thongngam	thongngam	PROPN
ejpam-6004	125	6	/	/	SYM
ejpam-6004	125	7	eur	eur	NOUN
ejpam-6004	125	8	.	.	PUNCT
ejpam-6004	126	1	j.	j.	PROPN
ejpam-6004	126	2	pure	pure	PROPN
ejpam-6004	126	3	appl	appl	PROPN
ejpam-6004	126	4	.	.	PROPN
ejpam-6004	126	5	math	math	PROPN
ejpam-6004	126	6	,	,	PUNCT
ejpam-6004	126	7	18	18	NUM
ejpam-6004	126	8	(	(	PUNCT
ejpam-6004	126	9	2	2	NUM
ejpam-6004	126	10	)	)	PUNCT
ejpam-6004	126	11	(	(	PUNCT
ejpam-6004	126	12	2025	2025	NUM
ejpam-6004	126	13	)	)	PUNCT
ejpam-6004	126	14	,	,	PUNCT
ejpam-6004	126	15	6004	6004	NUM
ejpam-6004	126	16	5	5	NUM
ejpam-6004	126	17	of	of	ADP
ejpam-6004	126	18	8	8	NUM
ejpam-6004	126	19	proof	proof	NOUN
ejpam-6004	126	20	.	.	PUNCT
ejpam-6004	127	1	let	let	VERB
ejpam-6004	127	2	l	l	NOUN
ejpam-6004	127	3	≥	≥	NUM
ejpam-6004	127	4	3	3	NUM
ejpam-6004	127	5	be	be	AUX
ejpam-6004	127	6	a	a	DET
ejpam-6004	127	7	positive	positive	ADJ
ejpam-6004	127	8	integer	integer	NOUN
ejpam-6004	127	9	.	.	PUNCT
ejpam-6004	128	1	suppose	suppose	VERB
ejpam-6004	128	2	x2	x2	PRON
ejpam-6004	128	3	−	−	PROPN
ejpam-6004	128	4	kxy	kxy	NOUN
ejpam-6004	128	5	+	+	CCONJ
ejpam-6004	128	6	ky2	ky2	NOUN
ejpam-6004	128	7	+	+	CCONJ
ejpam-6004	128	8	2l	2l	X
ejpam-6004	128	9	=	=	SYM
ejpam-6004	128	10	0	0	NUM
ejpam-6004	128	11	has	have	VERB
ejpam-6004	128	12	a	a	DET
ejpam-6004	128	13	solution	solution	NOUN
ejpam-6004	128	14	(	(	PUNCT
ejpam-6004	128	15	x	x	X
ejpam-6004	128	16	,	,	PUNCT
ejpam-6004	128	17	y	y	PROPN
ejpam-6004	128	18	)	)	PUNCT
ejpam-6004	128	19	where	where	SCONJ
ejpam-6004	128	20	gcd(x	gcd(x	PROPN
ejpam-6004	128	21	,	,	PUNCT
ejpam-6004	128	22	y	y	NOUN
ejpam-6004	128	23	)	)	PUNCT
ejpam-6004	128	24	=	=	SYM
ejpam-6004	129	1	1	1	X
ejpam-6004	129	2	.	.	PUNCT
ejpam-6004	129	3	by	by	ADP
ejpam-6004	129	4	lemma	lemma	PROPN
ejpam-6004	129	5	8	8	NUM
ejpam-6004	129	6	,	,	PUNCT
ejpam-6004	129	7	we	we	PRON
ejpam-6004	129	8	have	have	VERB
ejpam-6004	129	9	x	x	VERB
ejpam-6004	129	10	is	be	AUX
ejpam-6004	129	11	even	even	ADV
ejpam-6004	129	12	and	and	CCONJ
ejpam-6004	129	13	k	k	PROPN
ejpam-6004	130	1	=	=	SYM
ejpam-6004	130	2	4s	4s	NUM
ejpam-6004	130	3	for	for	ADP
ejpam-6004	130	4	some	some	DET
ejpam-6004	130	5	positive	positive	ADJ
ejpam-6004	130	6	integer	integer	NOUN
ejpam-6004	130	7	s.	s.	PROPN
ejpam-6004	130	8	now	now	ADV
ejpam-6004	130	9	,	,	PUNCT
ejpam-6004	130	10	let	let	VERB
ejpam-6004	130	11	x	x	PUNCT
ejpam-6004	130	12	=	=	SYM
ejpam-6004	130	13	2x′	2x′	NUM
ejpam-6004	130	14	for	for	ADP
ejpam-6004	130	15	some	some	DET
ejpam-6004	130	16	positive	positive	ADJ
ejpam-6004	130	17	integer	integer	NOUN
ejpam-6004	130	18	x′.	x′.	PROPN
ejpam-6004	130	19	thus	thus	ADV
ejpam-6004	130	20	(	(	PUNCT
ejpam-6004	130	21	2x′)2	2x′)2	NUM
ejpam-6004	130	22	−	−	X
ejpam-6004	130	23	k(2x′)y	k(2x′)y	NOUN
ejpam-6004	130	24	+	+	NUM
ejpam-6004	130	25	ky2	ky2	NOUN
ejpam-6004	130	26	+	+	CCONJ
ejpam-6004	130	27	2l	2l	NUM
ejpam-6004	130	28	=	=	SYM
ejpam-6004	130	29	0	0	X
ejpam-6004	130	30	.	.	PUNCT
ejpam-6004	131	1	we	we	PRON
ejpam-6004	131	2	now	now	ADV
ejpam-6004	131	3	have	have	VERB
ejpam-6004	131	4	x′2	x′2	VERB
ejpam-6004	131	5	−	−	PROPN
ejpam-6004	131	6	2sx′y	2sx′y	PROPN
ejpam-6004	132	1	+	+	CCONJ
ejpam-6004	132	2	sy2	sy2	ADJ
ejpam-6004	132	3	+	+	CCONJ
ejpam-6004	132	4	2l−2	2l−2	NUM
ejpam-6004	132	5	=	=	SYM
ejpam-6004	132	6	0	0	PUNCT
ejpam-6004	132	7	(	(	PUNCT
ejpam-6004	132	8	x′	x′	PROPN
ejpam-6004	132	9	−	−	PROPN
ejpam-6004	132	10	sy)2	sy)2	PROPN
ejpam-6004	132	11	−	−	PROPN
ejpam-6004	132	12	s(s−	s(s−	PROPN
ejpam-6004	132	13	1)y2	1)y2	NUM
ejpam-6004	132	14	=	=	SYM
ejpam-6004	132	15	−2l−2	−2l−2	PROPN
ejpam-6004	132	16	.	.	PUNCT
ejpam-6004	133	1	let	let	VERB
ejpam-6004	133	2	u	u	PRON
ejpam-6004	133	3	=	=	PUNCT
ejpam-6004	133	4	x′	x′	PROPN
ejpam-6004	134	1	−	−	PROPN
ejpam-6004	134	2	sy	sy	PROPN
ejpam-6004	134	3	and	and	CCONJ
ejpam-6004	134	4	v	v	NOUN
ejpam-6004	134	5	=	=	SYM
ejpam-6004	134	6	y.	y.	NOUN
ejpam-6004	134	7	then	then	ADV
ejpam-6004	134	8	u2	u2	PROPN
ejpam-6004	134	9	−	−	PROPN
ejpam-6004	134	10	s(s	s(s	PROPN
ejpam-6004	134	11	−	−	PROPN
ejpam-6004	134	12	1)v2	1)v2	PROPN
ejpam-6004	134	13	=	=	SYM
ejpam-6004	134	14	−2l−2	−2l−2	PROPN
ejpam-6004	134	15	.	.	PUNCT
ejpam-6004	135	1	since	since	SCONJ
ejpam-6004	135	2	gcd(x	gcd(x	PROPN
ejpam-6004	135	3	,	,	PUNCT
ejpam-6004	135	4	y	y	NOUN
ejpam-6004	135	5	)	)	PUNCT
ejpam-6004	135	6	=	=	SYM
ejpam-6004	135	7	1	1	NUM
ejpam-6004	135	8	,	,	PUNCT
ejpam-6004	135	9	we	we	PRON
ejpam-6004	135	10	have	have	VERB
ejpam-6004	135	11	gcd(u	gcd(u	PROPN
ejpam-6004	135	12	,	,	PUNCT
ejpam-6004	135	13	v	v	NOUN
ejpam-6004	135	14	)	)	PUNCT
ejpam-6004	135	15	=	=	SYM
ejpam-6004	136	1	gcd(x′	gcd(x′	PROPN
ejpam-6004	136	2	,	,	PUNCT
ejpam-6004	136	3	y	y	NOUN
ejpam-6004	136	4	)	)	PUNCT
ejpam-6004	136	5	=	=	SYM
ejpam-6004	137	1	1	1	X
ejpam-6004	137	2	.	.	PUNCT
ejpam-6004	137	3	now	now	ADV
ejpam-6004	137	4	for	for	ADP
ejpam-6004	137	5	the	the	DET
ejpam-6004	137	6	converse	converse	NOUN
ejpam-6004	137	7	,	,	PUNCT
ejpam-6004	137	8	suppose	suppose	VERB
ejpam-6004	137	9	u2	u2	PROPN
ejpam-6004	137	10	−	−	PROPN
ejpam-6004	137	11	s(s−	s(s−	PROPN
ejpam-6004	137	12	1)v2	1)v2	PROPN
ejpam-6004	137	13	=	=	SYM
ejpam-6004	137	14	−2l−2	−2l−2	PROPN
ejpam-6004	137	15	where	where	SCONJ
ejpam-6004	137	16	gcd(u	gcd(u	PROPN
ejpam-6004	137	17	,	,	PUNCT
ejpam-6004	137	18	v	v	NOUN
ejpam-6004	137	19	)	)	PUNCT
ejpam-6004	137	20	=	=	SYM
ejpam-6004	138	1	1	1	X
ejpam-6004	138	2	.	.	X
ejpam-6004	138	3	we	we	PRON
ejpam-6004	138	4	can	can	AUX
ejpam-6004	138	5	see	see	VERB
ejpam-6004	138	6	that	that	SCONJ
ejpam-6004	138	7	u	u	PRON
ejpam-6004	138	8	must	must	AUX
ejpam-6004	138	9	be	be	AUX
ejpam-6004	138	10	even	even	ADV
ejpam-6004	138	11	.	.	PUNCT
ejpam-6004	139	1	now	now	ADV
ejpam-6004	139	2	let	let	VERB
ejpam-6004	139	3	x	x	X
ejpam-6004	139	4	=	=	PUNCT
ejpam-6004	139	5	2x′	2x′	NUM
ejpam-6004	139	6	where	where	SCONJ
ejpam-6004	139	7	x′	x′	PROPN
ejpam-6004	139	8	=	=	X
ejpam-6004	139	9	u+	u+	NUM
ejpam-6004	139	10	sv	sv	NOUN
ejpam-6004	139	11	and	and	CCONJ
ejpam-6004	139	12	y	y	PROPN
ejpam-6004	139	13	=	=	PROPN
ejpam-6004	140	1	v.	v.	CCONJ
ejpam-6004	140	2	then	then	ADV
ejpam-6004	140	3	(	(	PUNCT
ejpam-6004	140	4	x′	x′	PROPN
ejpam-6004	140	5	−	−	PROPN
ejpam-6004	140	6	sy)2	sy)2	PROPN
ejpam-6004	141	1	−	−	PROPN
ejpam-6004	141	2	s(s−	s(s−	PROPN
ejpam-6004	141	3	1)y2	1)y2	NUM
ejpam-6004	141	4	=	=	SYM
ejpam-6004	141	5	−2l−2	−2l−2	PROPN
ejpam-6004	141	6	.	.	PUNCT
ejpam-6004	142	1	x′2	x′2	NOUN
ejpam-6004	143	1	−	−	PROPN
ejpam-6004	143	2	2sx′y	2sx′y	PROPN
ejpam-6004	144	1	+	+	CCONJ
ejpam-6004	145	1	sy2	sy2	ADJ
ejpam-6004	145	2	+	+	CCONJ
ejpam-6004	145	3	2l−2	2l−2	NUM
ejpam-6004	145	4	=	=	SYM
ejpam-6004	145	5	0	0	NUM
ejpam-6004	145	6	4x′2	4x′2	NUM
ejpam-6004	146	1	−	−	NOUN
ejpam-6004	146	2	8sx′y	8sx′y	NOUN
ejpam-6004	147	1	+	+	CCONJ
ejpam-6004	147	2	4sy2	4sy2	NUM
ejpam-6004	148	1	+	+	CCONJ
ejpam-6004	148	2	2l	2l	NUM
ejpam-6004	148	3	=	=	SYM
ejpam-6004	148	4	0	0	NUM
ejpam-6004	148	5	x2	x2	NOUN
ejpam-6004	148	6	−	−	PROPN
ejpam-6004	148	7	kxy	kxy	NOUN
ejpam-6004	148	8	+	+	CCONJ
ejpam-6004	148	9	ky2	ky2	NOUN
ejpam-6004	148	10	+	+	CCONJ
ejpam-6004	148	11	2l	2l	X
ejpam-6004	148	12	=	=	SYM
ejpam-6004	148	13	0	0	X
ejpam-6004	148	14	.	.	PUNCT
ejpam-6004	149	1	since	since	SCONJ
ejpam-6004	149	2	u	u	NOUN
ejpam-6004	149	3	is	be	AUX
ejpam-6004	149	4	even	even	ADV
ejpam-6004	149	5	and	and	CCONJ
ejpam-6004	149	6	v	v	NOUN
ejpam-6004	149	7	is	be	AUX
ejpam-6004	149	8	odd	odd	ADJ
ejpam-6004	149	9	,	,	PUNCT
ejpam-6004	149	10	we	we	PRON
ejpam-6004	149	11	have	have	VERB
ejpam-6004	149	12	gcd(x	gcd(x	PROPN
ejpam-6004	149	13	,	,	PUNCT
ejpam-6004	149	14	y	y	NOUN
ejpam-6004	149	15	)	)	PUNCT
ejpam-6004	149	16	=	=	SYM
ejpam-6004	150	1	gcd(2x′	gcd(2x′	PROPN
ejpam-6004	150	2	,	,	PUNCT
ejpam-6004	150	3	y	y	NOUN
ejpam-6004	150	4	)	)	PUNCT
ejpam-6004	150	5	=	=	VERB
ejpam-6004	151	1	gcd(2u	gcd(2u	NOUN
ejpam-6004	151	2	+	+	CCONJ
ejpam-6004	151	3	2sv	2sv	ADJ
ejpam-6004	151	4	,	,	PUNCT
ejpam-6004	151	5	v	v	NOUN
ejpam-6004	151	6	)	)	PUNCT
ejpam-6004	151	7	=	=	NOUN
ejpam-6004	151	8	gcd(2u	gcd(2u	NOUN
ejpam-6004	151	9	,	,	PUNCT
ejpam-6004	151	10	v	v	NOUN
ejpam-6004	151	11	)	)	PUNCT
ejpam-6004	151	12	=	=	SYM
ejpam-6004	152	1	1	1	X
ejpam-6004	152	2	.	.	PUNCT
ejpam-6004	152	3	lemma	lemma	PROPN
ejpam-6004	152	4	10	10	NUM
ejpam-6004	152	5	.	.	PUNCT
ejpam-6004	153	1	for	for	ADP
ejpam-6004	153	2	any	any	DET
ejpam-6004	153	3	non	non	ADJ
ejpam-6004	153	4	-	-	ADJ
ejpam-6004	153	5	negative	negative	ADJ
ejpam-6004	153	6	integer	integer	NOUN
ejpam-6004	153	7	l	l	NOUN
ejpam-6004	153	8	,	,	PUNCT
ejpam-6004	153	9	the	the	DET
ejpam-6004	153	10	diophantine	diophantine	NOUN
ejpam-6004	153	11	equation	equation	NOUN
ejpam-6004	153	12	x2−4xy+4y2	x2−4xy+4y2	ADP
ejpam-6004	153	13	+	+	ADJ
ejpam-6004	153	14	2l	2l	X
ejpam-6004	153	15	=	=	SYM
ejpam-6004	153	16	0	0	NUM
ejpam-6004	153	17	has	have	VERB
ejpam-6004	153	18	no	no	DET
ejpam-6004	153	19	solution	solution	NOUN
ejpam-6004	153	20	.	.	PUNCT
ejpam-6004	154	1	proof	proof	NOUN
ejpam-6004	154	2	.	.	PUNCT
ejpam-6004	155	1	suppose	suppose	VERB
ejpam-6004	155	2	that	that	SCONJ
ejpam-6004	155	3	x2	x2	PRON
ejpam-6004	155	4	−	−	PROPN
ejpam-6004	155	5	4xy	4xy	NOUN
ejpam-6004	156	1	+	+	CCONJ
ejpam-6004	156	2	4y2	4y2	NUM
ejpam-6004	156	3	+	+	CCONJ
ejpam-6004	156	4	2l	2l	NUM
ejpam-6004	156	5	=	=	SYM
ejpam-6004	156	6	0	0	X
ejpam-6004	156	7	.	.	PUNCT
ejpam-6004	157	1	we	we	PRON
ejpam-6004	157	2	can	can	AUX
ejpam-6004	157	3	see	see	VERB
ejpam-6004	157	4	that	that	PRON
