id	sid	tid	token	lemma	pos
easat-2965	1	1	edelweiss	edelweiss	PROPN
easat-2965	1	2	applied	apply	VERB
easat-2965	1	3	science	science	NOUN
easat-2965	1	4	and	and	CCONJ
easat-2965	1	5	technology	technology	NOUN
easat-2965	1	6	issn	issn	PROPN
easat-2965	1	7	:	:	PUNCT
easat-2965	1	8	2576	2576	NUM
easat-2965	1	9	-	-	SYM
easat-2965	1	10	8484	8484	NUM
easat-2965	1	11	vol	vol	NOUN
easat-2965	1	12	.	.	PROPN
easat-2965	1	13	8	8	NUM
easat-2965	1	14	,	,	PUNCT
easat-2965	1	15	no	no	INTJ
easat-2965	1	16	.	.	NOUN
easat-2965	1	17	6	6	NUM
easat-2965	1	18	,	,	PUNCT
easat-2965	1	19	4408	4408	NUM
easat-2965	1	20	-	-	SYM
easat-2965	1	21	4414	4414	NUM
easat-2965	1	22	2024	2024	NUM
easat-2965	1	23	publisher	publisher	NOUN
easat-2965	1	24	:	:	PUNCT
easat-2965	1	25	learning	learn	VERB
easat-2965	1	26	gate	gate	NOUN
easat-2965	1	27	doi	doi	PROPN
easat-2965	1	28	:	:	PUNCT
easat-2965	1	29	10.55214/25768484.v8i6.2965	10.55214/25768484.v8i6.2965	NUM
easat-2965	1	30	©	©	PROPN
easat-2965	1	31	2024	2024	NUM
easat-2965	1	32	by	by	ADP
easat-2965	1	33	the	the	DET
easat-2965	1	34	authors	author	NOUN
easat-2965	1	35	;	;	PUNCT
easat-2965	1	36	licensee	licensee	PROPN
easat-2965	1	37	learning	learning	NOUN
easat-2965	1	38	gate	gate	NOUN
easat-2965	1	39	©	©	PROPN
easat-2965	1	40	2024	2024	NUM
easat-2965	1	41	by	by	ADP
easat-2965	1	42	the	the	DET
easat-2965	1	43	authors	author	NOUN
easat-2965	1	44	;	;	PUNCT
easat-2965	1	45	licensee	licensee	PROPN
easat-2965	1	46	learning	learn	VERB
easat-2965	1	47	gate	gate	NOUN
easat-2965	1	48	*	*	PUNCT
easat-2965	1	49	correspondence	correspondence	NOUN
easat-2965	1	50	:	:	PUNCT
easat-2965	1	51	amarachandoul@yahoo.fr	amarachandoul@yahoo.fr	PROPN
easat-2965	1	52	solutions	solution	NOUN
easat-2965	1	53	of	of	ADP
easat-2965	1	54	the	the	DET
easat-2965	1	55	equation	equation	NOUN
easat-2965	1	56	𝒙𝟐	𝒙𝟐	NOUN
easat-2965	1	57	−	−	PROPN
easat-2965	1	58	(	(	PUNCT
easat-2965	1	59	𝒑𝟐𝒒𝟐	𝒑𝟐𝒒𝟐	NOUN
easat-2965	1	60	±	±	NOUN
easat-2965	1	61	𝐚𝐩)𝒚𝟐	𝐚𝐩)𝒚𝟐	NOUN
easat-2965	1	62	=	=	SYM
easat-2965	1	63	𝒌𝒕	𝒌𝒕	DET
easat-2965	1	64	amara	amara	PROPN
easat-2965	1	65	chandoul1	chandoul1	PROPN
easat-2965	1	66	*	*	PROPN
easat-2965	1	67	,	,	PUNCT
easat-2965	1	68	alanod	alanod	PROPN
easat-2965	1	69	sibih2	sibih2	NOUN
easat-2965	2	1	1department	1department	NUM
easat-2965	2	2	of	of	ADP
easat-2965	2	3	computer	computer	NOUN
easat-2965	2	4	science	science	NOUN
easat-2965	2	5	𝖠multimedia	𝖠multimedia	PROPN
easat-2965	2	6	,	,	PUNCT
easat-2965	2	7	higher	high	ADJ
easat-2965	2	8	institute	institute	NOUN
easat-2965	2	9	of	of	ADP
easat-2965	2	10	informatics	informatics	PROPN
easat-2965	2	11	𝖠multimedia	𝖠multimedia	PROPN
easat-2965	2	12	of	of	ADP
easat-2965	2	13	sfax	sfax	NOUN
easat-2965	2	14	,	,	PUNCT
easat-2965	2	15	tunis	tunis	VERB
easat-2965	2	16	road	road	NOUN
easat-2965	2	17	,	,	PUNCT
easat-2965	2	18	km	km	PROPN
easat-2965	2	19	10	10	NUM
easat-2965	2	20	,	,	PUNCT
easat-2965	2	21	al	al	PROPN
easat-2965	2	22	−ons	−ons	PROPN
easat-2965	2	23	,	,	PUNCT
easat-2965	2	24	b.	b.	PROPN
easat-2965	3	1	p.	p.	NOUN
easat-2965	3	2	242	242	NUM
easat-2965	3	3	,	,	PUNCT
easat-2965	3	4	3021	3021	NUM
easat-2965	3	5	,	,	PUNCT
easat-2965	3	6	tunisia	tunisia	NOUN
easat-2965	3	7	;	;	PUNCT
easat-2965	3	8	amarachandoul@yahoo.fr	amarachandoul@yahoo.fr	PROPN
easat-2965	3	9	(	(	PUNCT
easat-2965	3	10	a.c	a.c	PROPN
easat-2965	3	11	.	.	PUNCT
easat-2965	3	12	)	)	PUNCT
easat-2965	4	1	2department	2department	NUM
easat-2965	4	2	of	of	ADP
easat-2965	4	3	mathematics	mathematic	NOUN
easat-2965	4	4	,	,	PUNCT
easat-2965	4	5	jamoum	jamoum	PROPN
easat-2965	4	6	university	university	PROPN
easat-2965	4	7	college	college	NOUN
easat-2965	4	8	,	,	PUNCT
easat-2965	4	9	umm	umm	INTJ
easat-2965	4	10	al	al	PROPN
easat-2965	4	11	−qura	−qura	PROPN
easat-2965	4	12	university	university	PROPN
easat-2965	4	13	,	,	PUNCT
easat-2965	4	14	holly	holly	PROPN
easat-2965	4	15	makkah	makkah	PROPN
easat-2965	4	16	21955	21955	NUM
easat-2965	4	17	,	,	PUNCT
easat-2965	4	18	saudi	saudi	PROPN
easat-2965	4	19	arabia	arabia	PROPN
easat-2965	4	20	;	;	PUNCT
easat-2965	4	21	amsibih@uqu.edu.sa	amsibih@uqu.edu.sa	PROPN
easat-2965	4	22	(	(	PUNCT
easat-2965	4	23	a.s	a.s	PROPN
easat-2965	4	24	.	.	PROPN
easat-2965	4	25	)	)	PUNCT
easat-2965	4	26	.	.	PUNCT
easat-2965	5	1	abstract	abstract	ADV
easat-2965	5	2	:	:	PUNCT
easat-2965	5	3	in	in	ADP
easat-2965	5	4	this	this	DET
easat-2965	5	5	note	note	NOUN
easat-2965	5	6	,	,	PUNCT
easat-2965	5	7	the	the	DET
easat-2965	5	8	diophantine	diophantine	NOUN
easat-2965	5	9	equation	equation	NOUN
easat-2965	5	10	𝑥2	𝑥2	NOUN
easat-2965	5	11	−	−	PROPN
easat-2965	5	12	(	(	PUNCT
easat-2965	5	13	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	5	14	±	±	NUM
easat-2965	5	15	ap)𝑦2	ap)𝑦2	NOUN
easat-2965	6	1	=	=	PRON
easat-2965	6	2	𝑘𝑡	𝑘𝑡	PROPN
easat-2965	6	3	has	have	AUX
easat-2965	6	4	been	be	AUX
easat-2965	6	5	solved	solve	VERB
easat-2965	6	6	,	,	PUNCT
easat-2965	6	7	and	and	CCONJ
easat-2965	6	8	its	its	PRON
easat-2965	6	9	positive	positive	ADJ
easat-2965	6	10	integer	integer	NOUN
easat-2965	6	11	solutions	solution	NOUN
easat-2965	6	12	have	have	AUX
easat-2965	6	13	been	be	AUX
easat-2965	6	14	stated	state	VERB
easat-2965	6	15	in	in	ADP
easat-2965	6	16	terms	term	NOUN
easat-2965	6	17	of	of	ADP
easat-2965	6	18	generalized	generalized	ADJ
easat-2965	6	19	fibonacci	fibonacci	NOUN
easat-2965	6	20	,	,	PUNCT
easat-2965	6	21	generalized	generalized	ADJ
easat-2965	6	22	lucas	lucas	NOUN
easat-2965	6	23	,	,	PUNCT
easat-2965	6	24	generalized	generalized	ADJ
easat-2965	6	25	pell	pell	NOUN
easat-2965	6	26	,	,	PUNCT
easat-2965	6	27	and	and	CCONJ
easat-2965	6	28	generalized	generalize	VERB
easat-2965	6	29	pell	pell	NOUN
easat-2965	6	30	-	-	PUNCT
easat-2965	6	31	lucas	lucas	NOUN
easat-2965	6	32	sequences	sequence	NOUN
easat-2965	6	33	.	.	PUNCT
easat-2965	7	1	we	we	PRON
easat-2965	7	2	have	have	AUX
easat-2965	7	3	discovered	discover	VERB
easat-2965	7	4	units	unit	NOUN
easat-2965	7	5	for	for	ADP
easat-2965	7	6	𝑍[𝐷	𝑍[𝐷	NOUN
easat-2965	7	7	]	]	PUNCT
easat-2965	7	8	in	in	ADP
easat-2965	7	9	terms	term	NOUN
easat-2965	7	10	of	of	ADP
easat-2965	7	11	the	the	DET
easat-2965	7	12	above	above	ADJ
easat-2965	7	13	sequences	sequence	NOUN
easat-2965	7	14	of	of	ADP
easat-2965	7	15	numbers	number	NOUN
easat-2965	7	16	.	.	PUNCT
easat-2965	8	1	keywords	keyword	NOUN
easat-2965	8	2	:	:	PUNCT
easat-2965	8	3	diophantine	diophantine	VERB
easat-2965	8	4	equations	equation	NOUN
easat-2965	8	5	,	,	PUNCT
easat-2965	8	6	generalized	generalized	ADJ
easat-2965	8	7	fibonacci	fibonacci	NOUN
easat-2965	8	8	numbers	number	NOUN
easat-2965	8	9	,	,	PUNCT
easat-2965	8	10	generalized	generalized	ADJ
easat-2965	8	11	lucas	lucas	NOUN
easat-2965	8	12	numbers	number	NOUN
easat-2965	8	13	,	,	PUNCT
easat-2965	8	14	generalized	generalized	ADJ
easat-2965	8	15	pell	pell	NOUN
easat-2965	8	16	numbers	number	NOUN
easat-2965	8	17	,	,	PUNCT
easat-2965	8	18	generalized	generalized	ADJ
easat-2965	8	19	pell	pell	NOUN
easat-2965	8	20	–	–	PUNCT
easat-2965	8	21	lucas	lucas	NOUN
easat-2965	8	22	numbers	number	NOUN
easat-2965	8	23	,	,	PUNCT
easat-2965	8	24	pell	pell	PROPN
easat-2965	8	25	equations	equation	NOUN
easat-2965	8	26	.	.	PUNCT
easat-2965	9	1	1	1	X
easat-2965	9	2	.	.	X
easat-2965	9	3	introduction	introduction	NOUN
easat-2965	9	4	since	since	SCONJ
easat-2965	9	5	ancient	ancient	ADJ
easat-2965	9	6	times	time	NOUN
easat-2965	9	7	,	,	PUNCT
easat-2965	9	8	numerous	numerous	ADJ
easat-2965	9	9	mathematicians	mathematician	NOUN
easat-2965	9	10	have	have	AUX
easat-2965	9	11	studied	study	VERB
easat-2965	9	12	number	number	NOUN
easat-2965	9	13	sequences	sequence	NOUN
easat-2965	9	14	.	.	PUNCT
easat-2965	10	1	particular	particular	ADJ
easat-2965	10	2	attention	attention	NOUN
easat-2965	10	3	has	have	AUX
easat-2965	10	4	been	be	AUX
easat-2965	10	5	paid	pay	VERB
easat-2965	10	6	to	to	PART
easat-2965	10	7	recurrent	recurrent	VERB
easat-2965	10	8	sequences	sequence	NOUN
easat-2965	10	9	like	like	ADP
easat-2965	10	10	the	the	DET
easat-2965	10	11	fibonacci	fibonacci	PROPN
easat-2965	10	12	,	,	PUNCT
easat-2965	10	13	lucas	lucas	PROPN
easat-2965	10	14	,	,	PUNCT
easat-2965	10	15	pell	pell	NOUN
easat-2965	10	16	,	,	PUNCT
easat-2965	10	17	or	or	CCONJ
easat-2965	10	18	pell	pell	NOUN
easat-2965	10	19	-	-	PUNCT
easat-2965	10	20	lucas	lucas	NOUN
easat-2965	10	21	sequences	sequence	NOUN
easat-2965	10	22	,	,	PUNCT
easat-2965	10	23	which	which	PRON
easat-2965	10	24	are	be	AUX
easat-2965	10	25	described	describe	VERB
easat-2965	10	26	below	below	ADP
easat-2965	10	27	:	:	PUNCT
easat-2965	10	28	fibonacci	fibonacci	NOUN
easat-2965	10	29	number	number	NOUN
easat-2965	10	30	sequence	sequence	NOUN
easat-2965	10	31	𝐹𝑛+2	𝐹𝑛+2	NOUN
easat-2965	10	32	=	=	SYM
easat-2965	10	33	𝐹𝑛+1	𝐹𝑛+1	NOUN
easat-2965	11	1	+	+	NUM
easat-2965	11	2	𝐹𝑛	𝐹𝑛	PROPN
easat-2965	11	3	for	for	ADP
easat-2965	11	4	𝑛	𝑛	DET
easat-2965	11	5	≥	≥	NOUN
easat-2965	11	6	0	0	NUM
easat-2965	11	7	with	with	ADP
easat-2965	11	8	𝐹0	𝐹0	NOUN
easat-2965	11	9	=	=	SYM
easat-2965	11	10	0	0	NUM
easat-2965	11	11	and	and	CCONJ
easat-2965	11	12	𝐹1	𝐹1	NOUN
easat-2965	11	13	=	=	NOUN
easat-2965	11	14	1	1	X
easat-2965	11	15	.	.	PUNCT
easat-2965	11	16	lucas	lucas	PROPN
easat-2965	11	17	number	number	NOUN
easat-2965	11	18	sequence	sequence	NOUN
easat-2965	11	19	𝐿𝑛+2	𝐿𝑛+2	NOUN
easat-2965	12	1	=	=	SYM
easat-2965	13	1	𝐿𝑛+1	𝐿𝑛+1	X
easat-2965	14	1	+	+	CCONJ
easat-2965	14	2	𝐿𝑛	𝐿𝑛	PROPN
easat-2965	14	3	for	for	ADP
easat-2965	14	4	𝑛	𝑛	PRON
easat-2965	14	5	≥	≥	NOUN
easat-2965	14	6	0	0	NUM
easat-2965	14	7	with	with	ADP
easat-2965	14	8	𝐿0	𝐿0	ADJ
easat-2965	14	9	=	=	SYM
easat-2965	14	10	2	2	NUM
easat-2965	14	11	and	and	CCONJ
easat-2965	14	12	𝐿1	𝐿1	VERB
easat-2965	14	13	=	=	SYM
easat-2965	14	14	1	1	X
easat-2965	14	15	.	.	PUNCT
easat-2965	14	16	pell	pell	NOUN
easat-2965	14	17	number	number	NOUN
easat-2965	14	18	sequence	sequence	NOUN
easat-2965	14	19	φ𝑛+2	φ𝑛+2	X
easat-2965	14	20	=	=	SYM
easat-2965	14	21	2φ𝑛+1	2φ𝑛+1	PROPN
easat-2965	14	22	+	+	CCONJ
easat-2965	14	23	φ𝑛	φ𝑛	PROPN
easat-2965	14	24	for	for	ADP
easat-2965	14	25	𝑛	𝑛	DET
easat-2965	14	26	≥	≥	NOUN
easat-2965	14	27	0	0	NUM
easat-2965	14	28	with	with	ADP
easat-2965	14	29	φ0	φ0	PROPN
easat-2965	14	30	=	=	PUNCT
easat-2965	14	31	0	0	PROPN
easat-2965	14	32	and	and	CCONJ
easat-2965	14	33	φ1	φ1	NOUN
easat-2965	14	34	=	=	SYM
easat-2965	14	35	1	1	X
easat-2965	14	36	.	.	X
easat-2965	14	37	pell	pell	NOUN
easat-2965	14	38	-	-	PUNCT
easat-2965	14	39	lucas	lucas	PROPN
easat-2965	14	40	number	number	NOUN
easat-2965	14	41	sequence	sequence	NOUN
easat-2965	14	42	ψ𝑛+2	ψ𝑛+2	NOUN
easat-2965	14	43	=	=	SYM
easat-2965	14	44	2ψ𝑛+1	2ψ𝑛+1	PROPN
easat-2965	15	1	+	+	CCONJ
easat-2965	15	2	ψ𝑛	ψ𝑛	PRON
easat-2965	15	3	for	for	ADP
easat-2965	15	4	𝑛	𝑛	DET
easat-2965	15	5	≥	≥	NOUN
easat-2965	15	6	0	0	NUM
easat-2965	15	7	with	with	ADP
easat-2965	15	8	ψ0	ψ0	ADJ
easat-2965	15	9	=	=	SYM
easat-2965	15	10	2	2	NUM
easat-2965	15	11	and	and	CCONJ
easat-2965	15	12	ψ1	ψ1	NOUN
easat-2965	15	13	=	=	SYM
easat-2965	15	14	2	2	X
easat-2965	15	15	.	.	PUNCT
easat-2965	16	1	in	in	ADP
easat-2965	16	2	recent	recent	ADJ
easat-2965	16	3	years	year	NOUN
easat-2965	16	4	,	,	PUNCT
easat-2965	16	5	these	these	DET
easat-2965	16	6	sequences	sequence	NOUN
easat-2965	16	7	of	of	ADP
easat-2965	16	8	numbers	number	NOUN
easat-2965	16	9	have	have	VERB
easat-2965	16	10	significant	significant	ADJ
easat-2965	16	11	applications	application	NOUN
easat-2965	16	12	in	in	ADP
easat-2965	16	13	the	the	DET
easat-2965	16	14	fields	field	NOUN
easat-2965	16	15	including	include	VERB
easat-2965	16	16	statistics	statistic	NOUN
easat-2965	16	17	,	,	PUNCT
easat-2965	16	18	music	music	NOUN
easat-2965	16	19	,	,	PUNCT
easat-2965	16	20	coding	code	VERB
easat-2965	16	21	theory	theory	NOUN
easat-2965	16	22	,	,	PUNCT
easat-2965	16	23	cryptography	cryptography	NOUN
easat-2965	16	24	and	and	CCONJ
easat-2965	16	25	communication	communication	NOUN
easat-2965	16	26	systems	system	NOUN
easat-2965	16	27	.	.	PUNCT
easat-2965	17	1	[	[	X
easat-2965	17	2	2	2	NUM
easat-2965	17	3	]	]	PUNCT
easat-2965	17	4	,	,	PUNCT
easat-2965	17	5	[	[	X
easat-2965	17	6	3	3	NUM
easat-2965	17	7	]	]	PUNCT
easat-2965	17	8	,	,	PUNCT
easat-2965	17	9	[	[	X
easat-2965	17	10	4	4	NUM
easat-2965	17	11	]	]	PUNCT
easat-2965	17	12	.	.	PUNCT
easat-2965	18	1	as	as	SCONJ
easat-2965	18	2	seen	see	VERB
easat-2965	18	3	in	in	ADP
easat-2965	18	4	[	[	X
easat-2965	18	5	11,12,13	11,12,13	NUM
easat-2965	18	6	]	]	PUNCT
easat-2965	18	7	,	,	PUNCT
easat-2965	18	8	number	number	NOUN
easat-2965	18	9	sequences	sequence	NOUN
easat-2965	18	10	are	be	AUX
easat-2965	18	11	generalized	generalize	VERB
easat-2965	18	12	in	in	ADP
easat-2965	18	13	several	several	ADJ
easat-2965	18	14	ways	way	NOUN
easat-2965	18	15	.	.	PUNCT
easat-2965	19	1	one	one	PRON
easat-2965	19	2	can	can	AUX
easat-2965	19	3	see	see	VERB
easat-2965	19	4	some	some	DET
easat-2965	19	5	generalizations	generalization	NOUN
easat-2965	19	6	of	of	ADP
easat-2965	19	7	pell	pell	NOUN
easat-2965	19	8	and	and	CCONJ
easat-2965	19	9	pell	pell	ADJ
easat-2965	19	10	–	–	PUNCT
easat-2965	19	11	lucas	lucas	NOUN
easat-2965	19	12	numbers	number	NOUN
easat-2965	19	13	.	.	PUNCT
easat-2965	20	1	in	in	ADP
easat-2965	20	2	this	this	DET
easat-2965	20	3	paper	paper	NOUN
easat-2965	20	4	we	we	PRON
easat-2965	20	5	will	will	AUX
easat-2965	20	6	pay	pay	VERB
easat-2965	20	7	attention	attention	NOUN
easat-2965	20	8	to	to	ADP
easat-2965	20	9	the	the	DET
easat-2965	20	10	following	follow	VERB
easat-2965	20	11	generalization	generalization	NOUN
easat-2965	20	12	:	:	PUNCT
easat-2965	20	13	let	let	VERB
easat-2965	20	14	𝑠	𝑠	PROPN
easat-2965	20	15	and	and	CCONJ
easat-2965	20	16	𝑡	𝑡	PROPN
easat-2965	20	17	be	be	VERB
easat-2965	20	18	two	two	NUM
easat-2965	20	19	non	non	ADJ
easat-2965	20	20	-	-	ADJ
easat-2965	20	21	zero	zero	ADJ
easat-2965	20	22	integers	integer	NOUN
easat-2965	20	23	satisfying	satisfy	VERB
easat-2965	20	24	𝑠2	𝑠2	NOUN
easat-2965	20	25	+	+	CCONJ
easat-2965	20	26	4𝑡	4𝑡	NOUN
easat-2965	20	27	>	>	X
easat-2965	20	28	0	0	NUM
easat-2965	20	29	,	,	PUNCT
easat-2965	20	30	the	the	DET
easat-2965	20	31	generalized	generalized	ADJ
easat-2965	20	32	fibonacci	fibonacci	NOUN
easat-2965	20	33	and	and	CCONJ
easat-2965	20	34	generalized	generalized	ADJ
easat-2965	20	35	lucas	lucas	NOUN
easat-2965	20	36	sequences	sequence	NOUN
easat-2965	20	37	are	be	AUX
easat-2965	20	38	,	,	PUNCT
easat-2965	20	39	respectively	respectively	ADV
easat-2965	20	40	,	,	PUNCT
easat-2965	20	41	defined	define	VERB
easat-2965	20	42	as	as	ADP
easat-2965	20	43	:	:	PUNCT
