id	sid	tid	token	lemma	pos
ap-1821	1	1	acta	acta	PROPN
ap-1821	1	2	polytechnica	polytechnica	PROPN
ap-1821	1	3	acta	acta	PROPN
ap-1821	1	4	polytechnica	polytechnica	PROPN
ap-1821	1	5	53(4):322–328	53(4):322–328	PROPN
ap-1821	1	6	,	,	PUNCT
ap-1821	1	7	2013	2013	NUM
ap-1821	1	8	©	©	PROPN
ap-1821	1	9	czech	czech	PROPN
ap-1821	1	10	technical	technical	PROPN
ap-1821	1	11	university	university	PROPN
ap-1821	1	12	in	in	ADP
ap-1821	1	13	prague	prague	PROPN
ap-1821	1	14	,	,	PUNCT
ap-1821	1	15	2013	2013	NUM
ap-1821	1	16	available	available	ADJ
ap-1821	1	17	online	online	ADV
ap-1821	1	18	at	at	ADP
ap-1821	1	19	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	ADJ
ap-1821	1	20	continued	continue	VERB
ap-1821	1	21	fractions	fraction	NOUN
ap-1821	1	22	of	of	ADP
ap-1821	1	23	square	square	ADJ
ap-1821	1	24	roots	root	NOUN
ap-1821	1	25	of	of	ADP
ap-1821	1	26	natural	natural	ADJ
ap-1821	1	27	numbers	number	NOUN
ap-1821	1	28	ľubomíra	ľubomíra	ADJ
ap-1821	1	29	balkováa,∗	balkováa,∗	NOUN
ap-1821	1	30	,	,	PUNCT
ap-1821	1	31	aranka	aranka	PROPN
ap-1821	1	32	hruškováb	hruškováb	NOUN
ap-1821	1	33	a	a	DET
ap-1821	1	34	department	department	NOUN
ap-1821	1	35	of	of	ADP
ap-1821	1	36	mathematics	mathematic	NOUN
ap-1821	1	37	,	,	PUNCT
ap-1821	1	38	faculty	faculty	NOUN
ap-1821	1	39	of	of	ADP
ap-1821	1	40	nuclear	nuclear	ADJ
ap-1821	1	41	sciences	science	NOUN
ap-1821	1	42	and	and	CCONJ
ap-1821	1	43	physical	physical	ADJ
ap-1821	1	44	engineering	engineering	NOUN
ap-1821	1	45	,	,	PUNCT
ap-1821	1	46	czech	czech	PROPN
ap-1821	1	47	technical	technical	PROPN
ap-1821	1	48	university	university	PROPN
ap-1821	1	49	in	in	ADP
ap-1821	1	50	prague	prague	PROPN
ap-1821	1	51	,	,	PUNCT
ap-1821	1	52	trojanova	trojanova	X
ap-1821	1	53	13	13	NUM
ap-1821	1	54	,	,	PUNCT
ap-1821	1	55	120	120	NUM
ap-1821	1	56	00	00	NUM
ap-1821	1	57	praha	praha	PROPN
ap-1821	1	58	2	2	NUM
ap-1821	1	59	,	,	PUNCT
ap-1821	1	60	czech	czech	PROPN
ap-1821	1	61	republic	republic	PROPN
ap-1821	1	62	b	b	PROPN
ap-1821	1	63	gymnázium	gymnázium	PROPN
ap-1821	1	64	christiana	christiana	PROPN
ap-1821	1	65	dopplera	dopplera	PROPN
ap-1821	1	66	,	,	PUNCT
ap-1821	1	67	zborovská	zborovská	PROPN
ap-1821	1	68	45	45	NUM
ap-1821	1	69	,	,	PUNCT
ap-1821	1	70	150	150	NUM
ap-1821	1	71	00	00	NUM
ap-1821	1	72	praha	praha	PROPN
ap-1821	1	73	5	5	NUM
ap-1821	1	74	–	–	PUNCT
ap-1821	1	75	smíchov	smíchov	NOUN
ap-1821	1	76	,	,	PUNCT
ap-1821	1	77	czech	czech	PROPN
ap-1821	1	78	republic	republic	NOUN
ap-1821	1	79	∗	∗	NOUN
ap-1821	1	80	corresponding	correspond	VERB
ap-1821	1	81	author	author	NOUN
ap-1821	1	82	:	:	PUNCT
ap-1821	1	83	lubomira.balkova@fjfi.cvut.cz	lubomira.balkova@fjfi.cvut.cz	NOUN
ap-1821	1	84	abstract	abstract	ADJ
ap-1821	1	85	.	.	PUNCT
ap-1821	2	1	in	in	ADP
ap-1821	2	2	this	this	DET
ap-1821	2	3	paper	paper	NOUN
ap-1821	2	4	,	,	PUNCT
ap-1821	2	5	we	we	PRON
ap-1821	2	6	will	will	AUX
ap-1821	2	7	first	first	ADV
ap-1821	2	8	summarize	summarize	VERB
ap-1821	2	9	known	know	VERB
ap-1821	2	10	results	result	NOUN
ap-1821	2	11	concerning	concern	VERB
ap-1821	2	12	continued	continued	ADJ
ap-1821	2	13	fractions	fraction	NOUN
ap-1821	2	14	.	.	PUNCT
ap-1821	3	1	then	then	ADV
ap-1821	3	2	we	we	PRON
ap-1821	3	3	will	will	AUX
ap-1821	3	4	limit	limit	VERB
ap-1821	3	5	our	our	PRON
ap-1821	3	6	consideration	consideration	NOUN
ap-1821	3	7	to	to	ADP
ap-1821	3	8	continued	continue	VERB
ap-1821	3	9	fractions	fraction	NOUN
ap-1821	3	10	of	of	ADP
ap-1821	3	11	quadratic	quadratic	ADJ
ap-1821	3	12	numbers	number	NOUN
ap-1821	3	13	.	.	PUNCT
ap-1821	4	1	the	the	DET
ap-1821	4	2	second	second	ADJ
ap-1821	4	3	author	author	NOUN
ap-1821	4	4	describes	describe	VERB
ap-1821	4	5	periods	period	NOUN
ap-1821	4	6	and	and	CCONJ
ap-1821	4	7	sometimes	sometimes	ADV
ap-1821	4	8	the	the	DET
ap-1821	4	9	precise	precise	ADJ
ap-1821	4	10	form	form	NOUN
ap-1821	4	11	of	of	ADP
ap-1821	4	12	continued	continue	VERB
ap-1821	4	13	fractions	fraction	NOUN
ap-1821	4	14	of	of	ADP
ap-1821	4	15	√	√	NUM
ap-1821	4	16	n	n	CCONJ
ap-1821	4	17	,	,	PUNCT
ap-1821	4	18	where	where	SCONJ
ap-1821	4	19	n	n	PRON
ap-1821	4	20	is	be	AUX
ap-1821	4	21	a	a	DET
ap-1821	4	22	natural	natural	ADJ
ap-1821	4	23	number	number	NOUN
ap-1821	4	24	.	.	PUNCT
ap-1821	5	1	in	in	ADP
ap-1821	5	2	cases	case	NOUN
ap-1821	5	3	where	where	SCONJ
ap-1821	5	4	we	we	PRON
ap-1821	5	5	have	have	AUX
ap-1821	5	6	been	be	AUX
ap-1821	5	7	able	able	ADJ
ap-1821	5	8	to	to	PART
ap-1821	5	9	find	find	VERB
ap-1821	5	10	such	such	ADJ
ap-1821	5	11	results	result	NOUN
ap-1821	5	12	in	in	ADP
ap-1821	5	13	the	the	DET
ap-1821	5	14	literature	literature	NOUN
ap-1821	5	15	,	,	PUNCT
ap-1821	5	16	we	we	PRON
ap-1821	5	17	recall	recall	VERB
ap-1821	5	18	the	the	DET
ap-1821	5	19	original	original	ADJ
ap-1821	5	20	authors	author	NOUN
ap-1821	5	21	,	,	PUNCT
ap-1821	5	22	however	however	ADV
ap-1821	5	23	many	many	ADJ
ap-1821	5	24	results	result	NOUN
ap-1821	5	25	seem	seem	VERB
ap-1821	5	26	to	to	PART
ap-1821	5	27	be	be	AUX
ap-1821	5	28	new	new	ADJ
ap-1821	5	29	.	.	PUNCT
ap-1821	6	1	keywords	keyword	NOUN
ap-1821	6	2	:	:	PUNCT
ap-1821	6	3	continued	continue	VERB
ap-1821	6	4	fractions	fraction	NOUN
ap-1821	6	5	,	,	PUNCT
ap-1821	6	6	periodic	periodic	ADJ
ap-1821	6	7	continued	continued	ADJ
ap-1821	6	8	fractions	fraction	NOUN
ap-1821	6	9	,	,	PUNCT
ap-1821	6	10	periods	period	NOUN
ap-1821	6	11	of	of	ADP
ap-1821	6	12	continued	continued	ADJ
ap-1821	6	13	fractions	fraction	NOUN
ap-1821	6	14	,	,	PUNCT
ap-1821	6	15	quadratic	quadratic	ADJ
ap-1821	6	16	numbers	number	NOUN
ap-1821	6	17	.	.	PUNCT
ap-1821	7	1	1	1	X
ap-1821	7	2	.	.	X
ap-1821	7	3	introduction	introduction	NOUN
ap-1821	7	4	continued	continue	VERB
ap-1821	7	5	fractions	fraction	NOUN
ap-1821	7	6	have	have	VERB
ap-1821	7	7	a	a	DET
ap-1821	7	8	long	long	ADJ
ap-1821	7	9	history	history	NOUN
ap-1821	7	10	behind	behind	ADP
ap-1821	7	11	them	they	PRON
ap-1821	7	12	—	—	PUNCT
ap-1821	7	13	their	their	PRON
ap-1821	7	14	origin	origin	NOUN
ap-1821	7	15	may	may	AUX
ap-1821	7	16	go	go	VERB
ap-1821	7	17	back	back	ADV
ap-1821	7	18	to	to	ADP
ap-1821	7	19	the	the	DET
ap-1821	7	20	age	age	NOUN
ap-1821	7	21	of	of	ADP
ap-1821	7	22	euclid	euclid	PROPN
ap-1821	7	23	’s	’s	PART
ap-1821	7	24	algorithm	algorithm	NOUN
ap-1821	7	25	for	for	ADP
ap-1821	7	26	the	the	DET
ap-1821	7	27	greatest	great	ADJ
ap-1821	7	28	common	common	ADJ
ap-1821	7	29	divisor	divisor	NOUN
ap-1821	7	30	,	,	PUNCT
ap-1821	7	31	or	or	CCONJ
ap-1821	7	32	even	even	ADV
ap-1821	7	33	earlier	early	ADV
ap-1821	7	34	.	.	PUNCT
ap-1821	8	1	however	however	ADV
ap-1821	8	2	,	,	PUNCT
ap-1821	8	3	they	they	PRON
ap-1821	8	4	are	be	AUX
ap-1821	8	5	experiencing	experience	VERB
ap-1821	8	6	a	a	DET
ap-1821	8	7	revival	revival	NOUN
ap-1821	8	8	nowadays	nowadays	ADV
ap-1821	8	9	,	,	PUNCT
ap-1821	8	10	thanks	thank	NOUN
ap-1821	8	11	to	to	ADP
ap-1821	8	12	their	their	PRON
ap-1821	8	13	applications	application	NOUN
ap-1821	8	14	in	in	ADP
ap-1821	8	15	high	high	ADJ
ap-1821	8	16	-	-	PUNCT
ap-1821	8	17	speed	speed	NOUN
ap-1821	8	18	and	and	CCONJ
ap-1821	8	19	highaccuracy	highaccuracy	NOUN
ap-1821	8	20	computer	computer	NOUN
ap-1821	8	21	arithmetics	arithmetic	NOUN
ap-1821	8	22	.	.	PUNCT
ap-1821	9	1	some	some	PRON
ap-1821	9	2	of	of	ADP
ap-1821	9	3	the	the	DET
ap-1821	9	4	advantages	advantage	NOUN
ap-1821	9	5	of	of	ADP
ap-1821	9	6	continued	continue	VERB
ap-1821	9	7	fractions	fraction	NOUN
ap-1821	9	8	in	in	ADP
ap-1821	9	9	computer	computer	NOUN
ap-1821	9	10	arithmetics	arithmetic	NOUN
ap-1821	9	11	are	be	AUX
ap-1821	9	12	:	:	PUNCT
ap-1821	9	13	faster	fast	ADJ
ap-1821	9	14	division	division	NOUN
ap-1821	9	15	and	and	CCONJ
ap-1821	9	16	multiplication	multiplication	NOUN
ap-1821	9	17	than	than	ADP
ap-1821	9	18	with	with	ADP
ap-1821	9	19	positional	positional	ADJ
ap-1821	9	20	number	number	NOUN
ap-1821	9	21	representations	representation	NOUN
ap-1821	9	22	,	,	PUNCT
ap-1821	9	23	fast	fast	ADJ
ap-1821	9	24	and	and	CCONJ
ap-1821	9	25	precise	precise	ADJ
ap-1821	9	26	evaluation	evaluation	NOUN
ap-1821	9	27	of	of	ADP
ap-1821	9	28	trigonometric	trigonometric	ADJ
ap-1821	9	29	,	,	PUNCT
ap-1821	9	30	logarithmic	logarithmic	ADJ
ap-1821	9	31	and	and	CCONJ
ap-1821	9	32	other	other	ADJ
ap-1821	9	33	functions	function	NOUN
ap-1821	9	34	,	,	PUNCT
ap-1821	9	35	precise	precise	ADJ
ap-1821	9	36	representation	representation	NOUN
ap-1821	9	37	of	of	ADP
ap-1821	9	38	transcendental	transcendental	ADJ
ap-1821	9	39	numbers	number	NOUN
ap-1821	9	40	,	,	PUNCT
ap-1821	9	41	no	no	DET
ap-1821	9	42	roundoff	roundoff	NOUN
ap-1821	9	43	or	or	CCONJ
ap-1821	9	44	truncation	truncation	NOUN
ap-1821	9	45	errors	error	NOUN
ap-1821	9	46	(	(	PUNCT
ap-1821	9	47	[	[	X
ap-1821	9	48	6	6	NUM
ap-1821	9	49	]	]	PUNCT
ap-1821	9	50	,	,	PUNCT
ap-1821	9	51	kahan	kahan	PROPN
ap-1821	9	52	’s	’s	PART
ap-1821	9	53	method	method	NOUN
ap-1821	9	54	in	in	ADP
ap-1821	9	55	[	[	X
ap-1821	9	56	3	3	NUM
ap-1821	9	57	,	,	PUNCT
ap-1821	9	58	p.	p.	NOUN
ap-1821	9	59	179	179	NUM
ap-1821	9	60	]	]	PUNCT
ap-1821	9	61	)	)	PUNCT
ap-1821	9	62	.	.	PUNCT
ap-1821	10	1	2	2	X
ap-1821	10	2	.	.	NUM
ap-1821	10	3	continued	continue	VERB
ap-1821	10	4	fractions	fraction	NOUN
ap-1821	10	5	in	in	ADP
ap-1821	10	6	this	this	DET
ap-1821	10	7	section	section	NOUN
ap-1821	10	8	we	we	PRON
ap-1821	10	9	summarize	summarize	VERB
ap-1821	10	10	some	some	DET
ap-1821	10	11	basic	basic	ADJ
ap-1821	10	12	definitions	definition	NOUN
ap-1821	10	13	and	and	CCONJ
ap-1821	10	14	results	result	NOUN
ap-1821	10	15	that	that	PRON
ap-1821	10	16	can	can	AUX
ap-1821	10	17	be	be	AUX
ap-1821	10	18	found	find	VERB
ap-1821	10	19	in	in	ADP
ap-1821	10	20	any	any	DET
ap-1821	10	21	number	number	NOUN
ap-1821	10	22	theory	theory	NOUN
ap-1821	10	23	course	course	NOUN
ap-1821	11	1	[	[	X
ap-1821	11	2	1	1	NUM
ap-1821	11	3	,	,	PUNCT
ap-1821	11	4	2	2	NUM
ap-1821	11	5	,	,	PUNCT
ap-1821	11	6	4	4	NUM
ap-1821	11	7	]	]	PUNCT
ap-1821	11	8	.	.	PUNCT
ap-1821	12	1	we	we	PRON
ap-1821	12	2	use	use	VERB
ap-1821	12	3	bxc	bxc	NOUN
ap-1821	12	4	to	to	PART
ap-1821	12	5	denote	denote	VERB
ap-1821	12	6	the	the	DET
ap-1821	12	7	integer	integer	NOUN
ap-1821	12	8	part	part	NOUN
ap-1821	12	9	of	of	ADP
ap-1821	12	10	a	a	DET
ap-1821	12	11	real	real	ADJ
ap-1821	12	12	number	number	NOUN
ap-1821	12	13	x.	x.	NOUN
ap-1821	12	14	definition	definition	NOUN
ap-1821	12	15	2.1	2.1	NUM
ap-1821	12	16	.	.	PUNCT
ap-1821	13	1	the	the	DET
ap-1821	13	2	continued	continue	VERB
ap-1821	13	3	fraction	fraction	NOUN
ap-1821	13	4	(	(	PUNCT
ap-1821	13	5	expansion	expansion	NOUN
ap-1821	13	6	)	)	PUNCT
ap-1821	13	7	of	of	ADP
ap-1821	13	8	a	a	DET
ap-1821	13	9	real	real	ADJ
ap-1821	13	10	number	number	NOUN
ap-1821	13	11	x	x	PUNCT
ap-1821	13	12	is	be	AUX
ap-1821	13	13	the	the	DET
ap-1821	13	14	sequence	sequence	NOUN
ap-1821	13	15	of	of	ADP
ap-1821	13	16	integers	integer	NOUN
ap-1821	13	17	(	(	PUNCT
ap-1821	13	18	an)n∈n	an)n∈n	NUM
ap-1821	13	19	obtained	obtain	VERB
ap-1821	13	20	by	by	ADP
ap-1821	13	21	the	the	DET
ap-1821	13	22	following	follow	VERB
ap-1821	13	23	algorithm	algorithm	NOUN
ap-1821	13	24	x0	x0	PROPN
ap-1821	14	1	=	=	PUNCT
ap-1821	14	2	x	x	PROPN
ap-1821	14	3	,	,	PUNCT
ap-1821	14	4	an	an	DET
ap-1821	14	5	=	=	PUNCT
ap-1821	14	6	bxnc	bxnc	NOUN
ap-1821	14	7	,	,	PUNCT
ap-1821	14	8	xn+1	xn+1	PROPN
ap-1821	14	9	=	=	SYM
ap-1821	14	10	{	{	PUNCT
ap-1821	14	11	1	1	NUM
ap-1821	14	12	xn−an	xn−an	X
ap-1821	14	13	if	if	SCONJ
ap-1821	14	14	xn	xn	PROPN
ap-1821	14	15	/∈	/∈	PUNCT
ap-1821	15	1	z	z	NOUN
ap-1821	15	2	,	,	PUNCT
ap-1821	15	3	0	0	NUM
ap-1821	15	4	otherwise	otherwise	ADV
ap-1821	15	5	.	.	PUNCT
ap-1821	16	1	note	note	VERB
ap-1821	16	2	that	that	SCONJ
ap-1821	16	3	a0	a0	PROPN
ap-1821	16	4	∈	∈	PROPN
ap-1821	16	5	z	z	PROPN
ap-1821	16	6	and	and	CCONJ
ap-1821	16	7	an	an	DET
ap-1821	16	8	∈	∈	PROPN
ap-1821	16	9	n.	n.	NOUN
ap-1821	16	10	the	the	DET
ap-1821	16	11	algorithm	algorithm	NOUN
ap-1821	16	12	producing	produce	VERB
ap-1821	16	13	the	the	DET
ap-1821	16	14	continued	continue	VERB
ap-1821	16	15	fraction	fraction	NOUN
ap-1821	16	16	is	be	AUX
ap-1821	16	17	closely	closely	ADV
ap-1821	16	18	related	relate	VERB
ap-1821	16	19	to	to	ADP
ap-1821	16	20	the	the	DET
ap-1821	16	21	euclidean	euclidean	ADJ
ap-1821	16	22	algorithm	algorithm	NOUN
ap-1821	16	23	for	for	ADP
ap-1821	16	24	computing	compute	VERB
ap-1821	16	25	the	the	DET
ap-1821	16	26	greatest	great	ADJ
ap-1821	16	27	common	common	ADJ
ap-1821	16	28	divisor	divisor	NOUN
ap-1821	16	29	of	of	ADP
ap-1821	16	30	two	two	NUM
ap-1821	16	31	integers	integer	NOUN
ap-1821	16	32	.	.	PUNCT
ap-1821	17	1	it	it	PRON
ap-1821	17	2	is	be	AUX
ap-1821	17	3	thus	thus	ADV
ap-1821	17	4	readily	readily	ADV
ap-1821	17	5	seen	see	VERB
ap-1821	17	6	that	that	SCONJ
ap-1821	17	7	if	if	SCONJ
ap-1821	17	8	the	the	DET
ap-1821	17	9	number	number	NOUN
ap-1821	17	10	x	x	PUNCT
ap-1821	17	11	is	be	AUX
ap-1821	17	12	rational	rational	ADJ
ap-1821	17	13	,	,	PUNCT
ap-1821	17	14	then	then	ADV
ap-1821	17	15	the	the	DET
ap-1821	17	16	algorithm	algorithm	NOUN
ap-1821	17	17	eventually	eventually	ADV
ap-1821	17	18	produces	produce	VERB
ap-1821	17	19	zeroes	zero	NOUN
ap-1821	17	20	,	,	PUNCT
ap-1821	17	21	i.e.	i.e.	X
ap-1821	17	22	there	there	PRON
ap-1821	17	23	exists	exist	VERB
ap-1821	17	24	n	n	PRON
ap-1821	17	25	∈	∈	PROPN
ap-1821	17	26	n	n	PRON
ap-1821	17	27	such	such	ADJ
ap-1821	17	28	that	that	SCONJ
ap-1821	17	29	an	an	DET
ap-1821	17	30	=	=	NOUN
ap-1821	17	31	0	0	NUM
ap-1821	17	32	for	for	ADP
ap-1821	17	33	all	all	DET
ap-1821	17	34	n	n	CCONJ
ap-1821	17	35	>	>	PUNCT
ap-1821	17	36	n	n	NOUN
ap-1821	17	37	,	,	PUNCT
ap-1821	17	38	thus	thus	ADV
ap-1821	17	39	x	x	X
ap-1821	17	40	=	=	SYM
ap-1821	17	41	a0	a0	PROPN
ap-1821	17	42	+	+	CCONJ
ap-1821	17	43	1	1	NUM
ap-1821	17	44	a1	a1	NOUN
ap-1821	17	45	+	+	CCONJ
ap-1821	17	46	1	1	NUM
ap-1821	17	47	a2	a2	NOUN
ap-1821	17	48	+	+	ADP
ap-1821	17	49	1	1	NUM
ap-1821	17	50	.	.	PUNCT
ap-1821	17	51	.	.	PUNCT
ap-1821	17	52	.	.	PUNCT
ap-1821	18	1	+	+	CCONJ
ap-1821	18	2	1	1	NUM
ap-1821	18	3	an−1	an−1	ADJ
ap-1821	18	4	+	+	NUM
ap-1821	18	5	1	1	NUM
ap-1821	18	6	an	an	PRON
ap-1821	18	7	.	.	PUNCT
ap-1821	19	1	(	(	PUNCT
ap-1821	19	2	1	1	X
ap-1821	19	3	)	)	PUNCT
ap-1821	19	4	we	we	PRON
ap-1821	19	5	write	write	VERB
ap-1821	19	6	x	x	PUNCT
ap-1821	20	1	=	=	PUNCT
ap-1821	21	1	[	[	X
ap-1821	21	2	a0	a0	NOUN
ap-1821	21	3	,	,	PUNCT
ap-1821	21	4	.	.	PUNCT
ap-1821	21	5	.	.	PUNCT
ap-1821	22	1	.	.	PUNCT
ap-1821	23	1	,	,	PUNCT
ap-1821	23	2	an	an	DET
ap-1821	23	3	]	]	X
ap-1821	23	4	.	.	PUNCT
ap-1821	24	1	on	on	ADP
ap-1821	24	2	the	the	DET
ap-1821	24	3	other	other	ADJ
ap-1821	24	4	hand	hand	NOUN
ap-1821	24	5	,	,	PUNCT
ap-1821	24	6	if	if	SCONJ
ap-1821	24	7	we	we	PRON
ap-1821	24	8	want	want	VERB
ap-1821	24	9	to	to	PART
ap-1821	24	10	find	find	VERB
ap-1821	24	11	an	an	DET
ap-1821	24	12	expression	expression	NOUN
ap-1821	24	13	of	of	ADP
ap-1821	24	14	the	the	DET
ap-1821	24	15	form	form	NOUN
ap-1821	24	16	(	(	PUNCT
ap-1821	24	17	1	1	NUM
ap-1821	24	18	)	)	PUNCT
ap-1821	24	19	with	with	ADP
ap-1821	24	20	a0	a0	PROPN
ap-1821	24	21	∈	∈	PROPN
ap-1821	24	22	z	z	PROPN
ap-1821	24	23	and	and	CCONJ
ap-1821	24	24	an	an	DET
ap-1821	24	25	∈	∈	PROPN
ap-1821	24	26	n	n	CCONJ
ap-1821	24	27	\	\	NOUN
ap-1821	24	28	{	{	PUNCT
ap-1821	24	29	0	0	NUM
ap-1821	24	30	}	}	PUNCT
ap-1821	24	31	otherwise	otherwise	ADV
ap-1821	24	32	,	,	PUNCT
ap-1821	24	33	then	then	ADV
ap-1821	24	34	there	there	PRON
ap-1821	24	35	are	be	VERB
ap-1821	24	36	exactly	exactly	ADV
ap-1821	24	37	two	two	NUM
ap-1821	24	38	of	of	ADP
ap-1821	24	39	them	they	PRON
ap-1821	24	40	–	–	PUNCT
ap-1821	24	41	the	the	DET
ap-1821	24	42	continued	continue	VERB
ap-1821	24	43	fraction	fraction	NOUN
ap-1821	24	44	[	[	X
ap-1821	24	45	a0	a0	NOUN
ap-1821	24	46	,	,	PUNCT
ap-1821	24	47	.	.	PUNCT
ap-1821	24	48	.	.	PUNCT
ap-1821	25	1	.	.	PUNCT
ap-1821	26	1	,	,	PUNCT
ap-1821	26	2	an	an	DET
ap-1821	26	3	]	]	X
ap-1821	26	4	and	and	CCONJ
ap-1821	26	5	x	x	X
ap-1821	26	6	=	=	SYM
ap-1821	26	7	a0	a0	PROPN
ap-1821	26	8	+	+	CCONJ
ap-1821	26	9	1	1	NUM
ap-1821	26	10	a1	a1	NOUN
ap-1821	26	11	+	+	CCONJ
ap-1821	26	12	1	1	NUM
ap-1821	26	13	a2	a2	NOUN
ap-1821	26	14	+	+	ADP
ap-1821	26	15	1	1	NUM
ap-1821	26	16	.	.	PUNCT
ap-1821	26	17	.	.	PUNCT
ap-1821	26	18	.	.	PUNCT
ap-1821	27	1	+	+	CCONJ
ap-1821	27	2	1	1	NUM
ap-1821	27	3	an−1	an−1	ADJ
ap-1821	27	4	+	+	NUM
ap-1821	27	5	1	1	NUM
ap-1821	27	6	(	(	PUNCT
ap-1821	27	7	an	an	DET
ap-1821	27	8	−	−	PROPN
ap-1821	27	9	1	1	NUM
ap-1821	27	10	)	)	PUNCT
ap-1821	27	11	+	+	CCONJ
ap-1821	27	12	1	1	NUM
ap-1821	27	13	1	1	NUM
ap-1821	27	14	.	.	PUNCT
ap-1821	28	1	if	if	SCONJ
ap-1821	28	2	the	the	DET
ap-1821	28	3	number	number	NOUN
ap-1821	28	4	x	x	PUNCT
ap-1821	28	5	is	be	AUX
ap-1821	28	6	irrational	irrational	ADJ
ap-1821	28	7	,	,	PUNCT
