id	sid	tid	token	lemma	pos
ejpam-4508	1	1	european	european	PROPN
ejpam-4508	1	2	journal	journal	PROPN
ejpam-4508	1	3	of	of	ADP
ejpam-4508	1	4	pure	pure	ADJ
ejpam-4508	1	5	and	and	CCONJ
ejpam-4508	1	6	applied	apply	VERB
ejpam-4508	1	7	mathematics	mathematic	NOUN
ejpam-4508	1	8	vol	vol	NOUN
ejpam-4508	1	9	.	.	PROPN
ejpam-4508	2	1	15	15	NUM
ejpam-4508	2	2	,	,	PUNCT
ejpam-4508	2	3	no	no	INTJ
ejpam-4508	2	4	.	.	NOUN
ejpam-4508	2	5	4	4	NUM
ejpam-4508	2	6	,	,	PUNCT
ejpam-4508	2	7	2022	2022	NUM
ejpam-4508	2	8	,	,	PUNCT
ejpam-4508	2	9	1593	1593	NUM
ejpam-4508	2	10	-	-	SYM
ejpam-4508	2	11	1596	1596	NUM
ejpam-4508	2	12	issn	issn	VERB
ejpam-4508	2	13	1307	1307	NUM
ejpam-4508	2	14	-	-	SYM
ejpam-4508	2	15	5543	5543	NUM
ejpam-4508	2	16	–	–	PUNCT
ejpam-4508	3	1	ejpam.com	ejpam.com	X
ejpam-4508	3	2	published	publish	VERB
ejpam-4508	3	3	by	by	ADP
ejpam-4508	3	4	new	new	PROPN
ejpam-4508	3	5	york	york	PROPN
ejpam-4508	3	6	business	business	PROPN
ejpam-4508	3	7	global	global	PROPN
ejpam-4508	3	8	on	on	ADP
ejpam-4508	3	9	the	the	DET
ejpam-4508	3	10	diophantine	diophantine	NOUN
ejpam-4508	3	11	equation	equation	NOUN
ejpam-4508	3	12	(	(	PUNCT
ejpam-4508	3	13	p+	p+	NOUN
ejpam-4508	3	14	4n)x	4n)x	NUM
ejpam-4508	3	15	+	+	CCONJ
ejpam-4508	3	16	py	py	PROPN
ejpam-4508	3	17	=	=	SYM
ejpam-4508	3	18	z2	z2	PROPN
ejpam-4508	3	19	wachirarak	wachirarak	PROPN
ejpam-4508	3	20	orosram1,∗	orosram1,∗	PROPN
ejpam-4508	3	21	,	,	PUNCT
ejpam-4508	3	22	kitsanuphong	kitsanuphong	PROPN
ejpam-4508	3	23	makonwattana1	makonwattana1	PROPN
ejpam-4508	3	24	,	,	PUNCT
ejpam-4508	3	25	saichon	saichon	PROPN
ejpam-4508	3	26	khongsawat1	khongsawat1	PROPN
ejpam-4508	3	27	1department	1department	NUM
ejpam-4508	3	28	of	of	ADP
ejpam-4508	3	29	mathematics	mathematic	NOUN
ejpam-4508	3	30	,	,	PUNCT
ejpam-4508	3	31	faculty	faculty	NOUN
ejpam-4508	3	32	of	of	ADP
ejpam-4508	3	33	science	science	NOUN
ejpam-4508	3	34	,	,	PUNCT
ejpam-4508	3	35	buriram	buriram	NOUN
ejpam-4508	3	36	rajabhat	rajabhat	PROPN
ejpam-4508	3	37	university	university	NOUN
ejpam-4508	3	38	,	,	PUNCT
ejpam-4508	3	39	buriram	buriram	NOUN
ejpam-4508	3	40	31000	31000	NUM
ejpam-4508	3	41	,	,	PUNCT
ejpam-4508	3	42	thailand	thailand	PROPN
ejpam-4508	3	43	abstract	abstract	NOUN
ejpam-4508	3	44	.	.	PUNCT
ejpam-4508	4	1	in	in	ADP
ejpam-4508	4	2	this	this	DET
ejpam-4508	4	3	paper	paper	NOUN
ejpam-4508	4	4	,	,	PUNCT
ejpam-4508	4	5	we	we	PRON
ejpam-4508	4	6	study	study	VERB
ejpam-4508	4	7	the	the	DET
ejpam-4508	4	8	diophantine	diophantine	NOUN
ejpam-4508	4	9	equation	equation	NOUN
ejpam-4508	4	10	(	(	PUNCT
ejpam-4508	4	11	p	p	X
ejpam-4508	5	1	+	+	NOUN
ejpam-4508	5	2	4n)x	4n)x	NUM
ejpam-4508	6	1	+	+	CCONJ
ejpam-4508	6	2	py	py	PROPN
ejpam-4508	6	3	=	=	SYM
ejpam-4508	6	4	z2	z2	PROPN
ejpam-4508	6	5	,	,	PUNCT
ejpam-4508	6	6	where	where	SCONJ
ejpam-4508	6	7	n	n	PRON
ejpam-4508	6	8	is	be	AUX
ejpam-4508	6	9	a	a	DET
ejpam-4508	6	10	non	non	ADJ
ejpam-4508	6	11	-	-	ADJ
ejpam-4508	6	12	negative	negative	ADJ
ejpam-4508	6	13	integer	integer	NOUN
ejpam-4508	6	14	and	and	CCONJ
ejpam-4508	6	15	p	p	NOUN
ejpam-4508	6	16	,	,	PUNCT
ejpam-4508	6	17	p+4n	p+4n	PROPN
ejpam-4508	6	18	are	be	AUX
ejpam-4508	6	19	prime	prime	ADJ
ejpam-4508	6	20	numbers	number	NOUN
ejpam-4508	6	21	such	such	ADJ
ejpam-4508	6	22	that	that	SCONJ
ejpam-4508	6	23	p	p	PROPN
ejpam-4508	6	24	≡	≡	PROPN
ejpam-4508	6	25	7	7	NUM
ejpam-4508	6	26	(	(	PUNCT
ejpam-4508	6	27	mod	mod	PROPN
ejpam-4508	6	28	12	12	NUM
ejpam-4508	6	29	)	)	PUNCT
ejpam-4508	6	30	.	.	PUNCT
ejpam-4508	7	1	we	we	PRON
ejpam-4508	7	2	show	show	VERB
ejpam-4508	7	3	that	that	SCONJ
ejpam-4508	7	4	the	the	DET
ejpam-4508	7	5	non	non	ADJ
ejpam-4508	7	6	-	-	ADJ
ejpam-4508	7	7	negative	negative	ADJ
ejpam-4508	7	8	integer	integer	NOUN
ejpam-4508	7	9	solutions	solution	NOUN
ejpam-4508	7	10	of	of	ADP
ejpam-4508	7	11	such	such	ADJ
ejpam-4508	7	12	equation	equation	NOUN
ejpam-4508	7	13	are	be	AUX
ejpam-4508	7	14	(	(	PUNCT
ejpam-4508	7	15	x	x	NOUN
ejpam-4508	7	16	,	,	PUNCT
ejpam-4508	7	17	y	y	PROPN
ejpam-4508	7	18	,	,	PUNCT
ejpam-4508	7	19	z	z	NOUN
ejpam-4508	7	20	)	)	PUNCT
ejpam-4508	7	21	∈	∈	NOUN
ejpam-4508	7	22	{	{	PUNCT
ejpam-4508	7	23	(	(	PUNCT
ejpam-4508	7	24	0	0	NUM
ejpam-4508	7	25	,	,	PUNCT
ejpam-4508	7	26	1	1	NUM
ejpam-4508	7	27	,	,	PUNCT
ejpam-4508	7	28	√	√	PROPN
ejpam-4508	7	29	p+	p+	PROPN
ejpam-4508	7	30	1)}∪{(1	1)}∪{(1	NUM
ejpam-4508	7	31	,	,	PUNCT
ejpam-4508	7	32	0	0	NUM
ejpam-4508	7	33	,	,	PUNCT
ejpam-4508	7	34	2	2	NUM
ejpam-4508	7	35	√	√	NUM
ejpam-4508	7	36	n+	n+	ADP
ejpam-4508	7	37	p+1	p+1	NOUN
ejpam-4508	7	38	4	4	NUM
ejpam-4508	7	39	)	)	PUNCT
ejpam-4508	7	40	}	}	PUNCT
ejpam-4508	7	41	,	,	PUNCT
ejpam-4508	7	42	where	where	SCONJ
ejpam-4508	7	43	√	√	PRON
ejpam-4508	7	44	p+	p+	VERB
ejpam-4508	7	45	1	1	NUM
ejpam-4508	7	46	and	and	CCONJ
ejpam-4508	7	47	√	√	NUM
ejpam-4508	7	48	n+	n+	NUM
ejpam-4508	7	49	p+1	p+1	NOUN
ejpam-4508	7	50	4	4	NUM
ejpam-4508	7	51	are	be	AUX
ejpam-4508	7	52	integers	integer	NOUN
ejpam-4508	7	53	.	.	PUNCT
ejpam-4508	8	1	2020	2020	NUM
ejpam-4508	8	2	mathematics	mathematic	NOUN
ejpam-4508	8	3	subject	subject	NOUN
ejpam-4508	8	4	classifications	classification	NOUN
ejpam-4508	8	5	:	:	PUNCT
ejpam-4508	8	6	11d61	11d61	NUM
ejpam-4508	8	7	key	key	ADJ
ejpam-4508	8	8	words	word	NOUN
ejpam-4508	8	9	and	and	CCONJ
ejpam-4508	8	10	phrases	phrase	NOUN
ejpam-4508	8	11	:	:	PUNCT
ejpam-4508	8	12	exponential	exponential	ADJ
ejpam-4508	8	13	diophantine	diophantine	NOUN
ejpam-4508	8	14	equation	equation	NOUN
ejpam-4508	8	15	,	,	PUNCT
ejpam-4508	8	16	catalan	catalan	NOUN
ejpam-4508	8	17	’s	’s	PART
ejpam-4508	8	18	conjecture	conjecture	NOUN
ejpam-4508	8	19	1	1	NUM
ejpam-4508	8	20	.	.	PUNCT
ejpam-4508	8	21	introduction	introduction	NOUN
ejpam-4508	8	22	a	a	DET
ejpam-4508	8	23	problem	problem	NOUN
ejpam-4508	8	24	related	relate	VERB
ejpam-4508	8	25	to	to	ADP
ejpam-4508	8	26	the	the	DET
ejpam-4508	8	27	diophantine	diophantine	NOUN
ejpam-4508	8	28	equation	equation	NOUN
ejpam-4508	8	29	has	have	AUX
ejpam-4508	8	30	been	be	AUX
ejpam-4508	8	31	investigated	investigate	VERB
ejpam-4508	8	32	by	by	ADP
ejpam-4508	8	33	many	many	ADJ
ejpam-4508	8	34	researchers	researcher	NOUN
ejpam-4508	8	35	.	.	PUNCT
ejpam-4508	9	1	it	it	PRON
ejpam-4508	9	2	is	be	AUX
ejpam-4508	9	3	considered	consider	VERB
ejpam-4508	9	4	one	one	NUM
ejpam-4508	9	5	of	of	ADP
ejpam-4508	9	6	the	the	DET
ejpam-4508	9	7	significant	significant	ADJ
ejpam-4508	9	8	problems	problem	NOUN
ejpam-4508	9	9	in	in	ADP
ejpam-4508	9	10	elementary	elementary	ADJ
ejpam-4508	9	11	number	number	NOUN
ejpam-4508	9	12	theory	theory	NOUN
ejpam-4508	9	13	.	.	PUNCT
ejpam-4508	10	1	the	the	DET
ejpam-4508	10	2	proving	proving	NOUN
ejpam-4508	10	3	method	method	NOUN
ejpam-4508	10	4	mainly	mainly	ADV
ejpam-4508	10	5	uses	use	VERB
ejpam-4508	10	6	a	a	DET
ejpam-4508	10	7	property	property	NOUN
ejpam-4508	10	8	in	in	ADP
ejpam-4508	10	9	the	the	DET
ejpam-4508	10	10	integer	integer	NOUN
ejpam-4508	10	11	system	system	NOUN
ejpam-4508	10	12	and	and	CCONJ
ejpam-4508	10	13	algebraic	algebraic	ADJ
ejpam-4508	10	14	number	number	NOUN
ejpam-4508	10	15	theory	theory	NOUN
ejpam-4508	10	16	.	.	PUNCT
ejpam-4508	11	1	some	some	PRON
ejpam-4508	11	2	of	of	ADP
ejpam-4508	11	3	which	which	PRON
ejpam-4508	11	4	appear	appear	VERB
ejpam-4508	11	5	in	in	ADP
ejpam-4508	11	6	a	a	DET
ejpam-4508	11	7	higher	high	ADJ
ejpam-4508	11	8	system	system	NOUN
ejpam-4508	11	9	of	of	ADP
ejpam-4508	11	10	the	the	DET
ejpam-4508	11	11	integer	integer	NOUN
ejpam-4508	11	12	called	call	VERB
ejpam-4508	11	13	the	the	DET
ejpam-4508	11	14	ring	ring	NOUN
ejpam-4508	11	15	of	of	ADP
ejpam-4508	11	16	integers	integer	NOUN
ejpam-4508	11	17	.	.	PUNCT
ejpam-4508	12	1	in	in	ADP
ejpam-4508	12	2	2011	2011	NUM
ejpam-4508	12	3	,	,	PUNCT
ejpam-4508	12	4	suvarnamani	suvarnamani	NOUN
ejpam-4508	12	5	[	[	X
ejpam-4508	12	6	10	10	NUM
ejpam-4508	12	7	]	]	PUNCT
ejpam-4508	12	8	considered	consider	VERB
ejpam-4508	12	9	a	a	DET
ejpam-4508	12	10	diophantine	diophantine	NOUN
ejpam-4508	12	11	equation	equation	NOUN
ejpam-4508	12	12	2x	2x	NUM
ejpam-4508	12	13	+	+	CCONJ
ejpam-4508	12	14	py	py	X
ejpam-4508	12	15	=	=	SYM
ejpam-4508	12	16	z2	z2	PROPN
ejpam-4508	12	17	when	when	SCONJ
ejpam-4508	12	18	p	p	PROPN
ejpam-4508	12	19	>	>	X
ejpam-4508	12	20	2	2	NUM
ejpam-4508	12	21	and	and	CCONJ
ejpam-4508	12	22	p	p	NOUN
ejpam-4508	12	23	is	be	AUX
ejpam-4508	12	24	a	a	DET
ejpam-4508	12	25	prime	prime	ADJ
ejpam-4508	12	26	number	number	NOUN
ejpam-4508	12	27	.	.	PUNCT
ejpam-4508	13	1	the	the	DET
ejpam-4508	13	2	result	result	NOUN
ejpam-4508	13	3	showed	show	VERB
ejpam-4508	13	4	that	that	SCONJ
ejpam-4508	13	5	(	(	PUNCT
ejpam-4508	13	6	x	x	X
ejpam-4508	13	7	,	,	PUNCT
ejpam-4508	13	8	y	y	PROPN
ejpam-4508	13	9	,	,	PUNCT
ejpam-4508	13	10	z	z	NOUN
ejpam-4508	13	11	)	)	PUNCT
ejpam-4508	13	12	=	=	SYM
ejpam-4508	13	13	(	(	PUNCT
ejpam-4508	13	14	3	3	NUM
ejpam-4508	13	15	,	,	PUNCT
ejpam-4508	13	16	0	0	NUM
ejpam-4508	13	17	,	,	PUNCT
ejpam-4508	13	18	3	3	NUM
ejpam-4508	13	19	)	)	PUNCT
ejpam-4508	13	20	is	be	AUX
ejpam-4508	13	21	a	a	DET
ejpam-4508	13	22	solution	solution	NOUN
ejpam-4508	13	23	of	of	ADP
ejpam-4508	13	24	the	the	DET
ejpam-4508	13	25	equation	equation	NOUN
ejpam-4508	13	26	for	for	ADP
ejpam-4508	13	27	all	all	DET
ejpam-4508	13	28	prime	prime	NOUN
ejpam-4508	13	29	p	p	X
ejpam-4508	13	30	>	>	X
ejpam-4508	13	31	2	2	NUM
ejpam-4508	13	32	.	.	PUNCT
ejpam-4508	14	1	if	if	SCONJ
ejpam-4508	14	2	p	p	NOUN
ejpam-4508	14	3	=	=	NOUN
ejpam-4508	14	4	3	3	NUM
ejpam-4508	14	5	,	,	PUNCT
ejpam-4508	14	6	then	then	ADV
ejpam-4508	14	7	(	(	PUNCT
ejpam-4508	14	8	x	x	X
ejpam-4508	14	9	,	,	PUNCT
ejpam-4508	14	10	y	y	PROPN
ejpam-4508	14	11	,	,	PUNCT
ejpam-4508	14	12	z	z	NOUN
ejpam-4508	14	13	)	)	PUNCT
ejpam-4508	14	14	=	=	SYM
ejpam-4508	14	15	(	(	PUNCT
