id	sid	tid	token	lemma	pos
ejpam-1831	1	1	european	european	PROPN
ejpam-1831	1	2	journal	journal	PROPN
ejpam-1831	1	3	of	of	ADP
ejpam-1831	1	4	pure	pure	ADJ
ejpam-1831	1	5	and	and	CCONJ
ejpam-1831	1	6	applied	apply	VERB
ejpam-1831	1	7	mathematics	mathematic	NOUN
ejpam-1831	1	8	vol	vol	NOUN
ejpam-1831	1	9	.	.	PROPN
ejpam-1831	2	1	6	6	NUM
ejpam-1831	2	2	,	,	PUNCT
ejpam-1831	2	3	no	no	INTJ
ejpam-1831	2	4	.	.	NOUN
ejpam-1831	2	5	4	4	NUM
ejpam-1831	2	6	,	,	PUNCT
ejpam-1831	2	7	2013	2013	NUM
ejpam-1831	2	8	,	,	PUNCT
ejpam-1831	2	9	400	400	NUM
ejpam-1831	2	10	-	-	SYM
ejpam-1831	2	11	404	404	NUM
ejpam-1831	2	12	issn	issn	PROPN
ejpam-1831	2	13	1307	1307	NUM
ejpam-1831	2	14	-	-	SYM
ejpam-1831	2	15	5543	5543	NUM
ejpam-1831	2	16	–	–	PUNCT
ejpam-1831	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-1831	2	18	on	on	ADP
ejpam-1831	2	19	the	the	DET
ejpam-1831	2	20	number	number	NOUN
ejpam-1831	2	21	of	of	ADP
ejpam-1831	2	22	pairs	pair	NOUN
ejpam-1831	2	23	of	of	ADP
ejpam-1831	2	24	points	point	NOUN
ejpam-1831	2	25	in	in	ADP
ejpam-1831	2	26	a	a	DET
ejpam-1831	2	27	quadratic	quadratic	ADJ
ejpam-1831	2	28	equation	equation	NOUN
ejpam-1831	2	29	with	with	ADP
ejpam-1831	2	30	rational	rational	ADJ
ejpam-1831	2	31	distance	distance	NOUN
ejpam-1831	2	32	juan	juan	PROPN
ejpam-1831	2	33	sebastian	sebastian	PROPN
ejpam-1831	2	34	beleño	beleño	PROPN
ejpam-1831	2	35	diaz	diaz	PROPN
ejpam-1831	2	36	escuela	escuela	PROPN
ejpam-1831	2	37	de	de	PROPN
ejpam-1831	2	38	ingeniería	ingeniería	PROPN
ejpam-1831	2	39	de	de	PROPN
ejpam-1831	2	40	sistemas	sistemas	PROPN
ejpam-1831	2	41	e	e	PROPN
ejpam-1831	2	42	informática	informática	PROPN
ejpam-1831	2	43	,	,	PUNCT
ejpam-1831	2	44	facultad	facultad	PROPN
ejpam-1831	2	45	de	de	PROPN
ejpam-1831	2	46	ingenierías	ingenierías	PROPN
ejpam-1831	2	47	físico	físico	PROPN
ejpam-1831	2	48	-	-	PUNCT
ejpam-1831	2	49	mecánicas	mecánica	NOUN
ejpam-1831	2	50	,	,	PUNCT
ejpam-1831	2	51	industrial	industrial	ADJ
ejpam-1831	2	52	university	university	NOUN
ejpam-1831	2	53	of	of	ADP
ejpam-1831	2	54	santander	santander	NOUN
ejpam-1831	2	55	,	,	PUNCT
ejpam-1831	2	56	bucaramanga	bucaramanga	PROPN
ejpam-1831	2	57	,	,	PUNCT
ejpam-1831	2	58	colombia	colombia	PROPN
ejpam-1831	2	59	abstract	abstract	NOUN
ejpam-1831	2	60	.	.	PUNCT
ejpam-1831	3	1	in	in	ADP
ejpam-1831	3	2	this	this	DET
ejpam-1831	3	3	paper	paper	NOUN
ejpam-1831	3	4	is	be	AUX
ejpam-1831	3	5	shown	show	VERB
ejpam-1831	3	6	a	a	DET
ejpam-1831	3	7	solution	solution	NOUN
ejpam-1831	3	8	to	to	ADP
ejpam-1831	3	9	the	the	DET
ejpam-1831	3	10	number	number	NOUN
ejpam-1831	3	11	of	of	ADP
ejpam-1831	3	12	pair	pair	NOUN
ejpam-1831	3	13	of	of	ADP
ejpam-1831	3	14	points	point	NOUN
ejpam-1831	3	15	in	in	ADP
ejpam-1831	3	16	a	a	DET
ejpam-1831	3	17	quadratic	quadratic	ADJ
ejpam-1831	3	18	equation	equation	NOUN
ejpam-1831	3	19	with	with	ADP
ejpam-1831	3	20	rational	rational	ADJ
ejpam-1831	3	21	distance	distance	NOUN
ejpam-1831	3	22	,	,	PUNCT
ejpam-1831	3	23	this	this	DET
ejpam-1831	3	24	result	result	NOUN
ejpam-1831	3	25	have	have	VERB
ejpam-1831	3	26	an	an	DET
ejpam-1831	3	27	important	important	ADJ
ejpam-1831	3	28	impact	impact	NOUN
ejpam-1831	3	29	to	to	PART
ejpam-1831	3	30	solve	solve	VERB
ejpam-1831	3	31	the	the	DET
ejpam-1831	3	32	open	open	ADJ
ejpam-1831	3	33	problem	problem	NOUN
ejpam-1831	3	34	“	"	PUNCT
ejpam-1831	3	35	points	point	NOUN
ejpam-1831	3	36	on	on	ADP
ejpam-1831	3	37	a	a	DET
ejpam-1831	3	38	parabola	parabola	NOUN
ejpam-1831	3	39	”	"	PUNCT
ejpam-1831	4	1	[	[	X
ejpam-1831	4	2	3	3	NUM
ejpam-1831	4	3	]	]	PUNCT
ejpam-1831	4	4	proposed	propose	VERB
ejpam-1831	4	5	in	in	ADP
ejpam-1831	4	6	the	the	DET
ejpam-1831	4	7	center	center	NOUN
ejpam-1831	4	8	for	for	ADP
ejpam-1831	4	9	discrete	discrete	ADJ
ejpam-1831	4	10	mathematics	mathematic	NOUN
ejpam-1831	4	11	and	and	CCONJ
ejpam-1831	4	12	theoretical	theoretical	ADJ
ejpam-1831	4	13	computer	computer	NOUN
ejpam-1831	4	14	science	science	NOUN
ejpam-1831	4	15	(	(	PUNCT
ejpam-1831	4	16	dimacs	dimacs	PROPN
ejpam-1831	4	17	)	)	PUNCT
ejpam-1831	4	18	,	,	PUNCT
ejpam-1831	4	19	because	because	SCONJ
ejpam-1831	4	20	it	it	PRON
ejpam-1831	4	21	’s	’	VERB
ejpam-1831	4	22	an	an	DET
ejpam-1831	4	23	approach	approach	NOUN
ejpam-1831	4	24	to	to	PART
ejpam-1831	4	25	set	set	VERB
ejpam-1831	4	26	down	down	ADP
ejpam-1831	4	27	basis	basis	NOUN
ejpam-1831	4	28	in	in	ADP
ejpam-1831	4	29	the	the	DET
ejpam-1831	4	30	problem	problem	NOUN
ejpam-1831	4	31	.	.	PUNCT
ejpam-1831	5	1	2010	2010	NUM
ejpam-1831	5	2	mathematics	mathematic	NOUN
ejpam-1831	5	3	subject	subject	NOUN
ejpam-1831	5	4	classifications	classification	NOUN
ejpam-1831	5	5	:	:	PUNCT
ejpam-1831	5	6	97f40,11d09,41a20	97f40,11d09,41a20	NUM
ejpam-1831	5	7	key	key	ADJ
ejpam-1831	5	8	words	word	NOUN
ejpam-1831	5	9	and	and	CCONJ
ejpam-1831	5	10	phrases	phrase	NOUN
ejpam-1831	5	11	:	:	PUNCT
ejpam-1831	5	12	rational	rational	ADJ
ejpam-1831	5	13	numbers	number	NOUN
ejpam-1831	5	14	,	,	PUNCT
ejpam-1831	5	15	quadratic	quadratic	ADJ
ejpam-1831	5	16	equations	equation	NOUN
ejpam-1831	5	17	,	,	PUNCT
ejpam-1831	5	18	approximation	approximation	NOUN
ejpam-1831	5	19	by	by	ADP
ejpam-1831	5	20	rational	rational	ADJ
ejpam-1831	5	21	functions	function	NOUN
ejpam-1831	5	22	1	1	NUM
ejpam-1831	5	23	.	.	PUNCT
ejpam-1831	6	1	introduction	introduction	NOUN
ejpam-1831	6	2	define	define	VERB
ejpam-1831	6	3	f	f	X
ejpam-1831	6	4	:	:	PUNCT
ejpam-1831	6	5	ℜ	ℜ	PROPN
ejpam-1831	6	6	−→	−→	NOUN
ejpam-1831	6	7	ℜ	ℜ	NOUN
ejpam-1831	6	8	by	by	ADP
ejpam-1831	6	9	the	the	DET
ejpam-1831	6	10	function	function	NOUN
ejpam-1831	6	11	f	f	PROPN
ejpam-1831	6	12	(	(	PUNCT
ejpam-1831	6	13	x	x	X
ejpam-1831	6	14	)	)	PUNCT
ejpam-1831	6	15	=	=	SYM
ejpam-1831	6	16	ax2	ax2	NOUN
ejpam-1831	6	17	+	+	CCONJ
ejpam-1831	6	18	bx	bx	X
ejpam-1831	6	19	+	+	CCONJ
ejpam-1831	6	20	c	c	X
ejpam-1831	6	21	;	;	PUNCT
ejpam-1831	6	22	where	where	SCONJ
ejpam-1831	6	23	x	x	X
ejpam-1831	6	24	>	>	X
ejpam-1831	6	25	0	0	NUM
ejpam-1831	6	26	and	and	CCONJ
ejpam-1831	6	27	a	a	DET
ejpam-1831	6	28	,	,	PUNCT
ejpam-1831	6	29	b	b	NOUN
ejpam-1831	6	30	,	,	PUNCT
ejpam-1831	6	31	c	c	PROPN
ejpam-1831	6	32	∈	∈	PROPN
ejpam-1831	6	33	ℜ	ℜ	PROPN
ejpam-1831	6	34	with	with	ADP
ejpam-1831	6	35	a	a	DET
ejpam-1831	6	36	6=	6=	NUM
ejpam-1831	6	37	0	0	NUM
ejpam-1831	6	38	,	,	PUNCT
ejpam-1831	6	39	then	then	ADV
ejpam-1831	6	40	the	the	DET
ejpam-1831	6	41	question	question	NOUN
ejpam-1831	6	42	is	be	AUX
ejpam-1831	6	43	:	:	PUNCT
ejpam-1831	6	44	how	how	SCONJ
ejpam-1831	6	45	many	many	ADJ
