id	sid	tid	token	lemma	pos
ejpam-3952	1	1	european	european	PROPN
ejpam-3952	1	2	journal	journal	PROPN
ejpam-3952	1	3	of	of	ADP
ejpam-3952	1	4	pure	pure	ADJ
ejpam-3952	1	5	and	and	CCONJ
ejpam-3952	1	6	applied	apply	VERB
ejpam-3952	1	7	mathematics	mathematic	NOUN
ejpam-3952	1	8	vol	vol	NOUN
ejpam-3952	1	9	.	.	PUNCT
ejpam-3952	2	1	14	14	NUM
ejpam-3952	2	2	,	,	PUNCT
ejpam-3952	2	3	no	no	INTJ
ejpam-3952	2	4	.	.	NOUN
ejpam-3952	2	5	2	2	NUM
ejpam-3952	2	6	,	,	PUNCT
ejpam-3952	2	7	2021	2021	NUM
ejpam-3952	2	8	,	,	PUNCT
ejpam-3952	2	9	380	380	NUM
ejpam-3952	2	10	-	-	SYM
ejpam-3952	2	11	395	395	NUM
ejpam-3952	2	12	issn	issn	PROPN
ejpam-3952	2	13	1307	1307	NUM
ejpam-3952	2	14	-	-	SYM
ejpam-3952	2	15	5543	5543	NUM
ejpam-3952	2	16	–	–	PUNCT
ejpam-3952	2	17	ejpam.com	ejpam.com	X
ejpam-3952	2	18	published	publish	VERB
ejpam-3952	2	19	by	by	ADP
ejpam-3952	2	20	new	new	PROPN
ejpam-3952	2	21	york	york	PROPN
ejpam-3952	2	22	business	business	PROPN
ejpam-3952	2	23	global	global	ADJ
ejpam-3952	2	24	characterizations	characterization	NOUN
ejpam-3952	2	25	and	and	CCONJ
ejpam-3952	2	26	identities	identity	NOUN
ejpam-3952	2	27	for	for	ADP
ejpam-3952	2	28	isosceles	isoscele	NOUN
ejpam-3952	2	29	triangular	triangular	NOUN
ejpam-3952	2	30	numbers	number	NOUN
ejpam-3952	2	31	jiramate	jiramate	VERB
ejpam-3952	2	32	punpim1	punpim1	NOUN
ejpam-3952	2	33	,	,	PUNCT
ejpam-3952	2	34	somphong	somphong	NOUN
ejpam-3952	2	35	jitman1,∗	jitman1,∗	PROPN
ejpam-3952	2	36	1	1	NUM
ejpam-3952	2	37	department	department	NOUN
ejpam-3952	2	38	of	of	ADP
ejpam-3952	2	39	mathematics	mathematic	NOUN
ejpam-3952	2	40	,	,	PUNCT
ejpam-3952	2	41	faculty	faculty	NOUN
ejpam-3952	2	42	of	of	ADP
ejpam-3952	2	43	science	science	NOUN
ejpam-3952	2	44	,	,	PUNCT
ejpam-3952	2	45	silpakorn	silpakorn	VERB
ejpam-3952	2	46	university	university	PROPN
ejpam-3952	2	47	,	,	PUNCT
ejpam-3952	2	48	nakhon	nakhon	PROPN
ejpam-3952	2	49	pathom	pathom	PROPN
ejpam-3952	2	50	73000	73000	NUM
ejpam-3952	2	51	,	,	PUNCT
ejpam-3952	2	52	thailand	thailand	PROPN
ejpam-3952	2	53	abstract	abstract	NOUN
ejpam-3952	2	54	.	.	PUNCT
ejpam-3952	3	1	triangular	triangular	NOUN
ejpam-3952	3	2	numbers	number	NOUN
ejpam-3952	3	3	have	have	AUX
ejpam-3952	3	4	been	be	AUX
ejpam-3952	3	5	of	of	ADP
ejpam-3952	3	6	interest	interest	NOUN
ejpam-3952	3	7	and	and	CCONJ
ejpam-3952	3	8	continuously	continuously	ADV
ejpam-3952	3	9	studied	study	VERB
ejpam-3952	3	10	due	due	ADP
ejpam-3952	3	11	to	to	ADP
ejpam-3952	3	12	their	their	PRON
ejpam-3952	3	13	beautiful	beautiful	ADJ
ejpam-3952	3	14	representations	representation	NOUN
ejpam-3952	3	15	,	,	PUNCT
ejpam-3952	3	16	nice	nice	ADJ
ejpam-3952	3	17	properties	property	NOUN
ejpam-3952	3	18	,	,	PUNCT
ejpam-3952	3	19	and	and	CCONJ
ejpam-3952	3	20	various	various	ADJ
ejpam-3952	3	21	links	link	NOUN
ejpam-3952	3	22	with	with	ADP
ejpam-3952	3	23	other	other	ADJ
ejpam-3952	3	24	figurate	figurate	NOUN
ejpam-3952	3	25	numbers	number	NOUN
ejpam-3952	3	26	.	.	PUNCT
ejpam-3952	4	1	for	for	ADP
ejpam-3952	4	2	positive	positive	ADJ
ejpam-3952	4	3	integers	integer	NOUN
ejpam-3952	4	4	n	n	PRON
ejpam-3952	4	5	and	and	CCONJ
ejpam-3952	4	6	l	l	NOUN
ejpam-3952	4	7	,	,	PUNCT
ejpam-3952	4	8	the	the	DET
ejpam-3952	4	9	nth	nth	NOUN
ejpam-3952	4	10	l	l	NOUN
ejpam-3952	4	11	-	-	PUNCT
ejpam-3952	4	12	isosceles	isoscele	NOUN
ejpam-3952	4	13	triangular	triangular	NOUN
ejpam-3952	4	14	number	number	NOUN
ejpam-3952	4	15	is	be	AUX
ejpam-3952	4	16	a	a	DET
ejpam-3952	4	17	generalization	generalization	NOUN
ejpam-3952	4	18	of	of	ADP
ejpam-3952	4	19	triangular	triangular	NOUN
ejpam-3952	4	20	numbers	number	NOUN
ejpam-3952	4	21	defined	define	VERB
ejpam-3952	4	22	to	to	PART
ejpam-3952	4	23	be	be	AUX
ejpam-3952	4	24	the	the	DET
ejpam-3952	4	25	arithmetic	arithmetic	ADJ
ejpam-3952	4	26	sum	sum	NOUN
ejpam-3952	4	27	of	of	ADP
ejpam-3952	4	28	the	the	DET
ejpam-3952	4	29	form	form	NOUN
ejpam-3952	4	30	t	t	PROPN
ejpam-3952	4	31	(	(	PUNCT
ejpam-3952	4	32	n	n	X
ejpam-3952	4	33	,	,	PUNCT
ejpam-3952	4	34	l	l	NOUN
ejpam-3952	4	35	)	)	PUNCT
ejpam-3952	4	36	=	=	SYM
ejpam-3952	5	1	1	1	NUM
ejpam-3952	5	2	+	+	CCONJ
ejpam-3952	5	3	(	(	PUNCT
ejpam-3952	5	4	1	1	NUM
ejpam-3952	5	5	+	+	NUM
ejpam-3952	5	6	l	l	NOUN
ejpam-3952	5	7	)	)	PUNCT
ejpam-3952	6	1	+	+	CCONJ
ejpam-3952	6	2	(	(	PUNCT
ejpam-3952	6	3	1	1	NUM
ejpam-3952	6	4	+	+	NUM
ejpam-3952	6	5	2l	2l	NUM
ejpam-3952	6	6	)	)	PUNCT
ejpam-3952	6	7	+	+	CCONJ
ejpam-3952	6	8	·	·	PUNCT
ejpam-3952	6	9	·	·	PUNCT
ejpam-3952	6	10	·	·	PUNCT
ejpam-3952	6	11	+	+	CCONJ
ejpam-3952	6	12	(	(	PUNCT
ejpam-3952	6	13	1	1	NUM
ejpam-3952	6	14	+	+	CCONJ
ejpam-3952	6	15	(	(	PUNCT
ejpam-3952	6	16	n−	n−	NOUN
ejpam-3952	6	17	1)l	1)l	NUM
ejpam-3952	6	18	)	)	PUNCT
ejpam-3952	6	19	.	.	PUNCT
ejpam-3952	7	1	in	in	ADP
ejpam-3952	7	2	this	this	DET
ejpam-3952	7	3	paper	paper	NOUN
ejpam-3952	7	4	,	,	PUNCT
ejpam-3952	7	5	we	we	PRON
ejpam-3952	7	6	focus	focus	VERB
ejpam-3952	7	7	on	on	ADP
ejpam-3952	7	8	characterizations	characterization	NOUN
ejpam-3952	7	9	and	and	CCONJ
ejpam-3952	7	10	identities	identity	NOUN
ejpam-3952	7	11	for	for	ADP
ejpam-3952	7	12	isosceles	isoscele	NOUN
ejpam-3952	7	13	triangular	triangular	NOUN
ejpam-3952	7	14	numbers	number	NOUN
ejpam-3952	7	15	as	as	ADV
ejpam-3952	7	16	well	well	ADV
ejpam-3952	7	17	as	as	ADP
ejpam-3952	7	18	their	their	PRON
ejpam-3952	7	19	links	link	NOUN
ejpam-3952	7	20	with	with	ADP
ejpam-3952	7	21	other	other	ADJ
ejpam-3952	7	22	figurate	figurate	NOUN
ejpam-3952	7	23	numbers	number	NOUN
ejpam-3952	7	24	.	.	PUNCT
ejpam-3952	8	1	recursive	recursive	ADJ
ejpam-3952	8	2	formulas	formula	NOUN
ejpam-3952	8	3	for	for	ADP
ejpam-3952	8	4	constructions	construction	NOUN
ejpam-3952	8	5	of	of	ADP
ejpam-3952	8	6	isosceles	isoscele	NOUN
ejpam-3952	8	7	triangular	triangular	NOUN
ejpam-3952	8	8	numbers	number	NOUN
ejpam-3952	8	9	are	be	AUX
ejpam-3952	8	10	given	give	VERB
ejpam-3952	8	11	together	together	ADV
ejpam-3952	8	12	with	with	ADP
ejpam-3952	8	13	necessary	necessary	ADJ
ejpam-3952	8	14	and	and	CCONJ
ejpam-3952	8	15	sufficient	sufficient	ADJ
ejpam-3952	8	16	conditions	condition	NOUN
ejpam-3952	8	17	for	for	ADP
ejpam-3952	8	18	a	a	DET
ejpam-3952	8	19	positive	positive	ADJ
ejpam-3952	8	20	integer	integer	NOUN
ejpam-3952	8	21	to	to	PART
ejpam-3952	8	22	be	be	AUX
ejpam-3952	8	23	a	a	DET
ejpam-3952	8	24	sum	sum	NOUN
ejpam-3952	8	25	of	of	ADP
ejpam-3952	8	26	isosceles	isoscele	NOUN
ejpam-3952	8	27	triangular	triangular	NOUN
ejpam-3952	8	28	numbers	number	NOUN
ejpam-3952	8	29	.	.	PUNCT
ejpam-3952	9	1	various	various	ADJ
ejpam-3952	9	2	identities	identity	NOUN
ejpam-3952	9	3	for	for	ADP
ejpam-3952	9	4	isosceles	isoscele	NOUN
ejpam-3952	9	5	triangular	triangular	NOUN
ejpam-3952	9	6	numbers	number	NOUN
ejpam-3952	9	7	are	be	AUX
ejpam-3952	9	8	established	establish	VERB
ejpam-3952	9	9	.	.	PUNCT
ejpam-3952	10	1	results	result	NOUN
ejpam-3952	10	2	on	on	ADP
ejpam-3952	10	3	triangular	triangular	NOUN
ejpam-3952	10	4	numbers	number	NOUN
ejpam-3952	10	5	can	can	AUX
ejpam-3952	10	6	be	be	AUX
ejpam-3952	10	7	viewed	view	VERB
ejpam-3952	10	8	as	as	ADP
ejpam-3952	10	9	a	a	DET
ejpam-3952	10	10	special	special	ADJ
ejpam-3952	10	11	case	case	NOUN
ejpam-3952	10	12	.	.	PUNCT
ejpam-3952	11	1	2020	2020	NUM
ejpam-3952	11	2	mathematics	mathematic	NOUN
ejpam-3952	11	3	subject	subject	NOUN
ejpam-3952	11	4	classifications	classification	NOUN
ejpam-3952	11	5	:	:	PUNCT
ejpam-3952	11	6	11b99	11b99	NUM
ejpam-3952	11	7	,	,	PUNCT
ejpam-3952	11	8	11e25	11e25	PRON
ejpam-3952	11	9	key	key	ADJ
ejpam-3952	11	10	words	word	NOUN
ejpam-3952	11	11	and	and	CCONJ
ejpam-3952	11	12	phrases	phrase	NOUN
ejpam-3952	11	13	:	:	PUNCT
ejpam-3952	11	14	integer	integer	NOUN
ejpam-3952	11	15	sequences	sequence	NOUN
ejpam-3952	11	16	,	,	PUNCT
ejpam-3952	11	17	triangular	triangular	NOUN
ejpam-3952	11	18	numbers	number	NOUN
ejpam-3952	11	19	,	,	PUNCT
ejpam-3952	11	20	isosceles	isoscele	NOUN
ejpam-3952	11	21	triangular	triangular	NOUN
ejpam-3952	11	22	numbers	number	NOUN
ejpam-3952	11	23	,	,	PUNCT
ejpam-3952	11	24	polygonal	polygonal	ADJ
ejpam-3952	11	25	numbers	number	NOUN
ejpam-3952	11	26	,	,	PUNCT
ejpam-3952	11	27	identities	identity	NOUN
ejpam-3952	11	28	1	1	NUM
ejpam-3952	11	29	.	.	X
ejpam-3952	11	30	introduction	introduction	NOUN
ejpam-3952	11	31	a	a	DET
ejpam-3952	11	32	triangular	triangular	NOUN
ejpam-3952	11	33	number	number	NOUN
ejpam-3952	11	34	is	be	AUX
ejpam-3952	11	35	a	a	DET
ejpam-3952	11	36	number	number	NOUN
ejpam-3952	11	37	that	that	PRON
ejpam-3952	11	38	can	can	AUX
ejpam-3952	11	39	be	be	AUX
ejpam-3952	11	40	represented	represent	VERB
ejpam-3952	11	41	as	as	ADP
ejpam-3952	11	42	an	an	DET
ejpam-3952	11	43	equilateral	equilateral	ADJ
ejpam-3952	11	44	triangular	triangular	NOUN
ejpam-3952	11	45	arrangement	arrangement	NOUN
ejpam-3952	11	46	of	of	ADP
ejpam-3952	11	47	points	point	NOUN
ejpam-3952	11	48	equally	equally	ADV
ejpam-3952	11	49	spaced	space	VERB
ejpam-3952	11	50	.	.	PUNCT
ejpam-3952	12	1	precisely	precisely	ADV
ejpam-3952	12	2	,	,	PUNCT
ejpam-3952	12	3	the	the	DET
ejpam-3952	12	4	nth	nth	NOUN
ejpam-3952	12	5	triangular	triangular	NOUN
ejpam-3952	12	6	number	number	NOUN
ejpam-3952	12	7	is	be	AUX
ejpam-3952	12	8	the	the	DET
ejpam-3952	12	9	number	number	NOUN
ejpam-3952	12	10	of	of	ADP
ejpam-3952	12	11	points	point	NOUN
ejpam-3952	12	12	composing	compose	VERB
ejpam-3952	12	13	an	an	DET
ejpam-3952	12	14	equilateral	equilateral	ADJ
ejpam-3952	12	15	triangle	triangle	NOUN
ejpam-3952	12	16	with	with	ADP
ejpam-3952	12	17	n	n	NUM
ejpam-3952	12	18	points	point	NOUN
ejpam-3952	12	19	on	on	ADP
ejpam-3952	12	20	a	a	DET
ejpam-3952	12	21	side	side	NOUN
ejpam-3952	12	22	which	which	PRON
ejpam-3952	12	23	equals	equal	VERB
ejpam-3952	12	24	the	the	DET
ejpam-3952	12	25	sum	sum	NOUN
ejpam-3952	12	26	of	of	ADP
ejpam-3952	12	27	the	the	DET
ejpam-3952	12	28	n	n	DET
ejpam-3952	12	29	natural	natural	ADJ
ejpam-3952	12	30	numbers	number	NOUN
ejpam-3952	12	31	of	of	ADP
ejpam-3952	12	32	the	the	DET
ejpam-3952	12	33	form	form	NOUN
ejpam-3952	12	34	t	t	PROPN
ejpam-3952	12	35	(	(	PUNCT
ejpam-3952	12	36	n	n	CCONJ
ejpam-3952	12	37	)	)	PUNCT
ejpam-3952	12	38	=	=	SYM
ejpam-3952	12	39	1	1	NUM
ejpam-3952	12	40	+	+	NUM
ejpam-3952	12	41	2	2	NUM
ejpam-3952	12	42	+	+	NUM
ejpam-3952	12	43	3	3	NUM
ejpam-3952	12	44	+	+	NUM
ejpam-3952	12	45	·	·	PUNCT
ejpam-3952	12	46	·	·	PUNCT
ejpam-3952	12	47	·	·	PUNCT
ejpam-3952	13	1	+	+	NUM
ejpam-3952	13	2	n	n	PROPN
ejpam-3952	13	3	=	=	PUNCT
ejpam-3952	13	4	n(n	n(n	PROPN
ejpam-3952	13	5	+	+	CCONJ
ejpam-3952	13	6	1	1	NUM
ejpam-3952	13	7	)	)	PUNCT
ejpam-3952	13	8	2	2	NUM
ejpam-3952	13	9	.	.	PUNCT
ejpam-3952	14	1	(	(	PUNCT
ejpam-3952	14	2	1	1	X
ejpam-3952	14	3	)	)	PUNCT
ejpam-3952	14	4	the	the	DET
ejpam-3952	14	5	nth	nth	NOUN
ejpam-3952	14	6	triangular	triangular	NOUN
ejpam-3952	14	7	number	number	NOUN
ejpam-3952	14	8	can	can	AUX
ejpam-3952	14	9	be	be	AUX
ejpam-3952	14	10	represented	represent	VERB
ejpam-3952	14	11	as	as	ADP
ejpam-3952	14	12	points	point	NOUN
ejpam-3952	14	13	in	in	ADP
ejpam-3952	14	14	an	an	DET
ejpam-3952	14	15	equilateral	equilateral	ADJ
ejpam-3952	14	16	triangle	triangle	NOUN
ejpam-3952	14	17	as	as	ADP
ejpam-3952	14	18	in	in	ADP
ejpam-3952	14	19	figure	figure	NOUN
ejpam-3952	14	20	1	1	NUM
ejpam-3952	14	21	.	.	PUNCT
ejpam-3952	14	22	∗corresponding	∗corresponde	VERB
ejpam-3952	14	23	author	author	NOUN
ejpam-3952	14	24	.	.	PUNCT
ejpam-3952	15	1	doi	doi	NOUN
ejpam-3952	15	2	:	:	PUNCT
ejpam-3952	15	3	https://doi.org/10.29020/nybg.ejpam.v14i2.3952	https://doi.org/10.29020/nybg.ejpam.v14i2.3952	NOUN
ejpam-3952	15	4	email	email	NOUN
ejpam-3952	15	5	addresses	address	NOUN
ejpam-3952	15	6	:	:	PUNCT
ejpam-3952	15	7	punpim	punpim	VERB
ejpam-3952	15	8	j@su.ac.th	j@su.ac.th	PROPN
ejpam-3952	15	9	(	(	PUNCT
ejpam-3952	15	10	j.	j.	PROPN
ejpam-3952	15	11	punpim	punpim	PROPN
ejpam-3952	15	12	)	)	PUNCT
ejpam-3952	15	13	,	,	PUNCT
ejpam-3952	16	1	sjitman@gmail.com	sjitman@gmail.com	X
ejpam-3952	16	2	(	(	PUNCT
ejpam-3952	16	3	s.	s.	PROPN
ejpam-3952	16	4	jitman	jitman	PROPN
ejpam-3952	16	5	)	)	PUNCT
ejpam-3952	16	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3952	17	1	380	380	NUM
ejpam-3952	17	2	c	c	X
ejpam-3952	17	3	©	©	PROPN
ejpam-3952	17	4	2021	2021	NUM
ejpam-3952	17	5	ejpam	ejpam	VERB
ejpam-3952	17	6	all	all	DET
ejpam-3952	17	7	rights	right	NOUN
ejpam-3952	17	8	reserved	reserve	VERB
ejpam-3952	17	9	.	.	PUNCT
ejpam-3952	18	1	j.	j.	PROPN
ejpam-3952	18	2	punpim	punpim	PROPN
ejpam-3952	18	3	,	,	PUNCT
ejpam-3952	18	4	s.	s.	PROPN
ejpam-3952	18	5	jitman	jitman	PROPN
ejpam-3952	18	6	/	/	SYM
ejpam-3952	18	7	eur	eur	PROPN
ejpam-3952	18	8	.	.	PUNCT
ejpam-3952	19	1	j.	j.	PROPN
ejpam-3952	19	2	pure	pure	PROPN
ejpam-3952	19	3	appl	appl	PROPN
ejpam-3952	19	4	.	.	PROPN
ejpam-3952	19	5	math	math	PROPN
ejpam-3952	19	6	,	,	PUNCT
ejpam-3952	19	7	14	14	NUM
ejpam-3952	19	8	(	(	PUNCT
ejpam-3952	19	9	2	2	NUM
ejpam-3952	19	10	)	)	PUNCT
ejpam-3952	19	11	(	(	PUNCT
ejpam-3952	19	12	2021	2021	NUM
ejpam-3952	19	13	)	)	PUNCT
ejpam-3952	19	14	,	,	PUNCT
ejpam-3952	19	15	380	380	NUM
ejpam-3952	19	16	-	-	SYM
ejpam-3952	19	17	395	395	NUM
ejpam-3952	19	18	381	381	NUM
ejpam-3952	19	19	n	n	NUM
ejpam-3952	19	20	points	point	NOUN
ejpam-3952	19	21	n	n	NUM
ejpam-3952	19	22	points	point	NOUN
ejpam-3952	19	23	figure	figure	NOUN
ejpam-3952	19	24	1	1	NUM
ejpam-3952	19	25	:	:	PUNCT
ejpam-3952	19	26	triangular	triangular	NOUN
ejpam-3952	19	27	number	number	NOUN
ejpam-3952	19	28	t	t	PROPN
ejpam-3952	19	29	(	(	PUNCT
ejpam-3952	19	30	n	n	CCONJ
ejpam-3952	19	31	)	)	PUNCT
ejpam-3952	19	32	triangular	triangular	NOUN
ejpam-3952	19	33	numbers	number	NOUN
ejpam-3952	19	34	have	have	AUX
ejpam-3952	19	35	been	be	AUX
ejpam-3952	19	36	introduced	introduce	VERB
ejpam-3952	19	37	and	and	CCONJ
ejpam-3952	19	38	studied	study	VERB
ejpam-3952	19	39	since	since	SCONJ
ejpam-3952	19	40	the	the	DET
ejpam-3952	19	41	6th	6th	ADJ
ejpam-3952	19	42	century	century	NOUN
ejpam-3952	19	43	bc	bc	PROPN
ejpam-3952	19	44	.	.	PROPN
ejpam-3952	20	1	due	due	ADP
ejpam-3952	20	2	to	to	ADP
ejpam-3952	20	3	their	their	PRON
ejpam-3952	20	4	nice	nice	ADJ
ejpam-3952	20	5	properties	property	NOUN
ejpam-3952	20	6	and	and	CCONJ
ejpam-3952	20	7	wide	wide	ADJ
ejpam-3952	20	8	links	link	NOUN
ejpam-3952	20	9	with	with	ADP
ejpam-3952	20	10	other	other	ADJ
ejpam-3952	20	11	mathematical	mathematical	ADJ
ejpam-3952	20	12	objects	object	NOUN
ejpam-3952	20	13	,	,	PUNCT
ejpam-3952	20	14	such	such	ADJ
ejpam-3952	20	15	numbers	number	NOUN
ejpam-3952	20	16	have	have	AUX
ejpam-3952	20	17	been	be	AUX
ejpam-3952	20	18	extensively	extensively	ADV
ejpam-3952	20	19	studied	study	VERB
ejpam-3952	20	20	(	(	PUNCT
ejpam-3952	20	21	see	see	VERB
ejpam-3952	20	22	[	[	X
ejpam-3952	20	23	1–3	1–3	NOUN
ejpam-3952	20	24	]	]	PUNCT
ejpam-3952	20	25	and	and	CCONJ
ejpam-3952	20	26	references	reference	NOUN
ejpam-3952	20	27	therein	therein	ADV
ejpam-3952	20	28	)	)	PUNCT
ejpam-3952	20	29	.	.	PUNCT
ejpam-3952	21	1	subsequently	subsequently	ADV
ejpam-3952	21	2	,	,	PUNCT
ejpam-3952	21	3	characterizations	characterization	NOUN
ejpam-3952	21	4	of	of	ADP
ejpam-3952	21	5	triangular	triangular	NOUN
ejpam-3952	21	6	numbers	number	NOUN
ejpam-3952	21	7	and	and	CCONJ
ejpam-3952	21	8	interesting	interesting	ADJ
ejpam-3952	21	9	relationships	relationship	NOUN
ejpam-3952	21	10	among	among	ADP
ejpam-3952	21	11	triangular	triangular	NOUN
ejpam-3952	21	12	numbers	number	NOUN
ejpam-3952	21	13	and	and	CCONJ
ejpam-3952	21	14	other	other	ADJ
ejpam-3952	21	15	figurate	figurate	NOUN
ejpam-3952	21	16	numbers	number	NOUN
ejpam-3952	21	17	have	have	AUX
ejpam-3952	21	18	been	be	AUX
ejpam-3952	21	19	established	establish	VERB
ejpam-3952	21	20	(	(	PUNCT
ejpam-3952	21	21	see	see	VERB
ejpam-3952	21	22	[	[	X
ejpam-3952	21	23	2	2	NUM
ejpam-3952	21	24	,	,	PUNCT
ejpam-3952	21	25	4	4	NUM
ejpam-3952	21	26	,	,	PUNCT
ejpam-3952	21	27	10	10	NUM
ejpam-3952	21	28	]	]	NUM
ejpam-3952	21	29	)	)	PUNCT
ejpam-3952	21	30	.	.	PUNCT
ejpam-3952	22	1	various	various	ADJ
ejpam-3952	22	2	subsequences	subsequence	NOUN
ejpam-3952	22	3	of	of	ADP
ejpam-3952	22	4	triangular	triangular	NOUN
ejpam-3952	22	5	numbers	number	NOUN
ejpam-3952	22	6	with	with	ADP
ejpam-3952	22	7	nice	nice	ADJ
ejpam-3952	22	8	properties	property	NOUN
ejpam-3952	22	9	have	have	AUX
ejpam-3952	22	10	been	be	AUX
ejpam-3952	22	11	presented	present	VERB
ejpam-3952	22	12	in	in	ADP
ejpam-3952	22	13	[	[	X
ejpam-3952	22	14	2	2	NUM
ejpam-3952	22	15	,	,	PUNCT
ejpam-3952	22	16	8	8	NUM
ejpam-3952	22	17	,	,	PUNCT
ejpam-3952	22	18	11	11	NUM
ejpam-3952	22	19	]	]	PUNCT
ejpam-3952	22	20	.	.	PUNCT
ejpam-3952	23	1	identities	identity	NOUN
ejpam-3952	23	2	of	of	ADP
ejpam-3952	23	3	triangular	triangular	NOUN
ejpam-3952	23	4	numbers	number	NOUN
ejpam-3952	23	5	have	have	AUX
ejpam-3952	23	6	been	be	AUX
ejpam-3952	23	7	of	of	ADP
ejpam-3952	23	8	interest	interest	NOUN
ejpam-3952	23	9	and	and	CCONJ
ejpam-3952	23	10	studied	study	VERB
ejpam-3952	23	11	in	in	ADP
ejpam-3952	23	12	[	[	X
ejpam-3952	23	13	1–3	1–3	NOUN
ejpam-3952	23	14	,	,	PUNCT
ejpam-3952	23	15	7	7	NUM
ejpam-3952	23	16	,	,	PUNCT
ejpam-3952	23	17	9	9	NUM
ejpam-3952	23	18	,	,	PUNCT
ejpam-3952	23	19	12	12	NUM
ejpam-3952	23	20	]	]	PUNCT
ejpam-3952	23	21	.	.	PUNCT
ejpam-3952	24	1	in	in	ADP
ejpam-3952	24	2	[	[	X
ejpam-3952	24	3	5	5	NUM
ejpam-3952	24	4	]	]	PUNCT
ejpam-3952	24	5	,	,	PUNCT
ejpam-3952	24	6	an	an	DET
ejpam-3952	24	7	isosceles	isoscele	NOUN
ejpam-3952	24	8	triangular	triangular	NOUN
ejpam-3952	24	9	number	number	NOUN
ejpam-3952	24	10	has	have	AUX
ejpam-3952	24	11	been	be	AUX
ejpam-3952	24	12	introduced	introduce	VERB
ejpam-3952	24	13	as	as	ADP
ejpam-3952	24	14	a	a	DET
ejpam-3952	24	15	generalization	generalization	NOUN
ejpam-3952	24	16	of	of	ADP
ejpam-3952	24	17	triangular	triangular	NOUN
ejpam-3952	24	18	numbers	number	NOUN
ejpam-3952	24	19	and	and	CCONJ
ejpam-3952	24	20	it	it	PRON
ejpam-3952	24	21	is	be	AUX
ejpam-3952	24	22	defined	define	VERB
ejpam-3952	24	23	to	to	PART
ejpam-3952	24	24	be	be	AUX
ejpam-3952	24	25	a	a	DET
ejpam-3952	24	26	number	number	NOUN
ejpam-3952	24	27	that	that	PRON
ejpam-3952	24	28	can	can	AUX
ejpam-3952	24	29	be	be	AUX
ejpam-3952	24	30	represented	represent	VERB
ejpam-3952	24	31	as	as	ADP
ejpam-3952	24	32	an	an	DET
ejpam-3952	24	33	isosceles	isoscele	NOUN
ejpam-3952	24	34	triangular	triangular	NOUN
ejpam-3952	24	35	arrangement	arrangement	NOUN
ejpam-3952	24	36	of	of	ADP
ejpam-3952	24	37	points	point	NOUN
ejpam-3952	24	38	.	.	PUNCT
ejpam-3952	25	1	precisely	precisely	ADV
ejpam-3952	25	2	,	,	PUNCT
ejpam-3952	25	3	the	the	DET
ejpam-3952	25	4	nth	nth	NOUN
ejpam-3952	25	5	l	l	NOUN
ejpam-3952	25	6	-	-	PUNCT
ejpam-3952	25	7	isosceles	isoscele	NOUN
ejpam-3952	25	8	triangular	triangular	NOUN
ejpam-3952	25	9	number	number	NOUN
ejpam-3952	25	10	,	,	PUNCT
ejpam-3952	25	11	denoted	denote	VERB
ejpam-3952	25	12	by	by	ADP
ejpam-3952	25	13	t	t	PROPN
ejpam-3952	25	14	(	(	PUNCT
ejpam-3952	25	15	n	n	CCONJ
ejpam-3952	25	16	,	,	PUNCT
ejpam-3952	25	17	l	l	NOUN
ejpam-3952	25	18	)	)	PUNCT
ejpam-3952	25	19	,	,	PUNCT
ejpam-3952	25	20	is	be	AUX
ejpam-3952	25	21	defined	define	VERB
ejpam-3952	25	22	to	to	PART
ejpam-3952	25	23	be	be	AUX
ejpam-3952	25	24	the	the	DET
ejpam-3952	25	25	arithmetic	arithmetic	ADJ
ejpam-3952	25	26	sum	sum	NOUN
ejpam-3952	25	27	of	of	ADP
ejpam-3952	25	28	the	the	DET
ejpam-3952	25	29	form	form	NOUN
ejpam-3952	25	30	t	t	PROPN
ejpam-3952	25	31	(	(	PUNCT
ejpam-3952	25	32	n	n	X
ejpam-3952	25	33	,	,	PUNCT
ejpam-3952	25	34	l	l	NOUN
ejpam-3952	25	35	)	)	PUNCT
ejpam-3952	26	1	=	=	SYM
ejpam-3952	26	2	1	1	NUM
ejpam-3952	26	3	+	+	CCONJ
ejpam-3952	26	4	(	(	PUNCT
ejpam-3952	26	5	1	1	NUM
ejpam-3952	26	6	+	+	NUM
ejpam-3952	26	7	l	l	NOUN
ejpam-3952	26	8	)	)	PUNCT
ejpam-3952	27	1	+	+	CCONJ
ejpam-3952	27	2	(	(	PUNCT
ejpam-3952	27	3	1	1	NUM
ejpam-3952	27	4	+	+	NUM
ejpam-3952	27	5	2l	2l	NUM
ejpam-3952	27	6	)	)	PUNCT
ejpam-3952	27	7	+	+	CCONJ
ejpam-3952	27	8	·	·	PUNCT
ejpam-3952	27	9	·	·	PUNCT
ejpam-3952	27	10	·	·	PUNCT
ejpam-3952	27	11	+	+	CCONJ
ejpam-3952	27	12	(	(	PUNCT
ejpam-3952	27	13	1	1	NUM
ejpam-3952	27	14	+	+	CCONJ
ejpam-3952	27	15	(	(	PUNCT
ejpam-3952	27	16	n−	n−	NOUN
ejpam-3952	27	17	1)l	1)l	NUM
ejpam-3952	27	18	)	)	PUNCT
ejpam-3952	27	19	=	=	SYM
ejpam-3952	28	1	n	n	PROPN
ejpam-3952	28	2	+	+	CCONJ
ejpam-3952	28	3	n(n−	n(n−	PROPN
ejpam-3952	28	4	1)l	1)l	NUM
ejpam-3952	28	5	2	2	NUM
ejpam-3952	28	6	=	=	SYM
ejpam-3952	28	7	n	n	NOUN
ejpam-3952	28	8	+	+	NUM
ejpam-3952	28	9	lt	lt	ADJ
ejpam-3952	28	10	(	(	PUNCT
ejpam-3952	28	11	n−	n−	NOUN
ejpam-3952	28	12	1	1	NUM
ejpam-3952	28	13	)	)	PUNCT
ejpam-3952	28	14	.	.	PUNCT
ejpam-3952	29	1	alternatively	alternatively	ADV
ejpam-3952	29	2	,	,	PUNCT
ejpam-3952	29	3	an	an	DET
ejpam-3952	29	4	l	l	NOUN
ejpam-3952	29	5	-	-	PUNCT
ejpam-3952	29	6	isosceles	isoscele	NOUN
ejpam-3952	29	7	triangular	triangular	NOUN
ejpam-3952	29	8	number	number	NOUN
ejpam-3952	29	9	can	can	AUX
ejpam-3952	29	10	be	be	AUX
ejpam-3952	29	11	viewed	view	VERB
ejpam-3952	29	12	as	as	ADP
ejpam-3952	29	13	a	a	DET
ejpam-3952	29	14	special	special	ADJ
ejpam-3952	29	15	case	case	NOUN
ejpam-3952	29	16	of	of	ADP
ejpam-3952	29	17	generalized	generalized	ADJ
ejpam-3952	29	18	trapezoidal	trapezoidal	ADJ
ejpam-3952	29	19	numbers	number	NOUN
ejpam-3952	29	20	in	in	ADP
ejpam-3952	29	21	[	[	X
ejpam-3952	29	22	6	6	NUM
ejpam-3952	29	23	]	]	PUNCT
ejpam-3952	29	24	.	.	PUNCT
ejpam-3952	30	1	clearly	clearly	ADV
ejpam-3952	30	2	,	,	PUNCT
ejpam-3952	30	3	the	the	DET
ejpam-3952	30	4	isosceles	isoscele	NOUN
ejpam-3952	30	5	triangular	triangular	NOUN
ejpam-3952	30	6	number	number	NOUN
ejpam-3952	30	7	t	t	PROPN
ejpam-3952	30	8	(	(	PUNCT
ejpam-3952	30	9	n	n	CCONJ
ejpam-3952	30	10	,	,	PUNCT
ejpam-3952	30	11	l	l	NOUN
ejpam-3952	30	12	)	)	PUNCT
ejpam-3952	30	13	becomes	become	VERB
ejpam-3952	30	14	the	the	DET
ejpam-3952	30	15	nth	nth	NOUN
ejpam-3952	30	16	triangular	triangular	NOUN
ejpam-3952	30	17	number	number	NOUN
ejpam-3952	30	18	t	t	PROPN
ejpam-3952	30	19	(	(	PUNCT
ejpam-3952	30	20	n	n	CCONJ
ejpam-3952	30	21	)	)	PUNCT
ejpam-3952	30	22	whenever	whenever	SCONJ
ejpam-3952	30	23	l	l	NOUN
ejpam-3952	30	24	=	=	NOUN
ejpam-3952	30	25	1	1	X
ejpam-3952	30	26	.	.	PUNCT
ejpam-3952	31	1	in	in	ADP
ejpam-3952	31	2	the	the	DET
ejpam-3952	31	3	case	case	NOUN
ejpam-3952	31	4	where	where	SCONJ
ejpam-3952	31	5	l	l	NOUN
ejpam-3952	31	6	=	=	SYM
ejpam-3952	31	7	2	2	NUM
ejpam-3952	31	8	,	,	PUNCT
ejpam-3952	31	9	it	it	PRON
ejpam-3952	31	10	can	can	AUX
ejpam-3952	31	11	be	be	AUX
ejpam-3952	31	12	easily	easily	ADV
ejpam-3952	31	13	seen	see	VERB
ejpam-3952	31	14	that	that	SCONJ
ejpam-3952	31	15	t	t	PROPN
ejpam-3952	31	16	(	(	PUNCT
ejpam-3952	31	17	n	n	CCONJ
ejpam-3952	31	18	,	,	PUNCT
ejpam-3952	31	19	2	2	X
ejpam-3952	31	20	)	)	PUNCT
ejpam-3952	32	1	=	=	SYM
ejpam-3952	32	2	n2	n2	NOUN
ejpam-3952	32	3	is	be	AUX
ejpam-3952	32	4	a	a	DET
ejpam-3952	32	5	square	square	ADJ
ejpam-3952	32	6	number	number	NOUN
ejpam-3952	32	7	.	.	PUNCT
ejpam-3952	33	1	for	for	ADP
ejpam-3952	33	2	convenience	convenience	NOUN
ejpam-3952	33	3	,	,	PUNCT
ejpam-3952	33	4	the	the	DET
ejpam-3952	33	5	notions	notion	NOUN
ejpam-3952	33	6	of	of	ADP
ejpam-3952	33	7	the	the	DET
ejpam-3952	33	8	l	l	NOUN
ejpam-3952	33	9	-	-	PUNCT
ejpam-3952	33	10	isosceles	isoscele	NOUN
ejpam-3952	33	11	triangular	triangular	NOUN
ejpam-3952	33	12	number	number	NOUN
ejpam-3952	33	13	t	t	PROPN
ejpam-3952	33	14	(	(	PUNCT
ejpam-3952	33	15	0	0	NUM
ejpam-3952	33	16	,	,	PUNCT
ejpam-3952	33	17	l	l	NOUN
ejpam-3952	33	18	)	)	PUNCT
ejpam-3952	33	19	and	and	CCONJ
ejpam-3952	33	20	the	the	DET
ejpam-3952	33	21	triangular	triangular	NOUN
ejpam-3952	33	22	number	number	NOUN
ejpam-3952	33	23	t	t	PROPN
ejpam-3952	33	24	(	(	PUNCT
ejpam-3952	33	25	0	0	NUM
ejpam-3952	33	26	)	)	PUNCT
ejpam-3952	33	27	are	be	AUX
ejpam-3952	33	28	used	use	VERB
ejpam-3952	33	29	in	in	ADP
ejpam-3952	33	30	some	some	DET
ejpam-3952	33	31	contexts	context	NOUN
ejpam-3952	33	32	and	and	CCONJ
ejpam-3952	33	33	they	they	PRON
ejpam-3952	33	34	are	be	AUX
ejpam-3952	33	35	set	set	VERB
ejpam-3952	33	36	to	to	PART
ejpam-3952	33	37	be	be	AUX
ejpam-3952	33	38	zero	zero	NUM
ejpam-3952	33	39	.	.	PUNCT
ejpam-3952	34	1	in	in	ADP
ejpam-3952	34	2	[	[	X
ejpam-3952	34	3	5	5	NUM
ejpam-3952	34	4	]	]	PUNCT
ejpam-3952	34	5	,	,	PUNCT
ejpam-3952	34	6	parity	parity	NOUN
ejpam-3952	34	7	and	and	CCONJ
ejpam-3952	34	8	some	some	DET
ejpam-3952	34	9	properties	property	NOUN
ejpam-3952	34	10	of	of	ADP
ejpam-3952	34	11	isosceles	isoscele	NOUN
ejpam-3952	34	12	triangular	triangular	NOUN
ejpam-3952	34	13	numbers	number	NOUN
ejpam-3952	34	14	have	have	AUX
ejpam-3952	34	15	been	be	AUX
ejpam-3952	34	16	studied	study	VERB
ejpam-3952	34	17	.	.	PUNCT
ejpam-3952	35	1	the	the	DET
ejpam-3952	35	2	nth	nth	PROPN
ejpam-3952	35	3	l	l	NOUN
ejpam-3952	35	4	-	-	PUNCT
ejpam-3952	35	5	isosceles	isoscele	NOUN
ejpam-3952	35	6	triangular	triangular	NOUN
ejpam-3952	35	7	number	number	NOUN
ejpam-3952	35	8	t	t	PROPN
ejpam-3952	35	9	(	(	PUNCT
ejpam-3952	35	10	n	n	CCONJ
ejpam-3952	35	11	,	,	PUNCT
ejpam-3952	35	12	l	l	NOUN
ejpam-3952	35	13	)	)	PUNCT
ejpam-3952	35	14	can	can	AUX
ejpam-3952	35	15	be	be	AUX
ejpam-3952	35	16	represented	represent	VERB
ejpam-3952	35	17	as	as	ADP
ejpam-3952	35	18	an	an	DET
ejpam-3952	35	19	isosceles	isoscele	NOUN
ejpam-3952	35	20	triangular	triangular	NOUN
ejpam-3952	35	21	arrangement	arrangement	NOUN
ejpam-3952	35	22	of	of	ADP
ejpam-3952	35	23	points	point	NOUN
ejpam-3952	35	24	in	in	ADP
ejpam-3952	35	25	figure	figure	NOUN
ejpam-3952	35	26	2	2	NUM
ejpam-3952	35	27	.	.	PUNCT
ejpam-3952	35	28	illustrative	illustrative	ADJ
ejpam-3952	35	29	examples	example	NOUN
ejpam-3952	35	30	of	of	ADP
ejpam-3952	35	31	isosceles	isoscele	NOUN
ejpam-3952	35	32	triangular	triangular	NOUN
ejpam-3952	35	33	numbers	number	NOUN
ejpam-3952	35	34	t	t	PROPN
ejpam-3952	35	35	(	(	PUNCT
ejpam-3952	35	36	4	4	NUM
ejpam-3952	35	37	,	,	PUNCT
ejpam-3952	35	38	2	2	NUM
ejpam-3952	35	39	)	)	PUNCT
ejpam-3952	35	40	=	=	SYM
ejpam-3952	36	1	16	16	NUM
ejpam-3952	36	2	and	and	CCONJ
ejpam-3952	36	3	t	t	PROPN
ejpam-3952	36	4	(	(	PUNCT
ejpam-3952	36	5	4	4	NUM
ejpam-3952	36	6	,	,	PUNCT
ejpam-3952	36	7	3	3	NUM
ejpam-3952	36	8	)	)	PUNCT
ejpam-3952	37	1	=	=	SYM
ejpam-3952	37	2	22	22	NUM
ejpam-3952	37	3	are	be	AUX
ejpam-3952	37	4	given	give	VERB
ejpam-3952	37	5	in	in	ADP
ejpam-3952	37	6	example	example	NOUN
ejpam-3952	37	7	1	1	NUM
ejpam-3952	37	8	.	.	PUNCT
ejpam-3952	37	9	example	example	NOUN
ejpam-3952	38	1	1	1	NUM
ejpam-3952	38	2	.	.	PUNCT
ejpam-3952	39	1	the	the	DET
ejpam-3952	39	2	positive	positive	ADJ
ejpam-3952	39	3	integers	integer	NOUN
ejpam-3952	39	4	16	16	NUM
ejpam-3952	39	5	and	and	CCONJ
ejpam-3952	39	6	22	22	NUM
ejpam-3952	39	7	are	be	AUX
ejpam-3952	39	8	2	2	NUM
ejpam-3952	39	9	-	-	PUNCT
ejpam-3952	39	10	isosceles	isoscele	NOUN
ejpam-3952	39	11	and	and	CCONJ
ejpam-3952	39	12	3	3	NUM
ejpam-3952	39	13	-	-	PUNCT
ejpam-3952	39	14	isosceles	isoscele	NOUN
ejpam-3952	39	15	triangular	triangular	NOUN
ejpam-3952	39	16	numbers	number	NOUN
ejpam-3952	39	17	,	,	PUNCT
ejpam-3952	39	18	respectively	respectively	ADV
ejpam-3952	39	19	.	.	PUNCT
ejpam-3952	40	1	they	they	PRON
ejpam-3952	40	2	can	can	AUX
ejpam-3952	40	3	be	be	AUX
ejpam-3952	40	4	represented	represent	VERB
ejpam-3952	40	5	as	as	ADP
ejpam-3952	40	6	isosceles	isoscele	NOUN
ejpam-3952	40	7	triangular	triangular	NOUN
ejpam-3952	40	8	shapes	shape	NOUN
ejpam-3952	40	9	as	as	SCONJ
ejpam-3952	40	10	follows	follow	VERB
ejpam-3952	40	11	.	.	PUNCT
ejpam-3952	41	1	a	a	DET
ejpam-3952	41	2	polygonal	polygonal	ADJ
ejpam-3952	41	3	number	number	NOUN
ejpam-3952	41	4	is	be	AUX
ejpam-3952	41	5	a	a	DET
ejpam-3952	41	6	number	number	NOUN
ejpam-3952	41	7	represented	represent	VERB
ejpam-3952	41	8	as	as	ADP
ejpam-3952	41	9	points	point	NOUN
ejpam-3952	41	10	or	or	CCONJ
ejpam-3952	41	11	pebbles	pebble	NOUN
ejpam-3952	41	12	arranged	arrange	VERB
ejpam-3952	41	13	in	in	ADP
ejpam-3952	41	14	the	the	DET
ejpam-3952	41	15	shape	shape	NOUN
ejpam-3952	41	16	of	of	ADP
ejpam-3952	41	17	a	a	DET
ejpam-3952	41	18	regular	regular	ADJ
ejpam-3952	41	19	polygon	polygon	NOUN
ejpam-3952	41	20	(	(	PUNCT
ejpam-3952	41	21	see	see	VERB
ejpam-3952	41	22	[	[	X
ejpam-3952	41	23	2	2	NUM
ejpam-3952	41	24	]	]	NUM
ejpam-3952	41	25	)	)	PUNCT
ejpam-3952	41	26	.	.	PUNCT
ejpam-3952	42	1	for	for	ADP
ejpam-3952	42	2	positive	positive	ADJ
ejpam-3952	42	3	integers	integer	NOUN
ejpam-3952	42	4	n	n	PRON
ejpam-3952	42	5	and	and	CCONJ
ejpam-3952	42	6	m	m	PROPN
ejpam-3952	42	7	,	,	PUNCT
ejpam-3952	42	8	the	the	DET
ejpam-3952	42	9	nth	nth	NOUN
ejpam-3952	42	10	m	m	ADJ
ejpam-3952	42	11	-	-	ADJ
ejpam-3952	42	12	gonal	gonal	ADJ
ejpam-3952	42	13	number	number	NOUN
ejpam-3952	42	14	is	be	AUX
