id	sid	tid	token	lemma	pos
ejpam-4961	1	1	european	european	PROPN
ejpam-4961	1	2	journal	journal	PROPN
ejpam-4961	1	3	of	of	ADP
ejpam-4961	1	4	pure	pure	ADJ
ejpam-4961	1	5	and	and	CCONJ
ejpam-4961	1	6	applied	apply	VERB
ejpam-4961	1	7	mathematics	mathematic	NOUN
ejpam-4961	1	8	vol	vol	NOUN
ejpam-4961	1	9	.	.	PROPN
ejpam-4961	2	1	17	17	NUM
ejpam-4961	2	2	,	,	PUNCT
ejpam-4961	2	3	no	no	INTJ
ejpam-4961	2	4	.	.	NOUN
ejpam-4961	2	5	2	2	NUM
ejpam-4961	2	6	,	,	PUNCT
ejpam-4961	2	7	2024	2024	NUM
ejpam-4961	2	8	,	,	PUNCT
ejpam-4961	2	9	1146	1146	NUM
ejpam-4961	2	10	-	-	SYM
ejpam-4961	2	11	1154	1154	NUM
ejpam-4961	2	12	issn	issn	PROPN
ejpam-4961	2	13	1307	1307	NUM
ejpam-4961	2	14	-	-	SYM
ejpam-4961	2	15	5543	5543	NUM
ejpam-4961	2	16	–	–	PUNCT
ejpam-4961	2	17	ejpam.com	ejpam.com	X
ejpam-4961	2	18	published	publish	VERB
ejpam-4961	2	19	by	by	ADP
ejpam-4961	2	20	new	new	PROPN
ejpam-4961	2	21	york	york	PROPN
ejpam-4961	2	22	business	business	PROPN
ejpam-4961	2	23	global	global	PROPN
ejpam-4961	2	24	on	on	ADP
ejpam-4961	2	25	prime	prime	ADJ
ejpam-4961	2	26	counting	counting	NOUN
ejpam-4961	2	27	functions	function	NOUN
ejpam-4961	2	28	using	use	VERB
ejpam-4961	2	29	odd	odd	ADJ
ejpam-4961	2	30	k	k	ADJ
ejpam-4961	2	31	-	-	PUNCT
ejpam-4961	2	32	almost	almost	ADV
ejpam-4961	2	33	primes	prime	NOUN
ejpam-4961	2	34	t.	t.	PROPN
ejpam-4961	2	35	rashid1,2	rashid1,2	PROPN
ejpam-4961	2	36	,	,	PUNCT
ejpam-4961	2	37	m.	m.	NOUN
ejpam-4961	2	38	m.	m.	NOUN
ejpam-4961	2	39	m.	m.	PROPN
ejpam-4961	2	40	jaradat3,∗	jaradat3,∗	PROPN
ejpam-4961	2	41	,	,	PUNCT
ejpam-4961	2	42	e.	e.	PROPN
ejpam-4961	2	43	yolacan4	yolacan4	PROPN
ejpam-4961	2	44	,	,	PUNCT
ejpam-4961	3	1	h.	h.	PROPN
ejpam-4961	3	2	ahmad5	ahmad5	PROPN
ejpam-4961	3	3	1	1	NUM
ejpam-4961	3	4	department	department	NOUN
ejpam-4961	3	5	of	of	ADP
ejpam-4961	3	6	mathematical	mathematical	ADJ
ejpam-4961	3	7	sciences	science	NOUN
ejpam-4961	3	8	,	,	PUNCT
ejpam-4961	3	9	iust	iust	NOUN
ejpam-4961	3	10	,	,	PUNCT
ejpam-4961	3	11	awantipora	awantipora	PROPN
ejpam-4961	3	12	,	,	PUNCT
ejpam-4961	3	13	india	india	PROPN
ejpam-4961	3	14	,	,	PUNCT
ejpam-4961	3	15	192122	192122	NUM
ejpam-4961	3	16	2	2	NUM
ejpam-4961	3	17	department	department	NOUN
ejpam-4961	3	18	of	of	ADP
ejpam-4961	3	19	applied	apply	VERB
ejpam-4961	3	20	sciences	science	NOUN
ejpam-4961	3	21	and	and	CCONJ
ejpam-4961	3	22	humanities	humanity	NOUN
ejpam-4961	3	23	,	,	PUNCT
ejpam-4961	3	24	svkm	svkm	VERB
ejpam-4961	3	25	’s	’s	PART
ejpam-4961	3	26	,	,	PUNCT
ejpam-4961	3	27	nmims	nmim	NOUN
ejpam-4961	3	28	,	,	PUNCT
ejpam-4961	3	29	mukesh	mukesh	PROPN
ejpam-4961	3	30	patel	patel	PROPN
ejpam-4961	3	31	school	school	PROPN
ejpam-4961	3	32	of	of	ADP
ejpam-4961	3	33	technology	technology	NOUN
ejpam-4961	3	34	management	management	NOUN
ejpam-4961	3	35	and	and	CCONJ
ejpam-4961	3	36	engineering	engineering	NOUN
ejpam-4961	3	37	,	,	PUNCT
ejpam-4961	3	38	shirpur	shirpur	NOUN
ejpam-4961	3	39	,	,	PUNCT
ejpam-4961	3	40	india	india	PROPN
ejpam-4961	3	41	3	3	NUM
ejpam-4961	3	42	mathematics	mathematics	PROPN
ejpam-4961	3	43	program	program	NOUN
ejpam-4961	3	44	,	,	PUNCT
ejpam-4961	3	45	department	department	NOUN
ejpam-4961	3	46	of	of	ADP
ejpam-4961	3	47	mathematics	mathematics	PROPN
ejpam-4961	3	48	and	and	CCONJ
ejpam-4961	3	49	statistics	statistic	NOUN
ejpam-4961	3	50	,	,	PUNCT
ejpam-4961	3	51	college	college	NOUN
ejpam-4961	3	52	of	of	ADP
ejpam-4961	3	53	arts	art	NOUN
ejpam-4961	3	54	and	and	CCONJ
ejpam-4961	3	55	sciences	sciences	PROPN
ejpam-4961	3	56	,	,	PUNCT
ejpam-4961	3	57	qatar	qatar	PROPN
ejpam-4961	3	58	university	university	NOUN
ejpam-4961	3	59	,	,	PUNCT
ejpam-4961	3	60	2723	2723	NUM
ejpam-4961	3	61	,	,	PUNCT
ejpam-4961	3	62	doha	doha	PROPN
ejpam-4961	3	63	,	,	PUNCT
ejpam-4961	3	64	qatar	qatar	PROPN
ejpam-4961	3	65	4	4	NUM
ejpam-4961	3	66	department	department	NOUN
ejpam-4961	3	67	of	of	ADP
ejpam-4961	3	68	airframe	airframe	NOUN
ejpam-4961	3	69	and	and	CCONJ
ejpam-4961	3	70	powerplant	powerplant	ADJ
ejpam-4961	3	71	maintenance	maintenance	NOUN
ejpam-4961	3	72	,	,	PUNCT
ejpam-4961	3	73	school	school	NOUN
ejpam-4961	3	74	of	of	ADP
ejpam-4961	3	75	applied	apply	VERB
ejpam-4961	3	76	sciences	science	NOUN
ejpam-4961	3	77	,	,	PUNCT
ejpam-4961	3	78	cappadocia	cappadocia	PROPN
ejpam-4961	3	79	university	university	PROPN
ejpam-4961	3	80	,	,	PUNCT
ejpam-4961	3	81	mustafapasa	mustafapasa	PROPN
ejpam-4961	3	82	campus	campus	PROPN
ejpam-4961	3	83	,	,	PUNCT
ejpam-4961	3	84	ürgüp	ürgüp	NOUN
ejpam-4961	3	85	nevsehir	nevsehir	NOUN
ejpam-4961	3	86	,	,	PUNCT
ejpam-4961	3	87	turkey	turkey	NOUN
ejpam-4961	3	88	5	5	NUM
ejpam-4961	3	89	near	near	ADP
ejpam-4961	3	90	east	east	PROPN
ejpam-4961	3	91	university	university	PROPN
ejpam-4961	3	92	,	,	PUNCT
ejpam-4961	3	93	operational	operational	ADJ
ejpam-4961	3	94	research	research	NOUN
ejpam-4961	3	95	center	center	NOUN
ejpam-4961	3	96	in	in	ADP
ejpam-4961	3	97	healthcare	healthcare	PROPN
ejpam-4961	3	98	,	,	PUNCT
ejpam-4961	3	99	near	near	ADP
ejpam-4961	3	100	east	east	PROPN
ejpam-4961	3	101	boulevard	boulevard	PROPN
ejpam-4961	3	102	,	,	PUNCT
ejpam-4961	3	103	pc	pc	NOUN
ejpam-4961	3	104	:	:	PUNCT
ejpam-4961	3	105	99138	99138	NUM
ejpam-4961	3	106	nicosia	nicosia	PROPN
ejpam-4961	3	107	/	/	SYM
ejpam-4961	3	108	mersin	mersin	PROPN
ejpam-4961	3	109	10	10	NUM
ejpam-4961	3	110	,	,	PUNCT
ejpam-4961	3	111	turkey	turkey	NOUN
ejpam-4961	3	112	abstract	abstract	NOUN
ejpam-4961	3	113	.	.	PUNCT
ejpam-4961	4	1	this	this	DET
ejpam-4961	4	2	work	work	NOUN
ejpam-4961	4	3	takes	take	VERB
ejpam-4961	4	4	an	an	DET