ejpam-1831	6	46	pairs	pair	NOUN
ejpam-1831	6	47	of	of	ADP
ejpam-1831	6	48	points	point	NOUN
ejpam-1831	6	49	,	,	PUNCT
ejpam-1831	6	50	so	so	SCONJ
ejpam-1831	6	51	that	that	SCONJ
ejpam-1831	6	52	the	the	DET
ejpam-1831	6	53	distance	distance	NOUN
ejpam-1831	6	54	between	between	ADP
ejpam-1831	6	55	them	they	PRON
ejpam-1831	6	56	is	be	AUX
ejpam-1831	6	57	a	a	DET
ejpam-1831	6	58	rational	rational	ADJ
ejpam-1831	6	59	number	number	NOUN
ejpam-1831	6	60	?	?	PUNCT
ejpam-1831	6	61	,	,	PUNCT
ejpam-1831	6	62	although	although	SCONJ
ejpam-1831	6	63	exist	exist	VERB
ejpam-1831	6	64	some	some	DET
ejpam-1831	6	65	references	reference	NOUN
ejpam-1831	6	66	about	about	ADP
ejpam-1831	6	67	quadratic	quadratic	ADJ
ejpam-1831	6	68	equations	equation	NOUN
ejpam-1831	6	69	and	and	CCONJ
ejpam-1831	6	70	distances[1–3	distances[1–3	ADJ
ejpam-1831	6	71	,	,	PUNCT
ejpam-1831	6	72	6	6	NUM
ejpam-1831	6	73	,	,	PUNCT
ejpam-1831	6	74	7	7	NUM
ejpam-1831	6	75	,	,	PUNCT
ejpam-1831	6	76	9	9	NUM
ejpam-1831	6	77	,	,	PUNCT
ejpam-1831	6	78	10	10	NUM
ejpam-1831	6	79	]	]	PUNCT
ejpam-1831	6	80	,	,	PUNCT
ejpam-1831	6	81	there	there	PRON
ejpam-1831	6	82	is	be	VERB
ejpam-1831	6	83	no	no	DET
ejpam-1831	6	84	information	information	NOUN
ejpam-1831	6	85	about	about	ADP
ejpam-1831	6	86	this	this	DET
ejpam-1831	6	87	specifically	specifically	ADV
ejpam-1831	6	88	question	question	NOUN
ejpam-1831	6	89	and	and	CCONJ
ejpam-1831	6	90	the	the	DET
ejpam-1831	6	91	solution	solution	NOUN
ejpam-1831	6	92	of	of	ADP
ejpam-1831	6	93	this	this	DET
ejpam-1831	6	94	problem	problem	NOUN
ejpam-1831	6	95	allows	allow	VERB
ejpam-1831	6	96	to	to	PART
ejpam-1831	6	97	start	start	VERB
ejpam-1831	6	98	to	to	PART
ejpam-1831	6	99	solve	solve	VERB
ejpam-1831	6	100	the	the	DET
ejpam-1831	6	101	still	still	ADV
ejpam-1831	6	102	open	open	ADJ
ejpam-1831	6	103	problem	problem	NOUN
ejpam-1831	6	104	“	"	PUNCT
ejpam-1831	6	105	points	point	NOUN
ejpam-1831	6	106	on	on	ADP
ejpam-1831	6	107	a	a	DET
ejpam-1831	6	108	parabola	parabola	NOUN
ejpam-1831	6	109	”	"	PUNCT
ejpam-1831	7	1	[	[	X
ejpam-1831	7	2	3	3	NUM
ejpam-1831	7	3	]	]	PUNCT
ejpam-1831	7	4	,	,	PUNCT
ejpam-1831	7	5	that	that	SCONJ
ejpam-1831	7	6	it	it	PRON
ejpam-1831	7	7	’s	’	VERB
ejpam-1831	7	8	about	about	ADJ
ejpam-1831	7	9	to	to	PART
ejpam-1831	7	10	find	find	VERB
ejpam-1831	7	11	the	the	DET
ejpam-1831	7	12	maximum	maximum	ADJ
ejpam-1831	7	13	number	number	NOUN
ejpam-1831	7	14	of	of	ADP
ejpam-1831	7	15	points	point	NOUN
ejpam-1831	7	16	that	that	PRON
ejpam-1831	7	17	satisfies	satisfy	VERB
ejpam-1831	7	18	the	the	DET
ejpam-1831	7	19	condition	condition	NOUN
ejpam-1831	7	20	to	to	PART
ejpam-1831	7	21	have	have	VERB
ejpam-1831	7	22	a	a	DET
ejpam-1831	7	23	rational	rational	ADJ
ejpam-1831	7	24	distance	distance	NOUN
ejpam-1831	7	25	between	between	ADP
ejpam-1831	7	26	any	any	PRON
ejpam-1831	7	27	of	of	ADP
ejpam-1831	7	28	them	they	PRON
ejpam-1831	7	29	.	.	PUNCT
ejpam-1831	8	1	2	2	X
ejpam-1831	8	2	.	.	X
ejpam-1831	8	3	main	main	ADJ
ejpam-1831	8	4	result	result	NOUN
ejpam-1831	8	5	theorem	theorem	VERB
ejpam-1831	8	6	1	1	X
ejpam-1831	8	7	.	.	PUNCT
ejpam-1831	9	1	let	let	VERB
ejpam-1831	9	2	f	f	NOUN
ejpam-1831	9	3	:	:	PUNCT
ejpam-1831	9	4	ℜ	ℜ	PROPN
ejpam-1831	9	5	−→	−→	NOUN
ejpam-1831	9	6	ℜ	ℜ	ADJ
ejpam-1831	9	7	by	by	ADP
ejpam-1831	9	8	f	f	PROPN
ejpam-1831	9	9	(	(	PUNCT
ejpam-1831	9	10	x	x	NOUN
ejpam-1831	9	11	)	)	PUNCT
ejpam-1831	9	12	=	=	SYM
ejpam-1831	9	13	ax2	ax2	NOUN
ejpam-1831	9	14	+	+	CCONJ
ejpam-1831	9	15	bx	bx	X
ejpam-1831	10	1	+	+	CCONJ
ejpam-1831	10	2	c	c	X
ejpam-1831	10	3	;	;	PUNCT
ejpam-1831	10	4	where	where	SCONJ
ejpam-1831	10	5	x	x	SYM
ejpam-1831	10	6	∈	∈	PROPN
ejpam-1831	10	7	z+	z+	NUM
ejpam-1831	10	8	and	and	CCONJ
ejpam-1831	10	9	a	a	DET
ejpam-1831	10	10	,	,	PUNCT
ejpam-1831	10	11	b	b	NOUN
ejpam-1831	10	12	,	,	PUNCT
ejpam-1831	10	13	c	c	PROPN
ejpam-1831	10	14	∈	∈	PROPN
ejpam-1831	10	15	ℜ	ℜ	PROPN
ejpam-1831	10	16	with	with	ADP
ejpam-1831	10	17	a	a	DET
ejpam-1831	10	18	6=	6=	NUM
ejpam-1831	10	19	0	0	NUM
ejpam-1831	10	20	⇒exist	⇒exist	ADJ
ejpam-1831	10	21	infinite	infinite	ADJ
ejpam-1831	10	22	pairs	pair	NOUN
ejpam-1831	10	23	of	of	ADP
ejpam-1831	10	24	points	point	NOUN
ejpam-1831	10	25	within	within	ADP
ejpam-1831	10	26	the	the	DET
ejpam-1831	10	27	polynomial	polynomial	ADJ
ejpam-1831	10	28	,	,	PUNCT
ejpam-1831	10	29	where	where	SCONJ
ejpam-1831	10	30	the	the	DET
ejpam-1831	10	31	distance	distance	NOUN
ejpam-1831	10	32	between	between	ADP
ejpam-1831	10	33	them	they	PRON
ejpam-1831	10	34	is	be	AUX
ejpam-1831	10	35	a	a	DET
ejpam-1831	10	36	rational	rational	ADJ
ejpam-1831	10	37	number	number	NOUN
ejpam-1831	10	38	.	.	PUNCT
ejpam-1831	11	1	by	by	ADP
ejpam-1831	11	2	reductio	reductio	NOUN
ejpam-1831	11	3	ad	ad	NOUN
ejpam-1831	11	4	absurdum	absurdum	NOUN
ejpam-1831	11	5	,	,	PUNCT
ejpam-1831	11	6	we	we	PRON
ejpam-1831	11	7	suppose	suppose	VERB
ejpam-1831	11	8	that	that	SCONJ
ejpam-1831	11	9	the	the	DET
ejpam-1831	11	10	quadratic	quadratic	ADJ
ejpam-1831	11	11	equation	equation	NOUN
ejpam-1831	11	12	with	with	ADP
ejpam-1831	11	13	form	form	NOUN
ejpam-1831	11	14	f	f	X
ejpam-1831	11	15	(	(	PUNCT
ejpam-1831	11	16	x	x	NOUN
ejpam-1831	11	17	)	)	PUNCT
ejpam-1831	11	18	=	=	SYM
ejpam-1831	11	19	ax2	ax2	NOUN
ejpam-1831	11	20	+	+	CCONJ
ejpam-1831	11	21	bx	bx	PROPN
ejpam-1831	11	22	+	+	CCONJ
ejpam-1831	12	1	c	c	NOUN
ejpam-1831	12	2	have	have	AUX
ejpam-1831	12	3	finite	finite	VERB
ejpam-1831	12	4	pairs	pair	NOUN
ejpam-1831	12	5	of	of	ADP
ejpam-1831	12	6	points	point	NOUN
ejpam-1831	12	7	that	that	PRON
ejpam-1831	12	8	satisfies	satisfy	VERB
ejpam-1831	12	9	the	the	DET
ejpam-1831	12	10	condition	condition	NOUN
ejpam-1831	12	11	that	that	SCONJ
ejpam-1831	12	12	its	its	PRON
ejpam-1831	12	13	distance	distance	NOUN
ejpam-1831	12	14	is	be	AUX
ejpam-1831	12	15	a	a	DET
ejpam-1831	12	16	rational	rational	ADJ
ejpam-1831	12	17	number	number	NOUN
ejpam-1831	12	18	.	.	PUNCT
ejpam-1831	13	1	email	email	NOUN
ejpam-1831	13	2	address	address	NOUN
ejpam-1831	13	3	:	:	PUNCT
ejpam-1831	13	4	juan.beleno@correo.uis.edu.co	juan.beleno@correo.uis.edu.co	ADV
ejpam-1831	13	5	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1831	13	6	400	400	NUM
ejpam-1831	14	1	c	c	X
ejpam-1831	14	2	©	©	PROPN
ejpam-1831	14	3	2013	2013	NUM
ejpam-1831	14	4	ejpam	ejpam	NOUN
ejpam-1831	14	5	all	all	DET
ejpam-1831	14	6	rights	right	NOUN
ejpam-1831	14	7	reserved	reserve	VERB
ejpam-1831	14	8	.	.	PUNCT
ejpam-1831	15	1	j.	j.	PROPN
ejpam-1831	15	2	diaz	diaz	PROPN
ejpam-1831	15	3	/	/	SYM
ejpam-1831	15	4	eur	eur	PROPN