ap-1821	28	8	then	then	ADV
ap-1821	28	9	the	the	DET
ap-1821	28	10	sequence	sequence	NOUN
ap-1821	28	11	of	of	ADP
ap-1821	28	12	the	the	DET
ap-1821	28	13	so	so	ADV
ap-1821	28	14	-	-	PUNCT
ap-1821	28	15	called	call	VERB
ap-1821	28	16	convergents	convergent	NOUN
ap-1821	28	17	a0	a0	PROPN
ap-1821	28	18	,	,	PUNCT
ap-1821	28	19	a0	a0	PROPN
ap-1821	28	20	+	+	CCONJ
ap-1821	28	21	1	1	NUM
ap-1821	28	22	a1	a1	NOUN
ap-1821	28	23	,	,	PUNCT
ap-1821	28	24	.	.	PUNCT
ap-1821	28	25	.	.	PUNCT
ap-1821	29	1	.	.	PUNCT
ap-1821	30	1	,	,	PUNCT
ap-1821	30	2	a0	a0	PROPN
ap-1821	30	3	+	+	CCONJ
ap-1821	30	4	1	1	NUM
ap-1821	30	5	a1	a1	NOUN
ap-1821	30	6	+	+	CCONJ
ap-1821	30	7	1	1	NUM
ap-1821	30	8	a2	a2	NOUN
ap-1821	30	9	+	+	ADP
ap-1821	30	10	1	1	NUM
ap-1821	30	11	.	.	PUNCT
ap-1821	30	12	.	.	PUNCT
ap-1821	30	13	.	.	PUNCT
ap-1821	31	1	+	+	CCONJ
ap-1821	31	2	1	1	NUM
ap-1821	31	3	an−1	an−1	ADJ
ap-1821	31	4	+	+	NUM
ap-1821	31	5	1	1	NUM
ap-1821	31	6	an	an	DET
ap-1821	31	7	(	(	PUNCT
ap-1821	31	8	2	2	NUM
ap-1821	31	9	)	)	PUNCT
ap-1821	31	10	converges	converge	NOUN
ap-1821	31	11	to	to	ADP
ap-1821	31	12	x	x	PUNCT
ap-1821	31	13	for	for	ADP
ap-1821	31	14	n→∞.	n→∞.	PROPN
ap-1821	31	15	on	on	ADP
ap-1821	31	16	the	the	DET
ap-1821	31	17	other	other	ADJ
ap-1821	31	18	hand	hand	NOUN
ap-1821	31	19	,	,	PUNCT
ap-1821	31	20	every	every	DET
ap-1821	31	21	sequence	sequence	NOUN
ap-1821	31	22	of	of	ADP
ap-1821	31	23	rational	rational	ADJ
ap-1821	31	24	numbers	number	NOUN
ap-1821	31	25	of	of	ADP
ap-1821	31	26	the	the	DET
ap-1821	31	27	form	form	NOUN
ap-1821	31	28	(	(	PUNCT
ap-1821	31	29	2	2	NUM
ap-1821	31	30	)	)	PUNCT
ap-1821	31	31	with	with	ADP
ap-1821	31	32	322	322	NUM
ap-1821	31	33	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	ADJ
ap-1821	31	34	vol	vol	NOUN
ap-1821	31	35	.	.	PUNCT
ap-1821	32	1	53	53	NUM
ap-1821	32	2	no	no	NOUN
ap-1821	32	3	.	.	PUNCT
ap-1821	33	1	4/2013	4/2013	PROPN
ap-1821	33	2	continued	continue	VERB
ap-1821	33	3	fractions	fraction	NOUN
ap-1821	33	4	of	of	ADP
ap-1821	33	5	square	square	ADJ
ap-1821	33	6	roots	root	NOUN
ap-1821	33	7	of	of	ADP
ap-1821	33	8	natural	natural	ADJ
ap-1821	33	9	numbers	number	NOUN
ap-1821	33	10	a0	a0	PROPN
ap-1821	33	11	∈	∈	PROPN
ap-1821	33	12	z	z	PROPN
ap-1821	33	13	and	and	CCONJ
ap-1821	33	14	an	an	DET
ap-1821	33	15	∈	∈	PROPN
ap-1821	33	16	n	n	CCONJ
ap-1821	33	17	\	\	NOUN
ap-1821	33	18	{	{	PUNCT
ap-1821	33	19	0	0	NUM
ap-1821	33	20	}	}	PUNCT
ap-1821	33	21	converges	converge	NOUN
ap-1821	33	22	to	to	ADP
ap-1821	33	23	an	an	DET
ap-1821	33	24	irrational	irrational	ADJ
ap-1821	33	25	number	number	NOUN
ap-1821	33	26	(	(	PUNCT
ap-1821	33	27	and	and	CCONJ
ap-1821	33	28	for	for	ADP
ap-1821	33	29	every	every	DET
ap-1821	33	30	irrational	irrational	ADJ
ap-1821	33	31	number	number	NOUN
ap-1821	33	32	there	there	PRON
ap-1821	33	33	is	be	VERB
ap-1821	33	34	only	only	ADV
ap-1821	33	35	one	one	NUM
ap-1821	33	36	such	such	ADJ
ap-1821	33	37	sequence	sequence	NOUN
ap-1821	33	38	—	—	PUNCT
ap-1821	33	39	the	the	DET
ap-1821	33	40	sequence	sequence	NOUN
ap-1821	33	41	of	of	ADP
ap-1821	33	42	convergents	convergent	NOUN
ap-1821	33	43	)	)	PUNCT
ap-1821	33	44	.	.	PUNCT
ap-1821	34	1	we	we	PRON
ap-1821	34	2	write	write	VERB
ap-1821	34	3	x	x	PUNCT
ap-1821	34	4	=	=	PUNCT
ap-1821	35	1	[	[	X
ap-1821	35	2	a0	a0	NOUN
ap-1821	35	3	,	,	PUNCT
ap-1821	35	4	.	.	PUNCT
ap-1821	35	5	.	.	PUNCT
ap-1821	36	1	.	.	PUNCT
ap-1821	37	1	,	,	PUNCT
ap-1821	37	2	an	an	PRON
ap-1821	37	3	,	,	PUNCT
ap-1821	37	4	.	.	PUNCT
ap-1821	37	5	.	.	PUNCT
ap-1821	37	6	.	.	PUNCT
ap-1821	38	1	]	]	X
ap-1821	38	2	.	.	PUNCT
ap-1821	39	1	the	the	DET
ap-1821	39	2	convergents	convergent	NOUN
ap-1821	39	3	of	of	ADP
ap-1821	39	4	the	the	DET
ap-1821	39	5	continued	continue	VERB
ap-1821	39	6	fraction	fraction	NOUN
ap-1821	39	7	are	be	AUX
ap-1821	39	8	known	know	VERB
ap-1821	39	9	to	to	PART
ap-1821	39	10	represent	represent	VERB
ap-1821	39	11	irrational	irrational	ADJ
ap-1821	39	12	numbers	number	NOUN
ap-1821	39	13	better	well	ADJ
ap-1821	39	14	than	than	ADP
ap-1821	39	15	any	any	DET
ap-1821	39	16	other	other	ADJ
ap-1821	39	17	fractions	fraction	NOUN
ap-1821	39	18	.	.	PUNCT
ap-1821	40	1	theorem	theorem	VERB
ap-1821	40	2	2.2	2.2	NUM
ap-1821	40	3	(	(	PUNCT
ap-1821	40	4	lagrange	lagrange	NOUN
ap-1821	40	5	)	)	PUNCT
ap-1821	40	6	.	.	PUNCT
ap-1821	41	1	let	let	VERB
ap-1821	41	2	x	x	SYM
ap-1821	41	3	∈	∈	X
ap-1821	41	4	r\q	r\q	ADP
ap-1821	41	5	and	and	CCONJ
ap-1821	41	6	let	let	VERB
ap-1821	41	7	pn	pn	PROPN
ap-1821	41	8	qn	qn	PROPN
ap-1821	41	9	be	be	AUX
ap-1821	41	10	its	its	PRON
ap-1821	41	11	n	n	ADV
ap-1821	41	12	-	-	PUNCT
ap-1821	41	13	th	th	X
ap-1821	41	14	convergent	convergent	NOUN
ap-1821	41	15	(	(	PUNCT
ap-1821	41	16	where	where	SCONJ
ap-1821	41	17	pn	pn	PROPN
ap-1821	41	18	and	and	CCONJ
ap-1821	41	19	qn	qn	PROPN
ap-1821	41	20	are	be	AUX
ap-1821	41	21	coprime	coprime	ADJ
ap-1821	41	22	)	)	PUNCT
ap-1821	41	23	and	and	CCONJ
ap-1821	41	24	let	let	VERB
ap-1821	41	25	p	p	NOUN
ap-1821	41	26	q	q	VERB
ap-1821	41	27	with	with	ADP
ap-1821	41	28	p	p	NOUN
ap-1821	41	29	,	,	PUNCT
ap-1821	41	30	q	q	PROPN
ap-1821	41	31	∈	∈	PROPN
ap-1821	41	32	z	z	NOUN
ap-1821	41	33	be	be	AUX
ap-1821	41	34	distinct	distinct	ADJ
ap-1821	41	35	from	from	ADP
ap-1821	41	36	pn	pn	PROPN
ap-1821	41	37	qn	qn	PROPN
ap-1821	41	38	and	and	CCONJ
ap-1821	41	39	such	such	ADJ
ap-1821	41	40	that	that	SCONJ
ap-1821	41	41	0	0	NUM
ap-1821	41	42	<	<	X
ap-1821	41	43	q	q	X
ap-1821	41	44	≤	≤	NUM
ap-1821	41	45	qn	qn	NOUN
ap-1821	41	46	.	.	PUNCT
ap-1821	42	1	then∣∣∣x−	then∣∣∣x−	PROPN
ap-1821	42	2	pn	pn	PROPN
ap-1821	42	3	qn	qn	INTJ
ap-1821	42	4	∣∣∣	∣∣∣	NOUN
ap-1821	42	5	<	<	X
ap-1821	42	6	∣∣∣x−	∣∣∣x−	DET
ap-1821	42	7	p	p	X
ap-1821	42	8	q	q	PROPN
ap-1821	42	9	∣∣∣.	∣∣∣.	PROPN
ap-1821	42	10	it	it	PRON
ap-1821	42	11	is	be	AUX
ap-1821	42	12	also	also	ADV
ap-1821	42	13	known	know	VERB
ap-1821	42	14	how	how	SCONJ
ap-1821	42	15	well	well	ADV
ap-1821	42	16	continued	continue	VERB
ap-1821	42	17	fractions	fraction	NOUN
ap-1821	42	18	approximate	approximate	ADJ
ap-1821	42	19	irrational	irrational	ADJ
ap-1821	42	20	numbers	number	NOUN
ap-1821	42	21	.	.	PUNCT
ap-1821	43	1	theorem	theorem	VERB
ap-1821	43	2	2.3	2.3	NUM
ap-1821	43	3	.	.	PUNCT
ap-1821	44	1	let	let	VERB
ap-1821	44	2	x	x	PUNCT
ap-1821	44	3	∈	∈	PROPN
ap-1821	44	4	r	r	NOUN
ap-1821	44	5	\	\	NOUN
ap-1821	44	6	q	q	NOUN
ap-1821	44	7	and	and	CCONJ
ap-1821	44	8	let	let	VERB
ap-1821	44	9	pn	pn	PROPN
ap-1821	44	10	qn	qn	PROPN
ap-1821	44	11	be	be	AUX
ap-1821	44	12	its	its	PRON
ap-1821	44	13	nth	nth	NOUN
ap-1821	44	14	convergent	convergent	NOUN
ap-1821	44	15	(	(	PUNCT
ap-1821	44	16	where	where	SCONJ
ap-1821	44	17	pn	pn	PROPN
ap-1821	44	18	and	and	CCONJ
ap-1821	44	19	qn	qn	PROPN
ap-1821	44	20	are	be	AUX
ap-1821	44	21	coprime	coprime	ADJ
ap-1821	44	22	)	)	PUNCT
ap-1821	44	23	.	.	PUNCT
ap-1821	45	1	then	then	ADV
ap-1821	45	2	either∣∣∣x−	either∣∣∣x−	PROPN
ap-1821	45	3	pn	pn	AUX
ap-1821	45	4	qn	qn	INTJ
ap-1821	45	5	∣∣∣	∣∣∣	PROPN
ap-1821	45	6	<	<	X
ap-1821	45	7	1	1	NUM
ap-1821	45	8	2q2	2q2	NUM
ap-1821	45	9	n	n	ADP
ap-1821	45	10	or	or	CCONJ
ap-1821	45	11	∣∣∣x−	∣∣∣x−	PRON
ap-1821	45	12	pn+1	pn+1	NOUN
ap-1821	45	13	qn+1	qn+1	VERB
ap-1821	45	14	∣∣∣	∣∣∣	NOUN
ap-1821	45	15	<	<	X
ap-1821	45	16	1	1	NUM
ap-1821	45	17	2q2	2q2	NUM
ap-1821	45	18	n+1	n+1	NUM
ap-1821	45	19	.	.	PUNCT
ap-1821	46	1	and	and	CCONJ
ap-1821	46	2	in	in	ADP
ap-1821	46	3	a	a	DET
ap-1821	46	4	certain	certain	ADJ
ap-1821	46	5	way	way	NOUN
ap-1821	46	6	,	,	PUNCT
ap-1821	46	7	only	only	ADV
ap-1821	46	8	continued	continue	VERB
ap-1821	46	9	fractions	fraction	NOUN
ap-1821	46	10	get	get	VERB
ap-1821	46	11	very	very	ADV
ap-1821	46	12	close	close	ADJ
ap-1821	46	13	to	to	ADP
ap-1821	46	14	irrational	irrational	ADJ
ap-1821	46	15	numbers	number	NOUN
ap-1821	46	16	.	.	PUNCT
ap-1821	47	1	theorem	theorem	VERB
ap-1821	47	2	2.4	2.4	NUM
ap-1821	47	3	(	(	PUNCT
ap-1821	47	4	legendre	legendre	PROPN
ap-1821	47	5	)	)	PUNCT
ap-1821	47	6	.	.	PUNCT
ap-1821	48	1	let	let	VERB
ap-1821	48	2	x	x	PUNCT
ap-1821	48	3	∈	∈	PROPN
ap-1821	48	4	r	r	NOUN
ap-1821	48	5	\	\	NOUN
ap-1821	48	6	q	q	NOUN
ap-1821	48	7	and	and	CCONJ
ap-1821	48	8	let	let	VERB
ap-1821	48	9	p	p	NOUN
ap-1821	48	10	q	q	VERB
ap-1821	48	11	with	with	ADP
ap-1821	48	12	p	p	NOUN
ap-1821	48	13	,	,	PUNCT
ap-1821	48	14	q	q	PROPN
ap-1821	48	15	∈	∈	PROPN
ap-1821	48	16	z	z	NOUN
ap-1821	48	17	satisfy	satisfy	NOUN
ap-1821	48	18	|x	|x	NOUN
ap-1821	49	1	−	−	PROPN
ap-1821	49	2	p	p	NOUN
ap-1821	49	3	q	q	NOUN
ap-1821	50	1	|	|	ADV
ap-1821	50	2	<	<	X
ap-1821	50	3	1	1	NUM
ap-1821	50	4	2q2	2q2	NUM
ap-1821	50	5	.	.	PUNCT
ap-1821	51	1	then	then	ADV
ap-1821	51	2	p	p	X
ap-1821	51	3	q	q	PROPN
ap-1821	51	4	is	be	AUX
ap-1821	51	5	a	a	DET
ap-1821	51	6	convergent	convergent	NOUN
ap-1821	51	7	of	of	ADP
ap-1821	51	8	x.	x.	NOUN
ap-1821	51	9	2.1	2.1	NUM
ap-1821	51	10	.	.	PUNCT
ap-1821	52	1	continued	continue	VERB
ap-1821	52	2	fractions	fraction	NOUN
ap-1821	52	3	and	and	CCONJ
ap-1821	52	4	continuants	continuant	VERB
ap-1821	52	5	the	the	DET
ap-1821	52	6	convergents	convergent	NOUN
ap-1821	52	7	of	of	ADP
ap-1821	52	8	continued	continue	VERB
ap-1821	52	9	fractions	fraction	NOUN
ap-1821	52	10	are	be	AUX
ap-1821	52	11	closely	closely	ADV
ap-1821	52	12	related	relate	VERB
ap-1821	52	13	to	to	ADP
ap-1821	52	14	the	the	DET
ap-1821	52	15	so	so	ADV
ap-1821	52	16	-	-	PUNCT
ap-1821	52	17	called	call	VERB
ap-1821	52	18	continuants	continuant	NOUN
ap-1821	52	19	kn(x1	kn(x1	NOUN
ap-1821	52	20	,	,	PUNCT
ap-1821	52	21	.	.	PUNCT
ap-1821	52	22	.	.	PUNCT
ap-1821	53	1	.	.	PUNCT
ap-1821	54	1	,	,	PUNCT
ap-1821	54	2	xn	xn	PROPN
ap-1821	54	3	)	)	PUNCT
ap-1821	54	4	.	.	PUNCT
ap-1821	55	1	theorem	theorem	VERB
ap-1821	55	2	2.5	2.5	NUM
ap-1821	55	3	.	.	PUNCT
ap-1821	56	1	let	let	VERB
ap-1821	56	2	a0	a0	PROPN
ap-1821	56	3	∈	∈	PROPN
ap-1821	56	4	r	r	PROPN
ap-1821	56	5	,	,	PUNCT
ap-1821	56	6	ai	ai	VERB
ap-1821	56	7	>	>	X
ap-1821	56	8	0	0	NUM
ap-1821	56	9	,	,	PUNCT
ap-1821	56	10	i	i	PRON
ap-1821	56	11	∈	∈	PROPN
ap-1821	56	12	n.	n.	NOUN
ap-1821	56	13	then	then	ADV
ap-1821	56	14	it	it	PRON
ap-1821	56	15	holds	hold	VERB
ap-1821	56	16	a0	a0	NOUN
ap-1821	56	17	+	+	CCONJ
ap-1821	56	18	1	1	NUM
ap-1821	56	19	a1	a1	NOUN
ap-1821	56	20	+	+	CCONJ
ap-1821	56	21	1	1	NUM
ap-1821	56	22	.	.	PUNCT
ap-1821	56	23	.	.	PUNCT
ap-1821	56	24	.	.	PUNCT
ap-1821	57	1	+	+	CCONJ
ap-1821	57	2	1	1	NUM
ap-1821	57	3	an	an	DET
ap-1821	57	4	=	=	SYM
ap-1821	57	5	kn+1(a0	kn+1(a0	NOUN
ap-1821	57	6	,	,	PUNCT
ap-1821	57	7	a1	a1	NOUN
ap-1821	57	8	,	,	PUNCT
ap-1821	57	9	.	.	PUNCT
ap-1821	57	10	.	.	PUNCT
ap-1821	57	11	.	.	PUNCT
ap-1821	58	1	,	,	PUNCT
ap-1821	58	2	an	an	X
ap-1821	58	3	)	)	PUNCT
ap-1821	58	4	kn(a1	kn(a1	NOUN
ap-1821	58	5	,	,	PUNCT
ap-1821	58	6	.	.	PUNCT
ap-1821	58	7	.	.	PUNCT
ap-1821	59	1	.	.	PUNCT
ap-1821	60	1	,	,	PUNCT
ap-1821	60	2	an	an	X
ap-1821	60	3	)	)	PUNCT
ap-1821	60	4	,	,	PUNCT
ap-1821	60	5	where	where	SCONJ
ap-1821	60	6	the	the	DET
ap-1821	60	7	polynomial	polynomial	ADJ
ap-1821	60	8	kn(x1	kn(x1	NOUN
ap-1821	60	9	,	,	PUNCT
ap-1821	60	10	.	.	PUNCT
ap-1821	60	11	.	.	PUNCT
ap-1821	60	12	.	.	PUNCT
ap-1821	61	1	,	,	PUNCT
ap-1821	61	2	xn	xn	X
ap-1821	61	3	)	)	PUNCT
ap-1821	61	4	is	be	AUX
ap-1821	61	5	given	give	VERB
ap-1821	61	6	by	by	ADP
ap-1821	61	7	the	the	DET
ap-1821	61	8	recurrence	recurrence	NOUN
ap-1821	61	9	relation	relation	NOUN
ap-1821	61	10	k−1	k−1	PROPN
ap-1821	61	11	=	=	SYM
ap-1821	61	12	0	0	PROPN
ap-1821	61	13	,	,	PUNCT
ap-1821	61	14	k0	k0	PROPN
ap-1821	61	15	=	=	PROPN
ap-1821	61	16	1	1	NUM
ap-1821	61	17	and	and	CCONJ
ap-1821	61	18	for	for	ADP
ap-1821	61	19	n	n	PRON
ap-1821	61	20	≥	≥	NOUN
ap-1821	61	21	1	1	NUM
ap-1821	61	22	by	by	ADP
ap-1821	61	23	kn(x1	kn(x1	NUM
ap-1821	61	24	,	,	PUNCT
ap-1821	61	25	.	.	PUNCT
ap-1821	61	26	.	.	PUNCT
ap-1821	61	27	.	.	PUNCT
ap-1821	62	1	,	,	PUNCT
ap-1821	62	2	xn	xn	X
ap-1821	62	3	)	)	PUNCT
ap-1821	62	4	=	=	SYM
ap-1821	62	5	kn−2(x1	kn−2(x1	PROPN
ap-1821	62	6	,	,	PUNCT
ap-1821	62	7	.	.	PUNCT
ap-1821	62	8	.	.	PUNCT
ap-1821	62	9	.	.	PUNCT
ap-1821	63	1	,	,	PUNCT
ap-1821	63	2	xn−2	xn−2	PROPN
ap-1821	63	3	)	)	PUNCT
ap-1821	64	1	+	+	CCONJ
ap-1821	64	2	xnkn−1(x1	xnkn−1(x1	NOUN
ap-1821	64	3	,	,	PUNCT
ap-1821	64	4	.	.	PUNCT
ap-1821	64	5	.	.	PUNCT
ap-1821	65	1	.	.	PUNCT
ap-1821	66	1	,	,	PUNCT
ap-1821	66	2	xn−1	xn−1	PROPN
ap-1821	66	3	)	)	PUNCT
ap-1821	66	4	.	.	PUNCT
ap-1821	67	1	corollary	corollary	ADJ
ap-1821	67	2	2.6	2.6	NUM
ap-1821	67	3	.	.	PUNCT
ap-1821	68	1	let	let	VERB
ap-1821	68	2	[	[	X
ap-1821	68	3	a0	a0	NOUN
ap-1821	68	4	,	,	PUNCT
ap-1821	68	5	.	.	PUNCT
ap-1821	68	6	.	.	PUNCT
ap-1821	69	1	.	.	PUNCT
ap-1821	70	1	,	,	PUNCT
ap-1821	70	2	an	an	PRON
ap-1821	70	3	,	,	PUNCT
ap-1821	70	4	.	.	PUNCT
ap-1821	70	5	.	.	PUNCT
ap-1821	70	6	.	.	PUNCT
ap-1821	71	1	]	]	PUNCT
ap-1821	71	2	be	be	AUX
ap-1821	71	3	the	the	DET
ap-1821	71	4	continued	continue	VERB
ap-1821	71	5	fraction	fraction	NOUN
ap-1821	71	6	of	of	ADP
ap-1821	71	7	an	an	DET
ap-1821	71	8	irrational	irrational	ADJ
ap-1821	71	9	number	number	NOUN
ap-1821	71	10	x.	x.	NOUN
ap-1821	72	1	then	then	ADV
ap-1821	72	2	its	its	PRON
ap-1821	72	3	n	n	CCONJ
ap-1821	72	4	-	-	PUNCT
ap-1821	72	5	th	th	VERB
ap-1821	72	6	convergent	convergent	NOUN
ap-1821	72	7	pn	pn	PROPN
ap-1821	72	8	qn	qn	PROPN
ap-1821	72	9	satisfies	satisfie	NOUN
ap-1821	72	10	pn	pn	PROPN
ap-1821	72	11	=	=	SYM
ap-1821	72	12	kn+1(a0	kn+1(a0	X
ap-1821	72	13	,	,	PUNCT
ap-1821	72	14	.	.	PUNCT
ap-1821	72	15	.	.	PUNCT
ap-1821	73	1	.	.	PUNCT
ap-1821	74	1	,	,	PUNCT
ap-1821	74	2	an	an	X
ap-1821	74	3	)	)	PUNCT
ap-1821	74	4	,	,	PUNCT
ap-1821	74	5	qn	qn	NOUN
ap-1821	74	6	=	=	PUNCT
ap-1821	74	7	kn(a1	kn(a1	X
ap-1821	74	8	,	,	PUNCT
ap-1821	74	9	.	.	PUNCT
ap-1821	74	10	.	.	PUNCT
ap-1821	75	1	.	.	PUNCT
ap-1821	76	1	,	,	PUNCT
ap-1821	76	2	an	an	X
ap-1821	76	3	)	)	PUNCT
ap-1821	76	4	.	.	PUNCT
ap-1821	76	5	theorem	theorem	ADJ
ap-1821	76	6	2.7	2.7	NUM
ap-1821	76	7	.	.	PUNCT
ap-1821	77	1	for	for	ADP
ap-1821	77	2	every	every	DET
ap-1821	77	3	n	n	PRON
ap-1821	77	4	∈	∈	PROPN
ap-1821	77	5	n	n	NOUN
ap-1821	77	6	and	and	CCONJ
ap-1821	77	7	a1	a1	NOUN
ap-1821	77	8	,	,	PUNCT
ap-1821	77	9	.	.	PUNCT
ap-1821	77	10	.	.	PUNCT
ap-1821	77	11	.	.	PUNCT
ap-1821	78	1	,	,	PUNCT
ap-1821	78	2	an	an	DET
ap-1821	78	3	∈	∈	PROPN
ap-1821	78	4	r	r	NOUN
ap-1821	78	5	,	,	PUNCT
ap-1821	78	6	we	we	PRON
ap-1821	78	7	have	have	VERB
ap-1821	78	8	kn(a1	kn(a1	NOUN
ap-1821	78	9	,	,	PUNCT
ap-1821	78	10	.	.	PUNCT
ap-1821	78	11	.	.	PUNCT
ap-1821	79	1	.	.	PUNCT
ap-1821	80	1	,	,	PUNCT
ap-1821	80	2	an	an	X
ap-1821	80	3	)	)	PUNCT
ap-1821	80	4	=	=	SYM
ap-1821	80	5	kn(an	kn(an	PROPN
ap-1821	80	6	,	,	PUNCT
ap-1821	80	7	.	.	PUNCT
ap-1821	80	8	.	.	PUNCT
ap-1821	81	1	.	.	PUNCT
ap-1821	82	1	,	,	PUNCT
ap-1821	82	2	a1	a1	PROPN
ap-1821	82	3	)	)	PUNCT
ap-1821	82	4	.	.	PUNCT
ap-1821	83	1	2.2	2.2	NUM
ap-1821	83	2	.	.	PUNCT
ap-1821	84	1	continued	continue	VERB
ap-1821	84	2	fractions	fraction	NOUN
ap-1821	84	3	of	of	ADP
ap-1821	84	4	quadratic	quadratic	ADJ
ap-1821	84	5	numbers	number	NOUN
ap-1821	84	6	we	we	PRON
ap-1821	84	7	will	will	AUX
ap-1821	84	8	call	call	VERB
ap-1821	84	9	a	a	DET
ap-1821	84	10	quadratic	quadratic	ADJ
ap-1821	84	11	irrational	irrational	ADJ
ap-1821	84	12	an	an	DET
ap-1821	84	13	irrational	irrational	ADJ
ap-1821	84	14	root	root	NOUN
ap-1821	84	15	α	α	NOUN
ap-1821	84	16	of	of	ADP
ap-1821	84	17	a	a	DET
ap-1821	84	18	quadratic	quadratic	ADJ
ap-1821	84	19	equation	equation	NOUN
ap-1821	84	20	ax2	ax2	NOUN
ap-1821	84	21	+	+	CCONJ
ap-1821	84	22	bx+	bx+	NOUN
ap-1821	84	23	c	c	NOUN
ap-1821	84	24	=	=	SYM
ap-1821	84	25	0	0	PROPN
ap-1821	84	26	,	,	PUNCT
ap-1821	84	27	where	where	SCONJ
ap-1821	84	28	a	a	DET
ap-1821	84	29	,	,	PUNCT
ap-1821	84	30	b	b	NOUN
ap-1821	84	31	,	,	PUNCT
ap-1821	84	32	c	c	PROPN
ap-1821	84	33	∈	∈	PROPN
ap-1821	84	34	z.	z.	PROPN
ap-1821	84	35	the	the	DET
ap-1821	84	36	second	second	ADJ
ap-1821	84	37	root	root	NOUN
ap-1821	84	38	of	of	ADP
ap-1821	84	39	the	the	DET
ap-1821	84	40	equation	equation	NOUN
ap-1821	84	41	will	will	AUX
ap-1821	84	42	be	be	AUX
ap-1821	84	43	denoted	denote	VERB
ap-1821	84	44	α′	α′	NUM
ap-1821	84	45	and	and	CCONJ
ap-1821	84	46	called	call	VERB
ap-1821	84	47	the	the	DET
ap-1821	84	48	(	(	PUNCT
ap-1821	84	49	algebraic	algebraic	ADJ
ap-1821	84	50	)	)	PUNCT
ap-1821	84	51	conjugate	conjugate	NOUN
ap-1821	84	52	of	of	ADP
ap-1821	84	53	α	α	NOUN
ap-1821	84	54	.	.	PUNCT
ap-1821	85	1	in	in	ADP
ap-1821	85	2	order	order	NOUN
ap-1821	85	3	to	to	PART
ap-1821	85	4	state	state	VERB
ap-1821	85	5	the	the	DET
ap-1821	85	6	theorem	theorem	NOUN
ap-1821	85	7	describing	describe	VERB
ap-1821	85	8	continued	continue	VERB
ap-1821	85	9	fractions	fraction	NOUN
ap-1821	85	10	of	of	ADP
ap-1821	85	11	quadratic	quadratic	ADJ
ap-1821	85	12	irrationals	irrational	NOUN
ap-1821	85	13	,	,	PUNCT