easat-2965	20	44	𝐹𝑛+2(𝑠	𝐹𝑛+2(𝑠	ADJ
easat-2965	20	45	,	,	PUNCT
easat-2965	20	46	𝑡	𝑡	X
easat-2965	20	47	)	)	PUNCT
easat-2965	20	48	=	=	SYM
easat-2965	20	49	𝑠𝐹𝑛+1(𝑠	𝑠𝐹𝑛+1(𝑠	PROPN
easat-2965	20	50	,	,	PUNCT
easat-2965	20	51	𝑡	𝑡	PROPN
easat-2965	20	52	)	)	PUNCT
easat-2965	20	53	+	+	NUM
easat-2965	20	54	𝑡𝐹𝑛(𝑠	𝑡𝐹𝑛(𝑠	NOUN
easat-2965	20	55	,	,	PUNCT
easat-2965	20	56	𝑡	𝑡	PROPN
easat-2965	20	57	)	)	PUNCT
easat-2965	20	58	for	for	ADP
easat-2965	20	59	𝑛	𝑛	DET
easat-2965	20	60	≥	≥	NOUN
easat-2965	20	61	0	0	NUM
easat-2965	20	62	with	with	ADP
easat-2965	20	63	𝐹0(𝑠	𝐹0(𝑠	PROPN
easat-2965	20	64	,	,	PUNCT
easat-2965	20	65	𝑡	𝑡	NOUN
easat-2965	20	66	)	)	PUNCT
easat-2965	20	67	=	=	SYM
easat-2965	20	68	0	0	NUM
easat-2965	20	69	,	,	PUNCT
easat-2965	20	70	𝐹1(𝑠	𝐹1(𝑠	PROPN
easat-2965	20	71	,	,	PUNCT
easat-2965	20	72	𝑡	𝑡	X
easat-2965	20	73	)	)	PUNCT
easat-2965	20	74	=	=	SYM
easat-2965	20	75	1	1	NUM
easat-2965	20	76	,	,	PUNCT
easat-2965	20	77	and	and	CCONJ
easat-2965	20	78	𝐿𝑛+2(𝑠	𝐿𝑛+2(𝑠	NOUN
easat-2965	20	79	,	,	PUNCT
easat-2965	20	80	𝑡	𝑡	NOUN
easat-2965	20	81	)	)	PUNCT
easat-2965	20	82	=	=	SYM
easat-2965	21	1	𝑠𝐿𝑛+1(𝑠	𝑠𝐿𝑛+1(𝑠	PROPN
easat-2965	21	2	,	,	PUNCT
easat-2965	21	3	𝑡	𝑡	NOUN
easat-2965	21	4	)	)	PUNCT
easat-2965	21	5	+	+	CCONJ
easat-2965	21	6	𝑡𝐿𝑛(𝑠	𝑡𝐿𝑛(𝑠	PROPN
easat-2965	21	7	,	,	PUNCT
easat-2965	21	8	𝑡	𝑡	PROPN
easat-2965	21	9	)	)	PUNCT
easat-2965	21	10	for	for	ADP
easat-2965	21	11	𝑛	𝑛	DET
easat-2965	21	12	≥	≥	NOUN
easat-2965	21	13	0	0	NUM
easat-2965	21	14	with	with	ADP
easat-2965	21	15	𝐿0(𝑠	𝐿0(𝑠	NOUN
easat-2965	21	16	,	,	PUNCT
easat-2965	21	17	𝑡	𝑡	NOUN
easat-2965	21	18	)	)	PUNCT
easat-2965	21	19	=	=	SYM
easat-2965	21	20	2	2	NUM
easat-2965	21	21	,	,	PUNCT
easat-2965	21	22	𝐿1(𝑠	𝐿1(𝑠	PROPN
easat-2965	21	23	,	,	PUNCT
easat-2965	21	24	𝑡	𝑡	PROPN
easat-2965	21	25	)	)	PUNCT
easat-2965	21	26	=	=	SYM
easat-2965	21	27	1	1	NUM
easat-2965	21	28	,	,	PUNCT
easat-2965	21	29	binet	binet	NOUN
easat-2965	21	30	’s	’s	PART
easat-2965	21	31	formulae	formulae	NOUN
easat-2965	21	32	for	for	ADP
easat-2965	21	33	these	these	DET
easat-2965	21	34	sequences	sequence	NOUN
easat-2965	21	35	are	be	AUX
easat-2965	21	36	:	:	PUNCT
easat-2965	21	37	4409	4409	NUM
easat-2965	21	38	edelweiss	edelweiss	PROPN
easat-2965	21	39	applied	apply	VERB
easat-2965	21	40	science	science	NOUN
easat-2965	21	41	and	and	CCONJ
easat-2965	21	42	technology	technology	NOUN
easat-2965	21	43	issn	issn	PROPN
easat-2965	21	44	:	:	PUNCT
easat-2965	21	45	2576	2576	NUM
easat-2965	21	46	-	-	SYM
easat-2965	21	47	8484	8484	NUM
easat-2965	21	48	vol	vol	NOUN
easat-2965	21	49	.	.	PROPN
easat-2965	21	50	8	8	NUM
easat-2965	21	51	,	,	PUNCT
easat-2965	21	52	no	no	INTJ
easat-2965	21	53	.	.	NOUN
easat-2965	22	1	6	6	NUM
easat-2965	22	2	:	:	SYM
easat-2965	22	3	4408	4408	NUM
easat-2965	22	4	-	-	SYM
easat-2965	22	5	4414	4414	NUM
easat-2965	22	6	,	,	PUNCT
easat-2965	22	7	2024	2024	NUM
easat-2965	22	8	doi	doi	NOUN
easat-2965	22	9	:	:	PUNCT
easat-2965	22	10	10.55214/25768484.v8i6.2965	10.55214/25768484.v8i6.2965	NUM
easat-2965	22	11	©	©	PROPN
easat-2965	22	12	2024	2024	NUM
easat-2965	22	13	by	by	ADP
easat-2965	22	14	the	the	DET
easat-2965	22	15	authors	author	NOUN
easat-2965	22	16	;	;	PUNCT
easat-2965	22	17	licensee	licensee	PROPN
easat-2965	22	18	learning	learn	VERB
easat-2965	22	19	gate	gate	NOUN
easat-2965	22	20	𝐹𝑛(𝑠	𝐹𝑛(𝑠	PROPN
easat-2965	22	21	,	,	PUNCT
easat-2965	22	22	𝑡	𝑡	X
easat-2965	22	23	)	)	PUNCT
easat-2965	22	24	=	=	SYM
easat-2965	23	1	𝛼𝑛−	𝛼𝑛−	NUM
easat-2965	23	2	𝛽𝑛	𝛽𝑛	PROPN
easat-2965	23	3	𝛼−𝛽	𝛼−𝛽	NOUN
easat-2965	23	4	,	,	PUNCT
easat-2965	23	5	𝐿𝑛(𝑠	𝐿𝑛(𝑠	PROPN
easat-2965	23	6	,	,	PUNCT
easat-2965	23	7	𝑡	𝑡	PROPN
easat-2965	23	8	)	)	PUNCT
easat-2965	23	9	=	=	SYM
easat-2965	23	10	𝛼	𝛼	PART
easat-2965	23	11	𝑛	𝑛	PROPN
easat-2965	23	12	+	+	CCONJ
easat-2965	23	13	𝛽𝑛	𝛽𝑛	X
easat-2965	23	14	,	,	PUNCT
easat-2965	23	15	where	where	SCONJ
easat-2965	23	16	𝛼	𝛼	NOUN
easat-2965	23	17	and	and	CCONJ
easat-2965	23	18	𝛽	𝛽	NOUN
easat-2965	23	19	are	be	AUX
easat-2965	23	20	the	the	DET
easat-2965	23	21	roots	root	NOUN
easat-2965	23	22	of	of	ADP
easat-2965	23	23	equation	equation	NOUN
easat-2965	23	24	𝑥2	𝑥2	NOUN
easat-2965	23	25	−	−	PROPN
easat-2965	23	26	𝑠𝑥	𝑠𝑥	ADP
easat-2965	23	27	−	−	PROPN
easat-2965	23	28	𝑡	𝑡	PROPN
easat-2965	23	29	=	=	NOUN
easat-2965	23	30	0	0	NUM
easat-2965	23	31	.	.	PUNCT
easat-2965	24	1	𝛼	𝛼	PRON
easat-2965	24	2	and	and	CCONJ
easat-2965	24	3	𝛽	𝛽	NOUN
easat-2965	24	4	verify	verify	VERB
easat-2965	24	5	𝛼	𝛼	PRON
easat-2965	24	6	+	+	NOUN
easat-2965	24	7	𝛽	𝛽	NOUN
easat-2965	24	8	=	=	SYM
easat-2965	24	9	𝑠	𝑠	PROPN
easat-2965	24	10	,	,	PUNCT
easat-2965	24	11	𝛼	𝛼	PROPN
easat-2965	24	12	−	−	PROPN
easat-2965	24	13	𝛽	𝛽	NOUN
easat-2965	24	14	=	=	SYM
easat-2965	24	15	2√𝑠2	2√𝑠2	PROPN
easat-2965	24	16	+	+	CCONJ
easat-2965	24	17	4𝑡	4𝑡	ADJ
easat-2965	24	18	,	,	PUNCT
easat-2965	24	19	𝛼𝛽	𝛼𝛽	NOUN
easat-2965	24	20	=	=	SYM
easat-2965	24	21	−𝑡.	−𝑡.	NOUN
easat-2965	24	22	for	for	ADP
easat-2965	24	23	further	further	ADJ
easat-2965	24	24	details	detail	NOUN
easat-2965	24	25	of	of	ADP
easat-2965	24	26	these	these	DET
easat-2965	24	27	sequences	sequence	NOUN
easat-2965	24	28	,	,	PUNCT
easat-2965	24	29	see	see	VERB
easat-2965	24	30	[	[	X
easat-2965	24	31	5	5	NUM
easat-2965	24	32	]	]	PUNCT
easat-2965	24	33	,	,	PUNCT
easat-2965	24	34	[	[	X
easat-2965	24	35	6	6	NUM
easat-2965	24	36	]	]	PUNCT
easat-2965	24	37	,	,	PUNCT
easat-2965	25	1	[	[	X
easat-2965	25	2	7],[17	7],[17	X
easat-2965	25	3	]	]	PUNCT
easat-2965	25	4	,	,	PUNCT
easat-2965	25	5	[	[	X
easat-2965	25	6	18	18	NUM
easat-2965	25	7	]	]	PUNCT
easat-2965	25	8	,	,	PUNCT
easat-2965	25	9	[	[	X
easat-2965	25	10	19	19	NUM
easat-2965	25	11	]	]	PUNCT
easat-2965	25	12	,	,	PUNCT
easat-2965	26	1	[	[	X
easat-2965	26	2	20],[21	20],[21	NOUN
easat-2965	26	3	]	]	X
easat-2965	26	4	.	.	PUNCT
easat-2965	27	1	let	let	VERB
easat-2965	27	2	,	,	PUNCT
easat-2965	27	3	now	now	ADV
easat-2965	27	4	,	,	PUNCT
easat-2965	27	5	𝑘	𝑘	PROPN
easat-2965	27	6	and	and	CCONJ
easat-2965	27	7	ℎ	ℎ	NOUN
easat-2965	27	8	be	be	AUX
easat-2965	27	9	two	two	NUM
easat-2965	27	10	non	non	ADJ
easat-2965	27	11	-	-	ADJ
easat-2965	27	12	zero	zero	ADJ
easat-2965	27	13	integers	integer	NOUN
easat-2965	27	14	satisfying	satisfy	VERB
easat-2965	27	15	𝑘2	𝑘2	PROPN
easat-2965	28	1	+	+	CCONJ
easat-2965	28	2	ℎ	ℎ	PROPN
easat-2965	28	3	>	>	X
easat-2965	28	4	0	0	PROPN
easat-2965	28	5	,	,	PUNCT
easat-2965	28	6	the	the	DET
easat-2965	28	7	generalized	generalized	ADJ
easat-2965	28	8	pell	pell	NOUN
easat-2965	28	9	and	and	CCONJ
easat-2965	28	10	generalized	generalize	VERB
easat-2965	28	11	pell	pell	NOUN
easat-2965	28	12	-	-	PUNCT
easat-2965	28	13	lucas	lucas	NOUN
easat-2965	28	14	sequences	sequence	NOUN
easat-2965	28	15	are	be	AUX
easat-2965	28	16	,	,	PUNCT
easat-2965	28	17	respectively	respectively	ADV
easat-2965	28	18	,	,	PUNCT
easat-2965	28	19	defined	define	VERB
easat-2965	28	20	as	as	ADP
easat-2965	28	21	:	:	PUNCT
easat-2965	28	22	φ𝑛+2(𝑘	φ𝑛+2(𝑘	NOUN
easat-2965	28	23	,	,	PUNCT
easat-2965	28	24	ℎ	ℎ	PROPN
easat-2965	28	25	)	)	PUNCT
easat-2965	28	26	=	=	SYM
easat-2965	28	27	2𝑘φ𝑛+1(𝑘	2𝑘φ𝑛+1(𝑘	NUM
easat-2965	28	28	,	,	PUNCT
easat-2965	28	29	ℎ	ℎ	PROPN
easat-2965	28	30	)	)	PUNCT
easat-2965	29	1	+	+	CCONJ
easat-2965	29	2	ℎφ𝑛(𝑘	ℎφ𝑛(𝑘	ADJ
easat-2965	29	3	,	,	PUNCT
easat-2965	29	4	ℎ	ℎ	PROPN
easat-2965	29	5	)	)	PUNCT
easat-2965	29	6	for	for	ADP
easat-2965	29	7	𝑛	𝑛	DET
easat-2965	29	8	≥	≥	NOUN
easat-2965	29	9	0	0	NUM
easat-2965	29	10	with	with	ADP
easat-2965	29	11	φ0(𝑘	φ0(𝑘	PROPN
easat-2965	29	12	,	,	PUNCT
easat-2965	29	13	ℎ	ℎ	PROPN
easat-2965	29	14	)	)	PUNCT
easat-2965	29	15	=	=	SYM
easat-2965	29	16	0	0	NUM
easat-2965	29	17	and	and	CCONJ
easat-2965	29	18	φ1(𝑘	φ1(𝑘	ADP
easat-2965	29	19	,	,	PUNCT
easat-2965	29	20	ℎ	ℎ	PROPN
easat-2965	29	21	)	)	PUNCT
easat-2965	29	22	=	=	SYM
easat-2965	29	23	1	1	NUM
easat-2965	29	24	,	,	PUNCT
easat-2965	29	25	and	and	CCONJ
easat-2965	29	26	ψ𝑛+2(𝑘	ψ𝑛+2(𝑘	VERB
easat-2965	29	27	,	,	PUNCT
easat-2965	29	28	ℎ	ℎ	PROPN
easat-2965	29	29	)	)	PUNCT
easat-2965	29	30	=	=	SYM
easat-2965	29	31	2𝑘ψ𝑛+1(𝑘	2𝑘ψ𝑛+1(𝑘	NUM
easat-2965	29	32	,	,	PUNCT
easat-2965	29	33	ℎ	ℎ	PROPN
easat-2965	29	34	)	)	PUNCT
easat-2965	29	35	+	+	X
easat-2965	29	36	ℎψ𝑛(𝑘	ℎψ𝑛(𝑘	PROPN
easat-2965	29	37	,	,	PUNCT
easat-2965	29	38	ℎ	ℎ	PROPN
easat-2965	29	39	)	)	PUNCT
easat-2965	29	40	for	for	ADP
easat-2965	29	41	𝑛	𝑛	DET
easat-2965	29	42	≥	≥	NOUN
easat-2965	29	43	0	0	NUM
easat-2965	29	44	with	with	ADP
easat-2965	29	45	ψ0(𝑘	ψ0(𝑘	PROPN
easat-2965	29	46	,	,	PUNCT
easat-2965	29	47	ℎ	ℎ	PROPN
easat-2965	29	48	)	)	PUNCT
easat-2965	29	49	=	=	SYM
easat-2965	29	50	2	2	NUM
easat-2965	29	51	and	and	CCONJ
easat-2965	29	52	ψ1(𝑘	ψ1(𝑘	PROPN
easat-2965	29	53	,	,	PUNCT
easat-2965	29	54	ℎ	ℎ	PROPN
easat-2965	29	55	)	)	PUNCT
easat-2965	29	56	=	=	SYM
easat-2965	30	1	2𝑘	2𝑘	NUM
easat-2965	30	2	binet	binet	NOUN
easat-2965	30	3	’s	’s	PART
easat-2965	30	4	formulae	formulae	NOUN
easat-2965	30	5	for	for	ADP
easat-2965	30	6	these	these	DET
easat-2965	30	7	sequences	sequence	NOUN
easat-2965	30	8	are	be	AUX
easat-2965	30	9	:	:	PUNCT
easat-2965	30	10	φ𝑛(𝑘	φ𝑛(𝑘	NOUN
easat-2965	30	11	,	,	PUNCT
easat-2965	30	12	ℎ	ℎ	ADJ
easat-2965	30	13	)	)	PUNCT
easat-2965	30	14	=	=	NOUN
easat-2965	31	1	2𝑘	2𝑘	NUM
easat-2965	31	2	𝛼𝑛−	𝛼𝑛−	NUM
easat-2965	31	3	𝛽𝑛	𝛽𝑛	PROPN
easat-2965	31	4	𝛼−𝛽	𝛼−𝛽	NOUN
easat-2965	31	5	,	,	PUNCT
easat-2965	31	6	ψ𝑛(𝑘	ψ𝑛(𝑘	NOUN
easat-2965	31	7	,	,	PUNCT
easat-2965	31	8	ℎ	ℎ	PROPN
easat-2965	31	9	)	)	PUNCT
easat-2965	31	10	=	=	SYM
easat-2965	31	11	𝛼	𝛼	PART
easat-2965	31	12	𝑛	𝑛	PROPN
easat-2965	31	13	+	+	CCONJ
easat-2965	32	1	𝛽𝑛	𝛽𝑛	X
easat-2965	32	2	,	,	PUNCT
easat-2965	32	3	where	where	SCONJ
easat-2965	32	4	𝛼	𝛼	NOUN
easat-2965	32	5	and	and	CCONJ
easat-2965	32	6	𝛽	𝛽	NOUN
easat-2965	32	7	are	be	AUX
easat-2965	32	8	the	the	DET
easat-2965	32	9	roots	root	NOUN
easat-2965	32	10	of	of	ADP
easat-2965	32	11	equation	equation	NOUN
easat-2965	32	12	𝑥2	𝑥2	NOUN
easat-2965	33	1	−	−	PROPN
easat-2965	33	2	2𝑘𝑥	2𝑘𝑥	ADJ
easat-2965	33	3	−	−	NOUN
easat-2965	33	4	ℎ	ℎ	X
easat-2965	33	5	=	=	SYM
easat-2965	33	6	0	0	PROPN
easat-2965	33	7	.	.	PUNCT
easat-2965	34	1	𝛼	𝛼	PRON
easat-2965	34	2	and	and	CCONJ
easat-2965	34	3	𝛽	𝛽	NOUN
easat-2965	34	4	verify	verify	VERB
easat-2965	34	5	𝛼	𝛼	PRON
easat-2965	34	6	+	+	NOUN
easat-2965	34	7	𝛽	𝛽	NOUN
easat-2965	34	8	=	=	NOUN
easat-2965	34	9	2𝑘	2𝑘	NUM
easat-2965	34	10	,	,	PUNCT
easat-2965	34	11	𝛼	𝛼	PROPN
easat-2965	34	12	−	−	PROPN
easat-2965	34	13	𝛽	𝛽	NOUN
easat-2965	34	14	=	=	SYM
easat-2965	34	15	2√𝑘2	2√𝑘2	PROPN
easat-2965	34	16	+	+	CCONJ
easat-2965	34	17	ℎ	ℎ	PROPN
easat-2965	34	18	,	,	PUNCT
easat-2965	34	19	𝛼𝛽	𝛼𝛽	NOUN
easat-2965	34	20	=	=	SYM
easat-2965	34	21	−ℎ.	−ℎ.	PROPN
easat-2965	34	22	in	in	ADP
easat-2965	34	23	the	the	DET
easat-2965	34	24	plethora	plethora	NOUN
easat-2965	34	25	of	of	ADP
easat-2965	34	26	integer	integer	NOUN
easat-2965	34	27	sequences	sequence	NOUN
easat-2965	34	28	,	,	PUNCT
easat-2965	34	29	the	the	DET
easat-2965	34	30	fibonacci	fibonacci	PROPN
easat-2965	34	31	and	and	CCONJ
easat-2965	34	32	lucas	lucas	PROPN
easat-2965	34	33	sequences	sequence	NOUN
easat-2965	34	34	stand	stand	VERB
easat-2965	34	35	out	out	ADP
easat-2965	34	36	as	as	ADP
easat-2965	34	37	the	the	DET
easat-2965	34	38	two	two	NUM
easat-2965	34	39	brightest	bright	ADJ
easat-2965	34	40	.	.	PUNCT
easat-2965	35	1	they	they	PRON
easat-2965	35	2	have	have	AUX
easat-2965	35	3	captivated	captivate	VERB
easat-2965	35	4	both	both	CCONJ
easat-2965	35	5	amateur	amateur	ADJ
easat-2965	35	6	and	and	CCONJ
easat-2965	35	7	expert	expert	ADJ
easat-2965	35	8	mathematicians	mathematician	NOUN
easat-2965	35	9	for	for	ADP
easat-2965	35	10	ages	age	NOUN
easat-2965	35	11	,	,	PUNCT
easat-2965	35	12	and	and	CCONJ
easat-2965	35	13	they	they	PRON
easat-2965	35	14	never	never	ADV
easat-2965	35	15	cease	cease	VERB
easat-2965	35	16	to	to	PART
easat-2965	35	17	enchant	enchant	VERB
easat-2965	35	18	us	we	PRON
easat-2965	35	19	with	with	ADP
easat-2965	35	20	their	their	PRON
easat-2965	35	21	beauty	beauty	NOUN
easat-2965	35	22	,	,	PUNCT
easat-2965	35	23	myriad	myriad	ADJ
easat-2965	35	24	practical	practical	ADJ
easat-2965	35	25	uses	use	NOUN
easat-2965	35	26	,	,	PUNCT
easat-2965	35	27	and	and	CCONJ
easat-2965	35	28	omnipresent	omnipresent	NOUN
easat-2965	35	29	propensity	propensity	NOUN
easat-2965	35	30	to	to	PART
easat-2965	35	31	appear	appear	VERB
easat-2965	35	32	in	in	ADP
easat-2965	35	33	completely	completely	ADV
easat-2965	35	34	unexpected	unexpected	ADJ
easat-2965	35	35	and	and	CCONJ
easat-2965	35	36	unrelated	unrelated	ADJ
easat-2965	35	37	contexts	contexts	NOUN
easat-2965	35	38	.	.	PUNCT
easat-2965	36	1	they	they	PRON
easat-2965	36	2	continue	continue	VERB
easat-2965	36	3	to	to	PART
easat-2965	36	4	be	be	AUX
easat-2965	36	5	a	a	DET
easat-2965	36	6	fertile	fertile	ADJ
easat-2965	36	7	ground	ground	NOUN
easat-2965	36	8	for	for	ADP
easat-2965	36	9	creative	creative	ADJ
easat-2965	36	10	amateurs	amateur	NOUN
easat-2965	36	11	and	and	CCONJ
easat-2965	36	12	mathematicians	mathematician	NOUN
easat-2965	36	13	alike	alike	ADV
easat-2965	36	14	.	.	PUNCT
easat-2965	37	1	in	in	ADP
easat-2965	37	2	literature	literature	NOUN
easat-2965	37	3	,	,	PUNCT
easat-2965	37	4	fibonacci	fibonacci	PROPN
easat-2965	37	5	and	and	CCONJ
easat-2965	37	6	lucas	lucas	PROPN
easat-2965	37	7	numbers	number	NOUN
easat-2965	37	8	are	be	AUX
easat-2965	37	9	used	use	VERB
easat-2965	37	10	to	to	PART
easat-2965	37	11	resolve	resolve	VERB
easat-2965	37	12	many	many	ADJ
easat-2965	37	13	diophantine	diophantine	NOUN
easat-2965	37	14	equations	equation	NOUN
easat-2965	37	15	.	.	PUNCT
easat-2965	38	1	the	the	DET
easat-2965	38	2	quadratic	quadratic	ADJ
easat-2965	38	3	diophantine	diophantine	NOUN
easat-2965	38	4	equation	equation	NOUN
easat-2965	38	5	of	of	ADP
easat-2965	38	6	the	the	DET
easat-2965	38	7	form	form	NOUN
easat-2965	38	8	𝑥2	𝑥2	NOUN
easat-2965	38	9	−	−	PROPN
easat-2965	38	10	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	38	11	=	=	SYM
easat-2965	38	12	𝑁	𝑁	PROPN
easat-2965	38	13	,	,	PUNCT
easat-2965	38	14	where	where	SCONJ
easat-2965	38	15	𝐷	𝐷	PROPN
easat-2965	38	16	is	be	AUX
easat-2965	38	17	square	square	ADV
easat-2965	38	18	free	free	ADJ
easat-2965	38	19	,	,	PUNCT
easat-2965	38	20	generally	generally	ADV
easat-2965	38	21	known	know	VERB
easat-2965	38	22	as	as	ADP
easat-2965	38	23	pell	pell	NOUN
easat-2965	38	24	’s	’s	PART
easat-2965	38	25	equation	equation	NOUN
easat-2965	38	26	.	.	PUNCT
easat-2965	39	1	in	in	ADP
easat-2965	39	2	this	this	DET
easat-2965	39	3	case	case	NOUN
easat-2965	39	4	,	,	PUNCT
easat-2965	39	5	if	if	SCONJ