ejpam-3952	42	15	defined	define	VERB
ejpam-3952	42	16	to	to	PART
ejpam-3952	42	17	be	be	AUX
ejpam-3952	42	18	j.	j.	PROPN
ejpam-3952	42	19	punpim	punpim	PROPN
ejpam-3952	42	20	,	,	PUNCT
ejpam-3952	42	21	s.	s.	PROPN
ejpam-3952	42	22	jitman	jitman	PROPN
ejpam-3952	42	23	/	/	SYM
ejpam-3952	42	24	eur	eur	PROPN
ejpam-3952	42	25	.	.	PUNCT
ejpam-3952	43	1	j.	j.	PROPN
ejpam-3952	43	2	pure	pure	PROPN
ejpam-3952	43	3	appl	appl	PROPN
ejpam-3952	43	4	.	.	PROPN
ejpam-3952	43	5	math	math	PROPN
ejpam-3952	43	6	,	,	PUNCT
ejpam-3952	43	7	14	14	NUM
ejpam-3952	43	8	(	(	PUNCT
ejpam-3952	43	9	2	2	NUM
ejpam-3952	43	10	)	)	PUNCT
ejpam-3952	43	11	(	(	PUNCT
ejpam-3952	43	12	2021	2021	NUM
ejpam-3952	43	13	)	)	PUNCT
ejpam-3952	43	14	,	,	PUNCT
ejpam-3952	43	15	380	380	NUM
ejpam-3952	43	16	-	-	SYM
ejpam-3952	43	17	395	395	NUM
ejpam-3952	43	18	382	382	NUM
ejpam-3952	43	19	1	1	NUM
ejpam-3952	43	20	+	+	NUM
ejpam-3952	43	21	l	l	NOUN
ejpam-3952	43	22	points	point	NOUN
ejpam-3952	43	23	1	1	NUM
ejpam-3952	43	24	+	+	NUM
ejpam-3952	43	25	2l	2l	NUM
ejpam-3952	43	26	points	point	NOUN
ejpam-3952	43	27	1	1	NUM
ejpam-3952	43	28	+	+	CCONJ
ejpam-3952	43	29	(	(	PUNCT
ejpam-3952	43	30	n−	n−	NOUN
ejpam-3952	43	31	3)l	3)l	NOUN
ejpam-3952	43	32	points	point	VERB
ejpam-3952	43	33	1	1	NUM
ejpam-3952	43	34	+	+	CCONJ
ejpam-3952	43	35	(	(	PUNCT
ejpam-3952	43	36	n−	n−	NOUN
ejpam-3952	43	37	2)l	2)l	NOUN
ejpam-3952	43	38	points	point	VERB
ejpam-3952	43	39	1	1	NUM
ejpam-3952	43	40	+	+	CCONJ
ejpam-3952	43	41	(	(	PUNCT
ejpam-3952	43	42	n−	n−	NOUN
ejpam-3952	43	43	1)l	1)l	NUM
ejpam-3952	43	44	points	point	NOUN
ejpam-3952	43	45	n	n	NUM
ejpam-3952	43	46	points	point	NOUN
ejpam-3952	43	47	figure	figure	NOUN
ejpam-3952	43	48	2	2	NUM
ejpam-3952	43	49	:	:	PUNCT
ejpam-3952	43	50	isosceles	isoscele	NOUN
ejpam-3952	43	51	triangular	triangular	NOUN
ejpam-3952	43	52	number	number	NOUN
ejpam-3952	43	53	t	t	PROPN
ejpam-3952	43	54	(	(	PUNCT
ejpam-3952	43	55	n	n	CCONJ
ejpam-3952	43	56	,	,	PUNCT
ejpam-3952	43	57	l	l	NOUN
ejpam-3952	43	58	)	)	PUNCT
ejpam-3952	43	59	(	(	PUNCT
ejpam-3952	43	60	a	a	X
ejpam-3952	43	61	)	)	PUNCT
ejpam-3952	43	62	t	t	NOUN
ejpam-3952	43	63	(	(	PUNCT
ejpam-3952	43	64	4	4	NUM
ejpam-3952	43	65	,	,	PUNCT
ejpam-3952	43	66	2	2	NUM
ejpam-3952	43	67	)	)	PUNCT
ejpam-3952	43	68	=	=	SYM
ejpam-3952	43	69	16	16	NUM
ejpam-3952	43	70	(	(	PUNCT
ejpam-3952	43	71	b	b	NOUN
ejpam-3952	43	72	)	)	PUNCT
ejpam-3952	43	73	t	t	NOUN
ejpam-3952	43	74	(	(	PUNCT
ejpam-3952	43	75	4	4	NUM
ejpam-3952	43	76	,	,	PUNCT
ejpam-3952	43	77	3	3	NUM
ejpam-3952	43	78	)	)	PUNCT
ejpam-3952	43	79	=	=	SYM
ejpam-3952	43	80	22	22	NUM
ejpam-3952	43	81	figure	figure	NOUN
ejpam-3952	43	82	3	3	NUM
ejpam-3952	43	83	:	:	PUNCT
ejpam-3952	43	84	isosceles	isoscele	NOUN
ejpam-3952	43	85	triangular	triangular	NOUN
ejpam-3952	43	86	numbers	number	NOUN
ejpam-3952	43	87	t	t	PROPN
ejpam-3952	43	88	(	(	PUNCT
ejpam-3952	43	89	4	4	NUM
ejpam-3952	43	90	,	,	PUNCT
ejpam-3952	43	91	2	2	NUM
ejpam-3952	43	92	)	)	PUNCT
ejpam-3952	43	93	=	=	SYM
ejpam-3952	43	94	16	16	NUM
ejpam-3952	43	95	and	and	CCONJ
ejpam-3952	43	96	t	t	PROPN
ejpam-3952	43	97	(	(	PUNCT
ejpam-3952	43	98	4	4	NUM
ejpam-3952	43	99	,	,	PUNCT
ejpam-3952	43	100	3	3	NUM
ejpam-3952	43	101	)	)	PUNCT
ejpam-3952	43	102	=	=	SYM
ejpam-3952	43	103	22	22	NUM
ejpam-3952	43	104	p	p	NOUN
ejpam-3952	43	105	(	(	PUNCT
ejpam-3952	43	106	n	n	CCONJ
ejpam-3952	43	107	,	,	PUNCT
ejpam-3952	43	108	m	m	NOUN
ejpam-3952	43	109	)	)	PUNCT
ejpam-3952	43	110	=	=	SYM
ejpam-3952	44	1	1	1	NUM
ejpam-3952	44	2	+	+	CCONJ
ejpam-3952	44	3	(	(	PUNCT
ejpam-3952	44	4	1	1	NUM
ejpam-3952	44	5	+	+	CCONJ
ejpam-3952	44	6	(	(	PUNCT
ejpam-3952	44	7	m−	m−	PROPN
ejpam-3952	44	8	2	2	NUM
ejpam-3952	44	9	)	)	PUNCT
ejpam-3952	44	10	)	)	PUNCT
ejpam-3952	45	1	+	+	CCONJ
ejpam-3952	45	2	(	(	PUNCT
ejpam-3952	45	3	1	1	NUM
ejpam-3952	45	4	+	+	NUM
ejpam-3952	45	5	2(m−	2(m−	NUM
ejpam-3952	45	6	2	2	NUM
ejpam-3952	45	7	)	)	PUNCT
ejpam-3952	45	8	)	)	PUNCT
ejpam-3952	46	1	+	+	CCONJ
ejpam-3952	46	2	·	·	PUNCT
ejpam-3952	46	3	·	·	PUNCT
ejpam-3952	46	4	·	·	PUNCT
ejpam-3952	46	5	+	+	CCONJ
ejpam-3952	46	6	(	(	PUNCT
ejpam-3952	46	7	1	1	NUM
ejpam-3952	46	8	+	+	CCONJ
ejpam-3952	46	9	(	(	PUNCT
ejpam-3952	46	10	m−	m−	PROPN
ejpam-3952	46	11	2)(n−	2)(n−	NUM
ejpam-3952	46	12	1	1	NUM
ejpam-3952	46	13	)	)	PUNCT
ejpam-3952	46	14	)	)	PUNCT
ejpam-3952	47	1	=	=	SYM
ejpam-3952	48	1	n	n	PROPN
ejpam-3952	48	2	+	+	CCONJ
ejpam-3952	48	3	n(n−	n(n−	PROPN
ejpam-3952	48	4	1)(m−	1)(m−	NUM
ejpam-3952	48	5	2	2	NUM
ejpam-3952	48	6	)	)	PUNCT
ejpam-3952	48	7	2	2	NUM
ejpam-3952	48	8	.	.	PUNCT
ejpam-3952	49	1	it	it	PRON
ejpam-3952	49	2	is	be	AUX
ejpam-3952	49	3	not	not	PART
ejpam-3952	49	4	difficult	difficult	ADJ
ejpam-3952	49	5	to	to	PART
ejpam-3952	49	6	see	see	VERB
ejpam-3952	49	7	that	that	DET
ejpam-3952	49	8	t	t	PROPN
ejpam-3952	49	9	(	(	PUNCT
ejpam-3952	49	10	n	n	CCONJ
ejpam-3952	49	11	,	,	PUNCT
ejpam-3952	49	12	l	l	NOUN
ejpam-3952	49	13	)	)	PUNCT
ejpam-3952	50	1	=	=	SYM
ejpam-3952	50	2	p	p	X
ejpam-3952	50	3	(	(	PUNCT
ejpam-3952	50	4	n	n	X
ejpam-3952	50	5	,	,	PUNCT
ejpam-3952	50	6	l	l	PROPN
ejpam-3952	50	7	+	+	PROPN
ejpam-3952	50	8	2	2	NUM
ejpam-3952	50	9	)	)	PUNCT
ejpam-3952	50	10	,	,	PUNCT
ejpam-3952	50	11	i.e.	i.e.	X
ejpam-3952	50	12	,	,	PUNCT
ejpam-3952	50	13	an	an	DET
ejpam-3952	50	14	isosceles	isoscele	NOUN
ejpam-3952	50	15	triangular	triangular	NOUN
ejpam-3952	50	16	number	number	NOUN
ejpam-3952	50	17	is	be	AUX
ejpam-3952	50	18	actually	actually	ADV
ejpam-3952	50	19	a	a	DET
ejpam-3952	50	20	shifted	shift	VERB
ejpam-3952	50	21	version	version	NOUN
ejpam-3952	50	22	of	of	ADP
ejpam-3952	50	23	a	a	DET
ejpam-3952	50	24	polygonal	polygonal	ADJ
ejpam-3952	50	25	number	number	NOUN
ejpam-3952	50	26	.	.	PUNCT
ejpam-3952	51	1	however	however	ADV
ejpam-3952	51	2	,	,	PUNCT
ejpam-3952	51	3	in	in	ADP
ejpam-3952	51	4	this	this	DET
ejpam-3952	51	5	manuscript	manuscript	NOUN
ejpam-3952	51	6	,	,	PUNCT
ejpam-3952	51	7	the	the	DET
ejpam-3952	51	8	notion	notion	NOUN
ejpam-3952	51	9	of	of	ADP
ejpam-3952	51	10	isosceles	isoscele	NOUN
ejpam-3952	51	11	triangular	triangular	NOUN
ejpam-3952	51	12	numbers	number	NOUN
ejpam-3952	51	13	is	be	AUX
ejpam-3952	51	14	used	use	VERB
ejpam-3952	51	15	to	to	PART
ejpam-3952	51	16	present	present	VERB
ejpam-3952	51	17	the	the	DET
ejpam-3952	51	18	generality	generality	NOUN
ejpam-3952	51	19	of	of	ADP
ejpam-3952	51	20	triangular	triangular	NOUN
ejpam-3952	51	21	numbers	number	NOUN
ejpam-3952	51	22	and	and	CCONJ
ejpam-3952	51	23	nice	nice	ADJ
ejpam-3952	51	24	appearance	appearance	NOUN
ejpam-3952	51	25	of	of	ADP
ejpam-3952	51	26	some	some	DET
ejpam-3952	51	27	identities	identity	NOUN
ejpam-3952	51	28	.	.	PUNCT
ejpam-3952	52	1	as	as	SCONJ
ejpam-3952	52	2	mentioned	mention	VERB
ejpam-3952	52	3	above	above	ADV
ejpam-3952	52	4	,	,	PUNCT
ejpam-3952	52	5	various	various	ADJ
ejpam-3952	52	6	characterizations	characterization	NOUN
ejpam-3952	52	7	and	and	CCONJ
ejpam-3952	52	8	identities	identity	NOUN
ejpam-3952	52	9	for	for	ADP
ejpam-3952	52	10	triangular	triangular	NOUN
ejpam-3952	52	11	numbers	number	NOUN
ejpam-3952	52	12	have	have	AUX
ejpam-3952	52	13	been	be	AUX
ejpam-3952	52	14	established	establish	VERB
ejpam-3952	52	15	.	.	PUNCT
ejpam-3952	53	1	for	for	ADP
ejpam-3952	53	2	isosceles	isoscele	NOUN
ejpam-3952	53	3	triangular	triangular	NOUN
ejpam-3952	53	4	numbers	number	NOUN
ejpam-3952	53	5	,	,	PUNCT
ejpam-3952	53	6	only	only	ADV
ejpam-3952	53	7	a	a	DET
ejpam-3952	53	8	few	few	ADJ
ejpam-3952	53	9	works	work	NOUN
ejpam-3952	53	10	have	have	AUX
ejpam-3952	53	11	been	be	AUX
ejpam-3952	53	12	done	do	VERB
ejpam-3952	53	13	on	on	ADP
ejpam-3952	53	14	their	their	PRON
ejpam-3952	53	15	characterizations	characterization	NOUN
ejpam-3952	53	16	and	and	CCONJ
ejpam-3952	53	17	identities	identity	NOUN
ejpam-3952	53	18	.	.	PUNCT
ejpam-3952	54	1	it	it	PRON
ejpam-3952	54	2	is	be	AUX
ejpam-3952	54	3	therefore	therefore	ADV
ejpam-3952	54	4	of	of	ADP
ejpam-3952	54	5	interest	interest	NOUN
ejpam-3952	54	6	to	to	PART
ejpam-3952	54	7	investigate	investigate	VERB
ejpam-3952	54	8	such	such	ADJ
ejpam-3952	54	9	problems	problem	NOUN
ejpam-3952	54	10	.	.	PUNCT
ejpam-3952	55	1	the	the	DET
ejpam-3952	55	2	first	first	ADJ
ejpam-3952	55	3	goal	goal	NOUN
ejpam-3952	55	4	of	of	ADP
ejpam-3952	55	5	this	this	DET
ejpam-3952	55	6	paper	paper	NOUN
ejpam-3952	55	7	is	be	AUX
ejpam-3952	55	8	to	to	PART
ejpam-3952	55	9	generalize	generalize	VERB
ejpam-3952	55	10	the	the	DET
ejpam-3952	55	11	facts	fact	NOUN
ejpam-3952	55	12	(	(	PUNCT
ejpam-3952	55	13	see	see	VERB
ejpam-3952	55	14	[	[	X
ejpam-3952	55	15	3	3	NUM
ejpam-3952	55	16	]	]	PUNCT
ejpam-3952	55	17	)	)	PUNCT
ejpam-3952	56	1	that	that	SCONJ
ejpam-3952	56	2	“	"	PUNCT
ejpam-3952	56	3	a	a	DET
ejpam-3952	56	4	positive	positive	ADJ
ejpam-3952	56	5	integer	integer	NOUN
ejpam-3952	56	6	n	n	NUM
ejpam-3952	56	7	is	be	AUX
ejpam-3952	56	8	a	a	DET
ejpam-3952	56	9	triangular	triangular	NOUN
ejpam-3952	56	10	number	number	NOUN
ejpam-3952	56	11	if	if	SCONJ
ejpam-3952	56	12	and	and	CCONJ
ejpam-3952	56	13	only	only	ADV
ejpam-3952	57	1	if	if	SCONJ
ejpam-3952	57	2	9n	9n	NOUN
ejpam-3952	57	3	+	+	CCONJ
ejpam-3952	57	4	1	1	NUM
ejpam-3952	57	5	is	be	AUX
ejpam-3952	57	6	a	a	DET
ejpam-3952	57	7	triangular	triangular	ADJ
ejpam-3952	57	8	number	number	NOUN
ejpam-3952	57	9	”	"	PUNCT
ejpam-3952	57	10	and	and	CCONJ
ejpam-3952	57	11	“	"	PUNCT
ejpam-3952	57	12	n	n	PRON
ejpam-3952	57	13	is	be	AUX
ejpam-3952	57	14	a	a	DET
ejpam-3952	57	15	triangular	triangular	NOUN
ejpam-3952	57	16	number	number	NOUN
ejpam-3952	57	17	if	if	SCONJ
ejpam-3952	57	18	and	and	CCONJ
ejpam-3952	57	19	only	only	ADV
ejpam-3952	57	20	if	if	SCONJ
ejpam-3952	57	21	8n	8n	NOUN
ejpam-3952	57	22	+	+	CCONJ
ejpam-3952	57	23	1	1	NUM
ejpam-3952	57	24	is	be	AUX
ejpam-3952	57	25	square	square	ADJ
ejpam-3952	57	26	”	"	PUNCT
ejpam-3952	57	27	to	to	ADP
ejpam-3952	57	28	isosceles	isoscele	NOUN
ejpam-3952	57	29	triangular	triangular	NOUN
ejpam-3952	57	30	numbers	number	NOUN
ejpam-3952	57	31	.	.	PUNCT
ejpam-3952	58	1	secondly	secondly	ADV
ejpam-3952	58	2	,	,	PUNCT
ejpam-3952	58	3	we	we	PRON
ejpam-3952	58	4	aim	aim	VERB
ejpam-3952	58	5	to	to	PART
ejpam-3952	58	6	generalize	generalize	VERB
ejpam-3952	58	7	some	some	DET
ejpam-3952	58	8	identities	identity	NOUN
ejpam-3952	58	9	for	for	ADP
ejpam-3952	58	10	triangular	triangular	NOUN
ejpam-3952	58	11	numbers	number	NOUN
ejpam-3952	58	12	in	in	ADP
ejpam-3952	58	13	[	[	X
ejpam-3952	58	14	4	4	NUM
ejpam-3952	58	15	]	]	PUNCT
ejpam-3952	58	16	and	and	CCONJ
ejpam-3952	58	17	[	[	X
ejpam-3952	58	18	7	7	X
ejpam-3952	58	19	]	]	PUNCT
ejpam-3952	58	20	to	to	ADP
ejpam-3952	58	21	isosceles	isoscele	NOUN
ejpam-3952	58	22	triangular	triangular	NOUN
ejpam-3952	58	23	numbers	number	NOUN
ejpam-3952	58	24	.	.	PUNCT
ejpam-3952	59	1	to	to	ADP
ejpam-3952	59	2	the	the	DET
ejpam-3952	59	3	best	good	ADJ
ejpam-3952	59	4	of	of	ADP
ejpam-3952	59	5	our	our	PRON
ejpam-3952	59	6	knowledge	knowledge	NOUN
ejpam-3952	59	7	,	,	PUNCT
ejpam-3952	59	8	the	the	DET
ejpam-3952	59	9	characterizations	characterization	NOUN
ejpam-3952	59	10	and	and	CCONJ
ejpam-3952	59	11	identities	identity	NOUN
ejpam-3952	59	12	for	for	ADP
ejpam-3952	59	13	isosceles	isoscele	NOUN
ejpam-3952	59	14	triangular	triangular	NOUN
ejpam-3952	59	15	numbers	number	NOUN
ejpam-3952	59	16	presented	present	VERB
ejpam-3952	59	17	in	in	ADP
ejpam-3952	59	18	this	this	DET
ejpam-3952	59	19	paper	paper	NOUN
ejpam-3952	59	20	have	have	AUX
ejpam-3952	59	21	not	not	PART
ejpam-3952	59	22	been	be	AUX
ejpam-3952	59	23	established	establish	VERB
ejpam-3952	59	24	either	either	CCONJ
ejpam-3952	59	25	in	in	ADP
ejpam-3952	59	26	terms	term	NOUN
ejpam-3952	59	27	of	of	ADP
ejpam-3952	59	28	isosceles	isoscele	NOUN
ejpam-3952	59	29	triangular	triangular	NOUN
ejpam-3952	59	30	numbers	number	NOUN
ejpam-3952	59	31	or	or	CCONJ
ejpam-3952	59	32	polygonal	polygonal	ADJ
ejpam-3952	59	33	numbers	number	NOUN
ejpam-3952	59	34	.	.	PUNCT
ejpam-3952	60	1	the	the	DET
ejpam-3952	60	2	paper	paper	NOUN
ejpam-3952	60	3	is	be	AUX
ejpam-3952	60	4	organized	organize	VERB
ejpam-3952	60	5	as	as	SCONJ
ejpam-3952	60	6	follows	follow	VERB
ejpam-3952	60	7	.	.	PUNCT
ejpam-3952	61	1	characterizations	characterization	NOUN
ejpam-3952	61	2	of	of	ADP
ejpam-3952	61	3	isosceles	isoscele	NOUN
ejpam-3952	61	4	triangular	triangular	NOUN
ejpam-3952	61	5	numbers	number	NOUN
ejpam-3952	61	6	are	be	AUX
ejpam-3952	61	7	presented	present	VERB
ejpam-3952	61	8	in	in	ADP
ejpam-3952	61	9	section	section	NOUN
ejpam-3952	61	10	2	2	NUM
ejpam-3952	61	11	as	as	ADV
ejpam-3952	61	12	well	well	ADV
ejpam-3952	61	13	as	as	ADP
ejpam-3952	61	14	their	their	PRON
ejpam-3952	61	15	links	link	NOUN
ejpam-3952	61	16	with	with	ADP
ejpam-3952	61	17	other	other	ADJ
ejpam-3952	61	18	figurate	figurate	NOUN
ejpam-3952	61	19	numbers	number	NOUN
ejpam-3952	61	20	.	.	PUNCT
ejpam-3952	62	1	in	in	ADP
ejpam-3952	62	2	section	section	NOUN
ejpam-3952	62	3	3	3	NUM
ejpam-3952	62	4	,	,	PUNCT
ejpam-3952	62	5	some	some	DET
ejpam-3952	62	6	identities	identity	NOUN
ejpam-3952	62	7	for	for	ADP
ejpam-3952	62	8	isosceles	isoscele	NOUN
ejpam-3952	62	9	triangular	triangular	NOUN
ejpam-3952	62	10	numbers	number	NOUN
ejpam-3952	62	11	are	be	AUX
ejpam-3952	62	12	established	establish	VERB
ejpam-3952	62	13	as	as	ADP
ejpam-3952	62	14	generalizations	generalization	NOUN
ejpam-3952	62	15	of	of	ADP
ejpam-3952	62	16	triangular	triangular	NOUN
ejpam-3952	62	17	numbers	number	NOUN
ejpam-3952	62	18	.	.	PUNCT
ejpam-3952	63	1	summary	summary	NOUN
ejpam-3952	63	2	and	and	CCONJ
ejpam-3952	63	3	remarks	remark	NOUN
ejpam-3952	63	4	are	be	AUX
ejpam-3952	63	5	provided	provide	VERB
ejpam-3952	63	6	in	in	ADP
ejpam-3952	63	7	section	section	NOUN
ejpam-3952	63	8	4	4	NUM
ejpam-3952	63	9	.	.	PUNCT
ejpam-3952	63	10	j.	j.	PROPN
ejpam-3952	63	11	punpim	punpim	PROPN
ejpam-3952	63	12	,	,	PUNCT
ejpam-3952	63	13	s.	s.	PROPN
ejpam-3952	63	14	jitman	jitman	PROPN
ejpam-3952	63	15	/	/	SYM
ejpam-3952	63	16	eur	eur	PROPN
ejpam-3952	63	17	.	.	PUNCT
ejpam-3952	64	1	j.	j.	PROPN
ejpam-3952	64	2	pure	pure	PROPN
ejpam-3952	64	3	appl	appl	PROPN
ejpam-3952	64	4	.	.	PROPN
ejpam-3952	64	5	math	math	PROPN
ejpam-3952	64	6	,	,	PUNCT
ejpam-3952	64	7	14	14	NUM
ejpam-3952	64	8	(	(	PUNCT
ejpam-3952	64	9	2	2	NUM
ejpam-3952	64	10	)	)	PUNCT
ejpam-3952	64	11	(	(	PUNCT
ejpam-3952	64	12	2021	2021	NUM
ejpam-3952	64	13	)	)	PUNCT
ejpam-3952	64	14	,	,	PUNCT
ejpam-3952	64	15	380	380	NUM
ejpam-3952	64	16	-	-	SYM
ejpam-3952	64	17	395	395	NUM
ejpam-3952	64	18	383	383	NUM
ejpam-3952	64	19	2	2	NUM
ejpam-3952	64	20	.	.	PUNCT
ejpam-3952	65	1	characterizations	characterization	NOUN
ejpam-3952	65	2	of	of	ADP
ejpam-3952	65	3	isosceles	isoscele	NOUN
ejpam-3952	65	4	triangular	triangular	NOUN
ejpam-3952	65	5	numbers	number	NOUN
ejpam-3952	65	6	in	in	ADP
ejpam-3952	65	7	this	this	DET
ejpam-3952	65	8	section	section	NOUN
ejpam-3952	65	9	,	,	PUNCT
ejpam-3952	65	10	some	some	DET
ejpam-3952	65	11	characterizations	characterization	NOUN
ejpam-3952	65	12	of	of	ADP
ejpam-3952	65	13	isosceles	isoscele	NOUN
ejpam-3952	65	14	triangular	triangular	NOUN
ejpam-3952	65	15	numbers	number	NOUN
ejpam-3952	65	16	are	be	AUX
ejpam-3952	65	17	given	give	VERB
ejpam-3952	65	18	as	as	ADV
ejpam-3952	65	19	well	well	ADV
ejpam-3952	65	20	as	as	ADP
ejpam-3952	65	21	links	link	NOUN
ejpam-3952	65	22	with	with	ADP
ejpam-3952	65	23	other	other	ADJ
ejpam-3952	65	24	figurate	figurate	NOUN
ejpam-3952	65	25	numbers	number	NOUN
ejpam-3952	65	26	.	.	PUNCT
ejpam-3952	66	1	some	some	DET
ejpam-3952	66	2	classical	classical	ADJ
ejpam-3952	66	3	results	result	NOUN
ejpam-3952	66	4	on	on	ADP
ejpam-3952	66	5	the	the	DET
ejpam-3952	66	6	characterizations	characterization	NOUN
ejpam-3952	66	7	of	of	ADP
ejpam-3952	66	8	triangular	triangular	NOUN
ejpam-3952	66	9	numbers	number	NOUN
ejpam-3952	66	10	can	can	AUX
ejpam-3952	66	11	be	be	AUX
ejpam-3952	66	12	viewed	view	VERB
ejpam-3952	66	13	as	as	ADP
ejpam-3952	66	14	a	a	DET
ejpam-3952	66	15	special	special	ADJ
ejpam-3952	66	16	case	case	NOUN
ejpam-3952	66	17	.	.	PUNCT
ejpam-3952	67	1	2.1	2.1	NUM
ejpam-3952	67	2	.	.	PUNCT
ejpam-3952	67	3	characterizations	characterization	NOUN
ejpam-3952	67	4	and	and	CCONJ
ejpam-3952	67	5	recursive	recursive	ADJ
ejpam-3952	67	6	constructions	construction	NOUN
ejpam-3952	67	7	of	of	ADP
ejpam-3952	67	8	isosceles	isoscele	NOUN
ejpam-3952	67	9	triangular	triangular	NOUN
ejpam-3952	67	10	numbers	number	NOUN
ejpam-3952	67	11	in	in	ADP
ejpam-3952	67	12	this	this	DET
ejpam-3952	67	13	subsection	subsection	NOUN
ejpam-3952	67	14	,	,	PUNCT
ejpam-3952	67	15	we	we	PRON
ejpam-3952	67	16	focus	focus	VERB
ejpam-3952	67	17	on	on	ADP
ejpam-3952	67	18	a	a	DET
ejpam-3952	67	19	generalization	generalization	NOUN
ejpam-3952	67	20	of	of	ADP
ejpam-3952	67	21	the	the	DET
ejpam-3952	67	22	well	well	ADV
ejpam-3952	67	23	-	-	PUNCT
ejpam-3952	67	24	known	know	VERB
ejpam-3952	67	25	fact	fact	NOUN
ejpam-3952	67	26	“	"	PUNCT
ejpam-3952	67	27	a	a	DET
ejpam-3952	67	28	positive	positive	ADJ
ejpam-3952	67	29	integer	integer	NOUN
ejpam-3952	67	30	n	n	NUM
ejpam-3952	67	31	is	be	AUX
ejpam-3952	67	32	a	a	DET
ejpam-3952	67	33	triangular	triangular	NOUN
ejpam-3952	67	34	number	number	NOUN
ejpam-3952	67	35	if	if	SCONJ
ejpam-3952	67	36	and	and	CCONJ
ejpam-3952	68	1	only	only	ADV
ejpam-3952	68	2	if	if	SCONJ
ejpam-3952	68	3	9n	9n	NOUN
ejpam-3952	68	4	+	+	CCONJ
ejpam-3952	68	5	1	1	NUM
ejpam-3952	68	6	is	be	AUX
ejpam-3952	68	7	a	a	DET
ejpam-3952	68	8	triangular	triangular	ADJ
ejpam-3952	68	9	number	number	NOUN
ejpam-3952	68	10	”	"	PUNCT
ejpam-3952	68	11	in	in	ADP
ejpam-3952	68	12	[	[	X
ejpam-3952	68	13	3	3	NUM
ejpam-3952	68	14	]	]	PUNCT
ejpam-3952	68	15	.	.	PUNCT
ejpam-3952	69	1	in	in	ADP
ejpam-3952	69	2	the	the	DET
ejpam-3952	69	3	following	following	NOUN
ejpam-3952	69	4	theorem	theorem	NOUN
ejpam-3952	69	5	,	,	PUNCT
ejpam-3952	69	6	a	a	DET
ejpam-3952	69	7	characterization	characterization	NOUN
ejpam-3952	69	8	of	of	ADP
ejpam-3952	69	9	l	l	NOUN
ejpam-3952	69	10	-	-	PUNCT
ejpam-3952	69	11	isosceles	isoscele	NOUN
ejpam-3952	69	12	triangular	triangular	NOUN
ejpam-3952	69	13	numbers	number	NOUN
ejpam-3952	69	14	is	be	AUX
ejpam-3952	69	15	presented	present	VERB
ejpam-3952	69	16	.	.	PUNCT
ejpam-3952	70	1	a	a	DET
ejpam-3952	70	2	recursive	recursive	ADJ
ejpam-3952	70	3	construction	construction	NOUN
ejpam-3952	70	4	of	of	ADP
ejpam-3952	70	5	l	l	NOUN
ejpam-3952	70	6	-	-	PUNCT
ejpam-3952	70	7	isosceles	isoscele	NOUN
ejpam-3952	70	8	triangular	triangular	NOUN
ejpam-3952	70	9	numbers	number	NOUN
ejpam-3952	70	10	can	can	AUX
ejpam-3952	70	11	be	be	AUX
ejpam-3952	70	12	deduced	deduce	VERB
ejpam-3952	70	13	directly	directly	ADV
ejpam-3952	70	14	from	from	ADP
ejpam-3952	70	15	the	the	DET
ejpam-3952	70	16	theorem	theorem	PROPN
ejpam-3952	70	17	.	.	PUNCT
ejpam-3952	70	18	theorem	theorem	NOUN
ejpam-3952	70	19	1	1	NUM
ejpam-3952	70	20	.	.	PUNCT
ejpam-3952	71	1	let	let	VERB
ejpam-3952	71	2	n	n	NOUN
ejpam-3952	71	3	and	and	CCONJ
ejpam-3952	71	4	l	l	NOUN
ejpam-3952	71	5	be	be	AUX
ejpam-3952	71	6	positive	positive	ADJ
ejpam-3952	71	7	integers	integer	NOUN
ejpam-3952	71	8	.	.	PUNCT
ejpam-3952	72	1	let	let	VERB
ejpam-3952	72	2	r	r	NOUN
ejpam-3952	72	3	and	and	CCONJ
ejpam-3952	72	4	s	s	AUX
ejpam-3952	72	5	be	be	AUX
ejpam-3952	72	6	positive	positive	ADJ
ejpam-3952	72	7	integers	integer	NOUN
ejpam-3952	72	8	of	of	ADP
ejpam-3952	72	9	the	the	DET
ejpam-3952	72	10	form	form	NOUN
ejpam-3952	73	1	r	r	NOUN
ejpam-3952	73	2	=	=	SYM
ejpam-3952	73	3	(	(	PUNCT
ejpam-3952	73	4	2l+1)2	2l+1)2	NUM
ejpam-3952	73	5	and	and	CCONJ
ejpam-3952	73	6	s	s	NOUN
ejpam-3952	73	7	=	=	SYM
ejpam-3952	73	8	l3	l3	PROPN
ejpam-3952	73	9	−	−	PROPN
ejpam-3952	73	10	3l2	3l2	NUM
ejpam-3952	74	1	+	+	CCONJ
ejpam-3952	74	2	4	4	NUM
ejpam-3952	74	3	2	2	NUM
ejpam-3952	74	4	.	.	PUNCT
ejpam-3952	75	1	then	then	ADV
ejpam-3952	75	2	n	n	PRON
ejpam-3952	75	3	is	be	AUX
ejpam-3952	75	4	an	an	DET
ejpam-3952	75	5	l	l	NOUN
ejpam-3952	75	6	-	-	PUNCT
ejpam-3952	75	7	isosceles	isoscele	NOUN
ejpam-3952	75	8	triangular	triangular	NOUN
ejpam-3952	75	9	number	number	NOUN
ejpam-3952	75	10	if	if	SCONJ
ejpam-3952	75	11	and	and	CCONJ
ejpam-3952	75	12	only	only	ADV
ejpam-3952	75	13	if	if	SCONJ
ejpam-3952	75	14	rn	rn	PROPN
ejpam-3952	75	15	+	+	NOUN
ejpam-3952	75	16	s	s	X
ejpam-3952	75	17	is	be	AUX
ejpam-3952	75	18	an	an	DET
ejpam-3952	75	19	l	l	ADJ
ejpam-3952	75	20	-	-	PUNCT
ejpam-3952	75	21	isosceles	isoscele	NOUN
ejpam-3952	75	22	triangular	triangular	NOUN
ejpam-3952	75	23	number	number	NOUN
ejpam-3952	75	24	.	.	PUNCT
ejpam-3952	76	1	proof	proof	NOUN
ejpam-3952	76	2	.	.	PUNCT
ejpam-3952	77	1	assume	assume	VERB
ejpam-3952	77	2	that	that	SCONJ
ejpam-3952	77	3	n	n	PRON
ejpam-3952	77	4	is	be	AUX
ejpam-3952	77	5	an	an	DET
ejpam-3952	77	6	l	l	NOUN
ejpam-3952	77	7	-	-	PUNCT
ejpam-3952	77	8	isosceles	isoscele	NOUN
ejpam-3952	77	9	triangular	triangular	NOUN
ejpam-3952	77	10	number	number	NOUN
ejpam-3952	77	11	.	.	PUNCT
ejpam-3952	78	1	then	then	ADV
ejpam-3952	78	2	n	n	PROPN
ejpam-3952	78	3	=	=	SYM
ejpam-3952	78	4	n	n	PROPN
ejpam-3952	78	5	+	+	CCONJ
ejpam-3952	78	6	n(n−	n(n−	PROPN
ejpam-3952	78	7	1)l	1)l	NUM
ejpam-3952	78	8	2	2	NUM
ejpam-3952	78	9	for	for	ADP
ejpam-3952	78	10	some	some	DET
ejpam-3952	78	11	positive	positive	ADJ
ejpam-3952	78	12	integer	integer	NOUN
ejpam-3952	78	13	n.	n.	NOUN
ejpam-3952	78	14	it	it	PRON
ejpam-3952	78	15	follows	follow	VERB
ejpam-3952	78	16	that	that	SCONJ
ejpam-3952	78	17	rn	rn	PROPN
ejpam-3952	79	1	+	+	PROPN
ejpam-3952	79	2	s	s	PART
ejpam-3952	79	3	=	=	X
ejpam-3952	79	4	(	(	PUNCT
ejpam-3952	79	5	2l	2l	X
ejpam-3952	79	6	+	+	X
ejpam-3952	79	7	1)2	1)2	NUM
ejpam-3952	79	8	(	(	PUNCT
ejpam-3952	79	9	n	n	X
ejpam-3952	79	10	+	+	CCONJ
ejpam-3952	79	11	n(n−	n(n−	PROPN
ejpam-3952	79	12	1)l	1)l	NUM
ejpam-3952	79	13	2	2	NUM
ejpam-3952	79	14	)	)	PUNCT
ejpam-3952	79	15	+	+	CCONJ
ejpam-3952	79	16	l3	l3	PROPN
ejpam-3952	79	17	−	−	PROPN
ejpam-3952	79	18	3l2	3l2	NUM
ejpam-3952	79	19	+	+	CCONJ
ejpam-3952	79	20	4	4	NUM
ejpam-3952	79	21	2	2	NUM
ejpam-3952	79	22	=	=	SYM
ejpam-3952	79	23	n2l(2l	n2l(2l	PROPN
ejpam-3952	80	1	+	+	CCONJ
ejpam-3952	80	2	1)2	1)2	NUM
ejpam-3952	80	3	+	+	CCONJ
ejpam-3952	80	4	(	(	PUNCT
ejpam-3952	80	5	2−	2−	NUM
ejpam-3952	80	6	l)n(2l	l)n(2l	PROPN
ejpam-3952	80	7	+	+	CCONJ
ejpam-3952	80	8	1)2	1)2	NUM
ejpam-3952	80	9	+	+	NUM
ejpam-3952	80	10	l3	l3	NOUN
ejpam-3952	80	11	−	−	PROPN
ejpam-3952	80	12	3l2	3l2	NUM
ejpam-3952	80	13	+	+	CCONJ
ejpam-3952	80	14	4	4	NUM
ejpam-3952	80	15	2	2	NUM
ejpam-3952	80	16	=	=	SYM
ejpam-3952	80	17	4l3n2	4l3n2	NOUN
ejpam-3952	81	1	+	+	NUM
ejpam-3952	81	2	4l2n2	4l2n2	NOUN
ejpam-3952	82	1	+	+	CCONJ
ejpam-3952	82	2	ln2	ln2	ADJ
ejpam-3952	82	3	+	+	CCONJ
ejpam-3952	82	4	8l2n	8l2n	ADJ
ejpam-3952	82	5	+	+	CCONJ
ejpam-3952	82	6	8ln	8ln	ADJ
ejpam-3952	82	7	+	+	CCONJ
ejpam-3952	83	1	2n−	2n−	NUM
ejpam-3952	83	2	4l3n−	4l3n−	NUM
ejpam-3952	83	3	4l2n−	4l2n−	NUM
ejpam-3952	83	4	ln	ln	ADJ
ejpam-3952	83	5	+	+	CCONJ
ejpam-3952	83	6	l3	l3	NOUN
ejpam-3952	83	7	−	−	PROPN
ejpam-3952	83	8	3l2	3l2	NUM
ejpam-3952	83	9	+	+	CCONJ
ejpam-3952	83	10	4	4	NUM
ejpam-3952	83	11	2	2	NUM
ejpam-3952	83	12	=	=	SYM
ejpam-3952	83	13	(	(	PUNCT
ejpam-3952	83	14	4l3	4l3	NUM
ejpam-3952	83	15	+	+	SYM
ejpam-3952	83	16	4l2	4l2	NUM
ejpam-3952	84	1	+	+	CCONJ
ejpam-3952	84	2	l)n2	l)n2	ADJ
ejpam-3952	84	3	−	−	PROPN
ejpam-3952	84	4	(	(	PUNCT
ejpam-3952	84	5	4l3	4l3	NUM
ejpam-3952	84	6	−	−	NOUN
ejpam-3952	85	1	4l2	4l2	NUM
ejpam-3952	86	1	−	−	NOUN
ejpam-3952	86	2	7l	7l	NUM
ejpam-3952	86	3	−	−	NUM
ejpam-3952	86	4	2)n	2)n	NUM
ejpam-3952	87	1	+	+	CCONJ
ejpam-3952	87	2	(	(	PUNCT
ejpam-3952	87	3	l3	l3	PROPN
ejpam-3952	87	4	−	−	PROPN
ejpam-3952	87	5	3l2	3l2	NUM
ejpam-3952	87	6	+	+	CCONJ
ejpam-3952	87	7	4	4	NUM
ejpam-3952	87	8	)	)	SYM
ejpam-3952	87	9	2	2	NUM
ejpam-3952	87	10	=	=	SYM
ejpam-3952	87	11	l	l	NOUN
ejpam-3952	87	12	(	(	PUNCT
ejpam-3952	87	13	(	(	PUNCT
ejpam-3952	87	14	4l2	4l2	NUM
ejpam-3952	87	15	+	+	NUM
ejpam-3952	87	16	4l	4l	NOUN
ejpam-3952	87	17	+	+	CCONJ
ejpam-3952	87	18	1)n2	1)n2	NUM
ejpam-3952	87	19	−	−	PROPN
ejpam-3952	87	20	(	(	PUNCT
ejpam-3952	87	21	4l2	4l2	NUM
ejpam-3952	87	22	−	−	NOUN
ejpam-3952	87	23	6l	6l	NUM
ejpam-3952	87	24	−	−	ADV
ejpam-3952	87	25	4)n	4)n	NUM
ejpam-3952	87	26	+	+	CCONJ
ejpam-3952	87	27	(	(	PUNCT
ejpam-3952	87	28	l2	l2	NOUN
ejpam-3952	87	29	−	−	NOUN
ejpam-3952	87	30	4l	4l	NOUN
ejpam-3952	87	31	+	+	CCONJ
ejpam-3952	87	32	4	4	NUM
ejpam-3952	87	33	)	)	PUNCT
ejpam-3952	87	34	)	)	PUNCT
ejpam-3952	88	1	−	−	PROPN
ejpam-3952	89	1	(	(	PUNCT
ejpam-3952	89	2	l	l	NOUN
ejpam-3952	89	3	−	−	PROPN
ejpam-3952	89	4	2	2	NUM
ejpam-3952	89	5	)	)	PUNCT
ejpam-3952	89	6	(	(	PUNCT
ejpam-3952	89	7	(	(	PUNCT
ejpam-3952	89	8	2l	2l	NOUN
ejpam-3952	89	9	+	+	X
ejpam-3952	89	10	1)n−	1)n−	NUM
ejpam-3952	89	11	(	(	PUNCT
ejpam-3952	89	12	l	l	NOUN
ejpam-3952	89	13	−	−	PROPN
ejpam-3952	89	14	2	2	NUM
ejpam-3952	89	15	)	)	PUNCT
ejpam-3952	89	16	)	)	PUNCT
ejpam-3952	89	17	2	2	NUM
ejpam-3952	89	18	=	=	SYM
ejpam-3952	89	19	l	l	NOUN
ejpam-3952	89	20	(	(	PUNCT
ejpam-3952	89	21	(	(	PUNCT
ejpam-3952	89	22	2l	2l	NOUN
ejpam-3952	89	23	+	+	X
ejpam-3952	89	24	1)n−	1)n−	NUM
ejpam-3952	89	25	(	(	PUNCT
ejpam-3952	89	26	l	l	NOUN
ejpam-3952	89	27	−	−	PROPN
ejpam-3952	89	28	2))2	2))2	NUM
ejpam-3952	89	29	−	−	NOUN
ejpam-3952	89	30	(	(	PUNCT
ejpam-3952	89	31	l	l	NOUN
ejpam-3952	89	32	−	−	PROPN
ejpam-3952	89	33	2	2	NUM
ejpam-3952	89	34	)	)	PUNCT
ejpam-3952	89	35	(	(	PUNCT
ejpam-3952	89	36	(	(	PUNCT
ejpam-3952	89	37	2l	2l	NOUN
ejpam-3952	89	38	+	+	X
ejpam-3952	89	39	1)n−	1)n−	NUM
ejpam-3952	89	40	(	(	PUNCT
ejpam-3952	89	41	l	l	NOUN
ejpam-3952	89	42	−	−	PROPN
ejpam-3952	89	43	2	2	NUM
ejpam-3952	89	44	)	)	PUNCT
ejpam-3952	89	45	)	)	PUNCT
ejpam-3952	89	46	2	2	NUM
ejpam-3952	89	47	=	=	SYM
ejpam-3952	89	48	2	2	NUM
ejpam-3952	89	49	(	(	PUNCT
ejpam-3952	89	50	(	(	PUNCT
ejpam-3952	89	51	2l	2l	NOUN
ejpam-3952	89	52	+	+	X
ejpam-3952	89	53	1)n−	1)n−	NUM
ejpam-3952	89	54	(	(	PUNCT
ejpam-3952	89	55	l	l	NOUN
ejpam-3952	89	56	−	−	NOUN
ejpam-3952	89	57	2	2	NUM
ejpam-3952	89	58	)	)	PUNCT
ejpam-3952	89	59	)	)	PUNCT
ejpam-3952	90	1	+	+	CCONJ
ejpam-3952	90	2	l	l	NOUN
ejpam-3952	90	3	(	(	PUNCT
ejpam-3952	90	4	(	(	PUNCT
ejpam-3952	90	5	2l	2l	NOUN
ejpam-3952	90	6	+	+	X
ejpam-3952	90	7	1)n−	1)n−	NUM
ejpam-3952	90	8	(	(	PUNCT
ejpam-3952	90	9	l	l	NOUN
ejpam-3952	90	10	−	−	PROPN
ejpam-3952	90	11	2))2	2))2	NUM
ejpam-3952	90	12	−	−	NOUN
ejpam-3952	90	13	l	l	NOUN
ejpam-3952	90	14	(	(	PUNCT
ejpam-3952	90	15	(	(	PUNCT
ejpam-3952	90	16	2l	2l	NOUN
ejpam-3952	90	17	+	+	X
ejpam-3952	90	18	1)n−	1)n−	NUM
ejpam-3952	90	19	(	(	PUNCT
ejpam-3952	90	20	l	l	NOUN
ejpam-3952	90	21	−	−	PROPN
ejpam-3952	90	22	2	2	NUM
ejpam-3952	90	23	)	)	PUNCT
ejpam-3952	90	24	)	)	PUNCT
ejpam-3952	90	25	2	2	NUM
ejpam-3952	90	26	=	=	SYM
ejpam-3952	90	27	(	(	PUNCT
ejpam-3952	90	28	(	(	PUNCT
ejpam-3952	90	29	2l	2l	NOUN
ejpam-3952	90	30	+	+	X
ejpam-3952	90	31	1)n−	1)n−	NUM
ejpam-3952	90	32	(	(	PUNCT
ejpam-3952	90	33	l	l	NOUN
ejpam-3952	90	34	−	−	NOUN
ejpam-3952	90	35	2	2	NUM
ejpam-3952	90	36	)	)	PUNCT
ejpam-3952	90	37	)	)	PUNCT
ejpam-3952	91	1	+	+	CCONJ
ejpam-3952	91	2	(	(	PUNCT
ejpam-3952	91	3	(	(	PUNCT
ejpam-3952	91	4	2l	2l	NOUN
ejpam-3952	91	5	+	+	X
ejpam-3952	91	6	1)n−	1)n−	NUM
ejpam-3952	91	7	(	(	PUNCT
ejpam-3952	91	8	l	l	NOUN
ejpam-3952	91	9	−	−	NOUN
ejpam-3952	91	10	2	2	NUM
ejpam-3952	91	11	)	)	PUNCT
ejpam-3952	91	12	)	)	PUNCT
ejpam-3952	91	13	(	(	PUNCT
ejpam-3952	91	14	(	(	PUNCT
ejpam-3952	91	15	(	(	PUNCT
ejpam-3952	91	16	2l	2l	NOUN
ejpam-3952	91	17	+	+	X
ejpam-3952	91	18	1)n−	1)n−	NUM
ejpam-3952	91	19	(	(	PUNCT
ejpam-3952	91	20	l	l	NOUN
ejpam-3952	91	21	−	−	PROPN
ejpam-3952	91	22	2))−	2))−	NUM
ejpam-3952	91	23	1	1	NUM
ejpam-3952	91	24	)	)	PUNCT
ejpam-3952	91	25	l	l	NOUN
ejpam-3952	91	26	2	2	X
ejpam-3952	91	27	=	=	SYM
ejpam-3952	91	28	t	t	X
ejpam-3952	91	29	(	(	PUNCT
ejpam-3952	91	30	(	(	PUNCT
ejpam-3952	91	31	2l	2l	NOUN
ejpam-3952	91	32	+	+	X
ejpam-3952	91	33	1)n−	1)n−	NUM
ejpam-3952	91	34	(	(	PUNCT
ejpam-3952	91	35	l	l	NOUN
ejpam-3952	91	36	−	−	PROPN
ejpam-3952	91	37	2	2	NUM
ejpam-3952	91	38	)	)	PUNCT
ejpam-3952	91	39	,	,	PUNCT
ejpam-3952	91	40	l	l	NOUN
ejpam-3952	91	41	)	)	PUNCT
ejpam-3952	91	42	.	.	PUNCT
ejpam-3952	92	1	hence	hence	ADV
ejpam-3952	92	2	,	,	PUNCT
ejpam-3952	92	3	rn	rn	PROPN
ejpam-3952	92	4	+	+	PROPN
ejpam-3952	92	5	s	s	PART
ejpam-3952	92	6	is	be	AUX
ejpam-3952	92	7	an	an	DET
ejpam-3952	92	8	l	l	ADJ
ejpam-3952	92	9	-	-	PUNCT
ejpam-3952	92	10	isosceles	isoscele	NOUN
ejpam-3952	92	11	triangular	triangular	NOUN
ejpam-3952	92	12	number	number	NOUN
ejpam-3952	92	13	.	.	PUNCT
ejpam-3952	93	1	conversely	conversely	ADV
ejpam-3952	93	2	,	,	PUNCT
ejpam-3952	93	3	assume	assume	VERB