ejpam-6004	157	5	(	(	PUNCT
ejpam-6004	157	6	x−	x−	PROPN
ejpam-6004	157	7	2y)2	2y)2	NUM
ejpam-6004	157	8	=	=	SYM
ejpam-6004	157	9	−2l	−2l	PROPN
ejpam-6004	157	10	,	,	PUNCT
ejpam-6004	157	11	which	which	PRON
ejpam-6004	157	12	is	be	AUX
ejpam-6004	157	13	impossible	impossible	ADJ
ejpam-6004	157	14	.	.	PUNCT
ejpam-6004	158	1	hence	hence	ADV
ejpam-6004	158	2	x2	x2	INTJ
ejpam-6004	159	1	−	−	PROPN
ejpam-6004	159	2	4xy	4xy	NOUN
ejpam-6004	160	1	+	+	CCONJ
ejpam-6004	160	2	4y2	4y2	NUM
ejpam-6004	160	3	+	+	CCONJ
ejpam-6004	160	4	2l	2l	NUM
ejpam-6004	160	5	=	=	SYM
ejpam-6004	160	6	0	0	NUM
ejpam-6004	160	7	has	have	VERB
ejpam-6004	160	8	no	no	DET
ejpam-6004	160	9	solution	solution	NOUN
ejpam-6004	160	10	.	.	PUNCT
ejpam-6004	161	1	lemma	lemma	PROPN
ejpam-6004	161	2	11	11	NUM
ejpam-6004	161	3	.	.	PUNCT
ejpam-6004	162	1	for	for	ADP
ejpam-6004	162	2	positive	positive	ADJ
ejpam-6004	162	3	integers	integer	NOUN
ejpam-6004	162	4	l	l	NOUN
ejpam-6004	162	5	and	and	CCONJ
ejpam-6004	162	6	k	k	NOUN
ejpam-6004	162	7	,	,	PUNCT
ejpam-6004	162	8	if	if	SCONJ
ejpam-6004	162	9	the	the	DET
ejpam-6004	162	10	diophantine	diophantine	NOUN
ejpam-6004	162	11	equation	equation	NOUN
ejpam-6004	162	12	x2−kxy+ky2+l	x2−kxy+ky2+l	PUNCT
ejpam-6004	162	13	=	=	SYM
ejpam-6004	162	14	0	0	NUM
ejpam-6004	162	15	is	be	AUX
ejpam-6004	162	16	solvable	solvable	ADJ
ejpam-6004	162	17	,	,	PUNCT
ejpam-6004	162	18	then	then	ADV
ejpam-6004	162	19	k	k	X
ejpam-6004	162	20	>	>	X
ejpam-6004	162	21	4	4	X
ejpam-6004	162	22	.	.	PUNCT
ejpam-6004	163	1	proof	proof	NOUN
ejpam-6004	163	2	.	.	PUNCT
ejpam-6004	164	1	suppose	suppose	VERB
ejpam-6004	164	2	x2	x2	PRON
ejpam-6004	164	3	−	−	PROPN
ejpam-6004	164	4	kxy	kxy	NOUN
ejpam-6004	164	5	+	+	CCONJ
ejpam-6004	164	6	ky2	ky2	NOUN
ejpam-6004	164	7	+	+	CCONJ
ejpam-6004	164	8	l	l	NOUN
ejpam-6004	164	9	=	=	SYM
ejpam-6004	165	1	0	0	X
ejpam-6004	165	2	.	.	PUNCT
ejpam-6004	166	1	then	then	ADV
ejpam-6004	166	2	(	(	PUNCT
ejpam-6004	166	3	2x	2x	NUM
ejpam-6004	166	4	−	−	VERB
ejpam-6004	166	5	ky)2	ky)2	NOUN
ejpam-6004	166	6	+	+	CCONJ
ejpam-6004	166	7	y2(4k	y2(4k	ADJ
ejpam-6004	166	8	−	−	PROPN
ejpam-6004	166	9	k2	k2	NOUN
ejpam-6004	166	10	)	)	PUNCT
ejpam-6004	166	11	=	=	SYM
ejpam-6004	167	1	−4l	−4l	PROPN
ejpam-6004	167	2	.	.	PUNCT
ejpam-6004	168	1	this	this	PRON
ejpam-6004	168	2	implies	imply	VERB
ejpam-6004	168	3	that	that	SCONJ
ejpam-6004	168	4	4k	4k	PRON
ejpam-6004	168	5	−	−	PROPN
ejpam-6004	168	6	k2	k2	X
ejpam-6004	168	7	<	<	X
ejpam-6004	168	8	0	0	NUM
ejpam-6004	168	9	.	.	PUNCT
ejpam-6004	169	1	since	since	SCONJ
ejpam-6004	169	2	k	k	PROPN
ejpam-6004	169	3	>	>	X
ejpam-6004	169	4	0	0	PROPN
ejpam-6004	169	5	,	,	PUNCT
ejpam-6004	169	6	we	we	PRON
ejpam-6004	169	7	have	have	VERB
ejpam-6004	169	8	k	k	X
ejpam-6004	169	9	>	>	X
ejpam-6004	169	10	4	4	X
ejpam-6004	169	11	.	.	PUNCT
ejpam-6004	170	1	lemma	lemma	PROPN
ejpam-6004	170	2	12	12	NUM
ejpam-6004	170	3	.	.	PUNCT
ejpam-6004	171	1	let	let	VERB
ejpam-6004	171	2	s	s	PRON
ejpam-6004	171	3	and	and	CCONJ
ejpam-6004	171	4	l	l	NOUN
ejpam-6004	171	5	be	be	AUX
ejpam-6004	171	6	positive	positive	ADJ
ejpam-6004	171	7	integers	integer	NOUN
ejpam-6004	171	8	and	and	CCONJ
ejpam-6004	171	9	p	p	NOUN
ejpam-6004	171	10	be	be	AUX
ejpam-6004	171	11	an	an	DET
ejpam-6004	171	12	odd	odd	ADJ
ejpam-6004	171	13	prime	prime	NOUN
ejpam-6004	171	14	.	.	PUNCT
ejpam-6004	172	1	if	if	SCONJ
ejpam-6004	172	2	p	p	PROPN
ejpam-6004	172	3	|	|	ADP
ejpam-6004	172	4	s(s	s(s	PROPN
ejpam-6004	172	5	−	−	PROPN
ejpam-6004	172	6	1	1	NUM
ejpam-6004	172	7	)	)	PUNCT
ejpam-6004	172	8	and	and	CCONJ
ejpam-6004	172	9	either	either	DET
ejpam-6004	172	10	one	one	NUM
ejpam-6004	172	11	of	of	ADP
ejpam-6004	172	12	the	the	DET
ejpam-6004	172	13	following	follow	VERB
ejpam-6004	172	14	holds	hold	NOUN
ejpam-6004	172	15	:	:	PUNCT
ejpam-6004	172	16	s.	s.	PROPN
ejpam-6004	172	17	prugsapitak	prugsapitak	PROPN
ejpam-6004	172	18	,	,	PUNCT
ejpam-6004	172	19	n.	n.	PROPN
ejpam-6004	172	20	thongngam	thongngam	PROPN
ejpam-6004	172	21	/	/	SYM
ejpam-6004	172	22	eur	eur	NOUN
ejpam-6004	172	23	.	.	PUNCT
ejpam-6004	173	1	j.	j.	PROPN
ejpam-6004	173	2	pure	pure	PROPN
ejpam-6004	173	3	appl	appl	PROPN
ejpam-6004	173	4	.	.	PROPN
ejpam-6004	173	5	math	math	PROPN
ejpam-6004	173	6	,	,	PUNCT
ejpam-6004	173	7	18	18	NUM
ejpam-6004	173	8	(	(	PUNCT
ejpam-6004	173	9	2	2	NUM
ejpam-6004	173	10	)	)	PUNCT
ejpam-6004	173	11	(	(	PUNCT
ejpam-6004	173	12	2025	2025	NUM
ejpam-6004	173	13	)	)	PUNCT
ejpam-6004	173	14	,	,	PUNCT
ejpam-6004	173	15	6004	6004	NUM
ejpam-6004	173	16	6	6	NUM
ejpam-6004	173	17	of	of	ADP
ejpam-6004	173	18	8	8	NUM
ejpam-6004	173	19	(	(	PUNCT
ejpam-6004	173	20	i	i	NOUN
ejpam-6004	173	21	)	)	PUNCT
ejpam-6004	173	22	l	l	NOUN
ejpam-6004	173	23	is	be	AUX
ejpam-6004	173	24	odd	odd	ADJ
ejpam-6004	173	25	and	and	CCONJ
ejpam-6004	173	26	p	p	PRON
ejpam-6004	173	27	≡	≡	PROPN
ejpam-6004	173	28	5	5	NUM
ejpam-6004	173	29	(	(	PUNCT
ejpam-6004	173	30	mod	mod	NOUN
ejpam-6004	173	31	8)	8)	NUM
ejpam-6004	173	32	or	or	CCONJ
ejpam-6004	173	33	p	p	PRON
ejpam-6004	173	34	≡	≡	PROPN
ejpam-6004	173	35	7	7	NUM
ejpam-6004	173	36	(	(	PUNCT
ejpam-6004	173	37	mod	mod	NOUN
ejpam-6004	173	38	8)	8)	NUM
ejpam-6004	173	39	;	;	PUNCT
ejpam-6004	173	40	(	(	PUNCT
ejpam-6004	173	41	ii	ii	NOUN
ejpam-6004	173	42	)	)	PUNCT
ejpam-6004	173	43	l	l	NOUN
ejpam-6004	173	44	is	be	AUX
ejpam-6004	173	45	even	even	ADV
ejpam-6004	173	46	and	and	CCONJ
ejpam-6004	173	47	p	p	PROPN
ejpam-6004	173	48	≡	≡	PROPN
ejpam-6004	173	49	3	3	NUM
ejpam-6004	173	50	(	(	PUNCT
ejpam-6004	173	51	mod	mod	PROPN
ejpam-6004	173	52	8)	8)	NUM
ejpam-6004	173	53	or	or	CCONJ
ejpam-6004	173	54	p	p	PRON
ejpam-6004	173	55	≡	≡	PROPN
ejpam-6004	173	56	7	7	NUM
ejpam-6004	173	57	(	(	PUNCT
ejpam-6004	173	58	mod	mod	PROPN
ejpam-6004	173	59	8)	8)	NUM
ejpam-6004	173	60	;	;	PUNCT
ejpam-6004	173	61	then	then	ADV
ejpam-6004	173	62	u2	u2	PROPN
ejpam-6004	173	63	−	−	PROPN
ejpam-6004	173	64	s(s−	s(s−	PROPN
ejpam-6004	173	65	1)v2	1)v2	PROPN
ejpam-6004	173	66	=	=	SYM
ejpam-6004	173	67	−2l	−2l	PROPN
ejpam-6004	173	68	is	be	AUX
ejpam-6004	173	69	not	not	PART
ejpam-6004	173	70	solvable	solvable	ADJ
ejpam-6004	173	71	.	.	PUNCT
ejpam-6004	174	1	proof	proof	NOUN
ejpam-6004	174	2	.	.	PUNCT
ejpam-6004	175	1	since	since	SCONJ
ejpam-6004	175	2	p	p	PROPN
ejpam-6004	175	3	|	|	ADV
ejpam-6004	175	4	s(s−	s(s−	PROPN
ejpam-6004	175	5	1	1	NUM
ejpam-6004	175	6	)	)	PUNCT
ejpam-6004	175	7	,	,	PUNCT
ejpam-6004	175	8	we	we	PRON
ejpam-6004	175	9	obtain	obtain	VERB
ejpam-6004	175	10	that	that	DET
ejpam-6004	175	11	u2	u2	PROPN
ejpam-6004	175	12	≡	≡	PROPN
ejpam-6004	175	13	−2l	−2l	PROPN
ejpam-6004	175	14	(	(	PUNCT
ejpam-6004	175	15	mod	mod	PROPN
ejpam-6004	175	16	p	p	X
ejpam-6004	175	17	)	)	PUNCT
ejpam-6004	175	18	.	.	PUNCT
ejpam-6004	176	1	we	we	PRON
ejpam-6004	176	2	now	now	ADV
ejpam-6004	176	3	consider	consider	VERB
ejpam-6004	176	4	the	the	DET
ejpam-6004	176	5	legendre	legendre	NOUN
ejpam-6004	176	6	symbol	symbol	NOUN
ejpam-6004	176	7	(	(	PUNCT
ejpam-6004	176	8	−2l	−2l	PROPN
ejpam-6004	176	9	p	p	PROPN
ejpam-6004	176	10	)	)	PUNCT
ejpam-6004	176	11	.	.	PUNCT
ejpam-6004	177	1	we	we	PRON
ejpam-6004	177	2	have	have	VERB
ejpam-6004	177	3	(	(	PUNCT
ejpam-6004	177	4	−2l	−2l	PROPN
ejpam-6004	177	5	p	p	NOUN
ejpam-6004	177	6	)	)	PUNCT
ejpam-6004	177	7	=	=	PUNCT
ejpam-6004	177	8	(	(	PUNCT
ejpam-6004	177	9	−1	−1	NOUN
ejpam-6004	177	10	p	p	NOUN
ejpam-6004	177	11	)	)	PUNCT
ejpam-6004	177	12	(	(	PUNCT
ejpam-6004	177	13	2l	2l	NUM
ejpam-6004	177	14	p	p	NOUN
ejpam-6004	177	15	)	)	PUNCT
ejpam-6004	177	16	=	=	PUNCT
ejpam-6004	177	17	(	(	PUNCT
ejpam-6004	177	18	−1	−1	NOUN
ejpam-6004	177	19	p	p	NOUN
ejpam-6004	177	20	)	)	PUNCT
ejpam-6004	177	21	(	(	PUNCT
ejpam-6004	177	22	2	2	NUM
ejpam-6004	177	23	p	p	NOUN
ejpam-6004	177	24	)	)	PUNCT
ejpam-6004	178	1	l	l	NOUN
ejpam-6004	178	2	=	=	SYM
ejpam-6004	178	3	(	(	PUNCT
ejpam-6004	178	4	−1	−1	NOUN
ejpam-6004	178	5	)	)	PUNCT
ejpam-6004	178	6	(	(	PUNCT
ejpam-6004	178	7	p−1	p−1	PROPN
ejpam-6004	178	8	)	)	PUNCT
ejpam-6004	178	9	2	2	NUM
ejpam-6004	178	10	(	(	PUNCT
ejpam-6004	178	11	−1	−1	NOUN
ejpam-6004	178	12	)	)	PUNCT
ejpam-6004	178	13	(	(	PUNCT
ejpam-6004	178	14	p2−1)l	p2−1)l	NOUN
ejpam-6004	178	15	8	8	NUM
ejpam-6004	178	16	.	.	PUNCT
ejpam-6004	179	1	if	if	SCONJ
ejpam-6004	179	2	l	l	NOUN
ejpam-6004	179	3	is	be	AUX
ejpam-6004	179	4	odd	odd	ADJ
ejpam-6004	179	5	and	and	CCONJ
ejpam-6004	179	6	p	p	PRON
ejpam-6004	179	7	≡	≡	PROPN
ejpam-6004	179	8	5	5	NUM
ejpam-6004	179	9	,	,	PUNCT
ejpam-6004	179	10	7	7	NUM
ejpam-6004	179	11	(	(	PUNCT
ejpam-6004	179	12	mod	mod	NOUN
ejpam-6004	179	13	8)	8)	NUM
ejpam-6004	179	14	or	or	CCONJ
ejpam-6004	179	15	l	l	NOUN
ejpam-6004	179	16	is	be	AUX
ejpam-6004	179	17	even	even	ADV
ejpam-6004	179	18	and	and	CCONJ
ejpam-6004	179	19	p	p	PRON
ejpam-6004	179	20	≡	≡	PROPN
ejpam-6004	179	21	3	3	NUM
ejpam-6004	179	22	,	,	PUNCT
ejpam-6004	179	23	7	7	NUM
ejpam-6004	179	24	(	(	PUNCT
ejpam-6004	179	25	mod	mod	NOUN
ejpam-6004	179	26	8)	8)	NUM
ejpam-6004	179	27	,	,	PUNCT
ejpam-6004	179	28	then	then	ADV
ejpam-6004	179	29	it	it	PRON
ejpam-6004	179	30	is	be	AUX
ejpam-6004	179	31	easy	easy	ADJ
ejpam-6004	179	32	to	to	PART
ejpam-6004	179	33	see	see	VERB
ejpam-6004	179	34	that	that	PRON
ejpam-6004	179	35	(	(	PUNCT
ejpam-6004	179	36	−2l	−2l	PROPN
ejpam-6004	179	37	p	p	NOUN
ejpam-6004	179	38	)	)	PUNCT
ejpam-6004	180	1	=	=	SYM
ejpam-6004	180	2	−1	−1	NOUN
ejpam-6004	180	3	.	.	PUNCT
ejpam-6004	181	1	hence	hence	ADV
ejpam-6004	181	2	,	,	PUNCT
ejpam-6004	181	3	u2	u2	PROPN
ejpam-6004	181	4	≡	≡	PROPN