easat-2965	39	6	𝐷	𝐷	PROPN
easat-2965	39	7	is	be	AUX
easat-2965	39	8	square	square	ADV
easat-2965	39	9	free	free	ADJ
easat-2965	39	10	,	,	PUNCT
easat-2965	39	11	the	the	DET
easat-2965	39	12	continued	continue	VERB
easat-2965	39	13	fraction	fraction	NOUN
easat-2965	39	14	expansion	expansion	NOUN
easat-2965	39	15	of	of	ADP
easat-2965	39	16	𝐷	𝐷	PROPN
easat-2965	39	17	is	be	AUX
easat-2965	39	18	periodic	periodic	ADJ
easat-2965	39	19	and	and	CCONJ
easat-2965	39	20	it	it	PRON
easat-2965	39	21	is	be	AUX
easat-2965	39	22	given	give	VERB
easat-2965	39	23	by	by	ADP
easat-2965	39	24	𝐷	𝐷	PROPN
easat-2965	39	25	=	=	PUNCT
easat-2965	39	26	𝑎0	𝑎0	PROPN
easat-2965	39	27	;	;	PUNCT
easat-2965	39	28	𝑎1	𝑎1	INTJ
easat-2965	39	29	,	,	PUNCT
easat-2965	39	30	𝑎2	𝑎2	NOUN
easat-2965	39	31	,	,	PUNCT
easat-2965	39	32	⋯	⋯	PROPN
easat-2965	39	33	𝑎𝑛−1	𝑎𝑛−1	PROPN
easat-2965	39	34	,	,	PUNCT
easat-2965	39	35	𝑎𝑛̅̅	𝑎𝑛̅̅	ADJ
easat-2965	39	36	̅̅	̅̅	PROPN
easat-2965	39	37	̅̅	̅̅	PROPN
easat-2965	39	38	̅̅	̅̅	PROPN
easat-2965	39	39	̅̅	̅̅	PROPN
easat-2965	39	40	̅̅	̅̅	PROPN
easat-2965	39	41	̅̅	̅̅	PROPN
easat-2965	39	42	̅̅	̅̅	PROPN
easat-2965	39	43	̅̅	̅̅	PROPN
easat-2965	39	44	̅̅	̅̅	PROPN
easat-2965	39	45	̅̅	̅̅	PROPN
easat-2965	39	46	̅	̅	PROPN
easat-2965	39	47	,	,	PUNCT
easat-2965	39	48	where	where	SCONJ
easat-2965	39	49	𝑎𝑛	𝑎𝑛	NOUN
easat-2965	39	50	=	=	SYM
easat-2965	39	51	2𝑎0	2𝑎0	NUM
easat-2965	39	52	and	and	CCONJ
easat-2965	39	53	𝑛	𝑛	PROPN
easat-2965	39	54	is	be	AUX
easat-2965	39	55	the	the	DET
easat-2965	39	56	length	length	NOUN
easat-2965	39	57	of	of	ADP
easat-2965	39	58	period	period	NOUN
easat-2965	39	59	.	.	PUNCT
easat-2965	40	1	as	as	SCONJ
easat-2965	40	2	it	it	PRON
easat-2965	40	3	is	be	AUX
easat-2965	40	4	known	know	VERB
easat-2965	40	5	,	,	PUNCT
easat-2965	40	6	we	we	PRON
easat-2965	40	7	denote	denote	VERB
easat-2965	40	8	by	by	ADP
easat-2965	40	9	𝑝1	𝑝1	NOUN
easat-2965	40	10	𝑞1	𝑞1	PROPN
easat-2965	40	11	=	=	PUNCT
easat-2965	41	1	[	[	X
easat-2965	41	2	𝑎0	𝑎0	ADJ
easat-2965	41	3	;	;	PUNCT
easat-2965	41	4	𝑎1	𝑎1	INTJ
easat-2965	41	5	,	,	PUNCT
easat-2965	41	6	𝑎2	𝑎2	PROPN
easat-2965	41	7	,	,	PUNCT
easat-2965	41	8	⋯	⋯	PROPN
easat-2965	41	9	𝑎𝑙	𝑎𝑙	PROPN
easat-2965	41	10	]	]	X
easat-2965	41	11	the	the	DET
easat-2965	41	12	𝑙𝑡ℎ	𝑙𝑡ℎ	NOUN
easat-2965	41	13	convergent	convergent	NOUN
easat-2965	41	14	of	of	ADP
easat-2965	41	15	𝐷	𝐷	PROPN
easat-2965	41	16	,	,	PUNCT
easat-2965	41	17	for	for	ADP
easat-2965	41	18	𝑙	𝑙	DET
easat-2965	41	19	≥	≥	NOUN
easat-2965	41	20	0	0	NUM
easat-2965	41	21	.	.	PUNCT
easat-2965	42	1	in	in	ADP
easat-2965	42	2	(	(	PUNCT
easat-2965	42	3	5	5	NUM
easat-2965	42	4	)	)	PUNCT
easat-2965	42	5	,	,	PUNCT
easat-2965	42	6	we	we	PRON
easat-2965	42	7	have	have	AUX
easat-2965	42	8	obtained	obtain	VERB
easat-2965	42	9	some	some	DET
easat-2965	42	10	formulas	formula	NOUN
easat-2965	42	11	for	for	ADP
easat-2965	42	12	the	the	DET
easat-2965	42	13	integer	integer	NOUN
easat-2965	42	14	solutions	solution	NOUN
easat-2965	42	15	of	of	ADP
easat-2965	42	16	the	the	DET
easat-2965	42	17	pell	pell	NOUN
easat-2965	42	18	equation	equation	NOUN
easat-2965	42	19	𝑥2	𝑥2	NOUN
easat-2965	42	20	−	−	PROPN
easat-2965	42	21	𝑦2	𝑦2	PROPN
easat-2965	42	22	=	=	SYM
easat-2965	42	23	±𝑘2	±𝑘2	PROPN
easat-2965	42	24	for	for	ADP
easat-2965	42	25	all	all	DET
easat-2965	42	26	𝑘	𝑘	DET
easat-2965	42	27	≥	≥	NOUN
easat-2965	42	28	1	1	NUM
easat-2965	42	29	.	.	PUNCT
easat-2965	43	1	in	in	ADP
easat-2965	43	2	[	[	X
easat-2965	43	3	9	9	NUM
easat-2965	43	4	]	]	PUNCT
easat-2965	43	5	,	,	PUNCT
easat-2965	43	6	bala	bala	PROPN
easat-2965	43	7	and	and	CCONJ
easat-2965	43	8	mishra	mishra	PROPN
easat-2965	43	9	,	,	PUNCT
easat-2965	43	10	considered	consider	VERB
easat-2965	43	11	the	the	DET
easat-2965	43	12	solutions	solution	NOUN
easat-2965	43	13	of	of	ADP
easat-2965	43	14	two	two	NUM
easat-2965	43	15	diophantine	diophantine	NOUN
easat-2965	43	16	equations	equation	NOUN
easat-2965	43	17	𝑥2	𝑥2	NOUN
easat-2965	43	18	−	−	PROPN
easat-2965	43	19	(	(	PUNCT
easat-2965	43	20	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	43	21	±	±	NOUN
easat-2965	43	22	3p)𝑦2	3p)𝑦2	NUM
easat-2965	43	23	=	=	SYM
easat-2965	43	24	𝑘𝑡	𝑘𝑡	NOUN
easat-2965	43	25	and	and	CCONJ
easat-2965	43	26	𝑥2	𝑥2	PROPN
easat-2965	43	27	−	−	PROPN
easat-2965	43	28	(	(	PUNCT
easat-2965	43	29	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	43	30	±	±	NUM
easat-2965	43	31	5p)𝑦2	5p)𝑦2	NOUN
easat-2965	43	32	=	=	PUNCT
easat-2965	43	33	𝑘𝑡.	𝑘𝑡.	PROPN
easat-2965	43	34	they	they	PRON
easat-2965	43	35	generalized	generalize	VERB
easat-2965	43	36	a	a	DET
easat-2965	43	37	previous	previous	ADJ
easat-2965	43	38	result	result	NOUN
easat-2965	43	39	of	of	ADP
easat-2965	43	40	guney	guney	NOUN
easat-2965	44	1	[	[	X
easat-2965	44	2	10	10	NUM
easat-2965	44	3	]	]	PUNCT
easat-2965	44	4	,	,	PUNCT
easat-2965	44	5	who	who	PRON
easat-2965	44	6	found	find	VERB
easat-2965	44	7	all	all	DET
easat-2965	44	8	positive	positive	ADJ
easat-2965	44	9	integer	integer	NOUN
easat-2965	44	10	solutions	solution	NOUN
easat-2965	44	11	of	of	ADP
easat-2965	44	12	the	the	DET
easat-2965	44	13	equations	equation	NOUN
easat-2965	44	14	𝑥2	𝑥2	NOUN
easat-2965	44	15	−	−	PROPN
easat-2965	44	16	(	(	PUNCT
easat-2965	44	17	𝑎2𝑏2	𝑎2𝑏2	PROPN
easat-2965	44	18	+	+	NUM
easat-2965	44	19	2𝑏)𝑦2	2𝑏)𝑦2	NUM
easat-2965	44	20	=	=	SYM
easat-2965	44	21	𝑁	𝑁	NOUN
easat-2965	44	22	when	when	SCONJ
easat-2965	44	23	𝑁	𝑁	PROPN
easat-2965	44	24	∈	∈	PROPN
easat-2965	44	25	{	{	PUNCT
easat-2965	44	26	±1,±4	±1,±4	PROPN
easat-2965	44	27	}	}	PUNCT
easat-2965	44	28	in	in	ADP
easat-2965	44	29	terms	term	NOUN
easat-2965	44	30	of	of	ADP
easat-2965	44	31	generalized	generalized	ADJ
easat-2965	44	32	fibonacci	fibonacci	NOUN
easat-2965	44	33	and	and	CCONJ
easat-2965	44	34	lucas	lucas	PROPN
easat-2965	44	35	sequences	sequence	NOUN
easat-2965	44	36	.	.	PUNCT
easat-2965	45	1	in	in	ADP
easat-2965	45	2	this	this	DET
easat-2965	45	3	paper	paper	NOUN
easat-2965	45	4	,	,	PUNCT
easat-2965	45	5	we	we	PRON
easat-2965	45	6	generalize	generalize	VERB
easat-2965	45	7	the	the	DET
easat-2965	45	8	results	result	NOUN
easat-2965	45	9	of	of	ADP
easat-2965	45	10	bala	bala	PROPN
easat-2965	45	11	and	and	CCONJ
easat-2965	45	12	mishra	mishra	PROPN
easat-2965	46	1	[	[	X
easat-2965	46	2	9	9	NUM
easat-2965	46	3	]	]	PUNCT
easat-2965	46	4	,	,	PUNCT
easat-2965	46	5	by	by	ADP
easat-2965	46	6	solving	solve	VERB
easat-2965	46	7	𝑥2	𝑥2	PROPN
easat-2965	46	8	±	±	NOUN
easat-2965	46	9	(	(	PUNCT
easat-2965	46	10	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	46	11	−	−	NOUN
easat-2965	46	12	𝑎𝑝)𝑦2	𝑎𝑝)𝑦2	NOUN
easat-2965	46	13	=	=	PUNCT
easat-2965	46	14	𝑘𝑡	𝑘𝑡	NOUN
easat-2965	46	15	then	then	ADV
easat-2965	46	16	we	we	PRON
easat-2965	46	17	express	express	VERB
easat-2965	46	18	its	its	PRON
easat-2965	46	19	positive	positive	ADJ
easat-2965	46	20	integer	integer	NOUN
easat-2965	46	21	solutions	solution	NOUN
easat-2965	46	22	in	in	ADP
easat-2965	46	23	the	the	DET
easat-2965	46	24	current	current	ADJ
easat-2965	46	25	study	study	NOUN
easat-2965	46	26	.	.	PUNCT
easat-2965	47	1	using	use	VERB
easat-2965	47	2	generalized	generalized	ADJ
easat-2965	47	3	lucas	lucas	NOUN
easat-2965	47	4	,	,	PUNCT
easat-2965	47	5	generalized	generalized	ADJ
easat-2965	47	6	pell	pell	NOUN
easat-2965	47	7	,	,	PUNCT
easat-2965	47	8	and	and	CCONJ
easat-2965	47	9	generalized	generalize	VERB
easat-2965	47	10	pell	pell	NOUN
easat-2965	47	11	-	-	PUNCT
easat-2965	47	12	lucas	lucas	NOUN
easat-2965	47	13	sequences	sequence	NOUN
easat-2965	47	14	.	.	PUNCT
easat-2965	48	1	to	to	PART
easat-2965	48	2	do	do	VERB
easat-2965	48	3	,	,	PUNCT
easat-2965	48	4	we	we	PRON
easat-2965	48	5	need	need	VERB
easat-2965	48	6	the	the	DET
easat-2965	48	7	following	follow	VERB
easat-2965	48	8	theorems	theorem	NOUN
easat-2965	48	9	(	(	PUNCT
easat-2965	48	10	1,2,3,2.1	1,2,3,2.1	NUM
easat-2965	48	11	):	):	PUNCT
easat-2965	48	12	theorem	theorem	NOUN
easat-2965	48	13	1	1	NUM
easat-2965	48	14	,	,	PUNCT
easat-2965	48	15	there	there	PRON
easat-2965	48	16	is	be	VERB
easat-2965	48	17	no	no	DET
easat-2965	48	18	positive	positive	ADJ
easat-2965	48	19	integer	integer	NOUN
easat-2965	48	20	solution	solution	NOUN
easat-2965	48	21	to	to	ADP
easat-2965	48	22	the	the	DET
easat-2965	48	23	equation	equation	NOUN
easat-2965	48	24	𝑥2	𝑥2	NOUN
easat-2965	48	25	−	−	PROPN
easat-2965	48	26	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	48	27	=	=	NOUN
easat-2965	48	28	−1	−1	NOUN
easat-2965	48	29	if	if	SCONJ
easat-2965	48	30	the	the	DET
easat-2965	48	31	length	length	NOUN
easat-2965	48	32	of	of	ADP
easat-2965	48	33	the	the	DET
easat-2965	48	34	period	period	NOUN
easat-2965	48	35	of	of	ADP
easat-2965	48	36	√𝐷	√𝐷	NOUN
easat-2965	48	37	's	's	PART
easat-2965	48	38	continued	continue	VERB
easat-2965	48	39	fraction	fraction	NOUN
easat-2965	48	40	expansion	expansion	NOUN
easat-2965	48	41	is	be	AUX
easat-2965	48	42	even	even	ADV
easat-2965	48	43	,	,	PUNCT
easat-2965	48	44	while	while	SCONJ
easat-2965	48	45	the	the	DET
easat-2965	48	46	fundamental	fundamental	ADJ
easat-2965	48	47	solution	solution	NOUN
easat-2965	48	48	of	of	ADP
easat-2965	48	49	equation	equation	NOUN
easat-2965	48	50	𝑥2	𝑥2	NOUN
easat-2965	48	51	−	−	PROPN
easat-2965	48	52	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	48	53	=	=	SYM
easat-2965	48	54	1	1	NUM
easat-2965	48	55	is	be	AUX
easat-2965	48	56	𝑝𝑛−1	𝑝𝑛−1	PROPN
easat-2965	48	57	𝑞𝑛−1	𝑞𝑛−1	PROPN
easat-2965	48	58	.	.	PUNCT
easat-2965	49	1	4410	4410	NUM
easat-2965	49	2	edelweiss	edelweiss	PROPN
easat-2965	49	3	applied	apply	VERB
easat-2965	49	4	science	science	NOUN
easat-2965	49	5	and	and	CCONJ
easat-2965	49	6	technology	technology	NOUN
easat-2965	49	7	issn	issn	PROPN
easat-2965	49	8	:	:	PUNCT
easat-2965	49	9	2576	2576	NUM
easat-2965	49	10	-	-	SYM
easat-2965	49	11	8484	8484	NUM
easat-2965	49	12	vol	vol	NOUN
easat-2965	49	13	.	.	PROPN
easat-2965	49	14	8	8	NUM
easat-2965	49	15	,	,	PUNCT
easat-2965	49	16	no	no	INTJ
easat-2965	49	17	.	.	NOUN
easat-2965	50	1	6	6	NUM
easat-2965	50	2	:	:	SYM
easat-2965	50	3	4408	4408	NUM
easat-2965	50	4	-	-	SYM
easat-2965	50	5	4414	4414	NUM
easat-2965	50	6	,	,	PUNCT
easat-2965	50	7	2024	2024	NUM
easat-2965	50	8	doi	doi	NOUN
easat-2965	50	9	:	:	PUNCT
easat-2965	50	10	10.55214/25768484.v8i6.2965	10.55214/25768484.v8i6.2965	NUM
easat-2965	50	11	©	©	PROPN
easat-2965	50	12	2024	2024	NUM
easat-2965	50	13	by	by	ADP
easat-2965	50	14	the	the	DET
easat-2965	50	15	authors	author	NOUN
easat-2965	50	16	;	;	PUNCT
easat-2965	50	17	licensee	licensee	PROPN
easat-2965	50	18	learning	learning	PROPN
easat-2965	50	19	gate	gate	PROPN
easat-2965	50	20	theorem	theorem	VERB
easat-2965	50	21	2	2	NUM
easat-2965	50	22	,	,	PUNCT
easat-2965	50	23	there	there	PRON
easat-2965	50	24	are	be	VERB
easat-2965	50	25	infinitely	infinitely	ADV
easat-2965	50	26	many	many	ADJ
easat-2965	50	27	solutions	solution	NOUN
easat-2965	50	28	to	to	ADP
easat-2965	50	29	the	the	DET
easat-2965	50	30	pell	pell	NOUN
easat-2965	50	31	equation	equation	NOUN
easat-2965	50	32	𝑥2	𝑥2	NOUN
easat-2965	50	33	−	−	PROPN
easat-2965	50	34	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	50	35	=	=	NOUN
easat-2965	50	36	1	1	NUM
easat-2965	50	37	if	if	SCONJ
easat-2965	50	38	𝐷	𝐷	PROPN
easat-2965	50	39	is	be	AUX
easat-2965	50	40	a	a	DET
easat-2965	50	41	natural	natural	ADJ
easat-2965	50	42	number	number	NOUN
easat-2965	50	43	that	that	PRON
easat-2965	50	44	is	be	AUX
easat-2965	50	45	not	not	PART
easat-2965	50	46	a	a	DET
easat-2965	50	47	perfect	perfect	ADJ
easat-2965	50	48	square	square	NOUN
easat-2965	50	49	.	.	PUNCT
easat-2965	51	1	all	all	DET
easat-2965	51	2	positive	positive	ADJ
easat-2965	51	3	solutions	solution	NOUN
easat-2965	51	4	can	can	AUX
easat-2965	51	5	be	be	AUX
easat-2965	51	6	obtained	obtain	VERB
easat-2965	51	7	by	by	ADP
easat-2965	51	8	the	the	DET
easat-2965	51	9	formula	formula	NOUN
easat-2965	51	10	𝑥𝑛	𝑥𝑛	NOUN
easat-2965	51	11	+	+	CCONJ
easat-2965	51	12	𝑦𝑛√𝐷	𝑦𝑛√𝐷	VERB
easat-2965	51	13	=	=	SYM
easat-2965	51	14	(	(	PUNCT
easat-2965	51	15	𝑥1	𝑥1	NOUN
easat-2965	51	16	+	+	CCONJ
easat-2965	51	17	𝑦1√𝐷	𝑦1√𝐷	PROPN
easat-2965	51	18	)	)	PUNCT
easat-2965	51	19	𝑛	𝑛	NOUN
easat-2965	51	20	,	,	PUNCT
easat-2965	51	21	for	for	ADP
easat-2965	51	22	all	all	DET
easat-2965	51	23	𝑛	𝑛	PRON
easat-2965	51	24	>	>	X
easat-2965	51	25	1	1	NUM
easat-2965	51	26	,	,	PUNCT
easat-2965	51	27	where	where	SCONJ
easat-2965	51	28	(	(	PUNCT
easat-2965	51	29	𝑥1	𝑥1	NOUN
easat-2965	51	30	,	,	PUNCT
easat-2965	51	31	𝑦1	𝑦1	PROPN
easat-2965	51	32	)	)	PUNCT
easat-2965	51	33	is	be	AUX
easat-2965	51	34	fundamental	fundamental	ADJ
easat-2965	51	35	solution	solution	NOUN
easat-2965	51	36	of	of	ADP
easat-2965	51	37	equation	equation	NOUN
easat-2965	51	38	𝑥2	𝑥2	NOUN
easat-2965	51	39	−	−	PROPN
easat-2965	51	40	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	51	41	=	=	NOUN
easat-2965	51	42	1	1	X
easat-2965	51	43	.	.	X
easat-2965	51	44	theorem	theorem	NOUN
easat-2965	51	45	3	3	NUM
easat-2965	51	46	,	,	PUNCT
easat-2965	51	47	let	let	VERB
easat-2965	51	48	𝑁	𝑁	PROPN
easat-2965	51	49	be	be	AUX
easat-2965	51	50	a	a	DET
easat-2965	51	51	positive	positive	ADJ
easat-2965	51	52	integer	integer	NOUN
easat-2965	51	53	and	and	CCONJ
easat-2965	51	54	(	(	PUNCT
easat-2965	51	55	𝑢1	𝑢1	PROPN
easat-2965	51	56	,	,	PUNCT
easat-2965	51	57	𝑣1	𝑣1	PROPN
easat-2965	51	58	)	)	PUNCT
easat-2965	51	59	be	be	VERB
easat-2965	51	60	the	the	DET
easat-2965	51	61	fundamental	fundamental	ADJ
easat-2965	51	62	solution	solution	NOUN
easat-2965	51	63	of	of	ADP
easat-2965	51	64	equation	equation	NOUN
easat-2965	51	65	𝑥2	𝑥2	NOUN
easat-2965	51	66	−	−	PROPN
easat-2965	51	67	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	52	1	=	=	PUNCT
easat-2965	52	2	𝑁.	𝑁.	PROPN
easat-2965	52	3	then	then	ADV
easat-2965	52	4	all	all	DET
easat-2965	52	5	positive	positive	ADJ
easat-2965	52	6	solutions	solution	NOUN
easat-2965	52	7	of	of	ADP
easat-2965	52	8	𝑥2	𝑥2	NOUN
easat-2965	52	9	−	−	PROPN
easat-2965	52	10	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	52	11	=	=	NOUN
easat-2965	53	1	𝑁	𝑁	PROPN
easat-2965	53	2	can	can	AUX
easat-2965	53	3	be	be	AUX
easat-2965	53	4	obtained	obtain	VERB
easat-2965	53	5	by	by	ADP
easat-2965	53	6	the	the	DET
easat-2965	53	7	formula	formula	NOUN
easat-2965	53	8	𝑥𝑛	𝑥𝑛	NOUN
easat-2965	53	9	+	+	CCONJ
easat-2965	53	10	𝑦𝑛√𝐷	𝑦𝑛√𝐷	VERB
easat-2965	53	11	=	=	PUNCT
easat-2965	53	12	(	(	PUNCT
easat-2965	53	13	𝑢1	𝑢1	PROPN
easat-2965	53	14	+	+	CCONJ
easat-2965	53	15	𝑣1√𝐷)(𝑥1	𝑣1√𝐷)(𝑥1	PROPN
easat-2965	53	16	+	+	X
easat-2965	53	17	𝑦1√𝐷	𝑦1√𝐷	NOUN
easat-2965	53	18	)	)	PUNCT
easat-2965	53	19	𝑛	𝑛	NOUN
easat-2965	53	20	,	,	PUNCT
