id	sid	tid	token	lemma	pos
ejpam-3948	1	1	european	european	PROPN
ejpam-3948	1	2	journal	journal	PROPN
ejpam-3948	1	3	of	of	ADP
ejpam-3948	1	4	pure	pure	ADJ
ejpam-3948	1	5	and	and	CCONJ
ejpam-3948	1	6	applied	apply	VERB
ejpam-3948	1	7	mathematics	mathematic	NOUN
ejpam-3948	1	8	vol	vol	NOUN
ejpam-3948	1	9	.	.	PUNCT
ejpam-3948	2	1	14	14	NUM
ejpam-3948	2	2	,	,	PUNCT
ejpam-3948	2	3	no	no	INTJ
ejpam-3948	2	4	.	.	NOUN
ejpam-3948	2	5	2	2	NUM
ejpam-3948	2	6	,	,	PUNCT
ejpam-3948	2	7	2021	2021	NUM
ejpam-3948	2	8	,	,	PUNCT
ejpam-3948	2	9	396	396	NUM
ejpam-3948	2	10	-	-	SYM
ejpam-3948	2	11	403	403	NUM
ejpam-3948	2	12	issn	issn	PROPN
ejpam-3948	2	13	1307	1307	NUM
ejpam-3948	2	14	-	-	SYM
ejpam-3948	2	15	5543	5543	NUM
ejpam-3948	2	16	–	–	PUNCT
ejpam-3948	3	1	ejpam.com	ejpam.com	X
ejpam-3948	3	2	published	publish	VERB
ejpam-3948	3	3	by	by	ADP
ejpam-3948	3	4	new	new	PROPN
ejpam-3948	3	5	york	york	PROPN
ejpam-3948	3	6	business	business	PROPN
ejpam-3948	3	7	global	global	PROPN
ejpam-3948	3	8	on	on	ADP
ejpam-3948	3	9	the	the	DET
ejpam-3948	3	10	diophantine	diophantine	NOUN
ejpam-3948	3	11	equation	equation	NOUN
ejpam-3948	3	12	mx	mx	PROPN
ejpam-3948	3	13	p	p	PROPN
ejpam-3948	3	14	+	+	X
ejpam-3948	3	15	(	(	PUNCT
ejpam-3948	3	16	mq	mq	PROPN
ejpam-3948	3	17	+	+	NOUN
ejpam-3948	3	18	1)y	1)y	NUM
ejpam-3948	3	19	=	=	SYM
ejpam-3948	3	20	z2	z2	PROPN
ejpam-3948	3	21	william	william	PROPN
ejpam-3948	3	22	s.	s.	PROPN
ejpam-3948	3	23	gayo	gayo	PROPN
ejpam-3948	3	24	,	,	PUNCT
ejpam-3948	3	25	jr.1,2	jr.1,2	PROPN
ejpam-3948	3	26	,	,	PUNCT
ejpam-3948	3	27	jerico	jerico	PROPN
ejpam-3948	3	28	b.	b.	PROPN
ejpam-3948	3	29	bacani1,∗	bacani1,∗	PROPN
ejpam-3948	3	30	1department	1department	NUM
ejpam-3948	3	31	of	of	ADP
ejpam-3948	3	32	mathematics	mathematic	NOUN
ejpam-3948	3	33	and	and	CCONJ
ejpam-3948	3	34	computer	computer	NOUN
ejpam-3948	3	35	science	science	NOUN
ejpam-3948	3	36	,	,	PUNCT
ejpam-3948	3	37	college	college	NOUN
ejpam-3948	3	38	of	of	ADP
ejpam-3948	3	39	science	science	NOUN
ejpam-3948	3	40	,	,	PUNCT
ejpam-3948	3	41	university	university	NOUN
ejpam-3948	3	42	of	of	ADP
ejpam-3948	3	43	the	the	DET
ejpam-3948	3	44	philippines	philippines	PROPN
ejpam-3948	3	45	baguio	baguio	PROPN
ejpam-3948	3	46	,	,	PUNCT
ejpam-3948	3	47	baguio	baguio	PROPN
ejpam-3948	3	48	city	city	PROPN
ejpam-3948	3	49	2600	2600	NUM
ejpam-3948	3	50	,	,	PUNCT
ejpam-3948	3	51	benguet	benguet	PROPN
ejpam-3948	3	52	,	,	PUNCT
ejpam-3948	3	53	philippines	philippine	VERB
ejpam-3948	3	54	2general	2general	NUM
ejpam-3948	3	55	education	education	NOUN
ejpam-3948	3	56	department	department	NOUN
ejpam-3948	3	57	,	,	PUNCT
ejpam-3948	3	58	college	college	NOUN
ejpam-3948	3	59	of	of	ADP
ejpam-3948	3	60	arts	art	NOUN
ejpam-3948	3	61	and	and	CCONJ
ejpam-3948	3	62	sciences	sciences	PROPN
ejpam-3948	3	63	don	don	PROPN
ejpam-3948	3	64	mariano	mariano	PROPN
ejpam-3948	3	65	marcos	marcos	PROPN
ejpam-3948	3	66	memorial	memorial	PROPN
ejpam-3948	3	67	state	state	PROPN
ejpam-3948	3	68	university	university	PROPN
ejpam-3948	3	69	north	north	PROPN
ejpam-3948	3	70	la	la	PROPN
ejpam-3948	3	71	union	union	PROPN
ejpam-3948	3	72	campus	campus	PROPN
ejpam-3948	3	73	,	,	PUNCT
ejpam-3948	3	74	bacnotan	bacnotan	PROPN
ejpam-3948	3	75	2515	2515	NUM
ejpam-3948	3	76	,	,	PUNCT
ejpam-3948	3	77	la	la	PROPN
ejpam-3948	3	78	union	union	PROPN
ejpam-3948	3	79	,	,	PUNCT
ejpam-3948	3	80	philippines	philippine	NOUN
ejpam-3948	3	81	abstract	abstract	ADJ
ejpam-3948	3	82	.	.	PUNCT
ejpam-3948	4	1	in	in	ADP
ejpam-3948	4	2	this	this	DET
ejpam-3948	4	3	paper	paper	NOUN
ejpam-3948	4	4	,	,	PUNCT
ejpam-3948	4	5	we	we	PRON
ejpam-3948	4	6	study	study	VERB
ejpam-3948	4	7	and	and	CCONJ
ejpam-3948	4	8	solve	solve	VERB
ejpam-3948	4	9	the	the	DET
ejpam-3948	4	10	exponential	exponential	ADJ
ejpam-3948	4	11	diophantine	diophantine	NOUN
ejpam-3948	4	12	equation	equation	NOUN
ejpam-3948	4	13	of	of	ADP
ejpam-3948	4	14	the	the	DET
ejpam-3948	4	15	form	form	NOUN
ejpam-3948	5	1	mx	mx	PROPN
ejpam-3948	5	2	p	p	NOUN
ejpam-3948	5	3	+	+	X
ejpam-3948	5	4	(	(	PUNCT
ejpam-3948	5	5	mq	mq	PROPN
ejpam-3948	5	6	+	+	NOUN
ejpam-3948	5	7	1)y	1)y	NUM
ejpam-3948	5	8	=	=	SYM
ejpam-3948	5	9	z2	z2	PROPN
ejpam-3948	5	10	for	for	ADP
ejpam-3948	5	11	mersenne	mersenne	PROPN
ejpam-3948	5	12	primes	prime	NOUN
ejpam-3948	5	13	mp	mp	PROPN
ejpam-3948	5	14	and	and	CCONJ
ejpam-3948	5	15	mq	mq	PROPN
ejpam-3948	5	16	and	and	CCONJ
ejpam-3948	5	17	non	non	ADJ
ejpam-3948	5	18	-	-	ADJ
ejpam-3948	5	19	negative	negative	ADJ
ejpam-3948	5	20	integers	integer	NOUN
ejpam-3948	5	21	x	x	NOUN
ejpam-3948	5	22	,	,	PUNCT
ejpam-3948	5	23	y	y	PROPN
ejpam-3948	5	24	,	,	PUNCT
ejpam-3948	5	25	and	and	CCONJ
ejpam-3948	5	26	z.	z.	PROPN
ejpam-3948	6	1	we	we	PRON
ejpam-3948	6	2	use	use	VERB
ejpam-3948	6	3	elementary	elementary	ADJ
ejpam-3948	6	4	methods	method	NOUN
ejpam-3948	6	5	,	,	PUNCT
ejpam-3948	6	6	such	such	ADJ
ejpam-3948	6	7	as	as	ADP
ejpam-3948	6	8	the	the	DET
ejpam-3948	6	9	factoring	factoring	NOUN
ejpam-3948	6	10	method	method	NOUN
ejpam-3948	6	11	and	and	CCONJ
ejpam-3948	6	12	the	the	DET
ejpam-3948	6	13	modular	modular	ADJ
ejpam-3948	6	14	arithmetic	arithmetic	ADJ
ejpam-3948	6	15	method	method	NOUN
ejpam-3948	6	16	,	,	PUNCT
ejpam-3948	6	17	to	to	PART
ejpam-3948	6	18	prove	prove	VERB
ejpam-3948	6	19	our	our	PRON
ejpam-3948	6	20	research	research	NOUN
ejpam-3948	6	21	results	result	NOUN
ejpam-3948	6	22	.	.	PUNCT
ejpam-3948	7	1	several	several	ADJ
ejpam-3948	7	2	illustrations	illustration	NOUN
ejpam-3948	7	3	are	be	AUX
ejpam-3948	7	4	presented	present	VERB
ejpam-3948	7	5	,	,	PUNCT
ejpam-3948	7	6	as	as	ADV
ejpam-3948	7	7	well	well	ADV
ejpam-3948	7	8	as	as	ADP
ejpam-3948	7	9	cases	case	NOUN
ejpam-3948	7	10	where	where	SCONJ
ejpam-3948	7	11	solutions	solution	NOUN
ejpam-3948	7	12	to	to	ADP
ejpam-3948	7	13	the	the	DET
ejpam-3948	7	14	diophantine	diophantine	NOUN
ejpam-3948	7	15	equation	equation	NOUN
ejpam-3948	7	16	do	do	AUX
ejpam-3948	7	17	not	not	PART
ejpam-3948	7	18	exist	exist	VERB
ejpam-3948	7	19	.	.	PUNCT
ejpam-3948	8	1	2020	2020	NUM
ejpam-3948	8	2	mathematics	mathematic	NOUN
ejpam-3948	8	3	subject	subject	NOUN
ejpam-3948	8	4	classifications	classification	NOUN
ejpam-3948	8	5	:	:	PUNCT
ejpam-3948	8	6	11d61	11d61	NUM
ejpam-3948	8	7	,	,	PUNCT
ejpam-3948	8	8	11d72	11d72	NUM
ejpam-3948	8	9	,	,	PUNCT
ejpam-3948	8	10	11a41	11a41	NUM
ejpam-3948	8	11	key	key	ADJ
ejpam-3948	8	12	words	word	NOUN
ejpam-3948	8	13	and	and	CCONJ
ejpam-3948	8	14	phrases	phrase	NOUN
ejpam-3948	8	15	:	:	PUNCT
ejpam-3948	8	16	diophantine	diophantine	VERB
ejpam-3948	8	17	equation	equation	NOUN
ejpam-3948	8	18	,	,	PUNCT
ejpam-3948	8	19	exponential	exponential	ADJ
ejpam-3948	8	20	diophantine	diophantine	NOUN
ejpam-3948	8	21	equation	equation	NOUN
ejpam-3948	8	22	,	,	PUNCT
ejpam-3948	8	23	mersenne	mersenne	NOUN
ejpam-3948	8	24	primes	prime	NOUN
ejpam-3948	8	25	1	1	NUM
ejpam-3948	8	26	.	.	PUNCT
ejpam-3948	8	27	introduction	introduction	NOUN
ejpam-3948	8	28	a	a	DET
ejpam-3948	8	29	number	number	NOUN
ejpam-3948	8	30	of	of	ADP
ejpam-3948	8	31	researchers	researcher	NOUN
ejpam-3948	8	32	have	have	AUX
ejpam-3948	8	33	been	be	AUX
ejpam-3948	8	34	studying	study	VERB
ejpam-3948	8	35	the	the	DET
ejpam-3948	8	36	exponential	exponential	ADJ
ejpam-3948	8	37	diophantine	diophantine	NOUN
ejpam-3948	8	38	equations	equation	NOUN
ejpam-3948	8	39	of	of	ADP
ejpam-3948	8	40	the	the	DET
ejpam-3948	8	41	form	form	NOUN
ejpam-3948	8	42	ax	ax	NOUN
ejpam-3948	8	43	+	+	CCONJ
ejpam-3948	8	44	by	by	ADP
ejpam-3948	8	45	=	=	PROPN
ejpam-3948	8	46	z2	z2	PROPN
ejpam-3948	8	47	.	.	PUNCT
ejpam-3948	9	1	this	this	PRON
ejpam-3948	9	2	includes	include	VERB
ejpam-3948	9	3	aggarwal	aggarwal	NOUN
ejpam-3948	9	4	,	,	PUNCT
ejpam-3948	9	5	burshtein	burshtein	PROPN
ejpam-3948	9	6	,	,	PUNCT
ejpam-3948	9	7	kumar	kumar	PROPN
ejpam-3948	9	8	,	,	PUNCT
ejpam-3948	9	9	sroysang	sroysang	PROPN
ejpam-3948	9	10	,	,	PUNCT
ejpam-3948	9	11	rabago	rabago	PROPN
ejpam-3948	9	12	,	,	PUNCT
ejpam-3948	9	13	among	among	ADP
ejpam-3948	9	14	others	other	NOUN
ejpam-3948	9	15	(	(	PUNCT
ejpam-3948	9	16	cf	cf	NOUN
ejpam-3948	9	17	.	.	PUNCT
ejpam-3948	10	1	[	[	X
ejpam-3948	10	2	1	1	NUM
ejpam-3948	10	3	]	]	PUNCT
ejpam-3948	10	4	,	,	PUNCT
ejpam-3948	10	5	[	[	X
ejpam-3948	10	6	2	2	NUM
ejpam-3948	10	7	]	]	PUNCT
ejpam-3948	10	8	,	,	PUNCT
ejpam-3948	10	9	[	[	X
ejpam-3948	10	10	3	3	NUM
ejpam-3948	10	11	]	]	PUNCT
ejpam-3948	10	12	,	,	PUNCT
ejpam-3948	10	13	[	[	X
ejpam-3948	10	14	5	5	NUM
ejpam-3948	10	15	]	]	PUNCT
ejpam-3948	10	16	,	,	PUNCT
ejpam-3948	10	17	[	[	X
ejpam-3948	10	18	6	6	NUM
ejpam-3948	10	19	]	]	PUNCT
ejpam-3948	10	20	,	,	PUNCT
ejpam-3948	10	21	[	[	X
ejpam-3948	10	22	8	8	NUM
ejpam-3948	10	23	]	]	PUNCT
ejpam-3948	10	24	,	,	PUNCT
ejpam-3948	10	25	[	[	X
ejpam-3948	10	26	13	13	NUM
ejpam-3948	10	27	]	]	PUNCT
ejpam-3948	10	28	,	,	PUNCT
ejpam-3948	10	29	[	[	X
ejpam-3948	10	30	14	14	NUM
ejpam-3948	10	31	]	]	PUNCT
ejpam-3948	10	32	,	,	PUNCT
ejpam-3948	10	33	[	[	X
ejpam-3948	10	34	15	15	NUM
ejpam-3948	10	35	]	]	PUNCT
ejpam-3948	10	36	,	,	PUNCT
ejpam-3948	10	37	[	[	X
ejpam-3948	10	38	16	16	NUM
ejpam-3948	10	39	]	]	PUNCT
ejpam-3948	10	40	,	,	PUNCT
ejpam-3948	11	1	[	[	X
ejpam-3948	11	2	17	17	NUM
ejpam-3948	11	3	]	]	PUNCT
ejpam-3948	11	4	,	,	PUNCT
ejpam-3948	12	1	[	[	X
ejpam-3948	12	2	18	18	NUM
ejpam-3948	12	3	]	]	PUNCT
ejpam-3948	12	4	,	,	PUNCT
ejpam-3948	12	5	[	[	X
ejpam-3948	12	6	19	19	NUM
ejpam-3948	12	7	]	]	PUNCT
ejpam-3948	12	8	,	,	PUNCT
ejpam-3948	12	9	[	[	X
ejpam-3948	12	10	20	20	NUM
ejpam-3948	12	11	]	]	PUNCT
ejpam-3948	12	12	,	,	PUNCT
ejpam-3948	12	13	[	[	X
ejpam-3948	12	14	21	21	NUM
ejpam-3948	12	15	]	]	PUNCT
ejpam-3948	12	16	,	,	PUNCT
ejpam-3948	12	17	[	[	X
ejpam-3948	12	18	22	22	NUM
ejpam-3948	12	19	]	]	PUNCT
ejpam-3948	12	20	,	,	PUNCT
ejpam-3948	12	21	[	[	X
ejpam-3948	12	22	24	24	NUM
ejpam-3948	12	23	]	]	PUNCT
ejpam-3948	12	24	,	,	PUNCT
ejpam-3948	12	25	[	[	X
ejpam-3948	12	26	27	27	NUM
ejpam-3948	12	27	]	]	PUNCT
ejpam-3948	12	28	)	)	PUNCT
ejpam-3948	12	29	.	.	PUNCT
ejpam-3948	13	1	some	some	PRON
ejpam-3948	13	2	of	of	ADP
ejpam-3948	13	3	them	they	PRON
ejpam-3948	13	4	have	have	AUX
ejpam-3948	13	5	studied	study	VERB
ejpam-3948	13	6	these	these	DET
ejpam-3948	13	7	equations	equation	NOUN
ejpam-3948	13	8	in	in	ADP
ejpam-3948	13	9	relation	relation	NOUN
ejpam-3948	13	10	to	to	ADP
ejpam-3948	13	11	mersenne	mersenne	NOUN
ejpam-3948	13	12	primes	prime	NOUN
ejpam-3948	13	13	.	.	PUNCT
ejpam-3948	14	1	they	they	PRON
ejpam-3948	14	2	focused	focus	VERB
ejpam-3948	14	3	on	on	ADP
ejpam-3948	14	4	the	the	DET
ejpam-3948	14	5	case	case	NOUN
ejpam-3948	14	6	where	where	SCONJ
ejpam-3948	14	7	one	one	NUM
ejpam-3948	14	8	of	of	ADP
ejpam-3948	14	9	the	the	DET
ejpam-3948	14	10	bases	basis	NOUN
ejpam-3948	14	11	a	a	PRON
ejpam-3948	14	12	and	and	CCONJ
ejpam-3948	14	13	b	b	NOUN
ejpam-3948	14	14	is	be	AUX
ejpam-3948	14	15	a	a	DET
ejpam-3948	14	16	mersenne	mersenne	NOUN
ejpam-3948	14	17	prime	prime	NOUN
ejpam-3948	14	18	.	.	PUNCT
ejpam-3948	15	1	in	in	ADP
ejpam-3948	15	2	particular	particular	ADJ
ejpam-3948	15	3	,	,	PUNCT
ejpam-3948	15	4	some	some	PRON
ejpam-3948	15	5	considered	consider	VERB
ejpam-3948	15	6	m2	m2	PROPN
ejpam-3948	15	7	=	=	SYM
ejpam-3948	15	8	3	3	NUM
ejpam-3948	15	9	,	,	PUNCT
ejpam-3948	15	10	m3	m3	PROPN
ejpam-3948	15	11	=	=	SYM
ejpam-3948	15	12	7	7	NUM
ejpam-3948	15	13	and	and	CCONJ
ejpam-3948	15	14	m5	m5	NOUN
ejpam-3948	15	15	=	=	SYM
ejpam-3948	15	16	31	31	NUM
ejpam-3948	15	17	,	,	PUNCT
ejpam-3948	15	18	which	which	PRON
ejpam-3948	15	19	are	be	AUX
ejpam-3948	15	20	actually	actually	ADV
ejpam-3948	15	21	the	the	DET
ejpam-3948	15	22	first	first	ADJ
ejpam-3948	15	23	three	three	NUM
ejpam-3948	15	24	mersenne	mersenne	NOUN
ejpam-3948	15	25	prime	prime	ADJ
ejpam-3948	15	26	numbers	number	NOUN
ejpam-3948	15	27	.	.	PUNCT
ejpam-3948	16	1	records	record	NOUN
ejpam-3948	16	2	show	show	VERB
ejpam-3948	16	3	that	that	SCONJ
ejpam-3948	16	4	sroysang	sroysang	PROPN
ejpam-3948	17	1	[	[	X
ejpam-3948	17	2	25	25	NUM
ejpam-3948	17	3	]	]	PUNCT
ejpam-3948	17	4	proved	prove	VERB
ejpam-3948	17	5	that	that	SCONJ
ejpam-3948	17	6	the	the	DET
ejpam-3948	17	7	solutions	solution	NOUN
ejpam-3948	17	8	of	of	ADP
ejpam-3948	17	9	3x	3x	NUM
ejpam-3948	17	10	+	+	CCONJ
ejpam-3948	17	11	2y	2y	PROPN
ejpam-3948	17	12	=	=	SYM
ejpam-3948	17	13	z2	z2	NOUN
ejpam-3948	17	14	are	be	AUX
ejpam-3948	17	15	(	(	PUNCT
ejpam-3948	17	16	0	0	NUM
ejpam-3948	17	17	,	,	PUNCT
ejpam-3948	17	18	1	1	NUM
ejpam-3948	17	19	,	,	PUNCT
ejpam-3948	17	20	2	2	NUM
ejpam-3948	17	21	)	)	PUNCT
ejpam-3948	17	22	,	,	PUNCT
ejpam-3948	17	23	(	(	PUNCT
ejpam-3948	17	24	3	3	NUM
ejpam-3948	17	25	,	,	PUNCT
ejpam-3948	17	26	0	0	NUM
ejpam-3948	17	27	,	,	PUNCT
ejpam-3948	17	28	3	3	NUM
ejpam-3948	17	29	)	)	PUNCT
ejpam-3948	17	30	,	,	PUNCT
ejpam-3948	17	31	and	and	CCONJ
ejpam-3948	17	32	(	(	PUNCT
ejpam-3948	17	33	2	2	NUM
ejpam-3948	17	34	,	,	PUNCT
ejpam-3948	17	35	4	4	NUM
ejpam-3948	17	36	,	,	PUNCT
ejpam-3948	17	37	5	5	NUM
ejpam-3948	17	38	)	)	PUNCT
ejpam-3948	17	39	.	.	PUNCT
ejpam-3948	18	1	asthana	asthana	PROPN
ejpam-3948	18	2	and	and	CCONJ
ejpam-3948	18	3	singh	singh	PROPN
ejpam-3948	19	1	[	[	X
ejpam-3948	19	2	4	4	X
ejpam-3948	19	3	]	]	PUNCT
ejpam-3948	19	4	proved	prove	VERB
ejpam-3948	19	5	that	that	SCONJ
ejpam-3948	19	6	3x	3x	PRON
ejpam-3948	19	7	+	+	NOUN
ejpam-3948	19	8	13y	13y	NOUN
ejpam-3948	19	9	=	=	SYM
ejpam-3948	19	10	z2	z2	PROPN
ejpam-3948	19	11	has	have	VERB
ejpam-3948	19	12	exactly	exactly	ADV
ejpam-3948	19	13	four	four	NUM
ejpam-3948	19	14	non	non	ADJ
ejpam-3948	19	15	-	-	ADJ
ejpam-3948	19	16	negative	negative	ADJ
ejpam-3948	19	17	integer	integer	NOUN
ejpam-3948	19	18	solutions	solution	NOUN
ejpam-3948	19	19	,	,	PUNCT
ejpam-3948	19	20	and	and	CCONJ
ejpam-3948	19	21	these	these	PRON
ejpam-3948	19	22	are	be	AUX
ejpam-3948	19	23	(	(	PUNCT
ejpam-3948	19	24	1	1	NUM
ejpam-3948	19	25	,	,	PUNCT
ejpam-3948	19	26	0	0	NUM
ejpam-3948	19	27	,	,	PUNCT
ejpam-3948	19	28	2	2	NUM
ejpam-3948	19	29	)	)	PUNCT
ejpam-3948	19	30	,	,	PUNCT
ejpam-3948	19	31	(	(	PUNCT
ejpam-3948	19	32	1	1	NUM
ejpam-3948	19	33	,	,	PUNCT
ejpam-3948	19	34	1	1	NUM
ejpam-3948	19	35	,	,	PUNCT
ejpam-3948	19	36	4	4	NUM
ejpam-3948	19	37	)	)	PUNCT
ejpam-3948	19	38	,	,	PUNCT
ejpam-3948	19	39	(	(	PUNCT
ejpam-3948	19	40	3	3	NUM
ejpam-3948	19	41	,	,	PUNCT
ejpam-3948	19	42	2	2	NUM
ejpam-3948	19	43	,	,	PUNCT
ejpam-3948	19	44	14	14	NUM
ejpam-3948	19	45	)	)	PUNCT
ejpam-3948	19	46	and	and	CCONJ
ejpam-3948	19	47	(	(	PUNCT
ejpam-3948	19	48	5	5	NUM
ejpam-3948	19	49	,	,	PUNCT
ejpam-3948	19	50	1	1	NUM
ejpam-3948	19	51	,	,	PUNCT
ejpam-3948	19	52	6	6	NUM
ejpam-3948	19	53	)	)	PUNCT
ejpam-3948	19	54	.	.	PUNCT
ejpam-3948	20	1	rabago	rabago	VERB
ejpam-3948	21	1	[	[	X
ejpam-3948	21	2	16	16	NUM
ejpam-3948	21	3	]	]	PUNCT
ejpam-3948	21	4	proved	prove	VERB
ejpam-3948	21	5	that	that	SCONJ
ejpam-3948	21	6	the	the	DET
ejpam-3948	21	7	triples	triple	NOUN
ejpam-3948	21	8	(	(	PUNCT
ejpam-3948	21	9	4	4	NUM
ejpam-3948	21	10	,	,	PUNCT
ejpam-3948	21	11	1	1	NUM
ejpam-3948	21	12	,	,	PUNCT
ejpam-3948	21	13	10	10	NUM
ejpam-3948	21	14	)	)	PUNCT
ejpam-3948	21	15	and	and	CCONJ
ejpam-3948	21	16	(	(	PUNCT
ejpam-3948	21	17	1	1	NUM
ejpam-3948	21	18	,	,	PUNCT
ejpam-3948	21	19	0	0	NUM
ejpam-3948	21	20	,	,	PUNCT
ejpam-3948	21	21	2	2	NUM
ejpam-3948	21	22	)	)	PUNCT
ejpam-3948	21	23	are	be	AUX
ejpam-3948	21	24	the	the	DET
ejpam-3948	21	25	only	only	ADJ
ejpam-3948	21	26	solutions	solution	NOUN
ejpam-3948	21	27	to	to	ADP
ejpam-3948	21	28	the	the	DET
ejpam-3948	21	29	diophantine	diophantine	NOUN
ejpam-3948	21	30	equation	equation	NOUN
ejpam-3948	21	31	3x	3x	NUM
ejpam-3948	22	1	+	+	CCONJ
ejpam-3948	22	2	19y	19y	NOUN
ejpam-3948	22	3	=	=	SYM
ejpam-3948	22	4	z2	z2	PROPN
ejpam-3948	22	5	,	,	PUNCT
ejpam-3948	22	6	and	and	CCONJ
ejpam-3948	22	7	that	that	SCONJ
ejpam-3948	22	8	(	(	PUNCT
ejpam-3948	22	9	2	2	NUM
ejpam-3948	22	10	,	,	PUNCT
ejpam-3948	22	11	1	1	NUM
ejpam-3948	22	12	,	,	PUNCT
ejpam-3948	22	13	10	10	NUM
ejpam-3948	22	14	)	)	PUNCT
ejpam-3948	22	15	and	and	CCONJ
ejpam-3948	22	16	(	(	PUNCT
ejpam-3948	22	17	1	1	NUM
ejpam-3948	22	18	,	,	PUNCT
ejpam-3948	22	19	0	0	NUM
ejpam-3948	22	20	,	,	PUNCT
ejpam-3948	22	21	2	2	NUM
ejpam-3948	22	22	)	)	PUNCT
ejpam-3948	22	23	are	be	AUX
ejpam-3948	22	24	the	the	DET
ejpam-3948	22	25	only	only	ADV
ejpam-3948	22	26	two	two	NUM
ejpam-3948	22	27	solutions	solution	NOUN
ejpam-3948	22	28	to	to	ADP
ejpam-3948	22	29	3x	3x	PRON
ejpam-3948	22	30	+	+	CCONJ
ejpam-3948	22	31	91y	91y	NOUN
ejpam-3948	22	32	=	=	SYM
ejpam-3948	22	33	z2	z2	PROPN
