id	sid	tid	token	lemma	pos
ejpam-5133	1	1	european	european	PROPN
ejpam-5133	1	2	journal	journal	PROPN
ejpam-5133	1	3	of	of	ADP
ejpam-5133	1	4	pure	pure	ADJ
ejpam-5133	1	5	and	and	CCONJ
ejpam-5133	1	6	applied	apply	VERB
ejpam-5133	1	7	mathematics	mathematic	NOUN
ejpam-5133	1	8	vol	vol	NOUN
ejpam-5133	1	9	.	.	PROPN
ejpam-5133	2	1	17	17	NUM
ejpam-5133	2	2	,	,	PUNCT
ejpam-5133	2	3	no	no	INTJ
ejpam-5133	2	4	.	.	NOUN
ejpam-5133	2	5	2	2	NUM
ejpam-5133	2	6	,	,	PUNCT
ejpam-5133	2	7	2024	2024	NUM
ejpam-5133	2	8	,	,	PUNCT
ejpam-5133	2	9	676	676	NUM
ejpam-5133	2	10	-	-	SYM
ejpam-5133	2	11	689	689	NUM
ejpam-5133	2	12	issn	issn	PROPN
ejpam-5133	2	13	1307	1307	NUM
ejpam-5133	2	14	-	-	SYM
ejpam-5133	2	15	5543	5543	NUM
ejpam-5133	2	16	–	–	PUNCT
ejpam-5133	3	1	ejpam.com	ejpam.com	X
ejpam-5133	3	2	published	publish	VERB
ejpam-5133	3	3	by	by	ADP
ejpam-5133	3	4	new	new	PROPN
ejpam-5133	3	5	york	york	PROPN
ejpam-5133	3	6	business	business	PROPN
ejpam-5133	3	7	global	global	PROPN
ejpam-5133	3	8	a	a	DET
ejpam-5133	3	9	novel	novel	ADJ
ejpam-5133	3	10	approach	approach	NOUN
ejpam-5133	3	11	for	for	ADP
ejpam-5133	3	12	studying	study	VERB
ejpam-5133	3	13	pythagorean	pythagorean	PROPN
ejpam-5133	3	14	triples	triple	NOUN
ejpam-5133	3	15	suitable	suitable	ADJ
ejpam-5133	3	16	for	for	ADP
ejpam-5133	3	17	students	student	NOUN
ejpam-5133	3	18	at	at	ADP
ejpam-5133	3	19	all	all	DET
ejpam-5133	3	20	educational	educational	ADJ
ejpam-5133	3	21	levels	level	NOUN
ejpam-5133	3	22	roberto	roberto	VERB
ejpam-5133	3	23	amato	amato	PROPN
ejpam-5133	3	24	department	department	PROPN
ejpam-5133	3	25	of	of	ADP
ejpam-5133	3	26	engineering	engineering	PROPN
ejpam-5133	3	27	,	,	PUNCT
ejpam-5133	3	28	university	university	PROPN
ejpam-5133	3	29	of	of	ADP
ejpam-5133	3	30	messina	messina	PROPN
ejpam-5133	3	31	,	,	PUNCT
ejpam-5133	3	32	messina	messina	PROPN
ejpam-5133	3	33	,	,	PUNCT
ejpam-5133	3	34	italy	italy	PROPN
ejpam-5133	3	35	abstract	abstract	NOUN
ejpam-5133	3	36	.	.	PUNCT
ejpam-5133	4	1	this	this	DET
ejpam-5133	4	2	paper	paper	NOUN
ejpam-5133	4	3	seeks	seek	VERB
ejpam-5133	4	4	to	to	PART
ejpam-5133	4	5	showcase	showcase	VERB
ejpam-5133	4	6	how	how	SCONJ
ejpam-5133	4	7	a	a	DET
ejpam-5133	4	8	new	new	ADJ
ejpam-5133	4	9	approach	approach	NOUN
ejpam-5133	4	10	can	can	AUX
ejpam-5133	4	11	breathe	breathe	VERB
ejpam-5133	4	12	new	new	ADJ
ejpam-5133	4	13	life	life	NOUN
ejpam-5133	4	14	into	into	ADP
ejpam-5133	4	15	research	research	NOUN
ejpam-5133	4	16	within	within	ADP
ejpam-5133	4	17	the	the	DET
ejpam-5133	4	18	traditional	traditional	ADJ
ejpam-5133	4	19	domain	domain	NOUN
ejpam-5133	4	20	of	of	ADP
ejpam-5133	4	21	pythagorean	pythagorean	PROPN
ejpam-5133	4	22	triples	triple	NOUN
ejpam-5133	4	23	,	,	PUNCT
ejpam-5133	4	24	introducing	introduce	VERB
ejpam-5133	4	25	innovative	innovative	ADJ
ejpam-5133	4	26	applications	application	NOUN
ejpam-5133	4	27	to	to	PART
ejpam-5133	4	28	invigorate	invigorate	VERB
ejpam-5133	4	29	the	the	DET
ejpam-5133	4	30	field	field	NOUN
ejpam-5133	4	31	.	.	PUNCT
ejpam-5133	5	1	this	this	PRON
ejpam-5133	5	2	serves	serve	VERB
ejpam-5133	5	3	not	not	PART
ejpam-5133	5	4	only	only	ADV
ejpam-5133	5	5	as	as	ADP
ejpam-5133	5	6	an	an	DET
ejpam-5133	5	7	exemplar	exemplar	NOUN
ejpam-5133	5	8	but	but	CCONJ
ejpam-5133	5	9	also	also	ADV
ejpam-5133	5	10	as	as	ADP
ejpam-5133	5	11	a	a	DET
ejpam-5133	5	12	wellspring	wellspring	NOUN
ejpam-5133	5	13	of	of	ADP
ejpam-5133	5	14	inspiration	inspiration	NOUN
ejpam-5133	5	15	for	for	ADP
ejpam-5133	5	16	students	student	NOUN
ejpam-5133	5	17	at	at	ADP
ejpam-5133	5	18	both	both	DET
ejpam-5133	5	19	school	school	NOUN
ejpam-5133	5	20	and	and	CCONJ
ejpam-5133	5	21	university	university	NOUN
ejpam-5133	5	22	levels	level	NOUN
ejpam-5133	5	23	.	.	PUNCT
ejpam-5133	6	1	the	the	DET
ejpam-5133	6	2	demonstrations	demonstration	NOUN
ejpam-5133	6	3	will	will	AUX
ejpam-5133	6	4	underscore	underscore	VERB
ejpam-5133	6	5	that	that	PRON
ejpam-5133	6	6	,	,	PUNCT
ejpam-5133	6	7	with	with	ADP
ejpam-5133	6	8	fundamental	fundamental	ADJ
ejpam-5133	6	9	mathematical	mathematical	ADJ
ejpam-5133	6	10	concepts	concept	NOUN
ejpam-5133	6	11	and	and	CCONJ
ejpam-5133	6	12	unencumbered	unencumbered	ADJ
ejpam-5133	6	13	by	by	ADP
ejpam-5133	6	14	intricate	intricate	ADJ
ejpam-5133	6	15	calculations	calculation	NOUN
ejpam-5133	6	16	,	,	PUNCT
ejpam-5133	6	17	one	one	PRON
ejpam-5133	6	18	can	can	AUX
ejpam-5133	6	19	unveil	unveil	VERB
ejpam-5133	6	20	novel	novel	ADJ
ejpam-5133	6	21	results	result	NOUN
ejpam-5133	6	22	and	and	CCONJ
ejpam-5133	6	23	applications	application	NOUN
ejpam-5133	6	24	with	with	ADP
ejpam-5133	6	25	ease	ease	NOUN
ejpam-5133	6	26	.	.	PUNCT
ejpam-5133	7	1	the	the	DET
ejpam-5133	7	2	new	new	ADJ
ejpam-5133	7	3	results	result	NOUN
ejpam-5133	7	4	and	and	CCONJ
ejpam-5133	7	5	applications	application	NOUN
ejpam-5133	7	6	,	,	PUNCT
ejpam-5133	7	7	along	along	ADP
ejpam-5133	7	8	with	with	ADP
ejpam-5133	7	9	those	those	PRON
ejpam-5133	7	10	found	find	VERB
ejpam-5133	7	11	in	in	ADP
ejpam-5133	7	12	the	the	DET
ejpam-5133	7	13	preliminary	preliminary	ADJ
ejpam-5133	7	14	results	result	NOUN
ejpam-5133	7	15	section	section	NOUN
ejpam-5133	7	16	,	,	PUNCT
ejpam-5133	7	17	show	show	VERB
ejpam-5133	7	18	how	how	SCONJ
ejpam-5133	7	19	the	the	DET
ejpam-5133	7	20	field	field	NOUN
ejpam-5133	7	21	of	of	ADP
ejpam-5133	7	22	pythagorean	pythagorean	PROPN
ejpam-5133	7	23	triples	triple	NOUN
ejpam-5133	7	24	is	be	AUX
ejpam-5133	7	25	still	still	ADV
ejpam-5133	7	26	interesting	interesting	ADJ
ejpam-5133	7	27	and	and	CCONJ
ejpam-5133	7	28	stimulating	stimulate	VERB
ejpam-5133	7	29	to	to	PART
ejpam-5133	7	30	study	study	VERB
ejpam-5133	7	31	,	,	PUNCT
ejpam-5133	7	32	despite	despite	SCONJ
ejpam-5133	7	33	the	the	DET
ejpam-5133	7	34	centuries	century	NOUN
ejpam-5133	7	35	that	that	PRON
ejpam-5133	7	36	have	have	AUX
ejpam-5133	7	37	elapsed	elapse	VERB
ejpam-5133	7	38	.	.	PUNCT
ejpam-5133	8	1	2020	2020	NUM
ejpam-5133	8	2	mathematics	mathematic	NOUN
ejpam-5133	8	3	subject	subject	NOUN
ejpam-5133	8	4	classifications	classification	NOUN
ejpam-5133	8	5	:	:	PUNCT
ejpam-5133	8	6	11d61	11d61	NUM
ejpam-5133	8	7	,	,	PUNCT
ejpam-5133	8	8	97f99	97f99	NUM
ejpam-5133	8	9	,	,	PUNCT
ejpam-5133	8	10	97g99	97g99	NUM
ejpam-5133	8	11	,	,	PUNCT
ejpam-5133	8	12	97h99	97h99	NUM
ejpam-5133	8	13	,	,	PUNCT
ejpam-5133	8	14	97i99	97i99	NUM
ejpam-5133	8	15	key	key	ADJ
ejpam-5133	8	16	words	word	NOUN
ejpam-5133	8	17	and	and	CCONJ
ejpam-5133	8	18	phrases	phrase	NOUN
ejpam-5133	8	19	:	:	PUNCT
ejpam-5133	8	20	pythagorean	pythagorean	NOUN
ejpam-5133	8	21	triples	triple	NOUN
ejpam-5133	8	22	,	,	PUNCT
ejpam-5133	8	23	diophantine	diophantine	VERB
ejpam-5133	8	24	equations	equation	NOUN
ejpam-5133	8	25	,	,	PUNCT
ejpam-5133	8	26	algebric	algebric	ADJ
ejpam-5133	8	27	groups	group	NOUN
ejpam-5133	8	28	,	,	PUNCT
ejpam-5133	8	29	geometry	geometry	NOUN
ejpam-5133	8	30	1	1	NUM
ejpam-5133	8	31	.	.	PUNCT
ejpam-5133	9	1	introduction	introduction	NOUN
ejpam-5133	9	2	let	let	VERB
ejpam-5133	9	3	x	x	PRON
ejpam-5133	9	4	,	,	PUNCT
ejpam-5133	9	5	y	y	PROPN
ejpam-5133	9	6	,	,	PUNCT
ejpam-5133	9	7	and	and	CCONJ
ejpam-5133	9	8	z	z	NOUN
ejpam-5133	9	9	be	be	AUX
ejpam-5133	9	10	positive	positive	ADJ
ejpam-5133	9	11	integers	integer	NOUN
ejpam-5133	9	12	satisfying	satisfy	VERB
ejpam-5133	10	1	x2	x2	PROPN
ejpam-5133	11	1	+	+	CCONJ
ejpam-5133	11	2	y2	y2	NOUN
ejpam-5133	11	3	=	=	SYM
ejpam-5133	11	4	z2	z2	PROPN
ejpam-5133	11	5	.	.	PUNCT
ejpam-5133	12	1	such	such	DET
ejpam-5133	12	2	a	a	DET
ejpam-5133	12	3	triple	triple	ADJ
ejpam-5133	12	4	(	(	PUNCT
ejpam-5133	12	5	x	x	NOUN
ejpam-5133	12	6	,	,	PUNCT
ejpam-5133	12	7	y	y	PROPN
ejpam-5133	12	8	,	,	PUNCT
ejpam-5133	12	9	z	z	NOUN
ejpam-5133	12	10	)	)	PUNCT
ejpam-5133	12	11	is	be	AUX
ejpam-5133	12	12	called	call	VERB
ejpam-5133	12	13	a	a	DET
ejpam-5133	12	14	pythagorean	pythagorean	PROPN
ejpam-5133	12	15	triple	triple	NOUN
ejpam-5133	12	16	.	.	PUNCT
ejpam-5133	13	1	in	in	ADP
ejpam-5133	13	2	particular	particular	ADJ
ejpam-5133	13	3	,	,	PUNCT
ejpam-5133	13	4	if	if	SCONJ
ejpam-5133	13	5	x	x	X
ejpam-5133	13	6	,	,	PUNCT
ejpam-5133	13	7	y	y	PROPN
ejpam-5133	13	8	,	,	PUNCT
ejpam-5133	13	9	and	and	CCONJ
ejpam-5133	13	10	z	z	NOUN
ejpam-5133	13	11	are	be	AUX
ejpam-5133	13	12	coprime	coprime	ADJ
ejpam-5133	13	13	,	,	PUNCT
ejpam-5133	13	14	the	the	DET
ejpam-5133	13	15	triple	triple	ADJ
ejpam-5133	13	16	is	be	AUX
ejpam-5133	13	17	termed	term	VERB
ejpam-5133	13	18	a	a	DET
ejpam-5133	13	19	primitive	primitive	ADJ
ejpam-5133	13	20	pythagorean	pythagorean	NOUN
ejpam-5133	13	21	triple	triple	NOUN
ejpam-5133	13	22	.	.	PUNCT
ejpam-5133	14	1	pythagorean	pythagorean	PROPN
ejpam-5133	14	2	triples	triple	NOUN
ejpam-5133	14	3	owe	owe	VERB
ejpam-5133	14	4	their	their	PRON
ejpam-5133	14	5	name	name	NOUN
ejpam-5133	14	6	to	to	ADP
ejpam-5133	14	7	the	the	DET
ejpam-5133	14	8	greek	greek	ADJ
ejpam-5133	14	9	mathematician	mathematician	NOUN
ejpam-5133	14	10	pythagoras	pythagoras	PROPN
ejpam-5133	14	11	,	,	PUNCT
ejpam-5133	14	12	who	who	PRON
ejpam-5133	14	13	lived	live	VERB
ejpam-5133	14	14	in	in	ADP
ejpam-5133	14	15	the	the	DET
ejpam-5133	14	16	6th	6th	ADJ
ejpam-5133	14	17	century	century	NOUN
ejpam-5133	14	18	b.c	b.c	PROPN
ejpam-5133	14	19	.	.	PUNCT
ejpam-5133	15	1	pythagoras	pythagoras	PROPN
ejpam-5133	15	2	was	be	AUX
ejpam-5133	15	3	the	the	DET
ejpam-5133	15	4	founder	founder	NOUN
ejpam-5133	15	5	of	of	ADP
ejpam-5133	15	6	the	the	DET
ejpam-5133	15	7	philosophical	philosophical	ADJ
ejpam-5133	15	8	school	school	NOUN
ejpam-5133	15	9	known	know	VERB
ejpam-5133	15	10	as	as	ADP
ejpam-5133	15	11	pythagoreanism	pythagoreanism	NOUN
ejpam-5133	15	12	,	,	PUNCT
ejpam-5133	15	13	and	and	CCONJ
ejpam-5133	15	14	pythagorean	pythagorean	PROPN
ejpam-5133	15	15	triples	triple	NOUN
ejpam-5133	15	16	are	be	AUX
ejpam-5133	15	17	often	often	ADV
ejpam-5133	15	18	associated	associate	VERB
ejpam-5133	15	19	with	with	ADP
ejpam-5133	15	20	his	his	PRON
ejpam-5133	15	21	discoveries	discovery	NOUN
ejpam-5133	15	22	and	and	CCONJ
ejpam-5133	15	23	teachings	teaching	NOUN
ejpam-5133	15	24	.	.	PUNCT
ejpam-5133	16	1	according	accord	VERB
ejpam-5133	16	2	to	to	ADP
ejpam-5133	16	3	legend	legend	PROPN
ejpam-5133	16	4	,	,	PUNCT
ejpam-5133	16	5	pythagoras	pythagoras	PROPN
ejpam-5133	16	6	and	and	CCONJ
ejpam-5133	16	7	his	his	PRON
ejpam-5133	16	8	followers	follower	NOUN
ejpam-5133	16	9	became	become	VERB
ejpam-5133	16	10	interested	interested	ADJ
ejpam-5133	16	11	in	in	ADP
ejpam-5133	16	12	pythagorean	pythagorean	PROPN
ejpam-5133	16	13	triples	triple	NOUN
ejpam-5133	16	14	while	while	SCONJ
ejpam-5133	16	15	studying	study	VERB
ejpam-5133	16	16	numbers	number	NOUN
ejpam-5133	16	17	and	and	CCONJ
ejpam-5133	16	18	musical	musical	ADJ
ejpam-5133	16	19	proportions	proportion	NOUN
ejpam-5133	16	20	.	.	PUNCT
ejpam-5133	17	1	it	it	PRON
ejpam-5133	17	2	is	be	AUX
ejpam-5133	17	3	said	say	VERB
ejpam-5133	17	4	that	that	SCONJ
ejpam-5133	17	5	they	they	PRON
ejpam-5133	17	6	noticed	notice	VERB
ejpam-5133	17	7	certain	certain	ADJ
ejpam-5133	17	8	combinations	combination	NOUN
ejpam-5133	17	9	of	of	ADP
ejpam-5133	17	10	lengths	length	NOUN
ejpam-5133	17	11	of	of	ADP
ejpam-5133	17	12	musical	musical	ADJ
ejpam-5133	17	13	strings	string	NOUN
ejpam-5133	17	14	produced	produce	VERB
ejpam-5133	17	15	harmonic	harmonic	ADJ
ejpam-5133	17	16	sounds	sound	NOUN
ejpam-5133	17	17	,	,	PUNCT
ejpam-5133	17	18	and	and	CCONJ
ejpam-5133	17	19	these	these	DET
ejpam-5133	17	20	combinations	combination	NOUN
ejpam-5133	17	21	corresponded	correspond	VERB
ejpam-5133	17	22	to	to	ADP
ejpam-5133	17	23	pythagorean	pythagorean	VERB
ejpam-5133	17	24	triples	triple	NOUN
ejpam-5133	17	25	.	.	PUNCT
ejpam-5133	18	1	however	however	ADV
ejpam-5133	18	2	,	,	PUNCT
ejpam-5133	18	3	it	it	PRON
ejpam-5133	18	4	’s	’	VERB
ejpam-5133	18	5	important	important	ADJ
ejpam-5133	18	6	to	to	PART
ejpam-5133	18	7	note	note	VERB
ejpam-5133	18	8	that	that	SCONJ
ejpam-5133	18	9	pythagorean	pythagorean	PROPN
ejpam-5133	18	10	triples	triple	NOUN
ejpam-5133	18	11	were	be	AUX
ejpam-5133	18	12	not	not	PART
ejpam-5133	18	13	discovered	discover	VERB
ejpam-5133	18	14	or	or	CCONJ
ejpam-5133	18	15	introduced	introduce	VERB
ejpam-5133	18	16	by	by	ADP
ejpam-5133	18	17	pythagoras	pythagoras	PROPN
ejpam-5133	18	18	himself	himself	PRON
ejpam-5133	18	19	.	.	PUNCT
ejpam-5133	19	1	ancient	ancient	ADJ
ejpam-5133	19	2	doi	doi	NOUN
ejpam-5133	19	3	:	:	PUNCT
ejpam-5133	19	4	https://doi.org/10.29020/nybg.ejpam.v17i2.5133	https://doi.org/10.29020/nybg.ejpam.v17i2.5133	X
ejpam-5133	19	5	email	email	NOUN
ejpam-5133	19	6	address	address	NOUN
ejpam-5133	19	7	:	:	PUNCT
ejpam-5133	19	8	ramato@unime.it	ramato@unime.it	PROPN
ejpam-5133	19	9	(	(	PUNCT
ejpam-5133	19	10	r.	r.	PROPN
ejpam-5133	19	11	amato	amato	PROPN
ejpam-5133	19	12	)	)	PUNCT
ejpam-5133	19	13	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5133	20	1	676	676	NUM
ejpam-5133	20	2	©	©	ADP
ejpam-5133	20	3	2024	2024	NUM
ejpam-5133	20	4	ejpam	ejpam	NOUN
ejpam-5133	20	5	all	all	DET
ejpam-5133	20	6	rights	right	NOUN
ejpam-5133	20	7	reserved	reserve	VERB
ejpam-5133	20	8	.	.	PUNCT
ejpam-5133	21	1	r.	r.	PROPN
ejpam-5133	21	2	amato	amato	PROPN
ejpam-5133	21	3	/	/	SYM
ejpam-5133	21	4	eur	eur	PROPN
ejpam-5133	21	5	.	.	PUNCT
ejpam-5133	22	1	j.	j.	PROPN
ejpam-5133	22	2	pure	pure	PROPN
ejpam-5133	22	3	appl	appl	PROPN
ejpam-5133	22	4	.	.	PROPN
ejpam-5133	22	5	math	math	PROPN
ejpam-5133	22	6	,	,	PUNCT
ejpam-5133	22	7	17	17	NUM
ejpam-5133	22	8	(	(	PUNCT
ejpam-5133	22	9	2	2	NUM
ejpam-5133	22	10	)	)	PUNCT
ejpam-5133	22	11	(	(	PUNCT
ejpam-5133	22	12	2024	2024	NUM
ejpam-5133	22	13	)	)	PUNCT
ejpam-5133	22	14	,	,	PUNCT
ejpam-5133	22	15	676	676	NUM
ejpam-5133	22	16	-	-	SYM
ejpam-5133	22	17	689	689	NUM
ejpam-5133	22	18	677	677	NUM
ejpam-5133	22	19	babylonian	babylonian	ADJ
ejpam-5133	22	20	mathematicians	mathematician	NOUN
ejpam-5133	22	21	were	be	AUX
ejpam-5133	22	22	already	already	ADV
ejpam-5133	22	23	aware	aware	ADJ
ejpam-5133	22	24	of	of	ADP
ejpam-5133	22	25	some	some	DET
ejpam-5133	22	26	pythagorean	pythagorean	ADJ
ejpam-5133	22	27	triples	triple	NOUN
ejpam-5133	22	28	long	long	ADV
ejpam-5133	22	29	before	before	SCONJ
ejpam-5133	22	30	pythagoreanism	pythagoreanism	NOUN
ejpam-5133	22	31	became	become	VERB
ejpam-5133	22	32	famous	famous	ADJ
ejpam-5133	22	33	.	.	PUNCT
ejpam-5133	23	1	the	the	DET
ejpam-5133	23	2	earliest	early	ADJ
ejpam-5133	23	3	known	know	VERB
ejpam-5133	23	4	record	record	NOUN
ejpam-5133	23	5	of	of	ADP
ejpam-5133	23	6	the	the	DET
ejpam-5133	23	7	theorem	theorem	NOUN
ejpam-5133	23	8	is	be	AUX
ejpam-5133	23	9	found	find	VERB
ejpam-5133	23	10	in	in	ADP
ejpam-5133	23	11	an	an	DET
ejpam-5133	23	12	ancient	ancient	ADJ
ejpam-5133	23	13	babylonian	babylonian	ADJ
ejpam-5133	23	14	manuscript	manuscript	NOUN
ejpam-5133	23	15	called	call	VERB
ejpam-5133	23	16	plimpton	plimpton	PROPN
ejpam-5133	23	17	322	322	NUM
ejpam-5133	23	18	,	,	PUNCT
ejpam-5133	23	19	dating	date	VERB
ejpam-5133	23	20	back	back	ADV
ejpam-5133	23	21	about	about	ADV
ejpam-5133	23	22	1,800	1,800	NUM
ejpam-5133	23	23	years	year	NOUN
ejpam-5133	23	24	before	before	ADP
ejpam-5133	23	25	pythagoras	pythagoras	PROPN
ejpam-5133	23	26	.	.	PUNCT
ejpam-5133	24	1	this	this	DET
ejpam-5133	24	2	text	text	NOUN
ejpam-5133	24	3	provides	provide	VERB
ejpam-5133	24	4	a	a	DET
ejpam-5133	24	5	list	list	NOUN
ejpam-5133	24	6	of	of	ADP
ejpam-5133	24	7	pythagorean	pythagorean	PROPN
ejpam-5133	24	8	triples	triple	NOUN
ejpam-5133	24	9	and	and	CCONJ
ejpam-5133	24	10	demonstrates	demonstrate	VERB
ejpam-5133	24	11	the	the	DET
ejpam-5133	24	12	babylonians	babylonians	PROPN
ejpam-5133	24	13	’	’	PART
ejpam-5133	24	14	knowledge	knowledge	NOUN
ejpam-5133	24	15	of	of	ADP
ejpam-5133	24	16	pythagorean	pythagorean	PROPN
ejpam-5133	24	17	formulas	formula	NOUN
ejpam-5133	24	18	and	and	CCONJ
ejpam-5133	24	19	how	how	SCONJ
ejpam-5133	24	20	to	to	PART
ejpam-5133	24	21	use	use	VERB
ejpam-5133	24	22	them	they	PRON
ejpam-5133	24	23	.	.	PUNCT
ejpam-5133	25	1	nevertheless	nevertheless	ADV
ejpam-5133	25	2	,	,	PUNCT
ejpam-5133	25	3	it	it	PRON
ejpam-5133	25	4	was	be	AUX
ejpam-5133	25	5	pythagoras	pythagoras	PROPN
ejpam-5133	25	6	and	and	CCONJ
ejpam-5133	25	7	his	his	PRON
ejpam-5133	25	8	school	school	NOUN
ejpam-5133	25	9	that	that	PRON
ejpam-5133	25	10	placed	place	VERB
ejpam-5133	25	11	a	a	DET
ejpam-5133	25	12	particular	particular	ADJ
ejpam-5133	25	13	emphasis	emphasis	NOUN
ejpam-5133	25	14	on	on	ADP
ejpam-5133	25	15	these	these	DET
ejpam-5133	25	16	triples	triple	NOUN
ejpam-5133	25	17	and	and	CCONJ
ejpam-5133	25	18	discovered	discover	VERB
ejpam-5133	25	19	some	some	PRON
ejpam-5133	25	20	of	of	ADP
ejpam-5133	25	21	their	their	PRON
ejpam-5133	25	22	interesting	interesting	ADJ
ejpam-5133	25	23	properties	property	NOUN
ejpam-5133	25	24	.	.	PUNCT
ejpam-5133	26	1	pythagorean	pythagorean	PROPN
ejpam-5133	26	2	triples	triple	NOUN
ejpam-5133	26	3	were	be	AUX
ejpam-5133	26	4	extensively	extensively	ADV
ejpam-5133	26	5	studied	study	VERB
ejpam-5133	26	6	by	by	ADP
ejpam-5133	26	7	the	the	DET
ejpam-5133	26	8	pythagoreans	pythagorean	NOUN
ejpam-5133	26	9	and	and	CCONJ
ejpam-5133	26	10	the	the	DET
ejpam-5133	26	11	subsequent	subsequent	ADJ
ejpam-5133	26	12	pythagorean	pythagorean	PROPN
ejpam-5133	26	13	school	school	NOUN
ejpam-5133	26	14	.	.	PUNCT
ejpam-5133	27	1	they	they	PRON
ejpam-5133	27	2	sought	seek	VERB
ejpam-5133	27	3	to	to	PART
ejpam-5133	27	4	find	find	VERB
ejpam-5133	27	5	all	all	DET
ejpam-5133	27	6	possible	possible	ADJ
ejpam-5133	27	7	pythagorean	pythagorean	NOUN
ejpam-5133	27	8	triples	triple	NOUN
ejpam-5133	27	9	and	and	CCONJ
ejpam-5133	27	10	developed	develop	VERB
ejpam-5133	27	11	methods	method	NOUN
ejpam-5133	27	12	to	to	PART
ejpam-5133	27	13	generate	generate	VERB
ejpam-5133	27	14	new	new	ADJ
ejpam-5133	27	15	triples	triple	NOUN
ejpam-5133	27	16	.	.	PUNCT
ejpam-5133	28	1	the	the	DET
ejpam-5133	28	2	theory	theory	NOUN
ejpam-5133	28	3	of	of	ADP
ejpam-5133	28	4	pythagorean	pythagorean	PROPN
ejpam-5133	28	5	triples	triple	NOUN
ejpam-5133	28	6	was	be	AUX
ejpam-5133	28	7	further	far	ADV
ejpam-5133	28	8	developed	develop	VERB
ejpam-5133	28	9	by	by	ADP
ejpam-5133	28	10	mathematicians	mathematician	NOUN
ejpam-5133	28	11	from	from	ADP
ejpam-5133	28	12	various	various	ADJ
ejpam-5133	28	13	cultures	culture	NOUN
ejpam-5133	28	14	throughout	throughout	ADP
ejpam-5133	28	15	history	history	NOUN
ejpam-5133	28	16	.	.	PUNCT
ejpam-5133	29	1	for	for	ADP
ejpam-5133	29	2	instance	instance	NOUN
ejpam-5133	29	3	,	,	PUNCT
ejpam-5133	29	4	ancient	ancient	ADJ
ejpam-5133	29	5	indians	indians	PROPN
ejpam-5133	29	6	,	,	PUNCT
ejpam-5133	29	7	chinese	chinese	PROPN
ejpam-5133	29	8	,	,	PUNCT
ejpam-5133	29	9	arabs	arabs	PROPN
ejpam-5133	29	10	,	,	PUNCT
ejpam-5133	29	11	and	and	CCONJ
ejpam-5133	29	12	europeans	european	NOUN
ejpam-5133	29	13	contributed	contribute	VERB
ejpam-5133	29	14	to	to	ADP
ejpam-5133	29	15	further	further	ADJ
ejpam-5133	29	16	developments	development	NOUN
ejpam-5133	29	17	in	in	ADP
ejpam-5133	29	18	the	the	DET
ejpam-5133	29	19	theory	theory	NOUN
ejpam-5133	29	20	of	of	ADP
ejpam-5133	29	21	pythagorean	pythagorean	PROPN
ejpam-5133	29	22	triples	triple	NOUN
ejpam-5133	29	23	.	.	PUNCT
ejpam-5133	30	1	over	over	ADP
ejpam-5133	30	2	time	time	NOUN
ejpam-5133	30	3	,	,	PUNCT
ejpam-5133	30	4	numerous	numerous	ADJ
ejpam-5133	30	5	pythagorean	pythagorean	PROPN
ejpam-5133	30	6	triples	triple	NOUN
ejpam-5133	30	7	with	with	ADP
ejpam-5133	30	8	increasingly	increasingly	ADV
ejpam-5133	30	9	large	large	ADJ
ejpam-5133	30	10	values	value	NOUN
ejpam-5133	30	11	for	for	ADP
ejpam-5133	30	12	their	their	PRON
ejpam-5133	30	13	components	component	NOUN
ejpam-5133	30	14	have	have	AUX
ejpam-5133	30	15	been	be	AUX
ejpam-5133	30	16	discovered	discover	VERB
ejpam-5133	30	17	.	.	PUNCT
ejpam-5133	31	1	the	the	DET
ejpam-5133	31	2	most	most	ADV
ejpam-5133	31	3	famous	famous	ADJ
ejpam-5133	31	4	discovery	discovery	NOUN
ejpam-5133	31	5	attributed	attribute	VERB
ejpam-5133	31	6	to	to	ADP
ejpam-5133	31	7	pythagoras	pythagoras	PROPN
ejpam-5133	31	8	regarding	regard	VERB
ejpam-5133	31	9	pythagorean	pythagorean	PROPN
ejpam-5133	31	10	triples	triple	NOUN
ejpam-5133	31	11	is	be	AUX
ejpam-5133	31	12	the	the	DET
ejpam-5133	31	13	pythagorean	pythagorean	PROPN
ejpam-5133	31	14	theorem	theorem	PROPN
ejpam-5133	31	15	.	.	PUNCT
ejpam-5133	32	1	this	this	DET
ejpam-5133	32	2	theorem	theorem	NOUN
ejpam-5133	32	3	is	be	AUX
ejpam-5133	32	4	fundamental	fundamental	ADJ
ejpam-5133	32	5	in	in	ADP
ejpam-5133	32	6	geometry	geometry	NOUN
ejpam-5133	32	7	and	and	CCONJ
ejpam-5133	32	8	has	have	VERB
ejpam-5133	32	9	many	many	ADJ
ejpam-5133	32	10	practical	practical	ADJ
ejpam-5133	32	11	applications	application	NOUN
ejpam-5133	32	12	.	.	PUNCT
ejpam-5133	33	1	however	however	ADV
ejpam-5133	33	2	,	,	PUNCT
ejpam-5133	33	3	pythagoras	pythagoras	PROPN
ejpam-5133	33	4	himself	himself	PRON
ejpam-5133	33	5	may	may	AUX
ejpam-5133	33	6	not	not	PART
ejpam-5133	33	7	have	have	AUX
ejpam-5133	33	8	provided	provide	VERB
ejpam-5133	33	9	a	a	DET
ejpam-5133	33	10	general	general	ADJ
ejpam-5133	33	11	proof	proof	NOUN
ejpam-5133	33	12	of	of	ADP
ejpam-5133	33	13	the	the	DET
ejpam-5133	33	14	theorem	theorem	NOUN
ejpam-5133	33	15	,	,	PUNCT
ejpam-5133	33	16	and	and	CCONJ
ejpam-5133	33	17	many	many	ADJ
ejpam-5133	33	18	of	of	ADP
ejpam-5133	33	19	the	the	DET
ejpam-5133	33	20	proofs	proof	NOUN
ejpam-5133	33	21	attributed	attribute	VERB
ejpam-5133	33	22	to	to	ADP
ejpam-5133	33	23	him	he	PRON
ejpam-5133	33	24	may	may	AUX
ejpam-5133	33	25	have	have	AUX
ejpam-5133	33	26	been	be	AUX
ejpam-5133	33	27	developed	develop	VERB
ejpam-5133	33	28	later	later	ADV
ejpam-5133	33	29	by	by	ADP
ejpam-5133	33	30	his	his	PRON
ejpam-5133	33	31	followers	follower	NOUN
ejpam-5133	33	32	.	.	PUNCT
ejpam-5133	34	1	pythagorean	pythagorean	PROPN
ejpam-5133	34	2	triples	triple	NOUN
ejpam-5133	34	3	,	,	PUNCT
ejpam-5133	34	4	due	due	ADP
ejpam-5133	34	5	to	to	ADP
ejpam-5133	34	6	their	their	PRON
ejpam-5133	34	7	connection	connection	NOUN
ejpam-5133	34	8	with	with	ADP
ejpam-5133	34	9	the	the	DET
ejpam-5133	34	10	pythagorean	pythagorean	PROPN
ejpam-5133	34	11	theorem	theorem	PROPN
ejpam-5133	34	12	,	,	PUNCT
ejpam-5133	34	13	are	be	AUX
ejpam-5133	34	14	a	a	DET
ejpam-5133	34	15	fundamental	fundamental	ADJ
ejpam-5133	34	16	concept	concept	NOUN
ejpam-5133	34	17	with	with	ADP
ejpam-5133	34	18	multiple	multiple	ADJ
ejpam-5133	34	19	practical	practical	ADJ
ejpam-5133	34	20	applications	application	NOUN
ejpam-5133	34	21	in	in	ADP
ejpam-5133	34	22	different	different	ADJ
ejpam-5133	34	23	fields	field	NOUN
ejpam-5133	34	24	of	of	ADP
ejpam-5133	34	25	mathematics	mathematic	NOUN
ejpam-5133	34	26	and	and	CCONJ
ejpam-5133	34	27	applied	applied	ADJ
ejpam-5133	34	28	sciences	science	NOUN
ejpam-5133	34	29	.	.	PUNCT
ejpam-5133	35	1	pythagorean	pythagorean	PROPN
ejpam-5133	35	2	triples	triple	NOUN
ejpam-5133	35	3	continued	continue	VERB
ejpam-5133	35	4	to	to	PART
ejpam-5133	35	5	be	be	AUX
ejpam-5133	35	6	studied	study	VERB
ejpam-5133	35	7	and	and	CCONJ
ejpam-5133	35	8	explored	explore	VERB
ejpam-5133	35	9	by	by	ADP
ejpam-5133	35	10	numerous	numerous	ADJ
ejpam-5133	35	11	later	later	ADJ
ejpam-5133	35	12	mathematicians	mathematician	NOUN
ejpam-5133	35	13	,	,	PUNCT
ejpam-5133	35	14	such	such	ADJ
ejpam-5133	35	15	as	as	ADP
ejpam-5133	35	16	euclid	euclid	PROPN
ejpam-5133	35	17	and	and	CCONJ
ejpam-5133	35	18	diophantus	diophantus	NOUN
ejpam-5133	35	19	.	.	PUNCT
ejpam-5133	36	1	over	over	ADP
ejpam-5133	36	2	the	the	DET
ejpam-5133	36	3	centuries	century	NOUN
ejpam-5133	36	4	,	,	PUNCT
ejpam-5133	36	5	many	many	ADJ
ejpam-5133	36	6	interesting	interesting	ADJ
ejpam-5133	36	7	properties	property	NOUN
ejpam-5133	36	8	and	and	CCONJ
ejpam-5133	36	9	relationships	relationship	NOUN
ejpam-5133	36	10	regarding	regard	VERB
ejpam-5133	36	11	pythagorean	pythagorean	PROPN
ejpam-5133	36	12	triples	triple	NOUN
ejpam-5133	36	13	have	have	AUX
ejpam-5133	36	14	been	be	AUX
ejpam-5133	36	15	discovered	discover	VERB
ejpam-5133	36	16	,	,	PUNCT
ejpam-5133	36	17	and	and	CCONJ
ejpam-5133	36	18	their	their	PRON
ejpam-5133	36	19	importance	importance	NOUN
ejpam-5133	36	20	has	have	AUX
ejpam-5133	36	21	extended	extend	VERB
ejpam-5133	36	22	to	to	ADP
ejpam-5133	36	23	various	various	ADJ
ejpam-5133	36	24	mathematical	mathematical	ADJ
ejpam-5133	36	25	fields	field	NOUN
ejpam-5133	36	26	,	,	PUNCT
ejpam-5133	36	27	including	include	VERB
ejpam-5133	36	28	number	number	NOUN
ejpam-5133	36	29	theory	theory	NOUN
ejpam-5133	36	30	,	,	PUNCT
ejpam-5133	36	31	modular	modular	ADJ
ejpam-5133	36	32	arithmetic	arithmetic	ADJ
ejpam-5133	36	33	,	,	PUNCT
ejpam-5133	36	34	geometry	geometry	NOUN
ejpam-5133	36	35	,	,	PUNCT
ejpam-5133	36	36	and	and	CCONJ
ejpam-5133	36	37	graph	graph	NOUN
ejpam-5133	36	38	theory	theory	NOUN
ejpam-5133	36	39	.	.	PUNCT
ejpam-5133	37	1	pythagorean	pythagorean	PROPN
ejpam-5133	37	2	triples	triple	NOUN
ejpam-5133	37	3	and	and	CCONJ
ejpam-5133	37	4	the	the	DET
ejpam-5133	37	5	pythagorean	pythagorean	PROPN
ejpam-5133	37	6	theorem	theorem	NOUN
ejpam-5133	37	7	have	have	AUX
ejpam-5133	37	8	had	have	VERB
ejpam-5133	37	9	a	a	DET
ejpam-5133	37	10	lasting	lasting	ADJ
ejpam-5133	37	11	impact	impact	NOUN
ejpam-5133	37	12	on	on	ADP
ejpam-5133	37	13	mathematics	mathematic	NOUN
ejpam-5133	37	14	and	and	CCONJ
ejpam-5133	37	15	geometry	geometry	NOUN
ejpam-5133	37	16	and	and	CCONJ
ejpam-5133	37	17	have	have	AUX
ejpam-5133	37	18	been	be	AUX
ejpam-5133	37	19	extensively	extensively	ADV
ejpam-5133	37	20	studied	study	VERB
ejpam-5133	37	21	and	and	CCONJ
ejpam-5133	37	22	applied	apply	VERB
ejpam-5133	37	23	over	over	ADP
ejpam-5133	37	24	the	the	DET
ejpam-5133	37	25	centuries	century	NOUN
ejpam-5133	37	26	,	,	PUNCT
ejpam-5133	37	27	influencing	influence	VERB