easat-2965	53	21	for	for	ADP
easat-2965	53	22	all	all	DET
easat-2965	53	23	𝑛	𝑛	PRON
easat-2965	53	24	>	>	X
easat-2965	53	25	1	1	NUM
easat-2965	53	26	,	,	PUNCT
easat-2965	53	27	where	where	SCONJ
easat-2965	53	28	(	(	PUNCT
easat-2965	53	29	𝑥1	𝑥1	NOUN
easat-2965	53	30	,	,	PUNCT
easat-2965	53	31	𝑦1	𝑦1	PROPN
easat-2965	53	32	)	)	PUNCT
easat-2965	53	33	is	be	AUX
easat-2965	53	34	fundamental	fundamental	ADJ
easat-2965	53	35	solution	solution	NOUN
easat-2965	53	36	of	of	ADP
easat-2965	53	37	equation	equation	NOUN
easat-2965	53	38	𝑥2	𝑥2	NOUN
easat-2965	53	39	−	−	PROPN
easat-2965	53	40	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	53	41	=	=	NOUN
easat-2965	54	1	1	1	X
easat-2965	54	2	.	.	X
easat-2965	55	1	we	we	PRON
easat-2965	55	2	are	be	AUX
easat-2965	55	3	now	now	ADV
easat-2965	55	4	ready	ready	ADJ
easat-2965	55	5	to	to	PART
easat-2965	55	6	present	present	VERB
easat-2965	55	7	our	our	PRON
easat-2965	55	8	main	main	ADJ
easat-2965	55	9	results	result	NOUN
easat-2965	55	10	.	.	PUNCT
easat-2965	56	1	2	2	X
easat-2965	56	2	.	.	X
easat-2965	56	3	main	main	ADJ
easat-2965	56	4	results	result	NOUN
easat-2965	56	5	2.1	2.1	NUM
easat-2965	56	6	.	.	PUNCT
easat-2965	56	7	theorem	theorem	NOUN
easat-2965	56	8	let	let	VERB
easat-2965	56	9	𝐷	𝐷	NOUN
easat-2965	56	10	=	=	NOUN
easat-2965	56	11	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	56	12	−	−	PROPN
easat-2965	56	13	𝑎𝑝	𝑎𝑝	PROPN
easat-2965	56	14	,	,	PUNCT
easat-2965	56	15	with	with	ADP
easat-2965	56	16	𝑞	𝑞	X
easat-2965	56	17	>	>	X
easat-2965	56	18	𝑎	𝑎	X
easat-2965	56	19	being	be	AUX
easat-2965	56	20	a	a	DET
easat-2965	56	21	multiple	multiple	NOUN
easat-2965	56	22	of	of	ADP
easat-2965	56	23	𝑎	𝑎	NOUN
easat-2965	56	24	and	and	CCONJ
easat-2965	56	25	𝑝	𝑝	NOUN
easat-2965	56	26	,	,	PUNCT
easat-2965	56	27	𝑞	𝑞	X
easat-2965	56	28	being	be	AUX
easat-2965	56	29	positive	positive	ADJ
easat-2965	56	30	integers	integer	NOUN
easat-2965	56	31	chosen	choose	VERB
easat-2965	56	32	in	in	ADP
easat-2965	56	33	such	such	DET
easat-2965	56	34	a	a	DET
easat-2965	56	35	way	way	NOUN
easat-2965	56	36	that	that	PRON
easat-2965	56	37	𝐷	𝐷	NOUN
easat-2965	56	38	is	be	AUX
easat-2965	56	39	not	not	PART
easat-2965	56	40	a	a	DET
easat-2965	56	41	perfect	perfect	ADJ
easat-2965	56	42	square	square	NOUN
easat-2965	56	43	,	,	PUNCT
easat-2965	56	44	then	then	ADV
easat-2965	56	45	1	1	X
easat-2965	56	46	.	.	PUNCT
easat-2965	56	47	√𝐷	√𝐷	NOUN
easat-2965	57	1	=	=	PUNCT
easat-2965	57	2	√𝑝2𝑞2	√𝑝2𝑞2	PROPN
easat-2965	57	3	−	−	PROPN
easat-2965	57	4	𝑎𝑝	𝑎𝑝	NOUN
easat-2965	57	5	=	=	PUNCT
easat-2965	58	1	[	[	X
easat-2965	58	2	𝑝𝑞	𝑝𝑞	VERB
easat-2965	58	3	−	−	PROPN
easat-2965	58	4	1	1	NUM
easat-2965	58	5	;	;	PUNCT
easat-2965	58	6	1	1	NUM
easat-2965	58	7	,	,	PUNCT
easat-2965	58	8	2(𝑞−𝑎	2(𝑞−𝑎	NOUN
easat-2965	58	9	)	)	PUNCT
easat-2965	58	10	𝑎	𝑎	NOUN
easat-2965	58	11	,	,	PUNCT
easat-2965	58	12	1	1	NUM
easat-2965	58	13	,	,	PUNCT
easat-2965	58	14	29𝑝𝑞	29𝑝𝑞	NOUN
easat-2965	58	15	−	−	PROPN
easat-2965	58	16	1	1	NUM
easat-2965	58	17	)	)	PUNCT
easat-2965	58	18	̅̅	̅̅	PROPN
easat-2965	58	19	̅̅	̅̅	PROPN
easat-2965	58	20	̅̅	̅̅	PROPN
easat-2965	58	21	̅̅	̅̅	PROPN
easat-2965	58	22	̅̅	̅̅	PROPN
easat-2965	58	23	̅̅	̅̅	PROPN
easat-2965	58	24	̅̅	̅̅	PROPN
easat-2965	58	25	̅̅	̅̅	PROPN
easat-2965	58	26	̅̅	̅̅	PROPN
easat-2965	58	27	̅̅	̅̅	PROPN
easat-2965	58	28	̅̅	̅̅	PROPN
easat-2965	58	29	̅̅	̅̅	PROPN
easat-2965	58	30	̅̅	̅̅	PROPN
easat-2965	58	31	̅̅	̅̅	PROPN
easat-2965	58	32	]	]	X
easat-2965	58	33	.	.	PUNCT
easat-2965	59	1	2	2	X
easat-2965	59	2	.	.	X
easat-2965	59	3	the	the	DET
easat-2965	59	4	fundamental	fundamental	ADJ
easat-2965	59	5	solution	solution	NOUN
easat-2965	59	6	of	of	ADP
easat-2965	59	7	the	the	DET
easat-2965	59	8	pell	pell	NOUN
easat-2965	59	9	equation	equation	NOUN
easat-2965	59	10	𝑥2	𝑥2	NOUN
easat-2965	59	11	−	−	PROPN
easat-2965	59	12	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	59	13	=	=	SYM
easat-2965	59	14	1	1	X
easat-2965	59	15	is	be	AUX
easat-2965	59	16	(	(	PUNCT
easat-2965	59	17	𝑥1	𝑥1	NOUN
easat-2965	59	18	,	,	PUNCT
easat-2965	59	19	𝑦1	𝑦1	NOUN
easat-2965	59	20	)	)	PUNCT
easat-2965	59	21	=	=	PUNCT
easat-2965	59	22	(	(	PUNCT
easat-2965	59	23	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	59	24	𝑎	𝑎	NOUN
easat-2965	59	25	,	,	PUNCT
easat-2965	59	26	2𝑞	2𝑞	NOUN
easat-2965	59	27	𝑎	𝑎	NOUN
easat-2965	59	28	)	)	PUNCT
easat-2965	59	29	.	.	PUNCT
easat-2965	60	1	3	3	X
easat-2965	60	2	.	.	X
easat-2965	60	3	all	all	DET
easat-2965	60	4	positive	positive	ADJ
easat-2965	60	5	integer	integer	NOUN
easat-2965	60	6	solution	solution	NOUN
easat-2965	60	7	(	(	PUNCT
easat-2965	60	8	𝑥𝑛	𝑥𝑛	NOUN
easat-2965	60	9	,	,	PUNCT
easat-2965	60	10	𝑦𝑛	𝑦𝑛	NOUN
easat-2965	60	11	)	)	PUNCT
easat-2965	60	12	of	of	ADP
easat-2965	60	13	the	the	DET
easat-2965	60	14	pell	pell	NOUN
easat-2965	60	15	equation	equation	NOUN
easat-2965	60	16	𝑥2	𝑥2	NOUN
easat-2965	60	17	−	−	PROPN
easat-2965	60	18	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	60	19	=	=	SYM
easat-2965	60	20	1	1	NUM
easat-2965	60	21	are	be	AUX
easat-2965	60	22	provided	provide	VERB
easat-2965	60	23	by	by	ADP
easat-2965	60	24	𝑥𝑛	𝑥𝑛	NOUN
easat-2965	60	25	=	=	NOUN
easat-2965	60	26	𝛼𝑛+𝛽2	𝛼𝑛+𝛽2	NOUN
easat-2965	60	27	2	2	NUM
easat-2965	60	28	=	=	SYM
easat-2965	60	29	1	1	NUM
easat-2965	60	30	2	2	NUM
easat-2965	60	31	𝑙𝑛(𝑠	𝑙𝑛(𝑠	NOUN
easat-2965	60	32	,	,	PUNCT
easat-2965	60	33	𝑡	𝑡	X
easat-2965	60	34	)	)	PUNCT
easat-2965	60	35	=	=	SYM
easat-2965	60	36	1	1	NUM
easat-2965	60	37	2	2	NUM
easat-2965	60	38	𝑙𝑛	𝑙𝑛	NOUN
easat-2965	60	39	(	(	PUNCT
easat-2965	60	40	2	2	NUM
easat-2965	60	41	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	60	42	𝑎	𝑎	NOUN
easat-2965	60	43	,	,	PUNCT
easat-2965	60	44	−1	−1	NOUN
easat-2965	60	45	)	)	PUNCT
easat-2965	60	46	,	,	PUNCT
easat-2965	60	47	and	and	CCONJ
easat-2965	60	48	𝑦𝑛	𝑦𝑛	NOUN
easat-2965	60	49	=	=	ADJ
easat-2965	60	50	𝛼𝑛+𝛽2	𝛼𝑛+𝛽2	NOUN
easat-2965	60	51	2√𝐷	2√𝐷	NUM
easat-2965	60	52	=	=	SYM
easat-2965	60	53	2𝑞	2𝑞	NUM
easat-2965	60	54	𝑎	𝑎	DET
easat-2965	60	55	𝐹𝑛(𝑠	𝐹𝑛(𝑠	NOUN
easat-2965	60	56	,	,	PUNCT
easat-2965	60	57	𝑡	𝑡	NOUN
easat-2965	60	58	)	)	PUNCT
easat-2965	60	59	=	=	SYM
easat-2965	60	60	2𝑞	2𝑞	NOUN
easat-2965	60	61	𝑎	𝑎	X
easat-2965	60	62	𝐹𝑛	𝐹𝑛	PROPN
easat-2965	60	63	(	(	PUNCT
easat-2965	60	64	2	2	NUM
easat-2965	60	65	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	60	66	𝑎	𝑎	NOUN
easat-2965	60	67	,	,	PUNCT
easat-2965	60	68	−1	−1	NOUN
easat-2965	60	69	)	)	PUNCT
easat-2965	60	70	=	=	PUNCT
easat-2965	61	1	𝑞	𝑞	PROPN
easat-2965	61	2	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	61	3	φ𝑛	φ𝑛	PROPN
easat-2965	61	4	(	(	PUNCT
easat-2965	61	5	2	2	NUM
easat-2965	61	6	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	61	7	𝑎	𝑎	NOUN
easat-2965	61	8	,	,	PUNCT
easat-2965	61	9	−1	−1	NOUN
easat-2965	61	10	)	)	PUNCT
easat-2965	61	11	,	,	PUNCT
easat-2965	61	12	for	for	ADP
easat-2965	61	13	all	all	DET
easat-2965	61	14	𝑛	𝑛	PRON
easat-2965	61	15	≥	≥	NUM
easat-2965	61	16	1	1	NUM
easat-2965	61	17	.	.	NOUN
easat-2965	61	18	1	1	NUM
easat-2965	61	19	.	.	PUNCT
easat-2965	62	1	the	the	DET
easat-2965	62	2	fundamental	fundamental	ADJ
easat-2965	62	3	solution	solution	NOUN
easat-2965	62	4	of	of	ADP
easat-2965	62	5	pell	pell	ADJ
easat-2965	62	6	equation	equation	NOUN
easat-2965	62	7	𝑥2	𝑥2	NOUN
easat-2965	62	8	−	−	PROPN
easat-2965	62	9	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	62	10	=	=	PROPN
easat-2965	62	11	𝑘𝑡	𝑘𝑡	PROPN
easat-2965	62	12	is	be	AUX
easat-2965	62	13	(	(	PUNCT
easat-2965	62	14	𝑥1	𝑥1	NOUN
easat-2965	62	15	,	,	PUNCT
easat-2965	62	16	𝑦1	𝑦1	NOUN
easat-2965	62	17	)	)	PUNCT
easat-2965	62	18	=	=	SYM
easat-2965	62	19	(	(	PUNCT
easat-2965	62	20	2𝑝𝑞2	2𝑝𝑞2	NUM
easat-2965	62	21	−	−	NOUN
easat-2965	62	22	𝑎	𝑎	SYM
easat-2965	62	23	𝑎	𝑎	X
easat-2965	62	24	𝑘	𝑘	PRON
easat-2965	62	25	𝑡	𝑡	PROPN
easat-2965	62	26	2	2	NUM
easat-2965	62	27	,	,	PUNCT
easat-2965	62	28	2𝑞	2𝑞	NUM
easat-2965	62	29	𝑎	𝑎	VERB
easat-2965	62	30	𝑘	𝑘	X
easat-2965	62	31	𝑡	𝑡	PROPN
easat-2965	62	32	2	2	NUM
easat-2965	62	33	)	)	PUNCT
easat-2965	62	34	.	.	PUNCT
easat-2965	63	1	2	2	X
easat-2965	63	2	.	.	X
easat-2965	63	3	all	all	DET
easat-2965	63	4	positive	positive	ADJ
easat-2965	63	5	solutions	solution	NOUN
easat-2965	63	6	of	of	ADP
easat-2965	63	7	pell	pell	NOUN
easat-2965	63	8	equation	equation	NOUN
easat-2965	63	9	𝑥2	𝑥2	NOUN
easat-2965	63	10	−	−	PROPN
easat-2965	63	11	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	63	12	=	=	PROPN
easat-2965	63	13	𝑘𝑡is	𝑘𝑡is	NOUN
easat-2965	63	14	given	give	VERB
easat-2965	63	15	by	by	ADP
easat-2965	63	16	𝑥𝑛+1	𝑥𝑛+1	PROPN
easat-2965	63	17	=	=	SYM
easat-2965	63	18	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	63	19	𝑎	𝑎	NOUN
easat-2965	63	20	𝑘	𝑘	ADP
easat-2965	63	21	𝑡	𝑡	PROPN
easat-2965	63	22	2𝑥𝑛	2𝑥𝑛	ADJ
easat-2965	63	23	+	+	CCONJ
easat-2965	63	24	2𝑞	2𝑞	NUM
easat-2965	63	25	𝑝2𝑞2−𝑎𝑝	𝑝2𝑞2−𝑎𝑝	VERB
easat-2965	63	26	𝑎	𝑎	NOUN
easat-2965	63	27	𝑘	𝑘	ADP
easat-2965	63	28	𝑡	𝑡	NOUN
easat-2965	63	29	2𝑦𝑛	2𝑦𝑛	NOUN
easat-2965	63	30	and	and	CCONJ
easat-2965	63	31	𝑦𝑛+1	𝑦𝑛+1	NOUN
easat-2965	63	32	=	=	SYM
easat-2965	63	33	2𝑞	2𝑞	NUM
easat-2965	63	34	𝑎	𝑎	VERB
easat-2965	63	35	𝑘	𝑘	ADP
easat-2965	63	36	𝑡	𝑡	NOUN
easat-2965	63	37	2𝑥𝑛	2𝑥𝑛	NOUN
easat-2965	63	38	+	+	CCONJ
easat-2965	63	39	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	63	40	𝑎	𝑎	NOUN
easat-2965	63	41	𝑘	𝑘	ADP
easat-2965	63	42	𝑡	𝑡	NOUN
easat-2965	63	43	2𝑦𝑛	2𝑦𝑛	NOUN
easat-2965	63	44	for	for	ADP
easat-2965	63	45	all	all	DET
easat-2965	63	46	𝑛	𝑛	DET
easat-2965	63	47	≥	≥	NUM
easat-2965	63	48	1	1	NUM
easat-2965	63	49	proof	proof	NOUN
easat-2965	63	50	:	:	PUNCT
easat-2965	63	51	1	1	NUM
easat-2965	63	52	.	.	PUNCT
easat-2965	63	53	√d	√d	PUNCT
easat-2965	64	1	=	=	PUNCT
easat-2965	64	2	√p2q2	√p2q2	NOUN
easat-2965	64	3	−	−	PROPN
easat-2965	64	4	ap	ap	NOUN
easat-2965	64	5	=	=	PUNCT
easat-2965	64	6	(	(	PUNCT
easat-2965	64	7	pq	pq	INTJ
easat-2965	64	8	−	−	NOUN
easat-2965	64	9	1	1	NUM
easat-2965	64	10	)	)	PUNCT
easat-2965	64	11	+	+	CCONJ
easat-2965	64	12	√p2q2	√p2q2	NOUN
easat-2965	64	13	−	−	PROPN
easat-2965	64	14	ap	ap	NOUN
easat-2965	64	15	−	−	PROPN
easat-2965	64	16	(	(	PUNCT
easat-2965	64	17	pq	pq	INTJ
easat-2965	64	18	−	−	NOUN
easat-2965	64	19	1	1	NUM
easat-2965	64	20	)	)	PUNCT
easat-2965	64	21	4411	4411	NUM
easat-2965	64	22	edelweiss	edelweiss	PROPN
easat-2965	64	23	applied	apply	VERB
easat-2965	64	24	science	science	NOUN
easat-2965	64	25	and	and	CCONJ
easat-2965	64	26	technology	technology	NOUN
easat-2965	64	27	issn	issn	PROPN
easat-2965	64	28	:	:	PUNCT
easat-2965	64	29	2576	2576	NUM
easat-2965	64	30	-	-	SYM
easat-2965	64	31	8484	8484	NUM
easat-2965	64	32	vol	vol	NOUN
easat-2965	64	33	.	.	PROPN
easat-2965	64	34	8	8	NUM
easat-2965	64	35	,	,	PUNCT
easat-2965	64	36	no	no	INTJ
easat-2965	64	37	.	.	NOUN
easat-2965	65	1	6	6	NUM
easat-2965	65	2	:	:	SYM
easat-2965	65	3	4408	4408	NUM
easat-2965	65	4	-	-	SYM
easat-2965	65	5	4414	4414	NUM
easat-2965	65	6	,	,	PUNCT
easat-2965	65	7	2024	2024	NUM
easat-2965	65	8	doi	doi	NOUN
easat-2965	65	9	:	:	PUNCT
easat-2965	65	10	10.55214/25768484.v8i6.2965	10.55214/25768484.v8i6.2965	NUM
easat-2965	65	11	©	©	PROPN
easat-2965	65	12	2024	2024	NUM
easat-2965	65	13	by	by	ADP
easat-2965	65	14	the	the	DET
easat-2965	65	15	authors	author	NOUN
easat-2965	65	16	;	;	PUNCT
easat-2965	65	17	licensee	licensee	NOUN
easat-2965	65	18	learning	learning	NOUN
easat-2965	65	19	gate	gate	NOUN
easat-2965	65	20	=	=	PUNCT
easat-2965	65	21	(	(	PUNCT
easat-2965	65	22	pq	pq	INTJ
easat-2965	65	23	−	−	NOUN
easat-2965	65	24	1	1	NUM
easat-2965	65	25	)	)	PUNCT
easat-2965	65	26	+	+	CCONJ
easat-2965	65	27	1	1	NUM
easat-2965	65	28	√p2q2−ap+(pq−1	√p2q2−ap+(pq−1	NOUN
easat-2965	65	29	)	)	PUNCT
easat-2965	66	1	2pq−ap−1	2pq−ap−1	NUM
easat-2965	66	2	=	=	SYM
easat-2965	66	3	(	(	PUNCT
easat-2965	66	4	pq	pq	INTJ
easat-2965	66	5	−	−	NOUN
easat-2965	66	6	1	1	NUM
easat-2965	66	7	)	)	PUNCT
easat-2965	66	8	+	+	CCONJ
easat-2965	66	9	1	1	NUM
easat-2965	66	10	1	1	NUM
easat-2965	66	11	+	+	CCONJ
easat-2965	66	12	√p2q2−ap−(pq−ap	√p2q2−ap−(pq−ap	NOUN
easat-2965	66	13	)	)	PUNCT
easat-2965	66	14	2pq−ap−1	2pq−ap−1	NUM
easat-2965	67	1	=	=	SYM
easat-2965	67	2	(	(	PUNCT
easat-2965	67	3	pq	pq	INTJ
easat-2965	67	4	−	−	NOUN
easat-2965	67	5	1	1	NUM
easat-2965	67	6	)	)	PUNCT
easat-2965	67	7	+	+	CCONJ
easat-2965	67	8	1	1	NUM
easat-2965	67	9	1	1	NUM
easat-2965	67	10	+	+	NUM
easat-2965	67	11	1	1	NUM
easat-2965	67	12	√p2q2−ap+(pq−ap	√p2q2−ap+(pq−ap	NOUN
easat-2965	67	13	)	)	PUNCT
easat-2965	67	14	ap	ap	NOUN
easat-2965	67	15	=	=	PUNCT
easat-2965	68	1	(	(	PUNCT
easat-2965	68	2	pq	pq	INTJ
easat-2965	68	3	−	−	NOUN
easat-2965	68	4	1	1	NUM
easat-2965	68	5	)	)	PUNCT
easat-2965	68	6	+	+	CCONJ
easat-2965	68	7	1	1	NUM
easat-2965	68	8	1	1	NUM
easat-2965	68	9	+	+	NUM
easat-2965	68	10	1	1	NUM
easat-2965	68	11	2(q−a	2(q−a	NUM
easat-2965	68	12	)	)	PUNCT
easat-2965	68	13	a	a	DET
easat-2965	68	14	+	+	NUM
easat-2965	68	15	√p2q2−ap−(pq−ap	√p2q2−ap−(pq−ap	ADJ
easat-2965	68	16	)	)	PUNCT
easat-2965	68	17	ap	ap	NOUN
easat-2965	69	1	=	=	PUNCT
easat-2965	69	2	(	(	PUNCT
easat-2965	69	3	pq	pq	INTJ
easat-2965	69	4	−	−	NOUN
easat-2965	69	5	1	1	NUM
easat-2965	69	6	)	)	PUNCT
easat-2965	69	7	+	+	CCONJ
easat-2965	69	8	1	1	NUM
easat-2965	69	9	1	1	NUM
easat-2965	69	10	+	+	NUM
easat-2965	69	11	1	1	NUM
easat-2965	69	12	2(q−a	2(q−a	NUM
easat-2965	69	13	)	)	PUNCT
easat-2965	69	14	a	a	DET
easat-2965	69	15	+	+	NUM
easat-2965	69	16	1	1	NUM
easat-2965	69	17	1	1	NUM
easat-2965	69	18	+	+	NUM
easat-2965	69	19	1	1	NUM
easat-2965	69	20	2(pq−1)+√p2q2−ap−(pq−1	2(pq−1)+√p2q2−ap−(pq−1	NUM
easat-2965	69	21	)	)	PUNCT
easat-2965	70	1	=	=	PUNCT
easat-2965	71	1	[	[	X
easat-2965	71	2	pq	pq	INTJ
easat-2965	71	3	−	−	NOUN
easat-2965	71	4	1	1	NUM
easat-2965	71	5	;	;	PUNCT
easat-2965	71	6	1	1	NUM
easat-2965	71	7	,	,	PUNCT
easat-2965	71	8	2(q	2(q	NUM
easat-2965	71	9	−	−	NOUN
easat-2965	71	10	a	a	X
easat-2965	71	11	)	)	PUNCT
easat-2965	71	12	a	a	PRON
easat-2965	71	13	,	,	PUNCT
easat-2965	71	14	1,2(pq	1,2(pq	NUM
easat-2965	71	15	−	−	NOUN
easat-2965	71	16	1	1	NUM
easat-2965	71	17	)	)	PUNCT
easat-2965	71	18	̅̅	̅̅	PROPN
easat-2965	71	19	̅̅	̅̅	PROPN
easat-2965	71	20	̅̅	̅̅	PROPN
easat-2965	71	21	̅̅	̅̅	PROPN
easat-2965	71	22	̅̅	̅̅	PROPN
easat-2965	71	23	̅̅	̅̅	PROPN
easat-2965	71	24	̅̅	̅̅	PROPN
easat-2965	71	25	̅̅	̅̅	PROPN
easat-2965	71	26	̅̅	̅̅	PROPN
easat-2965	71	27	̅̅	̅̅	PROPN
easat-2965	71	28	̅̅	̅̅	PROPN
easat-2965	71	29	̅̅	̅̅	PROPN
easat-2965	71	30	̅̅	̅̅	PROPN
easat-2965	71	31	̅̅	̅̅	PROPN
easat-2965	71	32	̅̅	̅̅	PROPN
easat-2965	71	33	]	]	PUNCT
easat-2965	72	1	2	2	X
easat-2965	72	2	.	.	X
easat-2965	72	3	the	the	DET