ejpam-4508	14	16	4	4	NUM
ejpam-4508	14	17	,	,	PUNCT
ejpam-4508	14	18	2	2	NUM
ejpam-4508	14	19	,	,	PUNCT
ejpam-4508	14	20	5	5	NUM
ejpam-4508	14	21	)	)	PUNCT
ejpam-4508	14	22	is	be	AUX
ejpam-4508	14	23	also	also	ADV
ejpam-4508	14	24	a	a	DET
ejpam-4508	14	25	solution	solution	NOUN
ejpam-4508	14	26	of	of	ADP
ejpam-4508	14	27	the	the	DET
ejpam-4508	14	28	equation	equation	NOUN
ejpam-4508	14	29	.	.	PUNCT
ejpam-4508	15	1	if	if	SCONJ
ejpam-4508	15	2	p	p	NOUN
ejpam-4508	15	3	=	=	NOUN
ejpam-4508	15	4	1	1	NUM
ejpam-4508	15	5	+	+	NUM
ejpam-4508	15	6	2k+1	2k+1	NOUN
ejpam-4508	15	7	for	for	ADP
ejpam-4508	15	8	some	some	DET
ejpam-4508	15	9	non	non	ADJ
ejpam-4508	15	10	-	-	ADJ
ejpam-4508	15	11	negative	negative	ADJ
ejpam-4508	15	12	integer	integer	NOUN
ejpam-4508	15	13	k	k	NOUN
ejpam-4508	15	14	,	,	PUNCT
ejpam-4508	15	15	then	then	ADV
ejpam-4508	15	16	(	(	PUNCT
ejpam-4508	15	17	x	x	X
ejpam-4508	15	18	,	,	PUNCT
ejpam-4508	15	19	y	y	PROPN
ejpam-4508	15	20	,	,	PUNCT
ejpam-4508	15	21	z	z	NOUN
ejpam-4508	15	22	)	)	PUNCT
ejpam-4508	15	23	=	=	SYM
ejpam-4508	15	24	(	(	PUNCT
ejpam-4508	15	25	2k	2k	NUM
ejpam-4508	15	26	,	,	PUNCT
ejpam-4508	15	27	1	1	NUM
ejpam-4508	15	28	,	,	PUNCT
ejpam-4508	15	29	1	1	NUM
ejpam-4508	16	1	+	+	NUM
ejpam-4508	16	2	2k	2k	NUM
ejpam-4508	16	3	)	)	PUNCT
ejpam-4508	16	4	.	.	PUNCT
ejpam-4508	17	1	in	in	ADP
ejpam-4508	17	2	2012	2012	NUM
ejpam-4508	17	3	,	,	PUNCT
ejpam-4508	17	4	the	the	DET
ejpam-4508	17	5	diophantine	diophantine	NOUN
ejpam-4508	17	6	equation	equation	NOUN
ejpam-4508	17	7	4x	4x	NOUN
ejpam-4508	17	8	+	+	CCONJ
ejpam-4508	17	9	py	py	PROPN
ejpam-4508	17	10	=	=	SYM
ejpam-4508	17	11	z2	z2	PROPN
ejpam-4508	17	12	,	,	PUNCT
ejpam-4508	17	13	where	where	SCONJ
ejpam-4508	17	14	x	x	X
ejpam-4508	17	15	,	,	PUNCT
ejpam-4508	17	16	y	y	PROPN
ejpam-4508	17	17	and	and	CCONJ
ejpam-4508	17	18	z	z	PROPN
ejpam-4508	17	19	are	be	AUX
ejpam-4508	17	20	non	non	ADJ
ejpam-4508	17	21	-	-	ADJ
ejpam-4508	17	22	negative	negative	ADJ
ejpam-4508	17	23	integers	integer	NOUN
ejpam-4508	17	24	and	and	CCONJ
ejpam-4508	17	25	p	p	NOUN
ejpam-4508	17	26	is	be	AUX
ejpam-4508	17	27	a	a	DET
ejpam-4508	17	28	positive	positive	ADJ
ejpam-4508	17	29	prime	prime	ADJ
ejpam-4508	17	30	number	number	NOUN
ejpam-4508	17	31	was	be	AUX
ejpam-4508	17	32	studied	study	VERB
ejpam-4508	17	33	by	by	ADP
ejpam-4508	17	34	chotchaisthit	chotchaisthit	VERB
ejpam-4508	17	35	[	[	X
ejpam-4508	17	36	2	2	NUM
ejpam-4508	17	37	]	]	PUNCT
ejpam-4508	17	38	.	.	PUNCT
ejpam-4508	18	1	the	the	DET
ejpam-4508	18	2	study	study	NOUN
ejpam-4508	18	3	revealed	reveal	VERB
ejpam-4508	18	4	that	that	SCONJ
ejpam-4508	18	5	the	the	DET
ejpam-4508	18	6	equation	equation	NOUN
ejpam-4508	18	7	has	have	VERB
ejpam-4508	18	8	no	no	DET
ejpam-4508	18	9	non	non	ADJ
ejpam-4508	18	10	-	-	ADJ
ejpam-4508	18	11	negative	negative	ADJ
ejpam-4508	18	12	integer	integer	NOUN
ejpam-4508	18	13	solution	solution	NOUN
ejpam-4508	18	14	.	.	PUNCT
ejpam-4508	19	1	in	in	ADP
ejpam-4508	19	2	2014	2014	NUM
ejpam-4508	19	3	,	,	PUNCT
ejpam-4508	19	4	suvarnamani	suvarnamani	NOUN
ejpam-4508	19	5	[	[	X
ejpam-4508	19	6	11	11	NUM
ejpam-4508	19	7	]	]	PUNCT
ejpam-4508	19	8	proved	prove	VERB
ejpam-4508	19	9	that	that	SCONJ
ejpam-4508	19	10	the	the	DET
ejpam-4508	19	11	equation	equation	NOUN
ejpam-4508	19	12	px+(p+1)y	px+(p+1)y	NOUN
ejpam-4508	19	13	=	=	SYM
ejpam-4508	19	14	z2	z2	PROPN
ejpam-4508	19	15	has	have	VERB
ejpam-4508	19	16	a	a	DET
ejpam-4508	19	17	unique	unique	ADJ
ejpam-4508	19	18	non	non	ADJ
ejpam-4508	19	19	-	-	ADJ
ejpam-4508	19	20	negative	negative	ADJ
ejpam-4508	19	21	integer	integer	NOUN
ejpam-4508	19	22	solution	solution	NOUN
ejpam-4508	19	23	(	(	PUNCT
ejpam-4508	19	24	p	p	X
ejpam-4508	19	25	,	,	PUNCT
ejpam-4508	19	26	x	x	PROPN
ejpam-4508	19	27	,	,	PUNCT
ejpam-4508	19	28	y	y	PROPN
ejpam-4508	19	29	,	,	PUNCT
ejpam-4508	19	30	z	z	NOUN
ejpam-4508	19	31	)	)	PUNCT
ejpam-4508	19	32	=	=	SYM
ejpam-4508	19	33	(	(	PUNCT
ejpam-4508	19	34	3	3	NUM
ejpam-4508	19	35	,	,	PUNCT
ejpam-4508	19	36	1	1	NUM
ejpam-4508	19	37	,	,	PUNCT
ejpam-4508	19	38	0	0	NUM
ejpam-4508	19	39	,	,	PUNCT
ejpam-4508	19	40	2	2	NUM
ejpam-4508	19	41	)	)	PUNCT
ejpam-4508	19	42	when	when	SCONJ
ejpam-4508	19	43	p	p	NOUN
ejpam-4508	19	44	is	be	AUX
ejpam-4508	19	45	an	an	DET
ejpam-4508	19	46	odd	odd	ADJ
ejpam-4508	19	47	prime	prime	ADJ
ejpam-4508	19	48	number	number	NOUN
ejpam-4508	19	49	.	.	PUNCT
ejpam-4508	20	1	in	in	ADP
ejpam-4508	20	2	2016	2016	NUM
ejpam-4508	20	3	,	,	PUNCT
ejpam-4508	20	4	hoque	hoque	NOUN
ejpam-4508	20	5	[	[	X
ejpam-4508	20	6	6	6	NUM
ejpam-4508	20	7	]	]	PUNCT
ejpam-4508	20	8	proved	prove	VERB
ejpam-4508	20	9	that	that	SCONJ
ejpam-4508	20	10	there	there	PRON
ejpam-4508	20	11	are	be	VERB
ejpam-4508	20	12	exactly	exactly	ADV
ejpam-4508	20	13	two	two	NUM
ejpam-4508	20	14	solutions	solution	NOUN
ejpam-4508	20	15	to	to	ADP
ejpam-4508	20	16	(	(	PUNCT
ejpam-4508	20	17	mpq	mpq	X
ejpam-4508	20	18	)	)	PUNCT
ejpam-4508	20	19	x	x	PUNCT
ejpam-4508	21	1	+	+	PUNCT
ejpam-4508	21	2	(	(	PUNCT
ejpam-4508	21	3	mpq	mpq	X
ejpam-4508	21	4	+	+	NUM
ejpam-4508	21	5	1)y	1)y	NUM
ejpam-4508	21	6	=	=	SYM
ejpam-4508	21	7	z2	z2	PROPN
ejpam-4508	21	8	,	,	PUNCT
ejpam-4508	21	9	where	where	SCONJ
ejpam-4508	21	10	p	p	X
ejpam-4508	21	11	,	,	PUNCT
ejpam-4508	21	12	q	q	PROPN
ejpam-4508	21	13	∈	∈	PROPN
ejpam-4508	21	14	z	z	NOUN
ejpam-4508	21	15	such	such	ADJ
ejpam-4508	21	16	that	that	SCONJ
ejpam-4508	21	17	p	p	X
ejpam-4508	21	18	>	>	X
ejpam-4508	21	19	0	0	NUM
ejpam-4508	21	20	,	,	PUNCT
ejpam-4508	21	21	q	q	X
ejpam-4508	21	22	>	>	X
ejpam-4508	21	23	1	1	NUM
ejpam-4508	21	24	and	and	CCONJ
ejpam-4508	21	25	mpq	mpq	X
ejpam-4508	21	26	=	=	SYM
ejpam-4508	21	27	pq	pq	NOUN
ejpam-4508	21	28	−	−	NOUN
ejpam-4508	21	29	1	1	X
ejpam-4508	21	30	.	.	PUNCT
ejpam-4508	21	31	in	in	ADP
ejpam-4508	21	32	2018	2018	NUM
ejpam-4508	21	33	,	,	PUNCT
ejpam-4508	21	34	kumar	kumar	PROPN
ejpam-4508	21	35	et	et	PROPN
ejpam-4508	21	36	al	al	PROPN
ejpam-4508	21	37	.	.	PUNCT
ejpam-4508	22	1	[	[	X
ejpam-4508	22	2	7	7	X
ejpam-4508	22	3	]	]	PUNCT
ejpam-4508	22	4	showed	show	VERB
ejpam-4508	22	5	that	that	SCONJ
ejpam-4508	22	6	the	the	DET
ejpam-4508	22	7	non	non	ADJ
ejpam-4508	22	8	-	-	ADJ
ejpam-4508	22	9	linear	linear	ADJ
ejpam-4508	22	10	diophantine	diophantine	NOUN
ejpam-4508	22	11	equation	equation	NOUN
ejpam-4508	22	12	px	px	X
ejpam-4508	22	13	+	+	CCONJ
ejpam-4508	22	14	(	(	PUNCT
ejpam-4508	22	15	p+	p+	NOUN
ejpam-4508	22	16	6)y	6)y	NOUN
ejpam-4508	22	17	=	=	SYM
ejpam-4508	22	18	z2	z2	PROPN
ejpam-4508	22	19	has	have	VERB
ejpam-4508	22	20	no	no	DET
ejpam-4508	22	21	solution	solution	NOUN
ejpam-4508	22	22	.	.	PUNCT
ejpam-4508	23	1	moreover	moreover	ADV
ejpam-4508	23	2	,	,	PUNCT
ejpam-4508	23	3	fernando	fernando	PROPN
ejpam-4508	23	4	[	[	X
ejpam-4508	23	5	4	4	X
ejpam-4508	23	6	]	]	PUNCT
ejpam-4508	23	7	showed	show	VERB
ejpam-4508	23	8	∗corresponding	∗corresponde	VERB
ejpam-4508	23	9	author	author	NOUN
ejpam-4508	23	10	.	.	PUNCT
ejpam-4508	24	1	doi	doi	NOUN
ejpam-4508	24	2	:	:	PUNCT
ejpam-4508	24	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4508	https://doi.org/10.29020/nybg.ejpam.v15i4.4508	NUM
ejpam-4508	24	4	email	email	NOUN
ejpam-4508	24	5	addresses	address	NOUN
ejpam-4508	24	6	:	:	PUNCT
ejpam-4508	24	7	wachirarak.tc@bru.ac.th	wachirarak.tc@bru.ac.th	NUM
ejpam-4508	24	8	(	(	PUNCT
ejpam-4508	24	9	w.	w.	PROPN
ejpam-4508	24	10	orosram	orosram	PROPN
ejpam-4508	24	11	)	)	PUNCT
ejpam-4508	24	12	,	,	PUNCT
ejpam-4508	24	13	620112210001@bru.ac.th	620112210001@bru.ac.th	NOUN
ejpam-4508	24	14	(	(	PUNCT
ejpam-4508	24	15	k.	k.	NOUN
ejpam-4508	24	16	makonwattana)620112210034@bru.ac.th	makonwattana)620112210034@bru.ac.th	PROPN
ejpam-4508	24	17	(	(	PUNCT
ejpam-4508	24	18	s.	s.	PROPN
ejpam-4508	24	19	khongsawat	khongsawat	PROPN
ejpam-4508	24	20	)	)	PUNCT
ejpam-4508	24	21	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4508	24	22	1593	1593	NUM
ejpam-4508	25	1	©	©	PROPN
ejpam-4508	25	2	2022	2022	NUM
ejpam-4508	25	3	ejpam	ejpam	VERB
ejpam-4508	25	4	all	all	DET
ejpam-4508	25	5	rights	right	NOUN
ejpam-4508	25	6	reserved	reserve	VERB
ejpam-4508	25	7	.	.	PUNCT
ejpam-4508	26	1	w.	w.	PROPN
ejpam-4508	26	2	orosram	orosram	PROPN
ejpam-4508	26	3	,	,	PUNCT
ejpam-4508	26	4	k.	k.	PROPN
ejpam-4508	26	5	makonwattana	makonwattana	PROPN
ejpam-4508	26	6	,	,	PUNCT
ejpam-4508	26	7	s.	s.	PROPN
ejpam-4508	26	8	khongsawat	khongsawat	PROPN
ejpam-4508	26	9	/	/	SYM
ejpam-4508	26	10	eur	eur	PROPN
ejpam-4508	26	11	.	.	PUNCT
ejpam-4508	27	1	j.	j.	PROPN
ejpam-4508	27	2	pure	pure	PROPN
ejpam-4508	27	3	appl	appl	PROPN
ejpam-4508	27	4	.	.	PROPN
ejpam-4508	27	5	math	math	PROPN
ejpam-4508	27	6	,	,	PUNCT
ejpam-4508	27	7	15	15	NUM
ejpam-4508	27	8	(	(	PUNCT
ejpam-4508	27	9	4	4	NUM
ejpam-4508	27	10	)	)	PUNCT
ejpam-4508	27	11	(	(	PUNCT
ejpam-4508	27	12	2022	2022	NUM
ejpam-4508	27	13	)	)	PUNCT
ejpam-4508	27	14	,	,	PUNCT
ejpam-4508	27	15	1593	1593	NUM
ejpam-4508	27	16	-	-	SYM
ejpam-4508	27	17	1596	1596	NUM
ejpam-4508	27	18	1594	1594	NUM
ejpam-4508	27	19	that	that	SCONJ
ejpam-4508	27	20	a	a	DET
ejpam-4508	27	21	diophantine	diophantine	NOUN
ejpam-4508	27	22	equation	equation	NOUN
ejpam-4508	27	23	px	px	X
ejpam-4508	28	1	+	+	CCONJ
ejpam-4508	28	2	(	(	PUNCT
ejpam-4508	28	3	p	p	X
ejpam-4508	28	4	+	+	NUM
ejpam-4508	28	5	8)y	8)y	NOUN
ejpam-4508	28	6	=	=	SYM
ejpam-4508	28	7	z2	z2	PROPN
ejpam-4508	28	8	has	have	VERB
ejpam-4508	28	9	no	no	DET
ejpam-4508	28	10	positive	positive	ADJ
ejpam-4508	28	11	-	-	PUNCT
ejpam-4508	28	12	integer	integer	NOUN
ejpam-4508	28	13	solution	solution	NOUN
ejpam-4508	28	14	,	,	PUNCT
ejpam-4508	28	15	when	when	SCONJ
ejpam-4508	28	16	p	p	X
ejpam-4508	28	17	,	,	PUNCT
ejpam-4508	28	18	p	p	NOUN
ejpam-4508	28	19	+	+	NOUN
ejpam-4508	28	20	8	8	NUM
ejpam-4508	28	21	are	be	AUX
