id	sid	tid	token	lemma	pos
ejpam-6781	1	1	european	european	PROPN
ejpam-6781	1	2	journal	journal	PROPN
ejpam-6781	1	3	of	of	ADP
ejpam-6781	1	4	pure	pure	ADJ
ejpam-6781	1	5	and	and	CCONJ
ejpam-6781	1	6	applied	applied	ADJ
ejpam-6781	1	7	mathematics	mathematic	NOUN
ejpam-6781	1	8	2025	2025	NUM
ejpam-6781	1	9	,	,	PUNCT
ejpam-6781	1	10	vol	vol	NOUN
ejpam-6781	1	11	.	.	PROPN
ejpam-6781	1	12	18	18	NUM
ejpam-6781	1	13	,	,	PUNCT
ejpam-6781	1	14	issue	issue	NOUN
ejpam-6781	1	15	4	4	NUM
ejpam-6781	1	16	,	,	PUNCT
ejpam-6781	1	17	article	article	NOUN
ejpam-6781	1	18	number	number	NOUN
ejpam-6781	1	19	6781	6781	NUM
ejpam-6781	1	20	issn	issn	VERB
ejpam-6781	1	21	1307	1307	NUM
ejpam-6781	1	22	-	-	SYM
ejpam-6781	1	23	5543	5543	NUM
ejpam-6781	1	24	–	–	PUNCT
ejpam-6781	1	25	ejpam.com	ejpam.com	X
ejpam-6781	1	26	published	publish	VERB
ejpam-6781	1	27	by	by	ADP
ejpam-6781	1	28	new	new	PROPN
ejpam-6781	1	29	york	york	PROPN
ejpam-6781	1	30	business	business	PROPN
ejpam-6781	1	31	global	global	ADJ
ejpam-6781	1	32	classification	classification	NOUN
ejpam-6781	1	33	and	and	CCONJ
ejpam-6781	1	34	enumeration	enumeration	NOUN
ejpam-6781	1	35	of	of	ADP
ejpam-6781	1	36	primitive	primitive	ADJ
ejpam-6781	1	37	eisenstein	eisenstein	NOUN
ejpam-6781	1	38	triples	triple	NOUN
ejpam-6781	1	39	using	use	VERB
ejpam-6781	1	40	prime	prime	ADJ
ejpam-6781	1	41	factorization	factorization	NOUN
ejpam-6781	1	42	techniques	technique	NOUN
ejpam-6781	1	43	somphong	somphong	NOUN
ejpam-6781	1	44	jitman1	jitman1	PROPN
ejpam-6781	1	45	,	,	PUNCT
ejpam-6781	1	46	mohd	mohd	PROPN
ejpam-6781	1	47	sham	sham	PROPN
ejpam-6781	1	48	mohammad2	mohammad2	PROPN
ejpam-6781	1	49	,	,	PUNCT
ejpam-6781	1	50	ekkasit	ekkasit	VERB
ejpam-6781	1	51	sangwisut3,∗	sangwisut3,∗	NOUN
ejpam-6781	1	52	1	1	NUM
ejpam-6781	1	53	department	department	NOUN
ejpam-6781	1	54	of	of	ADP
ejpam-6781	1	55	mathematics	mathematic	NOUN
ejpam-6781	1	56	,	,	PUNCT
ejpam-6781	1	57	faculty	faculty	NOUN
ejpam-6781	1	58	of	of	ADP
ejpam-6781	1	59	science	science	NOUN
ejpam-6781	1	60	,	,	PUNCT
ejpam-6781	1	61	silpakorn	silpakorn	VERB
ejpam-6781	1	62	university	university	PROPN
ejpam-6781	1	63	,	,	PUNCT
ejpam-6781	1	64	nakhon	nakhon	PROPN
ejpam-6781	1	65	pathom	pathom	PROPN
ejpam-6781	1	66	,	,	PUNCT
ejpam-6781	1	67	thailand	thailand	PROPN
ejpam-6781	1	68	2	2	NUM
ejpam-6781	1	69	centre	centre	NOUN
ejpam-6781	1	70	for	for	ADP
ejpam-6781	1	71	mathematical	mathematical	ADJ
ejpam-6781	1	72	sciences	science	NOUN
ejpam-6781	1	73	,	,	PUNCT
ejpam-6781	1	74	universiti	universiti	PROPN
ejpam-6781	1	75	malaysia	malaysia	PROPN
ejpam-6781	1	76	pahang	pahang	PROPN
ejpam-6781	1	77	al	al	PROPN
ejpam-6781	1	78	-	-	PUNCT
ejpam-6781	1	79	sultan	sultan	PROPN
ejpam-6781	1	80	abdullah	abdullah	PROPN
ejpam-6781	1	81	,	,	PUNCT
ejpam-6781	1	82	lebuh	lebuh	PROPN
ejpam-6781	1	83	persiaran	persiaran	VERB
ejpam-6781	1	84	tan	tan	PROPN
ejpam-6781	1	85	khalil	khalil	PROPN
ejpam-6781	1	86	yaakob	yaakob	PROPN
ejpam-6781	1	87	,	,	PUNCT
ejpam-6781	1	88	kuantan	kuantan	PROPN
ejpam-6781	1	89	,	,	PUNCT
ejpam-6781	1	90	pahang	pahang	PROPN
ejpam-6781	1	91	,	,	PUNCT
ejpam-6781	1	92	malaysia	malaysia	PROPN
ejpam-6781	1	93	3	3	NUM
ejpam-6781	1	94	department	department	NOUN
ejpam-6781	1	95	of	of	ADP
ejpam-6781	1	96	mathematics	mathematic	NOUN
ejpam-6781	1	97	and	and	CCONJ
ejpam-6781	1	98	statistics	statistic	NOUN
ejpam-6781	1	99	,	,	PUNCT
ejpam-6781	1	100	faculty	faculty	NOUN
ejpam-6781	1	101	of	of	ADP
ejpam-6781	1	102	science	science	NOUN
ejpam-6781	1	103	and	and	CCONJ
ejpam-6781	1	104	digital	digital	ADJ
ejpam-6781	1	105	innovation	innovation	NOUN
ejpam-6781	1	106	,	,	PUNCT
ejpam-6781	1	107	thaksin	thaksin	PROPN
ejpam-6781	1	108	university	university	PROPN
ejpam-6781	1	109	,	,	PUNCT
ejpam-6781	1	110	phattalung	phattalung	NOUN
ejpam-6781	1	111	,	,	PUNCT
ejpam-6781	1	112	thailand	thailand	PROPN
ejpam-6781	1	113	abstract	abstract	NOUN
ejpam-6781	1	114	.	.	PUNCT
ejpam-6781	2	1	this	this	DET
ejpam-6781	2	2	study	study	NOUN
ejpam-6781	2	3	delves	delve	VERB
ejpam-6781	2	4	into	into	ADP
ejpam-6781	2	5	the	the	DET
ejpam-6781	2	6	concept	concept	NOUN
ejpam-6781	2	7	of	of	ADP
ejpam-6781	2	8	primitive	primitive	ADJ
ejpam-6781	2	9	eisenstein	eisenstein	NOUN
ejpam-6781	2	10	triples	triple	NOUN
ejpam-6781	2	11	,	,	PUNCT
ejpam-6781	2	12	defined	define	VERB
ejpam-6781	2	13	as	as	ADP
ejpam-6781	2	14	positive	positive	ADJ
ejpam-6781	2	15	integer	integer	NOUN
ejpam-6781	2	16	solutions	solution	NOUN
ejpam-6781	2	17	(	(	PUNCT
ejpam-6781	2	18	a	a	PRON
ejpam-6781	2	19	,	,	PUNCT
ejpam-6781	2	20	b	b	NOUN
ejpam-6781	2	21	,	,	PUNCT
ejpam-6781	2	22	c	c	NOUN
ejpam-6781	2	23	)	)	PUNCT
ejpam-6781	2	24	to	to	ADP
ejpam-6781	2	25	the	the	DET
ejpam-6781	2	26	quadratic	quadratic	ADJ
ejpam-6781	2	27	equation	equation	NOUN
ejpam-6781	2	28	a2	a2	PROPN
ejpam-6781	2	29	−	−	PROPN
ejpam-6781	2	30	ab	ab	PROPN
ejpam-6781	2	31	+	+	CCONJ
ejpam-6781	2	32	b2	b2	NOUN
ejpam-6781	2	33	=	=	PROPN
ejpam-6781	2	34	c2	c2	PROPN
ejpam-6781	2	35	,	,	PUNCT
ejpam-6781	2	36	subject	subject	ADJ
ejpam-6781	2	37	to	to	ADP
ejpam-6781	2	38	the	the	DET
ejpam-6781	2	39	condition	condition	NOUN
ejpam-6781	2	40	a	a	DET
ejpam-6781	2	41	<	<	X
ejpam-6781	2	42	c	c	X
ejpam-6781	2	43	<	<	X
ejpam-6781	2	44	b	b	PROPN
ejpam-6781	2	45	and	and	CCONJ
ejpam-6781	2	46	gcd(a	gcd(a	PROPN
ejpam-6781	2	47	,	,	PUNCT
ejpam-6781	2	48	b	b	PROPN
ejpam-6781	2	49	,	,	PUNCT
ejpam-6781	2	50	c	c	NOUN
ejpam-6781	2	51	)	)	PUNCT
ejpam-6781	2	52	=	=	SYM
ejpam-6781	3	1	1	1	X
ejpam-6781	3	2	.	.	PUNCT
ejpam-6781	4	1	we	we	PRON
ejpam-6781	4	2	classify	classify	VERB
ejpam-6781	4	3	these	these	DET
ejpam-6781	4	4	triples	triple	NOUN
ejpam-6781	4	5	according	accord	VERB
ejpam-6781	4	6	to	to	ADP
ejpam-6781	4	7	the	the	DET
ejpam-6781	4	8	prime	prime	ADJ
ejpam-6781	4	9	factorization	factorization	NOUN
ejpam-6781	4	10	of	of	ADP
ejpam-6781	4	11	the	the	DET
ejpam-6781	4	12	integer	integer	NOUN
ejpam-6781	4	13	c	c	NOUN
ejpam-6781	4	14	,	,	PUNCT
ejpam-6781	4	15	elucidating	elucidate	VERB
ejpam-6781	4	16	how	how	SCONJ
ejpam-6781	4	17	their	their	PRON
ejpam-6781	4	18	existence	existence	NOUN
ejpam-6781	4	19	is	be	AUX
ejpam-6781	4	20	intricately	intricately	ADV
ejpam-6781	4	21	linked	link	VERB
ejpam-6781	4	22	to	to	ADP
ejpam-6781	4	23	specific	specific	ADJ
ejpam-6781	4	24	congruence	congruence	NOUN
ejpam-6781	4	25	conditions	condition	NOUN
ejpam-6781	4	26	imposed	impose	VERB
ejpam-6781	4	27	on	on	ADP
ejpam-6781	4	28	the	the	DET
ejpam-6781	4	29	prime	prime	ADJ
ejpam-6781	4	30	divisors	divisor	NOUN
ejpam-6781	4	31	of	of	ADP
ejpam-6781	4	32	c.	c.	PROPN
ejpam-6781	4	33	furthermore	furthermore	ADV
ejpam-6781	4	34	,	,	PUNCT
ejpam-6781	4	35	we	we	PRON
ejpam-6781	4	36	establish	establish	VERB
ejpam-6781	4	37	a	a	DET
ejpam-6781	4	38	bijective	bijective	ADJ
ejpam-6781	4	39	correspondence	correspondence	NOUN
ejpam-6781	4	40	between	between	ADP
ejpam-6781	4	41	these	these	DET
ejpam-6781	4	42	triples	triple	NOUN
ejpam-6781	4	43	and	and	CCONJ
ejpam-6781	4	44	a	a	DET
ejpam-6781	4	45	certain	certain	ADJ
ejpam-6781	4	46	subset	subset	NOUN
ejpam-6781	4	47	of	of	ADP
ejpam-6781	4	48	the	the	DET
ejpam-6781	4	49	unit	unit	NOUN
ejpam-6781	4	50	circle	circle	NOUN
ejpam-6781	4	51	.	.	PUNCT
ejpam-6781	5	1	this	this	DET
ejpam-6781	5	2	correspondence	correspondence	NOUN
ejpam-6781	5	3	enables	enable	VERB
ejpam-6781	5	4	a	a	DET
ejpam-6781	5	5	comprehensive	comprehensive	ADJ
ejpam-6781	5	6	enumeration	enumeration	NOUN
ejpam-6781	5	7	of	of	ADP
ejpam-6781	5	8	the	the	DET
ejpam-6781	5	9	triples	triple	NOUN
ejpam-6781	5	10	and	and	CCONJ
ejpam-6781	5	11	precisely	precisely	ADV
ejpam-6781	5	12	characterizes	characterize	VERB
ejpam-6781	5	13	the	the	DET
ejpam-6781	5	14	conditions	condition	NOUN
ejpam-6781	5	15	under	under	ADP
ejpam-6781	5	16	which	which	PRON
ejpam-6781	5	17	such	such	ADJ
ejpam-6781	5	18	solutions	solution	NOUN
ejpam-6781	5	19	exist	exist	VERB
ejpam-6781	5	20	.	.	PUNCT
ejpam-6781	6	1	2020	2020	NUM
ejpam-6781	6	2	mathematics	mathematic	NOUN
ejpam-6781	6	3	subject	subject	NOUN
ejpam-6781	6	4	classifications	classification	NOUN
ejpam-6781	6	5	:	:	PUNCT
ejpam-6781	6	6	11d09	11d09	NUM
ejpam-6781	6	7	,	,	PUNCT
ejpam-6781	6	8	11d45	11d45	NUM
ejpam-6781	6	9	,	,	PUNCT
ejpam-6781	6	10	20k25	20k25	NUM
ejpam-6781	6	11	key	key	ADJ
ejpam-6781	6	12	words	word	NOUN
ejpam-6781	6	13	and	and	CCONJ
ejpam-6781	6	14	phrases	phrase	NOUN
ejpam-6781	6	15	:	:	PUNCT
ejpam-6781	6	16	eisenstein	eisenstein	NOUN
ejpam-6781	6	17	triples	triple	NOUN
ejpam-6781	6	18	,	,	PUNCT
ejpam-6781	6	19	eisenstein	eisenstein	NOUN
ejpam-6781	6	20	integers	integer	NOUN
ejpam-6781	6	21	,	,	PUNCT
ejpam-6781	6	22	abelian	abelian	ADJ
ejpam-6781	6	23	groups	group	NOUN
ejpam-6781	6	24	,	,	PUNCT
ejpam-6781	6	25	prime	prime	ADJ
ejpam-6781	6	26	number	number	NOUN
ejpam-6781	6	27	1	1	NUM
ejpam-6781	6	28	.	.	PUNCT
ejpam-6781	7	1	introduction	introduction	NOUN
ejpam-6781	7	2	the	the	DET
ejpam-6781	7	3	study	study	NOUN
ejpam-6781	7	4	of	of	ADP
ejpam-6781	7	5	positive	positive	ADJ
ejpam-6781	7	6	integer	integer	NOUN
ejpam-6781	7	7	solutions	solution	NOUN
ejpam-6781	7	8	to	to	ADP
ejpam-6781	7	9	the	the	DET
ejpam-6781	7	10	equation	equation	NOUN
ejpam-6781	7	11	a2	a2	PROPN
ejpam-6781	7	12	+	+	CCONJ
ejpam-6781	7	13	b2	b2	NOUN
ejpam-6781	7	14	=	=	SYM
ejpam-6781	7	15	c2	c2	PROPN
ejpam-6781	7	16	,	,	PUNCT
ejpam-6781	7	17	known	know	VERB
ejpam-6781	7	18	as	as	ADP
ejpam-6781	7	19	pythagorean	pythagorean	PROPN
ejpam-6781	7	20	triples	triple	NOUN
ejpam-6781	7	21	,	,	PUNCT
ejpam-6781	7	22	has	have	VERB
ejpam-6781	7	23	fascinated	fascinated	ADJ
ejpam-6781	7	24	mathematicians	mathematician	NOUN
ejpam-6781	7	25	for	for	ADP
ejpam-6781	7	26	centuries	century	NOUN
ejpam-6781	7	27	.	.	PUNCT
ejpam-6781	8	1	equivalently	equivalently	ADV
ejpam-6781	8	2	,	,	PUNCT
ejpam-6781	8	3	the	the	DET
ejpam-6781	8	4	integers	integer	NOUN
ejpam-6781	8	5	a	a	PRON
ejpam-6781	8	6	,	,	PUNCT
ejpam-6781	8	7	b	b	NOUN
ejpam-6781	8	8	and	and	CCONJ
ejpam-6781	8	9	c	c	PROPN
ejpam-6781	8	10	are	be	AUX
ejpam-6781	8	11	the	the	DET
ejpam-6781	8	12	lengths	length	NOUN
ejpam-6781	8	13	of	of	ADP
ejpam-6781	8	14	a	a	DET
ejpam-6781	8	15	right	right	ADJ
ejpam-6781	8	16	-	-	PUNCT
ejpam-6781	8	17	angled	angle	VERB
ejpam-6781	8	18	triangle	triangle	NOUN
ejpam-6781	8	19	.	.	PUNCT
ejpam-6781	9	1	these	these	DET
ejpam-6781	9	2	triples	triple	NOUN
ejpam-6781	9	3	,	,	PUNCT
ejpam-6781	9	4	characterized	characterize	VERB
ejpam-6781	9	5	by	by	ADP
ejpam-6781	9	6	positive	positive	ADJ
ejpam-6781	9	7	integers	integer	NOUN
ejpam-6781	9	8	a	a	DET
ejpam-6781	9	9	,	,	PUNCT
ejpam-6781	9	10	b	b	NOUN
ejpam-6781	9	11	and	and	CCONJ
ejpam-6781	9	12	c	c	X
ejpam-6781	9	13	satisfying	satisfy	VERB
ejpam-6781	9	14	the	the	DET
ejpam-6781	9	15	equation	equation	NOUN
ejpam-6781	9	16	,	,	PUNCT
ejpam-6781	9	17	serve	serve	VERB
ejpam-6781	9	18	as	as	ADP
ejpam-6781	9	19	a	a	DET
ejpam-6781	9	20	cornerstone	cornerstone	NOUN
ejpam-6781	9	21	of	of	ADP
ejpam-6781	9	22	number	number	NOUN
ejpam-6781	9	23	theory	theory	NOUN
ejpam-6781	9	24	and	and	CCONJ
ejpam-6781	9	25	geometry	geometry	NOUN
ejpam-6781	9	26	.	.	PUNCT
ejpam-6781	10	1	beyond	beyond	ADP
ejpam-6781	10	2	their	their	PRON
ejpam-6781	10	3	classical	classical	ADJ
ejpam-6781	10	4	role	role	NOUN
ejpam-6781	10	5	in	in	ADP
ejpam-6781	10	6	mathematics	mathematic	NOUN
ejpam-6781	10	7	,	,	PUNCT
ejpam-6781	10	8	pythagorean	pythagorean	PROPN
ejpam-6781	10	9	triples	triple	NOUN
ejpam-6781	10	10	have	have	VERB
ejpam-6781	10	11	deep	deep	ADJ
ejpam-6781	10	12	connections	connection	NOUN
ejpam-6781	10	13	to	to	ADP
ejpam-6781	10	14	algebraic	algebraic	ADJ
ejpam-6781	10	15	structures	structure	NOUN
ejpam-6781	10	16	,	,	PUNCT
ejpam-6781	10	17	modular	modular	ADJ
ejpam-6781	10	18	forms	form	NOUN
ejpam-6781	10	19	,	,	PUNCT
ejpam-6781	10	20	and	and	CCONJ
ejpam-6781	10	21	applications	application	NOUN
ejpam-6781	10	22	in	in	ADP
ejpam-6781	10	23	modern	modern	ADJ
ejpam-6781	10	24	cryptography	cryptography	NOUN
ejpam-6781	10	25	and	and	CCONJ
ejpam-6781	10	26	coding	code	VERB
ejpam-6781	10	27	theory	theory	NOUN
ejpam-6781	10	28	.	.	PUNCT
ejpam-6781	11	1	in	in	ADP
ejpam-6781	11	2	[	[	X
ejpam-6781	11	3	1	1	NUM
ejpam-6781	11	4	]	]	PUNCT
ejpam-6781	11	5	,	,	PUNCT
ejpam-6781	11	6	w.	w.	PROPN
ejpam-6781	11	7	sierpinski	sierpinski	PROPN
ejpam-6781	11	8	remarked	remark	VERB
ejpam-6781	11	9	that	that	SCONJ
ejpam-6781	11	10	“	"	PUNCT
ejpam-6781	11	11	it	it	PRON
ejpam-6781	11	12	would	would	AUX
ejpam-6781	11	13	be	be	AUX
ejpam-6781	11	14	more	more	ADV
ejpam-6781	11	15	difficult	difficult	ADJ
ejpam-6781	11	16	to	to	PART
ejpam-6781	11	17	prove	prove	VERB
ejpam-6781	11	18	the	the	DET
ejpam-6781	11	19	existence	existence	NOUN
ejpam-6781	11	20	of	of	ADP
ejpam-6781	11	21	an	an	DET
ejpam-6781	11	22	arbitrary	arbitrary	ADJ
ejpam-6781	11	23	number	number	NOUN
ejpam-6781	11	24	of	of	ADP
ejpam-6781	11	25	primitive	primitive	ADJ
ejpam-6781	11	26	pythagorean	pythagorean	ADJ
ejpam-6781	11	27	triples	triple	NOUN
ejpam-6781	11	28	with	with	ADP
ejpam-6781	11	29	the	the	DET
ejpam-6781	11	30	same	same	ADJ
ejpam-6781	11	31	hypotenuse	hypotenuse	NOUN
ejpam-6781	11	32	.	.	PUNCT
ejpam-6781	11	33	”	"	PUNCT
ejpam-6781	12	1	∗corresponding	∗corresponde	VERB
ejpam-6781	12	2	author	author	NOUN
ejpam-6781	12	3	.	.	PUNCT
ejpam-6781	13	1	doi	doi	NOUN
ejpam-6781	13	2	:	:	PUNCT
ejpam-6781	13	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6781	https://doi.org/10.29020/nybg.ejpam.v18i4.6781	ADP
ejpam-6781	13	4	email	email	NOUN
ejpam-6781	13	5	addresses	address	VERB
ejpam-6781	13	6	:	:	PUNCT
ejpam-6781	14	1	sjitman@gmail.com	sjitman@gmail.com	PROPN
ejpam-6781	14	2	(	(	PUNCT
ejpam-6781	14	3	s.	s.	PROPN
ejpam-6781	14	4	jitman	jitman	PROPN
ejpam-6781	14	5	)	)	PUNCT
ejpam-6781	14	6	,	,	PUNCT
ejpam-6781	14	7	mohadsham@umpsa.edu.my	mohadsham@umpsa.edu.my	NOUN
ejpam-6781	14	8	(	(	PUNCT
ejpam-6781	14	9	m.	m.	NOUN
ejpam-6781	14	10	sham	sham	PROPN
ejpam-6781	14	11	)	)	PUNCT
ejpam-6781	14	12	,	,	PUNCT
ejpam-6781	14	13	ekkasit@tsu.ac.th	ekkasit@tsu.ac.th	PROPN
ejpam-6781	14	14	(	(	PUNCT
ejpam-6781	14	15	e.	e.	PROPN
ejpam-6781	14	16	sangwisut	sangwisut	PROPN
ejpam-6781	14	17	)	)	PUNCT
ejpam-6781	14	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6781	15	1	1	1	NUM
ejpam-6781	15	2	copyright	copyright	NOUN
ejpam-6781	15	3	:	:	PUNCT
ejpam-6781	15	4	©	©	PROPN
ejpam-6781	15	5	2025	2025	NUM
ejpam-6781	15	6	the	the	DET
ejpam-6781	15	7	author(s	author(s	NOUN
ejpam-6781	15	8	)	)	PUNCT
ejpam-6781	15	9	.	.	PUNCT
ejpam-6781	16	1	(	(	PUNCT
ejpam-6781	16	2	cc	cc	NOUN
ejpam-6781	16	3	by	by	ADP
ejpam-6781	16	4	-	-	PUNCT
ejpam-6781	16	5	nc	nc	PROPN
ejpam-6781	16	6	4.0	4.0	NUM
ejpam-6781	16	7	)	)	PUNCT
ejpam-6781	16	8	s.	s.	PROPN
ejpam-6781	16	9	jitman	jitman	PROPN
ejpam-6781	16	10	,	,	PUNCT
ejpam-6781	16	11	m.	m.	NOUN
ejpam-6781	16	12	mohammad	mohammad	PROPN
ejpam-6781	16	13	,	,	PUNCT
ejpam-6781	16	14	e.	e.	PROPN
ejpam-6781	16	15	sangwisut	sangwisut	PROPN
ejpam-6781	16	16	/	/	SYM
ejpam-6781	16	17	eur	eur	PROPN
ejpam-6781	16	18	.	.	PUNCT
ejpam-6781	17	1	j.	j.	PROPN
ejpam-6781	17	2	pure	pure	PROPN
ejpam-6781	17	3	appl	appl	PROPN
ejpam-6781	17	4	.	.	PROPN
ejpam-6781	17	5	math	math	PROPN
ejpam-6781	17	6	,	,	PUNCT
ejpam-6781	17	7	18	18	NUM
ejpam-6781	17	8	(	(	PUNCT
ejpam-6781	17	9	4	4	NUM
ejpam-6781	17	10	)	)	PUNCT
ejpam-6781	17	11	(	(	PUNCT
ejpam-6781	17	12	2025	2025	NUM
ejpam-6781	17	13	)	)	PUNCT
ejpam-6781	17	14	,	,	PUNCT
ejpam-6781	17	15	6781	6781	NUM
ejpam-6781	17	16	2	2	NUM
ejpam-6781	17	17	of	of	ADP
ejpam-6781	17	18	16	16	NUM
ejpam-6781	17	19	subsequently	subsequently	ADV
ejpam-6781	17	20	,	,	PUNCT
ejpam-6781	17	21	in	in	ADP
ejpam-6781	17	22	[	[	PUNCT
ejpam-6781	17	23	2	2	NUM
ejpam-6781	17	24	]	]	PUNCT
ejpam-6781	17	25	,	,	PUNCT
ejpam-6781	17	26	ch	ch	NOUN
ejpam-6781	17	27	.	.	PUNCT
ejpam-6781	17	28	l.	l.	PROPN
ejpam-6781	17	29	shedd	shedd	PROPN
ejpam-6781	17	30	demonstrated	demonstrate	VERB
ejpam-6781	17	31	that	that	SCONJ
ejpam-6781	17	32	there	there	PRON
ejpam-6781	17	33	are	be	VERB
ejpam-6781	17	34	precisely	precisely	ADV
ejpam-6781	17	35	64	64	NUM
ejpam-6781	17	36	primitive	primitive	ADJ
ejpam-6781	17	37	pythagorean	pythagorean	ADJ
ejpam-6781	17	38	triples	triple	NOUN
ejpam-6781	17	39	with	with	ADP
ejpam-6781	17	40	the	the	DET
ejpam-6781	17	41	hypotenuse	hypotenuse	NOUN
ejpam-6781	17	42	c	c	NOUN
ejpam-6781	17	43	=	=	SYM
ejpam-6781	17	44	2	2	NUM
ejpam-6781	17	45	,	,	PUNCT
ejpam-6781	17	46	576	576	NUM
ejpam-6781	17	47	,	,	PUNCT
ejpam-6781	17	48	450	450	NUM
ejpam-6781	17	49	,	,	PUNCT
ejpam-6781	17	50	045	045	NUM
ejpam-6781	17	51	=	=	SYM
ejpam-6781	17	52	5	5	NUM
ejpam-6781	17	53	·	·	SYM
ejpam-6781	17	54	13	13	NUM
ejpam-6781	17	55	·	·	SYM
ejpam-6781	17	56	17	17	NUM
ejpam-6781	17	57	·	·	SYM
ejpam-6781	17	58	29	29	NUM
ejpam-6781	17	59	·	·	SYM
ejpam-6781	17	60	37	37	NUM
ejpam-6781	17	61	·	·	SYM
ejpam-6781	17	62	41	41	NUM
ejpam-6781	17	63	·	·	PUNCT
ejpam-6781	17	64	53	53	NUM
ejpam-6781	17	65	.	.	X
ejpam-6781	17	66	twenty	twenty	NUM
ejpam-6781	17	67	-	-	PUNCT
ejpam-6781	17	68	two	two	NUM
ejpam-6781	17	69	years	year	NOUN
ejpam-6781	17	70	later	later	ADV
ejpam-6781	17	71	,	,	PUNCT
ejpam-6781	17	72	this	this	DET
ejpam-6781	17	73	question	question	NOUN
ejpam-6781	17	74	was	be	AUX
ejpam-6781	17	75	revisited	revisit	VERB
ejpam-6781	17	76	by	by	ADP
ejpam-6781	17	77	e.	e.	PROPN
ejpam-6781	17	78	j.	j.	PROPN
ejpam-6781	17	79	eckert	eckert	PROPN
ejpam-6781	17	80	in	in	ADP
ejpam-6781	17	81	[	[	X
ejpam-6781	17	82	3	3	NUM
ejpam-6781	17	83	]	]	PUNCT
ejpam-6781	17	84	,	,	PUNCT
ejpam-6781	17	85	the	the	DET
ejpam-6781	17	86	group	group	NOUN
ejpam-6781	17	87	structure	structure	NOUN
ejpam-6781	17	88	of	of	ADP
ejpam-6781	17	89	the	the	DET
ejpam-6781	17	90	set	set	NOUN
ejpam-6781	17	91	of	of	ADP
ejpam-6781	17	92	primitive	primitive	ADJ
ejpam-6781	17	93	pythagorean	pythagorean	NOUN
ejpam-6781	17	94	triples	triple	NOUN
ejpam-6781	17	95	was	be	AUX
ejpam-6781	17	96	investigated	investigate	VERB
ejpam-6781	17	97	.	.	PUNCT
ejpam-6781	18	1	eckert	eckert	PROPN
ejpam-6781	18	2	provided	provide	VERB
ejpam-6781	18	3	necessary	necessary	ADJ
ejpam-6781	18	4	and	and	CCONJ
ejpam-6781	18	5	sufficient	sufficient	ADJ
ejpam-6781	18	6	conditions	condition	NOUN
ejpam-6781	18	7	for	for	ADP
ejpam-6781	18	8	the	the	DET
ejpam-6781	18	9	existence	existence	NOUN
ejpam-6781	18	10	of	of	ADP
ejpam-6781	18	11	primitive	primitive	ADJ
ejpam-6781	18	12	pythagorean	pythagorean	ADJ
ejpam-6781	18	13	triples	triple	NOUN
ejpam-6781	18	14	with	with	ADP
ejpam-6781	18	15	a	a	DET
ejpam-6781	18	16	given	give	VERB
ejpam-6781	18	17	hypotenuse	hypotenuse	NOUN
ejpam-6781	18	18	.	.	PUNCT
ejpam-6781	19	1	alternatively	alternatively	ADV
ejpam-6781	19	2	,	,	PUNCT
ejpam-6781	19	3	the	the	DET
ejpam-6781	19	4	set	set	NOUN
ejpam-6781	19	5	of	of	ADP
ejpam-6781	19	6	primitive	primitive	ADJ
ejpam-6781	19	7	pythagorean	pythagorean	NOUN
ejpam-6781	19	8	triples	triple	NOUN
ejpam-6781	19	9	can	can	AUX
ejpam-6781	19	10	be	be	AUX
ejpam-6781	19	11	identified	identify	VERB
ejpam-6781	19	12	with	with	ADP
ejpam-6781	19	13	the	the	DET
ejpam-6781	19	14	group	group	NOUN
ejpam-6781	19	15	of	of	ADP
ejpam-6781	19	16	rational	rational	ADJ
ejpam-6781	19	17	points	point	NOUN
ejpam-6781	19	18	on	on	ADP
ejpam-6781	19	19	the	the	DET
ejpam-6781	19	20	unit	unit	NOUN
ejpam-6781	19	21	circle	circle	NOUN
ejpam-6781	19	22	.	.	PUNCT
ejpam-6781	20	1	as	as	SCONJ
ejpam-6781	20	2	discussed	discuss	VERB
ejpam-6781	20	3	in	in	ADP
ejpam-6781	20	4	[	[	X
ejpam-6781	20	5	4	4	NUM
ejpam-6781	20	6	,	,	PUNCT
ejpam-6781	20	7	5	5	NUM
ejpam-6781	20	8	]	]	PUNCT
ejpam-6781	20	9	,	,	PUNCT
ejpam-6781	20	10	this	this	DET
ejpam-6781	20	11	group	group	NOUN
ejpam-6781	20	12	can	can	AUX
ejpam-6781	20	13	be	be	AUX
ejpam-6781	20	14	decomposed	decompose	VERB
ejpam-6781	20	15	into	into	ADP
ejpam-6781	20	16	a	a	DET
ejpam-6781	20	17	direct	direct	ADJ
ejpam-6781	20	18	sum	sum	NOUN
ejpam-6781	20	19	of	of	ADP
ejpam-6781	20	20	the	the	DET
ejpam-6781	20	21	unit	unit	NOUN
ejpam-6781	20	22	group	group	NOUN
ejpam-6781	20	23	of	of	ADP
ejpam-6781	20	24	the	the	DET
ejpam-6781	20	25	group	group	NOUN
ejpam-6781	20	26	of	of	ADP
ejpam-6781	20	27	rational	rational	ADJ
ejpam-6781	20	28	points	point	NOUN
ejpam-6781	20	29	on	on	ADP
ejpam-6781	20	30	the	the	DET
ejpam-6781	20	31	unit	unit	NOUN
ejpam-6781	20	32	circle	circle	NOUN
ejpam-6781	20	33	and	and	CCONJ
ejpam-6781	20	34	a	a	DET
ejpam-6781	20	35	free	free	ADJ
ejpam-6781	20	36	abelian	abelian	ADJ
ejpam-6781	20	37	group	group	NOUN
ejpam-6781	20	38	.	.	PUNCT
ejpam-6781	21	1	this	this	DET
ejpam-6781	21	2	group	group	NOUN
ejpam-6781	21	3	-	-	PUNCT
ejpam-6781	21	4	theoretic	theoretic	NOUN
ejpam-6781	21	5	framework	framework	NOUN
ejpam-6781	21	6	enables	enable	VERB
ejpam-6781	21	7	both	both	CCONJ
ejpam-6781	21	8	the	the	DET
ejpam-6781	21	9	characterization	characterization	NOUN
ejpam-6781	21	10	and	and	CCONJ
ejpam-6781	21	11	enumeration	enumeration	NOUN
ejpam-6781	21	12	of	of	ADP
ejpam-6781	21	13	primitive	primitive	ADJ
ejpam-6781	21	14	pythagorean	pythagorean	ADJ
ejpam-6781	21	15	triples	triple	NOUN
ejpam-6781	21	16	with	with	ADP
ejpam-6781	21	17	a	a	DET
ejpam-6781	21	18	given	give	VERB
ejpam-6781	21	19	hypotenuse	hypotenuse	NOUN
ejpam-6781	21	20	(	(	PUNCT
ejpam-6781	21	21	see	see	VERB
ejpam-6781	21	22	[	[	X
ejpam-6781	21	23	5	5	NUM
ejpam-6781	21	24	]	]	NUM
ejpam-6781	21	25	)	)	PUNCT
ejpam-6781	21	26	.	.	PUNCT
ejpam-6781	22	1	in	in	ADP
ejpam-6781	22	2	a	a	DET
ejpam-6781	22	3	similar	similar	ADJ
ejpam-6781	22	4	manner	manner	NOUN
ejpam-6781	22	5	,	,	PUNCT
ejpam-6781	22	6	eisenstein	eisenstein	NOUN
ejpam-6781	22	7	triples	triple	NOUN
ejpam-6781	22	8	(	(	PUNCT
ejpam-6781	22	9	60	60	NUM
ejpam-6781	22	10	-	-	PUNCT
ejpam-6781	22	11	degree	degree	NOUN
ejpam-6781	22	12	triples	triple	NOUN
ejpam-6781	22	13	)	)	PUNCT
ejpam-6781	22	14	arise	arise	NOUN
ejpam-6781	22	15	from	from	ADP
ejpam-6781	22	16	the	the	DET
ejpam-6781	22	17	analogous	analogous	ADJ
ejpam-6781	22	18	equation	equation	NOUN
ejpam-6781	22	19	a2	a2	PROPN
ejpam-6781	22	20	−	−	PROPN
ejpam-6781	22	21	ab	ab	PROPN
ejpam-6781	22	22	+	+	CCONJ
ejpam-6781	22	23	b2	b2	NOUN
ejpam-6781	22	24	=	=	PROPN
ejpam-6781	22	25	c2	c2	PROPN
ejpam-6781	22	26	.	.	PUNCT
ejpam-6781	23	1	these	these	DET
ejpam-6781	23	2	triples	triple	NOUN
ejpam-6781	23	3	correspond	correspond	VERB
ejpam-6781	23	4	to	to	ADP
ejpam-6781	23	5	triangles	triangle	NOUN
ejpam-6781	23	6	with	with	ADP
ejpam-6781	23	7	sides	side	NOUN
ejpam-6781	23	8	of	of	ADP
ejpam-6781	23	9	integer	integer	NOUN
ejpam-6781	23	10	lengths	length	NOUN
ejpam-6781	23	11	a	a	DET
ejpam-6781	23	12	,	,	PUNCT
ejpam-6781	23	13	b	b	NOUN
ejpam-6781	23	14	,	,	PUNCT
ejpam-6781	23	15	and	and	CCONJ
ejpam-6781	23	16	c	c	X
ejpam-6781	23	17	,	,	PUNCT
ejpam-6781	23	18	where	where	SCONJ
ejpam-6781	23	19	the	the	DET
ejpam-6781	23	20	angle	angle	NOUN
ejpam-6781	23	21	opposite	opposite	ADP
ejpam-6781	23	22	the	the	DET
ejpam-6781	23	23	side	side	NOUN
ejpam-6781	23	24	of	of	ADP
ejpam-6781	23	25	length	length	NOUN
ejpam-6781	23	26	c	c	PROPN
ejpam-6781	23	27	is	be	AUX
ejpam-6781	23	28	60	60	NUM
ejpam-6781	23	29	◦	◦	NOUN
ejpam-6781	23	30	.	.	PUNCT
ejpam-6781	24	1	a	a	DET
ejpam-6781	24	2	general	general	ADJ
ejpam-6781	24	3	method	method	NOUN
ejpam-6781	24	4	for	for	ADP
ejpam-6781	24	5	constructing	construct	VERB
ejpam-6781	24	6	all	all	DET
ejpam-6781	24	7	such	such	ADJ
ejpam-6781	24	8	triples	triple	NOUN
ejpam-6781	24	9	have	have	AUX
ejpam-6781	24	10	been	be	AUX
ejpam-6781	24	11	established	establish	VERB
ejpam-6781	24	12	(	(	PUNCT
ejpam-6781	24	13	see	see	VERB
ejpam-6781	24	14	[	[	X
ejpam-6781	24	15	6	6	NUM
ejpam-6781	24	16	]	]	NUM
ejpam-6781	24	17	)	)	PUNCT
ejpam-6781	24	18	.	.	PUNCT
ejpam-6781	25	1	previous	previous	ADJ
ejpam-6781	25	2	studies	study	NOUN
ejpam-6781	25	3	,	,	PUNCT
ejpam-6781	25	4	including	include	VERB
ejpam-6781	25	5	[	[	X
ejpam-6781	25	6	7	7	NUM
ejpam-6781	25	7	]	]	PUNCT
ejpam-6781	25	8	,	,	PUNCT
ejpam-6781	25	9	[	[	X
ejpam-6781	25	10	8	8	NUM
ejpam-6781	25	11	]	]	PUNCT
ejpam-6781	25	12	,	,	PUNCT
ejpam-6781	25	13	and	and	CCONJ
ejpam-6781	25	14	[	[	X
ejpam-6781	25	15	9	9	NUM
ejpam-6781	25	16	]	]	PUNCT
ejpam-6781	25	17	,	,	PUNCT
ejpam-6781	25	18	have	have	AUX
ejpam-6781	25	19	explored	explore	VERB
ejpam-6781	25	20	parametric	parametric	ADJ
ejpam-6781	25	21	equations	equation	NOUN
ejpam-6781	25	22	and	and	CCONJ
ejpam-6781	25	23	have	have	AUX
ejpam-6781	25	24	established	establish	VERB
ejpam-6781	25	25	relationships	relationship	NOUN
ejpam-6781	25	26	between	between	ADP
ejpam-6781	25	27	120	120	NUM
ejpam-6781	25	28	-	-	PUNCT
ejpam-6781	25	29	degree	degree	NOUN
ejpam-6781	25	30	triples	triple	NOUN
ejpam-6781	25	31	and	and	CCONJ
ejpam-6781	25	32	60	60	NUM
ejpam-6781	25	33	-	-	PUNCT
ejpam-6781	25	34	degree	degree	NOUN
ejpam-6781	25	35	triples	triple	NOUN
ejpam-6781	25	36	.	.	PUNCT
ejpam-6781	26	1	fundamental	fundamental	ADJ
ejpam-6781	26	2	properties	property	NOUN
ejpam-6781	26	3	of	of	ADP
ejpam-6781	26	4	eisenstein	eisenstein	NOUN
ejpam-6781	26	5	triples	triple	NOUN
ejpam-6781	26	6	have	have	AUX
ejpam-6781	26	7	been	be	AUX
ejpam-6781	26	8	presented	present	VERB
ejpam-6781	26	9	in	in	ADP
ejpam-6781	26	10	[	[	X
ejpam-6781	26	11	10	10	NUM
ejpam-6781	26	12	]	]	PUNCT
ejpam-6781	26	13	.	.	PUNCT
ejpam-6781	27	1	this	this	DET
ejpam-6781	27	2	paper	paper	NOUN
ejpam-6781	27	3	focuses	focus	VERB
ejpam-6781	27	4	on	on	ADP
ejpam-6781	27	5	the	the	DET
ejpam-6781	27	6	characterization	characterization	NOUN
ejpam-6781	27	7	and	and	CCONJ
ejpam-6781	27	8	enumeration	enumeration	NOUN
ejpam-6781	27	9	of	of	ADP
ejpam-6781	27	10	primitive	primitive	ADJ
ejpam-6781	27	11	eisenstein	eisenstein	NOUN
ejpam-6781	27	12	triples	triple	NOUN
ejpam-6781	27	13	for	for	ADP
ejpam-6781	27	14	a	a	DET
ejpam-6781	27	15	fixed	fix	VERB
ejpam-6781	27	16	hypotenuse	hypotenuse	NOUN
ejpam-6781	27	17	.	.	PUNCT
ejpam-6781	28	1	in	in	ADP
ejpam-6781	28	2	particular	particular	ADJ
ejpam-6781	28	3	,	,	PUNCT
ejpam-6781	28	4	the	the	DET
ejpam-6781	28	5	one	one	NUM
ejpam-6781	28	6	-	-	PUNCT
ejpam-6781	28	7	to	to	ADP
ejpam-6781	28	8	-	-	PUNCT
ejpam-6781	28	9	one	one	NUM
ejpam-6781	28	10	correspondence	correspondence	NOUN
ejpam-6781	28	11	between	between	ADP
ejpam-6781	28	12	these	these	DET
ejpam-6781	28	13	triples	triple	NOUN
ejpam-6781	28	14	and	and	CCONJ
ejpam-6781	28	15	the	the	DET
ejpam-6781	28	16	ω	ω	ADJ
ejpam-6781	28	17	-	-	ADJ
ejpam-6781	28	18	rational	rational	ADJ
ejpam-6781	28	19	points	point	NOUN
ejpam-6781	28	20	(	(	PUNCT
ejpam-6781	28	21	see	see	VERB
ejpam-6781	28	22	(	(	PUNCT
ejpam-6781	28	23	3	3	NUM
ejpam-6781	28	24	)	)	PUNCT
ejpam-6781	28	25	for	for	ADP
ejpam-6781	28	26	the	the	DET
ejpam-6781	28	27	definition	definition	NOUN
ejpam-6781	28	28	)	)	PUNCT
ejpam-6781	28	29	in	in	ADP
ejpam-6781	28	30	the	the	DET
ejpam-6781	28	31	second	second	ADJ
ejpam-6781	28	32	sextant	sextant	NOUN
ejpam-6781	28	33	of	of	ADP
ejpam-6781	28	34	the	the	DET
ejpam-6781	28	35	unit	unit	NOUN
ejpam-6781	28	36	circle	circle	NOUN
ejpam-6781	28	37	.	.	PUNCT
ejpam-6781	29	1	the	the	DET
ejpam-6781	29	2	paper	paper	NOUN
ejpam-6781	29	3	is	be	AUX
ejpam-6781	29	4	organized	organize	VERB
ejpam-6781	29	5	as	as	SCONJ
ejpam-6781	29	6	follows	follow	VERB
ejpam-6781	29	7	.	.	PUNCT
ejpam-6781	30	1	in	in	ADP
ejpam-6781	30	2	section	section	NOUN
ejpam-6781	30	3	2	2	NUM
ejpam-6781	30	4	,	,	PUNCT
ejpam-6781	30	5	the	the	DET
ejpam-6781	30	6	concept	concept	NOUN
ejpam-6781	30	7	and	and	CCONJ
ejpam-6781	30	8	basic	basic	ADJ
ejpam-6781	30	9	properties	property	NOUN
ejpam-6781	30	10	of	of	ADP
ejpam-6781	30	11	(	(	PUNCT
ejpam-6781	30	12	primitive	primitive	ADJ
ejpam-6781	30	13	)	)	PUNCT
ejpam-6781	30	14	eisenstein	eisenstein	NOUN
ejpam-6781	30	15	triples	triple	NOUN
ejpam-6781	30	16	are	be	AUX
ejpam-6781	30	17	recalled	recall	VERB
ejpam-6781	30	18	.	.	PUNCT
ejpam-6781	31	1	in	in	ADP
ejpam-6781	31	2	section	section	NOUN
ejpam-6781	31	3	3	3	NUM
ejpam-6781	31	4	,	,	PUNCT
ejpam-6781	31	5	the	the	DET
ejpam-6781	31	6	concept	concept	NOUN
ejpam-6781	31	7	of	of	ADP
ejpam-6781	31	8	the	the	DET
ejpam-6781	31	9	ω	ω	ADJ
ejpam-6781	31	10	-	-	ADJ
ejpam-6781	31	11	rational	rational	ADJ
ejpam-6781	31	12	unit	unit	NOUN
ejpam-6781	31	13	circle	circle	NOUN
ejpam-6781	31	14	is	be	AUX
ejpam-6781	31	15	introduced	introduce	VERB
ejpam-6781	31	16	together	together	ADV
ejpam-6781	31	17	with	with	ADP
ejpam-6781	31	18	a	a	DET
ejpam-6781	31	19	link	link	NOUN
ejpam-6781	31	20	between	between	ADP
ejpam-6781	31	21	eisenstein	eisenstein	NOUN
ejpam-6781	31	22	triples	triple	NOUN
ejpam-6781	31	23	and	and	CCONJ
ejpam-6781	31	24	points	point	NOUN
ejpam-6781	31	25	on	on	ADP
ejpam-6781	31	26	the	the	DET
ejpam-6781	31	27	ω	ω	ADJ
ejpam-6781	31	28	-	-	ADJ
ejpam-6781	31	29	rational	rational	ADJ
ejpam-6781	31	30	unit	unit	NOUN
ejpam-6781	31	31	circle	circle	NOUN
ejpam-6781	31	32	.	.	PUNCT
ejpam-6781	32	1	the	the	DET
ejpam-6781	32	2	group	group	NOUN
ejpam-6781	32	3	structure	structure	NOUN
ejpam-6781	32	4	of	of	ADP
ejpam-6781	32	5	the	the	DET
ejpam-6781	32	6	ω	ω	ADJ
ejpam-6781	32	7	-	-	ADJ
ejpam-6781	32	8	rational	rational	ADJ
ejpam-6781	32	9	unit	unit	NOUN
ejpam-6781	32	10	circle	circle	NOUN
ejpam-6781	32	11	is	be	AUX
ejpam-6781	32	12	presented	present	VERB
ejpam-6781	32	13	in	in	ADP
ejpam-6781	32	14	section	section	NOUN
ejpam-6781	32	15	4	4	NUM
ejpam-6781	32	16	.	.	PUNCT
ejpam-6781	32	17	based	base	VERB
ejpam-6781	32	18	on	on	ADP
ejpam-6781	32	19	this	this	DET
ejpam-6781	32	20	group	group	NOUN
ejpam-6781	32	21	structure	structure	NOUN
ejpam-6781	32	22	,	,	PUNCT
ejpam-6781	32	23	the	the	DET
ejpam-6781	32	24	characterization	characterization	NOUN
ejpam-6781	32	25	and	and	CCONJ
ejpam-6781	32	26	enumeration	enumeration	NOUN
ejpam-6781	32	27	of	of	ADP
ejpam-6781	32	28	primitive	primitive	ADJ
ejpam-6781	32	29	eisenstein	eisenstein	NOUN
ejpam-6781	32	30	triples	triple	NOUN
ejpam-6781	32	31	with	with	ADP
ejpam-6781	32	32	a	a	DET
ejpam-6781	32	33	fixed	fix	VERB
ejpam-6781	32	34	hypotenuse	hypotenuse	NOUN
ejpam-6781	32	35	are	be	AUX
ejpam-6781	32	36	established	establish	VERB
ejpam-6781	32	37	in	in	ADP
ejpam-6781	32	38	section	section	NOUN
ejpam-6781	32	39	5	5	NUM
ejpam-6781	32	40	.	.	PUNCT
ejpam-6781	33	1	the	the	DET
ejpam-6781	33	2	summary	summary	NOUN
ejpam-6781	33	3	is	be	AUX
ejpam-6781	33	4	given	give	VERB
ejpam-6781	33	5	in	in	ADP
ejpam-6781	33	6	section	section	NOUN
ejpam-6781	33	7	6	6	NUM
ejpam-6781	33	8	.	.	NOUN
ejpam-6781	33	9	2	2	NUM
ejpam-6781	33	10	.	.	X
ejpam-6781	33	11	eisenstein	eisenstein	NOUN
ejpam-6781	33	12	triples	triple	NOUN
ejpam-6781	33	13	in	in	ADP
ejpam-6781	33	14	this	this	DET
ejpam-6781	33	15	section	section	NOUN
ejpam-6781	33	16	,	,	PUNCT
ejpam-6781	33	17	some	some	DET
ejpam-6781	33	18	properties	property	NOUN
ejpam-6781	33	19	and	and	CCONJ