easat-2965	72	4	period	period	NOUN
easat-2965	72	5	length	length	NOUN
easat-2965	72	6	of	of	ADP
easat-2965	72	7	√𝐷	√𝐷	NOUN
easat-2965	72	8	's	's	PART
easat-2965	72	9	continued	continue	VERB
easat-2965	72	10	fraction	fraction	NOUN
easat-2965	72	11	expansion	expansion	NOUN
easat-2965	72	12	is	be	AUX
easat-2965	72	13	4	4	NUM
easat-2965	72	14	,	,	PUNCT
easat-2965	72	15	making	make	VERB
easat-2965	72	16	it	it	PRON
easat-2965	72	17	even	even	ADV
easat-2965	72	18	.	.	PUNCT
easat-2965	73	1	then	then	ADV
easat-2965	73	2	,	,	PUNCT
easat-2965	73	3	according	accord	VERB
easat-2965	73	4	to	to	ADP
easat-2965	73	5	theorem	theorem	ADJ
easat-2965	73	6	1.1	1.1	NUM
easat-2965	73	7	,	,	PUNCT
easat-2965	73	8	the	the	DET
easat-2965	73	9	fundamental	fundamental	ADJ
easat-2965	73	10	solution	solution	NOUN
easat-2965	73	11	of	of	ADP
easat-2965	73	12	equation	equation	NOUN
easat-2965	73	13	𝑥2	𝑥2	NOUN
easat-2965	73	14	−	−	PROPN
easat-2965	73	15	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	73	16	=	=	SYM
easat-2965	73	17	1	1	NUM
easat-2965	73	18	is	be	AUX
easat-2965	73	19	given	give	VERB
easat-2965	73	20	by	by	ADP
easat-2965	73	21	𝑝3	𝑝3	ADV
easat-2965	73	22	𝑎	𝑎	PROPN
easat-2965	73	23	𝑞3	𝑞3	NOUN
easat-2965	73	24	𝑎	𝑎	X
easat-2965	73	25	.	.	PUNCT
easat-2965	74	1	let	let	VERB
easat-2965	74	2	𝑝3	𝑝3	ADV
easat-2965	74	3	𝑞3	𝑞3	VERB
easat-2965	74	4	=	=	PUNCT
easat-2965	75	1	[	[	X
easat-2965	75	2	𝑝𝑞	𝑝𝑞	VERB
easat-2965	75	3	−	−	PROPN
easat-2965	75	4	1	1	NUM
easat-2965	75	5	;	;	PUNCT
easat-2965	75	6	1	1	NUM
easat-2965	75	7	,	,	PUNCT
easat-2965	75	8	2(𝑞−𝑎	2(𝑞−𝑎	NOUN
easat-2965	75	9	)	)	PUNCT
easat-2965	75	10	𝑎	𝑎	NOUN
easat-2965	75	11	,	,	PUNCT
easat-2965	75	12	1	1	NUM
easat-2965	75	13	]	]	PUNCT
easat-2965	75	14	=	=	SYM
easat-2965	75	15	(	(	PUNCT
easat-2965	75	16	𝑝𝑞	𝑝𝑞	NOUN
easat-2965	75	17	−	−	PROPN
easat-2965	75	18	1	1	NUM
easat-2965	75	19	)	)	PUNCT
easat-2965	75	20	+	+	CCONJ
easat-2965	75	21	1	1	NUM
easat-2965	75	22	1	1	NUM
easat-2965	75	23	+	+	NUM
easat-2965	75	24	2(𝑞−𝑎	2(𝑞−𝑎	NUM
easat-2965	75	25	)	)	PUNCT
easat-2965	75	26	𝑎	𝑎	PRON
easat-2965	75	27	+1	+1	NOUN
easat-2965	75	28	=	=	PUNCT
easat-2965	75	29	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	75	30	2𝑞	2𝑞	NOUN
easat-2965	75	31	.	.	PUNCT
easat-2965	76	1	then	then	ADV
easat-2965	76	2	,	,	PUNCT
easat-2965	76	3	(	(	PUNCT
easat-2965	76	4	𝑥1	𝑥1	NOUN
easat-2965	76	5	,	,	PUNCT
easat-2965	76	6	𝑦1	𝑦1	NOUN
easat-2965	76	7	)	)	PUNCT
easat-2965	76	8	=	=	PUNCT
easat-2965	76	9	(	(	PUNCT
easat-2965	76	10	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	76	11	𝑎	𝑎	NOUN
easat-2965	76	12	,	,	PUNCT
easat-2965	76	13	2𝑞	2𝑞	NOUN
easat-2965	76	14	𝑎	𝑎	NOUN
easat-2965	76	15	)	)	PUNCT
easat-2965	76	16	.	.	PUNCT
easat-2965	77	1	3	3	X
easat-2965	77	2	.	.	X
easat-2965	77	3	according	accord	VERB
easat-2965	77	4	to	to	ADP
easat-2965	77	5	theorem	theorem	ADJ
easat-2965	77	6	1.1	1.1	NUM
easat-2965	77	7	,	,	PUNCT
easat-2965	77	8	all	all	PRON
easat-2965	77	9	of	of	ADP
easat-2965	77	10	the	the	DET
easat-2965	77	11	positive	positive	ADJ
easat-2965	77	12	integral	integral	ADJ
easat-2965	77	13	solutions	solution	NOUN
easat-2965	77	14	to	to	ADP
easat-2965	77	15	the	the	DET
easat-2965	77	16	equation	equation	NOUN
easat-2965	77	17	𝑥2	𝑥2	NOUN
easat-2965	77	18	−	−	PROPN
easat-2965	77	19	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	77	20	=	=	SYM
easat-2965	77	21	1	1	NUM
easat-2965	77	22	is	be	AUX
easat-2965	77	23	provided	provide	VERB
easat-2965	77	24	by	by	ADP
easat-2965	77	25	𝑥𝑛	𝑥𝑛	PRON
easat-2965	77	26	+	+	NOUN
easat-2965	77	27	𝑦𝑛√𝐷	𝑦𝑛√𝐷	VERB
easat-2965	77	28	=	=	PUNCT
easat-2965	77	29	(	(	PUNCT
easat-2965	77	30	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	77	31	𝑎	𝑎	NOUN
easat-2965	77	32	+	+	NUM
easat-2965	77	33	2𝑞	2𝑞	NUM
easat-2965	77	34	𝑎	𝑎	PRON
easat-2965	77	35	√𝐷	√𝐷	NOUN
easat-2965	77	36	)	)	PUNCT
easat-2965	77	37	𝑛	𝑛	NOUN
easat-2965	77	38	,	,	PUNCT
easat-2965	77	39	and	and	CCONJ
easat-2965	77	40	𝑥𝑛	𝑥𝑛	PROPN
easat-2965	77	41	−	−	NOUN
easat-2965	77	42	𝑦𝑛√𝐷	𝑦𝑛√𝐷	NOUN
easat-2965	77	43	=	=	PUNCT
easat-2965	77	44	(	(	PUNCT
easat-2965	77	45	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	77	46	𝑎	𝑎	DET
easat-2965	77	47	−	−	NUM
easat-2965	77	48	2𝑞	2𝑞	NOUN
easat-2965	77	49	𝑎	𝑎	DET
easat-2965	77	50	√𝐷	√𝐷	NOUN
easat-2965	77	51	)	)	PUNCT
easat-2965	77	52	𝑛	𝑛	PROPN
easat-2965	77	53	set	set	VERB
easat-2965	77	54	𝛼	𝛼	NOUN
easat-2965	77	55	=	=	PUNCT
easat-2965	77	56	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	78	1	𝑎	𝑎	X
easat-2965	78	2	+	+	NUM
easat-2965	78	3	2𝑞	2𝑞	NUM
easat-2965	78	4	𝑎	𝑎	PRON
easat-2965	78	5	√𝐷	√𝐷	NOUN
easat-2965	78	6	and	and	CCONJ
easat-2965	78	7	𝛽	𝛽	NOUN
easat-2965	78	8	=	=	PUNCT
easat-2965	78	9	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	78	10	𝑎	𝑎	DET
easat-2965	78	11	−	−	NUM
easat-2965	78	12	2𝑞	2𝑞	NOUN
easat-2965	78	13	𝑎	𝑎	PRON
easat-2965	78	14	√𝐷	√𝐷	NOUN
easat-2965	78	15	,	,	PUNCT
easat-2965	78	16	then	then	ADV
easat-2965	78	17	𝛼	𝛼	X
easat-2965	78	18	+	+	X
easat-2965	78	19	𝛽	𝛽	NOUN
easat-2965	78	20	=	=	SYM
easat-2965	78	21	2	2	NUM
easat-2965	78	22	(	(	PUNCT
easat-2965	78	23	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	78	24	𝑎	𝑎	NOUN
easat-2965	78	25	)	)	PUNCT
easat-2965	78	26	and	and	CCONJ
easat-2965	78	27	𝑡	𝑡	PROPN
easat-2965	78	28	=	=	SYM
easat-2965	78	29	−𝛼𝛽	−𝛼𝛽	NUM
easat-2965	78	30	=	=	PUNCT
easat-2965	78	31	−1	−1	NOUN
easat-2965	78	32	.	.	PUNCT
easat-2965	79	1	thus	thus	ADV
easat-2965	79	2	,	,	PUNCT
easat-2965	79	3	for	for	ADP
easat-2965	79	4	all	all	DET
easat-2965	79	5	𝑛	𝑛	PRON
easat-2965	79	6	≥	≥	NOUN
easat-2965	79	7	1	1	NUM
easat-2965	79	8	.	.	PUNCT
easat-2965	79	9	𝑥𝑛	𝑥𝑛	PROPN
easat-2965	79	10	=	=	SYM
easat-2965	79	11	𝛼𝑛+𝛽𝑛	𝛼𝑛+𝛽𝑛	PROPN
easat-2965	79	12	2	2	NUM
easat-2965	79	13	=	=	SYM
easat-2965	79	14	1	1	NUM
easat-2965	79	15	2	2	NUM
easat-2965	79	16	𝐿𝑛(𝑠	𝐿𝑛(𝑠	NOUN
easat-2965	79	17	,	,	PUNCT
easat-2965	79	18	𝑡	𝑡	X
easat-2965	79	19	)	)	PUNCT
easat-2965	79	20	=	=	SYM
easat-2965	79	21	1	1	NUM
easat-2965	79	22	2	2	X
easat-2965	79	23	𝐿𝑛	𝐿𝑛	PROPN
easat-2965	79	24	(	(	PUNCT
easat-2965	79	25	2	2	NUM
easat-2965	79	26	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	79	27	𝑎	𝑎	NOUN
easat-2965	79	28	,	,	PUNCT
easat-2965	79	29	−1	−1	NOUN
easat-2965	79	30	)	)	PUNCT
easat-2965	79	31	=	=	SYM
easat-2965	79	32	1	1	NUM
easat-2965	79	33	2	2	NUM
easat-2965	79	34	ψ𝑛	ψ𝑛	PRON
easat-2965	79	35	(	(	PUNCT
easat-2965	79	36	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	79	37	𝑎	𝑎	NOUN
easat-2965	79	38	,	,	PUNCT
easat-2965	79	39	−1	−1	NOUN
easat-2965	79	40	)	)	PUNCT
easat-2965	79	41	,	,	PUNCT
easat-2965	79	42	4412	4412	NUM
easat-2965	79	43	edelweiss	edelweiss	PROPN
easat-2965	79	44	applied	apply	VERB
easat-2965	79	45	science	science	NOUN
easat-2965	79	46	and	and	CCONJ
easat-2965	79	47	technology	technology	NOUN
easat-2965	79	48	issn	issn	PROPN
easat-2965	79	49	:	:	PUNCT
easat-2965	79	50	2576	2576	NUM
easat-2965	79	51	-	-	SYM
easat-2965	79	52	8484	8484	NUM
easat-2965	79	53	vol	vol	NOUN
easat-2965	79	54	.	.	PROPN
easat-2965	80	1	8	8	NUM
easat-2965	80	2	,	,	PUNCT
easat-2965	80	3	no	no	INTJ
easat-2965	80	4	.	.	NOUN
easat-2965	81	1	6	6	NUM
easat-2965	81	2	:	:	SYM
easat-2965	81	3	4408	4408	NUM
easat-2965	81	4	-	-	SYM
easat-2965	81	5	4414	4414	NUM
easat-2965	81	6	,	,	PUNCT
easat-2965	81	7	2024	2024	NUM
easat-2965	81	8	doi	doi	NOUN
easat-2965	81	9	:	:	PUNCT
easat-2965	81	10	10.55214/25768484.v8i6.2965	10.55214/25768484.v8i6.2965	NUM
easat-2965	81	11	©	©	PROPN
easat-2965	81	12	2024	2024	NUM
easat-2965	81	13	by	by	ADP
easat-2965	81	14	the	the	DET
easat-2965	81	15	authors	author	NOUN
easat-2965	81	16	;	;	PUNCT
easat-2965	81	17	licensee	licensee	PROPN
easat-2965	81	18	learning	learning	NOUN
easat-2965	81	19	gate	gate	NOUN
easat-2965	81	20	and	and	CCONJ
easat-2965	81	21	𝑦𝑛	𝑦𝑛	ADJ
easat-2965	81	22	=	=	PUNCT
easat-2965	81	23	𝛼𝑛−𝛽𝑛	𝛼𝑛−𝛽𝑛	NOUN
easat-2965	82	1	2√𝐷	2√𝐷	NUM
easat-2965	82	2	=	=	SYM
easat-2965	82	3	2𝑞	2𝑞	NUM
easat-2965	82	4	𝑎	𝑎	DET
easat-2965	82	5	𝐹𝑛(𝑠	𝐹𝑛(𝑠	NOUN
easat-2965	82	6	,	,	PUNCT
easat-2965	82	7	𝑡	𝑡	NOUN
easat-2965	82	8	)	)	PUNCT
easat-2965	82	9	=	=	SYM
easat-2965	82	10	2𝑞	2𝑞	NOUN
easat-2965	82	11	𝑎	𝑎	X
easat-2965	82	12	𝐹𝑛	𝐹𝑛	PROPN
easat-2965	82	13	(	(	PUNCT
easat-2965	82	14	2	2	NUM
easat-2965	82	15	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	82	16	𝑎	𝑎	NOUN
easat-2965	82	17	,	,	PUNCT
easat-2965	82	18	−1	−1	NOUN
easat-2965	82	19	)	)	PUNCT
easat-2965	82	20	=	=	PUNCT
easat-2965	83	1	𝑞	𝑞	PROPN
easat-2965	83	2	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	83	3	ψ𝑛	ψ𝑛	NOUN
easat-2965	83	4	(	(	PUNCT
easat-2965	83	5	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	83	6	𝑎	𝑎	NOUN
easat-2965	83	7	,	,	PUNCT
easat-2965	83	8	−1	−1	NOUN
easat-2965	83	9	)	)	PUNCT
easat-2965	83	10	,	,	PUNCT
easat-2965	83	11	for	for	ADP
easat-2965	83	12	all	all	DET
easat-2965	83	13	𝑛	𝑛	DET
easat-2965	83	14	≥	≥	NUM
easat-2965	83	15	1	1	NUM
easat-2965	83	16	.	.	NOUN
easat-2965	83	17	4	4	NUM
easat-2965	83	18	.	.	X
easat-2965	84	1	it	it	PRON
easat-2965	84	2	is	be	AUX
easat-2965	84	3	easy	easy	ADJ
easat-2965	84	4	to	to	PART
easat-2965	84	5	verify	verify	VERB
easat-2965	84	6	(	(	PUNCT
easat-2965	84	7	4	4	NUM
easat-2965	84	8	)	)	PUNCT
easat-2965	84	9	from	from	ADP
easat-2965	84	10	(	(	PUNCT
easat-2965	84	11	1	1	NUM
easat-2965	84	12	)	)	PUNCT
easat-2965	84	13	.	.	PUNCT
easat-2965	85	1	5	5	X
easat-2965	85	2	.	.	X
easat-2965	85	3	the	the	DET
easat-2965	85	4	fifth	fifth	ADJ
easat-2965	85	5	part	part	NOUN
easat-2965	85	6	can	can	AUX
easat-2965	85	7	be	be	AUX
easat-2965	85	8	proved	prove	VERB
easat-2965	85	9	using	use	VERB
easat-2965	85	10	the	the	DET
easat-2965	85	11	theorem	theorem	NOUN
easat-2965	85	12	.	.	PROPN
easat-2965	85	13	2.2	2.2	NUM
easat-2965	85	14	.	.	PUNCT
easat-2965	85	15	corollary	corollary	ADJ
easat-2965	85	16	let	let	VERB
easat-2965	85	17	𝐷	𝐷	NOUN
easat-2965	85	18	=	=	NOUN
easat-2965	86	1	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	86	2	−	−	PROPN
easat-2965	86	3	𝑎𝑝	𝑎𝑝	PROPN
easat-2965	86	4	,	,	PUNCT
easat-2965	86	5	with	with	ADP
easat-2965	86	6	𝑞	𝑞	X
easat-2965	86	7	>	>	X
easat-2965	86	8	𝑎	𝑎	X
easat-2965	86	9	being	be	AUX
easat-2965	86	10	a	a	DET
easat-2965	86	11	multiple	multiple	NOUN
easat-2965	86	12	of	of	ADP
easat-2965	86	13	𝑎	𝑎	NOUN
easat-2965	86	14	and	and	CCONJ
easat-2965	86	15	𝑝	𝑝	NOUN
easat-2965	86	16	,	,	PUNCT
easat-2965	86	17	𝑞	𝑞	X
easat-2965	86	18	being	be	AUX
easat-2965	86	19	positive	positive	ADJ
easat-2965	86	20	integers	integer	NOUN
easat-2965	86	21	chosen	choose	VERB
easat-2965	86	22	in	in	ADP
easat-2965	86	23	such	such	DET
easat-2965	86	24	a	a	DET
easat-2965	86	25	way	way	NOUN
easat-2965	86	26	that	that	PRON
easat-2965	86	27	𝐷	𝐷	NOUN
easat-2965	86	28	is	be	AUX
easat-2965	86	29	not	not	PART
easat-2965	86	30	a	a	DET
easat-2965	86	31	perfect	perfect	ADJ
easat-2965	86	32	square	square	NOUN
easat-2965	86	33	,	,	PUNCT
easat-2965	86	34	then	then	ADV
easat-2965	86	35	,	,	PUNCT
easat-2965	86	36	the	the	DET
easat-2965	86	37	equation	equation	NOUN
easat-2965	86	38	𝑥2	𝑥2	NOUN
easat-2965	86	39	−	−	PROPN
easat-2965	86	40	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	86	41	=	=	SYM
easat-2965	86	42	1	1	NUM
easat-2965	86	43	has	have	VERB
easat-2965	86	44	no	no	DET
easat-2965	86	45	positive	positive	ADJ
easat-2965	86	46	integer	integer	NOUN
easat-2965	86	47	solution	solution	NOUN
easat-2965	86	48	.	.	PUNCT
easat-2965	87	1	proof	proof	NOUN
easat-2965	87	2	:	:	PUNCT
easat-2965	87	3	it	it	PRON
easat-2965	87	4	is	be	AUX
easat-2965	87	5	simple	simple	ADJ
easat-2965	87	6	to	to	PART
easat-2965	87	7	conclude	conclude	VERB
easat-2965	87	8	using	use	VERB
easat-2965	87	9	the	the	DET
easat-2965	87	10	theorem	theorem	NOUN
easat-2965	87	11	..	..	PUNCT
easat-2965	87	12	;	;	PUNCT
easat-2965	87	13	2.3	2.3	NUM
easat-2965	87	14	.	.	PUNCT
easat-2965	87	15	theorem	theorem	NOUN
easat-2965	87	16	1	1	NUM
easat-2965	87	17	.	.	PUNCT
easat-2965	88	1	let	let	VERB
easat-2965	88	2	𝐷	𝐷	NOUN
easat-2965	88	3	=	=	PUNCT
easat-2965	88	4	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	88	5	+	+	CCONJ
easat-2965	88	6	𝑎𝑝	𝑎𝑝	NOUN
easat-2965	88	7	,	,	PUNCT
easat-2965	88	8	with	with	ADP
easat-2965	88	9	𝑞	𝑞	X
easat-2965	88	10	>	>	X
easat-2965	88	11	𝑎	𝑎	X
easat-2965	88	12	being	be	AUX
easat-2965	88	13	a	a	DET
easat-2965	88	14	multiple	multiple	NOUN
easat-2965	88	15	of	of	ADP
easat-2965	88	16	𝑎	𝑎	NOUN
easat-2965	88	17	and	and	CCONJ
easat-2965	88	18	𝑝	𝑝	NOUN
easat-2965	88	19	,	,	PUNCT
easat-2965	88	20	𝑞	𝑞	X
easat-2965	88	21	being	be	AUX
easat-2965	88	22	positive	positive	ADJ
easat-2965	88	23	integers	integer	NOUN
easat-2965	88	24	chosen	choose	VERB
easat-2965	88	25	in	in	ADP
easat-2965	88	26	such	such	DET
easat-2965	88	27	a	a	DET
easat-2965	88	28	way	way	NOUN
easat-2965	88	29	that	that	PRON
easat-2965	88	30	𝐷	𝐷	NOUN
easat-2965	88	31	is	be	AUX
easat-2965	88	32	not	not	PART
easat-2965	88	33	a	a	DET
easat-2965	88	34	perfect	perfect	ADJ
easat-2965	88	35	square	square	NOUN
easat-2965	88	36	,	,	PUNCT
easat-2965	88	37	then	then	ADV
easat-2965	88	38	,	,	PUNCT
easat-2965	88	39	√𝐷	√𝐷	ADJ
easat-2965	88	40	=	=	SYM
easat-2965	88	41	√𝑝2𝑞2	√𝑝2𝑞2	NOUN
easat-2965	88	42	+	+	NUM
easat-2965	88	43	𝑎𝑝	𝑎𝑝	NOUN
easat-2965	88	44	=	=	PUNCT
easat-2965	89	1	[	[	X
easat-2965	89	2	𝑝𝑞	𝑝𝑞	NOUN
easat-2965	89	3	;	;	PUNCT
easat-2965	89	4	2𝑞	2𝑞	NUM
easat-2965	89	5	𝑎	𝑎	X
easat-2965	89	6	,	,	PUNCT
easat-2965	89	7	2𝑝𝑞	2𝑝𝑞	ADJ
easat-2965	89	8	̅̅	̅̅	PROPN
easat-2965	89	9	̅̅	̅̅	PROPN
easat-2965	89	10	̅̅	̅̅	PROPN
easat-2965	89	11	̅̅	̅̅	PROPN
easat-2965	89	12	̅	̅	PROPN
easat-2965	89	13	]	]	X
easat-2965	89	14	.	.	PUNCT
easat-2965	90	1	2	2	X
easat-2965	90	2	.	.	X
easat-2965	90	3	the	the	DET
easat-2965	90	4	fundamental	fundamental	ADJ
easat-2965	90	5	solution	solution	NOUN
easat-2965	90	6	of	of	ADP
easat-2965	90	7	the	the	DET
easat-2965	90	8	pell	pell	NOUN
easat-2965	90	9	equation	equation	NOUN
easat-2965	90	10	𝑥2	𝑥2	NOUN
easat-2965	90	11	−	−	PROPN
easat-2965	90	12	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	90	13	=	=	SYM
easat-2965	90	14	1	1	X
easat-2965	90	15	is	be	AUX
easat-2965	90	16	(	(	PUNCT
easat-2965	90	17	𝑥1	𝑥1	NOUN
easat-2965	90	18	,	,	PUNCT
easat-2965	90	19	𝑦1	𝑦1	NOUN
easat-2965	90	20	)	)	PUNCT
easat-2965	90	21	=	=	PUNCT
easat-2965	90	22	(	(	PUNCT
easat-2965	90	23	2𝑝𝑞2+𝑎	2𝑝𝑞2+𝑎	NUM
easat-2965	90	24	𝑎	𝑎	NOUN
easat-2965	90	25	,	,	PUNCT
easat-2965	90	26	2𝑞	2𝑞	NOUN
easat-2965	90	27	𝑎	𝑎	NOUN
easat-2965	90	28	)	)	PUNCT
easat-2965	90	29	.	.	PUNCT
easat-2965	91	1	3	3	X
easat-2965	91	2	.	.	X
easat-2965	91	3	all	all	DET
easat-2965	91	4	positive	positive	ADJ
easat-2965	91	5	integer	integer	NOUN