ejpam-1831	15	5	.	.	PUNCT
ejpam-1831	16	1	j.	j.	PROPN
ejpam-1831	16	2	pure	pure	PROPN
ejpam-1831	16	3	appl	appl	PROPN
ejpam-1831	16	4	.	.	PROPN
ejpam-1831	16	5	math	math	PROPN
ejpam-1831	16	6	,	,	PUNCT
ejpam-1831	16	7	6	6	NUM
ejpam-1831	16	8	(	(	PUNCT
ejpam-1831	16	9	2013	2013	NUM
ejpam-1831	16	10	)	)	PUNCT
ejpam-1831	16	11	,	,	PUNCT
ejpam-1831	16	12	400	400	NUM
ejpam-1831	16	13	-	-	SYM
ejpam-1831	16	14	404	404	NUM
ejpam-1831	16	15	401	401	NUM
ejpam-1831	16	16	select	select	ADJ
ejpam-1831	16	17	two	two	NUM
ejpam-1831	16	18	points	point	NOUN
ejpam-1831	16	19	in	in	ADP
ejpam-1831	16	20	the	the	DET
ejpam-1831	16	21	polynomial	polynomial	NOUN
ejpam-1831	16	22	:	:	PUNCT
ejpam-1831	16	23	(	(	PUNCT
ejpam-1831	16	24	r	r	NOUN
ejpam-1831	16	25	,	,	PUNCT
ejpam-1831	16	26	ar2	ar2	PROPN
ejpam-1831	16	27	+	+	X
ejpam-1831	16	28	br+	br+	ADJ
ejpam-1831	16	29	c	c	NOUN
ejpam-1831	16	30	)	)	PUNCT
ejpam-1831	16	31	and	and	CCONJ
ejpam-1831	16	32	(	(	PUNCT
ejpam-1831	16	33	s	s	PROPN
ejpam-1831	16	34	,	,	PUNCT
ejpam-1831	16	35	as2	as2	NOUN
ejpam-1831	16	36	+	+	NOUN
ejpam-1831	16	37	bs+	bs+	ADJ
ejpam-1831	16	38	c	c	NOUN
ejpam-1831	16	39	)	)	PUNCT
ejpam-1831	16	40	;	;	PUNCT
ejpam-1831	16	41	where	where	SCONJ
ejpam-1831	16	42	r	r	NOUN
ejpam-1831	16	43	,	,	PUNCT
ejpam-1831	16	44	s	s	PART
ejpam-1831	16	45	∈	∈	NOUN
ejpam-1831	16	46	z+	z+	NUM
ejpam-1831	16	47	define	define	VERB
ejpam-1831	16	48	the	the	DET
ejpam-1831	16	49	distance	distance	NOUN
ejpam-1831	16	50	function	function	NOUN
ejpam-1831	16	51	for	for	ADP
ejpam-1831	16	52	this	this	DET
ejpam-1831	16	53	case	case	NOUN
ejpam-1831	17	1	[	[	X
ejpam-1831	17	2	5	5	NUM
ejpam-1831	17	3	]	]	X
ejpam-1831	17	4	d	d	NOUN
ejpam-1831	17	5	=	=	SYM
ejpam-1831	17	6	p	p	X
ejpam-1831	17	7	(	(	PUNCT
ejpam-1831	17	8	s−	s−	PROPN
ejpam-1831	17	9	r)2	r)2	NOUN
ejpam-1831	17	10	+	+	CCONJ
ejpam-1831	17	11	(	(	PUNCT
ejpam-1831	17	12	as2	as2	PROPN
ejpam-1831	17	13	+	+	X
ejpam-1831	17	14	bs+	bs+	ADJ
ejpam-1831	17	15	c−	c−	NOUN
ejpam-1831	17	16	ar2−	ar2−	VERB
ejpam-1831	17	17	br	br	NOUN
ejpam-1831	17	18	−	−	PROPN
ejpam-1831	17	19	c)2	c)2	NOUN
ejpam-1831	17	20	cancel	cancel	VERB
ejpam-1831	17	21	the	the	DET
ejpam-1831	17	22	constants	constant	NOUN
ejpam-1831	17	23	:	:	PUNCT
ejpam-1831	17	24	c−	c−	NOUN
ejpam-1831	17	25	c	c	NOUN
ejpam-1831	17	26	=	=	SYM
ejpam-1831	17	27	0	0	PUNCT
ejpam-1831	18	1	d	d	NOUN
ejpam-1831	18	2	=	=	X
ejpam-1831	18	3	p	p	X
ejpam-1831	18	4	(	(	PUNCT
ejpam-1831	18	5	s−	s−	PROPN
ejpam-1831	18	6	r)2	r)2	NOUN
ejpam-1831	18	7	+	+	CCONJ
ejpam-1831	18	8	(	(	PUNCT
ejpam-1831	18	9	as2	as2	PROPN
ejpam-1831	18	10	+	+	PROPN
ejpam-1831	18	11	bs−	bs−	PROPN
ejpam-1831	18	12	ar2−	ar2−	VERB
ejpam-1831	18	13	br)2	br)2	ADP
ejpam-1831	18	14	factorize	factorize	NOUN
ejpam-1831	18	15	by	by	ADP
ejpam-1831	18	16	common	common	ADJ
ejpam-1831	18	17	factor	factor	NOUN
ejpam-1831	18	18	:	:	PUNCT
ejpam-1831	18	19	d	d	X
ejpam-1831	18	20	=	=	SYM
ejpam-1831	18	21	p	p	X
ejpam-1831	18	22	(	(	PUNCT
ejpam-1831	18	23	s−	s−	PROPN
ejpam-1831	18	24	r)2	r)2	NOUN
ejpam-1831	18	25	+	+	CCONJ
ejpam-1831	18	26	(	(	PUNCT
ejpam-1831	18	27	a(s2−	a(s2−	NOUN
ejpam-1831	18	28	r2	r2	NOUN
ejpam-1831	18	29	)	)	PUNCT
ejpam-1831	19	1	+	+	CCONJ
ejpam-1831	19	2	b(s−	b(s−	ADJ
ejpam-1831	19	3	r))2	r))2	NOUN
ejpam-1831	19	4	factorize	factorize	VERB
ejpam-1831	19	5	by	by	ADP
ejpam-1831	19	6	difference	difference	NOUN
ejpam-1831	19	7	of	of	ADP
ejpam-1831	19	8	squares	square	NOUN
ejpam-1831	19	9	:	:	PUNCT
ejpam-1831	20	1	d	d	X
ejpam-1831	20	2	=	=	SYM
ejpam-1831	20	3	p	p	X
ejpam-1831	20	4	(	(	PUNCT
ejpam-1831	20	5	s−	s−	PROPN
ejpam-1831	20	6	r)2	r)2	NOUN
ejpam-1831	20	7	+	+	CCONJ
ejpam-1831	20	8	(	(	PUNCT
ejpam-1831	20	9	a(s−	a(s−	X
ejpam-1831	20	10	r)(s+	r)(s+	NOUN
ejpam-1831	20	11	r	r	NOUN
ejpam-1831	20	12	)	)	PUNCT
ejpam-1831	20	13	+	+	CCONJ
ejpam-1831	20	14	b(s−	b(s−	ADJ
ejpam-1831	20	15	r))2	r))2	NOUN
ejpam-1831	20	16	again	again	ADV
ejpam-1831	20	17	,	,	PUNCT
ejpam-1831	20	18	factorize	factorize	VERB
ejpam-1831	20	19	by	by	ADP
ejpam-1831	20	20	common	common	ADJ
ejpam-1831	20	21	factor	factor	NOUN
ejpam-1831	20	22	:	:	PUNCT
ejpam-1831	20	23	d	d	X
ejpam-1831	20	24	=	=	SYM
ejpam-1831	20	25	p	p	X
ejpam-1831	20	26	(	(	PUNCT
ejpam-1831	20	27	s−	s−	PROPN
ejpam-1831	20	28	r)2	r)2	NOUN
ejpam-1831	20	29	+	+	CCONJ
ejpam-1831	20	30	(	(	PUNCT
ejpam-1831	20	31	(	(	PUNCT
ejpam-1831	20	32	s−	s−	NOUN
ejpam-1831	20	33	r)(a(s+	r)(a(s+	PROPN
ejpam-1831	20	34	r	r	NOUN
ejpam-1831	20	35	)	)	PUNCT
ejpam-1831	20	36	+	+	CCONJ
ejpam-1831	20	37	b))2	b))2	NOUN
ejpam-1831	20	38	(	(	PUNCT
ejpam-1831	20	39	1	1	NUM
ejpam-1831	20	40	)	)	PUNCT
ejpam-1831	20	41	now	now	ADV
ejpam-1831	20	42	define	define	VERB
ejpam-1831	21	1	d	d	NOUN
ejpam-1831	21	2	=	=	X
ejpam-1831	22	1	p	p	NOUN
ejpam-1831	22	2	q	q	NOUN
ejpam-1831	22	3	;	;	PUNCT
ejpam-1831	22	4	where	where	SCONJ
ejpam-1831	22	5	p	p	X
ejpam-1831	22	6	,	,	PUNCT
ejpam-1831	22	7	q	q	PROPN
ejpam-1831	22	8	∈	∈	PROPN
ejpam-1831	22	9	z	z	NOUN
ejpam-1831	22	10	and	and	CCONJ
ejpam-1831	22	11	q	q	NOUN
ejpam-1831	22	12	6=	6=	NUM
ejpam-1831	22	13	0	0	NUM
ejpam-1831	22	14	,	,	PUNCT
ejpam-1831	22	15	to	to	PART
ejpam-1831	22	16	find	find	VERB
ejpam-1831	22	17	all	all	DET
ejpam-1831	22	18	points	point	NOUN
ejpam-1831	22	19	where	where	SCONJ
ejpam-1831	22	20	the	the	DET
ejpam-1831	22	21	distance	distance	NOUN
ejpam-1831	22	22	between	between	ADP
ejpam-1831	22	23	them	they	PRON
ejpam-1831	22	24	is	be	AUX
ejpam-1831	22	25	a	a	DET
ejpam-1831	22	26	rational	rational	ADJ
ejpam-1831	22	27	number	number	NOUN
ejpam-1831	22	28	.	.	PUNCT
ejpam-1831	23	1	replace	replace	VERB
ejpam-1831	23	2	d	d	NOUN
ejpam-1831	23	3	in	in	ADP
ejpam-1831	23	4	the	the	DET
ejpam-1831	23	5	equation	equation	NOUN
ejpam-1831	23	6	by	by	ADP
ejpam-1831	23	7	(	(	PUNCT
ejpam-1831	23	8	1	1	NUM
ejpam-1831	23	9	)	)	PUNCT
ejpam-1831	23	10	.	.	PUNCT
ejpam-1831	24	1	p	p	X
ejpam-1831	24	2	q	q	NOUN
ejpam-1831	25	1	=	=	X
ejpam-1831	25	2	p	p	X
ejpam-1831	25	3	(	(	PUNCT
ejpam-1831	25	4	s−	s−	PROPN
ejpam-1831	25	5	r)2	r)2	NOUN
ejpam-1831	25	6	+	+	CCONJ
ejpam-1831	25	7	(	(	PUNCT
ejpam-1831	25	8	(	(	PUNCT
ejpam-1831	25	9	s−	s−	NOUN
ejpam-1831	25	10	r)(a(s+	r)(a(s+	PROPN
ejpam-1831	25	11	r	r	NOUN
ejpam-1831	25	12	)	)	PUNCT
ejpam-1831	25	13	+	+	CCONJ
ejpam-1831	25	14	b))2	b))2	NOUN
ejpam-1831	25	15	squaring	square	VERB
ejpam-1831	25	16	both	both	DET
ejpam-1831	25	17	sides	side	NOUN
ejpam-1831	25	18	:	:	PUNCT
ejpam-1831	25	19	p2	p2	PROPN
ejpam-1831	25	20	q2	q2	NOUN
ejpam-1831	25	21	=	=	SYM
ejpam-1831	25	22	(	(	PUNCT