ejpam-3948	22	34	.	.	PUNCT
ejpam-3948	22	35	sroysang	sroysang	PROPN
ejpam-3948	23	1	[	[	X
ejpam-3948	23	2	26	26	NUM
ejpam-3948	23	3	]	]	PUNCT
ejpam-3948	23	4	also	also	ADV
ejpam-3948	23	5	showed	show	VERB
ejpam-3948	23	6	that	that	SCONJ
ejpam-3948	23	7	the	the	DET
ejpam-3948	23	8	7x	7x	NUM
ejpam-3948	23	9	+	+	CCONJ
ejpam-3948	23	10	8y	8y	NUM
ejpam-3948	23	11	=	=	SYM
ejpam-3948	23	12	z2	z2	PROPN
ejpam-3948	23	13	has	have	VERB
ejpam-3948	23	14	the	the	DET
ejpam-3948	23	15	only	only	ADJ
ejpam-3948	23	16	solution	solution	NOUN
ejpam-3948	23	17	(	(	PUNCT
ejpam-3948	23	18	x	x	X
ejpam-3948	23	19	,	,	PUNCT
ejpam-3948	23	20	y	y	PROPN
ejpam-3948	23	21	,	,	PUNCT
ejpam-3948	23	22	z	z	NOUN
ejpam-3948	23	23	)	)	PUNCT
ejpam-3948	23	24	=	=	SYM
ejpam-3948	23	25	(	(	PUNCT
ejpam-3948	23	26	0	0	NUM
ejpam-3948	23	27	,	,	PUNCT
ejpam-3948	23	28	1	1	NUM
ejpam-3948	23	29	,	,	PUNCT
ejpam-3948	23	30	3	3	NUM
ejpam-3948	23	31	)	)	PUNCT
ejpam-3948	23	32	.	.	PUNCT
ejpam-3948	24	1	another	another	DET
ejpam-3948	24	2	work	work	NOUN
ejpam-3948	24	3	of	of	ADP
ejpam-3948	24	4	sroysang	sroysang	PROPN
ejpam-3948	24	5	[	[	X
ejpam-3948	24	6	23	23	NUM
ejpam-3948	24	7	]	]	PUNCT
ejpam-3948	24	8	shows	show	VERB
ejpam-3948	24	9	∗corresponding	∗corresponde	VERB
ejpam-3948	24	10	author	author	NOUN
ejpam-3948	24	11	.	.	PUNCT
ejpam-3948	25	1	doi	doi	NOUN
ejpam-3948	25	2	:	:	PUNCT
ejpam-3948	25	3	https://doi.org/10.29020/nybg.ejpam.v14i2.3948	https://doi.org/10.29020/nybg.ejpam.v14i2.3948	ADJ
ejpam-3948	25	4	email	email	NOUN
ejpam-3948	25	5	addresses	address	NOUN
ejpam-3948	25	6	:	:	PUNCT
ejpam-3948	25	7	wsgayo2@up.edu.ph	wsgayo2@up.edu.ph	PROPN
ejpam-3948	25	8	(	(	PUNCT
ejpam-3948	25	9	w.	w.	PROPN
ejpam-3948	25	10	s.	s.	PROPN
ejpam-3948	25	11	gayo	gayo	PROPN
ejpam-3948	25	12	,	,	PUNCT
ejpam-3948	25	13	jr	jr	PROPN
ejpam-3948	25	14	.	.	PROPN
ejpam-3948	25	15	)	)	PUNCT
ejpam-3948	25	16	,	,	PUNCT
ejpam-3948	26	1	jbbacani@up.edu.ph	jbbacani@up.edu.ph	PROPN
ejpam-3948	26	2	(	(	PUNCT
ejpam-3948	26	3	j.	j.	PROPN
ejpam-3948	26	4	b.	b.	PROPN
ejpam-3948	26	5	bacani	bacani	PROPN
ejpam-3948	26	6	)	)	PUNCT
ejpam-3948	26	7	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3948	27	1	396	396	NUM
ejpam-3948	27	2	c	c	X
ejpam-3948	27	3	©	©	PROPN
ejpam-3948	27	4	2021	2021	NUM
ejpam-3948	27	5	ejpam	ejpam	VERB
ejpam-3948	27	6	all	all	DET
ejpam-3948	27	7	rights	right	NOUN
ejpam-3948	27	8	reserved	reserve	VERB
ejpam-3948	27	9	.	.	PUNCT
ejpam-3948	28	1	w.	w.	PROPN
ejpam-3948	28	2	s.	s.	PROPN
ejpam-3948	28	3	gayo	gayo	PROPN
ejpam-3948	28	4	,	,	PUNCT
ejpam-3948	28	5	jr	jr	PROPN
ejpam-3948	28	6	.	.	PROPN
ejpam-3948	28	7	,	,	PUNCT
ejpam-3948	28	8	j.	j.	PROPN
ejpam-3948	28	9	b.	b.	PROPN
ejpam-3948	28	10	bacani	bacani	PROPN
ejpam-3948	28	11	/	/	SYM
ejpam-3948	28	12	eur	eur	PROPN
ejpam-3948	28	13	.	.	PUNCT
ejpam-3948	29	1	j.	j.	PROPN
ejpam-3948	29	2	pure	pure	PROPN
ejpam-3948	29	3	appl	appl	PROPN
ejpam-3948	29	4	.	.	PROPN
ejpam-3948	29	5	math	math	PROPN
ejpam-3948	29	6	,	,	PUNCT
ejpam-3948	29	7	14	14	NUM
ejpam-3948	29	8	(	(	PUNCT
ejpam-3948	29	9	2	2	NUM
ejpam-3948	29	10	)	)	PUNCT
ejpam-3948	29	11	(	(	PUNCT
ejpam-3948	29	12	2021	2021	NUM
ejpam-3948	29	13	)	)	PUNCT
ejpam-3948	29	14	,	,	PUNCT
ejpam-3948	29	15	396	396	NUM
ejpam-3948	29	16	-	-	SYM
ejpam-3948	29	17	403	403	NUM
ejpam-3948	29	18	397	397	NUM
ejpam-3948	29	19	that	that	PRON
ejpam-3948	29	20	the	the	DET
ejpam-3948	29	21	equation	equation	NOUN
ejpam-3948	29	22	31x	31x	NOUN
ejpam-3948	30	1	+	+	SYM
ejpam-3948	30	2	32y	32y	NOUN
ejpam-3948	30	3	=	=	SYM
ejpam-3948	30	4	z2	z2	PROPN
ejpam-3948	30	5	has	have	VERB
ejpam-3948	30	6	no	no	DET
ejpam-3948	30	7	non	non	ADJ
ejpam-3948	30	8	-	-	ADJ
ejpam-3948	30	9	negative	negative	ADJ
ejpam-3948	30	10	integer	integer	NOUN
ejpam-3948	30	11	solution	solution	NOUN
ejpam-3948	30	12	.	.	PUNCT
ejpam-3948	31	1	chotchaisthit	chotchaisthit	VERB
ejpam-3948	32	1	[	[	X
ejpam-3948	32	2	9	9	NUM
ejpam-3948	32	3	]	]	PUNCT
ejpam-3948	32	4	aimed	aim	VERB
ejpam-3948	32	5	to	to	PART
ejpam-3948	32	6	study	study	VERB
ejpam-3948	32	7	px	px	X
ejpam-3948	32	8	+	+	CCONJ
ejpam-3948	32	9	(	(	PUNCT
ejpam-3948	32	10	p+	p+	NOUN
ejpam-3948	32	11	1)y	1)y	NUM
ejpam-3948	32	12	=	=	SYM
ejpam-3948	32	13	z2	z2	PROPN
ejpam-3948	32	14	in	in	ADP
ejpam-3948	32	15	the	the	DET
ejpam-3948	32	16	set	set	NOUN
ejpam-3948	32	17	of	of	ADP
ejpam-3948	32	18	nonnegative	nonnegative	ADJ
ejpam-3948	32	19	integers	integer	NOUN
ejpam-3948	32	20	and	and	CCONJ
ejpam-3948	32	21	where	where	SCONJ
ejpam-3948	32	22	p	p	NOUN
ejpam-3948	32	23	is	be	AUX
ejpam-3948	32	24	a	a	DET
ejpam-3948	32	25	mersenne	mersenne	NOUN
ejpam-3948	32	26	prime	prime	NOUN
ejpam-3948	32	27	.	.	PUNCT
ejpam-3948	33	1	these	these	DET
ejpam-3948	33	2	works	work	NOUN
ejpam-3948	33	3	motivate	motivate	VERB
ejpam-3948	33	4	the	the	DET
ejpam-3948	33	5	researchers	researcher	NOUN
ejpam-3948	33	6	to	to	PART
ejpam-3948	33	7	study	study	VERB
ejpam-3948	33	8	the	the	DET
ejpam-3948	33	9	diophantine	diophantine	NOUN
ejpam-3948	33	10	equations	equation	NOUN
ejpam-3948	33	11	of	of	ADP
ejpam-3948	33	12	the	the	DET
ejpam-3948	33	13	form	form	NOUN
ejpam-3948	34	1	mx	mx	PROPN
ejpam-3948	34	2	p	p	NOUN
ejpam-3948	34	3	+	+	X
ejpam-3948	34	4	(	(	PUNCT
ejpam-3948	34	5	mq	mq	PROPN
ejpam-3948	34	6	+	+	CCONJ
ejpam-3948	34	7	1)y	1)y	NUM
ejpam-3948	34	8	=	=	SYM
ejpam-3948	34	9	z2	z2	PROPN
ejpam-3948	34	10	,	,	PUNCT
ejpam-3948	34	11	where	where	SCONJ
ejpam-3948	34	12	mp	mp	PROPN
ejpam-3948	34	13	and	and	CCONJ
ejpam-3948	34	14	mq	mq	PROPN
ejpam-3948	34	15	are	be	AUX
ejpam-3948	34	16	mersenne	mersenne	NOUN
ejpam-3948	34	17	primes	prime	NOUN
ejpam-3948	34	18	.	.	PUNCT
ejpam-3948	35	1	factoring	factoring	NOUN
ejpam-3948	35	2	and	and	CCONJ
ejpam-3948	35	3	modular	modular	ADJ
ejpam-3948	35	4	arithmetic	arithmetic	ADJ
ejpam-3948	35	5	methods	method	NOUN
ejpam-3948	35	6	are	be	AUX
ejpam-3948	35	7	the	the	DET
ejpam-3948	35	8	elementary	elementary	ADJ
ejpam-3948	35	9	methods	method	NOUN
ejpam-3948	35	10	used	use	VERB
ejpam-3948	35	11	in	in	ADP
ejpam-3948	35	12	the	the	DET
ejpam-3948	35	13	study	study	NOUN
ejpam-3948	35	14	.	.	PUNCT
ejpam-3948	36	1	using	use	VERB
ejpam-3948	36	2	the	the	DET
ejpam-3948	36	3	factoring	factoring	NOUN
ejpam-3948	36	4	method	method	NOUN
ejpam-3948	36	5	,	,	PUNCT
ejpam-3948	36	6	an	an	DET
ejpam-3948	36	7	equation	equation	NOUN
ejpam-3948	36	8	,	,	PUNCT
ejpam-3948	36	9	say	say	VERB
ejpam-3948	36	10	f(x1	f(x1	ADJ
ejpam-3948	36	11	,	,	PUNCT
ejpam-3948	36	12	x2	x2	PROPN
ejpam-3948	36	13	,	,	PUNCT
ejpam-3948	36	14	...	...	PUNCT
ejpam-3948	36	15	,	,	PUNCT
ejpam-3948	36	16	xn	xn	PROPN
ejpam-3948	36	17	)	)	PUNCT
ejpam-3948	37	1	=	=	SYM
ejpam-3948	37	2	0	0	NUM
ejpam-3948	37	3	,	,	PUNCT
ejpam-3948	37	4	will	will	AUX
ejpam-3948	37	5	be	be	AUX
ejpam-3948	37	6	written	write	VERB
ejpam-3948	37	7	as	as	ADP
ejpam-3948	37	8	f1(x	f1(x	NOUN
ejpam-3948	37	9	)	)	PUNCT
ejpam-3948	37	10	·	·	PUNCT
ejpam-3948	38	1	f2(x	f2(x	X
ejpam-3948	38	2	)	)	PUNCT
ejpam-3948	38	3	·	·	PUNCT
ejpam-3948	38	4	...	...	PUNCT
ejpam-3948	38	5	·	·	PUNCT
ejpam-3948	38	6	fk(x	fk(x	X
ejpam-3948	38	7	)	)	PUNCT
ejpam-3948	38	8	=	=	SYM
ejpam-3948	38	9	c	c	X
ejpam-3948	38	10	,	,	PUNCT
ejpam-3948	38	11	where	where	SCONJ
ejpam-3948	38	12	f1	f1	NOUN
ejpam-3948	38	13	,	,	PUNCT
ejpam-3948	38	14	f2	f2	PROPN
ejpam-3948	38	15	,	,	PUNCT
ejpam-3948	38	16	...	...	PUNCT
ejpam-3948	38	17	fk	fk	INTJ
ejpam-3948	38	18	∈	∈	PROPN
ejpam-3948	38	19	z[x	z[x	NOUN
ejpam-3948	38	20	]	]	X
ejpam-3948	38	21	,	,	PUNCT
ejpam-3948	38	22	x	x	SYM
ejpam-3948	38	23	=	=	SYM
ejpam-3948	38	24	(	(	PUNCT
ejpam-3948	38	25	x1	x1	PROPN
ejpam-3948	38	26	,	,	PUNCT
ejpam-3948	38	27	x2	x2	PROPN
ejpam-3948	38	28	,	,	PUNCT
ejpam-3948	38	29	...	...	PUNCT
ejpam-3948	38	30	,	,	PUNCT
ejpam-3948	38	31	xn	xn	PROPN
ejpam-3948	38	32	)	)	PUNCT
ejpam-3948	38	33	,	,	PUNCT
ejpam-3948	38	34	c	c	PROPN
ejpam-3948	38	35	∈	∈	PROPN
ejpam-3948	38	36	z.	z.	PROPN
ejpam-3948	38	37	given	give	VERB
ejpam-3948	38	38	the	the	DET
ejpam-3948	38	39	prime	prime	ADJ
ejpam-3948	38	40	factorization	factorization	NOUN
ejpam-3948	38	41	of	of	ADP
ejpam-3948	38	42	c	c	NOUN
ejpam-3948	38	43	,	,	PUNCT
ejpam-3948	38	44	we	we	PRON
ejpam-3948	38	45	obatin	obatin	VERB
ejpam-3948	38	46	finitely	finitely	ADV
ejpam-3948	38	47	many	many	ADJ
ejpam-3948	38	48	decompositions	decomposition	NOUN
ejpam-3948	38	49	into	into	ADP
ejpam-3948	38	50	k	k	PROPN
ejpam-3948	38	51	factorizations	factorization	NOUN
ejpam-3948	38	52	c1	c1	PROPN
ejpam-3948	38	53	,	,	PUNCT
ejpam-3948	38	54	c2	c2	PROPN
ejpam-3948	38	55	,	,	PUNCT
ejpam-3948	38	56	...	...	PUNCT
ejpam-3948	38	57	ck	ck	ADJ
ejpam-3948	38	58	.	.	PUNCT
ejpam-3948	39	1	every	every	DET
ejpam-3948	39	2	factorization	factorization	NOUN
ejpam-3948	39	3	yields	yield	VERB
ejpam-3948	39	4	a	a	DET
ejpam-3948	39	5	system	system	NOUN
ejpam-3948	39	6	of	of	ADP
ejpam-3948	39	7	equations	equation	NOUN
ejpam-3948	39	8			VERB
ejpam-3948	39	9	f1(x1	f1(x1	NOUN
ejpam-3948	39	10	,	,	PUNCT
ejpam-3948	39	11	x2	x2	PROPN
ejpam-3948	39	12	,	,	PUNCT
ejpam-3948	39	13	...	...	PUNCT
ejpam-3948	39	14	,	,	PUNCT
ejpam-3948	39	15	xn	xn	X
ejpam-3948	39	16	)	)	PUNCT
ejpam-3948	40	1	=	=	SYM
ejpam-3948	40	2	c1	c1	PROPN
ejpam-3948	40	3	f2(x1	f2(x1	PROPN
ejpam-3948	40	4	,	,	PUNCT
ejpam-3948	40	5	x2	x2	PROPN
ejpam-3948	40	6	,	,	PUNCT
ejpam-3948	40	7	...	...	PUNCT
ejpam-3948	40	8	,	,	PUNCT
ejpam-3948	40	9	xn	xn	X
ejpam-3948	40	10	)	)	PUNCT
ejpam-3948	41	1	=	=	SYM
ejpam-3948	41	2	c2	c2	PROPN
ejpam-3948	41	3	...	...	PUNCT
ejpam-3948	42	1	fk(x1	fk(x1	PROPN
ejpam-3948	42	2	,	,	PUNCT
ejpam-3948	42	3	x2	x2	PROPN
ejpam-3948	42	4	,	,	PUNCT
ejpam-3948	42	5	...	...	PUNCT
ejpam-3948	42	6	,	,	PUNCT
ejpam-3948	42	7	xn	xn	PROPN
ejpam-3948	42	8	)	)	PUNCT
ejpam-3948	42	9	=	=	SYM
ejpam-3948	43	1	ck	ck	ADJ
ejpam-3948	43	2	.	.	PUNCT
ejpam-3948	44	1	the	the	DET
ejpam-3948	44	2	complete	complete	ADJ
ejpam-3948	44	3	set	set	NOUN
ejpam-3948	44	4	of	of	ADP
ejpam-3948	44	5	solutions	solution	NOUN
ejpam-3948	44	6	is	be	AUX
ejpam-3948	44	7	obtained	obtain	VERB
ejpam-3948	44	8	by	by	ADP
ejpam-3948	44	9	solving	solve	VERB
ejpam-3948	44	10	all	all	DET
ejpam-3948	44	11	such	such	ADJ
ejpam-3948	44	12	systems	system	NOUN
ejpam-3948	44	13	.	.	PUNCT
ejpam-3948	45	1	the	the	DET
ejpam-3948	45	2	modular	modular	ADJ
ejpam-3948	45	3	arithmetic	arithmetic	ADJ
ejpam-3948	45	4	method	method	NOUN
ejpam-3948	45	5	,	,	PUNCT
ejpam-3948	45	6	on	on	ADP
ejpam-3948	45	7	the	the	DET
ejpam-3948	45	8	other	other	ADJ
ejpam-3948	45	9	hand	hand	NOUN
ejpam-3948	45	10	,	,	PUNCT
ejpam-3948	45	11	is	be	AUX
ejpam-3948	45	12	widely	widely	ADV
ejpam-3948	45	13	used	use	VERB
ejpam-3948	45	14	in	in	ADP
ejpam-3948	45	15	proving	prove	VERB
ejpam-3948	45	16	nonsolvability	nonsolvability	NOUN
ejpam-3948	45	17	of	of	ADP
ejpam-3948	45	18	a	a	DET
ejpam-3948	45	19	given	give	VERB
ejpam-3948	45	20	equation	equation	NOUN
ejpam-3948	45	21	or	or	CCONJ
ejpam-3948	45	22	at	at	ADP
ejpam-3948	45	23	least	least	ADJ
ejpam-3948	45	24	reducing	reduce	VERB
ejpam-3948	45	25	the	the	DET
ejpam-3948	45	26	set	set	NOUN
ejpam-3948	45	27	of	of	ADP
ejpam-3948	45	28	integers	integer	NOUN
ejpam-3948	45	29	where	where	SCONJ
ejpam-3948	45	30	we	we	PRON
ejpam-3948	45	31	can	can	AUX
ejpam-3948	45	32	find	find	VERB
ejpam-3948	45	33	possible	possible	ADJ
ejpam-3948	45	34	solutions	solution	NOUN
ejpam-3948	45	35	.	.	PUNCT
ejpam-3948	46	1	properties	property	NOUN
ejpam-3948	46	2	of	of	ADP
ejpam-3948	46	3	modular	modular	ADJ
ejpam-3948	46	4	arithmetic	arithmetic	NOUN
ejpam-3948	46	5	are	be	AUX
ejpam-3948	46	6	utilized	utilize	VERB
ejpam-3948	46	7	in	in	ADP
ejpam-3948	46	8	deriving	derive	VERB
ejpam-3948	46	9	the	the	DET
ejpam-3948	46	10	results	result	NOUN
ejpam-3948	46	11	.	.	PUNCT
ejpam-3948	47	1	for	for	ADP
ejpam-3948	47	2	interesting	interesting	ADJ
ejpam-3948	47	3	examples	example	NOUN
ejpam-3948	47	4	using	use	VERB
ejpam-3948	47	5	the	the	DET
ejpam-3948	47	6	above	above	ADJ
ejpam-3948	47	7	methods	method	NOUN
ejpam-3948	47	8	,	,	PUNCT
ejpam-3948	47	9	the	the	DET
ejpam-3948	47	10	reader	reader	NOUN
ejpam-3948	47	11	is	be	AUX
ejpam-3948	47	12	referred	refer	VERB
ejpam-3948	47	13	to	to	ADP
ejpam-3948	47	14	the	the	DET
ejpam-3948	47	15	book	book	NOUN
ejpam-3948	47	16	by	by	ADP
ejpam-3948	47	17	andreescu	andreescu	PROPN
ejpam-3948	47	18	et	et	PROPN
ejpam-3948	47	19	al	al	PROPN
ejpam-3948	47	20	.	.	PUNCT
ejpam-3948	48	1	[	[	X
ejpam-3948	48	2	28	28	NUM
ejpam-3948	48	3	]	]	PUNCT
ejpam-3948	48	4	.	.	PUNCT
ejpam-3948	49	1	2	2	X
ejpam-3948	49	2	.	.	X
ejpam-3948	49	3	main	main	ADJ
ejpam-3948	49	4	results	result	NOUN
ejpam-3948	49	5	the	the	DET
ejpam-3948	49	6	following	follow	VERB
ejpam-3948	49	7	definition	definition	NOUN
ejpam-3948	49	8	and	and	CCONJ
ejpam-3948	49	9	lemmas	lemma	NOUN
ejpam-3948	49	10	are	be	AUX
ejpam-3948	49	11	needed	need	VERB
ejpam-3948	49	12	in	in	ADP
ejpam-3948	49	13	this	this	DET
ejpam-3948	49	14	study	study	NOUN
ejpam-3948	49	15	.	.	PUNCT
ejpam-3948	50	1	definition	definition	NOUN
ejpam-3948	50	2	1	1	NUM
ejpam-3948	50	3	.	.	PUNCT
ejpam-3948	51	1	a	a	DET
ejpam-3948	51	2	mersenne	mersenne	NOUN
ejpam-3948	51	3	prime	prime	NOUN
ejpam-3948	51	4	is	be	AUX
ejpam-3948	51	5	a	a	DET
ejpam-3948	51	6	prime	prime	ADJ
ejpam-3948	51	7	number	number	NOUN
ejpam-3948	51	8	of	of	ADP
ejpam-3948	51	9	the	the	DET
ejpam-3948	51	10	form	form	NOUN
ejpam-3948	51	11	2p	2p	NUM
ejpam-3948	51	12	−	−	NOUN
ejpam-3948	51	13	1	1	NUM
ejpam-3948	51	14	,	,	PUNCT
ejpam-3948	51	15	where	where	SCONJ
ejpam-3948	51	16	p	p	NOUN
ejpam-3948	51	17	is	be	AUX
ejpam-3948	51	18	also	also	ADV
ejpam-3948	51	19	a	a	DET
ejpam-3948	51	20	prime	prime	ADJ
ejpam-3948	51	21	number	number	NOUN
ejpam-3948	51	22	.	.	PUNCT
ejpam-3948	52	1	lemma	lemma	PROPN
ejpam-3948	52	2	1	1	NUM
ejpam-3948	52	3	.	.	PUNCT
ejpam-3948	53	1	all	all	DET
ejpam-3948	53	2	mersenne	mersenne	NOUN
ejpam-3948	53	3	primes	prime	NOUN
ejpam-3948	53	4	are	be	AUX
ejpam-3948	53	5	congruent	congruent	ADJ
ejpam-3948	53	6	to	to	ADP
ejpam-3948	53	7	3	3	NUM
ejpam-3948	53	8	(	(	PUNCT
ejpam-3948	53	9	mod	mod	PROPN
ejpam-3948	53	10	4	4	NUM
ejpam-3948	53	11	)	)	PUNCT
ejpam-3948	53	12	.	.	PUNCT
ejpam-3948	54	1	proof	proof	NOUN
ejpam-3948	54	2	.	.	PUNCT
ejpam-3948	55	1	since	since	SCONJ
ejpam-3948	55	2	a	a	DET
ejpam-3948	55	3	mersenne	mersenne	NOUN
ejpam-3948	55	4	prime	prime	NOUN
ejpam-3948	55	5	is	be	AUX
ejpam-3948	55	6	of	of	ADP
ejpam-3948	55	7	the	the	DET
ejpam-3948	55	8	form	form	NOUN
ejpam-3948	55	9	2p	2p	NUM
ejpam-3948	55	10	−	−	NOUN
ejpam-3948	55	11	1	1	NUM
ejpam-3948	55	12	,	,	PUNCT
ejpam-3948	55	13	it	it	PRON
ejpam-3948	55	14	follows	follow	VERB
ejpam-3948	55	15	that	that	SCONJ
ejpam-3948	55	16	p	p	PRON
ejpam-3948	55	17	≥	≥	NUM
ejpam-3948	55	18	2	2	NUM
ejpam-3948	55	19	.	.	PUNCT
ejpam-3948	56	1	thus	thus	ADV
ejpam-3948	56	2	,	,	PUNCT
ejpam-3948	56	3	2p	2p	NUM
ejpam-3948	56	4	≡	≡	PROPN
ejpam-3948	56	5	0	0	NUM
ejpam-3948	57	1	(	(	PUNCT
ejpam-3948	57	2	mod	mod	PROPN
ejpam-3948	57	3	4	4	NUM
ejpam-3948	57	4	)	)	PUNCT
ejpam-3948	57	5	,	,	PUNCT
ejpam-3948	57	6	yielding	yield	VERB
ejpam-3948	57	7	2p	2p	NUM
ejpam-3948	57	8	−	−	PROPN
ejpam-3948	57	9	1	1	NUM
ejpam-3948	57	10	≡	≡	PROPN
ejpam-3948	57	11	−1	−1	NOUN
ejpam-3948	57	12	(	(	PUNCT
ejpam-3948	57	13	mod	mod	PROPN
ejpam-3948	57	14	4	4	NUM
ejpam-3948	57	15	)	)	PUNCT
ejpam-3948	57	16	or	or	CCONJ
ejpam-3948	57	17	3	3	NUM
ejpam-3948	57	18	(	(	PUNCT
ejpam-3948	57	19	mod	mod	NOUN
ejpam-3948	57	20	4	4	NUM
ejpam-3948	57	21	)	)	PUNCT
ejpam-3948	57	22	.	.	PUNCT
ejpam-3948	58	1	lemma	lemma	PROPN
ejpam-3948	58	2	2	2	NUM
ejpam-3948	58	3	(	(	PUNCT
ejpam-3948	58	4	mihailescu	mihailescu	PROPN
ejpam-3948	58	5	’s	’s	PART
ejpam-3948	58	6	theorem	theorem	NOUN
ejpam-3948	58	7	[	[	X
ejpam-3948	58	8	12	12	NUM
ejpam-3948	58	9	]	]	PUNCT
ejpam-3948	58	10	)	)	PUNCT
ejpam-3948	58	11	.	.	PUNCT
ejpam-3948	59	1	the	the	DET
ejpam-3948	59	2	quadruple	quadruple	NOUN
ejpam-3948	59	3	(	(	PUNCT
ejpam-3948	59	4	3	3	NUM
ejpam-3948	59	5	,	,	PUNCT
ejpam-3948	59	6	2	2	NUM
ejpam-3948	59	7	,	,	PUNCT
ejpam-3948	59	8	2	2	NUM
ejpam-3948	59	9	,	,	PUNCT
ejpam-3948	59	10	3	3	NUM
ejpam-3948	59	11	)	)	PUNCT
ejpam-3948	59	12	is	be	AUX
ejpam-3948	59	13	the	the	DET
ejpam-3948	59	14	unique	unique	ADJ
ejpam-3948	59	15	solution	solution	NOUN
ejpam-3948	59	16	for	for	ADP
ejpam-3948	59	17	the	the	DET
ejpam-3948	59	18	diophantine	diophantine	NOUN
ejpam-3948	59	19	equation	equation	NOUN
ejpam-3948	59	20	ax	ax	NOUN
ejpam-3948	59	21	−	−	NOUN
ejpam-3948	59	22	by	by	ADP
ejpam-3948	59	23	=	=	PROPN
ejpam-3948	59	24	1	1	NUM
ejpam-3948	59	25	,	,	PUNCT