ejpam-4961	4	5	interesting	interesting	ADJ
ejpam-4961	4	6	diversion	diversion	NOUN
ejpam-4961	4	7	,	,	PUNCT
ejpam-4961	4	8	revealing	reveal	VERB
ejpam-4961	4	9	the	the	DET
ejpam-4961	4	10	extraordinary	extraordinary	ADJ
ejpam-4961	4	11	capacity	capacity	NOUN
ejpam-4961	4	12	to	to	PART
ejpam-4961	4	13	determine	determine	VERB
ejpam-4961	4	14	the	the	DET
ejpam-4961	4	15	precise	precise	ADJ
ejpam-4961	4	16	number	number	NOUN
ejpam-4961	4	17	of	of	ADP
ejpam-4961	4	18	primes	prime	NOUN
ejpam-4961	4	19	in	in	ADP
ejpam-4961	4	20	a	a	DET
ejpam-4961	4	21	space	space	NOUN
ejpam-4961	4	22	tripled	triple	VERB
ejpam-4961	4	23	over	over	ADP
ejpam-4961	4	24	another	another	PRON
ejpam-4961	4	25	.	.	PUNCT
ejpam-4961	5	1	exploring	explore	VERB
ejpam-4961	5	2	the	the	DET
ejpam-4961	5	3	domain	domain	NOUN
ejpam-4961	5	4	of	of	ADP
ejpam-4961	5	5	k	k	NOUN
ejpam-4961	5	6	-	-	PUNCT
ejpam-4961	5	7	almost	almost	ADV
ejpam-4961	5	8	prime	prime	ADJ
ejpam-4961	5	9	numbers	number	NOUN
ejpam-4961	5	10	,	,	PUNCT
ejpam-4961	5	11	this	this	DET
ejpam-4961	5	12	paper	paper	NOUN
ejpam-4961	5	13	provides	provide	VERB
ejpam-4961	5	14	a	a	DET
ejpam-4961	5	15	clear	clear	ADJ
ejpam-4961	5	16	explanation	explanation	NOUN
ejpam-4961	5	17	of	of	ADP
ejpam-4961	5	18	the	the	DET
ejpam-4961	5	19	complex	complex	ADJ
ejpam-4961	5	20	idea	idea	NOUN
ejpam-4961	5	21	.	.	PUNCT
ejpam-4961	6	1	in	in	ADP
ejpam-4961	6	2	addition	addition	NOUN
ejpam-4961	6	3	to	to	ADP
ejpam-4961	6	4	outlining	outline	VERB
ejpam-4961	6	5	the	the	DET
ejpam-4961	6	6	conditions	condition	NOUN
ejpam-4961	6	7	under	under	ADP
ejpam-4961	6	8	which	which	PRON
ejpam-4961	6	9	odd	odd	ADJ
ejpam-4961	6	10	k	k	ADJ
ejpam-4961	6	11	-	-	PUNCT
ejpam-4961	6	12	almost	almost	ADV
ejpam-4961	6	13	prime	prime	ADJ
ejpam-4961	6	14	numbers	number	NOUN
ejpam-4961	6	15	must	must	AUX
ejpam-4961	6	16	exist	exist	VERB
ejpam-4961	6	17	,	,	PUNCT
ejpam-4961	6	18	it	it	PRON
ejpam-4961	6	19	presents	present	VERB
ejpam-4961	6	20	a	a	DET
ejpam-4961	6	21	novel	novel	ADJ
ejpam-4961	6	22	method	method	NOUN
ejpam-4961	6	23	for	for	ADP
ejpam-4961	6	24	figuring	figure	VERB
ejpam-4961	6	25	out	out	ADP
ejpam-4961	6	26	how	how	SCONJ
ejpam-4961	6	27	often	often	ADV
ejpam-4961	6	28	odd	odd	ADJ
ejpam-4961	6	29	numbers	number	NOUN
ejpam-4961	6	30	are	be	AUX
ejpam-4961	6	31	as	as	ADP
ejpam-4961	6	32	2	2	NUM
ejpam-4961	6	33	-	-	PUNCT
ejpam-4961	6	34	almost	almost	ADV
ejpam-4961	6	35	prime	prime	ADJ
ejpam-4961	6	36	,	,	PUNCT
ejpam-4961	6	37	3	3	NUM
ejpam-4961	6	38	-	-	PUNCT
ejpam-4961	6	39	almost	almost	ADV
ejpam-4961	6	40	prime	prime	ADJ
ejpam-4961	6	41	,	,	PUNCT
ejpam-4961	6	42	4	4	NUM
ejpam-4961	6	43	-	-	PUNCT
ejpam-4961	6	44	almost	almost	ADV
ejpam-4961	6	45	prime	prime	ADJ
ejpam-4961	6	46	,	,	PUNCT
ejpam-4961	6	47	and	and	CCONJ
ejpam-4961	6	48	so	so	ADV
ejpam-4961	6	49	on	on	ADV
ejpam-4961	6	50	,	,	PUNCT
ejpam-4961	6	51	up	up	ADP
ejpam-4961	6	52	to	to	ADP
ejpam-4961	6	53	a	a	DET
ejpam-4961	6	54	specified	specify	VERB
ejpam-4961	6	55	limit	limit	NOUN
ejpam-4961	6	56	n.	n.	NOUN
ejpam-4961	6	57	the	the	DET
ejpam-4961	6	58	work	work	NOUN
ejpam-4961	6	59	goes	go	VERB
ejpam-4961	6	60	one	one	NUM
ejpam-4961	6	61	step	step	NOUN
ejpam-4961	6	62	further	far	ADV
ejpam-4961	6	63	and	and	CCONJ
ejpam-4961	6	64	offers	offer	VERB
ejpam-4961	6	65	useful	useful	ADJ
ejpam-4961	6	66	advice	advice	NOUN
ejpam-4961	6	67	on	on	ADP
ejpam-4961	6	68	how	how	SCONJ
ejpam-4961	6	69	to	to	PART
ejpam-4961	6	70	use	use	VERB
ejpam-4961	6	71	these	these	DET
ejpam-4961	6	72	approaches	approach	NOUN
ejpam-4961	6	73	to	to	PART
ejpam-4961	6	74	precisely	precisely	ADV
ejpam-4961	6	75	calculate	calculate	VERB
ejpam-4961	6	76	the	the	DET
ejpam-4961	6	77	prime	prime	ADJ
ejpam-4961	6	78	counting	counting	NOUN
ejpam-4961	6	79	function	function	NOUN
ejpam-4961	6	80	,	,	PUNCT
ejpam-4961	6	81	π(n	π(n	PROPN
ejpam-4961	6	82	)	)	PUNCT
ejpam-4961	6	83	.	.	PUNCT
ejpam-4961	7	1	essentially	essentially	ADV
ejpam-4961	7	2	,	,	PUNCT
ejpam-4961	7	3	it	it	PRON
ejpam-4961	7	4	offers	offer	VERB
ejpam-4961	7	5	a	a	DET
ejpam-4961	7	6	comprehensive	comprehensive	ADJ
ejpam-4961	7	7	exploration	exploration	NOUN
ejpam-4961	7	8	of	of	ADP
ejpam-4961	7	9	the	the	DET
ejpam-4961	7	10	mathematical	mathematical	ADJ
ejpam-4961	7	11	fabric	fabric	NOUN
ejpam-4961	7	12	,	,	PUNCT
ejpam-4961	7	13	where	where	SCONJ
ejpam-4961	7	14	primes	prime	NOUN
ejpam-4961	7	15	reveal	reveal	VERB
ejpam-4961	7	16	their	their	PRON
ejpam-4961	7	17	mysteries	mystery	NOUN
ejpam-4961	7	18	in	in	ADP
ejpam-4961	7	19	both	both	CCONJ
ejpam-4961	7	20	large	large	ADJ
ejpam-4961	7	21	and	and	CCONJ
ejpam-4961	7	22	small	small	ADJ
ejpam-4961	7	23	spaces	space	NOUN
ejpam-4961	7	24	.	.	PUNCT
ejpam-4961	8	1	2020	2020	NUM
ejpam-4961	8	2	mathematics	mathematic	NOUN
ejpam-4961	8	3	subject	subject	NOUN
ejpam-4961	8	4	classifications	classification	NOUN
ejpam-4961	8	5	:	:	PUNCT
ejpam-4961	8	6	11y16	11y16	NUM
ejpam-4961	8	7	,	,	PUNCT
ejpam-4961	8	8	11y11,11y40	11y11,11y40	DET
ejpam-4961	8	9	key	key	ADJ
ejpam-4961	8	10	words	word	NOUN
ejpam-4961	8	11	and	and	CCONJ
ejpam-4961	8	12	phrases	phrase	NOUN
ejpam-4961	8	13	:	:	PUNCT
ejpam-4961	8	14	prime	prime	ADJ
ejpam-4961	8	15	counting	counting	NOUN
ejpam-4961	8	16	function	function	NOUN
ejpam-4961	8	17	,	,	PUNCT
ejpam-4961	8	18	odd	odd	ADJ
ejpam-4961	8	19	k	k	ADJ
ejpam-4961	8	20	-	-	PUNCT
ejpam-4961	8	21	almost	almost	ADV
ejpam-4961	8	22	primes	prime	NOUN
ejpam-4961	8	23	1	1	NUM
ejpam-4961	8	24	.	.	PUNCT
ejpam-4961	9	1	introduction	introduction	NOUN
ejpam-4961	9	2	greek	greek	NOUN