ejpam-1831	25	23	s−	s−	PROPN
ejpam-1831	25	24	r)2	r)2	VERB
ejpam-1831	25	25	+	+	CCONJ
ejpam-1831	25	26	(	(	PUNCT
ejpam-1831	25	27	(	(	PUNCT
ejpam-1831	25	28	s−	s−	NOUN
ejpam-1831	25	29	r)(a(s+	r)(a(s+	PROPN
ejpam-1831	25	30	r	r	NOUN
ejpam-1831	25	31	)	)	PUNCT
ejpam-1831	26	1	+	+	CCONJ
ejpam-1831	26	2	b))2	b))2	NOUN
ejpam-1831	26	3	reorganizing	reorganize	VERB
ejpam-1831	26	4	the	the	DET
ejpam-1831	26	5	equation	equation	NOUN
ejpam-1831	26	6	:	:	PUNCT
ejpam-1831	26	7	�	�	PROPN
ejpam-1831	26	8	p	p	PROPN
ejpam-1831	26	9	q	q	PROPN
ejpam-1831	26	10	�	�	PROPN
ejpam-1831	26	11	2	2	NUM
ejpam-1831	26	12	=	=	SYM
ejpam-1831	26	13	(	(	PUNCT
ejpam-1831	26	14	s−	s−	PROPN
ejpam-1831	26	15	r)2	r)2	VERB
ejpam-1831	26	16	+	+	CCONJ
ejpam-1831	26	17	(	(	PUNCT
ejpam-1831	26	18	(	(	PUNCT
ejpam-1831	26	19	s−	s−	NOUN
ejpam-1831	26	20	r)(a(s+	r)(a(s+	PROPN
ejpam-1831	26	21	r	r	NOUN
ejpam-1831	26	22	)	)	PUNCT
ejpam-1831	26	23	+	+	CCONJ
ejpam-1831	26	24	b))2	b))2	NOUN
ejpam-1831	26	25	(	(	PUNCT
ejpam-1831	26	26	2	2	NUM
ejpam-1831	26	27	)	)	PUNCT
ejpam-1831	26	28	without	without	ADP
ejpam-1831	26	29	loss	loss	NOUN
ejpam-1831	26	30	of	of	ADP
ejpam-1831	26	31	generality	generality	NOUN
ejpam-1831	26	32	,	,	PUNCT
ejpam-1831	26	33	we	we	PRON
ejpam-1831	26	34	will	will	AUX
ejpam-1831	26	35	use	use	VERB
ejpam-1831	26	36	the	the	DET
ejpam-1831	26	37	equation	equation	NOUN
ejpam-1831	26	38	[	[	X
ejpam-1831	26	39	5	5	NUM
ejpam-1831	26	40	]	]	PUNCT
ejpam-1831	26	41	(	(	PUNCT
ejpam-1831	26	42	5n)2	5n)2	NUM
ejpam-1831	26	43	=	=	SYM
ejpam-1831	26	44	(	(	PUNCT
ejpam-1831	26	45	−3n)2	−3n)2	X
ejpam-1831	26	46	+	+	CCONJ
ejpam-1831	26	47	(	(	PUNCT
ejpam-1831	26	48	4n)2	4n)2	NUM
ejpam-1831	26	49	(	(	PUNCT
ejpam-1831	26	50	3	3	NUM
ejpam-1831	26	51	)	)	PUNCT
ejpam-1831	27	1	where	where	SCONJ
ejpam-1831	27	2	n	n	X
ejpam-1831	27	3	∈	∈	PROPN
ejpam-1831	27	4	q	q	NOUN
ejpam-1831	27	5	,	,	PUNCT
ejpam-1831	27	6	to	to	PART
ejpam-1831	27	7	represent	represent	VERB
ejpam-1831	27	8	that	that	SCONJ
ejpam-1831	27	9	this	this	DET
ejpam-1831	27	10	family	family	NOUN
ejpam-1831	27	11	of	of	ADP
ejpam-1831	27	12	pairs	pair	NOUN
ejpam-1831	27	13	of	of	ADP
ejpam-1831	27	14	points	point	NOUN
ejpam-1831	27	15	is	be	AUX
ejpam-1831	27	16	infinite	infinite	ADJ
ejpam-1831	27	17	even	even	ADV
ejpam-1831	27	18	if	if	SCONJ
ejpam-1831	27	19	it	it	PRON
ejpam-1831	27	20	’s	’	VERB
ejpam-1831	27	21	a	a	DET
ejpam-1831	27	22	subset	subset	NOUN
ejpam-1831	27	23	of	of	ADP
ejpam-1831	27	24	all	all	DET
ejpam-1831	27	25	points	point	NOUN
ejpam-1831	27	26	that	that	PRON
ejpam-1831	27	27	satisfies	satisfy	VERB
ejpam-1831	27	28	the	the	DET
ejpam-1831	27	29	condition	condition	NOUN
ejpam-1831	27	30	to	to	PART
ejpam-1831	27	31	have	have	VERB
ejpam-1831	27	32	a	a	DET
ejpam-1831	27	33	rational	rational	ADJ
ejpam-1831	27	34	distance	distance	NOUN
ejpam-1831	27	35	between	between	ADP
ejpam-1831	27	36	them	they	PRON
ejpam-1831	27	37	inside	inside	ADP
ejpam-1831	27	38	the	the	DET
ejpam-1831	27	39	quadratic	quadratic	ADJ
ejpam-1831	27	40	equation	equation	NOUN
ejpam-1831	27	41	.	.	PUNCT
ejpam-1831	28	1	matching	match	VERB
ejpam-1831	28	2	the	the	DET
ejpam-1831	28	3	equations	equation	NOUN
ejpam-1831	28	4	(	(	PUNCT
ejpam-1831	28	5	2	2	NUM
ejpam-1831	28	6	)	)	PUNCT
ejpam-1831	28	7	and	and	CCONJ
ejpam-1831	28	8	(	(	PUNCT
ejpam-1831	28	9	3	3	X
ejpam-1831	28	10	)	)	PUNCT
ejpam-1831	28	11	p	p	NOUN
ejpam-1831	28	12	q	q	NOUN
ejpam-1831	29	1	=	=	SYM
ejpam-1831	29	2	5n	5n	NOUN
ejpam-1831	29	3	(	(	PUNCT
ejpam-1831	29	4	4	4	NUM
ejpam-1831	29	5	)	)	PUNCT
ejpam-1831	29	6	j.	j.	PROPN
ejpam-1831	29	7	diaz	diaz	PROPN
ejpam-1831	29	8	/	/	SYM
ejpam-1831	29	9	eur	eur	PROPN
ejpam-1831	29	10	.	.	PUNCT
ejpam-1831	30	1	j.	j.	PROPN
ejpam-1831	30	2	pure	pure	PROPN
ejpam-1831	30	3	appl	appl	PROPN
ejpam-1831	30	4	.	.	PROPN
ejpam-1831	30	5	math	math	PROPN
ejpam-1831	30	6	,	,	PUNCT
ejpam-1831	30	7	6	6	NUM
ejpam-1831	30	8	(	(	PUNCT
ejpam-1831	30	9	2013	2013	NUM
ejpam-1831	30	10	)	)	PUNCT
ejpam-1831	30	11	,	,	PUNCT
ejpam-1831	30	12	400	400	NUM
ejpam-1831	30	13	-	-	SYM
ejpam-1831	30	14	404	404	NUM
ejpam-1831	30	15	402	402	NUM
ejpam-1831	30	16	s−	s−	NOUN
ejpam-1831	30	17	r	r	NOUN
ejpam-1831	30	18	=	=	SYM
ejpam-1831	30	19	−3n	−3n	X
ejpam-1831	30	20	(	(	PUNCT
ejpam-1831	30	21	5	5	NUM
ejpam-1831	30	22	)	)	PUNCT
ejpam-1831	30	23	(	(	PUNCT
ejpam-1831	30	24	s−	s−	NOUN
ejpam-1831	30	25	r)(a(s+	r)(a(s+	PROPN
ejpam-1831	30	26	r	r	NOUN
ejpam-1831	30	27	)	)	PUNCT
ejpam-1831	30	28	+	+	NOUN
ejpam-1831	30	29	b	b	X
ejpam-1831	30	30	)	)	PUNCT
ejpam-1831	30	31	=	=	NOUN
ejpam-1831	31	1	4n	4n	NOUN
ejpam-1831	31	2	(	(	PUNCT
ejpam-1831	31	3	6	6	NUM
ejpam-1831	31	4	)	)	PUNCT
ejpam-1831	31	5	replace	replace	NOUN
ejpam-1831	31	6	(	(	PUNCT
ejpam-1831	31	7	5	5	NUM
ejpam-1831	31	8	)	)	PUNCT
ejpam-1831	31	9	in	in	ADP
ejpam-1831	31	10	(	(	PUNCT
ejpam-1831	31	11	6	6	NUM
ejpam-1831	31	12	):	):	PUNCT
ejpam-1831	31	13	−3n(a(s+	−3n(a(s+	NOUN
ejpam-1831	31	14	r	r	NOUN
ejpam-1831	31	15	)	)	PUNCT
ejpam-1831	31	16	+	+	NOUN
ejpam-1831	31	17	b	b	X
ejpam-1831	31	18	)	)	PUNCT
ejpam-1831	31	19	=	=	NOUN
ejpam-1831	31	20	4n	4n	NOUN
ejpam-1831	31	21	divide	divide	NOUN
ejpam-1831	31	22	by	by	ADP
ejpam-1831	31	23	3n	3n	NOUN
ejpam-1831	31	24	in	in	ADP
ejpam-1831	31	25	both	both	DET
ejpam-1831	31	26	sides	side	NOUN
ejpam-1831	31	27	:	:	PUNCT
ejpam-1831	31	28	a(s+	a(s+	PROPN
ejpam-1831	31	29	r	r	NOUN
ejpam-1831	31	30	)	)	PUNCT
ejpam-1831	32	1	+	+	NOUN
ejpam-1831	32	2	b	b	X
ejpam-1831	32	3	=	=	NOUN
ejpam-1831	32	4	−	−	PROPN
ejpam-1831	32	5	4	4	NUM
ejpam-1831	32	6	3	3	NUM
ejpam-1831	32	7	deduct	deduct	NOUN
ejpam-1831	32	8	b	b	NOUN
ejpam-1831	32	9	in	in	ADP
ejpam-1831	32	10	both	both	DET
ejpam-1831	32	11	sides	side	NOUN
ejpam-1831	32	12	:	:	PUNCT
ejpam-1831	32	13	a(s+	a(s+	PROPN
ejpam-1831	32	14	r	r	NOUN
ejpam-1831	32	15	)	)	PUNCT
ejpam-1831	32	16	=	=	NOUN
ejpam-1831	32	17	−	−	PROPN
ejpam-1831	32	18	4	4	NUM
ejpam-1831	32	19	3	3	NUM
ejpam-1831	32	20	−	−	PROPN
ejpam-1831	32	21	b	b	NOUN
ejpam-1831	32	22	divide	divide	NOUN
ejpam-1831	32	23	by	by	ADP
ejpam-1831	32	24	a	a	PRON
ejpam-1831	32	25	in	in	ADP
ejpam-1831	32	26	both	both	DET
ejpam-1831	32	27	sides	side	NOUN
ejpam-1831	32	28	:	:	PUNCT
ejpam-1831	32	29	s+	s+	PUNCT
ejpam-1831	32	30	r	r	NOUN
ejpam-1831	32	31	=	=	NOUN
ejpam-1831	32	32	−	−	PROPN
ejpam-1831	32	33	4	4	NUM
ejpam-1831	32	34	3	3	NUM
ejpam-1831	32	35	+	+	SYM
ejpam-1831	32	36	b	b	NOUN
ejpam-1831	33	1	a	a	DET
ejpam-1831	33	2	reorganizing	reorganize	VERB
ejpam-1831	33	3	the	the	DET
