id	sid	tid	token	lemma	pos
ejpam-4979	1	1	european	european	PROPN
ejpam-4979	1	2	journal	journal	PROPN
ejpam-4979	1	3	of	of	ADP
ejpam-4979	1	4	pure	pure	ADJ
ejpam-4979	1	5	and	and	CCONJ
ejpam-4979	1	6	applied	apply	VERB
ejpam-4979	1	7	mathematics	mathematic	NOUN
ejpam-4979	1	8	vol	vol	NOUN
ejpam-4979	1	9	.	.	PROPN
ejpam-4979	2	1	17	17	NUM
ejpam-4979	2	2	,	,	PUNCT
ejpam-4979	2	3	no	no	INTJ
ejpam-4979	2	4	.	.	NOUN
ejpam-4979	2	5	1	1	NUM
ejpam-4979	2	6	,	,	PUNCT
ejpam-4979	2	7	2024	2024	NUM
ejpam-4979	2	8	,	,	PUNCT
ejpam-4979	2	9	42	42	NUM
ejpam-4979	2	10	-	-	SYM
ejpam-4979	2	11	58	58	NUM
ejpam-4979	2	12	issn	issn	PROPN
ejpam-4979	2	13	1307	1307	NUM
ejpam-4979	2	14	-	-	SYM
ejpam-4979	2	15	5543	5543	NUM
ejpam-4979	2	16	–	–	PUNCT
ejpam-4979	2	17	ejpam.com	ejpam.com	X
ejpam-4979	2	18	published	publish	VERB
ejpam-4979	2	19	by	by	ADP
ejpam-4979	2	20	new	new	PROPN
ejpam-4979	2	21	york	york	PROPN
ejpam-4979	2	22	business	business	PROPN
ejpam-4979	2	23	global	global	ADJ
ejpam-4979	2	24	application	application	NOUN
ejpam-4979	2	25	of	of	ADP
ejpam-4979	2	26	the	the	DET
ejpam-4979	2	27	inclusion	inclusion	NOUN
ejpam-4979	2	28	-	-	PUNCT
ejpam-4979	2	29	exclusion	exclusion	NOUN
ejpam-4979	2	30	principle	principle	NOUN
ejpam-4979	2	31	to	to	ADP
ejpam-4979	2	32	prime	prime	ADJ
ejpam-4979	2	33	number	number	NOUN
ejpam-4979	2	34	subsequences	subsequence	VERB
ejpam-4979	2	35	michael	michael	PROPN
ejpam-4979	2	36	p.	p.	PROPN
ejpam-4979	3	1	may	may	AUX
ejpam-4979	3	2	department	department	PROPN
ejpam-4979	3	3	of	of	ADP
ejpam-4979	3	4	mechanical	mechanical	PROPN
ejpam-4979	3	5	&	&	CCONJ
ejpam-4979	3	6	aerospace	aerospace	PROPN
ejpam-4979	3	7	engineering	engineering	PROPN
ejpam-4979	3	8	,	,	PUNCT
ejpam-4979	3	9	missouri	missouri	PROPN
ejpam-4979	3	10	university	university	PROPN
ejpam-4979	3	11	of	of	ADP
ejpam-4979	3	12	science	science	NOUN
ejpam-4979	3	13	and	and	CCONJ
ejpam-4979	3	14	technology	technology	NOUN
ejpam-4979	3	15	,	,	PUNCT
ejpam-4979	3	16	rolla	rolla	PROPN
ejpam-4979	3	17	,	,	PUNCT
ejpam-4979	3	18	missouri	missouri	PROPN
ejpam-4979	3	19	,	,	PUNCT
ejpam-4979	3	20	usa	usa	PROPN
ejpam-4979	3	21	abstract	abstract	NOUN
ejpam-4979	3	22	.	.	PUNCT
ejpam-4979	4	1	we	we	PRON
ejpam-4979	4	2	apply	apply	VERB
ejpam-4979	4	3	the	the	DET
ejpam-4979	4	4	inclusion	inclusion	NOUN
ejpam-4979	4	5	-	-	PUNCT
ejpam-4979	4	6	exclusion	exclusion	NOUN
ejpam-4979	4	7	principle	principle	NOUN
ejpam-4979	4	8	to	to	ADP
ejpam-4979	4	9	a	a	DET
ejpam-4979	4	10	unique	unique	ADJ
ejpam-4979	4	11	pair	pair	NOUN
ejpam-4979	4	12	of	of	ADP
ejpam-4979	4	13	prime	prime	ADJ
ejpam-4979	4	14	number	number	NOUN
ejpam-4979	4	15	subsequences	subsequence	VERB
ejpam-4979	4	16	to	to	PART
ejpam-4979	4	17	determine	determine	VERB
ejpam-4979	4	18	whether	whether	SCONJ
ejpam-4979	4	19	these	these	DET
ejpam-4979	4	20	subsequences	subsequence	NOUN
ejpam-4979	4	21	form	form	VERB
ejpam-4979	4	22	a	a	DET
ejpam-4979	4	23	small	small	ADJ
ejpam-4979	4	24	set	set	NOUN
ejpam-4979	4	25	or	or	CCONJ
ejpam-4979	4	26	a	a	DET
ejpam-4979	4	27	large	large	ADJ
ejpam-4979	4	28	set	set	NOUN
ejpam-4979	4	29	and	and	CCONJ
ejpam-4979	4	30	thus	thus	ADV
ejpam-4979	4	31	whether	whether	SCONJ
ejpam-4979	4	32	the	the	DET
ejpam-4979	4	33	infinite	infinite	ADJ
ejpam-4979	4	34	sum	sum	NOUN
ejpam-4979	4	35	of	of	ADP
ejpam-4979	4	36	the	the	DET
ejpam-4979	4	37	inverse	inverse	NOUN
ejpam-4979	4	38	of	of	ADP
ejpam-4979	4	39	their	their	PRON
ejpam-4979	4	40	terms	term	NOUN
ejpam-4979	4	41	converges	converge	VERB
ejpam-4979	4	42	or	or	CCONJ
ejpam-4979	4	43	diverges	diverge	NOUN
ejpam-4979	4	44	.	.	PUNCT
ejpam-4979	5	1	in	in	ADP
ejpam-4979	5	2	this	this	DET
ejpam-4979	5	3	paper	paper	NOUN
ejpam-4979	5	4	,	,	PUNCT
ejpam-4979	5	5	we	we	PRON
ejpam-4979	5	6	analyze	analyze	VERB
ejpam-4979	5	7	the	the	DET
ejpam-4979	5	8	complementary	complementary	ADJ
ejpam-4979	5	9	prime	prime	ADJ
ejpam-4979	5	10	number	number	NOUN
ejpam-4979	5	11	subsequences	subsequence	VERB
ejpam-4979	5	12	p′	p′	NOUN
ejpam-4979	5	13	and	and	CCONJ
ejpam-4979	5	14	p′′	p′′	PROPN
ejpam-4979	5	15	as	as	ADV
ejpam-4979	5	16	well	well	ADV
ejpam-4979	5	17	as	as	ADP
ejpam-4979	5	18	revisit	revisit	VERB
ejpam-4979	5	19	the	the	DET
ejpam-4979	5	20	twin	twin	ADJ
ejpam-4979	5	21	prime	prime	ADJ
ejpam-4979	5	22	subsequence	subsequence	NOUN
ejpam-4979	5	23	p2	p2	NOUN
ejpam-4979	5	24	.	.	PUNCT
ejpam-4979	6	1	2020	2020	NUM
ejpam-4979	6	2	mathematics	mathematics	PROPN
ejpam-4979	6	3	subject	subject	NOUN
ejpam-4979	6	4	classifications	classification	NOUN
ejpam-4979	6	5	:	:	PUNCT
ejpam-4979	6	6	11a41	11a41	NUM
ejpam-4979	6	7	,	,	PUNCT
ejpam-4979	6	8	11l20	11l20	NUM
ejpam-4979	6	9	,	,	PUNCT
ejpam-4979	6	10	11b05	11b05	NUM
ejpam-4979	6	11	,	,	PUNCT
ejpam-4979	6	12	11k31	11k31	NUM
ejpam-4979	6	13	,	,	PUNCT
ejpam-4979	6	14	11n36	11n36	NUM
ejpam-4979	6	15	key	key	ADJ
ejpam-4979	6	16	words	word	NOUN
ejpam-4979	6	17	and	and	CCONJ
ejpam-4979	6	18	phrases	phrase	NOUN
ejpam-4979	6	19	:	:	PUNCT
ejpam-4979	6	20	prime	prime	ADJ
ejpam-4979	6	21	numbers	number	NOUN
ejpam-4979	6	22	,	,	PUNCT
ejpam-4979	6	23	higher	high	ADJ
ejpam-4979	6	24	-	-	PUNCT
ejpam-4979	6	25	order	order	NOUN
ejpam-4979	6	26	prime	prime	ADJ
ejpam-4979	6	27	number	number	NOUN
ejpam-4979	6	28	sequences	sequence	NOUN
ejpam-4979	6	29	,	,	PUNCT
ejpam-4979	6	30	inclusionexclusion	inclusionexclusion	NOUN
ejpam-4979	6	31	principle	principle	NOUN
ejpam-4979	6	32	,	,	PUNCT
ejpam-4979	6	33	sums	sum	NOUN
ejpam-4979	6	34	over	over	ADP
ejpam-4979	6	35	prime	prime	ADJ
ejpam-4979	6	36	reciprocals	reciprocal	NOUN
ejpam-4979	6	37	,	,	PUNCT
ejpam-4979	6	38	prime	prime	ADJ
ejpam-4979	6	39	density	density	NOUN
ejpam-4979	6	40	,	,	PUNCT
ejpam-4979	6	41	sieves	sieve	NOUN
ejpam-4979	6	42	1	1	NUM
ejpam-4979	6	43	.	.	PUNCT
ejpam-4979	7	1	the	the	DET
ejpam-4979	7	2	prime	prime	NOUN
ejpam-4979	7	3	subsequences	subsequence	VERB
ejpam-4979	7	4	p′	p′	NOUN
ejpam-4979	7	5	and	and	CCONJ
ejpam-4979	7	6	p′′	p′′	PROPN
ejpam-4979	7	7	the	the	DET
ejpam-4979	7	8	prime	prime	ADJ
ejpam-4979	7	9	number	number	NOUN
ejpam-4979	7	10	subsequence	subsequence	NOUN
ejpam-4979	8	1	[	[	X
ejpam-4979	8	2	4	4	NUM
ejpam-4979	8	3	]	]	PUNCT
ejpam-4979	8	4	p	p	X
ejpam-4979	8	5	′	′	NUM
ejpam-4979	8	6	=	=	SYM
ejpam-4979	8	7	{	{	PUNCT
ejpam-4979	8	8	p′	p′	NOUN
ejpam-4979	8	9	}	}	PUNCT
ejpam-4979	8	10	=	=	PUNCT
ejpam-4979	8	11	{	{	PUNCT
ejpam-4979	8	12	2	2	NUM
ejpam-4979	8	13	,	,	PUNCT
ejpam-4979	8	14	5	5	NUM
ejpam-4979	8	15	,	,	PUNCT
ejpam-4979	8	16	7	7	NUM
ejpam-4979	8	17	,	,	PUNCT
ejpam-4979	8	18	13	13	NUM
ejpam-4979	8	19	,	,	PUNCT
ejpam-4979	8	20	19	19	NUM
ejpam-4979	8	21	,	,	PUNCT
ejpam-4979	8	22	23	23	NUM
ejpam-4979	8	23	,	,	PUNCT
ejpam-4979	8	24	29	29	NUM
ejpam-4979	8	25	,	,	PUNCT
ejpam-4979	8	26	31	31	NUM
ejpam-4979	8	27	,	,	PUNCT
ejpam-4979	8	28	37	37	NUM
ejpam-4979	8	29	,	,	PUNCT
ejpam-4979	8	30	43	43	NUM
ejpam-4979	8	31	,	,	PUNCT
ejpam-4979	8	32	47	47	NUM
ejpam-4979	8	33	,	,	PUNCT
ejpam-4979	8	34	53	53	NUM
ejpam-4979	8	35	,	,	PUNCT
ejpam-4979	8	36	59	59	NUM
ejpam-4979	8	37	,	,	PUNCT
ejpam-4979	8	38	61	61	NUM
ejpam-4979	8	39	,	,	PUNCT
ejpam-4979	8	40	71	71	NUM
ejpam-4979	8	41	,	,	PUNCT
ejpam-4979	8	42	...	...	PUNCT
ejpam-4979	8	43	}	}	PUNCT
ejpam-4979	8	44	can	can	AUX
ejpam-4979	8	45	be	be	AUX
ejpam-4979	8	46	generated	generate	VERB
ejpam-4979	8	47	via	via	ADP
ejpam-4979	8	48	an	an	DET
ejpam-4979	8	49	alternating	alternate	VERB
ejpam-4979	8	50	sum	sum	NOUN
ejpam-4979	8	51	of	of	ADP
ejpam-4979	8	52	the	the	DET
ejpam-4979	8	53	prime	prime	ADJ
ejpam-4979	8	54	number	number	NOUN
ejpam-4979	8	55	subsequences	subsequence	NOUN
ejpam-4979	8	56	of	of	ADP
ejpam-4979	8	57	increasing	increase	VERB
ejpam-4979	8	58	order	order	NOUN
ejpam-4979	9	1	[	[	X
ejpam-4979	9	2	7	7	NUM
ejpam-4979	9	3	]	]	PUNCT
ejpam-4979	9	4	,	,	PUNCT
ejpam-4979	9	5	i.e.	i.e.	X
ejpam-4979	9	6	,	,	PUNCT
ejpam-4979	9	7	p	p	NOUN
ejpam-4979	9	8	′	′	NOUN
ejpam-4979	9	9	=	=	SYM
ejpam-4979	9	10	{	{	PUNCT
ejpam-4979	9	11	(	(	PUNCT
ejpam-4979	9	12	−1)n−1	−1)n−1	PRON
ejpam-4979	9	13	{	{	PUNCT
ejpam-4979	9	14	p(n	p(n	PROPN
ejpam-4979	9	15	)	)	PUNCT
ejpam-4979	9	16	}	}	PUNCT
ejpam-4979	9	17	}	}	PUNCT
ejpam-4979	9	18	∞	∞	NUM
ejpam-4979	9	19	n=1	n=1	PROPN
ejpam-4979	9	20	(	(	PUNCT
ejpam-4979	9	21	1	1	X
ejpam-4979	9	22	)	)	PUNCT
ejpam-4979	9	23	where	where	SCONJ
ejpam-4979	9	24	the	the	DET
ejpam-4979	9	25	right	right	ADJ
ejpam-4979	9	26	-	-	PUNCT
ejpam-4979	9	27	hand	hand	NOUN
ejpam-4979	9	28	side	side	NOUN
ejpam-4979	9	29	of	of	ADP
ejpam-4979	9	30	eq	eq	PROPN
ejpam-4979	9	31	.	.	PROPN
ejpam-4979	9	32	1	1	NUM
ejpam-4979	9	33	is	be	AUX
ejpam-4979	9	34	an	an	DET
ejpam-4979	9	35	expression	expression	NOUN
ejpam-4979	9	36	of	of	ADP
ejpam-4979	9	37	the	the	DET
ejpam-4979	9	38	alternating	alternate	VERB
ejpam-4979	9	39	sum	sum	NOUN
ejpam-4979	9	40	{	{	PUNCT
ejpam-4979	9	41	p(1	p(1	NOUN
ejpam-4979	9	42	)	)	PUNCT
ejpam-4979	9	43	}	}	PUNCT
ejpam-4979	9	44	−	−	PROPN
ejpam-4979	9	45	{	{	PUNCT
ejpam-4979	9	46	p(2	p(2	NOUN
ejpam-4979	9	47	)	)	PUNCT
ejpam-4979	9	48	}	}	PUNCT
ejpam-4979	9	49	+	+	CCONJ
ejpam-4979	9	50	{	{	PUNCT
ejpam-4979	9	51	p(3	p(3	NOUN
ejpam-4979	9	52	)	)	PUNCT
ejpam-4979	9	53	}	}	PUNCT
ejpam-4979	9	54	−	−	PROPN
ejpam-4979	9	55	{	{	PUNCT
ejpam-4979	9	56	p(4	p(4	NOUN
ejpam-4979	9	57	)	)	PUNCT
ejpam-4979	9	58	}	}	PUNCT
ejpam-4979	10	1	+	+	CCONJ
ejpam-4979	10	2	{	{	PUNCT
ejpam-4979	10	3	p(5	p(5	NOUN
ejpam-4979	10	4	)	)	PUNCT
ejpam-4979	10	5	}	}	PUNCT
ejpam-4979	10	6	−	−	PROPN
ejpam-4979	10	7	...	...	PUNCT
ejpam-4979	10	8	.	.	PUNCT
ejpam-4979	11	1	(	(	PUNCT
ejpam-4979	11	2	2	2	X
ejpam-4979	11	3	)	)	PUNCT
ejpam-4979	11	4	the	the	DET
ejpam-4979	11	5	prime	prime	ADJ
ejpam-4979	11	6	number	number	NOUN
ejpam-4979	11	7	subsequences	subsequence	NOUN
ejpam-4979	11	8	of	of	ADP
ejpam-4979	11	9	increasing	increase	VERB
ejpam-4979	11	10	order	order	NOUN
ejpam-4979	12	1	[	[	X
ejpam-4979	12	2	1	1	X
ejpam-4979	12	3	]	]	PUNCT
ejpam-4979	12	4	in	in	ADP
ejpam-4979	12	5	expression	expression	NOUN
ejpam-4979	12	6	2	2	NUM
ejpam-4979	12	7	are	be	AUX
ejpam-4979	12	8	defined	define	VERB
ejpam-4979	12	9	as	as	ADP
ejpam-4979	12	10	doi	doi	NOUN
ejpam-4979	12	11	:	:	PUNCT
ejpam-4979	12	12	https://doi.org/10.29020/nybg.ejpam.v17i1.4979	https://doi.org/10.29020/nybg.ejpam.v17i1.4979	NOUN
ejpam-4979	12	13	email	email	NOUN
ejpam-4979	12	14	address	address	NOUN
ejpam-4979	12	15	:	:	PUNCT
ejpam-4979	12	16	mike.may.bbi@gmail.com	mike.may.bbi@gmail.com	PROPN
ejpam-4979	12	17	(	(	PUNCT
ejpam-4979	12	18	m.	m.	PROPN
ejpam-4979	12	19	p.	p.	PROPN
ejpam-4979	12	20	may	may	AUX
ejpam-4979	12	21	)	)	PUNCT
ejpam-4979	12	22	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4979	13	1	42	42	NUM
ejpam-4979	13	2	©	©	ADP
ejpam-4979	13	3	2024	2024	NUM
ejpam-4979	13	4	ejpam	ejpam	NOUN
ejpam-4979	13	5	all	all	DET
ejpam-4979	13	6	rights	right	NOUN
ejpam-4979	13	7	reserved	reserve	VERB
ejpam-4979	13	8	.	.	PUNCT
ejpam-4979	14	1	m.	m.	NOUN
ejpam-4979	14	2	p.	p.	PROPN
ejpam-4979	14	3	may	may	AUX
ejpam-4979	14	4	/	/	SYM
ejpam-4979	14	5	eur	eur	PROPN
ejpam-4979	14	6	.	.	PUNCT
ejpam-4979	15	1	j.	j.	PROPN
ejpam-4979	15	2	pure	pure	PROPN
ejpam-4979	15	3	appl	appl	PROPN
ejpam-4979	15	4	.	.	PROPN
ejpam-4979	15	5	math	math	PROPN
ejpam-4979	15	6	,	,	PUNCT
ejpam-4979	15	7	17	17	NUM
ejpam-4979	15	8	(	(	PUNCT
ejpam-4979	15	9	1	1	NUM
ejpam-4979	15	10	)	)	PUNCT
ejpam-4979	15	11	(	(	PUNCT
ejpam-4979	15	12	2024	2024	NUM
ejpam-4979	15	13	)	)	PUNCT
ejpam-4979	15	14	,	,	PUNCT
ejpam-4979	15	15	42	42	NUM
ejpam-4979	15	16	-	-	SYM
ejpam-4979	15	17	58	58	NUM
ejpam-4979	15	18	43	43	NUM
ejpam-4979	15	19	{	{	PUNCT
ejpam-4979	15	20	p(1	p(1	NOUN
ejpam-4979	15	21	)	)	PUNCT
ejpam-4979	15	22	}	}	PUNCT
ejpam-4979	15	23	=	=	PUNCT
ejpam-4979	15	24	{	{	PUNCT
ejpam-4979	15	25	pn}∞n=1	pn}∞n=1	X
ejpam-4979	15	26	=	=	SYM
ejpam-4979	15	27	{	{	PUNCT
ejpam-4979	15	28	2	2	NUM
ejpam-4979	15	29	,	,	PUNCT
ejpam-4979	15	30	3	3	NUM
ejpam-4979	15	31	,	,	PUNCT
ejpam-4979	15	32	5	5	NUM
ejpam-4979	15	33	,	,	PUNCT
ejpam-4979	15	34	7	7	NUM
ejpam-4979	15	35	,	,	PUNCT
ejpam-4979	15	36	11	11	NUM
ejpam-4979	15	37	,	,	PUNCT
ejpam-4979	15	38	13	13	NUM
ejpam-4979	15	39	,	,	PUNCT
ejpam-4979	15	40	17	17	NUM
ejpam-4979	15	41	,	,	PUNCT
ejpam-4979	15	42	19	19	NUM
ejpam-4979	15	43	,	,	PUNCT
ejpam-4979	15	44	23	23	NUM
ejpam-4979	15	45	,	,	PUNCT
ejpam-4979	15	46	29	29	NUM
ejpam-4979	15	47	,	,	PUNCT
ejpam-4979	15	48	31	31	NUM
ejpam-4979	15	49	,	,	PUNCT
ejpam-4979	15	50	37	37	NUM
ejpam-4979	15	51	,	,	PUNCT
ejpam-4979	15	52	41	41	NUM
ejpam-4979	15	53	,	,	PUNCT
ejpam-4979	15	54	43	43	NUM
ejpam-4979	15	55	,	,	PUNCT
ejpam-4979	15	56	...	...	PUNCT
ejpam-4979	15	57	}	}	PUNCT
ejpam-4979	15	58	{	{	PUNCT
ejpam-4979	15	59	p(2	p(2	NOUN
ejpam-4979	15	60	)	)	PUNCT
ejpam-4979	15	61	}	}	PUNCT
ejpam-4979	15	62	=	=	SYM
ejpam-4979	15	63	{	{	PUNCT
ejpam-4979	15	64	ppn	ppn	NOUN
ejpam-4979	15	65	}	}	PUNCT
ejpam-4979	15	66	∞	∞	NUM
ejpam-4979	15	67	n=1	n=1	PROPN
ejpam-4979	15	68	=	=	PUNCT
ejpam-4979	15	69	{	{	PUNCT
ejpam-4979	15	70	3	3	NUM
ejpam-4979	15	71	,	,	PUNCT
ejpam-4979	15	72	5	5	NUM
ejpam-4979	15	73	,	,	PUNCT
ejpam-4979	15	74	11	11	NUM
ejpam-4979	15	75	,	,	PUNCT
ejpam-4979	15	76	17	17	NUM
ejpam-4979	15	77	,	,	PUNCT
ejpam-4979	15	78	31	31	NUM
ejpam-4979	15	79	,	,	PUNCT
ejpam-4979	15	80	41	41	NUM
ejpam-4979	15	81	,	,	PUNCT
ejpam-4979	15	82	59	59	NUM
ejpam-4979	15	83	,	,	PUNCT
ejpam-4979	15	84	67	67	NUM
ejpam-4979	15	85	,	,	PUNCT
ejpam-4979	15	86	83	83	NUM
ejpam-4979	15	87	,	,	PUNCT
ejpam-4979	15	88	109	109	NUM
ejpam-4979	15	89	,	,	PUNCT
ejpam-4979	15	90	127	127	NUM
ejpam-4979	15	91	,	,	PUNCT
ejpam-4979	15	92	...	...	PUNCT
ejpam-4979	15	93	}	}	PUNCT
ejpam-4979	15	94	{	{	PUNCT
ejpam-4979	15	95	p(3	p(3	NOUN
ejpam-4979	15	96	)	)	PUNCT
ejpam-4979	15	97	}	}	PUNCT
ejpam-4979	16	1	=	=	PRON
ejpam-4979	16	2	{	{	PUNCT
ejpam-4979	16	3	pppn	pppn	NOUN
ejpam-4979	16	4	}	}	PUNCT
ejpam-4979	16	5	∞	∞	NUM
ejpam-4979	16	6	n=1	n=1	VERB
ejpam-4979	16	7	=	=	PUNCT
ejpam-4979	16	8	{	{	PUNCT
ejpam-4979	16	9	5	5	NUM
ejpam-4979	16	10	,	,	PUNCT
ejpam-4979	16	11	11	11	NUM
ejpam-4979	16	12	,	,	PUNCT
ejpam-4979	16	13	31	31	NUM
ejpam-4979	16	14	,	,	PUNCT
ejpam-4979	16	15	59	59	NUM
ejpam-4979	16	16	,	,	PUNCT
ejpam-4979	16	17	127	127	NUM
ejpam-4979	16	18	,	,	PUNCT
ejpam-4979	16	19	179	179	NUM
ejpam-4979	16	20	,	,	PUNCT
ejpam-4979	16	21	277	277	NUM
ejpam-4979	16	22	,	,	PUNCT
ejpam-4979	16	23	331	331	NUM
ejpam-4979	16	24	,	,	PUNCT
ejpam-4979	16	25	...	...	PUNCT
ejpam-4979	16	26	}	}	PUNCT
ejpam-4979	16	27	{	{	PUNCT
ejpam-4979	16	28	p(4	p(4	NOUN
ejpam-4979	16	29	)	)	PUNCT
ejpam-4979	16	30	}	}	PUNCT
ejpam-4979	17	1	=	=	PRON
ejpam-4979	17	2	{	{	PUNCT
ejpam-4979	17	3	ppppn	ppppn	NOUN
ejpam-4979	17	4	}	}	PUNCT
ejpam-4979	17	5	∞	∞	NUM
ejpam-4979	17	6	n=1	n=1	PROPN
ejpam-4979	17	7	=	=	PUNCT
ejpam-4979	17	8	{	{	PUNCT
ejpam-4979	17	9	11	11	NUM
ejpam-4979	17	10	,	,	PUNCT
ejpam-4979	17	11	31	31	NUM
ejpam-4979	17	12	,	,	PUNCT
ejpam-4979	17	13	127	127	NUM
ejpam-4979	17	14	,	,	PUNCT
ejpam-4979	17	15	277	277	NUM
ejpam-4979	17	16	,	,	PUNCT
ejpam-4979	17	17	709	709	NUM
ejpam-4979	17	18	,	,	PUNCT
ejpam-4979	17	19	...	...	PUNCT
ejpam-4979	17	20	}	}	PUNCT
ejpam-4979	17	21	{	{	PUNCT
ejpam-4979	17	22	p(5	p(5	NOUN
ejpam-4979	17	23	)	)	PUNCT
ejpam-4979	17	24	}	}	PUNCT
ejpam-4979	17	25	=	=	SYM
ejpam-4979	17	26	{	{	PUNCT
ejpam-4979	17	27	pppppn	pppppn	NOUN
ejpam-4979	17	28	}	}	PUNCT
ejpam-4979	17	29	∞	∞	NUM
ejpam-4979	17	30	n=1	n=1	PROPN
ejpam-4979	17	31	=	=	PRON
ejpam-4979	17	32	{	{	PUNCT
ejpam-4979	17	33	31	31	NUM
ejpam-4979	17	34	,	,	PUNCT
ejpam-4979	17	35	127	127	NUM
ejpam-4979	17	36	,	,	PUNCT
ejpam-4979	17	37	709	709	NUM
ejpam-4979	17	38	,	,	PUNCT
ejpam-4979	17	39	...	...	PUNCT
ejpam-4979	17	40	}	}	PUNCT
ejpam-4979	17	41	and	and	CCONJ
ejpam-4979	17	42	so	so	ADV
ejpam-4979	17	43	on	on	ADV
ejpam-4979	17	44	and	and	CCONJ
ejpam-4979	17	45	so	so	ADV
ejpam-4979	17	46	forth	forth	ADV
ejpam-4979	17	47	.	.	PUNCT
ejpam-4979	18	1	thus	thus	ADV
ejpam-4979	18	2	,	,	PUNCT
ejpam-4979	18	3	the	the	DET
ejpam-4979	18	4	operation	operation	NOUN
ejpam-4979	18	5	performed	perform	VERB
ejpam-4979	18	6	on	on	ADP
ejpam-4979	18	7	the	the	DET
ejpam-4979	18	8	right	right	ADJ
ejpam-4979	18	9	-	-	PUNCT
ejpam-4979	18	10	hand	hand	NOUN
ejpam-4979	18	11	side	side	NOUN
ejpam-4979	18	12	of	of	ADP
ejpam-4979	18	13	eq	eq	PROPN
ejpam-4979	18	14	.	.	PROPN
ejpam-4979	18	15	1	1	NUM
ejpam-4979	18	16	denotes	denote	VERB
ejpam-4979	18	17	an	an	DET
ejpam-4979	18	18	infinite	infinite	ADJ
ejpam-4979	18	19	alternating	alternate	VERB
ejpam-4979	18	20	sum	sum	NOUN
ejpam-4979	18	21	of	of	ADP
ejpam-4979	18	22	the	the	DET
ejpam-4979	18	23	sets	set	NOUN
ejpam-4979	18	24	of	of	ADP
ejpam-4979	18	25	prime	prime	ADJ
ejpam-4979	18	26	number	number	NOUN
ejpam-4979	18	27	subsequences	subsequence	NOUN
ejpam-4979	18	28	of	of	ADP
ejpam-4979	18	29	increasing	increase	VERB
ejpam-4979	18	30	order	order	NOUN
ejpam-4979	18	31	.	.	PUNCT
ejpam-4979	19	1	the	the	DET
ejpam-4979	19	2	prime	prime	ADJ
ejpam-4979	19	3	number	number	NOUN
ejpam-4979	19	4	subsequence	subsequence	NOUN
ejpam-4979	19	5	p′	p′	NOUN
ejpam-4979	19	6	can	can	AUX
ejpam-4979	19	7	also	also	ADV
ejpam-4979	19	8	be	be	AUX
ejpam-4979	19	9	generated	generate	VERB
ejpam-4979	19	10	by	by	ADP
ejpam-4979	19	11	performing	perform	VERB
ejpam-4979	19	12	a	a	DET
ejpam-4979	19	13	structured	structured	ADJ
ejpam-4979	19	14	alternating	alternate	VERB
ejpam-4979	19	15	summation	summation	NOUN
ejpam-4979	19	16	of	of	ADP
ejpam-4979	19	17	the	the	DET
ejpam-4979	19	18	individual	individual	ADJ
ejpam-4979	19	19	elements	element	NOUN
ejpam-4979	19	20	across	across	ADP
ejpam-4979	19	21	the	the	DET
ejpam-4979	19	22	sets	set	NOUN
ejpam-4979	19	23	denoted	denote	VERB
ejpam-4979	19	24	on	on	ADP
ejpam-4979	19	25	the	the	DET
ejpam-4979	19	26	righthand	righthand	NOUN
ejpam-4979	19	27	side	side	NOUN
ejpam-4979	19	28	of	of	ADP
ejpam-4979	19	29	eq	eq	PROPN
ejpam-4979	19	30	.	.	PROPN
ejpam-4979	19	31	1	1	NUM
ejpam-4979	19	32	.	.	PUNCT
ejpam-4979	19	33	to	to	PART
ejpam-4979	19	34	illustrate	illustrate	VERB
ejpam-4979	19	35	this	this	PRON
ejpam-4979	19	36	,	,	PUNCT
ejpam-4979	19	37	we	we	PRON
ejpam-4979	19	38	arrange	arrange	VERB
ejpam-4979	19	39	the	the	DET
ejpam-4979	19	40	subsequences	subsequence	NOUN
ejpam-4979	19	41	of	of	ADP
ejpam-4979	19	42	increasing	increase	VERB
ejpam-4979	19	43	order	order	NOUN
ejpam-4979	20	1	[	[	X
ejpam-4979	20	2	1	1	X
ejpam-4979	20	3	]	]	PUNCT
ejpam-4979	20	4	in	in	ADP
ejpam-4979	20	5	expression	expression	NOUN
ejpam-4979	20	6	2	2	NUM
ejpam-4979	20	7	side	side	NOUN
ejpam-4979	20	8	-	-	PUNCT
ejpam-4979	20	9	by	by	ADP
ejpam-4979	20	10	-	-	PUNCT
ejpam-4979	20	11	side	side	NOUN
ejpam-4979	20	12	and	and	CCONJ
ejpam-4979	20	13	sum	sum	NOUN
ejpam-4979	20	14	elements	element	NOUN
ejpam-4979	20	15	laterally	laterally	ADV
ejpam-4979	20	16	across	across	ADP
ejpam-4979	20	17	the	the	DET
ejpam-4979	20	18	rows	row	NOUN
ejpam-4979	20	19	to	to	PART
ejpam-4979	20	20	create	create	VERB
ejpam-4979	20	21	the	the	DET
ejpam-4979	20	22	p′	p′	NOUN
ejpam-4979	20	23	subsequence	subsequence	VERB
ejpam-4979	20	24	term	term	NOUN
ejpam-4979	20	25	-	-	PUNCT
ejpam-4979	20	26	by	by	ADP
ejpam-4979	20	27	-	-	PUNCT
ejpam-4979	20	28	term	term	NOUN
ejpam-4979	20	29	as	as	SCONJ
ejpam-4979	20	30	follows	follow	VERB
ejpam-4979	20	31	:	:	PUNCT
ejpam-4979	20	32	table	table	NOUN
ejpam-4979	20	33	1	1	NUM
ejpam-4979	20	34	:	:	PUNCT
ejpam-4979	20	35	alternating	alternate	VERB
ejpam-4979	20	36	sum	sum	NOUN
ejpam-4979	20	37	of	of	ADP
ejpam-4979	20	38	p(n	p(n	PROPN
ejpam-4979	20	39	)	)	PUNCT
ejpam-4979	20	40	(	(	PUNCT
ejpam-4979	20	41	row	row	NOUN
ejpam-4979	20	42	)	)	PUNCT
ejpam-4979	21	1	+	+	NOUN
ejpam-4979	21	2	p(1	p(1	NOUN
ejpam-4979	21	3	)	)	PUNCT
ejpam-4979	21	4	−p(2	−p(2	NUM
ejpam-4979	21	5	)	)	PUNCT
ejpam-4979	22	1	+	+	NOUN
ejpam-4979	22	2	p(3	p(3	X
ejpam-4979	22	3	)	)	PUNCT
ejpam-4979	22	4	−p(4	−p(4	NOUN
ejpam-4979	22	5	)	)	PUNCT
ejpam-4979	23	1	+	+	NOUN
ejpam-4979	23	2	p(5	p(5	NOUN
ejpam-4979	23	3	)	)	PUNCT
ejpam-4979	23	4	−p(6	−p(6	NOUN
ejpam-4979	23	5	)	)	PUNCT
ejpam-4979	23	6	...	...	PUNCT
ejpam-4979	24	1	p′	p′	NOUN
ejpam-4979	24	2	(	(	PUNCT
ejpam-4979	24	3	1	1	NUM
ejpam-4979	24	4	)	)	SYM
ejpam-4979	24	5	2	2	NUM
ejpam-4979	24	6	−→	−→	NOUN
ejpam-4979	24	7	−→	−→	ADV
ejpam-4979	24	8	−→	−→	ADV
ejpam-4979	24	9	−→	−→	ADV
ejpam-4979	24	10	−→	−→	ADV
ejpam-4979	24	11	−→	−→	ADJ
ejpam-4979	24	12	2	2	NUM
ejpam-4979	24	13	(	(	PUNCT
ejpam-4979	24	14	2	2	NUM
ejpam-4979	24	15	)	)	PUNCT
ejpam-4979	24	16	3	3	NUM
ejpam-4979	24	17	3	3	NUM
ejpam-4979	24	18	−→	−→	NOUN
ejpam-4979	24	19	−→	−→	ADV
ejpam-4979	24	20	−→	−→	ADV
ejpam-4979	24	21	−→	−→	NOUN
ejpam-4979	24	22	−→	−→	NOUN
ejpam-4979	24	23	0	0	NUM
ejpam-4979	24	24	(	(	PUNCT
ejpam-4979	24	25	3	3	NUM
ejpam-4979	24	26	)	)	PUNCT
ejpam-4979	24	27	5	5	NUM
ejpam-4979	24	28	5	5	NUM
ejpam-4979	24	29	5	5	NUM
ejpam-4979	24	30	−→	−→	NOUN
ejpam-4979	24	31	−→	−→	ADV
ejpam-4979	24	32	−→	−→	ADV
ejpam-4979	24	33	−→	−→	NOUN
ejpam-4979	24	34	5	5	NUM
ejpam-4979	24	35	(	(	PUNCT
ejpam-4979	24	36	4	4	NUM
ejpam-4979	24	37	)	)	PUNCT
ejpam-4979	24	38	7	7	NUM
ejpam-4979	24	39	−→	−→	NOUN
ejpam-4979	24	40	−→	−→	ADV
ejpam-4979	24	41	−→	−→	ADV
ejpam-4979	24	42	−→	−→	ADV
ejpam-4979	24	43	−→	−→	ADJ
ejpam-4979	24	44	−→	−→	NOUN
ejpam-4979	24	45	7	7	NUM
ejpam-4979	24	46	(	(	PUNCT
ejpam-4979	24	47	5	5	NUM
ejpam-4979	24	48	)	)	PUNCT
ejpam-4979	24	49	11	11	NUM
ejpam-4979	24	50	11	11	NUM
ejpam-4979	24	51	11	11	NUM
ejpam-4979	24	52	11	11	NUM
ejpam-4979	24	53	−→	−→	NOUN
ejpam-4979	24	54	−→	−→	ADV
ejpam-4979	24	55	−→	−→	NOUN
ejpam-4979	24	56	0	0	NUM
ejpam-4979	24	57	(	(	PUNCT
ejpam-4979	24	58	6	6	NUM
ejpam-4979	24	59	)	)	PUNCT
ejpam-4979	24	60	13	13	NUM
ejpam-4979	24	61	−→	−→	NOUN
ejpam-4979	24	62	−→	−→	ADV
ejpam-4979	24	63	−→	−→	ADV
ejpam-4979	24	64	−→	−→	ADV
ejpam-4979	24	65	−→	−→	ADV
ejpam-4979	24	66	−→	−→	NOUN
ejpam-4979	24	67	13	13	NUM
ejpam-4979	24	68	(	(	PUNCT
ejpam-4979	24	69	7	7	NUM
ejpam-4979	24	70	)	)	PUNCT
ejpam-4979	24	71	17	17	NUM
ejpam-4979	24	72	17	17	NUM
ejpam-4979	24	73	−→	−→	ADV
ejpam-4979	24	74	−→	−→	ADV
ejpam-4979	24	75	−→	−→	ADV
ejpam-4979	24	76	−→	−→	NOUN
ejpam-4979	24	77	−→	−→	NOUN
ejpam-4979	24	78	0	0	NUM
ejpam-4979	24	79	(	(	PUNCT
ejpam-4979	24	80	8)	8)	NUM
ejpam-4979	24	81	19	19	NUM
ejpam-4979	24	82	−→	−→	NOUN
ejpam-4979	24	83	−→	−→	ADV
ejpam-4979	24	84	−→	−→	ADV
ejpam-4979	24	85	−→	−→	ADV
ejpam-4979	24	86	−→	−→	ADJ
ejpam-4979	24	87	−→	−→	NOUN
ejpam-4979	24	88	19	19	NUM
ejpam-4979	24	89	(	(	PUNCT
ejpam-4979	24	90	9	9	NUM
ejpam-4979	24	91	)	)	PUNCT