ejpam-3948	59	26	where	where	SCONJ
ejpam-3948	59	27	a	a	DET
ejpam-3948	59	28	,	,	PUNCT
ejpam-3948	59	29	b	b	NOUN
ejpam-3948	59	30	,	,	PUNCT
ejpam-3948	59	31	x	x	PUNCT
ejpam-3948	59	32	and	and	CCONJ
ejpam-3948	59	33	y	y	PROPN
ejpam-3948	59	34	are	be	AUX
ejpam-3948	59	35	integers	integer	NOUN
ejpam-3948	59	36	with	with	ADP
ejpam-3948	59	37	min{a	min{a	PROPN
ejpam-3948	59	38	,	,	PUNCT
ejpam-3948	59	39	b	b	PROPN
ejpam-3948	59	40	,	,	PUNCT
ejpam-3948	59	41	x	x	NOUN
ejpam-3948	59	42	,	,	PUNCT
ejpam-3948	59	43	y	y	PROPN
ejpam-3948	59	44	}	}	PUNCT
ejpam-3948	59	45	>	>	X
ejpam-3948	60	1	1	1	X
ejpam-3948	60	2	.	.	PUNCT
ejpam-3948	61	1	the	the	DET
ejpam-3948	61	2	main	main	ADJ
ejpam-3948	61	3	result	result	NOUN
ejpam-3948	61	4	for	for	ADP
ejpam-3948	61	5	this	this	DET
ejpam-3948	61	6	study	study	NOUN
ejpam-3948	61	7	is	be	AUX
ejpam-3948	61	8	stated	state	VERB
ejpam-3948	61	9	as	as	SCONJ
ejpam-3948	61	10	follows	follow	VERB
ejpam-3948	61	11	.	.	PUNCT
ejpam-3948	62	1	w.	w.	PROPN
ejpam-3948	62	2	s.	s.	PROPN
ejpam-3948	62	3	gayo	gayo	PROPN
ejpam-3948	62	4	,	,	PUNCT
ejpam-3948	62	5	jr	jr	PROPN
ejpam-3948	62	6	.	.	PROPN
ejpam-3948	62	7	,	,	PUNCT
ejpam-3948	62	8	j.	j.	PROPN
ejpam-3948	62	9	b.	b.	PROPN
ejpam-3948	62	10	bacani	bacani	PROPN
ejpam-3948	62	11	/	/	SYM
ejpam-3948	62	12	eur	eur	PROPN
ejpam-3948	62	13	.	.	PUNCT
ejpam-3948	63	1	j.	j.	PROPN
ejpam-3948	63	2	pure	pure	PROPN
ejpam-3948	63	3	appl	appl	PROPN
ejpam-3948	63	4	.	.	PROPN
ejpam-3948	63	5	math	math	PROPN
ejpam-3948	63	6	,	,	PUNCT
ejpam-3948	63	7	14	14	NUM
ejpam-3948	63	8	(	(	PUNCT
ejpam-3948	63	9	2	2	NUM
ejpam-3948	63	10	)	)	PUNCT
ejpam-3948	63	11	(	(	PUNCT
ejpam-3948	63	12	2021	2021	NUM
ejpam-3948	63	13	)	)	PUNCT
ejpam-3948	63	14	,	,	PUNCT
ejpam-3948	63	15	396	396	NUM
ejpam-3948	63	16	-	-	SYM
ejpam-3948	63	17	403	403	NUM
ejpam-3948	63	18	398	398	NUM
ejpam-3948	63	19	theorem	theorem	NOUN
ejpam-3948	63	20	1	1	NUM
ejpam-3948	63	21	.	.	PUNCT
ejpam-3948	64	1	every	every	DET
ejpam-3948	64	2	nonnegative	nonnegative	ADJ
ejpam-3948	64	3	integer	integer	NOUN
ejpam-3948	64	4	solution	solution	NOUN
ejpam-3948	64	5	(	(	PUNCT
ejpam-3948	64	6	mp	mp	PROPN
ejpam-3948	64	7	,	,	PUNCT
ejpam-3948	64	8	mq	mq	PROPN
ejpam-3948	64	9	,	,	PUNCT
ejpam-3948	64	10	x	x	PROPN
ejpam-3948	64	11	,	,	PUNCT
ejpam-3948	64	12	y	y	PROPN
ejpam-3948	64	13	,	,	PUNCT
ejpam-3948	64	14	z	z	NOUN
ejpam-3948	64	15	)	)	PUNCT
ejpam-3948	64	16	of	of	ADP
ejpam-3948	64	17	the	the	DET
ejpam-3948	64	18	diophantine	diophantine	NOUN
ejpam-3948	64	19	equation	equation	NOUN
ejpam-3948	64	20	mx	mx	PROPN
ejpam-3948	64	21	p	p	PROPN
ejpam-3948	64	22	+	+	X
ejpam-3948	64	23	(	(	PUNCT
ejpam-3948	64	24	mq	mq	PROPN
ejpam-3948	64	25	+	+	CCONJ
ejpam-3948	64	26	1)y	1)y	NUM
ejpam-3948	64	27	=	=	SYM
ejpam-3948	64	28	z2	z2	PROPN
ejpam-3948	64	29	,	,	PUNCT
ejpam-3948	64	30	where	where	SCONJ
ejpam-3948	64	31	mp	mp	PROPN
ejpam-3948	64	32	and	and	CCONJ
ejpam-3948	64	33	mq	mq	PROPN
ejpam-3948	64	34	are	be	AUX
ejpam-3948	64	35	mersenne	mersenne	NOUN
ejpam-3948	64	36	primes	prime	NOUN
ejpam-3948	64	37	,	,	PUNCT
ejpam-3948	64	38	takes	take	VERB
ejpam-3948	64	39	any	any	PRON
ejpam-3948	64	40	of	of	ADP
ejpam-3948	64	41	the	the	DET
ejpam-3948	64	42	following	follow	VERB
ejpam-3948	64	43	forms	form	NOUN
ejpam-3948	64	44	:	:	PUNCT
ejpam-3948	64	45	i.	i.	PROPN
ejpam-3948	64	46	(	(	PUNCT
ejpam-3948	64	47	mp	mp	PROPN
ejpam-3948	64	48	,	,	PUNCT
ejpam-3948	64	49	7	7	NUM
ejpam-3948	64	50	,	,	PUNCT
ejpam-3948	64	51	0	0	NUM
ejpam-3948	64	52	,	,	PUNCT
ejpam-3948	64	53	1	1	NUM
ejpam-3948	64	54	,	,	PUNCT
ejpam-3948	64	55	3	3	X
ejpam-3948	64	56	)	)	PUNCT
ejpam-3948	64	57	ii	ii	NOUN
ejpam-3948	64	58	.	.	PUNCT
ejpam-3948	65	1	(	(	PUNCT
ejpam-3948	65	2	3,mq	3,mq	NUM
ejpam-3948	65	3	,	,	PUNCT
ejpam-3948	65	4	1	1	NUM
ejpam-3948	65	5	,	,	PUNCT
ejpam-3948	65	6	0	0	NUM
ejpam-3948	65	7	,	,	PUNCT
ejpam-3948	65	8	2	2	X
ejpam-3948	65	9	)	)	PUNCT
ejpam-3948	65	10	iii	iii	NOUN
ejpam-3948	65	11	.	.	PUNCT
ejpam-3948	66	1	(	(	PUNCT
ejpam-3948	66	2	mp	mp	PROPN
ejpam-3948	66	3	,	,	PUNCT
ejpam-3948	66	4	mq	mq	PROPN
ejpam-3948	66	5	,	,	PUNCT
ejpam-3948	66	6	2	2	NUM
ejpam-3948	66	7	,	,	PUNCT
ejpam-3948	66	8	p+	p+	VERB
ejpam-3948	66	9	2	2	NUM
ejpam-3948	66	10	q	q	NOUN
ejpam-3948	66	11	,	,	PUNCT
ejpam-3948	66	12	2p	2p	NUM
ejpam-3948	66	13	+	+	SYM
ejpam-3948	66	14	1	1	X
ejpam-3948	66	15	)	)	PUNCT
ejpam-3948	66	16	proof	proof	NOUN
ejpam-3948	66	17	.	.	PUNCT
ejpam-3948	67	1	let	let	VERB
ejpam-3948	67	2	us	we	PRON
ejpam-3948	67	3	consider	consider	VERB
ejpam-3948	67	4	first	first	ADV
ejpam-3948	67	5	the	the	DET
ejpam-3948	67	6	case	case	NOUN
ejpam-3948	67	7	when	when	SCONJ
ejpam-3948	67	8	one	one	NUM
ejpam-3948	67	9	of	of	ADP
ejpam-3948	67	10	the	the	DET
ejpam-3948	67	11	exponents	exponent	NOUN
ejpam-3948	67	12	x	x	PUNCT
ejpam-3948	67	13	and	and	CCONJ
ejpam-3948	67	14	y	y	PROPN
ejpam-3948	67	15	is	be	AUX
ejpam-3948	67	16	zero	zero	NUM
ejpam-3948	67	17	.	.	PUNCT
ejpam-3948	68	1	if	if	SCONJ
ejpam-3948	68	2	x	x	PROPN
ejpam-3948	68	3	=	=	SYM
ejpam-3948	68	4	0	0	NUM
ejpam-3948	68	5	,	,	PUNCT
ejpam-3948	68	6	then	then	ADV
ejpam-3948	68	7	regardless	regardless	ADV
ejpam-3948	68	8	of	of	ADP
ejpam-3948	68	9	any	any	DET
ejpam-3948	68	10	mersenne	mersenne	PROPN
ejpam-3948	68	11	prime	prime	PROPN
ejpam-3948	68	12	mp	mp	PROPN
ejpam-3948	68	13	,	,	PUNCT
ejpam-3948	68	14	we	we	PRON
ejpam-3948	68	15	have	have	VERB
ejpam-3948	68	16	the	the	DET
ejpam-3948	68	17	equation	equation	NOUN
ejpam-3948	68	18	1	1	NUM
ejpam-3948	68	19	+	+	CCONJ
ejpam-3948	68	20	(	(	PUNCT
ejpam-3948	68	21	mq	mq	PROPN
ejpam-3948	68	22	+	+	CCONJ
ejpam-3948	68	23	1)y	1)y	NUM
ejpam-3948	68	24	=	=	SYM
ejpam-3948	68	25	z2	z2	PROPN
ejpam-3948	68	26	.	.	PUNCT
ejpam-3948	69	1	if	if	SCONJ
ejpam-3948	69	2	y	y	PROPN
ejpam-3948	69	3	=	=	SYM
ejpam-3948	69	4	0	0	PROPN
ejpam-3948	69	5	,	,	PUNCT
ejpam-3948	69	6	then	then	ADV
ejpam-3948	69	7	z2	z2	PROPN
ejpam-3948	69	8	=	=	SYM
ejpam-3948	69	9	2	2	NUM
ejpam-3948	69	10	,	,	PUNCT
ejpam-3948	69	11	not	not	PART
ejpam-3948	69	12	a	a	DET
ejpam-3948	69	13	perfect	perfect	ADJ
ejpam-3948	69	14	square	square	NOUN
ejpam-3948	69	15	.	.	PUNCT
ejpam-3948	70	1	if	if	SCONJ
ejpam-3948	70	2	y	y	PROPN
ejpam-3948	70	3	=	=	SYM
ejpam-3948	70	4	1	1	NUM
ejpam-3948	70	5	,	,	PUNCT
ejpam-3948	70	6	then	then	ADV
ejpam-3948	70	7	z2	z2	PROPN
ejpam-3948	70	8	−	−	PROPN
ejpam-3948	70	9	2q	2q	NOUN
ejpam-3948	70	10	=	=	SYM
ejpam-3948	70	11	1	1	X
ejpam-3948	70	12	.	.	PUNCT
ejpam-3948	71	1	by	by	ADP
ejpam-3948	71	2	mihailescu	mihailescu	PROPN
ejpam-3948	71	3	’s	’s	PART
ejpam-3948	71	4	theorem	theorem	NOUN
ejpam-3948	71	5	,	,	PUNCT
ejpam-3948	71	6	z	z	NOUN
ejpam-3948	71	7	=	=	SYM
ejpam-3948	71	8	3	3	NUM
ejpam-3948	71	9	and	and	CCONJ
ejpam-3948	71	10	q	q	NOUN
ejpam-3948	71	11	=	=	NOUN
ejpam-3948	71	12	3	3	X
ejpam-3948	71	13	.	.	PUNCT
ejpam-3948	71	14	then	then	ADV
ejpam-3948	71	15	,	,	PUNCT
ejpam-3948	71	16	mq	mq	PROPN
ejpam-3948	71	17	=	=	SYM
ejpam-3948	71	18	7	7	NUM
ejpam-3948	71	19	,	,	PUNCT
ejpam-3948	71	20	a	a	DET
ejpam-3948	71	21	mersenne	mersenne	NOUN
ejpam-3948	71	22	prime	prime	NOUN
ejpam-3948	71	23	and	and	CCONJ
ejpam-3948	71	24	thus	thus	ADV
ejpam-3948	71	25	(	(	PUNCT
ejpam-3948	71	26	mp	mp	PROPN
ejpam-3948	71	27	,	,	PUNCT
ejpam-3948	71	28	mq	mq	PROPN
ejpam-3948	71	29	,	,	PUNCT
ejpam-3948	71	30	x	x	PROPN
ejpam-3948	71	31	,	,	PUNCT
ejpam-3948	71	32	y	y	PROPN
ejpam-3948	71	33	,	,	PUNCT
ejpam-3948	71	34	z	z	NOUN
ejpam-3948	71	35	)	)	PUNCT
ejpam-3948	71	36	=	=	SYM
ejpam-3948	71	37	(	(	PUNCT
ejpam-3948	71	38	mp	mp	PROPN
ejpam-3948	71	39	,	,	PUNCT
ejpam-3948	71	40	7	7	NUM
ejpam-3948	71	41	,	,	PUNCT
ejpam-3948	71	42	0	0	NUM
ejpam-3948	71	43	,	,	PUNCT
ejpam-3948	71	44	1	1	NUM
ejpam-3948	71	45	,	,	PUNCT
ejpam-3948	71	46	3	3	NUM
ejpam-3948	71	47	)	)	PUNCT
ejpam-3948	71	48	for	for	ADP
ejpam-3948	71	49	any	any	DET
ejpam-3948	71	50	mersenne	mersenne	NOUN
ejpam-3948	71	51	prime	prime	PROPN
ejpam-3948	71	52	mp	mp	PROPN
ejpam-3948	71	53	is	be	AUX
ejpam-3948	71	54	a	a	DET
ejpam-3948	71	55	solution	solution	NOUN
ejpam-3948	71	56	.	.	PUNCT
ejpam-3948	72	1	if	if	SCONJ
ejpam-3948	72	2	y	y	PROPN
ejpam-3948	72	3	>	>	X
ejpam-3948	72	4	1	1	NUM
ejpam-3948	72	5	,	,	PUNCT
ejpam-3948	72	6	then	then	ADV
ejpam-3948	72	7	by	by	ADP
ejpam-3948	72	8	mihailescu	mihailescu	PROPN
ejpam-3948	72	9	’s	’s	PART
ejpam-3948	72	10	theorem	theorem	ADJ
ejpam-3948	72	11	,	,	PUNCT
ejpam-3948	72	12	2q	2q	NOUN
ejpam-3948	72	13	=	=	SYM
ejpam-3948	72	14	2	2	NUM
ejpam-3948	72	15	giving	give	VERB
ejpam-3948	72	16	us	we	PRON
ejpam-3948	72	17	q	q	NOUN
ejpam-3948	72	18	=	=	SYM
ejpam-3948	72	19	1	1	NUM
ejpam-3948	72	20	which	which	PRON
ejpam-3948	72	21	is	be	AUX
ejpam-3948	72	22	not	not	PART
ejpam-3948	72	23	possible	possible	ADJ
ejpam-3948	72	24	because	because	SCONJ
ejpam-3948	72	25	q	q	NOUN
ejpam-3948	72	26	must	must	AUX
ejpam-3948	72	27	be	be	AUX
ejpam-3948	72	28	a	a	DET
ejpam-3948	72	29	prime	prime	ADJ
ejpam-3948	72	30	number	number	NOUN
ejpam-3948	72	31	.	.	PUNCT
ejpam-3948	73	1	if	if	SCONJ
ejpam-3948	73	2	y	y	PROPN
ejpam-3948	73	3	=	=	SYM
ejpam-3948	73	4	0	0	PROPN
ejpam-3948	73	5	,	,	PUNCT
ejpam-3948	73	6	then	then	ADV
ejpam-3948	73	7	regardless	regardless	ADV
ejpam-3948	73	8	of	of	ADP
ejpam-3948	73	9	any	any	DET
ejpam-3948	73	10	mersenne	mersenne	NOUN
ejpam-3948	73	11	prime	prime	PROPN
ejpam-3948	73	12	mq	mq	PROPN
ejpam-3948	73	13	,	,	PUNCT
ejpam-3948	73	14	we	we	PRON
ejpam-3948	73	15	have	have	VERB
ejpam-3948	73	16	the	the	DET
ejpam-3948	73	17	equation	equation	NOUN
ejpam-3948	74	1	mx	mx	PROPN
ejpam-3948	74	2	p	p	PROPN
ejpam-3948	74	3	+	+	PROPN
ejpam-3948	74	4	1	1	NUM
ejpam-3948	74	5	=	=	SYM
ejpam-3948	74	6	z2	z2	PROPN
ejpam-3948	74	7	.	.	PUNCT
ejpam-3948	75	1	substituting	substitute	VERB
ejpam-3948	75	2	mp	mp	NOUN
ejpam-3948	75	3	=	=	NOUN
ejpam-3948	75	4	2p	2p	NUM
ejpam-3948	75	5	−	−	NOUN
ejpam-3948	75	6	1	1	NUM
ejpam-3948	75	7	to	to	ADP
ejpam-3948	75	8	the	the	DET
ejpam-3948	75	9	equation	equation	NOUN
ejpam-3948	75	10	above	above	ADV
ejpam-3948	75	11	will	will	AUX
ejpam-3948	75	12	lead	lead	VERB
ejpam-3948	75	13	to	to	ADP
ejpam-3948	75	14	the	the	DET
ejpam-3948	75	15	equation	equation	NOUN
ejpam-3948	75	16	(	(	PUNCT
ejpam-3948	75	17	2p	2p	NUM
ejpam-3948	75	18	−	−	NUM
ejpam-3948	75	19	1)x	1)x	NUM
ejpam-3948	75	20	+	+	CCONJ
ejpam-3948	75	21	1	1	NUM
ejpam-3948	75	22	=	=	SYM
ejpam-3948	75	23	z2	z2	PROPN
ejpam-3948	75	24	.	.	PUNCT
ejpam-3948	76	1	if	if	SCONJ
ejpam-3948	76	2	x	x	PROPN
ejpam-3948	76	3	=	=	SYM
ejpam-3948	76	4	0	0	NUM
ejpam-3948	76	5	,	,	PUNCT
ejpam-3948	76	6	then	then	ADV
ejpam-3948	76	7	z2	z2	PROPN
ejpam-3948	76	8	=	=	SYM
ejpam-3948	76	9	2	2	NUM
ejpam-3948	76	10	,	,	PUNCT
ejpam-3948	76	11	not	not	PART
ejpam-3948	76	12	a	a	DET
ejpam-3948	76	13	perfect	perfect	ADJ
ejpam-3948	76	14	square	square	NOUN
ejpam-3948	76	15	.	.	PUNCT
ejpam-3948	77	1	if	if	SCONJ
ejpam-3948	77	2	x	x	SYM
ejpam-3948	77	3	=	=	SYM
ejpam-3948	77	4	1	1	NUM
ejpam-3948	77	5	,	,	PUNCT
ejpam-3948	77	6	then	then	ADV
ejpam-3948	77	7	z2	z2	PROPN
ejpam-3948	77	8	=	=	SYM
ejpam-3948	77	9	2p	2p	NOUN
ejpam-3948	77	10	.	.	PUNCT
ejpam-3948	78	1	let	let	VERB
ejpam-3948	78	2	z	z	NOUN
ejpam-3948	78	3	=	=	SYM
ejpam-3948	78	4	2a	2a	NUM
ejpam-3948	78	5	.	.	PUNCT
ejpam-3948	79	1	then	then	ADV
ejpam-3948	79	2	,	,	PUNCT
ejpam-3948	79	3	22a	22a	NUM
ejpam-3948	79	4	=	=	SYM
ejpam-3948	79	5	2p	2p	NOUN
ejpam-3948	79	6	,	,	PUNCT
ejpam-3948	79	7	which	which	PRON
ejpam-3948	79	8	implies	imply	VERB
ejpam-3948	79	9	that	that	DET
ejpam-3948	79	10	2a	2a	NUM
ejpam-3948	79	11	=	=	PUNCT
ejpam-3948	80	1	p.	p.	NOUN
ejpam-3948	80	2	using	use	VERB
ejpam-3948	80	3	the	the	DET
ejpam-3948	80	4	primality	primality	NOUN
ejpam-3948	80	5	of	of	ADP
ejpam-3948	80	6	p	p	NOUN
ejpam-3948	80	7	,	,	PUNCT
ejpam-3948	80	8	we	we	PRON
ejpam-3948	80	9	get	get	VERB
ejpam-3948	80	10	a	a	DET
ejpam-3948	80	11	=	=	SYM
ejpam-3948	80	12	1	1	NUM
ejpam-3948	80	13	and	and	CCONJ
ejpam-3948	81	1	p	p	NOUN
ejpam-3948	81	2	=	=	ADJ
ejpam-3948	81	3	2	2	X
ejpam-3948	81	4	.	.	PUNCT
ejpam-3948	82	1	this	this	PRON
ejpam-3948	82	2	results	result	VERB
ejpam-3948	82	3	to	to	ADP
ejpam-3948	82	4	z	z	NOUN
ejpam-3948	82	5	=	=	SYM
ejpam-3948	82	6	2	2	NUM
ejpam-3948	82	7	and	and	CCONJ
ejpam-3948	82	8	mp	mp	NOUN
ejpam-3948	82	9	=	=	NOUN
ejpam-3948	82	10	2p	2p	NUM
ejpam-3948	82	11	−	−	NOUN
ejpam-3948	82	12	1	1	NUM
ejpam-3948	82	13	=	=	SYM
ejpam-3948	82	14	3	3	NUM
ejpam-3948	82	15	,	,	PUNCT
ejpam-3948	82	16	a	a	DET
ejpam-3948	82	17	mersenne	mersenne	NOUN
ejpam-3948	82	18	prime	prime	NOUN
ejpam-3948	82	19	.	.	PUNCT
ejpam-3948	83	1	hence	hence	ADV
ejpam-3948	83	2	,	,	PUNCT
ejpam-3948	83	3	(	(	PUNCT
ejpam-3948	83	4	mp	mp	PROPN
ejpam-3948	83	5	,	,	PUNCT
ejpam-3948	83	6	mq	mq	PROPN
ejpam-3948	83	7	,	,	PUNCT
ejpam-3948	83	8	x	x	NOUN
ejpam-3948	83	9	,	,	PUNCT
ejpam-3948	83	10	z	z	NOUN
ejpam-3948	83	11	)	)	PUNCT
ejpam-3948	83	12	=	=	SYM
ejpam-3948	83	13	(	(	PUNCT
ejpam-3948	83	14	3,mq	3,mq	NUM
ejpam-3948	83	15	,	,	PUNCT
ejpam-3948	83	16	1	1	NUM
ejpam-3948	83	17	,	,	PUNCT
ejpam-3948	83	18	0	0	NUM
ejpam-3948	83	19	,	,	PUNCT
ejpam-3948	83	20	2	2	NUM
ejpam-3948	83	21	)	)	PUNCT
ejpam-3948	83	22	,	,	PUNCT
ejpam-3948	83	23	for	for	ADP
ejpam-3948	83	24	any	any	DET
ejpam-3948	83	25	mersenne	mersenne	PROPN
ejpam-3948	83	26	prime	prime	PROPN
ejpam-3948	83	27	mq	mq	PROPN
ejpam-3948	83	28	,	,	PUNCT
ejpam-3948	83	29	is	be	AUX
ejpam-3948	83	30	a	a	DET
ejpam-3948	83	31	solution	solution	NOUN
ejpam-3948	83	32	.	.	PUNCT
ejpam-3948	84	1	if	if	SCONJ
ejpam-3948	84	2	x	x	PROPN
ejpam-3948	84	3	>	>	X
ejpam-3948	84	4	1	1	NUM
ejpam-3948	84	5	,	,	PUNCT
ejpam-3948	84	6	then	then	ADV
ejpam-3948	84	7	by	by	ADP
ejpam-3948	84	8	mihailescu	mihailescu	PROPN
ejpam-3948	84	9	’s	’s	PART
ejpam-3948	84	10	theorem	theorem	NOUN
ejpam-3948	84	11	,	,	PUNCT
ejpam-3948	84	12	z	z	NOUN
ejpam-3948	84	13	=	=	SYM
ejpam-3948	84	14	3	3	NUM
ejpam-3948	84	15	,	,	PUNCT
ejpam-3948	84	16	x	x	SYM
ejpam-3948	84	17	=	=	SYM
ejpam-3948	84	18	2	2	NUM
ejpam-3948	84	19	and	and	CCONJ
ejpam-3948	84	20	2p	2p	NUM
ejpam-3948	84	21	−	−	NOUN
ejpam-3948	84	22	1	1	NUM
ejpam-3948	84	23	=	=	SYM
ejpam-3948	84	24	2	2	NUM
ejpam-3948	84	25	,	,	PUNCT
ejpam-3948	84	26	a	a	DET
ejpam-3948	84	27	contradiction	contradiction	NOUN
ejpam-3948	84	28	.	.	PUNCT
ejpam-3948	85	1	we	we	PRON
ejpam-3948	85	2	are	be	AUX
ejpam-3948	85	3	now	now	ADV
ejpam-3948	85	4	left	leave	VERB
ejpam-3948	85	5	with	with	ADP
ejpam-3948	85	6	the	the	DET
ejpam-3948	85	7	case	case	NOUN
ejpam-3948	85	8	where	where	SCONJ
ejpam-3948	85	9	min{x	min{x	NOUN
ejpam-3948	85	10	,	,	PUNCT
ejpam-3948	85	11	y	y	PROPN
ejpam-3948	85	12	}	}	PUNCT
ejpam-3948	85	13	≥	≥	NOUN
ejpam-3948	85	14	1	1	NUM
ejpam-3948	85	15	.	.	PUNCT
ejpam-3948	86	1	we	we	PRON
ejpam-3948	86	2	note	note	VERB
ejpam-3948	86	3	that	that	SCONJ
ejpam-3948	86	4	all	all	DET
ejpam-3948	86	5	mersenne	mersenne	NOUN
ejpam-3948	86	6	primes	prime	NOUN
ejpam-3948	86	7	are	be	AUX
ejpam-3948	86	8	congruent	congruent	ADJ
ejpam-3948	86	9	to	to	ADP
ejpam-3948	86	10	3	3	NUM
ejpam-3948	86	11	(	(	PUNCT
ejpam-3948	86	12	mod	mod	PROPN
ejpam-3948	86	13	4	4	NUM
ejpam-3948	86	14	)	)	PUNCT
ejpam-3948	86	15	.	.	PUNCT
ejpam-3948	87	1	hence	hence	ADV
ejpam-3948	87	2	,	,	PUNCT
ejpam-3948	87	3	mp	mp	PROPN
ejpam-3948	87	4	≡	≡	PROPN
ejpam-3948	87	5	3	3	NUM
ejpam-3948	87	6	(	(	PUNCT
ejpam-3948	87	7	mod	mod	NOUN
ejpam-3948	87	8	4	4	NUM
ejpam-3948	87	9	)	)	PUNCT
ejpam-3948	87	10	and	and	CCONJ
ejpam-3948	87	11	mq	mq	PROPN
ejpam-3948	88	1	+	+	CCONJ
ejpam-3948	88	2	1	1	NUM
ejpam-3948	88	3	≡	≡	PROPN
ejpam-3948	88	4	0	0	NUM
ejpam-3948	88	5	(	(	PUNCT
ejpam-3948	88	6	mod	mod	PROPN
ejpam-3948	88	7	4	4	NUM
ejpam-3948	88	8	)	)	PUNCT
ejpam-3948	88	9	.	.	PUNCT
ejpam-3948	89	1	thus	thus	ADV
ejpam-3948	89	2	,	,	PUNCT
ejpam-3948	89	3	for	for	ADP
ejpam-3948	89	4	any	any	DET
ejpam-3948	89	5	positive	positive	ADJ
ejpam-3948	89	6	integer	integer	NOUN
ejpam-3948	89	7	y	y	PROPN
ejpam-3948	89	8	,	,	PUNCT
ejpam-3948	89	9	mx	mx	PROPN
ejpam-3948	90	1	p	p	PROPN
ejpam-3948	90	2	+	+	X
ejpam-3948	90	3	(	(	PUNCT
ejpam-3948	90	4	mq	mq	PROPN
ejpam-3948	90	5	+	+	CCONJ
ejpam-3948	90	6	1)y	1)y	NUM
ejpam-3948	90	7	≡	≡	PROPN
ejpam-3948	90	8	{	{	PUNCT
ejpam-3948	90	9	3	3	X
ejpam-3948	90	10	(	(	PUNCT
ejpam-3948	90	11	mod	mod	NOUN
ejpam-3948	90	12	4	4	NUM
ejpam-3948	90	13	)	)	PUNCT
ejpam-3948	90	14	for	for	ADP
ejpam-3948	90	15	odd	odd	ADJ
ejpam-3948	90	16	x	x	SYM
ejpam-3948	90	17	1	1	NUM
ejpam-3948	90	18	(	(	PUNCT
ejpam-3948	90	19	mod	mod	NOUN
ejpam-3948	90	20	4	4	NUM
ejpam-3948	90	21	)	)	PUNCT
ejpam-3948	90	22	for	for	ADP