ejpam-5133	37	28	disciplines	discipline	NOUN
ejpam-5133	37	29	ranging	range	VERB
ejpam-5133	37	30	from	from	ADP
ejpam-5133	37	31	trigonometry	trigonometry	NOUN
ejpam-5133	37	32	to	to	ADP
ejpam-5133	37	33	mathematical	mathematical	ADJ
ejpam-5133	37	34	analysis	analysis	NOUN
ejpam-5133	37	35	.	.	PUNCT
ejpam-5133	38	1	today	today	NOUN
ejpam-5133	38	2	,	,	PUNCT
ejpam-5133	38	3	pythagorean	pythagorean	PROPN
ejpam-5133	38	4	triples	triple	NOUN
ejpam-5133	38	5	continue	continue	VERB
ejpam-5133	38	6	to	to	PART
ejpam-5133	38	7	be	be	AUX
ejpam-5133	38	8	a	a	DET
ejpam-5133	38	9	subject	subject	NOUN
ejpam-5133	38	10	of	of	ADP
ejpam-5133	38	11	study	study	NOUN
ejpam-5133	38	12	and	and	CCONJ
ejpam-5133	38	13	appreciation	appreciation	NOUN
ejpam-5133	38	14	for	for	ADP
ejpam-5133	38	15	their	their	PRON
ejpam-5133	38	16	unique	unique	ADJ
ejpam-5133	38	17	properties	property	NOUN
ejpam-5133	38	18	.	.	PUNCT
ejpam-5133	39	1	they	they	PRON
ejpam-5133	39	2	are	be	AUX
ejpam-5133	39	3	used	use	VERB
ejpam-5133	39	4	in	in	ADP
ejpam-5133	39	5	various	various	ADJ
ejpam-5133	39	6	mathematical	mathematical	ADJ
ejpam-5133	39	7	contexts	context	NOUN
ejpam-5133	39	8	and	and	CCONJ
ejpam-5133	39	9	practical	practical	ADJ
ejpam-5133	39	10	applications	application	NOUN
ejpam-5133	39	11	,	,	PUNCT
ejpam-5133	39	12	such	such	ADJ
ejpam-5133	39	13	as	as	ADP
ejpam-5133	39	14	cryptography	cryptography	NOUN
ejpam-5133	39	15	,	,	PUNCT
ejpam-5133	39	16	random	random	ADJ
ejpam-5133	39	17	number	number	NOUN
ejpam-5133	39	18	generation	generation	NOUN
ejpam-5133	39	19	,	,	PUNCT
ejpam-5133	39	20	algorithm	algorithm	NOUN
ejpam-5133	39	21	design	design	NOUN
ejpam-5133	39	22	,	,	PUNCT
ejpam-5133	39	23	solving	solve	VERB
ejpam-5133	39	24	problems	problem	NOUN
ejpam-5133	39	25	involving	involve	VERB
ejpam-5133	39	26	right	right	ADJ
ejpam-5133	39	27	-	-	PUNCT
ejpam-5133	39	28	angled	angle	VERB
ejpam-5133	39	29	triangles	triangle	NOUN
ejpam-5133	39	30	,	,	PUNCT
ejpam-5133	39	31	number	number	NOUN
ejpam-5133	39	32	theory	theory	NOUN
ejpam-5133	39	33	,	,	PUNCT
ejpam-5133	39	34	and	and	CCONJ
ejpam-5133	39	35	many	many	ADJ
ejpam-5133	39	36	other	other	ADJ
ejpam-5133	39	37	fields	field	NOUN
ejpam-5133	39	38	such	such	ADJ
ejpam-5133	39	39	as	as	ADP
ejpam-5133	39	40	physics	physics	NOUN
ejpam-5133	39	41	,	,	PUNCT
ejpam-5133	39	42	engineering	engineering	NOUN
ejpam-5133	39	43	,	,	PUNCT
ejpam-5133	39	44	and	and	CCONJ
ejpam-5133	39	45	computer	computer	NOUN
ejpam-5133	39	46	science	science	NOUN
ejpam-5133	40	1	[	[	X
ejpam-5133	40	2	11	11	NUM
ejpam-5133	40	3	]	]	PUNCT
ejpam-5133	40	4	,	,	PUNCT
ejpam-5133	40	5	[	[	X
ejpam-5133	40	6	12	12	NUM
ejpam-5133	40	7	]	]	PUNCT
ejpam-5133	40	8	.	.	PUNCT
ejpam-5133	41	1	beyond	beyond	ADP
ejpam-5133	41	2	mathematics	mathematic	NOUN
ejpam-5133	41	3	,	,	PUNCT
ejpam-5133	41	4	pythagorean	pythagorean	PROPN
ejpam-5133	41	5	triples	triple	NOUN
ejpam-5133	41	6	have	have	AUX
ejpam-5133	41	7	been	be	AUX
ejpam-5133	41	8	of	of	ADP
ejpam-5133	41	9	interest	interest	NOUN
ejpam-5133	41	10	in	in	ADP
ejpam-5133	41	11	the	the	DET
ejpam-5133	41	12	history	history	NOUN
ejpam-5133	41	13	of	of	ADP
ejpam-5133	41	14	art	art	NOUN
ejpam-5133	41	15	and	and	CCONJ
ejpam-5133	41	16	culture	culture	NOUN
ejpam-5133	41	17	,	,	PUNCT
ejpam-5133	41	18	appearing	appear	VERB
ejpam-5133	41	19	in	in	ADP
ejpam-5133	41	20	various	various	ADJ
ejpam-5133	41	21	forms	form	NOUN
ejpam-5133	41	22	in	in	ADP
ejpam-5133	41	23	architecture	architecture	NOUN
ejpam-5133	41	24	,	,	PUNCT
ejpam-5133	41	25	music	music	NOUN
ejpam-5133	41	26	,	,	PUNCT
ejpam-5133	41	27	and	and	CCONJ
ejpam-5133	41	28	symbolism	symbolism	NOUN
ejpam-5133	41	29	.	.	PUNCT
ejpam-5133	42	1	pythagorean	pythagorean	PROPN
ejpam-5133	42	2	triples	triple	NOUN
ejpam-5133	42	3	have	have	AUX
ejpam-5133	42	4	also	also	ADV
ejpam-5133	42	5	played	play	VERB
ejpam-5133	42	6	a	a	DET
ejpam-5133	42	7	significant	significant	ADJ
ejpam-5133	42	8	role	role	NOUN
ejpam-5133	42	9	in	in	ADP
ejpam-5133	42	10	number	number	NOUN
ejpam-5133	42	11	theory	theory	NOUN
ejpam-5133	42	12	,	,	PUNCT
ejpam-5133	42	13	with	with	ADP
ejpam-5133	42	14	mathematicians	mathematician	NOUN
ejpam-5133	42	15	like	like	ADP
ejpam-5133	42	16	fermat	fermat	PROPN
ejpam-5133	42	17	and	and	CCONJ
ejpam-5133	42	18	euclid	euclid	PROPN
ejpam-5133	42	19	studying	study	VERB
ejpam-5133	42	20	their	their	PRON
ejpam-5133	42	21	properties	property	NOUN
ejpam-5133	42	22	.	.	PUNCT
ejpam-5133	43	1	for	for	ADP
ejpam-5133	43	2	example	example	NOUN
ejpam-5133	43	3	,	,	PUNCT
ejpam-5133	43	4	a	a	DET
ejpam-5133	43	5	common	common	ADJ
ejpam-5133	43	6	formula	formula	NOUN
ejpam-5133	43	7	to	to	PART
ejpam-5133	43	8	generate	generate	VERB
ejpam-5133	43	9	pythagorean	pythagorean	NOUN
ejpam-5133	43	10	triples	triple	NOUN
ejpam-5133	43	11	is	be	AUX
ejpam-5133	43	12	the	the	DET
ejpam-5133	43	13	following	follow	VERB
ejpam-5133	43	14	:	:	PUNCT
ejpam-5133	43	15	x	x	SYM
ejpam-5133	43	16	=	=	SYM
ejpam-5133	43	17	m2	m2	PROPN
ejpam-5133	43	18	−	−	PROPN
ejpam-5133	43	19	n2	n2	PROPN
ejpam-5133	43	20	,	,	PUNCT
ejpam-5133	43	21	y	y	PROPN
ejpam-5133	43	22	=	=	SYM
ejpam-5133	43	23	2mn	2mn	PROPN
ejpam-5133	43	24	,	,	PUNCT
ejpam-5133	43	25	z	z	PROPN
ejpam-5133	43	26	=	=	SYM
ejpam-5133	43	27	m2	m2	PROPN
ejpam-5133	43	28	+	+	CCONJ
ejpam-5133	43	29	n2	n2	PROPN
ejpam-5133	43	30	where	where	SCONJ
ejpam-5133	43	31	m	m	VERB
ejpam-5133	43	32	and	and	CCONJ
ejpam-5133	43	33	n	n	PRON
ejpam-5133	43	34	are	be	AUX
ejpam-5133	43	35	arbitrary	arbitrary	ADJ
ejpam-5133	43	36	positive	positive	ADJ
ejpam-5133	43	37	integers	integer	NOUN
ejpam-5133	43	38	with	with	ADP
ejpam-5133	43	39	m	m	PROPN
ejpam-5133	43	40	>	>	X
ejpam-5133	43	41	n	n	CCONJ
ejpam-5133	43	42	,	,	PUNCT
ejpam-5133	43	43	and	and	CCONJ
ejpam-5133	43	44	m	m	PROPN
ejpam-5133	43	45	,	,	PUNCT
ejpam-5133	43	46	n	n	PROPN
ejpam-5133	43	47	∈	∈	NOUN
ejpam-5133	43	48	n	n	CCONJ
ejpam-5133	43	49	[	[	X
ejpam-5133	43	50	10	10	NUM
ejpam-5133	43	51	]	]	PUNCT
ejpam-5133	43	52	.	.	PUNCT
ejpam-5133	44	1	the	the	DET
ejpam-5133	44	2	above	above	ADJ
ejpam-5133	44	3	r.	r.	PROPN
ejpam-5133	44	4	amato	amato	PROPN
ejpam-5133	44	5	/	/	SYM
ejpam-5133	44	6	eur	eur	PROPN
ejpam-5133	44	7	.	.	PUNCT
ejpam-5133	45	1	j.	j.	PROPN
ejpam-5133	45	2	pure	pure	PROPN
ejpam-5133	45	3	appl	appl	PROPN
ejpam-5133	45	4	.	.	PROPN
ejpam-5133	45	5	math	math	PROPN
ejpam-5133	45	6	,	,	PUNCT
ejpam-5133	45	7	17	17	NUM
ejpam-5133	45	8	(	(	PUNCT
ejpam-5133	45	9	2	2	NUM
ejpam-5133	45	10	)	)	PUNCT
ejpam-5133	45	11	(	(	PUNCT
ejpam-5133	45	12	2024	2024	NUM
ejpam-5133	45	13	)	)	PUNCT
ejpam-5133	45	14	,	,	PUNCT
ejpam-5133	45	15	676	676	NUM
ejpam-5133	45	16	-	-	SYM
ejpam-5133	45	17	689	689	NUM
ejpam-5133	45	18	678	678	NUM
ejpam-5133	45	19	formula	formula	NOUN
ejpam-5133	45	20	mentioned	mention	VERB
ejpam-5133	45	21	earlier	early	ADV
ejpam-5133	45	22	,	,	PUNCT
ejpam-5133	45	23	involving	involve	VERB
ejpam-5133	45	24	the	the	DET
ejpam-5133	45	25	numbers	number	NOUN
ejpam-5133	45	26	m	m	VERB
ejpam-5133	45	27	and	and	CCONJ
ejpam-5133	45	28	n	n	CCONJ
ejpam-5133	45	29	,	,	PUNCT
ejpam-5133	45	30	was	be	AUX
ejpam-5133	45	31	developed	develop	VERB
ejpam-5133	45	32	by	by	ADP
ejpam-5133	45	33	the	the	DET
ejpam-5133	45	34	euclid	euclid	PROPN
ejpam-5133	45	35	,	,	PUNCT
ejpam-5133	45	36	and	and	CCONJ
ejpam-5133	45	37	the	the	DET
ejpam-5133	45	38	majority	majority	NOUN
ejpam-5133	45	39	of	of	ADP
ejpam-5133	45	40	results	result	NOUN
ejpam-5133	45	41	on	on	ADP
ejpam-5133	45	42	pythagorean	pythagorean	PROPN
ejpam-5133	45	43	triples	triple	NOUN
ejpam-5133	45	44	are	be	AUX
ejpam-5133	45	45	due	due	ADJ
ejpam-5133	45	46	to	to	ADP
ejpam-5133	45	47	this	this	DET
ejpam-5133	45	48	formula	formula	NOUN
ejpam-5133	46	1	[	[	X
ejpam-5133	46	2	9	9	NUM
ejpam-5133	46	3	]	]	PUNCT
ejpam-5133	46	4	,	,	PUNCT
ejpam-5133	47	1	[	[	X
ejpam-5133	47	2	8	8	NUM
ejpam-5133	47	3	]	]	PUNCT
ejpam-5133	47	4	,	,	PUNCT
ejpam-5133	47	5	[	[	X
ejpam-5133	47	6	1	1	NUM
ejpam-5133	47	7	]	]	PUNCT
ejpam-5133	47	8	.	.	PUNCT
ejpam-5133	48	1	in	in	ADP
ejpam-5133	48	2	addition	addition	NOUN
ejpam-5133	48	3	to	to	ADP
ejpam-5133	48	4	the	the	DET
ejpam-5133	48	5	scientific	scientific	ADJ
ejpam-5133	48	6	aspects	aspect	NOUN
ejpam-5133	48	7	of	of	ADP
ejpam-5133	48	8	pythagorean	pythagorean	PROPN
ejpam-5133	48	9	triples	triple	NOUN
ejpam-5133	48	10	,	,	PUNCT
ejpam-5133	48	11	we	we	PRON
ejpam-5133	48	12	also	also	ADV
ejpam-5133	48	13	underscore	underscore	VERB
ejpam-5133	48	14	several	several	ADJ
ejpam-5133	48	15	valuable	valuable	ADJ
ejpam-5133	48	16	didactic	didactic	ADJ
ejpam-5133	48	17	perspectives	perspective	NOUN
ejpam-5133	48	18	that	that	PRON
ejpam-5133	48	19	enrich	enrich	VERB
ejpam-5133	48	20	the	the	DET
ejpam-5133	48	21	learning	learning	NOUN
ejpam-5133	48	22	experience	experience	NOUN
ejpam-5133	48	23	:	:	PUNCT
ejpam-5133	48	24	pythagorean	pythagorean	PROPN
ejpam-5133	48	25	triples	triple	NOUN
ejpam-5133	48	26	serve	serve	VERB
ejpam-5133	48	27	as	as	ADP
ejpam-5133	48	28	a	a	DET
ejpam-5133	48	29	tangible	tangible	ADJ
ejpam-5133	48	30	gateway	gateway	NOUN
ejpam-5133	48	31	to	to	ADP
ejpam-5133	48	32	the	the	DET
ejpam-5133	48	33	realm	realm	NOUN
ejpam-5133	48	34	of	of	ADP
ejpam-5133	48	35	number	number	NOUN
ejpam-5133	48	36	theory	theory	NOUN
ejpam-5133	48	37	.	.	PUNCT
ejpam-5133	49	1	by	by	ADP
ejpam-5133	49	2	actively	actively	ADV
ejpam-5133	49	3	engaging	engage	VERB
ejpam-5133	49	4	with	with	ADP
ejpam-5133	49	5	these	these	DET
ejpam-5133	49	6	triples	triple	NOUN
ejpam-5133	49	7	,	,	PUNCT
ejpam-5133	49	8	students	student	NOUN
ejpam-5133	49	9	can	can	AUX
ejpam-5133	49	10	develop	develop	VERB
ejpam-5133	49	11	a	a	DET
ejpam-5133	49	12	hands	hand	NOUN
ejpam-5133	49	13	-	-	PUNCT
ejpam-5133	49	14	on	on	ADP
ejpam-5133	49	15	understanding	understanding	NOUN
ejpam-5133	49	16	of	of	ADP
ejpam-5133	49	17	fundamental	fundamental	ADJ
ejpam-5133	49	18	concepts	concept	NOUN
ejpam-5133	49	19	in	in	ADP
ejpam-5133	49	20	number	number	NOUN
ejpam-5133	49	21	theory	theory	NOUN
ejpam-5133	49	22	,	,	PUNCT
ejpam-5133	49	23	such	such	ADJ
ejpam-5133	49	24	as	as	ADP
ejpam-5133	49	25	divisibility	divisibility	NOUN
ejpam-5133	49	26	,	,	PUNCT
ejpam-5133	49	27	prime	prime	ADJ
ejpam-5133	49	28	factorization	factorization	NOUN
ejpam-5133	49	29	,	,	PUNCT
ejpam-5133	49	30	and	and	CCONJ
ejpam-5133	49	31	the	the	DET
ejpam-5133	49	32	properties	property	NOUN
ejpam-5133	49	33	of	of	ADP
ejpam-5133	49	34	integers	integer	NOUN
ejpam-5133	49	35	.	.	PUNCT
ejpam-5133	50	1	delving	delve	VERB
ejpam-5133	50	2	into	into	ADP
ejpam-5133	50	3	pythagorean	pythagorean	PROPN
ejpam-5133	50	4	triples	triple	NOUN
ejpam-5133	50	5	provides	provide	VERB
ejpam-5133	50	6	an	an	DET
ejpam-5133	50	7	opportunity	opportunity	NOUN
ejpam-5133	50	8	to	to	PART
ejpam-5133	50	9	reinforce	reinforce	VERB
ejpam-5133	50	10	the	the	DET
ejpam-5133	50	11	geometric	geometric	ADJ
ejpam-5133	50	12	interpretation	interpretation	NOUN
ejpam-5133	50	13	of	of	ADP
ejpam-5133	50	14	the	the	DET
ejpam-5133	50	15	pythagorean	pythagorean	PROPN
ejpam-5133	50	16	theorem	theorem	PROPN
ejpam-5133	50	17	.	.	PROPN
ejpam-5133	51	1	through	through	ADP
ejpam-5133	51	2	visualizing	visualize	VERB
ejpam-5133	51	3	right	right	ADJ
ejpam-5133	51	4	-	-	PUNCT
ejpam-5133	51	5	angled	angle	VERB
ejpam-5133	51	6	triangles	triangle	NOUN
ejpam-5133	51	7	and	and	CCONJ
ejpam-5133	51	8	their	their	PRON
ejpam-5133	51	9	associated	associated	ADJ
ejpam-5133	51	10	triples	triple	NOUN
ejpam-5133	51	11	,	,	PUNCT
ejpam-5133	51	12	students	student	NOUN
ejpam-5133	51	13	gain	gain	VERB
ejpam-5133	51	14	insights	insight	NOUN
ejpam-5133	51	15	into	into	ADP
ejpam-5133	51	16	the	the	DET
ejpam-5133	51	17	geometric	geometric	ADJ
ejpam-5133	51	18	relationships	relationship	NOUN
ejpam-5133	51	19	embedded	embed	VERB
ejpam-5133	51	20	in	in	ADP
ejpam-5133	51	21	the	the	DET
ejpam-5133	51	22	theorem	theorem	NOUN
ejpam-5133	51	23	,	,	PUNCT
ejpam-5133	51	24	fostering	foster	VERB
ejpam-5133	51	25	a	a	DET
ejpam-5133	51	26	deeper	deep	ADJ
ejpam-5133	51	27	comprehension	comprehension	NOUN
ejpam-5133	51	28	of	of	ADP
ejpam-5133	51	29	its	its	PRON
ejpam-5133	51	30	principles	principle	NOUN
ejpam-5133	51	31	.	.	PUNCT
ejpam-5133	52	1	the	the	DET
ejpam-5133	52	2	exploration	exploration	NOUN
ejpam-5133	52	3	of	of	ADP
ejpam-5133	52	4	pythagorean	pythagorean	PROPN
ejpam-5133	52	5	triples	triple	NOUN
ejpam-5133	52	6	naturally	naturally	ADV
ejpam-5133	52	7	encourages	encourage	VERB
ejpam-5133	52	8	students	student	NOUN
ejpam-5133	52	9	to	to	PART
ejpam-5133	52	10	recognize	recognize	VERB
ejpam-5133	52	11	patterns	pattern	NOUN
ejpam-5133	52	12	within	within	ADP
ejpam-5133	52	13	numerical	numerical	ADJ
ejpam-5133	52	14	relationships	relationship	NOUN
ejpam-5133	52	15	.	.	PUNCT
ejpam-5133	53	1	analyzing	analyze	VERB
ejpam-5133	53	2	these	these	DET
ejpam-5133	53	3	triples	triple	NOUN
ejpam-5133	53	4	prompts	prompt	VERB
ejpam-5133	53	5	discussions	discussion	NOUN
ejpam-5133	53	6	about	about	ADP
ejpam-5133	53	7	the	the	DET
ejpam-5133	53	8	symmetry	symmetry	NOUN
ejpam-5133	53	9	inherent	inherent	ADJ
ejpam-5133	53	10	in	in	ADP
ejpam-5133	53	11	certain	certain	ADJ
ejpam-5133	53	12	configurations	configuration	NOUN
ejpam-5133	53	13	,	,	PUNCT
ejpam-5133	53	14	the	the	DET
ejpam-5133	53	15	distinctive	distinctive	ADJ
ejpam-5133	53	16	roles	role	NOUN
ejpam-5133	53	17	of	of	ADP
ejpam-5133	53	18	odd	odd	ADJ
ejpam-5133	53	19	and	and	CCONJ
ejpam-5133	53	20	even	even	ADV
ejpam-5133	53	21	numbers	number	NOUN
ejpam-5133	53	22	,	,	PUNCT
ejpam-5133	53	23	and	and	CCONJ
ejpam-5133	53	24	the	the	DET
ejpam-5133	53	25	impact	impact	NOUN
ejpam-5133	53	26	of	of	ADP
ejpam-5133	53	27	scaling	scale	VERB
ejpam-5133	53	28	factors	factor	NOUN
ejpam-5133	53	29	on	on	ADP
ejpam-5133	53	30	the	the	DET
ejpam-5133	53	31	generation	generation	NOUN
ejpam-5133	53	32	of	of	ADP
ejpam-5133	53	33	triples	triple	NOUN
ejpam-5133	53	34	.	.	PUNCT
ejpam-5133	54	1	this	this	DET
ejpam-5133	54	2	process	process	NOUN
ejpam-5133	54	3	cultivates	cultivate	VERB
ejpam-5133	54	4	analytical	analytical	ADJ
ejpam-5133	54	5	thinking	thinking	NOUN
ejpam-5133	54	6	and	and	CCONJ
ejpam-5133	54	7	the	the	DET
ejpam-5133	54	8	ability	ability	NOUN
ejpam-5133	54	9	to	to	PART
ejpam-5133	54	10	discern	discern	VERB
ejpam-5133	54	11	mathematical	mathematical	ADJ
ejpam-5133	54	12	patterns	pattern	NOUN
ejpam-5133	54	13	in	in	ADP
ejpam-5133	54	14	different	different	ADJ
ejpam-5133	54	15	contexts	contexts	NOUN
ejpam-5133	54	16	.	.	PUNCT
ejpam-5133	55	1	working	work	VERB
ejpam-5133	55	2	with	with	ADP
ejpam-5133	55	3	pythagorean	pythagorean	PROPN
ejpam-5133	55	4	triples	triple	NOUN
ejpam-5133	55	5	presents	present	VERB
ejpam-5133	55	6	students	student	NOUN
ejpam-5133	55	7	with	with	ADP
ejpam-5133	55	8	a	a	DET
ejpam-5133	55	9	variety	variety	NOUN
ejpam-5133	55	10	of	of	ADP
ejpam-5133	55	11	mathematical	mathematical	ADJ
ejpam-5133	55	12	scenarios	scenario	NOUN
ejpam-5133	55	13	that	that	PRON
ejpam-5133	55	14	require	require	VERB
ejpam-5133	55	15	creative	creative	ADJ
ejpam-5133	55	16	problem	problem	NOUN
ejpam-5133	55	17	-	-	PUNCT
ejpam-5133	55	18	solving	solve	VERB
ejpam-5133	55	19	approaches	approach	NOUN
ejpam-5133	55	20	.	.	PUNCT
ejpam-5133	56	1	as	as	SCONJ
ejpam-5133	56	2	they	they	PRON
ejpam-5133	56	3	investigate	investigate	VERB
ejpam-5133	56	4	unique	unique	ADJ
ejpam-5133	56	5	cases	case	NOUN
ejpam-5133	56	6	and	and	CCONJ
ejpam-5133	56	7	consider	consider	VERB
ejpam-5133	56	8	different	different	ADJ
ejpam-5133	56	9	parameterizations	parameterization	NOUN
ejpam-5133	56	10	,	,	PUNCT
ejpam-5133	56	11	students	student	NOUN
ejpam-5133	56	12	enhance	enhance	VERB
ejpam-5133	56	13	their	their	PRON
ejpam-5133	56	14	problem	problem	NOUN
ejpam-5133	56	15	-	-	PUNCT
ejpam-5133	56	16	solving	solve	VERB
ejpam-5133	56	17	skills	skill	NOUN
ejpam-5133	56	18	and	and	CCONJ
ejpam-5133	56	19	develop	develop	VERB
ejpam-5133	56	20	a	a	DET
ejpam-5133	56	21	robust	robust	ADJ
ejpam-5133	56	22	toolkit	toolkit	NOUN
ejpam-5133	56	23	for	for	ADP
ejpam-5133	56	24	addressing	address	VERB
ejpam-5133	56	25	mathematical	mathematical	ADJ
ejpam-5133	56	26	challenges	challenge	NOUN
ejpam-5133	56	27	.	.	PUNCT
ejpam-5133	57	1	exploring	explore	VERB
ejpam-5133	57	2	the	the	DET
ejpam-5133	57	3	origins	origin	NOUN
ejpam-5133	57	4	and	and	CCONJ
ejpam-5133	57	5	historical	historical	ADJ
ejpam-5133	57	6	significance	significance	NOUN
ejpam-5133	57	7	of	of	ADP
ejpam-5133	57	8	pythagorean	pythagorean	PROPN
ejpam-5133	57	9	triples	triple	NOUN
ejpam-5133	57	10	provides	provide	VERB
ejpam-5133	57	11	a	a	DET
ejpam-5133	57	12	broader	broad	ADJ
ejpam-5133	57	13	context	context	NOUN
ejpam-5133	57	14	for	for	ADP
ejpam-5133	57	15	their	their	PRON
ejpam-5133	57	16	study	study	NOUN
ejpam-5133	57	17	.	.	PUNCT
ejpam-5133	58	1	students	student	NOUN
ejpam-5133	58	2	can	can	AUX
ejpam-5133	58	3	appreciate	appreciate	VERB
ejpam-5133	58	4	the	the	DET
ejpam-5133	58	5	cultural	cultural	ADJ
ejpam-5133	58	6	contributions	contribution	NOUN
ejpam-5133	58	7	of	of	ADP
ejpam-5133	58	8	ancient	ancient	ADJ
ejpam-5133	58	9	mathematicians	mathematician	NOUN
ejpam-5133	58	10	like	like	ADP
ejpam-5133	58	11	pythagoras	pythagoras	PROPN
ejpam-5133	58	12	and	and	CCONJ
ejpam-5133	58	13	recognize	recognize	VERB
ejpam-5133	58	14	the	the	DET
ejpam-5133	58	15	enduring	endure	VERB
ejpam-5133	58	16	legacy	legacy	NOUN
ejpam-5133	58	17	of	of	ADP
ejpam-5133	58	18	these	these	DET
ejpam-5133	58	19	triples	triple	NOUN
ejpam-5133	58	20	in	in	ADP
ejpam-5133	58	21	various	various	ADJ
ejpam-5133	58	22	mathematical	mathematical	ADJ
ejpam-5133	58	23	and	and	CCONJ
ejpam-5133	58	24	scientific	scientific	ADJ
ejpam-5133	58	25	disciplines	discipline	NOUN
ejpam-5133	58	26	.	.	PUNCT
ejpam-5133	59	1	by	by	ADP
ejpam-5133	59	2	embracing	embrace	VERB
ejpam-5133	59	3	these	these	DET
ejpam-5133	59	4	didactic	didactic	ADJ
ejpam-5133	59	5	perspectives	perspective	NOUN
ejpam-5133	59	6	,	,	PUNCT
ejpam-5133	59	7	the	the	DET
ejpam-5133	59	8	study	study	NOUN
ejpam-5133	59	9	of	of	ADP
ejpam-5133	59	10	pythagorean	pythagorean	PROPN
ejpam-5133	59	11	triples	triple	NOUN
ejpam-5133	59	12	transcends	transcend	VERB
ejpam-5133	59	13	mere	mere	ADJ
ejpam-5133	59	14	mathematical	mathematical	ADJ
ejpam-5133	59	15	abstraction	abstraction	NOUN
ejpam-5133	59	16	,	,	PUNCT
ejpam-5133	59	17	offering	offer	VERB
ejpam-5133	59	18	a	a	DET
ejpam-5133	59	19	rich	rich	ADJ
ejpam-5133	59	20	and	and	CCONJ
ejpam-5133	59	21	interconnected	interconnected	ADJ
ejpam-5133	59	22	learning	learning	NOUN
ejpam-5133	59	23	experience	experience	NOUN
ejpam-5133	59	24	that	that	PRON
ejpam-5133	59	25	extends	extend	VERB
ejpam-5133	59	26	beyond	beyond	ADP
ejpam-5133	59	27	the	the	DET
ejpam-5133	59	28	confines	confine	NOUN
ejpam-5133	59	29	of	of	ADP
ejpam-5133	59	30	a	a	DET
ejpam-5133	59	31	single	single	ADJ
ejpam-5133	59	32	theorem	theorem	NOUN
ejpam-5133	59	33	.	.	PUNCT
ejpam-5133	60	1	this	this	DET
ejpam-5133	60	2	multifaceted	multifaceted	ADJ
ejpam-5133	60	3	approach	approach	NOUN
ejpam-5133	60	4	not	not	PART
ejpam-5133	60	5	only	only	ADV
ejpam-5133	60	6	deepens	deepen	VERB
ejpam-5133	60	7	students	student	NOUN
ejpam-5133	60	8	’	’	PART
ejpam-5133	60	9	understanding	understanding	NOUN
ejpam-5133	60	10	of	of	ADP
ejpam-5133	60	11	mathematical	mathematical	ADJ
ejpam-5133	60	12	concepts	concept	NOUN
ejpam-5133	60	13	but	but	CCONJ
ejpam-5133	60	14	also	also	ADV
ejpam-5133	60	15	nurtures	nurture	VERB
ejpam-5133	60	16	a	a	DET
ejpam-5133	60	17	broader	broad	ADJ
ejpam-5133	60	18	appreciation	appreciation	NOUN
ejpam-5133	60	19	for	for	ADP
ejpam-5133	60	20	the	the	DET
ejpam-5133	60	21	historical	historical	ADJ
ejpam-5133	60	22	,	,	PUNCT
ejpam-5133	60	23	cultural	cultural	ADJ
ejpam-5133	60	24	,	,	PUNCT
ejpam-5133	60	25	and	and	CCONJ
ejpam-5133	60	26	problem	problem	NOUN
ejpam-5133	60	27	-	-	PUNCT
ejpam-5133	60	28	solving	solve	VERB
ejpam-5133	60	29	dimensions	dimension	NOUN
ejpam-5133	60	30	of	of	ADP
ejpam-5133	60	31	mathematics	mathematic	NOUN
ejpam-5133	60	32	.	.	PUNCT
ejpam-5133	61	1	already	already	ADV
ejpam-5133	61	2	in	in	ADP
ejpam-5133	61	3	1981	1981	NUM
ejpam-5133	61	4	,	,	PUNCT
ejpam-5133	61	5	as	as	ADP
ejpam-5133	61	6	a	a	DET
ejpam-5133	61	7	student	student	NOUN
ejpam-5133	61	8	,	,	PUNCT
ejpam-5133	61	9	i	i	PRON
ejpam-5133	61	10	had	have	AUX
ejpam-5133	61	11	studied	study	VERB
ejpam-5133	61	12	how	how	SCONJ
ejpam-5133	61	13	to	to	PART
ejpam-5133	61	14	generate	generate	VERB
ejpam-5133	61	15	pythagorean	pythagorean	NOUN
ejpam-5133	61	16	triples	triple	NOUN
ejpam-5133	61	17	,	,	PUNCT
ejpam-5133	61	18	achieving	achieve	VERB
ejpam-5133	61	19	a	a	DET
ejpam-5133	61	20	preliminary	preliminary	ADJ
ejpam-5133	61	21	result	result	NOUN
ejpam-5133	61	22	[	[	X
ejpam-5133	61	23	2	2	NUM
ejpam-5133	61	24	]	]	PUNCT
ejpam-5133	61	25	.	.	PUNCT
ejpam-5133	62	1	after	after	ADP
ejpam-5133	62	2	many	many	ADJ
ejpam-5133	62	3	years	year	NOUN
ejpam-5133	62	4	,	,	PUNCT
ejpam-5133	62	5	returning	return	VERB
ejpam-5133	62	6	to	to	PART
ejpam-5133	62	7	study	study	VERB
ejpam-5133	62	8	the	the	DET
ejpam-5133	62	9	topic	topic	NOUN
ejpam-5133	62	10	,	,	PUNCT
ejpam-5133	62	11	i	i	PRON
ejpam-5133	62	12	found	find	VERB
ejpam-5133	62	13	a	a	DET
ejpam-5133	62	14	new	new	ADJ
ejpam-5133	62	15	and	and	CCONJ
ejpam-5133	62	16	comprehensive	comprehensive	ADJ
ejpam-5133	62	17	result	result	NOUN
ejpam-5133	62	18	[	[	X
ejpam-5133	62	19	3	3	X
ejpam-5133	62	20	]	]	PUNCT
ejpam-5133	62	21	that	that	PRON
ejpam-5133	62	22	is	be	AUX
ejpam-5133	62	23	suitable	suitable	ADJ
ejpam-5133	62	24	for	for	ADP
ejpam-5133	62	25	obtaining	obtain	VERB
ejpam-5133	62	26	new	new	ADJ
ejpam-5133	62	27	results	result	NOUN
ejpam-5133	62	28	and	and	CCONJ
ejpam-5133	62	29	applications	application	NOUN
ejpam-5133	62	30	in	in	ADP
ejpam-5133	62	31	fields	field	NOUN
ejpam-5133	62	32	such	such	ADJ
ejpam-5133	62	33	as	as	ADP
ejpam-5133	62	34	geometry	geometry	NOUN
ejpam-5133	62	35	,	,	PUNCT
ejpam-5133	62	36	trigonometry	trigonometry	NOUN
ejpam-5133	62	37	,	,	PUNCT
ejpam-5133	62	38	linear	linear	ADJ
ejpam-5133	62	39	algebra	algebra	NOUN
ejpam-5133	62	40	,	,	PUNCT
ejpam-5133	62	41	and	and	CCONJ
ejpam-5133	62	42	number	number	NOUN
ejpam-5133	62	43	theory	theory	NOUN
ejpam-5133	62	44	.	.	PUNCT
ejpam-5133	63	1	this	this	DET
ejpam-5133	63	2	paper	paper	NOUN
ejpam-5133	63	3	seeks	seek	VERB
ejpam-5133	63	4	to	to	PART
ejpam-5133	63	5	showcase	showcase	VERB
ejpam-5133	63	6	how	how	SCONJ
ejpam-5133	63	7	a	a	DET
ejpam-5133	63	8	new	new	ADJ
ejpam-5133	63	9	approach	approach	NOUN
ejpam-5133	63	10	can	can	AUX
ejpam-5133	63	11	breathe	breathe	VERB
ejpam-5133	63	12	new	new	ADJ
ejpam-5133	63	13	life	life	NOUN
ejpam-5133	63	14	into	into	ADP
ejpam-5133	63	15	research	research	NOUN
ejpam-5133	63	16	within	within	ADP
ejpam-5133	63	17	the	the	DET
ejpam-5133	63	18	traditional	traditional	ADJ
ejpam-5133	63	19	domain	domain	NOUN
ejpam-5133	63	20	of	of	ADP
ejpam-5133	63	21	pythagorean	pythagorean	PROPN
ejpam-5133	63	22	triples	triple	NOUN
ejpam-5133	63	23	,	,	PUNCT
ejpam-5133	63	24	introducing	introduce	VERB
ejpam-5133	63	25	innovative	innovative	ADJ
ejpam-5133	63	26	applications	application	NOUN
ejpam-5133	63	27	to	to	PART
ejpam-5133	63	28	invigorate	invigorate	VERB
ejpam-5133	63	29	the	the	DET
ejpam-5133	63	30	field	field	NOUN
ejpam-5133	63	31	.	.	PUNCT
ejpam-5133	64	1	this	this	PRON
ejpam-5133	64	2	serves	serve	VERB
ejpam-5133	64	3	not	not	PART
ejpam-5133	64	4	only	only	ADV
ejpam-5133	64	5	as	as	ADP
ejpam-5133	64	6	an	an	DET
ejpam-5133	64	7	exemplar	exemplar	NOUN
ejpam-5133	64	8	but	but	CCONJ
ejpam-5133	64	9	also	also	ADV
ejpam-5133	64	10	as	as	ADP
ejpam-5133	64	11	a	a	DET
ejpam-5133	64	12	wellspring	wellspring	NOUN
ejpam-5133	64	13	of	of	ADP
ejpam-5133	64	14	inspiration	inspiration	NOUN
ejpam-5133	64	15	for	for	ADP
ejpam-5133	64	16	students	student	NOUN
ejpam-5133	64	17	at	at	ADP
ejpam-5133	64	18	both	both	DET
ejpam-5133	64	19	school	school	NOUN
ejpam-5133	64	20	and	and	CCONJ
ejpam-5133	64	21	university	university	NOUN
ejpam-5133	64	22	levels	level	NOUN
ejpam-5133	64	23	.	.	PUNCT
ejpam-5133	65	1	the	the	DET
ejpam-5133	65	2	demonstration	demonstration	NOUN
ejpam-5133	65	3	will	will	AUX
ejpam-5133	65	4	underscore	underscore	VERB
ejpam-5133	65	5	that	that	PRON
ejpam-5133	65	6	,	,	PUNCT
ejpam-5133	65	7	with	with	ADP
ejpam-5133	65	8	fundamental	fundamental	ADJ
ejpam-5133	65	9	mathematical	mathematical	ADJ
ejpam-5133	65	10	concepts	concept	NOUN
ejpam-5133	65	11	and	and	CCONJ
ejpam-5133	65	12	unencumbered	unencumbered	ADJ
ejpam-5133	65	13	by	by	ADP
ejpam-5133	65	14	intricate	intricate	ADJ
ejpam-5133	65	15	calculations	calculation	NOUN
ejpam-5133	65	16	,	,	PUNCT
ejpam-5133	65	17	one	one	PRON
ejpam-5133	65	18	can	can	AUX
ejpam-5133	65	19	unveil	unveil	VERB
ejpam-5133	65	20	novel	novel	ADJ
ejpam-5133	65	21	results	result	NOUN
ejpam-5133	65	22	and	and	CCONJ
ejpam-5133	65	23	applications	application	NOUN
ejpam-5133	65	24	with	with	ADP
ejpam-5133	65	25	ease	ease	NOUN
ejpam-5133	65	26	.	.	PUNCT
ejpam-5133	66	1	r.	r.	PROPN
ejpam-5133	66	2	amato	amato	PROPN
ejpam-5133	66	3	/	/	SYM
ejpam-5133	66	4	eur	eur	PROPN
ejpam-5133	66	5	.	.	PUNCT
ejpam-5133	67	1	j.	j.	PROPN
ejpam-5133	67	2	pure	pure	PROPN
ejpam-5133	67	3	appl	appl	PROPN
ejpam-5133	67	4	.	.	PROPN
ejpam-5133	67	5	math	math	PROPN
ejpam-5133	67	6	,	,	PUNCT
ejpam-5133	67	7	17	17	NUM
ejpam-5133	67	8	(	(	PUNCT
ejpam-5133	67	9	2	2	NUM
ejpam-5133	67	10	)	)	PUNCT
ejpam-5133	67	11	(	(	PUNCT
ejpam-5133	67	12	2024	2024	NUM
ejpam-5133	67	13	)	)	PUNCT
ejpam-5133	67	14	,	,	PUNCT
ejpam-5133	67	15	676	676	NUM
ejpam-5133	67	16	-	-	SYM
ejpam-5133	67	17	689	689	NUM
ejpam-5133	67	18	679	679	NUM
ejpam-5133	67	19	2	2	NUM
ejpam-5133	67	20	.	.	PUNCT
ejpam-5133	68	1	prelimunary	prelimunary	ADJ
ejpam-5133	68	2	results	result	NOUN
ejpam-5133	68	3	let	let	VERB
ejpam-5133	68	4	us	we	PRON