ejpam-1831	33	4	equation	equation	NOUN
ejpam-1831	33	5	:	:	PUNCT
ejpam-1831	33	6	s+	s+	PUNCT
ejpam-1831	33	7	r	r	NOUN
ejpam-1831	33	8	=	=	NOUN
ejpam-1831	33	9	−	−	PROPN
ejpam-1831	33	10	4	4	NUM
ejpam-1831	33	11	3	3	NUM
ejpam-1831	33	12	+	+	SYM
ejpam-1831	33	13	3b	3b	NUM
ejpam-1831	33	14	3	3	NUM
ejpam-1831	33	15	a	a	DET
ejpam-1831	33	16	=	=	NOUN
ejpam-1831	33	17	−	−	ADP
ejpam-1831	33	18	4−3b	4−3b	NUM
ejpam-1831	33	19	3	3	NUM
ejpam-1831	33	20	a	a	DET
ejpam-1831	33	21	=	=	NOUN
ejpam-1831	33	22	−	−	NOUN
ejpam-1831	33	23	4−	4−	NUM
ejpam-1831	33	24	3b	3b	NOUN
ejpam-1831	33	25	3a	3a	NUM
ejpam-1831	33	26	(	(	PUNCT
ejpam-1831	33	27	7	7	X
ejpam-1831	33	28	)	)	PUNCT
ejpam-1831	33	29	define	define	VERB
ejpam-1831	33	30	j	j	PROPN
ejpam-1831	34	1	=	=	NOUN
ejpam-1831	34	2	−4−3b	−4−3b	PROPN
ejpam-1831	34	3	3a	3a	NUM
ejpam-1831	34	4	;	;	PUNCT
ejpam-1831	34	5	where	where	SCONJ
ejpam-1831	34	6	j	j	PROPN
ejpam-1831	34	7	∈	∈	PROPN
ejpam-1831	34	8	ℜ	ℜ	PROPN
ejpam-1831	34	9	and	and	CCONJ
ejpam-1831	34	10	replace	replace	VERB
ejpam-1831	34	11	in	in	ADP
ejpam-1831	34	12	the	the	DET
ejpam-1831	34	13	equation	equation	NOUN
ejpam-1831	34	14	(	(	PUNCT
ejpam-1831	34	15	7	7	NUM
ejpam-1831	34	16	)	)	PUNCT
ejpam-1831	34	17	.	.	PUNCT
ejpam-1831	35	1	s+	s+	ADV
ejpam-1831	35	2	r	r	NOUN
ejpam-1831	35	3	=	=	SYM
ejpam-1831	35	4	j	j	PROPN
ejpam-1831	35	5	(	(	PUNCT
ejpam-1831	35	6	8)	8)	NUM
ejpam-1831	35	7	do	do	NOUN
ejpam-1831	35	8	(	(	PUNCT
ejpam-1831	35	9	5)+(8	5)+(8	NUM
ejpam-1831	35	10	)	)	PUNCT
ejpam-1831	35	11	2s	2s	NOUN
ejpam-1831	35	12	=	=	SYM
ejpam-1831	35	13	j−	j−	PROPN
ejpam-1831	35	14	3n	3n	NUM
ejpam-1831	35	15	divide	divide	NOUN
ejpam-1831	35	16	by	by	ADP
ejpam-1831	35	17	2	2	NUM
ejpam-1831	35	18	in	in	ADP
ejpam-1831	35	19	both	both	DET
ejpam-1831	35	20	sides	side	NOUN
ejpam-1831	35	21	:	:	PUNCT
ejpam-1831	35	22	s	s	AUX
ejpam-1831	35	23	=	=	PUNCT
ejpam-1831	35	24	j−	j−	VERB
ejpam-1831	35	25	3n	3n	NUM
ejpam-1831	35	26	2	2	NUM
ejpam-1831	35	27	(	(	PUNCT
ejpam-1831	35	28	9	9	NUM
ejpam-1831	35	29	)	)	PUNCT
ejpam-1831	35	30	now	now	ADV
ejpam-1831	35	31	,	,	PUNCT
ejpam-1831	35	32	do	do	AUX
ejpam-1831	35	33	(	(	PUNCT
ejpam-1831	35	34	8)-(5	8)-(5	NUM
ejpam-1831	35	35	)	)	PUNCT
ejpam-1831	35	36	2r	2r	NUM
ejpam-1831	35	37	=	=	SYM
ejpam-1831	35	38	j+	j+	NUM
ejpam-1831	35	39	3n	3n	NUM
ejpam-1831	35	40	divide	divide	NOUN
ejpam-1831	35	41	by	by	ADP
ejpam-1831	35	42	2	2	NUM
ejpam-1831	35	43	in	in	ADP
ejpam-1831	35	44	both	both	DET
ejpam-1831	35	45	sides	side	NOUN
ejpam-1831	35	46	:	:	PUNCT
ejpam-1831	35	47	r	r	NOUN
ejpam-1831	35	48	=	=	PUNCT
ejpam-1831	35	49	j+	j+	NUM
ejpam-1831	35	50	3n	3n	NUM
ejpam-1831	35	51	2	2	NUM
ejpam-1831	35	52	(	(	PUNCT
ejpam-1831	35	53	10	10	NUM
ejpam-1831	35	54	)	)	PUNCT
ejpam-1831	35	55	the	the	DET
ejpam-1831	35	56	equations	equation	NOUN
ejpam-1831	35	57	(	(	PUNCT
ejpam-1831	35	58	9	9	NUM
ejpam-1831	35	59	)	)	PUNCT
ejpam-1831	35	60	and	and	CCONJ
ejpam-1831	35	61	(	(	PUNCT
ejpam-1831	35	62	10	10	NUM
ejpam-1831	35	63	)	)	PUNCT
ejpam-1831	35	64	present	present	VERB
ejpam-1831	35	65	some	some	DET
ejpam-1831	35	66	restriction	restriction	NOUN
ejpam-1831	35	67	:	:	PUNCT
ejpam-1831	35	68	j	j	PROPN
ejpam-1831	35	69	>	>	X
ejpam-1831	35	70	0	0	PUNCT
ejpam-1831	36	1	(	(	PUNCT
ejpam-1831	36	2	11	11	NUM
ejpam-1831	36	3	)	)	PUNCT
ejpam-1831	36	4	r	r	NOUN
ejpam-1831	36	5	>	>	X
ejpam-1831	36	6	0	0	PUNCT
ejpam-1831	37	1	(	(	PUNCT
ejpam-1831	37	2	12	12	NUM
ejpam-1831	37	3	)	)	PUNCT
ejpam-1831	37	4	j.	j.	PROPN
ejpam-1831	37	5	diaz	diaz	PROPN
ejpam-1831	37	6	/	/	SYM
ejpam-1831	37	7	eur	eur	PROPN
ejpam-1831	37	8	.	.	PUNCT
ejpam-1831	38	1	j.	j.	PROPN
ejpam-1831	38	2	pure	pure	PROPN
ejpam-1831	38	3	appl	appl	PROPN
ejpam-1831	38	4	.	.	PROPN
ejpam-1831	38	5	math	math	PROPN
ejpam-1831	38	6	,	,	PUNCT
ejpam-1831	38	7	6	6	NUM
ejpam-1831	38	8	(	(	PUNCT
ejpam-1831	38	9	2013	2013	NUM
ejpam-1831	38	10	)	)	PUNCT
ejpam-1831	38	11	,	,	PUNCT
ejpam-1831	38	12	400	400	NUM
ejpam-1831	38	13	-	-	SYM
ejpam-1831	38	14	404	404	NUM
ejpam-1831	38	15	403	403	NUM
ejpam-1831	38	16	s	s	NOUN
ejpam-1831	38	17	>	>	X
ejpam-1831	38	18	0	0	PUNCT
ejpam-1831	39	1	(	(	PUNCT
ejpam-1831	39	2	13	13	NUM
ejpam-1831	39	3	)	)	PUNCT
ejpam-1831	39	4	replace	replace	NOUN
ejpam-1831	39	5	(	(	PUNCT
ejpam-1831	39	6	10	10	NUM
ejpam-1831	39	7	)	)	PUNCT
ejpam-1831	39	8	in	in	ADP
ejpam-1831	39	9	(	(	PUNCT
ejpam-1831	39	10	12	12	NUM
ejpam-1831	39	11	)	)	PUNCT
ejpam-1831	39	12	j+	j+	NUM
ejpam-1831	39	13	3n	3n	NUM
ejpam-1831	39	14	2	2	NUM
ejpam-1831	39	15	>	>	SYM
ejpam-1831	39	16	0	0	NUM
ejpam-1831	39	17	multiply	multiply	NOUN
ejpam-1831	39	18	2	2	NUM
ejpam-1831	39	19	in	in	ADP
ejpam-1831	39	20	both	both	DET
ejpam-1831	39	21	sides	side	NOUN
ejpam-1831	39	22	:	:	PUNCT
ejpam-1831	39	23	j+	j+	NUM
ejpam-1831	39	24	3n	3n	NUM
ejpam-1831	39	25	>	>	SYM
ejpam-1831	39	26	0	0	NUM
ejpam-1831	39	27	deduct	deduct	NOUN
ejpam-1831	39	28	j	j	PROPN
ejpam-1831	39	29	in	in	ADP
ejpam-1831	39	30	both	both	DET
ejpam-1831	39	31	sides	side	NOUN
ejpam-1831	39	32	:	:	PUNCT
ejpam-1831	39	33	3n>−	3n>−	NUM
ejpam-1831	39	34	j	j	PROPN
ejpam-1831	39	35	divide	divide	NOUN
ejpam-1831	39	36	by	by	ADP
ejpam-1831	39	37	3	3	NUM
ejpam-1831	39	38	in	in	ADP
ejpam-1831	39	39	both	both	DET
ejpam-1831	39	40	sides	side	NOUN
ejpam-1831	39	41	n	n	CCONJ
ejpam-1831	39	42	>	>	X
ejpam-1831	39	43	−	−	PROPN
ejpam-1831	39	44	j	j	PROPN
ejpam-1831	39	45	3	3	NUM
ejpam-1831	39	46	(	(	PUNCT
ejpam-1831	39	47	14	14	NUM
ejpam-1831	39	48	)	)	PUNCT
ejpam-1831	39	49	now	now	ADV
ejpam-1831	39	50	,	,	PUNCT
ejpam-1831	39	51	replace	replace	VERB
ejpam-1831	39	52	(	(	PUNCT
ejpam-1831	39	53	9	9	NUM
ejpam-1831	39	54	)	)	PUNCT
ejpam-1831	39	55	in	in	ADP
ejpam-1831	39	56	(	(	PUNCT
ejpam-1831	39	57	13	13	NUM
ejpam-1831	39	58	)	)	PUNCT
ejpam-1831	39	59	j−	j−	VERB
ejpam-1831	39	60	3n	3n	NUM
ejpam-1831	39	61	2	2	NUM
ejpam-1831	39	62	>	>	SYM
ejpam-1831	39	63	0	0	NUM
ejpam-1831	39	64	multiply	multiply	NOUN
ejpam-1831	39	65	2	2	NUM
ejpam-1831	39	66	in	in	ADP
ejpam-1831	39	67	both	both	DET
ejpam-1831	39	68	sides	side	NOUN
ejpam-1831	39	69	:	:	PUNCT
ejpam-1831	39	70	j−	j−	VERB
ejpam-1831	39	71	3n	3n	NOUN
ejpam-1831	39	72	>	>	SYM
ejpam-1831	39	73	0	0	PUNCT
ejpam-1831	39	74	add	add	VERB
ejpam-1831	39	75	3n	3n	NOUN
ejpam-1831	39	76	in	in	ADP
ejpam-1831	39	77	both	both	DET
ejpam-1831	39	78	sides	side	NOUN
ejpam-1831	39	79	:	:	PUNCT
ejpam-1831	39	80	j	j	PROPN
ejpam-1831	39	81	>	>	X
ejpam-1831	39	82	3n	3n	NUM
ejpam-1831	39	83	divide	divide	NOUN
ejpam-1831	39	84	by	by	ADP
ejpam-1831	39	85	3	3	NUM