ejpam-3952	93	4	that	that	SCONJ
ejpam-3952	93	5	rn	rn	PROPN
ejpam-3952	94	1	+	+	PROPN
ejpam-3952	94	2	s	s	VERB
ejpam-3952	94	3	is	be	AUX
ejpam-3952	94	4	an	an	DET
ejpam-3952	94	5	l	l	ADJ
ejpam-3952	94	6	-	-	PUNCT
ejpam-3952	94	7	isosceles	isoscele	NOUN
ejpam-3952	94	8	triangular	triangular	NOUN
ejpam-3952	94	9	number	number	NOUN
ejpam-3952	94	10	.	.	PUNCT
ejpam-3952	95	1	then	then	ADV
ejpam-3952	95	2	rn	rn	PROPN
ejpam-3952	95	3	+	+	PROPN
ejpam-3952	95	4	s	s	PART
ejpam-3952	95	5	=	=	PUNCT
ejpam-3952	95	6	n	n	PROPN
ejpam-3952	95	7	+	+	CCONJ
ejpam-3952	95	8	n(n−	n(n−	PROPN
ejpam-3952	95	9	1)l	1)l	NUM
ejpam-3952	95	10	2	2	NUM
ejpam-3952	95	11	j.	j.	PROPN
ejpam-3952	95	12	punpim	punpim	PROPN
ejpam-3952	95	13	,	,	PUNCT
ejpam-3952	95	14	s.	s.	PROPN
ejpam-3952	95	15	jitman	jitman	PROPN
ejpam-3952	95	16	/	/	SYM
ejpam-3952	95	17	eur	eur	PROPN
ejpam-3952	95	18	.	.	PUNCT
ejpam-3952	96	1	j.	j.	PROPN
ejpam-3952	96	2	pure	pure	PROPN
ejpam-3952	96	3	appl	appl	PROPN
ejpam-3952	96	4	.	.	PROPN
ejpam-3952	96	5	math	math	PROPN
ejpam-3952	96	6	,	,	PUNCT
ejpam-3952	96	7	14	14	NUM
ejpam-3952	96	8	(	(	PUNCT
ejpam-3952	96	9	2	2	NUM
ejpam-3952	96	10	)	)	PUNCT
ejpam-3952	96	11	(	(	PUNCT
ejpam-3952	96	12	2021	2021	NUM
ejpam-3952	96	13	)	)	PUNCT
ejpam-3952	96	14	,	,	PUNCT
ejpam-3952	96	15	380	380	NUM
ejpam-3952	96	16	-	-	SYM
ejpam-3952	96	17	395	395	NUM
ejpam-3952	96	18	384	384	NUM
ejpam-3952	96	19	for	for	ADP
ejpam-3952	96	20	some	some	DET
ejpam-3952	96	21	positive	positive	ADJ
ejpam-3952	96	22	number	number	NOUN
ejpam-3952	96	23	n.	n.	NOUN
ejpam-3952	96	24	equivalently	equivalently	ADV
ejpam-3952	96	25	,	,	PUNCT
ejpam-3952	96	26	we	we	PRON
ejpam-3952	96	27	have	have	VERB
ejpam-3952	96	28	n	n	NOUN
ejpam-3952	96	29	=	=	SYM
ejpam-3952	96	30	2n	2n	X
ejpam-3952	97	1	+	+	CCONJ
ejpam-3952	97	2	ln2	ln2	ADJ
ejpam-3952	97	3	−	−	ADP
ejpam-3952	97	4	ln−	ln−	PROPN
ejpam-3952	97	5	2s	2s	NUM
ejpam-3952	97	6	2r	2r	NUM
ejpam-3952	97	7	=	=	SYM
ejpam-3952	97	8	2n	2n	NUM
ejpam-3952	98	1	+	+	CCONJ
ejpam-3952	98	2	ln2	ln2	ADJ
ejpam-3952	98	3	−	−	PROPN
ejpam-3952	98	4	ln−	ln−	PROPN
ejpam-3952	98	5	(	(	PUNCT
ejpam-3952	98	6	l3	l3	PROPN
ejpam-3952	98	7	−	−	PROPN
ejpam-3952	98	8	3l2	3l2	NUM
ejpam-3952	98	9	+	+	CCONJ
ejpam-3952	98	10	4	4	X
ejpam-3952	98	11	)	)	PUNCT
ejpam-3952	98	12	2(2l	2(2l	NUM
ejpam-3952	99	1	+	+	CCONJ
ejpam-3952	99	2	1)2	1)2	NUM
ejpam-3952	99	3	=	=	SYM
ejpam-3952	99	4	l	l	NOUN
ejpam-3952	99	5	(	(	PUNCT
ejpam-3952	99	6	n2	n2	NOUN
ejpam-3952	99	7	+	+	CCONJ
ejpam-3952	99	8	2n(l	2n(l	NOUN
ejpam-3952	99	9	−	−	PROPN
ejpam-3952	99	10	2	2	NUM
ejpam-3952	99	11	)	)	PUNCT
ejpam-3952	99	12	+	+	CCONJ
ejpam-3952	99	13	(	(	PUNCT
ejpam-3952	99	14	l	l	NOUN
ejpam-3952	99	15	−	−	PROPN
ejpam-3952	99	16	2)2	2)2	NUM
ejpam-3952	99	17	)	)	PUNCT
ejpam-3952	99	18	−	−	PROPN
ejpam-3952	100	1	(	(	PUNCT
ejpam-3952	100	2	l	l	NOUN
ejpam-3952	100	3	−	−	PROPN
ejpam-3952	101	1	2)(2l	2)(2l	NUM
ejpam-3952	102	1	+	+	SYM
ejpam-3952	102	2	1)(n	1)(n	NUM
ejpam-3952	102	3	+	+	CCONJ
ejpam-3952	102	4	l	l	NOUN
ejpam-3952	102	5	−	−	PROPN
ejpam-3952	102	6	2	2	NUM
ejpam-3952	102	7	)	)	PUNCT
ejpam-3952	102	8	2(2l	2(2l	NUM
ejpam-3952	103	1	+	+	CCONJ
ejpam-3952	103	2	1)2	1)2	NUM
ejpam-3952	103	3	=	=	SYM
ejpam-3952	103	4	l(n	l(n	NOUN
ejpam-3952	103	5	+	+	CCONJ
ejpam-3952	103	6	l	l	NOUN
ejpam-3952	103	7	−	−	PROPN
ejpam-3952	103	8	2)2	2)2	NUM
ejpam-3952	103	9	−	−	NOUN
ejpam-3952	103	10	(	(	PUNCT
ejpam-3952	103	11	l	l	NOUN
ejpam-3952	103	12	−	−	PROPN
ejpam-3952	104	1	2)(2l	2)(2l	NUM
ejpam-3952	105	1	+	+	SYM
ejpam-3952	105	2	1)(n	1)(n	NUM
ejpam-3952	105	3	+	+	CCONJ
ejpam-3952	105	4	l	l	NOUN
ejpam-3952	105	5	−	−	PROPN
ejpam-3952	105	6	2	2	NUM
ejpam-3952	105	7	)	)	PUNCT
ejpam-3952	105	8	2(2l	2(2l	NUM
ejpam-3952	106	1	+	+	CCONJ
ejpam-3952	106	2	1)2	1)2	NUM
ejpam-3952	106	3	=	=	SYM
ejpam-3952	106	4	l(n	l(n	NOUN
ejpam-3952	106	5	+	+	CCONJ
ejpam-3952	106	6	l	l	NOUN
ejpam-3952	106	7	−	−	PROPN
ejpam-3952	106	8	2)2	2)2	NUM
ejpam-3952	106	9	(	(	PUNCT
ejpam-3952	106	10	2l	2l	X
ejpam-3952	106	11	+	+	X
ejpam-3952	106	12	1)2	1)2	NUM
ejpam-3952	106	13	−	−	NOUN
ejpam-3952	106	14	(	(	PUNCT
ejpam-3952	106	15	l	l	NOUN
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ejpam-3952	106	17	2)(n	2)(n	NUM
ejpam-3952	106	18	+	+	NOUN
ejpam-3952	106	19	l	l	NOUN
ejpam-3952	106	20	−	−	NOUN
ejpam-3952	106	21	2	2	NUM
ejpam-3952	106	22	)	)	PUNCT
ejpam-3952	106	23	2l	2l	NOUN
ejpam-3952	106	24	+	+	CCONJ
ejpam-3952	106	25	1	1	NUM
ejpam-3952	106	26	2	2	NUM
ejpam-3952	106	27	=	=	SYM
ejpam-3952	106	28	2(n	2(n	NUM
ejpam-3952	106	29	+	+	CCONJ
ejpam-3952	106	30	l	l	NOUN
ejpam-3952	106	31	−	−	NOUN
ejpam-3952	106	32	2	2	NUM
ejpam-3952	106	33	)	)	PUNCT
ejpam-3952	106	34	2l	2l	NOUN
ejpam-3952	107	1	+	+	CCONJ
ejpam-3952	107	2	1	1	NUM
ejpam-3952	107	3	+	+	CCONJ
ejpam-3952	107	4	l(n	l(n	NOUN
ejpam-3952	107	5	+	+	CCONJ
ejpam-3952	107	6	l	l	NOUN
ejpam-3952	107	7	−	−	PROPN
ejpam-3952	107	8	2)2	2)2	NUM
ejpam-3952	107	9	(	(	PUNCT
ejpam-3952	107	10	2l	2l	X
ejpam-3952	107	11	+	+	X
ejpam-3952	107	12	1)2	1)2	NUM
ejpam-3952	107	13	−	−	NOUN
ejpam-3952	107	14	l(n	l(n	NOUN
ejpam-3952	107	15	+	+	CCONJ
ejpam-3952	107	16	l	l	NOUN
ejpam-3952	107	17	−	−	PROPN
ejpam-3952	107	18	2	2	NUM
ejpam-3952	107	19	)	)	PUNCT
ejpam-3952	107	20	2l	2l	NOUN
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ejpam-3952	107	22	1	1	NUM
ejpam-3952	107	23	2	2	NUM
ejpam-3952	107	24	=	=	SYM
ejpam-3952	107	25	n	n	NOUN
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ejpam-3952	107	27	l	l	NOUN
ejpam-3952	107	28	−	−	NOUN
ejpam-3952	107	29	2	2	NUM
ejpam-3952	107	30	2l	2l	NOUN
ejpam-3952	107	31	+	+	CCONJ
ejpam-3952	107	32	1	1	NUM
ejpam-3952	107	33	+	+	CCONJ
ejpam-3952	107	34	(	(	PUNCT
ejpam-3952	107	35	n	n	PROPN
ejpam-3952	107	36	+	+	CCONJ
ejpam-3952	107	37	l	l	NOUN
ejpam-3952	107	38	−	−	NOUN
ejpam-3952	107	39	2	2	NUM
ejpam-3952	107	40	2l	2l	NOUN
ejpam-3952	107	41	+	+	CCONJ
ejpam-3952	107	42	1	1	NUM
ejpam-3952	107	43	)	)	PUNCT
ejpam-3952	107	44	(	(	PUNCT
ejpam-3952	107	45	n	n	PROPN
ejpam-3952	107	46	+	+	CCONJ
ejpam-3952	107	47	l	l	NOUN
ejpam-3952	107	48	−	−	NOUN
ejpam-3952	107	49	2	2	NUM
ejpam-3952	107	50	2l	2l	NOUN
ejpam-3952	107	51	+	+	CCONJ
ejpam-3952	107	52	1	1	NUM
ejpam-3952	107	53	−	−	NUM
ejpam-3952	107	54	1	1	NUM
ejpam-3952	107	55	)	)	PUNCT
ejpam-3952	107	56	l	l	NOUN
ejpam-3952	107	57	2	2	NUM
ejpam-3952	107	58	.	.	PUNCT
ejpam-3952	108	1	since	since	SCONJ
ejpam-3952	108	2	n	n	NOUN
ejpam-3952	108	3	=	=	SYM
ejpam-3952	108	4	2n	2n	X
ejpam-3952	109	1	+	+	CCONJ
ejpam-3952	109	2	ln2	ln2	ADJ
ejpam-3952	109	3	−	−	PROPN
ejpam-3952	109	4	ln−	ln−	PROPN
ejpam-3952	109	5	(	(	PUNCT
ejpam-3952	109	6	l3	l3	PROPN
ejpam-3952	109	7	−	−	PROPN
ejpam-3952	109	8	3l2	3l2	NUM
ejpam-3952	109	9	+	+	CCONJ
ejpam-3952	109	10	4	4	X
ejpam-3952	109	11	)	)	PUNCT
ejpam-3952	109	12	2(2l	2(2l	NUM
ejpam-3952	110	1	+	+	CCONJ
ejpam-3952	110	2	1)2	1)2	NUM
ejpam-3952	110	3	=	=	SYM
ejpam-3952	110	4	(	(	PUNCT
ejpam-3952	110	5	ln−	ln−	NOUN
ejpam-3952	110	6	l2	l2	VERB
ejpam-3952	110	7	+	+	CCONJ
ejpam-3952	110	8	l	l	NOUN
ejpam-3952	111	1	+	+	NUM
ejpam-3952	111	2	2)(n	2)(n	NUM
ejpam-3952	111	3	+	+	CCONJ
ejpam-3952	111	4	l	l	NOUN
ejpam-3952	111	5	−	−	NOUN
ejpam-3952	112	1	2	2	X
ejpam-3952	112	2	)	)	PUNCT
ejpam-3952	112	3	2(2l	2(2l	NUM
ejpam-3952	113	1	+	+	CCONJ
ejpam-3952	113	2	1)2	1)2	NUM
ejpam-3952	113	3	is	be	AUX
ejpam-3952	113	4	a	a	DET
ejpam-3952	113	5	positive	positive	ADJ
ejpam-3952	113	6	integer	integer	NOUN
ejpam-3952	113	7	,	,	PUNCT
ejpam-3952	113	8	we	we	PRON
ejpam-3952	113	9	have	have	VERB
ejpam-3952	113	10	(	(	PUNCT
ejpam-3952	113	11	2l+1)2|(ln−	2l+1)2|(ln−	NUM
ejpam-3952	113	12	l2	l2	NOUN
ejpam-3952	113	13	+	+	CCONJ
ejpam-3952	113	14	l+2)(n+	l+2)(n+	PROPN
ejpam-3952	113	15	l−2	l−2	PROPN
ejpam-3952	113	16	)	)	PUNCT
ejpam-3952	113	17	.	.	PUNCT
ejpam-3952	114	1	if	if	SCONJ
ejpam-3952	114	2	(	(	PUNCT
ejpam-3952	114	3	2l+1)|(n+	2l+1)|(n+	NUM
ejpam-3952	114	4	l−2	l−2	NOUN
ejpam-3952	114	5	)	)	PUNCT
ejpam-3952	114	6	,	,	PUNCT
ejpam-3952	114	7	then	then	ADV
ejpam-3952	114	8	(	(	PUNCT
ejpam-3952	114	9	n	n	X
ejpam-3952	114	10	+	+	X
ejpam-3952	114	11	l	l	NOUN
ejpam-3952	114	12	−	−	NOUN
ejpam-3952	114	13	2	2	NUM
ejpam-3952	114	14	)	)	PUNCT
ejpam-3952	114	15	(	(	PUNCT
ejpam-3952	114	16	2l	2l	NOUN
ejpam-3952	114	17	+	+	CCONJ
ejpam-3952	114	18	1	1	X
ejpam-3952	114	19	)	)	PUNCT
ejpam-3952	114	20	∈	∈	PROPN
ejpam-3952	114	21	n.	n.	NOUN
ejpam-3952	114	22	suppose	suppose	VERB
ejpam-3952	114	23	that	that	SCONJ
ejpam-3952	114	24	(	(	PUNCT
ejpam-3952	114	25	2l	2l	NOUN
ejpam-3952	114	26	+	+	NOUN
ejpam-3952	114	27	1	1	NUM
ejpam-3952	114	28	)	)	PUNCT
ejpam-3952	114	29	(	(	PUNCT
ejpam-3952	114	30	n+	n+	NUM
ejpam-3952	114	31	l−	l−	NOUN
ejpam-3952	114	32	2	2	NUM
ejpam-3952	114	33	)	)	PUNCT
ejpam-3952	114	34	.	.	PUNCT
ejpam-3952	115	1	since	since	SCONJ
ejpam-3952	115	2	ln−	ln−	NOUN
ejpam-3952	115	3	l2	l2	VERB
ejpam-3952	115	4	+	+	CCONJ
ejpam-3952	115	5	l	l	NOUN
ejpam-3952	116	1	+	+	CCONJ
ejpam-3952	116	2	2	2	NUM
ejpam-3952	116	3	=	=	SYM
ejpam-3952	116	4	l(n+	l(n+	ADJ
ejpam-3952	116	5	l−	l−	PROPN
ejpam-3952	116	6	2)−	2)−	NUM
ejpam-3952	116	7	(	(	PUNCT
ejpam-3952	116	8	2l	2l	X
ejpam-3952	116	9	+	+	X
ejpam-3952	116	10	1)(l−	1)(l−	NUM
ejpam-3952	116	11	2	2	NUM
ejpam-3952	116	12	)	)	PUNCT
ejpam-3952	116	13	and	and	CCONJ
ejpam-3952	116	14	gcd(l	gcd(l	PROPN
ejpam-3952	116	15	,	,	PUNCT
ejpam-3952	116	16	2l+1	2l+1	PROPN
ejpam-3952	116	17	)	)	PUNCT
ejpam-3952	117	1	=	=	SYM
ejpam-3952	117	2	1	1	NUM
ejpam-3952	117	3	,	,	PUNCT
ejpam-3952	117	4	we	we	PRON
ejpam-3952	117	5	have	have	VERB
ejpam-3952	117	6	(	(	PUNCT
ejpam-3952	117	7	2l+1	2l+1	NOUN
ejpam-3952	117	8	)	)	PUNCT
ejpam-3952	117	9	l(n+	l(n+	X
ejpam-3952	118	1	l−2	l−2	NOUN
ejpam-3952	118	2	)	)	PUNCT
ejpam-3952	118	3	which	which	PRON
ejpam-3952	118	4	implies	imply	VERB
ejpam-3952	118	5	that	that	SCONJ
ejpam-3952	118	6	(	(	PUNCT
ejpam-3952	118	7	2l+1	2l+1	NOUN
ejpam-3952	118	8	)	)	PUNCT
ejpam-3952	118	9	(	(	PUNCT
ejpam-3952	118	10	ln−	ln−	NOUN
ejpam-3952	118	11	l2	l2	VERB
ejpam-3952	118	12	+	+	CCONJ
ejpam-3952	118	13	l+2	l+2	NOUN
ejpam-3952	118	14	)	)	PUNCT
ejpam-3952	118	15	.	.	PUNCT
ejpam-3952	119	1	hence	hence	ADV
ejpam-3952	119	2	,	,	PUNCT
ejpam-3952	119	3	(	(	PUNCT
ejpam-3952	119	4	2l+1)2	2l+1)2	NUM
ejpam-3952	119	5	(	(	PUNCT
ejpam-3952	119	6	ln−	ln−	NOUN
ejpam-3952	119	7	l2	l2	NOUN
ejpam-3952	119	8	+	+	CCONJ
ejpam-3952	119	9	l+2)(n+	l+2)(n+	PROPN
ejpam-3952	119	10	l−2	l−2	PROPN
ejpam-3952	119	11	)	)	PUNCT
ejpam-3952	119	12	which	which	PRON
ejpam-3952	119	13	is	be	AUX
ejpam-3952	119	14	a	a	DET
ejpam-3952	119	15	contradiction	contradiction	NOUN
ejpam-3952	119	16	.	.	PUNCT
ejpam-3952	120	1	then	then	ADV
ejpam-3952	120	2	(	(	PUNCT
ejpam-3952	120	3	2l+1)|(n+	2l+1)|(n+	NUM
ejpam-3952	120	4	l−2	l−2	NOUN
ejpam-3952	120	5	)	)	PUNCT
ejpam-3952	120	6	which	which	PRON
ejpam-3952	120	7	implies	imply	VERB
ejpam-3952	120	8	that	that	PRON
ejpam-3952	120	9	(	(	PUNCT
ejpam-3952	120	10	n	n	X
ejpam-3952	120	11	+	+	X
ejpam-3952	120	12	l	l	NOUN
ejpam-3952	120	13	−	−	NOUN
ejpam-3952	120	14	2	2	NUM
ejpam-3952	120	15	)	)	PUNCT
ejpam-3952	120	16	(	(	PUNCT
ejpam-3952	120	17	2l	2l	NOUN
ejpam-3952	120	18	+	+	CCONJ
ejpam-3952	120	19	1	1	X
ejpam-3952	120	20	)	)	PUNCT
ejpam-3952	120	21	is	be	AUX
ejpam-3952	120	22	a	a	DET
ejpam-3952	120	23	positive	positive	ADJ
ejpam-3952	120	24	integer	integer	NOUN
ejpam-3952	120	25	.	.	PUNCT
ejpam-3952	121	1	therefore	therefore	ADV
ejpam-3952	121	2	,	,	PUNCT
ejpam-3952	121	3	n	n	NOUN
ejpam-3952	121	4	=	=	SYM
ejpam-3952	121	5	n	n	PROPN
ejpam-3952	121	6	+	+	NOUN
ejpam-3952	121	7	l	l	NOUN
ejpam-3952	121	8	−	−	NOUN
ejpam-3952	121	9	2	2	NUM
ejpam-3952	121	10	2l	2l	NOUN
ejpam-3952	121	11	+	+	CCONJ
ejpam-3952	121	12	1	1	NUM
ejpam-3952	121	13	+	+	CCONJ
ejpam-3952	121	14	(	(	PUNCT
ejpam-3952	121	15	n	n	PROPN
ejpam-3952	121	16	+	+	CCONJ
ejpam-3952	121	17	l	l	NOUN
ejpam-3952	121	18	−	−	NOUN
ejpam-3952	121	19	2	2	NUM
ejpam-3952	121	20	2l	2l	NOUN
ejpam-3952	121	21	+	+	CCONJ
ejpam-3952	121	22	1	1	NUM
ejpam-3952	121	23	)	)	PUNCT
ejpam-3952	121	24	(	(	PUNCT
ejpam-3952	121	25	n	n	PROPN
ejpam-3952	121	26	+	+	CCONJ
ejpam-3952	121	27	l	l	NOUN
ejpam-3952	121	28	−	−	NOUN
ejpam-3952	121	29	2	2	NUM
ejpam-3952	121	30	2l	2l	NOUN
ejpam-3952	121	31	+	+	CCONJ
ejpam-3952	121	32	1	1	NUM
ejpam-3952	121	33	−	−	NUM
ejpam-3952	121	34	1	1	NUM
ejpam-3952	121	35	)	)	PUNCT
ejpam-3952	121	36	l	l	NOUN
ejpam-3952	121	37	2	2	X
ejpam-3952	121	38	=	=	SYM
ejpam-3952	121	39	t	t	X
ejpam-3952	121	40	(	(	PUNCT
ejpam-3952	121	41	n	n	PROPN
ejpam-3952	121	42	+	+	CCONJ
ejpam-3952	121	43	l	l	NOUN
ejpam-3952	121	44	−	−	NOUN
ejpam-3952	121	45	2	2	NUM
ejpam-3952	121	46	2l	2l	NOUN
ejpam-3952	121	47	+	+	CCONJ
ejpam-3952	121	48	1	1	NUM
ejpam-3952	121	49	,	,	PUNCT
ejpam-3952	121	50	l	l	NOUN
ejpam-3952	121	51	)	)	PUNCT
ejpam-3952	121	52	is	be	AUX
ejpam-3952	121	53	an	an	DET
ejpam-3952	121	54	l	l	NOUN
ejpam-3952	121	55	-	-	PUNCT
ejpam-3952	121	56	isosceles	isoscele	NOUN
ejpam-3952	121	57	triangular	triangular	NOUN
ejpam-3952	121	58	number	number	NOUN
ejpam-3952	121	59	.	.	PUNCT
ejpam-3952	122	1	by	by	ADP
ejpam-3952	122	2	setting	set	VERB
ejpam-3952	122	3	l	l	NOUN
ejpam-3952	122	4	=	=	SYM
ejpam-3952	122	5	1	1	NUM
ejpam-3952	122	6	,	,	PUNCT
ejpam-3952	122	7	we	we	PRON
ejpam-3952	122	8	have	have	VERB
ejpam-3952	122	9	the	the	DET
ejpam-3952	122	10	classical	classical	ADJ
ejpam-3952	122	11	fact	fact	NOUN
ejpam-3952	122	12	“	"	PUNCT
ejpam-3952	122	13	n	n	PRON
ejpam-3952	122	14	is	be	AUX
ejpam-3952	122	15	a	a	DET
ejpam-3952	122	16	triangular	triangular	NOUN
ejpam-3952	122	17	number	number	NOUN
ejpam-3952	122	18	if	if	SCONJ
ejpam-3952	122	19	and	and	CCONJ
ejpam-3952	122	20	only	only	ADV
ejpam-3952	123	1	if	if	SCONJ
ejpam-3952	123	2	9n	9n	NOUN
ejpam-3952	123	3	+	+	CCONJ
ejpam-3952	123	4	1	1	NUM
ejpam-3952	123	5	is	be	AUX
ejpam-3952	123	6	a	a	DET
ejpam-3952	123	7	triangular	triangular	ADJ
ejpam-3952	123	8	number	number	NOUN
ejpam-3952	123	9	”	"	PUNCT
ejpam-3952	123	10	.	.	PUNCT
ejpam-3952	124	1	for	for	ADP
ejpam-3952	124	2	l	l	PROPN
ejpam-3952	124	3	∈	∈	PROPN
ejpam-3952	124	4	{	{	PUNCT
ejpam-3952	124	5	2	2	NUM
ejpam-3952	124	6	,	,	PUNCT
ejpam-3952	124	7	3	3	NUM
ejpam-3952	124	8	,	,	PUNCT
ejpam-3952	124	9	4	4	NUM
ejpam-3952	124	10	}	}	PUNCT
ejpam-3952	124	11	,	,	PUNCT
ejpam-3952	124	12	we	we	PRON
ejpam-3952	124	13	have	have	VERB
ejpam-3952	124	14	the	the	DET
ejpam-3952	124	15	following	follow	VERB
ejpam-3952	124	16	results	result	NOUN
ejpam-3952	124	17	.	.	PUNCT
ejpam-3952	125	1	•	•	NUM
ejpam-3952	125	2	a	a	DET
ejpam-3952	125	3	positive	positive	ADJ
ejpam-3952	125	4	integer	integer	NOUN
ejpam-3952	125	5	n	n	X
ejpam-3952	125	6	is	be	AUX
ejpam-3952	125	7	2	2	NUM
ejpam-3952	125	8	-	-	PUNCT
ejpam-3952	125	9	isosceles	isoscele	NOUN
ejpam-3952	125	10	triangular	triangular	NOUN
ejpam-3952	125	11	if	if	SCONJ
ejpam-3952	125	12	and	and	CCONJ
ejpam-3952	125	13	only	only	ADV
ejpam-3952	125	14	if	if	SCONJ
ejpam-3952	125	15	25n	25n	PROPN
ejpam-3952	125	16	is	be	AUX
ejpam-3952	125	17	a	a	DET
ejpam-3952	125	18	2	2	NUM
ejpam-3952	125	19	-	-	PUNCT
ejpam-3952	125	20	isosceles	isoscele	NOUN
ejpam-3952	125	21	triangular	triangular	NOUN
ejpam-3952	125	22	number	number	NOUN
ejpam-3952	125	23	.	.	PUNCT
ejpam-3952	126	1	equivalently	equivalently	ADV
ejpam-3952	126	2	,	,	PUNCT
ejpam-3952	126	3	n	n	PRON
ejpam-3952	126	4	is	be	AUX
ejpam-3952	126	5	a	a	DET
ejpam-3952	126	6	square	square	ADJ
ejpam-3952	126	7	number	number	NOUN
ejpam-3952	126	8	.	.	PUNCT
ejpam-3952	127	1	•	•	NUM
ejpam-3952	127	2	a	a	DET
ejpam-3952	127	3	positive	positive	ADJ
ejpam-3952	127	4	integer	integer	NOUN
ejpam-3952	127	5	n	n	X
ejpam-3952	127	6	is	be	AUX
ejpam-3952	127	7	3	3	NUM
ejpam-3952	127	8	-	-	PUNCT
ejpam-3952	127	9	isosceles	isoscele	NOUN
ejpam-3952	127	10	triangular	triangular	NOUN
ejpam-3952	127	11	if	if	SCONJ
ejpam-3952	127	12	and	and	CCONJ
ejpam-3952	127	13	only	only	ADV
ejpam-3952	127	14	if	if	SCONJ
ejpam-3952	127	15	49n	49n	NUM
ejpam-3952	127	16	+	+	CCONJ
ejpam-3952	127	17	2	2	NUM
ejpam-3952	127	18	is	be	AUX
ejpam-3952	127	19	a	a	DET
ejpam-3952	127	20	3	3	NUM
ejpam-3952	127	21	-	-	PUNCT
ejpam-3952	127	22	isosceles	isoscele	NOUN
ejpam-3952	127	23	triangular	triangular	NOUN
ejpam-3952	127	24	number	number	NOUN
ejpam-3952	127	25	.	.	PUNCT
ejpam-3952	128	1	j.	j.	PROPN
ejpam-3952	128	2	punpim	punpim	PROPN
ejpam-3952	128	3	,	,	PUNCT
ejpam-3952	128	4	s.	s.	PROPN
ejpam-3952	128	5	jitman	jitman	PROPN
ejpam-3952	128	6	/	/	SYM
ejpam-3952	128	7	eur	eur	PROPN
ejpam-3952	128	8	.	.	PUNCT
ejpam-3952	129	1	j.	j.	PROPN
ejpam-3952	129	2	pure	pure	PROPN
ejpam-3952	129	3	appl	appl	PROPN
ejpam-3952	129	4	.	.	PROPN
ejpam-3952	129	5	math	math	PROPN
ejpam-3952	129	6	,	,	PUNCT
ejpam-3952	129	7	14	14	NUM
ejpam-3952	129	8	(	(	PUNCT
ejpam-3952	129	9	2	2	NUM
ejpam-3952	129	10	)	)	PUNCT
ejpam-3952	129	11	(	(	PUNCT
ejpam-3952	129	12	2021	2021	NUM
ejpam-3952	129	13	)	)	PUNCT
ejpam-3952	129	14	,	,	PUNCT
ejpam-3952	129	15	380	380	NUM
ejpam-3952	129	16	-	-	SYM
ejpam-3952	129	17	395	395	NUM
ejpam-3952	129	18	385	385	NUM
ejpam-3952	129	19	•	•	NOUN
ejpam-3952	129	20	a	a	DET
ejpam-3952	129	21	positive	positive	ADJ
ejpam-3952	129	22	integer	integer	NOUN
ejpam-3952	129	23	n	n	NUM
ejpam-3952	129	24	is	be	AUX
ejpam-3952	129	25	4	4	NUM
ejpam-3952	129	26	-	-	PUNCT
ejpam-3952	129	27	isosceles	isoscele	NOUN
ejpam-3952	129	28	triangular	triangular	NOUN
ejpam-3952	129	29	if	if	SCONJ
ejpam-3952	129	30	and	and	CCONJ
ejpam-3952	129	31	only	only	ADV
ejpam-3952	129	32	if	if	SCONJ
ejpam-3952	129	33	81n	81n	NOUN
ejpam-3952	129	34	+	+	CCONJ
ejpam-3952	129	35	10	10	NUM
ejpam-3952	129	36	is	be	AUX
ejpam-3952	129	37	a	a	DET
ejpam-3952	129	38	4	4	NUM
ejpam-3952	129	39	-	-	PUNCT
ejpam-3952	129	40	isosceles	isoscele	NOUN
ejpam-3952	129	41	triangular	triangular	NOUN
ejpam-3952	129	42	number	number	NOUN
ejpam-3952	129	43	.	.	PUNCT
ejpam-3952	130	1	in	in	ADP
ejpam-3952	130	2	general	general	ADJ
ejpam-3952	130	3	,	,	PUNCT
ejpam-3952	130	4	each	each	DET
ejpam-3952	130	5	positive	positive	ADJ
ejpam-3952	130	6	integer	integer	NOUN
ejpam-3952	130	7	l	l	PROPN
ejpam-3952	130	8	,	,	PUNCT
ejpam-3952	130	9	a	a	DET
ejpam-3952	130	10	recursive	recursive	ADJ
ejpam-3952	130	11	construction	construction	NOUN
ejpam-3952	130	12	of	of	ADP
ejpam-3952	130	13	l	l	NOUN
ejpam-3952	130	14	-	-	PUNCT
ejpam-3952	130	15	isosceles	isoscele	NOUN
ejpam-3952	130	16	triangular	triangular	NOUN
ejpam-3952	130	17	numbers	number	NOUN
ejpam-3952	130	18	can	can	AUX
ejpam-3952	130	19	be	be	AUX
ejpam-3952	130	20	deduced	deduce	VERB
ejpam-3952	130	21	directly	directly	ADV
ejpam-3952	130	22	from	from	ADP
ejpam-3952	130	23	theorem	theorem	ADJ
ejpam-3952	130	24	1	1	NUM
ejpam-3952	130	25	.	.	NUM
ejpam-3952	130	26	2.2	2.2	NUM
ejpam-3952	130	27	.	.	PUNCT
ejpam-3952	131	1	isosceles	isoscele	NOUN
ejpam-3952	131	2	triangular	triangular	NOUN
ejpam-3952	131	3	numbers	number	NOUN
ejpam-3952	131	4	and	and	CCONJ
ejpam-3952	131	5	square	square	ADJ
ejpam-3952	131	6	numbers	number	NOUN
ejpam-3952	131	7	in	in	ADP
ejpam-3952	131	8	this	this	DET
ejpam-3952	131	9	subsection	subsection	NOUN
ejpam-3952	132	1	,	,	PUNCT
ejpam-3952	132	2	we	we	PRON
ejpam-3952	132	3	focus	focus	VERB
ejpam-3952	132	4	on	on	ADP
ejpam-3952	132	5	a	a	DET
ejpam-3952	132	6	generalization	generalization	NOUN
ejpam-3952	132	7	of	of	ADP
ejpam-3952	132	8	the	the	DET
ejpam-3952	132	9	classical	classical	ADJ
ejpam-3952	132	10	fact	fact	NOUN
ejpam-3952	132	11	“	"	PUNCT
ejpam-3952	132	12	n	n	PRON
ejpam-3952	132	13	is	be	AUX
ejpam-3952	132	14	a	a	DET
ejpam-3952	132	15	triangular	triangular	NOUN
ejpam-3952	132	16	number	number	NOUN
ejpam-3952	132	17	if	if	SCONJ
ejpam-3952	132	18	and	and	CCONJ
ejpam-3952	132	19	only	only	ADV
ejpam-3952	132	20	if	if	SCONJ
ejpam-3952	132	21	8n	8n	NOUN
ejpam-3952	132	22	+	+	CCONJ
ejpam-3952	132	23	1	1	NUM
ejpam-3952	132	24	is	be	AUX
ejpam-3952	132	25	square	square	ADJ
ejpam-3952	132	26	”	"	PUNCT
ejpam-3952	132	27	(	(	PUNCT
ejpam-3952	132	28	see	see	VERB
ejpam-3952	132	29	[	[	X
ejpam-3952	132	30	3	3	NUM
ejpam-3952	132	31	]	]	NUM
ejpam-3952	132	32	)	)	PUNCT
ejpam-3952	132	33	.	.	PUNCT
ejpam-3952	133	1	necessary	necessary	ADJ
ejpam-3952	133	2	and	and	CCONJ
ejpam-3952	133	3	sufficient	sufficient	ADJ
ejpam-3952	133	4	conditions	condition	NOUN
ejpam-3952	133	5	for	for	ADP
ejpam-3952	133	6	an	an	DET
ejpam-3952	133	7	l	l	ADJ
ejpam-3952	133	8	-	-	PUNCT
ejpam-3952	133	9	isosceles	isoscele	NOUN
ejpam-3952	133	10	triangular	triangular	NOUN
ejpam-3952	133	11	number	number	NOUN
ejpam-3952	133	12	to	to	PART
ejpam-3952	133	13	be	be	AUX
ejpam-3952	133	14	square	square	ADJ
ejpam-3952	133	15	are	be	AUX
ejpam-3952	133	16	given	give	VERB
ejpam-3952	133	17	in	in	ADP
ejpam-3952	133	18	the	the	DET
ejpam-3952	133	19	following	follow	VERB
ejpam-3952	133	20	theorem	theorem	NOUN
ejpam-3952	133	21	.	.	PUNCT
ejpam-3952	133	22	theorem	theorem	NOUN
ejpam-3952	133	23	2	2	NUM
ejpam-3952	133	24	.	.	PUNCT
ejpam-3952	134	1	let	let	VERB
ejpam-3952	134	2	n	n	NOUN
ejpam-3952	134	3	and	and	CCONJ
ejpam-3952	134	4	l	l	NOUN
ejpam-3952	134	5	be	be	AUX
ejpam-3952	134	6	positive	positive	ADJ
ejpam-3952	134	7	integers	integer	NOUN
ejpam-3952	134	8	.	.	PUNCT
ejpam-3952	135	1	then	then	ADV
ejpam-3952	135	2	n	n	PRON
ejpam-3952	135	3	is	be	AUX
ejpam-3952	135	4	an	an	DET
ejpam-3952	135	5	l	l	NOUN
ejpam-3952	135	6	-	-	PUNCT
ejpam-3952	135	7	isosceles	isoscele	NOUN
ejpam-3952	135	8	triangular	triangular	NOUN
ejpam-3952	135	9	number	number	NOUN
ejpam-3952	135	10	if	if	SCONJ
ejpam-3952	135	11	and	and	CCONJ
ejpam-3952	135	12	only	only	ADV
ejpam-3952	135	13	if	if	SCONJ
ejpam-3952	135	14	8nl	8nl	PROPN
ejpam-3952	135	15	+	+	CCONJ
ejpam-3952	135	16	(	(	PUNCT
ejpam-3952	135	17	l	l	NOUN
ejpam-3952	135	18	−	−	PROPN
ejpam-3952	135	19	2)2	2)2	NUM
ejpam-3952	135	20	is	be	AUX
ejpam-3952	135	21	square	square	ADJ
ejpam-3952	135	22	and	and	CCONJ
ejpam-3952	135	23	√	√	NOUN
ejpam-3952	135	24	8nl	8nl	NOUN
ejpam-3952	136	1	+	+	CCONJ
ejpam-3952	136	2	(	(	PUNCT
ejpam-3952	136	3	l	l	NOUN
ejpam-3952	136	4	−	−	PROPN
ejpam-3952	136	5	2)2	2)2	NUM
ejpam-3952	136	6	≡	≡	PROPN
ejpam-3952	136	7	(	(	PUNCT
ejpam-3952	136	8	l	l	NOUN
ejpam-3952	136	9	+	+	NOUN
ejpam-3952	136	10	2	2	NUM
ejpam-3952	136	11	)	)	PUNCT
ejpam-3952	136	12	(	(	PUNCT
ejpam-3952	136	13	mod	mod	PROPN
ejpam-3952	136	14	2l	2l	NUM
ejpam-3952	136	15	)	)	PUNCT
ejpam-3952	136	16	.	.	PUNCT
ejpam-3952	137	1	proof	proof	NOUN
ejpam-3952	137	2	.	.	PUNCT
ejpam-3952	138	1	assume	assume	VERB
ejpam-3952	138	2	that	that	SCONJ
ejpam-3952	138	3	n	n	PRON
ejpam-3952	138	4	is	be	AUX
ejpam-3952	138	5	an	an	DET
ejpam-3952	138	6	l	l	NOUN
ejpam-3952	138	7	-	-	PUNCT
ejpam-3952	138	8	isosceles	isoscele	NOUN
ejpam-3952	138	9	triangular	triangular	NOUN
ejpam-3952	138	10	number	number	NOUN
ejpam-3952	138	11	.	.	PUNCT
ejpam-3952	139	1	then	then	ADV
ejpam-3952	139	2	n	n	PROPN
ejpam-3952	139	3	=	=	SYM
ejpam-3952	139	4	n	n	PROPN
ejpam-3952	139	5	+	+	CCONJ
ejpam-3952	139	6	n(n−	n(n−	PROPN
ejpam-3952	139	7	1)l	1)l	NUM
ejpam-3952	139	8	2	2	NUM
ejpam-3952	139	9	for	for	ADP
ejpam-3952	139	10	some	some	DET
ejpam-3952	139	11	positive	positive	ADJ
ejpam-3952	139	12	integer	integer	NOUN
ejpam-3952	139	13	n.	n.	NOUN
ejpam-3952	139	14	it	it	PRON
ejpam-3952	139	15	follows	follow	VERB
ejpam-3952	139	16	that	that	SCONJ
ejpam-3952	139	17	8nl	8nl	PROPN
ejpam-3952	139	18	+	+	CCONJ
ejpam-3952	139	19	(	(	PUNCT
ejpam-3952	139	20	l	l	NOUN
ejpam-3952	139	21	−	−	PROPN
ejpam-3952	139	22	2)2	2)2	NUM
ejpam-3952	139	23	=	=	SYM
ejpam-3952	139	24	8nl	8nl	NOUN
ejpam-3952	139	25	+	+	CCONJ
ejpam-3952	139	26	4n(n−	4n(n−	NUM
ejpam-3952	140	1	1)l2	1)l2	NUM
ejpam-3952	140	2	+	+	CCONJ
ejpam-3952	140	3	(	(	PUNCT
ejpam-3952	140	4	l	l	NOUN
ejpam-3952	140	5	−	−	PROPN
ejpam-3952	140	6	2)2	2)2	NUM
ejpam-3952	140	7	=	=	SYM
ejpam-3952	140	8	(	(	PUNCT
ejpam-3952	140	9	2nl)2	2nl)2	NUM
ejpam-3952	140	10	−	−	NOUN
ejpam-3952	140	11	2(2nl)(l	2(2nl)(l	NUM
ejpam-3952	140	12	−	−	NOUN
ejpam-3952	140	13	2	2	NUM
ejpam-3952	140	14	)	)	PUNCT
ejpam-3952	140	15	+	+	CCONJ
ejpam-3952	140	16	(	(	PUNCT
ejpam-3952	140	17	l	l	NOUN
ejpam-3952	140	18	−	−	PROPN
ejpam-3952	140	19	2)2	2)2	NUM
ejpam-3952	140	20	=	=	SYM
ejpam-3952	140	21	(	(	PUNCT
ejpam-3952	140	22	2nl	2nl	ADJ
ejpam-3952	140	23	−	−	PROPN
ejpam-3952	140	24	l	l	NOUN
ejpam-3952	140	25	+	+	CCONJ
ejpam-3952	140	26	2)2	2)2	NUM
ejpam-3952	140	27	is	be	AUX
ejpam-3952	140	28	square	square	ADJ
ejpam-3952	140	29	and	and	CCONJ
ejpam-3952	140	30	√	√	NOUN
ejpam-3952	140	31	8nl	8nl	NOUN
ejpam-3952	141	1	+	+	CCONJ
ejpam-3952	142	1	(	(	PUNCT
ejpam-3952	142	2	l	l	NOUN
ejpam-3952	142	3	−	−	PROPN
ejpam-3952	142	4	2)2	2)2	NUM
ejpam-3952	142	5	≡	≡	PROPN
ejpam-3952	142	6	2nl	2nl	NOUN
ejpam-3952	142	7	−	−	PROPN
ejpam-3952	143	1	l	l	NOUN
ejpam-3952	143	2	+	+	CCONJ
ejpam-3952	143	3	2	2	NUM
ejpam-3952	143	4	≡	≡	PROPN
ejpam-3952	143	5	l	l	NOUN
ejpam-3952	144	1	+	+	CCONJ
ejpam-3952	144	2	2	2	NUM
ejpam-3952	144	3	(	(	PUNCT
ejpam-3952	144	4	mod	mod	NOUN
ejpam-3952	144	5	2l	2l	NUM
ejpam-3952	144	6	)	)	PUNCT
ejpam-3952	144	7	.	.	PUNCT
ejpam-3952	145	1	conversely	conversely	ADV
ejpam-3952	145	2	,	,	PUNCT
ejpam-3952	145	3	assume	assume	VERB
ejpam-3952	145	4	that	that	SCONJ
ejpam-3952	145	5	8nl+(l−2)2	8nl+(l−2)2	NUM
ejpam-3952	145	6	is	be	AUX
ejpam-3952	145	7	square	square	ADJ
ejpam-3952	145	8	and	and	CCONJ
ejpam-3952	145	9	√	√	NOUN
ejpam-3952	145	10	8nl	8nl	NOUN
ejpam-3952	145	11	+	+	CCONJ
ejpam-3952	145	12	(	(	PUNCT
ejpam-3952	145	13	l	l	NOUN
ejpam-3952	145	14	−	−	PROPN
ejpam-3952	145	15	2)2	2)2	NUM
ejpam-3952	145	16	≡	≡	PROPN
ejpam-3952	145	17	(	(	PUNCT
ejpam-3952	145	18	l+2	l+2	NOUN
ejpam-3952	145	19	)	)	PUNCT
ejpam-3952	145	20	(	(	PUNCT
ejpam-3952	145	21	mod	mod	NOUN
ejpam-3952	145	22	2l	2l	NUM
ejpam-3952	145	23	)	)	PUNCT
ejpam-3952	145	24	.	.	PUNCT
ejpam-3952	146	1	then	then	ADV
ejpam-3952	146	2	there	there	PRON
ejpam-3952	146	3	exists	exist	VERB
ejpam-3952	146	4	a	a	DET
ejpam-3952	146	5	positive	positive	ADJ
ejpam-3952	146	6	integer	integer	NOUN
ejpam-3952	146	7	m	m	VERB
ejpam-3952	146	8	such	such	ADJ
ejpam-3952	146	9	that	that	DET
ejpam-3952	146	10	√	√	PROPN
ejpam-3952	146	11	8nl	8nl	NOUN
ejpam-3952	147	1	+	+	CCONJ
ejpam-3952	147	2	(	(	PUNCT
ejpam-3952	147	3	l	l	NOUN
ejpam-3952	147	4	−	−	PROPN
ejpam-3952	147	5	2)2	2)2	NUM
ejpam-3952	147	6	=	=	SYM
ejpam-3952	147	7	2lm+(l+2	2lm+(l+2	NOUN
ejpam-3952	147	8	)	)	PUNCT
ejpam-3952	147	9	.	.	PUNCT
ejpam-3952	148	1	it	it	PRON
ejpam-3952	148	2	follows	follow	VERB
ejpam-3952	148	3	that	that	SCONJ
ejpam-3952	148	4	8nl	8nl	PROPN
ejpam-3952	148	5	+	+	CCONJ
ejpam-3952	148	6	(	(	PUNCT
ejpam-3952	148	7	l	l	NOUN
ejpam-3952	148	8	−	−	PROPN
ejpam-3952	148	9	2)2	2)2	NUM
ejpam-3952	148	10	=	=	SYM
ejpam-3952	148	11	(	(	PUNCT
ejpam-3952	148	12	2lm	2lm	NOUN
ejpam-3952	149	1	+	+	CCONJ
ejpam-3952	149	2	(	(	PUNCT
ejpam-3952	149	3	l	l	NOUN
ejpam-3952	149	4	+	+	NOUN
ejpam-3952	149	5	2))2	2))2	NUM
ejpam-3952	149	6	=	=	SYM
ejpam-3952	149	7	4l2m2	4l2m2	NUM
ejpam-3952	150	1	+	+	PUNCT
ejpam-3952	150	2	4lm(l	4lm(l	NUM
ejpam-3952	150	3	+	+	CCONJ
ejpam-3952	150	4	2	2	NUM
ejpam-3952	150	5	)	)	PUNCT
ejpam-3952	151	1	+	+	CCONJ
ejpam-3952	151	2	(	(	PUNCT
ejpam-3952	151	3	l	l	NOUN
ejpam-3952	151	4	+	+	CCONJ
ejpam-3952	151	5	2)2	2)2	NUM
ejpam-3952	151	6	,	,	PUNCT
ejpam-3952	151	7	and	and	CCONJ
ejpam-3952	151	8	hence	hence	ADV
ejpam-3952	151	9	,	,	PUNCT
ejpam-3952	151	10	n	n	NOUN
ejpam-3952	151	11	=	=	SYM
ejpam-3952	151	12	4l2m2	4l2m2	NUM
ejpam-3952	152	1	+	+	PUNCT
ejpam-3952	152	2	4lm(l	4lm(l	NUM
ejpam-3952	152	3	+	+	CCONJ
ejpam-3952	152	4	2	2	NUM
ejpam-3952	152	5	)	)	PUNCT
ejpam-3952	153	1	+	+	CCONJ
ejpam-3952	153	2	(	(	PUNCT
ejpam-3952	153	3	l	l	NOUN
ejpam-3952	153	4	+	+	CCONJ
ejpam-3952	153	5	2)2	2)2	NUM
ejpam-3952	153	6	−	−	NOUN
ejpam-3952	153	7	(	(	PUNCT
ejpam-3952	153	8	l	l	NOUN
ejpam-3952	153	9	−	−	PROPN
ejpam-3952	153	10	2)2	2)2	NUM
ejpam-3952	153	11	8l	8l	NOUN
ejpam-3952	153	12	=	=	PUNCT
ejpam-3952	153	13	8ml	8ml	NOUN
ejpam-3952	153	14	+	+	CCONJ
ejpam-3952	153	15	8l	8l	NOUN
ejpam-3952	153	16	+	+	CCONJ
ejpam-3952	153	17	4(m	4(m	NUM
ejpam-3952	153	18	+	+	CCONJ
ejpam-3952	153	19	1)ml2	1)ml2	NUM
ejpam-3952	153	20	8l	8l	NOUN
ejpam-3952	153	21	=	=	SYM
ejpam-3952	153	22	(	(	PUNCT
ejpam-3952	153	23	m	m	VERB
ejpam-3952	153	24	+	+	ADJ
ejpam-3952	153	25	1	1	NUM
ejpam-3952	153	26	)	)	PUNCT
ejpam-3952	153	27	+	+	CCONJ
ejpam-3952	153	28	(	(	PUNCT
ejpam-3952	153	29	m	m	VERB
ejpam-3952	153	30	+	+	NUM
ejpam-3952	153	31	1)ml	1)ml	NUM
ejpam-3952	153	32	2	2	NUM
ejpam-3952	153	33	=	=	SYM
ejpam-3952	153	34	t	t	PROPN
ejpam-3952	153	35	(	(	PUNCT
ejpam-3952	153	36	m	m	VERB
ejpam-3952	153	37	+	+	ADJ
ejpam-3952	153	38	1	1	NUM
ejpam-3952	153	39	,	,	PUNCT
ejpam-3952	153	40	l	l	NOUN