ejpam-6004	181	5	−2l	−2l	PROPN
ejpam-6004	181	6	(	(	PUNCT
ejpam-6004	181	7	mod	mod	PROPN
ejpam-6004	181	8	p	p	X
ejpam-6004	181	9	)	)	PUNCT
ejpam-6004	181	10	has	have	VERB
ejpam-6004	181	11	no	no	DET
ejpam-6004	181	12	solution	solution	NOUN
ejpam-6004	181	13	.	.	PUNCT
ejpam-6004	182	1	this	this	PRON
ejpam-6004	182	2	implies	imply	VERB
ejpam-6004	182	3	that	that	PRON
ejpam-6004	182	4	u2	u2	PROPN
ejpam-6004	182	5	−	−	PROPN
ejpam-6004	182	6	s(s−	s(s−	PROPN
ejpam-6004	182	7	1)v2	1)v2	PROPN
ejpam-6004	182	8	=	=	SYM
ejpam-6004	182	9	−2l	−2l	PROPN
ejpam-6004	182	10	is	be	AUX
ejpam-6004	182	11	not	not	PART
ejpam-6004	182	12	solvable	solvable	ADJ
ejpam-6004	182	13	.	.	PUNCT
ejpam-6004	183	1	lemma	lemma	PROPN
ejpam-6004	183	2	13	13	NUM
ejpam-6004	183	3	.	.	PUNCT
ejpam-6004	184	1	for	for	ADP
ejpam-6004	184	2	any	any	DET
ejpam-6004	184	3	integer	integer	NOUN
ejpam-6004	184	4	l	l	NOUN
ejpam-6004	184	5	≥	≥	NUM
ejpam-6004	184	6	2	2	NUM
ejpam-6004	184	7	,	,	PUNCT
ejpam-6004	184	8	we	we	PRON
ejpam-6004	184	9	have	have	VERB
ejpam-6004	184	10	2l	2l	NOUN
ejpam-6004	184	11	+	+	CCONJ
ejpam-6004	184	12	4	4	NUM
ejpam-6004	184	13	∈	∈	NOUN
ejpam-6004	184	14	t	t	NOUN
ejpam-6004	184	15	′(2l	′(2l	NOUN
ejpam-6004	184	16	)	)	PUNCT
ejpam-6004	184	17	.	.	PUNCT
ejpam-6004	185	1	proof	proof	NOUN
ejpam-6004	185	2	.	.	PUNCT
ejpam-6004	186	1	given	give	VERB
ejpam-6004	186	2	that	that	SCONJ
ejpam-6004	186	3	(	(	PUNCT
ejpam-6004	186	4	u	u	NOUN
ejpam-6004	186	5	,	,	PUNCT
ejpam-6004	186	6	v	v	NOUN
ejpam-6004	186	7	)	)	PUNCT
ejpam-6004	186	8	=	=	SYM
ejpam-6004	186	9	(	(	PUNCT
ejpam-6004	186	10	2l−2	2l−2	NUM
ejpam-6004	186	11	,	,	PUNCT
ejpam-6004	186	12	1	1	NUM
ejpam-6004	186	13	)	)	PUNCT
ejpam-6004	186	14	is	be	AUX
ejpam-6004	186	15	a	a	DET
ejpam-6004	186	16	solution	solution	NOUN
ejpam-6004	186	17	to	to	ADP
ejpam-6004	186	18	the	the	DET
ejpam-6004	186	19	equation	equation	NOUN
ejpam-6004	186	20	u2−s(s−1)v2	u2−s(s−1)v2	NOUN
ejpam-6004	186	21	=	=	SYM
ejpam-6004	186	22	−2l−2	−2l−2	NOUN
ejpam-6004	186	23	with	with	ADP
ejpam-6004	186	24	s	s	NOUN
ejpam-6004	186	25	=	=	SYM
ejpam-6004	186	26	2l−2	2l−2	NUM
ejpam-6004	186	27	+	+	NOUN
ejpam-6004	186	28	1	1	NUM
ejpam-6004	186	29	,	,	PUNCT
ejpam-6004	186	30	lemma	lemma	PROPN
ejpam-6004	186	31	9	9	NUM
ejpam-6004	186	32	implies	imply	VERB
ejpam-6004	186	33	that	that	SCONJ
ejpam-6004	186	34	the	the	DET
ejpam-6004	186	35	diophantine	diophantine	NOUN
ejpam-6004	186	36	equation	equation	NOUN
ejpam-6004	186	37	x2−kxy+ky2	x2−kxy+ky2	PUNCT
ejpam-6004	187	1	+	+	ADJ
ejpam-6004	187	2	2l	2l	X
ejpam-6004	187	3	=	=	SYM
ejpam-6004	187	4	0	0	NUM
ejpam-6004	187	5	,	,	PUNCT
ejpam-6004	187	6	where	where	SCONJ
ejpam-6004	187	7	k	k	NOUN
ejpam-6004	187	8	=	=	PUNCT
ejpam-6004	187	9	2l	2l	NOUN
ejpam-6004	187	10	+	+	CCONJ
ejpam-6004	187	11	4	4	NUM
ejpam-6004	187	12	,	,	PUNCT
ejpam-6004	187	13	has	have	VERB
ejpam-6004	187	14	infinitely	infinitely	ADV
ejpam-6004	187	15	many	many	ADJ
ejpam-6004	187	16	coprime	coprime	ADJ
ejpam-6004	187	17	solutions	solution	NOUN
ejpam-6004	187	18	(	(	PUNCT
ejpam-6004	187	19	x	x	X
ejpam-6004	187	20	,	,	PUNCT
ejpam-6004	187	21	y	y	PROPN
ejpam-6004	187	22	)	)	PUNCT
ejpam-6004	187	23	.	.	PUNCT
ejpam-6004	188	1	therefore	therefore	ADV
ejpam-6004	188	2	,	,	PUNCT
ejpam-6004	188	3	2l	2l	NUM
ejpam-6004	188	4	+	+	CCONJ
ejpam-6004	188	5	4	4	NUM
ejpam-6004	188	6	∈	∈	NOUN
ejpam-6004	188	7	t	t	NOUN
ejpam-6004	188	8	′(2l	′(2l	NOUN
ejpam-6004	188	9	)	)	PUNCT
ejpam-6004	188	10	.	.	PUNCT
ejpam-6004	189	1	we	we	PRON
ejpam-6004	189	2	are	be	AUX
ejpam-6004	189	3	now	now	ADV
ejpam-6004	189	4	ready	ready	ADJ
ejpam-6004	189	5	to	to	PART
ejpam-6004	189	6	find	find	VERB
ejpam-6004	189	7	the	the	DET
ejpam-6004	189	8	sets	set	NOUN
ejpam-6004	189	9	t	t	PROPN
ejpam-6004	189	10	′(2n	′(2n	NOUN
ejpam-6004	189	11	)	)	PUNCT
ejpam-6004	189	12	for	for	ADP
ejpam-6004	189	13	3	3	NUM
ejpam-6004	189	14	≤	≤	NOUN
ejpam-6004	189	15	n	n	PRON
ejpam-6004	189	16	≤	≤	NOUN
ejpam-6004	189	17	7	7	NUM
ejpam-6004	189	18	.	.	PUNCT
ejpam-6004	190	1	we	we	PRON
ejpam-6004	190	2	first	first	ADV
ejpam-6004	190	3	mentioned	mention	VERB
ejpam-6004	190	4	the	the	DET
ejpam-6004	190	5	previous	previous	ADJ
ejpam-6004	190	6	results	result	NOUN
ejpam-6004	190	7	on	on	ADP
ejpam-6004	190	8	t	t	PROPN
ejpam-6004	190	9	′(1	′(1	PROPN
ejpam-6004	190	10	)	)	PUNCT
ejpam-6004	190	11	,	,	PUNCT
ejpam-6004	190	12	t	t	PROPN
ejpam-6004	190	13	′(2	′(2	PROPN
ejpam-6004	190	14	)	)	PUNCT
ejpam-6004	190	15	and	and	CCONJ
ejpam-6004	190	16	t	t	PROPN
ejpam-6004	190	17	′(4	′(4	PROPN
ejpam-6004	190	18	)	)	PUNCT
ejpam-6004	190	19	.	.	PUNCT
ejpam-6004	191	1	theorem	theorem	NOUN
ejpam-6004	191	2	2	2	NUM
ejpam-6004	191	3	.	.	NUM
ejpam-6004	191	4	t	t	PROPN
ejpam-6004	191	5	′(1	′(1	PROPN
ejpam-6004	191	6	)	)	PUNCT
ejpam-6004	191	7	=	=	PUNCT
ejpam-6004	191	8	{	{	PUNCT
ejpam-6004	191	9	5	5	NUM
ejpam-6004	191	10	}	}	PUNCT
ejpam-6004	191	11	.	.	PUNCT
ejpam-6004	192	1	proof	proof	NOUN
ejpam-6004	192	2	.	.	PUNCT
ejpam-6004	193	1	suppose	suppose	VERB
ejpam-6004	193	2	x2	x2	PRON
ejpam-6004	193	3	−	−	PROPN
ejpam-6004	193	4	kxy	kxy	NOUN
ejpam-6004	193	5	+	+	CCONJ
ejpam-6004	193	6	ky2	ky2	NOUN
ejpam-6004	193	7	+	+	CCONJ
ejpam-6004	193	8	1	1	NUM
ejpam-6004	193	9	=	=	SYM
ejpam-6004	193	10	0	0	NUM
ejpam-6004	194	1	where	where	SCONJ
ejpam-6004	194	2	gcd(x	gcd(x	PROPN
ejpam-6004	194	3	,	,	PUNCT
ejpam-6004	194	4	y	y	NOUN
ejpam-6004	194	5	)	)	PUNCT
ejpam-6004	194	6	=	=	SYM
ejpam-6004	194	7	1	1	X
ejpam-6004	194	8	.	.	PUNCT
ejpam-6004	194	9	then	then	ADV
ejpam-6004	194	10	(	(	PUNCT
ejpam-6004	194	11	2x−	2x−	NUM
ejpam-6004	194	12	ky)2	ky)2	NOUN
ejpam-6004	194	13	−	−	PROPN
ejpam-6004	194	14	(	(	PUNCT
ejpam-6004	194	15	(	(	PUNCT
ejpam-6004	194	16	k	k	X
ejpam-6004	194	17	−	−	PROPN
ejpam-6004	194	18	2)2	2)2	NUM
ejpam-6004	194	19	−	−	PROPN
ejpam-6004	194	20	4)y2	4)y2	NUM
ejpam-6004	194	21	=	=	SYM
ejpam-6004	194	22	−4	−4	X
ejpam-6004	194	23	.	.	PUNCT
ejpam-6004	195	1	by	by	ADP
ejpam-6004	195	2	lemma	lemma	PROPN
ejpam-6004	195	3	6	6	NUM
ejpam-6004	195	4	,	,	PUNCT
ejpam-6004	195	5	we	we	PRON
ejpam-6004	195	6	have	have	VERB
ejpam-6004	195	7	k	k	X
ejpam-6004	195	8	−	−	VERB
ejpam-6004	195	9	2	2	NUM
ejpam-6004	195	10	=	=	SYM
ejpam-6004	195	11	3	3	NUM
ejpam-6004	195	12	.	.	PUNCT
ejpam-6004	196	1	hence	hence	ADV
ejpam-6004	196	2	,	,	PUNCT
ejpam-6004	196	3	k	k	PROPN
ejpam-6004	196	4	=	=	SYM
ejpam-6004	196	5	5	5	NUM
ejpam-6004	196	6	as	as	SCONJ
ejpam-6004	196	7	desired	desire	VERB
ejpam-6004	196	8	.	.	PUNCT
ejpam-6004	197	1	theorem	theorem	NOUN
ejpam-6004	197	2	3	3	NUM
ejpam-6004	197	3	.	.	PUNCT
ejpam-6004	198	1	[	[	X
ejpam-6004	198	2	4	4	NUM
ejpam-6004	198	3	]	]	PUNCT
ejpam-6004	198	4	t	t	PROPN
ejpam-6004	198	5	′(2	′(2	PROPN
ejpam-6004	198	6	)	)	PUNCT
ejpam-6004	198	7	=	=	PUNCT
ejpam-6004	198	8	{	{	PUNCT
ejpam-6004	198	9	6	6	NUM
ejpam-6004	198	10	}	}	PUNCT
ejpam-6004	198	11	and	and	CCONJ
ejpam-6004	198	12	t	t	PROPN
ejpam-6004	198	13	′(4	′(4	PROPN
ejpam-6004	198	14	)	)	PUNCT
ejpam-6004	198	15	=	=	PUNCT
ejpam-6004	198	16	{	{	PUNCT
ejpam-6004	198	17	8	8	NUM
ejpam-6004	198	18	}	}	PUNCT
ejpam-6004	198	19	.	.	PUNCT
ejpam-6004	199	1	proof	proof	NOUN
ejpam-6004	199	2	.	.	PUNCT
ejpam-6004	200	1	the	the	DET
ejpam-6004	200	2	proof	proof	NOUN
ejpam-6004	200	3	of	of	ADP
ejpam-6004	200	4	this	this	DET
ejpam-6004	200	5	theorem	theorem	NOUN
ejpam-6004	200	6	can	can	AUX
ejpam-6004	200	7	be	be	AUX
ejpam-6004	200	8	found	find	VERB
ejpam-6004	200	9	in	in	ADP
ejpam-6004	200	10	theorem	theorem	NOUN
ejpam-6004	200	11	3.2	3.2	NUM
ejpam-6004	200	12	from	from	ADP
ejpam-6004	200	13	[	[	X
ejpam-6004	200	14	4	4	NUM
ejpam-6004	200	15	]	]	PUNCT
ejpam-6004	200	16	.	.	PUNCT
ejpam-6004	201	1	we	we	PRON
ejpam-6004	201	2	next	next	ADV
ejpam-6004	201	3	find	find	VERB
ejpam-6004	201	4	t	t	PROPN
ejpam-6004	201	5	′(8	′(8	PROPN
ejpam-6004	201	6	)	)	PUNCT
ejpam-6004	201	7	,	,	PUNCT
ejpam-6004	201	8	t	t	PROPN
ejpam-6004	201	9	′(16	′(16	NUM
ejpam-6004	201	10	)	)	PUNCT
ejpam-6004	201	11	,	,	PUNCT
ejpam-6004	201	12	t	t	PROPN
ejpam-6004	201	13	′(16	′(16	NUM
ejpam-6004	201	14	)	)	PUNCT
ejpam-6004	201	15	,	,	PUNCT
ejpam-6004	201	16	t	t	PROPN
ejpam-6004	201	17	′(32	′(32	PROPN
ejpam-6004	201	18	)	)	PUNCT
ejpam-6004	201	19	,	,	PUNCT
ejpam-6004	201	20	t	t	PROPN
ejpam-6004	201	21	′(64	′(64	NOUN
ejpam-6004	201	22	)	)	PUNCT
ejpam-6004	201	23	and	and	CCONJ
ejpam-6004	201	24	t	t	PROPN
ejpam-6004	201	25	′(128	′(128	NOUN
ejpam-6004	201	26	)	)	PUNCT
ejpam-6004	201	27	using	use	VERB
ejpam-6004	201	28	our	our	PRON
ejpam-6004	201	29	method	method	NOUN
ejpam-6004	201	30	.	.	PUNCT
ejpam-6004	202	1	theorem	theorem	ADJ
ejpam-6004	202	2	4	4	NUM
ejpam-6004	202	3	.	.	PUNCT
ejpam-6004	202	4	t	t	PROPN
ejpam-6004	202	5	′(8	′(8	PROPN
ejpam-6004	202	6	)	)	PUNCT
ejpam-6004	203	1	=	=	PRON
ejpam-6004	203	2	{	{	PUNCT
ejpam-6004	203	3	8	8	NUM
ejpam-6004	203	4	,	,	PUNCT
ejpam-6004	203	5	12	12	NUM
ejpam-6004	203	6	}	}	PUNCT
ejpam-6004	203	7	.	.	PUNCT
ejpam-6004	204	1	proof	proof	NOUN
ejpam-6004	204	2	.	.	PUNCT
ejpam-6004	205	1	suppose	suppose	VERB
ejpam-6004	205	2	x2−kxy+ky2	x2−kxy+ky2	PUNCT
ejpam-6004	206	1	+	+	NOUN
ejpam-6004	206	2	8	8	NUM
ejpam-6004	206	3	=	=	SYM
ejpam-6004	206	4	0	0	NUM
ejpam-6004	207	1	where	where	SCONJ
ejpam-6004	207	2	gcd(x	gcd(x	PROPN
ejpam-6004	207	3	,	,	PUNCT
ejpam-6004	207	4	y	y	NOUN
ejpam-6004	207	5	)	)	PUNCT
ejpam-6004	207	6	=	=	SYM
ejpam-6004	207	7	1	1	X
ejpam-6004	207	8	.	.	PUNCT
ejpam-6004	207	9	by	by	ADP
ejpam-6004	207	10	lemma	lemma	PROPN
ejpam-6004	207	11	9	9	NUM
ejpam-6004	207	12	,	,	PUNCT