ap-1821	85	14	we	we	PRON
ap-1821	85	15	need	need	VERB
ap-1821	85	16	to	to	PART
ap-1821	85	17	recall	recall	VERB
ap-1821	85	18	that	that	SCONJ
ap-1821	85	19	a	a	DET
ap-1821	85	20	continued	continue	VERB
ap-1821	85	21	fraction	fraction	NOUN
ap-1821	85	22	[	[	X
ap-1821	85	23	a0	a0	NOUN
ap-1821	85	24	,	,	PUNCT
ap-1821	85	25	.	.	PUNCT
ap-1821	85	26	.	.	PUNCT
ap-1821	86	1	.	.	PUNCT
ap-1821	87	1	,	,	PUNCT
ap-1821	87	2	an	an	PRON
ap-1821	87	3	,	,	PUNCT
ap-1821	87	4	.	.	PUNCT
ap-1821	87	5	.	.	PUNCT
ap-1821	87	6	.	.	PUNCT
ap-1821	88	1	]	]	PUNCT
ap-1821	88	2	is	be	AUX
ap-1821	88	3	called	call	VERB
ap-1821	88	4	eventually	eventually	ADV
ap-1821	88	5	periodic	periodic	ADJ
ap-1821	88	6	if	if	SCONJ
ap-1821	88	7	[	[	X
ap-1821	88	8	a0	a0	NOUN
ap-1821	88	9	,	,	PUNCT
ap-1821	88	10	.	.	PUNCT
ap-1821	88	11	.	.	PUNCT
ap-1821	89	1	.	.	PUNCT
ap-1821	90	1	,	,	PUNCT
ap-1821	90	2	an	an	PRON
ap-1821	90	3	,	,	PUNCT
ap-1821	90	4	.	.	PUNCT
ap-1821	90	5	.	.	PUNCT
ap-1821	90	6	.	.	PUNCT
ap-1821	90	7	]	]	PUNCT
ap-1821	91	1	=	=	PUNCT
ap-1821	92	1	[	[	X
ap-1821	92	2	a0	a0	NOUN
ap-1821	92	3	,	,	PUNCT
ap-1821	92	4	.	.	PUNCT
ap-1821	92	5	.	.	PUNCT
ap-1821	93	1	.	.	PUNCT
ap-1821	94	1	,	,	PUNCT
ap-1821	94	2	ak−1	ak−1	PROPN
ap-1821	94	3	,	,	PUNCT
ap-1821	94	4	ak	ak	PROPN
ap-1821	94	5	,	,	PUNCT
ap-1821	94	6	.	.	PUNCT
ap-1821	94	7	.	.	PUNCT
ap-1821	95	1	.	.	PUNCT
ap-1821	96	1	,	,	PUNCT
ap-1821	96	2	a	a	DET
ap-1821	96	3	`	`	PUNCT
ap-1821	96	4	]	]	PUNCT
ap-1821	96	5	starts	start	VERB
ap-1821	96	6	with	with	ADP
ap-1821	96	7	a	a	DET
ap-1821	96	8	preperiod	preperiod	NOUN
ap-1821	96	9	a0	a0	NOUN
ap-1821	96	10	,	,	PUNCT
ap-1821	96	11	.	.	PUNCT
ap-1821	96	12	.	.	PUNCT
ap-1821	97	1	.	.	PUNCT
ap-1821	98	1	,	,	PUNCT
ap-1821	98	2	ak−1	ak−1	INTJ
ap-1821	98	3	and	and	CCONJ
ap-1821	98	4	then	then	ADV
ap-1821	98	5	a	a	DET
ap-1821	98	6	period	period	NOUN
ap-1821	98	7	ak	ak	PROPN
ap-1821	98	8	,	,	PUNCT
ap-1821	98	9	.	.	PUNCT
ap-1821	98	10	.	.	PUNCT
ap-1821	99	1	.	.	PUNCT
ap-1821	100	1	,	,	PUNCT
ap-1821	100	2	a	a	PRON
ap-1821	100	3	`	`	PUNCT
ap-1821	100	4	is	be	AUX
ap-1821	100	5	repeated	repeat	VERB
ap-1821	100	6	an	an	DET
ap-1821	100	7	infinite	infinite	ADJ
ap-1821	100	8	number	number	NOUN
ap-1821	100	9	of	of	ADP
ap-1821	100	10	times	time	NOUN
ap-1821	100	11	.	.	PUNCT
ap-1821	101	1	it	it	PRON
ap-1821	101	2	is	be	AUX
ap-1821	101	3	called	call	VERB
ap-1821	101	4	purely	purely	ADV
ap-1821	101	5	periodic	periodic	ADJ
ap-1821	101	6	if	if	SCONJ
ap-1821	101	7	[	[	X
ap-1821	101	8	a0	a0	NOUN
ap-1821	101	9	,	,	PUNCT
ap-1821	101	10	.	.	PUNCT
ap-1821	101	11	.	.	PUNCT
ap-1821	102	1	.	.	PUNCT
ap-1821	103	1	,	,	PUNCT
ap-1821	103	2	an	an	PRON
ap-1821	103	3	,	,	PUNCT
ap-1821	103	4	.	.	PUNCT
ap-1821	103	5	.	.	PUNCT
ap-1821	103	6	.	.	PUNCT
ap-1821	103	7	]	]	PUNCT
ap-1821	104	1	=	=	PUNCT
ap-1821	105	1	[	[	X
ap-1821	105	2	a0	a0	NOUN
ap-1821	105	3	,	,	PUNCT
ap-1821	105	4	.	.	PUNCT
ap-1821	105	5	.	.	PUNCT
ap-1821	106	1	.	.	PUNCT
ap-1821	107	1	,	,	PUNCT
ap-1821	107	2	a	a	PRON
ap-1821	107	3	`	`	PUNCT
ap-1821	107	4	]	]	X
ap-1821	107	5	,	,	PUNCT
ap-1821	107	6	i.e.	i.e.	X
ap-1821	107	7	,	,	PUNCT
ap-1821	107	8	if	if	SCONJ
ap-1821	107	9	the	the	DET
ap-1821	107	10	preperiod	preperiod	NOUN
ap-1821	107	11	is	be	AUX
ap-1821	107	12	empty	empty	ADJ
ap-1821	107	13	.	.	PUNCT
ap-1821	108	1	theorem	theorem	VERB
ap-1821	108	2	2.8	2.8	NUM
ap-1821	108	3	(	(	PUNCT
ap-1821	108	4	lagrange	lagrange	NOUN
ap-1821	108	5	)	)	PUNCT
ap-1821	108	6	.	.	PUNCT
ap-1821	109	1	let	let	VERB
ap-1821	109	2	α	α	PRON
ap-1821	109	3	∈	∈	NOUN
ap-1821	109	4	r	r	NOUN
ap-1821	109	5	\	\	NOUN
ap-1821	109	6	q.	q.	NOUN
ap-1821	109	7	the	the	DET
ap-1821	109	8	continued	continue	VERB
ap-1821	109	9	fraction	fraction	NOUN
ap-1821	109	10	of	of	ADP
ap-1821	109	11	α	α	PROPN
ap-1821	109	12	is	be	AUX
ap-1821	109	13	eventually	eventually	ADV
ap-1821	109	14	periodic	periodic	ADJ
ap-1821	109	15	if	if	SCONJ
ap-1821	110	1	and	and	CCONJ
ap-1821	110	2	only	only	ADV
ap-1821	110	3	if	if	SCONJ
ap-1821	110	4	α	α	PRON
ap-1821	110	5	is	be	AUX
ap-1821	110	6	a	a	DET
ap-1821	110	7	quadratic	quadratic	ADJ
ap-1821	110	8	irrational	irrational	ADJ
ap-1821	110	9	.	.	PUNCT
ap-1821	111	1	theorem	theorem	ADJ
ap-1821	111	2	2.9	2.9	NUM
ap-1821	111	3	(	(	PUNCT
ap-1821	111	4	galois	galois	PROPN
ap-1821	111	5	)	)	PUNCT
ap-1821	111	6	.	.	PUNCT
ap-1821	112	1	let	let	VERB
ap-1821	112	2	α	α	PRON
ap-1821	112	3	be	be	AUX
ap-1821	112	4	a	a	DET
ap-1821	112	5	quadratic	quadratic	ADJ
ap-1821	112	6	irrational	irrational	ADJ
ap-1821	112	7	and	and	CCONJ
ap-1821	112	8	α′	α′	NUM
ap-1821	112	9	its	its	PRON
ap-1821	112	10	conjugate	conjugate	NOUN
ap-1821	112	11	.	.	PUNCT
ap-1821	113	1	the	the	DET
ap-1821	113	2	continued	continue	VERB
ap-1821	113	3	fraction	fraction	NOUN
ap-1821	113	4	of	of	ADP
ap-1821	113	5	α	α	PROPN
ap-1821	113	6	is	be	AUX
ap-1821	113	7	purely	purely	ADV
ap-1821	113	8	periodic	periodic	ADJ
ap-1821	113	9	if	if	SCONJ
ap-1821	113	10	and	and	CCONJ
ap-1821	113	11	only	only	ADV
ap-1821	113	12	if	if	SCONJ
ap-1821	113	13	α	α	PROPN
ap-1821	113	14	>	>	X
ap-1821	113	15	1	1	NUM
ap-1821	113	16	and	and	CCONJ
ap-1821	113	17	α′	α′	NUM
ap-1821	113	18	∈	∈	NOUN
ap-1821	113	19	(	(	PUNCT
ap-1821	113	20	−1	−1	NOUN
ap-1821	113	21	,	,	PUNCT
ap-1821	113	22	0	0	NUM
ap-1821	113	23	)	)	PUNCT
ap-1821	113	24	.	.	PUNCT
ap-1821	114	1	example	example	NOUN
ap-1821	115	1	2.10	2.10	NUM
ap-1821	115	2	.	.	PUNCT
ap-1821	116	1	let	let	VERB
ap-1821	116	2	α	α	NOUN
ap-1821	116	3	=	=	SYM
ap-1821	116	4	1	1	NUM
ap-1821	116	5	+	+	NUM
ap-1821	116	6	√	√	NUM
ap-1821	116	7	5	5	NUM
ap-1821	116	8	2	2	NUM
ap-1821	116	9	,	,	PUNCT
ap-1821	116	10	i.e.	i.e.	X
ap-1821	116	11	,	,	PUNCT
ap-1821	116	12	the	the	DET
ap-1821	116	13	so	so	ADV
ap-1821	116	14	-	-	PUNCT
ap-1821	116	15	called	call	VERB
ap-1821	116	16	golden	golden	ADJ
ap-1821	116	17	ratio	ratio	NOUN
ap-1821	116	18	,	,	PUNCT
ap-1821	116	19	then	then	ADV
ap-1821	116	20	it	it	PRON
ap-1821	116	21	is	be	AUX
ap-1821	116	22	the	the	DET
ap-1821	116	23	root	root	NOUN
ap-1821	116	24	of	of	ADP
ap-1821	116	25	x2−x−	x2−x−	X
ap-1821	116	26	1	1	NUM
ap-1821	116	27	=	=	SYM
ap-1821	116	28	0	0	NUM
ap-1821	116	29	and	and	CCONJ
ap-1821	116	30	α′	α′	NUM
ap-1821	116	31	=	=	SYM
ap-1821	117	1	1−	1−	NUM
ap-1821	117	2	√	√	NUM
ap-1821	117	3	5	5	NUM
ap-1821	117	4	2	2	NUM
ap-1821	117	5	∈	∈	NOUN
ap-1821	117	6	(	(	PUNCT
ap-1821	117	7	−1	−1	NOUN
ap-1821	117	8	,	,	PUNCT
ap-1821	117	9	0	0	NUM
ap-1821	117	10	)	)	PUNCT
ap-1821	117	11	.	.	PUNCT
ap-1821	118	1	the	the	DET
ap-1821	118	2	continued	continue	VERB
ap-1821	118	3	fraction	fraction	NOUN
ap-1821	118	4	of	of	ADP
ap-1821	118	5	α	α	PROPN
ap-1821	118	6	is	be	AUX
ap-1821	118	7	indeed	indeed	ADV
ap-1821	118	8	purely	purely	ADV
ap-1821	118	9	periodic	periodic	ADJ
ap-1821	118	10	since	since	SCONJ
ap-1821	118	11	α	α	NOUN
ap-1821	118	12	=	=	SYM
ap-1821	118	13	1	1	NUM
ap-1821	118	14	+	+	NUM
ap-1821	118	15	−1	−1	NOUN
ap-1821	119	1	+	+	CCONJ
ap-1821	119	2	√	√	NUM
ap-1821	119	3	5	5	NUM
ap-1821	119	4	2	2	NUM
ap-1821	119	5	=	=	SYM
ap-1821	119	6	1	1	NUM
ap-1821	119	7	+	+	CCONJ
ap-1821	119	8	1	1	NUM
ap-1821	119	9	1	1	NUM
ap-1821	119	10	+	+	NUM
ap-1821	119	11	√	√	NUM
ap-1821	119	12	5	5	NUM
ap-1821	119	13	2	2	NUM
ap-1821	119	14	=	=	SYM
ap-1821	119	15	1	1	NUM
ap-1821	119	16	+	+	NUM
ap-1821	119	17	1	1	NUM
ap-1821	119	18	α	α	NOUN
ap-1821	119	19	,	,	PUNCT
ap-1821	119	20	consequently	consequently	ADV
ap-1821	119	21	α	α	X
ap-1821	119	22	=	=	PUNCT
ap-1821	120	1	[	[	X
ap-1821	120	2	1	1	NUM
ap-1821	120	3	]	]	PUNCT
ap-1821	120	4	.	.	PUNCT
ap-1821	121	1	in	in	ADP
ap-1821	121	2	the	the	DET
ap-1821	121	3	sequel	sequel	NOUN
ap-1821	121	4	when	when	SCONJ
ap-1821	121	5	we	we	PRON
ap-1821	121	6	restrict	restrict	VERB
ap-1821	121	7	our	our	PRON
ap-1821	121	8	consideration	consideration	NOUN
ap-1821	121	9	to	to	ADP
ap-1821	121	10	square	square	ADJ
ap-1821	121	11	roots	root	NOUN
ap-1821	121	12	of	of	ADP
ap-1821	121	13	natural	natural	ADJ
ap-1821	121	14	numbers	number	NOUN
ap-1821	121	15	,	,	PUNCT
ap-1821	121	16	we	we	PRON
ap-1821	121	17	will	will	AUX
ap-1821	121	18	make	make	VERB
ap-1821	121	19	use	use	NOUN
ap-1821	121	20	of	of	ADP
ap-1821	121	21	the	the	DET
ap-1821	121	22	following	follow	VERB
ap-1821	121	23	lemma	lemma	PROPN
ap-1821	121	24	from	from	ADP
ap-1821	121	25	[	[	X
ap-1821	121	26	4	4	NUM
ap-1821	121	27	]	]	PUNCT
ap-1821	121	28	.	.	PUNCT
ap-1821	122	1	lemma	lemma	PROPN
ap-1821	122	2	2.11	2.11	NUM
ap-1821	122	3	.	.	PUNCT
ap-1821	123	1	let	let	VERB
ap-1821	123	2	α	α	PRON
ap-1821	123	3	be	be	AUX
ap-1821	123	4	a	a	DET
ap-1821	123	5	quadratic	quadratic	ADJ
ap-1821	123	6	irrational	irrational	ADJ
ap-1821	123	7	and	and	CCONJ
ap-1821	123	8	α′	α′	NUM
ap-1821	123	9	its	its	PRON
ap-1821	123	10	conjugate	conjugate	NOUN
ap-1821	123	11	.	.	PUNCT
ap-1821	124	1	if	if	SCONJ
ap-1821	124	2	α	α	PRON
ap-1821	124	3	has	have	VERB
ap-1821	124	4	a	a	DET
ap-1821	124	5	purely	purely	ADV
ap-1821	124	6	periodic	periodic	ADJ
ap-1821	124	7	continued	continued	ADJ
ap-1821	124	8	fraction	fraction	NOUN
ap-1821	124	9	[	[	X
ap-1821	124	10	a0	a0	NOUN
ap-1821	124	11	,	,	PUNCT
ap-1821	124	12	a1	a1	NOUN
ap-1821	124	13	,	,	PUNCT
ap-1821	124	14	.	.	PUNCT
ap-1821	124	15	.	.	PUNCT
ap-1821	125	1	.	.	PUNCT
ap-1821	126	1	,	,	PUNCT
ap-1821	126	2	an	an	PRON
ap-1821	126	3	]	]	X
ap-1821	126	4	,	,	PUNCT
ap-1821	126	5	then	then	ADV
ap-1821	126	6	−1	−1	VERB
ap-1821	126	7	α′	α′	NUM
ap-1821	126	8	=	=	PUNCT
ap-1821	127	1	[	[	X
ap-1821	127	2	an	an	X
ap-1821	127	3	,	,	PUNCT
ap-1821	127	4	.	.	PUNCT
ap-1821	127	5	.	.	PUNCT
ap-1821	128	1	.	.	PUNCT
ap-1821	129	1	,	,	PUNCT
ap-1821	129	2	a1	a1	PROPN
ap-1821	129	3	,	,	PUNCT
ap-1821	129	4	a0	a0	NOUN
ap-1821	129	5	]	]	PUNCT
ap-1821	129	6	.	.	PUNCT
ap-1821	130	1	3	3	X
ap-1821	130	2	.	.	NUM
ap-1821	130	3	continued	continue	VERB
ap-1821	130	4	fractions	fraction	NOUN
ap-1821	130	5	of	of	ADP
ap-1821	130	6	√	√	NOUN
ap-1821	130	7	n	n	CCONJ
ap-1821	130	8	let	let	VERB
ap-1821	130	9	us	we	PRON
ap-1821	130	10	consider	consider	VERB
ap-1821	130	11	n	n	PRON
ap-1821	130	12	∈	∈	PROPN
ap-1821	130	13	n	n	CCONJ
ap-1821	130	14	\	\	NOUN
ap-1821	130	15	{	{	PUNCT
ap-1821	130	16	0	0	NUM
ap-1821	130	17	}	}	PUNCT
ap-1821	130	18	.	.	PUNCT
ap-1821	131	1	if	if	SCONJ
ap-1821	131	2	n	n	NOUN
ap-1821	131	3	=	=	SYM
ap-1821	131	4	k2	k2	PROPN
ap-1821	131	5	for	for	ADP
ap-1821	131	6	some	some	DET
ap-1821	131	7	k	k	PROPN
ap-1821	131	8	∈	∈	PROPN
ap-1821	131	9	n	n	CCONJ
ap-1821	131	10	,	,	PUNCT
ap-1821	131	11	then	then	ADV
ap-1821	131	12	√	√	VERB
ap-1821	131	13	n	n	NOUN
ap-1821	131	14	=	=	SYM
ap-1821	131	15	k	k	PROPN
ap-1821	131	16	and	and	CCONJ
ap-1821	131	17	the	the	DET
ap-1821	131	18	continued	continue	VERB
ap-1821	131	19	fraction	fraction	NOUN
ap-1821	131	20	is√	is√	NOUN
ap-1821	131	21	n	n	NOUN
ap-1821	131	22	=	=	PUNCT
ap-1821	132	1	[	[	X
ap-1821	132	2	k	k	X
ap-1821	132	3	]	]	X
ap-1821	132	4	.	.	PUNCT
ap-1821	133	1	therefore	therefore	ADV
ap-1821	133	2	,	,	PUNCT
ap-1821	133	3	we	we	PRON
ap-1821	133	4	limit	limit	VERB
ap-1821	133	5	our	our	PRON
ap-1821	133	6	considerations	consideration	NOUN
ap-1821	133	7	to	to	ADP
ap-1821	133	8	n	n	PRON
ap-1821	133	9	∈	∈	NOUN
ap-1821	133	10	n	n	CCONJ
ap-1821	133	11	\	\	NOUN
ap-1821	133	12	{	{	PUNCT
ap-1821	133	13	0	0	NUM
ap-1821	133	14	}	}	PUNCT
ap-1821	133	15	which	which	PRON
ap-1821	133	16	is	be	AUX
ap-1821	133	17	not	not	PART
ap-1821	133	18	a	a	DET
ap-1821	133	19	square	square	NOUN
ap-1821	133	20	in	in	ADP
ap-1821	133	21	the	the	DET
ap-1821	133	22	sequel	sequel	NOUN
ap-1821	133	23	.	.	PUNCT
ap-1821	134	1	then	then	ADV
ap-1821	134	2	there	there	PRON
ap-1821	134	3	exists	exist	VERB
ap-1821	134	4	a	a	DET
ap-1821	134	5	unique	unique	ADJ
ap-1821	134	6	n	n	CCONJ
ap-1821	134	7	∈	∈	NOUN
ap-1821	134	8	n	n	CCONJ
ap-1821	134	9	\	\	NOUN
ap-1821	134	10	{	{	PUNCT
ap-1821	134	11	0	0	NUM
ap-1821	134	12	}	}	PUNCT
ap-1821	134	13	and	and	CCONJ
ap-1821	134	14	a	a	DET
ap-1821	134	15	unique	unique	ADJ
ap-1821	134	16	j	j	PROPN
ap-1821	134	17	∈	∈	PROPN
ap-1821	134	18	{	{	PUNCT
ap-1821	134	19	1	1	NUM
ap-1821	134	20	,	,	PUNCT
ap-1821	134	21	.	.	PUNCT
ap-1821	134	22	.	.	PUNCT
ap-1821	134	23	.	.	PUNCT
ap-1821	135	1	,	,	PUNCT
ap-1821	135	2	2n	2n	X
ap-1821	135	3	}	}	PUNCT
ap-1821	135	4	such	such	ADJ
ap-1821	135	5	that	that	SCONJ
ap-1821	135	6	n	n	NOUN
ap-1821	135	7	=	=	PUNCT
ap-1821	135	8	n2	n2	PROPN
ap-1821	135	9	+	+	CCONJ
ap-1821	135	10	j.	j.	PROPN
ap-1821	135	11	the	the	DET
ap-1821	135	12	proofs	proof	NOUN
ap-1821	135	13	of	of	ADP
ap-1821	135	14	the	the	DET
ap-1821	135	15	two	two	NUM
ap-1821	135	16	following	follow	VERB
ap-1821	135	17	theorems	theorem	NOUN
ap-1821	135	18	can	can	AUX
ap-1821	135	19	be	be	AUX
ap-1821	135	20	found	find	VERB
ap-1821	135	21	in	in	ADP
ap-1821	135	22	[	[	X
ap-1821	135	23	4	4	NUM
ap-1821	135	24	]	]	PUNCT
ap-1821	135	25	page	page	NOUN
ap-1821	135	26	15	15	NUM
ap-1821	135	27	.	.	PUNCT
ap-1821	136	1	however	however	ADV
ap-1821	136	2	we	we	PRON
ap-1821	136	3	repeat	repeat	VERB
ap-1821	136	4	them	they	PRON
ap-1821	136	5	here	here	ADV
ap-1821	136	6	since	since	SCONJ
ap-1821	136	7	they	they	PRON
ap-1821	136	8	follow	follow	VERB
ap-1821	136	9	almost	almost	ADV
ap-1821	136	10	immediately	immediately	ADV
ap-1821	136	11	from	from	ADP
ap-1821	136	12	the	the	DET
ap-1821	136	13	previous	previous	ADJ
ap-1821	136	14	statements	statement	NOUN
ap-1821	136	15	and	and	CCONJ
ap-1821	136	16	they	they	PRON
ap-1821	136	17	give	give	VERB
ap-1821	136	18	an	an	DET
ap-1821	136	19	insight	insight	NOUN
ap-1821	136	20	into	into	ADP
ap-1821	136	21	the	the	DET
ap-1821	136	22	form	form	NOUN
ap-1821	136	23	of	of	ADP
ap-1821	136	24	continued	continue	VERB
ap-1821	136	25	fractions	fraction	NOUN
ap-1821	136	26	of	of	ADP
ap-1821	136	27	quadratic	quadratic	ADJ
ap-1821	136	28	numbers	number	NOUN
ap-1821	136	29	.	.	PUNCT
ap-1821	137	1	323	323	NUM
ap-1821	137	2	ľ	ľ	NUM
ap-1821	137	3	.	.	PUNCT
ap-1821	138	1	balková	balková	PROPN
ap-1821	138	2	,	,	PUNCT
ap-1821	138	3	a.	a.	PROPN
ap-1821	138	4	hrušková	hrušková	PROPN
ap-1821	138	5	acta	acta	PROPN
ap-1821	138	6	polytechnica	polytechnica	PROPN
ap-1821	138	7	theorem	theorem	VERB
ap-1821	138	8	3.1	3.1	NUM
ap-1821	138	9	.	.	PUNCT
ap-1821	139	1	for	for	ADP
ap-1821	139	2	every	every	DET
ap-1821	139	3	n	n	CCONJ
ap-1821	139	4	∈	∈	PROPN
ap-1821	139	5	n	n	CCONJ
ap-1821	139	6	\	\	NOUN
ap-1821	139	7	{	{	PUNCT
ap-1821	139	8	0	0	NUM
ap-1821	139	9	}	}	PUNCT
ap-1821	139	10	and	and	CCONJ
ap-1821	139	11	every	every	DET
ap-1821	139	12	j	j	PROPN
ap-1821	139	13	∈	∈	PROPN
ap-1821	139	14	{	{	PUNCT
ap-1821	139	15	1	1	NUM
ap-1821	139	16	,	,	PUNCT
ap-1821	139	17	.	.	PUNCT
ap-1821	139	18	.	.	PUNCT
ap-1821	139	19	.	.	PUNCT
ap-1821	140	1	,	,	PUNCT
ap-1821	140	2	2n	2n	X
ap-1821	140	3	}	}	PUNCT
ap-1821	140	4	the	the	DET
ap-1821	140	5	continued	continue	VERB
ap-1821	140	6	fraction	fraction	NOUN
ap-1821	140	7	of	of	ADP
ap-1821	140	8	√	√	PROPN
ap-1821	140	9	n2	n2	NOUN
ap-1821	140	10	+	+	CCONJ
ap-1821	140	11	j	j	PROPN
ap-1821	140	12	is	be	AUX
ap-1821	140	13	of	of	ADP
ap-1821	140	14	the	the	DET
ap-1821	140	15	form	form	NOUN
ap-1821	140	16	[	[	X
ap-1821	140	17	n	n	CCONJ
ap-1821	140	18	,	,	PUNCT
ap-1821	140	19	a1	a1	NOUN
ap-1821	140	20	,	,	PUNCT
ap-1821	140	21	.	.	PUNCT
ap-1821	140	22	.	.	PUNCT
ap-1821	141	1	.	.	PUNCT
ap-1821	142	1	,	,	PUNCT
ap-1821	142	2	ar	ar	PROPN
ap-1821	142	3	,	,	PUNCT
ap-1821	142	4	2n	2n	NUM
ap-1821	142	5	]	]	X
ap-1821	142	6	,	,	PUNCT
ap-1821	142	7	where	where	SCONJ
ap-1821	142	8	a1	a1	NOUN
ap-1821	142	9	.	.	PUNCT
ap-1821	142	10	.	.	PUNCT
ap-1821	142	11	.	.	PUNCT
ap-1821	143	1	ar	ar	PROPN
ap-1821	143	2	is	be	AUX
ap-1821	143	3	a	a	DET
ap-1821	143	4	palindrome	palindrome	NOUN
ap-1821	143	5	.	.	PUNCT
ap-1821	144	1	proof	proof	NOUN
ap-1821	144	2	.	.	PUNCT
ap-1821	145	1	denote	denote	VERB
ap-1821	145	2	α	α	NOUN
ap-1821	145	3	=	=	SYM
ap-1821	145	4	n	n	PROPN
ap-1821	145	5	+	+	CCONJ
ap-1821	145	6	√	√	PROPN
ap-1821	145	7	n2	n2	NOUN
ap-1821	145	8	+	+	CCONJ
ap-1821	145	9	j.	j.	PROPN
ap-1821	145	10	then	then	ADV
ap-1821	145	11	α	α	PROPN
ap-1821	145	12	is	be	AUX
ap-1821	145	13	a	a	DET
ap-1821	145	14	quadratic	quadratic	ADJ
ap-1821	145	15	irrational	irrational	ADJ
ap-1821	145	16	greater	great	ADJ
ap-1821	145	17	than	than	ADP
ap-1821	145	18	1	1	NUM
ap-1821	145	19	and	and	CCONJ
ap-1821	145	20	α′	α′	NUM
ap-1821	145	21	=	=	SYM
ap-1821	146	1	n	n	NUM
ap-1821	146	2	−√	−√	NOUN
ap-1821	146	3	n2	n2	NOUN
ap-1821	146	4	+	+	CCONJ
ap-1821	146	5	j	j	PROPN
ap-1821	146	6	∈	∈	PROPN
ap-1821	146	7	(	(	PUNCT
ap-1821	146	8	−1	−1	NOUN
ap-1821	146	9	,	,	PUNCT
ap-1821	146	10	0	0	NUM
ap-1821	146	11	)	)	PUNCT
ap-1821	146	12	.	.	PUNCT
ap-1821	147	1	therefore	therefore	ADV
ap-1821	147	2	α	α	PROPN
ap-1821	147	3	has	have	AUX
ap-1821	147	4	by	by	ADP
ap-1821	147	5	theorem	theorem	NOUN
ap-1821	147	6	2.9	2.9	NUM
ap-1821	147	7	a	a	DET
ap-1821	147	8	purely	purely	ADV
ap-1821	147	9	periodic	periodic	ADJ
ap-1821	147	10	continued	continued	ADJ
ap-1821	147	11	fraction	fraction	NOUN
ap-1821	147	12	,	,	PUNCT
ap-1821	147	13	i.e.	i.e.	X
ap-1821	147	14	,	,	PUNCT
ap-1821	147	15	there	there	PRON
ap-1821	147	16	exist	exist	VERB
ap-1821	147	17	a1	a1	NOUN
ap-1821	147	18	,	,	PUNCT
ap-1821	147	19	.	.	PUNCT
ap-1821	147	20	.	.	PUNCT
ap-1821	148	1	.	.	PUNCT
ap-1821	149	1	,	,	PUNCT
ap-1821	149	2	ar	ar	PROPN
ap-1821	149	3	∈	∈	PROPN
ap-1821	149	4	n	n	PRON
ap-1821	149	5	such	such	ADJ
ap-1821	149	6	that	that	SCONJ
ap-1821	149	7	α	α	NOUN
ap-1821	149	8	=	=	PUNCT
ap-1821	150	1	[	[	X
ap-1821	150	2	2n	2n	NUM
ap-1821	150	3	,	,	PUNCT
ap-1821	150	4	a1	a1	NOUN
ap-1821	150	5	,	,	PUNCT
ap-1821	150	6	.	.	PUNCT
ap-1821	150	7	.	.	PUNCT
ap-1821	151	1	.	.	PUNCT
ap-1821	152	1	,	,	PUNCT
ap-1821	152	2	ar	ar	PROPN
ap-1821	152	3	]	]	PUNCT
ap-1821	152	4	.	.	PUNCT
ap-1821	153	1	it	it	PRON
ap-1821	153	2	is	be	AUX
ap-1821	153	3	thus	thus	ADV
ap-1821	153	4	evident	evident	ADJ
ap-1821	153	5	that	that	SCONJ
ap-1821	153	6	√	√	ADJ
ap-1821	153	7	n2	n2	NOUN
ap-1821	153	8	+	+	CCONJ
ap-1821	153	9	j	j	NOUN
ap-1821	153	10	=	=	SYM
ap-1821	154	1	[	[	X
ap-1821	154	2	n	n	CCONJ
ap-1821	154	3	,	,	PUNCT
ap-1821	154	4	a1	a1	NOUN
ap-1821	154	5	,	,	PUNCT
ap-1821	154	6	.	.	PUNCT
ap-1821	154	7	.	.	PUNCT
ap-1821	155	1	.	.	PUNCT
ap-1821	156	1	,	,	PUNCT
ap-1821	156	2	ar	ar	PROPN
ap-1821	156	3	,	,	PUNCT
ap-1821	156	4	2n	2n	NUM
ap-1821	156	5	]	]	PUNCT
ap-1821	156	6	.	.	PUNCT
ap-1821	157	1	it	it	PRON
ap-1821	157	2	remains	remain	VERB
ap-1821	157	3	to	to	PART
ap-1821	157	4	prove	prove	VERB
ap-1821	157	5	that	that	DET
ap-1821	157	6	a1	a1	NOUN
ap-1821	157	7	.	.	PUNCT
ap-1821	157	8	.	.	PUNCT
ap-1821	157	9	.	.	PUNCT
ap-1821	158	1	ar	ar	PROPN
ap-1821	158	2	is	be	AUX
ap-1821	158	3	a	a	DET
ap-1821	158	4	palindrome	palindrome	NOUN
ap-1821	158	5	.	.	PUNCT
ap-1821	159	1	according	accord	VERB
ap-1821	159	2	to	to	ADP
ap-1821	159	3	lemma	lemma	PROPN
ap-1821	159	4	2.11	2.11	NUM
ap-1821	159	5	the	the	DET
ap-1821	159	6	number	number	NOUN
ap-1821	159	7	−1	−1	NOUN
ap-1821	159	8	α′	α′	NUM
ap-1821	159	9	has	have	VERB
ap-1821	159	10	its	its	PRON
ap-1821	159	11	continued	continue	VERB
ap-1821	159	12	fraction	fraction	NOUN
ap-1821	159	13	equal	equal	ADJ
ap-1821	159	14	to	to	ADP
ap-1821	159	15	[	[	X
ap-1821	159	16	ar	ar	NOUN
ap-1821	159	17	,	,	PUNCT
ap-1821	159	18	.	.	PUNCT
ap-1821	159	19	.	.	PUNCT
ap-1821	160	1	.	.	PUNCT
ap-1821	161	1	,	,	PUNCT
ap-1821	161	2	a1	a1	NOUN
ap-1821	161	3	,	,	PUNCT
ap-1821	161	4	2n	2n	NUM
ap-1821	161	5	]	]	PUNCT
ap-1821	161	6	.	.	PUNCT
ap-1821	162	1	we	we	PRON
ap-1821	162	2	obtain	obtain	VERB
ap-1821	162	3	thus	thus	ADV
ap-1821	162	4	√	√	ADJ
ap-1821	162	5	n2	n2	NOUN
ap-1821	162	6	+	+	CCONJ
ap-1821	162	7	j	j	PROPN
ap-1821	162	8	=	=	SYM
ap-1821	162	9	n+	n+	ADP
ap-1821	162	10	1	1	NUM
ap-1821	162	11	−1	−1	NOUN
ap-1821	162	12	n−	n−	NOUN
ap-1821	162	13	√	√	PUNCT
ap-1821	162	14	n2+j	n2+j	PROPN
ap-1821	162	15	=	=	PUNCT
ap-1821	162	16	n+	n+	ADP
ap-1821	162	17	1	1	NUM
ap-1821	162	18	−1	−1	NOUN
ap-1821	162	19	α′	α′	NUM
ap-1821	162	20	=	=	PUNCT
ap-1821	163	1	[	[	X
ap-1821	163	2	n	n	CCONJ
ap-1821	163	3	,	,	PUNCT
ap-1821	163	4	ar	ar	PROPN
ap-1821	163	5	,	,	PUNCT
ap-1821	163	6	.	.	PUNCT
ap-1821	163	7	.	.	PUNCT
ap-1821	164	1	.	.	PUNCT
ap-1821	165	1	,	,	PUNCT
ap-1821	165	2	a1	a1	NOUN
ap-1821	165	3	,	,	PUNCT
ap-1821	165	4	2n	2n	NUM
ap-1821	165	5	]	]	PUNCT
ap-1821	165	6	.	.	PUNCT
ap-1821	166	1	since	since	SCONJ
ap-1821	166	2	the	the	DET
ap-1821	166	3	continued	continue	VERB
ap-1821	166	4	fraction	fraction	NOUN
ap-1821	166	5	of	of	ADP
ap-1821	166	6	irrational	irrational	ADJ
ap-1821	166	7	numbers	number	NOUN
ap-1821	166	8	is	be	AUX
ap-1821	166	9	unique	unique	ADJ
ap-1821	166	10	and	and	CCONJ
ap-1821	166	11	we	we	PRON
ap-1821	166	12	have√	have√	VERB
ap-1821	166	13	n2	n2	ADJ
ap-1821	166	14	+	+	CCONJ
ap-1821	166	15	j	j	NOUN
ap-1821	166	16	=	=	SYM
ap-1821	167	1	[	[	X
ap-1821	167	2	n	n	CCONJ
ap-1821	167	3	,	,	PUNCT
ap-1821	167	4	a1	a1	NOUN
ap-1821	167	5	,	,	PUNCT
ap-1821	167	6	.	.	PUNCT
ap-1821	167	7	.	.	PUNCT
ap-1821	168	1	.	.	PUNCT
ap-1821	169	1	,	,	PUNCT
ap-1821	169	2	ar	ar	PROPN
ap-1821	169	3	,	,	PUNCT
ap-1821	169	4	2n	2n	NUM
ap-1821	169	5	]	]	PUNCT
ap-1821	170	1	=	=	PUNCT
ap-1821	171	1	[	[	X
ap-1821	171	2	n	n	CCONJ
ap-1821	171	3	,	,	PUNCT
ap-1821	171	4	ar	ar	PROPN
ap-1821	171	5	,	,	PUNCT
ap-1821	171	6	.	.	PUNCT
ap-1821	171	7	.	.	PUNCT
ap-1821	172	1	.	.	PUNCT
ap-1821	173	1	,	,	PUNCT
ap-1821	173	2	a1	a1	NOUN
ap-1821	173	3	,	,	PUNCT
ap-1821	173	4	2n	2n	NUM
ap-1821	173	5	]	]	PUNCT
ap-1821	173	6	,	,	PUNCT
ap-1821	173	7	it	it	PRON
ap-1821	173	8	follows	follow	VERB
ap-1821	173	9	that	that	PRON
ap-1821	173	10	a1	a1	NOUN
ap-1821	173	11	=	=	SYM
ap-1821	173	12	ar	ar	PROPN
ap-1821	173	13	,	,	PUNCT
ap-1821	173	14	a2	a2	PROPN
ap-1821	173	15	=	=	SYM
ap-1821	173	16	ar−1	ar−1	PROPN
ap-1821	173	17	etc	etc	X
ap-1821	173	18	.	.	X
ap-1821	173	19	consequently	consequently	ADV
ap-1821	173	20	,	,	PUNCT
ap-1821	173	21	a1	a1	NOUN
ap-1821	173	22	.	.	PUNCT
ap-1821	173	23	.	.	PUNCT
ap-1821	173	24	.	.	PUNCT
ap-1821	174	1	ar	ar	PROPN
ap-1821	174	2	is	be	AUX
ap-1821	174	3	a	a	DET
ap-1821	174	4	palindrome	palindrome	NOUN
ap-1821	174	5	.	.	PUNCT
ap-1821	175	1	theorem	theorem	ADJ
ap-1821	175	2	3.2	3.2	NUM
ap-1821	175	3	.	.	PUNCT
ap-1821	176	1	the	the	DET
ap-1821	176	2	continued	continue	VERB
ap-1821	176	3	fraction	fraction	NOUN
ap-1821	176	4	of	of	ADP
ap-1821	176	5	the	the	DET
ap-1821	176	6	form	form	NOUN
ap-1821	176	7	[	[	X
ap-1821	176	8	n	n	CCONJ
ap-1821	176	9	,	,	PUNCT
ap-1821	176	10	a1	a1	NOUN
ap-1821	176	11	,	,	PUNCT
ap-1821	176	12	.	.	PUNCT
ap-1821	176	13	.	.	PUNCT
ap-1821	176	14	.	.	PUNCT
ap-1821	177	1	,	,	PUNCT
ap-1821	177	2	ar	ar	PROPN
ap-1821	177	3	,	,	PUNCT
ap-1821	177	4	2n	2n	NUM
ap-1821	177	5	]	]	X
ap-1821	177	6	,	,	PUNCT
ap-1821	177	7	where	where	SCONJ
ap-1821	177	8	a1	a1	NOUN
ap-1821	177	9	.	.	PUNCT
ap-1821	177	10	.	.	PUNCT
ap-1821	177	11	.	.	PUNCT
ap-1821	178	1	ar	ar	PROPN
ap-1821	178	2	is	be	AUX
ap-1821	178	3	a	a	DET
ap-1821	178	4	palindrome	palindrome	NOUN
ap-1821	178	5	,	,	PUNCT
ap-1821	178	6	corresponds	correspond	VERB
ap-1821	178	7	to	to	ADP
ap-1821	178	8	√	√	PROPN
ap-1821	178	9	n	n	PROPN
ap-1821	178	10	for	for	ADP
ap-1821	178	11	a	a	DET
ap-1821	178	12	rational	rational	ADJ
ap-1821	178	13	number	number	NOUN
ap-1821	178	14	n	n	NOUN
ap-1821	178	15	.	.	PUNCT
ap-1821	179	1	proof	proof	NOUN
ap-1821	179	2	.	.	PUNCT
ap-1821	180	1	denote	denote	VERB
ap-1821	180	2	by	by	ADP
ap-1821	180	3	x	x	SYM
ap-1821	180	4	the	the	DET
ap-1821	180	5	number	number	NOUN
ap-1821	180	6	whose	whose	DET
ap-1821	180	7	continued	continue	VERB
ap-1821	180	8	fraction	fraction	NOUN
ap-1821	180	9	equals	equal	VERB
ap-1821	180	10	[	[	X
ap-1821	180	11	n	n	CCONJ
ap-1821	180	12	,	,	PUNCT
ap-1821	180	13	a1	a1	NOUN
ap-1821	180	14	,	,	PUNCT
ap-1821	180	15	.	.	PUNCT
ap-1821	180	16	.	.	PUNCT
ap-1821	181	1	.	.	PUNCT
ap-1821	182	1	,	,	PUNCT
ap-1821	182	2	ar	ar	PROPN
ap-1821	182	3	,	,	PUNCT
ap-1821	182	4	2n	2n	NUM
ap-1821	182	5	]	]	X
ap-1821	182	6	,	,	PUNCT
ap-1821	182	7	i.e.	i.e.	X
ap-1821	182	8	,	,	PUNCT
ap-1821	182	9	x	x	SYM
ap-1821	182	10	=	=	SYM
ap-1821	182	11	n+	n+	X
ap-1821	182	12	1	1	NUM
ap-1821	182	13	a1	a1	NOUN
ap-1821	182	14	+	+	CCONJ
ap-1821	182	15	1	1	NUM
ap-1821	182	16	.	.	PUNCT
ap-1821	182	17	.	.	PUNCT
ap-1821	182	18	.	.	PUNCT
ap-1821	183	1	+	+	CCONJ
ap-1821	183	2	1	1	NUM
ap-1821	183	3	ar	ar	NOUN
ap-1821	183	4	+	+	NOUN
ap-1821	183	5	1	1	NUM
ap-1821	183	6	2n+	2n+	NUM
ap-1821	183	7	(	(	PUNCT
ap-1821	183	8	x−	x−	PROPN
ap-1821	183	9	n	n	CCONJ
ap-1821	183	10	)	)	PUNCT
ap-1821	183	11	.	.	PUNCT
ap-1821	184	1	hence	hence	ADV
ap-1821	184	2	by	by	ADP
ap-1821	184	3	theorem	theorem	ADJ
ap-1821	184	4	2.5	2.5	NUM
ap-1821	184	5	,	,	PUNCT
ap-1821	184	6	x−	x−	PROPN
ap-1821	184	7	n	n	PROPN
ap-1821	184	8	=	=	SYM
ap-1821	184	9	kr(a2	kr(a2	NOUN
ap-1821	184	10	,	,	PUNCT
ap-1821	184	11	.	.	PUNCT
ap-1821	184	12	.	.	PUNCT
ap-1821	184	13	.	.	PUNCT
ap-1821	185	1	,	,	PUNCT
ap-1821	185	2	ar	ar	PROPN
ap-1821	185	3	,	,	PUNCT
ap-1821	185	4	x+	x+	NUM
ap-1821	185	5	n	n	CCONJ
ap-1821	185	6	)	)	PUNCT
ap-1821	185	7	kr+1(a1	kr+1(a1	PROPN
ap-1821	185	8	,	,	PUNCT
ap-1821	185	9	.	.	PUNCT
ap-1821	185	10	.	.	PUNCT
ap-1821	186	1	.	.	PUNCT
ap-1821	187	1	,	,	PUNCT
ap-1821	187	2	ar	ar	PROPN
ap-1821	187	3	,	,	PUNCT
ap-1821	187	4	x+	x+	NUM
ap-1821	187	5	n	n	CCONJ
ap-1821	187	6	)	)	PUNCT
ap-1821	187	7	=	=	SYM
ap-1821	187	8	kr−2(a2	kr−2(a2	PROPN
ap-1821	187	9	,	,	PUNCT
ap-1821	187	10	.	.	PUNCT
ap-1821	187	11	.	.	PUNCT
ap-1821	188	1	.	.	PUNCT
ap-1821	189	1	,	,	PUNCT
ap-1821	189	2	ar−1	ar−1	PROPN
ap-1821	189	3	)	)	PUNCT
ap-1821	190	1	+	+	CCONJ
ap-1821	190	2	(	(	PUNCT
ap-1821	190	3	x+	x+	ADJ
ap-1821	190	4	n)kr−1(a2	n)kr−1(a2	PROPN
ap-1821	190	5	,	,	PUNCT
ap-1821	190	6	.	.	PUNCT
ap-1821	190	7	.	.	PUNCT
ap-1821	190	8	.	.	PUNCT
ap-1821	191	1	,	,	PUNCT
ap-1821	191	2	ar	ar	PROPN
ap-1821	191	3	)	)	PUNCT
ap-1821	191	4	kr−1(a1	kr−1(a1	PROPN
ap-1821	191	5	,	,	PUNCT
ap-1821	191	6	.	.	PUNCT
ap-1821	191	7	.	.	PUNCT
ap-1821	192	1	.	.	PUNCT
ap-1821	193	1	,	,	PUNCT
ap-1821	193	2	ar−1	ar−1	PROPN
ap-1821	193	3	)	)	PUNCT
ap-1821	194	1	+	+	CCONJ
ap-1821	194	2	(	(	PUNCT
ap-1821	194	3	x+	x+	ADJ
ap-1821	194	4	n)kr(a1	n)kr(a1	NOUN
ap-1821	194	5	,	,	PUNCT
ap-1821	194	6	.	.	PUNCT
ap-1821	194	7	.	.	PUNCT
ap-1821	194	8	.	.	PUNCT
ap-1821	195	1	,	,	PUNCT
ap-1821	195	2	ar	ar	PROPN
ap-1821	195	3	)	)	PUNCT
ap-1821	195	4	.	.	PUNCT
ap-1821	196	1	by	by	ADP
ap-1821	196	2	theorem	theorem	NOUN
ap-1821	196	3	2.7	2.7	NUM
ap-1821	196	4	and	and	CCONJ
ap-1821	196	5	since	since	SCONJ
ap-1821	196	6	a1	a1	NOUN
ap-1821	196	7	.	.	PUNCT
ap-1821	196	8	.	.	PUNCT
ap-1821	196	9	.	.	PUNCT
ap-1821	197	1	ar	ar	PROPN
ap-1821	197	2	is	be	AUX
ap-1821	197	3	a	a	DET
ap-1821	197	4	palindrome	palindrome	NOUN
ap-1821	197	5	,	,	PUNCT
ap-1821	197	6	we	we	PRON
ap-1821	197	7	have	have	VERB
ap-1821	197	8	kr−1(a1	kr−1(a1	PROPN
ap-1821	197	9	,	,	PUNCT
ap-1821	197	10	.	.	PUNCT
ap-1821	197	11	.	.	PUNCT
ap-1821	198	1	.	.	PUNCT
ap-1821	199	1	,	,	PUNCT
ap-1821	199	2	ar−1	ar−1	PROPN
ap-1821	199	3	)	)	PUNCT
ap-1821	200	1	=	=	SYM
ap-1821	200	2	kr−1(a2	kr−1(a2	PROPN
ap-1821	200	3	,	,	PUNCT
ap-1821	200	4	.	.	PUNCT
ap-1821	200	5	.	.	PUNCT
ap-1821	200	6	.	.	PUNCT
ap-1821	201	1	,	,	PUNCT
ap-1821	201	2	ar	ar	PROPN
ap-1821	201	3	)	)	PUNCT
ap-1821	201	4	.	.	PUNCT
ap-1821	202	1	consequently	consequently	ADV
ap-1821	202	2	,	,	PUNCT
ap-1821	202	3	we	we	PRON
ap-1821	202	4	obtain	obtain	VERB
ap-1821	202	5	x	x	X
ap-1821	202	6	=	=	PUNCT
ap-1821	202	7	√	√	PROPN
ap-1821	202	8	n2	n2	NOUN
ap-1821	202	9	+	+	CCONJ
ap-1821	202	10	2nkr−1(a1,	2nkr−1(a1,	NUM
ap-1821	202	11	...	...	PUNCT
ap-1821	202	12	,ar−1)+kr−2(a2,	,ar−1)+kr−2(a2,	PUNCT
ap-1821	202	13	...	...	PUNCT
ap-1821	202	14	,ar−1	,ar−1	PUNCT
ap-1821	202	15	)	)	PUNCT
ap-1821	202	16	kr(a1,	kr(a1,	PROPN
ap-1821	202	17	...	...	PUNCT
ap-1821	202	18	,ar	,ar	PUNCT
ap-1821	202	19	)	)	PUNCT
ap-1821	202	20	,	,	PUNCT
ap-1821	202	21	where	where	SCONJ
ap-1821	202	22	under	under	ADP
ap-1821	202	23	the	the	DET
ap-1821	202	24	square	square	ADJ
ap-1821	202	25	root	root	NOUN
ap-1821	202	26	,	,	PUNCT
ap-1821	202	27	there	there	PRON
ap-1821	202	28	is	be	VERB
ap-1821	202	29	certainly	certainly	ADV
ap-1821	202	30	a	a	DET
ap-1821	202	31	rational	rational	ADJ
ap-1821	202	32	number	number	NOUN
ap-1821	202	33	since	since	SCONJ
ap-1821	202	34	by	by	ADP
ap-1821	202	35	their	their	PRON
ap-1821	202	36	definition	definition	NOUN
ap-1821	202	37	,	,	PUNCT
ap-1821	202	38	continuants	continuant	NOUN
ap-1821	202	39	with	with	ADP
ap-1821	202	40	integer	integer	NOUN
ap-1821	202	41	variables	variable	NOUN
ap-1821	202	42	are	be	AUX
ap-1821	202	43	integers	integer	NOUN
ap-1821	202	44	.	.	PUNCT
ap-1821	203	1	in	in	ADP
ap-1821	203	2	the	the	DET
ap-1821	203	3	sequel	sequel	NOUN
ap-1821	203	4	,	,	PUNCT
ap-1821	203	5	let	let	VERB
ap-1821	203	6	us	we	PRON
ap-1821	203	7	study	study	VERB
ap-1821	203	8	the	the	DET
ap-1821	203	9	length	length	NOUN
ap-1821	203	10	of	of	ADP
ap-1821	203	11	the	the	DET
ap-1821	203	12	period	period	NOUN
ap-1821	203	13	and	and	CCONJ
ap-1821	203	14	the	the	DET
ap-1821	203	15	form	form	NOUN
ap-1821	203	16	of	of	ADP
ap-1821	203	17	the	the	DET
ap-1821	203	18	continued	continue	VERB
ap-1821	203	19	fraction	fraction	NOUN
ap-1821	203	20	of	of	ADP
ap-1821	203	21	√	√	NUM
ap-1821	203	22	n	n	PRON
ap-1821	203	23	=	=	NOUN
ap-1821	203	24	√	√	PROPN
ap-1821	203	25	n2	n2	NOUN
ap-1821	203	26	+	+	CCONJ
ap-1821	203	27	j	j	PROPN
ap-1821	203	28	in	in	ADP
ap-1821	203	29	dependence	dependence	NOUN
ap-1821	203	30	on	on	ADP
ap-1821	203	31	n	n	PROPN
ap-1821	203	32	and	and	CCONJ
ap-1821	203	33	j	j	NOUN
ap-1821	203	34	,	,	PUNCT
ap-1821	203	35	where	where	SCONJ
ap-1821	203	36	n	n	X
ap-1821	203	37	∈	∈	PROPN
ap-1821	203	38	n	n	CCONJ
ap-1821	203	39	\	\	NOUN
ap-1821	203	40	{	{	PUNCT
ap-1821	203	41	0	0	NUM
ap-1821	203	42	}	}	PUNCT
ap-1821	203	43	and	and	CCONJ
ap-1821	203	44	j	j	PROPN
ap-1821	203	45	∈	∈	PROPN
ap-1821	203	46	{	{	PUNCT
ap-1821	203	47	1	1	NUM
ap-1821	203	48	,	,	PUNCT
ap-1821	203	49	.	.	PUNCT
ap-1821	203	50	.	.	PUNCT
ap-1821	204	1	.	.	PUNCT
ap-1821	205	1	,	,	PUNCT
ap-1821	205	2	2n	2n	NUM
ap-1821	205	3	}	}	PUNCT
ap-1821	205	4	.	.	PUNCT
ap-1821	206	1	we	we	PRON
ap-1821	206	2	will	will	AUX
ap-1821	206	3	prove	prove	VERB
ap-1821	206	4	only	only	ADV
ap-1821	206	5	some	some	DET
ap-1821	206	6	observations	observation	NOUN
ap-1821	206	7	since	since	SCONJ
ap-1821	206	8	the	the	DET
ap-1821	206	9	proofs	proof	NOUN
ap-1821	206	10	are	be	AUX
ap-1821	206	11	quite	quite	ADV
ap-1821	206	12	technical	technical	ADJ
ap-1821	206	13	and	and	CCONJ
ap-1821	206	14	space	space	NOUN
ap-1821	206	15	-	-	PUNCT
ap-1821	206	16	demanding	demanding	NOUN
ap-1821	206	17	.	.	PUNCT
ap-1821	207	1	the	the	DET
ap-1821	207	2	rest	rest	NOUN
ap-1821	207	3	of	of	ADP
ap-1821	207	4	the	the	DET
ap-1821	207	5	proofs	proof	NOUN
ap-1821	207	6	may	may	AUX
ap-1821	207	7	be	be	AUX
ap-1821	207	8	found	find	VERB
ap-1821	207	9	in	in	ADP
ap-1821	207	10	[	[	X
ap-1821	207	11	5	5	NUM
ap-1821	207	12	]	]	PUNCT
ap-1821	207	13	.	.	PUNCT
ap-1821	208	1	in	in	ADP
ap-1821	208	2	table	table	NOUN
ap-1821	208	3	1	1	NUM
ap-1821	208	4	,	,	PUNCT
ap-1821	208	5	we	we	PRON
ap-1821	208	6	have	have	AUX
ap-1821	208	7	highlighted	highlight	VERB
ap-1821	208	8	all	all	DET
ap-1821	208	9	classes	class	NOUN
ap-1821	208	10	of	of	ADP
ap-1821	208	11	n	n	PROPN
ap-1821	208	12	and	and	CCONJ
ap-1821	208	13	j	j	PROPN
ap-1821	208	14	for	for	ADP
ap-1821	208	15	which	which	PRON
ap-1821	208	16	their	their	PRON
ap-1821	208	17	continued	continue	VERB
ap-1821	208	18	fractions	fraction	NOUN
ap-1821	208	19	of	of	ADP
ap-1821	208	20	√	√	NUM
ap-1821	208	21	n	n	NOUN
ap-1821	208	22	=	=	PUNCT
ap-1821	208	23	√	√	PROPN
ap-1821	208	24	n2	n2	NOUN
ap-1821	208	25	+	+	CCONJ
ap-1821	208	26	j	j	PROPN
ap-1821	208	27	have	have	AUX
ap-1821	208	28	been	be	AUX
ap-1821	208	29	described	describe	VERB
ap-1821	208	30	.	.	PUNCT
ap-1821	209	1	observation	observation	NOUN
ap-1821	209	2	3.3	3.3	NUM
ap-1821	209	3	.	.	PUNCT
ap-1821	210	1	the	the	DET
ap-1821	210	2	continued	continue	VERB
ap-1821	210	3	fraction	fraction	NOUN
ap-1821	210	4	of	of	ADP
ap-1821	210	5	√	√	PROPN
ap-1821	210	6	n	n	PRON
ap-1821	210	7	has	have	VERB
ap-1821	210	8	period	period	NOUN
ap-1821	210	9	of	of	ADP
ap-1821	210	10	length	length	NOUN
ap-1821	210	11	1	1	NUM
ap-1821	210	12	if	if	SCONJ
ap-1821	210	13	and	and	CCONJ
ap-1821	210	14	only	only	ADV
ap-1821	210	15	if	if	SCONJ
ap-1821	210	16	n	n	NOUN
ap-1821	210	17	=	=	PUNCT
ap-1821	210	18	n2	n2	NOUN
ap-1821	211	1	+	+	CCONJ
ap-1821	211	2	1	1	X
ap-1821	211	3	.	.	PUNCT
ap-1821	211	4	it	it	PRON
ap-1821	211	5	holds	hold	VERB
ap-1821	211	6	then	then	ADV
ap-1821	211	7	√	√	PROPN
ap-1821	211	8	n	n	NOUN
ap-1821	211	9	=	=	SYM
ap-1821	212	1	[	[	X
ap-1821	212	2	n	n	CCONJ
ap-1821	212	3	,	,	PUNCT
ap-1821	212	4	2n	2n	NUM
ap-1821	212	5	]	]	PUNCT
ap-1821	212	6	.	.	PUNCT
ap-1821	213	1	proof	proof	NOUN
ap-1821	213	2	.	.	PUNCT
ap-1821	214	1	this	this	DET
ap-1821	214	2	observation	observation	NOUN
ap-1821	214	3	has	have	AUX
ap-1821	214	4	already	already	ADV
ap-1821	214	5	been	be	AUX
ap-1821	214	6	made	make	VERB
ap-1821	214	7	in	in	ADP
ap-1821	214	8	[	[	X
ap-1821	214	9	7	7	NUM
ap-1821	214	10	]	]	PUNCT
ap-1821	214	11	.	.	PUNCT
ap-1821	215	1	(	(	PUNCT
ap-1821	215	2	⇐	⇐	NOUN
ap-1821	215	3	)	)	PUNCT
ap-1821	215	4	:	:	PUNCT
ap-1821	215	5	√	√	PUNCT
ap-1821	216	1	n2	n2	NOUN
ap-1821	216	2	+	+	CCONJ
ap-1821	216	3	1	1	NUM
ap-1821	216	4	=	=	SYM
ap-1821	216	5	n+	n+	NOUN
ap-1821	216	6	√	√	ADJ
ap-1821	216	7	n2	n2	NOUN
ap-1821	216	8	+	+	CCONJ
ap-1821	216	9	1−	1−	NUM
ap-1821	216	10	n	n	PRON
ap-1821	216	11	1	1	NUM
ap-1821	216	12	=	=	X
ap-1821	216	13	=	=	PUNCT
ap-1821	216	14	n+	n+	NUM
ap-1821	216	15	1	1	NUM
ap-1821	216	16	√	√	NOUN
ap-1821	216	17	n2	n2	NOUN
ap-1821	216	18	+	+	CCONJ
ap-1821	216	19	1	1	NUM
ap-1821	216	20	+	+	NUM
ap-1821	216	21	n	n	NOUN
ap-1821	216	22	=	=	SYM
ap-1821	216	23	n+	n+	ADP
ap-1821	216	24	1	1	NUM
ap-1821	216	25	2n+	2n+	NUM
ap-1821	216	26	√	√	PROPN
ap-1821	216	27	n2	n2	PROPN
ap-1821	216	28	+	+	CCONJ
ap-1821	216	29	1−	1−	NUM
ap-1821	216	30	n	n	DET
ap-1821	216	31	1	1	NUM
ap-1821	216	32	,	,	PUNCT
ap-1821	216	33	hence	hence	ADV
ap-1821	216	34	√	√	PROPN
ap-1821	216	35	n	n	NOUN
ap-1821	216	36	=	=	SYM
ap-1821	217	1	[	[	X
ap-1821	217	2	n	n	CCONJ
ap-1821	217	3	,	,	PUNCT
ap-1821	217	4	2n	2n	NUM
ap-1821	217	5	]	]	PUNCT
ap-1821	217	6	.	.	PUNCT
ap-1821	218	1	(	(	PUNCT
ap-1821	218	2	⇒	⇒	PROPN
ap-1821	218	3	)	)	PUNCT
ap-1821	218	4	:	:	PUNCT
ap-1821	218	5	if	if	SCONJ
ap-1821	218	6	the	the	DET
ap-1821	218	7	length	length	NOUN
ap-1821	218	8	of	of	ADP
ap-1821	218	9	the	the	DET
ap-1821	218	10	period	period	NOUN
ap-1821	218	11	equals	equal	VERB
ap-1821	218	12	1	1	NUM
ap-1821	218	13	,	,	PUNCT
ap-1821	218	14	then	then	ADV
ap-1821	218	15	by	by	ADP
ap-1821	218	16	theorem	theorem	NOUN
ap-1821	218	17	3.1	3.1	NUM
ap-1821	218	18	we	we	PRON
ap-1821	218	19	have	have	VERB
ap-1821	218	20	√	√	NUM
ap-1821	218	21	n	n	NOUN
ap-1821	218	22	=	=	SYM
ap-1821	219	1	[	[	X
ap-1821	219	2	n	n	CCONJ
ap-1821	219	3	,	,	PUNCT
ap-1821	219	4	2n].√	2n].√	NOUN
ap-1821	219	5	n2	n2	NOUN
ap-1821	219	6	+	+	CCONJ
ap-1821	219	7	j	j	PROPN
ap-1821	219	8	=	=	SYM
ap-1821	219	9	n+	n+	X
ap-1821	219	10	(	(	PUNCT
ap-1821	219	11	√	√	PROPN
ap-1821	219	12	n2	n2	NOUN
ap-1821	219	13	+	+	CCONJ
ap-1821	219	14	j	j	PROPN
ap-1821	219	15	−	−	PROPN
ap-1821	219	16	n	n	CCONJ
ap-1821	219	17	)	)	PUNCT
ap-1821	220	1	=	=	SYM
ap-1821	220	2	n+	n+	PUNCT
ap-1821	220	3	1	1	NUM
ap-1821	220	4	2n+	2n+	NUM
ap-1821	220	5	√	√	PROPN
ap-1821	220	6	n2	n2	PROPN
ap-1821	220	7	+	+	CCONJ
ap-1821	220	8	j	j	PROPN
ap-1821	220	9	−	−	PROPN
ap-1821	220	10	n	n	CCONJ
ap-1821	220	11	,	,	PUNCT
ap-1821	220	12	hence	hence	ADV
ap-1821	220	13	we	we	PRON
ap-1821	220	14	have√	have√	VERB
ap-1821	220	15	n2	n2	ADJ
ap-1821	220	16	+	+	CCONJ
ap-1821	220	17	j	j	PROPN
ap-1821	220	18	−	−	PROPN
ap-1821	220	19	n	n	NOUN
ap-1821	220	20	=	=	SYM
ap-1821	220	21	1	1	NUM
ap-1821	220	22	2n+	2n+	NUM
ap-1821	220	23	√	√	PROPN
ap-1821	220	24	n2	n2	PROPN
ap-1821	220	25	+	+	CCONJ
ap-1821	220	26	j	j	PROPN
ap-1821	220	27	−	−	PROPN
ap-1821	220	28	n	n	PROPN
ap-1821	220	29	,	,	PUNCT
ap-1821	220	30	j√	j√	PROPN
ap-1821	220	31	n2	n2	PROPN
ap-1821	220	32	+	+	CCONJ
ap-1821	220	33	j	j	PROPN
ap-1821	220	34	+	+	CCONJ
ap-1821	220	35	n	n	CCONJ
ap-1821	220	36	=	=	SYM
ap-1821	220	37	1√	1√	PROPN
ap-1821	220	38	n2	n2	NOUN
ap-1821	220	39	+	+	CCONJ
ap-1821	220	40	j	j	PROPN
ap-1821	220	41	+	+	CCONJ
ap-1821	220	42	n	n	CCONJ
ap-1821	220	43	,	,	PUNCT
ap-1821	220	44	j	j	PROPN
ap-1821	220	45	=	=	SYM
ap-1821	220	46	1	1	X
ap-1821	220	47	.	.	PUNCT
ap-1821	220	48	observation	observation	NOUN
ap-1821	220	49	3.4	3.4	NUM
ap-1821	220	50	.	.	PUNCT
ap-1821	221	1	the	the	DET
ap-1821	221	2	continued	continue	VERB
ap-1821	221	3	fraction	fraction	NOUN
ap-1821	221	4	of	of	ADP
ap-1821	221	5	√	√	PROPN
ap-1821	221	6	n	n	PRON
ap-1821	221	7	has	have	VERB
ap-1821	221	8	period	period	NOUN
ap-1821	221	9	of	of	ADP
ap-1821	221	10	length	length	NOUN
ap-1821	221	11	2	2	NUM
ap-1821	222	1	if	if	SCONJ
ap-1821	222	2	and	and	CCONJ
ap-1821	222	3	only	only	ADV
ap-1821	222	4	if	if	SCONJ
ap-1821	222	5	2n	2n	NUM
ap-1821	222	6	j	j	PROPN
ap-1821	222	7	is	be	AUX
ap-1821	222	8	an	an	DET
ap-1821	222	9	integer	integer	NOUN
ap-1821	222	10	.	.	PUNCT
ap-1821	223	1	it	it	PRON
ap-1821	223	2	holds	hold	VERB
ap-1821	223	3	then	then	ADV
ap-1821	223	4	√	√	PROPN
ap-1821	223	5	n	n	NOUN
ap-1821	223	6	=	=	SYM
ap-1821	224	1	[	[	X
ap-1821	224	2	n	n	CCONJ
ap-1821	224	3	,	,	PUNCT
ap-1821	224	4	2n	2n	NUM
ap-1821	224	5	j	j	PROPN
ap-1821	224	6	,	,	PUNCT
ap-1821	224	7	2n	2n	NUM
ap-1821	224	8	]	]	PUNCT
ap-1821	224	9	.	.	PUNCT
ap-1821	225	1	proof	proof	NOUN
ap-1821	225	2	.	.	PUNCT
ap-1821	226	1	(	(	PUNCT
ap-1821	226	2	⇐	⇐	ADJ
ap-1821	226	3	):	):	PUNCT
ap-1821	226	4	√	√	ADJ
ap-1821	226	5	n2	n2	NOUN
ap-1821	226	6	+	+	CCONJ
ap-1821	226	7	j	j	PROPN
ap-1821	226	8	=	=	SYM
ap-1821	226	9	n+	n+	PROPN
ap-1821	226	10	(	(	PUNCT
ap-1821	226	11	√	√	PROPN
ap-1821	226	12	n2	n2	NOUN
ap-1821	226	13	+	+	CCONJ
ap-1821	226	14	j	j	PROPN
ap-1821	226	15	−	−	PROPN
ap-1821	226	16	n	n	CCONJ
ap-1821	226	17	)	)	PUNCT
ap-1821	226	18	=	=	SYM
ap-1821	226	19	n+	n+	NUM
ap-1821	226	20	j√	j√	NOUN
ap-1821	226	21	n2	n2	PROPN
ap-1821	226	22	+	+	CCONJ
ap-1821	226	23	j	j	PROPN
ap-1821	226	24	+	+	CCONJ
ap-1821	226	25	n	n	PROPN
ap-1821	226	26	=	=	SYM
ap-1821	226	27	n+	n+	ADP
ap-1821	226	28	1	1	NUM
ap-1821	226	29	2n	2n	NUM
ap-1821	226	30	j	j	PROPN
ap-1821	226	31	+	+	CCONJ
ap-1821	226	32	√	√	PROPN
ap-1821	226	33	n2	n2	NOUN
ap-1821	226	34	+	+	CCONJ
ap-1821	226	35	j	j	PROPN
ap-1821	226	36	−	−	PROPN
ap-1821	226	37	n	n	PROPN
ap-1821	226	38	j	j	PROPN
ap-1821	226	39	,	,	PUNCT
ap-1821	226	40	=	=	SYM
ap-1821	226	41	n+	n+	ADP
ap-1821	226	42	1	1	NUM
ap-1821	226	43	2n	2n	NUM
ap-1821	226	44	j	j	PROPN
ap-1821	226	45	+	+	CCONJ
ap-1821	226	46	1√	1√	PROPN
ap-1821	226	47	n2	n2	NOUN
ap-1821	226	48	+	+	CCONJ
ap-1821	226	49	j	j	PROPN
ap-1821	226	50	+	+	CCONJ
ap-1821	226	51	n	n	CCONJ
ap-1821	226	52	,	,	PUNCT
ap-1821	226	53	=	=	SYM
ap-1821	226	54	n+	n+	X
ap-1821	226	55	1	1	NUM
ap-1821	226	56	2n	2n	NUM
ap-1821	226	57	j	j	PROPN
ap-1821	227	1	+	+	CCONJ
ap-1821	227	2	1	1	NUM
ap-1821	227	3	2n+	2n+	NUM
ap-1821	227	4	(	(	PUNCT
ap-1821	227	5	√	√	PROPN
ap-1821	227	6	n2	n2	NOUN
ap-1821	227	7	+	+	CCONJ
ap-1821	227	8	j	j	PROPN
ap-1821	227	9	−	−	PROPN
ap-1821	227	10	n	n	CCONJ
ap-1821	227	11	)	)	PUNCT
ap-1821	227	12	,	,	PUNCT
ap-1821	227	13	thus	thus	ADV
ap-1821	227	14	√	√	VERB
ap-1821	227	15	n	n	NOUN
ap-1821	227	16	=	=	SYM
ap-1821	228	1	[	[	X
ap-1821	228	2	n	n	CCONJ
ap-1821	228	3	,	,	PUNCT
ap-1821	228	4	2n	2n	NUM
ap-1821	228	5	j	j	PROPN
ap-1821	228	6	,	,	PUNCT
ap-1821	228	7	2n	2n	NUM
ap-1821	228	8	]	]	PUNCT
ap-1821	228	9	.	.	PUNCT
ap-1821	229	1	324	324	NUM
ap-1821	229	2	vol	vol	NOUN
ap-1821	229	3	.	.	PUNCT
ap-1821	230	1	53	53	NUM
ap-1821	230	2	no	no	NOUN
ap-1821	230	3	.	.	PUNCT
ap-1821	231	1	4/2013	4/2013	PROPN
ap-1821	231	2	continued	continue	VERB
ap-1821	231	3	fractions	fraction	NOUN
ap-1821	231	4	of	of	ADP
ap-1821	231	5	square	square	ADJ
ap-1821	231	6	roots	root	NOUN
ap-1821	231	7	of	of	ADP
ap-1821	231	8	natural	natural	ADJ
ap-1821	231	9	numbers	number	NOUN
ap-1821	231	10	table	table	NOUN
ap-1821	231	11	1	1	NUM
ap-1821	231	12	.	.	PUNCT
ap-1821	232	1	all	all	DET
ap-1821	232	2	classes	class	NOUN
ap-1821	232	3	of	of	ADP
ap-1821	232	4	n	n	PRON
ap-1821	232	5	≤	≤	ADV
ap-1821	232	6	40	40	NUM
ap-1821	232	7	(	(	PUNCT
ap-1821	232	8	first	first	ADJ
ap-1821	232	9	column	column	NOUN
ap-1821	232	10	)	)	PUNCT
ap-1821	232	11	and	and	CCONJ
ap-1821	232	12	j	j	PROPN
ap-1821	232	13	≤	≤	ADV
ap-1821	232	14	31	31	NUM
ap-1821	232	15	(	(	PUNCT
ap-1821	232	16	first	first	ADJ
ap-1821	232	17	row	row	NOUN
ap-1821	232	18	)	)	PUNCT
ap-1821	232	19	for	for	ADP
ap-1821	232	20	which	which	PRON
ap-1821	232	21	their	their	PRON
ap-1821	232	22	continued	continue	VERB
ap-1821	232	23	fractions	fraction	NOUN
ap-1821	232	24	of√	of√	X
ap-1821	233	1	n	n	CCONJ
ap-1821	233	2	=	=	PUNCT
ap-1821	233	3	√	√	PROPN
ap-1821	233	4	n2	n2	NOUN
ap-1821	233	5	+	+	CCONJ
ap-1821	233	6	j	j	PROPN
ap-1821	233	7	have	have	AUX
ap-1821	233	8	been	be	AUX
ap-1821	233	9	described	describe	VERB
ap-1821	233	10	are	be	AUX
ap-1821	233	11	highlighted	highlight	VERB
ap-1821	233	12	.	.	PUNCT
ap-1821	234	1	(	(	PUNCT
ap-1821	234	2	⇒	⇒	NOUN
ap-1821	234	3	):	):	PUNCT
ap-1821	234	4	if	if	SCONJ
ap-1821	234	5	the	the	DET
ap-1821	234	6	length	length	NOUN
ap-1821	234	7	of	of	ADP
ap-1821	234	8	the	the	DET
ap-1821	234	9	period	period	NOUN
ap-1821	234	10	equals	equal	VERB
ap-1821	234	11	2	2	NUM
ap-1821	234	12	,	,	PUNCT
ap-1821	234	13	then	then	ADV
ap-1821	234	14	by	by	ADP
ap-1821	234	15	theorem	theorem	NOUN
ap-1821	234	16	3.1	3.1	NUM
ap-1821	234	17	we	we	PRON
ap-1821	234	18	have	have	VERB
ap-1821	234	19	√	√	NUM
ap-1821	234	20	n	n	NOUN
ap-1821	234	21	=	=	SYM
ap-1821	235	1	[	[	X
ap-1821	235	2	n	n	CCONJ
ap-1821	235	3	,	,	PUNCT
ap-1821	235	4	x	x	NOUN
ap-1821	235	5	,	,	PUNCT
ap-1821	235	6	2n].√	2n].√	NOUN
ap-1821	235	7	n2	n2	NOUN
ap-1821	235	8	+	+	CCONJ
ap-1821	235	9	j	j	PROPN
ap-1821	235	10	=	=	SYM
ap-1821	235	11	n+	n+	PROPN
ap-1821	235	12	(	(	PUNCT
ap-1821	235	13	√	√	PROPN
ap-1821	235	14	n2	n2	NOUN
ap-1821	235	15	+	+	CCONJ
ap-1821	235	16	j	j	PROPN
ap-1821	235	17	−	−	PROPN
ap-1821	235	18	n	n	CCONJ
ap-1821	235	19	)	)	PUNCT
ap-1821	235	20	=	=	PUNCT
ap-1821	235	21	n+	n+	ADP
ap-1821	235	22	1	1	NUM
ap-1821	235	23	x+	x+	SYM
ap-1821	235	24	1	1	NUM
ap-1821	235	25	2n+	2n+	NUM
ap-1821	235	26	(	(	PUNCT
ap-1821	235	27	√	√	PROPN
ap-1821	235	28	n2	n2	NOUN
ap-1821	235	29	+	+	CCONJ
ap-1821	235	30	j	j	PROPN
ap-1821	235	31	−	−	PROPN
ap-1821	235	32	n	n	CCONJ
ap-1821	235	33	)	)	PUNCT
ap-1821	235	34	,	,	PUNCT
ap-1821	235	35	hence	hence	ADV
ap-1821	235	36	we	we	PRON
ap-1821	235	37	have√	have√	VERB
ap-1821	235	38	n2	n2	ADJ
ap-1821	235	39	+	+	CCONJ
ap-1821	235	40	j	j	PROPN
ap-1821	235	41	−	−	PROPN
ap-1821	235	42	n	n	NOUN
ap-1821	235	43	=	=	SYM
ap-1821	235	44	1	1	NUM
ap-1821	235	45	x	x	SYM
ap-1821	235	46	(	(	PUNCT
ap-1821	235	47	√	√	PROPN
ap-1821	235	48	n2	n2	NOUN
ap-1821	235	49	+	+	CCONJ
ap-1821	235	50	j	j	PROPN
ap-1821	235	51	+	+	CCONJ
ap-1821	235	52	n	n	CCONJ
ap-1821	235	53	)	)	PUNCT
ap-1821	235	54	+	+	CCONJ
ap-1821	235	55	1√	1√	PROPN
ap-1821	235	56	n2	n2	NOUN
ap-1821	235	57	+	+	CCONJ
ap-1821	235	58	j	j	PROPN
ap-1821	235	59	+	+	CCONJ
ap-1821	235	60	n	n	CCONJ
ap-1821	235	61	,	,	PUNCT
ap-1821	235	62	x	x	PUNCT
ap-1821	236	1	=	=	SYM
ap-1821	236	2	2n	2n	NUM
ap-1821	236	3	j	j	PROPN
ap-1821	236	4	.	.	PUNCT
ap-1821	237	1	observation	observation	NOUN
ap-1821	237	2	3.5	3.5	NUM
ap-1821	237	3	.	.	PUNCT
ap-1821	238	1	the	the	DET
ap-1821	238	2	continued	continue	VERB
ap-1821	238	3	fraction	fraction	NOUN
ap-1821	238	4	of	of	ADP
ap-1821	238	5	√	√	PROPN
ap-1821	238	6	n	n	PRON
ap-1821	238	7	has	have	VERB
ap-1821	238	8	period	period	NOUN
ap-1821	238	9	of	of	ADP
ap-1821	238	10	length	length	NOUN
ap-1821	238	11	3	3	NUM
ap-1821	238	12	if	if	SCONJ
ap-1821	238	13	and	and	CCONJ
ap-1821	239	1	only	only	ADV
ap-1821	239	2	if	if	SCONJ
ap-1821	239	3	j	j	PROPN
ap-1821	239	4	=	=	SYM
ap-1821	239	5	4ak	4ak	PROPN
ap-1821	240	1	+	+	CCONJ
ap-1821	240	2	1	1	NUM
ap-1821	240	3	and	and	CCONJ
ap-1821	240	4	n	n	PROPN
ap-1821	240	5	=	=	SYM
ap-1821	240	6	aj	aj	PROPN
ap-1821	241	1	+	+	CCONJ
ap-1821	241	2	k	k	PROPN
ap-1821	241	3	for	for	ADP
ap-1821	241	4	some	some	DET
ap-1821	241	5	a	a	PRON
ap-1821	241	6	,	,	PUNCT
ap-1821	241	7	k	k	PROPN
ap-1821	241	8	∈	∈	PROPN
ap-1821	241	9	n	n	CCONJ
ap-1821	241	10	,	,	PUNCT
ap-1821	241	11	a	a	DET
ap-1821	241	12	≥	≥	NOUN
ap-1821	241	13	1	1	NUM
ap-1821	241	14	,	,	PUNCT
ap-1821	241	15	k	k	X
ap-1821	241	16	≥	≥	NUM
ap-1821	241	17	1	1	NUM
ap-1821	241	18	.	.	PUNCT
ap-1821	242	1	it	it	PRON
ap-1821	242	2	holds	hold	VERB
ap-1821	242	3	then	then	ADV
ap-1821	242	4	√	√	PROPN
ap-1821	242	5	n	n	NOUN
ap-1821	242	6	=	=	SYM
ap-1821	243	1	[	[	X
ap-1821	243	2	n	n	CCONJ
ap-1821	243	3	,	,	PUNCT
ap-1821	243	4	2a	2a	NUM
ap-1821	243	5	,	,	PUNCT
ap-1821	243	6	2a	2a	NUM
ap-1821	243	7	,	,	PUNCT
ap-1821	243	8	2n	2n	NUM
ap-1821	243	9	]	]	PUNCT
ap-1821	243	10	and	and	CCONJ
ap-1821	243	11	5	5	NUM
ap-1821	243	12	≤	≤	NUM
ap-1821	243	13	j	j	PROPN
ap-1821	243	14	≤	≤	PROPN
ap-1821	243	15	n−	n−	PROPN
ap-1821	243	16	1	1	NUM
ap-1821	243	17	.	.	PUNCT
ap-1821	244	1	proof	proof	NOUN
ap-1821	244	2	.	.	PUNCT
ap-1821	245	1	(	(	PUNCT
ap-1821	245	2	⇒	⇒	PROPN
ap-1821	245	3	)	)	PUNCT
ap-1821	245	4	:	:	PUNCT
ap-1821	245	5	if	if	SCONJ
ap-1821	245	6	the	the	DET
ap-1821	245	7	length	length	NOUN
ap-1821	245	8	of	of	ADP
ap-1821	245	9	the	the	DET
ap-1821	245	10	period	period	NOUN
ap-1821	245	11	equals	equal	VERB
ap-1821	245	12	3	3	NUM
ap-1821	245	13	,	,	PUNCT
ap-1821	245	14	then	then	ADV
ap-1821	245	15	by	by	ADP
ap-1821	245	16	theorem	theorem	NOUN
ap-1821	245	17	3.1	3.1	NUM
ap-1821	245	18	we	we	PRON
ap-1821	245	19	have	have	VERB
ap-1821	245	20	√	√	NUM
ap-1821	245	21	n	n	NOUN
ap-1821	245	22	=	=	SYM
ap-1821	246	1	[	[	X
ap-1821	246	2	n	n	CCONJ
ap-1821	246	3	,	,	PUNCT
ap-1821	246	4	x	x	X
ap-1821	246	5	,	,	PUNCT
ap-1821	246	6	x	x	NOUN
ap-1821	246	7	,	,	PUNCT
ap-1821	246	8	2n].√	2n].√	NOUN
ap-1821	246	9	n2	n2	NOUN
ap-1821	246	10	+	+	CCONJ
ap-1821	246	11	j	j	NOUN
ap-1821	246	12	=	=	SYM
ap-1821	246	13	n+	n+	NUM
ap-1821	246	14	1	1	NUM
ap-1821	246	15	x+	x+	SYM
ap-1821	246	16	1	1	NUM
ap-1821	246	17	x+	x+	SYM
ap-1821	246	18	1	1	NUM
ap-1821	246	19	2n+	2n+	NUM
ap-1821	246	20	(	(	PUNCT
ap-1821	246	21	√	√	PROPN
ap-1821	246	22	n2	n2	NOUN
ap-1821	246	23	+	+	CCONJ
ap-1821	246	24	j	j	PROPN
ap-1821	246	25	−	−	PROPN
ap-1821	246	26	n	n	CCONJ
ap-1821	246	27	)	)	PUNCT
ap-1821	246	28	,	,	PUNCT
ap-1821	246	29	hence	hence	ADV
ap-1821	246	30	we	we	PRON
ap-1821	246	31	get	get	VERB
ap-1821	246	32	j	j	NOUN
ap-1821	246	33	=	=	SYM
ap-1821	246	34	2xn+	2xn+	NUM
ap-1821	246	35	1	1	NUM
ap-1821	246	36	x2	x2	NOUN
ap-1821	247	1	+	+	CCONJ
ap-1821	247	2	1	1	NUM
ap-1821	247	3	.	.	PUNCT
ap-1821	248	1	since	since	SCONJ
ap-1821	248	2	j	j	PROPN
ap-1821	248	3	is	be	AUX
ap-1821	248	4	an	an	DET
ap-1821	248	5	integer	integer	NOUN
ap-1821	248	6	,	,	PUNCT
ap-1821	248	7	x	x	PRON
ap-1821	248	8	must	must	AUX
ap-1821	248	9	be	be	AUX
ap-1821	248	10	even	even	ADV
ap-1821	248	11	.	.	PUNCT
ap-1821	249	1	furthermore	furthermore	ADV
ap-1821	249	2	,	,	PUNCT
ap-1821	249	3	as	as	ADP
ap-1821	249	4	j	j	PROPN
ap-1821	249	5	6=	6=	PROPN
ap-1821	249	6	1	1	NUM
ap-1821	249	7	by	by	ADP
ap-1821	249	8	observation	observation	NOUN
ap-1821	249	9	3.3	3.3	NUM
ap-1821	249	10	,	,	PUNCT
ap-1821	249	11	there	there	PRON
ap-1821	249	12	exists	exist	VERB
ap-1821	249	13	a	a	DET
ap-1821	249	14	≥	≥	NUM
ap-1821	249	15	1	1	NUM
ap-1821	249	16	such	such	ADJ
ap-1821	249	17	that	that	SCONJ
ap-1821	249	18	x	x	NOUN
ap-1821	249	19	=	=	SYM
ap-1821	249	20	2a	2a	NUM
ap-1821	249	21	.	.	PUNCT
ap-1821	250	1	it	it	PRON
ap-1821	250	2	follows	follow	VERB
ap-1821	250	3	then	then	ADV
ap-1821	250	4	from	from	ADP
ap-1821	250	5	j	j	PROPN
ap-1821	250	6	=	=	SYM
ap-1821	250	7	4an+	4an+	PROPN
ap-1821	250	8	1	1	NUM
ap-1821	250	9	4a2	4a2	NUM
ap-1821	251	1	+	+	CCONJ
ap-1821	251	2	1	1	NUM
ap-1821	251	3	that	that	PRON
ap-1821	251	4	n	n	AUX
ap-1821	251	5	=	=	SYM
ap-1821	251	6	aj	aj	PROPN
ap-1821	252	1	+	+	PROPN
ap-1821	252	2	j−1	j−1	PROPN
ap-1821	252	3	4a	4a	NOUN
ap-1821	252	4	.	.	PUNCT
ap-1821	253	1	since	since	SCONJ
ap-1821	253	2	n	n	NUM
ap-1821	253	3	is	be	AUX
ap-1821	253	4	an	an	DET
ap-1821	253	5	integer	integer	NOUN
ap-1821	253	6	,	,	PUNCT
ap-1821	253	7	we	we	PRON
ap-1821	253	8	obtain	obtain	VERB
ap-1821	253	9	finally	finally	ADV
ap-1821	253	10	j	j	X
ap-1821	254	1	=	=	PUNCT
ap-1821	254	2	4ak	4ak	PROPN
ap-1821	255	1	+	+	CCONJ
ap-1821	255	2	1	1	NUM
ap-1821	255	3	and	and	CCONJ
ap-1821	255	4	n	n	PROPN
ap-1821	255	5	=	=	SYM
ap-1821	255	6	aj	aj	PROPN
ap-1821	256	1	+	+	CCONJ
ap-1821	256	2	k	k	PROPN
ap-1821	256	3	for	for	ADP
ap-1821	256	4	some	some	DET
ap-1821	256	5	k	k	PROPN
ap-1821	256	6	≥	≥	NUM
ap-1821	256	7	1	1	NUM
ap-1821	256	8	.	.	PUNCT
ap-1821	257	1	it	it	PRON
ap-1821	257	2	is	be	AUX
ap-1821	257	3	easy	easy	ADJ
ap-1821	257	4	to	to	PART
ap-1821	257	5	verify	verify	VERB
ap-1821	257	6	that	that	SCONJ
ap-1821	257	7	j	j	PROPN
ap-1821	257	8	≥	≥	NUM
ap-1821	257	9	5	5	NUM
ap-1821	257	10	and	and	CCONJ
ap-1821	257	11	j	j	PROPN
ap-1821	257	12	≤	≤	PROPN
ap-1821	257	13	n−	n−	PROPN
ap-1821	257	14	1	1	NUM
ap-1821	257	15	.	.	PUNCT
ap-1821	258	1	(	(	PUNCT
ap-1821	258	2	⇐	⇐	PROPN
ap-1821	258	3	)	)	PUNCT
ap-1821	258	4	:	:	PUNCT
ap-1821	258	5	the	the	DET
ap-1821	258	6	reverse	reverse	ADJ
ap-1821	258	7	implication	implication	NOUN
ap-1821	258	8	is	be	AUX
ap-1821	258	9	only	only	ADV
ap-1821	258	10	an	an	DET
ap-1821	258	11	exercise	exercise	NOUN
ap-1821	258	12	in	in	ADP
ap-1821	258	13	manipulation	manipulation	NOUN
ap-1821	258	14	with	with	ADP
ap-1821	258	15	square	square	ADJ
ap-1821	258	16	roots	root	NOUN
ap-1821	258	17	and	and	CCONJ
ap-1821	258	18	integer	integer	NOUN
ap-1821	258	19	parts	part	NOUN
ap-1821	258	20	.	.	PUNCT
ap-1821	259	1	we	we	PRON
ap-1821	259	2	have	have	VERB
ap-1821	259	3	it	it	PRON
ap-1821	259	4	for	for	ADP
ap-1821	259	5	the	the	DET
ap-1821	259	6	reader	reader	NOUN
ap-1821	259	7	.	.	PUNCT
ap-1821	260	1	in	in	ADP
ap-1821	260	2	order	order	NOUN
ap-1821	260	3	to	to	PART
ap-1821	260	4	save	save	VERB
ap-1821	260	5	space	space	NOUN
ap-1821	260	6	in	in	ADP
ap-1821	260	7	the	the	DET
ap-1821	260	8	proofs	proof	NOUN
ap-1821	260	9	,	,	PUNCT
ap-1821	260	10	let	let	VERB
ap-1821	260	11	us	we	PRON
ap-1821	260	12	introduce	introduce	VERB
ap-1821	260	13	325	325	NUM
ap-1821	260	14	ľ	ľ	NUM
ap-1821	260	15	.	.	PUNCT
ap-1821	261	1	balková	balková	PROPN
ap-1821	261	2	,	,	PUNCT
ap-1821	261	3	a.	a.	PROPN
ap-1821	261	4	hrušková	hrušková	PROPN
ap-1821	261	5	acta	acta	PROPN
ap-1821	261	6	polytechnica	polytechnica	PROPN
ap-1821	261	7	n	n	PROPN
ap-1821	261	8	j	j	PROPN
ap-1821	261	9	√	√	PROPN
ap-1821	261	10	n	n	CCONJ
ap-1821	261	11	`	`	PUNCT
ap-1821	261	12	=	=	SYM
ap-1821	261	13	4	4	NUM
ap-1821	261	14	2k	2k	NUM
ap-1821	261	15	+	+	CCONJ
ap-1821	261	16	1	1	NUM
ap-1821	261	17	,	,	PUNCT
ap-1821	261	18	k	k	X
ap-1821	261	19	≥	≥	NUM
ap-1821	261	20	2	2	NUM
ap-1821	261	21	2n−	2n−	NUM
ap-1821	261	22	3	3	NUM
ap-1821	261	23	[	[	X
ap-1821	261	24	n	n	CCONJ
ap-1821	261	25	,	,	PUNCT
ap-1821	261	26	1	1	NUM
ap-1821	261	27	,	,	PUNCT
ap-1821	261	28	n−3	n−3	PROPN
ap-1821	261	29	2	2	NUM
ap-1821	261	30	,	,	PUNCT
ap-1821	261	31	1	1	NUM
ap-1821	261	32	,	,	PUNCT
ap-1821	261	33	2n	2n	NUM
ap-1821	261	34	]	]	PUNCT
ap-1821	261	35	2k	2k	NOUN
ap-1821	261	36	+	+	CCONJ
ap-1821	261	37	1	1	NUM
ap-1821	261	38	,	,	PUNCT
ap-1821	261	39	k	k	X
ap-1821	261	40	≥	≥	NUM
ap-1821	261	41	1	1	NUM
ap-1821	261	42	3n+1	3n+1	NUM
ap-1821	261	43	2	2	NUM
ap-1821	261	44	[	[	X
ap-1821	261	45	n	n	CCONJ
ap-1821	261	46	,	,	PUNCT
ap-1821	261	47	1	1	NUM
ap-1821	261	48	,	,	PUNCT
ap-1821	261	49	2	2	NUM
ap-1821	261	50	,	,	PUNCT
ap-1821	261	51	1	1	NUM
ap-1821	261	52	,	,	PUNCT
ap-1821	261	53	2n	2n	NUM
ap-1821	261	54	]	]	PUNCT
ap-1821	262	1	3k	3k	X
ap-1821	262	2	+	+	CCONJ
ap-1821	262	3	2	2	NUM
ap-1821	262	4	5n+2	5n+2	NUM
ap-1821	262	5	3	3	NUM
ap-1821	262	6	[	[	X
ap-1821	262	7	n	n	CCONJ
ap-1821	262	8	,	,	PUNCT
ap-1821	262	9	1	1	NUM
ap-1821	262	10	,	,	PUNCT
ap-1821	262	11	4	4	NUM
ap-1821	262	12	,	,	PUNCT
ap-1821	262	13	1	1	NUM
ap-1821	262	14	,	,	PUNCT
ap-1821	262	15	2n	2n	NUM
ap-1821	262	16	]	]	PUNCT
ap-1821	262	17	3k	3k	X
ap-1821	263	1	+	+	CCONJ
ap-1821	263	2	2	2	NUM
ap-1821	263	3	,	,	PUNCT
ap-1821	263	4	k	k	X
ap-1821	263	5	≥	≥	NUM
ap-1821	263	6	1	1	NUM
ap-1821	263	7	2n−	2n−	NUM
ap-1821	263	8	2	2	NUM
ap-1821	263	9	[	[	X
ap-1821	263	10	n	n	CCONJ
ap-1821	263	11	,	,	PUNCT
ap-1821	263	12	1	1	NUM
ap-1821	263	13	,	,	PUNCT
ap-1821	263	14	2n−4	2n−4	NUM
ap-1821	263	15	3	3	NUM
ap-1821	263	16	,	,	PUNCT
ap-1821	263	17	1	1	NUM
ap-1821	263	18	,	,	PUNCT
ap-1821	263	19	2n	2n	NUM
ap-1821	263	20	]	]	PUNCT
ap-1821	263	21	3k	3k	X
ap-1821	263	22	+	+	CCONJ
ap-1821	263	23	2	2	NUM
ap-1821	263	24	4n+1	4n+1	NOUN
ap-1821	263	25	3	3	NUM
ap-1821	263	26	[	[	X
ap-1821	263	27	n	n	CCONJ
ap-1821	263	28	,	,	PUNCT
ap-1821	263	29	1	1	NUM
ap-1821	263	30	,	,	PUNCT
ap-1821	263	31	1	1	NUM
ap-1821	263	32	,	,	PUNCT
ap-1821	263	33	1	1	NUM
ap-1821	263	34	,	,	PUNCT
ap-1821	263	35	2n	2n	NUM
ap-1821	263	36	]	]	PUNCT
ap-1821	263	37	5k	5k	NUM
ap-1821	263	38	+	+	CCONJ
ap-1821	263	39	2	2	NUM
ap-1821	263	40	,	,	PUNCT
ap-1821	263	41	k	k	X
ap-1821	263	42	≥	≥	NUM
ap-1821	263	43	1	1	NUM
ap-1821	263	44	n−	n−	NOUN
ap-1821	263	45	1	1	NUM
ap-1821	263	46	[	[	X
ap-1821	263	47	n	n	CCONJ
ap-1821	263	48	,	,	PUNCT
ap-1821	263	49	2	2	NUM
ap-1821	263	50	,	,	PUNCT
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ap-1821	265	23	1	1	NUM
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ap-1821	265	30	,	,	PUNCT
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ap-1821	265	47	1	1	NUM
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ap-1821	265	50	n	n	CCONJ
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ap-1821	265	52	n−1	n−1	PROPN
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ap-1821	265	54	,	,	PUNCT
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ap-1821	265	59	n−1	n−1	PROPN
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ap-1821	265	75	,	,	PUNCT
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ap-1821	266	44	1	1	NUM
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ap-1821	273	21	1	1	NUM
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ap-1821	273	23	2	2	NUM
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ap-1821	273	25	2n−3	2n−3	NUM
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ap-1821	273	27	,	,	PUNCT
ap-1821	273	28	2n	2n	NUM
ap-1821	273	29	]	]	PUNCT
ap-1821	273	30	`	`	PUNCT
ap-1821	273	31	=	=	SYM
ap-1821	273	32	10	10	NUM
ap-1821	273	33	6k	6k	NOUN
ap-1821	273	34	+	+	CCONJ
ap-1821	273	35	3	3	NUM
ap-1821	273	36	,	,	PUNCT
ap-1821	273	37	k	k	X
ap-1821	273	38	≥	≥	NUM
ap-1821	273	39	1	1	NUM
ap-1821	273	40	4n	4n	X
ap-1821	273	41	3	3	NUM
ap-1821	273	42	[	[	X
ap-1821	273	43	n	n	CCONJ
ap-1821	273	44	,	,	PUNCT
ap-1821	273	45	1	1	NUM
ap-1821	273	46	,	,	PUNCT
ap-1821	273	47	1	1	NUM
ap-1821	273	48	,	,	PUNCT
ap-1821	273	49	1	1	NUM
ap-1821	273	50	,	,	PUNCT
ap-1821	273	51	n−1	n−1	PROPN
ap-1821	273	52	2	2	NUM
ap-1821	273	53	,	,	PUNCT
ap-1821	273	54	6	6	NUM
ap-1821	273	55	,	,	PUNCT
ap-1821	273	56	n−1	n−1	PROPN
ap-1821	273	57	2	2	NUM
ap-1821	273	58	,	,	PUNCT
ap-1821	273	59	1	1	NUM
ap-1821	273	60	,	,	PUNCT
ap-1821	273	61	1	1	NUM
ap-1821	273	62	,	,	PUNCT
ap-1821	273	63	1	1	NUM
ap-1821	273	64	,	,	PUNCT
ap-1821	273	65	2n	2n	NUM
ap-1821	273	66	]	]	X
ap-1821	273	67	9k	9k	NUM
ap-1821	274	1	+	+	CCONJ
ap-1821	274	2	6	6	NUM
ap-1821	274	3	10n+3	10n+3	NUM
ap-1821	274	4	9	9	NUM
ap-1821	274	5	[	[	X
ap-1821	274	6	n	n	CCONJ
ap-1821	274	7	,	,	PUNCT
ap-1821	274	8	1	1	NUM
ap-1821	274	9	,	,	PUNCT
ap-1821	274	10	1	1	NUM
ap-1821	274	11	,	,	PUNCT
ap-1821	274	12	3	3	NUM
ap-1821	274	13	,	,	PUNCT
ap-1821	274	14	1	1	NUM
ap-1821	274	15	,	,	PUNCT
ap-1821	274	16	n−	n−	NOUN
ap-1821	274	17	1	1	NUM
ap-1821	274	18	,	,	PUNCT
ap-1821	274	19	1	1	NUM
ap-1821	274	20	,	,	PUNCT
ap-1821	274	21	3	3	NUM
ap-1821	274	22	,	,	PUNCT
ap-1821	274	23	1	1	NUM
ap-1821	274	24	,	,	PUNCT
ap-1821	274	25	1	1	NUM
ap-1821	274	26	,	,	PUNCT
ap-1821	274	27	2n	2n	NUM
ap-1821	274	28	]	]	PUNCT
ap-1821	274	29	10k	10k	NOUN
ap-1821	274	30	+	+	CCONJ
ap-1821	274	31	5	5	NUM
ap-1821	274	32	,	,	PUNCT
ap-1821	274	33	k	k	X
ap-1821	274	34	≥	≥	NUM
ap-1821	274	35	1	1	NUM
ap-1821	274	36	4n	4n	X
ap-1821	274	37	5	5	NUM
ap-1821	274	38	[	[	X
ap-1821	274	39	n	n	CCONJ
ap-1821	274	40	,	,	PUNCT
ap-1821	274	41	2	2	NUM
ap-1821	274	42	,	,	PUNCT
ap-1821	274	43	1	1	NUM
ap-1821	274	44	,	,	PUNCT
ap-1821	274	45	1	1	NUM
ap-1821	274	46	,	,	PUNCT
ap-1821	274	47	n−1	n−1	PROPN
ap-1821	274	48	2	2	NUM
ap-1821	274	49	,	,	PUNCT
ap-1821	274	50	10	10	NUM
ap-1821	274	51	,	,	PUNCT
ap-1821	274	52	n−1	n−1	PROPN
ap-1821	274	53	2	2	NUM
ap-1821	274	54	,	,	PUNCT
ap-1821	274	55	1	1	NUM
ap-1821	274	56	,	,	PUNCT
ap-1821	274	57	1	1	NUM
ap-1821	274	58	,	,	PUNCT
ap-1821	274	59	2	2	NUM
ap-1821	274	60	,	,	PUNCT
ap-1821	274	61	2n	2n	NUM
ap-1821	274	62	]	]	PUNCT
ap-1821	274	63	table	table	NOUN
ap-1821	274	64	2	2	NUM
ap-1821	274	65	.	.	PUNCT
ap-1821	275	1	lengths	length	NOUN
ap-1821	275	2	`	`	PUNCT
ap-1821	275	3	of	of	ADP
ap-1821	275	4	periods	period	NOUN
ap-1821	275	5	and	and	CCONJ
ap-1821	275	6	the	the	DET
ap-1821	275	7	form	form	NOUN
ap-1821	275	8	of	of	ADP
ap-1821	275	9	continued	continue	VERB
ap-1821	275	10	fractions	fraction	NOUN
ap-1821	275	11	for	for	ADP
ap-1821	275	12	several	several	ADJ
ap-1821	275	13	classes	class	NOUN
ap-1821	275	14	.	.	PUNCT
ap-1821	276	1	the	the	DET
ap-1821	276	2	following	follow	VERB
ap-1821	276	3	notation	notation	NOUN
ap-1821	276	4	〈	〈	PROPN
ap-1821	276	5	a0	a0	PROPN
ap-1821	276	6	,	,	PUNCT
ap-1821	276	7	a1	a1	NOUN
ap-1821	276	8	,	,	PUNCT
ap-1821	276	9	.	.	PUNCT
ap-1821	276	10	.	.	PUNCT
ap-1821	276	11	.	.	PUNCT
ap-1821	277	1	,	,	PUNCT
ap-1821	277	2	an−1	an−1	ADJ
ap-1821	277	3	,	,	PUNCT
ap-1821	277	4	an	an	DET
ap-1821	277	5	〉	〉	PROPN
ap-1821	277	6	=	=	SYM
ap-1821	277	7	a0	a0	PROPN
ap-1821	277	8	+	+	CCONJ
ap-1821	277	9	1	1	NUM
ap-1821	277	10	a1	a1	NOUN
ap-1821	277	11	+	+	CCONJ
ap-1821	277	12	1	1	NUM
ap-1821	277	13	.	.	PUNCT
ap-1821	277	14	.	.	PUNCT
ap-1821	277	15	.	.	PUNCT
ap-1821	278	1	+	+	CCONJ
ap-1821	278	2	1	1	NUM
ap-1821	278	3	an−1	an−1	ADJ
ap-1821	278	4	+	+	NUM
ap-1821	278	5	1	1	NUM
ap-1821	278	6	an	an	DET
ap-1821	278	7	,	,	PUNCT
ap-1821	278	8	where	where	SCONJ
ap-1821	278	9	ai	ai	VERB
ap-1821	278	10	∈	∈	PROPN
ap-1821	278	11	n	n	ADV
ap-1821	278	12	for	for	ADP
ap-1821	278	13	i	i	PRON
ap-1821	278	14	∈	∈	PROPN
ap-1821	278	15	{	{	PUNCT
ap-1821	278	16	0	0	NUM
ap-1821	278	17	,	,	PUNCT
ap-1821	278	18	1	1	NUM
ap-1821	278	19	,	,	PUNCT
ap-1821	278	20	2	2	NUM
ap-1821	278	21	,	,	PUNCT
ap-1821	278	22	.	.	PUNCT
ap-1821	278	23	.	.	PUNCT
ap-1821	278	24	.	.	PUNCT
ap-1821	279	1	,	,	PUNCT
ap-1821	280	1	n	n	CCONJ
ap-1821	280	2	−	−	PROPN
ap-1821	280	3	1	1	NUM
ap-1821	280	4	}	}	PUNCT
ap-1821	280	5	,	,	PUNCT
ap-1821	280	6	but	but	CCONJ
ap-1821	280	7	an	an	DET
ap-1821	280	8	∈	∈	PROPN
ap-1821	280	9	r.	r.	PROPN
ap-1821	280	10	observation	observation	NOUN
ap-1821	280	11	3.6	3.6	NUM
ap-1821	280	12	.	.	PUNCT
ap-1821	281	1	let	let	VERB
ap-1821	281	2	j	j	NOUN
ap-1821	281	3	=	=	NOUN
ap-1821	281	4	4	4	X
ap-1821	281	5	.	.	PUNCT
ap-1821	282	1	if	if	SCONJ
ap-1821	282	2	n	n	PRON
ap-1821	282	3	is	be	AUX
ap-1821	282	4	even	even	ADV
ap-1821	282	5	,	,	PUNCT
ap-1821	282	6	then	then	ADV
ap-1821	282	7	the	the	DET
ap-1821	282	8	length	length	NOUN
ap-1821	282	9	of	of	ADP
ap-1821	282	10	the	the	DET
ap-1821	282	11	period	period	NOUN
ap-1821	282	12	is	be	AUX
ap-1821	282	13	2	2	NUM
ap-1821	282	14	and	and	CCONJ
ap-1821	282	15	√	√	NOUN
ap-1821	282	16	n	n	NOUN
ap-1821	282	17	=	=	SYM
ap-1821	282	18	[	[	PUNCT
ap-1821	282	19	n	n	CCONJ
ap-1821	282	20	,	,	PUNCT
ap-1821	282	21	2n	2n	NUM
ap-1821	282	22	j	j	PROPN
ap-1821	282	23	,	,	PUNCT
ap-1821	282	24	2n	2n	NUM
ap-1821	282	25	]	]	PUNCT
ap-1821	282	26	.	.	PUNCT
ap-1821	283	1	if	if	SCONJ
ap-1821	283	2	n	n	PROPN
ap-1821	283	3	is	be	AUX
ap-1821	283	4	odd	odd	ADJ
ap-1821	283	5	,	,	PUNCT
ap-1821	283	6	then	then	ADV
ap-1821	283	7	the	the	DET
ap-1821	283	8	length	length	NOUN
ap-1821	283	9	of	of	ADP
ap-1821	283	10	the	the	DET
ap-1821	283	11	period	period	NOUN
ap-1821	283	12	is	be	AUX
ap-1821	283	13	5	5	NUM
ap-1821	283	14	and√	and√	NOUN
ap-1821	283	15	n	n	NOUN
ap-1821	283	16	=	=	SYM
ap-1821	283	17	[	[	PUNCT
ap-1821	283	18	n	n	CCONJ
ap-1821	283	19	,	,	PUNCT
ap-1821	283	20	n−1	n−1	PROPN
ap-1821	283	21	2	2	NUM
ap-1821	283	22	,	,	PUNCT
ap-1821	283	23	1	1	NUM
ap-1821	283	24	,	,	PUNCT
ap-1821	283	25	1	1	NUM
ap-1821	283	26	,	,	PUNCT
ap-1821	283	27	n−1	n−1	PROPN
ap-1821	283	28	2	2	NUM
ap-1821	283	29	,	,	PUNCT
ap-1821	283	30	2n	2n	NUM
ap-1821	283	31	]	]	PUNCT
ap-1821	283	32	.	.	PUNCT
ap-1821	284	1	proof	proof	NOUN
ap-1821	284	2	.	.	PUNCT
ap-1821	285	1	if	if	SCONJ
ap-1821	285	2	n	n	PRON
ap-1821	285	3	is	be	AUX
ap-1821	285	4	even	even	ADV
ap-1821	285	5	,	,	PUNCT
ap-1821	285	6	then	then	ADV
ap-1821	285	7	2n	2n	NUM
ap-1821	285	8	j	j	PROPN
ap-1821	285	9	is	be	AUX
ap-1821	285	10	an	an	DET
ap-1821	285	11	integer	integer	NOUN
ap-1821	285	12	and	and	CCONJ
ap-1821	285	13	the	the	DET
ap-1821	285	14	statement	statement	NOUN
ap-1821	285	15	is	be	AUX
ap-1821	285	16	a	a	DET
ap-1821	285	17	corollary	corollary	NOUN
ap-1821	285	18	of	of	ADP
ap-1821	285	19	observation	observation	NOUN
ap-1821	285	20	3.4	3.4	NUM
ap-1821	285	21	.	.	PUNCT
ap-1821	286	1	if	if	SCONJ
ap-1821	286	2	n	n	NOUN
ap-1821	286	3	is	be	AUX
ap-1821	286	4	odd	odd	ADJ
ap-1821	286	5	,	,	PUNCT
ap-1821	286	6	it	it	PRON
ap-1821	286	7	holds√	holds√	VERB
ap-1821	286	8	n2	n2	NOUN
ap-1821	286	9	+	+	CCONJ
ap-1821	286	10	4	4	NUM
ap-1821	286	11	=	=	SYM
ap-1821	286	12	n+	n+	X
ap-1821	286	13	(	(	PUNCT
ap-1821	286	14	√	√	PROPN
ap-1821	286	15	n2	n2	NOUN
ap-1821	286	16	+	+	CCONJ
ap-1821	286	17	4−	4−	NOUN
ap-1821	286	18	n	n	NOUN
ap-1821	286	19	)	)	PUNCT
ap-1821	286	20	=	=	SYM
ap-1821	286	21	〈	〈	PROPN
ap-1821	286	22	n	n	CCONJ
ap-1821	286	23	,	,	PUNCT
ap-1821	286	24	√	√	PROPN
ap-1821	286	25	n2	n2	NOUN
ap-1821	286	26	+	+	CCONJ
ap-1821	286	27	4	4	NUM
ap-1821	286	28	+	+	SYM
ap-1821	286	29	n	n	PRON
ap-1821	286	30	4	4	NUM
ap-1821	286	31	〉	〉	NOUN
ap-1821	286	32	=	=	X
ap-1821	286	33	〈	〈	PROPN
ap-1821	286	34	n	n	CCONJ
ap-1821	286	35	,	,	PUNCT
ap-1821	286	36	n−	n−	NOUN
ap-1821	286	37	1	1	NUM
ap-1821	286	38	2	2	NUM
ap-1821	286	39	,	,	PUNCT
ap-1821	286	40	√	√	NOUN
ap-1821	286	41	n2	n2	NOUN
ap-1821	286	42	+	+	CCONJ
ap-1821	286	43	4	4	NUM
ap-1821	286	44	+	+	NUM
ap-1821	286	45	n−	n−	NOUN
ap-1821	286	46	2	2	NUM
ap-1821	286	47	n	n	PRON
ap-1821	286	48	〉	〉	NOUN
ap-1821	286	49	=	=	SYM
ap-1821	286	50	〈	〈	PROPN
ap-1821	286	51	n	n	CCONJ
ap-1821	286	52	,	,	PUNCT
ap-1821	286	53	n−	n−	NOUN
ap-1821	286	54	1	1	NUM
ap-1821	286	55	2	2	NUM
ap-1821	286	56	,	,	PUNCT
ap-1821	286	57	1	1	NUM
ap-1821	286	58	,	,	PUNCT
ap-1821	286	59	√	√	ADJ
ap-1821	286	60	n2	n2	NOUN
ap-1821	286	61	+	+	CCONJ
ap-1821	286	62	4	4	NUM
ap-1821	286	63	+	+	SYM
ap-1821	286	64	2	2	NUM
ap-1821	286	65	n	n	PRON
ap-1821	286	66	〉	〉	NOUN
ap-1821	286	67	=	=	SYM
ap-1821	286	68	〈	〈	PROPN
ap-1821	286	69	n	n	CCONJ
ap-1821	286	70	,	,	PUNCT
ap-1821	286	71	n−	n−	NOUN
ap-1821	286	72	1	1	NUM
ap-1821	286	73	2	2	NUM
ap-1821	286	74	,	,	PUNCT
ap-1821	286	75	1	1	NUM
ap-1821	286	76	,	,	PUNCT
ap-1821	286	77	1	1	NUM
ap-1821	286	78	,	,	PUNCT
ap-1821	286	79	√	√	ADJ
ap-1821	286	80	n2	n2	NOUN
ap-1821	286	81	+	+	CCONJ
ap-1821	286	82	4	4	NUM
ap-1821	286	83	+	+	NUM
ap-1821	286	84	n−	n−	NOUN
ap-1821	286	85	2	2	NUM
ap-1821	286	86	4	4	NUM
ap-1821	286	87	〉	〉	NOUN
ap-1821	286	88	=	=	SYM
ap-1821	286	89	〈	〈	PROPN
ap-1821	286	90	n	n	CCONJ
ap-1821	286	91	,	,	PUNCT
ap-1821	286	92	n−	n−	NOUN
ap-1821	286	93	1	1	NUM
ap-1821	286	94	2	2	NUM
ap-1821	286	95	,	,	PUNCT
ap-1821	286	96	1	1	NUM
ap-1821	286	97	,	,	PUNCT
ap-1821	286	98	1	1	NUM
ap-1821	286	99	,	,	PUNCT
ap-1821	286	100	n−	n−	NOUN
ap-1821	286	101	1	1	NUM
ap-1821	286	102	2	2	NUM
ap-1821	286	103	,	,	PUNCT
ap-1821	286	104	2n+	2n+	NUM
ap-1821	286	105	(	(	PUNCT
ap-1821	286	106	√	√	NOUN
ap-1821	286	107	n2	n2	NOUN
ap-1821	286	108	+	+	CCONJ
ap-1821	286	109	4−	4−	NUM
ap-1821	286	110	n	n	CCONJ
ap-1821	286	111	)	)	PUNCT
ap-1821	286	112	〉	〉	NOUN
ap-1821	286	113	,	,	PUNCT
ap-1821	286	114	thus	thus	ADV
ap-1821	286	115	√	√	ADP
ap-1821	286	116	n	n	NOUN
ap-1821	286	117	=	=	SYM
ap-1821	286	118	[	[	X
ap-1821	286	119	n	n	CCONJ
ap-1821	286	120	,	,	PUNCT
ap-1821	286	121	n−1	n−1	PROPN
ap-1821	286	122	2	2	NUM
ap-1821	286	123	,	,	PUNCT
ap-1821	286	124	1	1	NUM
ap-1821	286	125	,	,	PUNCT
ap-1821	286	126	1	1	NUM
ap-1821	286	127	,	,	PUNCT
ap-1821	286	128	n−1	n−1	PROPN
ap-1821	286	129	2	2	NUM
ap-1821	286	130	,	,	PUNCT
ap-1821	286	131	2n	2n	NUM
ap-1821	286	132	]	]	PUNCT
ap-1821	286	133	.	.	PUNCT
ap-1821	287	1	observation	observation	NOUN
ap-1821	287	2	3.7	3.7	NUM
ap-1821	287	3	.	.	PUNCT
ap-1821	288	1	for	for	ADP
ap-1821	288	2	n	n	PROPN
ap-1821	288	3	>	>	SYM
ap-1821	288	4	1	1	NUM
ap-1821	288	5	and	and	CCONJ
ap-1821	288	6	j	j	NOUN
ap-1821	288	7	=	=	SYM
ap-1821	288	8	2n	2n	NUM
ap-1821	288	9	−	−	NOUN
ap-1821	288	10	1	1	NUM
ap-1821	288	11	the	the	DET
ap-1821	288	12	length	length	NOUN
ap-1821	288	13	of	of	ADP
ap-1821	288	14	the	the	DET
ap-1821	288	15	period	period	NOUN
ap-1821	288	16	is	be	AUX
ap-1821	288	17	4	4	NUM
ap-1821	288	18	and	and	CCONJ
ap-1821	288	19	the	the	DET
ap-1821	288	20	continued	continue	VERB
ap-1821	288	21	fraction	fraction	NOUN
ap-1821	288	22	is	be	AUX
ap-1821	288	23	then	then	ADV
ap-1821	288	24	√	√	PROPN
ap-1821	288	25	n	n	NOUN
ap-1821	288	26	=	=	SYM
ap-1821	289	1	[	[	X
ap-1821	289	2	n	n	CCONJ
ap-1821	289	3	,	,	PUNCT
ap-1821	289	4	1	1	NUM
ap-1821	289	5	,	,	PUNCT
ap-1821	289	6	n−	n−	NOUN
ap-1821	289	7	1	1	NUM
ap-1821	289	8	,	,	PUNCT
ap-1821	289	9	1	1	NUM
ap-1821	289	10	,	,	PUNCT
ap-1821	289	11	2n	2n	NUM
ap-1821	289	12	]	]	PUNCT
ap-1821	289	13	.	.	PUNCT
ap-1821	290	1	proof.√	proof.√	NOUN
ap-1821	290	2	n2	n2	NOUN
ap-1821	290	3	+	+	CCONJ
ap-1821	290	4	2n−	2n−	PROPN
ap-1821	290	5	1	1	NUM
ap-1821	290	6	=	=	SYM
ap-1821	290	7	n+	n+	X
ap-1821	290	8	(	(	PUNCT
ap-1821	290	9	√	√	PROPN
ap-1821	290	10	n2	n2	NOUN
ap-1821	290	11	+	+	CCONJ
ap-1821	291	1	2n−	2n−	PROPN
ap-1821	291	2	1−	1−	NUM
ap-1821	291	3	n	n	NOUN
ap-1821	291	4	)	)	PUNCT
ap-1821	291	5	=	=	SYM
ap-1821	291	6	〈	〈	PROPN
ap-1821	291	7	n	n	CCONJ
ap-1821	291	8	,	,	PUNCT
ap-1821	291	9	√	√	PROPN
ap-1821	291	10	n2	n2	NOUN
ap-1821	291	11	+	+	CCONJ
ap-1821	291	12	2n−	2n−	NUM
ap-1821	291	13	1	1	NUM
ap-1821	291	14	+	+	NUM
ap-1821	291	15	n	n	CCONJ
ap-1821	291	16	2n−	2n−	PROPN
ap-1821	291	17	1	1	NUM
ap-1821	291	18	〉	〉	NOUN
ap-1821	291	19	326	326	NUM
ap-1821	291	20	vol	vol	NOUN
ap-1821	291	21	.	.	PUNCT
ap-1821	292	1	53	53	NUM
ap-1821	292	2	no	no	NOUN
ap-1821	292	3	.	.	PUNCT
ap-1821	293	1	4/2013	4/2013	PROPN
ap-1821	293	2	continued	continue	VERB
ap-1821	293	3	fractions	fraction	NOUN
ap-1821	293	4	of	of	ADP
ap-1821	293	5	square	square	ADJ
ap-1821	293	6	roots	root	NOUN
ap-1821	293	7	of	of	ADP
ap-1821	293	8	natural	natural	ADJ
ap-1821	293	9	numbers	number	NOUN
ap-1821	293	10	=	=	PUNCT
ap-1821	293	11	〈	〈	NOUN
ap-1821	293	12	n	n	CCONJ
ap-1821	293	13	,	,	PUNCT
ap-1821	293	14	1	1	NUM
ap-1821	293	15	,	,	PUNCT
ap-1821	293	16	√	√	ADJ
ap-1821	293	17	n2	n2	NOUN
ap-1821	293	18	+	+	CCONJ
ap-1821	293	19	2n−	2n−	NUM
ap-1821	293	20	1	1	NUM
ap-1821	293	21	+	+	CCONJ
ap-1821	293	22	(	(	PUNCT
ap-1821	293	23	n−	n−	NOUN
ap-1821	293	24	1	1	NUM
ap-1821	293	25	)	)	PUNCT
ap-1821	293	26	2	2	NUM
ap-1821	293	27	〉	〉	NOUN
ap-1821	293	28	=	=	X
ap-1821	293	29	〈	〈	PROPN
ap-1821	293	30	n	n	CCONJ
ap-1821	293	31	,	,	PUNCT
ap-1821	293	32	1	1	NUM
ap-1821	293	33	,	,	PUNCT
ap-1821	293	34	n−	n−	NOUN
ap-1821	293	35	1	1	NUM
ap-1821	293	36	,	,	PUNCT
ap-1821	293	37	√	√	NOUN
ap-1821	293	38	n2	n2	NOUN
ap-1821	293	39	+	+	CCONJ
ap-1821	293	40	2n−	2n−	NUM
ap-1821	293	41	1	1	NUM
ap-1821	293	42	+	+	CCONJ
ap-1821	293	43	(	(	PUNCT
ap-1821	293	44	n−	n−	NOUN
ap-1821	293	45	1	1	NUM
ap-1821	293	46	)	)	PUNCT
ap-1821	293	47	2n−	2n−	NUM
ap-1821	293	48	1	1	NUM
ap-1821	293	49	〉	〉	NOUN
ap-1821	293	50	=	=	X
ap-1821	293	51	〈	〈	PROPN
ap-1821	293	52	n	n	CCONJ
ap-1821	293	53	,	,	PUNCT
ap-1821	293	54	1	1	NUM
ap-1821	293	55	,	,	PUNCT
ap-1821	293	56	n−	n−	NOUN
ap-1821	293	57	1	1	NUM
ap-1821	293	58	,	,	PUNCT
ap-1821	293	59	1	1	NUM
ap-1821	293	60	,	,	PUNCT
ap-1821	293	61	2n+	2n+	NUM
ap-1821	293	62	(	(	PUNCT
ap-1821	293	63	√	√	PROPN
ap-1821	293	64	n2	n2	NOUN
ap-1821	293	65	+	+	CCONJ
ap-1821	293	66	2n−	2n−	PROPN
ap-1821	293	67	1−	1−	NUM
ap-1821	293	68	n	n	NOUN
ap-1821	293	69	)	)	PUNCT
ap-1821	293	70	〉	〉	NOUN
ap-1821	293	71	,	,	PUNCT
ap-1821	293	72	hence	hence	ADV
ap-1821	293	73	√	√	VERB
ap-1821	293	74	n	n	NOUN
ap-1821	293	75	=	=	SYM
ap-1821	294	1	[	[	X
ap-1821	294	2	n	n	CCONJ
ap-1821	294	3	,	,	PUNCT
ap-1821	294	4	1	1	NUM
ap-1821	294	5	,	,	PUNCT
ap-1821	294	6	n−	n−	NOUN
ap-1821	294	7	1	1	NUM
ap-1821	294	8	,	,	PUNCT
ap-1821	294	9	1	1	NUM
ap-1821	294	10	,	,	PUNCT
ap-1821	294	11	2n	2n	NUM
ap-1821	294	12	]	]	PUNCT
ap-1821	294	13	observation	observation	NOUN
ap-1821	294	14	3.8	3.8	NUM
ap-1821	294	15	.	.	PUNCT
ap-1821	295	1	for	for	ADP
ap-1821	295	2	n	n	PROPN
ap-1821	295	3	>	>	SYM
ap-1821	295	4	3	3	NUM
ap-1821	295	5	and	and	CCONJ
ap-1821	295	6	j	j	PROPN
ap-1821	295	7	=	=	SYM
ap-1821	295	8	2n−	2n−	PROPN
ap-1821	295	9	3	3	NUM
ap-1821	295	10	,	,	PUNCT
ap-1821	295	11	either	either	CCONJ
ap-1821	295	12	the	the	DET
ap-1821	295	13	length	length	NOUN
ap-1821	295	14	of	of	ADP
ap-1821	295	15	the	the	DET
ap-1821	295	16	period	period	NOUN
ap-1821	295	17	is	be	AUX
ap-1821	295	18	4	4	NUM
ap-1821	295	19	if	if	SCONJ
ap-1821	295	20	n	n	NOUN
ap-1821	295	21	is	be	AUX
ap-1821	295	22	odd	odd	ADJ
ap-1821	295	23	and	and	CCONJ
ap-1821	295	24	the	the	DET
ap-1821	295	25	continued	continue	VERB
ap-1821	295	26	fraction	fraction	NOUN
ap-1821	295	27	is	be	AUX
ap-1821	295	28	then	then	ADV
ap-1821	295	29	√	√	PROPN
ap-1821	296	1	n	n	NOUN
ap-1821	296	2	=	=	SYM
ap-1821	296	3	[	[	PUNCT
ap-1821	296	4	n	n	CCONJ
ap-1821	296	5	,	,	PUNCT
ap-1821	296	6	1	1	NUM
ap-1821	296	7	,	,	PUNCT
ap-1821	296	8	n−3	n−3	PROPN
ap-1821	296	9	2	2	NUM
ap-1821	296	10	,	,	PUNCT
ap-1821	296	11	1	1	NUM
ap-1821	296	12	,	,	PUNCT
ap-1821	296	13	2n	2n	NUM
ap-1821	296	14	]	]	PUNCT
ap-1821	296	15	,	,	PUNCT
ap-1821	296	16	or	or	CCONJ
ap-1821	296	17	the	the	DET
ap-1821	296	18	length	length	NOUN
ap-1821	296	19	of	of	ADP
ap-1821	296	20	the	the	DET
ap-1821	296	21	period	period	NOUN
ap-1821	296	22	is	be	AUX
ap-1821	296	23	6	6	NUM
ap-1821	296	24	if	if	SCONJ
ap-1821	296	25	n	n	NOUN
ap-1821	296	26	is	be	AUX
ap-1821	296	27	even	even	ADV
ap-1821	296	28	and	and	CCONJ
ap-1821	296	29	the	the	DET
ap-1821	296	30	continued	continue	VERB
ap-1821	296	31	fraction	fraction	NOUN
ap-1821	296	32	is	be	AUX
ap-1821	296	33	then	then	ADV
ap-1821	296	34	√	√	PROPN
ap-1821	296	35	n	n	NOUN
ap-1821	296	36	=	=	SYM
ap-1821	297	1	[	[	X
ap-1821	297	2	n	n	CCONJ
ap-1821	297	3	,	,	PUNCT
ap-1821	297	4	1	1	NUM
ap-1821	297	5	,	,	PUNCT
ap-1821	297	6	n2	n2	ADJ
ap-1821	297	7	−	−	PROPN
ap-1821	297	8	1	1	NUM
ap-1821	297	9	,	,	PUNCT
ap-1821	297	10	2	2	NUM
ap-1821	297	11	,	,	PUNCT
ap-1821	297	12	n2	n2	ADJ
ap-1821	297	13	−	−	PROPN
ap-1821	297	14	1	1	NUM
ap-1821	297	15	,	,	PUNCT
ap-1821	297	16	1	1	NUM
ap-1821	297	17	,	,	PUNCT
ap-1821	297	18	2n	2n	NUM
ap-1821	297	19	]	]	PUNCT
ap-1821	297	20	.	.	PUNCT
ap-1821	298	1	proof	proof	NOUN
ap-1821	298	2	.	.	PUNCT
ap-1821	299	1	for	for	ADP
ap-1821	299	2	n	n	PROPN
ap-1821	299	3	odd:√	odd:√	PROPN
ap-1821	299	4	n2	n2	PROPN
ap-1821	300	1	+	+	CCONJ
ap-1821	301	1	2n−	2n−	PROPN
ap-1821	301	2	3	3	NUM
ap-1821	301	3	=	=	SYM
ap-1821	301	4	n+	n+	X
ap-1821	301	5	(	(	PUNCT
ap-1821	301	6	√	√	PROPN
ap-1821	301	7	n2	n2	NOUN
ap-1821	301	8	+	+	CCONJ
ap-1821	301	9	2n−	2n−	PROPN
ap-1821	301	10	3−	3−	NUM
ap-1821	301	11	n	n	NOUN
ap-1821	301	12	)	)	PUNCT
ap-1821	301	13	=	=	SYM
ap-1821	301	14	〈	〈	PROPN
ap-1821	301	15	n	n	CCONJ
ap-1821	301	16	,	,	PUNCT
ap-1821	301	17	√	√	PROPN
ap-1821	301	18	n2	n2	NOUN
ap-1821	301	19	+	+	CCONJ
ap-1821	301	20	2n−	2n−	NUM
ap-1821	301	21	3	3	NUM
ap-1821	301	22	+	+	NOUN
ap-1821	301	23	n	n	PROPN
ap-1821	301	24	2n−	2n−	PROPN
ap-1821	301	25	3	3	NUM
ap-1821	301	26	〉	〉	NOUN
ap-1821	301	27	=	=	X
ap-1821	301	28	〈	〈	PROPN
ap-1821	301	29	n	n	CCONJ
ap-1821	301	30	,	,	PUNCT
ap-1821	301	31	1	1	NUM
ap-1821	301	32	,	,	PUNCT
ap-1821	301	33	√	√	ADJ
ap-1821	301	34	n2	n2	NOUN
ap-1821	301	35	+	+	CCONJ
ap-1821	302	1	2n−	2n−	NUM
ap-1821	302	2	3−	3−	NUM
ap-1821	302	3	(	(	PUNCT
ap-1821	302	4	n−	n−	NOUN
ap-1821	302	5	3	3	NUM
ap-1821	302	6	)	)	PUNCT
ap-1821	302	7	4	4	NUM
ap-1821	302	8	〉	〉	NOUN
ap-1821	302	9	=	=	X
ap-1821	302	10	〈	〈	PROPN
ap-1821	302	11	n	n	CCONJ
ap-1821	302	12	,	,	PUNCT
ap-1821	302	13	1	1	NUM
ap-1821	302	14	,	,	PUNCT
ap-1821	302	15	n−	n−	NOUN
ap-1821	302	16	3	3	NUM
ap-1821	302	17	2	2	NUM
ap-1821	302	18	,	,	PUNCT
ap-1821	302	19	√	√	NOUN
ap-1821	302	20	n2	n2	NOUN
ap-1821	302	21	+	+	CCONJ
ap-1821	302	22	2n−	2n−	NUM
ap-1821	302	23	3	3	NUM
ap-1821	302	24	+	+	CCONJ
ap-1821	302	25	(	(	PUNCT
ap-1821	302	26	n−	n−	NOUN
ap-1821	302	27	3	3	NUM
ap-1821	302	28	)	)	PUNCT
ap-1821	302	29	2n−	2n−	PROPN
ap-1821	302	30	3	3	NUM
ap-1821	302	31	〉	〉	NOUN
ap-1821	302	32	=	=	X
ap-1821	302	33	〈	〈	PROPN
ap-1821	302	34	n	n	CCONJ
ap-1821	302	35	,	,	PUNCT
ap-1821	302	36	1	1	NUM
ap-1821	302	37	,	,	PUNCT
ap-1821	302	38	n−	n−	NOUN
ap-1821	302	39	3	3	NUM
ap-1821	302	40	2	2	NUM
ap-1821	302	41	,	,	PUNCT
ap-1821	302	42	1	1	NUM
ap-1821	302	43	,	,	PUNCT
ap-1821	302	44	2n+	2n+	NUM
ap-1821	302	45	(	(	PUNCT
ap-1821	302	46	√	√	PROPN
ap-1821	302	47	n2	n2	NOUN
ap-1821	302	48	+	+	CCONJ
ap-1821	302	49	2n−	2n−	PROPN
ap-1821	302	50	3−	3−	NUM
ap-1821	302	51	n	n	NOUN
ap-1821	302	52	)	)	PUNCT
ap-1821	302	53	〉	〉	NOUN
ap-1821	302	54	,	,	PUNCT
ap-1821	302	55	thus	thus	ADV
ap-1821	302	56	√	√	DET
ap-1821	302	57	n	n	NOUN
ap-1821	302	58	=	=	SYM
ap-1821	302	59	[	[	PUNCT
ap-1821	302	60	n	n	CCONJ
ap-1821	302	61	,	,	PUNCT
ap-1821	302	62	1	1	NUM
ap-1821	302	63	,	,	PUNCT
ap-1821	302	64	n−3	n−3	PROPN
ap-1821	302	65	2	2	NUM
ap-1821	302	66	,	,	PUNCT
ap-1821	302	67	1	1	NUM
ap-1821	302	68	,	,	PUNCT
ap-1821	302	69	2n	2n	NUM
ap-1821	302	70	]	]	PUNCT
ap-1821	302	71	.	.	PUNCT
ap-1821	303	1	for	for	ADP
ap-1821	303	2	n	n	CCONJ
ap-1821	303	3	even:√	even:√	ADJ
ap-1821	303	4	n2	n2	NOUN
ap-1821	303	5	+	+	CCONJ
ap-1821	303	6	2n−	2n−	PROPN
ap-1821	303	7	3	3	NUM
ap-1821	303	8	=	=	SYM
ap-1821	303	9	n+	n+	X
ap-1821	303	10	(	(	PUNCT
ap-1821	303	11	√	√	PROPN
ap-1821	303	12	n2	n2	NOUN
ap-1821	303	13	+	+	CCONJ
ap-1821	303	14	2n−	2n−	PROPN
ap-1821	303	15	3−	3−	NUM
ap-1821	303	16	n	n	NOUN
ap-1821	303	17	)	)	PUNCT
ap-1821	303	18	=	=	SYM
ap-1821	303	19	〈	〈	PROPN
ap-1821	303	20	n	n	CCONJ
ap-1821	303	21	,	,	PUNCT
ap-1821	303	22	√	√	PROPN
ap-1821	303	23	n2	n2	NOUN
ap-1821	303	24	+	+	CCONJ
ap-1821	303	25	2n−	2n−	NUM
ap-1821	303	26	3	3	NUM
ap-1821	303	27	+	+	NOUN
ap-1821	303	28	n	n	PROPN
ap-1821	303	29	2n−	2n−	PROPN
ap-1821	303	30	3	3	NUM
ap-1821	303	31	〉	〉	NOUN
ap-1821	303	32	=	=	X
ap-1821	303	33	〈	〈	PROPN
ap-1821	303	34	n	n	CCONJ
ap-1821	303	35	,	,	PUNCT
ap-1821	303	36	1	1	NUM
ap-1821	303	37	,	,	PUNCT
ap-1821	303	38	√	√	ADJ
ap-1821	303	39	n2	n2	NOUN
ap-1821	303	40	+	+	CCONJ
ap-1821	303	41	2n−	2n−	NUM
ap-1821	303	42	3−	3−	NUM
ap-1821	303	43	(	(	PUNCT
ap-1821	303	44	n−	n−	NOUN
ap-1821	303	45	3	3	NUM
ap-1821	303	46	)	)	PUNCT
ap-1821	303	47	2n−	2n−	PROPN
ap-1821	303	48	3	3	NUM
ap-1821	303	49	〉	〉	NOUN
ap-1821	303	50	=	=	X
ap-1821	303	51	〈	〈	PROPN
ap-1821	303	52	n	n	CCONJ
ap-1821	303	53	,	,	PUNCT
ap-1821	303	54	1	1	NUM
ap-1821	303	55	,	,	PUNCT
ap-1821	303	56	n2	n2	ADJ
ap-1821	303	57	−	−	PROPN
ap-1821	303	58	1	1	NUM
ap-1821	303	59	,	,	PUNCT
ap-1821	303	60	√	√	ADJ
ap-1821	303	61	n2	n2	NOUN
ap-1821	303	62	+	+	CCONJ
ap-1821	303	63	2n−	2n−	NUM
ap-1821	303	64	3	3	NUM
ap-1821	303	65	+	+	CCONJ
ap-1821	303	66	(	(	PUNCT
ap-1821	303	67	n−	n−	NOUN
ap-1821	303	68	1	1	NUM
ap-1821	303	69	)	)	PUNCT
ap-1821	303	70	n−	n−	NOUN
ap-1821	303	71	1	1	NUM
ap-1821	303	72	〉	〉	NOUN
ap-1821	303	73	=	=	PUNCT
ap-1821	303	74	〈	〈	PROPN
ap-1821	303	75	n	n	CCONJ
ap-1821	303	76	,	,	PUNCT
ap-1821	303	77	1	1	NUM
ap-1821	303	78	,	,	PUNCT
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ap-1821	303	80	−	−	PROPN
ap-1821	303	81	1	1	NUM
ap-1821	303	82	,	,	PUNCT
ap-1821	303	83	2	2	NUM
ap-1821	303	84	,	,	PUNCT
ap-1821	303	85	√	√	ADJ
ap-1821	303	86	n2	n2	NOUN
ap-1821	303	87	+	+	CCONJ
ap-1821	303	88	2n−	2n−	NUM
ap-1821	303	89	3	3	NUM
ap-1821	303	90	+	+	CCONJ
ap-1821	303	91	(	(	PUNCT
ap-1821	303	92	n−	n−	NOUN
ap-1821	303	93	1	1	NUM
ap-1821	303	94	)	)	PUNCT
ap-1821	303	95	4	4	NUM
ap-1821	303	96	〉	〉	NOUN
ap-1821	303	97	=	=	X
ap-1821	303	98	〈	〈	PROPN
ap-1821	303	99	n	n	CCONJ
ap-1821	303	100	,	,	PUNCT
ap-1821	303	101	1	1	NUM
ap-1821	303	102	,	,	PUNCT
ap-1821	303	103	n2	n2	ADJ
ap-1821	303	104	−	−	PROPN
ap-1821	303	105	1	1	NUM
ap-1821	303	106	,	,	PUNCT
ap-1821	303	107	2	2	NUM
ap-1821	303	108	,	,	PUNCT
ap-1821	303	109	n2	n2	ADJ
ap-1821	303	110	−	−	PROPN
ap-1821	303	111	1	1	NUM
ap-1821	303	112	,	,	PUNCT
ap-1821	303	113	√	√	ADJ
ap-1821	303	114	n2	n2	NOUN
ap-1821	303	115	+	+	CCONJ
ap-1821	303	116	2n−	2n−	NUM
ap-1821	303	117	3	3	NUM
ap-1821	303	118	+	+	CCONJ
ap-1821	303	119	(	(	PUNCT
ap-1821	303	120	n−	n−	NOUN
ap-1821	303	121	3	3	NUM
ap-1821	303	122	)	)	PUNCT
ap-1821	303	123	2n−	2n−	PROPN
ap-1821	303	124	3	3	NUM
ap-1821	303	125	〉	〉	NOUN
ap-1821	303	126	=	=	X
ap-1821	303	127	〈	〈	PROPN
ap-1821	303	128	n	n	CCONJ
ap-1821	303	129	,	,	PUNCT
ap-1821	303	130	1	1	NUM
ap-1821	303	131	,	,	PUNCT
ap-1821	303	132	n2	n2	ADJ
ap-1821	303	133	−	−	PROPN
ap-1821	303	134	1	1	NUM
ap-1821	303	135	,	,	PUNCT
ap-1821	303	136	2	2	NUM
ap-1821	303	137	,	,	PUNCT
ap-1821	303	138	n2	n2	ADJ
ap-1821	303	139	−	−	PROPN
ap-1821	303	140	1	1	NUM
ap-1821	303	141	,	,	PUNCT
ap-1821	303	142	1	1	NUM
ap-1821	303	143	,	,	PUNCT
ap-1821	303	144	2n+	2n+	NUM
ap-1821	303	145	(	(	PUNCT
ap-1821	303	146	√	√	NOUN
ap-1821	303	147	n2	n2	NOUN
ap-1821	303	148	+	+	CCONJ
ap-1821	303	149	2n−	2n−	PROPN
ap-1821	303	150	3−	3−	NUM
ap-1821	303	151	n	n	CCONJ
ap-1821	303	152	)	)	PUNCT
ap-1821	303	153	〉	〉	NOUN
ap-1821	303	154	.	.	PUNCT
ap-1821	304	1	thus	thus	ADV
ap-1821	304	2	√	√	DET
ap-1821	304	3	n	n	NOUN
ap-1821	304	4	=	=	SYM
ap-1821	305	1	[	[	X
ap-1821	305	2	n	n	CCONJ
ap-1821	305	3	,	,	PUNCT
ap-1821	305	4	1	1	NUM
ap-1821	305	5	,	,	PUNCT
ap-1821	305	6	n2	n2	ADJ
ap-1821	305	7	−	−	PROPN
ap-1821	305	8	1	1	NUM
ap-1821	305	9	,	,	PUNCT
ap-1821	305	10	2	2	NUM
ap-1821	305	11	,	,	PUNCT
ap-1821	305	12	n2	n2	ADJ
ap-1821	305	13	−	−	PROPN
ap-1821	305	14	1	1	NUM
ap-1821	305	15	,	,	PUNCT
ap-1821	305	16	1	1	NUM
ap-1821	305	17	,	,	PUNCT
ap-1821	305	18	2n	2n	NUM
ap-1821	305	19	]	]	PUNCT
ap-1821	305	20	.	.	PUNCT
ap-1821	306	1	the	the	DET
ap-1821	306	2	following	follow	VERB
ap-1821	306	3	table	table	NOUN
ap-1821	306	4	will	will	AUX
ap-1821	306	5	include	include	VERB
ap-1821	306	6	all	all	DET
ap-1821	306	7	remaining	remain	VERB
ap-1821	306	8	cases	case	NOUN
ap-1821	306	9	of	of	ADP
ap-1821	306	10	continued	continue	VERB
ap-1821	306	11	fractions	fraction	NOUN
ap-1821	306	12	of	of	ADP
ap-1821	306	13	√	√	PROPN
ap-1821	306	14	n2	n2	NOUN
ap-1821	306	15	+	+	CCONJ
ap-1821	306	16	j	j	NOUN
ap-1821	306	17	that	that	SCONJ
ap-1821	306	18	we	we	PRON
ap-1821	306	19	were	be	AUX
ap-1821	306	20	able	able	ADJ
ap-1821	306	21	to	to	PART
ap-1821	306	22	determine	determine	VERB
ap-1821	306	23	in	in	ADP
ap-1821	306	24	terms	term	NOUN
ap-1821	306	25	of	of	ADP
ap-1821	306	26	n	n	PRON
ap-1821	306	27	and	and	CCONJ
ap-1821	306	28	j.	j.	PROPN
ap-1821	306	29	observation	observation	PROPN
ap-1821	306	30	3.9	3.9	NUM
ap-1821	306	31	.	.	PUNCT
ap-1821	307	1	let	let	VERB
ap-1821	307	2	k	k	PROPN
ap-1821	307	3	∈	∈	PROPN
ap-1821	307	4	n.	n.	PROPN
ap-1821	307	5	let	let	VERB
ap-1821	307	6	us	we	PRON
ap-1821	307	7	summarize	summarize	VERB
ap-1821	307	8	in	in	ADP
ap-1821	307	9	table	table	NOUN
ap-1821	307	10	2	2	NUM
ap-1821	307	11	the	the	DET
ap-1821	307	12	lengths	length	NOUN
ap-1821	307	13	`	`	PUNCT
ap-1821	307	14	of	of	ADP
ap-1821	307	15	periods	period	NOUN
ap-1821	307	16	and	and	CCONJ
ap-1821	307	17	the	the	DET
ap-1821	307	18	form	form	NOUN
ap-1821	307	19	of	of	ADP
ap-1821	307	20	continued	continue	VERB
ap-1821	307	21	fractions	fraction	NOUN
ap-1821	307	22	for	for	ADP
ap-1821	307	23	several	several	ADJ
ap-1821	307	24	classes	class	NOUN
ap-1821	307	25	(	(	PUNCT
ap-1821	307	26	described	describe	VERB
ap-1821	307	27	in	in	ADP
ap-1821	307	28	an	an	DET
ap-1821	307	29	analogous	analogous	ADJ
ap-1821	307	30	way	way	NOUN
ap-1821	307	31	)	)	PUNCT
ap-1821	307	32	of	of	ADP
ap-1821	307	33	n	n	PROPN
ap-1821	307	34	and	and	CCONJ
ap-1821	307	35	j.	j.	PROPN
ap-1821	307	36	the	the	DET
ap-1821	307	37	next	next	ADJ
ap-1821	307	38	observation	observation	NOUN
ap-1821	307	39	was	be	AUX
ap-1821	307	40	made	make	VERB
ap-1821	307	41	in	in	ADP
ap-1821	307	42	a	a	DET
ap-1821	307	43	different	different	ADJ
ap-1821	307	44	way	way	NOUN
ap-1821	307	45	than	than	ADP
ap-1821	307	46	all	all	DET
ap-1821	307	47	previous	previous	ADJ
ap-1821	307	48	ones	one	NOUN
ap-1821	307	49	.	.	PUNCT
ap-1821	308	1	we	we	PRON
ap-1821	308	2	prescribed	prescribe	VERB
ap-1821	308	3	the	the	DET
ap-1821	308	4	form	form	NOUN
ap-1821	308	5	of	of	ADP
ap-1821	308	6	the	the	DET
ap-1821	308	7	continued	continue	VERB
ap-1821	308	8	fraction	fraction	NOUN
ap-1821	308	9	and	and	CCONJ
ap-1821	308	10	searched	search	VERB
ap-1821	308	11	for	for	ADP
ap-1821	308	12	√	√	NOUN
ap-1821	308	13	n	n	CCONJ
ap-1821	308	14	having	have	VERB
ap-1821	308	15	such	such	DET
ap-1821	308	16	a	a	DET
ap-1821	308	17	continued	continue	VERB
ap-1821	308	18	fraction	fraction	NOUN
ap-1821	308	19	.	.	PUNCT
ap-1821	309	1	observation	observation	NOUN
ap-1821	309	2	3.10	3.10	NUM
ap-1821	309	3	.	.	PUNCT
ap-1821	310	1	if	if	SCONJ
ap-1821	310	2	the	the	DET
ap-1821	310	3	period	period	NOUN
ap-1821	310	4	of	of	ADP
ap-1821	310	5	the	the	DET
ap-1821	310	6	continued	continue	VERB
ap-1821	310	7	fraction	fraction	NOUN
ap-1821	310	8	of	of	ADP
ap-1821	310	9	√	√	NUM
ap-1821	310	10	n	n	NOUN
ap-1821	310	11	=	=	PUNCT
ap-1821	310	12	√	√	PROPN
ap-1821	310	13	n2	n2	NOUN
ap-1821	310	14	+	+	CCONJ
ap-1821	310	15	j	j	PROPN
ap-1821	310	16	contains	contain	VERB
ap-1821	310	17	p	p	PROPN
ap-1821	310	18	≥	≥	NUM
ap-1821	310	19	1	1	NUM
ap-1821	310	20	ones	one	NOUN
ap-1821	310	21	as	as	ADP
ap-1821	310	22	its	its	PRON
ap-1821	310	23	palindromic	palindromic	ADJ
ap-1821	310	24	part	part	NOUN
ap-1821	310	25	,	,	PUNCT
ap-1821	310	26	i.e.	i.e.	X
ap-1821	310	27	,	,	PUNCT
ap-1821	310	28	√	√	ADP
ap-1821	310	29	n	n	NOUN
ap-1821	310	30	=	=	SYM
ap-1821	311	1	[	[	X
ap-1821	311	2	n	n	CCONJ
ap-1821	311	3	,	,	PUNCT
ap-1821	311	4	1	1	NUM
ap-1821	311	5	,	,	PUNCT
ap-1821	311	6	.	.	PUNCT
ap-1821	311	7	.	.	PUNCT
ap-1821	312	1	.	.	PUNCT
ap-1821	313	1	,	,	PUNCT
ap-1821	313	2	1︸	1︸	NUM
ap-1821	313	3	︷︷	︷︷	NOUN
ap-1821	313	4	︸	︸	X
ap-1821	313	5	p	p	X
ap-1821	313	6	,	,	PUNCT
ap-1821	313	7	2n	2n	NUM
ap-1821	313	8	]	]	PUNCT
ap-1821	313	9	then	then	ADV
ap-1821	313	10	n	n	PROPN
ap-1821	313	11	=	=	PUNCT
ap-1821	313	12	kfp	kfp	PROPN
ap-1821	313	13	+	+	CCONJ
ap-1821	313	14	fp+1	fp+1	NOUN
ap-1821	313	15	2	2	NUM
ap-1821	313	16	for	for	ADP
ap-1821	313	17	some	some	DET
ap-1821	313	18	k	k	PROPN
ap-1821	313	19	∈	∈	PROPN
ap-1821	313	20	n	n	CCONJ
ap-1821	313	21	,	,	PUNCT
ap-1821	313	22	p	p	X
ap-1821	313	23	+	+	PROPN
ap-1821	313	24	1	1	NUM
ap-1821	313	25	6=	6=	SYM
ap-1821	313	26	3	3	NUM
ap-1821	313	27	`	`	PUNCT
ap-1821	313	28	,	,	PUNCT
ap-1821	313	29	where	where	SCONJ
ap-1821	313	30	`	`	PUNCT
ap-1821	313	31	∈	∈	PROPN
ap-1821	313	32	n	n	CCONJ
ap-1821	313	33	,	,	PUNCT
ap-1821	313	34	and	and	CCONJ
ap-1821	313	35	j	j	PROPN
ap-1821	313	36	=	=	SYM
ap-1821	314	1	2nfp−1+fp−2	2nfp−1+fp−2	PROPN
ap-1821	314	2	fp	fp	INTJ
ap-1821	314	3	,	,	PUNCT
ap-1821	314	4	where	where	SCONJ
ap-1821	314	5	fn	fn	NOUN
ap-1821	314	6	denotes	denote	VERB
ap-1821	314	7	the	the	DET
ap-1821	314	8	nth	nth	NOUN
ap-1821	314	9	fibonacci	fibonacci	NOUN
ap-1821	314	10	number	number	NOUN
ap-1821	314	11	given	give	VERB
ap-1821	314	12	by	by	ADP
ap-1821	314	13	the	the	DET
ap-1821	314	14	recurrence	recurrence	NOUN
ap-1821	314	15	relation	relation	NOUN
ap-1821	314	16	f−1	f−1	PROPN
ap-1821	314	17	=	=	SYM
ap-1821	314	18	0	0	NUM
ap-1821	314	19	,	,	PUNCT
ap-1821	314	20	f0	f0	PROPN
ap-1821	314	21	=	=	SYM
ap-1821	314	22	1	1	NUM
ap-1821	314	23	and	and	CCONJ
ap-1821	314	24	fn	fn	NOUN
ap-1821	314	25	=	=	NOUN
ap-1821	314	26	fn−2	fn−2	PROPN
ap-1821	315	1	+	+	X
ap-1821	315	2	fn−1	fn−1	ADJ
ap-1821	315	3	for	for	ADP
ap-1821	315	4	all	all	DET
ap-1821	315	5	n	n	PRON
ap-1821	315	6	≥	≥	NOUN
ap-1821	315	7	1	1	NUM
ap-1821	315	8	.	.	PUNCT
ap-1821	316	1	proof	proof	NOUN
ap-1821	316	2	.	.	PUNCT
ap-1821	317	1	it	it	PRON
ap-1821	317	2	is	be	AUX
ap-1821	317	3	a	a	DET
ap-1821	317	4	direct	direct	ADJ
ap-1821	317	5	consequence	consequence	NOUN
ap-1821	317	6	of	of	ADP
ap-1821	317	7	the	the	DET
ap-1821	317	8	proof	proof	NOUN
ap-1821	317	9	of	of	ADP
ap-1821	317	10	theorem	theorem	ADJ
ap-1821	317	11	3.2	3.2	NUM
ap-1821	317	12	and	and	CCONJ
ap-1821	317	13	the	the	DET
ap-1821	317	14	definition	definition	NOUN
ap-1821	317	15	of	of	ADP
ap-1821	317	16	continuants	continuant	NOUN
ap-1821	317	17	.	.	PUNCT
ap-1821	318	1	the	the	DET
ap-1821	318	2	last	last	ADJ
ap-1821	318	3	observation	observation	NOUN
ap-1821	318	4	is	be	AUX
ap-1821	318	5	also	also	ADV
ap-1821	318	6	of	of	ADP
ap-1821	318	7	a	a	DET
ap-1821	318	8	different	different	ADJ
ap-1821	318	9	form	form	NOUN
ap-1821	318	10	than	than	ADP
ap-1821	318	11	the	the	DET
ap-1821	318	12	previous	previous	ADJ
ap-1821	318	13	ones	one	NOUN
ap-1821	318	14	since	since	SCONJ
ap-1821	318	15	j	j	PROPN
ap-1821	318	16	and	and	CCONJ
ap-1821	318	17	n	n	PROPN
ap-1821	318	18	depend	depend	VERB
ap-1821	318	19	on	on	ADP
ap-1821	318	20	two	two	NUM
ap-1821	318	21	parameters	parameter	NOUN
ap-1821	318	22	.	.	PUNCT
ap-1821	319	1	observation	observation	NOUN
ap-1821	319	2	3.11	3.11	NUM
ap-1821	319	3	.	.	PUNCT
ap-1821	320	1	let	let	VERB
ap-1821	320	2	n	n	PROPN
ap-1821	320	3	=	=	PUNCT
ap-1821	320	4	4ka+	4ka+	PROPN
ap-1821	320	5	2a	2a	NUM
ap-1821	320	6	,	,	PUNCT
ap-1821	320	7	where	where	SCONJ
ap-1821	320	8	k	k	X
ap-1821	320	9	,	,	PUNCT
ap-1821	320	10	a	a	DET
ap-1821	320	11	∈	∈	PROPN
ap-1821	320	12	n	n	CCONJ
ap-1821	320	13	,	,	PUNCT
ap-1821	320	14	k	k	PROPN
ap-1821	320	15	≥	≥	NUM
ap-1821	320	16	1	1	NUM
ap-1821	320	17	,	,	PUNCT
ap-1821	320	18	a	a	DET
ap-1821	320	19	≥	≥	NOUN
ap-1821	320	20	1	1	NUM
ap-1821	320	21	,	,	PUNCT
ap-1821	320	22	and	and	CCONJ
ap-1821	320	23	j	j	PROPN
ap-1821	320	24	=	=	SYM
ap-1821	320	25	8a	8a	PROPN
ap-1821	320	26	.	.	PUNCT
ap-1821	321	1	then	then	ADV
ap-1821	321	2	the	the	DET
ap-1821	321	3	continued	continue	VERB
ap-1821	321	4	fraction	fraction	NOUN
ap-1821	321	5	of	of	ADP
ap-1821	321	6	√	√	NUM
ap-1821	321	7	n	n	NOUN
ap-1821	321	8	=	=	PUNCT
ap-1821	321	9	√	√	PROPN
ap-1821	321	10	n2	n2	NOUN
ap-1821	321	11	+	+	CCONJ
ap-1821	321	12	j	j	PROPN
ap-1821	321	13	equals	equal	VERB
ap-1821	321	14	[	[	PUNCT
ap-1821	321	15	n	n	CCONJ
ap-1821	321	16	,	,	PUNCT
ap-1821	321	17	4n−	4n−	NUM
ap-1821	321	18	j	j	NOUN
ap-1821	321	19	2j	2j	NUM
ap-1821	321	20	,	,	PUNCT
ap-1821	321	21	1	1	NUM
ap-1821	321	22	,	,	PUNCT
ap-1821	321	23	1	1	NUM
ap-1821	321	24	,	,	PUNCT
ap-1821	321	25	n−	n−	NOUN
ap-1821	321	26	2	2	NUM
ap-1821	321	27	2	2	NUM
ap-1821	321	28	,	,	PUNCT
ap-1821	321	29	1	1	NUM
ap-1821	321	30	,	,	PUNCT
ap-1821	321	31	1	1	NUM
ap-1821	321	32	,	,	PUNCT
ap-1821	321	33	4n−	4n−	NUM
ap-1821	321	34	j	j	NOUN
ap-1821	321	35	2j	2j	NUM
ap-1821	321	36	,	,	PUNCT
ap-1821	321	37	2n	2n	NUM
ap-1821	321	38	]	]	PUNCT
ap-1821	321	39	.	.	PUNCT
ap-1821	322	1	proof	proof	NOUN
ap-1821	322	2	.	.	PUNCT
ap-1821	323	1	the	the	DET
ap-1821	323	2	proof	proof	NOUN
ap-1821	323	3	may	may	AUX
ap-1821	323	4	be	be	AUX
ap-1821	323	5	found	find	VERB
ap-1821	323	6	in	in	ADP
ap-1821	323	7	[	[	X
ap-1821	323	8	5	5	NUM
ap-1821	323	9	]	]	PUNCT
ap-1821	323	10	.	.	PUNCT
ap-1821	324	1	we	we	PRON
ap-1821	324	2	also	also	ADV
ap-1821	324	3	made	make	VERB
ap-1821	324	4	one	one	NUM
ap-1821	324	5	conjecture	conjecture	NOUN
ap-1821	324	6	that	that	PRON
ap-1821	324	7	turned	turn	VERB
ap-1821	324	8	out	out	ADP
ap-1821	324	9	to	to	PART
ap-1821	324	10	be	be	AUX
ap-1821	324	11	false	false	ADJ
ap-1821	324	12	.	.	PUNCT
ap-1821	325	1	conjecture	conjecture	NOUN
ap-1821	325	2	3.12	3.12	NUM
ap-1821	325	3	.	.	PUNCT
ap-1821	326	1	for	for	ADP
ap-1821	326	2	√	√	PROPN
ap-1821	326	3	n	n	CCONJ
ap-1821	326	4	the	the	DET
ap-1821	326	5	length	length	NOUN
ap-1821	326	6	of	of	ADP
ap-1821	326	7	the	the	DET
ap-1821	326	8	period	period	NOUN
ap-1821	326	9	of	of	ADP
ap-1821	326	10	the	the	DET
ap-1821	326	11	continued	continue	VERB
ap-1821	326	12	fraction	fraction	NOUN
ap-1821	326	13	is	be	AUX
ap-1821	326	14	less	less	ADJ
ap-1821	326	15	than	than	ADP
ap-1821	326	16	or	or	CCONJ
ap-1821	326	17	equal	equal	ADJ
ap-1821	326	18	to	to	ADP
ap-1821	326	19	2n	2n	NUM
ap-1821	326	20	.	.	PUNCT
ap-1821	327	1	this	this	DET
ap-1821	327	2	observation	observation	NOUN
ap-1821	327	3	was	be	AUX
ap-1821	327	4	made	make	VERB
ap-1821	327	5	when	when	SCONJ
ap-1821	327	6	contemplating	contemplate	VERB
ap-1821	327	7	a	a	DET
ap-1821	327	8	table	table	NOUN
ap-1821	327	9	of	of	ADP
ap-1821	327	10	periods	period	NOUN
ap-1821	327	11	of	of	ADP
ap-1821	327	12	√	√	NUM
ap-1821	327	13	n	n	NOUN
ap-1821	327	14	for	for	ADP
ap-1821	327	15	n	n	PRON
ap-1821	327	16	≤	≤	NOUN
ap-1821	327	17	1000	1000	NUM
ap-1821	327	18	.	.	PUNCT
ap-1821	328	1	however	however	ADV
ap-1821	328	2	,	,	PUNCT
ap-1821	328	3	in	in	ADP
ap-1821	328	4	[	[	X
ap-1821	328	5	8	8	NUM
ap-1821	328	6	]	]	X
ap-1821	328	7	it	it	PRON
ap-1821	328	8	is	be	AUX
ap-1821	328	9	shown	show	VERB
ap-1821	328	10	that	that	SCONJ
ap-1821	328	11	for	for	ADP
ap-1821	328	12	n	n	NOUN
ap-1821	328	13	=	=	SYM
ap-1821	328	14	1726	1726	NUM
ap-1821	328	15	with	with	ADP
ap-1821	328	16	n	n	NOUN
ap-1821	328	17	=	=	SYM
ap-1821	328	18	41	41	NUM
ap-1821	328	19	,	,	PUNCT
ap-1821	328	20	the	the	DET
ap-1821	328	21	period	period	NOUN
ap-1821	328	22	of	of	ADP
ap-1821	328	23	the	the	DET
ap-1821	328	24	continued	continue	VERB
ap-1821	328	25	fraction	fraction	NOUN
ap-1821	328	26	of	of	ADP
ap-1821	328	27	√	√	PROPN
ap-1821	328	28	n	n	NUM
ap-1821	328	29	is	be	AUX
ap-1821	328	30	of	of	ADP
ap-1821	328	31	length	length	NOUN
ap-1821	328	32	88	88	NUM
ap-1821	328	33	>	>	SYM
ap-1821	328	34	82	82	NUM
ap-1821	328	35	=	=	SYM
ap-1821	328	36	2n	2n	NUM
ap-1821	328	37	.	.	PUNCT
ap-1821	329	1	a	a	DET
ap-1821	329	2	rougher	rougher	ADJ
ap-1821	329	3	upper	upper	ADJ
ap-1821	329	4	bound	bind	VERB
ap-1821	329	5	comes	come	VERB
ap-1821	329	6	from	from	ADP
ap-1821	329	7	[	[	X
ap-1821	329	8	7	7	NUM
ap-1821	329	9	]	]	PUNCT
ap-1821	329	10	.	.	PUNCT
ap-1821	330	1	theorem	theorem	VERB
ap-1821	330	2	3.13	3.13	NUM
ap-1821	330	3	.	.	PUNCT
ap-1821	331	1	for	for	ADP
ap-1821	331	2	√	√	PROPN
ap-1821	331	3	n	n	CCONJ
ap-1821	331	4	the	the	DET
ap-1821	331	5	length	length	NOUN
ap-1821	331	6	of	of	ADP
ap-1821	331	7	the	the	DET
ap-1821	331	8	period	period	NOUN
ap-1821	331	9	of	of	ADP
ap-1821	331	10	the	the	DET
ap-1821	331	11	continued	continue	VERB
ap-1821	331	12	fraction	fraction	NOUN
ap-1821	331	13	is	be	AUX
ap-1821	331	14	less	less	ADJ
ap-1821	331	15	than	than	ADP
ap-1821	331	16	or	or	CCONJ
ap-1821	331	17	equal	equal	ADJ
ap-1821	331	18	to	to	ADP
ap-1821	331	19	2n	2n	NUM
ap-1821	331	20	.	.	PUNCT
ap-1821	332	1	let	let	VERB
ap-1821	332	2	us	we	PRON
ap-1821	332	3	terminate	terminate	VERB
ap-1821	332	4	with	with	ADP
ap-1821	332	5	two	two	NUM
ap-1821	332	6	conjectures	conjecture	NOUN
ap-1821	332	7	that	that	PRON
ap-1821	332	8	have	have	AUX
ap-1821	332	9	not	not	PART
ap-1821	332	10	been	be	AUX
ap-1821	332	11	proved	prove	VERB
ap-1821	332	12	yet	yet	ADV
ap-1821	332	13	.	.	PUNCT
ap-1821	333	1	conjecture	conjecture	VERB
ap-1821	333	2	3.14	3.14	NUM
ap-1821	333	3	.	.	PUNCT
ap-1821	334	1	no	no	DET
ap-1821	334	2	element	element	NOUN
ap-1821	334	3	of	of	ADP
ap-1821	334	4	the	the	DET
ap-1821	334	5	period	period	NOUN
ap-1821	334	6	of	of	ADP
ap-1821	334	7	√	√	NUM
ap-1821	334	8	n	n	CCONJ
ap-1821	334	9	apart	apart	ADV
ap-1821	334	10	from	from	ADP
ap-1821	334	11	the	the	DET
ap-1821	334	12	last	last	ADJ
ap-1821	334	13	one	one	NOUN
ap-1821	334	14	is	be	AUX
ap-1821	334	15	bigger	big	ADJ
ap-1821	334	16	than	than	ADP
ap-1821	334	17	n.	n.	NOUN
ap-1821	334	18	conjecture	conjecture	NOUN
ap-1821	334	19	3.15	3.15	NUM
ap-1821	334	20	.	.	PUNCT
ap-1821	335	1	there	there	PRON
ap-1821	335	2	is	be	VERB
ap-1821	335	3	no	no	DET
ap-1821	335	4	period	period	NOUN
ap-1821	335	5	of	of	ADP
ap-1821	335	6	an	an	DET
ap-1821	335	7	odd	odd	ADJ
ap-1821	335	8	length	length	NOUN
ap-1821	335	9	for	for	ADP
ap-1821	335	10	j	j	NOUN
ap-1821	335	11	=	=	SYM
ap-1821	335	12	4k	4k	NUM
ap-1821	335	13	+	+	NOUN
ap-1821	336	1	3	3	NUM
ap-1821	336	2	,	,	PUNCT
ap-1821	336	3	where	where	SCONJ
ap-1821	336	4	k	k	PROPN
ap-1821	336	5	∈	∈	PROPN
ap-1821	336	6	n.	n.	NOUN
ap-1821	336	7	acknowledgements	acknowledgement	VERB
ap-1821	336	8	the	the	DET
ap-1821	336	9	first	first	ADJ
ap-1821	336	10	author	author	NOUN
ap-1821	336	11	acknowledges	acknowledge	VERB
ap-1821	336	12	financial	financial	ADJ
ap-1821	336	13	support	support	NOUN
ap-1821	336	14	from	from	ADP
ap-1821	336	15	czech	czech	PROPN
ap-1821	336	16	science	science	NOUN
ap-1821	336	17	foundation	foundation	NOUN
ap-1821	336	18	grant	grant	VERB
ap-1821	336	19	13	13	NUM
ap-1821	336	20	-	-	PUNCT
ap-1821	336	21	03538s	03538s	NUM
ap-1821	336	22	.	.	PUNCT
ap-1821	337	1	references	reference	NOUN
ap-1821	337	2	[	[	X
ap-1821	337	3	1	1	NUM
ap-1821	337	4	]	]	X
ap-1821	337	5	klazar	klazar	NOUN
ap-1821	337	6	,	,	PUNCT
ap-1821	337	7	m.	m.	NOUN
ap-1821	337	8	:	:	PUNCT
ap-1821	337	9	introduction	introduction	NOUN
ap-1821	337	10	to	to	ADP
ap-1821	337	11	number	number	NOUN
ap-1821	337	12	theory	theory	NOUN
ap-1821	337	13	,	,	PUNCT
ap-1821	337	14	kam	kam	ADJ
ap-1821	337	15	-	-	PUNCT
ap-1821	337	16	dimatia	dimatia	PROPN
ap-1821	337	17	series	series	PROPN
ap-1821	337	18	782	782	NUM
ap-1821	337	19	,	,	PUNCT
ap-1821	337	20	2006	2006	NUM
ap-1821	337	21	.	.	PUNCT
ap-1821	338	1	[	[	X
ap-1821	338	2	2	2	NUM
ap-1821	338	3	]	]	PUNCT
ap-1821	338	4	masáková	masáková	NOUN
ap-1821	338	5	,	,	PUNCT
ap-1821	338	6	z.	z.	PROPN
ap-1821	338	7	,	,	PUNCT
ap-1821	338	8	pelantová	pelantová	PROPN
ap-1821	338	9	,	,	PUNCT
ap-1821	338	10	e.	e.	PROPN
ap-1821	338	11	:	:	PUNCT
ap-1821	338	12	teorie	teorie	PROPN
ap-1821	338	13	čísel	čísel	NOUN
ap-1821	338	14	,	,	PUNCT
ap-1821	338	15	skriptum	skriptum	NOUN
ap-1821	338	16	čvut	čvut	PROPN
ap-1821	338	17	,	,	PUNCT
ap-1821	338	18	2010	2010	NUM
ap-1821	338	19	.	.	PUNCT
ap-1821	339	1	[	[	X
ap-1821	339	2	3	3	NUM
ap-1821	339	3	]	]	X
ap-1821	339	4	muller	muller	NOUN
ap-1821	339	5	,	,	PUNCT
ap-1821	339	6	j.-m	j.-m	PROPN
ap-1821	339	7	.	.	PUNCT
ap-1821	339	8	:	:	PUNCT
ap-1821	340	1	elementary	elementary	ADJ
ap-1821	340	2	functions	function	NOUN
ap-1821	340	3	:	:	PUNCT
ap-1821	340	4	algorithms	algorithm	NOUN
ap-1821	340	5	and	and	CCONJ
ap-1821	340	6	implementation	implementation	NOUN
ap-1821	340	7	,	,	PUNCT
ap-1821	340	8	2nd	2nd	PROPN
ap-1821	340	9	edition	edition	NOUN
ap-1821	340	10	,	,	PUNCT
ap-1821	340	11	birkhäuser	birkhäuser	PROPN
ap-1821	340	12	boston	boston	PROPN
ap-1821	340	13	,	,	PUNCT
ap-1821	340	14	2006	2006	NUM
ap-1821	340	15	.	.	PUNCT
ap-1821	341	1	[	[	X
ap-1821	341	2	4	4	NUM
ap-1821	341	3	]	]	X
ap-1821	341	4	lauritzen	lauritzen	PROPN
ap-1821	341	5	,	,	PUNCT
ap-1821	341	6	n.	n.	NOUN
ap-1821	341	7	:	:	PUNCT
ap-1821	341	8	continued	continue	VERB
ap-1821	341	9	fractions	fraction	NOUN
ap-1821	341	10	and	and	CCONJ
ap-1821	341	11	factoring	factoring	NOUN
ap-1821	341	12	,	,	PUNCT
ap-1821	341	13	http://home.imf.au.dk/niels/cfracfact.pdf	http://home.imf.au.dk/niels/cfracfact.pdf	NOUN
ap-1821	341	14	,	,	PUNCT
ap-1821	341	15	2009	2009	NUM
ap-1821	341	16	.	.	PUNCT
ap-1821	342	1	327	327	NUM
ap-1821	342	2	ľ	ľ	X
ap-1821	342	3	.	.	PUNCT
ap-1821	343	1	balková	balková	PROPN
ap-1821	343	2	,	,	PUNCT
ap-1821	343	3	a.	a.	PROPN
ap-1821	343	4	hrušková	hrušková	PROPN
ap-1821	343	5	acta	acta	PROPN
ap-1821	343	6	polytechnica	polytechnica	PROPN
ap-1821	344	1	[	[	X
ap-1821	344	2	5	5	NUM
ap-1821	344	3	]	]	X
ap-1821	344	4	hrušková	hrušková	X
ap-1821	344	5	,	,	PUNCT
ap-1821	344	6	a.	a.	NOUN
ap-1821	344	7	:	:	PUNCT
ap-1821	344	8	řetězové	řetězové	PROPN
ap-1821	344	9	zlomky	zlomky	PROPN
ap-1821	344	10	kvadratických	kvadratických	PROPN
ap-1821	344	11	čísel	čísel	NOUN
ap-1821	344	12	,	,	PUNCT
ap-1821	344	13	soč	soč	NOUN
ap-1821	344	14	práce	práce	NOUN
ap-1821	344	15	,	,	PUNCT
ap-1821	344	16	http://bimbo.fjfi.cvut.cz/∼soc/	http://bimbo.fjfi.cvut.cz/∼soc/	NUM
ap-1821	344	17	,	,	PUNCT
ap-1821	344	18	2013–2014	2013–2014	NUM
ap-1821	345	1	[	[	X
ap-1821	345	2	6	6	NUM
ap-1821	345	3	]	]	PUNCT
ap-1821	345	4	seidensticker	seidensticker	NOUN
ap-1821	345	5	,	,	PUNCT
ap-1821	345	6	r.	r.	PROPN
ap-1821	345	7	:	:	PUNCT
ap-1821	345	8	continued	continue	VERB
ap-1821	345	9	fractions	fraction	NOUN
ap-1821	345	10	for	for	ADP
ap-1821	345	11	high	high	ADJ
ap-1821	345	12	-	-	PUNCT
ap-1821	345	13	speed	speed	NOUN
ap-1821	345	14	and	and	CCONJ
ap-1821	345	15	high	high	ADJ
ap-1821	345	16	-	-	PUNCT
ap-1821	345	17	accuracy	accuracy	NOUN
ap-1821	345	18	computer	computer	NOUN
ap-1821	345	19	arithmetic	arithmetic	NOUN
ap-1821	345	20	,	,	PUNCT
ap-1821	345	21	ieee	ieee	NOUN
ap-1821	345	22	symposium	symposium	NOUN
ap-1821	345	23	on	on	ADP
ap-1821	345	24	computer	computer	NOUN
ap-1821	345	25	arithmetic	arithmetic	ADJ
ap-1821	345	26	(	(	PUNCT
ap-1821	345	27	1983	1983	NUM
ap-1821	345	28	)	)	PUNCT
ap-1821	345	29	,	,	PUNCT
ap-1821	345	30	184–193	184–193	NUM
ap-1821	346	1	[	[	X
ap-1821	346	2	7	7	NUM
ap-1821	346	3	]	]	X
ap-1821	346	4	sierpinski	sierpinski	ADJ
ap-1821	346	5	,	,	PUNCT
ap-1821	346	6	w.	w.	PROPN
ap-1821	346	7	:	:	PUNCT
ap-1821	346	8	elementary	elementary	ADJ
ap-1821	346	9	theory	theory	NOUN
ap-1821	346	10	of	of	ADP
ap-1821	346	11	numbers	number	NOUN
ap-1821	346	12	,	,	PUNCT
ap-1821	346	13	panstwowe	panstwowe	PROPN
ap-1821	346	14	wydawnictwo	wydawnictwo	PROPN
ap-1821	346	15	naukowe	naukowe	PROPN
ap-1821	346	16	,	,	PUNCT
ap-1821	346	17	warszawa	warszawa	PROPN
ap-1821	346	18	,	,	PUNCT
ap-1821	346	19	1964	1964	NUM
ap-1821	346	20	.	.	PUNCT
ap-1821	347	1	[	[	X
ap-1821	347	2	8	8	NUM
ap-1821	347	3	]	]	X
ap-1821	347	4	hickerson	hickerson	NOUN
ap-1821	347	5	,	,	PUNCT
ap-1821	347	6	d.	d.	PROPN
ap-1821	347	7	r.	r.	PROPN
ap-1821	347	8	:	:	PUNCT
ap-1821	347	9	length	length	NOUN
ap-1821	347	10	of	of	ADP
ap-1821	347	11	period	period	NOUN
ap-1821	347	12	of	of	ADP
ap-1821	347	13	simple	simple	ADJ
ap-1821	347	14	continued	continue	VERB
ap-1821	347	15	fraction	fraction	NOUN
ap-1821	347	16	expansion	expansion	NOUN
ap-1821	347	17	of	of	ADP
ap-1821	347	18	√	√	PROPN
ap-1821	347	19	d	d	PROPN
ap-1821	347	20	,	,	PUNCT
ap-1821	347	21	pacific	pacific	PROPN
ap-1821	347	22	journal	journal	NOUN
ap-1821	347	23	of	of	ADP
ap-1821	347	24	mathematics	mathematics	PROPN
ap-1821	347	25	46(2	46(2	PROPN
ap-1821	347	26	)	)	PUNCT
ap-1821	347	27	(	(	PUNCT
ap-1821	347	28	1973	1973	NUM
ap-1821	347	29	)	)	PUNCT
ap-1821	347	30	,	,	PUNCT
ap-1821	347	31	429–432	429–432	NUM
ap-1821	347	32	328	328	NUM
ap-1821	347	33	acta	acta	PROPN
ap-1821	347	34	polytechnica	polytechnica	PROPN
ap-1821	347	35	53(4):322–328	53(4):322–328	PROPN
ap-1821	347	36	,	,	PUNCT
ap-1821	347	37	2013	2013	NUM
ap-1821	347	38	1	1	NUM
ap-1821	347	39	introduction	introduction	NOUN
ap-1821	347	40	2	2	NUM
ap-1821	347	41	continued	continued	ADJ
ap-1821	347	42	fractions	fraction	NOUN
ap-1821	347	43	2.1	2.1	NUM
ap-1821	347	44	continued	continue	VERB
ap-1821	347	45	fractions	fraction	NOUN
ap-1821	347	46	and	and	CCONJ
ap-1821	347	47	continuants	continuant	NOUN
ap-1821	347	48	2.2	2.2	NUM
ap-1821	347	49	continued	continue	VERB
ap-1821	347	50	fractions	fraction	NOUN
ap-1821	347	51	of	of	ADP
ap-1821	347	52	quadratic	quadratic	ADJ
ap-1821	347	53	numbers	number	NOUN
ap-1821	347	54	3	3	NUM
ap-1821	347	55	continued	continue	VERB
ap-1821	347	56	fractions	fraction	NOUN
ap-1821	347	57	of	of	ADP
ap-1821	347	58	sq	sq	PROPN
ap-1821	347	59	.	.	PROPN
ap-1821	347	60	root	root	NOUN
ap-1821	347	61	of	of	ADP
ap-1821	347	62	n	n	PRON
ap-1821	347	63	acknowledgements	acknowledgement	NOUN
ap-1821	347	64	references	reference	NOUN