ejpam-4508	28	22	primes	prime	NOUN
ejpam-4508	28	23	such	such	ADJ
ejpam-4508	28	24	that	that	SCONJ
ejpam-4508	28	25	p	p	X
ejpam-4508	28	26	>	>	X
ejpam-4508	28	27	3	3	X
ejpam-4508	28	28	.	.	PUNCT
ejpam-4508	28	29	in	in	ADP
ejpam-4508	28	30	2019	2019	NUM
ejpam-4508	28	31	,	,	PUNCT
ejpam-4508	28	32	kumar	kumar	PROPN
ejpam-4508	28	33	et	et	PROPN
ejpam-4508	28	34	al	al	PROPN
ejpam-4508	28	35	.	.	PUNCT
ejpam-4508	29	1	[	[	X
ejpam-4508	29	2	8	8	NUM
ejpam-4508	29	3	]	]	PUNCT
ejpam-4508	29	4	proved	prove	VERB
ejpam-4508	29	5	that	that	SCONJ
ejpam-4508	29	6	the	the	DET
ejpam-4508	29	7	solution	solution	NOUN
ejpam-4508	29	8	of	of	ADP
ejpam-4508	29	9	an	an	DET
ejpam-4508	29	10	exponential	exponential	ADJ
ejpam-4508	29	11	diophantine	diophantine	NOUN
ejpam-4508	29	12	equation	equation	NOUN
ejpam-4508	29	13	px	px	X
ejpam-4508	29	14	+	+	CCONJ
ejpam-4508	29	15	(	(	PUNCT
ejpam-4508	29	16	p	p	X
ejpam-4508	29	17	+	+	NOUN
ejpam-4508	29	18	12)y	12)y	NUM
ejpam-4508	29	19	=	=	SYM
ejpam-4508	29	20	z2	z2	PROPN
ejpam-4508	29	21	has	have	VERB
ejpam-4508	29	22	no	no	DET
ejpam-4508	29	23	non	non	ADJ
ejpam-4508	29	24	-	-	ADJ
ejpam-4508	29	25	negative	negative	ADJ
ejpam-4508	29	26	integer	integer	NOUN
ejpam-4508	29	27	solution	solution	NOUN
ejpam-4508	29	28	,	,	PUNCT
ejpam-4508	29	29	when	when	SCONJ
ejpam-4508	29	30	p	p	NOUN
ejpam-4508	29	31	and	and	CCONJ
ejpam-4508	29	32	p+12	p+12	NOUN
ejpam-4508	29	33	are	be	AUX
ejpam-4508	29	34	prime	prime	ADJ
ejpam-4508	29	35	numbers	number	NOUN
ejpam-4508	29	36	such	such	ADJ
ejpam-4508	29	37	that	that	SCONJ
ejpam-4508	29	38	p	p	NOUN
ejpam-4508	29	39	is	be	AUX
ejpam-4508	29	40	in	in	ADP
ejpam-4508	29	41	the	the	DET
ejpam-4508	29	42	form	form	NOUN
ejpam-4508	29	43	of	of	ADP
ejpam-4508	29	44	6n+1	6n+1	PROPN
ejpam-4508	29	45	.	.	PUNCT
ejpam-4508	30	1	in	in	ADP
ejpam-4508	30	2	2020	2020	NUM
ejpam-4508	30	3	,	,	PUNCT
ejpam-4508	30	4	burshtein	burshtein	ADV
ejpam-4508	30	5	[	[	X
ejpam-4508	30	6	1	1	X
ejpam-4508	30	7	]	]	PUNCT
ejpam-4508	30	8	proved	prove	VERB
ejpam-4508	30	9	that	that	SCONJ
ejpam-4508	30	10	a	a	DET
ejpam-4508	30	11	diophantine	diophantine	NOUN
ejpam-4508	30	12	equation	equation	NOUN
ejpam-4508	30	13	px+(p+12)y	px+(p+12)y	PROPN
ejpam-4508	30	14	=	=	PROPN
ejpam-4508	30	15	z2	z2	PROPN
ejpam-4508	30	16	has	have	VERB
ejpam-4508	30	17	no	no	DET
ejpam-4508	30	18	positive	positive	ADJ
ejpam-4508	30	19	integer	integer	NOUN
ejpam-4508	30	20	solution	solution	NOUN
ejpam-4508	30	21	(	(	PUNCT
ejpam-4508	30	22	x	x	X
ejpam-4508	30	23	,	,	PUNCT
ejpam-4508	30	24	y	y	PROPN
ejpam-4508	30	25	,	,	PUNCT
ejpam-4508	30	26	z	z	NOUN
ejpam-4508	30	27	)	)	PUNCT
ejpam-4508	30	28	,	,	PUNCT
ejpam-4508	30	29	when	when	SCONJ
ejpam-4508	30	30	p	p	NOUN
ejpam-4508	30	31	is	be	AUX
ejpam-4508	30	32	a	a	DET
ejpam-4508	30	33	prime	prime	ADJ
ejpam-4508	30	34	number	number	NOUN
ejpam-4508	30	35	such	such	ADJ
ejpam-4508	30	36	that	that	SCONJ
ejpam-4508	30	37	p+	p+	NOUN
ejpam-4508	30	38	5	5	NUM
ejpam-4508	30	39	=	=	NOUN
ejpam-4508	30	40	22u	22u	X
ejpam-4508	30	41	.	.	PUNCT
ejpam-4508	31	1	in	in	ADP
ejpam-4508	31	2	2021	2021	NUM
ejpam-4508	31	3	,	,	PUNCT
ejpam-4508	31	4	dokchan	dokchan	NOUN
ejpam-4508	31	5	and	and	CCONJ
ejpam-4508	31	6	pakapongpun	pakapongpun	VERB
ejpam-4508	31	7	[	[	X
ejpam-4508	31	8	3	3	NUM
ejpam-4508	31	9	]	]	PUNCT
ejpam-4508	31	10	studied	study	VERB
ejpam-4508	31	11	a	a	DET
ejpam-4508	31	12	diophantine	diophantine	NOUN
ejpam-4508	31	13	equation	equation	NOUN
ejpam-4508	31	14	px	px	X
ejpam-4508	31	15	+	+	CCONJ
ejpam-4508	31	16	(	(	PUNCT
ejpam-4508	31	17	p+	p+	NOUN
ejpam-4508	31	18	20)y	20)y	PROPN
ejpam-4508	31	19	=	=	SYM
ejpam-4508	31	20	z2	z2	PROPN
ejpam-4508	31	21	,	,	PUNCT
ejpam-4508	31	22	when	when	SCONJ
ejpam-4508	31	23	p	p	PROPN
ejpam-4508	31	24	and	and	CCONJ
ejpam-4508	31	25	p+	p+	PROPN
ejpam-4508	31	26	20	20	NUM
ejpam-4508	31	27	are	be	AUX
ejpam-4508	31	28	primes	prime	NOUN
ejpam-4508	31	29	and	and	CCONJ
ejpam-4508	31	30	showed	show	VERB
ejpam-4508	31	31	that	that	SCONJ
ejpam-4508	31	32	the	the	DET
ejpam-4508	31	33	equation	equation	NOUN
ejpam-4508	31	34	has	have	VERB
ejpam-4508	31	35	no	no	DET
ejpam-4508	31	36	positive	positive	ADJ
ejpam-4508	31	37	integer	integer	NOUN
ejpam-4508	31	38	solution	solution	NOUN
ejpam-4508	31	39	(	(	PUNCT
ejpam-4508	31	40	x	x	X
ejpam-4508	31	41	,	,	PUNCT
ejpam-4508	31	42	y	y	PROPN
ejpam-4508	31	43	,	,	PUNCT
ejpam-4508	31	44	z	z	NOUN
ejpam-4508	31	45	)	)	PUNCT
ejpam-4508	31	46	.	.	PUNCT
ejpam-4508	32	1	in	in	ADP
ejpam-4508	32	2	the	the	DET
ejpam-4508	32	3	same	same	ADJ
ejpam-4508	32	4	year	year	NOUN
ejpam-4508	32	5	,	,	PUNCT
ejpam-4508	32	6	gayo	gayo	NOUN
ejpam-4508	32	7	and	and	CCONJ
ejpam-4508	32	8	bacani	bacani	ADJ
ejpam-4508	33	1	[	[	X
ejpam-4508	33	2	5	5	NUM
ejpam-4508	33	3	]	]	PUNCT
ejpam-4508	33	4	solved	solve	VERB
ejpam-4508	33	5	the	the	DET
ejpam-4508	33	6	diophantine	diophantine	NOUN
ejpam-4508	33	7	equation	equation	NOUN
ejpam-4508	33	8	mx	mx	PROPN
ejpam-4508	34	1	p	p	PROPN
ejpam-4508	34	2	+	+	X
ejpam-4508	34	3	(	(	PUNCT
ejpam-4508	34	4	mq	mq	PROPN
ejpam-4508	34	5	+	+	NOUN
ejpam-4508	34	6	1)y	1)y	NUM
ejpam-4508	34	7	=	=	SYM
ejpam-4508	34	8	z2	z2	PROPN
ejpam-4508	34	9	when	when	SCONJ
ejpam-4508	34	10	mp	mp	PROPN
ejpam-4508	34	11	and	and	CCONJ
ejpam-4508	34	12	mq	mq	PROPN
ejpam-4508	34	13	are	be	AUX
ejpam-4508	34	14	mersenne	mersenne	NOUN
ejpam-4508	34	15	primes	prime	NOUN
ejpam-4508	34	16	.	.	PUNCT
ejpam-4508	35	1	in	in	ADP
ejpam-4508	35	2	this	this	DET
ejpam-4508	35	3	work	work	NOUN
ejpam-4508	35	4	,	,	PUNCT
ejpam-4508	35	5	we	we	PRON
ejpam-4508	35	6	give	give	VERB
ejpam-4508	35	7	solutions	solution	NOUN
ejpam-4508	35	8	of	of	ADP
ejpam-4508	35	9	the	the	DET
ejpam-4508	35	10	diophantine	diophantine	NOUN
ejpam-4508	35	11	equations	equation	NOUN
ejpam-4508	35	12	1+by	1+by	NUM
ejpam-4508	35	13	=	=	SYM
ejpam-4508	35	14	z2	z2	PROPN
ejpam-4508	35	15	,	,	PUNCT
ejpam-4508	35	16	1+(d+4t)x	1+(d+4t)x	PROPN
ejpam-4508	35	17	=	=	SYM
ejpam-4508	35	18	z2	z2	PROPN
ejpam-4508	35	19	where	where	SCONJ
ejpam-4508	35	20	b	b	PROPN
ejpam-4508	35	21	,	,	PUNCT
ejpam-4508	35	22	t	t	PROPN
ejpam-4508	35	23	,	,	PUNCT
ejpam-4508	35	24	d	d	X
ejpam-4508	35	25	are	be	AUX
ejpam-4508	35	26	positive	positive	ADJ
ejpam-4508	35	27	integers	integer	NOUN
ejpam-4508	35	28	.	.	PUNCT
ejpam-4508	36	1	then	then	ADV
ejpam-4508	36	2	,	,	PUNCT
ejpam-4508	36	3	we	we	PRON
ejpam-4508	36	4	extend	extend	VERB
ejpam-4508	36	5	to	to	ADP
ejpam-4508	36	6	the	the	DET
ejpam-4508	36	7	solutions	solution	NOUN
ejpam-4508	36	8	of	of	ADP
ejpam-4508	36	9	the	the	DET
ejpam-4508	36	10	diophantine	diophantine	NOUN
ejpam-4508	36	11	equation	equation	NOUN
ejpam-4508	36	12	(	(	PUNCT
ejpam-4508	36	13	p+4n)x+py	p+4n)x+py	ADJ
ejpam-4508	36	14	=	=	SYM
ejpam-4508	36	15	z2	z2	NUM
ejpam-4508	36	16	where	where	SCONJ
ejpam-4508	36	17	p	p	X
ejpam-4508	36	18	,	,	PUNCT
ejpam-4508	36	19	p+4n	p+4n	PROPN
ejpam-4508	36	20	are	be	AUX
ejpam-4508	36	21	prime	prime	ADJ
ejpam-4508	36	22	numbers	number	NOUN
ejpam-4508	36	23	such	such	ADJ
ejpam-4508	36	24	that	that	SCONJ
ejpam-4508	36	25	p	p	PROPN
ejpam-4508	36	26	≡	≡	PROPN
ejpam-4508	36	27	7	7	NUM
ejpam-4508	36	28	(	(	PUNCT
ejpam-4508	36	29	mod	mod	NOUN
ejpam-4508	36	30	12	12	NUM
ejpam-4508	36	31	)	)	PUNCT
ejpam-4508	36	32	and	and	CCONJ
ejpam-4508	36	33	n	n	PRON
ejpam-4508	36	34	is	be	AUX
ejpam-4508	36	35	a	a	DET
ejpam-4508	36	36	positive	positive	ADJ
ejpam-4508	36	37	integer	integer	NOUN
ejpam-4508	36	38	such	such	ADJ
ejpam-4508	36	39	that	that	SCONJ
ejpam-4508	36	40	n	n	NUM
ejpam-4508	36	41	≡	≡	PROPN
ejpam-4508	36	42	0	0	NUM
ejpam-4508	36	43	,	,	PUNCT
ejpam-4508	36	44	1	1	NUM
ejpam-4508	36	45	(	(	PUNCT
ejpam-4508	36	46	mod	mod	NOUN
ejpam-4508	36	47	3	3	NUM
ejpam-4508	36	48	)	)	PUNCT
ejpam-4508	36	49	.	.	PUNCT
ejpam-4508	37	1	2	2	X
ejpam-4508	37	2	.	.	X
ejpam-4508	37	3	main	main	ADJ
ejpam-4508	37	4	results	result	NOUN
ejpam-4508	37	5	proposition	proposition	NOUN
ejpam-4508	37	6	1	1	NUM
ejpam-4508	37	7	.	.	PUNCT
ejpam-4508	38	1	(	(	PUNCT
ejpam-4508	38	2	catalan	catalan	NOUN
ejpam-4508	38	3	’s	’s	PART
ejpam-4508	38	4	conjecture	conjecture	NOUN
ejpam-4508	38	5	)	)	PUNCT
ejpam-4508	38	6	(	(	PUNCT
ejpam-4508	38	7	a	a	PRON
ejpam-4508	38	8	,	,	PUNCT
ejpam-4508	38	9	b	b	NOUN
ejpam-4508	38	10	,	,	PUNCT
ejpam-4508	38	11	x	x	NOUN
ejpam-4508	38	12	,	,	PUNCT
ejpam-4508	38	13	y	y	NOUN
ejpam-4508	38	14	)	)	PUNCT
ejpam-4508	38	15	=	=	PUNCT
ejpam-4508	39	1	(	(	PUNCT
ejpam-4508	39	2	3	3	NUM
ejpam-4508	39	3	,	,	PUNCT
ejpam-4508	39	4	2	2	NUM
ejpam-4508	39	5	,	,	PUNCT
ejpam-4508	39	6	2	2	NUM
ejpam-4508	39	7	,	,	PUNCT
ejpam-4508	39	8	3	3	NUM
ejpam-4508	39	9	)	)	PUNCT
ejpam-4508	39	10	is	be	AUX
ejpam-4508	39	11	the	the	DET
ejpam-4508	39	12	unique	unique	ADJ
ejpam-4508	39	13	solution	solution	NOUN
ejpam-4508	39	14	of	of	ADP
ejpam-4508	39	15	the	the	DET
ejpam-4508	39	16	diophantine	diophantine	NOUN
ejpam-4508	39	17	equation	equation	NOUN
ejpam-4508	39	18	ax	ax	NOUN
ejpam-4508	39	19	−	−	NOUN
ejpam-4508	39	20	by	by	ADP
ejpam-4508	39	21	=	=	PROPN
ejpam-4508	39	22	1	1	NUM
ejpam-4508	39	23	,	,	PUNCT
ejpam-4508	39	24	where	where	SCONJ
ejpam-4508	39	25	a	a	DET
ejpam-4508	39	26	,	,	PUNCT
ejpam-4508	39	27	b	b	NOUN
ejpam-4508	39	28	,	,	PUNCT
ejpam-4508	39	29	x	x	PUNCT
ejpam-4508	39	30	and	and	CCONJ
ejpam-4508	39	31	y	y	PROPN
ejpam-4508	39	32	are	be	AUX
ejpam-4508	39	33	integers	integer	NOUN
ejpam-4508	39	34	such	such	ADJ
ejpam-4508	39	35	that	that	SCONJ
ejpam-4508	39	36	min{a	min{a	PROPN
ejpam-4508	39	37	,	,	PUNCT
ejpam-4508	39	38	b	b	PROPN
ejpam-4508	39	39	,	,	PUNCT
ejpam-4508	39	40	x	x	NOUN
ejpam-4508	39	41	,	,	PUNCT
ejpam-4508	39	42	y	y	PROPN
ejpam-4508	39	43	}	}	PUNCT
ejpam-4508	39	44	>	>	X
ejpam-4508	40	1	1	1	X
ejpam-4508	40	2	.	.	PUNCT
ejpam-4508	41	1	this	this	DET
ejpam-4508	41	2	proposition	proposition	NOUN