ejpam-4979	24	92	23	23	NUM
ejpam-4979	24	93	−→	−→	ADV
ejpam-4979	24	94	−→	−→	ADV
ejpam-4979	24	95	−→	−→	ADV
ejpam-4979	24	96	−→	−→	ADV
ejpam-4979	24	97	−→	−→	ADV
ejpam-4979	24	98	−→	−→	NOUN
ejpam-4979	24	99	23	23	NUM
ejpam-4979	24	100	(	(	PUNCT
ejpam-4979	24	101	10	10	NUM
ejpam-4979	24	102	)	)	PUNCT
ejpam-4979	24	103	29	29	NUM
ejpam-4979	24	104	−→	−→	ADV
ejpam-4979	24	105	−→	−→	ADV
ejpam-4979	24	106	−→	−→	ADV
ejpam-4979	24	107	−→	−→	ADV
ejpam-4979	24	108	−→	−→	ADV
ejpam-4979	24	109	−→	−→	ADJ
ejpam-4979	24	110	29	29	NUM
ejpam-4979	24	111	(	(	PUNCT
ejpam-4979	24	112	11	11	NUM
ejpam-4979	24	113	)	)	PUNCT
ejpam-4979	24	114	31	31	NUM
ejpam-4979	24	115	31	31	NUM
ejpam-4979	24	116	31	31	NUM
ejpam-4979	24	117	31	31	NUM
ejpam-4979	24	118	31	31	NUM
ejpam-4979	24	119	−→	−→	NOUN
ejpam-4979	24	120	−→	−→	NOUN
ejpam-4979	24	121	31	31	NUM
ejpam-4979	24	122	...	...	PUNCT
ejpam-4979	24	123	...	...	PUNCT
ejpam-4979	24	124	...	...	PUNCT
ejpam-4979	24	125	...	...	PUNCT
ejpam-4979	24	126	...	...	PUNCT
ejpam-4979	24	127	...	...	PUNCT
ejpam-4979	24	128	...	...	PUNCT
ejpam-4979	24	129	...	...	PUNCT
ejpam-4979	24	130	...	...	PUNCT
ejpam-4979	25	1	thus	thus	ADV
ejpam-4979	25	2	,	,	PUNCT
ejpam-4979	25	3	the	the	DET
ejpam-4979	25	4	infinite	infinite	ADJ
ejpam-4979	25	5	prime	prime	ADJ
ejpam-4979	25	6	number	number	NOUN
ejpam-4979	25	7	subsequence	subsequence	NOUN
ejpam-4979	25	8	p′	p′	NOUN
ejpam-4979	25	9	of	of	ADP
ejpam-4979	25	10	higher	high	ADJ
ejpam-4979	25	11	order	order	NOUN
ejpam-4979	25	12	[	[	X
ejpam-4979	25	13	4	4	X
ejpam-4979	25	14	]	]	PUNCT
ejpam-4979	25	15	that	that	PRON
ejpam-4979	25	16	emerges	emerge	VERB
ejpam-4979	25	17	in	in	ADP
ejpam-4979	25	18	the	the	DET
ejpam-4979	25	19	rightmost	rightmost	ADJ
ejpam-4979	25	20	column	column	NOUN
ejpam-4979	25	21	of	of	ADP
ejpam-4979	25	22	table	table	NOUN
ejpam-4979	25	23	1	1	NUM
ejpam-4979	25	24	is	be	AUX
ejpam-4979	25	25	p	p	NOUN
ejpam-4979	25	26	′	′	NOUN
ejpam-4979	25	27	=	=	PUNCT
ejpam-4979	25	28	{	{	PUNCT
ejpam-4979	25	29	p′	p′	NOUN
ejpam-4979	25	30	}	}	PUNCT
ejpam-4979	25	31	=	=	PUNCT
ejpam-4979	25	32	{	{	PUNCT
ejpam-4979	25	33	2	2	NUM
ejpam-4979	25	34	,	,	PUNCT
ejpam-4979	25	35	5	5	NUM
ejpam-4979	25	36	,	,	PUNCT
ejpam-4979	25	37	7	7	NUM
ejpam-4979	25	38	,	,	PUNCT
ejpam-4979	25	39	13	13	NUM
ejpam-4979	25	40	,	,	PUNCT
ejpam-4979	25	41	19	19	NUM
ejpam-4979	25	42	,	,	PUNCT
ejpam-4979	25	43	23	23	NUM
ejpam-4979	25	44	,	,	PUNCT
ejpam-4979	25	45	29	29	NUM
ejpam-4979	25	46	,	,	PUNCT
ejpam-4979	25	47	31	31	NUM
ejpam-4979	25	48	,	,	PUNCT
ejpam-4979	25	49	37	37	NUM
ejpam-4979	25	50	,	,	PUNCT
ejpam-4979	25	51	43	43	NUM
ejpam-4979	25	52	,	,	PUNCT
ejpam-4979	25	53	47	47	NUM
ejpam-4979	25	54	,	,	PUNCT
ejpam-4979	25	55	53	53	NUM
ejpam-4979	25	56	,	,	PUNCT
ejpam-4979	25	57	59	59	NUM
ejpam-4979	25	58	,	,	PUNCT
ejpam-4979	25	59	61	61	NUM
ejpam-4979	25	60	,	,	PUNCT
ejpam-4979	25	61	71	71	NUM
ejpam-4979	25	62	,	,	PUNCT
ejpam-4979	25	63	...	...	PUNCT
ejpam-4979	25	64	}	}	PUNCT
ejpam-4979	25	65	.	.	PUNCT
ejpam-4979	26	1	m.	m.	NOUN
ejpam-4979	26	2	p.	p.	NOUN
ejpam-4979	26	3	may	may	AUX
ejpam-4979	26	4	/	/	SYM
ejpam-4979	26	5	eur	eur	PROPN
ejpam-4979	26	6	.	.	PUNCT
ejpam-4979	27	1	j.	j.	PROPN
ejpam-4979	27	2	pure	pure	PROPN
ejpam-4979	27	3	appl	appl	PROPN
ejpam-4979	27	4	.	.	PROPN
ejpam-4979	27	5	math	math	PROPN
ejpam-4979	27	6	,	,	PUNCT
ejpam-4979	27	7	17	17	NUM
ejpam-4979	27	8	(	(	PUNCT
ejpam-4979	27	9	1	1	NUM
ejpam-4979	27	10	)	)	PUNCT
ejpam-4979	27	11	(	(	PUNCT
ejpam-4979	27	12	2024	2024	NUM
ejpam-4979	27	13	)	)	PUNCT
ejpam-4979	27	14	,	,	PUNCT
ejpam-4979	27	15	42	42	NUM
ejpam-4979	27	16	-	-	SYM
ejpam-4979	27	17	58	58	NUM
ejpam-4979	27	18	44	44	NUM
ejpam-4979	27	19	the	the	DET
ejpam-4979	27	20	prime	prime	ADJ
ejpam-4979	27	21	number	number	NOUN
ejpam-4979	27	22	subsequence	subsequence	NOUN
ejpam-4979	27	23	p′	p′	NOUN
ejpam-4979	27	24	can	can	AUX
ejpam-4979	27	25	also	also	ADV
ejpam-4979	27	26	be	be	AUX
ejpam-4979	27	27	generated	generate	VERB
ejpam-4979	27	28	by	by	ADP
ejpam-4979	27	29	performing	perform	VERB
ejpam-4979	27	30	a	a	DET
ejpam-4979	27	31	sieving	sieve	VERB
ejpam-4979	27	32	operation	operation	NOUN
ejpam-4979	27	33	on	on	ADP
ejpam-4979	27	34	the	the	DET
ejpam-4979	27	35	natural	natural	ADJ
ejpam-4979	27	36	numbers	number	NOUN
ejpam-4979	27	37	n	n	CCONJ
ejpam-4979	27	38	[	[	X
ejpam-4979	27	39	7	7	NUM
ejpam-4979	27	40	]	]	PUNCT
ejpam-4979	27	41	.	.	PUNCT
ejpam-4979	28	1	starting	start	VERB
ejpam-4979	28	2	with	with	ADP
ejpam-4979	28	3	n	n	NOUN
ejpam-4979	28	4	=	=	SYM
ejpam-4979	28	5	1	1	NUM
ejpam-4979	28	6	,	,	PUNCT
ejpam-4979	28	7	choose	choose	VERB
ejpam-4979	28	8	the	the	DET
ejpam-4979	28	9	prime	prime	ADJ
ejpam-4979	28	10	number	number	NOUN
ejpam-4979	28	11	with	with	ADP
ejpam-4979	28	12	subscript	subscript	NOUN
ejpam-4979	28	13	1	1	NUM
ejpam-4979	28	14	(	(	PUNCT
ejpam-4979	28	15	i.e.	i.e.	X
ejpam-4979	28	16	,	,	PUNCT
ejpam-4979	28	17	p1	p1	NOUN
ejpam-4979	28	18	=	=	NOUN
ejpam-4979	28	19	2	2	NUM
ejpam-4979	28	20	)	)	PUNCT
ejpam-4979	28	21	as	as	ADP
ejpam-4979	28	22	the	the	DET
ejpam-4979	28	23	first	first	ADJ
ejpam-4979	28	24	term	term	NOUN
ejpam-4979	28	25	of	of	ADP
ejpam-4979	28	26	the	the	DET
ejpam-4979	28	27	subsequence	subsequence	NOUN
ejpam-4979	28	28	and	and	CCONJ
ejpam-4979	28	29	eliminate	eliminate	VERB
ejpam-4979	28	30	that	that	DET
ejpam-4979	28	31	prime	prime	ADJ
ejpam-4979	28	32	number	number	NOUN
ejpam-4979	28	33	from	from	ADP
ejpam-4979	28	34	the	the	DET
ejpam-4979	28	35	natural	natural	ADJ
ejpam-4979	28	36	number	number	NOUN
ejpam-4979	28	37	line	line	NOUN
ejpam-4979	28	38	.	.	PUNCT
ejpam-4979	29	1	next	next	ADV
ejpam-4979	29	2	,	,	PUNCT
ejpam-4979	29	3	proceed	proceed	VERB
ejpam-4979	29	4	forward	forward	ADV
ejpam-4979	29	5	on	on	ADP
ejpam-4979	29	6	n	n	CCONJ
ejpam-4979	29	7	from	from	ADP
ejpam-4979	29	8	1	1	NUM
ejpam-4979	29	9	to	to	ADP
ejpam-4979	29	10	the	the	DET
ejpam-4979	29	11	next	next	ADJ
ejpam-4979	29	12	available	available	ADJ
ejpam-4979	29	13	natural	natural	ADJ
ejpam-4979	29	14	number	number	NOUN
ejpam-4979	29	15	.	.	PUNCT
ejpam-4979	30	1	since	since	SCONJ
ejpam-4979	30	2	2	2	NUM
ejpam-4979	30	3	was	be	AUX
ejpam-4979	30	4	eliminated	eliminate	VERB
ejpam-4979	30	5	from	from	ADP
ejpam-4979	30	6	the	the	DET
ejpam-4979	30	7	natural	natural	ADJ
ejpam-4979	30	8	number	number	NOUN
ejpam-4979	30	9	line	line	NOUN
ejpam-4979	30	10	in	in	ADP
ejpam-4979	30	11	the	the	DET
ejpam-4979	30	12	previous	previous	ADJ
ejpam-4979	30	13	step	step	NOUN
ejpam-4979	30	14	,	,	PUNCT
ejpam-4979	30	15	one	one	NUM
ejpam-4979	30	16	moves	move	VERB
ejpam-4979	30	17	forward	forward	ADV
ejpam-4979	30	18	to	to	ADP
ejpam-4979	30	19	the	the	DET
ejpam-4979	30	20	next	next	ADJ
ejpam-4979	30	21	available	available	ADJ
ejpam-4979	30	22	natural	natural	ADJ
ejpam-4979	30	23	number	number	NOUN
ejpam-4979	30	24	that	that	PRON
ejpam-4979	30	25	has	have	AUX
ejpam-4979	30	26	not	not	PART
ejpam-4979	30	27	been	be	AUX
ejpam-4979	30	28	eliminated	eliminate	VERB
ejpam-4979	30	29	,	,	PUNCT
ejpam-4979	30	30	which	which	PRON
ejpam-4979	30	31	is	be	AUX
ejpam-4979	30	32	3	3	NUM
ejpam-4979	30	33	.	.	PUNCT
ejpam-4979	31	1	the	the	DET
ejpam-4979	31	2	prime	prime	ADJ
ejpam-4979	31	3	number	number	NOUN
ejpam-4979	31	4	3	3	NUM
ejpam-4979	31	5	then	then	ADV
ejpam-4979	31	6	becomes	become	VERB
ejpam-4979	31	7	the	the	DET
ejpam-4979	31	8	subscript	subscript	NOUN
ejpam-4979	31	9	for	for	ADP
ejpam-4979	31	10	the	the	DET
ejpam-4979	31	11	next	next	ADJ
ejpam-4979	31	12	p′	p′	NOUN
ejpam-4979	31	13	term	term	NOUN
ejpam-4979	31	14	which	which	PRON
ejpam-4979	31	15	is	be	AUX
ejpam-4979	31	16	p3	p3	NOUN
ejpam-4979	31	17	=	=	SYM
ejpam-4979	31	18	5	5	NUM
ejpam-4979	31	19	,	,	PUNCT
ejpam-4979	31	20	and	and	CCONJ
ejpam-4979	31	21	5	5	NUM
ejpam-4979	31	22	is	be	AUX
ejpam-4979	31	23	then	then	ADV
ejpam-4979	31	24	eliminated	eliminate	VERB
ejpam-4979	31	25	from	from	ADP
ejpam-4979	31	26	the	the	DET
ejpam-4979	31	27	natural	natural	ADJ
ejpam-4979	31	28	number	number	NOUN
ejpam-4979	31	29	line	line	NOUN
ejpam-4979	31	30	,	,	PUNCT
ejpam-4979	31	31	and	and	CCONJ
ejpam-4979	31	32	so	so	ADV
ejpam-4979	31	33	on	on	ADV
ejpam-4979	31	34	and	and	CCONJ
ejpam-4979	31	35	so	so	ADV
ejpam-4979	31	36	forth	forth	ADV
ejpam-4979	31	37	.	.	PUNCT
ejpam-4979	32	1	such	such	DET
ejpam-4979	32	2	a	a	DET
ejpam-4979	32	3	sieving	sieve	VERB
ejpam-4979	32	4	operation	operation	NOUN
ejpam-4979	32	5	has	have	AUX
ejpam-4979	32	6	been	be	AUX
ejpam-4979	32	7	carried	carry	VERB
ejpam-4979	32	8	out	out	ADP
ejpam-4979	32	9	in	in	ADP
ejpam-4979	32	10	table	table	NOUN
ejpam-4979	32	11	2	2	NUM
ejpam-4979	32	12	for	for	ADP
ejpam-4979	32	13	the	the	DET
ejpam-4979	32	14	natural	natural	ADJ
ejpam-4979	32	15	numbers	number	NOUN
ejpam-4979	32	16	1	1	NUM
ejpam-4979	32	17	to	to	PART
ejpam-4979	32	18	100	100	NUM
ejpam-4979	32	19	:	:	PUNCT
ejpam-4979	32	20	table	table	NOUN
ejpam-4979	32	21	2	2	NUM
ejpam-4979	32	22	:	:	PUNCT
ejpam-4979	32	23	sieving	sieve	VERB
ejpam-4979	32	24	n	n	PART
ejpam-4979	32	25	to	to	PART
ejpam-4979	32	26	generate	generate	VERB
ejpam-4979	32	27	p′	p′	NOUN
ejpam-4979	32	28	1	1	NUM
ejpam-4979	32	29	2	2	NUM
ejpam-4979	32	30	3	3	NUM
ejpam-4979	32	31	4	4	NUM
ejpam-4979	32	32	5	5	NUM
ejpam-4979	32	33	6	6	NUM
ejpam-4979	32	34	7	7	NUM
ejpam-4979	32	35	8	8	NUM
ejpam-4979	32	36	9	9	NUM
ejpam-4979	32	37	10	10	NUM
ejpam-4979	32	38	11	11	NUM
ejpam-4979	32	39	12	12	NUM
ejpam-4979	32	40	13	13	NUM
ejpam-4979	32	41	14	14	NUM
ejpam-4979	32	42	15	15	NUM
ejpam-4979	32	43	16	16	NUM
ejpam-4979	32	44	17	17	NUM
ejpam-4979	32	45	18	18	NUM
ejpam-4979	32	46	19	19	NUM
ejpam-4979	32	47	20	20	NUM
ejpam-4979	32	48	21	21	NUM
ejpam-4979	32	49	22	22	NUM
ejpam-4979	32	50	23	23	NUM
ejpam-4979	32	51	24	24	NUM
ejpam-4979	32	52	25	25	NUM
ejpam-4979	32	53	26	26	NUM
ejpam-4979	32	54	27	27	NUM
ejpam-4979	32	55	28	28	NUM
ejpam-4979	32	56	29	29	NUM
ejpam-4979	32	57	30	30	NUM
ejpam-4979	32	58	31	31	NUM
ejpam-4979	32	59	32	32	NUM
ejpam-4979	32	60	33	33	NUM
ejpam-4979	32	61	34	34	NUM
ejpam-4979	32	62	35	35	NUM
ejpam-4979	32	63	36	36	NUM
ejpam-4979	32	64	37	37	NUM
ejpam-4979	32	65	38	38	NUM
ejpam-4979	32	66	39	39	NUM
ejpam-4979	32	67	40	40	NUM
ejpam-4979	32	68	41	41	NUM
ejpam-4979	32	69	42	42	NUM
ejpam-4979	32	70	43	43	NUM
ejpam-4979	32	71	44	44	NUM
ejpam-4979	32	72	45	45	NUM
ejpam-4979	32	73	46	46	NUM
ejpam-4979	32	74	47	47	NUM
ejpam-4979	32	75	48	48	NUM
ejpam-4979	32	76	49	49	NUM
ejpam-4979	32	77	50	50	NUM
ejpam-4979	32	78	51	51	NUM
ejpam-4979	32	79	52	52	NUM
ejpam-4979	32	80	53	53	NUM
ejpam-4979	32	81	54	54	NUM
ejpam-4979	32	82	55	55	NUM
ejpam-4979	32	83	56	56	NUM
ejpam-4979	32	84	57	57	NUM
ejpam-4979	32	85	58	58	NUM
ejpam-4979	32	86	59	59	NUM
ejpam-4979	32	87	60	60	NUM
ejpam-4979	32	88	61	61	NUM
ejpam-4979	32	89	62	62	NUM
ejpam-4979	32	90	63	63	NUM
ejpam-4979	32	91	64	64	NUM
ejpam-4979	32	92	65	65	NUM
ejpam-4979	32	93	66	66	NUM
ejpam-4979	32	94	67	67	NUM
ejpam-4979	32	95	68	68	NUM
ejpam-4979	32	96	69	69	NUM
ejpam-4979	32	97	70	70	NUM
ejpam-4979	32	98	71	71	NUM
ejpam-4979	32	99	72	72	NUM
ejpam-4979	32	100	73	73	NUM
ejpam-4979	32	101	74	74	NUM
ejpam-4979	32	102	75	75	NUM
ejpam-4979	32	103	76	76	NUM
ejpam-4979	32	104	77	77	NUM
ejpam-4979	32	105	78	78	NUM
ejpam-4979	32	106	79	79	NUM
ejpam-4979	32	107	80	80	NUM
ejpam-4979	32	108	81	81	NUM
ejpam-4979	32	109	82	82	NUM
ejpam-4979	32	110	83	83	NUM
ejpam-4979	32	111	84	84	NUM
ejpam-4979	32	112	85	85	NUM
ejpam-4979	32	113	86	86	NUM
ejpam-4979	32	114	87	87	NUM
ejpam-4979	32	115	88	88	NUM
ejpam-4979	32	116	89	89	NUM
ejpam-4979	32	117	90	90	NUM
ejpam-4979	32	118	91	91	NUM
ejpam-4979	33	1	92	92	NUM
ejpam-4979	33	2	93	93	NUM
ejpam-4979	33	3	94	94	NUM
ejpam-4979	33	4	95	95	NUM
ejpam-4979	33	5	96	96	NUM
ejpam-4979	33	6	97	97	NUM
ejpam-4979	33	7	98	98	NUM
ejpam-4979	33	8	99	99	NUM
ejpam-4979	33	9	100	100	NUM
ejpam-4979	33	10	thus	thus	ADV
ejpam-4979	33	11	,	,	PUNCT
ejpam-4979	33	12	we	we	PRON
ejpam-4979	33	13	may	may	AUX
ejpam-4979	33	14	also	also	ADV
ejpam-4979	33	15	designate	designate	VERB
ejpam-4979	33	16	p′	p′	NOUN
ejpam-4979	33	17	,	,	PUNCT
ejpam-4979	33	18	which	which	PRON
ejpam-4979	33	19	has	have	AUX
ejpam-4979	33	20	been	be	AUX
ejpam-4979	33	21	alternately	alternately	ADV
ejpam-4979	33	22	created	create	VERB
ejpam-4979	33	23	via	via	ADP
ejpam-4979	33	24	the	the	DET
ejpam-4979	33	25	sieving	sieve	VERB
ejpam-4979	33	26	operation	operation	NOUN
ejpam-4979	33	27	in	in	ADP
ejpam-4979	33	28	table	table	NOUN
ejpam-4979	33	29	2	2	NUM
ejpam-4979	33	30	,	,	PUNCT
ejpam-4979	33	31	by	by	ADP
ejpam-4979	33	32	the	the	DET
ejpam-4979	33	33	following	follow	VERB
ejpam-4979	33	34	notation	notation	NOUN
ejpam-4979	33	35	[	[	X
ejpam-4979	33	36	7	7	NUM
ejpam-4979	33	37	]	]	PUNCT
ejpam-4979	33	38	to	to	PART
ejpam-4979	33	39	indicate	indicate	VERB
ejpam-4979	33	40	that	that	SCONJ
ejpam-4979	33	41	the	the	DET
ejpam-4979	33	42	natural	natural	ADJ
ejpam-4979	33	43	numbers	number	NOUN
ejpam-4979	33	44	n	n	PRON
ejpam-4979	33	45	have	have	AUX
ejpam-4979	33	46	been	be	AUX
ejpam-4979	33	47	sieved	sieve	VERB
ejpam-4979	33	48	to	to	PART
ejpam-4979	33	49	produce	produce	VERB
ejpam-4979	33	50	this	this	DET
ejpam-4979	33	51	prime	prime	ADJ
ejpam-4979	33	52	number	number	NOUN
ejpam-4979	33	53	subsequence	subsequence	NOUN
ejpam-4979	33	54	:	:	PUNCT
ejpam-4979	33	55	⌊n⌋	⌊n⌋	X
ejpam-4979	33	56	=	=	SYM
ejpam-4979	33	57	p′	p′	X
ejpam-4979	33	58	=	=	SYM
ejpam-4979	33	59	{	{	PUNCT
ejpam-4979	33	60	2	2	NUM
ejpam-4979	33	61	,	,	PUNCT
ejpam-4979	33	62	5	5	NUM
ejpam-4979	33	63	,	,	PUNCT
ejpam-4979	33	64	7	7	NUM
ejpam-4979	33	65	,	,	PUNCT
ejpam-4979	33	66	13	13	NUM
ejpam-4979	33	67	,	,	PUNCT
ejpam-4979	33	68	19	19	NUM
ejpam-4979	33	69	,	,	PUNCT
ejpam-4979	33	70	23	23	NUM
ejpam-4979	33	71	,	,	PUNCT
ejpam-4979	33	72	29	29	NUM
ejpam-4979	33	73	,	,	PUNCT
ejpam-4979	33	74	31	31	NUM
ejpam-4979	33	75	,	,	PUNCT
ejpam-4979	33	76	37	37	NUM
ejpam-4979	33	77	,	,	PUNCT
ejpam-4979	33	78	43	43	NUM
ejpam-4979	33	79	,	,	PUNCT
ejpam-4979	33	80	47	47	NUM
ejpam-4979	33	81	,	,	PUNCT
ejpam-4979	33	82	53	53	NUM
ejpam-4979	33	83	,	,	PUNCT
ejpam-4979	33	84	59	59	NUM
ejpam-4979	33	85	,	,	PUNCT
ejpam-4979	33	86	61	61	NUM
ejpam-4979	33	87	,	,	PUNCT
ejpam-4979	33	88	71	71	NUM
ejpam-4979	33	89	,	,	PUNCT
ejpam-4979	33	90	...	...	PUNCT
ejpam-4979	33	91	}	}	PUNCT
ejpam-4979	33	92	.	.	PUNCT
ejpam-4979	34	1	regardless	regardless	ADV
ejpam-4979	34	2	of	of	ADP
ejpam-4979	34	3	which	which	DET
ejpam-4979	34	4	one	one	NUM
ejpam-4979	34	5	of	of	ADP
ejpam-4979	34	6	these	these	DET
ejpam-4979	34	7	three	three	NUM
ejpam-4979	34	8	methods	method	NOUN
ejpam-4979	34	9	is	be	AUX
ejpam-4979	34	10	used	use	VERB
ejpam-4979	34	11	to	to	PART
ejpam-4979	34	12	generate	generate	VERB
ejpam-4979	34	13	p′	p′	NOUN
ejpam-4979	34	14	,	,	PUNCT
ejpam-4979	34	15	when	when	SCONJ
ejpam-4979	34	16	the	the	DET
ejpam-4979	34	17	prime	prime	ADJ
ejpam-4979	34	18	numbers	number	NOUN
ejpam-4979	34	19	in	in	ADP
ejpam-4979	34	20	this	this	DET
ejpam-4979	34	21	unique	unique	ADJ
ejpam-4979	34	22	subsequence	subsequence	NOUN
ejpam-4979	34	23	are	be	AUX
ejpam-4979	34	24	applied	apply	VERB
ejpam-4979	34	25	as	as	ADP
ejpam-4979	34	26	indexes	index	NOUN
ejpam-4979	34	27	to	to	ADP
ejpam-4979	34	28	the	the	DET
ejpam-4979	34	29	set	set	NOUN
ejpam-4979	34	30	of	of	ADP
ejpam-4979	34	31	all	all	DET
ejpam-4979	34	32	prime	prime	ADJ
ejpam-4979	34	33	numbers	number	NOUN
ejpam-4979	34	34	p	p	X
ejpam-4979	34	35	,	,	PUNCT
ejpam-4979	34	36	one	one	NOUN
ejpam-4979	34	37	obtains	obtain	VERB
ejpam-4979	34	38	the	the	DET
ejpam-4979	34	39	next	next	ADJ
ejpam-4979	34	40	higher	high	ADJ
ejpam-4979	34	41	-	-	PUNCT
ejpam-4979	34	42	order	order	NOUN
ejpam-4979	34	43	prime	prime	ADJ
ejpam-4979	34	44	number	number	NOUN
ejpam-4979	34	45	subsequence	subsequence	NOUN
ejpam-4979	34	46	p′′	p′′	PROPN
ejpam-4979	34	47	[	[	X
ejpam-4979	34	48	3	3	NUM
ejpam-4979	34	49	]	]	X
ejpam-4979	34	50	:	:	PUNCT
ejpam-4979	34	51	p	p	X
ejpam-4979	34	52	′′	′′	PROPN
ejpam-4979	34	53	=	=	PRON
ejpam-4979	34	54	{	{	PUNCT
ejpam-4979	34	55	p′′	p′′	PROPN
ejpam-4979	34	56	}	}	PUNCT
ejpam-4979	34	57	=	=	PUNCT
ejpam-4979	34	58	{	{	PUNCT
ejpam-4979	34	59	3	3	NUM
ejpam-4979	34	60	,	,	PUNCT
ejpam-4979	34	61	11	11	NUM
ejpam-4979	34	62	,	,	PUNCT
ejpam-4979	34	63	17	17	NUM
ejpam-4979	34	64	,	,	PUNCT
ejpam-4979	34	65	41	41	NUM
ejpam-4979	34	66	,	,	PUNCT
ejpam-4979	34	67	67	67	NUM
ejpam-4979	34	68	,	,	PUNCT
ejpam-4979	34	69	83	83	NUM
ejpam-4979	34	70	,	,	PUNCT
ejpam-4979	34	71	109	109	NUM
ejpam-4979	34	72	,	,	PUNCT
ejpam-4979	34	73	127	127	NUM
ejpam-4979	34	74	,	,	PUNCT
ejpam-4979	34	75	157	157	NUM
ejpam-4979	34	76	,	,	PUNCT
ejpam-4979	34	77	191	191	NUM
ejpam-4979	34	78	,	,	PUNCT
ejpam-4979	34	79	211	211	NUM
ejpam-4979	34	80	,	,	PUNCT
ejpam-4979	34	81	241	241	NUM
ejpam-4979	34	82	,	,	PUNCT
ejpam-4979	34	83	...	...	PUNCT
ejpam-4979	34	84	}	}	PUNCT
ejpam-4979	34	85	.	.	PUNCT
ejpam-4979	35	1	by	by	ADP
ejpam-4979	35	2	definition	definition	NOUN
ejpam-4979	35	3	,	,	PUNCT
ejpam-4979	35	4	the	the	DET
ejpam-4979	35	5	sequence	sequence	NOUN
ejpam-4979	35	6	p′′	p′′	PROPN
ejpam-4979	35	7	can	can	AUX
ejpam-4979	35	8	also	also	ADV
ejpam-4979	35	9	be	be	AUX
ejpam-4979	35	10	generated	generate	VERB
ejpam-4979	35	11	via	via	ADP
ejpam-4979	35	12	the	the	DET
ejpam-4979	35	13	expression	expression	NOUN
ejpam-4979	35	14	[	[	X
ejpam-4979	35	15	7	7	X
ejpam-4979	35	16	]	]	X
ejpam-4979	36	1	p	p	X
ejpam-4979	36	2	′′	′′	PROPN
ejpam-4979	36	3	=	=	PRON
ejpam-4979	36	4	{	{	PUNCT
ejpam-4979	36	5	(	(	PUNCT
ejpam-4979	36	6	−1)n	−1)n	X
ejpam-4979	36	7	{	{	PUNCT
ejpam-4979	36	8	p(n	p(n	PROPN
ejpam-4979	36	9	)	)	PUNCT
ejpam-4979	36	10	}	}	PUNCT
ejpam-4979	36	11	}	}	PUNCT
ejpam-4979	36	12	∞	∞	NUM
ejpam-4979	36	13	n=2	n=2	X
ejpam-4979	36	14	(	(	PUNCT
ejpam-4979	36	15	3	3	NUM
ejpam-4979	36	16	)	)	PUNCT
ejpam-4979	36	17	where	where	SCONJ
ejpam-4979	36	18	an	an	DET
ejpam-4979	36	19	expansion	expansion	NOUN
ejpam-4979	36	20	of	of	ADP
ejpam-4979	36	21	the	the	DET
ejpam-4979	36	22	right	right	ADJ
ejpam-4979	36	23	-	-	PUNCT
ejpam-4979	36	24	hand	hand	NOUN
ejpam-4979	36	25	side	side	NOUN
ejpam-4979	36	26	of	of	ADP
ejpam-4979	36	27	eq	eq	PROPN
ejpam-4979	36	28	.	.	PROPN
ejpam-4979	36	29	3	3	NUM
ejpam-4979	36	30	reveals	reveal	VERB
ejpam-4979	36	31	the	the	DET
ejpam-4979	36	32	alternating	alternate	VERB
ejpam-4979	36	33	sum	sum	NOUN
ejpam-4979	36	34	{	{	PUNCT
ejpam-4979	36	35	p(2	p(2	NOUN
ejpam-4979	36	36	)	)	PUNCT
ejpam-4979	36	37	}	}	PUNCT
ejpam-4979	36	38	−	−	PROPN
ejpam-4979	36	39	{	{	PUNCT
ejpam-4979	36	40	p(3	p(3	NOUN
ejpam-4979	36	41	)	)	PUNCT
ejpam-4979	36	42	}	}	PUNCT
ejpam-4979	37	1	+	+	CCONJ
ejpam-4979	37	2	{	{	PUNCT
ejpam-4979	37	3	p(4	p(4	NOUN
ejpam-4979	37	4	)	)	PUNCT
ejpam-4979	37	5	}	}	PUNCT
ejpam-4979	37	6	−	−	PROPN
ejpam-4979	37	7	{	{	PUNCT
ejpam-4979	37	8	p(5	p(5	NOUN
ejpam-4979	37	9	)	)	PUNCT
ejpam-4979	37	10	}	}	PUNCT
ejpam-4979	38	1	+	+	CCONJ
ejpam-4979	38	2	{	{	PUNCT
ejpam-4979	38	3	p(6	p(6	PROPN
ejpam-4979	38	4	)	)	PUNCT
ejpam-4979	38	5	}	}	PUNCT
ejpam-4979	38	6	−	−	PROPN
ejpam-4979	38	7	...	...	PUNCT
ejpam-4979	38	8	.	.	PUNCT
ejpam-4979	39	1	m.	m.	NOUN
ejpam-4979	39	2	p.	p.	NOUN
ejpam-4979	39	3	may	may	AUX
ejpam-4979	39	4	/	/	SYM
ejpam-4979	39	5	eur	eur	PROPN
ejpam-4979	39	6	.	.	PUNCT
ejpam-4979	40	1	j.	j.	PROPN
ejpam-4979	40	2	pure	pure	PROPN
ejpam-4979	40	3	appl	appl	PROPN
ejpam-4979	40	4	.	.	PROPN
ejpam-4979	40	5	math	math	PROPN
ejpam-4979	40	6	,	,	PUNCT
ejpam-4979	40	7	17	17	NUM
ejpam-4979	40	8	(	(	PUNCT
ejpam-4979	40	9	1	1	NUM
ejpam-4979	40	10	)	)	PUNCT
ejpam-4979	40	11	(	(	PUNCT
ejpam-4979	40	12	2024	2024	NUM
ejpam-4979	40	13	)	)	PUNCT
ejpam-4979	40	14	,	,	PUNCT
ejpam-4979	40	15	42	42	NUM
ejpam-4979	40	16	-	-	SYM
ejpam-4979	40	17	58	58	NUM
ejpam-4979	40	18	45	45	NUM
ejpam-4979	40	19	the	the	DET
ejpam-4979	40	20	prime	prime	ADJ
ejpam-4979	40	21	number	number	NOUN
ejpam-4979	40	22	subsequence	subsequence	NOUN
ejpam-4979	40	23	of	of	ADP
ejpam-4979	40	24	higher	high	ADJ
ejpam-4979	40	25	order	order	NOUN
ejpam-4979	40	26	p′′	p′′	PROPN
ejpam-4979	40	27	can	can	AUX
ejpam-4979	40	28	also	also	ADV
ejpam-4979	40	29	be	be	AUX
ejpam-4979	40	30	generated	generate	VERB
ejpam-4979	40	31	by	by	ADP
ejpam-4979	40	32	performing	perform	VERB
ejpam-4979	40	33	the	the	DET
ejpam-4979	40	34	aforementioned	aforementioned	ADJ
ejpam-4979	40	35	sieving	sieve	VERB
ejpam-4979	40	36	operation	operation	NOUN
ejpam-4979	40	37	on	on	ADP
ejpam-4979	40	38	the	the	DET
ejpam-4979	40	39	set	set	NOUN
ejpam-4979	40	40	of	of	ADP
ejpam-4979	40	41	all	all	DET
ejpam-4979	40	42	prime	prime	ADJ
ejpam-4979	40	43	numbers	number	NOUN
ejpam-4979	40	44	p	p	NOUN
ejpam-4979	40	45	,	,	PUNCT
ejpam-4979	40	46	similar	similar	ADJ
ejpam-4979	40	47	to	to	ADP
ejpam-4979	40	48	how	how	SCONJ
ejpam-4979	40	49	the	the	DET
ejpam-4979	40	50	primes	prime	NOUN
ejpam-4979	40	51	p′	p′	NOUN
ejpam-4979	40	52	were	be	AUX
ejpam-4979	40	53	sifted	sift	VERB
ejpam-4979	40	54	from	from	ADP
ejpam-4979	40	55	the	the	DET
ejpam-4979	40	56	set	set	NOUN
ejpam-4979	40	57	of	of	ADP
ejpam-4979	40	58	all	all	DET
ejpam-4979	40	59	natural	natural	ADJ
ejpam-4979	40	60	numbers	number	NOUN
ejpam-4979	40	61	n.	n.	PROPN
ejpam-4979	40	62	furthermore	furthermore	ADV
ejpam-4979	40	63	,	,	PUNCT
ejpam-4979	40	64	it	it	PRON
ejpam-4979	40	65	has	have	AUX
ejpam-4979	40	66	been	be	AUX
ejpam-4979	40	67	shown	show	VERB
ejpam-4979	40	68	[	[	PUNCT
ejpam-4979	40	69	7	7	X
ejpam-4979	40	70	]	]	PUNCT
ejpam-4979	40	71	that	that	SCONJ
ejpam-4979	40	72	the	the	DET
ejpam-4979	40	73	subsequences	subsequence	NOUN
ejpam-4979	40	74	p′	p′	NOUN
ejpam-4979	40	75	and	and	CCONJ
ejpam-4979	40	76	p′′	p′′	PROPN
ejpam-4979	40	77	,	,	PUNCT
ejpam-4979	40	78	when	when	SCONJ
ejpam-4979	40	79	added	add	VERB
ejpam-4979	40	80	together	together	ADV
ejpam-4979	40	81	,	,	PUNCT
ejpam-4979	40	82	form	form	VERB
ejpam-4979	40	83	the	the	DET
ejpam-4979	40	84	entire	entire	ADJ
ejpam-4979	40	85	set	set	NOUN
ejpam-4979	40	86	of	of	ADP
ejpam-4979	40	87	prime	prime	ADJ
ejpam-4979	40	88	numbers	number	NOUN
ejpam-4979	40	89	p	p	X
ejpam-4979	40	90	:	:	PUNCT
ejpam-4979	40	91	p	p	X
ejpam-4979	40	92	=	=	X
ejpam-4979	40	93	p′	p′	NOUN
ejpam-4979	41	1	+	+	CCONJ
ejpam-4979	41	2	p′′.	p′′.	PROPN
ejpam-4979	41	3	(	(	PUNCT
ejpam-4979	41	4	4	4	NUM
ejpam-4979	41	5	)	)	PUNCT
ejpam-4979	41	6	we	we	PRON
ejpam-4979	41	7	sketch	sketch	VERB
ejpam-4979	41	8	a	a	DET
ejpam-4979	41	9	proof	proof	NOUN
ejpam-4979	41	10	of	of	ADP
ejpam-4979	41	11	eq	eq	PROPN
ejpam-4979	41	12	.	.	PROPN
ejpam-4979	41	13	4	4	NUM
ejpam-4979	41	14	here	here	ADV