ejpam-5133	68	5	review	review	VERB
ejpam-5133	68	6	recent	recent	ADJ
ejpam-5133	68	7	results	result	NOUN
ejpam-5133	68	8	concerning	concern	VERB
ejpam-5133	68	9	some	some	DET
ejpam-5133	68	10	relations	relation	NOUN
ejpam-5133	68	11	among	among	ADP
ejpam-5133	68	12	pythagorean	pythagorean	PROPN
ejpam-5133	68	13	triples	triple	NOUN
ejpam-5133	68	14	that	that	PRON
ejpam-5133	68	15	have	have	AUX
ejpam-5133	68	16	already	already	ADV
ejpam-5133	68	17	been	be	AUX
ejpam-5133	68	18	established	establish	VERB
ejpam-5133	68	19	.	.	PUNCT
ejpam-5133	69	1	the	the	DET
ejpam-5133	69	2	primary	primary	ADJ
ejpam-5133	69	3	tool	tool	NOUN
ejpam-5133	69	4	utilized	utilize	VERB
ejpam-5133	69	5	in	in	ADP
ejpam-5133	69	6	those	those	DET
ejpam-5133	69	7	works	work	NOUN
ejpam-5133	69	8	was	be	AUX
ejpam-5133	69	9	the	the	DET
ejpam-5133	69	10	fundamental	fundamental	ADJ
ejpam-5133	69	11	characterization	characterization	NOUN
ejpam-5133	69	12	of	of	ADP
ejpam-5133	69	13	pythagorean	pythagorean	PROPN
ejpam-5133	69	14	triples	triple	NOUN
ejpam-5133	69	15	through	through	ADP
ejpam-5133	69	16	a	a	DET
ejpam-5133	69	17	cathetus	cathetus	NOUN
ejpam-5133	69	18	.	.	PUNCT
ejpam-5133	70	1	this	this	DET
ejpam-5133	70	2	reads	read	VERB
ejpam-5133	70	3	as	as	SCONJ
ejpam-5133	70	4	follows	follow	VERB
ejpam-5133	70	5	.	.	PUNCT
ejpam-5133	71	1	theorem	theorem	NOUN
ejpam-5133	71	2	1	1	NUM
ejpam-5133	71	3	.	.	PUNCT
ejpam-5133	72	1	[	[	X
ejpam-5133	72	2	3	3	X
ejpam-5133	72	3	]	]	X
ejpam-5133	72	4	the	the	DET
ejpam-5133	72	5	triple	triple	ADJ
ejpam-5133	72	6	(	(	PUNCT
ejpam-5133	72	7	x	x	NOUN
ejpam-5133	72	8	,	,	PUNCT
ejpam-5133	72	9	y	y	PROPN
ejpam-5133	72	10	,	,	PUNCT
ejpam-5133	72	11	z	z	NOUN
ejpam-5133	72	12	)	)	PUNCT
ejpam-5133	72	13	is	be	AUX
ejpam-5133	72	14	a	a	DET
ejpam-5133	72	15	pythagorean	pythagorean	PROPN
ejpam-5133	72	16	triple	triple	NOUN
ejpam-5133	72	17	if	if	SCONJ
ejpam-5133	73	1	and	and	CCONJ
ejpam-5133	73	2	only	only	ADV
ejpam-5133	73	3	if	if	SCONJ
ejpam-5133	73	4	there	there	PRON
ejpam-5133	73	5	exists	exist	VERB
ejpam-5133	73	6	d	d	PROPN
ejpam-5133	73	7	∈	∈	PROPN
ejpam-5133	73	8	c(x	c(x	NOUN
ejpam-5133	73	9	)	)	PUNCT
ejpam-5133	74	1	such	such	ADJ
ejpam-5133	74	2	that	that	SCONJ
ejpam-5133	74	3	x	x	X
ejpam-5133	74	4	=	=	SYM
ejpam-5133	74	5	x	x	NOUN
ejpam-5133	74	6	,	,	PUNCT
ejpam-5133	74	7	y	y	PROPN
ejpam-5133	74	8	=	=	SYM
ejpam-5133	74	9	x2	x2	PROPN
ejpam-5133	74	10	2d	2d	NOUN
ejpam-5133	74	11	−	−	PROPN
ejpam-5133	74	12	d	d	SYM
ejpam-5133	74	13	2	2	NUM
ejpam-5133	74	14	,	,	PUNCT
ejpam-5133	74	15	z	z	NOUN
ejpam-5133	74	16	=	=	SYM
ejpam-5133	74	17	x2	x2	NUM
ejpam-5133	74	18	2d	2d	NOUN
ejpam-5133	75	1	+	+	CCONJ
ejpam-5133	75	2	d	d	SYM
ejpam-5133	75	3	2	2	NUM
ejpam-5133	75	4	(	(	PUNCT
ejpam-5133	75	5	1	1	NUM
ejpam-5133	75	6	)	)	PUNCT
ejpam-5133	75	7	with	with	ADP
ejpam-5133	75	8	x	x	SYM
ejpam-5133	75	9	positive	positive	ADJ
ejpam-5133	75	10	integer	integer	NOUN
ejpam-5133	75	11	,	,	PUNCT
ejpam-5133	75	12	and	and	CCONJ
ejpam-5133	75	13	where	where	SCONJ
ejpam-5133	75	14	c(x	c(x	NOUN
ejpam-5133	75	15	)	)	PUNCT
ejpam-5133	75	16	=	=	PUNCT
ejpam-5133	75	17			NOUN
ejpam-5133	75	18	d(x	d(x	PROPN
ejpam-5133	75	19	)	)	PUNCT
ejpam-5133	75	20	,	,	PUNCT
ejpam-5133	75	21	if	if	SCONJ
ejpam-5133	75	22	x	x	PRON
ejpam-5133	75	23	is	be	AUX
ejpam-5133	75	24	odd	odd	ADJ
ejpam-5133	75	25	,	,	PUNCT
ejpam-5133	75	26	d(x	d(x	PROPN
ejpam-5133	75	27	)	)	PUNCT
ejpam-5133	75	28	∩	∩	NOUN
ejpam-5133	75	29	p	p	X
ejpam-5133	75	30	(	(	PUNCT
ejpam-5133	75	31	x	x	NOUN
ejpam-5133	75	32	)	)	PUNCT
ejpam-5133	75	33	,	,	PUNCT
ejpam-5133	75	34	if	if	SCONJ
ejpam-5133	75	35	x	x	PRON
ejpam-5133	75	36	is	be	AUX
ejpam-5133	75	37	even	even	ADV
ejpam-5133	75	38	,	,	PUNCT
ejpam-5133	75	39	with	with	ADP
ejpam-5133	75	40	d(x	d(x	NOUN
ejpam-5133	75	41	)	)	PUNCT
ejpam-5133	75	42	=	=	PRON
ejpam-5133	76	1	{	{	PUNCT
ejpam-5133	76	2	d	d	X
ejpam-5133	76	3	∈	∈	PROPN
ejpam-5133	76	4	n	n	NOUN
ejpam-5133	76	5	:	:	PUNCT
ejpam-5133	76	6	d	d	X
ejpam-5133	76	7	≤	≤	NUM
ejpam-5133	76	8	x	x	PUNCT
ejpam-5133	76	9	with	with	ADP
ejpam-5133	76	10	d	d	NOUN
ejpam-5133	76	11	divisor	divisor	NOUN
ejpam-5133	76	12	of	of	ADP
ejpam-5133	76	13	x2	x2	PROPN
ejpam-5133	76	14	}	}	PUNCT
ejpam-5133	76	15	,	,	PUNCT
ejpam-5133	76	16	and	and	CCONJ
ejpam-5133	76	17	if	if	SCONJ
ejpam-5133	76	18	x	x	PRON
ejpam-5133	76	19	is	be	AUX
ejpam-5133	76	20	even	even	ADV
ejpam-5133	76	21	with	with	ADP
ejpam-5133	76	22	x	x	X
ejpam-5133	76	23	=	=	SYM
ejpam-5133	76	24	2nk	2nk	NOUN
ejpam-5133	76	25	,	,	PUNCT
ejpam-5133	76	26	n	n	PRON
ejpam-5133	76	27	∈	∈	PROPN
ejpam-5133	76	28	n	n	NOUN
ejpam-5133	76	29	and	and	CCONJ
ejpam-5133	76	30	k	k	PROPN
ejpam-5133	76	31	≥	≥	PROPN
ejpam-5133	76	32	1	1	NUM
ejpam-5133	76	33	odd	odd	ADJ
ejpam-5133	76	34	fixed	fix	VERB
ejpam-5133	76	35	,	,	PUNCT
ejpam-5133	76	36	with	with	ADP
ejpam-5133	76	37	p	p	PROPN
ejpam-5133	76	38	(	(	PUNCT
ejpam-5133	76	39	x	x	NOUN
ejpam-5133	76	40	)	)	PUNCT
ejpam-5133	76	41	=	=	SYM
ejpam-5133	77	1	{	{	PUNCT
ejpam-5133	77	2	d	d	X
ejpam-5133	77	3	∈	∈	PROPN
ejpam-5133	77	4	n	n	NOUN
ejpam-5133	77	5	:	:	PUNCT
ejpam-5133	78	1	d	d	X
ejpam-5133	78	2	=	=	SYM
ejpam-5133	78	3	2sl	2sl	NOUN
ejpam-5133	78	4	with	with	ADP
ejpam-5133	78	5	l	l	NOUN
ejpam-5133	78	6	divisor	divisor	NOUN
ejpam-5133	78	7	of	of	ADP
ejpam-5133	78	8	x2	x2	PROPN
ejpam-5133	78	9	and	and	CCONJ
ejpam-5133	78	10	s	s	PROPN
ejpam-5133	78	11	∈	∈	NOUN
ejpam-5133	78	12	{	{	PUNCT
ejpam-5133	78	13	1	1	NUM
ejpam-5133	78	14	,	,	PUNCT
ejpam-5133	78	15	2	2	NUM
ejpam-5133	78	16	,	,	PUNCT
ejpam-5133	78	17	...	...	PUNCT
ejpam-5133	78	18	,	,	PUNCT
ejpam-5133	78	19	2n−	2n−	PROPN
ejpam-5133	78	20	1	1	NUM
ejpam-5133	78	21	}	}	PUNCT
ejpam-5133	78	22	}	}	PUNCT
ejpam-5133	78	23	.	.	PUNCT
ejpam-5133	79	1	in	in	ADP
ejpam-5133	79	2	theorem	theorem	NOUN
ejpam-5133	79	3	(	(	PUNCT
ejpam-5133	79	4	1	1	NUM
ejpam-5133	79	5	)	)	PUNCT
ejpam-5133	79	6	x	x	X
ejpam-5133	79	7	is	be	AUX
ejpam-5133	79	8	a	a	DET
ejpam-5133	79	9	predetermined	predetermined	ADJ
ejpam-5133	79	10	integer	integer	NOUN
ejpam-5133	79	11	,	,	PUNCT
ejpam-5133	79	12	which	which	PRON
ejpam-5133	79	13	means	mean	VERB
ejpam-5133	79	14	finding	find	VERB
ejpam-5133	79	15	all	all	DET
ejpam-5133	79	16	right	right	ADJ
ejpam-5133	79	17	triangles	triangle	NOUN
ejpam-5133	79	18	whose	whose	DET
ejpam-5133	79	19	sides	side	NOUN
ejpam-5133	79	20	have	have	VERB
ejpam-5133	79	21	integer	integer	NOUN
ejpam-5133	79	22	measures	measure	NOUN
ejpam-5133	79	23	and	and	CCONJ
ejpam-5133	79	24	one	one	NUM
ejpam-5133	79	25	cathetus	cathetus	NOUN
ejpam-5133	79	26	is	be	AUX
ejpam-5133	79	27	predetermined	predetermine	VERB
ejpam-5133	79	28	.	.	PUNCT
ejpam-5133	80	1	theorem	theorem	NOUN
ejpam-5133	80	2	(	(	PUNCT
ejpam-5133	80	3	1	1	NUM
ejpam-5133	80	4	)	)	PUNCT
ejpam-5133	80	5	has	have	VERB
ejpam-5133	80	6	also	also	ADV
ejpam-5133	80	7	one	one	NUM
ejpam-5133	80	8	geometrical	geometrical	ADJ
ejpam-5133	80	9	interpretation	interpretation	NOUN
ejpam-5133	80	10	.	.	PUNCT
ejpam-5133	81	1	moreover	moreover	ADV
ejpam-5133	81	2	in	in	ADP
ejpam-5133	81	3	[	[	X
ejpam-5133	81	4	3	3	NUM
ejpam-5133	81	5	]	]	PUNCT
ejpam-5133	81	6	,	,	PUNCT
ejpam-5133	81	7	based	base	VERB
ejpam-5133	81	8	on	on	ADP
ejpam-5133	81	9	theorem	theorem	NOUN
ejpam-5133	81	10	(	(	PUNCT
ejpam-5133	81	11	1	1	NUM
ejpam-5133	81	12	)	)	PUNCT
ejpam-5133	81	13	,	,	PUNCT
ejpam-5133	81	14	we	we	PRON
ejpam-5133	81	15	have	have	AUX
ejpam-5133	81	16	proved	prove	VERB
ejpam-5133	81	17	the	the	DET
ejpam-5133	81	18	following	follow	VERB
ejpam-5133	81	19	theorem	theorem	NOUN
ejpam-5133	81	20	.	.	PUNCT
ejpam-5133	81	21	theorem	theorem	NOUN
ejpam-5133	81	22	2	2	NUM
ejpam-5133	81	23	.	.	PUNCT
ejpam-5133	82	1	[	[	X
ejpam-5133	82	2	3	3	X
ejpam-5133	82	3	]	]	PUNCT
ejpam-5133	82	4	each	each	DET
ejpam-5133	82	5	x	x	SYM
ejpam-5133	82	6	∈	∈	PROPN
ejpam-5133	82	7	n	n	PRON
ejpam-5133	82	8	can	can	AUX
ejpam-5133	82	9	be	be	AUX
ejpam-5133	82	10	found	find	VERB
ejpam-5133	82	11	as	as	ADP
ejpam-5133	82	12	cathetus	cathetus	NOUN
ejpam-5133	82	13	in	in	ADP
ejpam-5133	82	14	at	at	ADV
ejpam-5133	82	15	least	least	ADV
ejpam-5133	82	16	one	one	NUM
ejpam-5133	82	17	pythagorean	pythagorean	PROPN
ejpam-5133	82	18	triple	triple	NOUN
ejpam-5133	82	19	.	.	PUNCT
ejpam-5133	83	1	every	every	DET
ejpam-5133	83	2	x	x	SYM
ejpam-5133	83	3	∈	∈	PROPN
ejpam-5133	83	4	n	n	PRON
ejpam-5133	83	5	can	can	AUX
ejpam-5133	83	6	be	be	AUX
ejpam-5133	83	7	represented	represent	VERB
ejpam-5133	83	8	in	in	ADP
ejpam-5133	83	9	the	the	DET
ejpam-5133	83	10	form	form	NOUN
ejpam-5133	83	11	x	x	PUNCT
ejpam-5133	83	12	=	=	PUNCT
ejpam-5133	83	13	√	√	PROPN
ejpam-5133	83	14	z2	z2	PROPN
ejpam-5133	83	15	−	−	PROPN
ejpam-5133	83	16	y2	y2	PROPN
ejpam-5133	83	17	with	with	ADP
ejpam-5133	83	18	y	y	PROPN
ejpam-5133	83	19	,	,	PUNCT
ejpam-5133	83	20	z	z	PROPN
ejpam-5133	83	21	∈	∈	PROPN
ejpam-5133	83	22	n.	n.	NOUN
ejpam-5133	83	23	moreover	moreover	ADV
ejpam-5133	83	24	in	in	ADP
ejpam-5133	83	25	[	[	X
ejpam-5133	83	26	6	6	NUM
ejpam-5133	83	27	]	]	PUNCT
ejpam-5133	83	28	,	,	PUNCT
ejpam-5133	83	29	an	an	DET
ejpam-5133	83	30	analytic	analytic	ADJ
ejpam-5133	83	31	result	result	NOUN
ejpam-5133	83	32	was	be	AUX
ejpam-5133	83	33	found	find	VERB
ejpam-5133	83	34	that	that	PRON
ejpam-5133	83	35	characterizes	characterize	VERB
ejpam-5133	83	36	primitive	primitive	ADJ
ejpam-5133	83	37	pythagorean	pythagorean	ADJ
ejpam-5133	83	38	triples	triple	NOUN
ejpam-5133	83	39	through	through	ADP
ejpam-5133	83	40	a	a	DET
ejpam-5133	83	41	cathetus	cathetus	NOUN
ejpam-5133	83	42	.	.	PUNCT
ejpam-5133	84	1	this	this	DET
ejpam-5133	84	2	method	method	NOUN
ejpam-5133	84	3	,	,	PUNCT
ejpam-5133	84	4	which	which	PRON
ejpam-5133	84	5	differs	differ	VERB
ejpam-5133	84	6	from	from	ADP
ejpam-5133	84	7	euler	euler	PROPN
ejpam-5133	84	8	’s	’s	PART
ejpam-5133	84	9	formulas	formula	NOUN
ejpam-5133	84	10	,	,	PUNCT
ejpam-5133	84	11	offers	offer	VERB
ejpam-5133	84	12	the	the	DET
ejpam-5133	84	13	advantage	advantage	NOUN
ejpam-5133	84	14	of	of	ADP
ejpam-5133	84	15	easily	easily	ADV
ejpam-5133	84	16	identifying	identify	VERB
ejpam-5133	84	17	all	all	DET
ejpam-5133	84	18	primitive	primitive	ADJ
ejpam-5133	84	19	pythagorean	pythagorean	ADJ
ejpam-5133	84	20	triples	triple	NOUN
ejpam-5133	84	21	x	x	NOUN
ejpam-5133	84	22	,	,	PUNCT
ejpam-5133	84	23	y	y	PROPN
ejpam-5133	84	24	,	,	PUNCT
ejpam-5133	84	25	z	z	PROPN
ejpam-5133	84	26	∈	∈	PROPN
ejpam-5133	84	27	n	n	CCONJ
ejpam-5133	84	28	,	,	PUNCT
ejpam-5133	84	29	where	where	SCONJ
ejpam-5133	84	30	x	x	PRON
ejpam-5133	84	31	is	be	AUX
ejpam-5133	84	32	a	a	DET
ejpam-5133	84	33	predetermined	predetermined	ADJ
ejpam-5133	84	34	integer	integer	NOUN
ejpam-5133	84	35	.	.	PUNCT
ejpam-5133	85	1	this	this	DET
ejpam-5133	85	2	reads	read	VERB
ejpam-5133	85	3	as	as	SCONJ
ejpam-5133	85	4	follows	follow	VERB
ejpam-5133	85	5	.	.	PUNCT
ejpam-5133	86	1	theorem	theorem	NOUN
ejpam-5133	86	2	3	3	NUM
ejpam-5133	86	3	.	.	PUNCT
ejpam-5133	87	1	[	[	X
ejpam-5133	87	2	6	6	NUM
ejpam-5133	87	3	]	]	PUNCT
ejpam-5133	87	4	let	let	VERB
ejpam-5133	87	5	(	(	PUNCT
ejpam-5133	87	6	x	x	NOUN
ejpam-5133	87	7	,	,	PUNCT
ejpam-5133	87	8	y	y	PROPN
ejpam-5133	87	9	,	,	PUNCT
ejpam-5133	87	10	z	z	NOUN
ejpam-5133	87	11	)	)	PUNCT
ejpam-5133	87	12	be	be	VERB
ejpam-5133	87	13	all	all	DET
ejpam-5133	87	14	the	the	DET
ejpam-5133	87	15	pythagorean	pythagorean	PROPN
ejpam-5133	87	16	triples	triple	NOUN
ejpam-5133	87	17	generated	generate	VERB
ejpam-5133	87	18	by	by	ADP
ejpam-5133	87	19	any	any	DET
ejpam-5133	87	20	predetermined	predetermine	VERB
ejpam-5133	87	21	positive	positive	ADJ
ejpam-5133	87	22	integer	integer	NOUN
ejpam-5133	87	23	x	x	X
ejpam-5133	87	24	≥	≥	NOUN
ejpam-5133	87	25	1	1	NUM
ejpam-5133	87	26	using	use	VERB
ejpam-5133	87	27	(	(	PUNCT
ejpam-5133	87	28	1	1	NUM
ejpam-5133	87	29	)	)	PUNCT
ejpam-5133	87	30	,	,	PUNCT
ejpam-5133	87	31	d	d	PROPN
ejpam-5133	87	32	∈	∈	PROPN
ejpam-5133	87	33	c(x	c(x	NOUN
ejpam-5133	87	34	)	)	PUNCT
ejpam-5133	87	35	,	,	PUNCT
ejpam-5133	87	36	then	then	ADV
ejpam-5133	87	37	(	(	PUNCT
ejpam-5133	87	38	x	x	X
ejpam-5133	87	39	,	,	PUNCT
ejpam-5133	87	40	y	y	PROPN
ejpam-5133	87	41	,	,	PUNCT
ejpam-5133	87	42	z	z	NOUN
ejpam-5133	87	43	)	)	PUNCT
ejpam-5133	87	44	is	be	AUX
ejpam-5133	87	45	a	a	DET
ejpam-5133	87	46	primitive	primitive	ADJ
ejpam-5133	87	47	pythagorean	pythagorean	NOUN
ejpam-5133	87	48	triple	triple	NOUN
ejpam-5133	87	49	if	if	SCONJ
ejpam-5133	88	1	and	and	CCONJ
ejpam-5133	88	2	only	only	ADV
ejpam-5133	88	3	if	if	SCONJ
ejpam-5133	88	4	following	follow	VERB
ejpam-5133	88	5	both	both	DET
ejpam-5133	88	6	conditions	condition	NOUN
ejpam-5133	88	7	are	be	AUX
ejpam-5133	88	8	verified	verify	VERB
ejpam-5133	88	9	if	if	SCONJ
ejpam-5133	88	10	x	x	PRON
ejpam-5133	88	11	is	be	AUX
ejpam-5133	88	12	odd	odd	ADJ
ejpam-5133	88	13	then	then	ADV
ejpam-5133	88	14			PROPN
ejpam-5133	88	15	d	d	NOUN
ejpam-5133	88	16	is	be	AUX
ejpam-5133	88	17	a	a	DET
ejpam-5133	88	18	square	square	ADJ
ejpam-5133	88	19	x2	x2	NOUN
ejpam-5133	88	20	d	d	NOUN
ejpam-5133	88	21	with	with	ADP
ejpam-5133	88	22	d	d	PROPN
ejpam-5133	88	23	are	be	AUX
ejpam-5133	88	24	coprime	coprime	ADV
ejpam-5133	88	25	positive	positive	ADJ
ejpam-5133	88	26	odd	odd	ADJ
ejpam-5133	88	27	integers	integer	NOUN
ejpam-5133	88	28	r.	r.	PROPN
ejpam-5133	88	29	amato	amato	PROPN
ejpam-5133	88	30	/	/	SYM
ejpam-5133	88	31	eur	eur	PROPN
ejpam-5133	88	32	.	.	PUNCT
ejpam-5133	89	1	j.	j.	PROPN
ejpam-5133	89	2	pure	pure	PROPN
ejpam-5133	89	3	appl	appl	PROPN
ejpam-5133	89	4	.	.	PROPN
ejpam-5133	89	5	math	math	PROPN
ejpam-5133	89	6	,	,	PUNCT
ejpam-5133	89	7	17	17	NUM
ejpam-5133	89	8	(	(	PUNCT
ejpam-5133	89	9	2	2	NUM
ejpam-5133	89	10	)	)	PUNCT
ejpam-5133	89	11	(	(	PUNCT
ejpam-5133	89	12	2024	2024	NUM
ejpam-5133	89	13	)	)	PUNCT
ejpam-5133	89	14	,	,	PUNCT
ejpam-5133	89	15	676	676	NUM
ejpam-5133	89	16	-	-	SYM
ejpam-5133	89	17	689	689	NUM
ejpam-5133	89	18	680	680	NUM
ejpam-5133	89	19	if	if	SCONJ
ejpam-5133	89	20	x	x	PRON
ejpam-5133	89	21	is	be	AUX
ejpam-5133	89	22	even	even	ADV
ejpam-5133	89	23	then	then	ADV
ejpam-5133	89	24			PROPN
ejpam-5133	89	25	d	d	NOUN
ejpam-5133	89	26	2	2	NUM
ejpam-5133	89	27	is	be	AUX
ejpam-5133	89	28	a	a	DET
ejpam-5133	89	29	square	square	ADJ
ejpam-5133	89	30	x2	x2	ADJ
ejpam-5133	89	31	2d	2d	NOUN
ejpam-5133	89	32	with	with	ADP
ejpam-5133	89	33	d	d	PROPN
ejpam-5133	89	34	2	2	NUM
ejpam-5133	89	35	are	be	AUX
ejpam-5133	89	36	coprime	coprime	ADJ
ejpam-5133	89	37	positive	positive	ADJ
ejpam-5133	89	38	integers	integer	NOUN
ejpam-5133	89	39	of	of	ADP
ejpam-5133	89	40	different	different	ADJ
ejpam-5133	89	41	parities	parity	NOUN
ejpam-5133	89	42	.	.	PUNCT
ejpam-5133	90	1	we	we	PRON
ejpam-5133	90	2	remember	remember	VERB
ejpam-5133	90	3	that	that	SCONJ
ejpam-5133	90	4	the	the	DET
ejpam-5133	90	5	euclid	euclid	PROPN
ejpam-5133	90	6	’s	’s	PART
ejpam-5133	90	7	formulas	formula	NOUN
ejpam-5133	90	8	do	do	AUX
ejpam-5133	90	9	not	not	PART
ejpam-5133	90	10	give	give	VERB
ejpam-5133	90	11	all	all	DET
ejpam-5133	90	12	pythagorean	pythagorean	ADJ
ejpam-5133	90	13	triples	triple	NOUN
ejpam-5133	90	14	that	that	PRON
ejpam-5133	90	15	involves	involve	VERB
ejpam-5133	90	16	a	a	DET
ejpam-5133	90	17	predetermined	predetermine	VERB
ejpam-5133	90	18	positive	positive	ADJ
ejpam-5133	90	19	integer	integer	NOUN
ejpam-5133	90	20	x	x	NOUN
ejpam-5133	90	21	,	,	PUNCT
ejpam-5133	90	22	for	for	ADP
ejpam-5133	90	23	example	example	NOUN
ejpam-5133	90	24	the	the	DET
ejpam-5133	90	25	triples	triple	NOUN
ejpam-5133	90	26	(	(	PUNCT
ejpam-5133	90	27	12	12	NUM
ejpam-5133	90	28	,	,	PUNCT
ejpam-5133	90	29	9	9	NUM
ejpam-5133	90	30	,	,	PUNCT
ejpam-5133	90	31	15	15	NUM
ejpam-5133	90	32	)	)	PUNCT
ejpam-5133	90	33	,	,	PUNCT
ejpam-5133	90	34	(	(	PUNCT
ejpam-5133	90	35	33	33	NUM
ejpam-5133	90	36	,	,	PUNCT
ejpam-5133	90	37	180	180	NUM
ejpam-5133	90	38	,	,	PUNCT
ejpam-5133	90	39	183	183	NUM
ejpam-5133	90	40	)	)	PUNCT
ejpam-5133	90	41	and	and	CCONJ
ejpam-5133	90	42	(	(	PUNCT
ejpam-5133	90	43	33	33	NUM
ejpam-5133	90	44	,	,	PUNCT
ejpam-5133	90	45	44	44	NUM
ejpam-5133	90	46	,	,	PUNCT
ejpam-5133	90	47	55	55	NUM
ejpam-5133	90	48	)	)	PUNCT
ejpam-5133	90	49	.	.	PUNCT
ejpam-5133	91	1	moreover	moreover	ADV
ejpam-5133	91	2	it	it	PRON
ejpam-5133	91	3	can	can	AUX
ejpam-5133	91	4	be	be	AUX
ejpam-5133	91	5	laborious	laborious	ADJ
ejpam-5133	91	6	to	to	PART
ejpam-5133	91	7	find	find	VERB
ejpam-5133	91	8	m	m	PRON
ejpam-5133	91	9	and	and	CCONJ
ejpam-5133	91	10	n	n	PRON
ejpam-5133	91	11	such	such	ADJ
ejpam-5133	91	12	that	that	SCONJ
ejpam-5133	91	13	x	x	X
ejpam-5133	91	14	=	=	SYM
ejpam-5133	91	15	m2−n2	m2−n2	PROPN
ejpam-5133	91	16	,	,	PUNCT
ejpam-5133	91	17	while	while	SCONJ
ejpam-5133	91	18	using	use	VERB
ejpam-5133	91	19	theorem	theorem	NOUN
ejpam-5133	91	20	(	(	PUNCT
ejpam-5133	91	21	1	1	NUM
ejpam-5133	91	22	)	)	PUNCT
ejpam-5133	91	23	,	,	PUNCT
ejpam-5133	91	24	it	it	PRON
ejpam-5133	91	25	is	be	AUX
ejpam-5133	91	26	enough	enough	ADJ
ejpam-5133	91	27	to	to	PART
ejpam-5133	91	28	find	find	VERB
ejpam-5133	91	29	all	all	DET
ejpam-5133	91	30	the	the	DET
ejpam-5133	91	31	d	d	PROPN
ejpam-5133	91	32	∈	∈	PROPN
ejpam-5133	91	33	c(x	c(x	NOUN
ejpam-5133	91	34	)	)	PUNCT
ejpam-5133	91	35	to	to	PART
ejpam-5133	91	36	obtain	obtain	VERB
ejpam-5133	91	37	all	all	DET
ejpam-5133	91	38	pythagorean	pythagorean	PROPN
ejpam-5133	91	39	triples	triple	NOUN
ejpam-5133	91	40	.	.	PUNCT
ejpam-5133	92	1	in	in	ADP
ejpam-5133	92	2	particular	particular	ADJ
ejpam-5133	92	3	,	,	PUNCT
ejpam-5133	92	4	if	if	SCONJ
ejpam-5133	92	5	we	we	PRON
ejpam-5133	92	6	need	need	VERB
ejpam-5133	92	7	to	to	PART
ejpam-5133	92	8	find	find	VERB
ejpam-5133	92	9	all	all	DET
ejpam-5133	92	10	primitive	primitive	ADJ
ejpam-5133	92	11	pythagorean	pythagorean	ADJ
ejpam-5133	92	12	triples	triple	NOUN
ejpam-5133	92	13	that	that	PRON
ejpam-5133	92	14	involves	involve	VERB
ejpam-5133	92	15	a	a	DET
ejpam-5133	92	16	predetermined	predetermine	VERB
ejpam-5133	92	17	positive	positive	ADJ
ejpam-5133	92	18	integer	integer	NOUN
ejpam-5133	92	19	x	x	NOUN
ejpam-5133	92	20	,	,	PUNCT
ejpam-5133	92	21	now	now	ADV
ejpam-5133	92	22	we	we	PRON
ejpam-5133	92	23	can	can	AUX
ejpam-5133	92	24	use	use	VERB
ejpam-5133	92	25	only	only	ADV
ejpam-5133	92	26	the	the	DET
ejpam-5133	92	27	d	d	PROPN
ejpam-5133	92	28	∈	∈	PROPN
ejpam-5133	92	29	c(x	c(x	NOUN
ejpam-5133	92	30	)	)	PUNCT
ejpam-5133	92	31	that	that	PRON
ejpam-5133	92	32	satisfy	satisfy	VERB
ejpam-5133	92	33	the	the	DET
ejpam-5133	92	34	conditions	condition	NOUN
ejpam-5133	92	35	of	of	ADP
ejpam-5133	92	36	the	the	DET
ejpam-5133	92	37	theorem	theorem	NOUN
ejpam-5133	92	38	(	(	PUNCT
ejpam-5133	92	39	3	3	NUM
ejpam-5133	92	40	)	)	PUNCT
ejpam-5133	92	41	.	.	PUNCT
ejpam-5133	93	1	in	in	ADP
ejpam-5133	93	2	[	[	X
ejpam-5133	93	3	4	4	NUM
ejpam-5133	93	4	]	]	PUNCT
ejpam-5133	93	5	,	,	PUNCT
ejpam-5133	93	6	relations	relation	NOUN
ejpam-5133	93	7	were	be	AUX
ejpam-5133	93	8	established	establish	VERB
ejpam-5133	93	9	between	between	ADP
ejpam-5133	93	10	the	the	DET
ejpam-5133	93	11	primitive	primitive	ADJ
ejpam-5133	93	12	pythagorean	pythagorean	NOUN
ejpam-5133	93	13	triple	triple	ADJ
ejpam-5133	93	14	(	(	PUNCT
ejpam-5133	93	15	x	x	NOUN
ejpam-5133	93	16	,	,	PUNCT
ejpam-5133	93	17	y	y	PROPN
ejpam-5133	93	18	,	,	PUNCT
ejpam-5133	93	19	z	z	NOUN
ejpam-5133	93	20	)	)	PUNCT
ejpam-5133	93	21	generated	generate	VERB
ejpam-5133	93	22	by	by	ADP
ejpam-5133	93	23	any	any	DET
ejpam-5133	93	24	predetermined	predetermine	VERB
ejpam-5133	93	25	positive	positive	ADJ
ejpam-5133	93	26	odd	odd	ADJ
ejpam-5133	93	27	integer	integer	NOUN
ejpam-5133	93	28	x	x	X
ejpam-5133	93	29	and	and	CCONJ
ejpam-5133	93	30	the	the	DET
ejpam-5133	93	31	primitive	primitive	ADJ
ejpam-5133	93	32	pythagorean	pythagorean	NOUN
ejpam-5133	93	33	triple	triple	ADP
ejpam-5133	93	34	generated	generate	VERB
ejpam-5133	93	35	by	by	ADP
ejpam-5133	93	36	xm	xm	PROPN
ejpam-5133	93	37	with	with	ADP
ejpam-5133	93	38	m	m	PROPN
ejpam-5133	93	39	∈	∈	PROPN
ejpam-5133	93	40	n	n	NOUN
ejpam-5133	93	41	and	and	CCONJ
ejpam-5133	93	42	m	m	PROPN
ejpam-5133	93	43	≥	≥	NOUN
ejpam-5133	93	44	2	2	NUM
ejpam-5133	93	45	,	,	PUNCT
ejpam-5133	93	46	rispectively	rispectively	ADV
ejpam-5133	93	47	,	,	PUNCT
ejpam-5133	93	48	using	use	VERB
ejpam-5133	93	49	formulas	formula	NOUN
ejpam-5133	93	50	(	(	PUNCT
ejpam-5133	93	51	1	1	NUM
ejpam-5133	93	52	)	)	PUNCT
ejpam-5133	93	53	.	.	PUNCT
ejpam-5133	94	1	subsequently	subsequently	ADV
ejpam-5133	94	2	,	,	PUNCT
ejpam-5133	94	3	additional	additional	ADJ
ejpam-5133	94	4	relations	relation	NOUN
ejpam-5133	94	5	among	among	ADP
ejpam-5133	94	6	pythagorean	pythagorean	PROPN
ejpam-5133	94	7	triples	triple	NOUN
ejpam-5133	94	8	were	be	AUX
ejpam-5133	94	9	established	establish	VERB
ejpam-5133	94	10	in	in	ADP
ejpam-5133	94	11	[	[	X
ejpam-5133	94	12	5	5	NUM
ejpam-5133	94	13	]	]	PUNCT
ejpam-5133	94	14	.	.	PUNCT
ejpam-5133	95	1	the	the	DET
ejpam-5133	95	2	primary	primary	ADJ
ejpam-5133	95	3	tool	tool	NOUN
ejpam-5133	95	4	that	that	PRON
ejpam-5133	95	5	serves	serve	VERB
ejpam-5133	95	6	as	as	ADP
ejpam-5133	95	7	the	the	DET
ejpam-5133	95	8	foundation	foundation	NOUN
ejpam-5133	95	9	of	of	ADP
ejpam-5133	95	10	our	our	PRON
ejpam-5133	95	11	analysis	analysis	NOUN
ejpam-5133	95	12	is	be	AUX
ejpam-5133	95	13	theorem	theorem	ADJ
ejpam-5133	95	14	(	(	PUNCT
ejpam-5133	95	15	1	1	NUM
ejpam-5133	95	16	)	)	PUNCT
ejpam-5133	95	17	in	in	ADP
ejpam-5133	95	18	[	[	X
ejpam-5133	95	19	3	3	NUM
ejpam-5133	95	20	]	]	PUNCT
ejpam-5133	95	21	,	,	PUNCT
ejpam-5133	95	22	enabling	enable	VERB
ejpam-5133	95	23	the	the	DET
ejpam-5133	95	24	determination	determination	NOUN
ejpam-5133	95	25	of	of	ADP
ejpam-5133	95	26	relationships	relationship	NOUN
ejpam-5133	95	27	between	between	ADP
ejpam-5133	95	28	two	two	NUM
ejpam-5133	95	29	pythagorean	pythagorean	ADJ
ejpam-5133	95	30	triples	triple	NOUN
ejpam-5133	95	31	with	with	ADP
ejpam-5133	95	32	assigned	assign	VERB
ejpam-5133	95	33	catheti	catheti	ADJ
ejpam-5133	95	34	a	a	PRON
ejpam-5133	95	35	and	and	CCONJ
ejpam-5133	95	36	b	b	NOUN
ejpam-5133	95	37	,	,	PUNCT
ejpam-5133	95	38	and	and	CCONJ
ejpam-5133	95	39	the	the	DET
ejpam-5133	95	40	pythagorean	pythagorean	PROPN
ejpam-5133	95	41	triple	triple	NOUN
ejpam-5133	95	42	with	with	ADP
ejpam-5133	95	43	cathetus	cathetus	PROPN
ejpam-5133	95	44	a	a	DET
ejpam-5133	95	45	·	·	PUNCT
ejpam-5133	95	46	b.	b.	NOUN
ejpam-5133	96	1	this	this	PRON
ejpam-5133	96	2	reads	read	VERB
ejpam-5133	96	3	as	as	SCONJ
ejpam-5133	96	4	follows	follow	VERB
ejpam-5133	96	5	.	.	PUNCT
ejpam-5133	97	1	theorem	theorem	ADJ
ejpam-5133	97	2	4	4	NUM
ejpam-5133	97	3	.	.	PUNCT
ejpam-5133	98	1	[	[	X
ejpam-5133	98	2	5	5	NUM
ejpam-5133	98	3	]	]	X
ejpam-5133	98	4	let	let	VERB
ejpam-5133	98	5	(	(	PUNCT
ejpam-5133	98	6	a	a	DET
ejpam-5133	98	7	,	,	PUNCT
ejpam-5133	98	8	b	b	NOUN
ejpam-5133	98	9	,	,	PUNCT
ejpam-5133	98	10	c	c	NOUN
ejpam-5133	98	11	)	)	PUNCT
ejpam-5133	98	12	,	,	PUNCT
ejpam-5133	98	13	(	(	PUNCT
ejpam-5133	98	14	d	d	X
ejpam-5133	98	15	,	,	PUNCT
ejpam-5133	98	16	e	e	NOUN
ejpam-5133	98	17	,	,	PUNCT
ejpam-5133	98	18	f	f	PROPN
ejpam-5133	98	19	)	)	PUNCT
ejpam-5133	98	20	,	,	PUNCT
ejpam-5133	98	21	(	(	PUNCT
ejpam-5133	98	22	a·d	a·d	ADJ
ejpam-5133	98	23	,	,	PUNCT
ejpam-5133	98	24	y	y	PROPN
ejpam-5133	98	25	,	,	PUNCT
ejpam-5133	98	26	z	z	NOUN
ejpam-5133	98	27	)	)	PUNCT
ejpam-5133	98	28	be	be	VERB
ejpam-5133	98	29	the	the	DET
ejpam-5133	98	30	pythagorean	pythagorean	PROPN
ejpam-5133	98	31	triples	triple	NOUN
ejpam-5133	98	32	generated	generate	VERB
ejpam-5133	98	33	by	by	ADP
ejpam-5133	98	34	a	a	DET
ejpam-5133	98	35	,	,	PUNCT
ejpam-5133	98	36	d	d	NOUN
ejpam-5133	98	37	,	,	PUNCT
ejpam-5133	98	38	and	and	CCONJ
ejpam-5133	98	39	a·d	a·d	ADJ
ejpam-5133	98	40	,	,	PUNCT
ejpam-5133	98	41	respectively	respectively	ADV
ejpam-5133	98	42	using	use	VERB
ejpam-5133	98	43	(	(	PUNCT
ejpam-5133	98	44	1	1	NUM
ejpam-5133	98	45	)	)	PUNCT
ejpam-5133	98	46	with	with	ADP
ejpam-5133	98	47	c	c	PROPN
ejpam-5133	98	48	−	−	PROPN
ejpam-5133	98	49	b	b	NOUN
ejpam-5133	98	50	=	=	SYM
ejpam-5133	98	51	d1	d1	PROPN
ejpam-5133	98	52	∈	∈	PROPN
ejpam-5133	98	53	c(a	c(a	PROPN
ejpam-5133	98	54	)	)	PUNCT
ejpam-5133	98	55	,	,	PUNCT
ejpam-5133	98	56	f	f	PROPN
ejpam-5133	98	57	−	−	PROPN
ejpam-5133	98	58	e	e	PROPN
ejpam-5133	98	59	=	=	SYM
ejpam-5133	98	60	d2	d2	PROPN
ejpam-5133	98	61	∈	∈	PROPN
ejpam-5133	98	62	c(b	c(b	PROPN
ejpam-5133	98	63	)	)	PUNCT
ejpam-5133	98	64	,	,	PUNCT
ejpam-5133	98	65	and	and	CCONJ
ejpam-5133	98	66	z	z	NOUN
ejpam-5133	98	67	−	−	PROPN
ejpam-5133	98	68	y	y	PROPN
ejpam-5133	98	69	=	=	PROPN
ejpam-5133	98	70	d3	d3	PROPN
ejpam-5133	98	71	∈	∈	PROPN
ejpam-5133	98	72	c(a·d	c(a·d	PROPN
ejpam-5133	98	73	)	)	PUNCT
ejpam-5133	98	74	.	.	PUNCT
ejpam-5133	99	1	then	then	ADV
ejpam-5133	99	2	y	y	PROPN
ejpam-5133	99	3	=	=	SYM
ejpam-5133	99	4	ce	ce	PROPN
ejpam-5133	100	1	+	+	CCONJ
ejpam-5133	100	2	bf	bf	NOUN
ejpam-5133	101	1	,	,	PUNCT
ejpam-5133	101	2	z	z	PROPN
ejpam-5133	101	3	=	=	SYM
ejpam-5133	101	4	ce	ce	PROPN
ejpam-5133	102	1	+	+	CCONJ
ejpam-5133	102	2	bf	bf	NOUN
ejpam-5133	102	3	+	+	CCONJ
ejpam-5133	102	4	d1d2	d1d2	NOUN
ejpam-5133	102	5	,	,	PUNCT
ejpam-5133	102	6	and	and	CCONJ
ejpam-5133	102	7	also	also	ADV
ejpam-5133	102	8	y	y	PROPN
ejpam-5133	102	9	=	=	PUNCT
ejpam-5133	102	10	be	be	AUX
ejpam-5133	102	11	+	+	X
ejpam-5133	103	1	cf	cf	NOUN
ejpam-5133	103	2	−	−	NOUN
ejpam-5133	103	3	d1d2	d1d2	NOUN
ejpam-5133	103	4	,	,	PUNCT
ejpam-5133	103	5	z	z	NOUN
ejpam-5133	103	6	=	=	PUNCT
ejpam-5133	103	7	be	be	AUX
ejpam-5133	103	8	+	+	X
ejpam-5133	103	9	cf	cf	NOUN
ejpam-5133	103	10	with	with	ADP
ejpam-5133	103	11	d3	d3	PROPN
ejpam-5133	103	12	=	=	SYM
ejpam-5133	103	13	d1d2∈	d1d2∈	PROPN
ejpam-5133	103	14	c(a·d	c(a·d	PROPN
ejpam-5133	103	15	)	)	PUNCT
ejpam-5133	103	16	.	.	PUNCT
ejpam-5133	104	1	above	above	ADP
ejpam-5133	104	2	theorem	theorem	ADJ
ejpam-5133	104	3	introduces	introduce	VERB