ejpam-4961	9	3	mathematicians	mathematician	NOUN
ejpam-4961	9	4	were	be	AUX
ejpam-4961	9	5	the	the	DET
ejpam-4961	9	6	first	first	ADJ
ejpam-4961	9	7	to	to	PART
ejpam-4961	9	8	study	study	VERB
ejpam-4961	9	9	prime	prime	ADJ
ejpam-4961	9	10	numbers	number	NOUN
ejpam-4961	9	11	and	and	CCONJ
ejpam-4961	9	12	their	their	PRON
ejpam-4961	9	13	characteristics	characteristic	NOUN
ejpam-4961	9	14	in	in	ADP
ejpam-4961	9	15	depth	depth	NOUN
ejpam-4961	9	16	.	.	PUNCT
ejpam-4961	10	1	several	several	ADJ
ejpam-4961	10	2	significant	significant	ADJ
ejpam-4961	10	3	primes	prime	NOUN
ejpam-4961	10	4	-	-	PUNCT
ejpam-4961	10	5	related	relate	VERB
ejpam-4961	10	6	findings	finding	NOUN
ejpam-4961	10	7	had	have	AUX
ejpam-4961	10	8	been	be	AUX
ejpam-4961	10	9	established	establish	VERB
ejpam-4961	10	10	by	by	ADP
ejpam-4961	10	11	the	the	DET
ejpam-4961	10	12	time	time	NOUN
ejpam-4961	10	13	euclid	euclid	PROPN
ejpam-4961	10	14	’s	’s	PART
ejpam-4961	10	15	elements	element	NOUN
ejpam-4961	10	16	,	,	PUNCT
ejpam-4961	10	17	which	which	PRON
ejpam-4961	10	18	was	be	AUX
ejpam-4961	10	19	written	write	VERB
ejpam-4961	10	20	around	around	ADP
ejpam-4961	10	21	300	300	NUM
ejpam-4961	10	22	bc	bc	NOUN
ejpam-4961	10	23	.	.	PUNCT
ejpam-4961	11	1	one	one	NUM
ejpam-4961	11	2	of	of	ADP
ejpam-4961	11	3	the	the	DET
ejpam-4961	11	4	numerous	numerous	ADJ
ejpam-4961	11	5	unresolved	unresolved	ADJ
ejpam-4961	11	6	issues	issue	NOUN
ejpam-4961	11	7	in	in	ADP
ejpam-4961	11	8	number	number	NOUN
ejpam-4961	11	9	theory	theory	NOUN
ejpam-4961	11	10	is	be	AUX
ejpam-4961	11	11	the	the	DET
ejpam-4961	11	12	prime	prime	ADJ
ejpam-4961	11	13	calculating	calculate	VERB
ejpam-4961	11	14	function	function	NOUN
ejpam-4961	11	15	π(n	π(n	PROPN
ejpam-4961	11	16	)	)	PUNCT
ejpam-4961	11	17	(	(	PUNCT
ejpam-4961	11	18	number	number	NOUN
ejpam-4961	11	19	of	of	ADP
ejpam-4961	11	20	primes	prime	NOUN
ejpam-4961	11	21	≤	≤	NUM
ejpam-4961	11	22	n	n	CCONJ
ejpam-4961	11	23	)	)	PUNCT
ejpam-4961	11	24	.	.	PUNCT
ejpam-4961	12	1	the	the	DET
ejpam-4961	12	2	prime	prime	ADJ
ejpam-4961	12	3	counting	counting	NOUN
ejpam-4961	12	4	function	function	NOUN
ejpam-4961	12	5	can	can	AUX
ejpam-4961	12	6	be	be	AUX
ejpam-4961	12	7	expressed	express	VERB
ejpam-4961	12	8	by	by	ADP
ejpam-4961	12	9	legendre	legendre	PROPN
ejpam-4961	12	10	’s	’s	PART
ejpam-4961	12	11	formula	formula	NOUN
ejpam-4961	12	12	,	,	PUNCT
ejpam-4961	12	13	lehmer	lehmer	NOUN
ejpam-4961	12	14	’s	’s	PART
ejpam-4961	12	15	formula	formula	NOUN
ejpam-4961	12	16	,	,	PUNCT
ejpam-4961	12	17	∗corresponding	∗corresponde	VERB
ejpam-4961	12	18	author	author	NOUN
ejpam-4961	12	19	.	.	PUNCT
ejpam-4961	13	1	doi	doi	NOUN
ejpam-4961	13	2	:	:	PUNCT
ejpam-4961	13	3	https://doi.org/10.29020/nybg.ejpam.v17i2.4961	https://doi.org/10.29020/nybg.ejpam.v17i2.4961	ADJ
ejpam-4961	13	4	email	email	NOUN
ejpam-4961	13	5	addresses	address	NOUN
ejpam-4961	13	6	:	:	PUNCT
ejpam-4961	13	7	mmjst4@qu.edu.qa	mmjst4@qu.edu.qa	PROPN
ejpam-4961	13	8	(	(	PUNCT
ejpam-4961	13	9	m.	m.	NOUN
ejpam-4961	13	10	m.	m.	PROPN
ejpam-4961	13	11	m.	m.	PROPN
ejpam-4961	13	12	jaradat	jaradat	PROPN
ejpam-4961	13	13	)	)	PUNCT
ejpam-4961	13	14	,	,	PUNCT
ejpam-4961	13	15	tawseefrashid123@gmail.com	tawseefrashid123@gmail.com	X
ejpam-4961	13	16	(	(	PUNCT
ejpam-4961	13	17	t.	t.	PROPN
ejpam-4961	13	18	rashid	rashid	PROPN
ejpam-4961	13	19	)	)	PUNCT
ejpam-4961	13	20	,	,	PUNCT
ejpam-4961	13	21	yolacanesra@gmail.com	yolacanesra@gmail.com	X
ejpam-4961	13	22	(	(	PUNCT
ejpam-4961	13	23	e.	e.	PROPN
ejpam-4961	13	24	yolacan	yolacan	PROPN
ejpam-4961	13	25	)	)	PUNCT
ejpam-4961	13	26	,	,	PUNCT
ejpam-4961	13	27	hijaz555@gmail.com	hijaz555@gmail.com	X
ejpam-4961	13	28	(	(	PUNCT
ejpam-4961	13	29	h.	h.	PROPN
ejpam-4961	13	30	ahmad	ahmad	PROPN
ejpam-4961	13	31	)	)	PUNCT
ejpam-4961	13	32	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4961	13	33	1146	1146	NUM
ejpam-4961	14	1	©	©	PROPN
ejpam-4961	14	2	2024	2024	NUM
ejpam-4961	14	3	ejpam	ejpam	NOUN
ejpam-4961	14	4	all	all	DET
ejpam-4961	14	5	rights	right	NOUN
ejpam-4961	14	6	reserved	reserve	VERB
ejpam-4961	14	7	.	.	PUNCT
ejpam-4961	15	1	m.	m.	NOUN
ejpam-4961	15	2	m.	m.	PROPN
ejpam-4961	15	3	m.	m.	PROPN
ejpam-4961	15	4	jaradat	jaradat	PROPN
ejpam-4961	15	5	et	et	PROPN
ejpam-4961	15	6	al	al	PROPN
ejpam-4961	15	7	.	.	PUNCT
ejpam-4961	15	8	/	/	SYM
ejpam-4961	15	9	eur	eur	PROPN
ejpam-4961	15	10	.	.	PUNCT
ejpam-4961	16	1	j.	j.	PROPN
ejpam-4961	16	2	pure	pure	PROPN
ejpam-4961	16	3	appl	appl	PROPN
ejpam-4961	16	4	.	.	PROPN
ejpam-4961	16	5	math	math	PROPN
ejpam-4961	16	6	,	,	PUNCT
ejpam-4961	16	7	17	17	NUM
ejpam-4961	16	8	(	(	PUNCT
ejpam-4961	16	9	2	2	NUM
ejpam-4961	16	10	)	)	PUNCT
ejpam-4961	16	11	(	(	PUNCT
ejpam-4961	16	12	2024	2024	NUM
ejpam-4961	16	13	)	)	PUNCT
ejpam-4961	16	14	,	,	PUNCT
ejpam-4961	16	15	1146	1146	NUM
ejpam-4961	16	16	-	-	SYM
ejpam-4961	16	17	1154	1154	NUM
ejpam-4961	16	18	1147	1147	NUM
ejpam-4961	16	19	mapes	mapes	PROPN
ejpam-4961	16	20	’	'	PUNCT
ejpam-4961	16	21	method	method	NOUN
ejpam-4961	16	22	,	,	PUNCT
ejpam-4961	16	23	or	or	CCONJ
ejpam-4961	16	24	meissel	meissel	NOUN
ejpam-4961	16	25	’s	’s	PART
ejpam-4961	16	26	formula	formula	NOUN
ejpam-4961	16	27	.	.	PUNCT
ejpam-4961	17	1	the	the	DET
ejpam-4961	17	2	prime	prime	ADJ
ejpam-4961	17	3	counting	counting	NOUN
ejpam-4961	17	4	function	function	NOUN
ejpam-4961	17	5	has	have	VERB
ejpam-4961	17	6	a	a	DET
ejpam-4961	17	7	large	large	ADJ
ejpam-4961	17	8	body	body	NOUN
ejpam-4961	17	9	of	of	ADP
ejpam-4961	17	10	literature	literature	NOUN
ejpam-4961	17	11	.	.	PUNCT
ejpam-4961	18	1	different	different	ADJ