ejpam-6004	207	13	it	it	PRON
ejpam-6004	207	14	suffices	suffice	VERB
ejpam-6004	207	15	to	to	PART
ejpam-6004	207	16	consider	consider	VERB
ejpam-6004	207	17	the	the	DET
ejpam-6004	207	18	diophantine	diophantine	NOUN
ejpam-6004	207	19	equation	equation	NOUN
ejpam-6004	207	20	u2−s(s−1)v2	u2−s(s−1)v2	NOUN
ejpam-6004	207	21	=	=	PUNCT
ejpam-6004	207	22	−2	−2	NOUN
ejpam-6004	207	23	.	.	PUNCT
ejpam-6004	208	1	if	if	SCONJ
ejpam-6004	208	2	s	s	PRON
ejpam-6004	208	3	=	=	SYM
ejpam-6004	208	4	2	2	NUM
ejpam-6004	208	5	,	,	PUNCT
ejpam-6004	208	6	then	then	ADV
ejpam-6004	208	7	u2−2v2	u2−2v2	PROPN
ejpam-6004	208	8	=	=	ADJ
ejpam-6004	208	9	−2	−2	PROPN
ejpam-6004	208	10	.	.	PUNCT
ejpam-6004	209	1	since	since	SCONJ
ejpam-6004	209	2	(	(	PUNCT
ejpam-6004	209	3	4	4	NUM
ejpam-6004	209	4	,	,	PUNCT
ejpam-6004	209	5	3	3	NUM
ejpam-6004	209	6	)	)	PUNCT
ejpam-6004	209	7	is	be	AUX
ejpam-6004	209	8	a	a	DET
ejpam-6004	209	9	solution	solution	NOUN
ejpam-6004	209	10	of	of	ADP
ejpam-6004	209	11	the	the	DET
ejpam-6004	209	12	equation	equation	NOUN
ejpam-6004	209	13	u2−2v2	u2−2v2	NOUN
ejpam-6004	209	14	=	=	NOUN
ejpam-6004	209	15	−2	−2	NOUN
ejpam-6004	209	16	.	.	PUNCT
ejpam-6004	210	1	thus	thus	ADV
ejpam-6004	210	2	we	we	PRON
ejpam-6004	210	3	obtain	obtain	VERB
ejpam-6004	210	4	k	k	X
ejpam-6004	211	1	=	=	NOUN
ejpam-6004	212	1	8	8	X
ejpam-6004	212	2	.	.	PUNCT
ejpam-6004	213	1	if	if	SCONJ
ejpam-6004	213	2	s	s	PROPN
ejpam-6004	213	3	≥	≥	NOUN
ejpam-6004	213	4	3	3	NUM
ejpam-6004	213	5	,	,	PUNCT
ejpam-6004	213	6	by	by	ADP
ejpam-6004	213	7	lemma	lemma	PROPN
ejpam-6004	213	8	7	7	NUM
ejpam-6004	213	9	and	and	CCONJ
ejpam-6004	213	10	lemma	lemma	PROPN
ejpam-6004	213	11	9	9	NUM
ejpam-6004	213	12	,	,	PUNCT
ejpam-6004	213	13	it	it	PRON
ejpam-6004	213	14	suffices	suffice	VERB
ejpam-6004	213	15	to	to	PART
ejpam-6004	213	16	consider	consider	VERB
ejpam-6004	213	17	the	the	DET
ejpam-6004	213	18	diophantine	diophantine	NOUN
ejpam-6004	213	19	equation	equation	NOUN
ejpam-6004	213	20	u2	u2	NOUN
ejpam-6004	213	21	−	−	PROPN
ejpam-6004	213	22	s(s−	s(s−	PROPN
ejpam-6004	214	1	1)v2	1)v2	PROPN
ejpam-6004	214	2	=	=	SYM
ejpam-6004	214	3	−2	−2	PROPN
ejpam-6004	214	4	for	for	ADP
ejpam-6004	214	5	s	s	NOUN
ejpam-6004	214	6	≤	≤	NOUN
ejpam-6004	214	7	3	3	NUM
ejpam-6004	214	8	.	.	PUNCT
ejpam-6004	215	1	this	this	PRON
ejpam-6004	215	2	implies	imply	VERB
ejpam-6004	215	3	that	that	SCONJ
ejpam-6004	215	4	s	s	VERB
ejpam-6004	215	5	=	=	ADJ
ejpam-6004	215	6	3	3	X
ejpam-6004	215	7	.	.	PUNCT
ejpam-6004	216	1	it	it	PRON
ejpam-6004	216	2	is	be	AUX
ejpam-6004	216	3	easy	easy	ADJ
ejpam-6004	216	4	to	to	PART
ejpam-6004	216	5	see	see	VERB
ejpam-6004	216	6	that	that	PRON
ejpam-6004	216	7	(	(	PUNCT
ejpam-6004	216	8	2	2	NUM
ejpam-6004	216	9	,	,	PUNCT
ejpam-6004	216	10	1	1	NUM
ejpam-6004	216	11	)	)	PUNCT
ejpam-6004	216	12	is	be	AUX
ejpam-6004	216	13	a	a	DET
ejpam-6004	216	14	solution	solution	NOUN
ejpam-6004	216	15	of	of	ADP
ejpam-6004	216	16	the	the	DET
ejpam-6004	216	17	equation	equation	NOUN
ejpam-6004	216	18	u2	u2	PROPN
ejpam-6004	216	19	−	−	PROPN
ejpam-6004	216	20	6v2	6v2	NUM
ejpam-6004	216	21	=	=	SYM
ejpam-6004	216	22	−2	−2	NOUN
ejpam-6004	216	23	.	.	PUNCT
ejpam-6004	217	1	thus	thus	ADV
ejpam-6004	217	2	we	we	PRON
ejpam-6004	217	3	obtain	obtain	VERB
ejpam-6004	217	4	k	k	PROPN
ejpam-6004	217	5	=	=	NOUN
ejpam-6004	217	6	12	12	NUM
ejpam-6004	217	7	.	.	PUNCT
ejpam-6004	218	1	hence	hence	ADV
ejpam-6004	218	2	t	t	PROPN
ejpam-6004	218	3	′(8	′(8	PROPN
ejpam-6004	218	4	)	)	PUNCT
ejpam-6004	219	1	=	=	PRON
ejpam-6004	219	2	{	{	PUNCT
ejpam-6004	219	3	8	8	NUM
ejpam-6004	219	4	,	,	PUNCT
ejpam-6004	219	5	12	12	NUM
ejpam-6004	219	6	}	}	PUNCT
ejpam-6004	219	7	.	.	PUNCT
ejpam-6004	220	1	theorem	theorem	NOUN
ejpam-6004	220	2	5	5	NUM
ejpam-6004	220	3	.	.	PUNCT
ejpam-6004	220	4	t	t	PROPN
ejpam-6004	220	5	′(16	′(16	PROPN
ejpam-6004	220	6	)	)	PUNCT
ejpam-6004	220	7	=	=	PRON
ejpam-6004	220	8	{	{	PUNCT
ejpam-6004	220	9	20	20	NUM
ejpam-6004	220	10	}	}	PUNCT
ejpam-6004	220	11	.	.	PUNCT
ejpam-6004	221	1	proof	proof	NOUN
ejpam-6004	221	2	.	.	PUNCT
ejpam-6004	222	1	by	by	ADP
ejpam-6004	222	2	lemma	lemma	PROPN
ejpam-6004	222	3	7	7	NUM
ejpam-6004	222	4	and	and	CCONJ
ejpam-6004	222	5	lemma	lemma	PROPN
ejpam-6004	222	6	9	9	NUM
ejpam-6004	222	7	,	,	PUNCT
ejpam-6004	222	8	it	it	PRON
ejpam-6004	222	9	suffices	suffice	VERB
ejpam-6004	222	10	to	to	PART
ejpam-6004	222	11	consider	consider	VERB
ejpam-6004	222	12	the	the	DET
ejpam-6004	222	13	diophantine	diophantine	NOUN
ejpam-6004	222	14	equation	equation	NOUN
ejpam-6004	222	15	u2	u2	NOUN
ejpam-6004	222	16	−	−	PROPN
ejpam-6004	222	17	s(s−	s(s−	PROPN
ejpam-6004	222	18	1)v2	1)v2	PROPN
ejpam-6004	222	19	=	=	SYM
ejpam-6004	222	20	−22	−22	PROPN
ejpam-6004	222	21	for	for	ADP
ejpam-6004	222	22	s	s	NOUN
ejpam-6004	222	23	≤	≤	NOUN
ejpam-6004	222	24	5	5	NUM
ejpam-6004	222	25	.	.	PUNCT
ejpam-6004	222	26	by	by	ADP
ejpam-6004	222	27	lemma	lemma	PROPN
ejpam-6004	222	28	12	12	NUM
ejpam-6004	222	29	,	,	PUNCT
ejpam-6004	222	30	u2	u2	PROPN
ejpam-6004	222	31	−	−	PROPN
ejpam-6004	222	32	s(s−	s(s−	PROPN
ejpam-6004	222	33	1)v2	1)v2	PROPN
ejpam-6004	222	34	=	=	SYM
ejpam-6004	222	35	−22	−22	PROPN
ejpam-6004	222	36	is	be	AUX
ejpam-6004	222	37	not	not	PART
ejpam-6004	222	38	solvable	solvable	ADJ
ejpam-6004	222	39	for	for	ADP
ejpam-6004	222	40	s	s	NOUN
ejpam-6004	222	41	=	=	SYM
ejpam-6004	222	42	1	1	NUM
ejpam-6004	222	43	,	,	PUNCT
ejpam-6004	222	44	3	3	NUM
ejpam-6004	222	45	and	and	CCONJ
ejpam-6004	222	46	4	4	NUM
ejpam-6004	222	47	.	.	X
ejpam-6004	222	48	for	for	ADP
ejpam-6004	222	49	s	s	NOUN
ejpam-6004	222	50	=	=	SYM
ejpam-6004	222	51	2	2	NUM
ejpam-6004	222	52	,	,	PUNCT
ejpam-6004	222	53	it	it	PRON
ejpam-6004	222	54	is	be	AUX
ejpam-6004	222	55	easy	easy	ADJ
ejpam-6004	222	56	to	to	PART
ejpam-6004	222	57	see	see	VERB
ejpam-6004	222	58	that	that	SCONJ
ejpam-6004	222	59	if	if	SCONJ
ejpam-6004	222	60	u2	u2	PROPN
ejpam-6004	222	61	−	−	PROPN
ejpam-6004	222	62	2v2	2v2	NUM
ejpam-6004	222	63	=	=	SYM
ejpam-6004	222	64	−4	−4	PROPN
ejpam-6004	222	65	,	,	PUNCT
ejpam-6004	222	66	then	then	ADV
ejpam-6004	222	67	u	u	NOUN
ejpam-6004	222	68	is	be	AUX
ejpam-6004	222	69	even	even	ADV
ejpam-6004	222	70	.	.	PUNCT
ejpam-6004	223	1	thus	thus	ADV
ejpam-6004	223	2	4	4	NUM
ejpam-6004	223	3	|	|	NOUN
ejpam-6004	223	4	2v2	2v2	NUM
ejpam-6004	223	5	and	and	CCONJ
ejpam-6004	223	6	this	this	PRON
ejpam-6004	223	7	implies	imply	VERB
ejpam-6004	223	8	that	that	SCONJ
ejpam-6004	223	9	v	v	NOUN
ejpam-6004	223	10	is	be	AUX
ejpam-6004	223	11	even	even	ADV
ejpam-6004	223	12	.	.	PUNCT
ejpam-6004	224	1	hence	hence	ADV
ejpam-6004	224	2	2	2	NUM
ejpam-6004	224	3	|	|	ADV
ejpam-6004	224	4	gcd(u	gcd(u	NOUN
ejpam-6004	224	5	,	,	PUNCT
ejpam-6004	224	6	v	v	NOUN
ejpam-6004	224	7	)	)	PUNCT
ejpam-6004	224	8	and	and	CCONJ
ejpam-6004	224	9	thus	thus	ADV
ejpam-6004	224	10	8	8	NUM
ejpam-6004	224	11	̸∈	̸∈	PROPN
ejpam-6004	224	12	t	t	PROPN
ejpam-6004	224	13	′(16	′(16	NUM
ejpam-6004	224	14	)	)	PUNCT
ejpam-6004	224	15	.	.	PUNCT
ejpam-6004	225	1	for	for	ADP
ejpam-6004	225	2	s	s	NOUN
ejpam-6004	225	3	=	=	SYM
ejpam-6004	225	4	5	5	NUM
ejpam-6004	225	5	,	,	PUNCT
ejpam-6004	225	6	we	we	PRON
ejpam-6004	225	7	see	see	VERB
ejpam-6004	225	8	that	that	SCONJ
ejpam-6004	225	9	(	(	PUNCT
ejpam-6004	225	10	4	4	NUM
ejpam-6004	225	11	,	,	PUNCT
ejpam-6004	225	12	1	1	NUM
ejpam-6004	225	13	)	)	PUNCT
ejpam-6004	225	14	is	be	AUX
ejpam-6004	225	15	a	a	DET
ejpam-6004	225	16	solution	solution	NOUN
ejpam-6004	225	17	of	of	ADP
ejpam-6004	225	18	u2	u2	NOUN
ejpam-6004	225	19	−	−	PROPN
ejpam-6004	225	20	20v2	20v2	NUM
ejpam-6004	225	21	=	=	SYM
ejpam-6004	225	22	−4	−4	PROPN
ejpam-6004	225	23	.	.	PUNCT
ejpam-6004	226	1	thus	thus	ADV
ejpam-6004	226	2	by	by	ADP
ejpam-6004	226	3	lemma	lemma	PROPN
ejpam-6004	226	4	9	9	NUM
ejpam-6004	226	5	,	,	PUNCT
ejpam-6004	226	6	we	we	PRON
ejpam-6004	226	7	obtain	obtain	VERB
ejpam-6004	226	8	that	that	DET
ejpam-6004	226	9	t	t	PROPN
ejpam-6004	226	10	′(16	′(16	NOUN
ejpam-6004	226	11	)	)	PUNCT
ejpam-6004	226	12	=	=	PRON
ejpam-6004	226	13	{	{	PUNCT
ejpam-6004	226	14	20	20	NUM
ejpam-6004	226	15	}	}	PUNCT
ejpam-6004	226	16	as	as	SCONJ
ejpam-6004	226	17	desired	desire	VERB
ejpam-6004	226	18	.	.	PUNCT
ejpam-6004	227	1	s.	s.	PROPN
ejpam-6004	227	2	prugsapitak	prugsapitak	PROPN
ejpam-6004	227	3	,	,	PUNCT
ejpam-6004	227	4	n.	n.	PROPN
ejpam-6004	227	5	thongngam	thongngam	PROPN
ejpam-6004	227	6	/	/	SYM
ejpam-6004	227	7	eur	eur	NOUN
ejpam-6004	227	8	.	.	PUNCT
ejpam-6004	228	1	j.	j.	PROPN
ejpam-6004	228	2	pure	pure	PROPN
ejpam-6004	228	3	appl	appl	PROPN
ejpam-6004	228	4	.	.	PROPN
ejpam-6004	228	5	math	math	PROPN
ejpam-6004	228	6	,	,	PUNCT
ejpam-6004	228	7	18	18	NUM
ejpam-6004	228	8	(	(	PUNCT
ejpam-6004	228	9	2	2	NUM
ejpam-6004	228	10	)	)	PUNCT
ejpam-6004	228	11	(	(	PUNCT
ejpam-6004	228	12	2025	2025	NUM
ejpam-6004	228	13	)	)	PUNCT
ejpam-6004	228	14	,	,	PUNCT
ejpam-6004	228	15	6004	6004	NUM
ejpam-6004	228	16	7	7	NUM
ejpam-6004	228	17	of	of	ADP
ejpam-6004	228	18	8	8	NUM
ejpam-6004	228	19	theorem	theorem	VERB
ejpam-6004	228	20	6	6	NUM
ejpam-6004	228	21	.	.	PUNCT
ejpam-6004	228	22	t	t	PROPN
ejpam-6004	228	23	′(32	′(32	PROPN
ejpam-6004	228	24	)	)	PUNCT
ejpam-6004	228	25	=	=	PUNCT
ejpam-6004	228	26	{	{	PUNCT
ejpam-6004	228	27	16	16	NUM
ejpam-6004	228	28	,	,	PUNCT
ejpam-6004	228	29	36	36	NUM
ejpam-6004	228	30	}	}	PUNCT
ejpam-6004	228	31	.	.	PUNCT
ejpam-6004	229	1	proof	proof	NOUN
ejpam-6004	229	2	.	.	PUNCT
ejpam-6004	230	1	by	by	ADP
ejpam-6004	230	2	lemma	lemma	PROPN
ejpam-6004	230	3	7	7	NUM