ejpam-5133	104	4	one	one	NUM
ejpam-5133	104	5	suitable	suitable	ADJ
ejpam-5133	104	6	binary	binary	ADJ
ejpam-5133	104	7	operation	operation	NOUN
ejpam-5133	104	8	in	in	ADP
ejpam-5133	104	9	the	the	DET
ejpam-5133	104	10	set	set	NOUN
ejpam-5133	104	11	of	of	ADP
ejpam-5133	104	12	pythagorean	pythagorean	PROPN
ejpam-5133	104	13	triples	triple	NOUN
ejpam-5133	104	14	.	.	PUNCT
ejpam-5133	105	1	in	in	ADP
ejpam-5133	105	2	[	[	X
ejpam-5133	105	3	7	7	NUM
ejpam-5133	105	4	]	]	PUNCT
ejpam-5133	105	5	,	,	PUNCT
ejpam-5133	105	6	thanks	thank	NOUN
ejpam-5133	105	7	to	to	ADP
ejpam-5133	105	8	theorem	theorem	NOUN
ejpam-5133	105	9	(	(	PUNCT
ejpam-5133	105	10	4	4	NUM
ejpam-5133	105	11	)	)	PUNCT
ejpam-5133	105	12	,	,	PUNCT
ejpam-5133	105	13	we	we	PRON
ejpam-5133	105	14	found	find	VERB
ejpam-5133	105	15	suitable	suitable	ADJ
ejpam-5133	105	16	binary	binary	ADJ
ejpam-5133	105	17	operations	operation	NOUN
ejpam-5133	105	18	on	on	ADP
ejpam-5133	105	19	the	the	DET
ejpam-5133	105	20	set	set	NOUN
ejpam-5133	105	21	of	of	ADP
ejpam-5133	105	22	pythagorean	pythagorean	PROPN
ejpam-5133	105	23	triples	triple	NOUN
ejpam-5133	105	24	,	,	PUNCT
ejpam-5133	105	25	obtaining	obtain	VERB
ejpam-5133	105	26	two	two	NUM
ejpam-5133	105	27	commutative	commutative	ADJ
ejpam-5133	105	28	infinite	infinite	ADJ
ejpam-5133	105	29	groups	group	NOUN
ejpam-5133	105	30	,	,	PUNCT
ejpam-5133	105	31	one	one	NUM
ejpam-5133	105	32	with	with	ADP
ejpam-5133	105	33	elements	element	NOUN
ejpam-5133	105	34	in	in	ADP
ejpam-5133	105	35	q	q	PROPN
ejpam-5133	105	36	and	and	CCONJ
ejpam-5133	105	37	the	the	DET
ejpam-5133	105	38	other	other	ADJ
ejpam-5133	105	39	with	with	ADP
ejpam-5133	105	40	elements	element	NOUN
ejpam-5133	105	41	in	in	ADP
ejpam-5133	105	42	z.	z.	PROPN
ejpam-5133	105	43	additionally	additionally	ADV
ejpam-5133	105	44	,	,	PUNCT
ejpam-5133	105	45	we	we	PRON
ejpam-5133	105	46	obtained	obtain	VERB
ejpam-5133	105	47	a	a	DET
ejpam-5133	105	48	commutative	commutative	ADJ
ejpam-5133	105	49	infinite	infinite	ADJ
ejpam-5133	105	50	monoid	monoid	NOUN
ejpam-5133	105	51	with	with	ADP
ejpam-5133	105	52	elements	element	NOUN
ejpam-5133	105	53	in	in	ADP
ejpam-5133	105	54	n	n	PRON
ejpam-5133	105	55	or	or	CCONJ
ejpam-5133	105	56	in	in	ADP
ejpam-5133	105	57	z.	z.	PROPN
ejpam-5133	105	58	in	in	ADP
ejpam-5133	105	59	particular	particular	ADJ
ejpam-5133	105	60	,	,	PUNCT
ejpam-5133	105	61	on	on	ADP
ejpam-5133	105	62	the	the	DET
ejpam-5133	105	63	set	set	NOUN
ejpam-5133	105	64	of	of	ADP
ejpam-5133	105	65	primitive	primitive	ADJ
ejpam-5133	105	66	pythagorean	pythagorean	NOUN
ejpam-5133	105	67	triples	triple	NOUN
ejpam-5133	105	68	,	,	PUNCT
ejpam-5133	105	69	we	we	PRON
ejpam-5133	105	70	established	establish	VERB
ejpam-5133	105	71	two	two	NUM
ejpam-5133	105	72	commutative	commutative	ADJ
ejpam-5133	105	73	infinite	infinite	ADJ
ejpam-5133	105	74	groups	group	NOUN
ejpam-5133	105	75	,	,	PUNCT
ejpam-5133	105	76	one	one	NUM
ejpam-5133	105	77	with	with	ADP
ejpam-5133	105	78	elements	element	NOUN
ejpam-5133	105	79	in	in	ADP
ejpam-5133	105	80	q	q	PROPN
ejpam-5133	105	81	and	and	CCONJ
ejpam-5133	105	82	the	the	DET
ejpam-5133	105	83	other	other	ADJ
ejpam-5133	105	84	with	with	ADP
ejpam-5133	105	85	elements	element	NOUN
ejpam-5133	105	86	in	in	ADP
ejpam-5133	105	87	z.	z.	PROPN
ejpam-5133	106	1	all	all	DET
ejpam-5133	106	2	previous	previous	ADJ
ejpam-5133	106	3	results	result	NOUN
ejpam-5133	106	4	were	be	AUX
ejpam-5133	106	5	obtained	obtain	VERB
ejpam-5133	106	6	without	without	ADP
ejpam-5133	106	7	advanced	advanced	ADJ
ejpam-5133	106	8	techniques	technique	NOUN
ejpam-5133	106	9	and	and	CCONJ
ejpam-5133	106	10	this	this	PRON
ejpam-5133	106	11	can	can	AUX
ejpam-5133	106	12	be	be	AUX
ejpam-5133	106	13	a	a	DET
ejpam-5133	106	14	virtue	virtue	NOUN
ejpam-5133	106	15	to	to	PART
ejpam-5133	106	16	reach	reach	VERB
ejpam-5133	106	17	a	a	DET
ejpam-5133	106	18	wider	wide	ADJ
ejpam-5133	106	19	audience	audience	NOUN
ejpam-5133	106	20	of	of	ADP
ejpam-5133	106	21	readers	reader	NOUN
ejpam-5133	106	22	,	,	PUNCT
ejpam-5133	106	23	including	include	VERB
ejpam-5133	106	24	students	student	NOUN
ejpam-5133	106	25	in	in	ADP
ejpam-5133	106	26	schools	school	NOUN
ejpam-5133	106	27	and	and	CCONJ
ejpam-5133	106	28	universities	university	NOUN
ejpam-5133	106	29	.	.	PUNCT
ejpam-5133	107	1	r.	r.	PROPN
ejpam-5133	107	2	amato	amato	PROPN
ejpam-5133	107	3	/	/	SYM
ejpam-5133	107	4	eur	eur	PROPN
ejpam-5133	107	5	.	.	PUNCT
ejpam-5133	108	1	j.	j.	PROPN
ejpam-5133	108	2	pure	pure	PROPN
ejpam-5133	108	3	appl	appl	PROPN
ejpam-5133	108	4	.	.	PROPN
ejpam-5133	108	5	math	math	PROPN
ejpam-5133	108	6	,	,	PUNCT
ejpam-5133	108	7	17	17	NUM
ejpam-5133	108	8	(	(	PUNCT
ejpam-5133	108	9	2	2	NUM
ejpam-5133	108	10	)	)	PUNCT
ejpam-5133	108	11	(	(	PUNCT
ejpam-5133	108	12	2024	2024	NUM
ejpam-5133	108	13	)	)	PUNCT
ejpam-5133	108	14	,	,	PUNCT
ejpam-5133	108	15	676	676	NUM
ejpam-5133	108	16	-	-	SYM
ejpam-5133	108	17	689	689	NUM
ejpam-5133	108	18	681	681	NUM
ejpam-5133	108	19	3	3	NUM
ejpam-5133	108	20	.	.	PUNCT
ejpam-5133	108	21	applications	application	NOUN
ejpam-5133	108	22	and	and	CCONJ
ejpam-5133	108	23	results	result	NOUN
ejpam-5133	108	24	in	in	ADP
ejpam-5133	108	25	this	this	DET
ejpam-5133	108	26	section	section	NOUN
ejpam-5133	108	27	,	,	PUNCT
ejpam-5133	108	28	we	we	PRON
ejpam-5133	108	29	want	want	VERB
ejpam-5133	108	30	to	to	PART
ejpam-5133	108	31	study	study	VERB
ejpam-5133	108	32	some	some	DET
ejpam-5133	108	33	applications	application	NOUN
ejpam-5133	108	34	and	and	CCONJ
ejpam-5133	108	35	results	result	NOUN
ejpam-5133	108	36	in	in	ADP
ejpam-5133	108	37	fields	field	NOUN
ejpam-5133	108	38	such	such	ADJ
ejpam-5133	108	39	as	as	ADP
ejpam-5133	108	40	geometry	geometry	NOUN
ejpam-5133	108	41	,	,	PUNCT
ejpam-5133	108	42	trigonometry	trigonometry	NOUN
ejpam-5133	108	43	,	,	PUNCT
ejpam-5133	108	44	linear	linear	ADJ
ejpam-5133	108	45	algebra	algebra	NOUN
ejpam-5133	108	46	and	and	CCONJ
ejpam-5133	108	47	number	number	NOUN
ejpam-5133	108	48	theory	theory	NOUN
ejpam-5133	108	49	.	.	PUNCT
ejpam-5133	109	1	we	we	PRON
ejpam-5133	109	2	will	will	AUX
ejpam-5133	109	3	obtain	obtain	VERB
ejpam-5133	109	4	new	new	ADJ
ejpam-5133	109	5	relations	relation	NOUN
ejpam-5133	109	6	taking	take	VERB
ejpam-5133	109	7	into	into	ADP
ejpam-5133	109	8	account	account	NOUN
ejpam-5133	109	9	results	result	NOUN
ejpam-5133	109	10	seen	see	VERB
ejpam-5133	109	11	in	in	ADP
ejpam-5133	109	12	preliminar	preliminar	PROPN
ejpam-5133	109	13	results	result	NOUN
ejpam-5133	109	14	section	section	NOUN
ejpam-5133	109	15	,	,	PUNCT
ejpam-5133	109	16	and	and	CCONJ
ejpam-5133	109	17	often	often	ADV
ejpam-5133	109	18	using	use	VERB
ejpam-5133	109	19	only	only	ADV
ejpam-5133	109	20	a	a	DET
ejpam-5133	109	21	predeterminatus	predeterminatus	NOUN
ejpam-5133	109	22	x	x	PUNCT
ejpam-5133	109	23	and	and	CCONJ
ejpam-5133	109	24	d.	d.	PROPN
ejpam-5133	109	25	let	let	VERB
ejpam-5133	109	26	’s	’s	NOUN
ejpam-5133	109	27	begin	begin	VERB
ejpam-5133	109	28	to	to	PART
ejpam-5133	109	29	notice	notice	VERB
ejpam-5133	109	30	that	that	SCONJ
ejpam-5133	109	31	,	,	PUNCT
ejpam-5133	109	32	the	the	DET
ejpam-5133	109	33	formulas	formula	NOUN
ejpam-5133	109	34	of	of	ADP
ejpam-5133	109	35	theorem	theorem	NOUN
ejpam-5133	109	36	(	(	PUNCT
ejpam-5133	109	37	1	1	NUM
ejpam-5133	109	38	)	)	PUNCT
ejpam-5133	109	39	satisfy	satisfy	VERB
ejpam-5133	109	40	the	the	DET
ejpam-5133	109	41	ralation	ralation	NOUN
ejpam-5133	109	42	x2+y2	x2+y2	PUNCT
ejpam-5133	110	1	=	=	PROPN
ejpam-5133	110	2	z2	z2	PROPN
ejpam-5133	110	3	for	for	ADP
ejpam-5133	110	4	every	every	DET
ejpam-5133	110	5	x	x	PROPN
ejpam-5133	110	6	,	,	PUNCT
ejpam-5133	110	7	d	d	PROPN
ejpam-5133	110	8	∈	∈	PROPN
ejpam-5133	110	9	r.	r.	NOUN
ejpam-5133	110	10	it	it	PRON
ejpam-5133	110	11	is	be	AUX
ejpam-5133	110	12	easy	easy	ADJ
ejpam-5133	110	13	to	to	PART
ejpam-5133	110	14	see	see	VERB
ejpam-5133	110	15	that	that	SCONJ
ejpam-5133	110	16	,	,	PUNCT
ejpam-5133	110	17	if	if	SCONJ
ejpam-5133	110	18	x	x	X
ejpam-5133	110	19	,	,	PUNCT
ejpam-5133	110	20	d	d	PROPN
ejpam-5133	110	21	∈	∈	NOUN
ejpam-5133	110	22	r	r	NOUN
ejpam-5133	110	23	then	then	ADV
ejpam-5133	110	24	we	we	PRON
ejpam-5133	110	25	obtain	obtain	VERB
ejpam-5133	110	26	all	all	DET
ejpam-5133	110	27	pythagorean	pythagorean	ADJ
ejpam-5133	110	28	triples	triple	NOUN
ejpam-5133	110	29	in	in	ADP
ejpam-5133	110	30	r	r	NOUN
ejpam-5133	110	31	,	,	PUNCT
ejpam-5133	110	32	that	that	ADV
ejpam-5133	110	33	is	is	ADV
ejpam-5133	110	34	,	,	PUNCT
ejpam-5133	110	35	also	also	ADV
ejpam-5133	110	36	also	also	ADV
ejpam-5133	110	37	y	y	PROPN
ejpam-5133	110	38	,	,	PUNCT
ejpam-5133	110	39	z	z	PROPN
ejpam-5133	110	40	∈	∈	PROPN
ejpam-5133	110	41	r.	r.	PROPN
ejpam-5133	110	42	moreover	moreover	ADV
ejpam-5133	110	43	,	,	PUNCT
ejpam-5133	110	44	if	if	SCONJ
ejpam-5133	110	45	x	x	X
ejpam-5133	110	46	,	,	PUNCT
ejpam-5133	110	47	d	d	PROPN
ejpam-5133	110	48	∈	∈	NOUN
ejpam-5133	110	49	r	r	NOUN
ejpam-5133	110	50	are	be	AUX
ejpam-5133	110	51	positive	positive	ADJ
ejpam-5133	110	52	,	,	PUNCT
ejpam-5133	110	53	with	with	ADP
ejpam-5133	110	54	d	d	PROPN
ejpam-5133	110	55	≤	≤	NUM
ejpam-5133	110	56	x	x	PUNCT
ejpam-5133	110	57	,	,	PUNCT
ejpam-5133	110	58	then	then	ADV
ejpam-5133	110	59	also	also	ADV
ejpam-5133	110	60	y	y	PROPN
ejpam-5133	110	61	,	,	PUNCT
ejpam-5133	110	62	z	z	NOUN
ejpam-5133	110	63	∈	∈	NOUN
ejpam-5133	110	64	r	r	NOUN
ejpam-5133	110	65	are	be	AUX
ejpam-5133	110	66	positive	positive	ADJ
ejpam-5133	110	67	.	.	PUNCT
ejpam-5133	111	1	if	if	SCONJ
ejpam-5133	111	2	x	x	PRON
ejpam-5133	111	3	is	be	AUX
ejpam-5133	111	4	a	a	DET
ejpam-5133	111	5	positive	positive	ADJ
ejpam-5133	111	6	integer	integer	NOUN
ejpam-5133	111	7	and	and	CCONJ
ejpam-5133	111	8	d	d	PROPN
ejpam-5133	111	9	∈	∈	PROPN
ejpam-5133	111	10	c(x	c(x	NOUN
ejpam-5133	111	11	)	)	PUNCT
ejpam-5133	111	12	,	,	PUNCT
ejpam-5133	111	13	we	we	PRON
ejpam-5133	111	14	want	want	VERB
ejpam-5133	111	15	to	to	PART
ejpam-5133	111	16	obtain	obtain	VERB
ejpam-5133	111	17	directly	directly	ADV
ejpam-5133	111	18	area	area	VERB
ejpam-5133	111	19	a	a	PRON
ejpam-5133	111	20	,	,	PUNCT
ejpam-5133	111	21	perimeter	perimeter	NOUN
ejpam-5133	111	22	p	p	NOUN
ejpam-5133	111	23	and	and	CCONJ
ejpam-5133	111	24	inradius	inradius	PROPN
ejpam-5133	111	25	r	r	NOUN
ejpam-5133	111	26	of	of	ADP
ejpam-5133	111	27	all	all	DET
ejpam-5133	111	28	right	right	ADV
ejpam-5133	111	29	-	-	PUNCT
ejpam-5133	111	30	angled	angle	VERB
ejpam-5133	111	31	triangle	triangle	NOUN
ejpam-5133	111	32	having	have	VERB
ejpam-5133	111	33	only	only	ADV
ejpam-5133	111	34	a	a	DET
ejpam-5133	111	35	predeterminatus	predeterminatus	NOUN
ejpam-5133	111	36	positive	positive	ADJ
ejpam-5133	111	37	integer	integer	NOUN
ejpam-5133	111	38	cathetus	cathetus	PROPN
ejpam-5133	111	39	x.	x.	NOUN
ejpam-5133	112	1	we	we	PRON
ejpam-5133	112	2	have	have	VERB
ejpam-5133	112	3	the	the	DET
ejpam-5133	112	4	following	follow	VERB
ejpam-5133	112	5	remark	remark	NOUN
ejpam-5133	112	6	.	.	PUNCT
ejpam-5133	113	1	remark	remark	PROPN
ejpam-5133	113	2	1	1	NUM
ejpam-5133	113	3	.	.	PUNCT
ejpam-5133	114	1	in	in	ADP
ejpam-5133	114	2	a	a	DET
ejpam-5133	114	3	right	right	ADJ
ejpam-5133	114	4	-	-	PUNCT
ejpam-5133	114	5	angled	angle	VERB
ejpam-5133	114	6	triangle	triangle	NOUN
ejpam-5133	114	7	,	,	PUNCT
ejpam-5133	114	8	with	with	ADP
ejpam-5133	114	9	a	a	DET
ejpam-5133	114	10	predeterminatus	predeterminatus	NOUN
ejpam-5133	114	11	cathetus	cathetus	NOUN
ejpam-5133	114	12	x	x	SYM
ejpam-5133	114	13	∈	∈	PROPN
ejpam-5133	114	14	n	n	CCONJ
ejpam-5133	114	15	,	,	PUNCT
ejpam-5133	114	16	we	we	PRON
ejpam-5133	114	17	have	have	AUX
ejpam-5133	114	18	,	,	PUNCT
ejpam-5133	114	19	regard	regard	VERB
ejpam-5133	114	20	to	to	ADP
ejpam-5133	114	21	area	area	VERB
ejpam-5133	114	22	a	a	PRON
ejpam-5133	114	23	,	,	PUNCT
ejpam-5133	114	24	perimeter	perimeter	NOUN
ejpam-5133	114	25	p	p	NOUN
ejpam-5133	114	26	and	and	CCONJ
ejpam-5133	114	27	inradius	inradius	PROPN
ejpam-5133	114	28	r	r	NOUN
ejpam-5133	114	29	,	,	PUNCT
ejpam-5133	114	30	the	the	DET
ejpam-5133	114	31	following	follow	VERB
ejpam-5133	114	32	fundamental	fundamental	ADJ
ejpam-5133	114	33	formulas	formula	NOUN
ejpam-5133	114	34	a	a	DET
ejpam-5133	114	35	=	=	SYM
ejpam-5133	114	36	x(x2	x(x2	PROPN
ejpam-5133	114	37	−	−	PROPN
ejpam-5133	114	38	d2	d2	PROPN
ejpam-5133	114	39	)	)	PUNCT
ejpam-5133	114	40	4d	4d	NOUN
ejpam-5133	114	41	,	,	PUNCT
ejpam-5133	114	42	(	(	PUNCT
ejpam-5133	114	43	2	2	X
ejpam-5133	114	44	)	)	PUNCT
ejpam-5133	114	45	p	p	NOUN
ejpam-5133	114	46	=	=	PUNCT
ejpam-5133	114	47	x+	x+	NUM
ejpam-5133	115	1	x2	x2	PROPN
ejpam-5133	115	2	d	d	PROPN
ejpam-5133	115	3	,	,	PUNCT
ejpam-5133	115	4	(	(	PUNCT
ejpam-5133	115	5	3	3	X
ejpam-5133	115	6	)	)	PUNCT
ejpam-5133	115	7	r	r	NOUN
ejpam-5133	115	8	=	=	SYM
ejpam-5133	115	9	x−	x−	PROPN
ejpam-5133	115	10	d	d	PROPN
ejpam-5133	115	11	2	2	NUM
ejpam-5133	115	12	.	.	PUNCT
ejpam-5133	116	1	(	(	PUNCT
ejpam-5133	116	2	4	4	NUM
ejpam-5133	116	3	)	)	PUNCT
ejpam-5133	116	4	with	with	ADP
ejpam-5133	116	5	d	d	PROPN
ejpam-5133	116	6	∈	∈	PROPN
ejpam-5133	116	7	c(x	c(x	NOUN
ejpam-5133	116	8	)	)	PUNCT
ejpam-5133	116	9	.	.	PUNCT
ejpam-5133	117	1	formulas	formula	NOUN
ejpam-5133	117	2	(	(	PUNCT
ejpam-5133	117	3	2	2	NUM
ejpam-5133	117	4	)	)	PUNCT
ejpam-5133	117	5	and	and	CCONJ
ejpam-5133	117	6	(	(	PUNCT
ejpam-5133	117	7	3	3	X
ejpam-5133	117	8	)	)	PUNCT
ejpam-5133	117	9	follow	follow	VERB
ejpam-5133	117	10	directly	directly	ADV
ejpam-5133	117	11	from	from	ADP
ejpam-5133	117	12	(	(	PUNCT
ejpam-5133	117	13	1	1	NUM
ejpam-5133	117	14	)	)	PUNCT
ejpam-5133	117	15	.	.	PUNCT
ejpam-5133	118	1	to	to	PART
ejpam-5133	118	2	find	find	VERB
ejpam-5133	118	3	formula	formula	NOUN
ejpam-5133	118	4	(	(	PUNCT
ejpam-5133	118	5	4	4	NUM
ejpam-5133	118	6	)	)	PUNCT
ejpam-5133	118	7	,	,	PUNCT
ejpam-5133	118	8	it	it	PRON
ejpam-5133	118	9	suffices	suffice	VERB
ejpam-5133	118	10	that	that	SCONJ
ejpam-5133	118	11	we	we	PRON
ejpam-5133	118	12	consider	consider	VERB
ejpam-5133	118	13	the	the	DET
ejpam-5133	118	14	known	know	VERB
ejpam-5133	118	15	formula	formula	NOUN
ejpam-5133	118	16	r	r	NOUN
ejpam-5133	118	17	=	=	SYM
ejpam-5133	118	18	2a	2a	NUM
ejpam-5133	118	19	p	p	NOUN
ejpam-5133	118	20	,	,	PUNCT
ejpam-5133	118	21	obtaining	obtain	VERB
ejpam-5133	118	22	r	r	NOUN
ejpam-5133	118	23	=	=	SYM
ejpam-5133	118	24	2a	2a	NUM
ejpam-5133	118	25	p	p	NOUN
ejpam-5133	118	26	=	=	SYM
ejpam-5133	118	27	2x(x2	2x(x2	NUM
ejpam-5133	118	28	−	−	PROPN
ejpam-5133	118	29	d2	d2	PROPN
ejpam-5133	118	30	)	)	PUNCT
ejpam-5133	118	31	4d	4d	NOUN
ejpam-5133	118	32	x+	x+	PUNCT
ejpam-5133	119	1	x2	x2	PROPN
ejpam-5133	119	2	d	d	NOUN
ejpam-5133	119	3	=	=	SYM
ejpam-5133	119	4	x(x2	x(x2	PROPN
ejpam-5133	119	5	−	−	PROPN
ejpam-5133	119	6	d2	d2	PROPN
ejpam-5133	119	7	)	)	PUNCT
ejpam-5133	119	8	2(xd+	2(xd+	NUM
ejpam-5133	119	9	x2	x2	NOUN
ejpam-5133	119	10	)	)	PUNCT
ejpam-5133	119	11	=	=	SYM
ejpam-5133	119	12	x(x−	x(x−	PROPN
ejpam-5133	119	13	d)(x+	d)(x+	VERB
ejpam-5133	119	14	d	d	NOUN
ejpam-5133	119	15	)	)	PUNCT
ejpam-5133	120	1	2x(x+	2x(x+	PROPN
ejpam-5133	120	2	d	d	X
ejpam-5133	120	3	)	)	PUNCT
ejpam-5133	120	4	=	=	PUNCT
ejpam-5133	121	1	x−	x−	PROPN
ejpam-5133	121	2	d	d	PROPN
ejpam-5133	121	3	2	2	NUM
ejpam-5133	121	4	.	.	PUNCT
ejpam-5133	122	1	moreover	moreover	ADV
ejpam-5133	122	2	,	,	PUNCT
ejpam-5133	122	3	if	if	SCONJ
ejpam-5133	122	4	x	x	PRON
ejpam-5133	122	5	is	be	AUX
ejpam-5133	122	6	a	a	DET
ejpam-5133	122	7	positive	positive	ADJ
ejpam-5133	122	8	integer	integer	NOUN
ejpam-5133	122	9	then	then	ADV
ejpam-5133	122	10	d	d	PROPN
ejpam-5133	122	11	∈	∈	PROPN
ejpam-5133	122	12	c(x),and	c(x),and	PROPN
ejpam-5133	122	13	since	since	SCONJ
ejpam-5133	122	14	x	x	PROPN
ejpam-5133	122	15	and	and	CCONJ
ejpam-5133	122	16	d	d	AUX
ejpam-5133	122	17	have	have	VERB
ejpam-5133	122	18	the	the	DET
ejpam-5133	122	19	same	same	ADJ
ejpam-5133	122	20	parity	parity	NOUN
ejpam-5133	122	21	,	,	PUNCT
ejpam-5133	122	22	we	we	PRON
ejpam-5133	122	23	obtain	obtain	VERB
ejpam-5133	122	24	also	also	ADV
ejpam-5133	122	25	the	the	DET
ejpam-5133	122	26	known	know	VERB
ejpam-5133	122	27	result	result	NOUN
ejpam-5133	122	28	that	that	SCONJ
ejpam-5133	122	29	r	r	NOUN
ejpam-5133	122	30	is	be	AUX
ejpam-5133	122	31	an	an	DET
ejpam-5133	122	32	integer	integer	NOUN
ejpam-5133	122	33	.	.	PUNCT
ejpam-5133	123	1	if	if	SCONJ
ejpam-5133	123	2	we	we	PRON
ejpam-5133	123	3	have	have	VERB
ejpam-5133	123	4	a	a	DET
ejpam-5133	123	5	positive	positive	ADJ
ejpam-5133	123	6	x	x	SYM
ejpam-5133	123	7	∈	∈	PROPN
ejpam-5133	123	8	r	r	NOUN
ejpam-5133	123	9	,	,	PUNCT
ejpam-5133	123	10	(	(	PUNCT
ejpam-5133	123	11	2	2	NUM
ejpam-5133	123	12	)	)	PUNCT
ejpam-5133	123	13	,	,	PUNCT
ejpam-5133	123	14	(	(	PUNCT
ejpam-5133	123	15	3	3	X
ejpam-5133	123	16	)	)	PUNCT
ejpam-5133	123	17	and	and	CCONJ
ejpam-5133	123	18	(	(	PUNCT
ejpam-5133	123	19	4	4	X
ejpam-5133	123	20	)	)	PUNCT
ejpam-5133	123	21	hold	hold	VERB
ejpam-5133	123	22	,	,	PUNCT
ejpam-5133	123	23	with	with	ADP
ejpam-5133	123	24	d	d	PROPN
ejpam-5133	123	25	=	=	SYM
ejpam-5133	123	26	z	z	NOUN
ejpam-5133	123	27	−	−	PROPN
ejpam-5133	123	28	y.	y.	NOUN
ejpam-5133	123	29	for	for	ADP
ejpam-5133	123	30	example	example	NOUN
ejpam-5133	123	31	,	,	PUNCT
ejpam-5133	123	32	we	we	PRON
ejpam-5133	123	33	can	can	AUX
ejpam-5133	123	34	find	find	VERB
ejpam-5133	123	35	the	the	DET
ejpam-5133	123	36	sides	side	NOUN
ejpam-5133	123	37	of	of	ADP
ejpam-5133	123	38	a	a	DET
ejpam-5133	123	39	right	right	ADJ
ejpam-5133	123	40	-	-	PUNCT
ejpam-5133	123	41	angled	angle	VERB
ejpam-5133	123	42	triangle	triangle	NOUN
ejpam-5133	123	43	,	,	PUNCT
ejpam-5133	123	44	knowing	know	VERB
ejpam-5133	123	45	one	one	NUM
ejpam-5133	123	46	cathetus	cathetus	NOUN
ejpam-5133	123	47	x	x	X
ejpam-5133	123	48	and	and	CCONJ
ejpam-5133	123	49	the	the	DET
ejpam-5133	123	50	perimeter	perimeter	PROPN
ejpam-5133	123	51	p.	p.	PROPN
ejpam-5133	123	52	from	from	ADP
ejpam-5133	123	53	(	(	PUNCT
ejpam-5133	123	54	3	3	NUM
ejpam-5133	123	55	)	)	PUNCT
ejpam-5133	123	56	,	,	PUNCT
ejpam-5133	123	57	we	we	PRON
ejpam-5133	123	58	have	have	VERB
ejpam-5133	123	59	d	d	NOUN
ejpam-5133	123	60	=	=	SYM
ejpam-5133	124	1	x2	x2	NOUN
ejpam-5133	124	2	p−	p−	NOUN
ejpam-5133	124	3	x	x	PUNCT
ejpam-5133	124	4	,	,	PUNCT
ejpam-5133	124	5	that	that	PRON
ejpam-5133	124	6	substituted	substitute	VERB
ejpam-5133	124	7	into	into	ADP
ejpam-5133	124	8	the	the	DET
ejpam-5133	124	9	formulas	formula	NOUN
ejpam-5133	124	10	of	of	ADP
ejpam-5133	124	11	theorem	theorem	NOUN
ejpam-5133	124	12	(	(	PUNCT
ejpam-5133	124	13	1	1	NUM
ejpam-5133	124	14	)	)	PUNCT
ejpam-5133	124	15	,	,	PUNCT
ejpam-5133	124	16	provides	provide	VERB
ejpam-5133	124	17	the	the	DET
ejpam-5133	124	18	values	value	NOUN
ejpam-5133	124	19	of	of	ADP
ejpam-5133	124	20	y	y	PROPN
ejpam-5133	124	21	and	and	CCONJ
ejpam-5133	124	22	z.	z.	PROPN
ejpam-5133	124	23	other	other	ADJ
ejpam-5133	124	24	example	example	NOUN
ejpam-5133	124	25	,	,	PUNCT
ejpam-5133	124	26	we	we	PRON
ejpam-5133	124	27	can	can	AUX
ejpam-5133	124	28	find	find	VERB
ejpam-5133	124	29	the	the	DET
ejpam-5133	124	30	sides	side	NOUN
ejpam-5133	124	31	of	of	ADP
ejpam-5133	124	32	a	a	DET
ejpam-5133	124	33	right	right	ADJ
ejpam-5133	124	34	-	-	PUNCT
ejpam-5133	124	35	angled	angle	VERB
ejpam-5133	124	36	triangle	triangle	NOUN
ejpam-5133	124	37	,	,	PUNCT
ejpam-5133	124	38	knowing	know	VERB
ejpam-5133	124	39	one	one	NUM
ejpam-5133	124	40	cathetus	cathetus	NOUN
ejpam-5133	124	41	x	x	X
ejpam-5133	124	42	and	and	CCONJ
ejpam-5133	124	43	the	the	DET
ejpam-5133	124	44	inradius	inradius	PROPN
ejpam-5133	124	45	r.	r.	PROPN
ejpam-5133	124	46	from	from	ADP
ejpam-5133	124	47	(	(	PUNCT
ejpam-5133	124	48	4	4	X
ejpam-5133	124	49	)	)	PUNCT
ejpam-5133	124	50	we	we	PRON
ejpam-5133	124	51	have	have	VERB
ejpam-5133	124	52	d	d	NOUN
ejpam-5133	124	53	=	=	SYM
ejpam-5133	124	54	x−2r	x−2r	NOUN
ejpam-5133	124	55	,	,	PUNCT
ejpam-5133	124	56	that	that	PRON
ejpam-5133	124	57	substituted	substitute	VERB
ejpam-5133	124	58	into	into	ADP
ejpam-5133	124	59	the	the	DET
ejpam-5133	124	60	formulas	formula	NOUN
ejpam-5133	124	61	of	of	ADP
ejpam-5133	124	62	theorem	theorem	NOUN
ejpam-5133	124	63	(	(	PUNCT
ejpam-5133	124	64	1	1	NUM
ejpam-5133	124	65	)	)	PUNCT
ejpam-5133	124	66	,	,	PUNCT
ejpam-5133	124	67	provides	provide	VERB
ejpam-5133	124	68	the	the	DET
ejpam-5133	124	69	values	value	NOUN
ejpam-5133	124	70	of	of	ADP
ejpam-5133	124	71	y	y	PROPN
ejpam-5133	124	72	and	and	CCONJ
ejpam-5133	124	73	z.	z.	PROPN
ejpam-5133	124	74	for	for	ADP
ejpam-5133	124	75	both	both	DET
ejpam-5133	124	76	examples	example	NOUN
ejpam-5133	124	77	,	,	PUNCT
ejpam-5133	124	78	this	this	PRON
ejpam-5133	124	79	avoids	avoid	VERB
ejpam-5133	124	80	forming	form	VERB
ejpam-5133	124	81	relationships	relationship	NOUN
ejpam-5133	124	82	and	and	CCONJ
ejpam-5133	124	83	solving	solve	VERB
ejpam-5133	124	84	systems	system	NOUN
ejpam-5133	124	85	of	of	ADP
ejpam-5133	124	86	equations	equation	NOUN
ejpam-5133	124	87	using	use	VERB
ejpam-5133	124	88	the	the	DET
ejpam-5133	124	89	classical	classical	ADJ
ejpam-5133	124	90	method	method	NOUN
ejpam-5133	124	91	.	.	PUNCT
ejpam-5133	125	1	r.	r.	PROPN
ejpam-5133	125	2	amato	amato	PROPN
ejpam-5133	125	3	/	/	SYM
ejpam-5133	125	4	eur	eur	PROPN
ejpam-5133	125	5	.	.	PUNCT
ejpam-5133	126	1	j.	j.	PROPN
ejpam-5133	126	2	pure	pure	PROPN
ejpam-5133	126	3	appl	appl	PROPN
ejpam-5133	126	4	.	.	PROPN
ejpam-5133	126	5	math	math	PROPN
ejpam-5133	126	6	,	,	PUNCT
ejpam-5133	126	7	17	17	NUM
ejpam-5133	126	8	(	(	PUNCT
ejpam-5133	126	9	2	2	NUM
ejpam-5133	126	10	)	)	PUNCT
ejpam-5133	126	11	(	(	PUNCT
ejpam-5133	126	12	2024	2024	NUM
ejpam-5133	126	13	)	)	PUNCT
ejpam-5133	126	14	,	,	PUNCT
ejpam-5133	126	15	676	676	NUM
ejpam-5133	126	16	-	-	SYM
ejpam-5133	126	17	689	689	NUM
ejpam-5133	126	18	682	682	NUM
ejpam-5133	126	19	let	let	VERB
ejpam-5133	126	20	the	the	DET
ejpam-5133	126	21	pythagorean	pythagorean	PROPN
ejpam-5133	126	22	triangle	triangle	NOUN
ejpam-5133	126	23	abc	abc	PROPN
ejpam-5133	126	24	be	be	AUX
ejpam-5133	126	25	depicted	depict	VERB
ejpam-5133	126	26	in	in	ADP
ejpam-5133	126	27	figure	figure	NOUN
ejpam-5133	126	28	(	(	PUNCT
ejpam-5133	126	29	1	1	NUM
ejpam-5133	126	30	)	)	PUNCT
ejpam-5133	126	31	.	.	PUNCT
ejpam-5133	127	1	we	we	PRON
ejpam-5133	127	2	consider	consider	VERB
ejpam-5133	127	3	lines	line	NOUN
ejpam-5133	127	4	oa	oa	ADP
ejpam-5133	127	5	,	,	PUNCT
ejpam-5133	127	6	ob	ob	INTJ
ejpam-5133	127	7	,	,	PUNCT
ejpam-5133	127	8	oc	oc	NOUN
ejpam-5133	127	9	from	from	ADP
ejpam-5133	127	10	the	the	DET
ejpam-5133	127	11	incentre	incentre	NOUN
ejpam-5133	127	12	to	to	ADP
ejpam-5133	127	13	the	the	DET
ejpam-5133	127	14	vertices	vertex	NOUN
ejpam-5133	127	15	,	,	PUNCT
ejpam-5133	127	16	and	and	CCONJ
ejpam-5133	127	17	x	x	X
ejpam-5133	127	18	=	=	SYM
ejpam-5133	127	19	ab	ab	PROPN
ejpam-5133	127	20	.	.	PUNCT
ejpam-5133	128	1	the	the	DET
ejpam-5133	128	2	following	follow	VERB
ejpam-5133	128	3	theorem	theorem	ADJ
ejpam-5133	128	4	holds	hold	NOUN
ejpam-5133	128	5	.	.	PUNCT
ejpam-5133	128	6	theorem	theorem	NOUN
ejpam-5133	128	7	5	5	NUM
ejpam-5133	128	8	.	.	PUNCT
ejpam-5133	129	1	in	in	ADP
ejpam-5133	129	2	a	a	DET
ejpam-5133	129	3	right	right	ADJ
ejpam-5133	129	4	-	-	PUNCT
ejpam-5133	129	5	angled	angle	VERB
ejpam-5133	129	6	triangle	triangle	NOUN
ejpam-5133	129	7	,	,	PUNCT
ejpam-5133	129	8	with	with	ADP
ejpam-5133	129	9	a	a	DET
ejpam-5133	129	10	predeterminatus	predeterminatus	NOUN
ejpam-5133	129	11	cathetus	cathetus	NOUN
ejpam-5133	129	12	x	x	SYM
ejpam-5133	129	13	∈	∈	PROPN
ejpam-5133	129	14	r	r	NOUN
ejpam-5133	129	15	,	,	PUNCT
ejpam-5133	129	16	we	we	PRON
ejpam-5133	129	17	have	have	VERB
ejpam-5133	129	18	the	the	DET
ejpam-5133	129	19	following	follow	VERB
ejpam-5133	129	20	relation	relation	NOUN
ejpam-5133	129	21	among	among	ADP
ejpam-5133	129	22	lines	line	NOUN
ejpam-5133	129	23	oa	oa	ADP
ejpam-5133	129	24	,	,	PUNCT
ejpam-5133	129	25	ob	ob	INTJ
ejpam-5133	129	26	,	,	PUNCT
ejpam-5133	129	27	oc	oc	NOUN
ejpam-5133	129	28	from	from	ADP
ejpam-5133	129	29	the	the	DET
ejpam-5133	129	30	incentre	incentre	NOUN
ejpam-5133	129	31	to	to	ADP
ejpam-5133	129	32	the	the	DET
ejpam-5133	129	33	vertices	vertex	NOUN
ejpam-5133	129	34	oa	oa	X
ejpam-5133	129	35	·	·	PUNCT
ejpam-5133	129	36	ob	ob	NOUN
ejpam-5133	129	37	=	=	SYM
ejpam-5133	129	38	d	d	PROPN
ejpam-5133	129	39	·	·	PUNCT
ejpam-5133	129	40	oc	oc	X
ejpam-5133	129	41	(	(	PUNCT
ejpam-5133	129	42	5	5	NUM
ejpam-5133	129	43	)	)	PUNCT
ejpam-5133	129	44	with	with	ADP
ejpam-5133	129	45	oa	oa	NOUN
ejpam-5133	129	46	,	,	PUNCT
ejpam-5133	129	47	ob	ob	ADP
ejpam-5133	129	48	<	<	X
ejpam-5133	129	49	oc	oc	NOUN
ejpam-5133	129	50	,	,	PUNCT
ejpam-5133	129	51	and	and	CCONJ
ejpam-5133	129	52	d	d	NOUN
ejpam-5133	129	53	=	=	SYM
ejpam-5133	129	54	z	z	X
ejpam-5133	129	55	−	−	PROPN
ejpam-5133	129	56	y.	y.	NOUN
ejpam-5133	129	57	figure	figure	NOUN
ejpam-5133	129	58	1	1	NUM
ejpam-5133	129	59	:	:	PUNCT
ejpam-5133	129	60	proof	proof	NOUN
ejpam-5133	129	61	.	.	PUNCT
ejpam-5133	130	1	let	let	VERB
ejpam-5133	130	2	’s	’s	NOUN
ejpam-5133	130	3	begin	begin	VERB
ejpam-5133	130	4	to	to	PART
ejpam-5133	130	5	notice	notice	VERB
ejpam-5133	130	6	that	that	SCONJ
ejpam-5133	130	7	eb	eb	PROPN
ejpam-5133	130	8	=	=	PUNCT
ejpam-5133	130	9	x+	x+	PUNCT
ejpam-5133	131	1	d	d	NOUN
ejpam-5133	131	2	2	2	NUM
ejpam-5133	131	3	,	,	PUNCT
ejpam-5133	131	4	while	while	SCONJ
ejpam-5133	131	5	from	from	ADP
ejpam-5133	131	6	theorem	theorem	ADJ
ejpam-5133	131	7	2.1	2.1	NUM
ejpam-5133	131	8	and	and	CCONJ
ejpam-5133	131	9	(	(	PUNCT
ejpam-5133	131	10	4	4	X
ejpam-5133	131	11	)	)	PUNCT
ejpam-5133	131	12	we	we	PRON
ejpam-5133	131	13	have	have	VERB
ejpam-5133	131	14	fc	fc	PROPN
ejpam-5133	132	1	=	=	PUNCT
ejpam-5133	132	2	ac	ac	PROPN
ejpam-5133	132	3	−	−	NOUN
ejpam-5133	132	4	r	r	NOUN
ejpam-5133	132	5	=	=	SYM
ejpam-5133	132	6	x2	x2	PROPN
ejpam-5133	133	1	−	−	PROPN
ejpam-5133	133	2	d2	d2	PROPN