ejpam-4961	18	2	mathematicians	mathematician	NOUN
ejpam-4961	18	3	made	make	VERB
ejpam-4961	18	4	several	several	ADJ
ejpam-4961	18	5	attempts	attempt	NOUN
ejpam-4961	18	6	at	at	ADP
ejpam-4961	18	7	the	the	DET
ejpam-4961	18	8	prime	prime	ADJ
ejpam-4961	18	9	counting	counting	NOUN
ejpam-4961	18	10	function	function	NOUN
ejpam-4961	18	11	by	by	ADP
ejpam-4961	18	12	using	use	VERB
ejpam-4961	18	13	analytic	analytic	ADJ
ejpam-4961	18	14	and	and	CCONJ
ejpam-4961	18	15	algebriac	algebriac	ADJ
ejpam-4961	18	16	approaches	approach	NOUN
ejpam-4961	18	17	like	like	ADP
ejpam-4961	18	18	[	[	X
ejpam-4961	18	19	1–7	1–7	NUM
ejpam-4961	18	20	,	,	PUNCT
ejpam-4961	18	21	9–12	9–12	PROPN
ejpam-4961	18	22	]	]	PUNCT
ejpam-4961	18	23	.	.	PUNCT
ejpam-4961	19	1	no	no	DET
ejpam-4961	19	2	one	one	NOUN
ejpam-4961	19	3	provided	provide	VERB
ejpam-4961	19	4	the	the	DET
ejpam-4961	19	5	precise	precise	ADJ
ejpam-4961	19	6	result	result	NOUN
ejpam-4961	19	7	using	use	VERB
ejpam-4961	19	8	any	any	PRON
ejpam-4961	19	9	of	of	ADP
ejpam-4961	19	10	these	these	DET
ejpam-4961	19	11	methods	method	NOUN
ejpam-4961	19	12	,	,	PUNCT
ejpam-4961	19	13	which	which	PRON
ejpam-4961	19	14	is	be	AUX
ejpam-4961	19	15	a	a	DET
ejpam-4961	19	16	common	common	ADJ
ejpam-4961	19	17	problem	problem	NOUN
ejpam-4961	19	18	.	.	PUNCT
ejpam-4961	20	1	on	on	ADP
ejpam-4961	20	2	the	the	DET
ejpam-4961	20	3	other	other	ADJ
ejpam-4961	20	4	hand	hand	NOUN
ejpam-4961	20	5	,	,	PUNCT
ejpam-4961	20	6	if	if	SCONJ
ejpam-4961	20	7	a	a	DET
ejpam-4961	20	8	number	number	NOUN
ejpam-4961	20	9	is	be	AUX
ejpam-4961	20	10	the	the	DET
ejpam-4961	20	11	product	product	NOUN
ejpam-4961	20	12	of	of	ADP
ejpam-4961	20	13	exactly	exactly	ADV
ejpam-4961	20	14	k	k	ADJ
ejpam-4961	20	15	prime	prime	ADJ
ejpam-4961	20	16	numbers	number	NOUN
ejpam-4961	20	17	,	,	PUNCT
ejpam-4961	20	18	that	that	PRON
ejpam-4961	20	19	are	be	AUX
ejpam-4961	20	20	the	the	DET
ejpam-4961	20	21	same	same	ADJ
ejpam-4961	20	22	or	or	CCONJ
ejpam-4961	20	23	distinct	distinct	ADJ
ejpam-4961	20	24	,	,	PUNCT
ejpam-4961	20	25	it	it	PRON
ejpam-4961	20	26	is	be	AUX
ejpam-4961	20	27	said	say	VERB
ejpam-4961	20	28	to	to	PART
ejpam-4961	20	29	be	be	AUX
ejpam-4961	20	30	k	k	ADJ
ejpam-4961	20	31	-	-	ADJ
ejpam-4961	20	32	almost	almost	ADV
ejpam-4961	20	33	prime	prime	ADJ
ejpam-4961	20	34	.	.	PUNCT
ejpam-4961	21	1	the	the	DET
ejpam-4961	21	2	”	"	PUNCT
ejpam-4961	21	3	1	1	NUM
ejpam-4961	21	4	-	-	PUNCT
ejpam-4961	21	5	almost	almost	ADV
ejpam-4961	21	6	prime	prime	ADJ
ejpam-4961	21	7	”	"	PUNCT
ejpam-4961	21	8	numbers	number	NOUN
ejpam-4961	21	9	are	be	AUX
ejpam-4961	21	10	equivalent	equivalent	ADJ
ejpam-4961	21	11	to	to	ADP
ejpam-4961	21	12	the	the	DET
ejpam-4961	21	13	primes	prime	NOUN
ejpam-4961	21	14	,	,	PUNCT
ejpam-4961	21	15	whereas	whereas	SCONJ
ejpam-4961	21	16	the	the	DET
ejpam-4961	21	17	”	"	PUNCT
ejpam-4961	21	18	2	2	NUM
ejpam-4961	21	19	-	-	PUNCT
ejpam-4961	21	20	almost	almost	ADV
ejpam-4961	21	21	prime	prime	ADJ
ejpam-4961	21	22	”	"	PUNCT
ejpam-4961	21	23	numbers	number	NOUN
ejpam-4961	21	24	are	be	AUX
ejpam-4961	21	25	equivalent	equivalent	ADJ
ejpam-4961	21	26	to	to	ADP
ejpam-4961	21	27	semiprimes	semiprime	NOUN
ejpam-4961	21	28	.	.	PUNCT
ejpam-4961	22	1	these	these	DET
ejpam-4961	22	2	numbers	number	NOUN
ejpam-4961	22	3	are	be	AUX
ejpam-4961	22	4	referred	refer	VERB
ejpam-4961	22	5	to	to	ADP
ejpam-4961	22	6	as	as	ADP
ejpam-4961	22	7	primes	prime	NOUN
ejpam-4961	22	8	,	,	PUNCT
ejpam-4961	22	9	biprimes	biprime	NOUN
ejpam-4961	22	10	,	,	PUNCT
ejpam-4961	22	11	triprimes	triprime	NOUN
ejpam-4961	22	12	,	,	PUNCT
ejpam-4961	22	13	etc	etc	X
ejpam-4961	22	14	.	.	X
ejpam-4961	22	15	by	by	ADP
ejpam-4961	22	16	conway	conway	PROPN
ejpam-4961	22	17	et	et	PROPN
ejpam-4961	22	18	al	al	PROPN
ejpam-4961	22	19	.	.	PUNCT
ejpam-4961	23	1	[	[	X
ejpam-4961	23	2	8	8	NUM
ejpam-4961	23	3	]	]	PUNCT
ejpam-4961	23	4	.	.	PUNCT
ejpam-4961	24	1	the	the	DET
ejpam-4961	24	2	formulas	formula	NOUN
ejpam-4961	24	3	for	for	ADP
ejpam-4961	24	4	the	the	DET
ejpam-4961	24	5	number	number	NOUN
ejpam-4961	24	6	of	of	ADP
ejpam-4961	24	7	k	k	NOUN
ejpam-4961	24	8	-	-	PUNCT
ejpam-4961	24	9	almost	almost	ADV
ejpam-4961	24	10	-	-	PUNCT
ejpam-4961	24	11	primes	prime	NOUN
ejpam-4961	24	12	less	less	ADJ
ejpam-4961	24	13	than	than	ADP
ejpam-4961	24	14	or	or	CCONJ
ejpam-4961	24	15	equal	equal	ADJ
ejpam-4961	24	16	to	to	ADP
ejpam-4961	24	17	n	n	NUM
ejpam-4961	24	18	are	be	AUX
ejpam-4961	24	19	listed	list	VERB
ejpam-4961	24	20	below	below	ADV
ejpam-4961	24	21	.	.	PUNCT
ejpam-4961	25	1	π(2)(n	π(2)(n	X
ejpam-4961	25	2	)	)	PUNCT
ejpam-4961	25	3	=	=	SYM
ejpam-4961	26	1	π(n	π(n	PROPN
ejpam-4961	26	2	1	1	NUM
ejpam-4961	26	3	2	2	NUM
ejpam-4961	26	4	)	)	PUNCT
ejpam-4961	26	5	∑	∑	ADP
ejpam-4961	26	6	i=1	i=1	PROPN
ejpam-4961	26	7	[	[	PUNCT
ejpam-4961	26	8	π	π	PROPN
ejpam-4961	26	9	(	(	PUNCT
ejpam-4961	26	10	n	n	NUM
ejpam-4961	26	11	pi	pi	NOUN
ejpam-4961	26	12	)	)	PUNCT
ejpam-4961	26	13	−	−	NOUN
ejpam-4961	26	14	i+	i+	NUM
ejpam-4961	26	15	1	1	NUM
ejpam-4961	26	16	]	]	PUNCT
ejpam-4961	26	17	,	,	PUNCT
ejpam-4961	26	18	(	(	PUNCT
ejpam-4961	26	19	1	1	X
ejpam-4961	26	20	)	)	PUNCT
ejpam-4961	26	21	π(3)(n	π(3)(n	NOUN
ejpam-4961	26	22	)	)	PUNCT
ejpam-4961	27	1	=	=	SYM
ejpam-4961	27	2	π(n	π(n	PROPN
ejpam-4961	27	3	1	1	NUM
ejpam-4961	27	4	3	3	NUM
ejpam-4961	27	5	)	)	PUNCT
ejpam-4961	27	6	∑	∑	ADP
ejpam-4961	27	7	i=1	i=1	PROPN
ejpam-4961	27	8	π	π	PROPN
ejpam-4961	27	9	(	(	PUNCT
ejpam-4961	27	10	n	n	CCONJ
ejpam-4961	27	11	pi	pi	NOUN
ejpam-4961	27	12	)	)	PUNCT
ejpam-4961	28	1	1	1	NUM