ejpam-6004	230	4	and	and	CCONJ
ejpam-6004	230	5	lemma	lemma	PROPN
ejpam-6004	230	6	9	9	NUM
ejpam-6004	230	7	,	,	PUNCT
ejpam-6004	230	8	it	it	PRON
ejpam-6004	230	9	suffices	suffice	VERB
ejpam-6004	230	10	to	to	PART
ejpam-6004	230	11	consider	consider	VERB
ejpam-6004	230	12	the	the	DET
ejpam-6004	230	13	diophantine	diophantine	NOUN
ejpam-6004	230	14	equation	equation	NOUN
ejpam-6004	230	15	u2	u2	NOUN
ejpam-6004	230	16	−	−	PROPN
ejpam-6004	230	17	s(s−	s(s−	PROPN
ejpam-6004	230	18	1)v2	1)v2	PROPN
ejpam-6004	230	19	=	=	SYM
ejpam-6004	230	20	−23	−23	PROPN
ejpam-6004	230	21	for	for	ADP
ejpam-6004	230	22	s	s	PROPN
ejpam-6004	230	23	≤	≤	ADJ
ejpam-6004	230	24	9	9	NUM
ejpam-6004	230	25	.	.	PUNCT
ejpam-6004	231	1	by	by	ADP
ejpam-6004	231	2	lemma	lemma	PROPN
ejpam-6004	231	3	12	12	NUM
ejpam-6004	231	4	,	,	PUNCT
ejpam-6004	231	5	u2	u2	PROPN
ejpam-6004	231	6	−	−	PROPN
ejpam-6004	231	7	s(s−	s(s−	PROPN
ejpam-6004	231	8	1)v2	1)v2	PROPN
ejpam-6004	231	9	=	=	SYM
ejpam-6004	231	10	−23	−23	PROPN
ejpam-6004	231	11	is	be	AUX
ejpam-6004	231	12	not	not	PART
ejpam-6004	231	13	solvable	solvable	ADJ
ejpam-6004	231	14	for	for	ADP
ejpam-6004	231	15	s	s	NOUN
ejpam-6004	231	16	=	=	SYM
ejpam-6004	231	17	1	1	NUM
ejpam-6004	231	18	,	,	PUNCT
ejpam-6004	231	19	5	5	NUM
ejpam-6004	231	20	,	,	PUNCT
ejpam-6004	231	21	6	6	NUM
ejpam-6004	231	22	,	,	PUNCT
ejpam-6004	231	23	7	7	NUM
ejpam-6004	231	24	and	and	CCONJ
ejpam-6004	231	25	8	8	NUM
ejpam-6004	231	26	.	.	X
ejpam-6004	232	1	for	for	ADP
ejpam-6004	232	2	s	s	NOUN
ejpam-6004	232	3	=	=	SYM
ejpam-6004	232	4	2	2	NUM
ejpam-6004	232	5	,	,	PUNCT
ejpam-6004	232	6	3	3	NUM
ejpam-6004	232	7	,	,	PUNCT
ejpam-6004	232	8	we	we	PRON
ejpam-6004	232	9	can	can	AUX
ejpam-6004	232	10	see	see	VERB
ejpam-6004	232	11	that	that	SCONJ
ejpam-6004	232	12	if	if	SCONJ
ejpam-6004	232	13	u2−s(s−1)v2	u2−s(s−1)v2	PROPN
ejpam-6004	232	14	=	=	SYM
ejpam-6004	232	15	−8	−8	NOUN
ejpam-6004	232	16	,	,	PUNCT
ejpam-6004	232	17	then	then	ADV
ejpam-6004	232	18	u	u	NOUN
ejpam-6004	232	19	and	and	CCONJ
ejpam-6004	232	20	v	v	NOUN
ejpam-6004	232	21	are	be	AUX
ejpam-6004	232	22	both	both	PRON
ejpam-6004	232	23	even	even	ADV
ejpam-6004	232	24	.	.	PUNCT
ejpam-6004	233	1	thus	thus	ADV
ejpam-6004	233	2	all	all	DET
ejpam-6004	233	3	solutions	solution	NOUN
ejpam-6004	233	4	of	of	ADP
ejpam-6004	233	5	u2	u2	PROPN
ejpam-6004	233	6	−	−	PROPN
ejpam-6004	233	7	s(s−	s(s−	PROPN
ejpam-6004	233	8	1)v2	1)v2	PROPN
ejpam-6004	233	9	=	=	SYM
ejpam-6004	233	10	−8	−8	NOUN
ejpam-6004	233	11	are	be	AUX
ejpam-6004	233	12	not	not	PART
ejpam-6004	233	13	relatively	relatively	ADV
ejpam-6004	233	14	prime	prime	ADJ
ejpam-6004	233	15	.	.	PUNCT
ejpam-6004	234	1	we	we	PRON
ejpam-6004	234	2	see	see	VERB
ejpam-6004	234	3	that	that	SCONJ
ejpam-6004	234	4	(	(	PUNCT
ejpam-6004	234	5	2	2	NUM
ejpam-6004	234	6	,	,	PUNCT
ejpam-6004	234	7	1	1	NUM
ejpam-6004	234	8	)	)	PUNCT
ejpam-6004	234	9	and	and	CCONJ
ejpam-6004	234	10	(	(	PUNCT
ejpam-6004	234	11	8	8	NUM
ejpam-6004	234	12	,	,	PUNCT
ejpam-6004	234	13	1	1	NUM
ejpam-6004	234	14	)	)	PUNCT
ejpam-6004	234	15	are	be	AUX
ejpam-6004	234	16	solutions	solution	NOUN
ejpam-6004	234	17	of	of	ADP
ejpam-6004	234	18	u2	u2	NOUN
ejpam-6004	234	19	−	−	PROPN
ejpam-6004	234	20	s(s−	s(s−	PROPN
ejpam-6004	235	1	1)v2	1)v2	PROPN
ejpam-6004	235	2	=	=	SYM
ejpam-6004	235	3	−8	−8	PROPN
ejpam-6004	235	4	for	for	ADP
ejpam-6004	235	5	s	s	NOUN
ejpam-6004	235	6	=	=	SYM
ejpam-6004	235	7	4	4	NUM
ejpam-6004	235	8	and	and	CCONJ
ejpam-6004	235	9	s	s	NOUN
ejpam-6004	235	10	=	=	NOUN
ejpam-6004	235	11	9	9	NUM
ejpam-6004	235	12	respectively	respectively	ADV
ejpam-6004	235	13	.	.	PUNCT
ejpam-6004	236	1	hence	hence	ADV
ejpam-6004	236	2	by	by	ADP
ejpam-6004	236	3	lemma	lemma	PROPN
ejpam-6004	236	4	9	9	NUM
ejpam-6004	236	5	,	,	PUNCT
ejpam-6004	236	6	we	we	PRON
ejpam-6004	236	7	obtain	obtain	VERB
ejpam-6004	236	8	that	that	DET
ejpam-6004	236	9	t	t	PROPN
ejpam-6004	236	10	′(32	′(32	PROPN
ejpam-6004	236	11	)	)	PUNCT
ejpam-6004	236	12	=	=	PUNCT
ejpam-6004	236	13	{	{	PUNCT
ejpam-6004	236	14	16	16	NUM
ejpam-6004	236	15	,	,	PUNCT
ejpam-6004	236	16	36	36	NUM
ejpam-6004	236	17	}	}	PUNCT
ejpam-6004	236	18	as	as	SCONJ
ejpam-6004	236	19	desired	desire	VERB
ejpam-6004	236	20	.	.	PUNCT
ejpam-6004	237	1	theorem	theorem	VERB
ejpam-6004	237	2	7	7	NUM
ejpam-6004	237	3	.	.	PUNCT
ejpam-6004	237	4	t	t	NOUN
ejpam-6004	237	5	′(64	′(64	PROPN
ejpam-6004	237	6	)	)	PUNCT
ejpam-6004	237	7	=	=	SYM
ejpam-6004	237	8	{	{	PUNCT
ejpam-6004	237	9	20	20	NUM
ejpam-6004	237	10	,	,	PUNCT
ejpam-6004	237	11	68	68	NUM
ejpam-6004	237	12	}	}	PUNCT
ejpam-6004	237	13	.	.	PUNCT
ejpam-6004	238	1	proof	proof	NOUN
ejpam-6004	238	2	.	.	PUNCT
ejpam-6004	239	1	by	by	ADP
ejpam-6004	239	2	lemma	lemma	PROPN
ejpam-6004	239	3	7	7	NUM
ejpam-6004	239	4	and	and	CCONJ
ejpam-6004	239	5	lemma	lemma	PROPN
ejpam-6004	239	6	9	9	NUM
ejpam-6004	239	7	,	,	PUNCT
ejpam-6004	239	8	it	it	PRON
ejpam-6004	239	9	suffices	suffice	VERB
ejpam-6004	239	10	to	to	PART
ejpam-6004	239	11	consider	consider	VERB
ejpam-6004	239	12	the	the	DET
ejpam-6004	239	13	diophantine	diophantine	NOUN
ejpam-6004	239	14	equation	equation	NOUN
ejpam-6004	239	15	u2	u2	NOUN
ejpam-6004	239	16	−	−	PROPN
ejpam-6004	239	17	s(s−	s(s−	PROPN
ejpam-6004	239	18	1)v2	1)v2	NUM
ejpam-6004	239	19	=	=	SYM
ejpam-6004	239	20	−24	−24	ADP
ejpam-6004	239	21	for	for	ADP
ejpam-6004	239	22	s	s	PROPN
ejpam-6004	239	23	≤	≤	NOUN
ejpam-6004	239	24	17	17	NUM
ejpam-6004	239	25	.	.	PUNCT
ejpam-6004	240	1	by	by	ADP
ejpam-6004	240	2	lemma	lemma	PROPN
ejpam-6004	240	3	12	12	NUM
ejpam-6004	240	4	,	,	PUNCT
ejpam-6004	240	5	u2	u2	PROPN
ejpam-6004	240	6	−	−	PROPN
ejpam-6004	241	1	s(s−	s(s−	PROPN
ejpam-6004	242	1	1)v2	1)v2	NUM
ejpam-6004	242	2	=	=	PUNCT
ejpam-6004	243	1	−24	−24	ADV
ejpam-6004	243	2	is	be	AUX
ejpam-6004	243	3	not	not	PART
ejpam-6004	243	4	solvable	solvable	ADJ
ejpam-6004	243	5	for	for	ADP
ejpam-6004	243	6	s	s	NOUN
ejpam-6004	243	7	=	=	SYM
ejpam-6004	243	8	1	1	NUM
ejpam-6004	243	9	,	,	PUNCT
ejpam-6004	243	10	3	3	NUM
ejpam-6004	243	11	,	,	PUNCT
ejpam-6004	243	12	4	4	NUM
ejpam-6004	243	13	,	,	PUNCT
ejpam-6004	243	14	6	6	NUM
ejpam-6004	243	15	,	,	PUNCT
ejpam-6004	243	16	7	7	NUM
ejpam-6004	243	17	,	,	PUNCT
ejpam-6004	243	18	8	8	NUM
ejpam-6004	243	19	,	,	PUNCT
ejpam-6004	243	20	9	9	NUM
ejpam-6004	243	21	,	,	PUNCT
ejpam-6004	243	22	10	10	NUM
ejpam-6004	243	23	,	,	PUNCT
ejpam-6004	243	24	11	11	NUM
ejpam-6004	243	25	,	,	PUNCT
ejpam-6004	243	26	12	12	NUM
ejpam-6004	243	27	,	,	PUNCT
ejpam-6004	243	28	13	13	NUM
ejpam-6004	243	29	,	,	PUNCT
ejpam-6004	243	30	14	14	NUM
ejpam-6004	243	31	,	,	PUNCT
ejpam-6004	243	32	15	15	NUM
ejpam-6004	243	33	and	and	CCONJ
ejpam-6004	243	34	16	16	NUM
ejpam-6004	243	35	.	.	PUNCT
ejpam-6004	244	1	for	for	ADP
ejpam-6004	244	2	s	s	NOUN
ejpam-6004	244	3	=	=	SYM
ejpam-6004	244	4	2	2	NUM
ejpam-6004	244	5	,	,	PUNCT
ejpam-6004	244	6	it	it	PRON
ejpam-6004	244	7	is	be	AUX
ejpam-6004	244	8	easy	easy	ADJ
ejpam-6004	244	9	to	to	PART
ejpam-6004	244	10	see	see	VERB
ejpam-6004	244	11	that	that	SCONJ
ejpam-6004	244	12	if	if	SCONJ
ejpam-6004	244	13	u2	u2	NOUN
ejpam-6004	244	14	−	−	PROPN
ejpam-6004	244	15	2v2	2v2	NUM
ejpam-6004	244	16	=	=	SYM
ejpam-6004	244	17	−16	−16	PROPN
ejpam-6004	244	18	,	,	PUNCT
ejpam-6004	244	19	then	then	ADV
ejpam-6004	244	20	u	u	NOUN
ejpam-6004	244	21	is	be	AUX
ejpam-6004	244	22	even	even	ADV
ejpam-6004	244	23	.	.	PUNCT
ejpam-6004	245	1	thus	thus	ADV
ejpam-6004	245	2	4	4	NUM
ejpam-6004	245	3	|	|	NOUN
ejpam-6004	245	4	2v2	2v2	NUM
ejpam-6004	245	5	and	and	CCONJ
ejpam-6004	245	6	this	this	PRON
ejpam-6004	245	7	implies	imply	VERB
ejpam-6004	245	8	that	that	SCONJ
ejpam-6004	245	9	v	v	NOUN
ejpam-6004	245	10	is	be	AUX
ejpam-6004	245	11	even	even	ADV
ejpam-6004	245	12	.	.	PUNCT
ejpam-6004	246	1	hence	hence	ADV
ejpam-6004	246	2	,	,	PUNCT
ejpam-6004	246	3	2	2	NUM
ejpam-6004	246	4	|	|	ADV
ejpam-6004	246	5	gcd(u	gcd(u	NOUN
ejpam-6004	246	6	,	,	PUNCT
ejpam-6004	246	7	v	v	NOUN
ejpam-6004	246	8	)	)	PUNCT
ejpam-6004	246	9	and	and	CCONJ
ejpam-6004	246	10	8	8	NUM
ejpam-6004	246	11	̸∈	̸∈	PROPN
ejpam-6004	246	12	t	t	PROPN
ejpam-6004	246	13	′(64	′(64	NOUN
ejpam-6004	246	14	)	)	PUNCT
ejpam-6004	246	15	.	.	PUNCT
ejpam-6004	247	1	we	we	PRON
ejpam-6004	247	2	see	see	VERB
ejpam-6004	247	3	that	that	SCONJ
ejpam-6004	247	4	(	(	PUNCT
ejpam-6004	247	5	2	2	NUM
ejpam-6004	247	6	,	,	PUNCT
ejpam-6004	247	7	1	1	NUM
ejpam-6004	247	8	)	)	PUNCT
ejpam-6004	247	9	and	and	CCONJ
ejpam-6004	247	10	(	(	PUNCT
ejpam-6004	247	11	16	16	NUM
ejpam-6004	247	12	,	,	PUNCT
ejpam-6004	247	13	1	1	NUM
ejpam-6004	247	14	)	)	PUNCT
ejpam-6004	247	15	are	be	AUX
ejpam-6004	247	16	solutions	solution	NOUN
ejpam-6004	247	17	of	of	ADP
ejpam-6004	247	18	u2	u2	PROPN
ejpam-6004	247	19	−	−	PROPN
ejpam-6004	247	20	s(s	s(s	PROPN
ejpam-6004	247	21	−	−	PROPN
ejpam-6004	248	1	1)v2	1)v2	PROPN
ejpam-6004	248	2	=	=	SYM
ejpam-6004	248	3	−16	−16	PROPN
ejpam-6004	248	4	for	for	ADP
ejpam-6004	248	5	s	s	NOUN
ejpam-6004	248	6	=	=	SYM
ejpam-6004	248	7	5	5	NUM
ejpam-6004	248	8	and	and	CCONJ
ejpam-6004	248	9	s	s	X
ejpam-6004	248	10	=	=	NOUN
ejpam-6004	248	11	17	17	NUM
ejpam-6004	248	12	respectively	respectively	ADV
ejpam-6004	248	13	.	.	PUNCT
ejpam-6004	249	1	hence	hence	ADV
ejpam-6004	249	2	by	by	ADP
ejpam-6004	249	3	lemma	lemma	PROPN
ejpam-6004	249	4	9	9	NUM
ejpam-6004	249	5	,	,	PUNCT
ejpam-6004	249	6	we	we	PRON
ejpam-6004	249	7	obtain	obtain	VERB
ejpam-6004	249	8	that	that	DET
ejpam-6004	249	9	t	t	NOUN
ejpam-6004	249	10	′(64	′(64	NOUN
ejpam-6004	249	11	)	)	PUNCT