ejpam-4979	41	15	:	:	PUNCT
ejpam-4979	41	16	proof	proof	NOUN
ejpam-4979	41	17	.	.	PUNCT
ejpam-4979	42	1	it	it	PRON
ejpam-4979	42	2	has	have	AUX
ejpam-4979	42	3	been	be	AUX
ejpam-4979	42	4	shown	show	VERB
ejpam-4979	42	5	[	[	PUNCT
ejpam-4979	42	6	7	7	X
ejpam-4979	42	7	]	]	PUNCT
ejpam-4979	42	8	that	that	DET
ejpam-4979	42	9	p′	p′	NOUN
ejpam-4979	42	10	=	=	SYM
ejpam-4979	42	11	{	{	PUNCT
ejpam-4979	42	12	(	(	PUNCT
ejpam-4979	42	13	−1)n−1	−1)n−1	PRON
ejpam-4979	42	14	{	{	PUNCT
ejpam-4979	42	15	p(n	p(n	PROPN
ejpam-4979	42	16	)	)	PUNCT
ejpam-4979	42	17	}	}	PUNCT
ejpam-4979	42	18	}	}	PUNCT
ejpam-4979	42	19	∞	∞	NUM
ejpam-4979	42	20	n=1	n=1	PROPN
ejpam-4979	42	21	=	=	PRON
ejpam-4979	42	22	{	{	PUNCT
ejpam-4979	42	23	p(1	p(1	NOUN
ejpam-4979	42	24	)	)	PUNCT
ejpam-4979	42	25	}	}	PUNCT
ejpam-4979	42	26	−	−	PROPN
ejpam-4979	42	27	{	{	PUNCT
ejpam-4979	42	28	p(2	p(2	NOUN
ejpam-4979	42	29	)	)	PUNCT
ejpam-4979	42	30	}	}	PUNCT
ejpam-4979	43	1	+	+	CCONJ
ejpam-4979	43	2	{	{	PUNCT
ejpam-4979	43	3	p(3	p(3	NOUN
ejpam-4979	43	4	)	)	PUNCT
ejpam-4979	43	5	}	}	PUNCT
ejpam-4979	43	6	−	−	PROPN
ejpam-4979	43	7	...	...	PUNCT
ejpam-4979	43	8	and	and	CCONJ
ejpam-4979	43	9	p′′	p′′	PROPN
ejpam-4979	43	10	=	=	PRON
ejpam-4979	43	11	{	{	PUNCT
ejpam-4979	43	12	(	(	PUNCT
ejpam-4979	43	13	−1)n	−1)n	X
ejpam-4979	43	14	{	{	PUNCT
ejpam-4979	43	15	p(n	p(n	PROPN
ejpam-4979	43	16	)	)	PUNCT
ejpam-4979	43	17	}	}	PUNCT
ejpam-4979	43	18	}	}	PUNCT
ejpam-4979	43	19	∞	∞	X
ejpam-4979	43	20	n=2	n=2	X
ejpam-4979	43	21	=	=	NOUN
ejpam-4979	43	22	{	{	PUNCT
ejpam-4979	43	23	p(2	p(2	NOUN
ejpam-4979	43	24	)	)	PUNCT
ejpam-4979	43	25	}	}	PUNCT
ejpam-4979	43	26	−	−	PROPN
ejpam-4979	43	27	{	{	PUNCT
ejpam-4979	43	28	p(3	p(3	NOUN
ejpam-4979	43	29	)	)	PUNCT
ejpam-4979	43	30	}	}	PUNCT
ejpam-4979	44	1	+	+	CCONJ
ejpam-4979	44	2	{	{	PUNCT
ejpam-4979	44	3	p(4	p(4	NOUN
ejpam-4979	44	4	)	)	PUNCT
ejpam-4979	44	5	}	}	PUNCT
ejpam-4979	44	6	−	−	PROPN
ejpam-4979	44	7	...	...	PUNCT
ejpam-4979	44	8	.	.	PUNCT
ejpam-4979	45	1	therefore	therefore	ADV
ejpam-4979	45	2	,	,	PUNCT
ejpam-4979	45	3	p′	p′	NOUN
ejpam-4979	45	4	+	+	CCONJ
ejpam-4979	45	5	p′′	p′′	PROPN
ejpam-4979	45	6	=	=	SYM
ejpam-4979	45	7	{	{	PUNCT
ejpam-4979	45	8	p(1	p(1	NOUN
ejpam-4979	45	9	)	)	PUNCT
ejpam-4979	45	10	}	}	PUNCT
ejpam-4979	45	11	−	−	PROPN
ejpam-4979	45	12	{	{	PUNCT
ejpam-4979	45	13	p(2	p(2	NOUN
ejpam-4979	45	14	)	)	PUNCT
ejpam-4979	45	15	}	}	PUNCT
ejpam-4979	46	1	+	+	CCONJ
ejpam-4979	46	2	{	{	PUNCT
ejpam-4979	46	3	p(3	p(3	NOUN
ejpam-4979	46	4	)	)	PUNCT
ejpam-4979	46	5	}	}	PUNCT
ejpam-4979	46	6	−	−	PROPN
ejpam-4979	46	7	...	...	PUNCT
ejpam-4979	47	1	+	+	CCONJ
ejpam-4979	47	2	{	{	PUNCT
ejpam-4979	47	3	p(2	p(2	NOUN
ejpam-4979	47	4	)	)	PUNCT
ejpam-4979	47	5	}	}	PUNCT
ejpam-4979	47	6	−	−	PROPN
ejpam-4979	47	7	{	{	PUNCT
ejpam-4979	47	8	p(3	p(3	NOUN
ejpam-4979	47	9	)	)	PUNCT
ejpam-4979	47	10	}	}	PUNCT
ejpam-4979	48	1	+	+	CCONJ
ejpam-4979	48	2	{	{	PUNCT
ejpam-4979	48	3	p(4	p(4	NOUN
ejpam-4979	48	4	)	)	PUNCT
ejpam-4979	48	5	}	}	PUNCT
ejpam-4979	48	6	−	−	PROPN
ejpam-4979	48	7	...	...	PUNCT
ejpam-4979	49	1	=	=	PRON
ejpam-4979	49	2	{	{	PUNCT
ejpam-4979	49	3	p(1	p(1	NOUN
ejpam-4979	49	4	)	)	PUNCT
ejpam-4979	49	5	}	}	PUNCT
ejpam-4979	49	6	=	=	PUNCT
ejpam-4979	50	1	p.	p.	NOUN
ejpam-4979	50	2	an	an	DET
ejpam-4979	50	3	interesting	interesting	ADJ
ejpam-4979	50	4	property	property	NOUN
ejpam-4979	50	5	was	be	AUX
ejpam-4979	50	6	observed	observe	VERB
ejpam-4979	50	7	in	in	ADP
ejpam-4979	50	8	the	the	DET
ejpam-4979	50	9	relationship	relationship	NOUN
ejpam-4979	50	10	between	between	ADP
ejpam-4979	50	11	the	the	DET
ejpam-4979	50	12	set	set	NOUN
ejpam-4979	50	13	of	of	ADP
ejpam-4979	50	14	all	all	DET
ejpam-4979	50	15	prime	prime	ADJ
ejpam-4979	50	16	numbers	number	NOUN
ejpam-4979	50	17	p	p	NOUN
ejpam-4979	50	18	and	and	CCONJ
ejpam-4979	50	19	the	the	DET
ejpam-4979	50	20	complementary	complementary	ADJ
ejpam-4979	50	21	prime	prime	ADJ
ejpam-4979	50	22	number	number	NOUN
ejpam-4979	50	23	sets	set	VERB
ejpam-4979	50	24	p′	p′	NOUN
ejpam-4979	50	25	and	and	CCONJ
ejpam-4979	50	26	p′′.	p′′.	PROPN
ejpam-4979	50	27	since	since	SCONJ
ejpam-4979	50	28	p′′	p′′	PROPN
ejpam-4979	50	29	=	=	PROPN
ejpam-4979	50	30	pp′	pp′	PROPN
ejpam-4979	50	31	,	,	PUNCT
ejpam-4979	50	32	eq	eq	NOUN
ejpam-4979	50	33	.	.	PROPN
ejpam-4979	50	34	4	4	NUM
ejpam-4979	50	35	can	can	AUX
ejpam-4979	50	36	be	be	AUX
ejpam-4979	50	37	rewritten	rewrite	VERB
ejpam-4979	50	38	as	as	ADP
ejpam-4979	50	39	p′′=	p′′=	NOUN
ejpam-4979	50	40	p−	p−	X
ejpam-4979	50	41	{	{	PUNCT
ejpam-4979	50	42	2	2	NUM
ejpam-4979	50	43	,	,	PUNCT
ejpam-4979	50	44	5	5	NUM
ejpam-4979	50	45	,	,	PUNCT
ejpam-4979	50	46	7	7	NUM
ejpam-4979	50	47	,	,	PUNCT
ejpam-4979	50	48	13	13	NUM
ejpam-4979	50	49	,	,	PUNCT
ejpam-4979	50	50	19	19	NUM
ejpam-4979	50	51	,	,	PUNCT
ejpam-4979	50	52	23	23	NUM
ejpam-4979	50	53	,	,	PUNCT
ejpam-4979	50	54	29	29	NUM
ejpam-4979	50	55	,	,	PUNCT
ejpam-4979	50	56	...	...	PUNCT
ejpam-4979	50	57	}	}	PUNCT
ejpam-4979	50	58	=	=	PRON
ejpam-4979	50	59	{	{	PUNCT
ejpam-4979	51	1	p	p	NOUN
ejpam-4979	51	2	2	2	NUM
ejpam-4979	51	3	,	,	PUNCT
ejpam-4979	51	4	p	p	NOUN
ejpam-4979	51	5	5	5	NUM
ejpam-4979	51	6	,	,	PUNCT
ejpam-4979	51	7	p	p	NOUN
ejpam-4979	51	8	7	7	NUM
ejpam-4979	51	9	,	,	PUNCT
ejpam-4979	51	10	p	p	NOUN
ejpam-4979	51	11	13	13	NUM
ejpam-4979	51	12	,	,	PUNCT
ejpam-4979	51	13	p	p	NOUN
ejpam-4979	51	14	19	19	NUM
ejpam-4979	51	15	,	,	PUNCT
ejpam-4979	51	16	p	p	NOUN
ejpam-4979	51	17	23	23	NUM
ejpam-4979	51	18	,	,	PUNCT
ejpam-4979	51	19	p	p	NOUN
ejpam-4979	51	20	29	29	NUM
ejpam-4979	51	21	,	,	PUNCT
ejpam-4979	51	22	...	...	PUNCT
ejpam-4979	51	23	}	}	PUNCT
ejpam-4979	52	1	=	=	PUNCT
ejpam-4979	52	2	pp′	pp′	NOUN
ejpam-4979	52	3	where	where	SCONJ
ejpam-4979	52	4	the	the	DET
ejpam-4979	52	5	prime	prime	ADJ
ejpam-4979	52	6	numbers	number	NOUN
ejpam-4979	52	7	of	of	ADP
ejpam-4979	52	8	the	the	DET
ejpam-4979	52	9	subsequence	subsequence	NOUN
ejpam-4979	52	10	p′	p′	NOUN
ejpam-4979	52	11	form	form	VERB
ejpam-4979	52	12	the	the	DET
ejpam-4979	52	13	indexes	index	NOUN
ejpam-4979	52	14	for	for	ADP
ejpam-4979	52	15	the	the	DET
ejpam-4979	52	16	complement	complement	NOUN
ejpam-4979	52	17	set	set	NOUN
ejpam-4979	52	18	of	of	ADP
ejpam-4979	52	19	primes	prime	NOUN
ejpam-4979	52	20	p′′	p′′	PROPN
ejpam-4979	52	21	such	such	ADJ
ejpam-4979	52	22	that	that	SCONJ
ejpam-4979	52	23	p′′	p′′	PROPN
ejpam-4979	52	24	=	=	PROPN
ejpam-4979	52	25	pp′	pp′	NOUN
ejpam-4979	52	26	=	=	SYM
ejpam-4979	52	27	{	{	PUNCT
ejpam-4979	52	28	pk	pk	NOUN
ejpam-4979	52	29	|	|	ADV
ejpam-4979	52	30	k	k	PROPN
ejpam-4979	52	31	∈	∈	PROPN
ejpam-4979	52	32	p′	p′	PROPN
ejpam-4979	52	33	}	}	PUNCT
ejpam-4979	52	34	.	.	PUNCT
ejpam-4979	53	1	m.	m.	NOUN
ejpam-4979	53	2	p.	p.	NOUN
ejpam-4979	53	3	may	may	AUX
ejpam-4979	53	4	/	/	SYM
ejpam-4979	53	5	eur	eur	PROPN
ejpam-4979	53	6	.	.	PUNCT
ejpam-4979	54	1	j.	j.	PROPN
ejpam-4979	54	2	pure	pure	PROPN
ejpam-4979	54	3	appl	appl	PROPN
ejpam-4979	54	4	.	.	PROPN
ejpam-4979	54	5	math	math	PROPN
ejpam-4979	54	6	,	,	PUNCT
ejpam-4979	54	7	17	17	NUM
ejpam-4979	54	8	(	(	PUNCT
ejpam-4979	54	9	1	1	NUM
ejpam-4979	54	10	)	)	PUNCT
ejpam-4979	54	11	(	(	PUNCT
ejpam-4979	54	12	2024	2024	NUM
ejpam-4979	54	13	)	)	PUNCT
ejpam-4979	54	14	,	,	PUNCT
ejpam-4979	54	15	42	42	NUM
ejpam-4979	54	16	-	-	SYM
ejpam-4979	54	17	58	58	NUM
ejpam-4979	54	18	46	46	NUM
ejpam-4979	54	19	2	2	NUM
ejpam-4979	54	20	.	.	PUNCT
ejpam-4979	55	1	asymptotic	asymptotic	ADJ
ejpam-4979	55	2	densities	density	NOUN
ejpam-4979	55	3	of	of	ADP
ejpam-4979	55	4	p′	p′	NOUN
ejpam-4979	55	5	and	and	CCONJ
ejpam-4979	55	6	p′′	p′′	PROPN
ejpam-4979	55	7	we	we	PRON
ejpam-4979	55	8	will	will	AUX
ejpam-4979	55	9	now	now	ADV
ejpam-4979	55	10	derive	derive	VERB
ejpam-4979	55	11	the	the	DET
ejpam-4979	55	12	asymptotic	asymptotic	ADJ
ejpam-4979	55	13	densities	density	NOUN
ejpam-4979	55	14	for	for	ADP
ejpam-4979	55	15	the	the	DET
ejpam-4979	55	16	prime	prime	ADJ
ejpam-4979	55	17	number	number	NOUN
ejpam-4979	55	18	subsequences	subsequence	VERB
ejpam-4979	55	19	p′	p′	NOUN
ejpam-4979	55	20	and	and	CCONJ
ejpam-4979	55	21	p′′	p′′	PROPN
ejpam-4979	55	22	assuming	assume	VERB
ejpam-4979	55	23	that	that	SCONJ
ejpam-4979	55	24	1/	1/	NUM
ejpam-4979	55	25	lnn	lnn	NOUN
ejpam-4979	55	26	is	be	AUX
ejpam-4979	55	27	the	the	DET
ejpam-4979	55	28	asymptotic	asymptotic	ADJ
ejpam-4979	55	29	density	density	NOUN
ejpam-4979	55	30	of	of	ADP
ejpam-4979	55	31	the	the	DET
ejpam-4979	55	32	set	set	NOUN
ejpam-4979	55	33	of	of	ADP
ejpam-4979	55	34	all	all	DET
ejpam-4979	55	35	prime	prime	ADJ
ejpam-4979	55	36	numbers	number	NOUN
ejpam-4979	55	37	p	p	NOUN
ejpam-4979	55	38	as	as	ADP
ejpam-4979	55	39	n	n	PROPN
ejpam-4979	55	40	→	→	SYM
ejpam-4979	55	41	∞.	∞.	PROPN
ejpam-4979	55	42	we	we	PRON
ejpam-4979	55	43	approach	approach	VERB
ejpam-4979	55	44	this	this	DET
ejpam-4979	55	45	task	task	NOUN
ejpam-4979	55	46	by	by	ADP
ejpam-4979	55	47	alternately	alternately	ADV
ejpam-4979	55	48	adding	add	VERB
ejpam-4979	55	49	and	and	CCONJ
ejpam-4979	55	50	subtracting	subtract	VERB
ejpam-4979	55	51	the	the	DET
ejpam-4979	55	52	prime	prime	ADJ
ejpam-4979	55	53	number	number	NOUN
ejpam-4979	55	54	densities	density	NOUN
ejpam-4979	55	55	(	(	PUNCT
ejpam-4979	55	56	or	or	CCONJ
ejpam-4979	55	57	“	"	PUNCT
ejpam-4979	55	58	probabilities	probability	NOUN
ejpam-4979	55	59	”	"	PUNCT
ejpam-4979	55	60	)	)	PUNCT
ejpam-4979	55	61	of	of	ADP
ejpam-4979	55	62	the	the	DET
ejpam-4979	55	63	prime	prime	ADJ
ejpam-4979	55	64	number	number	NOUN
ejpam-4979	55	65	subsequences	subsequence	NOUN
ejpam-4979	55	66	of	of	ADP
ejpam-4979	55	67	increasing	increase	VERB
ejpam-4979	55	68	order	order	NOUN
ejpam-4979	55	69	,	,	PUNCT
ejpam-4979	55	70	also	also	ADV
ejpam-4979	55	71	known	know	VERB
ejpam-4979	55	72	as	as	ADP
ejpam-4979	55	73	”	"	PUNCT
ejpam-4979	55	74	superprimes	superprime	NOUN
ejpam-4979	55	75	”	"	PUNCT
ejpam-4979	56	1	[	[	X
ejpam-4979	56	2	1	1	NUM
ejpam-4979	56	3	]	]	PUNCT
ejpam-4979	56	4	,	,	PUNCT
ejpam-4979	56	5	to	to	PART
ejpam-4979	56	6	arrive	arrive	VERB
ejpam-4979	56	7	at	at	ADP
ejpam-4979	56	8	values	value	NOUN
ejpam-4979	56	9	for	for	ADP
ejpam-4979	56	10	the	the	DET
ejpam-4979	56	11	asymptotic	asymptotic	ADJ
ejpam-4979	56	12	densities	density	NOUN
ejpam-4979	56	13	for	for	ADP
ejpam-4979	56	14	p′	p′	NOUN
ejpam-4979	57	1	and	and	CCONJ
ejpam-4979	57	2	p′′.	p′′.	PRON
ejpam-4979	57	3	we	we	PRON
ejpam-4979	57	4	begin	begin	VERB
ejpam-4979	57	5	by	by	ADP
ejpam-4979	57	6	recalling	recall	VERB
ejpam-4979	57	7	[	[	X
ejpam-4979	57	8	7	7	NUM
ejpam-4979	57	9	]	]	PUNCT
ejpam-4979	57	10	that	that	SCONJ
ejpam-4979	57	11	the	the	DET
ejpam-4979	57	12	prime	prime	ADJ
ejpam-4979	57	13	number	number	NOUN
ejpam-4979	57	14	subsequence	subsequence	NOUN
ejpam-4979	57	15	p′	p′	NOUN
ejpam-4979	57	16	is	be	AUX
ejpam-4979	57	17	formed	form	VERB
ejpam-4979	57	18	by	by	ADP
ejpam-4979	57	19	the	the	DET
ejpam-4979	57	20	alternating	alternate	VERB
ejpam-4979	57	21	series	series	NOUN
ejpam-4979	57	22	p′	p′	NOUN
ejpam-4979	57	23	=	=	SYM
ejpam-4979	57	24	{	{	PUNCT
ejpam-4979	57	25	(	(	PUNCT
ejpam-4979	57	26	−1)n−1	−1)n−1	PRON
ejpam-4979	57	27	{	{	PUNCT
ejpam-4979	57	28	p(n	p(n	PROPN
ejpam-4979	57	29	)	)	PUNCT
ejpam-4979	57	30	}	}	PUNCT
ejpam-4979	57	31	}	}	PUNCT
ejpam-4979	57	32	∞	∞	NUM
ejpam-4979	57	33	n=1	n=1	PROPN
ejpam-4979	57	34	=	=	PRON
ejpam-4979	57	35	{	{	PUNCT
ejpam-4979	57	36	p(1	p(1	NOUN
ejpam-4979	57	37	)	)	PUNCT
ejpam-4979	57	38	}	}	PUNCT
ejpam-4979	57	39	−	−	PROPN
ejpam-4979	57	40	{	{	PUNCT
ejpam-4979	57	41	p(2	p(2	NOUN
ejpam-4979	57	42	)	)	PUNCT
ejpam-4979	57	43	}	}	PUNCT
ejpam-4979	58	1	+	+	CCONJ
ejpam-4979	58	2	{	{	PUNCT
ejpam-4979	58	3	p(3	p(3	NOUN
ejpam-4979	58	4	)	)	PUNCT
ejpam-4979	58	5	}	}	PUNCT
ejpam-4979	58	6	−	−	PROPN
ejpam-4979	58	7	...	...	PUNCT
ejpam-4979	59	1	where	where	SCONJ
ejpam-4979	59	2	{	{	PUNCT
ejpam-4979	59	3	p(k	p(k	NOUN
ejpam-4979	59	4	)	)	PUNCT
ejpam-4979	59	5	}	}	PUNCT
ejpam-4979	59	6	=	=	PRON
ejpam-4979	59	7	{	{	PUNCT
ejpam-4979	59	8	pp	pp	ADV
ejpam-4979	59	9	...	...	PUNCT
ejpam-4979	59	10	pn	pn	NOUN
ejpam-4979	59	11	}	}	PUNCT
ejpam-4979	59	12	(	(	PUNCT
ejpam-4979	59	13	p	p	NOUN
ejpam-4979	59	14	“	"	PUNCT
ejpam-4979	59	15	k	k	ADJ
ejpam-4979	59	16	”	"	PUNCT
ejpam-4979	59	17	times	time	NOUN
ejpam-4979	59	18	)	)	PUNCT
ejpam-4979	59	19	.	.	PUNCT
ejpam-4979	60	1	broughan	broughan	PROPN
ejpam-4979	60	2	and	and	CCONJ
ejpam-4979	60	3	barnett	barnett	PROPN
ejpam-4979	60	4	have	have	AUX
ejpam-4979	60	5	shown	show	VERB
ejpam-4979	60	6	[	[	PUNCT
ejpam-4979	60	7	1	1	X
ejpam-4979	60	8	]	]	PUNCT
ejpam-4979	60	9	that	that	SCONJ
ejpam-4979	60	10	for	for	ADP
ejpam-4979	60	11	the	the	DET
ejpam-4979	60	12	general	general	ADJ
ejpam-4979	60	13	case	case	NOUN
ejpam-4979	60	14	of	of	ADP
ejpam-4979	60	15	higher	high	ADJ
ejpam-4979	60	16	-	-	PUNCT
ejpam-4979	60	17	order	order	NOUN
ejpam-4979	60	18	“	"	PUNCT
ejpam-4979	60	19	superprimes	superprime	NOUN
ejpam-4979	60	20	”	"	PUNCT
ejpam-4979	60	21	pp	pp	ADJ
ejpam-4979	60	22	...	...	PUNCT
ejpam-4979	60	23	pk	pk	NOUN
ejpam-4979	60	24	,	,	PUNCT
ejpam-4979	60	25	the	the	DET
ejpam-4979	60	26	asymptotic	asymptotic	ADJ
ejpam-4979	60	27	density	density	NOUN
ejpam-4979	60	28	is	be	AUX
ejpam-4979	60	29	approximately	approximately	ADV
ejpam-4979	60	30	n	n	ADV
ejpam-4979	60	31	pp	pp	ADV
ejpam-4979	60	32	...	...	PUNCT
ejpam-4979	60	33	pn	pn	VERB
ejpam-4979	60	34	∼	∼	NOUN
ejpam-4979	60	35	n	n	CCONJ
ejpam-4979	60	36	n	n	CCONJ
ejpam-4979	60	37	(	(	PUNCT
ejpam-4979	60	38	lnn)k	lnn)k	PROPN
ejpam-4979	60	39	∼	∼	NOUN
ejpam-4979	60	40	1	1	NUM
ejpam-4979	60	41	(	(	PUNCT
ejpam-4979	60	42	lnn)k	lnn)k	PROPN
ejpam-4979	60	43	for	for	ADP
ejpam-4979	60	44	large	large	ADJ
ejpam-4979	60	45	n	n	CCONJ
ejpam-4979	60	46	∈	∈	PROPN
ejpam-4979	60	47	n.	n.	NOUN
ejpam-4979	60	48	now	now	ADV
ejpam-4979	60	49	,	,	PUNCT
ejpam-4979	60	50	assuming	assume	VERB
ejpam-4979	60	51	that	that	SCONJ
ejpam-4979	60	52	1/	1/	NUM
ejpam-4979	60	53	lnn	lnn	NOUN
ejpam-4979	60	54	is	be	AUX
ejpam-4979	60	55	the	the	DET
ejpam-4979	60	56	asymptotic	asymptotic	ADJ
ejpam-4979	60	57	density	density	NOUN
ejpam-4979	60	58	for	for	ADP
ejpam-4979	60	59	the	the	DET
ejpam-4979	60	60	set	set	NOUN
ejpam-4979	60	61	of	of	ADP
ejpam-4979	60	62	all	all	DET
ejpam-4979	60	63	prime	prime	ADJ
ejpam-4979	60	64	numbers	number	NOUN
ejpam-4979	60	65	p	p	X
ejpam-4979	60	66	,	,	PUNCT
ejpam-4979	60	67	we	we	PRON
ejpam-4979	60	68	derive	derive	VERB
ejpam-4979	60	69	an	an	DET
ejpam-4979	60	70	expression	expression	NOUN
ejpam-4979	60	71	for	for	ADP
ejpam-4979	60	72	the	the	DET
ejpam-4979	60	73	density	density	NOUN
ejpam-4979	60	74	d′	d′	NUM
ejpam-4979	60	75	for	for	ADP
ejpam-4979	60	76	the	the	DET
ejpam-4979	60	77	prime	prime	ADJ
ejpam-4979	60	78	number	number	NOUN
ejpam-4979	60	79	subsequence	subsequence	VERB
ejpam-4979	60	80	p′	p′	NOUN
ejpam-4979	60	81	at	at	ADP
ejpam-4979	60	82	∞.	∞.	PROPN
ejpam-4979	60	83	we	we	PRON
ejpam-4979	60	84	begin	begin	VERB
ejpam-4979	60	85	with	with	ADP
ejpam-4979	60	86	the	the	DET
ejpam-4979	60	87	geometric	geometric	ADJ
ejpam-4979	60	88	series	series	NOUN
ejpam-4979	60	89	s	s	PART
ejpam-4979	60	90	=	=	SYM
ejpam-4979	60	91	1−	1−	NUM
ejpam-4979	60	92	x+	x+	NUM
ejpam-4979	61	1	x2	x2	INTJ
ejpam-4979	62	1	−	−	NOUN
ejpam-4979	63	1	x3	x3	PROPN
ejpam-4979	63	2	+	+	CCONJ
ejpam-4979	63	3	x4	x4	PROPN
ejpam-4979	63	4	−	−	PROPN
ejpam-4979	63	5	x5	x5	PROPN
ejpam-4979	63	6	+	+	CCONJ
ejpam-4979	63	7	...	...	PUNCT
ejpam-4979	64	1	=	=	SYM
ejpam-4979	64	2	1	1	NUM
ejpam-4979	64	3	1	1	NUM
ejpam-4979	64	4	+	+	CCONJ
ejpam-4979	64	5	x	x	SYM
ejpam-4979	64	6	(	(	PUNCT
ejpam-4979	64	7	|x|	|x|	X
ejpam-4979	64	8	<	<	X
ejpam-4979	64	9	1	1	NUM
ejpam-4979	64	10	)	)	PUNCT
ejpam-4979	64	11	.	.	PUNCT
ejpam-4979	65	1	then	then	ADV
ejpam-4979	65	2	let	let	VERB
ejpam-4979	65	3	t	t	NOUN
ejpam-4979	65	4	′	′	NUM
ejpam-4979	65	5	=	=	PUNCT
ejpam-4979	66	1	1−	1−	NUM
ejpam-4979	66	2	s	s	VERB
ejpam-4979	66	3	so	so	SCONJ
ejpam-4979	66	4	that	that	SCONJ
ejpam-4979	66	5	t	t	NOUN
ejpam-4979	67	1	′	′	NUM
ejpam-4979	68	1	=	=	PUNCT
ejpam-4979	68	2	x−	x−	PROPN
ejpam-4979	68	3	x2	x2	PROPN
ejpam-4979	69	1	+	+	CCONJ
ejpam-4979	69	2	x3	x3	ADJ
ejpam-4979	69	3	−	−	PROPN
ejpam-4979	70	1	x4	x4	PROPN
ejpam-4979	70	2	+	+	NUM
ejpam-4979	70	3	x5	x5	NOUN
ejpam-4979	70	4	+	+	CCONJ
ejpam-4979	70	5	...	...	PUNCT
ejpam-4979	71	1	=	=	SYM
ejpam-4979	71	2	−1	−1	NOUN
ejpam-4979	71	3	1	1	NUM
ejpam-4979	71	4	+	+	CCONJ
ejpam-4979	71	5	x	x	SYM
ejpam-4979	71	6	+	+	NUM
ejpam-4979	71	7	1	1	NUM
ejpam-4979	71	8	=	=	SYM
ejpam-4979	71	9	x	x	SYM
ejpam-4979	71	10	1	1	NUM
ejpam-4979	71	11	+	+	CCONJ
ejpam-4979	71	12	x	x	X
ejpam-4979	71	13	.	.	PUNCT
ejpam-4979	72	1	now	now	ADV
ejpam-4979	72	2	substitute	substitute	VERB
ejpam-4979	72	3	1	1	NUM
ejpam-4979	72	4	lnn	lnn	NOUN
ejpam-4979	72	5	for	for	SCONJ
ejpam-4979	72	6	x	x	PART
ejpam-4979	72	7	to	to	PART
ejpam-4979	72	8	get	get	VERB
ejpam-4979	72	9	m.	m.	NOUN
ejpam-4979	72	10	p.	p.	NOUN
ejpam-4979	72	11	may	may	AUX
ejpam-4979	72	12	/	/	SYM
ejpam-4979	72	13	eur	eur	PROPN
ejpam-4979	72	14	.	.	PUNCT
ejpam-4979	73	1	j.	j.	PROPN
ejpam-4979	73	2	pure	pure	PROPN
ejpam-4979	73	3	appl	appl	PROPN
ejpam-4979	73	4	.	.	PROPN
ejpam-4979	73	5	math	math	PROPN
ejpam-4979	73	6	,	,	PUNCT
ejpam-4979	73	7	17	17	NUM
ejpam-4979	73	8	(	(	PUNCT
ejpam-4979	73	9	1	1	NUM
ejpam-4979	73	10	)	)	PUNCT
ejpam-4979	73	11	(	(	PUNCT
ejpam-4979	73	12	2024	2024	NUM
ejpam-4979	73	13	)	)	PUNCT
ejpam-4979	73	14	,	,	PUNCT
ejpam-4979	73	15	42	42	NUM
ejpam-4979	73	16	-	-	SYM
ejpam-4979	73	17	58	58	NUM
ejpam-4979	73	18	47	47	NUM
ejpam-4979	73	19	1	1	NUM
ejpam-4979	73	20	lnn	lnn	NOUN
ejpam-4979	73	21	1	1	NUM
ejpam-4979	73	22	+	+	SYM
ejpam-4979	73	23	1	1	NUM
ejpam-4979	73	24	lnn	lnn	NOUN
ejpam-4979	73	25	=	=	SYM
ejpam-4979	73	26	1	1	NUM
ejpam-4979	73	27	lnn+	lnn+	NOUN
ejpam-4979	73	28	1	1	NUM
ejpam-4979	73	29	so	so	SCONJ
ejpam-4979	73	30	that	that	SCONJ
ejpam-4979	73	31	we	we	PRON
ejpam-4979	73	32	have	have	AUX
ejpam-4979	73	33	d′≈	d′≈	VERB
ejpam-4979	73	34	1	1	NUM
ejpam-4979	73	35	lnn	lnn	NOUN
ejpam-4979	73	36	−	−	NOUN
ejpam-4979	73	37	1	1	NUM
ejpam-4979	73	38	(	(	PUNCT
ejpam-4979	73	39	lnn)2	lnn)2	NOUN
ejpam-4979	73	40	+	+	CCONJ
ejpam-4979	73	41	1	1	NUM
ejpam-4979	73	42	(	(	PUNCT
ejpam-4979	73	43	lnn)3	lnn)3	NOUN
ejpam-4979	73	44	−	−	PROPN
ejpam-4979	73	45	1	1	NUM
ejpam-4979	73	46	(	(	PUNCT
ejpam-4979	73	47	lnn)4	lnn)4	PROPN
ejpam-4979	73	48	+	+	NUM
ejpam-4979	73	49	...	...	PUNCT
ejpam-4979	73	50	(	(	PUNCT
ejpam-4979	73	51	5	5	X
ejpam-4979	73	52	)	)	PUNCT
ejpam-4979	73	53	=	=	SYM
ejpam-4979	73	54	1	1	NUM
ejpam-4979	73	55	lnn+	lnn+	NOUN
ejpam-4979	73	56	1	1	NUM
ejpam-4979	73	57	.	.	PUNCT
ejpam-4979	74	1	(	(	PUNCT
ejpam-4979	74	2	6	6	NUM
ejpam-4979	74	3	)	)	PUNCT
ejpam-4979	74	4	similarly	similarly	ADV
ejpam-4979	74	5	,	,	PUNCT
ejpam-4979	74	6	we	we	PRON
ejpam-4979	74	7	derive	derive	VERB
ejpam-4979	74	8	the	the	DET
ejpam-4979	74	9	asymptotic	asymptotic	ADJ
ejpam-4979	74	10	density	density	NOUN
ejpam-4979	74	11	for	for	ADP
ejpam-4979	74	12	the	the	DET
ejpam-4979	74	13	prime	prime	ADJ
ejpam-4979	74	14	number	number	NOUN
ejpam-4979	74	15	subsequence	subsequence	NOUN
ejpam-4979	74	16	p′′.	p′′.	PRON
ejpam-4979	74	17	when	when	SCONJ
ejpam-4979	74	18	we	we	PRON
ejpam-4979	74	19	set	set	VERB
ejpam-4979	74	20	t	t	NOUN
ejpam-4979	75	1	′′	′′	PROPN
ejpam-4979	75	2	=	=	SYM
ejpam-4979	75	3	s	s	PART
ejpam-4979	75	4	−	−	X
ejpam-4979	75	5	(	(	PUNCT
ejpam-4979	75	6	1−	1−	NUM
ejpam-4979	75	7	x	x	X
ejpam-4979	75	8	)	)	PUNCT
ejpam-4979	75	9	we	we	PRON
ejpam-4979	75	10	have	have	VERB
ejpam-4979	75	11	t	t	NOUN
ejpam-4979	75	12	′′	′′	PROPN
ejpam-4979	75	13	=	=	SYM
ejpam-4979	75	14	s	s	PART
ejpam-4979	75	15	−	−	X
ejpam-4979	75	16	(	(	PUNCT
ejpam-4979	75	17	1−	1−	NUM
ejpam-4979	75	18	x	x	NOUN
ejpam-4979	75	19	)	)	PUNCT
ejpam-4979	76	1	=	=	SYM
ejpam-4979	77	1	x2	x2	PROPN
ejpam-4979	78	1	−	−	NOUN
ejpam-4979	79	1	x3	x3	PROPN
ejpam-4979	79	2	+	+	CCONJ
ejpam-4979	79	3	x4	x4	PROPN
ejpam-4979	79	4	−	−	PROPN
ejpam-4979	79	5	x5	x5	PROPN
ejpam-4979	79	6	+	+	CCONJ
ejpam-4979	79	7	...	...	PUNCT
ejpam-4979	80	1	=	=	SYM
ejpam-4979	80	2	1	1	NUM
ejpam-4979	80	3	1	1	NUM
ejpam-4979	80	4	+	+	CCONJ
ejpam-4979	80	5	x	x	SYM
ejpam-4979	80	6	−	−	NOUN
ejpam-4979	80	7	1	1	NUM
ejpam-4979	80	8	+	+	CCONJ
ejpam-4979	80	9	x	x	SYM
ejpam-4979	80	10	=	=	SYM
ejpam-4979	80	11	x2	x2	NOUN
ejpam-4979	80	12	1	1	NUM
ejpam-4979	80	13	+	+	CCONJ
ejpam-4979	80	14	x	x	X
ejpam-4979	80	15	.	.	PUNCT
ejpam-4979	81	1	now	now	ADV
ejpam-4979	81	2	substitute	substitute	VERB
ejpam-4979	81	3	1	1	NUM
ejpam-4979	81	4	lnn	lnn	NOUN
ejpam-4979	81	5	for	for	SCONJ
ejpam-4979	81	6	x	x	PART
ejpam-4979	81	7	to	to	PART
ejpam-4979	81	8	get	get	VERB
ejpam-4979	81	9	(	(	PUNCT
ejpam-4979	81	10	1	1	NUM
ejpam-4979	81	11	lnn	lnn	NOUN
ejpam-4979	81	12	)	)	PUNCT
ejpam-4979	82	1	2	2	NUM
ejpam-4979	82	2	1	1	NUM
ejpam-4979	82	3	+	+	SYM
ejpam-4979	82	4	1	1	NUM
ejpam-4979	82	5	lnn	lnn	NOUN
ejpam-4979	82	6	=	=	SYM
ejpam-4979	82	7	1	1	NUM
ejpam-4979	82	8	lnn(lnn+	lnn(lnn+	PROPN
ejpam-4979	82	9	1	1	NUM
ejpam-4979	82	10	)	)	PUNCT
ejpam-4979	82	11	so	so	SCONJ
ejpam-4979	82	12	that	that	SCONJ
ejpam-4979	82	13	d′′≈	d′′≈	NOUN
ejpam-4979	82	14	1	1	NUM
ejpam-4979	82	15	(	(	PUNCT
ejpam-4979	82	16	lnn)2	lnn)2	NOUN
ejpam-4979	82	17	−	−	PROPN
ejpam-4979	82	18	1	1	NUM
ejpam-4979	82	19	(	(	PUNCT
ejpam-4979	82	20	lnn)3	lnn)3	NOUN
ejpam-4979	82	21	+	+	NUM
ejpam-4979	82	22	1	1	NUM
ejpam-4979	82	23	(	(	PUNCT
ejpam-4979	82	24	lnn)4	lnn)4	PROPN
ejpam-4979	82	25	−	−	PROPN
ejpam-4979	82	26	1	1	NUM
ejpam-4979	82	27	(	(	PUNCT
ejpam-4979	82	28	lnn)5	lnn)5	PROPN
ejpam-4979	82	29	+	+	CCONJ
ejpam-4979	82	30	...	...	PUNCT
ejpam-4979	82	31	(	(	PUNCT
ejpam-4979	82	32	7	7	X
ejpam-4979	82	33	)	)	PUNCT
ejpam-4979	82	34	=	=	SYM
ejpam-4979	82	35	1	1	NUM
ejpam-4979	82	36	lnn(lnn+	lnn(lnn+	PROPN
ejpam-4979	82	37	1	1	NUM
ejpam-4979	82	38	)	)	PUNCT
ejpam-4979	82	39	.	.	PUNCT
ejpam-4979	83	1	(	(	PUNCT
ejpam-4979	83	2	8)	8)	NUM
ejpam-4979	83	3	based	base	VERB
ejpam-4979	83	4	on	on	ADP
ejpam-4979	83	5	our	our	PRON
ejpam-4979	83	6	assumption	assumption	NOUN
ejpam-4979	83	7	that	that	SCONJ
ejpam-4979	83	8	1/	1/	NUM
ejpam-4979	83	9	lnn	lnn	NOUN
ejpam-4979	83	10	is	be	AUX
ejpam-4979	83	11	the	the	DET
ejpam-4979	83	12	asymptotic	asymptotic	ADJ
ejpam-4979	83	13	density	density	NOUN
ejpam-4979	83	14	of	of	ADP
ejpam-4979	83	15	the	the	DET
ejpam-4979	83	16	set	set	NOUN
ejpam-4979	83	17	of	of	ADP
ejpam-4979	83	18	all	all	DET
ejpam-4979	83	19	prime	prime	ADJ
ejpam-4979	83	20	numbers	number	NOUN
ejpam-4979	83	21	p	p	NOUN
ejpam-4979	83	22	as	as	ADP