ejpam-1831	39	86	in	in	ADP
ejpam-1831	39	87	both	both	DET
ejpam-1831	39	88	sides	side	NOUN
ejpam-1831	39	89	:	:	PUNCT
ejpam-1831	39	90	j	j	PROPN
ejpam-1831	39	91	3	3	NUM
ejpam-1831	39	92	>	>	SYM
ejpam-1831	39	93	n	n	PROPN
ejpam-1831	39	94	(	(	PUNCT
ejpam-1831	39	95	15	15	NUM
ejpam-1831	39	96	)	)	PUNCT
ejpam-1831	39	97	from	from	ADP
ejpam-1831	39	98	(	(	PUNCT
ejpam-1831	39	99	14	14	NUM
ejpam-1831	39	100	)	)	PUNCT
ejpam-1831	39	101	and	and	CCONJ
ejpam-1831	39	102	(	(	PUNCT
ejpam-1831	39	103	15	15	NUM
ejpam-1831	39	104	)	)	PUNCT
ejpam-1831	39	105	−	−	PROPN
ejpam-1831	40	1	j	j	NOUN
ejpam-1831	40	2	3	3	NUM
ejpam-1831	40	3	<	<	X
ejpam-1831	40	4	n	n	CCONJ
ejpam-1831	40	5	<	<	X
ejpam-1831	40	6	j	j	PROPN
ejpam-1831	40	7	3	3	NUM
ejpam-1831	40	8	(	(	PUNCT
ejpam-1831	40	9	16	16	NUM
ejpam-1831	40	10	)	)	PUNCT
ejpam-1831	40	11	lemma	lemma	PROPN
ejpam-1831	40	12	1	1	NUM
ejpam-1831	40	13	.	.	PUNCT
ejpam-1831	41	1	the	the	DET
ejpam-1831	41	2	set	set	NOUN
ejpam-1831	41	3	q	q	NOUN
ejpam-1831	41	4	∩	∩	ADJ
ejpam-1831	41	5	�	�	PROPN
ejpam-1831	41	6	−	−	PROPN
ejpam-1831	41	7	j	j	PROPN
ejpam-1831	41	8	3	3	NUM
ejpam-1831	41	9	,	,	PUNCT
ejpam-1831	41	10	j	j	PROPN
ejpam-1831	41	11	3	3	NUM
ejpam-1831	41	12	�	�	PROPN
ejpam-1831	41	13	is	be	AUX
ejpam-1831	41	14	countably	countably	ADV
ejpam-1831	41	15	infinite	infinite	ADJ
ejpam-1831	41	16	.	.	PUNCT
ejpam-1831	42	1	proof	proof	NOUN
ejpam-1831	42	2	.	.	PUNCT
ejpam-1831	43	1	let	let	VERB
ejpam-1831	43	2	s	s	PRON
ejpam-1831	43	3	∈	∈	PROPN
ejpam-1831	43	4	q	q	NOUN
ejpam-1831	43	5	∩	∩	ADJ
ejpam-1831	43	6	�	�	PROPN
ejpam-1831	43	7	−	−	PROPN
ejpam-1831	43	8	j	j	PROPN
ejpam-1831	43	9	3	3	NUM
ejpam-1831	43	10	,	,	PUNCT
ejpam-1831	43	11	j	j	PROPN
ejpam-1831	43	12	3	3	NUM
ejpam-1831	43	13	�	�	PROPN
ejpam-1831	43	14	,	,	PUNCT
ejpam-1831	43	15	then	then	ADV
ejpam-1831	43	16	each	each	PRON
ejpam-1831	43	17	s	s	VERB
ejpam-1831	43	18	will	will	AUX
ejpam-1831	43	19	be	be	AUX
ejpam-1831	43	20	written	write	VERB
ejpam-1831	43	21	in	in	ADP
ejpam-1831	43	22	the	the	DET
ejpam-1831	43	23	(	(	PUNCT
ejpam-1831	43	24	unique	unique	ADJ
ejpam-1831	43	25	)	)	PUNCT
ejpam-1831	43	26	form	form	NOUN
ejpam-1831	43	27	p	p	NOUN
ejpam-1831	43	28	q	q	NOUN
ejpam-1831	43	29	,	,	PUNCT
ejpam-1831	43	30	where	where	SCONJ
ejpam-1831	43	31	p	p	X
ejpam-1831	43	32	,	,	PUNCT
ejpam-1831	43	33	q	q	PROPN
ejpam-1831	43	34	∈	∈	PROPN
ejpam-1831	43	35	z+	z+	NUM
ejpam-1831	43	36	and	and	CCONJ
ejpam-1831	43	37	have	have	VERB
ejpam-1831	43	38	no	no	DET
ejpam-1831	43	39	common	common	ADJ
ejpam-1831	43	40	divisor	divisor	NOUN
ejpam-1831	43	41	other	other	ADJ
ejpam-1831	43	42	than	than	ADP
ejpam-1831	43	43	1	1	NUM
ejpam-1831	43	44	[	[	SYM
ejpam-1831	43	45	8	8	NUM
ejpam-1831	43	46	,	,	PUNCT
ejpam-1831	43	47	11	11	NUM
ejpam-1831	43	48	]	]	PUNCT
ejpam-1831	43	49	.	.	PUNCT
ejpam-1831	44	1	now	now	ADV
ejpam-1831	44	2	,	,	PUNCT
ejpam-1831	44	3	define	define	VERB
ejpam-1831	44	4	f	f	X
ejpam-1831	44	5	:	:	PUNCT
ejpam-1831	44	6	q	q	X
ejpam-1831	44	7	∩	∩	X
ejpam-1831	44	8	�	�	PROPN
ejpam-1831	44	9	−	−	PROPN
ejpam-1831	44	10	j	j	PROPN
ejpam-1831	44	11	3	3	NUM
ejpam-1831	44	12	,	,	PUNCT
ejpam-1831	44	13	j	j	PROPN
ejpam-1831	44	14	3	3	NUM
ejpam-1831	44	15	�	�	PROPN
ejpam-1831	44	16	→	→	SYM
ejpam-1831	44	17	z+	z+	NUM
ejpam-1831	44	18	×	×	NOUN
ejpam-1831	44	19	z+	z+	NUM
ejpam-1831	44	20	by	by	ADP
ejpam-1831	44	21	f	f	PROPN
ejpam-1831	44	22	�	�	PROPN
ejpam-1831	45	1	p	p	PROPN
ejpam-1831	45	2	q	q	X
ejpam-1831	45	3	�	�	PROPN
ejpam-1831	45	4	=	=	SYM
ejpam-1831	45	5	(	(	PUNCT
ejpam-1831	45	6	p	p	X
ejpam-1831	45	7	,	,	PUNCT
ejpam-1831	45	8	q	q	NOUN
ejpam-1831	45	9	)	)	PUNCT
ejpam-1831	45	10	,	,	PUNCT
ejpam-1831	45	11	and	and	CCONJ
ejpam-1831	45	12	let	let	VERB
ejpam-1831	45	13	k	k	NOUN
ejpam-1831	45	14	=	=	PUNCT
ejpam-1831	45	15	range	range	NOUN
ejpam-1831	45	16	f	f	X
ejpam-1831	45	17	.	.	PUNCT
ejpam-1831	46	1	for	for	ADP
ejpam-1831	46	2	p	p	PROPN
ejpam-1831	46	3	q	q	PROPN
ejpam-1831	46	4	,	,	PUNCT
ejpam-1831	46	5	u	u	NOUN
ejpam-1831	46	6	v	v	ADP
ejpam-1831	46	7	∈	∈	PROPN
ejpam-1831	46	8	q	q	NOUN
ejpam-1831	46	9	∩	∩	ADJ
ejpam-1831	46	10	�	�	PROPN
ejpam-1831	46	11	−	−	PROPN
ejpam-1831	46	12	j	j	PROPN
ejpam-1831	46	13	3	3	NUM
ejpam-1831	46	14	,	,	PUNCT
ejpam-1831	46	15	j	j	PROPN
ejpam-1831	46	16	3	3	NUM
ejpam-1831	46	17	�	�	PROPN
ejpam-1831	46	18	,	,	PUNCT
ejpam-1831	46	19	we	we	PRON
ejpam-1831	46	20	find	find	VERB
ejpam-1831	46	21	that	that	SCONJ
ejpam-1831	46	22	f	f	PROPN
ejpam-1831	46	23	�	�	PROPN
ejpam-1831	46	24	p	p	X
ejpam-1831	46	25	q	q	PROPN
ejpam-1831	46	26	�	�	PROPN
ejpam-1831	46	27	=	=	SYM
ejpam-1831	46	28	f	f	PROPN
ejpam-1831	46	29	�	�	PROPN
ejpam-1831	46	30	u	u	PROPN
ejpam-1831	46	31	v	v	PROPN
ejpam-1831	46	32	�	�	PROPN
ejpam-1831	46	33	⇒	⇒	NOUN
ejpam-1831	46	34	(	(	PUNCT
ejpam-1831	46	35	p	p	X
ejpam-1831	46	36	,	,	PUNCT
ejpam-1831	46	37	q	q	NOUN
ejpam-1831	46	38	)	)	PUNCT
ejpam-1831	47	1	=	=	SYM
ejpam-1831	47	2	(	(	PUNCT
ejpam-1831	47	3	u	u	NOUN
ejpam-1831	47	4	,	,	PUNCT
ejpam-1831	47	5	v	v	NOUN
ejpam-1831	47	6	)	)	PUNCT
ejpam-1831	47	7	⇒	⇒	NOUN
ejpam-1831	47	8	p	p	X
ejpam-1831	47	9	=	=	PUNCT
ejpam-1831	47	10	u	u	NOUN
ejpam-1831	47	11	and	and	CCONJ
ejpam-1831	47	12	q	q	NOUN
ejpam-1831	47	13	=	=	SYM
ejpam-1831	47	14	v	v	ADJ
ejpam-1831	47	15	⇒	⇒	X
ejpam-1831	47	16	�	�	PROPN
ejpam-1831	48	1	p	p	X
ejpam-1831	48	2	q	q	PROPN
ejpam-1831	48	3	�	�	PROPN
ejpam-1831	48	4	=	=	SYM
ejpam-1831	48	5	�	�	PROPN
ejpam-1831	48	6	u	u	PROPN
ejpam-1831	48	7	v	v	PROPN
ejpam-1831	48	8	�	�	PROPN
ejpam-1831	48	9	,	,	PUNCT
ejpam-1831	48	10	so	so	CCONJ
ejpam-1831	48	11	f	f	PROPN
ejpam-1831	48	12	is	be	AUX
ejpam-1831	48	13	a	a	DET
ejpam-1831	48	14	one	one	NUM
ejpam-1831	48	15	-	-	PUNCT
ejpam-1831	48	16	to	to	ADP
ejpam-1831	48	17	-	-	PUNCT
ejpam-1831	48	18	one	one	NUM
ejpam-1831	48	19	function	function	NOUN
ejpam-1831	48	20	.	.	PUNCT
ejpam-1831	49	1	therefore	therefore	ADV
ejpam-1831	49	2	|q∩	|q∩	PROPN
ejpam-1831	49	3	�	�	PROPN
ejpam-1831	50	1	−	−	PROPN
ejpam-1831	50	2	j	j	PROPN
ejpam-1831	50	3	3	3	NUM
ejpam-1831	50	4	,	,	PUNCT
ejpam-1831	50	5	j	j	PROPN
ejpam-1831	50	6	3	3	NUM
ejpam-1831	50	7	�	�	PROPN
ejpam-1831	50	8	|=	|=	PUNCT
ejpam-1831	50	9	|k	|k	NOUN
ejpam-1831	50	10	|	|	ADV
ejpam-1831	50	11	,	,	PUNCT
ejpam-1831	50	12	a	a	DET
ejpam-1831	50	13	subset	subset	NOUN
ejpam-1831	50	14	of	of	ADP
ejpam-1831	50	15	the	the	DET
ejpam-1831	50	16	countable	countable	ADJ
ejpam-1831	50	17	set	set	VERB
ejpam-1831	50	18	z+×	z+×	PROPN
ejpam-1831	50	19	z+	z+	NUM
ejpam-1831	50	20	(	(	PUNCT
ejpam-1831	50	21	by	by	ADP