ejpam-4961	28	2	2∑	2∑	NUM
ejpam-4961	28	3	j=1	j=1	NOUN
ejpam-4961	28	4	[	[	PUNCT
ejpam-4961	28	5	π	π	X
ejpam-4961	28	6	(	(	PUNCT
ejpam-4961	28	7	n	n	PRON
ejpam-4961	28	8	pipj	pipj	NOUN
ejpam-4961	28	9	)	)	PUNCT
ejpam-4961	28	10	−	−	PROPN
ejpam-4961	29	1	j	j	NOUN
ejpam-4961	30	1	+	+	CCONJ
ejpam-4961	30	2	1	1	NUM
ejpam-4961	30	3	]	]	PUNCT
ejpam-4961	30	4	,	,	PUNCT
ejpam-4961	30	5	(	(	PUNCT
ejpam-4961	30	6	2	2	NUM
ejpam-4961	30	7	)	)	PUNCT
ejpam-4961	30	8	π(4)(n	π(4)(n	NUM
ejpam-4961	30	9	)	)	PUNCT
ejpam-4961	31	1	=	=	SYM
ejpam-4961	31	2	π(n	π(n	PROPN
ejpam-4961	31	3	1	1	NUM
ejpam-4961	31	4	4	4	NUM
ejpam-4961	31	5	)	)	PUNCT
ejpam-4961	31	6	∑	∑	PROPN
ejpam-4961	31	7	i=1	i=1	PROPN
ejpam-4961	31	8	π	π	PROPN
ejpam-4961	31	9	(	(	PUNCT
ejpam-4961	31	10	n	n	NUM
ejpam-4961	31	11	pi	pi	NOUN
ejpam-4961	31	12	)	)	PUNCT
ejpam-4961	32	1	1	1	NUM
ejpam-4961	32	2	3∑	3∑	NUM
ejpam-4961	32	3	j=1	j=1	NOUN
ejpam-4961	32	4	π	π	PROPN
ejpam-4961	32	5	(	(	PUNCT
ejpam-4961	32	6	n	n	CCONJ
ejpam-4961	32	7	pipj	pipj	NOUN
ejpam-4961	32	8	)	)	PUNCT
ejpam-4961	33	1	1	1	NUM
ejpam-4961	33	2	2∑	2∑	NUM
ejpam-4961	33	3	k	k	X
ejpam-4961	33	4	=	=	PRON
ejpam-4961	33	5	j	j	X
ejpam-4961	33	6	[	[	PUNCT
ejpam-4961	33	7	π	π	PROPN
ejpam-4961	33	8	(	(	PUNCT
ejpam-4961	33	9	n	n	ADV
ejpam-4961	33	10	pipjpk	pipjpk	VERB
ejpam-4961	33	11	)	)	PUNCT
ejpam-4961	33	12	−	−	PROPN
ejpam-4961	34	1	k	k	X
ejpam-4961	35	1	+	+	CCONJ
ejpam-4961	35	2	1	1	X
ejpam-4961	35	3	]	]	PUNCT
ejpam-4961	35	4	(	(	PUNCT
ejpam-4961	35	5	3	3	NUM
ejpam-4961	35	6	)	)	PUNCT
ejpam-4961	35	7	and	and	CCONJ
ejpam-4961	35	8	so	so	ADV
ejpam-4961	35	9	forth	forth	ADV
ejpam-4961	35	10	,	,	PUNCT
ejpam-4961	35	11	where	where	SCONJ
ejpam-4961	35	12	pn	pn	PROPN
ejpam-4961	35	13	is	be	AUX
ejpam-4961	35	14	the	the	DET
ejpam-4961	35	15	nth	nth	ADJ
ejpam-4961	35	16	prime	prime	NOUN
ejpam-4961	35	17	.	.	PUNCT
ejpam-4961	36	1	π(x	π(x	NOUN
ejpam-4961	36	2	)	)	PUNCT
ejpam-4961	36	3	is	be	AUX
ejpam-4961	36	4	the	the	DET
ejpam-4961	36	5	prime	prime	ADJ
ejpam-4961	36	6	counting	counting	NOUN
ejpam-4961	36	7	function	function	NOUN
ejpam-4961	36	8	and	and	CCONJ
ejpam-4961	36	9	π(t)(n	π(t)(n	NUM
ejpam-4961	36	10	)	)	PUNCT
ejpam-4961	36	11	denotes	denote	NOUN
ejpam-4961	36	12	t	t	PROPN
ejpam-4961	36	13	-	-	PUNCT
ejpam-4961	36	14	almost	almost	ADV
ejpam-4961	36	15	prime	prime	ADJ
ejpam-4961	36	16	function	function	NOUN
ejpam-4961	36	17	.	.	PUNCT
ejpam-4961	37	1	noel	noel	PROPN
ejpam-4961	37	2	,	,	PUNCT
ejpam-4961	37	3	panos	pano	NOUN
ejpam-4961	37	4	,	,	PUNCT
ejpam-4961	37	5	and	and	CCONJ
ejpam-4961	37	6	wilson	wilson	PROPN
ejpam-4961	37	7	separately	separately	ADV
ejpam-4961	37	8	introduced	introduce	VERB
ejpam-4961	37	9	these	these	DET
ejpam-4961	37	10	formulations	formulation	NOUN
ejpam-4961	37	11	for	for	ADP
ejpam-4961	37	12	the	the	DET
ejpam-4961	37	13	first	first	ADJ
ejpam-4961	37	14	time	time	NOUN
ejpam-4961	37	15	in	in	ADP
ejpam-4961	37	16	2006	2006	NUM
ejpam-4961	37	17	.	.	PUNCT
ejpam-4961	38	1	the	the	DET
ejpam-4961	38	2	problem	problem	NOUN
ejpam-4961	38	3	with	with	ADP
ejpam-4961	38	4	the	the	DET
ejpam-4961	38	5	calculations	calculation	NOUN
ejpam-4961	38	6	above	above	ADV
ejpam-4961	38	7	is	be	AUX
ejpam-4961	38	8	that	that	SCONJ
ejpam-4961	38	9	they	they	PRON
ejpam-4961	38	10	count	count	VERB
ejpam-4961	38	11	both	both	CCONJ
ejpam-4961	38	12	even	even	ADV
ejpam-4961	38	13	and	and	CCONJ
ejpam-4961	38	14	odd	odd	ADJ
ejpam-4961	38	15	k	k	ADJ
ejpam-4961	38	16	-	-	PUNCT
ejpam-4961	38	17	almost	almost	ADV
ejpam-4961	38	18	primes	prime	NOUN
ejpam-4961	38	19	that	that	PRON
ejpam-4961	38	20	are	be	AUX
ejpam-4961	38	21	less	less	ADJ
ejpam-4961	38	22	than	than	ADP
ejpam-4961	38	23	or	or	CCONJ
ejpam-4961	38	24	equal	equal	ADJ
ejpam-4961	38	25	to	to	ADP
ejpam-4961	38	26	n	n	NOUN
ejpam-4961	38	27	and	and	CCONJ
ejpam-4961	38	28	require	require	VERB
ejpam-4961	38	29	the	the	DET
ejpam-4961	38	30	list	list	NOUN
ejpam-4961	38	31	of	of	ADP
ejpam-4961	38	32	primes	prime	NOUN
ejpam-4961	38	33	up	up	ADP
ejpam-4961	38	34	to	to	ADP
ejpam-4961	38	35	n	n	DET
ejpam-4961	38	36	2	2	NUM
ejpam-4961	38	37	.	.	PUNCT
ejpam-4961	39	1	this	this	DET
ejpam-4961	39	2	paper	paper	NOUN
ejpam-4961	39	3	’s	’s	PART
ejpam-4961	39	4	main	main	ADJ
ejpam-4961	39	5	goal	goal	NOUN
ejpam-4961	39	6	is	be	AUX
ejpam-4961	39	7	to	to	PART
ejpam-4961	39	8	locate	locate	VERB
ejpam-4961	39	9	the	the	DET
ejpam-4961	39	10	odd	odd	ADJ
ejpam-4961	39	11	k	k	ADJ
ejpam-4961	39	12	-	-	PUNCT
ejpam-4961	39	13	almost	almost	ADV
ejpam-4961	39	14	primes	prime	NOUN
ejpam-4961	39	15	,	,	PUNCT
ejpam-4961	39	16	which	which	PRON
ejpam-4961	39	17	requires	require	VERB
ejpam-4961	39	18	a	a	DET
ejpam-4961	39	19	prime	prime	ADJ
ejpam-4961	39	20	list	list	NOUN
ejpam-4961	39	21	up	up	ADP
ejpam-4961	39	22	to	to	ADP
ejpam-4961	39	23	n	n	PRON
ejpam-4961	39	24	3	3	NUM
ejpam-4961	39	25	,	,	PUNCT
ejpam-4961	39	26	which	which	PRON
ejpam-4961	39	27	is	be	AUX
ejpam-4961	39	28	a	a	DET
ejpam-4961	39	29	much	much	ADV
ejpam-4961	39	30	smaller	small	ADJ
ejpam-4961	39	31	interval	interval	NOUN
ejpam-4961	39	32	than	than	ADP
ejpam-4961	39	33	n	n	ADV
ejpam-4961	39	34	2	2	NUM
ejpam-4961	39	35	.	.	PUNCT
ejpam-4961	40	1	2	2	X
ejpam-4961	40	2	.	.	X
ejpam-4961	40	3	main	main	ADJ
ejpam-4961	40	4	result	result	NOUN
ejpam-4961	40	5	before	before	ADP
ejpam-4961	40	6	proceeding	proceed	VERB
ejpam-4961	40	7	,	,	PUNCT
ejpam-4961	40	8	it	it	PRON
ejpam-4961	40	9	is	be	AUX
ejpam-4961	40	10	worth	worth	ADJ
ejpam-4961	40	11	mentioning	mention	VERB
ejpam-4961	40	12	that	that	SCONJ
ejpam-4961	40	13	we	we	PRON