ejpam-3952	153	41	)	)	PUNCT
ejpam-3952	153	42	.	.	PUNCT
ejpam-3952	154	1	therefore	therefore	ADV
ejpam-3952	154	2	,	,	PUNCT
ejpam-3952	154	3	n	n	PRON
ejpam-3952	154	4	is	be	AUX
ejpam-3952	154	5	an	an	DET
ejpam-3952	154	6	l	l	NOUN
ejpam-3952	154	7	-	-	PUNCT
ejpam-3952	154	8	isosceles	isoscele	NOUN
ejpam-3952	154	9	triangular	triangular	NOUN
ejpam-3952	154	10	number	number	NOUN
ejpam-3952	154	11	as	as	SCONJ
ejpam-3952	154	12	desired	desire	VERB
ejpam-3952	154	13	.	.	PUNCT
ejpam-3952	155	1	for	for	ADP
ejpam-3952	155	2	l	l	NOUN
ejpam-3952	155	3	=	=	SYM
ejpam-3952	155	4	1	1	NUM
ejpam-3952	155	5	,	,	PUNCT
ejpam-3952	155	6	8nl	8nl	NOUN
ejpam-3952	155	7	+	+	CCONJ
ejpam-3952	155	8	(	(	PUNCT
ejpam-3952	155	9	l	l	NOUN
ejpam-3952	155	10	−	−	PROPN
ejpam-3952	155	11	2)2	2)2	NUM
ejpam-3952	155	12	=	=	SYM
ejpam-3952	155	13	8n	8n	NOUN
ejpam-3952	155	14	+	+	CCONJ
ejpam-3952	155	15	1	1	NUM
ejpam-3952	155	16	is	be	AUX
ejpam-3952	155	17	always	always	ADV
ejpam-3952	155	18	odd	odd	ADJ
ejpam-3952	155	19	.	.	PUNCT
ejpam-3952	156	1	hence	hence	ADV
ejpam-3952	156	2	,	,	PUNCT
ejpam-3952	156	3	if	if	SCONJ
ejpam-3952	156	4	8nl	8nl	PROPN
ejpam-3952	156	5	+	+	CCONJ
ejpam-3952	156	6	(	(	PUNCT
ejpam-3952	156	7	l	l	NOUN
ejpam-3952	156	8	−	−	PROPN
ejpam-3952	156	9	2)2	2)2	NUM
ejpam-3952	156	10	is	be	AUX
ejpam-3952	156	11	square	square	ADJ
ejpam-3952	156	12	,	,	PUNCT
ejpam-3952	156	13	then	then	ADV
ejpam-3952	156	14	√	√	VERB
ejpam-3952	156	15	8nl	8nl	NOUN
ejpam-3952	156	16	+	+	CCONJ
ejpam-3952	156	17	(	(	PUNCT
ejpam-3952	156	18	l	l	NOUN
ejpam-3952	156	19	−	−	PROPN
ejpam-3952	156	20	2)2	2)2	NUM
ejpam-3952	156	21	≡	≡	ADJ
ejpam-3952	156	22	√	√	ADJ
ejpam-3952	156	23	8n	8n	NOUN
ejpam-3952	156	24	+	+	CCONJ
ejpam-3952	156	25	1	1	NUM
ejpam-3952	156	26	≡	≡	PROPN
ejpam-3952	156	27	1	1	NUM
ejpam-3952	156	28	≡	≡	PROPN
ejpam-3952	156	29	(	(	PUNCT
ejpam-3952	156	30	l+2	l+2	NOUN
ejpam-3952	156	31	)	)	PUNCT
ejpam-3952	156	32	(	(	PUNCT
ejpam-3952	156	33	mod	mod	NOUN
ejpam-3952	156	34	2l	2l	NUM
ejpam-3952	156	35	)	)	PUNCT
ejpam-3952	156	36	.	.	PUNCT
ejpam-3952	157	1	using	use	VERB
ejpam-3952	157	2	theorem	theorem	NOUN
ejpam-3952	157	3	2	2	NUM
ejpam-3952	157	4	,	,	PUNCT
ejpam-3952	157	5	the	the	DET
ejpam-3952	157	6	following	follow	VERB
ejpam-3952	157	7	well	well	ADV
ejpam-3952	157	8	-	-	PUNCT
ejpam-3952	157	9	know	know	NOUN
ejpam-3952	157	10	result	result	NOUN
ejpam-3952	157	11	can	can	AUX
ejpam-3952	157	12	be	be	AUX
ejpam-3952	157	13	derived	derive	VERB
ejpam-3952	157	14	immediately	immediately	ADV
ejpam-3952	157	15	.	.	PUNCT
ejpam-3952	158	1	j.	j.	PROPN
ejpam-3952	158	2	punpim	punpim	PROPN
ejpam-3952	158	3	,	,	PUNCT
ejpam-3952	158	4	s.	s.	PROPN
ejpam-3952	158	5	jitman	jitman	PROPN
ejpam-3952	158	6	/	/	SYM
ejpam-3952	158	7	eur	eur	PROPN
ejpam-3952	158	8	.	.	PUNCT
ejpam-3952	159	1	j.	j.	PROPN
ejpam-3952	159	2	pure	pure	PROPN
ejpam-3952	159	3	appl	appl	PROPN
ejpam-3952	159	4	.	.	PROPN
ejpam-3952	159	5	math	math	PROPN
ejpam-3952	159	6	,	,	PUNCT
ejpam-3952	159	7	14	14	NUM
ejpam-3952	159	8	(	(	PUNCT
ejpam-3952	159	9	2	2	NUM
ejpam-3952	159	10	)	)	PUNCT
ejpam-3952	159	11	(	(	PUNCT
ejpam-3952	159	12	2021	2021	NUM
ejpam-3952	159	13	)	)	PUNCT
ejpam-3952	159	14	,	,	PUNCT
ejpam-3952	159	15	380	380	NUM
ejpam-3952	159	16	-	-	SYM
ejpam-3952	159	17	395	395	NUM
ejpam-3952	159	18	386	386	NUM
ejpam-3952	159	19	corollary	corollary	ADJ
ejpam-3952	159	20	1	1	NUM
ejpam-3952	159	21	(	(	PUNCT
ejpam-3952	159	22	[	[	X
ejpam-3952	159	23	3	3	NUM
ejpam-3952	159	24	]	]	PUNCT
ejpam-3952	159	25	)	)	PUNCT
ejpam-3952	159	26	.	.	PUNCT
ejpam-3952	160	1	let	let	VERB
ejpam-3952	160	2	n	n	PRON
ejpam-3952	160	3	be	be	AUX
ejpam-3952	160	4	a	a	DET
ejpam-3952	160	5	positive	positive	ADJ
ejpam-3952	160	6	integer	integer	NOUN
ejpam-3952	160	7	.	.	PUNCT
ejpam-3952	161	1	then	then	ADV
ejpam-3952	161	2	n	n	PRON
ejpam-3952	161	3	is	be	AUX
ejpam-3952	161	4	a	a	DET
ejpam-3952	161	5	triangular	triangular	NOUN
ejpam-3952	161	6	number	number	NOUN
ejpam-3952	161	7	if	if	SCONJ
ejpam-3952	161	8	and	and	CCONJ
ejpam-3952	161	9	only	only	ADV
ejpam-3952	161	10	if	if	SCONJ
ejpam-3952	161	11	8n	8n	NOUN
ejpam-3952	161	12	+	+	CCONJ
ejpam-3952	161	13	1	1	NUM
ejpam-3952	161	14	is	be	AUX
ejpam-3952	161	15	square	square	ADJ
ejpam-3952	161	16	.	.	PUNCT
ejpam-3952	162	1	in	in	ADP
ejpam-3952	162	2	general	general	ADJ
ejpam-3952	162	3	,	,	PUNCT
ejpam-3952	162	4	we	we	PRON
ejpam-3952	162	5	have	have	VERB
ejpam-3952	162	6	the	the	DET
ejpam-3952	162	7	following	follow	VERB
ejpam-3952	162	8	results	result	NOUN
ejpam-3952	162	9	on	on	ADP
ejpam-3952	162	10	the	the	DET
ejpam-3952	162	11	sum	sum	NOUN
ejpam-3952	162	12	of	of	ADP
ejpam-3952	162	13	k	k	PROPN
ejpam-3952	162	14	l	l	ADJ
ejpam-3952	162	15	-	-	PUNCT
ejpam-3952	162	16	isosceles	isoscele	NOUN
ejpam-3952	162	17	triangular	triangular	NOUN
ejpam-3952	162	18	numbers	number	NOUN
ejpam-3952	162	19	.	.	PUNCT
ejpam-3952	163	1	theorem	theorem	NOUN
ejpam-3952	163	2	3	3	X
ejpam-3952	163	3	.	.	PUNCT
ejpam-3952	164	1	let	let	VERB
ejpam-3952	164	2	l	l	NOUN
ejpam-3952	164	3	,	,	PUNCT
ejpam-3952	164	4	k	k	NOUN
ejpam-3952	164	5	,	,	PUNCT
ejpam-3952	164	6	and	and	CCONJ
ejpam-3952	164	7	n	n	PRON
ejpam-3952	164	8	be	be	AUX
ejpam-3952	164	9	positive	positive	ADJ
ejpam-3952	164	10	integers	integer	NOUN
ejpam-3952	164	11	.	.	PUNCT
ejpam-3952	165	1	then	then	ADV
ejpam-3952	165	2	n	n	PRON
ejpam-3952	165	3	is	be	AUX
ejpam-3952	165	4	a	a	DET
ejpam-3952	165	5	sum	sum	NOUN
ejpam-3952	165	6	of	of	ADP
ejpam-3952	165	7	k	k	PROPN
ejpam-3952	165	8	l	l	ADJ
ejpam-3952	165	9	-	-	PUNCT
ejpam-3952	165	10	isosceles	isoscele	NOUN
ejpam-3952	165	11	triangular	triangular	NOUN
ejpam-3952	165	12	numbers	number	NOUN
ejpam-3952	165	13	if	if	SCONJ
ejpam-3952	165	14	and	and	CCONJ
ejpam-3952	165	15	only	only	ADV
ejpam-3952	165	16	if	if	SCONJ
ejpam-3952	165	17	there	there	PRON
ejpam-3952	165	18	exist	exist	VERB
ejpam-3952	165	19	positive	positive	ADJ
ejpam-3952	165	20	integers	integer	NOUN
ejpam-3952	165	21	u1	u1	NOUN
ejpam-3952	165	22	,	,	PUNCT
ejpam-3952	165	23	u2	u2	NOUN
ejpam-3952	165	24	,	,	PUNCT
ejpam-3952	165	25	.	.	PUNCT
ejpam-3952	165	26	.	.	PUNCT
ejpam-3952	166	1	.	.	PUNCT
ejpam-3952	167	1	,	,	PUNCT
ejpam-3952	167	2	uk	uk	PROPN
ejpam-3952	167	3	such	such	ADJ
ejpam-3952	167	4	that	that	SCONJ
ejpam-3952	167	5	(	(	PUNCT
ejpam-3952	167	6	i	i	NOUN
ejpam-3952	167	7	)	)	PUNCT
ejpam-3952	167	8	8ln	8ln	ADJ
ejpam-3952	168	1	+	+	CCONJ
ejpam-3952	168	2	k(l	k(l	NOUN
ejpam-3952	168	3	−	−	PROPN
ejpam-3952	168	4	2)2	2)2	NUM
ejpam-3952	168	5	=	=	SYM
ejpam-3952	168	6	u21	u21	PROPN
ejpam-3952	168	7	+	+	CCONJ
ejpam-3952	168	8	u22	u22	PROPN
ejpam-3952	168	9	+	+	PRON
ejpam-3952	168	10	·	·	PUNCT
ejpam-3952	168	11	·	·	PUNCT
ejpam-3952	168	12	·	·	PUNCT
ejpam-3952	168	13	+	+	NUM
ejpam-3952	168	14	u2k	u2k	NOUN
ejpam-3952	168	15	,	,	PUNCT
ejpam-3952	168	16	and	and	CCONJ
ejpam-3952	168	17	(	(	PUNCT
ejpam-3952	168	18	ii	ii	NOUN
ejpam-3952	168	19	)	)	PUNCT
ejpam-3952	168	20	ui	ui	PROPN
ejpam-3952	169	1	≡	≡	PROPN
ejpam-3952	169	2	l	l	PROPN
ejpam-3952	170	1	+	+	CCONJ
ejpam-3952	170	2	2	2	NUM
ejpam-3952	170	3	(	(	PUNCT
ejpam-3952	170	4	mod	mod	NOUN
ejpam-3952	170	5	2l	2l	NUM
ejpam-3952	170	6	)	)	PUNCT
ejpam-3952	170	7	for	for	ADP
ejpam-3952	170	8	all	all	DET
ejpam-3952	170	9	i	i	PRON
ejpam-3952	170	10	=	=	NOUN
ejpam-3952	170	11	1	1	NUM
ejpam-3952	170	12	,	,	PUNCT
ejpam-3952	170	13	2	2	NUM
ejpam-3952	170	14	,	,	PUNCT
ejpam-3952	170	15	.	.	PUNCT
ejpam-3952	170	16	.	.	PUNCT
ejpam-3952	170	17	.	.	PUNCT
ejpam-3952	171	1	,	,	PUNCT
ejpam-3952	171	2	k.	k.	PROPN
ejpam-3952	171	3	proof	proof	PROPN
ejpam-3952	171	4	.	.	PUNCT
ejpam-3952	172	1	assume	assume	VERB
ejpam-3952	172	2	that	that	SCONJ
ejpam-3952	172	3	n	n	PRON
ejpam-3952	172	4	is	be	AUX
ejpam-3952	172	5	a	a	DET
ejpam-3952	172	6	sum	sum	NOUN
ejpam-3952	172	7	of	of	ADP
ejpam-3952	172	8	k	k	PROPN
ejpam-3952	172	9	l	l	ADJ
ejpam-3952	172	10	-	-	PUNCT
ejpam-3952	172	11	isosceles	isoscele	NOUN
ejpam-3952	172	12	triangular	triangular	NOUN
ejpam-3952	172	13	numbers	number	NOUN
ejpam-3952	172	14	.	.	PUNCT
ejpam-3952	173	1	then	then	ADV
ejpam-3952	173	2	n	n	PROPN
ejpam-3952	173	3	=	=	SYM
ejpam-3952	173	4	k∑	k∑	PROPN
ejpam-3952	173	5	i=1	i=1	PROPN
ejpam-3952	173	6	t	t	PROPN
ejpam-3952	173	7	(	(	PUNCT
ejpam-3952	173	8	mi	mi	PROPN
ejpam-3952	173	9	,	,	PUNCT
ejpam-3952	173	10	l	l	NOUN
ejpam-3952	173	11	)	)	PUNCT
ejpam-3952	173	12	for	for	ADP
ejpam-3952	173	13	some	some	DET
ejpam-3952	173	14	positive	positive	ADJ
ejpam-3952	173	15	integers	integer	NOUN
ejpam-3952	173	16	mi	mi	PROPN
ejpam-3952	173	17	.	.	PROPN
ejpam-3952	174	1	it	it	PRON
ejpam-3952	174	2	follows	follow	VERB
ejpam-3952	174	3	that	that	SCONJ
ejpam-3952	174	4	n	n	PROPN
ejpam-3952	174	5	=	=	SYM
ejpam-3952	174	6	k∑	k∑	PROPN
ejpam-3952	174	7	i=1	i=1	PROPN
ejpam-3952	175	1	(	(	PUNCT
ejpam-3952	175	2	mi	mi	PROPN
ejpam-3952	175	3	+	+	CCONJ
ejpam-3952	175	4	mi(mi	mi(mi	PROPN
ejpam-3952	175	5	−	−	PROPN
ejpam-3952	175	6	1)l	1)l	NUM
ejpam-3952	175	7	2	2	NUM
ejpam-3952	175	8	)	)	PUNCT
ejpam-3952	175	9	=	=	VERB
ejpam-3952	176	1	k∑	k∑	VERB
ejpam-3952	176	2	i=1	i=1	PROPN
ejpam-3952	177	1	2mi	2mi	PROPN
ejpam-3952	178	1	+	+	CCONJ
ejpam-3952	178	2	m2	m2	PROPN
ejpam-3952	178	3	i	i	PROPN
ejpam-3952	178	4	l	l	PROPN
ejpam-3952	178	5	−mil	−mil	X
ejpam-3952	178	6	2	2	NUM
ejpam-3952	178	7	and	and	CCONJ
ejpam-3952	178	8	8ln	8ln	ADJ
ejpam-3952	179	1	+	+	CCONJ
ejpam-3952	179	2	k(l	k(l	NOUN
ejpam-3952	179	3	−	−	PROPN
ejpam-3952	179	4	2)2	2)2	NUM
ejpam-3952	179	5	=	=	SYM
ejpam-3952	179	6	k(l	k(l	NOUN
ejpam-3952	179	7	−	−	PROPN
ejpam-3952	179	8	2)2	2)2	NUM
ejpam-3952	179	9	+	+	NUM
ejpam-3952	179	10	4l	4l	NOUN
ejpam-3952	179	11	k∑	k∑	VERB
ejpam-3952	179	12	i=1	i=1	PROPN
ejpam-3952	180	1	(	(	PUNCT
ejpam-3952	180	2	2mi	2mi	PROPN
ejpam-3952	180	3	+	+	NUM
ejpam-3952	180	4	m2	m2	PROPN
ejpam-3952	180	5	i	i	PROPN
ejpam-3952	180	6	l	l	NOUN
ejpam-3952	180	7	−mil	−mil	NOUN
ejpam-3952	180	8	)	)	PUNCT
ejpam-3952	181	1	=	=	PRON
ejpam-3952	181	2	k∑	k∑	VERB
ejpam-3952	181	3	i=1	i=1	X
ejpam-3952	182	1	(	(	PUNCT
ejpam-3952	182	2	4m2	4m2	NUM
ejpam-3952	182	3	i	i	NOUN
ejpam-3952	182	4	l	l	NOUN
ejpam-3952	182	5	2	2	NUM
ejpam-3952	182	6	−	−	NOUN
ejpam-3952	182	7	4mil(l	4mil(l	NUM
ejpam-3952	182	8	−	−	NOUN
ejpam-3952	182	9	2	2	NUM
ejpam-3952	182	10	)	)	PUNCT
ejpam-3952	182	11	+	+	CCONJ
ejpam-3952	182	12	(	(	PUNCT
ejpam-3952	182	13	l	l	NOUN
ejpam-3952	182	14	−	−	PROPN
ejpam-3952	182	15	2)2	2)2	NUM
ejpam-3952	182	16	)	)	PUNCT
ejpam-3952	183	1	=	=	SYM
ejpam-3952	183	2	k∑	k∑	PROPN
ejpam-3952	183	3	i=1	i=1	X
ejpam-3952	184	1	(	(	PUNCT
ejpam-3952	184	2	2mil	2mil	NUM
ejpam-3952	184	3	−	−	PROPN
ejpam-3952	184	4	(	(	PUNCT
ejpam-3952	184	5	l	l	NOUN
ejpam-3952	184	6	−	−	PROPN
ejpam-3952	184	7	2))2	2))2	NUM
ejpam-3952	184	8	.	.	PUNCT
ejpam-3952	185	1	for	for	ADP
ejpam-3952	185	2	each	each	DET
ejpam-3952	185	3	i	i	PRON
ejpam-3952	185	4	∈	∈	PROPN
ejpam-3952	185	5	{	{	PUNCT
ejpam-3952	185	6	1	1	NUM
ejpam-3952	185	7	,	,	PUNCT
ejpam-3952	185	8	2	2	NUM
ejpam-3952	185	9	,	,	PUNCT
ejpam-3952	185	10	.	.	PUNCT
ejpam-3952	185	11	.	.	PUNCT
ejpam-3952	185	12	.	.	PUNCT
ejpam-3952	186	1	,	,	PUNCT
ejpam-3952	186	2	k	k	X
ejpam-3952	186	3	}	}	PUNCT
ejpam-3952	186	4	,	,	PUNCT
ejpam-3952	186	5	let	let	VERB
ejpam-3952	186	6	ui	ui	NOUN
ejpam-3952	186	7	=	=	SYM
ejpam-3952	187	1	2mil	2mil	NUM
ejpam-3952	187	2	−	−	PROPN
ejpam-3952	187	3	(	(	PUNCT
ejpam-3952	187	4	l	l	NOUN
ejpam-3952	187	5	−	−	PROPN
ejpam-3952	188	1	2	2	NUM
ejpam-3952	188	2	)	)	PUNCT
ejpam-3952	188	3	.	.	PUNCT
ejpam-3952	189	1	it	it	PRON
ejpam-3952	189	2	follows	follow	VERB
ejpam-3952	189	3	that	that	SCONJ
ejpam-3952	189	4	8ln	8ln	ADJ
ejpam-3952	190	1	+	+	CCONJ
ejpam-3952	190	2	k(l	k(l	NOUN
ejpam-3952	190	3	−	−	PROPN
ejpam-3952	190	4	2)2	2)2	NUM
ejpam-3952	190	5	=	=	SYM
ejpam-3952	190	6	u21	u21	PROPN
ejpam-3952	190	7	+	+	CCONJ
ejpam-3952	190	8	u22	u22	PROPN
ejpam-3952	190	9	+	+	PRON
ejpam-3952	190	10	·	·	PUNCT
ejpam-3952	190	11	·	·	PUNCT
ejpam-3952	190	12	·	·	PUNCT
ejpam-3952	190	13	+	+	CCONJ
ejpam-3952	190	14	u2k	u2k	PRON
ejpam-3952	190	15	and	and	CCONJ
ejpam-3952	190	16	ui	ui	PROPN
ejpam-3952	190	17	≡	≡	PROPN
ejpam-3952	190	18	2lmi	2lmi	PROPN
ejpam-3952	190	19	−	−	PROPN
ejpam-3952	190	20	(	(	PUNCT
ejpam-3952	190	21	l	l	NOUN
ejpam-3952	190	22	−	−	PROPN
ejpam-3952	190	23	2	2	X
ejpam-3952	190	24	)	)	PUNCT
ejpam-3952	190	25	≡	≡	PROPN
ejpam-3952	190	26	l	l	NOUN
ejpam-3952	191	1	+	+	CCONJ
ejpam-3952	191	2	2	2	NUM
ejpam-3952	191	3	(	(	PUNCT
ejpam-3952	191	4	mod	mod	NOUN
ejpam-3952	191	5	2l	2l	NUM
ejpam-3952	191	6	)	)	PUNCT
ejpam-3952	191	7	for	for	ADP
ejpam-3952	191	8	all	all	PRON
ejpam-3952	191	9	i	i	PRON
ejpam-3952	191	10	∈	∈	PROPN
ejpam-3952	191	11	{	{	PUNCT
ejpam-3952	191	12	1	1	NUM
ejpam-3952	191	13	,	,	PUNCT
ejpam-3952	191	14	2	2	NUM
ejpam-3952	191	15	,	,	PUNCT
ejpam-3952	191	16	.	.	PUNCT
ejpam-3952	191	17	.	.	PUNCT
ejpam-3952	191	18	.	.	PUNCT
ejpam-3952	192	1	,	,	PUNCT
ejpam-3952	192	2	k	k	X
ejpam-3952	192	3	}	}	PUNCT
ejpam-3952	192	4	.	.	PUNCT
ejpam-3952	193	1	conversely	conversely	ADV
ejpam-3952	193	2	,	,	PUNCT
ejpam-3952	193	3	assume	assume	VERB
ejpam-3952	193	4	that	that	SCONJ
ejpam-3952	193	5	there	there	PRON
ejpam-3952	193	6	exist	exist	VERB
ejpam-3952	193	7	positive	positive	ADJ
ejpam-3952	193	8	integers	integer	NOUN
ejpam-3952	193	9	u1	u1	NOUN
ejpam-3952	193	10	,	,	PUNCT
ejpam-3952	193	11	u2	u2	NOUN
ejpam-3952	193	12	,	,	PUNCT
ejpam-3952	193	13	.	.	PUNCT
ejpam-3952	193	14	.	.	PUNCT
ejpam-3952	193	15	.	.	PUNCT
ejpam-3952	194	1	,	,	PUNCT
ejpam-3952	194	2	uk	uk	PROPN
ejpam-3952	194	3	such	such	ADJ
ejpam-3952	194	4	that	that	PRON
ejpam-3952	194	5	8ln	8ln	ADJ
ejpam-3952	194	6	+	+	CCONJ
ejpam-3952	194	7	k(l	k(l	NOUN
ejpam-3952	194	8	−	−	PROPN
ejpam-3952	194	9	2)2	2)2	NUM
ejpam-3952	194	10	=	=	SYM
ejpam-3952	194	11	u21	u21	PROPN
ejpam-3952	194	12	+	+	CCONJ
ejpam-3952	194	13	u22	u22	PROPN
ejpam-3952	194	14	+	+	PRON
ejpam-3952	194	15	·	·	PUNCT
ejpam-3952	194	16	·	·	PUNCT
ejpam-3952	194	17	·	·	PUNCT
ejpam-3952	194	18	+	+	CCONJ
ejpam-3952	194	19	u2k	u2k	PRON
ejpam-3952	194	20	and	and	CCONJ
ejpam-3952	194	21	ui	ui	PROPN
ejpam-3952	194	22	≡	≡	PROPN
ejpam-3952	194	23	l	l	PROPN
ejpam-3952	195	1	+	+	CCONJ
ejpam-3952	195	2	2	2	NUM
ejpam-3952	195	3	(	(	PUNCT
ejpam-3952	195	4	mod	mod	NOUN
ejpam-3952	195	5	2l	2l	NUM
ejpam-3952	195	6	)	)	PUNCT
ejpam-3952	195	7	for	for	ADP
ejpam-3952	195	8	all	all	DET
ejpam-3952	195	9	i	i	PRON
ejpam-3952	195	10	=	=	NOUN
ejpam-3952	195	11	1	1	NUM
ejpam-3952	195	12	,	,	PUNCT
ejpam-3952	195	13	2	2	NUM
ejpam-3952	195	14	,	,	PUNCT
ejpam-3952	195	15	.	.	PUNCT
ejpam-3952	195	16	.	.	PUNCT
ejpam-3952	195	17	.	.	PUNCT
ejpam-3952	196	1	,	,	PUNCT
ejpam-3952	196	2	k.	k.	PROPN
ejpam-3952	196	3	for	for	ADP
ejpam-3952	196	4	each	each	DET
ejpam-3952	196	5	i	i	PRON
ejpam-3952	196	6	∈	∈	PROPN
ejpam-3952	196	7	{	{	PUNCT
ejpam-3952	196	8	1	1	NUM
ejpam-3952	196	9	,	,	PUNCT
ejpam-3952	196	10	2	2	NUM
ejpam-3952	196	11	,	,	PUNCT
ejpam-3952	196	12	.	.	PUNCT
ejpam-3952	196	13	.	.	PUNCT
ejpam-3952	197	1	.	.	PUNCT
ejpam-3952	198	1	,	,	PUNCT
ejpam-3952	198	2	k	k	X
ejpam-3952	198	3	}	}	PUNCT
ejpam-3952	198	4	,	,	PUNCT
ejpam-3952	198	5	let	let	VERB
ejpam-3952	198	6	mi	mi	PROPN
ejpam-3952	198	7	=	=	PROPN
ejpam-3952	198	8	ui	ui	PROPN
ejpam-3952	199	1	+	+	CCONJ
ejpam-3952	199	2	(	(	PUNCT
ejpam-3952	199	3	l	l	NOUN
ejpam-3952	199	4	−	−	PROPN
ejpam-3952	199	5	2	2	NUM
ejpam-3952	199	6	)	)	PUNCT
ejpam-3952	199	7	2l	2l	NOUN
ejpam-3952	199	8	.	.	PUNCT
ejpam-3952	200	1	since	since	SCONJ
ejpam-3952	200	2	ui	ui	PROPN
ejpam-3952	200	3	≡	≡	PROPN
ejpam-3952	200	4	l	l	PROPN
ejpam-3952	201	1	+	+	CCONJ
ejpam-3952	201	2	2	2	NUM
ejpam-3952	201	3	(	(	PUNCT
ejpam-3952	201	4	mod	mod	NOUN
ejpam-3952	201	5	2l	2l	NUM
ejpam-3952	201	6	)	)	PUNCT
ejpam-3952	201	7	,	,	PUNCT
ejpam-3952	201	8	it	it	PRON
ejpam-3952	201	9	follows	follow	VERB
ejpam-3952	201	10	that	that	SCONJ
ejpam-3952	201	11	mi	mi	PROPN
ejpam-3952	201	12	is	be	AUX
ejpam-3952	201	13	a	a	DET
ejpam-3952	201	14	positive	positive	ADJ
ejpam-3952	201	15	integer	integer	NOUN
ejpam-3952	201	16	.	.	PUNCT
ejpam-3952	202	1	it	it	PRON
ejpam-3952	202	2	is	be	AUX
ejpam-3952	202	3	not	not	PART
ejpam-3952	202	4	difficult	difficult	ADJ
ejpam-3952	202	5	to	to	PART
ejpam-3952	202	6	verify	verify	VERB
ejpam-3952	202	7	that	that	PRON
ejpam-3952	202	8	n	n	NOUN
ejpam-3952	202	9	=	=	SYM
ejpam-3952	202	10	k∑	k∑	PROPN
ejpam-3952	203	1	i=1	i=1	PROPN
ejpam-3952	204	1	t	t	PROPN
ejpam-3952	205	1	(	(	PUNCT
ejpam-3952	206	1	mi	mi	PROPN
ejpam-3952	206	2	,	,	PUNCT
ejpam-3952	206	3	l	l	NOUN
ejpam-3952	206	4	)	)	PUNCT
ejpam-3952	206	5	.	.	PUNCT
ejpam-3952	207	1	for	for	ADP
ejpam-3952	207	2	k	k	PROPN
ejpam-3952	207	3	=	=	SYM
ejpam-3952	207	4	1	1	NUM
ejpam-3952	207	5	,	,	PUNCT
ejpam-3952	207	6	theorem	theorem	VERB
ejpam-3952	207	7	3	3	NUM
ejpam-3952	207	8	becomes	become	VERB
ejpam-3952	207	9	theorem	theorem	ADJ
ejpam-3952	207	10	2	2	NUM
ejpam-3952	207	11	.	.	PUNCT
ejpam-3952	207	12	by	by	ADP
ejpam-3952	207	13	setting	set	VERB
ejpam-3952	207	14	l	l	NOUN
ejpam-3952	207	15	=	=	SYM
ejpam-3952	207	16	1	1	NUM
ejpam-3952	207	17	in	in	ADP
ejpam-3952	207	18	theorem	theorem	NOUN
ejpam-3952	207	19	3	3	NUM
ejpam-3952	207	20	,	,	PUNCT
ejpam-3952	207	21	we	we	PRON
ejpam-3952	207	22	have	have	VERB
ejpam-3952	207	23	the	the	DET
ejpam-3952	207	24	following	follow	VERB
ejpam-3952	207	25	well	well	ADV
ejpam-3952	207	26	-	-	PUNCT
ejpam-3952	207	27	known	know	VERB
ejpam-3952	207	28	result	result	NOUN
ejpam-3952	207	29	.	.	PUNCT
ejpam-3952	208	1	j.	j.	PROPN
ejpam-3952	208	2	punpim	punpim	PROPN
ejpam-3952	208	3	,	,	PUNCT
ejpam-3952	208	4	s.	s.	PROPN
ejpam-3952	208	5	jitman	jitman	PROPN
ejpam-3952	208	6	/	/	SYM
ejpam-3952	208	7	eur	eur	PROPN
ejpam-3952	208	8	.	.	PUNCT
ejpam-3952	209	1	j.	j.	PROPN
ejpam-3952	209	2	pure	pure	PROPN
ejpam-3952	209	3	appl	appl	PROPN
ejpam-3952	209	4	.	.	PROPN
ejpam-3952	209	5	math	math	PROPN
ejpam-3952	209	6	,	,	PUNCT
ejpam-3952	209	7	14	14	NUM
ejpam-3952	209	8	(	(	PUNCT
ejpam-3952	209	9	2	2	NUM
ejpam-3952	209	10	)	)	PUNCT
ejpam-3952	209	11	(	(	PUNCT
ejpam-3952	209	12	2021	2021	NUM
ejpam-3952	209	13	)	)	PUNCT
ejpam-3952	209	14	,	,	PUNCT
ejpam-3952	209	15	380	380	NUM
ejpam-3952	209	16	-	-	SYM
ejpam-3952	209	17	395	395	NUM
ejpam-3952	209	18	387	387	NUM
ejpam-3952	209	19	corollary	corollary	ADJ
ejpam-3952	209	20	2	2	NUM
ejpam-3952	209	21	(	(	PUNCT
ejpam-3952	209	22	[	[	X
ejpam-3952	209	23	3	3	NUM
ejpam-3952	209	24	]	]	PUNCT
ejpam-3952	209	25	)	)	PUNCT
ejpam-3952	209	26	.	.	PUNCT
ejpam-3952	210	1	let	let	VERB
ejpam-3952	210	2	k	k	NOUN
ejpam-3952	210	3	and	and	CCONJ
ejpam-3952	210	4	n	n	CCONJ
ejpam-3952	210	5	be	be	VERB
ejpam-3952	210	6	positive	positive	ADJ
ejpam-3952	210	7	integers	integer	NOUN
ejpam-3952	210	8	.	.	PUNCT
ejpam-3952	211	1	then	then	ADV
ejpam-3952	211	2	n	n	PRON
ejpam-3952	211	3	is	be	AUX
ejpam-3952	211	4	a	a	DET
ejpam-3952	211	5	sum	sum	NOUN
ejpam-3952	211	6	of	of	ADP
ejpam-3952	211	7	k	k	PROPN
ejpam-3952	211	8	triangular	triangular	NOUN
ejpam-3952	211	9	numbers	number	NOUN
ejpam-3952	211	10	if	if	SCONJ
ejpam-3952	211	11	and	and	CCONJ
ejpam-3952	211	12	only	only	ADV
ejpam-3952	211	13	if	if	SCONJ
ejpam-3952	211	14	8n	8n	NOUN
ejpam-3952	211	15	+	+	CCONJ
ejpam-3952	211	16	k	k	PROPN
ejpam-3952	211	17	is	be	AUX
ejpam-3952	211	18	a	a	DET
ejpam-3952	211	19	sum	sum	NOUN
ejpam-3952	211	20	of	of	ADP
ejpam-3952	211	21	k	k	PROPN
ejpam-3952	211	22	odd	odd	ADJ
ejpam-3952	211	23	squares	square	NOUN
ejpam-3952	211	24	.	.	PUNCT
ejpam-3952	212	1	from	from	ADP
ejpam-3952	212	2	theorem	theorem	ADJ
ejpam-3952	212	3	3	3	NUM
ejpam-3952	212	4	,	,	PUNCT
ejpam-3952	212	5	a	a	DET
ejpam-3952	212	6	positive	positive	ADJ
ejpam-3952	212	7	integer	integer	NOUN
ejpam-3952	212	8	n	n	NUM
ejpam-3952	212	9	is	be	AUX
ejpam-3952	212	10	a	a	DET
ejpam-3952	212	11	sum	sum	NOUN
ejpam-3952	212	12	of	of	ADP
ejpam-3952	212	13	two	two	NUM
ejpam-3952	212	14	l	l	ADJ
ejpam-3952	212	15	-	-	PUNCT
ejpam-3952	212	16	isosceles	isoscele	NOUN
ejpam-3952	212	17	triangular	triangular	NOUN
ejpam-3952	212	18	numbers	number	NOUN
ejpam-3952	212	19	if	if	SCONJ
ejpam-3952	212	20	and	and	CCONJ
ejpam-3952	212	21	only	only	ADV
ejpam-3952	212	22	if	if	SCONJ
ejpam-3952	212	23	there	there	PRON
ejpam-3952	212	24	exist	exist	VERB
ejpam-3952	212	25	positive	positive	ADJ
ejpam-3952	212	26	integers	integer	NOUN
ejpam-3952	212	27	u	u	NOUN
ejpam-3952	212	28	and	and	CCONJ
ejpam-3952	212	29	v	v	ADP
ejpam-3952	212	30	such	such	ADJ
ejpam-3952	212	31	that	that	PRON
ejpam-3952	212	32	8ln	8ln	ADJ
ejpam-3952	213	1	+	+	PROPN
ejpam-3952	213	2	2(l−	2(l−	NUM
ejpam-3952	213	3	2)2	2)2	NUM
ejpam-3952	213	4	=	=	SYM
ejpam-3952	213	5	u2	u2	NOUN
ejpam-3952	213	6	+	+	CCONJ
ejpam-3952	213	7	v2	v2	PROPN
ejpam-3952	213	8	and	and	CCONJ
ejpam-3952	213	9	u	u	PROPN
ejpam-3952	213	10	≡	≡	PROPN
ejpam-3952	213	11	v	v	ADP
ejpam-3952	213	12	≡	≡	PROPN
ejpam-3952	213	13	l	l	PROPN
ejpam-3952	214	1	+	+	CCONJ
ejpam-3952	214	2	2	2	NUM
ejpam-3952	214	3	(	(	PUNCT
ejpam-3952	214	4	mod	mod	NOUN
ejpam-3952	214	5	2l	2l	NUM
ejpam-3952	214	6	)	)	PUNCT
ejpam-3952	214	7	.	.	PUNCT
ejpam-3952	215	1	an	an	DET
ejpam-3952	215	2	alternative	alternative	ADJ
ejpam-3952	215	3	characterization	characterization	NOUN
ejpam-3952	215	4	for	for	ADP
ejpam-3952	215	5	this	this	DET
ejpam-3952	215	6	case	case	NOUN
ejpam-3952	215	7	is	be	AUX
ejpam-3952	215	8	given	give	VERB
ejpam-3952	215	9	in	in	ADP
ejpam-3952	215	10	the	the	DET
ejpam-3952	215	11	next	next	ADJ
ejpam-3952	215	12	theorem	theorem	PROPN
ejpam-3952	215	13	.	.	PUNCT
ejpam-3952	215	14	theorem	theorem	NOUN
ejpam-3952	215	15	4	4	NUM
ejpam-3952	215	16	.	.	PUNCT
ejpam-3952	216	1	let	let	VERB
ejpam-3952	216	2	l	l	NOUN
ejpam-3952	216	3	and	and	CCONJ
ejpam-3952	216	4	n	n	CCONJ
ejpam-3952	216	5	be	be	AUX
ejpam-3952	216	6	positive	positive	ADJ
ejpam-3952	216	7	integers	integer	NOUN
ejpam-3952	216	8	.	.	PUNCT
ejpam-3952	217	1	then	then	ADV
ejpam-3952	217	2	n	n	PRON
ejpam-3952	217	3	is	be	AUX
ejpam-3952	217	4	a	a	DET
ejpam-3952	217	5	sum	sum	NOUN
ejpam-3952	217	6	of	of	ADP
ejpam-3952	217	7	two	two	NUM
ejpam-3952	217	8	l	l	ADJ
ejpam-3952	217	9	-	-	PUNCT
ejpam-3952	217	10	isosceles	isoscele	NOUN
ejpam-3952	217	11	triangular	triangular	NOUN
ejpam-3952	217	12	numbers	number	NOUN
ejpam-3952	217	13	if	if	SCONJ
ejpam-3952	217	14	and	and	CCONJ
ejpam-3952	217	15	only	only	ADV
ejpam-3952	217	16	if	if	SCONJ
ejpam-3952	217	17	there	there	PRON
ejpam-3952	217	18	exist	exist	VERB
ejpam-3952	217	19	positive	positive	ADJ
ejpam-3952	217	20	integers	integer	NOUN
ejpam-3952	217	21	u	u	NOUN
ejpam-3952	217	22	and	and	CCONJ
ejpam-3952	217	23	v	v	ADP
ejpam-3952	217	24	such	such	ADJ
ejpam-3952	217	25	that	that	SCONJ
ejpam-3952	217	26	(	(	PUNCT
ejpam-3952	217	27	i	i	NOUN
ejpam-3952	217	28	)	)	PUNCT
ejpam-3952	217	29	4ln	4ln	NOUN
ejpam-3952	218	1	+	+	PUNCT
ejpam-3952	218	2	(	(	PUNCT
ejpam-3952	218	3	l	l	NOUN
ejpam-3952	218	4	−	−	PROPN
ejpam-3952	218	5	2)2	2)2	NUM
ejpam-3952	218	6	=	=	SYM
ejpam-3952	218	7	u2	u2	PROPN
ejpam-3952	218	8	+	+	CCONJ
ejpam-3952	218	9	v2	v2	PROPN
ejpam-3952	218	10	,	,	PUNCT
ejpam-3952	218	11	and	and	CCONJ
ejpam-3952	218	12	(	(	PUNCT
ejpam-3952	218	13	ii	ii	NOUN
ejpam-3952	218	14	)	)	PUNCT
ejpam-3952	218	15	u	u	NOUN
ejpam-3952	218	16	+	+	PROPN
ejpam-3952	218	17	v	v	NUM
ejpam-3952	218	18	≡	≡	PROPN
ejpam-3952	218	19	l	l	NOUN
ejpam-3952	219	1	+	+	CCONJ
ejpam-3952	219	2	2	2	NUM
ejpam-3952	219	3	(	(	PUNCT
ejpam-3952	219	4	mod	mod	NOUN
ejpam-3952	219	5	2l	2l	NUM
ejpam-3952	219	6	)	)	PUNCT
ejpam-3952	219	7	and	and	CCONJ
ejpam-3952	219	8	u−	u−	PROPN
ejpam-3952	219	9	v	v	ADP
ejpam-3952	219	10	≡	≡	PROPN
ejpam-3952	219	11	l	l	NOUN
ejpam-3952	220	1	+	+	CCONJ
ejpam-3952	220	2	2	2	NUM
ejpam-3952	220	3	(	(	PUNCT
ejpam-3952	220	4	mod	mod	NOUN
ejpam-3952	220	5	2l	2l	NUM
ejpam-3952	220	6	)	)	PUNCT
ejpam-3952	220	7	.	.	PUNCT
ejpam-3952	221	1	proof	proof	NOUN
ejpam-3952	221	2	.	.	PUNCT
ejpam-3952	222	1	assume	assume	VERB
ejpam-3952	222	2	that	that	SCONJ
ejpam-3952	222	3	n	n	PRON
ejpam-3952	222	4	is	be	AUX
ejpam-3952	222	5	a	a	DET
ejpam-3952	222	6	sum	sum	NOUN
ejpam-3952	222	7	of	of	ADP
ejpam-3952	222	8	two	two	NUM
ejpam-3952	222	9	l	l	ADJ
ejpam-3952	222	10	-	-	PUNCT
ejpam-3952	222	11	isosceles	isoscele	NOUN
ejpam-3952	222	12	triangular	triangular	NOUN
ejpam-3952	222	13	numbers	number	NOUN
ejpam-3952	222	14	.	.	PUNCT
ejpam-3952	223	1	then	then	ADV
ejpam-3952	223	2	n	n	PROPN
ejpam-3952	223	3	=	=	SYM
ejpam-3952	223	4	t	t	PROPN
ejpam-3952	223	5	(	(	PUNCT
ejpam-3952	223	6	m	m	PROPN
ejpam-3952	223	7	,	,	PUNCT
ejpam-3952	223	8	l	l	NOUN
ejpam-3952	223	9	)	)	PUNCT
ejpam-3952	224	1	+	+	NUM
ejpam-3952	224	2	t	t	PROPN
ejpam-3952	224	3	(	(	PUNCT
ejpam-3952	224	4	n	n	CCONJ
ejpam-3952	224	5	,	,	PUNCT
ejpam-3952	224	6	l	l	NOUN
ejpam-3952	224	7	)	)	PUNCT
ejpam-3952	224	8	for	for	ADP
ejpam-3952	224	9	some	some	DET
ejpam-3952	224	10	positive	positive	ADJ
ejpam-3952	224	11	integers	integer	NOUN
ejpam-3952	224	12	m	m	VERB
ejpam-3952	224	13	and	and	CCONJ
ejpam-3952	224	14	n	n	CCONJ
ejpam-3952	224	15	,	,	PUNCT
ejpam-3952	224	16	i.e.	i.e.	X
ejpam-3952	224	17	,	,	PUNCT
ejpam-3952	224	18	n	n	PROPN
ejpam-3952	224	19	=	=	SYM
ejpam-3952	224	20	m	m	VERB
ejpam-3952	224	21	+	+	NUM
ejpam-3952	224	22	m(m−	m(m−	PROPN
ejpam-3952	224	23	1)l	1)l	NUM
ejpam-3952	224	24	2	2	NUM
ejpam-3952	224	25	+	+	CCONJ
ejpam-3952	224	26	n	n	PROPN
ejpam-3952	224	27	+	+	CCONJ
ejpam-3952	224	28	n(n−	n(n−	PROPN
ejpam-3952	224	29	1)l	1)l	NUM
ejpam-3952	224	30	2	2	NUM
ejpam-3952	224	31	=	=	SYM
ejpam-3952	224	32	2	2	NUM
ejpam-3952	224	33	m	m	NOUN
ejpam-3952	224	34	+	+	NUM
ejpam-3952	224	35	m(m−	m(m−	PROPN
ejpam-3952	224	36	1)l	1)l	NUM
ejpam-3952	224	37	+	+	CCONJ
ejpam-3952	224	38	2n	2n	NUM
ejpam-3952	225	1	+	+	CCONJ
ejpam-3952	225	2	n(n−	n(n−	PROPN
ejpam-3952	225	3	1)l	1)l	NUM
ejpam-3952	225	4	2	2	NUM
ejpam-3952	225	5	.	.	PUNCT
ejpam-3952	226	1	it	it	PRON
ejpam-3952	226	2	follows	follow	VERB
ejpam-3952	226	3	that	that	PRON
ejpam-3952	226	4	4ln	4ln	NOUN
ejpam-3952	227	1	+	+	CCONJ
ejpam-3952	227	2	(	(	PUNCT
ejpam-3952	227	3	l	l	NOUN
ejpam-3952	227	4	−	−	PROPN
ejpam-3952	227	5	2)2	2)2	NUM
ejpam-3952	227	6	=	=	SYM
ejpam-3952	227	7	2l	2l	NOUN
ejpam-3952	227	8	(	(	PUNCT
ejpam-3952	227	9	2	2	NUM
ejpam-3952	227	10	m	m	NOUN
ejpam-3952	227	11	+	+	NUM
ejpam-3952	227	12	m(m−	m(m−	PROPN
ejpam-3952	227	13	1)l	1)l	NUM
ejpam-3952	227	14	+	+	CCONJ
ejpam-3952	227	15	2n	2n	NUM
ejpam-3952	227	16	+	+	CCONJ
ejpam-3952	227	17	n(n−	n(n−	PROPN
ejpam-3952	227	18	1)l	1)l	NUM
ejpam-3952	227	19	)	)	PUNCT
ejpam-3952	228	1	+	+	CCONJ
ejpam-3952	228	2	(	(	PUNCT
ejpam-3952	228	3	l	l	NOUN
ejpam-3952	228	4	−	−	PROPN
ejpam-3952	228	5	2)2	2)2	NUM
ejpam-3952	228	6	=	=	SYM
ejpam-3952	228	7	4ml	4ml	NOUN
ejpam-3952	228	8	+	+	CCONJ
ejpam-3952	228	9	2m2l2	2m2l2	NUM
ejpam-3952	228	10	−	−	NOUN
ejpam-3952	228	11	2ml2	2ml2	NUM
ejpam-3952	228	12	+	+	CCONJ
ejpam-3952	228	13	4nl	4nl	NOUN
ejpam-3952	228	14	+	+	CCONJ
ejpam-3952	228	15	2n2l2	2n2l2	NUM
ejpam-3952	228	16	−	−	NOUN
ejpam-3952	228	17	2nl2	2nl2	NUM
ejpam-3952	229	1	+	+	CCONJ
ejpam-3952	229	2	(	(	PUNCT
ejpam-3952	229	3	l	l	NOUN
ejpam-3952	229	4	−	−	PROPN
ejpam-3952	229	5	2)2	2)2	NUM
ejpam-3952	229	6	=	=	SYM
ejpam-3952	229	7	(	(	PUNCT
ejpam-3952	229	8	lm)2	lm)2	PROPN
ejpam-3952	229	9	+	+	CCONJ
ejpam-3952	229	10	(	(	PUNCT
ejpam-3952	229	11	ln)2	ln)2	NOUN
ejpam-3952	229	12	+	+	CCONJ
ejpam-3952	229	13	(	(	PUNCT
ejpam-3952	229	14	l	l	NOUN
ejpam-3952	229	15	−	−	PROPN
ejpam-3952	229	16	2)2	2)2	NUM
ejpam-3952	229	17	+	+	PUNCT
ejpam-3952	229	18	2l2mn−	2l2mn−	NUM
ejpam-3952	229	19	2(l	2(l	NUM
ejpam-3952	229	20	−	−	PROPN
ejpam-3952	229	21	2)lm	2)lm	NUM
ejpam-3952	229	22	−	−	NOUN
ejpam-3952	229	23	2(l	2(l	NUM
ejpam-3952	229	24	−	−	PROPN
ejpam-3952	229	25	2)ln	2)ln	PROPN
ejpam-3952	230	1	+	+	CCONJ
ejpam-3952	230	2	(	(	PUNCT
ejpam-3952	230	3	lm)2	lm)2	VERB
ejpam-3952	230	4	−	−	PROPN
ejpam-3952	230	5	2mnl2	2mnl2	NUM
ejpam-3952	231	1	+	+	CCONJ
ejpam-3952	231	2	(	(	PUNCT
ejpam-3952	231	3	ln)2	ln)2	NOUN
ejpam-3952	231	4	=	=	PUNCT
ejpam-3952	231	5	(	(	PUNCT
ejpam-3952	231	6	l(m	l(m	PROPN
ejpam-3952	231	7	+	+	NUM
ejpam-3952	231	8	n)−	n)−	PROPN
ejpam-3952	231	9	(	(	PUNCT
ejpam-3952	231	10	l	l	NOUN
ejpam-3952	231	11	−	−	PROPN
ejpam-3952	231	12	2))2	2))2	NUM
ejpam-3952	231	13	+	+	CCONJ
ejpam-3952	231	14	(	(	PUNCT
ejpam-3952	231	15	l(m−	l(m−	PROPN
ejpam-3952	231	16	n))2	n))2	NOUN
ejpam-3952	231	17	.	.	PUNCT
ejpam-3952	231	18	let	let	VERB
ejpam-3952	231	19	u	u	NOUN
ejpam-3952	231	20	=	=	PUNCT
ejpam-3952	231	21	l(m	l(m	PROPN
ejpam-3952	232	1	+	+	NUM
ejpam-3952	232	2	n)−	n)−	PROPN
ejpam-3952	232	3	(	(	PUNCT
ejpam-3952	232	4	l−	l−	NOUN
ejpam-3952	232	5	2	2	NUM
ejpam-3952	232	6	)	)	PUNCT
ejpam-3952	232	7	and	and	CCONJ
ejpam-3952	232	8	v	v	X