ejpam-3948	90	23	even	even	ADV
ejpam-3948	90	24	x.	x.	NOUN
ejpam-3948	90	25	because	because	SCONJ
ejpam-3948	90	26	z2	z2	PROPN
ejpam-3948	90	27	≡	≡	PROPN
ejpam-3948	90	28	1	1	NUM
ejpam-3948	90	29	(	(	PUNCT
ejpam-3948	90	30	mod	mod	NOUN
ejpam-3948	90	31	4	4	NUM
ejpam-3948	90	32	)	)	PUNCT
ejpam-3948	90	33	,	,	PUNCT
ejpam-3948	90	34	we	we	PRON
ejpam-3948	90	35	can	can	AUX
ejpam-3948	90	36	say	say	VERB
ejpam-3948	90	37	that	that	SCONJ
ejpam-3948	90	38	x	x	PRON
ejpam-3948	90	39	is	be	AUX
ejpam-3948	90	40	even	even	ADV
ejpam-3948	90	41	.	.	PUNCT
ejpam-3948	91	1	thus	thus	ADV
ejpam-3948	91	2	,	,	PUNCT
ejpam-3948	91	3	there	there	PRON
ejpam-3948	91	4	exists	exist	VERB
ejpam-3948	91	5	a	a	DET
ejpam-3948	91	6	positive	positive	ADJ
ejpam-3948	91	7	integer	integer	NOUN
ejpam-3948	91	8	k	k	PROPN
ejpam-3948	91	9	such	such	ADJ
ejpam-3948	91	10	that	that	SCONJ
ejpam-3948	91	11	x	x	PROPN
ejpam-3948	91	12	=	=	PUNCT
ejpam-3948	91	13	2k	2k	NUM
ejpam-3948	91	14	.	.	PUNCT
ejpam-3948	92	1	so	so	ADV
ejpam-3948	92	2	,	,	PUNCT
ejpam-3948	92	3	m2k	m2k	PROPN
ejpam-3948	92	4	p	p	X
ejpam-3948	92	5	+	+	X
ejpam-3948	92	6	(	(	PUNCT
ejpam-3948	92	7	mq	mq	PROPN
ejpam-3948	92	8	+	+	CCONJ
ejpam-3948	92	9	1)y	1)y	NUM
ejpam-3948	92	10	=	=	SYM
ejpam-3948	92	11	z2	z2	PROPN
ejpam-3948	92	12	.	.	PUNCT
ejpam-3948	93	1	by	by	ADP
ejpam-3948	93	2	substituting	substitute	VERB
ejpam-3948	93	3	mq	mq	NOUN
ejpam-3948	93	4	=	=	SYM
ejpam-3948	93	5	2q	2q	NUM
ejpam-3948	93	6	−	−	NOUN
ejpam-3948	93	7	1	1	NUM
ejpam-3948	93	8	for	for	ADP
ejpam-3948	93	9	some	some	DET
ejpam-3948	93	10	prime	prime	ADJ
ejpam-3948	93	11	q	q	NOUN
ejpam-3948	93	12	,	,	PUNCT
ejpam-3948	93	13	we	we	PRON
ejpam-3948	93	14	get	get	VERB
ejpam-3948	93	15	the	the	DET
ejpam-3948	93	16	equation	equation	NOUN
ejpam-3948	93	17	(	(	PUNCT
ejpam-3948	93	18	mp	mp	PROPN
ejpam-3948	93	19	)	)	PUNCT
ejpam-3948	93	20	2k	2k	NOUN
ejpam-3948	93	21	+	+	CCONJ
ejpam-3948	93	22	2qy	2qy	NOUN
ejpam-3948	93	23	=	=	SYM
ejpam-3948	93	24	z2	z2	PROPN
ejpam-3948	93	25	.	.	PUNCT
ejpam-3948	94	1	it	it	PRON
ejpam-3948	94	2	can	can	AUX
ejpam-3948	94	3	be	be	AUX
ejpam-3948	94	4	expressed	express	VERB
ejpam-3948	94	5	as	as	ADP
ejpam-3948	94	6	z2−	z2−	PROPN
ejpam-3948	94	7	(	(	PUNCT
ejpam-3948	94	8	mp	mp	NOUN
ejpam-3948	94	9	)	)	PUNCT
ejpam-3948	94	10	2k	2k	NOUN
ejpam-3948	94	11	=	=	SYM
ejpam-3948	94	12	2qy	2qy	NOUN
ejpam-3948	94	13	.	.	PUNCT
ejpam-3948	95	1	factoring	factor	VERB
ejpam-3948	95	2	the	the	DET
ejpam-3948	95	3	left	left	ADJ
ejpam-3948	95	4	side	side	NOUN
ejpam-3948	95	5	of	of	ADP
ejpam-3948	95	6	the	the	DET
ejpam-3948	95	7	equation	equation	NOUN
ejpam-3948	95	8	leads	lead	VERB
ejpam-3948	95	9	to	to	ADP
ejpam-3948	95	10	(	(	PUNCT
ejpam-3948	95	11	z	z	PROPN
ejpam-3948	95	12	+	+	PROPN
ejpam-3948	95	13	mk	mk	PROPN
ejpam-3948	95	14	p	p	NOUN
ejpam-3948	95	15	)	)	PUNCT
ejpam-3948	95	16	(	(	PUNCT
ejpam-3948	95	17	z	z	NOUN
ejpam-3948	95	18	−mk	−mk	PROPN
ejpam-3948	95	19	p	p	NOUN
ejpam-3948	95	20	)	)	PUNCT
ejpam-3948	96	1	=	=	SYM
ejpam-3948	96	2	2qy	2qy	NOUN
ejpam-3948	96	3	.	.	PUNCT
ejpam-3948	97	1	w.	w.	PROPN
ejpam-3948	97	2	s.	s.	PROPN
ejpam-3948	97	3	gayo	gayo	PROPN
ejpam-3948	97	4	,	,	PUNCT
ejpam-3948	97	5	jr	jr	PROPN
ejpam-3948	97	6	.	.	PROPN
ejpam-3948	97	7	,	,	PUNCT
ejpam-3948	97	8	j.	j.	PROPN
ejpam-3948	97	9	b.	b.	PROPN
ejpam-3948	97	10	bacani	bacani	PROPN
ejpam-3948	97	11	/	/	SYM
ejpam-3948	97	12	eur	eur	PROPN
ejpam-3948	97	13	.	.	PUNCT
ejpam-3948	98	1	j.	j.	PROPN
ejpam-3948	98	2	pure	pure	PROPN
ejpam-3948	98	3	appl	appl	PROPN
ejpam-3948	98	4	.	.	PROPN
ejpam-3948	98	5	math	math	PROPN
ejpam-3948	98	6	,	,	PUNCT
ejpam-3948	98	7	14	14	NUM
ejpam-3948	98	8	(	(	PUNCT
ejpam-3948	98	9	2	2	NUM
ejpam-3948	98	10	)	)	PUNCT
ejpam-3948	98	11	(	(	PUNCT
ejpam-3948	98	12	2021	2021	NUM
ejpam-3948	98	13	)	)	PUNCT
ejpam-3948	98	14	,	,	PUNCT
ejpam-3948	98	15	396	396	NUM
ejpam-3948	98	16	-	-	SYM
ejpam-3948	98	17	403	403	NUM
ejpam-3948	98	18	399	399	NUM
ejpam-3948	98	19	there	there	PRON
ejpam-3948	98	20	exist	exist	VERB
ejpam-3948	98	21	nonnegative	nonnegative	ADJ
ejpam-3948	98	22	integers	integer	NOUN
ejpam-3948	98	23	α	α	NOUN
ejpam-3948	98	24	and	and	CCONJ
ejpam-3948	98	25	β	β	X
ejpam-3948	98	26	with	with	ADP
ejpam-3948	98	27	α	α	PROPN
ejpam-3948	98	28	>	>	X
ejpam-3948	98	29	β	β	X
ejpam-3948	98	30	and	and	CCONJ
ejpam-3948	98	31	α	α	PROPN
ejpam-3948	98	32	+	+	X
ejpam-3948	98	33	β	β	X
ejpam-3948	98	34	=	=	SYM
ejpam-3948	98	35	2qy	2qy	NOUN
ejpam-3948	98	36	such	such	ADJ
ejpam-3948	98	37	that	that	SCONJ
ejpam-3948	98	38	(	(	PUNCT
ejpam-3948	98	39	z	z	NOUN
ejpam-3948	98	40	+	+	NUM
ejpam-3948	98	41	mk	mk	PROPN
ejpam-3948	98	42	p	p	NOUN
ejpam-3948	98	43	)	)	PUNCT
ejpam-3948	98	44	(	(	PUNCT
ejpam-3948	98	45	z	z	NOUN
ejpam-3948	98	46	−mk	−mk	PROPN
ejpam-3948	98	47	p	p	NOUN
ejpam-3948	98	48	)	)	PUNCT
ejpam-3948	98	49	=	=	SYM
ejpam-3948	98	50	2α+β	2α+β	NUM
ejpam-3948	98	51	.	.	PUNCT
ejpam-3948	99	1	this	this	PRON
ejpam-3948	99	2	implies	imply	VERB
ejpam-3948	99	3	that	that	SCONJ
ejpam-3948	99	4	(	(	PUNCT
ejpam-3948	99	5	z	z	X
ejpam-3948	99	6	+	+	NUM
ejpam-3948	99	7	mk	mk	PROPN
ejpam-3948	99	8	p	p	NOUN
ejpam-3948	99	9	)	)	PUNCT
ejpam-3948	100	1	=	=	SYM
ejpam-3948	100	2	2α	2α	NOUN
ejpam-3948	100	3	and	and	CCONJ
ejpam-3948	100	4	(	(	PUNCT
ejpam-3948	100	5	z	z	NOUN
ejpam-3948	100	6	−mk	−mk	PROPN
ejpam-3948	100	7	p	p	NOUN
ejpam-3948	100	8	)	)	PUNCT
ejpam-3948	101	1	=	=	SYM
ejpam-3948	101	2	2β	2β	NOUN
ejpam-3948	101	3	,	,	PUNCT
ejpam-3948	101	4	which	which	PRON
ejpam-3948	101	5	gives	give	VERB
ejpam-3948	101	6	2mk	2mk	ADJ
ejpam-3948	101	7	p	p	NOUN
ejpam-3948	101	8	=	=	NOUN
ejpam-3948	101	9	2β(2α−β	2β(2α−β	NUM
ejpam-3948	101	10	−	−	NOUN
ejpam-3948	101	11	1	1	NUM
ejpam-3948	101	12	)	)	PUNCT
ejpam-3948	101	13	.	.	PUNCT
ejpam-3948	102	1	equating	equate	VERB
ejpam-3948	102	2	the	the	DET
ejpam-3948	102	3	odd	odd	ADJ
ejpam-3948	102	4	parts	part	NOUN
ejpam-3948	102	5	and	and	CCONJ
ejpam-3948	102	6	the	the	DET
ejpam-3948	102	7	even	even	ADJ
ejpam-3948	102	8	parts	part	NOUN
ejpam-3948	102	9	leads	lead	VERB
ejpam-3948	102	10	to	to	ADP
ejpam-3948	102	11	the	the	DET
ejpam-3948	102	12	system	system	NOUN
ejpam-3948	102	13	{	{	PUNCT
ejpam-3948	102	14	2β	2β	NOUN
ejpam-3948	102	15	=	=	SYM
ejpam-3948	102	16	2	2	NUM
ejpam-3948	102	17	2α−β	2α−β	NUM
ejpam-3948	102	18	−	−	NOUN
ejpam-3948	102	19	1	1	NUM
ejpam-3948	102	20	=	=	SYM
ejpam-3948	102	21	mk	mk	PROPN
ejpam-3948	102	22	p	p	NOUN
ejpam-3948	102	23	.	.	PUNCT
ejpam-3948	103	1	the	the	DET
ejpam-3948	103	2	first	first	ADJ
ejpam-3948	103	3	equation	equation	NOUN
ejpam-3948	103	4	implies	imply	VERB
ejpam-3948	103	5	that	that	SCONJ
ejpam-3948	103	6	β	β	NOUN
ejpam-3948	103	7	=	=	SYM
ejpam-3948	103	8	1	1	X
ejpam-3948	103	9	.	.	PUNCT
ejpam-3948	103	10	then	then	ADV
ejpam-3948	103	11	,	,	PUNCT
ejpam-3948	103	12	the	the	DET
ejpam-3948	103	13	second	second	ADJ
ejpam-3948	103	14	equation	equation	NOUN
ejpam-3948	103	15	becomes	become	VERB
ejpam-3948	103	16	2α−1	2α−1	NUM
ejpam-3948	103	17	−mk	−mk	NOUN
ejpam-3948	103	18	p	p	NOUN
ejpam-3948	103	19	=	=	NOUN
ejpam-3948	103	20	1	1	X
ejpam-3948	103	21	.	.	PUNCT
ejpam-3948	104	1	if	if	SCONJ
ejpam-3948	104	2	k	k	PROPN
ejpam-3948	104	3	>	>	X
ejpam-3948	104	4	1	1	NUM
ejpam-3948	104	5	and	and	CCONJ
ejpam-3948	104	6	α	α	X
ejpam-3948	104	7	>	>	X
ejpam-3948	104	8	2	2	NUM
ejpam-3948	104	9	,	,	PUNCT
ejpam-3948	104	10	there	there	PRON
ejpam-3948	104	11	is	be	VERB
ejpam-3948	104	12	no	no	DET
ejpam-3948	104	13	solution	solution	NOUN
ejpam-3948	104	14	by	by	ADP
ejpam-3948	104	15	mihailescu	mihailescu	PROPN
ejpam-3948	104	16	’s	’s	PART
ejpam-3948	104	17	theorem	theorem	NOUN
ejpam-3948	104	18	.	.	PUNCT
ejpam-3948	105	1	if	if	SCONJ
ejpam-3948	105	2	α	α	PRON
ejpam-3948	105	3	=	=	SYM
ejpam-3948	105	4	2	2	NUM
ejpam-3948	105	5	,	,	PUNCT
ejpam-3948	105	6	then	then	ADV
ejpam-3948	105	7	mk	mk	PROPN
ejpam-3948	105	8	p	p	NOUN
ejpam-3948	106	1	=	=	PROPN
ejpam-3948	107	1	1	1	X
ejpam-3948	107	2	.	.	PUNCT
ejpam-3948	108	1	this	this	PRON
ejpam-3948	108	2	gives	give	VERB
ejpam-3948	108	3	the	the	DET
ejpam-3948	108	4	value	value	NOUN
ejpam-3948	108	5	k	k	PROPN
ejpam-3948	108	6	=	=	SYM
ejpam-3948	108	7	0	0	PROPN
ejpam-3948	108	8	,	,	PUNCT
ejpam-3948	108	9	a	a	DET
ejpam-3948	108	10	contradiction	contradiction	NOUN
ejpam-3948	108	11	to	to	ADP
ejpam-3948	108	12	k	k	NOUN
ejpam-3948	108	13	being	be	AUX
ejpam-3948	108	14	positive	positive	ADJ
ejpam-3948	108	15	.	.	PUNCT
ejpam-3948	109	1	if	if	SCONJ
ejpam-3948	109	2	k	k	PROPN
ejpam-3948	109	3	=	=	SYM
ejpam-3948	109	4	1	1	NUM
ejpam-3948	109	5	,	,	PUNCT
ejpam-3948	109	6	then	then	ADV
ejpam-3948	109	7	x	x	NOUN
ejpam-3948	109	8	=	=	SYM
ejpam-3948	109	9	2	2	NUM
ejpam-3948	109	10	,	,	PUNCT
ejpam-3948	109	11	z	z	NOUN
ejpam-3948	109	12	=	=	SYM
ejpam-3948	109	13	2p+1	2p+1	PROPN
ejpam-3948	109	14	and	and	CCONJ
ejpam-3948	109	15	2α−1−mp	2α−1−mp	NUM
ejpam-3948	109	16	=	=	SYM
ejpam-3948	109	17	1	1	NUM
ejpam-3948	109	18	or	or	CCONJ
ejpam-3948	109	19	in	in	ADP
ejpam-3948	109	20	equivalent	equivalent	ADJ
ejpam-3948	109	21	form	form	NOUN
ejpam-3948	109	22	2α−1	2α−1	NUM
ejpam-3948	109	23	=	=	SYM
ejpam-3948	109	24	2p	2p	NOUN
ejpam-3948	109	25	.	.	PUNCT
ejpam-3948	110	1	this	this	PRON
ejpam-3948	110	2	implies	imply	VERB
ejpam-3948	110	3	that	that	SCONJ
ejpam-3948	110	4	α	α	PROPN
ejpam-3948	110	5	=	=	PUNCT
ejpam-3948	110	6	p+1	p+1	NOUN
ejpam-3948	110	7	.	.	PUNCT
ejpam-3948	111	1	since	since	SCONJ
ejpam-3948	111	2	α	α	PROPN
ejpam-3948	111	3	+	+	X
ejpam-3948	111	4	β	β	X
ejpam-3948	111	5	=	=	SYM
ejpam-3948	111	6	qy	qy	PROPN
ejpam-3948	111	7	and	and	CCONJ
ejpam-3948	111	8	β	β	X
ejpam-3948	111	9	=	=	SYM
ejpam-3948	111	10	1	1	NUM
ejpam-3948	111	11	,	,	PUNCT
ejpam-3948	111	12	it	it	PRON
ejpam-3948	111	13	follows	follow	VERB
ejpam-3948	111	14	that	that	SCONJ
ejpam-3948	111	15	p	p	PROPN
ejpam-3948	111	16	+	+	NOUN
ejpam-3948	111	17	2	2	NUM
ejpam-3948	111	18	=	=	SYM
ejpam-3948	111	19	qy	qy	NOUN
ejpam-3948	111	20	or	or	CCONJ
ejpam-3948	111	21	y	y	PROPN
ejpam-3948	111	22	=	=	PRON
ejpam-3948	111	23	p+	p+	PROPN
ejpam-3948	111	24	2	2	NUM
ejpam-3948	111	25	q	q	NOUN
ejpam-3948	111	26	.	.	PUNCT
ejpam-3948	112	1	if	if	SCONJ
ejpam-3948	112	2	q|p	q|p	ADV
ejpam-3948	112	3	+	+	CCONJ
ejpam-3948	112	4	2	2	NUM
ejpam-3948	112	5	,	,	PUNCT
ejpam-3948	112	6	then	then	ADV
ejpam-3948	112	7	we	we	PRON
ejpam-3948	112	8	have	have	VERB
ejpam-3948	112	9	the	the	DET
ejpam-3948	112	10	set	set	NOUN
ejpam-3948	112	11	of	of	ADP
ejpam-3948	112	12	solutions	solution	NOUN
ejpam-3948	112	13	{	{	PUNCT
ejpam-3948	112	14	(	(	PUNCT
ejpam-3948	112	15	mp	mp	PROPN
ejpam-3948	112	16	,	,	PUNCT
ejpam-3948	112	17	mq	mq	PROPN
ejpam-3948	112	18	,	,	PUNCT
ejpam-3948	112	19	x	x	PROPN
ejpam-3948	112	20	,	,	PUNCT
ejpam-3948	112	21	y	y	PROPN
ejpam-3948	112	22	,	,	PUNCT
ejpam-3948	112	23	z	z	NOUN
ejpam-3948	112	24	)	)	PUNCT
ejpam-3948	112	25	}	}	PUNCT
ejpam-3948	112	26	=	=	SYM
ejpam-3948	112	27	{	{	PUNCT
ejpam-3948	112	28	(	(	PUNCT
ejpam-3948	112	29	mp	mp	PROPN
ejpam-3948	112	30	,	,	PUNCT
ejpam-3948	112	31	mq	mq	PROPN
ejpam-3948	112	32	,	,	PUNCT
ejpam-3948	112	33	2	2	NUM
ejpam-3948	112	34	,	,	PUNCT
ejpam-3948	112	35	p+	p+	VERB
ejpam-3948	112	36	2	2	NUM
ejpam-3948	112	37	q	q	NOUN
ejpam-3948	112	38	,	,	PUNCT
ejpam-3948	112	39	2p	2p	NUM
ejpam-3948	112	40	+	+	SYM
ejpam-3948	112	41	1	1	NUM
ejpam-3948	112	42	)	)	PUNCT
ejpam-3948	112	43	}	}	PUNCT
ejpam-3948	112	44	.	.	PUNCT
ejpam-3948	113	1	by	by	ADP
ejpam-3948	113	2	theorem	theorem	NOUN
ejpam-3948	113	3	1	1	NUM
ejpam-3948	113	4	,	,	PUNCT
ejpam-3948	113	5	the	the	DET
ejpam-3948	113	6	positive	positive	ADJ
ejpam-3948	113	7	integer	integer	NOUN
ejpam-3948	113	8	solutions	solution	NOUN
ejpam-3948	113	9	of	of	ADP
ejpam-3948	113	10	mx	mx	PROPN
ejpam-3948	113	11	p	p	PROPN
ejpam-3948	113	12	+	+	X
ejpam-3948	113	13	(	(	PUNCT
ejpam-3948	113	14	mq	mq	PROPN
ejpam-3948	113	15	+	+	NOUN
ejpam-3948	113	16	1)y	1)y	NUM
ejpam-3948	113	17	=	=	SYM
ejpam-3948	113	18	z2	z2	PROPN
ejpam-3948	113	19	are	be	AUX
ejpam-3948	113	20	given	give	VERB
ejpam-3948	113	21	by	by	ADP
ejpam-3948	113	22	(	(	PUNCT
ejpam-3948	113	23	mp	mp	PROPN
ejpam-3948	113	24	,	,	PUNCT
ejpam-3948	113	25	mq	mq	PROPN
ejpam-3948	113	26	,	,	PUNCT
ejpam-3948	113	27	x	x	PROPN
ejpam-3948	113	28	,	,	PUNCT
ejpam-3948	113	29	y	y	PROPN
ejpam-3948	113	30	,	,	PUNCT
ejpam-3948	113	31	z	z	NOUN
ejpam-3948	113	32	)	)	PUNCT
ejpam-3948	113	33	=	=	SYM
ejpam-3948	113	34	(	(	PUNCT
ejpam-3948	113	35	mp	mp	PROPN
ejpam-3948	113	36	,	,	PUNCT
ejpam-3948	113	37	mq	mq	PROPN
ejpam-3948	113	38	,	,	PUNCT
ejpam-3948	113	39	2	2	NUM
ejpam-3948	113	40	,	,	PUNCT
ejpam-3948	113	41	p+	p+	VERB
ejpam-3948	113	42	2	2	NUM
ejpam-3948	113	43	q	q	NOUN
ejpam-3948	113	44	,	,	PUNCT
ejpam-3948	113	45	2p	2p	NUM
ejpam-3948	113	46	+	+	CCONJ
ejpam-3948	113	47	1	1	NUM
ejpam-3948	113	48	)	)	PUNCT
ejpam-3948	113	49	.	.	PUNCT
ejpam-3948	114	1	given	give	VERB
ejpam-3948	114	2	the	the	DET
ejpam-3948	114	3	mersenne	mersenne	PROPN
ejpam-3948	114	4	prime	prime	PROPN
ejpam-3948	114	5	mp	mp	PROPN
ejpam-3948	114	6	,	,	PUNCT
ejpam-3948	114	7	the	the	DET
ejpam-3948	114	8	solutions	solution	NOUN
ejpam-3948	114	9	can	can	AUX
ejpam-3948	114	10	be	be	AUX
ejpam-3948	114	11	found	find	VERB
ejpam-3948	114	12	by	by	ADP
ejpam-3948	114	13	finding	find	VERB
ejpam-3948	114	14	all	all	DET
ejpam-3948	114	15	primes	prime	NOUN
ejpam-3948	114	16	q	q	NOUN
ejpam-3948	114	17	that	that	PRON
ejpam-3948	114	18	divide	divide	VERB
ejpam-3948	114	19	p	p	NOUN
ejpam-3948	114	20	+	+	NOUN
ejpam-3948	114	21	2	2	X
ejpam-3948	114	22	.	.	X
ejpam-3948	115	1	it	it	PRON
ejpam-3948	115	2	should	should	AUX
ejpam-3948	115	3	be	be	AUX
ejpam-3948	115	4	checked	check	VERB
ejpam-3948	115	5	also	also	ADV
ejpam-3948	115	6	if	if	SCONJ
ejpam-3948	115	7	the	the	DET
ejpam-3948	115	8	corresponding	corresponding	ADJ
ejpam-3948	115	9	mersenne	mersenne	NOUN
ejpam-3948	115	10	number	number	NOUN
ejpam-3948	115	11	2q−	2q−	PROPN
ejpam-3948	115	12	1	1	NUM
ejpam-3948	115	13	is	be	AUX
ejpam-3948	115	14	a	a	DET
ejpam-3948	115	15	mersenne	mersenne	NOUN
ejpam-3948	115	16	prime	prime	NOUN
ejpam-3948	115	17	.	.	PUNCT
ejpam-3948	116	1	the	the	DET
ejpam-3948	116	2	number	number	NOUN
ejpam-3948	116	3	of	of	ADP
ejpam-3948	116	4	solutions	solution	NOUN
ejpam-3948	116	5	depends	depend	VERB
ejpam-3948	116	6	on	on	ADP
ejpam-3948	116	7	how	how	SCONJ
ejpam-3948	116	8	many	many	ADJ
ejpam-3948	116	9	primes	prime	NOUN
ejpam-3948	116	10	q	q	NOUN
ejpam-3948	116	11	that	that	DET
ejpam-3948	116	12	divide	divide	NOUN
ejpam-3948	116	13	p+	p+	VERB
ejpam-3948	116	14	2	2	NUM
ejpam-3948	116	15	such	such	ADJ
ejpam-3948	116	16	that	that	SCONJ
ejpam-3948	116	17	mq	mq	PROPN
ejpam-3948	116	18	is	be	AUX
ejpam-3948	116	19	a	a	DET
ejpam-3948	116	20	mersenne	mersenne	NOUN
ejpam-3948	116	21	prime	prime	NOUN
ejpam-3948	116	22	.	.	PUNCT
ejpam-3948	117	1	let	let	VERB
ejpam-3948	117	2	us	we	PRON
ejpam-3948	117	3	take	take	VERB
ejpam-3948	117	4	take	take	VERB
ejpam-3948	117	5	the	the	DET
ejpam-3948	117	6	case	case	NOUN
ejpam-3948	117	7	of	of	ADP
ejpam-3948	117	8	m3	m3	PROPN
ejpam-3948	117	9	=	=	SYM
ejpam-3948	117	10	7	7	NUM
ejpam-3948	117	11	and	and	CCONJ
ejpam-3948	117	12	m13	m13	NUM
ejpam-3948	117	13	=	=	SYM
ejpam-3948	117	14	8191	8191	NUM
ejpam-3948	117	15	.	.	PUNCT
ejpam-3948	118	1	example	example	NOUN
ejpam-3948	118	2	1	1	NUM
ejpam-3948	118	3	.	.	PUNCT
ejpam-3948	119	1	find	find	VERB
ejpam-3948	119	2	the	the	DET
ejpam-3948	119	3	positive	positive	ADJ
ejpam-3948	119	4	integer	integer	NOUN
ejpam-3948	119	5	solution	solution	NOUN
ejpam-3948	119	6	of	of	ADP
ejpam-3948	119	7	7x	7x	NUM
ejpam-3948	119	8	+	+	CCONJ
ejpam-3948	119	9	(	(	PUNCT
ejpam-3948	119	10	mq	mq	PROPN
ejpam-3948	119	11	+	+	CCONJ
ejpam-3948	119	12	1)y	1)y	NUM
ejpam-3948	119	13	=	=	SYM
ejpam-3948	119	14	z2	z2	PROPN
ejpam-3948	119	15	,	,	PUNCT
ejpam-3948	119	16	where	where	SCONJ
ejpam-3948	119	17	mq	mq	PROPN
ejpam-3948	119	18	is	be	AUX
ejpam-3948	119	19	a	a	DET
ejpam-3948	119	20	mersenne	mersenne	NOUN
ejpam-3948	119	21	prime	prime	NOUN
ejpam-3948	119	22	.	.	PUNCT
ejpam-3948	120	1	solution	solution	NOUN
ejpam-3948	120	2	.	.	PUNCT
ejpam-3948	121	1	theorem	theorem	ADJ
ejpam-3948	121	2	1	1	NUM
ejpam-3948	121	3	asserts	assert	VERB
ejpam-3948	121	4	that	that	SCONJ
ejpam-3948	121	5	(	(	PUNCT
ejpam-3948	121	6	mq	mq	NOUN
ejpam-3948	121	7	,	,	PUNCT
ejpam-3948	121	8	x	x	PROPN
ejpam-3948	121	9	,	,	PUNCT
ejpam-3948	121	10	y	y	PROPN
ejpam-3948	121	11	,	,	PUNCT
ejpam-3948	121	12	z	z	NOUN
ejpam-3948	121	13	)	)	PUNCT
ejpam-3948	121	14	=	=	SYM
ejpam-3948	121	15	(	(	PUNCT
ejpam-3948	121	16	mq	mq	PROPN
ejpam-3948	121	17	,	,	PUNCT
ejpam-3948	121	18	2	2	NUM
ejpam-3948	121	19	,	,	PUNCT
ejpam-3948	121	20	4	4	NUM
ejpam-3948	121	21	q	q	NOUN
ejpam-3948	121	22	,	,	PUNCT
ejpam-3948	121	23	9	9	NUM
ejpam-3948	121	24	)	)	PUNCT
ejpam-3948	121	25	if	if	SCONJ
ejpam-3948	121	26	q|4	q|4	PROPN
ejpam-3948	121	27	.	.	PUNCT
ejpam-3948	122	1	the	the	DET
ejpam-3948	122	2	only	only	ADJ
ejpam-3948	122	3	prime	prime	NOUN
ejpam-3948	122	4	q	q	NOUN
ejpam-3948	122	5	that	that	PRON
ejpam-3948	122	6	divides	divide	VERB
ejpam-3948	122	7	4	4	NUM
ejpam-3948	122	8	is	be	AUX
ejpam-3948	122	9	2	2	NUM
ejpam-3948	122	10	,	,	PUNCT
ejpam-3948	122	11	and	and	CCONJ
ejpam-3948	122	12	it	it	PRON
ejpam-3948	122	13	happens	happen	VERB
ejpam-3948	122	14	that	that	SCONJ
ejpam-3948	122	15	m2	m2	PROPN
ejpam-3948	122	16	=	=	PRON
ejpam-3948	122	17	3	3	NUM
ejpam-3948	122	18	is	be	AUX
ejpam-3948	122	19	a	a	DET
ejpam-3948	122	20	mersenne	mersenne	NOUN
ejpam-3948	122	21	prime	prime	NOUN
ejpam-3948	122	22	.	.	PUNCT
ejpam-3948	123	1	in	in	ADP
ejpam-3948	123	2	conclusion	conclusion	NOUN
ejpam-3948	123	3	,	,	PUNCT
ejpam-3948	123	4	(	(	PUNCT
ejpam-3948	123	5	3	3	NUM
ejpam-3948	123	6	,	,	PUNCT
ejpam-3948	123	7	2	2	NUM
ejpam-3948	123	8	,	,	PUNCT
ejpam-3948	123	9	2	2	NUM
ejpam-3948	123	10	,	,	PUNCT
ejpam-3948	123	11	8)	8)	NUM
ejpam-3948	123	12	is	be	AUX
ejpam-3948	123	13	the	the	DET
ejpam-3948	123	14	unique	unique	ADJ
ejpam-3948	123	15	positive	positive	ADJ
ejpam-3948	123	16	integer	integer	NOUN
ejpam-3948	123	17	solution	solution	NOUN
ejpam-3948	123	18	.	.	PUNCT
ejpam-3948	124	1	example	example	NOUN
ejpam-3948	125	1	2	2	NUM
ejpam-3948	125	2	.	.	PUNCT
ejpam-3948	125	3	find	find	VERB
ejpam-3948	125	4	the	the	DET
ejpam-3948	125	5	positive	positive	ADJ
ejpam-3948	125	6	integer	integer	NOUN
ejpam-3948	125	7	solution	solution	NOUN
ejpam-3948	125	8	of	of	ADP
ejpam-3948	125	9	8191x	8191x	NUM
ejpam-3948	125	10	+	+	CCONJ
ejpam-3948	125	11	(	(	PUNCT
ejpam-3948	125	12	mq	mq	PROPN
ejpam-3948	125	13	+	+	NOUN
ejpam-3948	125	14	1)y	1)y	NUM
ejpam-3948	125	15	=	=	SYM
ejpam-3948	125	16	z2	z2	PROPN
ejpam-3948	125	17	,	,	PUNCT
ejpam-3948	125	18	where	where	SCONJ
ejpam-3948	125	19	mq	mq	PROPN
ejpam-3948	125	20	is	be	AUX
ejpam-3948	125	21	a	a	DET
ejpam-3948	125	22	mersenne	mersenne	NOUN
ejpam-3948	125	23	prime	prime	NOUN
ejpam-3948	125	24	.	.	PUNCT
ejpam-3948	126	1	solution	solution	NOUN
ejpam-3948	126	2	.	.	PUNCT
ejpam-3948	127	1	theorem	theorem	ADJ
ejpam-3948	127	2	1	1	NUM
ejpam-3948	127	3	guarantees	guarantee	NOUN
ejpam-3948	127	4	that	that	SCONJ
ejpam-3948	127	5	(	(	PUNCT
ejpam-3948	127	6	mq	mq	NOUN
ejpam-3948	127	7	,	,	PUNCT
ejpam-3948	127	8	x	x	PROPN
ejpam-3948	127	9	,	,	PUNCT
ejpam-3948	127	10	y	y	PROPN
ejpam-3948	127	11	,	,	PUNCT
ejpam-3948	127	12	z	z	NOUN
ejpam-3948	127	13	)	)	PUNCT
ejpam-3948	127	14	=	=	SYM
ejpam-3948	127	15	(	(	PUNCT
ejpam-3948	127	16	mq	mq	PROPN
ejpam-3948	127	17	,	,	PUNCT
ejpam-3948	127	18	2	2	NUM
ejpam-3948	127	19	,	,	PUNCT
ejpam-3948	127	20	15	15	NUM
ejpam-3948	127	21	q	q	NOUN
ejpam-3948	127	22	,	,	PUNCT
ejpam-3948	127	23	8193	8193	NUM
ejpam-3948	127	24	)	)	PUNCT
ejpam-3948	127	25	if	if	SCONJ
ejpam-3948	127	26	q|15	q|15	VERB
ejpam-3948	127	27	.	.	PUNCT
ejpam-3948	128	1	the	the	DET
ejpam-3948	128	2	primes	prime	NOUN
ejpam-3948	128	3	that	that	PRON
ejpam-3948	128	4	divide	divide	VERB
ejpam-3948	128	5	15	15	NUM
ejpam-3948	128	6	are	be	AUX
ejpam-3948	128	7	3	3	NUM
ejpam-3948	128	8	and	and	CCONJ
ejpam-3948	128	9	5	5	NUM
ejpam-3948	128	10	.	.	X
ejpam-3948	129	1	if	if	SCONJ
ejpam-3948	129	2	q	q	NOUN
ejpam-3948	129	3	=	=	SYM
ejpam-3948	129	4	3	3	NUM
ejpam-3948	129	5	,	,	PUNCT
ejpam-3948	129	6	then	then	ADV
ejpam-3948	129	7	y	y	PROPN
ejpam-3948	129	8	=	=	SYM
ejpam-3948	129	9	5	5	NUM
ejpam-3948	129	10	and	and	CCONJ
ejpam-3948	129	11	mq	mq	NOUN
ejpam-3948	129	12	=	=	SYM
ejpam-3948	129	13	7	7	NUM
ejpam-3948	129	14	,	,	PUNCT
ejpam-3948	129	15	a	a	DET
ejpam-3948	129	16	mersenne	mersenne	NOUN
ejpam-3948	129	17	prime	prime	NOUN
ejpam-3948	129	18	.	.	PUNCT
ejpam-3948	130	1	hence	hence	ADV
ejpam-3948	130	2	,	,	PUNCT
ejpam-3948	130	3	we	we	PRON
ejpam-3948	130	4	have	have	VERB
ejpam-3948	130	5	(	(	PUNCT
ejpam-3948	130	6	7	7	NUM
ejpam-3948	130	7	,	,	PUNCT
ejpam-3948	130	8	2	2	NUM
ejpam-3948	130	9	,	,	PUNCT
ejpam-3948	130	10	5	5	NUM
ejpam-3948	130	11	,	,	PUNCT
ejpam-3948	130	12	8193	8193	NUM
ejpam-3948	130	13	)	)	PUNCT
ejpam-3948	130	14	as	as	ADP
ejpam-3948	130	15	a	a	DET
ejpam-3948	130	16	solution	solution	NOUN
ejpam-3948	130	17	.	.	PUNCT
ejpam-3948	131	1	if	if	SCONJ
ejpam-3948	131	2	q	q	NOUN
ejpam-3948	131	3	=	=	SYM
ejpam-3948	131	4	5	5	NUM
ejpam-3948	131	5	,	,	PUNCT
ejpam-3948	131	6	then	then	ADV
ejpam-3948	131	7	y	y	PROPN
ejpam-3948	131	8	=	=	SYM
ejpam-3948	131	9	5	5	NUM
ejpam-3948	131	10	and	and	CCONJ
ejpam-3948	131	11	mq	mq	NOUN
ejpam-3948	131	12	=	=	SYM
ejpam-3948	131	13	31	31	NUM
ejpam-3948	131	14	.	.	PUNCT
ejpam-3948	132	1	thus	thus	ADV
ejpam-3948	132	2	,	,	PUNCT
ejpam-3948	132	3	(	(	PUNCT
ejpam-3948	132	4	31	31	NUM
ejpam-3948	132	5	,	,	PUNCT
ejpam-3948	132	6	2	2	NUM
ejpam-3948	132	7	,	,	PUNCT
ejpam-3948	132	8	3	3	NUM
ejpam-3948	132	9	,	,	PUNCT
ejpam-3948	132	10	8193	8193	NUM
ejpam-3948	132	11	)	)	PUNCT
ejpam-3948	132	12	is	be	AUX
ejpam-3948	132	13	another	another	DET
ejpam-3948	132	14	solution	solution	NOUN
ejpam-3948	132	15	.	.	PUNCT
ejpam-3948	133	1	let	let	VERB
ejpam-3948	133	2	us	we	PRON
ejpam-3948	133	3	also	also	ADV
ejpam-3948	133	4	solve	solve	VERB
ejpam-3948	133	5	some	some	DET
ejpam-3948	133	6	examples	example	NOUN
ejpam-3948	133	7	where	where	SCONJ
ejpam-3948	133	8	mp	mp	PROPN
ejpam-3948	133	9	and	and	CCONJ
ejpam-3948	133	10	mq	mq	PROPN
ejpam-3948	133	11	are	be	AUX
ejpam-3948	133	12	given	give	VERB
ejpam-3948	133	13	.	.	PUNCT
ejpam-3948	134	1	consider	consider	VERB
ejpam-3948	134	2	the	the	DET
ejpam-3948	134	3	equations	equation	NOUN
ejpam-3948	134	4	3x	3x	X
ejpam-3948	134	5	+	+	CCONJ
ejpam-3948	134	6	8y	8y	X
ejpam-3948	134	7	=	=	SYM
ejpam-3948	134	8	z2	z2	NOUN
ejpam-3948	134	9	and	and	CCONJ
ejpam-3948	134	10	31x	31x	NOUN
ejpam-3948	134	11	+	+	NUM
ejpam-3948	134	12	128y	128y	NOUN
ejpam-3948	134	13	=	=	SYM
ejpam-3948	134	14	z2	z2	PROPN
ejpam-3948	134	15	.	.	PUNCT
ejpam-3948	135	1	w.	w.	PROPN
ejpam-3948	135	2	s.	s.	PROPN
ejpam-3948	135	3	gayo	gayo	PROPN
ejpam-3948	135	4	,	,	PUNCT
ejpam-3948	135	5	jr	jr	PROPN
ejpam-3948	135	6	.	.	PROPN
ejpam-3948	135	7	,	,	PUNCT
ejpam-3948	135	8	j.	j.	PROPN
ejpam-3948	135	9	b.	b.	PROPN
ejpam-3948	135	10	bacani	bacani	PROPN
ejpam-3948	135	11	/	/	SYM
ejpam-3948	135	12	eur	eur	PROPN
ejpam-3948	135	13	.	.	PUNCT
ejpam-3948	136	1	j.	j.	PROPN
ejpam-3948	136	2	pure	pure	PROPN
ejpam-3948	136	3	appl	appl	PROPN
ejpam-3948	136	4	.	.	PROPN
ejpam-3948	136	5	math	math	PROPN
ejpam-3948	136	6	,	,	PUNCT
ejpam-3948	136	7	14	14	NUM
ejpam-3948	136	8	(	(	PUNCT
ejpam-3948	136	9	2	2	NUM
ejpam-3948	136	10	)	)	PUNCT
ejpam-3948	136	11	(	(	PUNCT
ejpam-3948	136	12	2021	2021	NUM
ejpam-3948	136	13	)	)	PUNCT
ejpam-3948	136	14	,	,	PUNCT
ejpam-3948	136	15	396	396	NUM
ejpam-3948	136	16	-	-	SYM
ejpam-3948	136	17	403	403	NUM
ejpam-3948	136	18	400	400	NUM
ejpam-3948	136	19	example	example	NOUN
ejpam-3948	136	20	3	3	NUM
ejpam-3948	136	21	.	.	PUNCT
ejpam-3948	136	22	find	find	VERB
ejpam-3948	136	23	the	the	DET
ejpam-3948	136	24	positive	positive	ADJ
ejpam-3948	136	25	integer	integer	NOUN
ejpam-3948	136	26	solution	solution	NOUN
ejpam-3948	136	27	of	of	ADP
ejpam-3948	136	28	3x	3x	PROPN
ejpam-3948	136	29	+	+	CCONJ
ejpam-3948	136	30	8y	8y	ADJ
ejpam-3948	136	31	=	=	SYM
ejpam-3948	136	32	z2	z2	PROPN
ejpam-3948	136	33	.	.	PUNCT
ejpam-3948	136	34	solution	solution	NOUN
ejpam-3948	136	35	.	.	PUNCT
ejpam-3948	137	1	here	here	ADV
ejpam-3948	137	2	,	,	PUNCT
ejpam-3948	137	3	mp	mp	PROPN
ejpam-3948	137	4	=	=	SYM
ejpam-3948	137	5	3	3	NUM
ejpam-3948	137	6	,	,	PUNCT
ejpam-3948	137	7	where	where	SCONJ
ejpam-3948	137	8	p	p	NOUN
ejpam-3948	137	9	=	=	SYM
ejpam-3948	137	10	2	2	NUM
ejpam-3948	137	11	and	and	CCONJ
ejpam-3948	137	12	mq	mq	NOUN
ejpam-3948	137	13	=	=	SYM
ejpam-3948	137	14	7	7	NUM
ejpam-3948	137	15	,	,	PUNCT
ejpam-3948	137	16	where	where	SCONJ
ejpam-3948	137	17	q	q	NOUN
ejpam-3948	137	18	=	=	SYM
ejpam-3948	137	19	3	3	X
ejpam-3948	137	20	.	.	PUNCT
ejpam-3948	137	21	theorem	theorem	ADJ
ejpam-3948	137	22	1	1	NUM
ejpam-3948	137	23	guarantees	guarantee	VERB
ejpam-3948	137	24	that	that	SCONJ
ejpam-3948	137	25	a	a	DET
ejpam-3948	137	26	positive	positive	ADJ
ejpam-3948	137	27	integer	integer	NOUN
ejpam-3948	137	28	solution	solution	NOUN
ejpam-3948	137	29	exists	exist	VERB
ejpam-3948	137	30	if	if	SCONJ
ejpam-3948	137	31	q|p	q|p	ADJ
ejpam-3948	137	32	+	+	CCONJ
ejpam-3948	137	33	2	2	X
ejpam-3948	137	34	.	.	PUNCT
ejpam-3948	137	35	since	since	SCONJ
ejpam-3948	137	36	3	3	NUM
ejpam-3948	137	37	4	4	NUM
ejpam-3948	137	38	,	,	PUNCT
ejpam-3948	137	39	it	it	PRON
ejpam-3948	137	40	follows	follow	VERB
ejpam-3948	137	41	that	that	SCONJ
ejpam-3948	137	42	there	there	PRON
ejpam-3948	137	43	is	be	VERB
ejpam-3948	137	44	no	no	DET
ejpam-3948	137	45	positive	positive	ADJ
ejpam-3948	137	46	integer	integer	NOUN
ejpam-3948	137	47	solution	solution	NOUN
ejpam-3948	137	48	.	.	PUNCT
ejpam-3948	138	1	example	example	NOUN
ejpam-3948	139	1	4	4	NUM
ejpam-3948	139	2	.	.	PUNCT
ejpam-3948	139	3	find	find	VERB
ejpam-3948	139	4	the	the	DET
ejpam-3948	139	5	positive	positive	ADJ
ejpam-3948	139	6	integer	integer	NOUN
ejpam-3948	139	7	solution	solution	NOUN
ejpam-3948	139	8	of	of	ADP
ejpam-3948	139	9	31x	31x	NOUN
ejpam-3948	139	10	+	+	NUM
ejpam-3948	139	11	128y	128y	NOUN
ejpam-3948	139	12	=	=	SYM
ejpam-3948	139	13	z2	z2	PROPN
ejpam-3948	139	14	.	.	PUNCT
ejpam-3948	139	15	solution	solution	NOUN
ejpam-3948	139	16	.	.	PUNCT
ejpam-3948	140	1	here	here	ADV
ejpam-3948	140	2	,	,	PUNCT
ejpam-3948	140	3	mp	mp	PROPN
ejpam-3948	140	4	=	=	PROPN
ejpam-3948	140	5	31	31	NUM
ejpam-3948	140	6	,	,	PUNCT
ejpam-3948	140	7	where	where	SCONJ
ejpam-3948	140	8	p	p	NOUN
ejpam-3948	140	9	=	=	SYM
ejpam-3948	140	10	5	5	NUM
ejpam-3948	140	11	and	and	CCONJ
ejpam-3948	140	12	mq	mq	PROPN
ejpam-3948	140	13	=	=	SYM
ejpam-3948	140	14	127	127	NUM
ejpam-3948	140	15	,	,	PUNCT
ejpam-3948	140	16	where	where	SCONJ
ejpam-3948	140	17	q	q	NOUN
ejpam-3948	140	18	=	=	SYM
ejpam-3948	140	19	7	7	X
ejpam-3948	140	20	.	.	PUNCT
ejpam-3948	140	21	theorem	theorem	ADJ
ejpam-3948	140	22	1	1	NUM
ejpam-3948	140	23	asserts	assert	VERB
ejpam-3948	140	24	that	that	SCONJ
ejpam-3948	140	25	(	(	PUNCT
ejpam-3948	140	26	x	x	X
ejpam-3948	140	27	,	,	PUNCT
ejpam-3948	140	28	y	y	PROPN
ejpam-3948	140	29	,	,	PUNCT
ejpam-3948	140	30	z	z	NOUN
ejpam-3948	140	31	)	)	PUNCT
ejpam-3948	140	32	=	=	SYM
ejpam-3948	140	33	(	(	PUNCT
ejpam-3948	140	34	2	2	NUM
ejpam-3948	140	35	,	,	PUNCT
ejpam-3948	140	36	p+	p+	VERB
ejpam-3948	140	37	2	2	NUM
ejpam-3948	140	38	q	q	NOUN
ejpam-3948	140	39	,	,	PUNCT
ejpam-3948	140	40	9	9	NUM
ejpam-3948	140	41	)	)	PUNCT
ejpam-3948	140	42	is	be	AUX
ejpam-3948	140	43	a	a	DET
ejpam-3948	140	44	solution	solution	NOUN
ejpam-3948	140	45	if	if	SCONJ
ejpam-3948	140	46	q|p+	q|p+	PROPN
ejpam-3948	140	47	2	2	NUM
ejpam-3948	140	48	.	.	PUNCT
ejpam-3948	141	1	hence	hence	ADV
ejpam-3948	141	2	,	,	PUNCT
ejpam-3948	141	3	(	(	PUNCT
ejpam-3948	141	4	x	x	X
ejpam-3948	141	5	,	,	PUNCT
ejpam-3948	141	6	y	y	PROPN
ejpam-3948	141	7	,	,	PUNCT
ejpam-3948	141	8	z	z	NOUN
ejpam-3948	141	9	)	)	PUNCT
ejpam-3948	141	10	=	=	SYM
ejpam-3948	141	11	(	(	PUNCT
ejpam-3948	141	12	2	2	NUM
ejpam-3948	141	13	,	,	PUNCT
ejpam-3948	141	14	1	1	NUM
ejpam-3948	141	15	,	,	PUNCT
ejpam-3948	141	16	9	9	NUM
ejpam-3948	141	17	)	)	PUNCT
ejpam-3948	141	18	is	be	AUX
ejpam-3948	141	19	the	the	DET
ejpam-3948	141	20	unique	unique	ADJ
ejpam-3948	141	21	solution	solution	NOUN
ejpam-3948	141	22	.	.	PUNCT
ejpam-3948	142	1	3	3	X
ejpam-3948	142	2	.	.	X
ejpam-3948	142	3	conclusion	conclusion	NOUN
ejpam-3948	142	4	and	and	CCONJ
ejpam-3948	142	5	recommendation	recommendation	NOUN
ejpam-3948	142	6	in	in	ADP
ejpam-3948	142	7	this	this	DET
ejpam-3948	142	8	work	work	NOUN
ejpam-3948	142	9	,	,	PUNCT
ejpam-3948	142	10	using	use	VERB
ejpam-3948	142	11	the	the	DET
ejpam-3948	142	12	factoring	factoring	NOUN
ejpam-3948	142	13	and	and	CCONJ
ejpam-3948	142	14	modular	modular	ADJ
ejpam-3948	142	15	arithmetic	arithmetic	ADJ
ejpam-3948	142	16	methods	method	NOUN
ejpam-3948	142	17	,	,	PUNCT
ejpam-3948	142	18	the	the	DET
ejpam-3948	142	19	mihailescu	mihailescu	NOUN
ejpam-3948	142	20	’s	’s	PART
ejpam-3948	142	21	theorem	theorem	NOUN
ejpam-3948	142	22	,	,	PUNCT
ejpam-3948	142	23	and	and	CCONJ
ejpam-3948	142	24	the	the	DET
ejpam-3948	142	25	fact	fact	NOUN
ejpam-3948	142	26	that	that	SCONJ
ejpam-3948	142	27	every	every	DET
ejpam-3948	142	28	mersenne	mersenne	NOUN
ejpam-3948	142	29	prime	prime	NOUN
ejpam-3948	142	30	is	be	AUX
ejpam-3948	142	31	of	of	ADP
ejpam-3948	142	32	the	the	DET
ejpam-3948	142	33	form	form	NOUN
ejpam-3948	142	34	4k+3	4k+3	PROPN
ejpam-3948	142	35	,	,	PUNCT
ejpam-3948	142	36	we	we	PRON
ejpam-3948	142	37	were	be	AUX
ejpam-3948	142	38	able	able	ADJ
ejpam-3948	142	39	to	to	PART
ejpam-3948	142	40	show	show	VERB
ejpam-3948	142	41	that	that	SCONJ
ejpam-3948	142	42	the	the	DET
ejpam-3948	142	43	diophantine	diophantine	NOUN
ejpam-3948	142	44	equation	equation	NOUN
ejpam-3948	142	45	mx	mx	PROPN
ejpam-3948	142	46	p	p	PROPN
ejpam-3948	142	47	+	+	X
ejpam-3948	142	48	(	(	PUNCT
ejpam-3948	142	49	mq	mq	PROPN
ejpam-3948	142	50	+	+	CCONJ
ejpam-3948	142	51	1)y	1)y	NUM
ejpam-3948	142	52	=	=	SYM
ejpam-3948	142	53	z2	z2	PROPN
ejpam-3948	142	54	,	,	PUNCT
ejpam-3948	142	55	where	where	SCONJ
ejpam-3948	142	56	mp	mp	PROPN
ejpam-3948	142	57	and	and	CCONJ
ejpam-3948	142	58	mq	mq	PROPN
ejpam-3948	142	59	are	be	AUX
ejpam-3948	142	60	mersenne	mersenne	NOUN
ejpam-3948	142	61	primes	prime	NOUN
ejpam-3948	142	62	,	,	PUNCT
ejpam-3948	142	63	have	have	VERB
ejpam-3948	142	64	the	the	DET
ejpam-3948	142	65	following	follow	VERB
ejpam-3948	142	66	nonnegative	nonnegative	ADJ
ejpam-3948	142	67	integer	integer	NOUN
ejpam-3948	142	68	solutions	solution	NOUN
ejpam-3948	142	69	(	(	PUNCT
ejpam-3948	142	70	mp	mp	PROPN
ejpam-3948	142	71	,	,	PUNCT
ejpam-3948	142	72	mq	mq	PROPN
ejpam-3948	142	73	,	,	PUNCT
ejpam-3948	142	74	x	x	PROPN
ejpam-3948	142	75	,	,	PUNCT
ejpam-3948	142	76	y	y	PROPN
ejpam-3948	142	77	,	,	PUNCT
ejpam-3948	142	78	z	z	NOUN
ejpam-3948	142	79	)	)	PUNCT
ejpam-3948	142	80	,	,	PUNCT
ejpam-3948	142	81	namely	namely	ADV
ejpam-3948	142	82	(	(	PUNCT
ejpam-3948	142	83	mp	mp	PROPN
ejpam-3948	142	84	,	,	PUNCT
ejpam-3948	142	85	7	7	NUM
ejpam-3948	142	86	,	,	PUNCT
ejpam-3948	142	87	0	0	NUM
ejpam-3948	142	88	,	,	PUNCT
ejpam-3948	142	89	1	1	NUM
ejpam-3948	142	90	,	,	PUNCT
ejpam-3948	142	91	3	3	NUM
ejpam-3948	142	92	)	)	PUNCT
ejpam-3948	142	93	,	,	PUNCT
ejpam-3948	142	94	(	(	PUNCT
ejpam-3948	142	95	3,mq	3,mq	NUM
ejpam-3948	142	96	,	,	PUNCT
ejpam-3948	142	97	1	1	NUM
ejpam-3948	142	98	,	,	PUNCT
ejpam-3948	142	99	0	0	NUM
ejpam-3948	142	100	,	,	PUNCT
ejpam-3948	142	101	2	2	NUM
ejpam-3948	142	102	)	)	PUNCT
ejpam-3948	142	103	and	and	CCONJ
ejpam-3948	142	104	(	(	PUNCT
ejpam-3948	142	105	mp	mp	PROPN
ejpam-3948	142	106	,	,	PUNCT
ejpam-3948	142	107	mq	mq	PROPN
ejpam-3948	142	108	,	,	PUNCT
ejpam-3948	142	109	2	2	NUM
ejpam-3948	142	110	,	,	PUNCT
ejpam-3948	142	111	p+	p+	VERB
ejpam-3948	142	112	2	2	NUM
ejpam-3948	142	113	q	q	NOUN
ejpam-3948	142	114	,	,	PUNCT
ejpam-3948	142	115	2p	2p	NUM
ejpam-3948	142	116	+	+	CCONJ
ejpam-3948	142	117	1	1	NUM
ejpam-3948	142	118	)	)	PUNCT
ejpam-3948	142	119	.	.	PUNCT
ejpam-3948	143	1	the	the	DET
ejpam-3948	143	2	following	follow	VERB
ejpam-3948	143	3	table	table	NOUN
ejpam-3948	143	4	presents	present	VERB
ejpam-3948	143	5	some	some	DET
ejpam-3948	143	6	positive	positive	ADJ
ejpam-3948	143	7	integer	integer	NOUN
ejpam-3948	143	8	solutions	solution	NOUN
ejpam-3948	143	9	of	of	ADP
ejpam-3948	143	10	the	the	DET
ejpam-3948	143	11	diophantine	diophantine	NOUN
ejpam-3948	143	12	equation	equation	NOUN
ejpam-3948	143	13	mx	mx	PROPN
ejpam-3948	144	1	p	p	PROPN
ejpam-3948	144	2	+	+	X
ejpam-3948	144	3	(	(	PUNCT
ejpam-3948	144	4	mq	mq	PROPN
ejpam-3948	144	5	+	+	NOUN
ejpam-3948	144	6	1)y	1)y	NUM
ejpam-3948	144	7	=	=	SYM
ejpam-3948	144	8	z2	z2	PROPN
ejpam-3948	144	9	for	for	ADP
ejpam-3948	144	10	the	the	DET
ejpam-3948	144	11	first	first	ADJ
ejpam-3948	144	12	five	five	NUM
ejpam-3948	144	13	mersenne	mersenne	NOUN
ejpam-3948	144	14	primes	prime	NOUN
ejpam-3948	144	15	mp	mp	PROPN
ejpam-3948	144	16	.	.	PROPN
ejpam-3948	144	17	table	table	PROPN
ejpam-3948	144	18	1	1	NUM
ejpam-3948	144	19	.	.	PUNCT
ejpam-3948	145	1	some	some	DET
ejpam-3948	145	2	positive	positive	ADJ
ejpam-3948	145	3	integer	integer	NOUN
ejpam-3948	145	4	solutions	solution	NOUN
ejpam-3948	145	5	of	of	ADP
ejpam-3948	145	6	mx	mx	PROPN
ejpam-3948	145	7	p	p	PROPN
ejpam-3948	145	8	+	+	X
ejpam-3948	145	9	(	(	PUNCT
ejpam-3948	145	10	mq	mq	PROPN
ejpam-3948	145	11	+	+	NOUN
ejpam-3948	145	12	1)y	1)y	NUM
ejpam-3948	145	13	=	=	SYM
ejpam-3948	145	14	z2	z2	PROPN
ejpam-3948	145	15	mp	mp	PROPN
ejpam-3948	145	16	p	p	PROPN
ejpam-3948	145	17	p+	p+	PROPN
ejpam-3948	145	18	2	2	NUM
ejpam-3948	145	19	q	q	PROPN
ejpam-3948	145	20	y	y	PROPN
ejpam-3948	145	21	mq	mq	PROPN
ejpam-3948	145	22	(	(	PUNCT
ejpam-3948	145	23	mp	mp	PROPN
ejpam-3948	145	24	,	,	PUNCT
ejpam-3948	145	25	mq	mq	PROPN
ejpam-3948	145	26	,	,	PUNCT
ejpam-3948	145	27	x	x	PROPN
ejpam-3948	145	28	,	,	PUNCT
ejpam-3948	145	29	y	y	PROPN
ejpam-3948	145	30	,	,	PUNCT
ejpam-3948	145	31	z	z	NOUN
ejpam-3948	145	32	)	)	PUNCT
ejpam-3948	145	33	3	3	NUM
ejpam-3948	145	34	2	2	NUM
ejpam-3948	145	35	4	4	NUM
ejpam-3948	145	36	2	2	NUM
ejpam-3948	145	37	2	2	NUM
ejpam-3948	145	38	3	3	NUM
ejpam-3948	145	39	(	(	PUNCT
ejpam-3948	145	40	3,3,2,2,5	3,3,2,2,5	NUM
ejpam-3948	145	41	)	)	PUNCT
ejpam-3948	145	42	7	7	NUM
ejpam-3948	145	43	3	3	NUM
ejpam-3948	145	44	5	5	NUM
ejpam-3948	145	45	5	5	NUM
ejpam-3948	145	46	1	1	NUM
ejpam-3948	145	47	31	31	NUM
ejpam-3948	145	48	(	(	PUNCT
ejpam-3948	145	49	7,31,2,1,9	7,31,2,1,9	NUM
ejpam-3948	145	50	)	)	PUNCT
ejpam-3948	145	51	31	31	NUM
ejpam-3948	145	52	5	5	NUM
ejpam-3948	145	53	7	7	NUM
ejpam-3948	145	54	7	7	NUM
ejpam-3948	145	55	1	1	NUM
ejpam-3948	145	56	127	127	NUM
ejpam-3948	145	57	(	(	PUNCT
ejpam-3948	145	58	31,127,2,1,33	31,127,2,1,33	NUM
ejpam-3948	145	59	)	)	PUNCT
ejpam-3948	145	60	127	127	NUM
ejpam-3948	145	61	7	7	NUM
ejpam-3948	145	62	9	9	NUM
ejpam-3948	145	63	3	3	NUM
ejpam-3948	145	64	3	3	NUM
ejpam-3948	145	65	7	7	NUM
ejpam-3948	145	66	(	(	PUNCT
ejpam-3948	145	67	127,7,2,3,129	127,7,2,3,129	NUM
ejpam-3948	145	68	)	)	PUNCT
ejpam-3948	145	69	8191	8191	NUM
ejpam-3948	145	70	13	13	NUM
ejpam-3948	145	71	15	15	NUM
ejpam-3948	145	72	3	3	NUM
ejpam-3948	145	73	5	5	NUM
ejpam-3948	145	74	7	7	NUM
ejpam-3948	145	75	(	(	PUNCT
ejpam-3948	145	76	8191,7,2,5,8193	8191,7,2,5,8193	NUM
ejpam-3948	145	77	)	)	PUNCT
ejpam-3948	145	78	8191	8191	NUM
ejpam-3948	145	79	13	13	NUM
ejpam-3948	145	80	15	15	NUM
ejpam-3948	145	81	5	5	NUM
ejpam-3948	145	82	3	3	NUM
ejpam-3948	145	83	31	31	NUM
ejpam-3948	145	84	(	(	PUNCT
ejpam-3948	145	85	8191,31,2,3,8193	8191,31,2,3,8193	NUM
ejpam-3948	145	86	)	)	PUNCT
ejpam-3948	145	87	the	the	DET
ejpam-3948	145	88	next	next	ADJ
ejpam-3948	145	89	table	table	NOUN
ejpam-3948	145	90	presents	present	VERB
ejpam-3948	145	91	some	some	DET
ejpam-3948	145	92	particular	particular	ADJ
ejpam-3948	145	93	cases	case	NOUN
ejpam-3948	145	94	of	of	ADP
ejpam-3948	145	95	the	the	DET
ejpam-3948	145	96	exponential	exponential	ADJ
ejpam-3948	145	97	diophantine	diophantine	NOUN
ejpam-3948	145	98	equation	equation	NOUN
ejpam-3948	146	1	mx	mx	PROPN
ejpam-3948	146	2	p	p	PROPN
ejpam-3948	146	3	+	+	X
ejpam-3948	146	4	(	(	PUNCT
ejpam-3948	146	5	mq	mq	PROPN
ejpam-3948	146	6	+	+	CCONJ
ejpam-3948	146	7	1)y	1)y	NUM
ejpam-3948	146	8	=	=	SYM
ejpam-3948	146	9	z2	z2	PROPN
ejpam-3948	146	10	,	,	PUNCT
ejpam-3948	146	11	wherein	wherein	SCONJ
ejpam-3948	146	12	no	no	DET
ejpam-3948	146	13	solutions	solution	NOUN
ejpam-3948	146	14	can	can	AUX
ejpam-3948	146	15	be	be	AUX
ejpam-3948	146	16	obtained	obtain	VERB
ejpam-3948	146	17	.	.	PUNCT
ejpam-3948	147	1	the	the	DET
ejpam-3948	147	2	unsolvability	unsolvability	NOUN
ejpam-3948	147	3	of	of	ADP
ejpam-3948	147	4	these	these	DET
ejpam-3948	147	5	equations	equation	NOUN
ejpam-3948	147	6	is	be	AUX
ejpam-3948	147	7	achieved	achieve	VERB
ejpam-3948	147	8	because	because	SCONJ
ejpam-3948	147	9	the	the	DET
ejpam-3948	147	10	prime	prime	NOUN
ejpam-3948	147	11	q	q	PROPN
ejpam-3948	147	12	fails	fail	VERB
ejpam-3948	147	13	to	to	PART
ejpam-3948	147	14	divide	divide	VERB
ejpam-3948	147	15	p+	p+	NOUN
ejpam-3948	147	16	2	2	NUM
ejpam-3948	147	17	.	.	PUNCT
ejpam-3948	147	18	table	table	NOUN
ejpam-3948	147	19	2	2	NUM
ejpam-3948	147	20	.	.	X
ejpam-3948	147	21	list	list	NOUN
ejpam-3948	147	22	of	of	ADP
ejpam-3948	147	23	some	some	DET
ejpam-3948	147	24	unsolvable	unsolvable	ADJ
ejpam-3948	147	25	cases	case	NOUN
ejpam-3948	147	26	of	of	ADP
ejpam-3948	147	27	mx	mx	NOUN
ejpam-3948	147	28	p	p	PROPN
ejpam-3948	148	1	+	+	X
ejpam-3948	148	2	(	(	PUNCT
ejpam-3948	148	3	mq	mq	PROPN
ejpam-3948	148	4	+	+	NOUN
ejpam-3948	148	5	1)y	1)y	NUM
ejpam-3948	148	6	=	=	SYM
ejpam-3948	148	7	z2	z2	PROPN
ejpam-3948	148	8	mp	mp	PROPN
ejpam-3948	148	9	p	p	PROPN
ejpam-3948	148	10	mq	mq	PROPN
ejpam-3948	148	11	q	q	PROPN
ejpam-3948	148	12	p+	p+	PROPN
ejpam-3948	149	1	2	2	NUM
ejpam-3948	149	2	mx	mx	NOUN
ejpam-3948	149	3	p	p	NOUN
ejpam-3948	150	1	+	+	X
ejpam-3948	150	2	(	(	PUNCT
ejpam-3948	150	3	mq	mq	PROPN
ejpam-3948	150	4	+	+	NOUN
ejpam-3948	150	5	1)y	1)y	NUM
ejpam-3948	150	6	=	=	SYM
ejpam-3948	150	7	z2	z2	PROPN
ejpam-3948	150	8	3	3	NUM
ejpam-3948	150	9	2	2	NUM
ejpam-3948	150	10	127	127	NUM
ejpam-3948	150	11	5	5	NUM
ejpam-3948	150	12	4	4	NUM
ejpam-3948	150	13	3x	3x	NUM
ejpam-3948	150	14	+	+	CCONJ
ejpam-3948	150	15	128y	128y	NOUN
ejpam-3948	150	16	=	=	SYM
ejpam-3948	150	17	z2	z2	PROPN
ejpam-3948	150	18	7	7	NUM
ejpam-3948	150	19	3	3	NUM
ejpam-3948	150	20	7	7	NUM
ejpam-3948	150	21	3	3	NUM
ejpam-3948	150	22	5	5	NUM
ejpam-3948	150	23	7x	7x	NUM
ejpam-3948	150	24	+	+	CCONJ
ejpam-3948	150	25	8y	8y	X
ejpam-3948	150	26	=	=	SYM
ejpam-3948	150	27	z2	z2	NOUN
ejpam-3948	150	28	31	31	NUM
ejpam-3948	150	29	5	5	NUM
ejpam-3948	150	30	7	7	NUM
ejpam-3948	150	31	3	3	NUM
ejpam-3948	150	32	7	7	NUM
ejpam-3948	150	33	31x	31x	NOUN
ejpam-3948	150	34	+	+	CCONJ
ejpam-3948	150	35	8y	8y	X
ejpam-3948	150	36	=	=	SYM
ejpam-3948	150	37	z2	z2	NOUN
ejpam-3948	150	38	127	127	NUM
ejpam-3948	150	39	7	7	NUM
ejpam-3948	150	40	31	31	NUM
ejpam-3948	150	41	5	5	NUM
ejpam-3948	150	42	9	9	NUM
ejpam-3948	150	43	127x	127x	NOUN
ejpam-3948	150	44	+	+	CCONJ
ejpam-3948	150	45	32y	32y	NOUN
ejpam-3948	150	46	=	=	SYM
ejpam-3948	150	47	z2	z2	PROPN
ejpam-3948	150	48	8191	8191	NUM
ejpam-3948	150	49	13	13	NUM
ejpam-3948	150	50	127	127	NUM
ejpam-3948	150	51	7	7	NUM
ejpam-3948	150	52	15	15	NUM
ejpam-3948	150	53	8191x	8191x	NOUN
ejpam-3948	150	54	+	+	CCONJ
ejpam-3948	150	55	128y	128y	NOUN
ejpam-3948	150	56	=	=	SYM
ejpam-3948	150	57	z2	z2	PROPN
ejpam-3948	150	58	references	reference	VERB
ejpam-3948	150	59	401	401	NUM
ejpam-3948	150	60	the	the	DET
ejpam-3948	150	61	results	result	NOUN
ejpam-3948	150	62	presented	present	VERB
ejpam-3948	150	63	in	in	ADP
ejpam-3948	150	64	this	this	DET
ejpam-3948	150	65	study	study	NOUN
ejpam-3948	150	66	contribute	contribute	VERB
ejpam-3948	150	67	to	to	ADP
ejpam-3948	150	68	the	the	DET
ejpam-3948	150	69	repository	repository	NOUN
ejpam-3948	150	70	of	of	ADP
ejpam-3948	150	71	knowlege	knowlege	NOUN
ejpam-3948	150	72	in	in	ADP
ejpam-3948	150	73	the	the	DET
ejpam-3948	150	74	theory	theory	NOUN
ejpam-3948	150	75	of	of	ADP
ejpam-3948	150	76	numbers	number	NOUN
ejpam-3948	150	77	,	,	PUNCT
ejpam-3948	150	78	especially	especially	ADV
ejpam-3948	150	79	in	in	ADP
ejpam-3948	150	80	solving	solve	VERB
ejpam-3948	150	81	exponential	exponential	ADJ
ejpam-3948	150	82	diophantine	diophantine	NOUN
ejpam-3948	150	83	equations	equation	NOUN
ejpam-3948	150	84	.	.	PUNCT
ejpam-3948	151	1	for	for	ADP
ejpam-3948	151	2	possible	possible	ADJ
ejpam-3948	151	3	extensions	extension	NOUN
ejpam-3948	151	4	,	,	PUNCT
ejpam-3948	151	5	the	the	DET
ejpam-3948	151	6	reader	reader	NOUN
ejpam-3948	151	7	may	may	AUX
ejpam-3948	151	8	try	try	VERB
ejpam-3948	151	9	to	to	PART
ejpam-3948	151	10	solve	solve	VERB
ejpam-3948	151	11	the	the	DET
ejpam-3948	151	12	following	follow	VERB
ejpam-3948	151	13	diophantine	diophantine	NOUN
ejpam-3948	151	14	equations	equation	NOUN
ejpam-3948	151	15	in	in	ADP
ejpam-3948	151	16	n0	n0	NOUN
ejpam-3948	151	17	:	:	PUNCT
ejpam-3948	151	18	(	(	PUNCT
ejpam-3948	151	19	i	i	NOUN
ejpam-3948	151	20	)	)	PUNCT
ejpam-3948	151	21	mx	mx	PROPN
ejpam-3948	152	1	p	p	PROPN
ejpam-3948	152	2	+	+	X
ejpam-3948	152	3	(	(	PUNCT
ejpam-3948	152	4	mq	mq	NOUN
ejpam-3948	152	5	+	+	CCONJ
ejpam-3948	152	6	k)y	k)y	X
ejpam-3948	152	7	=	=	SYM
ejpam-3948	152	8	z2	z2	PROPN
ejpam-3948	152	9	,	,	PUNCT
ejpam-3948	152	10	where	where	SCONJ
ejpam-3948	152	11	k	k	PROPN
ejpam-3948	152	12	≥	≥	PROPN
ejpam-3948	152	13	1	1	NUM
ejpam-3948	152	14	,	,	PUNCT
ejpam-3948	152	15	and	and	CCONJ
ejpam-3948	152	16	mp	mp	PROPN
ejpam-3948	152	17	and	and	CCONJ
ejpam-3948	152	18	mq	mq	PROPN
ejpam-3948	152	19	are	be	AUX
ejpam-3948	152	20	mersenne	mersenne	NOUN
ejpam-3948	152	21	primes	prime	NOUN
ejpam-3948	152	22	;	;	PUNCT
ejpam-3948	152	23	(	(	PUNCT
ejpam-3948	152	24	ii	ii	NOUN
ejpam-3948	152	25	)	)	PUNCT
ejpam-3948	152	26	mx	mx	NOUN
ejpam-3948	153	1	p	p	NOUN
ejpam-3948	153	2	+	+	X
ejpam-3948	153	3	(	(	PUNCT
ejpam-3948	153	4	mq	mq	PROPN
ejpam-3948	153	5	+	+	NOUN
ejpam-3948	153	6	1)y	1)y	NUM
ejpam-3948	153	7	=	=	SYM
ejpam-3948	153	8	zn	zn	PROPN
ejpam-3948	153	9	,	,	PUNCT
ejpam-3948	153	10	where	where	SCONJ
ejpam-3948	153	11	n	n	PRON
ejpam-3948	153	12	≥	≥	NOUN
ejpam-3948	153	13	1	1	NUM
ejpam-3948	153	14	,	,	PUNCT
ejpam-3948	153	15	and	and	CCONJ
ejpam-3948	153	16	mp	mp	PROPN
ejpam-3948	153	17	and	and	CCONJ
ejpam-3948	153	18	mq	mq	PROPN
ejpam-3948	153	19	are	be	AUX
ejpam-3948	153	20	mersenne	mersenne	NOUN
ejpam-3948	153	21	primes	prime	NOUN
ejpam-3948	153	22	;	;	PUNCT
ejpam-3948	153	23	and	and	CCONJ
ejpam-3948	153	24	(	(	PUNCT
ejpam-3948	153	25	iii	iii	NOUN
ejpam-3948	153	26	)	)	PUNCT
ejpam-3948	153	27	(	(	PUNCT
ejpam-3948	153	28	mq	mq	NOUN
ejpam-3948	153	29	−	−	NUM
ejpam-3948	153	30	1)x	1)x	NUM
ejpam-3948	154	1	+	+	ADP
ejpam-3948	154	2	my	my	PRON
ejpam-3948	154	3	p	p	NOUN
ejpam-3948	154	4	+	+	X
ejpam-3948	154	5	(	(	PUNCT
ejpam-3948	154	6	mq	mq	PROPN
ejpam-3948	154	7	+	+	CCONJ
ejpam-3948	154	8	1)z	1)z	NOUN
ejpam-3948	154	9	=	=	SYM
ejpam-3948	154	10	w2	w2	NOUN
ejpam-3948	154	11	,	,	PUNCT
ejpam-3948	154	12	where	where	SCONJ
ejpam-3948	154	13	mp	mp	PROPN
ejpam-3948	154	14	and	and	CCONJ
ejpam-3948	154	15	mq	mq	PROPN
ejpam-3948	154	16	are	be	AUX
ejpam-3948	154	17	mersenne	mersenne	NOUN
ejpam-3948	154	18	primes	prime	NOUN
ejpam-3948	154	19	.	.	PUNCT
ejpam-3948	155	1	since	since	SCONJ
ejpam-3948	155	2	the	the	DET
ejpam-3948	155	3	equation	equation	NOUN
ejpam-3948	155	4	under	under	ADP
ejpam-3948	155	5	consideration	consideration	NOUN
ejpam-3948	155	6	in	in	ADP
ejpam-3948	155	7	this	this	DET
ejpam-3948	155	8	study	study	NOUN
ejpam-3948	155	9	is	be	AUX
ejpam-3948	155	10	equivalent	equivalent	ADJ
ejpam-3948	155	11	to	to	ADP
ejpam-3948	155	12	(	(	PUNCT
ejpam-3948	155	13	2p−1)x+2qy	2p−1)x+2qy	PROPN
ejpam-3948	155	14	=	=	SYM
ejpam-3948	155	15	z2	z2	PROPN
ejpam-3948	155	16	,	,	PUNCT
ejpam-3948	155	17	the	the	DET
ejpam-3948	155	18	reader	reader	NOUN
ejpam-3948	155	19	may	may	AUX
ejpam-3948	155	20	get	get	VERB
ejpam-3948	155	21	additional	additional	ADJ
ejpam-3948	155	22	results	result	NOUN
ejpam-3948	155	23	when	when	SCONJ
ejpam-3948	155	24	compared	compare	VERB
ejpam-3948	155	25	to	to	ADP
ejpam-3948	155	26	or	or	CCONJ
ejpam-3948	155	27	combined	combine	VERB
ejpam-3948	155	28	with	with	ADP
ejpam-3948	155	29	results	result	NOUN
ejpam-3948	155	30	on	on	ADP
ejpam-3948	155	31	similar	similar	ADJ
ejpam-3948	155	32	/	/	SYM
ejpam-3948	155	33	related	relate	VERB
ejpam-3948	155	34	diophantine	diophantine	NOUN
ejpam-3948	155	35	equations	equation	NOUN
ejpam-3948	155	36	,	,	PUNCT
ejpam-3948	155	37	such	such	ADJ
ejpam-3948	155	38	as	as	ADP
ejpam-3948	155	39	x2	x2	PROPN
ejpam-3948	155	40	−	−	PROPN
ejpam-3948	155	41	2r	2r	NUM
ejpam-3948	155	42	=	=	SYM
ejpam-3948	155	43	pn	pn	PROPN
ejpam-3948	156	1	[	[	X
ejpam-3948	156	2	29	29	NUM
ejpam-3948	156	3	]	]	PUNCT
ejpam-3948	156	4	and	and	CCONJ
ejpam-3948	157	1	x2	x2	INTJ
ejpam-3948	157	2	−	−	PROPN
ejpam-3948	158	1	d	d	NOUN
ejpam-3948	158	2	=	=	SYM
ejpam-3948	158	3	pn	pn	PROPN
ejpam-3948	158	4	(	(	PUNCT
ejpam-3948	158	5	cf	cf	NOUN
ejpam-3948	158	6	.	.	PUNCT
ejpam-3948	159	1	[	[	X
ejpam-3948	159	2	10	10	NUM
ejpam-3948	159	3	]	]	PUNCT
ejpam-3948	159	4	,	,	PUNCT
ejpam-3948	159	5	[	[	X
ejpam-3948	159	6	11	11	NUM
ejpam-3948	159	7	]	]	X
ejpam-3948	160	1	[	[	X
ejpam-3948	160	2	31	31	NUM
ejpam-3948	160	3	]	]	PUNCT
ejpam-3948	160	4	,	,	PUNCT
ejpam-3948	160	5	[	[	X
ejpam-3948	160	6	30	30	NUM
ejpam-3948	160	7	]	]	PUNCT
ejpam-3948	160	8	)	)	PUNCT
ejpam-3948	160	9	.	.	PUNCT
ejpam-3948	161	1	lastly	lastly	ADV
ejpam-3948	161	2	,	,	PUNCT
ejpam-3948	161	3	to	to	PART
ejpam-3948	161	4	find	find	VERB
ejpam-3948	161	5	other	other	ADJ
ejpam-3948	161	6	results	result	NOUN
ejpam-3948	161	7	for	for	ADP
ejpam-3948	161	8	the	the	DET
ejpam-3948	161	9	equation	equation	NOUN
ejpam-3948	161	10	under	under	ADP
ejpam-3948	161	11	consideration	consideration	NOUN
ejpam-3948	161	12	and	and	CCONJ
ejpam-3948	161	13	the	the	DET
ejpam-3948	161	14	suggested	suggest	VERB
ejpam-3948	161	15	equations	equation	NOUN
ejpam-3948	161	16	above	above	ADV
ejpam-3948	161	17	,	,	PUNCT
ejpam-3948	161	18	the	the	DET
ejpam-3948	161	19	reader	reader	NOUN
ejpam-3948	161	20	might	might	AUX
ejpam-3948	161	21	get	get	VERB
ejpam-3948	161	22	interested	interested	ADJ
ejpam-3948	161	23	in	in	ADP
ejpam-3948	161	24	applying	apply	VERB
ejpam-3948	161	25	other	other	ADJ
ejpam-3948	161	26	methods	method	NOUN
ejpam-3948	161	27	such	such	ADJ
ejpam-3948	161	28	as	as	ADP
ejpam-3948	161	29	the	the	DET
ejpam-3948	161	30	linear	linear	ADJ
ejpam-3948	161	31	forms	form	NOUN
ejpam-3948	161	32	in	in	ADP
ejpam-3948	161	33	logarithms	logarithm	NOUN
ejpam-3948	161	34	,	,	PUNCT
ejpam-3948	161	35	like	like	ADP
ejpam-3948	161	36	what	what	PRON
ejpam-3948	161	37	was	be	AUX
ejpam-3948	161	38	done	do	VERB
ejpam-3948	161	39	in	in	ADP
ejpam-3948	161	40	the	the	DET
ejpam-3948	161	41	paper	paper	NOUN
ejpam-3948	161	42	by	by	ADP
ejpam-3948	161	43	bugeaud	bugeaud	NOUN
ejpam-3948	162	1	[	[	X
ejpam-3948	162	2	7	7	NUM
ejpam-3948	162	3	]	]	PUNCT
ejpam-3948	162	4	.	.	PUNCT
ejpam-3948	163	1	acknowledgements	acknowledgement	NOUN
ejpam-3948	163	2	the	the	DET
ejpam-3948	163	3	authors	author	NOUN
ejpam-3948	163	4	would	would	AUX
ejpam-3948	163	5	like	like	VERB
ejpam-3948	163	6	to	to	PART
ejpam-3948	163	7	thank	thank	VERB
ejpam-3948	163	8	the	the	DET
ejpam-3948	163	9	university	university	NOUN
ejpam-3948	163	10	of	of	ADP
ejpam-3948	163	11	the	the	DET
ejpam-3948	163	12	philippines	philippine	NOUN
ejpam-3948	163	13	baguio	baguio	VERB
ejpam-3948	163	14	for	for	ADP
ejpam-3948	163	15	the	the	DET
ejpam-3948	163	16	support	support	NOUN
ejpam-3948	163	17	given	give	VERB
ejpam-3948	163	18	in	in	ADP
ejpam-3948	163	19	the	the	DET
ejpam-3948	163	20	conduct	conduct	NOUN
ejpam-3948	163	21	of	of	ADP
ejpam-3948	163	22	this	this	DET
ejpam-3948	163	23	study	study	NOUN
ejpam-3948	163	24	and	and	CCONJ
ejpam-3948	163	25	its	its	PRON
ejpam-3948	163	26	publication	publication	NOUN
ejpam-3948	163	27	.	.	PUNCT
ejpam-3948	164	1	the	the	DET
ejpam-3948	164	2	authors	author	NOUN
ejpam-3948	164	3	would	would	AUX
ejpam-3948	164	4	also	also	ADV
ejpam-3948	164	5	like	like	VERB
ejpam-3948	164	6	to	to	PART
ejpam-3948	164	7	thank	thank	VERB
ejpam-3948	164	8	the	the	DET
ejpam-3948	164	9	referees	referee	NOUN
ejpam-3948	164	10	for	for	ADP
ejpam-3948	164	11	sharing	share	VERB
ejpam-3948	164	12	their	their	PRON
ejpam-3948	164	13	time	time	NOUN
ejpam-3948	164	14	and	and	CCONJ
ejpam-3948	164	15	expertise	expertise	NOUN
ejpam-3948	164	16	in	in	ADP
ejpam-3948	164	17	reviewing	review	VERB
ejpam-3948	164	18	the	the	DET
ejpam-3948	164	19	paper	paper	NOUN
ejpam-3948	164	20	.	.	PUNCT
ejpam-3948	165	1	the	the	DET
ejpam-3948	165	2	comments	comment	NOUN
ejpam-3948	165	3	and	and	CCONJ
ejpam-3948	165	4	suggestions	suggestion	NOUN
ejpam-3948	165	5	are	be	AUX
ejpam-3948	165	6	very	very	ADV
ejpam-3948	165	7	helpful	helpful	ADJ
ejpam-3948	165	8	to	to	PART
ejpam-3948	165	9	have	have	VERB
ejpam-3948	165	10	an	an	DET
ejpam-3948	165	11	improved	improved	ADJ
ejpam-3948	165	12	manuscript	manuscript	NOUN
ejpam-3948	165	13	.	.	PUNCT
ejpam-3948	166	1	references	reference	NOUN
ejpam-3948	166	2	[	[	X
ejpam-3948	166	3	1	1	NUM
ejpam-3948	166	4	]	]	PUNCT
ejpam-3948	166	5	l	l	PROPN
ejpam-3948	166	6	chaudhary	chaudhary	PROPN
ejpam-3948	166	7	a	a	DET
ejpam-3948	166	8	kumar	kumar	PROPN
ejpam-3948	166	9	and	and	CCONJ
ejpam-3948	166	10	s	s	PROPN