easat-2965	91	6	solution	solution	NOUN
easat-2965	91	7	(	(	PUNCT
easat-2965	91	8	𝑥𝑛	𝑥𝑛	NOUN
easat-2965	91	9	,	,	PUNCT
easat-2965	91	10	𝑦𝑛	𝑦𝑛	NOUN
easat-2965	91	11	)	)	PUNCT
easat-2965	91	12	of	of	ADP
easat-2965	91	13	the	the	DET
easat-2965	91	14	pell	pell	NOUN
easat-2965	91	15	equation	equation	NOUN
easat-2965	91	16	𝑥2	𝑥2	NOUN
easat-2965	91	17	−	−	PROPN
easat-2965	91	18	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	91	19	=	=	SYM
easat-2965	91	20	1	1	NUM
easat-2965	91	21	are	be	AUX
easat-2965	91	22	provided	provide	VERB
easat-2965	91	23	by	by	ADP
easat-2965	91	24	𝑥𝑛	𝑥𝑛	PROPN
easat-2965	91	25	=	=	SYM
easat-2965	91	26	𝛼𝑛+𝛽𝑛	𝛼𝑛+𝛽𝑛	PROPN
easat-2965	91	27	2	2	NUM
easat-2965	91	28	=	=	SYM
easat-2965	91	29	1	1	NUM
easat-2965	91	30	2	2	NUM
easat-2965	91	31	𝐿𝑛(𝑠	𝐿𝑛(𝑠	NOUN
easat-2965	91	32	,	,	PUNCT
easat-2965	91	33	𝑡	𝑡	X
easat-2965	91	34	)	)	PUNCT
easat-2965	91	35	=	=	SYM
easat-2965	91	36	1	1	NUM
easat-2965	91	37	2	2	X
easat-2965	91	38	𝐿𝑛	𝐿𝑛	PROPN
easat-2965	91	39	(	(	PUNCT
easat-2965	91	40	2	2	NUM
easat-2965	91	41	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	91	42	𝑎	𝑎	NOUN
easat-2965	91	43	,	,	PUNCT
easat-2965	91	44	−1	−1	NOUN
easat-2965	91	45	)	)	PUNCT
easat-2965	91	46	=	=	SYM
easat-2965	91	47	1	1	NUM
easat-2965	91	48	2	2	NUM
easat-2965	91	49	ψ𝑛	ψ𝑛	PRON
easat-2965	91	50	(	(	PUNCT
easat-2965	91	51	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	91	52	𝑎	𝑎	NOUN
easat-2965	91	53	,	,	PUNCT
easat-2965	91	54	−1	−1	NOUN
easat-2965	91	55	)	)	PUNCT
easat-2965	91	56	,	,	PUNCT
easat-2965	91	57	and	and	CCONJ
easat-2965	91	58	𝑦𝑛	𝑦𝑛	ADP
easat-2965	91	59	=	=	SYM
easat-2965	91	60	𝛼𝑛−𝛽𝑛	𝛼𝑛−𝛽𝑛	NOUN
easat-2965	91	61	2√𝐷	2√𝐷	NUM
easat-2965	91	62	=	=	SYM
easat-2965	91	63	2𝑞	2𝑞	NUM
easat-2965	91	64	𝑎	𝑎	DET
easat-2965	91	65	𝐹𝑛(𝑠	𝐹𝑛(𝑠	NOUN
easat-2965	91	66	,	,	PUNCT
easat-2965	91	67	𝑡	𝑡	NOUN
easat-2965	91	68	)	)	PUNCT
easat-2965	91	69	=	=	SYM
easat-2965	91	70	2𝑞	2𝑞	NOUN
easat-2965	91	71	𝑎	𝑎	X
easat-2965	91	72	𝐹𝑛	𝐹𝑛	PROPN
easat-2965	91	73	(	(	PUNCT
easat-2965	91	74	2	2	NUM
easat-2965	91	75	2𝑝𝑞2+𝑎	2𝑝𝑞2+𝑎	NUM
easat-2965	91	76	𝑎	𝑎	NOUN
easat-2965	91	77	,	,	PUNCT
easat-2965	91	78	−1	−1	NOUN
easat-2965	91	79	)	)	PUNCT
easat-2965	91	80	=	=	PUNCT
easat-2965	92	1	𝑞	𝑞	PROPN
easat-2965	92	2	2𝑝𝑞2−𝑎	2𝑝𝑞2−𝑎	NUM
easat-2965	92	3	φ𝑛	φ𝑛	PROPN
easat-2965	92	4	(	(	PUNCT
easat-2965	92	5	2	2	NUM
easat-2965	92	6	2𝑝𝑞2+𝑎	2𝑝𝑞2+𝑎	NUM
easat-2965	92	7	𝑎	𝑎	NOUN
easat-2965	92	8	,	,	PUNCT
easat-2965	92	9	−1	−1	NOUN
easat-2965	92	10	)	)	PUNCT
easat-2965	92	11	,	,	PUNCT
easat-2965	92	12	for	for	ADP
easat-2965	92	13	all	all	DET
easat-2965	92	14	𝑛	𝑛	DET
easat-2965	92	15	≥	≥	NUM
easat-2965	92	16	1	1	NUM
easat-2965	92	17	.	.	NOUN
easat-2965	92	18	4	4	NUM
easat-2965	92	19	.	.	PUNCT
easat-2965	93	1	the	the	DET
easat-2965	93	2	fundamental	fundamental	ADJ
easat-2965	93	3	solution	solution	NOUN
easat-2965	93	4	of	of	ADP
easat-2965	93	5	pell	pell	ADJ
easat-2965	93	6	equation	equation	NOUN
easat-2965	93	7	𝑥2	𝑥2	NOUN
easat-2965	93	8	−	−	PROPN
easat-2965	93	9	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	93	10	=	=	PROPN
easat-2965	93	11	𝑘𝑡	𝑘𝑡	PROPN
easat-2965	93	12	is	be	AUX
easat-2965	93	13	(	(	PUNCT
easat-2965	93	14	𝑥1	𝑥1	NOUN
easat-2965	93	15	,	,	PUNCT
easat-2965	93	16	𝑦1	𝑦1	NOUN
easat-2965	93	17	)	)	PUNCT
easat-2965	93	18	=	=	PUNCT
easat-2965	94	1	(	(	PUNCT
easat-2965	94	2	2𝑝𝑞2+𝑎	2𝑝𝑞2+𝑎	NUM
easat-2965	94	3	𝑎	𝑎	X
easat-2965	94	4	𝑘	𝑘	PRON
easat-2965	94	5	𝑡	𝑡	PROPN
easat-2965	94	6	2	2	NUM
easat-2965	94	7	,	,	PUNCT
easat-2965	94	8	2𝑞	2𝑞	NUM
easat-2965	94	9	𝑎	𝑎	VERB
easat-2965	94	10	𝑘	𝑘	X
easat-2965	94	11	𝑡	𝑡	PROPN
easat-2965	94	12	2	2	NUM
easat-2965	94	13	)	)	PUNCT
easat-2965	94	14	.	.	PUNCT
easat-2965	95	1	5	5	X
easat-2965	95	2	.	.	X
easat-2965	95	3	all	all	DET
easat-2965	95	4	positive	positive	ADJ
easat-2965	95	5	solutions	solution	NOUN
easat-2965	95	6	of	of	ADP
easat-2965	95	7	pell	pell	NOUN
easat-2965	95	8	equation	equation	NOUN
easat-2965	95	9	𝑥2	𝑥2	NOUN
easat-2965	95	10	−	−	PROPN
easat-2965	95	11	𝐷𝑦2	𝐷𝑦2	NOUN
easat-2965	95	12	=	=	PROPN
easat-2965	96	1	𝑘𝑡	𝑘𝑡	PROPN
easat-2965	96	2	are	be	AUX
easat-2965	96	3	given	give	VERB
easat-2965	96	4	by	by	ADP
easat-2965	96	5	𝑥𝑛+1	𝑥𝑛+1	PROPN
easat-2965	96	6	=	=	SYM
easat-2965	96	7	2𝑝𝑞2	2𝑝𝑞2	PROPN
easat-2965	97	1	+	+	CCONJ
easat-2965	97	2	𝑎	𝑎	X
easat-2965	97	3	𝑎	𝑎	X
easat-2965	97	4	𝑘	𝑘	ADP
easat-2965	97	5	𝑡	𝑡	NOUN
easat-2965	97	6	2𝑥𝑛	2𝑥𝑛	ADJ
easat-2965	97	7	+	+	CCONJ
easat-2965	97	8	2𝑞	2𝑞	NUM
easat-2965	97	9	𝑝2𝑞2	𝑝2𝑞2	VERB
easat-2965	97	10	+	+	CCONJ
easat-2965	97	11	𝑎𝑝	𝑎𝑝	PROPN
easat-2965	97	12	𝑎	𝑎	X
easat-2965	97	13	𝑘	𝑘	ADP
easat-2965	97	14	𝑡	𝑡	NOUN
easat-2965	97	15	2𝑦𝑛	2𝑦𝑛	NOUN
easat-2965	97	16	and	and	CCONJ
easat-2965	97	17	𝑦𝑛+1	𝑦𝑛+1	NOUN
easat-2965	97	18	=	=	SYM
easat-2965	97	19	2𝑞	2𝑞	NUM
easat-2965	97	20	𝑎	𝑎	VERB
easat-2965	97	21	𝑘	𝑘	ADP
easat-2965	97	22	𝑡	𝑡	NOUN
easat-2965	97	23	2𝑥𝑛	2𝑥𝑛	NOUN
easat-2965	98	1	+	+	CCONJ
easat-2965	98	2	2𝑝𝑞2	2𝑝𝑞2	NUM
easat-2965	99	1	+	+	CCONJ
easat-2965	99	2	𝑎	𝑎	X
easat-2965	99	3	𝑎	𝑎	X
easat-2965	99	4	𝑘	𝑘	PRON
easat-2965	99	5	𝑡	𝑡	NOUN
easat-2965	99	6	2𝑦𝑛	2𝑦𝑛	NOUN
easat-2965	99	7	for	for	ADP
easat-2965	99	8	all	all	DET
easat-2965	99	9	𝑛	𝑛	PRON
easat-2965	99	10	≥	≥	NUM
easat-2965	99	11	1	1	NUM
easat-2965	99	12	.	.	PUNCT
easat-2965	100	1	proof	proof	NOUN
easat-2965	100	2	:	:	PUNCT
easat-2965	100	3	the	the	DET
easat-2965	100	4	proof	proof	NOUN
easat-2965	100	5	is	be	AUX
easat-2965	100	6	similar	similar	ADJ
easat-2965	100	7	to	to	ADP
easat-2965	100	8	theorem	theorem	VERB
easat-2965	100	9	1	1	NUM
easat-2965	100	10	.	.	SYM
easat-2965	100	11	4413	4413	NUM
easat-2965	100	12	edelweiss	edelweiss	PROPN
easat-2965	100	13	applied	apply	VERB
easat-2965	100	14	science	science	NOUN
easat-2965	100	15	and	and	CCONJ
easat-2965	100	16	technology	technology	NOUN
easat-2965	100	17	issn	issn	PROPN
easat-2965	100	18	:	:	PUNCT
easat-2965	100	19	2576	2576	NUM
easat-2965	100	20	-	-	SYM
easat-2965	100	21	8484	8484	NUM
easat-2965	100	22	vol	vol	NOUN
easat-2965	100	23	.	.	PROPN
easat-2965	101	1	8	8	NUM
easat-2965	101	2	,	,	PUNCT
easat-2965	101	3	no	no	INTJ
easat-2965	101	4	.	.	NOUN
easat-2965	102	1	6	6	NUM
easat-2965	102	2	:	:	SYM
easat-2965	102	3	4408	4408	NUM
easat-2965	102	4	-	-	SYM
easat-2965	102	5	4414	4414	NUM
easat-2965	102	6	,	,	PUNCT
easat-2965	102	7	2024	2024	NUM
easat-2965	102	8	doi	doi	NOUN
easat-2965	102	9	:	:	PUNCT
easat-2965	102	10	10.55214/25768484.v8i6.2965	10.55214/25768484.v8i6.2965	NUM
easat-2965	102	11	©	©	PROPN
easat-2965	102	12	2024	2024	NUM
easat-2965	102	13	by	by	ADP
easat-2965	102	14	the	the	DET
easat-2965	102	15	authors	author	NOUN
easat-2965	102	16	;	;	PUNCT
easat-2965	102	17	licensee	licensee	PROPN
easat-2965	102	18	learning	learning	NOUN
easat-2965	102	19	gate	gate	NOUN
easat-2965	102	20	3	3	PROPN
easat-2965	102	21	.	.	PUNCT
easat-2965	102	22	applications	application	NOUN
easat-2965	102	23	one	one	NUM
easat-2965	102	24	application	application	NOUN
easat-2965	102	25	of	of	ADP
easat-2965	102	26	our	our	PRON
easat-2965	102	27	results	result	NOUN
easat-2965	102	28	is	be	AUX
easat-2965	102	29	to	to	PART
easat-2965	102	30	find	find	VERB
easat-2965	102	31	the	the	DET
easat-2965	102	32	units	unit	NOUN
easat-2965	102	33	of	of	ADP
easat-2965	102	34	ℤ[√𝐷	ℤ[√𝐷	NOUN
easat-2965	102	35	]	]	PUNCT
easat-2965	102	36	.	.	PUNCT
easat-2965	103	1	the	the	DET
easat-2965	103	2	unity	unity	NOUN
easat-2965	103	3	,	,	PUNCT
easat-2965	103	4	in𝑐	in𝑐	NOUN
easat-2965	103	5	,	,	PUNCT
easat-2965	103	6	is	be	AUX
easat-2965	103	7	1and	1and	NUM
easat-2965	103	8	units	unit	NOUN
easat-2965	103	9	are	be	AUX
easat-2965	103	10	those	those	DET
easat-2965	103	11	invertible	invertible	ADJ
easat-2965	103	12	elements	element	NOUN
easat-2965	103	13	.	.	PUNCT
easat-2965	104	1	in	in	ADP
easat-2965	104	2	the	the	DET
easat-2965	104	3	following	following	NOUN
easat-2965	104	4	,	,	PUNCT
easat-2965	104	5	we	we	PRON
easat-2965	104	6	will	will	AUX
easat-2965	104	7	find	find	VERB
easat-2965	104	8	units	unit	NOUN
easat-2965	104	9	of	of	ADP
easat-2965	104	10	ℤ[√𝐷	ℤ[√𝐷	NOUN
easat-2965	104	11	]	]	PUNCT
easat-2965	104	12	,	,	PUNCT
easat-2965	104	13	where	where	SCONJ
easat-2965	104	14	𝐷	𝐷	NOUN
easat-2965	104	15	=	=	SYM
easat-2965	104	16	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	104	17	±	±	NUM
easat-2965	104	18	𝑎𝑝	𝑎𝑝	PROPN
easat-2965	104	19	,	,	PUNCT
easat-2965	104	20	with	with	ADP
easat-2965	104	21	𝑞	𝑞	X
easat-2965	104	22	>	>	X
easat-2965	104	23	𝑎	𝑎	X
easat-2965	104	24	being	be	AUX
easat-2965	104	25	a	a	DET
easat-2965	104	26	multiple	multiple	NOUN
easat-2965	104	27	of	of	ADP
easat-2965	104	28	𝑎	𝑎	NOUN
easat-2965	104	29	and	and	CCONJ
easat-2965	104	30	𝑝	𝑝	NOUN
easat-2965	104	31	,	,	PUNCT
easat-2965	104	32	𝑞	𝑞	X
easat-2965	104	33	being	be	AUX
easat-2965	104	34	positive	positive	ADJ
easat-2965	104	35	integers	integer	NOUN
easat-2965	104	36	chosen	choose	VERB
easat-2965	104	37	in	in	ADP
easat-2965	104	38	such	such	DET
easat-2965	104	39	a	a	DET
easat-2965	104	40	way	way	NOUN
easat-2965	104	41	that	that	PRON
easat-2965	104	42	𝐷	𝐷	NOUN
easat-2965	104	43	is	be	AUX
easat-2965	104	44	not	not	PART
easat-2965	104	45	a	a	DET
easat-2965	104	46	perfect	perfect	ADJ
easat-2965	104	47	square	square	NOUN
easat-2965	104	48	.	.	PUNCT
easat-2965	105	1	let	let	VERB
easat-2965	105	2	𝑢	𝑢	PRON
easat-2965	105	3	+	+	X
easat-2965	105	4	𝑣√𝐷	𝑣√𝐷	PROPN
easat-2965	105	5	,	,	PUNCT
easat-2965	105	6	where	where	SCONJ
easat-2965	105	7	𝑢	𝑢	X
easat-2965	105	8	,	,	PUNCT
easat-2965	105	9	𝑣	𝑣	PRON
easat-2965	105	10	∈	∈	NOUN
easat-2965	105	11	ℤ	ℤ	PRON
easat-2965	105	12	be	be	VERB
easat-2965	105	13	a	a	DET
easat-2965	105	14	unit	unit	NOUN
easat-2965	105	15	element	element	NOUN
easat-2965	105	16	in	in	ADP
easat-2965	105	17	ℤ[√𝐷	ℤ[√𝐷	PROPN
easat-2965	105	18	]	]	PUNCT
easat-2965	105	19	,	,	PUNCT
easat-2965	105	20	then	then	ADV
easat-2965	105	21	there	there	PRON
easat-2965	105	22	exists	exist	VERB
easat-2965	105	23	some	some	DET
easat-2965	105	24	𝑐	𝑐	PROPN
easat-2965	105	25	+	+	CCONJ
easat-2965	105	26	𝑑√𝐷	𝑑√𝐷	PROPN
easat-2965	105	27	∈	∈	PROPN
easat-2965	105	28	ℤ[√𝐷	ℤ[√𝐷	NOUN
easat-2965	105	29	]	]	PUNCT
easat-2965	105	30	such	such	ADJ
easat-2965	105	31	that	that	SCONJ
easat-2965	105	32	(	(	PUNCT
easat-2965	105	33	𝑢	𝑢	X
easat-2965	105	34	+	+	X
easat-2965	105	35	𝑣√𝐷)(𝑐	𝑣√𝐷)(𝑐	NOUN
easat-2965	105	36	+	+	CCONJ
easat-2965	105	37	𝑑√𝐷	𝑑√𝐷	PROPN
easat-2965	105	38	)	)	PUNCT
easat-2965	105	39	=	=	SYM
easat-2965	105	40	1	1	X
easat-2965	105	41	.	.	PUNCT
easat-2965	106	1	so	so	ADV
easat-2965	106	2	,	,	PUNCT
easat-2965	106	3	𝑐	𝑐	PROPN
easat-2965	106	4	+	+	CCONJ
easat-2965	106	5	𝑑√𝐷	𝑑√𝐷	PROPN
easat-2965	106	6	=	=	SYM
easat-2965	106	7	1	1	NUM
easat-2965	106	8	𝑢+𝑣√𝐷	𝑢+𝑣√𝐷	NOUN
easat-2965	106	9	=	=	SYM
easat-2965	106	10	𝑢−𝑣√𝐷	𝑢−𝑣√𝐷	NOUN
easat-2965	106	11	𝑢2−𝑣2𝐷	𝑢2−𝑣2𝐷	NOUN
easat-2965	106	12	,	,	PUNCT
easat-2965	106	13	then	then	ADV
easat-2965	106	14	if	if	SCONJ
easat-2965	106	15	we	we	PRON
easat-2965	106	16	consider	consider	VERB
easat-2965	106	17	the	the	DET
easat-2965	106	18	quantity	quantity	NOUN
easat-2965	106	19	𝑁(𝑢	𝑁(𝑢	X
easat-2965	106	20	+	+	CCONJ
easat-2965	106	21	𝑣√𝐷	𝑣√𝐷	NUM
easat-2965	106	22	)	)	PUNCT
easat-2965	106	23	=	=	SYM
easat-2965	106	24	𝑢2	𝑢2	PROPN
easat-2965	106	25	−	−	PROPN
easat-2965	106	26	𝑣2𝐷	𝑣2𝐷	NOUN
easat-2965	106	27	,	,	PUNCT
easat-2965	106	28	one	one	PRON
easat-2965	106	29	can	can	AUX
easat-2965	106	30	verify	verify	VERB
easat-2965	106	31	that	that	SCONJ
easat-2965	106	32	𝑁	𝑁	PROPN
easat-2965	106	33	is	be	AUX
easat-2965	106	34	multiplicative	multiplicative	ADJ
easat-2965	106	35	,	,	PUNCT
easat-2965	106	36	and	and	CCONJ
easat-2965	106	37	then	then	ADV
easat-2965	106	38	if	if	SCONJ
easat-2965	106	39	𝑢	𝑢	PRON
easat-2965	106	40	+	+	VERB
easat-2965	106	41	𝑣√𝐷	𝑣√𝐷	PROPN
easat-2965	106	42	is	be	AUX
easat-2965	106	43	invertible	invertible	ADJ
easat-2965	106	44	,	,	PUNCT
easat-2965	106	45	𝑁	𝑁	PROPN
easat-2965	106	46	must	must	AUX
easat-2965	106	47	be	be	AUX
easat-2965	106	48	invertible	invertible	ADJ
easat-2965	106	49	too	too	ADV
easat-2965	106	50	,	,	PUNCT
easat-2965	106	51	so	so	CCONJ
easat-2965	106	52	it	it	PRON
easat-2965	106	53	is	be	AUX
easat-2965	106	54	either	either	CCONJ
easat-2965	106	55	1	1	NUM
easat-2965	106	56	or	or	CCONJ
easat-2965	106	57	−1	−1	NOUN
easat-2965	106	58	.	.	PUNCT
easat-2965	107	1	then	then	ADV
easat-2965	107	2	,	,	PUNCT
easat-2965	107	3	we	we	PRON
easat-2965	107	4	get	get	VERB
easat-2965	107	5	𝑢2	𝑢2	PROPN
easat-2965	107	6	−𝐷𝑣2	−𝐷𝑣2	PROPN
easat-2965	107	7	=	=	PUNCT
easat-2965	107	8	±1	±1	VERB
easat-2965	107	9	.	.	PUNCT
easat-2965	108	1	further	further	ADJ
easat-2965	108	2	𝑢2	𝑢2	PROPN
easat-2965	108	3	−𝐷𝑣2	−𝐷𝑣2	PROPN
easat-2965	109	1	=	=	PUNCT
easat-2965	109	2	−1	−1	NOUN
easat-2965	109	3	has	have	VERB
easat-2965	109	4	no	no	DET
easat-2965	109	5	positive	positive	ADJ
easat-2965	109	6	solution	solution	NOUN
easat-2965	109	7	.	.	PUNCT
easat-2965	110	1	to	to	PART
easat-2965	110	2	solve	solve	VERB
easat-2965	110	3	𝑢2	𝑢2	PROPN
easat-2965	110	4	−	−	PROPN
easat-2965	110	5	𝐷𝑣2	𝐷𝑣2	NOUN
easat-2965	110	6	=	=	NOUN
easat-2965	110	7	1	1	NUM
easat-2965	110	8	,	,	PUNCT
easat-2965	110	9	we	we	PRON
easat-2965	110	10	apply	apply	VERB
easat-2965	110	11	our	our	PRON
easat-2965	110	12	theorems	theorem	NOUN
easat-2965	110	13	.	.	PUNCT
easat-2965	111	1	let	let	VERB
easat-2965	111	2	for	for	ADP
easat-2965	111	3	example	example	NOUN
easat-2965	111	4	,	,	PUNCT
easat-2965	111	5	𝑝	𝑝	NOUN
easat-2965	111	6	=	=	SYM
easat-2965	111	7	2	2	NUM
easat-2965	111	8	,	,	PUNCT
easat-2965	111	9	𝑞	𝑞	X
easat-2965	111	10	=	=	NOUN
easat-2965	111	11	14	14	NUM
easat-2965	111	12	,	,	PUNCT
easat-2965	111	13	𝑎	𝑎	NOUN
easat-2965	111	14	=	=	SYM
easat-2965	111	15	7	7	NUM
easat-2965	111	16	,	,	PUNCT
easat-2965	111	17	then	then	ADV
easat-2965	111	18	𝐷	𝐷	PROPN
easat-2965	111	19	=	=	SYM
easat-2965	112	1	22(14)2	22(14)2	NUM
easat-2965	112	2	−	−	NOUN
easat-2965	112	3	7.14	7.14	NUM
easat-2965	112	4	=	=	SYM
easat-2965	112	5	686	686	NUM
easat-2965	112	6	which	which	PRON
easat-2965	112	7	is	be	AUX
easat-2965	112	8	,	,	PUNCT
easat-2965	112	9	clearly	clearly	ADV
easat-2965	112	10	,	,	PUNCT
easat-2965	112	11	not	not	PART
easat-2965	112	12	a	a	DET
easat-2965	112	13	perfect	perfect	ADJ
easat-2965	112	14	square	square	NOUN
easat-2965	112	15	.	.	PUNCT
easat-2965	113	1	fundamental	fundamental	ADJ
easat-2965	113	2	solution	solution	NOUN
easat-2965	113	3	of	of	ADP
easat-2965	113	4	𝑥2	𝑥2	NOUN
easat-2965	113	5	−	−	PROPN
easat-2965	113	6	868𝑦2	868𝑦2	NUM
easat-2965	113	7	=	=	SYM
easat-2965	113	8	1	1	NUM
easat-2965	113	9	is	be	AUX
easat-2965	113	10	(	(	PUNCT
easat-2965	113	11	2.2(14)2−7	2.2(14)2−7	NUM