ejpam-6004	249	12	=	=	SYM
ejpam-6004	249	13	{	{	PUNCT
ejpam-6004	249	14	20	20	NUM
ejpam-6004	249	15	,	,	PUNCT
ejpam-6004	249	16	68	68	NUM
ejpam-6004	249	17	}	}	PUNCT
ejpam-6004	249	18	as	as	SCONJ
ejpam-6004	249	19	desired	desire	VERB
ejpam-6004	249	20	.	.	PUNCT
ejpam-6004	250	1	theorem	theorem	ADJ
ejpam-6004	250	2	8	8	NUM
ejpam-6004	250	3	.	.	PUNCT
ejpam-6004	250	4	t	t	PROPN
ejpam-6004	250	5	′(128	′(128	NOUN
ejpam-6004	250	6	)	)	PUNCT
ejpam-6004	251	1	=	=	PRON
ejpam-6004	251	2	{	{	PUNCT
ejpam-6004	251	3	132	132	NUM
ejpam-6004	251	4	}	}	PUNCT
ejpam-6004	251	5	.	.	PUNCT
ejpam-6004	252	1	proof	proof	NOUN
ejpam-6004	252	2	.	.	PUNCT
ejpam-6004	253	1	by	by	ADP
ejpam-6004	253	2	lemma	lemma	PROPN
ejpam-6004	253	3	7	7	NUM
ejpam-6004	253	4	and	and	CCONJ
ejpam-6004	253	5	lemma	lemma	PROPN
ejpam-6004	253	6	9	9	NUM
ejpam-6004	253	7	,	,	PUNCT
ejpam-6004	253	8	it	it	PRON
ejpam-6004	253	9	suffices	suffice	VERB
ejpam-6004	253	10	to	to	PART
ejpam-6004	253	11	consider	consider	VERB
ejpam-6004	253	12	the	the	DET
ejpam-6004	253	13	diophantine	diophantine	NOUN
ejpam-6004	253	14	equation	equation	NOUN
ejpam-6004	253	15	u2	u2	NOUN
ejpam-6004	253	16	−	−	PROPN
ejpam-6004	253	17	s(s−	s(s−	PROPN
ejpam-6004	253	18	1)v2	1)v2	NUM
ejpam-6004	253	19	=	=	SYM
ejpam-6004	253	20	−25	−25	NOUN
ejpam-6004	253	21	for	for	ADP
ejpam-6004	253	22	s	s	PROPN
ejpam-6004	253	23	≤	≤	NUM
ejpam-6004	253	24	33	33	NUM
ejpam-6004	253	25	so	so	SCONJ
ejpam-6004	253	26	we	we	PRON
ejpam-6004	253	27	will	will	AUX
ejpam-6004	253	28	show	show	VERB
ejpam-6004	253	29	that	that	SCONJ
ejpam-6004	253	30	the	the	DET
ejpam-6004	253	31	above	above	ADJ
ejpam-6004	253	32	equation	equation	NOUN
ejpam-6004	253	33	has	have	VERB
ejpam-6004	253	34	a	a	DET
ejpam-6004	253	35	solution	solution	NOUN
ejpam-6004	253	36	(	(	PUNCT
ejpam-6004	253	37	u	u	NOUN
ejpam-6004	253	38	,	,	PUNCT
ejpam-6004	253	39	v	v	NOUN
ejpam-6004	253	40	)	)	PUNCT
ejpam-6004	253	41	where	where	SCONJ
ejpam-6004	253	42	gcd(u	gcd(u	PROPN
ejpam-6004	253	43	,	,	PUNCT
ejpam-6004	253	44	v	v	NOUN
ejpam-6004	253	45	)	)	PUNCT
ejpam-6004	253	46	=	=	SYM
ejpam-6004	253	47	1	1	NUM
ejpam-6004	253	48	if	if	SCONJ
ejpam-6004	253	49	and	and	CCONJ
ejpam-6004	253	50	only	only	ADV
ejpam-6004	253	51	if	if	SCONJ
ejpam-6004	253	52	s	s	X
ejpam-6004	253	53	=	=	NOUN
ejpam-6004	253	54	33	33	NUM
ejpam-6004	253	55	.	.	PUNCT
ejpam-6004	254	1	by	by	ADP
ejpam-6004	254	2	lemma	lemma	PROPN
ejpam-6004	254	3	12	12	NUM
ejpam-6004	254	4	,	,	PUNCT
ejpam-6004	254	5	u2	u2	PROPN
ejpam-6004	254	6	−	−	PROPN
ejpam-6004	254	7	s(s	s(s	PROPN
ejpam-6004	254	8	−	−	PROPN
ejpam-6004	255	1	1)v2	1)v2	PROPN
ejpam-6004	255	2	=	=	SYM
ejpam-6004	255	3	−25	−25	NOUN
ejpam-6004	255	4	is	be	AUX
ejpam-6004	255	5	not	not	PART
ejpam-6004	255	6	solvable	solvable	ADJ
ejpam-6004	255	7	for	for	ADP
ejpam-6004	255	8	s	s	NOUN
ejpam-6004	255	9	=	=	SYM
ejpam-6004	255	10	1	1	NUM
ejpam-6004	255	11	,	,	PUNCT
ejpam-6004	255	12	5	5	NUM
ejpam-6004	255	13	,	,	PUNCT
ejpam-6004	255	14	6	6	NUM
ejpam-6004	255	15	,	,	PUNCT
ejpam-6004	255	16	7	7	NUM
ejpam-6004	255	17	,	,	PUNCT
ejpam-6004	255	18	8	8	NUM
ejpam-6004	255	19	,	,	PUNCT
ejpam-6004	255	20	10	10	NUM
ejpam-6004	255	21	,	,	PUNCT
ejpam-6004	255	22	11	11	NUM
ejpam-6004	255	23	,	,	PUNCT
ejpam-6004	255	24	13	13	NUM
ejpam-6004	255	25	,	,	PUNCT
ejpam-6004	255	26	14	14	NUM
ejpam-6004	255	27	,	,	PUNCT
ejpam-6004	255	28	15	15	NUM
ejpam-6004	255	29	,	,	PUNCT
ejpam-6004	255	30	16	16	NUM
ejpam-6004	255	31	,	,	PUNCT
ejpam-6004	255	32	20	20	NUM
ejpam-6004	255	33	,	,	PUNCT
ejpam-6004	255	34	21	21	NUM
ejpam-6004	255	35	,	,	PUNCT
ejpam-6004	255	36	22	22	NUM
ejpam-6004	255	37	,	,	PUNCT
ejpam-6004	255	38	23	23	NUM
ejpam-6004	255	39	,	,	PUNCT
ejpam-6004	255	40	24	24	NUM
ejpam-6004	255	41	,	,	PUNCT
ejpam-6004	255	42	25	25	NUM
ejpam-6004	255	43	,	,	PUNCT
ejpam-6004	255	44	26	26	NUM
ejpam-6004	255	45	,	,	PUNCT
ejpam-6004	255	46	27	27	NUM
ejpam-6004	255	47	,	,	PUNCT
ejpam-6004	255	48	28	28	NUM
ejpam-6004	255	49	,	,	PUNCT
ejpam-6004	255	50	29	29	NUM
ejpam-6004	255	51	,	,	PUNCT
ejpam-6004	255	52	30	30	NUM
ejpam-6004	255	53	,	,	PUNCT
ejpam-6004	255	54	31	31	NUM
ejpam-6004	255	55	and	and	CCONJ
ejpam-6004	255	56	32	32	NUM
ejpam-6004	255	57	.	.	PUNCT
ejpam-6004	256	1	for	for	ADP
ejpam-6004	256	2	s	s	NOUN
ejpam-6004	256	3	=	=	SYM
ejpam-6004	256	4	2	2	NUM
ejpam-6004	256	5	,	,	PUNCT
ejpam-6004	256	6	3	3	NUM
ejpam-6004	256	7	,	,	PUNCT
ejpam-6004	256	8	18	18	NUM
ejpam-6004	256	9	,	,	PUNCT
ejpam-6004	256	10	19	19	NUM
ejpam-6004	256	11	,	,	PUNCT
ejpam-6004	256	12	we	we	PRON
ejpam-6004	256	13	can	can	AUX
ejpam-6004	256	14	see	see	VERB
ejpam-6004	256	15	that	that	SCONJ
ejpam-6004	256	16	if	if	SCONJ
ejpam-6004	256	17	u2	u2	PROPN
ejpam-6004	256	18	−	−	PROPN
ejpam-6004	256	19	s(s	s(s	PROPN
ejpam-6004	256	20	−	−	PROPN
ejpam-6004	256	21	1)v2	1)v2	PROPN
ejpam-6004	256	22	=	=	SYM
ejpam-6004	256	23	−32	−32	X
ejpam-6004	256	24	,	,	PUNCT
ejpam-6004	256	25	then	then	ADV
ejpam-6004	256	26	u	u	NOUN
ejpam-6004	256	27	is	be	AUX
ejpam-6004	256	28	even	even	ADV
ejpam-6004	256	29	.	.	PUNCT
ejpam-6004	257	1	thus	thus	ADV
ejpam-6004	257	2	4	4	NUM
ejpam-6004	257	3	|	|	NOUN
ejpam-6004	257	4	2v2	2v2	NUM
ejpam-6004	257	5	and	and	CCONJ
ejpam-6004	257	6	this	this	PRON
ejpam-6004	257	7	implies	imply	VERB
ejpam-6004	257	8	that	that	SCONJ
ejpam-6004	257	9	v	v	NOUN
ejpam-6004	257	10	is	be	AUX
ejpam-6004	257	11	even	even	ADV
ejpam-6004	257	12	.	.	PUNCT
ejpam-6004	258	1	hence	hence	ADV
ejpam-6004	258	2	,	,	PUNCT
ejpam-6004	258	3	2	2	NUM
ejpam-6004	258	4	|	|	ADV
ejpam-6004	258	5	gcd(u	gcd(u	NOUN
ejpam-6004	258	6	,	,	PUNCT
ejpam-6004	258	7	v	v	NOUN
ejpam-6004	258	8	)	)	PUNCT
ejpam-6004	258	9	.	.	PUNCT
ejpam-6004	259	1	thus	thus	ADV
ejpam-6004	259	2	all	all	DET
ejpam-6004	259	3	solutions	solution	NOUN
ejpam-6004	259	4	of	of	ADP
ejpam-6004	259	5	u2	u2	PROPN
ejpam-6004	259	6	−	−	PROPN
ejpam-6004	259	7	s(s−	s(s−	PROPN
ejpam-6004	259	8	1)v2	1)v2	NUM
ejpam-6004	259	9	=	=	SYM
ejpam-6004	259	10	−32	−32	X
ejpam-6004	259	11	are	be	AUX
ejpam-6004	259	12	not	not	PART
ejpam-6004	259	13	relatively	relatively	ADV
ejpam-6004	259	14	prime	prime	ADJ
ejpam-6004	259	15	.	.	PUNCT
ejpam-6004	260	1	we	we	PRON
ejpam-6004	260	2	next	next	ADV
ejpam-6004	260	3	show	show	VERB
ejpam-6004	260	4	that	that	SCONJ
ejpam-6004	260	5	all	all	DET
ejpam-6004	260	6	solutions	solution	NOUN
ejpam-6004	260	7	of	of	ADP
ejpam-6004	260	8	u2	u2	PROPN
ejpam-6004	260	9	−	−	PROPN
ejpam-6004	260	10	s(s	s(s	PROPN
ejpam-6004	260	11	−	−	PROPN
ejpam-6004	260	12	1)v2	1)v2	PROPN
ejpam-6004	260	13	=	=	SYM
ejpam-6004	260	14	−32	−32	PROPN
ejpam-6004	260	15	,	,	PUNCT
ejpam-6004	260	16	where	where	SCONJ
ejpam-6004	260	17	s	s	VERB
ejpam-6004	260	18	=	=	SYM
ejpam-6004	260	19	4	4	NUM
ejpam-6004	260	20	,	,	PUNCT
ejpam-6004	260	21	9	9	NUM
ejpam-6004	260	22	are	be	AUX
ejpam-6004	260	23	not	not	PART
ejpam-6004	260	24	relatively	relatively	ADV
ejpam-6004	260	25	prime	prime	ADJ
ejpam-6004	260	26	.	.	PUNCT
ejpam-6004	261	1	for	for	ADP
ejpam-6004	261	2	s	s	NOUN
ejpam-6004	261	3	=	=	SYM
ejpam-6004	261	4	4	4	NUM
ejpam-6004	261	5	,	,	PUNCT
ejpam-6004	261	6	we	we	PRON
ejpam-6004	261	7	can	can	AUX
ejpam-6004	261	8	see	see	VERB
ejpam-6004	261	9	that	that	SCONJ
ejpam-6004	261	10	if	if	SCONJ
ejpam-6004	261	11	u2	u2	PROPN
ejpam-6004	261	12	−	−	PROPN
ejpam-6004	261	13	12v2	12v2	NUM
ejpam-6004	261	14	=	=	SYM
ejpam-6004	261	15	−32	−32	X
ejpam-6004	261	16	,	,	PUNCT
ejpam-6004	261	17	then	then	ADV
ejpam-6004	261	18	u	u	NOUN
ejpam-6004	261	19	is	be	AUX
ejpam-6004	261	20	even	even	ADV
ejpam-6004	261	21	.	.	PUNCT
ejpam-6004	262	1	let	let	VERB
ejpam-6004	262	2	u	u	PRON
ejpam-6004	262	3	=	=	PROPN
ejpam-6004	262	4	2	2	NUM
ejpam-6004	262	5	m	m	NOUN
ejpam-6004	262	6	for	for	ADP
ejpam-6004	262	7	some	some	DET
ejpam-6004	262	8	positive	positive	ADJ
ejpam-6004	262	9	integer	integer	NOUN
ejpam-6004	262	10	m.	m.	NOUN
ejpam-6004	262	11	we	we	PRON
ejpam-6004	262	12	have	have	VERB
ejpam-6004	262	13	4m2	4m2	NUM
ejpam-6004	263	1	−	−	PROPN
ejpam-6004	264	1	12v2	12v2	NUM
ejpam-6004	264	2	=	=	SYM
ejpam-6004	264	3	−32	−32	X
ejpam-6004	264	4	.	.	PUNCT
ejpam-6004	265	1	thus	thus	ADV
ejpam-6004	265	2	m2	m2	PROPN
ejpam-6004	265	3	−	−	PROPN
ejpam-6004	265	4	3v2	3v2	NUM
ejpam-6004	265	5	=	=	SYM
ejpam-6004	265	6	−8	−8	X
ejpam-6004	265	7	.	.	PUNCT
ejpam-6004	266	1	it	it	PRON
ejpam-6004	266	2	is	be	AUX
ejpam-6004	266	3	easy	easy	ADJ
ejpam-6004	266	4	to	to	PART
ejpam-6004	266	5	see	see	VERB
ejpam-6004	266	6	that	that	SCONJ
ejpam-6004	266	7	m	m	PROPN
ejpam-6004	266	8	and	and	CCONJ
ejpam-6004	266	9	v	v	AUX
ejpam-6004	266	10	have	have	VERB
ejpam-6004	266	11	the	the	DET
ejpam-6004	266	12	same	same	ADJ
ejpam-6004	266	13	parities	parity	NOUN
ejpam-6004	266	14	.	.	PUNCT
ejpam-6004	267	1	if	if	SCONJ
ejpam-6004	267	2	m	m	PROPN
ejpam-6004	267	3	and	and	CCONJ
ejpam-6004	267	4	v	v	NOUN
ejpam-6004	267	5	are	be	AUX
ejpam-6004	267	6	odd	odd	ADJ
ejpam-6004	267	7	,	,	PUNCT
ejpam-6004	267	8	then	then	ADV
ejpam-6004	267	9	0	0	NUM
ejpam-6004	267	10	≡	≡	PROPN
ejpam-6004	267	11	−8	−8	PROPN
ejpam-6004	267	12	≡	≡	PROPN
ejpam-6004	267	13	m2	m2	PROPN
ejpam-6004	268	1	−	−	PROPN
ejpam-6004	268	2	3v2	3v2	NUM
ejpam-6004	268	3	≡	≡	PROPN
ejpam-6004	268	4	6	6	NUM
ejpam-6004	268	5	(	(	PUNCT
ejpam-6004	268	6	mod	mod	PROPN
ejpam-6004	268	7	8)	8)	NUM
ejpam-6004	268	8	.	.	PUNCT
ejpam-6004	269	1	this	this	PRON
ejpam-6004	269	2	is	be	AUX
ejpam-6004	269	3	a	a	DET
ejpam-6004	269	4	contradiction	contradiction	NOUN
ejpam-6004	269	5	.	.	PUNCT
ejpam-6004	270	1	this	this	PRON