ejpam-4961	40	14	have	have	AUX
ejpam-4961	40	15	used	use	VERB
ejpam-4961	40	16	the	the	DET
ejpam-4961	40	17	following	following	ADJ
ejpam-4961	40	18	notations	notation	NOUN
ejpam-4961	40	19	.	.	PUNCT
ejpam-4961	41	1	1	1	NUM
ejpam-4961	41	2	.	.	X
ejpam-4961	41	3	π(n	π(n	NUM
ejpam-4961	41	4	)	)	PUNCT
ejpam-4961	41	5	denotes	denote	VERB
ejpam-4961	41	6	the	the	DET
ejpam-4961	41	7	number	number	NOUN
ejpam-4961	41	8	of	of	ADP
ejpam-4961	41	9	primes	prime	NOUN
ejpam-4961	41	10	≤	≤	NUM
ejpam-4961	41	11	n.	n.	NOUN
ejpam-4961	41	12	2	2	NUM
ejpam-4961	41	13	.	.	PUNCT
ejpam-4961	42	1	π(n	π(n	NUM
ejpam-4961	42	2	)	)	PUNCT
ejpam-4961	42	3	denotes	denote	VERB
ejpam-4961	42	4	the	the	DET
ejpam-4961	42	5	number	number	NOUN
ejpam-4961	42	6	of	of	ADP
ejpam-4961	42	7	odd	odd	ADJ
ejpam-4961	42	8	primes	prime	NOUN
ejpam-4961	42	9	≤	≤	NUM
ejpam-4961	42	10	n.	n.	NOUN
ejpam-4961	42	11	clearly	clearly	ADV
ejpam-4961	42	12	,	,	PUNCT
ejpam-4961	42	13	for	for	ADP
ejpam-4961	42	14	any	any	DET
ejpam-4961	42	15	n	n	PRON
ejpam-4961	42	16	π(n	π(n	PROPN
ejpam-4961	42	17	)	)	PUNCT
ejpam-4961	43	1	=	=	PRON
ejpam-4961	43	2	π(n)−	π(n)−	PROPN
ejpam-4961	43	3	1	1	X
ejpam-4961	43	4	.	.	PUNCT
ejpam-4961	43	5	(	(	PUNCT
ejpam-4961	43	6	4	4	NUM
ejpam-4961	43	7	)	)	PUNCT
ejpam-4961	43	8	3	3	NUM
ejpam-4961	43	9	.	.	X
ejpam-4961	43	10	πm(n	πm(n	NOUN
ejpam-4961	43	11	)	)	PUNCT
ejpam-4961	43	12	denotes	denote	VERB
ejpam-4961	43	13	the	the	DET
ejpam-4961	43	14	number	number	NOUN
ejpam-4961	43	15	of	of	ADP
ejpam-4961	43	16	odd	odd	ADJ
ejpam-4961	43	17	primes	prime	NOUN
ejpam-4961	43	18	≥	≥	NOUN
ejpam-4961	43	19	m	m	PROPN
ejpam-4961	43	20	and	and	CCONJ
ejpam-4961	43	21	≤	≤	NUM
ejpam-4961	43	22	n.	n.	NOUN
ejpam-4961	43	23	clearly	clearly	ADV
ejpam-4961	43	24	,	,	PUNCT
ejpam-4961	43	25	πm(n	πm(n	NOUN
ejpam-4961	43	26	)	)	PUNCT
ejpam-4961	43	27	=	=	SYM
ejpam-4961	43	28	π(n)−	π(n)−	PROPN
ejpam-4961	43	29	π(m	π(m	PROPN
ejpam-4961	43	30	)	)	PUNCT
ejpam-4961	43	31	+	+	NUM
ejpam-4961	43	32	1	1	NUM
ejpam-4961	43	33	,	,	PUNCT
ejpam-4961	43	34	if	if	SCONJ
ejpam-4961	43	35	m	m	PROPN
ejpam-4961	43	36	is	be	AUX
ejpam-4961	44	1	odd	odd	ADJ
ejpam-4961	44	2	prime	prime	ADJ
ejpam-4961	44	3	m.	m.	NOUN
ejpam-4961	44	4	m.	m.	NOUN
ejpam-4961	44	5	m.	m.	PROPN
ejpam-4961	44	6	jaradat	jaradat	PROPN
ejpam-4961	44	7	et	et	PROPN
ejpam-4961	44	8	al	al	PROPN
ejpam-4961	44	9	.	.	PUNCT
ejpam-4961	44	10	/	/	SYM
ejpam-4961	44	11	eur	eur	PROPN
ejpam-4961	44	12	.	.	PUNCT
ejpam-4961	45	1	j.	j.	PROPN
ejpam-4961	45	2	pure	pure	PROPN
ejpam-4961	45	3	appl	appl	PROPN
ejpam-4961	45	4	.	.	PROPN
ejpam-4961	45	5	math	math	PROPN
ejpam-4961	45	6	,	,	PUNCT
ejpam-4961	45	7	17	17	NUM
ejpam-4961	45	8	(	(	PUNCT
ejpam-4961	45	9	2	2	NUM
ejpam-4961	45	10	)	)	PUNCT
ejpam-4961	45	11	(	(	PUNCT
ejpam-4961	45	12	2024	2024	NUM
ejpam-4961	45	13	)	)	PUNCT
ejpam-4961	45	14	,	,	PUNCT
ejpam-4961	45	15	1146	1146	NUM
ejpam-4961	45	16	-	-	SYM
ejpam-4961	45	17	1154	1154	NUM
ejpam-4961	45	18	1148	1148	NUM
ejpam-4961	45	19	=	=	SYM
ejpam-4961	45	20	π(n)−	π(n)−	PROPN
ejpam-4961	45	21	π(m	π(m	PROPN
ejpam-4961	45	22	)	)	PUNCT
ejpam-4961	45	23	,	,	PUNCT
ejpam-4961	45	24	otherwise	otherwise	ADV
ejpam-4961	45	25	4	4	NUM
ejpam-4961	45	26	.	.	X
ejpam-4961	45	27	π(k)(n	π(k)(n	NUM
ejpam-4961	45	28	)	)	PUNCT
ejpam-4961	45	29	denotes	denote	VERB
ejpam-4961	45	30	the	the	DET
ejpam-4961	45	31	number	number	NOUN
ejpam-4961	45	32	of	of	ADP
ejpam-4961	45	33	odd	odd	ADJ
ejpam-4961	45	34	kalmost	kalmost	ADJ
ejpam-4961	45	35	primes	prime	NOUN
ejpam-4961	45	36	≤	≤	NUM
ejpam-4961	46	1	n.	n.	NOUN
ejpam-4961	46	2	we	we	PRON
ejpam-4961	46	3	can	can	AUX
ejpam-4961	46	4	determine	determine	VERB
ejpam-4961	46	5	the	the	DET
ejpam-4961	46	6	number	number	NOUN
ejpam-4961	46	7	of	of	ADP
ejpam-4961	46	8	2	2	NUM
ejpam-4961	46	9	-	-	PUNCT
ejpam-4961	46	10	almost	almost	ADV
ejpam-4961	46	11	primes	prime	NOUN
ejpam-4961	46	12	,	,	PUNCT
ejpam-4961	46	13	3	3	NUM
ejpam-4961	46	14	-	-	PUNCT
ejpam-4961	46	15	almost	almost	ADV
ejpam-4961	46	16	primes	prime	NOUN
ejpam-4961	46	17	,	,	PUNCT
ejpam-4961	46	18	4	4	NUM
ejpam-4961	46	19	-	-	PUNCT
ejpam-4961	46	20	almost	almost	ADV
ejpam-4961	46	21	primes	prime	NOUN
ejpam-4961	46	22	,	,	PUNCT
ejpam-4961	46	23	etc	etc	X
ejpam-4961	46	24	.	.	X
ejpam-4961	46	25	,	,	PUNCT
ejpam-4961	46	26	using	use	VERB
ejpam-4961	46	27	equations	equation	NOUN
ejpam-4961	46	28	(	(	PUNCT
ejpam-4961	46	29	1),(2	1),(2	NUM
ejpam-4961	46	30	)	)	PUNCT
ejpam-4961	46	31	and	and	CCONJ
ejpam-4961	46	32	(	(	PUNCT
ejpam-4961	46	33	3	3	NUM
ejpam-4961	46	34	)	)	PUNCT
ejpam-4961	46	35	.	.	PUNCT
ejpam-4961	47	1	these	these	DET
ejpam-4961	47	2	almost	almost	ADV
ejpam-4961	47	3	prime	prime	ADJ
ejpam-4961	47	4	numbers	number	NOUN
ejpam-4961	47	5	fall	fall	VERB
ejpam-4961	47	6	into	into	ADP
ejpam-4961	47	7	the	the	DET
ejpam-4961	47	8	even	even	ADJ
ejpam-4961	47	9	and	and	CCONJ
ejpam-4961	47	10	odd	odd	ADJ
ejpam-4961	47	11	categories	category	NOUN
ejpam-4961	47	12	.	.	PUNCT
ejpam-4961	48	1	in	in	ADP
ejpam-4961	48	2	this	this	DET
ejpam-4961	48	3	article	article	NOUN
ejpam-4961	48	4	,	,	PUNCT
ejpam-4961	48	5	we	we	PRON
ejpam-4961	48	6	only	only	ADV
ejpam-4961	48	7	focus	focus	VERB
ejpam-4961	48	8	on	on	ADP
ejpam-4961	48	9	odd	odd	ADJ
ejpam-4961	48	10	almost	almost	ADV
ejpam-4961	48	11	primes	prime	NOUN
ejpam-4961	48	12	,	,	PUNCT
ejpam-4961	48	13	as	as	SCONJ
ejpam-4961	48	14	every	every	DET
ejpam-4961	48	15	odd	odd	ADJ
ejpam-4961	48	16	composite	composite	ADJ
ejpam-4961	48	17	number	number	NOUN
ejpam-4961	48	18	is	be	AUX
ejpam-4961	48	19	an	an	DET