easat-2965	113	12	7	7	NUM
easat-2965	113	13	,	,	PUNCT
easat-2965	113	14	2.14	2.14	NUM
easat-2965	113	15	7	7	NUM
easat-2965	113	16	)	)	PUNCT
easat-2965	113	17	=	=	SYM
easat-2965	113	18	(	(	PUNCT
easat-2965	113	19	195,4	195,4	NOUN
easat-2965	113	20	)	)	PUNCT
easat-2965	113	21	.	.	PUNCT
easat-2965	114	1	all	all	DET
easat-2965	114	2	positive	positive	ADJ
easat-2965	114	3	integer	integer	NOUN
easat-2965	114	4	solutions	solution	NOUN
easat-2965	114	5	of	of	ADP
easat-2965	114	6	𝑥2	𝑥2	NOUN
easat-2965	114	7	−	−	PROPN
easat-2965	114	8	868𝑦2	868𝑦2	NUM
easat-2965	114	9	=	=	SYM
easat-2965	114	10	1	1	NUM
easat-2965	114	11	are	be	AUX
easat-2965	114	12	{	{	PUNCT
easat-2965	114	13	𝑥𝑛	𝑥𝑛	NOUN
easat-2965	114	14	=	=	SYM
easat-2965	114	15	1	1	NUM
easat-2965	114	16	2	2	NUM
easat-2965	114	17	𝐿𝑛(390,−1	𝐿𝑛(390,−1	PROPN
easat-2965	114	18	)	)	PUNCT
easat-2965	114	19	=	=	SYM
easat-2965	114	20	1	1	NUM
easat-2965	114	21	2	2	NUM
easat-2965	114	22	ψ𝑛(195,−1	ψ𝑛(195,−1	NOUN
easat-2965	114	23	)	)	PUNCT
easat-2965	114	24	𝑦𝑛	𝑦𝑛	NOUN
easat-2965	114	25	=	=	NOUN
easat-2965	114	26	4𝐹𝑛(390,−1	4𝐹𝑛(390,−1	NOUN
easat-2965	114	27	)	)	PUNCT
easat-2965	114	28	=	=	SYM
easat-2965	114	29	1	1	NUM
easat-2965	114	30	195	195	NUM
easat-2965	114	31	φ𝑛(195,−1	φ𝑛(195,−1	NOUN
easat-2965	114	32	)	)	PUNCT
easat-2965	114	33	.	.	PUNCT
easat-2965	115	1	we	we	PRON
easat-2965	115	2	will	will	AUX
easat-2965	115	3	then	then	ADV
easat-2965	115	4	be	be	AUX
easat-2965	115	5	required	require	VERB
easat-2965	115	6	to	to	PART
easat-2965	115	7	find	find	VERB
easat-2965	115	8	𝐹𝑛	𝐹𝑛	PROPN
easat-2965	115	9	and	and	CCONJ
easat-2965	115	10	𝐿𝑛	𝐿𝑛	PROPN
easat-2965	115	11	for	for	ADP
easat-2965	115	12	all	all	DET
easat-2965	115	13	𝑛	𝑛	PRON
easat-2965	115	14	>	>	X
easat-2965	116	1	1	1	X
easat-2965	116	2	.	.	PUNCT
easat-2965	117	1	let	let	VERB
easat-2965	117	2	{	{	PUNCT
easat-2965	117	3	𝐿𝑛(390,−1	𝐿𝑛(390,−1	ADJ
easat-2965	117	4	)	)	PUNCT
easat-2965	117	5	=	=	SYM
easat-2965	118	1	390	390	NUM
easat-2965	118	2	l𝑛−1(390,−1	l𝑛−1(390,−1	NOUN
easat-2965	118	3	)	)	PUNCT
easat-2965	118	4	−	−	PROPN
easat-2965	119	1	l𝑛−2(390,−1	l𝑛−2(390,−1	PROPN
easat-2965	119	2	)	)	PUNCT
easat-2965	119	3	,	,	PUNCT
easat-2965	119	4	∀	∀	X
easat-2965	120	1	𝑛	𝑛	VERB
easat-2965	120	2	>	>	X
easat-2965	120	3	2	2	NUM
easat-2965	120	4	,	,	PUNCT
easat-2965	120	5	𝐿0(390,−1	𝐿0(390,−1	PRON
easat-2965	120	6	)	)	PUNCT
easat-2965	120	7	=	=	SYM
easat-2965	120	8	2	2	NUM
easat-2965	120	9	,	,	PUNCT
easat-2965	120	10	𝐿1(390,−1	𝐿1(390,−1	NOUN
easat-2965	120	11	)	)	PUNCT
easat-2965	120	12	=	=	SYM
easat-2965	121	1	390	390	NUM
easat-2965	121	2	𝐹𝑛(390,−1	𝐹𝑛(390,−1	NOUN
easat-2965	121	3	)	)	PUNCT
easat-2965	121	4	=	=	SYM
easat-2965	121	5	390	390	NUM
easat-2965	121	6	f𝑛−1(390,−1	f𝑛−1(390,−1	PROPN
easat-2965	121	7	)	)	PUNCT
easat-2965	121	8	−	−	PROPN
easat-2965	122	1	𝐹𝑛−2(390,−1	𝐹𝑛−2(390,−1	NOUN
easat-2965	122	2	)	)	PUNCT
easat-2965	122	3	∀	∀	X
easat-2965	123	1	𝑛	𝑛	ADP
easat-2965	123	2	>	>	X
easat-2965	123	3	2	2	NUM
easat-2965	123	4	,	,	PUNCT
easat-2965	123	5	𝐹0(390,−1	𝐹0(390,−1	NUM
easat-2965	123	6	)	)	PUNCT
easat-2965	123	7	=	=	SYM
easat-2965	123	8	0	0	NUM
easat-2965	123	9	,	,	PUNCT
easat-2965	123	10	𝐹1(390,−1	𝐹1(390,−1	NUM
easat-2965	123	11	)	)	PUNCT
easat-2965	123	12	=	=	SYM
easat-2965	123	13	1	1	NUM
easat-2965	123	14	so	so	ADV
easat-2965	123	15	,	,	PUNCT
easat-2965	123	16	195	195	NUM
easat-2965	123	17	+	+	NOUN
easat-2965	123	18	4√686	4√686	NUM
easat-2965	123	19	$	$	SYM
easat-2965	123	20	195	195	NUM
easat-2965	123	21	+	+	SYM
easat-2965	123	22	4	4	NUM
easat-2965	123	23	is	be	AUX
easat-2965	123	24	a	a	DET
easat-2965	123	25	unit	unit	NOUN
easat-2965	123	26	of	of	ADP
easat-2965	123	27	ℤ[√686	ℤ[√686	NOUN
easat-2965	123	28	]	]	PUNCT
easat-2965	123	29	.	.	PUNCT
easat-2965	124	1	as	as	ADP
easat-2965	124	2	𝐿2(390,−1	𝐿2(390,−1	ADV
easat-2965	124	3	)	)	PUNCT
easat-2965	124	4	=	=	SYM
easat-2965	124	5	152098	152098	NUM
easat-2965	124	6	and	and	CCONJ
easat-2965	124	7	𝐹2(390,−1	𝐹2(390,−1	ADJ
easat-2965	124	8	)	)	PUNCT
easat-2965	125	1	=	=	SYM
easat-2965	125	2	390	390	NUM
easat-2965	125	3	,	,	PUNCT
easat-2965	125	4	then	then	ADV
easat-2965	125	5	76049	76049	NUM
easat-2965	125	6	+	+	CCONJ
easat-2965	125	7	1560√686	1560√686	NUM
easat-2965	125	8	is	be	AUX
easat-2965	125	9	another	another	DET
easat-2965	125	10	unit	unit	NOUN
easat-2965	125	11	of	of	ADP
easat-2965	125	12	ℤ[√686	ℤ[√686	NOUN
easat-2965	125	13	]	]	PUNCT
easat-2965	125	14	,	,	PUNCT
easat-2965	125	15	and	and	CCONJ
easat-2965	125	16	so	so	ADV
easat-2965	125	17	on	on	ADV
easat-2965	125	18	.	.	PUNCT
easat-2965	126	1	we	we	PRON
easat-2965	126	2	can	can	AUX
easat-2965	126	3	clearly	clearly	ADV
easat-2965	126	4	express	express	VERB
easat-2965	126	5	units	unit	NOUN
easat-2965	126	6	of	of	ADP
easat-2965	126	7	ℤ[√686	ℤ[√686	NOUN
easat-2965	126	8	]	]	PUNCT
easat-2965	126	9	in	in	ADP
easat-2965	126	10	terms	term	NOUN
easat-2965	126	11	of	of	ADP
easat-2965	126	12	generalized	generalized	ADJ
easat-2965	126	13	fibonacci	fibonacci	NOUN
easat-2965	126	14	,	,	PUNCT
easat-2965	126	15	generalized	generalized	ADJ
easat-2965	126	16	lucas	lucas	NOUN
easat-2965	126	17	,	,	PUNCT
easat-2965	126	18	generalized	generalized	ADJ
easat-2965	126	19	pell	pell	NOUN
easat-2965	126	20	,	,	PUNCT
easat-2965	126	21	and	and	CCONJ
easat-2965	126	22	generalized	generalized	ADJ
easat-2965	126	23	pelllucas	pelllucas	NOUN
easat-2965	126	24	numbers	number	NOUN
easat-2965	126	25	.	.	PUNCT
easat-2965	127	1	4	4	X
easat-2965	127	2	.	.	X
easat-2965	127	3	conclusion	conclusion	NOUN
easat-2965	127	4	the	the	DET
easat-2965	127	5	diophantine	diophantine	NOUN
easat-2965	127	6	equation	equation	NOUN
easat-2965	127	7	𝑥2	𝑥2	NOUN
easat-2965	127	8	−	−	PROPN
easat-2965	127	9	(	(	PUNCT
easat-2965	127	10	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	127	11	±	±	NOUN
easat-2965	127	12	𝑎𝑝)𝑦2	𝑎𝑝)𝑦2	NOUN
easat-2965	127	13	=	=	PUNCT
easat-2965	127	14	𝑘𝑡	𝑘𝑡	PROPN
easat-2965	127	15	has	have	AUX
easat-2965	127	16	been	be	AUX
easat-2965	127	17	solved	solve	VERB
easat-2965	127	18	,	,	PUNCT
easat-2965	127	19	and	and	CCONJ
easat-2965	127	20	its	its	PRON
easat-2965	127	21	positive	positive	ADJ
easat-2965	127	22	integer	integer	NOUN
easat-2965	127	23	solutions	solution	NOUN
easat-2965	127	24	have	have	AUX
easat-2965	127	25	been	be	AUX
easat-2965	127	26	stated	state	VERB
easat-2965	127	27	in	in	ADP
easat-2965	127	28	terms	term	NOUN
easat-2965	127	29	of	of	ADP
easat-2965	127	30	generalized	generalized	ADJ
easat-2965	127	31	fibonacci	fibonacci	NOUN
easat-2965	127	32	,	,	PUNCT
easat-2965	127	33	generalized	generalized	ADJ
easat-2965	127	34	lucas	lucas	NOUN
easat-2965	127	35	,	,	PUNCT
easat-2965	127	36	generalized	generalized	ADJ
easat-2965	127	37	pell	pell	NOUN
easat-2965	127	38	,	,	PUNCT
easat-2965	127	39	and	and	CCONJ
easat-2965	127	40	generalized	generalize	VERB
easat-2965	127	41	pell	pell	NOUN
easat-2965	127	42	-	-	PUNCT
easat-2965	127	43	lucas	lucas	NOUN
easat-2965	127	44	sequences	sequence	NOUN
easat-2965	127	45	.	.	PUNCT
easat-2965	128	1	we	we	PRON
easat-2965	128	2	have	have	AUX
easat-2965	128	3	discovered	discover	VERB
easat-2965	128	4	units	unit	NOUN
easat-2965	128	5	for	for	ADP
easat-2965	128	6	ℤ[√𝐷	ℤ[√𝐷	NOUN
easat-2965	128	7	]	]	PUNCT
easat-2965	128	8	in	in	ADP
easat-2965	128	9	terms	term	NOUN
easat-2965	128	10	of	of	ADP
easat-2965	128	11	the	the	DET
easat-2965	128	12	above	above	ADJ
easat-2965	128	13	sequences	sequence	NOUN
easat-2965	128	14	of	of	ADP
easat-2965	128	15	numbers	number	NOUN
easat-2965	128	16	.	.	PUNCT
easat-2965	129	1	4414	4414	NUM
easat-2965	129	2	edelweiss	edelweiss	PROPN
easat-2965	129	3	applied	apply	VERB
easat-2965	129	4	science	science	NOUN
easat-2965	129	5	and	and	CCONJ
easat-2965	129	6	technology	technology	NOUN
easat-2965	129	7	issn	issn	PROPN
easat-2965	129	8	:	:	PUNCT
easat-2965	129	9	2576	2576	NUM
easat-2965	129	10	-	-	SYM
easat-2965	129	11	8484	8484	NUM
easat-2965	129	12	vol	vol	NOUN
easat-2965	129	13	.	.	PROPN
easat-2965	129	14	8	8	NUM
easat-2965	129	15	,	,	PUNCT
easat-2965	129	16	no	no	INTJ
easat-2965	129	17	.	.	NOUN
easat-2965	130	1	6	6	NUM
easat-2965	130	2	:	:	SYM
easat-2965	130	3	4408	4408	NUM
easat-2965	130	4	-	-	SYM
easat-2965	130	5	4414	4414	NUM
easat-2965	130	6	,	,	PUNCT
easat-2965	130	7	2024	2024	NUM
easat-2965	130	8	doi	doi	NOUN
easat-2965	130	9	:	:	PUNCT
easat-2965	130	10	10.55214/25768484.v8i6.2965	10.55214/25768484.v8i6.2965	NUM
easat-2965	130	11	©	©	PROPN
easat-2965	130	12	2024	2024	NUM
easat-2965	130	13	by	by	ADP
easat-2965	130	14	the	the	DET
easat-2965	130	15	authors	author	NOUN
easat-2965	130	16	;	;	PUNCT
easat-2965	131	1	licensee	licensee	PROPN
easat-2965	131	2	learning	learn	VERB
easat-2965	131	3	gate	gate	NOUN
easat-2965	131	4	copyright	copyright	NOUN
easat-2965	131	5	:	:	PUNCT
easat-2965	131	6	©	©	PROPN
easat-2965	131	7	2024	2024	NUM
easat-2965	131	8	by	by	ADP
easat-2965	131	9	the	the	DET
easat-2965	131	10	authors	author	NOUN
easat-2965	131	11	.	.	PUNCT
easat-2965	132	1	this	this	DET
easat-2965	132	2	article	article	NOUN
easat-2965	132	3	is	be	AUX
easat-2965	132	4	an	an	DET
easat-2965	132	5	open	open	ADJ
easat-2965	132	6	access	access	NOUN
easat-2965	132	7	article	article	NOUN
easat-2965	132	8	distributed	distribute	VERB
easat-2965	132	9	under	under	ADP
easat-2965	132	10	the	the	DET
easat-2965	132	11	terms	term	NOUN
easat-2965	132	12	and	and	CCONJ
easat-2965	132	13	conditions	condition	NOUN
easat-2965	132	14	of	of	ADP
easat-2965	132	15	the	the	DET
easat-2965	132	16	creative	creative	ADJ
easat-2965	132	17	commons	common	NOUN
easat-2965	132	18	attribution	attribution	NOUN
easat-2965	132	19	(	(	PUNCT
easat-2965	132	20	cc	cc	NOUN
easat-2965	132	21	by	by	ADP
easat-2965	132	22	)	)	PUNCT
easat-2965	132	23	license	license	NOUN
easat-2965	132	24	(	(	PUNCT
easat-2965	132	25	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-2965	132	26	)	)	PUNCT
easat-2965	132	27	.	.	PUNCT
easat-2965	133	1	references	reference	NOUN
easat-2965	134	1	[	[	X
easat-2965	134	2	1	1	X
easat-2965	134	3	]	]	X
easat-2965	134	4	j.w	j.w	PROPN
easat-2965	134	5	.	.	PUNCT
easat-2965	134	6	leveque	leveque	ADJ
easat-2965	134	7	,	,	PUNCT
easat-2965	134	8	topics	topic	NOUN
easat-2965	134	9	in	in	ADP
easat-2965	134	10	number	number	NOUN
easat-2965	134	11	theory	theory	NOUN
easat-2965	134	12	1	1	NUM
easat-2965	134	13	,	,	PUNCT
easat-2965	134	14	2	2	NUM
easat-2965	134	15	,	,	PUNCT
easat-2965	134	16	dover	dover	PROPN
easat-2965	134	17	publications	publication	NOUN
easat-2965	134	18	,	,	PUNCT
easat-2965	134	19	2002	2002	NUM
easat-2965	134	20	.	.	PUNCT
easat-2965	135	1	[	[	X
easat-2965	135	2	2	2	X
easat-2965	135	3	]	]	PUNCT
easat-2965	135	4	t.	t.	PROPN
easat-2965	135	5	nagell	nagell	PROPN
easat-2965	135	6	,	,	PUNCT
easat-2965	135	7	introduction	introduction	NOUN
easat-2965	135	8	to	to	ADP
easat-2965	135	9	number	number	NOUN
easat-2965	135	10	theory	theory	NOUN
easat-2965	135	11	,	,	PUNCT
easat-2965	135	12	chelsea	chelsea	PROPN
easat-2965	135	13	publishing	publishing	PROPN
easat-2965	135	14	company	company	NOUN
easat-2965	135	15	,	,	PUNCT
easat-2965	135	16	new	new	PROPN
easat-2965	135	17	york	york	PROPN
easat-2965	135	18	,	,	PUNCT
easat-2965	135	19	1981	1981	NUM
easat-2965	135	20	.	.	PUNCT
easat-2965	136	1	[	[	X
easat-2965	136	2	3	3	X
easat-2965	136	3	]	]	X
easat-2965	136	4	d.	d.	PROPN
easat-2965	136	5	redmond	redmond	PROPN
easat-2965	136	6	,	,	PUNCT
easat-2965	136	7	number	number	NOUN
easat-2965	136	8	theory	theory	NOUN
easat-2965	136	9	:	:	PUNCT
easat-2965	136	10	an	an	DET
easat-2965	136	11	introduction	introduction	NOUN
easat-2965	136	12	to	to	ADP
easat-2965	136	13	pure	pure	ADJ
easat-2965	136	14	and	and	CCONJ
easat-2965	136	15	applied	applied	ADJ
easat-2965	136	16	mathematics	mathematic	NOUN
easat-2965	136	17	201	201	NUM
easat-2965	136	18	,	,	PUNCT
easat-2965	136	19	crc	crc	NOUN
easat-2965	136	20	press	press	NOUN
easat-2965	136	21	,	,	PUNCT
easat-2965	136	22	1996	1996	NUM
easat-2965	136	23	.	.	PUNCT
easat-2965	137	1	[	[	X
easat-2965	137	2	4	4	NUM
easat-2965	137	3	]	]	X
easat-2965	137	4	d.	d.	PROPN
easat-2965	137	5	redmond	redmond	PROPN
easat-2965	137	6	,	,	PUNCT
easat-2965	137	7	number	number	NOUN
easat-2965	137	8	theory	theory	NOUN
easat-2965	137	9	:	:	PUNCT
easat-2965	137	10	an	an	DET
easat-2965	137	11	introduction	introduction	NOUN
easat-2965	137	12	,	,	PUNCT
easat-2965	137	13	markel	markel	PROPN
easat-2965	137	14	dekker	dekker	PROPN
easat-2965	137	15	.	.	PUNCT
easat-2965	138	1	inc	inc	PROPN
easat-2965	138	2	.	.	PROPN
easat-2965	138	3	,	,	PUNCT
easat-2965	138	4	new	new	PROPN
easat-2965	138	5	york,1996	york,1996	NOUN
easat-2965	138	6	.	.	PUNCT
easat-2965	139	1	[	[	X
easat-2965	139	2	5	5	NUM
easat-2965	139	3	]	]	PUNCT
easat-2965	139	4	a.	a.	NOUN
easat-2965	139	5	chandoul	chandoul	PROPN
easat-2965	139	6	,	,	PUNCT
easat-2965	139	7	“	"	PUNCT
easat-2965	139	8	the	the	DET
easat-2965	139	9	pell	pell	NOUN
easat-2965	139	10	equation	equation	NOUN
easat-2965	139	11	,	,	PUNCT
easat-2965	139	12	”	"	PUNCT
easat-2965	139	13	research	research	NOUN
easat-2965	139	14	journal	journal	NOUN
easat-2965	139	15	of	of	ADP
easat-2965	139	16	pure	pure	ADJ
easat-2965	139	17	algebra	algebra	NOUN
easat-2965	139	18	,	,	PUNCT
easat-2965	139	19	vol	vol	NOUN
easat-2965	139	20	.	.	PROPN
easat-2965	139	21	1	1	NUM
easat-2965	139	22	,	,	PUNCT
easat-2965	139	23	no	no	INTJ
easat-2965	139	24	.	.	NOUN
easat-2965	139	25	2	2	NUM
easat-2965	139	26	,	,	PUNCT
easat-2965	139	27	2011	2011	NUM
easat-2965	139	28	,	,	PUNCT
easat-2965	139	29	pp	pp	ADJ
easat-2965	139	30	.	.	PUNCT
easat-2965	140	1	11	11	NUM
easat-2965	140	2	-	-	SYM
easat-2965	140	3	15	15	NUM
easat-2965	140	4	.	.	PUNCT
easat-2965	141	1	[	[	X
easat-2965	141	2	6	6	NUM
easat-2965	141	3	]	]	PUNCT
easat-2965	141	4	sankari	sankari	PROPN
easat-2965	141	5	,	,	PUNCT
easat-2965	141	6	h.	h.	PROPN
easat-2965	141	7	,	,	PUNCT
easat-2965	141	8	&	&	CCONJ
easat-2965	141	9	abdo	abdo	PROPN
easat-2965	141	10	,	,	PUNCT
easat-2965	141	11	a	a	PRON
easat-2965	141	12	,	,	PUNCT
easat-2965	141	13	the	the	DET
easat-2965	141	14	polynomial	polynomial	ADJ
easat-2965	141	15	solutions	solution	NOUN
easat-2965	141	16	of	of	ADP
easat-2965	141	17	quadratic	quadratic	ADJ
easat-2965	141	18	diophantine	diophantine	NOUN
easat-2965	141	19	equation	equation	NOUN
easat-2965	141	20	𝑥2	𝑥2	NOUN
easat-2965	141	21	−	−	PROPN
easat-2965	141	22	𝑝(𝑡)𝑦2	𝑝(𝑡)𝑦2	NOUN
easat-2965	141	23	+	+	CCONJ
easat-2965	141	24	2𝑘(𝑡)𝑥	2𝑘(𝑡)𝑥	NUM
easat-2965	141	25	+	+	NUM
easat-2965	141	26	2𝑝(𝑡)𝑙(𝑡)𝑦	2𝑝(𝑡)𝑙(𝑡)𝑦	NUM
easat-2965	141	27	=	=	SYM
easat-2965	141	28	0	0	NUM