ejpam-3952	232	9	=	=	SYM
ejpam-3952	232	10	l(m−	l(m−	PROPN
ejpam-3952	232	11	n	n	CCONJ
ejpam-3952	232	12	)	)	PUNCT
ejpam-3952	232	13	.	.	PUNCT
ejpam-3952	233	1	we	we	PRON
ejpam-3952	233	2	therefore	therefore	ADV
ejpam-3952	233	3	have	have	VERB
ejpam-3952	233	4	4ln	4ln	NOUN
ejpam-3952	233	5	+	+	CCONJ
ejpam-3952	233	6	(	(	PUNCT
ejpam-3952	233	7	l−	l−	PROPN
ejpam-3952	233	8	2)2	2)2	NUM
ejpam-3952	233	9	=	=	SYM
ejpam-3952	233	10	u2	u2	PROPN
ejpam-3952	233	11	+	+	CCONJ
ejpam-3952	233	12	v2	v2	PROPN
ejpam-3952	233	13	,	,	PUNCT
ejpam-3952	233	14	u	u	NOUN
ejpam-3952	233	15	+	+	PROPN
ejpam-3952	233	16	v	v	NUM
ejpam-3952	233	17	≡	≡	PROPN
ejpam-3952	233	18	2lm−	2lm−	NUM
ejpam-3952	233	19	l	l	NOUN
ejpam-3952	233	20	+	+	SYM
ejpam-3952	233	21	2	2	NUM
ejpam-3952	233	22	≡	≡	PROPN
ejpam-3952	233	23	l	l	NOUN
ejpam-3952	234	1	+	+	CCONJ
ejpam-3952	234	2	2	2	NUM
ejpam-3952	234	3	(	(	PUNCT
ejpam-3952	234	4	mod	mod	NOUN
ejpam-3952	234	5	2l	2l	NUM
ejpam-3952	234	6	)	)	PUNCT
ejpam-3952	234	7	,	,	PUNCT
ejpam-3952	234	8	and	and	CCONJ
ejpam-3952	234	9	u−	u−	PROPN
ejpam-3952	234	10	v	v	NUM
ejpam-3952	234	11	≡	≡	PROPN
ejpam-3952	234	12	2ln−	2ln−	NUM
ejpam-3952	234	13	l	l	NOUN
ejpam-3952	234	14	+	+	SYM
ejpam-3952	234	15	2	2	NUM
ejpam-3952	234	16	≡	≡	PROPN
ejpam-3952	234	17	l	l	NOUN
ejpam-3952	235	1	+	+	CCONJ
ejpam-3952	235	2	2	2	NUM
ejpam-3952	235	3	(	(	PUNCT
ejpam-3952	235	4	mod	mod	NOUN
ejpam-3952	235	5	2l	2l	NUM
ejpam-3952	235	6	)	)	PUNCT
ejpam-3952	235	7	.	.	PUNCT
ejpam-3952	236	1	conversely	conversely	ADV
ejpam-3952	236	2	,	,	PUNCT
ejpam-3952	236	3	assume	assume	VERB
ejpam-3952	236	4	that	that	SCONJ
ejpam-3952	236	5	there	there	PRON
ejpam-3952	236	6	exist	exist	VERB
ejpam-3952	236	7	positive	positive	ADJ
ejpam-3952	236	8	integers	integer	NOUN
ejpam-3952	236	9	u	u	NOUN
ejpam-3952	236	10	and	and	CCONJ
ejpam-3952	236	11	v	v	ADP
ejpam-3952	236	12	such	such	ADJ
ejpam-3952	236	13	that	that	DET
ejpam-3952	236	14	4ln	4ln	NOUN
ejpam-3952	237	1	+	+	CCONJ
ejpam-3952	237	2	(	(	PUNCT
ejpam-3952	237	3	l	l	NOUN
ejpam-3952	237	4	−	−	PROPN
ejpam-3952	237	5	2)2	2)2	NUM
ejpam-3952	237	6	=	=	SYM
ejpam-3952	237	7	u2	u2	PROPN
ejpam-3952	237	8	+	+	CCONJ
ejpam-3952	237	9	v2	v2	PROPN
ejpam-3952	237	10	,	,	PUNCT
ejpam-3952	237	11	u+v	u+v	NUM
ejpam-3952	237	12	≡	≡	PROPN
ejpam-3952	237	13	l+2	l+2	PROPN
ejpam-3952	237	14	(	(	PUNCT
ejpam-3952	237	15	mod	mod	PROPN
ejpam-3952	237	16	2l	2l	NUM
ejpam-3952	237	17	)	)	PUNCT
ejpam-3952	237	18	,	,	PUNCT
ejpam-3952	237	19	and	and	CCONJ
ejpam-3952	237	20	u−v	u−v	NUM
ejpam-3952	237	21	≡	≡	PROPN
ejpam-3952	237	22	l+2	l+2	PROPN
ejpam-3952	237	23	(	(	PUNCT
ejpam-3952	237	24	mod	mod	PROPN
ejpam-3952	237	25	2l	2l	NUM
ejpam-3952	237	26	)	)	PUNCT
ejpam-3952	237	27	.	.	PUNCT
ejpam-3952	238	1	without	without	ADP
ejpam-3952	238	2	loss	loss	NOUN
ejpam-3952	238	3	of	of	ADP
ejpam-3952	238	4	generality	generality	NOUN
ejpam-3952	238	5	,	,	PUNCT
ejpam-3952	238	6	assume	assume	VERB
ejpam-3952	238	7	that	that	SCONJ
ejpam-3952	238	8	u	u	PRON
ejpam-3952	238	9	≥	≥	X
ejpam-3952	238	10	v.	v.	ADV
ejpam-3952	238	11	let	let	VERB
ejpam-3952	238	12	m	m	VERB
ejpam-3952	238	13	=	=	VERB
ejpam-3952	238	14	u	u	PROPN
ejpam-3952	238	15	+	+	X
ejpam-3952	238	16	v	v	NOUN
ejpam-3952	238	17	+	+	CCONJ
ejpam-3952	238	18	(	(	PUNCT
ejpam-3952	238	19	l	l	NOUN
ejpam-3952	238	20	−	−	PROPN
ejpam-3952	238	21	2	2	NUM
ejpam-3952	238	22	)	)	PUNCT
ejpam-3952	238	23	2l	2l	NOUN
ejpam-3952	238	24	and	and	CCONJ
ejpam-3952	238	25	n	n	CCONJ
ejpam-3952	238	26	=	=	NOUN
ejpam-3952	238	27	u−	u−	PROPN
ejpam-3952	238	28	v	v	NOUN
ejpam-3952	238	29	+	+	CCONJ
ejpam-3952	238	30	(	(	PUNCT
ejpam-3952	238	31	l	l	NOUN
ejpam-3952	238	32	−	−	PROPN
ejpam-3952	238	33	2	2	NUM
ejpam-3952	238	34	)	)	PUNCT
ejpam-3952	238	35	2l	2l	NOUN
ejpam-3952	238	36	.	.	PUNCT
ejpam-3952	239	1	since	since	SCONJ
ejpam-3952	239	2	u	u	PROPN
ejpam-3952	239	3	+	+	PROPN
ejpam-3952	239	4	v	v	ADP
ejpam-3952	239	5	≡	≡	PROPN
ejpam-3952	239	6	l	l	NOUN
ejpam-3952	239	7	+	+	CCONJ
ejpam-3952	239	8	2	2	NUM
ejpam-3952	239	9	(	(	PUNCT
ejpam-3952	239	10	mod	mod	NOUN
ejpam-3952	239	11	2l	2l	NUM
ejpam-3952	239	12	)	)	PUNCT
ejpam-3952	239	13	and	and	CCONJ
ejpam-3952	239	14	u−	u−	PROPN
ejpam-3952	239	15	v	v	ADP
ejpam-3952	239	16	≡	≡	PROPN
ejpam-3952	239	17	l+	l+	X
ejpam-3952	239	18	2	2	NUM
ejpam-3952	239	19	(	(	PUNCT
ejpam-3952	239	20	mod	mod	NOUN
ejpam-3952	239	21	2l	2l	NUM
ejpam-3952	239	22	)	)	PUNCT
ejpam-3952	239	23	,	,	PUNCT
ejpam-3952	239	24	it	it	PRON
ejpam-3952	239	25	follows	follow	VERB
ejpam-3952	239	26	that	that	SCONJ
ejpam-3952	239	27	m	m	VERB
ejpam-3952	239	28	and	and	CCONJ
ejpam-3952	239	29	n	n	PRON
ejpam-3952	239	30	are	be	AUX
ejpam-3952	239	31	positive	positive	ADJ
ejpam-3952	239	32	integers	integer	NOUN
ejpam-3952	239	33	.	.	PUNCT
ejpam-3952	240	1	it	it	PRON
ejpam-3952	240	2	is	be	AUX
ejpam-3952	240	3	not	not	PART
ejpam-3952	240	4	difficult	difficult	ADJ
ejpam-3952	240	5	to	to	PART
ejpam-3952	240	6	verify	verify	VERB
ejpam-3952	240	7	that	that	PRON
ejpam-3952	240	8	n	n	NOUN
ejpam-3952	240	9	=	=	SYM
ejpam-3952	240	10	t	t	PROPN
ejpam-3952	240	11	(	(	PUNCT
ejpam-3952	240	12	m	m	PROPN
ejpam-3952	240	13	,	,	PUNCT
ejpam-3952	240	14	l	l	NOUN
ejpam-3952	240	15	)	)	PUNCT
ejpam-3952	241	1	+	+	NUM
ejpam-3952	241	2	t	t	PROPN
ejpam-3952	241	3	(	(	PUNCT
ejpam-3952	241	4	n	n	CCONJ
ejpam-3952	241	5	,	,	PUNCT
ejpam-3952	241	6	l	l	NOUN
ejpam-3952	241	7	)	)	PUNCT
ejpam-3952	241	8	.	.	PUNCT
ejpam-3952	242	1	j.	j.	PROPN
ejpam-3952	242	2	punpim	punpim	PROPN
ejpam-3952	242	3	,	,	PUNCT
ejpam-3952	242	4	s.	s.	PROPN
ejpam-3952	242	5	jitman	jitman	PROPN
ejpam-3952	242	6	/	/	SYM
ejpam-3952	242	7	eur	eur	PROPN
ejpam-3952	242	8	.	.	PUNCT
ejpam-3952	243	1	j.	j.	PROPN
ejpam-3952	243	2	pure	pure	PROPN
ejpam-3952	243	3	appl	appl	PROPN
ejpam-3952	243	4	.	.	PROPN
ejpam-3952	243	5	math	math	PROPN
ejpam-3952	243	6	,	,	PUNCT
ejpam-3952	243	7	14	14	NUM
ejpam-3952	243	8	(	(	PUNCT
ejpam-3952	243	9	2	2	NUM
ejpam-3952	243	10	)	)	PUNCT
ejpam-3952	243	11	(	(	PUNCT
ejpam-3952	243	12	2021	2021	NUM
ejpam-3952	243	13	)	)	PUNCT
ejpam-3952	243	14	,	,	PUNCT
ejpam-3952	243	15	380	380	NUM
ejpam-3952	243	16	-	-	SYM
ejpam-3952	243	17	395	395	NUM
ejpam-3952	243	18	388	388	NUM
ejpam-3952	243	19	3	3	NUM
ejpam-3952	243	20	.	.	PUNCT
ejpam-3952	244	1	identities	identity	NOUN
ejpam-3952	244	2	in	in	ADP
ejpam-3952	244	3	this	this	DET
ejpam-3952	244	4	section	section	NOUN
ejpam-3952	244	5	,	,	PUNCT
ejpam-3952	244	6	we	we	PRON
ejpam-3952	244	7	focus	focus	VERB
ejpam-3952	244	8	on	on	ADP
ejpam-3952	244	9	generalizations	generalization	NOUN
ejpam-3952	244	10	of	of	ADP
ejpam-3952	244	11	classical	classical	ADJ
ejpam-3952	244	12	identities	identity	NOUN
ejpam-3952	244	13	for	for	ADP
ejpam-3952	244	14	triangular	triangular	NOUN
ejpam-3952	244	15	numbers	number	NOUN
ejpam-3952	244	16	(	(	PUNCT
ejpam-3952	244	17	see	see	VERB
ejpam-3952	244	18	,	,	PUNCT
ejpam-3952	244	19	for	for	ADP
ejpam-3952	244	20	example	example	NOUN
ejpam-3952	244	21	,	,	PUNCT
ejpam-3952	244	22	[	[	X
ejpam-3952	244	23	1–3	1–3	NOUN
ejpam-3952	244	24	,	,	PUNCT
ejpam-3952	244	25	7	7	NUM
ejpam-3952	244	26	,	,	PUNCT
ejpam-3952	244	27	9	9	NUM
ejpam-3952	244	28	,	,	PUNCT
ejpam-3952	244	29	12	12	NUM
ejpam-3952	244	30	]	]	PUNCT
ejpam-3952	244	31	)	)	PUNCT
ejpam-3952	244	32	.	.	PUNCT
ejpam-3952	245	1	each	each	DET
ejpam-3952	245	2	identity	identity	NOUN
ejpam-3952	245	3	for	for	ADP
ejpam-3952	245	4	l	l	NOUN
ejpam-3952	245	5	-	-	PUNCT
ejpam-3952	245	6	isosceles	isoscele	NOUN
ejpam-3952	245	7	triangular	triangular	NOUN
ejpam-3952	245	8	numbers	number	NOUN
ejpam-3952	245	9	is	be	AUX
ejpam-3952	245	10	given	give	VERB
ejpam-3952	245	11	and	and	CCONJ
ejpam-3952	245	12	followed	follow	VERB
ejpam-3952	245	13	by	by	ADP
ejpam-3952	245	14	the	the	DET
ejpam-3952	245	15	corresponding	correspond	VERB
ejpam-3952	245	16	identity	identity	NOUN
ejpam-3952	245	17	for	for	ADP
ejpam-3952	245	18	triangular	triangular	NOUN
ejpam-3952	245	19	numbers	number	NOUN
ejpam-3952	245	20	.	.	PUNCT
ejpam-3952	246	1	the	the	DET
ejpam-3952	246	2	identity	identity	NOUN
ejpam-3952	246	3	concerning	concern	VERB
ejpam-3952	246	4	the	the	DET
ejpam-3952	246	5	(	(	PUNCT
ejpam-3952	246	6	m	m	PROPN
ejpam-3952	246	7	+	+	ADJ
ejpam-3952	246	8	n)th	n)th	PROPN
ejpam-3952	246	9	l	l	ADJ
ejpam-3952	246	10	-	-	PUNCT
ejpam-3952	246	11	isosceles	isoscele	NOUN
ejpam-3952	246	12	triangular	triangular	NOUN
ejpam-3952	246	13	numbers	number	NOUN
ejpam-3952	246	14	is	be	AUX
ejpam-3952	246	15	given	give	VERB
ejpam-3952	246	16	in	in	ADP
ejpam-3952	246	17	the	the	DET
ejpam-3952	246	18	next	next	ADJ
ejpam-3952	246	19	theorem	theorem	PROPN
ejpam-3952	246	20	.	.	PUNCT
ejpam-3952	246	21	theorem	theorem	NOUN
ejpam-3952	246	22	5	5	NUM
ejpam-3952	247	1	.	.	PUNCT
ejpam-3952	247	2	t	t	PROPN
ejpam-3952	247	3	(	(	PUNCT
ejpam-3952	247	4	m	m	PROPN
ejpam-3952	247	5	+	+	NUM
ejpam-3952	247	6	n	n	CCONJ
ejpam-3952	247	7	,	,	PUNCT
ejpam-3952	247	8	l	l	NOUN
ejpam-3952	247	9	)	)	PUNCT
ejpam-3952	247	10	=	=	SYM
ejpam-3952	247	11	t	t	PROPN
ejpam-3952	247	12	(	(	PUNCT
ejpam-3952	247	13	m	m	PROPN
ejpam-3952	247	14	,	,	PUNCT
ejpam-3952	247	15	l	l	NOUN
ejpam-3952	247	16	)	)	PUNCT
ejpam-3952	248	1	+	+	NUM
ejpam-3952	248	2	t	t	PROPN
ejpam-3952	248	3	(	(	PUNCT
ejpam-3952	248	4	n	n	CCONJ
ejpam-3952	248	5	,	,	PUNCT
ejpam-3952	248	6	l	l	NOUN
ejpam-3952	248	7	)	)	PUNCT
ejpam-3952	248	8	+	+	CCONJ
ejpam-3952	248	9	lmn	lmn	NOUN
ejpam-3952	248	10	for	for	ADP
ejpam-3952	248	11	all	all	DET
ejpam-3952	248	12	positive	positive	ADJ
ejpam-3952	248	13	integers	integer	NOUN
ejpam-3952	248	14	l	l	NOUN
ejpam-3952	248	15	,	,	PUNCT
ejpam-3952	248	16	m	m	PROPN
ejpam-3952	248	17	,	,	PUNCT
ejpam-3952	248	18	and	and	CCONJ
ejpam-3952	248	19	n.	n.	NOUN
ejpam-3952	248	20	proof	proof	NOUN
ejpam-3952	248	21	.	.	PUNCT
ejpam-3952	249	1	let	let	VERB
ejpam-3952	249	2	l	l	NOUN
ejpam-3952	249	3	,	,	PUNCT
ejpam-3952	249	4	m	m	PROPN
ejpam-3952	249	5	,	,	PUNCT
ejpam-3952	249	6	and	and	CCONJ
ejpam-3952	249	7	n	n	CCONJ
ejpam-3952	249	8	be	be	AUX
ejpam-3952	249	9	positive	positive	ADJ
ejpam-3952	249	10	integers	integer	NOUN
ejpam-3952	249	11	.	.	PUNCT
ejpam-3952	250	1	then	then	ADV
ejpam-3952	250	2	t	t	PROPN
ejpam-3952	250	3	(	(	PUNCT
ejpam-3952	250	4	m	m	PROPN
ejpam-3952	250	5	+	+	NUM
ejpam-3952	250	6	n	n	CCONJ
ejpam-3952	250	7	,	,	PUNCT
ejpam-3952	250	8	l	l	NOUN
ejpam-3952	250	9	)	)	PUNCT
ejpam-3952	250	10	=	=	PUNCT
ejpam-3952	251	1	m	m	VERB
ejpam-3952	251	2	+	+	NUM
ejpam-3952	251	3	n	n	PROPN
ejpam-3952	251	4	+	+	NOUN
ejpam-3952	251	5	m	m	PROPN
ejpam-3952	251	6	+	+	CCONJ
ejpam-3952	251	7	n(m	n(m	PROPN
ejpam-3952	251	8	+	+	NUM
ejpam-3952	251	9	n−	n−	PROPN
ejpam-3952	251	10	1)l	1)l	NOUN
ejpam-3952	251	11	2	2	NUM
ejpam-3952	251	12	=	=	SYM
ejpam-3952	251	13	m	m	VERB
ejpam-3952	251	14	+	+	NOUN
ejpam-3952	251	15	n	n	PROPN
ejpam-3952	251	16	+	+	CCONJ
ejpam-3952	251	17	(	(	PUNCT
ejpam-3952	251	18	m2	m2	PROPN
ejpam-3952	251	19	−m	−m	PROPN
ejpam-3952	251	20	+	+	CCONJ
ejpam-3952	251	21	n2	n2	ADJ
ejpam-3952	251	22	−	−	PROPN
ejpam-3952	251	23	n	n	PROPN
ejpam-3952	251	24	+	+	CCONJ
ejpam-3952	251	25	2mn)l	2mn)l	PROPN
ejpam-3952	251	26	2	2	NUM
ejpam-3952	251	27	=	=	SYM
ejpam-3952	251	28	2	2	NUM
ejpam-3952	251	29	m	m	NOUN
ejpam-3952	251	30	+	+	NOUN
ejpam-3952	251	31	2n	2n	NUM
ejpam-3952	252	1	+	+	PUNCT
ejpam-3952	252	2	m2l	m2l	NOUN
ejpam-3952	252	3	−ml	−ml	NOUN
ejpam-3952	252	4	+	+	CCONJ
ejpam-3952	252	5	n2l	n2l	NOUN
ejpam-3952	252	6	−	−	ADP
ejpam-3952	252	7	nl	nl	NOUN
ejpam-3952	252	8	+	+	CCONJ
ejpam-3952	252	9	2lmn	2lmn	NUM
ejpam-3952	252	10	2	2	NUM
ejpam-3952	252	11	=	=	SYM
ejpam-3952	252	12	2	2	NUM
ejpam-3952	252	13	m	m	NOUN
ejpam-3952	252	14	+	+	NUM
ejpam-3952	252	15	m(m−	m(m−	PROPN
ejpam-3952	252	16	1)l	1)l	NUM
ejpam-3952	252	17	+	+	CCONJ
ejpam-3952	252	18	2n	2n	NUM
ejpam-3952	252	19	+	+	CCONJ
ejpam-3952	252	20	n(n−	n(n−	PROPN
ejpam-3952	252	21	1)l	1)l	NOUN
ejpam-3952	252	22	+	+	CCONJ
ejpam-3952	252	23	2lmn	2lmn	NUM
ejpam-3952	252	24	2	2	NUM
ejpam-3952	252	25	=	=	SYM
ejpam-3952	252	26	m	m	VERB
ejpam-3952	252	27	+	+	NUM
ejpam-3952	252	28	m(m−	m(m−	PROPN
ejpam-3952	252	29	1)l	1)l	NUM
ejpam-3952	252	30	2	2	NUM
ejpam-3952	252	31	+	+	CCONJ
ejpam-3952	252	32	n	n	PROPN
ejpam-3952	252	33	+	+	CCONJ
ejpam-3952	252	34	n(n−	n(n−	PROPN
ejpam-3952	252	35	1)l	1)l	NUM
ejpam-3952	252	36	2	2	NUM
ejpam-3952	252	37	+	+	CCONJ
ejpam-3952	252	38	lmn	lmn	PROPN
ejpam-3952	252	39	=	=	PROPN
ejpam-3952	252	40	t	t	PROPN
ejpam-3952	252	41	(	(	PUNCT
ejpam-3952	252	42	m	m	PROPN
ejpam-3952	252	43	,	,	PUNCT
ejpam-3952	252	44	l	l	NOUN
ejpam-3952	252	45	)	)	PUNCT
ejpam-3952	253	1	+	+	NUM
ejpam-3952	253	2	t	t	PROPN
ejpam-3952	253	3	(	(	PUNCT
ejpam-3952	253	4	n	n	CCONJ
ejpam-3952	253	5	,	,	PUNCT
ejpam-3952	253	6	l	l	NOUN
ejpam-3952	253	7	)	)	PUNCT
ejpam-3952	253	8	+	+	CCONJ
ejpam-3952	253	9	lmn	lmn	NOUN
ejpam-3952	253	10	.	.	PUNCT
ejpam-3952	254	1	as	as	SCONJ
ejpam-3952	254	2	desired	desire	VERB
ejpam-3952	254	3	,	,	PUNCT
ejpam-3952	254	4	we	we	PRON
ejpam-3952	254	5	have	have	VERB
ejpam-3952	254	6	t	t	PROPN
ejpam-3952	254	7	(	(	PUNCT
ejpam-3952	254	8	m	m	PROPN
ejpam-3952	254	9	+	+	NUM
ejpam-3952	254	10	n	n	CCONJ
ejpam-3952	254	11	,	,	PUNCT
ejpam-3952	254	12	l	l	NOUN
ejpam-3952	254	13	)	)	PUNCT
ejpam-3952	255	1	=	=	SYM
ejpam-3952	255	2	t	t	PROPN
ejpam-3952	255	3	(	(	PUNCT
ejpam-3952	255	4	m	m	PROPN
ejpam-3952	255	5	,	,	PUNCT
ejpam-3952	255	6	l	l	NOUN
ejpam-3952	255	7	)	)	PUNCT
ejpam-3952	256	1	+	+	NUM
ejpam-3952	256	2	t	t	PROPN
ejpam-3952	256	3	(	(	PUNCT
ejpam-3952	256	4	n	n	CCONJ
ejpam-3952	256	5	,	,	PUNCT
ejpam-3952	256	6	l	l	NOUN
ejpam-3952	256	7	)	)	PUNCT
ejpam-3952	256	8	+	+	CCONJ
ejpam-3952	256	9	lmn	lmn	NOUN
ejpam-3952	256	10	.	.	PUNCT
ejpam-3952	256	11	by	by	ADP
ejpam-3952	256	12	setting	set	VERB
ejpam-3952	256	13	l	l	NOUN
ejpam-3952	256	14	=	=	SYM
ejpam-3952	256	15	1	1	NUM
ejpam-3952	256	16	,	,	PUNCT
ejpam-3952	256	17	we	we	PRON
ejpam-3952	256	18	have	have	VERB
ejpam-3952	256	19	the	the	DET
ejpam-3952	256	20	classical	classical	ADJ
ejpam-3952	256	21	identity	identity	NOUN
ejpam-3952	256	22	t	t	NOUN
ejpam-3952	256	23	(	(	PUNCT
ejpam-3952	256	24	m	m	PROPN
ejpam-3952	256	25	+	+	NOUN
ejpam-3952	256	26	n	n	CCONJ
ejpam-3952	256	27	)	)	PUNCT
ejpam-3952	256	28	=	=	SYM
ejpam-3952	256	29	t	t	PROPN
ejpam-3952	256	30	(	(	PUNCT
ejpam-3952	256	31	m	m	PROPN
ejpam-3952	256	32	)	)	PUNCT
ejpam-3952	257	1	+	+	CCONJ
ejpam-3952	257	2	t	t	PROPN
ejpam-3952	257	3	(	(	PUNCT
ejpam-3952	257	4	n	n	CCONJ
ejpam-3952	257	5	)	)	PUNCT
ejpam-3952	257	6	+	+	CCONJ
ejpam-3952	257	7	mn	mn	NOUN
ejpam-3952	257	8	in	in	ADP
ejpam-3952	257	9	[	[	X
ejpam-3952	257	10	12	12	NUM
ejpam-3952	257	11	,	,	PUNCT
ejpam-3952	257	12	equation	equation	NOUN
ejpam-3952	257	13	(	(	PUNCT
ejpam-3952	257	14	4a	4a	NOUN
ejpam-3952	257	15	)	)	PUNCT
ejpam-3952	257	16	]	]	PUNCT
ejpam-3952	257	17	.	.	PUNCT
ejpam-3952	258	1	in	in	ADP
ejpam-3952	258	2	the	the	DET
ejpam-3952	258	3	following	following	NOUN
ejpam-3952	258	4	theorem	theorem	NOUN
ejpam-3952	258	5	,	,	PUNCT
ejpam-3952	258	6	the	the	DET
ejpam-3952	258	7	identity	identity	NOUN
ejpam-3952	258	8	concerning	concern	VERB
ejpam-3952	258	9	the	the	DET
ejpam-3952	258	10	mnth	mnth	ADJ
ejpam-3952	258	11	l	l	ADJ
ejpam-3952	258	12	-	-	PUNCT
ejpam-3952	258	13	isosceles	isoscele	NOUN
ejpam-3952	258	14	triangular	triangular	NOUN
ejpam-3952	258	15	numbers	number	NOUN
ejpam-3952	258	16	is	be	AUX
ejpam-3952	258	17	presented	present	VERB
ejpam-3952	258	18	.	.	PUNCT
ejpam-3952	259	1	theorem	theorem	ADJ
ejpam-3952	259	2	6	6	NUM
ejpam-3952	259	3	.	.	PUNCT
ejpam-3952	260	1	t	t	PROPN
ejpam-3952	260	2	(	(	PUNCT
ejpam-3952	260	3	mn	mn	PROPN
ejpam-3952	260	4	,	,	PUNCT
ejpam-3952	260	5	l	l	NOUN
ejpam-3952	260	6	)	)	PUNCT
ejpam-3952	260	7	=	=	SYM
ejpam-3952	260	8	t	t	PROPN
ejpam-3952	260	9	(	(	PUNCT
ejpam-3952	260	10	m	m	PROPN
ejpam-3952	260	11	,	,	PUNCT
ejpam-3952	260	12	l)t	l)t	X
ejpam-3952	260	13	(	(	PUNCT
ejpam-3952	260	14	n	n	CCONJ
ejpam-3952	260	15	,	,	PUNCT
ejpam-3952	260	16	l)+(2−l)lt	l)+(2−l)lt	PROPN
ejpam-3952	260	17	(	(	PUNCT
ejpam-3952	260	18	m−	m−	PROPN
ejpam-3952	260	19	1)t	1)t	PROPN
ejpam-3952	260	20	(	(	PUNCT
ejpam-3952	260	21	n−	n−	NOUN
ejpam-3952	260	22	1	1	NUM
ejpam-3952	260	23	)	)	PUNCT
ejpam-3952	260	24	for	for	ADP
ejpam-3952	260	25	all	all	DET
ejpam-3952	260	26	positive	positive	ADJ
ejpam-3952	260	27	integers	integer	NOUN
ejpam-3952	260	28	m	m	PRON
ejpam-3952	260	29	,	,	PUNCT
ejpam-3952	260	30	n	n	CCONJ
ejpam-3952	260	31	,	,	PUNCT
ejpam-3952	260	32	and	and	CCONJ
ejpam-3952	260	33	l.	l.	PROPN
ejpam-3952	260	34	proof	proof	NOUN
ejpam-3952	260	35	.	.	PUNCT
ejpam-3952	261	1	let	let	VERB
ejpam-3952	261	2	m	m	PRON
ejpam-3952	261	3	,	,	PUNCT
ejpam-3952	261	4	n	n	CCONJ
ejpam-3952	261	5	,	,	PUNCT
ejpam-3952	261	6	and	and	CCONJ
ejpam-3952	261	7	l	l	NOUN
ejpam-3952	261	8	be	be	AUX
ejpam-3952	261	9	positive	positive	ADJ
ejpam-3952	261	10	integers	integer	NOUN
ejpam-3952	261	11	.	.	PUNCT
ejpam-3952	262	1	using	use	VERB
ejpam-3952	262	2	a	a	DET
ejpam-3952	262	3	direct	direct	ADJ
ejpam-3952	262	4	calculation	calculation	NOUN
ejpam-3952	262	5	,	,	PUNCT
ejpam-3952	262	6	we	we	PRON
ejpam-3952	262	7	have	have	VERB
ejpam-3952	262	8	t	t	PROPN
ejpam-3952	262	9	(	(	PUNCT
ejpam-3952	262	10	mn	mn	PROPN
ejpam-3952	262	11	,	,	PUNCT
ejpam-3952	262	12	l	l	NOUN
ejpam-3952	262	13	)	)	PUNCT
ejpam-3952	262	14	=	=	SYM
ejpam-3952	262	15	mn	mn	PROPN
ejpam-3952	263	1	+	+	CCONJ
ejpam-3952	263	2	mn(mn−	mn(mn−	NOUN
ejpam-3952	263	3	1)l	1)l	NUM
ejpam-3952	263	4	2	2	NUM
ejpam-3952	263	5	=	=	SYM
ejpam-3952	263	6	4mn	4mn	NOUN
ejpam-3952	263	7	+	+	CCONJ
ejpam-3952	263	8	2m2n2l	2m2n2l	NUM
ejpam-3952	263	9	−	−	NOUN
ejpam-3952	263	10	2mnl	2mnl	NUM
ejpam-3952	263	11	4	4	NUM
ejpam-3952	263	12	=	=	SYM
ejpam-3952	263	13	(	(	PUNCT
ejpam-3952	263	14	2	2	NUM
ejpam-3952	263	15	m	m	NOUN
ejpam-3952	263	16	+	+	NUM
ejpam-3952	263	17	m2l	m2l	NOUN
ejpam-3952	263	18	−ml)(2n	−ml)(2n	NOUN
ejpam-3952	263	19	+	+	CCONJ
ejpam-3952	263	20	n2l	n2l	PROPN
ejpam-3952	263	21	−	−	ADP
ejpam-3952	263	22	nl	nl	NOUN
ejpam-3952	263	23	)	)	PUNCT
ejpam-3952	263	24	4	4	NUM
ejpam-3952	264	1	+	+	CCONJ
ejpam-3952	264	2	(	(	PUNCT
ejpam-3952	264	3	2l	2l	NUM
ejpam-3952	264	4	−	−	NOUN
ejpam-3952	264	5	l2)(m2	l2)(m2	NOUN
ejpam-3952	265	1	−m)(n2	−m)(n2	PUNCT
ejpam-3952	265	2	−	−	PROPN
ejpam-3952	265	3	n	n	CCONJ
ejpam-3952	265	4	)	)	PUNCT
ejpam-3952	265	5	4	4	NUM
ejpam-3952	265	6	=	=	SYM
ejpam-3952	265	7	(	(	PUNCT
ejpam-3952	265	8	m	m	VERB
ejpam-3952	265	9	+	+	NUM
ejpam-3952	265	10	m(m−	m(m−	PROPN
ejpam-3952	265	11	1)l	1)l	NUM
ejpam-3952	265	12	2	2	NUM
ejpam-3952	265	13	)	)	PUNCT
ejpam-3952	265	14	(	(	PUNCT
ejpam-3952	265	15	n	n	PROPN
ejpam-3952	265	16	+	+	CCONJ
ejpam-3952	265	17	n(n−	n(n−	PROPN
ejpam-3952	265	18	1)l	1)l	NUM
ejpam-3952	265	19	2	2	NUM
ejpam-3952	265	20	)	)	PUNCT
ejpam-3952	265	21	+	+	CCONJ
ejpam-3952	265	22	(	(	PUNCT
ejpam-3952	265	23	2−	2−	NUM
ejpam-3952	265	24	l)l	l)l	NOUN
ejpam-3952	265	25	(	(	PUNCT
ejpam-3952	265	26	(	(	PUNCT
ejpam-3952	265	27	m−	m−	PROPN
ejpam-3952	265	28	1)m	1)m	NUM
ejpam-3952	265	29	2	2	NUM
ejpam-3952	265	30	)	)	PUNCT
ejpam-3952	265	31	(	(	PUNCT
ejpam-3952	265	32	(	(	PUNCT
ejpam-3952	265	33	n−	n−	NOUN
ejpam-3952	265	34	1)n	1)n	NOUN
ejpam-3952	265	35	2	2	NUM
ejpam-3952	265	36	)	)	PUNCT
ejpam-3952	265	37	=	=	SYM
ejpam-3952	265	38	t	t	PROPN
ejpam-3952	265	39	(	(	PUNCT
ejpam-3952	265	40	m	m	PROPN
ejpam-3952	265	41	,	,	PUNCT
ejpam-3952	265	42	l)t	l)t	X
ejpam-3952	265	43	(	(	PUNCT
ejpam-3952	265	44	n	n	X
ejpam-3952	265	45	,	,	PUNCT
ejpam-3952	265	46	l	l	NOUN
ejpam-3952	265	47	)	)	PUNCT
ejpam-3952	266	1	+	+	CCONJ
ejpam-3952	266	2	(	(	PUNCT
ejpam-3952	266	3	2−	2−	NUM
ejpam-3952	266	4	l)lt	l)lt	PROPN
ejpam-3952	266	5	(	(	PUNCT
ejpam-3952	266	6	m−	m−	PROPN
ejpam-3952	266	7	1)t	1)t	PROPN
ejpam-3952	266	8	(	(	PUNCT
ejpam-3952	266	9	n−	n−	NOUN
ejpam-3952	266	10	1	1	NUM
ejpam-3952	266	11	)	)	PUNCT
ejpam-3952	266	12	.	.	PUNCT
ejpam-3952	267	1	j.	j.	PROPN
ejpam-3952	267	2	punpim	punpim	PROPN
ejpam-3952	267	3	,	,	PUNCT
ejpam-3952	267	4	s.	s.	PROPN
ejpam-3952	267	5	jitman	jitman	PROPN
ejpam-3952	267	6	/	/	SYM
ejpam-3952	267	7	eur	eur	PROPN
ejpam-3952	267	8	.	.	PUNCT
ejpam-3952	268	1	j.	j.	PROPN
ejpam-3952	268	2	pure	pure	PROPN
ejpam-3952	268	3	appl	appl	PROPN
ejpam-3952	268	4	.	.	PROPN
ejpam-3952	268	5	math	math	PROPN
ejpam-3952	268	6	,	,	PUNCT
ejpam-3952	268	7	14	14	NUM
ejpam-3952	268	8	(	(	PUNCT
ejpam-3952	268	9	2	2	NUM
ejpam-3952	268	10	)	)	PUNCT
ejpam-3952	268	11	(	(	PUNCT
ejpam-3952	268	12	2021	2021	NUM
ejpam-3952	268	13	)	)	PUNCT
ejpam-3952	268	14	,	,	PUNCT
ejpam-3952	268	15	380	380	NUM
ejpam-3952	268	16	-	-	SYM
ejpam-3952	268	17	395	395	NUM
ejpam-3952	268	18	389	389	NUM
ejpam-3952	268	19	this	this	PRON
ejpam-3952	268	20	completes	complete	VERB
ejpam-3952	268	21	the	the	DET
ejpam-3952	268	22	proof	proof	NOUN
ejpam-3952	268	23	.	.	PUNCT
ejpam-3952	269	1	the	the	DET
ejpam-3952	269	2	classical	classical	ADJ
ejpam-3952	269	3	identity	identity	NOUN
ejpam-3952	269	4	t	t	PROPN
ejpam-3952	269	5	(	(	PUNCT
ejpam-3952	269	6	mn	mn	PROPN
ejpam-3952	269	7	)	)	PUNCT
ejpam-3952	269	8	=	=	SYM
ejpam-3952	269	9	t	t	PROPN
ejpam-3952	269	10	(	(	PUNCT
ejpam-3952	269	11	m)t	m)t	PROPN
ejpam-3952	269	12	(	(	PUNCT
ejpam-3952	269	13	n	n	CCONJ
ejpam-3952	269	14	)	)	PUNCT
ejpam-3952	270	1	+	+	CCONJ
ejpam-3952	270	2	t	t	PROPN
ejpam-3952	270	3	(	(	PUNCT
ejpam-3952	270	4	m−	m−	PROPN
ejpam-3952	270	5	1)t	1)t	PROPN
ejpam-3952	270	6	(	(	PUNCT
ejpam-3952	270	7	n−	n−	NOUN
ejpam-3952	270	8	1	1	NUM
ejpam-3952	270	9	)	)	PUNCT
ejpam-3952	270	10	for	for	ADP
ejpam-3952	270	11	triangular	triangular	NOUN
ejpam-3952	270	12	numbers	number	NOUN
ejpam-3952	270	13	in	in	ADP
ejpam-3952	270	14	[	[	X
ejpam-3952	270	15	12	12	NUM
ejpam-3952	270	16	,	,	PUNCT
ejpam-3952	270	17	equation	equation	NOUN
ejpam-3952	270	18	(	(	PUNCT
ejpam-3952	270	19	17a	17a	PROPN
ejpam-3952	270	20	)	)	PUNCT
ejpam-3952	270	21	]	]	PUNCT
ejpam-3952	270	22	follows	follow	VERB
ejpam-3952	270	23	easily	easily	ADV
ejpam-3952	270	24	when	when	SCONJ
ejpam-3952	270	25	l	l	NOUN
ejpam-3952	270	26	=	=	NOUN
ejpam-3952	271	1	1	1	X
ejpam-3952	271	2	.	.	X
ejpam-3952	272	1	one	one	NUM
ejpam-3952	272	2	of	of	ADP
ejpam-3952	272	3	the	the	DET
ejpam-3952	272	4	classical	classical	ADJ
ejpam-3952	272	5	facts	fact	NOUN
ejpam-3952	272	6	about	about	ADP
ejpam-3952	272	7	triangular	triangular	NOUN
ejpam-3952	272	8	numbers	number	NOUN
ejpam-3952	272	9	is	be	AUX
ejpam-3952	272	10	that	that	SCONJ
ejpam-3952	272	11	a	a	DET
ejpam-3952	272	12	sum	sum	NOUN
ejpam-3952	272	13	of	of	ADP
ejpam-3952	272	14	two	two	NUM
ejpam-3952	272	15	consecutive	consecutive	ADJ
ejpam-3952	272	16	triangular	triangular	NOUN
ejpam-3952	272	17	numbers	number	NOUN
ejpam-3952	272	18	is	be	AUX
ejpam-3952	272	19	square	square	ADJ
ejpam-3952	272	20	(	(	PUNCT
ejpam-3952	272	21	see	see	VERB
ejpam-3952	272	22	[	[	X
ejpam-3952	272	23	12	12	NUM
ejpam-3952	272	24	,	,	PUNCT
ejpam-3952	272	25	equation	equation	NOUN
ejpam-3952	272	26	(	(	PUNCT
ejpam-3952	272	27	1	1	NUM
ejpam-3952	272	28	)	)	PUNCT
ejpam-3952	272	29	]	]	PUNCT
ejpam-3952	272	30	)	)	PUNCT
ejpam-3952	272	31	.	.	PUNCT
ejpam-3952	273	1	precisely	precisely	ADV
ejpam-3952	273	2	,	,	PUNCT
ejpam-3952	273	3	t	t	PROPN
ejpam-3952	273	4	(	(	PUNCT
ejpam-3952	273	5	n)+t	n)+t	PROPN
ejpam-3952	273	6	(	(	PUNCT
ejpam-3952	273	7	n+1	n+1	PROPN
ejpam-3952	273	8	)	)	PUNCT
ejpam-3952	273	9	=	=	SYM
ejpam-3952	274	1	(	(	PUNCT
ejpam-3952	274	2	n+1)2	n+1)2	PROPN
ejpam-3952	274	3	is	be	AUX
ejpam-3952	274	4	square	square	ADJ
ejpam-3952	274	5	for	for	ADP
ejpam-3952	274	6	all	all	DET
ejpam-3952	274	7	positive	positive	ADJ
ejpam-3952	274	8	integers	integer	NOUN
ejpam-3952	274	9	n.	n.	VERB
ejpam-3952	274	10	in	in	ADP
ejpam-3952	274	11	the	the	DET
ejpam-3952	274	12	following	following	NOUN
ejpam-3952	274	13	theorem	theorem	NOUN
ejpam-3952	274	14	,	,	PUNCT
ejpam-3952	274	15	we	we	PRON
ejpam-3952	274	16	prove	prove	VERB
ejpam-3952	274	17	that	that	SCONJ
ejpam-3952	274	18	l	l	NOUN
ejpam-3952	274	19	=	=	SYM
ejpam-3952	274	20	1	1	NUM
ejpam-3952	274	21	is	be	AUX
ejpam-3952	274	22	the	the	DET
ejpam-3952	274	23	necessary	necessary	ADJ
ejpam-3952	274	24	and	and	CCONJ
ejpam-3952	274	25	sufficient	sufficient	ADJ
ejpam-3952	274	26	condition	condition	NOUN
ejpam-3952	274	27	for	for	ADP
ejpam-3952	274	28	the	the	DET
ejpam-3952	274	29	sum	sum	NOUN
ejpam-3952	274	30	t	t	PROPN
ejpam-3952	274	31	(	(	PUNCT
ejpam-3952	274	32	n	n	CCONJ
ejpam-3952	274	33	,	,	PUNCT
ejpam-3952	274	34	l	l	NOUN
ejpam-3952	274	35	)	)	PUNCT
ejpam-3952	275	1	+	+	NUM
ejpam-3952	275	2	t	t	PROPN
ejpam-3952	275	3	(	(	PUNCT
ejpam-3952	275	4	n	n	PROPN
ejpam-3952	275	5	+	+	CCONJ
ejpam-3952	275	6	1	1	NUM
ejpam-3952	275	7	,	,	PUNCT
ejpam-3952	275	8	l	l	NOUN
ejpam-3952	275	9	)	)	PUNCT
ejpam-3952	275	10	to	to	PART
ejpam-3952	275	11	be	be	AUX
ejpam-3952	275	12	square	square	ADJ
ejpam-3952	275	13	for	for	SCONJ
ejpam-3952	275	14	all	all	DET
ejpam-3952	275	15	positive	positive	ADJ
ejpam-3952	275	16	integers	integer	NOUN
ejpam-3952	275	17	n.	n.	NOUN
ejpam-3952	275	18	theorem	theorem	VERB
ejpam-3952	275	19	7	7	NUM
ejpam-3952	275	20	.	.	PUNCT
ejpam-3952	276	1	let	let	VERB
ejpam-3952	276	2	l	l	NOUN
ejpam-3952	276	3	be	be	AUX
ejpam-3952	276	4	a	a	DET
ejpam-3952	276	5	positive	positive	ADJ
ejpam-3952	276	6	integer	integer	NOUN
ejpam-3952	276	7	.	.	PUNCT
ejpam-3952	277	1	then	then	ADV
ejpam-3952	277	2	t	t	PROPN
ejpam-3952	277	3	(	(	PUNCT
ejpam-3952	277	4	n	n	CCONJ
ejpam-3952	277	5	,	,	PUNCT
ejpam-3952	277	6	l)+t	l)+t	PROPN
ejpam-3952	277	7	(	(	PUNCT
ejpam-3952	277	8	n+1	n+1	PROPN
ejpam-3952	277	9	,	,	PUNCT
ejpam-3952	277	10	l	l	NOUN
ejpam-3952	277	11	)	)	PUNCT
ejpam-3952	277	12	is	be	AUX
ejpam-3952	277	13	square	square	ADJ
ejpam-3952	277	14	for	for	ADP
ejpam-3952	277	15	all	all	DET
ejpam-3952	277	16	positive	positive	ADJ
ejpam-3952	277	17	integers	integer	NOUN
ejpam-3952	277	18	n	n	CCONJ
ejpam-3952	277	19	if	if	SCONJ
ejpam-3952	277	20	and	and	CCONJ
ejpam-3952	277	21	only	only	ADV
ejpam-3952	277	22	if	if	SCONJ
ejpam-3952	277	23	l	l	NOUN
ejpam-3952	277	24	=	=	NOUN
ejpam-3952	278	1	1	1	X
ejpam-3952	278	2	.	.	PUNCT
ejpam-3952	278	3	proof	proof	NOUN
ejpam-3952	278	4	.	.	PUNCT
ejpam-3952	279	1	assume	assume	VERB
ejpam-3952	279	2	that	that	SCONJ
ejpam-3952	279	3	l	l	NOUN
ejpam-3952	279	4	≥	≥	NUM
ejpam-3952	279	5	2	2	NUM
ejpam-3952	279	6	.	.	PUNCT
ejpam-3952	280	1	first	first	ADV
ejpam-3952	280	2	,	,	PUNCT
ejpam-3952	280	3	we	we	PRON
ejpam-3952	280	4	note	note	VERB
ejpam-3952	280	5	that	that	SCONJ
ejpam-3952	280	6	t	t	PROPN
ejpam-3952	280	7	(	(	PUNCT
ejpam-3952	280	8	n	n	CCONJ
ejpam-3952	280	9	,	,	PUNCT
ejpam-3952	280	10	l	l	NOUN
ejpam-3952	280	11	)	)	PUNCT
ejpam-3952	281	1	+	+	NUM
ejpam-3952	281	2	t	t	PROPN
ejpam-3952	281	3	(	(	PUNCT
ejpam-3952	281	4	n	n	PROPN
ejpam-3952	281	5	+	+	CCONJ
ejpam-3952	281	6	1	1	NUM
ejpam-3952	281	7	,	,	PUNCT
ejpam-3952	281	8	l	l	NOUN
ejpam-3952	281	9	)	)	PUNCT
ejpam-3952	281	10	=	=	SYM
ejpam-3952	281	11	n2l	n2l	NOUN
ejpam-3952	281	12	+	+	X
ejpam-3952	281	13	2n	2n	NUM
ejpam-3952	282	1	+	+	CCONJ
ejpam-3952	282	2	1	1	NUM
ejpam-3952	282	3	for	for	ADP
ejpam-3952	282	4	all	all	DET
ejpam-3952	282	5	positive	positive	ADJ
ejpam-3952	282	6	integers	integer	NOUN
ejpam-3952	282	7	n.	n.	VERB
ejpam-3952	282	8	it	it	PRON
ejpam-3952	282	9	is	be	AUX
ejpam-3952	282	10	not	not	PART
ejpam-3952	282	11	difficult	difficult	ADJ
ejpam-3952	282	12	to	to	PART
ejpam-3952	282	13	see	see	VERB
ejpam-3952	282	14	that	that	DET
ejpam-3952	282	15	t	t	NOUN
ejpam-3952	282	16	(	(	PUNCT
ejpam-3952	282	17	1	1	NUM
ejpam-3952	282	18	,	,	PUNCT
ejpam-3952	282	19	l	l	NOUN
ejpam-3952	282	20	)	)	PUNCT
ejpam-3952	283	1	+	+	NUM
ejpam-3952	283	2	t	t	X
ejpam-3952	283	3	(	(	PUNCT
ejpam-3952	283	4	1	1	NUM
ejpam-3952	283	5	+	+	NUM
ejpam-3952	283	6	1	1	NUM
ejpam-3952	283	7	,	,	PUNCT
ejpam-3952	283	8	l	l	NOUN
ejpam-3952	283	9	)	)	PUNCT
ejpam-3952	284	1	=	=	SYM
ejpam-3952	284	2	l	l	NOUN
ejpam-3952	285	1	+	+	NOUN
ejpam-3952	285	2	3	3	NUM
ejpam-3952	285	3	is	be	AUX
ejpam-3952	285	4	square	square	ADJ
ejpam-3952	285	5	if	if	SCONJ
ejpam-3952	285	6	and	and	CCONJ
ejpam-3952	285	7	only	only	ADV
ejpam-3952	285	8	if	if	SCONJ
ejpam-3952	285	9	22(l	22(l	NUM
ejpam-3952	285	10	+	+	CCONJ
ejpam-3952	285	11	3	3	X
ejpam-3952	285	12	)	)	PUNCT
ejpam-3952	285	13	is	be	AUX
ejpam-3952	285	14	square	square	ADJ
ejpam-3952	285	15	.	.	PUNCT
ejpam-3952	286	1	for	for	ADP
ejpam-3952	286	2	l	l	PROPN
ejpam-3952	286	3	≥	≥	NUM
ejpam-3952	286	4	2	2	NUM
ejpam-3952	286	5	,	,	PUNCT
ejpam-3952	286	6	if	if	SCONJ
ejpam-3952	286	7	t	t	PROPN
ejpam-3952	286	8	(	(	PUNCT
ejpam-3952	286	9	1	1	NUM
ejpam-3952	286	10	,	,	PUNCT
ejpam-3952	286	11	l	l	NOUN
ejpam-3952	286	12	)	)	PUNCT