ejpam-4979	83	23	n	n	PROPN
ejpam-4979	83	24	→	→	SYM
ejpam-4979	83	25	∞	∞	PROPN
ejpam-4979	83	26	,	,	PUNCT
ejpam-4979	83	27	eqs	eqs	PROPN
ejpam-4979	83	28	.	.	PROPN
ejpam-4979	83	29	6	6	NUM
ejpam-4979	83	30	and	and	CCONJ
ejpam-4979	83	31	8	8	NUM
ejpam-4979	83	32	provide	provide	VERB
ejpam-4979	83	33	us	we	PRON
ejpam-4979	83	34	with	with	ADP
ejpam-4979	83	35	the	the	DET
ejpam-4979	83	36	densities	density	NOUN
ejpam-4979	83	37	(	(	PUNCT
ejpam-4979	83	38	or	or	CCONJ
ejpam-4979	83	39	probabilities	probability	NOUN
ejpam-4979	83	40	m.	m.	NOUN
ejpam-4979	83	41	p.	p.	NOUN
ejpam-4979	83	42	may	may	AUX
ejpam-4979	83	43	/	/	SYM
ejpam-4979	83	44	eur	eur	PROPN
ejpam-4979	83	45	.	.	PUNCT
ejpam-4979	84	1	j.	j.	PROPN
ejpam-4979	84	2	pure	pure	PROPN
ejpam-4979	84	3	appl	appl	PROPN
ejpam-4979	84	4	.	.	PROPN
ejpam-4979	84	5	math	math	PROPN
ejpam-4979	84	6	,	,	PUNCT
ejpam-4979	84	7	17	17	NUM
ejpam-4979	84	8	(	(	PUNCT
ejpam-4979	84	9	1	1	NUM
ejpam-4979	84	10	)	)	PUNCT
ejpam-4979	84	11	(	(	PUNCT
ejpam-4979	84	12	2024	2024	NUM
ejpam-4979	84	13	)	)	PUNCT
ejpam-4979	84	14	,	,	PUNCT
ejpam-4979	84	15	42	42	NUM
ejpam-4979	84	16	-	-	SYM
ejpam-4979	84	17	58	58	NUM
ejpam-4979	84	18	48	48	NUM
ejpam-4979	84	19	of	of	ADP
ejpam-4979	84	20	occurrence	occurrence	NOUN
ejpam-4979	84	21	)	)	PUNCT
ejpam-4979	84	22	of	of	ADP
ejpam-4979	84	23	the	the	DET
ejpam-4979	84	24	primes	prime	NOUN
ejpam-4979	84	25	in	in	ADP
ejpam-4979	84	26	the	the	DET
ejpam-4979	84	27	complementary	complementary	ADJ
ejpam-4979	84	28	sets	set	NOUN
ejpam-4979	84	29	p′	p′	NOUN
ejpam-4979	84	30	and	and	CCONJ
ejpam-4979	84	31	p′′	p′′	PROPN
ejpam-4979	84	32	,	,	PUNCT
ejpam-4979	84	33	respectively	respectively	ADV
ejpam-4979	84	34	,	,	PUNCT
ejpam-4979	84	35	as	as	ADP
ejpam-4979	84	36	n	n	PRON
ejpam-4979	84	37	approaches	approach	NOUN
ejpam-4979	84	38	∞.	∞.	PROPN
ejpam-4979	84	39	thus	thus	ADV
ejpam-4979	84	40	,	,	PUNCT
ejpam-4979	84	41	the	the	DET
ejpam-4979	84	42	average	average	ADJ
ejpam-4979	84	43	gap	gap	NOUN
ejpam-4979	84	44	size	size	NOUN
ejpam-4979	84	45	g′	g′	NOUN
ejpam-4979	84	46	between	between	ADP
ejpam-4979	84	47	prime	prime	ADJ
ejpam-4979	84	48	numbers	number	NOUN
ejpam-4979	84	49	in	in	ADP
ejpam-4979	84	50	the	the	DET
ejpam-4979	84	51	subsequence	subsequence	NOUN
ejpam-4979	84	52	p′	p′	NOUN
ejpam-4979	84	53	on	on	ADP
ejpam-4979	84	54	the	the	DET
ejpam-4979	84	55	natural	natural	ADJ
ejpam-4979	84	56	number	number	NOUN
ejpam-4979	84	57	line	line	NOUN
ejpam-4979	84	58	as	as	ADP
ejpam-4979	84	59	n	n	PROPN
ejpam-4979	84	60	→	→	SYM
ejpam-4979	84	61	∞	∞	PROPN
ejpam-4979	84	62	is	be	AUX
ejpam-4979	84	63	the	the	DET
ejpam-4979	84	64	inverse	inverse	NOUN
ejpam-4979	84	65	of	of	ADP
ejpam-4979	84	66	the	the	DET
ejpam-4979	84	67	density	density	NOUN
ejpam-4979	84	68	d′	d′	NUM
ejpam-4979	84	69	of	of	ADP
ejpam-4979	84	70	p′	p′	NOUN
ejpam-4979	84	71	such	such	ADJ
ejpam-4979	84	72	that	that	DET
ejpam-4979	84	73	g′	g′	NOUN
ejpam-4979	84	74	=	=	NOUN
ejpam-4979	84	75	1	1	NUM
ejpam-4979	84	76	d′	d′	NUM
ejpam-4979	84	77	≈	≈	NOUN
ejpam-4979	84	78	1	1	NUM
ejpam-4979	84	79	1	1	NUM
ejpam-4979	84	80	lnn	lnn	NOUN
ejpam-4979	84	81	−	−	NOUN
ejpam-4979	84	82	1	1	NUM
ejpam-4979	84	83	(	(	PUNCT
ejpam-4979	84	84	lnn)2	lnn)2	NOUN
ejpam-4979	84	85	+	+	CCONJ
ejpam-4979	84	86	1	1	NUM
ejpam-4979	84	87	(	(	PUNCT
ejpam-4979	84	88	lnn)3	lnn)3	NOUN
ejpam-4979	84	89	−	−	PROPN
ejpam-4979	84	90	1	1	NUM
ejpam-4979	84	91	(	(	PUNCT
ejpam-4979	84	92	lnn)4	lnn)4	PROPN
ejpam-4979	84	93	+	+	CCONJ
ejpam-4979	84	94	...	...	PUNCT
ejpam-4979	85	1	=	=	SYM
ejpam-4979	85	2	lnn+	lnn+	NOUN
ejpam-4979	85	3	1	1	X
ejpam-4979	85	4	.	.	PUNCT
ejpam-4979	85	5	similarly	similarly	ADV
ejpam-4979	85	6	,	,	PUNCT
ejpam-4979	85	7	the	the	DET
ejpam-4979	85	8	average	average	ADJ
ejpam-4979	85	9	gap	gap	NOUN
ejpam-4979	85	10	size	size	NOUN
ejpam-4979	85	11	g′′	g′′	PROPN
ejpam-4979	85	12	between	between	ADP
ejpam-4979	85	13	prime	prime	ADJ
ejpam-4979	85	14	numbers	number	NOUN
ejpam-4979	85	15	in	in	ADP
ejpam-4979	85	16	the	the	DET
ejpam-4979	85	17	subsequence	subsequence	NOUN
ejpam-4979	85	18	p′′	p′′	PROPN
ejpam-4979	85	19	on	on	ADP
ejpam-4979	85	20	the	the	DET
ejpam-4979	85	21	natural	natural	ADJ
ejpam-4979	85	22	number	number	NOUN
ejpam-4979	85	23	line	line	NOUN
ejpam-4979	85	24	as	as	ADP
ejpam-4979	85	25	n	n	PROPN
ejpam-4979	85	26	→	→	SYM
ejpam-4979	85	27	∞	∞	PROPN
ejpam-4979	85	28	is	be	AUX
ejpam-4979	85	29	the	the	DET
ejpam-4979	85	30	inverse	inverse	NOUN
ejpam-4979	85	31	of	of	ADP
ejpam-4979	85	32	the	the	DET
ejpam-4979	85	33	density	density	NOUN
ejpam-4979	85	34	d′′	d′′	PROPN
ejpam-4979	85	35	of	of	ADP
ejpam-4979	85	36	p′′	p′′	PROPN
ejpam-4979	85	37	such	such	ADJ
ejpam-4979	85	38	that	that	PRON
ejpam-4979	85	39	g′′	g′′	PROPN
ejpam-4979	85	40	=	=	SYM
ejpam-4979	85	41	1	1	NUM
ejpam-4979	85	42	d′′	d′′	NOUN
ejpam-4979	85	43	≈	≈	PROPN
ejpam-4979	85	44	1	1	NUM
ejpam-4979	85	45	1	1	NUM
ejpam-4979	85	46	(	(	PUNCT
ejpam-4979	85	47	lnn)2	lnn)2	PROPN
ejpam-4979	85	48	−	−	PROPN
ejpam-4979	85	49	1	1	NUM
ejpam-4979	85	50	(	(	PUNCT
ejpam-4979	85	51	lnn)3	lnn)3	NOUN
ejpam-4979	85	52	+	+	NUM
ejpam-4979	85	53	1	1	NUM
ejpam-4979	85	54	(	(	PUNCT
ejpam-4979	85	55	lnn)4	lnn)4	PROPN
ejpam-4979	85	56	−	−	PROPN
ejpam-4979	85	57	1	1	NUM
ejpam-4979	85	58	(	(	PUNCT
ejpam-4979	85	59	lnn)5	lnn)5	PROPN
ejpam-4979	85	60	+	+	NUM
ejpam-4979	85	61	...	...	PUNCT
ejpam-4979	86	1	=	=	SYM
ejpam-4979	86	2	lnn(lnn+	lnn(lnn+	PROPN
ejpam-4979	86	3	1	1	NUM
ejpam-4979	86	4	)	)	PUNCT
ejpam-4979	86	5	.	.	PUNCT
ejpam-4979	87	1	since	since	SCONJ
ejpam-4979	87	2	it	it	PRON
ejpam-4979	87	3	has	have	AUX
ejpam-4979	87	4	been	be	AUX
ejpam-4979	87	5	shown	show	VERB
ejpam-4979	87	6	via	via	ADP
ejpam-4979	87	7	the	the	DET
ejpam-4979	87	8	sieving	sieve	VERB
ejpam-4979	87	9	operation	operation	NOUN
ejpam-4979	88	1	[	[	X
ejpam-4979	88	2	7	7	X
ejpam-4979	88	3	]	]	PUNCT
ejpam-4979	88	4	that	that	SCONJ
ejpam-4979	88	5	the	the	DET
ejpam-4979	88	6	prime	prime	ADJ
ejpam-4979	88	7	number	number	NOUN
ejpam-4979	88	8	subsequence	subsequence	NOUN
ejpam-4979	88	9	p′	p′	NOUN
ejpam-4979	88	10	has	have	VERB
ejpam-4979	88	11	fewer	few	ADJ
ejpam-4979	88	12	primes	prime	NOUN
ejpam-4979	88	13	than	than	ADP
ejpam-4979	88	14	the	the	DET
ejpam-4979	88	15	set	set	NOUN
ejpam-4979	88	16	of	of	ADP
ejpam-4979	88	17	all	all	DET
ejpam-4979	88	18	prime	prime	ADJ
ejpam-4979	88	19	numbers	number	NOUN
ejpam-4979	88	20	p	p	X
ejpam-4979	88	21	,	,	PUNCT
ejpam-4979	88	22	it	it	PRON
ejpam-4979	88	23	intuitively	intuitively	ADV
ejpam-4979	88	24	follows	follow	VERB
ejpam-4979	88	25	that	that	SCONJ
ejpam-4979	88	26	the	the	DET
ejpam-4979	88	27	average	average	ADJ
ejpam-4979	88	28	gap	gap	NOUN
ejpam-4979	88	29	size	size	NOUN
ejpam-4979	88	30	for	for	ADP
ejpam-4979	88	31	p′	p′	NOUN
ejpam-4979	88	32	will	will	AUX
ejpam-4979	88	33	always	always	ADV
ejpam-4979	88	34	be	be	AUX
ejpam-4979	88	35	larger	large	ADJ
ejpam-4979	88	36	than	than	ADP
ejpam-4979	88	37	the	the	DET
ejpam-4979	88	38	gap	gap	NOUN
ejpam-4979	88	39	size	size	NOUN
ejpam-4979	88	40	for	for	ADP
ejpam-4979	88	41	p	p	NOUN
ejpam-4979	88	42	and	and	CCONJ
ejpam-4979	88	43	that	that	SCONJ
ejpam-4979	88	44	the	the	DET
ejpam-4979	88	45	larger	large	ADJ
ejpam-4979	88	46	gap	gap	NOUN
ejpam-4979	88	47	size	size	NOUN
ejpam-4979	88	48	for	for	ADP
ejpam-4979	88	49	p′	p′	NOUN
ejpam-4979	88	50	results	result	NOUN
ejpam-4979	88	51	from	from	ADP
ejpam-4979	88	52	omitting	omit	VERB
ejpam-4979	88	53	the	the	DET
ejpam-4979	88	54	count	count	NOUN
ejpam-4979	88	55	of	of	ADP
ejpam-4979	88	56	the	the	DET
ejpam-4979	88	57	prime	prime	ADJ
ejpam-4979	88	58	numbers	number	NOUN
ejpam-4979	88	59	p′′	p′′	PROPN
ejpam-4979	88	60	on	on	ADP
ejpam-4979	88	61	n.	n.	PROPN
ejpam-4979	88	62	3	3	NUM
ejpam-4979	88	63	.	.	PUNCT
ejpam-4979	88	64	π′(x	π′(x	NOUN
ejpam-4979	88	65	)	)	PUNCT
ejpam-4979	88	66	and	and	CCONJ
ejpam-4979	88	67	π′′(x	π′′(x	NOUN
ejpam-4979	88	68	)	)	PUNCT
ejpam-4979	88	69	we	we	PRON
ejpam-4979	88	70	have	have	AUX
ejpam-4979	88	71	shown	show	VERB
ejpam-4979	88	72	that	that	SCONJ
ejpam-4979	88	73	when	when	SCONJ
ejpam-4979	88	74	we	we	PRON
ejpam-4979	88	75	remove	remove	VERB
ejpam-4979	88	76	the	the	DET
ejpam-4979	88	77	prime	prime	ADJ
ejpam-4979	88	78	number	number	NOUN
ejpam-4979	88	79	subsequence	subsequence	NOUN
ejpam-4979	88	80	p′′	p′′	PROPN
ejpam-4979	88	81	from	from	ADP
ejpam-4979	88	82	the	the	DET
ejpam-4979	88	83	set	set	NOUN
ejpam-4979	88	84	of	of	ADP
ejpam-4979	88	85	all	all	DET
ejpam-4979	88	86	prime	prime	ADJ
ejpam-4979	88	87	numbers	number	NOUN
ejpam-4979	88	88	p	p	X
ejpam-4979	88	89	,	,	PUNCT
ejpam-4979	88	90	we	we	PRON
ejpam-4979	88	91	create	create	VERB
ejpam-4979	88	92	the	the	DET
ejpam-4979	88	93	prime	prime	ADJ
ejpam-4979	88	94	number	number	NOUN
ejpam-4979	88	95	subsequence	subsequence	NOUN
ejpam-4979	88	96	p′	p′	NOUN
ejpam-4979	89	1	[	[	X
ejpam-4979	89	2	7	7	NUM
ejpam-4979	89	3	]	]	PUNCT
ejpam-4979	89	4	.	.	PUNCT
ejpam-4979	90	1	thus	thus	ADV
ejpam-4979	90	2	,	,	PUNCT
ejpam-4979	90	3	we	we	PRON
ejpam-4979	90	4	define	define	VERB
ejpam-4979	90	5	the	the	DET
ejpam-4979	90	6	prime	prime	ADJ
ejpam-4979	90	7	number	number	NOUN
ejpam-4979	90	8	count	count	NOUN
ejpam-4979	90	9	for	for	ADP
ejpam-4979	90	10	the	the	DET
ejpam-4979	90	11	sequences	sequence	NOUN
ejpam-4979	90	12	p′	p′	NOUN
ejpam-4979	90	13	and	and	CCONJ
ejpam-4979	90	14	p′′	p′′	PROPN
ejpam-4979	90	15	up	up	ADP
ejpam-4979	90	16	to	to	ADP
ejpam-4979	90	17	x	x	PUNCT
ejpam-4979	90	18	as	as	ADP
ejpam-4979	90	19	π′(x	π′(x	NOUN
ejpam-4979	90	20	)	)	PUNCT
ejpam-4979	90	21	=	=	SYM
ejpam-4979	90	22	|p′(x)|	|p′(x)|	NOUN
ejpam-4979	90	23	and	and	CCONJ
ejpam-4979	90	24	π′′(x	π′′(x	NOUN
ejpam-4979	90	25	)	)	PUNCT
ejpam-4979	90	26	=	=	PUNCT
ejpam-4979	91	1	|p′′(x)|	|p′′(x)|	VERB
ejpam-4979	91	2	where	where	SCONJ
ejpam-4979	91	3	|p′(x)|	|p′(x)|	PRON
ejpam-4979	91	4	and	and	CCONJ
ejpam-4979	91	5	|p′′(x)|	|p′′(x)|	NOUN
ejpam-4979	91	6	represent	represent	VERB
ejpam-4979	91	7	the	the	DET
ejpam-4979	91	8	cardinality	cardinality	NOUN
ejpam-4979	91	9	of	of	ADP
ejpam-4979	91	10	the	the	DET
ejpam-4979	91	11	prime	prime	ADJ
ejpam-4979	91	12	number	number	NOUN
ejpam-4979	91	13	subsequences	subsequence	VERB
ejpam-4979	91	14	p′	p′	NOUN
ejpam-4979	91	15	and	and	CCONJ
ejpam-4979	91	16	p′′	p′′	PROPN
ejpam-4979	91	17	up	up	ADP
ejpam-4979	91	18	to	to	PART
ejpam-4979	91	19	x.	x.	VERB
ejpam-4979	91	20	however	however	ADV
ejpam-4979	91	21	,	,	PUNCT
ejpam-4979	91	22	since	since	SCONJ
ejpam-4979	91	23	neither	neither	PRON
ejpam-4979	91	24	π′(x	π′(x	NOUN
ejpam-4979	91	25	)	)	PUNCT
ejpam-4979	91	26	nor	nor	CCONJ
ejpam-4979	91	27	π′′(x	π′′(x	NOUN
ejpam-4979	91	28	)	)	PUNCT
ejpam-4979	91	29	have	have	AUX
ejpam-4979	91	30	been	be	AUX
ejpam-4979	91	31	shown	show	VERB
ejpam-4979	91	32	up	up	ADP
ejpam-4979	91	33	to	to	ADP
ejpam-4979	91	34	this	this	DET
ejpam-4979	91	35	point	point	NOUN
ejpam-4979	91	36	to	to	PART
ejpam-4979	91	37	be	be	AUX
ejpam-4979	91	38	calculable	calculable	ADJ
ejpam-4979	91	39	without	without	ADP
ejpam-4979	91	40	manually	manually	ADV
ejpam-4979	91	41	counting	count	VERB
ejpam-4979	91	42	each	each	DET
ejpam-4979	91	43	term	term	NOUN
ejpam-4979	91	44	up	up	ADP
ejpam-4979	91	45	to	to	ADP
ejpam-4979	91	46	x	x	SYM
ejpam-4979	91	47	,	,	PUNCT
ejpam-4979	91	48	we	we	PRON
ejpam-4979	91	49	will	will	AUX
ejpam-4979	91	50	begin	begin	VERB
ejpam-4979	91	51	by	by	ADP
ejpam-4979	91	52	generating	generate	VERB
ejpam-4979	91	53	an	an	DET
ejpam-4979	91	54	estimate	estimate	NOUN
ejpam-4979	91	55	of	of	ADP
ejpam-4979	91	56	the	the	DET
ejpam-4979	91	57	count	count	NOUN
ejpam-4979	91	58	π(x	π(x	ADP
ejpam-4979	91	59	)	)	PUNCT
ejpam-4979	91	60	of	of	ADP
ejpam-4979	91	61	set	set	NOUN
ejpam-4979	91	62	of	of	ADP
ejpam-4979	91	63	all	all	DET
ejpam-4979	91	64	primes	prime	NOUN
ejpam-4979	91	65	p	p	NOUN
ejpam-4979	91	66	up	up	ADP
ejpam-4979	91	67	to	to	PART
ejpam-4979	91	68	x	x	PUNCT
ejpam-4979	91	69	via	via	ADP
ejpam-4979	91	70	the	the	DET
ejpam-4979	91	71	inclusion	inclusion	NOUN
ejpam-4979	91	72	-	-	PUNCT
ejpam-4979	91	73	exclusion	exclusion	NOUN
ejpam-4979	91	74	principle	principle	NOUN
ejpam-4979	91	75	and	and	CCONJ
ejpam-4979	91	76	then	then	ADV
ejpam-4979	91	77	perform	perform	VERB
ejpam-4979	91	78	an	an	DET
ejpam-4979	91	79	operation	operation	NOUN
ejpam-4979	91	80	on	on	ADP
ejpam-4979	91	81	that	that	DET
ejpam-4979	91	82	result	result	NOUN
ejpam-4979	91	83	to	to	PART
ejpam-4979	91	84	reduce	reduce	VERB
ejpam-4979	91	85	the	the	DET
ejpam-4979	91	86	count	count	NOUN
ejpam-4979	91	87	of	of	ADP
ejpam-4979	91	88	all	all	DET
ejpam-4979	91	89	primes	prime	NOUN
ejpam-4979	91	90	down	down	ADP
ejpam-4979	91	91	to	to	ADP
ejpam-4979	91	92	π′(x	π′(x	NOUN
ejpam-4979	91	93	)	)	PUNCT
ejpam-4979	91	94	and	and	CCONJ
ejpam-4979	91	95	π′′(x	π′′(x	NOUN
ejpam-4979	91	96	)	)	PUNCT
ejpam-4979	91	97	.	.	PUNCT
ejpam-4979	92	1	m.	m.	NOUN
ejpam-4979	92	2	p.	p.	NOUN
ejpam-4979	92	3	may	may	AUX
ejpam-4979	92	4	/	/	SYM
ejpam-4979	92	5	eur	eur	PROPN
ejpam-4979	92	6	.	.	PUNCT
ejpam-4979	93	1	j.	j.	PROPN
ejpam-4979	93	2	pure	pure	PROPN
ejpam-4979	93	3	appl	appl	PROPN
ejpam-4979	93	4	.	.	PROPN
ejpam-4979	93	5	math	math	PROPN
ejpam-4979	93	6	,	,	PUNCT
ejpam-4979	93	7	17	17	NUM
ejpam-4979	93	8	(	(	PUNCT
ejpam-4979	93	9	1	1	NUM
ejpam-4979	93	10	)	)	PUNCT
ejpam-4979	93	11	(	(	PUNCT
ejpam-4979	93	12	2024	2024	NUM
ejpam-4979	93	13	)	)	PUNCT
ejpam-4979	93	14	,	,	PUNCT
ejpam-4979	93	15	42	42	NUM
ejpam-4979	93	16	-	-	SYM
ejpam-4979	93	17	58	58	NUM
ejpam-4979	93	18	49	49	NUM
ejpam-4979	93	19	4	4	NUM
ejpam-4979	93	20	.	.	PUNCT
ejpam-4979	94	1	π(x	π(x	NOUN
ejpam-4979	94	2	)	)	PUNCT
ejpam-4979	95	1	via	via	ADP
ejpam-4979	95	2	the	the	DET
ejpam-4979	95	3	inclusion	inclusion	NOUN
ejpam-4979	95	4	-	-	PUNCT
ejpam-4979	95	5	exclusion	exclusion	NOUN
ejpam-4979	95	6	principle	principle	NOUN
ejpam-4979	95	7	to	to	PART
ejpam-4979	95	8	calculate	calculate	VERB
ejpam-4979	95	9	π(x	π(x	ADP
ejpam-4979	95	10	)	)	PUNCT
ejpam-4979	95	11	,	,	PUNCT
ejpam-4979	95	12	we	we	PRON
ejpam-4979	95	13	invoke	invoke	VERB
ejpam-4979	95	14	the	the	DET
ejpam-4979	95	15	inclusion	inclusion	NOUN
ejpam-4979	95	16	-	-	PUNCT
ejpam-4979	95	17	exclusion	exclusion	NOUN
ejpam-4979	95	18	principle	principle	NOUN
ejpam-4979	95	19	[	[	X
ejpam-4979	95	20	8	8	X
ejpam-4979	95	21	]	]	X
ejpam-4979	96	1	[	[	X
ejpam-4979	96	2	6	6	NUM
ejpam-4979	96	3	]	]	PUNCT
ejpam-4979	96	4	.	.	PUNCT
ejpam-4979	97	1	let	let	VERB
ejpam-4979	97	2	r	r	NOUN
ejpam-4979	97	3	represent	represent	VERB
ejpam-4979	97	4	the	the	DET
ejpam-4979	97	5	number	number	NOUN
ejpam-4979	97	6	of	of	ADP
ejpam-4979	97	7	primes	prime	NOUN
ejpam-4979	97	8	less	less	ADJ
ejpam-4979	97	9	than	than	ADP
ejpam-4979	97	10	√	√	PROPN
ejpam-4979	98	1	x.	x.	NOUN
ejpam-4979	99	1	then	then	ADV
ejpam-4979	99	2	let	let	VERB
ejpam-4979	99	3	p	p	NOUN
ejpam-4979	99	4	=	=	X
ejpam-4979	99	5	{	{	PUNCT
ejpam-4979	99	6	n	n	NOUN
ejpam-4979	99	7	∈	∈	PROPN
ejpam-4979	99	8	n	n	CCONJ
ejpam-4979	99	9	|1	|1	PRON
ejpam-4979	99	10	<	<	X
ejpam-4979	99	11	n	n	PRON
ejpam-4979	99	12	≤	≤	NOUN
ejpam-4979	99	13	x	x	X
ejpam-4979	99	14	}	}	PUNCT
ejpam-4979	99	15	such	such	ADJ
ejpam-4979	99	16	that	that	SCONJ
ejpam-4979	99	17	n	n	PRON
ejpam-4979	99	18	is	be	AUX
ejpam-4979	99	19	not	not	PART
ejpam-4979	99	20	a	a	DET
ejpam-4979	99	21	multiple	multiple	NOUN
ejpam-4979	99	22	of	of	ADP
ejpam-4979	99	23	p1	p1	NOUN
ejpam-4979	99	24	,	,	PUNCT
ejpam-4979	99	25	p2	p2	NOUN
ejpam-4979	99	26	,	,	PUNCT
ejpam-4979	99	27	...	...	PUNCT
ejpam-4979	99	28	,	,	PUNCT
ejpam-4979	99	29	pr	pr	X
ejpam-4979	99	30	.	.	PUNCT
ejpam-4979	100	1	if	if	SCONJ
ejpam-4979	100	2	a(x	a(x	NOUN
ejpam-4979	100	3	,	,	PUNCT
ejpam-4979	100	4	r	r	NOUN
ejpam-4979	100	5	)	)	PUNCT
ejpam-4979	100	6	represents	represent	VERB
ejpam-4979	100	7	the	the	DET
ejpam-4979	100	8	cardinality	cardinality	NOUN
ejpam-4979	100	9	of	of	ADP
ejpam-4979	100	10	p	p	NOUN
ejpam-4979	100	11	,	,	PUNCT
ejpam-4979	100	12	then	then	ADV
ejpam-4979	100	13	it	it	PRON
ejpam-4979	100	14	follows	follow	VERB
ejpam-4979	100	15	that	that	SCONJ
ejpam-4979	100	16	the	the	DET
ejpam-4979	100	17	number	number	NOUN
ejpam-4979	100	18	of	of	ADP
ejpam-4979	100	19	primes	prime	NOUN
ejpam-4979	100	20	≤	≤	NUM
ejpam-4979	100	21	x	x	PUNCT
ejpam-4979	100	22	is	be	AUX
ejpam-4979	100	23	π(x	π(x	ADP
ejpam-4979	100	24	)	)	PUNCT
ejpam-4979	100	25	≤	≤	NOUN
ejpam-4979	100	26	r	r	NOUN
ejpam-4979	100	27	+	+	NOUN
ejpam-4979	100	28	a(x	a(x	NOUN
ejpam-4979	100	29	,	,	PUNCT
ejpam-4979	100	30	r	r	NOUN
ejpam-4979	100	31	)	)	PUNCT
ejpam-4979	100	32	.	.	PUNCT
ejpam-4979	101	1	now	now	ADV
ejpam-4979	101	2	,	,	PUNCT
ejpam-4979	101	3	let	let	VERB
ejpam-4979	101	4	mi	mi	PROPN
ejpam-4979	101	5	be	be	AUX
ejpam-4979	101	6	the	the	DET
ejpam-4979	101	7	set	set	NOUN
ejpam-4979	101	8	of	of	ADP
ejpam-4979	101	9	integers	integer	NOUN
ejpam-4979	101	10	from	from	ADP
ejpam-4979	101	11	1	1	NUM
ejpam-4979	101	12	to	to	ADP
ejpam-4979	101	13	n	n	PRON
ejpam-4979	101	14	which	which	PRON
ejpam-4979	101	15	are	be	AUX
ejpam-4979	101	16	multiples	multiple	NOUN
ejpam-4979	101	17	of	of	ADP
ejpam-4979	101	18	pi	pi	NOUN
ejpam-4979	101	19	,	,	PUNCT
ejpam-4979	101	20	and	and	CCONJ
ejpam-4979	101	21	let	let	VERB
ejpam-4979	101	22	mij	mij	NOUN
ejpam-4979	101	23	be	be	AUX
ejpam-4979	101	24	the	the	DET
ejpam-4979	101	25	set	set	NOUN
ejpam-4979	101	26	of	of	ADP
ejpam-4979	101	27	integers	integer	NOUN
ejpam-4979	101	28	from	from	ADP
ejpam-4979	101	29	1	1	NUM
ejpam-4979	101	30	to	to	ADP
ejpam-4979	101	31	n	n	PROPN
ejpam-4979	101	32	that	that	PRON
ejpam-4979	101	33	are	be	AUX
ejpam-4979	101	34	multiples	multiple	NOUN
ejpam-4979	101	35	of	of	ADP
ejpam-4979	101	36	both	both	DET
ejpam-4979	101	37	pi	pi	NOUN
ejpam-4979	101	38	and	and	CCONJ
ejpam-4979	101	39	pj	pj	PROPN
ejpam-4979	101	40	.	.	PUNCT
ejpam-4979	102	1	then	then	ADV
ejpam-4979	102	2	,	,	PUNCT
ejpam-4979	102	3	mij	mij	PROPN
ejpam-4979	102	4	=	=	SYM
ejpam-4979	102	5	mi	mi	PROPN
ejpam-4979	102	6	∩mj	∩mj	PROPN
ejpam-4979	102	7	so	so	SCONJ
ejpam-4979	102	8	that	that	PRON
ejpam-4979	102	9	|mi|	|mi|	NOUN
ejpam-4979	102	10	=	=	PUNCT
ejpam-4979	102	11	⌊	⌊	PUNCT
ejpam-4979	102	12	x	x	SYM
ejpam-4979	102	13	pi	pi	NOUN
ejpam-4979	102	14	⌋	⌋	NOUN
ejpam-4979	102	15	and	and	CCONJ
ejpam-4979	102	16	|mij	|mij	PROPN
ejpam-4979	102	17	|	|	NOUN
ejpam-4979	102	18	=	=	SYM
ejpam-4979	102	19	⌊	⌊	NOUN
ejpam-4979	102	20	x	x	PUNCT
ejpam-4979	102	21	pipj	pipj	NOUN
ejpam-4979	102	22	⌋.	⌋.	PRON
ejpam-4979	102	23	then	then	ADV
ejpam-4979	102	24	it	it	PRON
ejpam-4979	102	25	follows	follow	VERB
ejpam-4979	102	26	by	by	ADP
ejpam-4979	102	27	the	the	DET
ejpam-4979	102	28	inclusion	inclusion	NOUN
ejpam-4979	102	29	-	-	PUNCT
ejpam-4979	102	30	exclusion	exclusion	NOUN
ejpam-4979	102	31	principle	principle	NOUN
ejpam-4979	102	32	that	that	SCONJ
ejpam-4979	102	33	a(x	a(x	NOUN
ejpam-4979	102	34	,	,	PUNCT
ejpam-4979	102	35	r	r	NOUN
ejpam-4979	102	36	)	)	PUNCT
ejpam-4979	102	37	=	=	SYM
ejpam-4979	102	38	⌊x⌋	⌊x⌋	PUNCT
ejpam-4979	102	39	−	−	PUNCT
ejpam-4979	102	40	r∑	r∑	NOUN
ejpam-4979	103	1	i=1	i=1	X
ejpam-4979	103	2	⌊	⌊	PROPN
ejpam-4979	103	3	x	x	X
ejpam-4979	103	4	pi	pi	NOUN
ejpam-4979	103	5	⌋+	⌋+	X
ejpam-4979	104	1	r∑	r∑	NOUN
ejpam-4979	105	1	i	i	X
ejpam-4979	105	2	<	<	X
ejpam-4979	105	3	j≤r	j≤r	PROPN
ejpam-4979	105	4	⌊	⌊	PART
ejpam-4979	105	5	x	x	PUNCT
ejpam-4979	105	6	pipj	pipj	NOUN
ejpam-4979	105	7	⌋	⌋	NOUN
ejpam-4979	105	8	−	−	PROPN
ejpam-4979	105	9	.	.	PUNCT
ejpam-4979	105	10	.	.	PUNCT
ejpam-4979	106	1	.+	.+	NOUN
ejpam-4979	106	2	(	(	PUNCT
ejpam-4979	106	3	−1)r⌊	−1)r⌊	NOUN
ejpam-4979	106	4	x	x	SYM
ejpam-4979	106	5	p1p2	p1p2	PROPN
ejpam-4979	106	6	·	·	PUNCT
ejpam-4979	106	7	·	·	PUNCT
ejpam-4979	106	8	·	·	PUNCT
ejpam-4979	107	1	pr	pr	INTJ
ejpam-4979	107	2	⌋.	⌋.	X
ejpam-4979	107	3	(	(	PUNCT
ejpam-4979	107	4	9	9	X
ejpam-4979	107	5	)	)	PUNCT
ejpam-4979	107	6	if	if	SCONJ
ejpam-4979	107	7	we	we	PRON
ejpam-4979	107	8	approximate	approximate	VERB
ejpam-4979	107	9	the	the	DET
ejpam-4979	107	10	rhs	rhs	PROPN
ejpam-4979	107	11	of	of	ADP
ejpam-4979	107	12	eq	eq	PROPN
ejpam-4979	107	13	.	.	PROPN
ejpam-4979	107	14	9	9	NUM
ejpam-4979	107	15	by	by	ADP
ejpam-4979	107	16	ignoring	ignore	VERB
ejpam-4979	107	17	the	the	DET
ejpam-4979	107	18	round	round	ADJ
ejpam-4979	107	19	-	-	PUNCT
ejpam-4979	107	20	downs	down	NOUN
ejpam-4979	107	21	,	,	PUNCT
ejpam-4979	107	22	then	then	ADV
ejpam-4979	107	23	we	we	PRON
ejpam-4979	107	24	have	have	VERB
ejpam-4979	107	25	x−	x−	NOUN
ejpam-4979	107	26	r∑	r∑	NOUN
ejpam-4979	107	27	i=1	i=1	PROPN
ejpam-4979	107	28	x	x	PUNCT
ejpam-4979	108	1	pi	pi	NOUN
ejpam-4979	108	2	+	+	CCONJ
ejpam-4979	108	3	r∑	r∑	NOUN
ejpam-4979	108	4	i	i	X
ejpam-4979	108	5	<	<	X
ejpam-4979	108	6	j≤r	j≤r	PROPN
ejpam-4979	108	7	x	x	X
ejpam-4979	108	8	pipj	pipj	NOUN
ejpam-4979	108	9	−	−	PROPN
ejpam-4979	108	10	.	.	PUNCT
ejpam-4979	108	11	.	.	PUNCT
ejpam-4979	109	1	.+	.+	NOUN
ejpam-4979	109	2	(	(	PUNCT
ejpam-4979	109	3	−1)r	−1)r	X
ejpam-4979	109	4	x	x	SYM
ejpam-4979	109	5	p1p2	p1p2	PROPN
ejpam-4979	109	6	·	·	PUNCT
ejpam-4979	109	7	·	·	PUNCT
ejpam-4979	109	8	·	·	PUNCT
ejpam-4979	109	9	pr	pr	X
ejpam-4979	109	10	with	with	ADP
ejpam-4979	109	11	an	an	DET
ejpam-4979	109	12	error	error	NOUN
ejpam-4979	109	13	of	of	ADP
ejpam-4979	109	14	at	at	ADV
ejpam-4979	109	15	most	most	ADV
ejpam-4979	109	16	1	1	NUM
ejpam-4979	109	17	+	+	CCONJ
ejpam-4979	109	18	(	(	PUNCT
ejpam-4979	109	19	r	r	NOUN
ejpam-4979	109	20	1	1	NUM
ejpam-4979	109	21	)	)	PUNCT
ejpam-4979	109	22	+	+	CCONJ
ejpam-4979	109	23	(	(	PUNCT
ejpam-4979	109	24	r	r	NOUN
ejpam-4979	109	25	2	2	NUM
ejpam-4979	109	26	)	)	PUNCT
ejpam-4979	109	27	+	+	CCONJ
ejpam-4979	109	28	.	.	PUNCT
ejpam-4979	109	29	.	.	PUNCT
ejpam-4979	110	1	.+	.+	NOUN
ejpam-4979	110	2	(	(	PUNCT
ejpam-4979	110	3	r	r	NOUN
ejpam-4979	110	4	r	r	NOUN
ejpam-4979	110	5	)	)	PUNCT
ejpam-4979	110	6	=	=	SYM
ejpam-4979	110	7	2r	2r	NUM
ejpam-4979	110	8	.	.	PUNCT
ejpam-4979	111	1	thus	thus	ADV
ejpam-4979	111	2	,	,	PUNCT
ejpam-4979	111	3	we	we	PRON
ejpam-4979	111	4	now	now	ADV
ejpam-4979	111	5	have	have	VERB
ejpam-4979	111	6	for	for	ADP
ejpam-4979	111	7	our	our	PRON
ejpam-4979	111	8	estimate	estimate	NOUN
ejpam-4979	111	9	of	of	ADP
ejpam-4979	111	10	the	the	DET
ejpam-4979	111	11	number	number	NOUN
ejpam-4979	111	12	of	of	ADP
ejpam-4979	111	13	primes	prime	NOUN
ejpam-4979	111	14	less	less	ADJ
ejpam-4979	111	15	than	than	ADP
ejpam-4979	111	16	or	or	CCONJ
ejpam-4979	111	17	equal	equal	ADJ
ejpam-4979	111	18	to	to	ADP
ejpam-4979	111	19	x	x	PRON
ejpam-4979	111	20	as	as	ADP
ejpam-4979	111	21	π(x	π(x	NOUN
ejpam-4979	111	22	)	)	PUNCT
ejpam-4979	111	23	≤	≤	NOUN
ejpam-4979	111	24	r	r	NOUN
ejpam-4979	111	25	+	+	NOUN
ejpam-4979	112	1	x	x	SYM
ejpam-4979	112	2	·	·	PUNCT
ejpam-4979	112	3	r∏	r∏	NOUN
ejpam-4979	112	4	i=1	i=1	X
ejpam-4979	113	1	(	(	PUNCT
ejpam-4979	113	2	1−	1−	NUM
ejpam-4979	113	3	1	1	NUM
ejpam-4979	113	4	pi	pi	NOUN
ejpam-4979	113	5	)	)	PUNCT
ejpam-4979	114	1	+	+	CCONJ
ejpam-4979	114	2	2r	2r	NUM
ejpam-4979	114	3	.	.	PUNCT
ejpam-4979	115	1	(	(	PUNCT
ejpam-4979	115	2	10	10	NUM
ejpam-4979	115	3	)	)	PUNCT
ejpam-4979	115	4	we	we	PRON
ejpam-4979	115	5	now	now	ADV
ejpam-4979	115	6	want	want	VERB
ejpam-4979	115	7	to	to	PART
ejpam-4979	115	8	choose	choose	VERB
ejpam-4979	115	9	r	r	NOUN
ejpam-4979	115	10	relatively	relatively	ADV
ejpam-4979	115	11	small	small	ADJ
ejpam-4979	115	12	compared	compare	VERB
ejpam-4979	115	13	to	to	PART
ejpam-4979	115	14	x.	x.	VERB
ejpam-4979	115	15	in	in	ADP
ejpam-4979	115	16	order	order	NOUN
ejpam-4979	115	17	to	to	PART
ejpam-4979	115	18	do	do	AUX
ejpam-4979	115	19	so	so	ADV
ejpam-4979	115	20	,	,	PUNCT
ejpam-4979	115	21	we	we	PRON
ejpam-4979	115	22	need	need	VERB
ejpam-4979	115	23	a	a	DET
ejpam-4979	115	24	good	good	ADJ
ejpam-4979	115	25	estimate	estimate	NOUN
ejpam-4979	115	26	of	of	ADP
ejpam-4979	115	27	the	the	DET
ejpam-4979	115	28	coefficient	coefficient	NOUN
ejpam-4979	115	29	of	of	ADP
ejpam-4979	115	30	x	x	PUNCT
ejpam-4979	115	31	in	in	ADP
ejpam-4979	115	32	the	the	DET
ejpam-4979	115	33	middle	middle	ADJ
ejpam-4979	115	34	term	term	NOUN
ejpam-4979	115	35	on	on	ADP
ejpam-4979	115	36	the	the	DET
ejpam-4979	115	37	rhs	rhs	PROPN
ejpam-4979	115	38	of	of	ADP
ejpam-4979	115	39	10	10	NUM
ejpam-4979	115	40	in	in	ADP
ejpam-4979	115	41	terms	term	NOUN
ejpam-4979	115	42	of	of	ADP
ejpam-4979	115	43	r.	r.	PROPN
ejpam-4979	115	44	theorem	theorem	NOUN
ejpam-4979	115	45	1	1	X
ejpam-4979	115	46	.	.	PUNCT
ejpam-4979	116	1	if	if	SCONJ
ejpam-4979	116	2	x	x	X
ejpam-4979	116	3	≥	≥	NUM
ejpam-4979	116	4	2	2	NUM
ejpam-4979	116	5	,	,	PUNCT
ejpam-4979	116	6	then	then	ADV
ejpam-4979	116	7	∏	∏	PROPN
ejpam-4979	116	8	p≤x	p≤x	PROPN
ejpam-4979	116	9	(	(	PUNCT
ejpam-4979	116	10	1−	1−	NUM
ejpam-4979	116	11	1	1	NUM
ejpam-4979	116	12	p	p	NOUN
ejpam-4979	116	13	)	)	PUNCT
ejpam-4979	116	14	<	<	X
ejpam-4979	116	15	1	1	NUM
ejpam-4979	116	16	lnx	lnx	NOUN
ejpam-4979	116	17	.	.	PUNCT
ejpam-4979	117	1	m.	m.	NOUN
ejpam-4979	117	2	p.	p.	PROPN
ejpam-4979	117	3	may	may	AUX
ejpam-4979	117	4	/	/	SYM
ejpam-4979	117	5	eur	eur	PROPN
ejpam-4979	117	6	.	.	PUNCT
ejpam-4979	118	1	j.	j.	PROPN
ejpam-4979	118	2	pure	pure	PROPN
ejpam-4979	118	3	appl	appl	PROPN
ejpam-4979	118	4	.	.	PROPN
ejpam-4979	118	5	math	math	PROPN
ejpam-4979	118	6	,	,	PUNCT
ejpam-4979	118	7	17	17	NUM
ejpam-4979	118	8	(	(	PUNCT
ejpam-4979	118	9	1	1	NUM
ejpam-4979	118	10	)	)	PUNCT
ejpam-4979	118	11	(	(	PUNCT
ejpam-4979	118	12	2024	2024	NUM
ejpam-4979	118	13	)	)	PUNCT
ejpam-4979	118	14	,	,	PUNCT
ejpam-4979	118	15	42	42	NUM
ejpam-4979	118	16	-	-	SYM
ejpam-4979	118	17	58	58	NUM
ejpam-4979	118	18	50	50	NUM
ejpam-4979	118	19	proof	proof	NOUN
ejpam-4979	118	20	.	.	PUNCT
ejpam-4979	119	1	∏	∏	PROPN
ejpam-4979	119	2	p≤x	p≤x	PROPN
ejpam-4979	119	3	1	1	NUM
ejpam-4979	119	4	1−	1−	NUM
ejpam-4979	119	5	1	1	NUM
ejpam-4979	119	6	p	p	NOUN
ejpam-4979	119	7	=	=	SYM
ejpam-4979	119	8	∏	∏	PROPN
ejpam-4979	119	9	p≤x	p≤x	PROPN
ejpam-4979	119	10	(	(	PUNCT
ejpam-4979	119	11	1	1	NUM
ejpam-4979	119	12	+	+	SYM
ejpam-4979	119	13	1	1	NUM
ejpam-4979	119	14	p	p	NOUN
ejpam-4979	119	15	+	+	NOUN
ejpam-4979	119	16	1	1	NUM
ejpam-4979	119	17	p2	p2	NOUN
ejpam-4979	119	18	+	+	X
ejpam-4979	119	19	.	.	PUNCT
ejpam-4979	119	20	.	.	PUNCT
ejpam-4979	119	21	.	.	PUNCT
ejpam-4979	119	22	)	)	PUNCT
ejpam-4979	119	23	.	.	PUNCT
ejpam-4979	120	1	now	now	ADV
ejpam-4979	120	2	,	,	PUNCT
ejpam-4979	120	3	∏	∏	PROPN
ejpam-4979	120	4	p≤x	p≤x	PROPN
ejpam-4979	120	5	1	1	NUM
ejpam-4979	120	6	1−	1−	NUM
ejpam-4979	120	7	1	1	NUM
ejpam-4979	120	8	p	p	NOUN
ejpam-4979	120	9	>	>	X
ejpam-4979	120	10	n∑	n∑	INTJ
ejpam-4979	121	1	k=1	k=1	NOUN
ejpam-4979	121	2	1	1	NUM
ejpam-4979	121	3	k	k	X
ejpam-4979	121	4	>	>	X
ejpam-4979	121	5	∫	∫	PROPN
ejpam-4979	121	6	⌈x⌉	⌈x⌉	NOUN
ejpam-4979	121	7	1	1	NUM
ejpam-4979	121	8	du	du	PROPN
ejpam-4979	121	9	u	u	PROPN
ejpam-4979	121	10	>	>	X
ejpam-4979	121	11	lnx	lnx	PROPN
ejpam-4979	121	12	.	.	PUNCT
ejpam-4979	122	1	∴	∴	PROPN
ejpam-4979	122	2	∏	∏	PROPN
ejpam-4979	122	3	p≤x	p≤x	PROPN
ejpam-4979	122	4	(	(	PUNCT
ejpam-4979	122	5	1−	1−	NUM
ejpam-4979	122	6	1	1	NUM
ejpam-4979	122	7	p	p	NOUN
ejpam-4979	122	8	)	)	PUNCT
ejpam-4979	122	9	<	<	X
ejpam-4979	122	10	1	1	NUM
ejpam-4979	122	11	lnx	lnx	PROPN
ejpam-4979	122	12	⇒	⇒	PROPN
ejpam-4979	123	1	r∏	r∏	PROPN
ejpam-4979	123	2	i=1	i=1	PROPN
ejpam-4979	124	1	(	(	PUNCT
ejpam-4979	124	2	1−	1−	NUM
ejpam-4979	124	3	1	1	NUM
ejpam-4979	124	4	pi	pi	NOUN
ejpam-4979	124	5	)	)	PUNCT
ejpam-4979	125	1	<	<	X
ejpam-4979	125	2	1	1	NUM
ejpam-4979	125	3	ln	ln	ADJ
ejpam-4979	125	4	pr	pr	NOUN
ejpam-4979	125	5	.	.	PUNCT
ejpam-4979	126	1	we	we	PRON
ejpam-4979	126	2	now	now	ADV
ejpam-4979	126	3	have	have	VERB
ejpam-4979	126	4	as	as	ADP
ejpam-4979	126	5	our	our	PRON
ejpam-4979	126	6	estimate	estimate	NOUN
ejpam-4979	126	7	of	of	ADP
ejpam-4979	126	8	π(x	π(x	NOUN
ejpam-4979	126	9	)	)	PUNCT
ejpam-4979	126	10	,	,	PUNCT
ejpam-4979	126	11	π(x	π(x	NOUN
ejpam-4979	126	12	)	)	PUNCT
ejpam-4979	126	13	≤	≤	NOUN
ejpam-4979	127	1	r	r	NOUN
ejpam-4979	127	2	+	+	NOUN
ejpam-4979	127	3	x	x	SYM
ejpam-4979	127	4	ln	ln	ADJ
ejpam-4979	127	5	pr	pr	NOUN
ejpam-4979	128	1	+	+	CCONJ
ejpam-4979	128	2	2r	2r	NUM
ejpam-4979	128	3	.	.	PUNCT
ejpam-4979	129	1	(	(	PUNCT
ejpam-4979	129	2	11	11	NUM
ejpam-4979	129	3	)	)	SYM
ejpam-4979	129	4	5	5	NUM
ejpam-4979	129	5	.	.	PUNCT
ejpam-4979	129	6	estimating	estimate	VERB
ejpam-4979	129	7	π′′(x	π′′(x	NOUN
ejpam-4979	129	8	)	)	PUNCT
ejpam-4979	129	9	in	in	ADP
ejpam-4979	129	10	order	order	NOUN
ejpam-4979	129	11	to	to	PART
ejpam-4979	129	12	calculate	calculate	VERB
ejpam-4979	129	13	an	an	DET
ejpam-4979	129	14	estimate	estimate	NOUN
ejpam-4979	129	15	of	of	ADP
ejpam-4979	129	16	π′′(x	π′′(x	NOUN
ejpam-4979	129	17	)	)	PUNCT
ejpam-4979	129	18	,	,	PUNCT
ejpam-4979	129	19	we	we	PRON
ejpam-4979	129	20	begin	begin	VERB
ejpam-4979	129	21	by	by	ADP
ejpam-4979	129	22	taking	take	VERB
ejpam-4979	129	23	a	a	DET
ejpam-4979	129	24	look	look	NOUN
ejpam-4979	129	25	at	at	ADP
ejpam-4979	129	26	the	the	DET
ejpam-4979	129	27	coefficient	coefficient	NOUN
ejpam-4979	129	28	of	of	ADP
ejpam-4979	129	29	x	x	PUNCT
ejpam-4979	129	30	in	in	ADP
ejpam-4979	129	31	the	the	DET
ejpam-4979	129	32	middle	middle	ADJ
ejpam-4979	129	33	term	term	NOUN
ejpam-4979	129	34	on	on	ADP
ejpam-4979	129	35	the	the	DET
ejpam-4979	129	36	rhs	rhs	PROPN
ejpam-4979	129	37	of	of	ADP
ejpam-4979	129	38	10	10	NUM
ejpam-4979	129	39	.	.	PUNCT
ejpam-4979	130	1	we	we	PRON
ejpam-4979	130	2	can	can	AUX
ejpam-4979	130	3	write	write	VERB
ejpam-4979	130	4	that	that	DET
ejpam-4979	130	5	coefficient	coefficient	NOUN
ejpam-4979	130	6	as	as	ADP
ejpam-4979	130	7	r∏	r∏	PROPN
ejpam-4979	130	8	i=1	i=1	PROPN
ejpam-4979	131	1	(	(	PUNCT
ejpam-4979	131	2	1−	1−	NUM
ejpam-4979	131	3	1	1	NUM
ejpam-4979	131	4	pi	pi	NOUN
ejpam-4979	131	5	)	)	PUNCT
ejpam-4979	132	1	=	=	PUNCT
ejpam-4979	133	1	r′∏	r′∏	PRON
ejpam-4979	133	2	i=1	i=1	X
ejpam-4979	134	1	(	(	PUNCT
ejpam-4979	134	2	1−	1−	NUM
ejpam-4979	134	3	1	1	NUM
ejpam-4979	134	4	p′i	p′i	NOUN
ejpam-4979	134	5	)	)	PUNCT
ejpam-4979	134	6	·	·	PUNCT
ejpam-4979	134	7	r′′∏	r′′∏	NUM
ejpam-4979	135	1	i=1	i=1	PROPN
ejpam-4979	135	2	(	(	PUNCT
ejpam-4979	135	3	1−	1−	NUM
ejpam-4979	135	4	1	1	NUM
ejpam-4979	135	5	p′′i	p′′i	PROPN
ejpam-4979	135	6	)	)	PUNCT
ejpam-4979	135	7	(	(	PUNCT
ejpam-4979	135	8	12	12	NUM
ejpam-4979	135	9	)	)	PUNCT
ejpam-4979	135	10	where	where	SCONJ
ejpam-4979	135	11	r′	r′	NOUN
ejpam-4979	135	12	is	be	AUX
ejpam-4979	135	13	the	the	DET
ejpam-4979	135	14	number	number	NOUN
ejpam-4979	135	15	of	of	ADP
ejpam-4979	135	16	p′	p′	NOUN
ejpam-4979	135	17	<	<	X
ejpam-4979	135	18	the	the	DET
ejpam-4979	135	19	number	number	NOUN
ejpam-4979	135	20	of	of	ADP
ejpam-4979	135	21	the	the	DET
ejpam-4979	135	22	first	first	ADJ
ejpam-4979	135	23	r	r	NOUN
ejpam-4979	135	24	primes	prime	NOUN
ejpam-4979	135	25	≤	≤	NUM
ejpam-4979	135	26	√	√	DET
ejpam-4979	135	27	x	x	SYM
ejpam-4979	135	28	,	,	PUNCT
ejpam-4979	135	29	and	and	CCONJ
ejpam-4979	135	30	r′′	r′′	VERB
ejpam-4979	135	31	is	be	AUX
ejpam-4979	135	32	the	the	DET
ejpam-4979	135	33	number	number	NOUN
ejpam-4979	135	34	of	of	ADP
ejpam-4979	135	35	p′′	p′′	PROPN
ejpam-4979	135	36	<	<	X
ejpam-4979	135	37	the	the	DET
ejpam-4979	135	38	number	number	NOUN
ejpam-4979	135	39	of	of	ADP
ejpam-4979	135	40	the	the	DET
ejpam-4979	135	41	first	first	ADJ
ejpam-4979	135	42	r	r	NOUN
ejpam-4979	135	43	primes	prime	NOUN
ejpam-4979	135	44	≤	≤	NUM
ejpam-4979	135	45	√	√	NUM
ejpam-4979	135	46	x	x	PUNCT
ejpam-4979	135	47	such	such	ADJ
ejpam-4979	135	48	that	that	PRON
ejpam-4979	135	49	r′	r′	PROPN
ejpam-4979	136	1	+	+	CCONJ
ejpam-4979	136	2	r′′	r′′	VERB
ejpam-4979	136	3	=	=	PUNCT
ejpam-4979	136	4	r.	r.	NOUN
ejpam-4979	136	5	we	we	PRON
ejpam-4979	136	6	found	find	VERB
ejpam-4979	136	7	that	that	SCONJ
ejpam-4979	136	8	one	one	PRON
ejpam-4979	136	9	can	can	AUX
ejpam-4979	136	10	not	not	PART
ejpam-4979	136	11	simply	simply	ADV
ejpam-4979	136	12	divide	divide	VERB
ejpam-4979	136	13	the	the	DET
ejpam-4979	136	14	product	product	NOUN
ejpam-4979	136	15	on	on	ADP
ejpam-4979	136	16	the	the	DET
ejpam-4979	136	17	lhs	lhs	PROPN
ejpam-4979	136	18	of	of	ADP
ejpam-4979	136	19	eq	eq	PROPN
ejpam-4979	136	20	.	.	PROPN
ejpam-4979	136	21	12	12	NUM
ejpam-4979	136	22	by	by	ADP
ejpam-4979	136	23	either	either	DET
ejpam-4979	136	24	product	product	NOUN
ejpam-4979	136	25	on	on	ADP
ejpam-4979	136	26	the	the	DET
ejpam-4979	136	27	rhs	rhs	PROPN
ejpam-4979	136	28	of	of	ADP
ejpam-4979	136	29	eq	eq	PROPN
ejpam-4979	136	30	.	.	PROPN
ejpam-4979	136	31	12	12	NUM
ejpam-4979	136	32	and	and	CCONJ
ejpam-4979	136	33	expect	expect	VERB
ejpam-4979	136	34	the	the	DET
ejpam-4979	136	35	quotient	quotient	NOUN
ejpam-4979	136	36	to	to	PART
ejpam-4979	136	37	represent	represent	VERB
ejpam-4979	136	38	a	a	DET
ejpam-4979	136	39	pure	pure	ADJ
ejpam-4979	136	40	count	count	NOUN
ejpam-4979	136	41	of	of	ADP
ejpam-4979	136	42	p′	p′	NOUN
ejpam-4979	136	43	or	or	CCONJ
ejpam-4979	136	44	p′′	p′′	PROPN
ejpam-4979	136	45	≤	≤	PROPN
ejpam-4979	136	46	x.	x.	PUNCT
ejpam-4979	136	47	therefore	therefore	ADV
ejpam-4979	136	48	,	,	PUNCT
ejpam-4979	136	49	we	we	PRON
ejpam-4979	136	50	must	must	AUX
ejpam-4979	136	51	approach	approach	VERB
ejpam-4979	136	52	the	the	DET
ejpam-4979	136	53	problem	problem	NOUN
ejpam-4979	136	54	from	from	ADP
ejpam-4979	136	55	a	a	DET
ejpam-4979	136	56	different	different	ADJ
ejpam-4979	136	57	direction	direction	NOUN
ejpam-4979	136	58	;	;	PUNCT
ejpam-4979	136	59	i.e.	i.e.	X
ejpam-4979	136	60	,	,	PUNCT
ejpam-4979	136	61	we	we	PRON
ejpam-4979	136	62	must	must	AUX
ejpam-4979	136	63	find	find	VERB
ejpam-4979	136	64	another	another	DET
ejpam-4979	136	65	way	way	NOUN
ejpam-4979	136	66	to	to	PART
ejpam-4979	136	67	reduce	reduce	VERB
ejpam-4979	136	68	the	the	DET
ejpam-4979	136	69	coefficient	coefficient	NOUN
ejpam-4979	136	70	of	of	ADP
ejpam-4979	136	71	x	x	PUNCT
ejpam-4979	136	72	on	on	ADP
ejpam-4979	136	73	the	the	DET
ejpam-4979	136	74	rhs	rhs	PROPN
ejpam-4979	136	75	of	of	ADP
ejpam-4979	136	76	10	10	NUM
ejpam-4979	136	77	such	such	ADJ
ejpam-4979	136	78	that	that	SCONJ
ejpam-4979	136	79	the	the	DET
ejpam-4979	136	80	estimate	estimate	NOUN
ejpam-4979	136	81	will	will	AUX
ejpam-4979	136	82	leave	leave	VERB
ejpam-4979	136	83	the	the	DET
ejpam-4979	136	84	count	count	NOUN
ejpam-4979	136	85	of	of	ADP
ejpam-4979	136	86	p′′	p′′	PROPN
ejpam-4979	136	87	only	only	ADV
ejpam-4979	136	88	with	with	ADP
ejpam-4979	136	89	no	no	DET
ejpam-4979	136	90	p′	p′	NOUN
ejpam-4979	136	91	and	and	CCONJ
ejpam-4979	136	92	no	no	DET
ejpam-4979	136	93	composites	composite	NOUN
ejpam-4979	136	94	remaining	remain	VERB
ejpam-4979	136	95	when	when	SCONJ
ejpam-4979	136	96	the	the	DET
ejpam-4979	136	97	coefficient	coefficient	NOUN
ejpam-4979	136	98	is	be	AUX
ejpam-4979	136	99	multiplied	multiply	VERB
ejpam-4979	136	100	by	by	ADP
ejpam-4979	136	101	x.	x.	PROPN
ejpam-4979	136	102	hence	hence	ADV
ejpam-4979	136	103	,	,	PUNCT
ejpam-4979	136	104	we	we	PRON
ejpam-4979	136	105	model	model	VERB
ejpam-4979	136	106	the	the	DET
ejpam-4979	136	107	inequality	inequality	NOUN
ejpam-4979	136	108	in	in	ADP
ejpam-4979	136	109	theorem	theorem	NOUN
ejpam-4979	136	110	1	1	NUM
ejpam-4979	136	111	as	as	ADP
ejpam-4979	136	112	r∏	r∏	PROPN
ejpam-4979	136	113	i=1	i=1	PROPN
ejpam-4979	137	1	(	(	PUNCT
ejpam-4979	137	2	1−	1−	NUM
ejpam-4979	137	3	1	1	NUM
ejpam-4979	137	4	pi	pi	NOUN
ejpam-4979	137	5	)	)	PUNCT
ejpam-4979	138	1	<	<	X
ejpam-4979	138	2	1	1	NUM
ejpam-4979	138	3	ln	ln	NOUN
ejpam-4979	138	4	pr	pr	NOUN
ejpam-4979	138	5	=	=	NOUN
ejpam-4979	138	6	1	1	NUM
ejpam-4979	138	7	lnj	lnj	PROPN
ejpam-4979	138	8	pr	pr	NOUN
ejpam-4979	138	9	·	·	PUNCT
ejpam-4979	138	10	lnk	lnk	ADJ
ejpam-4979	138	11	pr	pr	NOUN
ejpam-4979	138	12	.	.	PUNCT
ejpam-4979	139	1	(	(	PUNCT
ejpam-4979	139	2	13	13	NUM
ejpam-4979	139	3	)	)	PUNCT
ejpam-4979	139	4	it	it	PRON
ejpam-4979	139	5	was	be	AUX
ejpam-4979	139	6	found	find	VERB
ejpam-4979	139	7	that	that	SCONJ
ejpam-4979	139	8	if	if	SCONJ
ejpam-4979	139	9	the	the	DET
ejpam-4979	139	10	last	last	ADJ
ejpam-4979	139	11	term	term	NOUN
ejpam-4979	139	12	on	on	ADP
ejpam-4979	139	13	the	the	DET
ejpam-4979	139	14	rhs	rhs	PROPN
ejpam-4979	139	15	of	of	ADP
ejpam-4979	139	16	eq	eq	PROPN
ejpam-4979	139	17	.	.	PROPN
ejpam-4979	139	18	13	13	NUM
ejpam-4979	139	19	is	be	AUX
ejpam-4979	139	20	multiplied	multiply	VERB
ejpam-4979	139	21	by	by	ADP
ejpam-4979	139	22	either	either	PRON
ejpam-4979	139	23	lnj	lnj	PROPN
ejpam-4979	139	24	pr	pr	NOUN
ejpam-4979	139	25	or	or	CCONJ
ejpam-4979	139	26	lnk	lnk	ADJ
ejpam-4979	139	27	pr	pr	NOUN
ejpam-4979	139	28	,	,	PUNCT
ejpam-4979	139	29	then	then	ADV
ejpam-4979	139	30	the	the	DET
ejpam-4979	139	31	resultant	resultant	NOUN
ejpam-4979	139	32	value	value	NOUN
ejpam-4979	139	33	is	be	AUX
ejpam-4979	139	34	greater	great	ADJ
ejpam-4979	139	35	than	than	ADP
ejpam-4979	139	36	1	1	NUM
ejpam-4979	139	37	ln	ln	NOUN
ejpam-4979	139	38	pr	pr	NOUN
ejpam-4979	139	39	m.	m.	NOUN
ejpam-4979	140	1	p.	p.	PROPN
ejpam-4979	140	2	may	may	AUX
ejpam-4979	140	3	/	/	SYM
ejpam-4979	140	4	eur	eur	PROPN
ejpam-4979	140	5	.	.	PUNCT
ejpam-4979	141	1	j.	j.	PROPN
ejpam-4979	141	2	pure	pure	PROPN
ejpam-4979	141	3	appl	appl	PROPN
ejpam-4979	141	4	.	.	PROPN
ejpam-4979	141	5	math	math	PROPN
ejpam-4979	141	6	,	,	PUNCT
ejpam-4979	141	7	17	17	NUM
ejpam-4979	141	8	(	(	PUNCT
ejpam-4979	141	9	1	1	NUM
ejpam-4979	141	10	)	)	PUNCT
ejpam-4979	141	11	(	(	PUNCT
ejpam-4979	141	12	2024	2024	NUM
ejpam-4979	141	13	)	)	PUNCT
ejpam-4979	141	14	,	,	PUNCT
ejpam-4979	141	15	42	42	NUM
ejpam-4979	141	16	-	-	SYM
ejpam-4979	141	17	58	58	NUM
ejpam-4979	141	18	51	51	NUM
ejpam-4979	141	19	which	which	PRON
ejpam-4979	141	20	is	be	AUX
ejpam-4979	141	21	counterintuitive	counterintuitive	ADJ
ejpam-4979	141	22	to	to	ADP
ejpam-4979	141	23	our	our	PRON
ejpam-4979	141	24	proof	proof	NOUN
ejpam-4979	141	25	that	that	SCONJ
ejpam-4979	141	26	the	the	DET
ejpam-4979	141	27	complementary	complementary	ADJ
ejpam-4979	141	28	prime	prime	ADJ
ejpam-4979	141	29	number	number	NOUN
ejpam-4979	141	30	subsequences	subsequence	VERB
ejpam-4979	141	31	p′	p′	NOUN
ejpam-4979	141	32	and	and	CCONJ
ejpam-4979	141	33	p′′	p′′	PROPN
ejpam-4979	141	34	add	add	VERB
ejpam-4979	141	35	to	to	PART
ejpam-4979	141	36	form	form	VERB
ejpam-4979	141	37	the	the	DET
ejpam-4979	141	38	complete	complete	ADJ
ejpam-4979	141	39	set	set	NOUN
ejpam-4979	141	40	of	of	ADP
ejpam-4979	141	41	prime	prime	ADJ
ejpam-4979	141	42	numbers	number	NOUN
ejpam-4979	142	1	p.	p.	NOUN
ejpam-4979	142	2	multiplying	multiply	VERB
ejpam-4979	142	3	the	the	DET
ejpam-4979	142	4	last	last	ADJ
ejpam-4979	142	5	term	term	NOUN
ejpam-4979	142	6	of	of	ADP
ejpam-4979	142	7	eq	eq	NOUN
ejpam-4979	142	8	.	.	PROPN
ejpam-4979	142	9	13	13	NUM
ejpam-4979	142	10	by	by	ADP
ejpam-4979	142	11	either	either	CCONJ
ejpam-4979	142	12	lnj	lnj	PROPN
ejpam-4979	142	13	pr	pr	NOUN
ejpam-4979	142	14	or	or	CCONJ
ejpam-4979	142	15	lnk	lnk	NOUN
ejpam-4979	142	16	pr	pr	NOUN
ejpam-4979	142	17	actually	actually	ADV
ejpam-4979	142	18	increases	increase	VERB
ejpam-4979	142	19	the	the	DET
ejpam-4979	142	20	cardinality	cardinality	NOUN
ejpam-4979	142	21	of	of	ADP
ejpam-4979	142	22	p′(x	p′(x	NOUN
ejpam-4979	142	23	)	)	PUNCT
ejpam-4979	142	24	or	or	CCONJ
ejpam-4979	142	25	p′′(x	p′′(x	NOUN
ejpam-4979	142	26	)	)	PUNCT
ejpam-4979	142	27	to	to	PART
ejpam-4979	142	28	be	be	AUX
ejpam-4979	142	29	greater	great	ADJ
ejpam-4979	142	30	than	than	ADP
ejpam-4979	142	31	the	the	DET
ejpam-4979	142	32	cardinality	cardinality	NOUN
ejpam-4979	142	33	of	of	ADP
ejpam-4979	142	34	the	the	DET
ejpam-4979	142	35	entire	entire	ADJ
ejpam-4979	142	36	set	set	NOUN
ejpam-4979	142	37	of	of	ADP
ejpam-4979	142	38	prime	prime	ADJ
ejpam-4979	142	39	numbers	number	NOUN
ejpam-4979	142	40	p(x	p(x	PROPN
ejpam-4979	142	41	)	)	PUNCT
ejpam-4979	142	42	when	when	SCONJ
ejpam-4979	142	43	that	that	DET
ejpam-4979	142	44	coefficient	coefficient	NOUN
ejpam-4979	142	45	is	be	AUX
ejpam-4979	142	46	multiplied	multiply	VERB
ejpam-4979	142	47	by	by	ADP
ejpam-4979	142	48	x	x	PUNCT
ejpam-4979	142	49	on	on	ADP
ejpam-4979	142	50	the	the	DET
ejpam-4979	142	51	rhs	rhs	PROPN
ejpam-4979	142	52	of	of	ADP
ejpam-4979	142	53	10	10	NUM
ejpam-4979	142	54	.	.	PUNCT
ejpam-4979	143	1	hence	hence	ADV
ejpam-4979	143	2	our	our	PRON
ejpam-4979	143	3	motivation	motivation	NOUN
ejpam-4979	143	4	to	to	PART
ejpam-4979	143	5	approach	approach	VERB
ejpam-4979	143	6	the	the	DET
ejpam-4979	143	7	solution	solution	NOUN
ejpam-4979	143	8	from	from	ADP
ejpam-4979	143	9	a	a	DET
ejpam-4979	143	10	different	different	ADJ
ejpam-4979	143	11	direction	direction	NOUN
ejpam-4979	143	12	.	.	PUNCT
ejpam-4979	144	1	in	in	ADP
ejpam-4979	144	2	that	that	DET
ejpam-4979	144	3	light	light	NOUN
ejpam-4979	144	4	,	,	PUNCT
ejpam-4979	144	5	it	it	PRON
ejpam-4979	144	6	was	be	AUX
ejpam-4979	144	7	found	find	VERB
ejpam-4979	144	8	that	that	SCONJ
ejpam-4979	144	9	if	if	SCONJ
ejpam-4979	144	10	we	we	PRON
ejpam-4979	144	11	let	let	VERB
ejpam-4979	144	12	1	1	NUM
ejpam-4979	144	13	lnj	lnj	PROPN
ejpam-4979	144	14	pr	pr	NOUN
ejpam-4979	144	15	·	·	PUNCT
ejpam-4979	144	16	lnk	lnk	ADJ
ejpam-4979	144	17	pr	pr	NOUN
ejpam-4979	144	18	=	=	SYM
ejpam-4979	144	19	1	1	NUM
ejpam-4979	144	20	ln	ln	NOUN
ejpam-4979	144	21	pr	pr	NOUN
ejpam-4979	144	22	·	·	PUNCT
ejpam-4979	145	1	[	[	PUNCT
ejpam-4979	145	2	j	j	X
ejpam-4979	145	3	+	+	CCONJ
ejpam-4979	145	4	k	k	X
ejpam-4979	145	5	]	]	X
ejpam-4979	145	6	(	(	PUNCT
ejpam-4979	145	7	14	14	NUM
ejpam-4979	145	8	)	)	PUNCT
ejpam-4979	145	9	we	we	PRON
ejpam-4979	145	10	can	can	AUX
ejpam-4979	145	11	then	then	ADV
ejpam-4979	145	12	subtract	subtract	VERB
ejpam-4979	145	13	j	j	PROPN
ejpam-4979	145	14	ln	ln	ADJ
ejpam-4979	145	15	pr	pr	NOUN
ejpam-4979	145	16	or	or	CCONJ
ejpam-4979	145	17	k	k	X
ejpam-4979	145	18	ln	ln	ADJ
ejpam-4979	145	19	pr	pr	NOUN
ejpam-4979	145	20	from	from	ADP
ejpam-4979	145	21	the	the	DET
ejpam-4979	145	22	lhs	lhs	PROPN
ejpam-4979	145	23	of	of	ADP
ejpam-4979	145	24	eq	eq	PROPN
ejpam-4979	145	25	.	.	PROPN
ejpam-4979	145	26	14	14	NUM
ejpam-4979	145	27	(	(	PUNCT
ejpam-4979	145	28	or	or	CCONJ
ejpam-4979	145	29	from	from	ADP
ejpam-4979	145	30	the	the	DET
ejpam-4979	145	31	rhs	rhs	PROPN
ejpam-4979	145	32	of	of	ADP
ejpam-4979	145	33	eq	eq	PROPN
ejpam-4979	145	34	.	.	PROPN
ejpam-4979	145	35	13	13	NUM
ejpam-4979	145	36	)	)	PUNCT
ejpam-4979	145	37	and	and	CCONJ
ejpam-4979	145	38	obtain	obtain	VERB
ejpam-4979	145	39	the	the	DET
ejpam-4979	145	40	proper	proper	ADJ
ejpam-4979	145	41	coefficient	coefficient	NOUN
ejpam-4979	145	42	to	to	PART
ejpam-4979	145	43	multiply	multiply	VERB
ejpam-4979	145	44	times	time	NOUN
ejpam-4979	145	45	x	x	PUNCT
ejpam-4979	145	46	on	on	ADP
ejpam-4979	145	47	the	the	DET
ejpam-4979	145	48	rhs	rhs	PROPN
ejpam-4979	145	49	of	of	ADP
ejpam-4979	145	50	11	11	NUM
ejpam-4979	145	51	to	to	PART
ejpam-4979	145	52	obtain	obtain	VERB
ejpam-4979	145	53	the	the	DET
ejpam-4979	145	54	correct	correct	ADJ
ejpam-4979	145	55	estimate	estimate	NOUN
ejpam-4979	145	56	of	of	ADP
ejpam-4979	145	57	the	the	DET
ejpam-4979	145	58	quantity	quantity	NOUN
ejpam-4979	145	59	of	of	ADP
ejpam-4979	145	60	p′(x	p′(x	NOUN
ejpam-4979	145	61	)	)	PUNCT
ejpam-4979	145	62	or	or	CCONJ
ejpam-4979	145	63	p′′(x	p′′(x	NOUN
ejpam-4979	145	64	)	)	PUNCT
ejpam-4979	145	65	depending	depend	VERB
ejpam-4979	145	66	upon	upon	SCONJ
ejpam-4979	145	67	which	which	PRON
ejpam-4979	145	68	of	of	ADP
ejpam-4979	145	69	these	these	DET
ejpam-4979	145	70	quantities	quantity	NOUN
ejpam-4979	145	71	is	be	AUX
ejpam-4979	145	72	subtracted	subtract	VERB
ejpam-4979	145	73	from	from	ADP
ejpam-4979	145	74	the	the	DET
ejpam-4979	145	75	rhs	rhs	PROPN
ejpam-4979	145	76	of	of	ADP
ejpam-4979	145	77	eq	eq	PROPN
ejpam-4979	145	78	.	.	PROPN
ejpam-4979	145	79	14	14	NUM
ejpam-4979	145	80	.	.	PUNCT
ejpam-4979	146	1	so	so	ADV
ejpam-4979	146	2	the	the	DET
ejpam-4979	146	3	task	task	NOUN
ejpam-4979	146	4	at	at	ADP
ejpam-4979	146	5	hand	hand	NOUN
ejpam-4979	146	6	is	be	AUX
ejpam-4979	146	7	to	to	PART
ejpam-4979	146	8	find	find	VERB
ejpam-4979	146	9	a	a	DET
ejpam-4979	146	10	j	j	NOUN
ejpam-4979	146	11	and	and	CCONJ
ejpam-4979	146	12	a	a	DET
ejpam-4979	146	13	k	k	NOUN
ejpam-4979	146	14	that	that	PRON
ejpam-4979	146	15	will	will	AUX
ejpam-4979	146	16	satisfy	satisfy	VERB
ejpam-4979	146	17	both	both	DET
ejpam-4979	146	18	sides	side	NOUN
ejpam-4979	146	19	of	of	ADP
ejpam-4979	146	20	eq	eq	NOUN
ejpam-4979	146	21	.	.	PROPN
ejpam-4979	146	22	14	14	NUM
ejpam-4979	146	23	.	.	PUNCT
ejpam-4979	147	1	to	to	ADP
ejpam-4979	147	2	that	that	DET
ejpam-4979	147	3	end	end	NOUN
ejpam-4979	147	4	,	,	PUNCT
ejpam-4979	147	5	it	it	PRON
ejpam-4979	147	6	is	be	AUX
ejpam-4979	147	7	seen	see	VERB
ejpam-4979	147	8	in	in	ADP
ejpam-4979	147	9	eq	eq	NOUN
ejpam-4979	147	10	.	.	PROPN
ejpam-4979	147	11	14	14	NUM
ejpam-4979	147	12	that	that	PRON
ejpam-4979	147	13	j	j	PROPN
ejpam-4979	147	14	and	and	CCONJ
ejpam-4979	147	15	k	k	PROPN
ejpam-4979	147	16	must	must	AUX
ejpam-4979	147	17	sum	sum	VERB
ejpam-4979	147	18	to	to	ADP
ejpam-4979	147	19	unity	unity	NOUN
ejpam-4979	147	20	on	on	ADP
ejpam-4979	147	21	both	both	DET
ejpam-4979	147	22	sides	side	NOUN
ejpam-4979	147	23	of	of	ADP
ejpam-4979	147	24	the	the	DET
ejpam-4979	147	25	equation	equation	NOUN
ejpam-4979	147	26	to	to	PART
ejpam-4979	147	27	make	make	VERB
ejpam-4979	147	28	this	this	DET
ejpam-4979	147	29	approach	approach	NOUN
ejpam-4979	147	30	work	work	NOUN
ejpam-4979	147	31	.	.	PUNCT
ejpam-4979	148	1	in	in	ADP
ejpam-4979	148	2	order	order	NOUN
ejpam-4979	148	3	to	to	PART
ejpam-4979	148	4	do	do	AUX
ejpam-4979	148	5	so	so	ADV
ejpam-4979	148	6	,	,	PUNCT
ejpam-4979	148	7	we	we	PRON
ejpam-4979	148	8	recall	recall	VERB
ejpam-4979	148	9	the	the	DET
ejpam-4979	148	10	asymptotic	asymptotic	ADJ
ejpam-4979	148	11	densities	density	NOUN
ejpam-4979	148	12	that	that	PRON
ejpam-4979	148	13	we	we	PRON
ejpam-4979	148	14	derived	derive	VERB
ejpam-4979	148	15	earlier	early	ADV
ejpam-4979	148	16	for	for	ADP
ejpam-4979	148	17	π′(x	π′(x	PRON
ejpam-4979	148	18	)	)	PUNCT
ejpam-4979	148	19	and	and	CCONJ
ejpam-4979	148	20	π′′(x	π′′(x	NOUN
ejpam-4979	148	21	)	)	PUNCT
ejpam-4979	148	22	as	as	ADP
ejpam-4979	148	23	π′(x	π′(x	NOUN
ejpam-4979	148	24	)	)	PUNCT
ejpam-4979	148	25	∼	∼	NOUN
ejpam-4979	148	26	1	1	NUM
ejpam-4979	148	27	lnn+	lnn+	SYM
ejpam-4979	148	28	1	1	NUM
ejpam-4979	148	29	and	and	CCONJ
ejpam-4979	148	30	π′′(x	π′′(x	NOUN
ejpam-4979	148	31	)	)	PUNCT
ejpam-4979	148	32	∼	∼	NOUN
ejpam-4979	148	33	1	1	NUM
ejpam-4979	148	34	lnn(lnn+	lnn(lnn+	ADP
ejpam-4979	148	35	1	1	NUM
ejpam-4979	148	36	)	)	PUNCT
ejpam-4979	148	37	.	.	PUNCT
ejpam-4979	149	1	now	now	ADV
ejpam-4979	149	2	,	,	PUNCT
ejpam-4979	149	3	since	since	SCONJ
ejpam-4979	149	4	j	j	PROPN
ejpam-4979	150	1	+	+	CCONJ
ejpam-4979	150	2	k	k	NOUN
ejpam-4979	150	3	=	=	SYM
ejpam-4979	150	4	1	1	NUM
ejpam-4979	150	5	must	must	AUX
ejpam-4979	150	6	hold	hold	VERB
ejpam-4979	150	7	true	true	ADJ
ejpam-4979	150	8	to	to	PART
ejpam-4979	150	9	satisfy	satisfy	VERB
ejpam-4979	150	10	eq	eq	ADP
ejpam-4979	150	11	.	.	PROPN
ejpam-4979	150	12	14	14	NUM
ejpam-4979	150	13	,	,	PUNCT
ejpam-4979	150	14	we	we	PRON
ejpam-4979	150	15	let	let	VERB
ejpam-4979	150	16	j	j	NOUN
ejpam-4979	150	17	=	=	SYM
ejpam-4979	150	18	1	1	NUM
ejpam-4979	150	19	lnn(lnn+	lnn(lnn+	PROPN
ejpam-4979	150	20	1	1	NUM
ejpam-4979	150	21	)	)	SYM
ejpam-4979	150	22	1	1	NUM
ejpam-4979	150	23	lnn	lnn	NOUN
ejpam-4979	150	24	=	=	SYM
ejpam-4979	150	25	lnn	lnn	NOUN
ejpam-4979	150	26	lnn(lnn+	lnn(lnn+	PROPN
ejpam-4979	150	27	1	1	NUM
ejpam-4979	150	28	)	)	PUNCT
ejpam-4979	150	29	and	and	CCONJ
ejpam-4979	150	30	k	k	NOUN
ejpam-4979	150	31	=	=	SYM
ejpam-4979	150	32	1	1	NUM
ejpam-4979	150	33	lnn+	lnn+	SYM
ejpam-4979	150	34	1	1	NUM
ejpam-4979	150	35	1	1	NUM
ejpam-4979	150	36	lnn	lnn	NOUN
ejpam-4979	150	37	=	=	SYM
ejpam-4979	150	38	lnn	lnn	NOUN
ejpam-4979	150	39	lnn+	lnn+	ADJ
ejpam-4979	150	40	1	1	NUM
ejpam-4979	150	41	such	such	ADJ
ejpam-4979	150	42	that	that	PRON
ejpam-4979	150	43	j	j	PROPN
ejpam-4979	151	1	=	=	PUNCT
ejpam-4979	152	1	the	the	DET
ejpam-4979	152	2	ratio	ratio	NOUN
ejpam-4979	152	3	of	of	ADP
ejpam-4979	152	4	the	the	DET
ejpam-4979	152	5	asymptotic	asymptotic	ADJ
ejpam-4979	152	6	density	density	NOUN
ejpam-4979	152	7	of	of	ADP
ejpam-4979	152	8	the	the	DET
ejpam-4979	152	9	prime	prime	ADJ
ejpam-4979	152	10	subsequence	subsequence	NOUN
ejpam-4979	152	11	p′′	p′′	PROPN
ejpam-4979	152	12	divided	divide	VERB
ejpam-4979	152	13	by	by	ADP
ejpam-4979	152	14	the	the	DET
ejpam-4979	152	15	asymptotic	asymptotic	ADJ
ejpam-4979	152	16	density	density	NOUN
ejpam-4979	152	17	of	of	ADP
ejpam-4979	152	18	the	the	DET
ejpam-4979	152	19	set	set	NOUN
ejpam-4979	152	20	of	of	ADP
ejpam-4979	152	21	all	all	DET
ejpam-4979	152	22	primes	prime	NOUN
ejpam-4979	152	23	p	p	X
ejpam-4979	152	24	;	;	PUNCT
ejpam-4979	152	25	and	and	CCONJ
ejpam-4979	152	26	k	k	X
ejpam-4979	153	1	=	=	PUNCT
ejpam-4979	153	2	the	the	DET
ejpam-4979	153	3	ratio	ratio	NOUN
ejpam-4979	153	4	of	of	ADP
ejpam-4979	153	5	the	the	DET
ejpam-4979	153	6	asymptotic	asymptotic	ADJ
ejpam-4979	153	7	density	density	NOUN
ejpam-4979	153	8	of	of	ADP
ejpam-4979	153	9	the	the	DET
ejpam-4979	153	10	prime	prime	ADJ
ejpam-4979	153	11	subsequence	subsequence	NOUN
ejpam-4979	153	12	p′	p′	NOUN
ejpam-4979	153	13	to	to	ADP
ejpam-4979	153	14	the	the	DET
ejpam-4979	153	15	asymptotic	asymptotic	ADJ
ejpam-4979	153	16	density	density	NOUN
ejpam-4979	153	17	of	of	ADP
ejpam-4979	153	18	all	all	DET
ejpam-4979	153	19	primes	prime	NOUN
ejpam-4979	153	20	p.	p.	NOUN
ejpam-4979	153	21	we	we	PRON
ejpam-4979	153	22	now	now	ADV
ejpam-4979	153	23	introduce	introduce	VERB
ejpam-4979	153	24	a	a	DET
ejpam-4979	153	25	lemma	lemma	PROPN
ejpam-4979	153	26	:	:	PUNCT
ejpam-4979	153	27	lemma	lemma	PROPN
ejpam-4979	153	28	1	1	NUM
ejpam-4979	153	29	.	.	PUNCT
ejpam-4979	154	1	for	for	ADP
ejpam-4979	154	2	x	x	SYM
ejpam-4979	154	3	>	>	X
ejpam-4979	154	4	1	1	NUM
ejpam-4979	154	5	,	,	PUNCT
ejpam-4979	154	6	j	j	PROPN
ejpam-4979	154	7	+	+	CCONJ
ejpam-4979	154	8	k	k	NOUN
ejpam-4979	154	9	=	=	SYM
ejpam-4979	154	10	1	1	X
ejpam-4979	154	11	.	.	PUNCT
ejpam-4979	154	12	m.	m.	NOUN
ejpam-4979	154	13	p.	p.	PROPN
ejpam-4979	154	14	may	may	AUX
ejpam-4979	154	15	/	/	SYM
ejpam-4979	154	16	eur	eur	PROPN
ejpam-4979	154	17	.	.	PUNCT
ejpam-4979	155	1	j.	j.	PROPN
ejpam-4979	155	2	pure	pure	PROPN
ejpam-4979	155	3	appl	appl	PROPN
ejpam-4979	155	4	.	.	PROPN
ejpam-4979	155	5	math	math	PROPN
ejpam-4979	155	6	,	,	PUNCT
ejpam-4979	155	7	17	17	NUM
ejpam-4979	155	8	(	(	PUNCT
ejpam-4979	155	9	1	1	NUM
ejpam-4979	155	10	)	)	PUNCT
ejpam-4979	155	11	(	(	PUNCT
ejpam-4979	155	12	2024	2024	NUM
ejpam-4979	155	13	)	)	PUNCT
ejpam-4979	155	14	,	,	PUNCT
ejpam-4979	155	15	42	42	NUM
ejpam-4979	155	16	-	-	SYM
ejpam-4979	155	17	58	58	NUM
ejpam-4979	155	18	52	52	NUM
ejpam-4979	155	19	proof	proof	NOUN
ejpam-4979	155	20	.	.	PUNCT
ejpam-4979	156	1	j	j	PROPN
ejpam-4979	157	1	+	+	CCONJ
ejpam-4979	157	2	k	k	X
ejpam-4979	157	3	=	=	PUNCT
ejpam-4979	157	4	lnx	lnx	PROPN
ejpam-4979	157	5	lnx+	lnx+	PROPN
ejpam-4979	157	6	1	1	NUM
ejpam-4979	157	7	+	+	CCONJ
ejpam-4979	157	8	lnx	lnx	PROPN
ejpam-4979	157	9	lnx(lnx+	lnx(lnx+	PROPN
ejpam-4979	157	10	1	1	NUM
ejpam-4979	157	11	)	)	PUNCT
ejpam-4979	157	12	=	=	SYM
ejpam-4979	157	13	lnx(lnx+	lnx(lnx+	ADJ
ejpam-4979	157	14	1	1	NUM
ejpam-4979	157	15	)	)	PUNCT
ejpam-4979	157	16	+	+	CCONJ
ejpam-4979	157	17	lnx	lnx	PROPN
ejpam-4979	157	18	lnx(lnx+	lnx(lnx+	PROPN
ejpam-4979	157	19	1	1	NUM
ejpam-4979	157	20	)	)	PUNCT
ejpam-4979	157	21	lnx(lnx+	lnx(lnx+	PROPN
ejpam-4979	157	22	1)2	1)2	NUM
ejpam-4979	157	23	=	=	SYM
ejpam-4979	157	24	lnx(lnx+	lnx(lnx+	PROPN
ejpam-4979	157	25	1)(lnx+	1)(lnx+	PROPN
ejpam-4979	157	26	1	1	NUM
ejpam-4979	157	27	)	)	PUNCT
ejpam-4979	157	28	lnx(lnx+	lnx(lnx+	PROPN
ejpam-4979	157	29	1)2	1)2	NUM
ejpam-4979	157	30	=	=	SYM
ejpam-4979	157	31	1	1	X
ejpam-4979	157	32	.	.	PUNCT
ejpam-4979	158	1	since	since	SCONJ
ejpam-4979	158	2	j	j	PROPN
ejpam-4979	158	3	+	+	CCONJ
ejpam-4979	158	4	k	k	NOUN
ejpam-4979	158	5	=	=	SYM
ejpam-4979	158	6	1	1	NUM
ejpam-4979	158	7	is	be	AUX
ejpam-4979	158	8	valid	valid	ADJ
ejpam-4979	158	9	in	in	ADP
ejpam-4979	158	10	the	the	DET
ejpam-4979	158	11	lemma	lemma	PROPN
ejpam-4979	158	12	for	for	ADP
ejpam-4979	158	13	all	all	PRON
ejpam-4979	158	14	x	x	SYM
ejpam-4979	158	15	>	>	X
ejpam-4979	158	16	1	1	NUM
ejpam-4979	158	17	,	,	PUNCT
ejpam-4979	158	18	we	we	PRON
ejpam-4979	158	19	have	have	AUX
ejpam-4979	158	20	established	establish	VERB
ejpam-4979	158	21	that	that	SCONJ
ejpam-4979	158	22	1	1	NUM
ejpam-4979	158	23	lnj	lnj	PROPN
ejpam-4979	158	24	pr	pr	NOUN
ejpam-4979	158	25	·	·	PUNCT
ejpam-4979	158	26	lnk	lnk	ADJ
ejpam-4979	158	27	pr	pr	NOUN
ejpam-4979	158	28	=	=	SYM
ejpam-4979	158	29	1	1	NUM
ejpam-4979	158	30	ln	ln	NOUN
ejpam-4979	158	31	pr	pr	NOUN
ejpam-4979	158	32	·	·	PUNCT
ejpam-4979	159	1	[	[	PUNCT
ejpam-4979	159	2	j	j	X
ejpam-4979	159	3	+	+	CCONJ
ejpam-4979	159	4	k	k	X
ejpam-4979	159	5	]	]	X
ejpam-4979	159	6	=	=	PUNCT
ejpam-4979	159	7	1	1	NUM
ejpam-4979	159	8	ln	ln	NOUN
ejpam-4979	159	9	pr	pr	NOUN
ejpam-4979	159	10	·	·	PUNCT
ejpam-4979	159	11	[	[	PUNCT
ejpam-4979	159	12	ln	ln	ADJ
ejpam-4979	159	13	pr	pr	NOUN
ejpam-4979	159	14	ln	ln	ADJ
ejpam-4979	159	15	pr(ln	pr(ln	NOUN
ejpam-4979	159	16	pr	pr	NOUN
ejpam-4979	159	17	+	+	CCONJ
ejpam-4979	159	18	1	1	NUM
ejpam-4979	159	19	)	)	PUNCT
ejpam-4979	159	20	+	+	CCONJ
ejpam-4979	159	21	ln	ln	ADJ
ejpam-4979	159	22	pr	pr	NOUN
ejpam-4979	159	23	ln	ln	ADJ
ejpam-4979	159	24	pr	pr	NOUN
ejpam-4979	159	25	+	+	CCONJ
ejpam-4979	159	26	1	1	NUM
ejpam-4979	159	27	]	]	PUNCT
ejpam-4979	159	28	.	.	PUNCT
ejpam-4979	160	1	thus	thus	ADV
ejpam-4979	160	2	,	,	PUNCT
ejpam-4979	160	3	in	in	ADP
ejpam-4979	160	4	harmony	harmony	NOUN
ejpam-4979	160	5	with	with	ADP
ejpam-4979	160	6	the	the	DET
ejpam-4979	160	7	lemma	lemma	PROPN
ejpam-4979	160	8	,	,	PUNCT
ejpam-4979	160	9	we	we	PRON
ejpam-4979	160	10	have	have	AUX
ejpam-4979	160	11	π′′(x	π′′(x	NOUN
ejpam-4979	160	12	)	)	PUNCT
ejpam-4979	160	13	≤	≤	NOUN
ejpam-4979	160	14	r′′	r′′	VERB
ejpam-4979	160	15	+	+	CCONJ
ejpam-4979	160	16	x	x	SYM
ejpam-4979	160	17	ln	ln	ADJ
ejpam-4979	160	18	pr(ln	pr(ln	NOUN
ejpam-4979	160	19	pr	pr	X
ejpam-4979	161	1	+	+	CCONJ
ejpam-4979	161	2	1	1	NUM
ejpam-4979	161	3	)	)	PUNCT
ejpam-4979	162	1	+	+	SYM
ejpam-4979	162	2	2r	2r	NUM
ejpam-4979	162	3	′′	′′	PROPN
ejpam-4979	162	4	(	(	PUNCT
ejpam-4979	162	5	15	15	NUM
ejpam-4979	162	6	)	)	PUNCT
ejpam-4979	162	7	where	where	SCONJ
ejpam-4979	162	8	r′′	r′′	VERB
ejpam-4979	162	9	=	=	PUNCT
ejpam-4979	162	10	the	the	DET
ejpam-4979	162	11	number	number	NOUN
ejpam-4979	162	12	of	of	ADP
ejpam-4979	162	13	p′′	p′′	PROPN
ejpam-4979	162	14	≤	≤	PROPN
ejpam-4979	162	15	pr	pr	NOUN
ejpam-4979	162	16	and	and	CCONJ
ejpam-4979	162	17	2r	2r	NUM
ejpam-4979	162	18	′′	′′	PROPN
ejpam-4979	162	19	=	=	PRON
ejpam-4979	162	20	the	the	DET
ejpam-4979	162	21	maximum	maximum	ADJ
ejpam-4979	162	22	error	error	NOUN
ejpam-4979	162	23	resulting	result	VERB
ejpam-4979	162	24	from	from	ADP
ejpam-4979	162	25	the	the	DET
ejpam-4979	162	26	main	main	ADJ
ejpam-4979	162	27	term	term	NOUN
ejpam-4979	162	28	.	.	PUNCT
ejpam-4979	163	1	we	we	PRON
ejpam-4979	163	2	now	now	ADV
ejpam-4979	163	3	proceed	proceed	VERB
ejpam-4979	163	4	with	with	ADP
ejpam-4979	163	5	our	our	PRON
ejpam-4979	163	6	estimate	estimate	NOUN
ejpam-4979	163	7	of	of	ADP
ejpam-4979	163	8	15	15	NUM
ejpam-4979	163	9	.	.	PUNCT
ejpam-4979	164	1	we	we	PRON
ejpam-4979	164	2	know	know	VERB
ejpam-4979	164	3	that	that	SCONJ
ejpam-4979	164	4	r′′	r′′	VERB
ejpam-4979	164	5	≤	≤	NOUN
ejpam-4979	164	6	⌊r	⌊r	CCONJ
ejpam-4979	164	7	2	2	NUM
ejpam-4979	164	8	⌋	⌋	NOUN
ejpam-4979	164	9	for	for	ADP
ejpam-4979	164	10	p′′	p′′	PROPN
ejpam-4979	164	11	>	>	X
ejpam-4979	164	12	2	2	NUM
ejpam-4979	164	13	because	because	SCONJ
ejpam-4979	164	14	it	it	PRON
ejpam-4979	164	15	was	be	AUX
ejpam-4979	164	16	proven	prove	VERB
ejpam-4979	164	17	that	that	SCONJ
ejpam-4979	164	18	there	there	PRON
ejpam-4979	164	19	are	be	VERB
ejpam-4979	164	20	fewer	few	ADJ
ejpam-4979	164	21	primes	prime	NOUN
ejpam-4979	164	22	in	in	ADP
ejpam-4979	164	23	the	the	DET
ejpam-4979	164	24	subsequence	subsequence	NOUN
ejpam-4979	164	25	p′′	p′′	PROPN
ejpam-4979	164	26	than	than	ADP
ejpam-4979	164	27	in	in	ADP
ejpam-4979	164	28	the	the	DET
ejpam-4979	164	29	complementary	complementary	ADJ
ejpam-4979	164	30	prime	prime	ADJ
ejpam-4979	164	31	subsequence	subsequence	PROPN
ejpam-4979	164	32	p′	p′	NOUN
ejpam-4979	164	33	(	(	PUNCT
ejpam-4979	164	34	recall	recall	VERB
ejpam-4979	164	35	eq	eq	ADP
ejpam-4979	164	36	.	.	PROPN
ejpam-4979	164	37	4	4	NUM
ejpam-4979	164	38	)	)	PUNCT
ejpam-4979	164	39	.	.	PUNCT
ejpam-4979	165	1	thus	thus	ADV
ejpam-4979	165	2	,	,	PUNCT
ejpam-4979	165	3	π′′(x)≤	π′′(x)≤	PROPN
ejpam-4979	165	4	r′′	r′′	VERB
ejpam-4979	165	5	+	+	NOUN
ejpam-4979	165	6	x	x	SYM
ejpam-4979	165	7	ln	ln	ADJ
ejpam-4979	165	8	pr(ln	pr(ln	NOUN
ejpam-4979	165	9	pr	pr	X
ejpam-4979	165	10	+	+	CCONJ
ejpam-4979	165	11	1	1	NUM
ejpam-4979	165	12	)	)	PUNCT
ejpam-4979	165	13	+	+	SYM
ejpam-4979	166	1	2r	2r	NUM
ejpam-4979	166	2	′′	′′	PROPN
ejpam-4979	166	3	(	(	PUNCT
ejpam-4979	166	4	15	15	NUM
ejpam-4979	166	5	)	)	PUNCT
ejpam-4979	166	6	<	<	X
ejpam-4979	166	7	r′′	r′′	VERB
ejpam-4979	166	8	+	+	NOUN
ejpam-4979	166	9	x	x	SYM
ejpam-4979	166	10	ln	ln	PROPN
ejpam-4979	166	11	r(ln	r(ln	PROPN
ejpam-4979	166	12	r	r	NOUN
ejpam-4979	166	13	+	+	NOUN
ejpam-4979	166	14	1	1	NUM
ejpam-4979	166	15	)	)	PUNCT
ejpam-4979	166	16	+	+	SYM
ejpam-4979	167	1	2r	2r	NUM
ejpam-4979	167	2	′′	′′	NOUN
ejpam-4979	167	3	(	(	PUNCT
ejpam-4979	167	4	r	r	NOUN
ejpam-4979	167	5	<	<	X
ejpam-4979	167	6	pr	pr	NOUN
ejpam-4979	167	7	)	)	PUNCT
ejpam-4979	167	8	≤	≤	NOUN
ejpam-4979	167	9	⌊r	⌊r	ADP
ejpam-4979	167	10	2	2	NUM
ejpam-4979	167	11	⌋+	⌋+	NOUN
ejpam-4979	167	12	x	x	X
ejpam-4979	167	13	ln	ln	X
ejpam-4979	167	14	r(ln	r(ln	PROPN
ejpam-4979	167	15	r	r	NOUN
ejpam-4979	167	16	+	+	NOUN
ejpam-4979	167	17	1	1	NUM
ejpam-4979	167	18	)	)	PUNCT
ejpam-4979	167	19	+	+	NOUN
ejpam-4979	167	20	2⌊	2⌊	NUM
ejpam-4979	167	21	r	r	NOUN
ejpam-4979	167	22	2	2	NUM
ejpam-4979	167	23	⌋	⌋	NOUN
ejpam-4979	167	24	(	(	PUNCT
ejpam-4979	167	25	⌊r	⌊r	NOUN
ejpam-4979	167	26	2	2	NUM
ejpam-4979	167	27	⌋	⌋	NOUN
ejpam-4979	167	28	≥	≥	NOUN
ejpam-4979	167	29	r′′	r′′	VERB
ejpam-4979	167	30	)	)	PUNCT
ejpam-4979	167	31	<	<	X
ejpam-4979	167	32	x	x	X
ejpam-4979	167	33	ln	ln	PROPN
ejpam-4979	167	34	r(ln	r(ln	PROPN
ejpam-4979	167	35	r	r	NOUN
ejpam-4979	167	36	+	+	NOUN
ejpam-4979	167	37	1	1	NUM
ejpam-4979	167	38	)	)	PUNCT
ejpam-4979	167	39	+	+	NOUN
ejpam-4979	168	1	2⌊	2⌊	NUM
ejpam-4979	168	2	r	r	NOUN
ejpam-4979	168	3	2	2	NUM
ejpam-4979	168	4	⌋+1	⌋+1	NOUN
ejpam-4979	168	5	(	(	PUNCT
ejpam-4979	168	6	2⌊	2⌊	NUM
ejpam-4979	168	7	r	r	NOUN
ejpam-4979	168	8	2	2	NUM
ejpam-4979	168	9	⌋	⌋	NOUN
ejpam-4979	168	10	>	>	X
ejpam-4979	168	11	⌊r	⌊r	NOUN
ejpam-4979	168	12	2	2	NUM
ejpam-4979	168	13	⌋	⌋	NOUN
ejpam-4979	168	14	)	)	PUNCT
ejpam-4979	168	15	<	<	X
ejpam-4979	169	1	x	x	X
ejpam-4979	169	2	ln	ln	PROPN
ejpam-4979	169	3	r(ln	r(ln	PROPN
ejpam-4979	169	4	r	r	NOUN
ejpam-4979	169	5	+	+	NOUN
ejpam-4979	169	6	1	1	NUM
ejpam-4979	169	7	)	)	PUNCT
ejpam-4979	169	8	+	+	CCONJ
ejpam-4979	169	9	2	2	NUM
ejpam-4979	169	10	r	r	NOUN
ejpam-4979	169	11	2	2	NUM
ejpam-4979	169	12	+1	+1	NOUN
ejpam-4979	169	13	(	(	PUNCT
ejpam-4979	169	14	r	r	NOUN
ejpam-4979	169	15	2	2	NUM
ejpam-4979	169	16	>	>	PUNCT
ejpam-4979	169	17	⌊r	⌊r	NOUN
ejpam-4979	169	18	2	2	NUM
ejpam-4979	169	19	⌋	⌋	NOUN
ejpam-4979	169	20	)	)	PUNCT
ejpam-4979	169	21	.	.	PUNCT
ejpam-4979	170	1	m.	m.	NOUN
ejpam-4979	170	2	p.	p.	NOUN
ejpam-4979	170	3	may	may	AUX
ejpam-4979	170	4	/	/	SYM
ejpam-4979	170	5	eur	eur	PROPN
ejpam-4979	170	6	.	.	PUNCT
ejpam-4979	171	1	j.	j.	PROPN
ejpam-4979	171	2	pure	pure	PROPN
ejpam-4979	171	3	appl	appl	PROPN
ejpam-4979	171	4	.	.	PROPN
ejpam-4979	171	5	math	math	PROPN
ejpam-4979	171	6	,	,	PUNCT
ejpam-4979	171	7	17	17	NUM
ejpam-4979	171	8	(	(	PUNCT
ejpam-4979	171	9	1	1	NUM
ejpam-4979	171	10	)	)	PUNCT
ejpam-4979	171	11	(	(	PUNCT
ejpam-4979	171	12	2024	2024	NUM
ejpam-4979	171	13	)	)	PUNCT
ejpam-4979	171	14	,	,	PUNCT
ejpam-4979	171	15	42	42	NUM
ejpam-4979	171	16	-	-	SYM
ejpam-4979	171	17	58	58	NUM
ejpam-4979	171	18	53	53	NUM
ejpam-4979	171	19	now	now	ADV
ejpam-4979	171	20	,	,	PUNCT
ejpam-4979	171	21	let	let	VERB
ejpam-4979	171	22	r	r	NOUN
ejpam-4979	171	23	=	=	PUNCT
ejpam-4979	171	24	xm	xm	PROPN
ejpam-4979	171	25	such	such	ADJ
ejpam-4979	171	26	that	that	SCONJ
ejpam-4979	171	27	m	m	VERB
ejpam-4979	171	28	=	=	SYM
ejpam-4979	171	29	1	1	NUM
ejpam-4979	171	30	c	c	NOUN
ejpam-4979	171	31	·	·	PUNCT
ejpam-4979	171	32	ln	ln	ADJ
ejpam-4979	171	33	lnx	lnx	NOUN
ejpam-4979	171	34	for	for	ADP
ejpam-4979	171	35	some	some	DET
ejpam-4979	171	36	positive	positive	ADJ
ejpam-4979	171	37	constant	constant	ADJ
ejpam-4979	171	38	c.	c.	NOUN
ejpam-4979	171	39	we	we	PRON
ejpam-4979	171	40	now	now	ADV
ejpam-4979	171	41	have	have	VERB
ejpam-4979	171	42	π′′(x	π′′(x	NOUN
ejpam-4979	171	43	)	)	PUNCT
ejpam-4979	171	44	<	<	X
ejpam-4979	171	45	x	x	PUNCT
ejpam-4979	171	46	lnxm(lnxm	lnxm(lnxm	NOUN
ejpam-4979	171	47	+	+	CCONJ
ejpam-4979	171	48	1	1	NUM
ejpam-4979	171	49	)	)	PUNCT
ejpam-4979	171	50	+	+	CCONJ
ejpam-4979	171	51	2	2	NUM
ejpam-4979	171	52	1	1	NUM
ejpam-4979	171	53	2	2	NUM
ejpam-4979	171	54	(	(	PUNCT
ejpam-4979	171	55	xm+2	xm+2	PROPN
ejpam-4979	171	56	)	)	PUNCT
ejpam-4979	171	57	(	(	PUNCT
ejpam-4979	171	58	m	m	VERB
ejpam-4979	171	59	=	=	SYM
ejpam-4979	171	60	1	1	NUM
ejpam-4979	171	61	c	c	NOUN
ejpam-4979	171	62	·	·	PUNCT
ejpam-4979	171	63	ln	ln	ADJ
ejpam-4979	171	64	lnx	lnx	NOUN
ejpam-4979	171	65	)	)	PUNCT
ejpam-4979	172	1	=	=	PUNCT
ejpam-4979	173	1	x	x	X
ejpam-4979	173	2	(	(	PUNCT
ejpam-4979	173	3	lnx	lnx	PROPN
ejpam-4979	173	4	c	c	PROPN
ejpam-4979	173	5	·	·	PUNCT
ejpam-4979	173	6	ln	ln	ADJ
ejpam-4979	173	7	lnx	lnx	PROPN
ejpam-4979	173	8	)	)	PUNCT
ejpam-4979	173	9	2	2	PROPN
ejpam-4979	174	1	+	+	CCONJ
ejpam-4979	174	2	lnx	lnx	PROPN
ejpam-4979	174	3	c	c	PROPN
ejpam-4979	174	4	·	·	PUNCT
ejpam-4979	174	5	ln	ln	ADJ
ejpam-4979	174	6	lnx	lnx	NOUN
ejpam-4979	174	7	+	+	CCONJ
ejpam-4979	174	8	2	2	NUM
ejpam-4979	174	9	1	1	NUM
ejpam-4979	174	10	2	2	NUM
ejpam-4979	174	11	(	(	PUNCT
ejpam-4979	174	12	xm+2	xm+2	NOUN
ejpam-4979	174	13	)	)	PUNCT
ejpam-4979	175	1	=	=	PUNCT
ejpam-4979	175	2	x	x	PUNCT
ejpam-4979	175	3	c	c	X
ejpam-4979	175	4	·	·	PUNCT
ejpam-4979	175	5	ln	ln	ADJ
ejpam-4979	175	6	lnx	lnx	PROPN
ejpam-4979	175	7	·	·	PUNCT
ejpam-4979	176	1	ln2	ln2	ADJ
ejpam-4979	176	2	x+	x+	NUM
ejpam-4979	176	3	(	(	PUNCT
ejpam-4979	176	4	c	c	NOUN
ejpam-4979	176	5	·	·	PUNCT
ejpam-4979	176	6	ln	ln	ADJ
ejpam-4979	176	7	lnx)2	lnx)2	PROPN
ejpam-4979	176	8	·	·	PUNCT
ejpam-4979	176	9	lnx	lnx	X
ejpam-4979	176	10	(	(	PUNCT
ejpam-4979	176	11	c	c	X
ejpam-4979	176	12	·	·	PUNCT
ejpam-4979	176	13	ln	ln	PROPN
ejpam-4979	176	14	lnx)3	lnx)3	NOUN
ejpam-4979	177	1	+	+	CCONJ
ejpam-4979	177	2	2	2	NUM
ejpam-4979	177	3	1	1	NUM
ejpam-4979	177	4	2	2	NUM
ejpam-4979	177	5	(	(	PUNCT
ejpam-4979	177	6	xm+2	xm+2	NOUN
ejpam-4979	177	7	)	)	PUNCT
ejpam-4979	177	8	=	=	PUNCT
ejpam-4979	177	9	x	x	SYM
ejpam-4979	177	10	·	·	PUNCT
ejpam-4979	177	11	(	(	PUNCT
ejpam-4979	177	12	c	c	X
ejpam-4979	177	13	·	·	PUNCT
ejpam-4979	177	14	ln	ln	ADJ
ejpam-4979	177	15	lnx)2	lnx)2	PROPN
ejpam-4979	178	1	ln2	ln2	ADJ
ejpam-4979	178	2	x+	x+	X
ejpam-4979	178	3	c	c	X
ejpam-4979	178	4	·	·	PUNCT
ejpam-4979	178	5	ln	ln	ADJ
ejpam-4979	178	6	lnx	lnx	PROPN
ejpam-4979	178	7	·	·	PUNCT
ejpam-4979	178	8	lnx	lnx	PROPN
ejpam-4979	178	9	+	+	CCONJ
ejpam-4979	178	10	2	2	NUM
ejpam-4979	178	11	1	1	NUM
ejpam-4979	178	12	2	2	NUM
ejpam-4979	178	13	(	(	PUNCT
ejpam-4979	178	14	xm+2	xm+2	X
ejpam-4979	178	15	)	)	PUNCT
ejpam-4979	178	16	<	<	X
ejpam-4979	178	17	c	c	X
ejpam-4979	178	18	·	·	PUNCT
ejpam-4979	178	19	x	x	SYM
ejpam-4979	178	20	·	·	PUNCT
ejpam-4979	179	1	(	(	PUNCT
ejpam-4979	179	2	ln	ln	NOUN
ejpam-4979	179	3	lnx)2	lnx)2	PROPN
ejpam-4979	179	4	ln2	ln2	ADJ
ejpam-4979	179	5	x+	x+	X
ejpam-4979	179	6	c	c	X
ejpam-4979	179	7	·	·	PUNCT
ejpam-4979	179	8	ln	ln	ADJ
ejpam-4979	179	9	lnx	lnx	PROPN
ejpam-4979	179	10	·	·	PUNCT
ejpam-4979	179	11	lnx	lnx	PROPN
ejpam-4979	179	12	+	+	CCONJ
ejpam-4979	179	13	2	2	NUM
ejpam-4979	179	14	1	1	NUM
ejpam-4979	179	15	2	2	NUM
ejpam-4979	179	16	(	(	PUNCT
ejpam-4979	179	17	xm+2	xm+2	PROPN
ejpam-4979	179	18	)	)	PUNCT
ejpam-4979	179	19	.	.	PUNCT
ejpam-4979	180	1	since	since	SCONJ
ejpam-4979	180	2	ln	ln	PROPN
ejpam-4979	180	3	lnx	lnx	PROPN
ejpam-4979	180	4	·	·	PUNCT
ejpam-4979	180	5	lnx	lnx	PROPN
ejpam-4979	180	6	<	<	X
ejpam-4979	180	7	c	c	X
ejpam-4979	180	8	·	·	PUNCT
ejpam-4979	180	9	ln	ln	ADJ
ejpam-4979	180	10	lnx	lnx	PROPN
ejpam-4979	180	11	·	·	PUNCT
ejpam-4979	180	12	lnx	lnx	PROPN
ejpam-4979	180	13	for	for	ADP
ejpam-4979	180	14	c	c	PROPN
ejpam-4979	180	15	≥	≥	NUM
ejpam-4979	180	16	1	1	NUM
ejpam-4979	180	17	,	,	PUNCT
ejpam-4979	180	18	and	and	CCONJ
ejpam-4979	180	19	since	since	SCONJ
ejpam-4979	180	20	2	2	NUM
ejpam-4979	180	21	1	1	NUM
ejpam-4979	180	22	2	2	NUM
ejpam-4979	180	23	(	(	PUNCT
ejpam-4979	180	24	xm+2	xm+2	NOUN
ejpam-4979	180	25	)	)	PUNCT
ejpam-4979	180	26	≪	≪	VERB
ejpam-4979	180	27	than	than	ADP
ejpam-4979	180	28	the	the	DET
ejpam-4979	180	29	main	main	ADJ
ejpam-4979	180	30	term	term	NOUN
ejpam-4979	180	31	when	when	SCONJ
ejpam-4979	180	32	c	c	X
ejpam-4979	180	33	≥	≥	NOUN
ejpam-4979	180	34	5	5	NUM
ejpam-4979	180	35	,	,	PUNCT
ejpam-4979	180	36	we	we	PRON
ejpam-4979	180	37	finally	finally	ADV
ejpam-4979	180	38	arrive	arrive	VERB
ejpam-4979	180	39	at	at	ADP
ejpam-4979	180	40	π′′(x	π′′(x	NOUN
ejpam-4979	180	41	)	)	PUNCT
ejpam-4979	180	42	<	<	X
ejpam-4979	180	43	c	c	X
ejpam-4979	180	44	·	·	PUNCT
ejpam-4979	180	45	x	x	SYM
ejpam-4979	180	46	·	·	PUNCT
ejpam-4979	180	47	(	(	PUNCT
ejpam-4979	180	48	ln	ln	NOUN
ejpam-4979	180	49	lnx)2	lnx)2	PROPN
ejpam-4979	180	50	(	(	PUNCT
ejpam-4979	180	51	lnx)2	lnx)2	PROPN
ejpam-4979	180	52	+	+	CCONJ
ejpam-4979	180	53	ln	ln	ADJ
ejpam-4979	180	54	lnx	lnx	PROPN
ejpam-4979	180	55	·	·	PUNCT
ejpam-4979	180	56	lnx	lnx	PROPN
ejpam-4979	180	57	.	.	PUNCT
ejpam-4979	181	1	(	(	PUNCT
ejpam-4979	181	2	16	16	NUM
ejpam-4979	181	3	)	)	PUNCT
ejpam-4979	181	4	thus	thus	ADV
ejpam-4979	181	5	,	,	PUNCT
ejpam-4979	181	6	we	we	PRON
ejpam-4979	181	7	see	see	VERB
ejpam-4979	181	8	that	that	PRON
ejpam-4979	181	9	for	for	ADP
ejpam-4979	181	10	some	some	DET
ejpam-4979	181	11	positive	positive	ADJ
ejpam-4979	181	12	constant	constant	ADJ
ejpam-4979	181	13	c	c	NOUN
ejpam-4979	181	14	<	<	X
ejpam-4979	181	15	+	+	PROPN
ejpam-4979	181	16	∞	∞	PROPN
ejpam-4979	181	17	,	,	PUNCT
ejpam-4979	181	18	the	the	DET
ejpam-4979	181	19	sum	sum	NOUN
ejpam-4979	181	20	of	of	ADP
ejpam-4979	181	21	the	the	DET
ejpam-4979	181	22	reciprocals	reciprocal	NOUN
ejpam-4979	181	23	of	of	ADP
ejpam-4979	181	24	the	the	DET
ejpam-4979	181	25	infinite	infinite	ADJ
ejpam-4979	181	26	subsequence	subsequence	NOUN
ejpam-4979	181	27	of	of	ADP
ejpam-4979	181	28	prime	prime	ADJ
ejpam-4979	181	29	numbers	number	NOUN
ejpam-4979	181	30	p′′	p′′	PROPN
ejpam-4979	181	31	converges	converge	VERB
ejpam-4979	181	32	,	,	PUNCT
ejpam-4979	181	33	and	and	CCONJ
ejpam-4979	181	34	this	this	PRON
ejpam-4979	181	35	is	be	AUX
ejpam-4979	181	36	confirmed	confirm	VERB
ejpam-4979	181	37	when	when	SCONJ
ejpam-4979	181	38	we	we	PRON
ejpam-4979	181	39	compare	compare	VERB
ejpam-4979	181	40	16	16	NUM
ejpam-4979	181	41	to	to	ADP
ejpam-4979	181	42	the	the	DET
ejpam-4979	181	43	count	count	NOUN
ejpam-4979	181	44	of	of	ADP
ejpam-4979	181	45	p2	p2	PROPN
ejpam-4979	181	46	≤	≤	NUM
ejpam-4979	181	47	x	x	SYM
ejpam-4979	181	48	(	(	PUNCT
ejpam-4979	181	49	see	see	VERB
ejpam-4979	181	50	20	20	NUM
ejpam-4979	181	51	)	)	PUNCT
ejpam-4979	181	52	.	.	PUNCT
ejpam-4979	182	1	6	6	X
ejpam-4979	182	2	.	.	X
ejpam-4979	182	3	estimating	estimate	VERB
ejpam-4979	182	4	π′(x	π′(x	NOUN
ejpam-4979	182	5	)	)	PUNCT
ejpam-4979	182	6	in	in	ADP
ejpam-4979	182	7	order	order	NOUN
ejpam-4979	182	8	to	to	PART
ejpam-4979	182	9	calculate	calculate	VERB
ejpam-4979	182	10	an	an	DET
ejpam-4979	182	11	estimate	estimate	NOUN
ejpam-4979	182	12	of	of	ADP
ejpam-4979	182	13	π′(x	π′(x	NOUN
ejpam-4979	182	14	)	)	PUNCT
ejpam-4979	182	15	,	,	PUNCT
ejpam-4979	182	16	we	we	PRON
ejpam-4979	182	17	invoke	invoke	VERB
ejpam-4979	182	18	the	the	DET
ejpam-4979	182	19	lemma	lemma	PROPN
ejpam-4979	182	20	and	and	CCONJ
ejpam-4979	182	21	begin	begin	VERB
ejpam-4979	182	22	with	with	ADP
ejpam-4979	182	23	π′(x	π′(x	NOUN
ejpam-4979	182	24	)	)	PUNCT
ejpam-4979	182	25	≤	≤	NUM
ejpam-4979	182	26	r′	r′	NOUN
ejpam-4979	183	1	+	+	CCONJ
ejpam-4979	183	2	x	x	SYM
ejpam-4979	183	3	ln	ln	ADJ
ejpam-4979	183	4	pr	pr	NOUN
ejpam-4979	183	5	+	+	CCONJ
ejpam-4979	183	6	1	1	NUM
ejpam-4979	183	7	+	+	NUM
ejpam-4979	183	8	2r	2r	NUM
ejpam-4979	183	9	′	′	NUM
ejpam-4979	183	10	.	.	PUNCT
ejpam-4979	184	1	(	(	PUNCT
ejpam-4979	184	2	17	17	NUM
ejpam-4979	184	3	)	)	PUNCT
ejpam-4979	184	4	proceeding	proceeding	NOUN
ejpam-4979	184	5	as	as	ADP
ejpam-4979	184	6	before	before	ADV
ejpam-4979	184	7	,	,	PUNCT
ejpam-4979	184	8	we	we	PRON
ejpam-4979	184	9	know	know	VERB
ejpam-4979	184	10	that	that	SCONJ
ejpam-4979	184	11	r′	r′	NUM
ejpam-4979	184	12	≤	≤	X
ejpam-4979	184	13	⌊r⌋	⌊r⌋	PUNCT
ejpam-4979	184	14	for	for	ADP
ejpam-4979	184	15	p′	p′	NOUN
ejpam-4979	184	16	≥	≥	NOUN
ejpam-4979	184	17	2	2	NUM
ejpam-4979	184	18	since	since	SCONJ
ejpam-4979	184	19	there	there	PRON
ejpam-4979	184	20	are	be	VERB
ejpam-4979	184	21	fewer	few	ADJ
ejpam-4979	184	22	primes	prime	NOUN
ejpam-4979	184	23	in	in	ADP
ejpam-4979	184	24	the	the	DET
ejpam-4979	184	25	subsequence	subsequence	NOUN
ejpam-4979	184	26	p′	p′	NOUN
ejpam-4979	184	27	than	than	ADP
ejpam-4979	184	28	in	in	ADP
ejpam-4979	184	29	the	the	DET
ejpam-4979	184	30	set	set	NOUN
ejpam-4979	184	31	of	of	ADP
ejpam-4979	184	32	all	all	DET
ejpam-4979	184	33	prime	prime	ADJ
ejpam-4979	184	34	numbers	number	NOUN
ejpam-4979	184	35	p.	p.	NOUN
ejpam-4979	184	36	thus	thus	ADV
ejpam-4979	184	37	,	,	PUNCT
ejpam-4979	184	38	m.	m.	NOUN
ejpam-4979	184	39	p.	p.	PROPN
ejpam-4979	184	40	may	may	AUX
ejpam-4979	184	41	/	/	SYM
ejpam-4979	184	42	eur	eur	PROPN
ejpam-4979	184	43	.	.	PUNCT
ejpam-4979	185	1	j.	j.	PROPN
ejpam-4979	185	2	pure	pure	PROPN
ejpam-4979	185	3	appl	appl	PROPN
ejpam-4979	185	4	.	.	PROPN
ejpam-4979	185	5	math	math	PROPN
ejpam-4979	185	6	,	,	PUNCT
ejpam-4979	185	7	17	17	NUM
ejpam-4979	185	8	(	(	PUNCT
ejpam-4979	185	9	1	1	NUM
ejpam-4979	185	10	)	)	PUNCT
ejpam-4979	185	11	(	(	PUNCT
ejpam-4979	185	12	2024	2024	NUM
ejpam-4979	185	13	)	)	PUNCT
ejpam-4979	185	14	,	,	PUNCT
ejpam-4979	185	15	42	42	NUM
ejpam-4979	185	16	-	-	SYM
ejpam-4979	185	17	58	58	NUM
ejpam-4979	185	18	54	54	NUM
ejpam-4979	185	19	π′(x)≤	π′(x)≤	PUNCT
ejpam-4979	185	20	r′	r′	PROPN
ejpam-4979	186	1	+	+	X
ejpam-4979	186	2	x	x	SYM
ejpam-4979	186	3	ln	ln	ADJ
ejpam-4979	186	4	pr	pr	NOUN
ejpam-4979	186	5	+	+	CCONJ
ejpam-4979	186	6	1	1	NUM
ejpam-4979	186	7	+	+	NUM
ejpam-4979	186	8	2r	2r	NUM
ejpam-4979	186	9	′	′	NUM
ejpam-4979	186	10	(	(	PUNCT
ejpam-4979	186	11	17	17	NUM
ejpam-4979	186	12	)	)	PUNCT
ejpam-4979	186	13	<	<	X
ejpam-4979	186	14	r′	r′	PROPN
ejpam-4979	187	1	+	+	NUM
ejpam-4979	187	2	x	x	SYM
ejpam-4979	187	3	ln	ln	ADJ
ejpam-4979	187	4	r	r	NOUN
ejpam-4979	187	5	+	+	CCONJ
ejpam-4979	187	6	1	1	NUM
ejpam-4979	187	7	+	+	NUM
ejpam-4979	187	8	2r	2r	NUM
ejpam-4979	187	9	′	′	NUM
ejpam-4979	188	1	(	(	PUNCT
ejpam-4979	188	2	r	r	NOUN
ejpam-4979	188	3	<	<	X
ejpam-4979	188	4	pr	pr	NOUN
ejpam-4979	188	5	)	)	PUNCT
ejpam-4979	188	6	≤	≤	NOUN
ejpam-4979	188	7	⌊r⌋+	⌊r⌋+	NOUN
ejpam-4979	188	8	x	x	X
ejpam-4979	188	9	ln	ln	NOUN
ejpam-4979	188	10	r	r	NOUN
ejpam-4979	188	11	+	+	CCONJ
ejpam-4979	188	12	1	1	NUM
ejpam-4979	188	13	+	+	NUM
ejpam-4979	188	14	2⌊r⌋	2⌊r⌋	NUM
ejpam-4979	188	15	(	(	PUNCT
ejpam-4979	188	16	⌊r⌋	⌊r⌋	PUNCT
ejpam-4979	188	17	≥	≥	NUM
ejpam-4979	188	18	r′	r′	NUM
ejpam-4979	188	19	)	)	PUNCT
ejpam-4979	189	1	<	<	X
ejpam-4979	189	2	x	x	X
ejpam-4979	189	3	ln	ln	NOUN
ejpam-4979	189	4	r	r	NOUN
ejpam-4979	189	5	+	+	CCONJ
ejpam-4979	189	6	1	1	NUM
ejpam-4979	189	7	+	+	NUM
ejpam-4979	189	8	2⌊r⌋+1	2⌊r⌋+1	NUM
ejpam-4979	189	9	(	(	PUNCT
ejpam-4979	189	10	2⌊r⌋	2⌊r⌋	NOUN
ejpam-4979	189	11	>	>	X
ejpam-4979	189	12	⌊r⌋	⌊r⌋	PUNCT
ejpam-4979	189	13	)	)	PUNCT
ejpam-4979	189	14	<	<	X
ejpam-4979	190	1	x	x	X
ejpam-4979	190	2	ln	ln	NOUN
ejpam-4979	190	3	r	r	NOUN
ejpam-4979	190	4	+	+	CCONJ
ejpam-4979	190	5	1	1	NUM
ejpam-4979	190	6	+	+	NUM
ejpam-4979	190	7	2r+1	2r+1	NOUN
ejpam-4979	190	8	(	(	PUNCT
ejpam-4979	190	9	r	r	NOUN
ejpam-4979	190	10	>	>	PUNCT
ejpam-4979	190	11	⌊r⌋	⌊r⌋	NUM
ejpam-4979	190	12	)	)	PUNCT
ejpam-4979	190	13	.	.	PUNCT
ejpam-4979	191	1	now	now	ADV
ejpam-4979	191	2	,	,	PUNCT
ejpam-4979	191	3	let	let	VERB
ejpam-4979	191	4	r	r	NOUN
ejpam-4979	191	5	=	=	PUNCT
ejpam-4979	191	6	xm	xm	PROPN
ejpam-4979	191	7	such	such	ADJ
ejpam-4979	191	8	that	that	SCONJ
ejpam-4979	191	9	m	m	VERB
ejpam-4979	191	10	=	=	SYM
ejpam-4979	191	11	1	1	NUM
ejpam-4979	191	12	c	c	NOUN
ejpam-4979	191	13	·	·	PUNCT
ejpam-4979	191	14	ln	ln	ADJ
ejpam-4979	191	15	lnx	lnx	NOUN
ejpam-4979	191	16	for	for	ADP
ejpam-4979	191	17	some	some	DET
ejpam-4979	191	18	positive	positive	ADJ
ejpam-4979	191	19	constant	constant	ADJ
ejpam-4979	191	20	c.	c.	NOUN
ejpam-4979	191	21	we	we	PRON
ejpam-4979	191	22	now	now	ADV
ejpam-4979	191	23	have	have	VERB
ejpam-4979	191	24	π′(x	π′(x	NOUN
ejpam-4979	191	25	)	)	PUNCT
ejpam-4979	191	26	<	<	X
ejpam-4979	191	27	x	x	PUNCT
ejpam-4979	191	28	lnxm	lnxm	PROPN
ejpam-4979	191	29	+	+	CCONJ
ejpam-4979	191	30	1	1	NUM
ejpam-4979	191	31	+	+	NUM
ejpam-4979	191	32	2x	2x	NUM
ejpam-4979	191	33	m+1	m+1	NUM
ejpam-4979	191	34	(	(	PUNCT
ejpam-4979	191	35	m	m	NOUN
ejpam-4979	191	36	=	=	SYM
ejpam-4979	191	37	1	1	NUM
ejpam-4979	191	38	c	c	NOUN
ejpam-4979	191	39	·	·	PUNCT
ejpam-4979	191	40	ln	ln	ADJ
ejpam-4979	191	41	lnx	lnx	NOUN
ejpam-4979	191	42	)	)	PUNCT
ejpam-4979	192	1	=	=	PUNCT
ejpam-4979	192	2	x	x	PUNCT
ejpam-4979	192	3	lnx	lnx	PROPN
ejpam-4979	192	4	c	c	PROPN
ejpam-4979	192	5	·	·	PUNCT
ejpam-4979	192	6	ln	ln	ADJ
ejpam-4979	192	7	lnx	lnx	NOUN
ejpam-4979	192	8	+	+	CCONJ
ejpam-4979	192	9	1	1	NUM
ejpam-4979	192	10	+	+	NUM
ejpam-4979	192	11	2x	2x	NUM
ejpam-4979	192	12	m+1	m+1	NUM
ejpam-4979	192	13	=	=	PUNCT
ejpam-4979	192	14	x	x	PUNCT
ejpam-4979	192	15	lnx+	lnx+	PROPN
ejpam-4979	192	16	c	c	PROPN
ejpam-4979	192	17	·	·	PUNCT
ejpam-4979	192	18	ln	ln	ADJ
ejpam-4979	192	19	lnx	lnx	PROPN
ejpam-4979	192	20	c	c	PROPN
ejpam-4979	192	21	·	·	PUNCT
ejpam-4979	192	22	ln	ln	ADJ
ejpam-4979	192	23	lnx	lnx	PROPN
ejpam-4979	192	24	+	+	CCONJ
ejpam-4979	192	25	2x	2x	NUM
ejpam-4979	192	26	m+1	m+1	NUM
ejpam-4979	192	27	=	=	PUNCT
ejpam-4979	192	28	x	x	SYM
ejpam-4979	192	29	·	·	PUNCT
ejpam-4979	192	30	c	c	X
ejpam-4979	192	31	·	·	PUNCT
ejpam-4979	192	32	ln	ln	ADJ
ejpam-4979	192	33	lnx	lnx	PROPN
ejpam-4979	192	34	lnx+	lnx+	PROPN
ejpam-4979	192	35	c	c	PROPN
ejpam-4979	192	36	·	·	PUNCT
ejpam-4979	192	37	ln	ln	ADJ
ejpam-4979	192	38	lnx	lnx	PROPN
ejpam-4979	192	39	+	+	CCONJ
ejpam-4979	192	40	2x	2x	NUM
ejpam-4979	192	41	m+1	m+1	PRON
ejpam-4979	192	42	<	<	X
ejpam-4979	192	43	c	c	X
ejpam-4979	192	44	·	·	PUNCT
ejpam-4979	192	45	x	x	PUNCT
ejpam-4979	192	46	·	·	PUNCT
ejpam-4979	192	47	ln	ln	ADJ
ejpam-4979	192	48	lnx	lnx	PROPN
ejpam-4979	192	49	lnx+	lnx+	PROPN
ejpam-4979	192	50	c	c	PROPN
ejpam-4979	192	51	·	·	PUNCT
ejpam-4979	192	52	ln	ln	ADJ
ejpam-4979	192	53	lnx	lnx	PROPN
ejpam-4979	192	54	+	+	CCONJ
ejpam-4979	192	55	2x	2x	NUM
ejpam-4979	192	56	m+1	m+1	NOUN
ejpam-4979	192	57	.	.	PUNCT
ejpam-4979	193	1	since	since	SCONJ
ejpam-4979	193	2	ln	ln	PROPN
ejpam-4979	193	3	lnx	lnx	PROPN
ejpam-4979	193	4	<	<	X
ejpam-4979	193	5	c	c	X
ejpam-4979	193	6	·	·	PUNCT
ejpam-4979	193	7	ln	ln	ADJ
ejpam-4979	193	8	lnx	lnx	PROPN
ejpam-4979	193	9	for	for	ADP
ejpam-4979	193	10	c	c	PROPN
ejpam-4979	193	11	≥	≥	NUM
ejpam-4979	193	12	1	1	NUM
ejpam-4979	193	13	,	,	PUNCT
ejpam-4979	193	14	and	and	CCONJ
ejpam-4979	193	15	since	since	SCONJ
ejpam-4979	193	16	2x	2x	NUM
ejpam-4979	193	17	m+1	m+1	PRON
ejpam-4979	193	18	≪	≪	PUNCT
ejpam-4979	193	19	than	than	ADP
ejpam-4979	193	20	the	the	DET
ejpam-4979	193	21	main	main	ADJ
ejpam-4979	193	22	term	term	NOUN
ejpam-4979	193	23	when	when	SCONJ
ejpam-4979	193	24	c	c	X
ejpam-4979	193	25	≥	≥	NOUN
ejpam-4979	193	26	5	5	NUM
ejpam-4979	193	27	,	,	PUNCT
ejpam-4979	193	28	we	we	PRON
ejpam-4979	193	29	finally	finally	ADV
ejpam-4979	193	30	arrive	arrive	VERB
ejpam-4979	193	31	at	at	ADP
ejpam-4979	193	32	π′(x	π′(x	NOUN
ejpam-4979	193	33	)	)	PUNCT
ejpam-4979	193	34	<	<	X
ejpam-4979	193	35	c	c	X
ejpam-4979	193	36	·	·	PUNCT
ejpam-4979	193	37	x	x	PUNCT
ejpam-4979	193	38	·	·	PUNCT
ejpam-4979	193	39	ln	ln	ADJ
ejpam-4979	193	40	lnx	lnx	PROPN
ejpam-4979	193	41	lnx+	lnx+	PROPN
ejpam-4979	193	42	ln	ln	PROPN
ejpam-4979	193	43	lnx	lnx	PROPN
ejpam-4979	193	44	.	.	PUNCT
ejpam-4979	194	1	since	since	SCONJ
ejpam-4979	194	2	we	we	PRON
ejpam-4979	194	3	’ve	’ve	AUX
ejpam-4979	194	4	shown	show	VERB
ejpam-4979	194	5	that	that	SCONJ
ejpam-4979	194	6	p′′	p′′	PROPN
ejpam-4979	194	7	is	be	AUX
ejpam-4979	194	8	a	a	DET
ejpam-4979	194	9	small	small	ADJ
ejpam-4979	194	10	set	set	NOUN
ejpam-4979	194	11	in	in	ADP
ejpam-4979	194	12	that	that	SCONJ
ejpam-4979	194	13	the	the	DET
ejpam-4979	194	14	infinite	infinite	ADJ
ejpam-4979	194	15	sum	sum	NOUN
ejpam-4979	194	16	of	of	ADP
ejpam-4979	194	17	its	its	PRON
ejpam-4979	194	18	reciprocals	reciprocal	NOUN
ejpam-4979	194	19	converges	converge	NOUN
ejpam-4979	194	20	,	,	PUNCT
ejpam-4979	194	21	and	and	CCONJ
ejpam-4979	194	22	since	since	SCONJ
ejpam-4979	194	23	it	it	PRON
ejpam-4979	194	24	is	be	AUX
ejpam-4979	194	25	known	know	VERB
ejpam-4979	194	26	that	that	SCONJ
ejpam-4979	194	27	the	the	DET
ejpam-4979	194	28	sum	sum	NOUN
ejpam-4979	194	29	of	of	ADP
ejpam-4979	194	30	the	the	DET
ejpam-4979	194	31	reciprocals	reciprocal	NOUN
ejpam-4979	194	32	of	of	ADP
ejpam-4979	194	33	the	the	DET
ejpam-4979	194	34	set	set	NOUN
ejpam-4979	194	35	of	of	ADP
ejpam-4979	194	36	all	all	DET
ejpam-4979	194	37	prime	prime	ADJ
ejpam-4979	194	38	numbers	number	NOUN
ejpam-4979	194	39	p	p	NOUN
ejpam-4979	194	40	diverges	diverge	NOUN
ejpam-4979	194	41	,	,	PUNCT
ejpam-4979	194	42	we	we	PRON
ejpam-4979	194	43	can	can	AUX
ejpam-4979	194	44	deduce	deduce	VERB
ejpam-4979	194	45	from	from	ADP
ejpam-4979	194	46	the	the	DET
ejpam-4979	194	47	relation	relation	NOUN
ejpam-4979	194	48	p	p	NOUN
ejpam-4979	194	49	=	=	PROPN
ejpam-4979	194	50	p′	p′	NOUN
ejpam-4979	194	51	+	+	CCONJ
ejpam-4979	194	52	p′′	p′′	PROPN
ejpam-4979	194	53	that	that	SCONJ
ejpam-4979	194	54	the	the	DET
ejpam-4979	194	55	prime	prime	ADJ
ejpam-4979	194	56	number	number	NOUN
ejpam-4979	194	57	subsequence	subsequence	PROPN
ejpam-4979	194	58	p′	p′	NOUN
ejpam-4979	194	59	is	be	AUX
ejpam-4979	194	60	a	a	DET
ejpam-4979	194	61	large	large	ADJ
ejpam-4979	194	62	set	set	NOUN
ejpam-4979	194	63	and	and	CCONJ
ejpam-4979	194	64	that	that	SCONJ
ejpam-4979	194	65	the	the	DET
ejpam-4979	194	66	infinite	infinite	ADJ
ejpam-4979	194	67	sum	sum	NOUN
ejpam-4979	194	68	of	of	ADP
ejpam-4979	194	69	its	its	PRON
ejpam-4979	194	70	reciprocals	reciprocal	NOUN
ejpam-4979	194	71	diverges	diverge	VERB
ejpam-4979	194	72	.	.	PUNCT
ejpam-4979	195	1	m.	m.	NOUN
ejpam-4979	195	2	p.	p.	PROPN
ejpam-4979	195	3	may	may	AUX
ejpam-4979	195	4	/	/	SYM
ejpam-4979	195	5	eur	eur	PROPN
ejpam-4979	195	6	.	.	PUNCT
ejpam-4979	196	1	j.	j.	PROPN
ejpam-4979	196	2	pure	pure	PROPN
ejpam-4979	196	3	appl	appl	PROPN
ejpam-4979	196	4	.	.	PROPN
ejpam-4979	196	5	math	math	PROPN
ejpam-4979	196	6	,	,	PUNCT
ejpam-4979	196	7	17	17	NUM
ejpam-4979	196	8	(	(	PUNCT
ejpam-4979	196	9	1	1	NUM
ejpam-4979	196	10	)	)	PUNCT
ejpam-4979	196	11	(	(	PUNCT
ejpam-4979	196	12	2024	2024	NUM
ejpam-4979	196	13	)	)	PUNCT
ejpam-4979	196	14	,	,	PUNCT
ejpam-4979	196	15	42	42	NUM
ejpam-4979	196	16	-	-	SYM
ejpam-4979	196	17	58	58	NUM
ejpam-4979	196	18	55	55	NUM
ejpam-4979	196	19	7	7	NUM
ejpam-4979	196	20	.	.	PUNCT
ejpam-4979	196	21	estimating	estimate	VERB
ejpam-4979	196	22	π2(x	π2(x	X
ejpam-4979	196	23	)	)	PUNCT
ejpam-4979	196	24	we	we	PRON
ejpam-4979	196	25	now	now	ADV
ejpam-4979	196	26	take	take	VERB
ejpam-4979	196	27	a	a	DET
ejpam-4979	196	28	look	look	NOUN
ejpam-4979	196	29	at	at	ADP
ejpam-4979	196	30	how	how	SCONJ
ejpam-4979	196	31	the	the	DET
ejpam-4979	196	32	twin	twin	ADJ
ejpam-4979	196	33	prime	prime	ADJ
ejpam-4979	196	34	count	count	NOUN
ejpam-4979	196	35	π2(x	π2(x	X
ejpam-4979	196	36	)	)	PUNCT
ejpam-4979	196	37	can	can	AUX
ejpam-4979	196	38	be	be	AUX
ejpam-4979	196	39	estimated	estimate	VERB
ejpam-4979	196	40	using	use	VERB
ejpam-4979	196	41	the	the	DET
ejpam-4979	196	42	technique	technique	NOUN
ejpam-4979	196	43	heretofore	heretofore	ADV
ejpam-4979	196	44	disclosed	disclose	VERB
ejpam-4979	196	45	.	.	PUNCT
ejpam-4979	197	1	if	if	SCONJ
ejpam-4979	197	2	we	we	PRON
ejpam-4979	197	3	assume	assume	VERB
ejpam-4979	197	4	that	that	SCONJ
ejpam-4979	197	5	π2(x	π2(x	NOUN
ejpam-4979	197	6	)	)	PUNCT
ejpam-4979	197	7	∼	∼	NOUN
ejpam-4979	197	8	c	c	NOUN
ejpam-4979	197	9	ln2	ln2	NOUN
ejpam-4979	197	10	x	x	PUNCT
ejpam-4979	197	11	for	for	ADP
ejpam-4979	197	12	some	some	DET
ejpam-4979	197	13	positive	positive	ADJ
ejpam-4979	197	14	constant	constant	ADJ
ejpam-4979	197	15	c	c	NOUN
ejpam-4979	197	16	[	[	X
ejpam-4979	197	17	2	2	NUM
ejpam-4979	197	18	]	]	PUNCT
ejpam-4979	197	19	,	,	PUNCT
ejpam-4979	197	20	we	we	PRON
ejpam-4979	197	21	can	can	AUX
ejpam-4979	197	22	then	then	ADV
ejpam-4979	197	23	model	model	VERB
ejpam-4979	197	24	j	j	PROPN
ejpam-4979	197	25	found	find	VERB
ejpam-4979	197	26	on	on	ADP
ejpam-4979	197	27	the	the	DET
ejpam-4979	197	28	rhs	rhs	PROPN
ejpam-4979	197	29	of	of	ADP
ejpam-4979	197	30	eq	eq	PROPN
ejpam-4979	197	31	.	.	PROPN
ejpam-4979	197	32	14	14	NUM
ejpam-4979	197	33	as	as	ADP
ejpam-4979	197	34	j	j	PROPN
ejpam-4979	197	35	=	=	SYM
ejpam-4979	197	36	c	c	X
ejpam-4979	197	37	ln2	ln2	ADJ
ejpam-4979	197	38	pr	pr	VERB
ejpam-4979	197	39	1	1	NUM
ejpam-4979	197	40	ln	ln	NOUN
ejpam-4979	197	41	pr	pr	NOUN
ejpam-4979	197	42	=	=	PUNCT
ejpam-4979	197	43	c	c	X
ejpam-4979	197	44	·	·	PUNCT
ejpam-4979	197	45	1	1	NUM
ejpam-4979	197	46	ln	ln	NOUN
ejpam-4979	197	47	pr	pr	NOUN
ejpam-4979	197	48	and	and	CCONJ
ejpam-4979	197	49	we	we	PRON
ejpam-4979	197	50	can	can	AUX
ejpam-4979	197	51	model	model	VERB
ejpam-4979	197	52	k	k	PROPN
ejpam-4979	197	53	found	find	VERB
ejpam-4979	197	54	on	on	ADP
ejpam-4979	197	55	the	the	DET
ejpam-4979	197	56	rhs	rhs	PROPN
ejpam-4979	197	57	of	of	ADP
ejpam-4979	197	58	eq	eq	PROPN
ejpam-4979	197	59	.	.	PROPN
ejpam-4979	197	60	14	14	NUM
ejpam-4979	197	61	as	as	ADP
ejpam-4979	197	62	k	k	PROPN
ejpam-4979	197	63	=	=	PROPN
ejpam-4979	197	64	1−	1−	NUM
ejpam-4979	197	65	c	c	X
ejpam-4979	197	66	·	·	PUNCT
ejpam-4979	197	67	1	1	NUM
ejpam-4979	197	68	ln	ln	NOUN
ejpam-4979	197	69	pr	pr	NOUN
ejpam-4979	197	70	such	such	ADJ
ejpam-4979	197	71	that	that	DET
ejpam-4979	197	72	j	j	PROPN
ejpam-4979	198	1	=	=	PUNCT
ejpam-4979	198	2	the	the	DET
ejpam-4979	198	3	ratio	ratio	NOUN
ejpam-4979	198	4	of	of	ADP
ejpam-4979	198	5	the	the	DET
ejpam-4979	198	6	asymptotic	asymptotic	ADJ
ejpam-4979	198	7	density	density	NOUN
ejpam-4979	198	8	of	of	ADP
ejpam-4979	198	9	the	the	DET
ejpam-4979	198	10	twin	twin	ADJ
ejpam-4979	198	11	prime	prime	ADJ
ejpam-4979	198	12	subsequence	subsequence	NOUN
ejpam-4979	198	13	p2	p2	NOUN
ejpam-4979	198	14	divided	divide	VERB
ejpam-4979	198	15	by	by	ADP
ejpam-4979	198	16	the	the	DET
ejpam-4979	198	17	asymptotic	asymptotic	ADJ
ejpam-4979	198	18	density	density	NOUN
ejpam-4979	198	19	of	of	ADP
ejpam-4979	198	20	the	the	DET
ejpam-4979	198	21	set	set	NOUN
ejpam-4979	198	22	of	of	ADP
ejpam-4979	198	23	all	all	DET
ejpam-4979	198	24	primes	prime	NOUN
ejpam-4979	198	25	p	p	X
ejpam-4979	198	26	;	;	PUNCT
ejpam-4979	198	27	and	and	CCONJ
ejpam-4979	198	28	k	k	X
ejpam-4979	198	29	=	=	PUNCT
ejpam-4979	199	1	the	the	DET
ejpam-4979	199	2	ratio	ratio	NOUN
ejpam-4979	199	3	of	of	ADP
ejpam-4979	199	4	the	the	DET
ejpam-4979	199	5	asymptotic	asymptotic	ADJ
ejpam-4979	199	6	density	density	NOUN
ejpam-4979	199	7	of	of	ADP
ejpam-4979	199	8	the	the	DET
ejpam-4979	199	9	set	set	NOUN
ejpam-4979	199	10	of	of	ADP
ejpam-4979	199	11	remaining	remain	VERB
ejpam-4979	199	12	prime	prime	ADJ
ejpam-4979	199	13	numbers	number	NOUN
ejpam-4979	199	14	[	[	X
ejpam-4979	199	15	p	p	X
ejpam-4979	199	16	−	−	PROPN
ejpam-4979	199	17	p2	p2	NOUN
ejpam-4979	199	18	]	]	PUNCT
ejpam-4979	199	19	to	to	ADP
ejpam-4979	199	20	the	the	DET
ejpam-4979	199	21	asymptotic	asymptotic	ADJ
ejpam-4979	199	22	density	density	NOUN
ejpam-4979	199	23	of	of	ADP
ejpam-4979	199	24	all	all	DET
ejpam-4979	199	25	primes	prime	NOUN
ejpam-4979	199	26	p	p	NOUN
ejpam-4979	199	27	so	so	SCONJ
ejpam-4979	199	28	that	that	SCONJ
ejpam-4979	199	29	j	j	PROPN
ejpam-4979	200	1	+	+	CCONJ
ejpam-4979	200	2	k	k	X
ejpam-4979	200	3	=	=	SYM
ejpam-4979	200	4	1	1	X
ejpam-4979	200	5	.	.	PUNCT
ejpam-4979	201	1	we	we	PRON
ejpam-4979	201	2	can	can	AUX
ejpam-4979	201	3	then	then	ADV
ejpam-4979	201	4	model	model	VERB
ejpam-4979	201	5	π2(x	π2(x	PROPN
ejpam-4979	201	6	)	)	PUNCT
ejpam-4979	201	7	as	as	ADP
ejpam-4979	201	8	π2(x)≤	π2(x)≤	NUM
ejpam-4979	201	9	r2	r2	PROPN
ejpam-4979	201	10	+	+	CCONJ
ejpam-4979	201	11	x	x	SYM
ejpam-4979	201	12	·	·	PUNCT
ejpam-4979	201	13	1	1	NUM
ejpam-4979	201	14	ln	ln	NOUN
ejpam-4979	201	15	pr	pr	NOUN
ejpam-4979	201	16	·	·	PUNCT
ejpam-4979	202	1	[	[	PUNCT
ejpam-4979	202	2	c	c	X
ejpam-4979	202	3	·	·	PUNCT
ejpam-4979	202	4	1	1	NUM
ejpam-4979	202	5	ln	ln	NOUN
ejpam-4979	202	6	pr	pr	NOUN
ejpam-4979	202	7	]	]	PUNCT
ejpam-4979	203	1	+	+	CCONJ
ejpam-4979	203	2	2r2	2r2	NUM
ejpam-4979	203	3	(	(	PUNCT
ejpam-4979	203	4	18	18	NUM
ejpam-4979	203	5	)	)	PUNCT
ejpam-4979	203	6	=	=	SYM
ejpam-4979	203	7	r2	r2	PROPN
ejpam-4979	203	8	+	+	CCONJ
ejpam-4979	203	9	x	x	X
ejpam-4979	203	10	·	·	PUNCT
ejpam-4979	203	11	c	c	X
ejpam-4979	203	12	·	·	PUNCT
ejpam-4979	203	13	1	1	NUM
ejpam-4979	203	14	ln2	ln2	ADJ
ejpam-4979	203	15	pr	pr	NOUN
ejpam-4979	203	16	+	+	CCONJ
ejpam-4979	203	17	2r2	2r2	NUM
ejpam-4979	203	18	(	(	PUNCT
ejpam-4979	203	19	19	19	NUM
ejpam-4979	203	20	)	)	PUNCT
ejpam-4979	203	21	where	where	SCONJ
ejpam-4979	203	22	r2	r2	NOUN
ejpam-4979	203	23	=	=	PUNCT
ejpam-4979	203	24	the	the	DET
ejpam-4979	203	25	number	number	NOUN
ejpam-4979	203	26	of	of	ADP
ejpam-4979	203	27	p2	p2	PROPN
ejpam-4979	203	28	≤	≤	PUNCT
ejpam-4979	203	29	pr	pr	NOUN
ejpam-4979	203	30	and	and	CCONJ
ejpam-4979	203	31	2r2	2r2	NUM
ejpam-4979	203	32	=	=	NOUN
ejpam-4979	203	33	the	the	DET
ejpam-4979	203	34	maximum	maximum	ADJ
ejpam-4979	203	35	error	error	NOUN
ejpam-4979	203	36	resulting	result	VERB
ejpam-4979	203	37	from	from	ADP
ejpam-4979	203	38	the	the	DET
ejpam-4979	203	39	main	main	ADJ
ejpam-4979	203	40	term	term	NOUN
ejpam-4979	203	41	.	.	PUNCT
ejpam-4979	204	1	similar	similar	ADJ
ejpam-4979	204	2	to	to	ADP
ejpam-4979	204	3	r′′	r′′	VERB
ejpam-4979	204	4	,	,	PUNCT
ejpam-4979	204	5	we	we	PRON
ejpam-4979	204	6	know	know	VERB
ejpam-4979	204	7	that	that	SCONJ
ejpam-4979	204	8	r2	r2	PROPN
ejpam-4979	204	9	≤	≤	ADV
ejpam-4979	204	10	⌊r	⌊r	CCONJ
ejpam-4979	204	11	2	2	NUM
ejpam-4979	204	12	⌋	⌋	NOUN
ejpam-4979	204	13	for	for	ADP
ejpam-4979	204	14	p2	p2	PROPN
ejpam-4979	204	15	>	>	X
ejpam-4979	204	16	2	2	NUM
ejpam-4979	204	17	because	because	SCONJ
ejpam-4979	204	18	there	there	PRON
ejpam-4979	204	19	are	be	VERB
ejpam-4979	204	20	fewer	few	ADJ
ejpam-4979	204	21	twin	twin	ADJ
ejpam-4979	204	22	primes	prime	NOUN
ejpam-4979	204	23	p2	p2	NOUN
ejpam-4979	204	24	than	than	ADP
ejpam-4979	204	25	half	half	DET
ejpam-4979	204	26	the	the	DET
ejpam-4979	204	27	count	count	NOUN
ejpam-4979	204	28	of	of	ADP
ejpam-4979	204	29	all	all	DET
ejpam-4979	204	30	prime	prime	ADJ
ejpam-4979	204	31	numbers	number	NOUN
ejpam-4979	204	32	p.	p.	NOUN
ejpam-4979	204	33	thus	thus	ADV
ejpam-4979	204	34	,	,	PUNCT
ejpam-4979	204	35	π2(x)≤	π2(x)≤	NUM
ejpam-4979	204	36	r2	r2	PROPN
ejpam-4979	204	37	+	+	CCONJ
ejpam-4979	204	38	x	x	X
ejpam-4979	204	39	·	·	PUNCT
ejpam-4979	204	40	c	c	X
ejpam-4979	204	41	ln2	ln2	ADJ
ejpam-4979	204	42	pr	pr	NOUN
ejpam-4979	205	1	+	+	CCONJ
ejpam-4979	205	2	2r2	2r2	NUM
ejpam-4979	205	3	(	(	PUNCT
ejpam-4979	205	4	19	19	NUM
ejpam-4979	205	5	)	)	PUNCT
ejpam-4979	205	6	<	<	X
ejpam-4979	205	7	r2	r2	PROPN
ejpam-4979	205	8	+	+	CCONJ
ejpam-4979	206	1	x	x	X
ejpam-4979	206	2	·	·	PUNCT
ejpam-4979	206	3	c	c	NOUN
ejpam-4979	206	4	ln2	ln2	ADJ
ejpam-4979	206	5	r	r	NOUN
ejpam-4979	206	6	+	+	CCONJ
ejpam-4979	206	7	2r2	2r2	NUM
ejpam-4979	206	8	(	(	PUNCT
ejpam-4979	206	9	r	r	NOUN
ejpam-4979	206	10	<	<	X
ejpam-4979	206	11	pr	pr	NOUN
ejpam-4979	206	12	)	)	PUNCT
ejpam-4979	206	13	≤	≤	NOUN
ejpam-4979	206	14	⌊r	⌊r	ADP
ejpam-4979	207	1	2	2	NUM
ejpam-4979	207	2	⌋+	⌋+	NOUN
ejpam-4979	207	3	x	x	X
ejpam-4979	207	4	·	·	PUNCT
ejpam-4979	207	5	c	c	NOUN
ejpam-4979	208	1	ln2	ln2	ADJ
ejpam-4979	208	2	r	r	NOUN
ejpam-4979	208	3	+	+	CCONJ
ejpam-4979	208	4	2⌊	2⌊	NUM
ejpam-4979	208	5	r	r	NOUN
ejpam-4979	208	6	2	2	NUM
ejpam-4979	208	7	⌋	⌋	NOUN
ejpam-4979	208	8	(	(	PUNCT
ejpam-4979	208	9	⌊r	⌊r	NOUN
ejpam-4979	208	10	2	2	NUM
ejpam-4979	208	11	⌋	⌋	NOUN
ejpam-4979	208	12	≥	≥	NUM
ejpam-4979	208	13	r2	r2	PROPN
ejpam-4979	208	14	)	)	PUNCT
ejpam-4979	208	15	<	<	X
ejpam-4979	208	16	x	x	X
ejpam-4979	208	17	·	·	PUNCT
ejpam-4979	208	18	c	c	NOUN
ejpam-4979	209	1	ln2	ln2	ADJ
ejpam-4979	209	2	r	r	NOUN
ejpam-4979	209	3	+	+	CCONJ
ejpam-4979	209	4	2⌊	2⌊	NUM
ejpam-4979	209	5	r	r	NOUN
ejpam-4979	209	6	2	2	NUM
ejpam-4979	209	7	⌋+1	⌋+1	NOUN
ejpam-4979	209	8	(	(	PUNCT
ejpam-4979	209	9	2⌊	2⌊	NUM
ejpam-4979	209	10	r	r	NOUN
ejpam-4979	209	11	2	2	NUM
ejpam-4979	209	12	⌋	⌋	NOUN
ejpam-4979	209	13	>	>	X
ejpam-4979	209	14	⌊r	⌊r	NOUN
ejpam-4979	209	15	2	2	NUM
ejpam-4979	209	16	⌋	⌋	NOUN
ejpam-4979	209	17	)	)	PUNCT
ejpam-4979	210	1	m.	m.	NOUN
ejpam-4979	210	2	p.	p.	NOUN
ejpam-4979	210	3	may	may	AUX
ejpam-4979	210	4	/	/	SYM
ejpam-4979	210	5	eur	eur	PROPN
ejpam-4979	210	6	.	.	PUNCT
ejpam-4979	211	1	j.	j.	PROPN
ejpam-4979	211	2	pure	pure	PROPN
ejpam-4979	211	3	appl	appl	PROPN
ejpam-4979	211	4	.	.	PROPN
ejpam-4979	211	5	math	math	PROPN
ejpam-4979	211	6	,	,	PUNCT
ejpam-4979	211	7	17	17	NUM
ejpam-4979	211	8	(	(	PUNCT
ejpam-4979	211	9	1	1	NUM
ejpam-4979	211	10	)	)	PUNCT
ejpam-4979	211	11	(	(	PUNCT
ejpam-4979	211	12	2024	2024	NUM