easat-2965	141	29	,	,	PUNCT
easat-2965	141	30	journal	journal	NOUN
easat-2965	141	31	of	of	ADP
easat-2965	141	32	mathematics	mathematic	NOUN
easat-2965	141	33	,	,	PUNCT
easat-2965	141	34	2021	2021	NUM
easat-2965	141	35	,	,	PUNCT
easat-2965	141	36	1	1	NUM
easat-2965	141	37	-	-	SYM
easat-2965	141	38	6	6	NUM
easat-2965	141	39	.	.	PUNCT
easat-2965	142	1	[	[	X
easat-2965	142	2	7	7	NUM
easat-2965	142	3	]	]	X
easat-2965	142	4	h	h	NOUN
easat-2965	142	5	utz	utz	NOUN
easat-2965	142	6	,	,	PUNCT
easat-2965	142	7	w.	w.	PROPN
easat-2965	142	8	r	r	PROPN
easat-2965	142	9	,	,	PUNCT
easat-2965	142	10	positive	positive	ADJ
easat-2965	142	11	solutions	solution	NOUN
easat-2965	142	12	of	of	ADP
easat-2965	142	13	the	the	DET
easat-2965	142	14	diophantine	diophantine	NOUN
easat-2965	142	15	equation	equation	NOUN
easat-2965	142	16	,	,	PUNCT
easat-2965	142	17	international	international	ADJ
easat-2965	142	18	journal	journal	NOUN
easat-2965	142	19	of	of	ADP
easat-2965	142	20	mathematics	mathematics	PROPN
easat-2965	142	21	and	and	CCONJ
easat-2965	142	22	mathematical	mathematical	ADJ
easat-2965	142	23	sciences	science	NOUN
easat-2965	142	24	,	,	PUNCT
easat-2965	142	25	5	5	NUM
easat-2965	142	26	,	,	PUNCT
easat-2965	142	27	311	311	NUM
easat-2965	142	28	-	-	SYM
easat-2965	142	29	314	314	NUM
easat-2965	142	30	[	[	SYM
easat-2965	142	31	8	8	NUM
easat-2965	142	32	]	]	PUNCT
easat-2965	142	33	akbik	akbik	NOUN
easat-2965	142	34	,	,	PUNCT
easat-2965	142	35	s	s	AUX
easat-2965	142	36	,	,	PUNCT
easat-2965	142	37	on	on	ADP
easat-2965	142	38	a	a	DET
easat-2965	142	39	class	class	NOUN
easat-2965	142	40	of	of	ADP
easat-2965	142	41	diophantine	diophantine	NOUN
easat-2965	142	42	equations	equation	NOUN
easat-2965	142	43	.	.	PUNCT
easat-2965	143	1	international	international	ADJ
easat-2965	143	2	journal	journal	PROPN
easat-2965	143	3	of	of	ADP
easat-2965	143	4	mathematics	mathematics	PROPN
easat-2965	143	5	and	and	CCONJ
easat-2965	143	6	mathematical	mathematical	ADJ
easat-2965	143	7	sciences	science	NOUN
easat-2965	143	8	,	,	PUNCT
easat-2965	143	9	29	29	NUM
easat-2965	143	10	,	,	PUNCT
easat-2965	143	11	545553	545553	NUM
easat-2965	143	12	.	.	PUNCT
easat-2965	144	1	[	[	X
easat-2965	144	2	9	9	NUM
easat-2965	144	3	]	]	X
easat-2965	144	4	bala	bala	PROPN
easat-2965	144	5	,	,	PUNCT
easat-2965	144	6	r.	r.	PROPN
easat-2965	144	7	,	,	PUNCT
easat-2965	144	8	&	&	CCONJ
easat-2965	144	9	mishra	mishra	PROPN
easat-2965	144	10	,	,	PUNCT
easat-2965	144	11	v.	v.	ADP
easat-2965	144	12	solutions	solution	NOUN
easat-2965	144	13	of	of	ADP
easat-2965	144	14	equations	equation	NOUN
easat-2965	144	15	𝑥2	𝑥2	NOUN
easat-2965	144	16	−	−	PROPN
easat-2965	144	17	(	(	PUNCT
easat-2965	144	18	𝑝2𝑞2	𝑝2𝑞2	NOUN
easat-2965	144	19	±	±	NUM
easat-2965	144	20	3𝑝)𝑦2	3𝑝)𝑦2	NUM
easat-2965	144	21	=	=	SYM
easat-2965	144	22	𝑘𝑡.	𝑘𝑡.	PROPN
easat-2965	144	23	examples	example	NOUN
easat-2965	144	24	and	and	CCONJ
easat-2965	144	25	counterexamples	counterexample	NOUN
easat-2965	144	26	,	,	PUNCT
easat-2965	144	27	2	2	NUM
easat-2965	144	28	.	.	PUNCT
easat-2965	145	1	[	[	X
easat-2965	145	2	10	10	NUM
easat-2965	145	3	]	]	X
easat-2965	145	4	guney	guney	NOUN
easat-2965	145	5	,	,	PUNCT
easat-2965	145	6	m.	m.	NOUN
easat-2965	145	7	solutions	solution	NOUN
easat-2965	145	8	of	of	ADP
easat-2965	145	9	the	the	DET
easat-2965	145	10	pell	pell	NOUN
easat-2965	145	11	equations	equation	NOUN
easat-2965	145	12	𝑥2	𝑥2	NOUN
easat-2965	145	13	−	−	PROPN
easat-2965	145	14	(	(	PUNCT
easat-2965	145	15	𝑎2𝑏2	𝑎2𝑏2	PROPN
easat-2965	145	16	+	+	NUM
easat-2965	145	17	2𝑏)𝑦2	2𝑏)𝑦2	NUM
easat-2965	145	18	=	=	SYM
easat-2965	145	19	𝑁	𝑁	NOUN
easat-2965	145	20	when	when	SCONJ
easat-2965	145	21	𝑁	𝑁	PROPN
easat-2965	145	22	∈	∈	PROPN
easat-2965	145	23	{	{	PUNCT
easat-2965	145	24	±1,±4	±1,±4	PROPN
easat-2965	145	25	}	}	PUNCT
easat-2965	145	26	.	.	PUNCT
easat-2965	146	1	mathematica	mathematica	PROPN
easat-2965	146	2	aeterna	aeterna	PROPN
easat-2965	146	3	,	,	PUNCT
easat-2965	146	4	2(7	2(7	NUM
easat-2965	146	5	)	)	PUNCT
easat-2965	146	6	,	,	PUNCT
easat-2965	146	7	629	629	NUM
easat-2965	146	8	-	-	SYM
easat-2965	146	9	638	638	NUM
easat-2965	146	10	.	.	PUNCT
easat-2965	147	1	[	[	X
easat-2965	147	2	11	11	NUM
easat-2965	147	3	]	]	X
easat-2965	147	4	edson	edson	PROPN
easat-2965	147	5	,	,	PUNCT
easat-2965	147	6	m.	m.	NOUN
easat-2965	147	7	,	,	PUNCT
easat-2965	147	8	&	&	CCONJ
easat-2965	147	9	yayenie	yayenie	PROPN
easat-2965	147	10	,	,	PUNCT
easat-2965	147	11	o.	o.	PROPN
easat-2965	147	12	(	(	PUNCT
easat-2965	147	13	2009	2009	NUM
easat-2965	147	14	)	)	PUNCT
easat-2965	147	15	.	.	PUNCT
easat-2965	148	1	a	a	DET
easat-2965	148	2	new	new	ADJ
easat-2965	148	3	generalization	generalization	NOUN
easat-2965	148	4	of	of	ADP
easat-2965	148	5	fibonacci	fibonacci	PROPN
easat-2965	148	6	sequence	sequence	NOUN
easat-2965	148	7	&	&	CCONJ
easat-2965	148	8	extended	extend	VERB
easat-2965	148	9	binet	binet	NOUN
easat-2965	148	10	's	's	PART
easat-2965	148	11	formula	formula	NOUN
easat-2965	148	12	.	.	PUNCT
easat-2965	149	1	[	[	X
easat-2965	149	2	12	12	NUM
easat-2965	149	3	]	]	PUNCT
easat-2965	149	4	s.	s.	PROPN
easat-2965	149	5	falcon	falcon	PROPN
easat-2965	149	6	,	,	PUNCT
easat-2965	149	7	a.	a.	NOUN
easat-2965	149	8	plaza	plaza	PROPN
easat-2965	149	9	,	,	PUNCT
easat-2965	149	10	the	the	DET
easat-2965	149	11	𝑘-fibonacci	𝑘-fibonacci	NOUN
easat-2965	149	12	sequence	sequence	NOUN
easat-2965	149	13	and	and	CCONJ
easat-2965	149	14	the	the	DET
easat-2965	149	15	pascal	pascal	ADJ
easat-2965	149	16	2	2	NUM
easat-2965	149	17	-	-	PUNCT
easat-2965	149	18	triangle	triangle	NOUN
easat-2965	149	19	,	,	PUNCT
easat-2965	149	20	chaos	chaos	NOUN
easat-2965	149	21	,	,	PUNCT
easat-2965	149	22	solitons	soliton	NOUN
easat-2965	149	23	and	and	CCONJ
easat-2965	149	24	fractals	fractal	NOUN
easat-2965	149	25	33(2007	33(2007	NUM
easat-2965	149	26	)	)	PUNCT
easat-2965	149	27	,	,	PUNCT
easat-2965	149	28	38	38	NUM
easat-2965	149	29	-	-	SYM
easat-2965	149	30	49	49	NUM
easat-2965	149	31	.	.	PUNCT
easat-2965	150	1	[	[	X
easat-2965	150	2	13	13	NUM
easat-2965	150	3	]	]	PUNCT
easat-2965	150	4	a.	a.	NOUN
easat-2965	150	5	f.	f.	PROPN
easat-2965	150	6	horadam	horadam	PROPN
easat-2965	150	7	,	,	PUNCT
easat-2965	150	8	a	a	DET
easat-2965	150	9	generalized	generalized	ADJ
easat-2965	150	10	fibonacci	fibonacci	NOUN
easat-2965	150	11	sequence	sequence	NOUN
easat-2965	150	12	,	,	PUNCT
easat-2965	150	13	amer	amer	PROPN
easat-2965	150	14	.	.	PROPN
easat-2965	150	15	math	math	PROPN
easat-2965	150	16	.	.	PUNCT
easat-2965	151	1	monthly	monthly	ADJ
easat-2965	151	2	68(1961	68(1961	NUM
easat-2965	151	3	)	)	PUNCT
easat-2965	151	4	,	,	PUNCT
easat-2965	151	5	455	455	NUM
easat-2965	151	6	-	-	SYM
easat-2965	151	7	459	459	NUM
easat-2965	152	1	[	[	X
easat-2965	152	2	14	14	NUM
easat-2965	152	3	]	]	X
easat-2965	152	4	d.	d.	PROPN
easat-2965	152	5	v.	v.	PROPN
easat-2965	152	6	jaiswal	jaiswal	PROPN
easat-2965	152	7	,	,	PUNCT
easat-2965	152	8	on	on	ADP
easat-2965	152	9	a	a	DET
easat-2965	152	10	generalized	generalized	ADJ
easat-2965	152	11	fibonacci	fibonacci	NOUN
easat-2965	152	12	sequence	sequence	NOUN
easat-2965	152	13	,	,	PUNCT
easat-2965	152	14	labdevj	labdevj	PROPN
easat-2965	152	15	.	.	PUNCT
easat-2965	153	1	sci	sci	PROPN
easat-2965	153	2	.	.	PUNCT
easat-2965	153	3	tech	tech	PROPN
easat-2965	153	4	.	.	PUNCT
easat-2965	154	1	part	part	NOUN
easat-2965	154	2	a	a	DET
easat-2965	154	3	7(1969	7(1969	NOUN
easat-2965	154	4	)	)	PUNCT
easat-2965	154	5	,	,	PUNCT
easat-2965	154	6	67	67	NUM
easat-2965	154	7	-	-	SYM
easat-2965	154	8	71	71	NUM
easat-2965	154	9	.	.	PUNCT
easat-2965	155	1	[	[	X
easat-2965	155	2	15	15	NUM
easat-2965	155	3	]	]	X
easat-2965	155	4	a.	a.	NOUN
easat-2965	155	5	t.	t.	PROPN
easat-2965	155	6	krassimir	krassimir	PROPN
easat-2965	155	7	,	,	PUNCT
easat-2965	155	8	a.	a.	PROPN
easat-2965	155	9	c.	c.	PROPN
easat-2965	155	10	liliya	liliya	PROPN
easat-2965	155	11	,	,	PUNCT
easat-2965	155	12	s.	s.	PROPN
easat-2965	155	13	d.	d.	PROPN
easat-2965	155	14	dimitar	dimitar	PROPN
easat-2965	155	15	,	,	PUNCT
easat-2965	155	16	a	a	DET
easat-2965	155	17	new	new	ADJ
easat-2965	155	18	perspective	perspective	NOUN
easat-2965	155	19	to	to	ADP
easat-2965	155	20	the	the	DET
easat-2965	155	21	generalization	generalization	NOUN
easat-2965	155	22	of	of	ADP
easat-2965	155	23	the	the	DET
easat-2965	155	24	fibonacci	fibonacci	NOUN
easat-2965	155	25	sequence	sequence	NOUN
easat-2965	155	26	,	,	PUNCT
easat-2965	155	27	fibonacci	fibonacci	NOUN
easat-2965	155	28	quart	quart	NOUN
easat-2965	155	29	.	.	PUNCT
easat-2965	156	1	23(1985	23(1985	NUM
easat-2965	156	2	)	)	PUNCT
easat-2965	156	3	,	,	PUNCT
easat-2965	156	4	no	no	INTJ
easat-2965	156	5	.	.	NOUN
easat-2965	156	6	1	1	NUM
easat-2965	156	7	,	,	PUNCT
easat-2965	156	8	21	21	NUM
easat-2965	156	9	-	-	SYM
easat-2965	156	10	28	28	NUM
easat-2965	157	1	[	[	X
easat-2965	157	2	16	16	NUM
easat-2965	157	3	]	]	PUNCT
easat-2965	157	4	l.	l.	PROPN
easat-2965	157	5	j.	j.	PROPN
easat-2965	157	6	zai	zai	PROPN
easat-2965	157	7	,	,	PUNCT
easat-2965	157	8	l.j	l.j	PROPN
easat-2965	157	9	.	.	PROPN
easat-2965	157	10	sheng	sheng	PROPN
easat-2965	157	11	,	,	PUNCT
easat-2965	157	12	some	some	DET
easat-2965	157	13	properties	property	NOUN
easat-2965	157	14	of	of	ADP
easat-2965	157	15	the	the	DET
easat-2965	157	16	generalization	generalization	NOUN
easat-2965	157	17	of	of	ADP
easat-2965	157	18	the	the	DET
easat-2965	157	19	fibonacci	fibonacci	NOUN
easat-2965	157	20	sequence	sequence	NOUN
easat-2965	157	21	,	,	PUNCT
easat-2965	157	22	fibonacci	fibonacci	NOUN
easat-2965	157	23	quart	quart	NOUN
easat-2965	157	24	.	.	PUNCT
easat-2965	158	1	25(1987	25(1987	NUM
easat-2965	158	2	)	)	PUNCT
easat-2965	158	3	,	,	PUNCT
easat-2965	158	4	no	no	INTJ
easat-2965	158	5	.	.	NOUN
easat-2965	158	6	2	2	NUM
easat-2965	158	7	,	,	PUNCT
easat-2965	158	8	111	111	NUM
easat-2965	158	9	-	-	SYM
easat-2965	158	10	117	117	NUM
easat-2965	158	11	.	.	PUNCT
easat-2965	159	1	[	[	X
easat-2965	159	2	17	17	NUM
easat-2965	159	3	]	]	X
easat-2965	159	4	kalman	kalman	PROPN
easat-2965	159	5	,	,	PUNCT
easat-2965	159	6	d.	d.	PROPN
easat-2965	159	7	,	,	PUNCT
easat-2965	159	8	mena	mena	PROPN
easat-2965	159	9	r.	r.	PROPN
easat-2965	159	10	,	,	PUNCT
easat-2965	159	11	the	the	DET
easat-2965	159	12	fibonacci	fibonacci	NOUN
easat-2965	159	13	numbers	number	NOUN
easat-2965	159	14	exposed	expose	VERB
easat-2965	159	15	,	,	PUNCT
easat-2965	159	16	mathematics	mathematics	PROPN
easat-2965	159	17	magazine	magazine	NOUN
easat-2965	159	18	76(2003	76(2003	NOUN
easat-2965	159	19	)	)	PUNCT
easat-2965	159	20	,	,	PUNCT
easat-2965	159	21	167	167	NUM
easat-2965	159	22	-	-	SYM
easat-2965	159	23	181	181	NUM
easat-2965	159	24	.	.	PUNCT
easat-2965	160	1	[	[	X
easat-2965	160	2	18	18	NUM
easat-2965	160	3	]	]	X
easat-2965	160	4	mcdaniel	mcdaniel	PROPN
easat-2965	160	5	,	,	PUNCT
easat-2965	160	6	w.l	w.l	PROPN
easat-2965	160	7	.	.	PROPN
easat-2965	160	8	,	,	PUNCT
easat-2965	160	9	diophantine	diophantine	VERB
easat-2965	160	10	representation	representation	NOUN
easat-2965	160	11	of	of	ADP
easat-2965	160	12	lucas	lucas	PROPN
easat-2965	160	13	sequences	sequence	NOUN
easat-2965	160	14	,	,	PUNCT
easat-2965	160	15	the	the	DET
easat-2965	160	16	fibonacci	fibonacci	NOUN
easat-2965	160	17	quarterly	quarterly	ADV
easat-2965	160	18	33	33	NUM
easat-2965	160	19	(	(	PUNCT
easat-2965	160	20	1995	1995	NUM
easat-2965	160	21	)	)	PUNCT
easat-2965	160	22	,	,	PUNCT
easat-2965	160	23	58	58	NUM
easat-2965	160	24	-	-	SYM
easat-2965	160	25	63	63	NUM
easat-2965	160	26	.	.	PUNCT
easat-2965	161	1	[	[	X
easat-2965	161	2	19	19	NUM
easat-2965	161	3	]	]	SYM
easat-2965	161	4	melham	melham	PROPN
easat-2965	161	5	,	,	PUNCT
easat-2965	161	6	r.	r.	PROPN
easat-2965	161	7	,	,	PUNCT
easat-2965	161	8	conics	conic	NOUN
easat-2965	161	9	which	which	PRON
easat-2965	161	10	characterize	characterize	VERB
easat-2965	161	11	certain	certain	ADJ
easat-2965	161	12	lucas	lucas	PROPN
easat-2965	161	13	sequences	sequence	NOUN
easat-2965	161	14	,	,	PUNCT
easat-2965	161	15	the	the	DET
easat-2965	161	16	fibonacci	fibonacci	NOUN
easat-2965	161	17	quarterly	quarterly	ADV
easat-2965	161	18	35	35	NUM
easat-2965	161	19	(	(	PUNCT
easat-2965	161	20	1997	1997	NUM
easat-2965	161	21	)	)	PUNCT
easat-2965	161	22	,	,	PUNCT
easat-2965	161	23	248	248	NUM
easat-2965	161	24	-	-	SYM
easat-2965	161	25	251	251	NUM
easat-2965	161	26	.	.	PUNCT
easat-2965	162	1	[	[	X
easat-2965	162	2	20	20	NUM
easat-2965	162	3	]	]	PUNCT
easat-2965	162	4	ribenboim	ribenboim	NOUN
easat-2965	162	5	,	,	PUNCT
easat-2965	162	6	p.	p.	NOUN
easat-2965	162	7	,	,	PUNCT
easat-2965	162	8	my	my	PRON
easat-2965	162	9	numbers	number	NOUN
easat-2965	162	10	,	,	PUNCT
easat-2965	162	11	my	my	PRON
easat-2965	162	12	friends	friend	NOUN
easat-2965	162	13	,	,	PUNCT
easat-2965	162	14	springer	springer	NOUN
easat-2965	162	15	-	-	PUNCT
easat-2965	162	16	verlag	verlag	PROPN
easat-2965	162	17	new	new	PROPN
easat-2965	162	18	york	york	PROPN
easat-2965	162	19	,	,	PUNCT
easat-2965	162	20	inc	inc	PROPN
easat-2965	162	21	.	.	PROPN
easat-2965	162	22	,	,	PUNCT
easat-2965	162	23	2000	2000	NUM
easat-2965	162	24	.	.	PUNCT
easat-2965	163	1	[	[	X
easat-2965	163	2	21	21	NUM
easat-2965	163	3	]	]	PUNCT
easat-2965	163	4	robinowitz	robinowitz	NOUN
easat-2965	163	5	,	,	PUNCT
easat-2965	163	6	s.	s.	PROPN
easat-2965	163	7	,	,	PUNCT
easat-2965	163	8	algorithmic	algorithmic	ADJ
easat-2965	163	9	manipulation	manipulation	NOUN
easat-2965	163	10	of	of	ADP
easat-2965	163	11	fibonacci	fibonacci	NOUN
easat-2965	163	12	identities	identity	NOUN
easat-2965	163	13	,	,	PUNCT
easat-2965	163	14	in	in	ADP
easat-2965	163	15	:	:	PUNCT
easat-2965	163	16	application	application	NOUN
easat-2965	163	17	of	of	ADP
easat-2965	163	18	fibonacci	fibonacci	NOUN
easat-2965	163	19	numbers	number	NOUN
easat-2965	163	20	,	,	PUNCT
easat-2965	163	21	vol	vol	NOUN
easat-2965	163	22	.	.	PROPN
easat-2965	163	23	6	6	NUM
easat-2965	163	24	,	,	PUNCT
easat-2965	163	25	kluwer	kluwer	NOUN
easat-2965	163	26	academic	academic	ADJ
easat-2965	163	27	pub	pub	NOUN
easat-2965	163	28	.	.	PUNCT
easat-2965	163	29	,	,	PUNCT
easat-2965	163	30	dordrect	dordrect	PROPN
easat-2965	163	31	,	,	PUNCT
easat-2965	163	32	the	the	DET
easat-2965	163	33	netherlands	netherlands	PROPN
easat-2965	163	34	,	,	PUNCT
easat-2965	163	35	1996	1996	NUM
easat-2965	163	36	,	,	PUNCT
easat-2965	163	37	pp	pp	ADV
easat-2965	163	38	.	.	PUNCT
easat-2965	164	1	389	389	NUM
easat-2965	164	2	-	-	SYM
easat-2965	164	3	408	408	NUM
easat-2965	164	4	.	.	PUNCT
easat-2965	165	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN