id	sid	tid	token	lemma	pos
ejpam-4936	1	1	european	european	PROPN
ejpam-4936	1	2	journal	journal	PROPN
ejpam-4936	1	3	of	of	ADP
ejpam-4936	1	4	pure	pure	ADJ
ejpam-4936	1	5	and	and	CCONJ
ejpam-4936	1	6	applied	apply	VERB
ejpam-4936	1	7	mathematics	mathematic	NOUN
ejpam-4936	1	8	vol	vol	NOUN
ejpam-4936	1	9	.	.	PUNCT
ejpam-4936	2	1	16	16	NUM
ejpam-4936	2	2	,	,	PUNCT
ejpam-4936	2	3	no	no	INTJ
ejpam-4936	2	4	.	.	NOUN
ejpam-4936	2	5	4	4	NUM
ejpam-4936	2	6	,	,	PUNCT
ejpam-4936	2	7	2023	2023	NUM
ejpam-4936	2	8	,	,	PUNCT
ejpam-4936	2	9	2066	2066	NUM
ejpam-4936	2	10	-	-	SYM
ejpam-4936	2	11	2081	2081	NUM
ejpam-4936	2	12	issn	issn	PROPN
ejpam-4936	2	13	1307	1307	NUM
ejpam-4936	2	14	-	-	SYM
ejpam-4936	2	15	5543	5543	NUM
ejpam-4936	2	16	–	–	PUNCT
ejpam-4936	2	17	ejpam.com	ejpam.com	X
ejpam-4936	2	18	published	publish	VERB
ejpam-4936	2	19	by	by	ADP
ejpam-4936	2	20	new	new	PROPN
ejpam-4936	2	21	york	york	PROPN
ejpam-4936	2	22	business	business	PROPN
ejpam-4936	2	23	global	global	PROPN
ejpam-4936	2	24	on	on	ADP
ejpam-4936	2	25	the	the	DET
ejpam-4936	2	26	diophantine	diophantine	NOUN
ejpam-4936	2	27	equations	equation	NOUN
ejpam-4936	2	28	ax	ax	NOUN
ejpam-4936	2	29	+	+	CCONJ
ejpam-4936	2	30	by	by	ADP
ejpam-4936	2	31	+	+	CCONJ
ejpam-4936	2	32	cz	cz	NOUN
ejpam-4936	2	33	=	=	SYM
ejpam-4936	2	34	w2	w2	NOUN
ejpam-4936	2	35	kittipong	kittipong	PROPN
ejpam-4936	2	36	laipaporn1	laipaporn1	PROPN
ejpam-4936	2	37	,	,	PUNCT
ejpam-4936	2	38	saowapak	saowapak	PROPN
ejpam-4936	2	39	kaewchay1	kaewchay1	PROPN
ejpam-4936	2	40	,	,	PUNCT
ejpam-4936	2	41	adisak	adisak	PROPN
ejpam-4936	2	42	karnbanjong1,∗	karnbanjong1,∗	NOUN
ejpam-4936	2	43	1	1	NUM
ejpam-4936	2	44	department	department	NOUN
ejpam-4936	2	45	of	of	ADP
ejpam-4936	2	46	mathematics	mathematic	NOUN
ejpam-4936	2	47	and	and	CCONJ
ejpam-4936	2	48	statistics	statistic	NOUN
ejpam-4936	2	49	,	,	PUNCT
ejpam-4936	2	50	center	center	NOUN
ejpam-4936	2	51	of	of	ADP
ejpam-4936	2	52	excellence	excellence	NOUN
ejpam-4936	2	53	for	for	ADP
ejpam-4936	2	54	ecoinformatics	ecoinformatic	NOUN
ejpam-4936	2	55	,	,	PUNCT
ejpam-4936	2	56	school	school	NOUN
ejpam-4936	2	57	of	of	ADP
ejpam-4936	2	58	science	science	NOUN
ejpam-4936	2	59	,	,	PUNCT
ejpam-4936	2	60	walailak	walailak	ADJ
ejpam-4936	2	61	university	university	NOUN
ejpam-4936	2	62	,	,	PUNCT
ejpam-4936	2	63	nakhon	nakhon	PROPN
ejpam-4936	2	64	si	si	PROPN
ejpam-4936	2	65	thammarat	thammarat	PROPN
ejpam-4936	2	66	,	,	PUNCT
ejpam-4936	2	67	80160	80160	NUM
ejpam-4936	2	68	,	,	PUNCT
ejpam-4936	2	69	thailand	thailand	PROPN
ejpam-4936	2	70	abstract	abstract	NOUN
ejpam-4936	2	71	.	.	PUNCT
ejpam-4936	3	1	over	over	ADP
ejpam-4936	3	2	the	the	DET
ejpam-4936	3	3	past	past	ADJ
ejpam-4936	3	4	decade	decade	NOUN
ejpam-4936	3	5	,	,	PUNCT
ejpam-4936	3	6	exponential	exponential	ADJ
ejpam-4936	3	7	diophantine	diophantine	NOUN
ejpam-4936	3	8	equations	equation	NOUN
ejpam-4936	3	9	of	of	ADP
ejpam-4936	3	10	the	the	DET
ejpam-4936	3	11	form	form	NOUN
ejpam-4936	3	12	ax	ax	NOUN
ejpam-4936	3	13	+	+	CCONJ
ejpam-4936	3	14	by	by	ADP
ejpam-4936	3	15	=	=	PUNCT
ejpam-4936	3	16	wn	wn	PROPN
ejpam-4936	3	17	have	have	AUX
ejpam-4936	3	18	been	be	AUX
ejpam-4936	3	19	studied	study	VERB
ejpam-4936	3	20	as	as	SCONJ
ejpam-4936	3	21	if	if	SCONJ
ejpam-4936	3	22	they	they	PRON
ejpam-4936	3	23	were	be	AUX
ejpam-4936	3	24	a	a	DET
ejpam-4936	3	25	phenomenon	phenomenon	NOUN
ejpam-4936	3	26	.	.	PUNCT
ejpam-4936	4	1	in	in	ADP
ejpam-4936	4	2	particular	particular	ADJ
ejpam-4936	4	3	,	,	PUNCT
ejpam-4936	4	4	numerous	numerous	ADJ
ejpam-4936	4	5	articles	article	NOUN
ejpam-4936	4	6	have	have	AUX
ejpam-4936	4	7	focused	focus	VERB
ejpam-4936	4	8	on	on	ADP
ejpam-4936	4	9	the	the	DET
ejpam-4936	4	10	cases	case	NOUN
ejpam-4936	4	11	where	where	SCONJ
ejpam-4936	4	12	n	n	NOUN
ejpam-4936	4	13	=	=	SYM
ejpam-4936	4	14	2	2	NUM
ejpam-4936	4	15	or	or	CCONJ
ejpam-4936	4	16	n	n	NOUN
ejpam-4936	4	17	=	=	SYM
ejpam-4936	4	18	4	4	NUM
ejpam-4936	4	19	and	and	CCONJ
ejpam-4936	4	20	2	2	NUM
ejpam-4936	4	21	≤	≤	NOUN
ejpam-4936	4	22	a	a	PRON
ejpam-4936	4	23	,	,	PUNCT
ejpam-4936	4	24	b	b	NOUN
ejpam-4936	4	25	≤	≤	ADV
ejpam-4936	4	26	200	200	NUM
ejpam-4936	4	27	.	.	PUNCT
ejpam-4936	5	1	however	however	ADV
ejpam-4936	5	2	,	,	PUNCT
ejpam-4936	5	3	these	these	DET
ejpam-4936	5	4	articles	article	NOUN
ejpam-4936	5	5	are	be	AUX
ejpam-4936	5	6	primarily	primarily	ADV
ejpam-4936	5	7	concerned	concerned	ADJ
ejpam-4936	5	8	with	with	ADP
ejpam-4936	5	9	determining	determine	VERB
ejpam-4936	5	10	whether	whether	SCONJ
ejpam-4936	5	11	the	the	DET
ejpam-4936	5	12	left	leave	VERB
ejpam-4936	5	13	-	-	PUNCT
ejpam-4936	5	14	hand	hand	NOUN
ejpam-4936	5	15	side	side	NOUN
ejpam-4936	5	16	of	of	ADP
ejpam-4936	5	17	the	the	DET
ejpam-4936	5	18	equation	equation	NOUN
ejpam-4936	5	19	needs	need	VERB
ejpam-4936	5	20	to	to	PART
ejpam-4936	5	21	consist	consist	VERB
ejpam-4936	5	22	of	of	ADP
ejpam-4936	5	23	more	more	ADJ
ejpam-4936	5	24	than	than	ADP
ejpam-4936	5	25	two	two	NUM
ejpam-4936	5	26	exponentials	exponential	NOUN
ejpam-4936	5	27	.	.	PUNCT
ejpam-4936	6	1	therefore	therefore	ADV
ejpam-4936	6	2	,	,	PUNCT
ejpam-4936	6	3	in	in	ADP
ejpam-4936	6	4	this	this	DET
ejpam-4936	6	5	article	article	NOUN
ejpam-4936	6	6	,	,	PUNCT
ejpam-4936	6	7	we	we	PRON
ejpam-4936	6	8	investigate	investigate	VERB
ejpam-4936	6	9	the	the	DET
ejpam-4936	6	10	exponential	exponential	ADJ
ejpam-4936	6	11	diophantine	diophantine	NOUN
ejpam-4936	6	12	equation	equation	NOUN
ejpam-4936	6	13	in	in	ADP
ejpam-4936	6	14	the	the	DET
ejpam-4936	6	15	form	form	NOUN
ejpam-4936	6	16	ax	ax	NOUN
ejpam-4936	6	17	+	+	CCONJ
ejpam-4936	6	18	by	by	ADP
ejpam-4936	6	19	+	+	CCONJ
ejpam-4936	6	20	cz	cz	NOUN
ejpam-4936	6	21	=	=	SYM
ejpam-4936	6	22	w2	w2	NOUN
ejpam-4936	6	23	,	,	PUNCT
ejpam-4936	6	24	using	use	VERB
ejpam-4936	6	25	only	only	ADJ
ejpam-4936	6	26	elementary	elementary	ADJ
ejpam-4936	6	27	tools	tool	NOUN
ejpam-4936	6	28	related	relate	VERB
ejpam-4936	6	29	to	to	ADP
ejpam-4936	6	30	modulo	modulo	ADJ
ejpam-4936	6	31	concepts	concept	NOUN
ejpam-4936	6	32	.	.	PUNCT
ejpam-4936	7	1	we	we	PRON
ejpam-4936	7	2	present	present	VERB
ejpam-4936	7	3	three	three	NUM
ejpam-4936	7	4	theorems	theorem	NOUN
ejpam-4936	7	5	in	in	ADP
ejpam-4936	7	6	which	which	PRON
ejpam-4936	7	7	the	the	DET
ejpam-4936	7	8	variables	variable	NOUN
ejpam-4936	7	9	a	a	PRON
ejpam-4936	7	10	,	,	PUNCT
ejpam-4936	7	11	b	b	PROPN
ejpam-4936	7	12	and	and	CCONJ
ejpam-4936	7	13	c	c	PROPN
ejpam-4936	7	14	vary	vary	VERB
ejpam-4936	7	15	under	under	ADP
ejpam-4936	7	16	certain	certain	ADJ
ejpam-4936	7	17	conditions	condition	NOUN
ejpam-4936	7	18	,	,	PUNCT
ejpam-4936	7	19	and	and	CCONJ
ejpam-4936	7	20	three	three	NUM
ejpam-4936	7	21	additional	additional	ADJ
ejpam-4936	7	22	theorems	theorem	NOUN
ejpam-4936	7	23	where	where	SCONJ
ejpam-4936	7	24	the	the	DET
ejpam-4936	7	25	variable	variable	NOUN
ejpam-4936	7	26	c	c	PROPN
ejpam-4936	7	27	is	be	AUX
ejpam-4936	7	28	fixed	fix	VERB
ejpam-4936	7	29	at	at	ADP
ejpam-4936	7	30	7	7	NUM
ejpam-4936	7	31	.	.	PUNCT
ejpam-4936	8	1	furthermore	furthermore	ADV
ejpam-4936	8	2	,	,	PUNCT
ejpam-4936	8	3	if	if	SCONJ
ejpam-4936	8	4	we	we	PRON
ejpam-4936	8	5	restrict	restrict	VERB
ejpam-4936	8	6	our	our	PRON
ejpam-4936	8	7	parameters	parameter	NOUN
ejpam-4936	8	8	a	a	DET
ejpam-4936	8	9	,	,	PUNCT
ejpam-4936	8	10	b	b	PROPN
ejpam-4936	8	11	and	and	CCONJ
ejpam-4936	8	12	c	c	NOUN
ejpam-4936	8	13	to	to	ADP
ejpam-4936	8	14	2	2	NUM
ejpam-4936	8	15	≤	≤	NOUN
ejpam-4936	8	16	a	a	DET
ejpam-4936	8	17	≤	≤	NUM
ejpam-4936	8	18	b	b	NOUN
ejpam-4936	8	19	≤	≤	NUM
ejpam-4936	8	20	c	c	NOUN
ejpam-4936	8	21	≤	≤	NUM
ejpam-4936	8	22	20	20	NUM
ejpam-4936	8	23	,	,	PUNCT
ejpam-4936	8	24	then	then	ADV
ejpam-4936	8	25	1,330	1,330	NUM
ejpam-4936	8	26	equations	equation	NOUN
ejpam-4936	8	27	have	have	AUX
ejpam-4936	8	28	been	be	AUX
ejpam-4936	8	29	considered	consider	VERB
ejpam-4936	8	30	.	.	PUNCT
ejpam-4936	9	1	our	our	PRON
ejpam-4936	9	2	results	result	NOUN
ejpam-4936	9	3	confirm	confirm	VERB
ejpam-4936	9	4	that	that	SCONJ
ejpam-4936	9	5	135	135	NUM
ejpam-4936	9	6	of	of	ADP
ejpam-4936	9	7	these	these	DET
ejpam-4936	9	8	equations	equation	NOUN
ejpam-4936	9	9	have	have	AUX
ejpam-4936	9	10	been	be	AUX
ejpam-4936	9	11	fully	fully	ADV
ejpam-4936	9	12	clarified	clarify	VERB
ejpam-4936	9	13	.	.	PUNCT
ejpam-4936	10	1	2020	2020	NUM
ejpam-4936	10	2	mathematics	mathematic	NOUN
ejpam-4936	10	3	subject	subject	NOUN
ejpam-4936	10	4	classifications	classification	NOUN
ejpam-4936	10	5	:	:	PUNCT
ejpam-4936	10	6	11a07	11a07	NUM
ejpam-4936	10	7	key	key	ADJ
ejpam-4936	10	8	words	word	NOUN
ejpam-4936	10	9	and	and	CCONJ
ejpam-4936	10	10	phrases	phrase	NOUN
ejpam-4936	10	11	:	:	PUNCT
ejpam-4936	10	12	exponential	exponential	ADJ
ejpam-4936	10	13	diophantine	diophantine	NOUN
ejpam-4936	10	14	equation	equation	NOUN
ejpam-4936	10	15	,	,	PUNCT
ejpam-4936	10	16	modulo	modulo	PROPN
ejpam-4936	10	17	1	1	NUM
ejpam-4936	10	18	.	.	X
ejpam-4936	11	1	introduction	introduction	NOUN
ejpam-4936	11	2	many	many	ADJ
ejpam-4936	11	3	mathematicians	mathematician	NOUN
ejpam-4936	11	4	have	have	AUX
ejpam-4936	11	5	proposed	propose	VERB
ejpam-4936	11	6	generalized	generalized	ADJ
ejpam-4936	11	7	forms	form	NOUN
ejpam-4936	11	8	of	of	ADP
ejpam-4936	11	9	the	the	DET
ejpam-4936	11	10	diophantine	diophantine	NOUN
ejpam-4936	11	11	equation	equation	NOUN
ejpam-4936	11	12	in	in	ADP
ejpam-4936	11	13	various	various	ADJ
ejpam-4936	11	14	ways	way	NOUN
ejpam-4936	11	15	.	.	PUNCT
ejpam-4936	12	1	one	one	NUM
ejpam-4936	12	2	example	example	NOUN
ejpam-4936	12	3	is	be	AUX
ejpam-4936	12	4	the	the	DET
ejpam-4936	12	5	exponential	exponential	ADJ
ejpam-4936	12	6	diophantine	diophantine	NOUN
ejpam-4936	12	7	equation	equation	NOUN
ejpam-4936	12	8	3x	3x	NUM
ejpam-4936	13	1	+	+	CCONJ
ejpam-4936	13	2	4y	4y	NUM
ejpam-4936	13	3	=	=	SYM
ejpam-4936	13	4	5z	5z	NOUN
ejpam-4936	13	5	,	,	PUNCT
ejpam-4936	13	6	which	which	PRON
ejpam-4936	13	7	has	have	VERB
ejpam-4936	13	8	no	no	DET
ejpam-4936	13	9	solutions	solution	NOUN
ejpam-4936	13	10	for	for	ADP
ejpam-4936	13	11	any	any	DET
ejpam-4936	13	12	natural	natural	ADJ
ejpam-4936	13	13	numbers	number	NOUN
ejpam-4936	13	14	x	x	NOUN
ejpam-4936	13	15	,	,	PUNCT
ejpam-4936	13	16	y	y	PROPN
ejpam-4936	13	17	,	,	PUNCT
ejpam-4936	13	18	z	z	NOUN
ejpam-4936	13	19	except	except	SCONJ
ejpam-4936	13	20	when	when	SCONJ
ejpam-4936	13	21	x	x	PROPN
ejpam-4936	13	22	=	=	SYM
ejpam-4936	13	23	y	y	NOUN
ejpam-4936	13	24	=	=	PUNCT
ejpam-4936	13	25	z	z	NOUN
ejpam-4936	13	26	=	=	SYM
ejpam-4936	13	27	2	2	X
ejpam-4936	13	28	.	.	PUNCT
ejpam-4936	14	1	this	this	PRON
ejpam-4936	14	2	was	be	AUX
ejpam-4936	14	3	proven	prove	VERB
ejpam-4936	14	4	in	in	ADP
ejpam-4936	14	5	1956	1956	NUM
ejpam-4936	14	6	by	by	ADP
ejpam-4936	14	7	w.	w.	PROPN
ejpam-4936	14	8	sierpinski	sierpinski	PROPN
ejpam-4936	14	9	,	,	PUNCT
ejpam-4936	14	10	[	[	X
ejpam-4936	14	11	17	17	NUM
ejpam-4936	14	12	]	]	PUNCT
ejpam-4936	14	13	.	.	PUNCT
ejpam-4936	15	1	in	in	ADP
ejpam-4936	15	2	the	the	DET
ejpam-4936	15	3	same	same	ADJ
ejpam-4936	15	4	year	year	NOUN
ejpam-4936	15	5	,	,	PUNCT
ejpam-4936	15	6	l.	l.	PROPN
ejpam-4936	15	7	jamanowicz	jamanowicz	PROPN
ejpam-4936	15	8	published	publish	VERB
ejpam-4936	15	9	an	an	DET
ejpam-4936	15	10	article	article	NOUN
ejpam-4936	15	11	that	that	PRON
ejpam-4936	15	12	seemed	seem	VERB
ejpam-4936	15	13	to	to	PART
ejpam-4936	15	14	follow	follow	VERB
ejpam-4936	15	15	in	in	ADP
ejpam-4936	15	16	w.	w.	PROPN
ejpam-4936	15	17	sierpinski	sierpinski	PROPN
ejpam-4936	15	18	’s	’s	PART
ejpam-4936	15	19	footsteps	footstep	NOUN
ejpam-4936	15	20	by	by	ADP
ejpam-4936	15	21	selecting	select	VERB
ejpam-4936	15	22	other	other	ADJ
ejpam-4936	15	23	equations	equation	NOUN
ejpam-4936	15	24	such	such	ADJ
ejpam-4936	15	25	as	as	ADP
ejpam-4936	15	26	5x	5x	NOUN
ejpam-4936	15	27	+	+	NOUN
ejpam-4936	15	28	12y	12y	NOUN
ejpam-4936	15	29	=	=	SYM
ejpam-4936	15	30	13z	13z	NOUN
ejpam-4936	15	31	,	,	PUNCT
ejpam-4936	15	32	7x	7x	NUM
ejpam-4936	15	33	+	+	NUM
ejpam-4936	15	34	24y	24y	NOUN
ejpam-4936	15	35	=	=	SYM
ejpam-4936	15	36	25z	25z	NOUN
ejpam-4936	15	37	,	,	PUNCT
ejpam-4936	15	38	9x	9x	X
ejpam-4936	15	39	+	+	CCONJ
ejpam-4936	15	40	40y	40y	NOUN
ejpam-4936	15	41	=	=	SYM
ejpam-4936	15	42	41z	41z	NOUN
ejpam-4936	15	43	,	,	PUNCT
ejpam-4936	15	44	and	and	CCONJ
ejpam-4936	15	45	11x	11x	NOUN
ejpam-4936	15	46	+	+	CCONJ
ejpam-4936	15	47	60y	60y	NOUN
ejpam-4936	15	48	=	=	SYM
ejpam-4936	15	49	61z[see	61z[see	NOUN
ejpam-4936	15	50	13	13	NUM
ejpam-4936	15	51	]	]	PUNCT
ejpam-4936	15	52	.	.	PUNCT
ejpam-4936	16	1	since	since	SCONJ
ejpam-4936	16	2	then	then	ADV
ejpam-4936	16	3	,	,	PUNCT
ejpam-4936	16	4	many	many	ADJ
ejpam-4936	16	5	mathematicians	mathematician	NOUN
ejpam-4936	16	6	have	have	AUX
ejpam-4936	16	7	explored	explore	VERB
ejpam-4936	16	8	variations	variation	NOUN
ejpam-4936	16	9	by	by	ADP
ejpam-4936	16	10	changing	change	VERB
ejpam-4936	16	11	the	the	DET
ejpam-4936	16	12	base	base	NOUN
ejpam-4936	16	13	values	value	NOUN
ejpam-4936	16	14	a	a	DET
ejpam-4936	16	15	,	,	PUNCT
ejpam-4936	16	16	b	b	NOUN
ejpam-4936	16	17	,	,	PUNCT
ejpam-4936	16	18	and	and	CCONJ
ejpam-4936	16	19	c	c	X
ejpam-4936	16	20	in	in	ADP
ejpam-4936	16	21	the	the	DET
ejpam-4936	16	22	exponential	exponential	ADJ
ejpam-4936	16	23	diophantine	diophantine	NOUN
ejpam-4936	16	24	equation	equation	NOUN
ejpam-4936	16	25	ax	ax	NOUN
ejpam-4936	16	26	+	+	CCONJ
ejpam-4936	16	27	by	by	ADP
ejpam-4936	16	28	=	=	NOUN
ejpam-4936	16	29	cz	cz	NOUN
ejpam-4936	16	30	.	.	PUNCT
ejpam-4936	17	1	for	for	ADP
ejpam-4936	17	2	example	example	NOUN
ejpam-4936	17	3	,	,	PUNCT
ejpam-4936	17	4	in	in	ADP
ejpam-4936	17	5	2005	2005	NUM
ejpam-4936	17	6	,	,	PUNCT
ejpam-4936	17	7	d.	d.	PROPN
ejpam-4936	17	8	acu	acu	PROPN
ejpam-4936	18	1	[	[	X
ejpam-4936	18	2	1	1	X
ejpam-4936	18	3	]	]	PUNCT
ejpam-4936	18	4	considered	consider	VERB
ejpam-4936	18	5	three	three	NUM
ejpam-4936	18	6	cases	case	NOUN
ejpam-4936	18	7	:	:	PUNCT
ejpam-4936	18	8	case	case	NOUN
ejpam-4936	18	9	1	1	NUM
ejpam-4936	18	10	where	where	SCONJ
ejpam-4936	18	11	a	a	PRON
ejpam-4936	18	12	=	=	SYM
ejpam-4936	18	13	b	b	NOUN
ejpam-4936	18	14	=	=	SYM
ejpam-4936	18	15	c	c	NOUN
ejpam-4936	18	16	=	=	SYM
ejpam-4936	18	17	p	p	NOUN
ejpam-4936	18	18	,	,	PUNCT
ejpam-4936	18	19	case	case	NOUN
ejpam-4936	18	20	2	2	NUM
ejpam-4936	18	21	where	where	SCONJ
ejpam-4936	18	22	a	a	DET
ejpam-4936	18	23	=	=	SYM
ejpam-4936	18	24	b	b	NOUN
ejpam-4936	18	25	=	=	SYM
ejpam-4936	18	26	p	p	PROPN
ejpam-4936	18	27	and	and	CCONJ
ejpam-4936	18	28	c	c	NOUN
ejpam-4936	18	29	=	=	SYM
ejpam-4936	18	30	2p	2p	NUM
ejpam-4936	18	31	,	,	PUNCT
ejpam-4936	18	32	and	and	CCONJ
ejpam-4936	18	33	the	the	DET
ejpam-4936	18	34	last	last	ADJ
ejpam-4936	18	35	case	case	NOUN
ejpam-4936	18	36	where	where	SCONJ
ejpam-4936	18	37	a	a	DET
ejpam-4936	18	38	=	=	SYM
ejpam-4936	18	39	p	p	NOUN
ejpam-4936	18	40	,	,	PUNCT
ejpam-4936	18	41	b	b	X
ejpam-4936	18	42	=	=	SYM
ejpam-4936	18	43	q	q	NOUN
ejpam-4936	18	44	,	,	PUNCT
ejpam-4936	18	45	and	and	CCONJ
ejpam-4936	18	46	c	c	X
ejpam-4936	18	47	=	=	SYM
ejpam-4936	18	48	pq	pq	PROPN
ejpam-4936	18	49	,	,	PUNCT
ejpam-4936	18	50	with	with	ADP
ejpam-4936	18	51	p	p	NOUN
ejpam-4936	18	52	and	and	CCONJ
ejpam-4936	18	53	q	q	ADJ
ejpam-4936	18	54	being	be	AUX
ejpam-4936	18	55	prime	prime	ADJ
ejpam-4936	18	56	numbers	number	NOUN
ejpam-4936	18	57	.	.	PUNCT
ejpam-4936	19	1	furthermore	furthermore	ADV
ejpam-4936	19	2	,	,	PUNCT
ejpam-4936	19	3	he	he	PRON
ejpam-4936	19	4	proposed	propose	VERB
ejpam-4936	19	5	the	the	DET
ejpam-4936	19	6	alternative	alternative	ADJ
ejpam-4936	19	7	form	form	NOUN
ejpam-4936	19	8	2x	2x	NUM
ejpam-4936	19	9	+	+	CCONJ
ejpam-4936	19	10	5y	5y	NOUN
ejpam-4936	19	11	=	=	SYM
ejpam-4936	19	12	z2	z2	PROPN
ejpam-4936	19	13	in	in	ADP
ejpam-4936	19	14	2007	2007	NUM
ejpam-4936	19	15	,	,	PUNCT
ejpam-4936	19	16	and	and	CCONJ
ejpam-4936	19	17	noted	note	VERB
ejpam-4936	19	18	that	that	SCONJ
ejpam-4936	19	19	catalan	catalan	NOUN
ejpam-4936	19	20	’s	’s	PART
ejpam-4936	19	21	conjecture	conjecture	NOUN
ejpam-4936	19	22	was	be	AUX
ejpam-4936	19	23	an	an	DET
ejpam-4936	19	24	important	important	ADJ
ejpam-4936	19	25	tool	tool	NOUN
ejpam-4936	19	26	for	for	ADP
ejpam-4936	19	27	finding	find	VERB
ejpam-4936	19	28	solutions,[see	solutions,[see	NOUN
ejpam-4936	19	29	2	2	NUM
ejpam-4936	19	30	]	]	PUNCT
ejpam-4936	19	31	.	.	PUNCT
ejpam-4936	20	1	it	it	PRON
ejpam-4936	20	2	’s	’	VERB
ejpam-4936	20	3	also	also	ADV
ejpam-4936	20	4	worth	worth	ADJ
ejpam-4936	20	5	noting	note	VERB
ejpam-4936	20	6	that	that	SCONJ
ejpam-4936	20	7	the	the	DET
ejpam-4936	20	8	exponential	exponential	ADJ
ejpam-4936	20	9	term	term	NOUN
ejpam-4936	20	10	cz	cz	NOUN
ejpam-4936	20	11	was	be	AUX
ejpam-4936	20	12	interchanged	interchange	VERB
ejpam-4936	20	13	with	with	ADP
ejpam-4936	20	14	the	the	DET
ejpam-4936	20	15	polynomial	polynomial	ADJ
ejpam-4936	20	16	term	term	NOUN
ejpam-4936	20	17	z2	z2	NOUN
ejpam-4936	20	18	.	.	PUNCT
ejpam-4936	21	1	∗corresponding	∗corresponde	VERB
ejpam-4936	21	2	author	author	NOUN
ejpam-4936	21	3	.	.	PUNCT
ejpam-4936	22	1	doi	doi	NOUN
ejpam-4936	22	2	:	:	PUNCT
ejpam-4936	22	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4936	https://doi.org/10.29020/nybg.ejpam.v16i4.4936	NOUN
ejpam-4936	22	4	email	email	NOUN
ejpam-4936	22	5	addresses	address	NOUN
ejpam-4936	22	6	:	:	PUNCT
ejpam-4936	22	7	lkittipo@wu.ac.th	lkittipo@wu.ac.th	PROPN
ejpam-4936	22	8	(	(	PUNCT
ejpam-4936	22	9	k.	k.	PROPN
ejpam-4936	22	10	laipaporn	laipaporn	PROPN
ejpam-4936	22	11	)	)	PUNCT
ejpam-4936	22	12	,	,	PUNCT
ejpam-4936	22	13	saowapak.ka@mail.wu.ac.th	saowapak.ka@mail.wu.ac.th	ADJ
ejpam-4936	22	14	(	(	PUNCT
ejpam-4936	22	15	s.	s.	PROPN
ejpam-4936	22	16	kaewchay	kaewchay	PROPN
ejpam-4936	22	17	)	)	PUNCT
ejpam-4936	22	18	,	,	PUNCT
ejpam-4936	22	19	kadisak@mail.wu.ac.th	kadisak@mail.wu.ac.th	PROPN
ejpam-4936	22	20	(	(	PUNCT
ejpam-4936	22	21	a.	a.	NOUN
ejpam-4936	22	22	karnbanjong	karnbanjong	PROPN
ejpam-4936	22	23	)	)	PUNCT
ejpam-4936	22	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4936	22	25	2066	2066	NUM
ejpam-4936	23	1	©	©	PROPN
ejpam-4936	23	2	2023	2023	NUM
ejpam-4936	23	3	ejpam	ejpam	NOUN
ejpam-4936	23	4	all	all	DET
ejpam-4936	23	5	rights	right	NOUN
ejpam-4936	23	6	reserved	reserve	VERB
ejpam-4936	23	7	.	.	PUNCT
ejpam-4936	24	1	k.	k.	PROPN
ejpam-4936	24	2	laipaporn	laipaporn	PROPN
ejpam-4936	24	3	,	,	PUNCT
ejpam-4936	24	4	s.	s.	PROPN
ejpam-4936	24	5	kaewchay	kaewchay	PROPN
ejpam-4936	24	6	,	,	PUNCT
ejpam-4936	24	7	a.	a.	PROPN
ejpam-4936	24	8	karnbanjong	karnbanjong	PROPN
ejpam-4936	24	9	/	/	SYM
ejpam-4936	24	10	eur	eur	PROPN
ejpam-4936	24	11	.	.	PUNCT
ejpam-4936	25	1	j.	j.	PROPN
ejpam-4936	25	2	pure	pure	PROPN
ejpam-4936	25	3	appl	appl	PROPN
ejpam-4936	25	4	.	.	PROPN
ejpam-4936	25	5	math	math	PROPN
ejpam-4936	25	6	,	,	PUNCT
ejpam-4936	25	7	16	16	NUM
ejpam-4936	25	8	(	(	PUNCT
ejpam-4936	25	9	4	4	NUM
ejpam-4936	25	10	)	)	PUNCT
ejpam-4936	25	11	(	(	PUNCT
ejpam-4936	25	12	2023	2023	NUM
ejpam-4936	25	13	)	)	PUNCT
ejpam-4936	25	14	,	,	PUNCT
ejpam-4936	25	15	2066	2066	NUM
ejpam-4936	25	16	-	-	SYM
ejpam-4936	25	17	2081	2081	NUM
ejpam-4936	25	18	2067	2067	NUM
ejpam-4936	25	19	since	since	SCONJ
ejpam-4936	25	20	2007	2007	NUM
ejpam-4936	25	21	,	,	PUNCT
ejpam-4936	25	22	hundreds	hundred	NOUN
ejpam-4936	25	23	of	of	ADP
ejpam-4936	25	24	articles	article	NOUN
ejpam-4936	25	25	inspired	inspire	VERB
ejpam-4936	25	26	by	by	ADP
ejpam-4936	25	27	the	the	DET
ejpam-4936	25	28	equation	equation	NOUN
ejpam-4936	25	29	2x	2x	NUM
ejpam-4936	25	30	+	+	CCONJ
ejpam-4936	25	31	5y	5y	NOUN
ejpam-4936	25	32	=	=	SYM
ejpam-4936	25	33	z2	z2	NOUN
ejpam-4936	25	34	have	have	AUX
ejpam-4936	25	35	been	be	AUX
ejpam-4936	25	36	published	publish	VERB
ejpam-4936	25	37	.	.	PUNCT
ejpam-4936	26	1	many	many	ADJ
ejpam-4936	26	2	were	be	AUX
ejpam-4936	26	3	authored	author	VERB
ejpam-4936	26	4	by	by	ADP
ejpam-4936	26	5	b.	b.	PROPN
ejpam-4936	26	6	sroysang	sroysang	PROPN
ejpam-4936	26	7	,	,	PUNCT
ejpam-4936	26	8	as	as	SCONJ
ejpam-4936	26	9	seen	see	VERB
ejpam-4936	26	10	in	in	ADP
ejpam-4936	26	11	references	reference	NOUN
ejpam-4936	26	12	[	[	X
ejpam-4936	26	13	18–20	18–20	NUM
ejpam-4936	26	14	]	]	PUNCT
ejpam-4936	26	15	,	,	PUNCT
ejpam-4936	26	16	among	among	ADP
ejpam-4936	26	17	others	other	NOUN
ejpam-4936	26	18	listed	list	VERB
ejpam-4936	26	19	in	in	ADP
ejpam-4936	26	20	[	[	X
ejpam-4936	26	21	5	5	NUM
ejpam-4936	26	22	,	,	PUNCT
ejpam-4936	26	23	9	9	NUM
ejpam-4936	26	24	,	,	PUNCT
ejpam-4936	26	25	16	16	NUM
ejpam-4936	26	26	,	,	PUNCT
ejpam-4936	26	27	22	22	NUM
ejpam-4936	26	28	]	]	PUNCT
ejpam-4936	26	29	.	.	PUNCT
ejpam-4936	27	1	in	in	ADP
ejpam-4936	27	2	addition	addition	NOUN
ejpam-4936	27	3	,	,	PUNCT
ejpam-4936	27	4	there	there	PRON
ejpam-4936	27	5	are	be	VERB
ejpam-4936	27	6	exponential	exponential	ADJ
ejpam-4936	27	7	diophantine	diophantine	NOUN
ejpam-4936	27	8	equations	equation	NOUN
ejpam-4936	27	9	similar	similar	ADJ
ejpam-4936	27	10	to	to	ADP
ejpam-4936	27	11	2x	2x	NUM
ejpam-4936	27	12	+	+	CCONJ
ejpam-4936	27	13	5y	5y	NOUN
ejpam-4936	27	14	=	=	SYM
ejpam-4936	27	15	z2	z2	NOUN
ejpam-4936	27	16	but	but	CCONJ
ejpam-4936	27	17	with	with	ADP
ejpam-4936	27	18	more	more	ADJ
ejpam-4936	27	19	than	than	ADP
ejpam-4936	27	20	three	three	NUM
ejpam-4936	27	21	variables	variable	NOUN
ejpam-4936	27	22	.	.	PUNCT
ejpam-4936	28	1	some	some	PRON
ejpam-4936	28	2	of	of	ADP
ejpam-4936	28	3	these	these	PRON
ejpam-4936	28	4	are	be	AUX
ejpam-4936	28	5	showcased	showcase	VERB
ejpam-4936	28	6	in	in	ADP
ejpam-4936	28	7	table	table	NOUN
ejpam-4936	28	8	1	1	NUM
ejpam-4936	28	9	.	.	PUNCT
ejpam-4936	29	1	ever	ever	ADV
ejpam-4936	29	2	since	since	SCONJ
ejpam-4936	29	3	d.	d.	PROPN
ejpam-4936	29	4	acu	acu	PROPN
ejpam-4936	29	5	introduced	introduce	VERB
ejpam-4936	29	6	the	the	DET
ejpam-4936	29	7	equation	equation	NOUN
ejpam-4936	29	8	2x	2x	NUM
ejpam-4936	30	1	+	+	CCONJ
ejpam-4936	30	2	5y	5y	NOUN
ejpam-4936	30	3	=	=	SYM
ejpam-4936	30	4	z2	z2	PROPN
ejpam-4936	30	5	in	in	ADP
ejpam-4936	30	6	2007	2007	NUM
ejpam-4936	30	7	,	,	PUNCT
ejpam-4936	30	8	researchers	researcher	NOUN
ejpam-4936	30	9	have	have	AUX
ejpam-4936	30	10	explored	explore	VERB
ejpam-4936	30	11	alternative	alternative	ADJ
ejpam-4936	30	12	forms	form	NOUN
ejpam-4936	30	13	by	by	ADP
ejpam-4936	30	14	altering	alter	VERB
ejpam-4936	30	15	the	the	DET
ejpam-4936	30	16	base	base	NOUN
ejpam-4936	30	17	numbers	number	NOUN
ejpam-4936	30	18	2	2	NUM
ejpam-4936	30	19	and	and	CCONJ
ejpam-4936	30	20	5	5	NUM
ejpam-4936	30	21	.	.	X
ejpam-4936	31	1	they	they	PRON
ejpam-4936	31	2	have	have	AUX
ejpam-4936	31	3	also	also	ADV
ejpam-4936	31	4	tried	try	VERB
ejpam-4936	31	5	to	to	PART
ejpam-4936	31	6	generalize	generalize	VERB
ejpam-4936	31	7	the	the	DET
ejpam-4936	31	8	equation	equation	NOUN
ejpam-4936	31	9	,	,	PUNCT
ejpam-4936	31	10	creating	create	VERB
ejpam-4936	31	11	new	new	ADJ
ejpam-4936	31	12	forms	form	NOUN
ejpam-4936	31	13	and	and	CCONJ
ejpam-4936	31	14	investigating	investigate	VERB
ejpam-4936	31	15	their	their	PRON
ejpam-4936	31	16	solutions	solution	NOUN
ejpam-4936	31	17	.	.	PUNCT
ejpam-4936	32	1	interestingly	interestingly	ADV
ejpam-4936	32	2	,	,	PUNCT
ejpam-4936	32	3	there	there	PRON
ejpam-4936	32	4	are	be	VERB
ejpam-4936	32	5	few	few	ADJ
ejpam-4936	32	6	equations	equation	NOUN
ejpam-4936	32	7	like	like	ADP
ejpam-4936	32	8	3x	3x	NUM
ejpam-4936	33	1	+	+	CCONJ
ejpam-4936	33	2	5y	5y	NOUN
ejpam-4936	33	3	+	+	X
ejpam-4936	33	4	7z	7z	NOUN
ejpam-4936	33	5	=	=	SYM
ejpam-4936	33	6	w2	w2	NOUN
ejpam-4936	33	7	that	that	PRON
ejpam-4936	33	8	consider	consider	VERB
ejpam-4936	33	9	three	three	NUM
ejpam-4936	33	10	base	base	NOUN
ejpam-4936	33	11	numbers	number	NOUN
ejpam-4936	33	12	for	for	ADP
ejpam-4936	33	13	the	the	DET
ejpam-4936	33	14	exponential	exponential	ADJ
ejpam-4936	33	15	terms	term	NOUN
ejpam-4936	33	16	.	.	PUNCT
ejpam-4936	34	1	this	this	DET
ejpam-4936	34	2	particular	particular	ADJ
ejpam-4936	34	3	equation	equation	NOUN
ejpam-4936	34	4	was	be	AUX
ejpam-4936	34	5	established	establish	VERB
ejpam-4936	34	6	by	by	ADP
ejpam-4936	34	7	j.	j.	PROPN
ejpam-4936	34	8	b.	b.	PROPN
ejpam-4936	34	9	bacani	bacani	PROPN
ejpam-4936	34	10	and	and	CCONJ
ejpam-4936	34	11	j.	j.	PROPN
ejpam-4936	34	12	f.	f.	PROPN
ejpam-4936	34	13	t.	t.	PROPN
ejpam-4936	34	14	rabago	rabago	PROPN
ejpam-4936	34	15	,	,	PUNCT
ejpam-4936	34	16	and	and	CCONJ
ejpam-4936	34	17	serves	serve	VERB
ejpam-4936	34	18	as	as	ADP
ejpam-4936	34	19	the	the	DET
ejpam-4936	34	20	inspiration	inspiration	NOUN
ejpam-4936	34	21	for	for	ADP
ejpam-4936	34	22	this	this	DET
ejpam-4936	34	23	article	article	NOUN
ejpam-4936	34	24	.	.	PUNCT
ejpam-4936	35	1	here	here	ADV
ejpam-4936	35	2	,	,	PUNCT
ejpam-4936	35	3	we	we	PRON
ejpam-4936	35	4	focus	focus	VERB
ejpam-4936	35	5	on	on	ADP
ejpam-4936	35	6	the	the	DET
ejpam-4936	35	7	form	form	NOUN
ejpam-4936	35	8	ax	ax	NOUN
ejpam-4936	35	9	+	+	CCONJ
ejpam-4936	35	10	by	by	ADP
ejpam-4936	35	11	+	+	CCONJ
ejpam-4936	35	12	cz	cz	NOUN
ejpam-4936	35	13	=	=	SYM
ejpam-4936	35	14	w2	w2	NOUN
ejpam-4936	35	15	under	under	ADP
ejpam-4936	35	16	certain	certain	ADJ
ejpam-4936	35	17	conditions	condition	NOUN
ejpam-4936	35	18	for	for	ADP
ejpam-4936	35	19	a	a	DET
ejpam-4936	35	20	,	,	PUNCT
ejpam-4936	35	21	b	b	NOUN
ejpam-4936	35	22	,	,	PUNCT
ejpam-4936	35	23	c	c	NOUN
ejpam-4936	35	24	,	,	PUNCT
ejpam-4936	35	25	x	x	NOUN
ejpam-4936	35	26	,	,	PUNCT
ejpam-4936	35	27	y	y	PROPN
ejpam-4936	35	28	,	,	PUNCT
ejpam-4936	35	29	z	z	PROPN
ejpam-4936	35	30	,	,	PUNCT
ejpam-4936	35	31	and	and	CCONJ
ejpam-4936	35	32	w	w	NOUN
ejpam-4936	35	33	,	,	PUNCT
ejpam-4936	35	34	using	use	VERB
ejpam-4936	35	35	only	only	ADJ
ejpam-4936	35	36	elementary	elementary	ADJ
ejpam-4936	35	37	tools	tool	NOUN
ejpam-4936	35	38	related	relate	VERB
ejpam-4936	35	39	to	to	ADP
ejpam-4936	35	40	modulo	modulo	ADJ
ejpam-4936	35	41	concepts	concept	NOUN
ejpam-4936	35	42	.	.	PUNCT
ejpam-4936	36	1	most	most	ADJ
ejpam-4936	36	2	equations	equation	NOUN
ejpam-4936	36	3	in	in	ADP
ejpam-4936	36	4	table	table	NOUN
ejpam-4936	36	5	1	1	NUM
ejpam-4936	36	6	have	have	VERB
ejpam-4936	36	7	more	more	ADJ
ejpam-4936	36	8	than	than	ADP
ejpam-4936	36	9	three	three	NUM
ejpam-4936	36	10	variables	variable	NOUN
ejpam-4936	36	11	,	,	PUNCT
ejpam-4936	36	12	but	but	CCONJ
ejpam-4936	36	13	they	they	PRON
ejpam-4936	36	14	still	still	ADV
ejpam-4936	36	15	involve	involve	VERB
ejpam-4936	36	16	only	only	ADV
ejpam-4936	36	17	two	two	NUM
ejpam-4936	36	18	base	base	NOUN
ejpam-4936	36	19	numbers	number	NOUN
ejpam-4936	36	20	.	.	PUNCT
ejpam-4936	37	1	even	even	ADV
ejpam-4936	37	2	the	the	DET
ejpam-4936	37	3	equation	equation	NOUN
ejpam-4936	37	4	px	px	X
ejpam-4936	37	5	+	+	CCONJ
ejpam-4936	37	6	(	(	PUNCT
ejpam-4936	37	7	p+	p+	NOUN
ejpam-4936	37	8	1)y	1)y	NUM
ejpam-4936	37	9	+	+	CCONJ
ejpam-4936	37	10	(	(	PUNCT
ejpam-4936	37	11	p+	p+	NOUN
ejpam-4936	37	12	2)z	2)z	NUM
ejpam-4936	37	13	=	=	SYM
ejpam-4936	37	14	m2	m2	PROPN
ejpam-4936	37	15	,	,	PUNCT
ejpam-4936	37	16	cited	cite	VERB
ejpam-4936	37	17	in	in	ADP
ejpam-4936	37	18	[	[	X
ejpam-4936	37	19	6	6	NUM
ejpam-4936	37	20	]	]	PUNCT
ejpam-4936	37	21	,	,	PUNCT
ejpam-4936	37	22	appears	appear	VERB
ejpam-4936	37	23	to	to	PART
ejpam-4936	37	24	have	have	VERB
ejpam-4936	37	25	three	three	NUM
ejpam-4936	37	26	exponential	exponential	ADJ
ejpam-4936	37	27	terms	term	NOUN
ejpam-4936	37	28	,	,	PUNCT
ejpam-4936	37	29	but	but	CCONJ
ejpam-4936	37	30	its	its	PRON
ejpam-4936	37	31	parameters	parameter	NOUN
ejpam-4936	37	32	x	x	SYM
ejpam-4936	37	33	,	,	PUNCT
ejpam-4936	37	34	y	y	PROPN
ejpam-4936	37	35	,	,	PUNCT
ejpam-4936	37	36	and	and	CCONJ
ejpam-4936	37	37	z	z	NOUN
ejpam-4936	37	38	are	be	AUX
ejpam-4936	37	39	restricted	restrict	VERB
ejpam-4936	37	40	to	to	ADP
ejpam-4936	37	41	the	the	DET
ejpam-4936	37	42	set	set	NOUN
ejpam-4936	37	43	{	{	PUNCT
ejpam-4936	37	44	1	1	NUM
ejpam-4936	37	45	,	,	PUNCT
ejpam-4936	37	46	2	2	NUM
ejpam-4936	37	47	,	,	PUNCT
ejpam-4936	37	48	3	3	NUM
ejpam-4936	37	49	}	}	PUNCT
ejpam-4936	37	50	.	.	PUNCT
ejpam-4936	38	1	table	table	NOUN
ejpam-4936	38	2	1	1	NUM
ejpam-4936	38	3	:	:	PUNCT
ejpam-4936	38	4	example	example	NOUN
ejpam-4936	38	5	of	of	ADP
ejpam-4936	38	6	the	the	DET
ejpam-4936	38	7	exponential	exponential	ADJ
ejpam-4936	38	8	diophantine	diophantine	NOUN
ejpam-4936	38	9	equations	equation	NOUN
ejpam-4936	38	10	during	during	ADP
ejpam-4936	38	11	the	the	DET
ejpam-4936	38	12	past	past	ADJ
ejpam-4936	38	13	ten	ten	NUM
ejpam-4936	38	14	years	year	NOUN
ejpam-4936	38	15	which	which	PRON
ejpam-4936	38	16	were	be	AUX
ejpam-4936	38	17	considered	consider	VERB
ejpam-4936	38	18	more	more	ADJ
ejpam-4936	38	19	than	than	ADP
ejpam-4936	38	20	three	three	NUM
ejpam-4936	38	21	variables	variable	NOUN
ejpam-4936	38	22	.	.	PUNCT
ejpam-4936	39	1	ref	ref	NOUN
ejpam-4936	39	2	.	.	PUNCT
ejpam-4936	40	1	author	author	NOUN
ejpam-4936	40	2	equation	equation	NOUN
ejpam-4936	40	3	[	[	X
ejpam-4936	40	4	3	3	X
ejpam-4936	40	5	]	]	X
ejpam-4936	40	6	j.b	j.b	PROPN
ejpam-4936	40	7	.	.	PUNCT
ejpam-4936	40	8	bacani	bacani	PROPN
ejpam-4936	40	9	and	and	CCONJ
ejpam-4936	40	10	j.f.t	j.f.t	NOUN
ejpam-4936	40	11	.	.	PUNCT
ejpam-4936	41	1	rabago	rabago	VERB
ejpam-4936	41	2	3x	3x	PRON
ejpam-4936	42	1	+	+	CCONJ
ejpam-4936	42	2	5y	5y	NOUN
ejpam-4936	42	3	+	+	X
ejpam-4936	42	4	7z	7z	NOUN
ejpam-4936	42	5	=	=	SYM
ejpam-4936	42	6	w2	w2	NOUN
ejpam-4936	43	1	[	[	X
ejpam-4936	43	2	4	4	NUM
ejpam-4936	43	3	]	]	X
ejpam-4936	43	4	j.b	j.b	PROPN
ejpam-4936	43	5	.	.	PUNCT
ejpam-4936	43	6	bacani	bacani	PROPN
ejpam-4936	43	7	and	and	CCONJ
ejpam-4936	43	8	j.f.t	j.f.t	NOUN
ejpam-4936	43	9	.	.	PUNCT
ejpam-4936	44	1	rabago	rabago	PROPN
ejpam-4936	44	2	px	px	PROPN
ejpam-4936	44	3	+	+	CCONJ
ejpam-4936	44	4	qy	qy	NOUN
ejpam-4936	44	5	=	=	PROPN
ejpam-4936	44	6	z2	z2	PROPN
ejpam-4936	44	7	[	[	X
ejpam-4936	44	8	6	6	NUM
ejpam-4936	44	9	]	]	PUNCT
ejpam-4936	44	10	nechemia	nechemia	NOUN
ejpam-4936	44	11	burshtein	burshtein	ADP
ejpam-4936	44	12	px	px	PROPN
ejpam-4936	44	13	+	+	CCONJ
ejpam-4936	44	14	(	(	PUNCT
ejpam-4936	44	15	p+	p+	NOUN
ejpam-4936	44	16	1)y	1)y	NUM
ejpam-4936	44	17	+	+	CCONJ
ejpam-4936	44	18	(	(	PUNCT
ejpam-4936	44	19	p+	p+	NOUN
ejpam-4936	44	20	2)z	2)z	NUM
ejpam-4936	45	1	=	=	SYM
ejpam-4936	45	2	m2	m2	PROPN
ejpam-4936	46	1	[	[	X
ejpam-4936	46	2	7	7	NUM
ejpam-4936	46	3	]	]	SYM
ejpam-4936	46	4	nechemia	nechemia	NOUN
ejpam-4936	46	5	burshtein	burshtein	ADP
ejpam-4936	46	6	p3	p3	PROPN
ejpam-4936	46	7	+	+	CCONJ
ejpam-4936	46	8	qy	qy	NOUN
ejpam-4936	46	9	=	=	SYM
ejpam-4936	46	10	z3	z3	PROPN
ejpam-4936	47	1	[	[	X
ejpam-4936	47	2	10	10	NUM
ejpam-4936	47	3	]	]	X
ejpam-4936	47	4	r.	r.	PROPN
ejpam-4936	47	5	dokchan	dokchan	PROPN
ejpam-4936	47	6	and	and	CCONJ
ejpam-4936	47	7	a.	a.	NOUN
ejpam-4936	47	8	pakapongpun	pakapongpun	NOUN
ejpam-4936	47	9	px	px	PROPN
ejpam-4936	47	10	+	+	CCONJ
ejpam-4936	47	11	(	(	PUNCT
ejpam-4936	47	12	p+	p+	NOUN
ejpam-4936	47	13	20)y	20)y	PROPN
ejpam-4936	47	14	=	=	SYM
ejpam-4936	47	15	z2	z2	PROPN
ejpam-4936	48	1	[	[	X
ejpam-4936	48	2	14	14	NUM
ejpam-4936	48	3	]	]	PUNCT
ejpam-4936	48	4	k.	k.	PROPN
ejpam-4936	49	1	laipaporn	laipaporn	PROPN
ejpam-4936	49	2	,	,	PUNCT
ejpam-4936	49	3	s.	s.	PROPN
ejpam-4936	49	4	wananiyakul	wananiyakul	VERB
ejpam-4936	49	5	3x	3x	PROPN
ejpam-4936	50	1	+	+	CCONJ
ejpam-4936	50	2	p5y	p5y	NOUN
ejpam-4936	50	3	=	=	SYM
ejpam-4936	50	4	z2	z2	NOUN
ejpam-4936	50	5	and	and	CCONJ
ejpam-4936	50	6	p.	p.	NOUN
ejpam-4936	50	7	khachorncharoenkul	khachorncharoenkul	NOUN
ejpam-4936	51	1	[	[	X
ejpam-4936	51	2	21	21	NUM
ejpam-4936	51	3	]	]	X
ejpam-4936	51	4	s.	s.	PROPN
ejpam-4936	51	5	subburam	subburam	PROPN
ejpam-4936	51	6	lax	lax	PROPN
ejpam-4936	52	1	+	+	CCONJ
ejpam-4936	52	2	mby	mby	X
ejpam-4936	52	3	=	=	PUNCT
ejpam-4936	53	1	ncz	ncz	PROPN
ejpam-4936	54	1	[	[	X
ejpam-4936	54	2	23	23	NUM
ejpam-4936	54	3	]	]	PUNCT
ejpam-4936	54	4	a.	a.	NOUN
ejpam-4936	54	5	suvarnamani	suvarnamani	PROPN
ejpam-4936	54	6	px	px	PROPN
ejpam-4936	54	7	+	+	CCONJ
ejpam-4936	54	8	(	(	PUNCT
ejpam-4936	54	9	p+	p+	NOUN
ejpam-4936	54	10	1)y	1)y	NUM
ejpam-4936	54	11	=	=	SYM
ejpam-4936	54	12	z2	z2	PROPN
ejpam-4936	54	13	2	2	NUM
ejpam-4936	54	14	.	.	PUNCT
ejpam-4936	54	15	main	main	ADJ
ejpam-4936	54	16	theorem	theorem	NOUN
ejpam-4936	54	17	our	our	PRON
ejpam-4936	54	18	main	main	ADJ
ejpam-4936	54	19	results	result	NOUN
ejpam-4936	54	20	are	be	AUX
ejpam-4936	54	21	divided	divide	VERB
ejpam-4936	54	22	into	into	ADP
ejpam-4936	54	23	two	two	NUM
ejpam-4936	54	24	groups	group	NOUN
ejpam-4936	54	25	.	.	PUNCT
ejpam-4936	55	1	in	in	ADP
ejpam-4936	55	2	the	the	DET
ejpam-4936	55	3	first	first	ADJ
ejpam-4936	55	4	group	group	NOUN
ejpam-4936	55	5	,	,	PUNCT
ejpam-4936	55	6	we	we	PRON
ejpam-4936	55	7	focus	focus	VERB
ejpam-4936	55	8	on	on	ADP
ejpam-4936	55	9	the	the	DET
ejpam-4936	55	10	equation	equation	NOUN
ejpam-4936	55	11	ax+by+cz	ax+by+cz	NOUN
ejpam-4936	55	12	=	=	PROPN
ejpam-4936	55	13	w2	w2	NOUN
ejpam-4936	55	14	with	with	ADP
ejpam-4936	55	15	the	the	DET
ejpam-4936	55	16	variable	variable	NOUN
ejpam-4936	55	17	c	c	NOUN
ejpam-4936	55	18	fixed	fix	VERB
ejpam-4936	55	19	at	at	ADP
ejpam-4936	55	20	7	7	NUM
ejpam-4936	55	21	,	,	PUNCT
ejpam-4936	55	22	while	while	SCONJ
ejpam-4936	55	23	varying	vary	VERB
ejpam-4936	55	24	(	(	PUNCT
ejpam-4936	55	25	a	a	DET
ejpam-4936	55	26	,	,	PUNCT
ejpam-4936	55	27	b	b	NOUN
ejpam-4936	55	28	)	)	PUNCT
ejpam-4936	55	29	in	in	ADP
ejpam-4936	55	30	specific	specific	ADJ
ejpam-4936	55	31	cases	case	NOUN
ejpam-4936	55	32	.	.	PUNCT
ejpam-4936	56	1	these	these	DET
ejpam-4936	56	2	cases	case	NOUN
ejpam-4936	56	3	consider	consider	VERB
ejpam-4936	56	4	all	all	DET
ejpam-4936	56	5	elements	element	NOUN
ejpam-4936	56	6	in	in	ADP
ejpam-4936	56	7	the	the	DET
ejpam-4936	56	8	set	set	NOUN
ejpam-4936	56	9	a	a	PRON
ejpam-4936	56	10	,	,	PUNCT
ejpam-4936	56	11	which	which	PRON
ejpam-4936	56	12	includes	include	VERB
ejpam-4936	56	13	(	(	PUNCT
ejpam-4936	56	14	3	3	NUM
ejpam-4936	56	15	,	,	PUNCT
ejpam-4936	56	16	4	4	NUM
ejpam-4936	56	17	)	)	PUNCT
ejpam-4936	56	18	,	,	PUNCT
ejpam-4936	56	19	(	(	PUNCT
ejpam-4936	56	20	9	9	NUM
ejpam-4936	56	21	,	,	PUNCT
ejpam-4936	56	22	4	4	NUM
ejpam-4936	56	23	)	)	PUNCT
ejpam-4936	56	24	,	,	PUNCT
ejpam-4936	56	25	(	(	PUNCT
ejpam-4936	56	26	3	3	NUM
ejpam-4936	56	27	,	,	PUNCT
ejpam-4936	56	28	16	16	NUM
ejpam-4936	56	29	)	)	PUNCT
ejpam-4936	56	30	,	,	PUNCT
ejpam-4936	56	31	(	(	PUNCT
ejpam-4936	56	32	9	9	NUM
ejpam-4936	56	33	,	,	PUNCT
ejpam-4936	56	34	16	16	NUM
ejpam-4936	56	35	)	)	PUNCT
ejpam-4936	56	36	,	,	PUNCT
ejpam-4936	56	37	(	(	PUNCT
ejpam-4936	56	38	4	4	NUM
ejpam-4936	56	39	,	,	PUNCT
ejpam-4936	56	40	6	6	NUM
ejpam-4936	56	41	)	)	PUNCT
ejpam-4936	56	42	,	,	PUNCT
ejpam-4936	56	43	(	(	PUNCT
ejpam-4936	56	44	16	16	NUM
ejpam-4936	56	45	,	,	PUNCT
ejpam-4936	56	46	6	6	NUM
ejpam-4936	56	47	)	)	PUNCT
ejpam-4936	56	48	,	,	PUNCT
ejpam-4936	56	49	(	(	PUNCT
ejpam-4936	56	50	9	9	NUM
ejpam-4936	56	51	,	,	PUNCT
ejpam-4936	56	52	10	10	NUM
ejpam-4936	56	53	)	)	PUNCT
ejpam-4936	56	54	,	,	PUNCT
ejpam-4936	56	55	and	and	CCONJ
ejpam-4936	56	56	(	(	PUNCT
ejpam-4936	56	57	6	6	NUM
ejpam-4936	56	58	,	,	PUNCT
ejpam-4936	56	59	10	10	NUM
ejpam-4936	56	60	)	)	PUNCT
ejpam-4936	56	61	.	.	PUNCT
ejpam-4936	57	1	this	this	PRON
ejpam-4936	57	2	is	be	AUX
ejpam-4936	57	3	discussed	discuss	VERB
ejpam-4936	57	4	in	in	ADP
ejpam-4936	57	5	theorems	theorem	NOUN
ejpam-4936	57	6	1–4	1–4	PROPN
ejpam-4936	57	7	and	and	CCONJ
ejpam-4936	57	8	corollaries	corollary	NOUN
ejpam-4936	57	9	1–2	1–2	NUM
ejpam-4936	57	10	.	.	PUNCT
ejpam-4936	58	1	the	the	DET
ejpam-4936	58	2	second	second	ADJ
ejpam-4936	58	3	group	group	NOUN
ejpam-4936	58	4	consists	consist	VERB
ejpam-4936	58	5	of	of	ADP
ejpam-4936	58	6	theorems	theorem	NOUN
ejpam-4936	58	7	5–7	5–7	NUM
ejpam-4936	58	8	and	and	CCONJ
ejpam-4936	58	9	corollary	corollary	ADJ
ejpam-4936	58	10	3	3	NUM
ejpam-4936	58	11	,	,	PUNCT
ejpam-4936	58	12	where	where	SCONJ
ejpam-4936	58	13	all	all	DET
ejpam-4936	58	14	variables	variable	NOUN
ejpam-4936	58	15	a	a	DET
ejpam-4936	58	16	,	,	PUNCT
ejpam-4936	58	17	b	b	NOUN
ejpam-4936	58	18	,	,	PUNCT
ejpam-4936	58	19	and	and	CCONJ
ejpam-4936	58	20	c	c	NOUN
ejpam-4936	58	21	are	be	AUX
ejpam-4936	58	22	allowed	allow	VERB
ejpam-4936	58	23	to	to	PART
ejpam-4936	58	24	vary	vary	VERB
ejpam-4936	58	25	under	under	ADP
ejpam-4936	58	26	certain	certain	ADJ
ejpam-4936	58	27	conditions	condition	NOUN
ejpam-4936	58	28	.	.	PUNCT
ejpam-4936	59	1	additionally	additionally	ADV
ejpam-4936	59	2	,	,	PUNCT
ejpam-4936	59	3	we	we	PRON
ejpam-4936	59	4	introduce	introduce	VERB
ejpam-4936	59	5	an	an	DET
ejpam-4936	59	6	auxiliary	auxiliary	ADJ
ejpam-4936	59	7	result	result	NOUN
ejpam-4936	59	8	,	,	PUNCT
ejpam-4936	59	9	referred	refer	VERB
ejpam-4936	59	10	to	to	ADP
ejpam-4936	59	11	as	as	ADP
ejpam-4936	59	12	’	'	PUNCT
ejpam-4936	59	13	result	result	NOUN
ejpam-4936	59	14	1	1	NUM
ejpam-4936	59	15	,	,	PUNCT
ejpam-4936	59	16	’	'	PUNCT
ejpam-4936	59	17	which	which	PRON
ejpam-4936	59	18	is	be	AUX
ejpam-4936	59	19	utilized	utilize	VERB
ejpam-4936	59	20	in	in	ADP
ejpam-4936	59	21	sections	section	NOUN
ejpam-4936	59	22	1	1	NUM
ejpam-4936	59	23	and	and	CCONJ
ejpam-4936	59	24	2	2	NUM
ejpam-4936	59	25	.	.	X
ejpam-4936	60	1	lemma	lemma	PROPN
ejpam-4936	60	2	1	1	NUM
ejpam-4936	60	3	.	.	PUNCT
ejpam-4936	61	1	for	for	ADP
ejpam-4936	61	2	any	any	DET
ejpam-4936	61	3	non	non	ADJ
ejpam-4936	61	4	-	-	ADJ
ejpam-4936	61	5	negative	negative	ADJ
ejpam-4936	61	6	integers	integer	NOUN
ejpam-4936	61	7	w	w	ADP
ejpam-4936	61	8	and	and	CCONJ
ejpam-4936	61	9	z	z	NOUN
ejpam-4936	61	10	,	,	PUNCT
ejpam-4936	61	11	the	the	DET
ejpam-4936	61	12	equation	equation	NOUN
ejpam-4936	61	13	5	5	NUM
ejpam-4936	61	14	+	+	CCONJ
ejpam-4936	61	15	7z	7z	PROPN
ejpam-4936	61	16	=	=	SYM
ejpam-4936	61	17	w2	w2	NOUN
ejpam-4936	61	18	has	have	VERB
ejpam-4936	61	19	no	no	DET
ejpam-4936	61	20	solution	solution	NOUN
ejpam-4936	61	21	.	.	PUNCT
ejpam-4936	62	1	k.	k.	PROPN
ejpam-4936	62	2	laipaporn	laipaporn	PROPN
ejpam-4936	62	3	,	,	PUNCT
ejpam-4936	62	4	s.	s.	PROPN
ejpam-4936	62	5	kaewchay	kaewchay	PROPN
ejpam-4936	62	6	,	,	PUNCT
ejpam-4936	62	7	a.	a.	PROPN
ejpam-4936	62	8	karnbanjong	karnbanjong	PROPN
ejpam-4936	62	9	/	/	SYM
ejpam-4936	62	10	eur	eur	PROPN
ejpam-4936	62	11	.	.	PUNCT
ejpam-4936	63	1	j.	j.	PROPN
ejpam-4936	63	2	pure	pure	PROPN
ejpam-4936	63	3	appl	appl	PROPN
ejpam-4936	63	4	.	.	PROPN
ejpam-4936	63	5	math	math	PROPN
ejpam-4936	63	6	,	,	PUNCT
ejpam-4936	63	7	16	16	NUM
ejpam-4936	63	8	(	(	PUNCT
ejpam-4936	63	9	4	4	NUM
ejpam-4936	63	10	)	)	PUNCT
ejpam-4936	63	11	(	(	PUNCT
ejpam-4936	63	12	2023	2023	NUM
ejpam-4936	63	13	)	)	PUNCT
ejpam-4936	63	14	,	,	PUNCT
ejpam-4936	63	15	2066	2066	NUM
ejpam-4936	63	16	-	-	SYM
ejpam-4936	63	17	2081	2081	NUM
ejpam-4936	63	18	2068	2068	NUM
ejpam-4936	63	19	proof	proof	NOUN
ejpam-4936	63	20	.	.	PUNCT
ejpam-4936	64	1	it	it	PRON
ejpam-4936	64	2	is	be	AUX
ejpam-4936	64	3	clear	clear	ADJ
ejpam-4936	64	4	that	that	SCONJ
ejpam-4936	64	5	w	w	NOUN
ejpam-4936	64	6	has	have	VERB
ejpam-4936	64	7	to	to	PART
ejpam-4936	64	8	be	be	AUX
ejpam-4936	64	9	even	even	ADV
ejpam-4936	64	10	.	.	PUNCT
ejpam-4936	65	1	so	so	ADV
ejpam-4936	65	2	,	,	PUNCT
ejpam-4936	65	3	we	we	PRON
ejpam-4936	65	4	have	have	VERB
ejpam-4936	65	5	that	that	PRON
ejpam-4936	65	6	z	z	NOUN
ejpam-4936	65	7	is	be	AUX
ejpam-4936	65	8	odd	odd	ADJ
ejpam-4936	65	9	because	because	SCONJ
ejpam-4936	65	10	of	of	ADP
ejpam-4936	65	11	0	0	NUM
ejpam-4936	65	12	≡	≡	PROPN
ejpam-4936	65	13	w2	w2	NOUN
ejpam-4936	65	14	≡	≡	PROPN
ejpam-4936	65	15	5	5	NUM
ejpam-4936	66	1	+	+	CCONJ
ejpam-4936	66	2	7z	7z	PROPN
ejpam-4936	66	3	≡	≡	PROPN
ejpam-4936	66	4	1	1	NUM
ejpam-4936	66	5	+	+	CCONJ
ejpam-4936	66	6	(	(	PUNCT
ejpam-4936	66	7	−1)z	−1)z	PROPN
ejpam-4936	66	8	≡	≡	PROPN
ejpam-4936	66	9	{	{	PUNCT
ejpam-4936	66	10	0	0	NUM
ejpam-4936	66	11	(	(	PUNCT
ejpam-4936	66	12	mod	mod	PROPN
ejpam-4936	66	13	4	4	X
ejpam-4936	66	14	)	)	PUNCT
ejpam-4936	66	15	if	if	SCONJ
ejpam-4936	66	16	z	z	NOUN
ejpam-4936	66	17	is	be	AUX
ejpam-4936	66	18	odd	odd	ADJ
ejpam-4936	66	19	,	,	PUNCT
ejpam-4936	66	20	2	2	NUM
ejpam-4936	66	21	(	(	PUNCT
ejpam-4936	66	22	mod	mod	NOUN
ejpam-4936	66	23	4	4	X
ejpam-4936	66	24	)	)	PUNCT
ejpam-4936	66	25	if	if	SCONJ
ejpam-4936	66	26	z	z	NOUN
ejpam-4936	66	27	is	be	AUX
ejpam-4936	66	28	even	even	ADV
ejpam-4936	66	29	.	.	PUNCT
ejpam-4936	67	1	if	if	SCONJ
ejpam-4936	67	2	z	z	NOUN
ejpam-4936	67	3	=	=	NOUN
ejpam-4936	67	4	1	1	NUM
ejpam-4936	67	5	then	then	ADV
ejpam-4936	67	6	w	w	NOUN
ejpam-4936	67	7	is	be	AUX
ejpam-4936	67	8	not	not	PART
ejpam-4936	67	9	integer	integer	ADJ
ejpam-4936	67	10	,	,	PUNCT
ejpam-4936	67	11	so	so	SCONJ
ejpam-4936	67	12	we	we	PRON
ejpam-4936	67	13	can	can	AUX
ejpam-4936	67	14	let	let	VERB
ejpam-4936	67	15	z	z	NOUN
ejpam-4936	67	16	=	=	SYM
ejpam-4936	67	17	2k	2k	NUM
ejpam-4936	67	18	+	+	CCONJ
ejpam-4936	67	19	1	1	NUM
ejpam-4936	67	20	for	for	ADP
ejpam-4936	67	21	some	some	DET
ejpam-4936	67	22	integer	integer	NOUN
ejpam-4936	67	23	k	k	PROPN
ejpam-4936	67	24	≥	≥	NUM
ejpam-4936	67	25	1	1	NUM
ejpam-4936	67	26	.	.	PUNCT
ejpam-4936	68	1	then	then	ADV
ejpam-4936	68	2	the	the	DET
ejpam-4936	68	3	eqaution	eqaution	NOUN
ejpam-4936	68	4	5	5	NUM
ejpam-4936	68	5	+	+	CCONJ
ejpam-4936	68	6	7z	7z	NOUN
ejpam-4936	68	7	=	=	SYM
ejpam-4936	68	8	w2	w2	NOUN
ejpam-4936	68	9	becomes	become	VERB
ejpam-4936	68	10	(	(	PUNCT
ejpam-4936	68	11	w	w	NOUN
ejpam-4936	68	12	−	−	PROPN
ejpam-4936	68	13	2)(w	2)(w	NUM
ejpam-4936	68	14	+	+	NOUN
ejpam-4936	68	15	2	2	NUM
ejpam-4936	68	16	)	)	PUNCT
ejpam-4936	68	17	=	=	NOUN
ejpam-4936	68	18	(	(	PUNCT
ejpam-4936	68	19	7	7	NUM
ejpam-4936	68	20	+	+	NUM
ejpam-4936	68	21	1)(72k	1)(72k	NUM
ejpam-4936	69	1	−	−	NUM
ejpam-4936	69	2	72k−1	72k−1	NUM
ejpam-4936	69	3	+	+	CCONJ
ejpam-4936	69	4	·	·	PUNCT
ejpam-4936	69	5	·	·	PUNCT
ejpam-4936	69	6	·	·	PUNCT
ejpam-4936	70	1	−	−	NOUN
ejpam-4936	70	2	7	7	NUM
ejpam-4936	71	1	+	+	CCONJ
ejpam-4936	71	2	1	1	NUM
ejpam-4936	71	3	)	)	PUNCT
ejpam-4936	71	4	so	so	SCONJ
ejpam-4936	71	5	8|(w	8|(w	NUM
ejpam-4936	71	6	−	−	PROPN
ejpam-4936	71	7	2)(w	2)(w	NUM
ejpam-4936	71	8	+	+	NOUN
ejpam-4936	71	9	2	2	NUM
ejpam-4936	71	10	)	)	PUNCT
ejpam-4936	71	11	.	.	PUNCT
ejpam-4936	72	1	since	since	SCONJ
ejpam-4936	72	2	2|(w	2|(w	NUM
ejpam-4936	72	3	−	−	PROPN
ejpam-4936	72	4	2	2	NUM
ejpam-4936	72	5	)	)	PUNCT
ejpam-4936	72	6	,	,	PUNCT
ejpam-4936	72	7	2|(w	2|(w	NUM
ejpam-4936	72	8	+	+	SYM
ejpam-4936	72	9	2	2	NUM
ejpam-4936	72	10	)	)	PUNCT
ejpam-4936	72	11	,	,	PUNCT
ejpam-4936	72	12	and	and	CCONJ
ejpam-4936	72	13	w	w	PROPN
ejpam-4936	72	14	+	+	NUM
ejpam-4936	72	15	2	2	NUM
ejpam-4936	72	16	=	=	SYM
ejpam-4936	72	17	(	(	PUNCT
ejpam-4936	72	18	w	w	NOUN
ejpam-4936	72	19	−	−	NOUN
ejpam-4936	72	20	2	2	NUM
ejpam-4936	72	21	)	)	PUNCT
ejpam-4936	72	22	+	+	NOUN
ejpam-4936	72	23	4	4	NUM
ejpam-4936	72	24	,	,	PUNCT
ejpam-4936	72	25	we	we	PRON
ejpam-4936	72	26	conclude	conclude	VERB
ejpam-4936	72	27	that	that	SCONJ
ejpam-4936	72	28	4|(w	4|(w	NUM
ejpam-4936	72	29	−	−	PROPN
ejpam-4936	72	30	2	2	NUM
ejpam-4936	72	31	)	)	PUNCT
ejpam-4936	72	32	and	and	CCONJ
ejpam-4936	72	33	4|(w	4|(w	NUM
ejpam-4936	72	34	+	+	NOUN
ejpam-4936	72	35	2	2	NUM
ejpam-4936	72	36	)	)	PUNCT
ejpam-4936	72	37	.	.	PUNCT
ejpam-4936	73	1	this	this	PRON
ejpam-4936	73	2	implies	imply	VERB
ejpam-4936	73	3	that	that	SCONJ
ejpam-4936	73	4	0	0	NUM
ejpam-4936	73	5	≡	≡	PROPN
ejpam-4936	73	6	w2	w2	NOUN
ejpam-4936	73	7	−	−	PROPN
ejpam-4936	73	8	4	4	NUM
ejpam-4936	73	9	≡	≡	PROPN
ejpam-4936	73	10	72k+1	72k+1	NUM
ejpam-4936	74	1	+	+	CCONJ
ejpam-4936	74	2	1	1	NUM
ejpam-4936	74	3	≡	≡	PROPN
ejpam-4936	74	4	8	8	NUM
ejpam-4936	74	5	(	(	PUNCT
ejpam-4936	74	6	mod	mod	PROPN
ejpam-4936	74	7	16	16	NUM
ejpam-4936	74	8	)	)	PUNCT
ejpam-4936	74	9	,	,	PUNCT
ejpam-4936	74	10	so	so	CCONJ
ejpam-4936	74	11	it	it	PRON
ejpam-4936	74	12	is	be	AUX
ejpam-4936	74	13	a	a	DET
ejpam-4936	74	14	contratiction	contratiction	NOUN
ejpam-4936	74	15	.	.	PUNCT
ejpam-4936	75	1	hence	hence	ADV
ejpam-4936	75	2	the	the	DET
ejpam-4936	75	3	equation	equation	NOUN
ejpam-4936	75	4	5	5	NUM
ejpam-4936	75	5	+	+	CCONJ
ejpam-4936	75	6	7z	7z	PROPN
ejpam-4936	75	7	=	=	SYM
ejpam-4936	75	8	w2	w2	NOUN
ejpam-4936	75	9	has	have	VERB
ejpam-4936	75	10	no	no	DET
ejpam-4936	75	11	solution	solution	NOUN
ejpam-4936	75	12	.	.	PUNCT
ejpam-4936	76	1	with	with	ADP
ejpam-4936	76	2	the	the	DET
ejpam-4936	76	3	previous	previous	ADJ
ejpam-4936	76	4	lemma	lemma	PROPN
ejpam-4936	76	5	,	,	PUNCT
ejpam-4936	76	6	we	we	PRON
ejpam-4936	76	7	are	be	AUX
ejpam-4936	76	8	ready	ready	ADJ
ejpam-4936	76	9	to	to	PART
ejpam-4936	76	10	solve	solve	VERB
ejpam-4936	76	11	the	the	DET
ejpam-4936	76	12	following	follow	VERB
ejpam-4936	76	13	equation	equation	NOUN
ejpam-4936	76	14	.	.	PUNCT
ejpam-4936	77	1	2.1	2.1	NUM
ejpam-4936	77	2	.	.	PUNCT
ejpam-4936	78	1	the	the	DET
ejpam-4936	78	2	exponential	exponential	ADJ
ejpam-4936	78	3	diophantine	diophantine	NOUN
ejpam-4936	78	4	equation	equation	NOUN
ejpam-4936	78	5	ax	ax	NOUN
ejpam-4936	78	6	+	+	X
ejpam-4936	78	7	by	by	ADP
ejpam-4936	78	8	+	+	CCONJ
ejpam-4936	78	9	7z	7z	PROPN
ejpam-4936	78	10	=	=	SYM
ejpam-4936	78	11	w2	w2	NOUN
ejpam-4936	78	12	first	first	ADV
ejpam-4936	78	13	,	,	PUNCT
ejpam-4936	78	14	we	we	PRON
ejpam-4936	78	15	note	note	VERB
ejpam-4936	78	16	from	from	ADP
ejpam-4936	78	17	[	[	X
ejpam-4936	78	18	3	3	NUM
ejpam-4936	78	19	]	]	PUNCT
ejpam-4936	78	20	,	,	PUNCT
ejpam-4936	78	21	j.	j.	PROPN
ejpam-4936	78	22	b.	b.	PROPN
ejpam-4936	78	23	bacani	bacani	PROPN
ejpam-4936	78	24	and	and	CCONJ
ejpam-4936	78	25	j.	j.	PROPN
ejpam-4936	78	26	f.	f.	PROPN
ejpam-4936	78	27	t.	t.	PROPN
ejpam-4936	78	28	rabago	rabago	PROPN
ejpam-4936	78	29	that	that	SCONJ
ejpam-4936	78	30	they	they	PRON
ejpam-4936	78	31	focused	focus	VERB
ejpam-4936	78	32	on	on	ADP
ejpam-4936	78	33	the	the	DET
ejpam-4936	78	34	equation	equation	NOUN
ejpam-4936	78	35	3x	3x	NUM
ejpam-4936	79	1	+	+	CCONJ
ejpam-4936	79	2	5y	5y	NOUN
ejpam-4936	79	3	+	+	X
ejpam-4936	79	4	7z	7z	NOUN
ejpam-4936	79	5	=	=	SYM
ejpam-4936	79	6	w2	w2	NOUN
ejpam-4936	79	7	.	.	PUNCT
ejpam-4936	80	1	in	in	ADP
ejpam-4936	80	2	case	case	NOUN
ejpam-4936	80	3	x	x	X
ejpam-4936	80	4	=	=	PUNCT
ejpam-4936	80	5	y	y	PROPN
ejpam-4936	80	6	=	=	SYM
ejpam-4936	80	7	0	0	PROPN
ejpam-4936	80	8	,	,	PUNCT
ejpam-4936	80	9	the	the	DET
ejpam-4936	80	10	equation	equation	NOUN
ejpam-4936	80	11	becomes	become	VERB
ejpam-4936	80	12	to	to	ADP
ejpam-4936	80	13	2	2	NUM
ejpam-4936	80	14	+	+	CCONJ
ejpam-4936	80	15	7z	7z	NOUN
ejpam-4936	80	16	=	=	SYM
ejpam-4936	80	17	w2	w2	NOUN
ejpam-4936	80	18	that	that	PRON
ejpam-4936	80	19	already	already	ADV
ejpam-4936	80	20	considered	consider	VERB
ejpam-4936	80	21	that	that	SCONJ
ejpam-4936	80	22	why	why	SCONJ
ejpam-4936	80	23	we	we	PRON
ejpam-4936	80	24	examined	examine	VERB
ejpam-4936	80	25	our	our	PRON
ejpam-4936	80	26	theorems	theorem	NOUN
ejpam-4936	80	27	in	in	ADP
ejpam-4936	80	28	this	this	DET
ejpam-4936	80	29	section	section	NOUN
ejpam-4936	80	30	over	over	ADP
ejpam-4936	80	31	the	the	DET
ejpam-4936	80	32	set	set	NOUN
ejpam-4936	80	33	u	u	NOUN
ejpam-4936	80	34	=	=	PROPN
ejpam-4936	80	35	n4	n4	PROPN
ejpam-4936	80	36	0	0	PUNCT
ejpam-4936	81	1	−	−	NOUN
ejpam-4936	81	2	{	{	PUNCT
ejpam-4936	81	3	(	(	PUNCT
ejpam-4936	81	4	0	0	NUM
ejpam-4936	81	5	,	,	PUNCT
ejpam-4936	81	6	0	0	NUM
ejpam-4936	81	7	,	,	PUNCT
ejpam-4936	81	8	z	z	NOUN
ejpam-4936	81	9	,	,	PUNCT
ejpam-4936	81	10	w)|z	w)|z	X
ejpam-4936	81	11	,	,	PUNCT
ejpam-4936	81	12	w	w	PROPN
ejpam-4936	81	13	∈	∈	PROPN
ejpam-4936	81	14	n0	n0	PROPN
ejpam-4936	81	15	}	}	PUNCT
ejpam-4936	81	16	where	where	SCONJ
ejpam-4936	81	17	n0	n0	PROPN
ejpam-4936	81	18	is	be	AUX
ejpam-4936	81	19	the	the	DET
ejpam-4936	81	20	set	set	NOUN
ejpam-4936	81	21	of	of	ADP
ejpam-4936	81	22	all	all	DET
ejpam-4936	81	23	non	non	ADJ
ejpam-4936	81	24	-	-	ADJ
ejpam-4936	81	25	negative	negative	ADJ
ejpam-4936	81	26	integers	integer	NOUN
ejpam-4936	81	27	.	.	PUNCT
ejpam-4936	82	1	from	from	ADP
ejpam-4936	82	2	now	now	ADV
ejpam-4936	82	3	on	on	ADV
ejpam-4936	82	4	,	,	PUNCT
ejpam-4936	82	5	it	it	PRON
ejpam-4936	82	6	causes	cause	VERB
ejpam-4936	82	7	us	we	PRON
ejpam-4936	82	8	to	to	PART
ejpam-4936	82	9	investigate	investigate	VERB
ejpam-4936	82	10	x	x	PUNCT
ejpam-4936	82	11	and	and	CCONJ
ejpam-4936	82	12	y	y	PROPN
ejpam-4936	82	13	are	be	AUX
ejpam-4936	82	14	unequal	unequal	ADJ
ejpam-4936	82	15	to	to	ADP
ejpam-4936	82	16	zero	zero	NUM
ejpam-4936	82	17	simultaneously	simultaneously	ADV
ejpam-4936	82	18	.	.	PUNCT
ejpam-4936	83	1	then	then	ADV
ejpam-4936	83	2	we	we	PRON
ejpam-4936	83	3	have	have	VERB
ejpam-4936	83	4	the	the	DET
ejpam-4936	83	5	results	result	NOUN
ejpam-4936	83	6	of	of	ADP
ejpam-4936	83	7	the	the	DET
ejpam-4936	83	8	exponential	exponential	ADJ
ejpam-4936	83	9	diophantine	diophantine	NOUN
ejpam-4936	83	10	equation	equation	NOUN
ejpam-4936	83	11	ax	ax	NOUN
ejpam-4936	83	12	+	+	X
ejpam-4936	83	13	by	by	ADP
ejpam-4936	83	14	+	+	CCONJ
ejpam-4936	83	15	7z	7z	PROPN
ejpam-4936	83	16	=	=	SYM
ejpam-4936	83	17	w2	w2	NOUN
ejpam-4936	83	18	where	where	SCONJ
ejpam-4936	83	19	a	a	PRON
ejpam-4936	83	20	and	and	CCONJ
ejpam-4936	83	21	b	b	NOUN
ejpam-4936	83	22	are	be	AUX
ejpam-4936	83	23	positive	positive	ADJ
ejpam-4936	83	24	integers	integer	NOUN
ejpam-4936	83	25	and	and	CCONJ
ejpam-4936	83	26	the	the	DET
ejpam-4936	83	27	variables	variable	NOUN
ejpam-4936	83	28	(	(	PUNCT
ejpam-4936	83	29	x	x	X
ejpam-4936	83	30	,	,	PUNCT
ejpam-4936	83	31	y	y	PROPN
ejpam-4936	83	32	,	,	PUNCT
ejpam-4936	83	33	z	z	PROPN
ejpam-4936	83	34	,	,	PUNCT
ejpam-4936	83	35	w	w	NOUN
ejpam-4936	83	36	)	)	PUNCT
ejpam-4936	83	37	are	be	AUX
ejpam-4936	83	38	elements	element	NOUN
ejpam-4936	83	39	in	in	ADP
ejpam-4936	83	40	u	u	NOUN
ejpam-4936	83	41	as	as	SCONJ
ejpam-4936	83	42	follows	follow	VERB
ejpam-4936	83	43	:	:	PUNCT
ejpam-4936	83	44	theorem	theorem	NOUN
ejpam-4936	83	45	1	1	NUM
ejpam-4936	83	46	.	.	PUNCT
ejpam-4936	84	1	the	the	DET
ejpam-4936	84	2	equation	equation	NOUN
ejpam-4936	84	3	3x	3x	NUM
ejpam-4936	85	1	+	+	CCONJ
ejpam-4936	85	2	4y	4y	PRON
ejpam-4936	85	3	+	+	CCONJ
ejpam-4936	85	4	7z	7z	PROPN
ejpam-4936	85	5	=	=	SYM
ejpam-4936	85	6	w2	w2	NOUN
ejpam-4936	85	7	(	(	PUNCT
ejpam-4936	85	8	1	1	NUM
ejpam-4936	85	9	)	)	PUNCT
ejpam-4936	85	10	has	have	VERB
ejpam-4936	85	11	no	no	DET
ejpam-4936	85	12	solution	solution	NOUN
ejpam-4936	85	13	for	for	ADP
ejpam-4936	85	14	any	any	DET
ejpam-4936	85	15	(	(	PUNCT
ejpam-4936	85	16	x	x	NOUN
ejpam-4936	85	17	,	,	PUNCT
ejpam-4936	85	18	y	y	PROPN
ejpam-4936	85	19	,	,	PUNCT
ejpam-4936	85	20	z	z	PROPN
ejpam-4936	85	21	,	,	PUNCT
ejpam-4936	85	22	w	w	NOUN
ejpam-4936	85	23	)	)	PUNCT
ejpam-4936	85	24	∈	∈	PROPN
ejpam-4936	85	25	u	u	NOUN
ejpam-4936	85	26	.	.	PUNCT
ejpam-4936	86	1	proof	proof	NOUN
ejpam-4936	86	2	.	.	PUNCT
ejpam-4936	87	1	the	the	DET
ejpam-4936	87	2	proof	proof	NOUN
ejpam-4936	87	3	of	of	ADP
ejpam-4936	87	4	the	the	DET
ejpam-4936	87	5	theorem	theorem	NOUN
ejpam-4936	87	6	is	be	AUX
ejpam-4936	87	7	separated	separate	VERB
ejpam-4936	87	8	into	into	ADP
ejpam-4936	87	9	3	3	NUM
ejpam-4936	87	10	cases	case	NOUN
ejpam-4936	87	11	.	.	PUNCT
ejpam-4936	88	1	case	case	NOUN
ejpam-4936	88	2	1	1	NUM
ejpam-4936	88	3	z	z	NOUN
ejpam-4936	88	4	=	=	NOUN
ejpam-4936	88	5	0	0	X
ejpam-4936	88	6	.	.	PUNCT
ejpam-4936	89	1	then	then	ADV
ejpam-4936	89	2	the	the	DET
ejpam-4936	89	3	equation	equation	NOUN
ejpam-4936	89	4	(	(	PUNCT
ejpam-4936	89	5	1	1	X
ejpam-4936	89	6	)	)	PUNCT
ejpam-4936	89	7	becomes	become	VERB
ejpam-4936	89	8	3x	3x	PROPN
ejpam-4936	90	1	+	+	CCONJ
ejpam-4936	90	2	4y	4y	PRON
ejpam-4936	90	3	+	+	CCONJ
ejpam-4936	90	4	1	1	NUM
ejpam-4936	90	5	=	=	NOUN
ejpam-4936	90	6	w2	w2	NOUN
ejpam-4936	90	7	.	.	PUNCT
ejpam-4936	91	1	(	(	PUNCT
ejpam-4936	91	2	2	2	X
ejpam-4936	91	3	)	)	PUNCT
ejpam-4936	91	4	we	we	PRON
ejpam-4936	91	5	note	note	VERB
ejpam-4936	91	6	that	that	SCONJ
ejpam-4936	91	7	3x	3x	PRON
ejpam-4936	92	1	+	+	CCONJ
ejpam-4936	92	2	4y	4y	PRON
ejpam-4936	92	3	+	+	CCONJ
ejpam-4936	92	4	1	1	NUM
ejpam-4936	92	5	≡	≡	PROPN
ejpam-4936	92	6	{	{	PUNCT
ejpam-4936	92	7	2	2	NUM
ejpam-4936	92	8	(	(	PUNCT
ejpam-4936	92	9	mod	mod	NOUN
ejpam-4936	92	10	3	3	NUM
ejpam-4936	92	11	)	)	PUNCT
ejpam-4936	92	12	for	for	ADP
ejpam-4936	92	13	x	x	X
ejpam-4936	92	14	≥	≥	NUM
ejpam-4936	92	15	1	1	NUM
ejpam-4936	92	16	and	and	CCONJ
ejpam-4936	92	17	y	y	PROPN
ejpam-4936	92	18	≥	≥	NOUN
ejpam-4936	92	19	0	0	NUM
ejpam-4936	92	20	2	2	NUM
ejpam-4936	92	21	(	(	PUNCT
ejpam-4936	92	22	mod	mod	NOUN
ejpam-4936	92	23	4	4	NUM
ejpam-4936	92	24	)	)	PUNCT
ejpam-4936	92	25	for	for	ADP
ejpam-4936	92	26	x	x	SYM
ejpam-4936	92	27	=	=	SYM
ejpam-4936	92	28	0	0	NUM
ejpam-4936	92	29	and	and	CCONJ
ejpam-4936	92	30	y	y	PROPN
ejpam-4936	92	31	≥	≥	NUM
ejpam-4936	92	32	1	1	NUM
ejpam-4936	92	33	(	(	PUNCT
ejpam-4936	92	34	3	3	NUM
ejpam-4936	92	35	)	)	PUNCT
ejpam-4936	92	36	since	since	SCONJ
ejpam-4936	92	37	w2	w2	PROPN
ejpam-4936	92	38	≡	≡	PROPN
ejpam-4936	92	39	0	0	NUM
ejpam-4936	92	40	,	,	PUNCT
ejpam-4936	92	41	1	1	NUM
ejpam-4936	92	42	(	(	PUNCT
ejpam-4936	92	43	mod	mod	NOUN
ejpam-4936	92	44	3	3	NUM
ejpam-4936	92	45	)	)	PUNCT
ejpam-4936	92	46	and	and	CCONJ
ejpam-4936	92	47	w2	w2	PROPN
ejpam-4936	92	48	≡	≡	PROPN
ejpam-4936	92	49	0	0	NUM
ejpam-4936	92	50	,	,	PUNCT
ejpam-4936	92	51	1	1	NUM
ejpam-4936	92	52	(	(	PUNCT
ejpam-4936	92	53	mod	mod	NOUN
ejpam-4936	92	54	4	4	NUM
ejpam-4936	92	55	)	)	PUNCT
ejpam-4936	92	56	,	,	PUNCT
ejpam-4936	92	57	we	we	PRON
ejpam-4936	92	58	conclude	conclude	VERB
ejpam-4936	92	59	that	that	SCONJ
ejpam-4936	92	60	3x	3x	PROPN
ejpam-4936	93	1	+	+	CCONJ
ejpam-4936	93	2	4y	4y	PRON
ejpam-4936	93	3	+	+	CCONJ
ejpam-4936	93	4	1	1	X
ejpam-4936	93	5	=	=	NOUN
ejpam-4936	93	6	w2	w2	NOUN
ejpam-4936	93	7	has	have	VERB
ejpam-4936	93	8	no	no	DET
ejpam-4936	93	9	solution	solution	NOUN
ejpam-4936	93	10	for	for	ADP
ejpam-4936	93	11	all	all	DET
ejpam-4936	93	12	x	x	NOUN
ejpam-4936	93	13	,	,	PUNCT
ejpam-4936	93	14	y	y	PROPN
ejpam-4936	93	15	,	,	PUNCT
ejpam-4936	93	16	w	w	PROPN
ejpam-4936	93	17	∈	∈	PROPN
ejpam-4936	93	18	n0	n0	PROPN
ejpam-4936	93	19	.	.	PUNCT
ejpam-4936	93	20	case	case	NOUN
ejpam-4936	93	21	2	2	NUM
ejpam-4936	93	22	z	z	NOUN
ejpam-4936	93	23	=	=	SYM
ejpam-4936	93	24	1	1	X
ejpam-4936	93	25	.	.	X
ejpam-4936	94	1	for	for	ADP
ejpam-4936	94	2	any	any	DET
ejpam-4936	94	3	x	x	SYM
ejpam-4936	94	4	≥	≥	NUM
ejpam-4936	94	5	1	1	NUM
ejpam-4936	94	6	and	and	CCONJ
ejpam-4936	94	7	y	y	PROPN
ejpam-4936	94	8	≥	≥	NUM
ejpam-4936	94	9	0	0	NUM
ejpam-4936	94	10	,	,	PUNCT
ejpam-4936	94	11	we	we	PRON
ejpam-4936	94	12	see	see	VERB
ejpam-4936	94	13	that	that	SCONJ
ejpam-4936	94	14	3x	3x	PRON
ejpam-4936	95	1	+	+	CCONJ
ejpam-4936	95	2	4y	4y	PRON
ejpam-4936	95	3	+	+	CCONJ
ejpam-4936	95	4	7	7	NUM
ejpam-4936	95	5	≡	≡	PROPN
ejpam-4936	95	6	2	2	NUM
ejpam-4936	95	7	(	(	PUNCT
ejpam-4936	95	8	mod	mod	NOUN
ejpam-4936	95	9	3	3	NUM
ejpam-4936	95	10	)	)	PUNCT
ejpam-4936	95	11	.	.	PUNCT
ejpam-4936	96	1	again	again	ADV
ejpam-4936	96	2	,	,	PUNCT
ejpam-4936	96	3	w2	w2	NOUN
ejpam-4936	96	4	≡	≡	PROPN
ejpam-4936	96	5	0	0	NUM
ejpam-4936	96	6	,	,	PUNCT
ejpam-4936	96	7	1	1	NUM
ejpam-4936	96	8	(	(	PUNCT
ejpam-4936	96	9	mod	mod	NOUN
ejpam-4936	96	10	3	3	NUM
ejpam-4936	96	11	)	)	PUNCT
ejpam-4936	96	12	and	and	CCONJ
ejpam-4936	96	13	this	this	DET
ejpam-4936	96	14	fact	fact	NOUN
ejpam-4936	96	15	forces	force	VERB
ejpam-4936	96	16	the	the	DET
ejpam-4936	96	17	equation	equation	NOUN
ejpam-4936	96	18	3x+4y+7	3x+4y+7	NUM
ejpam-4936	96	19	=	=	PROPN
ejpam-4936	96	20	w2	w2	NOUN
ejpam-4936	96	21	has	have	VERB
ejpam-4936	96	22	no	no	DET
ejpam-4936	96	23	solution	solution	NOUN
ejpam-4936	96	24	.	.	PUNCT
ejpam-4936	97	1	now	now	ADV
ejpam-4936	97	2	,	,	PUNCT
ejpam-4936	97	3	we	we	PRON
ejpam-4936	97	4	remain	remain	VERB
ejpam-4936	97	5	to	to	PART
ejpam-4936	97	6	consider	consider	VERB
ejpam-4936	97	7	x	x	X
ejpam-4936	97	8	=	=	SYM
ejpam-4936	97	9	0	0	NUM
ejpam-4936	97	10	and	and	CCONJ
ejpam-4936	97	11	y	y	PROPN
ejpam-4936	97	12	≥	≥	NUM
ejpam-4936	97	13	1	1	NUM
ejpam-4936	97	14	and	and	CCONJ
ejpam-4936	97	15	then	then	ADV
ejpam-4936	97	16	the	the	DET
ejpam-4936	97	17	equation	equation	NOUN
ejpam-4936	97	18	(	(	PUNCT
ejpam-4936	97	19	1	1	X
ejpam-4936	97	20	)	)	PUNCT
ejpam-4936	97	21	is	be	AUX
ejpam-4936	97	22	in	in	ADP
ejpam-4936	97	23	the	the	DET
ejpam-4936	97	24	form	form	NOUN
ejpam-4936	97	25	8	8	NUM
ejpam-4936	97	26	=	=	SYM
ejpam-4936	97	27	(	(	PUNCT
ejpam-4936	97	28	w	w	NOUN
ejpam-4936	97	29	−	−	PROPN
ejpam-4936	97	30	2y)(w	2y)(w	NUM
ejpam-4936	97	31	+	+	X
ejpam-4936	97	32	2y	2y	NUM
ejpam-4936	97	33	)	)	PUNCT
ejpam-4936	97	34	.	.	PUNCT
ejpam-4936	98	1	so	so	ADV
ejpam-4936	98	2	,	,	PUNCT
ejpam-4936	98	3	we	we	PRON
ejpam-4936	98	4	can	can	AUX
ejpam-4936	98	5	let	let	VERB
ejpam-4936	98	6	w	w	ADP
ejpam-4936	98	7	−	−	NOUN
ejpam-4936	98	8	2y	2y	NUM
ejpam-4936	98	9	=	=	SYM
ejpam-4936	98	10	2u	2u	NOUN
ejpam-4936	98	11	and	and	CCONJ
ejpam-4936	98	12	w	w	NOUN
ejpam-4936	98	13	+	+	ADJ
ejpam-4936	98	14	2y	2y	PROPN
ejpam-4936	98	15	=	=	SYM
ejpam-4936	98	16	23−u	23−u	NUM
ejpam-4936	98	17	where	where	SCONJ
ejpam-4936	98	18	u	u	NOUN
ejpam-4936	98	19	=	=	NOUN
ejpam-4936	98	20	0	0	NUM
ejpam-4936	98	21	or	or	CCONJ
ejpam-4936	98	22	u	u	X
ejpam-4936	98	23	=	=	NOUN
ejpam-4936	98	24	1	1	NUM
ejpam-4936	98	25	.	.	PUNCT
ejpam-4936	99	1	since	since	SCONJ
ejpam-4936	99	2	2w	2w	NUM
ejpam-4936	99	3	=	=	SYM
ejpam-4936	99	4	{	{	PUNCT
ejpam-4936	99	5	9	9	NUM
ejpam-4936	99	6	if	if	SCONJ
ejpam-4936	99	7	u	u	NOUN
ejpam-4936	99	8	=	=	NOUN
ejpam-4936	99	9	0	0	NUM
ejpam-4936	99	10	6	6	NUM
ejpam-4936	99	11	if	if	SCONJ
ejpam-4936	99	12	u	u	NOUN
ejpam-4936	99	13	=	=	NOUN
ejpam-4936	99	14	1	1	NUM
ejpam-4936	99	15	and	and	CCONJ
ejpam-4936	99	16	w	w	PROPN
ejpam-4936	99	17	has	have	AUX
ejpam-4936	99	18	to	to	PART
ejpam-4936	99	19	be	be	AUX
ejpam-4936	99	20	integer	integer	NOUN
ejpam-4936	99	21	,	,	PUNCT
ejpam-4936	99	22	we	we	PRON
ejpam-4936	99	23	have	have	VERB
ejpam-4936	99	24	w	w	NOUN
ejpam-4936	99	25	=	=	SYM
ejpam-4936	99	26	3	3	NUM
ejpam-4936	99	27	and	and	CCONJ
ejpam-4936	99	28	y	y	PROPN
ejpam-4936	99	29	=	=	SYM
ejpam-4936	99	30	0	0	PROPN
ejpam-4936	99	31	.	.	PUNCT
ejpam-4936	100	1	it	it	PRON
ejpam-4936	100	2	contradics	contradic	VERB
ejpam-4936	100	3	to	to	ADP
ejpam-4936	100	4	y	y	PROPN
ejpam-4936	100	5	≥	≥	PROPN
ejpam-4936	100	6	1	1	NUM
ejpam-4936	100	7	.	.	PUNCT
ejpam-4936	101	1	k.	k.	PROPN
ejpam-4936	101	2	laipaporn	laipaporn	PROPN
ejpam-4936	101	3	,	,	PUNCT
ejpam-4936	101	4	s.	s.	PROPN
ejpam-4936	101	5	kaewchay	kaewchay	PROPN
ejpam-4936	101	6	,	,	PUNCT
ejpam-4936	101	7	a.	a.	PROPN
ejpam-4936	101	8	karnbanjong	karnbanjong	PROPN
ejpam-4936	101	9	/	/	SYM
ejpam-4936	101	10	eur	eur	PROPN
ejpam-4936	101	11	.	.	PUNCT
ejpam-4936	102	1	j.	j.	PROPN
ejpam-4936	102	2	pure	pure	PROPN
ejpam-4936	102	3	appl	appl	PROPN
ejpam-4936	102	4	.	.	PROPN
ejpam-4936	102	5	math	math	PROPN
ejpam-4936	102	6	,	,	PUNCT
ejpam-4936	102	7	16	16	NUM
ejpam-4936	102	8	(	(	PUNCT
ejpam-4936	102	9	4	4	NUM
ejpam-4936	102	10	)	)	PUNCT
ejpam-4936	102	11	(	(	PUNCT
ejpam-4936	102	12	2023	2023	NUM
ejpam-4936	102	13	)	)	PUNCT
ejpam-4936	102	14	,	,	PUNCT
ejpam-4936	102	15	2066	2066	NUM
ejpam-4936	102	16	-	-	SYM
ejpam-4936	102	17	2081	2081	NUM
ejpam-4936	102	18	2069	2069	NUM
ejpam-4936	102	19	case	case	NOUN
ejpam-4936	102	20	3	3	NUM
ejpam-4936	102	21	z	z	NOUN
ejpam-4936	102	22	≥	≥	NUM
ejpam-4936	102	23	2	2	NUM
ejpam-4936	102	24	.	.	PUNCT
ejpam-4936	103	1	first	first	ADV
ejpam-4936	103	2	,	,	PUNCT
ejpam-4936	103	3	we	we	PRON
ejpam-4936	103	4	can	can	AUX
ejpam-4936	103	5	see	see	VERB
ejpam-4936	103	6	that	that	SCONJ
ejpam-4936	103	7	3x	3x	PROPN
ejpam-4936	104	1	+	+	CCONJ
ejpam-4936	104	2	4y	4y	PRON
ejpam-4936	104	3	+	+	CCONJ
ejpam-4936	104	4	7z	7z	PROPN
ejpam-4936	104	5	≡	≡	PROPN
ejpam-4936	104	6	2	2	NUM
ejpam-4936	104	7	(	(	PUNCT
ejpam-4936	104	8	mod	mod	NOUN
ejpam-4936	104	9	3	3	NUM
ejpam-4936	104	10	)	)	PUNCT
ejpam-4936	104	11	for	for	ADP
ejpam-4936	104	12	any	any	DET
ejpam-4936	104	13	x	x	SYM
ejpam-4936	104	14	≥	≥	NUM
ejpam-4936	104	15	1	1	NUM
ejpam-4936	104	16	and	and	CCONJ
ejpam-4936	104	17	y	y	PROPN
ejpam-4936	104	18	≥	≥	NUM
ejpam-4936	104	19	0	0	NUM
ejpam-4936	104	20	,	,	PUNCT
ejpam-4936	104	21	but	but	CCONJ
ejpam-4936	104	22	w2	w2	NOUN
ejpam-4936	104	23	̸≡	̸≡	NOUN
ejpam-4936	104	24	2	2	NUM
ejpam-4936	104	25	(	(	PUNCT
ejpam-4936	104	26	mod	mod	NOUN
ejpam-4936	104	27	3	3	NUM
ejpam-4936	104	28	)	)	PUNCT
ejpam-4936	104	29	,	,	PUNCT
ejpam-4936	104	30	so	so	CCONJ
ejpam-4936	104	31	the	the	DET
ejpam-4936	104	32	equation	equation	NOUN
ejpam-4936	104	33	(	(	PUNCT
ejpam-4936	104	34	1	1	X
ejpam-4936	104	35	)	)	PUNCT
ejpam-4936	104	36	has	have	VERB
ejpam-4936	104	37	no	no	DET
ejpam-4936	104	38	solution	solution	NOUN
ejpam-4936	104	39	.	.	PUNCT
ejpam-4936	105	1	next	next	ADV
ejpam-4936	105	2	,	,	PUNCT
ejpam-4936	105	3	we	we	PRON
ejpam-4936	105	4	consider	consider	VERB
ejpam-4936	105	5	x	x	X
ejpam-4936	105	6	=	=	SYM
ejpam-4936	105	7	0	0	NUM
ejpam-4936	105	8	and	and	CCONJ
ejpam-4936	105	9	y	y	PROPN
ejpam-4936	105	10	=	=	SYM
ejpam-4936	105	11	1	1	X
ejpam-4936	105	12	.	.	PUNCT
ejpam-4936	105	13	by	by	ADP
ejpam-4936	105	14	lemma	lemma	PROPN
ejpam-4936	105	15	1	1	NUM
ejpam-4936	105	16	,	,	PUNCT
ejpam-4936	105	17	we	we	PRON
ejpam-4936	105	18	know	know	VERB
ejpam-4936	105	19	that	that	SCONJ
ejpam-4936	105	20	the	the	DET
ejpam-4936	105	21	equation	equation	NOUN
ejpam-4936	105	22	(	(	PUNCT
ejpam-4936	105	23	1	1	X
ejpam-4936	105	24	)	)	PUNCT
ejpam-4936	105	25	has	have	VERB
ejpam-4936	105	26	no	no	DET
ejpam-4936	105	27	solution	solution	NOUN
ejpam-4936	105	28	.	.	PUNCT
ejpam-4936	106	1	finally	finally	ADV
ejpam-4936	106	2	,	,	PUNCT
ejpam-4936	106	3	we	we	PRON
ejpam-4936	106	4	assume	assume	VERB
ejpam-4936	106	5	that	that	SCONJ
ejpam-4936	106	6	x	x	X
ejpam-4936	106	7	=	=	SYM
ejpam-4936	106	8	0	0	NUM
ejpam-4936	106	9	and	and	CCONJ
ejpam-4936	106	10	y	y	PROPN
ejpam-4936	106	11	≥	≥	NUM
ejpam-4936	106	12	2	2	NUM
ejpam-4936	106	13	.	.	PUNCT
ejpam-4936	107	1	so	so	ADV
ejpam-4936	107	2	w	w	NOUN
ejpam-4936	107	3	is	be	AUX
ejpam-4936	107	4	even	even	ADV
ejpam-4936	107	5	.	.	PUNCT
ejpam-4936	108	1	it	it	PRON
ejpam-4936	108	2	implies	imply	VERB
ejpam-4936	108	3	that	that	SCONJ
ejpam-4936	108	4	0	0	NUM
ejpam-4936	108	5	≡	≡	PROPN
ejpam-4936	108	6	w2	w2	NOUN
ejpam-4936	108	7	≡	≡	PROPN
ejpam-4936	108	8	1	1	NUM
ejpam-4936	109	1	+	+	NUM
ejpam-4936	109	2	4y	4y	PRON
ejpam-4936	109	3	+	+	CCONJ
ejpam-4936	109	4	7z	7z	PROPN
ejpam-4936	109	5	≡	≡	PROPN
ejpam-4936	109	6	1	1	NUM
ejpam-4936	109	7	+	+	CCONJ
ejpam-4936	109	8	(	(	PUNCT
ejpam-4936	109	9	−1)z	−1)z	X
ejpam-4936	109	10	(	(	PUNCT
ejpam-4936	109	11	mod	mod	PROPN
ejpam-4936	109	12	4	4	NUM
ejpam-4936	109	13	)	)	PUNCT
ejpam-4936	109	14	and	and	CCONJ
ejpam-4936	109	15	then	then	ADV
ejpam-4936	109	16	z	z	PROPN
ejpam-4936	109	17	has	have	VERB
ejpam-4936	109	18	to	to	PART
ejpam-4936	109	19	be	be	AUX
ejpam-4936	109	20	odd	odd	ADJ
ejpam-4936	109	21	.	.	PUNCT
ejpam-4936	110	1	again	again	ADV
ejpam-4936	110	2	w2	w2	PROPN
ejpam-4936	110	3	≡	≡	PROPN
ejpam-4936	110	4	1	1	NUM
ejpam-4936	110	5	+	+	PROPN
ejpam-4936	110	6	4y	4y	NOUN
ejpam-4936	110	7	+7z	+7z	NUM
ejpam-4936	110	8	≡	≡	PROPN
ejpam-4936	110	9	1	1	NUM
ejpam-4936	110	10	+	+	NOUN
ejpam-4936	110	11	0	0	NUM
ejpam-4936	110	12	+	+	ADJ
ejpam-4936	110	13	7	7	NUM
ejpam-4936	110	14	≡	≡	ADJ
ejpam-4936	110	15	8	8	NUM
ejpam-4936	110	16	(	(	PUNCT
ejpam-4936	110	17	mod	mod	PROPN
ejpam-4936	110	18	16	16	NUM
ejpam-4936	110	19	)	)	PUNCT
ejpam-4936	110	20	but	but	CCONJ
ejpam-4936	110	21	we	we	PRON
ejpam-4936	110	22	have	have	VERB
ejpam-4936	110	23	the	the	DET
ejpam-4936	110	24	fact	fact	NOUN
ejpam-4936	110	25	that	that	SCONJ
ejpam-4936	110	26	w2	w2	NOUN
ejpam-4936	110	27	≡	≡	PROPN
ejpam-4936	110	28	0	0	NUM
ejpam-4936	110	29	,	,	PUNCT
ejpam-4936	110	30	1	1	NUM
ejpam-4936	110	31	,	,	PUNCT
ejpam-4936	110	32	4	4	NUM
ejpam-4936	110	33	,	,	PUNCT
ejpam-4936	110	34	9	9	NUM
ejpam-4936	110	35	(	(	PUNCT
ejpam-4936	110	36	mod	mod	PROPN
ejpam-4936	110	37	16	16	NUM
ejpam-4936	110	38	)	)	PUNCT
ejpam-4936	110	39	which	which	PRON
ejpam-4936	110	40	is	be	AUX
ejpam-4936	110	41	a	a	DET
ejpam-4936	110	42	contradiction	contradiction	NOUN
ejpam-4936	110	43	.	.	PUNCT
ejpam-4936	111	1	hence	hence	ADV
ejpam-4936	111	2	3x	3x	NUM
ejpam-4936	112	1	+	+	CCONJ
ejpam-4936	112	2	4y	4y	PRON
ejpam-4936	113	1	+	+	CCONJ
ejpam-4936	113	2	7z	7z	PROPN
ejpam-4936	113	3	=	=	SYM
ejpam-4936	113	4	w2	w2	NOUN
ejpam-4936	113	5	has	have	VERB
ejpam-4936	113	6	no	no	DET
ejpam-4936	113	7	solution	solution	NOUN
ejpam-4936	113	8	for	for	ADP
ejpam-4936	113	9	any	any	DET
ejpam-4936	113	10	(	(	PUNCT
ejpam-4936	113	11	x	x	NOUN
ejpam-4936	113	12	,	,	PUNCT
ejpam-4936	113	13	y	y	PROPN
ejpam-4936	113	14	,	,	PUNCT
ejpam-4936	113	15	z	z	PROPN
ejpam-4936	113	16	,	,	PUNCT
ejpam-4936	113	17	w	w	NOUN
ejpam-4936	113	18	)	)	PUNCT
ejpam-4936	113	19	∈	∈	PROPN
ejpam-4936	113	20	u	u	NOUN
ejpam-4936	113	21	.	.	PUNCT
ejpam-4936	114	1	corollary	corollary	ADJ
ejpam-4936	114	2	1	1	NUM
ejpam-4936	114	3	.	.	PUNCT
ejpam-4936	115	1	each	each	PRON
ejpam-4936	115	2	of	of	ADP
ejpam-4936	115	3	the	the	DET
ejpam-4936	115	4	following	follow	VERB
ejpam-4936	115	5	equations	equation	NOUN
ejpam-4936	115	6	:	:	PUNCT
ejpam-4936	115	7	9x	9x	X
ejpam-4936	115	8	+	+	CCONJ
ejpam-4936	115	9	4y	4y	PRON
ejpam-4936	115	10	+	+	CCONJ
ejpam-4936	115	11	7z	7z	NOUN
ejpam-4936	115	12	=	=	SYM
ejpam-4936	115	13	w2	w2	NOUN
ejpam-4936	115	14	,	,	PUNCT
ejpam-4936	115	15	(	(	PUNCT
ejpam-4936	115	16	4	4	X
ejpam-4936	115	17	)	)	PUNCT
ejpam-4936	115	18	3x	3x	PROPN
ejpam-4936	115	19	+	+	CCONJ
ejpam-4936	115	20	16y	16y	NOUN
ejpam-4936	115	21	+	+	CCONJ
ejpam-4936	115	22	7z	7z	NOUN
ejpam-4936	115	23	=	=	SYM
ejpam-4936	115	24	w2	w2	NOUN
ejpam-4936	115	25	(	(	PUNCT
ejpam-4936	115	26	5	5	NUM
ejpam-4936	115	27	)	)	PUNCT
ejpam-4936	115	28	and	and	CCONJ
ejpam-4936	115	29	9x	9x	X
ejpam-4936	115	30	+	+	CCONJ
ejpam-4936	115	31	16y	16y	NOUN
ejpam-4936	115	32	+	+	CCONJ
ejpam-4936	115	33	7z	7z	PROPN
ejpam-4936	115	34	=	=	SYM
ejpam-4936	115	35	w2	w2	NOUN
ejpam-4936	115	36	(	(	PUNCT
ejpam-4936	115	37	6	6	NUM
ejpam-4936	115	38	)	)	PUNCT
ejpam-4936	115	39	have	have	VERB
ejpam-4936	115	40	no	no	DET
ejpam-4936	115	41	solution	solution	NOUN
ejpam-4936	115	42	for	for	ADP
ejpam-4936	115	43	any	any	DET
ejpam-4936	115	44	(	(	PUNCT
ejpam-4936	115	45	x	x	NOUN
ejpam-4936	115	46	,	,	PUNCT
ejpam-4936	115	47	y	y	PROPN
ejpam-4936	115	48	,	,	PUNCT
ejpam-4936	115	49	z	z	PROPN
ejpam-4936	115	50	,	,	PUNCT
ejpam-4936	115	51	w	w	NOUN
ejpam-4936	115	52	)	)	PUNCT
ejpam-4936	115	53	∈	∈	PROPN
ejpam-4936	115	54	u	u	NOUN
ejpam-4936	115	55	.	.	PUNCT
ejpam-4936	116	1	proof	proof	NOUN
ejpam-4936	116	2	.	.	PUNCT
ejpam-4936	117	1	suppose	suppose	VERB
ejpam-4936	117	2	that	that	SCONJ
ejpam-4936	117	3	(	(	PUNCT
ejpam-4936	117	4	x0	x0	PROPN
ejpam-4936	117	5	,	,	PUNCT
ejpam-4936	117	6	y0	y0	PROPN
ejpam-4936	117	7	,	,	PUNCT
ejpam-4936	117	8	z0	z0	NOUN
ejpam-4936	117	9	,	,	PUNCT
ejpam-4936	117	10	w0	w0	PROPN
ejpam-4936	117	11	)	)	PUNCT
ejpam-4936	117	12	∈	∈	PROPN
ejpam-4936	117	13	u	u	NOUN
ejpam-4936	117	14	is	be	AUX
ejpam-4936	117	15	a	a	DET
ejpam-4936	117	16	solution	solution	NOUN
ejpam-4936	117	17	of	of	ADP
ejpam-4936	117	18	the	the	DET
ejpam-4936	117	19	equation	equation	NOUN
ejpam-4936	117	20	(	(	PUNCT
ejpam-4936	117	21	4	4	NUM
ejpam-4936	117	22	)	)	PUNCT
ejpam-4936	117	23	.	.	PUNCT
ejpam-4936	118	1	then	then	ADV
ejpam-4936	118	2	w2	w2	NOUN
ejpam-4936	118	3	0	0	NUM
ejpam-4936	118	4	=	=	SYM
ejpam-4936	119	1	32x0	32x0	NUM
ejpam-4936	119	2	+	+	NUM
ejpam-4936	119	3	4y0	4y0	NUM
ejpam-4936	120	1	+	+	CCONJ
ejpam-4936	120	2	7z0	7z0	NUM
ejpam-4936	120	3	,	,	PUNCT
ejpam-4936	120	4	i.e.	i.e.	X
ejpam-4936	120	5	(	(	PUNCT
ejpam-4936	120	6	2x0	2x0	NUM
ejpam-4936	120	7	,	,	PUNCT
ejpam-4936	120	8	y0	y0	NOUN
ejpam-4936	120	9	,	,	PUNCT
ejpam-4936	120	10	z0	z0	NOUN
ejpam-4936	120	11	,	,	PUNCT
ejpam-4936	120	12	w0	w0	PROPN
ejpam-4936	120	13	)	)	PUNCT
ejpam-4936	120	14	is	be	AUX
ejpam-4936	120	15	a	a	DET
ejpam-4936	120	16	solution	solution	NOUN
ejpam-4936	120	17	of	of	ADP
ejpam-4936	120	18	the	the	DET
ejpam-4936	120	19	equation	equation	NOUN
ejpam-4936	120	20	(	(	PUNCT
ejpam-4936	120	21	1	1	NUM
ejpam-4936	120	22	)	)	PUNCT
ejpam-4936	120	23	.	.	PUNCT
ejpam-4936	121	1	it	it	PRON
ejpam-4936	121	2	contradicts	contradict	VERB
ejpam-4936	121	3	to	to	ADP
ejpam-4936	121	4	1	1	NUM
ejpam-4936	121	5	.	.	PUNCT
ejpam-4936	121	6	with	with	ADP
ejpam-4936	121	7	the	the	DET
ejpam-4936	121	8	same	same	ADJ
ejpam-4936	121	9	trace	trace	NOUN
ejpam-4936	121	10	of	of	ADP
ejpam-4936	121	11	the	the	DET
ejpam-4936	121	12	proof	proof	NOUN
ejpam-4936	121	13	of	of	ADP
ejpam-4936	121	14	equation	equation	NOUN
ejpam-4936	121	15	(	(	PUNCT
ejpam-4936	121	16	4	4	NUM
ejpam-4936	121	17	)	)	PUNCT
ejpam-4936	121	18	,	,	PUNCT
ejpam-4936	121	19	we	we	PRON
ejpam-4936	121	20	can	can	AUX
ejpam-4936	121	21	conclude	conclude	VERB
ejpam-4936	121	22	that	that	DET
ejpam-4936	121	23	equation	equation	NOUN
ejpam-4936	121	24	(	(	PUNCT
ejpam-4936	121	25	5	5	NUM
ejpam-4936	121	26	)	)	PUNCT
ejpam-4936	121	27	and	and	CCONJ
ejpam-4936	121	28	equation	equation	NOUN
ejpam-4936	121	29	(	(	PUNCT
ejpam-4936	121	30	6	6	NUM
ejpam-4936	121	31	)	)	PUNCT
ejpam-4936	121	32	have	have	VERB
ejpam-4936	121	33	no	no	DET
ejpam-4936	121	34	solution	solution	NOUN
ejpam-4936	121	35	for	for	ADP
ejpam-4936	121	36	any	any	DET
ejpam-4936	121	37	(	(	PUNCT
ejpam-4936	121	38	x	x	NOUN
ejpam-4936	121	39	,	,	PUNCT
ejpam-4936	121	40	y	y	PROPN
ejpam-4936	121	41	,	,	PUNCT
ejpam-4936	121	42	z	z	PROPN
ejpam-4936	121	43	,	,	PUNCT
ejpam-4936	121	44	w	w	NOUN
ejpam-4936	121	45	)	)	PUNCT
ejpam-4936	121	46	∈	∈	PROPN
ejpam-4936	121	47	u	u	NOUN
ejpam-4936	121	48	.	.	PUNCT
ejpam-4936	122	1	theorem	theorem	NOUN
ejpam-4936	122	2	2	2	NUM
ejpam-4936	122	3	.	.	PUNCT
ejpam-4936	123	1	the	the	DET
ejpam-4936	123	2	equation	equation	NOUN
ejpam-4936	123	3	4x	4x	NOUN
ejpam-4936	123	4	+	+	CCONJ
ejpam-4936	123	5	6y	6y	NOUN
ejpam-4936	123	6	+	+	CCONJ
ejpam-4936	123	7	7z	7z	NOUN
ejpam-4936	123	8	=	=	SYM
ejpam-4936	123	9	w2	w2	NOUN
ejpam-4936	123	10	(	(	PUNCT
ejpam-4936	123	11	7	7	NUM
ejpam-4936	123	12	)	)	PUNCT
ejpam-4936	123	13	has	have	VERB
ejpam-4936	123	14	no	no	DET
ejpam-4936	123	15	solution	solution	NOUN
ejpam-4936	123	16	for	for	ADP
ejpam-4936	123	17	all	all	DET
ejpam-4936	123	18	(	(	PUNCT
ejpam-4936	123	19	x	x	NOUN
ejpam-4936	123	20	,	,	PUNCT
ejpam-4936	123	21	y	y	PROPN
ejpam-4936	123	22	,	,	PUNCT
ejpam-4936	123	23	z	z	PROPN
ejpam-4936	123	24	,	,	PUNCT
ejpam-4936	123	25	w	w	NOUN
ejpam-4936	123	26	)	)	PUNCT
ejpam-4936	123	27	∈	∈	PROPN
ejpam-4936	123	28	u	u	NOUN
ejpam-4936	123	29	.	.	PUNCT
ejpam-4936	124	1	proof	proof	NOUN
ejpam-4936	124	2	.	.	PUNCT
ejpam-4936	125	1	the	the	DET
ejpam-4936	125	2	proof	proof	NOUN
ejpam-4936	125	3	of	of	ADP
ejpam-4936	125	4	2	2	NUM
ejpam-4936	125	5	follows	follow	VERB
ejpam-4936	125	6	from	from	ADP
ejpam-4936	125	7	the	the	DET
ejpam-4936	125	8	footprint	footprint	NOUN
ejpam-4936	125	9	of	of	ADP
ejpam-4936	125	10	1	1	NUM
ejpam-4936	125	11	by	by	ADP
ejpam-4936	125	12	using	use	VERB
ejpam-4936	125	13	modulo	modulo	PROPN
ejpam-4936	125	14	3	3	NUM
ejpam-4936	125	15	,	,	PUNCT
ejpam-4936	125	16	4	4	NUM
ejpam-4936	125	17	and	and	CCONJ
ejpam-4936	125	18	16	16	NUM
ejpam-4936	125	19	but	but	CCONJ
ejpam-4936	125	20	dividing	divide	VERB
ejpam-4936	125	21	the	the	DET
ejpam-4936	125	22	value	value	NOUN
ejpam-4936	125	23	z	z	NOUN
ejpam-4936	125	24	to	to	ADP
ejpam-4936	125	25	two	two	NUM
ejpam-4936	125	26	cases	case	NOUN
ejpam-4936	125	27	.	.	PUNCT
ejpam-4936	126	1	case	case	NOUN
ejpam-4936	126	2	1	1	NUM
ejpam-4936	126	3	z	z	NOUN
ejpam-4936	126	4	=	=	NOUN
ejpam-4936	126	5	0	0	X
ejpam-4936	126	6	.	.	PUNCT
ejpam-4936	127	1	since	since	SCONJ
ejpam-4936	127	2	w2	w2	NOUN
ejpam-4936	127	3	̸≡	̸≡	PROPN
ejpam-4936	127	4	2	2	NUM
ejpam-4936	127	5	(	(	PUNCT
ejpam-4936	127	6	mod	mod	NOUN
ejpam-4936	127	7	3	3	NUM
ejpam-4936	127	8	)	)	PUNCT
ejpam-4936	127	9	but	but	CCONJ
ejpam-4936	127	10	4x	4x	NOUN
ejpam-4936	127	11	+	+	CCONJ
ejpam-4936	127	12	6y	6y	NOUN
ejpam-4936	127	13	+	+	X
ejpam-4936	127	14	1	1	NUM
ejpam-4936	127	15	≡	≡	PROPN
ejpam-4936	127	16	2	2	NUM
ejpam-4936	127	17	(	(	PUNCT
ejpam-4936	127	18	mod	mod	NOUN
ejpam-4936	127	19	3	3	NUM
ejpam-4936	127	20	)	)	PUNCT
ejpam-4936	127	21	for	for	ADP
ejpam-4936	127	22	all	all	PRON
ejpam-4936	127	23	x	x	PRON
ejpam-4936	127	24	≥	≥	NUM
ejpam-4936	127	25	0	0	NUM
ejpam-4936	127	26	and	and	CCONJ
ejpam-4936	127	27	y	y	PROPN
ejpam-4936	127	28	≥	≥	NUM
ejpam-4936	127	29	1	1	NUM
ejpam-4936	127	30	,	,	PUNCT
ejpam-4936	127	31	the	the	DET
ejpam-4936	127	32	equation	equation	NOUN
ejpam-4936	127	33	(	(	PUNCT
ejpam-4936	127	34	7	7	X
ejpam-4936	127	35	)	)	PUNCT
ejpam-4936	127	36	has	have	VERB
ejpam-4936	127	37	no	no	DET
ejpam-4936	127	38	solution	solution	NOUN
ejpam-4936	127	39	.	.	PUNCT
ejpam-4936	128	1	so	so	ADV
ejpam-4936	128	2	,	,	PUNCT
ejpam-4936	128	3	we	we	PRON
ejpam-4936	128	4	remain	remain	VERB
ejpam-4936	128	5	to	to	PART
ejpam-4936	128	6	proof	proof	NOUN
ejpam-4936	128	7	the	the	DET
ejpam-4936	128	8	case	case	NOUN
ejpam-4936	128	9	x	x	X
ejpam-4936	128	10	≥	≥	NUM
ejpam-4936	128	11	1	1	NUM
ejpam-4936	128	12	and	and	CCONJ
ejpam-4936	128	13	y	y	PROPN
ejpam-4936	128	14	=	=	SYM
ejpam-4936	128	15	0	0	PROPN
ejpam-4936	128	16	,	,	PUNCT
ejpam-4936	128	17	the	the	DET
ejpam-4936	128	18	equation	equation	NOUN
ejpam-4936	128	19	(	(	PUNCT
ejpam-4936	128	20	7	7	X
ejpam-4936	128	21	)	)	PUNCT
ejpam-4936	128	22	becomes	become	VERB
ejpam-4936	128	23	to	to	ADP
ejpam-4936	128	24	4x	4x	NUM
ejpam-4936	128	25	+	+	CCONJ
ejpam-4936	128	26	2	2	NUM
ejpam-4936	128	27	=	=	NOUN
ejpam-4936	128	28	w2	w2	NOUN
ejpam-4936	128	29	.	.	PUNCT
ejpam-4936	129	1	by	by	ADP
ejpam-4936	129	2	the	the	DET
ejpam-4936	129	3	fact	fact	NOUN
ejpam-4936	129	4	that	that	SCONJ
ejpam-4936	129	5	x	x	X
ejpam-4936	129	6	≥	≥	NUM
ejpam-4936	129	7	1	1	NUM
ejpam-4936	129	8	,	,	PUNCT
ejpam-4936	129	9	we	we	PRON
ejpam-4936	129	10	have	have	AUX
ejpam-4936	129	11	w	w	NOUN
ejpam-4936	129	12	is	be	AUX
ejpam-4936	129	13	even	even	ADV
ejpam-4936	129	14	and	and	CCONJ
ejpam-4936	129	15	then	then	ADV
ejpam-4936	129	16	w2	w2	PROPN
ejpam-4936	129	17	≡	≡	PROPN
ejpam-4936	129	18	0	0	PUNCT
ejpam-4936	130	1	(	(	PUNCT
ejpam-4936	130	2	mod	mod	PROPN
ejpam-4936	130	3	4	4	NUM
ejpam-4936	130	4	)	)	PUNCT
ejpam-4936	130	5	that	that	PRON
ejpam-4936	130	6	contradicts	contradict	VERB
ejpam-4936	130	7	to	to	ADP
ejpam-4936	130	8	w2	w2	PROPN
ejpam-4936	130	9	≡	≡	PROPN
ejpam-4936	130	10	4x	4x	PROPN
ejpam-4936	131	1	+	+	CCONJ
ejpam-4936	131	2	2	2	NUM
ejpam-4936	131	3	≡	≡	PROPN
ejpam-4936	131	4	2	2	NUM
ejpam-4936	131	5	(	(	PUNCT
ejpam-4936	131	6	mod	mod	NOUN
ejpam-4936	131	7	4	4	NUM
ejpam-4936	131	8	)	)	PUNCT
ejpam-4936	131	9	.	.	PUNCT
ejpam-4936	132	1	case	case	NOUN
ejpam-4936	132	2	2	2	NUM
ejpam-4936	132	3	z	z	NOUN
ejpam-4936	132	4	≥	≥	NUM
ejpam-4936	132	5	1	1	NUM
ejpam-4936	132	6	.	.	PUNCT
ejpam-4936	133	1	again	again	ADV
ejpam-4936	133	2	w2	w2	NOUN
ejpam-4936	133	3	̸≡	̸≡	PROPN
ejpam-4936	133	4	2	2	NUM
ejpam-4936	133	5	(	(	PUNCT
ejpam-4936	133	6	mod	mod	NOUN
ejpam-4936	133	7	3	3	NUM
ejpam-4936	133	8	)	)	PUNCT
ejpam-4936	133	9	and	and	CCONJ
ejpam-4936	133	10	4x	4x	NOUN
ejpam-4936	133	11	+	+	NUM
ejpam-4936	133	12	6y	6y	NOUN
ejpam-4936	133	13	+	+	CCONJ
ejpam-4936	133	14	7z	7z	NUM
ejpam-4936	133	15	≡	≡	PROPN
ejpam-4936	133	16	2	2	NUM
ejpam-4936	133	17	(	(	PUNCT
ejpam-4936	133	18	mod	mod	NOUN
ejpam-4936	133	19	3	3	NUM
ejpam-4936	133	20	)	)	PUNCT
ejpam-4936	133	21	for	for	ADP
ejpam-4936	133	22	any	any	DET
ejpam-4936	133	23	x	x	SYM
ejpam-4936	133	24	≥	≥	NOUN
ejpam-4936	133	25	0	0	NUM
ejpam-4936	133	26	and	and	CCONJ
ejpam-4936	133	27	y	y	PROPN
ejpam-4936	133	28	≥	≥	NUM
ejpam-4936	133	29	1	1	NUM
ejpam-4936	133	30	.	.	PUNCT
ejpam-4936	134	1	so	so	ADV
ejpam-4936	134	2	the	the	DET
ejpam-4936	134	3	equation	equation	NOUN
ejpam-4936	134	4	(	(	PUNCT
ejpam-4936	134	5	7	7	X
ejpam-4936	134	6	)	)	PUNCT
ejpam-4936	134	7	has	have	VERB
ejpam-4936	134	8	no	no	DET
ejpam-4936	134	9	solution	solution	NOUN
ejpam-4936	134	10	.	.	PUNCT
ejpam-4936	135	1	next	next	ADV
ejpam-4936	135	2	,	,	PUNCT
ejpam-4936	135	3	we	we	PRON
ejpam-4936	135	4	have	have	VERB
ejpam-4936	135	5	to	to	PART
ejpam-4936	135	6	consider	consider	VERB
ejpam-4936	135	7	only	only	ADV
ejpam-4936	135	8	subcase	subcase	NOUN
ejpam-4936	135	9	x	x	PUNCT
ejpam-4936	135	10	≥	≥	NUM
ejpam-4936	135	11	1	1	NUM
ejpam-4936	135	12	and	and	CCONJ
ejpam-4936	136	1	y	y	PROPN
ejpam-4936	136	2	=	=	SYM
ejpam-4936	136	3	0	0	PROPN
ejpam-4936	136	4	.	.	PUNCT
ejpam-4936	137	1	if	if	SCONJ
ejpam-4936	137	2	x	x	SYM
ejpam-4936	137	3	=	=	NOUN
ejpam-4936	137	4	1	1	NUM
ejpam-4936	137	5	then	then	ADV
ejpam-4936	137	6	the	the	DET
ejpam-4936	137	7	equation	equation	NOUN
ejpam-4936	137	8	(	(	PUNCT
ejpam-4936	137	9	7	7	X
ejpam-4936	137	10	)	)	PUNCT
ejpam-4936	137	11	has	have	VERB
ejpam-4936	137	12	also	also	ADV
ejpam-4936	137	13	no	no	DET
ejpam-4936	137	14	solution	solution	NOUN
ejpam-4936	137	15	by	by	ADP
ejpam-4936	137	16	1	1	NUM
ejpam-4936	137	17	.	.	PUNCT
ejpam-4936	138	1	now	now	ADV
ejpam-4936	138	2	,	,	PUNCT
ejpam-4936	138	3	we	we	PRON
ejpam-4936	138	4	focus	focus	VERB
ejpam-4936	138	5	on	on	ADP
ejpam-4936	138	6	the	the	DET
ejpam-4936	138	7	equation	equation	NOUN
ejpam-4936	138	8	4x+1	4x+1	NOUN
ejpam-4936	138	9	+	+	PROPN
ejpam-4936	138	10	7z	7z	PROPN
ejpam-4936	138	11	=	=	SYM
ejpam-4936	138	12	w2	w2	NOUN
ejpam-4936	138	13	for	for	ADP
ejpam-4936	138	14	x	x	X
ejpam-4936	138	15	≥	≥	NUM
ejpam-4936	138	16	2	2	NUM
ejpam-4936	138	17	.	.	PUNCT
ejpam-4936	139	1	since	since	SCONJ
ejpam-4936	139	2	w2	w2	PROPN
ejpam-4936	139	3	≡	≡	PROPN
ejpam-4936	139	4	0	0	NUM
ejpam-4936	139	5	,	,	PUNCT
ejpam-4936	139	6	1	1	NUM
ejpam-4936	139	7	,	,	PUNCT
ejpam-4936	139	8	4	4	NUM
ejpam-4936	139	9	,	,	PUNCT
ejpam-4936	139	10	9	9	NUM
ejpam-4936	139	11	(	(	PUNCT
ejpam-4936	139	12	mod	mod	NOUN
ejpam-4936	139	13	16	16	NUM
ejpam-4936	139	14	)	)	PUNCT
ejpam-4936	139	15	and	and	CCONJ
ejpam-4936	139	16	4x	4x	NUM
ejpam-4936	140	1	+	+	CCONJ
ejpam-4936	140	2	1	1	NUM
ejpam-4936	141	1	+	+	NUM
ejpam-4936	141	2	7z	7z	PROPN
ejpam-4936	141	3	≡	≡	PROPN
ejpam-4936	141	4	{	{	PUNCT
ejpam-4936	141	5	2	2	NUM
ejpam-4936	141	6	(	(	PUNCT
ejpam-4936	141	7	mod	mod	NOUN
ejpam-4936	141	8	16	16	NUM
ejpam-4936	141	9	)	)	PUNCT
ejpam-4936	141	10	if	if	SCONJ
ejpam-4936	141	11	z	z	NOUN
ejpam-4936	141	12	is	be	AUX
ejpam-4936	141	13	even	even	ADV
ejpam-4936	141	14	,	,	PUNCT
ejpam-4936	141	15	8	8	NUM
ejpam-4936	141	16	(	(	PUNCT
ejpam-4936	141	17	mod	mod	NOUN
ejpam-4936	141	18	16	16	NUM
ejpam-4936	141	19	)	)	PUNCT
ejpam-4936	141	20	if	if	SCONJ
ejpam-4936	141	21	z	z	NOUN
ejpam-4936	141	22	is	be	AUX
ejpam-4936	141	23	odd	odd	ADJ
ejpam-4936	141	24	.	.	PUNCT
ejpam-4936	142	1	we	we	PRON
ejpam-4936	142	2	have	have	VERB
ejpam-4936	142	3	4x	4x	NOUN
ejpam-4936	143	1	+	+	CCONJ
ejpam-4936	143	2	1	1	NUM
ejpam-4936	143	3	+	+	CCONJ
ejpam-4936	143	4	7z	7z	PROPN
ejpam-4936	143	5	̸≡	̸≡	PROPN
ejpam-4936	143	6	w2	w2	PROPN
ejpam-4936	143	7	(	(	PUNCT
ejpam-4936	143	8	mod	mod	PROPN
ejpam-4936	143	9	16	16	NUM
ejpam-4936	143	10	)	)	PUNCT
ejpam-4936	143	11	.	.	PUNCT
ejpam-4936	144	1	thus	thus	ADV
ejpam-4936	144	2	the	the	DET
ejpam-4936	144	3	equation	equation	NOUN
ejpam-4936	144	4	(	(	PUNCT
ejpam-4936	144	5	7	7	X
ejpam-4936	144	6	)	)	PUNCT
ejpam-4936	144	7	has	have	VERB
ejpam-4936	144	8	no	no	DET
ejpam-4936	144	9	solution	solution	NOUN
ejpam-4936	144	10	.	.	PUNCT
ejpam-4936	145	1	k.	k.	PROPN
ejpam-4936	145	2	laipaporn	laipaporn	PROPN
ejpam-4936	145	3	,	,	PUNCT
ejpam-4936	145	4	s.	s.	PROPN
ejpam-4936	145	5	kaewchay	kaewchay	PROPN
ejpam-4936	145	6	,	,	PUNCT
ejpam-4936	145	7	a.	a.	PROPN
ejpam-4936	145	8	karnbanjong	karnbanjong	PROPN
ejpam-4936	145	9	/	/	SYM
ejpam-4936	145	10	eur	eur	PROPN
ejpam-4936	145	11	.	.	PUNCT
ejpam-4936	146	1	j.	j.	PROPN
ejpam-4936	146	2	pure	pure	PROPN
ejpam-4936	146	3	appl	appl	PROPN
ejpam-4936	146	4	.	.	PROPN
ejpam-4936	146	5	math	math	PROPN
ejpam-4936	146	6	,	,	PUNCT
ejpam-4936	146	7	16	16	NUM
ejpam-4936	146	8	(	(	PUNCT
ejpam-4936	146	9	4	4	NUM
ejpam-4936	146	10	)	)	PUNCT
ejpam-4936	146	11	(	(	PUNCT
ejpam-4936	146	12	2023	2023	NUM
ejpam-4936	146	13	)	)	PUNCT
ejpam-4936	146	14	,	,	PUNCT
ejpam-4936	146	15	2066	2066	NUM
ejpam-4936	146	16	-	-	SYM
ejpam-4936	146	17	2081	2081	NUM
ejpam-4936	146	18	2070	2070	NUM
ejpam-4936	146	19	corollary	corollary	NOUN
ejpam-4936	146	20	2	2	NUM
ejpam-4936	146	21	.	.	PUNCT
ejpam-4936	147	1	the	the	DET
ejpam-4936	147	2	equation	equation	NOUN
ejpam-4936	147	3	16x	16x	NOUN
ejpam-4936	147	4	+	+	CCONJ
ejpam-4936	147	5	6y	6y	NOUN
ejpam-4936	147	6	+	+	CCONJ
ejpam-4936	147	7	7z	7z	NOUN
ejpam-4936	147	8	=	=	SYM
ejpam-4936	147	9	w2	w2	NOUN
ejpam-4936	147	10	(	(	PUNCT
ejpam-4936	147	11	8)	8)	NUM
ejpam-4936	147	12	has	have	VERB
ejpam-4936	147	13	no	no	DET
ejpam-4936	147	14	solution	solution	NOUN
ejpam-4936	147	15	for	for	ADP
ejpam-4936	147	16	any	any	DET
ejpam-4936	147	17	(	(	PUNCT
ejpam-4936	147	18	x	x	NOUN
ejpam-4936	147	19	,	,	PUNCT
ejpam-4936	147	20	y	y	PROPN
ejpam-4936	147	21	,	,	PUNCT
ejpam-4936	147	22	z	z	PROPN
ejpam-4936	147	23	,	,	PUNCT
ejpam-4936	147	24	w	w	NOUN
ejpam-4936	147	25	)	)	PUNCT
ejpam-4936	147	26	∈	∈	PROPN
ejpam-4936	147	27	u	u	NOUN
ejpam-4936	147	28	.	.	PUNCT
ejpam-4936	148	1	proof	proof	NOUN
ejpam-4936	148	2	.	.	PUNCT
ejpam-4936	149	1	with	with	ADP
ejpam-4936	149	2	the	the	DET
ejpam-4936	149	3	same	same	ADJ
ejpam-4936	149	4	trace	trace	NOUN
ejpam-4936	149	5	of	of	ADP
ejpam-4936	149	6	1	1	NUM
ejpam-4936	149	7	,	,	PUNCT
ejpam-4936	149	8	we	we	PRON
ejpam-4936	149	9	can	can	AUX
ejpam-4936	149	10	conclude	conclude	VERB
ejpam-4936	149	11	that	that	DET
ejpam-4936	149	12	equation	equation	NOUN
ejpam-4936	149	13	(	(	PUNCT
ejpam-4936	149	14	8)	8)	NUM
ejpam-4936	149	15	has	have	VERB
ejpam-4936	149	16	no	no	DET
ejpam-4936	149	17	solution	solution	NOUN
ejpam-4936	149	18	by	by	ADP
ejpam-4936	149	19	using	use	VERB
ejpam-4936	149	20	2	2	NUM
ejpam-4936	149	21	.	.	PUNCT
ejpam-4936	149	22	theorem	theorem	NOUN
ejpam-4936	149	23	3	3	NUM
ejpam-4936	149	24	.	.	PUNCT
ejpam-4936	150	1	the	the	DET
ejpam-4936	150	2	equation	equation	NOUN
ejpam-4936	150	3	9x	9x	NOUN
ejpam-4936	150	4	+	+	CCONJ
ejpam-4936	150	5	10y	10y	PROPN
ejpam-4936	150	6	+	+	CCONJ
ejpam-4936	150	7	7z	7z	PROPN
ejpam-4936	150	8	=	=	SYM
ejpam-4936	150	9	w2	w2	NOUN
ejpam-4936	150	10	(	(	PUNCT
ejpam-4936	150	11	9	9	NUM
ejpam-4936	150	12	)	)	PUNCT
ejpam-4936	150	13	has	have	VERB
ejpam-4936	150	14	no	no	DET
ejpam-4936	150	15	solution	solution	NOUN
ejpam-4936	150	16	for	for	ADP
ejpam-4936	150	17	any	any	DET
ejpam-4936	150	18	(	(	PUNCT
ejpam-4936	150	19	x	x	NOUN
ejpam-4936	150	20	,	,	PUNCT
ejpam-4936	150	21	y	y	PROPN
ejpam-4936	150	22	,	,	PUNCT
ejpam-4936	150	23	z	z	PROPN
ejpam-4936	150	24	,	,	PUNCT
ejpam-4936	150	25	w	w	NOUN
ejpam-4936	150	26	)	)	PUNCT
ejpam-4936	150	27	∈	∈	PROPN
ejpam-4936	150	28	u	u	NOUN
ejpam-4936	150	29	−	−	PROPN
ejpam-4936	150	30	t	t	PROPN
ejpam-4936	150	31	,	,	PUNCT
ejpam-4936	150	32	where	where	SCONJ
ejpam-4936	150	33	t	t	NOUN
ejpam-4936	150	34	=	=	SYM
ejpam-4936	150	35	{	{	PUNCT
ejpam-4936	150	36	(	(	PUNCT
ejpam-4936	150	37	0	0	NUM
ejpam-4936	150	38	,	,	PUNCT
ejpam-4936	150	39	3	3	NUM
ejpam-4936	150	40	,	,	PUNCT
ejpam-4936	150	41	z	z	PROPN
ejpam-4936	150	42	,	,	PUNCT
ejpam-4936	150	43	w)|z	w)|z	X
ejpam-4936	150	44	≥	≥	NOUN
ejpam-4936	150	45	2	2	NUM
ejpam-4936	150	46	and	and	CCONJ
ejpam-4936	150	47	z	z	NOUN
ejpam-4936	150	48	is	be	AUX
ejpam-4936	150	49	odd	odd	ADJ
ejpam-4936	150	50	}	}	PUNCT
ejpam-4936	150	51	.	.	PUNCT
ejpam-4936	151	1	proof	proof	NOUN
ejpam-4936	151	2	.	.	PUNCT
ejpam-4936	152	1	for	for	ADP
ejpam-4936	152	2	proving	prove	VERB
ejpam-4936	152	3	this	this	DET
ejpam-4936	152	4	theorem	theorem	NOUN
ejpam-4936	152	5	,	,	PUNCT
ejpam-4936	152	6	we	we	PRON
ejpam-4936	152	7	still	still	ADV
ejpam-4936	152	8	use	use	VERB
ejpam-4936	152	9	modulo	modulo	NOUN
ejpam-4936	152	10	3	3	NUM
ejpam-4936	152	11	and	and	CCONJ
ejpam-4936	152	12	16	16	NUM
ejpam-4936	152	13	that	that	PRON
ejpam-4936	152	14	play	play	VERB
ejpam-4936	152	15	the	the	DET
ejpam-4936	152	16	main	main	ADJ
ejpam-4936	152	17	role	role	NOUN
ejpam-4936	152	18	to	to	PART
ejpam-4936	152	19	verify	verify	VERB
ejpam-4936	152	20	the	the	DET
ejpam-4936	152	21	existence	existence	NOUN
ejpam-4936	152	22	of	of	ADP
ejpam-4936	152	23	its	its	PRON
ejpam-4936	152	24	solution	solution	NOUN
ejpam-4936	152	25	and	and	CCONJ
ejpam-4936	152	26	also	also	ADV
ejpam-4936	152	27	use	use	VERB
ejpam-4936	152	28	modulo	modulo	NOUN
ejpam-4936	152	29	5	5	NUM
ejpam-4936	152	30	and	and	CCONJ
ejpam-4936	152	31	10	10	NUM
ejpam-4936	152	32	in	in	ADP
ejpam-4936	152	33	some	some	DET
ejpam-4936	152	34	subcaes	subcaes	NOUN
ejpam-4936	152	35	of	of	ADP
ejpam-4936	152	36	the	the	DET
ejpam-4936	152	37	variable	variable	ADJ
ejpam-4936	152	38	z.	z.	PROPN
ejpam-4936	152	39	case	case	NOUN
ejpam-4936	152	40	1	1	NUM
ejpam-4936	152	41	z	z	NOUN
ejpam-4936	152	42	=	=	SYM
ejpam-4936	152	43	0	0	X
ejpam-4936	152	44	.	.	PUNCT
ejpam-4936	153	1	note	note	VERB
ejpam-4936	153	2	that	that	SCONJ
ejpam-4936	153	3	9x	9x	NOUN
ejpam-4936	153	4	+	+	CCONJ
ejpam-4936	153	5	10y	10y	X
ejpam-4936	154	1	+	+	CCONJ
ejpam-4936	154	2	1	1	NUM
ejpam-4936	154	3	≡	≡	PROPN
ejpam-4936	154	4	{	{	PUNCT
ejpam-4936	154	5	2	2	NUM
ejpam-4936	154	6	(	(	PUNCT
ejpam-4936	154	7	mod	mod	NOUN
ejpam-4936	154	8	3	3	X
ejpam-4936	154	9	)	)	PUNCT
ejpam-4936	154	10	if	if	SCONJ
ejpam-4936	154	11	x	x	X
ejpam-4936	154	12	≥	≥	VERB
ejpam-4936	154	13	1	1	NUM
ejpam-4936	154	14	and	and	CCONJ
ejpam-4936	154	15	y	y	PROPN
ejpam-4936	154	16	≥	≥	NUM
ejpam-4936	154	17	0	0	NUM
ejpam-4936	154	18	,	,	PUNCT
ejpam-4936	154	19	2	2	NUM
ejpam-4936	154	20	(	(	PUNCT
ejpam-4936	154	21	mod	mod	PROPN
ejpam-4936	154	22	10	10	NUM
ejpam-4936	154	23	)	)	PUNCT
ejpam-4936	154	24	if	if	SCONJ
ejpam-4936	154	25	x	x	PROPN
ejpam-4936	154	26	=	=	SYM
ejpam-4936	154	27	0	0	NUM
ejpam-4936	154	28	and	and	CCONJ
ejpam-4936	154	29	y	y	PROPN
ejpam-4936	154	30	≥	≥	NUM
ejpam-4936	154	31	1	1	NUM
ejpam-4936	154	32	,	,	PUNCT
ejpam-4936	154	33	and	and	CCONJ
ejpam-4936	154	34	we	we	PRON
ejpam-4936	154	35	know	know	VERB
ejpam-4936	154	36	that	that	SCONJ
ejpam-4936	154	37	neither	neither	PRON
ejpam-4936	154	38	w2	w2	NOUN
ejpam-4936	154	39	̸≡	̸≡	NOUN
ejpam-4936	154	40	2	2	NUM
ejpam-4936	154	41	(	(	PUNCT
ejpam-4936	154	42	mod	mod	NOUN
ejpam-4936	154	43	3	3	NUM
ejpam-4936	154	44	)	)	PUNCT
ejpam-4936	154	45	nor	nor	CCONJ
ejpam-4936	154	46	w2	w2	NOUN
ejpam-4936	154	47	̸≡	̸≡	NOUN
ejpam-4936	154	48	2	2	NUM
ejpam-4936	154	49	(	(	PUNCT
ejpam-4936	154	50	mod	mod	PROPN
ejpam-4936	154	51	10	10	NUM
ejpam-4936	154	52	)	)	PUNCT
ejpam-4936	154	53	.	.	PUNCT
ejpam-4936	155	1	then	then	ADV
ejpam-4936	155	2	it	it	PRON
ejpam-4936	155	3	is	be	AUX
ejpam-4936	155	4	clear	clear	ADJ
ejpam-4936	155	5	that	that	SCONJ
ejpam-4936	155	6	the	the	DET
ejpam-4936	155	7	equation	equation	NOUN
ejpam-4936	155	8	(	(	PUNCT
ejpam-4936	155	9	9	9	NUM
ejpam-4936	155	10	)	)	PUNCT
ejpam-4936	155	11	has	have	VERB
ejpam-4936	155	12	no	no	DET
ejpam-4936	155	13	solution	solution	NOUN
ejpam-4936	155	14	on	on	ADP
ejpam-4936	155	15	u	u	PRON
ejpam-4936	155	16	if	if	SCONJ
ejpam-4936	155	17	z	z	NOUN
ejpam-4936	155	18	=	=	NOUN
ejpam-4936	155	19	0	0	X
ejpam-4936	155	20	.	.	PUNCT
ejpam-4936	155	21	case	case	NOUN
ejpam-4936	155	22	2	2	NUM
ejpam-4936	155	23	z	z	NOUN
ejpam-4936	155	24	=	=	NOUN
ejpam-4936	155	25	1	1	X
ejpam-4936	155	26	.	.	PUNCT
ejpam-4936	155	27	with	with	ADP
ejpam-4936	155	28	the	the	DET
ejpam-4936	155	29	same	same	ADJ
ejpam-4936	155	30	fashion	fashion	NOUN
ejpam-4936	155	31	in	in	ADP
ejpam-4936	155	32	case	case	NOUN
ejpam-4936	155	33	1	1	NUM
ejpam-4936	155	34	,	,	PUNCT
ejpam-4936	155	35	we	we	PRON
ejpam-4936	155	36	note	note	VERB
ejpam-4936	155	37	that	that	SCONJ
ejpam-4936	155	38	9x	9x	NOUN
ejpam-4936	155	39	+	+	CCONJ
ejpam-4936	155	40	10y	10y	X
ejpam-4936	155	41	+	+	CCONJ
ejpam-4936	155	42	7	7	NUM
ejpam-4936	155	43	≡	≡	PROPN
ejpam-4936	155	44	{	{	PUNCT
ejpam-4936	155	45	2	2	NUM
ejpam-4936	155	46	(	(	PUNCT
ejpam-4936	155	47	mod	mod	NOUN
ejpam-4936	155	48	3	3	X
ejpam-4936	155	49	)	)	PUNCT
ejpam-4936	155	50	if	if	SCONJ
ejpam-4936	155	51	x	x	X
ejpam-4936	155	52	≥	≥	VERB
ejpam-4936	155	53	1	1	NUM
ejpam-4936	155	54	and	and	CCONJ
ejpam-4936	155	55	y	y	PROPN
ejpam-4936	155	56	≥	≥	NUM
ejpam-4936	155	57	0	0	NUM
ejpam-4936	155	58	,	,	PUNCT
ejpam-4936	155	59	3	3	NUM
ejpam-4936	155	60	(	(	PUNCT
ejpam-4936	155	61	mod	mod	NOUN
ejpam-4936	155	62	5	5	NUM
ejpam-4936	155	63	)	)	PUNCT
ejpam-4936	155	64	if	if	SCONJ
ejpam-4936	155	65	x	x	PROPN
ejpam-4936	155	66	=	=	SYM
ejpam-4936	155	67	0	0	NUM
ejpam-4936	155	68	and	and	CCONJ
ejpam-4936	155	69	y	y	PROPN
ejpam-4936	155	70	≥	≥	PROPN
ejpam-4936	155	71	1	1	NUM
ejpam-4936	155	72	.	.	PUNCT
ejpam-4936	156	1	since	since	SCONJ
ejpam-4936	156	2	w2	w2	PROPN
ejpam-4936	156	3	≡	≡	PROPN
ejpam-4936	156	4	2	2	NUM
ejpam-4936	156	5	(	(	PUNCT
ejpam-4936	156	6	mod	mod	NOUN
ejpam-4936	156	7	3	3	NUM
ejpam-4936	156	8	)	)	PUNCT
ejpam-4936	156	9	and	and	CCONJ
ejpam-4936	156	10	w2	w2	NOUN
ejpam-4936	156	11	̸≡	̸≡	PROPN
ejpam-4936	156	12	3	3	NUM
ejpam-4936	156	13	(	(	PUNCT
ejpam-4936	156	14	mod	mod	NOUN
ejpam-4936	156	15	5	5	NUM
ejpam-4936	156	16	)	)	PUNCT
ejpam-4936	156	17	the	the	DET
ejpam-4936	156	18	equation	equation	NOUN
ejpam-4936	156	19	9x	9x	NOUN
ejpam-4936	156	20	+	+	CCONJ
ejpam-4936	156	21	10y	10y	X
ejpam-4936	156	22	+	+	CCONJ
ejpam-4936	156	23	7	7	NUM
ejpam-4936	156	24	=	=	NOUN
ejpam-4936	156	25	w2	w2	NOUN
ejpam-4936	156	26	has	have	VERB
ejpam-4936	156	27	no	no	DET
ejpam-4936	156	28	solution	solution	NOUN
ejpam-4936	156	29	on	on	ADP
ejpam-4936	156	30	u	u	PRON
ejpam-4936	156	31	if	if	SCONJ
ejpam-4936	156	32	z	z	NOUN
ejpam-4936	156	33	=	=	NOUN
ejpam-4936	156	34	1	1	X
ejpam-4936	156	35	.	.	PUNCT
ejpam-4936	156	36	case	case	NOUN
ejpam-4936	156	37	3	3	NUM
ejpam-4936	156	38	z	z	NOUN
ejpam-4936	156	39	≥	≥	NUM
ejpam-4936	156	40	2	2	NUM
ejpam-4936	156	41	.	.	PUNCT
ejpam-4936	157	1	if	if	SCONJ
ejpam-4936	157	2	x	x	X
ejpam-4936	157	3	≥	≥	VERB
ejpam-4936	157	4	1	1	NUM
ejpam-4936	157	5	and	and	CCONJ
ejpam-4936	157	6	y	y	PROPN
ejpam-4936	157	7	≥	≥	PROPN
ejpam-4936	157	8	0	0	NUM
ejpam-4936	157	9	then	then	ADV
ejpam-4936	157	10	it	it	PRON
ejpam-4936	157	11	is	be	AUX
ejpam-4936	157	12	easy	easy	ADJ
ejpam-4936	157	13	to	to	PART
ejpam-4936	157	14	see	see	VERB
ejpam-4936	157	15	that	that	DET
ejpam-4936	157	16	9x	9x	NOUN
ejpam-4936	157	17	+	+	CCONJ
ejpam-4936	157	18	10y	10y	PROPN
ejpam-4936	157	19	+	+	CCONJ
ejpam-4936	157	20	7z	7z	PROPN
ejpam-4936	157	21	=	=	SYM
ejpam-4936	157	22	w2	w2	NOUN
ejpam-4936	157	23	has	have	VERB
ejpam-4936	157	24	no	no	DET
ejpam-4936	157	25	solution	solution	NOUN
ejpam-4936	157	26	by	by	ADP
ejpam-4936	157	27	examing	exame	VERB
ejpam-4936	157	28	with	with	ADP
ejpam-4936	157	29	modulo	modulo	NOUN
ejpam-4936	157	30	3	3	NUM
ejpam-4936	157	31	.	.	PUNCT
ejpam-4936	158	1	now	now	ADV
ejpam-4936	158	2	,	,	PUNCT
ejpam-4936	158	3	we	we	PRON
ejpam-4936	158	4	remain	remain	VERB
ejpam-4936	158	5	to	to	PART
ejpam-4936	158	6	investigate	investigate	VERB
ejpam-4936	158	7	the	the	DET
ejpam-4936	158	8	last	last	ADJ
ejpam-4936	158	9	subcase	subcase	NOUN
ejpam-4936	158	10	x	x	PUNCT
ejpam-4936	159	1	=	=	SYM
ejpam-4936	159	2	0	0	NUM
ejpam-4936	159	3	and	and	CCONJ
ejpam-4936	159	4	y	y	PROPN
ejpam-4936	159	5	≥	≥	NUM
ejpam-4936	159	6	1	1	NUM
ejpam-4936	159	7	.	.	PUNCT
ejpam-4936	160	1	then	then	ADV
ejpam-4936	160	2	the	the	DET
ejpam-4936	160	3	equation	equation	NOUN
ejpam-4936	160	4	(	(	PUNCT
ejpam-4936	160	5	9	9	X
ejpam-4936	160	6	)	)	PUNCT
ejpam-4936	160	7	becomes	become	VERB
ejpam-4936	160	8	1	1	NUM
ejpam-4936	160	9	+	+	NUM
ejpam-4936	160	10	10y	10y	NUM
ejpam-4936	160	11	+	+	CCONJ
ejpam-4936	160	12	7z	7z	PROPN
ejpam-4936	160	13	=	=	SYM
ejpam-4936	160	14	w2	w2	NOUN
ejpam-4936	160	15	.	.	PUNCT
ejpam-4936	161	1	(	(	PUNCT
ejpam-4936	161	2	10	10	NUM
ejpam-4936	161	3	)	)	PUNCT
ejpam-4936	161	4	note	note	VERB
ejpam-4936	161	5	that	that	SCONJ
ejpam-4936	161	6	1	1	NUM
ejpam-4936	162	1	+	+	NUM
ejpam-4936	162	2	7z	7z	PROPN
ejpam-4936	162	3	≡	≡	PROPN
ejpam-4936	162	4	{	{	PUNCT
ejpam-4936	162	5	2	2	NUM
ejpam-4936	162	6	(	(	PUNCT
ejpam-4936	162	7	mod	mod	NOUN
ejpam-4936	162	8	16	16	NUM
ejpam-4936	162	9	)	)	PUNCT
ejpam-4936	162	10	if	if	SCONJ
ejpam-4936	162	11	z	z	NOUN
ejpam-4936	162	12	is	be	AUX
ejpam-4936	162	13	even	even	ADV
ejpam-4936	162	14	,	,	PUNCT
ejpam-4936	162	15	8	8	NUM
ejpam-4936	162	16	(	(	PUNCT
ejpam-4936	162	17	mod	mod	NOUN
ejpam-4936	162	18	16	16	NUM
ejpam-4936	162	19	)	)	PUNCT
ejpam-4936	162	20	if	if	SCONJ
ejpam-4936	162	21	z	z	NOUN
ejpam-4936	162	22	is	be	AUX
ejpam-4936	162	23	odd	odd	ADJ
ejpam-4936	162	24	,	,	PUNCT
ejpam-4936	162	25	and	and	CCONJ
ejpam-4936	162	26	10y	10y	PROPN
ejpam-4936	162	27	≡	≡	PROPN
ejpam-4936	162	28			NOUN
ejpam-4936	162	29	10	10	NUM
ejpam-4936	162	30	(	(	PUNCT
ejpam-4936	162	31	mod	mod	PROPN
ejpam-4936	162	32	16	16	NUM
ejpam-4936	162	33	)	)	PUNCT
ejpam-4936	162	34	if	if	SCONJ
ejpam-4936	162	35	y	y	PROPN
ejpam-4936	162	36	=	=	SYM
ejpam-4936	162	37	1	1	NUM
ejpam-4936	162	38	,	,	PUNCT
ejpam-4936	162	39	4	4	NUM
ejpam-4936	162	40	(	(	PUNCT
ejpam-4936	162	41	mod	mod	PROPN
ejpam-4936	162	42	16	16	NUM
ejpam-4936	162	43	)	)	PUNCT
ejpam-4936	162	44	if	if	SCONJ
ejpam-4936	162	45	y	y	PROPN
ejpam-4936	162	46	=	=	SYM
ejpam-4936	162	47	2	2	NUM
ejpam-4936	162	48	,	,	PUNCT
ejpam-4936	162	49	8	8	NUM
ejpam-4936	162	50	(	(	PUNCT
ejpam-4936	162	51	mod	mod	NOUN
ejpam-4936	162	52	16	16	NUM
ejpam-4936	162	53	)	)	PUNCT
ejpam-4936	162	54	if	if	SCONJ
ejpam-4936	162	55	y	y	PROPN
ejpam-4936	162	56	=	=	SYM
ejpam-4936	162	57	3	3	NUM
ejpam-4936	162	58	,	,	PUNCT
ejpam-4936	162	59	0	0	NUM
ejpam-4936	162	60	(	(	PUNCT
ejpam-4936	162	61	mod	mod	PROPN
ejpam-4936	162	62	16	16	NUM
ejpam-4936	162	63	)	)	PUNCT
ejpam-4936	162	64	if	if	SCONJ
ejpam-4936	162	65	y	y	PROPN
ejpam-4936	162	66	≥	≥	NUM
ejpam-4936	162	67	4	4	NUM
ejpam-4936	162	68	.	.	PUNCT
ejpam-4936	163	1	k.	k.	PROPN
ejpam-4936	163	2	laipaporn	laipaporn	PROPN
ejpam-4936	163	3	,	,	PUNCT
ejpam-4936	163	4	s.	s.	PROPN
ejpam-4936	163	5	kaewchay	kaewchay	PROPN
ejpam-4936	163	6	,	,	PUNCT
ejpam-4936	163	7	a.	a.	PROPN
ejpam-4936	163	8	karnbanjong	karnbanjong	PROPN
ejpam-4936	163	9	/	/	SYM
ejpam-4936	163	10	eur	eur	PROPN
ejpam-4936	163	11	.	.	PUNCT
ejpam-4936	164	1	j.	j.	PROPN
ejpam-4936	164	2	pure	pure	PROPN
ejpam-4936	164	3	appl	appl	PROPN
ejpam-4936	164	4	.	.	PROPN
ejpam-4936	164	5	math	math	PROPN
ejpam-4936	164	6	,	,	PUNCT
ejpam-4936	164	7	16	16	NUM
ejpam-4936	164	8	(	(	PUNCT
ejpam-4936	164	9	4	4	NUM
ejpam-4936	164	10	)	)	PUNCT
ejpam-4936	164	11	(	(	PUNCT
ejpam-4936	164	12	2023	2023	NUM
ejpam-4936	164	13	)	)	PUNCT
ejpam-4936	164	14	,	,	PUNCT
ejpam-4936	164	15	2066	2066	NUM
ejpam-4936	164	16	-	-	SYM
ejpam-4936	164	17	2081	2081	NUM
ejpam-4936	164	18	2071	2071	NUM
ejpam-4936	164	19	so	so	CCONJ
ejpam-4936	164	20	1	1	NUM
ejpam-4936	164	21	+	+	NOUN
ejpam-4936	164	22	10y	10y	NUM
ejpam-4936	164	23	+	+	CCONJ
ejpam-4936	164	24	7z	7z	PROPN
ejpam-4936	164	25	≡	≡	PROPN
ejpam-4936	164	26	{	{	PUNCT
ejpam-4936	164	27	2	2	NUM
ejpam-4936	164	28	,	,	PUNCT
ejpam-4936	164	29	6	6	NUM
ejpam-4936	164	30	,	,	PUNCT
ejpam-4936	164	31	10	10	NUM
ejpam-4936	164	32	,	,	PUNCT
ejpam-4936	164	33	12	12	NUM
ejpam-4936	164	34	(	(	PUNCT
ejpam-4936	164	35	mod	mod	PROPN
ejpam-4936	164	36	16	16	NUM
ejpam-4936	164	37	)	)	PUNCT
ejpam-4936	164	38	if	if	SCONJ
ejpam-4936	164	39	z	z	NOUN
ejpam-4936	164	40	is	be	AUX
ejpam-4936	164	41	even	even	ADV
ejpam-4936	164	42	and	and	CCONJ
ejpam-4936	164	43	y	y	PROPN
ejpam-4936	164	44	≥	≥	NUM
ejpam-4936	164	45	1	1	NUM
ejpam-4936	164	46	,	,	PUNCT
ejpam-4936	164	47	2	2	NUM
ejpam-4936	164	48	,	,	PUNCT
ejpam-4936	164	49	8	8	NUM
ejpam-4936	164	50	,	,	PUNCT
ejpam-4936	164	51	12	12	NUM
ejpam-4936	164	52	(	(	PUNCT
ejpam-4936	164	53	mod	mod	PROPN
ejpam-4936	164	54	16	16	NUM
ejpam-4936	164	55	)	)	PUNCT
ejpam-4936	164	56	if	if	SCONJ
ejpam-4936	164	57	z	z	NOUN
ejpam-4936	164	58	is	be	AUX
ejpam-4936	164	59	odd	odd	ADJ
ejpam-4936	164	60	,	,	PUNCT
ejpam-4936	164	61	y	y	PROPN
ejpam-4936	164	62	≥	≥	NUM
ejpam-4936	164	63	1	1	NUM
ejpam-4936	164	64	and	and	CCONJ
ejpam-4936	164	65	y	y	PROPN
ejpam-4936	164	66	̸=	̸=	PROPN
ejpam-4936	164	67	3	3	NUM
ejpam-4936	164	68	,	,	PUNCT
ejpam-4936	164	69	but	but	CCONJ
ejpam-4936	164	70	w2	w2	PROPN
ejpam-4936	164	71	≡	≡	PROPN
ejpam-4936	164	72	0	0	NUM
ejpam-4936	164	73	,	,	PUNCT
ejpam-4936	164	74	1	1	NUM
ejpam-4936	164	75	,	,	PUNCT
ejpam-4936	164	76	4	4	NUM
ejpam-4936	164	77	,	,	PUNCT
ejpam-4936	164	78	9	9	NUM
ejpam-4936	164	79	(	(	PUNCT
ejpam-4936	164	80	mod	mod	PROPN
ejpam-4936	164	81	16	16	NUM
ejpam-4936	164	82	)	)	PUNCT
ejpam-4936	164	83	,	,	PUNCT
ejpam-4936	164	84	this	this	PRON
ejpam-4936	164	85	leads	lead	VERB
ejpam-4936	164	86	us	we	PRON
ejpam-4936	164	87	to	to	PART
ejpam-4936	164	88	complete	complete	VERB
ejpam-4936	164	89	the	the	DET
ejpam-4936	164	90	proof	proof	NOUN
ejpam-4936	164	91	that	that	SCONJ
ejpam-4936	164	92	the	the	DET
ejpam-4936	164	93	equation	equation	NOUN
ejpam-4936	164	94	(	(	PUNCT
ejpam-4936	164	95	9	9	NUM
ejpam-4936	164	96	)	)	PUNCT
ejpam-4936	164	97	has	have	AUX
ejpam-4936	164	98	no	no	DET
ejpam-4936	164	99	solution	solution	NOUN
ejpam-4936	164	100	on	on	ADP
ejpam-4936	164	101	u−t	u−t	PROPN
ejpam-4936	164	102	if	if	SCONJ
ejpam-4936	164	103	z	z	PROPN
ejpam-4936	164	104	≥	≥	NOUN
ejpam-4936	164	105	2	2	NUM
ejpam-4936	164	106	.	.	PUNCT
ejpam-4936	165	1	from	from	ADP
ejpam-4936	165	2	here	here	ADV
ejpam-4936	165	3	we	we	PRON
ejpam-4936	165	4	obtain	obtain	VERB
ejpam-4936	165	5	the	the	DET
ejpam-4936	165	6	equation	equation	NOUN
ejpam-4936	165	7	9x+10y+7z	9x+10y+7z	NUM
ejpam-4936	165	8	=	=	SYM
ejpam-4936	165	9	w2	w2	NOUN
ejpam-4936	165	10	has	have	VERB
ejpam-4936	165	11	no	no	DET
ejpam-4936	165	12	solution	solution	NOUN
ejpam-4936	165	13	for	for	ADP
ejpam-4936	165	14	any	any	DET
ejpam-4936	165	15	(	(	PUNCT
ejpam-4936	165	16	x	x	NOUN
ejpam-4936	165	17	,	,	PUNCT
ejpam-4936	165	18	y	y	PROPN
ejpam-4936	165	19	,	,	PUNCT
ejpam-4936	165	20	z	z	PROPN
ejpam-4936	165	21	,	,	PUNCT
ejpam-4936	165	22	w	w	NOUN
ejpam-4936	165	23	)	)	PUNCT
ejpam-4936	165	24	∈	∈	PROPN
ejpam-4936	165	25	u	u	NOUN
ejpam-4936	165	26	−	−	PROPN
ejpam-4936	165	27	t	t	PROPN
ejpam-4936	165	28	.	.	PUNCT
ejpam-4936	166	1	theorem	theorem	ADJ
ejpam-4936	166	2	4	4	NUM
ejpam-4936	166	3	.	.	PUNCT
ejpam-4936	167	1	the	the	DET
ejpam-4936	167	2	equation	equation	NOUN
ejpam-4936	167	3	6x	6x	NOUN
ejpam-4936	167	4	+	+	X
ejpam-4936	167	5	10y	10y	NUM
ejpam-4936	167	6	+	+	CCONJ
ejpam-4936	167	7	7z	7z	PROPN
ejpam-4936	167	8	=	=	SYM
ejpam-4936	167	9	w2	w2	NOUN
ejpam-4936	167	10	(	(	PUNCT
ejpam-4936	167	11	11	11	NUM
ejpam-4936	167	12	)	)	PUNCT
ejpam-4936	167	13	has	have	VERB
ejpam-4936	167	14	no	no	DET
ejpam-4936	167	15	solution	solution	NOUN
ejpam-4936	167	16	for	for	ADP
ejpam-4936	167	17	any	any	DET
ejpam-4936	167	18	(	(	PUNCT
ejpam-4936	167	19	x	x	NOUN
ejpam-4936	167	20	,	,	PUNCT
ejpam-4936	167	21	y	y	PROPN
ejpam-4936	167	22	,	,	PUNCT
ejpam-4936	167	23	z	z	PROPN
ejpam-4936	167	24	,	,	PUNCT
ejpam-4936	167	25	w	w	NOUN
ejpam-4936	167	26	)	)	PUNCT
ejpam-4936	167	27	∈	∈	PROPN
ejpam-4936	167	28	u	u	NOUN
ejpam-4936	167	29	−	−	PROPN
ejpam-4936	167	30	t	t	NOUN
ejpam-4936	167	31	where	where	SCONJ
ejpam-4936	167	32	t	t	NOUN
ejpam-4936	167	33	=	=	SYM
ejpam-4936	167	34	{	{	PUNCT
ejpam-4936	167	35	(	(	PUNCT
ejpam-4936	167	36	0	0	NUM
ejpam-4936	167	37	,	,	PUNCT
ejpam-4936	167	38	3	3	NUM
ejpam-4936	167	39	,	,	PUNCT
ejpam-4936	167	40	z	z	PROPN
ejpam-4936	167	41	,	,	PUNCT
ejpam-4936	167	42	w)|z	w)|z	X
ejpam-4936	167	43	≥	≥	NOUN
ejpam-4936	167	44	2	2	NUM
ejpam-4936	167	45	and	and	CCONJ
ejpam-4936	167	46	z	z	NOUN
ejpam-4936	167	47	is	be	AUX
ejpam-4936	167	48	odd	odd	ADJ
ejpam-4936	167	49	}	}	PUNCT
ejpam-4936	167	50	.	.	PUNCT
ejpam-4936	168	1	proof	proof	NOUN
ejpam-4936	168	2	.	.	PUNCT
ejpam-4936	169	1	note	note	VERB
ejpam-4936	169	2	that	that	SCONJ
ejpam-4936	169	3	,	,	PUNCT
ejpam-4936	169	4	if	if	SCONJ
ejpam-4936	169	5	x	x	X
ejpam-4936	169	6	≥	≥	NUM
ejpam-4936	169	7	1	1	NUM
ejpam-4936	169	8	and	and	CCONJ
ejpam-4936	169	9	y	y	PROPN
ejpam-4936	169	10	≥	≥	NOUN
ejpam-4936	169	11	0	0	NUM
ejpam-4936	170	1	then	then	ADV
ejpam-4936	170	2	6x	6x	NUM
ejpam-4936	170	3	+	+	SYM
ejpam-4936	170	4	10y	10y	NUM
ejpam-4936	170	5	+	+	CCONJ
ejpam-4936	170	6	7z	7z	PROPN
ejpam-4936	170	7	≡	≡	PROPN
ejpam-4936	170	8	2	2	NUM
ejpam-4936	170	9	(	(	PUNCT
ejpam-4936	170	10	mod	mod	NOUN
ejpam-4936	170	11	3	3	NUM
ejpam-4936	170	12	)	)	PUNCT
ejpam-4936	170	13	for	for	ADP
ejpam-4936	170	14	all	all	DET
ejpam-4936	170	15	z	z	NOUN
ejpam-4936	170	16	≥	≥	NOUN
ejpam-4936	170	17	0	0	NUM
ejpam-4936	170	18	.	.	PUNCT
ejpam-4936	171	1	from	from	ADP
ejpam-4936	171	2	this	this	DET
ejpam-4936	171	3	fact	fact	NOUN
ejpam-4936	171	4	and	and	CCONJ
ejpam-4936	171	5	the	the	DET
ejpam-4936	171	6	fact	fact	NOUN
ejpam-4936	171	7	that	that	SCONJ
ejpam-4936	171	8	w2	w2	NOUN
ejpam-4936	171	9	̸≡	̸≡	NOUN
ejpam-4936	171	10	2	2	NUM
ejpam-4936	171	11	(	(	PUNCT
ejpam-4936	171	12	mod	mod	NOUN
ejpam-4936	171	13	3	3	NUM
ejpam-4936	171	14	)	)	PUNCT
ejpam-4936	171	15	,	,	PUNCT
ejpam-4936	171	16	we	we	PRON
ejpam-4936	171	17	conclude	conclude	VERB
ejpam-4936	171	18	that	that	SCONJ
ejpam-4936	171	19	the	the	DET
ejpam-4936	171	20	equation	equation	NOUN
ejpam-4936	171	21	(	(	PUNCT
ejpam-4936	171	22	11	11	NUM
ejpam-4936	171	23	)	)	PUNCT
ejpam-4936	171	24	has	have	VERB
ejpam-4936	171	25	no	no	DET
ejpam-4936	171	26	solution	solution	NOUN
ejpam-4936	171	27	for	for	ADP
ejpam-4936	171	28	x	x	SYM
ejpam-4936	171	29	̸=	̸=	PROPN
ejpam-4936	171	30	0	0	NUM
ejpam-4936	171	31	.	.	PUNCT
ejpam-4936	172	1	now	now	ADV
ejpam-4936	172	2	,	,	PUNCT
ejpam-4936	172	3	it	it	PRON
ejpam-4936	172	4	leads	lead	VERB
ejpam-4936	172	5	us	we	PRON
ejpam-4936	172	6	to	to	PART
ejpam-4936	172	7	consider	consider	VERB
ejpam-4936	172	8	the	the	DET
ejpam-4936	172	9	equation	equation	NOUN
ejpam-4936	172	10	(	(	PUNCT
ejpam-4936	172	11	11	11	NUM
ejpam-4936	172	12	)	)	PUNCT
ejpam-4936	172	13	in	in	ADP
ejpam-4936	172	14	the	the	DET
ejpam-4936	172	15	form	form	NOUN
ejpam-4936	172	16	1	1	NUM
ejpam-4936	172	17	+	+	NOUN
ejpam-4936	172	18	10y	10y	NUM
ejpam-4936	172	19	+	+	CCONJ
ejpam-4936	172	20	7z	7z	PROPN
ejpam-4936	172	21	=	=	SYM
ejpam-4936	172	22	w2	w2	NOUN
ejpam-4936	172	23	for	for	ADP
ejpam-4936	172	24	any	any	DET
ejpam-4936	172	25	y	y	PROPN
ejpam-4936	172	26	≥	≥	NUM
ejpam-4936	172	27	1	1	NUM
ejpam-4936	172	28	,	,	PUNCT
ejpam-4936	172	29	z	z	NOUN
ejpam-4936	172	30	and	and	CCONJ
ejpam-4936	172	31	w	w	PROPN
ejpam-4936	172	32	are	be	AUX
ejpam-4936	172	33	non	non	ADJ
ejpam-4936	172	34	-	-	ADJ
ejpam-4936	172	35	negative	negative	ADJ
ejpam-4936	172	36	.	.	PUNCT
ejpam-4936	173	1	for	for	ADP
ejpam-4936	173	2	the	the	DET
ejpam-4936	173	3	case	case	NOUN
ejpam-4936	173	4	z	z	NOUN
ejpam-4936	173	5	=	=	SYM
ejpam-4936	173	6	0	0	NUM
ejpam-4936	173	7	or	or	CCONJ
ejpam-4936	173	8	z	z	NOUN
ejpam-4936	173	9	=	=	SYM
ejpam-4936	173	10	1	1	NUM
ejpam-4936	173	11	,	,	PUNCT
ejpam-4936	173	12	we	we	PRON
ejpam-4936	173	13	get	get	VERB
ejpam-4936	173	14	1	1	NUM
ejpam-4936	173	15	+	+	NOUN
ejpam-4936	173	16	10y	10y	NUM
ejpam-4936	173	17	+	+	CCONJ
ejpam-4936	173	18	7z	7z	NUM
ejpam-4936	173	19	≡	≡	PROPN
ejpam-4936	173	20	2	2	NUM
ejpam-4936	173	21	or	or	CCONJ
ejpam-4936	173	22	3	3	NUM
ejpam-4936	173	23	(	(	PUNCT
ejpam-4936	173	24	mod	mod	NOUN
ejpam-4936	173	25	5	5	NUM
ejpam-4936	173	26	)	)	PUNCT
ejpam-4936	173	27	.	.	PUNCT
ejpam-4936	174	1	but	but	CCONJ
ejpam-4936	174	2	w2	w2	NOUN
ejpam-4936	174	3	̸≡	̸≡	PROPN
ejpam-4936	174	4	2	2	NUM
ejpam-4936	174	5	,	,	PUNCT
ejpam-4936	174	6	3	3	NUM
ejpam-4936	174	7	(	(	PUNCT
ejpam-4936	174	8	mod	mod	NOUN
ejpam-4936	174	9	5	5	NUM
ejpam-4936	174	10	)	)	PUNCT
ejpam-4936	174	11	.	.	PUNCT
ejpam-4936	175	1	so	so	ADV
ejpam-4936	175	2	,	,	PUNCT
ejpam-4936	175	3	we	we	PRON
ejpam-4936	175	4	remain	remain	VERB
ejpam-4936	175	5	to	to	PART
ejpam-4936	175	6	examine	examine	VERB
ejpam-4936	175	7	the	the	DET
ejpam-4936	175	8	equation	equation	NOUN
ejpam-4936	175	9	1	1	NUM
ejpam-4936	175	10	+	+	NUM
ejpam-4936	175	11	10y	10y	NUM
ejpam-4936	175	12	+	+	CCONJ
ejpam-4936	175	13	7z	7z	PROPN
ejpam-4936	175	14	=	=	SYM
ejpam-4936	175	15	w2	w2	NOUN
ejpam-4936	175	16	in	in	ADP
ejpam-4936	175	17	the	the	DET
ejpam-4936	175	18	case	case	NOUN
ejpam-4936	175	19	of	of	ADP
ejpam-4936	175	20	y	y	PROPN
ejpam-4936	175	21	≥	≥	NUM
ejpam-4936	175	22	1	1	NUM
ejpam-4936	175	23	,	,	PUNCT
ejpam-4936	176	1	z	z	NOUN
ejpam-4936	176	2	≥	≥	NUM
ejpam-4936	176	3	2	2	NUM
ejpam-4936	176	4	and	and	CCONJ
ejpam-4936	176	5	w	w	NOUN
ejpam-4936	176	6	≥	≥	NOUN
ejpam-4936	176	7	0.that	0.that	NUM
ejpam-4936	176	8	is	be	AUX
ejpam-4936	176	9	the	the	DET
ejpam-4936	176	10	same	same	ADJ
ejpam-4936	176	11	equation	equation	NOUN
ejpam-4936	176	12	(	(	PUNCT
ejpam-4936	176	13	10	10	NUM
ejpam-4936	176	14	)	)	PUNCT
ejpam-4936	176	15	in	in	ADP
ejpam-4936	176	16	3	3	NUM
ejpam-4936	176	17	and	and	CCONJ
ejpam-4936	176	18	we	we	PRON
ejpam-4936	176	19	immediately	immediately	ADV
ejpam-4936	176	20	conclude	conclude	VERB
ejpam-4936	176	21	that	that	SCONJ
ejpam-4936	176	22	the	the	DET
ejpam-4936	176	23	equation	equation	NOUN
ejpam-4936	176	24	(	(	PUNCT
ejpam-4936	176	25	11	11	NUM
ejpam-4936	176	26	)	)	PUNCT
ejpam-4936	176	27	has	have	VERB
ejpam-4936	176	28	no	no	DET
ejpam-4936	176	29	solution	solution	NOUN
ejpam-4936	176	30	for	for	ADP
ejpam-4936	176	31	all	all	DET
ejpam-4936	176	32	(	(	PUNCT
ejpam-4936	176	33	x	x	NOUN
ejpam-4936	176	34	,	,	PUNCT
ejpam-4936	176	35	y	y	PROPN
ejpam-4936	176	36	,	,	PUNCT
ejpam-4936	176	37	z	z	PROPN
ejpam-4936	176	38	,	,	PUNCT
ejpam-4936	176	39	w	w	NOUN
ejpam-4936	176	40	)	)	PUNCT
ejpam-4936	176	41	∈	∈	PROPN
ejpam-4936	176	42	u	u	NOUN
ejpam-4936	176	43	−	−	PROPN
ejpam-4936	176	44	t	t	PROPN
ejpam-4936	176	45	.	.	PUNCT
ejpam-4936	177	1	2.2	2.2	NUM
ejpam-4936	177	2	.	.	PUNCT
ejpam-4936	178	1	the	the	DET
ejpam-4936	178	2	exponential	exponential	ADJ
ejpam-4936	178	3	diophantine	diophantine	NOUN
ejpam-4936	178	4	equation	equation	NOUN
ejpam-4936	178	5	ax	ax	NOUN
ejpam-4936	178	6	+	+	CCONJ
ejpam-4936	178	7	by	by	ADP
ejpam-4936	178	8	+	+	CCONJ
ejpam-4936	178	9	cz	cz	NOUN
ejpam-4936	178	10	=	=	NOUN
ejpam-4936	178	11	w2	w2	NOUN
ejpam-4936	178	12	in	in	ADP
ejpam-4936	178	13	this	this	DET
ejpam-4936	178	14	section	section	NOUN
ejpam-4936	178	15	,	,	PUNCT
ejpam-4936	178	16	we	we	PRON
ejpam-4936	178	17	present	present	VERB
ejpam-4936	178	18	our	our	PRON
ejpam-4936	178	19	results	result	NOUN
ejpam-4936	178	20	and	and	CCONJ
ejpam-4936	178	21	discussion	discussion	NOUN
ejpam-4936	178	22	for	for	ADP
ejpam-4936	178	23	ax	ax	NOUN
ejpam-4936	178	24	+	+	CCONJ
ejpam-4936	178	25	by	by	ADP
ejpam-4936	178	26	+	+	CCONJ
ejpam-4936	178	27	cz	cz	NOUN
ejpam-4936	178	28	=	=	SYM
ejpam-4936	178	29	w2	w2	NOUN
ejpam-4936	178	30	,	,	PUNCT
ejpam-4936	178	31	where	where	SCONJ
ejpam-4936	178	32	all	all	DET
ejpam-4936	178	33	bases	basis	NOUN
ejpam-4936	178	34	of	of	ADP
ejpam-4936	178	35	the	the	DET
ejpam-4936	178	36	exponential	exponential	NOUN
ejpam-4936	178	37	are	be	AUX
ejpam-4936	178	38	in	in	ADP
ejpam-4936	178	39	terms	term	NOUN
ejpam-4936	178	40	of	of	ADP
ejpam-4936	178	41	variables	variable	NOUN
ejpam-4936	178	42	except	except	SCONJ
ejpam-4936	178	43	7	7	NUM
ejpam-4936	178	44	and	and	CCONJ
ejpam-4936	178	45	3	3	NUM
ejpam-4936	178	46	.	.	X
ejpam-4936	178	47	theorem	theorem	NOUN
ejpam-4936	178	48	5	5	NUM
ejpam-4936	178	49	.	.	PUNCT
ejpam-4936	179	1	the	the	DET
ejpam-4936	179	2	diophantine	diophantine	NOUN
ejpam-4936	179	3	equation	equation	NOUN
ejpam-4936	179	4	ax	ax	NOUN
ejpam-4936	179	5	+	+	CCONJ
ejpam-4936	179	6	by	by	ADP
ejpam-4936	179	7	+	+	CCONJ
ejpam-4936	179	8	cz	cz	NOUN
ejpam-4936	179	9	=	=	NOUN
ejpam-4936	179	10	w2	w2	NOUN
ejpam-4936	179	11	has	have	VERB
ejpam-4936	179	12	no	no	DET
ejpam-4936	179	13	solution	solution	NOUN
ejpam-4936	179	14	when	when	SCONJ
ejpam-4936	179	15	a	a	DET
ejpam-4936	179	16	,	,	PUNCT
ejpam-4936	179	17	b	b	NOUN
ejpam-4936	179	18	and	and	CCONJ
ejpam-4936	179	19	c	c	AUX
ejpam-4936	179	20	satisfy	satisfy	VERB
ejpam-4936	179	21	one	one	NUM
ejpam-4936	179	22	of	of	ADP
ejpam-4936	179	23	the	the	DET
ejpam-4936	179	24	following	following	ADJ
ejpam-4936	179	25	conditions	condition	NOUN
ejpam-4936	179	26	:	:	PUNCT
ejpam-4936	179	27	(	(	PUNCT
ejpam-4936	179	28	i	i	NOUN
ejpam-4936	179	29	)	)	PUNCT
ejpam-4936	179	30	a	a	DET
ejpam-4936	179	31	,	,	PUNCT
ejpam-4936	179	32	b	b	NOUN
ejpam-4936	179	33	,	,	PUNCT
ejpam-4936	179	34	c	c	PROPN
ejpam-4936	179	35	≡	≡	PROPN
ejpam-4936	179	36	1	1	NUM
ejpam-4936	179	37	(	(	PUNCT
ejpam-4936	179	38	mod	mod	NOUN
ejpam-4936	179	39	4	4	NUM
ejpam-4936	179	40	)	)	PUNCT
ejpam-4936	179	41	.	.	PUNCT
ejpam-4936	180	1	(	(	PUNCT
ejpam-4936	180	2	ii	ii	NOUN
ejpam-4936	180	3	)	)	PUNCT
ejpam-4936	180	4	a	a	DET
ejpam-4936	180	5	,	,	PUNCT
ejpam-4936	180	6	b	b	NOUN
ejpam-4936	180	7	,	,	PUNCT
ejpam-4936	180	8	c	c	PROPN
ejpam-4936	180	9	≡	≡	PROPN
ejpam-4936	180	10	1	1	NUM
ejpam-4936	180	11	(	(	PUNCT
ejpam-4936	180	12	mod	mod	NOUN
ejpam-4936	180	13	5	5	NUM
ejpam-4936	180	14	)	)	PUNCT
ejpam-4936	180	15	.	.	PUNCT
ejpam-4936	181	1	(	(	PUNCT
ejpam-4936	181	2	iii	iii	X
ejpam-4936	181	3	)	)	PUNCT
ejpam-4936	181	4	a	a	DET
ejpam-4936	181	5	≡	≡	PROPN
ejpam-4936	181	6	0	0	PUNCT
ejpam-4936	182	1	(	(	PUNCT
ejpam-4936	182	2	mod	mod	NOUN
ejpam-4936	182	3	4	4	NUM
ejpam-4936	182	4	)	)	PUNCT
ejpam-4936	182	5	and	and	CCONJ
ejpam-4936	182	6	b	b	NOUN
ejpam-4936	182	7	,	,	PUNCT
ejpam-4936	182	8	c	c	PROPN
ejpam-4936	182	9	≡	≡	PROPN
ejpam-4936	182	10	1	1	NUM
ejpam-4936	182	11	(	(	PUNCT
ejpam-4936	182	12	mod	mod	NOUN
ejpam-4936	182	13	4	4	NUM
ejpam-4936	182	14	)	)	PUNCT
ejpam-4936	182	15	.	.	PUNCT
ejpam-4936	183	1	(	(	PUNCT
ejpam-4936	183	2	iv	iv	X
ejpam-4936	183	3	)	)	PUNCT
ejpam-4936	183	4	a	a	DET
ejpam-4936	183	5	≡	≡	PROPN
ejpam-4936	183	6	0	0	PUNCT
ejpam-4936	183	7	(	(	PUNCT
ejpam-4936	183	8	mod	mod	NOUN
ejpam-4936	183	9	5	5	NUM
ejpam-4936	183	10	)	)	PUNCT
ejpam-4936	183	11	and	and	CCONJ
ejpam-4936	183	12	b	b	NOUN
ejpam-4936	183	13	,	,	PUNCT
ejpam-4936	183	14	c	c	PROPN
ejpam-4936	183	15	≡	≡	PROPN
ejpam-4936	183	16	1	1	NUM
ejpam-4936	183	17	(	(	PUNCT
ejpam-4936	183	18	mod	mod	NOUN
ejpam-4936	183	19	5	5	NUM
ejpam-4936	183	20	)	)	PUNCT
ejpam-4936	183	21	.	.	PUNCT
ejpam-4936	184	1	(	(	PUNCT
ejpam-4936	184	2	v	v	NOUN
ejpam-4936	184	3	)	)	PUNCT
ejpam-4936	184	4	a	a	DET
ejpam-4936	184	5	≡	≡	PROPN
ejpam-4936	184	6	3	3	NUM
ejpam-4936	184	7	(	(	PUNCT
ejpam-4936	184	8	mod	mod	PROPN
ejpam-4936	184	9	8)	8)	NUM
ejpam-4936	184	10	and	and	CCONJ
ejpam-4936	184	11	b	b	NOUN
ejpam-4936	184	12	,	,	PUNCT
ejpam-4936	184	13	c	c	PROPN
ejpam-4936	184	14	≡	≡	PROPN
ejpam-4936	184	15	1	1	NUM
ejpam-4936	184	16	(	(	PUNCT
ejpam-4936	184	17	mod	mod	PROPN
ejpam-4936	184	18	8)	8)	NUM
ejpam-4936	184	19	.	.	PUNCT
ejpam-4936	185	1	(	(	PUNCT
ejpam-4936	185	2	vi	vi	NOUN
ejpam-4936	185	3	)	)	PUNCT
ejpam-4936	185	4	a	a	DET
ejpam-4936	185	5	≡	≡	PROPN
ejpam-4936	185	6	1	1	NUM
ejpam-4936	185	7	(	(	PUNCT
ejpam-4936	185	8	mod	mod	PROPN
ejpam-4936	185	9	8)	8)	NUM
ejpam-4936	185	10	and	and	CCONJ
ejpam-4936	185	11	b	b	NOUN
ejpam-4936	185	12	,	,	PUNCT
ejpam-4936	185	13	c	c	PROPN
ejpam-4936	185	14	≡	≡	PROPN
ejpam-4936	185	15	3	3	NUM
ejpam-4936	185	16	(	(	PUNCT
ejpam-4936	185	17	mod	mod	NOUN
ejpam-4936	185	18	8)	8)	NUM
ejpam-4936	185	19	.	.	PUNCT
ejpam-4936	186	1	proof	proof	NOUN
ejpam-4936	186	2	.	.	PUNCT
ejpam-4936	187	1	(	(	PUNCT
ejpam-4936	187	2	i	i	NOUN
ejpam-4936	187	3	)	)	PUNCT
ejpam-4936	187	4	suppose	suppose	VERB
ejpam-4936	187	5	that	that	SCONJ
ejpam-4936	187	6	a	a	DET
ejpam-4936	187	7	,	,	PUNCT
ejpam-4936	187	8	b	b	NOUN
ejpam-4936	187	9	,	,	PUNCT
ejpam-4936	187	10	c	c	PROPN
ejpam-4936	187	11	≡	≡	PROPN
ejpam-4936	187	12	1	1	NUM
ejpam-4936	187	13	(	(	PUNCT
ejpam-4936	187	14	mod	mod	NOUN
ejpam-4936	187	15	4	4	NUM
ejpam-4936	187	16	)	)	PUNCT
ejpam-4936	187	17	.	.	PUNCT
ejpam-4936	188	1	for	for	ADP
ejpam-4936	188	2	any	any	DET
ejpam-4936	188	3	nonnegative	nonnegative	ADJ
ejpam-4936	188	4	integers	integer	NOUN
ejpam-4936	188	5	x	x	NOUN
ejpam-4936	188	6	,	,	PUNCT
ejpam-4936	188	7	y	y	PROPN
ejpam-4936	188	8	,	,	PUNCT
ejpam-4936	188	9	z	z	PROPN
ejpam-4936	188	10	and	and	CCONJ
ejpam-4936	188	11	w	w	PROPN
ejpam-4936	188	12	,	,	PUNCT
ejpam-4936	188	13	it	it	PRON
ejpam-4936	188	14	is	be	AUX
ejpam-4936	188	15	obvious	obvious	ADJ
ejpam-4936	188	16	that	that	SCONJ
ejpam-4936	188	17	ax	ax	NOUN
ejpam-4936	188	18	+	+	CCONJ
ejpam-4936	188	19	by	by	ADP
ejpam-4936	188	20	+	+	CCONJ
ejpam-4936	188	21	cz	cz	PROPN
ejpam-4936	188	22	≡	≡	PROPN
ejpam-4936	188	23	3	3	NUM
ejpam-4936	188	24	(	(	PUNCT
ejpam-4936	188	25	mod	mod	NOUN
ejpam-4936	188	26	4	4	NUM
ejpam-4936	188	27	)	)	PUNCT
ejpam-4936	188	28	,	,	PUNCT
ejpam-4936	188	29	but	but	CCONJ
ejpam-4936	188	30	w2	w2	PROPN
ejpam-4936	188	31	≡	≡	PROPN
ejpam-4936	188	32	0	0	NUM
ejpam-4936	188	33	,	,	PUNCT
ejpam-4936	188	34	1	1	NUM
ejpam-4936	188	35	(	(	PUNCT
ejpam-4936	188	36	mod	mod	NOUN
ejpam-4936	188	37	4	4	NUM
ejpam-4936	188	38	)	)	PUNCT
ejpam-4936	188	39	.	.	PUNCT
ejpam-4936	189	1	this	this	PRON
ejpam-4936	189	2	means	mean	VERB
ejpam-4936	189	3	that	that	SCONJ
ejpam-4936	189	4	ax	ax	NOUN
ejpam-4936	189	5	+	+	X
ejpam-4936	189	6	by	by	ADP
ejpam-4936	189	7	+	+	CCONJ
ejpam-4936	189	8	cz	cz	PROPN
ejpam-4936	189	9	̸≡	̸≡	PROPN
ejpam-4936	189	10	w2	w2	NOUN
ejpam-4936	189	11	(	(	PUNCT
ejpam-4936	189	12	mod	mod	PROPN
ejpam-4936	189	13	4	4	NUM
ejpam-4936	189	14	)	)	PUNCT
ejpam-4936	189	15	.	.	PUNCT
ejpam-4936	190	1	hence	hence	ADV
ejpam-4936	190	2	,	,	PUNCT
ejpam-4936	190	3	the	the	DET
ejpam-4936	190	4	diophantine	diophantine	NOUN
ejpam-4936	190	5	equation	equation	NOUN
ejpam-4936	190	6	has	have	VERB
ejpam-4936	190	7	no	no	DET
ejpam-4936	190	8	solution	solution	NOUN
ejpam-4936	190	9	.	.	PUNCT
ejpam-4936	191	1	k.	k.	PROPN
ejpam-4936	191	2	laipaporn	laipaporn	PROPN
ejpam-4936	191	3	,	,	PUNCT
ejpam-4936	191	4	s.	s.	PROPN
ejpam-4936	191	5	kaewchay	kaewchay	PROPN
ejpam-4936	191	6	,	,	PUNCT
ejpam-4936	191	7	a.	a.	PROPN
ejpam-4936	191	8	karnbanjong	karnbanjong	PROPN
ejpam-4936	191	9	/	/	SYM
ejpam-4936	191	10	eur	eur	PROPN
ejpam-4936	191	11	.	.	PUNCT
ejpam-4936	192	1	j.	j.	PROPN
ejpam-4936	192	2	pure	pure	PROPN
ejpam-4936	192	3	appl	appl	PROPN
ejpam-4936	192	4	.	.	PROPN
ejpam-4936	192	5	math	math	PROPN
ejpam-4936	192	6	,	,	PUNCT
ejpam-4936	192	7	16	16	NUM
ejpam-4936	192	8	(	(	PUNCT
ejpam-4936	192	9	4	4	NUM
ejpam-4936	192	10	)	)	PUNCT
ejpam-4936	192	11	(	(	PUNCT
ejpam-4936	192	12	2023	2023	NUM
ejpam-4936	192	13	)	)	PUNCT
ejpam-4936	192	14	,	,	PUNCT
ejpam-4936	192	15	2066	2066	NUM
ejpam-4936	192	16	-	-	SYM
ejpam-4936	192	17	2081	2081	NUM
ejpam-4936	192	18	2072	2072	NUM
ejpam-4936	192	19	(	(	PUNCT
ejpam-4936	192	20	ii	ii	NOUN
ejpam-4936	192	21	)	)	PUNCT
ejpam-4936	192	22	suppose	suppose	VERB
ejpam-4936	192	23	that	that	SCONJ
ejpam-4936	192	24	a	a	DET
ejpam-4936	192	25	,	,	PUNCT
ejpam-4936	192	26	b	b	NOUN
ejpam-4936	192	27	,	,	PUNCT
ejpam-4936	192	28	c	c	PROPN
ejpam-4936	192	29	≡	≡	PROPN
ejpam-4936	192	30	1	1	NUM
ejpam-4936	192	31	(	(	PUNCT
ejpam-4936	192	32	mod	mod	NOUN
ejpam-4936	192	33	5	5	NUM
ejpam-4936	192	34	)	)	PUNCT
ejpam-4936	192	35	.	.	PUNCT
ejpam-4936	193	1	given	give	VERB
ejpam-4936	193	2	that	that	DET
ejpam-4936	193	3	w2	w2	NOUN
ejpam-4936	193	4	≡	≡	PROPN
ejpam-4936	193	5	0	0	NUM
ejpam-4936	193	6	,	,	PUNCT
ejpam-4936	193	7	1	1	NUM
ejpam-4936	193	8	,	,	PUNCT
ejpam-4936	193	9	4	4	NUM
ejpam-4936	193	10	(	(	PUNCT
ejpam-4936	193	11	mod	mod	NOUN
ejpam-4936	193	12	5	5	NUM
ejpam-4936	193	13	)	)	PUNCT
ejpam-4936	193	14	and	and	CCONJ
ejpam-4936	193	15	by	by	ADP
ejpam-4936	193	16	the	the	DET
ejpam-4936	193	17	same	same	ADJ
ejpam-4936	193	18	trace	trace	NOUN
ejpam-4936	193	19	of	of	ADP
ejpam-4936	193	20	(	(	PUNCT
ejpam-4936	193	21	i	i	PROPN
ejpam-4936	193	22	)	)	PUNCT
ejpam-4936	193	23	,	,	PUNCT
ejpam-4936	193	24	we	we	PRON
ejpam-4936	193	25	can	can	AUX
ejpam-4936	193	26	conclude	conclude	VERB
ejpam-4936	193	27	that	that	SCONJ
ejpam-4936	193	28	the	the	DET
ejpam-4936	193	29	diophantine	diophantine	NOUN
ejpam-4936	193	30	equation	equation	NOUN
ejpam-4936	193	31	has	have	VERB
ejpam-4936	193	32	no	no	DET
ejpam-4936	193	33	solution	solution	NOUN
ejpam-4936	193	34	.	.	PUNCT
ejpam-4936	194	1	(	(	PUNCT
ejpam-4936	194	2	iii	iii	X
ejpam-4936	194	3	)	)	PUNCT
ejpam-4936	194	4	suppose	suppose	VERB
ejpam-4936	194	5	that	that	SCONJ
ejpam-4936	194	6	a	a	DET
ejpam-4936	194	7	≡	≡	PROPN
ejpam-4936	194	8	0	0	PUNCT
ejpam-4936	194	9	(	(	PUNCT
ejpam-4936	194	10	mod	mod	NOUN
ejpam-4936	194	11	4	4	NUM
ejpam-4936	194	12	)	)	PUNCT
ejpam-4936	194	13	and	and	CCONJ
ejpam-4936	194	14	b	b	NOUN
ejpam-4936	194	15	,	,	PUNCT
ejpam-4936	194	16	c	c	PROPN
ejpam-4936	194	17	≡	≡	PROPN
ejpam-4936	194	18	1	1	NUM
ejpam-4936	194	19	(	(	PUNCT
ejpam-4936	194	20	mod	mod	NOUN
ejpam-4936	194	21	4	4	NUM
ejpam-4936	194	22	)	)	PUNCT
ejpam-4936	194	23	.	.	PUNCT
ejpam-4936	195	1	for	for	ADP
ejpam-4936	195	2	any	any	DET
ejpam-4936	195	3	nonnegative	nonnegative	ADJ
ejpam-4936	195	4	integers	integer	NOUN
ejpam-4936	195	5	x	x	X
ejpam-4936	195	6	,	,	PUNCT
ejpam-4936	195	7	y	y	PROPN
ejpam-4936	195	8	and	and	CCONJ
ejpam-4936	195	9	z	z	PROPN
ejpam-4936	195	10	,	,	PUNCT
ejpam-4936	195	11	we	we	PRON
ejpam-4936	195	12	know	know	VERB
ejpam-4936	195	13	that	that	SCONJ
ejpam-4936	195	14	ax	ax	NOUN
ejpam-4936	195	15	≡	≡	PROPN
ejpam-4936	195	16	{	{	PUNCT
ejpam-4936	195	17	1	1	NUM
ejpam-4936	195	18	(	(	PUNCT
ejpam-4936	195	19	mod	mod	NOUN
ejpam-4936	195	20	4	4	X
ejpam-4936	195	21	)	)	PUNCT
ejpam-4936	195	22	if	if	SCONJ
ejpam-4936	195	23	x	x	PROPN
ejpam-4936	196	1	=	=	SYM
ejpam-4936	196	2	0	0	SYM
ejpam-4936	196	3	0	0	NUM
ejpam-4936	196	4	(	(	PUNCT
ejpam-4936	196	5	mod	mod	PROPN
ejpam-4936	196	6	4	4	X
ejpam-4936	196	7	)	)	PUNCT
ejpam-4936	196	8	if	if	SCONJ
ejpam-4936	196	9	x	x	X
ejpam-4936	196	10	≥	≥	NOUN
ejpam-4936	196	11	1	1	NUM
ejpam-4936	196	12	,	,	PUNCT
ejpam-4936	196	13	and	and	CCONJ
ejpam-4936	196	14	by	by	ADV
ejpam-4936	196	15	,	,	PUNCT
ejpam-4936	196	16	cz	cz	PROPN
ejpam-4936	196	17	≡	≡	PROPN
ejpam-4936	196	18	1	1	NUM
ejpam-4936	196	19	(	(	PUNCT
ejpam-4936	196	20	mod	mod	NOUN
ejpam-4936	196	21	4	4	NUM
ejpam-4936	196	22	)	)	PUNCT
ejpam-4936	196	23	.	.	PUNCT
ejpam-4936	197	1	thus	thus	ADV
ejpam-4936	197	2	,	,	PUNCT
ejpam-4936	197	3	ax	ax	NOUN
ejpam-4936	197	4	+	+	CCONJ
ejpam-4936	197	5	by	by	ADP
ejpam-4936	197	6	+	+	CCONJ
ejpam-4936	197	7	cz	cz	PROPN
ejpam-4936	197	8	≡	≡	PROPN
ejpam-4936	197	9	{	{	PUNCT
ejpam-4936	197	10	3	3	NUM
ejpam-4936	197	11	(	(	PUNCT
ejpam-4936	197	12	mod	mod	NOUN
ejpam-4936	197	13	4	4	X
ejpam-4936	197	14	)	)	PUNCT
ejpam-4936	197	15	if	if	SCONJ
ejpam-4936	197	16	x	x	PROPN
ejpam-4936	197	17	=	=	SYM
ejpam-4936	197	18	0	0	SYM
ejpam-4936	197	19	2	2	NUM
ejpam-4936	197	20	(	(	PUNCT
ejpam-4936	197	21	mod	mod	NOUN
ejpam-4936	197	22	4	4	X
ejpam-4936	197	23	)	)	PUNCT
ejpam-4936	197	24	if	if	SCONJ
ejpam-4936	197	25	x	x	X
ejpam-4936	197	26	≥	≥	NOUN
ejpam-4936	197	27	1	1	NUM
ejpam-4936	197	28	,	,	PUNCT
ejpam-4936	197	29	for	for	ADP
ejpam-4936	197	30	any	any	DET
ejpam-4936	197	31	nonnegative	nonnegative	ADJ
ejpam-4936	197	32	integers	integer	NOUN
ejpam-4936	197	33	x	x	X
ejpam-4936	197	34	,	,	PUNCT
ejpam-4936	197	35	y	y	PROPN
ejpam-4936	197	36	and	and	CCONJ
ejpam-4936	197	37	z.	z.	PROPN
ejpam-4936	197	38	since	since	SCONJ
ejpam-4936	197	39	w2	w2	PROPN
ejpam-4936	197	40	≡	≡	PROPN
ejpam-4936	197	41	0	0	NUM
ejpam-4936	197	42	,	,	PUNCT
ejpam-4936	197	43	1	1	NUM
ejpam-4936	197	44	(	(	PUNCT
ejpam-4936	197	45	mod	mod	NOUN
ejpam-4936	197	46	4	4	NUM
ejpam-4936	197	47	)	)	PUNCT
ejpam-4936	197	48	,	,	PUNCT
ejpam-4936	197	49	the	the	DET
ejpam-4936	197	50	diophantine	diophantine	NOUN
ejpam-4936	197	51	equation	equation	NOUN
ejpam-4936	197	52	has	have	VERB
ejpam-4936	197	53	no	no	DET
ejpam-4936	197	54	solution	solution	NOUN
ejpam-4936	197	55	.	.	PUNCT
ejpam-4936	198	1	(	(	PUNCT
ejpam-4936	198	2	iv	iv	X
ejpam-4936	198	3	)	)	PUNCT
ejpam-4936	198	4	by	by	ADP
ejpam-4936	198	5	the	the	DET
ejpam-4936	198	6	same	same	ADJ
ejpam-4936	198	7	trace	trace	NOUN
ejpam-4936	198	8	of	of	ADP
ejpam-4936	198	9	(	(	PUNCT
ejpam-4936	198	10	i	i	NOUN
ejpam-4936	198	11	)	)	PUNCT
ejpam-4936	198	12	and	and	CCONJ
ejpam-4936	198	13	(	(	PUNCT
ejpam-4936	198	14	ii	ii	NOUN
ejpam-4936	198	15	)	)	PUNCT
ejpam-4936	198	16	,	,	PUNCT
ejpam-4936	198	17	we	we	PRON
ejpam-4936	198	18	can	can	AUX
ejpam-4936	198	19	conclude	conclude	VERB
ejpam-4936	198	20	that	that	SCONJ
ejpam-4936	198	21	the	the	DET
ejpam-4936	198	22	diophantine	diophantine	NOUN
ejpam-4936	198	23	equation	equation	NOUN
ejpam-4936	198	24	has	have	VERB
ejpam-4936	198	25	no	no	DET
ejpam-4936	198	26	solution	solution	NOUN
ejpam-4936	198	27	.	.	PUNCT
ejpam-4936	199	1	(	(	PUNCT
ejpam-4936	199	2	v	v	NOUN
ejpam-4936	199	3	)	)	PUNCT
ejpam-4936	199	4	suppose	suppose	VERB
ejpam-4936	199	5	that	that	SCONJ
ejpam-4936	199	6	a	a	DET
ejpam-4936	199	7	≡	≡	PROPN
ejpam-4936	199	8	3	3	NUM
ejpam-4936	199	9	(	(	PUNCT
ejpam-4936	199	10	mod	mod	PROPN
ejpam-4936	199	11	8)	8)	NUM
ejpam-4936	199	12	and	and	CCONJ
ejpam-4936	199	13	b	b	NOUN
ejpam-4936	199	14	,	,	PUNCT
ejpam-4936	199	15	c	c	PROPN
ejpam-4936	199	16	≡	≡	PROPN
ejpam-4936	199	17	1	1	NUM
ejpam-4936	199	18	(	(	PUNCT
ejpam-4936	199	19	mod	mod	PROPN
ejpam-4936	199	20	8)	8)	NUM
ejpam-4936	199	21	.	.	PUNCT
ejpam-4936	199	22	for	for	ADP
ejpam-4936	199	23	any	any	DET
ejpam-4936	199	24	nonnegative	nonnegative	ADJ
ejpam-4936	199	25	integers	integer	NOUN
ejpam-4936	199	26	x	x	X
ejpam-4936	199	27	,	,	PUNCT
ejpam-4936	199	28	y	y	PROPN
ejpam-4936	199	29	and	and	CCONJ
ejpam-4936	199	30	z	z	PROPN
ejpam-4936	199	31	,	,	PUNCT
ejpam-4936	199	32	we	we	PRON
ejpam-4936	199	33	can	can	AUX
ejpam-4936	199	34	see	see	VERB
ejpam-4936	199	35	that	that	DET
ejpam-4936	199	36	ax	ax	NOUN
ejpam-4936	199	37	≡	≡	PROPN
ejpam-4936	199	38	{	{	PUNCT
ejpam-4936	199	39	1	1	NUM
ejpam-4936	199	40	(	(	PUNCT
ejpam-4936	199	41	mod	mod	PROPN
ejpam-4936	199	42	8)	8)	NUM
ejpam-4936	199	43	if	if	SCONJ
ejpam-4936	199	44	x	x	PRON
ejpam-4936	199	45	is	be	AUX
ejpam-4936	199	46	even	even	ADV
ejpam-4936	199	47	3	3	NUM
ejpam-4936	199	48	(	(	PUNCT
ejpam-4936	199	49	mod	mod	PROPN
ejpam-4936	199	50	8)	8)	NUM
ejpam-4936	199	51	if	if	SCONJ
ejpam-4936	199	52	x	x	PRON
ejpam-4936	199	53	is	be	AUX
ejpam-4936	199	54	odd	odd	ADJ
ejpam-4936	199	55	,	,	PUNCT
ejpam-4936	199	56	and	and	CCONJ
ejpam-4936	199	57	by	by	ADV
ejpam-4936	199	58	,	,	PUNCT
ejpam-4936	199	59	cz	cz	PROPN
ejpam-4936	199	60	≡	≡	PROPN
ejpam-4936	199	61	1	1	NUM
ejpam-4936	199	62	(	(	PUNCT
ejpam-4936	199	63	mod	mod	PROPN
ejpam-4936	199	64	8)	8)	NUM
ejpam-4936	199	65	.	.	PUNCT
ejpam-4936	200	1	thus	thus	ADV
ejpam-4936	200	2	,	,	PUNCT
ejpam-4936	200	3	ax	ax	NOUN
ejpam-4936	200	4	+	+	CCONJ
ejpam-4936	200	5	by	by	ADP
ejpam-4936	200	6	+	+	CCONJ
ejpam-4936	200	7	cz	cz	PROPN
ejpam-4936	200	8	≡	≡	PROPN
ejpam-4936	200	9	{	{	PUNCT
ejpam-4936	200	10	3	3	NUM
ejpam-4936	200	11	(	(	PUNCT
ejpam-4936	200	12	mod	mod	PROPN
ejpam-4936	200	13	8)	8)	NUM
ejpam-4936	200	14	if	if	SCONJ
ejpam-4936	200	15	x	x	PRON
ejpam-4936	200	16	is	be	AUX
ejpam-4936	200	17	even	even	ADV
ejpam-4936	200	18	5	5	NUM
ejpam-4936	200	19	(	(	PUNCT
ejpam-4936	200	20	mod	mod	PROPN
ejpam-4936	200	21	8)	8)	NUM
ejpam-4936	200	22	if	if	SCONJ
ejpam-4936	200	23	x	x	PRON
ejpam-4936	200	24	is	be	AUX
ejpam-4936	200	25	odd	odd	ADJ
ejpam-4936	200	26	,	,	PUNCT
ejpam-4936	200	27	for	for	ADP
ejpam-4936	200	28	any	any	DET
ejpam-4936	200	29	nonnegative	nonnegative	ADJ
ejpam-4936	200	30	integers	integer	NOUN
ejpam-4936	200	31	x	x	X
ejpam-4936	200	32	,	,	PUNCT
ejpam-4936	200	33	y	y	PROPN
ejpam-4936	200	34	and	and	CCONJ
ejpam-4936	200	35	z.	z.	PROPN
ejpam-4936	200	36	since	since	SCONJ
ejpam-4936	200	37	w2	w2	PROPN
ejpam-4936	200	38	≡	≡	PROPN
ejpam-4936	200	39	0	0	NUM
ejpam-4936	200	40	,	,	PUNCT
ejpam-4936	200	41	1	1	NUM
ejpam-4936	200	42	,	,	PUNCT
ejpam-4936	200	43	4	4	NUM
ejpam-4936	200	44	(	(	PUNCT
ejpam-4936	200	45	mod	mod	PROPN
ejpam-4936	200	46	8)	8)	NUM
ejpam-4936	200	47	,	,	PUNCT
ejpam-4936	200	48	the	the	DET
ejpam-4936	200	49	diophantine	diophantine	NOUN
ejpam-4936	200	50	equation	equation	NOUN
ejpam-4936	200	51	has	have	VERB
ejpam-4936	200	52	no	no	DET
ejpam-4936	200	53	solution	solution	NOUN
ejpam-4936	200	54	.	.	PUNCT
ejpam-4936	201	1	(	(	PUNCT
ejpam-4936	201	2	vi	vi	X
ejpam-4936	201	3	)	)	PUNCT
ejpam-4936	201	4	suppose	suppose	VERB
ejpam-4936	201	5	that	that	SCONJ
ejpam-4936	201	6	a	a	DET
ejpam-4936	201	7	≡	≡	PROPN
ejpam-4936	201	8	1	1	NUM
ejpam-4936	201	9	(	(	PUNCT
ejpam-4936	201	10	mod	mod	PROPN
ejpam-4936	201	11	8)	8)	NUM
ejpam-4936	201	12	and	and	CCONJ
ejpam-4936	201	13	b	b	NOUN
ejpam-4936	201	14	,	,	PUNCT
ejpam-4936	201	15	c	c	PROPN
ejpam-4936	201	16	≡	≡	PROPN
ejpam-4936	201	17	3	3	NUM
ejpam-4936	201	18	(	(	PUNCT
ejpam-4936	201	19	mod	mod	PROPN
ejpam-4936	201	20	8)	8)	NUM
ejpam-4936	201	21	.	.	PUNCT
ejpam-4936	202	1	first	first	ADV
ejpam-4936	202	2	,	,	PUNCT
ejpam-4936	202	3	we	we	PRON
ejpam-4936	202	4	note	note	VERB
ejpam-4936	202	5	that	that	SCONJ
ejpam-4936	202	6	by	by	ADP
ejpam-4936	202	7	≡	≡	PROPN
ejpam-4936	202	8	{	{	PUNCT
ejpam-4936	202	9	1	1	NUM
ejpam-4936	202	10	(	(	PUNCT
ejpam-4936	202	11	mod	mod	PROPN
ejpam-4936	202	12	8)	8)	NUM
ejpam-4936	202	13	if	if	SCONJ
ejpam-4936	202	14	x	x	PRON
ejpam-4936	202	15	is	be	AUX
ejpam-4936	202	16	even	even	ADV
ejpam-4936	202	17	3	3	NUM
ejpam-4936	202	18	(	(	PUNCT
ejpam-4936	202	19	mod	mod	PROPN
ejpam-4936	202	20	8)	8)	NUM
ejpam-4936	202	21	if	if	SCONJ
ejpam-4936	202	22	x	x	PRON
ejpam-4936	202	23	is	be	AUX
ejpam-4936	202	24	odd	odd	ADJ
ejpam-4936	202	25	,	,	PUNCT
ejpam-4936	202	26	cz	cz	NOUN
ejpam-4936	202	27	≡	≡	PROPN
ejpam-4936	202	28	{	{	PUNCT
ejpam-4936	202	29	1	1	NUM
ejpam-4936	202	30	(	(	PUNCT
ejpam-4936	202	31	mod	mod	PROPN
ejpam-4936	202	32	8)	8)	NUM
ejpam-4936	202	33	if	if	SCONJ
ejpam-4936	202	34	x	x	PRON
ejpam-4936	202	35	is	be	AUX
ejpam-4936	202	36	even	even	ADV
ejpam-4936	202	37	3	3	NUM
ejpam-4936	202	38	(	(	PUNCT
ejpam-4936	202	39	mod	mod	PROPN
ejpam-4936	202	40	8)	8)	NUM
ejpam-4936	202	41	if	if	SCONJ
ejpam-4936	202	42	x	x	PRON
ejpam-4936	202	43	is	be	AUX
ejpam-4936	202	44	odd	odd	ADJ
ejpam-4936	202	45	,	,	PUNCT
ejpam-4936	202	46	ax	ax	NOUN
ejpam-4936	202	47	≡	≡	PROPN
ejpam-4936	202	48	1	1	NUM
ejpam-4936	202	49	(	(	PUNCT
ejpam-4936	202	50	mod	mod	PROPN
ejpam-4936	202	51	8)	8)	NUM
ejpam-4936	202	52	and	and	CCONJ
ejpam-4936	202	53	w2	w2	NOUN
ejpam-4936	202	54	≡	≡	PROPN
ejpam-4936	202	55	0	0	NUM
ejpam-4936	202	56	,	,	PUNCT
ejpam-4936	202	57	1	1	NUM
ejpam-4936	202	58	,	,	PUNCT
ejpam-4936	202	59	4	4	NUM
ejpam-4936	202	60	(	(	PUNCT
ejpam-4936	202	61	mod	mod	PROPN
ejpam-4936	202	62	8)	8)	NUM
ejpam-4936	202	63	for	for	ADP
ejpam-4936	202	64	any	any	DET
ejpam-4936	202	65	nonnegative	nonnegative	ADJ
ejpam-4936	202	66	integers	integer	NOUN
ejpam-4936	202	67	w	w	VERB
ejpam-4936	202	68	,	,	PUNCT
ejpam-4936	202	69	x	x	NOUN
ejpam-4936	202	70	,	,	PUNCT
ejpam-4936	202	71	y	y	PROPN
ejpam-4936	202	72	and	and	CCONJ
ejpam-4936	202	73	z.	z.	PROPN
ejpam-4936	202	74	next	next	ADV
ejpam-4936	202	75	,	,	PUNCT
ejpam-4936	202	76	we	we	PRON
ejpam-4936	202	77	separate	separate	VERB
ejpam-4936	202	78	the	the	DET
ejpam-4936	202	79	proof	proof	NOUN
ejpam-4936	202	80	into	into	ADP
ejpam-4936	202	81	four	four	NUM
ejpam-4936	202	82	cases	case	NOUN
ejpam-4936	202	83	.	.	PUNCT
ejpam-4936	203	1	k.	k.	PROPN
ejpam-4936	203	2	laipaporn	laipaporn	PROPN
ejpam-4936	203	3	,	,	PUNCT
ejpam-4936	203	4	s.	s.	PROPN
ejpam-4936	203	5	kaewchay	kaewchay	PROPN
ejpam-4936	203	6	,	,	PUNCT
ejpam-4936	203	7	a.	a.	PROPN
ejpam-4936	203	8	karnbanjong	karnbanjong	PROPN
ejpam-4936	203	9	/	/	SYM
ejpam-4936	203	10	eur	eur	PROPN
ejpam-4936	203	11	.	.	PUNCT
ejpam-4936	204	1	j.	j.	PROPN
ejpam-4936	204	2	pure	pure	PROPN
ejpam-4936	204	3	appl	appl	PROPN
ejpam-4936	204	4	.	.	PROPN
ejpam-4936	204	5	math	math	PROPN
ejpam-4936	204	6	,	,	PUNCT
ejpam-4936	204	7	16	16	NUM
ejpam-4936	204	8	(	(	PUNCT
ejpam-4936	204	9	4	4	NUM
ejpam-4936	204	10	)	)	PUNCT
ejpam-4936	204	11	(	(	PUNCT
ejpam-4936	204	12	2023	2023	NUM
ejpam-4936	204	13	)	)	PUNCT
ejpam-4936	204	14	,	,	PUNCT
ejpam-4936	204	15	2066	2066	NUM
ejpam-4936	204	16	-	-	SYM
ejpam-4936	204	17	2081	2081	NUM
ejpam-4936	204	18	2073	2073	NUM
ejpam-4936	204	19	case	case	NOUN
ejpam-4936	204	20	1	1	NUM
ejpam-4936	204	21	.	.	PUNCT
ejpam-4936	205	1	here	here	ADV
ejpam-4936	205	2	,	,	PUNCT
ejpam-4936	205	3	y	y	PROPN
ejpam-4936	205	4	=	=	SYM
ejpam-4936	205	5	0	0	PROPN
ejpam-4936	205	6	and	and	CCONJ
ejpam-4936	205	7	z	z	NOUN
ejpam-4936	205	8	=	=	NOUN
ejpam-4936	205	9	0	0	X
ejpam-4936	205	10	.	.	PUNCT
ejpam-4936	206	1	since	since	SCONJ
ejpam-4936	206	2	ax	ax	NOUN
ejpam-4936	206	3	+	+	CCONJ
ejpam-4936	206	4	by	by	ADP
ejpam-4936	206	5	+	+	CCONJ
ejpam-4936	206	6	cz	cz	PROPN
ejpam-4936	206	7	≡	≡	PROPN
ejpam-4936	206	8	3	3	NUM
ejpam-4936	206	9	(	(	PUNCT
ejpam-4936	206	10	mod	mod	PROPN
ejpam-4936	206	11	8)	8)	NUM
ejpam-4936	206	12	and	and	CCONJ
ejpam-4936	206	13	w2	w2	NOUN
ejpam-4936	206	14	̸≡	̸≡	PROPN
ejpam-4936	206	15	3	3	NUM
ejpam-4936	206	16	(	(	PUNCT
ejpam-4936	206	17	mod	mod	PROPN
ejpam-4936	206	18	8)	8)	NUM
ejpam-4936	206	19	,	,	PUNCT
ejpam-4936	206	20	the	the	DET
ejpam-4936	206	21	diophantine	diophantine	NOUN
ejpam-4936	206	22	equation	equation	NOUN
ejpam-4936	206	23	has	have	VERB
ejpam-4936	206	24	no	no	DET
ejpam-4936	206	25	solution	solution	NOUN
ejpam-4936	206	26	.	.	PUNCT
ejpam-4936	207	1	case	case	NOUN
ejpam-4936	207	2	2	2	NUM
ejpam-4936	207	3	.	.	PUNCT
ejpam-4936	208	1	here	here	ADV
ejpam-4936	208	2	,	,	PUNCT
ejpam-4936	208	3	y	y	PROPN
ejpam-4936	208	4	=	=	PUNCT
ejpam-4936	208	5	0	0	PROPN
ejpam-4936	208	6	and	and	CCONJ
ejpam-4936	208	7	z	z	NOUN
ejpam-4936	208	8	>	>	X
ejpam-4936	208	9	0	0	X
ejpam-4936	208	10	.	.	PUNCT
ejpam-4936	209	1	since	since	SCONJ
ejpam-4936	209	2	ax	ax	NOUN
ejpam-4936	209	3	+	+	CCONJ
ejpam-4936	209	4	1	1	NUM
ejpam-4936	209	5	+	+	CCONJ
ejpam-4936	209	6	cz	cz	ADJ
ejpam-4936	209	7	≡	≡	PROPN
ejpam-4936	209	8	{	{	PUNCT
ejpam-4936	209	9	3	3	NUM
ejpam-4936	209	10	(	(	PUNCT
ejpam-4936	209	11	mod	mod	PROPN
ejpam-4936	209	12	8)	8)	NUM
ejpam-4936	209	13	if	if	SCONJ
ejpam-4936	209	14	x	x	PRON
ejpam-4936	209	15	is	be	AUX
ejpam-4936	209	16	even	even	ADV
ejpam-4936	209	17	5	5	NUM
ejpam-4936	209	18	(	(	PUNCT
ejpam-4936	209	19	mod	mod	PROPN
ejpam-4936	209	20	8)	8)	NUM
ejpam-4936	209	21	if	if	SCONJ
ejpam-4936	209	22	x	x	PRON
ejpam-4936	209	23	is	be	AUX
ejpam-4936	209	24	odd	odd	ADJ
ejpam-4936	209	25	,	,	PUNCT
ejpam-4936	209	26	and	and	CCONJ
ejpam-4936	209	27	w2	w2	NOUN
ejpam-4936	209	28	̸≡	̸≡	NOUN
ejpam-4936	209	29	3	3	NUM
ejpam-4936	209	30	,	,	PUNCT
ejpam-4936	209	31	5	5	NUM
ejpam-4936	209	32	(	(	PUNCT
ejpam-4936	209	33	mod	mod	PROPN
ejpam-4936	209	34	8)	8)	NUM
ejpam-4936	209	35	,	,	PUNCT
ejpam-4936	209	36	the	the	DET
ejpam-4936	209	37	diophantine	diophantine	NOUN
ejpam-4936	209	38	equation	equation	NOUN
ejpam-4936	209	39	has	have	VERB
ejpam-4936	209	40	no	no	DET
ejpam-4936	209	41	solution	solution	NOUN
ejpam-4936	209	42	.	.	PUNCT
ejpam-4936	210	1	case	case	NOUN
ejpam-4936	210	2	3	3	X
ejpam-4936	210	3	.	.	PUNCT
ejpam-4936	211	1	here	here	ADV
ejpam-4936	211	2	,	,	PUNCT
ejpam-4936	211	3	y	y	PROPN
ejpam-4936	211	4	>	>	X
ejpam-4936	211	5	0	0	PUNCT
ejpam-4936	212	1	and	and	CCONJ
ejpam-4936	212	2	z	z	NOUN
ejpam-4936	212	3	=	=	NOUN
ejpam-4936	212	4	0	0	X
ejpam-4936	212	5	.	.	PUNCT
ejpam-4936	213	1	the	the	DET
ejpam-4936	213	2	proof	proof	NOUN
ejpam-4936	213	3	of	of	ADP
ejpam-4936	213	4	this	this	DET
ejpam-4936	213	5	case	case	NOUN
ejpam-4936	213	6	is	be	AUX
ejpam-4936	213	7	the	the	DET
ejpam-4936	213	8	same	same	ADJ
ejpam-4936	213	9	as	as	ADP
ejpam-4936	213	10	in	in	ADP
ejpam-4936	213	11	case	case	NOUN
ejpam-4936	213	12	2	2	NUM
ejpam-4936	213	13	.	.	PUNCT
ejpam-4936	213	14	case	case	NOUN
ejpam-4936	213	15	4	4	NUM
ejpam-4936	213	16	.	.	PUNCT
ejpam-4936	214	1	here	here	ADV
ejpam-4936	214	2	,	,	PUNCT
ejpam-4936	214	3	y	y	PROPN
ejpam-4936	214	4	>	>	X
ejpam-4936	214	5	0	0	PUNCT
ejpam-4936	215	1	and	and	CCONJ
ejpam-4936	215	2	z	z	NOUN
ejpam-4936	215	3	>	>	X
ejpam-4936	216	1	0	0	X
ejpam-4936	216	2	.	.	PUNCT
ejpam-4936	217	1	then	then	ADV
ejpam-4936	217	2	,	,	PUNCT
ejpam-4936	217	3	by	by	ADP
ejpam-4936	217	4	+	+	CCONJ
ejpam-4936	217	5	cz	cz	PROPN
ejpam-4936	217	6	≡	≡	PROPN
ejpam-4936	217	7			ADV
ejpam-4936	217	8	6	6	NUM
ejpam-4936	217	9	(	(	PUNCT
ejpam-4936	217	10	mod	mod	PROPN
ejpam-4936	217	11	8)	8)	NUM
ejpam-4936	217	12	if	if	SCONJ
ejpam-4936	217	13	y	y	PROPN
ejpam-4936	217	14	and	and	CCONJ
ejpam-4936	217	15	z	z	PROPN
ejpam-4936	217	16	are	be	AUX
ejpam-4936	217	17	odd	odd	ADJ
ejpam-4936	217	18	,	,	PUNCT
ejpam-4936	217	19	2	2	NUM
ejpam-4936	217	20	(	(	PUNCT
ejpam-4936	217	21	mod	mod	PROPN
ejpam-4936	217	22	8)	8)	NUM
ejpam-4936	217	23	if	if	SCONJ
ejpam-4936	217	24	y	y	PROPN
ejpam-4936	217	25	and	and	CCONJ
ejpam-4936	217	26	z	z	NOUN
ejpam-4936	217	27	are	be	AUX
ejpam-4936	217	28	even	even	ADV
ejpam-4936	217	29	,	,	PUNCT
ejpam-4936	217	30	4	4	NUM
ejpam-4936	217	31	(	(	PUNCT
ejpam-4936	217	32	mod	mod	PROPN
ejpam-4936	217	33	8)	8)	NUM
ejpam-4936	217	34	if	if	SCONJ
ejpam-4936	217	35	otherwise	otherwise	ADV
ejpam-4936	217	36	.	.	PUNCT
ejpam-4936	218	1	however	however	ADV
ejpam-4936	218	2	,	,	PUNCT
ejpam-4936	218	3	w2	w2	NOUN
ejpam-4936	218	4	≡	≡	PROPN
ejpam-4936	218	5	0	0	NUM
ejpam-4936	218	6	,	,	PUNCT
ejpam-4936	218	7	1	1	NUM
ejpam-4936	218	8	,	,	PUNCT
ejpam-4936	218	9	4	4	NUM
ejpam-4936	218	10	(	(	PUNCT
ejpam-4936	218	11	mod	mod	NOUN
ejpam-4936	218	12	8)	8)	NUM
ejpam-4936	218	13	,	,	PUNCT
ejpam-4936	218	14	so	so	ADV
ejpam-4936	218	15	,	,	PUNCT
ejpam-4936	218	16	the	the	DET
ejpam-4936	218	17	diophantine	diophantine	NOUN
ejpam-4936	218	18	equation	equation	NOUN
ejpam-4936	218	19	has	have	VERB
ejpam-4936	218	20	no	no	DET
ejpam-4936	218	21	solution	solution	NOUN
ejpam-4936	218	22	.	.	PUNCT
ejpam-4936	219	1	theorem	theorem	VERB
ejpam-4936	219	2	6	6	NUM
ejpam-4936	219	3	.	.	PUNCT
ejpam-4936	220	1	if	if	SCONJ
ejpam-4936	220	2	a	a	DET
ejpam-4936	220	3	+	+	ADJ
ejpam-4936	220	4	2	2	NUM
ejpam-4936	220	5	is	be	AUX
ejpam-4936	220	6	a	a	DET
ejpam-4936	220	7	perfect	perfect	ADJ
ejpam-4936	220	8	square	square	ADJ
ejpam-4936	220	9	number	number	NOUN
ejpam-4936	220	10	such	such	ADJ
ejpam-4936	220	11	that	that	SCONJ
ejpam-4936	220	12	a	a	DET
ejpam-4936	220	13	≡	≡	PROPN
ejpam-4936	220	14	2	2	NUM
ejpam-4936	220	15	(	(	PUNCT
ejpam-4936	220	16	mod	mod	NOUN
ejpam-4936	220	17	36	36	NUM
ejpam-4936	220	18	)	)	PUNCT
ejpam-4936	220	19	and	and	CCONJ
ejpam-4936	220	20	b	b	NOUN
ejpam-4936	220	21	,	,	PUNCT
ejpam-4936	220	22	c	c	PROPN
ejpam-4936	220	23	≡	≡	PROPN
ejpam-4936	220	24	9	9	NUM
ejpam-4936	220	25	(	(	PUNCT
ejpam-4936	220	26	mod	mod	PROPN
ejpam-4936	220	27	36	36	NUM
ejpam-4936	220	28	)	)	PUNCT
ejpam-4936	220	29	,	,	PUNCT
ejpam-4936	220	30	then	then	ADV
ejpam-4936	220	31	the	the	DET
ejpam-4936	220	32	solutions	solution	NOUN
ejpam-4936	220	33	of	of	ADP
ejpam-4936	220	34	diophantine	diophantine	NOUN
ejpam-4936	220	35	equation	equation	NOUN
ejpam-4936	220	36	ax	ax	NOUN
ejpam-4936	220	37	+	+	CCONJ
ejpam-4936	220	38	by	by	ADP
ejpam-4936	220	39	+	+	CCONJ
ejpam-4936	220	40	cz	cz	NOUN
ejpam-4936	220	41	=	=	NOUN
ejpam-4936	220	42	w2	w2	NOUN
ejpam-4936	220	43	are	be	AUX
ejpam-4936	220	44	(	(	PUNCT
ejpam-4936	220	45	x	x	NOUN
ejpam-4936	220	46	,	,	PUNCT
ejpam-4936	220	47	y	y	PROPN
ejpam-4936	220	48	,	,	PUNCT
ejpam-4936	220	49	z	z	PROPN
ejpam-4936	220	50	,	,	PUNCT
ejpam-4936	220	51	w	w	NOUN
ejpam-4936	220	52	)	)	PUNCT
ejpam-4936	220	53	=	=	SYM
ejpam-4936	220	54	(	(	PUNCT
ejpam-4936	220	55	1	1	NUM
ejpam-4936	220	56	,	,	PUNCT
ejpam-4936	220	57	0	0	NUM
ejpam-4936	220	58	,	,	PUNCT
ejpam-4936	220	59	0	0	NUM
ejpam-4936	220	60	,	,	PUNCT
ejpam-4936	220	61	√	√	NUM
ejpam-4936	220	62	a+	a+	PUNCT
ejpam-4936	220	63	2	2	NUM
ejpam-4936	220	64	)	)	PUNCT
ejpam-4936	220	65	.	.	PUNCT
ejpam-4936	221	1	proof	proof	NOUN
ejpam-4936	221	2	.	.	PUNCT
ejpam-4936	222	1	note	note	VERB
ejpam-4936	222	2	that	that	SCONJ
ejpam-4936	222	3	a	a	DET
ejpam-4936	222	4	≡	≡	PROPN
ejpam-4936	222	5	2	2	NUM
ejpam-4936	222	6	(	(	PUNCT
ejpam-4936	222	7	mod	mod	NOUN
ejpam-4936	222	8	4	4	NUM
ejpam-4936	222	9	)	)	PUNCT
ejpam-4936	222	10	,	,	PUNCT
ejpam-4936	222	11	b	b	X
ejpam-4936	222	12	,	,	PUNCT
ejpam-4936	222	13	c	c	PROPN
ejpam-4936	222	14	≡	≡	PROPN
ejpam-4936	222	15	1	1	NUM
ejpam-4936	222	16	(	(	PUNCT
ejpam-4936	222	17	mod	mod	NOUN
ejpam-4936	222	18	4	4	NUM
ejpam-4936	222	19	)	)	PUNCT
ejpam-4936	222	20	,	,	PUNCT
ejpam-4936	222	21	a	a	DET
ejpam-4936	222	22	≡	≡	PROPN
ejpam-4936	222	23	2	2	NUM
ejpam-4936	222	24	(	(	PUNCT
ejpam-4936	222	25	mod	mod	NOUN
ejpam-4936	222	26	9	9	NUM
ejpam-4936	222	27	)	)	PUNCT
ejpam-4936	222	28	and	and	CCONJ
ejpam-4936	222	29	b	b	NOUN
ejpam-4936	222	30	,	,	PUNCT
ejpam-4936	222	31	c	c	PROPN
ejpam-4936	222	32	≡	≡	PROPN
ejpam-4936	222	33	0	0	PUNCT
ejpam-4936	222	34	(	(	PUNCT
ejpam-4936	222	35	mod	mod	NOUN
ejpam-4936	222	36	9	9	NUM
ejpam-4936	222	37	)	)	PUNCT
ejpam-4936	222	38	.	.	PUNCT
ejpam-4936	223	1	since	since	SCONJ
ejpam-4936	223	2	1+by+cz	1+by+cz	NUM
ejpam-4936	223	3	≡	≡	PROPN
ejpam-4936	223	4	3	3	NUM
ejpam-4936	223	5	(	(	PUNCT
ejpam-4936	223	6	mod	mod	NOUN
ejpam-4936	223	7	4	4	NUM
ejpam-4936	223	8	)	)	PUNCT
ejpam-4936	223	9	and	and	CCONJ
ejpam-4936	223	10	w2	w2	PROPN
ejpam-4936	223	11	≡	≡	PROPN
ejpam-4936	223	12	0	0	NUM
ejpam-4936	223	13	,	,	PUNCT
ejpam-4936	223	14	1	1	NUM
ejpam-4936	223	15	(	(	PUNCT
ejpam-4936	223	16	mod	mod	NOUN
ejpam-4936	223	17	4	4	NUM
ejpam-4936	223	18	)	)	PUNCT
ejpam-4936	223	19	,	,	PUNCT
ejpam-4936	223	20	equation	equation	NOUN
ejpam-4936	223	21	ax+by+cz	ax+by+cz	NOUN
ejpam-4936	223	22	=	=	PROPN
ejpam-4936	223	23	w2	w2	NOUN
ejpam-4936	223	24	has	have	VERB
ejpam-4936	223	25	no	no	DET
ejpam-4936	223	26	solution	solution	NOUN
ejpam-4936	223	27	for	for	ADP
ejpam-4936	223	28	the	the	DET
ejpam-4936	223	29	case	case	NOUN
ejpam-4936	223	30	x	x	PUNCT
ejpam-4936	224	1	=	=	NOUN
ejpam-4936	224	2	0	0	X
ejpam-4936	224	3	.	.	PUNCT
ejpam-4936	225	1	if	if	SCONJ
ejpam-4936	225	2	x	x	X
ejpam-4936	225	3	≥	≥	NUM
ejpam-4936	225	4	2	2	NUM
ejpam-4936	225	5	,	,	PUNCT
ejpam-4936	225	6	then	then	ADV
ejpam-4936	225	7	ax	ax	NOUN
ejpam-4936	225	8	+	+	CCONJ
ejpam-4936	225	9	by	by	ADP
ejpam-4936	225	10	+	+	CCONJ
ejpam-4936	225	11	cz	cz	PROPN
ejpam-4936	225	12	≡	≡	PROPN
ejpam-4936	225	13	2	2	NUM
ejpam-4936	225	14	(	(	PUNCT
ejpam-4936	225	15	mod	mod	NOUN
ejpam-4936	225	16	4	4	NUM
ejpam-4936	225	17	)	)	PUNCT
ejpam-4936	225	18	,	,	PUNCT
ejpam-4936	225	19	which	which	PRON
ejpam-4936	225	20	contradicts	contradict	VERB
ejpam-4936	225	21	w2	w2	NOUN
ejpam-4936	225	22	≡	≡	PROPN
ejpam-4936	225	23	0	0	NUM
ejpam-4936	225	24	,	,	PUNCT
ejpam-4936	225	25	1	1	NUM
ejpam-4936	225	26	(	(	PUNCT
ejpam-4936	225	27	mod	mod	NOUN
ejpam-4936	225	28	4	4	NUM
ejpam-4936	225	29	)	)	PUNCT
ejpam-4936	225	30	.	.	PUNCT
ejpam-4936	226	1	thus	thus	ADV
ejpam-4936	226	2	,	,	PUNCT
ejpam-4936	226	3	in	in	ADP
ejpam-4936	226	4	this	this	DET
ejpam-4936	226	5	case	case	NOUN
ejpam-4936	226	6	,	,	PUNCT
ejpam-4936	226	7	x	x	X
ejpam-4936	226	8	≥	≥	NOUN
ejpam-4936	226	9	2	2	NUM
ejpam-4936	226	10	,	,	PUNCT
ejpam-4936	226	11	and	and	CCONJ
ejpam-4936	226	12	there	there	PRON
ejpam-4936	226	13	is	be	VERB
ejpam-4936	226	14	no	no	DET
ejpam-4936	226	15	solution	solution	NOUN
ejpam-4936	226	16	.	.	PUNCT
ejpam-4936	227	1	we	we	PRON
ejpam-4936	227	2	now	now	ADV
ejpam-4936	227	3	consider	consider	VERB
ejpam-4936	227	4	that	that	PRON
ejpam-4936	227	5	for	for	ADP
ejpam-4936	227	6	x	x	SYM
ejpam-4936	227	7	=	=	SYM
ejpam-4936	227	8	1	1	NUM
ejpam-4936	227	9	,	,	PUNCT
ejpam-4936	227	10	y	y	PROPN
ejpam-4936	227	11	and	and	CCONJ
ejpam-4936	227	12	z	z	PROPN
ejpam-4936	227	13	are	be	AUX
ejpam-4936	227	14	nonzero	nonzero	NOUN
ejpam-4936	227	15	at	at	ADP
ejpam-4936	227	16	the	the	DET
ejpam-4936	227	17	same	same	ADJ
ejpam-4936	227	18	time	time	NOUN
ejpam-4936	227	19	.	.	PUNCT
ejpam-4936	228	1	because	because	SCONJ
ejpam-4936	228	2	a+	a+	PUNCT
ejpam-4936	228	3	by	by	ADP
ejpam-4936	228	4	+	+	CCONJ
ejpam-4936	228	5	cz	cz	PROPN
ejpam-4936	228	6	≡	≡	PROPN
ejpam-4936	228	7	{	{	PUNCT
ejpam-4936	228	8	2	2	NUM
ejpam-4936	228	9	(	(	PUNCT
ejpam-4936	228	10	mod	mod	NOUN
ejpam-4936	228	11	9	9	NUM
ejpam-4936	228	12	)	)	PUNCT
ejpam-4936	228	13	if	if	SCONJ
ejpam-4936	228	14	y	y	PROPN
ejpam-4936	228	15	,	,	PUNCT
ejpam-4936	228	16	z	z	NOUN
ejpam-4936	228	17	≥	≥	NUM
ejpam-4936	228	18	1	1	NUM
ejpam-4936	228	19	,	,	PUNCT
ejpam-4936	228	20	3	3	NUM
ejpam-4936	228	21	(	(	PUNCT
ejpam-4936	228	22	mod	mod	NOUN
ejpam-4936	228	23	9	9	NUM
ejpam-4936	228	24	)	)	PUNCT
ejpam-4936	228	25	if	if	SCONJ
ejpam-4936	228	26	either	either	CCONJ
ejpam-4936	228	27	y	y	PROPN
ejpam-4936	228	28	or	or	CCONJ
ejpam-4936	228	29	z	z	PROPN
ejpam-4936	228	30	is	be	AUX
ejpam-4936	228	31	zero	zero	NUM
ejpam-4936	228	32	,	,	PUNCT
ejpam-4936	228	33	and	and	CCONJ
ejpam-4936	228	34	w2	w2	PROPN
ejpam-4936	228	35	≡	≡	PROPN
ejpam-4936	228	36	0	0	NUM
ejpam-4936	228	37	,	,	PUNCT
ejpam-4936	228	38	1	1	NUM
ejpam-4936	228	39	,	,	PUNCT
ejpam-4936	228	40	4	4	NUM
ejpam-4936	228	41	,	,	PUNCT
ejpam-4936	228	42	7	7	NUM
ejpam-4936	228	43	(	(	PUNCT
ejpam-4936	228	44	mod	mod	NOUN
ejpam-4936	228	45	9	9	NUM
ejpam-4936	228	46	)	)	PUNCT
ejpam-4936	228	47	,	,	PUNCT
ejpam-4936	228	48	a	a	DET
ejpam-4936	228	49	solution	solution	NOUN
ejpam-4936	228	50	still	still	ADV
ejpam-4936	228	51	does	do	AUX
ejpam-4936	228	52	not	not	PART
ejpam-4936	228	53	exist	exist	VERB
ejpam-4936	228	54	.	.	PUNCT
ejpam-4936	229	1	for	for	ADP
ejpam-4936	229	2	the	the	DET
ejpam-4936	229	3	last	last	ADJ
ejpam-4936	229	4	case	case	NOUN
ejpam-4936	229	5	,	,	PUNCT
ejpam-4936	229	6	that	that	ADV
ejpam-4936	229	7	is	is	ADV
ejpam-4936	229	8	,	,	PUNCT
ejpam-4936	229	9	x	x	PUNCT
ejpam-4936	229	10	=	=	SYM
ejpam-4936	229	11	1	1	NUM
ejpam-4936	229	12	,	,	PUNCT
ejpam-4936	229	13	y	y	NOUN
ejpam-4936	230	1	=	=	PUNCT
ejpam-4936	230	2	z	z	NOUN
ejpam-4936	230	3	=	=	SYM
ejpam-4936	230	4	0	0	NUM
ejpam-4936	230	5	,	,	PUNCT
ejpam-4936	230	6	we	we	PRON
ejpam-4936	230	7	see	see	VERB
ejpam-4936	230	8	that	that	DET
ejpam-4936	230	9	ax	ax	NOUN
ejpam-4936	230	10	+	+	X
ejpam-4936	230	11	by	by	ADP
ejpam-4936	230	12	+	+	NOUN
ejpam-4936	230	13	cz	cz	NOUN
ejpam-4936	230	14	=	=	SYM
ejpam-4936	230	15	a+	a+	PRON
ejpam-4936	230	16	2	2	NUM
ejpam-4936	230	17	is	be	AUX
ejpam-4936	230	18	a	a	DET
ejpam-4936	230	19	perfect	perfect	ADJ
ejpam-4936	230	20	square	square	ADJ
ejpam-4936	230	21	number	number	NOUN
ejpam-4936	230	22	.	.	PUNCT
ejpam-4936	231	1	hence	hence	ADV
ejpam-4936	231	2	,	,	PUNCT
ejpam-4936	231	3	the	the	DET
ejpam-4936	231	4	solution	solution	NOUN
ejpam-4936	231	5	of	of	ADP
ejpam-4936	231	6	equation	equation	NOUN
ejpam-4936	231	7	ax	ax	NOUN
ejpam-4936	231	8	+	+	CCONJ
ejpam-4936	231	9	by	by	ADP
ejpam-4936	231	10	+	+	CCONJ
ejpam-4936	231	11	cz	cz	NOUN
ejpam-4936	231	12	=	=	NOUN
ejpam-4936	231	13	w2	w2	NOUN
ejpam-4936	231	14	is	be	AUX
ejpam-4936	231	15	(	(	PUNCT
ejpam-4936	231	16	1	1	NUM
ejpam-4936	231	17	,	,	PUNCT
ejpam-4936	231	18	0	0	NUM
ejpam-4936	231	19	,	,	PUNCT
ejpam-4936	231	20	0	0	NUM
ejpam-4936	231	21	,	,	PUNCT
ejpam-4936	231	22	√	√	NUM
ejpam-4936	231	23	a+	a+	PUNCT
ejpam-4936	231	24	2	2	NUM
ejpam-4936	231	25	)	)	PUNCT
ejpam-4936	231	26	.	.	PUNCT
ejpam-4936	232	1	theorem	theorem	VERB
ejpam-4936	232	2	7	7	NUM
ejpam-4936	232	3	.	.	PUNCT
ejpam-4936	233	1	if	if	SCONJ
ejpam-4936	233	2	b	b	X
ejpam-4936	233	3	,	,	PUNCT
ejpam-4936	233	4	c	c	PROPN
ejpam-4936	233	5	≡	≡	PROPN
ejpam-4936	233	6	5	5	NUM
ejpam-4936	233	7	(	(	PUNCT
ejpam-4936	233	8	mod	mod	PROPN
ejpam-4936	233	9	20	20	NUM
ejpam-4936	233	10	)	)	PUNCT
ejpam-4936	233	11	,	,	PUNCT
ejpam-4936	233	12	then	then	ADV
ejpam-4936	233	13	the	the	DET
ejpam-4936	233	14	solution	solution	NOUN
ejpam-4936	233	15	of	of	ADP
ejpam-4936	233	16	the	the	DET
ejpam-4936	233	17	diophantine	diophantine	NOUN
ejpam-4936	233	18	equation	equation	NOUN
ejpam-4936	233	19	2x+by+	2x+by+	NUM
ejpam-4936	233	20	cz	cz	NOUN
ejpam-4936	233	21	=	=	PROPN
ejpam-4936	233	22	w2	w2	NOUN
ejpam-4936	233	23	is	be	AUX
ejpam-4936	233	24	(	(	PUNCT
ejpam-4936	233	25	x	x	NOUN
ejpam-4936	233	26	,	,	PUNCT
ejpam-4936	233	27	y	y	PROPN
ejpam-4936	233	28	,	,	PUNCT
ejpam-4936	233	29	z	z	PROPN
ejpam-4936	233	30	,	,	PUNCT
ejpam-4936	233	31	w	w	NOUN
ejpam-4936	233	32	)	)	PUNCT
ejpam-4936	233	33	=	=	SYM
ejpam-4936	233	34	(	(	PUNCT
ejpam-4936	233	35	1	1	NUM
ejpam-4936	233	36	,	,	PUNCT
ejpam-4936	233	37	0	0	NUM
ejpam-4936	233	38	,	,	PUNCT
ejpam-4936	233	39	0	0	NUM
ejpam-4936	233	40	,	,	PUNCT
ejpam-4936	233	41	2	2	NUM
ejpam-4936	233	42	)	)	PUNCT
ejpam-4936	233	43	.	.	PUNCT
ejpam-4936	234	1	proof	proof	NOUN
ejpam-4936	234	2	.	.	PUNCT
ejpam-4936	235	1	first	first	ADV
ejpam-4936	235	2	,	,	PUNCT
ejpam-4936	235	3	we	we	PRON
ejpam-4936	235	4	note	note	VERB
ejpam-4936	235	5	that	that	PRON
ejpam-4936	235	6	b	b	NOUN
ejpam-4936	235	7	,	,	PUNCT
ejpam-4936	235	8	c	c	PROPN
ejpam-4936	235	9	≡	≡	PROPN
ejpam-4936	235	10	1	1	NUM
ejpam-4936	235	11	(	(	PUNCT
ejpam-4936	235	12	mod	mod	NOUN
ejpam-4936	235	13	4	4	NUM
ejpam-4936	235	14	)	)	PUNCT
ejpam-4936	235	15	and	and	CCONJ
ejpam-4936	235	16	b	b	NOUN
ejpam-4936	235	17	,	,	PUNCT
ejpam-4936	235	18	c	c	PROPN
ejpam-4936	235	19	≡	≡	PROPN
ejpam-4936	235	20	0	0	PUNCT
ejpam-4936	236	1	(	(	PUNCT
ejpam-4936	236	2	mod	mod	PROPN
ejpam-4936	236	3	5	5	NUM
ejpam-4936	236	4	)	)	PUNCT
ejpam-4936	236	5	since	since	SCONJ
ejpam-4936	236	6	b	b	NUM
ejpam-4936	236	7	,	,	PUNCT
ejpam-4936	236	8	c	c	PROPN
ejpam-4936	236	9	≡	≡	PROPN
ejpam-4936	236	10	5	5	NUM
ejpam-4936	236	11	(	(	PUNCT
ejpam-4936	236	12	mod	mod	PROPN
ejpam-4936	236	13	20	20	NUM
ejpam-4936	236	14	)	)	PUNCT
ejpam-4936	236	15	.	.	PUNCT
ejpam-4936	237	1	if	if	SCONJ
ejpam-4936	237	2	x	x	PROPN
ejpam-4936	237	3	=	=	SYM
ejpam-4936	237	4	0	0	NUM
ejpam-4936	237	5	,	,	PUNCT
ejpam-4936	237	6	then	then	ADV
ejpam-4936	237	7	we	we	PRON
ejpam-4936	237	8	odd	odd	ADJ
ejpam-4936	237	9	value	value	NOUN
ejpam-4936	237	10	,	,	PUNCT
ejpam-4936	237	11	which	which	PRON
ejpam-4936	237	12	leads	lead	VERB
ejpam-4936	237	13	to	to	ADP
ejpam-4936	237	14	w2	w2	NOUN
ejpam-4936	237	15	≡	≡	PROPN
ejpam-4936	237	16	1	1	NUM
ejpam-4936	237	17	(	(	PUNCT
ejpam-4936	237	18	mod	mod	NOUN
ejpam-4936	237	19	4	4	NUM
ejpam-4936	237	20	)	)	PUNCT
ejpam-4936	237	21	.	.	PUNCT
ejpam-4936	238	1	however	however	ADV
ejpam-4936	238	2	,	,	PUNCT
ejpam-4936	238	3	1	1	NUM
ejpam-4936	238	4	+	+	CCONJ
ejpam-4936	238	5	by	by	ADP
ejpam-4936	238	6	+	+	CCONJ
ejpam-4936	238	7	cz	cz	PROPN
ejpam-4936	238	8	≡	≡	PROPN
ejpam-4936	238	9	3	3	NUM
ejpam-4936	238	10	(	(	PUNCT
ejpam-4936	238	11	mod	mod	NOUN
ejpam-4936	238	12	4	4	NUM
ejpam-4936	238	13	)	)	PUNCT
ejpam-4936	238	14	,	,	PUNCT
ejpam-4936	238	15	contradicts	contradict	VERB
ejpam-4936	238	16	the	the	DET
ejpam-4936	238	17	equation	equation	NOUN
ejpam-4936	238	18	2x	2x	NUM
ejpam-4936	238	19	+	+	CCONJ
ejpam-4936	238	20	by	by	ADP
ejpam-4936	238	21	+	+	CCONJ
ejpam-4936	238	22	cz	cz	NOUN
ejpam-4936	238	23	=	=	SYM
ejpam-4936	238	24	w2	w2	NOUN
ejpam-4936	238	25	,	,	PUNCT
ejpam-4936	238	26	which	which	PRON
ejpam-4936	238	27	has	have	VERB
ejpam-4936	238	28	a	a	DET
ejpam-4936	238	29	solution	solution	NOUN
ejpam-4936	238	30	in	in	ADP
ejpam-4936	238	31	this	this	DET
ejpam-4936	238	32	case	case	NOUN
ejpam-4936	238	33	.	.	PUNCT
ejpam-4936	239	1	moreover	moreover	ADV
ejpam-4936	239	2	,	,	PUNCT
ejpam-4936	239	3	for	for	ADP
ejpam-4936	239	4	the	the	DET
ejpam-4936	239	5	case	case	NOUN
ejpam-4936	239	6	x	x	X
ejpam-4936	239	7	≥	≥	NUM
ejpam-4936	239	8	2	2	NUM
ejpam-4936	239	9	,	,	PUNCT
ejpam-4936	239	10	there	there	PRON
ejpam-4936	239	11	is	be	VERB
ejpam-4936	239	12	no	no	DET
ejpam-4936	239	13	solution	solution	NOUN
ejpam-4936	239	14	for	for	ADP
ejpam-4936	239	15	nonnegative	nonnegative	ADJ
ejpam-4936	239	16	integer	integer	NOUN
ejpam-4936	239	17	(	(	PUNCT
ejpam-4936	239	18	x	x	NOUN
ejpam-4936	239	19	,	,	PUNCT
ejpam-4936	239	20	y	y	PROPN
ejpam-4936	239	21	,	,	PUNCT
ejpam-4936	239	22	z	z	PROPN
ejpam-4936	239	23	,	,	PUNCT
ejpam-4936	239	24	w	w	PROPN
ejpam-4936	239	25	)	)	PUNCT
ejpam-4936	239	26	because	because	SCONJ
ejpam-4936	239	27	of	of	ADP
ejpam-4936	239	28	2x	2x	NUM
ejpam-4936	239	29	+	+	CCONJ
ejpam-4936	239	30	by	by	ADP
ejpam-4936	239	31	+	+	CCONJ
ejpam-4936	239	32	cz	cz	PROPN
ejpam-4936	239	33	≡	≡	PROPN
ejpam-4936	239	34	2	2	NUM
ejpam-4936	239	35	(	(	PUNCT
ejpam-4936	239	36	mod	mod	NOUN
ejpam-4936	239	37	4	4	NUM
ejpam-4936	239	38	)	)	PUNCT
ejpam-4936	239	39	and	and	CCONJ
ejpam-4936	239	40	w2	w2	PROPN
ejpam-4936	239	41	≡	≡	PROPN
ejpam-4936	239	42	0	0	NUM
ejpam-4936	239	43	,	,	PUNCT
ejpam-4936	239	44	1	1	NUM
ejpam-4936	239	45	(	(	PUNCT
ejpam-4936	239	46	mod	mod	NOUN
ejpam-4936	239	47	4	4	NUM
ejpam-4936	239	48	)	)	PUNCT
ejpam-4936	239	49	.	.	PUNCT
ejpam-4936	240	1	now	now	ADV
ejpam-4936	240	2	,	,	PUNCT
ejpam-4936	240	3	it	it	PRON
ejpam-4936	240	4	remains	remain	VERB
ejpam-4936	240	5	to	to	PART
ejpam-4936	240	6	consider	consider	VERB
ejpam-4936	240	7	only	only	ADV
ejpam-4936	240	8	the	the	DET
ejpam-4936	240	9	case	case	NOUN
ejpam-4936	240	10	x	x	X
ejpam-4936	241	1	=	=	SYM
ejpam-4936	241	2	1	1	X
ejpam-4936	241	3	.	.	PUNCT
ejpam-4936	242	1	the	the	DET
ejpam-4936	242	2	result	result	NOUN
ejpam-4936	242	3	of	of	ADP
ejpam-4936	242	4	2	2	NUM
ejpam-4936	242	5	+	+	CCONJ
ejpam-4936	242	6	by	by	ADP
ejpam-4936	242	7	+	+	CCONJ
ejpam-4936	242	8	cz	cz	PROPN
ejpam-4936	242	9	≡	≡	PROPN
ejpam-4936	242	10	{	{	PUNCT
ejpam-4936	242	11	3	3	NUM
ejpam-4936	242	12	(	(	PUNCT
ejpam-4936	242	13	mod	mod	NOUN
ejpam-4936	242	14	5	5	NUM
ejpam-4936	242	15	)	)	PUNCT
ejpam-4936	242	16	if	if	SCONJ
ejpam-4936	242	17	y	y	PROPN
ejpam-4936	242	18	or	or	CCONJ
ejpam-4936	242	19	z	z	NOUN
ejpam-4936	242	20	are	be	AUX
ejpam-4936	242	21	zero	zero	NUM
ejpam-4936	242	22	,	,	PUNCT
ejpam-4936	242	23	2	2	NUM
ejpam-4936	242	24	(	(	PUNCT
ejpam-4936	242	25	mod	mod	NOUN
ejpam-4936	242	26	5	5	NUM
ejpam-4936	242	27	)	)	PUNCT
ejpam-4936	242	28	if	if	SCONJ
ejpam-4936	242	29	y	y	PROPN
ejpam-4936	242	30	or	or	CCONJ
ejpam-4936	242	31	z	z	NOUN
ejpam-4936	242	32	are	be	AUX
ejpam-4936	242	33	positive	positive	ADJ
ejpam-4936	242	34	,	,	PUNCT
ejpam-4936	242	35	k.	k.	PROPN
ejpam-4936	242	36	laipaporn	laipaporn	PROPN
ejpam-4936	242	37	,	,	PUNCT
ejpam-4936	242	38	s.	s.	PROPN
ejpam-4936	242	39	kaewchay	kaewchay	PROPN
ejpam-4936	242	40	,	,	PUNCT
ejpam-4936	242	41	a.	a.	PROPN
ejpam-4936	242	42	karnbanjong	karnbanjong	PROPN
ejpam-4936	242	43	/	/	SYM
ejpam-4936	242	44	eur	eur	PROPN
ejpam-4936	242	45	.	.	PUNCT
ejpam-4936	243	1	j.	j.	PROPN
ejpam-4936	243	2	pure	pure	PROPN
ejpam-4936	243	3	appl	appl	PROPN
ejpam-4936	243	4	.	.	PROPN
ejpam-4936	243	5	math	math	PROPN
ejpam-4936	243	6	,	,	PUNCT
ejpam-4936	243	7	16	16	NUM
ejpam-4936	243	8	(	(	PUNCT
ejpam-4936	243	9	4	4	NUM
ejpam-4936	243	10	)	)	PUNCT
ejpam-4936	243	11	(	(	PUNCT
ejpam-4936	243	12	2023	2023	NUM
ejpam-4936	243	13	)	)	PUNCT
ejpam-4936	243	14	,	,	PUNCT
ejpam-4936	243	15	2066	2066	NUM
ejpam-4936	243	16	-	-	SYM
ejpam-4936	243	17	2081	2081	NUM
ejpam-4936	243	18	2074	2074	NUM
ejpam-4936	243	19	and	and	CCONJ
ejpam-4936	243	20	w2	w2	PROPN
ejpam-4936	243	21	≡	≡	PROPN
ejpam-4936	243	22	0	0	NUM
ejpam-4936	243	23	,	,	PUNCT
ejpam-4936	243	24	1	1	NUM
ejpam-4936	243	25	,	,	PUNCT
ejpam-4936	243	26	4	4	NUM
ejpam-4936	243	27	(	(	PUNCT
ejpam-4936	243	28	mod	mod	NOUN
ejpam-4936	243	29	5	5	NUM
ejpam-4936	243	30	)	)	PUNCT
ejpam-4936	243	31	,	,	PUNCT
ejpam-4936	243	32	we	we	PRON
ejpam-4936	243	33	conclude	conclude	VERB
ejpam-4936	243	34	that	that	SCONJ
ejpam-4936	243	35	2+by+cz	2+by+cz	NUM
ejpam-4936	243	36	=	=	NOUN
ejpam-4936	243	37	w2	w2	NOUN
ejpam-4936	243	38	has	have	VERB
ejpam-4936	243	39	no	no	DET
ejpam-4936	243	40	solution	solution	NOUN
ejpam-4936	243	41	for	for	ADP
ejpam-4936	243	42	y	y	PROPN
ejpam-4936	243	43	,	,	PUNCT
ejpam-4936	243	44	z	z	PROPN
ejpam-4936	243	45	,	,	PUNCT
ejpam-4936	243	46	w	w	PROPN
ejpam-4936	243	47	∈	∈	PROPN
ejpam-4936	243	48	n0	n0	NUM
ejpam-4936	243	49	,	,	PUNCT
ejpam-4936	243	50	and	and	CCONJ
ejpam-4936	243	51	y	y	PROPN
ejpam-4936	243	52	and	and	CCONJ
ejpam-4936	243	53	z	z	PROPN
ejpam-4936	243	54	are	be	AUX
ejpam-4936	243	55	not	not	PART
ejpam-4936	243	56	all	all	DET
ejpam-4936	243	57	zero	zero	NUM
ejpam-4936	243	58	simultaneously	simultaneously	ADV
ejpam-4936	243	59	.	.	PUNCT
ejpam-4936	244	1	hence	hence	ADV
ejpam-4936	244	2	,	,	PUNCT
ejpam-4936	244	3	2x+by+cz	2x+by+cz	NUM
ejpam-4936	244	4	=	=	PROPN
ejpam-4936	244	5	w2	w2	NOUN
ejpam-4936	244	6	has	have	VERB
ejpam-4936	244	7	only	only	ADV
ejpam-4936	244	8	one	one	NUM
ejpam-4936	244	9	solution	solution	NOUN
ejpam-4936	244	10	(	(	PUNCT
ejpam-4936	244	11	x	x	X
ejpam-4936	244	12	,	,	PUNCT
ejpam-4936	244	13	y	y	PROPN
ejpam-4936	244	14	,	,	PUNCT
ejpam-4936	244	15	z	z	PROPN
ejpam-4936	244	16	,	,	PUNCT
ejpam-4936	244	17	w	w	NOUN
ejpam-4936	244	18	)	)	PUNCT
ejpam-4936	244	19	=	=	SYM
ejpam-4936	244	20	(	(	PUNCT
ejpam-4936	244	21	1	1	NUM
ejpam-4936	244	22	,	,	PUNCT
ejpam-4936	244	23	0	0	NUM
ejpam-4936	244	24	,	,	PUNCT
ejpam-4936	244	25	0	0	NUM
ejpam-4936	244	26	,	,	PUNCT
ejpam-4936	244	27	2	2	NUM
ejpam-4936	244	28	)	)	PUNCT
ejpam-4936	244	29	.	.	PUNCT
ejpam-4936	245	1	corollary	corollary	ADJ
ejpam-4936	245	2	3	3	NUM
ejpam-4936	245	3	.	.	PUNCT
ejpam-4936	246	1	for	for	ADP
ejpam-4936	246	2	any	any	DET
ejpam-4936	246	3	nonnegative	nonnegative	ADJ
ejpam-4936	246	4	integers	integer	NOUN
ejpam-4936	246	5	a	a	DET
ejpam-4936	246	6	,	,	PUNCT
ejpam-4936	246	7	b	b	NOUN
ejpam-4936	246	8	,	,	PUNCT
ejpam-4936	246	9	c	c	NOUN
ejpam-4936	246	10	,	,	PUNCT
ejpam-4936	246	11	x	x	NOUN
ejpam-4936	246	12	,	,	PUNCT
ejpam-4936	246	13	y	y	PROPN
ejpam-4936	246	14	,	,	PUNCT
ejpam-4936	246	15	z	z	PROPN
ejpam-4936	246	16	and	and	CCONJ
ejpam-4936	246	17	w	w	PROPN
ejpam-4936	246	18	,	,	PUNCT
ejpam-4936	246	19	if	if	SCONJ
ejpam-4936	246	20	a	a	DET
ejpam-4936	246	21	+	+	NOUN
ejpam-4936	246	22	2	2	NUM
ejpam-4936	246	23	is	be	AUX
ejpam-4936	246	24	a	a	DET
ejpam-4936	246	25	perfect	perfect	ADJ
ejpam-4936	246	26	square	square	ADJ
ejpam-4936	246	27	number	number	NOUN
ejpam-4936	246	28	,	,	PUNCT
ejpam-4936	246	29	such	such	ADJ
ejpam-4936	246	30	that	that	SCONJ
ejpam-4936	246	31	a	a	DET
ejpam-4936	246	32	≡	≡	PROPN
ejpam-4936	246	33	2	2	NUM
ejpam-4936	246	34	(	(	PUNCT
ejpam-4936	246	35	mod	mod	NOUN
ejpam-4936	246	36	20	20	NUM
ejpam-4936	246	37	)	)	PUNCT
ejpam-4936	246	38	and	and	CCONJ
ejpam-4936	246	39	b	b	NOUN
ejpam-4936	246	40	,	,	PUNCT
ejpam-4936	246	41	c	c	PROPN
ejpam-4936	246	42	≡	≡	PROPN
ejpam-4936	246	43	5	5	NUM
ejpam-4936	246	44	(	(	PUNCT
ejpam-4936	246	45	mod	mod	PROPN
ejpam-4936	246	46	20	20	NUM
ejpam-4936	246	47	)	)	PUNCT
ejpam-4936	246	48	,	,	PUNCT
ejpam-4936	246	49	then	then	ADV
ejpam-4936	246	50	the	the	DET
ejpam-4936	246	51	solutions	solution	NOUN
ejpam-4936	246	52	of	of	ADP
ejpam-4936	246	53	the	the	DET
ejpam-4936	246	54	diophantine	diophantine	NOUN
ejpam-4936	246	55	equation	equation	NOUN
ejpam-4936	246	56	ax	ax	NOUN
ejpam-4936	246	57	+	+	CCONJ
ejpam-4936	246	58	by	by	ADP
ejpam-4936	246	59	+	+	CCONJ
ejpam-4936	246	60	cz	cz	NOUN
ejpam-4936	246	61	=	=	NOUN
ejpam-4936	246	62	w2	w2	NOUN
ejpam-4936	246	63	are	be	AUX
ejpam-4936	246	64	(	(	PUNCT
ejpam-4936	246	65	x	x	NOUN
ejpam-4936	246	66	,	,	PUNCT
ejpam-4936	246	67	y	y	PROPN
ejpam-4936	246	68	,	,	PUNCT
ejpam-4936	246	69	z	z	PROPN
ejpam-4936	246	70	,	,	PUNCT
ejpam-4936	246	71	w	w	NOUN
ejpam-4936	246	72	)	)	PUNCT
ejpam-4936	246	73	=	=	SYM
ejpam-4936	246	74	(	(	PUNCT
ejpam-4936	246	75	1	1	NUM
ejpam-4936	246	76	,	,	PUNCT
ejpam-4936	246	77	0	0	NUM
ejpam-4936	246	78	,	,	PUNCT
ejpam-4936	246	79	0	0	NUM
ejpam-4936	246	80	,	,	PUNCT
ejpam-4936	246	81	√	√	NUM
ejpam-4936	246	82	a+	a+	PUNCT
ejpam-4936	246	83	2	2	NUM
ejpam-4936	246	84	)	)	PUNCT
ejpam-4936	246	85	.	.	PUNCT
ejpam-4936	247	1	proof	proof	NOUN
ejpam-4936	247	2	.	.	PUNCT
ejpam-4936	248	1	the	the	DET
ejpam-4936	248	2	proof	proof	NOUN
ejpam-4936	248	3	of	of	ADP
ejpam-4936	248	4	corollary	corollary	ADJ
ejpam-4936	248	5	3	3	NUM
ejpam-4936	248	6	follows	follow	VERB
ejpam-4936	248	7	from	from	ADP
ejpam-4936	248	8	the	the	DET
ejpam-4936	248	9	same	same	ADJ
ejpam-4936	248	10	manner	manner	NOUN
ejpam-4936	248	11	as	as	ADP
ejpam-4936	248	12	7	7	NUM
ejpam-4936	248	13	.	.	PUNCT
ejpam-4936	248	14	code	code	NOUN
ejpam-4936	248	15	listing	list	VERB
ejpam-4936	248	16	1	1	NUM
ejpam-4936	248	17	:	:	PUNCT
ejpam-4936	248	18	python	python	PROPN
ejpam-4936	248	19	source	source	NOUN
ejpam-4936	248	20	code	code	NOUN
ejpam-4936	248	21	.	.	PUNCT
ejpam-4936	249	1	import	import	NOUN
ejpam-4936	249	2	math	math	PROPN
ejpam-4936	249	3	import	import	NOUN
ejpam-4936	249	4	collections	collection	NOUN
ejpam-4936	249	5	def	def	PROPN
ejpam-4936	249	6	is_perfact_square(n	is_perfact_square(n	PROPN
ejpam-4936	249	7	):	):	PUNCT
ejpam-4936	249	8	i	i	PRON
ejpam-4936	249	9	=	=	NOUN
ejpam-4936	249	10	1	1	NUM
ejpam-4936	249	11	while	while	SCONJ
ejpam-4936	249	12	i<=math.floor(n	i<=math.floor(n	NOUN
ejpam-4936	249	13	*	*	PUNCT
ejpam-4936	249	14	*	*	SYM
ejpam-4936	249	15	0.5	0.5	NUM
ejpam-4936	249	16	):	):	PUNCT
ejpam-4936	249	17	if	if	SCONJ
ejpam-4936	249	18	i*i==n	i*i==n	VERB
ejpam-4936	249	19	:	:	PUNCT
ejpam-4936	249	20	return	return	VERB
ejpam-4936	249	21	true	true	ADJ
ejpam-4936	249	22	i	i	NOUN
ejpam-4936	249	23	=	=	PUNCT
ejpam-4936	249	24	i+1	i+1	PRON
ejpam-4936	249	25	return	return	VERB
ejpam-4936	249	26	false	false	ADJ
ejpam-4936	249	27	def	def	ADJ
ejpam-4936	249	28	check_theorem(a	check_theorem(a	NOUN
ejpam-4936	249	29	,	,	PUNCT
ejpam-4936	249	30	b	b	NOUN
ejpam-4936	249	31	,	,	PUNCT
ejpam-4936	249	32	c	c	NOUN
ejpam-4936	249	33	):	):	PUNCT
ejpam-4936	249	34	theorem_dict	theorem_dict	PROPN
ejpam-4936	249	35	=	=	SYM
ejpam-4936	249	36	{	{	PUNCT
ejpam-4936	249	37	(	(	PUNCT
ejpam-4936	249	38	3,4,7):"theoremm	3,4,7):"theoremm	NUM
ejpam-4936	249	39	1	1	NUM
ejpam-4936	249	40	"	"	PUNCT
ejpam-4936	249	41	,	,	PUNCT
ejpam-4936	249	42	(	(	PUNCT
ejpam-4936	249	43	9,4,7):"corollary	9,4,7):"corollary	NOUN
ejpam-4936	249	44	1	1	NUM
ejpam-4936	249	45	"	"	PUNCT
ejpam-4936	249	46	,	,	PUNCT
ejpam-4936	249	47	(	(	PUNCT
ejpam-4936	249	48	3,16	3,16	X
ejpam-4936	249	49	,	,	PUNCT
ejpam-4936	249	50	7):"corollary	7):"corollary	PROPN
ejpam-4936	249	51	1	1	NUM
ejpam-4936	249	52	"	"	PUNCT
ejpam-4936	249	53	,	,	PUNCT
ejpam-4936	249	54	(	(	PUNCT
ejpam-4936	249	55	9,16	9,16	NUM
ejpam-4936	249	56	,	,	PUNCT
ejpam-4936	249	57	7):"corollary	7):"corollary	NOUN
ejpam-4936	249	58	1	1	NUM
ejpam-4936	249	59	"	"	PUNCT
ejpam-4936	249	60	,	,	PUNCT
ejpam-4936	249	61	(	(	PUNCT
ejpam-4936	249	62	4,6,7):"theoremm	4,6,7):"theoremm	NOUN
ejpam-4936	249	63	2	2	NUM
ejpam-4936	249	64	"	"	PUNCT
ejpam-4936	249	65	,	,	PUNCT
ejpam-4936	249	66	(	(	PUNCT
ejpam-4936	249	67	16	16	NUM
ejpam-4936	249	68	,	,	PUNCT
ejpam-4936	249	69	6,7):"corollary	6,7):"corollary	NUM
ejpam-4936	249	70	2	2	NUM
ejpam-4936	249	71	"	"	PUNCT
ejpam-4936	249	72	,	,	PUNCT
ejpam-4936	249	73	(	(	PUNCT
ejpam-4936	249	74	9,10	9,10	NUM
ejpam-4936	249	75	,	,	PUNCT
ejpam-4936	249	76	7):"theoremm	7):"theoremm	PROPN
ejpam-4936	249	77	3	3	NUM
ejpam-4936	249	78	"	"	PUNCT
ejpam-4936	249	79	,	,	PUNCT
ejpam-4936	249	80	(	(	PUNCT
ejpam-4936	249	81	6,10	6,10	NUM
ejpam-4936	249	82	,	,	PUNCT
ejpam-4936	249	83	7):"theoremm	7):"theoremm	NUM
ejpam-4936	249	84	4	4	NUM
ejpam-4936	249	85	"	"	PUNCT
ejpam-4936	249	86	}	}	PUNCT
ejpam-4936	249	87	list_of_thm	list_of_thm	PROPN
ejpam-4936	249	88	=	=	PUNCT
ejpam-4936	250	1	[	[	X
ejpam-4936	250	2	]	]	X
ejpam-4936	250	3	if	if	SCONJ
ejpam-4936	250	4	(	(	PUNCT
ejpam-4936	250	5	a	a	DET
ejpam-4936	250	6	,	,	PUNCT
ejpam-4936	250	7	b	b	NOUN
ejpam-4936	250	8	,	,	PUNCT
ejpam-4936	250	9	c	c	NOUN
ejpam-4936	250	10	)	)	PUNCT
ejpam-4936	250	11	in	in	ADP
ejpam-4936	250	12	theorem_dict.keys	theorem_dict.key	NOUN
ejpam-4936	250	13	(	(	PUNCT
ejpam-4936	250	14	):	):	PUNCT
ejpam-4936	250	15	list_of_thm.append(theorem_dict[(a	list_of_thm.append(theorem_dict[(a	PROPN
ejpam-4936	250	16	,	,	PUNCT
ejpam-4936	250	17	b	b	PROPN
ejpam-4936	250	18	,	,	PUNCT
ejpam-4936	250	19	c	c	NOUN
ejpam-4936	250	20	)	)	PUNCT
ejpam-4936	250	21	]	]	PUNCT
ejpam-4936	250	22	)	)	PUNCT
ejpam-4936	251	1	if	if	SCONJ
ejpam-4936	251	2	b%4==1	b%4==1	PRON
ejpam-4936	251	3	and	and	CCONJ
ejpam-4936	251	4	c%4==1	c%4==1	PROPN
ejpam-4936	251	5	:	:	PUNCT
ejpam-4936	251	6	if	if	SCONJ
ejpam-4936	251	7	a%4==1	a%4==1	NOUN
ejpam-4936	251	8	:	:	PUNCT
ejpam-4936	251	9	list_of_thm.append("theoremm	list_of_thm.append("theoremm	NOUN
ejpam-4936	251	10	5(i	5(i	NOUN
ejpam-4936	251	11	)	)	PUNCT
ejpam-4936	251	12	"	"	PUNCT
ejpam-4936	251	13	)	)	PUNCT
ejpam-4936	251	14	elif	elif	PROPN
ejpam-4936	251	15	a%4==0	a%4==0	ADJ
ejpam-4936	251	16	:	:	PUNCT
ejpam-4936	251	17	list_of_thm.append("theoremm	list_of_thm.append("theoremm	NOUN
ejpam-4936	251	18	5(iii	5(iii	NUM
ejpam-4936	251	19	)	)	PUNCT
ejpam-4936	251	20	"	"	PUNCT
ejpam-4936	251	21	)	)	PUNCT
ejpam-4936	251	22	if	if	SCONJ
ejpam-4936	251	23	b%5==1	b%5==1	NOUN
ejpam-4936	251	24	and	and	CCONJ
ejpam-4936	251	25	c%5==1	c%5==1	NOUN
ejpam-4936	251	26	:	:	PUNCT
ejpam-4936	251	27	if	if	SCONJ
ejpam-4936	251	28	a%5==1	a%5==1	NOUN
ejpam-4936	251	29	:	:	PUNCT
ejpam-4936	251	30	list_of_thm.append("theoremm	list_of_thm.append("theoremm	PROPN
ejpam-4936	251	31	5(ii	5(ii	NUM
ejpam-4936	251	32	)	)	PUNCT
ejpam-4936	251	33	"	"	PUNCT
ejpam-4936	251	34	)	)	PUNCT
ejpam-4936	252	1	elif	elif	PROPN
ejpam-4936	252	2	a%5==0	a%5==0	ADJ
ejpam-4936	252	3	:	:	PUNCT
ejpam-4936	252	4	list_of_thm.append("theoremm	list_of_thm.append("theoremm	NOUN
ejpam-4936	252	5	5(iv	5(iv	NUM
ejpam-4936	252	6	)	)	PUNCT
ejpam-4936	252	7	"	"	PUNCT
ejpam-4936	252	8	)	)	PUNCT
ejpam-4936	253	1	if	if	SCONJ
ejpam-4936	253	2	a%8==3	a%8==3	PRON
ejpam-4936	253	3	and	and	CCONJ
ejpam-4936	253	4	b%8==1	b%8==1	PRON
ejpam-4936	253	5	and	and	CCONJ
ejpam-4936	253	6	c%8==1	c%8==1	NOUN
ejpam-4936	253	7	:	:	PUNCT
ejpam-4936	253	8	list_of_thm.append("theoremm	list_of_thm.append("theoremm	NOUN
ejpam-4936	253	9	5(v	5(v	NUM
ejpam-4936	253	10	)	)	PUNCT
ejpam-4936	253	11	"	"	PUNCT
ejpam-4936	253	12	)	)	PUNCT
ejpam-4936	253	13	if	if	SCONJ
ejpam-4936	253	14	a%8==1	a%8==1	PROPN
ejpam-4936	253	15	and	and	CCONJ
ejpam-4936	253	16	b%8==3	b%8==3	PROPN
ejpam-4936	253	17	and	and	CCONJ
ejpam-4936	253	18	c%8==3	c%8==3	PROPN
ejpam-4936	253	19	:	:	PUNCT
ejpam-4936	253	20	list_of_thm.append("theoremm	list_of_thm.append("theoremm	PROPN
ejpam-4936	253	21	5(vi	5(vi	NUM
ejpam-4936	253	22	)	)	PUNCT
ejpam-4936	253	23	"	"	PUNCT
ejpam-4936	253	24	)	)	PUNCT
ejpam-4936	253	25	if	if	SCONJ
ejpam-4936	253	26	a==2	a==2	PROPN
ejpam-4936	253	27	and	and	CCONJ
ejpam-4936	253	28	b%20==5	b%20==5	PROPN
ejpam-4936	253	29	and	and	CCONJ
ejpam-4936	253	30	c%20==5	c%20==5	PROPN
ejpam-4936	253	31	:	:	PUNCT
ejpam-4936	253	32	list_of_thm.append("theoremm	list_of_thm.append("theoremm	NOUN
ejpam-4936	253	33	7	7	NUM
ejpam-4936	253	34	"	"	PUNCT
ejpam-4936	253	35	)	)	PUNCT
ejpam-4936	253	36	if	if	SCONJ
ejpam-4936	253	37	is_perfact_square(a+2	is_perfact_square(a+2	PROPN
ejpam-4936	253	38	):	):	PUNCT
ejpam-4936	253	39	if	if	SCONJ
ejpam-4936	253	40	a%36==2	a%36==2	PROPN
ejpam-4936	253	41	and	and	CCONJ
ejpam-4936	253	42	b%36==9	b%36==9	PROPN
ejpam-4936	253	43	and	and	CCONJ
ejpam-4936	253	44	c%36==9	c%36==9	ADJ
ejpam-4936	253	45	:	:	PUNCT
ejpam-4936	253	46	list_of_thm.append("ttheoremm	list_of_thm.append("ttheoremm	PROPN
ejpam-4936	253	47	6	6	NUM
ejpam-4936	253	48	"	"	PUNCT
ejpam-4936	253	49	)	)	PUNCT
ejpam-4936	253	50	if	if	SCONJ
ejpam-4936	253	51	a%20==2	a%20==2	PROPN
ejpam-4936	253	52	and	and	CCONJ
ejpam-4936	253	53	b%20==5	b%20==5	PROPN
ejpam-4936	253	54	and	and	CCONJ
ejpam-4936	253	55	c%20==5	c%20==5	PROPN
ejpam-4936	253	56	:	:	PUNCT
ejpam-4936	253	57	list_of_thm.append("corollary	list_of_thm.append("corollary	ADJ
ejpam-4936	253	58	3	3	NUM
ejpam-4936	253	59	"	"	PUNCT
ejpam-4936	253	60	)	)	PUNCT
ejpam-4936	253	61	return	return	VERB
ejpam-4936	253	62	list_of_thm	list_of_thm	PROPN
ejpam-4936	253	63	if	if	SCONJ
ejpam-4936	253	64	_	_	PUNCT
ejpam-4936	253	65	_	_	PUNCT
ejpam-4936	253	66	name	name	VERB
ejpam-4936	253	67	_	_	PUNCT
ejpam-4936	253	68	_	_	PUNCT
ejpam-4936	254	1	=	=	PUNCT
ejpam-4936	254	2	=	=	PUNCT
ejpam-4936	254	3	"	"	PUNCT
ejpam-4936	254	4	_	_	PUNCT
ejpam-4936	254	5	_	_	PUNCT
ejpam-4936	255	1	main	main	ADJ
ejpam-4936	255	2	_	_	PUNCT
ejpam-4936	255	3	_	_	PUNCT
ejpam-4936	255	4	"	"	PUNCT
ejpam-4936	255	5	:	:	PUNCT
ejpam-4936	255	6	my_dict	my_dict	NOUN
ejpam-4936	255	7	=	=	PUNCT
ejpam-4936	255	8	dict	dict	PROPN
ejpam-4936	255	9	(	(	PUNCT
ejpam-4936	255	10	)	)	PUNCT
ejpam-4936	255	11	base_min	base_min	NUM
ejpam-4936	255	12	,	,	PUNCT
ejpam-4936	255	13	base_max	base_max	NUM
ejpam-4936	255	14	=	=	SYM
ejpam-4936	255	15	2	2	NUM
ejpam-4936	255	16	,	,	PUNCT
ejpam-4936	255	17	20	20	NUM
ejpam-4936	255	18	k.	k.	NOUN
ejpam-4936	255	19	laipaporn	laipaporn	PROPN
ejpam-4936	255	20	,	,	PUNCT
ejpam-4936	255	21	s.	s.	PROPN
ejpam-4936	255	22	kaewchay	kaewchay	PROPN
ejpam-4936	255	23	,	,	PUNCT
ejpam-4936	255	24	a.	a.	PROPN
ejpam-4936	255	25	karnbanjong	karnbanjong	PROPN
ejpam-4936	255	26	/	/	SYM
ejpam-4936	255	27	eur	eur	PROPN
ejpam-4936	255	28	.	.	PUNCT
ejpam-4936	256	1	j.	j.	PROPN
ejpam-4936	256	2	pure	pure	PROPN
ejpam-4936	256	3	appl	appl	PROPN
ejpam-4936	256	4	.	.	PROPN
ejpam-4936	256	5	math	math	PROPN
ejpam-4936	256	6	,	,	PUNCT
ejpam-4936	256	7	16	16	NUM
ejpam-4936	256	8	(	(	PUNCT
ejpam-4936	256	9	4	4	NUM
ejpam-4936	256	10	)	)	PUNCT
ejpam-4936	256	11	(	(	PUNCT
ejpam-4936	256	12	2023	2023	NUM
ejpam-4936	256	13	)	)	PUNCT
ejpam-4936	256	14	,	,	PUNCT
ejpam-4936	256	15	2066	2066	NUM
ejpam-4936	256	16	-	-	SYM
ejpam-4936	256	17	2081	2081	NUM
ejpam-4936	256	18	2075	2075	NUM
ejpam-4936	256	19	for	for	ADP
ejpam-4936	256	20	a	a	PRON
ejpam-4936	256	21	in	in	ADP
ejpam-4936	256	22	range(base_min	range(base_min	NOUN
ejpam-4936	256	23	,	,	PUNCT
ejpam-4936	256	24	base_max	base_max	NUM
ejpam-4936	256	25	+	+	CCONJ
ejpam-4936	256	26	1	1	NUM
ejpam-4936	256	27	):	):	PUNCT
ejpam-4936	256	28	for	for	ADP
ejpam-4936	256	29	b	b	PROPN
ejpam-4936	256	30	in	in	ADP
ejpam-4936	256	31	range(base_min	range(base_min	NOUN
ejpam-4936	256	32	,	,	PUNCT
ejpam-4936	256	33	base_max	base_max	NUM
ejpam-4936	256	34	+	+	CCONJ
ejpam-4936	256	35	1	1	NUM
ejpam-4936	256	36	):	):	PUNCT
ejpam-4936	256	37	for	for	ADP
ejpam-4936	256	38	c	c	PROPN
ejpam-4936	256	39	in	in	ADP
ejpam-4936	256	40	range(base_min	range(base_min	NOUN
ejpam-4936	256	41	,	,	PUNCT
ejpam-4936	256	42	base_max	base_max	NUM
ejpam-4936	256	43	+	+	CCONJ
ejpam-4936	256	44	1	1	NUM
ejpam-4936	256	45	):	):	PUNCT
ejpam-4936	256	46	if	if	SCONJ
ejpam-4936	256	47	len(check_theorem(a	len(check_theorem(a	PROPN
ejpam-4936	256	48	,	,	PUNCT
ejpam-4936	256	49	b	b	NOUN
ejpam-4936	256	50	,	,	PUNCT
ejpam-4936	256	51	c))>0	c))>0	NOUN
ejpam-4936	256	52	:	:	PUNCT
ejpam-4936	256	53	l	l	X
ejpam-4936	257	1	=	=	PUNCT
ejpam-4936	258	1	[	[	X
ejpam-4936	258	2	a	a	PRON
ejpam-4936	258	3	,	,	PUNCT
ejpam-4936	258	4	b	b	NOUN
ejpam-4936	258	5	,	,	PUNCT
ejpam-4936	258	6	c	c	NOUN
ejpam-4936	258	7	]	]	X
ejpam-4936	258	8	l.sort	l.sort	NOUN
ejpam-4936	258	9	(	(	PUNCT
ejpam-4936	258	10	)	)	PUNCT
ejpam-4936	258	11	my_dict[tuple(l	my_dict[tuple(l	PROPN
ejpam-4936	258	12	)	)	PUNCT
ejpam-4936	258	13	]	]	PUNCT
ejpam-4936	259	1	=	=	SYM
ejpam-4936	259	2	check_theorem(a	check_theorem(a	PROPN
ejpam-4936	259	3	,	,	PUNCT
ejpam-4936	259	4	b	b	NOUN
ejpam-4936	259	5	,	,	PUNCT
ejpam-4936	259	6	c	c	NOUN
ejpam-4936	259	7	)	)	PUNCT
ejpam-4936	259	8	ordered_my_dict	ordered_my_dict	PROPN
ejpam-4936	259	9	=	=	PUNCT
ejpam-4936	259	10	collections.ordereddict(sorted(my_dict.items	collections.ordereddict(sorted(my_dict.item	NOUN
ejpam-4936	259	11	(	(	PUNCT
ejpam-4936	259	12	)	)	PUNCT
ejpam-4936	259	13	)	)	PUNCT
ejpam-4936	259	14	)	)	PUNCT
ejpam-4936	260	1	print(f"no.\	print(f"no.\	PROPN
ejpam-4936	260	2	tcase\treference	tcase\treference	NOUN
ejpam-4936	260	3	of	of	ADP
ejpam-4936	260	4	theorem	theorem	NOUN
ejpam-4936	260	5	"	"	PUNCT
ejpam-4936	260	6	)	)	PUNCT
ejpam-4936	260	7	k	k	X
ejpam-4936	261	1	=	=	SYM
ejpam-4936	261	2	1	1	NUM
ejpam-4936	261	3	for	for	ADP
ejpam-4936	261	4	abc	abc	PROPN
ejpam-4936	261	5	in	in	ADP
ejpam-4936	261	6	ordered_my_dict	ordered_my_dict	PROPN
ejpam-4936	261	7	:	:	PUNCT
ejpam-4936	261	8	print(f"{k}\t{abc}\t	print(f"{k}\t{abc}\t	X
ejpam-4936	261	9	{	{	PUNCT
ejpam-4936	261	10	’	'	PUNCT
ejpam-4936	261	11	,	,	PUNCT
ejpam-4936	261	12	’	'	PUNCT
ejpam-4936	261	13	.join(ordered_my_dict[abc	.join(ordered_my_dict[abc	PROPN
ejpam-4936	261	14	]	]	X
ejpam-4936	261	15	)	)	PUNCT
ejpam-4936	261	16	}	}	PUNCT
ejpam-4936	261	17	"	"	PUNCT
ejpam-4936	261	18	)	)	PUNCT
ejpam-4936	262	1	k	k	X
ejpam-4936	262	2	=	=	NOUN
ejpam-4936	262	3	k+1	k+1	PRON
ejpam-4936	262	4	2.3	2.3	NUM
ejpam-4936	262	5	.	.	PUNCT
ejpam-4936	263	1	conclusion	conclusion	NOUN
ejpam-4936	263	2	in	in	ADP
ejpam-4936	263	3	summary	summary	NOUN
ejpam-4936	263	4	,	,	PUNCT
ejpam-4936	263	5	this	this	DET
ejpam-4936	263	6	article	article	NOUN
ejpam-4936	263	7	began	begin	VERB
ejpam-4936	263	8	by	by	ADP
ejpam-4936	263	9	extending	extend	VERB
ejpam-4936	263	10	the	the	DET
ejpam-4936	263	11	number	number	NOUN
ejpam-4936	263	12	of	of	ADP
ejpam-4936	263	13	exponential	exponential	ADJ
ejpam-4936	263	14	bases	basis	NOUN
ejpam-4936	263	15	in	in	ADP
ejpam-4936	263	16	diophantine	diophantine	NOUN
ejpam-4936	263	17	equations	equation	NOUN
ejpam-4936	263	18	from	from	ADP
ejpam-4936	263	19	two	two	NUM
ejpam-4936	263	20	—	—	PUNCT
ejpam-4936	263	21	represented	represent	VERB
ejpam-4936	263	22	as	as	ADP
ejpam-4936	263	23	ax	ax	NOUN
ejpam-4936	263	24	+	+	CCONJ
ejpam-4936	263	25	by	by	ADP
ejpam-4936	263	26	=	=	PROPN
ejpam-4936	263	27	z2	z2	PROPN
ejpam-4936	263	28	—	—	PUNCT
ejpam-4936	263	29	to	to	ADP
ejpam-4936	263	30	three	three	NUM
ejpam-4936	263	31	,	,	PUNCT
ejpam-4936	263	32	represented	represent	VERB
ejpam-4936	263	33	as	as	ADP
ejpam-4936	263	34	ax	ax	NOUN
ejpam-4936	263	35	+	+	CCONJ
ejpam-4936	263	36	by	by	ADP
ejpam-4936	263	37	+	+	CCONJ
ejpam-4936	263	38	cz	cz	NOUN
ejpam-4936	263	39	=	=	NOUN
ejpam-4936	263	40	w2	w2	NOUN
ejpam-4936	263	41	.	.	PUNCT
ejpam-4936	264	1	most	most	ADJ
ejpam-4936	264	2	of	of	ADP
ejpam-4936	264	3	our	our	PRON
ejpam-4936	264	4	results	result	NOUN
ejpam-4936	264	5	focus	focus	VERB
ejpam-4936	264	6	on	on	ADP
ejpam-4936	264	7	situations	situation	NOUN
ejpam-4936	264	8	where	where	SCONJ
ejpam-4936	264	9	the	the	DET
ejpam-4936	264	10	equation	equation	NOUN
ejpam-4936	264	11	ax	ax	NOUN
ejpam-4936	264	12	+	+	CCONJ
ejpam-4936	264	13	by	by	ADP
ejpam-4936	264	14	+	+	CCONJ
ejpam-4936	264	15	cz	cz	NOUN
ejpam-4936	264	16	=	=	NOUN
ejpam-4936	264	17	w2	w2	NOUN
ejpam-4936	264	18	has	have	VERB
ejpam-4936	264	19	no	no	DET
ejpam-4936	264	20	solution	solution	NOUN
ejpam-4936	264	21	.	.	PUNCT
ejpam-4936	265	1	some	some	PRON
ejpam-4936	265	2	of	of	ADP
ejpam-4936	265	3	these	these	DET
ejpam-4936	265	4	cases	case	NOUN
ejpam-4936	265	5	relate	relate	VERB
ejpam-4936	265	6	to	to	ADP
ejpam-4936	265	7	conditions	condition	NOUN
ejpam-4936	265	8	where	where	SCONJ
ejpam-4936	265	9	(	(	PUNCT
ejpam-4936	265	10	a	a	DET
ejpam-4936	265	11	,	,	PUNCT
ejpam-4936	265	12	b	b	NOUN
ejpam-4936	265	13	)	)	PUNCT
ejpam-4936	265	14	is	be	AUX
ejpam-4936	265	15	in	in	ADP
ejpam-4936	265	16	the	the	DET
ejpam-4936	265	17	set	set	NOUN
ejpam-4936	265	18	containing	contain	VERB
ejpam-4936	265	19	pairs	pair	NOUN
ejpam-4936	265	20	like	like	ADP
ejpam-4936	265	21	(	(	PUNCT
ejpam-4936	265	22	3	3	NUM
ejpam-4936	265	23	,	,	PUNCT
ejpam-4936	265	24	4	4	NUM
ejpam-4936	265	25	)	)	PUNCT
ejpam-4936	265	26	,	,	PUNCT
ejpam-4936	265	27	(	(	PUNCT
ejpam-4936	265	28	9	9	NUM
ejpam-4936	265	29	,	,	PUNCT
ejpam-4936	265	30	4	4	NUM
ejpam-4936	265	31	)	)	PUNCT
ejpam-4936	265	32	,	,	PUNCT
ejpam-4936	265	33	(	(	PUNCT
ejpam-4936	265	34	3	3	NUM
ejpam-4936	265	35	,	,	PUNCT
ejpam-4936	265	36	16	16	NUM
ejpam-4936	265	37	)	)	PUNCT
ejpam-4936	265	38	,	,	PUNCT
ejpam-4936	265	39	(	(	PUNCT
ejpam-4936	265	40	9	9	NUM
ejpam-4936	265	41	,	,	PUNCT
ejpam-4936	265	42	16	16	NUM
ejpam-4936	265	43	)	)	PUNCT
ejpam-4936	265	44	,	,	PUNCT
ejpam-4936	265	45	(	(	PUNCT
ejpam-4936	265	46	4	4	NUM
ejpam-4936	265	47	,	,	PUNCT
ejpam-4936	265	48	6	6	NUM
ejpam-4936	265	49	)	)	PUNCT
ejpam-4936	265	50	,	,	PUNCT
ejpam-4936	265	51	(	(	PUNCT
ejpam-4936	265	52	16	16	NUM
ejpam-4936	265	53	,	,	PUNCT
ejpam-4936	265	54	6	6	NUM
ejpam-4936	265	55	)	)	PUNCT
ejpam-4936	265	56	,	,	PUNCT
ejpam-4936	265	57	(	(	PUNCT
ejpam-4936	265	58	9	9	NUM
ejpam-4936	265	59	,	,	PUNCT
ejpam-4936	265	60	10	10	NUM
ejpam-4936	265	61	)	)	PUNCT
ejpam-4936	265	62	,	,	PUNCT
ejpam-4936	265	63	and	and	CCONJ
ejpam-4936	265	64	(	(	PUNCT
ejpam-4936	265	65	6	6	NUM
ejpam-4936	265	66	,	,	PUNCT
ejpam-4936	265	67	10	10	NUM
ejpam-4936	265	68	)	)	PUNCT
ejpam-4936	265	69	,	,	PUNCT
ejpam-4936	265	70	along	along	ADP
ejpam-4936	265	71	with	with	ADP
ejpam-4936	265	72	c	c	NOUN
ejpam-4936	265	73	being	be	AUX
ejpam-4936	265	74	equal	equal	ADJ
ejpam-4936	265	75	to	to	ADP
ejpam-4936	265	76	7	7	NUM
ejpam-4936	265	77	.	.	PUNCT
ejpam-4936	266	1	other	other	ADJ
ejpam-4936	266	2	results	result	NOUN
ejpam-4936	266	3	are	be	AUX
ejpam-4936	266	4	categorized	categorize	VERB
ejpam-4936	266	5	as	as	SCONJ
ejpam-4936	266	6	follows	follow	VERB
ejpam-4936	266	7	:	:	PUNCT
ejpam-4936	266	8	(	(	PUNCT
ejpam-4936	266	9	i	i	NOUN
ejpam-4936	266	10	)	)	PUNCT
ejpam-4936	266	11	a	a	DET
ejpam-4936	266	12	,	,	PUNCT
ejpam-4936	266	13	b	b	NOUN
ejpam-4936	266	14	,	,	PUNCT
ejpam-4936	266	15	c	c	PROPN
ejpam-4936	266	16	≡	≡	PROPN
ejpam-4936	266	17	1	1	NUM
ejpam-4936	266	18	(	(	PUNCT
ejpam-4936	266	19	mod	mod	NOUN
ejpam-4936	266	20	4	4	NUM
ejpam-4936	266	21	)	)	PUNCT
ejpam-4936	266	22	.	.	PUNCT
ejpam-4936	267	1	(	(	PUNCT
ejpam-4936	267	2	ii	ii	NOUN
ejpam-4936	267	3	)	)	PUNCT
ejpam-4936	267	4	a	a	DET
ejpam-4936	267	5	,	,	PUNCT
ejpam-4936	267	6	b	b	NOUN
ejpam-4936	267	7	,	,	PUNCT
ejpam-4936	267	8	c	c	PROPN
ejpam-4936	267	9	≡	≡	PROPN
ejpam-4936	267	10	1	1	NUM
ejpam-4936	267	11	(	(	PUNCT
ejpam-4936	267	12	mod	mod	NOUN
ejpam-4936	267	13	5	5	NUM
ejpam-4936	267	14	)	)	PUNCT
ejpam-4936	267	15	.	.	PUNCT
ejpam-4936	268	1	(	(	PUNCT
ejpam-4936	268	2	iii	iii	X
ejpam-4936	268	3	)	)	PUNCT
ejpam-4936	268	4	a	a	DET
ejpam-4936	268	5	≡	≡	PROPN
ejpam-4936	268	6	0	0	PUNCT
ejpam-4936	269	1	(	(	PUNCT
ejpam-4936	269	2	mod	mod	NOUN
ejpam-4936	269	3	4	4	NUM
ejpam-4936	269	4	)	)	PUNCT
ejpam-4936	269	5	and	and	CCONJ
ejpam-4936	269	6	b	b	NOUN
ejpam-4936	269	7	,	,	PUNCT
ejpam-4936	269	8	c	c	PROPN
ejpam-4936	269	9	≡	≡	PROPN
ejpam-4936	269	10	1	1	NUM
ejpam-4936	269	11	(	(	PUNCT
ejpam-4936	269	12	mod	mod	NOUN
ejpam-4936	269	13	4	4	NUM
ejpam-4936	269	14	)	)	PUNCT
ejpam-4936	269	15	.	.	PUNCT
ejpam-4936	270	1	(	(	PUNCT
ejpam-4936	270	2	iv	iv	X
ejpam-4936	270	3	)	)	PUNCT
ejpam-4936	270	4	a	a	DET
ejpam-4936	270	5	≡	≡	PROPN
ejpam-4936	270	6	0	0	PUNCT
ejpam-4936	270	7	(	(	PUNCT
ejpam-4936	270	8	mod	mod	NOUN
ejpam-4936	270	9	5	5	NUM
ejpam-4936	270	10	)	)	PUNCT
ejpam-4936	270	11	and	and	CCONJ
ejpam-4936	270	12	b	b	NOUN
ejpam-4936	270	13	,	,	PUNCT
ejpam-4936	270	14	c	c	PROPN
ejpam-4936	270	15	≡	≡	PROPN
ejpam-4936	270	16	1	1	NUM
ejpam-4936	270	17	(	(	PUNCT
ejpam-4936	270	18	mod	mod	NOUN
ejpam-4936	270	19	5	5	NUM
ejpam-4936	270	20	)	)	PUNCT
ejpam-4936	270	21	.	.	PUNCT
ejpam-4936	271	1	(	(	PUNCT
ejpam-4936	271	2	v	v	NOUN
ejpam-4936	271	3	)	)	PUNCT
ejpam-4936	271	4	a	a	DET
ejpam-4936	271	5	≡	≡	PROPN
ejpam-4936	271	6	3	3	NUM
ejpam-4936	271	7	(	(	PUNCT
ejpam-4936	271	8	mod	mod	PROPN
ejpam-4936	271	9	8)	8)	NUM
ejpam-4936	271	10	and	and	CCONJ
ejpam-4936	271	11	b	b	NOUN
ejpam-4936	271	12	,	,	PUNCT
ejpam-4936	271	13	c	c	PROPN
ejpam-4936	271	14	≡	≡	PROPN
ejpam-4936	271	15	1	1	NUM
ejpam-4936	271	16	(	(	PUNCT
ejpam-4936	271	17	mod	mod	PROPN
ejpam-4936	271	18	8)	8)	NUM
ejpam-4936	271	19	.	.	PUNCT
ejpam-4936	272	1	(	(	PUNCT
ejpam-4936	272	2	vi	vi	NOUN
ejpam-4936	272	3	)	)	PUNCT
ejpam-4936	272	4	a	a	DET
ejpam-4936	272	5	≡	≡	PROPN
ejpam-4936	272	6	1	1	NUM
ejpam-4936	272	7	(	(	PUNCT
ejpam-4936	272	8	mod	mod	PROPN
ejpam-4936	272	9	8)	8)	NUM
ejpam-4936	272	10	and	and	CCONJ
ejpam-4936	272	11	b	b	NOUN
ejpam-4936	272	12	,	,	PUNCT
ejpam-4936	272	13	c	c	PROPN
ejpam-4936	272	14	≡	≡	PROPN
ejpam-4936	272	15	3	3	NUM
ejpam-4936	272	16	(	(	PUNCT
ejpam-4936	272	17	mod	mod	PROPN
ejpam-4936	272	18	8)	8)	NUM
ejpam-4936	272	19	.	.	PUNCT
ejpam-4936	273	1	some	some	DET
ejpam-4936	273	2	equations	equation	NOUN
ejpam-4936	273	3	of	of	ADP
ejpam-4936	273	4	the	the	DET
ejpam-4936	273	5	form	form	NOUN
ejpam-4936	273	6	ax	ax	NOUN
ejpam-4936	273	7	+	+	CCONJ
ejpam-4936	273	8	by	by	ADP
ejpam-4936	273	9	+	+	CCONJ
ejpam-4936	273	10	cz	cz	NOUN
ejpam-4936	273	11	=	=	NOUN
ejpam-4936	273	12	w2	w2	NOUN
ejpam-4936	273	13	do	do	AUX
ejpam-4936	273	14	have	have	VERB
ejpam-4936	273	15	solutions	solution	NOUN
ejpam-4936	273	16	,	,	PUNCT
ejpam-4936	273	17	as	as	SCONJ
ejpam-4936	273	18	discussed	discuss	VERB
ejpam-4936	273	19	in	in	ADP
ejpam-4936	273	20	theorem	theorem	ADJ
ejpam-4936	273	21	6	6	NUM
ejpam-4936	273	22	,	,	PUNCT
ejpam-4936	273	23	7	7	NUM
ejpam-4936	273	24	,	,	PUNCT
ejpam-4936	273	25	and	and	CCONJ
ejpam-4936	273	26	corollary	corollary	ADJ
ejpam-4936	273	27	3	3	NUM
ejpam-4936	273	28	.	.	PUNCT
ejpam-4936	274	1	our	our	PRON
ejpam-4936	274	2	research	research	NOUN
ejpam-4936	274	3	yielded	yield	VERB
ejpam-4936	274	4	135	135	NUM
ejpam-4936	274	5	equations	equation	NOUN
ejpam-4936	274	6	that	that	PRON
ejpam-4936	274	7	have	have	AUX
ejpam-4936	274	8	been	be	AUX
ejpam-4936	274	9	clarified	clarify	VERB
ejpam-4936	274	10	from	from	ADP
ejpam-4936	274	11	a	a	DET
ejpam-4936	274	12	total	total	NOUN
ejpam-4936	274	13	of	of	ADP
ejpam-4936	274	14	1,330	1,330	NUM
ejpam-4936	274	15	equations	equation	NOUN
ejpam-4936	274	16	,	,	PUNCT
ejpam-4936	274	17	assuming	assume	VERB
ejpam-4936	274	18	we	we	PRON
ejpam-4936	274	19	limit	limit	VERB
ejpam-4936	274	20	all	all	DET
ejpam-4936	274	21	variables	variable	NOUN
ejpam-4936	274	22	a	a	DET
ejpam-4936	274	23	,	,	PUNCT
ejpam-4936	274	24	b	b	NOUN
ejpam-4936	274	25	,	,	PUNCT
ejpam-4936	274	26	and	and	CCONJ
ejpam-4936	274	27	c	c	X
ejpam-4936	274	28	to	to	PART
ejpam-4936	274	29	range	range	VERB
ejpam-4936	274	30	from	from	ADP
ejpam-4936	274	31	2	2	NUM
ejpam-4936	274	32	to	to	ADP
ejpam-4936	274	33	20	20	NUM
ejpam-4936	274	34	.	.	PUNCT
ejpam-4936	275	1	we	we	PRON
ejpam-4936	275	2	used	use	VERB
ejpam-4936	275	3	python	python	PROPN
ejpam-4936	275	4	code	code	NOUN
ejpam-4936	275	5	,	,	PUNCT
ejpam-4936	275	6	as	as	SCONJ
ejpam-4936	275	7	shown	show	VERB
ejpam-4936	275	8	in	in	ADP
ejpam-4936	275	9	code	code	NOUN
ejpam-4936	275	10	listing	list	VERB
ejpam-4936	275	11	1	1	NUM
ejpam-4936	275	12	,	,	PUNCT
ejpam-4936	275	13	to	to	PART
ejpam-4936	275	14	count	count	VERB
ejpam-4936	275	15	the	the	DET
ejpam-4936	275	16	number	number	NOUN
ejpam-4936	275	17	of	of	ADP
ejpam-4936	275	18	equations	equation	NOUN
ejpam-4936	275	19	that	that	PRON
ejpam-4936	275	20	satisfy	satisfy	VERB
ejpam-4936	275	21	our	our	PRON
ejpam-4936	275	22	theorems	theorem	NOUN
ejpam-4936	275	23	and	and	CCONJ
ejpam-4936	275	24	corollaries	corollary	NOUN
ejpam-4936	275	25	.	.	PUNCT
ejpam-4936	276	1	specifically	specifically	ADV
ejpam-4936	276	2	,	,	PUNCT
ejpam-4936	276	3	we	we	PRON
ejpam-4936	276	4	offer	offer	VERB
ejpam-4936	276	5	code	code	NOUN
ejpam-4936	276	6	that	that	PRON
ejpam-4936	276	7	counts	count	VERB
ejpam-4936	276	8	all	all	DET
ejpam-4936	276	9	equations	equation	NOUN
ejpam-4936	276	10	of	of	ADP
ejpam-4936	276	11	the	the	DET
ejpam-4936	276	12	form	form	NOUN
ejpam-4936	276	13	ax	ax	NOUN
ejpam-4936	276	14	+	+	CCONJ
ejpam-4936	276	15	by	by	ADP
ejpam-4936	276	16	+	+	CCONJ
ejpam-4936	276	17	cz	cz	NOUN
ejpam-4936	276	18	=	=	NOUN
ejpam-4936	276	19	w2	w2	NOUN
ejpam-4936	276	20	that	that	PRON
ejpam-4936	276	21	are	be	AUX
ejpam-4936	276	22	related	relate	VERB
ejpam-4936	276	23	to	to	ADP
ejpam-4936	276	24	our	our	PRON
ejpam-4936	276	25	results	result	NOUN
ejpam-4936	276	26	in	in	ADP
ejpam-4936	276	27	the	the	DET
ejpam-4936	276	28	theorems	theorem	NOUN
ejpam-4936	276	29	and	and	CCONJ
ejpam-4936	276	30	corollaries	corollary	NOUN
ejpam-4936	276	31	.	.	PUNCT
ejpam-4936	277	1	the	the	DET
ejpam-4936	277	2	accompanying	accompany	VERB
ejpam-4936	277	3	table	table	NOUN
ejpam-4936	277	4	lists	list	VERB
ejpam-4936	277	5	the	the	DET
ejpam-4936	277	6	exponential	exponential	ADJ
ejpam-4936	277	7	diophantine	diophantine	NOUN
ejpam-4936	277	8	equations	equation	NOUN
ejpam-4936	277	9	with	with	ADP
ejpam-4936	277	10	variables	variable	NOUN
ejpam-4936	277	11	a	a	DET
ejpam-4936	277	12	,	,	PUNCT
ejpam-4936	277	13	b	b	NOUN
ejpam-4936	277	14	,	,	PUNCT
ejpam-4936	277	15	and	and	CCONJ
ejpam-4936	277	16	c	c	AUX
ejpam-4936	277	17	limited	limit	VERB
ejpam-4936	277	18	to	to	ADP
ejpam-4936	277	19	the	the	DET
ejpam-4936	277	20	range	range	NOUN
ejpam-4936	277	21	from	from	ADP
ejpam-4936	277	22	2	2	NUM
ejpam-4936	277	23	to	to	ADP
ejpam-4936	277	24	20	20	NUM
ejpam-4936	277	25	.	.	PUNCT
ejpam-4936	278	1	for	for	ADP
ejpam-4936	278	2	future	future	ADJ
ejpam-4936	278	3	work	work	NOUN
ejpam-4936	278	4	,	,	PUNCT
ejpam-4936	278	5	we	we	PRON
ejpam-4936	278	6	aim	aim	VERB
ejpam-4936	278	7	to	to	PART
ejpam-4936	278	8	combine	combine	VERB
ejpam-4936	278	9	this	this	DET
ejpam-4936	278	10	research	research	NOUN
ejpam-4936	278	11	with	with	ADP
ejpam-4936	278	12	a	a	DET
ejpam-4936	278	13	new	new	ADJ
ejpam-4936	278	14	form	form	NOUN
ejpam-4936	278	15	of	of	ADP
ejpam-4936	278	16	diophantine	diophantine	NOUN
ejpam-4936	278	17	equation	equation	NOUN
ejpam-4936	278	18	,	,	PUNCT
ejpam-4936	278	19	as	as	SCONJ
ejpam-4936	278	20	seen	see	VERB
ejpam-4936	278	21	in	in	ADP
ejpam-4936	278	22	reference	reference	NOUN
ejpam-4936	278	23	[	[	X
ejpam-4936	278	24	8	8	NUM
ejpam-4936	278	25	]	]	PUNCT
ejpam-4936	278	26	,	,	PUNCT
ejpam-4936	278	27	or	or	CCONJ
ejpam-4936	278	28	we	we	PRON
ejpam-4936	278	29	will	will	AUX
ejpam-4936	278	30	expand	expand	VERB
ejpam-4936	278	31	our	our	PRON
ejpam-4936	278	32	research	research	NOUN
ejpam-4936	278	33	using	use	VERB
ejpam-4936	278	34	other	other	ADJ
ejpam-4936	278	35	techniques	technique	NOUN
ejpam-4936	278	36	,	,	PUNCT
ejpam-4936	278	37	such	such	ADJ
ejpam-4936	278	38	as	as	ADP
ejpam-4936	278	39	transforming	transform	VERB
ejpam-4936	278	40	the	the	DET
ejpam-4936	278	41	equation	equation	NOUN
ejpam-4936	278	42	into	into	ADP
ejpam-4936	278	43	an	an	DET
ejpam-4936	278	44	elliptic	elliptic	ADJ
ejpam-4936	278	45	curve[12	curve[12	NOUN
ejpam-4936	278	46	]	]	PUNCT
ejpam-4936	278	47	,	,	PUNCT
ejpam-4936	278	48	extending	extend	VERB
ejpam-4936	278	49	into	into	ADP
ejpam-4936	278	50	the	the	DET
ejpam-4936	278	51	quadratic	quadratic	ADJ
ejpam-4936	278	52	field	field	NOUN
ejpam-4936	278	53	[	[	X
ejpam-4936	278	54	see	see	VERB
ejpam-4936	278	55	details	detail	NOUN
ejpam-4936	278	56	11	11	NUM
ejpam-4936	278	57	]	]	PUNCT
ejpam-4936	278	58	,	,	PUNCT
ejpam-4936	278	59	or	or	CCONJ
ejpam-4936	278	60	using	use	VERB
ejpam-4936	278	61	legendre	legendre	PROPN
ejpam-4936	278	62	symbols	symbol	NOUN
ejpam-4936	278	63	with	with	ADP
ejpam-4936	278	64	related	related	ADJ
ejpam-4936	278	65	theorems,[see	theorems,[see	NOUN
ejpam-4936	278	66	also	also	ADV
ejpam-4936	278	67	15	15	NUM
ejpam-4936	278	68	,	,	PUNCT
ejpam-4936	278	69	24	24	NUM
ejpam-4936	278	70	]	]	PUNCT
ejpam-4936	278	71	.	.	PUNCT
ejpam-4936	279	1	k.	k.	PROPN
ejpam-4936	279	2	laipaporn	laipaporn	PROPN
ejpam-4936	279	3	,	,	PUNCT
ejpam-4936	279	4	s.	s.	PROPN
ejpam-4936	279	5	kaewchay	kaewchay	PROPN
ejpam-4936	279	6	,	,	PUNCT
ejpam-4936	279	7	a.	a.	PROPN
ejpam-4936	279	8	karnbanjong	karnbanjong	PROPN
ejpam-4936	279	9	/	/	SYM
ejpam-4936	279	10	eur	eur	PROPN
ejpam-4936	279	11	.	.	PUNCT
ejpam-4936	280	1	j.	j.	PROPN
ejpam-4936	280	2	pure	pure	PROPN
ejpam-4936	280	3	appl	appl	PROPN
ejpam-4936	280	4	.	.	PROPN
ejpam-4936	280	5	math	math	PROPN
ejpam-4936	280	6	,	,	PUNCT
ejpam-4936	280	7	16	16	NUM
ejpam-4936	280	8	(	(	PUNCT
ejpam-4936	280	9	4	4	NUM
ejpam-4936	280	10	)	)	PUNCT
ejpam-4936	280	11	(	(	PUNCT
ejpam-4936	280	12	2023	2023	NUM
ejpam-4936	280	13	)	)	PUNCT
ejpam-4936	280	14	,	,	PUNCT
ejpam-4936	280	15	2066	2066	NUM
ejpam-4936	280	16	-	-	SYM
ejpam-4936	280	17	2081	2081	NUM
ejpam-4936	280	18	2076	2076	NUM
ejpam-4936	280	19	table	table	NOUN
ejpam-4936	280	20	2	2	NUM
ejpam-4936	280	21	:	:	PUNCT
ejpam-4936	280	22	the	the	DET
ejpam-4936	280	23	list	list	NOUN
ejpam-4936	280	24	of	of	ADP
ejpam-4936	280	25	the	the	DET
ejpam-4936	280	26	equations	equation	NOUN
ejpam-4936	280	27	ax	ax	NOUN
ejpam-4936	280	28	+	+	CCONJ
ejpam-4936	280	29	by	by	ADP
ejpam-4936	280	30	+	+	CCONJ
ejpam-4936	280	31	cz	cz	NOUN
ejpam-4936	280	32	=	=	SYM
ejpam-4936	280	33	w2	w2	NOUN
ejpam-4936	280	34	where	where	SCONJ
ejpam-4936	280	35	2	2	NUM
ejpam-4936	280	36	≤	≤	NOUN
ejpam-4936	280	37	a	a	DET
ejpam-4936	280	38	≤	≤	NUM
ejpam-4936	280	39	b	b	NOUN
ejpam-4936	280	40	≤	≤	NUM
ejpam-4936	280	41	c	c	NOUN
ejpam-4936	280	42	≤	≤	NUM
ejpam-4936	280	43	20	20	NUM
ejpam-4936	280	44	and	and	CCONJ
ejpam-4936	280	45	all	all	DET
ejpam-4936	280	46	variables	variable	NOUN
ejpam-4936	280	47	a	a	PRON
ejpam-4936	280	48	,	,	PUNCT
ejpam-4936	280	49	b	b	PROPN
ejpam-4936	280	50	and	and	CCONJ
ejpam-4936	280	51	c	c	PROPN
ejpam-4936	280	52	are	be	AUX
ejpam-4936	280	53	satisfied	satisfied	ADJ
ejpam-4936	280	54	our	our	PRON
ejpam-4936	280	55	results	result	NOUN
ejpam-4936	280	56	.	.	PUNCT
ejpam-4936	281	1	no	no	INTJ
ejpam-4936	281	2	.	.	PUNCT
ejpam-4936	282	1	ax	ax	NOUN
ejpam-4936	282	2	+	+	CCONJ
ejpam-4936	282	3	by	by	ADP
ejpam-4936	282	4	+	+	CCONJ
ejpam-4936	282	5	cz	cz	NOUN
ejpam-4936	282	6	=	=	NOUN
ejpam-4936	282	7	w2	w2	NOUN
ejpam-4936	282	8	result	result	NOUN
ejpam-4936	282	9	ref	ref	NOUN
ejpam-4936	282	10	.	.	PUNCT
ejpam-4936	283	1	of	of	ADP
ejpam-4936	283	2	theorem	theorem	ADJ
ejpam-4936	283	3	1	1	NUM
ejpam-4936	283	4	2x	2x	NUM
ejpam-4936	283	5	+	+	CCONJ
ejpam-4936	283	6	5y	5y	NOUN
ejpam-4936	283	7	+	+	CCONJ
ejpam-4936	283	8	5z	5z	NOUN
ejpam-4936	283	9	=	=	SYM
ejpam-4936	283	10	w2	w2	NOUN
ejpam-4936	283	11	(	(	PUNCT
ejpam-4936	283	12	1	1	NUM
ejpam-4936	283	13	,	,	PUNCT
ejpam-4936	283	14	0	0	NUM
ejpam-4936	283	15	,	,	PUNCT
ejpam-4936	283	16	0	0	NUM
ejpam-4936	283	17	,	,	PUNCT
ejpam-4936	283	18	2	2	NUM
ejpam-4936	283	19	)	)	PUNCT
ejpam-4936	283	20	7	7	NUM
ejpam-4936	283	21	,	,	PUNCT
ejpam-4936	283	22	corollary	corollary	ADJ
ejpam-4936	283	23	3	3	NUM
ejpam-4936	283	24	2	2	NUM
ejpam-4936	283	25	2x	2x	NUM
ejpam-4936	283	26	+	+	CCONJ
ejpam-4936	283	27	9y	9y	NOUN
ejpam-4936	283	28	+	+	CCONJ
ejpam-4936	283	29	9z	9z	NUM
ejpam-4936	283	30	=	=	SYM
ejpam-4936	283	31	w2	w2	NOUN
ejpam-4936	283	32	(	(	PUNCT
ejpam-4936	283	33	1	1	NUM
ejpam-4936	283	34	,	,	PUNCT
ejpam-4936	283	35	0	0	NUM
ejpam-4936	283	36	,	,	PUNCT
ejpam-4936	283	37	0	0	NUM
ejpam-4936	283	38	,	,	PUNCT
ejpam-4936	283	39	2	2	NUM
ejpam-4936	283	40	)	)	PUNCT
ejpam-4936	283	41	6	6	NUM
ejpam-4936	283	42	3	3	NUM
ejpam-4936	283	43	3x	3x	NUM
ejpam-4936	284	1	+	+	CCONJ
ejpam-4936	284	2	3y	3y	NUM
ejpam-4936	284	3	+	+	CCONJ
ejpam-4936	284	4	9z	9z	NUM
ejpam-4936	284	5	=	=	SYM
ejpam-4936	284	6	w2	w2	NOUN
ejpam-4936	284	7	no	no	DET
ejpam-4936	284	8	solution	solution	NOUN
ejpam-4936	284	9	5(vi	5(vi	NUM
ejpam-4936	284	10	)	)	PUNCT
ejpam-4936	284	11	4	4	NUM
ejpam-4936	284	12	3x	3x	NUM
ejpam-4936	285	1	+	+	CCONJ
ejpam-4936	285	2	3y	3y	NUM
ejpam-4936	285	3	+	+	X
ejpam-4936	285	4	17z	17z	NOUN
ejpam-4936	285	5	=	=	SYM
ejpam-4936	285	6	w2	w2	NOUN
ejpam-4936	285	7	no	no	DET
ejpam-4936	285	8	solution	solution	NOUN
ejpam-4936	285	9	5(vi	5(vi	NUM
ejpam-4936	285	10	)	)	PUNCT
ejpam-4936	285	11	5	5	NUM
ejpam-4936	285	12	3x	3x	NUM
ejpam-4936	286	1	+	+	CCONJ
ejpam-4936	286	2	4y	4y	PRON
ejpam-4936	286	3	+	+	CCONJ
ejpam-4936	286	4	7z	7z	NOUN
ejpam-4936	286	5	=	=	SYM
ejpam-4936	286	6	w2	w2	NOUN
ejpam-4936	286	7	no	no	DET
ejpam-4936	286	8	solution	solution	NOUN
ejpam-4936	286	9	1	1	NUM
ejpam-4936	286	10	6	6	NUM
ejpam-4936	286	11	3x	3x	NUM
ejpam-4936	286	12	+	+	CCONJ
ejpam-4936	286	13	7y	7y	NOUN
ejpam-4936	286	14	+	+	NOUN
ejpam-4936	286	15	16z	16z	NOUN
ejpam-4936	286	16	=	=	SYM
ejpam-4936	286	17	w2	w2	PROPN
ejpam-4936	286	18	no	no	DET
ejpam-4936	286	19	solution	solution	NOUN
ejpam-4936	286	20	corollary	corollary	NOUN
ejpam-4936	286	21	1	1	NUM
ejpam-4936	286	22	7	7	NUM
ejpam-4936	286	23	3x	3x	NUM
ejpam-4936	286	24	+	+	CCONJ
ejpam-4936	286	25	9y	9y	NOUN
ejpam-4936	286	26	+	+	CCONJ
ejpam-4936	286	27	9z	9z	NUM
ejpam-4936	286	28	=	=	SYM
ejpam-4936	286	29	w2	w2	NOUN
ejpam-4936	286	30	no	no	DET
ejpam-4936	286	31	solution	solution	NOUN
ejpam-4936	286	32	5(v	5(v	NUM
ejpam-4936	286	33	)	)	PUNCT
ejpam-4936	286	34	8	8	NUM
ejpam-4936	286	35	3x	3x	NUM
ejpam-4936	287	1	+	+	CCONJ
ejpam-4936	287	2	9y	9y	NOUN
ejpam-4936	287	3	+	+	CCONJ
ejpam-4936	287	4	11z	11z	NOUN
ejpam-4936	287	5	=	=	SYM
ejpam-4936	287	6	w2	w2	NOUN
ejpam-4936	287	7	no	no	DET
ejpam-4936	287	8	solution	solution	NOUN
ejpam-4936	287	9	5(vi	5(vi	NUM
ejpam-4936	287	10	)	)	PUNCT
ejpam-4936	287	11	9	9	NUM
ejpam-4936	287	12	3x	3x	NUM
ejpam-4936	288	1	+	+	CCONJ
ejpam-4936	288	2	9y	9y	NOUN
ejpam-4936	288	3	+	+	NUM
ejpam-4936	288	4	17z	17z	NOUN
ejpam-4936	288	5	=	=	SYM
ejpam-4936	288	6	w2	w2	NOUN
ejpam-4936	288	7	no	no	DET
ejpam-4936	288	8	solution	solution	NOUN
ejpam-4936	288	9	5(v	5(v	NUM
ejpam-4936	288	10	)	)	PUNCT
ejpam-4936	288	11	10	10	NUM
ejpam-4936	288	12	3x	3x	NUM
ejpam-4936	289	1	+	+	CCONJ
ejpam-4936	289	2	9y	9y	NOUN
ejpam-4936	289	3	+	+	NUM
ejpam-4936	289	4	19z	19z	NOUN
ejpam-4936	289	5	=	=	SYM
ejpam-4936	289	6	w2	w2	NOUN
ejpam-4936	289	7	no	no	DET
ejpam-4936	289	8	solution	solution	NOUN
ejpam-4936	289	9	5(vi	5(vi	NUM
ejpam-4936	289	10	)	)	PUNCT
ejpam-4936	289	11	11	11	NUM
ejpam-4936	289	12	3x	3x	NUM
ejpam-4936	289	13	+	+	SYM
ejpam-4936	289	14	11y	11y	NOUN
ejpam-4936	289	15	+	+	CCONJ
ejpam-4936	289	16	17z	17z	X
ejpam-4936	289	17	=	=	SYM
ejpam-4936	289	18	w2	w2	NOUN
ejpam-4936	289	19	no	no	DET
ejpam-4936	289	20	solution	solution	NOUN
ejpam-4936	289	21	5(vi	5(vi	NUM
ejpam-4936	289	22	)	)	PUNCT
ejpam-4936	289	23	12	12	NUM
ejpam-4936	289	24	3x	3x	NUM
ejpam-4936	289	25	+	+	NOUN
ejpam-4936	289	26	17y	17y	NOUN
ejpam-4936	289	27	+	+	CCONJ
ejpam-4936	289	28	17z	17z	NOUN
ejpam-4936	289	29	=	=	SYM
ejpam-4936	289	30	w2	w2	NOUN
ejpam-4936	289	31	no	no	DET
ejpam-4936	289	32	solution	solution	NOUN
ejpam-4936	289	33	5(v	5(v	NUM
ejpam-4936	289	34	)	)	PUNCT
ejpam-4936	289	35	13	13	NUM
ejpam-4936	289	36	3x	3x	NUM
ejpam-4936	289	37	+	+	NOUN
ejpam-4936	289	38	17y	17y	NOUN
ejpam-4936	289	39	+	+	NUM
ejpam-4936	289	40	19z	19z	NOUN
ejpam-4936	289	41	=	=	SYM
ejpam-4936	289	42	w2	w2	NOUN
ejpam-4936	289	43	no	no	DET
ejpam-4936	289	44	solution	solution	NOUN
ejpam-4936	289	45	5(vi	5(vi	NUM
ejpam-4936	289	46	)	)	PUNCT
ejpam-4936	289	47	14	14	NUM
ejpam-4936	289	48	4x	4x	NOUN
ejpam-4936	290	1	+	+	NOUN
ejpam-4936	290	2	5y	5y	NOUN
ejpam-4936	290	3	+	+	CCONJ
ejpam-4936	290	4	5z	5z	NOUN
ejpam-4936	290	5	=	=	SYM
ejpam-4936	290	6	w2	w2	NOUN
ejpam-4936	290	7	no	no	DET
ejpam-4936	290	8	solution	solution	NOUN
ejpam-4936	290	9	5(iii	5(iii	NUM
ejpam-4936	290	10	)	)	PUNCT
ejpam-4936	290	11	15	15	NUM
ejpam-4936	290	12	4x	4x	NOUN
ejpam-4936	290	13	+	+	NOUN
ejpam-4936	290	14	5y	5y	NOUN
ejpam-4936	290	15	+	+	CCONJ
ejpam-4936	290	16	9z	9z	NUM
ejpam-4936	290	17	=	=	SYM
ejpam-4936	290	18	w2	w2	NOUN
ejpam-4936	290	19	no	no	DET
ejpam-4936	290	20	solution	solution	NOUN
ejpam-4936	290	21	5(iii	5(iii	NUM
ejpam-4936	290	22	)	)	PUNCT
ejpam-4936	290	23	16	16	NUM
ejpam-4936	290	24	4x	4x	NOUN
ejpam-4936	290	25	+	+	CCONJ
ejpam-4936	290	26	5y	5y	NOUN
ejpam-4936	290	27	+	+	NUM
ejpam-4936	290	28	13z	13z	NOUN
ejpam-4936	290	29	=	=	SYM
ejpam-4936	290	30	w2	w2	NOUN
ejpam-4936	290	31	no	no	DET
ejpam-4936	290	32	solution	solution	NOUN
ejpam-4936	290	33	5(iii	5(iii	NUM
ejpam-4936	290	34	)	)	PUNCT
ejpam-4936	290	35	17	17	NUM
ejpam-4936	290	36	4x	4x	NOUN
ejpam-4936	290	37	+	+	CCONJ
ejpam-4936	290	38	5y	5y	NOUN
ejpam-4936	290	39	+	+	NUM
ejpam-4936	290	40	17z	17z	NOUN
ejpam-4936	290	41	=	=	SYM
ejpam-4936	290	42	w2	w2	NOUN
ejpam-4936	290	43	no	no	DET
ejpam-4936	290	44	solution	solution	NOUN
ejpam-4936	290	45	5(iii	5(iii	NUM
ejpam-4936	290	46	)	)	PUNCT
ejpam-4936	290	47	18	18	NUM
ejpam-4936	290	48	4x	4x	NOUN
ejpam-4936	290	49	+	+	NUM
ejpam-4936	290	50	6y	6y	NOUN
ejpam-4936	290	51	+	+	CCONJ
ejpam-4936	290	52	7z	7z	NOUN
ejpam-4936	290	53	=	=	SYM
ejpam-4936	290	54	w2	w2	NOUN
ejpam-4936	290	55	no	no	DET
ejpam-4936	290	56	solution	solution	NOUN
ejpam-4936	290	57	2	2	NUM
ejpam-4936	290	58	19	19	NUM
ejpam-4936	290	59	4x	4x	NOUN
ejpam-4936	290	60	+	+	CCONJ
ejpam-4936	290	61	7y	7y	NOUN
ejpam-4936	290	62	+	+	CCONJ
ejpam-4936	290	63	9z	9z	NUM
ejpam-4936	290	64	=	=	SYM
ejpam-4936	290	65	w2	w2	NOUN
ejpam-4936	290	66	no	no	DET
ejpam-4936	290	67	solution	solution	NOUN
ejpam-4936	290	68	corollary	corollary	NOUN
ejpam-4936	290	69	1	1	NUM
ejpam-4936	290	70	20	20	NUM
ejpam-4936	290	71	4x	4x	NOUN
ejpam-4936	290	72	+	+	CCONJ
ejpam-4936	290	73	9y	9y	NOUN
ejpam-4936	290	74	+	+	CCONJ
ejpam-4936	290	75	9z	9z	NUM
ejpam-4936	290	76	=	=	SYM
ejpam-4936	290	77	w2	w2	NOUN
ejpam-4936	290	78	no	no	DET
ejpam-4936	290	79	solution	solution	NOUN
ejpam-4936	290	80	5(iii	5(iii	NUM
ejpam-4936	290	81	)	)	PUNCT
ejpam-4936	290	82	21	21	NUM
ejpam-4936	290	83	4x	4x	NOUN
ejpam-4936	290	84	+	+	CCONJ
ejpam-4936	290	85	9y	9y	NOUN
ejpam-4936	290	86	+	+	SYM
ejpam-4936	290	87	13z	13z	NOUN
ejpam-4936	290	88	=	=	SYM
ejpam-4936	290	89	w2	w2	NOUN
ejpam-4936	290	90	no	no	DET
ejpam-4936	290	91	solution	solution	NOUN
ejpam-4936	290	92	5(iii	5(iii	NUM
ejpam-4936	290	93	)	)	PUNCT
ejpam-4936	290	94	22	22	NUM
ejpam-4936	290	95	4x	4x	NOUN
ejpam-4936	290	96	+	+	CCONJ
ejpam-4936	290	97	9y	9y	NOUN
ejpam-4936	290	98	+	+	NUM
ejpam-4936	290	99	17z	17z	NOUN
ejpam-4936	290	100	=	=	SYM
ejpam-4936	290	101	w2	w2	NOUN
ejpam-4936	290	102	no	no	DET
ejpam-4936	290	103	solution	solution	NOUN
ejpam-4936	290	104	5(iii	5(iii	NUM
ejpam-4936	290	105	)	)	PUNCT
ejpam-4936	290	106	23	23	NUM
ejpam-4936	290	107	4x	4x	NOUN
ejpam-4936	290	108	+	+	SYM
ejpam-4936	290	109	13y	13y	NOUN
ejpam-4936	290	110	+	+	CCONJ
ejpam-4936	290	111	13z	13z	NOUN
ejpam-4936	290	112	=	=	SYM
ejpam-4936	290	113	w2	w2	NOUN
ejpam-4936	290	114	no	no	DET
ejpam-4936	290	115	solution	solution	NOUN
ejpam-4936	290	116	5(iii	5(iii	NUM
ejpam-4936	290	117	)	)	PUNCT
ejpam-4936	290	118	24	24	NUM
ejpam-4936	290	119	4x	4x	NOUN
ejpam-4936	290	120	+	+	NOUN
ejpam-4936	290	121	13y	13y	NOUN
ejpam-4936	290	122	+	+	CCONJ
ejpam-4936	290	123	17z	17z	NOUN
ejpam-4936	290	124	=	=	SYM
ejpam-4936	290	125	w2	w2	NOUN
ejpam-4936	290	126	no	no	DET
ejpam-4936	290	127	solution	solution	NOUN
ejpam-4936	290	128	5(iii	5(iii	NUM
ejpam-4936	290	129	)	)	PUNCT
ejpam-4936	290	130	25	25	NUM
ejpam-4936	290	131	4x	4x	NOUN
ejpam-4936	290	132	+	+	ADJ
ejpam-4936	290	133	17y	17y	NOUN
ejpam-4936	290	134	+	+	SYM
ejpam-4936	290	135	17z	17z	NOUN
ejpam-4936	290	136	=	=	SYM
ejpam-4936	290	137	w2	w2	NOUN
ejpam-4936	290	138	no	no	DET
ejpam-4936	290	139	solution	solution	NOUN
ejpam-4936	290	140	5(iii	5(iii	NUM
ejpam-4936	290	141	)	)	PUNCT
ejpam-4936	290	142	26	26	NUM
ejpam-4936	290	143	5x	5x	NOUN
ejpam-4936	290	144	+	+	CCONJ
ejpam-4936	290	145	5y	5y	NOUN
ejpam-4936	290	146	+	+	CCONJ
ejpam-4936	290	147	5z	5z	NOUN
ejpam-4936	290	148	=	=	SYM
ejpam-4936	290	149	w2	w2	NOUN
ejpam-4936	290	150	no	no	DET
ejpam-4936	290	151	solution	solution	NOUN
ejpam-4936	290	152	5(i	5(i	NOUN
ejpam-4936	290	153	)	)	PUNCT
ejpam-4936	290	154	27	27	NUM
ejpam-4936	290	155	5x	5x	NOUN
ejpam-4936	290	156	+	+	CCONJ
ejpam-4936	290	157	5y	5y	NOUN
ejpam-4936	290	158	+	+	NUM
ejpam-4936	290	159	8z	8z	ADJ
ejpam-4936	290	160	=	=	PROPN
ejpam-4936	290	161	w2	w2	NOUN
ejpam-4936	290	162	no	no	DET
ejpam-4936	290	163	solution	solution	NOUN
ejpam-4936	290	164	5(iii	5(iii	NUM
ejpam-4936	290	165	)	)	PUNCT
ejpam-4936	290	166	28	28	NUM
ejpam-4936	290	167	5x	5x	NOUN
ejpam-4936	290	168	+	+	CCONJ
ejpam-4936	290	169	5y	5y	NOUN
ejpam-4936	290	170	+	+	CCONJ
ejpam-4936	290	171	9z	9z	NUM
ejpam-4936	290	172	=	=	SYM
ejpam-4936	290	173	w2	w2	NOUN
ejpam-4936	290	174	no	no	DET
ejpam-4936	290	175	solution	solution	NOUN
ejpam-4936	290	176	5(i	5(i	NOUN
ejpam-4936	290	177	)	)	PUNCT
ejpam-4936	290	178	29	29	NUM
ejpam-4936	290	179	5x	5x	NOUN
ejpam-4936	290	180	+	+	CCONJ
ejpam-4936	290	181	5y	5y	NOUN
ejpam-4936	290	182	+	+	X
ejpam-4936	290	183	12z	12z	NOUN
ejpam-4936	290	184	=	=	SYM
ejpam-4936	290	185	w2	w2	NOUN
ejpam-4936	290	186	no	no	DET
ejpam-4936	290	187	solution	solution	NOUN
ejpam-4936	290	188	5(iii	5(iii	NUM
ejpam-4936	290	189	)	)	PUNCT
ejpam-4936	290	190	30	30	NUM
ejpam-4936	290	191	5x	5x	NOUN
ejpam-4936	290	192	+	+	CCONJ
ejpam-4936	290	193	5y	5y	NOUN
ejpam-4936	290	194	+	+	NUM
ejpam-4936	290	195	13z	13z	NOUN
ejpam-4936	290	196	=	=	SYM
ejpam-4936	290	197	w2	w2	NOUN
ejpam-4936	290	198	no	no	DET
ejpam-4936	290	199	solution	solution	NOUN
ejpam-4936	290	200	5(i	5(i	NOUN
ejpam-4936	290	201	)	)	PUNCT
ejpam-4936	290	202	31	31	NUM
ejpam-4936	290	203	5x	5x	NOUN
ejpam-4936	290	204	+	+	CCONJ
ejpam-4936	290	205	5y	5y	NOUN
ejpam-4936	290	206	+	+	NUM
ejpam-4936	290	207	16z	16z	NOUN
ejpam-4936	290	208	=	=	SYM
ejpam-4936	290	209	w2	w2	PROPN
ejpam-4936	290	210	no	no	DET
ejpam-4936	290	211	solution	solution	NOUN
ejpam-4936	290	212	5(iii	5(iii	NUM
ejpam-4936	290	213	)	)	PUNCT
ejpam-4936	290	214	32	32	NUM
ejpam-4936	290	215	5x	5x	NOUN
ejpam-4936	290	216	+	+	CCONJ
ejpam-4936	290	217	5y	5y	NOUN
ejpam-4936	290	218	+	+	NUM
ejpam-4936	290	219	17z	17z	NOUN
ejpam-4936	290	220	=	=	SYM
ejpam-4936	290	221	w2	w2	NOUN
ejpam-4936	290	222	no	no	DET
ejpam-4936	290	223	solution	solution	NOUN
ejpam-4936	290	224	5(i	5(i	NOUN
ejpam-4936	290	225	)	)	PUNCT
ejpam-4936	290	226	33	33	NUM
ejpam-4936	290	227	5x	5x	NOUN
ejpam-4936	290	228	+	+	CCONJ
ejpam-4936	290	229	5y	5y	NOUN
ejpam-4936	290	230	+	+	NUM
ejpam-4936	290	231	20z	20z	NOUN
ejpam-4936	290	232	=	=	NOUN
ejpam-4936	290	233	w2	w2	NOUN
ejpam-4936	290	234	no	no	DET
ejpam-4936	290	235	solution	solution	NOUN
ejpam-4936	290	236	5(iii	5(iii	NUM
ejpam-4936	290	237	)	)	PUNCT
ejpam-4936	290	238	34	34	NUM
ejpam-4936	290	239	5x	5x	NOUN
ejpam-4936	290	240	+	+	CCONJ
ejpam-4936	290	241	6y	6y	NOUN
ejpam-4936	290	242	+	+	CCONJ
ejpam-4936	290	243	6z	6z	NOUN
ejpam-4936	290	244	=	=	PROPN
ejpam-4936	290	245	w2	w2	NOUN
ejpam-4936	290	246	no	no	DET
ejpam-4936	290	247	solution	solution	NOUN
ejpam-4936	290	248	5(iv	5(iv	NUM
ejpam-4936	290	249	)	)	PUNCT
ejpam-4936	290	250	35	35	NUM
ejpam-4936	290	251	5x	5x	NOUN
ejpam-4936	290	252	+	+	CCONJ
ejpam-4936	290	253	6y	6y	NOUN
ejpam-4936	290	254	+	+	CCONJ
ejpam-4936	290	255	11z	11z	NOUN
ejpam-4936	290	256	=	=	SYM
ejpam-4936	290	257	w2	w2	NOUN
ejpam-4936	290	258	no	no	DET
ejpam-4936	290	259	solution	solution	NOUN
ejpam-4936	290	260	5(iv	5(iv	NUM
ejpam-4936	290	261	)	)	PUNCT
ejpam-4936	290	262	k.	k.	NOUN
ejpam-4936	291	1	laipaporn	laipaporn	PROPN
ejpam-4936	291	2	,	,	PUNCT
ejpam-4936	291	3	s.	s.	PROPN
ejpam-4936	291	4	kaewchay	kaewchay	PROPN
ejpam-4936	291	5	,	,	PUNCT
ejpam-4936	291	6	a.	a.	PROPN
ejpam-4936	291	7	karnbanjong	karnbanjong	PROPN
ejpam-4936	291	8	/	/	SYM
ejpam-4936	291	9	eur	eur	PROPN
ejpam-4936	291	10	.	.	PUNCT
ejpam-4936	292	1	j.	j.	PROPN
ejpam-4936	292	2	pure	pure	PROPN
ejpam-4936	292	3	appl	appl	PROPN
ejpam-4936	292	4	.	.	PROPN
ejpam-4936	292	5	math	math	PROPN
ejpam-4936	292	6	,	,	PUNCT
ejpam-4936	292	7	16	16	NUM
ejpam-4936	292	8	(	(	PUNCT
ejpam-4936	292	9	4	4	NUM
ejpam-4936	292	10	)	)	PUNCT
ejpam-4936	292	11	(	(	PUNCT
ejpam-4936	292	12	2023	2023	NUM
ejpam-4936	292	13	)	)	PUNCT
ejpam-4936	292	14	,	,	PUNCT
ejpam-4936	292	15	2066	2066	NUM
ejpam-4936	292	16	-	-	SYM
ejpam-4936	292	17	2081	2081	NUM
ejpam-4936	292	18	2077	2077	NUM
ejpam-4936	292	19	table	table	NOUN
ejpam-4936	292	20	3	3	NUM
ejpam-4936	292	21	:	:	PUNCT
ejpam-4936	292	22	the	the	DET
ejpam-4936	292	23	list	list	NOUN
ejpam-4936	292	24	of	of	ADP
ejpam-4936	292	25	the	the	DET
ejpam-4936	292	26	equations	equation	NOUN
ejpam-4936	292	27	ax	ax	NOUN
ejpam-4936	292	28	+	+	CCONJ
ejpam-4936	292	29	by	by	ADP
ejpam-4936	292	30	+	+	CCONJ
ejpam-4936	292	31	cz	cz	NOUN
ejpam-4936	292	32	=	=	SYM
ejpam-4936	292	33	w2	w2	NOUN
ejpam-4936	292	34	where	where	SCONJ
ejpam-4936	292	35	2	2	NUM
ejpam-4936	292	36	≤	≤	NOUN
ejpam-4936	292	37	a	a	DET
ejpam-4936	292	38	≤	≤	NUM
ejpam-4936	292	39	b	b	NOUN
ejpam-4936	292	40	≤	≤	NUM
ejpam-4936	292	41	c	c	NOUN
ejpam-4936	292	42	≤	≤	NUM
ejpam-4936	292	43	20	20	NUM
ejpam-4936	292	44	and	and	CCONJ
ejpam-4936	292	45	all	all	DET
ejpam-4936	292	46	variables	variable	NOUN
ejpam-4936	292	47	a	a	PRON
ejpam-4936	292	48	,	,	PUNCT
ejpam-4936	292	49	b	b	PROPN
ejpam-4936	292	50	and	and	CCONJ
ejpam-4936	292	51	c	c	PROPN
ejpam-4936	292	52	are	be	AUX
ejpam-4936	292	53	satisfied	satisfied	ADJ
ejpam-4936	292	54	our	our	PRON
ejpam-4936	292	55	results	result	NOUN
ejpam-4936	292	56	.	.	PUNCT
ejpam-4936	293	1	no	no	INTJ
ejpam-4936	293	2	.	.	PUNCT
ejpam-4936	294	1	ax	ax	NOUN
ejpam-4936	294	2	+	+	CCONJ
ejpam-4936	294	3	by	by	ADP
ejpam-4936	294	4	+	+	CCONJ
ejpam-4936	294	5	cz	cz	NOUN
ejpam-4936	294	6	=	=	NOUN
ejpam-4936	294	7	w2	w2	NOUN
ejpam-4936	294	8	result	result	NOUN
ejpam-4936	294	9	ref	ref	NOUN
ejpam-4936	294	10	.	.	PUNCT
ejpam-4936	295	1	of	of	ADP
ejpam-4936	295	2	theorem	theorem	ADJ
ejpam-4936	295	3	36	36	NUM
ejpam-4936	295	4	5x	5x	NOUN
ejpam-4936	295	5	+	+	CCONJ
ejpam-4936	295	6	6y	6y	NOUN
ejpam-4936	295	7	+	+	NUM
ejpam-4936	295	8	16z	16z	NOUN
ejpam-4936	295	9	=	=	SYM
ejpam-4936	295	10	w2	w2	PROPN
ejpam-4936	295	11	no	no	DET
ejpam-4936	295	12	solution	solution	NOUN
ejpam-4936	295	13	5(iv	5(iv	NUM
ejpam-4936	295	14	)	)	PUNCT
ejpam-4936	295	15	37	37	NUM
ejpam-4936	295	16	5x	5x	NOUN
ejpam-4936	295	17	+	+	CCONJ
ejpam-4936	295	18	8y	8y	NUM
ejpam-4936	295	19	+	+	CCONJ
ejpam-4936	295	20	9z	9z	NUM
ejpam-4936	295	21	=	=	SYM
ejpam-4936	295	22	w2	w2	NOUN
ejpam-4936	295	23	no	no	DET
ejpam-4936	295	24	solution	solution	NOUN
ejpam-4936	295	25	5(iii	5(iii	NUM
ejpam-4936	295	26	)	)	PUNCT
ejpam-4936	295	27	38	38	NUM
ejpam-4936	295	28	5x	5x	NOUN
ejpam-4936	295	29	+	+	CCONJ
ejpam-4936	295	30	8y	8y	ADJ
ejpam-4936	295	31	+	+	NUM
ejpam-4936	295	32	13z	13z	NOUN
ejpam-4936	295	33	=	=	SYM
ejpam-4936	295	34	w2	w2	NOUN
ejpam-4936	295	35	no	no	DET
ejpam-4936	295	36	solution	solution	NOUN
ejpam-4936	295	37	5(iii	5(iii	NUM
ejpam-4936	295	38	)	)	PUNCT
ejpam-4936	295	39	39	39	NUM
ejpam-4936	295	40	5x	5x	NOUN
ejpam-4936	295	41	+	+	CCONJ
ejpam-4936	295	42	8y	8y	ADJ
ejpam-4936	295	43	+	+	X
ejpam-4936	295	44	17z	17z	NOUN
ejpam-4936	295	45	=	=	SYM
ejpam-4936	295	46	w2	w2	NOUN
ejpam-4936	295	47	no	no	DET
ejpam-4936	295	48	solution	solution	NOUN
ejpam-4936	295	49	5(iii	5(iii	NUM
ejpam-4936	295	50	)	)	PUNCT
ejpam-4936	295	51	40	40	NUM
ejpam-4936	295	52	5x	5x	NOUN
ejpam-4936	295	53	+	+	CCONJ
ejpam-4936	295	54	9y	9y	NOUN
ejpam-4936	295	55	+	+	CCONJ
ejpam-4936	295	56	9z	9z	NUM
ejpam-4936	295	57	=	=	SYM
ejpam-4936	295	58	w2	w2	NOUN
ejpam-4936	295	59	no	no	DET
ejpam-4936	295	60	solution	solution	NOUN
ejpam-4936	295	61	5(i	5(i	NOUN
ejpam-4936	295	62	)	)	PUNCT
ejpam-4936	295	63	41	41	NUM
ejpam-4936	295	64	5x	5x	NOUN
ejpam-4936	295	65	+	+	CCONJ
ejpam-4936	295	66	9y	9y	NOUN
ejpam-4936	295	67	+	+	SYM
ejpam-4936	295	68	12z	12z	NOUN
ejpam-4936	295	69	=	=	SYM
ejpam-4936	295	70	w2	w2	NOUN
ejpam-4936	295	71	no	no	DET
ejpam-4936	295	72	solution	solution	NOUN
ejpam-4936	295	73	5(iii	5(iii	NUM
ejpam-4936	295	74	)	)	PUNCT
ejpam-4936	295	75	42	42	NUM
ejpam-4936	295	76	5x	5x	NOUN
ejpam-4936	295	77	+	+	CCONJ
ejpam-4936	295	78	9y	9y	NOUN
ejpam-4936	295	79	+	+	SYM
ejpam-4936	295	80	13z	13z	NOUN
ejpam-4936	295	81	=	=	SYM
ejpam-4936	295	82	w2	w2	NOUN
ejpam-4936	295	83	no	no	DET
ejpam-4936	295	84	solution	solution	NOUN
ejpam-4936	295	85	5(i	5(i	NOUN
ejpam-4936	295	86	)	)	PUNCT
ejpam-4936	295	87	43	43	NUM
ejpam-4936	295	88	5x	5x	NOUN
ejpam-4936	295	89	+	+	CCONJ
ejpam-4936	295	90	9y	9y	NOUN
ejpam-4936	295	91	+	+	NUM
ejpam-4936	295	92	16z	16z	NOUN
ejpam-4936	295	93	=	=	SYM
ejpam-4936	295	94	w2	w2	NOUN
ejpam-4936	295	95	no	no	DET
ejpam-4936	295	96	solution	solution	NOUN
ejpam-4936	295	97	5(iii	5(iii	NUM
ejpam-4936	295	98	)	)	PUNCT
ejpam-4936	295	99	44	44	NUM
ejpam-4936	295	100	5x	5x	NOUN
ejpam-4936	295	101	+	+	CCONJ
ejpam-4936	295	102	9y	9y	NOUN
ejpam-4936	295	103	+	+	NUM
ejpam-4936	295	104	17z	17z	NOUN
ejpam-4936	295	105	=	=	SYM
ejpam-4936	295	106	w2	w2	NOUN
ejpam-4936	295	107	no	no	DET
ejpam-4936	295	108	solution	solution	NOUN
ejpam-4936	295	109	5(i	5(i	NOUN
ejpam-4936	295	110	)	)	PUNCT
ejpam-4936	295	111	45	45	NUM
ejpam-4936	295	112	5x	5x	NOUN
ejpam-4936	295	113	+	+	CCONJ
ejpam-4936	295	114	9y	9y	NOUN
ejpam-4936	295	115	+	+	CCONJ
ejpam-4936	295	116	20z	20z	NOUN
ejpam-4936	295	117	=	=	NOUN
ejpam-4936	295	118	w2	w2	NOUN
ejpam-4936	295	119	no	no	DET
ejpam-4936	295	120	solution	solution	NOUN
ejpam-4936	295	121	5(iii	5(iii	NUM
ejpam-4936	295	122	)	)	PUNCT
ejpam-4936	295	123	46	46	NUM
ejpam-4936	295	124	5x	5x	NOUN
ejpam-4936	295	125	+	+	SYM
ejpam-4936	295	126	11y	11y	NOUN
ejpam-4936	295	127	+	+	CCONJ
ejpam-4936	295	128	11z	11z	NOUN
ejpam-4936	295	129	=	=	SYM
ejpam-4936	295	130	w2	w2	NOUN
ejpam-4936	295	131	no	no	DET
ejpam-4936	295	132	solution	solution	NOUN
ejpam-4936	295	133	5(iv	5(iv	NUM
ejpam-4936	295	134	)	)	PUNCT
ejpam-4936	295	135	47	47	NUM
ejpam-4936	295	136	5x	5x	NOUN
ejpam-4936	295	137	+	+	SYM
ejpam-4936	295	138	11y	11y	NOUN
ejpam-4936	295	139	+	+	CCONJ
ejpam-4936	295	140	16z	16z	NOUN
ejpam-4936	295	141	=	=	SYM
ejpam-4936	295	142	w2	w2	PROPN
ejpam-4936	295	143	no	no	DET
ejpam-4936	295	144	solution	solution	NOUN
ejpam-4936	295	145	5(iv	5(iv	NUM
ejpam-4936	295	146	)	)	PUNCT
ejpam-4936	295	147	48	48	NUM
ejpam-4936	295	148	5x	5x	NOUN
ejpam-4936	295	149	+	+	NUM
ejpam-4936	295	150	12y	12y	NOUN
ejpam-4936	295	151	+	+	CCONJ
ejpam-4936	295	152	13z	13z	NOUN
ejpam-4936	295	153	=	=	SYM
ejpam-4936	295	154	w2	w2	NOUN
ejpam-4936	295	155	no	no	DET
ejpam-4936	295	156	solution	solution	NOUN
ejpam-4936	295	157	5(iii	5(iii	NUM
ejpam-4936	295	158	)	)	PUNCT
ejpam-4936	295	159	49	49	NUM
ejpam-4936	295	160	5x	5x	NOUN
ejpam-4936	295	161	+	+	NUM
ejpam-4936	295	162	12y	12y	NOUN
ejpam-4936	295	163	+	+	CCONJ
ejpam-4936	295	164	17z	17z	NOUN
ejpam-4936	295	165	=	=	SYM
ejpam-4936	295	166	w2	w2	NOUN
ejpam-4936	295	167	no	no	DET
ejpam-4936	295	168	solution	solution	NOUN
ejpam-4936	295	169	5(iii	5(iii	NUM
ejpam-4936	295	170	)	)	PUNCT
ejpam-4936	295	171	50	50	NUM
ejpam-4936	295	172	5x	5x	NOUN
ejpam-4936	295	173	+	+	SYM
ejpam-4936	295	174	13y	13y	NOUN
ejpam-4936	295	175	+	+	CCONJ
ejpam-4936	295	176	13z	13z	NOUN
ejpam-4936	295	177	=	=	SYM
ejpam-4936	295	178	w2	w2	NOUN
ejpam-4936	295	179	no	no	DET
ejpam-4936	295	180	solution	solution	NOUN
ejpam-4936	295	181	5(i	5(i	NOUN
ejpam-4936	295	182	)	)	PUNCT
ejpam-4936	295	183	51	51	NUM
ejpam-4936	295	184	5x	5x	NOUN
ejpam-4936	295	185	+	+	CCONJ
ejpam-4936	295	186	13y	13y	NOUN
ejpam-4936	295	187	+	+	CCONJ
ejpam-4936	295	188	16z	16z	NOUN
ejpam-4936	295	189	=	=	SYM
ejpam-4936	295	190	w2	w2	PROPN
ejpam-4936	295	191	no	no	DET
ejpam-4936	295	192	solution	solution	NOUN
ejpam-4936	295	193	5(iii	5(iii	NUM
ejpam-4936	295	194	)	)	PUNCT
ejpam-4936	295	195	52	52	NUM
ejpam-4936	295	196	5x	5x	NOUN
ejpam-4936	295	197	+	+	SYM
ejpam-4936	295	198	13y	13y	NOUN
ejpam-4936	295	199	+	+	NUM
ejpam-4936	295	200	17z	17z	NOUN
ejpam-4936	295	201	=	=	SYM
ejpam-4936	295	202	w2	w2	NOUN
ejpam-4936	295	203	no	no	DET
ejpam-4936	295	204	solution	solution	NOUN
ejpam-4936	295	205	5(i	5(i	NOUN
ejpam-4936	295	206	)	)	PUNCT
ejpam-4936	295	207	53	53	NUM
ejpam-4936	295	208	5x	5x	NOUN
ejpam-4936	295	209	+	+	SYM
ejpam-4936	295	210	13y	13y	NOUN
ejpam-4936	295	211	+	+	NUM
ejpam-4936	295	212	20z	20z	NOUN
ejpam-4936	295	213	=	=	NOUN
ejpam-4936	295	214	w2	w2	NOUN
ejpam-4936	295	215	no	no	DET
ejpam-4936	295	216	solution	solution	NOUN
ejpam-4936	295	217	5(iii	5(iii	NUM
ejpam-4936	295	218	)	)	PUNCT
ejpam-4936	295	219	54	54	NUM
ejpam-4936	295	220	5x	5x	NOUN
ejpam-4936	295	221	+	+	CCONJ
ejpam-4936	295	222	16y	16y	NUM
ejpam-4936	295	223	+	+	CCONJ
ejpam-4936	295	224	16z	16z	NOUN
ejpam-4936	295	225	=	=	SYM
ejpam-4936	295	226	w2	w2	PROPN
ejpam-4936	295	227	no	no	DET
ejpam-4936	295	228	solution	solution	NOUN
ejpam-4936	295	229	5(iv	5(iv	NUM
ejpam-4936	295	230	)	)	PUNCT
ejpam-4936	295	231	55	55	NUM
ejpam-4936	295	232	5x	5x	NOUN
ejpam-4936	295	233	+	+	CCONJ
ejpam-4936	295	234	16y	16y	NOUN
ejpam-4936	295	235	+	+	CCONJ
ejpam-4936	295	236	17z	17z	NOUN
ejpam-4936	295	237	=	=	SYM
ejpam-4936	295	238	w2	w2	NOUN
ejpam-4936	295	239	no	no	DET
ejpam-4936	295	240	solution	solution	NOUN
ejpam-4936	295	241	5(iii	5(iii	NUM
ejpam-4936	295	242	)	)	PUNCT
ejpam-4936	295	243	56	56	NUM
ejpam-4936	295	244	5x	5x	NOUN
ejpam-4936	295	245	+	+	NOUN
ejpam-4936	295	246	17y	17y	NOUN
ejpam-4936	295	247	+	+	CCONJ
ejpam-4936	295	248	17z	17z	NOUN
ejpam-4936	295	249	=	=	SYM
ejpam-4936	295	250	w2	w2	NOUN
ejpam-4936	295	251	no	no	DET
ejpam-4936	295	252	solution	solution	NOUN
ejpam-4936	295	253	5(i	5(i	NOUN
ejpam-4936	295	254	)	)	PUNCT
ejpam-4936	295	255	57	57	NUM
ejpam-4936	295	256	5x	5x	NOUN
ejpam-4936	295	257	+	+	NOUN
ejpam-4936	295	258	17y	17y	NOUN
ejpam-4936	295	259	+	+	NUM
ejpam-4936	295	260	20z	20z	NOUN
ejpam-4936	295	261	=	=	NOUN
ejpam-4936	295	262	w2	w2	NOUN
ejpam-4936	295	263	no	no	DET
ejpam-4936	295	264	solution	solution	NOUN
ejpam-4936	295	265	5(iii	5(iii	NUM
ejpam-4936	295	266	)	)	PUNCT
ejpam-4936	295	267	58	58	NUM
ejpam-4936	295	268	6x	6x	NOUN
ejpam-4936	295	269	+	+	NUM
ejpam-4936	295	270	6y	6y	NOUN
ejpam-4936	295	271	+	+	CCONJ
ejpam-4936	295	272	6z	6z	NOUN
ejpam-4936	295	273	=	=	PROPN
ejpam-4936	295	274	w2	w2	NOUN
ejpam-4936	295	275	no	no	DET
ejpam-4936	295	276	solution	solution	NOUN
ejpam-4936	295	277	5(ii	5(ii	NUM
ejpam-4936	295	278	)	)	PUNCT
ejpam-4936	295	279	59	59	NUM
ejpam-4936	295	280	6x	6x	NOUN
ejpam-4936	295	281	+	+	CCONJ
ejpam-4936	295	282	6y	6y	NOUN
ejpam-4936	295	283	+	+	X
ejpam-4936	295	284	10z	10z	NOUN
ejpam-4936	295	285	=	=	SYM
ejpam-4936	295	286	w2	w2	NOUN
ejpam-4936	295	287	no	no	DET
ejpam-4936	295	288	solution	solution	NOUN
ejpam-4936	295	289	5(iv	5(iv	NUM
ejpam-4936	295	290	)	)	PUNCT
ejpam-4936	295	291	60	60	NUM
ejpam-4936	295	292	6x	6x	NOUN
ejpam-4936	295	293	+	+	NUM
ejpam-4936	295	294	6y	6y	NOUN
ejpam-4936	295	295	+	+	CCONJ
ejpam-4936	295	296	11z	11z	NOUN
ejpam-4936	295	297	=	=	SYM
ejpam-4936	295	298	w2	w2	NOUN
ejpam-4936	295	299	no	no	DET
ejpam-4936	295	300	solution	solution	NOUN
ejpam-4936	295	301	5(ii	5(ii	NUM
ejpam-4936	295	302	)	)	PUNCT
ejpam-4936	295	303	61	61	NUM
ejpam-4936	295	304	6x	6x	NOUN
ejpam-4936	295	305	+	+	NUM
ejpam-4936	295	306	6y	6y	NOUN
ejpam-4936	295	307	+	+	CCONJ
ejpam-4936	295	308	15z	15z	X
ejpam-4936	295	309	=	=	SYM
ejpam-4936	295	310	w2	w2	NOUN
ejpam-4936	295	311	no	no	DET
ejpam-4936	295	312	solution	solution	NOUN
ejpam-4936	295	313	5(iv	5(iv	NUM
ejpam-4936	295	314	)	)	PUNCT
ejpam-4936	295	315	62	62	NUM
ejpam-4936	295	316	6x	6x	NOUN
ejpam-4936	295	317	+	+	NUM
ejpam-4936	295	318	6y	6y	NOUN
ejpam-4936	295	319	+	+	NUM
ejpam-4936	295	320	16z	16z	NOUN
ejpam-4936	295	321	=	=	SYM
ejpam-4936	295	322	w2	w2	PROPN
ejpam-4936	295	323	no	no	DET
ejpam-4936	295	324	solution	solution	NOUN
ejpam-4936	295	325	5(ii	5(ii	NUM
ejpam-4936	295	326	)	)	PUNCT
ejpam-4936	295	327	63	63	NUM
ejpam-4936	295	328	6x	6x	NOUN
ejpam-4936	295	329	+	+	CCONJ
ejpam-4936	295	330	6y	6y	NOUN
ejpam-4936	295	331	+	+	NUM
ejpam-4936	295	332	20z	20z	NOUN
ejpam-4936	295	333	=	=	SYM
ejpam-4936	295	334	w2	w2	NOUN
ejpam-4936	295	335	no	no	DET
ejpam-4936	295	336	solution	solution	NOUN
ejpam-4936	295	337	5(iv	5(iv	NUM
ejpam-4936	295	338	)	)	PUNCT
ejpam-4936	295	339	64	64	NUM
ejpam-4936	295	340	6x	6x	NOUN
ejpam-4936	295	341	+	+	CCONJ
ejpam-4936	295	342	7y	7y	ADJ
ejpam-4936	295	343	+	+	ADJ
ejpam-4936	295	344	10z	10z	NOUN
ejpam-4936	295	345	=	=	SYM
ejpam-4936	295	346	w2	w2	NOUN
ejpam-4936	295	347	no	no	DET
ejpam-4936	295	348	solution	solution	NOUN
ejpam-4936	295	349	4	4	NUM
ejpam-4936	295	350	65	65	NUM
ejpam-4936	295	351	6x	6x	NOUN
ejpam-4936	295	352	+	+	CCONJ
ejpam-4936	295	353	7y	7y	NOUN
ejpam-4936	295	354	+	+	NOUN
ejpam-4936	295	355	16z	16z	NOUN
ejpam-4936	295	356	=	=	SYM
ejpam-4936	295	357	w2	w2	PROPN
ejpam-4936	295	358	no	no	DET
ejpam-4936	295	359	solution	solution	NOUN
ejpam-4936	295	360	corollary	corollary	NOUN
ejpam-4936	295	361	2	2	NUM
ejpam-4936	295	362	66	66	NUM
ejpam-4936	295	363	6x	6x	NOUN
ejpam-4936	295	364	+	+	CCONJ
ejpam-4936	295	365	10y	10y	NUM
ejpam-4936	295	366	+	+	CCONJ
ejpam-4936	295	367	11z	11z	NOUN
ejpam-4936	295	368	=	=	SYM
ejpam-4936	295	369	w2	w2	NOUN
ejpam-4936	295	370	no	no	DET
ejpam-4936	295	371	solution	solution	NOUN
ejpam-4936	295	372	5(iv	5(iv	NUM
ejpam-4936	295	373	)	)	PUNCT
ejpam-4936	295	374	67	67	NUM
ejpam-4936	295	375	6x	6x	NOUN
ejpam-4936	295	376	+	+	CCONJ
ejpam-4936	295	377	10y	10y	PROPN
ejpam-4936	295	378	+	+	SYM
ejpam-4936	295	379	16z	16z	NOUN
ejpam-4936	295	380	=	=	SYM
ejpam-4936	295	381	w2	w2	PROPN
ejpam-4936	295	382	no	no	DET
ejpam-4936	295	383	solution	solution	NOUN
ejpam-4936	295	384	5(iv	5(iv	NUM
ejpam-4936	295	385	)	)	PUNCT
ejpam-4936	295	386	68	68	NUM
ejpam-4936	295	387	6x	6x	NOUN
ejpam-4936	295	388	+	+	SYM
ejpam-4936	295	389	11y	11y	NOUN
ejpam-4936	295	390	+	+	CCONJ
ejpam-4936	295	391	11z	11z	NOUN
ejpam-4936	295	392	=	=	SYM
ejpam-4936	295	393	w2	w2	NOUN
ejpam-4936	295	394	no	no	DET
ejpam-4936	295	395	solution	solution	NOUN
ejpam-4936	295	396	5(ii	5(ii	NUM
ejpam-4936	295	397	)	)	PUNCT
ejpam-4936	295	398	69	69	NUM
ejpam-4936	295	399	6x	6x	NOUN
ejpam-4936	295	400	+	+	NOUN
ejpam-4936	295	401	11y	11y	NOUN
ejpam-4936	295	402	+	+	CCONJ
ejpam-4936	295	403	15z	15z	NOUN
ejpam-4936	295	404	=	=	SYM
ejpam-4936	295	405	w2	w2	NOUN
ejpam-4936	295	406	no	no	DET
ejpam-4936	295	407	solution	solution	NOUN
ejpam-4936	295	408	5(iv	5(iv	NUM
ejpam-4936	295	409	)	)	PUNCT
ejpam-4936	295	410	70	70	NUM
ejpam-4936	295	411	6x	6x	NOUN
ejpam-4936	295	412	+	+	NOUN
ejpam-4936	295	413	11y	11y	NOUN
ejpam-4936	295	414	+	+	CCONJ
ejpam-4936	295	415	16z	16z	NOUN
ejpam-4936	295	416	=	=	SYM
ejpam-4936	295	417	w2	w2	PROPN
ejpam-4936	295	418	no	no	DET
ejpam-4936	295	419	solution	solution	NOUN
ejpam-4936	295	420	5(ii	5(ii	NUM
ejpam-4936	295	421	)	)	PUNCT
ejpam-4936	295	422	71	71	NUM
ejpam-4936	295	423	6x	6x	NOUN
ejpam-4936	295	424	+	+	SYM
ejpam-4936	295	425	11y	11y	NOUN
ejpam-4936	295	426	+	+	NUM
ejpam-4936	295	427	20z	20z	NOUN
ejpam-4936	295	428	=	=	NOUN
ejpam-4936	295	429	w2	w2	NOUN
ejpam-4936	295	430	no	no	DET
ejpam-4936	295	431	solution	solution	NOUN
ejpam-4936	295	432	5(iv	5(iv	NUM
ejpam-4936	295	433	)	)	PUNCT
ejpam-4936	295	434	72	72	NUM
ejpam-4936	295	435	6x	6x	NOUN
ejpam-4936	295	436	+	+	CCONJ
ejpam-4936	295	437	15y	15y	ADJ
ejpam-4936	295	438	+	+	NUM
ejpam-4936	295	439	16z	16z	NOUN
ejpam-4936	295	440	=	=	SYM
ejpam-4936	295	441	w2	w2	PROPN
ejpam-4936	295	442	no	no	DET
ejpam-4936	295	443	solution	solution	NOUN
ejpam-4936	295	444	5(iv	5(iv	NUM
ejpam-4936	295	445	)	)	PUNCT
ejpam-4936	295	446	73	73	NUM
ejpam-4936	295	447	6x	6x	NOUN
ejpam-4936	295	448	+	+	ADJ
ejpam-4936	295	449	16y	16y	NUM
ejpam-4936	295	450	+	+	CCONJ
ejpam-4936	295	451	16z	16z	NOUN
ejpam-4936	295	452	=	=	SYM
ejpam-4936	295	453	w2	w2	PROPN
ejpam-4936	295	454	no	no	DET
ejpam-4936	295	455	solution	solution	NOUN
ejpam-4936	295	456	5(ii	5(ii	NUM
ejpam-4936	295	457	)	)	PUNCT
ejpam-4936	295	458	k.	k.	PROPN
ejpam-4936	296	1	laipaporn	laipaporn	PROPN
ejpam-4936	296	2	,	,	PUNCT
ejpam-4936	296	3	s.	s.	PROPN
ejpam-4936	296	4	kaewchay	kaewchay	PROPN
ejpam-4936	296	5	,	,	PUNCT
ejpam-4936	296	6	a.	a.	PROPN
ejpam-4936	296	7	karnbanjong	karnbanjong	PROPN
ejpam-4936	296	8	/	/	SYM
ejpam-4936	296	9	eur	eur	PROPN
ejpam-4936	296	10	.	.	PUNCT
ejpam-4936	297	1	j.	j.	PROPN
ejpam-4936	297	2	pure	pure	PROPN
ejpam-4936	297	3	appl	appl	PROPN
ejpam-4936	297	4	.	.	PROPN
ejpam-4936	297	5	math	math	PROPN
ejpam-4936	297	6	,	,	PUNCT
ejpam-4936	297	7	16	16	NUM
ejpam-4936	297	8	(	(	PUNCT
ejpam-4936	297	9	4	4	NUM
ejpam-4936	297	10	)	)	PUNCT
ejpam-4936	297	11	(	(	PUNCT
ejpam-4936	297	12	2023	2023	NUM
ejpam-4936	297	13	)	)	PUNCT
ejpam-4936	297	14	,	,	PUNCT
ejpam-4936	297	15	2066	2066	NUM
ejpam-4936	297	16	-	-	SYM
ejpam-4936	297	17	2081	2081	NUM
ejpam-4936	297	18	2078	2078	NUM
ejpam-4936	297	19	table	table	NOUN
ejpam-4936	297	20	4	4	NUM
ejpam-4936	297	21	:	:	PUNCT
ejpam-4936	297	22	the	the	DET
ejpam-4936	297	23	list	list	NOUN
ejpam-4936	297	24	of	of	ADP
ejpam-4936	297	25	the	the	DET
ejpam-4936	297	26	equations	equation	NOUN
ejpam-4936	297	27	ax	ax	NOUN
ejpam-4936	297	28	+	+	CCONJ
ejpam-4936	297	29	by	by	ADP
ejpam-4936	297	30	+	+	CCONJ
ejpam-4936	297	31	cz	cz	NOUN
ejpam-4936	297	32	=	=	SYM
ejpam-4936	297	33	w2	w2	NOUN
ejpam-4936	297	34	where	where	SCONJ
ejpam-4936	297	35	2	2	NUM
ejpam-4936	297	36	≤	≤	NOUN
ejpam-4936	297	37	a	a	DET
ejpam-4936	297	38	≤	≤	NUM
ejpam-4936	297	39	b	b	NOUN
ejpam-4936	297	40	≤	≤	NUM
ejpam-4936	297	41	c	c	NOUN
ejpam-4936	297	42	≤	≤	NUM
ejpam-4936	297	43	20	20	NUM
ejpam-4936	297	44	and	and	CCONJ
ejpam-4936	297	45	all	all	DET
ejpam-4936	297	46	variables	variable	NOUN
ejpam-4936	297	47	a	a	PRON
ejpam-4936	297	48	,	,	PUNCT
ejpam-4936	297	49	b	b	PROPN
ejpam-4936	297	50	and	and	CCONJ
ejpam-4936	297	51	c	c	PROPN
ejpam-4936	297	52	are	be	AUX
ejpam-4936	297	53	satisfied	satisfied	ADJ
ejpam-4936	297	54	our	our	PRON
ejpam-4936	297	55	results	result	NOUN
ejpam-4936	297	56	.	.	PUNCT
ejpam-4936	298	1	no	no	INTJ
ejpam-4936	298	2	.	.	PUNCT
ejpam-4936	299	1	ax	ax	NOUN
ejpam-4936	299	2	+	+	CCONJ
ejpam-4936	299	3	by	by	ADP
ejpam-4936	299	4	+	+	CCONJ
ejpam-4936	299	5	cz	cz	NOUN
ejpam-4936	299	6	=	=	NOUN
ejpam-4936	299	7	w2	w2	NOUN
ejpam-4936	299	8	result	result	NOUN
ejpam-4936	299	9	ref	ref	NOUN
ejpam-4936	299	10	.	.	PUNCT
ejpam-4936	300	1	of	of	ADP
ejpam-4936	300	2	theorem	theorem	ADJ
ejpam-4936	300	3	74	74	NUM
ejpam-4936	300	4	6x	6x	NUM
ejpam-4936	300	5	+	+	ADJ
ejpam-4936	300	6	16y	16y	NOUN
ejpam-4936	300	7	+	+	CCONJ
ejpam-4936	300	8	20z	20z	NOUN
ejpam-4936	300	9	=	=	NOUN
ejpam-4936	300	10	w2	w2	NOUN
ejpam-4936	300	11	no	no	DET
ejpam-4936	300	12	solution	solution	NOUN
ejpam-4936	300	13	5(iv	5(iv	NUM
ejpam-4936	300	14	)	)	PUNCT
ejpam-4936	300	15	75	75	NUM
ejpam-4936	300	16	7x	7x	NUM
ejpam-4936	300	17	+	+	NUM
ejpam-4936	300	18	9y	9y	NOUN
ejpam-4936	300	19	+	+	CCONJ
ejpam-4936	300	20	10z	10z	NOUN
ejpam-4936	300	21	=	=	SYM
ejpam-4936	300	22	w2	w2	NOUN
ejpam-4936	300	23	no	no	DET
ejpam-4936	300	24	solution	solution	NOUN
ejpam-4936	300	25	3	3	NUM
ejpam-4936	300	26	76	76	NUM
ejpam-4936	300	27	7x	7x	NUM
ejpam-4936	300	28	+	+	CCONJ
ejpam-4936	300	29	9y	9y	NOUN
ejpam-4936	300	30	+	+	NUM
ejpam-4936	300	31	16z	16z	NOUN
ejpam-4936	300	32	=	=	SYM
ejpam-4936	300	33	w2	w2	PROPN
ejpam-4936	300	34	no	no	DET
ejpam-4936	300	35	solution	solution	NOUN
ejpam-4936	300	36	corollary	corollary	NOUN
ejpam-4936	300	37	1	1	NUM
ejpam-4936	300	38	77	77	NUM
ejpam-4936	300	39	8x	8x	NOUN
ejpam-4936	300	40	+	+	CCONJ
ejpam-4936	300	41	9y	9y	NOUN
ejpam-4936	300	42	+	+	CCONJ
ejpam-4936	300	43	9z	9z	NUM
ejpam-4936	300	44	=	=	SYM
ejpam-4936	300	45	w2	w2	NOUN
ejpam-4936	300	46	no	no	DET
ejpam-4936	300	47	solution	solution	NOUN
ejpam-4936	300	48	5(iii	5(iii	NUM
ejpam-4936	300	49	)	)	PUNCT
ejpam-4936	300	50	78	78	NUM
ejpam-4936	300	51	8x	8x	NOUN
ejpam-4936	301	1	+	+	CCONJ
ejpam-4936	301	2	9y	9y	NOUN
ejpam-4936	301	3	+	+	SYM
ejpam-4936	301	4	13z	13z	NOUN
ejpam-4936	301	5	=	=	SYM
ejpam-4936	301	6	w2	w2	NOUN
ejpam-4936	301	7	no	no	DET
ejpam-4936	301	8	solution	solution	NOUN
ejpam-4936	301	9	5(iii	5(iii	NUM
ejpam-4936	301	10	)	)	PUNCT
ejpam-4936	301	11	79	79	NUM
ejpam-4936	301	12	8x	8x	NOUN
ejpam-4936	302	1	+	+	CCONJ
ejpam-4936	302	2	9y	9y	NOUN
ejpam-4936	302	3	+	+	NUM
ejpam-4936	302	4	17z	17z	NOUN
ejpam-4936	302	5	=	=	SYM
ejpam-4936	302	6	w2	w2	NOUN
ejpam-4936	302	7	no	no	DET
ejpam-4936	302	8	solution	solution	NOUN
ejpam-4936	302	9	5(iii	5(iii	NUM
ejpam-4936	302	10	)	)	PUNCT
ejpam-4936	302	11	80	80	NUM
ejpam-4936	302	12	8x	8x	NOUN
ejpam-4936	302	13	+	+	SYM
ejpam-4936	302	14	13y	13y	NOUN
ejpam-4936	302	15	+	+	CCONJ
ejpam-4936	302	16	13z	13z	NOUN
ejpam-4936	302	17	=	=	SYM
ejpam-4936	302	18	w2	w2	NOUN
ejpam-4936	302	19	no	no	DET
ejpam-4936	302	20	solution	solution	NOUN
ejpam-4936	302	21	5(iii	5(iii	NUM
ejpam-4936	302	22	)	)	PUNCT
ejpam-4936	302	23	81	81	NUM
ejpam-4936	302	24	8x	8x	NOUN
ejpam-4936	302	25	+	+	CCONJ
ejpam-4936	302	26	13y	13y	NOUN
ejpam-4936	302	27	+	+	NUM
ejpam-4936	302	28	17z	17z	NOUN
ejpam-4936	302	29	=	=	SYM
ejpam-4936	302	30	w2	w2	NOUN
ejpam-4936	302	31	no	no	DET
ejpam-4936	302	32	solution	solution	NOUN
ejpam-4936	302	33	5(iii	5(iii	NUM
ejpam-4936	302	34	)	)	PUNCT
ejpam-4936	302	35	82	82	NUM
ejpam-4936	302	36	8x	8x	NOUN
ejpam-4936	302	37	+	+	X
ejpam-4936	302	38	17y	17y	NOUN
ejpam-4936	302	39	+	+	CCONJ
ejpam-4936	302	40	17z	17z	NOUN
ejpam-4936	302	41	=	=	SYM
ejpam-4936	302	42	w2	w2	NOUN
ejpam-4936	302	43	no	no	DET
ejpam-4936	302	44	solution	solution	NOUN
ejpam-4936	302	45	5(iii	5(iii	NUM
ejpam-4936	302	46	)	)	PUNCT
ejpam-4936	302	47	83	83	NUM
ejpam-4936	302	48	9x	9x	NOUN
ejpam-4936	303	1	+	+	CCONJ
ejpam-4936	303	2	9y	9y	NOUN
ejpam-4936	303	3	+	+	CCONJ
ejpam-4936	303	4	9z	9z	NUM
ejpam-4936	303	5	=	=	SYM
ejpam-4936	303	6	w2	w2	NOUN
ejpam-4936	303	7	no	no	DET
ejpam-4936	303	8	solution	solution	NOUN
ejpam-4936	303	9	5(i	5(i	NOUN
ejpam-4936	303	10	)	)	PUNCT
ejpam-4936	303	11	84	84	NUM
ejpam-4936	303	12	9x	9x	NOUN
ejpam-4936	303	13	+	+	CCONJ
ejpam-4936	303	14	9y	9y	NOUN
ejpam-4936	303	15	+	+	CCONJ
ejpam-4936	303	16	11z	11z	NOUN
ejpam-4936	303	17	=	=	SYM
ejpam-4936	303	18	w2	w2	NOUN
ejpam-4936	303	19	no	no	DET
ejpam-4936	303	20	solution	solution	NOUN
ejpam-4936	303	21	5(v	5(v	NUM
ejpam-4936	303	22	)	)	PUNCT
ejpam-4936	303	23	85	85	NUM
ejpam-4936	303	24	9x	9x	NOUN
ejpam-4936	304	1	+	+	CCONJ
ejpam-4936	304	2	9y	9y	NOUN
ejpam-4936	304	3	+	+	SYM
ejpam-4936	304	4	12z	12z	NOUN
ejpam-4936	304	5	=	=	SYM
ejpam-4936	304	6	w2	w2	NOUN
ejpam-4936	304	7	no	no	DET
ejpam-4936	304	8	solution	solution	NOUN
ejpam-4936	304	9	5(iii	5(iii	NUM
ejpam-4936	304	10	)	)	PUNCT
ejpam-4936	304	11	86	86	NUM
ejpam-4936	304	12	9x	9x	NOUN
ejpam-4936	305	1	+	+	CCONJ
ejpam-4936	305	2	9y	9y	NOUN
ejpam-4936	305	3	+	+	SYM
ejpam-4936	305	4	13z	13z	NOUN
ejpam-4936	305	5	=	=	SYM
ejpam-4936	305	6	w2	w2	NOUN
ejpam-4936	305	7	no	no	DET
ejpam-4936	305	8	solution	solution	NOUN
ejpam-4936	305	9	5(i	5(i	NOUN
ejpam-4936	305	10	)	)	PUNCT
ejpam-4936	305	11	87	87	NUM
ejpam-4936	305	12	9x	9x	NOUN
ejpam-4936	305	13	+	+	CCONJ
ejpam-4936	305	14	9y	9y	NOUN
ejpam-4936	305	15	+	+	NUM
ejpam-4936	305	16	16z	16z	NOUN
ejpam-4936	305	17	=	=	SYM
ejpam-4936	305	18	w2	w2	NOUN
ejpam-4936	305	19	no	no	DET
ejpam-4936	305	20	solution	solution	NOUN
ejpam-4936	305	21	5(iii	5(iii	NUM
ejpam-4936	305	22	)	)	PUNCT
ejpam-4936	305	23	88	88	NUM
ejpam-4936	305	24	9x	9x	NOUN
ejpam-4936	305	25	+	+	CCONJ
ejpam-4936	305	26	9y	9y	NOUN
ejpam-4936	305	27	+	+	NUM
ejpam-4936	305	28	17z	17z	NOUN
ejpam-4936	305	29	=	=	SYM
ejpam-4936	305	30	w2	w2	NOUN
ejpam-4936	305	31	no	no	DET
ejpam-4936	305	32	solution	solution	NOUN
ejpam-4936	305	33	5(i	5(i	NOUN
ejpam-4936	305	34	)	)	PUNCT
ejpam-4936	305	35	89	89	NUM
ejpam-4936	305	36	9x	9x	NOUN
ejpam-4936	306	1	+	+	CCONJ
ejpam-4936	306	2	9y	9y	NOUN
ejpam-4936	306	3	+	+	NUM
ejpam-4936	306	4	19z	19z	NOUN
ejpam-4936	307	1	=	=	SYM
ejpam-4936	307	2	w2	w2	NOUN
ejpam-4936	307	3	no	no	DET
ejpam-4936	307	4	solution	solution	NOUN
ejpam-4936	307	5	5(v	5(v	NUM
ejpam-4936	307	6	)	)	PUNCT
ejpam-4936	307	7	90	90	NUM
ejpam-4936	307	8	9x	9x	NOUN
ejpam-4936	307	9	+	+	CCONJ
ejpam-4936	307	10	9y	9y	NOUN
ejpam-4936	307	11	+	+	CCONJ
ejpam-4936	307	12	20z	20z	NOUN
ejpam-4936	307	13	=	=	NOUN
ejpam-4936	307	14	w2	w2	NOUN
ejpam-4936	307	15	no	no	DET
ejpam-4936	307	16	solution	solution	NOUN
ejpam-4936	307	17	5(iii	5(iii	NUM
ejpam-4936	307	18	)	)	PUNCT
ejpam-4936	307	19	91	91	NUM
ejpam-4936	307	20	9x	9x	NOUN
ejpam-4936	307	21	+	+	CCONJ
ejpam-4936	307	22	11y	11y	NOUN
ejpam-4936	307	23	+	+	CCONJ
ejpam-4936	307	24	11z	11z	NOUN
ejpam-4936	307	25	=	=	SYM
ejpam-4936	307	26	w2	w2	NOUN
ejpam-4936	307	27	no	no	DET
ejpam-4936	307	28	solution	solution	NOUN
ejpam-4936	307	29	5(vi	5(vi	NUM
ejpam-4936	307	30	)	)	PUNCT
ejpam-4936	307	31	92	92	NUM
ejpam-4936	307	32	9x	9x	NOUN
ejpam-4936	307	33	+	+	CCONJ
ejpam-4936	307	34	11y	11y	NOUN
ejpam-4936	307	35	+	+	CCONJ
ejpam-4936	307	36	17z	17z	X
ejpam-4936	307	37	=	=	SYM
ejpam-4936	307	38	w2	w2	NOUN
ejpam-4936	307	39	no	no	DET
ejpam-4936	307	40	solution	solution	NOUN
ejpam-4936	307	41	5(v	5(v	NUM
ejpam-4936	307	42	)	)	PUNCT
ejpam-4936	307	43	93	93	NUM
ejpam-4936	307	44	9x	9x	NUM
ejpam-4936	307	45	+	+	CCONJ
ejpam-4936	307	46	11y	11y	NOUN
ejpam-4936	307	47	+	+	NUM
ejpam-4936	307	48	19z	19z	NOUN
ejpam-4936	307	49	=	=	SYM
ejpam-4936	307	50	w2	w2	NOUN
ejpam-4936	307	51	no	no	DET
ejpam-4936	307	52	solution	solution	NOUN
ejpam-4936	307	53	5(vi	5(vi	NUM
ejpam-4936	307	54	)	)	PUNCT
ejpam-4936	307	55	94	94	NUM
ejpam-4936	307	56	9x	9x	NUM
ejpam-4936	307	57	+	+	CCONJ
ejpam-4936	307	58	12y	12y	NOUN
ejpam-4936	307	59	+	+	CCONJ
ejpam-4936	307	60	13z	13z	NOUN
ejpam-4936	307	61	=	=	SYM
ejpam-4936	307	62	w2	w2	NOUN
ejpam-4936	307	63	no	no	DET
ejpam-4936	307	64	solution	solution	NOUN
ejpam-4936	307	65	5(iii	5(iii	NUM
ejpam-4936	307	66	)	)	PUNCT
ejpam-4936	307	67	95	95	NUM
ejpam-4936	307	68	9x	9x	NOUN
ejpam-4936	307	69	+	+	CCONJ
ejpam-4936	307	70	12y	12y	NOUN
ejpam-4936	307	71	+	+	CCONJ
ejpam-4936	307	72	17z	17z	NOUN
ejpam-4936	307	73	=	=	SYM
ejpam-4936	307	74	w2	w2	NOUN
ejpam-4936	307	75	no	no	DET
ejpam-4936	307	76	solution	solution	NOUN
ejpam-4936	307	77	5(iii	5(iii	NUM
ejpam-4936	307	78	)	)	PUNCT
ejpam-4936	307	79	96	96	NUM
ejpam-4936	307	80	9x	9x	NOUN
ejpam-4936	307	81	+	+	CCONJ
ejpam-4936	307	82	13y	13y	NOUN
ejpam-4936	307	83	+	+	CCONJ
ejpam-4936	307	84	13z	13z	NOUN
ejpam-4936	307	85	=	=	SYM
ejpam-4936	307	86	w2	w2	NOUN
ejpam-4936	307	87	no	no	DET
ejpam-4936	307	88	solution	solution	NOUN
ejpam-4936	307	89	5(i	5(i	NOUN
ejpam-4936	307	90	)	)	PUNCT
ejpam-4936	307	91	97	97	NUM
ejpam-4936	307	92	9x	9x	NOUN
ejpam-4936	307	93	+	+	CCONJ
ejpam-4936	307	94	13y	13y	NUM
ejpam-4936	307	95	+	+	CCONJ
ejpam-4936	307	96	16z	16z	NOUN
ejpam-4936	307	97	=	=	SYM
ejpam-4936	307	98	w2	w2	PROPN
ejpam-4936	307	99	no	no	DET
ejpam-4936	307	100	solution	solution	NOUN
ejpam-4936	307	101	5(iii	5(iii	NUM
ejpam-4936	307	102	)	)	PUNCT
ejpam-4936	307	103	98	98	NUM
ejpam-4936	307	104	9x	9x	NOUN
ejpam-4936	307	105	+	+	CCONJ
ejpam-4936	307	106	13y	13y	NOUN
ejpam-4936	307	107	+	+	NUM
ejpam-4936	307	108	17z	17z	NOUN
ejpam-4936	307	109	=	=	SYM
ejpam-4936	307	110	w2	w2	NOUN
ejpam-4936	307	111	no	no	DET
ejpam-4936	307	112	solution	solution	NOUN
ejpam-4936	307	113	5(i	5(i	NOUN
ejpam-4936	307	114	)	)	PUNCT
ejpam-4936	307	115	99	99	NUM
ejpam-4936	307	116	9x	9x	NOUN
ejpam-4936	307	117	+	+	CCONJ
ejpam-4936	307	118	13y	13y	NOUN
ejpam-4936	307	119	+	+	NUM
ejpam-4936	307	120	20z	20z	NOUN
ejpam-4936	307	121	=	=	NOUN
ejpam-4936	307	122	w2	w2	NOUN
ejpam-4936	307	123	no	no	DET
ejpam-4936	307	124	solution	solution	NOUN
ejpam-4936	307	125	5(iii	5(iii	NUM
ejpam-4936	307	126	)	)	PUNCT
ejpam-4936	307	127	100	100	NUM
ejpam-4936	307	128	9x	9x	NUM
ejpam-4936	307	129	+	+	CCONJ
ejpam-4936	307	130	16y	16y	NOUN
ejpam-4936	307	131	+	+	CCONJ
ejpam-4936	307	132	17z	17z	NOUN
ejpam-4936	307	133	=	=	SYM
ejpam-4936	307	134	w2	w2	NOUN
ejpam-4936	307	135	no	no	DET
ejpam-4936	307	136	solution	solution	NOUN
ejpam-4936	307	137	5(iii	5(iii	NUM
ejpam-4936	307	138	)	)	PUNCT
ejpam-4936	307	139	101	101	NUM
ejpam-4936	307	140	9x	9x	X
ejpam-4936	307	141	+	+	CCONJ
ejpam-4936	307	142	17y	17y	NOUN
ejpam-4936	307	143	+	+	CCONJ
ejpam-4936	307	144	17z	17z	NOUN
ejpam-4936	307	145	=	=	SYM
ejpam-4936	307	146	w2	w2	NOUN
ejpam-4936	307	147	no	no	DET
ejpam-4936	307	148	solution	solution	NOUN
ejpam-4936	307	149	5(i	5(i	NOUN
ejpam-4936	307	150	)	)	PUNCT
ejpam-4936	307	151	102	102	NUM
ejpam-4936	307	152	9x	9x	X
ejpam-4936	307	153	+	+	CCONJ
ejpam-4936	307	154	17y	17y	NOUN
ejpam-4936	307	155	+	+	NUM
ejpam-4936	307	156	19z	19z	NOUN
ejpam-4936	307	157	=	=	SYM
ejpam-4936	307	158	w2	w2	NOUN
ejpam-4936	307	159	no	no	DET
ejpam-4936	307	160	solution	solution	NOUN
ejpam-4936	307	161	5(v	5(v	NUM
ejpam-4936	307	162	)	)	PUNCT
ejpam-4936	307	163	103	103	NUM
ejpam-4936	307	164	9x	9x	X
ejpam-4936	307	165	+	+	SYM
ejpam-4936	307	166	17y	17y	NOUN
ejpam-4936	307	167	+	+	NUM
ejpam-4936	307	168	20z	20z	NOUN
ejpam-4936	307	169	=	=	NOUN
ejpam-4936	307	170	w2	w2	NOUN
ejpam-4936	307	171	no	no	DET
ejpam-4936	307	172	solution	solution	NOUN
ejpam-4936	307	173	5(iii	5(iii	NUM
ejpam-4936	307	174	)	)	PUNCT
ejpam-4936	307	175	104	104	NUM
ejpam-4936	307	176	9x	9x	NOUN
ejpam-4936	307	177	+	+	CCONJ
ejpam-4936	307	178	19y	19y	NOUN
ejpam-4936	307	179	+	+	CCONJ
ejpam-4936	307	180	19z	19z	NOUN
ejpam-4936	307	181	=	=	SYM
ejpam-4936	307	182	w2	w2	NOUN
ejpam-4936	307	183	no	no	DET
ejpam-4936	307	184	solution	solution	NOUN
ejpam-4936	307	185	5(vi	5(vi	NUM
ejpam-4936	307	186	)	)	PUNCT
ejpam-4936	307	187	105	105	NUM
ejpam-4936	307	188	10x	10x	NUM
ejpam-4936	307	189	+	+	SYM
ejpam-4936	307	190	11y	11y	NOUN
ejpam-4936	307	191	+	+	SYM
ejpam-4936	307	192	11z	11z	NOUN
ejpam-4936	307	193	=	=	SYM
ejpam-4936	307	194	w2	w2	NOUN
ejpam-4936	307	195	no	no	DET
ejpam-4936	307	196	solution	solution	NOUN
ejpam-4936	307	197	5(iv	5(iv	NUM
ejpam-4936	307	198	)	)	PUNCT
ejpam-4936	307	199	106	106	NUM
ejpam-4936	307	200	10x	10x	NUM
ejpam-4936	307	201	+	+	SYM
ejpam-4936	307	202	11y	11y	NOUN
ejpam-4936	307	203	+	+	SYM
ejpam-4936	307	204	16z	16z	NOUN
ejpam-4936	307	205	=	=	SYM
ejpam-4936	307	206	w2	w2	PROPN
ejpam-4936	307	207	no	no	DET
ejpam-4936	307	208	solution	solution	NOUN
ejpam-4936	307	209	5(iv	5(iv	NUM
ejpam-4936	307	210	)	)	PUNCT
ejpam-4936	307	211	107	107	NUM
ejpam-4936	307	212	10x	10x	NUM
ejpam-4936	307	213	+	+	CCONJ
ejpam-4936	307	214	16y	16y	NUM
ejpam-4936	307	215	+	+	CCONJ
ejpam-4936	307	216	16z	16z	NOUN
ejpam-4936	307	217	=	=	SYM
ejpam-4936	307	218	w2	w2	PROPN
ejpam-4936	307	219	no	no	DET
ejpam-4936	307	220	solution	solution	NOUN
ejpam-4936	307	221	5(iv	5(iv	NUM
ejpam-4936	307	222	)	)	PUNCT
ejpam-4936	307	223	108	108	NUM
ejpam-4936	307	224	11x	11x	NOUN
ejpam-4936	307	225	+	+	SYM
ejpam-4936	307	226	11y	11y	NOUN
ejpam-4936	307	227	+	+	CCONJ
ejpam-4936	307	228	11z	11z	NOUN
ejpam-4936	307	229	=	=	SYM
ejpam-4936	307	230	w2	w2	NOUN
ejpam-4936	307	231	no	no	DET
ejpam-4936	307	232	solution	solution	NOUN
ejpam-4936	307	233	5(ii	5(ii	NUM
ejpam-4936	307	234	)	)	PUNCT
ejpam-4936	307	235	109	109	NUM
ejpam-4936	307	236	11x	11x	NOUN
ejpam-4936	307	237	+	+	SYM
ejpam-4936	307	238	11y	11y	NOUN
ejpam-4936	307	239	+	+	CCONJ
ejpam-4936	307	240	15z	15z	NOUN
ejpam-4936	307	241	=	=	SYM
ejpam-4936	307	242	w2	w2	NOUN
ejpam-4936	307	243	no	no	DET
ejpam-4936	307	244	solution	solution	NOUN
ejpam-4936	307	245	5(iv	5(iv	NUM
ejpam-4936	307	246	)	)	PUNCT
ejpam-4936	307	247	110	110	NUM
ejpam-4936	307	248	11x	11x	NOUN
ejpam-4936	307	249	+	+	SYM
ejpam-4936	307	250	11y	11y	NOUN
ejpam-4936	307	251	+	+	CCONJ
ejpam-4936	307	252	16z	16z	NOUN
ejpam-4936	307	253	=	=	SYM
ejpam-4936	307	254	w2	w2	PROPN
ejpam-4936	307	255	no	no	DET
ejpam-4936	307	256	solution	solution	NOUN
ejpam-4936	307	257	5(ii	5(ii	NUM
ejpam-4936	307	258	)	)	PUNCT
ejpam-4936	307	259	references	reference	NOUN
ejpam-4936	307	260	2079	2079	NUM
ejpam-4936	307	261	table	table	NOUN
ejpam-4936	307	262	5	5	NUM
ejpam-4936	307	263	:	:	PUNCT
ejpam-4936	307	264	the	the	DET
ejpam-4936	307	265	list	list	NOUN
ejpam-4936	307	266	of	of	ADP
ejpam-4936	307	267	the	the	DET
ejpam-4936	307	268	equations	equation	NOUN
ejpam-4936	307	269	ax	ax	NOUN
ejpam-4936	307	270	+	+	CCONJ
ejpam-4936	307	271	by	by	ADP
ejpam-4936	307	272	+	+	CCONJ
ejpam-4936	307	273	cz	cz	NOUN
ejpam-4936	307	274	=	=	SYM
ejpam-4936	307	275	w2	w2	NOUN
ejpam-4936	307	276	where	where	SCONJ
ejpam-4936	307	277	2	2	NUM
ejpam-4936	307	278	≤	≤	NOUN
ejpam-4936	307	279	a	a	DET
ejpam-4936	307	280	≤	≤	NUM
ejpam-4936	307	281	b	b	NOUN
ejpam-4936	307	282	≤	≤	NUM
ejpam-4936	307	283	c	c	NOUN
ejpam-4936	307	284	≤	≤	NUM
ejpam-4936	307	285	20	20	NUM
ejpam-4936	307	286	and	and	CCONJ
ejpam-4936	307	287	all	all	DET
ejpam-4936	307	288	variables	variable	NOUN
ejpam-4936	307	289	a	a	PRON
ejpam-4936	307	290	,	,	PUNCT
ejpam-4936	307	291	b	b	PROPN
ejpam-4936	307	292	and	and	CCONJ
ejpam-4936	307	293	c	c	PROPN
ejpam-4936	307	294	are	be	AUX
ejpam-4936	307	295	satisfied	satisfied	ADJ
ejpam-4936	307	296	our	our	PRON
ejpam-4936	307	297	results	result	NOUN
ejpam-4936	307	298	.	.	PUNCT
ejpam-4936	308	1	no	no	INTJ
ejpam-4936	308	2	.	.	PUNCT
ejpam-4936	309	1	ax	ax	NOUN
ejpam-4936	309	2	+	+	CCONJ
ejpam-4936	309	3	by	by	ADP
ejpam-4936	309	4	+	+	CCONJ
ejpam-4936	309	5	cz	cz	NOUN
ejpam-4936	309	6	=	=	NOUN
ejpam-4936	309	7	w2	w2	NOUN
ejpam-4936	309	8	result	result	NOUN
ejpam-4936	309	9	ref	ref	NOUN
ejpam-4936	309	10	.	.	PUNCT
ejpam-4936	310	1	of	of	ADP
ejpam-4936	310	2	theorem	theorem	ADJ
ejpam-4936	310	3	111	111	NUM
ejpam-4936	310	4	11x	11x	NOUN
ejpam-4936	310	5	+	+	X
ejpam-4936	310	6	11y	11y	NOUN
ejpam-4936	310	7	+	+	CCONJ
ejpam-4936	310	8	17z	17z	X
ejpam-4936	310	9	=	=	SYM
ejpam-4936	310	10	w2	w2	NOUN
ejpam-4936	310	11	no	no	DET
ejpam-4936	310	12	solution	solution	NOUN
ejpam-4936	310	13	5(vi	5(vi	NUM
ejpam-4936	310	14	)	)	PUNCT
ejpam-4936	310	15	112	112	NUM
ejpam-4936	310	16	11x	11x	NOUN
ejpam-4936	310	17	+	+	SYM
ejpam-4936	310	18	11y	11y	NOUN
ejpam-4936	310	19	+	+	NUM
ejpam-4936	310	20	20z	20z	NOUN
ejpam-4936	310	21	=	=	NOUN
ejpam-4936	310	22	w2	w2	NOUN
ejpam-4936	310	23	no	no	DET
ejpam-4936	310	24	solution	solution	NOUN
ejpam-4936	310	25	5(iv	5(iv	NUM
ejpam-4936	310	26	)	)	PUNCT
ejpam-4936	310	27	113	113	NUM
ejpam-4936	310	28	11x	11x	NOUN
ejpam-4936	310	29	+	+	CCONJ
ejpam-4936	310	30	15y	15y	ADJ
ejpam-4936	310	31	+	+	NUM
ejpam-4936	310	32	16z	16z	NOUN
ejpam-4936	310	33	=	=	SYM
ejpam-4936	310	34	w2	w2	PROPN
ejpam-4936	310	35	no	no	DET
ejpam-4936	310	36	solution	solution	NOUN
ejpam-4936	310	37	5(iv	5(iv	NUM
ejpam-4936	310	38	)	)	PUNCT
ejpam-4936	310	39	114	114	NUM
ejpam-4936	310	40	11x	11x	NOUN
ejpam-4936	310	41	+	+	CCONJ
ejpam-4936	310	42	16y	16y	NUM
ejpam-4936	310	43	+	+	CCONJ
ejpam-4936	310	44	16z	16z	NOUN
ejpam-4936	310	45	=	=	SYM
ejpam-4936	310	46	w2	w2	PROPN
ejpam-4936	310	47	no	no	DET
ejpam-4936	310	48	solution	solution	NOUN
ejpam-4936	310	49	5(ii	5(ii	NUM
ejpam-4936	310	50	)	)	PUNCT
ejpam-4936	310	51	115	115	NUM
ejpam-4936	310	52	11x	11x	NOUN
ejpam-4936	310	53	+	+	CCONJ
ejpam-4936	310	54	16y	16y	NUM
ejpam-4936	310	55	+	+	CCONJ
ejpam-4936	310	56	20z	20z	NOUN
ejpam-4936	310	57	=	=	NOUN
ejpam-4936	310	58	w2	w2	NOUN
ejpam-4936	310	59	no	no	DET
ejpam-4936	310	60	solution	solution	NOUN
ejpam-4936	310	61	5(iv	5(iv	NUM
ejpam-4936	310	62	)	)	PUNCT
ejpam-4936	310	63	116	116	NUM
ejpam-4936	310	64	11x	11x	NOUN
ejpam-4936	310	65	+	+	ADJ
ejpam-4936	310	66	17y	17y	NOUN
ejpam-4936	310	67	+	+	CCONJ
ejpam-4936	310	68	17z	17z	NOUN
ejpam-4936	310	69	=	=	SYM
ejpam-4936	310	70	w2	w2	NOUN
ejpam-4936	310	71	no	no	DET
ejpam-4936	310	72	solution	solution	NOUN
ejpam-4936	310	73	5(v	5(v	NUM
ejpam-4936	310	74	)	)	PUNCT
ejpam-4936	310	75	117	117	NUM
ejpam-4936	310	76	11x	11x	NOUN
ejpam-4936	310	77	+	+	ADJ
ejpam-4936	310	78	17y	17y	NOUN
ejpam-4936	310	79	+	+	NUM
ejpam-4936	310	80	19z	19z	NOUN
ejpam-4936	310	81	=	=	SYM
ejpam-4936	310	82	w2	w2	NOUN
ejpam-4936	310	83	no	no	DET
ejpam-4936	310	84	solution	solution	NOUN
ejpam-4936	310	85	5(vi	5(vi	NUM
ejpam-4936	310	86	)	)	PUNCT
ejpam-4936	310	87	118	118	NUM
ejpam-4936	310	88	12x	12x	NOUN
ejpam-4936	310	89	+	+	CCONJ
ejpam-4936	310	90	13y	13y	NOUN
ejpam-4936	310	91	+	+	CCONJ
ejpam-4936	310	92	13z	13z	NOUN
ejpam-4936	310	93	=	=	SYM
ejpam-4936	310	94	w2	w2	NOUN
ejpam-4936	310	95	no	no	DET
ejpam-4936	310	96	solution	solution	NOUN
ejpam-4936	310	97	5(iii	5(iii	NUM
ejpam-4936	310	98	)	)	PUNCT
ejpam-4936	310	99	119	119	NUM
ejpam-4936	310	100	12x	12x	NOUN
ejpam-4936	310	101	+	+	CCONJ
ejpam-4936	310	102	13y	13y	NOUN
ejpam-4936	310	103	+	+	CCONJ
ejpam-4936	310	104	17z	17z	NOUN
ejpam-4936	310	105	=	=	SYM
ejpam-4936	310	106	w2	w2	NOUN
ejpam-4936	310	107	no	no	DET
ejpam-4936	310	108	solution	solution	NOUN
ejpam-4936	310	109	5(iii	5(iii	NUM
ejpam-4936	310	110	)	)	PUNCT
ejpam-4936	310	111	120	120	NUM
ejpam-4936	310	112	12x	12x	NOUN
ejpam-4936	310	113	+	+	ADJ
ejpam-4936	310	114	17y	17y	NOUN
ejpam-4936	310	115	+	+	CCONJ
ejpam-4936	310	116	17z	17z	NOUN
ejpam-4936	310	117	=	=	SYM
ejpam-4936	310	118	w2	w2	NOUN
ejpam-4936	310	119	no	no	DET
ejpam-4936	310	120	solution	solution	NOUN
ejpam-4936	310	121	5(iii	5(iii	NUM
ejpam-4936	310	122	)	)	PUNCT
ejpam-4936	310	123	121	121	NUM
ejpam-4936	310	124	13x	13x	NUM
ejpam-4936	310	125	+	+	CCONJ
ejpam-4936	310	126	13y	13y	NOUN
ejpam-4936	310	127	+	+	CCONJ
ejpam-4936	310	128	13z	13z	NOUN
ejpam-4936	310	129	=	=	SYM
ejpam-4936	310	130	w2	w2	NOUN
ejpam-4936	310	131	no	no	DET
ejpam-4936	310	132	solution	solution	NOUN
ejpam-4936	310	133	5(i	5(i	NOUN
ejpam-4936	310	134	)	)	PUNCT
ejpam-4936	310	135	122	122	NUM
ejpam-4936	310	136	13x	13x	NUM
ejpam-4936	310	137	+	+	CCONJ
ejpam-4936	310	138	13y	13y	NOUN
ejpam-4936	310	139	+	+	CCONJ
ejpam-4936	310	140	16z	16z	NOUN
ejpam-4936	310	141	=	=	SYM
ejpam-4936	310	142	w2	w2	PROPN
ejpam-4936	310	143	no	no	DET
ejpam-4936	310	144	solution	solution	NOUN
ejpam-4936	310	145	5(iii	5(iii	NUM
ejpam-4936	310	146	)	)	PUNCT
ejpam-4936	310	147	123	123	NUM
ejpam-4936	310	148	13x	13x	NUM
ejpam-4936	310	149	+	+	SYM
ejpam-4936	310	150	13y	13y	NOUN
ejpam-4936	310	151	+	+	NUM
ejpam-4936	310	152	17z	17z	NOUN
ejpam-4936	310	153	=	=	SYM
ejpam-4936	310	154	w2	w2	NOUN
ejpam-4936	310	155	no	no	DET
ejpam-4936	310	156	solution	solution	NOUN
ejpam-4936	310	157	5(i	5(i	NOUN
ejpam-4936	310	158	)	)	PUNCT
ejpam-4936	310	159	124	124	NUM
ejpam-4936	310	160	13x	13x	NUM
ejpam-4936	310	161	+	+	CCONJ
ejpam-4936	310	162	13y	13y	NOUN
ejpam-4936	310	163	+	+	NUM
ejpam-4936	310	164	20z	20z	NOUN
ejpam-4936	310	165	=	=	NOUN
ejpam-4936	310	166	w2	w2	NOUN
ejpam-4936	310	167	no	no	DET
ejpam-4936	310	168	solution	solution	NOUN
ejpam-4936	310	169	5(iii	5(iii	NUM
ejpam-4936	310	170	)	)	PUNCT
ejpam-4936	310	171	125	125	NUM
ejpam-4936	310	172	13x	13x	NUM
ejpam-4936	310	173	+	+	CCONJ
ejpam-4936	310	174	16y	16y	NOUN
ejpam-4936	310	175	+	+	CCONJ
ejpam-4936	310	176	17z	17z	NOUN
ejpam-4936	310	177	=	=	SYM
ejpam-4936	310	178	w2	w2	NOUN
ejpam-4936	310	179	no	no	DET
ejpam-4936	310	180	solution	solution	NOUN
ejpam-4936	310	181	5(iii	5(iii	NUM
ejpam-4936	310	182	)	)	PUNCT
ejpam-4936	310	183	126	126	NUM
ejpam-4936	310	184	13x	13x	NUM
ejpam-4936	310	185	+	+	SYM
ejpam-4936	310	186	17y	17y	NOUN
ejpam-4936	310	187	+	+	SYM
ejpam-4936	310	188	17z	17z	NOUN
ejpam-4936	310	189	=	=	SYM
ejpam-4936	310	190	w2	w2	NOUN
ejpam-4936	310	191	no	no	DET
ejpam-4936	310	192	solution	solution	NOUN
ejpam-4936	310	193	5(i	5(i	NOUN
ejpam-4936	310	194	)	)	PUNCT
ejpam-4936	310	195	127	127	NUM
ejpam-4936	310	196	13x	13x	NUM
ejpam-4936	310	197	+	+	SYM
ejpam-4936	310	198	17y	17y	NOUN
ejpam-4936	310	199	+	+	NUM
ejpam-4936	310	200	20z	20z	NOUN
ejpam-4936	310	201	=	=	NOUN
ejpam-4936	310	202	w2	w2	NOUN
ejpam-4936	310	203	no	no	DET
ejpam-4936	310	204	solution	solution	NOUN
ejpam-4936	310	205	5(iii	5(iii	NUM
ejpam-4936	310	206	)	)	PUNCT
ejpam-4936	310	207	128	128	NUM
ejpam-4936	310	208	15x	15x	NUM
ejpam-4936	310	209	+	+	CCONJ
ejpam-4936	310	210	16y	16y	NUM
ejpam-4936	310	211	+	+	CCONJ
ejpam-4936	310	212	16z	16z	NOUN
ejpam-4936	310	213	=	=	SYM
ejpam-4936	310	214	w2	w2	PROPN
ejpam-4936	310	215	no	no	DET
ejpam-4936	310	216	solution	solution	NOUN
ejpam-4936	310	217	5(iv	5(iv	NUM
ejpam-4936	310	218	)	)	PUNCT
ejpam-4936	310	219	129	129	NUM
ejpam-4936	310	220	16x	16x	NOUN
ejpam-4936	310	221	+	+	CCONJ
ejpam-4936	310	222	16y	16y	NUM
ejpam-4936	310	223	+	+	CCONJ
ejpam-4936	310	224	16z	16z	NOUN
ejpam-4936	310	225	=	=	SYM
ejpam-4936	310	226	w2	w2	PROPN
ejpam-4936	310	227	no	no	DET
ejpam-4936	310	228	solution	solution	NOUN
ejpam-4936	310	229	5(ii	5(ii	NUM
ejpam-4936	310	230	)	)	PUNCT
ejpam-4936	310	231	130	130	NUM
ejpam-4936	310	232	16x	16x	NOUN
ejpam-4936	310	233	+	+	CCONJ
ejpam-4936	310	234	16y	16y	NUM
ejpam-4936	310	235	+	+	CCONJ
ejpam-4936	310	236	20z	20z	NOUN
ejpam-4936	310	237	=	=	NOUN
ejpam-4936	310	238	w2	w2	NOUN
ejpam-4936	310	239	no	no	DET
ejpam-4936	310	240	solution	solution	NOUN
ejpam-4936	310	241	5(iv	5(iv	NUM
ejpam-4936	310	242	)	)	PUNCT
ejpam-4936	310	243	131	131	NUM
ejpam-4936	310	244	16x	16x	NOUN
ejpam-4936	310	245	+	+	CCONJ
ejpam-4936	310	246	17y	17y	NOUN
ejpam-4936	310	247	+	+	CCONJ
ejpam-4936	310	248	17z	17z	NOUN
ejpam-4936	310	249	=	=	SYM
ejpam-4936	310	250	w2	w2	NOUN
ejpam-4936	310	251	no	no	DET
ejpam-4936	310	252	solution	solution	NOUN
ejpam-4936	310	253	5(iii	5(iii	NUM
ejpam-4936	310	254	)	)	PUNCT
ejpam-4936	310	255	132	132	NUM
ejpam-4936	310	256	17x	17x	NOUN
ejpam-4936	310	257	+	+	NUM
ejpam-4936	310	258	17y	17y	NOUN
ejpam-4936	310	259	+	+	SYM
ejpam-4936	310	260	17z	17z	NOUN
ejpam-4936	310	261	=	=	SYM
ejpam-4936	310	262	w2	w2	NOUN
ejpam-4936	310	263	no	no	DET
ejpam-4936	310	264	solution	solution	NOUN
ejpam-4936	310	265	5(i	5(i	NOUN
ejpam-4936	310	266	)	)	PUNCT
ejpam-4936	310	267	133	133	NUM
ejpam-4936	310	268	17x	17x	NOUN
ejpam-4936	310	269	+	+	NUM
ejpam-4936	310	270	17y	17y	NOUN
ejpam-4936	310	271	+	+	NUM
ejpam-4936	310	272	19z	19z	NOUN
ejpam-4936	310	273	=	=	SYM
ejpam-4936	310	274	w2	w2	NOUN
ejpam-4936	310	275	no	no	DET
ejpam-4936	310	276	solution	solution	NOUN
ejpam-4936	310	277	5(v	5(v	NUM
ejpam-4936	310	278	)	)	PUNCT
ejpam-4936	310	279	134	134	NUM
ejpam-4936	310	280	17x	17x	NOUN
ejpam-4936	310	281	+	+	NUM
ejpam-4936	310	282	17y	17y	NOUN
ejpam-4936	310	283	+	+	NUM
ejpam-4936	310	284	20z	20z	NOUN
ejpam-4936	310	285	=	=	NOUN
ejpam-4936	310	286	w2	w2	NOUN
ejpam-4936	310	287	no	no	DET
ejpam-4936	310	288	solution	solution	NOUN
ejpam-4936	310	289	5(iii	5(iii	NUM
ejpam-4936	310	290	)	)	PUNCT
ejpam-4936	310	291	135	135	NUM
ejpam-4936	310	292	17x	17x	NOUN
ejpam-4936	310	293	+	+	CCONJ
ejpam-4936	310	294	19y	19y	NOUN
ejpam-4936	310	295	+	+	CCONJ
ejpam-4936	310	296	19z	19z	NOUN
ejpam-4936	310	297	=	=	SYM
ejpam-4936	310	298	w2	w2	NOUN
ejpam-4936	310	299	no	no	DET
ejpam-4936	310	300	solution	solution	NOUN
ejpam-4936	310	301	5(vi	5(vi	NUM
ejpam-4936	310	302	)	)	PUNCT
ejpam-4936	310	303	acknowledgements	acknowledgement	NOUN
ejpam-4936	310	304	the	the	DET
ejpam-4936	310	305	authors	author	NOUN
ejpam-4936	310	306	would	would	AUX
ejpam-4936	310	307	like	like	VERB
ejpam-4936	310	308	to	to	PART
ejpam-4936	310	309	thank	thank	VERB
ejpam-4936	310	310	pimchanok	pimchanok	PROPN
ejpam-4936	310	311	pimton	pimton	NOUN
ejpam-4936	310	312	and	and	CCONJ
ejpam-4936	310	313	jutarat	jutarat	PROPN
ejpam-4936	310	314	wiriyadamrikul	wiriyadamrikul	PROPN
ejpam-4936	310	315	for	for	ADP
ejpam-4936	310	316	her	her	PRON
ejpam-4936	310	317	useful	useful	ADJ
ejpam-4936	310	318	suggestions	suggestion	NOUN
ejpam-4936	310	319	.	.	PUNCT
ejpam-4936	311	1	this	this	DET
ejpam-4936	311	2	paper	paper	NOUN
ejpam-4936	311	3	is	be	AUX
ejpam-4936	311	4	funded	fund	VERB
ejpam-4936	311	5	by	by	ADP
ejpam-4936	311	6	walailak	walailak	ADJ
ejpam-4936	311	7	university	university	NOUN
ejpam-4936	311	8	.	.	PUNCT
ejpam-4936	312	1	references	reference	NOUN
ejpam-4936	312	2	[	[	X
ejpam-4936	312	3	1	1	NUM
ejpam-4936	312	4	]	]	X
ejpam-4936	312	5	d.	d.	PROPN
ejpam-4936	312	6	acu	acu	PROPN
ejpam-4936	312	7	.	.	PUNCT
ejpam-4936	313	1	on	on	ADP
ejpam-4936	313	2	the	the	DET
ejpam-4936	313	3	diophantine	diophantine	NOUN
ejpam-4936	313	4	equation	equation	NOUN
ejpam-4936	313	5	of	of	ADP
ejpam-4936	313	6	type	type	NOUN
ejpam-4936	313	7	ax	ax	NOUN
ejpam-4936	313	8	+	+	CCONJ
ejpam-4936	313	9	by	by	ADP
ejpam-4936	313	10	=	=	NOUN
ejpam-4936	313	11	cz	cz	NOUN
ejpam-4936	313	12	.	.	PUNCT
ejpam-4936	313	13	general	general	ADJ
ejpam-4936	313	14	mathematics	mathematics	PROPN
ejpam-4936	313	15	,	,	PUNCT
ejpam-4936	313	16	13(1):67–72	13(1):67–72	NUM
ejpam-4936	313	17	,	,	PUNCT
ejpam-4936	313	18	2005	2005	NUM
ejpam-4936	313	19	.	.	PUNCT
ejpam-4936	314	1	[	[	X
ejpam-4936	314	2	2	2	NUM
ejpam-4936	314	3	]	]	X
ejpam-4936	314	4	d.	d.	PROPN
ejpam-4936	314	5	acu	acu	PROPN
ejpam-4936	314	6	.	.	PUNCT
ejpam-4936	315	1	on	on	ADP
ejpam-4936	315	2	the	the	DET
ejpam-4936	315	3	diophantine	diophantine	NOUN
ejpam-4936	315	4	equation	equation	NOUN
ejpam-4936	315	5	2x+5y	2x+5y	NUM
ejpam-4936	315	6	=	=	SYM
ejpam-4936	315	7	z2	z2	PROPN
ejpam-4936	315	8	.	.	PUNCT
ejpam-4936	316	1	general	general	ADJ
ejpam-4936	316	2	mathematics	mathematic	NOUN
ejpam-4936	316	3	,	,	PUNCT
ejpam-4936	316	4	15(4):145	15(4):145	NUM
ejpam-4936	316	5	–	–	PUNCT
ejpam-4936	316	6	148	148	NUM
ejpam-4936	316	7	,	,	PUNCT
ejpam-4936	316	8	2007	2007	NUM
ejpam-4936	316	9	.	.	PUNCT
ejpam-4936	317	1	references	reference	NOUN
ejpam-4936	317	2	2080	2080	NUM
ejpam-4936	317	3	[	[	X
ejpam-4936	317	4	3	3	X
ejpam-4936	317	5	]	]	PUNCT
ejpam-4936	317	6	j.	j.	PROPN
ejpam-4936	317	7	b.	b.	PROPN
ejpam-4936	317	8	bacani	bacani	PROPN
ejpam-4936	317	9	and	and	CCONJ
ejpam-4936	317	10	j.	j.	PROPN
ejpam-4936	317	11	f.	f.	PROPN
ejpam-4936	317	12	t.	t.	PROPN
ejpam-4936	317	13	rabago	rabago	PROPN
ejpam-4936	317	14	.	.	PUNCT
ejpam-4936	318	1	on	on	ADP
ejpam-4936	318	2	the	the	DET
ejpam-4936	318	3	diophantine	diophantine	NOUN
ejpam-4936	318	4	equation	equation	NOUN
ejpam-4936	318	5	3x	3x	NOUN
ejpam-4936	318	6	+	+	CCONJ
ejpam-4936	318	7	5y	5y	NOUN
ejpam-4936	318	8	+	+	X
ejpam-4936	318	9	7z	7z	NOUN
ejpam-4936	318	10	=	=	SYM
ejpam-4936	318	11	w2	w2	PROPN
ejpam-4936	318	12	.	.	PUNCT
ejpam-4936	319	1	konuralp	konuralp	PROPN
ejpam-4936	319	2	journal	journal	PROPN
ejpam-4936	319	3	of	of	ADP
ejpam-4936	319	4	mathematics	mathematic	NOUN
ejpam-4936	319	5	,	,	PUNCT
ejpam-4936	319	6	22(1):64–69	22(1):64–69	NUM
ejpam-4936	319	7	,	,	PUNCT
ejpam-4936	319	8	2014	2014	NUM
ejpam-4936	319	9	.	.	PUNCT
ejpam-4936	320	1	[	[	X
ejpam-4936	320	2	4	4	X
ejpam-4936	320	3	]	]	PUNCT
ejpam-4936	320	4	j.	j.	PROPN
ejpam-4936	320	5	b.	b.	PROPN
ejpam-4936	320	6	bacani	bacani	PROPN
ejpam-4936	320	7	and	and	CCONJ
ejpam-4936	320	8	j.	j.	PROPN
ejpam-4936	320	9	f.	f.	PROPN
ejpam-4936	320	10	t.	t.	PROPN
ejpam-4936	320	11	rabago	rabago	PROPN
ejpam-4936	320	12	.	.	PUNCT
ejpam-4936	321	1	the	the	DET
ejpam-4936	321	2	complete	complete	ADJ
ejpam-4936	321	3	set	set	NOUN
ejpam-4936	321	4	of	of	ADP
ejpam-4936	321	5	solutions	solution	NOUN
ejpam-4936	321	6	of	of	ADP
ejpam-4936	321	7	the	the	DET
ejpam-4936	321	8	diophantine	diophantine	NOUN
ejpam-4936	321	9	equation	equation	NOUN
ejpam-4936	321	10	px	px	X
ejpam-4936	321	11	+	+	CCONJ
ejpam-4936	321	12	qy	qy	NOUN
ejpam-4936	321	13	=	=	PROPN
ejpam-4936	321	14	z2	z2	PROPN
ejpam-4936	321	15	for	for	ADP
ejpam-4936	321	16	twin	twin	ADJ
ejpam-4936	321	17	primes	prime	NOUN
ejpam-4936	321	18	p	p	NOUN
ejpam-4936	321	19	and	and	CCONJ
ejpam-4936	321	20	q.	q.	PROPN
ejpam-4936	321	21	international	international	PROPN
ejpam-4936	321	22	journal	journal	PROPN
ejpam-4936	321	23	of	of	ADP
ejpam-4936	321	24	pure	pure	ADJ
ejpam-4936	321	25	and	and	CCONJ
ejpam-4936	321	26	applied	applied	ADJ
ejpam-4936	321	27	mathematics	mathematic	NOUN
ejpam-4936	321	28	,	,	PUNCT
ejpam-4936	321	29	104(4):517–521	104(4):517–521	NUM
ejpam-4936	321	30	,	,	PUNCT
ejpam-4936	321	31	2015	2015	NUM
ejpam-4936	321	32	.	.	PUNCT
ejpam-4936	322	1	[	[	X
ejpam-4936	322	2	5	5	NUM
ejpam-4936	322	3	]	]	X
ejpam-4936	322	4	n.	n.	NOUN
ejpam-4936	322	5	burshtein	burshtein	PROPN
ejpam-4936	322	6	.	.	PUNCT
ejpam-4936	323	1	on	on	ADP
ejpam-4936	323	2	solutions	solution	NOUN
ejpam-4936	323	3	of	of	ADP
ejpam-4936	323	4	the	the	DET
ejpam-4936	323	5	diophantine	diophantine	NOUN
ejpam-4936	323	6	equation	equation	NOUN
ejpam-4936	323	7	11x	11x	NOUN
ejpam-4936	323	8	+	+	CCONJ
ejpam-4936	323	9	23y	23y	NUM
ejpam-4936	323	10	=	=	SYM
ejpam-4936	323	11	z2	z2	PROPN
ejpam-4936	323	12	with	with	ADP
ejpam-4936	323	13	consecutive	consecutive	ADJ
ejpam-4936	323	14	positive	positive	ADJ
ejpam-4936	323	15	integers	integer	NOUN
ejpam-4936	323	16	x	x	VERB
ejpam-4936	323	17	,	,	PUNCT
ejpam-4936	323	18	y.	y.	NOUN
ejpam-4936	323	19	annals	annal	NOUN
ejpam-4936	323	20	of	of	ADP
ejpam-4936	323	21	pure	pure	ADJ
ejpam-4936	323	22	and	and	CCONJ
ejpam-4936	323	23	applied	applied	ADJ
ejpam-4936	323	24	mathematics	mathematic	NOUN
ejpam-4936	323	25	,	,	PUNCT
ejpam-4936	323	26	20(2):57–61	20(2):57–61	NUM
ejpam-4936	323	27	,	,	PUNCT
ejpam-4936	323	28	2019	2019	NUM
ejpam-4936	323	29	.	.	PUNCT
ejpam-4936	324	1	[	[	X
ejpam-4936	324	2	6	6	NUM
ejpam-4936	324	3	]	]	X
ejpam-4936	324	4	n.	n.	NOUN
ejpam-4936	324	5	burshtein	burshtein	PROPN
ejpam-4936	324	6	.	.	PUNCT
ejpam-4936	325	1	solutions	solution	NOUN
ejpam-4936	325	2	of	of	ADP
ejpam-4936	325	3	the	the	DET
ejpam-4936	325	4	diophantine	diophantine	NOUN
ejpam-4936	325	5	equation	equation	NOUN
ejpam-4936	325	6	px	px	X
ejpam-4936	325	7	+	+	CCONJ
ejpam-4936	325	8	(	(	PUNCT
ejpam-4936	325	9	p	p	X
ejpam-4936	325	10	+	+	NOUN
ejpam-4936	325	11	1)y	1)y	NUM
ejpam-4936	325	12	+	+	CCONJ
ejpam-4936	325	13	(	(	PUNCT
ejpam-4936	325	14	p	p	X
ejpam-4936	325	15	+	+	NOUN
ejpam-4936	325	16	2)z	2)z	NUM
ejpam-4936	325	17	=	=	SYM
ejpam-4936	325	18	m2	m2	PROPN
ejpam-4936	325	19	for	for	ADP
ejpam-4936	325	20	primes	prime	NOUN
ejpam-4936	325	21	p	p	NOUN
ejpam-4936	325	22	≥	≥	NUM
ejpam-4936	325	23	2	2	NUM
ejpam-4936	325	24	when	when	SCONJ
ejpam-4936	325	25	1	1	NUM
ejpam-4936	325	26	≤	≤	NUM
ejpam-4936	325	27	x	x	X
ejpam-4936	325	28	,	,	PUNCT
ejpam-4936	325	29	y	y	PROPN
ejpam-4936	325	30	,	,	PUNCT
ejpam-4936	325	31	z	z	NOUN
ejpam-4936	325	32	≤	≤	NOUN
ejpam-4936	325	33	2	2	NUM
ejpam-4936	325	34	.	.	PUNCT
ejpam-4936	326	1	pure	pure	ADJ
ejpam-4936	326	2	and	and	CCONJ
ejpam-4936	326	3	applied	applied	ADJ
ejpam-4936	326	4	mathematics	mathematic	NOUN
ejpam-4936	326	5	,	,	PUNCT
ejpam-4936	326	6	22(1):41–49	22(1):41–49	NUM
ejpam-4936	326	7	,	,	PUNCT
ejpam-4936	326	8	2020	2020	NUM
ejpam-4936	326	9	.	.	PUNCT
ejpam-4936	327	1	[	[	X
ejpam-4936	327	2	7	7	X
ejpam-4936	327	3	]	]	X
ejpam-4936	327	4	n.	n.	NOUN
ejpam-4936	327	5	burshtein	burshtein	PROPN
ejpam-4936	327	6	.	.	PUNCT
ejpam-4936	328	1	all	all	DET
ejpam-4936	328	2	the	the	DET
ejpam-4936	328	3	solutions	solution	NOUN
ejpam-4936	328	4	of	of	ADP
ejpam-4936	328	5	the	the	DET
ejpam-4936	328	6	diophantine	diophantine	NOUN
ejpam-4936	328	7	equation	equation	NOUN
ejpam-4936	328	8	p3+qy	p3+qy	NOUN
ejpam-4936	328	9	=	=	NOUN
ejpam-4936	328	10	z3	z3	PROPN
ejpam-4936	328	11	with	with	ADP
ejpam-4936	328	12	distinct	distinct	ADJ
ejpam-4936	328	13	odd	odd	ADJ
ejpam-4936	328	14	primes	prime	NOUN
ejpam-4936	328	15	p	p	X
ejpam-4936	328	16	,	,	PUNCT
ejpam-4936	328	17	q	q	PROPN
ejpam-4936	328	18	when	when	SCONJ
ejpam-4936	328	19	y	y	PROPN
ejpam-4936	328	20	>	>	X
ejpam-4936	328	21	3	3	X
ejpam-4936	328	22	.	.	PUNCT
ejpam-4936	328	23	journal	journal	NOUN
ejpam-4936	328	24	of	of	ADP
ejpam-4936	328	25	mathematics	mathematics	PROPN
ejpam-4936	328	26	and	and	CCONJ
ejpam-4936	328	27	informatics	informatic	NOUN
ejpam-4936	328	28	,	,	PUNCT
ejpam-4936	328	29	20:1–4	20:1–4	NOUN
ejpam-4936	328	30	,	,	PUNCT
ejpam-4936	328	31	2021	2021	NUM
ejpam-4936	328	32	.	.	PUNCT
ejpam-4936	329	1	[	[	X
ejpam-4936	329	2	8	8	NUM
ejpam-4936	329	3	]	]	X
ejpam-4936	329	4	r.	r.	PROPN
ejpam-4936	329	5	chinram	chinram	PROPN
ejpam-4936	329	6	,	,	PUNCT
ejpam-4936	329	7	k.	k.	PROPN
ejpam-4936	329	8	sirikantisophon	sirikantisophon	PROPN
ejpam-4936	329	9	,	,	PUNCT
ejpam-4936	329	10	and	and	CCONJ
ejpam-4936	329	11	s.	s.	PROPN
ejpam-4936	329	12	kaewchay	kaewchay	PROPN
ejpam-4936	329	13	.	.	PUNCT
ejpam-4936	330	1	positive	positive	ADJ
ejpam-4936	330	2	integer	integer	NOUN
ejpam-4936	330	3	solutions	solution	NOUN
ejpam-4936	330	4	of	of	ADP
ejpam-4936	330	5	the	the	DET
ejpam-4936	330	6	diophantine	diophantine	NOUN
ejpam-4936	330	7	equation	equation	NOUN
ejpam-4936	330	8	1	1	NUM
ejpam-4936	330	9	/	/	SYM
ejpam-4936	330	10	x+	x+	ADJ
ejpam-4936	330	11	2	2	NUM
ejpam-4936	330	12	/	/	SYM
ejpam-4936	330	13	y+	y+	NUM
ejpam-4936	330	14	3	3	NUM
ejpam-4936	330	15	/	/	SYM
ejpam-4936	330	16	z	z	NOUN
ejpam-4936	330	17	=	=	SYM
ejpam-4936	330	18	1/3	1/3	PROPN
ejpam-4936	330	19	.	.	PUNCT
ejpam-4936	331	1	international	international	ADJ
ejpam-4936	331	2	journal	journal	NOUN
ejpam-4936	331	3	of	of	ADP
ejpam-4936	331	4	mathematics	mathematic	NOUN
ejpam-4936	331	5	and	and	CCONJ
ejpam-4936	331	6	computer	computer	NOUN
ejpam-4936	331	7	science	science	NOUN
ejpam-4936	331	8	,	,	PUNCT
ejpam-4936	331	9	17(3):1051–1059	17(3):1051–1059	NUM
ejpam-4936	331	10	,	,	PUNCT
ejpam-4936	331	11	2022	2022	NUM
ejpam-4936	331	12	.	.	PUNCT
ejpam-4936	332	1	[	[	X
ejpam-4936	332	2	9	9	NUM
ejpam-4936	332	3	]	]	PUNCT
ejpam-4936	332	4	s.	s.	PROPN
ejpam-4936	332	5	chotchaisthit	chotchaisthit	VERB
ejpam-4936	332	6	.	.	PUNCT
ejpam-4936	333	1	on	on	ADP
ejpam-4936	333	2	the	the	DET
ejpam-4936	333	3	diophantine	diophantine	NOUN
ejpam-4936	333	4	equation	equation	NOUN
ejpam-4936	333	5	2x	2x	NUM
ejpam-4936	333	6	+	+	CCONJ
ejpam-4936	333	7	11y	11y	NOUN
ejpam-4936	333	8	=	=	SYM
ejpam-4936	333	9	z2	z2	PROPN
ejpam-4936	333	10	.	.	PUNCT
ejpam-4936	334	1	maejo	maejo	PROPN
ejpam-4936	334	2	international	international	PROPN
ejpam-4936	334	3	journal	journal	PROPN
ejpam-4936	334	4	of	of	ADP
ejpam-4936	334	5	science	science	NOUN
ejpam-4936	334	6	and	and	CCONJ
ejpam-4936	334	7	technology	technology	NOUN
ejpam-4936	334	8	,	,	PUNCT
ejpam-4936	334	9	7(2):291–293	7(2):291–293	NOUN
ejpam-4936	334	10	,	,	PUNCT
ejpam-4936	334	11	2013	2013	NUM
ejpam-4936	334	12	.	.	PUNCT
ejpam-4936	335	1	[	[	X
ejpam-4936	335	2	10	10	NUM
ejpam-4936	335	3	]	]	X
ejpam-4936	335	4	r.	r.	PROPN
ejpam-4936	335	5	dokchan	dokchan	PROPN
ejpam-4936	335	6	and	and	CCONJ
ejpam-4936	335	7	a.	a.	NOUN
ejpam-4936	335	8	pakapongpun	pakapongpun	NOUN
ejpam-4936	335	9	.	.	PUNCT
ejpam-4936	336	1	on	on	ADP
ejpam-4936	336	2	the	the	DET
ejpam-4936	336	3	diophantine	diophantine	NOUN
ejpam-4936	336	4	equation	equation	NOUN
ejpam-4936	336	5	px	px	X
ejpam-4936	336	6	+	+	CCONJ
ejpam-4936	336	7	(	(	PUNCT
ejpam-4936	336	8	p+	p+	NOUN
ejpam-4936	336	9	20)y	20)y	PROPN
ejpam-4936	336	10	=	=	SYM
ejpam-4936	336	11	z2	z2	PROPN
ejpam-4936	336	12	where	where	SCONJ
ejpam-4936	336	13	p	p	PROPN
ejpam-4936	336	14	and	and	CCONJ
ejpam-4936	336	15	p+	p+	PROPN
ejpam-4936	336	16	20	20	NUM
ejpam-4936	336	17	are	be	AUX
ejpam-4936	336	18	primes	prime	NOUN
ejpam-4936	336	19	.	.	PUNCT
ejpam-4936	337	1	international	international	ADJ
ejpam-4936	337	2	journal	journal	PROPN
ejpam-4936	337	3	of	of	ADP
ejpam-4936	337	4	mathematics	mathematic	NOUN
ejpam-4936	337	5	and	and	CCONJ
ejpam-4936	337	6	computer	computer	NOUN
ejpam-4936	337	7	science	science	NOUN
ejpam-4936	337	8	,	,	PUNCT
ejpam-4936	337	9	16(1):179–183	16(1):179–183	NUM
ejpam-4936	337	10	,	,	PUNCT
ejpam-4936	337	11	2021	2021	NUM
ejpam-4936	337	12	.	.	PUNCT
ejpam-4936	338	1	[	[	X
ejpam-4936	338	2	11	11	NUM
ejpam-4936	338	3	]	]	PUNCT
ejpam-4936	338	4	a.	a.	NOUN
ejpam-4936	338	5	hamttat	hamttat	NOUN
ejpam-4936	338	6	.	.	PUNCT
ejpam-4936	339	1	on	on	ADP
ejpam-4936	339	2	the	the	DET
ejpam-4936	339	3	diophantine	diophantine	NOUN
ejpam-4936	339	4	equation	equation	NOUN
ejpam-4936	339	5	1	1	NUM
ejpam-4936	339	6	+	+	SYM
ejpam-4936	339	7	5x2	5x2	NUM
ejpam-4936	339	8	=	=	SYM
ejpam-4936	339	9	3yn	3yn	NOUN
ejpam-4936	339	10	,	,	PUNCT
ejpam-4936	339	11	n	n	X
ejpam-4936	339	12	≥	≥	NOUN
ejpam-4936	339	13	3	3	NUM
ejpam-4936	339	14	.	.	PUNCT
ejpam-4936	339	15	journal	journal	PROPN
ejpam-4936	339	16	of	of	ADP
ejpam-4936	339	17	applied	applied	ADJ
ejpam-4936	339	18	and	and	CCONJ
ejpam-4936	339	19	computational	computational	ADJ
ejpam-4936	339	20	mathematics	mathematic	NOUN
ejpam-4936	339	21	,	,	PUNCT
ejpam-4936	339	22	4(6):1–2	4(6):1–2	NUM
ejpam-4936	339	23	,	,	PUNCT
ejpam-4936	339	24	2015	2015	NUM
ejpam-4936	339	25	.	.	PUNCT
ejpam-4936	340	1	[	[	X
ejpam-4936	340	2	12	12	NUM
ejpam-4936	340	3	]	]	PUNCT
ejpam-4936	340	4	a.	a.	NOUN
ejpam-4936	340	5	hoque	hoque	NOUN
ejpam-4936	340	6	.	.	PUNCT
ejpam-4936	341	1	on	on	ADP
ejpam-4936	341	2	the	the	DET
ejpam-4936	341	3	exponential	exponential	ADJ
ejpam-4936	341	4	diophantine	diophantine	NOUN
ejpam-4936	341	5	equation	equation	NOUN
ejpam-4936	341	6	x2	x2	PROPN
ejpam-4936	342	1	+	+	PROPN
ejpam-4936	342	2	5a13b17c	5a13b17c	PROPN
ejpam-4936	342	3	=	=	SYM
ejpam-4936	342	4	2myn	2myn	PROPN
ejpam-4936	342	5	.	.	PUNCT
ejpam-4936	343	1	math.nt	math.nt	PROPN
ejpam-4936	343	2	,	,	PUNCT
ejpam-4936	343	3	pages	page	NOUN
ejpam-4936	343	4	1–12	1–12	PROPN
ejpam-4936	343	5	,	,	PUNCT
ejpam-4936	343	6	2021	2021	NUM
ejpam-4936	343	7	.	.	PUNCT
ejpam-4936	344	1	[	[	X
ejpam-4936	344	2	13	13	NUM
ejpam-4936	344	3	]	]	PUNCT
ejpam-4936	344	4	l.	l.	PROPN
ejpam-4936	344	5	jeśmanowicz	jeśmanowicz	PROPN
ejpam-4936	344	6	.	.	PUNCT
ejpam-4936	345	1	several	several	ADJ
ejpam-4936	345	2	remarks	remark	NOUN
ejpam-4936	345	3	on	on	ADP
ejpam-4936	345	4	pythagorean	pythagorean	PROPN
ejpam-4936	345	5	numbers	number	NOUN
ejpam-4936	345	6	.	.	PUNCT
ejpam-4936	346	1	wiadom	wiadom	PROPN
ejpam-4936	346	2	.	.	PUNCT
ejpam-4936	347	1	math	math	NOUN
ejpam-4936	347	2	.	.	PUNCT
ejpam-4936	347	3	,	,	PUNCT
ejpam-4936	347	4	1:196	1:196	NUM
ejpam-4936	347	5	–	–	PUNCT
ejpam-4936	347	6	202	202	NUM
ejpam-4936	347	7	,	,	PUNCT
ejpam-4936	347	8	1955/56	1955/56	NUM
ejpam-4936	347	9	.	.	PUNCT
ejpam-4936	348	1	[	[	X
ejpam-4936	348	2	14	14	NUM
ejpam-4936	348	3	]	]	PUNCT
ejpam-4936	348	4	k.	k.	PROPN
ejpam-4936	349	1	laipaporn	laipaporn	PROPN
ejpam-4936	349	2	,	,	PUNCT
ejpam-4936	349	3	s.	s.	PROPN
ejpam-4936	349	4	wananiyakul	wananiyakul	PROPN
ejpam-4936	349	5	,	,	PUNCT
ejpam-4936	349	6	and	and	CCONJ
ejpam-4936	349	7	p.	p.	PROPN
ejpam-4936	349	8	khachorncharoenkul	khachorncharoenkul	PROPN
ejpam-4936	349	9	.	.	PUNCT
ejpam-4936	350	1	on	on	ADP
ejpam-4936	350	2	the	the	DET
ejpam-4936	350	3	diophantine	diophantine	NOUN
ejpam-4936	350	4	equation	equation	NOUN
ejpam-4936	350	5	3x	3x	NOUN
ejpam-4936	350	6	+	+	CCONJ
ejpam-4936	350	7	p5y	p5y	NOUN
ejpam-4936	350	8	=	=	SYM
ejpam-4936	350	9	z2	z2	PROPN
ejpam-4936	350	10	.	.	PROPN
ejpam-4936	350	11	walailak	walailak	ADJ
ejpam-4936	350	12	journal	journal	NOUN
ejpam-4936	350	13	of	of	ADP
ejpam-4936	350	14	science	science	NOUN
ejpam-4936	350	15	and	and	CCONJ
ejpam-4936	350	16	technology	technology	NOUN
ejpam-4936	350	17	,	,	PUNCT
ejpam-4936	350	18	16(9):647–653	16(9):647–653	NUM
ejpam-4936	350	19	,	,	PUNCT
ejpam-4936	350	20	2019	2019	NUM
ejpam-4936	350	21	.	.	PUNCT
ejpam-4936	351	1	[	[	X
ejpam-4936	351	2	15	15	NUM
ejpam-4936	351	3	]	]	X
ejpam-4936	351	4	r.	r.	PROPN
ejpam-4936	351	5	j.	j.	PROPN
ejpam-4936	351	6	s.	s.	PROPN
ejpam-4936	351	7	mina	mina	PROPN
ejpam-4936	351	8	and	and	CCONJ
ejpam-4936	351	9	j.	j.	PROPN
ejpam-4936	351	10	b.	b.	PROPN
ejpam-4936	351	11	bacani	bacani	PROPN
ejpam-4936	351	12	.	.	PUNCT
ejpam-4936	352	1	non	non	ADJ
ejpam-4936	352	2	-	-	NOUN
ejpam-4936	352	3	existence	existence	NOUN
ejpam-4936	352	4	of	of	ADP
ejpam-4936	352	5	solutions	solution	NOUN
ejpam-4936	352	6	of	of	ADP
ejpam-4936	352	7	diophantine	diophantine	NOUN
ejpam-4936	352	8	equations	equation	NOUN
ejpam-4936	352	9	of	of	ADP
ejpam-4936	352	10	the	the	DET
ejpam-4936	352	11	form	form	NOUN
ejpam-4936	352	12	px	px	X
ejpam-4936	352	13	+	+	CCONJ
ejpam-4936	352	14	qy	qy	NOUN
ejpam-4936	352	15	=	=	SYM
ejpam-4936	352	16	z2n	z2n	PROPN
ejpam-4936	352	17	.	.	PUNCT
ejpam-4936	353	1	mathematics	mathematic	NOUN
ejpam-4936	353	2	and	and	CCONJ
ejpam-4936	353	3	statistics	statistic	NOUN
ejpam-4936	353	4	,	,	PUNCT
ejpam-4936	353	5	7(3):78–81	7(3):78–81	NUM
ejpam-4936	353	6	,	,	PUNCT
ejpam-4936	353	7	2019	2019	NUM
ejpam-4936	353	8	.	.	PUNCT
ejpam-4936	354	1	[	[	X
ejpam-4936	354	2	16	16	NUM
ejpam-4936	354	3	]	]	PUNCT
ejpam-4936	354	4	b.	b.	PROPN
ejpam-4936	354	5	r.	r.	PROPN
ejpam-4936	354	6	sangam	sangam	PROPN
ejpam-4936	354	7	.	.	PUNCT
ejpam-4936	355	1	on	on	ADP
ejpam-4936	355	2	the	the	DET
ejpam-4936	355	3	diophantine	diophantine	NOUN
ejpam-4936	355	4	equations	equation	NOUN
ejpam-4936	355	5	3x	3x	NUM
ejpam-4936	355	6	+6y	+6y	PROPN
ejpam-4936	355	7	=	=	SYM
ejpam-4936	355	8	z2	z2	PROPN
ejpam-4936	355	9	and	and	CCONJ
ejpam-4936	355	10	5x	5x	NOUN
ejpam-4936	355	11	+8y	+8y	NUM
ejpam-4936	355	12	=	=	SYM
ejpam-4936	355	13	z2	z2	PROPN
ejpam-4936	355	14	.	.	PUNCT
ejpam-4936	356	1	annals	annal	NOUN
ejpam-4936	356	2	of	of	ADP
ejpam-4936	356	3	pure	pure	ADJ
ejpam-4936	356	4	and	and	CCONJ
ejpam-4936	356	5	applied	applied	ADJ
ejpam-4936	356	6	mathematics	mathematic	NOUN
ejpam-4936	356	7	,	,	PUNCT
ejpam-4936	356	8	22(1):7–11	22(1):7–11	NUM
ejpam-4936	356	9	,	,	PUNCT
ejpam-4936	356	10	2020	2020	NUM
ejpam-4936	356	11	.	.	PUNCT
ejpam-4936	357	1	references	reference	NOUN
ejpam-4936	357	2	2081	2081	NUM
ejpam-4936	358	1	[	[	X
ejpam-4936	358	2	17	17	NUM
ejpam-4936	358	3	]	]	PUNCT
ejpam-4936	358	4	w.	w.	PROPN
ejpam-4936	358	5	sierpiński	sierpiński	PROPN
ejpam-4936	358	6	.	.	PUNCT
ejpam-4936	359	1	on	on	ADP
ejpam-4936	359	2	the	the	DET
ejpam-4936	359	3	equation	equation	NOUN
ejpam-4936	359	4	3x+4y	3x+4y	NUM
ejpam-4936	359	5	=	=	SYM
ejpam-4936	359	6	5z	5z	NOUN
ejpam-4936	359	7	.	.	PUNCT
ejpam-4936	360	1	wiadom	wiadom	NOUN
ejpam-4936	360	2	.	.	PUNCT
ejpam-4936	361	1	math	math	NOUN
ejpam-4936	361	2	.	.	PUNCT
ejpam-4936	361	3	,	,	PUNCT
ejpam-4936	361	4	1(1):194–195	1(1):194–195	NUM
ejpam-4936	361	5	,	,	PUNCT
ejpam-4936	361	6	1955/56	1955/56	NUM
ejpam-4936	361	7	.	.	PUNCT
ejpam-4936	362	1	[	[	X
ejpam-4936	362	2	18	18	NUM
ejpam-4936	362	3	]	]	X
ejpam-4936	362	4	b.	b.	PROPN
ejpam-4936	362	5	sroysang	sroysang	PROPN
ejpam-4936	362	6	.	.	PUNCT
ejpam-4936	363	1	on	on	ADP
ejpam-4936	363	2	the	the	DET
ejpam-4936	363	3	diophantine	diophantine	NOUN
ejpam-4936	363	4	equation	equation	NOUN
ejpam-4936	363	5	31x	31x	NOUN
ejpam-4936	363	6	+	+	SYM
ejpam-4936	363	7	32y	32y	NUM
ejpam-4936	363	8	=	=	SYM
ejpam-4936	363	9	z2	z2	PROPN
ejpam-4936	363	10	.	.	PUNCT
ejpam-4936	364	1	international	international	ADJ
ejpam-4936	364	2	journal	journal	NOUN
ejpam-4936	364	3	of	of	ADP
ejpam-4936	364	4	pure	pure	ADJ
ejpam-4936	364	5	and	and	CCONJ
ejpam-4936	364	6	applied	applied	ADJ
ejpam-4936	364	7	mathematics	mathematic	NOUN
ejpam-4936	364	8	,	,	PUNCT
ejpam-4936	364	9	81(4):609–612	81(4):609–612	PROPN
ejpam-4936	364	10	,	,	PUNCT
ejpam-4936	364	11	2012	2012	NUM
ejpam-4936	364	12	.	.	PUNCT
ejpam-4936	365	1	[	[	X
ejpam-4936	365	2	19	19	NUM
ejpam-4936	365	3	]	]	PUNCT
ejpam-4936	365	4	b.	b.	PROPN
ejpam-4936	365	5	sroysang	sroysang	PROPN
ejpam-4936	365	6	.	.	PUNCT
ejpam-4936	366	1	on	on	ADP
ejpam-4936	366	2	the	the	DET
ejpam-4936	366	3	diophantine	diophantine	NOUN
ejpam-4936	366	4	equation	equation	NOUN
ejpam-4936	366	5	5x	5x	NOUN
ejpam-4936	366	6	+	+	CCONJ
ejpam-4936	366	7	7y	7y	NOUN
ejpam-4936	366	8	=	=	SYM
ejpam-4936	366	9	z2	z2	PROPN
ejpam-4936	366	10	.	.	PUNCT
ejpam-4936	366	11	international	international	ADJ
ejpam-4936	366	12	journal	journal	NOUN
ejpam-4936	366	13	of	of	ADP
ejpam-4936	366	14	pure	pure	ADJ
ejpam-4936	366	15	and	and	CCONJ
ejpam-4936	366	16	applied	applied	ADJ
ejpam-4936	366	17	mathematics	mathematic	NOUN
ejpam-4936	366	18	,	,	PUNCT
ejpam-4936	366	19	89(1):115–118	89(1):115–118	NUM
ejpam-4936	366	20	,	,	PUNCT
ejpam-4936	366	21	2013	2013	NUM
ejpam-4936	366	22	.	.	PUNCT
ejpam-4936	367	1	[	[	X
ejpam-4936	367	2	20	20	NUM
ejpam-4936	367	3	]	]	PUNCT
ejpam-4936	367	4	b.	b.	PROPN
ejpam-4936	367	5	sroysang	sroysang	PROPN
ejpam-4936	367	6	.	.	PUNCT
ejpam-4936	368	1	on	on	ADP
ejpam-4936	368	2	the	the	DET
ejpam-4936	368	3	diophantine	diophantine	NOUN
ejpam-4936	368	4	equation	equation	NOUN
ejpam-4936	368	5	5x	5x	NUM
ejpam-4936	368	6	+	+	CCONJ
ejpam-4936	368	7	43y	43y	NOUN
ejpam-4936	368	8	=	=	SYM
ejpam-4936	368	9	z2	z2	PROPN
ejpam-4936	368	10	.	.	PUNCT
ejpam-4936	369	1	international	international	ADJ
ejpam-4936	369	2	journal	journal	NOUN
ejpam-4936	369	3	of	of	ADP
ejpam-4936	369	4	pure	pure	ADJ
ejpam-4936	369	5	and	and	CCONJ
ejpam-4936	369	6	applied	applied	ADJ
ejpam-4936	369	7	mathematics	mathematic	NOUN
ejpam-4936	369	8	,	,	PUNCT
ejpam-4936	369	9	91(4):537–540	91(4):537–540	PROPN
ejpam-4936	369	10	,	,	PUNCT
ejpam-4936	369	11	2014	2014	NUM
ejpam-4936	369	12	.	.	PUNCT
ejpam-4936	370	1	[	[	X
ejpam-4936	370	2	21	21	NUM
ejpam-4936	370	3	]	]	X
ejpam-4936	370	4	s.	s.	PROPN
ejpam-4936	370	5	subburam	subburam	PROPN
ejpam-4936	370	6	.	.	PUNCT
ejpam-4936	371	1	on	on	ADP
ejpam-4936	371	2	the	the	DET
ejpam-4936	371	3	diophantine	diophantine	NOUN
ejpam-4936	371	4	equation	equation	NOUN
ejpam-4936	371	5	lax	lax	NOUN
ejpam-4936	371	6	+	+	CCONJ
ejpam-4936	371	7	mby	mby	X
ejpam-4936	371	8	=	=	SYM
ejpam-4936	371	9	ncz	ncz	PROPN
ejpam-4936	371	10	.	.	PUNCT
ejpam-4936	372	1	research	research	NOUN
ejpam-4936	372	2	in	in	ADP
ejpam-4936	372	3	number	number	NOUN
ejpam-4936	372	4	theory	theory	NOUN
ejpam-4936	372	5	,	,	PUNCT
ejpam-4936	372	6	4(25):24–26	4(25):24–26	NOUN
ejpam-4936	372	7	,	,	PUNCT
ejpam-4936	372	8	2018	2018	NUM
ejpam-4936	372	9	.	.	PUNCT
ejpam-4936	373	1	[	[	X
ejpam-4936	373	2	22	22	NUM
ejpam-4936	373	3	]	]	PUNCT
ejpam-4936	373	4	a.	a.	NOUN
ejpam-4936	373	5	sugandha	sugandha	NOUN
ejpam-4936	373	6	,	,	PUNCT
ejpam-4936	373	7	a.	a.	PROPN
ejpam-4936	373	8	prabowo	prabowo	PROPN
ejpam-4936	373	9	a.	a.	PROPN
ejpam-4936	373	10	tripena	tripena	PROPN
ejpam-4936	373	11	,	,	PUNCT
ejpam-4936	373	12	and	and	CCONJ
ejpam-4936	373	13	f.	f.	PROPN
ejpam-4936	373	14	sukono	sukono	PROPN
ejpam-4936	373	15	.	.	PUNCT
ejpam-4936	374	1	nonlinear	nonlinear	ADJ
ejpam-4936	374	2	diophantine	diophantine	NOUN
ejpam-4936	374	3	equation	equation	NOUN
ejpam-4936	374	4	11x	11x	NOUN
ejpam-4936	374	5	+	+	CCONJ
ejpam-4936	374	6	13y	13y	NOUN
ejpam-4936	374	7	=	=	SYM
ejpam-4936	374	8	z2	z2	PROPN
ejpam-4936	374	9	.	.	PUNCT
ejpam-4936	375	1	materials	material	NOUN
ejpam-4936	375	2	science	science	NOUN
ejpam-4936	375	3	and	and	CCONJ
ejpam-4936	375	4	engineering	engineering	NOUN
ejpam-4936	375	5	,	,	PUNCT
ejpam-4936	375	6	332:1–4	332:1–4	NOUN
ejpam-4936	375	7	,	,	PUNCT
ejpam-4936	375	8	2018	2018	NUM
ejpam-4936	375	9	.	.	PUNCT
ejpam-4936	376	1	[	[	X
ejpam-4936	376	2	23	23	NUM
ejpam-4936	376	3	]	]	PUNCT
ejpam-4936	376	4	a.	a.	NOUN
ejpam-4936	376	5	suvarnamani	suvarnamani	PROPN
ejpam-4936	376	6	.	.	PUNCT
ejpam-4936	377	1	on	on	ADP
ejpam-4936	377	2	the	the	DET
ejpam-4936	377	3	diophantine	diophantine	NOUN
ejpam-4936	377	4	equation	equation	NOUN
ejpam-4936	377	5	px	px	X
ejpam-4936	377	6	+	+	CCONJ
ejpam-4936	377	7	(	(	PUNCT
ejpam-4936	377	8	p	p	X
ejpam-4936	377	9	+	+	NUM
ejpam-4936	377	10	1)y	1)y	NUM
ejpam-4936	377	11	=	=	SYM
ejpam-4936	377	12	z2	z2	PROPN
ejpam-4936	377	13	.	.	PROPN
ejpam-4936	377	14	journal	journal	PROPN
ejpam-4936	377	15	of	of	ADP
ejpam-4936	377	16	pure	pure	ADJ
ejpam-4936	377	17	and	and	CCONJ
ejpam-4936	377	18	applied	applied	ADJ
ejpam-4936	377	19	mathematics	mathematic	NOUN
ejpam-4936	377	20	,	,	PUNCT
ejpam-4936	377	21	94(5):689–692	94(5):689–692	NUM
ejpam-4936	377	22	,	,	PUNCT
ejpam-4936	377	23	2014	2014	NUM
ejpam-4936	377	24	.	.	PUNCT
ejpam-4936	378	1	[	[	X
ejpam-4936	378	2	24	24	NUM
ejpam-4936	378	3	]	]	X
ejpam-4936	378	4	s.	s.	PROPN
ejpam-4936	378	5	vesarachasart	vesarachasart	PROPN
ejpam-4936	378	6	,	,	PUNCT
ejpam-4936	378	7	a.	a.	NOUN
ejpam-4936	378	8	makuwing	makuwing	NOUN
ejpam-4936	378	9	,	,	PUNCT
ejpam-4936	378	10	n.	n.	NOUN
ejpam-4936	378	11	wasurakka	wasurakka	NOUN
ejpam-4936	378	12	,	,	PUNCT
ejpam-4936	378	13	and	and	CCONJ
ejpam-4936	378	14	k.	k.	PROPN
ejpam-4936	378	15	phattharachittra	phattharachittra	PROPN
ejpam-4936	378	16	.	.	PUNCT
ejpam-4936	379	1	on	on	ADP
ejpam-4936	379	2	the	the	DET
ejpam-4936	379	3	diophantine	diophantine	NOUN
ejpam-4936	379	4	equation	equation	NOUN
ejpam-4936	379	5	3x	3x	NUM
ejpam-4936	379	6	+	+	CCONJ
ejpam-4936	379	7	5y7z	5y7z	NOUN
ejpam-4936	379	8	=	=	SYM
ejpam-4936	379	9	u2	u2	PROPN
ejpam-4936	379	10	.	.	PROPN
ejpam-4936	379	11	thai	thai	PROPN
ejpam-4936	379	12	journal	journal	PROPN
ejpam-4936	379	13	of	of	ADP
ejpam-4936	379	14	science	science	NOUN
ejpam-4936	379	15	and	and	CCONJ
ejpam-4936	379	16	technology	technology	NOUN
ejpam-4936	379	17	,	,	PUNCT
ejpam-4936	379	18	8(3):220–225	8(3):220–225	NUM
ejpam-4936	379	19	,	,	PUNCT
ejpam-4936	379	20	2018	2018	NUM
ejpam-4936	379	21	.	.	PUNCT