ejpam-3952	286	13	+	+	NUM
ejpam-3952	286	14	t	t	X
ejpam-3952	286	15	(	(	PUNCT
ejpam-3952	286	16	1	1	NUM
ejpam-3952	286	17	+	+	NUM
ejpam-3952	286	18	1	1	NUM
ejpam-3952	286	19	,	,	PUNCT
ejpam-3952	286	20	l	l	NOUN
ejpam-3952	286	21	)	)	PUNCT
ejpam-3952	286	22	=	=	SYM
ejpam-3952	286	23	l	l	NOUN
ejpam-3952	286	24	+	+	NOUN
ejpam-3952	286	25	3	3	NUM
ejpam-3952	286	26	is	be	AUX
ejpam-3952	286	27	square	square	ADJ
ejpam-3952	286	28	,	,	PUNCT
ejpam-3952	286	29	then	then	ADV
ejpam-3952	286	30	l	l	PROPN
ejpam-3952	286	31	≥	≥	NUM
ejpam-3952	286	32	6	6	NUM
ejpam-3952	286	33	and	and	CCONJ
ejpam-3952	286	34	22(l	22(l	NUM
ejpam-3952	287	1	+	+	CCONJ
ejpam-3952	287	2	3	3	X
ejpam-3952	287	3	)	)	PUNCT
ejpam-3952	287	4	≥	≥	NOUN
ejpam-3952	287	5	36	36	NUM
ejpam-3952	287	6	.	.	PUNCT
ejpam-3952	288	1	hence	hence	ADV
ejpam-3952	288	2	,	,	PUNCT
ejpam-3952	288	3	t	t	PROPN
ejpam-3952	288	4	(	(	PUNCT
ejpam-3952	288	5	2	2	NUM
ejpam-3952	288	6	,	,	PUNCT
ejpam-3952	288	7	l	l	NOUN
ejpam-3952	288	8	)	)	PUNCT
ejpam-3952	288	9	+	+	NUM
ejpam-3952	288	10	t	t	X
ejpam-3952	288	11	(	(	PUNCT
ejpam-3952	288	12	2	2	NUM
ejpam-3952	288	13	+	+	SYM
ejpam-3952	288	14	1	1	NUM
ejpam-3952	288	15	,	,	PUNCT
ejpam-3952	288	16	l	l	NOUN
ejpam-3952	288	17	)	)	PUNCT
ejpam-3952	288	18	=	=	SYM
ejpam-3952	288	19	4l	4l	NOUN
ejpam-3952	288	20	+	+	CCONJ
ejpam-3952	288	21	5	5	NUM
ejpam-3952	288	22	=	=	SYM
ejpam-3952	288	23	22(l	22(l	NUM
ejpam-3952	288	24	+	+	CCONJ
ejpam-3952	288	25	3	3	X
ejpam-3952	288	26	)	)	PUNCT
ejpam-3952	288	27	−	−	NOUN
ejpam-3952	288	28	7	7	NUM
ejpam-3952	288	29	can	can	AUX
ejpam-3952	288	30	not	not	PART
ejpam-3952	288	31	be	be	AUX
ejpam-3952	288	32	square	square	ADJ
ejpam-3952	288	33	.	.	PUNCT
ejpam-3952	289	1	therefore	therefore	ADV
ejpam-3952	289	2	,	,	PUNCT
ejpam-3952	289	3	t	t	PROPN
ejpam-3952	289	4	(	(	PUNCT
ejpam-3952	289	5	n	n	CCONJ
ejpam-3952	289	6	,	,	PUNCT
ejpam-3952	289	7	l	l	NOUN
ejpam-3952	289	8	)	)	PUNCT
ejpam-3952	289	9	+	+	NUM
ejpam-3952	289	10	t	t	PROPN
ejpam-3952	289	11	(	(	PUNCT
ejpam-3952	289	12	n	n	PROPN
ejpam-3952	289	13	+	+	CCONJ
ejpam-3952	289	14	1	1	NUM
ejpam-3952	289	15	,	,	PUNCT
ejpam-3952	289	16	l	l	NOUN
ejpam-3952	289	17	)	)	PUNCT
ejpam-3952	289	18	is	be	AUX
ejpam-3952	289	19	non	non	ADJ
ejpam-3952	289	20	-	-	ADJ
ejpam-3952	289	21	square	square	ADJ
ejpam-3952	289	22	for	for	ADP
ejpam-3952	289	23	n	n	NOUN
ejpam-3952	289	24	=	=	SYM
ejpam-3952	289	25	1	1	NUM
ejpam-3952	289	26	or	or	CCONJ
ejpam-3952	289	27	n	n	NOUN
ejpam-3952	289	28	=	=	SYM
ejpam-3952	289	29	2	2	NUM
ejpam-3952	289	30	.	.	PUNCT
ejpam-3952	290	1	the	the	DET
ejpam-3952	290	2	converse	converse	NOUN
ejpam-3952	290	3	follows	follow	VERB
ejpam-3952	290	4	directly	directly	ADV
ejpam-3952	290	5	from	from	ADP
ejpam-3952	290	6	[	[	X
ejpam-3952	290	7	12	12	NUM
ejpam-3952	290	8	,	,	PUNCT
ejpam-3952	290	9	equation	equation	NOUN
ejpam-3952	290	10	(	(	PUNCT
ejpam-3952	290	11	1	1	NUM
ejpam-3952	290	12	)	)	PUNCT
ejpam-3952	290	13	]	]	PUNCT
ejpam-3952	290	14	.	.	PUNCT
ejpam-3952	291	1	next	next	ADV
ejpam-3952	291	2	,	,	PUNCT
ejpam-3952	291	3	we	we	PRON
ejpam-3952	291	4	focus	focus	VERB
ejpam-3952	291	5	on	on	ADP
ejpam-3952	291	6	some	some	DET
ejpam-3952	291	7	identities	identity	NOUN
ejpam-3952	291	8	for	for	ADP
ejpam-3952	291	9	isosceles	isoscele	NOUN
ejpam-3952	291	10	triangular	triangular	NOUN
ejpam-3952	291	11	numbers	number	NOUN
ejpam-3952	291	12	induced	induce	VERB
ejpam-3952	291	13	by	by	ADP
ejpam-3952	291	14	a	a	DET
ejpam-3952	291	15	recurrence	recurrence	NOUN
ejpam-3952	291	16	relation	relation	NOUN
ejpam-3952	291	17	.	.	PUNCT
ejpam-3952	292	1	theorem	theorem	VERB
ejpam-3952	292	2	8	8	NUM
ejpam-3952	292	3	.	.	PUNCT
ejpam-3952	293	1	nt	not	PART
ejpam-3952	293	2	(	(	PUNCT
ejpam-3952	293	3	n	n	PROPN
ejpam-3952	293	4	+	+	NUM
ejpam-3952	293	5	1	1	NUM
ejpam-3952	293	6	,	,	PUNCT
ejpam-3952	293	7	l)−	l)−	PROPN
ejpam-3952	293	8	(	(	PUNCT
ejpam-3952	293	9	l	l	NOUN
ejpam-3952	293	10	−	−	PROPN
ejpam-3952	293	11	1)n	1)n	X
ejpam-3952	293	12	=	=	SYM
ejpam-3952	293	13	(	(	PUNCT
ejpam-3952	293	14	n	n	CCONJ
ejpam-3952	293	15	+	+	NUM
ejpam-3952	293	16	2)t	2)t	NUM
ejpam-3952	293	17	(	(	PUNCT
ejpam-3952	293	18	n	n	X
ejpam-3952	293	19	,	,	PUNCT
ejpam-3952	293	20	l	l	NOUN
ejpam-3952	293	21	)	)	PUNCT
ejpam-3952	293	22	for	for	ADP
ejpam-3952	293	23	all	all	DET
ejpam-3952	293	24	positive	positive	ADJ
ejpam-3952	293	25	integers	integer	NOUN
ejpam-3952	293	26	n	n	PRON
ejpam-3952	293	27	and	and	CCONJ
ejpam-3952	293	28	l.	l.	PROPN
ejpam-3952	293	29	proof	proof	NOUN
ejpam-3952	293	30	.	.	PUNCT
ejpam-3952	294	1	let	let	VERB
ejpam-3952	294	2	n	n	NOUN
ejpam-3952	294	3	and	and	CCONJ
ejpam-3952	294	4	l	l	NOUN
ejpam-3952	294	5	be	be	AUX
ejpam-3952	294	6	positive	positive	ADJ
ejpam-3952	294	7	integers	integer	NOUN
ejpam-3952	294	8	.	.	PUNCT
ejpam-3952	295	1	then	then	ADV
ejpam-3952	295	2	nt	not	PART
ejpam-3952	295	3	(	(	PUNCT
ejpam-3952	295	4	n	n	PROPN
ejpam-3952	295	5	+	+	NUM
ejpam-3952	295	6	1	1	NUM
ejpam-3952	295	7	,	,	PUNCT
ejpam-3952	295	8	l)−	l)−	PROPN
ejpam-3952	295	9	(	(	PUNCT
ejpam-3952	295	10	l	l	NOUN
ejpam-3952	295	11	−	−	PROPN
ejpam-3952	295	12	1)n	1)n	X
ejpam-3952	295	13	=	=	SYM
ejpam-3952	295	14	n	n	PROPN
ejpam-3952	295	15	(	(	PUNCT
ejpam-3952	295	16	(	(	PUNCT
ejpam-3952	295	17	n	n	X
ejpam-3952	295	18	+	+	NUM
ejpam-3952	295	19	1	1	NUM
ejpam-3952	295	20	)	)	PUNCT
ejpam-3952	295	21	+	+	CCONJ
ejpam-3952	295	22	(	(	PUNCT
ejpam-3952	295	23	n	n	X
ejpam-3952	295	24	+	+	CCONJ
ejpam-3952	295	25	1)nl	1)nl	NUM
ejpam-3952	295	26	2	2	NUM
ejpam-3952	295	27	)	)	PUNCT
ejpam-3952	295	28	−	−	PROPN
ejpam-3952	296	1	nl	nl	NOUN
ejpam-3952	296	2	+	+	CCONJ
ejpam-3952	296	3	n	n	CCONJ
ejpam-3952	296	4	=	=	SYM
ejpam-3952	296	5	2n2	2n2	NUM
ejpam-3952	296	6	+	+	CCONJ
ejpam-3952	296	7	2n	2n	NUM
ejpam-3952	296	8	+	+	CCONJ
ejpam-3952	296	9	n3l	n3l	ADV
ejpam-3952	296	10	+	+	CCONJ
ejpam-3952	296	11	n2l	n2l	NOUN
ejpam-3952	296	12	−	−	NUM
ejpam-3952	296	13	2nl	2nl	NOUN
ejpam-3952	297	1	+	+	CCONJ
ejpam-3952	297	2	2n	2n	NUM
ejpam-3952	297	3	2	2	NUM
ejpam-3952	297	4	=	=	SYM
ejpam-3952	297	5	2n2	2n2	NUM
ejpam-3952	297	6	+	+	CCONJ
ejpam-3952	297	7	n3l	n3l	ADV
ejpam-3952	297	8	−	−	PROPN
ejpam-3952	297	9	n2l	n2l	NOUN
ejpam-3952	297	10	+	+	CCONJ
ejpam-3952	297	11	4n	4n	X
ejpam-3952	297	12	+	+	CCONJ
ejpam-3952	297	13	2n2l	2n2l	NUM
ejpam-3952	298	1	−	−	NOUN
ejpam-3952	298	2	2nl	2nl	NOUN
ejpam-3952	298	3	2	2	X
ejpam-3952	298	4	=	=	SYM
ejpam-3952	298	5	(	(	PUNCT
ejpam-3952	298	6	n	n	NOUN
ejpam-3952	298	7	+	+	CCONJ
ejpam-3952	298	8	2	2	NUM
ejpam-3952	298	9	)	)	PUNCT
ejpam-3952	298	10	(	(	PUNCT
ejpam-3952	298	11	2n	2n	X
ejpam-3952	298	12	+	+	CCONJ
ejpam-3952	298	13	n2l	n2l	PROPN
ejpam-3952	298	14	−	−	PROPN
ejpam-3952	298	15	nl	nl	NOUN
ejpam-3952	298	16	2	2	NUM
ejpam-3952	298	17	)	)	PUNCT
ejpam-3952	298	18	=	=	PUNCT
ejpam-3952	298	19	(	(	PUNCT
ejpam-3952	298	20	n	n	PROPN
ejpam-3952	298	21	+	+	CCONJ
ejpam-3952	298	22	2	2	NUM
ejpam-3952	298	23	)	)	PUNCT
ejpam-3952	298	24	(	(	PUNCT
ejpam-3952	298	25	n	n	X
ejpam-3952	298	26	+	+	CCONJ
ejpam-3952	298	27	n(n−	n(n−	PROPN
ejpam-3952	298	28	1)l	1)l	NUM
ejpam-3952	298	29	2	2	NUM
ejpam-3952	298	30	)	)	PUNCT
ejpam-3952	298	31	=	=	SYM
ejpam-3952	298	32	(	(	PUNCT
ejpam-3952	298	33	n	n	CCONJ
ejpam-3952	298	34	+	+	NUM
ejpam-3952	298	35	2)t	2)t	NUM
ejpam-3952	298	36	(	(	PUNCT
ejpam-3952	298	37	n	n	X
ejpam-3952	298	38	,	,	PUNCT
ejpam-3952	298	39	l	l	NOUN
ejpam-3952	298	40	)	)	PUNCT
ejpam-3952	298	41	.	.	PUNCT
ejpam-3952	299	1	hence	hence	ADV
ejpam-3952	299	2	,	,	PUNCT
ejpam-3952	299	3	nt	not	PART
ejpam-3952	299	4	(	(	PUNCT
ejpam-3952	299	5	n	n	X
ejpam-3952	299	6	+	+	NUM
ejpam-3952	299	7	1	1	NUM
ejpam-3952	299	8	,	,	PUNCT
ejpam-3952	299	9	l)−	l)−	PROPN
ejpam-3952	299	10	(	(	PUNCT
ejpam-3952	299	11	l	l	NOUN
ejpam-3952	299	12	−	−	PROPN
ejpam-3952	299	13	1)n	1)n	X
ejpam-3952	299	14	=	=	SYM
ejpam-3952	299	15	(	(	PUNCT
ejpam-3952	299	16	n	n	CCONJ
ejpam-3952	299	17	+	+	NUM
ejpam-3952	299	18	2)t	2)t	NUM
ejpam-3952	299	19	(	(	PUNCT
ejpam-3952	299	20	n	n	X
ejpam-3952	299	21	,	,	PUNCT
ejpam-3952	299	22	l	l	NOUN
ejpam-3952	299	23	)	)	PUNCT
ejpam-3952	299	24	as	as	SCONJ
ejpam-3952	299	25	desired	desire	VERB
ejpam-3952	299	26	.	.	PUNCT
ejpam-3952	300	1	by	by	ADP
ejpam-3952	300	2	setting	set	VERB
ejpam-3952	300	3	l	l	NOUN
ejpam-3952	300	4	=	=	SYM
ejpam-3952	300	5	1	1	NUM
ejpam-3952	300	6	,	,	PUNCT
ejpam-3952	300	7	the	the	DET
ejpam-3952	300	8	identity	identity	NOUN
ejpam-3952	300	9	nt	not	PART
ejpam-3952	300	10	(	(	PUNCT
ejpam-3952	300	11	n	n	PROPN
ejpam-3952	300	12	+	+	NOUN
ejpam-3952	300	13	1	1	NUM
ejpam-3952	300	14	)	)	PUNCT
ejpam-3952	300	15	=	=	SYM
ejpam-3952	300	16	(	(	PUNCT
ejpam-3952	300	17	n	n	CCONJ
ejpam-3952	300	18	+	+	NUM
ejpam-3952	300	19	2)t	2)t	NUM
ejpam-3952	300	20	(	(	PUNCT
ejpam-3952	300	21	n	n	CCONJ
ejpam-3952	300	22	)	)	PUNCT
ejpam-3952	300	23	for	for	ADP
ejpam-3952	300	24	triangular	triangular	NOUN
ejpam-3952	300	25	numbers	number	NOUN
ejpam-3952	300	26	in	in	ADP
ejpam-3952	300	27	[	[	X
ejpam-3952	300	28	7	7	NUM
ejpam-3952	300	29	,	,	PUNCT
ejpam-3952	300	30	equation	equation	NOUN
ejpam-3952	300	31	(	(	PUNCT
ejpam-3952	300	32	1.7	1.7	NUM
ejpam-3952	300	33	)	)	PUNCT
ejpam-3952	300	34	]	]	PUNCT
ejpam-3952	300	35	follows	follow	VERB
ejpam-3952	300	36	.	.	PUNCT
ejpam-3952	301	1	theorem	theorem	ADJ
ejpam-3952	301	2	9	9	NUM
ejpam-3952	301	3	.	.	PUNCT
ejpam-3952	301	4	t	t	PROPN
ejpam-3952	301	5	(	(	PUNCT
ejpam-3952	301	6	2n	2n	NUM
ejpam-3952	302	1	+	+	CCONJ
ejpam-3952	302	2	1	1	NUM
ejpam-3952	302	3	,	,	PUNCT
ejpam-3952	302	4	l	l	NOUN
ejpam-3952	302	5	)	)	PUNCT
ejpam-3952	302	6	−	−	PROPN
ejpam-3952	302	7	t	t	PROPN
ejpam-3952	302	8	(	(	PUNCT
ejpam-3952	302	9	2n	2n	NUM
ejpam-3952	302	10	,	,	PUNCT
ejpam-3952	302	11	l	l	NOUN
ejpam-3952	302	12	)	)	PUNCT
ejpam-3952	302	13	=	=	SYM
ejpam-3952	302	14	t	t	PROPN
ejpam-3952	302	15	(	(	PUNCT
ejpam-3952	302	16	n	n	PROPN
ejpam-3952	302	17	+	+	CCONJ
ejpam-3952	302	18	1	1	NUM
ejpam-3952	302	19	,	,	PUNCT
ejpam-3952	302	20	l	l	NOUN
ejpam-3952	302	21	)	)	PUNCT
ejpam-3952	302	22	−	−	PROPN
ejpam-3952	302	23	t	t	PROPN
ejpam-3952	302	24	(	(	PUNCT
ejpam-3952	302	25	n−	n−	NOUN
ejpam-3952	302	26	1	1	NUM
ejpam-3952	302	27	,	,	PUNCT
ejpam-3952	302	28	l	l	NOUN
ejpam-3952	302	29	)	)	PUNCT
ejpam-3952	303	1	+	+	CCONJ
ejpam-3952	303	2	(	(	PUNCT
ejpam-3952	303	3	l	l	NOUN
ejpam-3952	303	4	−	−	PROPN
ejpam-3952	303	5	1	1	NUM
ejpam-3952	303	6	)	)	PUNCT
ejpam-3952	303	7	for	for	ADP
ejpam-3952	303	8	all	all	DET
ejpam-3952	303	9	positive	positive	ADJ
ejpam-3952	303	10	integers	integer	NOUN
ejpam-3952	303	11	n	n	PRON
ejpam-3952	303	12	and	and	CCONJ
ejpam-3952	303	13	l.	l.	PROPN
ejpam-3952	303	14	j.	j.	PROPN
ejpam-3952	303	15	punpim	punpim	PROPN
ejpam-3952	303	16	,	,	PUNCT
ejpam-3952	303	17	s.	s.	PROPN
ejpam-3952	303	18	jitman	jitman	PROPN
ejpam-3952	303	19	/	/	SYM
ejpam-3952	303	20	eur	eur	PROPN
ejpam-3952	303	21	.	.	PUNCT
ejpam-3952	304	1	j.	j.	PROPN
ejpam-3952	304	2	pure	pure	PROPN
ejpam-3952	304	3	appl	appl	PROPN
ejpam-3952	304	4	.	.	PROPN
ejpam-3952	304	5	math	math	PROPN
ejpam-3952	304	6	,	,	PUNCT
ejpam-3952	304	7	14	14	NUM
ejpam-3952	304	8	(	(	PUNCT
ejpam-3952	304	9	2	2	NUM
ejpam-3952	304	10	)	)	PUNCT
ejpam-3952	304	11	(	(	PUNCT
ejpam-3952	304	12	2021	2021	NUM
ejpam-3952	304	13	)	)	PUNCT
ejpam-3952	304	14	,	,	PUNCT
ejpam-3952	304	15	380	380	NUM
ejpam-3952	304	16	-	-	SYM
ejpam-3952	304	17	395	395	NUM
ejpam-3952	304	18	390	390	NUM
ejpam-3952	304	19	proof	proof	NOUN
ejpam-3952	304	20	.	.	PUNCT
ejpam-3952	305	1	let	let	VERB
ejpam-3952	305	2	n	n	NOUN
ejpam-3952	305	3	and	and	CCONJ
ejpam-3952	305	4	l	l	NOUN
ejpam-3952	305	5	be	be	AUX
ejpam-3952	305	6	positive	positive	ADJ
ejpam-3952	305	7	integers	integer	NOUN
ejpam-3952	305	8	.	.	PUNCT
ejpam-3952	306	1	then	then	ADV
ejpam-3952	306	2	t	t	PROPN
ejpam-3952	306	3	(	(	PUNCT
ejpam-3952	306	4	2n	2n	NUM
ejpam-3952	306	5	+	+	CCONJ
ejpam-3952	306	6	1	1	NUM
ejpam-3952	306	7	,	,	PUNCT
ejpam-3952	306	8	l)−	l)−	PROPN
ejpam-3952	306	9	t	t	PROPN
ejpam-3952	306	10	(	(	PUNCT
ejpam-3952	306	11	2n	2n	NUM
ejpam-3952	306	12	,	,	PUNCT
ejpam-3952	306	13	l	l	NOUN
ejpam-3952	306	14	)	)	PUNCT
ejpam-3952	306	15	=	=	SYM
ejpam-3952	306	16	2n	2n	NUM
ejpam-3952	307	1	+	+	CCONJ
ejpam-3952	307	2	1	1	NUM
ejpam-3952	307	3	+	+	CCONJ
ejpam-3952	307	4	(	(	PUNCT
ejpam-3952	307	5	2n	2n	NUM
ejpam-3952	307	6	+	+	X
ejpam-3952	307	7	1)(2n)l	1)(2n)l	NUM
ejpam-3952	307	8	2	2	NUM
ejpam-3952	307	9	−	−	NOUN
ejpam-3952	308	1	2n−	2n−	PROPN
ejpam-3952	308	2	(	(	PUNCT
ejpam-3952	308	3	2n)(2n−	2n)(2n−	NUM
ejpam-3952	308	4	1)l	1)l	NUM
ejpam-3952	308	5	2	2	NUM
ejpam-3952	308	6	=	=	NOUN
ejpam-3952	308	7	4n	4n	NOUN
ejpam-3952	308	8	+	+	CCONJ
ejpam-3952	308	9	2	2	NUM
ejpam-3952	308	10	+	+	NUM
ejpam-3952	308	11	4n2l	4n2l	NOUN
ejpam-3952	308	12	+	+	CCONJ
ejpam-3952	308	13	2nl	2nl	ADJ
ejpam-3952	308	14	−	−	PROPN
ejpam-3952	309	1	4n−	4n−	ADJ
ejpam-3952	309	2	4n2l	4n2l	NOUN
ejpam-3952	309	3	+	+	CCONJ
ejpam-3952	309	4	2nl	2nl	ADJ
ejpam-3952	309	5	2	2	NUM
ejpam-3952	309	6	=	=	SYM
ejpam-3952	309	7	2n	2n	NUM
ejpam-3952	310	1	+	+	CCONJ
ejpam-3952	310	2	2	2	NUM
ejpam-3952	310	3	+	+	NUM
ejpam-3952	310	4	n2l	n2l	NOUN
ejpam-3952	310	5	+	+	CCONJ
ejpam-3952	310	6	nl	nl	NOUN
ejpam-3952	310	7	−	−	PROPN
ejpam-3952	310	8	2n	2n	NUM
ejpam-3952	310	9	+	+	CCONJ
ejpam-3952	310	10	2−	2−	NUM
ejpam-3952	310	11	n2l	n2l	NOUN
ejpam-3952	310	12	+	+	NUM
ejpam-3952	310	13	3nl	3nl	ADJ
ejpam-3952	310	14	−	−	ADP
ejpam-3952	310	15	2l	2l	NOUN
ejpam-3952	310	16	+	+	CCONJ
ejpam-3952	310	17	2l	2l	NUM
ejpam-3952	310	18	−	−	NOUN
ejpam-3952	310	19	2	2	NUM
ejpam-3952	310	20	2	2	NUM
ejpam-3952	310	21	=	=	SYM
ejpam-3952	310	22	n	n	NOUN
ejpam-3952	310	23	+	+	NOUN
ejpam-3952	310	24	1	1	NUM
ejpam-3952	310	25	+	+	CCONJ
ejpam-3952	310	26	(	(	PUNCT
ejpam-3952	310	27	n	n	PROPN
ejpam-3952	310	28	+	+	CCONJ
ejpam-3952	310	29	1)(n)l	1)(n)l	NUM
ejpam-3952	310	30	2	2	NUM
ejpam-3952	310	31	−	−	PROPN
ejpam-3952	310	32	(	(	PUNCT
ejpam-3952	310	33	n−	n−	PROPN
ejpam-3952	310	34	1)−	1)−	PROPN
ejpam-3952	310	35	(	(	PUNCT
ejpam-3952	310	36	n−	n−	NOUN
ejpam-3952	310	37	1)(n−	1)(n−	NUM
ejpam-3952	310	38	2)l	2)l	NUM
ejpam-3952	310	39	2	2	NUM
ejpam-3952	310	40	+	+	CCONJ
ejpam-3952	310	41	l	l	NOUN
ejpam-3952	310	42	−	−	NOUN
ejpam-3952	310	43	1	1	NUM
ejpam-3952	310	44	=	=	SYM
ejpam-3952	310	45	t	t	PROPN
ejpam-3952	310	46	(	(	PUNCT
ejpam-3952	310	47	n	n	PROPN
ejpam-3952	310	48	+	+	NUM
ejpam-3952	310	49	1	1	NUM
ejpam-3952	310	50	,	,	PUNCT
ejpam-3952	310	51	l)−	l)−	PROPN
ejpam-3952	310	52	t	t	PROPN
ejpam-3952	310	53	(	(	PUNCT
ejpam-3952	310	54	n−	n−	NOUN
ejpam-3952	310	55	1	1	NUM
ejpam-3952	310	56	,	,	PUNCT
ejpam-3952	310	57	l	l	NOUN
ejpam-3952	310	58	)	)	PUNCT
ejpam-3952	311	1	+	+	CCONJ
ejpam-3952	311	2	(	(	PUNCT
ejpam-3952	311	3	l	l	NOUN
ejpam-3952	311	4	−	−	NOUN
ejpam-3952	311	5	1	1	NUM
ejpam-3952	311	6	)	)	PUNCT
ejpam-3952	311	7	.	.	PUNCT
ejpam-3952	312	1	this	this	PRON
ejpam-3952	312	2	completes	complete	VERB
ejpam-3952	312	3	the	the	DET
ejpam-3952	312	4	proof	proof	NOUN
ejpam-3952	312	5	.	.	PUNCT
ejpam-3952	313	1	from	from	ADP
ejpam-3952	313	2	the	the	DET
ejpam-3952	313	3	theorem	theorem	NOUN
ejpam-3952	313	4	above	above	ADV
ejpam-3952	313	5	,	,	PUNCT
ejpam-3952	313	6	the	the	DET
ejpam-3952	313	7	relation	relation	NOUN
ejpam-3952	313	8	t	t	PROPN
ejpam-3952	313	9	(	(	PUNCT
ejpam-3952	313	10	2n	2n	NUM
ejpam-3952	313	11	+	+	PROPN
ejpam-3952	313	12	1)−	1)−	PROPN
ejpam-3952	313	13	t	t	PROPN
ejpam-3952	313	14	(	(	PUNCT
ejpam-3952	313	15	2n	2n	NUM
ejpam-3952	313	16	)	)	PUNCT
ejpam-3952	313	17	=	=	SYM
ejpam-3952	313	18	t	t	PROPN
ejpam-3952	313	19	(	(	PUNCT
ejpam-3952	313	20	n	n	PROPN
ejpam-3952	313	21	+	+	PROPN
ejpam-3952	313	22	1)−	1)−	PROPN
ejpam-3952	313	23	t	t	NOUN
ejpam-3952	313	24	(	(	PUNCT
ejpam-3952	313	25	n−	n−	NOUN
ejpam-3952	313	26	1	1	NUM
ejpam-3952	313	27	)	)	PUNCT
ejpam-3952	313	28	in	in	ADP
ejpam-3952	313	29	[	[	X
ejpam-3952	313	30	9	9	NUM
ejpam-3952	313	31	,	,	PUNCT
ejpam-3952	313	32	equation	equation	NOUN
ejpam-3952	313	33	(	(	PUNCT
ejpam-3952	313	34	7.14	7.14	NUM
ejpam-3952	313	35	)	)	PUNCT
ejpam-3952	313	36	]	]	PUNCT
ejpam-3952	313	37	can	can	AUX
ejpam-3952	313	38	be	be	AUX
ejpam-3952	313	39	obtained	obtain	VERB
ejpam-3952	313	40	directly	directly	ADV
ejpam-3952	313	41	when	when	SCONJ
ejpam-3952	313	42	l	l	NOUN
ejpam-3952	313	43	=	=	SYM
ejpam-3952	313	44	1	1	X
ejpam-3952	313	45	.	.	PUNCT
ejpam-3952	313	46	theorem	theorem	VERB
ejpam-3952	313	47	10	10	NUM
ejpam-3952	313	48	.	.	PUNCT
ejpam-3952	314	1	t	t	PROPN
ejpam-3952	314	2	(	(	PUNCT
ejpam-3952	314	3	2n	2n	NUM
ejpam-3952	314	4	,	,	PUNCT
ejpam-3952	314	5	l	l	NOUN
ejpam-3952	314	6	)	)	PUNCT
ejpam-3952	314	7	=	=	SYM
ejpam-3952	315	1	3	3	NUM
ejpam-3952	315	2	t	t	NOUN
ejpam-3952	315	3	(	(	PUNCT
ejpam-3952	315	4	n	n	X
ejpam-3952	315	5	,	,	PUNCT
ejpam-3952	315	6	l	l	NOUN
ejpam-3952	315	7	)	)	PUNCT
ejpam-3952	316	1	+	+	CCONJ
ejpam-3952	316	2	lt	lt	NOUN
ejpam-3952	316	3	(	(	PUNCT
ejpam-3952	316	4	n−	n−	NOUN
ejpam-3952	316	5	1	1	NUM
ejpam-3952	316	6	)	)	PUNCT
ejpam-3952	316	7	+	+	CCONJ
ejpam-3952	316	8	(	(	PUNCT
ejpam-3952	316	9	l−	l−	PROPN
ejpam-3952	316	10	1)n	1)n	NUM
ejpam-3952	316	11	for	for	ADP
ejpam-3952	316	12	all	all	DET
ejpam-3952	316	13	positive	positive	ADJ
ejpam-3952	316	14	integers	integer	NOUN
ejpam-3952	316	15	n	n	PRON
ejpam-3952	316	16	and	and	CCONJ
ejpam-3952	316	17	l.	l.	PROPN
ejpam-3952	316	18	proof	proof	NOUN
ejpam-3952	316	19	.	.	PUNCT
ejpam-3952	317	1	let	let	VERB
ejpam-3952	317	2	n	n	NOUN
ejpam-3952	317	3	and	and	CCONJ
ejpam-3952	317	4	l	l	NOUN
ejpam-3952	317	5	be	be	AUX
ejpam-3952	317	6	positive	positive	ADJ
ejpam-3952	317	7	integers	integer	NOUN
ejpam-3952	317	8	.	.	PUNCT
ejpam-3952	318	1	then	then	ADV
ejpam-3952	318	2	3	3	NUM
ejpam-3952	318	3	t	t	NOUN
ejpam-3952	318	4	(	(	PUNCT
ejpam-3952	318	5	n	n	X
ejpam-3952	318	6	,	,	PUNCT
ejpam-3952	318	7	l	l	NOUN
ejpam-3952	318	8	)	)	PUNCT
ejpam-3952	319	1	+	+	CCONJ
ejpam-3952	319	2	lt	lt	NOUN
ejpam-3952	319	3	(	(	PUNCT
ejpam-3952	319	4	n−	n−	NOUN
ejpam-3952	319	5	1	1	NUM
ejpam-3952	319	6	)	)	PUNCT
ejpam-3952	319	7	+	+	CCONJ
ejpam-3952	319	8	(	(	PUNCT
ejpam-3952	319	9	l	l	NOUN
ejpam-3952	319	10	−	−	PROPN
ejpam-3952	319	11	1)n	1)n	X
ejpam-3952	319	12	=	=	SYM
ejpam-3952	319	13	3	3	NUM
ejpam-3952	319	14	(	(	PUNCT
ejpam-3952	319	15	n	n	PROPN
ejpam-3952	319	16	+	+	CCONJ
ejpam-3952	319	17	n(n−	n(n−	PROPN
ejpam-3952	319	18	1)l	1)l	NUM
ejpam-3952	319	19	2	2	NUM
ejpam-3952	319	20	)	)	PUNCT
ejpam-3952	319	21	+	+	CCONJ
ejpam-3952	319	22	l	l	NOUN
ejpam-3952	319	23	(	(	PUNCT
ejpam-3952	319	24	(	(	PUNCT
ejpam-3952	319	25	n−	n−	NOUN
ejpam-3952	319	26	1)n	1)n	NOUN
ejpam-3952	319	27	2	2	NUM
ejpam-3952	319	28	)	)	PUNCT
ejpam-3952	319	29	+	+	CCONJ
ejpam-3952	319	30	(	(	PUNCT
ejpam-3952	319	31	l	l	NOUN
ejpam-3952	319	32	−	−	PROPN
ejpam-3952	319	33	1)n	1)n	NOUN
ejpam-3952	319	34	=	=	SYM
ejpam-3952	319	35	6n	6n	NOUN
ejpam-3952	320	1	+	+	CCONJ
ejpam-3952	320	2	3n2l	3n2l	NUM
ejpam-3952	321	1	−	−	NOUN
ejpam-3952	321	2	3nl	3nl	ADJ
ejpam-3952	321	3	+	+	CCONJ
ejpam-3952	321	4	n2l	n2l	NOUN
ejpam-3952	321	5	−	−	ADP
ejpam-3952	321	6	nl	nl	NOUN
ejpam-3952	321	7	+	+	CCONJ
ejpam-3952	321	8	2nl	2nl	ADJ
ejpam-3952	321	9	−	−	ADP
ejpam-3952	321	10	2n	2n	NUM
ejpam-3952	321	11	2	2	NUM
ejpam-3952	321	12	=	=	SYM
ejpam-3952	321	13	4n	4n	NOUN
ejpam-3952	321	14	+	+	CCONJ
ejpam-3952	321	15	(	(	PUNCT
ejpam-3952	321	16	4n2	4n2	NUM
ejpam-3952	321	17	−	−	NOUN
ejpam-3952	321	18	2n)l	2n)l	NUM
ejpam-3952	321	19	2	2	NUM
ejpam-3952	321	20	=	=	SYM
ejpam-3952	321	21	2n	2n	NUM
ejpam-3952	321	22	+	+	CCONJ
ejpam-3952	321	23	2n(2n−	2n(2n−	NUM
ejpam-3952	321	24	1)l	1)l	NUM
ejpam-3952	321	25	2	2	NUM
ejpam-3952	321	26	=	=	SYM
ejpam-3952	321	27	t	t	PROPN
ejpam-3952	321	28	(	(	PUNCT
ejpam-3952	321	29	2n	2n	NUM
ejpam-3952	321	30	,	,	PUNCT
ejpam-3952	321	31	l	l	NOUN
ejpam-3952	321	32	)	)	PUNCT
ejpam-3952	321	33	.	.	PUNCT
ejpam-3952	322	1	as	as	SCONJ
ejpam-3952	322	2	desired	desire	VERB
ejpam-3952	322	3	,	,	PUNCT
ejpam-3952	322	4	we	we	PRON
ejpam-3952	322	5	have	have	VERB
ejpam-3952	322	6	t	t	PROPN
ejpam-3952	322	7	(	(	PUNCT
ejpam-3952	322	8	2n	2n	NUM
ejpam-3952	322	9	,	,	PUNCT
ejpam-3952	322	10	l	l	NOUN
ejpam-3952	322	11	)	)	PUNCT
ejpam-3952	322	12	=	=	SYM
ejpam-3952	323	1	3	3	NUM
ejpam-3952	323	2	t	t	NOUN
ejpam-3952	323	3	(	(	PUNCT
ejpam-3952	323	4	n	n	X
ejpam-3952	323	5	,	,	PUNCT
ejpam-3952	323	6	l	l	NOUN
ejpam-3952	323	7	)	)	PUNCT
ejpam-3952	324	1	+	+	CCONJ
ejpam-3952	324	2	lt	lt	NOUN
ejpam-3952	324	3	(	(	PUNCT
ejpam-3952	324	4	n−	n−	NOUN
ejpam-3952	324	5	1	1	NUM
ejpam-3952	324	6	)	)	PUNCT
ejpam-3952	324	7	+	+	CCONJ
ejpam-3952	324	8	(	(	PUNCT
ejpam-3952	324	9	l	l	NOUN
ejpam-3952	324	10	−	−	PROPN
ejpam-3952	324	11	1)n	1)n	X
ejpam-3952	324	12	.	.	PUNCT
ejpam-3952	325	1	lemma	lemma	PROPN
ejpam-3952	325	2	1	1	NUM
ejpam-3952	325	3	.	.	PUNCT
ejpam-3952	326	1	lt	lt	PRON
ejpam-3952	326	2	(	(	PUNCT
ejpam-3952	326	3	n−	n−	NOUN
ejpam-3952	326	4	1	1	NUM
ejpam-3952	326	5	)	)	PUNCT
ejpam-3952	326	6	+	+	CCONJ
ejpam-3952	326	7	(	(	PUNCT
ejpam-3952	326	8	l−	l−	PROPN
ejpam-3952	326	9	1)n	1)n	PROPN
ejpam-3952	326	10	=	=	SYM
ejpam-3952	326	11	t	t	PROPN
ejpam-3952	326	12	(	(	PUNCT
ejpam-3952	326	13	n−	n−	NOUN
ejpam-3952	326	14	1	1	NUM
ejpam-3952	326	15	)	)	PUNCT
ejpam-3952	327	1	+	+	CCONJ
ejpam-3952	327	2	(	(	PUNCT
ejpam-3952	327	3	l−	l−	PROPN
ejpam-3952	327	4	1)t	1)t	PROPN
ejpam-3952	327	5	(	(	PUNCT
ejpam-3952	327	6	n	n	CCONJ
ejpam-3952	327	7	)	)	PUNCT
ejpam-3952	327	8	for	for	ADP
ejpam-3952	327	9	all	all	DET
ejpam-3952	327	10	positive	positive	ADJ
ejpam-3952	327	11	integers	integer	NOUN
ejpam-3952	327	12	n	n	PRON
ejpam-3952	327	13	and	and	CCONJ
ejpam-3952	327	14	l.	l.	PROPN
ejpam-3952	327	15	proof	proof	NOUN
ejpam-3952	327	16	.	.	PUNCT
ejpam-3952	328	1	let	let	VERB
ejpam-3952	328	2	n	n	NOUN
ejpam-3952	328	3	and	and	CCONJ
ejpam-3952	328	4	l	l	NOUN
ejpam-3952	328	5	be	be	AUX
ejpam-3952	328	6	positive	positive	ADJ
ejpam-3952	328	7	integers	integer	NOUN
ejpam-3952	328	8	.	.	PUNCT
ejpam-3952	329	1	then	then	ADV
ejpam-3952	329	2	we	we	PRON
ejpam-3952	329	3	have	have	VERB
ejpam-3952	329	4	t	t	PROPN
ejpam-3952	329	5	(	(	PUNCT
ejpam-3952	329	6	n−	n−	NOUN
ejpam-3952	329	7	1	1	NUM
ejpam-3952	329	8	)	)	PUNCT
ejpam-3952	330	1	+	+	CCONJ
ejpam-3952	330	2	(	(	PUNCT
ejpam-3952	330	3	l	l	NOUN
ejpam-3952	330	4	−	−	PROPN
ejpam-3952	330	5	1)t	1)t	PROPN
ejpam-3952	330	6	(	(	PUNCT
ejpam-3952	330	7	n	n	CCONJ
ejpam-3952	330	8	)	)	PUNCT
ejpam-3952	330	9	=	=	SYM
ejpam-3952	330	10	(	(	PUNCT
ejpam-3952	330	11	n−	n−	NOUN
ejpam-3952	330	12	1)n	1)n	NOUN
ejpam-3952	330	13	2	2	NUM
ejpam-3952	330	14	+	+	CCONJ
ejpam-3952	330	15	(	(	PUNCT
ejpam-3952	330	16	l	l	NOUN
ejpam-3952	330	17	−	−	PROPN
ejpam-3952	330	18	1	1	NUM
ejpam-3952	330	19	)	)	PUNCT
ejpam-3952	330	20	(	(	PUNCT
ejpam-3952	330	21	n(n	n(n	NOUN
ejpam-3952	330	22	+	+	CCONJ
ejpam-3952	330	23	1	1	NUM
ejpam-3952	330	24	)	)	SYM
ejpam-3952	330	25	2	2	NUM
ejpam-3952	330	26	)	)	PUNCT
ejpam-3952	330	27	=	=	SYM
ejpam-3952	330	28	n2	n2	NOUN
ejpam-3952	330	29	−	−	PROPN
ejpam-3952	330	30	n	n	PROPN
ejpam-3952	330	31	+	+	CCONJ
ejpam-3952	330	32	n2l	n2l	X
ejpam-3952	330	33	+	+	CCONJ
ejpam-3952	330	34	nl	nl	PROPN
ejpam-3952	330	35	−	−	PROPN
ejpam-3952	330	36	n2	n2	NOUN
ejpam-3952	330	37	−	−	PROPN
ejpam-3952	330	38	n	n	CCONJ
ejpam-3952	330	39	2	2	NUM
ejpam-3952	330	40	=	=	SYM
ejpam-3952	330	41	n2l	n2l	PROPN
ejpam-3952	330	42	−	−	PROPN
ejpam-3952	330	43	nl	nl	PROPN
ejpam-3952	330	44	+	+	CCONJ
ejpam-3952	330	45	2nl	2nl	ADJ
ejpam-3952	330	46	−	−	ADP
ejpam-3952	330	47	2n	2n	NUM
ejpam-3952	330	48	2	2	NUM
ejpam-3952	330	49	=	=	SYM
ejpam-3952	330	50	l	l	NOUN
ejpam-3952	330	51	(	(	PUNCT
ejpam-3952	330	52	(	(	PUNCT
ejpam-3952	330	53	n−	n−	NOUN
ejpam-3952	330	54	1)n	1)n	NOUN
ejpam-3952	330	55	2	2	NUM
ejpam-3952	330	56	)	)	PUNCT
ejpam-3952	330	57	+	+	CCONJ
ejpam-3952	330	58	(	(	PUNCT
ejpam-3952	330	59	l	l	NOUN
ejpam-3952	330	60	−	−	PROPN
ejpam-3952	330	61	1)n	1)n	PROPN
ejpam-3952	330	62	j.	j.	PROPN
ejpam-3952	330	63	punpim	punpim	PROPN
ejpam-3952	330	64	,	,	PUNCT
ejpam-3952	330	65	s.	s.	PROPN
ejpam-3952	330	66	jitman	jitman	PROPN
ejpam-3952	330	67	/	/	SYM
ejpam-3952	330	68	eur	eur	PROPN
ejpam-3952	330	69	.	.	PUNCT
ejpam-3952	331	1	j.	j.	PROPN
ejpam-3952	331	2	pure	pure	PROPN
ejpam-3952	331	3	appl	appl	PROPN
ejpam-3952	331	4	.	.	PROPN
ejpam-3952	331	5	math	math	PROPN
ejpam-3952	331	6	,	,	PUNCT
ejpam-3952	331	7	14	14	NUM
ejpam-3952	331	8	(	(	PUNCT
ejpam-3952	331	9	2	2	NUM
ejpam-3952	331	10	)	)	PUNCT
ejpam-3952	331	11	(	(	PUNCT
ejpam-3952	331	12	2021	2021	NUM
ejpam-3952	331	13	)	)	PUNCT
ejpam-3952	331	14	,	,	PUNCT
ejpam-3952	331	15	380	380	NUM
ejpam-3952	331	16	-	-	SYM
ejpam-3952	331	17	395	395	NUM
ejpam-3952	331	18	391	391	NUM
ejpam-3952	331	19	=	=	SYM
ejpam-3952	331	20	lt	lt	NOUN
ejpam-3952	331	21	(	(	PUNCT
ejpam-3952	331	22	n−	n−	NOUN
ejpam-3952	331	23	1	1	NUM
ejpam-3952	331	24	)	)	PUNCT
ejpam-3952	331	25	+	+	CCONJ
ejpam-3952	331	26	(	(	PUNCT
ejpam-3952	331	27	l	l	NOUN
ejpam-3952	331	28	−	−	PROPN
ejpam-3952	331	29	1)n	1)n	PUNCT
ejpam-3952	331	30	as	as	SCONJ
ejpam-3952	331	31	desired	desire	VERB
ejpam-3952	331	32	.	.	PUNCT
ejpam-3952	332	1	from	from	ADP
ejpam-3952	332	2	theorem	theorem	ADJ
ejpam-3952	332	3	10	10	NUM
ejpam-3952	332	4	and	and	CCONJ
ejpam-3952	332	5	lemma	lemma	PROPN
ejpam-3952	332	6	1	1	NUM
ejpam-3952	332	7	,	,	PUNCT
ejpam-3952	332	8	we	we	PRON
ejpam-3952	332	9	have	have	VERB
ejpam-3952	332	10	the	the	DET
ejpam-3952	332	11	following	follow	VERB
ejpam-3952	332	12	identity	identity	NOUN
ejpam-3952	332	13	.	.	PUNCT
ejpam-3952	333	1	corollary	corollary	ADJ
ejpam-3952	333	2	3	3	NUM
ejpam-3952	333	3	.	.	PUNCT
ejpam-3952	334	1	t	t	PROPN
ejpam-3952	334	2	(	(	PUNCT
ejpam-3952	334	3	2n	2n	NUM
ejpam-3952	334	4	,	,	PUNCT
ejpam-3952	334	5	l	l	NOUN
ejpam-3952	334	6	)	)	PUNCT
ejpam-3952	334	7	=	=	SYM
ejpam-3952	335	1	3	3	NUM
ejpam-3952	335	2	t	t	NOUN
ejpam-3952	335	3	(	(	PUNCT
ejpam-3952	335	4	n	n	X
ejpam-3952	335	5	,	,	PUNCT
ejpam-3952	335	6	l	l	NOUN
ejpam-3952	335	7	)	)	PUNCT
ejpam-3952	336	1	+	+	NUM
ejpam-3952	336	2	t	t	PROPN
ejpam-3952	336	3	(	(	PUNCT
ejpam-3952	336	4	n−	n−	NOUN
ejpam-3952	336	5	1	1	NUM
ejpam-3952	336	6	)	)	PUNCT
ejpam-3952	336	7	+	+	CCONJ
ejpam-3952	336	8	(	(	PUNCT
ejpam-3952	336	9	l	l	NOUN
ejpam-3952	336	10	−	−	PROPN
ejpam-3952	336	11	1)t	1)t	PROPN
ejpam-3952	336	12	(	(	PUNCT
ejpam-3952	336	13	n	n	CCONJ
ejpam-3952	336	14	)	)	PUNCT
ejpam-3952	336	15	for	for	ADP
ejpam-3952	336	16	all	all	DET
ejpam-3952	336	17	positive	positive	ADJ
ejpam-3952	336	18	integers	integer	NOUN
ejpam-3952	336	19	n	n	VERB
ejpam-3952	336	20	and	and	CCONJ
ejpam-3952	336	21	l.	l.	NOUN
ejpam-3952	336	22	by	by	ADP
ejpam-3952	336	23	setting	set	VERB
ejpam-3952	336	24	l	l	NOUN
ejpam-3952	336	25	=	=	SYM
ejpam-3952	336	26	1	1	NUM
ejpam-3952	336	27	,	,	PUNCT
ejpam-3952	336	28	we	we	PRON
ejpam-3952	336	29	have	have	VERB
ejpam-3952	336	30	t	t	PROPN
ejpam-3952	336	31	(	(	PUNCT
ejpam-3952	336	32	2n	2n	NUM
ejpam-3952	336	33	)	)	PUNCT
ejpam-3952	336	34	=	=	PUNCT
ejpam-3952	337	1	3	3	NUM
ejpam-3952	337	2	t	t	NOUN
ejpam-3952	337	3	(	(	PUNCT
ejpam-3952	337	4	n	n	CCONJ
ejpam-3952	337	5	)	)	PUNCT
ejpam-3952	338	1	+	+	CCONJ
ejpam-3952	338	2	t	t	PROPN
ejpam-3952	338	3	(	(	PUNCT
ejpam-3952	338	4	n	n	CCONJ
ejpam-3952	338	5	−	−	PROPN
ejpam-3952	338	6	1	1	NUM
ejpam-3952	338	7	)	)	PUNCT
ejpam-3952	338	8	for	for	ADP
ejpam-3952	338	9	all	all	DET
ejpam-3952	338	10	positive	positive	ADJ
ejpam-3952	338	11	integers	integer	NOUN
ejpam-3952	338	12	n	n	CCONJ
ejpam-3952	338	13	as	as	ADP
ejpam-3952	338	14	in	in	ADP
ejpam-3952	338	15	[	[	X
ejpam-3952	338	16	7	7	NUM
ejpam-3952	338	17	,	,	PUNCT
ejpam-3952	338	18	equation	equation	NOUN
ejpam-3952	338	19	(	(	PUNCT
ejpam-3952	338	20	1.12	1.12	NUM
ejpam-3952	338	21	)	)	PUNCT
ejpam-3952	338	22	]	]	PUNCT
ejpam-3952	338	23	.	.	PUNCT
ejpam-3952	339	1	theorem	theorem	ADJ
ejpam-3952	339	2	11	11	NUM
ejpam-3952	339	3	.	.	PUNCT
ejpam-3952	340	1	t	t	PROPN
ejpam-3952	340	2	(	(	PUNCT
ejpam-3952	340	3	2n	2n	NUM
ejpam-3952	340	4	+	+	CCONJ
ejpam-3952	340	5	1	1	NUM
ejpam-3952	340	6	,	,	PUNCT
ejpam-3952	340	7	l	l	NOUN
ejpam-3952	340	8	)	)	PUNCT
ejpam-3952	340	9	=	=	SYM
ejpam-3952	341	1	3	3	NUM
ejpam-3952	341	2	t	t	NOUN
ejpam-3952	341	3	(	(	PUNCT
ejpam-3952	341	4	n	n	X
ejpam-3952	341	5	,	,	PUNCT
ejpam-3952	341	6	l	l	NOUN
ejpam-3952	341	7	)	)	PUNCT
ejpam-3952	342	1	+	+	NUM
ejpam-3952	342	2	t	t	PROPN
ejpam-3952	342	3	(	(	PUNCT
ejpam-3952	342	4	n	n	PROPN
ejpam-3952	342	5	+	+	CCONJ
ejpam-3952	342	6	1	1	NUM
ejpam-3952	342	7	,	,	PUNCT
ejpam-3952	342	8	l	l	NOUN
ejpam-3952	342	9	)	)	PUNCT
ejpam-3952	343	1	+	+	CCONJ
ejpam-3952	343	2	2(l	2(l	NUM
ejpam-3952	343	3	−	−	PROPN
ejpam-3952	343	4	1)n	1)n	NUM
ejpam-3952	343	5	for	for	ADP