ejpam-6781	33	20	geometric	geometric	ADJ
ejpam-6781	33	21	interpretation	interpretation	NOUN
ejpam-6781	33	22	of	of	ADP
ejpam-6781	33	23	eisenstein	eisenstein	PROPN
ejpam-6781	33	24	triples	triple	NOUN
ejpam-6781	33	25	are	be	AUX
ejpam-6781	33	26	recalled	recall	VERB
ejpam-6781	33	27	in	in	ADP
ejpam-6781	33	28	terms	term	NOUN
ejpam-6781	33	29	of	of	ADP
ejpam-6781	33	30	triangles	triangle	NOUN
ejpam-6781	33	31	with	with	ADP
ejpam-6781	33	32	a	a	DET
ejpam-6781	33	33	60	60	NUM
ejpam-6781	33	34	◦	◦	NOUN
ejpam-6781	33	35	angle	angle	NOUN
ejpam-6781	33	36	.	.	PUNCT
ejpam-6781	34	1	key	key	ADJ
ejpam-6781	34	2	results	result	NOUN
ejpam-6781	34	3	from	from	ADP
ejpam-6781	34	4	[	[	X
ejpam-6781	34	5	10	10	NUM
ejpam-6781	34	6	]	]	PUNCT
ejpam-6781	34	7	concerning	concern	VERB
ejpam-6781	34	8	the	the	DET
ejpam-6781	34	9	classification	classification	NOUN
ejpam-6781	34	10	of	of	ADP
ejpam-6781	34	11	these	these	DET
ejpam-6781	34	12	triples	triple	NOUN
ejpam-6781	34	13	and	and	CCONJ
ejpam-6781	34	14	the	the	DET
ejpam-6781	34	15	constraints	constraint	NOUN
ejpam-6781	34	16	on	on	ADP
ejpam-6781	34	17	the	the	DET
ejpam-6781	34	18	values	value	NOUN
ejpam-6781	34	19	of	of	ADP
ejpam-6781	34	20	c	c	PROPN
ejpam-6781	34	21	are	be	AUX
ejpam-6781	34	22	presented	present	VERB
ejpam-6781	34	23	.	.	PUNCT
ejpam-6781	35	1	in	in	ADP
ejpam-6781	35	2	addition	addition	NOUN
ejpam-6781	35	3	,	,	PUNCT
ejpam-6781	35	4	a	a	DET
ejpam-6781	35	5	useful	useful	ADJ
ejpam-6781	35	6	partition	partition	NOUN
ejpam-6781	35	7	of	of	ADP
ejpam-6781	35	8	the	the	DET
ejpam-6781	35	9	set	set	NOUN
ejpam-6781	35	10	of	of	ADP
ejpam-6781	35	11	all	all	DET
ejpam-6781	35	12	primitive	primitive	ADJ
ejpam-6781	35	13	eisenstein	eisenstein	NOUN
ejpam-6781	35	14	triples	triple	NOUN
ejpam-6781	35	15	is	be	AUX
ejpam-6781	35	16	introduced	introduce	VERB
ejpam-6781	35	17	.	.	PUNCT
ejpam-6781	36	1	an	an	DET
ejpam-6781	36	2	eisenstein	eisenstein	NOUN
ejpam-6781	36	3	triple	triple	NOUN
ejpam-6781	36	4	is	be	AUX
ejpam-6781	36	5	a	a	DET
ejpam-6781	36	6	triple	triple	ADJ
ejpam-6781	36	7	(	(	PUNCT
ejpam-6781	36	8	a	a	DET
ejpam-6781	36	9	,	,	PUNCT
ejpam-6781	36	10	b	b	NOUN
ejpam-6781	36	11	,	,	PUNCT
ejpam-6781	36	12	c	c	NOUN
ejpam-6781	36	13	)	)	PUNCT
ejpam-6781	36	14	of	of	ADP
ejpam-6781	36	15	positive	positive	ADJ
ejpam-6781	36	16	integers	integer	NOUN
ejpam-6781	36	17	with	with	ADP
ejpam-6781	36	18	a	a	DET
ejpam-6781	36	19	<	<	X
ejpam-6781	36	20	c	c	X
ejpam-6781	36	21	<	<	X
ejpam-6781	36	22	b	b	X
ejpam-6781	36	23	,	,	PUNCT
ejpam-6781	36	24	that	that	PRON
ejpam-6781	36	25	satisfies	satisfy	VERB
ejpam-6781	36	26	the	the	DET
ejpam-6781	36	27	equation	equation	NOUN
ejpam-6781	36	28	a2	a2	PROPN
ejpam-6781	36	29	−	−	PROPN
ejpam-6781	36	30	ab	ab	PROPN
ejpam-6781	36	31	+	+	CCONJ
ejpam-6781	36	32	b2	b2	NOUN
ejpam-6781	36	33	=	=	PROPN
ejpam-6781	36	34	c2	c2	PROPN
ejpam-6781	36	35	.	.	PUNCT
ejpam-6781	37	1	(	(	PUNCT
ejpam-6781	37	2	1	1	X
ejpam-6781	37	3	)	)	PUNCT
ejpam-6781	37	4	the	the	DET
ejpam-6781	37	5	eisenstein	eisenstein	PROPN
ejpam-6781	37	6	triple	triple	NOUN
ejpam-6781	37	7	is	be	AUX
ejpam-6781	37	8	said	say	VERB
ejpam-6781	37	9	to	to	PART
ejpam-6781	37	10	be	be	AUX
ejpam-6781	37	11	primitive	primitive	ADJ
ejpam-6781	37	12	if	if	SCONJ
ejpam-6781	37	13	gcd(a	gcd(a	PROPN
ejpam-6781	37	14	,	,	PUNCT
ejpam-6781	37	15	b	b	PROPN
ejpam-6781	37	16	,	,	PUNCT
ejpam-6781	37	17	c	c	NOUN
ejpam-6781	37	18	)	)	PUNCT
ejpam-6781	37	19	=	=	SYM
ejpam-6781	38	1	1	1	X
ejpam-6781	38	2	.	.	PUNCT
ejpam-6781	39	1	the	the	DET
ejpam-6781	39	2	law	law	NOUN
ejpam-6781	39	3	of	of	ADP
ejpam-6781	39	4	cosines	cosine	NOUN
ejpam-6781	39	5	states	state	VERB
ejpam-6781	39	6	that	that	SCONJ
ejpam-6781	39	7	for	for	ADP
ejpam-6781	39	8	any	any	DET
ejpam-6781	39	9	triangle	triangle	NOUN
ejpam-6781	39	10	with	with	ADP
ejpam-6781	39	11	sides	side	NOUN
ejpam-6781	39	12	a	a	PRON
ejpam-6781	39	13	,	,	PUNCT
ejpam-6781	39	14	b	b	PROPN
ejpam-6781	39	15	and	and	CCONJ
ejpam-6781	39	16	c	c	NOUN
ejpam-6781	39	17	,	,	PUNCT
ejpam-6781	39	18	and	and	CCONJ
ejpam-6781	39	19	θ	θ	PROPN
ejpam-6781	39	20	is	be	AUX
ejpam-6781	39	21	the	the	DET
ejpam-6781	39	22	angle	angle	NOUN
ejpam-6781	39	23	θ	θ	PROPN
ejpam-6781	39	24	opposite	opposite	ADJ
ejpam-6781	39	25	side	side	NOUN
ejpam-6781	39	26	c	c	NOUN
ejpam-6781	39	27	,	,	PUNCT
ejpam-6781	39	28	the	the	DET
ejpam-6781	39	29	following	follow	VERB
ejpam-6781	39	30	equation	equation	NOUN
ejpam-6781	39	31	holds	hold	VERB
ejpam-6781	39	32	:	:	PUNCT
ejpam-6781	39	33	a2	a2	PROPN
ejpam-6781	39	34	−	−	PROPN
ejpam-6781	39	35	2ab	2ab	NOUN
ejpam-6781	39	36	cos	cos	PROPN
ejpam-6781	39	37	θ	θ	PROPN
ejpam-6781	39	38	+	+	CCONJ
ejpam-6781	39	39	b2	b2	NOUN
ejpam-6781	39	40	=	=	PROPN
ejpam-6781	39	41	c2	c2	PROPN
ejpam-6781	39	42	.	.	PUNCT
ejpam-6781	40	1	(	(	PUNCT
ejpam-6781	40	2	2	2	X
ejpam-6781	40	3	)	)	PUNCT
ejpam-6781	40	4	s.	s.	PROPN
ejpam-6781	40	5	jitman	jitman	PROPN
ejpam-6781	40	6	,	,	PUNCT
ejpam-6781	40	7	m.	m.	NOUN
ejpam-6781	40	8	mohammad	mohammad	PROPN
ejpam-6781	40	9	,	,	PUNCT
ejpam-6781	40	10	e.	e.	PROPN
ejpam-6781	40	11	sangwisut	sangwisut	PROPN
ejpam-6781	40	12	/	/	SYM
ejpam-6781	40	13	eur	eur	PROPN
ejpam-6781	40	14	.	.	PUNCT
ejpam-6781	41	1	j.	j.	PROPN
ejpam-6781	41	2	pure	pure	PROPN
ejpam-6781	41	3	appl	appl	PROPN
ejpam-6781	41	4	.	.	PROPN
ejpam-6781	41	5	math	math	PROPN
ejpam-6781	41	6	,	,	PUNCT
ejpam-6781	41	7	18	18	NUM
ejpam-6781	41	8	(	(	PUNCT
ejpam-6781	41	9	4	4	NUM
ejpam-6781	41	10	)	)	PUNCT
ejpam-6781	41	11	(	(	PUNCT
ejpam-6781	41	12	2025	2025	NUM
ejpam-6781	41	13	)	)	PUNCT
ejpam-6781	41	14	,	,	PUNCT
ejpam-6781	41	15	6781	6781	NUM
ejpam-6781	41	16	3	3	NUM
ejpam-6781	41	17	of	of	ADP
ejpam-6781	41	18	16	16	NUM
ejpam-6781	41	19	when	when	SCONJ
ejpam-6781	41	20	θ	θ	PROPN
ejpam-6781	41	21	is	be	AUX
ejpam-6781	41	22	60	60	NUM
ejpam-6781	41	23	◦	◦	NOUN
ejpam-6781	41	24	,	,	PUNCT
ejpam-6781	41	25	(	(	PUNCT
ejpam-6781	41	26	2	2	X
ejpam-6781	41	27	)	)	PUNCT
ejpam-6781	41	28	simplifies	simplifie	NOUN
ejpam-6781	41	29	the	the	DET
ejpam-6781	41	30	form	form	NOUN
ejpam-6781	41	31	given	give	VERB
ejpam-6781	41	32	in	in	ADP
ejpam-6781	41	33	(	(	PUNCT
ejpam-6781	41	34	1	1	NUM
ejpam-6781	41	35	)	)	PUNCT
ejpam-6781	41	36	,	,	PUNCT
ejpam-6781	41	37	which	which	PRON
ejpam-6781	41	38	is	be	AUX
ejpam-6781	41	39	the	the	DET
ejpam-6781	41	40	equation	equation	NOUN
ejpam-6781	41	41	of	of	ADP
ejpam-6781	41	42	an	an	DET
ejpam-6781	41	43	eisenstein	eisenstein	NOUN
ejpam-6781	41	44	triple	triple	NOUN
ejpam-6781	41	45	(	(	PUNCT
ejpam-6781	41	46	see	see	VERB
ejpam-6781	41	47	figure	figure	NOUN
ejpam-6781	41	48	1	1	NUM
ejpam-6781	41	49	)	)	PUNCT
ejpam-6781	41	50	.	.	PUNCT
ejpam-6781	42	1	b	b	X
ejpam-6781	42	2	c	c	PROPN
ejpam-6781	42	3	a	a	DET
ejpam-6781	42	4	ca	ca	NOUN
ejpam-6781	42	5	b	b	SYM
ejpam-6781	42	6	60	60	NUM
ejpam-6781	42	7	◦	◦	NOUN
ejpam-6781	42	8	figure	figure	NOUN
ejpam-6781	42	9	1	1	NUM
ejpam-6781	42	10	:	:	PUNCT
ejpam-6781	42	11	triangles	triangle	NOUN
ejpam-6781	42	12	containing	contain	VERB
ejpam-6781	42	13	a	a	DET
ejpam-6781	42	14	60	60	NUM
ejpam-6781	42	15	◦	◦	NOUN
ejpam-6781	42	16	angle	angle	NOUN
ejpam-6781	42	17	.	.	PUNCT
ejpam-6781	43	1	for	for	ADP
ejpam-6781	43	2	examples	example	NOUN
ejpam-6781	43	3	:	:	PUNCT
ejpam-6781	43	4	•	•	ADP
ejpam-6781	43	5	the	the	DET
ejpam-6781	43	6	triples	triple	NOUN
ejpam-6781	43	7	(	(	PUNCT
ejpam-6781	43	8	3	3	NUM
ejpam-6781	43	9	,	,	PUNCT
ejpam-6781	43	10	8	8	NUM
ejpam-6781	43	11	,	,	PUNCT
ejpam-6781	43	12	7	7	NUM
ejpam-6781	43	13	)	)	PUNCT
ejpam-6781	43	14	and	and	CCONJ
ejpam-6781	43	15	(	(	PUNCT
ejpam-6781	43	16	5	5	NUM
ejpam-6781	43	17	,	,	PUNCT
ejpam-6781	43	18	8	8	NUM
ejpam-6781	43	19	,	,	PUNCT
ejpam-6781	43	20	7	7	NUM
ejpam-6781	43	21	)	)	PUNCT
ejpam-6781	43	22	are	be	AUX
ejpam-6781	43	23	primitive	primitive	ADJ
ejpam-6781	43	24	eisenstein	eisenstein	NOUN
ejpam-6781	43	25	triples	triple	NOUN
ejpam-6781	43	26	.	.	PUNCT
ejpam-6781	44	1	•	•	NUM
ejpam-6781	44	2	the	the	DET
ejpam-6781	44	3	eisenstein	eisenstein	PROPN
ejpam-6781	44	4	triples	triple	NOUN
ejpam-6781	44	5	(	(	PUNCT
ejpam-6781	44	6	6	6	NUM
ejpam-6781	44	7	,	,	PUNCT
ejpam-6781	44	8	16	16	NUM
ejpam-6781	44	9	,	,	PUNCT
ejpam-6781	44	10	14	14	NUM
ejpam-6781	44	11	)	)	PUNCT
ejpam-6781	44	12	and	and	CCONJ
ejpam-6781	44	13	(	(	PUNCT
ejpam-6781	44	14	10	10	NUM
ejpam-6781	44	15	,	,	PUNCT
ejpam-6781	44	16	16	16	NUM
ejpam-6781	44	17	,	,	PUNCT
ejpam-6781	44	18	14	14	NUM
ejpam-6781	44	19	)	)	PUNCT
ejpam-6781	44	20	are	be	AUX
ejpam-6781	44	21	not	not	PART
ejpam-6781	44	22	primitive	primitive	ADJ
ejpam-6781	44	23	.	.	PUNCT
ejpam-6781	45	1	we	we	PRON
ejpam-6781	45	2	note	note	VERB
ejpam-6781	45	3	that	that	SCONJ
ejpam-6781	45	4	every	every	DET
ejpam-6781	45	5	non	non	ADJ
ejpam-6781	45	6	-	-	ADJ
ejpam-6781	45	7	primitive	primitive	ADJ
ejpam-6781	45	8	eisenstein	eisenstein	NOUN
ejpam-6781	45	9	triple	triple	NOUN
ejpam-6781	45	10	can	can	AUX
ejpam-6781	45	11	be	be	AUX
ejpam-6781	45	12	expressed	express	VERB
ejpam-6781	45	13	as	as	ADP
ejpam-6781	45	14	a	a	DET
ejpam-6781	45	15	positive	positive	ADJ
ejpam-6781	45	16	multiple	multiple	NOUN
ejpam-6781	45	17	of	of	ADP
ejpam-6781	45	18	a	a	DET
ejpam-6781	45	19	primitive	primitive	ADJ
ejpam-6781	45	20	one	one	NUM
ejpam-6781	45	21	.	.	PUNCT
ejpam-6781	46	1	it	it	PRON
ejpam-6781	46	2	is	be	AUX
ejpam-6781	46	3	therefore	therefore	ADV
ejpam-6781	46	4	sufficient	sufficient	ADJ
ejpam-6781	46	5	to	to	PART
ejpam-6781	46	6	focus	focus	VERB
ejpam-6781	46	7	primarily	primarily	ADV
ejpam-6781	46	8	on	on	ADP
ejpam-6781	46	9	the	the	DET
ejpam-6781	46	10	study	study	NOUN
ejpam-6781	46	11	of	of	ADP
ejpam-6781	46	12	primitive	primitive	ADJ
ejpam-6781	46	13	eisenstein	eisenstein	NOUN
ejpam-6781	46	14	triples	triple	NOUN
ejpam-6781	46	15	.	.	PUNCT
ejpam-6781	47	1	the	the	DET
ejpam-6781	47	2	concept	concept	NOUN
ejpam-6781	47	3	of	of	ADP
ejpam-6781	47	4	conjugate	conjugate	ADJ
ejpam-6781	47	5	eisenstein	eisenstein	NOUN
ejpam-6781	47	6	triples	triple	NOUN
ejpam-6781	47	7	is	be	AUX
ejpam-6781	47	8	given	give	VERB
ejpam-6781	47	9	as	as	SCONJ
ejpam-6781	47	10	follows	follow	NOUN
ejpam-6781	47	11	.	.	PUNCT
ejpam-6781	48	1	theorem	theorem	ADJ
ejpam-6781	48	2	1	1	NUM
ejpam-6781	48	3	(	(	PUNCT
ejpam-6781	48	4	[	[	X
ejpam-6781	48	5	10	10	NUM
ejpam-6781	48	6	]	]	NUM
ejpam-6781	48	7	)	)	PUNCT
ejpam-6781	48	8	.	.	PUNCT
ejpam-6781	49	1	if	if	SCONJ
ejpam-6781	49	2	(	(	PUNCT
ejpam-6781	49	3	a	a	DET
ejpam-6781	49	4	,	,	PUNCT
ejpam-6781	49	5	b	b	NOUN
ejpam-6781	49	6	,	,	PUNCT
ejpam-6781	49	7	c	c	NOUN
ejpam-6781	49	8	)	)	PUNCT
ejpam-6781	49	9	is	be	AUX
ejpam-6781	49	10	a	a	DET
ejpam-6781	49	11	primitive	primitive	ADJ
ejpam-6781	49	12	eisenstein	eisenstein	NOUN
ejpam-6781	49	13	triple	triple	NOUN
ejpam-6781	49	14	,	,	PUNCT
ejpam-6781	49	15	then	then	ADV
ejpam-6781	49	16	(	(	PUNCT
ejpam-6781	49	17	b	b	X
ejpam-6781	49	18	−	−	PROPN
ejpam-6781	49	19	a	a	PROPN
ejpam-6781	49	20	,	,	PUNCT
ejpam-6781	49	21	b	b	NOUN
ejpam-6781	49	22	,	,	PUNCT
ejpam-6781	49	23	c	c	NOUN
ejpam-6781	49	24	)	)	PUNCT
ejpam-6781	49	25	is	be	AUX
ejpam-6781	49	26	also	also	ADV
ejpam-6781	49	27	a	a	DET
ejpam-6781	49	28	primitive	primitive	ADJ
ejpam-6781	49	29	eisenstein	eisenstein	NOUN
ejpam-6781	49	30	triple	triple	NOUN
ejpam-6781	49	31	.	.	PUNCT
ejpam-6781	50	1	a	a	DET
ejpam-6781	50	2	primitive	primitive	ADJ
ejpam-6781	50	3	eisenstein	eisenstein	NOUN
ejpam-6781	50	4	triple	triple	NOUN
ejpam-6781	50	5	(	(	PUNCT
ejpam-6781	50	6	b−	b−	PROPN
ejpam-6781	50	7	a	a	PROPN
ejpam-6781	50	8	,	,	PUNCT
ejpam-6781	50	9	b	b	NOUN
ejpam-6781	50	10	,	,	PUNCT
ejpam-6781	50	11	c	c	NOUN
ejpam-6781	50	12	)	)	PUNCT
ejpam-6781	50	13	is	be	AUX
ejpam-6781	50	14	called	call	VERB
ejpam-6781	50	15	a	a	DET
ejpam-6781	50	16	conjugate	conjugate	NOUN
ejpam-6781	50	17	of	of	ADP
ejpam-6781	50	18	the	the	DET
ejpam-6781	50	19	primitive	primitive	ADJ
ejpam-6781	50	20	eisenstein	eisenstein	NOUN
ejpam-6781	50	21	triple	triple	NOUN
ejpam-6781	50	22	(	(	PUNCT
ejpam-6781	50	23	a	a	PRON
ejpam-6781	50	24	,	,	PUNCT
ejpam-6781	50	25	b	b	NOUN
ejpam-6781	50	26	,	,	PUNCT
ejpam-6781	50	27	c	c	NOUN
ejpam-6781	50	28	)	)	PUNCT
ejpam-6781	50	29	(	(	PUNCT
ejpam-6781	50	30	see	see	VERB
ejpam-6781	50	31	figure	figure	NOUN
ejpam-6781	50	32	2	2	NUM
ejpam-6781	50	33	)	)	PUNCT
ejpam-6781	50	34	.	.	PUNCT
ejpam-6781	51	1	a	a	DET
ejpam-6781	51	2	b	b	X
ejpam-6781	51	3	c	c	NOUN
ejpam-6781	51	4	d	d	PROPN
ejpam-6781	51	5	b	b	PROPN
ejpam-6781	51	6	a	a	DET
ejpam-6781	51	7	b−	b−	PROPN
ejpam-6781	51	8	a	a	DET
ejpam-6781	51	9	bc	bc	PROPN
ejpam-6781	51	10	60	60	NUM
ejpam-6781	51	11	◦	◦	NOUN
ejpam-6781	51	12	60	60	NUM
ejpam-6781	51	13	◦	◦	NOUN
ejpam-6781	51	14	figure	figure	NOUN
ejpam-6781	51	15	2	2	NUM
ejpam-6781	51	16	:	:	PUNCT
ejpam-6781	51	17	an	an	DET
ejpam-6781	51	18	eisenstein	eisenstein	NOUN
ejpam-6781	51	19	triple	triple	NOUN
ejpam-6781	51	20	and	and	CCONJ
ejpam-6781	51	21	its	its	PRON
ejpam-6781	51	22	conjugate	conjugate	NOUN
ejpam-6781	51	23	the	the	DET
ejpam-6781	51	24	length	length	NOUN
ejpam-6781	51	25	of	of	ADP
ejpam-6781	51	26	the	the	DET
ejpam-6781	51	27	side	side	NOUN
ejpam-6781	51	28	c	c	NOUN
ejpam-6781	51	29	of	of	ADP
ejpam-6781	51	30	a	a	DET
ejpam-6781	51	31	primitive	primitive	ADJ
ejpam-6781	51	32	eisenstein	eisenstein	NOUN
ejpam-6781	51	33	triple	triple	NOUN
ejpam-6781	51	34	(	(	PUNCT
ejpam-6781	51	35	a	a	DET
ejpam-6781	51	36	,	,	PUNCT
ejpam-6781	51	37	b	b	NOUN
ejpam-6781	51	38	,	,	PUNCT
ejpam-6781	51	39	c	c	NOUN
ejpam-6781	51	40	)	)	PUNCT
ejpam-6781	51	41	is	be	AUX
ejpam-6781	51	42	subject	subject	ADJ
ejpam-6781	51	43	to	to	ADP
ejpam-6781	51	44	the	the	DET
ejpam-6781	51	45	following	follow	VERB
ejpam-6781	51	46	restriction	restriction	NOUN
ejpam-6781	51	47	.	.	PUNCT
ejpam-6781	52	1	theorem	theorem	NOUN
ejpam-6781	52	2	2	2	NUM
ejpam-6781	52	3	(	(	PUNCT
ejpam-6781	52	4	[	[	X
ejpam-6781	52	5	10	10	NUM
ejpam-6781	52	6	,	,	PUNCT
ejpam-6781	52	7	theorem	theorem	VERB
ejpam-6781	52	8	1	1	NUM
ejpam-6781	52	9	]	]	PUNCT
ejpam-6781	52	10	)	)	PUNCT
ejpam-6781	52	11	.	.	PUNCT
ejpam-6781	53	1	if	if	SCONJ
ejpam-6781	53	2	(	(	PUNCT
ejpam-6781	53	3	a	a	DET
ejpam-6781	53	4	,	,	PUNCT
ejpam-6781	53	5	b	b	NOUN
ejpam-6781	53	6	,	,	PUNCT
ejpam-6781	53	7	c	c	NOUN
ejpam-6781	53	8	)	)	PUNCT
ejpam-6781	53	9	is	be	AUX
ejpam-6781	53	10	a	a	DET
ejpam-6781	53	11	primitive	primitive	ADJ
ejpam-6781	53	12	eisenstein	eisenstein	NOUN
ejpam-6781	53	13	triple	triple	NOUN
ejpam-6781	53	14	,	,	PUNCT
ejpam-6781	53	15	then	then	ADV
ejpam-6781	53	16	c	c	PROPN
ejpam-6781	53	17	is	be	AUX
ejpam-6781	53	18	neither	neither	CCONJ
ejpam-6781	53	19	a	a	DET
ejpam-6781	53	20	multiple	multiple	NOUN
ejpam-6781	53	21	of	of	ADP
ejpam-6781	53	22	2	2	NUM
ejpam-6781	53	23	nor	nor	CCONJ
ejpam-6781	53	24	3	3	NUM
ejpam-6781	53	25	.	.	PUNCT
ejpam-6781	54	1	more	more	ADV
ejpam-6781	54	2	generally	generally	ADV
ejpam-6781	54	3	,	,	PUNCT
ejpam-6781	54	4	the	the	DET
ejpam-6781	54	5	only	only	ADJ
ejpam-6781	54	6	prime	prime	ADJ
ejpam-6781	54	7	factors	factor	NOUN
ejpam-6781	54	8	of	of	ADP
ejpam-6781	54	9	c	c	NOUN
ejpam-6781	54	10	are	be	AUX
ejpam-6781	54	11	primes	prime	NOUN
ejpam-6781	54	12	of	of	ADP
ejpam-6781	54	13	the	the	DET
ejpam-6781	54	14	form	form	NOUN
ejpam-6781	54	15	6k	6k	NOUN
ejpam-6781	54	16	+	+	NOUN
ejpam-6781	54	17	1	1	X
ejpam-6781	54	18	.	.	PUNCT
ejpam-6781	54	19	s.	s.	PROPN
ejpam-6781	54	20	jitman	jitman	PROPN
ejpam-6781	54	21	,	,	PUNCT
ejpam-6781	54	22	m.	m.	NOUN
ejpam-6781	54	23	mohammad	mohammad	PROPN
ejpam-6781	54	24	,	,	PUNCT
ejpam-6781	54	25	e.	e.	PROPN
ejpam-6781	54	26	sangwisut	sangwisut	PROPN
ejpam-6781	54	27	/	/	SYM
ejpam-6781	54	28	eur	eur	PROPN
ejpam-6781	54	29	.	.	PUNCT
ejpam-6781	55	1	j.	j.	PROPN
ejpam-6781	55	2	pure	pure	PROPN
ejpam-6781	55	3	appl	appl	PROPN
ejpam-6781	55	4	.	.	PROPN
ejpam-6781	55	5	math	math	PROPN
ejpam-6781	55	6	,	,	PUNCT
ejpam-6781	55	7	18	18	NUM
ejpam-6781	55	8	(	(	PUNCT
ejpam-6781	55	9	4	4	NUM
ejpam-6781	55	10	)	)	PUNCT
ejpam-6781	55	11	(	(	PUNCT
ejpam-6781	55	12	2025	2025	NUM
ejpam-6781	55	13	)	)	PUNCT
ejpam-6781	55	14	,	,	PUNCT
ejpam-6781	55	15	6781	6781	NUM
ejpam-6781	55	16	4	4	NUM
ejpam-6781	55	17	of	of	ADP
ejpam-6781	55	18	16	16	NUM
ejpam-6781	55	19	based	base	VERB
ejpam-6781	55	20	on	on	ADP
ejpam-6781	55	21	theorem	theorem	NOUN
ejpam-6781	55	22	2	2	NUM
ejpam-6781	55	23	,	,	PUNCT
ejpam-6781	55	24	we	we	PRON
ejpam-6781	55	25	introduce	introduce	VERB
ejpam-6781	55	26	the	the	DET
ejpam-6781	55	27	following	follow	VERB
ejpam-6781	55	28	useful	useful	ADJ
ejpam-6781	55	29	partition	partition	NOUN
ejpam-6781	55	30	on	on	ADP
ejpam-6781	55	31	the	the	DET
ejpam-6781	55	32	set	set	NOUN
ejpam-6781	55	33	of	of	ADP
ejpam-6781	55	34	primitive	primitive	ADJ
ejpam-6781	55	35	eisenstein	eisenstein	NOUN
ejpam-6781	55	36	triples	triple	NOUN
ejpam-6781	55	37	.	.	PUNCT
ejpam-6781	56	1	let	let	VERB
ejpam-6781	56	2	et	et	NOUN
ejpam-6781	56	3	denote	denote	VERB
ejpam-6781	56	4	the	the	DET
ejpam-6781	56	5	set	set	NOUN
ejpam-6781	56	6	of	of	ADP
ejpam-6781	56	7	all	all	DET
ejpam-6781	56	8	primitive	primitive	ADJ
ejpam-6781	56	9	eisenstein	eisenstein	NOUN
ejpam-6781	56	10	triples	triple	NOUN
ejpam-6781	56	11	.	.	PUNCT
ejpam-6781	57	1	for	for	ADP
ejpam-6781	57	2	each	each	DET
ejpam-6781	57	3	positive	positive	ADJ
ejpam-6781	57	4	integer	integer	NOUN
ejpam-6781	57	5	c	c	NOUN
ejpam-6781	57	6	,	,	PUNCT
ejpam-6781	57	7	let	let	VERB
ejpam-6781	57	8	etc	etc	PRON
ejpam-6781	57	9	denote	denote	VERB
ejpam-6781	57	10	the	the	DET
ejpam-6781	57	11	subset	subset	NOUN
ejpam-6781	57	12	of	of	ADP
ejpam-6781	57	13	et	et	NOUN
ejpam-6781	57	14	containing	contain	VERB
ejpam-6781	57	15	triples	triple	NOUN
ejpam-6781	57	16	whose	whose	DET
ejpam-6781	57	17	side	side	NOUN
ejpam-6781	57	18	opposite	opposite	ADP
ejpam-6781	57	19	an	an	DET
ejpam-6781	57	20	angle	angle	NOUN
ejpam-6781	57	21	60	60	NUM
ejpam-6781	57	22	◦	◦	NOUN
ejpam-6781	57	23	is	be	AUX
ejpam-6781	57	24	c.	c.	NOUN
ejpam-6781	57	25	from	from	ADP
ejpam-6781	57	26	theorem	theorem	NOUN
ejpam-6781	57	27	2	2	NUM
ejpam-6781	57	28	,	,	PUNCT
ejpam-6781	57	29	the	the	DET
ejpam-6781	57	30	integer	integer	NOUN
ejpam-6781	57	31	c	c	PROPN
ejpam-6781	57	32	in	in	ADP
ejpam-6781	57	33	every	every	DET
ejpam-6781	57	34	primitive	primitive	ADJ
ejpam-6781	57	35	eisenstein	eisenstein	NOUN
ejpam-6781	57	36	triple	triple	NOUN
ejpam-6781	57	37	must	must	AUX
ejpam-6781	57	38	have	have	VERB
ejpam-6781	57	39	only	only	ADJ
ejpam-6781	57	40	prime	prime	ADJ
ejpam-6781	57	41	factors	factor	NOUN
ejpam-6781	57	42	of	of	ADP
ejpam-6781	57	43	the	the	DET
ejpam-6781	57	44	form	form	NOUN
ejpam-6781	57	45	6k	6k	NOUN
ejpam-6781	58	1	+	+	CCONJ
ejpam-6781	58	2	1	1	NUM
ejpam-6781	58	3	for	for	ADP
ejpam-6781	58	4	some	some	DET
ejpam-6781	58	5	non	non	ADJ
ejpam-6781	58	6	-	-	ADJ
ejpam-6781	58	7	negative	negative	ADJ
ejpam-6781	58	8	integer	integer	NOUN
ejpam-6781	58	9	k.	k.	PROPN
ejpam-6781	59	1	this	this	PRON
ejpam-6781	59	2	yields	yield	VERB
ejpam-6781	59	3	the	the	DET
ejpam-6781	59	4	following	follow	VERB
ejpam-6781	59	5	partition	partition	NOUN
ejpam-6781	59	6	et	et	NOUN
ejpam-6781	59	7	=	=	NOUN
ejpam-6781	60	1	⊔	⊔	VERB
ejpam-6781	60	2	c>1	c>1	NOUN
ejpam-6781	60	3	etc	etc	X
ejpam-6781	60	4	=	=	X
ejpam-6781	60	5	et7	et7	NOUN
ejpam-6781	60	6	⊔et13	⊔et13	PROPN
ejpam-6781	60	7	⊔et19	⊔et19	NUM
ejpam-6781	60	8	⊔	⊔	PROPN
ejpam-6781	60	9	.	.	PUNCT
ejpam-6781	60	10	.	.	PUNCT
ejpam-6781	60	11	.	.	PUNCT
ejpam-6781	61	1	,	,	PUNCT
ejpam-6781	61	2	where	where	SCONJ
ejpam-6781	61	3	the	the	DET
ejpam-6781	61	4	union	union	NOUN
ejpam-6781	61	5	is	be	AUX
ejpam-6781	61	6	taken	take	VERB
ejpam-6781	61	7	over	over	ADP
ejpam-6781	61	8	all	all	DET
ejpam-6781	61	9	values	value	NOUN
ejpam-6781	61	10	of	of	ADP
ejpam-6781	61	11	c	c	NOUN
ejpam-6781	61	12	whose	whose	DET
ejpam-6781	61	13	prime	prime	ADJ
ejpam-6781	61	14	factors	factor	NOUN
ejpam-6781	61	15	are	be	AUX
ejpam-6781	61	16	congruence	congruence	NOUN
ejpam-6781	61	17	to	to	ADP
ejpam-6781	61	18	1	1	NUM
ejpam-6781	61	19	modulo	modulo	NOUN
ejpam-6781	61	20	6	6	NUM
ejpam-6781	61	21	.	.	NOUN
ejpam-6781	62	1	3	3	NUM
ejpam-6781	62	2	.	.	X
ejpam-6781	63	1	from	from	ADP
ejpam-6781	63	2	eisenstein	eisenstein	NOUN
ejpam-6781	63	3	triples	triple	NOUN
ejpam-6781	63	4	to	to	ADP
ejpam-6781	63	5	the	the	DET
ejpam-6781	63	6	ω	ω	ADJ
ejpam-6781	63	7	-	-	ADJ
ejpam-6781	63	8	rational	rational	ADJ
ejpam-6781	63	9	unit	unit	NOUN
ejpam-6781	63	10	circle	circle	NOUN
ejpam-6781	63	11	this	this	DET
ejpam-6781	63	12	section	section	NOUN
ejpam-6781	63	13	presents	present	VERB
ejpam-6781	63	14	a	a	DET
ejpam-6781	63	15	connection	connection	NOUN
ejpam-6781	63	16	between	between	ADP
ejpam-6781	63	17	eisenstein	eisenstein	NOUN
ejpam-6781	63	18	triples	triple	NOUN
ejpam-6781	63	19	and	and	CCONJ
ejpam-6781	63	20	points	point	NOUN
ejpam-6781	63	21	on	on	ADP
ejpam-6781	63	22	the	the	DET
ejpam-6781	63	23	ωrational	ωrational	ADJ
ejpam-6781	63	24	unit	unit	NOUN
ejpam-6781	63	25	circle	circle	NOUN
ejpam-6781	63	26	(	(	PUNCT
ejpam-6781	63	27	see	see	VERB
ejpam-6781	63	28	(	(	PUNCT
ejpam-6781	63	29	3	3	NUM
ejpam-6781	63	30	)	)	PUNCT
ejpam-6781	63	31	for	for	ADP
ejpam-6781	63	32	the	the	DET
ejpam-6781	63	33	definition	definition	NOUN
ejpam-6781	63	34	)	)	PUNCT
ejpam-6781	63	35	.	.	PUNCT
ejpam-6781	64	1	using	use	VERB
ejpam-6781	64	2	the	the	DET
ejpam-6781	64	3	unique	unique	ADJ
ejpam-6781	64	4	factorization	factorization	NOUN
ejpam-6781	64	5	property	property	NOUN
ejpam-6781	64	6	of	of	ADP
ejpam-6781	64	7	eisenstein	eisenstein	NOUN
ejpam-6781	64	8	integers	integer	NOUN
ejpam-6781	64	9	,	,	PUNCT
ejpam-6781	64	10	we	we	PRON
ejpam-6781	64	11	demonstrate	demonstrate	VERB
ejpam-6781	64	12	how	how	SCONJ
ejpam-6781	64	13	each	each	DET
ejpam-6781	64	14	primitive	primitive	ADJ
ejpam-6781	64	15	eisenstein	eisenstein	NOUN
ejpam-6781	64	16	triple	triple	NOUN
ejpam-6781	64	17	is	be	AUX
ejpam-6781	64	18	mapped	map	VERB
ejpam-6781	64	19	to	to	ADP
ejpam-6781	64	20	an	an	DET
ejpam-6781	64	21	ω	ω	ADJ
ejpam-6781	64	22	-	-	ADJ
ejpam-6781	64	23	rational	rational	ADJ
ejpam-6781	64	24	point	point	NOUN
ejpam-6781	64	25	in	in	ADP
ejpam-6781	64	26	the	the	DET
ejpam-6781	64	27	unit	unit	NOUN
ejpam-6781	64	28	circle	circle	NOUN
ejpam-6781	64	29	in	in	ADP
ejpam-6781	64	30	the	the	DET
ejpam-6781	64	31	complex	complex	ADJ
ejpam-6781	64	32	plane	plane	NOUN
ejpam-6781	64	33	.	.	PUNCT
ejpam-6781	65	1	this	this	DET
ejpam-6781	65	2	mapping	mapping	NOUN
ejpam-6781	65	3	not	not	PART
ejpam-6781	65	4	only	only	ADV
ejpam-6781	65	5	highlights	highlight	VERB
ejpam-6781	65	6	the	the	DET
ejpam-6781	65	7	geometric	geometric	ADJ
ejpam-6781	65	8	structure	structure	NOUN
ejpam-6781	65	9	of	of	ADP
ejpam-6781	65	10	eisenstein	eisenstein	NOUN
ejpam-6781	65	11	triples	triple	NOUN
ejpam-6781	65	12	but	but	CCONJ
ejpam-6781	65	13	also	also	ADV
ejpam-6781	65	14	establishes	establish	VERB
ejpam-6781	65	15	a	a	DET
ejpam-6781	65	16	link	link	NOUN
ejpam-6781	65	17	between	between	ADP
ejpam-6781	65	18	their	their	PRON
ejpam-6781	65	19	algebraic	algebraic	ADJ
ejpam-6781	65	20	and	and	CCONJ
ejpam-6781	65	21	geometric	geometric	ADJ
ejpam-6781	65	22	representations	representation	NOUN
ejpam-6781	65	23	.	.	PUNCT
ejpam-6781	66	1	eisenstein	eisenstein	PROPN
ejpam-6781	66	2	integers	integer	NOUN
ejpam-6781	66	3	are	be	AUX
ejpam-6781	66	4	complex	complex	ADJ
ejpam-6781	66	5	numbers	number	NOUN
ejpam-6781	66	6	of	of	ADP
ejpam-6781	66	7	the	the	DET
ejpam-6781	66	8	form	form	NOUN
ejpam-6781	66	9	z	z	NOUN
ejpam-6781	66	10	=	=	PUNCT
ejpam-6781	66	11	a	a	DET
ejpam-6781	66	12	+	+	X
ejpam-6781	66	13	bω	bω	NOUN
ejpam-6781	66	14	,	,	PUNCT
ejpam-6781	66	15	where	where	SCONJ
ejpam-6781	66	16	a	a	DET
ejpam-6781	66	17	,	,	PUNCT
ejpam-6781	66	18	b	b	PROPN
ejpam-6781	66	19	∈	∈	PROPN
ejpam-6781	66	20	z	z	NOUN
ejpam-6781	66	21	,	,	PUNCT
ejpam-6781	66	22	and	and	CCONJ
ejpam-6781	66	23	ω	ω	X
ejpam-6781	66	24	=	=	SYM
ejpam-6781	66	25	e	e	PROPN
ejpam-6781	66	26	2πi	2πi	NOUN
ejpam-6781	66	27	3	3	X
ejpam-6781	66	28	=	=	SYM
ejpam-6781	66	29	−1	−1	NOUN
ejpam-6781	66	30	+	+	CCONJ
ejpam-6781	66	31	√	√	NOUN
ejpam-6781	66	32	3i	3i	NOUN
ejpam-6781	66	33	2	2	NUM
ejpam-6781	66	34	is	be	AUX
ejpam-6781	66	35	a	a	DET
ejpam-6781	66	36	primitive	primitive	ADJ
ejpam-6781	66	37	cube	cube	NOUN
ejpam-6781	66	38	root	root	NOUN
ejpam-6781	66	39	of	of	ADP
ejpam-6781	66	40	unity	unity	NOUN
ejpam-6781	66	41	.	.	PUNCT
ejpam-6781	67	1	we	we	PRON
ejpam-6781	67	2	note	note	VERB
ejpam-6781	67	3	that	that	SCONJ
ejpam-6781	67	4	ω	ω	PROPN
ejpam-6781	67	5	and	and	CCONJ
ejpam-6781	67	6	ω	ω	NUM
ejpam-6781	67	7	are	be	AUX
ejpam-6781	67	8	roots	root	NOUN
ejpam-6781	67	9	of	of	ADP
ejpam-6781	67	10	the	the	DET
ejpam-6781	67	11	polynomial	polynomial	ADJ
ejpam-6781	67	12	x2	x2	PROPN
ejpam-6781	68	1	+	+	CCONJ
ejpam-6781	68	2	x	x	SYM
ejpam-6781	68	3	+	+	NUM
ejpam-6781	68	4	1	1	NUM
ejpam-6781	68	5	,	,	PUNCT
ejpam-6781	68	6	where	where	SCONJ
ejpam-6781	68	7	ω	ω	PROPN
ejpam-6781	68	8	is	be	AUX
ejpam-6781	68	9	the	the	DET
ejpam-6781	68	10	complex	complex	ADJ
ejpam-6781	68	11	conjugate	conjugate	NOUN
ejpam-6781	68	12	of	of	ADP
ejpam-6781	68	13	ω	ω	NOUN
ejpam-6781	68	14	.	.	PUNCT
ejpam-6781	69	1	it	it	PRON
ejpam-6781	69	2	is	be	AUX
ejpam-6781	69	3	easily	easily	ADV
ejpam-6781	69	4	seen	see	VERB
ejpam-6781	69	5	that	that	SCONJ
ejpam-6781	69	6	ω3	ω3	NOUN
ejpam-6781	69	7	=	=	SYM
ejpam-6781	69	8	ω3	ω3	NOUN
ejpam-6781	69	9	=	=	SYM
ejpam-6781	69	10	1	1	NUM
ejpam-6781	69	11	,	,	PUNCT
ejpam-6781	69	12	ω2	ω2	NOUN
ejpam-6781	69	13	=	=	SYM
ejpam-6781	69	14	ω	ω	PROPN
ejpam-6781	69	15	=	=	PUNCT
ejpam-6781	69	16	−1	−1	NOUN
ejpam-6781	69	17	−	−	PROPN
ejpam-6781	69	18	ω	ω	PROPN
ejpam-6781	69	19	,	,	PUNCT
ejpam-6781	69	20	ω	ω	PROPN
ejpam-6781	69	21	·	·	PUNCT
ejpam-6781	69	22	ω	ω	NUM
ejpam-6781	69	23	=	=	SYM
ejpam-6781	69	24	1	1	NUM
ejpam-6781	69	25	and	and	CCONJ
ejpam-6781	69	26	ω	ω	NUM
ejpam-6781	69	27	+	+	CCONJ
ejpam-6781	69	28	ω	ω	NUM
ejpam-6781	69	29	=	=	SYM
ejpam-6781	69	30	−1	−1	NOUN
ejpam-6781	69	31	.	.	PUNCT
ejpam-6781	70	1	the	the	DET
ejpam-6781	70	2	set	set	NOUN
ejpam-6781	70	3	of	of	ADP
ejpam-6781	70	4	all	all	DET
ejpam-6781	70	5	eisenstein	eisenstein	NOUN
ejpam-6781	70	6	integers	integer	NOUN
ejpam-6781	70	7	,	,	PUNCT
ejpam-6781	70	8	denoted	denote	VERB
ejpam-6781	70	9	z[ω	z[ω	PROPN
ejpam-6781	70	10	]	]	PUNCT
ejpam-6781	70	11	,	,	PUNCT
ejpam-6781	70	12	forms	form	VERB
ejpam-6781	70	13	a	a	DET
ejpam-6781	70	14	commutative	commutative	ADJ
ejpam-6781	70	15	ring	ring	NOUN
ejpam-6781	70	16	with	with	ADP
ejpam-6781	70	17	identity	identity	NOUN
ejpam-6781	70	18	under	under	ADP
ejpam-6781	70	19	the	the	DET
ejpam-6781	70	20	usual	usual	ADJ
ejpam-6781	70	21	addition	addition	NOUN
ejpam-6781	70	22	and	and	CCONJ
ejpam-6781	70	23	multiplication	multiplication	NOUN
ejpam-6781	70	24	of	of	ADP
ejpam-6781	70	25	complex	complex	ADJ
ejpam-6781	70	26	numbers	number	NOUN
ejpam-6781	70	27	.	.	PUNCT
ejpam-6781	71	1	the	the	DET
ejpam-6781	71	2	unit	unit	NOUN
ejpam-6781	71	3	group	group	NOUN
ejpam-6781	71	4	of	of	ADP
ejpam-6781	71	5	z[ω	z[ω	PROPN
ejpam-6781	71	6	]	]	PUNCT
ejpam-6781	71	7	is	be	AUX
ejpam-6781	71	8	u	u	NOUN
ejpam-6781	71	9	=	=	PUNCT
ejpam-6781	71	10	{	{	PUNCT
ejpam-6781	71	11	1	1	NUM
ejpam-6781	71	12	,	,	PUNCT
ejpam-6781	71	13	ω	ω	PROPN
ejpam-6781	71	14	,	,	PUNCT
ejpam-6781	71	15	ω2,−1,−ω,−ω2	ω2,−1,−ω,−ω2	X
ejpam-6781	71	16	}	}	PUNCT
ejpam-6781	71	17	=	=	PUNCT
ejpam-6781	71	18	⟨−ω⟩	⟨−ω⟩	NOUN
ejpam-6781	71	19	,	,	PUNCT
ejpam-6781	71	20	which	which	PRON
ejpam-6781	71	21	is	be	AUX
ejpam-6781	71	22	isomorphic	isomorphic	ADJ
ejpam-6781	71	23	to	to	ADP
ejpam-6781	71	24	the	the	DET
ejpam-6781	71	25	cyclic	cyclic	ADJ
ejpam-6781	71	26	group	group	NOUN
ejpam-6781	71	27	of	of	ADP
ejpam-6781	71	28	order	order	NOUN
ejpam-6781	71	29	6	6	NUM
ejpam-6781	71	30	.	.	PUNCT
ejpam-6781	72	1	the	the	DET
ejpam-6781	72	2	norm	norm	NOUN
ejpam-6781	72	3	of	of	ADP
ejpam-6781	72	4	an	an	DET
ejpam-6781	72	5	eisenstein	eisenstein	NOUN
ejpam-6781	72	6	integer	integer	NOUN
ejpam-6781	72	7	z	z	PROPN
ejpam-6781	72	8	=	=	PUNCT
ejpam-6781	72	9	a	a	DET
ejpam-6781	72	10	+	+	NUM
ejpam-6781	72	11	bω	bω	NOUN
ejpam-6781	72	12	is	be	AUX
ejpam-6781	72	13	defined	define	VERB
ejpam-6781	72	14	by	by	ADP
ejpam-6781	72	15	n(z	n(z	NOUN
ejpam-6781	72	16	)	)	PUNCT
ejpam-6781	72	17	=	=	PUNCT
ejpam-6781	73	1	√	√	NUM
ejpam-6781	73	2	zz	zz	X
ejpam-6781	74	1	=	=	PUNCT
ejpam-6781	74	2	√	√	PROPN
ejpam-6781	74	3	(	(	PUNCT
ejpam-6781	74	4	a	a	DET
ejpam-6781	74	5	+	+	X
ejpam-6781	74	6	bω)(a	bω)(a	PROPN
ejpam-6781	74	7	+	+	CCONJ
ejpam-6781	74	8	bω	bω	NOUN
ejpam-6781	74	9	)	)	PUNCT
ejpam-6781	74	10	=	=	SYM
ejpam-6781	74	11	√	√	PROPN
ejpam-6781	74	12	a2	a2	PROPN
ejpam-6781	74	13	−	−	PROPN
ejpam-6781	74	14	ab	ab	PROPN
ejpam-6781	74	15	+	+	PROPN
ejpam-6781	74	16	b2	b2	NOUN
ejpam-6781	74	17	.	.	PUNCT
ejpam-6781	75	1	for	for	ADP
ejpam-6781	75	2	two	two	NUM
ejpam-6781	75	3	eisenstein	eisenstein	NOUN
ejpam-6781	75	4	integers	integer	NOUN
ejpam-6781	75	5	z	z	PROPN
ejpam-6781	75	6	and	and	CCONJ
ejpam-6781	75	7	z′	z′	PROPN
ejpam-6781	75	8	,	,	PUNCT
ejpam-6781	75	9	the	the	DET
ejpam-6781	75	10	norm	norm	NOUN
ejpam-6781	75	11	satisfies	satisfy	VERB
ejpam-6781	75	12	n(zz′	n(zz′	ADV
ejpam-6781	75	13	)	)	PUNCT
ejpam-6781	75	14	=	=	SYM
ejpam-6781	75	15	n(z)n(z′	n(z)n(z′	NOUN
ejpam-6781	75	16	)	)	PUNCT
ejpam-6781	75	17	.	.	PUNCT
ejpam-6781	76	1	two	two	NUM
ejpam-6781	76	2	eisenstein	eisenstein	PROPN
ejpam-6781	76	3	integers	integer	NOUN
ejpam-6781	76	4	z	z	PROPN
ejpam-6781	76	5	and	and	CCONJ
ejpam-6781	76	6	z′	z′	NUM
ejpam-6781	76	7	are	be	AUX
ejpam-6781	76	8	said	say	VERB
ejpam-6781	76	9	to	to	PART
ejpam-6781	76	10	be	be	AUX
ejpam-6781	76	11	associates	associate	NOUN
ejpam-6781	76	12	if	if	SCONJ
ejpam-6781	76	13	z	z	NOUN
ejpam-6781	76	14	=	=	SYM
ejpam-6781	76	15	δ	δ	X
ejpam-6781	76	16	·	·	PUNCT
ejpam-6781	76	17	z′	z′	NUM
ejpam-6781	76	18	for	for	ADP
ejpam-6781	76	19	some	some	DET
ejpam-6781	76	20	unit	unit	NOUN
ejpam-6781	76	21	δ	δ	PROPN
ejpam-6781	76	22	∈	∈	PROPN
ejpam-6781	76	23	u	u	PROPN
ejpam-6781	76	24	.	.	PUNCT
ejpam-6781	77	1	the	the	DET
ejpam-6781	77	2	ring	ring	NOUN
ejpam-6781	77	3	z[ω	z[ω	PROPN
ejpam-6781	77	4	]	]	PART
ejpam-6781	77	5	is	be	AUX
ejpam-6781	77	6	a	a	DET
ejpam-6781	77	7	unique	unique	ADJ
ejpam-6781	77	8	factorization	factorization	NOUN
ejpam-6781	77	9	domain	domain	NOUN
ejpam-6781	77	10	(	(	PUNCT
ejpam-6781	77	11	ufd	ufd	PROPN
ejpam-6781	77	12	)	)	PUNCT
ejpam-6781	77	13	,	,	PUNCT
ejpam-6781	77	14	meaning	mean	VERB
ejpam-6781	77	15	every	every	DET
ejpam-6781	77	16	nonzero	nonzero	ADJ
ejpam-6781	77	17	,	,	PUNCT
ejpam-6781	77	18	non	non	ADJ
ejpam-6781	77	19	-	-	ADJ
ejpam-6781	77	20	unit	unit	ADJ
ejpam-6781	77	21	element	element	NOUN
ejpam-6781	77	22	has	have	VERB
ejpam-6781	77	23	a	a	DET
ejpam-6781	77	24	unique	unique	ADJ
ejpam-6781	77	25	factorization	factorization	NOUN
ejpam-6781	77	26	into	into	ADP