ejpam-4508	41	3	was	be	AUX
ejpam-4508	41	4	proved	prove	VERB
ejpam-4508	41	5	in	in	ADP
ejpam-4508	41	6	2004	2004	NUM
ejpam-4508	41	7	by	by	ADP
ejpam-4508	41	8	mihailescu	mihailescu	NOUN
ejpam-4508	42	1	[	[	X
ejpam-4508	42	2	9	9	NUM
ejpam-4508	42	3	]	]	PUNCT
ejpam-4508	42	4	.	.	PUNCT
ejpam-4508	43	1	lemma	lemma	PROPN
ejpam-4508	43	2	1	1	X
ejpam-4508	43	3	.	.	PUNCT
ejpam-4508	44	1	let	let	VERB
ejpam-4508	44	2	b	b	X
ejpam-4508	44	3	be	be	AUX
ejpam-4508	44	4	a	a	DET
ejpam-4508	44	5	positive	positive	ADJ
ejpam-4508	44	6	integer	integer	NOUN
ejpam-4508	44	7	.	.	PUNCT
ejpam-4508	45	1	the	the	DET
ejpam-4508	45	2	non	non	ADJ
ejpam-4508	45	3	-	-	ADJ
ejpam-4508	45	4	negative	negative	ADJ
ejpam-4508	45	5	integer	integer	NOUN
ejpam-4508	45	6	solutions	solution	NOUN
ejpam-4508	45	7	to	to	ADP
ejpam-4508	45	8	the	the	DET
ejpam-4508	45	9	diophantine	diophantine	NOUN
ejpam-4508	45	10	equation	equation	NOUN
ejpam-4508	45	11	1	1	NUM
ejpam-4508	45	12	+	+	CCONJ
ejpam-4508	45	13	by	by	ADP
ejpam-4508	45	14	=	=	PROPN
ejpam-4508	45	15	z2	z2	PROPN
ejpam-4508	45	16	is	be	AUX
ejpam-4508	45	17	(	(	PUNCT
ejpam-4508	45	18	y	y	PROPN
ejpam-4508	45	19	,	,	PUNCT
ejpam-4508	45	20	z	z	NOUN
ejpam-4508	45	21	)	)	PUNCT
ejpam-4508	45	22	=	=	SYM
ejpam-4508	45	23	(	(	PUNCT
ejpam-4508	45	24	1	1	NUM
ejpam-4508	45	25	,	,	PUNCT
ejpam-4508	45	26	√	√	NUM
ejpam-4508	45	27	b+	b+	ADP
ejpam-4508	45	28	1	1	NUM
ejpam-4508	45	29	)	)	PUNCT
ejpam-4508	45	30	if	if	SCONJ
ejpam-4508	45	31	√	√	NUM
ejpam-4508	45	32	b+	b+	ADP
ejpam-4508	45	33	1	1	NUM
ejpam-4508	45	34	)	)	PUNCT
ejpam-4508	45	35	is	be	AUX
ejpam-4508	45	36	a	a	DET
ejpam-4508	45	37	positive	positive	ADJ
ejpam-4508	45	38	integer	integer	NOUN
ejpam-4508	45	39	.	.	PUNCT
ejpam-4508	46	1	proof	proof	NOUN
ejpam-4508	46	2	.	.	PUNCT
ejpam-4508	47	1	let	let	VERB
ejpam-4508	47	2	b	b	X
ejpam-4508	47	3	be	be	AUX
ejpam-4508	47	4	a	a	DET
ejpam-4508	47	5	positive	positive	ADJ
ejpam-4508	47	6	integer	integer	NOUN
ejpam-4508	47	7	.	.	PUNCT
ejpam-4508	48	1	we	we	PRON
ejpam-4508	48	2	have	have	VERB
ejpam-4508	48	3	z2−by	z2−by	NUM
ejpam-4508	48	4	=	=	SYM
ejpam-4508	48	5	1	1	X
ejpam-4508	48	6	.	.	PUNCT
ejpam-4508	49	1	by	by	ADP
ejpam-4508	49	2	proposition	proposition	NOUN
ejpam-4508	49	3	1	1	NUM
ejpam-4508	49	4	,	,	PUNCT
ejpam-4508	49	5	it	it	PRON
ejpam-4508	49	6	is	be	AUX
ejpam-4508	49	7	sufficient	sufficient	ADJ
ejpam-4508	49	8	to	to	PART
ejpam-4508	49	9	consider	consider	VERB
ejpam-4508	49	10	the	the	DET
ejpam-4508	49	11	case	case	NOUN
ejpam-4508	49	12	b	b	NOUN
ejpam-4508	49	13	=	=	SYM
ejpam-4508	49	14	1	1	NUM
ejpam-4508	49	15	,	,	PUNCT
ejpam-4508	49	16	z	z	NOUN
ejpam-4508	49	17	≤	≤	NOUN
ejpam-4508	49	18	1	1	NUM
ejpam-4508	49	19	or	or	CCONJ
ejpam-4508	49	20	y	y	PROPN
ejpam-4508	49	21	≤	≤	NUM
ejpam-4508	49	22	1	1	NUM
ejpam-4508	49	23	.	.	PUNCT
ejpam-4508	50	1	hence	hence	ADV
ejpam-4508	50	2	,	,	PUNCT
ejpam-4508	50	3	it	it	PRON
ejpam-4508	50	4	remains	remain	VERB
ejpam-4508	50	5	to	to	PART
ejpam-4508	50	6	consider	consider	VERB
ejpam-4508	50	7	the	the	DET
ejpam-4508	50	8	following	follow	VERB
ejpam-4508	50	9	cases	case	NOUN
ejpam-4508	50	10	of	of	ADP
ejpam-4508	50	11	b	b	NOUN
ejpam-4508	50	12	,	,	PUNCT
ejpam-4508	50	13	y	y	PROPN
ejpam-4508	50	14	and	and	CCONJ
ejpam-4508	50	15	z.	z.	PROPN
ejpam-4508	51	1	if	if	SCONJ
ejpam-4508	51	2	b	b	PROPN
ejpam-4508	51	3	=	=	SYM
ejpam-4508	51	4	1	1	NUM
ejpam-4508	51	5	,	,	PUNCT
ejpam-4508	51	6	then	then	ADV
ejpam-4508	51	7	we	we	PRON
ejpam-4508	51	8	have	have	VERB
ejpam-4508	51	9	z2	z2	NOUN
ejpam-4508	51	10	=	=	SYM
ejpam-4508	51	11	2	2	NUM
ejpam-4508	51	12	,	,	PUNCT
ejpam-4508	51	13	which	which	PRON
ejpam-4508	51	14	is	be	AUX
ejpam-4508	51	15	impossible	impossible	ADJ
ejpam-4508	51	16	.	.	PUNCT
ejpam-4508	52	1	if	if	SCONJ
ejpam-4508	52	2	z	z	NOUN
ejpam-4508	52	3	=	=	SYM
ejpam-4508	52	4	0	0	NUM
ejpam-4508	52	5	or	or	CCONJ
ejpam-4508	52	6	z	z	NOUN
ejpam-4508	52	7	=	=	SYM
ejpam-4508	52	8	1	1	NUM
ejpam-4508	52	9	,	,	PUNCT
ejpam-4508	52	10	then	then	ADV
ejpam-4508	52	11	there	there	PRON
ejpam-4508	52	12	is	be	VERB
ejpam-4508	52	13	no	no	DET
ejpam-4508	52	14	solution	solution	NOUN
ejpam-4508	52	15	.	.	PUNCT
ejpam-4508	53	1	if	if	SCONJ
ejpam-4508	53	2	y	y	PROPN
ejpam-4508	53	3	=	=	SYM
ejpam-4508	53	4	0	0	PROPN
ejpam-4508	53	5	,	,	PUNCT
ejpam-4508	53	6	then	then	ADV
ejpam-4508	53	7	we	we	PRON
ejpam-4508	53	8	have	have	VERB
ejpam-4508	53	9	z2	z2	NOUN
ejpam-4508	53	10	=	=	SYM
ejpam-4508	53	11	2	2	NUM
ejpam-4508	53	12	which	which	PRON
ejpam-4508	53	13	is	be	AUX
ejpam-4508	53	14	impossible	impossible	ADJ
ejpam-4508	53	15	.	.	PUNCT
ejpam-4508	54	1	if	if	SCONJ
ejpam-4508	54	2	y	y	PROPN
ejpam-4508	54	3	=	=	SYM
ejpam-4508	54	4	1	1	NUM
ejpam-4508	54	5	,	,	PUNCT
ejpam-4508	54	6	then	then	ADV
ejpam-4508	54	7	we	we	PRON
ejpam-4508	54	8	have	have	VERB
ejpam-4508	54	9	z2	z2	NOUN
ejpam-4508	54	10	=	=	SYM
ejpam-4508	54	11	b+	b+	X
ejpam-4508	54	12	1	1	NUM
ejpam-4508	54	13	or	or	CCONJ
ejpam-4508	54	14	z	z	NOUN
ejpam-4508	54	15	=	=	SYM
ejpam-4508	54	16	√	√	NUM
ejpam-4508	54	17	b+	b+	ADP
ejpam-4508	54	18	1	1	NUM
ejpam-4508	54	19	.	.	PUNCT
ejpam-4508	55	1	thus	thus	ADV
ejpam-4508	55	2	,	,	PUNCT
ejpam-4508	55	3	we	we	PRON
ejpam-4508	55	4	have	have	VERB
ejpam-4508	55	5	(	(	PUNCT
ejpam-4508	55	6	y	y	NOUN
ejpam-4508	55	7	,	,	PUNCT
ejpam-4508	55	8	z	z	NOUN
ejpam-4508	55	9	)	)	PUNCT
ejpam-4508	55	10	=	=	SYM
ejpam-4508	55	11	(	(	PUNCT
ejpam-4508	55	12	1	1	NUM
ejpam-4508	55	13	,	,	PUNCT
ejpam-4508	55	14	√	√	NUM
ejpam-4508	55	15	b+	b+	ADP
ejpam-4508	55	16	1	1	NUM
ejpam-4508	55	17	)	)	PUNCT
ejpam-4508	55	18	.	.	PUNCT
ejpam-4508	56	1	corollary	corollary	ADJ
ejpam-4508	56	2	1	1	NUM
ejpam-4508	56	3	.	.	PUNCT
ejpam-4508	57	1	let	let	VERB
ejpam-4508	57	2	p	p	PRON
ejpam-4508	57	3	be	be	AUX
ejpam-4508	57	4	a	a	DET
ejpam-4508	57	5	prime	prime	ADJ
ejpam-4508	57	6	number	number	NOUN
ejpam-4508	57	7	such	such	ADJ
ejpam-4508	58	1	that	that	SCONJ
ejpam-4508	58	2	p	p	PROPN
ejpam-4508	58	3	≡	≡	PROPN
ejpam-4508	58	4	7	7	NUM
ejpam-4508	58	5	(	(	PUNCT
ejpam-4508	58	6	mod	mod	PROPN
ejpam-4508	58	7	12	12	NUM
ejpam-4508	58	8	)	)	PUNCT
ejpam-4508	58	9	.	.	PUNCT
ejpam-4508	59	1	the	the	DET
ejpam-4508	59	2	non	non	ADJ
ejpam-4508	59	3	-	-	ADJ
ejpam-4508	59	4	negative	negative	ADJ
ejpam-4508	59	5	integer	integer	NOUN
ejpam-4508	59	6	solutions	solution	NOUN
ejpam-4508	59	7	to	to	ADP
ejpam-4508	59	8	the	the	DET
ejpam-4508	59	9	diophantine	diophantine	NOUN
ejpam-4508	59	10	equation	equation	NOUN
ejpam-4508	59	11	1+py	1+py	NOUN
ejpam-4508	59	12	=	=	SYM
ejpam-4508	59	13	z2	z2	PROPN
ejpam-4508	59	14	is	be	AUX
ejpam-4508	59	15	(	(	PUNCT
ejpam-4508	59	16	y	y	PROPN
ejpam-4508	59	17	,	,	PUNCT
ejpam-4508	59	18	z	z	NOUN
ejpam-4508	59	19	)	)	PUNCT
ejpam-4508	59	20	=	=	SYM
ejpam-4508	59	21	(	(	PUNCT
ejpam-4508	59	22	1	1	NUM
ejpam-4508	59	23	,	,	PUNCT
ejpam-4508	59	24	√	√	PROPN
ejpam-4508	59	25	p+	p+	VERB
ejpam-4508	59	26	1	1	NUM
ejpam-4508	59	27	)	)	PUNCT
ejpam-4508	59	28	if	if	SCONJ
ejpam-4508	59	29	√	√	PRON
ejpam-4508	59	30	p+	p+	VERB
ejpam-4508	59	31	1	1	NUM
ejpam-4508	59	32	)	)	PUNCT
ejpam-4508	59	33	is	be	AUX
ejpam-4508	59	34	a	a	DET
ejpam-4508	59	35	positive	positive	ADJ
ejpam-4508	59	36	integer	integer	NOUN
ejpam-4508	59	37	.	.	PUNCT
ejpam-4508	60	1	lemma	lemma	PROPN
ejpam-4508	60	2	2	2	X
ejpam-4508	60	3	.	.	PUNCT
ejpam-4508	61	1	let	let	VERB
ejpam-4508	61	2	t	t	PROPN
ejpam-4508	62	1	and	and	CCONJ
ejpam-4508	62	2	d	d	NOUN
ejpam-4508	62	3	be	be	AUX
ejpam-4508	62	4	positive	positive	ADJ
ejpam-4508	62	5	integers	integer	NOUN
ejpam-4508	62	6	.	.	PUNCT
ejpam-4508	63	1	the	the	DET
ejpam-4508	63	2	non	non	ADJ
ejpam-4508	63	3	-	-	ADJ
ejpam-4508	63	4	negative	negative	ADJ
ejpam-4508	63	5	integer	integer	NOUN
ejpam-4508	63	6	solutions	solution	NOUN
ejpam-4508	63	7	of	of	ADP
ejpam-4508	63	8	the	the	DET
ejpam-4508	63	9	diophantine	diophantine	NOUN
ejpam-4508	63	10	equation	equation	NOUN
ejpam-4508	63	11	1	1	NUM
ejpam-4508	63	12	+	+	CCONJ
ejpam-4508	63	13	(	(	PUNCT
ejpam-4508	63	14	d	d	X
ejpam-4508	63	15	+	+	NOUN
ejpam-4508	63	16	4t)x	4t)x	PROPN
ejpam-4508	63	17	=	=	SYM
ejpam-4508	63	18	z2	z2	PROPN
ejpam-4508	63	19	is	be	AUX
ejpam-4508	63	20	(	(	PUNCT
ejpam-4508	63	21	x	x	X
ejpam-4508	63	22	,	,	PUNCT
ejpam-4508	63	23	z	z	NOUN
ejpam-4508	63	24	)	)	PUNCT
ejpam-4508	63	25	=	=	SYM
ejpam-4508	63	26	(	(	PUNCT
ejpam-4508	63	27	1	1	NUM
ejpam-4508	63	28	,	,	PUNCT
ejpam-4508	63	29	2	2	NUM
ejpam-4508	63	30	√	√	NUM
ejpam-4508	63	31	t+	t+	NOUN
ejpam-4508	63	32	d+1	d+1	PROPN
ejpam-4508	63	33	4	4	NUM
ejpam-4508	63	34	)	)	PUNCT
ejpam-4508	63	35	if	if	SCONJ
ejpam-4508	63	36	√	√	ADJ
ejpam-4508	63	37	t+	t+	PUNCT
ejpam-4508	63	38	d+1	d+1	NOUN
ejpam-4508	63	39	4	4	NUM
ejpam-4508	63	40	is	be	AUX
ejpam-4508	63	41	a	a	DET
ejpam-4508	63	42	positive	positive	ADJ
ejpam-4508	63	43	integer	integer	NOUN
ejpam-4508	63	44	.	.	PUNCT
ejpam-4508	64	1	proof	proof	NOUN
ejpam-4508	64	2	.	.	PUNCT
ejpam-4508	65	1	let	let	VERB
ejpam-4508	65	2	t	t	PROPN
ejpam-4508	65	3	,	,	PUNCT
ejpam-4508	65	4	d	d	X
ejpam-4508	65	5	be	be	AUX
ejpam-4508	65	6	positive	positive	ADJ
ejpam-4508	65	7	integers	integer	NOUN
ejpam-4508	66	1	such	such	ADJ
ejpam-4508	66	2	that	that	PRON
ejpam-4508	66	3	√	√	PROPN
ejpam-4508	66	4	t+	t+	NOUN
ejpam-4508	66	5	d+1	d+1	NOUN
ejpam-4508	66	6	4	4	NUM
ejpam-4508	66	7	is	be	AUX
ejpam-4508	66	8	a	a	DET
ejpam-4508	66	9	positive	positive	ADJ
ejpam-4508	66	10	integer	integer	NOUN
ejpam-4508	66	11	.	.	PUNCT
ejpam-4508	67	1	we	we	PRON
ejpam-4508	67	2	have	have	VERB
ejpam-4508	67	3	z2	z2	NOUN
ejpam-4508	67	4	−	−	PROPN
ejpam-4508	68	1	(	(	PUNCT
ejpam-4508	68	2	d	d	PROPN
ejpam-4508	68	3	+	+	NOUN
ejpam-4508	68	4	4t)x	4t)x	NUM
ejpam-4508	68	5	=	=	SYM
ejpam-4508	68	6	1	1	X
ejpam-4508	68	7	.	.	PUNCT
ejpam-4508	68	8	by	by	ADP