ejpam-3952	343	6	all	all	DET
ejpam-3952	343	7	positive	positive	ADJ
ejpam-3952	343	8	integers	integer	NOUN
ejpam-3952	343	9	n	n	PRON
ejpam-3952	343	10	and	and	CCONJ
ejpam-3952	343	11	l.	l.	PROPN
ejpam-3952	343	12	proof	proof	NOUN
ejpam-3952	343	13	.	.	PUNCT
ejpam-3952	344	1	let	let	VERB
ejpam-3952	344	2	n	n	NOUN
ejpam-3952	344	3	and	and	CCONJ
ejpam-3952	344	4	l	l	NOUN
ejpam-3952	344	5	be	be	AUX
ejpam-3952	344	6	positive	positive	ADJ
ejpam-3952	344	7	integers	integer	NOUN
ejpam-3952	344	8	.	.	PUNCT
ejpam-3952	345	1	then	then	ADV
ejpam-3952	345	2	t	t	PROPN
ejpam-3952	345	3	(	(	PUNCT
ejpam-3952	345	4	2n	2n	NUM
ejpam-3952	345	5	+	+	CCONJ
ejpam-3952	345	6	1	1	NUM
ejpam-3952	345	7	,	,	PUNCT
ejpam-3952	345	8	l	l	NOUN
ejpam-3952	345	9	)	)	PUNCT
ejpam-3952	345	10	=	=	SYM
ejpam-3952	345	11	2n	2n	NUM
ejpam-3952	346	1	+	+	CCONJ
ejpam-3952	346	2	1	1	NUM
ejpam-3952	346	3	+	+	CCONJ
ejpam-3952	346	4	(	(	PUNCT
ejpam-3952	346	5	2n	2n	NUM
ejpam-3952	346	6	+	+	CCONJ
ejpam-3952	346	7	1)(2n)l	1)(2n)l	NUM
ejpam-3952	346	8	2	2	NUM
ejpam-3952	346	9	=	=	NOUN
ejpam-3952	346	10	4n	4n	NOUN
ejpam-3952	346	11	+	+	CCONJ
ejpam-3952	346	12	2	2	NUM
ejpam-3952	346	13	+	+	NUM
ejpam-3952	346	14	4n2l	4n2l	NOUN
ejpam-3952	346	15	+	+	CCONJ
ejpam-3952	346	16	2nl	2nl	ADJ
ejpam-3952	346	17	2	2	NUM
ejpam-3952	346	18	=	=	SYM
ejpam-3952	346	19	6n	6n	NOUN
ejpam-3952	347	1	+	+	CCONJ
ejpam-3952	347	2	3n2l	3n2l	NUM
ejpam-3952	348	1	−	−	NOUN
ejpam-3952	348	2	3nl	3nl	ADJ
ejpam-3952	349	1	+	+	X
ejpam-3952	349	2	2n	2n	NUM
ejpam-3952	349	3	+	+	CCONJ
ejpam-3952	349	4	2	2	NUM
ejpam-3952	349	5	+	+	NUM
ejpam-3952	349	6	n2l	n2l	NOUN
ejpam-3952	349	7	+	+	CCONJ
ejpam-3952	349	8	nl	nl	PROPN
ejpam-3952	349	9	+	+	CCONJ
ejpam-3952	349	10	4nl	4nl	ADJ
ejpam-3952	349	11	−	−	NOUN
ejpam-3952	349	12	4n	4n	X
ejpam-3952	349	13	2	2	NUM
ejpam-3952	349	14	=	=	SYM
ejpam-3952	349	15	3	3	NUM
ejpam-3952	349	16	(	(	PUNCT
ejpam-3952	349	17	n	n	PROPN
ejpam-3952	349	18	+	+	CCONJ
ejpam-3952	349	19	n(n−	n(n−	PROPN
ejpam-3952	349	20	1)l	1)l	NUM
ejpam-3952	349	21	2	2	NUM
ejpam-3952	349	22	)	)	PUNCT
ejpam-3952	349	23	+	+	CCONJ
ejpam-3952	349	24	(	(	PUNCT
ejpam-3952	349	25	n	n	PROPN
ejpam-3952	349	26	+	+	CCONJ
ejpam-3952	349	27	1	1	NUM
ejpam-3952	349	28	+	+	CCONJ
ejpam-3952	349	29	(	(	PUNCT
ejpam-3952	349	30	n	n	PROPN
ejpam-3952	349	31	+	+	CCONJ
ejpam-3952	349	32	1)(n)l	1)(n)l	NUM
ejpam-3952	349	33	2	2	NUM
ejpam-3952	349	34	)	)	PUNCT
ejpam-3952	349	35	+	+	CCONJ
ejpam-3952	349	36	2ln−	2ln−	NUM
ejpam-3952	349	37	2n	2n	NUM
ejpam-3952	349	38	=	=	SYM
ejpam-3952	349	39	3	3	NUM
ejpam-3952	349	40	t	t	NOUN
ejpam-3952	349	41	(	(	PUNCT
ejpam-3952	349	42	n	n	X
ejpam-3952	349	43	,	,	PUNCT
ejpam-3952	349	44	l	l	NOUN
ejpam-3952	349	45	)	)	PUNCT
ejpam-3952	349	46	+	+	NUM
ejpam-3952	349	47	t	t	PROPN
ejpam-3952	349	48	(	(	PUNCT
ejpam-3952	349	49	n	n	PROPN
ejpam-3952	349	50	+	+	CCONJ
ejpam-3952	349	51	1	1	NUM
ejpam-3952	349	52	,	,	PUNCT
ejpam-3952	349	53	l	l	NOUN
ejpam-3952	349	54	)	)	PUNCT
ejpam-3952	349	55	+	+	CCONJ
ejpam-3952	349	56	2(l	2(l	NUM
ejpam-3952	349	57	−	−	PROPN
ejpam-3952	349	58	1)n	1)n	NUM
ejpam-3952	349	59	.	.	PUNCT
ejpam-3952	350	1	hence	hence	ADV
ejpam-3952	350	2	,	,	PUNCT
ejpam-3952	350	3	the	the	DET
ejpam-3952	350	4	proof	proof	NOUN
ejpam-3952	350	5	is	be	AUX
ejpam-3952	350	6	completed	complete	VERB
ejpam-3952	350	7	.	.	PUNCT
ejpam-3952	351	1	theorem	theorem	NOUN
ejpam-3952	351	2	12	12	NUM
ejpam-3952	351	3	.	.	PUNCT
ejpam-3952	352	1	lt	lt	PRON
ejpam-3952	352	2	(	(	PUNCT
ejpam-3952	352	3	2n	2n	NUM
ejpam-3952	352	4	+	+	CCONJ
ejpam-3952	352	5	1	1	NUM
ejpam-3952	352	6	,	,	PUNCT
ejpam-3952	352	7	l	l	NOUN
ejpam-3952	352	8	)	)	PUNCT
ejpam-3952	352	9	=	=	SYM
ejpam-3952	352	10	3lt	3lt	ADJ
ejpam-3952	352	11	(	(	PUNCT
ejpam-3952	352	12	n	n	CCONJ
ejpam-3952	352	13	,	,	PUNCT
ejpam-3952	352	14	l)+	l)+	PROPN
ejpam-3952	352	15	lt	lt	PROPN
ejpam-3952	352	16	(	(	PUNCT
ejpam-3952	352	17	n	n	PROPN
ejpam-3952	352	18	+	+	NUM
ejpam-3952	352	19	1	1	NUM
ejpam-3952	352	20	,	,	PUNCT
ejpam-3952	352	21	l)+4nt	l)+4nt	X
ejpam-3952	352	22	(	(	PUNCT
ejpam-3952	352	23	l	l	NOUN
ejpam-3952	352	24	−	−	PROPN
ejpam-3952	352	25	1	1	NUM
ejpam-3952	352	26	)	)	PUNCT
ejpam-3952	352	27	for	for	ADP
ejpam-3952	352	28	all	all	DET
ejpam-3952	352	29	positive	positive	ADJ
ejpam-3952	352	30	integers	integer	NOUN
ejpam-3952	352	31	n	n	PRON
ejpam-3952	352	32	and	and	CCONJ
ejpam-3952	352	33	l.	l.	PROPN
ejpam-3952	352	34	proof	proof	NOUN
ejpam-3952	352	35	.	.	PUNCT
ejpam-3952	353	1	let	let	VERB
ejpam-3952	353	2	n	n	NOUN
ejpam-3952	353	3	and	and	CCONJ
ejpam-3952	353	4	l	l	NOUN
ejpam-3952	353	5	be	be	AUX
ejpam-3952	353	6	positive	positive	ADJ
ejpam-3952	353	7	integers	integer	NOUN
ejpam-3952	353	8	.	.	PUNCT
ejpam-3952	354	1	then	then	ADV
ejpam-3952	354	2	we	we	PRON
ejpam-3952	354	3	have	have	VERB
ejpam-3952	354	4	lt	lt	PRON
ejpam-3952	354	5	(	(	PUNCT
ejpam-3952	354	6	2n	2n	X
ejpam-3952	354	7	+	+	CCONJ
ejpam-3952	354	8	1	1	NUM
ejpam-3952	354	9	,	,	PUNCT
ejpam-3952	354	10	l	l	NOUN
ejpam-3952	354	11	)	)	PUNCT
ejpam-3952	354	12	=	=	SYM
ejpam-3952	354	13	l	l	NOUN
ejpam-3952	354	14	(	(	PUNCT
ejpam-3952	354	15	2n	2n	NUM
ejpam-3952	355	1	+	+	CCONJ
ejpam-3952	355	2	1	1	NUM
ejpam-3952	355	3	+	+	CCONJ
ejpam-3952	355	4	(	(	PUNCT
ejpam-3952	355	5	2n	2n	NUM
ejpam-3952	355	6	+	+	X
ejpam-3952	355	7	1)(2n)l	1)(2n)l	NUM
ejpam-3952	355	8	2	2	NUM
ejpam-3952	355	9	)	)	PUNCT
ejpam-3952	355	10	=	=	SYM
ejpam-3952	356	1	4nl	4nl	NOUN
ejpam-3952	356	2	+	+	CCONJ
ejpam-3952	356	3	2l	2l	NOUN
ejpam-3952	356	4	+	+	X
ejpam-3952	356	5	4n2l2	4n2l2	NUM
ejpam-3952	357	1	+	+	CCONJ
ejpam-3952	357	2	2nl2	2nl2	NUM
ejpam-3952	357	3	2	2	NUM
ejpam-3952	357	4	=	=	NOUN
ejpam-3952	357	5	6nl	6nl	NOUN
ejpam-3952	357	6	+	+	CCONJ
ejpam-3952	357	7	3n2l2	3n2l2	NUM
ejpam-3952	357	8	−	−	NOUN
ejpam-3952	357	9	3nl2	3nl2	NUM
ejpam-3952	357	10	+	+	NUM
ejpam-3952	357	11	2nl	2nl	ADJ
ejpam-3952	357	12	+	+	CCONJ
ejpam-3952	357	13	2l	2l	NOUN
ejpam-3952	357	14	+	+	X
ejpam-3952	357	15	n2l2	n2l2	NOUN
ejpam-3952	358	1	+	+	NUM
ejpam-3952	358	2	nl2	nl2	NOUN
ejpam-3952	359	1	+	+	NOUN
ejpam-3952	359	2	4nl2	4nl2	NUM
ejpam-3952	359	3	−	−	ADV
ejpam-3952	359	4	4nl	4nl	NOUN
ejpam-3952	359	5	2	2	NUM
ejpam-3952	359	6	=	=	SYM
ejpam-3952	359	7	3	3	NUM
ejpam-3952	359	8	(	(	PUNCT
ejpam-3952	359	9	n	n	PROPN
ejpam-3952	359	10	+	+	CCONJ
ejpam-3952	359	11	n(n−	n(n−	PROPN
ejpam-3952	359	12	1)l	1)l	NUM
ejpam-3952	359	13	2	2	NUM
ejpam-3952	359	14	)	)	PUNCT
ejpam-3952	360	1	+	+	CCONJ
ejpam-3952	360	2	l	l	NOUN
ejpam-3952	360	3	(	(	PUNCT
ejpam-3952	360	4	n	n	PROPN
ejpam-3952	360	5	+	+	CCONJ
ejpam-3952	360	6	1	1	NUM
ejpam-3952	360	7	+	+	CCONJ
ejpam-3952	360	8	(	(	PUNCT
ejpam-3952	360	9	n	n	PROPN
ejpam-3952	360	10	+	+	CCONJ
ejpam-3952	360	11	1)(n)l	1)(n)l	NUM
ejpam-3952	360	12	2	2	NUM
ejpam-3952	360	13	)	)	PUNCT
ejpam-3952	360	14	+	+	CCONJ
ejpam-3952	360	15	4n	4n	X
ejpam-3952	360	16	(	(	PUNCT
ejpam-3952	360	17	(	(	PUNCT
ejpam-3952	360	18	l	l	NOUN
ejpam-3952	360	19	−	−	PROPN
ejpam-3952	360	20	1)l	1)l	NUM
ejpam-3952	360	21	2	2	NUM
ejpam-3952	360	22	)	)	PUNCT
ejpam-3952	360	23	=	=	SYM
ejpam-3952	360	24	3lt	3lt	ADJ
ejpam-3952	360	25	(	(	PUNCT
ejpam-3952	360	26	n	n	CCONJ
ejpam-3952	360	27	,	,	PUNCT
ejpam-3952	360	28	l	l	NOUN
ejpam-3952	360	29	)	)	PUNCT
ejpam-3952	361	1	+	+	CCONJ
ejpam-3952	361	2	lt	lt	NOUN
ejpam-3952	361	3	(	(	PUNCT
ejpam-3952	361	4	n	n	PROPN
ejpam-3952	361	5	+	+	CCONJ
ejpam-3952	361	6	1	1	NUM
ejpam-3952	361	7	,	,	PUNCT
ejpam-3952	361	8	l	l	NOUN
ejpam-3952	361	9	)	)	PUNCT
ejpam-3952	362	1	+	+	NUM
ejpam-3952	362	2	4nt	4nt	NOUN
ejpam-3952	362	3	(	(	PUNCT
ejpam-3952	362	4	l	l	NOUN
ejpam-3952	362	5	−	−	PROPN
ejpam-3952	362	6	1	1	NUM
ejpam-3952	362	7	)	)	PUNCT
ejpam-3952	362	8	as	as	SCONJ
ejpam-3952	362	9	required	require	VERB
ejpam-3952	362	10	.	.	PUNCT
ejpam-3952	363	1	from	from	ADP
ejpam-3952	363	2	the	the	DET
ejpam-3952	363	3	two	two	NUM
ejpam-3952	363	4	theorems	theorem	NOUN
ejpam-3952	363	5	above	above	ADV
ejpam-3952	363	6	,	,	PUNCT
ejpam-3952	363	7	the	the	DET
ejpam-3952	363	8	well	well	ADV
ejpam-3952	363	9	-	-	PUNCT
ejpam-3952	363	10	known	know	VERB
ejpam-3952	363	11	relation	relation	NOUN
ejpam-3952	363	12	t	t	NOUN
ejpam-3952	363	13	(	(	PUNCT
ejpam-3952	363	14	2n	2n	X
ejpam-3952	363	15	+	+	CCONJ
ejpam-3952	363	16	1	1	X
ejpam-3952	363	17	)	)	PUNCT
ejpam-3952	363	18	=	=	SYM
ejpam-3952	363	19	3	3	NUM
ejpam-3952	363	20	t	t	NOUN
ejpam-3952	363	21	(	(	PUNCT
ejpam-3952	363	22	n	n	CCONJ
ejpam-3952	363	23	)	)	PUNCT
ejpam-3952	364	1	+	+	CCONJ
ejpam-3952	364	2	t	t	PROPN
ejpam-3952	364	3	(	(	PUNCT
ejpam-3952	364	4	n	n	PROPN
ejpam-3952	364	5	+	+	CCONJ
ejpam-3952	364	6	1	1	NUM
ejpam-3952	364	7	)	)	PUNCT
ejpam-3952	364	8	derived	derive	VERB
ejpam-3952	364	9	from	from	ADP
ejpam-3952	364	10	[	[	X
ejpam-3952	364	11	9	9	NUM
ejpam-3952	364	12	,	,	PUNCT
ejpam-3952	364	13	equations	equation	NOUN
ejpam-3952	364	14	(	(	PUNCT
ejpam-3952	364	15	7.14	7.14	NUM
ejpam-3952	364	16	)	)	PUNCT
ejpam-3952	364	17	and	and	CCONJ
ejpam-3952	364	18	(	(	PUNCT
ejpam-3952	364	19	7.22	7.22	NUM
ejpam-3952	364	20	)	)	PUNCT
ejpam-3952	364	21	]	]	PUNCT
ejpam-3952	364	22	can	can	AUX
ejpam-3952	364	23	be	be	AUX
ejpam-3952	364	24	obtained	obtain	VERB
ejpam-3952	364	25	by	by	ADP
ejpam-3952	364	26	setting	set	VERB
ejpam-3952	364	27	l	l	NOUN
ejpam-3952	364	28	=	=	SYM
ejpam-3952	364	29	1	1	X
ejpam-3952	364	30	.	.	PUNCT
ejpam-3952	364	31	j.	j.	PROPN
ejpam-3952	364	32	punpim	punpim	PROPN
ejpam-3952	364	33	,	,	PUNCT
ejpam-3952	364	34	s.	s.	PROPN
ejpam-3952	364	35	jitman	jitman	PROPN
ejpam-3952	364	36	/	/	SYM
ejpam-3952	364	37	eur	eur	PROPN
ejpam-3952	364	38	.	.	PUNCT
ejpam-3952	365	1	j.	j.	PROPN
ejpam-3952	365	2	pure	pure	PROPN
ejpam-3952	365	3	appl	appl	PROPN
ejpam-3952	365	4	.	.	PROPN
ejpam-3952	365	5	math	math	PROPN
ejpam-3952	365	6	,	,	PUNCT
ejpam-3952	365	7	14	14	NUM
ejpam-3952	365	8	(	(	PUNCT
ejpam-3952	365	9	2	2	NUM
ejpam-3952	365	10	)	)	PUNCT
ejpam-3952	365	11	(	(	PUNCT
ejpam-3952	365	12	2021	2021	NUM
ejpam-3952	365	13	)	)	PUNCT
ejpam-3952	365	14	,	,	PUNCT
ejpam-3952	365	15	380	380	NUM
ejpam-3952	365	16	-	-	SYM
ejpam-3952	365	17	395	395	NUM
ejpam-3952	365	18	392	392	NUM
ejpam-3952	365	19	theorem	theorem	VERB
ejpam-3952	365	20	13	13	NUM
ejpam-3952	365	21	.	.	PUNCT
ejpam-3952	366	1	n2	n2	PROPN
ejpam-3952	366	2	t	t	PROPN
ejpam-3952	366	3	(	(	PUNCT
ejpam-3952	366	4	k	k	PROPN
ejpam-3952	366	5	−	−	PROPN
ejpam-3952	366	6	1	1	NUM
ejpam-3952	366	7	,	,	PUNCT
ejpam-3952	366	8	l)+kt	l)+kt	X
ejpam-3952	366	9	(	(	PUNCT
ejpam-3952	366	10	n	n	X
ejpam-3952	366	11	,	,	PUNCT
ejpam-3952	366	12	l	l	NOUN
ejpam-3952	366	13	)	)	PUNCT
ejpam-3952	366	14	=	=	SYM
ejpam-3952	366	15	t	t	PROPN
ejpam-3952	366	16	(	(	PUNCT
ejpam-3952	366	17	nk	nk	PROPN
ejpam-3952	366	18	,	,	PUNCT
ejpam-3952	366	19	l)−n2(k−1)(l−1	l)−n2(k−1)(l−1	NOUN
ejpam-3952	366	20	)	)	PUNCT
ejpam-3952	366	21	for	for	ADP
ejpam-3952	366	22	all	all	DET
ejpam-3952	366	23	positive	positive	ADJ
ejpam-3952	366	24	integers	integer	NOUN
ejpam-3952	366	25	n	n	CCONJ
ejpam-3952	366	26	,	,	PUNCT
ejpam-3952	366	27	k	k	NOUN
ejpam-3952	366	28	,	,	PUNCT
ejpam-3952	366	29	and	and	CCONJ
ejpam-3952	366	30	l.	l.	PROPN
ejpam-3952	366	31	proof	proof	NOUN
ejpam-3952	366	32	.	.	PUNCT
ejpam-3952	367	1	let	let	VERB
ejpam-3952	367	2	n	n	CCONJ
ejpam-3952	367	3	,	,	PUNCT
ejpam-3952	367	4	k	k	NOUN
ejpam-3952	367	5	,	,	PUNCT
ejpam-3952	367	6	and	and	CCONJ
ejpam-3952	367	7	l	l	NOUN
ejpam-3952	367	8	be	be	AUX
ejpam-3952	367	9	positive	positive	ADJ
ejpam-3952	367	10	integers	integer	NOUN
ejpam-3952	367	11	.	.	PUNCT
ejpam-3952	368	1	then	then	ADV
ejpam-3952	368	2	n2	n2	PROPN
ejpam-3952	368	3	t	t	PROPN
ejpam-3952	368	4	(	(	PUNCT
ejpam-3952	368	5	k	k	PROPN
ejpam-3952	368	6	−	−	PROPN
ejpam-3952	368	7	1	1	NUM
ejpam-3952	368	8	,	,	PUNCT
ejpam-3952	368	9	l	l	NOUN
ejpam-3952	368	10	)	)	PUNCT
ejpam-3952	369	1	+	+	CCONJ
ejpam-3952	369	2	kt	kt	PROPN
ejpam-3952	369	3	(	(	PUNCT
ejpam-3952	369	4	n	n	CCONJ
ejpam-3952	369	5	,	,	PUNCT
ejpam-3952	369	6	l	l	NOUN
ejpam-3952	369	7	)	)	PUNCT
ejpam-3952	369	8	=	=	SYM
ejpam-3952	369	9	n2	n2	NOUN
ejpam-3952	369	10	(	(	PUNCT
ejpam-3952	369	11	k	k	NOUN
ejpam-3952	369	12	−	−	PROPN
ejpam-3952	369	13	1	1	NUM
ejpam-3952	370	1	+	+	CCONJ
ejpam-3952	370	2	(	(	PUNCT
ejpam-3952	370	3	k	k	PROPN
ejpam-3952	370	4	−	−	PROPN
ejpam-3952	370	5	1)(k	1)(k	NUM
ejpam-3952	370	6	−	−	PROPN
ejpam-3952	370	7	2)l	2)l	PROPN
ejpam-3952	370	8	2	2	NUM
ejpam-3952	370	9	)	)	PUNCT
ejpam-3952	371	1	+	+	CCONJ
ejpam-3952	371	2	k	k	X
ejpam-3952	371	3	(	(	PUNCT
ejpam-3952	371	4	n	n	X
ejpam-3952	371	5	+	+	CCONJ
ejpam-3952	371	6	n(n−	n(n−	PROPN
ejpam-3952	371	7	1)l	1)l	NUM
ejpam-3952	371	8	2	2	NUM
ejpam-3952	371	9	)	)	PUNCT
ejpam-3952	372	1	=	=	SYM
ejpam-3952	372	2	2n2k	2n2k	NOUN
ejpam-3952	373	1	−	−	ADP
ejpam-3952	373	2	2n2	2n2	NUM
ejpam-3952	374	1	+	+	CCONJ
ejpam-3952	374	2	n2k2l	n2k2l	NUM
ejpam-3952	374	3	−	−	PROPN
ejpam-3952	374	4	3n2kl	3n2kl	NUM
ejpam-3952	374	5	+	+	SYM
ejpam-3952	374	6	2n2l	2n2l	NUM
ejpam-3952	375	1	+	+	CCONJ
ejpam-3952	375	2	2nk	2nk	ADJ
ejpam-3952	375	3	+	+	CCONJ
ejpam-3952	375	4	n2kl	n2kl	NOUN
ejpam-3952	375	5	−	−	NOUN
ejpam-3952	375	6	nkl	nkl	ADV
ejpam-3952	375	7	2	2	NUM
ejpam-3952	375	8	=	=	NOUN
ejpam-3952	375	9	2nk	2nk	NOUN
ejpam-3952	375	10	+	+	CCONJ
ejpam-3952	375	11	n2k2l	n2k2l	NUM
ejpam-3952	375	12	−	−	NOUN
ejpam-3952	375	13	nkl	nkl	ADP
ejpam-3952	375	14	−	−	PROPN
ejpam-3952	375	15	2n2kl	2n2kl	NUM
ejpam-3952	375	16	+	+	CCONJ
ejpam-3952	375	17	2n2k	2n2k	NOUN
ejpam-3952	376	1	+	+	CCONJ
ejpam-3952	376	2	2n2l	2n2l	NUM
ejpam-3952	377	1	−	−	ADP
ejpam-3952	377	2	2n2	2n2	NUM
ejpam-3952	378	1	2	2	NUM
ejpam-3952	378	2	=	=	NOUN
ejpam-3952	378	3	2nk	2nk	NOUN
ejpam-3952	378	4	+	+	CCONJ
ejpam-3952	378	5	n2k2l	n2k2l	NUM
ejpam-3952	378	6	−	−	NOUN
ejpam-3952	379	1	nkl	nkl	ADJ
ejpam-3952	379	2	2	2	NUM
ejpam-3952	379	3	−	−	NOUN
ejpam-3952	379	4	n2kl	n2kl	NOUN
ejpam-3952	379	5	+	+	CCONJ
ejpam-3952	379	6	n2k	n2k	CCONJ
ejpam-3952	379	7	+	+	CCONJ
ejpam-3952	379	8	n2l	n2l	NOUN
ejpam-3952	379	9	−	−	NOUN
ejpam-3952	379	10	n2	n2	NOUN
ejpam-3952	379	11	=	=	PROPN
ejpam-3952	379	12	nk	nk	PROPN
ejpam-3952	380	1	+	+	NUM
ejpam-3952	380	2	nk(nk	nk(nk	NOUN
ejpam-3952	380	3	−	−	PROPN
ejpam-3952	380	4	1)l	1)l	NUM
ejpam-3952	380	5	2	2	NUM
ejpam-3952	380	6	−	−	NOUN
ejpam-3952	380	7	n2(k	n2(k	NOUN
ejpam-3952	380	8	−	−	PROPN
ejpam-3952	380	9	1)(l	1)(l	NUM
ejpam-3952	380	10	−	−	NOUN
ejpam-3952	380	11	1	1	NUM
ejpam-3952	380	12	)	)	PUNCT
ejpam-3952	380	13	=	=	SYM
ejpam-3952	380	14	t	t	PROPN
ejpam-3952	380	15	(	(	PUNCT
ejpam-3952	380	16	nk	nk	PROPN
ejpam-3952	380	17	,	,	PUNCT
ejpam-3952	380	18	l)−	l)−	PROPN
ejpam-3952	380	19	n2(k	n2(k	PROPN
ejpam-3952	381	1	−	−	PROPN
ejpam-3952	381	2	1)(l	1)(l	NUM
ejpam-3952	381	3	−	−	NOUN
ejpam-3952	381	4	1	1	NUM
ejpam-3952	381	5	)	)	PUNCT
ejpam-3952	381	6	.	.	PUNCT
ejpam-3952	382	1	therefore	therefore	ADV
ejpam-3952	382	2	,	,	PUNCT
ejpam-3952	382	3	we	we	PRON
ejpam-3952	382	4	have	have	VERB
ejpam-3952	382	5	n2	n2	PROPN
ejpam-3952	382	6	t	t	PROPN
ejpam-3952	382	7	(	(	PUNCT
ejpam-3952	382	8	k	k	PROPN
ejpam-3952	382	9	−	−	PROPN
ejpam-3952	382	10	1	1	NUM
ejpam-3952	382	11	,	,	PUNCT
ejpam-3952	382	12	l	l	NOUN
ejpam-3952	382	13	)	)	PUNCT
ejpam-3952	383	1	+	+	CCONJ
ejpam-3952	383	2	kt	kt	PROPN
ejpam-3952	383	3	(	(	PUNCT
ejpam-3952	383	4	n	n	CCONJ
ejpam-3952	383	5	,	,	PUNCT
ejpam-3952	383	6	l	l	NOUN
ejpam-3952	383	7	)	)	PUNCT
ejpam-3952	383	8	=	=	SYM
ejpam-3952	383	9	t	t	PROPN
ejpam-3952	383	10	(	(	PUNCT
ejpam-3952	383	11	nk	nk	PROPN
ejpam-3952	383	12	,	,	PUNCT
ejpam-3952	383	13	l)−	l)−	PROPN
ejpam-3952	383	14	n2(k	n2(k	PROPN
ejpam-3952	383	15	−	−	PROPN
ejpam-3952	383	16	1)(l	1)(l	NUM
ejpam-3952	383	17	−	−	NOUN
ejpam-3952	383	18	1	1	NUM
ejpam-3952	383	19	)	)	PUNCT
ejpam-3952	383	20	as	as	SCONJ
ejpam-3952	383	21	desired	desire	VERB
ejpam-3952	383	22	.	.	PUNCT
ejpam-3952	384	1	by	by	ADP
ejpam-3952	384	2	setting	set	VERB
ejpam-3952	384	3	l	l	NOUN
ejpam-3952	384	4	=	=	SYM
ejpam-3952	384	5	1	1	NUM
ejpam-3952	384	6	,	,	PUNCT
ejpam-3952	384	7	the	the	DET
ejpam-3952	384	8	identity	identity	NOUN
ejpam-3952	384	9	n2	n2	NOUN
ejpam-3952	384	10	t	t	PROPN
ejpam-3952	384	11	(	(	PUNCT
ejpam-3952	384	12	k	k	NOUN
ejpam-3952	384	13	−	−	PROPN
ejpam-3952	384	14	1	1	NUM
ejpam-3952	384	15	)	)	PUNCT
ejpam-3952	384	16	+	+	CCONJ
ejpam-3952	384	17	kt	kt	PROPN
ejpam-3952	384	18	(	(	PUNCT
ejpam-3952	384	19	n	n	CCONJ
ejpam-3952	384	20	)	)	PUNCT
ejpam-3952	384	21	=	=	SYM
ejpam-3952	384	22	t	t	PROPN
ejpam-3952	384	23	(	(	PUNCT
ejpam-3952	384	24	nk	nk	PROPN
ejpam-3952	384	25	)	)	PUNCT
ejpam-3952	384	26	for	for	ADP
ejpam-3952	384	27	triangular	triangular	NOUN
ejpam-3952	384	28	numbers	number	NOUN
ejpam-3952	384	29	follows	follow	VERB
ejpam-3952	384	30	.	.	PUNCT
ejpam-3952	385	1	theorem	theorem	ADJ
ejpam-3952	385	2	14	14	NUM
ejpam-3952	385	3	.	.	PUNCT
ejpam-3952	386	1	n2	n2	PROPN
ejpam-3952	386	2	t	t	PROPN
ejpam-3952	386	3	(	(	PUNCT
ejpam-3952	386	4	k	k	PROPN
ejpam-3952	386	5	−	−	PROPN
ejpam-3952	386	6	1	1	NUM
ejpam-3952	386	7	,	,	PUNCT
ejpam-3952	386	8	l	l	NOUN
ejpam-3952	386	9	)	)	PUNCT
ejpam-3952	387	1	+	+	CCONJ
ejpam-3952	387	2	kt	kt	PROPN
ejpam-3952	387	3	(	(	PUNCT
ejpam-3952	387	4	n−	n−	NOUN
ejpam-3952	387	5	1	1	NUM
ejpam-3952	387	6	,	,	PUNCT
ejpam-3952	387	7	l	l	NOUN
ejpam-3952	387	8	)	)	PUNCT
ejpam-3952	387	9	=	=	SYM
ejpam-3952	387	10	t	t	PROPN
ejpam-3952	387	11	(	(	PUNCT
ejpam-3952	387	12	nk	nk	PROPN
ejpam-3952	387	13	−	−	PROPN
ejpam-3952	387	14	1	1	NUM
ejpam-3952	387	15	,	,	PUNCT
ejpam-3952	387	16	l)−	l)−	PROPN
ejpam-3952	387	17	(	(	PUNCT
ejpam-3952	387	18	n2	n2	PROPN
ejpam-3952	387	19	−	−	PROPN
ejpam-3952	387	20	1)(k	1)(k	NUM
ejpam-3952	388	1	−	−	NUM
ejpam-3952	388	2	1)(l	1)(l	NUM
ejpam-3952	388	3	−	−	NOUN
ejpam-3952	388	4	1	1	NUM
ejpam-3952	388	5	)	)	PUNCT
ejpam-3952	388	6	for	for	ADP
ejpam-3952	388	7	all	all	DET
ejpam-3952	388	8	positive	positive	ADJ
ejpam-3952	388	9	integers	integer	NOUN
ejpam-3952	388	10	n	n	CCONJ
ejpam-3952	388	11	,	,	PUNCT
ejpam-3952	388	12	k	k	NOUN
ejpam-3952	388	13	,	,	PUNCT
ejpam-3952	388	14	and	and	CCONJ
ejpam-3952	388	15	l.	l.	PROPN
ejpam-3952	388	16	proof	proof	NOUN
ejpam-3952	388	17	.	.	PUNCT
ejpam-3952	389	1	let	let	VERB
ejpam-3952	389	2	n	n	CCONJ
ejpam-3952	389	3	,	,	PUNCT
ejpam-3952	389	4	k	k	NOUN
ejpam-3952	389	5	,	,	PUNCT
ejpam-3952	389	6	and	and	CCONJ
ejpam-3952	389	7	l	l	NOUN
ejpam-3952	389	8	be	be	AUX
ejpam-3952	389	9	positive	positive	ADJ
ejpam-3952	389	10	integers	integer	NOUN
ejpam-3952	389	11	.	.	PUNCT
ejpam-3952	390	1	then	then	ADV
ejpam-3952	390	2	n2	n2	PROPN
ejpam-3952	390	3	t	t	PROPN
ejpam-3952	390	4	(	(	PUNCT
ejpam-3952	390	5	k	k	PROPN
ejpam-3952	390	6	−	−	PROPN
ejpam-3952	390	7	1	1	NUM
ejpam-3952	390	8	,	,	PUNCT
ejpam-3952	390	9	l	l	NOUN
ejpam-3952	390	10	)	)	PUNCT
ejpam-3952	391	1	+	+	CCONJ
ejpam-3952	391	2	kt	kt	PROPN
ejpam-3952	391	3	(	(	PUNCT
ejpam-3952	391	4	n−	n−	NOUN
ejpam-3952	391	5	1	1	NUM
ejpam-3952	391	6	,	,	PUNCT
ejpam-3952	391	7	l	l	NOUN
ejpam-3952	391	8	)	)	PUNCT
ejpam-3952	391	9	=	=	SYM
ejpam-3952	391	10	n2	n2	NOUN
ejpam-3952	391	11	(	(	PUNCT
ejpam-3952	391	12	k	k	NOUN
ejpam-3952	391	13	−	−	PROPN
ejpam-3952	391	14	1	1	NUM
ejpam-3952	392	1	+	+	CCONJ
ejpam-3952	392	2	(	(	PUNCT
ejpam-3952	392	3	k	k	PROPN
ejpam-3952	392	4	−	−	PROPN
ejpam-3952	392	5	1)(k	1)(k	NUM
ejpam-3952	392	6	−	−	PROPN
ejpam-3952	392	7	2)l	2)l	PROPN
ejpam-3952	392	8	2	2	NUM
ejpam-3952	392	9	)	)	PUNCT
ejpam-3952	393	1	+	+	CCONJ
ejpam-3952	393	2	k	k	X
ejpam-3952	393	3	(	(	PUNCT
ejpam-3952	393	4	n−	n−	NOUN
ejpam-3952	393	5	1	1	NUM
ejpam-3952	393	6	+	+	CCONJ
ejpam-3952	393	7	(	(	PUNCT
ejpam-3952	393	8	n−	n−	NOUN
ejpam-3952	393	9	1)(n−	1)(n−	NUM
ejpam-3952	393	10	2)l	2)l	NOUN
ejpam-3952	393	11	2	2	NUM
ejpam-3952	393	12	)	)	PUNCT
ejpam-3952	394	1	=	=	SYM
ejpam-3952	394	2	2n2k	2n2k	NOUN
ejpam-3952	395	1	−	−	ADP
ejpam-3952	395	2	2n2	2n2	NUM
ejpam-3952	396	1	+	+	CCONJ
ejpam-3952	396	2	n2k2l	n2k2l	NUM
ejpam-3952	396	3	−	−	PROPN
ejpam-3952	396	4	3n2kl	3n2kl	NUM
ejpam-3952	396	5	+	+	SYM
ejpam-3952	396	6	2n2l	2n2l	NUM
ejpam-3952	397	1	+	+	CCONJ
ejpam-3952	397	2	2nk	2nk	ADJ
ejpam-3952	397	3	−	−	PROPN
ejpam-3952	397	4	2k	2k	NOUN
ejpam-3952	397	5	+	+	CCONJ
ejpam-3952	397	6	n2kl	n2kl	NOUN
ejpam-3952	397	7	−	−	PROPN
ejpam-3952	398	1	3nkl	3nkl	NUM
ejpam-3952	398	2	+	+	CCONJ
ejpam-3952	398	3	2kl	2kl	ADJ
ejpam-3952	398	4	2	2	NUM
ejpam-3952	398	5	=	=	NOUN
ejpam-3952	398	6	2nk	2nk	ADJ
ejpam-3952	398	7	−	−	PROPN
ejpam-3952	398	8	2	2	NUM
ejpam-3952	398	9	+	+	CCONJ
ejpam-3952	398	10	n2k2l	n2k2l	NUM
ejpam-3952	398	11	−	−	ADP
ejpam-3952	399	1	3nkl	3nkl	NUM
ejpam-3952	399	2	+	+	NUM
ejpam-3952	399	3	2l	2l	NUM
ejpam-3952	399	4	−	−	PROPN
ejpam-3952	399	5	2n2kl	2n2kl	NUM
ejpam-3952	399	6	+	+	SYM
ejpam-3952	399	7	2n2l	2n2l	NUM
ejpam-3952	400	1	+	+	CCONJ
ejpam-3952	400	2	2kl	2kl	ADJ
ejpam-3952	400	3	−	−	NOUN
ejpam-3952	400	4	2l	2l	NOUN
ejpam-3952	400	5	+	+	CCONJ
ejpam-3952	400	6	2nk	2nk	ADJ
ejpam-3952	400	7	−	−	NOUN
ejpam-3952	400	8	2n2	2n2	NUM
ejpam-3952	401	1	−	−	NOUN
ejpam-3952	401	2	2k	2k	NOUN
ejpam-3952	401	3	+	+	CCONJ
ejpam-3952	401	4	2	2	NUM
ejpam-3952	401	5	2	2	NUM
ejpam-3952	401	6	=	=	NOUN
ejpam-3952	401	7	2nk	2nk	ADJ
ejpam-3952	401	8	−	−	PROPN
ejpam-3952	401	9	2	2	NUM
ejpam-3952	401	10	+	+	CCONJ
ejpam-3952	401	11	n2k2l	n2k2l	NUM
ejpam-3952	401	12	−	−	ADP
ejpam-3952	402	1	3nkl	3nkl	NUM
ejpam-3952	402	2	+	+	NUM
ejpam-3952	402	3	2l	2l	NUM
ejpam-3952	402	4	2	2	NUM
ejpam-3952	402	5	−	−	NOUN
ejpam-3952	402	6	(	(	PUNCT
ejpam-3952	402	7	n2kl	n2kl	NOUN
ejpam-3952	402	8	−	−	PROPN
ejpam-3952	402	9	n2l	n2l	NOUN
ejpam-3952	402	10	−	−	PROPN
ejpam-3952	402	11	kl	kl	NOUN
ejpam-3952	402	12	+	+	CCONJ
ejpam-3952	402	13	l	l	NOUN
ejpam-3952	402	14	−	−	X
ejpam-3952	403	1	n2k	n2k	NOUN
ejpam-3952	403	2	+	+	CCONJ
ejpam-3952	403	3	n2	n2	ADJ
ejpam-3952	403	4	+	+	CCONJ
ejpam-3952	403	5	k	k	X
ejpam-3952	403	6	−	−	NOUN
ejpam-3952	403	7	1	1	NUM
ejpam-3952	403	8	)	)	PUNCT
ejpam-3952	403	9	=	=	SYM
ejpam-3952	404	1	(	(	PUNCT
ejpam-3952	404	2	nk	nk	INTJ
ejpam-3952	404	3	−	−	PROPN
ejpam-3952	404	4	1	1	NUM
ejpam-3952	404	5	)	)	PUNCT
ejpam-3952	404	6	+	+	CCONJ
ejpam-3952	404	7	(	(	PUNCT
ejpam-3952	404	8	nk	nk	PROPN
ejpam-3952	404	9	−	−	PROPN
ejpam-3952	404	10	1)(nk	1)(nk	NUM
ejpam-3952	404	11	−	−	NUM
ejpam-3952	404	12	2)l	2)l	PROPN
ejpam-3952	404	13	2	2	NUM
ejpam-3952	404	14	−	−	NOUN
ejpam-3952	404	15	(	(	PUNCT
ejpam-3952	404	16	n2	n2	ADJ
ejpam-3952	404	17	−	−	PROPN
ejpam-3952	404	18	1)(k	1)(k	NUM
ejpam-3952	404	19	−	−	NUM
ejpam-3952	404	20	1)(l	1)(l	NUM
ejpam-3952	404	21	−	−	NOUN
ejpam-3952	404	22	1	1	NUM
ejpam-3952	404	23	)	)	PUNCT
ejpam-3952	404	24	=	=	SYM
ejpam-3952	404	25	t	t	PROPN
ejpam-3952	404	26	(	(	PUNCT
ejpam-3952	404	27	nk	nk	PROPN
ejpam-3952	404	28	−	−	PROPN
ejpam-3952	404	29	1	1	NUM
ejpam-3952	404	30	,	,	PUNCT
ejpam-3952	404	31	l)−	l)−	PROPN
ejpam-3952	404	32	(	(	PUNCT
ejpam-3952	404	33	n2	n2	PROPN
ejpam-3952	404	34	−	−	PROPN
ejpam-3952	404	35	1)(k	1)(k	NUM
ejpam-3952	404	36	−	−	NUM
ejpam-3952	404	37	1)(l	1)(l	NUM
ejpam-3952	404	38	−	−	NOUN
ejpam-3952	404	39	1	1	NUM
ejpam-3952	404	40	)	)	PUNCT
ejpam-3952	404	41	as	as	SCONJ
ejpam-3952	404	42	desired	desire	VERB
ejpam-3952	404	43	.	.	PUNCT
ejpam-3952	405	1	for	for	ADP
ejpam-3952	405	2	l	l	NOUN
ejpam-3952	405	3	=	=	SYM
ejpam-3952	405	4	1	1	NUM
ejpam-3952	405	5	,	,	PUNCT
ejpam-3952	405	6	we	we	PRON
ejpam-3952	405	7	have	have	VERB
ejpam-3952	405	8	the	the	DET
ejpam-3952	405	9	identity	identity	NOUN
ejpam-3952	405	10	n2	n2	NOUN
ejpam-3952	405	11	t	t	PROPN
ejpam-3952	405	12	(	(	PUNCT
ejpam-3952	405	13	k	k	NOUN
ejpam-3952	405	14	−	−	PROPN
ejpam-3952	405	15	1	1	NUM
ejpam-3952	405	16	)	)	PUNCT
ejpam-3952	406	1	+	+	CCONJ
ejpam-3952	406	2	kt	kt	PROPN
ejpam-3952	406	3	(	(	PUNCT
ejpam-3952	406	4	n−	n−	NOUN
ejpam-3952	406	5	1	1	NUM
ejpam-3952	406	6	)	)	PUNCT
ejpam-3952	406	7	=	=	SYM
ejpam-3952	406	8	t	t	PROPN
ejpam-3952	406	9	(	(	PUNCT
ejpam-3952	406	10	nk	nk	PROPN
ejpam-3952	406	11	−	−	PROPN
ejpam-3952	406	12	1	1	NUM
ejpam-3952	406	13	)	)	PUNCT
ejpam-3952	406	14	for	for	ADP
ejpam-3952	406	15	all	all	DET
ejpam-3952	406	16	positive	positive	ADJ
ejpam-3952	406	17	integers	integer	NOUN
ejpam-3952	406	18	k	k	PROPN
ejpam-3952	406	19	and	and	CCONJ
ejpam-3952	406	20	n.	n.	NOUN
ejpam-3952	406	21	in	in	ADP
ejpam-3952	406	22	the	the	DET
ejpam-3952	406	23	following	follow	VERB
ejpam-3952	406	24	theorems	theorem	NOUN
ejpam-3952	406	25	,	,	PUNCT
ejpam-3952	406	26	some	some	DET
ejpam-3952	406	27	identities	identity	NOUN
ejpam-3952	406	28	concerning	concern	VERB
ejpam-3952	406	29	squares	square	NOUN
ejpam-3952	406	30	of	of	ADP
ejpam-3952	406	31	l	l	NOUN
ejpam-3952	406	32	-	-	PUNCT
ejpam-3952	406	33	isosceles	isoscele	NOUN
ejpam-3952	406	34	triangular	triangular	NOUN
ejpam-3952	406	35	numbers	number	NOUN
ejpam-3952	406	36	are	be	AUX
ejpam-3952	406	37	presented	present	VERB
ejpam-3952	406	38	.	.	PUNCT
ejpam-3952	407	1	j.	j.	PROPN
ejpam-3952	407	2	punpim	punpim	PROPN
ejpam-3952	407	3	,	,	PUNCT
ejpam-3952	407	4	s.	s.	PROPN
ejpam-3952	407	5	jitman	jitman	PROPN
ejpam-3952	407	6	/	/	SYM
ejpam-3952	407	7	eur	eur	PROPN
ejpam-3952	407	8	.	.	PUNCT
ejpam-3952	408	1	j.	j.	PROPN
ejpam-3952	408	2	pure	pure	PROPN
ejpam-3952	408	3	appl	appl	PROPN
ejpam-3952	408	4	.	.	PROPN
ejpam-3952	408	5	math	math	PROPN
ejpam-3952	408	6	,	,	PUNCT
ejpam-3952	408	7	14	14	NUM
ejpam-3952	408	8	(	(	PUNCT
ejpam-3952	408	9	2	2	NUM
ejpam-3952	408	10	)	)	PUNCT
ejpam-3952	408	11	(	(	PUNCT
ejpam-3952	408	12	2021	2021	NUM
ejpam-3952	408	13	)	)	PUNCT
ejpam-3952	408	14	,	,	PUNCT
ejpam-3952	408	15	380	380	NUM
ejpam-3952	408	16	-	-	SYM
ejpam-3952	408	17	395	395	NUM
ejpam-3952	408	18	393	393	NUM
ejpam-3952	408	19	theorem	theorem	NOUN
ejpam-3952	408	20	15	15	NUM
ejpam-3952	408	21	.	.	PUNCT
ejpam-3952	409	1	t	t	PROPN
ejpam-3952	409	2	(	(	PUNCT
ejpam-3952	409	3	n	n	PROPN
ejpam-3952	409	4	+	+	CCONJ
ejpam-3952	409	5	1	1	NUM
ejpam-3952	409	6	,	,	PUNCT
ejpam-3952	409	7	l)2	l)2	PROPN
ejpam-3952	409	8	−	−	PROPN
ejpam-3952	409	9	t	t	NOUN
ejpam-3952	409	10	(	(	PUNCT
ejpam-3952	409	11	n	n	CCONJ
ejpam-3952	409	12	,	,	PUNCT
ejpam-3952	409	13	l)2	l)2	NOUN
ejpam-3952	409	14	=	=	PUNCT
ejpam-3952	409	15	(	(	PUNCT
ejpam-3952	409	16	n	n	PROPN
ejpam-3952	409	17	+	+	CCONJ
ejpam-3952	409	18	1)3	1)3	PROPN
ejpam-3952	409	19	+	+	CCONJ
ejpam-3952	409	20	(	(	PUNCT
ejpam-3952	409	21	l	l	NOUN
ejpam-3952	409	22	−	−	PROPN
ejpam-3952	409	23	1)n	1)n	NUM
ejpam-3952	409	24	(	(	PUNCT
ejpam-3952	409	25	(	(	PUNCT
ejpam-3952	409	26	l	l	NOUN
ejpam-3952	409	27	+	+	NOUN
ejpam-3952	409	28	1)n2	1)n2	NUM
ejpam-3952	409	29	+	+	NUM
ejpam-3952	409	30	3n	3n	NUM
ejpam-3952	409	31	+	+	SYM
ejpam-3952	409	32	1	1	NUM
ejpam-3952	409	33	)	)	PUNCT
ejpam-3952	409	34	for	for	ADP
ejpam-3952	409	35	all	all	DET
ejpam-3952	409	36	positive	positive	ADJ
ejpam-3952	409	37	integers	integer	NOUN
ejpam-3952	409	38	l	l	NOUN
ejpam-3952	409	39	and	and	CCONJ
ejpam-3952	409	40	n.	n.	PROPN
ejpam-3952	409	41	proof	proof	NOUN
ejpam-3952	409	42	.	.	PUNCT
ejpam-3952	410	1	let	let	VERB
ejpam-3952	410	2	l	l	NOUN
ejpam-3952	410	3	and	and	CCONJ
ejpam-3952	410	4	n	n	CCONJ
ejpam-3952	410	5	be	be	AUX
ejpam-3952	410	6	positive	positive	ADJ
ejpam-3952	410	7	integers	integer	NOUN
ejpam-3952	410	8	.	.	PUNCT
ejpam-3952	411	1	then	then	ADV
ejpam-3952	411	2	t	t	PROPN
ejpam-3952	411	3	(	(	PUNCT
ejpam-3952	411	4	n	n	PROPN
ejpam-3952	411	5	+	+	CCONJ
ejpam-3952	411	6	1	1	NUM
ejpam-3952	411	7	,	,	PUNCT
ejpam-3952	411	8	l)2	l)2	PROPN
ejpam-3952	411	9	−	−	PROPN
ejpam-3952	411	10	t	t	NOUN
ejpam-3952	411	11	(	(	PUNCT
ejpam-3952	411	12	n	n	CCONJ
ejpam-3952	411	13	,	,	PUNCT
ejpam-3952	411	14	l)2	l)2	NOUN
ejpam-3952	411	15	=	=	PUNCT
ejpam-3952	411	16	(	(	PUNCT
ejpam-3952	411	17	(	(	PUNCT
ejpam-3952	411	18	n	n	X
ejpam-3952	411	19	+	+	NUM
ejpam-3952	411	20	1	1	NUM
ejpam-3952	411	21	)	)	PUNCT
ejpam-3952	411	22	+	+	CCONJ
ejpam-3952	411	23	(	(	PUNCT
ejpam-3952	411	24	n	n	X
ejpam-3952	411	25	+	+	CCONJ
ejpam-3952	411	26	1)nl	1)nl	NUM
ejpam-3952	411	27	2	2	NUM
ejpam-3952	411	28	)	)	PUNCT
ejpam-3952	411	29	2	2	NUM
ejpam-3952	411	30	−	−	PROPN
ejpam-3952	411	31	(	(	PUNCT
ejpam-3952	411	32	n	n	PROPN
ejpam-3952	411	33	+	+	CCONJ
ejpam-3952	411	34	n(n−	n(n−	PROPN
ejpam-3952	411	35	1)l	1)l	NUM
ejpam-3952	411	36	2	2	NUM
ejpam-3952	411	37	)	)	SYM
ejpam-3952	411	38	2	2	NUM
ejpam-3952	411	39	=	=	SYM
ejpam-3952	411	40	4n2	4n2	PROPN
ejpam-3952	412	1	+	+	CCONJ
ejpam-3952	412	2	8n	8n	NOUN
ejpam-3952	412	3	+	+	CCONJ
ejpam-3952	412	4	4n3l	4n3l	NOUN
ejpam-3952	412	5	+	+	CCONJ
ejpam-3952	412	6	8n2l	8n2l	NUM
ejpam-3952	413	1	+	+	CCONJ
ejpam-3952	413	2	4n	4n	NOUN
ejpam-3952	413	3	+	+	CCONJ
ejpam-3952	413	4	4	4	NUM
ejpam-3952	413	5	+	+	NUM
ejpam-3952	413	6	4nl	4nl	NOUN
ejpam-3952	413	7	+	+	CCONJ
ejpam-3952	413	8	n4l2	n4l2	PROPN