ejpam-4961	48	20	odd	odd	ADJ
ejpam-4961	48	21	kalmost	kalmost	ADJ
ejpam-4961	48	22	prime	prime	NOUN
ejpam-4961	48	23	.	.	PUNCT
ejpam-4961	49	1	first	first	ADV
ejpam-4961	49	2	,	,	PUNCT
ejpam-4961	49	3	we	we	PRON
ejpam-4961	49	4	talk	talk	VERB
ejpam-4961	49	5	about	about	ADP
ejpam-4961	49	6	the	the	DET
ejpam-4961	49	7	circumstances	circumstance	NOUN
ejpam-4961	49	8	in	in	ADP
ejpam-4961	49	9	which	which	PRON
ejpam-4961	49	10	an	an	DET
ejpam-4961	49	11	odd	odd	ADJ
ejpam-4961	49	12	k	k	ADJ
ejpam-4961	49	13	-	-	PUNCT
ejpam-4961	49	14	almost	almost	ADV
ejpam-4961	49	15	prime	prime	ADJ
ejpam-4961	49	16	≤	≤	PUNCT
ejpam-4961	49	17	n	n	PRON
ejpam-4961	49	18	exists	exist	VERB
ejpam-4961	49	19	for	for	ADP
ejpam-4961	49	20	any	any	DET
ejpam-4961	49	21	n.	n.	NOUN
ejpam-4961	49	22	the	the	DET
ejpam-4961	49	23	prerequisite	prerequisite	NOUN
ejpam-4961	49	24	is	be	AUX
ejpam-4961	49	25	that	that	PRON
ejpam-4961	49	26	(	(	PUNCT
ejpam-4961	49	27	n	n	CCONJ
ejpam-4961	49	28	)	)	PUNCT
ejpam-4961	50	1	1	1	NUM
ejpam-4961	50	2	k	k	NOUN
ejpam-4961	50	3	≥	≥	NUM
ejpam-4961	50	4	3	3	NUM
ejpam-4961	50	5	.	.	X
ejpam-4961	51	1	for	for	ADP
ejpam-4961	51	2	instance	instance	NOUN
ejpam-4961	51	3	,	,	PUNCT
ejpam-4961	51	4	if	if	SCONJ
ejpam-4961	51	5	n	n	PROPN
ejpam-4961	51	6	=	=	SYM
ejpam-4961	51	7	100	100	NUM
ejpam-4961	51	8	,	,	PUNCT
ejpam-4961	51	9	then	then	ADV
ejpam-4961	51	10	100	100	NUM
ejpam-4961	51	11	1	1	NUM
ejpam-4961	51	12	2	2	NUM
ejpam-4961	51	13	,	,	PUNCT
ejpam-4961	51	14	100	100	NUM
ejpam-4961	51	15	1	1	NUM
ejpam-4961	51	16	3	3	NUM
ejpam-4961	51	17	,	,	PUNCT
ejpam-4961	51	18	100	100	NUM
ejpam-4961	51	19	1	1	NUM
ejpam-4961	51	20	4	4	NUM
ejpam-4961	51	21	are	be	AUX
ejpam-4961	51	22	≥	≥	NOUN
ejpam-4961	51	23	3	3	NUM
ejpam-4961	51	24	but	but	CCONJ
ejpam-4961	51	25	100	100	NUM
ejpam-4961	51	26	1	1	NUM
ejpam-4961	51	27	5	5	NUM
ejpam-4961	51	28	<	<	SYM
ejpam-4961	51	29	3	3	NUM
ejpam-4961	51	30	.	.	PUNCT
ejpam-4961	52	1	thus	thus	ADV
ejpam-4961	52	2	,	,	PUNCT
ejpam-4961	52	3	upto	upto	NOUN
ejpam-4961	52	4	100	100	NUM
ejpam-4961	52	5	,	,	PUNCT
ejpam-4961	52	6	there	there	PRON
ejpam-4961	52	7	are	be	VERB
ejpam-4961	52	8	odd	odd	ADJ
ejpam-4961	52	9	2	2	NUM
ejpam-4961	52	10	-	-	PUNCT
ejpam-4961	52	11	almost	almost	ADV
ejpam-4961	52	12	primes	prime	NOUN
ejpam-4961	52	13	,	,	PUNCT
ejpam-4961	52	14	3	3	NUM
ejpam-4961	52	15	-	-	PUNCT
ejpam-4961	52	16	almost	almost	ADV
ejpam-4961	52	17	primes	prime	NOUN
ejpam-4961	52	18	,	,	PUNCT
ejpam-4961	52	19	4	4	NUM
ejpam-4961	52	20	-	-	PUNCT
ejpam-4961	52	21	almost	almost	ADV
ejpam-4961	52	22	primes	prime	NOUN
ejpam-4961	52	23	but	but	CCONJ
ejpam-4961	52	24	not	not	PART
ejpam-4961	52	25	odd	odd	ADJ
ejpam-4961	52	26	5	5	NUM
ejpam-4961	52	27	-	-	PUNCT
ejpam-4961	52	28	almost	almost	ADV
ejpam-4961	52	29	primes	prime	NOUN
ejpam-4961	52	30	and	and	CCONJ
ejpam-4961	52	31	higher	high	ADJ
ejpam-4961	52	32	.	.	PUNCT
ejpam-4961	53	1	every	every	DET
ejpam-4961	53	2	odd	odd	ADJ
ejpam-4961	53	3	composite	composite	ADJ
ejpam-4961	53	4	number	number	NOUN
ejpam-4961	53	5	,	,	PUNCT
ejpam-4961	53	6	according	accord	VERB
ejpam-4961	53	7	to	to	ADP
ejpam-4961	53	8	the	the	DET
ejpam-4961	53	9	fundamental	fundamental	ADJ
ejpam-4961	53	10	theorem	theorem	NOUN
ejpam-4961	53	11	of	of	ADP
ejpam-4961	53	12	arithmetic	arithmetic	NOUN
ejpam-4961	53	13	,	,	PUNCT
ejpam-4961	53	14	is	be	AUX
ejpam-4961	53	15	actually	actually	ADV
ejpam-4961	53	16	an	an	DET
ejpam-4961	53	17	odd	odd	ADJ
ejpam-4961	53	18	k	k	ADJ
ejpam-4961	53	19	-	-	PUNCT
ejpam-4961	53	20	almost	almost	ADV
ejpam-4961	53	21	prime	prime	ADJ
ejpam-4961	53	22	.	.	PUNCT
ejpam-4961	54	1	for	for	ADP
ejpam-4961	54	2	any	any	DET
ejpam-4961	54	3	natural	natural	ADJ
ejpam-4961	54	4	number	number	NOUN
ejpam-4961	54	5	n	n	CCONJ
ejpam-4961	54	6	,	,	PUNCT
ejpam-4961	54	7	we	we	PRON
ejpam-4961	54	8	know	know	VERB
ejpam-4961	54	9	that	that	SCONJ
ejpam-4961	54	10	there	there	PRON
ejpam-4961	54	11	are	be	VERB
ejpam-4961	54	12	even	even	ADV
ejpam-4961	54	13	numbers	number	NOUN
ejpam-4961	54	14	that	that	PRON
ejpam-4961	54	15	are	be	AUX
ejpam-4961	54	16	n	n	ADV
ejpam-4961	54	17	2	2	NUM
ejpam-4961	54	18	(	(	PUNCT
ejpam-4961	54	19	if	if	SCONJ
ejpam-4961	54	20	n	n	PRON
ejpam-4961	54	21	is	be	AUX
ejpam-4961	54	22	even	even	ADV
ejpam-4961	54	23	)	)	PUNCT
ejpam-4961	54	24	and	and	CCONJ
ejpam-4961	54	25	n−1	n−1	PROPN
ejpam-4961	54	26	2	2	NUM
ejpam-4961	54	27	(	(	PUNCT
ejpam-4961	54	28	if	if	SCONJ
ejpam-4961	54	29	n	n	PRON
ejpam-4961	54	30	is	be	AUX
ejpam-4961	54	31	odd	odd	ADJ
ejpam-4961	54	32	)	)	PUNCT
ejpam-4961	54	33	.	.	PUNCT
ejpam-4961	55	1	the	the	DET
ejpam-4961	55	2	other	other	ADJ
ejpam-4961	55	3	odd	odd	ADJ
ejpam-4961	55	4	numbers	number	NOUN
ejpam-4961	55	5	are	be	AUX
ejpam-4961	55	6	prime	prime	ADJ
ejpam-4961	55	7	and	and	CCONJ
ejpam-4961	55	8	composite	composite	ADJ
ejpam-4961	55	9	.	.	PUNCT
ejpam-4961	56	1	every	every	DET
ejpam-4961	56	2	odd	odd	ADJ
ejpam-4961	56	3	composite	composite	NOUN
ejpam-4961	56	4	is	be	AUX
ejpam-4961	56	5	an	an	DET
ejpam-4961	56	6	odd	odd	ADJ
ejpam-4961	56	7	k	k	ADJ
ejpam-4961	56	8	-	-	PUNCT
ejpam-4961	56	9	almost	almost	ADV
ejpam-4961	56	10	prime	prime	ADJ
ejpam-4961	56	11	,	,	PUNCT
ejpam-4961	56	12	as	as	SCONJ
ejpam-4961	56	13	we	we	PRON
ejpam-4961	56	14	noted	note	VERB
ejpam-4961	56	15	before	before	ADV
ejpam-4961	56	16	.	.	PUNCT
ejpam-4961	57	1	thus	thus	ADV
ejpam-4961	57	2	,	,	PUNCT
ejpam-4961	57	3	we	we	PRON
ejpam-4961	57	4	can	can	AUX
ejpam-4961	57	5	get	get	VERB
ejpam-4961	57	6	the	the	DET
ejpam-4961	57	7	exact	exact	ADJ
ejpam-4961	57	8	value	value	NOUN