ejpam-6004	270	2	implies	imply	VERB
ejpam-6004	270	3	that	that	SCONJ
ejpam-6004	270	4	m	m	VERB
ejpam-6004	270	5	and	and	CCONJ
ejpam-6004	270	6	v	v	NOUN
ejpam-6004	270	7	are	be	AUX
ejpam-6004	270	8	even	even	ADV
ejpam-6004	270	9	.	.	PUNCT
ejpam-6004	271	1	so	so	ADV
ejpam-6004	271	2	are	be	AUX
ejpam-6004	271	3	u	u	NOUN
ejpam-6004	271	4	and	and	CCONJ
ejpam-6004	271	5	v.	v.	ADP
ejpam-6004	271	6	thus	thus	ADV
ejpam-6004	271	7	all	all	DET
ejpam-6004	271	8	solutions	solution	NOUN
ejpam-6004	271	9	of	of	ADP
ejpam-6004	271	10	u2	u2	NOUN
ejpam-6004	271	11	−	−	PROPN
ejpam-6004	271	12	12v2	12v2	NUM
ejpam-6004	271	13	=	=	SYM
ejpam-6004	271	14	−32	−32	X
ejpam-6004	271	15	are	be	AUX
ejpam-6004	271	16	not	not	PART
ejpam-6004	271	17	relatively	relatively	ADV
ejpam-6004	271	18	prime	prime	ADJ
ejpam-6004	271	19	.	.	PUNCT
ejpam-6004	272	1	for	for	ADP
ejpam-6004	272	2	s	s	NOUN
ejpam-6004	272	3	=	=	SYM
ejpam-6004	272	4	9	9	NUM
ejpam-6004	272	5	,	,	PUNCT
ejpam-6004	272	6	we	we	PRON
ejpam-6004	272	7	can	can	AUX
ejpam-6004	272	8	see	see	VERB
ejpam-6004	272	9	that	that	SCONJ
ejpam-6004	272	10	if	if	SCONJ
ejpam-6004	272	11	u2	u2	PROPN
ejpam-6004	272	12	−	−	PROPN
ejpam-6004	273	1	72v2	72v2	NUM
ejpam-6004	273	2	=	=	SYM
ejpam-6004	274	1	−32	−32	X
ejpam-6004	274	2	,	,	PUNCT
ejpam-6004	274	3	then	then	ADV
ejpam-6004	274	4	u	u	NOUN
ejpam-6004	274	5	is	be	AUX
ejpam-6004	274	6	even	even	ADV
ejpam-6004	274	7	and	and	CCONJ
ejpam-6004	274	8	4	4	NUM
ejpam-6004	274	9	|	|	ADV
ejpam-6004	274	10	u.	u.	AUX
ejpam-6004	274	11	let	let	VERB
ejpam-6004	274	12	u	u	PRON
ejpam-6004	274	13	=	=	NOUN
ejpam-6004	274	14	4	4	NUM
ejpam-6004	274	15	m	m	NOUN
ejpam-6004	274	16	for	for	ADP
ejpam-6004	274	17	some	some	DET
ejpam-6004	274	18	positive	positive	ADJ
ejpam-6004	274	19	integer	integer	NOUN
ejpam-6004	274	20	m.	m.	NOUN
ejpam-6004	274	21	we	we	PRON
ejpam-6004	274	22	have	have	VERB
ejpam-6004	274	23	16m2	16m2	NUM
ejpam-6004	274	24	−	−	NOUN
ejpam-6004	275	1	72v2	72v2	NUM
ejpam-6004	275	2	=	=	SYM
ejpam-6004	275	3	−32	−32	NOUN
ejpam-6004	275	4	.	.	PUNCT
ejpam-6004	276	1	hence	hence	ADV
ejpam-6004	276	2	2m2	2m2	NUM
ejpam-6004	276	3	−	−	NUM
ejpam-6004	276	4	9v2	9v2	NUM
ejpam-6004	276	5	=	=	SYM
ejpam-6004	276	6	−4	−4	X
ejpam-6004	276	7	.	.	PUNCT
ejpam-6004	277	1	it	it	PRON
ejpam-6004	277	2	is	be	AUX
ejpam-6004	277	3	easy	easy	ADJ
ejpam-6004	277	4	to	to	PART
ejpam-6004	277	5	see	see	VERB
ejpam-6004	277	6	that	that	PRON
ejpam-6004	277	7	v	v	NOUN
ejpam-6004	277	8	is	be	AUX
ejpam-6004	277	9	even	even	ADV
ejpam-6004	277	10	.	.	PUNCT
ejpam-6004	278	1	thus	thus	ADV
ejpam-6004	278	2	all	all	DET
ejpam-6004	278	3	solutions	solution	NOUN
ejpam-6004	278	4	of	of	ADP
ejpam-6004	278	5	u2	u2	NOUN
ejpam-6004	278	6	−	−	PROPN
ejpam-6004	278	7	72v2	72v2	NUM
ejpam-6004	278	8	=	=	PUNCT
ejpam-6004	278	9	−32	−32	X
ejpam-6004	278	10	are	be	AUX
ejpam-6004	278	11	not	not	PART
ejpam-6004	278	12	relatively	relatively	ADV
ejpam-6004	278	13	prime	prime	ADJ
ejpam-6004	278	14	.	.	PUNCT
ejpam-6004	279	1	similarly	similarly	ADV
ejpam-6004	279	2	,	,	PUNCT
ejpam-6004	279	3	we	we	PRON
ejpam-6004	279	4	can	can	AUX
ejpam-6004	279	5	show	show	VERB
ejpam-6004	279	6	that	that	SCONJ
ejpam-6004	279	7	for	for	ADP
ejpam-6004	279	8	s	s	NOUN
ejpam-6004	279	9	=	=	SYM
ejpam-6004	279	10	12	12	NUM
ejpam-6004	279	11	,	,	PUNCT
ejpam-6004	279	12	17	17	NUM
ejpam-6004	279	13	,	,	PUNCT
ejpam-6004	279	14	all	all	DET
ejpam-6004	279	15	solutions	solution	NOUN
ejpam-6004	279	16	of	of	ADP
ejpam-6004	279	17	u2	u2	PROPN
ejpam-6004	279	18	−	−	PROPN
ejpam-6004	279	19	s(s−	s(s−	PROPN
ejpam-6004	279	20	1)v2	1)v2	NUM
ejpam-6004	279	21	=	=	SYM
ejpam-6004	279	22	−32	−32	X
ejpam-6004	279	23	are	be	AUX
ejpam-6004	279	24	not	not	PART
ejpam-6004	279	25	relatively	relatively	ADV
ejpam-6004	279	26	prime	prime	ADJ
ejpam-6004	279	27	.	.	PUNCT
ejpam-6004	280	1	for	for	ADP
ejpam-6004	280	2	s	s	NOUN
ejpam-6004	280	3	=	=	SYM
ejpam-6004	280	4	33	33	NUM
ejpam-6004	280	5	,	,	PUNCT
ejpam-6004	280	6	we	we	PRON
ejpam-6004	280	7	see	see	VERB
ejpam-6004	280	8	that	that	SCONJ
ejpam-6004	280	9	(	(	PUNCT
ejpam-6004	280	10	32	32	NUM
ejpam-6004	280	11	,	,	PUNCT
ejpam-6004	280	12	1	1	NUM
ejpam-6004	280	13	)	)	PUNCT
ejpam-6004	280	14	is	be	AUX
ejpam-6004	280	15	a	a	DET
ejpam-6004	280	16	solution	solution	NOUN
ejpam-6004	280	17	of	of	ADP
ejpam-6004	280	18	u2−s(s−1)v2	u2−s(s−1)v2	NOUN
ejpam-6004	280	19	=	=	SYM
ejpam-6004	280	20	−32	−32	NOUN
ejpam-6004	280	21	.	.	PUNCT
ejpam-6004	281	1	hence	hence	ADV
ejpam-6004	281	2	by	by	ADP
ejpam-6004	281	3	lemma	lemma	PROPN
ejpam-6004	281	4	9	9	NUM
ejpam-6004	281	5	,	,	PUNCT
ejpam-6004	281	6	we	we	PRON
ejpam-6004	281	7	obtain	obtain	VERB
ejpam-6004	281	8	that	that	DET
ejpam-6004	281	9	t	t	NOUN
ejpam-6004	281	10	′(128	′(128	NOUN
ejpam-6004	281	11	)	)	PUNCT
ejpam-6004	282	1	=	=	PRON
ejpam-6004	282	2	{	{	PUNCT
ejpam-6004	282	3	132	132	NUM
ejpam-6004	282	4	}	}	PUNCT
ejpam-6004	282	5	as	as	SCONJ
ejpam-6004	282	6	desired	desire	VERB
ejpam-6004	282	7	.	.	PUNCT
ejpam-6004	283	1	s.	s.	PROPN
ejpam-6004	283	2	prugsapitak	prugsapitak	PROPN
ejpam-6004	283	3	,	,	PUNCT
ejpam-6004	283	4	n.	n.	PROPN
ejpam-6004	283	5	thongngam	thongngam	PROPN
ejpam-6004	283	6	/	/	SYM
ejpam-6004	283	7	eur	eur	NOUN
ejpam-6004	283	8	.	.	PUNCT
ejpam-6004	284	1	j.	j.	PROPN
ejpam-6004	284	2	pure	pure	PROPN
ejpam-6004	284	3	appl	appl	PROPN
ejpam-6004	284	4	.	.	PROPN
ejpam-6004	284	5	math	math	PROPN
ejpam-6004	284	6	,	,	PUNCT
ejpam-6004	284	7	18	18	NUM
ejpam-6004	284	8	(	(	PUNCT
ejpam-6004	284	9	2	2	NUM
ejpam-6004	284	10	)	)	PUNCT
ejpam-6004	284	11	(	(	PUNCT
ejpam-6004	284	12	2025	2025	NUM
ejpam-6004	284	13	)	)	PUNCT
ejpam-6004	284	14	,	,	PUNCT
ejpam-6004	284	15	6004	6004	NUM
ejpam-6004	284	16	8	8	NUM
ejpam-6004	284	17	of	of	ADP
ejpam-6004	284	18	8	8	NUM
ejpam-6004	284	19	4	4	NUM
ejpam-6004	284	20	.	.	PUNCT
ejpam-6004	284	21	conclusion	conclusion	NOUN
ejpam-6004	284	22	by	by	ADP
ejpam-6004	284	23	applying	apply	VERB
ejpam-6004	284	24	our	our	PRON
ejpam-6004	284	25	method	method	NOUN
ejpam-6004	285	1	,	,	PUNCT
ejpam-6004	285	2	we	we	PRON
ejpam-6004	285	3	can	can	AUX
ejpam-6004	285	4	efficiently	efficiently	ADV
ejpam-6004	285	5	compute	compute	VERB
ejpam-6004	285	6	t	t	PROPN
ejpam-6004	285	7	′(2n	′(2n	PROPN
ejpam-6004	285	8	)	)	PUNCT
ejpam-6004	285	9	for	for	ADP
ejpam-6004	285	10	n	n	NOUN
ejpam-6004	285	11	=	=	SYM
ejpam-6004	285	12	3	3	NUM
ejpam-6004	285	13	,	,	PUNCT
ejpam-6004	285	14	4	4	NUM
ejpam-6004	285	15	,	,	PUNCT
ejpam-6004	285	16	5	5	NUM
ejpam-6004	285	17	,	,	PUNCT
ejpam-6004	285	18	6	6	NUM
ejpam-6004	285	19	and	and	CCONJ
ejpam-6004	285	20	7	7	NUM
ejpam-6004	285	21	.	.	X
ejpam-6004	286	1	this	this	PRON
ejpam-6004	286	2	,	,	PUNCT
ejpam-6004	286	3	in	in	ADP
ejpam-6004	286	4	turn	turn	NOUN
ejpam-6004	286	5	,	,	PUNCT
ejpam-6004	286	6	reveals	reveal	VERB
ejpam-6004	286	7	the	the	DET
ejpam-6004	286	8	values	value	NOUN
ejpam-6004	286	9	of	of	ADP
ejpam-6004	286	10	t	t	PROPN
ejpam-6004	286	11	(	(	PUNCT
ejpam-6004	286	12	2n	2n	NUM
ejpam-6004	286	13	)	)	PUNCT
ejpam-6004	286	14	for	for	ADP
ejpam-6004	286	15	that	that	DET
ejpam-6004	286	16	range	range	NOUN
ejpam-6004	286	17	.	.	PUNCT
ejpam-6004	287	1	n	n	PROPN
ejpam-6004	287	2	t	t	PROPN
ejpam-6004	287	3	′(2n	′(2n	PROPN
ejpam-6004	287	4	)	)	PUNCT
ejpam-6004	287	5	t	t	PROPN
ejpam-6004	287	6	(	(	PUNCT
ejpam-6004	287	7	2n	2n	NUM
ejpam-6004	287	8	)	)	PUNCT
ejpam-6004	287	9	0	0	PUNCT
ejpam-6004	288	1	{	{	PUNCT
ejpam-6004	288	2	5	5	NUM
ejpam-6004	288	3	}	}	PUNCT
ejpam-6004	288	4	{	{	PUNCT
ejpam-6004	288	5	5	5	NUM
ejpam-6004	288	6	}	}	SYM
ejpam-6004	288	7	1	1	NUM
ejpam-6004	288	8	{	{	PUNCT
ejpam-6004	288	9	6	6	NUM
ejpam-6004	288	10	}	}	PUNCT
ejpam-6004	288	11	{	{	PUNCT
ejpam-6004	288	12	5,6	5,6	NUM
ejpam-6004	288	13	}	}	SYM
ejpam-6004	288	14	2	2	NUM
ejpam-6004	288	15	{	{	PUNCT
ejpam-6004	288	16	8	8	NUM
ejpam-6004	288	17	}	}	PUNCT
ejpam-6004	288	18	{	{	PUNCT
ejpam-6004	288	19	5,6,8	5,6,8	NOUN
ejpam-6004	288	20	}	}	SYM
ejpam-6004	288	21	3	3	NUM
ejpam-6004	288	22	{	{	PUNCT
ejpam-6004	288	23	8,12	8,12	NUM
ejpam-6004	288	24	}	}	PUNCT
ejpam-6004	288	25	{	{	PUNCT
ejpam-6004	288	26	5,6,8,12	5,6,8,12	NUM
ejpam-6004	288	27	}	}	PUNCT
ejpam-6004	288	28	4	4	NUM
ejpam-6004	288	29	{	{	SYM
ejpam-6004	288	30	20	20	NUM
ejpam-6004	288	31	}	}	PUNCT
ejpam-6004	288	32	{	{	PUNCT
ejpam-6004	288	33	5,6,8,12,20	5,6,8,12,20	NOUN
ejpam-6004	288	34	}	}	PUNCT
ejpam-6004	288	35	5	5	NUM
ejpam-6004	288	36	{	{	PUNCT
ejpam-6004	288	37	16,36	16,36	NUM
ejpam-6004	288	38	}	}	PUNCT
ejpam-6004	288	39	{	{	PUNCT
ejpam-6004	288	40	5,6,8,12,16,20,36	5,6,8,12,16,20,36	NUM
ejpam-6004	288	41	}	}	PUNCT
ejpam-6004	288	42	6	6	NUM
ejpam-6004	288	43	{	{	PUNCT
ejpam-6004	288	44	20,68	20,68	NUM
ejpam-6004	288	45	}	}	PUNCT
ejpam-6004	288	46	{	{	PUNCT
ejpam-6004	288	47	5,6,8,12,16,20,36,68	5,6,8,12,16,20,36,68	NUM
ejpam-6004	288	48	}	}	PUNCT
ejpam-6004	288	49	7	7	NUM
ejpam-6004	288	50	{	{	PUNCT
ejpam-6004	288	51	132	132	NUM
ejpam-6004	288	52	}	}	PUNCT
ejpam-6004	288	53	{	{	PUNCT
ejpam-6004	288	54	5,6,8,12,16,20,36,68,132	5,6,8,12,16,20,36,68,132	NUM
ejpam-6004	288	55	}	}	PUNCT
ejpam-6004	288	56	acknowledgements	acknowledgement	NOUN
ejpam-6004	288	57	we	we	PRON
ejpam-6004	288	58	thank	thank	VERB
ejpam-6004	288	59	the	the	DET
ejpam-6004	288	60	referee	referee	NOUN
ejpam-6004	288	61	for	for	ADP
ejpam-6004	288	62	valuable	valuable	ADJ
ejpam-6004	288	63	comments	comment	NOUN
ejpam-6004	288	64	and	and	CCONJ
ejpam-6004	288	65	suggestions	suggestion	NOUN
ejpam-6004	288	66	.	.	PUNCT
ejpam-6004	289	1	references	reference	NOUN