ejpam-1831	50	22	theorem	theorem	ADJ
ejpam-1831	50	23	a3.5	a3.5	PROPN
ejpam-1831	50	24	in	in	ADP
ejpam-1831	50	25	[	[	X
ejpam-1831	50	26	4	4	X
ejpam-1831	50	27	]	]	PUNCT
ejpam-1831	50	28	we	we	PRON
ejpam-1831	50	29	know	know	VERB
ejpam-1831	50	30	that	that	SCONJ
ejpam-1831	50	31	z+	z+	NUM
ejpam-1831	50	32	×	×	NOUN
ejpam-1831	50	33	z+	z+	NUM
ejpam-1831	50	34	is	be	AUX
ejpam-1831	50	35	countable	countable	ADJ
ejpam-1831	50	36	)	)	PUNCT
ejpam-1831	50	37	.	.	PUNCT
ejpam-1831	51	1	from	from	ADP
ejpam-1831	51	2	[	[	X
ejpam-1831	51	3	4	4	NUM
ejpam-1831	51	4	,	,	PUNCT
ejpam-1831	51	5	theorem	theorem	VERB
ejpam-1831	51	6	a3.5	a3.5	PROPN
ejpam-1831	51	7	]	]	X
ejpam-1831	51	8	it	it	PRON
ejpam-1831	51	9	now	now	ADV
ejpam-1831	51	10	follows	follow	VERB
ejpam-1831	51	11	that	that	SCONJ
ejpam-1831	51	12	the	the	DET
ejpam-1831	51	13	set	set	NOUN
ejpam-1831	51	14	q	q	NOUN
ejpam-1831	51	15	∩	∩	ADJ
ejpam-1831	51	16	�	�	PROPN
ejpam-1831	51	17	−	−	PROPN
ejpam-1831	51	18	j	j	PROPN
ejpam-1831	51	19	3	3	NUM
ejpam-1831	51	20	,	,	PUNCT
ejpam-1831	51	21	j	j	PROPN
ejpam-1831	51	22	3	3	NUM
ejpam-1831	51	23	�	�	PROPN
ejpam-1831	51	24	is	be	AUX
ejpam-1831	51	25	countable	countable	ADJ
ejpam-1831	51	26	.	.	PUNCT
ejpam-1831	52	1	define	define	VERB
ejpam-1831	52	2	g	g	NOUN
ejpam-1831	52	3	:	:	PUNCT
ejpam-1831	52	4	z+→	z+→	PROPN
ejpam-1831	52	5	q	q	NOUN
ejpam-1831	52	6	∩	∩	X
ejpam-1831	52	7	�	�	PROPN
ejpam-1831	53	1	−	−	PROPN
ejpam-1831	53	2	j	j	PROPN
ejpam-1831	53	3	3	3	NUM
ejpam-1831	53	4	,	,	PUNCT
ejpam-1831	53	5	j	j	PROPN
ejpam-1831	53	6	3	3	NUM
ejpam-1831	53	7	�	�	PROPN
ejpam-1831	53	8	by	by	ADP
ejpam-1831	53	9	g(x	g(x	NOUN
ejpam-1831	53	10	)	)	PUNCT
ejpam-1831	54	1	=	=	SYM
ejpam-1831	55	1	−	−	PROPN
ejpam-1831	55	2	j(x−1	j(x−1	NOUN
ejpam-1831	55	3	)	)	PUNCT
ejpam-1831	55	4	3(x+1	3(x+1	NUM
ejpam-1831	55	5	)	)	PUNCT
ejpam-1831	55	6	;	;	PUNCT
ejpam-1831	55	7	where	where	SCONJ
ejpam-1831	55	8	x	x	SYM
ejpam-1831	55	9	∈	∈	PROPN
ejpam-1831	55	10	z+1	z+1	PROPN
ejpam-1831	55	11	and	and	CCONJ
ejpam-1831	55	12	let	let	VERB
ejpam-1831	55	13	l	l	NOUN
ejpam-1831	55	14	=	=	PUNCT
ejpam-1831	55	15	range	range	NOUN
ejpam-1831	55	16	g.	g.	NOUN
ejpam-1831	55	17	for	for	ADP
ejpam-1831	55	18	c	c	PROPN
ejpam-1831	55	19	,	,	PUNCT
ejpam-1831	55	20	d	d	PROPN
ejpam-1831	55	21	∈	∈	PROPN
ejpam-1831	55	22	z+	z+	NUM
ejpam-1831	55	23	,	,	PUNCT
ejpam-1831	55	24	we	we	PRON
ejpam-1831	55	25	find	find	VERB
ejpam-1831	55	26	that	that	SCONJ
ejpam-1831	55	27	if	if	SCONJ
ejpam-1831	55	28	g(c	g(c	VERB
ejpam-1831	55	29	)	)	PUNCT
ejpam-1831	55	30	=	=	PUNCT
ejpam-1831	56	1	g(d)⇒	g(d)⇒	NOUN
ejpam-1831	56	2	c	c	NOUN
ejpam-1831	56	3	=	=	SYM
ejpam-1831	56	4	d	d	PROPN
ejpam-1831	56	5	,	,	PUNCT
ejpam-1831	56	6	so	so	SCONJ
ejpam-1831	56	7	g	g	PROPN
ejpam-1831	56	8	is	be	AUX
ejpam-1831	56	9	a	a	DET
ejpam-1831	56	10	one	one	NUM
ejpam-1831	56	11	-	-	PUNCT
ejpam-1831	56	12	to	to	ADP
ejpam-1831	56	13	-	-	PUNCT
ejpam-1831	56	14	one	one	NUM
ejpam-1831	56	15	function	function	NOUN
ejpam-1831	56	16	.	.	PUNCT
ejpam-1831	57	1	consequently	consequently	ADV
ejpam-1831	57	2	|z+|	|z+|	PUNCT
ejpam-1831	57	3	=	=	PUNCT
ejpam-1831	57	4	|l|	|l|	NOUN
ejpam-1831	57	5	,	,	PUNCT
ejpam-1831	57	6	then	then	ADV
ejpam-1831	57	7	we	we	PRON
ejpam-1831	57	8	can	can	AUX
ejpam-1831	57	9	notice	notice	VERB
ejpam-1831	57	10	that	that	SCONJ
ejpam-1831	57	11	l	l	NOUN
ejpam-1831	57	12	is	be	AUX
ejpam-1831	57	13	countably	countably	ADV
ejpam-1831	57	14	infinite	infinite	ADJ
ejpam-1831	57	15	,	,	PUNCT
ejpam-1831	57	16	but	but	CCONJ
ejpam-1831	57	17	l	l	NOUN
ejpam-1831	57	18	∈	∈	PROPN
ejpam-1831	57	19	q	q	NOUN
ejpam-1831	57	20	∩	∩	ADJ
ejpam-1831	57	21	�	�	PROPN
ejpam-1831	57	22	−	−	PROPN
ejpam-1831	57	23	j	j	PROPN
ejpam-1831	57	24	3	3	NUM
ejpam-1831	57	25	,	,	PUNCT
ejpam-1831	57	26	j	j	PROPN
ejpam-1831	57	27	3	3	NUM
ejpam-1831	57	28	�	�	PROPN
ejpam-1831	57	29	,	,	PUNCT
ejpam-1831	57	30	therefore	therefore	ADV
ejpam-1831	57	31	q	q	X
ejpam-1831	57	32	∩	∩	ADJ
ejpam-1831	57	33	�	�	PROPN
ejpam-1831	57	34	−	−	PROPN
ejpam-1831	57	35	j	j	PROPN
ejpam-1831	57	36	3	3	NUM
ejpam-1831	57	37	,	,	PUNCT
ejpam-1831	57	38	j	j	PROPN
ejpam-1831	57	39	3	3	NUM
ejpam-1831	57	40	�	�	PROPN
ejpam-1831	57	41	is	be	AUX
ejpam-1831	57	42	also	also	ADV
ejpam-1831	57	43	infinite	infinite	ADJ
ejpam-1831	57	44	.	.	PUNCT
ejpam-1831	58	1	references	reference	NOUN
ejpam-1831	58	2	404	404	NUM
ejpam-1831	58	3	now	now	ADV
ejpam-1831	58	4	,	,	PUNCT
ejpam-1831	58	5	it	it	PRON
ejpam-1831	58	6	’s	’s	AUX
ejpam-1831	58	7	known	know	VERB
ejpam-1831	58	8	that	that	SCONJ
ejpam-1831	58	9	n	n	PRON
ejpam-1831	58	10	∈	∈	PROPN
ejpam-1831	58	11	q	q	NOUN
ejpam-1831	58	12	∩	∩	X
ejpam-1831	58	13	�	�	PROPN
ejpam-1831	59	1	−	−	PROPN
ejpam-1831	59	2	j	j	PROPN
ejpam-1831	59	3	3	3	NUM
ejpam-1831	59	4	,	,	PUNCT
ejpam-1831	59	5	j	j	PROPN
ejpam-1831	59	6	3	3	NUM
ejpam-1831	59	7	�	�	PROPN
ejpam-1831	59	8	,	,	PUNCT
ejpam-1831	59	9	then	then	ADV
ejpam-1831	59	10	r	r	NOUN
ejpam-1831	59	11	,	,	PUNCT
ejpam-1831	59	12	s	s	PART
ejpam-1831	59	13	could	could	AUX
ejpam-1831	59	14	take	take	VERB
ejpam-1831	59	15	infinities	infinity	NOUN
ejpam-1831	59	16	values	value	NOUN
ejpam-1831	59	17	,	,	PUNCT
ejpam-1831	59	18	where	where	SCONJ
ejpam-1831	59	19	the	the	DET
ejpam-1831	59	20	distance	distance	NOUN
ejpam-1831	59	21	between	between	ADP
ejpam-1831	59	22	them	they	PRON
ejpam-1831	59	23	is	be	AUX
ejpam-1831	59	24	a	a	DET
ejpam-1831	59	25	rational	rational	ADJ
ejpam-1831	59	26	number	number	NOUN
ejpam-1831	59	27	,	,	PUNCT
ejpam-1831	59	28	because	because	SCONJ
ejpam-1831	59	29	they	they	PRON
ejpam-1831	59	30	depend	depend	VERB
ejpam-1831	59	31	of	of	ADP
ejpam-1831	59	32	n.	n.	NOUN
ejpam-1831	59	33	references	reference	NOUN
ejpam-1831	59	34	[	[	X
ejpam-1831	59	35	1	1	X
ejpam-1831	59	36	]	]	PUNCT
ejpam-1831	59	37	p	p	NOUN
ejpam-1831	59	38	binder	binder	NOUN
ejpam-1831	59	39	.	.	PUNCT
ejpam-1831	60	1	theories	theory	NOUN
ejpam-1831	60	2	of	of	ADP
ejpam-1831	60	3	almost	almost	ADV
ejpam-1831	60	4	everything	everything	PRON
ejpam-1831	60	5	.	.	PUNCT
ejpam-1831	61	1	nature	nature	NOUN
ejpam-1831	61	2	,	,	PUNCT
ejpam-1831	61	3	455:884–885	455:884–885	NUM
ejpam-1831	61	4	,	,	PUNCT
ejpam-1831	61	5	2008	2008	NUM
ejpam-1831	61	6	.	.	PUNCT
ejpam-1831	62	1	[	[	X
ejpam-1831	62	2	2	2	NUM
ejpam-1831	62	3	]	]	PUNCT
ejpam-1831	62	4	e	e	NOUN
ejpam-1831	62	5	deza	deza	NOUN
ejpam-1831	62	6	and	and	CCONJ
ejpam-1831	62	7	m	m	PROPN
ejpam-1831	62	8	deza	deza	ADV
ejpam-1831	62	9	.	.	PUNCT
ejpam-1831	63	1	dictionary	dictionary	NOUN