ejpam-3948	166	11	aggarwal	aggarwal	NOUN
ejpam-3948	166	12	.	.	PUNCT
ejpam-3948	167	1	on	on	ADP
ejpam-3948	167	2	the	the	DET
ejpam-3948	167	3	exponential	exponential	ADJ
ejpam-3948	167	4	diophantine	diophantine	NOUN
ejpam-3948	167	5	equation	equation	NOUN
ejpam-3948	167	6	601p+	601p+	NUM
ejpam-3948	167	7	619q	619q	NOUN
ejpam-3948	167	8	=	=	SYM
ejpam-3948	167	9	r2	r2	PROPN
ejpam-3948	167	10	.	.	PUNCT
ejpam-3948	168	1	international	international	ADJ
ejpam-3948	168	2	journal	journal	PROPN
ejpam-3948	168	3	of	of	ADP
ejpam-3948	168	4	interdisciplinary	interdisciplinary	ADJ
ejpam-3948	168	5	global	global	ADJ
ejpam-3948	168	6	studies	study	NOUN
ejpam-3948	168	7	,	,	PUNCT
ejpam-3948	168	8	14:29–30	14:29–30	NUM
ejpam-3948	168	9	,	,	PUNCT
ejpam-3948	168	10	2020	2020	NUM
ejpam-3948	168	11	.	.	PUNCT
ejpam-3948	169	1	[	[	X
ejpam-3948	169	2	2	2	NUM
ejpam-3948	169	3	]	]	PUNCT
ejpam-3948	169	4	s	s	PART
ejpam-3948	169	5	aggarwal	aggarwal	NOUN
ejpam-3948	169	6	.	.	PUNCT
ejpam-3948	170	1	on	on	ADP
ejpam-3948	170	2	the	the	DET
ejpam-3948	170	3	existence	existence	NOUN
ejpam-3948	170	4	of	of	ADP
ejpam-3948	170	5	solution	solution	NOUN
ejpam-3948	170	6	of	of	ADP
ejpam-3948	170	7	diophantine	diophantine	NOUN
ejpam-3948	170	8	equation	equation	NOUN
ejpam-3948	170	9	193x	193x	PROPN
ejpam-3948	170	10	+	+	CCONJ
ejpam-3948	170	11	211y	211y	NUM
ejpam-3948	170	12	=	=	SYM
ejpam-3948	170	13	z2	z2	PROPN
ejpam-3948	170	14	.	.	PROPN
ejpam-3948	170	15	journal	journal	PROPN
ejpam-3948	170	16	of	of	ADP
ejpam-3948	170	17	advanced	advanced	ADJ
ejpam-3948	170	18	research	research	NOUN
ejpam-3948	170	19	in	in	ADP
ejpam-3948	170	20	applied	apply	VERB
ejpam-3948	170	21	mathematics	mathematic	NOUN
ejpam-3948	170	22	and	and	CCONJ
ejpam-3948	170	23	statistics	statistic	NOUN
ejpam-3948	170	24	,	,	PUNCT
ejpam-3948	170	25	5:1–2	5:1–2	NUM
ejpam-3948	170	26	,	,	PUNCT
ejpam-3948	170	27	2020	2020	NUM
ejpam-3948	170	28	.	.	PUNCT
ejpam-3948	171	1	[	[	X
ejpam-3948	171	2	3	3	NUM
ejpam-3948	171	3	]	]	SYM
ejpam-3948	171	4	s	s	NOUN
ejpam-3948	171	5	aggarwal	aggarwal	NOUN
ejpam-3948	171	6	and	and	CCONJ
ejpam-3948	171	7	n	n	NOUN
ejpam-3948	171	8	sharma	sharma	NOUN
ejpam-3948	171	9	.	.	PUNCT
ejpam-3948	172	1	on	on	ADP
ejpam-3948	172	2	the	the	DET
ejpam-3948	172	3	non	non	ADJ
ejpam-3948	172	4	-	-	ADJ
ejpam-3948	172	5	linear	linear	ADJ
ejpam-3948	172	6	diophantine	diophantine	NOUN
ejpam-3948	172	7	equation	equation	NOUN
ejpam-3948	172	8	379x+397y	379x+397y	NUM
ejpam-3948	172	9	=	=	SYM
ejpam-3948	172	10	z2	z2	PROPN
ejpam-3948	172	11	.	.	PUNCT
ejpam-3948	173	1	open	open	PROPN
ejpam-3948	173	2	journal	journal	PROPN
ejpam-3948	173	3	of	of	ADP
ejpam-3948	173	4	mathematical	mathematical	ADJ
ejpam-3948	173	5	sciences	science	NOUN
ejpam-3948	173	6	,	,	PUNCT
ejpam-3948	173	7	4	4	NUM
ejpam-3948	173	8	,	,	PUNCT
ejpam-3948	173	9	2020	2020	NUM
ejpam-3948	173	10	.	.	PUNCT
ejpam-3948	174	1	[	[	X
ejpam-3948	174	2	4	4	NUM
ejpam-3948	174	3	]	]	X
ejpam-3948	174	4	s	s	VERB
ejpam-3948	174	5	asthana	asthana	PROPN
ejpam-3948	174	6	and	and	CCONJ
ejpam-3948	174	7	m	m	PROPN
ejpam-3948	174	8	m	m	VERB
ejpam-3948	174	9	singh	singh	ADJ
ejpam-3948	174	10	.	.	PUNCT
ejpam-3948	175	1	on	on	ADP
ejpam-3948	175	2	the	the	DET
ejpam-3948	175	3	diophantine	diophantine	NOUN
ejpam-3948	175	4	equation	equation	NOUN
ejpam-3948	175	5	3x	3x	NUM
ejpam-3948	175	6	+	+	CCONJ
ejpam-3948	175	7	13y	13y	NOUN
ejpam-3948	175	8	=	=	SYM
ejpam-3948	175	9	z2	z2	PROPN
ejpam-3948	175	10	.	.	PUNCT
ejpam-3948	176	1	int	int	PROPN
ejpam-3948	176	2	.	.	PUNCT
ejpam-3948	177	1	j.	j.	PROPN
ejpam-3948	177	2	pure	pure	PROPN
ejpam-3948	177	3	appl	appl	PROPN
ejpam-3948	177	4	.	.	PUNCT
ejpam-3948	177	5	math	math	PROPN
ejpam-3948	177	6	.	.	PUNCT
ejpam-3948	177	7	,	,	PUNCT
ejpam-3948	177	8	114:301–304	114:301–304	NUM
ejpam-3948	177	9	,	,	PUNCT
ejpam-3948	177	10	2017	2017	NUM
ejpam-3948	177	11	.	.	PUNCT
ejpam-3948	178	1	[	[	X
ejpam-3948	178	2	5	5	X
ejpam-3948	178	3	]	]	PUNCT
ejpam-3948	178	4	j	j	PROPN
ejpam-3948	178	5	b	b	PROPN
ejpam-3948	178	6	bacani	bacani	PROPN
ejpam-3948	178	7	and	and	CCONJ
ejpam-3948	178	8	j	j	PROPN
ejpam-3948	178	9	f	f	PROPN
ejpam-3948	178	10	t	t	PROPN
ejpam-3948	178	11	rabago	rabago	PROPN
ejpam-3948	178	12	.	.	PUNCT
ejpam-3948	179	1	the	the	DET
ejpam-3948	179	2	complete	complete	ADJ
ejpam-3948	179	3	set	set	NOUN
ejpam-3948	179	4	of	of	ADP
ejpam-3948	179	5	solutions	solution	NOUN
ejpam-3948	179	6	of	of	ADP
ejpam-3948	179	7	the	the	DET
ejpam-3948	179	8	diophantine	diophantine	NOUN
ejpam-3948	179	9	equation	equation	NOUN
ejpam-3948	179	10	px+qy	px+qy	ADJ
ejpam-3948	179	11	=	=	SYM
ejpam-3948	179	12	z2	z2	PROPN
ejpam-3948	179	13	for	for	ADP
ejpam-3948	179	14	twin	twin	ADJ
ejpam-3948	179	15	primes	prime	NOUN
ejpam-3948	179	16	p	p	NOUN
ejpam-3948	179	17	and	and	CCONJ
ejpam-3948	179	18	q.	q.	PROPN
ejpam-3948	179	19	int	int	PROPN
ejpam-3948	179	20	.	.	PUNCT
ejpam-3948	180	1	j.	j.	PROPN
ejpam-3948	180	2	pure	pure	PROPN
ejpam-3948	180	3	appl	appl	PROPN
ejpam-3948	180	4	.	.	PUNCT
ejpam-3948	180	5	math	math	PROPN
ejpam-3948	180	6	.	.	PUNCT
ejpam-3948	180	7	,	,	PUNCT
ejpam-3948	180	8	104:517–521	104:517–521	NUM
ejpam-3948	180	9	,	,	PUNCT
ejpam-3948	180	10	2015	2015	NUM
ejpam-3948	180	11	.	.	PUNCT
ejpam-3948	181	1	[	[	X
ejpam-3948	181	2	6	6	NUM
ejpam-3948	181	3	]	]	X
ejpam-3948	181	4	k	k	PROPN
ejpam-3948	181	5	bhatnagar	bhatnagar	PROPN
ejpam-3948	181	6	and	and	CCONJ
ejpam-3948	181	7	s	s	NOUN
ejpam-3948	181	8	aggarwal	aggarwal	NOUN
ejpam-3948	181	9	.	.	PUNCT
ejpam-3948	182	1	on	on	ADP
ejpam-3948	182	2	the	the	DET
ejpam-3948	182	3	exponential	exponential	ADJ
ejpam-3948	182	4	diophantine	diophantine	NOUN
ejpam-3948	182	5	equation	equation	NOUN
ejpam-3948	182	6	421p+439q	421p+439q	NOUN
ejpam-3948	182	7	=	=	PUNCT
ejpam-3948	182	8	r2	r2	PROPN
ejpam-3948	182	9	.	.	PUNCT
ejpam-3948	183	1	international	international	ADJ
ejpam-3948	183	2	journal	journal	PROPN
ejpam-3948	183	3	of	of	ADP
ejpam-3948	183	4	interdisciplinary	interdisciplinary	ADJ
ejpam-3948	183	5	global	global	ADJ
ejpam-3948	183	6	studies	study	NOUN
ejpam-3948	183	7	,	,	PUNCT
ejpam-3948	183	8	14:128–129	14:128–129	NUM
ejpam-3948	183	9	,	,	PUNCT
ejpam-3948	183	10	2020	2020	NUM
ejpam-3948	183	11	.	.	PUNCT
ejpam-3948	184	1	references	reference	NOUN
ejpam-3948	184	2	402	402	NUM
ejpam-3948	185	1	[	[	X
ejpam-3948	185	2	7	7	NUM
ejpam-3948	185	3	]	]	X
ejpam-3948	185	4	y	y	PROPN
ejpam-3948	185	5	bugeaud	bugeaud	PROPN
ejpam-3948	185	6	.	.	PUNCT
ejpam-3948	186	1	on	on	ADP
ejpam-3948	186	2	the	the	DET
ejpam-3948	186	3	diophantine	diophantine	NOUN
ejpam-3948	186	4	equation	equation	NOUN
ejpam-3948	186	5	x2	x2	PROPN
ejpam-3948	186	6	−	−	PROPN
ejpam-3948	186	7	pm	pm	NOUN
ejpam-3948	186	8	=	=	SYM
ejpam-3948	186	9	±yn	±yn	NOUN
ejpam-3948	186	10	.	.	PUNCT
ejpam-3948	187	1	acta	acta	PROPN
ejpam-3948	187	2	arith	arith	PROPN
ejpam-3948	187	3	.	.	PUNCT
ejpam-3948	187	4	,	,	PUNCT
ejpam-3948	188	1	80:213–223	80:213–223	PROPN
ejpam-3948	188	2	,	,	PUNCT
ejpam-3948	188	3	1997	1997	NUM
ejpam-3948	188	4	.	.	PUNCT
ejpam-3948	189	1	[	[	X
ejpam-3948	189	2	8	8	NUM
ejpam-3948	189	3	]	]	PUNCT
ejpam-3948	189	4	n	n	PRON
ejpam-3948	189	5	burshtein	burshtein	ADV
ejpam-3948	189	6	.	.	PUNCT
ejpam-3948	190	1	all	all	DET
ejpam-3948	190	2	the	the	DET
ejpam-3948	190	3	solutions	solution	NOUN
ejpam-3948	190	4	of	of	ADP
ejpam-3948	190	5	the	the	DET
ejpam-3948	190	6	diophantine	diophantine	NOUN
ejpam-3948	190	7	equation	equation	NOUN
ejpam-3948	190	8	px	px	X
ejpam-3948	190	9	+	+	CCONJ
ejpam-3948	190	10	(	(	PUNCT
ejpam-3948	190	11	p+	p+	NOUN
ejpam-3948	190	12	4)y	4)y	PROPN
ejpam-3948	190	13	=	=	SYM
ejpam-3948	190	14	z2	z2	PROPN
ejpam-3948	190	15	when	when	SCONJ
ejpam-3948	190	16	p	p	X
ejpam-3948	190	17	,	,	PUNCT
ejpam-3948	190	18	(	(	PUNCT
ejpam-3948	190	19	p	p	NOUN
ejpam-3948	190	20	+	+	NOUN
ejpam-3948	190	21	4	4	NUM
ejpam-3948	190	22	)	)	PUNCT
ejpam-3948	190	23	are	be	AUX
ejpam-3948	190	24	primes	prime	NOUN
ejpam-3948	190	25	and	and	CCONJ
ejpam-3948	190	26	x	x	PUNCT
ejpam-3948	191	1	+	+	CCONJ
ejpam-3948	191	2	y	y	NOUN
ejpam-3948	191	3	=	=	SYM
ejpam-3948	191	4	2	2	NUM
ejpam-3948	191	5	,	,	PUNCT
ejpam-3948	191	6	3	3	NUM
ejpam-3948	191	7	,	,	PUNCT
ejpam-3948	191	8	4	4	NUM
ejpam-3948	191	9	.	.	PUNCT
ejpam-3948	191	10	annals	annal	NOUN
ejpam-3948	191	11	of	of	ADP
ejpam-3948	191	12	pure	pure	ADJ
ejpam-3948	191	13	and	and	CCONJ
ejpam-3948	191	14	applied	applied	ADJ
ejpam-3948	191	15	mathematics	mathematic	NOUN
ejpam-3948	191	16	,	,	PUNCT
ejpam-3948	191	17	1:241–244	1:241–244	NOUN
ejpam-3948	191	18	,	,	PUNCT
ejpam-3948	191	19	2018	2018	NUM
ejpam-3948	191	20	.	.	PUNCT
ejpam-3948	192	1	[	[	X
ejpam-3948	192	2	9	9	NUM
ejpam-3948	192	3	]	]	X
ejpam-3948	192	4	s	s	AUX
ejpam-3948	192	5	chotchaisthit	chotchaisthit	VERB
ejpam-3948	192	6	.	.	PUNCT
ejpam-3948	193	1	on	on	ADP
ejpam-3948	193	2	the	the	DET
ejpam-3948	193	3	diophantine	diophantine	NOUN
ejpam-3948	193	4	equation	equation	NOUN
ejpam-3948	193	5	px+(p+1)y	px+(p+1)y	NOUN
ejpam-3948	193	6	=	=	SYM
ejpam-3948	193	7	z2	z2	PROPN
ejpam-3948	193	8	where	where	SCONJ
ejpam-3948	193	9	p	p	NOUN
ejpam-3948	193	10	is	be	AUX
ejpam-3948	193	11	a	a	DET
ejpam-3948	193	12	mersenne	mersenne	NOUN
ejpam-3948	193	13	prime	prime	NOUN
ejpam-3948	193	14	.	.	PUNCT
ejpam-3948	194	1	int	int	NOUN
ejpam-3948	194	2	.	.	PUNCT
ejpam-3948	195	1	j.	j.	PROPN
ejpam-3948	195	2	pure	pure	PROPN
ejpam-3948	195	3	appl	appl	PROPN
ejpam-3948	195	4	.	.	PUNCT
ejpam-3948	195	5	math	math	PROPN
ejpam-3948	195	6	.	.	PUNCT
ejpam-3948	195	7	,	,	PUNCT
ejpam-3948	195	8	88:169–172	88:169–172	NUM
ejpam-3948	195	9	,	,	PUNCT
ejpam-3948	195	10	2013	2013	NUM
ejpam-3948	195	11	.	.	PUNCT
ejpam-3948	196	1	[	[	X
ejpam-3948	196	2	10	10	NUM
ejpam-3948	196	3	]	]	X
ejpam-3948	196	4	m	m	VERB
ejpam-3948	196	5	h	h	NOUN
ejpam-3948	196	6	le	le	X
ejpam-3948	196	7	.	.	PUNCT
ejpam-3948	197	1	on	on	ADP
ejpam-3948	197	2	the	the	DET
ejpam-3948	197	3	number	number	NOUN
ejpam-3948	197	4	of	of	ADP
ejpam-3948	197	5	solutions	solution	NOUN
ejpam-3948	197	6	of	of	ADP
ejpam-3948	197	7	diophantine	diophantine	NOUN
ejpam-3948	197	8	equations	equation	NOUN
ejpam-3948	197	9	x2−d	x2−d	PROPN
ejpam-3948	197	10	=	=	SYM
ejpam-3948	197	11	pn	pn	PROPN
ejpam-3948	197	12	.	.	PROPN
ejpam-3948	197	13	j.	j.	PROPN
ejpam-3948	197	14	math	math	PROPN
ejpam-3948	197	15	.	.	PUNCT
ejpam-3948	197	16	,	,	PUNCT
ejpam-3948	197	17	34:378–387	34:378–387	PROPN
ejpam-3948	197	18	,	,	PUNCT
ejpam-3948	197	19	1991	1991	NUM
ejpam-3948	197	20	.	.	PUNCT
ejpam-3948	198	1	[	[	X
ejpam-3948	198	2	11	11	NUM
ejpam-3948	198	3	]	]	X
ejpam-3948	198	4	m	m	VERB
ejpam-3948	198	5	h	h	NOUN
ejpam-3948	198	6	le	le	X
ejpam-3948	198	7	.	.	PUNCT
ejpam-3948	199	1	on	on	ADP
ejpam-3948	199	2	the	the	DET
ejpam-3948	199	3	number	number	NOUN
ejpam-3948	199	4	of	of	ADP
ejpam-3948	199	5	solutions	solution	NOUN
ejpam-3948	199	6	of	of	ADP
ejpam-3948	199	7	the	the	DET
ejpam-3948	199	8	generalized	generalize	VERB
ejpam-3948	199	9	ramanujan	ramanujan	PROPN
ejpam-3948	199	10	-	-	PUNCT
ejpam-3948	199	11	nagell	nagell	PROPN
ejpam-3948	199	12	equation	equation	NOUN
ejpam-3948	200	1	x2	x2	PROPN
ejpam-3948	200	2	−d	−d	PROPN
ejpam-3948	200	3	=	=	SYM
ejpam-3948	200	4	pn	pn	PROPN
ejpam-3948	200	5	.	.	PROPN
ejpam-3948	200	6	publ	publ	PROPN
ejpam-3948	200	7	.	.	PUNCT
ejpam-3948	201	1	math	math	NOUN
ejpam-3948	201	2	.	.	PUNCT
ejpam-3948	202	1	debr	debr	PROPN
ejpam-3948	202	2	.	.	PUNCT
ejpam-3948	203	1	,	,	PUNCT
ejpam-3948	203	2	45:239–254	45:239–254	PROPN
ejpam-3948	203	3	,	,	PUNCT
ejpam-3948	203	4	1994	1994	NUM
ejpam-3948	203	5	.	.	PUNCT
ejpam-3948	204	1	[	[	X
ejpam-3948	204	2	12	12	NUM
ejpam-3948	204	3	]	]	X
ejpam-3948	204	4	s	s	VERB
ejpam-3948	204	5	mihailescu	mihailescu	NOUN
ejpam-3948	204	6	.	.	PUNCT
ejpam-3948	205	1	primary	primary	ADJ
ejpam-3948	205	2	cyclotomic	cyclotomic	ADJ
ejpam-3948	205	3	units	unit	NOUN
ejpam-3948	205	4	and	and	CCONJ
ejpam-3948	205	5	a	a	DET
ejpam-3948	205	6	proof	proof	NOUN
ejpam-3948	205	7	of	of	ADP
ejpam-3948	205	8	catalan?s	catalan?s	PROPN
ejpam-3948	205	9	conjecture	conjecture	NOUN
ejpam-3948	205	10	.	.	PUNCT
ejpam-3948	206	1	j.	j.	PROPN
ejpam-3948	206	2	reine	reine	PROPN
ejpam-3948	206	3	angew	angew	PROPN
ejpam-3948	206	4	math	math	PROPN
ejpam-3948	206	5	.	.	PUNCT
ejpam-3948	206	6	,	,	PUNCT
ejpam-3948	206	7	572:167–195	572:167–195	NUM
ejpam-3948	206	8	,	,	PUNCT
ejpam-3948	206	9	2004	2004	NUM
ejpam-3948	206	10	.	.	PUNCT
ejpam-3948	207	1	[	[	X
ejpam-3948	207	2	13	13	NUM
ejpam-3948	207	3	]	]	X
ejpam-3948	207	4	k	k	PROPN
ejpam-3948	207	5	bhatnagar	bhatnagar	PROPN
ejpam-3948	207	6	p	p	PROPN
ejpam-3948	207	7	goel	goel	PROPN
ejpam-3948	207	8	and	and	CCONJ
ejpam-3948	207	9	s	s	NOUN
ejpam-3948	207	10	aggarwal	aggarwal	NOUN
ejpam-3948	207	11	.	.	PUNCT
ejpam-3948	208	1	on	on	ADP
ejpam-3948	208	2	the	the	DET
ejpam-3948	208	3	exponential	exponential	ADJ
ejpam-3948	208	4	diophantine	diophantine	NOUN
ejpam-3948	208	5	equation	equation	NOUN
ejpam-3948	208	6	mp	mp	PROPN
ejpam-3948	208	7	5	5	NUM
ejpam-3948	209	1	+	+	NOUN
ejpam-3948	209	2	m	m	VERB
ejpam-3948	209	3	q	q	ADJ
ejpam-3948	209	4	7	7	NUM
ejpam-3948	209	5	=	=	SYM
ejpam-3948	209	6	r2	r2	PROPN
ejpam-3948	209	7	.	.	PUNCT
ejpam-3948	210	1	international	international	ADJ
ejpam-3948	210	2	journal	journal	PROPN
ejpam-3948	210	3	of	of	ADP
ejpam-3948	210	4	interdisciplinary	interdisciplinary	ADJ
ejpam-3948	210	5	global	global	ADJ
ejpam-3948	210	6	studies	study	NOUN
ejpam-3948	210	7	,	,	PUNCT
ejpam-3948	210	8	14:170–171	14:170–171	NUM
ejpam-3948	210	9	,	,	PUNCT
ejpam-3948	210	10	2020	2020	NUM
ejpam-3948	210	11	.	.	PUNCT
ejpam-3948	211	1	[	[	X
ejpam-3948	211	2	14	14	NUM
ejpam-3948	211	3	]	]	X
ejpam-3948	211	4	j	j	PROPN
ejpam-3948	211	5	f	f	PROPN
ejpam-3948	211	6	t	t	PROPN
ejpam-3948	211	7	rabago	rabago	PROPN
ejpam-3948	211	8	.	.	PUNCT
ejpam-3948	212	1	a	a	DET
ejpam-3948	212	2	note	note	NOUN
ejpam-3948	212	3	on	on	ADP
ejpam-3948	212	4	two	two	NUM
ejpam-3948	212	5	diophantine	diophantine	NOUN
ejpam-3948	212	6	equations	equation	NOUN
ejpam-3948	212	7	17x+19y	17x+19y	NUM
ejpam-3948	212	8	=	=	SYM
ejpam-3948	212	9	z2	z2	PROPN
ejpam-3948	212	10	and	and	CCONJ
ejpam-3948	212	11	71x+73y	71x+73y	NUM
ejpam-3948	212	12	=	=	SYM
ejpam-3948	212	13	z2	z2	PROPN
ejpam-3948	212	14	.	.	PUNCT
ejpam-3948	212	15	math	math	PROPN
ejpam-3948	212	16	.	.	PUNCT
ejpam-3948	213	1	j.	j.	PROPN
ejpam-3948	213	2	interdisciplinary	interdisciplinary	PROPN
ejpam-3948	213	3	sci	sci	PROPN
ejpam-3948	213	4	.	.	PROPN
ejpam-3948	213	5	,	,	PUNCT
ejpam-3948	213	6	2:19–24	2:19–24	NUM
ejpam-3948	213	7	,	,	PUNCT
ejpam-3948	213	8	2013	2013	NUM
ejpam-3948	213	9	.	.	PUNCT
ejpam-3948	214	1	[	[	X
ejpam-3948	214	2	15	15	NUM
ejpam-3948	214	3	]	]	X
ejpam-3948	214	4	j	j	PROPN
ejpam-3948	214	5	f	f	PROPN
ejpam-3948	214	6	t	t	PROPN
ejpam-3948	214	7	rabago	rabago	PROPN
ejpam-3948	214	8	.	.	PUNCT
ejpam-3948	215	1	more	more	ADJ
ejpam-3948	215	2	on	on	ADP
ejpam-3948	215	3	diophantine	diophantine	NOUN
ejpam-3948	215	4	equations	equation	NOUN
ejpam-3948	215	5	of	of	ADP
ejpam-3948	215	6	type	type	NOUN
ejpam-3948	215	7	px	px	PROPN
ejpam-3948	215	8	+	+	CCONJ
ejpam-3948	215	9	qy	qy	NOUN
ejpam-3948	215	10	=	=	PROPN
ejpam-3948	215	11	z2	z2	PROPN
ejpam-3948	215	12	.	.	PUNCT
ejpam-3948	215	13	int	int	PROPN
ejpam-3948	215	14	.	.	PUNCT
ejpam-3948	216	1	j.	j.	PROPN
ejpam-3948	216	2	math	math	PROPN
ejpam-3948	216	3	.	.	PUNCT
ejpam-3948	217	1	sci	sci	PROPN
ejpam-3948	217	2	.	.	PROPN
ejpam-3948	217	3	comp	comp	PROPN
ejpam-3948	217	4	.	.	PUNCT
ejpam-3948	217	5	,	,	PUNCT
ejpam-3948	217	6	3:15–16	3:15–16	NUM
ejpam-3948	217	7	,	,	PUNCT
ejpam-3948	217	8	2013	2013	NUM
ejpam-3948	217	9	.	.	PUNCT
ejpam-3948	218	1	[	[	X
ejpam-3948	218	2	16	16	NUM
ejpam-3948	218	3	]	]	X
ejpam-3948	218	4	j	j	PROPN
ejpam-3948	218	5	f	f	PROPN
ejpam-3948	218	6	t	t	PROPN
ejpam-3948	218	7	rabago	rabago	PROPN
ejpam-3948	218	8	.	.	PUNCT
ejpam-3948	219	1	on	on	ADP
ejpam-3948	219	2	two	two	NUM
ejpam-3948	219	3	diophantine	diophantine	NOUN
ejpam-3948	219	4	equations	equation	NOUN
ejpam-3948	219	5	3x	3x	NUM
ejpam-3948	219	6	+	+	CCONJ
ejpam-3948	219	7	19y	19y	NOUN
ejpam-3948	219	8	=	=	SYM
ejpam-3948	219	9	z2	z2	PROPN
ejpam-3948	219	10	and	and	CCONJ
ejpam-3948	219	11	3x	3x	NUM
ejpam-3948	219	12	+	+	NUM
ejpam-3948	219	13	91y	91y	NOUN
ejpam-3948	219	14	=	=	SYM
ejpam-3948	219	15	z2	z2	PROPN
ejpam-3948	219	16	.	.	PUNCT
ejpam-3948	219	17	int	int	PROPN
ejpam-3948	219	18	.	.	PUNCT
ejpam-3948	220	1	j.	j.	PROPN
ejpam-3948	220	2	math	math	PROPN
ejpam-3948	220	3	.	.	PUNCT
ejpam-3948	221	1	sci	sci	PROPN