ejpam-3952	413	9	+	+	CCONJ
ejpam-3952	413	10	2n3l2	2n3l2	NUM
ejpam-3952	413	11	+	+	NUM
ejpam-3952	413	12	n2l2	n2l2	PROPN
ejpam-3952	413	13	4	4	NUM
ejpam-3952	413	14	−	−	NUM
ejpam-3952	413	15	4n2	4n2	NUM
ejpam-3952	414	1	+	+	CCONJ
ejpam-3952	414	2	4n3l	4n3l	NOUN
ejpam-3952	414	3	−	−	NOUN
ejpam-3952	415	1	4n2l	4n2l	NOUN
ejpam-3952	416	1	+	+	CCONJ
ejpam-3952	417	1	n4l2	n4l2	ADV
ejpam-3952	417	2	−	−	PROPN
ejpam-3952	417	3	2n3l2	2n3l2	NUM
ejpam-3952	418	1	+	+	CCONJ
ejpam-3952	418	2	n2l2	n2l2	PROPN
ejpam-3952	419	1	4	4	NUM
ejpam-3952	419	2	=	=	NOUN
ejpam-3952	419	3	8n	8n	NOUN
ejpam-3952	419	4	+	+	CCONJ
ejpam-3952	419	5	12n2l	12n2l	NUM
ejpam-3952	419	6	+	+	CCONJ
ejpam-3952	419	7	4	4	NUM
ejpam-3952	419	8	+	+	NUM
ejpam-3952	419	9	4nl	4nl	NOUN
ejpam-3952	419	10	+	+	CCONJ
ejpam-3952	419	11	4n3l2	4n3l2	NOUN
ejpam-3952	419	12	4	4	NUM
ejpam-3952	419	13	=	=	SYM
ejpam-3952	419	14	2n	2n	NUM
ejpam-3952	420	1	+	+	CCONJ
ejpam-3952	420	2	3n2l	3n2l	NUM
ejpam-3952	420	3	+	+	CCONJ
ejpam-3952	420	4	1	1	NUM
ejpam-3952	420	5	+	+	NUM
ejpam-3952	420	6	nl	nl	NOUN
ejpam-3952	420	7	+	+	CCONJ
ejpam-3952	420	8	n3l2	n3l2	NOUN
ejpam-3952	420	9	=	=	SYM
ejpam-3952	420	10	(	(	PUNCT
ejpam-3952	420	11	n3	n3	NOUN
ejpam-3952	420	12	+	+	CCONJ
ejpam-3952	420	13	3n2	3n2	NUM
ejpam-3952	420	14	+	+	NUM
ejpam-3952	420	15	3n	3n	NUM
ejpam-3952	420	16	+	+	CCONJ
ejpam-3952	420	17	1	1	NUM
ejpam-3952	420	18	)	)	PUNCT
ejpam-3952	420	19	+	+	CCONJ
ejpam-3952	420	20	(	(	PUNCT
ejpam-3952	420	21	nl	nl	PROPN
ejpam-3952	420	22	−	−	PROPN
ejpam-3952	420	23	n)(n2l	n)(n2l	NUM
ejpam-3952	420	24	+	+	CCONJ
ejpam-3952	420	25	n2	n2	ADJ
ejpam-3952	420	26	+	+	CCONJ
ejpam-3952	420	27	3n	3n	NUM
ejpam-3952	420	28	+	+	SYM
ejpam-3952	420	29	1	1	NUM
ejpam-3952	420	30	)	)	PUNCT
ejpam-3952	420	31	=	=	SYM
ejpam-3952	420	32	(	(	PUNCT
ejpam-3952	420	33	n	n	PROPN
ejpam-3952	420	34	+	+	CCONJ
ejpam-3952	420	35	1)3	1)3	PROPN
ejpam-3952	420	36	+	+	CCONJ
ejpam-3952	420	37	(	(	PUNCT
ejpam-3952	420	38	l	l	NOUN
ejpam-3952	420	39	−	−	PROPN
ejpam-3952	420	40	1)n((l	1)n((l	NUM
ejpam-3952	421	1	+	+	CCONJ
ejpam-3952	421	2	1)n2	1)n2	NUM
ejpam-3952	421	3	+	+	NUM
ejpam-3952	421	4	3n	3n	NUM
ejpam-3952	421	5	+	+	SYM
ejpam-3952	421	6	1	1	NUM
ejpam-3952	421	7	)	)	PUNCT
ejpam-3952	421	8	.	.	PUNCT
ejpam-3952	422	1	hence	hence	ADV
ejpam-3952	422	2	,	,	PUNCT
ejpam-3952	422	3	t	t	PROPN
ejpam-3952	422	4	(	(	PUNCT
ejpam-3952	422	5	n	n	PROPN
ejpam-3952	422	6	+	+	CCONJ
ejpam-3952	422	7	1	1	NUM
ejpam-3952	422	8	,	,	PUNCT
ejpam-3952	422	9	l)2	l)2	PROPN
ejpam-3952	422	10	−	−	PROPN
ejpam-3952	422	11	t	t	NOUN
ejpam-3952	422	12	(	(	PUNCT
ejpam-3952	422	13	n	n	CCONJ
ejpam-3952	422	14	,	,	PUNCT
ejpam-3952	422	15	l)2	l)2	NOUN
ejpam-3952	422	16	=	=	PUNCT
ejpam-3952	422	17	(	(	PUNCT
ejpam-3952	422	18	n	n	PROPN
ejpam-3952	422	19	+	+	CCONJ
ejpam-3952	422	20	1)3	1)3	PROPN
ejpam-3952	422	21	+	+	CCONJ
ejpam-3952	422	22	(	(	PUNCT
ejpam-3952	422	23	l	l	NOUN
ejpam-3952	422	24	−	−	PROPN
ejpam-3952	422	25	1)n	1)n	NUM
ejpam-3952	422	26	(	(	PUNCT
ejpam-3952	422	27	(	(	PUNCT
ejpam-3952	422	28	l	l	NOUN
ejpam-3952	422	29	+	+	NOUN
ejpam-3952	422	30	1)n2	1)n2	NUM
ejpam-3952	422	31	+	+	NUM
ejpam-3952	422	32	3n	3n	NUM
ejpam-3952	422	33	+	+	SYM
ejpam-3952	422	34	1	1	NUM
ejpam-3952	422	35	)	)	PUNCT
ejpam-3952	422	36	as	as	SCONJ
ejpam-3952	422	37	desired	desire	VERB
ejpam-3952	422	38	.	.	PUNCT
ejpam-3952	423	1	the	the	DET
ejpam-3952	423	2	identity	identity	NOUN
ejpam-3952	423	3	t	t	NOUN
ejpam-3952	423	4	(	(	PUNCT
ejpam-3952	423	5	n	n	PROPN
ejpam-3952	423	6	+	+	NOUN
ejpam-3952	423	7	1)2−t	1)2−t	NUM
ejpam-3952	423	8	(	(	PUNCT
ejpam-3952	423	9	n)2	n)2	NOUN
ejpam-3952	423	10	=	=	PUNCT
ejpam-3952	423	11	(	(	PUNCT
ejpam-3952	423	12	n+1)3	n+1)3	PROPN
ejpam-3952	423	13	for	for	ADP
ejpam-3952	423	14	triangular	triangular	NOUN
ejpam-3952	423	15	numbers	number	NOUN
ejpam-3952	423	16	in	in	ADP
ejpam-3952	423	17	[	[	X
ejpam-3952	423	18	7	7	NUM
ejpam-3952	423	19	,	,	PUNCT
ejpam-3952	423	20	equation	equation	NOUN
ejpam-3952	423	21	(	(	PUNCT
ejpam-3952	423	22	1.5	1.5	NUM
ejpam-3952	423	23	)	)	PUNCT
ejpam-3952	423	24	]	]	PUNCT
ejpam-3952	423	25	is	be	AUX
ejpam-3952	423	26	a	a	DET
ejpam-3952	423	27	special	special	ADJ
ejpam-3952	423	28	case	case	NOUN
ejpam-3952	423	29	of	of	ADP
ejpam-3952	423	30	the	the	DET
ejpam-3952	423	31	above	above	ADJ
ejpam-3952	423	32	theorem	theorem	NOUN
ejpam-3952	423	33	where	where	SCONJ
ejpam-3952	423	34	l	l	NOUN
ejpam-3952	423	35	=	=	NOUN
ejpam-3952	423	36	1	1	X
ejpam-3952	423	37	.	.	PUNCT
ejpam-3952	423	38	theorem	theorem	VERB
ejpam-3952	423	39	16	16	NUM
ejpam-3952	423	40	.	.	PUNCT
ejpam-3952	424	1	t	t	PROPN
ejpam-3952	424	2	(	(	PUNCT
ejpam-3952	424	3	n	n	CCONJ
ejpam-3952	424	4	,	,	PUNCT
ejpam-3952	424	5	l)2	l)2	PROPN
ejpam-3952	424	6	+	+	NUM
ejpam-3952	424	7	t	t	PROPN
ejpam-3952	424	8	(	(	PUNCT
ejpam-3952	424	9	n−	n−	NOUN
ejpam-3952	424	10	1	1	NUM
ejpam-3952	424	11	,	,	PUNCT
ejpam-3952	424	12	l)2	l)2	NOUN
ejpam-3952	424	13	=	=	PUNCT
ejpam-3952	424	14	lt	lt	PROPN
ejpam-3952	424	15	(	(	PUNCT
ejpam-3952	424	16	n2	n2	ADJ
ejpam-3952	424	17	,	,	PUNCT
ejpam-3952	424	18	l)−	l)−	PROPN
ejpam-3952	424	19	(	(	PUNCT
ejpam-3952	424	20	2n2	2n2	NUM
ejpam-3952	424	21	−	−	NUM
ejpam-3952	424	22	2n	2n	NUM
ejpam-3952	424	23	+	+	CCONJ
ejpam-3952	424	24	1)(2	1)(2	NUM
ejpam-3952	424	25	t	t	NOUN
ejpam-3952	424	26	(	(	PUNCT
ejpam-3952	424	27	l	l	NOUN
ejpam-3952	424	28	,	,	PUNCT
ejpam-3952	424	29	n−	n−	PROPN
ejpam-3952	424	30	1)−	1)−	PROPN
ejpam-3952	424	31	(	(	PUNCT
ejpam-3952	424	32	l	l	NOUN
ejpam-3952	424	33	+	+	PROPN
ejpam-3952	424	34	1	1	NUM
ejpam-3952	424	35	)	)	PUNCT
ejpam-3952	424	36	)	)	PUNCT
ejpam-3952	424	37	for	for	ADP
ejpam-3952	424	38	all	all	DET
ejpam-3952	424	39	positive	positive	ADJ
ejpam-3952	424	40	integers	integer	NOUN
ejpam-3952	424	41	n	n	PRON
ejpam-3952	424	42	and	and	CCONJ
ejpam-3952	424	43	l.	l.	PROPN
ejpam-3952	424	44	proof	proof	NOUN
ejpam-3952	424	45	.	.	PUNCT
ejpam-3952	425	1	let	let	VERB
ejpam-3952	425	2	n	n	NOUN
ejpam-3952	425	3	and	and	CCONJ
ejpam-3952	425	4	l	l	NOUN
ejpam-3952	425	5	be	be	AUX
ejpam-3952	425	6	positive	positive	ADJ
ejpam-3952	425	7	integers	integer	NOUN
ejpam-3952	425	8	.	.	PUNCT
ejpam-3952	426	1	then	then	ADV
ejpam-3952	426	2	t	t	PROPN
ejpam-3952	426	3	(	(	PUNCT
ejpam-3952	426	4	n	n	CCONJ
ejpam-3952	426	5	,	,	PUNCT
ejpam-3952	426	6	l)2	l)2	PROPN
ejpam-3952	426	7	+	+	NUM
ejpam-3952	426	8	t	t	PROPN
ejpam-3952	426	9	(	(	PUNCT
ejpam-3952	426	10	n−	n−	NOUN
ejpam-3952	426	11	1	1	NUM
ejpam-3952	426	12	,	,	PUNCT
ejpam-3952	426	13	l)2	l)2	NOUN
ejpam-3952	426	14	=	=	PUNCT
ejpam-3952	426	15	(	(	PUNCT
ejpam-3952	426	16	n	n	PROPN
ejpam-3952	426	17	+	+	CCONJ
ejpam-3952	426	18	n(n−	n(n−	PROPN
ejpam-3952	426	19	1)l	1)l	NUM
ejpam-3952	426	20	2	2	NUM
ejpam-3952	426	21	)	)	SYM
ejpam-3952	426	22	2	2	NUM
ejpam-3952	426	23	+	+	CCONJ
ejpam-3952	426	24	(	(	PUNCT
ejpam-3952	426	25	n−	n−	NOUN
ejpam-3952	426	26	1	1	NUM
ejpam-3952	426	27	+	+	CCONJ
ejpam-3952	426	28	(	(	PUNCT
ejpam-3952	426	29	n−	n−	NOUN
ejpam-3952	426	30	1)(n−	1)(n−	NUM
ejpam-3952	426	31	2)l	2)l	NUM
ejpam-3952	426	32	2	2	NUM
ejpam-3952	426	33	)	)	PUNCT
ejpam-3952	426	34	2	2	NUM
ejpam-3952	426	35	=	=	SYM
ejpam-3952	426	36	(	(	PUNCT
ejpam-3952	426	37	2n	2n	X
ejpam-3952	426	38	+	+	CCONJ
ejpam-3952	426	39	n2l	n2l	PROPN
ejpam-3952	426	40	−	−	PROPN
ejpam-3952	426	41	nl	nl	NOUN
ejpam-3952	426	42	2	2	NUM
ejpam-3952	426	43	)	)	PUNCT
ejpam-3952	426	44	2	2	NUM
ejpam-3952	426	45	+	+	CCONJ
ejpam-3952	426	46	(	(	PUNCT
ejpam-3952	426	47	2n−	2n−	NUM
ejpam-3952	426	48	2	2	NUM
ejpam-3952	426	49	+	+	NUM
ejpam-3952	426	50	n2l	n2l	NOUN
ejpam-3952	426	51	−	−	NUM
ejpam-3952	426	52	3nl	3nl	NOUN
ejpam-3952	427	1	+	+	X
ejpam-3952	427	2	2l	2l	NUM
ejpam-3952	427	3	2	2	NUM
ejpam-3952	427	4	)	)	SYM
ejpam-3952	427	5	2	2	NUM
ejpam-3952	427	6	=	=	SYM
ejpam-3952	427	7	8n2	8n2	NUM
ejpam-3952	428	1	+	+	CCONJ
ejpam-3952	428	2	8n3l	8n3l	ADJ
ejpam-3952	428	3	−	−	NUM
ejpam-3952	428	4	20n2l	20n2l	NUM
ejpam-3952	429	1	+	+	CCONJ
ejpam-3952	430	1	2n4l2	2n4l2	NUM
ejpam-3952	430	2	−	−	NOUN
ejpam-3952	430	3	8n3l2	8n3l2	NUM
ejpam-3952	430	4	+	+	CCONJ
ejpam-3952	430	5	14n2l2	14n2l2	PROPN
ejpam-3952	430	6	+	+	CCONJ
ejpam-3952	430	7	20nl	20nl	ADJ
ejpam-3952	430	8	−	−	NOUN
ejpam-3952	430	9	8n	8n	NOUN
ejpam-3952	430	10	+	+	CCONJ
ejpam-3952	430	11	4−	4−	NOUN
ejpam-3952	430	12	8l	8l	NOUN
ejpam-3952	430	13	−	−	PROPN
ejpam-3952	430	14	12nl2	12nl2	NUM
ejpam-3952	431	1	+	+	CCONJ
ejpam-3952	431	2	4l2	4l2	NUM
ejpam-3952	431	3	4	4	NUM
ejpam-3952	431	4	=	=	SYM
ejpam-3952	431	5	4n2	4n2	PROPN
ejpam-3952	432	1	+	+	NUM
ejpam-3952	432	2	4n3l	4n3l	NOUN
ejpam-3952	432	3	−	−	NOUN
ejpam-3952	433	1	10n2l	10n2l	NUM
ejpam-3952	434	1	+	+	CCONJ
ejpam-3952	434	2	n4l2	n4l2	PROPN
ejpam-3952	434	3	−	−	PROPN
ejpam-3952	434	4	4n3l2	4n3l2	NOUN
ejpam-3952	435	1	+	+	CCONJ
ejpam-3952	435	2	7n2l2	7n2l2	PROPN
ejpam-3952	436	1	+	+	CCONJ
ejpam-3952	436	2	10nl	10nl	ADJ
ejpam-3952	436	3	−	−	NOUN
ejpam-3952	436	4	4n	4n	NOUN
ejpam-3952	436	5	+	+	CCONJ
ejpam-3952	436	6	2−	2−	NUM
ejpam-3952	436	7	4l	4l	NOUN
ejpam-3952	436	8	−	−	NOUN
ejpam-3952	436	9	6nl2	6nl2	NUM
ejpam-3952	436	10	+	+	CCONJ
ejpam-3952	436	11	2l2	2l2	NUM
ejpam-3952	436	12	2	2	NUM
ejpam-3952	436	13	=	=	SYM
ejpam-3952	436	14	(	(	PUNCT
ejpam-3952	436	15	2n2l	2n2l	NUM
ejpam-3952	436	16	+	+	CCONJ
ejpam-3952	437	1	n4l2	n4l2	PROPN
ejpam-3952	437	2	−	−	NOUN
ejpam-3952	437	3	n2l2	n2l2	NOUN
ejpam-3952	437	4	2	2	NUM
ejpam-3952	437	5	)	)	PUNCT
ejpam-3952	437	6	−	−	PROPN
ejpam-3952	437	7	(	(	PUNCT
ejpam-3952	437	8	2n2	2n2	NUM
ejpam-3952	437	9	−	−	NUM
ejpam-3952	437	10	2n	2n	NUM
ejpam-3952	438	1	+	+	CCONJ
ejpam-3952	438	2	1	1	X
ejpam-3952	438	3	)	)	PUNCT
ejpam-3952	438	4	(	(	PUNCT
ejpam-3952	438	5	4l	4l	NOUN
ejpam-3952	438	6	+	+	CCONJ
ejpam-3952	438	7	2nl2	2nl2	NUM
ejpam-3952	438	8	−	−	NUM
ejpam-3952	438	9	2l2	2l2	NUM
ejpam-3952	438	10	−	−	PROPN
ejpam-3952	438	11	2nl	2nl	NOUN
ejpam-3952	439	1	+	+	CCONJ
ejpam-3952	439	2	2l	2l	NUM
ejpam-3952	439	3	−	−	NUM
ejpam-3952	439	4	2l	2l	NOUN
ejpam-3952	439	5	−	−	PROPN
ejpam-3952	439	6	2	2	NUM
ejpam-3952	439	7	2	2	NUM
ejpam-3952	439	8	)	)	PUNCT
ejpam-3952	439	9	=	=	SYM
ejpam-3952	439	10	l	l	NOUN
ejpam-3952	439	11	(	(	PUNCT
ejpam-3952	439	12	n2	n2	NOUN
ejpam-3952	439	13	+	+	CCONJ
ejpam-3952	439	14	n2(n2	n2(n2	NUM
ejpam-3952	439	15	−	−	PROPN
ejpam-3952	439	16	1)l	1)l	NUM
ejpam-3952	439	17	2	2	NUM
ejpam-3952	439	18	)	)	PUNCT
ejpam-3952	439	19	−	−	PROPN
ejpam-3952	439	20	(	(	PUNCT
ejpam-3952	439	21	2n2	2n2	NUM
ejpam-3952	439	22	−	−	NUM
ejpam-3952	439	23	2n	2n	NUM
ejpam-3952	439	24	+	+	CCONJ
ejpam-3952	439	25	1	1	X
ejpam-3952	439	26	)	)	PUNCT
ejpam-3952	439	27	(	(	PUNCT
ejpam-3952	439	28	2(l	2(l	NUM
ejpam-3952	439	29	+	+	CCONJ
ejpam-3952	439	30	l(l	l(l	NOUN
ejpam-3952	439	31	−	−	NOUN
ejpam-3952	439	32	1)(n−	1)(n−	NUM
ejpam-3952	439	33	1	1	NUM
ejpam-3952	439	34	)	)	PUNCT
ejpam-3952	439	35	2	2	NUM
ejpam-3952	439	36	)	)	PUNCT
ejpam-3952	439	37	−	−	NOUN
ejpam-3952	439	38	l	l	NOUN
ejpam-3952	439	39	−	−	NOUN
ejpam-3952	439	40	1	1	NUM
ejpam-3952	439	41	)	)	PUNCT
ejpam-3952	439	42	=	=	SYM
ejpam-3952	439	43	lt	lt	PROPN
ejpam-3952	439	44	(	(	PUNCT
ejpam-3952	439	45	n2	n2	ADJ
ejpam-3952	439	46	,	,	PUNCT
ejpam-3952	439	47	l)−	l)−	PROPN
ejpam-3952	439	48	(	(	PUNCT
ejpam-3952	439	49	2n2	2n2	NUM
ejpam-3952	439	50	−	−	NUM
ejpam-3952	439	51	2n	2n	NUM
ejpam-3952	439	52	+	+	CCONJ
ejpam-3952	439	53	1)(2	1)(2	NUM
ejpam-3952	439	54	t	t	NOUN
ejpam-3952	439	55	(	(	PUNCT
ejpam-3952	439	56	l	l	NOUN
ejpam-3952	439	57	,	,	PUNCT
ejpam-3952	439	58	n−	n−	PROPN
ejpam-3952	439	59	1)−	1)−	PROPN
ejpam-3952	439	60	(	(	PUNCT
ejpam-3952	439	61	l	l	NOUN
ejpam-3952	439	62	+	+	PROPN
ejpam-3952	439	63	1	1	NUM
ejpam-3952	439	64	)	)	PUNCT
ejpam-3952	439	65	)	)	PUNCT
ejpam-3952	439	66	.	.	PUNCT
ejpam-3952	440	1	the	the	DET
ejpam-3952	440	2	proof	proof	NOUN
ejpam-3952	440	3	is	be	AUX
ejpam-3952	440	4	therefore	therefore	ADV
ejpam-3952	440	5	completed	complete	VERB
ejpam-3952	440	6	.	.	PUNCT
ejpam-3952	441	1	the	the	DET
ejpam-3952	441	2	identity	identity	NOUN
ejpam-3952	441	3	t	t	PROPN
ejpam-3952	441	4	(	(	PUNCT
ejpam-3952	441	5	n)2	n)2	PROPN
ejpam-3952	441	6	+	+	X
ejpam-3952	442	1	t	t	PROPN
ejpam-3952	443	1	(	(	PUNCT
ejpam-3952	443	2	n−	n−	NOUN
ejpam-3952	443	3	1)2	1)2	NUM
ejpam-3952	443	4	=	=	SYM
ejpam-3952	443	5	t	t	PROPN
ejpam-3952	443	6	(	(	PUNCT
ejpam-3952	443	7	n2	n2	NOUN
ejpam-3952	443	8	)	)	PUNCT
ejpam-3952	443	9	in	in	ADP
ejpam-3952	443	10	[	[	X
ejpam-3952	443	11	4	4	NUM
ejpam-3952	443	12	,	,	PUNCT
ejpam-3952	443	13	equation	equation	NOUN
ejpam-3952	443	14	(	(	PUNCT
ejpam-3952	443	15	19a	19a	NUM
ejpam-3952	443	16	)	)	PUNCT
ejpam-3952	443	17	]	]	PUNCT
ejpam-3952	443	18	is	be	AUX
ejpam-3952	443	19	a	a	DET
ejpam-3952	443	20	special	special	ADJ
ejpam-3952	443	21	case	case	NOUN
ejpam-3952	443	22	of	of	ADP
ejpam-3952	443	23	the	the	DET
ejpam-3952	443	24	theorem	theorem	NOUN
ejpam-3952	443	25	where	where	SCONJ
ejpam-3952	443	26	l	l	NOUN
ejpam-3952	443	27	=	=	NOUN
ejpam-3952	443	28	1	1	X
ejpam-3952	443	29	.	.	PUNCT
ejpam-3952	443	30	references	reference	NOUN
ejpam-3952	443	31	394	394	NUM
ejpam-3952	443	32	4	4	NUM
ejpam-3952	443	33	.	.	PUNCT
ejpam-3952	443	34	conclusion	conclusion	NOUN
ejpam-3952	443	35	and	and	CCONJ
ejpam-3952	443	36	remarks	remark	VERB
ejpam-3952	443	37	generalizations	generalization	NOUN
ejpam-3952	443	38	of	of	ADP
ejpam-3952	443	39	triangular	triangular	NOUN
ejpam-3952	443	40	numbers	number	NOUN
ejpam-3952	443	41	have	have	AUX
ejpam-3952	443	42	been	be	AUX
ejpam-3952	443	43	studied	study	VERB
ejpam-3952	443	44	in	in	ADP
ejpam-3952	443	45	terms	term	NOUN
ejpam-3952	443	46	of	of	ADP
ejpam-3952	443	47	isosceles	isoscele	NOUN
ejpam-3952	443	48	triangular	triangular	NOUN
ejpam-3952	443	49	numbers	number	NOUN
ejpam-3952	443	50	.	.	PUNCT
ejpam-3952	444	1	some	some	DET
ejpam-3952	444	2	characterizations	characterization	NOUN
ejpam-3952	444	3	of	of	ADP
ejpam-3952	444	4	such	such	ADJ
ejpam-3952	444	5	numbers	number	NOUN
ejpam-3952	444	6	are	be	AUX
ejpam-3952	444	7	given	give	VERB
ejpam-3952	444	8	as	as	ADV
ejpam-3952	444	9	well	well	ADV
ejpam-3952	444	10	as	as	ADP
ejpam-3952	444	11	links	link	NOUN
ejpam-3952	444	12	with	with	ADP
ejpam-3952	444	13	square	square	ADJ
ejpam-3952	444	14	numbers	number	NOUN
ejpam-3952	444	15	.	.	PUNCT
ejpam-3952	445	1	some	some	DET
ejpam-3952	445	2	identities	identity	NOUN
ejpam-3952	445	3	concerning	concern	VERB
ejpam-3952	445	4	isosceles	isoscele	NOUN
ejpam-3952	445	5	triangular	triangular	NOUN
ejpam-3952	445	6	numbers	number	NOUN
ejpam-3952	445	7	have	have	AUX
ejpam-3952	445	8	been	be	AUX
ejpam-3952	445	9	established	establish	VERB
ejpam-3952	445	10	as	as	ADP
ejpam-3952	445	11	generalizations	generalization	NOUN
ejpam-3952	445	12	of	of	ADP
ejpam-3952	445	13	classical	classical	ADJ
ejpam-3952	445	14	identities	identity	NOUN
ejpam-3952	445	15	for	for	ADP
ejpam-3952	445	16	triangular	triangular	NOUN
ejpam-3952	445	17	numbers	number	NOUN
ejpam-3952	445	18	.	.	PUNCT
ejpam-3952	446	1	it	it	PRON
ejpam-3952	446	2	would	would	AUX
ejpam-3952	446	3	be	be	AUX
ejpam-3952	446	4	interesting	interesting	ADJ
ejpam-3952	446	5	to	to	PART
ejpam-3952	446	6	give	give	VERB
ejpam-3952	446	7	generalizations	generalization	NOUN
ejpam-3952	446	8	of	of	ADP
ejpam-3952	446	9	other	other	ADJ
ejpam-3952	446	10	identities	identity	NOUN
ejpam-3952	446	11	for	for	ADP
ejpam-3952	446	12	triangular	triangular	NOUN
ejpam-3952	446	13	numbers	number	NOUN
ejpam-3952	446	14	in	in	ADP
ejpam-3952	446	15	terms	term	NOUN
ejpam-3952	446	16	of	of	ADP
ejpam-3952	446	17	isosceles	isoscele	NOUN
ejpam-3952	446	18	triangular	triangular	NOUN
ejpam-3952	446	19	numbers	number	NOUN
ejpam-3952	446	20	.	.	PUNCT
ejpam-3952	447	1	the	the	DET
ejpam-3952	447	2	study	study	NOUN
ejpam-3952	447	3	of	of	ADP
ejpam-3952	447	4	relationships	relationship	NOUN
ejpam-3952	447	5	among	among	ADP
ejpam-3952	447	6	isosceles	isoscele	NOUN
ejpam-3952	447	7	triangular	triangular	NOUN
ejpam-3952	447	8	numbers	number	NOUN
ejpam-3952	447	9	and	and	CCONJ
ejpam-3952	447	10	various	various	ADJ
ejpam-3952	447	11	types	type	NOUN
ejpam-3952	447	12	of	of	ADP
ejpam-3952	447	13	figurate	figurate	NOUN
ejpam-3952	447	14	is	be	AUX
ejpam-3952	447	15	an	an	DET
ejpam-3952	447	16	interesting	interesting	ADJ
ejpam-3952	447	17	problem	problem	NOUN
ejpam-3952	447	18	as	as	ADV
ejpam-3952	447	19	well	well	ADV
ejpam-3952	447	20	.	.	PUNCT
ejpam-3952	448	1	acknowledgements	acknowledgement	NOUN
ejpam-3952	448	2	the	the	DET
ejpam-3952	448	3	authors	author	NOUN
ejpam-3952	448	4	would	would	AUX
ejpam-3952	448	5	like	like	VERB
ejpam-3952	448	6	to	to	PART
ejpam-3952	448	7	thank	thank	VERB
ejpam-3952	448	8	the	the	DET
ejpam-3952	448	9	referees	referee	NOUN
ejpam-3952	448	10	and	and	CCONJ
ejpam-3952	448	11	the	the	DET
ejpam-3952	448	12	editor	editor	NOUN
ejpam-3952	448	13	for	for	ADP
ejpam-3952	448	14	their	their	PRON
ejpam-3952	448	15	useful	useful	ADJ
ejpam-3952	448	16	and	and	CCONJ
ejpam-3952	448	17	insightful	insightful	ADJ
ejpam-3952	448	18	comments	comment	NOUN
ejpam-3952	448	19	.	.	PUNCT
ejpam-3952	449	1	s.	s.	PROPN
ejpam-3952	449	2	jitman	jitman	PROPN
ejpam-3952	449	3	was	be	AUX
ejpam-3952	449	4	supported	support	VERB
ejpam-3952	449	5	by	by	ADP
ejpam-3952	449	6	the	the	DET
ejpam-3952	449	7	thailand	thailand	PROPN
ejpam-3952	449	8	research	research	PROPN
ejpam-3952	449	9	fund	fund	PROPN
ejpam-3952	449	10	under	under	ADP
ejpam-3952	449	11	research	research	NOUN
ejpam-3952	449	12	grant	grant	NOUN
ejpam-3952	449	13	rsa6280042	rsa6280042	NOUN
ejpam-3952	449	14	.	.	PUNCT
ejpam-3952	450	1	references	reference	NOUN
ejpam-3952	450	2	[	[	X
ejpam-3952	450	3	1	1	X
ejpam-3952	450	4	]	]	X
ejpam-3952	450	5	p	p	X
ejpam-3952	450	6	j	j	PROPN
ejpam-3952	450	7	berana	berana	PROPN
ejpam-3952	450	8	,	,	PUNCT
ejpam-3952	450	9	j	j	PROPN
ejpam-3952	450	10	montalbo	montalbo	PROPN
ejpam-3952	450	11	,	,	PUNCT
ejpam-3952	450	12	and	and	CCONJ
ejpam-3952	450	13	d	d	ADP
ejpam-3952	450	14	magpantay	magpantay	NOUN
ejpam-3952	450	15	.	.	PUNCT
ejpam-3952	451	1	on	on	ADP
ejpam-3952	451	2	triangular	triangular	NOUN
ejpam-3952	451	3	and	and	CCONJ
ejpam-3952	451	4	trapezoidal	trapezoidal	ADJ
ejpam-3952	451	5	numbers	number	NOUN
ejpam-3952	451	6	.	.	PUNCT
ejpam-3952	452	1	asia	asia	PROPN
ejpam-3952	452	2	pacific	pacific	PROPN
ejpam-3952	452	3	journal	journal	PROPN
ejpam-3952	452	4	of	of	ADP
ejpam-3952	452	5	multidisciplinary	multidisciplinary	ADJ
ejpam-3952	452	6	research	research	NOUN
ejpam-3952	452	7	,	,	PUNCT
ejpam-3952	452	8	3:76–81	3:76–81	NUM
ejpam-3952	452	9	,	,	PUNCT
ejpam-3952	452	10	2015	2015	NUM
ejpam-3952	452	11	.	.	PUNCT
ejpam-3952	453	1	http	http	ADJ
ejpam-3952	453	2	:	:	PUNCT
ejpam-3952	453	3	//www.apjmr.com	//www.apjmr.com	PUNCT
ejpam-3952	453	4	/	/	SYM
ejpam-3952	453	5	wp	wp	PROPN
ejpam-3952	453	6	-	-	PUNCT
ejpam-3952	453	7	content	content	NOUN
ejpam-3952	453	8	/	/	SYM
ejpam-3952	453	9	uploads/2015/11	uploads/2015/11	NOUN
ejpam-3952	453	10	/	/	SYM
ejpam-3952	453	11	apjmr-2015	apjmr-2015	ADJ
ejpam-3952	453	12	-	-	PUNCT
ejpam-3952	453	13	3.4.5.11.pdf	3.4.5.11.pdf	NOUN
ejpam-3952	453	14	.	.	PUNCT
ejpam-3952	454	1	[	[	X
ejpam-3952	454	2	2	2	NUM
ejpam-3952	454	3	]	]	PUNCT
ejpam-3952	454	4	e	e	NOUN
ejpam-3952	454	5	deza	deza	NOUN
ejpam-3952	454	6	and	and	CCONJ
ejpam-3952	454	7	m	m	VERB
ejpam-3952	454	8	m	m	VERB
ejpam-3952	454	9	deza	deza	ADV
ejpam-3952	454	10	.	.	PUNCT
ejpam-3952	455	1	figurate	figurate	ADJ
ejpam-3952	455	2	numbers	number	NOUN
ejpam-3952	455	3	.	.	PUNCT
ejpam-3952	456	1	world	world	NOUN
ejpam-3952	456	2	scientific	scientific	ADJ
ejpam-3952	456	3	publishing	publishing	NOUN
ejpam-3952	456	4	company	company	NOUN
ejpam-3952	456	5	,	,	PUNCT
ejpam-3952	456	6	singapore	singapore	PROPN
ejpam-3952	456	7	,	,	PUNCT
ejpam-3952	456	8	2012	2012	NUM
ejpam-3952	456	9	.	.	PUNCT
ejpam-3952	457	1	[	[	X
ejpam-3952	457	2	3	3	X
ejpam-3952	457	3	]	]	PUNCT
ejpam-3952	457	4	a	a	DET
ejpam-3952	457	5	s	s	NOUN
ejpam-3952	457	6	garge	garge	NOUN
ejpam-3952	457	7	and	and	CCONJ
ejpam-3952	457	8	s	s	VERB
ejpam-3952	457	9	a	a	DET
ejpam-3952	457	10	shirali	shirali	ADJ
ejpam-3952	457	11	.	.	PUNCT
ejpam-3952	457	12	triangular	triangular	NOUN
ejpam-3952	457	13	numbers	number	NOUN
ejpam-3952	457	14	.	.	PUNCT
ejpam-3952	458	1	resonance	resonance	NOUN
ejpam-3952	458	2	,	,	PUNCT
ejpam-3952	458	3	17:672–681	17:672–681	NUM
ejpam-3952	458	4	,	,	PUNCT
ejpam-3952	458	5	2012	2012	NUM
ejpam-3952	458	6	.	.	PUNCT
ejpam-3952	459	1	https://www.ias.ac.in/article/fulltext/reso/017/07/0672-0681	https://www.ias.ac.in/article/fulltext/reso/017/07/0672-0681	NOUN
ejpam-3952	459	2	.	.	PUNCT
ejpam-3952	460	1	[	[	X
ejpam-3952	460	2	4	4	X
ejpam-3952	460	3	]	]	X
ejpam-3952	460	4	h	h	NOUN
ejpam-3952	460	5	hindin	hindin	PROPN
ejpam-3952	460	6	.	.	PUNCT
ejpam-3952	461	1	stars	star	NOUN
ejpam-3952	461	2	,	,	PUNCT
ejpam-3952	461	3	hexes	hex	NOUN
ejpam-3952	461	4	,	,	PUNCT
ejpam-3952	461	5	triangular	triangular	NOUN
ejpam-3952	461	6	numbers	number	NOUN
ejpam-3952	461	7	and	and	CCONJ
ejpam-3952	461	8	pythagorean	pythagorean	PROPN
ejpam-3952	461	9	triples	triple	NOUN
ejpam-3952	461	10	.	.	PUNCT
ejpam-3952	462	1	journal	journal	NOUN
ejpam-3952	462	2	of	of	ADP
ejpam-3952	462	3	recreational	recreational	ADJ
ejpam-3952	462	4	mathematics	mathematic	NOUN
ejpam-3952	462	5	,	,	PUNCT
ejpam-3952	462	6	16:191–193	16:191–193	NUM
ejpam-3952	462	7	,	,	PUNCT
ejpam-3952	462	8	1983	1983	NUM
ejpam-3952	462	9	-	-	SYM
ejpam-3952	462	10	1984	1984	NUM
ejpam-3952	462	11	.	.	PUNCT
ejpam-3952	463	1	https://oeis.org/a006062/a006062	https://oeis.org/a006062/a006062	PROPN
ejpam-3952	463	2	.	.	PUNCT
ejpam-3952	464	1	pdf	pdf	NOUN
ejpam-3952	464	2	.	.	PUNCT
ejpam-3952	465	1	[	[	X
ejpam-3952	465	2	5	5	NUM
ejpam-3952	465	3	]	]	PUNCT
ejpam-3952	465	4	s	s	VERB
ejpam-3952	465	5	jitman	jitman	NOUN
ejpam-3952	465	6	,	,	PUNCT
ejpam-3952	465	7	k	k	PROPN
ejpam-3952	465	8	awachai	awachai	PROPN
ejpam-3952	465	9	,	,	PUNCT
ejpam-3952	465	10	and	and	CCONJ
ejpam-3952	465	11	p	p	NOUN
ejpam-3952	465	12	tanla	tanla	NOUN
ejpam-3952	465	13	.	.	PUNCT
ejpam-3952	466	1	isosceles	isoscele	NOUN
ejpam-3952	466	2	triangular	triangular	NOUN
ejpam-3952	466	3	numbers	number	NOUN
ejpam-3952	466	4	.	.	PUNCT
ejpam-3952	467	1	mj	mj	PROPN
ejpam-3952	467	2	-	-	PUNCT
ejpam-3952	467	3	math	math	NOUN
ejpam-3952	467	4	,	,	PUNCT
ejpam-3952	467	5	62:39–49	62:39–49	NUM
ejpam-3952	467	6	,	,	PUNCT
ejpam-3952	467	7	2017	2017	NUM
ejpam-3952	467	8	.	.	PUNCT
ejpam-3952	468	1	https://ph02.tci-thaijo.org/index.php/mjmath/article/	https://ph02.tci-thaijo.org/index.php/mjmath/article/	NOUN
ejpam-3952	468	2	view/15740	view/15740	ADV
ejpam-3952	468	3	.	.	PUNCT
ejpam-3952	469	1	[	[	X
ejpam-3952	469	2	6	6	NUM
ejpam-3952	469	3	]	]	PUNCT
ejpam-3952	469	4	s	s	PART
ejpam-3952	469	5	jitman	jitman	NOUN
ejpam-3952	469	6	and	and	CCONJ
ejpam-3952	469	7	c	c	PROPN
ejpam-3952	469	8	phongthai	phongthai	ADJ
ejpam-3952	469	9	.	.	PUNCT
ejpam-3952	470	1	on	on	ADP
ejpam-3952	470	2	the	the	DET
ejpam-3952	470	3	characterization	characterization	NOUN
ejpam-3952	470	4	and	and	CCONJ
ejpam-3952	470	5	enumeration	enumeration	NOUN
ejpam-3952	470	6	of	of	ADP
ejpam-3952	470	7	some	some	DET
ejpam-3952	470	8	generalized	generalized	ADJ
ejpam-3952	470	9	trapezoidal	trapezoidal	ADJ
ejpam-3952	470	10	numbers	number	NOUN
ejpam-3952	470	11	.	.	PUNCT
ejpam-3952	471	1	international	international	ADJ
ejpam-3952	471	2	journal	journal	NOUN
ejpam-3952	471	3	of	of	ADP
ejpam-3952	471	4	mathematics	mathematics	PROPN
ejpam-3952	471	5	and	and	CCONJ
ejpam-3952	471	6	mathematical	mathematical	ADJ
ejpam-3952	471	7	sciences	science	NOUN
ejpam-3952	471	8	,	,	PUNCT
ejpam-3952	471	9	2017:4515249	2017:4515249	NUM
ejpam-3952	471	10	,	,	PUNCT
ejpam-3952	471	11	2017	2017	NUM
ejpam-3952	471	12	.	.	PUNCT
ejpam-3952	472	1	https://doi.org/10.1155/2017/4515249	https://doi.org/10.1155/2017/4515249	NOUN
ejpam-3952	472	2	.	.	PUNCT
ejpam-3952	473	1	[	[	X
ejpam-3952	473	2	7	7	X
ejpam-3952	473	3	]	]	SYM
ejpam-3952	473	4	v	v	NOUN
ejpam-3952	473	5	e	e	X
ejpam-3952	473	6	hoggatt	hoggatt	PROPN
ejpam-3952	473	7	jr	jr	PROPN
ejpam-3952	473	8	and	and	CCONJ
ejpam-3952	473	9	m	m	PROPN
ejpam-3952	473	10	bicknell	bicknell	NOUN
ejpam-3952	473	11	.	.	PUNCT
ejpam-3952	474	1	triangular	triangular	NOUN
ejpam-3952	474	2	numbers	number	NOUN
ejpam-3952	474	3	.	.	PUNCT
ejpam-3952	475	1	the	the	DET
ejpam-3952	475	2	fibonacci	fibonacci	NOUN
ejpam-3952	475	3	quarterly	quarterly	PROPN
ejpam-3952	475	4	,	,	PUNCT
ejpam-3952	475	5	12:221–230	12:221–230	PROPN
ejpam-3952	475	6	,	,	PUNCT
ejpam-3952	475	7	1974	1974	NUM
ejpam-3952	475	8	.	.	PUNCT
ejpam-3952	476	1	https://www.fq.math.ca/issues/12-3.pdf	https://www.fq.math.ca/issues/12-3.pdf	PROPN
ejpam-3952	476	2	.	.	PUNCT
ejpam-3952	477	1	[	[	X
ejpam-3952	477	2	8	8	NUM
ejpam-3952	477	3	]	]	SYM
ejpam-3952	477	4	w	w	PROPN
ejpam-3952	477	5	l	l	PROPN
ejpam-3952	477	6	mcdaniel	mcdaniel	PROPN
ejpam-3952	477	7	.	.	PROPN
ejpam-3952	477	8	triangular	triangular	NOUN
ejpam-3952	477	9	numbers	number	NOUN
ejpam-3952	477	10	in	in	ADP
ejpam-3952	477	11	the	the	DET
ejpam-3952	477	12	pell	pell	NOUN
ejpam-3952	477	13	sequence	sequence	NOUN
ejpam-3952	477	14	.	.	PUNCT
ejpam-3952	478	1	the	the	DET
ejpam-3952	478	2	fibonacci	fibonacci	NOUN
ejpam-3952	478	3	quarterly	quarterly	PROPN
ejpam-3952	478	4	,	,	PUNCT
ejpam-3952	478	5	34:105–107	34:105–107	NUM
ejpam-3952	478	6	,	,	PUNCT
ejpam-3952	478	7	1996	1996	NUM
ejpam-3952	478	8	.	.	PUNCT
ejpam-3952	479	1	https://www.fq.math.ca/scanned/34-2/mcdaniel.pdf	https://www.fq.math.ca/scanned/34-2/mcdaniel.pdf	PROPN
ejpam-3952	479	2	.	.	PUNCT
ejpam-3952	480	1	references	reference	NOUN
ejpam-3952	480	2	395	395	NUM
ejpam-3952	480	3	[	[	X
ejpam-3952	480	4	9	9	NUM
ejpam-3952	480	5	]	]	X
ejpam-3952	480	6	r	r	NOUN
ejpam-3952	480	7	nelson	nelson	PROPN
ejpam-3952	480	8	.	.	PUNCT
ejpam-3952	481	1	a	a	DET
ejpam-3952	481	2	brief	brief	ADJ
ejpam-3952	481	3	journey	journey	NOUN
ejpam-3952	481	4	in	in	ADP
ejpam-3952	481	5	discrete	discrete	ADJ
ejpam-3952	481	6	mathematics	mathematic	NOUN
ejpam-3952	481	7	.	.	PUNCT
ejpam-3952	482	1	springer	springer	PROPN
ejpam-3952	482	2	,	,	PUNCT
ejpam-3952	482	3	cham	cham	PROPN
ejpam-3952	482	4	,	,	PUNCT
ejpam-3952	482	5	2020	2020	NUM
ejpam-3952	482	6	.	.	PUNCT
ejpam-3952	483	1	https	https	NOUN
ejpam-3952	483	2	:	:	PUNCT
ejpam-3952	483	3	//doi.org/10.1007/978	//doi.org/10.1007/978	NUM
ejpam-3952	483	4	-	-	PUNCT
ejpam-3952	483	5	3	3	NUM
ejpam-3952	483	6	-	-	PUNCT
ejpam-3952	483	7	030	030	NUM
ejpam-3952	483	8	-	-	PUNCT
ejpam-3952	483	9	37861	37861	NUM
ejpam-3952	483	10	-	-	SYM
ejpam-3952	483	11	5	5	NUM
ejpam-3952	483	12	.	.	PUNCT
ejpam-3952	484	1	[	[	X
ejpam-3952	484	2	10	10	NUM
ejpam-3952	484	3	]	]	X
ejpam-3952	484	4	j	j	PROPN
ejpam-3952	484	5	l	l	PROPN
ejpam-3952	484	6	pietempol	pietempol	NOUN
ejpam-3952	484	7	.	.	PUNCT
ejpam-3952	485	1	square	square	ADJ
ejpam-3952	485	2	triangular	triangular	NOUN
ejpam-3952	485	3	numbers	number	NOUN
ejpam-3952	485	4	.	.	PUNCT
ejpam-3952	486	1	the	the	DET
ejpam-3952	486	2	american	american	PROPN
ejpam-3952	486	3	mathematical	mathematical	PROPN
ejpam-3952	486	4	monthly	monthly	ADV
ejpam-3952	486	5	,	,	PUNCT
ejpam-3952	486	6	169:168–169	169:168–169	NUM
ejpam-3952	486	7	,	,	PUNCT
ejpam-3952	486	8	1962	1962	NUM
ejpam-3952	486	9	.	.	PUNCT
ejpam-3952	487	1	[	[	X
ejpam-3952	487	2	11	11	NUM
ejpam-3952	487	3	]	]	PUNCT
ejpam-3952	487	4	n	n	PRON
ejpam-3952	487	5	j	j	PROPN
ejpam-3952	487	6	a	a	DET
ejpam-3952	487	7	sloane	sloane	NOUN
ejpam-3952	487	8	.	.	PUNCT
ejpam-3952	488	1	the	the	DET
ejpam-3952	488	2	on	on	ADP
ejpam-3952	488	3	-	-	PUNCT
ejpam-3952	488	4	line	line	NOUN
ejpam-3952	488	5	encyclopedia	encyclopedia	NOUN
ejpam-3952	488	6	of	of	ADP
ejpam-3952	488	7	integer	integer	NOUN
ejpam-3952	488	8	sequences	sequence	NOUN
ejpam-3952	488	9	.	.	PUNCT
ejpam-3952	489	1	http://oeis.org	http://oeis.org	PROPN
ejpam-3952	489	2	,	,	PUNCT
ejpam-3952	489	3	2021	2021	NUM
ejpam-3952	489	4	.	.	PUNCT
ejpam-3952	490	1	[	[	X
ejpam-3952	490	2	12	12	NUM
ejpam-3952	490	3	]	]	PUNCT
ejpam-3952	490	4	t	t	PROPN
ejpam-3952	490	5	j	j	PROPN
ejpam-3952	490	6	trotter	trotter	PROPN
ejpam-3952	490	7	.	.	PUNCT
ejpam-3952	491	1	some	some	DET
ejpam-3952	491	2	identities	identity	NOUN
ejpam-3952	491	3	for	for	ADP
ejpam-3952	491	4	the	the	DET
ejpam-3952	491	5	triangular	triangular	NOUN
ejpam-3952	491	6	numbers	number	NOUN
ejpam-3952	491	7	.	.	PUNCT
ejpam-3952	492	1	journal	journal	NOUN
ejpam-3952	492	2	of	of	ADP
ejpam-3952	492	3	recreational	recreational	ADJ
ejpam-3952	492	4	mathematics	mathematic	NOUN
ejpam-3952	492	5	,	,	PUNCT
ejpam-3952	492	6	6:128–135	6:128–135	NOUN
ejpam-3952	492	7	,	,	PUNCT
ejpam-3952	492	8	1973	1973	NUM
ejpam-3952	492	9	.	.	PUNCT