ejpam-1831	63	2	of	of	ADP
ejpam-1831	63	3	distances	distance	NOUN
ejpam-1831	63	4	.	.	PUNCT
ejpam-1831	64	1	elsevier	elsevier	NOUN
ejpam-1831	64	2	,	,	PUNCT
ejpam-1831	64	3	2006	2006	NUM
ejpam-1831	64	4	.	.	PUNCT
ejpam-1831	65	1	[	[	X
ejpam-1831	65	2	3	3	NUM
ejpam-1831	65	3	]	]	X
ejpam-1831	65	4	dimacs	dimac	NOUN
ejpam-1831	65	5	.	.	PUNCT
ejpam-1831	66	1	geometry	geometry	NOUN
ejpam-1831	66	2	/	/	SYM
ejpam-1831	66	3	number	number	NOUN
ejpam-1831	66	4	theory	theory	NOUN
ejpam-1831	66	5	open	open	ADJ
ejpam-1831	66	6	problems	problem	NOUN
ejpam-1831	66	7	.	.	PUNCT
ejpam-1831	66	8	,	,	PUNCT
ejpam-1831	66	9	1996	1996	NUM
ejpam-1831	66	10	.	.	PUNCT
ejpam-1831	67	1	[	[	X
ejpam-1831	67	2	online	online	X
ejpam-1831	67	3	]	]	X
ejpam-1831	67	4	,	,	PUNCT
ejpam-1831	67	5	(	(	PUNCT
ejpam-1831	67	6	accessed	access	VERB
ejpam-1831	67	7	:	:	PUNCT
ejpam-1831	67	8	10	10	NUM
ejpam-1831	67	9	january	january	NOUN
ejpam-1831	67	10	2013	2013	NUM
ejpam-1831	67	11	)	)	PUNCT
ejpam-1831	67	12	,	,	PUNCT
ejpam-1831	67	13	http://dimacs.rutgers.edu/~hochberg/undopen/geomnum/	http://dimacs.rutgers.edu/~hochberg/undopen/geomnum/	PROPN
ejpam-1831	67	14	geomnum.html	geomnum.html	PROPN
ejpam-1831	67	15	.	.	PUNCT
ejpam-1831	68	1	[	[	X
ejpam-1831	68	2	4	4	NUM
ejpam-1831	68	3	]	]	X
ejpam-1831	68	4	r	r	NOUN
ejpam-1831	68	5	grimaldi	grimaldi	NOUN
ejpam-1831	68	6	.	.	PUNCT
ejpam-1831	69	1	discrete	discrete	VERB
ejpam-1831	69	2	and	and	CCONJ
ejpam-1831	69	3	combinatorial	combinatorial	ADJ
ejpam-1831	69	4	mathematics	mathematic	NOUN
ejpam-1831	69	5	:	:	PUNCT
ejpam-1831	69	6	an	an	DET
ejpam-1831	69	7	applied	applied	ADJ
ejpam-1831	69	8	introduction	introduction	NOUN
ejpam-1831	69	9	.	.	PUNCT
ejpam-1831	70	1	addisonwesley	addisonwesley	PROPN
ejpam-1831	70	2	,	,	PUNCT
ejpam-1831	70	3	united	united	PROPN
ejpam-1831	70	4	states	states	PROPN
ejpam-1831	70	5	of	of	ADP
ejpam-1831	70	6	america	america	PROPN
ejpam-1831	70	7	,	,	PUNCT
ejpam-1831	70	8	2003	2003	NUM
ejpam-1831	70	9	.	.	PUNCT
ejpam-1831	71	1	[	[	X
ejpam-1831	71	2	5	5	NUM
ejpam-1831	71	3	]	]	X
ejpam-1831	71	4	hardy	hardy	ADJ
ejpam-1831	71	5	and	and	CCONJ
ejpam-1831	71	6	wright	wright	PROPN
ejpam-1831	71	7	.	.	PUNCT
ejpam-1831	72	1	an	an	DET
ejpam-1831	72	2	introduction	introduction	NOUN
ejpam-1831	72	3	to	to	ADP
ejpam-1831	72	4	the	the	DET
ejpam-1831	72	5	theory	theory	NOUN
ejpam-1831	72	6	of	of	ADP
ejpam-1831	72	7	numbers	number	NOUN
ejpam-1831	72	8	.	.	PUNCT
ejpam-1831	73	1	oxford	oxford	PROPN
ejpam-1831	73	2	science	science	PROPN
ejpam-1831	73	3	publications	publication	NOUN
ejpam-1831	73	4	,	,	PUNCT
ejpam-1831	73	5	1980	1980	NUM
ejpam-1831	73	6	.	.	PUNCT
ejpam-1831	74	1	[	[	X
ejpam-1831	74	2	6	6	NUM
ejpam-1831	74	3	]	]	X
ejpam-1831	74	4	m	m	VERB
ejpam-1831	74	5	hazewinkel	hazewinkel	ADJ
ejpam-1831	74	6	.	.	PUNCT
ejpam-1831	75	1	encyclopedia	encyclopedia	NOUN
ejpam-1831	75	2	of	of	ADP
ejpam-1831	75	3	mathematics	mathematic	NOUN
ejpam-1831	75	4	.	.	PUNCT
ejpam-1831	76	1	springer	springer	NOUN
ejpam-1831	76	2	,	,	PUNCT
ejpam-1831	76	3	2001	2001	NUM
ejpam-1831	76	4	.	.	PUNCT
ejpam-1831	77	1	[	[	X
ejpam-1831	77	2	7	7	X
ejpam-1831	77	3	]	]	X
ejpam-1831	77	4	h	h	PROPN
ejpam-1831	77	5	heaton	heaton	PROPN
ejpam-1831	77	6	.	.	PUNCT
ejpam-1831	78	1	a	a	DET
ejpam-1831	78	2	method	method	NOUN
ejpam-1831	78	3	of	of	ADP
ejpam-1831	78	4	solving	solve	VERB
ejpam-1831	78	5	quadratic	quadratic	ADJ
ejpam-1831	78	6	equations	equation	NOUN
ejpam-1831	78	7	.	.	PUNCT
ejpam-1831	79	1	the	the	DET
ejpam-1831	79	2	american	american	PROPN
ejpam-1831	79	3	mathematical	mathematical	PROPN
ejpam-1831	79	4	monthly	monthly	ADV
ejpam-1831	79	5	,	,	PUNCT
ejpam-1831	79	6	3:236–237	3:236–237	NUM
ejpam-1831	79	7	,	,	PUNCT
ejpam-1831	79	8	1896	1896	NUM
ejpam-1831	79	9	.	.	PUNCT
ejpam-1831	80	1	[	[	X
ejpam-1831	80	2	8	8	NUM
ejpam-1831	80	3	]	]	X
ejpam-1831	80	4	d	d	X
ejpam-1831	80	5	knuth	knuth	PROPN
ejpam-1831	80	6	.	.	PUNCT
ejpam-1831	81	1	seminumerical	seminumerical	ADJ
ejpam-1831	81	2	algorithms	algorithm	NOUN
ejpam-1831	81	3	.	.	PUNCT
ejpam-1831	82	1	the	the	DET
ejpam-1831	82	2	art	art	NOUN
ejpam-1831	82	3	of	of	ADP
ejpam-1831	82	4	computer	computer	NOUN
ejpam-1831	82	5	programming	programming	NOUN
ejpam-1831	82	6	.	.	PUNCT
ejpam-1831	83	1	addison	addison	PROPN
ejpam-1831	83	2	-	-	PUNCT
ejpam-1831	83	3	wesley	wesley	PROPN
ejpam-1831	83	4	,	,	PUNCT
ejpam-1831	83	5	reading	reading	NOUN
ejpam-1831	83	6	,	,	PUNCT
ejpam-1831	83	7	massachusetts	massachusetts	PROPN
ejpam-1831	83	8	,	,	PUNCT
ejpam-1831	83	9	1975	1975	NUM
ejpam-1831	83	10	.	.	PUNCT
ejpam-1831	84	1	[	[	X
ejpam-1831	84	2	9	9	NUM
ejpam-1831	84	3	]	]	PUNCT
ejpam-1831	84	4	a	a	DET
ejpam-1831	84	5	lenstra	lenstra	NOUN
ejpam-1831	84	6	,	,	PUNCT
ejpam-1831	84	7	h	h	NOUN
ejpam-1831	84	8	lenstra	lenstra	NOUN
ejpam-1831	84	9	,	,	PUNCT
ejpam-1831	84	10	and	and	CCONJ
ejpam-1831	84	11	l	l	PROPN
ejpam-1831	84	12	lovàsz	lovàsz	NOUN
ejpam-1831	84	13	.	.	PUNCT
ejpam-1831	85	1	factoring	factoring	NOUN
ejpam-1831	85	2	polynomials	polynomial	NOUN
ejpam-1831	85	3	with	with	ADP
ejpam-1831	85	4	rational	rational	ADJ
ejpam-1831	85	5	coefficients	coefficient	NOUN
ejpam-1831	85	6	.	.	PUNCT
ejpam-1831	86	1	mathematische	mathematische	PROPN
ejpam-1831	86	2	annalen	annalen	PROPN
ejpam-1831	86	3	,	,	PUNCT
ejpam-1831	86	4	261:515–534	261:515–534	NUM
ejpam-1831	86	5	,	,	PUNCT
ejpam-1831	86	6	1982	1982	NUM
ejpam-1831	86	7	.	.	PUNCT
ejpam-1831	87	1	[	[	X
ejpam-1831	87	2	10	10	NUM
ejpam-1831	87	3	]	]	X
ejpam-1831	87	4	e	e	X
ejpam-1831	87	5	lockwood	lockwood	PROPN
ejpam-1831	87	6	.	.	PUNCT
ejpam-1831	88	1	a	a	DET
ejpam-1831	88	2	book	book	NOUN
ejpam-1831	88	3	of	of	ADP
ejpam-1831	88	4	curves	curve	NOUN
ejpam-1831	88	5	.	.	PUNCT
ejpam-1831	89	1	cambridge	cambridge	PROPN
ejpam-1831	89	2	university	university	PROPN
ejpam-1831	89	3	press	press	NOUN
ejpam-1831	89	4	,	,	PUNCT
ejpam-1831	89	5	1961	1961	NUM
ejpam-1831	89	6	.	.	PUNCT
ejpam-1831	90	1	[	[	X
ejpam-1831	90	2	11	11	NUM
ejpam-1831	90	3	]	]	X
ejpam-1831	90	4	j	j	PROPN
ejpam-1831	90	5	mayberry	mayberry	PROPN
ejpam-1831	90	6	.	.	PUNCT
ejpam-1831	91	1	encyclopedia	encyclopedia	NOUN
ejpam-1831	91	2	of	of	ADP
ejpam-1831	91	3	mathematics	mathematic	NOUN
ejpam-1831	91	4	and	and	CCONJ
ejpam-1831	91	5	its	its	PRON
ejpam-1831	91	6	applications	application	NOUN
ejpam-1831	91	7	.	.	PUNCT
ejpam-1831	92	1	cambridge	cambridge	PROPN
ejpam-1831	92	2	university	university	PROPN
ejpam-1831	92	3	press	press	NOUN
ejpam-1831	92	4	,	,	PUNCT
ejpam-1831	92	5	2000	2000	NUM
ejpam-1831	92	6	.	.	PUNCT