ejpam-4508	68	9	proposition	proposition	NOUN
ejpam-4508	68	10	1	1	NUM
ejpam-4508	68	11	,	,	PUNCT
ejpam-4508	68	12	it	it	PRON
ejpam-4508	68	13	is	be	AUX
ejpam-4508	68	14	sufficient	sufficient	ADJ
ejpam-4508	68	15	to	to	PART
ejpam-4508	68	16	consider	consider	VERB
ejpam-4508	68	17	only	only	ADV
ejpam-4508	68	18	the	the	DET
ejpam-4508	68	19	case	case	NOUN
ejpam-4508	68	20	that	that	SCONJ
ejpam-4508	68	21	z	z	NOUN
ejpam-4508	68	22	≤	≤	NOUN
ejpam-4508	68	23	1	1	NUM
ejpam-4508	68	24	or	or	CCONJ
ejpam-4508	68	25	x	x	SYM
ejpam-4508	68	26	≤	≤	NUM
ejpam-4508	68	27	1	1	NUM
ejpam-4508	68	28	.	.	PUNCT
ejpam-4508	69	1	hence	hence	ADV
ejpam-4508	69	2	,	,	PUNCT
ejpam-4508	69	3	we	we	PRON
ejpam-4508	69	4	consider	consider	VERB
ejpam-4508	69	5	the	the	DET
ejpam-4508	69	6	following	follow	VERB
ejpam-4508	69	7	cases	case	NOUN
ejpam-4508	69	8	of	of	ADP
ejpam-4508	69	9	z	z	NOUN
ejpam-4508	69	10	and	and	CCONJ
ejpam-4508	69	11	x.	x.	NOUN
ejpam-4508	69	12	for	for	ADP
ejpam-4508	69	13	z	z	NOUN
ejpam-4508	69	14	=	=	SYM
ejpam-4508	69	15	0	0	NUM
ejpam-4508	69	16	and	and	CCONJ
ejpam-4508	69	17	z	z	NOUN
ejpam-4508	69	18	=	=	SYM
ejpam-4508	69	19	1	1	NUM
ejpam-4508	69	20	,	,	PUNCT
ejpam-4508	69	21	there	there	PRON
ejpam-4508	69	22	is	be	VERB
ejpam-4508	69	23	no	no	DET
ejpam-4508	69	24	solution	solution	NOUN
ejpam-4508	69	25	.	.	PUNCT
ejpam-4508	70	1	if	if	SCONJ
ejpam-4508	70	2	x	x	PROPN
ejpam-4508	70	3	=	=	SYM
ejpam-4508	70	4	0	0	NUM
ejpam-4508	70	5	,	,	PUNCT
ejpam-4508	70	6	then	then	ADV
ejpam-4508	70	7	we	we	PRON
ejpam-4508	70	8	have	have	VERB
ejpam-4508	70	9	z2	z2	NOUN
ejpam-4508	70	10	=	=	SYM
ejpam-4508	70	11	2	2	NUM
ejpam-4508	70	12	,	,	PUNCT
ejpam-4508	70	13	which	which	PRON
ejpam-4508	70	14	is	be	AUX
ejpam-4508	70	15	impossible	impossible	ADJ
ejpam-4508	70	16	.	.	PUNCT
ejpam-4508	71	1	if	if	SCONJ
ejpam-4508	71	2	x	x	SYM
ejpam-4508	71	3	=	=	SYM
ejpam-4508	71	4	1	1	NUM
ejpam-4508	71	5	,	,	PUNCT
ejpam-4508	71	6	then	then	ADV
ejpam-4508	71	7	we	we	PRON
ejpam-4508	71	8	have	have	VERB
ejpam-4508	71	9	w.	w.	PROPN
ejpam-4508	71	10	orosram	orosram	PROPN
ejpam-4508	71	11	,	,	PUNCT
ejpam-4508	71	12	k.	k.	PROPN
ejpam-4508	71	13	makonwattana	makonwattana	PROPN
ejpam-4508	71	14	,	,	PUNCT
ejpam-4508	71	15	s.	s.	PROPN
ejpam-4508	71	16	khongsawat	khongsawat	PROPN
ejpam-4508	71	17	/	/	SYM
ejpam-4508	71	18	eur	eur	PROPN
ejpam-4508	71	19	.	.	PUNCT
ejpam-4508	72	1	j.	j.	PROPN
ejpam-4508	72	2	pure	pure	PROPN
ejpam-4508	72	3	appl	appl	PROPN
ejpam-4508	72	4	.	.	PROPN
ejpam-4508	72	5	math	math	PROPN
ejpam-4508	72	6	,	,	PUNCT
ejpam-4508	72	7	15	15	NUM
ejpam-4508	72	8	(	(	PUNCT
ejpam-4508	72	9	4	4	NUM
ejpam-4508	72	10	)	)	PUNCT
ejpam-4508	72	11	(	(	PUNCT
ejpam-4508	72	12	2022	2022	NUM
ejpam-4508	72	13	)	)	PUNCT
ejpam-4508	72	14	,	,	PUNCT
ejpam-4508	72	15	1593	1593	NUM
ejpam-4508	72	16	-	-	SYM
ejpam-4508	72	17	1596	1596	NUM
ejpam-4508	72	18	1595	1595	NUM
ejpam-4508	72	19	z2	z2	NOUN
ejpam-4508	72	20	=	=	SYM
ejpam-4508	72	21	4	4	NUM
ejpam-4508	72	22	t	t	NOUN
ejpam-4508	72	23	+	+	CCONJ
ejpam-4508	73	1	d	d	X
ejpam-4508	73	2	+	+	NOUN
ejpam-4508	73	3	1	1	NUM
ejpam-4508	73	4	.	.	PUNCT
ejpam-4508	74	1	thus	thus	ADV
ejpam-4508	74	2	z	z	X
ejpam-4508	74	3	=	=	SYM
ejpam-4508	74	4	2	2	NUM
ejpam-4508	74	5	√	√	NUM
ejpam-4508	74	6	t+	t+	PUNCT
ejpam-4508	74	7	d+1	d+1	PROPN
ejpam-4508	74	8	4	4	NUM
ejpam-4508	74	9	where	where	SCONJ
ejpam-4508	74	10	√	√	VERB
ejpam-4508	74	11	t+	t+	PUNCT
ejpam-4508	74	12	d+1	d+1	NOUN
ejpam-4508	74	13	4	4	NUM
ejpam-4508	74	14	is	be	AUX
ejpam-4508	74	15	a	a	DET
ejpam-4508	74	16	positive	positive	ADJ
ejpam-4508	74	17	integer	integer	NOUN
ejpam-4508	74	18	.	.	PUNCT
ejpam-4508	75	1	therefore	therefore	ADV
ejpam-4508	75	2	,	,	PUNCT
ejpam-4508	75	3	(	(	PUNCT
ejpam-4508	75	4	x	x	X
ejpam-4508	75	5	,	,	PUNCT
ejpam-4508	75	6	z	z	NOUN
ejpam-4508	75	7	)	)	PUNCT
ejpam-4508	75	8	=	=	SYM
ejpam-4508	75	9	(	(	PUNCT
ejpam-4508	75	10	1	1	NUM
ejpam-4508	75	11	,	,	PUNCT
ejpam-4508	75	12	2	2	NUM
ejpam-4508	75	13	√	√	NUM
ejpam-4508	75	14	t+	t+	NOUN
ejpam-4508	75	15	d+1	d+1	PROPN
ejpam-4508	75	16	4	4	NUM
ejpam-4508	75	17	)	)	PUNCT
ejpam-4508	75	18	.	.	PUNCT
ejpam-4508	76	1	corollary	corollary	ADJ
ejpam-4508	76	2	2	2	NUM
ejpam-4508	76	3	.	.	PUNCT
ejpam-4508	77	1	let	let	VERB
ejpam-4508	77	2	n	n	PRON
ejpam-4508	77	3	be	be	AUX
ejpam-4508	77	4	a	a	DET
ejpam-4508	77	5	positive	positive	ADJ
ejpam-4508	77	6	integer	integer	NOUN
ejpam-4508	77	7	and	and	CCONJ
ejpam-4508	77	8	p	p	X
ejpam-4508	77	9	,	,	PUNCT
ejpam-4508	77	10	(	(	PUNCT
ejpam-4508	77	11	p+4n	p+4n	NOUN
ejpam-4508	77	12	)	)	PUNCT
ejpam-4508	77	13	be	be	VERB
ejpam-4508	77	14	prime	prime	ADJ
ejpam-4508	77	15	numbers	number	NOUN
ejpam-4508	77	16	such	such	ADJ
ejpam-4508	77	17	that	that	SCONJ
ejpam-4508	77	18	n	n	NUM
ejpam-4508	77	19	≡	≡	PROPN
ejpam-4508	77	20	0	0	NUM
ejpam-4508	77	21	,	,	PUNCT
ejpam-4508	77	22	1	1	NUM
ejpam-4508	77	23	(	(	PUNCT
ejpam-4508	77	24	mod	mod	NOUN
ejpam-4508	77	25	3	3	NUM
ejpam-4508	77	26	)	)	PUNCT
ejpam-4508	77	27	and	and	CCONJ
ejpam-4508	77	28	p	p	PRON
ejpam-4508	77	29	≡	≡	PROPN
ejpam-4508	77	30	7	7	NUM
ejpam-4508	77	31	(	(	PUNCT
ejpam-4508	77	32	mod	mod	PROPN
ejpam-4508	77	33	12	12	NUM
ejpam-4508	77	34	)	)	PUNCT
ejpam-4508	77	35	.	.	PUNCT
ejpam-4508	78	1	the	the	DET
ejpam-4508	78	2	non	non	ADJ
ejpam-4508	78	3	-	-	ADJ
ejpam-4508	78	4	negative	negative	ADJ
ejpam-4508	78	5	integer	integer	NOUN
ejpam-4508	78	6	solutions	solution	NOUN
ejpam-4508	78	7	of	of	ADP
ejpam-4508	78	8	the	the	DET
ejpam-4508	78	9	diophantine	diophantine	NOUN
ejpam-4508	78	10	equation	equation	NOUN
ejpam-4508	78	11	1	1	NUM
ejpam-4508	78	12	+	+	CCONJ
ejpam-4508	78	13	(	(	PUNCT
ejpam-4508	78	14	p+4n)x	p+4n)x	NOUN
ejpam-4508	78	15	=	=	SYM
ejpam-4508	78	16	z2	z2	PROPN
ejpam-4508	78	17	is	be	AUX
ejpam-4508	78	18	(	(	PUNCT
ejpam-4508	78	19	x	x	X
ejpam-4508	78	20	,	,	PUNCT
ejpam-4508	78	21	z	z	NOUN
ejpam-4508	78	22	)	)	PUNCT
ejpam-4508	78	23	=	=	SYM
ejpam-4508	78	24	(	(	PUNCT
ejpam-4508	78	25	1	1	NUM
ejpam-4508	78	26	,	,	PUNCT
ejpam-4508	78	27	2	2	NUM
ejpam-4508	78	28	√	√	NUM
ejpam-4508	78	29	n+	n+	ADP
ejpam-4508	78	30	p+1	p+1	NOUN
ejpam-4508	78	31	4	4	NUM
ejpam-4508	78	32	)	)	PUNCT
ejpam-4508	78	33	if	if	SCONJ
ejpam-4508	78	34	√	√	PROPN
ejpam-4508	78	35	n+	n+	ADP
ejpam-4508	78	36	p+1	p+1	NOUN
ejpam-4508	78	37	4	4	NUM
ejpam-4508	78	38	is	be	AUX
ejpam-4508	78	39	a	a	DET
ejpam-4508	78	40	positive	positive	ADJ
ejpam-4508	78	41	integer	integer	NOUN
ejpam-4508	78	42	.	.	PUNCT
ejpam-4508	79	1	theorem	theorem	NOUN
ejpam-4508	79	2	1	1	NUM
ejpam-4508	79	3	.	.	PUNCT
ejpam-4508	80	1	let	let	VERB
ejpam-4508	80	2	n	n	PRON
ejpam-4508	80	3	be	be	AUX
ejpam-4508	80	4	a	a	DET
ejpam-4508	80	5	positive	positive	ADJ
ejpam-4508	80	6	integer	integer	NOUN
ejpam-4508	81	1	such	such	ADJ
ejpam-4508	81	2	that	that	SCONJ
ejpam-4508	81	3	n	n	NUM
ejpam-4508	81	4	≡	≡	PROPN
ejpam-4508	81	5	0	0	NUM
ejpam-4508	81	6	,	,	PUNCT
ejpam-4508	81	7	1	1	NUM
ejpam-4508	81	8	(	(	PUNCT
ejpam-4508	81	9	mod	mod	NOUN
ejpam-4508	81	10	3	3	NUM
ejpam-4508	81	11	)	)	PUNCT
ejpam-4508	81	12	,	,	PUNCT
ejpam-4508	81	13	p	p	PROPN
ejpam-4508	81	14	≡	≡	PROPN
ejpam-4508	81	15	7	7	NUM
ejpam-4508	81	16	(	(	PUNCT
ejpam-4508	81	17	mod	mod	PROPN
ejpam-4508	81	18	12	12	NUM
ejpam-4508	81	19	)	)	PUNCT
ejpam-4508	81	20	.	.	PUNCT
ejpam-4508	82	1	if	if	SCONJ
ejpam-4508	82	2	√	√	PROPN
ejpam-4508	82	3	p+	p+	VERB
ejpam-4508	82	4	1	1	NUM
ejpam-4508	82	5	and	and	CCONJ
ejpam-4508	82	6	√	√	NUM
ejpam-4508	82	7	n+	n+	VERB
ejpam-4508	82	8	p+1	p+1	NOUN
ejpam-4508	82	9	4	4	NUM
ejpam-4508	82	10	are	be	AUX
ejpam-4508	82	11	also	also	ADV
ejpam-4508	82	12	integers	integer	NOUN
ejpam-4508	82	13	,	,	PUNCT
ejpam-4508	82	14	then	then	ADV
ejpam-4508	82	15	all	all	PRON
ejpam-4508	82	16	of	of	ADP
ejpam-4508	82	17	the	the	DET
ejpam-4508	82	18	non	non	ADJ
ejpam-4508	82	19	-	-	ADJ
ejpam-4508	82	20	negative	negative	ADJ
ejpam-4508	82	21	integer	integer	NOUN
ejpam-4508	82	22	solutions	solution	NOUN
ejpam-4508	82	23	to	to	ADP
ejpam-4508	82	24	the	the	DET
ejpam-4508	82	25	diophantine	diophantine	NOUN
ejpam-4508	82	26	equation	equation	NOUN
ejpam-4508	82	27	(	(	PUNCT
ejpam-4508	82	28	p	p	X
ejpam-4508	82	29	+	+	NOUN
ejpam-4508	82	30	4n)x	4n)x	NUM
ejpam-4508	83	1	+	+	CCONJ
ejpam-4508	83	2	py	py	NOUN
ejpam-4508	83	3	=	=	SYM
ejpam-4508	83	4	z2	z2	PROPN
ejpam-4508	83	5	are	be	AUX
ejpam-4508	83	6	given	give	VERB
ejpam-4508	83	7	by	by	ADP
ejpam-4508	83	8	(	(	PUNCT
ejpam-4508	83	9	x	x	PROPN
ejpam-4508	83	10	,	,	PUNCT
ejpam-4508	83	11	y	y	PROPN
ejpam-4508	83	12	,	,	PUNCT
ejpam-4508	83	13	z	z	NOUN
ejpam-4508	83	14	)	)	PUNCT
ejpam-4508	83	15	∈	∈	NOUN
ejpam-4508	83	16	{	{	PUNCT
ejpam-4508	83	17	(	(	PUNCT
ejpam-4508	83	18	0	0	NUM
ejpam-4508	83	19	,	,	PUNCT
ejpam-4508	83	20	1	1	NUM
ejpam-4508	83	21	,	,	PUNCT
ejpam-4508	83	22	√	√	PROPN
ejpam-4508	83	23	p+	p+	ADJ
ejpam-4508	83	24	1	1	NUM
ejpam-4508	83	25	)	)	PUNCT
ejpam-4508	83	26	}	}	PUNCT
ejpam-4508	83	27	∪	∪	X
ejpam-4508	83	28	{	{	PUNCT
ejpam-4508	83	29	(	(	PUNCT
ejpam-4508	83	30	1	1	NUM
ejpam-4508	83	31	,	,	PUNCT
ejpam-4508	83	32	0	0	NUM
ejpam-4508	83	33	,	,	PUNCT
ejpam-4508	83	34	2	2	NUM
ejpam-4508	83	35	√	√	NUM
ejpam-4508	83	36	n+	n+	ADP
ejpam-4508	83	37	p+1	p+1	NOUN
ejpam-4508	83	38	4	4	NUM
ejpam-4508	83	39	)	)	PUNCT
ejpam-4508	83	40	}	}	PUNCT
ejpam-4508	83	41	,	,	PUNCT
ejpam-4508	83	42	where	where	SCONJ
ejpam-4508	83	43	p	p	PROPN
ejpam-4508	83	44	and	and	CCONJ
ejpam-4508	83	45	p+	p+	PROPN
ejpam-4508	83	46	4n	4n	NOUN
ejpam-4508	83	47	are	be	AUX
ejpam-4508	83	48	prime	prime	ADJ
ejpam-4508	83	49	number	number	NOUN
ejpam-4508	83	50	.	.	PUNCT
ejpam-4508	84	1	proof	proof	NOUN
ejpam-4508	84	2	.	.	PUNCT
ejpam-4508	85	1	since	since	SCONJ
ejpam-4508	85	2	p	p	NOUN
ejpam-4508	85	3	is	be	AUX
ejpam-4508	85	4	a	a	DET
ejpam-4508	85	5	prime	prime	ADJ
ejpam-4508	85	6	number	number	NOUN
ejpam-4508	85	7	such	such	ADJ
ejpam-4508	85	8	that	that	SCONJ
ejpam-4508	85	9	p	p	PROPN
ejpam-4508	85	10	≡	≡	PROPN
ejpam-4508	85	11	7	7	NUM
ejpam-4508	85	12	(	(	PUNCT
ejpam-4508	85	13	mod	mod	PROPN
ejpam-4508	85	14	12	12	NUM
ejpam-4508	85	15	)	)	PUNCT
ejpam-4508	85	16	,	,	PUNCT
ejpam-4508	85	17	it	it	PRON
ejpam-4508	85	18	is	be	AUX
ejpam-4508	85	19	clear	clear	ADJ
ejpam-4508	85	20	that	that	SCONJ
ejpam-4508	85	21	p	p	PROPN
ejpam-4508	85	22	≡	≡	PROPN
ejpam-4508	85	23	3	3	NUM
ejpam-4508	85	24	(	(	PUNCT
ejpam-4508	85	25	mod	mod	NOUN
ejpam-4508	85	26	4	4	NUM
ejpam-4508	85	27	)	)	PUNCT
ejpam-4508	85	28	and	and	CCONJ
ejpam-4508	85	29	p	p	PROPN
ejpam-4508	85	30	≡	≡	PROPN
ejpam-4508	85	31	1	1	NUM
ejpam-4508	85	32	(	(	PUNCT
ejpam-4508	85	33	mod	mod	NOUN
ejpam-4508	85	34	3	3	NUM
ejpam-4508	85	35	)	)	PUNCT
ejpam-4508	85	36	.	.	PUNCT
ejpam-4508	86	1	let	let	AUX
ejpam-4508	86	2	(	(	PUNCT
ejpam-4508	86	3	x	x	X
ejpam-4508	86	4	,	,	PUNCT
ejpam-4508	86	5	y	y	PROPN
ejpam-4508	86	6	,	,	PUNCT
ejpam-4508	86	7	z	z	NOUN
ejpam-4508	86	8	)	)	PUNCT
ejpam-4508	86	9	be	be	AUX
ejpam-4508	86	10	a	a	DET
ejpam-4508	86	11	non	non	ADJ
ejpam-4508	86	12	-	-	ADJ
ejpam-4508	86	13	negative	negative	ADJ
ejpam-4508	86	14	integer	integer	NOUN
ejpam-4508	86	15	solution	solution	NOUN
ejpam-4508	86	16	of	of	ADP
ejpam-4508	86	17	(	(	PUNCT
ejpam-4508	86	18	p+4n)x+	p+4n)x+	NOUN
ejpam-4508	86	19	py	py	NOUN
ejpam-4508	86	20	=	=	SYM
ejpam-4508	86	21	z2	z2	PROPN
ejpam-4508	86	22	.	.	PUNCT
ejpam-4508	87	1	if	if	SCONJ
ejpam-4508	87	2	x	x	PROPN
ejpam-4508	87	3	=	=	SYM
ejpam-4508	87	4	0	0	NUM
ejpam-4508	87	5	or	or	CCONJ
ejpam-4508	87	6	y	y	PROPN
ejpam-4508	87	7	=	=	SYM
ejpam-4508	87	8	0	0	PROPN
ejpam-4508	87	9	,	,	PUNCT
ejpam-4508	87	10	then	then	ADV
ejpam-4508	87	11	(	(	PUNCT
ejpam-4508	87	12	x	x	X
ejpam-4508	87	13	,	,	PUNCT
ejpam-4508	87	14	y	y	PROPN
ejpam-4508	87	15	,	,	PUNCT
ejpam-4508	87	16	z	z	NOUN
ejpam-4508	87	17	)	)	PUNCT
ejpam-4508	87	18	=	=	SYM
ejpam-4508	87	19	(	(	PUNCT
ejpam-4508	87	20	0	0	NUM
ejpam-4508	87	21	,	,	PUNCT
ejpam-4508	87	22	1	1	NUM
ejpam-4508	87	23	,	,	PUNCT
ejpam-4508	87	24	√	√	PROPN
ejpam-4508	87	25	p+	p+	VERB
ejpam-4508	87	26	1	1	NUM
ejpam-4508	87	27	)	)	PUNCT
ejpam-4508	87	28	or	or	CCONJ
ejpam-4508	87	29	(	(	PUNCT
ejpam-4508	87	30	x	x	X
ejpam-4508	87	31	,	,	PUNCT
ejpam-4508	87	32	y	y	PROPN
ejpam-4508	87	33	,	,	PUNCT
ejpam-4508	87	34	z	z	NOUN
ejpam-4508	87	35	)	)	PUNCT
ejpam-4508	87	36	=	=	SYM
ejpam-4508	87	37	(	(	PUNCT
ejpam-4508	87	38	1	1	NUM
ejpam-4508	87	39	,	,	PUNCT
ejpam-4508	87	40	0	0	NUM
ejpam-4508	87	41	,	,	PUNCT
ejpam-4508	87	42	2	2	NUM
ejpam-4508	87	43	√	√	NUM
ejpam-4508	88	1	n+	n+	ADP
ejpam-4508	88	2	p+1	p+1	NOUN
ejpam-4508	88	3	4	4	NUM
ejpam-4508	88	4	)	)	PUNCT
ejpam-4508	88	5	.	.	PUNCT
ejpam-4508	89	1	suppose	suppose	VERB
ejpam-4508	89	2	x	x	PUNCT
ejpam-4508	89	3	>	>	X
ejpam-4508	89	4	0	0	PUNCT
ejpam-4508	89	5	and	and	CCONJ
ejpam-4508	89	6	y	y	PROPN
ejpam-4508	89	7	>	>	X
ejpam-4508	89	8	0	0	X
ejpam-4508	89	9	.	.	PUNCT
ejpam-4508	90	1	we	we	PRON
ejpam-4508	90	2	consider	consider	VERB
ejpam-4508	90	3	the	the	DET
ejpam-4508	90	4	following	follow	VERB
ejpam-4508	90	5	cases	case	NOUN
ejpam-4508	90	6	.	.	PUNCT
ejpam-4508	91	1	case	case	NOUN
ejpam-4508	91	2	1	1	NUM
ejpam-4508	91	3	.	.	NUM
ejpam-4508	91	4	x	x	PUNCT
ejpam-4508	91	5	and	and	CCONJ
ejpam-4508	91	6	y	y	PROPN
ejpam-4508	91	7	are	be	AUX
ejpam-4508	91	8	even	even	ADV
ejpam-4508	91	9	numbers	number	NOUN
ejpam-4508	91	10	.	.	PUNCT
ejpam-4508	92	1	since	since	SCONJ
ejpam-4508	92	2	(	(	PUNCT
ejpam-4508	92	3	p	p	X
ejpam-4508	92	4	+	+	NOUN
ejpam-4508	92	5	4n)x	4n)x	NUM
ejpam-4508	92	6	+	+	CCONJ
ejpam-4508	92	7	py	py	PROPN
ejpam-4508	92	8	=	=	SYM
ejpam-4508	92	9	z2	z2	PROPN
ejpam-4508	92	10	,	,	PUNCT
ejpam-4508	92	11	it	it	PRON
ejpam-4508	92	12	follows	follow	VERB
ejpam-4508	92	13	that	that	SCONJ
ejpam-4508	92	14	z	z	NOUN
ejpam-4508	92	15	is	be	AUX
ejpam-4508	92	16	even	even	ADV
ejpam-4508	92	17	.	.	PUNCT
ejpam-4508	93	1	so	so	ADV
ejpam-4508	93	2	z2	z2	PROPN
ejpam-4508	93	3	≡	≡	PROPN
ejpam-4508	93	4	0	0	PUNCT
ejpam-4508	94	1	(	(	PUNCT
ejpam-4508	94	2	mod	mod	PROPN
ejpam-4508	94	3	4	4	NUM
ejpam-4508	94	4	)	)	PUNCT
ejpam-4508	94	5	.	.	PUNCT
ejpam-4508	95	1	note	note	VERB
ejpam-4508	95	2	that	that	SCONJ
ejpam-4508	95	3	(	(	PUNCT
ejpam-4508	95	4	p	p	NOUN
ejpam-4508	95	5	+	+	CCONJ
ejpam-4508	95	6	4n)x	4n)x	NUM
ejpam-4508	95	7	≡	≡	ADJ
ejpam-4508	95	8	1	1	NUM
ejpam-4508	95	9	(	(	PUNCT
ejpam-4508	95	10	mod	mod	NOUN
ejpam-4508	95	11	4	4	NUM
ejpam-4508	95	12	)	)	PUNCT
ejpam-4508	95	13	and	and	CCONJ
ejpam-4508	95	14	py	py	PROPN
ejpam-4508	95	15	≡	≡	PROPN
ejpam-4508	95	16	1	1	NUM
ejpam-4508	95	17	(	(	PUNCT
ejpam-4508	95	18	mod	mod	NOUN
ejpam-4508	95	19	4	4	NUM
ejpam-4508	95	20	)	)	PUNCT
ejpam-4508	95	21	.	.	PUNCT
ejpam-4508	96	1	thus	thus	ADV
ejpam-4508	96	2	(	(	PUNCT
ejpam-4508	96	3	p+	p+	NOUN
ejpam-4508	96	4	4n)x	4n)x	NUM
ejpam-4508	96	5	+	+	CCONJ
ejpam-4508	97	1	py	py	X
ejpam-4508	97	2	≡	≡	PROPN
ejpam-4508	97	3	2	2	NUM
ejpam-4508	97	4	(	(	PUNCT
ejpam-4508	97	5	mod	mod	NOUN
ejpam-4508	97	6	4	4	NUM
ejpam-4508	97	7	)	)	PUNCT
ejpam-4508	97	8	which	which	PRON
ejpam-4508	97	9	contradicts	contradict	VERB
ejpam-4508	97	10	with	with	ADP
ejpam-4508	97	11	z2	z2	PROPN
ejpam-4508	97	12	≡	≡	PROPN
ejpam-4508	97	13	0	0	PUNCT
ejpam-4508	98	1	(	(	PUNCT
ejpam-4508	98	2	mod	mod	PROPN
ejpam-4508	98	3	4	4	NUM
ejpam-4508	98	4	)	)	PUNCT
ejpam-4508	98	5	.	.	PUNCT
ejpam-4508	99	1	case	case	NOUN
ejpam-4508	99	2	2	2	NUM
ejpam-4508	99	3	.	.	NUM
ejpam-4508	99	4	x	x	PUNCT
ejpam-4508	99	5	and	and	CCONJ
ejpam-4508	99	6	y	y	PROPN
ejpam-4508	99	7	are	be	AUX
ejpam-4508	99	8	odd	odd	ADJ
ejpam-4508	99	9	numbers	number	NOUN
ejpam-4508	99	10	.	.	PUNCT
ejpam-4508	100	1	since	since	SCONJ
ejpam-4508	100	2	(	(	PUNCT
ejpam-4508	100	3	p+4n)x	p+4n)x	PROPN
ejpam-4508	100	4	≡	≡	PROPN
ejpam-4508	100	5	3	3	NUM
ejpam-4508	100	6	(	(	PUNCT
ejpam-4508	100	7	mod	mod	NOUN
ejpam-4508	100	8	4	4	NUM
ejpam-4508	100	9	)	)	PUNCT
ejpam-4508	100	10	and	and	CCONJ
ejpam-4508	100	11	py	py	PROPN
ejpam-4508	100	12	≡	≡	PROPN
ejpam-4508	100	13	3	3	NUM
ejpam-4508	100	14	(	(	PUNCT
ejpam-4508	100	15	mod	mod	NOUN
ejpam-4508	100	16	4	4	NUM
ejpam-4508	100	17	)	)	PUNCT
ejpam-4508	100	18	,	,	PUNCT
ejpam-4508	100	19	it	it	PRON
ejpam-4508	100	20	follows	follow	VERB
ejpam-4508	100	21	that	that	SCONJ
ejpam-4508	100	22	(	(	PUNCT
ejpam-4508	100	23	p+	p+	NOUN
ejpam-4508	100	24	4n)x	4n)x	NUM
ejpam-4508	100	25	+	+	CCONJ
ejpam-4508	100	26	py	py	X
ejpam-4508	100	27	≡	≡	PROPN
ejpam-4508	100	28	2	2	NUM
ejpam-4508	100	29	(	(	PUNCT
ejpam-4508	100	30	mod	mod	NOUN
ejpam-4508	100	31	4	4	NUM
ejpam-4508	100	32	)	)	PUNCT
ejpam-4508	100	33	which	which	PRON
ejpam-4508	100	34	contradicts	contradict	VERB
ejpam-4508	100	35	with	with	ADP
ejpam-4508	100	36	z2	z2	PROPN
ejpam-4508	100	37	≡	≡	PROPN
ejpam-4508	100	38	0	0	PUNCT
ejpam-4508	101	1	(	(	PUNCT
ejpam-4508	101	2	mod	mod	PROPN
ejpam-4508	101	3	4	4	NUM
ejpam-4508	101	4	)	)	PUNCT
ejpam-4508	101	5	.	.	PUNCT
ejpam-4508	102	1	case	case	NOUN
ejpam-4508	102	2	3	3	NUM
ejpam-4508	102	3	.	.	NUM
ejpam-4508	102	4	x	x	PUNCT
ejpam-4508	102	5	is	be	AUX
ejpam-4508	102	6	an	an	DET
ejpam-4508	102	7	even	even	ADJ
ejpam-4508	102	8	number	number	NOUN
ejpam-4508	102	9	and	and	CCONJ
ejpam-4508	102	10	y	y	PROPN
ejpam-4508	102	11	is	be	AUX
ejpam-4508	102	12	an	an	DET
ejpam-4508	102	13	odd	odd	ADJ
ejpam-4508	102	14	number	number	NOUN
ejpam-4508	102	15	.	.	PUNCT
ejpam-4508	103	1	let	let	VERB
ejpam-4508	103	2	x	x	PUNCT
ejpam-4508	103	3	=	=	SYM
ejpam-4508	103	4	2k	2k	NUM
ejpam-4508	103	5	,	,	PUNCT
ejpam-4508	103	6	k	k	PROPN
ejpam-4508	103	7	≥	≥	NUM
ejpam-4508	103	8	1	1	NUM
ejpam-4508	103	9	and	and	CCONJ
ejpam-4508	103	10	y	y	PROPN
ejpam-4508	103	11	=	=	SYM
ejpam-4508	103	12	2s+1	2s+1	PROPN
ejpam-4508	103	13	,	,	PUNCT
ejpam-4508	103	14	s	s	VERB
ejpam-4508	103	15	≥	≥	NOUN
ejpam-4508	103	16	0	0	NUM
ejpam-4508	103	17	.	.	PUNCT
ejpam-4508	104	1	we	we	PRON
ejpam-4508	104	2	have	have	VERB
ejpam-4508	104	3	(	(	PUNCT
ejpam-4508	104	4	p+4n)2k+p2s+1	p+4n)2k+p2s+1	PROPN
ejpam-4508	104	5	=	=	SYM
ejpam-4508	104	6	z2	z2	PROPN
ejpam-4508	104	7	,	,	PUNCT
ejpam-4508	104	8	or	or	CCONJ
ejpam-4508	104	9	equivalently	equivalently	ADV
ejpam-4508	104	10	p2s+1	p2s+1	NOUN
ejpam-4508	104	11	=	=	NOUN
ejpam-4508	104	12	z2−(p+4n)2k	z2−(p+4n)2k	NOUN
ejpam-4508	104	13	=	=	PUNCT
ejpam-4508	105	1	[	[	X
ejpam-4508	105	2	z	z	X
ejpam-4508	105	3	+	+	X
ejpam-4508	105	4	(	(	PUNCT
ejpam-4508	105	5	p	p	X
ejpam-4508	105	6	+	+	PROPN
ejpam-4508	105	7	4n)k][z	4n)k][z	NUM
ejpam-4508	105	8	−	−	NOUN
ejpam-4508	106	1	(	(	PUNCT
ejpam-4508	106	2	p	p	X
ejpam-4508	106	3	+	+	ADJ
ejpam-4508	106	4	4n)k	4n)k	NUM
ejpam-4508	106	5	]	]	PUNCT
ejpam-4508	106	6	.	.	PUNCT
ejpam-4508	107	1	thus	thus	ADV
ejpam-4508	107	2	,	,	PUNCT
ejpam-4508	107	3	there	there	PRON
ejpam-4508	107	4	exist	exist	VERB
ejpam-4508	107	5	non	non	ADJ
ejpam-4508	107	6	-	-	ADJ
ejpam-4508	107	7	negative	negative	ADJ
ejpam-4508	107	8	integers	integer	NOUN
ejpam-4508	107	9	α	α	NOUN
ejpam-4508	107	10	,	,	PUNCT
ejpam-4508	107	11	β	β	PROPN
ejpam-4508	107	12	such	such	ADJ
ejpam-4508	107	13	that	that	DET
ejpam-4508	107	14	pα	pα	NOUN
ejpam-4508	107	15	=	=	PUNCT
ejpam-4508	107	16	z+(p+4n)k	z+(p+4n)k	NOUN
ejpam-4508	107	17	and	and	CCONJ
ejpam-4508	107	18	pβ	pβ	ADP
ejpam-4508	107	19	=	=	ADJ
ejpam-4508	107	20	z−	z−	X
ejpam-4508	107	21	(	(	PUNCT
ejpam-4508	107	22	p+4n)k	p+4n)k	NOUN
ejpam-4508	107	23	,	,	PUNCT
ejpam-4508	107	24	where	where	SCONJ
ejpam-4508	107	25	α	α	NOUN
ejpam-4508	107	26	>	>	X
ejpam-4508	107	27	β	β	X
ejpam-4508	107	28	and	and	CCONJ
ejpam-4508	107	29	α+β	α+β	NUM
ejpam-4508	107	30	=	=	SYM
ejpam-4508	107	31	2s+1	2s+1	PROPN
ejpam-4508	107	32	.	.	PUNCT
ejpam-4508	108	1	then	then	ADV