ejpam-6004	289	2	[	[	X
ejpam-6004	289	3	1	1	NUM
ejpam-6004	289	4	]	]	PUNCT
ejpam-6004	289	5	r.	r.	PROPN
ejpam-6004	289	6	keskin	keskin	PROPN
ejpam-6004	289	7	,	,	PUNCT
ejpam-6004	289	8	o.	o.	PROPN
ejpam-6004	289	9	karaatli	karaatli	PROPN
ejpam-6004	289	10	,	,	PUNCT
ejpam-6004	289	11	and	and	CCONJ
ejpam-6004	289	12	z.	z.	PROPN
ejpam-6004	289	13	siar	siar	PROPN
ejpam-6004	289	14	.	.	PUNCT
ejpam-6004	290	1	on	on	ADP
ejpam-6004	290	2	the	the	DET
ejpam-6004	290	3	diophantine	diophantine	NOUN
ejpam-6004	290	4	equation	equation	NOUN
ejpam-6004	290	5	x2−kxy+y2	x2−kxy+y2	PROPN
ejpam-6004	291	1	+	+	PROPN
ejpam-6004	291	2	2n	2n	X
ejpam-6004	291	3	=	=	SYM
ejpam-6004	291	4	0	0	X
ejpam-6004	291	5	.	.	PUNCT
ejpam-6004	291	6	miskolc	miskolc	ADJ
ejpam-6004	291	7	mathematical	mathematical	ADJ
ejpam-6004	291	8	notes	note	NOUN
ejpam-6004	291	9	,	,	PUNCT
ejpam-6004	291	10	13:375–388	13:375–388	NUM
ejpam-6004	291	11	,	,	PUNCT
ejpam-6004	291	12	2012	2012	NUM
ejpam-6004	291	13	.	.	PUNCT
ejpam-6004	292	1	[	[	X
ejpam-6004	292	2	2	2	NUM
ejpam-6004	292	3	]	]	PUNCT
ejpam-6004	292	4	r.	r.	PROPN
ejpam-6004	292	5	keskin	keskin	PROPN
ejpam-6004	292	6	,	,	PUNCT
ejpam-6004	292	7	z.	z.	PROPN
ejpam-6004	292	8	siar	siar	PROPN
ejpam-6004	292	9	,	,	PUNCT
ejpam-6004	292	10	and	and	CCONJ
ejpam-6004	292	11	o.	o.	PROPN
ejpam-6004	292	12	karaatli	karaatli	PROPN
ejpam-6004	292	13	.	.	PUNCT
ejpam-6004	293	1	on	on	ADP
ejpam-6004	293	2	the	the	DET
ejpam-6004	293	3	diophantine	diophantine	NOUN
ejpam-6004	293	4	equation	equation	NOUN
ejpam-6004	293	5	x2−kxy+y2−2n	x2−kxy+y2−2n	PUNCT
ejpam-6004	293	6	=	=	PUNCT
ejpam-6004	293	7	0	0	X
ejpam-6004	293	8	.	.	PUNCT
ejpam-6004	294	1	czechoslovak	czechoslovak	PROPN
ejpam-6004	294	2	mathematical	mathematical	PROPN
ejpam-6004	294	3	journal	journal	PROPN
ejpam-6004	294	4	,	,	PUNCT
ejpam-6004	294	5	63:783–797	63:783–797	PROPN
ejpam-6004	294	6	,	,	PUNCT
ejpam-6004	294	7	2013	2013	NUM
ejpam-6004	294	8	.	.	PUNCT
ejpam-6004	295	1	[	[	X
ejpam-6004	295	2	3	3	X
ejpam-6004	295	3	]	]	X
ejpam-6004	295	4	r.	r.	PROPN
ejpam-6004	295	5	boumahdi	boumahdi	PROPN
ejpam-6004	295	6	,	,	PUNCT
ejpam-6004	295	7	o.	o.	PROPN
ejpam-6004	295	8	kihel	kihel	PROPN
ejpam-6004	295	9	,	,	PUNCT
ejpam-6004	295	10	and	and	CCONJ
ejpam-6004	295	11	s.	s.	PROPN
ejpam-6004	295	12	mavechan	mavechan	PROPN
ejpam-6004	295	13	.	.	PUNCT
ejpam-6004	296	1	proof	proof	NOUN
ejpam-6004	296	2	of	of	ADP
ejpam-6004	296	3	the	the	DET
ejpam-6004	296	4	conjecture	conjecture	NOUN
ejpam-6004	296	5	of	of	ADP
ejpam-6004	296	6	keskin	keskin	PROPN
ejpam-6004	296	7	,	,	PUNCT
ejpam-6004	296	8	siar	siar	NOUN
ejpam-6004	296	9	and	and	CCONJ
ejpam-6004	296	10	karaatli	karaatli	PROPN
ejpam-6004	296	11	.	.	PROPN
ejpam-6004	297	1	annales	annales	PROPN
ejpam-6004	297	2	fennici	fennici	PROPN
ejpam-6004	297	3	mathematici	mathematici	PROPN
ejpam-6004	297	4	,	,	PUNCT
ejpam-6004	297	5	43:557–561	43:557–561	PROPN
ejpam-6004	297	6	,	,	PUNCT
ejpam-6004	297	7	2018	2018	NUM
ejpam-6004	297	8	.	.	PUNCT
ejpam-6004	298	1	[	[	X
ejpam-6004	298	2	4	4	X
ejpam-6004	298	3	]	]	X
ejpam-6004	298	4	o.	o.	PROPN
ejpam-6004	298	5	karaatli	karaatli	PROPN
ejpam-6004	298	6	and	and	CCONJ
ejpam-6004	298	7	z.	z.	PROPN
ejpam-6004	298	8	siar	siar	PROPN
ejpam-6004	298	9	.	.	PUNCT
ejpam-6004	299	1	on	on	ADP
ejpam-6004	299	2	the	the	DET
ejpam-6004	299	3	diophantine	diophantine	NOUN
ejpam-6004	299	4	equation	equation	NOUN
ejpam-6004	299	5	x2	x2	NOUN
ejpam-6004	299	6	−	−	PROPN
ejpam-6004	299	7	kxy	kxy	NOUN
ejpam-6004	299	8	+	+	CCONJ
ejpam-6004	299	9	ky2	ky2	NOUN
ejpam-6004	299	10	+	+	CCONJ
ejpam-6004	299	11	ly	ly	X
ejpam-6004	299	12	=	=	SYM
ejpam-6004	299	13	0	0	NUM
ejpam-6004	299	14	,	,	PUNCT
ejpam-6004	299	15	l	l	PROPN
ejpam-6004	299	16	∈	∈	PROPN
ejpam-6004	299	17	{	{	PUNCT
ejpam-6004	299	18	1	1	NUM
ejpam-6004	299	19	,	,	PUNCT
ejpam-6004	299	20	2	2	NUM
ejpam-6004	299	21	,	,	PUNCT
ejpam-6004	299	22	4	4	NUM
ejpam-6004	299	23	,	,	PUNCT
ejpam-6004	299	24	8	8	NUM
ejpam-6004	299	25	}	}	PUNCT
ejpam-6004	299	26	.	.	PUNCT
ejpam-6004	300	1	afr	afr	PROPN
ejpam-6004	300	2	.	.	PUNCT
ejpam-6004	301	1	diaspora	diaspora	PROPN
ejpam-6004	301	2	j.	j.	PROPN
ejpam-6004	301	3	math	math	PROPN
ejpam-6004	301	4	.	.	PUNCT
ejpam-6004	301	5	,	,	PUNCT
ejpam-6004	301	6	14:24–29	14:24–29	NUM
ejpam-6004	301	7	,	,	PUNCT
ejpam-6004	301	8	2012	2012	NUM
ejpam-6004	301	9	.	.	PUNCT
ejpam-6004	302	1	[	[	X
ejpam-6004	302	2	5	5	X
ejpam-6004	302	3	]	]	PUNCT
ejpam-6004	302	4	s.	s.	PROPN
ejpam-6004	302	5	mavechan	mavechan	PROPN
ejpam-6004	302	6	.	.	PUNCT
ejpam-6004	303	1	on	on	ADP
ejpam-6004	303	2	the	the	DET
ejpam-6004	303	3	diophantine	diophantine	NOUN
ejpam-6004	303	4	equation	equation	NOUN
ejpam-6004	303	5	x2	x2	NOUN
ejpam-6004	303	6	−	−	PROPN
ejpam-6004	303	7	kxy	kxy	NOUN
ejpam-6004	303	8	+	+	CCONJ
ejpam-6004	303	9	y2	y2	PROPN
ejpam-6004	303	10	+	+	CCONJ
ejpam-6004	303	11	ly	ly	X
ejpam-6004	303	12	=	=	SYM
ejpam-6004	303	13	0	0	NUM
ejpam-6004	303	14	,	,	PUNCT
ejpam-6004	303	15	l	l	NOUN
ejpam-6004	303	16	=	=	SYM
ejpam-6004	303	17	2n	2n	NUM
ejpam-6004	303	18	.	.	PUNCT
ejpam-6004	304	1	an	an	PRON
ejpam-6004	304	2	.	.	PUNCT
ejpam-6004	304	3	univ	univ	PROPN
ejpam-6004	304	4	.	.	PUNCT
ejpam-6004	305	1	vest	vest	PROPN
ejpam-6004	305	2	timiss	timiss	PROPN
ejpam-6004	305	3	.	.	PUNCT
ejpam-6004	305	4	ser	ser	PROPN
ejpam-6004	305	5	.	.	PUNCT
ejpam-6004	306	1	mat	mat	NOUN
ejpam-6004	306	2	-	-	PUNCT
ejpam-6004	306	3	inform	inform	NOUN
ejpam-6004	306	4	.	.	PUNCT
ejpam-6004	306	5	,	,	PUNCT
ejpam-6004	306	6	55:115–118	55:115–118	NUM
ejpam-6004	306	7	,	,	PUNCT
ejpam-6004	306	8	2017	2017	NUM
ejpam-6004	306	9	.	.	PUNCT
ejpam-6004	307	1	[	[	X
ejpam-6004	307	2	6	6	NUM
ejpam-6004	307	3	]	]	PUNCT
ejpam-6004	307	4	s.	s.	PROPN
ejpam-6004	307	5	alkabouss	alkabouss	PROPN
ejpam-6004	307	6	,	,	PUNCT
ejpam-6004	307	7	b.	b.	PROPN
ejpam-6004	307	8	benseba	benseba	PROPN
ejpam-6004	307	9	,	,	PUNCT
ejpam-6004	307	10	n.	n.	PROPN
ejpam-6004	307	11	berbara	berbara	PROPN
ejpam-6004	307	12	s.	s.	PROPN
ejpam-6004	307	13	earp	earp	PROPN
ejpam-6004	307	14	-	-	PUNCT
ejpam-6004	307	15	lynch	lynch	PROPN
ejpam-6004	307	16	,	,	PUNCT
ejpam-6004	307	17	and	and	CCONJ
ejpam-6004	307	18	f.	f.	PROPN
ejpam-6004	307	19	luca	luca	PROPN
ejpam-6004	307	20	.	.	PUNCT
ejpam-6004	308	1	a	a	DET
ejpam-6004	308	2	note	note	NOUN
ejpam-6004	308	3	on	on	ADP
ejpam-6004	308	4	the	the	DET
ejpam-6004	308	5	diophantine	diophantine	NOUN
ejpam-6004	308	6	equation	equation	NOUN
ejpam-6004	308	7	x2	x2	NOUN
ejpam-6004	308	8	−	−	PROPN
ejpam-6004	308	9	kxy	kxy	NOUN
ejpam-6004	308	10	+	+	CCONJ
ejpam-6004	308	11	ky2	ky2	NOUN
ejpam-6004	308	12	+	+	CCONJ
ejpam-6004	308	13	ly	ly	X
ejpam-6004	308	14	=	=	SYM
ejpam-6004	308	15	0	0	PROPN
ejpam-6004	308	16	.	.	PROPN
ejpam-6004	308	17	mathematica	mathematica	PROPN
ejpam-6004	308	18	,	,	PUNCT
ejpam-6004	308	19	63:151–157	63:151–157	PROPN
ejpam-6004	308	20	,	,	PUNCT
ejpam-6004	308	21	2021	2021	NUM
ejpam-6004	308	22	.	.	PUNCT
ejpam-6004	309	1	[	[	X
ejpam-6004	309	2	7	7	X
ejpam-6004	309	3	]	]	X
ejpam-6004	309	4	s.	s.	PROPN
ejpam-6004	309	5	prugsapitak	prugsapitak	PROPN
ejpam-6004	309	6	and	and	CCONJ
ejpam-6004	309	7	n.	n.	PROPN
ejpam-6004	309	8	thongngam	thongngam	NOUN
ejpam-6004	309	9	.	.	PUNCT
ejpam-6004	310	1	on	on	ADP
ejpam-6004	310	2	the	the	DET
ejpam-6004	310	3	diophantine	diophantine	NOUN
ejpam-6004	310	4	equation	equation	NOUN
ejpam-6004	310	5	x2−kxy+ky2	x2−kxy+ky2	PUNCT
ejpam-6004	311	1	+	+	ADJ
ejpam-6004	311	2	3ny	3ny	ADJ
ejpam-6004	311	3	=	=	SYM
ejpam-6004	311	4	0	0	NUM
ejpam-6004	311	5	,	,	PUNCT
ejpam-6004	311	6	n	n	NOUN
ejpam-6004	311	7	=	=	SYM
ejpam-6004	311	8	1	1	NUM
ejpam-6004	311	9	,	,	PUNCT
ejpam-6004	311	10	2	2	NUM
ejpam-6004	311	11	,	,	PUNCT
ejpam-6004	311	12	3	3	NUM
ejpam-6004	311	13	.	.	NOUN
ejpam-6004	311	14	thai	thai	PROPN
ejpam-6004	311	15	journal	journal	PROPN
ejpam-6004	311	16	of	of	ADP
ejpam-6004	311	17	mathematics	mathematic	NOUN
ejpam-6004	311	18	,	,	PUNCT
ejpam-6004	311	19	22:503–508	22:503–508	NUM
ejpam-6004	311	20	,	,	PUNCT
ejpam-6004	311	21	2024	2024	NUM
ejpam-6004	311	22	.	.	PUNCT
ejpam-6004	312	1	[	[	X
ejpam-6004	312	2	8	8	X
ejpam-6004	312	3	]	]	PUNCT
ejpam-6004	312	4	t.	t.	PROPN
ejpam-6004	312	5	nagell	nagell	PROPN
ejpam-6004	312	6	.	.	PUNCT
ejpam-6004	313	1	introduction	introduction	NOUN
ejpam-6004	313	2	to	to	ADP
ejpam-6004	313	3	number	number	NOUN
ejpam-6004	313	4	theory	theory	NOUN
ejpam-6004	313	5	.	.	PUNCT
ejpam-6004	314	1	chelsea	chelsea	PROPN
ejpam-6004	314	2	,	,	PUNCT
ejpam-6004	314	3	1981	1981	NUM
ejpam-6004	314	4	.	.	PUNCT
ejpam-6004	315	1	[	[	X
ejpam-6004	315	2	9	9	NUM
ejpam-6004	315	3	]	]	X
ejpam-6004	315	4	r.	r.	PROPN
ejpam-6004	315	5	keskin	keskin	PROPN
ejpam-6004	315	6	and	and	CCONJ
ejpam-6004	315	7	b.	b.	PROPN
ejpam-6004	315	8	demirturk	demirturk	PROPN
ejpam-6004	315	9	.	.	PUNCT
ejpam-6004	316	1	solutions	solution	NOUN
ejpam-6004	316	2	of	of	ADP
ejpam-6004	316	3	some	some	DET
ejpam-6004	316	4	diophantine	diophantine	NOUN
ejpam-6004	316	5	equations	equation	NOUN
ejpam-6004	316	6	using	use	VERB
ejpam-6004	316	7	generalized	generalized	ADJ
ejpam-6004	316	8	fibonacci	fibonacci	NOUN
ejpam-6004	316	9	and	and	CCONJ
ejpam-6004	316	10	lucas	lucas	PROPN
ejpam-6004	316	11	sequences	sequences	PROPN
ejpam-6004	316	12	.	.	PUNCT
ejpam-6004	317	1	ars	ars	PROPN
ejpam-6004	317	2	combinatoria	combinatoria	PROPN
ejpam-6004	317	3	,	,	PUNCT
ejpam-6004	317	4	111:161–179	111:161–179	NUM
ejpam-6004	317	5	,	,	PUNCT
ejpam-6004	317	6	2013	2013	NUM
ejpam-6004	317	7	.	.	PUNCT