ejpam-5133	133	3	2d	2d	PROPN
ejpam-5133	133	4	−	−	PROPN
ejpam-5133	133	5	x−	x−	PROPN
ejpam-5133	134	1	d	d	NOUN
ejpam-5133	134	2	2	2	NUM
ejpam-5133	134	3	=	=	SYM
ejpam-5133	134	4	x−	x−	PROPN
ejpam-5133	135	1	d	d	X
ejpam-5133	135	2	2d	2d	PROPN
ejpam-5133	135	3	·	·	PUNCT
ejpam-5133	135	4	x.	x.	NOUN
ejpam-5133	135	5	applying	apply	VERB
ejpam-5133	135	6	the	the	DET
ejpam-5133	135	7	pythagorean	pythagorean	PROPN
ejpam-5133	135	8	theorem	theorem	PROPN
ejpam-5133	135	9	,	,	PUNCT
ejpam-5133	135	10	we	we	PRON
ejpam-5133	135	11	obtain	obtain	VERB
ejpam-5133	135	12	oa	oa	ADP
ejpam-5133	135	13	=	=	PUNCT
ejpam-5133	135	14	x−	x−	PROPN
ejpam-5133	136	1	d	d	PROPN
ejpam-5133	136	2	2	2	NUM
ejpam-5133	136	3	·	·	PUNCT
ejpam-5133	136	4	√	√	NUM
ejpam-5133	136	5	2	2	NUM
ejpam-5133	136	6	,	,	PUNCT
ejpam-5133	136	7	ob	ob	NOUN
ejpam-5133	136	8	=	=	PUNCT
ejpam-5133	136	9	√	√	PROPN
ejpam-5133	136	10	x2	x2	NOUN
ejpam-5133	137	1	+	+	PUNCT
ejpam-5133	137	2	d2√	d2√	PROPN
ejpam-5133	137	3	2	2	NUM
ejpam-5133	137	4	,	,	PUNCT
ejpam-5133	137	5	oc	oc	ADP
ejpam-5133	137	6	=	=	PUNCT
ejpam-5133	137	7	x−	x−	PROPN
ejpam-5133	138	1	d	d	X
ejpam-5133	138	2	2d	2d	PROPN
ejpam-5133	138	3	·	·	PUNCT
ejpam-5133	138	4	√	√	NUM
ejpam-5133	138	5	x2	x2	PROPN
ejpam-5133	139	1	+	+	CCONJ
ejpam-5133	139	2	d2	d2	PROPN
ejpam-5133	139	3	from	from	ADP
ejpam-5133	139	4	wich	wich	PROPN
ejpam-5133	139	5	(	(	PUNCT
ejpam-5133	139	6	5	5	X
ejpam-5133	139	7	)	)	PUNCT
ejpam-5133	139	8	easily	easily	ADV
ejpam-5133	139	9	follows	follow	VERB
ejpam-5133	139	10	,	,	PUNCT
ejpam-5133	139	11	and	and	CCONJ
ejpam-5133	139	12	consequently	consequently	ADV
ejpam-5133	139	13	,	,	PUNCT
ejpam-5133	139	14	theorem	theorem	ADJ
ejpam-5133	139	15	(	(	PUNCT
ejpam-5133	139	16	5	5	NUM
ejpam-5133	139	17	)	)	PUNCT
ejpam-5133	139	18	is	be	AUX
ejpam-5133	139	19	proved	prove	VERB
ejpam-5133	139	20	.	.	PUNCT
ejpam-5133	140	1	obviously	obviously	ADV
ejpam-5133	140	2	if	if	SCONJ
ejpam-5133	140	3	x	x	PRON
ejpam-5133	140	4	is	be	AUX
ejpam-5133	140	5	a	a	DET
ejpam-5133	140	6	positive	positive	ADJ
ejpam-5133	140	7	integer	integer	NOUN
ejpam-5133	140	8	we	we	PRON
ejpam-5133	140	9	have	have	VERB
ejpam-5133	140	10	d	d	PROPN
ejpam-5133	140	11	∈	∈	PROPN
ejpam-5133	140	12	c(x	c(x	NOUN
ejpam-5133	140	13	)	)	PUNCT
ejpam-5133	140	14	.	.	PUNCT
ejpam-5133	141	1	r.	r.	PROPN
ejpam-5133	141	2	amato	amato	PROPN
ejpam-5133	141	3	/	/	SYM
ejpam-5133	141	4	eur	eur	PROPN
ejpam-5133	141	5	.	.	PUNCT
ejpam-5133	142	1	j.	j.	PROPN
ejpam-5133	142	2	pure	pure	PROPN
ejpam-5133	142	3	appl	appl	PROPN
ejpam-5133	142	4	.	.	PROPN
ejpam-5133	142	5	math	math	PROPN
ejpam-5133	142	6	,	,	PUNCT
ejpam-5133	142	7	17	17	NUM
ejpam-5133	142	8	(	(	PUNCT
ejpam-5133	142	9	2	2	NUM
ejpam-5133	142	10	)	)	PUNCT
ejpam-5133	142	11	(	(	PUNCT
ejpam-5133	142	12	2024	2024	NUM
ejpam-5133	142	13	)	)	PUNCT
ejpam-5133	142	14	,	,	PUNCT
ejpam-5133	142	15	676	676	NUM
ejpam-5133	142	16	-	-	SYM
ejpam-5133	142	17	689	689	NUM
ejpam-5133	142	18	683	683	NUM
ejpam-5133	142	19	figure	figure	NOUN
ejpam-5133	142	20	2	2	NUM
ejpam-5133	142	21	:	:	PUNCT
ejpam-5133	142	22	let	let	VERB
ejpam-5133	142	23	x	x	PRON
ejpam-5133	142	24	,	,	PUNCT
ejpam-5133	142	25	y	y	PROPN
ejpam-5133	142	26	,	,	PUNCT
ejpam-5133	142	27	z	z	NOUN
ejpam-5133	142	28	∈	∈	NOUN
ejpam-5133	142	29	r	r	NOUN
ejpam-5133	142	30	be	be	VERB
ejpam-5133	142	31	positive	positive	ADJ
ejpam-5133	142	32	satisfying	satisfy	VERB
ejpam-5133	142	33	x2	x2	NOUN
ejpam-5133	143	1	+	+	CCONJ
ejpam-5133	143	2	y2	y2	NOUN
ejpam-5133	143	3	=	=	SYM
ejpam-5133	143	4	z2	z2	PROPN
ejpam-5133	143	5	.	.	PROPN
ejpam-5133	144	1	from	from	ADP
ejpam-5133	144	2	theorem	theorem	NOUN
ejpam-5133	144	3	(	(	PUNCT
ejpam-5133	144	4	1	1	NUM
ejpam-5133	144	5	)	)	PUNCT
ejpam-5133	144	6	,	,	PUNCT
ejpam-5133	144	7	this	this	DET
ejpam-5133	144	8	triple	triple	NOUN
ejpam-5133	144	9	is	be	AUX
ejpam-5133	144	10	generated	generate	VERB
ejpam-5133	144	11	by	by	ADP
ejpam-5133	144	12	x	x	PUNCT
ejpam-5133	144	13	with	with	ADP
ejpam-5133	144	14	d	d	PROPN
ejpam-5133	144	15	=	=	SYM
ejpam-5133	144	16	z	z	NOUN
ejpam-5133	144	17	−	−	PROPN
ejpam-5133	144	18	y	y	PROPN
ejpam-5133	144	19	and	and	CCONJ
ejpam-5133	144	20	y	y	PROPN
ejpam-5133	144	21	with	with	ADP
ejpam-5133	144	22	d′	d′	NUM
ejpam-5133	144	23	=	=	PUNCT
ejpam-5133	145	1	z	z	NOUN
ejpam-5133	145	2	−	−	NOUN
ejpam-5133	145	3	x	x	PUNCT
ejpam-5133	145	4	respectively	respectively	ADV
ejpam-5133	145	5	.	.	PUNCT
ejpam-5133	146	1	we	we	PRON
ejpam-5133	146	2	want	want	VERB
ejpam-5133	146	3	to	to	PART
ejpam-5133	146	4	study	study	VERB
ejpam-5133	146	5	the	the	DET
ejpam-5133	146	6	relation	relation	NOUN
ejpam-5133	147	1	betwen	betwen	PROPN
ejpam-5133	147	2	d	d	PROPN
ejpam-5133	147	3	and	and	CCONJ
ejpam-5133	147	4	d′	d′	PRON
ejpam-5133	147	5	used	use	VERB
ejpam-5133	147	6	to	to	PART
ejpam-5133	147	7	obtain	obtain	VERB
ejpam-5133	147	8	the	the	DET
ejpam-5133	147	9	same	same	ADJ
ejpam-5133	147	10	triple	triple	NOUN
ejpam-5133	147	11	.	.	PUNCT
ejpam-5133	148	1	from	from	ADP
ejpam-5133	148	2	formulas	formula	NOUN
ejpam-5133	148	3	(	(	PUNCT
ejpam-5133	148	4	1	1	NUM
ejpam-5133	148	5	)	)	PUNCT
ejpam-5133	148	6	,	,	PUNCT
ejpam-5133	148	7	we	we	PRON
ejpam-5133	148	8	obtain	obtain	VERB
ejpam-5133	148	9	d′	d′	X
ejpam-5133	148	10	=	=	SYM
ejpam-5133	149	1	x2	x2	PROPN
ejpam-5133	150	1	+	+	NUM
ejpam-5133	150	2	d2	d2	PROPN
ejpam-5133	150	3	2d	2d	NOUN
ejpam-5133	150	4	−	−	NOUN
ejpam-5133	150	5	x	x	SYM
ejpam-5133	150	6	=	=	SYM
ejpam-5133	150	7	(	(	PUNCT
ejpam-5133	150	8	x−	x−	PROPN
ejpam-5133	150	9	d)2	d)2	PROPN
ejpam-5133	150	10	2d	2d	NUM
ejpam-5133	150	11	.	.	PUNCT
ejpam-5133	151	1	(	(	PUNCT
ejpam-5133	151	2	6	6	NUM
ejpam-5133	151	3	)	)	PUNCT
ejpam-5133	151	4	that	that	PRON
ejpam-5133	151	5	is	be	AUX
ejpam-5133	151	6	the	the	DET
ejpam-5133	151	7	the	the	DET
ejpam-5133	151	8	relation	relation	NOUN
ejpam-5133	151	9	betwen	betwen	PROPN
ejpam-5133	151	10	d	d	PROPN
ejpam-5133	151	11	and	and	CCONJ
ejpam-5133	151	12	d′.	d′.	AUX
ejpam-5133	151	13	let	let	VERB
ejpam-5133	151	14	the	the	DET
ejpam-5133	151	15	pythagorean	pythagorean	PROPN
ejpam-5133	151	16	triangle	triangle	NOUN
ejpam-5133	151	17	abc	abc	PROPN
ejpam-5133	151	18	be	be	AUX
ejpam-5133	151	19	depicted	depict	VERB
ejpam-5133	151	20	in	in	ADP
ejpam-5133	151	21	figure	figure	NOUN
ejpam-5133	151	22	(	(	PUNCT
ejpam-5133	151	23	2	2	NUM
ejpam-5133	151	24	)	)	PUNCT
ejpam-5133	151	25	,	,	PUNCT
ejpam-5133	151	26	and	and	CCONJ
ejpam-5133	151	27	oh	oh	INTJ
ejpam-5133	151	28	the	the	DET
ejpam-5133	151	29	line	line	NOUN
ejpam-5133	151	30	from	from	ADP
ejpam-5133	151	31	centre	centre	NOUN
ejpam-5133	151	32	of	of	ADP
ejpam-5133	151	33	incircle	incircle	NOUN
ejpam-5133	151	34	to	to	PART
ejpam-5133	151	35	centre	centre	VERB
ejpam-5133	151	36	of	of	ADP
ejpam-5133	151	37	circumcircle	circumcircle	NOUN
ejpam-5133	151	38	and	and	CCONJ
ejpam-5133	151	39	x	x	X
ejpam-5133	151	40	=	=	SYM
ejpam-5133	151	41	ab	ab	PROPN
ejpam-5133	151	42	.	.	PUNCT
ejpam-5133	152	1	the	the	DET
ejpam-5133	152	2	following	follow	VERB
ejpam-5133	152	3	theorem	theorem	ADJ
ejpam-5133	152	4	holds	hold	NOUN
ejpam-5133	152	5	.	.	PUNCT
ejpam-5133	152	6	theorem	theorem	NOUN
ejpam-5133	152	7	6	6	NUM
ejpam-5133	152	8	.	.	PUNCT
ejpam-5133	153	1	in	in	ADP
ejpam-5133	153	2	a	a	DET
ejpam-5133	153	3	right	right	ADJ
ejpam-5133	153	4	-	-	PUNCT
ejpam-5133	153	5	angled	angle	VERB
ejpam-5133	153	6	triangle	triangle	NOUN
ejpam-5133	153	7	,	,	PUNCT
ejpam-5133	153	8	with	with	ADP
ejpam-5133	153	9	a	a	DET
ejpam-5133	153	10	predeterminatus	predeterminatus	NOUN
ejpam-5133	153	11	cathetus	cathetus	NOUN
ejpam-5133	153	12	x	x	SYM
ejpam-5133	153	13	∈	∈	PROPN
ejpam-5133	153	14	r	r	NOUN
ejpam-5133	153	15	the	the	DET
ejpam-5133	153	16	line	line	NOUN
ejpam-5133	153	17	from	from	ADP
ejpam-5133	153	18	centre	centre	NOUN
ejpam-5133	153	19	of	of	ADP
ejpam-5133	153	20	incircle	incircle	NOUN
ejpam-5133	153	21	to	to	PART
ejpam-5133	153	22	centre	centre	VERB
ejpam-5133	153	23	of	of	ADP
ejpam-5133	153	24	circumcircle	circumcircle	NOUN
ejpam-5133	153	25	,	,	PUNCT
ejpam-5133	153	26	is	be	AUX
ejpam-5133	153	27	given	give	VERB
ejpam-5133	153	28	from	from	ADP
ejpam-5133	153	29	oh	oh	NOUN
ejpam-5133	153	30	=	=	SYM
ejpam-5133	153	31	√	√	PROPN
ejpam-5133	153	32	(	(	PUNCT
ejpam-5133	153	33	x−	x−	PROPN
ejpam-5133	153	34	d)4	d)4	PROPN
ejpam-5133	153	35	+	+	CCONJ
ejpam-5133	153	36	4d4	4d4	NUM
ejpam-5133	153	37	(	(	PUNCT
ejpam-5133	153	38	4d)4	4d)4	NUM
ejpam-5133	153	39	(	(	PUNCT
ejpam-5133	153	40	7	7	NUM
ejpam-5133	153	41	)	)	PUNCT
ejpam-5133	153	42	and	and	CCONJ
ejpam-5133	153	43	also	also	ADV
ejpam-5133	153	44	oh	oh	INTJ
ejpam-5133	153	45	=	=	SYM
ejpam-5133	153	46	√	√	PROPN
ejpam-5133	153	47	(	(	PUNCT
ejpam-5133	153	48	d	d	NOUN
ejpam-5133	153	49	2	2	NUM
ejpam-5133	153	50	)	)	SYM
ejpam-5133	153	51	2	2	NUM
ejpam-5133	153	52	+	+	CCONJ
ejpam-5133	153	53	(	(	PUNCT
ejpam-5133	153	54	d′	d′	NOUN
ejpam-5133	153	55	2	2	NUM
ejpam-5133	153	56	)	)	SYM
ejpam-5133	153	57	2	2	NUM
ejpam-5133	153	58	(	(	PUNCT
ejpam-5133	153	59	8)	8)	NUM
ejpam-5133	153	60	with	with	ADP
ejpam-5133	153	61	d	d	PROPN
ejpam-5133	153	62	=	=	SYM
ejpam-5133	153	63	z	z	NOUN
ejpam-5133	154	1	−	−	PROPN
ejpam-5133	154	2	y	y	PROPN
ejpam-5133	154	3	and	and	CCONJ
ejpam-5133	154	4	d′	d′	NUM
ejpam-5133	154	5	=	=	PUNCT
ejpam-5133	155	1	z	z	NOUN
ejpam-5133	155	2	−	−	PROPN
ejpam-5133	155	3	x.	x.	PROPN
ejpam-5133	155	4	r.	r.	PROPN
ejpam-5133	155	5	amato	amato	PROPN
ejpam-5133	155	6	/	/	SYM
ejpam-5133	155	7	eur	eur	PROPN
ejpam-5133	155	8	.	.	PUNCT
ejpam-5133	156	1	j.	j.	PROPN
ejpam-5133	156	2	pure	pure	PROPN
ejpam-5133	156	3	appl	appl	PROPN
ejpam-5133	156	4	.	.	PROPN
ejpam-5133	156	5	math	math	PROPN
ejpam-5133	156	6	,	,	PUNCT
ejpam-5133	156	7	17	17	NUM
ejpam-5133	156	8	(	(	PUNCT
ejpam-5133	156	9	2	2	NUM
ejpam-5133	156	10	)	)	PUNCT
ejpam-5133	156	11	(	(	PUNCT
ejpam-5133	156	12	2024	2024	NUM
ejpam-5133	156	13	)	)	PUNCT
ejpam-5133	156	14	,	,	PUNCT
ejpam-5133	156	15	676	676	NUM
ejpam-5133	156	16	-	-	SYM
ejpam-5133	156	17	689	689	NUM
ejpam-5133	156	18	684	684	NUM
ejpam-5133	156	19	proof	proof	NOUN
ejpam-5133	156	20	.	.	PUNCT
ejpam-5133	157	1	let	let	VERB
ejpam-5133	157	2	’s	’s	NOUN
ejpam-5133	157	3	begin	begin	VERB
ejpam-5133	157	4	to	to	PART
ejpam-5133	157	5	notice	notice	VERB
ejpam-5133	157	6	that	that	SCONJ
ejpam-5133	158	1	oi	oi	ADV
ejpam-5133	158	2	=	=	PUNCT
ejpam-5133	158	3	x	x	SYM
ejpam-5133	158	4	2	2	NUM
ejpam-5133	158	5	−	−	NOUN
ejpam-5133	158	6	x−	x−	NOUN
ejpam-5133	158	7	d	d	NOUN
ejpam-5133	158	8	2	2	NUM
ejpam-5133	158	9	=	=	SYM
ejpam-5133	158	10	d	d	NOUN
ejpam-5133	158	11	2	2	NUM
ejpam-5133	158	12	while	while	SCONJ
ejpam-5133	158	13	ih	ih	NOUN
ejpam-5133	158	14	=	=	NOUN
ejpam-5133	158	15	1	1	NUM
ejpam-5133	158	16	2	2	NUM
ejpam-5133	158	17	(	(	PUNCT
ejpam-5133	158	18	x2	x2	PROPN
ejpam-5133	158	19	2d	2d	PROPN
ejpam-5133	159	1	−	−	PROPN
ejpam-5133	159	2	d	d	NOUN
ejpam-5133	159	3	2	2	NUM
ejpam-5133	159	4	)	)	PUNCT
ejpam-5133	159	5	−	−	PROPN
ejpam-5133	159	6	x−	x−	PROPN
ejpam-5133	160	1	d	d	NOUN
ejpam-5133	160	2	2	2	NUM
ejpam-5133	160	3	=	=	SYM
ejpam-5133	160	4	(	(	PUNCT
ejpam-5133	160	5	x−	x−	PROPN
ejpam-5133	160	6	d)2	d)2	PROPN
ejpam-5133	160	7	4d	4d	NUM
ejpam-5133	160	8	.	.	PUNCT
ejpam-5133	161	1	applying	apply	VERB
ejpam-5133	161	2	the	the	DET
ejpam-5133	161	3	pythagorean	pythagorean	PROPN
ejpam-5133	161	4	theorem	theorem	PROPN
ejpam-5133	161	5	,	,	PUNCT
ejpam-5133	161	6	we	we	PRON
ejpam-5133	161	7	obtain	obtain	VERB
ejpam-5133	161	8	(	(	PUNCT
ejpam-5133	161	9	7	7	NUM
ejpam-5133	161	10	)	)	PUNCT
ejpam-5133	161	11	,	,	PUNCT
ejpam-5133	161	12	and	and	CCONJ
ejpam-5133	161	13	for	for	ADP
ejpam-5133	161	14	(	(	PUNCT
ejpam-5133	161	15	6	6	NUM
ejpam-5133	161	16	)	)	PUNCT
ejpam-5133	161	17	also	also	ADV
ejpam-5133	161	18	(	(	PUNCT
ejpam-5133	161	19	8)	8)	NUM
ejpam-5133	161	20	.	.	PUNCT
ejpam-5133	161	21	consequently	consequently	ADV
ejpam-5133	161	22	,	,	PUNCT
ejpam-5133	161	23	theorem	theorem	ADJ
ejpam-5133	161	24	(	(	PUNCT
ejpam-5133	161	25	6	6	NUM
ejpam-5133	161	26	)	)	PUNCT
ejpam-5133	161	27	is	be	AUX
ejpam-5133	161	28	proved	prove	VERB
ejpam-5133	161	29	.	.	PUNCT
ejpam-5133	162	1	obviously	obviously	ADV
ejpam-5133	162	2	if	if	SCONJ
ejpam-5133	162	3	x	x	PRON
ejpam-5133	162	4	is	be	AUX
ejpam-5133	162	5	a	a	DET
ejpam-5133	162	6	positive	positive	ADJ
ejpam-5133	162	7	integer	integer	NOUN
ejpam-5133	162	8	we	we	PRON
ejpam-5133	162	9	have	have	VERB
ejpam-5133	162	10	d	d	PROPN
ejpam-5133	162	11	∈	∈	PROPN
ejpam-5133	162	12	c(x	c(x	NOUN
ejpam-5133	162	13	)	)	PUNCT
ejpam-5133	162	14	and	and	CCONJ
ejpam-5133	162	15	d′	d′	NUM
ejpam-5133	162	16	∈	∈	PROPN
ejpam-5133	162	17	c(y	c(y	PROPN
ejpam-5133	162	18	)	)	PUNCT
ejpam-5133	162	19	.	.	PUNCT
ejpam-5133	163	1	let	let	VERB
ejpam-5133	163	2	the	the	DET
ejpam-5133	163	3	pythagorean	pythagorean	PROPN
ejpam-5133	163	4	triangle	triangle	NOUN
ejpam-5133	163	5	abc	abc	PROPN
ejpam-5133	163	6	be	be	AUX
ejpam-5133	163	7	depicted	depict	VERB
ejpam-5133	163	8	in	in	ADP
ejpam-5133	163	9	figure	figure	NOUN
ejpam-5133	163	10	(	(	PUNCT
ejpam-5133	163	11	2	2	NUM
ejpam-5133	163	12	)	)	PUNCT
ejpam-5133	163	13	.	.	PUNCT
ejpam-5133	164	1	to	to	PART
ejpam-5133	164	2	obtain	obtain	VERB
ejpam-5133	164	3	trigonometric	trigonometric	ADJ
ejpam-5133	164	4	formulas	formula	NOUN
ejpam-5133	164	5	,	,	PUNCT
ejpam-5133	164	6	using	use	VERB
ejpam-5133	164	7	only	only	ADV
ejpam-5133	164	8	a	a	DET
ejpam-5133	164	9	predeterminatus	predeterminatus	NOUN
ejpam-5133	164	10	cathetus	cathetus	NOUN
ejpam-5133	164	11	x	x	SYM
ejpam-5133	164	12	∈	∈	PROPN
ejpam-5133	164	13	n	n	CCONJ
ejpam-5133	164	14	,	,	PUNCT
ejpam-5133	164	15	with	with	ADP
ejpam-5133	164	16	d	d	PROPN
ejpam-5133	164	17	∈	∈	PROPN
ejpam-5133	164	18	c(x	c(x	NOUN
ejpam-5133	164	19	)	)	PUNCT
ejpam-5133	164	20	,	,	PUNCT
ejpam-5133	164	21	we	we	PRON
ejpam-5133	164	22	have	have	VERB
ejpam-5133	164	23	the	the	DET
ejpam-5133	164	24	following	follow	VERB
ejpam-5133	164	25	remark	remark	NOUN
ejpam-5133	164	26	.	.	PUNCT
ejpam-5133	165	1	remark	remark	PROPN
ejpam-5133	165	2	2	2	NUM
ejpam-5133	165	3	.	.	PUNCT
ejpam-5133	166	1	in	in	ADP
ejpam-5133	166	2	a	a	DET
ejpam-5133	166	3	right	right	ADJ
ejpam-5133	166	4	-	-	PUNCT
ejpam-5133	166	5	angled	angle	VERB
ejpam-5133	166	6	triangle	triangle	NOUN
ejpam-5133	166	7	,	,	PUNCT
ejpam-5133	166	8	with	with	ADP
ejpam-5133	166	9	a	a	DET
ejpam-5133	166	10	predeterminatus	predeterminatus	NOUN
ejpam-5133	166	11	cathetus	cathetus	NOUN
ejpam-5133	166	12	x	x	SYM
ejpam-5133	166	13	∈	∈	PROPN
ejpam-5133	166	14	n	n	X
ejpam-5133	166	15	for	for	ADP
ejpam-5133	166	16	the	the	DET
ejpam-5133	166	17	angle	angle	NOUN
ejpam-5133	166	18	apposite	apposite	ADV
ejpam-5133	166	19	to	to	ADP
ejpam-5133	166	20	x	x	PROPN
ejpam-5133	166	21	and	and	CCONJ
ejpam-5133	166	22	the	the	DET
ejpam-5133	166	23	acute	acute	PROPN
ejpam-5133	166	24	angle	angle	NOUN
ejpam-5133	166	25	adjacent	adjacent	ADJ
ejpam-5133	166	26	to	to	ADP
ejpam-5133	166	27	x	x	SYM
ejpam-5133	166	28	,	,	PUNCT
ejpam-5133	166	29	we	we	PRON
ejpam-5133	166	30	have	have	VERB
ejpam-5133	166	31	the	the	DET
ejpam-5133	166	32	following	follow	VERB
ejpam-5133	166	33	trigonometric	trigonometric	ADJ
ejpam-5133	166	34	formulas	formula	NOUN
ejpam-5133	166	35	sin(ĉ	sin(ĉ	NOUN
ejpam-5133	166	36	)	)	PUNCT
ejpam-5133	166	37	=	=	SYM
ejpam-5133	166	38	cos(b̂	cos(b̂	ADJ
ejpam-5133	166	39	)	)	PUNCT
ejpam-5133	166	40	=	=	SYM
ejpam-5133	166	41	2xd	2xd	ADJ
ejpam-5133	166	42	x2	x2	PROPN
ejpam-5133	167	1	+	+	CCONJ
ejpam-5133	167	2	d2	d2	PROPN
ejpam-5133	167	3	,	,	PUNCT
ejpam-5133	167	4	cos(ĉ	cos(ĉ	PROPN
ejpam-5133	167	5	)	)	PUNCT
ejpam-5133	167	6	=	=	SYM
ejpam-5133	167	7	sin(b̂	sin(b̂	PROPN
ejpam-5133	167	8	)	)	PUNCT
ejpam-5133	167	9	=	=	SYM
ejpam-5133	168	1	x2	x2	PROPN
ejpam-5133	169	1	−	−	PROPN
ejpam-5133	169	2	d2	d2	PROPN
ejpam-5133	169	3	x2	x2	PROPN
ejpam-5133	169	4	+	+	CCONJ
ejpam-5133	169	5	d2	d2	PROPN
ejpam-5133	169	6	,	,	PUNCT
ejpam-5133	169	7	(	(	PUNCT
ejpam-5133	169	8	9	9	X
ejpam-5133	169	9	)	)	PUNCT
ejpam-5133	169	10	sin	sin	NOUN
ejpam-5133	169	11	ˆ	ˆ	PROPN
ejpam-5133	169	12	(	(	PUNCT
ejpam-5133	169	13	c	c	NOUN
ejpam-5133	169	14	2	2	NUM
ejpam-5133	169	15	)	)	PUNCT
ejpam-5133	169	16	=	=	SYM
ejpam-5133	169	17	d√	d√	PROPN
ejpam-5133	169	18	x2	x2	PROPN
ejpam-5133	169	19	+	+	CCONJ
ejpam-5133	169	20	d2	d2	PROPN
ejpam-5133	169	21	,	,	PUNCT
ejpam-5133	169	22	cos	cos	PROPN
ejpam-5133	169	23	ˆ	ˆ	PROPN
ejpam-5133	169	24	(	(	PUNCT
ejpam-5133	169	25	c	c	NOUN
ejpam-5133	169	26	2	2	NUM
ejpam-5133	169	27	)	)	PUNCT
ejpam-5133	169	28	=	=	SYM
ejpam-5133	169	29	x√	x√	X
ejpam-5133	169	30	x2	x2	PROPN
ejpam-5133	169	31	+	+	CCONJ
ejpam-5133	169	32	d2	d2	PROPN
ejpam-5133	169	33	,	,	PUNCT
ejpam-5133	169	34	tan	tan	NOUN
ejpam-5133	169	35	ˆ	ˆ	ADJ
ejpam-5133	169	36	(	(	PUNCT
ejpam-5133	169	37	c	c	NOUN
ejpam-5133	169	38	2	2	NUM
ejpam-5133	169	39	)	)	PUNCT
ejpam-5133	169	40	=	=	PUNCT
ejpam-5133	170	1	d	d	NOUN
ejpam-5133	170	2	x	x	X
ejpam-5133	170	3	,	,	PUNCT
ejpam-5133	170	4	(	(	PUNCT
ejpam-5133	170	5	10	10	NUM
ejpam-5133	170	6	)	)	PUNCT
ejpam-5133	170	7	sin	sin	NOUN
ejpam-5133	170	8	ˆ	ˆ	PROPN
ejpam-5133	170	9	(	(	PUNCT
ejpam-5133	170	10	b	b	NOUN
ejpam-5133	170	11	2	2	NUM
ejpam-5133	170	12	)	)	PUNCT
ejpam-5133	171	1	=	=	SYM
ejpam-5133	171	2	x−	x−	PROPN
ejpam-5133	171	3	d√	d√	PROPN
ejpam-5133	171	4	2	2	NUM
ejpam-5133	171	5	·	·	PUNCT
ejpam-5133	171	6	√	√	NUM
ejpam-5133	171	7	x2	x2	PROPN
ejpam-5133	171	8	+	+	CCONJ
ejpam-5133	171	9	d2	d2	PROPN
ejpam-5133	171	10	,	,	PUNCT
ejpam-5133	171	11	cos	cos	PROPN
ejpam-5133	171	12	ˆ	ˆ	PROPN
ejpam-5133	171	13	(	(	PUNCT
ejpam-5133	171	14	b	b	NOUN
ejpam-5133	171	15	2	2	NUM
ejpam-5133	171	16	)	)	PUNCT
ejpam-5133	171	17	=	=	SYM
ejpam-5133	171	18	x+	x+	PROPN
ejpam-5133	171	19	d√	d√	PROPN
ejpam-5133	171	20	2	2	NUM
ejpam-5133	171	21	·	·	PUNCT
ejpam-5133	171	22	√	√	NUM
ejpam-5133	171	23	x2	x2	PROPN
ejpam-5133	171	24	+	+	CCONJ
ejpam-5133	171	25	d2	d2	PROPN
ejpam-5133	171	26	,	,	PUNCT
ejpam-5133	171	27	tan	tan	NOUN
ejpam-5133	171	28	ˆ	ˆ	PROPN
ejpam-5133	171	29	(	(	PUNCT
ejpam-5133	171	30	b	b	NOUN
ejpam-5133	171	31	2	2	NUM
ejpam-5133	171	32	)	)	PUNCT
ejpam-5133	171	33	=	=	PUNCT
ejpam-5133	171	34	x−	x−	PROPN
ejpam-5133	172	1	d	d	NOUN
ejpam-5133	172	2	x+	x+	PROPN
ejpam-5133	172	3	d	d	X
ejpam-5133	172	4	(	(	PUNCT
ejpam-5133	172	5	11	11	NUM
ejpam-5133	172	6	)	)	PUNCT
ejpam-5133	172	7	with	with	ADP
ejpam-5133	172	8	d	d	PROPN
ejpam-5133	172	9	∈	∈	PROPN
ejpam-5133	172	10	c(x	c(x	NOUN
ejpam-5133	172	11	)	)	PUNCT
ejpam-5133	172	12	.	.	PUNCT
ejpam-5133	173	1	let	let	VERB
ejpam-5133	173	2	’s	’s	NOUN
ejpam-5133	173	3	begin	begin	VERB
ejpam-5133	173	4	to	to	PART
ejpam-5133	173	5	notice	notice	VERB
ejpam-5133	173	6	that	that	SCONJ
ejpam-5133	173	7	from	from	ADP
ejpam-5133	173	8	formulas	formula	NOUN
ejpam-5133	173	9	of	of	ADP
ejpam-5133	173	10	theorem	theorem	NOUN
ejpam-5133	173	11	(	(	PUNCT
ejpam-5133	173	12	1	1	X
ejpam-5133	173	13	)	)	PUNCT
ejpam-5133	173	14	we	we	PRON
ejpam-5133	173	15	have	have	VERB
ejpam-5133	173	16	(	(	PUNCT
ejpam-5133	173	17	9	9	NUM
ejpam-5133	173	18	)	)	PUNCT
ejpam-5133	173	19	.	.	PUNCT
ejpam-5133	174	1	applying	apply	VERB
ejpam-5133	174	2	the	the	DET
ejpam-5133	174	3	halfangle	halfangle	NOUN
ejpam-5133	174	4	formulas	formula	NOUN
ejpam-5133	174	5	,	,	PUNCT
ejpam-5133	174	6	we	we	PRON
ejpam-5133	174	7	obtain	obtain	VERB
ejpam-5133	174	8	(	(	PUNCT
ejpam-5133	174	9	10	10	NUM
ejpam-5133	174	10	)	)	PUNCT
ejpam-5133	174	11	and	and	CCONJ
ejpam-5133	174	12	(	(	PUNCT
ejpam-5133	174	13	11	11	NUM
ejpam-5133	174	14	)	)	PUNCT
ejpam-5133	174	15	.	.	PUNCT
ejpam-5133	175	1	moreover	moreover	ADV
ejpam-5133	175	2	,	,	PUNCT
ejpam-5133	175	3	if	if	SCONJ
ejpam-5133	175	4	we	we	PRON
ejpam-5133	175	5	have	have	VERB
ejpam-5133	175	6	a	a	DET
ejpam-5133	175	7	positive	positive	ADJ
ejpam-5133	175	8	x	x	SYM
ejpam-5133	175	9	∈	∈	PROPN
ejpam-5133	175	10	r	r	NOUN
ejpam-5133	175	11	,	,	PUNCT
ejpam-5133	175	12	(	(	PUNCT
ejpam-5133	175	13	9	9	NUM
ejpam-5133	175	14	)	)	PUNCT
ejpam-5133	175	15	,	,	PUNCT
ejpam-5133	175	16	(	(	PUNCT
ejpam-5133	175	17	10	10	NUM
ejpam-5133	175	18	)	)	PUNCT
ejpam-5133	175	19	and	and	CCONJ
ejpam-5133	175	20	(	(	PUNCT
ejpam-5133	175	21	11	11	X
ejpam-5133	175	22	)	)	PUNCT
ejpam-5133	175	23	hold	hold	VERB
ejpam-5133	175	24	,	,	PUNCT
ejpam-5133	175	25	with	with	ADP
ejpam-5133	175	26	d	d	PROPN
ejpam-5133	175	27	=	=	SYM
ejpam-5133	175	28	z	z	NOUN
ejpam-5133	176	1	−	−	PROPN
ejpam-5133	176	2	y.	y.	NOUN
ejpam-5133	176	3	we	we	PRON
ejpam-5133	176	4	consider	consider	VERB
ejpam-5133	176	5	a	a	DET
ejpam-5133	176	6	rectangle	rectangle	NOUN
ejpam-5133	176	7	with	with	ADP
ejpam-5133	176	8	sides	side	NOUN
ejpam-5133	176	9	and	and	CCONJ
ejpam-5133	176	10	diagonal	diagonal	ADJ
ejpam-5133	176	11	given	give	VERB
ejpam-5133	176	12	by	by	ADP
ejpam-5133	176	13	x	x	PROPN
ejpam-5133	176	14	,	,	PUNCT
ejpam-5133	176	15	y	y	PROPN
ejpam-5133	176	16	,	,	PUNCT
ejpam-5133	176	17	and	and	CCONJ
ejpam-5133	176	18	z	z	NOUN
ejpam-5133	176	19	∈	∈	PROPN
ejpam-5133	176	20	n	n	CCONJ
ejpam-5133	176	21	,	,	PUNCT
ejpam-5133	176	22	respectively	respectively	ADV
ejpam-5133	176	23	.	.	PUNCT
ejpam-5133	177	1	let	let	VERB
ejpam-5133	177	2	the	the	DET
ejpam-5133	177	3	parallelepiped	parallelepipe	VERB
ejpam-5133	177	4	have	have	VERB
ejpam-5133	177	5	edges	edge	NOUN
ejpam-5133	177	6	and	and	CCONJ
ejpam-5133	177	7	spatial	spatial	ADJ
ejpam-5133	177	8	diagonal	diagonal	NOUN
ejpam-5133	177	9	given	give	VERB
ejpam-5133	177	10	by	by	ADP
ejpam-5133	177	11	x	x	PROPN
ejpam-5133	177	12	,	,	PUNCT
ejpam-5133	177	13	y	y	PROPN
ejpam-5133	177	14	,	,	PUNCT
ejpam-5133	177	15	w	w	PROPN
ejpam-5133	177	16	,	,	PUNCT
ejpam-5133	177	17	and	and	CCONJ
ejpam-5133	177	18	t	t	PROPN
ejpam-5133	177	19	∈	∈	PROPN
ejpam-5133	177	20	n	n	CCONJ
ejpam-5133	177	21	,	,	PUNCT
ejpam-5133	177	22	respectively	respectively	ADV
ejpam-5133	177	23	.	.	PUNCT
ejpam-5133	178	1	we	we	PRON
ejpam-5133	178	2	want	want	VERB
ejpam-5133	178	3	obtain	obtain	VERB
ejpam-5133	178	4	all	all	DET
ejpam-5133	178	5	parallelepipeds	parallelepiped	NOUN
ejpam-5133	178	6	with	with	ADP
ejpam-5133	178	7	x	x	PROPN
ejpam-5133	178	8	,	,	PUNCT
ejpam-5133	178	9	y	y	PROPN
ejpam-5133	178	10	,	,	PUNCT
ejpam-5133	178	11	z	z	PROPN
ejpam-5133	178	12	,	,	PUNCT
ejpam-5133	178	13	w	w	PROPN
ejpam-5133	178	14	,	,	PUNCT
ejpam-5133	178	15	and	and	CCONJ
ejpam-5133	178	16	t	t	PROPN
ejpam-5133	178	17	∈	∈	PROPN
ejpam-5133	178	18	n	n	CCONJ
ejpam-5133	178	19	,	,	PUNCT
ejpam-5133	178	20	where	where	SCONJ
ejpam-5133	178	21	x	x	PRON
ejpam-5133	178	22	is	be	AUX
ejpam-5133	178	23	a	a	DET
ejpam-5133	178	24	predetermined	predetermined	ADJ
ejpam-5133	178	25	integer	integer	NOUN
ejpam-5133	178	26	.	.	PUNCT
ejpam-5133	179	1	the	the	DET
ejpam-5133	179	2	following	follow	VERB
ejpam-5133	179	3	theorem	theorem	ADJ
ejpam-5133	179	4	holds	hold	NOUN
ejpam-5133	179	5	.	.	PUNCT
ejpam-5133	179	6	theorem	theorem	NOUN
ejpam-5133	179	7	7	7	NUM
ejpam-5133	179	8	.	.	PUNCT
ejpam-5133	180	1	let	let	VERB
ejpam-5133	180	2	a	a	DET
ejpam-5133	180	3	rectangle	rectangle	NOUN
ejpam-5133	180	4	have	have	VERB
ejpam-5133	180	5	sides	side	NOUN
ejpam-5133	180	6	x	x	X
ejpam-5133	180	7	,	,	PUNCT
ejpam-5133	180	8	y	y	PROPN
ejpam-5133	180	9	∈	∈	PROPN
ejpam-5133	180	10	n	n	PRON
ejpam-5133	180	11	and	and	CCONJ
ejpam-5133	180	12	diagonal	diagonal	ADJ
ejpam-5133	180	13	given	give	VERB
ejpam-5133	180	14	by	by	ADP
ejpam-5133	180	15	z	z	PROPN
ejpam-5133	180	16	∈	∈	PROPN
ejpam-5133	180	17	n	n	CCONJ
ejpam-5133	180	18	,	,	PUNCT
ejpam-5133	180	19	and	and	CCONJ
ejpam-5133	180	20	let	let	VERB
ejpam-5133	180	21	a	a	DET
ejpam-5133	180	22	parallelepiped	parallelepipe	VERB
ejpam-5133	180	23	have	have	VERB
ejpam-5133	180	24	edges	edge	NOUN
ejpam-5133	180	25	x	x	NOUN
ejpam-5133	180	26	,	,	PUNCT
ejpam-5133	180	27	y	y	PROPN
ejpam-5133	180	28	,	,	PUNCT
ejpam-5133	180	29	w	w	PROPN
ejpam-5133	180	30	∈	∈	PROPN
ejpam-5133	180	31	n	n	PRON
ejpam-5133	180	32	and	and	CCONJ
ejpam-5133	180	33	spatial	spatial	ADJ
ejpam-5133	180	34	diagonal	diagonal	NOUN
ejpam-5133	180	35	given	give	VERB
ejpam-5133	180	36	by	by	ADP
ejpam-5133	180	37	t	t	PROPN
ejpam-5133	180	38	∈	∈	PROPN
ejpam-5133	180	39	n	n	CCONJ
ejpam-5133	180	40	,	,	PUNCT
ejpam-5133	180	41	respectively	respectively	ADV
ejpam-5133	180	42	,	,	PUNCT
ejpam-5133	180	43	where	where	SCONJ
ejpam-5133	180	44	x	x	PRON
ejpam-5133	180	45	is	be	AUX
ejpam-5133	180	46	a	a	DET
ejpam-5133	180	47	predetermined	predetermined	ADJ
ejpam-5133	180	48	integer	integer	NOUN
ejpam-5133	180	49	.	.	PUNCT
ejpam-5133	181	1	the	the	DET
ejpam-5133	181	2	quadruple	quadruple	NOUN
ejpam-5133	181	3	(	(	PUNCT
ejpam-5133	181	4	x	x	X
ejpam-5133	181	5	,	,	PUNCT
ejpam-5133	181	6	y	y	PROPN
ejpam-5133	181	7	,	,	PUNCT
ejpam-5133	181	8	w	w	PROPN
ejpam-5133	181	9	,	,	PUNCT
ejpam-5133	181	10	t	t	PROPN
ejpam-5133	181	11	)	)	PUNCT
ejpam-5133	181	12	and	and	CCONJ
ejpam-5133	181	13	z	z	AUX
ejpam-5133	181	14	satisfy	satisfy	VERB
ejpam-5133	181	15	the	the	DET
ejpam-5133	181	16	conditions	condition	NOUN
ejpam-5133	182	1	x2	x2	NOUN
ejpam-5133	183	1	+	+	CCONJ
ejpam-5133	183	2	y2	y2	NOUN
ejpam-5133	183	3	=	=	SYM
ejpam-5133	183	4	z2	z2	PROPN
ejpam-5133	183	5	and	and	CCONJ
ejpam-5133	183	6	x2	x2	PROPN
ejpam-5133	184	1	+	+	CCONJ
ejpam-5133	184	2	y2	y2	PROPN
ejpam-5133	184	3	+	+	CCONJ
ejpam-5133	184	4	w2	w2	NOUN
ejpam-5133	184	5	=	=	SYM
ejpam-5133	184	6	t2	t2	PROPN
ejpam-5133	184	7	(	(	PUNCT
ejpam-5133	184	8	12	12	NUM
ejpam-5133	184	9	)	)	PUNCT
ejpam-5133	184	10	r.	r.	PROPN
ejpam-5133	184	11	amato	amato	PROPN
ejpam-5133	184	12	/	/	SYM
ejpam-5133	184	13	eur	eur	PROPN
ejpam-5133	184	14	.	.	PUNCT
ejpam-5133	185	1	j.	j.	PROPN
ejpam-5133	185	2	pure	pure	PROPN
ejpam-5133	185	3	appl	appl	PROPN
ejpam-5133	185	4	.	.	PROPN
ejpam-5133	185	5	math	math	PROPN
ejpam-5133	185	6	,	,	PUNCT
ejpam-5133	185	7	17	17	NUM
ejpam-5133	185	8	(	(	PUNCT
ejpam-5133	185	9	2	2	NUM
ejpam-5133	185	10	)	)	PUNCT