ejpam-4979	211	13	)	)	PUNCT
ejpam-4979	211	14	,	,	PUNCT
ejpam-4979	211	15	42	42	NUM
ejpam-4979	211	16	-	-	SYM
ejpam-4979	211	17	58	58	NUM
ejpam-4979	211	18	56	56	NUM
ejpam-4979	211	19	<	<	X
ejpam-4979	211	20	x	x	X
ejpam-4979	211	21	·	·	PUNCT
ejpam-4979	211	22	c	c	NOUN
ejpam-4979	212	1	ln2	ln2	ADJ
ejpam-4979	212	2	r	r	NOUN
ejpam-4979	212	3	+	+	CCONJ
ejpam-4979	212	4	2	2	NUM
ejpam-4979	212	5	r	r	NOUN
ejpam-4979	212	6	2	2	NUM
ejpam-4979	212	7	+1	+1	NOUN
ejpam-4979	212	8	(	(	PUNCT
ejpam-4979	212	9	r	r	NOUN
ejpam-4979	212	10	2	2	NUM
ejpam-4979	212	11	>	>	PUNCT
ejpam-4979	212	12	⌊r	⌊r	NOUN
ejpam-4979	212	13	2	2	NUM
ejpam-4979	212	14	⌋	⌋	NOUN
ejpam-4979	212	15	)	)	PUNCT
ejpam-4979	212	16	.	.	PUNCT
ejpam-4979	213	1	now	now	ADV
ejpam-4979	213	2	,	,	PUNCT
ejpam-4979	213	3	let	let	VERB
ejpam-4979	213	4	r	r	NOUN
ejpam-4979	213	5	=	=	PUNCT
ejpam-4979	213	6	xm	xm	PROPN
ejpam-4979	213	7	such	such	ADJ
ejpam-4979	213	8	that	that	SCONJ
ejpam-4979	213	9	m	m	VERB
ejpam-4979	213	10	=	=	SYM
ejpam-4979	213	11	1	1	NUM
ejpam-4979	213	12	c	c	NOUN
ejpam-4979	213	13	·	·	PUNCT
ejpam-4979	213	14	ln	ln	ADJ
ejpam-4979	213	15	lnx	lnx	NOUN
ejpam-4979	213	16	for	for	ADP
ejpam-4979	213	17	some	some	DET
ejpam-4979	213	18	positive	positive	ADJ
ejpam-4979	213	19	constant	constant	ADJ
ejpam-4979	213	20	c.	c.	NOUN
ejpam-4979	213	21	we	we	PRON
ejpam-4979	213	22	now	now	ADV
ejpam-4979	213	23	have	have	VERB
ejpam-4979	213	24	π2(x	π2(x	X
ejpam-4979	213	25	)	)	PUNCT
ejpam-4979	213	26	<	<	X
ejpam-4979	213	27	x	x	X
ejpam-4979	213	28	·	·	PUNCT
ejpam-4979	213	29	c	c	X
ejpam-4979	213	30	ln2	ln2	PROPN
ejpam-4979	213	31	xm	xm	PROPN
ejpam-4979	214	1	+	+	CCONJ
ejpam-4979	214	2	2	2	NUM
ejpam-4979	214	3	1	1	NUM
ejpam-4979	214	4	2	2	NUM
ejpam-4979	214	5	(	(	PUNCT
ejpam-4979	214	6	xm+2	xm+2	NOUN
ejpam-4979	214	7	)	)	PUNCT
ejpam-4979	214	8	=	=	SYM
ejpam-4979	214	9	x	x	PUNCT
ejpam-4979	214	10	·	·	PUNCT
ejpam-4979	214	11	c	c	X
ejpam-4979	214	12	(	(	PUNCT
ejpam-4979	214	13	lnx	lnx	PROPN
ejpam-4979	214	14	c	c	PROPN
ejpam-4979	214	15	·	·	PUNCT
ejpam-4979	214	16	ln	ln	ADJ
ejpam-4979	214	17	lnx	lnx	PROPN
ejpam-4979	214	18	)	)	PUNCT
ejpam-4979	214	19	2	2	NUM
ejpam-4979	215	1	+	+	CCONJ
ejpam-4979	215	2	2	2	NUM
ejpam-4979	215	3	1	1	NUM
ejpam-4979	215	4	2	2	NUM
ejpam-4979	215	5	(	(	PUNCT
ejpam-4979	215	6	xm+2	xm+2	NOUN
ejpam-4979	215	7	)	)	PUNCT
ejpam-4979	215	8	=	=	PUNCT
ejpam-4979	215	9	x	x	PUNCT
ejpam-4979	215	10	·	·	PUNCT
ejpam-4979	215	11	c	c	X
ejpam-4979	215	12	·	·	PUNCT
ejpam-4979	215	13	(	(	PUNCT
ejpam-4979	215	14	c	c	X
ejpam-4979	215	15	·	·	PUNCT
ejpam-4979	215	16	ln	ln	ADJ
ejpam-4979	215	17	lnx)2	lnx)2	PROPN
ejpam-4979	215	18	(	(	PUNCT
ejpam-4979	215	19	lnx)2	lnx)2	PROPN
ejpam-4979	215	20	+	+	CCONJ
ejpam-4979	215	21	2	2	NUM
ejpam-4979	215	22	1	1	NUM
ejpam-4979	215	23	2	2	NUM
ejpam-4979	215	24	(	(	PUNCT
ejpam-4979	215	25	xm+2	xm+2	NOUN
ejpam-4979	215	26	)	)	PUNCT
ejpam-4979	215	27	=	=	PUNCT
ejpam-4979	216	1	c	c	X
ejpam-4979	216	2	·	·	PUNCT
ejpam-4979	216	3	x	x	SYM
ejpam-4979	216	4	·	·	PUNCT
ejpam-4979	216	5	(	(	PUNCT
ejpam-4979	216	6	ln	ln	PROPN
ejpam-4979	216	7	lnx	lnx	PROPN
ejpam-4979	216	8	)	)	PUNCT
ejpam-4979	216	9	2	2	NUM
ejpam-4979	216	10	(	(	PUNCT
ejpam-4979	216	11	lnx)2	lnx)2	PROPN
ejpam-4979	216	12	+	+	CCONJ
ejpam-4979	216	13	2	2	NUM
ejpam-4979	216	14	1	1	NUM
ejpam-4979	216	15	2	2	NUM
ejpam-4979	216	16	(	(	PUNCT
ejpam-4979	216	17	xm+2	xm+2	NOUN
ejpam-4979	216	18	)	)	PUNCT
ejpam-4979	216	19	.	.	PUNCT
ejpam-4979	217	1	and	and	CCONJ
ejpam-4979	217	2	since	since	SCONJ
ejpam-4979	217	3	2	2	NUM
ejpam-4979	217	4	1	1	NUM
ejpam-4979	217	5	2	2	NUM
ejpam-4979	217	6	(	(	PUNCT
ejpam-4979	217	7	xm+2	xm+2	NOUN
ejpam-4979	217	8	)	)	PUNCT
ejpam-4979	217	9	≪	≪	VERB
ejpam-4979	217	10	than	than	ADP
ejpam-4979	217	11	the	the	DET
ejpam-4979	217	12	main	main	ADJ
ejpam-4979	217	13	term	term	NOUN
ejpam-4979	217	14	for	for	ADP
ejpam-4979	217	15	c	c	PROPN
ejpam-4979	217	16	≥	≥	NUM
ejpam-4979	217	17	5	5	NUM
ejpam-4979	217	18	,	,	PUNCT
ejpam-4979	217	19	we	we	PRON
ejpam-4979	217	20	arrive	arrive	VERB
ejpam-4979	217	21	at	at	ADP
ejpam-4979	217	22	π2(x	π2(x	NOUN
ejpam-4979	217	23	)	)	PUNCT
ejpam-4979	217	24	<	<	X
ejpam-4979	217	25	c	c	X
ejpam-4979	217	26	·	·	PUNCT
ejpam-4979	217	27	x	x	SYM
ejpam-4979	217	28	·	·	PUNCT
ejpam-4979	217	29	(	(	PUNCT
ejpam-4979	217	30	ln	ln	PROPN
ejpam-4979	217	31	lnx	lnx	PROPN
ejpam-4979	217	32	)	)	PUNCT
ejpam-4979	217	33	2	2	NUM
ejpam-4979	217	34	(	(	PUNCT
ejpam-4979	217	35	lnx)2	lnx)2	PROPN
ejpam-4979	217	36	.	.	PUNCT
ejpam-4979	218	1	(	(	PUNCT
ejpam-4979	218	2	20	20	NUM
ejpam-4979	218	3	)	)	PUNCT
ejpam-4979	218	4	thus	thus	ADV
ejpam-4979	218	5	,	,	PUNCT
ejpam-4979	218	6	it	it	PRON
ejpam-4979	218	7	is	be	AUX
ejpam-4979	218	8	confirmed	confirm	VERB
ejpam-4979	218	9	via	via	ADP
ejpam-4979	218	10	this	this	DET
ejpam-4979	218	11	approach	approach	NOUN
ejpam-4979	219	1	that	that	SCONJ
ejpam-4979	219	2	for	for	ADP
ejpam-4979	219	3	some	some	DET
ejpam-4979	219	4	positive	positive	ADJ
ejpam-4979	219	5	constant	constant	ADJ
ejpam-4979	219	6	c	c	NOUN
ejpam-4979	219	7	<	<	X
ejpam-4979	219	8	+	+	PROPN
ejpam-4979	219	9	∞	∞	PROPN
ejpam-4979	219	10	,	,	PUNCT
ejpam-4979	219	11	the	the	DET
ejpam-4979	219	12	sum	sum	NOUN
ejpam-4979	219	13	of	of	ADP
ejpam-4979	219	14	the	the	DET
ejpam-4979	219	15	reciprocals	reciprocal	NOUN
ejpam-4979	219	16	of	of	ADP
ejpam-4979	219	17	the	the	DET
ejpam-4979	219	18	twin	twin	ADJ
ejpam-4979	219	19	primes	prime	NOUN
ejpam-4979	219	20	p2	p2	NOUN
ejpam-4979	219	21	converges	converge	NOUN
ejpam-4979	219	22	.	.	PUNCT
ejpam-4979	220	1	further	far	ADV
ejpam-4979	220	2	,	,	PUNCT
ejpam-4979	220	3	when	when	SCONJ
ejpam-4979	220	4	we	we	PRON
ejpam-4979	220	5	compare	compare	VERB
ejpam-4979	220	6	the	the	DET
ejpam-4979	220	7	inequality	inequality	NOUN
ejpam-4979	220	8	20	20	NUM
ejpam-4979	220	9	with	with	ADP
ejpam-4979	220	10	the	the	DET
ejpam-4979	220	11	inequality	inequality	NOUN
ejpam-4979	220	12	for	for	ADP
ejpam-4979	220	13	p′′	p′′	PROPN
ejpam-4979	220	14	in	in	ADP
ejpam-4979	220	15	16	16	NUM
ejpam-4979	220	16	,	,	PUNCT
ejpam-4979	220	17	we	we	PRON
ejpam-4979	220	18	see	see	VERB
ejpam-4979	220	19	that	that	SCONJ
ejpam-4979	220	20	the	the	DET
ejpam-4979	220	21	count	count	NOUN
ejpam-4979	220	22	of	of	ADP
ejpam-4979	220	23	p′′	p′′	PROPN
ejpam-4979	220	24	≤	≤	PROPN
ejpam-4979	220	25	x	x	X
ejpam-4979	220	26	,	,	PUNCT
ejpam-4979	220	27	or	or	CCONJ
ejpam-4979	220	28	π′′(x	π′′(x	NOUN
ejpam-4979	220	29	)	)	PUNCT
ejpam-4979	220	30	,	,	PUNCT
ejpam-4979	220	31	is	be	AUX
ejpam-4979	220	32	less	less	ADJ
ejpam-4979	220	33	than	than	ADP
ejpam-4979	220	34	the	the	DET
ejpam-4979	220	35	count	count	NOUN
ejpam-4979	220	36	of	of	ADP
ejpam-4979	220	37	twin	twin	ADJ
ejpam-4979	220	38	primes	prime	NOUN
ejpam-4979	220	39	p2	p2	X
ejpam-4979	220	40	≤	≤	NUM
ejpam-4979	220	41	x	x	NOUN
ejpam-4979	220	42	,	,	PUNCT
ejpam-4979	220	43	or	or	CCONJ
ejpam-4979	220	44	π2(x	π2(x	NUM
ejpam-4979	220	45	)	)	PUNCT
ejpam-4979	220	46	.	.	PUNCT
ejpam-4979	221	1	8	8	X
ejpam-4979	221	2	.	.	X
ejpam-4979	221	3	mathematica	mathematica	PROPN
ejpam-4979	221	4	calculations	calculations	PROPN
ejpam-4979	221	5	mathematica	mathematica	PROPN
ejpam-4979	221	6	[	[	X
ejpam-4979	221	7	5	5	NUM
ejpam-4979	221	8	]	]	PUNCT
ejpam-4979	221	9	was	be	AUX
ejpam-4979	221	10	programmed	program	VERB
ejpam-4979	221	11	to	to	PART
ejpam-4979	221	12	calculate	calculate	VERB
ejpam-4979	221	13	the	the	DET
ejpam-4979	221	14	sum	sum	NOUN
ejpam-4979	221	15	of	of	ADP
ejpam-4979	221	16	the	the	DET
ejpam-4979	221	17	reciprocals	reciprocal	NOUN
ejpam-4979	221	18	of	of	ADP
ejpam-4979	221	19	p′′	p′′	PROPN
ejpam-4979	221	20	and	and	CCONJ
ejpam-4979	221	21	p2	p2	NOUN
ejpam-4979	221	22	for	for	ADP
ejpam-4979	221	23	various	various	ADJ
ejpam-4979	221	24	ranges	range	NOUN
ejpam-4979	221	25	of	of	ADP
ejpam-4979	221	26	x	x	X
ejpam-4979	221	27	up	up	ADP
ejpam-4979	221	28	to	to	ADP
ejpam-4979	221	29	10e6	10e6	NUM
ejpam-4979	221	30	,	,	PUNCT
ejpam-4979	221	31	and	and	CCONJ
ejpam-4979	221	32	a	a	DET
ejpam-4979	221	33	table	table	NOUN
ejpam-4979	221	34	of	of	ADP
ejpam-4979	221	35	the	the	DET
ejpam-4979	221	36	computations	computation	NOUN
ejpam-4979	221	37	appears	appear	VERB
ejpam-4979	221	38	below	below	ADV
ejpam-4979	221	39	.	.	PUNCT
ejpam-4979	222	1	table	table	NOUN
ejpam-4979	222	2	3	3	NUM
ejpam-4979	222	3	reveals	reveal	VERB
ejpam-4979	222	4	that	that	SCONJ
ejpam-4979	222	5	through	through	ADP
ejpam-4979	222	6	the	the	DET
ejpam-4979	222	7	ranges	range	NOUN
ejpam-4979	222	8	of	of	ADP
ejpam-4979	222	9	x	x	PUNCT
ejpam-4979	222	10	calculated	calculate	VERB
ejpam-4979	222	11	,	,	PUNCT
ejpam-4979	222	12	the	the	DET
ejpam-4979	222	13	sum	sum	NOUN
ejpam-4979	222	14	of	of	ADP
ejpam-4979	222	15	the	the	DET
ejpam-4979	222	16	reciprocals	reciprocal	NOUN
ejpam-4979	222	17	of	of	ADP
ejpam-4979	222	18	p′′	p′′	PROPN
ejpam-4979	222	19	is	be	AUX
ejpam-4979	222	20	smaller	small	ADJ
ejpam-4979	222	21	than	than	ADP
ejpam-4979	222	22	the	the	DET
ejpam-4979	222	23	sum	sum	NOUN
ejpam-4979	222	24	of	of	ADP
ejpam-4979	222	25	the	the	DET
ejpam-4979	222	26	reciprocals	reciprocal	NOUN
ejpam-4979	222	27	for	for	ADP
ejpam-4979	222	28	the	the	DET
ejpam-4979	222	29	twin	twin	ADJ
ejpam-4979	222	30	primes	prime	NOUN
ejpam-4979	222	31	p2	p2	NOUN
ejpam-4979	222	32	,	,	PUNCT
ejpam-4979	222	33	both	both	PRON
ejpam-4979	222	34	of	of	ADP
ejpam-4979	222	35	which	which	PRON
ejpam-4979	222	36	converge	converge	VERB
ejpam-4979	222	37	at	at	ADP
ejpam-4979	222	38	∞.	∞.	PROPN
ejpam-4979	222	39	references	reference	VERB
ejpam-4979	222	40	57	57	NUM
ejpam-4979	222	41	table	table	NOUN
ejpam-4979	222	42	3	3	NUM
ejpam-4979	222	43	:	:	PUNCT
ejpam-4979	222	44	p′′(x	p′′(x	PROPN
ejpam-4979	222	45	)	)	PUNCT
ejpam-4979	222	46	and	and	CCONJ
ejpam-4979	222	47	p2(x	p2(x	X
ejpam-4979	222	48	)	)	PUNCT
ejpam-4979	222	49	reciprocal	reciprocal	ADJ
ejpam-4979	222	50	sums	sum	NOUN
ejpam-4979	222	51	x	x	SYM
ejpam-4979	222	52	∑	∑	ADP
ejpam-4979	222	53	1	1	NUM
ejpam-4979	222	54	p′′(x	p′′(x	NOUN
ejpam-4979	222	55	)	)	PUNCT
ejpam-4979	222	56	∑	∑	ADV
ejpam-4979	222	57	1	1	NUM
ejpam-4979	222	58	p2(x	p2(x	NOUN
ejpam-4979	222	59	)	)	PUNCT
ejpam-4979	222	60	1e02	1e02	NOUN
ejpam-4979	223	1	0.534430	0.534430	NUM
ejpam-4979	224	1	1.28989	1.28989	NUM
ejpam-4979	224	2	1e03	1e03	NUM
ejpam-4979	225	1	0.606479	0.606479	NUM
ejpam-4979	225	2	1.40995	1.40995	NUM
ejpam-4979	225	3	1e04	1e04	NOUN
ejpam-4979	226	1	0.644283	0.644283	NUM
ejpam-4979	226	2	1.47370	1.47370	NUM
ejpam-4979	226	3	1e05	1e05	NUM
ejpam-4979	226	4	0.668046	0.668046	NUM
ejpam-4979	226	5	1.51443	1.51443	NUM
ejpam-4979	226	6	1e06	1e06	NUM
ejpam-4979	226	7	0.683968	0.683968	NUM
ejpam-4979	226	8	1.54268	1.54268	NUM
ejpam-4979	226	9	2e06	2e06	NUM
ejpam-4979	226	10	0.687789	0.687789	NUM
ejpam-4979	226	11	1.54950	1.54950	NUM
ejpam-4979	226	12	3e06	3e06	NUM
ejpam-4979	226	13	0.689858	0.689858	NUM
ejpam-4979	226	14	1.55321	1.55321	NUM
ejpam-4979	226	15	4e06	4e06	NOUN
ejpam-4979	226	16	0.691258	0.691258	NUM
ejpam-4979	226	17	1.55573	1.55573	NUM
ejpam-4979	226	18	5e06	5e06	NUM
ejpam-4979	226	19	0.692310	0.692310	NUM
ejpam-4979	226	20	1.55763	1.55763	NUM
ejpam-4979	226	21	6e06	6e06	NUM
ejpam-4979	226	22	0.693139	0.693139	NUM
ejpam-4979	226	23	1.55915	1.55915	NUM
ejpam-4979	226	24	7e06	7e06	NUM
ejpam-4979	226	25	0.693834	0.693834	NUM
ejpam-4979	226	26	1.56040	1.56040	NUM
ejpam-4979	226	27	8e06	8e06	NOUN
ejpam-4979	226	28	0.694421	0.694421	NUM
ejpam-4979	226	29	1.56148	1.56148	NUM
ejpam-4979	226	30	9e06	9e06	NOUN
ejpam-4979	226	31	0.694932	0.694932	NUM
ejpam-4979	226	32	1.56240	1.56240	NUM
ejpam-4979	226	33	10e6	10e6	NUM
ejpam-4979	226	34	0.695379	0.695379	NUM
ejpam-4979	226	35	1.56322	1.56322	NUM
ejpam-4979	226	36	9	9	NUM
ejpam-4979	226	37	.	.	PUNCT
ejpam-4979	226	38	conclusion	conclusion	NOUN
ejpam-4979	226	39	in	in	ADP
ejpam-4979	226	40	this	this	DET
ejpam-4979	226	41	paper	paper	NOUN
ejpam-4979	227	1	,	,	PUNCT
ejpam-4979	227	2	we	we	PRON
ejpam-4979	227	3	applied	apply	VERB
ejpam-4979	227	4	the	the	DET
ejpam-4979	227	5	inclusion	inclusion	NOUN
ejpam-4979	227	6	-	-	PUNCT
ejpam-4979	227	7	exclusion	exclusion	NOUN
ejpam-4979	227	8	principle	principle	NOUN
ejpam-4979	227	9	to	to	ADP
ejpam-4979	227	10	the	the	DET
ejpam-4979	227	11	complementary	complementary	ADJ
ejpam-4979	227	12	prime	prime	ADJ
ejpam-4979	227	13	number	number	NOUN
ejpam-4979	227	14	subsequences	subsequence	VERB
ejpam-4979	227	15	p′	p′	NOUN
ejpam-4979	227	16	and	and	CCONJ
ejpam-4979	227	17	p′′	p′′	PROPN
ejpam-4979	227	18	to	to	PART
ejpam-4979	227	19	derive	derive	VERB
ejpam-4979	227	20	the	the	DET
ejpam-4979	227	21	respective	respective	ADJ
ejpam-4979	227	22	prime	prime	ADJ
ejpam-4979	227	23	counting	counting	NOUN
ejpam-4979	227	24	functions	function	NOUN
ejpam-4979	227	25	π′(x	π′(x	NOUN
ejpam-4979	227	26	)	)	PUNCT
ejpam-4979	227	27	and	and	CCONJ
ejpam-4979	227	28	π′′(x	π′′(x	NOUN
ejpam-4979	227	29	)	)	PUNCT
ejpam-4979	227	30	to	to	PART
ejpam-4979	227	31	determine	determine	VERB
ejpam-4979	227	32	whether	whether	SCONJ
ejpam-4979	227	33	these	these	DET
ejpam-4979	227	34	subsequences	subsequence	NOUN
ejpam-4979	227	35	form	form	VERB
ejpam-4979	227	36	a	a	DET
ejpam-4979	227	37	small	small	ADJ
ejpam-4979	227	38	set	set	NOUN
ejpam-4979	227	39	or	or	CCONJ
ejpam-4979	227	40	a	a	DET
ejpam-4979	227	41	large	large	ADJ
ejpam-4979	227	42	set	set	NOUN
ejpam-4979	227	43	and	and	CCONJ
ejpam-4979	227	44	thus	thus	ADV
ejpam-4979	227	45	whether	whether	SCONJ
ejpam-4979	227	46	the	the	DET
ejpam-4979	227	47	infinite	infinite	ADJ
ejpam-4979	227	48	sum	sum	NOUN
ejpam-4979	227	49	of	of	ADP
ejpam-4979	227	50	the	the	DET
ejpam-4979	227	51	inverse	inverse	NOUN
ejpam-4979	227	52	of	of	ADP
ejpam-4979	227	53	their	their	PRON
ejpam-4979	227	54	terms	term	NOUN
ejpam-4979	227	55	converges	converge	VERB
ejpam-4979	227	56	or	or	CCONJ
ejpam-4979	227	57	diverges	diverge	NOUN
ejpam-4979	227	58	.	.	PUNCT
ejpam-4979	228	1	in	in	ADP
ejpam-4979	228	2	this	this	DET
ejpam-4979	228	3	study	study	NOUN
ejpam-4979	228	4	,	,	PUNCT
ejpam-4979	228	5	we	we	PRON
ejpam-4979	228	6	concluded	conclude	VERB
ejpam-4979	228	7	that	that	SCONJ
ejpam-4979	228	8	the	the	DET
ejpam-4979	228	9	sum	sum	NOUN
ejpam-4979	228	10	of	of	ADP
ejpam-4979	228	11	the	the	DET
ejpam-4979	228	12	reciprocals	reciprocal	NOUN
ejpam-4979	228	13	of	of	ADP
ejpam-4979	228	14	the	the	DET
ejpam-4979	228	15	prime	prime	ADJ
ejpam-4979	228	16	number	number	NOUN
ejpam-4979	228	17	subsequence	subsequence	VERB
ejpam-4979	228	18	p′	p′	NOUN
ejpam-4979	228	19	diverges	diverge	VERB
ejpam-4979	228	20	,	,	PUNCT
ejpam-4979	228	21	similar	similar	ADJ
ejpam-4979	228	22	to	to	ADP
ejpam-4979	228	23	that	that	PRON
ejpam-4979	228	24	for	for	ADP
ejpam-4979	228	25	the	the	DET
ejpam-4979	228	26	set	set	NOUN
ejpam-4979	228	27	of	of	ADP
ejpam-4979	228	28	all	all	DET
ejpam-4979	228	29	prime	prime	ADJ
ejpam-4979	228	30	numbers	number	NOUN
ejpam-4979	228	31	p	p	X
ejpam-4979	228	32	,	,	PUNCT
ejpam-4979	228	33	while	while	SCONJ
ejpam-4979	228	34	the	the	DET
ejpam-4979	228	35	sum	sum	NOUN
ejpam-4979	228	36	of	of	ADP
ejpam-4979	228	37	the	the	DET
ejpam-4979	228	38	reciprocals	reciprocal	NOUN
ejpam-4979	228	39	of	of	ADP
ejpam-4979	228	40	the	the	DET
ejpam-4979	228	41	prime	prime	ADJ
ejpam-4979	228	42	number	number	NOUN
ejpam-4979	228	43	subsequences	subsequence	VERB
ejpam-4979	228	44	p′′	p′′	PROPN
ejpam-4979	228	45	converges	converge	NOUN
ejpam-4979	228	46	,	,	PUNCT
ejpam-4979	228	47	similar	similar	ADJ
ejpam-4979	228	48	to	to	ADP
ejpam-4979	228	49	that	that	PRON
ejpam-4979	228	50	for	for	ADP
ejpam-4979	228	51	the	the	DET
ejpam-4979	228	52	set	set	NOUN
ejpam-4979	228	53	of	of	ADP
ejpam-4979	228	54	all	all	DET
ejpam-4979	228	55	twin	twin	ADJ
ejpam-4979	228	56	primes	prime	NOUN
ejpam-4979	228	57	p2	p2	NOUN
ejpam-4979	228	58	.	.	PUNCT
ejpam-4979	229	1	references	reference	NOUN
ejpam-4979	229	2	[	[	X
ejpam-4979	229	3	1	1	NUM
ejpam-4979	229	4	]	]	X
ejpam-4979	229	5	ka	ka	PROPN
ejpam-4979	229	6	broughan	broughan	PROPN
ejpam-4979	229	7	and	and	CCONJ
ejpam-4979	229	8	ar	ar	PROPN
ejpam-4979	229	9	barnett	barnett	PROPN
ejpam-4979	229	10	.	.	PUNCT
ejpam-4979	230	1	on	on	ADP
ejpam-4979	230	2	the	the	DET
ejpam-4979	230	3	subsequence	subsequence	NOUN
ejpam-4979	230	4	of	of	ADP
ejpam-4979	230	5	primes	prime	NOUN
ejpam-4979	230	6	having	have	VERB
ejpam-4979	230	7	prime	prime	ADJ
ejpam-4979	230	8	subscripts	subscript	NOUN
ejpam-4979	230	9	.	.	PUNCT
ejpam-4979	231	1	journal	journal	NOUN
ejpam-4979	231	2	of	of	ADP
ejpam-4979	231	3	integer	integer	PROPN
ejpam-4979	231	4	sequences	sequence	NOUN
ejpam-4979	231	5	,	,	PUNCT
ejpam-4979	231	6	12(09.2.3	12(09.2.3	NUM
ejpam-4979	231	7	)	)	PUNCT
ejpam-4979	231	8	,	,	PUNCT
ejpam-4979	231	9	2009	2009	NUM
ejpam-4979	231	10	.	.	PUNCT
ejpam-4979	232	1	[	[	X
ejpam-4979	232	2	2	2	NUM
ejpam-4979	232	3	]	]	PUNCT
ejpam-4979	232	4	gh	gh	PROPN
ejpam-4979	232	5	hardy	hardy	ADJ
ejpam-4979	232	6	and	and	CCONJ
ejpam-4979	232	7	je	je	PROPN
ejpam-4979	232	8	littlewood	littlewood	PROPN
ejpam-4979	232	9	.	.	PUNCT
ejpam-4979	233	1	some	some	DET
ejpam-4979	233	2	problems	problem	NOUN
ejpam-4979	233	3	of	of	ADP
ejpam-4979	233	4	’	'	PUNCT
ejpam-4979	233	5	partitio	partitio	NOUN
ejpam-4979	233	6	numerorum	numerorum	ADJ
ejpam-4979	233	7	’	'	PUNCT
ejpam-4979	233	8	iii	iii	NOUN
ejpam-4979	233	9	:	:	PUNCT
ejpam-4979	233	10	on	on	ADP
ejpam-4979	233	11	the	the	DET
ejpam-4979	233	12	expression	expression	NOUN
ejpam-4979	233	13	of	of	ADP
ejpam-4979	233	14	a	a	DET
ejpam-4979	233	15	number	number	NOUN
ejpam-4979	233	16	as	as	ADP
ejpam-4979	233	17	a	a	DET
ejpam-4979	233	18	sum	sum	NOUN
ejpam-4979	233	19	of	of	ADP
ejpam-4979	233	20	primes	prime	NOUN
ejpam-4979	233	21	.	.	PUNCT
ejpam-4979	234	1	acta	acta	PROPN
ejpam-4979	234	2	math	math	PROPN
ejpam-4979	234	3	,	,	PUNCT
ejpam-4979	234	4	44:1–70	44:1–70	NUM
ejpam-4979	234	5	,	,	PUNCT
ejpam-4979	234	6	1923	1923	NUM
ejpam-4979	234	7	.	.	PUNCT
ejpam-4979	235	1	[	[	X
ejpam-4979	235	2	3	3	NUM
ejpam-4979	235	3	]	]	X
ejpam-4979	235	4	oeis	oeis	PROPN
ejpam-4979	235	5	foundation	foundation	PROPN
ejpam-4979	235	6	inc	inc	PROPN
ejpam-4979	235	7	.	.	PROPN
ejpam-4979	236	1	entry	entry	NOUN
ejpam-4979	236	2	a262275	a262275	PROPN
ejpam-4979	236	3	in	in	ADP
ejpam-4979	236	4	the	the	DET
ejpam-4979	236	5	on	on	ADP
ejpam-4979	236	6	-	-	PUNCT
ejpam-4979	236	7	line	line	NOUN
ejpam-4979	236	8	encyclopedia	encyclopedia	NOUN
ejpam-4979	236	9	of	of	ADP
ejpam-4979	236	10	integer	integer	NOUN
ejpam-4979	236	11	sequences	sequence	NOUN
ejpam-4979	236	12	.	.	PUNCT
ejpam-4979	237	1	http://oeis.org/a262275	http://oeis.org/a262275	PROPN
ejpam-4979	237	2	,	,	PUNCT
ejpam-4979	237	3	2023	2023	NUM
ejpam-4979	237	4	.	.	PUNCT
ejpam-4979	238	1	[	[	X
ejpam-4979	238	2	4	4	NUM
ejpam-4979	238	3	]	]	X
ejpam-4979	238	4	oeis	oeis	PROPN
ejpam-4979	238	5	foundation	foundation	PROPN
ejpam-4979	238	6	inc	inc	PROPN
ejpam-4979	238	7	.	.	PROPN
ejpam-4979	239	1	entry	entry	NOUN
ejpam-4979	239	2	a333242	a333242	NOUN
ejpam-4979	239	3	in	in	ADP
ejpam-4979	239	4	the	the	DET
ejpam-4979	239	5	on	on	ADP
ejpam-4979	239	6	-	-	PUNCT
ejpam-4979	239	7	line	line	NOUN
ejpam-4979	239	8	encyclopedia	encyclopedia	NOUN
ejpam-4979	239	9	of	of	ADP
ejpam-4979	239	10	integer	integer	NOUN
ejpam-4979	239	11	sequences	sequence	NOUN
ejpam-4979	239	12	.	.	PUNCT
ejpam-4979	240	1	http://oeis.org/a333242	http://oeis.org/a333242	NOUN
ejpam-4979	240	2	,	,	PUNCT
ejpam-4979	240	3	2023	2023	NUM
ejpam-4979	240	4	.	.	PUNCT
ejpam-4979	241	1	[	[	X
ejpam-4979	241	2	5	5	NUM
ejpam-4979	241	3	]	]	PUNCT
ejpam-4979	241	4	wolfram	wolfram	PROPN
ejpam-4979	241	5	research	research	PROPN
ejpam-4979	241	6	,	,	PUNCT
ejpam-4979	241	7	inc	inc	PROPN
ejpam-4979	241	8	.	.	PROPN
ejpam-4979	241	9	mathematica	mathematica	PROPN
ejpam-4979	241	10	,	,	PUNCT
ejpam-4979	241	11	version	version	NOUN
ejpam-4979	241	12	13.3	13.3	NUM
ejpam-4979	241	13	.	.	PUNCT
ejpam-4979	242	1	champaign	champaign	PROPN
ejpam-4979	242	2	,	,	PUNCT
ejpam-4979	242	3	il	il	PROPN
ejpam-4979	242	4	,	,	PUNCT
ejpam-4979	242	5	2023	2023	NUM
ejpam-4979	242	6	.	.	PUNCT
ejpam-4979	243	1	[	[	X
ejpam-4979	243	2	6	6	NUM
ejpam-4979	243	3	]	]	PUNCT
ejpam-4979	243	4	wj	wj	X
ejpam-4979	243	5	leveque	leveque	ADJ
ejpam-4979	243	6	.	.	PUNCT
ejpam-4979	244	1	fundamentals	fundamental	NOUN
ejpam-4979	244	2	of	of	ADP
ejpam-4979	244	3	number	number	NOUN
ejpam-4979	244	4	theory	theory	NOUN
ejpam-4979	244	5	.	.	PUNCT
ejpam-4979	245	1	dover	dover	PROPN
ejpam-4979	245	2	publications	publication	NOUN
ejpam-4979	245	3	,	,	PUNCT
ejpam-4979	245	4	new	new	PROPN
ejpam-4979	245	5	york	york	PROPN
ejpam-4979	245	6	,	,	PUNCT
ejpam-4979	245	7	new	new	PROPN
ejpam-4979	245	8	york	york	PROPN
ejpam-4979	245	9	,	,	PUNCT
ejpam-4979	245	10	1977	1977	NUM
ejpam-4979	245	11	.	.	PUNCT
ejpam-4979	246	1	references	reference	NOUN
ejpam-4979	246	2	58	58	NUM
ejpam-4979	247	1	[	[	X
ejpam-4979	247	2	7	7	NUM
ejpam-4979	247	3	]	]	X
ejpam-4979	247	4	mp	mp	NOUN
ejpam-4979	247	5	may	may	AUX
ejpam-4979	247	6	.	.	PUNCT
ejpam-4979	248	1	properties	property	NOUN
ejpam-4979	248	2	of	of	ADP
ejpam-4979	248	3	higher	high	ADJ
ejpam-4979	248	4	-	-	PUNCT
ejpam-4979	248	5	order	order	NOUN
ejpam-4979	248	6	prime	prime	ADJ
ejpam-4979	248	7	number	number	NOUN
ejpam-4979	248	8	sequences	sequence	NOUN
ejpam-4979	248	9	.	.	PUNCT
ejpam-4979	249	1	missouri	missouri	PROPN
ejpam-4979	249	2	journal	journal	PROPN
ejpam-4979	249	3	of	of	ADP
ejpam-4979	249	4	mathematical	mathematical	ADJ
ejpam-4979	249	5	sciences	sciences	PROPN
ejpam-4979	249	6	,	,	PUNCT
ejpam-4979	249	7	32(2):158–170	32(2):158–170	PROPN
ejpam-4979	249	8	,	,	PUNCT
ejpam-4979	249	9	2020	2020	NUM
ejpam-4979	249	10	.	.	PUNCT
ejpam-4979	250	1	[	[	X
ejpam-4979	250	2	8	8	NUM
ejpam-4979	250	3	]	]	X
ejpam-4979	250	4	j	j	PROPN
ejpam-4979	250	5	mckernan	mckernan	PROPN
ejpam-4979	250	6	.	.	PUNCT
ejpam-4979	251	1	math	math	PROPN
ejpam-4979	251	2	104b	104b	PROPN
ejpam-4979	251	3	lecture	lecture	VERB
ejpam-4979	251	4	5	5	NUM
ejpam-4979	251	5	notes	note	NOUN
ejpam-4979	251	6	.	.	PUNCT
ejpam-4979	252	1	university	university	NOUN
ejpam-4979	252	2	of	of	ADP
ejpam-4979	252	3	california	california	PROPN
ejpam-4979	252	4	-	-	PUNCT
ejpam-4979	252	5	san	san	PROPN
ejpam-4979	252	6	diego	diego	PROPN
ejpam-4979	252	7	,	,	PUNCT
ejpam-4979	252	8	https://mathweb.ucsd.edu/˜jmckerna/teaching/17-18/winter/104b/lectures.html	https://mathweb.ucsd.edu/˜jmckerna/teaching/17-18/winter/104b/lectures.html	PROPN
ejpam-4979	252	9	,	,	PUNCT
ejpam-4979	252	10	2017	2017	NUM
ejpam-4979	252	11	.	.	PUNCT