ejpam-4508	108	2	,	,	PUNCT
ejpam-4508	108	3	we	we	PRON
ejpam-4508	108	4	have	have	VERB
ejpam-4508	108	5	2(p	2(p	NUM
ejpam-4508	108	6	+	+	CCONJ
ejpam-4508	108	7	4n)k	4n)k	NUM
ejpam-4508	108	8	=	=	SYM
ejpam-4508	108	9	pβ(pα−β	pβ(pα−β	NOUN
ejpam-4508	108	10	−	−	NOUN
ejpam-4508	108	11	1	1	NUM
ejpam-4508	108	12	)	)	PUNCT
ejpam-4508	108	13	.	.	PUNCT
ejpam-4508	109	1	this	this	PRON
ejpam-4508	109	2	implies	imply	VERB
ejpam-4508	109	3	that	that	SCONJ
ejpam-4508	109	4	β	β	X
ejpam-4508	109	5	=	=	SYM
ejpam-4508	109	6	0	0	X
ejpam-4508	109	7	.	.	PUNCT
ejpam-4508	110	1	we	we	PRON
ejpam-4508	110	2	have	have	VERB
ejpam-4508	110	3	2(p	2(p	NUM
ejpam-4508	110	4	+	+	CCONJ
ejpam-4508	110	5	4n)k	4n)k	NUM
ejpam-4508	110	6	=	=	SYM
ejpam-4508	110	7	(	(	PUNCT
ejpam-4508	110	8	p2s+1	p2s+1	NOUN
ejpam-4508	110	9	−	−	NOUN
ejpam-4508	110	10	1	1	NUM
ejpam-4508	110	11	)	)	PUNCT
ejpam-4508	110	12	,	,	PUNCT
ejpam-4508	110	13	which	which	PRON
ejpam-4508	110	14	is	be	AUX
ejpam-4508	110	15	impossible	impossible	ADJ
ejpam-4508	110	16	because	because	SCONJ
ejpam-4508	110	17	2(p+	2(p+	NUM
ejpam-4508	110	18	4n)k	4n)k	NUM
ejpam-4508	110	19	≡	≡	PROPN
ejpam-4508	110	20	1	1	NUM
ejpam-4508	110	21	,	,	PUNCT
ejpam-4508	110	22	2	2	NUM
ejpam-4508	110	23	(	(	PUNCT
ejpam-4508	110	24	mod	mod	NOUN
ejpam-4508	110	25	3	3	NUM
ejpam-4508	110	26	)	)	PUNCT
ejpam-4508	110	27	but	but	CCONJ
ejpam-4508	110	28	(	(	PUNCT
ejpam-4508	110	29	p2s+1	p2s+1	NOUN
ejpam-4508	110	30	−	−	NOUN
ejpam-4508	110	31	1	1	NUM
ejpam-4508	110	32	)	)	PUNCT
ejpam-4508	110	33	≡	≡	PROPN
ejpam-4508	110	34	0	0	PUNCT
ejpam-4508	110	35	(	(	PUNCT
ejpam-4508	110	36	mod	mod	NOUN
ejpam-4508	110	37	3	3	NUM
ejpam-4508	110	38	)	)	PUNCT
ejpam-4508	110	39	.	.	PUNCT
ejpam-4508	111	1	case	case	NOUN
ejpam-4508	111	2	4	4	NUM
ejpam-4508	111	3	.	.	PUNCT
ejpam-4508	111	4	x	x	PRON
ejpam-4508	111	5	is	be	AUX
ejpam-4508	111	6	an	an	DET
ejpam-4508	111	7	odd	odd	ADJ
ejpam-4508	111	8	number	number	NOUN
ejpam-4508	111	9	and	and	CCONJ
ejpam-4508	111	10	y	y	PROPN
ejpam-4508	111	11	is	be	AUX
ejpam-4508	111	12	an	an	DET
ejpam-4508	111	13	even	even	ADJ
ejpam-4508	111	14	number	number	NOUN
ejpam-4508	111	15	.	.	PUNCT
ejpam-4508	112	1	let	let	VERB
ejpam-4508	112	2	x	x	PUNCT
ejpam-4508	112	3	=	=	PUNCT
ejpam-4508	112	4	2k	2k	NUM
ejpam-4508	112	5	+	+	CCONJ
ejpam-4508	112	6	1	1	NUM
ejpam-4508	113	1	,	,	PUNCT
ejpam-4508	113	2	k	k	X
ejpam-4508	113	3	≥	≥	X
ejpam-4508	113	4	0	0	NUM
ejpam-4508	113	5	and	and	CCONJ
ejpam-4508	113	6	y	y	PROPN
ejpam-4508	113	7	=	=	SYM
ejpam-4508	113	8	2s	2s	X
ejpam-4508	113	9	,	,	PUNCT
ejpam-4508	113	10	s	s	PART
ejpam-4508	113	11	≥	≥	NOUN
ejpam-4508	113	12	1	1	NUM
ejpam-4508	113	13	.	.	PUNCT
ejpam-4508	114	1	we	we	PRON
ejpam-4508	114	2	have	have	VERB
ejpam-4508	114	3	(	(	PUNCT
ejpam-4508	114	4	p+	p+	PROPN
ejpam-4508	114	5	4n)2k+1	4n)2k+1	NUM
ejpam-4508	114	6	+	+	NUM
ejpam-4508	114	7	p2s	p2s	PROPN
ejpam-4508	114	8	=	=	SYM
ejpam-4508	114	9	z2	z2	PROPN
ejpam-4508	114	10	,	,	PUNCT
ejpam-4508	114	11	or	or	CCONJ
ejpam-4508	114	12	equivalently	equivalently	ADV
ejpam-4508	114	13	(	(	PUNCT
ejpam-4508	114	14	p+	p+	NOUN
ejpam-4508	114	15	4n)2k+1	4n)2k+1	X
ejpam-4508	114	16	=	=	SYM
ejpam-4508	114	17	z2	z2	PROPN
ejpam-4508	114	18	−	−	PROPN
ejpam-4508	114	19	p2s	p2s	PROPN
ejpam-4508	114	20	=	=	SYM
ejpam-4508	114	21	(	(	PUNCT
ejpam-4508	114	22	z	z	NOUN
ejpam-4508	114	23	+	+	NOUN
ejpam-4508	114	24	ps)(z	ps)(z	PROPN
ejpam-4508	114	25	−	−	NUM
ejpam-4508	114	26	ps	ps	NOUN
ejpam-4508	114	27	)	)	PUNCT
ejpam-4508	114	28	.	.	PUNCT
ejpam-4508	115	1	thus	thus	ADV
ejpam-4508	115	2	,	,	PUNCT
ejpam-4508	115	3	there	there	PRON
ejpam-4508	115	4	exist	exist	VERB
ejpam-4508	115	5	non	non	ADJ
ejpam-4508	115	6	-	-	ADJ
ejpam-4508	115	7	negative	negative	ADJ
ejpam-4508	115	8	integer	integer	NOUN
ejpam-4508	115	9	α	α	NOUN
ejpam-4508	115	10	,	,	PUNCT
ejpam-4508	115	11	β	β	PROPN
ejpam-4508	115	12	such	such	ADJ
ejpam-4508	115	13	that	that	SCONJ
ejpam-4508	115	14	(	(	PUNCT
ejpam-4508	115	15	p+	p+	NOUN
ejpam-4508	115	16	4n)α	4n)α	NOUN
ejpam-4508	115	17	=	=	SYM
ejpam-4508	116	1	z	z	PROPN
ejpam-4508	117	1	+	+	CCONJ
ejpam-4508	117	2	ps	ps	PROPN
ejpam-4508	117	3	and	and	CCONJ
ejpam-4508	117	4	(	(	PUNCT
ejpam-4508	117	5	p	p	X
ejpam-4508	117	6	+	+	NOUN
ejpam-4508	117	7	4n)β	4n)β	ADJ
ejpam-4508	117	8	=	=	SYM
ejpam-4508	117	9	z	z	NOUN
ejpam-4508	117	10	−	−	PROPN
ejpam-4508	117	11	ps	ps	INTJ
ejpam-4508	117	12	where	where	SCONJ
ejpam-4508	117	13	α	α	NOUN
ejpam-4508	117	14	>	>	X
ejpam-4508	117	15	β	β	X
ejpam-4508	117	16	and	and	CCONJ
ejpam-4508	117	17	α	α	PROPN
ejpam-4508	117	18	+	+	X
ejpam-4508	117	19	β	β	X
ejpam-4508	117	20	=	=	SYM
ejpam-4508	117	21	2k	2k	NOUN
ejpam-4508	117	22	+	+	CCONJ
ejpam-4508	117	23	1	1	X
ejpam-4508	117	24	.	.	PUNCT
ejpam-4508	118	1	then	then	ADV
ejpam-4508	118	2	,	,	PUNCT
ejpam-4508	118	3	we	we	PRON
ejpam-4508	118	4	have	have	VERB
ejpam-4508	118	5	2(p)s	2(p)s	NOUN
ejpam-4508	118	6	=	=	SYM
ejpam-4508	118	7	(	(	PUNCT
ejpam-4508	118	8	p	p	X
ejpam-4508	118	9	+	+	NOUN
ejpam-4508	118	10	4n)β[(p	4n)β[(p	NUM
ejpam-4508	119	1	+	+	CCONJ
ejpam-4508	119	2	4n)α−β	4n)α−β	NUM
ejpam-4508	119	3	−	−	NOUN
ejpam-4508	119	4	1	1	NUM
ejpam-4508	119	5	]	]	PUNCT
ejpam-4508	119	6	.	.	PUNCT
ejpam-4508	120	1	this	this	PRON
ejpam-4508	120	2	implies	imply	VERB
ejpam-4508	120	3	that	that	SCONJ
ejpam-4508	120	4	β	β	X
ejpam-4508	120	5	=	=	SYM
ejpam-4508	120	6	0	0	X
ejpam-4508	120	7	.	.	PUNCT
ejpam-4508	121	1	we	we	PRON
ejpam-4508	121	2	have	have	VERB
ejpam-4508	121	3	2(p)s	2(p)s	NOUN
ejpam-4508	121	4	=	=	SYM
ejpam-4508	121	5	(	(	PUNCT
ejpam-4508	121	6	p	p	X
ejpam-4508	122	1	+	+	NOUN
ejpam-4508	122	2	4n)2k+1	4n)2k+1	NUM
ejpam-4508	122	3	−	−	NOUN
ejpam-4508	122	4	1	1	NUM
ejpam-4508	122	5	,	,	PUNCT
ejpam-4508	122	6	which	which	PRON
ejpam-4508	122	7	is	be	AUX
ejpam-4508	122	8	impossible	impossible	ADJ
ejpam-4508	122	9	because	because	SCONJ
ejpam-4508	122	10	2(p)s	2(p)s	ADJ
ejpam-4508	122	11	≡	≡	PROPN
ejpam-4508	122	12	2	2	NUM
ejpam-4508	122	13	(	(	PUNCT
ejpam-4508	122	14	mod	mod	NOUN
ejpam-4508	122	15	3	3	NUM
ejpam-4508	122	16	)	)	PUNCT
ejpam-4508	122	17	but	but	CCONJ
ejpam-4508	122	18	(	(	PUNCT
ejpam-4508	122	19	p+	p+	NOUN
ejpam-4508	122	20	4n)2k+1	4n)2k+1	NUM
ejpam-4508	122	21	−	−	NOUN
ejpam-4508	122	22	1	1	NUM
ejpam-4508	122	23	≡	≡	PROPN
ejpam-4508	122	24	0	0	NUM
ejpam-4508	122	25	,	,	PUNCT
ejpam-4508	122	26	1	1	NUM
ejpam-4508	122	27	(	(	PUNCT
ejpam-4508	122	28	mod	mod	NOUN
ejpam-4508	122	29	3	3	NUM
ejpam-4508	122	30	)	)	PUNCT
ejpam-4508	122	31	.	.	PUNCT
ejpam-4508	123	1	acknowledgements	acknowledgement	NOUN
ejpam-4508	123	2	the	the	DET
ejpam-4508	123	3	authors	author	NOUN
ejpam-4508	123	4	wish	wish	VERB
ejpam-4508	123	5	to	to	PART
ejpam-4508	123	6	thank	thank	VERB
ejpam-4508	123	7	the	the	DET
ejpam-4508	123	8	referees	referee	NOUN
ejpam-4508	123	9	for	for	ADP
ejpam-4508	123	10	their	their	PRON
ejpam-4508	123	11	kind	kind	ADJ
ejpam-4508	123	12	suggestions	suggestion	NOUN
ejpam-4508	123	13	and	and	CCONJ
ejpam-4508	123	14	comments	comment	NOUN
ejpam-4508	123	15	to	to	PART
ejpam-4508	123	16	improve	improve	VERB
ejpam-4508	123	17	the	the	DET
ejpam-4508	123	18	article	article	NOUN
ejpam-4508	123	19	.	.	PUNCT
ejpam-4508	124	1	references	reference	NOUN
ejpam-4508	124	2	1596	1596	NUM
ejpam-4508	124	3	references	reference	NOUN
ejpam-4508	124	4	[	[	X
ejpam-4508	124	5	1	1	NUM
ejpam-4508	124	6	]	]	X
ejpam-4508	124	7	n.	n.	NOUN
ejpam-4508	124	8	burshtein	burshtein	PROPN
ejpam-4508	124	9	.	.	PUNCT
ejpam-4508	125	1	on	on	ADP
ejpam-4508	125	2	the	the	DET
ejpam-4508	125	3	diophantine	diophantine	NOUN
ejpam-4508	125	4	equation	equation	NOUN
ejpam-4508	125	5	px	px	X
ejpam-4508	125	6	+	+	CCONJ
ejpam-4508	125	7	(	(	PUNCT
ejpam-4508	125	8	p	p	X
ejpam-4508	125	9	+	+	NOUN
ejpam-4508	125	10	12)y	12)y	NUM
ejpam-4508	125	11	=	=	SYM
ejpam-4508	125	12	z2	z2	NOUN
ejpam-4508	125	13	when	when	SCONJ
ejpam-4508	125	14	p	p	PROPN
ejpam-4508	125	15	+	+	NOUN
ejpam-4508	125	16	5	5	NUM
ejpam-4508	125	17	=	=	NOUN
ejpam-4508	125	18	22u	22u	X
ejpam-4508	125	19	.	.	PUNCT
ejpam-4508	125	20	annals	annal	NOUN
ejpam-4508	125	21	of	of	ADP
ejpam-4508	125	22	pure	pure	ADJ
ejpam-4508	125	23	and	and	CCONJ
ejpam-4508	125	24	applied	applied	ADJ
ejpam-4508	125	25	mathematic	mathematic	ADJ
ejpam-4508	125	26	,	,	PUNCT
ejpam-4508	125	27	18(1):41–44	18(1):41–44	NUM
ejpam-4508	125	28	,	,	PUNCT
ejpam-4508	125	29	2020	2020	NUM
ejpam-4508	125	30	.	.	PUNCT
ejpam-4508	126	1	[	[	X
ejpam-4508	126	2	2	2	X
ejpam-4508	126	3	]	]	PUNCT
ejpam-4508	126	4	s.	s.	PROPN
ejpam-4508	126	5	chotchaisthit	chotchaisthit	VERB
ejpam-4508	126	6	.	.	PUNCT
ejpam-4508	127	1	on	on	ADP
ejpam-4508	127	2	the	the	DET
ejpam-4508	127	3	diophantine	diophantine	NOUN
ejpam-4508	127	4	equation	equation	NOUN
ejpam-4508	127	5	4x	4x	NOUN
ejpam-4508	127	6	+	+	CCONJ
ejpam-4508	127	7	py	py	PROPN
ejpam-4508	127	8	=	=	PROPN
ejpam-4508	127	9	z2	z2	PROPN
ejpam-4508	127	10	where	where	SCONJ
ejpam-4508	127	11	p	p	NOUN
ejpam-4508	127	12	is	be	AUX
ejpam-4508	127	13	a	a	DET
ejpam-4508	127	14	prime	prime	ADJ
ejpam-4508	127	15	number	number	NOUN
ejpam-4508	127	16	.	.	PUNCT
ejpam-4508	128	1	american	american	PROPN
ejpam-4508	128	2	journal	journal	PROPN
ejpam-4508	128	3	mathematics	mathematics	PROPN
ejpam-4508	128	4	and	and	CCONJ
ejpam-4508	128	5	sciences	science	NOUN
ejpam-4508	128	6	,	,	PUNCT
ejpam-4508	128	7	1(1):191–193	1(1):191–193	NUM
ejpam-4508	128	8	,	,	PUNCT
ejpam-4508	128	9	2012	2012	NUM
ejpam-4508	128	10	.	.	PUNCT
ejpam-4508	129	1	[	[	X
ejpam-4508	129	2	3	3	X
ejpam-4508	129	3	]	]	X
ejpam-4508	129	4	r.	r.	PROPN
ejpam-4508	129	5	dokchan	dokchan	PROPN
ejpam-4508	129	6	and	and	CCONJ
ejpam-4508	129	7	a.	a.	NOUN
ejpam-4508	129	8	pakapongpun	pakapongpun	NOUN
ejpam-4508	129	9	.	.	PUNCT
ejpam-4508	130	1	on	on	ADP
ejpam-4508	130	2	the	the	DET