ejpam-5133	185	11	(	(	PUNCT
ejpam-5133	185	12	2024	2024	NUM
ejpam-5133	185	13	)	)	PUNCT
ejpam-5133	185	14	,	,	PUNCT
ejpam-5133	185	15	676	676	NUM
ejpam-5133	185	16	-	-	SYM
ejpam-5133	185	17	689	689	NUM
ejpam-5133	185	18	685	685	NUM
ejpam-5133	185	19	if	if	SCONJ
ejpam-5133	185	20	and	and	CCONJ
ejpam-5133	185	21	only	only	ADV
ejpam-5133	185	22	if	if	SCONJ
ejpam-5133	185	23	x	x	SYM
ejpam-5133	185	24	=	=	SYM
ejpam-5133	185	25	x	x	NOUN
ejpam-5133	185	26	,	,	PUNCT
ejpam-5133	185	27	y	y	PROPN
ejpam-5133	185	28	=	=	SYM
ejpam-5133	185	29	x2	x2	PROPN
ejpam-5133	185	30	−	−	PROPN
ejpam-5133	185	31	d2	d2	PROPN
ejpam-5133	185	32	2d	2d	PROPN
ejpam-5133	185	33	,	,	PUNCT
ejpam-5133	185	34	w	w	NOUN
ejpam-5133	185	35	=	=	SYM
ejpam-5133	185	36	(	(	PUNCT
ejpam-5133	185	37	x2	x2	PROPN
ejpam-5133	185	38	+	+	NUM
ejpam-5133	185	39	d2	d2	PROPN
ejpam-5133	185	40	2d	2d	NOUN
ejpam-5133	185	41	)	)	PUNCT
ejpam-5133	185	42	2	2	NUM
ejpam-5133	185	43	−	−	NOUN
ejpam-5133	185	44	(	(	PUNCT
ejpam-5133	185	45	d∗)2	d∗)2	PROPN
ejpam-5133	185	46	2d∗	2d∗	NUM
ejpam-5133	185	47	,	,	PUNCT
ejpam-5133	185	48	t	t	NOUN
ejpam-5133	185	49	=	=	PUNCT
ejpam-5133	185	50	(	(	PUNCT
ejpam-5133	185	51	x2	x2	PROPN
ejpam-5133	185	52	+	+	NUM
ejpam-5133	185	53	d2	d2	PROPN
ejpam-5133	185	54	2d	2d	NOUN
ejpam-5133	185	55	)	)	PUNCT
ejpam-5133	185	56	2	2	NUM
ejpam-5133	186	1	+	+	CCONJ
ejpam-5133	186	2	(	(	PUNCT
ejpam-5133	186	3	d∗)2	d∗)2	PROPN
ejpam-5133	186	4	2d∗	2d∗	NUM
ejpam-5133	186	5	(	(	PUNCT
ejpam-5133	186	6	13	13	NUM
ejpam-5133	186	7	)	)	PUNCT
ejpam-5133	186	8	with	with	ADP
ejpam-5133	186	9	d	d	PROPN
ejpam-5133	186	10	∈	∈	PROPN
ejpam-5133	186	11	c(x	c(x	NOUN
ejpam-5133	186	12	)	)	PUNCT
ejpam-5133	186	13	and	and	CCONJ
ejpam-5133	186	14	d	d	X
ejpam-5133	186	15	*	*	PUNCT
ejpam-5133	186	16	∈	∈	PROPN
ejpam-5133	186	17	c(z	c(z	NOUN
ejpam-5133	186	18	)	)	PUNCT
ejpam-5133	186	19	.	.	PUNCT
ejpam-5133	187	1	proof	proof	NOUN
ejpam-5133	187	2	.	.	PUNCT
ejpam-5133	188	1	let	let	VERB
ejpam-5133	188	2	’s	’s	NOUN
ejpam-5133	188	3	begin	begin	VERB
ejpam-5133	188	4	to	to	PART
ejpam-5133	188	5	notice	notice	VERB
ejpam-5133	188	6	that	that	SCONJ
ejpam-5133	188	7	for	for	ADP
ejpam-5133	188	8	a	a	DET
ejpam-5133	188	9	predeteminatus	predeteminatus	NOUN
ejpam-5133	188	10	x	x	SYM
ejpam-5133	188	11	∈	∈	PROPN
ejpam-5133	188	12	n	n	CCONJ
ejpam-5133	188	13	,	,	PUNCT
ejpam-5133	188	14	f	f	PROPN
ejpam-5133	188	15	or	or	CCONJ
ejpam-5133	188	16	theorem	theorem	ADJ
ejpam-5133	188	17	(	(	PUNCT
ejpam-5133	188	18	2	2	NUM
ejpam-5133	188	19	)	)	PUNCT
ejpam-5133	188	20	there	there	PRON
ejpam-5133	188	21	exists	exist	VERB
ejpam-5133	188	22	almost	almost	ADV
ejpam-5133	188	23	a	a	PRON
ejpam-5133	188	24	pair	pair	NOUN
ejpam-5133	188	25	y	y	NOUN
ejpam-5133	188	26	,	,	PUNCT
ejpam-5133	188	27	z	z	NOUN
ejpam-5133	188	28	∈	∈	PROPN
ejpam-5133	189	1	n	n	PRON
ejpam-5133	190	1	such	such	ADJ
ejpam-5133	190	2	that	that	PRON
ejpam-5133	190	3	x2+y2	x2+y2	PROPN
ejpam-5133	190	4	=	=	PROPN
ejpam-5133	190	5	z2	z2	PROPN
ejpam-5133	190	6	with	with	ADP
ejpam-5133	190	7	z−y	z−y	PROPN
ejpam-5133	190	8	=	=	PUNCT
ejpam-5133	190	9	d	d	PUNCT
ejpam-5133	190	10	∈	∈	PROPN
ejpam-5133	190	11	c(x	c(x	NOUN
ejpam-5133	190	12	)	)	PUNCT
ejpam-5133	190	13	.	.	PUNCT
ejpam-5133	191	1	also	also	ADV
ejpam-5133	191	2	,	,	PUNCT
ejpam-5133	191	3	for	for	ADP
ejpam-5133	191	4	theorem	theorem	NOUN
ejpam-5133	191	5	(	(	PUNCT
ejpam-5133	191	6	2	2	X
ejpam-5133	191	7	)	)	PUNCT
ejpam-5133	191	8	there	there	PRON
ejpam-5133	191	9	exists	exist	VERB
ejpam-5133	191	10	almost	almost	ADV
ejpam-5133	191	11	a	a	PRON
ejpam-5133	191	12	pair	pair	NOUN
ejpam-5133	191	13	w	w	PROPN
ejpam-5133	191	14	,	,	PUNCT
ejpam-5133	191	15	t	t	PROPN
ejpam-5133	191	16	∈	∈	PROPN
ejpam-5133	191	17	n	n	PRON
ejpam-5133	191	18	such	such	ADJ
ejpam-5133	191	19	that	that	DET
ejpam-5133	191	20	z2+w2	z2+w2	PROPN
ejpam-5133	191	21	=	=	SYM
ejpam-5133	191	22	t2	t2	PROPN
ejpam-5133	191	23	with	with	ADP
ejpam-5133	191	24	t−w	t−w	NOUN
ejpam-5133	191	25	=	=	SYM
ejpam-5133	191	26	d∗	d∗	PROPN
ejpam-5133	191	27	∈	∈	PROPN
ejpam-5133	191	28	c(z	c(z	NOUN
ejpam-5133	191	29	)	)	PUNCT
ejpam-5133	191	30	,	,	PUNCT
ejpam-5133	191	31	and	and	CCONJ
ejpam-5133	191	32	then	then	ADV
ejpam-5133	191	33	there	there	PRON
ejpam-5133	191	34	exist	exist	VERB
ejpam-5133	191	35	x	x	NOUN
ejpam-5133	191	36	,	,	PUNCT
ejpam-5133	191	37	y	y	PROPN
ejpam-5133	191	38	,	,	PUNCT
ejpam-5133	191	39	w	w	PROPN
ejpam-5133	191	40	,	,	PUNCT
ejpam-5133	191	41	and	and	CCONJ
ejpam-5133	191	42	t	t	PROPN
ejpam-5133	191	43	∈	∈	PROPN
ejpam-5133	191	44	n	n	PRON
ejpam-5133	191	45	such	such	ADJ
ejpam-5133	191	46	that	that	SCONJ
ejpam-5133	191	47	x2	x2	PROPN
ejpam-5133	192	1	+	+	CCONJ
ejpam-5133	192	2	y2	y2	PROPN
ejpam-5133	192	3	+	+	CCONJ
ejpam-5133	192	4	w2	w2	NOUN
ejpam-5133	192	5	=	=	SYM
ejpam-5133	192	6	t2	t2	PROPN
ejpam-5133	192	7	with	with	ADP
ejpam-5133	192	8	z	z	PROPN
ejpam-5133	192	9	∈	∈	PROPN
ejpam-5133	192	10	n.	n.	NOUN
ejpam-5133	192	11	to	to	PART
ejpam-5133	192	12	obtain	obtain	VERB
ejpam-5133	192	13	formulas	formula	NOUN
ejpam-5133	192	14	(	(	PUNCT
ejpam-5133	192	15	13	13	NUM
ejpam-5133	192	16	)	)	PUNCT
ejpam-5133	192	17	,	,	PUNCT
ejpam-5133	192	18	using	use	VERB
ejpam-5133	192	19	formulas	formula	NOUN
ejpam-5133	192	20	(	(	PUNCT
ejpam-5133	192	21	1	1	NUM
ejpam-5133	192	22	)	)	PUNCT
ejpam-5133	192	23	to	to	ADP
ejpam-5133	192	24	a	a	DET
ejpam-5133	192	25	predeteminatus	predeteminatus	NOUN
ejpam-5133	192	26	x	x	PUNCT
ejpam-5133	192	27	∈	∈	NOUN
ejpam-5133	192	28	n	n	PRON
ejpam-5133	192	29	we	we	PRON
ejpam-5133	192	30	obtain	obtain	VERB
ejpam-5133	192	31	y	y	PROPN
ejpam-5133	192	32	and	and	CCONJ
ejpam-5133	192	33	z	z	NOUN
ejpam-5133	192	34	such	such	ADJ
ejpam-5133	192	35	that	that	SCONJ
ejpam-5133	192	36	x2	x2	PROPN
ejpam-5133	193	1	+	+	CCONJ
ejpam-5133	193	2	y2	y2	NOUN
ejpam-5133	193	3	=	=	SYM
ejpam-5133	193	4	z2	z2	PROPN
ejpam-5133	193	5	with	with	ADP
ejpam-5133	193	6	d	d	PROPN
ejpam-5133	193	7	∈	∈	PROPN
ejpam-5133	193	8	c(x	c(x	NOUN
ejpam-5133	193	9	)	)	PUNCT
ejpam-5133	193	10	,	,	PUNCT
ejpam-5133	193	11	and	and	CCONJ
ejpam-5133	193	12	re	re	VERB
ejpam-5133	193	13	-	-	VERB
ejpam-5133	193	14	applying	apply	VERB
ejpam-5133	193	15	same	same	ADJ
ejpam-5133	193	16	formulas	formula	NOUN
ejpam-5133	193	17	to	to	ADP
ejpam-5133	193	18	z	z	NOUN
ejpam-5133	193	19	=	=	SYM
ejpam-5133	193	20	x2	x2	PROPN
ejpam-5133	193	21	+	+	NUM
ejpam-5133	193	22	d2	d2	PROPN
ejpam-5133	193	23	2d	2d	NOUN
ejpam-5133	193	24	,	,	PUNCT
ejpam-5133	193	25	considering	consider	VERB
ejpam-5133	193	26	it	it	PRON
ejpam-5133	193	27	as	as	ADP
ejpam-5133	193	28	prededeterminatus	prededeterminatus	VERB
ejpam-5133	193	29	positive	positive	ADJ
ejpam-5133	193	30	integer	integer	NOUN
ejpam-5133	193	31	,	,	PUNCT
ejpam-5133	193	32	with	with	ADP
ejpam-5133	193	33	d∗	d∗	PROPN
ejpam-5133	193	34	∈	∈	PROPN
ejpam-5133	193	35	c(z	c(z	PROPN
ejpam-5133	193	36	)	)	PUNCT
ejpam-5133	193	37	,	,	PUNCT
ejpam-5133	193	38	we	we	PRON
ejpam-5133	193	39	obtain	obtain	VERB
ejpam-5133	193	40	also	also	ADV
ejpam-5133	193	41	w	w	PROPN
ejpam-5133	193	42	and	and	CCONJ
ejpam-5133	193	43	t	t	PROPN
ejpam-5133	193	44	that	that	PRON
ejpam-5133	193	45	togheter	togheter	VERB
ejpam-5133	193	46	with	with	ADP
ejpam-5133	193	47	y	y	PROPN
ejpam-5133	193	48	satisfy	satisfy	NOUN
ejpam-5133	193	49	(	(	PUNCT
ejpam-5133	193	50	12	12	NUM
ejpam-5133	193	51	)	)	PUNCT
ejpam-5133	193	52	.	.	PUNCT
ejpam-5133	194	1	to	to	PART
ejpam-5133	194	2	prove	prove	VERB
ejpam-5133	194	3	that	that	SCONJ
ejpam-5133	194	4	(	(	PUNCT
ejpam-5133	194	5	13	13	NUM
ejpam-5133	194	6	)	)	PUNCT
ejpam-5133	194	7	gives	give	VERB
ejpam-5133	194	8	every	every	DET
ejpam-5133	194	9	x2	x2	PROPN
ejpam-5133	195	1	+	+	CCONJ
ejpam-5133	195	2	y2	y2	PROPN
ejpam-5133	195	3	+	+	ADJ
ejpam-5133	195	4	w2	w2	NOUN
ejpam-5133	195	5	=	=	SYM
ejpam-5133	195	6	t2	t2	PROPN
ejpam-5133	195	7	with	with	ADP
ejpam-5133	195	8	z	z	PROPN
ejpam-5133	195	9	∈	∈	PROPN
ejpam-5133	195	10	n	n	CCONJ
ejpam-5133	195	11	,	,	PUNCT
ejpam-5133	195	12	it	it	PRON
ejpam-5133	195	13	suffices	suffice	VERB
ejpam-5133	195	14	apply	apply	VERB
ejpam-5133	195	15	theorem	theorem	NOUN
ejpam-5133	195	16	(	(	PUNCT
ejpam-5133	195	17	1	1	NUM
ejpam-5133	195	18	)	)	PUNCT
ejpam-5133	195	19	to	to	ADP
ejpam-5133	195	20	the	the	DET
ejpam-5133	195	21	triples	triple	NOUN
ejpam-5133	195	22	x2	x2	NOUN
ejpam-5133	196	1	+	+	CCONJ
ejpam-5133	196	2	y2	y2	NOUN
ejpam-5133	196	3	=	=	SYM
ejpam-5133	196	4	z2	z2	PROPN
ejpam-5133	196	5	and	and	CCONJ
ejpam-5133	196	6	z2	z2	PROPN
ejpam-5133	196	7	+	+	CCONJ
ejpam-5133	196	8	w2	w2	NOUN
ejpam-5133	196	9	=	=	SYM
ejpam-5133	196	10	t2	t2	PROPN
ejpam-5133	196	11	.	.	PUNCT
ejpam-5133	197	1	consequently	consequently	ADV
ejpam-5133	197	2	,	,	PUNCT
ejpam-5133	197	3	theorem	theorem	ADJ
ejpam-5133	197	4	(	(	PUNCT
ejpam-5133	197	5	7	7	NUM
ejpam-5133	197	6	)	)	PUNCT
ejpam-5133	197	7	is	be	AUX
ejpam-5133	197	8	proved	prove	VERB
ejpam-5133	197	9	.	.	PUNCT
ejpam-5133	198	1	we	we	PRON
ejpam-5133	198	2	observe	observe	VERB
ejpam-5133	198	3	that	that	SCONJ
ejpam-5133	198	4	if	if	SCONJ
ejpam-5133	198	5	c(x	c(x	NOUN
ejpam-5133	198	6	)	)	PUNCT
ejpam-5133	198	7	=	=	PRON
ejpam-5133	198	8	{	{	PUNCT
ejpam-5133	198	9	1	1	NUM
ejpam-5133	198	10	,	,	PUNCT
ejpam-5133	198	11	x	x	NOUN
ejpam-5133	198	12	}	}	PUNCT
ejpam-5133	198	13	and	and	CCONJ
ejpam-5133	198	14	c(z	c(z	NUM
ejpam-5133	198	15	)	)	PUNCT
ejpam-5133	198	16	=	=	PRON
ejpam-5133	198	17	{	{	PUNCT
ejpam-5133	198	18	1	1	NUM
ejpam-5133	198	19	,	,	PUNCT
ejpam-5133	198	20	z	z	NOUN
ejpam-5133	198	21	}	}	PUNCT
ejpam-5133	198	22	,	,	PUNCT
ejpam-5133	198	23	that	that	PRON
ejpam-5133	198	24	is	is	ADV
ejpam-5133	198	25	x	x	PUNCT
ejpam-5133	198	26	and	and	CCONJ
ejpam-5133	198	27	z	z	NOUN
ejpam-5133	198	28	are	be	AUX
ejpam-5133	198	29	prime	prime	ADJ
ejpam-5133	198	30	numbers	number	NOUN
ejpam-5133	198	31	,	,	PUNCT
ejpam-5133	198	32	then	then	ADV
ejpam-5133	198	33	we	we	PRON
ejpam-5133	198	34	have	have	VERB
ejpam-5133	198	35	one	one	NUM
ejpam-5133	198	36	unique	unique	ADJ
ejpam-5133	198	37	(	(	PUNCT
ejpam-5133	198	38	x	x	NOUN
ejpam-5133	198	39	,	,	PUNCT
ejpam-5133	198	40	y	y	PROPN
ejpam-5133	198	41	,	,	PUNCT
ejpam-5133	198	42	w	w	PROPN
ejpam-5133	198	43	,	,	PUNCT
ejpam-5133	198	44	t	t	PROPN
ejpam-5133	198	45	)	)	PUNCT
ejpam-5133	198	46	that	that	PRON
ejpam-5133	198	47	satisfies	satisfy	VERB
ejpam-5133	198	48	(	(	PUNCT
ejpam-5133	198	49	12	12	NUM
ejpam-5133	198	50	)	)	PUNCT
ejpam-5133	198	51	,	,	PUNCT
ejpam-5133	198	52	remembering	remember	VERB
ejpam-5133	198	53	that	that	SCONJ
ejpam-5133	198	54	for	for	ADP
ejpam-5133	198	55	d	d	PROPN
ejpam-5133	198	56	=	=	SYM
ejpam-5133	198	57	x	x	PROPN
ejpam-5133	198	58	and	and	CCONJ
ejpam-5133	198	59	d∗	d∗	VERB
ejpam-5133	198	60	=	=	SYM
ejpam-5133	198	61	z	z	NOUN
ejpam-5133	198	62	we	we	PRON
ejpam-5133	198	63	obtain	obtain	VERB
ejpam-5133	198	64	trivial	trivial	ADJ
ejpam-5133	198	65	triples	triple	NOUN
ejpam-5133	198	66	.	.	PUNCT
ejpam-5133	199	1	consequently	consequently	ADV
ejpam-5133	199	2	,	,	PUNCT
ejpam-5133	199	3	if	if	SCONJ
ejpam-5133	199	4	x	x	X
ejpam-5133	199	5	or	or	CCONJ
ejpam-5133	199	6	z	z	NOUN
ejpam-5133	199	7	are	be	AUX
ejpam-5133	199	8	not	not	PART
ejpam-5133	199	9	prime	prime	ADJ
ejpam-5133	199	10	number	number	NOUN
ejpam-5133	199	11	,	,	PUNCT
ejpam-5133	199	12	then	then	ADV
ejpam-5133	199	13	we	we	PRON
ejpam-5133	199	14	obtain	obtain	VERB
ejpam-5133	199	15	more	more	ADJ
ejpam-5133	199	16	(	(	PUNCT
ejpam-5133	199	17	y	y	NOUN
ejpam-5133	199	18	,	,	PUNCT
ejpam-5133	199	19	w	w	PROPN
ejpam-5133	199	20	,	,	PUNCT
ejpam-5133	199	21	t	t	PROPN
ejpam-5133	199	22	)	)	PUNCT
ejpam-5133	199	23	that	that	PRON
ejpam-5133	199	24	togheter	togheter	VERB
ejpam-5133	199	25	x	x	SYM
ejpam-5133	199	26	satisfy	satisfy	VERB
ejpam-5133	199	27	(	(	PUNCT
ejpam-5133	199	28	12	12	NUM
ejpam-5133	199	29	)	)	PUNCT
ejpam-5133	199	30	.	.	PUNCT
ejpam-5133	200	1	more	more	ADJ
ejpam-5133	200	2	in	in	ADP
ejpam-5133	200	3	general	general	ADJ
ejpam-5133	200	4	,	,	PUNCT
ejpam-5133	200	5	there	there	PRON
ejpam-5133	200	6	exists	exist	VERB
ejpam-5133	200	7	the	the	DET
ejpam-5133	200	8	following	follow	VERB
ejpam-5133	200	9	theorem	theorem	PROPN
ejpam-5133	200	10	.	.	PUNCT
ejpam-5133	200	11	theorem	theorem	NOUN
ejpam-5133	200	12	8	8	NUM
ejpam-5133	200	13	.	.	PUNCT
ejpam-5133	201	1	let	let	VERB
ejpam-5133	201	2	a1	a1	NOUN
ejpam-5133	201	3	be	be	AUX
ejpam-5133	201	4	a	a	DET
ejpam-5133	201	5	predetermined	predetermined	ADJ
ejpam-5133	201	6	integer	integer	NOUN
ejpam-5133	201	7	.	.	PUNCT
ejpam-5133	202	1	there	there	PRON
ejpam-5133	202	2	exist	exist	VERB
ejpam-5133	202	3	at	at	ADV
ejpam-5133	202	4	least	least	ADV
ejpam-5133	202	5	one	one	NUM
ejpam-5133	202	6	pythagorean	pythagorean	NOUN
ejpam-5133	202	7	n	n	CCONJ
ejpam-5133	202	8	-	-	PUNCT
ejpam-5133	202	9	uple	uple	NOUN
ejpam-5133	202	10	of	of	ADP
ejpam-5133	202	11	integers	integer	NOUN
ejpam-5133	202	12	(	(	PUNCT
ejpam-5133	202	13	a1	a1	NOUN
ejpam-5133	202	14	,	,	PUNCT
ejpam-5133	202	15	a2	a2	PROPN
ejpam-5133	202	16	,	,	PUNCT
ejpam-5133	202	17	...	...	PUNCT
ejpam-5133	202	18	,	,	PUNCT
ejpam-5133	202	19	an	an	X
ejpam-5133	202	20	)	)	PUNCT
ejpam-5133	202	21	such	such	ADJ
ejpam-5133	202	22	that	that	DET
ejpam-5133	202	23	a21	a21	NOUN
ejpam-5133	202	24	+	+	CCONJ
ejpam-5133	202	25	n−1∑	n−1∑	ADJ
ejpam-5133	202	26	i=2	i=2	NOUN
ejpam-5133	202	27	a2i	a2i	PUNCT
ejpam-5133	202	28	=	=	PUNCT
ejpam-5133	202	29	a2n	a2n	PROPN
ejpam-5133	202	30	(	(	PUNCT
ejpam-5133	202	31	14	14	NUM
ejpam-5133	202	32	)	)	PUNCT
ejpam-5133	202	33	proof	proof	NOUN
ejpam-5133	202	34	.	.	PUNCT
ejpam-5133	203	1	from	from	ADP
ejpam-5133	203	2	theorem	theorem	NOUN
ejpam-5133	203	3	(	(	PUNCT
ejpam-5133	203	4	7	7	NUM
ejpam-5133	203	5	)	)	PUNCT
ejpam-5133	203	6	,	,	PUNCT
ejpam-5133	203	7	we	we	PRON
ejpam-5133	203	8	know	know	VERB
ejpam-5133	203	9	that	that	SCONJ
ejpam-5133	203	10	for	for	ADP
ejpam-5133	203	11	a	a	DET
ejpam-5133	203	12	predetermined	predetermine	VERB
ejpam-5133	203	13	integer	integer	NOUN
ejpam-5133	203	14	a1	a1	NOUN
ejpam-5133	203	15	there	there	ADV
ejpam-5133	203	16	exist	exist	VERB
ejpam-5133	203	17	a2	a2	PROPN
ejpam-5133	203	18	,	,	PUNCT
ejpam-5133	203	19	a3	a3	NOUN
ejpam-5133	203	20	,	,	PUNCT
ejpam-5133	203	21	b	b	X
ejpam-5133	203	22	∈	∈	PROPN
ejpam-5133	203	23	n	n	PRON
ejpam-5133	203	24	such	such	ADJ
ejpam-5133	203	25	that	that	DET
ejpam-5133	203	26	a21	a21	NOUN
ejpam-5133	203	27	+	+	CCONJ
ejpam-5133	203	28	a22	a22	PROPN
ejpam-5133	203	29	+	+	CCONJ
ejpam-5133	203	30	a23	a23	PROPN
ejpam-5133	203	31	=	=	PUNCT
ejpam-5133	203	32	b2	b2	PROPN
ejpam-5133	203	33	.	.	PUNCT
ejpam-5133	204	1	from	from	ADP
ejpam-5133	204	2	theorem	theorem	NOUN
ejpam-5133	204	3	(	(	PUNCT
ejpam-5133	204	4	2	2	NUM
ejpam-5133	204	5	)	)	PUNCT
ejpam-5133	204	6	,	,	PUNCT
ejpam-5133	204	7	there	there	PRON
ejpam-5133	204	8	exists	exist	VERB
ejpam-5133	204	9	almost	almost	ADV
ejpam-5133	204	10	a	a	DET
ejpam-5133	204	11	couple	couple	NOUN
ejpam-5133	204	12	(	(	PUNCT
ejpam-5133	204	13	a4	a4	PROPN
ejpam-5133	204	14	,	,	PUNCT
ejpam-5133	204	15	c	c	NOUN
ejpam-5133	204	16	)	)	PUNCT
ejpam-5133	204	17	,	,	PUNCT
ejpam-5133	204	18	a4	a4	PROPN
ejpam-5133	204	19	,	,	PUNCT
ejpam-5133	204	20	c	c	PROPN
ejpam-5133	204	21	∈	∈	PROPN
ejpam-5133	204	22	n	n	CCONJ
ejpam-5133	204	23	,	,	PUNCT
ejpam-5133	204	24	such	such	ADJ
ejpam-5133	204	25	that	that	DET
ejpam-5133	204	26	b2	b2	NOUN
ejpam-5133	204	27	=	=	PROPN
ejpam-5133	204	28	c2	c2	PROPN
ejpam-5133	204	29	−	−	PROPN
ejpam-5133	204	30	a24	a24	PROPN
ejpam-5133	204	31	and	and	CCONJ
ejpam-5133	204	32	then	then	ADV
ejpam-5133	204	33	a21	a21	PROPN
ejpam-5133	205	1	+	+	CCONJ
ejpam-5133	205	2	a22	a22	PROPN
ejpam-5133	205	3	+	+	CCONJ
ejpam-5133	205	4	a23	a23	PROPN
ejpam-5133	205	5	+	+	CCONJ
ejpam-5133	205	6	a24	a24	PROPN
ejpam-5133	205	7	=	=	PROPN
ejpam-5133	205	8	c2	c2	PROPN
ejpam-5133	205	9	.	.	PUNCT
ejpam-5133	206	1	iterating	iterate	VERB
ejpam-5133	206	2	the	the	DET
ejpam-5133	206	3	procedure	procedure	NOUN
ejpam-5133	206	4	n−	n−	NOUN
ejpam-5133	206	5	5	5	NUM
ejpam-5133	206	6	times	time	NOUN
ejpam-5133	206	7	,	,	PUNCT
ejpam-5133	206	8	then	then	ADV
ejpam-5133	206	9	we	we	PRON
ejpam-5133	206	10	obtain	obtain	VERB
ejpam-5133	206	11	(	(	PUNCT
ejpam-5133	206	12	14	14	NUM
ejpam-5133	206	13	)	)	PUNCT
ejpam-5133	206	14	.	.	PUNCT
ejpam-5133	207	1	consequently	consequently	ADV
ejpam-5133	207	2	,	,	PUNCT
ejpam-5133	207	3	theorem	theorem	NOUN
ejpam-5133	207	4	(	(	PUNCT
ejpam-5133	207	5	8)	8)	NUM
ejpam-5133	207	6	is	be	AUX
ejpam-5133	207	7	proved	prove	VERB
ejpam-5133	207	8	.	.	PUNCT
ejpam-5133	208	1	it	it	PRON
ejpam-5133	208	2	is	be	AUX
ejpam-5133	208	3	interesting	interesting	ADJ
ejpam-5133	208	4	to	to	PART
ejpam-5133	208	5	note	note	VERB
ejpam-5133	208	6	that	that	SCONJ
ejpam-5133	208	7	we	we	PRON
ejpam-5133	208	8	can	can	AUX
ejpam-5133	208	9	iterate	iterate	VERB
ejpam-5133	208	10	the	the	DET
ejpam-5133	208	11	previous	previous	ADJ
ejpam-5133	208	12	procedure	procedure	NOUN
ejpam-5133	208	13	infinitely	infinitely	ADV
ejpam-5133	208	14	.	.	PUNCT
ejpam-5133	209	1	for	for	ADP
ejpam-5133	209	2	this	this	DET
ejpam-5133	209	3	reason	reason	NOUN
ejpam-5133	209	4	we	we	PRON
ejpam-5133	209	5	have	have	VERB
ejpam-5133	209	6	the	the	DET
ejpam-5133	209	7	following	follow	VERB
ejpam-5133	209	8	corollary	corollary	NOUN
ejpam-5133	209	9	.	.	PUNCT
ejpam-5133	210	1	corollary	corollary	ADJ
ejpam-5133	210	2	1	1	NUM
ejpam-5133	210	3	.	.	PUNCT
ejpam-5133	211	1	for	for	ADP
ejpam-5133	211	2	every	every	DET
ejpam-5133	211	3	predetermined	predetermine	VERB
ejpam-5133	211	4	integer	integer	NOUN
ejpam-5133	211	5	a1	a1	NOUN
ejpam-5133	211	6	,	,	PUNCT
ejpam-5133	211	7	there	there	PRON
ejpam-5133	211	8	exists	exist	VERB
ejpam-5133	211	9	at	at	ADP
ejpam-5133	211	10	least	least	ADV
ejpam-5133	211	11	one	one	NUM
ejpam-5133	211	12	b	b	NOUN
ejpam-5133	211	13	∈	∈	NOUN
ejpam-5133	211	14	n	n	NOUN
ejpam-5133	211	15	and	and	CCONJ
ejpam-5133	211	16	one	one	NUM
ejpam-5133	211	17	infinite	infinite	ADJ
ejpam-5133	211	18	set	set	NOUN
ejpam-5133	211	19	of	of	ADP
ejpam-5133	211	20	integers	integer	NOUN
ejpam-5133	211	21	ai	ai	VERB
ejpam-5133	211	22	,	,	PUNCT
ejpam-5133	211	23	i	i	PRON
ejpam-5133	211	24	=	=	NOUN
ejpam-5133	211	25	1	1	NUM
ejpam-5133	211	26	,	,	PUNCT
ejpam-5133	211	27	2	2	NUM
ejpam-5133	211	28	,	,	PUNCT
ejpam-5133	211	29	...	...	PUNCT
ejpam-5133	211	30	,	,	PUNCT
ejpam-5133	211	31	∞	∞	PROPN
ejpam-5133	211	32	,	,	PUNCT
ejpam-5133	211	33	such	such	ADJ
ejpam-5133	211	34	that	that	DET
ejpam-5133	211	35	a21	a21	NOUN
ejpam-5133	211	36	+	+	CCONJ
ejpam-5133	211	37	∞∑	∞∑	NUM
ejpam-5133	211	38	i=2	i=2	NOUN
ejpam-5133	211	39	a2i	a2i	NOUN
ejpam-5133	211	40	=	=	SYM
ejpam-5133	211	41	b2	b2	PROPN
ejpam-5133	211	42	.	.	PUNCT
ejpam-5133	212	1	r.	r.	PROPN
ejpam-5133	212	2	amato	amato	PROPN
ejpam-5133	212	3	/	/	SYM
ejpam-5133	212	4	eur	eur	PROPN
ejpam-5133	212	5	.	.	PUNCT
ejpam-5133	213	1	j.	j.	PROPN
ejpam-5133	213	2	pure	pure	PROPN
ejpam-5133	213	3	appl	appl	PROPN
ejpam-5133	213	4	.	.	PROPN
ejpam-5133	213	5	math	math	PROPN
ejpam-5133	213	6	,	,	PUNCT
ejpam-5133	213	7	17	17	NUM
ejpam-5133	213	8	(	(	PUNCT
ejpam-5133	213	9	2	2	NUM
ejpam-5133	213	10	)	)	PUNCT
ejpam-5133	213	11	(	(	PUNCT
ejpam-5133	213	12	2024	2024	NUM
ejpam-5133	213	13	)	)	PUNCT
ejpam-5133	213	14	,	,	PUNCT
ejpam-5133	213	15	676	676	NUM
ejpam-5133	213	16	-	-	SYM
ejpam-5133	213	17	689	689	NUM
ejpam-5133	213	18	686	686	NUM
ejpam-5133	213	19	let	let	VERB
ejpam-5133	213	20	g	g	NOUN
ejpam-5133	213	21	be	be	AUX
ejpam-5133	213	22	one	one	NUM
ejpam-5133	213	23	set	set	NOUN
ejpam-5133	213	24	of	of	ADP
ejpam-5133	213	25	2x2	2x2	NUM
ejpam-5133	213	26	symmetric	symmetric	ADJ
ejpam-5133	213	27	commuting	commuting	NOUN
ejpam-5133	213	28	matrices	matrix	NOUN
ejpam-5133	213	29	with	with	ADP
ejpam-5133	213	30	non	non	ADJ
ejpam-5133	213	31	-	-	ADJ
ejpam-5133	213	32	zero	zero	NUM
ejpam-5133	213	33	determinant	determinant	ADJ
ejpam-5133	213	34	,	,	PUNCT
ejpam-5133	213	35	defined	define	VERB
ejpam-5133	213	36	in	in	ADP
ejpam-5133	213	37	n	n	CCONJ
ejpam-5133	213	38	,	,	PUNCT
ejpam-5133	213	39	in	in	ADP
ejpam-5133	213	40	the	the	DET
ejpam-5133	213	41	following	follow	VERB
ejpam-5133	213	42	form	form	NOUN
ejpam-5133	213	43	:	:	PUNCT
ejpam-5133	213	44	g	g	NOUN
ejpam-5133	213	45	=	=	SYM
ejpam-5133	213	46	{	{	PUNCT
ejpam-5133	213	47	(	(	PUNCT
ejpam-5133	213	48	c	c	NOUN
ejpam-5133	213	49	b	b	PROPN
ejpam-5133	213	50	b	b	PROPN
ejpam-5133	213	51	c	c	PROPN
ejpam-5133	213	52	)	)	PUNCT
ejpam-5133	213	53	suchthat	suchthat	VERB
ejpam-5133	213	54	c2	c2	PROPN
ejpam-5133	213	55	−	−	PROPN
ejpam-5133	213	56	b2	b2	NOUN
ejpam-5133	213	57	=	=	PROPN
ejpam-5133	213	58	a2	a2	PROPN
ejpam-5133	213	59	,	,	PUNCT
ejpam-5133	213	60	with	with	ADP
ejpam-5133	213	61	a	a	DET
ejpam-5133	213	62	,	,	PUNCT
ejpam-5133	213	63	b	b	NOUN
ejpam-5133	213	64	,	,	PUNCT
ejpam-5133	213	65	c	c	PROPN
ejpam-5133	213	66	∈	∈	PROPN
ejpam-5133	213	67	n	n	CCONJ
ejpam-5133	213	68	,	,	PUNCT
ejpam-5133	213	69	a	a	DET
ejpam-5133	213	70	̸=	̸=	PROPN
ejpam-5133	213	71	0	0	NUM
ejpam-5133	213	72	}	}	PUNCT
ejpam-5133	213	73	.	.	PUNCT
ejpam-5133	214	1	let	let	VERB
ejpam-5133	214	2	(	(	PUNCT
ejpam-5133	214	3	a	a	DET
ejpam-5133	214	4	,	,	PUNCT
ejpam-5133	214	5	b	b	NOUN
ejpam-5133	214	6	,	,	PUNCT
ejpam-5133	214	7	c	c	NOUN
ejpam-5133	214	8	)	)	PUNCT
ejpam-5133	214	9	,	,	PUNCT
ejpam-5133	214	10	(	(	PUNCT
ejpam-5133	214	11	d	d	X
ejpam-5133	214	12	,	,	PUNCT
ejpam-5133	214	13	e	e	NOUN
ejpam-5133	214	14	,	,	PUNCT
ejpam-5133	214	15	f	f	PROPN
ejpam-5133	214	16	)	)	PUNCT
ejpam-5133	214	17	,	,	PUNCT
ejpam-5133	214	18	(	(	PUNCT
ejpam-5133	214	19	a·d	a·d	ADJ
ejpam-5133	214	20	,	,	PUNCT
ejpam-5133	214	21	y	y	PROPN
ejpam-5133	214	22	,	,	PUNCT
ejpam-5133	214	23	z	z	NOUN
ejpam-5133	214	24	)	)	PUNCT
ejpam-5133	214	25	be	be	VERB
ejpam-5133	214	26	the	the	DET
ejpam-5133	214	27	pythagorean	pythagorean	PROPN
ejpam-5133	214	28	triples	triple	NOUN
ejpam-5133	214	29	generated	generate	VERB
ejpam-5133	214	30	by	by	ADP
ejpam-5133	214	31	a	a	DET
ejpam-5133	214	32	,	,	PUNCT
ejpam-5133	214	33	d	d	NOUN
ejpam-5133	214	34	,	,	PUNCT
ejpam-5133	214	35	a·d	a·d	ADJ
ejpam-5133	214	36	,	,	PUNCT
ejpam-5133	214	37	respectively	respectively	ADV
ejpam-5133	214	38	using	use	VERB
ejpam-5133	214	39	(	(	PUNCT
ejpam-5133	214	40	1	1	NUM
ejpam-5133	214	41	)	)	PUNCT
ejpam-5133	214	42	.	.	PUNCT
ejpam-5133	215	1	taking	take	VERB
ejpam-5133	215	2	into	into	ADP
ejpam-5133	215	3	account	account	NOUN
ejpam-5133	215	4	theorem	theorem	NOUN
ejpam-5133	215	5	(	(	PUNCT
ejpam-5133	215	6	4	4	NUM
ejpam-5133	215	7	)	)	PUNCT
ejpam-5133	215	8	,	,	PUNCT
ejpam-5133	215	9	we	we	PRON
ejpam-5133	215	10	have	have	VERB
ejpam-5133	215	11	that	that	DET
ejpam-5133	215	12	y	y	PROPN
ejpam-5133	215	13	=	=	PUNCT
ejpam-5133	215	14	bf	bf	PROPN
ejpam-5133	215	15	+	+	CCONJ
ejpam-5133	215	16	ce	ce	PROPN
ejpam-5133	215	17	and	and	CCONJ
ejpam-5133	215	18	z	z	NOUN
ejpam-5133	215	19	=	=	SYM
ejpam-5133	215	20	be+	be+	PROPN
ejpam-5133	215	21	cf	cf	NOUN
ejpam-5133	215	22	.	.	PUNCT
ejpam-5133	216	1	the	the	DET
ejpam-5133	216	2	following	follow	VERB
ejpam-5133	216	3	theorem	theorem	ADJ
ejpam-5133	216	4	holds	hold	NOUN
ejpam-5133	216	5	.	.	PUNCT
ejpam-5133	216	6	theorem	theorem	NOUN
ejpam-5133	216	7	9	9	NUM
ejpam-5133	216	8	.	.	PUNCT
ejpam-5133	217	1	let	let	VERB
ejpam-5133	217	2	(	(	PUNCT
ejpam-5133	217	3	a	a	DET
ejpam-5133	217	4	,	,	PUNCT
ejpam-5133	217	5	b	b	NOUN
ejpam-5133	217	6	,	,	PUNCT
ejpam-5133	217	7	c	c	NOUN
ejpam-5133	217	8	)	)	PUNCT
ejpam-5133	217	9	,	,	PUNCT
ejpam-5133	217	10	(	(	PUNCT
ejpam-5133	217	11	d	d	X
ejpam-5133	217	12	,	,	PUNCT
ejpam-5133	217	13	e	e	NOUN
ejpam-5133	217	14	,	,	PUNCT
ejpam-5133	217	15	f	f	PROPN
ejpam-5133	217	16	)	)	PUNCT
ejpam-5133	217	17	,	,	PUNCT
ejpam-5133	217	18	and	and	CCONJ
ejpam-5133	217	19	(	(	PUNCT
ejpam-5133	217	20	ad	ad	NOUN
ejpam-5133	217	21	,	,	PUNCT
ejpam-5133	217	22	bf+ce	bf+ce	NOUN
ejpam-5133	217	23	,	,	PUNCT
ejpam-5133	217	24	be+cf	be+cf	PROPN
ejpam-5133	217	25	)	)	PUNCT
ejpam-5133	217	26	be	be	AUX
ejpam-5133	217	27	pythagorean	pythagorean	NOUN
ejpam-5133	217	28	triples	triple	NOUN
ejpam-5133	217	29	.	.	PUNCT
ejpam-5133	218	1	we	we	PRON
ejpam-5133	218	2	consider	consider	VERB
ejpam-5133	218	3	a	a	DET
ejpam-5133	218	4	=	=	X
ejpam-5133	218	5	(	(	PUNCT
ejpam-5133	218	6	c	c	PROPN
ejpam-5133	218	7	b	b	PROPN
ejpam-5133	218	8	b	b	PROPN
ejpam-5133	218	9	c	c	PROPN
ejpam-5133	218	10	)	)	PUNCT
ejpam-5133	218	11	,	,	PUNCT
ejpam-5133	218	12	b	b	X
ejpam-5133	218	13	=	=	PRON
ejpam-5133	218	14	(	(	PUNCT
ejpam-5133	218	15	f	f	X
ejpam-5133	218	16	e	e	X
ejpam-5133	218	17	e	e	X
ejpam-5133	218	18	f	f	PROPN
ejpam-5133	218	19	)	)	PUNCT
ejpam-5133	218	20	,	,	PUNCT
ejpam-5133	218	21	c	c	X
ejpam-5133	218	22	=	=	PUNCT
ejpam-5133	218	23	(	(	PUNCT
ejpam-5133	218	24	be	be	AUX
ejpam-5133	218	25	+	+	NUM
ejpam-5133	218	26	cf	cf	NOUN
ejpam-5133	218	27	bf	bf	NOUN
ejpam-5133	219	1	+	+	CCONJ
ejpam-5133	219	2	ce	ce	PROPN
ejpam-5133	219	3	bf	bf	NOUN
ejpam-5133	219	4	+	+	CCONJ
ejpam-5133	219	5	ce	ce	AUX
ejpam-5133	219	6	be	be	AUX
ejpam-5133	219	7	+	+	X
ejpam-5133	219	8	cf	cf	NOUN
ejpam-5133	219	9	)	)	PUNCT
ejpam-5133	219	10	∈	∈	PROPN
ejpam-5133	219	11	g.	g.	NOUN
ejpam-5133	219	12	in	in	ADP
ejpam-5133	219	13	the	the	DET
ejpam-5133	219	14	set	set	NOUN
ejpam-5133	219	15	of	of	ADP
ejpam-5133	219	16	pythagorean	pythagorean	PROPN
ejpam-5133	219	17	triples	triple	NOUN
ejpam-5133	219	18	,	,	PUNCT
ejpam-5133	219	19	the	the	DET
ejpam-5133	219	20	binary	binary	PROPN
ejpam-5133	219	21	operation	operation	NOUN
ejpam-5133	219	22	(	(	PUNCT
ejpam-5133	219	23	a	a	DET
ejpam-5133	219	24	,	,	PUNCT
ejpam-5133	219	25	b	b	NOUN
ejpam-5133	219	26	,	,	PUNCT
ejpam-5133	219	27	c	c	NOUN
ejpam-5133	219	28	)	)	PUNCT
ejpam-5133	219	29	·	·	PUNCT
ejpam-5133	220	1	(	(	PUNCT
ejpam-5133	220	2	d	d	X
ejpam-5133	220	3	,	,	PUNCT
ejpam-5133	220	4	e	e	NOUN