ejpam-3948	221	2	.	.	PROPN
ejpam-3948	221	3	comp	comp	PROPN
ejpam-3948	221	4	.	.	PUNCT
ejpam-3948	221	5	,	,	PUNCT
ejpam-3948	221	6	3:28–29	3:28–29	PROPN
ejpam-3948	221	7	,	,	PUNCT
ejpam-3948	221	8	2013	2013	NUM
ejpam-3948	221	9	.	.	PUNCT
ejpam-3948	222	1	[	[	X
ejpam-3948	222	2	17	17	NUM
ejpam-3948	222	3	]	]	SYM
ejpam-3948	222	4	s	s	PROPN
ejpam-3948	222	5	d	d	X
ejpam-3948	222	6	sharma	sharma	PROPN
ejpam-3948	222	7	s	s	PART
ejpam-3948	222	8	aggarwal	aggarwal	NOUN
ejpam-3948	222	9	and	and	CCONJ
ejpam-3948	222	10	n	n	NOUN
ejpam-3948	222	11	sharma	sharma	NOUN
ejpam-3948	222	12	.	.	PUNCT
ejpam-3948	223	1	on	on	ADP
ejpam-3948	223	2	the	the	DET
ejpam-3948	223	3	non	non	ADJ
ejpam-3948	223	4	-	-	ADJ
ejpam-3948	223	5	linear	linear	ADJ
ejpam-3948	223	6	diophantine	diophantine	NOUN
ejpam-3948	223	7	equation	equation	NOUN
ejpam-3948	223	8	313x	313x	NOUN
ejpam-3948	223	9	+	+	CCONJ
ejpam-3948	223	10	331y	331y	PROPN
ejpam-3948	223	11	=	=	SYM
ejpam-3948	223	12	z2	z2	PROPN
ejpam-3948	223	13	.	.	PROPN
ejpam-3948	223	14	journal	journal	PROPN
ejpam-3948	223	15	of	of	ADP
ejpam-3948	223	16	advanced	advanced	ADJ
ejpam-3948	223	17	research	research	NOUN
ejpam-3948	223	18	in	in	ADP
ejpam-3948	223	19	applied	apply	VERB
ejpam-3948	223	20	mathematics	mathematic	NOUN
ejpam-3948	223	21	and	and	CCONJ
ejpam-3948	223	22	statistics	statistic	NOUN
ejpam-3948	223	23	,	,	PUNCT
ejpam-3948	223	24	5:3–5	5:3–5	PROPN
ejpam-3948	223	25	,	,	PUNCT
ejpam-3948	223	26	2020	2020	NUM
ejpam-3948	223	27	.	.	PUNCT
ejpam-3948	224	1	[	[	X
ejpam-3948	224	2	18	18	NUM
ejpam-3948	224	3	]	]	PUNCT
ejpam-3948	224	4	a	a	DET
ejpam-3948	224	5	kumar	kumar	PROPN
ejpam-3948	224	6	s	s	PROPN
ejpam-3948	224	7	kumar	kumar	PROPN
ejpam-3948	224	8	,	,	PUNCT
ejpam-3948	224	9	k	k	PROPN
ejpam-3948	224	10	bhatnagar	bhatnagar	PROPN
ejpam-3948	224	11	and	and	CCONJ
ejpam-3948	224	12	s	s	NOUN
ejpam-3948	224	13	aggarwal	aggarwal	NOUN
ejpam-3948	224	14	.	.	PUNCT
ejpam-3948	225	1	on	on	ADP
ejpam-3948	225	2	the	the	DET
ejpam-3948	225	3	exponential	exponential	ADJ
ejpam-3948	225	4	diophantine	diophantine	NOUN
ejpam-3948	225	5	equation	equation	NOUN
ejpam-3948	225	6	(	(	PUNCT
ejpam-3948	225	7	22m+1	22m+1	NUM
ejpam-3948	225	8	−	−	NOUN
ejpam-3948	225	9	1	1	NUM
ejpam-3948	225	10	)	)	PUNCT
ejpam-3948	225	11	+	+	CCONJ
ejpam-3948	225	12	(	(	PUNCT
ejpam-3948	225	13	6r	6r	NUM
ejpam-3948	225	14	+	+	CCONJ
ejpam-3948	225	15	1)n	1)n	X
ejpam-3948	225	16	=	=	SYM
ejpam-3948	225	17	ω2	ω2	ADJ
ejpam-3948	225	18	.	.	PUNCT
ejpam-3948	226	1	international	international	ADJ
ejpam-3948	226	2	journal	journal	PROPN
ejpam-3948	226	3	of	of	ADP
ejpam-3948	226	4	interdisciplinary	interdisciplinary	ADJ
ejpam-3948	226	5	global	global	ADJ
ejpam-3948	226	6	studies	study	NOUN
ejpam-3948	226	7	,	,	PUNCT
ejpam-3948	226	8	14:183–184	14:183–184	PROPN
ejpam-3948	226	9	,	,	PUNCT
ejpam-3948	226	10	2020	2020	NUM
ejpam-3948	226	11	.	.	PUNCT
ejpam-3948	227	1	[	[	X
ejpam-3948	227	2	19	19	NUM
ejpam-3948	227	3	]	]	PUNCT
ejpam-3948	227	4	a	a	DET
ejpam-3948	227	5	kumar	kumar	PROPN
ejpam-3948	227	6	s	s	PROPN
ejpam-3948	227	7	kumar	kumar	PROPN
ejpam-3948	227	8	,	,	PUNCT
ejpam-3948	227	9	k	k	PROPN
ejpam-3948	227	10	bhatnagar	bhatnagar	PROPN
ejpam-3948	227	11	and	and	CCONJ
ejpam-3948	227	12	s	s	NOUN
ejpam-3948	227	13	aggarwal	aggarwal	NOUN
ejpam-3948	227	14	.	.	PUNCT
ejpam-3948	228	1	on	on	ADP
ejpam-3948	228	2	the	the	DET
ejpam-3948	228	3	exponential	exponential	ADJ
ejpam-3948	228	4	diophantine	diophantine	NOUN
ejpam-3948	228	5	equation	equation	NOUN
ejpam-3948	228	6	72	72	NUM
ejpam-3948	228	7	m	m	VERB
ejpam-3948	228	8	+	+	CCONJ
ejpam-3948	228	9	(	(	PUNCT
ejpam-3948	228	10	6r	6r	NUM
ejpam-3948	228	11	+	+	CCONJ
ejpam-3948	228	12	1)n	1)n	X
ejpam-3948	228	13	=	=	SYM
ejpam-3948	228	14	z2	z2	PROPN
ejpam-3948	228	15	.	.	PUNCT
ejpam-3948	228	16	international	international	ADJ
ejpam-3948	228	17	journal	journal	PROPN
ejpam-3948	228	18	of	of	ADP
ejpam-3948	228	19	interdisciplinary	interdisciplinary	ADJ
ejpam-3948	228	20	global	global	ADJ
ejpam-3948	228	21	studies	study	NOUN
ejpam-3948	228	22	,	,	PUNCT
ejpam-3948	228	23	14:181–182	14:181–182	NUM
ejpam-3948	228	24	,	,	PUNCT
ejpam-3948	228	25	2020	2020	NUM
ejpam-3948	228	26	.	.	PUNCT
ejpam-3948	229	1	[	[	X
ejpam-3948	229	2	20	20	NUM
ejpam-3948	229	3	]	]	SYM
ejpam-3948	229	4	s	s	PART
ejpam-3948	229	5	gupta	gupta	PROPN
ejpam-3948	229	6	s	s	PART
ejpam-3948	229	7	kumar	kumar	PROPN
ejpam-3948	229	8	and	and	CCONJ
ejpam-3948	229	9	h	h	PROPN
ejpam-3948	229	10	kishan	kishan	PROPN
ejpam-3948	229	11	.	.	PUNCT
ejpam-3948	230	1	on	on	ADP
ejpam-3948	230	2	the	the	DET
ejpam-3948	230	3	non	non	ADJ
ejpam-3948	230	4	-	-	ADJ
ejpam-3948	230	5	linear	linear	ADJ
ejpam-3948	230	6	diophantine	diophantine	NOUN
ejpam-3948	230	7	equations	equation	NOUN
ejpam-3948	230	8	31x+41y	31x+41y	NUM
ejpam-3948	230	9	=	=	SYM
ejpam-3948	230	10	z2	z2	PROPN
ejpam-3948	230	11	and	and	CCONJ
ejpam-3948	230	12	61x	61x	NOUN
ejpam-3948	230	13	+	+	SYM
ejpam-3948	230	14	71y	71y	X
ejpam-3948	230	15	=	=	SYM
ejpam-3948	230	16	z2	z2	PROPN
ejpam-3948	230	17	.	.	PUNCT
ejpam-3948	230	18	annals	annal	NOUN
ejpam-3948	230	19	of	of	ADP
ejpam-3948	230	20	pure	pure	ADJ
ejpam-3948	230	21	and	and	CCONJ
ejpam-3948	230	22	applied	applied	ADJ
ejpam-3948	230	23	mathematics	mathematic	NOUN
ejpam-3948	230	24	,	,	PUNCT
ejpam-3948	230	25	18:185–188	18:185–188	PROPN
ejpam-3948	230	26	,	,	PUNCT
ejpam-3948	230	27	2018	2018	NUM
ejpam-3948	230	28	.	.	PUNCT
ejpam-3948	231	1	references	reference	NOUN
ejpam-3948	231	2	403	403	NUM
ejpam-3948	232	1	[	[	X
ejpam-3948	232	2	21	21	NUM
ejpam-3948	232	3	]	]	X
ejpam-3948	232	4	s	s	PART
ejpam-3948	232	5	gupta	gupta	PROPN
ejpam-3948	232	6	s	s	PART
ejpam-3948	232	7	kumar	kumar	PROPN
ejpam-3948	232	8	and	and	CCONJ
ejpam-3948	232	9	h	h	PROPN
ejpam-3948	232	10	kishan	kishan	PROPN
ejpam-3948	232	11	.	.	PUNCT
ejpam-3948	233	1	on	on	ADP
ejpam-3948	233	2	the	the	DET
ejpam-3948	233	3	non	non	ADJ
ejpam-3948	233	4	-	-	ADJ
ejpam-3948	233	5	linear	linear	ADJ
ejpam-3948	233	6	diophantine	diophantine	NOUN
ejpam-3948	233	7	equations	equation	NOUN
ejpam-3948	233	8	61x+67y	61x+67y	NUM
ejpam-3948	233	9	=	=	SYM
ejpam-3948	233	10	z2	z2	PROPN
ejpam-3948	233	11	and	and	CCONJ
ejpam-3948	233	12	67x	67x	NUM
ejpam-3948	233	13	+	+	NUM
ejpam-3948	233	14	73y	73y	NOUN
ejpam-3948	233	15	=	=	SYM
ejpam-3948	233	16	z2	z2	PROPN
ejpam-3948	233	17	.	.	PUNCT
ejpam-3948	233	18	annals	annal	NOUN
ejpam-3948	233	19	of	of	ADP
ejpam-3948	233	20	pure	pure	ADJ
ejpam-3948	233	21	and	and	CCONJ
ejpam-3948	233	22	applied	applied	ADJ
ejpam-3948	233	23	mathematics	mathematic	NOUN
ejpam-3948	233	24	,	,	PUNCT
ejpam-3948	233	25	18:91–94	18:91–94	NUM
ejpam-3948	233	26	,	,	PUNCT
ejpam-3948	233	27	2018	2018	NUM
ejpam-3948	233	28	.	.	PUNCT
ejpam-3948	234	1	[	[	X
ejpam-3948	234	2	22	22	NUM
ejpam-3948	234	3	]	]	SYM
ejpam-3948	234	4	b	b	X
ejpam-3948	234	5	sroysang	sroysang	PROPN
ejpam-3948	234	6	.	.	PUNCT
ejpam-3948	235	1	more	more	ADJ
ejpam-3948	235	2	on	on	ADP
ejpam-3948	235	3	the	the	DET
ejpam-3948	235	4	diophantine	diophantine	NOUN
ejpam-3948	235	5	equation	equation	NOUN
ejpam-3948	235	6	8x	8x	NOUN
ejpam-3948	235	7	+	+	CCONJ
ejpam-3948	235	8	19y	19y	NOUN
ejpam-3948	235	9	=	=	SYM
ejpam-3948	235	10	z2	z2	PROPN
ejpam-3948	235	11	.	.	PUNCT
ejpam-3948	236	1	int	int	PROPN
ejpam-3948	236	2	.	.	PUNCT
ejpam-3948	237	1	j.	j.	PROPN
ejpam-3948	237	2	pure	pure	PROPN
ejpam-3948	237	3	appl	appl	PROPN
ejpam-3948	237	4	.	.	PUNCT
ejpam-3948	237	5	math	math	PROPN
ejpam-3948	237	6	.	.	PUNCT
ejpam-3948	237	7	,	,	PUNCT
ejpam-3948	238	1	81:601–604	81:601–604	PROPN
ejpam-3948	238	2	,	,	PUNCT
ejpam-3948	238	3	2012	2012	NUM
ejpam-3948	238	4	.	.	PUNCT
ejpam-3948	239	1	[	[	X
ejpam-3948	239	2	23	23	NUM
ejpam-3948	239	3	]	]	SYM
ejpam-3948	239	4	b	b	X
ejpam-3948	239	5	sroysang	sroysang	PROPN
ejpam-3948	239	6	.	.	PUNCT
ejpam-3948	240	1	on	on	ADP
ejpam-3948	240	2	the	the	DET
ejpam-3948	240	3	diophantine	diophantine	NOUN
ejpam-3948	240	4	equation	equation	NOUN
ejpam-3948	240	5	31x	31x	NOUN
ejpam-3948	240	6	+	+	SYM
ejpam-3948	240	7	32y	32y	NOUN
ejpam-3948	240	8	=	=	SYM
ejpam-3948	240	9	z2	z2	PROPN
ejpam-3948	240	10	.	.	PUNCT
ejpam-3948	240	11	int	int	PROPN
ejpam-3948	240	12	.	.	PUNCT
ejpam-3948	241	1	j.	j.	PROPN
ejpam-3948	241	2	pure	pure	PROPN
ejpam-3948	241	3	appl	appl	PROPN
ejpam-3948	241	4	.	.	PUNCT
ejpam-3948	241	5	math	math	PROPN
ejpam-3948	241	6	.	.	PUNCT
ejpam-3948	241	7	,	,	PUNCT
ejpam-3948	241	8	81:609–612	81:609–612	NUM
ejpam-3948	241	9	,	,	PUNCT
ejpam-3948	241	10	2012	2012	NUM
ejpam-3948	241	11	.	.	PUNCT
ejpam-3948	242	1	[	[	X
ejpam-3948	242	2	24	24	NUM
ejpam-3948	242	3	]	]	SYM
ejpam-3948	242	4	b	b	X
ejpam-3948	242	5	sroysang	sroysang	PROPN
ejpam-3948	242	6	.	.	PUNCT
ejpam-3948	243	1	on	on	ADP
ejpam-3948	243	2	the	the	DET
ejpam-3948	243	3	diophantine	diophantine	NOUN
ejpam-3948	243	4	equation	equation	NOUN
ejpam-3948	243	5	3x	3x	NUM
ejpam-3948	243	6	+	+	CCONJ
ejpam-3948	243	7	5y	5y	NOUN
ejpam-3948	243	8	=	=	SYM
ejpam-3948	243	9	z2	z2	PROPN
ejpam-3948	243	10	.	.	PUNCT
ejpam-3948	244	1	int	int	PROPN
ejpam-3948	244	2	.	.	PUNCT
ejpam-3948	245	1	j.	j.	PROPN
ejpam-3948	245	2	pure	pure	PROPN
ejpam-3948	245	3	appl	appl	PROPN
ejpam-3948	245	4	.	.	PUNCT
ejpam-3948	245	5	math	math	PROPN
ejpam-3948	245	6	.	.	PUNCT
ejpam-3948	245	7	,	,	PUNCT
ejpam-3948	245	8	81:605–608	81:605–608	NUM
ejpam-3948	245	9	,	,	PUNCT
ejpam-3948	245	10	2012	2012	NUM
ejpam-3948	245	11	.	.	PUNCT
ejpam-3948	246	1	[	[	X
ejpam-3948	246	2	25	25	NUM
ejpam-3948	246	3	]	]	SYM
ejpam-3948	246	4	b	b	X
ejpam-3948	246	5	sroysang	sroysang	PROPN
ejpam-3948	246	6	.	.	PUNCT
ejpam-3948	247	1	more	more	ADJ
ejpam-3948	247	2	on	on	ADP
ejpam-3948	247	3	the	the	DET
ejpam-3948	247	4	diophantine	diophantine	NOUN
ejpam-3948	247	5	equation	equation	NOUN
ejpam-3948	247	6	2x	2x	NUM
ejpam-3948	247	7	+	+	CCONJ
ejpam-3948	247	8	3y	3y	ADJ
ejpam-3948	247	9	=	=	SYM
ejpam-3948	247	10	z2	z2	PROPN
ejpam-3948	247	11	.	.	PUNCT
ejpam-3948	247	12	int	int	PROPN
ejpam-3948	247	13	.	.	PUNCT
ejpam-3948	248	1	j.	j.	PROPN
ejpam-3948	248	2	pure	pure	PROPN
ejpam-3948	248	3	appl	appl	PROPN
ejpam-3948	248	4	.	.	PUNCT
ejpam-3948	248	5	math	math	PROPN
ejpam-3948	248	6	.	.	PUNCT
ejpam-3948	248	7	,	,	PUNCT
ejpam-3948	248	8	84:133–137	84:133–137	NOUN
ejpam-3948	248	9	,	,	PUNCT
ejpam-3948	248	10	2013	2013	NUM
ejpam-3948	248	11	.	.	PUNCT
ejpam-3948	249	1	[	[	X
ejpam-3948	249	2	26	26	NUM
ejpam-3948	249	3	]	]	SYM
ejpam-3948	249	4	b	b	X
ejpam-3948	249	5	sroysang	sroysang	PROPN
ejpam-3948	249	6	.	.	PUNCT
ejpam-3948	250	1	on	on	ADP
ejpam-3948	250	2	the	the	DET
ejpam-3948	250	3	diophantine	diophantine	NOUN
ejpam-3948	250	4	equation	equation	NOUN
ejpam-3948	250	5	7x	7x	NUM
ejpam-3948	250	6	+	+	CCONJ
ejpam-3948	250	7	8y	8y	NUM
ejpam-3948	250	8	=	=	SYM
ejpam-3948	250	9	z2	z2	PROPN
ejpam-3948	250	10	.	.	PUNCT
ejpam-3948	250	11	int	int	PROPN
ejpam-3948	250	12	.	.	PUNCT
ejpam-3948	251	1	j.	j.	PROPN
ejpam-3948	251	2	pure	pure	PROPN
ejpam-3948	251	3	appl	appl	PROPN
ejpam-3948	251	4	.	.	PUNCT
ejpam-3948	251	5	math	math	PROPN
ejpam-3948	251	6	.	.	PUNCT
ejpam-3948	251	7	,	,	PUNCT
ejpam-3948	251	8	84:111–114	84:111–114	PROPN
ejpam-3948	251	9	,	,	PUNCT
ejpam-3948	251	10	2013	2013	NUM
ejpam-3948	251	11	.	.	PUNCT
ejpam-3948	252	1	[	[	X
ejpam-3948	252	2	27	27	NUM
ejpam-3948	252	3	]	]	SYM
ejpam-3948	252	4	b	b	X
ejpam-3948	252	5	sroysang	sroysang	PROPN
ejpam-3948	252	6	.	.	PUNCT
ejpam-3948	253	1	on	on	ADP
ejpam-3948	253	2	the	the	DET
ejpam-3948	253	3	diophantine	diophantine	NOUN
ejpam-3948	253	4	equation	equation	NOUN
ejpam-3948	253	5	8x	8x	NOUN
ejpam-3948	253	6	+	+	CCONJ
ejpam-3948	253	7	13y	13y	NOUN
ejpam-3948	253	8	=	=	SYM
ejpam-3948	253	9	z2	z2	PROPN
ejpam-3948	253	10	.	.	PUNCT
ejpam-3948	254	1	int	int	PROPN
ejpam-3948	254	2	.	.	PUNCT
ejpam-3948	255	1	j.	j.	PROPN
ejpam-3948	255	2	pure	pure	PROPN
ejpam-3948	255	3	appl	appl	PROPN
ejpam-3948	255	4	.	.	PUNCT
ejpam-3948	255	5	math	math	PROPN
ejpam-3948	255	6	.	.	PUNCT
ejpam-3948	255	7	,	,	PUNCT
ejpam-3948	255	8	90:69–72	90:69–72	NUM
ejpam-3948	255	9	,	,	PUNCT
ejpam-3948	255	10	2014	2014	NUM
ejpam-3948	255	11	.	.	PUNCT
ejpam-3948	256	1	[	[	X
ejpam-3948	256	2	28	28	NUM
ejpam-3948	256	3	]	]	X
ejpam-3948	256	4	d	d	X
ejpam-3948	256	5	andrica	andrica	PROPN
ejpam-3948	256	6	t	t	PROPN
ejpam-3948	256	7	andreescu	andreescu	NOUN
ejpam-3948	256	8	and	and	CCONJ
ejpam-3948	256	9	i	i	PRON
ejpam-3948	256	10	cucurezeanu	cucurezeanu	NOUN
ejpam-3948	256	11	.	.	PUNCT
ejpam-3948	257	1	an	an	DET
ejpam-3948	257	2	introduction	introduction	NOUN
ejpam-3948	257	3	to	to	PART
ejpam-3948	257	4	diophantine	diophantine	VERB
ejpam-3948	257	5	equations	equation	NOUN
ejpam-3948	257	6	(	(	PUNCT
ejpam-3948	257	7	a	a	DET
ejpam-3948	257	8	problem	problem	NOUN
ejpam-3948	257	9	-	-	PUNCT
ejpam-3948	257	10	based	base	VERB
ejpam-3948	257	11	approach	approach	NOUN
ejpam-3948	257	12	)	)	PUNCT
ejpam-3948	257	13	.	.	PUNCT
ejpam-3948	258	1	springer	springer	NOUN
ejpam-3948	258	2	,	,	PUNCT
ejpam-3948	258	3	new	new	PROPN
ejpam-3948	258	4	york	york	PROPN
ejpam-3948	258	5	,	,	PUNCT
ejpam-3948	258	6	2010	2010	NUM
ejpam-3948	258	7	.	.	PUNCT
ejpam-3948	259	1	[	[	X
ejpam-3948	259	2	29	29	NUM
ejpam-3948	259	3	]	]	PUNCT
ejpam-3948	259	4	t	t	PROPN
ejpam-3948	259	5	wang	wang	PROPN
ejpam-3948	259	6	and	and	CCONJ
ejpam-3948	259	7	y	y	PROPN
ejpam-3948	259	8	jiang	jiang	PROPN
ejpam-3948	259	9	.	.	PUNCT
ejpam-3948	260	1	on	on	ADP
ejpam-3948	260	2	the	the	DET
ejpam-3948	260	3	number	number	NOUN
ejpam-3948	260	4	of	of	ADP
ejpam-3948	260	5	positive	positive	ADJ
ejpam-3948	260	6	integer	integer	NOUN
ejpam-3948	260	7	solutions	solution	NOUN
ejpam-3948	260	8	(	(	PUNCT
ejpam-3948	260	9	x	x	NOUN
ejpam-3948	260	10	,	,	PUNCT
ejpam-3948	260	11	n	n	CCONJ
ejpam-3948	260	12	)	)	PUNCT
ejpam-3948	260	13	of	of	ADP
ejpam-3948	260	14	the	the	DET
ejpam-3948	260	15	generalized	generalized	ADJ
ejpam-3948	260	16	ramanujan	ramanujan	PROPN
ejpam-3948	260	17	-	-	PUNCT
ejpam-3948	260	18	nagell	nagell	PROPN
ejpam-3948	260	19	equation	equation	NOUN
ejpam-3948	260	20	x2−	x2−	PROPN
ejpam-3948	260	21	2r	2r	NUM
ejpam-3948	261	1	=	=	SYM
ejpam-3948	261	2	pn	pn	PROPN
ejpam-3948	261	3	.	.	PUNCT
ejpam-3948	262	1	(	(	PUNCT
ejpam-3948	262	2	norwegian	norwegian	PROPN
ejpam-3948	262	3	)	)	PUNCT
ejpam-3948	262	4	norsk	norsk	PROPN
ejpam-3948	262	5	mat	mat	PROPN
ejpam-3948	262	6	.	.	PUNCT
ejpam-3948	262	7	tidsskr	tidsskr	PROPN
ejpam-3948	262	8	.	.	PROPN
ejpam-3948	262	9	,	,	PUNCT
ejpam-3948	262	10	25:17–20	25:17–20	NUM
ejpam-3948	262	11	,	,	PUNCT
ejpam-3948	262	12	1943	1943	NUM
ejpam-3948	262	13	.	.	PUNCT
ejpam-3948	263	1	[	[	X
ejpam-3948	263	2	30	30	NUM
ejpam-3948	263	3	]	]	X
ejpam-3948	263	4	j	j	PROPN
ejpam-3948	263	5	m	m	PROPN
ejpam-3948	263	6	yang	yang	PROPN
ejpam-3948	263	7	.	.	PUNCT
ejpam-3948	264	1	the	the	DET
ejpam-3948	264	2	number	number	NOUN
ejpam-3948	264	3	of	of	ADP
ejpam-3948	264	4	solutions	solution	NOUN
ejpam-3948	264	5	of	of	ADP
ejpam-3948	264	6	the	the	DET
ejpam-3948	264	7	generalized	generalize	VERB
ejpam-3948	264	8	ramanujan	ramanujan	PROPN
ejpam-3948	264	9	-	-	PUNCT
ejpam-3948	264	10	nagell	nagell	PROPN
ejpam-3948	264	11	equation	equation	NOUN
ejpam-3948	264	12	x2	x2	PROPN
ejpam-3948	264	13	−d	−d	VERB
ejpam-3948	264	14	=	=	PUNCT
ejpam-3948	265	1	3n	3n	NUM
ejpam-3948	265	2	.	.	PUNCT
ejpam-3948	266	1	j.	j.	PROPN
ejpam-3948	266	2	math	math	PROPN
ejpam-3948	266	3	.	.	PUNCT
ejpam-3948	266	4	,	,	PUNCT
ejpam-3948	267	1	51:351–356	51:351–356	PROPN
ejpam-3948	267	2	,	,	PUNCT
ejpam-3948	267	3	2008	2008	NUM
ejpam-3948	267	4	.	.	PUNCT
ejpam-3948	268	1	[	[	X
ejpam-3948	268	2	31	31	NUM
ejpam-3948	268	3	]	]	X
ejpam-3948	268	4	p	p	PROPN
ejpam-3948	268	5	z	z	PROPN
ejpam-3948	268	6	yuan	yuan	NOUN
ejpam-3948	268	7	.	.	PUNCT
ejpam-3948	269	1	on	on	ADP
ejpam-3948	269	2	the	the	DET
ejpam-3948	269	3	number	number	NOUN
ejpam-3948	269	4	of	of	ADP
ejpam-3948	269	5	solutions	solution	NOUN
ejpam-3948	269	6	of	of	ADP
ejpam-3948	269	7	x2	x2	PROPN
ejpam-3948	269	8	−d	−d	PROPN
ejpam-3948	269	9	=	=	SYM
ejpam-3948	269	10	pn	pn	PROPN
ejpam-3948	269	11	.	.	PROPN
ejpam-3948	269	12	j.	j.	PROPN
ejpam-3948	269	13	sichuan	sichuan	PROPN
ejpam-3948	269	14	univ	univ	PROPN
ejpam-3948	269	15	.	.	PUNCT
ejpam-3948	270	1	nat	nat	PROPN
ejpam-3948	270	2	.	.	PUNCT
ejpam-3948	271	1	sci	sci	PROPN
ejpam-3948	271	2	.	.	PUNCT
ejpam-3948	272	1	ed	ed	PROPN
ejpam-3948	272	2	.	.	PROPN
ejpam-3948	272	3	,	,	PUNCT
ejpam-3948	273	1	35:311–316	35:311–316	NUM
ejpam-3948	273	2	,	,	PUNCT
ejpam-3948	273	3	1998	1998	NUM
ejpam-3948	273	4	.	.	PUNCT