ejpam-6781	77	27	a	a	DET
ejpam-6781	77	28	product	product	NOUN
ejpam-6781	77	29	of	of	ADP
ejpam-6781	77	30	irreducible	irreducible	ADJ
ejpam-6781	77	31	elements	element	NOUN
ejpam-6781	77	32	,	,	PUNCT
ejpam-6781	77	33	up	up	ADP
ejpam-6781	77	34	to	to	ADP
ejpam-6781	77	35	the	the	DET
ejpam-6781	77	36	rearrangement	rearrangement	NOUN
ejpam-6781	77	37	of	of	ADP
ejpam-6781	77	38	the	the	DET
ejpam-6781	77	39	factors	factor	NOUN
ejpam-6781	77	40	and	and	CCONJ
ejpam-6781	77	41	the	the	DET
ejpam-6781	77	42	replacement	replacement	NOUN
ejpam-6781	77	43	of	of	ADP
ejpam-6781	77	44	any	any	DET
ejpam-6781	77	45	irreducible	irreducible	ADJ
ejpam-6781	77	46	element	element	NOUN
ejpam-6781	77	47	with	with	ADP
ejpam-6781	77	48	one	one	NUM
ejpam-6781	77	49	of	of	ADP
ejpam-6781	77	50	its	its	PRON
ejpam-6781	77	51	associates	associate	NOUN
ejpam-6781	77	52	.	.	PUNCT
ejpam-6781	78	1	for	for	ADP
ejpam-6781	78	2	more	more	ADJ
ejpam-6781	78	3	information	information	NOUN
ejpam-6781	78	4	on	on	ADP
ejpam-6781	78	5	eisenstein	eisenstein	NOUN
ejpam-6781	78	6	integers	integer	NOUN
ejpam-6781	78	7	,	,	PUNCT
ejpam-6781	78	8	the	the	DET
ejpam-6781	78	9	reader	reader	NOUN
ejpam-6781	78	10	may	may	AUX
ejpam-6781	78	11	refer	refer	VERB
ejpam-6781	78	12	to	to	ADP
ejpam-6781	78	13	[	[	X
ejpam-6781	78	14	11–13	11–13	NUM
ejpam-6781	78	15	]	]	PUNCT
ejpam-6781	78	16	and	and	CCONJ
ejpam-6781	78	17	[	[	X
ejpam-6781	78	18	14	14	NUM
ejpam-6781	78	19	]	]	PUNCT
ejpam-6781	78	20	.	.	PUNCT
ejpam-6781	79	1	let	let	AUX
ejpam-6781	79	2	g(r	g(r	PRON
ejpam-6781	79	3	)	)	PUNCT
ejpam-6781	79	4	denote	denote	VERB
ejpam-6781	79	5	the	the	DET
ejpam-6781	79	6	unit	unit	NOUN
ejpam-6781	79	7	circle	circle	NOUN
ejpam-6781	79	8	in	in	ADP
ejpam-6781	79	9	the	the	DET
ejpam-6781	79	10	complex	complex	ADJ
ejpam-6781	79	11	plane	plane	NOUN
ejpam-6781	79	12	.	.	PUNCT
ejpam-6781	80	1	since	since	SCONJ
ejpam-6781	80	2	n(z	n(z	NOUN
ejpam-6781	80	3	)	)	PUNCT
ejpam-6781	80	4	equals	equal	VERB
ejpam-6781	80	5	the	the	DET
ejpam-6781	80	6	euclidean	euclidean	ADJ
ejpam-6781	80	7	norm	norm	NOUN
ejpam-6781	80	8	of	of	ADP
ejpam-6781	80	9	z	z	NOUN
ejpam-6781	80	10	for	for	ADP
ejpam-6781	80	11	all	all	DET
ejpam-6781	80	12	complex	complex	ADJ
ejpam-6781	80	13	numbers	number	NOUN
ejpam-6781	80	14	z	z	PROPN
ejpam-6781	80	15	,	,	PUNCT
ejpam-6781	80	16	the	the	DET
ejpam-6781	80	17	elements	element	NOUN
ejpam-6781	80	18	in	in	ADP
ejpam-6781	80	19	g(r	g(r	NOUN
ejpam-6781	80	20	)	)	PUNCT
ejpam-6781	80	21	can	can	AUX
ejpam-6781	80	22	be	be	AUX
ejpam-6781	80	23	viewed	view	VERB
ejpam-6781	80	24	as	as	ADP
ejpam-6781	80	25	their	their	PRON
ejpam-6781	80	26	ωexpansions	ωexpansion	NOUN
ejpam-6781	80	27	.	.	PUNCT
ejpam-6781	81	1	precisely	precisely	ADV
ejpam-6781	81	2	,	,	PUNCT
ejpam-6781	81	3	g(r	g(r	PROPN
ejpam-6781	81	4	)	)	PUNCT
ejpam-6781	81	5	:	:	PUNCT
ejpam-6781	82	1	=	=	PRON
ejpam-6781	82	2	{	{	PUNCT
ejpam-6781	82	3	ζ	ζ	NOUN
ejpam-6781	82	4	=	=	SYM
ejpam-6781	82	5	u	u	NOUN
ejpam-6781	82	6	+	+	NOUN
ejpam-6781	82	7	vω	vω	INTJ
ejpam-6781	82	8	∈	∈	PROPN
ejpam-6781	82	9	c	c	PROPN
ejpam-6781	82	10	∣∣∣u	∣∣∣u	PROPN
ejpam-6781	82	11	,	,	PUNCT
ejpam-6781	82	12	v	v	NOUN
ejpam-6781	82	13	∈	∈	PROPN
ejpam-6781	82	14	r	r	NOUN
ejpam-6781	82	15	,	,	PUNCT
ejpam-6781	82	16	n(ζ	n(ζ	PROPN
ejpam-6781	82	17	)	)	PUNCT
ejpam-6781	82	18	=	=	SYM
ejpam-6781	82	19	√	√	PROPN
ejpam-6781	82	20	u2	u2	NOUN
ejpam-6781	82	21	−	−	PROPN
ejpam-6781	82	22	uv	uv	NOUN
ejpam-6781	82	23	+	+	CCONJ
ejpam-6781	82	24	v2	v2	NOUN
ejpam-6781	82	25	=	=	SYM
ejpam-6781	82	26	1	1	NUM
ejpam-6781	82	27	}	}	PUNCT
ejpam-6781	82	28	.	.	PUNCT
ejpam-6781	83	1	s.	s.	PROPN
ejpam-6781	83	2	jitman	jitman	PROPN
ejpam-6781	83	3	,	,	PUNCT
ejpam-6781	83	4	m.	m.	NOUN
ejpam-6781	83	5	mohammad	mohammad	PROPN
ejpam-6781	83	6	,	,	PUNCT
ejpam-6781	83	7	e.	e.	PROPN
ejpam-6781	83	8	sangwisut	sangwisut	PROPN
ejpam-6781	83	9	/	/	SYM
ejpam-6781	83	10	eur	eur	PROPN
ejpam-6781	83	11	.	.	PUNCT
ejpam-6781	84	1	j.	j.	PROPN
ejpam-6781	84	2	pure	pure	PROPN
ejpam-6781	84	3	appl	appl	PROPN
ejpam-6781	84	4	.	.	PROPN
ejpam-6781	84	5	math	math	PROPN
ejpam-6781	84	6	,	,	PUNCT
ejpam-6781	84	7	18	18	NUM
ejpam-6781	84	8	(	(	PUNCT
ejpam-6781	84	9	4	4	NUM
ejpam-6781	84	10	)	)	PUNCT
ejpam-6781	84	11	(	(	PUNCT
ejpam-6781	84	12	2025	2025	NUM
ejpam-6781	84	13	)	)	PUNCT
ejpam-6781	84	14	,	,	PUNCT
ejpam-6781	84	15	6781	6781	NUM
ejpam-6781	84	16	5	5	NUM
ejpam-6781	84	17	of	of	ADP
ejpam-6781	84	18	16	16	NUM
ejpam-6781	84	19	the	the	DET
ejpam-6781	84	20	set	set	VERB
ejpam-6781	84	21	g(r	g(r	NOUN
ejpam-6781	84	22	)	)	PUNCT
ejpam-6781	84	23	forms	form	VERB
ejpam-6781	84	24	an	an	DET
ejpam-6781	84	25	abelian	abelian	ADJ
ejpam-6781	84	26	group	group	NOUN
ejpam-6781	84	27	under	under	ADP
ejpam-6781	84	28	the	the	DET
ejpam-6781	84	29	standard	standard	ADJ
ejpam-6781	84	30	multiplication	multiplication	NOUN
ejpam-6781	84	31	in	in	ADP
ejpam-6781	84	32	c.	c.	PROPN
ejpam-6781	84	33	moreover	moreover	ADV
ejpam-6781	84	34	,	,	PUNCT
ejpam-6781	84	35	n(1	n(1	NOUN
ejpam-6781	84	36	)	)	PUNCT
ejpam-6781	84	37	=	=	SYM
ejpam-6781	84	38	1	1	NUM
ejpam-6781	84	39	,	,	PUNCT
ejpam-6781	84	40	n(ζ1ζ2	n(ζ1ζ2	NUM
ejpam-6781	84	41	)	)	PUNCT
ejpam-6781	84	42	=	=	SYM
ejpam-6781	84	43	n(ζ1)n(ζ2	n(ζ1)n(ζ2	PROPN
ejpam-6781	84	44	)	)	PUNCT
ejpam-6781	84	45	,	,	PUNCT
ejpam-6781	84	46	and	and	CCONJ
ejpam-6781	84	47	n(ζ−1	n(ζ−1	NUM
ejpam-6781	84	48	)	)	PUNCT
ejpam-6781	84	49	=	=	SYM
ejpam-6781	84	50	n(ζ)−1	n(ζ)−1	PROPN
ejpam-6781	84	51	.	.	PUNCT
ejpam-6781	85	1	let	let	VERB
ejpam-6781	85	2	g(q	g(q	NOUN
ejpam-6781	85	3	)	)	PUNCT
ejpam-6781	85	4	be	be	AUX
ejpam-6781	85	5	the	the	DET
ejpam-6781	85	6	subset	subset	NOUN
ejpam-6781	85	7	of	of	ADP
ejpam-6781	85	8	g(r	g(r	PROPN
ejpam-6781	85	9	)	)	PUNCT
ejpam-6781	85	10	of	of	ADP
ejpam-6781	85	11	the	the	DET
ejpam-6781	85	12	form	form	NOUN
ejpam-6781	85	13	g(q	g(q	NOUN
ejpam-6781	85	14	)	)	PUNCT
ejpam-6781	85	15	:	:	PUNCT
ejpam-6781	86	1	=	=	SYM
ejpam-6781	86	2	{	{	PUNCT
ejpam-6781	86	3	ζ	ζ	NOUN
ejpam-6781	86	4	=	=	SYM
ejpam-6781	86	5	u	u	NOUN
ejpam-6781	86	6	+	+	PROPN
ejpam-6781	86	7	vω	vω	PROPN
ejpam-6781	86	8	∣∣∣u	∣∣∣u	PROPN
ejpam-6781	86	9	,	,	PUNCT
ejpam-6781	86	10	v	v	NOUN
ejpam-6781	86	11	∈	∈	PROPN
ejpam-6781	86	12	q	q	NOUN
ejpam-6781	86	13	,	,	PUNCT
ejpam-6781	86	14	n(ζ	n(ζ	PROPN
ejpam-6781	86	15	)	)	PUNCT
ejpam-6781	86	16	=	=	SYM
ejpam-6781	86	17	√	√	PROPN
ejpam-6781	86	18	u2	u2	NOUN
ejpam-6781	86	19	−	−	PROPN
ejpam-6781	87	1	uv	uv	NOUN
ejpam-6781	87	2	+	+	CCONJ
ejpam-6781	87	3	v2	v2	NOUN
ejpam-6781	87	4	=	=	SYM
ejpam-6781	87	5	1	1	NUM
ejpam-6781	87	6	}	}	PUNCT
ejpam-6781	87	7	.	.	PUNCT
ejpam-6781	88	1	(	(	PUNCT
ejpam-6781	88	2	3	3	X
ejpam-6781	88	3	)	)	PUNCT
ejpam-6781	88	4	the	the	DET
ejpam-6781	88	5	set	set	NOUN
ejpam-6781	88	6	g(q	g(q	NOUN
ejpam-6781	88	7	)	)	PUNCT
ejpam-6781	88	8	is	be	AUX
ejpam-6781	88	9	called	call	VERB
ejpam-6781	88	10	the	the	DET
ejpam-6781	88	11	ω	ω	ADJ
ejpam-6781	88	12	-	-	ADJ
ejpam-6781	88	13	rational	rational	ADJ
ejpam-6781	88	14	unit	unit	NOUN
ejpam-6781	88	15	circle	circle	NOUN
ejpam-6781	88	16	and	and	CCONJ
ejpam-6781	88	17	each	each	DET
ejpam-6781	88	18	element	element	NOUN
ejpam-6781	88	19	in	in	ADP
ejpam-6781	88	20	g(q	g(q	NOUN
ejpam-6781	88	21	)	)	PUNCT
ejpam-6781	88	22	is	be	AUX
ejpam-6781	88	23	called	call	VERB
ejpam-6781	88	24	an	an	DET
ejpam-6781	88	25	ω	ω	ADJ
ejpam-6781	88	26	-	-	ADJ
ejpam-6781	88	27	rational	rational	ADJ
ejpam-6781	88	28	point	point	NOUN
ejpam-6781	88	29	.	.	PUNCT
ejpam-6781	89	1	we	we	PRON
ejpam-6781	89	2	observe	observe	VERB
ejpam-6781	89	3	that	that	SCONJ
ejpam-6781	89	4	although	although	SCONJ
ejpam-6781	89	5	i	i	PRON
ejpam-6781	89	6	∈	∈	PROPN
ejpam-6781	89	7	g(r	g(r	PROPN
ejpam-6781	89	8	)	)	PUNCT
ejpam-6781	89	9	but	but	CCONJ
ejpam-6781	89	10	it	it	PRON
ejpam-6781	89	11	is	be	AUX
ejpam-6781	89	12	not	not	PART
ejpam-6781	89	13	in	in	ADP
ejpam-6781	89	14	g(q	g(q	NOUN
ejpam-6781	89	15	)	)	PUNCT
ejpam-6781	89	16	.	.	PUNCT
ejpam-6781	90	1	it	it	PRON
ejpam-6781	90	2	is	be	AUX
ejpam-6781	90	3	interesting	interesting	ADJ
ejpam-6781	90	4	to	to	PART
ejpam-6781	90	5	investigate	investigate	VERB
ejpam-6781	90	6	properties	property	NOUN
ejpam-6781	90	7	further	far	ADV
ejpam-6781	90	8	.	.	PUNCT
ejpam-6781	91	1	proposition	proposition	NOUN
ejpam-6781	91	2	1	1	NUM
ejpam-6781	91	3	.	.	PUNCT
ejpam-6781	92	1	the	the	DET
ejpam-6781	92	2	set	set	NOUN
ejpam-6781	92	3	g(q	g(q	NOUN
ejpam-6781	92	4	)	)	PUNCT
ejpam-6781	92	5	is	be	AUX
ejpam-6781	92	6	a	a	DET
ejpam-6781	92	7	subgroup	subgroup	NOUN
ejpam-6781	92	8	of	of	ADP
ejpam-6781	92	9	g(r	g(r	PROPN
ejpam-6781	92	10	)	)	PUNCT
ejpam-6781	92	11	.	.	PUNCT
ejpam-6781	93	1	proof	proof	NOUN
ejpam-6781	93	2	.	.	PUNCT
ejpam-6781	94	1	it	it	PRON
ejpam-6781	94	2	is	be	AUX
ejpam-6781	94	3	easy	easy	ADJ
ejpam-6781	94	4	to	to	PART
ejpam-6781	94	5	see	see	VERB
ejpam-6781	94	6	that	that	SCONJ
ejpam-6781	94	7	1	1	NUM
ejpam-6781	94	8	=	=	SYM
ejpam-6781	94	9	1	1	NUM
ejpam-6781	94	10	+	+	SYM
ejpam-6781	94	11	0	0	NUM
ejpam-6781	94	12	·	·	PUNCT
ejpam-6781	94	13	ω	ω	NUM
ejpam-6781	94	14	∈	∈	PROPN
ejpam-6781	94	15	g(q	g(q	NOUN
ejpam-6781	94	16	)	)	PUNCT
ejpam-6781	94	17	which	which	PRON
ejpam-6781	94	18	implies	imply	VERB
ejpam-6781	94	19	that	that	SCONJ
ejpam-6781	94	20	g(q	g(q	NOUN
ejpam-6781	94	21	)	)	PUNCT
ejpam-6781	94	22	̸=	̸=	PROPN
ejpam-6781	94	23	∅.	∅.	VERB
ejpam-6781	94	24	for	for	ADP
ejpam-6781	94	25	elements	element	NOUN
ejpam-6781	94	26	u1	u1	NOUN
ejpam-6781	94	27	+	+	CCONJ
ejpam-6781	94	28	v1ω	v1ω	ADP
ejpam-6781	94	29	,	,	PUNCT
ejpam-6781	94	30	u2	u2	PROPN
ejpam-6781	94	31	+	+	CCONJ
ejpam-6781	94	32	v2ω	v2ω	PROPN
ejpam-6781	94	33	∈	∈	PROPN
ejpam-6781	94	34	g(q	g(q	NOUN
ejpam-6781	94	35	)	)	PUNCT
ejpam-6781	94	36	,	,	PUNCT
ejpam-6781	94	37	the	the	DET
ejpam-6781	94	38	product	product	NOUN
ejpam-6781	94	39	(	(	PUNCT
ejpam-6781	94	40	u1	u1	NOUN
ejpam-6781	94	41	+	+	CCONJ
ejpam-6781	94	42	v1ω)(u2	v1ω)(u2	NOUN
ejpam-6781	94	43	+	+	CCONJ
ejpam-6781	94	44	v2ω)−1	v2ω)−1	NOUN
ejpam-6781	94	45	=	=	SYM
ejpam-6781	94	46	u1+v1ω	u1+v1ω	NOUN
ejpam-6781	94	47	u2+v2ω	u2+v2ω	PROPN
ejpam-6781	94	48	can	can	AUX
ejpam-6781	94	49	be	be	AUX
ejpam-6781	94	50	expressed	express	VERB
ejpam-6781	94	51	as	as	ADP
ejpam-6781	94	52	u1+v1ω	u1+v1ω	NOUN
ejpam-6781	94	53	u2+v2ω	u2+v2ω	PROPN
ejpam-6781	94	54	=	=	SYM
ejpam-6781	94	55	(	(	PUNCT
ejpam-6781	94	56	u1+v1ω)(u2+v2ω	u1+v1ω)(u2+v2ω	ADJ
ejpam-6781	94	57	)	)	PUNCT
ejpam-6781	94	58	u2	u2	PROPN
ejpam-6781	94	59	2−u2v2+v22	2−u2v2+v22	PROPN
ejpam-6781	94	60	.	.	PUNCT
ejpam-6781	95	1	simplifying	simplify	VERB
ejpam-6781	95	2	the	the	DET
ejpam-6781	95	3	numerator	numerator	NOUN
ejpam-6781	95	4	and	and	CCONJ
ejpam-6781	95	5	denominator	denominator	NOUN
ejpam-6781	95	6	yields	yield	NOUN
ejpam-6781	95	7	(	(	PUNCT
ejpam-6781	95	8	u1u2+v1v2−u1v2)+(v1u2−u1v2)ω	u1u2+v1v2−u1v2)+(v1u2−u1v2)ω	ADP
ejpam-6781	95	9	u2	u2	PROPN
ejpam-6781	95	10	2−u2v2+v22	2−u2v2+v22	PROPN
ejpam-6781	95	11	.	.	PUNCT
ejpam-6781	96	1	this	this	PRON
ejpam-6781	96	2	can	can	AUX
ejpam-6781	96	3	be	be	AUX
ejpam-6781	96	4	written	write	VERB
ejpam-6781	96	5	as	as	ADP
ejpam-6781	96	6	u1u2+v1v2−u1v2	u1u2+v1v2−u1v2	PROPN
ejpam-6781	96	7	u2	u2	PROPN
ejpam-6781	96	8	2−u2v2+v22	2−u2v2+v22	PROPN
ejpam-6781	96	9	+	+	CCONJ
ejpam-6781	96	10	v1u2−u1v2	v1u2−u1v2	PROPN
ejpam-6781	96	11	u2	u2	PROPN
ejpam-6781	96	12	2−u2v2+v22	2−u2v2+v22	NUM
ejpam-6781	96	13	ω	ω	PROPN
ejpam-6781	96	14	.	.	PUNCT
ejpam-6781	97	1	since	since	SCONJ
ejpam-6781	97	2	u1	u1	NOUN
ejpam-6781	97	3	,	,	PUNCT
ejpam-6781	97	4	v1	v1	NOUN
ejpam-6781	97	5	,	,	PUNCT
ejpam-6781	97	6	u2	u2	NOUN
ejpam-6781	97	7	,	,	PUNCT
ejpam-6781	97	8	v2	v2	PROPN
ejpam-6781	97	9	∈	∈	PROPN
ejpam-6781	97	10	q	q	NOUN
ejpam-6781	97	11	,	,	PUNCT
ejpam-6781	97	12	the	the	DET
ejpam-6781	97	13	result	result	NOUN
ejpam-6781	97	14	is	be	AUX
ejpam-6781	97	15	in	in	ADP
ejpam-6781	97	16	g(q	g(q	NOUN
ejpam-6781	97	17	)	)	PUNCT
ejpam-6781	97	18	.	.	PUNCT
ejpam-6781	98	1	for	for	ADP
ejpam-6781	98	2	a	a	DET
ejpam-6781	98	3	given	give	VERB
ejpam-6781	98	4	primitive	primitive	ADJ
ejpam-6781	98	5	eisenstein	eisenstein	NOUN
ejpam-6781	98	6	triple	triple	NOUN
ejpam-6781	98	7	(	(	PUNCT
ejpam-6781	98	8	a	a	DET
ejpam-6781	98	9	,	,	PUNCT
ejpam-6781	98	10	b	b	NOUN
ejpam-6781	98	11	,	,	PUNCT
ejpam-6781	98	12	c	c	NOUN
ejpam-6781	98	13	)	)	PUNCT
ejpam-6781	98	14	,	,	PUNCT
ejpam-6781	98	15	define	define	VERB
ejpam-6781	98	16	the	the	DET
ejpam-6781	98	17	associated	associated	ADJ
ejpam-6781	98	18	eisenstein	eisenstein	NOUN
ejpam-6781	98	19	integer	integer	PROPN
ejpam-6781	98	20	z	z	NOUN
ejpam-6781	99	1	:	:	PUNCT
ejpam-6781	99	2	=	=	PUNCT
ejpam-6781	99	3	a	a	DET
ejpam-6781	99	4	+	+	X
ejpam-6781	99	5	bω	bω	NOUN
ejpam-6781	99	6	,	,	PUNCT
ejpam-6781	99	7	whose	whose	DET
ejpam-6781	99	8	norm	norm	NOUN
ejpam-6781	99	9	is	be	AUX
ejpam-6781	99	10	given	give	VERB
ejpam-6781	99	11	by	by	ADP
ejpam-6781	99	12	n(z	n(z	NOUN
ejpam-6781	99	13	)	)	PUNCT
ejpam-6781	99	14	=	=	SYM
ejpam-6781	99	15	√	√	NUM
ejpam-6781	99	16	a2	a2	PROPN
ejpam-6781	99	17	−	−	PROPN
ejpam-6781	99	18	ab	ab	PROPN
ejpam-6781	99	19	+	+	CCONJ
ejpam-6781	99	20	b2	b2	NOUN
ejpam-6781	99	21	=	=	SYM
ejpam-6781	99	22	c.	c.	NOUN
ejpam-6781	99	23	it	it	PRON
ejpam-6781	99	24	follows	follow	VERB
ejpam-6781	99	25	immediately	immediately	ADV
ejpam-6781	99	26	that	that	SCONJ
ejpam-6781	99	27	2a	2a	NUM
ejpam-6781	99	28	̸=	̸=	PROPN
ejpam-6781	99	29	b.	b.	PROPN
ejpam-6781	99	30	otherwise	otherwise	ADV
ejpam-6781	99	31	,	,	PUNCT
ejpam-6781	99	32	we	we	PRON
ejpam-6781	99	33	would	would	AUX
ejpam-6781	99	34	have	have	VERB
ejpam-6781	99	35	c	c	NOUN
ejpam-6781	99	36	=	=	SYM
ejpam-6781	99	37	√	√	PROPN
ejpam-6781	99	38	3a	3a	NUM
ejpam-6781	99	39	,	,	PUNCT
ejpam-6781	99	40	which	which	PRON
ejpam-6781	99	41	contradicts	contradict	VERB
ejpam-6781	99	42	the	the	DET
ejpam-6781	99	43	assumption	assumption	NOUN
ejpam-6781	99	44	that	that	SCONJ
ejpam-6781	99	45	c	c	PROPN
ejpam-6781	99	46	is	be	AUX
ejpam-6781	99	47	an	an	DET
ejpam-6781	99	48	integer	integer	NOUN
ejpam-6781	99	49	.	.	PUNCT
ejpam-6781	100	1	the	the	DET
ejpam-6781	100	2	corresponding	corresponding	ADJ
ejpam-6781	100	3	point	point	NOUN
ejpam-6781	100	4	in	in	ADP
ejpam-6781	100	5	the	the	DET
ejpam-6781	100	6	unit	unit	NOUN
ejpam-6781	100	7	circle	circle	NOUN
ejpam-6781	100	8	is	be	AUX
ejpam-6781	100	9	the	the	DET
ejpam-6781	100	10	normalized	normalize	VERB
ejpam-6781	100	11	complex	complex	ADJ
ejpam-6781	100	12	number	number	NOUN
ejpam-6781	100	13	ζ	ζ	NOUN
ejpam-6781	100	14	:	:	PUNCT
ejpam-6781	100	15	=	=	SYM
ejpam-6781	100	16	z	z	NOUN
ejpam-6781	100	17	n(z	n(z	NOUN
ejpam-6781	100	18	)	)	PUNCT
ejpam-6781	100	19	=	=	PUNCT
ejpam-6781	100	20	a	a	DET
ejpam-6781	100	21	c	c	NOUN
ejpam-6781	101	1	+	+	PROPN
ejpam-6781	101	2	b	b	PROPN
ejpam-6781	101	3	c	c	X
ejpam-6781	101	4	ω	ω	PROPN
ejpam-6781	101	5	=	=	PROPN
ejpam-6781	102	1	2a−	2a−	NUM
ejpam-6781	102	2	b	b	PROPN
ejpam-6781	102	3	2c	2c	NUM
ejpam-6781	102	4	+	+	CCONJ
ejpam-6781	102	5	b	b	NOUN
ejpam-6781	102	6	√	√	NUM
ejpam-6781	102	7	3	3	NUM
ejpam-6781	102	8	2c	2c	NUM
ejpam-6781	102	9	i	i	PRON
ejpam-6781	102	10	∈	∈	PROPN
ejpam-6781	102	11	g(r	g(r	PROPN
ejpam-6781	102	12	)	)	PUNCT
ejpam-6781	102	13	.	.	PUNCT
ejpam-6781	103	1	since	since	SCONJ
ejpam-6781	103	2	a	a	DET
ejpam-6781	103	3	c	c	NOUN
ejpam-6781	103	4	and	and	CCONJ
ejpam-6781	103	5	b	b	PROPN
ejpam-6781	103	6	c	c	NOUN
ejpam-6781	103	7	are	be	AUX
ejpam-6781	103	8	rational	rational	ADJ
ejpam-6781	103	9	numbers	number	NOUN
ejpam-6781	103	10	,	,	PUNCT
ejpam-6781	103	11	ζ	ζ	NOUN
ejpam-6781	103	12	is	be	AUX
ejpam-6781	103	13	an	an	DET
ejpam-6781	103	14	ω	ω	ADJ
ejpam-6781	103	15	-	-	ADJ
ejpam-6781	103	16	rational	rational	ADJ
ejpam-6781	103	17	point	point	NOUN
ejpam-6781	103	18	in	in	ADP
ejpam-6781	103	19	g(q	g(q	NOUN
ejpam-6781	103	20	)	)	PUNCT
ejpam-6781	103	21	.	.	PUNCT
ejpam-6781	104	1	consequently	consequently	ADV
ejpam-6781	104	2	,	,	PUNCT
ejpam-6781	104	3	each	each	DET
ejpam-6781	104	4	primitive	primitive	ADJ
ejpam-6781	104	5	eisenstein	eisenstein	NOUN
ejpam-6781	104	6	triple	triple	ADV
ejpam-6781	104	7	in	in	ADP
ejpam-6781	104	8	et	et	NOUN
ejpam-6781	104	9	gives	give	VERB
ejpam-6781	104	10	rise	rise	NOUN
ejpam-6781	104	11	to	to	ADP
ejpam-6781	104	12	an	an	DET
ejpam-6781	104	13	ω	ω	ADJ
ejpam-6781	104	14	-	-	ADJ
ejpam-6781	104	15	rational	rational	ADJ
ejpam-6781	104	16	point	point	NOUN
ejpam-6781	104	17	in	in	ADP
ejpam-6781	104	18	the	the	DET
ejpam-6781	104	19	ω	ω	ADJ
ejpam-6781	104	20	-	-	ADJ
ejpam-6781	104	21	rational	rational	ADJ
ejpam-6781	104	22	unit	unit	NOUN
ejpam-6781	104	23	circle	circle	NOUN
ejpam-6781	104	24	g(q	g(q	PROPN
ejpam-6781	104	25	)	)	PUNCT
ejpam-6781	104	26	illustrated	illustrate	VERB
ejpam-6781	104	27	in	in	ADP
ejpam-6781	104	28	figure	figure	NOUN
ejpam-6781	104	29	3	3	NUM
ejpam-6781	104	30	.	.	PUNCT
ejpam-6781	105	1	next	next	ADV
ejpam-6781	105	2	,	,	PUNCT
ejpam-6781	105	3	we	we	PRON
ejpam-6781	105	4	show	show	VERB
ejpam-6781	105	5	that	that	SCONJ
ejpam-6781	105	6	ζ	ζ	NOUN
ejpam-6781	105	7	lies	lie	VERB
ejpam-6781	105	8	in	in	ADP
ejpam-6781	105	9	the	the	DET
ejpam-6781	105	10	second	second	ADJ
ejpam-6781	105	11	sextant	sextant	NOUN
ejpam-6781	105	12	.	.	PUNCT
ejpam-6781	106	1	since	since	SCONJ
ejpam-6781	106	2	im(ζ	im(ζ	NOUN
ejpam-6781	106	3	)	)	PUNCT
ejpam-6781	106	4	>	>	X
ejpam-6781	106	5	0	0	NUM
ejpam-6781	106	6	,	,	PUNCT
ejpam-6781	106	7	it	it	PRON
ejpam-6781	106	8	follows	follow	VERB
ejpam-6781	106	9	that	that	SCONJ
ejpam-6781	106	10	ζ	ζ	NOUN
ejpam-6781	106	11	lies	lie	VERB
ejpam-6781	106	12	in	in	ADP
ejpam-6781	106	13	the	the	DET
ejpam-6781	106	14	upper	upper	ADJ
ejpam-6781	106	15	half	half	ADJ
ejpam-6781	106	16	plane	plane	NOUN
ejpam-6781	106	17	.	.	PUNCT
ejpam-6781	107	1	we	we	PRON
ejpam-6781	107	2	note	note	VERB
ejpam-6781	107	3	that	that	SCONJ
ejpam-6781	107	4	the	the	DET
ejpam-6781	107	5	slope	slope	NOUN
ejpam-6781	107	6	of	of	ADP
ejpam-6781	107	7	the	the	DET
ejpam-6781	107	8	line	line	NOUN
ejpam-6781	107	9	from	from	ADP
ejpam-6781	107	10	the	the	DET
ejpam-6781	107	11	origin	origin	NOUN
ejpam-6781	107	12	to	to	ADP
ejpam-6781	107	13	ζ	ζ	NOUN
ejpam-6781	107	14	is	be	AUX
ejpam-6781	107	15	m	m	NOUN
ejpam-6781	107	16	=	=	SYM
ejpam-6781	107	17	b	b	NOUN
ejpam-6781	107	18	√	√	NUM
ejpam-6781	107	19	3	3	NUM
ejpam-6781	107	20	2a−	2a−	NUM
ejpam-6781	107	21	b	b	NUM
ejpam-6781	107	22	which	which	PRON
ejpam-6781	107	23	implies	imply	VERB
ejpam-6781	107	24	that	that	SCONJ
ejpam-6781	107	25	m	m	VERB
ejpam-6781	107	26	>	>	X
ejpam-6781	107	27	√	√	NUM
ejpam-6781	107	28	3	3	NUM
ejpam-6781	107	29	if	if	SCONJ
ejpam-6781	107	30	2a	2a	NUM
ejpam-6781	107	31	>	>	X
ejpam-6781	107	32	b	b	NOUN
ejpam-6781	107	33	,	,	PUNCT
ejpam-6781	107	34	or	or	CCONJ
ejpam-6781	107	35	m	m	VERB
ejpam-6781	107	36	<	<	X
ejpam-6781	107	37	−	−	NOUN
ejpam-6781	107	38	√	√	NUM
ejpam-6781	107	39	3	3	NUM
ejpam-6781	107	40	if	if	SCONJ
ejpam-6781	107	41	2a	2a	NUM
ejpam-6781	107	42	<	<	X
ejpam-6781	107	43	b.	b.	PROPN
ejpam-6781	107	44	in	in	ADP
ejpam-6781	107	45	both	both	DET
ejpam-6781	107	46	cases	case	NOUN
ejpam-6781	107	47	,	,	PUNCT
ejpam-6781	107	48	ζ	ζ	NOUN
ejpam-6781	107	49	lies	lie	VERB
ejpam-6781	107	50	in	in	ADP
ejpam-6781	107	51	the	the	DET
ejpam-6781	107	52	second	second	ADJ
ejpam-6781	107	53	sextant	sextant	NOUN
ejpam-6781	107	54	.	.	PUNCT
ejpam-6781	108	1	let	let	VERB
ejpam-6781	108	2	ζ	ζ	NOUN
ejpam-6781	108	3	:	:	PUNCT
ejpam-6781	108	4	=	=	SYM
ejpam-6781	108	5	u	u	NOUN
ejpam-6781	108	6	+	+	CCONJ
ejpam-6781	108	7	vω	vω	PRON
ejpam-6781	108	8	∈	∈	PROPN
ejpam-6781	108	9	g(q	g(q	NOUN
ejpam-6781	108	10	)	)	PUNCT
ejpam-6781	108	11	∖	∖	NOUN
ejpam-6781	108	12	{	{	PUNCT
ejpam-6781	108	13	ω,−ω	ω,−ω	AUX
ejpam-6781	108	14	}	}	PUNCT
ejpam-6781	108	15	be	be	AUX
ejpam-6781	108	16	in	in	ADP
ejpam-6781	108	17	the	the	DET
ejpam-6781	108	18	second	second	ADJ
ejpam-6781	108	19	sextant	sextant	NOUN
ejpam-6781	108	20	.	.	PUNCT
ejpam-6781	109	1	let	let	VERB
ejpam-6781	109	2	d	d	PART
ejpam-6781	109	3	denote	denote	VERB
ejpam-6781	109	4	the	the	DET
ejpam-6781	109	5	least	least	ADV
ejpam-6781	109	6	common	common	ADJ
ejpam-6781	109	7	multiple	multiple	NOUN
ejpam-6781	109	8	of	of	ADP
ejpam-6781	109	9	the	the	DET
ejpam-6781	109	10	denominators	denominator	NOUN
ejpam-6781	109	11	of	of	ADP
ejpam-6781	109	12	u	u	PROPN
ejpam-6781	109	13	and	and	CCONJ
ejpam-6781	109	14	v.	v.	CCONJ
ejpam-6781	109	15	then	then	ADV
ejpam-6781	109	16	dζ	dζ	PROPN
ejpam-6781	109	17	=	=	SYM
ejpam-6781	109	18	du	du	PROPN
ejpam-6781	110	1	+	+	CCONJ
ejpam-6781	110	2	dvω	dvω	PROPN
ejpam-6781	110	3	∈	∈	PROPN
ejpam-6781	110	4	z[ω	z[ω	PROPN
ejpam-6781	110	5	]	]	PUNCT
ejpam-6781	110	6	which	which	PRON
ejpam-6781	110	7	ensures	ensure	VERB
ejpam-6781	110	8	that	that	SCONJ
ejpam-6781	110	9	the	the	DET
ejpam-6781	110	10	coordinates	coordinate	NOUN
ejpam-6781	110	11	of	of	ADP
ejpam-6781	110	12	the	the	DET
ejpam-6781	110	13	associated	associated	ADJ
ejpam-6781	110	14	eisenstein	eisenstein	PROPN
ejpam-6781	110	15	triple	triple	ADV
ejpam-6781	110	16	(	(	PUNCT
ejpam-6781	110	17	du	du	PROPN
ejpam-6781	110	18	,	,	PUNCT
ejpam-6781	110	19	dv	dv	PROPN
ejpam-6781	110	20	,	,	PUNCT
ejpam-6781	110	21	d	d	NOUN
ejpam-6781	110	22	)	)	PUNCT
ejpam-6781	110	23	are	be	AUX
ejpam-6781	110	24	integral	integral	ADJ
ejpam-6781	110	25	.	.	PUNCT
ejpam-6781	111	1	alternatively	alternatively	ADV
ejpam-6781	111	2	,	,	PUNCT
ejpam-6781	111	3	we	we	PRON
ejpam-6781	111	4	may	may	AUX
ejpam-6781	111	5	consider	consider	VERB
ejpam-6781	111	6	the	the	DET
ejpam-6781	111	7	primitive	primitive	ADJ
ejpam-6781	111	8	representation	representation	NOUN
ejpam-6781	111	9	of	of	ADP
ejpam-6781	111	10	the	the	DET
ejpam-6781	111	11	triple	triple	ADJ
ejpam-6781	111	12	(	(	PUNCT
ejpam-6781	111	13	du	du	PROPN
ejpam-6781	111	14	,	,	PUNCT
ejpam-6781	111	15	dv	dv	PROPN
ejpam-6781	111	16	,	,	PUNCT
ejpam-6781	111	17	d	d	PROPN
ejpam-6781	111	18	)	)	PUNCT
ejpam-6781	111	19	by	by	ADP
ejpam-6781	111	20	dividing	divide	VERB
ejpam-6781	111	21	by	by	ADP
ejpam-6781	111	22	the	the	DET
ejpam-6781	111	23	greatest	great	ADJ
ejpam-6781	111	24	common	common	ADJ
ejpam-6781	111	25	divisor	divisor	NOUN
ejpam-6781	111	26	.	.	PUNCT
ejpam-6781	112	1	s.	s.	PROPN
ejpam-6781	112	2	jitman	jitman	PROPN
ejpam-6781	112	3	,	,	PUNCT
ejpam-6781	112	4	m.	m.	NOUN
ejpam-6781	112	5	mohammad	mohammad	PROPN
ejpam-6781	112	6	,	,	PUNCT
ejpam-6781	112	7	e.	e.	PROPN
ejpam-6781	112	8	sangwisut	sangwisut	PROPN
ejpam-6781	112	9	/	/	SYM
ejpam-6781	112	10	eur	eur	PROPN
ejpam-6781	112	11	.	.	PUNCT
ejpam-6781	113	1	j.	j.	PROPN
ejpam-6781	113	2	pure	pure	PROPN
ejpam-6781	113	3	appl	appl	PROPN
ejpam-6781	113	4	.	.	PROPN
ejpam-6781	113	5	math	math	PROPN
ejpam-6781	113	6	,	,	PUNCT
ejpam-6781	113	7	18	18	NUM
ejpam-6781	113	8	(	(	PUNCT
ejpam-6781	113	9	4	4	NUM
ejpam-6781	113	10	)	)	PUNCT
ejpam-6781	113	11	(	(	PUNCT
ejpam-6781	113	12	2025	2025	NUM
ejpam-6781	113	13	)	)	PUNCT
ejpam-6781	113	14	,	,	PUNCT
ejpam-6781	113	15	6781	6781	NUM
ejpam-6781	113	16	6	6	NUM
ejpam-6781	113	17	of	of	ADP
ejpam-6781	113	18	16	16	NUM
ejpam-6781	113	19	figure	figure	NOUN
ejpam-6781	113	20	3	3	NUM
ejpam-6781	113	21	:	:	PUNCT
ejpam-6781	113	22	geometric	geometric	ADJ
ejpam-6781	113	23	projection	projection	NOUN
ejpam-6781	113	24	of	of	ADP
ejpam-6781	113	25	the	the	DET
ejpam-6781	113	26	eisenstein	eisenstein	PROPN
ejpam-6781	113	27	integer	integer	NOUN
ejpam-6781	113	28	a+	a+	PUNCT
ejpam-6781	113	29	bω	bω	NOUN
ejpam-6781	113	30	into	into	ADP
ejpam-6781	113	31	the	the	DET
ejpam-6781	113	32	ω	ω	ADJ
ejpam-6781	113	33	-	-	ADJ
ejpam-6781	113	34	rational	rational	ADJ
ejpam-6781	113	35	unit	unit	NOUN
ejpam-6781	113	36	circle	circle	NOUN
ejpam-6781	113	37	by	by	ADP
ejpam-6781	113	38	normalization	normalization	NOUN
ejpam-6781	113	39	.	.	PUNCT
ejpam-6781	114	1	for	for	ADP
ejpam-6781	114	2	example	example	NOUN
ejpam-6781	114	3	,	,	PUNCT
ejpam-6781	114	4	take	take	VERB
ejpam-6781	114	5	ζ	ζ	NOUN
ejpam-6781	114	6	=	=	SYM
ejpam-6781	114	7	18	18	NUM
ejpam-6781	114	8	42	42	NUM
ejpam-6781	114	9	+	+	CCONJ
ejpam-6781	114	10	120	120	NUM
ejpam-6781	114	11	105ω	105ω	NUM
ejpam-6781	114	12	∈	∈	PROPN
ejpam-6781	114	13	g(q	g(q	NOUN
ejpam-6781	114	14	)	)	PUNCT
ejpam-6781	114	15	.	.	PUNCT
ejpam-6781	115	1	then	then	ADV
ejpam-6781	115	2	the	the	DET
ejpam-6781	115	3	least	least	ADV
ejpam-6781	115	4	common	common	ADJ
ejpam-6781	115	5	multiple	multiple	NOUN
ejpam-6781	115	6	of	of	ADP
ejpam-6781	115	7	the	the	DET
ejpam-6781	115	8	denominators	denominator	NOUN
ejpam-6781	115	9	is	be	AUX
ejpam-6781	115	10	d	d	NOUN
ejpam-6781	115	11	=	=	PUNCT
ejpam-6781	115	12	lcm(42	lcm(42	NOUN
ejpam-6781	115	13	,	,	PUNCT
ejpam-6781	115	14	105	105	NUM
ejpam-6781	115	15	)	)	PUNCT
ejpam-6781	115	16	=	=	SYM
ejpam-6781	115	17	210	210	NUM
ejpam-6781	115	18	which	which	PRON
ejpam-6781	115	19	implies	imply	VERB
ejpam-6781	115	20	that	that	SCONJ
ejpam-6781	115	21	210ζ	210ζ	PROPN
ejpam-6781	115	22	=	=	SYM
ejpam-6781	116	1	90	90	NUM
ejpam-6781	116	2	+	+	NUM
ejpam-6781	116	3	240ω	240ω	PROPN
ejpam-6781	116	4	∈	∈	PROPN
ejpam-6781	116	5	z[ω	z[ω	PROPN
ejpam-6781	116	6	]	]	PUNCT
ejpam-6781	116	7	.	.	PUNCT
ejpam-6781	117	1	hence	hence	ADV
ejpam-6781	117	2	,	,	PUNCT
ejpam-6781	117	3	the	the	DET
ejpam-6781	117	4	corresponding	correspond	VERB
ejpam-6781	117	5	eisenstein	eisenstein	NOUN
ejpam-6781	117	6	triple	triple	NOUN
ejpam-6781	117	7	is	be	AUX
ejpam-6781	117	8	(	(	PUNCT
ejpam-6781	117	9	90	90	NUM
ejpam-6781	117	10	,	,	PUNCT
ejpam-6781	117	11	240	240	NUM
ejpam-6781	117	12	,	,	PUNCT
ejpam-6781	117	13	210	210	NUM
ejpam-6781	117	14	)	)	PUNCT
ejpam-6781	117	15	whose	whose	DET
ejpam-6781	117	16	primitive	primitive	ADJ
ejpam-6781	117	17	representative	representative	NOUN
ejpam-6781	117	18	(	(	PUNCT
ejpam-6781	117	19	after	after	ADP
ejpam-6781	117	20	dividing	divide	VERB
ejpam-6781	117	21	by	by	ADP
ejpam-6781	117	22	the	the	DET
ejpam-6781	117	23	greatest	great	ADJ
ejpam-6781	117	24	common	common	ADJ
ejpam-6781	117	25	divisor	divisor	NOUN
ejpam-6781	117	26	30	30	NUM
ejpam-6781	117	27	)	)	PUNCT
ejpam-6781	117	28	is	be	AUX
ejpam-6781	117	29	(	(	PUNCT
ejpam-6781	117	30	3	3	NUM
ejpam-6781	117	31	,	,	PUNCT
ejpam-6781	117	32	8	8	NUM
ejpam-6781	117	33	,	,	PUNCT
ejpam-6781	117	34	7	7	X
ejpam-6781	117	35	)	)	PUNCT
ejpam-6781	117	36	∈	∈	PROPN
ejpam-6781	117	37	et	et	NOUN
ejpam-6781	117	38	.	.	PUNCT
ejpam-6781	118	1	the	the	DET
ejpam-6781	118	2	set	set	NOUN
ejpam-6781	118	3	of	of	ADP
ejpam-6781	118	4	associate	associate	ADJ
ejpam-6781	118	5	elements	element	NOUN
ejpam-6781	118	6	of	of	ADP
ejpam-6781	118	7	ζ	ζ	NOUN
ejpam-6781	118	8	∈	∈	PROPN
ejpam-6781	118	9	g(q	g(q	NOUN
ejpam-6781	118	10	)	)	PUNCT
ejpam-6781	118	11	is	be	AUX
ejpam-6781	118	12	denoted	denote	VERB
ejpam-6781	118	13	by	by	ADP
ejpam-6781	118	14	uζ	uζ	PROPN
ejpam-6781	118	15	=	=	X
ejpam-6781	118	16	{	{	PUNCT
ejpam-6781	118	17	ζ	ζ	NOUN
ejpam-6781	118	18	,	,	PUNCT
ejpam-6781	118	19	(	(	PUNCT
ejpam-6781	118	20	−ω)ζ	−ω)ζ	NOUN
ejpam-6781	118	21	,	,	PUNCT
ejpam-6781	118	22	(	(	PUNCT
ejpam-6781	118	23	−ω)2ζ	−ω)2ζ	NUM
ejpam-6781	118	24	,	,	PUNCT
ejpam-6781	118	25	(	(	PUNCT
ejpam-6781	118	26	−ω)3ζ	−ω)3ζ	NOUN
ejpam-6781	118	27	,	,	PUNCT
ejpam-6781	118	28	(	(	PUNCT
ejpam-6781	118	29	−ω)4ζ	−ω)4ζ	NUM
ejpam-6781	118	30	,	,	PUNCT
ejpam-6781	118	31	(	(	PUNCT
ejpam-6781	118	32	−ω)5ζ	−ω)5ζ	NUM
ejpam-6781	118	33	}	}	PUNCT
ejpam-6781	118	34	(	(	PUNCT
ejpam-6781	118	35	4	4	NUM
ejpam-6781	118	36	)	)	PUNCT
ejpam-6781	118	37	which	which	PRON
ejpam-6781	118	38	represents	represent	VERB
ejpam-6781	118	39	six	six	NUM
ejpam-6781	118	40	distinct	distinct	ADJ
ejpam-6781	118	41	points	point	NOUN
ejpam-6781	118	42	on	on	ADP
ejpam-6781	118	43	g(q	g(q	NOUN
ejpam-6781	118	44	)	)	PUNCT
ejpam-6781	118	45	,	,	PUNCT
ejpam-6781	118	46	each	each	PRON
ejpam-6781	118	47	located	locate	VERB
ejpam-6781	118	48	in	in	ADP
ejpam-6781	118	49	a	a	DET
ejpam-6781	118	50	different	different	ADJ
ejpam-6781	118	51	sextant	sextant	NOUN
ejpam-6781	118	52	(	(	PUNCT
ejpam-6781	118	53	see	see	VERB
ejpam-6781	118	54	figure	figure	NOUN
ejpam-6781	118	55	4	4	NUM
ejpam-6781	118	56	)	)	PUNCT
ejpam-6781	118	57	.	.	PUNCT
ejpam-6781	119	1	in	in	ADP
ejpam-6781	119	2	other	other	ADJ
ejpam-6781	119	3	words	word	NOUN
ejpam-6781	119	4	,	,	PUNCT
ejpam-6781	119	5	multiplying	multiply	VERB
ejpam-6781	119	6	ζ	ζ	NOUN
ejpam-6781	119	7	by	by	ADP
ejpam-6781	119	8	−ω	−ω	ADJ
ejpam-6781	119	9	results	result	NOUN
ejpam-6781	119	10	in	in	ADP
ejpam-6781	119	11	a	a	DET
ejpam-6781	119	12	60	60	NUM
ejpam-6781	119	13	◦	◦	NOUN
ejpam-6781	119	14	clockwise	clockwise	NOUN
ejpam-6781	119	15	rotation	rotation	NOUN
ejpam-6781	119	16	of	of	ADP
ejpam-6781	119	17	ζ	ζ	PROPN
ejpam-6781	119	18	.	.	PUNCT
ejpam-6781	119	19	figure	figure	NOUN
ejpam-6781	119	20	4	4	NUM
ejpam-6781	119	21	:	:	PUNCT
ejpam-6781	119	22	the	the	DET
ejpam-6781	119	23	associate	associate	ADJ
ejpam-6781	119	24	elements	element	NOUN
ejpam-6781	119	25	of	of	ADP
ejpam-6781	119	26	ζ	ζ	NOUN
ejpam-6781	119	27	in	in	ADP
ejpam-6781	119	28	g(q	g(q	NOUN
ejpam-6781	119	29	)	)	PUNCT
ejpam-6781	119	30	.	.	PUNCT
ejpam-6781	120	1	s.	s.	PROPN
ejpam-6781	120	2	jitman	jitman	PROPN
ejpam-6781	120	3	,	,	PUNCT
ejpam-6781	120	4	m.	m.	NOUN
ejpam-6781	120	5	mohammad	mohammad	PROPN
ejpam-6781	120	6	,	,	PUNCT
ejpam-6781	120	7	e.	e.	PROPN
ejpam-6781	120	8	sangwisut	sangwisut	PROPN
ejpam-6781	120	9	/	/	SYM
ejpam-6781	120	10	eur	eur	PROPN
ejpam-6781	120	11	.	.	PUNCT
ejpam-6781	121	1	j.	j.	PROPN
ejpam-6781	121	2	pure	pure	PROPN
ejpam-6781	121	3	appl	appl	PROPN
ejpam-6781	121	4	.	.	PROPN
ejpam-6781	121	5	math	math	PROPN
ejpam-6781	121	6	,	,	PUNCT
ejpam-6781	121	7	18	18	NUM
ejpam-6781	121	8	(	(	PUNCT
ejpam-6781	121	9	4	4	NUM
ejpam-6781	121	10	)	)	PUNCT
ejpam-6781	121	11	(	(	PUNCT
ejpam-6781	121	12	2025	2025	NUM
ejpam-6781	121	13	)	)	PUNCT
ejpam-6781	121	14	,	,	PUNCT
ejpam-6781	121	15	6781	6781	NUM
ejpam-6781	121	16	7	7	NUM
ejpam-6781	121	17	of	of	ADP
ejpam-6781	121	18	16	16	NUM
ejpam-6781	121	19	let	let	VERB
ejpam-6781	121	20	ζ	ζ	NOUN
ejpam-6781	121	21	be	be	AUX
ejpam-6781	121	22	a	a	DET
ejpam-6781	121	23	point	point	NOUN
ejpam-6781	121	24	in	in	ADP
ejpam-6781	121	25	g(q)∖u	g(q)∖u	NOUN
ejpam-6781	121	26	.	.	PUNCT
ejpam-6781	122	1	let	let	VERB
ejpam-6781	122	2	f	f	PRON
ejpam-6781	122	3	br	br	VERB
ejpam-6781	122	4	a	a	DET
ejpam-6781	122	5	function	function	NOUN
ejpam-6781	122	6	responsible	responsible	ADJ
ejpam-6781	122	7	for	for	ADP
ejpam-6781	122	8	rotating	rotate	VERB
ejpam-6781	122	9	ζ	ζ	NOUN
ejpam-6781	122	10	clockwise	clockwise	NOUN
ejpam-6781	122	11	into	into	ADP
ejpam-6781	122	12	the	the	DET
ejpam-6781	122	13	second	second	ADJ
ejpam-6781	122	14	sextant	sextant	NOUN
ejpam-6781	122	15	.	.	PUNCT
ejpam-6781	123	1	precisely	precisely	ADV
ejpam-6781	123	2	,	,	PUNCT
ejpam-6781	123	3	f(ζ	f(ζ	NOUN
ejpam-6781	123	4	)	)	PUNCT
ejpam-6781	124	1	=	=	PUNCT
ejpam-6781	124	2	(	(	PUNCT
ejpam-6781	124	3	−ω)i−2ζ	−ω)i−2ζ	VERB
ejpam-6781	124	4	for	for	ADP
ejpam-6781	124	5	all	all	DET
ejpam-6781	124	6	ζ	ζ	NOUN
ejpam-6781	124	7	in	in	ADP
ejpam-6781	124	8	the	the	DET
ejpam-6781	124	9	ith	ith	PROPN
ejpam-6781	124	10	sextant	sextant	NOUN
ejpam-6781	124	11	(	(	PUNCT
ejpam-6781	124	12	for	for	ADP
ejpam-6781	124	13	1	1	NUM
ejpam-6781	124	14	≤	≤	NUM
ejpam-6781	124	15	i	i	PRON
ejpam-6781	124	16	≤	≤	ADV
ejpam-6781	124	17	6	6	NUM
ejpam-6781	124	18	)	)	PUNCT
ejpam-6781	124	19	.	.	PUNCT
ejpam-6781	125	1	this	this	DET
ejpam-6781	125	2	operation	operation	NOUN
ejpam-6781	125	3	ensures	ensure	VERB
ejpam-6781	125	4	that	that	SCONJ
ejpam-6781	125	5	f(ζ	f(ζ	NOUN
ejpam-6781	125	6	)	)	PUNCT
ejpam-6781	125	7	locates	locate	VERB
ejpam-6781	125	8	in	in	ADP
ejpam-6781	125	9	the	the	DET
ejpam-6781	125	10	second	second	ADJ
ejpam-6781	125	11	sextant	sextant	NOUN
ejpam-6781	125	12	.	.	PUNCT
ejpam-6781	126	1	next	next	ADV
ejpam-6781	126	2	,	,	PUNCT
ejpam-6781	126	3	let	let	VERB
ejpam-6781	126	4	g	g	PRON
ejpam-6781	126	5	be	be	AUX
ejpam-6781	126	6	a	a	DET
ejpam-6781	126	7	function	function	NOUN
ejpam-6781	126	8	that	that	PRON
ejpam-6781	126	9	maps	map	VERB
ejpam-6781	126	10	points	point	NOUN
ejpam-6781	126	11	on	on	ADP
ejpam-6781	126	12	the	the	DET
ejpam-6781	126	13	second	second	ADJ
ejpam-6781	126	14	sextant	sextant	NOUN
ejpam-6781	126	15	to	to	ADP
ejpam-6781	126	16	the	the	DET
ejpam-6781	126	17	set	set	NOUN
ejpam-6781	126	18	of	of	ADP
ejpam-6781	126	19	eisenstein	eisenstein	PROPN
ejpam-6781	126	20	triples	triple	NOUN
ejpam-6781	126	21	et	et	NOUN
ejpam-6781	126	22	.	.	PUNCT
ejpam-6781	127	1	finally	finally	ADV
ejpam-6781	127	2	,	,	PUNCT
ejpam-6781	127	3	let	let	VERB
ejpam-6781	127	4	et	et	NOUN
ejpam-6781	127	5	=	=	SYM
ejpam-6781	127	6	g	g	PROPN
ejpam-6781	127	7	◦	◦	NOUN
ejpam-6781	127	8	f	f	X
ejpam-6781	127	9	(	(	PUNCT
ejpam-6781	127	10	5	5	X
ejpam-6781	127	11	)	)	PUNCT
ejpam-6781	127	12	be	be	AUX
ejpam-6781	127	13	a	a	DET
ejpam-6781	127	14	composite	composite	ADJ
ejpam-6781	127	15	function	function	NOUN
ejpam-6781	127	16	that	that	PRON
ejpam-6781	127	17	maps	map	VERB
ejpam-6781	127	18	any	any	DET
ejpam-6781	127	19	point	point	NOUN
ejpam-6781	127	20	ζ	ζ	NOUN
ejpam-6781	127	21	∈	∈	PROPN
ejpam-6781	127	22	g(q	g(q	NOUN
ejpam-6781	127	23	)	)	PUNCT
ejpam-6781	127	24	∖	∖	X
ejpam-6781	127	25	u	u	NOUN
ejpam-6781	127	26	to	to	ADP
ejpam-6781	127	27	et	et	PRON
ejpam-6781	127	28	by	by	ADP
ejpam-6781	127	29	first	first	ADV
ejpam-6781	127	30	rotating	rotate	VERB
ejpam-6781	127	31	ζ	ζ	NOUN
ejpam-6781	127	32	clockwise	clockwise	NOUN
ejpam-6781	127	33	into	into	ADP
ejpam-6781	127	34	the	the	DET
ejpam-6781	127	35	second	second	ADJ
ejpam-6781	127	36	sextant	sextant	NOUN
ejpam-6781	127	37	and	and	CCONJ
ejpam-6781	127	38	then	then	ADV
ejpam-6781	127	39	applying	apply	VERB
ejpam-6781	127	40	g.	g.	NOUN
ejpam-6781	127	41	we	we	PRON
ejpam-6781	127	42	demonstrate	demonstrate	VERB
ejpam-6781	127	43	the	the	DET
ejpam-6781	127	44	infinitude	infinitude	NOUN
ejpam-6781	127	45	of	of	ADP
ejpam-6781	127	46	the	the	DET
ejpam-6781	127	47	set	set	NOUN
ejpam-6781	127	48	of	of	ADP
ejpam-6781	127	49	primitive	primitive	ADJ
ejpam-6781	127	50	eisenstein	eisenstein	NOUN
ejpam-6781	127	51	triples	triple	NOUN
ejpam-6781	127	52	by	by	ADP
ejpam-6781	127	53	showing	show	VERB
ejpam-6781	127	54	that	that	SCONJ
ejpam-6781	127	55	g(q	g(q	NOUN
ejpam-6781	127	56	)	)	PUNCT
ejpam-6781	127	57	contains	contain	VERB
ejpam-6781	127	58	infinitely	infinitely	ADV
ejpam-6781	127	59	many	many	ADJ
ejpam-6781	127	60	points	point	NOUN
ejpam-6781	127	61	.	.	PUNCT
ejpam-6781	128	1	proposition	proposition	NOUN
ejpam-6781	128	2	2	2	NUM
ejpam-6781	128	3	.	.	PUNCT
ejpam-6781	129	1	the	the	DET
ejpam-6781	129	2	set	set	NOUN
ejpam-6781	129	3	of	of	ADP
ejpam-6781	129	4	primitive	primitive	ADJ
ejpam-6781	129	5	eisenstein	eisenstein	NOUN
ejpam-6781	129	6	triples	triple	NOUN
ejpam-6781	129	7	is	be	AUX
ejpam-6781	129	8	infinite	infinite	ADJ
ejpam-6781	129	9	.	.	PUNCT
ejpam-6781	130	1	proof	proof	NOUN
ejpam-6781	130	2	.	.	PUNCT
ejpam-6781	131	1	from	from	ADP
ejpam-6781	131	2	(	(	PUNCT
ejpam-6781	131	3	5	5	NUM
ejpam-6781	131	4	)	)	PUNCT
ejpam-6781	131	5	,	,	PUNCT
ejpam-6781	131	6	it	it	PRON
ejpam-6781	131	7	is	be	AUX
ejpam-6781	131	8	not	not	PART
ejpam-6781	131	9	difficult	difficult	ADJ
ejpam-6781	131	10	to	to	PART
ejpam-6781	131	11	see	see	VERB
ejpam-6781	131	12	that	that	SCONJ
ejpam-6781	131	13	the	the	DET
ejpam-6781	131	14	function	function	NOUN
ejpam-6781	131	15	et	et	NOUN
ejpam-6781	131	16	:	:	PUNCT
ejpam-6781	131	17	g(q	g(q	NUM
ejpam-6781	131	18	)	)	PUNCT
ejpam-6781	131	19	∖	∖	X
ejpam-6781	131	20	u	u	NOUN
ejpam-6781	131	21	→	→	X
ejpam-6781	131	22	et	et	PROPN
ejpam-6781	131	23	is	be	AUX
ejpam-6781	131	24	surjective	surjective	ADJ
ejpam-6781	131	25	and	and	CCONJ
ejpam-6781	131	26	it	it	PRON
ejpam-6781	131	27	is	be	AUX
ejpam-6781	131	28	a	a	DET
ejpam-6781	131	29	6	6	NUM
ejpam-6781	131	30	-	-	PUNCT
ejpam-6781	131	31	to-1	to-1	ADP
ejpam-6781	131	32	map	map	NOUN
ejpam-6781	131	33	.	.	PUNCT
ejpam-6781	132	1	to	to	PART
ejpam-6781	132	2	complete	complete	VERB
ejpam-6781	132	3	the	the	DET
ejpam-6781	132	4	proof	proof	NOUN
ejpam-6781	132	5	,	,	PUNCT
ejpam-6781	132	6	it	it	PRON
ejpam-6781	132	7	suffices	suffice	VERB
ejpam-6781	132	8	to	to	PART
ejpam-6781	132	9	show	show	VERB
ejpam-6781	132	10	that	that	SCONJ
ejpam-6781	132	11	g(q	g(q	NOUN
ejpam-6781	132	12	)	)	PUNCT
ejpam-6781	132	13	has	have	VERB
ejpam-6781	132	14	infinite	infinite	ADJ
ejpam-6781	132	15	order	order	NOUN
ejpam-6781	132	16	.	.	PUNCT
ejpam-6781	133	1	for	for	ADP
ejpam-6781	133	2	each	each	DET
ejpam-6781	133	3	element	element	NOUN
ejpam-6781	133	4	m	m	PROPN
ejpam-6781	133	5	∈	∈	PROPN
ejpam-6781	133	6	q	q	AUX
ejpam-6781	133	7	,	,	PUNCT
ejpam-6781	133	8	let	let	VERB
ejpam-6781	133	9	zm	zm	PROPN
ejpam-6781	133	10	=	=	PROPN
ejpam-6781	133	11	1−2	1−2	NUM
ejpam-6781	133	12	m	m	PROPN
ejpam-6781	133	13	1−m+m2	1−m+m2	NUM
ejpam-6781	133	14	+	+	NUM
ejpam-6781	133	15	1−m2	1−m2	NUM
ejpam-6781	133	16	1−m+m2ω	1−m+m2ω	NUM
ejpam-6781	133	17	.	.	PUNCT
ejpam-6781	134	1	then	then	ADV
ejpam-6781	134	2	1−2	1−2	NUM
ejpam-6781	134	3	m	m	PROPN
ejpam-6781	134	4	1−m+m2	1−m+m2	NUM
ejpam-6781	134	5	,	,	PUNCT
ejpam-6781	134	6	1−m2	1−m2	NUM
ejpam-6781	134	7	1−m+m2	1−m+m2	NUM
ejpam-6781	134	8	∈	∈	NOUN
ejpam-6781	134	9	q	q	NOUN
ejpam-6781	134	10	and	and	CCONJ
ejpam-6781	134	11	n(zm	n(zm	ADV
ejpam-6781	134	12	)	)	PUNCT
ejpam-6781	134	13	=	=	SYM
ejpam-6781	134	14	√	√	PROPN
ejpam-6781	134	15	(	(	PUNCT
ejpam-6781	134	16	1−2	1−2	NUM
ejpam-6781	134	17	m	m	PROPN
ejpam-6781	134	18	1−m+m2	1−m+m2	NUM
ejpam-6781	134	19	)	)	PUNCT
ejpam-6781	134	20	2	2	NUM
ejpam-6781	134	21	−	−	PROPN
ejpam-6781	134	22	(	(	PUNCT
ejpam-6781	134	23	1−2	1−2	NUM
ejpam-6781	134	24	m	m	PROPN
ejpam-6781	134	25	1−m+m2	1−m+m2	NUM
ejpam-6781	134	26	)	)	PUNCT
ejpam-6781	134	27	(	(	PUNCT
ejpam-6781	134	28	1−m2	1−m2	NUM
ejpam-6781	134	29	1−m+m2	1−m+m2	NUM
ejpam-6781	134	30	)	)	PUNCT
ejpam-6781	135	1	+	+	CCONJ
ejpam-6781	135	2	(	(	PUNCT
ejpam-6781	135	3	1−m2	1−m2	NUM
ejpam-6781	135	4	1−m+m2	1−m+m2	NUM
ejpam-6781	135	5	)	)	PUNCT
ejpam-6781	135	6	2	2	NUM
ejpam-6781	135	7	=	=	SYM
ejpam-6781	135	8	1	1	X
ejpam-6781	135	9	.	.	PUNCT
ejpam-6781	136	1	it	it	PRON
ejpam-6781	136	2	follows	follow	VERB
ejpam-6781	136	3	that	that	SCONJ
ejpam-6781	136	4	zm	zm	PROPN
ejpam-6781	136	5	∈	∈	PROPN
ejpam-6781	136	6	g(q	g(q	PROPN
ejpam-6781	136	7	)	)	PUNCT
ejpam-6781	136	8	for	for	ADP
ejpam-6781	136	9	all	all	DET
ejpam-6781	136	10	m	m	PROPN
ejpam-6781	136	11	∈	∈	NOUN
ejpam-6781	136	12	q.	q.	NOUN
ejpam-6781	136	13	since	since	SCONJ
ejpam-6781	136	14	zm	zm	PROPN
ejpam-6781	136	15	̸=	̸=	PROPN
ejpam-6781	136	16	zn	zn	PROPN
ejpam-6781	136	17	for	for	ADP
ejpam-6781	136	18	all	all	DET
ejpam-6781	136	19	m	m	PROPN
ejpam-6781	136	20	̸=	̸=	PROPN
ejpam-6781	136	21	n	n	ADP
ejpam-6781	136	22	∈	∈	PROPN
ejpam-6781	136	23	q	q	NOUN
ejpam-6781	136	24	,	,	PUNCT
ejpam-6781	136	25	g(q	g(q	X
ejpam-6781	136	26	)	)	PUNCT
ejpam-6781	136	27	contains	contain	VERB
ejpam-6781	136	28	infinitely	infinitely	ADV
ejpam-6781	136	29	many	many	ADJ
ejpam-6781	136	30	ω	ω	ADJ
ejpam-6781	136	31	-	-	ADJ
ejpam-6781	136	32	rational	rational	ADJ
ejpam-6781	136	33	points	point	NOUN
ejpam-6781	136	34	.	.	PUNCT
ejpam-6781	137	1	hence	hence	ADV
ejpam-6781	137	2	,	,	PUNCT
ejpam-6781	137	3	g(q	g(q	X
ejpam-6781	137	4	)	)	PUNCT
ejpam-6781	137	5	has	have	VERB
ejpam-6781	137	6	infinite	infinite	ADJ
ejpam-6781	137	7	order	order	NOUN
ejpam-6781	137	8	.	.	PUNCT
ejpam-6781	138	1	table	table	NOUN
ejpam-6781	138	2	1	1	NUM
ejpam-6781	138	3	illustrates	illustrate	VERB
ejpam-6781	138	4	a	a	DET
ejpam-6781	138	5	behavior	behavior	NOUN
ejpam-6781	138	6	of	of	ADP
ejpam-6781	138	7	the	the	DET
ejpam-6781	138	8	element	element	NOUN
ejpam-6781	138	9	ζ	ζ	NOUN
ejpam-6781	138	10	=	=	SYM
ejpam-6781	138	11	3	3	NUM
ejpam-6781	138	12	7	7	NUM
ejpam-6781	138	13	+	+	SYM
ejpam-6781	138	14	8	8	NUM
ejpam-6781	138	15	7ω	7ω	NOUN
ejpam-6781	138	16	in	in	ADP
ejpam-6781	138	17	g(q	g(q	NOUN
ejpam-6781	138	18	)	)	PUNCT
ejpam-6781	138	19	and	and	CCONJ
ejpam-6781	138	20	their	their	PRON
ejpam-6781	138	21	successive	successive	ADJ
ejpam-6781	138	22	powers	power	NOUN
ejpam-6781	138	23	ζn	ζn	ADP
ejpam-6781	138	24	for	for	ADP
ejpam-6781	138	25	all	all	DET
ejpam-6781	138	26	integers	integer	NOUN
ejpam-6781	138	27	−3	−3	PROPN
ejpam-6781	138	28	≤	≤	PROPN
ejpam-6781	138	29	n	n	CCONJ
ejpam-6781	138	30	≤	≤	NOUN
ejpam-6781	138	31	3	3	NUM
ejpam-6781	138	32	.	.	NOUN
ejpam-6781	138	33	•	•	NUM
ejpam-6781	138	34	the	the	DET
ejpam-6781	138	35	corresponding	corresponding	ADJ
ejpam-6781	138	36	values	value	NOUN
ejpam-6781	138	37	of	of	ADP
ejpam-6781	138	38	ζn	ζn	PRON
ejpam-6781	138	39	in	in	ADP
ejpam-6781	138	40	the	the	DET
ejpam-6781	138	41	second	second	ADJ
ejpam-6781	138	42	column	column	NOUN
ejpam-6781	138	43	.	.	PUNCT
ejpam-6781	139	1	•	•	NUM
ejpam-6781	139	2	the	the	DET
ejpam-6781	139	3	associated	associated	ADJ
ejpam-6781	139	4	eisenstein	eisenstein	NOUN
ejpam-6781	139	5	triples	triple	NOUN
ejpam-6781	139	6	et(ζn	et(ζn	NOUN
ejpam-6781	139	7	)	)	PUNCT
ejpam-6781	140	1	=	=	SYM
ejpam-6781	140	2	(	(	PUNCT
ejpam-6781	140	3	an	an	PRON
ejpam-6781	140	4	,	,	PUNCT
ejpam-6781	140	5	bn	bn	NOUN
ejpam-6781	140	6	,	,	PUNCT
ejpam-6781	140	7	cn	cn	PROPN
ejpam-6781	140	8	)	)	PUNCT
ejpam-6781	140	9	in	in	ADP
ejpam-6781	140	10	the	the	DET
ejpam-6781	140	11	third	third	ADJ
ejpam-6781	140	12	column	column	NOUN
ejpam-6781	140	13	.	.	PUNCT
ejpam-6781	141	1	4	4	X
ejpam-6781	141	2	.	.	X
ejpam-6781	141	3	prime	prime	ADJ
ejpam-6781	141	4	numbers	number	NOUN
ejpam-6781	141	5	and	and	CCONJ
ejpam-6781	141	6	the	the	DET
ejpam-6781	141	7	abelian	abelian	ADJ
ejpam-6781	141	8	group	group	NOUN
ejpam-6781	141	9	structure	structure	NOUN
ejpam-6781	141	10	this	this	DET
ejpam-6781	141	11	section	section	NOUN
ejpam-6781	141	12	examines	examine	VERB
ejpam-6781	141	13	the	the	DET
ejpam-6781	141	14	structure	structure	NOUN
ejpam-6781	141	15	of	of	ADP
ejpam-6781	141	16	prime	prime	ADJ
ejpam-6781	141	17	numbers	number	NOUN
ejpam-6781	141	18	in	in	ADP
ejpam-6781	141	19	the	the	DET
ejpam-6781	141	20	ring	ring	NOUN
ejpam-6781	141	21	z[ω	z[ω	PROPN
ejpam-6781	141	22	]	]	PUNCT
ejpam-6781	141	23	,	,	PUNCT
ejpam-6781	141	24	focusing	focus	VERB
ejpam-6781	141	25	on	on	ADP
ejpam-6781	141	26	their	their	PRON
ejpam-6781	141	27	factorization	factorization	NOUN
ejpam-6781	141	28	and	and	CCONJ
ejpam-6781	141	29	classification	classification	NOUN
ejpam-6781	141	30	based	base	VERB
ejpam-6781	141	31	on	on	ADP
ejpam-6781	141	32	residue	residue	NOUN
ejpam-6781	141	33	classes	class	NOUN
ejpam-6781	141	34	modulo	modulo	VERB
ejpam-6781	141	35	3	3	NUM
ejpam-6781	141	36	.	.	PUNCT
ejpam-6781	141	37	based	base	VERB
ejpam-6781	141	38	on	on	ADP
ejpam-6781	141	39	the	the	DET
ejpam-6781	141	40	unique	unique	ADJ
ejpam-6781	141	41	factorization	factorization	NOUN
ejpam-6781	141	42	in	in	ADP
ejpam-6781	141	43	this	this	DET
ejpam-6781	141	44	ring	ring	NOUN
ejpam-6781	141	45	,	,	PUNCT
ejpam-6781	141	46	it	it	PRON
ejpam-6781	141	47	can	can	AUX
ejpam-6781	141	48	be	be	AUX
ejpam-6781	141	49	shown	show	VERB
ejpam-6781	141	50	that	that	SCONJ
ejpam-6781	141	51	the	the	DET
ejpam-6781	141	52	group	group	NOUN
ejpam-6781	141	53	g(q	g(q	NOUN
ejpam-6781	141	54	)	)	PUNCT
ejpam-6781	141	55	can	can	AUX
ejpam-6781	141	56	be	be	AUX
ejpam-6781	141	57	decomposed	decompose	VERB
ejpam-6781	141	58	as	as	ADP
ejpam-6781	141	59	a	a	DET
ejpam-6781	141	60	direct	direct	ADJ
ejpam-6781	141	61	sum	sum	NOUN
ejpam-6781	141	62	of	of	ADP
ejpam-6781	141	63	the	the	DET
ejpam-6781	141	64	unit	unit	NOUN
ejpam-6781	141	65	group	group	NOUN
ejpam-6781	141	66	and	and	CCONJ
ejpam-6781	141	67	a	a	DET
ejpam-6781	141	68	free	free	ADJ
ejpam-6781	141	69	abelian	abelian	ADJ
ejpam-6781	141	70	group	group	NOUN
ejpam-6781	141	71	.	.	PUNCT
ejpam-6781	142	1	s.	s.	PROPN
ejpam-6781	142	2	jitman	jitman	PROPN
ejpam-6781	142	3	,	,	PUNCT
ejpam-6781	142	4	m.	m.	NOUN
ejpam-6781	142	5	mohammad	mohammad	PROPN
ejpam-6781	142	6	,	,	PUNCT
ejpam-6781	142	7	e.	e.	PROPN
ejpam-6781	142	8	sangwisut	sangwisut	PROPN
ejpam-6781	142	9	/	/	SYM
ejpam-6781	142	10	eur	eur	PROPN
ejpam-6781	142	11	.	.	PUNCT
ejpam-6781	143	1	j.	j.	PROPN
ejpam-6781	143	2	pure	pure	PROPN
ejpam-6781	143	3	appl	appl	PROPN
ejpam-6781	143	4	.	.	PROPN
ejpam-6781	143	5	math	math	PROPN
ejpam-6781	143	6	,	,	PUNCT
ejpam-6781	143	7	18	18	NUM
ejpam-6781	143	8	(	(	PUNCT
ejpam-6781	143	9	4	4	NUM
ejpam-6781	143	10	)	)	PUNCT
ejpam-6781	143	11	(	(	PUNCT
ejpam-6781	143	12	2025	2025	NUM
ejpam-6781	143	13	)	)	PUNCT
ejpam-6781	143	14	,	,	PUNCT
ejpam-6781	143	15	6781	6781	NUM
ejpam-6781	143	16	8	8	NUM
ejpam-6781	143	17	of	of	ADP
ejpam-6781	143	18	16	16	NUM
ejpam-6781	143	19	n	n	CCONJ
ejpam-6781	143	20	ζn	ζn	PRON
ejpam-6781	143	21	et(ζn	et(ζn	NOUN
ejpam-6781	143	22	)	)	PUNCT
ejpam-6781	144	1	=	=	PRON
ejpam-6781	144	2	(	(	PUNCT
ejpam-6781	144	3	an	an	PRON
ejpam-6781	144	4	,	,	PUNCT
ejpam-6781	144	5	bn	bn	NOUN
ejpam-6781	144	6	,	,	PUNCT
ejpam-6781	144	7	cn	cn	PROPN
ejpam-6781	144	8	)	)	PUNCT
ejpam-6781	144	9	−3	−3	PROPN
ejpam-6781	145	1	323	323	NUM
ejpam-6781	145	2	343	343	NUM
ejpam-6781	145	3	+	+	CCONJ
ejpam-6781	145	4	360	360	NUM
ejpam-6781	145	5	343ω	343ω	NOUN
ejpam-6781	145	6	(	(	PUNCT
ejpam-6781	145	7	323	323	NUM
ejpam-6781	145	8	,	,	PUNCT
ejpam-6781	145	9	360	360	NUM
ejpam-6781	145	10	,	,	PUNCT
ejpam-6781	145	11	343	343	NUM
ejpam-6781	145	12	)	)	PUNCT
ejpam-6781	145	13	−2	−2	NOUN
ejpam-6781	145	14	−39	−39	NOUN
ejpam-6781	145	15	49	49	NUM
ejpam-6781	145	16	+	+	CCONJ
ejpam-6781	145	17	16	16	NUM
ejpam-6781	145	18	49ω	49ω	NOUN
ejpam-6781	145	19	(	(	PUNCT
ejpam-6781	145	20	16	16	NUM
ejpam-6781	145	21	,	,	PUNCT
ejpam-6781	145	22	55	55	NUM
ejpam-6781	145	23	,	,	PUNCT
ejpam-6781	145	24	49	49	NUM
ejpam-6781	145	25	)	)	PUNCT
ejpam-6781	145	26	−1	−1	NOUN
ejpam-6781	146	1	−5	−5	ADV
ejpam-6781	146	2	7	7	NUM
ejpam-6781	146	3	−	−	NUM
ejpam-6781	146	4	8	8	NUM
ejpam-6781	146	5	7ω	7ω	NOUN
ejpam-6781	146	6	(	(	PUNCT
ejpam-6781	146	7	5	5	NUM
ejpam-6781	146	8	,	,	PUNCT
ejpam-6781	146	9	8	8	NUM
ejpam-6781	146	10	,	,	PUNCT
ejpam-6781	146	11	7	7	NUM
ejpam-6781	146	12	)	)	SYM
ejpam-6781	146	13	1	1	NUM
ejpam-6781	146	14	3	3	NUM
ejpam-6781	146	15	7	7	NUM
ejpam-6781	146	16	+	+	SYM
ejpam-6781	146	17	8	8	NUM
ejpam-6781	146	18	7ω	7ω	NOUN
ejpam-6781	146	19	(	(	PUNCT
ejpam-6781	146	20	3	3	NUM
ejpam-6781	146	21	,	,	PUNCT
ejpam-6781	146	22	8	8	NUM
ejpam-6781	146	23	,	,	PUNCT
ejpam-6781	146	24	7	7	NUM
ejpam-6781	146	25	)	)	PUNCT
ejpam-6781	146	26	2	2	NUM
ejpam-6781	146	27	−55	−55	VERB
ejpam-6781	146	28	49	49	NUM
ejpam-6781	146	29	−	−	NOUN
ejpam-6781	146	30	16	16	NUM
ejpam-6781	146	31	49ω	49ω	NOUN
ejpam-6781	146	32	(	(	PUNCT
ejpam-6781	146	33	39	39	NUM
ejpam-6781	146	34	,	,	PUNCT
ejpam-6781	146	35	55	55	NUM
ejpam-6781	146	36	,	,	PUNCT
ejpam-6781	146	37	49	49	NUM
ejpam-6781	146	38	)	)	PUNCT
ejpam-6781	146	39	3	3	NUM
ejpam-6781	146	40	−	−	PROPN
ejpam-6781	146	41	37	37	NUM
ejpam-6781	146	42	343	343	NUM
ejpam-6781	146	43	−	−	NUM
ejpam-6781	146	44	360	360	NUM
ejpam-6781	146	45	343ω	343ω	PROPN
ejpam-6781	146	46	(	(	PUNCT
ejpam-6781	146	47	37	37	NUM
ejpam-6781	146	48	,	,	PUNCT
ejpam-6781	146	49	360	360	NUM
ejpam-6781	146	50	,	,	PUNCT
ejpam-6781	146	51	343	343	NUM
ejpam-6781	146	52	)	)	PUNCT
ejpam-6781	146	53	table	table	NOUN
ejpam-6781	146	54	1	1	NUM
ejpam-6781	146	55	:	:	PUNCT
ejpam-6781	146	56	powers	power	NOUN
ejpam-6781	146	57	of	of	ADP
ejpam-6781	146	58	ζ	ζ	NOUN
ejpam-6781	146	59	=	=	SYM
ejpam-6781	146	60	3	3	NUM
ejpam-6781	146	61	7	7	NUM
ejpam-6781	146	62	+	+	CCONJ
ejpam-6781	146	63	8	8	NUM
ejpam-6781	146	64	7	7	NUM
ejpam-6781	146	65	ω	ω	NOUN
ejpam-6781	146	66	in	in	ADP
ejpam-6781	146	67	g(q	g(q	NOUN
ejpam-6781	146	68	)	)	PUNCT
ejpam-6781	146	69	and	and	CCONJ
ejpam-6781	146	70	their	their	PRON
ejpam-6781	146	71	associated	associate	VERB
ejpam-6781	146	72	primitive	primitive	ADJ
ejpam-6781	146	73	eisenstein	eisenstein	NOUN
ejpam-6781	146	74	triples	triple	NOUN
ejpam-6781	146	75	.	.	PUNCT
ejpam-6781	147	1	let	let	VERB
ejpam-6781	147	2	p	p	PRON
ejpam-6781	147	3	denote	denote	VERB
ejpam-6781	147	4	the	the	DET
ejpam-6781	147	5	set	set	NOUN
ejpam-6781	147	6	of	of	ADP
ejpam-6781	147	7	positive	positive	ADJ
ejpam-6781	147	8	prime	prime	ADJ
ejpam-6781	147	9	numbers	number	NOUN
ejpam-6781	147	10	.	.	PUNCT
ejpam-6781	148	1	for	for	ADP
ejpam-6781	148	2	i	i	PROPN
ejpam-6781	148	3	∈	∈	PROPN
ejpam-6781	148	4	{	{	PUNCT
ejpam-6781	148	5	1	1	NUM
ejpam-6781	148	6	,	,	PUNCT
ejpam-6781	148	7	2	2	NUM
ejpam-6781	148	8	,	,	PUNCT
ejpam-6781	148	9	3	3	NUM
ejpam-6781	148	10	}	}	PUNCT
ejpam-6781	148	11	,	,	PUNCT
ejpam-6781	148	12	let	let	VERB
ejpam-6781	148	13	pi	pi	NOUN
ejpam-6781	148	14	denote	denote	VERB
ejpam-6781	148	15	the	the	DET
ejpam-6781	148	16	set	set	NOUN
ejpam-6781	148	17	of	of	ADP
ejpam-6781	148	18	prime	prime	ADJ
ejpam-6781	148	19	numbers	number	NOUN
ejpam-6781	148	20	congruence	congruence	NOUN
ejpam-6781	148	21	i	i	PRON
ejpam-6781	148	22	modulo	modulo	VERB
ejpam-6781	148	23	3	3	X
ejpam-6781	148	24	.	.	PUNCT
ejpam-6781	149	1	then	then	ADV
ejpam-6781	149	2	p	p	X
ejpam-6781	149	3	can	can	AUX
ejpam-6781	149	4	be	be	AUX
ejpam-6781	149	5	partitioned	partition	VERB
ejpam-6781	149	6	into	into	ADP
ejpam-6781	149	7	residue	residue	NOUN
ejpam-6781	149	8	classes	class	NOUN
ejpam-6781	149	9	modulo	modulo	VERB
ejpam-6781	149	10	3	3	NUM
ejpam-6781	149	11	as	as	SCONJ
ejpam-6781	149	12	follows	follow	VERB
ejpam-6781	149	13	:	:	PUNCT
ejpam-6781	149	14	p	p	X
ejpam-6781	149	15	=	=	PROPN
ejpam-6781	149	16	p1	p1	PROPN
ejpam-6781	149	17	⊔	⊔	PROPN
ejpam-6781	149	18	p2	p2	PROPN
ejpam-6781	149	19	⊔	⊔	PROPN
ejpam-6781	149	20	p3	p3	PROPN
ejpam-6781	149	21	=	=	PUNCT
ejpam-6781	149	22	{	{	PUNCT
ejpam-6781	149	23	7	7	NUM
ejpam-6781	149	24	,	,	PUNCT
ejpam-6781	149	25	13	13	NUM
ejpam-6781	149	26	,	,	PUNCT
ejpam-6781	149	27	19	19	NUM
ejpam-6781	149	28	,	,	PUNCT
ejpam-6781	149	29	31	31	NUM
ejpam-6781	149	30	,	,	PUNCT
ejpam-6781	149	31	.	.	PUNCT
ejpam-6781	149	32	.	.	PUNCT
ejpam-6781	149	33	.	.	PUNCT
ejpam-6781	150	1	}	}	PUNCT
ejpam-6781	150	2	⊔	⊔	INTJ
ejpam-6781	150	3	{	{	PUNCT
ejpam-6781	150	4	2	2	NUM
ejpam-6781	150	5	,	,	PUNCT
ejpam-6781	150	6	5	5	NUM
ejpam-6781	150	7	,	,	PUNCT
ejpam-6781	150	8	11	11	NUM
ejpam-6781	150	9	,	,	PUNCT
ejpam-6781	150	10	17	17	NUM
ejpam-6781	150	11	,	,	PUNCT
ejpam-6781	150	12	.	.	PUNCT
ejpam-6781	150	13	.	.	PUNCT
ejpam-6781	150	14	.	.	PUNCT
ejpam-6781	151	1	}	}	PUNCT
ejpam-6781	151	2	⊔	⊔	INTJ
ejpam-6781	151	3	{	{	PUNCT
ejpam-6781	151	4	3	3	NUM
ejpam-6781	151	5	}	}	PUNCT
ejpam-6781	151	6	.	.	PUNCT
ejpam-6781	152	1	(	(	PUNCT
ejpam-6781	152	2	6	6	NUM
ejpam-6781	152	3	)	)	PUNCT
ejpam-6781	152	4	while	while	SCONJ
ejpam-6781	152	5	p3	p3	PROPN
ejpam-6781	152	6	is	be	AUX
ejpam-6781	152	7	a	a	DET
ejpam-6781	152	8	singleton	singleton	NOUN
ejpam-6781	152	9	,	,	PUNCT
ejpam-6781	152	10	it	it	PRON
ejpam-6781	152	11	can	can	AUX
ejpam-6781	152	12	be	be	AUX
ejpam-6781	152	13	seen	see	VERB
ejpam-6781	152	14	that	that	SCONJ
ejpam-6781	152	15	p1	p1	NOUN
ejpam-6781	152	16	and	and	CCONJ
ejpam-6781	152	17	p2	p2	PROPN
ejpam-6781	152	18	are	be	AUX
ejpam-6781	152	19	infinite	infinite	ADJ
ejpam-6781	152	20	sets	set	NOUN
ejpam-6781	152	21	,	,	PUNCT
ejpam-6781	152	22	for	for	ADP
ejpam-6781	152	23	each	each	DET
ejpam-6781	152	24	p	p	PROPN
ejpam-6781	152	25	∈	∈	PROPN
ejpam-6781	152	26	p1	p1	NOUN
ejpam-6781	152	27	,	,	PUNCT
ejpam-6781	152	28	we	we	PRON
ejpam-6781	152	29	have	have	VERB
ejpam-6781	152	30	p	p	X
ejpam-6781	152	31	≡	≡	PROPN
ejpam-6781	152	32	1	1	NUM
ejpam-6781	152	33	(	(	PUNCT
ejpam-6781	152	34	mod	mod	NOUN
ejpam-6781	152	35	3	3	NUM
ejpam-6781	152	36	)	)	PUNCT
ejpam-6781	152	37	which	which	PRON
ejpam-6781	152	38	can	can	AUX
ejpam-6781	152	39	be	be	AUX
ejpam-6781	152	40	expressed	express	VERB
ejpam-6781	152	41	as	as	ADP
ejpam-6781	152	42	p	p	PROPN
ejpam-6781	152	43	=	=	PROPN
ejpam-6781	152	44	a2	a2	PROPN
ejpam-6781	152	45	−	−	PROPN
ejpam-6781	152	46	ab	ab	PROPN
ejpam-6781	152	47	+	+	CCONJ
ejpam-6781	152	48	b2	b2	NOUN
ejpam-6781	152	49	for	for	ADP
ejpam-6781	152	50	some	some	DET
ejpam-6781	152	51	integers	integer	NOUN
ejpam-6781	152	52	a	a	PRON
ejpam-6781	152	53	and	and	CCONJ
ejpam-6781	152	54	b	b	NOUN
ejpam-6781	152	55	(	(	PUNCT
ejpam-6781	152	56	see	see	VERB
ejpam-6781	152	57	[	[	X
ejpam-6781	152	58	15	15	NUM
ejpam-6781	152	59	]	]	NUM
ejpam-6781	152	60	)	)	PUNCT
ejpam-6781	152	61	.	.	PUNCT
ejpam-6781	153	1	this	this	PRON
ejpam-6781	153	2	allows	allow	VERB
ejpam-6781	153	3	us	we	PRON
ejpam-6781	153	4	to	to	PART
ejpam-6781	153	5	express	express	VERB
ejpam-6781	153	6	the	the	DET
ejpam-6781	153	7	factorization	factorization	NOUN
ejpam-6781	153	8	p	p	NOUN
ejpam-6781	153	9	=	=	X
ejpam-6781	153	10	(	(	PUNCT
ejpam-6781	153	11	a	a	DET
ejpam-6781	153	12	+	+	X
ejpam-6781	153	13	bω)(a	bω)(a	PROPN
ejpam-6781	153	14	+	+	CCONJ
ejpam-6781	153	15	bω	bω	NOUN
ejpam-6781	153	16	)	)	PUNCT
ejpam-6781	153	17	(	(	PUNCT
ejpam-6781	153	18	see	see	VERB
ejpam-6781	153	19	[	[	X
ejpam-6781	153	20	11	11	NUM
ejpam-6781	153	21	,	,	PUNCT
ejpam-6781	153	22	12	12	NUM
ejpam-6781	153	23	]	]	PUNCT
ejpam-6781	153	24	)	)	PUNCT
ejpam-6781	153	25	.	.	PUNCT
ejpam-6781	154	1	let	let	VERB
ejpam-6781	154	2	q	q	NOUN
ejpam-6781	154	3	:	:	PUNCT
ejpam-6781	154	4	=	=	PUNCT
ejpam-6781	155	1	a	a	DET
ejpam-6781	155	2	+	+	NOUN
ejpam-6781	155	3	bω	bω	NOUN
ejpam-6781	155	4	.	.	PUNCT
ejpam-6781	156	1	then	then	ADV
ejpam-6781	156	2	the	the	DET
ejpam-6781	156	3	conjugate	conjugate	NOUN
ejpam-6781	156	4	of	of	ADP
ejpam-6781	156	5	q	q	NOUN
ejpam-6781	156	6	is	be	AUX
ejpam-6781	156	7	q	q	NOUN
ejpam-6781	156	8	=	=	PUNCT
ejpam-6781	156	9	a	a	DET
ejpam-6781	156	10	+	+	NOUN
ejpam-6781	156	11	bω	bω	NOUN
ejpam-6781	156	12	.	.	PUNCT
ejpam-6781	157	1	let	let	VERB
ejpam-6781	157	2	ζp	ζp	PRON
ejpam-6781	157	3	be	be	AUX
ejpam-6781	157	4	the	the	DET
ejpam-6781	157	5	complex	complex	ADJ
ejpam-6781	157	6	number	number	NOUN
ejpam-6781	157	7	of	of	ADP
ejpam-6781	157	8	the	the	DET
ejpam-6781	157	9	form	form	NOUN
ejpam-6781	157	10	ζp	ζp	X
ejpam-6781	157	11	=	=	X
ejpam-6781	157	12	q	q	NOUN
ejpam-6781	157	13	q	q	X
ejpam-6781	158	1	=	=	PUNCT
ejpam-6781	158	2	a	a	PRON
ejpam-6781	158	3	+	+	NUM
ejpam-6781	158	4	bω	bω	X
ejpam-6781	158	5	a	a	DET
ejpam-6781	158	6	+	+	NUM
ejpam-6781	158	7	bω	bω	NOUN
ejpam-6781	158	8	.	.	PUNCT
ejpam-6781	159	1	(	(	PUNCT
ejpam-6781	159	2	7	7	NUM
ejpam-6781	159	3	)	)	PUNCT
ejpam-6781	159	4	since	since	SCONJ
ejpam-6781	159	5	ζp	ζp	PROPN
ejpam-6781	159	6	has	have	VERB
ejpam-6781	159	7	rational	rational	ADJ
ejpam-6781	159	8	coordinates	coordinate	NOUN
ejpam-6781	159	9	and	and	CCONJ
ejpam-6781	159	10	its	its	PRON
ejpam-6781	159	11	norm	norm	NOUN
ejpam-6781	159	12	satisfies	satisfie	NOUN
ejpam-6781	159	13	n(ζp	n(ζp	NOUN
ejpam-6781	159	14	)	)	PUNCT
ejpam-6781	159	15	=	=	SYM
ejpam-6781	159	16	n(a+bω	n(a+bω	NUM
ejpam-6781	159	17	)	)	PUNCT
ejpam-6781	159	18	n(a+bω	n(a+bω	NUM
ejpam-6781	159	19	)	)	PUNCT
ejpam-6781	159	20	=	=	SYM
ejpam-6781	159	21	√	√	ADP
ejpam-6781	159	22	a2−ab+b2√	a2−ab+b2√	PROPN
ejpam-6781	159	23	a2−ab+b2	a2−ab+b2	PROPN
ejpam-6781	159	24	=	=	SYM
ejpam-6781	159	25	1	1	NUM
ejpam-6781	159	26	,	,	PUNCT
ejpam-6781	159	27	it	it	PRON
ejpam-6781	159	28	follows	follow	VERB
ejpam-6781	159	29	that	that	SCONJ
ejpam-6781	159	30	ζp	ζp	PROPN
ejpam-6781	159	31	belongs	belong	VERB
ejpam-6781	159	32	to	to	ADP
ejpam-6781	159	33	the	the	DET
ejpam-6781	159	34	group	group	NOUN
ejpam-6781	159	35	g(q	g(q	PROPN
ejpam-6781	159	36	)	)	PUNCT
ejpam-6781	159	37	.	.	PUNCT
ejpam-6781	160	1	example	example	NOUN
ejpam-6781	161	1	1	1	X
ejpam-6781	161	2	.	.	PUNCT
ejpam-6781	161	3	let	let	VERB
ejpam-6781	161	4	p	p	NOUN
ejpam-6781	161	5	=	=	ADJ
ejpam-6781	161	6	7	7	X
ejpam-6781	161	7	.	.	PUNCT
ejpam-6781	162	1	then	then	ADV
ejpam-6781	162	2	7	7	NUM
ejpam-6781	162	3	=	=	SYM
ejpam-6781	162	4	(	(	PUNCT
ejpam-6781	162	5	1	1	NUM
ejpam-6781	162	6	+	+	NUM
ejpam-6781	162	7	3ω	3ω	NUM
ejpam-6781	162	8	)	)	PUNCT
ejpam-6781	162	9	·	·	PUNCT
ejpam-6781	163	1	(	(	PUNCT
ejpam-6781	163	2	1	1	NUM
ejpam-6781	163	3	+	+	NUM
ejpam-6781	163	4	3ω	3ω	NUM
ejpam-6781	163	5	)	)	PUNCT
ejpam-6781	163	6	=	=	SYM
ejpam-6781	164	1	q	q	X
ejpam-6781	164	2	·	·	PUNCT
ejpam-6781	165	1	q	q	INTJ
ejpam-6781	165	2	,	,	PUNCT
ejpam-6781	165	3	where	where	SCONJ
ejpam-6781	165	4	q	q	NOUN
ejpam-6781	165	5	=	=	SYM
ejpam-6781	165	6	1	1	NUM
ejpam-6781	165	7	+	+	CCONJ
ejpam-6781	165	8	3ω	3ω	NUM
ejpam-6781	165	9	and	and	CCONJ
ejpam-6781	165	10	q	q	NOUN
ejpam-6781	165	11	=	=	SYM
ejpam-6781	165	12	1	1	NUM
ejpam-6781	165	13	+	+	NUM
ejpam-6781	165	14	3ω	3ω	NUM
ejpam-6781	165	15	.	.	PUNCT
ejpam-6781	166	1	it	it	PRON
ejpam-6781	166	2	follows	follow	VERB
ejpam-6781	166	3	that	that	SCONJ
ejpam-6781	166	4	ζ7	ζ7	NOUN
ejpam-6781	166	5	=	=	PUNCT
ejpam-6781	166	6	q	q	NOUN
ejpam-6781	166	7	q	q	NOUN
ejpam-6781	166	8	=	=	NOUN
ejpam-6781	166	9	1	1	NUM
ejpam-6781	166	10	+	+	NUM
ejpam-6781	166	11	3ω	3ω	NUM
ejpam-6781	166	12	1	1	NUM
ejpam-6781	167	1	+	+	NUM
ejpam-6781	167	2	3ω	3ω	NOUN
ejpam-6781	167	3	is	be	AUX
ejpam-6781	167	4	an	an	DET
ejpam-6781	167	5	element	element	NOUN
ejpam-6781	167	6	in	in	ADP
ejpam-6781	167	7	g(q	g(q	NOUN
ejpam-6781	167	8	)	)	PUNCT
ejpam-6781	167	9	.	.	PUNCT
ejpam-6781	168	1	similarly	similarly	ADV
ejpam-6781	168	2	,	,	PUNCT
ejpam-6781	168	3	for	for	ADP
ejpam-6781	168	4	p	p	NOUN
ejpam-6781	168	5	=	=	SYM
ejpam-6781	168	6	13	13	NUM
ejpam-6781	168	7	,	,	PUNCT
ejpam-6781	168	8	we	we	PRON
ejpam-6781	168	9	write	write	VERB
ejpam-6781	168	10	13	13	NUM
ejpam-6781	168	11	=	=	SYM
ejpam-6781	168	12	(	(	PUNCT
ejpam-6781	168	13	1	1	NUM
ejpam-6781	168	14	+	+	NUM
ejpam-6781	168	15	4ω)(1	4ω)(1	NUM
ejpam-6781	168	16	+	+	CCONJ
ejpam-6781	168	17	4ω	4ω	NOUN
ejpam-6781	168	18	)	)	PUNCT
ejpam-6781	168	19	,	,	PUNCT
ejpam-6781	168	20	where	where	SCONJ
ejpam-6781	168	21	q	q	NOUN
ejpam-6781	168	22	=	=	SYM
ejpam-6781	168	23	1	1	NUM
ejpam-6781	168	24	+	+	NUM
ejpam-6781	168	25	4ω	4ω	NOUN
ejpam-6781	168	26	and	and	CCONJ
ejpam-6781	168	27	q	q	NOUN
ejpam-6781	168	28	=	=	NOUN
ejpam-6781	168	29	1	1	NUM
ejpam-6781	168	30	+	+	NUM
ejpam-6781	168	31	4ω	4ω	NOUN
ejpam-6781	168	32	.	.	PUNCT
ejpam-6781	169	1	hence	hence	ADV
ejpam-6781	169	2	,	,	PUNCT
ejpam-6781	169	3	ζ13	ζ13	VERB
ejpam-6781	169	4	=	=	SYM
ejpam-6781	169	5	1	1	NUM
ejpam-6781	169	6	+	+	NUM
ejpam-6781	169	7	4ω	4ω	NOUN
ejpam-6781	169	8	1	1	NUM
ejpam-6781	169	9	+	+	NUM
ejpam-6781	169	10	4ω	4ω	NUM
ejpam-6781	169	11	∈	∈	PROPN
ejpam-6781	169	12	g(q	g(q	NOUN
ejpam-6781	169	13	)	)	PUNCT
ejpam-6781	169	14	.	.	PUNCT
ejpam-6781	170	1	we	we	PRON
ejpam-6781	170	2	note	note	VERB
ejpam-6781	170	3	that	that	SCONJ
ejpam-6781	170	4	z[ω	z[ω	PROPN
ejpam-6781	170	5	]	]	PUNCT
ejpam-6781	170	6	is	be	AUX
ejpam-6781	170	7	a	a	DET
ejpam-6781	170	8	unique	unique	ADJ
ejpam-6781	170	9	factorization	factorization	NOUN
ejpam-6781	170	10	domain	domain	NOUN
ejpam-6781	170	11	with	with	ADP
ejpam-6781	170	12	the	the	DET
ejpam-6781	170	13	unit	unit	NOUN
ejpam-6781	170	14	group	group	NOUN
ejpam-6781	170	15	u	u	NOUN
ejpam-6781	170	16	=	=	PUNCT
ejpam-6781	170	17	{	{	PUNCT
ejpam-6781	170	18	1	1	NUM
ejpam-6781	170	19	,	,	PUNCT
ejpam-6781	170	20	ω	ω	PROPN
ejpam-6781	170	21	,	,	PUNCT
ejpam-6781	170	22	ω2	ω2	ADJ
ejpam-6781	170	23	,	,	PUNCT
ejpam-6781	170	24	−1,−ω,−ω2	−1,−ω,−ω2	X
ejpam-6781	170	25	}	}	PUNCT
ejpam-6781	170	26	.	.	PUNCT
ejpam-6781	171	1	the	the	DET
ejpam-6781	171	2	classification	classification	NOUN
ejpam-6781	171	3	of	of	ADP
ejpam-6781	171	4	the	the	DET
ejpam-6781	171	5	irreducible	irreducible	ADJ
ejpam-6781	171	6	elements	element	NOUN
ejpam-6781	171	7	in	in	ADP
ejpam-6781	171	8	z[ω	z[ω	NOUN
ejpam-6781	171	9	]	]	PUNCT
ejpam-6781	171	10	is	be	AUX
ejpam-6781	171	11	recalled	recall	VERB
ejpam-6781	171	12	as	as	SCONJ
ejpam-6781	171	13	follows	follow	VERB
ejpam-6781	171	14	.	.	PUNCT
ejpam-6781	172	1	for	for	ADP
ejpam-6781	172	2	more	more	ADJ
ejpam-6781	172	3	information	information	NOUN
ejpam-6781	172	4	on	on	ADP
ejpam-6781	172	5	irreducible	irreducible	ADJ
ejpam-6781	172	6	elements	element	NOUN
ejpam-6781	172	7	in	in	ADP
ejpam-6781	172	8	z[ω	z[ω	NOUN
ejpam-6781	172	9	]	]	PUNCT
ejpam-6781	172	10	,	,	PUNCT
ejpam-6781	172	11	the	the	DET
ejpam-6781	172	12	reader	reader	NOUN
ejpam-6781	172	13	may	may	AUX
ejpam-6781	172	14	refer	refer	VERB
ejpam-6781	172	15	to	to	ADP
ejpam-6781	172	16	[	[	X
ejpam-6781	172	17	14	14	NUM
ejpam-6781	172	18	]	]	PUNCT
ejpam-6781	172	19	and	and	CCONJ
ejpam-6781	172	20	[	[	X
ejpam-6781	172	21	12	12	NUM
ejpam-6781	172	22	,	,	PUNCT
ejpam-6781	172	23	p.	p.	NOUN
ejpam-6781	172	24	110	110	NUM
ejpam-6781	172	25	]	]	PUNCT
ejpam-6781	172	26	.	.	PUNCT
ejpam-6781	173	1	based	base	VERB
ejpam-6781	173	2	on	on	ADP
ejpam-6781	173	3	the	the	DET
ejpam-6781	173	4	partition	partition	NOUN
ejpam-6781	173	5	in	in	ADP
ejpam-6781	173	6	(	(	PUNCT
ejpam-6781	173	7	6	6	NUM
ejpam-6781	173	8	)	)	PUNCT
ejpam-6781	173	9	of	of	ADP
ejpam-6781	173	10	p	p	NOUN
ejpam-6781	173	11	,	,	PUNCT
ejpam-6781	173	12	there	there	PRON
ejpam-6781	173	13	are	be	VERB
ejpam-6781	173	14	of	of	ADP
ejpam-6781	173	15	three	three	NUM
ejpam-6781	173	16	types	type	NOUN
ejpam-6781	173	17	of	of	ADP
ejpam-6781	173	18	partitions	partition	NOUN
ejpam-6781	173	19	.	.	PUNCT
ejpam-6781	174	1	•	•	NUM
ejpam-6781	174	2	for	for	ADP
ejpam-6781	174	3	each	each	DET
ejpam-6781	174	4	p	p	PROPN
ejpam-6781	174	5	∈	∈	PROPN
ejpam-6781	174	6	p1	p1	NOUN
ejpam-6781	174	7	,	,	PUNCT
ejpam-6781	174	8	the	the	DET
ejpam-6781	174	9	irreducible	irreducible	ADJ
ejpam-6781	174	10	factorization	factorization	NOUN
ejpam-6781	174	11	of	of	ADP
ejpam-6781	174	12	p	p	NOUN
ejpam-6781	174	13	in	in	ADP
ejpam-6781	174	14	z[ω	z[ω	NOUN
ejpam-6781	174	15	]	]	PUNCT
ejpam-6781	174	16	is	be	AUX
ejpam-6781	174	17	given	give	VERB
ejpam-6781	174	18	by	by	ADP
ejpam-6781	174	19	p	p	NOUN
ejpam-6781	174	20	=	=	X
ejpam-6781	174	21	(	(	PUNCT
ejpam-6781	174	22	a	a	DET
ejpam-6781	174	23	+	+	X
ejpam-6781	174	24	bω)(a	bω)(a	PROPN
ejpam-6781	174	25	+	+	CCONJ
ejpam-6781	174	26	bω	bω	NOUN
ejpam-6781	174	27	)	)	PUNCT
ejpam-6781	174	28	,	,	PUNCT
ejpam-6781	174	29	where	where	SCONJ
ejpam-6781	174	30	a	a	DET
ejpam-6781	174	31	+	+	NUM
ejpam-6781	174	32	bω	bω	NOUN
ejpam-6781	174	33	and	and	CCONJ
ejpam-6781	174	34	a	a	DET
ejpam-6781	174	35	+	+	NUM
ejpam-6781	174	36	bω	bω	NOUN
ejpam-6781	174	37	are	be	AUX
ejpam-6781	174	38	not	not	PART
ejpam-6781	174	39	associated	associate	VERB
ejpam-6781	174	40	.	.	PUNCT
ejpam-6781	175	1	in	in	ADP
ejpam-6781	175	2	this	this	DET
ejpam-6781	175	3	case	case	NOUN
ejpam-6781	175	4	,	,	PUNCT
ejpam-6781	175	5	we	we	PRON
ejpam-6781	175	6	have	have	VERB
ejpam-6781	175	7	u(a	u(a	PROPN
ejpam-6781	175	8	+	+	SYM
ejpam-6781	175	9	bω	bω	NOUN
ejpam-6781	175	10	)	)	PUNCT
ejpam-6781	175	11	̸=	̸=	PROPN
ejpam-6781	175	12	u(a	u(a	PROPN
ejpam-6781	175	13	+	+	CCONJ
ejpam-6781	175	14	bω	bω	NOUN
ejpam-6781	175	15	)	)	PUNCT
ejpam-6781	175	16	and	and	CCONJ
ejpam-6781	175	17	n(a	n(a	PROPN
ejpam-6781	175	18	+	+	CCONJ
ejpam-6781	175	19	bω	bω	NOUN
ejpam-6781	175	20	)	)	PUNCT
ejpam-6781	175	21	=	=	SYM
ejpam-6781	175	22	n(a	n(a	PROPN
ejpam-6781	175	23	+	+	CCONJ
ejpam-6781	175	24	bω	bω	NOUN
ejpam-6781	175	25	)	)	PUNCT
ejpam-6781	175	26	=	=	SYM
ejpam-6781	176	1	√	√	PROPN
ejpam-6781	176	2	p.	p.	NOUN
ejpam-6781	176	3	s.	s.	PROPN
ejpam-6781	176	4	jitman	jitman	PROPN
ejpam-6781	176	5	,	,	PUNCT
ejpam-6781	176	6	m.	m.	NOUN
ejpam-6781	176	7	mohammad	mohammad	PROPN
ejpam-6781	176	8	,	,	PUNCT
ejpam-6781	176	9	e.	e.	PROPN
ejpam-6781	176	10	sangwisut	sangwisut	PROPN
ejpam-6781	176	11	/	/	SYM
ejpam-6781	176	12	eur	eur	PROPN
ejpam-6781	176	13	.	.	PUNCT
ejpam-6781	177	1	j.	j.	PROPN
ejpam-6781	177	2	pure	pure	PROPN
ejpam-6781	177	3	appl	appl	PROPN
ejpam-6781	177	4	.	.	PROPN
ejpam-6781	177	5	math	math	PROPN
ejpam-6781	177	6	,	,	PUNCT
ejpam-6781	177	7	18	18	NUM
ejpam-6781	177	8	(	(	PUNCT
ejpam-6781	177	9	4	4	NUM
ejpam-6781	177	10	)	)	PUNCT
ejpam-6781	177	11	(	(	PUNCT
ejpam-6781	177	12	2025	2025	NUM
ejpam-6781	177	13	)	)	PUNCT
ejpam-6781	177	14	,	,	PUNCT
ejpam-6781	177	15	6781	6781	NUM
ejpam-6781	177	16	9	9	NUM
ejpam-6781	177	17	of	of	ADP
ejpam-6781	177	18	16	16	NUM
ejpam-6781	177	19	•	•	NOUN
ejpam-6781	177	20	for	for	ADP
ejpam-6781	177	21	each	each	DET
ejpam-6781	177	22	p	p	NOUN
ejpam-6781	177	23	∈	∈	NOUN
ejpam-6781	177	24	p2	p2	NOUN
ejpam-6781	178	1	,	,	PUNCT
ejpam-6781	178	2	p	p	NOUN
ejpam-6781	178	3	is	be	AUX
ejpam-6781	178	4	an	an	DET
ejpam-6781	178	5	irreducible	irreducible	ADJ
ejpam-6781	178	6	element	element	NOUN
ejpam-6781	178	7	in	in	ADP
ejpam-6781	178	8	z[ω	z[ω	PROPN
ejpam-6781	178	9	]	]	PUNCT
ejpam-6781	178	10	and	and	CCONJ
ejpam-6781	178	11	n(p	n(p	NUM
ejpam-6781	178	12	)	)	PUNCT
ejpam-6781	179	1	=	=	PUNCT
ejpam-6781	180	1	p.	p.	NOUN
ejpam-6781	180	2	•	•	NOUN
ejpam-6781	180	3	for	for	ADP
ejpam-6781	180	4	the	the	DET
ejpam-6781	180	5	positive	positive	ADJ
ejpam-6781	180	6	prime	prime	ADJ
ejpam-6781	180	7	integer	integer	NOUN
ejpam-6781	180	8	p	p	NOUN
ejpam-6781	180	9	=	=	NOUN
ejpam-6781	180	10	3	3	NUM
ejpam-6781	180	11	,	,	PUNCT
ejpam-6781	180	12	its	its	PRON
ejpam-6781	180	13	factorization	factorization	NOUN
ejpam-6781	180	14	is	be	AUX
ejpam-6781	180	15	3	3	NUM
ejpam-6781	180	16	=	=	SYM
ejpam-6781	180	17	(	(	PUNCT
ejpam-6781	180	18	1	1	NUM
ejpam-6781	180	19	+	+	NUM
ejpam-6781	180	20	2ω)(1	2ω)(1	NUM
ejpam-6781	180	21	+	+	CCONJ
ejpam-6781	180	22	2ω	2ω	NUM
ejpam-6781	180	23	)	)	PUNCT
ejpam-6781	180	24	=	=	PUNCT
ejpam-6781	181	1	−(1	−(1	NOUN
ejpam-6781	181	2	+	+	NUM
ejpam-6781	181	3	2ω)2	2ω)2	NUM
ejpam-6781	181	4	.	.	PUNCT
ejpam-6781	182	1	in	in	ADP
ejpam-6781	182	2	this	this	DET
ejpam-6781	182	3	case	case	NOUN
ejpam-6781	182	4	,	,	PUNCT
ejpam-6781	182	5	1	1	NUM
ejpam-6781	182	6	+	+	CCONJ
ejpam-6781	182	7	2ω	2ω	NUM
ejpam-6781	182	8	and	and	CCONJ
ejpam-6781	182	9	1	1	NUM
ejpam-6781	182	10	+	+	NUM
ejpam-6781	182	11	2ω	2ω	NUM
ejpam-6781	182	12	are	be	AUX
ejpam-6781	182	13	associated	associate	VERB
ejpam-6781	182	14	and	and	CCONJ
ejpam-6781	182	15	1	1	NUM
ejpam-6781	182	16	+	+	NUM
ejpam-6781	182	17	2ω	2ω	NUM
ejpam-6781	182	18	is	be	AUX
ejpam-6781	182	19	irreducible	irreducible	ADJ
ejpam-6781	182	20	in	in	ADP
ejpam-6781	182	21	z[ω	z[ω	NOUN
ejpam-6781	182	22	]	]	PUNCT
ejpam-6781	182	23	.	.	PUNCT
ejpam-6781	183	1	furthermore	furthermore	ADV
ejpam-6781	183	2	,	,	PUNCT
ejpam-6781	183	3	n(1	n(1	NOUN
ejpam-6781	183	4	+	+	NOUN
ejpam-6781	183	5	2ω	2ω	NUM
ejpam-6781	183	6	)	)	PUNCT
ejpam-6781	183	7	=	=	SYM
ejpam-6781	184	1	√	√	NUM
ejpam-6781	184	2	3	3	X
ejpam-6781	184	3	.	.	PUNCT
ejpam-6781	185	1	as	as	SCONJ
ejpam-6781	185	2	the	the	DET
ejpam-6781	185	3	ring	ring	NOUN
ejpam-6781	185	4	z[ω	z[ω	PROPN
ejpam-6781	185	5	]	]	PUNCT
ejpam-6781	185	6	contains	contain	VERB
ejpam-6781	185	7	the	the	DET
ejpam-6781	185	8	ring	ring	NOUN
ejpam-6781	185	9	z	z	PROPN
ejpam-6781	185	10	,	,	PUNCT
ejpam-6781	185	11	this	this	PRON
ejpam-6781	185	12	implies	imply	VERB
ejpam-6781	185	13	that	that	SCONJ
ejpam-6781	185	14	prime	prime	ADJ
ejpam-6781	185	15	integers	integer	NOUN
ejpam-6781	185	16	in	in	ADP
ejpam-6781	185	17	p1	p1	PROPN
ejpam-6781	185	18	and	and	CCONJ
ejpam-6781	185	19	p3	p3	PROPN
ejpam-6781	185	20	can	can	AUX
ejpam-6781	185	21	be	be	AUX
ejpam-6781	185	22	factored	factor	VERB
ejpam-6781	185	23	further	far	ADV
ejpam-6781	185	24	.	.	PUNCT
ejpam-6781	186	1	define	define	VERB
ejpam-6781	186	2	an	an	DET
ejpam-6781	186	3	equivalence	equivalence	NOUN
ejpam-6781	186	4	relation	relation	NOUN
ejpam-6781	186	5	on	on	ADP
ejpam-6781	186	6	the	the	DET
ejpam-6781	186	7	set	set	NOUN
ejpam-6781	186	8	z[ω	z[ω	PROPN
ejpam-6781	186	9	]	]	PUNCT
ejpam-6781	186	10	by	by	ADP
ejpam-6781	186	11	z	z	NOUN
ejpam-6781	186	12	∼	∼	NOUN
ejpam-6781	186	13	z′	z′	NOUN
ejpam-6781	186	14	if	if	SCONJ
ejpam-6781	186	15	and	and	CCONJ
ejpam-6781	186	16	only	only	ADV
ejpam-6781	186	17	if	if	SCONJ
ejpam-6781	186	18	z	z	NOUN
ejpam-6781	186	19	and	and	CCONJ
ejpam-6781	186	20	z′	z′	NUM
ejpam-6781	186	21	are	be	AUX
ejpam-6781	186	22	associated	associate	VERB
ejpam-6781	186	23	.	.	PUNCT
ejpam-6781	187	1	each	each	DET
ejpam-6781	187	2	equivalence	equivalence	NOUN
ejpam-6781	187	3	class	class	NOUN
ejpam-6781	187	4	under	under	ADP
ejpam-6781	187	5	the	the	DET
ejpam-6781	187	6	relation	relation	NOUN
ejpam-6781	187	7	∼	∼	NOUN
ejpam-6781	187	8	consists	consist	VERB
ejpam-6781	187	9	of	of	ADP
ejpam-6781	187	10	all	all	DET
ejpam-6781	187	11	elements	element	NOUN
ejpam-6781	187	12	in	in	ADP
ejpam-6781	187	13	z[ω	z[ω	NOUN
ejpam-6781	187	14	]	]	PUNCT
ejpam-6781	187	15	that	that	PRON
ejpam-6781	187	16	are	be	AUX
ejpam-6781	187	17	associated	associate	VERB
ejpam-6781	187	18	elements	element	NOUN
ejpam-6781	187	19	of	of	ADP
ejpam-6781	187	20	z.	z.	PROPN
ejpam-6781	187	21	precisely	precisely	ADV
ejpam-6781	187	22	,	,	PUNCT
ejpam-6781	187	23	the	the	DET
ejpam-6781	187	24	equivalence	equivalence	NOUN
ejpam-6781	187	25	class	class	NOUN
ejpam-6781	187	26	containing	contain	VERB
ejpam-6781	187	27	z	z	NOUN
ejpam-6781	187	28	is	be	AUX
ejpam-6781	187	29	uz	uz	NOUN
ejpam-6781	187	30	=	=	PUNCT
ejpam-6781	187	31	{	{	PUNCT
ejpam-6781	187	32	z	z	NOUN
ejpam-6781	187	33	,	,	PUNCT
ejpam-6781	187	34	(	(	PUNCT
ejpam-6781	187	35	−ω)z	−ω)z	NOUN
ejpam-6781	187	36	,	,	PUNCT
ejpam-6781	187	37	(	(	PUNCT
ejpam-6781	187	38	−ω)2z	−ω)2z	X
ejpam-6781	187	39	,	,	PUNCT
ejpam-6781	187	40	(	(	PUNCT
ejpam-6781	187	41	−ω)3z	−ω)3z	NUM
ejpam-6781	187	42	,	,	PUNCT
ejpam-6781	187	43	(	(	PUNCT
ejpam-6781	187	44	−ω)4z	−ω)4z	NUM
ejpam-6781	187	45	,	,	PUNCT
ejpam-6781	187	46	(	(	PUNCT
ejpam-6781	187	47	−ω)5z	−ω)5z	ADP
ejpam-6781	187	48	}	}	PUNCT
ejpam-6781	187	49	,	,	PUNCT
ejpam-6781	187	50	which	which	PRON
ejpam-6781	187	51	represents	represent	VERB
ejpam-6781	187	52	points	point	NOUN
ejpam-6781	187	53	on	on	ADP
ejpam-6781	187	54	the	the	DET
ejpam-6781	187	55	circle	circle	NOUN
ejpam-6781	187	56	of	of	ADP
ejpam-6781	187	57	radius	radius	NOUN
ejpam-6781	187	58	n(z	n(z	NOUN
ejpam-6781	187	59	)	)	PUNCT
ejpam-6781	187	60	in	in	ADP
ejpam-6781	187	61	different	different	ADJ
ejpam-6781	187	62	sextants	sextant	NOUN
ejpam-6781	187	63	(	(	PUNCT
ejpam-6781	187	64	see	see	VERB
ejpam-6781	187	65	figure	figure	NOUN
ejpam-6781	187	66	4	4	NUM
ejpam-6781	187	67	)	)	PUNCT
ejpam-6781	187	68	.	.	PUNCT
ejpam-6781	188	1	consequently	consequently	ADV
ejpam-6781	188	2	,	,	PUNCT
ejpam-6781	188	3	the	the	DET
ejpam-6781	188	4	partition	partition	NOUN
ejpam-6781	188	5	of	of	ADP
ejpam-6781	188	6	p	p	NOUN
ejpam-6781	188	7	in	in	ADP
ejpam-6781	188	8	(	(	PUNCT
ejpam-6781	188	9	6	6	NUM
ejpam-6781	188	10	)	)	PUNCT
ejpam-6781	188	11	is	be	AUX
ejpam-6781	188	12	defined	define	VERB
ejpam-6781	188	13	as	as	SCONJ
ejpam-6781	188	14	follows	follow	VERB
ejpam-6781	188	15	.	.	PUNCT
ejpam-6781	189	1	let	let	VERB
ejpam-6781	189	2	q	q	NOUN
ejpam-6781	189	3	denote	denote	VERB
ejpam-6781	189	4	the	the	DET
ejpam-6781	189	5	set	set	NOUN
ejpam-6781	189	6	of	of	ADP
ejpam-6781	189	7	irreducible	irreducible	ADJ
ejpam-6781	189	8	elements	element	NOUN
ejpam-6781	189	9	in	in	ADP
ejpam-6781	189	10	the	the	DET
ejpam-6781	189	11	ring	ring	NOUN
ejpam-6781	189	12	z[ω	z[ω	PROPN
ejpam-6781	189	13	]	]	PUNCT
ejpam-6781	189	14	.	.	PUNCT
ejpam-6781	190	1	the	the	DET
ejpam-6781	190	2	q	q	PROPN
ejpam-6781	190	3	has	have	VERB
ejpam-6781	190	4	a	a	DET
ejpam-6781	190	5	partition	partition	NOUN
ejpam-6781	190	6	of	of	ADP
ejpam-6781	190	7	the	the	DET
ejpam-6781	190	8	form	form	NOUN
ejpam-6781	190	9	q	q	NOUN
ejpam-6781	190	10	:	:	PUNCT
ejpam-6781	191	1	=	=	PROPN
ejpam-6781	191	2	q1	q1	PROPN
ejpam-6781	191	3	⊔q1	⊔q1	PROPN
ejpam-6781	191	4	⊔q2	⊔q2	PROPN
ejpam-6781	192	1	⊔q3	⊔q3	PROPN
ejpam-6781	192	2	(	(	PUNCT
ejpam-6781	192	3	8)	8)	NUM
ejpam-6781	192	4	with	with	ADP
ejpam-6781	192	5	the	the	DET
ejpam-6781	192	6	following	follow	VERB
ejpam-6781	192	7	conditions	condition	NOUN
ejpam-6781	192	8	.	.	PUNCT
ejpam-6781	193	1	(	(	PUNCT
ejpam-6781	193	2	i	i	NOUN
ejpam-6781	193	3	)	)	PUNCT
ejpam-6781	193	4	for	for	ADP
ejpam-6781	193	5	each	each	DET
ejpam-6781	193	6	p	p	PROPN
ejpam-6781	193	7	∈	∈	PROPN
ejpam-6781	193	8	p1	p1	NOUN
ejpam-6781	193	9	,	,	PUNCT
ejpam-6781	193	10	if	if	SCONJ
ejpam-6781	193	11	the	the	DET
ejpam-6781	193	12	factorization	factorization	NOUN
ejpam-6781	193	13	of	of	ADP
ejpam-6781	193	14	p	p	NOUN
ejpam-6781	193	15	in	in	ADP
ejpam-6781	193	16	z[ω	z[ω	NOUN
ejpam-6781	193	17	]	]	PUNCT
ejpam-6781	193	18	is	be	AUX
ejpam-6781	193	19	p	p	NOUN
ejpam-6781	193	20	=	=	ADJ
ejpam-6781	193	21	q	q	X
ejpam-6781	193	22	·	·	PUNCT
ejpam-6781	193	23	q	q	X
ejpam-6781	193	24	,	,	PUNCT
ejpam-6781	193	25	the	the	DET
ejpam-6781	193	26	elements	element	NOUN
ejpam-6781	193	27	q	q	PROPN
ejpam-6781	193	28	and	and	CCONJ
ejpam-6781	193	29	q	q	PROPN
ejpam-6781	193	30	are	be	AUX
ejpam-6781	193	31	assigned	assign	VERB
ejpam-6781	193	32	to	to	ADP
ejpam-6781	193	33	the	the	DET
ejpam-6781	193	34	sets	set	NOUN
ejpam-6781	193	35	q1	q1	PROPN
ejpam-6781	193	36	and	and	CCONJ
ejpam-6781	193	37	q1	q1	PROPN
ejpam-6781	193	38	,	,	PUNCT
ejpam-6781	193	39	respectively	respectively	ADV
ejpam-6781	193	40	.	.	PUNCT
ejpam-6781	194	1	(	(	PUNCT
ejpam-6781	194	2	ii	ii	NOUN
ejpam-6781	194	3	)	)	PUNCT
ejpam-6781	194	4	q2	q2	NOUN
ejpam-6781	194	5	:	:	PUNCT
ejpam-6781	194	6	=	=	PUNCT
ejpam-6781	194	7	p2	p2	X
ejpam-6781	194	8	.	.	PUNCT
ejpam-6781	195	1	(	(	PUNCT
ejpam-6781	195	2	iii	iii	NOUN
ejpam-6781	195	3	)	)	PUNCT
ejpam-6781	195	4	q3	q3	NOUN
ejpam-6781	195	5	:	:	PUNCT
ejpam-6781	195	6	=	=	SYM
ejpam-6781	195	7	{	{	PUNCT
ejpam-6781	195	8	1	1	NUM
ejpam-6781	195	9	+	+	NUM
ejpam-6781	195	10	2ω	2ω	NUM
ejpam-6781	195	11	}	}	PUNCT
ejpam-6781	195	12	.	.	PUNCT
ejpam-6781	196	1	it	it	PRON
ejpam-6781	196	2	follows	follow	VERB
ejpam-6781	196	3	that	that	SCONJ
ejpam-6781	196	4	every	every	DET
ejpam-6781	196	5	nonzero	nonzero	PROPN
ejpam-6781	196	6	element	element	NOUN
ejpam-6781	196	7	z	z	PROPN
ejpam-6781	196	8	∈	∈	PROPN
ejpam-6781	196	9	z[ω	z[ω	PROPN
ejpam-6781	196	10	]	]	PUNCT
ejpam-6781	196	11	can	can	AUX
ejpam-6781	196	12	be	be	AUX
ejpam-6781	196	13	uniquely	uniquely	ADV
ejpam-6781	196	14	expressed	express	VERB
ejpam-6781	196	15	in	in	ADP
ejpam-6781	196	16	the	the	DET
ejpam-6781	196	17	form	form	NOUN
ejpam-6781	197	1	z	z	NOUN
ejpam-6781	197	2	=	=	PUNCT
ejpam-6781	197	3	δ	δ	X
ejpam-6781	197	4	·	·	PUNCT
ejpam-6781	197	5	k∏	k∏	NOUN
ejpam-6781	197	6	i=1	i=1	PROPN
ejpam-6781	197	7	qeii	qeii	ADV
ejpam-6781	197	8	,	,	PUNCT
ejpam-6781	197	9	(	(	PUNCT
ejpam-6781	197	10	9	9	NUM
ejpam-6781	197	11	)	)	PUNCT
ejpam-6781	197	12	where	where	SCONJ
ejpam-6781	197	13	δ	δ	PROPN
ejpam-6781	197	14	∈	∈	PROPN
ejpam-6781	197	15	u	u	PROPN
ejpam-6781	197	16	,	,	PUNCT
ejpam-6781	197	17	k	k	PROPN
ejpam-6781	197	18	≥	≥	PROPN
ejpam-6781	197	19	0	0	NUM
ejpam-6781	197	20	,	,	PUNCT
ejpam-6781	197	21	qi	qi	PROPN
ejpam-6781	197	22	is	be	AUX
ejpam-6781	197	23	an	an	DET
ejpam-6781	197	24	irreducible	irreducible	ADJ
ejpam-6781	197	25	element	element	NOUN
ejpam-6781	197	26	in	in	ADP
ejpam-6781	197	27	q	q	NOUN
ejpam-6781	197	28	,	,	PUNCT
ejpam-6781	197	29	and	and	CCONJ
ejpam-6781	197	30	ei	ei	X
ejpam-6781	197	31	≥	≥	NUM
ejpam-6781	197	32	1	1	NUM
ejpam-6781	197	33	is	be	AUX
ejpam-6781	197	34	an	an	DET
ejpam-6781	197	35	integer	integer	NOUN
ejpam-6781	197	36	.	.	PUNCT
ejpam-6781	198	1	this	this	DET
ejpam-6781	198	2	factorization	factorization	NOUN
ejpam-6781	198	3	is	be	AUX
ejpam-6781	198	4	unique	unique	ADJ
ejpam-6781	198	5	up	up	ADP
ejpam-6781	198	6	to	to	ADP
ejpam-6781	198	7	the	the	DET
ejpam-6781	198	8	rearangement	rearangement	NOUN
ejpam-6781	198	9	of	of	ADP
ejpam-6781	198	10	the	the	DET
ejpam-6781	198	11	factors	factor	NOUN
ejpam-6781	198	12	and	and	CCONJ
ejpam-6781	198	13	the	the	DET
ejpam-6781	198	14	replacement	replacement	NOUN
ejpam-6781	198	15	of	of	ADP
ejpam-6781	198	16	any	any	DET
ejpam-6781	198	17	irreducible	irreducible	ADJ
ejpam-6781	198	18	element	element	NOUN
ejpam-6781	198	19	with	with	ADP
ejpam-6781	198	20	one	one	NUM
ejpam-6781	198	21	of	of	ADP
ejpam-6781	198	22	its	its	PRON
ejpam-6781	198	23	associates	associate	NOUN
ejpam-6781	198	24	.	.	PUNCT
ejpam-6781	199	1	next	next	ADV
ejpam-6781	199	2	,	,	PUNCT
ejpam-6781	199	3	the	the	DET
ejpam-6781	199	4	decomposition	decomposition	NOUN
ejpam-6781	199	5	of	of	ADP
ejpam-6781	199	6	g(q	g(q	NOUN
ejpam-6781	199	7	)	)	PUNCT
ejpam-6781	199	8	will	will	AUX
ejpam-6781	199	9	be	be	AUX
ejpam-6781	199	10	presented	present	VERB
ejpam-6781	199	11	using	use	VERB
ejpam-6781	199	12	the	the	DET
ejpam-6781	199	13	field	field	NOUN
ejpam-6781	199	14	of	of	ADP
ejpam-6781	199	15	fractions	fraction	NOUN
ejpam-6781	199	16	of	of	ADP
ejpam-6781	199	17	z[ω	z[ω	NOUN
ejpam-6781	199	18	]	]	PUNCT
ejpam-6781	199	19	.	.	PUNCT
ejpam-6781	200	1	let	let	VERB
ejpam-6781	200	2	q[ω	q[ω	X
ejpam-6781	200	3	]	]	X
ejpam-6781	201	1	=	=	PRON
ejpam-6781	201	2	{	{	PUNCT
ejpam-6781	201	3	u	u	NOUN
ejpam-6781	201	4	+	+	X
ejpam-6781	201	5	vω	vω	INTJ
ejpam-6781	201	6	|	|	ADV
ejpam-6781	201	7	u	u	NOUN
ejpam-6781	201	8	,	,	PUNCT
ejpam-6781	201	9	v	v	PROPN
ejpam-6781	201	10	∈	∈	ADJ
ejpam-6781	201	11	q	q	NOUN
ejpam-6781	201	12	}	}	PUNCT
ejpam-6781	201	13	⊆	⊆	NUM
ejpam-6781	201	14	c.	c.	NOUN
ejpam-6781	201	15	then	then	ADV
ejpam-6781	201	16	q[ω	q[ω	CCONJ
ejpam-6781	201	17	]	]	X
ejpam-6781	201	18	is	be	AUX
ejpam-6781	201	19	the	the	DET
ejpam-6781	201	20	field	field	NOUN
ejpam-6781	201	21	of	of	ADP
ejpam-6781	201	22	fractions	fraction	NOUN
ejpam-6781	201	23	of	of	ADP
ejpam-6781	201	24	z[ω	z[ω	NOUN
ejpam-6781	201	25	]	]	PUNCT
ejpam-6781	201	26	.	.	PUNCT
ejpam-6781	202	1	analogous	analogous	ADJ
ejpam-6781	202	2	to	to	ADP
ejpam-6781	202	3	(	(	PUNCT
ejpam-6781	202	4	9	9	NUM
ejpam-6781	202	5	)	)	PUNCT
ejpam-6781	202	6	,	,	PUNCT
ejpam-6781	202	7	each	each	DET
ejpam-6781	202	8	element	element	NOUN
ejpam-6781	202	9	z	z	PROPN
ejpam-6781	202	10	∈	∈	PROPN
ejpam-6781	202	11	q[ω	q[ω	NOUN
ejpam-6781	202	12	]	]	X
ejpam-6781	202	13	∖	∖	X
ejpam-6781	202	14	{	{	PUNCT
ejpam-6781	202	15	0	0	NUM
ejpam-6781	202	16	}	}	PUNCT
ejpam-6781	202	17	can	can	AUX
ejpam-6781	202	18	be	be	AUX
ejpam-6781	202	19	uniquely	uniquely	ADV
ejpam-6781	202	20	expressed	express	VERB
ejpam-6781	202	21	in	in	ADP
ejpam-6781	202	22	the	the	DET
ejpam-6781	202	23	form	form	NOUN
ejpam-6781	203	1	z	z	NOUN
ejpam-6781	203	2	=	=	PUNCT
ejpam-6781	203	3	δ	δ	X
ejpam-6781	203	4	·	·	PUNCT
ejpam-6781	203	5	k∏	k∏	NOUN
ejpam-6781	203	6	i=1	i=1	PROPN
ejpam-6781	203	7	qeii	qeii	ADV
ejpam-6781	203	8	,	,	PUNCT
ejpam-6781	203	9	(	(	PUNCT
ejpam-6781	203	10	10	10	NUM
ejpam-6781	203	11	)	)	PUNCT
ejpam-6781	203	12	where	where	SCONJ
ejpam-6781	203	13	δ	δ	PROPN
ejpam-6781	203	14	∈	∈	PROPN
ejpam-6781	203	15	u	u	PROPN
ejpam-6781	203	16	,	,	PUNCT
ejpam-6781	203	17	k	k	PROPN
ejpam-6781	203	18	≥	≥	PROPN
ejpam-6781	203	19	0	0	NUM
ejpam-6781	203	20	,	,	PUNCT
ejpam-6781	203	21	qi	qi	PROPN
ejpam-6781	203	22	is	be	AUX
ejpam-6781	203	23	an	an	DET
ejpam-6781	203	24	irreducible	irreducible	ADJ
ejpam-6781	203	25	element	element	NOUN
ejpam-6781	203	26	in	in	ADP
ejpam-6781	203	27	q	q	NOUN
ejpam-6781	203	28	,	,	PUNCT
ejpam-6781	203	29	and	and	CCONJ
ejpam-6781	203	30	ei	ei	ADP
ejpam-6781	203	31	∈	∈	PROPN
ejpam-6781	203	32	z	z	NOUN
ejpam-6781	203	33	∖	∖	X
ejpam-6781	203	34	{	{	PUNCT
ejpam-6781	203	35	0	0	NUM
ejpam-6781	203	36	}	}	PUNCT
ejpam-6781	203	37	.	.	PUNCT
ejpam-6781	204	1	the	the	DET
ejpam-6781	204	2	following	follow	VERB
ejpam-6781	204	3	theorem	theorem	NOUN
ejpam-6781	204	4	describes	describe	VERB
ejpam-6781	204	5	the	the	DET
ejpam-6781	204	6	decomposition	decomposition	NOUN
ejpam-6781	204	7	of	of	ADP
ejpam-6781	204	8	g(q	g(q	NOUN
ejpam-6781	204	9	)	)	PUNCT
ejpam-6781	204	10	.	.	PUNCT
ejpam-6781	205	1	s.	s.	PROPN
ejpam-6781	205	2	jitman	jitman	PROPN
ejpam-6781	205	3	,	,	PUNCT
ejpam-6781	205	4	m.	m.	NOUN
ejpam-6781	205	5	mohammad	mohammad	PROPN
ejpam-6781	205	6	,	,	PUNCT
ejpam-6781	205	7	e.	e.	PROPN
ejpam-6781	205	8	sangwisut	sangwisut	PROPN
ejpam-6781	205	9	/	/	SYM
ejpam-6781	205	10	eur	eur	PROPN
ejpam-6781	205	11	.	.	PUNCT
ejpam-6781	206	1	j.	j.	PROPN
ejpam-6781	206	2	pure	pure	PROPN
ejpam-6781	206	3	appl	appl	PROPN
ejpam-6781	206	4	.	.	PROPN
ejpam-6781	206	5	math	math	PROPN
ejpam-6781	206	6	,	,	PUNCT
ejpam-6781	206	7	18	18	NUM
ejpam-6781	206	8	(	(	PUNCT
ejpam-6781	206	9	4	4	NUM
ejpam-6781	206	10	)	)	PUNCT
ejpam-6781	206	11	(	(	PUNCT
ejpam-6781	206	12	2025	2025	NUM
ejpam-6781	206	13	)	)	PUNCT
ejpam-6781	206	14	,	,	PUNCT
ejpam-6781	206	15	6781	6781	NUM
ejpam-6781	206	16	10	10	NUM
ejpam-6781	206	17	of	of	ADP
ejpam-6781	206	18	16	16	NUM
ejpam-6781	206	19	theorem	theorem	NOUN
ejpam-6781	206	20	3	3	NUM
ejpam-6781	206	21	.	.	PUNCT
ejpam-6781	207	1	the	the	DET
ejpam-6781	207	2	abelian	abelian	ADJ
ejpam-6781	207	3	group	group	NOUN
ejpam-6781	207	4	g(q	g(q	PROPN
ejpam-6781	207	5	)	)	PUNCT
ejpam-6781	207	6	is	be	AUX
ejpam-6781	207	7	decomposed	decompose	VERB
ejpam-6781	207	8	as	as	ADP
ejpam-6781	207	9	an	an	DET
ejpam-6781	207	10	internal	internal	ADJ
ejpam-6781	207	11	direct	direct	ADJ
ejpam-6781	207	12	sum	sum	NOUN
ejpam-6781	207	13	:	:	PUNCT
ejpam-6781	207	14	g(q	g(q	NUM
ejpam-6781	207	15	)	)	PUNCT
ejpam-6781	208	1	=	=	SYM
ejpam-6781	208	2	u	u	PROPN
ejpam-6781	208	3	⊕	⊕	PROPN
ejpam-6781	208	4	f	f	PROPN
ejpam-6781	208	5	,	,	PUNCT
ejpam-6781	208	6	where	where	SCONJ
ejpam-6781	208	7	u	u	NOUN
ejpam-6781	208	8	=	=	PUNCT
ejpam-6781	208	9	{	{	PUNCT
ejpam-6781	208	10	1	1	NUM
ejpam-6781	208	11	,	,	PUNCT
ejpam-6781	208	12	ω	ω	PROPN
ejpam-6781	208	13	,	,	PUNCT
ejpam-6781	208	14	ω2,−1,−ω,−ω2	ω2,−1,−ω,−ω2	X
ejpam-6781	208	15	}	}	PUNCT
ejpam-6781	208	16	is	be	AUX
ejpam-6781	208	17	the	the	DET
ejpam-6781	208	18	unit	unit	NOUN
ejpam-6781	208	19	group	group	NOUN
ejpam-6781	208	20	of	of	ADP
ejpam-6781	208	21	z[ω	z[ω	PROPN
ejpam-6781	208	22	]	]	PUNCT
ejpam-6781	208	23	,	,	PUNCT
ejpam-6781	208	24	and	and	CCONJ
ejpam-6781	208	25	f	f	PROPN
ejpam-6781	208	26	is	be	AUX
ejpam-6781	208	27	a	a	DET
ejpam-6781	208	28	free	free	ADJ
ejpam-6781	208	29	abelian	abelian	ADJ
ejpam-6781	208	30	group	group	NOUN
ejpam-6781	208	31	with	with	ADP
ejpam-6781	208	32	basis	basis	NOUN
ejpam-6781	208	33	given	give	VERB
ejpam-6781	208	34	by	by	ADP
ejpam-6781	208	35	the	the	DET
ejpam-6781	208	36	collection	collection	NOUN
ejpam-6781	208	37	{	{	PUNCT
ejpam-6781	208	38	ζp	ζp	PROPN
ejpam-6781	208	39	|	|	ADV
ejpam-6781	208	40	p	p	PROPN
ejpam-6781	208	41	∈	∈	PROPN
ejpam-6781	208	42	p1	p1	NOUN
ejpam-6781	208	43	}	}	PUNCT
ejpam-6781	208	44	,	,	PUNCT
ejpam-6781	208	45	where	where	SCONJ
ejpam-6781	208	46	p1	p1	PROPN
ejpam-6781	208	47	is	be	AUX
ejpam-6781	208	48	the	the	DET
ejpam-6781	208	49	set	set	NOUN
ejpam-6781	208	50	of	of	ADP
ejpam-6781	208	51	prime	prime	ADJ
ejpam-6781	208	52	integers	integer	NOUN
ejpam-6781	208	53	congruent	congruent	ADJ
ejpam-6781	208	54	to	to	ADP
ejpam-6781	208	55	1	1	NUM
ejpam-6781	208	56	(	(	PUNCT
ejpam-6781	208	57	mod	mod	NOUN
ejpam-6781	208	58	3	3	NUM
ejpam-6781	208	59	)	)	PUNCT
ejpam-6781	208	60	.	.	PUNCT
ejpam-6781	209	1	proof	proof	NOUN
ejpam-6781	209	2	.	.	PUNCT
ejpam-6781	210	1	let	let	VERB
ejpam-6781	210	2	z	z	NOUN
ejpam-6781	210	3	∈	∈	PROPN
ejpam-6781	210	4	g(q	g(q	NOUN
ejpam-6781	210	5	)	)	PUNCT
ejpam-6781	210	6	.	.	PUNCT
ejpam-6781	211	1	then	then	ADV
ejpam-6781	211	2	z	z	PROPN
ejpam-6781	211	3	∈	∈	PROPN
ejpam-6781	211	4	q[ω	q[ω	NOUN
ejpam-6781	211	5	]	]	X
ejpam-6781	211	6	which	which	PRON
ejpam-6781	211	7	has	have	VERB
ejpam-6781	211	8	a	a	DET
ejpam-6781	211	9	unique	unique	ADJ
ejpam-6781	211	10	factorization	factorization	NOUN
ejpam-6781	211	11	in	in	ADP
ejpam-6781	211	12	the	the	DET
ejpam-6781	211	13	form	form	NOUN
ejpam-6781	211	14	of	of	ADP
ejpam-6781	211	15	(	(	PUNCT
ejpam-6781	211	16	10	10	NUM
ejpam-6781	211	17	)	)	PUNCT
ejpam-6781	211	18	.	.	PUNCT
ejpam-6781	212	1	rearrange	rearrange	VERB
ejpam-6781	212	2	the	the	DET
ejpam-6781	212	3	factorization	factorization	NOUN
ejpam-6781	212	4	using	use	VERB
ejpam-6781	212	5	the	the	DET
ejpam-6781	212	6	set	set	NOUN
ejpam-6781	212	7	of	of	ADP
ejpam-6781	212	8	irreducible	irreducible	ADJ
ejpam-6781	212	9	elements	element	NOUN
ejpam-6781	212	10	q	q	PUNCT
ejpam-6781	212	11	in	in	ADP
ejpam-6781	212	12	(	(	PUNCT
ejpam-6781	212	13	8)	8)	NUM
ejpam-6781	212	14	,	,	PUNCT
ejpam-6781	212	15	we	we	PRON
ejpam-6781	212	16	have	have	VERB
ejpam-6781	212	17	z	z	NOUN
ejpam-6781	212	18	=	=	SYM
ejpam-6781	212	19	δ	δ	X
ejpam-6781	212	20	·	·	PUNCT
ejpam-6781	212	21	(	(	PUNCT
ejpam-6781	212	22	r∏	r∏	NOUN
ejpam-6781	212	23	i=1	i=1	PROPN
ejpam-6781	213	1	qcii	qcii	PROPN
ejpam-6781	213	2	q	q	PROPN
ejpam-6781	213	3	di	di	NOUN
ejpam-6781	213	4	i	i	PROPN
ejpam-6781	213	5	)	)	PUNCT
ejpam-6781	213	6	·	·	PUNCT
ejpam-6781	214	1	(	(	PUNCT
ejpam-6781	214	2	s∏	s∏	PROPN
ejpam-6781	214	3	i=1	i=1	PRON
ejpam-6781	214	4	q̃j	q̃j	VERB
ejpam-6781	214	5	ej	ej	PROPN
ejpam-6781	214	6	)	)	PUNCT
ejpam-6781	214	7	·	·	PUNCT
ejpam-6781	215	1	(	(	PUNCT
ejpam-6781	215	2	1	1	NUM
ejpam-6781	215	3	+	+	SYM
ejpam-6781	215	4	2ω)t	2ω)t	NUM
ejpam-6781	215	5	,	,	PUNCT
ejpam-6781	215	6	(	(	PUNCT
ejpam-6781	215	7	11	11	NUM
ejpam-6781	215	8	)	)	PUNCT
ejpam-6781	215	9	where	where	SCONJ
ejpam-6781	215	10	δ	δ	PROPN
ejpam-6781	215	11	∈	∈	PROPN
ejpam-6781	215	12	u	u	PROPN
ejpam-6781	215	13	,	,	PUNCT
ejpam-6781	215	14	q1	q1	PROPN
ejpam-6781	215	15	,	,	PUNCT
ejpam-6781	215	16	.	.	PUNCT
ejpam-6781	215	17	.	.	PUNCT
ejpam-6781	215	18	.	.	PUNCT
ejpam-6781	216	1	,	,	PUNCT
ejpam-6781	216	2	qr	qr	PROPN
ejpam-6781	216	3	∈	∈	PROPN
ejpam-6781	216	4	q1	q1	PROPN
ejpam-6781	216	5	,	,	PUNCT
ejpam-6781	216	6	q1	q1	PROPN
ejpam-6781	216	7	,	,	PUNCT
ejpam-6781	216	8	.	.	PUNCT
ejpam-6781	216	9	.	.	PUNCT
ejpam-6781	216	10	.	.	PUNCT
ejpam-6781	217	1	,	,	PUNCT
ejpam-6781	217	2	qr	qr	PROPN
ejpam-6781	217	3	∈	∈	PROPN
ejpam-6781	217	4	q1	q1	PROPN
ejpam-6781	217	5	,	,	PUNCT
ejpam-6781	217	6	q̃1	q̃1	PROPN
ejpam-6781	217	7	,	,	PUNCT
ejpam-6781	217	8	.	.	PUNCT
ejpam-6781	217	9	.	.	PUNCT
ejpam-6781	217	10	.	.	PUNCT
ejpam-6781	218	1	,	,	PUNCT
ejpam-6781	218	2	q̃s	q̃s	PROPN
ejpam-6781	218	3	∈	∈	PROPN
ejpam-6781	218	4	q2	q2	NOUN
ejpam-6781	218	5	,	,	PUNCT
ejpam-6781	218	6	and	and	CCONJ
ejpam-6781	218	7	the	the	DET
ejpam-6781	218	8	exponents	exponent	NOUN
ejpam-6781	218	9	c1	c1	PROPN
ejpam-6781	218	10	,	,	PUNCT
ejpam-6781	218	11	.	.	PUNCT
ejpam-6781	218	12	.	.	PUNCT
ejpam-6781	218	13	.	.	PUNCT
ejpam-6781	219	1	,	,	PUNCT
ejpam-6781	219	2	cr	cr	X
ejpam-6781	219	3	,	,	PUNCT
ejpam-6781	219	4	d1	d1	PROPN
ejpam-6781	219	5	,	,	PUNCT
ejpam-6781	219	6	.	.	PUNCT
ejpam-6781	219	7	.	.	PUNCT
ejpam-6781	219	8	.	.	PUNCT
ejpam-6781	220	1	,	,	PUNCT
ejpam-6781	220	2	dr	dr	PROPN
ejpam-6781	220	3	,	,	PUNCT
ejpam-6781	220	4	e1	e1	PROPN
ejpam-6781	220	5	,	,	PUNCT
ejpam-6781	220	6	.	.	PUNCT
ejpam-6781	220	7	.	.	PUNCT
ejpam-6781	221	1	.	.	PUNCT
ejpam-6781	222	1	,	,	PUNCT
ejpam-6781	222	2	es	es	NOUN
ejpam-6781	222	3	and	and	CCONJ
ejpam-6781	222	4	t	t	PROPN
ejpam-6781	222	5	are	be	AUX
ejpam-6781	222	6	integers	integer	NOUN
ejpam-6781	222	7	.	.	PUNCT
ejpam-6781	223	1	since	since	SCONJ
ejpam-6781	223	2	z	z	PROPN
ejpam-6781	223	3	is	be	AUX
ejpam-6781	223	4	an	an	DET
ejpam-6781	223	5	ω	ω	ADJ
ejpam-6781	223	6	-	-	ADJ
ejpam-6781	223	7	rational	rational	ADJ
ejpam-6781	223	8	point	point	NOUN
ejpam-6781	223	9	in	in	ADP
ejpam-6781	223	10	the	the	DET
ejpam-6781	223	11	ω	ω	ADJ
ejpam-6781	223	12	-	-	ADJ
ejpam-6781	223	13	rational	rational	ADJ
ejpam-6781	223	14	unit	unit	NOUN
ejpam-6781	223	15	circle	circle	NOUN
ejpam-6781	223	16	,	,	PUNCT
ejpam-6781	223	17	we	we	PRON
ejpam-6781	223	18	have	have	VERB
ejpam-6781	223	19	n(z	n(z	NOUN
ejpam-6781	223	20	)	)	PUNCT
ejpam-6781	223	21	=	=	SYM
ejpam-6781	224	1	1	1	X
ejpam-6781	224	2	.	.	PUNCT
ejpam-6781	224	3	expanding	expand	VERB
ejpam-6781	224	4	n(z	n(z	NOUN
ejpam-6781	224	5	)	)	PUNCT
ejpam-6781	224	6	using	use	VERB
ejpam-6781	224	7	its	its	PRON
ejpam-6781	224	8	factorization	factorization	NOUN
ejpam-6781	224	9	,	,	PUNCT
ejpam-6781	224	10	it	it	PRON
ejpam-6781	224	11	follows	follow	VERB
ejpam-6781	224	12	that	that	SCONJ
ejpam-6781	224	13	1	1	NUM
ejpam-6781	224	14	=	=	SYM
ejpam-6781	224	15	n(z	n(z	NOUN
ejpam-6781	224	16	)	)	PUNCT
ejpam-6781	224	17	=	=	SYM
ejpam-6781	224	18	n(δ	n(δ	PROPN
ejpam-6781	224	19	)	)	PUNCT
ejpam-6781	224	20	·	·	PUNCT
ejpam-6781	224	21	n	n	CCONJ
ejpam-6781	224	22	(	(	PUNCT
ejpam-6781	224	23	r∏	r∏	PROPN
ejpam-6781	224	24	i=1	i=1	PROPN
ejpam-6781	225	1	qcii	qcii	PROPN
ejpam-6781	225	2	q	q	PROPN
ejpam-6781	225	3	di	di	NOUN
ejpam-6781	225	4	i	i	PROPN
ejpam-6781	225	5	)	)	PUNCT
ejpam-6781	225	6	·	·	PUNCT
ejpam-6781	225	7	n	n	CCONJ
ejpam-6781	225	8	(	(	PUNCT
ejpam-6781	225	9	s∏	s∏	PROPN
ejpam-6781	225	10	i=1	i=1	PRON
ejpam-6781	225	11	q̃j	q̃j	VERB
ejpam-6781	225	12	ej	ej	PROPN
ejpam-6781	225	13	)	)	PUNCT
ejpam-6781	225	14	·	·	PUNCT
ejpam-6781	226	1	n(1	n(1	NOUN
ejpam-6781	226	2	+	+	NOUN
ejpam-6781	226	3	2ω)t	2ω)t	NUM
ejpam-6781	226	4	=	=	SYM
ejpam-6781	226	5	(	(	PUNCT
ejpam-6781	226	6	r∏	r∏	NOUN
ejpam-6781	226	7	i=1	i=1	PRON
ejpam-6781	226	8	n(qi	n(qi	PROPN
ejpam-6781	226	9	)	)	PUNCT
ejpam-6781	226	10	cin(qi	cin(qi	NOUN
ejpam-6781	226	11	)	)	PUNCT
ejpam-6781	226	12	di	di	NOUN
ejpam-6781	226	13	)	)	PUNCT
ejpam-6781	226	14	(	(	PUNCT
ejpam-6781	226	15	s∏	s∏	PROPN
ejpam-6781	226	16	i=1	i=1	PRON
ejpam-6781	226	17	n	n	PROPN
ejpam-6781	226	18	(	(	PUNCT
ejpam-6781	226	19	q̃j	q̃j	NOUN
ejpam-6781	226	20	)	)	PUNCT
ejpam-6781	226	21	ej	ej	NOUN
ejpam-6781	226	22	)	)	PUNCT
ejpam-6781	226	23	·	·	PUNCT
ejpam-6781	227	1	(	(	PUNCT
ejpam-6781	227	2	√	√	NUM
ejpam-6781	227	3	3	3	NUM
ejpam-6781	227	4	)	)	PUNCT
ejpam-6781	227	5	t	t	NOUN
ejpam-6781	227	6	=	=	SYM
ejpam-6781	227	7	(	(	PUNCT
ejpam-6781	227	8	r∏	r∏	NOUN
ejpam-6781	227	9	i=1	i=1	PRON
ejpam-6781	227	10	n(qi	n(qi	PROPN
ejpam-6781	227	11	)	)	PUNCT
ejpam-6781	227	12	cin(qi	cin(qi	NOUN
ejpam-6781	227	13	)	)	PUNCT
ejpam-6781	227	14	di	di	NOUN
ejpam-6781	227	15	)	)	PUNCT
ejpam-6781	227	16	(	(	PUNCT
ejpam-6781	228	1	s∏	s∏	PROPN
ejpam-6781	228	2	i=1	i=1	PRON
ejpam-6781	228	3	n	n	PROPN
ejpam-6781	228	4	(	(	PUNCT
ejpam-6781	228	5	q̃j	q̃j	NOUN
ejpam-6781	228	6	)	)	PUNCT
ejpam-6781	228	7	ej	ej	NOUN
ejpam-6781	228	8	)	)	PUNCT
ejpam-6781	228	9	·	·	PUNCT
ejpam-6781	229	1	(	(	PUNCT
ejpam-6781	229	2	√	√	NUM
ejpam-6781	229	3	3	3	NUM
ejpam-6781	229	4	)	)	PUNCT
ejpam-6781	229	5	t	t	NOUN
ejpam-6781	229	6	.	.	PUNCT
ejpam-6781	230	1	we	we	PRON
ejpam-6781	230	2	note	note	VERB
ejpam-6781	230	3	that	that	SCONJ
ejpam-6781	230	4	n(qi	n(qi	X
ejpam-6781	230	5	)	)	PUNCT
ejpam-6781	230	6	=	=	SYM
ejpam-6781	230	7	n(qi	n(qi	NOUN
ejpam-6781	230	8	)	)	PUNCT
ejpam-6781	230	9	for	for	ADP
ejpam-6781	230	10	qi	qi	PROPN
ejpam-6781	230	11	∈	∈	PROPN
ejpam-6781	230	12	q1	q1	PROPN
ejpam-6781	230	13	and	and	CCONJ
ejpam-6781	230	14	qi	qi	PROPN
ejpam-6781	230	15	∈	∈	PROPN
ejpam-6781	230	16	q1	q1	PROPN
ejpam-6781	230	17	.	.	PUNCT
ejpam-6781	231	1	since	since	SCONJ
ejpam-6781	231	2	n(z	n(z	NOUN
ejpam-6781	231	3	)	)	PUNCT
ejpam-6781	231	4	=	=	SYM
ejpam-6781	231	5	1	1	NUM
ejpam-6781	231	6	,	,	PUNCT
ejpam-6781	231	7	we	we	PRON
ejpam-6781	231	8	deduce	deduce	VERB
ejpam-6781	231	9	that	that	DET
ejpam-6781	231	10	ci	ci	NOUN
ejpam-6781	231	11	=	=	SYM
ejpam-6781	231	12	−di	−di	PROPN
ejpam-6781	231	13	,	,	PUNCT
ejpam-6781	231	14	and	and	CCONJ
ejpam-6781	231	15	ej	ej	PROPN
ejpam-6781	231	16	and	and	CCONJ
ejpam-6781	231	17	t	t	PROPN
ejpam-6781	231	18	are	be	AUX
ejpam-6781	231	19	zero	zero	NUM
ejpam-6781	231	20	.	.	PUNCT
ejpam-6781	232	1	substituting	substitute	VERB
ejpam-6781	232	2	these	these	DET
ejpam-6781	232	3	constrains	constrain	NOUN
ejpam-6781	232	4	into	into	ADP
ejpam-6781	232	5	(	(	PUNCT
ejpam-6781	232	6	11	11	NUM
ejpam-6781	232	7	)	)	PUNCT
ejpam-6781	232	8	,	,	PUNCT
ejpam-6781	232	9	it	it	PRON
ejpam-6781	232	10	can	can	AUX
ejpam-6781	232	11	be	be	AUX
ejpam-6781	232	12	concluded	conclude	VERB
ejpam-6781	232	13	that	that	SCONJ
ejpam-6781	232	14	z	z	NOUN
ejpam-6781	232	15	=	=	PUNCT
ejpam-6781	232	16	δ	δ	X
ejpam-6781	232	17	·	·	PUNCT
ejpam-6781	232	18	(	(	PUNCT
ejpam-6781	232	19	r∏	r∏	NOUN
ejpam-6781	232	20	i=1	i=1	PROPN
ejpam-6781	233	1	qcii	qcii	PROPN
ejpam-6781	233	2	q	q	PROPN
ejpam-6781	233	3	−ci	−ci	PROPN
ejpam-6781	233	4	i	i	NOUN
ejpam-6781	233	5	)	)	PUNCT
ejpam-6781	234	1	=	=	PUNCT
ejpam-6781	234	2	δ	δ	X
ejpam-6781	234	3	·	·	PUNCT
ejpam-6781	235	1	r∏	r∏	NOUN
ejpam-6781	235	2	i=1	i=1	PROPN
ejpam-6781	236	1	(	(	PUNCT
ejpam-6781	236	2	qi	qi	PROPN
ejpam-6781	236	3	qi	qi	PROPN
ejpam-6781	236	4	)	)	PUNCT
ejpam-6781	236	5	ci	ci	PROPN
ejpam-6781	236	6	.	.	PUNCT
ejpam-6781	237	1	by	by	ADP
ejpam-6781	237	2	setting	set	VERB
ejpam-6781	237	3	ζi	ζi	PROPN
ejpam-6781	237	4	=	=	PUNCT
ejpam-6781	237	5	qi	qi	PROPN
ejpam-6781	237	6	qi	qi	NOUN
ejpam-6781	237	7	as	as	ADP
ejpam-6781	237	8	in	in	ADP
ejpam-6781	237	9	(	(	PUNCT
ejpam-6781	237	10	7	7	NUM
ejpam-6781	237	11	)	)	PUNCT
ejpam-6781	237	12	,	,	PUNCT
ejpam-6781	237	13	we	we	PRON
ejpam-6781	237	14	further	far	ADV
ejpam-6781	237	15	deduced	deduce	VERB
ejpam-6781	237	16	that	that	SCONJ
ejpam-6781	237	17	z	z	NOUN
ejpam-6781	237	18	=	=	PUNCT
ejpam-6781	237	19	δ	δ	X
ejpam-6781	238	1	·	·	PUNCT
ejpam-6781	238	2	r∏	r∏	NOUN
ejpam-6781	238	3	i=1	i=1	PROPN
ejpam-6781	239	1	ζcii	ζcii	PROPN
ejpam-6781	239	2	∈	∈	PROPN
ejpam-6781	239	3	u	u	PROPN
ejpam-6781	239	4	⊕	⊕	PROPN
ejpam-6781	239	5	f.	f.	PROPN
ejpam-6781	239	6	this	this	PRON
ejpam-6781	239	7	completes	complete	VERB
ejpam-6781	239	8	the	the	DET
ejpam-6781	239	9	proof	proof	NOUN
ejpam-6781	239	10	.	.	PUNCT
ejpam-6781	240	1	s.	s.	PROPN
ejpam-6781	240	2	jitman	jitman	PROPN
ejpam-6781	240	3	,	,	PUNCT
ejpam-6781	240	4	m.	m.	NOUN
ejpam-6781	240	5	mohammad	mohammad	PROPN
ejpam-6781	240	6	,	,	PUNCT
ejpam-6781	240	7	e.	e.	PROPN
ejpam-6781	240	8	sangwisut	sangwisut	PROPN
ejpam-6781	240	9	/	/	SYM
ejpam-6781	240	10	eur	eur	PROPN
ejpam-6781	240	11	.	.	PUNCT
ejpam-6781	241	1	j.	j.	PROPN
ejpam-6781	241	2	pure	pure	PROPN
ejpam-6781	241	3	appl	appl	PROPN
ejpam-6781	241	4	.	.	PROPN
ejpam-6781	241	5	math	math	PROPN
ejpam-6781	241	6	,	,	PUNCT
ejpam-6781	241	7	18	18	NUM
ejpam-6781	241	8	(	(	PUNCT
ejpam-6781	241	9	4	4	NUM
ejpam-6781	241	10	)	)	PUNCT
ejpam-6781	241	11	(	(	PUNCT
ejpam-6781	241	12	2025	2025	NUM
ejpam-6781	241	13	)	)	PUNCT
ejpam-6781	241	14	,	,	PUNCT
ejpam-6781	241	15	6781	6781	NUM
ejpam-6781	241	16	11	11	NUM
ejpam-6781	241	17	of	of	ADP
ejpam-6781	241	18	16	16	NUM
ejpam-6781	241	19	5	5	NUM
ejpam-6781	241	20	.	.	PUNCT
ejpam-6781	241	21	characterization	characterization	NOUN
ejpam-6781	241	22	and	and	CCONJ
ejpam-6781	241	23	enumeration	enumeration	NOUN
ejpam-6781	241	24	of	of	ADP
ejpam-6781	241	25	primitive	primitive	ADJ
ejpam-6781	241	26	eisenstein	eisenstein	NOUN
ejpam-6781	241	27	triples	triple	NOUN
ejpam-6781	241	28	in	in	ADP
ejpam-6781	241	29	this	this	DET
ejpam-6781	241	30	section	section	NOUN
ejpam-6781	241	31	,	,	PUNCT
ejpam-6781	241	32	we	we	PRON
ejpam-6781	241	33	focus	focus	VERB
ejpam-6781	241	34	on	on	ADP
ejpam-6781	241	35	the	the	DET
ejpam-6781	241	36	characterization	characterization	NOUN
ejpam-6781	241	37	and	and	CCONJ
ejpam-6781	241	38	enumeration	enumeration	NOUN
ejpam-6781	241	39	of	of	ADP
ejpam-6781	241	40	primitive	primitive	ADJ
ejpam-6781	241	41	eisenstein	eisenstein	NOUN
ejpam-6781	241	42	triples	triple	NOUN
ejpam-6781	241	43	.	.	PUNCT
ejpam-6781	242	1	we	we	PRON
ejpam-6781	242	2	begin	begin	VERB
ejpam-6781	242	3	with	with	ADP
ejpam-6781	242	4	a	a	DET
ejpam-6781	242	5	formula	formula	NOUN
ejpam-6781	242	6	for	for	ADP
ejpam-6781	242	7	the	the	DET
ejpam-6781	242	8	hypotenuse	hypotenuse	NOUN
ejpam-6781	242	9	c	c	PROPN
ejpam-6781	242	10	of	of	ADP
ejpam-6781	242	11	an	an	DET
ejpam-6781	242	12	eisenstein	eisenstein	PROPN
ejpam-6781	242	13	triple	triple	NOUN
ejpam-6781	242	14	.	.	PUNCT
ejpam-6781	243	1	next	next	ADV
ejpam-6781	243	2	,	,	PUNCT
ejpam-6781	243	3	we	we	PRON
ejpam-6781	243	4	present	present	VERB
ejpam-6781	243	5	necessary	necessary	ADJ
ejpam-6781	243	6	and	and	CCONJ
ejpam-6781	243	7	sufficient	sufficient	ADJ
ejpam-6781	243	8	conditions	condition	NOUN
ejpam-6781	243	9	for	for	ADP
ejpam-6781	243	10	the	the	DET
ejpam-6781	243	11	existence	existence	NOUN
ejpam-6781	243	12	of	of	ADP
ejpam-6781	243	13	primitive	primitive	ADJ
ejpam-6781	243	14	eisenstein	eisenstein	NOUN
ejpam-6781	243	15	triples	triple	NOUN
ejpam-6781	243	16	with	with	ADP
ejpam-6781	243	17	a	a	DET
ejpam-6781	243	18	given	give	VERB
ejpam-6781	243	19	hypotenuse	hypotenuse	NOUN
ejpam-6781	243	20	c	c	NOUN
ejpam-6781	243	21	,	,	PUNCT
ejpam-6781	243	22	highlighting	highlight	VERB
ejpam-6781	243	23	the	the	DET
ejpam-6781	243	24	relationship	relationship	NOUN
ejpam-6781	243	25	between	between	ADP
ejpam-6781	243	26	the	the	DET
ejpam-6781	243	27	prime	prime	ADJ
ejpam-6781	243	28	factorization	factorization	NOUN
ejpam-6781	243	29	of	of	ADP
ejpam-6781	243	30	c	c	PROPN
ejpam-6781	243	31	and	and	CCONJ
ejpam-6781	243	32	the	the	DET
ejpam-6781	243	33	structure	structure	NOUN
ejpam-6781	243	34	of	of	ADP
ejpam-6781	243	35	the	the	DET
ejpam-6781	243	36	corresponding	correspond	VERB
ejpam-6781	243	37	triples	triple	NOUN
ejpam-6781	243	38	.	.	PUNCT
ejpam-6781	244	1	this	this	PRON
ejpam-6781	244	2	allows	allow	VERB
ejpam-6781	244	3	us	we	PRON
ejpam-6781	244	4	to	to	PART
ejpam-6781	244	5	establish	establish	VERB
ejpam-6781	244	6	the	the	DET
ejpam-6781	244	7	enumeration	enumeration	NOUN
ejpam-6781	244	8	of	of	ADP
ejpam-6781	244	9	primitive	primitive	ADJ
ejpam-6781	244	10	eisenstein	eisenstein	NOUN
ejpam-6781	244	11	triples	triple	NOUN
ejpam-6781	244	12	with	with	ADP
ejpam-6781	244	13	a	a	DET
ejpam-6781	244	14	given	give	VERB
ejpam-6781	244	15	hypotenuse	hypotenuse	NOUN
ejpam-6781	244	16	c.	c.	NOUN
ejpam-6781	244	17	finally	finally	ADV
ejpam-6781	244	18	,	,	PUNCT
ejpam-6781	244	19	we	we	PRON
ejpam-6781	244	20	provide	provide	VERB
ejpam-6781	244	21	examples	example	NOUN
ejpam-6781	244	22	that	that	PRON
ejpam-6781	244	23	demonstrate	demonstrate	VERB
ejpam-6781	244	24	how	how	SCONJ
ejpam-6781	244	25	these	these	DET
ejpam-6781	244	26	results	result	NOUN
ejpam-6781	244	27	can	can	AUX
ejpam-6781	244	28	be	be	AUX
ejpam-6781	244	29	applied	apply	VERB
ejpam-6781	244	30	to	to	ADP
ejpam-6781	244	31	specific	specific	ADJ
ejpam-6781	244	32	values	value	NOUN
ejpam-6781	244	33	of	of	ADP
ejpam-6781	244	34	c	c	NOUN
ejpam-6781	244	35	,	,	PUNCT
ejpam-6781	244	36	illustrating	illustrate	VERB
ejpam-6781	244	37	the	the	DET
ejpam-6781	244	38	number	number	NOUN
ejpam-6781	244	39	of	of	ADP
ejpam-6781	244	40	primitive	primitive	ADJ
ejpam-6781	244	41	eisenstein	eisenstein	NOUN
ejpam-6781	244	42	triples	triple	NOUN
ejpam-6781	244	43	associated	associate	VERB
ejpam-6781	244	44	with	with	ADP
ejpam-6781	244	45	particular	particular	ADJ
ejpam-6781	244	46	integers	integer	NOUN
ejpam-6781	244	47	.	.	PUNCT
ejpam-6781	245	1	lemma	lemma	PROPN
ejpam-6781	245	2	1	1	NUM
ejpam-6781	245	3	.	.	PUNCT
ejpam-6781	246	1	for	for	ADP
ejpam-6781	246	2	a	a	DET
ejpam-6781	246	3	positive	positive	ADJ
ejpam-6781	246	4	integer	integer	NOUN
ejpam-6781	246	5	k	k	NOUN
ejpam-6781	246	6	,	,	PUNCT
ejpam-6781	246	7	distinct	distinct	ADJ
ejpam-6781	246	8	primes	prime	NOUN
ejpam-6781	246	9	p1	p1	NOUN
ejpam-6781	246	10	,	,	PUNCT
ejpam-6781	246	11	.	.	PUNCT
ejpam-6781	246	12	.	.	PUNCT
ejpam-6781	246	13	.	.	PUNCT
ejpam-6781	247	1	,	,	PUNCT
ejpam-6781	247	2	pk	pk	NOUN
ejpam-6781	247	3	in	in	ADP
ejpam-6781	247	4	p1	p1	PROPN
ejpam-6781	247	5	,	,	PUNCT
ejpam-6781	247	6	nonzero	nonzero	PROPN
ejpam-6781	247	7	integers	integer	VERB
ejpam-6781	247	8	n1	n1	NOUN
ejpam-6781	247	9	,	,	PUNCT
ejpam-6781	247	10	.	.	PUNCT
ejpam-6781	247	11	.	.	PUNCT
ejpam-6781	248	1	.	.	PUNCT
ejpam-6781	249	1	,	,	PUNCT
ejpam-6781	249	2	nk	nk	PROPN
ejpam-6781	249	3	,	,	PUNCT
ejpam-6781	249	4	and	and	CCONJ
ejpam-6781	249	5	signs	sign	NOUN
ejpam-6781	249	6	ϵ1	ϵ1	VERB
ejpam-6781	249	7	,	,	PUNCT
ejpam-6781	249	8	.	.	PUNCT
ejpam-6781	249	9	.	.	PUNCT
ejpam-6781	250	1	.	.	PUNCT
ejpam-6781	251	1	,	,	PUNCT
ejpam-6781	252	1	ϵk	ϵk	X
ejpam-6781	252	2	∈	∈	PROPN
ejpam-6781	252	3	{	{	PUNCT
ejpam-6781	252	4	±1	±1	NOUN
ejpam-6781	252	5	}	}	PUNCT
ejpam-6781	252	6	,	,	PUNCT
ejpam-6781	252	7	define	define	VERB
ejpam-6781	252	8	(	(	PUNCT
ejpam-6781	252	9	a	a	PRON
ejpam-6781	252	10	,	,	PUNCT
ejpam-6781	252	11	b	b	NOUN
ejpam-6781	252	12	,	,	PUNCT
ejpam-6781	252	13	c	c	NOUN
ejpam-6781	252	14	)	)	PUNCT
ejpam-6781	252	15	:	:	PUNCT
ejpam-6781	252	16	=	=	SYM
ejpam-6781	252	17	et	et	NOUN
ejpam-6781	252	18	(	(	PUNCT
ejpam-6781	252	19	k∏	k∏	PROPN
ejpam-6781	252	20	i=1	i=1	PROPN
ejpam-6781	252	21	ζϵi·ni	ζϵi·ni	PROPN
ejpam-6781	252	22	pi	pi	NOUN
ejpam-6781	252	23	)	)	PUNCT
ejpam-6781	252	24	∈	∈	PROPN
ejpam-6781	252	25	et	et	NOUN
ejpam-6781	252	26	.	.	PUNCT
ejpam-6781	253	1	then	then	ADV
ejpam-6781	253	2	c	c	X
ejpam-6781	253	3	=	=	PUNCT
ejpam-6781	253	4	pn1	pn1	PROPN
ejpam-6781	253	5	1	1	NUM
ejpam-6781	253	6	.	.	PUNCT
ejpam-6781	253	7	.	.	PUNCT
ejpam-6781	253	8	.	.	PUNCT
ejpam-6781	254	1	pnk	pnk	PROPN
ejpam-6781	255	1	k	k	X
ejpam-6781	255	2	.	.	PUNCT
ejpam-6781	256	1	proof	proof	NOUN
ejpam-6781	256	2	.	.	PUNCT
ejpam-6781	257	1	let	let	VERB
ejpam-6781	257	2	ζ	ζ	NOUN
ejpam-6781	257	3	=	=	PUNCT
ejpam-6781	258	1	∏k	∏k	X
ejpam-6781	258	2	i=1	i=1	PROPN
ejpam-6781	258	3	ζ	ζ	NOUN
ejpam-6781	258	4	ϵi·ni	ϵi·ni	NUM
ejpam-6781	258	5	pi	pi	NOUN
ejpam-6781	258	6	.	.	PUNCT
ejpam-6781	259	1	by	by	ADP
ejpam-6781	259	2	the	the	DET
ejpam-6781	259	3	defining	define	VERB
ejpam-6781	259	4	ζpi	ζpi	NOUN
ejpam-6781	259	5	=	=	SYM
ejpam-6781	259	6	qi	qi	PROPN
ejpam-6781	259	7	qi	qi	NOUN
ejpam-6781	259	8	given	give	VERB
ejpam-6781	259	9	in	in	ADP
ejpam-6781	259	10	(	(	PUNCT
ejpam-6781	259	11	7	7	NUM
ejpam-6781	259	12	)	)	PUNCT
ejpam-6781	259	13	,	,	PUNCT
ejpam-6781	259	14	it	it	PRON
ejpam-6781	259	15	can	can	AUX
ejpam-6781	259	16	be	be	AUX
ejpam-6781	259	17	rewritten	rewrite	VERB
ejpam-6781	259	18	in	in	ADP
ejpam-6781	259	19	the	the	DET
ejpam-6781	259	20	form	form	NOUN
ejpam-6781	259	21	of	of	ADP
ejpam-6781	259	22	ζ	ζ	NOUN
ejpam-6781	259	23	=	=	PUNCT
ejpam-6781	259	24	k∏	k∏	PROPN
ejpam-6781	259	25	i=1	i=1	NOUN
ejpam-6781	259	26	ζϵi·ni	ζϵi·ni	PROPN
ejpam-6781	259	27	pi	pi	NOUN
ejpam-6781	259	28	=	=	PUNCT
ejpam-6781	259	29	k∏	k∏	PROPN
ejpam-6781	259	30	i=1	i=1	PROPN
ejpam-6781	260	1	(	(	PUNCT
ejpam-6781	260	2	qi	qi	PROPN
ejpam-6781	260	3	qi	qi	PROPN
ejpam-6781	260	4	)	)	PUNCT
ejpam-6781	260	5	ϵi·ni	ϵi·ni	PROPN
ejpam-6781	260	6	.	.	PUNCT
ejpam-6781	261	1	since	since	SCONJ
ejpam-6781	261	2	p1	p1	PROPN
ejpam-6781	261	3	,	,	PUNCT
ejpam-6781	261	4	.	.	PUNCT
ejpam-6781	261	5	.	.	PUNCT
ejpam-6781	261	6	.	.	PUNCT
ejpam-6781	262	1	,	,	PUNCT
ejpam-6781	262	2	pk	pk	NOUN
ejpam-6781	262	3	are	be	AUX
ejpam-6781	262	4	in	in	ADP
ejpam-6781	262	5	p1	p1	NOUN
ejpam-6781	262	6	,	,	PUNCT
ejpam-6781	262	7	we	we	PRON
ejpam-6781	262	8	have	have	VERB
ejpam-6781	262	9	the	the	DET
ejpam-6781	262	10	factorization	factorization	NOUN
ejpam-6781	262	11	pi	pi	NOUN
ejpam-6781	263	1	=	=	SYM
ejpam-6781	263	2	qi	qi	PROPN
ejpam-6781	263	3	·	·	PROPN
ejpam-6781	263	4	qi	qi	PROPN
ejpam-6781	263	5	over	over	ADP
ejpam-6781	263	6	z[ω	z[ω	PROPN
ejpam-6781	263	7	]	]	PUNCT
ejpam-6781	263	8	for	for	ADP
ejpam-6781	263	9	all	all	DET
ejpam-6781	263	10	i	i	PRON
ejpam-6781	263	11	=	=	NOUN
ejpam-6781	263	12	1	1	NUM
ejpam-6781	263	13	,	,	PUNCT
ejpam-6781	263	14	.	.	PUNCT
ejpam-6781	263	15	.	.	PUNCT
ejpam-6781	263	16	.	.	PUNCT
ejpam-6781	264	1	,	,	PUNCT
ejpam-6781	264	2	k.	k.	PROPN
ejpam-6781	265	1	it	it	PRON
ejpam-6781	265	2	follows	follow	VERB
ejpam-6781	265	3	that	that	SCONJ
ejpam-6781	265	4	c	c	NOUN
ejpam-6781	265	5	=	=	SYM
ejpam-6781	265	6	k∏	k∏	PROPN
ejpam-6781	265	7	i=1	i=1	PROPN
ejpam-6781	265	8	pni	pni	NOUN
ejpam-6781	266	1	i	i	NOUN
ejpam-6781	266	2	=	=	SYM
ejpam-6781	266	3	k∏	k∏	PROPN
ejpam-6781	266	4	i=1	i=1	PROPN
ejpam-6781	266	5	(	(	PUNCT
ejpam-6781	266	6	qi	qi	PROPN
ejpam-6781	266	7	·	·	SYM
ejpam-6781	266	8	qi	qi	PROPN
ejpam-6781	266	9	)	)	PUNCT
ejpam-6781	266	10	ni	ni	PROPN
ejpam-6781	266	11	.	.	PUNCT
ejpam-6781	267	1	let	let	VERB
ejpam-6781	267	2	z	z	NOUN
ejpam-6781	267	3	:	:	PUNCT
ejpam-6781	267	4	=	=	PUNCT
ejpam-6781	267	5	ζ	ζ	X
ejpam-6781	267	6	·	·	PUNCT
ejpam-6781	267	7	c	c	X
ejpam-6781	268	1	=	=	PUNCT
ejpam-6781	268	2	k∏	k∏	PROPN
ejpam-6781	268	3	i=1	i=1	PROPN
ejpam-6781	268	4	(	(	PUNCT
ejpam-6781	268	5	qi	qi	PROPN
ejpam-6781	268	6	qi	qi	PROPN
ejpam-6781	268	7	)	)	PUNCT
ejpam-6781	268	8	ϵi·ni	ϵi·ni	PUNCT
ejpam-6781	268	9	·	·	PUNCT
ejpam-6781	268	10	(	(	PUNCT
ejpam-6781	268	11	qi	qi	X
ejpam-6781	268	12	·	·	SYM
ejpam-6781	268	13	qi	qi	PROPN
ejpam-6781	268	14	)	)	PUNCT
ejpam-6781	268	15	ni	ni	NOUN
ejpam-6781	268	16	=	=	PROPN
ejpam-6781	268	17	k∏	k∏	PROPN
ejpam-6781	268	18	i=1	i=1	PROPN
ejpam-6781	268	19	q	q	X
ejpam-6781	268	20	(	(	PUNCT
ejpam-6781	268	21	ϵi+1)·ni	ϵi+1)·ni	X
ejpam-6781	268	22	i	i	PRON
ejpam-6781	268	23	q	q	X
ejpam-6781	268	24	(	(	PUNCT
ejpam-6781	268	25	ϵi−1)·ni	ϵi−1)·ni	X
ejpam-6781	268	26	i	i	PRON
ejpam-6781	268	27	=	=	SYM
ejpam-6781	268	28	k∏	k∏	PROPN
ejpam-6781	268	29	i=1	i=1	PROPN
ejpam-6781	268	30	q̃i	q̃i	VERB
ejpam-6781	268	31	2ni	2ni	NOUN
ejpam-6781	268	32	,	,	PUNCT
ejpam-6781	268	33	where	where	SCONJ
ejpam-6781	268	34	q̃i	q̃i	PROPN
ejpam-6781	268	35	is	be	AUX
ejpam-6781	268	36	defined	define	VERB
ejpam-6781	268	37	to	to	PART
ejpam-6781	268	38	be	be	AUX
ejpam-6781	268	39	q̃i	q̃i	VERB
ejpam-6781	268	40	=	=	PUNCT
ejpam-6781	268	41	{	{	PUNCT
ejpam-6781	268	42	qi	qi	X
ejpam-6781	268	43	if	if	SCONJ
ejpam-6781	268	44	ϵi	ϵi	X
ejpam-6781	268	45	=	=	SYM
ejpam-6781	268	46	1	1	NUM
ejpam-6781	268	47	,	,	PUNCT
ejpam-6781	268	48	qi	qi	PRON
ejpam-6781	268	49	if	if	SCONJ
ejpam-6781	268	50	ϵi	ϵi	X
ejpam-6781	268	51	=	=	PUNCT
ejpam-6781	268	52	−1	−1	NOUN
ejpam-6781	268	53	.	.	PUNCT
ejpam-6781	269	1	hence	hence	ADV
ejpam-6781	269	2	,	,	PUNCT
ejpam-6781	269	3	z	z	PROPN
ejpam-6781	269	4	∈	∈	PROPN
ejpam-6781	269	5	z[ω	z[ω	PROPN
ejpam-6781	269	6	]	]	PUNCT
ejpam-6781	269	7	and	and	CCONJ
ejpam-6781	269	8	its	its	PRON
ejpam-6781	269	9	norm	norm	NOUN
ejpam-6781	269	10	satisfies	satisfy	VERB
ejpam-6781	269	11	n(z	n(z	NOUN
ejpam-6781	269	12	)	)	PUNCT
ejpam-6781	269	13	=	=	SYM
ejpam-6781	270	1	n	n	NOUN
ejpam-6781	270	2	(	(	PUNCT
ejpam-6781	270	3	∏k	∏k	X
ejpam-6781	270	4	i=1	i=1	PROPN
ejpam-6781	270	5	q̃i	q̃i	VERB
ejpam-6781	270	6	2ni	2ni	NOUN
ejpam-6781	270	7	)	)	PUNCT
ejpam-6781	271	1	=	=	PUNCT
ejpam-6781	272	1	∏k	∏k	X
ejpam-6781	272	2	i=1	i=1	X
ejpam-6781	272	3	p	p	X
ejpam-6781	272	4	ni	ni	PROPN
ejpam-6781	273	1	i	i	PROPN
ejpam-6781	273	2	=	=	PROPN
ejpam-6781	273	3	c.	c.	NOUN
ejpam-6781	273	4	we	we	PRON
ejpam-6781	273	5	can	can	AUX
ejpam-6781	273	6	choose	choose	VERB
ejpam-6781	273	7	δ	δ	PROPN
ejpam-6781	273	8	∈	∈	PROPN
ejpam-6781	273	9	u	u	NOUN
ejpam-6781	273	10	such	such	ADJ
ejpam-6781	273	11	that	that	SCONJ
ejpam-6781	273	12	the	the	DET
ejpam-6781	273	13	associate	associate	NOUN
ejpam-6781	273	14	δz	δz	VERB
ejpam-6781	273	15	=	=	PUNCT
ejpam-6781	273	16	a	a	DET
ejpam-6781	273	17	+	+	NUM
ejpam-6781	273	18	bω	bω	NOUN
ejpam-6781	273	19	is	be	AUX
ejpam-6781	273	20	in	in	ADP
ejpam-6781	273	21	the	the	DET
ejpam-6781	273	22	second	second	ADJ
ejpam-6781	273	23	sextant	sextant	NOUN
ejpam-6781	273	24	.	.	PUNCT
ejpam-6781	274	1	hence	hence	ADV
ejpam-6781	274	2	,	,	PUNCT
ejpam-6781	274	3	(	(	PUNCT
ejpam-6781	274	4	a	a	DET
ejpam-6781	274	5	,	,	PUNCT
ejpam-6781	274	6	b	b	NOUN
ejpam-6781	274	7	,	,	PUNCT
ejpam-6781	274	8	c	c	NOUN
ejpam-6781	274	9	)	)	PUNCT
ejpam-6781	274	10	forms	form	VERB
ejpam-6781	274	11	an	an	DET
ejpam-6781	274	12	eisenstein	eisenstein	NOUN
ejpam-6781	274	13	triple	triple	NOUN
ejpam-6781	274	14	.	.	PUNCT
ejpam-6781	275	1	next	next	ADV
ejpam-6781	275	2	,	,	PUNCT
ejpam-6781	275	3	we	we	PRON
ejpam-6781	275	4	show	show	VERB
ejpam-6781	275	5	that	that	SCONJ
ejpam-6781	275	6	the	the	DET
ejpam-6781	275	7	eisenstein	eisenstein	PROPN
ejpam-6781	275	8	triple	triple	NOUN
ejpam-6781	275	9	(	(	PUNCT
ejpam-6781	275	10	a	a	DET
ejpam-6781	275	11	,	,	PUNCT
ejpam-6781	275	12	b	b	NOUN
ejpam-6781	275	13	,	,	PUNCT
ejpam-6781	275	14	c	c	NOUN
ejpam-6781	275	15	)	)	PUNCT
ejpam-6781	275	16	is	be	AUX
ejpam-6781	275	17	primitive	primitive	ADJ
ejpam-6781	275	18	.	.	PUNCT
ejpam-6781	276	1	for	for	ADP
ejpam-6781	276	2	contrary	contrary	ADJ
ejpam-6781	276	3	,	,	PUNCT
ejpam-6781	276	4	we	we	PRON
ejpam-6781	276	5	suppose	suppose	VERB
ejpam-6781	276	6	that	that	SCONJ
ejpam-6781	276	7	there	there	PRON
ejpam-6781	276	8	exists	exist	VERB
ejpam-6781	276	9	pi	pi	NOUN
ejpam-6781	276	10	such	such	ADJ
ejpam-6781	276	11	that	that	DET
ejpam-6781	276	12	pi	pi	NOUN
ejpam-6781	277	1	|	|	ADV
ejpam-6781	277	2	a	a	PRON
ejpam-6781	277	3	and	and	CCONJ
ejpam-6781	277	4	pi	pi	NOUN
ejpam-6781	277	5	|	|	PROPN
ejpam-6781	277	6	b.	b.	PROPN
ejpam-6781	278	1	then	then	ADV
ejpam-6781	278	2	a	a	DET
ejpam-6781	278	3	=	=	X
ejpam-6781	278	4	pia	pia	PROPN
ejpam-6781	278	5	′	′	NOUN
ejpam-6781	278	6	and	and	CCONJ
ejpam-6781	278	7	b	b	X
ejpam-6781	278	8	=	=	X
ejpam-6781	278	9	pib	pib	PROPN
ejpam-6781	278	10	′	′	NOUN
ejpam-6781	278	11	for	for	ADP
ejpam-6781	278	12	some	some	DET
ejpam-6781	278	13	integers	integer	NOUN
ejpam-6781	278	14	a′	a′	PRON
ejpam-6781	278	15	and	and	CCONJ
ejpam-6781	278	16	b′.	b′.	PROPN
ejpam-6781	278	17	substituting	substitute	VERB
ejpam-6781	278	18	these	these	DET
ejpam-6781	278	19	expressions	expression	NOUN
ejpam-6781	278	20	into	into	ADP
ejpam-6781	278	21	δz	δz	PRON
ejpam-6781	278	22	,	,	PUNCT
ejpam-6781	278	23	we	we	PRON
ejpam-6781	278	24	obtain	obtain	VERB
ejpam-6781	278	25	δz	δz	ADP
ejpam-6781	278	26	=	=	PUNCT
ejpam-6781	278	27	a	a	DET
ejpam-6781	278	28	+	+	NUM
ejpam-6781	278	29	bω	bω	NOUN
ejpam-6781	278	30	=	=	PROPN
ejpam-6781	278	31	pia	pia	PROPN
ejpam-6781	279	1	′	′	PROPN
ejpam-6781	279	2	+	+	CCONJ
ejpam-6781	279	3	pib	pib	ADJ
ejpam-6781	279	4	′ω	′ω	NOUN
ejpam-6781	279	5	=	=	SYM
ejpam-6781	279	6	pi	pi	NOUN
ejpam-6781	279	7	(	(	PUNCT
ejpam-6781	279	8	a′	a′	X
ejpam-6781	279	9	+	+	SYM
ejpam-6781	279	10	b′ω	b′ω	ADJ
ejpam-6781	279	11	)	)	PUNCT
ejpam-6781	279	12	which	which	PRON
ejpam-6781	279	13	implies	imply	VERB
ejpam-6781	279	14	that	that	DET
ejpam-6781	279	15	pi	pi	NOUN
ejpam-6781	279	16	|	|	ADV
ejpam-6781	279	17	z.	z.	PROPN
ejpam-6781	279	18	since	since	SCONJ
ejpam-6781	279	19	z	z	PROPN
ejpam-6781	279	20	=	=	PUNCT
ejpam-6781	280	1	∏k	∏k	X
ejpam-6781	280	2	i=1	i=1	PROPN
ejpam-6781	280	3	q̃i	q̃i	VERB
ejpam-6781	280	4	2ni	2ni	ADJ
ejpam-6781	280	5	and	and	CCONJ
ejpam-6781	280	6	pi	pi	NOUN
ejpam-6781	280	7	=	=	SYM
ejpam-6781	280	8	qi	qi	PROPN
ejpam-6781	280	9	·	·	SYM
ejpam-6781	280	10	qi	qi	PROPN
ejpam-6781	280	11	,	,	PUNCT
ejpam-6781	280	12	qi	qi	PROPN
ejpam-6781	280	13	and	and	CCONJ
ejpam-6781	280	14	qi	qi	PROPN
ejpam-6781	280	15	appear	appear	VERB
ejpam-6781	280	16	s.	s.	PROPN
ejpam-6781	280	17	jitman	jitman	PROPN
ejpam-6781	280	18	,	,	PUNCT
ejpam-6781	280	19	m.	m.	NOUN
ejpam-6781	280	20	mohammad	mohammad	PROPN
ejpam-6781	280	21	,	,	PUNCT
ejpam-6781	280	22	e.	e.	PROPN
ejpam-6781	280	23	sangwisut	sangwisut	PROPN
ejpam-6781	280	24	/	/	SYM
ejpam-6781	280	25	eur	eur	PROPN
ejpam-6781	280	26	.	.	PUNCT
ejpam-6781	281	1	j.	j.	PROPN
ejpam-6781	281	2	pure	pure	PROPN
ejpam-6781	281	3	appl	appl	PROPN
ejpam-6781	281	4	.	.	PROPN
ejpam-6781	281	5	math	math	PROPN
ejpam-6781	281	6	,	,	PUNCT
ejpam-6781	281	7	18	18	NUM
ejpam-6781	281	8	(	(	PUNCT
ejpam-6781	281	9	4	4	NUM
ejpam-6781	281	10	)	)	PUNCT
ejpam-6781	281	11	(	(	PUNCT
ejpam-6781	281	12	2025	2025	NUM
ejpam-6781	281	13	)	)	PUNCT
ejpam-6781	281	14	,	,	PUNCT
ejpam-6781	281	15	6781	6781	NUM
ejpam-6781	281	16	12	12	NUM
ejpam-6781	281	17	of	of	ADP
ejpam-6781	281	18	16	16	NUM
ejpam-6781	281	19	as	as	ADP
ejpam-6781	281	20	factors	factor	NOUN
ejpam-6781	281	21	in	in	ADP
ejpam-6781	281	22	z.	z.	PROPN
ejpam-6781	282	1	this	this	PRON
ejpam-6781	282	2	is	be	AUX
ejpam-6781	282	3	a	a	DET
ejpam-6781	282	4	contradiction	contradiction	NOUN
ejpam-6781	282	5	.	.	PUNCT
ejpam-6781	283	1	consequently	consequently	ADV
ejpam-6781	283	2	,	,	PUNCT
ejpam-6781	283	3	there	there	ADV
ejpam-6781	283	4	a	a	PRON
ejpam-6781	283	5	and	and	CCONJ
ejpam-6781	283	6	b	b	NOUN
ejpam-6781	283	7	has	have	VERB
ejpam-6781	283	8	no	no	DET
ejpam-6781	283	9	common	common	ADJ
ejpam-6781	283	10	prime	prime	ADJ
ejpam-6781	283	11	divisors	divisor	NOUN
ejpam-6781	283	12	.	.	PUNCT
ejpam-6781	284	1	as	as	ADP
ejpam-6781	284	2	a	a	DET
ejpam-6781	284	3	result	result	NOUN
ejpam-6781	284	4	,	,	PUNCT
ejpam-6781	284	5	the	the	DET
ejpam-6781	284	6	eisenstein	eisenstein	NOUN
ejpam-6781	284	7	triple	triple	NOUN
ejpam-6781	284	8	(	(	PUNCT
ejpam-6781	284	9	a	a	DET
ejpam-6781	284	10	,	,	PUNCT
ejpam-6781	284	11	b	b	NOUN
ejpam-6781	284	12	,	,	PUNCT
ejpam-6781	284	13	c	c	NOUN
ejpam-6781	284	14	)	)	PUNCT
ejpam-6781	284	15	must	must	AUX
ejpam-6781	284	16	be	be	AUX
ejpam-6781	284	17	primitive	primitive	ADJ
ejpam-6781	284	18	.	.	PUNCT
ejpam-6781	285	1	the	the	DET
ejpam-6781	285	2	following	follow	VERB
ejpam-6781	285	3	theorem	theorem	NOUN
ejpam-6781	285	4	provides	provide	VERB
ejpam-6781	285	5	a	a	DET
ejpam-6781	285	6	classification	classification	NOUN
ejpam-6781	285	7	of	of	ADP
ejpam-6781	285	8	primitive	primitive	ADJ
ejpam-6781	285	9	eisenstein	eisenstein	NOUN
ejpam-6781	285	10	triples	triple	NOUN
ejpam-6781	285	11	based	base	VERB
ejpam-6781	285	12	on	on	ADP
ejpam-6781	285	13	the	the	DET
ejpam-6781	285	14	prime	prime	ADJ
ejpam-6781	285	15	factorization	factorization	NOUN
ejpam-6781	285	16	of	of	ADP
ejpam-6781	285	17	the	the	DET
ejpam-6781	285	18	hypotenuse	hypotenuse	NOUN
ejpam-6781	285	19	c.	c.	NOUN
ejpam-6781	285	20	theorem	theorem	VERB
ejpam-6781	285	21	4	4	NUM
ejpam-6781	285	22	.	.	PUNCT
ejpam-6781	286	1	let	let	VERB
ejpam-6781	286	2	c	c	PRON
ejpam-6781	286	3	>	>	X
ejpam-6781	286	4	1	1	NUM
ejpam-6781	286	5	be	be	AUX
ejpam-6781	286	6	an	an	DET
ejpam-6781	286	7	integer	integer	NOUN
ejpam-6781	286	8	with	with	ADP
ejpam-6781	286	9	prime	prime	ADJ
ejpam-6781	286	10	factorization	factorization	NOUN
ejpam-6781	286	11	c	c	NOUN
ejpam-6781	286	12	=	=	PUNCT
ejpam-6781	286	13	pn1	pn1	PROPN
ejpam-6781	286	14	1	1	NUM
ejpam-6781	286	15	.	.	PUNCT
ejpam-6781	286	16	.	.	PUNCT
ejpam-6781	286	17	.	.	PUNCT
ejpam-6781	287	1	pnk	pnk	PROPN
ejpam-6781	288	1	k	k	NOUN
ejpam-6781	288	2	,	,	PUNCT
ejpam-6781	288	3	where	where	SCONJ
ejpam-6781	288	4	k	k	PROPN
ejpam-6781	288	5	is	be	AUX
ejpam-6781	288	6	a	a	DET
ejpam-6781	288	7	positive	positive	ADJ
ejpam-6781	288	8	integer	integer	NOUN
ejpam-6781	288	9	,	,	PUNCT
ejpam-6781	288	10	p1	p1	NOUN
ejpam-6781	288	11	,	,	PUNCT
ejpam-6781	288	12	.	.	PUNCT
ejpam-6781	288	13	.	.	PUNCT
ejpam-6781	288	14	.	.	PUNCT
ejpam-6781	289	1	,	,	PUNCT
ejpam-6781	289	2	pk	pk	NOUN
ejpam-6781	289	3	are	be	AUX
ejpam-6781	289	4	distinct	distinct	ADJ
ejpam-6781	289	5	prime	prime	ADJ
ejpam-6781	289	6	numbers	number	NOUN
ejpam-6781	289	7	,	,	PUNCT
ejpam-6781	289	8	and	and	CCONJ
ejpam-6781	289	9	n1	n1	NOUN
ejpam-6781	289	10	,	,	PUNCT
ejpam-6781	289	11	.	.	PUNCT
ejpam-6781	289	12	.	.	PUNCT
ejpam-6781	290	1	.	.	PUNCT
ejpam-6781	291	1	,	,	PUNCT
ejpam-6781	291	2	nk	nk	PROPN
ejpam-6781	291	3	are	be	AUX
ejpam-6781	291	4	positive	positive	ADJ
ejpam-6781	291	5	exponents	exponent	NOUN
ejpam-6781	291	6	.	.	PUNCT
ejpam-6781	292	1	then	then	ADV
ejpam-6781	292	2	exactly	exactly	ADV
ejpam-6781	292	3	one	one	NUM
ejpam-6781	292	4	of	of	ADP
ejpam-6781	292	5	the	the	DET
ejpam-6781	292	6	following	following	ADJ
ejpam-6781	292	7	statements	statement	NOUN
ejpam-6781	292	8	holds	hold	VERB
ejpam-6781	292	9	:	:	PUNCT
ejpam-6781	292	10	(	(	PUNCT
ejpam-6781	292	11	i	i	NOUN
ejpam-6781	292	12	)	)	PUNCT
ejpam-6781	292	13	if	if	SCONJ
ejpam-6781	292	14	pi	pi	NOUN
ejpam-6781	292	15	≡	≡	PROPN
ejpam-6781	292	16	1	1	NUM
ejpam-6781	292	17	(	(	PUNCT
ejpam-6781	292	18	mod	mod	NOUN
ejpam-6781	292	19	3	3	NUM
ejpam-6781	292	20	)	)	PUNCT
ejpam-6781	292	21	for	for	ADP
ejpam-6781	292	22	all	all	DET
ejpam-6781	292	23	i	i	PRON
ejpam-6781	292	24	=	=	NOUN
ejpam-6781	292	25	1	1	NUM
ejpam-6781	292	26	,	,	PUNCT
ejpam-6781	292	27	.	.	PUNCT
ejpam-6781	292	28	.	.	PUNCT
ejpam-6781	292	29	.	.	PUNCT
ejpam-6781	293	1	,	,	PUNCT
ejpam-6781	293	2	k	k	NOUN
ejpam-6781	293	3	,	,	PUNCT
ejpam-6781	293	4	then	then	ADV
ejpam-6781	293	5	the	the	DET
ejpam-6781	293	6	map	map	NOUN
ejpam-6781	293	7	et|z	et|z	NOUN
ejpam-6781	293	8	:	:	PUNCT
ejpam-6781	293	9	z	z	X
ejpam-6781	293	10	→	→	PUNCT
ejpam-6781	293	11	etc	etc	X
ejpam-6781	293	12	is	be	AUX
ejpam-6781	293	13	a	a	DET
ejpam-6781	293	14	bijection	bijection	NOUN
ejpam-6781	293	15	,	,	PUNCT
ejpam-6781	293	16	where	where	SCONJ
ejpam-6781	293	17	z	z	NOUN
ejpam-6781	293	18	=	=	PRON
ejpam-6781	293	19	{	{	PUNCT
ejpam-6781	293	20	∏k	∏k	NOUN
ejpam-6781	293	21	i=1	i=1	PROPN
ejpam-6781	293	22	ζ	ζ	NOUN
ejpam-6781	293	23	ϵi·ni	ϵi·ni	NUM
ejpam-6781	293	24	pi	pi	NOUN
ejpam-6781	293	25	|	|	ADV
ejpam-6781	293	26	ϵ1	ϵ1	ADJ
ejpam-6781	293	27	,	,	PUNCT
ejpam-6781	293	28	.	.	PUNCT
ejpam-6781	293	29	.	.	PUNCT
ejpam-6781	294	1	.	.	PUNCT
ejpam-6781	295	1	,	,	PUNCT
ejpam-6781	295	2	ϵk	ϵk	X
ejpam-6781	295	3	∈	∈	PROPN
ejpam-6781	295	4	{	{	PUNCT
ejpam-6781	295	5	±1	±1	NOUN
ejpam-6781	295	6	}	}	PUNCT
ejpam-6781	295	7	}	}	PUNCT
ejpam-6781	295	8	,	,	PUNCT
ejpam-6781	295	9	and	and	CCONJ
ejpam-6781	295	10	ζpi	ζpi	PROPN
ejpam-6781	295	11	is	be	AUX
ejpam-6781	295	12	a	a	DET
ejpam-6781	295	13	complex	complex	ADJ
ejpam-6781	295	14	number	number	NOUN
ejpam-6781	295	15	defined	define	VERB
ejpam-6781	295	16	in	in	ADP
ejpam-6781	295	17	(	(	PUNCT
ejpam-6781	295	18	7	7	NUM
ejpam-6781	295	19	)	)	PUNCT
ejpam-6781	295	20	for	for	ADP
ejpam-6781	295	21	the	the	DET
ejpam-6781	295	22	prime	prime	ADJ
ejpam-6781	295	23	pi	pi	NOUN
ejpam-6781	295	24	.	.	PUNCT
ejpam-6781	296	1	(	(	PUNCT
ejpam-6781	296	2	ii	ii	NOUN
ejpam-6781	296	3	)	)	PUNCT
ejpam-6781	296	4	otherwise	otherwise	ADV
ejpam-6781	296	5	,	,	PUNCT
ejpam-6781	296	6	the	the	DET
ejpam-6781	296	7	set	set	VERB
ejpam-6781	296	8	etc	etc	X
ejpam-6781	296	9	is	be	AUX
ejpam-6781	296	10	empty	empty	ADJ
ejpam-6781	296	11	.	.	PUNCT
ejpam-6781	297	1	proof	proof	NOUN
ejpam-6781	297	2	.	.	PUNCT
ejpam-6781	298	1	to	to	PART
ejpam-6781	298	2	prove	prove	VERB
ejpam-6781	298	3	1	1	NUM
ejpam-6781	298	4	)	)	PUNCT
ejpam-6781	298	5	,	,	PUNCT
ejpam-6781	298	6	assume	assume	VERB
ejpam-6781	298	7	the	the	DET
ejpam-6781	298	8	notations	notation	NOUN
ejpam-6781	298	9	as	as	ADP
ejpam-6781	298	10	in	in	ADP
ejpam-6781	298	11	section	section	NOUN
ejpam-6781	298	12	3	3	NUM
ejpam-6781	298	13	.	.	PUNCT
ejpam-6781	298	14	consider	consider	VERB
ejpam-6781	298	15	the	the	DET
ejpam-6781	298	16	surjective	surjective	ADJ
ejpam-6781	298	17	function	function	NOUN
ejpam-6781	298	18	et	et	NOUN
ejpam-6781	298	19	:	:	PUNCT
ejpam-6781	298	20	g(q	g(q	NUM
ejpam-6781	298	21	)	)	PUNCT
ejpam-6781	298	22	∖	∖	X
ejpam-6781	298	23	u	u	NOUN
ejpam-6781	298	24	→	→	SYM
ejpam-6781	298	25	et	et	NOUN
ejpam-6781	298	26	defined	define	VERB
ejpam-6781	298	27	in	in	ADP
ejpam-6781	298	28	(	(	PUNCT
ejpam-6781	298	29	5	5	NUM
ejpam-6781	298	30	)	)	PUNCT
ejpam-6781	298	31	.	.	PUNCT
ejpam-6781	299	1	let	let	VERB
ejpam-6781	299	2	ω	ω	PRON
ejpam-6781	299	3	be	be	AUX
ejpam-6781	299	4	a	a	DET
ejpam-6781	299	5	relation	relation	NOUN
ejpam-6781	299	6	on	on	ADP
ejpam-6781	299	7	g(q	g(q	NOUN
ejpam-6781	299	8	)	)	PUNCT
ejpam-6781	299	9	∖	∖	X
ejpam-6781	299	10	u	u	NOUN
ejpam-6781	299	11	given	give	VERB
ejpam-6781	299	12	by	by	ADP
ejpam-6781	299	13	ζ1ωζ2	ζ1ωζ2	PROPN
ejpam-6781	299	14	if	if	SCONJ
ejpam-6781	299	15	and	and	CCONJ
ejpam-6781	299	16	only	only	ADV
ejpam-6781	299	17	if	if	SCONJ
ejpam-6781	299	18	ζ−1	ζ−1	PROPN
ejpam-6781	299	19	1	1	NUM
ejpam-6781	299	20	ζ2	ζ2	NOUN
ejpam-6781	299	21	∈	∈	PROPN
ejpam-6781	299	22	u	u	NOUN
ejpam-6781	299	23	.	.	PUNCT
ejpam-6781	300	1	then	then	ADV
ejpam-6781	300	2	ω	ω	PROPN
ejpam-6781	300	3	is	be	AUX
ejpam-6781	300	4	an	an	DET
ejpam-6781	300	5	equivalence	equivalence	NOUN
ejpam-6781	300	6	relation	relation	NOUN
ejpam-6781	300	7	and	and	CCONJ
ejpam-6781	300	8	we	we	PRON
ejpam-6781	300	9	denote	denote	VERB
ejpam-6781	300	10	the	the	DET
ejpam-6781	300	11	corresponding	corresponding	ADJ
ejpam-6781	300	12	quotient	quotient	NOUN
ejpam-6781	300	13	set	set	VERB
ejpam-6781	300	14	by	by	ADP
ejpam-6781	300	15	(	(	PUNCT
ejpam-6781	300	16	g(q	g(q	NOUN
ejpam-6781	300	17	)	)	PUNCT
ejpam-6781	300	18	∖	∖	X
ejpam-6781	300	19	u)/ω	u)/ω	PROPN
ejpam-6781	300	20	.	.	PUNCT
ejpam-6781	301	1	consequently	consequently	ADV
ejpam-6781	301	2	,	,	PUNCT
ejpam-6781	301	3	the	the	DET
ejpam-6781	301	4	induced	induced	ADJ
ejpam-6781	301	5	function	function	NOUN
ejpam-6781	301	6	et	et	NOUN
ejpam-6781	301	7	:	:	PUNCT
ejpam-6781	301	8	(	(	PUNCT
ejpam-6781	301	9	g(q	g(q	NOUN
ejpam-6781	301	10	)	)	PUNCT
ejpam-6781	301	11	∖	∖	X
ejpam-6781	301	12	u)/ω	u)/ω	VERB
ejpam-6781	301	13	→	→	PUNCT
ejpam-6781	301	14	et	et	NOUN
ejpam-6781	301	15	is	be	AUX
ejpam-6781	301	16	bicjective	bicjective	ADJ
ejpam-6781	301	17	.	.	PUNCT
ejpam-6781	302	1	by	by	ADP
ejpam-6781	302	2	the	the	DET
ejpam-6781	302	3	theorem	theorem	NOUN
ejpam-6781	302	4	3	3	NUM
ejpam-6781	302	5	,	,	PUNCT
ejpam-6781	302	6	we	we	PRON
ejpam-6781	302	7	have	have	VERB
ejpam-6781	302	8	g(q	g(q	NOUN
ejpam-6781	302	9	)	)	PUNCT
ejpam-6781	302	10	=	=	SYM
ejpam-6781	302	11	u	u	PROPN
ejpam-6781	302	12	⊕	⊕	PROPN
ejpam-6781	302	13	f	f	PROPN
ejpam-6781	302	14	,	,	PUNCT
ejpam-6781	302	15	which	which	PRON
ejpam-6781	302	16	implies	imply	VERB
ejpam-6781	302	17	that	that	SCONJ
ejpam-6781	302	18	g(q	g(q	NOUN
ejpam-6781	302	19	)	)	PUNCT
ejpam-6781	302	20	∖	∖	X
ejpam-6781	302	21	u	u	NOUN
ejpam-6781	302	22	=	=	PUNCT
ejpam-6781	302	23	(	(	PUNCT
ejpam-6781	302	24	u	u	NOUN
ejpam-6781	302	25	⊕	⊕	PROPN
ejpam-6781	302	26	f	f	PROPN
ejpam-6781	302	27	)	)	PUNCT
ejpam-6781	302	28	∖	∖	X
ejpam-6781	302	29	u	u	NOUN
ejpam-6781	302	30	=	=	PROPN
ejpam-6781	302	31	u	u	PROPN
ejpam-6781	302	32	⊕	⊕	PROPN
ejpam-6781	302	33	(	(	PUNCT
ejpam-6781	302	34	f	f	PROPN
ejpam-6781	302	35	∖	∖	X
ejpam-6781	302	36	{	{	PUNCT
ejpam-6781	302	37	1	1	NUM
ejpam-6781	302	38	}	}	PUNCT
ejpam-6781	302	39	)	)	PUNCT
ejpam-6781	302	40	.	.	PUNCT
ejpam-6781	303	1	since	since	SCONJ
ejpam-6781	303	2	every	every	DET
ejpam-6781	303	3	element	element	NOUN
ejpam-6781	303	4	in	in	ADP
ejpam-6781	303	5	u	u	PROPN
ejpam-6781	303	6	⊕	⊕	PROPN
ejpam-6781	303	7	f	f	PROPN
ejpam-6781	303	8	is	be	AUX
ejpam-6781	303	9	uniquely	uniquely	ADV
ejpam-6781	303	10	of	of	ADP
ejpam-6781	303	11	the	the	DET
ejpam-6781	303	12	form	form	NOUN
ejpam-6781	303	13	uf	uf	NOUN
ejpam-6781	303	14	with	with	ADP
ejpam-6781	303	15	u	u	PROPN
ejpam-6781	303	16	∈	∈	PROPN
ejpam-6781	303	17	u	u	NOUN
ejpam-6781	303	18	and	and	CCONJ
ejpam-6781	303	19	f	f	PROPN
ejpam-6781	303	20	∈	∈	PROPN
ejpam-6781	303	21	f	f	PROPN
ejpam-6781	303	22	,	,	PUNCT
ejpam-6781	303	23	the	the	DET
ejpam-6781	303	24	elements	element	NOUN
ejpam-6781	303	25	not	not	PART
ejpam-6781	303	26	in	in	ADP
ejpam-6781	303	27	u	u	NOUN
ejpam-6781	303	28	are	be	AUX
ejpam-6781	303	29	exactly	exactly	ADV
ejpam-6781	303	30	those	those	PRON
ejpam-6781	303	31	with	with	ADP
ejpam-6781	303	32	f	f	PROPN
ejpam-6781	303	33	̸=	̸=	PROPN
ejpam-6781	303	34	1	1	NUM
ejpam-6781	303	35	.	.	PUNCT
ejpam-6781	304	1	hence	hence	ADV
ejpam-6781	304	2	,	,	PUNCT
ejpam-6781	304	3	(	(	PUNCT
ejpam-6781	304	4	u	u	NOUN
ejpam-6781	304	5	⊕	⊕	PROPN
ejpam-6781	304	6	f	f	PROPN
ejpam-6781	304	7	)	)	PUNCT
ejpam-6781	304	8	\	\	PROPN
ejpam-6781	304	9	u	u	NOUN
ejpam-6781	304	10	=	=	PUNCT
ejpam-6781	304	11	{	{	PUNCT
ejpam-6781	304	12	uf	uf	INTJ
ejpam-6781	304	13	|	|	ADV
ejpam-6781	304	14	u	u	PROPN
ejpam-6781	304	15	∈	∈	PROPN
ejpam-6781	304	16	u	u	PROPN
ejpam-6781	304	17	,	,	PUNCT
ejpam-6781	304	18	f	f	PROPN
ejpam-6781	304	19	∈	∈	PROPN
ejpam-6781	304	20	f	f	PROPN
ejpam-6781	304	21	\	\	PROPN
ejpam-6781	304	22	{	{	PUNCT
ejpam-6781	304	23	1	1	NUM
ejpam-6781	304	24	}	}	PUNCT
ejpam-6781	304	25	}	}	PUNCT
ejpam-6781	304	26	=	=	SYM
ejpam-6781	304	27	u	u	PROPN
ejpam-6781	304	28	⊕	⊕	PROPN
ejpam-6781	304	29	(	(	PUNCT
ejpam-6781	304	30	f	f	PROPN
ejpam-6781	304	31	\	\	PROPN
ejpam-6781	304	32	{	{	PUNCT
ejpam-6781	304	33	1	1	NUM
ejpam-6781	304	34	}	}	PUNCT
ejpam-6781	304	35	)	)	PUNCT
ejpam-6781	304	36	.	.	PUNCT
ejpam-6781	305	1	thus	thus	ADV
ejpam-6781	305	2	,	,	PUNCT
ejpam-6781	305	3	the	the	DET
ejpam-6781	305	4	quotient	quotient	NOUN
ejpam-6781	305	5	sets	set	NOUN
ejpam-6781	305	6	are	be	AUX
ejpam-6781	305	7	identical	identical	ADJ
ejpam-6781	305	8	:	:	PUNCT
ejpam-6781	305	9	(	(	PUNCT
ejpam-6781	305	10	g(q	g(q	NOUN
ejpam-6781	305	11	)	)	PUNCT
ejpam-6781	305	12	∖	∖	X
ejpam-6781	305	13	u)/ω	u)/ω	PROPN
ejpam-6781	306	1	=	=	PRON
ejpam-6781	306	2	(	(	PUNCT
ejpam-6781	306	3	u	u	PROPN
ejpam-6781	306	4	⊕	⊕	PROPN
ejpam-6781	306	5	(	(	PUNCT
ejpam-6781	306	6	f	f	PROPN
ejpam-6781	306	7	∖	∖	X
ejpam-6781	306	8	{	{	PUNCT
ejpam-6781	306	9	1}))/ω	1}))/ω	PROPN
ejpam-6781	306	10	=	=	SYM
ejpam-6781	306	11	f	f	PROPN
ejpam-6781	306	12	∖	∖	X
ejpam-6781	306	13	{	{	PUNCT
ejpam-6781	306	14	1	1	NUM
ejpam-6781	306	15	}	}	PUNCT
ejpam-6781	306	16	.	.	PUNCT
ejpam-6781	307	1	this	this	PRON
ejpam-6781	307	2	establishes	establish	VERB
ejpam-6781	307	3	the	the	DET
ejpam-6781	307	4	one	one	NUM
ejpam-6781	307	5	-	-	PUNCT
ejpam-6781	307	6	to	to	ADP
ejpam-6781	307	7	-	-	PUNCT
ejpam-6781	307	8	one	one	NUM
ejpam-6781	307	9	correspondence	correspondence	NOUN
ejpam-6781	307	10	et	et	NOUN
ejpam-6781	307	11	:	:	PUNCT
ejpam-6781	307	12	f	f	X
ejpam-6781	307	13	∖	∖	X
ejpam-6781	307	14	{	{	PUNCT
ejpam-6781	307	15	1	1	NUM
ejpam-6781	307	16	}	}	PUNCT
ejpam-6781	307	17	→	→	SYM
ejpam-6781	307	18	et	et	NOUN
ejpam-6781	307	19	.	.	PUNCT
ejpam-6781	308	1	now	now	ADV
ejpam-6781	308	2	,	,	PUNCT
ejpam-6781	308	3	let	let	VERB
ejpam-6781	308	4	p1	p1	NOUN
ejpam-6781	308	5	,	,	PUNCT
ejpam-6781	308	6	.	.	PUNCT
ejpam-6781	308	7	.	.	PUNCT
ejpam-6781	309	1	.	.	PUNCT
ejpam-6781	310	1	,	,	PUNCT
ejpam-6781	310	2	pk	pk	NOUN
ejpam-6781	310	3	be	be	AUX
ejpam-6781	310	4	distinct	distinct	ADJ
ejpam-6781	310	5	primes	prime	NOUN
ejpam-6781	310	6	in	in	ADP
ejpam-6781	310	7	p1	p1	PROPN
ejpam-6781	310	8	and	and	CCONJ
ejpam-6781	310	9	let	let	VERB
ejpam-6781	310	10	n1	n1	NOUN
ejpam-6781	310	11	,	,	PUNCT
ejpam-6781	310	12	.	.	PUNCT
ejpam-6781	310	13	.	.	PUNCT
ejpam-6781	311	1	.	.	PUNCT
ejpam-6781	312	1	,	,	PUNCT
ejpam-6781	312	2	nk	nk	PROPN
ejpam-6781	312	3	be	be	VERB
ejpam-6781	312	4	positive	positive	ADJ
ejpam-6781	312	5	integers	integer	NOUN
ejpam-6781	312	6	.	.	PUNCT
ejpam-6781	313	1	let	let	VERB
ejpam-6781	313	2	z	z	NOUN
ejpam-6781	313	3	:	:	PUNCT
ejpam-6781	313	4	=	=	X
ejpam-6781	313	5	{	{	PUNCT
ejpam-6781	313	6	k∏	k∏	PROPN
ejpam-6781	313	7	i=1	i=1	PROPN
ejpam-6781	313	8	ζϵi·ni	ζϵi·ni	PROPN
ejpam-6781	313	9	pi	pi	NOUN
ejpam-6781	313	10	|	|	ADV
ejpam-6781	313	11	ϵ1	ϵ1	ADJ
ejpam-6781	313	12	,	,	PUNCT
ejpam-6781	313	13	.	.	PUNCT
ejpam-6781	313	14	.	.	PUNCT
ejpam-6781	314	1	.	.	PUNCT
ejpam-6781	315	1	,	,	PUNCT
ejpam-6781	315	2	ϵk	ϵk	X
ejpam-6781	315	3	∈	∈	PROPN
ejpam-6781	315	4	{	{	PUNCT
ejpam-6781	315	5	±1	±1	NOUN
ejpam-6781	315	6	}	}	PUNCT
ejpam-6781	315	7	}	}	PUNCT
ejpam-6781	315	8	.	.	PUNCT
ejpam-6781	316	1	s.	s.	PROPN
ejpam-6781	316	2	jitman	jitman	PROPN
ejpam-6781	316	3	,	,	PUNCT
ejpam-6781	316	4	m.	m.	NOUN
ejpam-6781	316	5	mohammad	mohammad	PROPN
ejpam-6781	316	6	,	,	PUNCT
ejpam-6781	316	7	e.	e.	PROPN
ejpam-6781	316	8	sangwisut	sangwisut	PROPN
ejpam-6781	316	9	/	/	SYM
ejpam-6781	316	10	eur	eur	PROPN
ejpam-6781	316	11	.	.	PUNCT
ejpam-6781	317	1	j.	j.	PROPN
ejpam-6781	317	2	pure	pure	PROPN
ejpam-6781	317	3	appl	appl	PROPN
ejpam-6781	317	4	.	.	PROPN
ejpam-6781	317	5	math	math	PROPN
ejpam-6781	317	6	,	,	PUNCT
ejpam-6781	317	7	18	18	NUM
ejpam-6781	317	8	(	(	PUNCT
ejpam-6781	317	9	4	4	NUM
ejpam-6781	317	10	)	)	PUNCT
ejpam-6781	317	11	(	(	PUNCT
ejpam-6781	317	12	2025	2025	NUM
ejpam-6781	317	13	)	)	PUNCT
ejpam-6781	317	14	,	,	PUNCT
ejpam-6781	317	15	6781	6781	NUM
ejpam-6781	317	16	13	13	NUM
ejpam-6781	317	17	of	of	ADP
ejpam-6781	317	18	16	16	NUM
ejpam-6781	317	19	then	then	ADV
ejpam-6781	317	20	z	z	PROPN
ejpam-6781	317	21	⊆	⊆	NUM
ejpam-6781	317	22	f	f	PROPN
ejpam-6781	317	23	∖	∖	X
ejpam-6781	317	24	{	{	PUNCT
ejpam-6781	317	25	1	1	NUM
ejpam-6781	317	26	}	}	PUNCT
ejpam-6781	317	27	and	and	CCONJ
ejpam-6781	317	28	the	the	DET
ejpam-6781	317	29	restriction	restriction	NOUN
ejpam-6781	317	30	map	map	NOUN
ejpam-6781	317	31	et	et	PROPN
ejpam-6781	317	32	|z	|z	PROPN
ejpam-6781	317	33	:	:	PUNCT
ejpam-6781	317	34	z	z	X
ejpam-6781	317	35	→	→	SYM
ejpam-6781	317	36	et	et	NOUN
ejpam-6781	317	37	is	be	AUX
ejpam-6781	317	38	injective	injective	ADJ
ejpam-6781	317	39	,	,	PUNCT
ejpam-6781	317	40	and	and	CCONJ
ejpam-6781	317	41	the	the	DET
ejpam-6781	317	42	image	image	NOUN
ejpam-6781	317	43	of	of	ADP
ejpam-6781	317	44	z	z	PROPN
ejpam-6781	317	45	is	be	AUX
ejpam-6781	317	46	etc	etc	X
ejpam-6781	317	47	(	(	PUNCT
ejpam-6781	317	48	see	see	VERB
ejpam-6781	317	49	lemma	lemma	PROPN
ejpam-6781	317	50	1	1	NUM
ejpam-6781	317	51	)	)	PUNCT
ejpam-6781	317	52	.	.	PUNCT
ejpam-6781	318	1	therefore	therefore	ADV
ejpam-6781	318	2	,	,	PUNCT
ejpam-6781	318	3	et	et	PROPN
ejpam-6781	318	4	|z	|z	PROPN
ejpam-6781	318	5	:	:	PUNCT
ejpam-6781	318	6	z	z	X
ejpam-6781	318	7	→	→	PUNCT
ejpam-6781	318	8	etc	etc	X
ejpam-6781	318	9	is	be	AUX
ejpam-6781	318	10	a	a	DET
ejpam-6781	318	11	bijection	bijection	NOUN
ejpam-6781	318	12	.	.	PUNCT
ejpam-6781	319	1	from	from	ADP
ejpam-6781	319	2	theorem	theorem	ADJ
ejpam-6781	319	3	2	2	NUM
ejpam-6781	319	4	,	,	PUNCT
ejpam-6781	319	5	2	2	NUM
ejpam-6781	319	6	)	)	PUNCT
ejpam-6781	319	7	follows	follow	VERB
ejpam-6781	319	8	immediately	immediately	ADV
ejpam-6781	319	9	.	.	PUNCT
ejpam-6781	320	1	corollary	corollary	ADJ
ejpam-6781	320	2	1	1	NUM
ejpam-6781	320	3	.	.	PUNCT
ejpam-6781	321	1	let	let	VERB
ejpam-6781	321	2	c	c	PRON
ejpam-6781	321	3	>	>	X
ejpam-6781	321	4	1	1	NUM
ejpam-6781	321	5	be	be	AUX
ejpam-6781	321	6	an	an	DET
ejpam-6781	321	7	integer	integer	NOUN
ejpam-6781	321	8	with	with	ADP
ejpam-6781	321	9	prime	prime	ADJ
ejpam-6781	321	10	factorization	factorization	NOUN
ejpam-6781	321	11	as	as	SCONJ
ejpam-6781	321	12	described	describe	VERB
ejpam-6781	321	13	in	in	ADP
ejpam-6781	321	14	theorem	theorem	ADJ
ejpam-6781	321	15	4	4	NUM
ejpam-6781	321	16	.	.	PUNCT
ejpam-6781	322	1	then	then	ADV
ejpam-6781	322	2	,	,	PUNCT
ejpam-6781	322	3	exactly	exactly	ADV
ejpam-6781	322	4	one	one	NUM
ejpam-6781	322	5	of	of	ADP
ejpam-6781	322	6	the	the	DET
ejpam-6781	322	7	following	follow	VERB
ejpam-6781	322	8	holds	hold	VERB
ejpam-6781	322	9	:	:	PUNCT
ejpam-6781	322	10	(	(	PUNCT
ejpam-6781	322	11	i	i	NOUN
ejpam-6781	322	12	)	)	PUNCT
ejpam-6781	322	13	if	if	SCONJ
ejpam-6781	322	14	pi	pi	NOUN
ejpam-6781	322	15	≡	≡	PROPN
ejpam-6781	322	16	1	1	NUM
ejpam-6781	322	17	(	(	PUNCT
ejpam-6781	322	18	mod	mod	PROPN
ejpam-6781	322	19	6	6	NUM
ejpam-6781	322	20	)	)	PUNCT
ejpam-6781	322	21	for	for	ADP
ejpam-6781	322	22	all	all	DET
ejpam-6781	322	23	i	i	PRON
ejpam-6781	322	24	=	=	NOUN
ejpam-6781	322	25	1	1	NUM
ejpam-6781	322	26	,	,	PUNCT
ejpam-6781	322	27	.	.	PUNCT
ejpam-6781	322	28	.	.	PUNCT
ejpam-6781	322	29	.	.	PUNCT
ejpam-6781	323	1	,	,	PUNCT
ejpam-6781	323	2	k	k	NOUN
ejpam-6781	323	3	,	,	PUNCT
ejpam-6781	323	4	then	then	ADV
ejpam-6781	323	5	there	there	PRON
ejpam-6781	323	6	are	be	VERB
ejpam-6781	323	7	2k	2k	NUM
ejpam-6781	323	8	primitive	primitive	ADJ
ejpam-6781	323	9	eisenstein	eisenstein	NOUN
ejpam-6781	323	10	triples	triple	NOUN
ejpam-6781	323	11	with	with	ADP
ejpam-6781	323	12	hypotenuse	hypotenuse	NOUN
ejpam-6781	323	13	c.	c.	PROPN
ejpam-6781	323	14	(	(	PUNCT
ejpam-6781	323	15	ii	ii	PROPN
ejpam-6781	323	16	)	)	PUNCT
ejpam-6781	323	17	otherwise	otherwise	ADV
ejpam-6781	323	18	,	,	PUNCT
ejpam-6781	323	19	there	there	PRON
ejpam-6781	323	20	are	be	VERB
ejpam-6781	323	21	no	no	DET
ejpam-6781	323	22	primitive	primitive	ADJ
ejpam-6781	323	23	eisenstein	eisenstein	NOUN
ejpam-6781	323	24	triples	triple	NOUN
ejpam-6781	323	25	with	with	ADP
ejpam-6781	323	26	hypotenuse	hypotenuse	NOUN
ejpam-6781	323	27	c.	c.	NOUN
ejpam-6781	323	28	proof	proof	NOUN
ejpam-6781	323	29	.	.	PUNCT
ejpam-6781	324	1	the	the	DET
ejpam-6781	324	2	first	first	ADJ
ejpam-6781	324	3	statement	statement	NOUN
ejpam-6781	324	4	follows	follow	VERB
ejpam-6781	324	5	form	form	VERB
ejpam-6781	324	6	the	the	DET
ejpam-6781	324	7	one	one	NUM
ejpam-6781	324	8	-	-	PUNCT
ejpam-6781	324	9	to	to	ADP
ejpam-6781	324	10	-	-	PUNCT
ejpam-6781	324	11	one	one	NUM
ejpam-6781	324	12	correspondence	correspondence	NOUN
ejpam-6781	324	13	established	establish	VERB
ejpam-6781	324	14	in	in	ADP
ejpam-6781	324	15	1	1	NUM
ejpam-6781	324	16	)	)	PUNCT
ejpam-6781	324	17	of	of	ADP
ejpam-6781	324	18	theorem	theorem	ADJ
ejpam-6781	324	19	4	4	NUM
ejpam-6781	324	20	.	.	PUNCT
ejpam-6781	325	1	precisely	precisely	ADV
ejpam-6781	325	2	,	,	PUNCT
ejpam-6781	325	3	|etc|	|etc|	NOUN
ejpam-6781	325	4	=	=	SYM
ejpam-6781	325	5	∣∣∣∣∣	∣∣∣∣∣	PROPN
ejpam-6781	325	6	{	{	PUNCT
ejpam-6781	325	7	k∏	k∏	PROPN
ejpam-6781	325	8	i=1	i=1	PROPN
ejpam-6781	325	9	ζϵi·ni	ζϵi·ni	PROPN
ejpam-6781	325	10	pi	pi	NOUN
ejpam-6781	325	11	|	|	ADV
ejpam-6781	325	12	ϵ1	ϵ1	ADJ
ejpam-6781	325	13	,	,	PUNCT
ejpam-6781	325	14	.	.	PUNCT
ejpam-6781	325	15	.	.	PUNCT
ejpam-6781	326	1	.	.	PUNCT
ejpam-6781	327	1	,	,	PUNCT
ejpam-6781	327	2	ϵk	ϵk	X
ejpam-6781	327	3	∈	∈	PROPN
ejpam-6781	327	4	{	{	PUNCT
ejpam-6781	327	5	±1	±1	NOUN
ejpam-6781	327	6	}	}	PUNCT
ejpam-6781	327	7	}	}	PUNCT
ejpam-6781	327	8	∣∣∣∣∣	∣∣∣∣∣	PROPN
ejpam-6781	327	9	=	=	SYM
ejpam-6781	327	10	2k	2k	NUM
ejpam-6781	327	11	.	.	PUNCT
ejpam-6781	328	1	the	the	DET
ejpam-6781	328	2	second	second	ADJ
ejpam-6781	328	3	statement	statement	NOUN
ejpam-6781	328	4	can	can	AUX
ejpam-6781	328	5	be	be	AUX
ejpam-6781	328	6	deduced	deduce	VERB
ejpam-6781	328	7	directly	directly	ADV
ejpam-6781	328	8	from	from	ADP
ejpam-6781	328	9	2	2	NUM
ejpam-6781	328	10	)	)	PUNCT
ejpam-6781	328	11	of	of	ADP
ejpam-6781	328	12	theorem	theorem	ADJ
ejpam-6781	328	13	4	4	NUM
ejpam-6781	328	14	.	.	PUNCT
ejpam-6781	329	1	the	the	DET
ejpam-6781	329	2	following	follow	VERB
ejpam-6781	329	3	examples	example	NOUN
ejpam-6781	329	4	illustrate	illustrate	VERB
ejpam-6781	329	5	the	the	DET
ejpam-6781	329	6	application	application	NOUN
ejpam-6781	329	7	of	of	ADP
ejpam-6781	329	8	theorem	theorem	ADJ
ejpam-6781	329	9	4	4	NUM
ejpam-6781	329	10	and	and	CCONJ
ejpam-6781	329	11	corollary	corollary	ADJ
ejpam-6781	329	12	1	1	NUM
ejpam-6781	329	13	to	to	PART
ejpam-6781	329	14	determine	determine	VERB
ejpam-6781	329	15	the	the	DET
ejpam-6781	329	16	primitive	primitive	ADJ
ejpam-6781	329	17	eisenstein	eisenstein	NOUN
ejpam-6781	329	18	triples	triple	NOUN
ejpam-6781	329	19	for	for	ADP
ejpam-6781	329	20	specific	specific	ADJ
ejpam-6781	329	21	hypotenuses	hypotenuse	NOUN
ejpam-6781	329	22	.	.	PUNCT
ejpam-6781	329	23	example	example	NOUN
ejpam-6781	330	1	2	2	NUM
ejpam-6781	330	2	.	.	PUNCT
ejpam-6781	330	3	let	let	VERB
ejpam-6781	330	4	c	c	NOUN
ejpam-6781	330	5	=	=	SYM
ejpam-6781	330	6	49	49	NUM
ejpam-6781	330	7	.	.	PUNCT
ejpam-6781	331	1	the	the	DET
ejpam-6781	331	2	prime	prime	ADJ
ejpam-6781	331	3	factorization	factorization	NOUN
ejpam-6781	331	4	is	be	AUX
ejpam-6781	331	5	c	c	NOUN
ejpam-6781	331	6	=	=	SYM
ejpam-6781	331	7	72	72	NUM
ejpam-6781	331	8	which	which	PRON
ejpam-6781	331	9	7	7	NUM
ejpam-6781	331	10	∈	∈	PROPN
ejpam-6781	331	11	p1	p1	NOUN
ejpam-6781	331	12	.	.	PUNCT
ejpam-6781	332	1	by	by	ADP
ejpam-6781	332	2	example	example	NOUN
ejpam-6781	332	3	1	1	NUM
ejpam-6781	332	4	,	,	PUNCT
ejpam-6781	332	5	we	we	PRON
ejpam-6781	332	6	have	have	AUX
ejpam-6781	332	7	ζ7	ζ7	VERB
ejpam-6781	332	8	=	=	PUNCT
ejpam-6781	332	9	1	1	NUM
ejpam-6781	332	10	+	+	NOUN
ejpam-6781	332	11	3ω	3ω	NOUN
ejpam-6781	332	12	1	1	NUM
ejpam-6781	332	13	+	+	NOUN
ejpam-6781	332	14	3ω	3ω	NOUN
ejpam-6781	332	15	=	=	SYM
ejpam-6781	333	1	−8	−8	X
ejpam-6781	333	2	7	7	NUM
ejpam-6781	333	3	−	−	SYM
ejpam-6781	333	4	3	3	NUM
ejpam-6781	333	5	7ω	7ω	NOUN
ejpam-6781	333	6	and	and	CCONJ
ejpam-6781	333	7	ζ−1	ζ−1	PROPN
ejpam-6781	333	8	7	7	NUM
ejpam-6781	334	1	=	=	SYM
ejpam-6781	334	2	1	1	NUM
ejpam-6781	334	3	+	+	NOUN
ejpam-6781	334	4	3ω	3ω	NOUN
ejpam-6781	334	5	1	1	NUM
ejpam-6781	334	6	+	+	NOUN
ejpam-6781	334	7	3ω	3ω	NOUN
ejpam-6781	334	8	=	=	SYM
ejpam-6781	335	1	−5	−5	ADP
ejpam-6781	335	2	7	7	NUM
ejpam-6781	335	3	+	+	SYM
ejpam-6781	335	4	3	3	NUM
ejpam-6781	335	5	7ω	7ω	NOUN
ejpam-6781	335	6	.	.	PUNCT
ejpam-6781	336	1	we	we	PRON
ejpam-6781	336	2	have	have	VERB
ejpam-6781	336	3	the	the	DET
ejpam-6781	336	4	set	set	NOUN
ejpam-6781	336	5	z	z	NOUN
ejpam-6781	336	6	=	=	NOUN
ejpam-6781	336	7	{	{	PUNCT
ejpam-6781	336	8	ζϵ·27	ζϵ·27	NOUN
ejpam-6781	336	9	|	|	ADV
ejpam-6781	336	10	ϵ	ϵ	PROPN
ejpam-6781	336	11	∈	∈	PROPN
ejpam-6781	336	12	{	{	PUNCT
ejpam-6781	336	13	±1	±1	NOUN
ejpam-6781	336	14	}	}	PUNCT
ejpam-6781	336	15	}	}	PUNCT
ejpam-6781	336	16	=	=	SYM
ejpam-6781	336	17	{	{	PUNCT
ejpam-6781	336	18	ζ27	ζ27	ADV
ejpam-6781	336	19	,	,	PUNCT
ejpam-6781	336	20	ζ	ζ	NOUN
ejpam-6781	336	21	−2	−2	NOUN
ejpam-6781	336	22	7	7	NUM
ejpam-6781	336	23	}	}	PUNCT
ejpam-6781	336	24	.	.	PUNCT
ejpam-6781	337	1	by	by	ADP
ejpam-6781	337	2	theorem	theorem	NOUN
ejpam-6781	337	3	4	4	NUM
ejpam-6781	337	4	,	,	PUNCT
ejpam-6781	337	5	there	there	PRON
ejpam-6781	337	6	are	be	VERB
ejpam-6781	337	7	two	two	NUM
ejpam-6781	337	8	cases	case	NOUN
ejpam-6781	337	9	:	:	PUNCT
ejpam-6781	337	10	case	case	NOUN
ejpam-6781	337	11	ϵ1	ϵ1	NOUN
ejpam-6781	337	12	=	=	NOUN
ejpam-6781	337	13	1	1	NUM
ejpam-6781	337	14	:	:	PUNCT
ejpam-6781	337	15	we	we	PRON
ejpam-6781	337	16	compute	compute	VERB
ejpam-6781	337	17	ζ27	ζ27	VERB
ejpam-6781	337	18	:	:	PUNCT
ejpam-6781	337	19	ζ27	ζ27	ADV
ejpam-6781	337	20	=	=	SYM
ejpam-6781	337	21	(	(	PUNCT
ejpam-6781	337	22	−8	−8	ADP
ejpam-6781	337	23	7	7	NUM
ejpam-6781	337	24	−	−	NOUN
ejpam-6781	337	25	3	3	NUM
ejpam-6781	337	26	7	7	NUM
ejpam-6781	337	27	ω	ω	NOUN
ejpam-6781	337	28	)	)	PUNCT
ejpam-6781	337	29	2	2	NUM
ejpam-6781	337	30	=	=	SYM
ejpam-6781	337	31	55	55	NUM
ejpam-6781	337	32	49	49	NUM
ejpam-6781	337	33	+	+	CCONJ
ejpam-6781	337	34	39	39	NUM
ejpam-6781	337	35	49	49	NUM
ejpam-6781	337	36	ω	ω	NOUN
ejpam-6781	337	37	.	.	PUNCT
ejpam-6781	338	1	this	this	PRON
ejpam-6781	338	2	lies	lie	VERB
ejpam-6781	338	3	in	in	ADP
ejpam-6781	338	4	the	the	DET
ejpam-6781	338	5	sixth	sixth	ADJ
ejpam-6781	338	6	sextant	sextant	NOUN
ejpam-6781	338	7	.	.	PUNCT
ejpam-6781	339	1	its	its	PRON
ejpam-6781	339	2	associate	associate	NOUN
ejpam-6781	339	3	in	in	ADP
ejpam-6781	339	4	the	the	DET
ejpam-6781	339	5	second	second	ADJ
ejpam-6781	339	6	sextant	sextant	NOUN
ejpam-6781	339	7	is	be	AUX
ejpam-6781	339	8	16	16	NUM
ejpam-6781	339	9	49	49	NUM
ejpam-6781	339	10	+	+	CCONJ
ejpam-6781	339	11	55	55	NUM
ejpam-6781	339	12	49ω	49ω	NOUN
ejpam-6781	339	13	.	.	PUNCT
ejpam-6781	340	1	the	the	DET
ejpam-6781	340	2	corresponding	corresponding	ADJ
ejpam-6781	340	3	primitive	primitive	ADJ
ejpam-6781	340	4	eisenstein	eisenstein	NOUN
ejpam-6781	340	5	triples	triple	NOUN
ejpam-6781	340	6	is	be	AUX
ejpam-6781	340	7	et	et	NOUN
ejpam-6781	340	8	(	(	PUNCT
ejpam-6781	340	9	16	16	NUM
ejpam-6781	340	10	49	49	NUM
ejpam-6781	340	11	+	+	SYM
ejpam-6781	340	12	55	55	NUM
ejpam-6781	340	13	49ω	49ω	NOUN
ejpam-6781	340	14	)	)	PUNCT
ejpam-6781	341	1	=	=	PUNCT
ejpam-6781	341	2	(	(	PUNCT
ejpam-6781	341	3	16	16	NUM
ejpam-6781	341	4	,	,	PUNCT
ejpam-6781	341	5	55	55	NUM
ejpam-6781	341	6	,	,	PUNCT
ejpam-6781	341	7	49	49	NUM
ejpam-6781	341	8	)	)	PUNCT
ejpam-6781	341	9	.	.	PUNCT
ejpam-6781	342	1	case	case	NOUN
ejpam-6781	342	2	ϵ1	ϵ1	NOUN
ejpam-6781	342	3	=	=	SYM
ejpam-6781	342	4	−1	−1	NOUN
ejpam-6781	342	5	:	:	PUNCT
ejpam-6781	342	6	we	we	PRON
ejpam-6781	342	7	compute	compute	VERB
ejpam-6781	342	8	ζ−2	ζ−2	PROPN
ejpam-6781	342	9	7	7	NUM
ejpam-6781	342	10	:	:	PUNCT
ejpam-6781	343	1	ζ−2	ζ−2	PROPN
ejpam-6781	343	2	7	7	NUM
ejpam-6781	343	3	=	=	SYM
ejpam-6781	343	4	(	(	PUNCT
ejpam-6781	343	5	−5	−5	ADV
ejpam-6781	343	6	7	7	NUM
ejpam-6781	343	7	+	+	CCONJ
ejpam-6781	343	8	3	3	NUM
ejpam-6781	343	9	7	7	NUM
ejpam-6781	343	10	ω	ω	NOUN
ejpam-6781	343	11	)	)	PUNCT
ejpam-6781	343	12	2	2	NUM
ejpam-6781	343	13	=	=	SYM
ejpam-6781	343	14	16	16	NUM
ejpam-6781	343	15	49	49	NUM
ejpam-6781	343	16	−	−	NOUN
ejpam-6781	343	17	39	39	NUM
ejpam-6781	343	18	49	49	NUM
ejpam-6781	343	19	ω	ω	NOUN
ejpam-6781	343	20	.	.	PUNCT
ejpam-6781	344	1	this	this	PRON
ejpam-6781	344	2	lies	lie	VERB
ejpam-6781	344	3	in	in	ADP
ejpam-6781	344	4	the	the	DET
ejpam-6781	344	5	fifth	fifth	ADJ
ejpam-6781	344	6	sextant	sextant	NOUN
ejpam-6781	344	7	.	.	PUNCT
ejpam-6781	345	1	its	its	PRON
ejpam-6781	345	2	associate	associate	NOUN
ejpam-6781	345	3	in	in	ADP
ejpam-6781	345	4	the	the	DET
ejpam-6781	345	5	second	second	ADJ
ejpam-6781	345	6	sextant	sextant	NOUN
ejpam-6781	345	7	is	be	AUX
ejpam-6781	345	8	39	39	NUM
ejpam-6781	345	9	49	49	NUM
ejpam-6781	345	10	+	+	SYM
ejpam-6781	345	11	55	55	NUM
ejpam-6781	345	12	49ω	49ω	NOUN
ejpam-6781	345	13	.	.	PUNCT
ejpam-6781	346	1	the	the	DET
ejpam-6781	346	2	corresponding	corresponding	ADJ
ejpam-6781	346	3	primitive	primitive	ADJ
ejpam-6781	346	4	eisenstein	eisenstein	NOUN
ejpam-6781	346	5	triples	triple	NOUN
ejpam-6781	346	6	is	be	AUX
ejpam-6781	346	7	et	et	NOUN
ejpam-6781	346	8	(	(	PUNCT
ejpam-6781	346	9	39	39	NUM
ejpam-6781	346	10	49	49	NUM
ejpam-6781	346	11	+	+	SYM
ejpam-6781	346	12	55	55	NUM
ejpam-6781	346	13	49ω	49ω	NOUN
ejpam-6781	346	14	)	)	PUNCT
ejpam-6781	347	1	=	=	PUNCT
ejpam-6781	347	2	(	(	PUNCT
ejpam-6781	347	3	39	39	NUM
ejpam-6781	347	4	,	,	PUNCT
ejpam-6781	347	5	55	55	NUM
ejpam-6781	347	6	,	,	PUNCT
ejpam-6781	347	7	49	49	NUM
ejpam-6781	347	8	)	)	PUNCT
ejpam-6781	347	9	.	.	PUNCT
ejpam-6781	348	1	consequently	consequently	ADV
ejpam-6781	348	2	,	,	PUNCT
ejpam-6781	348	3	et49	et49	PROPN
ejpam-6781	348	4	=	=	PRON
ejpam-6781	348	5	{	{	PUNCT
ejpam-6781	348	6	(	(	PUNCT
ejpam-6781	348	7	16	16	NUM
ejpam-6781	348	8	,	,	PUNCT
ejpam-6781	348	9	55	55	NUM
ejpam-6781	348	10	,	,	PUNCT
ejpam-6781	348	11	49	49	NUM
ejpam-6781	348	12	)	)	PUNCT
ejpam-6781	348	13	,	,	PUNCT
ejpam-6781	348	14	(	(	PUNCT
ejpam-6781	348	15	39	39	NUM
ejpam-6781	348	16	,	,	PUNCT
ejpam-6781	348	17	55	55	NUM
ejpam-6781	348	18	,	,	PUNCT
ejpam-6781	348	19	49	49	NUM
ejpam-6781	348	20	)	)	PUNCT
ejpam-6781	348	21	}	}	PUNCT
ejpam-6781	348	22	.	.	PUNCT
ejpam-6781	349	1	s.	s.	PROPN
ejpam-6781	349	2	jitman	jitman	PROPN
ejpam-6781	349	3	,	,	PUNCT
ejpam-6781	349	4	m.	m.	NOUN
ejpam-6781	349	5	mohammad	mohammad	PROPN
ejpam-6781	349	6	,	,	PUNCT
ejpam-6781	349	7	e.	e.	PROPN
ejpam-6781	349	8	sangwisut	sangwisut	PROPN
ejpam-6781	349	9	/	/	SYM
ejpam-6781	349	10	eur	eur	PROPN
ejpam-6781	349	11	.	.	PUNCT
ejpam-6781	350	1	j.	j.	PROPN
ejpam-6781	350	2	pure	pure	PROPN
ejpam-6781	350	3	appl	appl	PROPN
ejpam-6781	350	4	.	.	PROPN
ejpam-6781	350	5	math	math	PROPN
ejpam-6781	350	6	,	,	PUNCT
ejpam-6781	350	7	18	18	NUM
ejpam-6781	350	8	(	(	PUNCT
ejpam-6781	350	9	4	4	NUM
ejpam-6781	350	10	)	)	PUNCT
ejpam-6781	350	11	(	(	PUNCT
ejpam-6781	350	12	2025	2025	NUM
ejpam-6781	350	13	)	)	PUNCT
ejpam-6781	350	14	,	,	PUNCT
ejpam-6781	350	15	6781	6781	NUM
ejpam-6781	350	16	14	14	NUM
ejpam-6781	350	17	of	of	ADP
ejpam-6781	350	18	16	16	NUM
ejpam-6781	350	19	example	example	NOUN
ejpam-6781	350	20	3	3	NUM
ejpam-6781	350	21	.	.	PUNCT
ejpam-6781	351	1	let	let	VERB
ejpam-6781	351	2	c	c	NOUN
ejpam-6781	351	3	=	=	SYM
ejpam-6781	351	4	91	91	NUM
ejpam-6781	351	5	.	.	PUNCT
ejpam-6781	352	1	then	then	ADV
ejpam-6781	352	2	the	the	DET
ejpam-6781	352	3	prime	prime	ADJ
ejpam-6781	352	4	factorization	factorization	NOUN
ejpam-6781	352	5	is	be	AUX
ejpam-6781	352	6	c	c	NOUN
ejpam-6781	352	7	=	=	SYM
ejpam-6781	352	8	7	7	NUM
ejpam-6781	352	9	·	·	SYM
ejpam-6781	352	10	13	13	NUM
ejpam-6781	352	11	,	,	PUNCT
ejpam-6781	352	12	where	where	SCONJ
ejpam-6781	352	13	7	7	NUM
ejpam-6781	352	14	,	,	PUNCT
ejpam-6781	352	15	13	13	NUM
ejpam-6781	352	16	∈	∈	PROPN
ejpam-6781	352	17	p1	p1	NOUN
ejpam-6781	352	18	.	.	PUNCT
ejpam-6781	353	1	from	from	ADP
ejpam-6781	353	2	example	example	NOUN
ejpam-6781	353	3	1	1	NUM
ejpam-6781	353	4	,	,	PUNCT
ejpam-6781	353	5	we	we	PRON
ejpam-6781	353	6	have	have	AUX
ejpam-6781	353	7	ζ7	ζ7	VERB
ejpam-6781	353	8	=	=	SYM
ejpam-6781	353	9	1	1	NUM
ejpam-6781	354	1	+	+	NUM
ejpam-6781	354	2	3ω	3ω	NUM
ejpam-6781	354	3	1	1	NUM
ejpam-6781	354	4	+	+	CCONJ
ejpam-6781	354	5	3ω	3ω	NOUN
ejpam-6781	354	6	,	,	PUNCT
ejpam-6781	354	7	and	and	CCONJ
ejpam-6781	354	8	ζ13	ζ13	VERB
ejpam-6781	354	9	=	=	SYM
ejpam-6781	354	10	1	1	NUM
ejpam-6781	354	11	+	+	NUM
ejpam-6781	354	12	4ω	4ω	NOUN
ejpam-6781	354	13	1	1	NUM
ejpam-6781	354	14	+	+	NUM
ejpam-6781	354	15	4ω	4ω	NOUN
ejpam-6781	354	16	.	.	PUNCT
ejpam-6781	355	1	then	then	ADV
ejpam-6781	355	2	z	z	X
ejpam-6781	355	3	=	=	SYM
ejpam-6781	355	4	{	{	PUNCT
ejpam-6781	355	5	ζϵ17	ζϵ17	PROPN
ejpam-6781	355	6	ζϵ213	ζϵ213	PROPN
ejpam-6781	355	7	|	|	ADV
ejpam-6781	355	8	ϵ1	ϵ1	VERB
ejpam-6781	355	9	,	,	PUNCT
ejpam-6781	355	10	ϵ2	ϵ2	PROPN
ejpam-6781	355	11	∈	∈	PROPN
ejpam-6781	355	12	{	{	PUNCT
ejpam-6781	355	13	±	±	NOUN
ejpam-6781	355	14	}	}	PUNCT
ejpam-6781	355	15	}	}	PUNCT
ejpam-6781	355	16	=	=	SYM
ejpam-6781	355	17	{	{	PUNCT
ejpam-6781	355	18	ζ7ζ13	ζ7ζ13	PROPN
ejpam-6781	355	19	,	,	PUNCT
ejpam-6781	355	20	ζ−1	ζ−1	PROPN
ejpam-6781	355	21	7	7	NUM
ejpam-6781	355	22	ζ13	ζ13	NOUN
ejpam-6781	355	23	,	,	PUNCT
ejpam-6781	356	1	ζ7ζ	ζ7ζ	PROPN
ejpam-6781	356	2	−1	−1	NOUN
ejpam-6781	356	3	13	13	NUM
ejpam-6781	356	4	,	,	PUNCT
ejpam-6781	356	5	ζ−1	ζ−1	PROPN
ejpam-6781	356	6	7	7	NUM
ejpam-6781	356	7	ζ−1	ζ−1	PROPN
ejpam-6781	356	8	13	13	NUM
ejpam-6781	356	9	}	}	PUNCT
ejpam-6781	356	10	.	.	PUNCT
ejpam-6781	357	1	by	by	ADP
ejpam-6781	357	2	theorem	theorem	NOUN
ejpam-6781	357	3	4	4	NUM
ejpam-6781	357	4	,	,	PUNCT
ejpam-6781	357	5	we	we	PRON
ejpam-6781	357	6	analyze	analyze	VERB
ejpam-6781	357	7	the	the	DET
ejpam-6781	357	8	following	follow	VERB
ejpam-6781	357	9	four	four	NUM
ejpam-6781	357	10	cases	case	NOUN
ejpam-6781	357	11	:	:	PUNCT
ejpam-6781	357	12	case	case	NOUN
ejpam-6781	357	13	1	1	NUM
ejpam-6781	357	14	:	:	PUNCT
ejpam-6781	357	15	ϵ1	ϵ1	ADJ
ejpam-6781	357	16	=	=	SYM
ejpam-6781	357	17	1	1	NUM
ejpam-6781	357	18	,	,	PUNCT
ejpam-6781	357	19	ϵ2	ϵ2	NOUN
ejpam-6781	357	20	=	=	NOUN
ejpam-6781	357	21	1	1	X
ejpam-6781	357	22	.	.	X
ejpam-6781	357	23	compute	compute	NOUN
ejpam-6781	357	24	ζ7	ζ7	NOUN
ejpam-6781	357	25	·	·	PUNCT
ejpam-6781	357	26	ζ13	ζ13	NOUN
ejpam-6781	357	27	:	:	PUNCT
ejpam-6781	357	28	ζ7	ζ7	NOUN
ejpam-6781	357	29	·	·	PUNCT
ejpam-6781	357	30	ζ13	ζ13	VERB
ejpam-6781	357	31	=	=	SYM
ejpam-6781	357	32	1	1	NUM
ejpam-6781	357	33	+	+	NUM
ejpam-6781	357	34	3ω	3ω	NUM
ejpam-6781	357	35	1	1	NUM
ejpam-6781	358	1	+	+	CCONJ
ejpam-6781	358	2	3ω	3ω	NUM
ejpam-6781	358	3	·	·	PUNCT
ejpam-6781	358	4	1	1	NUM
ejpam-6781	359	1	+	+	NUM
ejpam-6781	359	2	4ω	4ω	NOUN
ejpam-6781	359	3	1	1	NUM
ejpam-6781	359	4	+	+	NUM
ejpam-6781	359	5	4ω	4ω	NOUN
ejpam-6781	359	6	=	=	SYM
ejpam-6781	359	7	96	96	NUM
ejpam-6781	359	8	91	91	NUM
ejpam-6781	359	9	+	+	NUM
ejpam-6781	359	10	85	85	NUM
ejpam-6781	359	11	91	91	NUM
ejpam-6781	359	12	ω	ω	NOUN
ejpam-6781	359	13	.	.	PUNCT
ejpam-6781	360	1	this	this	PRON
ejpam-6781	360	2	lies	lie	VERB
ejpam-6781	360	3	in	in	ADP
ejpam-6781	360	4	the	the	DET
ejpam-6781	360	5	sixth	sixth	ADJ
ejpam-6781	360	6	sextant	sextant	NOUN
ejpam-6781	360	7	.	.	PUNCT
ejpam-6781	361	1	its	its	PRON
ejpam-6781	361	2	associate	associate	NOUN
ejpam-6781	361	3	in	in	ADP
ejpam-6781	361	4	the	the	DET
ejpam-6781	361	5	second	second	ADJ
ejpam-6781	361	6	sextant	sextant	NOUN
ejpam-6781	361	7	is	be	AUX
ejpam-6781	361	8	11	11	NUM
ejpam-6781	361	9	91	91	NUM
ejpam-6781	361	10	+	+	CCONJ
ejpam-6781	361	11	96	96	NUM
ejpam-6781	361	12	91ω	91ω	NOUN
ejpam-6781	361	13	.	.	PUNCT
ejpam-6781	362	1	the	the	DET
ejpam-6781	362	2	corresponding	corresponding	ADJ
ejpam-6781	362	3	primitive	primitive	ADJ
ejpam-6781	362	4	eisenstein	eisenstein	NOUN
ejpam-6781	362	5	triples	triple	NOUN
ejpam-6781	362	6	is	be	AUX
ejpam-6781	362	7	et	et	NOUN
ejpam-6781	362	8	(	(	PUNCT
ejpam-6781	362	9	11	11	NUM
ejpam-6781	362	10	91	91	NUM
ejpam-6781	362	11	+	+	CCONJ
ejpam-6781	362	12	96	96	NUM
ejpam-6781	362	13	91ω	91ω	NOUN
ejpam-6781	362	14	)	)	PUNCT
ejpam-6781	363	1	=	=	PUNCT
ejpam-6781	363	2	(	(	PUNCT
ejpam-6781	363	3	11	11	NUM
ejpam-6781	363	4	,	,	PUNCT
ejpam-6781	363	5	96	96	NUM
ejpam-6781	363	6	,	,	PUNCT
ejpam-6781	363	7	91	91	NUM
ejpam-6781	363	8	)	)	PUNCT
ejpam-6781	363	9	.	.	PUNCT
ejpam-6781	364	1	case	case	NOUN
ejpam-6781	364	2	2	2	NUM
ejpam-6781	364	3	:	:	PUNCT
ejpam-6781	364	4	ϵ1	ϵ1	ADJ
ejpam-6781	364	5	=	=	SYM
ejpam-6781	364	6	−1	−1	NOUN
ejpam-6781	364	7	,	,	PUNCT
ejpam-6781	364	8	ϵ2	ϵ2	PROPN
ejpam-6781	364	9	=	=	NOUN
ejpam-6781	364	10	1	1	X
ejpam-6781	364	11	.	.	X
ejpam-6781	364	12	compute	compute	VERB
ejpam-6781	364	13	ζ−1	ζ−1	PROPN
ejpam-6781	364	14	7	7	NUM
ejpam-6781	364	15	·	·	PUNCT
ejpam-6781	364	16	ζ13	ζ13	NOUN
ejpam-6781	364	17	:	:	PUNCT
ejpam-6781	364	18	ζ−1	ζ−1	PROPN
ejpam-6781	364	19	7	7	NUM
ejpam-6781	364	20	·	·	PUNCT
ejpam-6781	364	21	ζ13	ζ13	VERB
ejpam-6781	364	22	=	=	SYM
ejpam-6781	364	23	1	1	NUM
ejpam-6781	364	24	+	+	NUM
ejpam-6781	364	25	3ω	3ω	NUM
ejpam-6781	364	26	1	1	NUM
ejpam-6781	364	27	+	+	CCONJ
ejpam-6781	364	28	3ω	3ω	NUM
ejpam-6781	364	29	·	·	PUNCT
ejpam-6781	364	30	1	1	NUM
ejpam-6781	365	1	+	+	NUM
ejpam-6781	365	2	4ω	4ω	NOUN
ejpam-6781	365	3	1	1	NUM
ejpam-6781	365	4	+	+	NUM
ejpam-6781	365	5	4ω	4ω	NOUN
ejpam-6781	365	6	=	=	SYM
ejpam-6781	365	7	99	99	NUM
ejpam-6781	365	8	91	91	NUM
ejpam-6781	365	9	+	+	NUM
ejpam-6781	365	10	19	19	NUM
ejpam-6781	365	11	91	91	NUM
ejpam-6781	365	12	ω	ω	NOUN
ejpam-6781	365	13	.	.	PUNCT
ejpam-6781	366	1	this	this	PRON
ejpam-6781	366	2	lies	lie	VERB
ejpam-6781	366	3	in	in	ADP
ejpam-6781	366	4	the	the	DET
ejpam-6781	366	5	sixth	sixth	ADJ
ejpam-6781	366	6	sextant	sextant	NOUN
ejpam-6781	366	7	.	.	PUNCT
ejpam-6781	367	1	its	its	PRON
ejpam-6781	367	2	associate	associate	NOUN
ejpam-6781	367	3	in	in	ADP
ejpam-6781	367	4	the	the	DET
ejpam-6781	367	5	second	second	ADJ
ejpam-6781	367	6	sextant	sextant	NOUN
ejpam-6781	367	7	is	be	AUX
ejpam-6781	367	8	80	80	NUM
ejpam-6781	367	9	91	91	NUM
ejpam-6781	367	10	+	+	NUM
ejpam-6781	367	11	99	99	NUM
ejpam-6781	367	12	91ω	91ω	NOUN
ejpam-6781	367	13	.	.	PUNCT
ejpam-6781	368	1	the	the	DET
ejpam-6781	368	2	corresponding	corresponding	ADJ
ejpam-6781	368	3	primitive	primitive	ADJ
ejpam-6781	368	4	eisenstein	eisenstein	NOUN
ejpam-6781	368	5	triples	triple	NOUN
ejpam-6781	368	6	is	be	AUX
ejpam-6781	368	7	et	et	NOUN
ejpam-6781	368	8	(	(	PUNCT
ejpam-6781	368	9	80	80	NUM
ejpam-6781	368	10	91	91	NUM
ejpam-6781	368	11	+	+	NUM
ejpam-6781	368	12	99	99	NUM
ejpam-6781	368	13	91ω	91ω	NOUN
ejpam-6781	368	14	)	)	PUNCT
ejpam-6781	369	1	=	=	PUNCT
ejpam-6781	369	2	(	(	PUNCT
ejpam-6781	369	3	80	80	NUM
ejpam-6781	369	4	,	,	PUNCT
ejpam-6781	369	5	99	99	NUM
ejpam-6781	369	6	,	,	PUNCT
ejpam-6781	369	7	91	91	NUM
ejpam-6781	369	8	)	)	PUNCT
ejpam-6781	369	9	.	.	PUNCT
ejpam-6781	370	1	case	case	NOUN
ejpam-6781	370	2	3	3	NUM
ejpam-6781	370	3	:	:	PUNCT
ejpam-6781	370	4	ϵ1	ϵ1	ADJ
ejpam-6781	370	5	=	=	SYM
ejpam-6781	370	6	1	1	NUM
ejpam-6781	370	7	,	,	PUNCT
ejpam-6781	370	8	ϵ2	ϵ2	NOUN
ejpam-6781	370	9	=	=	SYM
ejpam-6781	370	10	−1	−1	NOUN
ejpam-6781	370	11	.	.	PUNCT
ejpam-6781	371	1	compute	compute	NOUN
ejpam-6781	371	2	ζ7	ζ7	NOUN
ejpam-6781	371	3	·	·	PUNCT
ejpam-6781	371	4	ζ13	ζ13	NOUN
ejpam-6781	371	5	:	:	PUNCT
ejpam-6781	371	6	ζ7	ζ7	VERB
ejpam-6781	371	7	·	·	PUNCT
ejpam-6781	371	8	ζ−1	ζ−1	PROPN
ejpam-6781	371	9	13	13	NUM
ejpam-6781	371	10	=	=	SYM
ejpam-6781	371	11	1	1	NUM
ejpam-6781	371	12	+	+	NUM
ejpam-6781	371	13	3ω	3ω	NUM
ejpam-6781	371	14	1	1	NUM
ejpam-6781	371	15	+	+	CCONJ
ejpam-6781	371	16	3ω	3ω	NUM
ejpam-6781	371	17	·	·	PUNCT
ejpam-6781	371	18	1	1	NUM
ejpam-6781	372	1	+	+	NUM
ejpam-6781	372	2	4ω	4ω	NOUN
ejpam-6781	372	3	1	1	NUM
ejpam-6781	372	4	+	+	NUM
ejpam-6781	372	5	4ω	4ω	NOUN
ejpam-6781	372	6	=	=	SYM
ejpam-6781	372	7	80	80	NUM
ejpam-6781	372	8	91	91	NUM
ejpam-6781	372	9	−	−	NUM
ejpam-6781	372	10	19	19	NUM
ejpam-6781	372	11	91	91	NUM
ejpam-6781	372	12	ω	ω	NOUN
ejpam-6781	372	13	.	.	PUNCT
ejpam-6781	373	1	this	this	PRON
ejpam-6781	373	2	lies	lie	VERB
ejpam-6781	373	3	in	in	ADP
ejpam-6781	373	4	the	the	DET
ejpam-6781	373	5	fifth	fifth	ADJ
ejpam-6781	373	6	sextant	sextant	NOUN
ejpam-6781	373	7	.	.	PUNCT
ejpam-6781	374	1	its	its	PRON
ejpam-6781	374	2	associate	associate	NOUN
ejpam-6781	374	3	in	in	ADP
ejpam-6781	374	4	the	the	DET
ejpam-6781	374	5	second	second	ADJ
ejpam-6781	374	6	sextant	sextant	NOUN
ejpam-6781	374	7	is	be	AUX
ejpam-6781	374	8	19	19	NUM
ejpam-6781	374	9	91	91	NUM
ejpam-6781	374	10	+	+	NUM
ejpam-6781	374	11	99	99	NUM
ejpam-6781	374	12	91ω	91ω	NOUN
ejpam-6781	374	13	.	.	PUNCT
ejpam-6781	375	1	the	the	DET
ejpam-6781	375	2	corresponding	corresponding	ADJ
ejpam-6781	375	3	primitive	primitive	ADJ
ejpam-6781	375	4	eisenstein	eisenstein	NOUN
ejpam-6781	375	5	triples	triple	NOUN
ejpam-6781	375	6	is	be	AUX
ejpam-6781	375	7	et	et	NOUN
ejpam-6781	375	8	(	(	PUNCT
ejpam-6781	375	9	19	19	NUM
ejpam-6781	375	10	91	91	NUM
ejpam-6781	375	11	+	+	NUM
ejpam-6781	375	12	99	99	NUM
ejpam-6781	375	13	91ω	91ω	NOUN
ejpam-6781	375	14	)	)	PUNCT
ejpam-6781	376	1	=	=	PUNCT
ejpam-6781	376	2	(	(	PUNCT
ejpam-6781	376	3	19	19	NUM
ejpam-6781	376	4	,	,	PUNCT
ejpam-6781	376	5	99	99	NUM
ejpam-6781	376	6	,	,	PUNCT
ejpam-6781	376	7	91	91	NUM
ejpam-6781	376	8	)	)	PUNCT
ejpam-6781	376	9	.	.	PUNCT
ejpam-6781	377	1	case	case	NOUN
ejpam-6781	377	2	4	4	NUM
ejpam-6781	377	3	:	:	PUNCT
ejpam-6781	377	4	ϵ1	ϵ1	ADJ
ejpam-6781	377	5	=	=	SYM
ejpam-6781	377	6	−1	−1	NOUN
ejpam-6781	377	7	,	,	PUNCT
ejpam-6781	377	8	ϵ2	ϵ2	NOUN
ejpam-6781	377	9	=	=	SYM
ejpam-6781	377	10	−1	−1	NOUN
ejpam-6781	377	11	.	.	PUNCT
ejpam-6781	378	1	compute	compute	NOUN
ejpam-6781	378	2	ζ7	ζ7	NOUN
ejpam-6781	378	3	·	·	PUNCT
ejpam-6781	378	4	ζ13	ζ13	VERB
ejpam-6781	378	5	:	:	PUNCT
ejpam-6781	378	6	ζ−1	ζ−1	PROPN
ejpam-6781	378	7	7	7	NUM
ejpam-6781	378	8	·	·	PUNCT
ejpam-6781	378	9	ζ−1	ζ−1	PROPN
ejpam-6781	378	10	13	13	NUM
ejpam-6781	378	11	=	=	SYM
ejpam-6781	378	12	1	1	NUM
ejpam-6781	378	13	+	+	NUM
ejpam-6781	378	14	3ω	3ω	NUM
ejpam-6781	378	15	1	1	NUM
ejpam-6781	378	16	+	+	CCONJ
ejpam-6781	378	17	3ω	3ω	NUM
ejpam-6781	378	18	·	·	PUNCT
ejpam-6781	378	19	1	1	NUM
ejpam-6781	379	1	+	+	NUM
ejpam-6781	379	2	4ω	4ω	NOUN
ejpam-6781	379	3	1	1	NUM
ejpam-6781	379	4	+	+	NUM
ejpam-6781	379	5	4ω	4ω	NOUN
ejpam-6781	379	6	=	=	SYM
ejpam-6781	379	7	11	11	NUM
ejpam-6781	379	8	91	91	NUM
ejpam-6781	379	9	−	−	NUM
ejpam-6781	379	10	85	85	NUM
ejpam-6781	379	11	91	91	NUM
ejpam-6781	379	12	ω	ω	NOUN
ejpam-6781	379	13	.	.	PUNCT
ejpam-6781	380	1	this	this	PRON
ejpam-6781	380	2	lies	lie	VERB
ejpam-6781	380	3	in	in	ADP
ejpam-6781	380	4	the	the	DET
ejpam-6781	380	5	fifth	fifth	ADJ
ejpam-6781	380	6	sextant	sextant	NOUN
ejpam-6781	380	7	.	.	PUNCT
ejpam-6781	381	1	its	its	PRON
ejpam-6781	381	2	associate	associate	NOUN
ejpam-6781	381	3	in	in	ADP
ejpam-6781	381	4	the	the	DET
ejpam-6781	381	5	second	second	ADJ
ejpam-6781	381	6	sextant	sextant	NOUN
ejpam-6781	381	7	is	be	AUX
ejpam-6781	381	8	85	85	NUM
ejpam-6781	381	9	91	91	NUM
ejpam-6781	381	10	+	+	CCONJ
ejpam-6781	381	11	96	96	NUM
ejpam-6781	381	12	91ω	91ω	NOUN
ejpam-6781	381	13	.	.	PUNCT
ejpam-6781	382	1	the	the	DET
ejpam-6781	382	2	corresponding	corresponding	ADJ
ejpam-6781	382	3	primitive	primitive	ADJ
ejpam-6781	382	4	eisenstein	eisenstein	NOUN
ejpam-6781	382	5	triples	triple	NOUN
ejpam-6781	382	6	is	be	AUX
ejpam-6781	382	7	et	et	NOUN
ejpam-6781	382	8	(	(	PUNCT
ejpam-6781	382	9	85	85	NUM
ejpam-6781	382	10	91	91	NUM
ejpam-6781	382	11	+	+	NUM
ejpam-6781	382	12	96	96	NUM
ejpam-6781	382	13	91ω	91ω	NOUN
ejpam-6781	382	14	)	)	PUNCT
ejpam-6781	382	15	=	=	PUNCT
ejpam-6781	382	16	(	(	PUNCT
ejpam-6781	382	17	85	85	NUM
ejpam-6781	382	18	,	,	PUNCT
ejpam-6781	382	19	96	96	NUM
ejpam-6781	382	20	,	,	PUNCT
ejpam-6781	382	21	91	91	NUM
ejpam-6781	382	22	)	)	PUNCT
ejpam-6781	382	23	.	.	PUNCT
ejpam-6781	383	1	therefore	therefore	ADV
ejpam-6781	383	2	,	,	PUNCT
ejpam-6781	383	3	et49	et49	PROPN
ejpam-6781	383	4	=	=	PRON
ejpam-6781	383	5	{	{	PUNCT
ejpam-6781	383	6	(	(	PUNCT
ejpam-6781	383	7	11	11	NUM
ejpam-6781	383	8	,	,	PUNCT
ejpam-6781	383	9	96	96	NUM
ejpam-6781	383	10	,	,	PUNCT
ejpam-6781	383	11	91	91	NUM
ejpam-6781	383	12	)	)	PUNCT
ejpam-6781	383	13	,	,	PUNCT
ejpam-6781	383	14	(	(	PUNCT
ejpam-6781	383	15	85	85	NUM
ejpam-6781	383	16	,	,	PUNCT
ejpam-6781	383	17	96	96	NUM
ejpam-6781	383	18	,	,	PUNCT
ejpam-6781	383	19	91	91	NUM
ejpam-6781	383	20	)	)	PUNCT
ejpam-6781	383	21	,	,	PUNCT
ejpam-6781	383	22	(	(	PUNCT
ejpam-6781	383	23	80	80	NUM
ejpam-6781	383	24	,	,	PUNCT
ejpam-6781	383	25	99	99	NUM
ejpam-6781	383	26	,	,	PUNCT
ejpam-6781	383	27	91	91	NUM
ejpam-6781	383	28	)	)	PUNCT
ejpam-6781	383	29	,	,	PUNCT
ejpam-6781	383	30	(	(	PUNCT
ejpam-6781	383	31	19	19	NUM
ejpam-6781	383	32	,	,	PUNCT
ejpam-6781	383	33	99	99	NUM
ejpam-6781	383	34	,	,	PUNCT
ejpam-6781	383	35	91	91	NUM
ejpam-6781	383	36	)	)	PUNCT
ejpam-6781	383	37	}	}	PUNCT
ejpam-6781	383	38	.	.	PUNCT
ejpam-6781	384	1	6	6	X
ejpam-6781	384	2	.	.	X
ejpam-6781	384	3	conclusion	conclusion	NOUN
ejpam-6781	384	4	this	this	DET
ejpam-6781	384	5	paper	paper	NOUN
ejpam-6781	384	6	develops	develop	VERB
ejpam-6781	384	7	a	a	DET
ejpam-6781	384	8	unified	unified	ADJ
ejpam-6781	384	9	framework	framework	NOUN
ejpam-6781	384	10	for	for	ADP
ejpam-6781	384	11	primitive	primitive	ADJ
ejpam-6781	384	12	eisenstein	eisenstein	NOUN
ejpam-6781	384	13	triples	triple	NOUN
ejpam-6781	384	14	,	,	PUNCT
ejpam-6781	384	15	those	those	DET
ejpam-6781	384	16	integer	integer	NOUN
ejpam-6781	384	17	triangles	triangle	NOUN
ejpam-6781	384	18	with	with	ADP
ejpam-6781	384	19	a	a	DET
ejpam-6781	384	20	60	60	NUM
ejpam-6781	384	21	–	–	PUNCT
ejpam-6781	384	22	degree	degree	NOUN
ejpam-6781	384	23	angle	angle	NOUN
ejpam-6781	384	24	and	and	CCONJ
ejpam-6781	384	25	side	side	NOUN
ejpam-6781	384	26	lengths	length	NOUN
ejpam-6781	384	27	satisfying	satisfy	VERB
ejpam-6781	384	28	the	the	DET
ejpam-6781	384	29	classical	classical	ADJ
ejpam-6781	384	30	eisenstein	eisenstein	PROPN
ejpam-6781	384	31	relation	relation	PROPN
ejpam-6781	384	32	.	.	PUNCT
ejpam-6781	385	1	the	the	DET
ejpam-6781	385	2	key	key	ADJ
ejpam-6781	385	3	idea	idea	NOUN
ejpam-6781	385	4	is	be	AUX
ejpam-6781	385	5	to	to	PART
ejpam-6781	385	6	translate	translate	VERB
ejpam-6781	385	7	triples	triple	NOUN
ejpam-6781	385	8	into	into	ADP
ejpam-6781	385	9	points	point	NOUN
ejpam-6781	385	10	on	on	ADP
ejpam-6781	385	11	the	the	DET
ejpam-6781	385	12	ω	ω	ADJ
ejpam-6781	385	13	–	–	PUNCT
ejpam-6781	385	14	rational	rational	ADJ
ejpam-6781	385	15	unit	unit	NOUN
ejpam-6781	385	16	circle	circle	NOUN
ejpam-6781	385	17	and	and	CCONJ
ejpam-6781	385	18	back	back	ADV
ejpam-6781	385	19	.	.	PUNCT
ejpam-6781	386	1	using	use	VERB
ejpam-6781	386	2	unique	unique	ADJ
ejpam-6781	386	3	factorization	factorization	NOUN
ejpam-6781	386	4	in	in	ADP
ejpam-6781	386	5	the	the	DET
ejpam-6781	386	6	ring	ring	NOUN
ejpam-6781	386	7	of	of	ADP
ejpam-6781	386	8	eisenstein	eisenstein	PROPN
ejpam-6781	386	9	integers	integer	NOUN
ejpam-6781	386	10	,	,	PUNCT
ejpam-6781	386	11	we	we	PRON
ejpam-6781	386	12	show	show	VERB
ejpam-6781	386	13	that	that	SCONJ
ejpam-6781	386	14	the	the	DET
ejpam-6781	386	15	ω	ω	ADJ
ejpam-6781	386	16	–	–	PUNCT
ejpam-6781	386	17	rational	rational	ADJ
ejpam-6781	386	18	unit	unit	NOUN
ejpam-6781	386	19	circle	circle	NOUN
ejpam-6781	386	20	splits	split	VERB
ejpam-6781	386	21	cleanly	cleanly	ADV
ejpam-6781	386	22	into	into	ADP
ejpam-6781	386	23	two	two	NUM
ejpam-6781	386	24	parts	part	NOUN
ejpam-6781	386	25	:	:	PUNCT
ejpam-6781	386	26	a	a	DET
ejpam-6781	386	27	finite	finite	ADJ
ejpam-6781	386	28	set	set	NOUN
ejpam-6781	386	29	of	of	ADP
ejpam-6781	386	30	six	six	NUM
ejpam-6781	386	31	units	unit	NOUN
ejpam-6781	386	32	and	and	CCONJ
ejpam-6781	386	33	a	a	DET
ejpam-6781	386	34	free	free	ADJ
ejpam-6781	386	35	abelian	abelian	ADJ
ejpam-6781	386	36	group	group	NOUN
ejpam-6781	386	37	with	with	ADP
ejpam-6781	386	38	basis	basis	NOUN
ejpam-6781	386	39	given	give	VERB
ejpam-6781	386	40	by	by	ADP
ejpam-6781	386	41	the	the	DET
ejpam-6781	386	42	collection	collection	NOUN
ejpam-6781	386	43	{	{	PUNCT
ejpam-6781	386	44	ζp	ζp	PROPN
ejpam-6781	386	45	|	|	ADV
ejpam-6781	386	46	p	p	PROPN
ejpam-6781	386	47	∈	∈	PROPN
ejpam-6781	386	48	p1	p1	NOUN
ejpam-6781	386	49	}	}	PUNCT
ejpam-6781	386	50	,	,	PUNCT
ejpam-6781	386	51	where	where	SCONJ
ejpam-6781	386	52	p1	p1	PROPN
ejpam-6781	386	53	is	be	AUX
ejpam-6781	386	54	the	the	DET
ejpam-6781	386	55	set	set	NOUN
ejpam-6781	386	56	of	of	ADP
ejpam-6781	386	57	primes	prime	NOUN
ejpam-6781	386	58	congruent	congruent	ADJ
ejpam-6781	386	59	to	to	ADP
ejpam-6781	386	60	one	one	NUM
ejpam-6781	386	61	modulo	modulo	NOUN
ejpam-6781	386	62	three	three	NUM
ejpam-6781	386	63	.	.	PUNCT
ejpam-6781	387	1	every	every	DET
ejpam-6781	387	2	ω	ω	ADJ
ejpam-6781	387	3	–	–	PUNCT
ejpam-6781	387	4	rational	rational	ADJ
ejpam-6781	387	5	point	point	NOUN
ejpam-6781	387	6	can	can	AUX
ejpam-6781	387	7	be	be	AUX
ejpam-6781	387	8	written	write	VERB
ejpam-6781	387	9	uniquely	uniquely	ADV
ejpam-6781	387	10	as	as	ADP
ejpam-6781	387	11	a	a	DET
ejpam-6781	387	12	unit	unit	NOUN
ejpam-6781	387	13	times	time	VERB
ejpam-6781	387	14	a	a	DET
ejpam-6781	387	15	product	product	NOUN
ejpam-6781	387	16	of	of	ADP
ejpam-6781	387	17	s.	s.	PROPN
ejpam-6781	387	18	jitman	jitman	PROPN
ejpam-6781	387	19	,	,	PUNCT
ejpam-6781	387	20	m.	m.	NOUN
ejpam-6781	387	21	mohammad	mohammad	PROPN
ejpam-6781	387	22	,	,	PUNCT
ejpam-6781	387	23	e.	e.	PROPN
ejpam-6781	387	24	sangwisut	sangwisut	PROPN
ejpam-6781	387	25	/	/	SYM
ejpam-6781	387	26	eur	eur	PROPN
ejpam-6781	387	27	.	.	PUNCT
ejpam-6781	388	1	j.	j.	PROPN
ejpam-6781	388	2	pure	pure	PROPN
ejpam-6781	388	3	appl	appl	PROPN
ejpam-6781	388	4	.	.	PROPN
ejpam-6781	388	5	math	math	PROPN
ejpam-6781	388	6	,	,	PUNCT
ejpam-6781	388	7	18	18	NUM
ejpam-6781	388	8	(	(	PUNCT
ejpam-6781	388	9	4	4	NUM
ejpam-6781	388	10	)	)	PUNCT
ejpam-6781	388	11	(	(	PUNCT
ejpam-6781	388	12	2025	2025	NUM
ejpam-6781	388	13	)	)	PUNCT
ejpam-6781	388	14	,	,	PUNCT
ejpam-6781	388	15	6781	6781	NUM
ejpam-6781	388	16	15	15	NUM
ejpam-6781	388	17	of	of	ADP
ejpam-6781	388	18	16	16	NUM
ejpam-6781	388	19	these	these	DET
ejpam-6781	388	20	generators	generator	NOUN
ejpam-6781	388	21	.	.	PUNCT
ejpam-6781	389	1	choosing	choose	VERB
ejpam-6781	389	2	a	a	DET
ejpam-6781	389	3	canonical	canonical	ADJ
ejpam-6781	389	4	representative	representative	NOUN
ejpam-6781	389	5	in	in	ADP
ejpam-6781	389	6	the	the	DET
ejpam-6781	389	7	second	second	ADJ
ejpam-6781	389	8	sextant	sextant	NOUN
ejpam-6781	389	9	and	and	CCONJ
ejpam-6781	389	10	clearing	clearing	NOUN
ejpam-6781	389	11	denominators	denominator	NOUN
ejpam-6781	389	12	gives	give	VERB
ejpam-6781	389	13	a	a	DET
ejpam-6781	389	14	direct	direct	ADJ
ejpam-6781	389	15	and	and	CCONJ
ejpam-6781	389	16	lossless	lossless	ADJ
ejpam-6781	389	17	way	way	NOUN
ejpam-6781	389	18	to	to	PART
ejpam-6781	389	19	pass	pass	VERB
ejpam-6781	389	20	from	from	ADP
ejpam-6781	389	21	points	point	NOUN
ejpam-6781	389	22	to	to	ADP
ejpam-6781	389	23	primitive	primitive	ADJ
ejpam-6781	389	24	triples	triple	NOUN
ejpam-6781	389	25	.	.	PUNCT
ejpam-6781	390	1	this	this	DET
ejpam-6781	390	2	perspective	perspective	NOUN
ejpam-6781	390	3	yields	yield	VERB
ejpam-6781	390	4	sharp	sharp	ADJ
ejpam-6781	390	5	existence	existence	NOUN
ejpam-6781	390	6	and	and	CCONJ
ejpam-6781	390	7	counting	counting	NOUN
ejpam-6781	390	8	results	result	NOUN
ejpam-6781	390	9	for	for	ADP
ejpam-6781	390	10	a	a	DET
ejpam-6781	390	11	fixed	fix	VERB
ejpam-6781	390	12	hypotenuse	hypotenuse	NOUN
ejpam-6781	390	13	.	.	PUNCT
ejpam-6781	391	1	write	write	VERB
ejpam-6781	391	2	the	the	DET
ejpam-6781	391	3	hypotenuse	hypotenuse	NOUN
ejpam-6781	391	4	as	as	ADP
ejpam-6781	391	5	a	a	DET
ejpam-6781	391	6	product	product	NOUN
ejpam-6781	391	7	of	of	ADP
ejpam-6781	391	8	rational	rational	ADJ
ejpam-6781	391	9	primes	prime	NOUN
ejpam-6781	391	10	.	.	PUNCT
ejpam-6781	392	1	primitive	primitive	ADJ
ejpam-6781	392	2	eisenstein	eisenstein	NOUN
ejpam-6781	392	3	triples	triple	NOUN
ejpam-6781	392	4	exist	exist	VERB
ejpam-6781	392	5	exactly	exactly	ADV
ejpam-6781	392	6	when	when	SCONJ
ejpam-6781	392	7	every	every	DET
ejpam-6781	392	8	prime	prime	ADJ
ejpam-6781	392	9	factor	factor	NOUN
ejpam-6781	392	10	is	be	AUX
ejpam-6781	392	11	congruent	congruent	ADJ
ejpam-6781	392	12	to	to	ADP
ejpam-6781	392	13	one	one	NUM
ejpam-6781	392	14	modulo	modulo	NOUN
ejpam-6781	392	15	three	three	NUM
ejpam-6781	392	16	.	.	PUNCT
ejpam-6781	393	1	in	in	ADP
ejpam-6781	393	2	that	that	DET
ejpam-6781	393	3	case	case	NOUN
ejpam-6781	393	4	,	,	PUNCT
ejpam-6781	393	5	if	if	SCONJ
ejpam-6781	393	6	there	there	PRON
ejpam-6781	393	7	are	be	VERB
ejpam-6781	393	8	k	k	PROPN
ejpam-6781	393	9	distinct	distinct	ADJ
ejpam-6781	393	10	such	such	ADJ
ejpam-6781	393	11	primes	prime	NOUN
ejpam-6781	393	12	,	,	PUNCT
ejpam-6781	393	13	the	the	DET
ejpam-6781	393	14	number	number	NOUN
ejpam-6781	393	15	of	of	ADP
ejpam-6781	393	16	primitive	primitive	ADJ
ejpam-6781	393	17	triples	triple	NOUN
ejpam-6781	393	18	with	with	ADP
ejpam-6781	393	19	that	that	DET
ejpam-6781	393	20	hypotenuse	hypotenuse	NOUN
ejpam-6781	393	21	is	be	AUX
ejpam-6781	393	22	exactly	exactly	ADV
ejpam-6781	393	23	two	two	NUM
ejpam-6781	393	24	to	to	ADP
ejpam-6781	393	25	the	the	DET
ejpam-6781	393	26	power	power	NOUN
ejpam-6781	393	27	k.	k.	PROPN
ejpam-6781	393	28	practically	practically	ADV
ejpam-6781	393	29	,	,	PUNCT
ejpam-6781	393	30	the	the	DET
ejpam-6781	393	31	test	test	NOUN
ejpam-6781	393	32	for	for	ADP
ejpam-6781	393	33	existence	existence	NOUN
ejpam-6781	393	34	reduces	reduce	VERB
ejpam-6781	393	35	to	to	ADP
ejpam-6781	393	36	a	a	DET
ejpam-6781	393	37	single	single	ADJ
ejpam-6781	393	38	inspection	inspection	NOUN
ejpam-6781	393	39	of	of	ADP
ejpam-6781	393	40	the	the	DET
ejpam-6781	393	41	prime	prime	ADJ
ejpam-6781	393	42	factorization	factorization	NOUN
ejpam-6781	393	43	,	,	PUNCT
ejpam-6781	393	44	and	and	CCONJ
ejpam-6781	393	45	enumeration	enumeration	NOUN
ejpam-6781	393	46	follows	follow	VERB
ejpam-6781	393	47	immediately	immediately	ADV
ejpam-6781	393	48	.	.	PUNCT
ejpam-6781	394	1	algorithmically	algorithmically	ADV
ejpam-6781	394	2	,	,	PUNCT
ejpam-6781	394	3	recovering	recover	VERB
ejpam-6781	394	4	the	the	DET
ejpam-6781	394	5	side	side	NOUN
ejpam-6781	394	6	lengths	length	NOUN
ejpam-6781	394	7	from	from	ADP
ejpam-6781	394	8	a	a	DET
ejpam-6781	394	9	chosen	choose	VERB
ejpam-6781	394	10	omega	omega	NOUN
ejpam-6781	394	11	–	–	PUNCT
ejpam-6781	394	12	rational	rational	ADJ
ejpam-6781	394	13	point	point	NOUN
ejpam-6781	394	14	—	—	PUNCT
ejpam-6781	394	15	or	or	CCONJ
ejpam-6781	394	16	directly	directly	ADV
ejpam-6781	394	17	from	from	ADP
ejpam-6781	394	18	the	the	DET
ejpam-6781	394	19	factorization	factorization	NOUN
ejpam-6781	394	20	of	of	ADP
ejpam-6781	394	21	the	the	DET
ejpam-6781	394	22	hypotenuse	hypotenuse	NOUN
ejpam-6781	394	23	—	—	PUNCT
ejpam-6781	394	24	enables	enable	VERB
ejpam-6781	394	25	efficient	efficient	ADJ
ejpam-6781	394	26	generation	generation	NOUN
ejpam-6781	394	27	at	at	ADP
ejpam-6781	394	28	scale	scale	NOUN
ejpam-6781	394	29	.	.	PUNCT
ejpam-6781	395	1	it	it	PRON
ejpam-6781	395	2	would	would	AUX
ejpam-6781	395	3	be	be	AUX
ejpam-6781	395	4	interesting	interesting	ADJ
ejpam-6781	395	5	to	to	PART
ejpam-6781	395	6	investigate	investigate	VERB
ejpam-6781	395	7	whether	whether	SCONJ
ejpam-6781	395	8	the	the	DET
ejpam-6781	395	9	distribution	distribution	NOUN
ejpam-6781	395	10	and	and	CCONJ
ejpam-6781	395	11	averaging	averaging	NOUN
ejpam-6781	395	12	of	of	ADP
ejpam-6781	395	13	counts	count	NOUN
ejpam-6781	395	14	for	for	ADP
ejpam-6781	395	15	a	a	DET
ejpam-6781	395	16	fixed	fix	VERB
ejpam-6781	395	17	hypotenuse	hypotenuse	NOUN
ejpam-6781	395	18	admit	admit	VERB
ejpam-6781	395	19	a	a	DET
ejpam-6781	395	20	generating	generate	VERB
ejpam-6781	395	21	-	-	PUNCT
ejpam-6781	395	22	function	function	NOUN
ejpam-6781	395	23	treatment	treatment	NOUN
ejpam-6781	395	24	in	in	ADP
ejpam-6781	395	25	which	which	PRON
ejpam-6781	395	26	stirling	stirling	NOUN
ejpam-6781	395	27	numbers	number	NOUN
ejpam-6781	395	28	of	of	ADP
ejpam-6781	395	29	the	the	DET
ejpam-6781	395	30	second	second	ADJ
ejpam-6781	395	31	kind	kind	NOUN
ejpam-6781	395	32	or	or	CCONJ
ejpam-6781	395	33	whitney	whitney	NOUN
ejpam-6781	395	34	numbers	number	NOUN
ejpam-6781	395	35	arise	arise	VERB
ejpam-6781	395	36	naturally	naturally	ADV
ejpam-6781	395	37	;	;	PUNCT
ejpam-6781	395	38	we	we	PRON
ejpam-6781	395	39	leave	leave	VERB
ejpam-6781	395	40	this	this	PRON
ejpam-6781	395	41	as	as	ADP
ejpam-6781	395	42	future	future	ADJ
ejpam-6781	395	43	work	work	NOUN
ejpam-6781	395	44	(	(	PUNCT
ejpam-6781	395	45	see	see	VERB
ejpam-6781	395	46	,	,	PUNCT
ejpam-6781	395	47	e.g.	e.g.	ADV
ejpam-6781	395	48	,	,	PUNCT
ejpam-6781	395	49	[	[	X
ejpam-6781	395	50	16	16	NUM
ejpam-6781	395	51	,	,	PUNCT
ejpam-6781	395	52	17	17	NUM
ejpam-6781	395	53	]	]	PUNCT
ejpam-6781	395	54	)	)	PUNCT
ejpam-6781	395	55	.	.	PUNCT
ejpam-6781	396	1	acknowledgements	acknowledgement	NOUN
ejpam-6781	396	2	s.	s.	PROPN
ejpam-6781	396	3	jitmam	jitmam	PROPN
ejpam-6781	396	4	was	be	AUX
ejpam-6781	396	5	supported	support	VERB
ejpam-6781	396	6	by	by	ADP
ejpam-6781	396	7	the	the	DET
ejpam-6781	396	8	national	national	PROPN
ejpam-6781	396	9	research	research	PROPN
ejpam-6781	396	10	council	council	PROPN
ejpam-6781	396	11	of	of	ADP
ejpam-6781	396	12	thailand	thailand	PROPN
ejpam-6781	396	13	and	and	CCONJ
ejpam-6781	396	14	silpakorn	silpakorn	VERB
ejpam-6781	396	15	university	university	NOUN
ejpam-6781	396	16	under	under	ADP
ejpam-6781	396	17	research	research	NOUN
ejpam-6781	396	18	grant	grant	NOUN
ejpam-6781	396	19	n42a650381	n42a650381	PRON
ejpam-6781	396	20	.	.	PUNCT
ejpam-6781	396	21	references	reference	NOUN
ejpam-6781	396	22	[	[	X
ejpam-6781	396	23	1	1	NUM
ejpam-6781	396	24	]	]	X
ejpam-6781	396	25	wac	wac	NOUN
ejpam-6781	396	26	law	law	NOUN
ejpam-6781	396	27	sierpiński	sierpiński	NOUN
ejpam-6781	396	28	.	.	PUNCT
ejpam-6781	397	1	pythagorean	pythagorean	PROPN
ejpam-6781	397	2	triangles	triangles	PROPN
ejpam-6781	397	3	.	.	PUNCT
ejpam-6781	398	1	dover	dover	PROPN
ejpam-6781	398	2	publications	publications	PROPN
ejpam-6781	398	3	,	,	PUNCT
ejpam-6781	398	4	mineola	mineola	PROPN
ejpam-6781	398	5	,	,	PUNCT
ejpam-6781	398	6	ny	ny	PROPN
ejpam-6781	398	7	,	,	PUNCT
ejpam-6781	398	8	2011	2011	NUM
ejpam-6781	398	9	.	.	PUNCT
ejpam-6781	399	1	[	[	X
ejpam-6781	399	2	2	2	NUM
ejpam-6781	399	3	]	]	X
ejpam-6781	399	4	charles	charles	PROPN
ejpam-6781	399	5	l.	l.	PROPN
ejpam-6781	399	6	shedd	shedd	PROPN
ejpam-6781	399	7	.	.	PUNCT
ejpam-6781	400	1	a	a	DET
ejpam-6781	400	2	hypotenuse	hypotenuse	NOUN
ejpam-6781	400	3	common	common	ADJ
ejpam-6781	400	4	to	to	ADP
ejpam-6781	400	5	64	64	NUM
ejpam-6781	400	6	primitive	primitive	ADJ
ejpam-6781	400	7	right	right	ADJ
ejpam-6781	400	8	triangles	triangle	NOUN
ejpam-6781	400	9	.	.	PUNCT
ejpam-6781	401	1	scripta	scripta	PROPN
ejpam-6781	401	2	mathematica	mathematica	PROPN
ejpam-6781	401	3	,	,	PUNCT
ejpam-6781	401	4	15:131–132	15:131–132	PROPN
ejpam-6781	401	5	,	,	PUNCT
ejpam-6781	401	6	1949	1949	NUM
ejpam-6781	401	7	.	.	PUNCT
ejpam-6781	402	1	[	[	X
ejpam-6781	402	2	3	3	X
ejpam-6781	402	3	]	]	X
ejpam-6781	402	4	e.	e.	PROPN
ejpam-6781	402	5	eckert	eckert	PROPN
ejpam-6781	402	6	.	.	PUNCT
ejpam-6781	403	1	the	the	DET
ejpam-6781	403	2	group	group	NOUN
ejpam-6781	403	3	of	of	ADP
ejpam-6781	403	4	primitive	primitive	ADJ
ejpam-6781	403	5	pythagorean	pythagorean	PROPN
ejpam-6781	403	6	triangles	triangle	NOUN
ejpam-6781	403	7	.	.	PUNCT
ejpam-6781	404	1	mathematics	mathematic	NOUN
ejpam-6781	404	2	magazine	magazine	PROPN
ejpam-6781	404	3	,	,	PUNCT
ejpam-6781	404	4	57:22–27	57:22–27	PROPN
ejpam-6781	404	5	,	,	PUNCT
ejpam-6781	404	6	1984	1984	NUM
ejpam-6781	404	7	.	.	PUNCT
ejpam-6781	405	1	[	[	X
ejpam-6781	405	2	4	4	NUM
ejpam-6781	405	3	]	]	X
ejpam-6781	405	4	lin	lin	PROPN
ejpam-6781	405	5	tan	tan	PROPN
ejpam-6781	405	6	.	.	PUNCT
ejpam-6781	406	1	the	the	DET
ejpam-6781	406	2	group	group	NOUN
ejpam-6781	406	3	of	of	ADP
ejpam-6781	406	4	rational	rational	ADJ
ejpam-6781	406	5	points	point	NOUN
ejpam-6781	406	6	on	on	ADP
ejpam-6781	406	7	the	the	DET
ejpam-6781	406	8	unit	unit	NOUN
ejpam-6781	406	9	circle	circle	NOUN
ejpam-6781	406	10	.	.	PUNCT
ejpam-6781	407	1	mathematics	mathematics	PROPN
ejpam-6781	407	2	magazine	magazine	PROPN
ejpam-6781	407	3	,	,	PUNCT
ejpam-6781	407	4	69(3):163–171	69(3):163–171	PROPN
ejpam-6781	407	5	,	,	PUNCT
ejpam-6781	407	6	1996	1996	NUM
ejpam-6781	407	7	.	.	PUNCT
ejpam-6781	408	1	[	[	X
ejpam-6781	408	2	5	5	NUM
ejpam-6781	408	3	]	]	PUNCT
ejpam-6781	408	4	amnon	amnon	NOUN
ejpam-6781	408	5	yekutieli	yekutieli	PROPN
ejpam-6781	408	6	.	.	PUNCT
ejpam-6781	409	1	pythagorean	pythagorean	PROPN
ejpam-6781	409	2	triples	triple	NOUN
ejpam-6781	409	3	,	,	PUNCT
ejpam-6781	409	4	complex	complex	ADJ
ejpam-6781	409	5	numbers	number	NOUN
ejpam-6781	409	6	,	,	PUNCT
ejpam-6781	409	7	abelian	abelian	ADJ
ejpam-6781	409	8	groups	group	NOUN
ejpam-6781	409	9	and	and	CCONJ
ejpam-6781	409	10	prime	prime	ADJ
ejpam-6781	409	11	numbers	number	NOUN
ejpam-6781	409	12	.	.	PUNCT
ejpam-6781	410	1	the	the	DET
ejpam-6781	410	2	american	american	PROPN
ejpam-6781	410	3	mathematical	mathematical	PROPN
ejpam-6781	410	4	monthly	monthly	PROPN
ejpam-6781	410	5	,	,	PUNCT
ejpam-6781	410	6	130(4):321–334	130(4):321–334	NUM
ejpam-6781	410	7	,	,	PUNCT
ejpam-6781	410	8	2023	2023	NUM
ejpam-6781	410	9	.	.	PUNCT
ejpam-6781	411	1	[	[	X
ejpam-6781	411	2	6	6	NUM
ejpam-6781	411	3	]	]	X
ejpam-6781	411	4	bob	bob	NOUN
ejpam-6781	411	5	burn	burn	NOUN
ejpam-6781	411	6	.	.	PUNCT
ejpam-6781	412	1	87.23	87.23	NUM
ejpam-6781	412	2	triangles	triangle	NOUN
ejpam-6781	412	3	with	with	ADP
ejpam-6781	412	4	a	a	DET
ejpam-6781	412	5	60	60	NUM
ejpam-6781	412	6	◦	◦	NOUN
ejpam-6781	412	7	angle	angle	NOUN
ejpam-6781	412	8	and	and	CCONJ
ejpam-6781	412	9	sides	side	NOUN
ejpam-6781	412	10	of	of	ADP
ejpam-6781	412	11	integer	integer	NOUN
ejpam-6781	412	12	length	length	NOUN
ejpam-6781	412	13	.	.	PUNCT
ejpam-6781	413	1	the	the	DET
ejpam-6781	413	2	mathematical	mathematical	ADJ
ejpam-6781	413	3	gazette	gazette	NOUN
ejpam-6781	413	4	,	,	PUNCT
ejpam-6781	413	5	87(508):148–153	87(508):148–153	PROPN
ejpam-6781	413	6	,	,	PUNCT
ejpam-6781	413	7	2003	2003	NUM
ejpam-6781	413	8	.	.	PUNCT
ejpam-6781	414	1	[	[	X
ejpam-6781	414	2	7	7	X
ejpam-6781	414	3	]	]	X
ejpam-6781	414	4	john	john	PROPN
ejpam-6781	414	5	gilder	gilder	PROPN
ejpam-6781	414	6	.	.	PUNCT
ejpam-6781	415	1	integer	integer	NOUN
ejpam-6781	415	2	-	-	PUNCT
ejpam-6781	415	3	sided	side	VERB
ejpam-6781	415	4	triangles	triangle	NOUN
ejpam-6781	415	5	with	with	ADP
ejpam-6781	415	6	an	an	DET
ejpam-6781	415	7	angle	angle	NOUN
ejpam-6781	415	8	of	of	ADP
ejpam-6781	415	9	60	60	NUM
ejpam-6781	415	10	◦	◦	NOUN
ejpam-6781	415	11	.	.	PUNCT
ejpam-6781	416	1	the	the	DET
ejpam-6781	416	2	mathematical	mathematical	ADJ
ejpam-6781	416	3	gazette	gazette	NOUN
ejpam-6781	416	4	,	,	PUNCT
ejpam-6781	416	5	66:261–266	66:261–266	PROPN
ejpam-6781	416	6	,	,	PUNCT
ejpam-6781	416	7	1982	1982	NUM
ejpam-6781	416	8	.	.	PUNCT
ejpam-6781	417	1	[	[	X
ejpam-6781	417	2	8	8	NUM
ejpam-6781	417	3	]	]	PUNCT
ejpam-6781	417	4	emrys	emry	NOUN
ejpam-6781	417	5	read	read	VERB
ejpam-6781	417	6	.	.	PUNCT
ejpam-6781	418	1	on	on	ADP
ejpam-6781	418	2	integer	integer	NOUN
ejpam-6781	418	3	-	-	PUNCT
ejpam-6781	418	4	sided	side	VERB
ejpam-6781	418	5	triangles	triangle	NOUN
ejpam-6781	418	6	containing	contain	VERB
ejpam-6781	418	7	angles	angle	NOUN
ejpam-6781	418	8	of	of	ADP
ejpam-6781	418	9	120	120	NUM
ejpam-6781	418	10	◦	◦	NOUN
ejpam-6781	418	11	or	or	CCONJ
ejpam-6781	418	12	60	60	NUM
ejpam-6781	418	13	◦	◦	NOUN
ejpam-6781	418	14	.	.	PUNCT
ejpam-6781	419	1	the	the	DET
ejpam-6781	419	2	mathematical	mathematical	ADJ
ejpam-6781	419	3	gazette	gazette	NOUN
ejpam-6781	419	4	,	,	PUNCT
ejpam-6781	419	5	90:299–305	90:299–305	NUM
ejpam-6781	419	6	,	,	PUNCT
ejpam-6781	419	7	2006	2006	NUM
ejpam-6781	419	8	.	.	PUNCT
ejpam-6781	420	1	[	[	X
ejpam-6781	420	2	9	9	NUM
ejpam-6781	420	3	]	]	X
ejpam-6781	420	4	keith	keith	PROPN
ejpam-6781	420	5	selkirk	selkirk	PROPN
ejpam-6781	420	6	.	.	PUNCT
ejpam-6781	421	1	integer	integer	NOUN
ejpam-6781	421	2	-	-	PUNCT
ejpam-6781	421	3	sided	side	VERB
ejpam-6781	421	4	triangles	triangle	NOUN
ejpam-6781	421	5	with	with	ADP
ejpam-6781	421	6	an	an	DET
ejpam-6781	421	7	angle	angle	NOUN
ejpam-6781	421	8	of	of	ADP
ejpam-6781	421	9	120	120	NUM
ejpam-6781	421	10	◦	◦	NOUN
ejpam-6781	421	11	.	.	PUNCT
ejpam-6781	422	1	the	the	DET
ejpam-6781	422	2	mathematical	mathematical	ADJ
ejpam-6781	422	3	gazette	gazette	NOUN
ejpam-6781	422	4	,	,	PUNCT
ejpam-6781	422	5	67(442):251–255	67(442):251–255	PROPN
ejpam-6781	422	6	,	,	PUNCT
ejpam-6781	422	7	1983	1983	NUM
ejpam-6781	422	8	.	.	PUNCT
ejpam-6781	423	1	[	[	X
ejpam-6781	423	2	10	10	NUM
ejpam-6781	423	3	]	]	X
ejpam-6781	423	4	russell	russell	PROPN
ejpam-6781	423	5	a.	a.	PROPN
ejpam-6781	423	6	gordon	gordon	PROPN
ejpam-6781	423	7	.	.	PUNCT
ejpam-6781	424	1	properties	property	NOUN
ejpam-6781	424	2	of	of	ADP
ejpam-6781	424	3	eisenstein	eisenstein	NOUN
ejpam-6781	424	4	triples	triple	NOUN
ejpam-6781	424	5	.	.	PUNCT
ejpam-6781	425	1	mathematics	mathematic	NOUN
ejpam-6781	425	2	magazine	magazine	NOUN
ejpam-6781	425	3	,	,	PUNCT
ejpam-6781	425	4	85:12–25	85:12–25	PROPN
ejpam-6781	425	5	,	,	PUNCT
ejpam-6781	425	6	2012	2012	NUM
ejpam-6781	425	7	.	.	PUNCT
ejpam-6781	426	1	[	[	X
ejpam-6781	426	2	11	11	NUM
ejpam-6781	426	3	]	]	PUNCT
ejpam-6781	426	4	j.	j.	PROPN
ejpam-6781	426	5	h.	h.	PROPN
ejpam-6781	426	6	conway	conway	PROPN
ejpam-6781	426	7	and	and	CCONJ
ejpam-6781	426	8	r.	r.	PROPN
ejpam-6781	426	9	guy	guy	PROPN
ejpam-6781	426	10	.	.	PUNCT
ejpam-6781	427	1	the	the	DET
ejpam-6781	427	2	book	book	NOUN
ejpam-6781	427	3	of	of	ADP
ejpam-6781	427	4	numbers	number	NOUN
ejpam-6781	427	5	.	.	PUNCT
ejpam-6781	428	1	springer	springer	NOUN
ejpam-6781	428	2	,	,	PUNCT
ejpam-6781	428	3	new	new	PROPN
ejpam-6781	428	4	york	york	PROPN
ejpam-6781	428	5	,	,	PUNCT
ejpam-6781	428	6	ny	ny	PROPN
ejpam-6781	428	7	,	,	PUNCT
ejpam-6781	428	8	2012	2012	NUM
ejpam-6781	428	9	.	.	PUNCT
ejpam-6781	429	1	s.	s.	PROPN
ejpam-6781	429	2	jitman	jitman	PROPN
ejpam-6781	429	3	,	,	PUNCT
ejpam-6781	429	4	m.	m.	NOUN
ejpam-6781	429	5	mohammad	mohammad	PROPN
ejpam-6781	429	6	,	,	PUNCT
ejpam-6781	429	7	e.	e.	PROPN
ejpam-6781	429	8	sangwisut	sangwisut	PROPN
ejpam-6781	429	9	/	/	SYM
ejpam-6781	429	10	eur	eur	PROPN
ejpam-6781	429	11	.	.	PUNCT
ejpam-6781	430	1	j.	j.	PROPN
ejpam-6781	430	2	pure	pure	PROPN
ejpam-6781	430	3	appl	appl	PROPN
ejpam-6781	430	4	.	.	PROPN
ejpam-6781	430	5	math	math	PROPN
ejpam-6781	430	6	,	,	PUNCT
ejpam-6781	430	7	18	18	NUM
ejpam-6781	430	8	(	(	PUNCT
ejpam-6781	430	9	4	4	NUM
ejpam-6781	430	10	)	)	PUNCT
ejpam-6781	430	11	(	(	PUNCT
ejpam-6781	430	12	2025	2025	NUM
ejpam-6781	430	13	)	)	PUNCT
ejpam-6781	430	14	,	,	PUNCT
ejpam-6781	430	15	6781	6781	NUM
ejpam-6781	430	16	16	16	NUM
ejpam-6781	430	17	of	of	ADP
ejpam-6781	430	18	16	16	NUM
ejpam-6781	431	1	[	[	X
ejpam-6781	431	2	12	12	NUM
ejpam-6781	431	3	]	]	PUNCT
ejpam-6781	431	4	k.	k.	PROPN
ejpam-6781	431	5	ireland	ireland	PROPN
ejpam-6781	431	6	and	and	CCONJ
ejpam-6781	431	7	m.	m.	PROPN
ejpam-6781	431	8	rosen	rosen	PROPN
ejpam-6781	431	9	.	.	PUNCT
ejpam-6781	432	1	a	a	DET
ejpam-6781	432	2	classical	classical	ADJ
ejpam-6781	432	3	introduction	introduction	NOUN
ejpam-6781	432	4	to	to	ADP
ejpam-6781	432	5	modern	modern	ADJ
ejpam-6781	432	6	number	number	NOUN
ejpam-6781	432	7	theory	theory	NOUN
ejpam-6781	432	8	.	.	PUNCT
ejpam-6781	433	1	springer	springer	NOUN
ejpam-6781	433	2	,	,	PUNCT
ejpam-6781	433	3	new	new	PROPN
ejpam-6781	433	4	york	york	PROPN
ejpam-6781	433	5	,	,	PUNCT
ejpam-6781	433	6	ny	ny	PROPN
ejpam-6781	433	7	,	,	PUNCT
ejpam-6781	433	8	2	2	NUM
ejpam-6781	433	9	edition	edition	NOUN
ejpam-6781	433	10	,	,	PUNCT
ejpam-6781	433	11	1990	1990	NUM
ejpam-6781	433	12	.	.	PUNCT
ejpam-6781	434	1	[	[	X
ejpam-6781	434	2	13	13	NUM
ejpam-6781	434	3	]	]	X
ejpam-6781	434	4	r.	r.	PROPN
ejpam-6781	434	5	takloo	takloo	NOUN
ejpam-6781	434	6	-	-	PUNCT
ejpam-6781	434	7	bighash	bighash	NOUN
ejpam-6781	434	8	.	.	PUNCT
ejpam-6781	435	1	a	a	DET
ejpam-6781	435	2	pythagorean	pythagorean	ADJ
ejpam-6781	435	3	introduction	introduction	NOUN
ejpam-6781	435	4	to	to	ADP
ejpam-6781	435	5	number	number	NOUN
ejpam-6781	435	6	theory	theory	NOUN
ejpam-6781	435	7	:	:	PUNCT
ejpam-6781	435	8	right	right	ADJ
ejpam-6781	435	9	triangles	triangle	NOUN
ejpam-6781	435	10	,	,	PUNCT
ejpam-6781	435	11	sums	sum	NOUN
ejpam-6781	435	12	of	of	ADP
ejpam-6781	435	13	squares	square	NOUN
ejpam-6781	435	14	,	,	PUNCT
ejpam-6781	435	15	and	and	CCONJ
ejpam-6781	435	16	arithmetic	arithmetic	ADJ
ejpam-6781	435	17	.	.	PUNCT
ejpam-6781	436	1	undergraduate	undergraduate	ADJ
ejpam-6781	436	2	texts	text	NOUN
ejpam-6781	436	3	in	in	ADP
ejpam-6781	436	4	mathematics	mathematic	NOUN
ejpam-6781	436	5	.	.	PUNCT
ejpam-6781	437	1	springer	springer	NOUN
ejpam-6781	437	2	international	international	ADJ
ejpam-6781	437	3	publishing	publishing	NOUN
ejpam-6781	437	4	,	,	PUNCT
ejpam-6781	437	5	2018	2018	NUM
ejpam-6781	437	6	.	.	PUNCT
ejpam-6781	438	1	[	[	X
ejpam-6781	438	2	14	14	NUM
ejpam-6781	438	3	]	]	X
ejpam-6781	438	4	philip	philip	PROPN
ejpam-6781	438	5	p.	p.	PROPN
ejpam-6781	438	6	west	west	PROPN
ejpam-6781	438	7	and	and	CCONJ
ejpam-6781	438	8	brian	brian	PROPN
ejpam-6781	438	9	d.	d.	PROPN
ejpam-6781	438	10	sittinger	sittinger	PROPN
ejpam-6781	438	11	.	.	PUNCT
ejpam-6781	439	1	a	a	DET
ejpam-6781	439	2	further	further	ADJ
ejpam-6781	439	3	stroll	stroll	NOUN
ejpam-6781	439	4	into	into	ADP
ejpam-6781	439	5	the	the	DET
ejpam-6781	439	6	eisenstein	eisenstein	NOUN
ejpam-6781	439	7	primes	prime	NOUN
ejpam-6781	439	8	.	.	PUNCT
ejpam-6781	440	1	the	the	DET
ejpam-6781	440	2	american	american	PROPN
ejpam-6781	440	3	mathematical	mathematical	PROPN
ejpam-6781	440	4	monthly	monthly	PROPN
ejpam-6781	440	5	,	,	PUNCT
ejpam-6781	440	6	124:609–620	124:609–620	NUM
ejpam-6781	440	7	,	,	PUNCT
ejpam-6781	440	8	2017	2017	NUM
ejpam-6781	440	9	.	.	PUNCT
ejpam-6781	441	1	[	[	X
ejpam-6781	441	2	15	15	NUM
ejpam-6781	441	3	]	]	X
ejpam-6781	441	4	kamal	kamal	PROPN
ejpam-6781	441	5	bahmanpour	bahmanpour	PROPN
ejpam-6781	441	6	.	.	PUNCT
ejpam-6781	442	1	prime	prime	ADJ
ejpam-6781	442	2	numbers	number	NOUN
ejpam-6781	442	3	p	p	NOUN
ejpam-6781	442	4	with	with	ADP
ejpam-6781	442	5	expression	expression	NOUN
ejpam-6781	442	6	p	p	X
ejpam-6781	442	7	=	=	PROPN
ejpam-6781	442	8	a2	a2	PROPN
ejpam-6781	442	9	±	±	PROPN
ejpam-6781	442	10	ab±	ab±	PROPN
ejpam-6781	442	11	b2	b2	PROPN
ejpam-6781	442	12	.	.	PUNCT
ejpam-6781	443	1	journal	journal	NOUN
ejpam-6781	443	2	of	of	ADP
ejpam-6781	443	3	number	number	NOUN
ejpam-6781	443	4	theory	theory	NOUN
ejpam-6781	443	5	,	,	PUNCT
ejpam-6781	443	6	166:208–218	166:208–218	NUM
ejpam-6781	443	7	,	,	PUNCT
ejpam-6781	443	8	2016	2016	NUM
ejpam-6781	443	9	.	.	PUNCT
ejpam-6781	444	1	[	[	X
ejpam-6781	444	2	16	16	NUM
ejpam-6781	444	3	]	]	X
ejpam-6781	444	4	d.	d.	PROPN
ejpam-6781	444	5	s.	s.	PROPN
ejpam-6781	444	6	kim	kim	PROPN
ejpam-6781	444	7	and	and	CCONJ
ejpam-6781	444	8	t.	t.	PROPN
ejpam-6781	444	9	kim	kim	PROPN
ejpam-6781	444	10	.	.	PUNCT
ejpam-6781	445	1	moment	moment	NOUN
ejpam-6781	445	2	representations	representation	NOUN
ejpam-6781	445	3	of	of	ADP
ejpam-6781	445	4	fully	fully	ADV
ejpam-6781	445	5	degenerate	degenerate	ADJ
ejpam-6781	445	6	bernoulli	bernoulli	NOUN
ejpam-6781	445	7	and	and	CCONJ
ejpam-6781	445	8	degenerate	degenerate	ADJ
ejpam-6781	445	9	euler	euler	NOUN
ejpam-6781	445	10	polynomials	polynomial	NOUN
ejpam-6781	445	11	.	.	PUNCT
ejpam-6781	446	1	russian	russian	ADJ
ejpam-6781	446	2	journal	journal	PROPN
ejpam-6781	446	3	of	of	ADP
ejpam-6781	446	4	mathematical	mathematical	ADJ
ejpam-6781	446	5	physics	physics	NOUN
ejpam-6781	446	6	,	,	PUNCT
ejpam-6781	446	7	32(31):682	32(31):682	NUM
ejpam-6781	446	8	–	–	PUNCT
ejpam-6781	446	9	690	690	NUM
ejpam-6781	446	10	,	,	PUNCT
ejpam-6781	446	11	2024	2024	NUM
ejpam-6781	446	12	.	.	PUNCT
ejpam-6781	447	1	[	[	X
ejpam-6781	447	2	17	17	NUM
ejpam-6781	447	3	]	]	PUNCT
ejpam-6781	447	4	t.	t.	PROPN
ejpam-6781	447	5	kim	kim	PROPN
ejpam-6781	447	6	and	and	CCONJ
ejpam-6781	447	7	d.	d.	PROPN
ejpam-6781	447	8	s.	s.	PROPN
ejpam-6781	447	9	kim	kim	PROPN
ejpam-6781	447	10	.	.	PUNCT
ejpam-6781	448	1	spivey	spivey	PROPN
ejpam-6781	448	2	-	-	PUNCT
ejpam-6781	448	3	type	type	NOUN
ejpam-6781	448	4	recurrence	recurrence	NOUN
ejpam-6781	448	5	relations	relation	NOUN
ejpam-6781	448	6	for	for	ADP
ejpam-6781	448	7	degenerate	degenerate	ADJ
ejpam-6781	448	8	bell	bell	NOUN
ejpam-6781	448	9	and	and	CCONJ
ejpam-6781	448	10	dowling	dowling	NOUN
ejpam-6781	448	11	polynomials	polynomial	NOUN
ejpam-6781	448	12	.	.	PUNCT
ejpam-6781	449	1	russian	russian	ADJ
ejpam-6781	449	2	journal	journal	PROPN
ejpam-6781	449	3	of	of	ADP
ejpam-6781	449	4	mathematical	mathematical	ADJ
ejpam-6781	449	5	physics	physics	NOUN
ejpam-6781	449	6	,	,	PUNCT
ejpam-6781	449	7	32(2):288–296	32(2):288–296	PROPN
ejpam-6781	449	8	,	,	PUNCT
ejpam-6781	449	9	2025	2025	NUM
ejpam-6781	449	10	.	.	PUNCT