ejpam-5133	220	5	,	,	PUNCT
ejpam-5133	220	6	f	f	X
ejpam-5133	220	7	)	)	PUNCT
ejpam-5133	220	8	=	=	SYM
ejpam-5133	220	9	(	(	PUNCT
ejpam-5133	220	10	ad	ad	NOUN
ejpam-5133	220	11	,	,	PUNCT
ejpam-5133	220	12	bf	bf	NOUN
ejpam-5133	220	13	+	+	CCONJ
ejpam-5133	220	14	ce	ce	PROPN
ejpam-5133	220	15	,	,	PUNCT
ejpam-5133	220	16	be+	be+	PROPN
ejpam-5133	220	17	cf	cf	NOUN
ejpam-5133	220	18	)	)	PUNCT
ejpam-5133	220	19	is	be	AUX
ejpam-5133	220	20	equivalent	equivalent	ADJ
ejpam-5133	220	21	,	,	PUNCT
ejpam-5133	220	22	in	in	ADP
ejpam-5133	220	23	the	the	DET
ejpam-5133	220	24	set	set	NOUN
ejpam-5133	220	25	g	g	NOUN
ejpam-5133	220	26	,	,	PUNCT
ejpam-5133	220	27	to	to	ADP
ejpam-5133	220	28	the	the	DET
ejpam-5133	220	29	matrix	matrix	NOUN
ejpam-5133	220	30	multiplication	multiplication	NOUN
ejpam-5133	220	31	a·b	a·b	NOUN
ejpam-5133	220	32	=	=	SYM
ejpam-5133	220	33	b·a	b·a	NOUN
ejpam-5133	220	34	=	=	SYM
ejpam-5133	220	35	c	c	NOUN
ejpam-5133	220	36	and	and	CCONJ
ejpam-5133	220	37	det(a)·det(b)=det(c	det(a)·det(b)=det(c	NUM
ejpam-5133	220	38	)	)	PUNCT
ejpam-5133	220	39	,	,	PUNCT
ejpam-5133	220	40	and	and	CCONJ
ejpam-5133	220	41	the	the	DET
ejpam-5133	220	42	identity	identity	NOUN
ejpam-5133	220	43	matrix	matrix	NOUN
ejpam-5133	220	44	corresponds	correspond	VERB
ejpam-5133	220	45	to	to	ADP
ejpam-5133	220	46	the	the	DET
ejpam-5133	220	47	the	the	DET
ejpam-5133	220	48	identity	identity	NOUN
ejpam-5133	220	49	element	element	NOUN
ejpam-5133	220	50	(	(	PUNCT
ejpam-5133	220	51	1	1	NUM
ejpam-5133	220	52	,	,	PUNCT
ejpam-5133	220	53	0	0	NUM
ejpam-5133	220	54	,	,	PUNCT
ejpam-5133	220	55	1	1	NUM
ejpam-5133	220	56	)	)	PUNCT
ejpam-5133	220	57	.	.	PUNCT
ejpam-5133	221	1	proof	proof	NOUN
ejpam-5133	221	2	.	.	PUNCT
ejpam-5133	222	1	it	it	PRON
ejpam-5133	222	2	suffices	suffice	VERB
ejpam-5133	222	3	to	to	PART
ejpam-5133	222	4	apply	apply	VERB
ejpam-5133	222	5	the	the	DET
ejpam-5133	222	6	properties	property	NOUN
ejpam-5133	222	7	of	of	ADP
ejpam-5133	222	8	the	the	DET
ejpam-5133	222	9	product	product	NOUN
ejpam-5133	222	10	between	between	ADP
ejpam-5133	222	11	matrices	matrix	NOUN
ejpam-5133	222	12	and	and	CCONJ
ejpam-5133	222	13	,	,	PUNCT
ejpam-5133	222	14	to	to	PART
ejpam-5133	222	15	obtain	obtain	VERB
ejpam-5133	222	16	the	the	DET
ejpam-5133	222	17	identity	identity	NOUN
ejpam-5133	222	18	matrix	matrix	NOUN
ejpam-5133	222	19	,	,	PUNCT
ejpam-5133	222	20	use	use	VERB
ejpam-5133	222	21	the	the	DET
ejpam-5133	222	22	trivial	trivial	ADJ
ejpam-5133	222	23	pythagorean	pythagorean	NOUN
ejpam-5133	222	24	triple	triple	ADJ
ejpam-5133	222	25	(	(	PUNCT
ejpam-5133	222	26	1	1	NUM
ejpam-5133	222	27	,	,	PUNCT
ejpam-5133	222	28	0	0	NUM
ejpam-5133	222	29	,	,	PUNCT
ejpam-5133	222	30	1	1	NUM
ejpam-5133	222	31	)	)	PUNCT
ejpam-5133	222	32	.	.	PUNCT
ejpam-5133	223	1	consequently	consequently	ADV
ejpam-5133	223	2	,	,	PUNCT
ejpam-5133	223	3	theorem	theorem	ADJ
ejpam-5133	223	4	(	(	PUNCT
ejpam-5133	223	5	9	9	NUM
ejpam-5133	223	6	)	)	PUNCT
ejpam-5133	223	7	is	be	AUX
ejpam-5133	223	8	proved	prove	VERB
ejpam-5133	223	9	.	.	PUNCT
ejpam-5133	224	1	we	we	PRON
ejpam-5133	224	2	observe	observe	VERB
ejpam-5133	224	3	that	that	SCONJ
ejpam-5133	224	4	if	if	SCONJ
ejpam-5133	224	5	a	a	DET
ejpam-5133	224	6	,	,	PUNCT
ejpam-5133	224	7	b	b	NOUN
ejpam-5133	224	8	,	,	PUNCT
ejpam-5133	224	9	c	c	PROPN
ejpam-5133	224	10	∈	∈	PROPN
ejpam-5133	224	11	q	q	AUX
ejpam-5133	224	12	,	,	PUNCT
ejpam-5133	224	13	being	be	AUX
ejpam-5133	224	14	det(a	det(a	PROPN
ejpam-5133	224	15	)	)	PUNCT
ejpam-5133	224	16	=	=	SYM
ejpam-5133	224	17	c2	c2	PROPN
ejpam-5133	224	18	−	−	PROPN
ejpam-5133	224	19	b2	b2	NOUN
ejpam-5133	224	20	=	=	SYM
ejpam-5133	224	21	a2	a2	PROPN
ejpam-5133	224	22	̸=	̸=	PROPN
ejpam-5133	224	23	0	0	NUM
ejpam-5133	224	24	,	,	PUNCT
ejpam-5133	224	25	then	then	ADV
ejpam-5133	224	26	there	there	PRON
ejpam-5133	224	27	exists	exist	VERB
ejpam-5133	224	28	the	the	DET
ejpam-5133	224	29	inverse	inverse	NOUN
ejpam-5133	224	30	matrix	matrix	NOUN
ejpam-5133	224	31	of	of	ADP
ejpam-5133	224	32	a	a	PRON
ejpam-5133	224	33	in	in	ADP
ejpam-5133	224	34	g.	g.	NOUN
ejpam-5133	224	35	consequently	consequently	ADV
ejpam-5133	224	36	it	it	PRON
ejpam-5133	224	37	is	be	AUX
ejpam-5133	224	38	easy	easy	ADJ
ejpam-5133	224	39	to	to	PART
ejpam-5133	224	40	obtain	obtain	VERB
ejpam-5133	224	41	that	that	SCONJ
ejpam-5133	224	42	the	the	DET
ejpam-5133	224	43	set	set	NOUN
ejpam-5133	224	44	of	of	ADP
ejpam-5133	224	45	pythagorean	pythagorean	PROPN
ejpam-5133	224	46	triples	triple	NOUN
ejpam-5133	224	47	is	be	AUX
ejpam-5133	224	48	a	a	DET
ejpam-5133	224	49	commutative	commutative	ADJ
ejpam-5133	224	50	group	group	NOUN
ejpam-5133	224	51	in	in	ADP
ejpam-5133	224	52	q	q	NOUN
ejpam-5133	224	53	,	,	PUNCT
ejpam-5133	224	54	as	as	SCONJ
ejpam-5133	224	55	already	already	ADV
ejpam-5133	224	56	seen	see	VERB
ejpam-5133	224	57	in	in	ADP
ejpam-5133	224	58	[	[	X
ejpam-5133	224	59	7	7	NUM
ejpam-5133	224	60	]	]	PUNCT
ejpam-5133	224	61	but	but	CCONJ
ejpam-5133	224	62	with	with	ADP
ejpam-5133	224	63	different	different	ADJ
ejpam-5133	224	64	approach	approach	NOUN
ejpam-5133	224	65	.	.	PUNCT
ejpam-5133	225	1	if	if	SCONJ
ejpam-5133	225	2	a	a	DET
ejpam-5133	225	3	,	,	PUNCT
ejpam-5133	225	4	b	b	NOUN
ejpam-5133	225	5	,	,	PUNCT
ejpam-5133	225	6	c	c	PROPN
ejpam-5133	225	7	∈	∈	PROPN
ejpam-5133	225	8	z	z	NOUN
ejpam-5133	226	1	it	it	PRON
ejpam-5133	226	2	is	be	AUX
ejpam-5133	226	3	needed	need	VERB
ejpam-5133	226	4	to	to	PART
ejpam-5133	226	5	deal	deal	VERB
ejpam-5133	226	6	with	with	ADP
ejpam-5133	226	7	and	and	CCONJ
ejpam-5133	226	8	introduce	introduce	VERB
ejpam-5133	226	9	particular	particular	ADJ
ejpam-5133	226	10	endomorphisms	endomorphism	NOUN
ejpam-5133	226	11	to	to	PART
ejpam-5133	226	12	obtain	obtain	VERB
ejpam-5133	226	13	results	result	NOUN
ejpam-5133	226	14	also	also	ADV
ejpam-5133	226	15	in	in	ADP
ejpam-5133	226	16	z.	z.	PROPN
ejpam-5133	226	17	in	in	ADP
ejpam-5133	226	18	particular	particular	ADJ
ejpam-5133	226	19	to	to	PART
ejpam-5133	226	20	study	study	VERB
ejpam-5133	226	21	the	the	DET
ejpam-5133	226	22	set	set	NOUN
ejpam-5133	226	23	of	of	ADP
ejpam-5133	226	24	primitive	primitive	ADJ
ejpam-5133	226	25	pythagorean	pythagorean	ADJ
ejpam-5133	226	26	triples	triple	NOUN
ejpam-5133	226	27	to	to	PART
ejpam-5133	226	28	obtain	obtain	VERB
ejpam-5133	226	29	a	a	DET
ejpam-5133	226	30	commutative	commutative	ADJ
ejpam-5133	226	31	group	group	NOUN
ejpam-5133	226	32	with	with	ADP
ejpam-5133	226	33	elements	element	NOUN
ejpam-5133	226	34	in	in	ADP
ejpam-5133	226	35	q	q	NOUN
ejpam-5133	226	36	or	or	CCONJ
ejpam-5133	226	37	in	in	ADP
ejpam-5133	226	38	z.	z.	PROPN
ejpam-5133	226	39	,	,	PUNCT
ejpam-5133	226	40	therefore	therefore	ADV
ejpam-5133	226	41	,	,	PUNCT
ejpam-5133	226	42	from	from	ADP
ejpam-5133	226	43	the	the	DET
ejpam-5133	226	44	beginning	beginning	NOUN
ejpam-5133	226	45	,	,	PUNCT
ejpam-5133	226	46	to	to	PART
ejpam-5133	226	47	have	have	VERB
ejpam-5133	226	48	results	result	NOUN
ejpam-5133	226	49	we	we	PRON
ejpam-5133	226	50	preferred	prefer	VERB
ejpam-5133	226	51	in	in	ADP
ejpam-5133	226	52	[	[	X
ejpam-5133	226	53	7	7	NUM
ejpam-5133	226	54	]	]	PUNCT
ejpam-5133	226	55	to	to	PART
ejpam-5133	226	56	give	give	VERB
ejpam-5133	226	57	an	an	DET
ejpam-5133	226	58	approach	approach	NOUN
ejpam-5133	226	59	that	that	PRON
ejpam-5133	226	60	was	be	AUX
ejpam-5133	226	61	simpler	simple	ADJ
ejpam-5133	226	62	and	and	CCONJ
ejpam-5133	226	63	suitable	suitable	ADJ
ejpam-5133	226	64	for	for	ADP
ejpam-5133	226	65	a	a	DET
ejpam-5133	226	66	wider	wide	ADJ
ejpam-5133	226	67	audience	audience	NOUN
ejpam-5133	226	68	.	.	PUNCT
ejpam-5133	227	1	in	in	ADP
ejpam-5133	227	2	order	order	NOUN
ejpam-5133	227	3	to	to	PART
ejpam-5133	227	4	investigate	investigate	VERB
ejpam-5133	227	5	a	a	DET
ejpam-5133	227	6	possible	possible	ADJ
ejpam-5133	227	7	addition	addition	NOUN
ejpam-5133	227	8	operation	operation	NOUN
ejpam-5133	227	9	among	among	ADP
ejpam-5133	227	10	pythagorean	pythagorean	PROPN
ejpam-5133	227	11	triples	triple	NOUN
ejpam-5133	227	12	,	,	PUNCT
ejpam-5133	227	13	we	we	PRON
ejpam-5133	227	14	have	have	VERB
ejpam-5133	227	15	the	the	DET
ejpam-5133	227	16	following	follow	VERB
ejpam-5133	227	17	theorem	theorem	VERB
ejpam-5133	227	18	.	.	PUNCT
ejpam-5133	227	19	theorem	theorem	PROPN
ejpam-5133	227	20	10	10	NUM
ejpam-5133	227	21	.	.	PUNCT
ejpam-5133	228	1	let	let	VERB
ejpam-5133	228	2	(	(	PUNCT
ejpam-5133	228	3	x	x	X
ejpam-5133	228	4	,	,	PUNCT
ejpam-5133	228	5	x2	x2	PROPN
ejpam-5133	228	6	−	−	PROPN
ejpam-5133	228	7	1	1	NUM
ejpam-5133	228	8	2	2	NUM
ejpam-5133	228	9	,	,	PUNCT
ejpam-5133	228	10	x2	x2	PROPN
ejpam-5133	229	1	+	+	CCONJ
ejpam-5133	229	2	1	1	NUM
ejpam-5133	229	3	2	2	NUM
ejpam-5133	229	4	)	)	PUNCT
ejpam-5133	229	5	and	and	CCONJ
ejpam-5133	229	6	(	(	PUNCT
ejpam-5133	229	7	y	y	NOUN
ejpam-5133	229	8	,	,	PUNCT
ejpam-5133	229	9	y2	y2	PROPN
ejpam-5133	229	10	−	−	PROPN
ejpam-5133	229	11	1	1	NUM
ejpam-5133	229	12	2	2	NUM
ejpam-5133	229	13	,	,	PUNCT
ejpam-5133	229	14	y2	y2	INTJ
ejpam-5133	230	1	+	+	CCONJ
ejpam-5133	230	2	1	1	NUM
ejpam-5133	230	3	2	2	NUM
ejpam-5133	230	4	)	)	PUNCT
ejpam-5133	230	5	be	be	AUX
ejpam-5133	230	6	two	two	NUM
ejpam-5133	230	7	pythagorean	pythagorean	ADJ
ejpam-5133	230	8	triples	triple	NOUN
ejpam-5133	230	9	genetated	genetate	VERB
ejpam-5133	230	10	by	by	ADP
ejpam-5133	230	11	odd	odd	ADJ
ejpam-5133	230	12	integers	integer	NOUN
ejpam-5133	230	13	x	x	PUNCT
ejpam-5133	230	14	and	and	CCONJ
ejpam-5133	230	15	y	y	PROPN
ejpam-5133	230	16	,	,	PUNCT
ejpam-5133	230	17	rispectively	rispectively	ADV
ejpam-5133	230	18	,	,	PUNCT
ejpam-5133	230	19	using	use	VERB
ejpam-5133	230	20	(	(	PUNCT
ejpam-5133	230	21	1	1	NUM
ejpam-5133	230	22	)	)	PUNCT
ejpam-5133	230	23	with	with	ADP
ejpam-5133	230	24	d	d	PROPN
ejpam-5133	230	25	=	=	SYM
ejpam-5133	230	26	1	1	NUM
ejpam-5133	230	27	∈	∈	PROPN
ejpam-5133	230	28	c(x	c(x	NOUN
ejpam-5133	230	29	)	)	PUNCT
ejpam-5133	230	30	and	and	CCONJ
ejpam-5133	230	31	c(y	c(y	PROPN
ejpam-5133	230	32	)	)	PUNCT
ejpam-5133	230	33	.	.	PUNCT
ejpam-5133	231	1	one	one	NUM
ejpam-5133	231	2	addition	addition	NOUN
ejpam-5133	231	3	operation	operation	NOUN
ejpam-5133	231	4	between	between	ADP
ejpam-5133	231	5	the	the	DET
ejpam-5133	231	6	two	two	NUM
ejpam-5133	231	7	pythagorean	pythagorean	NOUN
ejpam-5133	231	8	triples	triple	NOUN
ejpam-5133	231	9	is	be	AUX
ejpam-5133	231	10	given	give	VERB
ejpam-5133	231	11	by	by	ADP
ejpam-5133	231	12	the	the	DET
ejpam-5133	231	13	pythagorean	pythagorean	PROPN
ejpam-5133	231	14	triple	triple	NOUN
ejpam-5133	231	15	(	(	PUNCT
ejpam-5133	231	16	x+	x+	PROPN
ejpam-5133	231	17	y	y	PROPN
ejpam-5133	231	18	,	,	PUNCT
ejpam-5133	231	19	x2	x2	PROPN
ejpam-5133	231	20	−	−	NOUN
ejpam-5133	231	21	1	1	NUM
ejpam-5133	231	22	2	2	NUM
ejpam-5133	231	23	+	+	CCONJ
ejpam-5133	231	24	y2	y2	NOUN
ejpam-5133	231	25	−	−	NOUN
ejpam-5133	231	26	1	1	NUM
ejpam-5133	231	27	2	2	NUM
ejpam-5133	231	28	−	−	PROPN
ejpam-5133	231	29	(	(	PUNCT
ejpam-5133	231	30	x−	x−	PROPN
ejpam-5133	231	31	y	y	PROPN
ejpam-5133	231	32	2	2	NUM
ejpam-5133	231	33	)	)	PUNCT
ejpam-5133	231	34	2	2	NUM
ejpam-5133	231	35	,	,	PUNCT
ejpam-5133	231	36	x2	x2	PROPN
ejpam-5133	232	1	+	+	CCONJ
ejpam-5133	232	2	1	1	NUM
ejpam-5133	232	3	2	2	NUM
ejpam-5133	232	4	+	+	CCONJ
ejpam-5133	232	5	y2	y2	NOUN
ejpam-5133	233	1	+	+	CCONJ
ejpam-5133	233	2	1	1	NUM
ejpam-5133	233	3	2	2	NUM
ejpam-5133	233	4	−	−	PROPN
ejpam-5133	233	5	(	(	PUNCT
ejpam-5133	233	6	x−	x−	PROPN
ejpam-5133	233	7	y	y	PROPN
ejpam-5133	233	8	2	2	NUM
ejpam-5133	233	9	)	)	PUNCT
ejpam-5133	233	10	2	2	NUM
ejpam-5133	233	11	)	)	PUNCT
ejpam-5133	233	12	.	.	PUNCT
ejpam-5133	234	1	(	(	PUNCT
ejpam-5133	234	2	15	15	X
ejpam-5133	234	3	)	)	PUNCT
ejpam-5133	234	4	r.	r.	PROPN
ejpam-5133	234	5	amato	amato	PROPN
ejpam-5133	234	6	/	/	SYM
ejpam-5133	234	7	eur	eur	PROPN
ejpam-5133	234	8	.	.	PUNCT
ejpam-5133	235	1	j.	j.	PROPN
ejpam-5133	235	2	pure	pure	PROPN
ejpam-5133	235	3	appl	appl	PROPN
ejpam-5133	235	4	.	.	PROPN
ejpam-5133	235	5	math	math	PROPN
ejpam-5133	235	6	,	,	PUNCT
ejpam-5133	235	7	17	17	NUM
ejpam-5133	235	8	(	(	PUNCT
ejpam-5133	235	9	2	2	NUM
ejpam-5133	235	10	)	)	PUNCT
ejpam-5133	235	11	(	(	PUNCT
ejpam-5133	235	12	2024	2024	NUM
ejpam-5133	235	13	)	)	PUNCT
ejpam-5133	235	14	,	,	PUNCT
ejpam-5133	235	15	676	676	NUM
ejpam-5133	235	16	-	-	SYM
ejpam-5133	235	17	689	689	NUM
ejpam-5133	235	18	687	687	NUM
ejpam-5133	235	19	proof	proof	NOUN
ejpam-5133	235	20	.	.	PUNCT
ejpam-5133	236	1	we	we	PRON
ejpam-5133	236	2	note	note	VERB
ejpam-5133	236	3	that	that	SCONJ
ejpam-5133	236	4	x	x	X
ejpam-5133	236	5	+	+	CCONJ
ejpam-5133	236	6	y	y	NOUN
ejpam-5133	236	7	is	be	AUX
ejpam-5133	236	8	even	even	ADV
ejpam-5133	236	9	and	and	CCONJ
ejpam-5133	236	10	considering	consider	VERB
ejpam-5133	236	11	the	the	DET
ejpam-5133	236	12	pythagorean	pythagorean	PROPN
ejpam-5133	236	13	triple	triple	NOUN
ejpam-5133	236	14	generated	generate	VERB
ejpam-5133	236	15	by	by	ADP
ejpam-5133	236	16	x+	x+	PROPN
ejpam-5133	236	17	y	y	PROPN
ejpam-5133	236	18	,	,	PUNCT
ejpam-5133	236	19	using	use	VERB
ejpam-5133	236	20	(	(	PUNCT
ejpam-5133	236	21	1	1	NUM
ejpam-5133	236	22	)	)	PUNCT
ejpam-5133	236	23	with	with	ADP
ejpam-5133	236	24	d	d	PROPN
ejpam-5133	236	25	=	=	SYM
ejpam-5133	236	26	2	2	NUM
ejpam-5133	236	27	∈	∈	NOUN
ejpam-5133	236	28	c(x+	c(x+	NOUN
ejpam-5133	236	29	y	y	PROPN
ejpam-5133	236	30	)	)	PUNCT
ejpam-5133	236	31	,	,	PUNCT
ejpam-5133	236	32	we	we	PRON
ejpam-5133	236	33	obtain	obtain	VERB
ejpam-5133	236	34	(	(	PUNCT
ejpam-5133	236	35	x+	x+	PROPN
ejpam-5133	236	36	y	y	NOUN
ejpam-5133	236	37	,	,	PUNCT
ejpam-5133	236	38	(	(	PUNCT
ejpam-5133	236	39	x+	x+	X
ejpam-5133	236	40	y)2	y)2	NOUN
ejpam-5133	236	41	−	−	NOUN
ejpam-5133	236	42	4	4	NUM
ejpam-5133	236	43	4	4	NUM
ejpam-5133	236	44	,	,	PUNCT
ejpam-5133	236	45	(	(	PUNCT
ejpam-5133	236	46	x+	x+	X
ejpam-5133	236	47	y)2	y)2	NOUN
ejpam-5133	236	48	+	+	CCONJ
ejpam-5133	236	49	4	4	NUM
ejpam-5133	236	50	4	4	NUM
ejpam-5133	236	51	)	)	PUNCT
ejpam-5133	236	52	.	.	PUNCT
ejpam-5133	237	1	to	to	PART
ejpam-5133	237	2	verify	verify	VERB
ejpam-5133	237	3	that	that	SCONJ
ejpam-5133	237	4	(	(	PUNCT
ejpam-5133	237	5	15	15	NUM
ejpam-5133	237	6	)	)	PUNCT
ejpam-5133	237	7	is	be	AUX
ejpam-5133	237	8	a	a	DET
ejpam-5133	237	9	pythagorean	pythagorean	PROPN
ejpam-5133	237	10	triple	triple	NOUN
ejpam-5133	237	11	,	,	PUNCT
ejpam-5133	237	12	for	for	ADP
ejpam-5133	237	13	theorem	theorem	NOUN
ejpam-5133	237	14	(	(	PUNCT
ejpam-5133	237	15	1	1	NUM
ejpam-5133	237	16	)	)	PUNCT
ejpam-5133	237	17	,	,	PUNCT
ejpam-5133	237	18	then	then	ADV
ejpam-5133	237	19	it	it	PRON
ejpam-5133	237	20	suffices	suffice	VERB
ejpam-5133	237	21	to	to	PART
ejpam-5133	237	22	prove	prove	VERB
ejpam-5133	237	23	(	(	PUNCT
ejpam-5133	237	24	x+	x+	X
ejpam-5133	237	25	y)2	y)2	NOUN
ejpam-5133	237	26	−	−	NOUN
ejpam-5133	237	27	4	4	NUM
ejpam-5133	237	28	4	4	NUM
ejpam-5133	237	29	=	=	SYM
ejpam-5133	237	30	x2	x2	NOUN
ejpam-5133	237	31	−	−	NOUN
ejpam-5133	237	32	1	1	NUM
ejpam-5133	237	33	2	2	NUM
ejpam-5133	237	34	+	+	CCONJ
ejpam-5133	238	1	y2	y2	NOUN
ejpam-5133	238	2	−	−	NOUN
ejpam-5133	238	3	1	1	NUM
ejpam-5133	238	4	2	2	NUM
ejpam-5133	238	5	−	−	PROPN
ejpam-5133	238	6	(	(	PUNCT
ejpam-5133	238	7	x−	x−	PROPN
ejpam-5133	238	8	y	y	PROPN
ejpam-5133	238	9	2	2	NUM
ejpam-5133	238	10	)	)	PUNCT
ejpam-5133	238	11	2	2	NUM
ejpam-5133	238	12	and	and	CCONJ
ejpam-5133	238	13	(	(	PUNCT
ejpam-5133	238	14	x+	x+	X
ejpam-5133	238	15	y)2	y)2	NOUN
ejpam-5133	238	16	+	+	CCONJ
ejpam-5133	238	17	4	4	NUM
ejpam-5133	238	18	4	4	NUM
ejpam-5133	238	19	=	=	SYM
ejpam-5133	238	20	x2	x2	NOUN
ejpam-5133	239	1	+	+	CCONJ
ejpam-5133	239	2	1	1	NUM
ejpam-5133	239	3	2	2	NUM
ejpam-5133	239	4	+	+	CCONJ
ejpam-5133	239	5	y2	y2	NOUN
ejpam-5133	240	1	+	+	CCONJ
ejpam-5133	240	2	1	1	NUM
ejpam-5133	240	3	2	2	NUM
ejpam-5133	240	4	−	−	PROPN
ejpam-5133	240	5	(	(	PUNCT
ejpam-5133	240	6	x−	x−	PROPN
ejpam-5133	240	7	y	y	PROPN
ejpam-5133	240	8	2	2	NUM
ejpam-5133	240	9	)	)	PUNCT
ejpam-5133	240	10	2	2	NUM
ejpam-5133	240	11	.	.	PUNCT
ejpam-5133	241	1	it	it	PRON
ejpam-5133	241	2	is	be	AUX
ejpam-5133	241	3	easy	easy	ADJ
ejpam-5133	241	4	to	to	PART
ejpam-5133	241	5	see	see	VERB
ejpam-5133	241	6	that	that	SCONJ
ejpam-5133	241	7	above	above	ADP
ejpam-5133	241	8	equations	equation	NOUN
ejpam-5133	241	9	are	be	AUX
ejpam-5133	241	10	identities	identity	NOUN
ejpam-5133	241	11	and	and	CCONJ
ejpam-5133	241	12	consequently	consequently	ADV
ejpam-5133	241	13	,	,	PUNCT
ejpam-5133	241	14	theorem	theorem	ADJ
ejpam-5133	241	15	(	(	PUNCT
ejpam-5133	241	16	10	10	NUM
ejpam-5133	241	17	)	)	PUNCT
ejpam-5133	241	18	is	be	AUX
ejpam-5133	241	19	proved	prove	VERB
ejpam-5133	241	20	.	.	PUNCT
ejpam-5133	242	1	we	we	PRON
ejpam-5133	242	2	observe	observe	VERB
ejpam-5133	242	3	that	that	SCONJ
ejpam-5133	242	4	(	(	PUNCT
ejpam-5133	242	5	15	15	NUM
ejpam-5133	242	6	)	)	PUNCT
ejpam-5133	242	7	is	be	AUX
ejpam-5133	242	8	applicable	applicable	ADJ
ejpam-5133	242	9	only	only	ADV
ejpam-5133	242	10	to	to	ADP
ejpam-5133	242	11	every	every	DET
ejpam-5133	242	12	pair	pair	NOUN
ejpam-5133	242	13	of	of	ADP
ejpam-5133	242	14	pythagorean	pythagorean	PROPN
ejpam-5133	242	15	triples	triple	NOUN
ejpam-5133	242	16	generated	generate	VERB
ejpam-5133	242	17	by	by	ADP
ejpam-5133	242	18	odd	odd	ADJ
ejpam-5133	242	19	integers	integer	NOUN
ejpam-5133	242	20	x	x	PUNCT
ejpam-5133	242	21	and	and	CCONJ
ejpam-5133	242	22	y	y	PROPN
ejpam-5133	242	23	,	,	PUNCT
ejpam-5133	242	24	respectively	respectively	ADV
ejpam-5133	242	25	,	,	PUNCT
ejpam-5133	242	26	using	use	VERB
ejpam-5133	242	27	(	(	PUNCT
ejpam-5133	242	28	1	1	NUM
ejpam-5133	242	29	)	)	PUNCT
ejpam-5133	242	30	with	with	ADP
ejpam-5133	242	31	d	d	PROPN
ejpam-5133	242	32	=	=	SYM
ejpam-5133	242	33	1	1	NUM
ejpam-5133	242	34	,	,	PUNCT
ejpam-5133	242	35	obtaining	obtain	VERB
ejpam-5133	242	36	one	one	NUM
ejpam-5133	242	37	pythagorean	pythagorean	PROPN
ejpam-5133	242	38	triple	triple	ADP
ejpam-5133	242	39	generated	generate	VERB
ejpam-5133	242	40	by	by	ADP
ejpam-5133	242	41	the	the	DET
ejpam-5133	242	42	even	even	ADJ
ejpam-5133	242	43	integer	integer	NOUN
ejpam-5133	242	44	x	x	PUNCT
ejpam-5133	243	1	+	+	CCONJ
ejpam-5133	243	2	y	y	PROPN
ejpam-5133	243	3	with	with	ADP
ejpam-5133	243	4	d	d	PROPN
ejpam-5133	243	5	=	=	SYM
ejpam-5133	243	6	2	2	NUM
ejpam-5133	243	7	.	.	PUNCT
ejpam-5133	243	8	currently	currently	ADV
ejpam-5133	243	9	,	,	PUNCT
ejpam-5133	243	10	the	the	DET
ejpam-5133	243	11	addition	addition	NOUN
ejpam-5133	243	12	operation	operation	NOUN
ejpam-5133	243	13	is	be	AUX
ejpam-5133	243	14	not	not	PART
ejpam-5133	243	15	iterable	iterable	ADJ
ejpam-5133	243	16	,	,	PUNCT
ejpam-5133	243	17	meaning	mean	VERB
ejpam-5133	243	18	that	that	SCONJ
ejpam-5133	243	19	the	the	DET
ejpam-5133	243	20	obtained	obtain	VERB
ejpam-5133	243	21	triple	triple	NOUN
ejpam-5133	243	22	can	can	AUX
ejpam-5133	243	23	not	not	PART
ejpam-5133	243	24	be	be	AUX
ejpam-5133	243	25	summed	sum	VERB
ejpam-5133	243	26	with	with	ADP
ejpam-5133	243	27	another	another	DET
ejpam-5133	243	28	pythagorean	pythagorean	NOUN
ejpam-5133	243	29	triple	triple	NOUN
ejpam-5133	243	30	to	to	PART
ejpam-5133	243	31	result	result	VERB
ejpam-5133	243	32	in	in	ADP
ejpam-5133	243	33	another	another	DET
ejpam-5133	243	34	pythagorean	pythagorean	PROPN
ejpam-5133	243	35	triple	triple	NOUN
ejpam-5133	243	36	.	.	PUNCT
ejpam-5133	244	1	additionally	additionally	ADV
ejpam-5133	244	2	,	,	PUNCT
ejpam-5133	244	3	there	there	PRON
ejpam-5133	244	4	is	be	VERB
ejpam-5133	244	5	a	a	DET
ejpam-5133	244	6	lack	lack	NOUN
ejpam-5133	244	7	of	of	ADP
ejpam-5133	244	8	identity	identity	NOUN
ejpam-5133	244	9	and	and	CCONJ
ejpam-5133	244	10	opposite	opposite	ADJ
ejpam-5133	244	11	elements	element	NOUN
ejpam-5133	244	12	.	.	PUNCT
ejpam-5133	245	1	anyway	anyway	ADV
ejpam-5133	245	2	,	,	PUNCT
ejpam-5133	245	3	it	it	PRON
ejpam-5133	245	4	’s	’	VERB
ejpam-5133	245	5	a	a	DET
ejpam-5133	245	6	start	start	NOUN
ejpam-5133	245	7	to	to	PART
ejpam-5133	245	8	explore	explore	VERB
ejpam-5133	245	9	addition	addition	NOUN
ejpam-5133	245	10	operation	operation	NOUN
ejpam-5133	245	11	in	in	ADP
ejpam-5133	245	12	the	the	DET
ejpam-5133	245	13	set	set	NOUN
ejpam-5133	245	14	of	of	ADP
ejpam-5133	245	15	pythagorean	pythagorean	PROPN
ejpam-5133	245	16	triples	triple	NOUN
ejpam-5133	245	17	.	.	PUNCT
ejpam-5133	246	1	as	as	ADP
ejpam-5133	246	2	a	a	DET
ejpam-5133	246	3	consequence	consequence	NOUN
ejpam-5133	246	4	of	of	ADP
ejpam-5133	246	5	theorem	theorem	NOUN
ejpam-5133	246	6	(	(	PUNCT
ejpam-5133	246	7	1	1	NUM
ejpam-5133	246	8	)	)	PUNCT
ejpam-5133	246	9	and	and	CCONJ
ejpam-5133	246	10	(	(	PUNCT
ejpam-5133	246	11	3	3	X
ejpam-5133	246	12	)	)	PUNCT
ejpam-5133	246	13	let	let	VERB
ejpam-5133	246	14	us	we	PRON
ejpam-5133	246	15	state	state	VERB
ejpam-5133	246	16	the	the	DET
ejpam-5133	246	17	following	follow	VERB
ejpam-5133	246	18	theorem	theorem	PROPN
ejpam-5133	246	19	.	.	PUNCT
ejpam-5133	246	20	theorem	theorem	NOUN
ejpam-5133	246	21	11	11	NUM
ejpam-5133	246	22	.	.	PUNCT
ejpam-5133	247	1	every	every	DET
ejpam-5133	247	2	prime	prime	ADJ
ejpam-5133	247	3	number	number	NOUN
ejpam-5133	247	4	is	be	AUX
ejpam-5133	247	5	present	present	ADJ
ejpam-5133	247	6	as	as	ADP
ejpam-5133	247	7	cathetus	cathetus	NOUN
ejpam-5133	247	8	in	in	ADP
ejpam-5133	247	9	only	only	ADV
ejpam-5133	247	10	one	one	NUM
ejpam-5133	247	11	pythagorean	pythagorean	PROPN
ejpam-5133	247	12	triple	triple	NOUN
ejpam-5133	247	13	,	,	PUNCT
ejpam-5133	247	14	and	and	CCONJ
ejpam-5133	247	15	this	this	PRON
ejpam-5133	247	16	is	be	AUX
ejpam-5133	247	17	a	a	DET
ejpam-5133	247	18	primitive	primitive	ADJ
ejpam-5133	247	19	pythagorean	pythagorean	NOUN
ejpam-5133	247	20	triple	triple	NOUN
ejpam-5133	247	21	.	.	PUNCT
ejpam-5133	248	1	proof	proof	NOUN
ejpam-5133	248	2	.	.	PUNCT
ejpam-5133	249	1	from	from	ADP
ejpam-5133	249	2	theorem	theorem	NOUN
ejpam-5133	249	3	(	(	PUNCT
ejpam-5133	249	4	1	1	NUM
ejpam-5133	249	5	)	)	PUNCT
ejpam-5133	249	6	,	,	PUNCT
ejpam-5133	249	7	we	we	PRON
ejpam-5133	249	8	have	have	VERB
ejpam-5133	249	9	that	that	DET
ejpam-5133	249	10	c(x	c(x	NOUN
ejpam-5133	249	11	)	)	PUNCT
ejpam-5133	249	12	=	=	PUNCT
ejpam-5133	250	1	{	{	PUNCT
ejpam-5133	250	2	1	1	NUM
ejpam-5133	250	3	,	,	PUNCT
ejpam-5133	250	4	x	x	NOUN
ejpam-5133	250	5	}	}	PUNCT
ejpam-5133	250	6	,	,	PUNCT
ejpam-5133	250	7	for	for	SCONJ
ejpam-5133	250	8	d	d	PROPN
ejpam-5133	250	9	=	=	PUNCT
ejpam-5133	250	10	x	x	SYM
ejpam-5133	250	11	the	the	DET
ejpam-5133	250	12	triple	triple	NOUN
ejpam-5133	250	13	is	be	AUX
ejpam-5133	250	14	trivial	trivial	ADJ
ejpam-5133	250	15	,	,	PUNCT
ejpam-5133	250	16	and	and	CCONJ
ejpam-5133	250	17	there	there	PRON
ejpam-5133	250	18	exists	exist	VERB
ejpam-5133	250	19	a	a	DET
ejpam-5133	250	20	unique	unique	ADJ
ejpam-5133	250	21	pythagorean	pythagorean	NOUN
ejpam-5133	250	22	triple	triple	ADV
ejpam-5133	250	23	generated	generate	VERB
ejpam-5133	250	24	by	by	ADP
ejpam-5133	250	25	x	x	PUNCT
ejpam-5133	250	26	for	for	ADP
ejpam-5133	250	27	d	d	PROPN
ejpam-5133	250	28	=	=	SYM
ejpam-5133	250	29	1	1	NUM
ejpam-5133	250	30	.	.	PUNCT
ejpam-5133	250	31	from	from	ADP
ejpam-5133	250	32	theorem	theorem	NOUN
ejpam-5133	250	33	(	(	PUNCT
ejpam-5133	250	34	3	3	NUM
ejpam-5133	250	35	)	)	PUNCT
ejpam-5133	250	36	,	,	PUNCT
ejpam-5133	250	37	we	we	PRON
ejpam-5133	250	38	have	have	VERB
ejpam-5133	250	39	that	that	SCONJ
ejpam-5133	250	40	this	this	DET
ejpam-5133	250	41	triple	triple	NOUN
ejpam-5133	250	42	is	be	AUX
ejpam-5133	250	43	primitive	primitive	ADJ
ejpam-5133	250	44	,	,	PUNCT
ejpam-5133	250	45	and	and	CCONJ
ejpam-5133	250	46	consequently	consequently	ADV
ejpam-5133	250	47	,	,	PUNCT
ejpam-5133	250	48	theorem	theorem	ADJ
ejpam-5133	250	49	(	(	PUNCT
ejpam-5133	250	50	11	11	NUM
ejpam-5133	250	51	)	)	PUNCT
ejpam-5133	250	52	is	be	AUX
ejpam-5133	250	53	proved	prove	VERB
ejpam-5133	250	54	.	.	PUNCT
ejpam-5133	251	1	at	at	ADP
ejpam-5133	251	2	last	last	ADV
ejpam-5133	251	3	,	,	PUNCT
ejpam-5133	251	4	let	let	VERB
ejpam-5133	251	5	f	f	PRON
ejpam-5133	251	6	:	:	PUNCT
ejpam-5133	251	7	]	]	X
ejpam-5133	251	8	0,∞	0,∞	NOUN
ejpam-5133	251	9	[	[	PUNCT
ejpam-5133	251	10	→	→	SYM
ejpam-5133	251	11	r	r	NOUN
ejpam-5133	251	12	be	be	AUX
ejpam-5133	251	13	such	such	ADJ
ejpam-5133	251	14	that	that	SCONJ
ejpam-5133	251	15	f(x	f(x	NOUN
ejpam-5133	251	16	)	)	PUNCT
ejpam-5133	251	17	=	=	PUNCT
ejpam-5133	252	1	2x	2x	NUM
ejpam-5133	252	2	,	,	PUNCT
ejpam-5133	252	3	∀	∀	X
ejpam-5133	252	4	y	y	NOUN
ejpam-5133	252	5	,	,	PUNCT
ejpam-5133	252	6	z	z	NOUN
ejpam-5133	252	7	∈	∈	PROPN
ejpam-5133	252	8	]	]	X
ejpam-5133	252	9	0,∞	0,∞	NOUN
ejpam-5133	253	1	[	[	X
ejpam-5133	253	2	,	,	PUNCT
ejpam-5133	253	3	as	as	SCONJ
ejpam-5133	253	4	depicted	depict	VERB
ejpam-5133	253	5	in	in	ADP
ejpam-5133	253	6	figure	figure	NOUN
ejpam-5133	253	7	(	(	PUNCT
ejpam-5133	253	8	3	3	NUM
ejpam-5133	253	9	)	)	PUNCT
ejpam-5133	253	10	.	.	PUNCT
ejpam-5133	254	1	we	we	PRON
ejpam-5133	254	2	have	have	VERB
ejpam-5133	254	3	the	the	DET
ejpam-5133	254	4	following	follow	VERB
ejpam-5133	254	5	remark	remark	NOUN
ejpam-5133	254	6	.	.	PUNCT
ejpam-5133	255	1	remark	remark	PROPN
ejpam-5133	255	2	3	3	NUM
ejpam-5133	255	3	.	.	PUNCT
ejpam-5133	256	1	let	let	VERB
ejpam-5133	256	2	f	f	PROPN
ejpam-5133	256	3	:	:	PUNCT
ejpam-5133	256	4	]	]	X
ejpam-5133	256	5	0,∞	0,∞	NOUN
ejpam-5133	256	6	[	[	PUNCT
ejpam-5133	256	7	→	→	SYM
ejpam-5133	256	8	r	r	NOUN
ejpam-5133	256	9	defined	define	VERB
ejpam-5133	256	10	as	as	ADP
ejpam-5133	256	11	f(x	f(x	PROPN
ejpam-5133	256	12	)	)	PUNCT
ejpam-5133	256	13	=	=	PUNCT
ejpam-5133	257	1	2x	2x	NUM
ejpam-5133	257	2	,	,	PUNCT
ejpam-5133	257	3	∀	∀	X
ejpam-5133	257	4	y	y	NOUN
ejpam-5133	257	5	,	,	PUNCT
ejpam-5133	257	6	z	z	NOUN
ejpam-5133	257	7	∈	∈	PROPN
ejpam-5133	257	8	]	]	X
ejpam-5133	257	9	0,∞	0,∞	NOUN
ejpam-5133	258	1	[	[	X
ejpam-5133	258	2	,	,	PUNCT
ejpam-5133	258	3	we	we	PRON
ejpam-5133	258	4	have∫	have∫	VERB