ejpam-4508	130	3	diophantine	diophantine	NOUN
ejpam-4508	130	4	px	px	X
ejpam-4508	130	5	+	+	CCONJ
ejpam-4508	130	6	(	(	PUNCT
ejpam-4508	130	7	p	p	X
ejpam-4508	130	8	+	+	NOUN
ejpam-4508	130	9	20)y	20)y	NUM
ejpam-4508	130	10	=	=	SYM
ejpam-4508	130	11	z2	z2	PROPN
ejpam-4508	130	12	when	when	SCONJ
ejpam-4508	130	13	p	p	PROPN
ejpam-4508	130	14	and	and	CCONJ
ejpam-4508	130	15	p+	p+	PROPN
ejpam-4508	130	16	20	20	NUM
ejpam-4508	130	17	are	be	AUX
ejpam-4508	130	18	primes	prime	NOUN
ejpam-4508	130	19	.	.	PUNCT
ejpam-4508	131	1	international	international	ADJ
ejpam-4508	131	2	journal	journal	PROPN
ejpam-4508	131	3	of	of	ADP
ejpam-4508	131	4	mathematics	mathematic	NOUN
ejpam-4508	131	5	and	and	CCONJ
ejpam-4508	131	6	computer	computer	NOUN
ejpam-4508	131	7	science	science	NOUN
ejpam-4508	131	8	,	,	PUNCT
ejpam-4508	131	9	16(1):179–183	16(1):179–183	NUM
ejpam-4508	131	10	,	,	PUNCT
ejpam-4508	131	11	2021	2021	NUM
ejpam-4508	131	12	.	.	PUNCT
ejpam-4508	132	1	[	[	X
ejpam-4508	132	2	4	4	NUM
ejpam-4508	132	3	]	]	X
ejpam-4508	132	4	n.	n.	NOUN
ejpam-4508	132	5	fernando	fernando	PROPN
ejpam-4508	132	6	.	.	PUNCT
ejpam-4508	133	1	on	on	ADP
ejpam-4508	133	2	the	the	DET
ejpam-4508	133	3	solvability	solvability	NOUN
ejpam-4508	133	4	of	of	ADP
ejpam-4508	133	5	the	the	DET
ejpam-4508	133	6	diophantine	diophantine	NOUN
ejpam-4508	133	7	equation	equation	NOUN
ejpam-4508	133	8	px+(p+8)y	px+(p+8)y	NOUN
ejpam-4508	133	9	=	=	SYM
ejpam-4508	133	10	z2	z2	PROPN
ejpam-4508	133	11	when	when	SCONJ
ejpam-4508	133	12	p	p	PROPN
ejpam-4508	133	13	>	>	X
ejpam-4508	133	14	3	3	NUM
ejpam-4508	133	15	and	and	CCONJ
ejpam-4508	133	16	p	p	PRON
ejpam-4508	133	17	+	+	NOUN
ejpam-4508	133	18	8	8	NUM
ejpam-4508	133	19	are	be	AUX
ejpam-4508	133	20	primes	prime	NOUN
ejpam-4508	133	21	.	.	PUNCT
ejpam-4508	134	1	annals	annal	NOUN
ejpam-4508	134	2	of	of	ADP
ejpam-4508	134	3	pure	pure	ADJ
ejpam-4508	134	4	and	and	CCONJ
ejpam-4508	134	5	applied	applied	ADJ
ejpam-4508	134	6	mathematics	mathematic	NOUN
ejpam-4508	134	7	,	,	PUNCT
ejpam-4508	134	8	18(1):9–13	18(1):9–13	NUM
ejpam-4508	134	9	,	,	PUNCT
ejpam-4508	134	10	2018	2018	NUM
ejpam-4508	134	11	.	.	PUNCT
ejpam-4508	135	1	[	[	X
ejpam-4508	135	2	5	5	X
ejpam-4508	135	3	]	]	PUNCT
ejpam-4508	135	4	w.	w.	PROPN
ejpam-4508	135	5	s.	s.	PROPN
ejpam-4508	135	6	gayo	gayo	PROPN
ejpam-4508	135	7	and	and	CCONJ
ejpam-4508	135	8	j.	j.	PROPN
ejpam-4508	135	9	b.	b.	PROPN
ejpam-4508	135	10	bacani	bacani	PROPN
ejpam-4508	135	11	.	.	PUNCT
ejpam-4508	136	1	on	on	ADP
ejpam-4508	136	2	the	the	DET
ejpam-4508	136	3	diophantine	diophantine	NOUN
ejpam-4508	136	4	equation	equation	NOUN
ejpam-4508	136	5	mx	mx	PROPN
ejpam-4508	136	6	p	p	PROPN
ejpam-4508	136	7	+	+	X
ejpam-4508	136	8	(	(	PUNCT
ejpam-4508	136	9	mq	mq	PROPN
ejpam-4508	136	10	+	+	CCONJ
ejpam-4508	136	11	1)y	1)y	NUM
ejpam-4508	136	12	=	=	SYM
ejpam-4508	136	13	z2	z2	PROPN
ejpam-4508	136	14	.	.	PUNCT
ejpam-4508	136	15	european	european	PROPN
ejpam-4508	136	16	journal	journal	PROPN
ejpam-4508	136	17	of	of	ADP
ejpam-4508	136	18	pure	pure	ADJ
ejpam-4508	136	19	and	and	CCONJ
ejpam-4508	136	20	applied	applied	ADJ
ejpam-4508	136	21	mathematics	mathematic	NOUN
ejpam-4508	136	22	,	,	PUNCT
ejpam-4508	136	23	14(2):396–403	14(2):396–403	NUM
ejpam-4508	136	24	,	,	PUNCT
ejpam-4508	136	25	2021	2021	NUM
ejpam-4508	136	26	.	.	PUNCT
ejpam-4508	137	1	[	[	X
ejpam-4508	137	2	6	6	NUM
ejpam-4508	137	3	]	]	PUNCT
ejpam-4508	137	4	a.	a.	NOUN
ejpam-4508	137	5	hoquea	hoquea	NOUN
ejpam-4508	137	6	.	.	PUNCT
ejpam-4508	138	1	on	on	ADP
ejpam-4508	138	2	the	the	DET
ejpam-4508	138	3	diophantine	diophantine	NOUN
ejpam-4508	138	4	equation	equation	NOUN
ejpam-4508	138	5	(	(	PUNCT
ejpam-4508	138	6	mpq	mpq	X
ejpam-4508	138	7	)	)	PUNCT
ejpam-4508	139	1	x+(mpq+1)y	x+(mpq+1)y	PROPN
ejpam-4508	139	2	=	=	SYM
ejpam-4508	139	3	z2	z2	PROPN
ejpam-4508	139	4	.	.	PUNCT
ejpam-4508	140	1	european	european	PROPN
ejpam-4508	140	2	journal	journal	PROPN
ejpam-4508	140	3	of	of	ADP
ejpam-4508	140	4	pure	pure	ADJ
ejpam-4508	140	5	and	and	CCONJ
ejpam-4508	140	6	applied	applied	ADJ
ejpam-4508	140	7	mathematics	mathematic	NOUN
ejpam-4508	140	8	,	,	PUNCT
ejpam-4508	140	9	9(2):240–243	9(2):240–243	NUM
ejpam-4508	140	10	,	,	PUNCT
ejpam-4508	140	11	2016	2016	NUM
ejpam-4508	140	12	.	.	PUNCT
ejpam-4508	141	1	[	[	X
ejpam-4508	141	2	7	7	X
ejpam-4508	141	3	]	]	PUNCT
ejpam-4508	141	4	s.	s.	PROPN
ejpam-4508	141	5	kumar	kumar	PROPN
ejpam-4508	141	6	,	,	PUNCT
ejpam-4508	141	7	s.	s.	PROPN
ejpam-4508	141	8	gupta	gupta	PROPN
ejpam-4508	141	9	and	and	CCONJ
ejpam-4508	141	10	h.	h.	PROPN
ejpam-4508	141	11	kishan	kishan	PROPN
ejpam-4508	141	12	.	.	PUNCT
ejpam-4508	142	1	on	on	ADP
ejpam-4508	142	2	the	the	DET
ejpam-4508	142	3	non	non	ADJ
ejpam-4508	142	4	-	-	ADJ
ejpam-4508	142	5	linear	linear	ADJ
ejpam-4508	142	6	diophantine	diophantine	NOUN
ejpam-4508	142	7	equation	equation	NOUN
ejpam-4508	142	8	px	px	X
ejpam-4508	142	9	+	+	CCONJ
ejpam-4508	142	10	(	(	PUNCT
ejpam-4508	142	11	p+	p+	NOUN
ejpam-4508	142	12	6)y	6)y	X
ejpam-4508	142	13	=	=	SYM
ejpam-4508	142	14	z2	z2	PROPN
ejpam-4508	142	15	.	.	PUNCT
ejpam-4508	142	16	annals	annal	NOUN
ejpam-4508	142	17	of	of	ADP
ejpam-4508	142	18	pure	pure	ADJ
ejpam-4508	142	19	and	and	CCONJ
ejpam-4508	142	20	applied	applied	ADJ
ejpam-4508	142	21	mathematics	mathematic	NOUN
ejpam-4508	142	22	,	,	PUNCT
ejpam-4508	142	23	18(1):125–128	18(1):125–128	NUM
ejpam-4508	142	24	,	,	PUNCT
ejpam-4508	142	25	2018	2018	NUM
ejpam-4508	142	26	.	.	PUNCT
ejpam-4508	143	1	[	[	X
ejpam-4508	143	2	8	8	NUM
ejpam-4508	143	3	]	]	PUNCT
ejpam-4508	143	4	s.	s.	PROPN
ejpam-4508	143	5	kumar	kumar	PROPN
ejpam-4508	143	6	,	,	PUNCT
ejpam-4508	143	7	s.	s.	PROPN
ejpam-4508	143	8	gupta	gupta	PROPN
ejpam-4508	143	9	and	and	CCONJ
ejpam-4508	143	10	h.	h.	PROPN
ejpam-4508	143	11	kishan	kishan	PROPN
ejpam-4508	143	12	.	.	PUNCT
ejpam-4508	144	1	on	on	ADP
ejpam-4508	144	2	the	the	DET
ejpam-4508	144	3	solution	solution	NOUN
ejpam-4508	144	4	of	of	ADP
ejpam-4508	144	5	exponential	exponential	ADJ
ejpam-4508	144	6	diophantine	diophantine	NOUN
ejpam-4508	144	7	equation	equation	NOUN
ejpam-4508	144	8	px	px	X
ejpam-4508	144	9	+	+	CCONJ
ejpam-4508	144	10	(	(	PUNCT
ejpam-4508	144	11	p	p	X
ejpam-4508	144	12	+	+	NUM
ejpam-4508	144	13	8)y	8)y	NOUN
ejpam-4508	144	14	=	=	SYM
ejpam-4508	144	15	z2	z2	PROPN
ejpam-4508	144	16	.	.	PUNCT
ejpam-4508	145	1	international	international	ADJ
ejpam-4508	145	2	journal	journal	PROPN
ejpam-4508	145	3	of	of	ADP
ejpam-4508	145	4	mathematics	mathematic	NOUN
ejpam-4508	145	5	and	and	CCONJ
ejpam-4508	145	6	computer	computer	NOUN
ejpam-4508	145	7	science	science	NOUN
ejpam-4508	145	8	,	,	PUNCT
ejpam-4508	145	9	11(1):1–19	11(1):1–19	NUM
ejpam-4508	145	10	,	,	PUNCT
ejpam-4508	145	11	2019	2019	NUM
ejpam-4508	145	12	.	.	PUNCT
ejpam-4508	146	1	[	[	X
ejpam-4508	146	2	9	9	NUM
ejpam-4508	146	3	]	]	PUNCT
ejpam-4508	146	4	p.	p.	NOUN
ejpam-4508	146	5	mihailescu	mihailescu	PROPN
ejpam-4508	146	6	.	.	PUNCT
ejpam-4508	147	1	primary	primary	ADJ
ejpam-4508	147	2	cyclotomic	cyclotomic	ADJ
ejpam-4508	147	3	units	unit	NOUN
ejpam-4508	147	4	and	and	CCONJ
ejpam-4508	147	5	a	a	DET
ejpam-4508	147	6	proof	proof	NOUN
ejpam-4508	147	7	of	of	ADP
ejpam-4508	147	8	catalan	catalan	NOUN
ejpam-4508	147	9	’s	’s	PART
ejpam-4508	147	10	conjecture	conjecture	NOUN
ejpam-4508	147	11	.	.	PUNCT
ejpam-4508	148	1	journal	journal	PROPN
ejpam-4508	148	2	für	für	AUX
ejpam-4508	148	3	die	die	VERB
ejpam-4508	148	4	reine	reine	PROPN
ejpam-4508	148	5	and	and	CCONJ
ejpam-4508	148	6	angewandte	angewandte	PROPN
ejpam-4508	148	7	mathematik	mathematik	PROPN
ejpam-4508	148	8	,	,	PUNCT
ejpam-4508	148	9	572:167–195	572:167–195	NUM
ejpam-4508	148	10	,	,	PUNCT
ejpam-4508	148	11	2004	2004	NUM
ejpam-4508	148	12	.	.	PUNCT
ejpam-4508	149	1	[	[	X
ejpam-4508	149	2	10	10	NUM
ejpam-4508	149	3	]	]	PUNCT
ejpam-4508	149	4	a.	a.	NOUN
ejpam-4508	149	5	suvarnamani	suvarnamani	PROPN
ejpam-4508	149	6	.	.	PUNCT
ejpam-4508	150	1	on	on	ADP
ejpam-4508	150	2	the	the	DET
ejpam-4508	150	3	diophantine	diophantine	NOUN
ejpam-4508	150	4	equation	equation	NOUN
ejpam-4508	150	5	px	px	X
ejpam-4508	150	6	+	+	CCONJ
ejpam-4508	150	7	(	(	PUNCT
ejpam-4508	150	8	p	p	X
ejpam-4508	150	9	+	+	NUM
ejpam-4508	150	10	1)y	1)y	NUM
ejpam-4508	150	11	=	=	SYM
ejpam-4508	150	12	z2	z2	PROPN
ejpam-4508	150	13	.	.	PUNCT
ejpam-4508	151	1	international	international	ADJ
ejpam-4508	151	2	journal	journal	PROPN
ejpam-4508	151	3	of	of	ADP
ejpam-4508	151	4	mathematical	mathematical	ADJ
ejpam-4508	151	5	sciences	science	NOUN
ejpam-4508	151	6	and	and	CCONJ
ejpam-4508	151	7	applications	application	NOUN
ejpam-4508	151	8	,	,	PUNCT
ejpam-4508	151	9	1(3):1415–1419	1(3):1415–1419	NOUN
ejpam-4508	151	10	,	,	PUNCT
ejpam-4508	151	11	2011	2011	NUM
ejpam-4508	151	12	.	.	PUNCT
ejpam-4508	152	1	[	[	X
ejpam-4508	152	2	11	11	NUM
ejpam-4508	152	3	]	]	PUNCT
ejpam-4508	152	4	a.	a.	NOUN
ejpam-4508	152	5	suvarnamani	suvarnamani	PROPN
ejpam-4508	152	6	.	.	PUNCT
ejpam-4508	153	1	on	on	ADP
ejpam-4508	153	2	the	the	DET
ejpam-4508	153	3	diophantine	diophantine	NOUN
ejpam-4508	153	4	equation	equation	NOUN
ejpam-4508	153	5	px	px	X
ejpam-4508	153	6	+	+	CCONJ
ejpam-4508	153	7	(	(	PUNCT
ejpam-4508	153	8	p	p	X
ejpam-4508	153	9	+	+	NUM
ejpam-4508	153	10	1)y	1)y	NUM
ejpam-4508	153	11	=	=	SYM
ejpam-4508	153	12	z2	z2	PROPN
ejpam-4508	153	13	.	.	PUNCT
ejpam-4508	154	1	international	international	ADJ
ejpam-4508	154	2	journal	journal	NOUN
ejpam-4508	154	3	of	of	ADP
ejpam-4508	154	4	pure	pure	ADJ
ejpam-4508	154	5	and	and	CCONJ
ejpam-4508	154	6	applied	applied	ADJ
ejpam-4508	154	7	mathematics	mathematic	NOUN
ejpam-4508	154	8	,	,	PUNCT
ejpam-4508	154	9	94(5):689–692	94(5):689–692	NUM
ejpam-4508	154	10	,	,	PUNCT
ejpam-4508	154	11	2014	2014	NUM
ejpam-4508	154	12	.	.	PUNCT