ejpam-5133	258	5	z	z	VERB
ejpam-5133	258	6	y	y	PROPN
ejpam-5133	258	7	2x	2x	NUM
ejpam-5133	258	8	dx	dx	PROPN
ejpam-5133	258	9	=	=	PROPN
ejpam-5133	258	10	d2	d2	PROPN
ejpam-5133	258	11	+	+	CCONJ
ejpam-5133	258	12	2	2	NUM
ejpam-5133	258	13	yd	yd	NOUN
ejpam-5133	258	14	(	(	PUNCT
ejpam-5133	258	15	16	16	NUM
ejpam-5133	258	16	)	)	PUNCT
ejpam-5133	258	17	with	with	ADP
ejpam-5133	258	18	d	d	PROPN
ejpam-5133	258	19	=	=	SYM
ejpam-5133	258	20	z	z	NOUN
ejpam-5133	258	21	−	−	PROPN
ejpam-5133	258	22	y.	y.	PROPN
ejpam-5133	258	23	r.	r.	PROPN
ejpam-5133	258	24	amato	amato	PROPN
ejpam-5133	258	25	/	/	SYM
ejpam-5133	258	26	eur	eur	PROPN
ejpam-5133	258	27	.	.	PUNCT
ejpam-5133	259	1	j.	j.	PROPN
ejpam-5133	259	2	pure	pure	PROPN
ejpam-5133	259	3	appl	appl	PROPN
ejpam-5133	259	4	.	.	PROPN
ejpam-5133	259	5	math	math	PROPN
ejpam-5133	259	6	,	,	PUNCT
ejpam-5133	259	7	17	17	NUM
ejpam-5133	259	8	(	(	PUNCT
ejpam-5133	259	9	2	2	NUM
ejpam-5133	259	10	)	)	PUNCT
ejpam-5133	259	11	(	(	PUNCT
ejpam-5133	259	12	2024	2024	NUM
ejpam-5133	259	13	)	)	PUNCT
ejpam-5133	259	14	,	,	PUNCT
ejpam-5133	259	15	676	676	NUM
ejpam-5133	259	16	-	-	SYM
ejpam-5133	259	17	689	689	NUM
ejpam-5133	259	18	688	688	NUM
ejpam-5133	259	19	figure	figure	NOUN
ejpam-5133	259	20	3	3	NUM
ejpam-5133	259	21	:	:	PUNCT
ejpam-5133	259	22	the	the	DET
ejpam-5133	259	23	above	above	ADJ
ejpam-5133	259	24	integral	integral	ADJ
ejpam-5133	259	25	is	be	AUX
ejpam-5133	259	26	the	the	DET
ejpam-5133	259	27	area	area	NOUN
ejpam-5133	259	28	of	of	ADP
ejpam-5133	259	29	right	right	ADV
ejpam-5133	259	30	-	-	PUNCT
ejpam-5133	259	31	angled	angle	VERB
ejpam-5133	259	32	trapezoid	trapezoid	ADJ
ejpam-5133	259	33	abcd	abcd	PROPN
ejpam-5133	259	34	given	give	VERB
ejpam-5133	259	35	by	by	ADP
ejpam-5133	259	36	the	the	DET
ejpam-5133	259	37	sum	sum	NOUN
ejpam-5133	259	38	of	of	ADP
ejpam-5133	259	39	the	the	DET
ejpam-5133	259	40	areas	area	NOUN
ejpam-5133	259	41	of	of	ADP
ejpam-5133	259	42	the	the	DET
ejpam-5133	259	43	rectangle	rectangle	NOUN
ejpam-5133	259	44	abed	abe	VERB
ejpam-5133	259	45	and	and	CCONJ
ejpam-5133	259	46	the	the	DET
ejpam-5133	259	47	triangle	triangle	NOUN
ejpam-5133	259	48	ecd	ecd	PROPN
ejpam-5133	259	49	.	.	PUNCT
ejpam-5133	260	1	consequently	consequently	ADV
ejpam-5133	260	2	,	,	PUNCT
ejpam-5133	260	3	we	we	PRON
ejpam-5133	260	4	have	have	VERB
ejpam-5133	260	5	(	(	PUNCT
ejpam-5133	260	6	16	16	NUM
ejpam-5133	260	7	)	)	PUNCT
ejpam-5133	260	8	.	.	PUNCT
ejpam-5133	261	1	observing	observe	VERB
ejpam-5133	261	2	that	that	SCONJ
ejpam-5133	261	3	the	the	DET
ejpam-5133	261	4	result	result	NOUN
ejpam-5133	261	5	of	of	ADP
ejpam-5133	261	6	this	this	DET
ejpam-5133	261	7	integral	integral	NOUN
ejpam-5133	261	8	is	be	AUX
ejpam-5133	261	9	also	also	ADV
ejpam-5133	261	10	in	in	ADP
ejpam-5133	261	11	the	the	DET
ejpam-5133	261	12	form	form	NOUN
ejpam-5133	261	13	z2	z2	NOUN
ejpam-5133	261	14	−	−	PROPN
ejpam-5133	261	15	y2	y2	NOUN
ejpam-5133	261	16	=	=	SYM
ejpam-5133	261	17	x2	x2	PROPN
ejpam-5133	261	18	,	,	PUNCT
ejpam-5133	261	19	then	then	ADV
ejpam-5133	261	20	we	we	PRON
ejpam-5133	261	21	get	get	VERB
ejpam-5133	261	22	x2	x2	NOUN
ejpam-5133	261	23	=	=	SYM
ejpam-5133	261	24	d2	d2	PROPN
ejpam-5133	261	25	+2	+2	PROPN
ejpam-5133	261	26	yd	yd	NOUN
ejpam-5133	261	27	.	.	PUNCT
ejpam-5133	262	1	from	from	ADP
ejpam-5133	262	2	last	last	ADJ
ejpam-5133	262	3	equation	equation	NOUN
ejpam-5133	262	4	,	,	PUNCT
ejpam-5133	262	5	and	and	CCONJ
ejpam-5133	262	6	taking	take	VERB
ejpam-5133	262	7	into	into	ADP
ejpam-5133	262	8	account	account	NOUN
ejpam-5133	262	9	that	that	SCONJ
ejpam-5133	262	10	z	z	NOUN
ejpam-5133	262	11	=	=	PUNCT
ejpam-5133	262	12	y	y	PROPN
ejpam-5133	262	13	+	+	CCONJ
ejpam-5133	262	14	d.	d.	PROPN
ejpam-5133	262	15	we	we	PRON
ejpam-5133	262	16	have	have	VERB
ejpam-5133	262	17	(	(	PUNCT
ejpam-5133	262	18	1	1	NUM
ejpam-5133	262	19	)	)	PUNCT
ejpam-5133	262	20	.	.	PUNCT
ejpam-5133	263	1	in	in	ADP
ejpam-5133	263	2	this	this	DET
ejpam-5133	263	3	way	way	NOUN
ejpam-5133	263	4	,	,	PUNCT
ejpam-5133	263	5	we	we	PRON
ejpam-5133	263	6	find	find	VERB
ejpam-5133	263	7	another	another	DET
ejpam-5133	263	8	geometric	geometric	ADJ
ejpam-5133	263	9	interpretation	interpretation	NOUN
ejpam-5133	263	10	of	of	ADP
ejpam-5133	263	11	(	(	PUNCT
ejpam-5133	263	12	1	1	NUM
ejpam-5133	263	13	)	)	PUNCT
ejpam-5133	263	14	,	,	PUNCT
ejpam-5133	263	15	different	different	ADJ
ejpam-5133	263	16	from	from	ADP
ejpam-5133	263	17	that	that	PRON
ejpam-5133	263	18	found	find	VERB
ejpam-5133	263	19	in	in	ADP
ejpam-5133	263	20	theorem	theorem	NOUN
ejpam-5133	263	21	(	(	PUNCT
ejpam-5133	263	22	1	1	NUM
ejpam-5133	263	23	)	)	PUNCT
ejpam-5133	263	24	,	,	PUNCT
ejpam-5133	263	25	and	and	CCONJ
ejpam-5133	263	26	it	it	PRON
ejpam-5133	263	27	holds	hold	VERB
ejpam-5133	263	28	∀	∀	NOUN
ejpam-5133	263	29	x	x	NOUN
ejpam-5133	263	30	,	,	PUNCT
ejpam-5133	263	31	y	y	PROPN
ejpam-5133	263	32	,	,	PUNCT
ejpam-5133	263	33	z	z	NOUN
ejpam-5133	263	34	∈	∈	PROPN
ejpam-5133	263	35	]	]	X
ejpam-5133	263	36	0,∞	0,∞	NOUN
ejpam-5133	264	1	[	[	X
ejpam-5133	264	2	.	.	PUNCT
ejpam-5133	265	1	obviously	obviously	ADV
ejpam-5133	265	2	,	,	PUNCT
ejpam-5133	265	3	if	if	SCONJ
ejpam-5133	265	4	x	x	PROPN
ejpam-5133	265	5	,	,	PUNCT
ejpam-5133	265	6	y	y	PROPN
ejpam-5133	265	7	,	,	PUNCT
ejpam-5133	265	8	z	z	NOUN
ejpam-5133	265	9	∈n	∈n	NOUN
ejpam-5133	265	10	then	then	ADV
ejpam-5133	265	11	d	d	PROPN
ejpam-5133	265	12	∈	∈	PROPN
ejpam-5133	265	13	c(x	c(x	NOUN
ejpam-5133	265	14	)	)	PUNCT
ejpam-5133	265	15	.	.	PUNCT
ejpam-5133	266	1	4	4	X
ejpam-5133	266	2	.	.	X
ejpam-5133	266	3	conclusion	conclusion	NOUN
ejpam-5133	266	4	and	and	CCONJ
ejpam-5133	266	5	remarks	remark	VERB
ejpam-5133	266	6	the	the	DET
ejpam-5133	266	7	discovery	discovery	NOUN
ejpam-5133	266	8	of	of	ADP
ejpam-5133	266	9	the	the	DET
ejpam-5133	266	10	parametrization	parametrization	NOUN
ejpam-5133	266	11	and	and	CCONJ
ejpam-5133	266	12	relations	relation	NOUN
ejpam-5133	266	13	among	among	ADP
ejpam-5133	266	14	pythagorean	pythagorean	PROPN
ejpam-5133	266	15	triples	triple	NOUN
ejpam-5133	266	16	that	that	PRON
ejpam-5133	266	17	we	we	PRON
ejpam-5133	266	18	have	have	AUX
ejpam-5133	266	19	found	find	VERB
ejpam-5133	266	20	shows	show	VERB
ejpam-5133	266	21	the	the	DET
ejpam-5133	266	22	fundamental	fundamental	ADJ
ejpam-5133	266	23	role	role	NOUN
ejpam-5133	266	24	of	of	ADP
ejpam-5133	266	25	d	d	PROPN
ejpam-5133	266	26	∈	∈	PROPN
ejpam-5133	266	27	c(x	c(x	NOUN
ejpam-5133	266	28	)	)	PUNCT
ejpam-5133	266	29	and	and	CCONJ
ejpam-5133	266	30	characterizing	characterize	VERB
ejpam-5133	266	31	the	the	DET
ejpam-5133	266	32	results	result	NOUN
ejpam-5133	266	33	.	.	PUNCT
ejpam-5133	267	1	this	this	DET
ejpam-5133	267	2	approach	approach	NOUN
ejpam-5133	267	3	could	could	AUX
ejpam-5133	267	4	be	be	AUX
ejpam-5133	267	5	employed	employ	VERB
ejpam-5133	267	6	to	to	PART
ejpam-5133	267	7	study	study	VERB
ejpam-5133	267	8	further	further	ADJ
ejpam-5133	267	9	relationships	relationship	NOUN
ejpam-5133	267	10	among	among	ADP
ejpam-5133	267	11	pythagorean	pythagorean	PROPN
ejpam-5133	267	12	triples	triple	NOUN
ejpam-5133	267	13	.	.	PUNCT
ejpam-5133	268	1	for	for	ADP
ejpam-5133	268	2	instance	instance	NOUN
ejpam-5133	268	3	,	,	PUNCT
ejpam-5133	268	4	it	it	PRON
ejpam-5133	268	5	might	might	AUX
ejpam-5133	268	6	be	be	AUX
ejpam-5133	268	7	used	use	VERB
ejpam-5133	268	8	to	to	PART
ejpam-5133	268	9	find	find	VERB
ejpam-5133	268	10	a	a	DET
ejpam-5133	268	11	suitable	suitable	ADJ
ejpam-5133	268	12	addition	addition	NOUN
ejpam-5133	268	13	operation	operation	NOUN
ejpam-5133	268	14	between	between	ADP
ejpam-5133	268	15	pythagorean	pythagorean	PROPN
ejpam-5133	268	16	triples	triple	NOUN
ejpam-5133	268	17	which	which	PRON
ejpam-5133	268	18	could	could	AUX
ejpam-5133	268	19	allow	allow	VERB
ejpam-5133	268	20	us	we	PRON
ejpam-5133	268	21	,	,	PUNCT
ejpam-5133	268	22	in	in	ADP
ejpam-5133	268	23	turn	turn	NOUN
ejpam-5133	268	24	,	,	PUNCT
ejpam-5133	268	25	to	to	PART
ejpam-5133	268	26	define	define	VERB
ejpam-5133	268	27	a	a	DET
ejpam-5133	268	28	vector	vector	NOUN
ejpam-5133	268	29	space	space	NOUN
ejpam-5133	268	30	of	of	ADP
ejpam-5133	268	31	pythagorean	pythagorean	PROPN
ejpam-5133	268	32	triples	triple	NOUN
ejpam-5133	268	33	.	.	PUNCT
ejpam-5133	269	1	this	this	DET
ejpam-5133	269	2	way	way	NOUN
ejpam-5133	269	3	could	could	AUX
ejpam-5133	269	4	be	be	AUX
ejpam-5133	269	5	used	use	VERB
ejpam-5133	269	6	to	to	PART
ejpam-5133	269	7	study	study	VERB
ejpam-5133	269	8	other	other	ADJ
ejpam-5133	269	9	problems	problem	NOUN
ejpam-5133	269	10	,	,	PUNCT
ejpam-5133	269	11	some	some	PRON
ejpam-5133	269	12	of	of	ADP
ejpam-5133	269	13	which	which	PRON
ejpam-5133	269	14	are	be	AUX
ejpam-5133	269	15	still	still	ADV
ejpam-5133	269	16	open	open	ADJ
ejpam-5133	269	17	.	.	PUNCT
ejpam-5133	270	1	one	one	NUM
ejpam-5133	270	2	of	of	ADP
ejpam-5133	270	3	the	the	DET
ejpam-5133	270	4	next	next	ADJ
ejpam-5133	270	5	steps	step	NOUN
ejpam-5133	270	6	could	could	AUX
ejpam-5133	270	7	be	be	AUX
ejpam-5133	270	8	to	to	PART
ejpam-5133	270	9	study	study	VERB
ejpam-5133	270	10	the	the	DET
ejpam-5133	270	11	parametrizations	parametrization	NOUN
ejpam-5133	270	12	of	of	ADP
ejpam-5133	270	13	pythagorean	pythagorean	PROPN
ejpam-5133	270	14	quadruples	quadruple	NOUN
ejpam-5133	270	15	,	,	PUNCT
ejpam-5133	270	16	looking	look	VERB
ejpam-5133	270	17	for	for	ADP
ejpam-5133	270	18	a	a	DET
ejpam-5133	270	19	representation	representation	NOUN
ejpam-5133	270	20	similar	similar	ADJ
ejpam-5133	270	21	to	to	ADP
ejpam-5133	270	22	(	(	PUNCT
ejpam-5133	270	23	1	1	NUM
ejpam-5133	270	24	)	)	PUNCT
ejpam-5133	270	25	and	and	CCONJ
ejpam-5133	270	26	finding	find	VERB
ejpam-5133	270	27	all	all	DET
ejpam-5133	270	28	pythagorean	pythagorean	PROPN
ejpam-5133	270	29	quadruples	quadruple	NOUN
ejpam-5133	270	30	.	.	PUNCT
ejpam-5133	271	1	mainly	mainly	ADV
ejpam-5133	271	2	it	it	PRON
ejpam-5133	271	3	will	will	AUX
ejpam-5133	271	4	be	be	AUX
ejpam-5133	271	5	interesting	interesting	ADJ
ejpam-5133	271	6	to	to	PART
ejpam-5133	271	7	find	find	VERB
ejpam-5133	271	8	other	other	ADJ
ejpam-5133	271	9	parametrizations	parametrization	NOUN
ejpam-5133	271	10	,	,	PUNCT
ejpam-5133	271	11	relations	relation	NOUN
ejpam-5133	271	12	and	and	CCONJ
ejpam-5133	271	13	characterizations	characterization	NOUN
ejpam-5133	271	14	regard	regard	VERB
ejpam-5133	271	15	to	to	ADP
ejpam-5133	271	16	the	the	DET
ejpam-5133	271	17	pythagorean	pythagorean	PROPN
ejpam-5133	271	18	n	n	CCONJ
ejpam-5133	271	19	-	-	PUNCT
ejpam-5133	271	20	uples	uple	NOUN
ejpam-5133	271	21	dependent	dependent	ADJ
ejpam-5133	271	22	by	by	ADP
ejpam-5133	271	23	d	d	PROPN
ejpam-5133	271	24	∈	∈	PROPN
ejpam-5133	271	25	c(x	c(x	NOUN
ejpam-5133	271	26	)	)	PUNCT
ejpam-5133	271	27	.	.	PUNCT
ejpam-5133	272	1	the	the	DET
ejpam-5133	272	2	new	new	ADJ
ejpam-5133	272	3	results	result	NOUN
ejpam-5133	272	4	and	and	CCONJ
ejpam-5133	272	5	applications	application	NOUN
ejpam-5133	272	6	,	,	PUNCT
ejpam-5133	272	7	along	along	ADP
ejpam-5133	272	8	with	with	ADP
ejpam-5133	272	9	those	those	PRON
ejpam-5133	272	10	found	find	VERB
ejpam-5133	272	11	in	in	ADP
ejpam-5133	272	12	the	the	DET
ejpam-5133	272	13	preliminary	preliminary	ADJ
ejpam-5133	272	14	results	result	NOUN
ejpam-5133	272	15	section	section	NOUN
ejpam-5133	272	16	,	,	PUNCT
ejpam-5133	272	17	show	show	VERB
ejpam-5133	272	18	how	how	SCONJ
ejpam-5133	272	19	the	the	DET
ejpam-5133	272	20	field	field	NOUN
ejpam-5133	272	21	of	of	ADP
ejpam-5133	272	22	pythagorean	pythagorean	PROPN
ejpam-5133	272	23	triples	triple	NOUN
ejpam-5133	272	24	is	be	AUX
ejpam-5133	272	25	still	still	ADV
ejpam-5133	272	26	interesting	interesting	ADJ
ejpam-5133	272	27	and	and	CCONJ
ejpam-5133	272	28	stimulating	stimulate	VERB
ejpam-5133	272	29	to	to	PART
ejpam-5133	272	30	study	study	VERB
ejpam-5133	272	31	,	,	PUNCT
ejpam-5133	272	32	despite	despite	SCONJ
ejpam-5133	272	33	the	the	DET
ejpam-5133	272	34	centuries	century	NOUN
ejpam-5133	272	35	that	that	PRON
ejpam-5133	272	36	have	have	AUX
ejpam-5133	272	37	elapsed	elapse	VERB
ejpam-5133	272	38	.	.	PUNCT
ejpam-5133	273	1	in	in	ADP
ejpam-5133	273	2	every	every	DET
ejpam-5133	273	3	case	case	NOUN
ejpam-5133	273	4	,	,	PUNCT
ejpam-5133	273	5	this	this	PRON
ejpam-5133	273	6	is	be	AUX
ejpam-5133	273	7	a	a	DET
ejpam-5133	273	8	new	new	ADJ
ejpam-5133	273	9	approach	approach	NOUN
ejpam-5133	273	10	to	to	PART
ejpam-5133	273	11	study	study	VERB
ejpam-5133	273	12	the	the	DET
ejpam-5133	273	13	pythagorean	pythagorean	PROPN
ejpam-5133	273	14	triples	triple	NOUN
ejpam-5133	273	15	for	for	ADP
ejpam-5133	273	16	students	student	NOUN
ejpam-5133	273	17	in	in	ADP
ejpam-5133	273	18	schools	school	NOUN
ejpam-5133	273	19	and	and	CCONJ
ejpam-5133	273	20	universities	university	NOUN
ejpam-5133	273	21	.	.	PUNCT
ejpam-5133	274	1	references	reference	NOUN
ejpam-5133	274	2	689	689	NUM
ejpam-5133	274	3	acknowledgements	acknowledgement	NOUN
ejpam-5133	274	4	the	the	DET
ejpam-5133	274	5	author	author	NOUN
ejpam-5133	274	6	is	be	AUX
ejpam-5133	274	7	member	member	NOUN
ejpam-5133	274	8	of	of	ADP
ejpam-5133	274	9	the	the	DET
ejpam-5133	274	10	gruppo	gruppo	PROPN
ejpam-5133	274	11	nazionale	nazionale	NOUN
ejpam-5133	274	12	per	per	ADP
ejpam-5133	274	13	l’analisi	l’analisi	PROPN
ejpam-5133	274	14	matematica	matematica	PROPN
ejpam-5133	274	15	,	,	PUNCT
ejpam-5133	274	16	la	la	X
ejpam-5133	274	17	probabilità	probabilità	PROPN
ejpam-5133	274	18	e	e	X
ejpam-5133	274	19	le	le	X
ejpam-5133	274	20	loro	loro	X
ejpam-5133	274	21	applicazioni	applicazioni	PROPN
ejpam-5133	274	22	(	(	PUNCT
ejpam-5133	274	23	gnampa	gnampa	NOUN
ejpam-5133	274	24	)	)	PUNCT
ejpam-5133	274	25	of	of	ADP
ejpam-5133	274	26	the	the	DET
ejpam-5133	274	27	istituto	istituto	PROPN
ejpam-5133	274	28	nazionale	nazionale	PROPN
ejpam-5133	274	29	di	di	PROPN
ejpam-5133	274	30	alta	alta	PROPN
ejpam-5133	274	31	matematica	matematica	PROPN
ejpam-5133	274	32	(	(	PUNCT
ejpam-5133	274	33	indam	indam	NOUN
ejpam-5133	274	34	)	)	PUNCT
ejpam-5133	274	35	.	.	PUNCT
ejpam-5133	275	1	the	the	DET
ejpam-5133	275	2	paper	paper	NOUN
ejpam-5133	275	3	is	be	AUX
ejpam-5133	275	4	supported	support	VERB
ejpam-5133	275	5	by	by	ADP
ejpam-5133	275	6	prin	prin	PROPN
ejpam-5133	275	7	2022	2022	NUM
ejpam-5133	275	8	progetti	progetti	PROPN
ejpam-5133	275	9	di	di	X
ejpam-5133	275	10	ricerca	ricerca	PROPN
ejpam-5133	275	11	di	di	PROPN
ejpam-5133	275	12	rilevante	rilevante	PROPN
ejpam-5133	275	13	interesse	interesse	PROPN
ejpam-5133	275	14	nazionale	nazionale	PROPN
ejpam-5133	275	15	,	,	PUNCT
ejpam-5133	275	16	”	"	PUNCT
ejpam-5133	275	17	nonlinear	nonlinear	ADJ
ejpam-5133	275	18	differential	differential	ADJ
ejpam-5133	275	19	problems	problem	NOUN
ejpam-5133	275	20	with	with	ADP
ejpam-5133	275	21	applications	application	NOUN
ejpam-5133	275	22	to	to	ADP
ejpam-5133	275	23	real	real	ADJ
ejpam-5133	275	24	phenomena	phenomenon	NOUN
ejpam-5133	275	25	”	"	PUNCT
ejpam-5133	275	26	,	,	PUNCT
ejpam-5133	275	27	(	(	PUNCT
ejpam-5133	275	28	2022zxztn2	2022zxztn2	NUM
ejpam-5133	275	29	)	)	PUNCT
ejpam-5133	275	30	.	.	PUNCT
ejpam-5133	276	1	references	reference	NOUN
ejpam-5133	276	2	[	[	X
ejpam-5133	276	3	1	1	NUM
ejpam-5133	276	4	]	]	PUNCT
ejpam-5133	276	5	a.	a.	NOUN
ejpam-5133	276	6	a.	a.	NOUN
ejpam-5133	276	7	adigun	adigun	PROPN
ejpam-5133	276	8	,	,	PUNCT
ejpam-5133	276	9	k.	k.	PROPN
ejpam-5133	276	10	o.	o.	PROPN
ejpam-5133	276	11	jimon	jimon	PROPN
ejpam-5133	276	12	,	,	PUNCT
ejpam-5133	276	13	y.	y.	PROPN
ejpam-5133	276	14	o.	o.	PROPN
ejpam-5133	276	15	adebayo	adebayo	PROPN
ejpam-5133	276	16	,	,	PUNCT
ejpam-5133	276	17	and	and	CCONJ
ejpam-5133	276	18	m.	m.	NOUN
ejpam-5133	276	19	o.	o.	PROPN
ejpam-5133	276	20	kolawole	kolawole	PROPN
ejpam-5133	276	21	.	.	PUNCT
ejpam-5133	277	1	review	review	NOUN
ejpam-5133	277	2	of	of	ADP
ejpam-5133	277	3	pythagorean	pythagorean	PROPN
ejpam-5133	277	4	triple	triple	ADV
ejpam-5133	277	5	based	base	VERB
ejpam-5133	277	6	cryptography	cryptography	NOUN
ejpam-5133	277	7	system	system	NOUN
ejpam-5133	277	8	for	for	ADP
ejpam-5133	277	9	information	information	NOUN
ejpam-5133	277	10	security	security	NOUN
ejpam-5133	277	11	.	.	PUNCT
ejpam-5133	278	1	university	university	NOUN
ejpam-5133	278	2	of	of	ADP
ejpam-5133	278	3	pitesti	pitesti	PROPN
ejpam-5133	278	4	scientific	scientific	ADJ
ejpam-5133	278	5	bulletin	bulletin	NOUN
ejpam-5133	278	6	,	,	PUNCT
ejpam-5133	278	7	series	series	NOUN
ejpam-5133	278	8	:	:	PUNCT
ejpam-5133	278	9	electronics	electronic	NOUN
ejpam-5133	278	10	and	and	CCONJ
ejpam-5133	278	11	computers	computer	NOUN
ejpam-5133	278	12	science	science	NOUN
ejpam-5133	278	13	,	,	PUNCT
ejpam-5133	278	14	21(1	21(1	NUM
ejpam-5133	278	15	)	)	PUNCT
ejpam-5133	278	16	,	,	PUNCT
ejpam-5133	278	17	2021	2021	NUM
ejpam-5133	278	18	.	.	PUNCT
ejpam-5133	279	1	[	[	X
ejpam-5133	279	2	2	2	NUM
ejpam-5133	279	3	]	]	X
ejpam-5133	279	4	r	r	NOUN
ejpam-5133	279	5	amato	amato	PROPN
ejpam-5133	279	6	.	.	PUNCT
ejpam-5133	279	7	sulla	sulla	PROPN
ejpam-5133	279	8	determinazione	determinazione	NOUN
ejpam-5133	279	9	delle	delle	PROPN
ejpam-5133	279	10	terne	terne	PROPN
ejpam-5133	279	11	pitagoriche	pitagoriche	PROPN
ejpam-5133	279	12	.	.	PUNCT
ejpam-5133	280	1	atti	atti	PROPN
ejpam-5133	280	2	della	della	PROPN
ejpam-5133	280	3	società	società	PROPN
ejpam-5133	280	4	peloritana	peloritana	PROPN
ejpam-5133	280	5	di	di	PROPN
ejpam-5133	280	6	scienze	scienze	PROPN
ejpam-5133	280	7	fisiche	fisiche	PROPN
ejpam-5133	280	8	matematiche	matematiche	PROPN
ejpam-5133	280	9	e	e	X
ejpam-5133	280	10	naturali	naturali	PROPN
ejpam-5133	280	11	,	,	PUNCT
ejpam-5133	280	12	27:3–8	27:3–8	NUM
ejpam-5133	280	13	,	,	PUNCT
ejpam-5133	280	14	1981	1981	NUM
ejpam-5133	280	15	.	.	PUNCT
ejpam-5133	281	1	[	[	X
ejpam-5133	281	2	3	3	NUM
ejpam-5133	281	3	]	]	X
ejpam-5133	281	4	r	r	NOUN
ejpam-5133	281	5	amato	amato	PROPN
ejpam-5133	281	6	.	.	PUNCT
ejpam-5133	282	1	a	a	DET
ejpam-5133	282	2	characterization	characterization	NOUN
ejpam-5133	282	3	of	of	ADP
ejpam-5133	282	4	pythagorean	pythagorean	PROPN
ejpam-5133	282	5	triples	triple	NOUN
ejpam-5133	282	6	.	.	PUNCT
ejpam-5133	283	1	jp	jp	PROPN
ejpam-5133	283	2	journal	journal	PROPN
ejpam-5133	283	3	of	of	ADP
ejpam-5133	283	4	algebra	algebra	PROPN
ejpam-5133	283	5	,	,	PUNCT
ejpam-5133	283	6	number	number	NOUN
ejpam-5133	283	7	theory	theory	NOUN
ejpam-5133	283	8	and	and	CCONJ
ejpam-5133	283	9	applications	application	NOUN
ejpam-5133	283	10	,	,	PUNCT
ejpam-5133	283	11	39(2):221–230	39(2):221–230	PROPN
ejpam-5133	283	12	,	,	PUNCT
ejpam-5133	283	13	2017	2017	NUM
ejpam-5133	283	14	.	.	PUNCT
ejpam-5133	284	1	[	[	X
ejpam-5133	284	2	4	4	NUM
ejpam-5133	284	3	]	]	X
ejpam-5133	284	4	r	r	X
ejpam-5133	284	5	amato	amato	PROPN
ejpam-5133	284	6	.	.	PUNCT
ejpam-5133	285	1	a	a	DET
ejpam-5133	285	2	note	note	NOUN
ejpam-5133	285	3	on	on	ADP
ejpam-5133	285	4	pythagorean	pythagorean	PROPN
ejpam-5133	285	5	triples	triple	NOUN
ejpam-5133	285	6	.	.	PUNCT
ejpam-5133	286	1	international	international	ADJ
ejpam-5133	286	2	journal	journal	PROPN
ejpam-5133	286	3	of	of	ADP
ejpam-5133	286	4	mathematics	mathematic	NOUN
ejpam-5133	286	5	and	and	CCONJ
ejpam-5133	286	6	computer	computer	NOUN
ejpam-5133	286	7	science	science	NOUN
ejpam-5133	286	8	,	,	PUNCT
ejpam-5133	286	9	15(2):485–490	15(2):485–490	PROPN
ejpam-5133	286	10	,	,	PUNCT
ejpam-5133	286	11	2020	2020	NUM
ejpam-5133	286	12	.	.	PUNCT
ejpam-5133	287	1	[	[	X
ejpam-5133	287	2	5	5	NUM
ejpam-5133	287	3	]	]	X
ejpam-5133	287	4	r	r	NOUN
ejpam-5133	287	5	amato	amato	PROPN
ejpam-5133	287	6	.	.	PUNCT
ejpam-5133	288	1	some	some	DET
ejpam-5133	288	2	relations	relation	NOUN
ejpam-5133	288	3	among	among	ADP
ejpam-5133	288	4	pythagorean	pythagorean	PROPN
ejpam-5133	288	5	triples	triple	NOUN
ejpam-5133	288	6	.	.	PUNCT
ejpam-5133	289	1	international	international	ADJ
ejpam-5133	289	2	journal	journal	PROPN
ejpam-5133	289	3	of	of	ADP
ejpam-5133	289	4	mathematics	mathematic	NOUN
ejpam-5133	289	5	and	and	CCONJ
ejpam-5133	289	6	computer	computer	NOUN
ejpam-5133	289	7	science	science	NOUN
ejpam-5133	289	8	,	,	PUNCT
ejpam-5133	289	9	16(1):143–147	16(1):143–147	PROPN
ejpam-5133	289	10	,	,	PUNCT
ejpam-5133	289	11	2021	2021	NUM
ejpam-5133	289	12	.	.	PUNCT
ejpam-5133	290	1	[	[	X
ejpam-5133	290	2	6	6	NUM
ejpam-5133	290	3	]	]	X
ejpam-5133	290	4	r	r	NOUN
ejpam-5133	290	5	amato	amato	PROPN
ejpam-5133	290	6	.	.	PUNCT
ejpam-5133	291	1	a	a	DET
ejpam-5133	291	2	characterization	characterization	NOUN
ejpam-5133	291	3	of	of	ADP
ejpam-5133	291	4	primitive	primitive	ADJ
ejpam-5133	291	5	pythagorean	pythagorean	ADJ
ejpam-5133	291	6	triples	triple	NOUN
ejpam-5133	291	7	.	.	PUNCT
ejpam-5133	292	1	palestine	palestine	PROPN
ejpam-5133	292	2	journal	journal	PROPN
ejpam-5133	292	3	of	of	ADP
ejpam-5133	292	4	mathematics	mathematics	PROPN
ejpam-5133	292	5	,	,	PUNCT
ejpam-5133	292	6	12(2):524–529	12(2):524–529	NOUN
ejpam-5133	292	7	,	,	PUNCT
ejpam-5133	292	8	2023	2023	NUM
ejpam-5133	292	9	.	.	PUNCT
ejpam-5133	293	1	[	[	X
ejpam-5133	293	2	7	7	NUM
ejpam-5133	293	3	]	]	X
ejpam-5133	293	4	r	r	NOUN
ejpam-5133	293	5	amato	amato	PROPN
ejpam-5133	293	6	.	.	PUNCT
ejpam-5133	293	7	groups	group	NOUN
ejpam-5133	293	8	and	and	CCONJ
ejpam-5133	293	9	monoid	monoid	NOUN
ejpam-5133	293	10	in	in	ADP
ejpam-5133	293	11	the	the	DET
ejpam-5133	293	12	set	set	NOUN
ejpam-5133	293	13	of	of	ADP
ejpam-5133	293	14	pythagorean	pythagorean	PROPN
ejpam-5133	293	15	triples	triple	NOUN
ejpam-5133	293	16	.	.	PUNCT
ejpam-5133	294	1	integers	integer	NOUN
ejpam-5133	294	2	,	,	PUNCT
ejpam-5133	294	3	24(a5	24(a5	ADV
ejpam-5133	294	4	)	)	PUNCT
ejpam-5133	294	5	,	,	PUNCT
ejpam-5133	294	6	2024	2024	NUM
ejpam-5133	294	7	.	.	PUNCT
ejpam-5133	295	1	[	[	X
ejpam-5133	295	2	8	8	NUM
ejpam-5133	295	3	]	]	X
ejpam-5133	295	4	r	r	NOUN
ejpam-5133	295	5	a	a	DET
ejpam-5133	295	6	beauregard	beauregard	PROPN
ejpam-5133	295	7	and	and	CCONJ
ejpam-5133	295	8	e	e	NOUN
ejpam-5133	295	9	r	r	NOUN
ejpam-5133	295	10	suryanarayan	suryanarayan	NOUN
ejpam-5133	295	11	.	.	PUNCT
ejpam-5133	296	1	proof	proof	NOUN
ejpam-5133	296	2	without	without	ADP
ejpam-5133	296	3	words	word	NOUN
ejpam-5133	296	4	:	:	PUNCT
ejpam-5133	296	5	parametric	parametric	ADJ
ejpam-5133	296	6	representation	representation	NOUN
ejpam-5133	296	7	of	of	ADP
ejpam-5133	296	8	primitive	primitive	ADJ
ejpam-5133	296	9	pythagorean	pythagorean	ADJ
ejpam-5133	296	10	triples	triple	NOUN
ejpam-5133	296	11	.	.	PUNCT
ejpam-5133	297	1	mathematics	mathematic	NOUN
ejpam-5133	297	2	magazine	magazine	NOUN
ejpam-5133	297	3	,	,	PUNCT
ejpam-5133	297	4	69(3):189–189	69(3):189–189	PROPN
ejpam-5133	297	5	,	,	PUNCT
ejpam-5133	297	6	1996	1996	NUM
ejpam-5133	297	7	.	.	PUNCT
ejpam-5133	298	1	[	[	X
ejpam-5133	298	2	9	9	NUM
ejpam-5133	298	3	]	]	X
ejpam-5133	298	4	j	j	PROPN
ejpam-5133	298	5	kocik	kocik	PROPN
ejpam-5133	298	6	.	.	PUNCT
ejpam-5133	299	1	clifford	clifford	PROPN
ejpam-5133	299	2	algebras	algebras	PROPN
ejpam-5133	299	3	and	and	CCONJ
ejpam-5133	299	4	euclid	euclid	PROPN
ejpam-5133	299	5	’s	’s	PART
ejpam-5133	299	6	parametrization	parametrization	NOUN
ejpam-5133	299	7	of	of	ADP
ejpam-5133	299	8	pythagorean	pythagorean	PROPN
ejpam-5133	299	9	triples	triple	NOUN
ejpam-5133	299	10	.	.	PUNCT
ejpam-5133	300	1	advances	advance	NOUN
ejpam-5133	300	2	in	in	ADP
ejpam-5133	300	3	applied	apply	VERB
ejpam-5133	300	4	clifford	clifford	PROPN
ejpam-5133	300	5	algebras	algebras	PROPN
ejpam-5133	300	6	,	,	PUNCT
ejpam-5133	300	7	17:71–93	17:71–93	NUM
ejpam-5133	300	8	,	,	PUNCT
ejpam-5133	300	9	2007	2007	NUM
ejpam-5133	300	10	.	.	PUNCT
ejpam-5133	301	1	[	[	X
ejpam-5133	301	2	10	10	NUM
ejpam-5133	301	3	]	]	X
ejpam-5133	301	4	w	w	NOUN
ejpam-5133	301	5	sierpinski	sierpinski	ADJ
ejpam-5133	301	6	.	.	PUNCT
ejpam-5133	302	1	elementary	elementary	ADJ
ejpam-5133	302	2	theory	theory	NOUN
ejpam-5133	302	3	of	of	ADP
ejpam-5133	302	4	numbers	number	NOUN
ejpam-5133	302	5	.	.	PUNCT
ejpam-5133	303	1	pwn	pwn	NOUN
ejpam-5133	303	2	-	-	PUNCT
ejpam-5133	303	3	polish	polish	ADJ
ejpam-5133	303	4	scientific	scientific	ADJ
ejpam-5133	303	5	publishersa	publishersa	NOUN
ejpam-5133	303	6	,	,	PUNCT
ejpam-5133	303	7	warszawa	warszawa	PROPN
ejpam-5133	303	8	,	,	PUNCT
ejpam-5133	303	9	1988	1988	NUM
ejpam-5133	303	10	.	.	PUNCT
ejpam-5133	304	1	[	[	X
ejpam-5133	304	2	11	11	NUM
ejpam-5133	304	3	]	]	X
ejpam-5133	304	4	r	r	NOUN
ejpam-5133	304	5	sivaraman	sivaraman	NOUN
ejpam-5133	304	6	.	.	PUNCT
ejpam-5133	305	1	pascal	pascal	ADJ
ejpam-5133	305	2	triangle	triangle	NOUN
ejpam-5133	305	3	and	and	CCONJ
ejpam-5133	305	4	pythagorean	pythagorean	PROPN
ejpam-5133	305	5	triples	triple	NOUN
ejpam-5133	305	6	.	.	PUNCT
ejpam-5133	306	1	advances	advance	NOUN
ejpam-5133	306	2	in	in	ADP
ejpam-5133	306	3	applied	apply	VERB
ejpam-5133	306	4	clifford	clifford	PROPN
ejpam-5133	306	5	algebras	algebras	PROPN
ejpam-5133	306	6	,	,	PUNCT
ejpam-5133	306	7	8(8):75–80	8(8):75–80	NUM
ejpam-5133	306	8	,	,	PUNCT
ejpam-5133	306	9	2021	2021	NUM
ejpam-5133	306	10	.	.	PUNCT
ejpam-5133	307	1	[	[	X
ejpam-5133	307	2	12	12	NUM
ejpam-5133	307	3	]	]	PUNCT
ejpam-5133	307	4	t	t	PROPN
ejpam-5133	307	5	srinivas	srinivas	PROPN
ejpam-5133	307	6	.	.	PUNCT
ejpam-5133	308	1	symmetric	symmetric	ADJ
ejpam-5133	308	2	key	key	ADJ
ejpam-5133	308	3	generation	generation	NOUN
ejpam-5133	308	4	and	and	CCONJ
ejpam-5133	308	5	tree	tree	NOUN
ejpam-5133	308	6	construction	construction	NOUN
ejpam-5133	308	7	in	in	ADP
ejpam-5133	308	8	cryptosystem	cryptosystem	NOUN
ejpam-5133	308	9	based	base	VERB
ejpam-5133	308	10	on	on	ADP
ejpam-5133	308	11	pythagorean	pythagorean	PROPN
ejpam-5133	308	12	and	and	CCONJ
ejpam-5133	308	13	reciprocal	reciprocal	ADJ
ejpam-5133	308	14	pythagorean	pythagorean	PROPN
ejpam-5133	308	15	triples	triple	NOUN
ejpam-5133	308	16	.	.	PUNCT
ejpam-5133	308	17	qeios	qeio	NOUN
ejpam-5133	308	18	,	,	PUNCT
ejpam-5133	308	19	2023	2023	NUM
ejpam-5133	308	20	.	.	PUNCT