ejpam-4961	57	9	of	of	ADP
ejpam-4961	57	10	π(n	π(n	PROPN
ejpam-4961	57	11	)	)	PUNCT
ejpam-4961	57	12	by	by	ADP
ejpam-4961	57	13	deducting	deduct	VERB
ejpam-4961	57	14	the	the	DET
ejpam-4961	57	15	number	number	NOUN
ejpam-4961	57	16	of	of	ADP
ejpam-4961	57	17	even	even	ADJ
ejpam-4961	57	18	numbers	number	NOUN
ejpam-4961	57	19	and	and	CCONJ
ejpam-4961	57	20	k	k	NOUN
ejpam-4961	57	21	-	-	PUNCT
ejpam-4961	57	22	almost	almost	ADV
ejpam-4961	57	23	primes	prime	NOUN
ejpam-4961	57	24	from	from	ADP
ejpam-4961	57	25	the	the	DET
ejpam-4961	57	26	number	number	NOUN
ejpam-4961	57	27	n.	n.	NOUN
ejpam-4961	57	28	theorem	theorem	VERB
ejpam-4961	57	29	1	1	NUM
ejpam-4961	57	30	.	.	X
ejpam-4961	58	1	for	for	ADP
ejpam-4961	58	2	any	any	DET
ejpam-4961	58	3	natural	natural	ADJ
ejpam-4961	58	4	number	number	NOUN
ejpam-4961	58	5	n	n	CCONJ
ejpam-4961	58	6	,	,	PUNCT
ejpam-4961	58	7	the	the	DET
ejpam-4961	58	8	number	number	NOUN
ejpam-4961	58	9	of	of	ADP
ejpam-4961	58	10	odd	odd	ADJ
ejpam-4961	58	11	2	2	NUM
ejpam-4961	58	12	-	-	PUNCT
ejpam-4961	58	13	almost	almost	ADV
ejpam-4961	58	14	primes	prime	NOUN
ejpam-4961	58	15	is	be	AUX
ejpam-4961	58	16	given	give	VERB
ejpam-4961	58	17	by	by	ADP
ejpam-4961	58	18	π(2)(n	π(2)(n	NOUN
ejpam-4961	58	19	)	)	PUNCT
ejpam-4961	58	20	=	=	PUNCT
ejpam-4961	59	1	t∑	t∑	PROPN
ejpam-4961	59	2	i=1	i=1	PROPN
ejpam-4961	59	3	π	π	PROPN
ejpam-4961	59	4	(	(	PUNCT
ejpam-4961	59	5	n	n	ADV
ejpam-4961	59	6	pi+1	pi+1	NOUN
ejpam-4961	59	7	)	)	PUNCT
ejpam-4961	59	8	−	−	PROPN
ejpam-4961	59	9	t(t−	t(t−	NOUN
ejpam-4961	59	10	1	1	NUM
ejpam-4961	59	11	)	)	PUNCT
ejpam-4961	59	12	2	2	NUM
ejpam-4961	59	13	,	,	PUNCT
ejpam-4961	59	14	(	(	PUNCT
ejpam-4961	59	15	5	5	NUM
ejpam-4961	59	16	)	)	PUNCT
ejpam-4961	59	17	where	where	SCONJ
ejpam-4961	59	18	t	t	NOUN
ejpam-4961	59	19	=	=	SYM
ejpam-4961	59	20	π(n	π(n	PROPN
ejpam-4961	59	21	1	1	NUM
ejpam-4961	59	22	2	2	NUM
ejpam-4961	59	23	)	)	PUNCT
ejpam-4961	59	24	and	and	CCONJ
ejpam-4961	59	25	pi	pi	VERB
ejpam-4961	59	26	the	the	DET
ejpam-4961	59	27	ith	ith	ADJ
ejpam-4961	59	28	prime	prime	ADJ
ejpam-4961	59	29	number	number	NOUN
ejpam-4961	59	30	.	.	PUNCT
ejpam-4961	60	1	proof	proof	NOUN
ejpam-4961	60	2	.	.	PUNCT
ejpam-4961	61	1	we	we	PRON
ejpam-4961	61	2	will	will	AUX
ejpam-4961	61	3	prove	prove	VERB
ejpam-4961	61	4	the	the	DET
ejpam-4961	61	5	formula	formula	NOUN
ejpam-4961	61	6	(	(	PUNCT
ejpam-4961	61	7	5	5	NUM
ejpam-4961	61	8	)	)	PUNCT
ejpam-4961	61	9	in	in	ADP
ejpam-4961	61	10	two	two	NUM
ejpam-4961	61	11	different	different	ADJ
ejpam-4961	61	12	ways	way	NOUN
ejpam-4961	61	13	.	.	PUNCT
ejpam-4961	62	1	the	the	DET
ejpam-4961	62	2	sieve	sieve	NOUN
ejpam-4961	62	3	method	method	NOUN
ejpam-4961	62	4	is	be	AUX
ejpam-4961	62	5	used	use	VERB
ejpam-4961	62	6	to	to	PART
ejpam-4961	62	7	obtain	obtain	VERB
ejpam-4961	62	8	the	the	DET
ejpam-4961	62	9	ist	ist	ADJ
ejpam-4961	62	10	proof	proof	NOUN
ejpam-4961	62	11	of	of	ADP
ejpam-4961	62	12	the	the	DET
ejpam-4961	62	13	formula	formula	NOUN
ejpam-4961	62	14	(	(	PUNCT
ejpam-4961	62	15	5	5	NUM
ejpam-4961	62	16	)	)	PUNCT
ejpam-4961	62	17	.	.	PUNCT
ejpam-4961	63	1	for	for	ADP
ejpam-4961	63	2	any	any	DET
ejpam-4961	63	3	natural	natural	ADJ
ejpam-4961	63	4	number	number	NOUN
ejpam-4961	63	5	n	n	CCONJ
ejpam-4961	63	6	,	,	PUNCT
ejpam-4961	63	7	let	let	VERB
ejpam-4961	63	8	u	u	PRON
ejpam-4961	63	9	be	be	AUX
ejpam-4961	63	10	an	an	DET
ejpam-4961	63	11	odd	odd	ADJ
ejpam-4961	63	12	2	2	NUM
ejpam-4961	63	13	-	-	PUNCT
ejpam-4961	63	14	almost	almost	ADV
ejpam-4961	63	15	primes	prime	NOUN
ejpam-4961	63	16	≤	≤	NUM
ejpam-4961	64	1	n.	n.	NOUN
ejpam-4961	64	2	then	then	ADV
ejpam-4961	64	3	u	u	PROPN
ejpam-4961	64	4	=	=	PROPN
ejpam-4961	64	5	pq	pq	PROPN
ejpam-4961	64	6	,	,	PUNCT
ejpam-4961	64	7	where	where	SCONJ
ejpam-4961	64	8	p	p	X
ejpam-4961	64	9	,	,	PUNCT
ejpam-4961	64	10	q	q	X
ejpam-4961	64	11	are	be	AUX
ejpam-4961	64	12	odd	odd	ADJ
ejpam-4961	64	13	prime	prime	ADJ
ejpam-4961	64	14	numbers	number	NOUN
ejpam-4961	64	15	.	.	PUNCT
ejpam-4961	65	1	in	in	ADP
ejpam-4961	65	2	the	the	DET
ejpam-4961	65	3	beginning	beginning	NOUN
ejpam-4961	65	4	,	,	PUNCT
ejpam-4961	65	5	we	we	PRON
ejpam-4961	65	6	set	set	VERB
ejpam-4961	65	7	p	p	NOUN
ejpam-4961	65	8	=	=	NOUN
ejpam-4961	65	9	3	3	NUM
ejpam-4961	65	10	and	and	CCONJ
ejpam-4961	65	11	change	change	VERB
ejpam-4961	65	12	q	q	PUNCT
ejpam-4961	65	13	from	from	ADP
ejpam-4961	65	14	3	3	NUM
ejpam-4961	65	15	,	,	PUNCT
ejpam-4961	65	16	5	5	NUM
ejpam-4961	65	17	,	,	PUNCT
ejpam-4961	65	18	7	7	NUM
ejpam-4961	65	19	,	,	PUNCT
ejpam-4961	65	20	...	...	PUNCT
ejpam-4961	65	21	up	up	ADP
ejpam-4961	65	22	to	to	ADP
ejpam-4961	65	23	the	the	DET
ejpam-4961	65	24	prime	prime	NOUN
ejpam-4961	65	25	z	z	NOUN
ejpam-4961	65	26	such	such	ADJ
ejpam-4961	65	27	that	that	SCONJ
ejpam-4961	65	28	z	z	NOUN
ejpam-4961	65	29	≤	≤	NOUN
ejpam-4961	65	30	n	n	DET
ejpam-4961	65	31	3	3	NUM
ejpam-4961	65	32	.	.	PUNCT
ejpam-4961	66	1	following	follow	VERB
ejpam-4961	66	2	that	that	PRON
ejpam-4961	66	3	,	,	PUNCT
ejpam-4961	66	4	π(n3	π(n3	NOUN
ejpam-4961	66	5	)	)	PUNCT
ejpam-4961	66	6	gives	give	VERB
ejpam-4961	66	7	the	the	DET
ejpam-4961	66	8	total	total	ADJ
ejpam-4961	66	9	number	number	NOUN
ejpam-4961	66	10	of	of	ADP
ejpam-4961	66	11	odd	odd	ADJ
ejpam-4961	66	12	2	2	NUM
ejpam-4961	66	13	-	-	PUNCT
ejpam-4961	66	14	almost	almost	ADV
ejpam-4961	66	15	primes	prime	NOUN
ejpam-4961	66	16	whose	whose	DET
ejpam-4961	66	17	only	only	ADJ
ejpam-4961	66	18	factor	factor	NOUN
ejpam-4961	66	19	is	be	AUX
ejpam-4961	66	20	3	3	NUM
ejpam-4961	66	21	.	.	PUNCT
ejpam-4961	67	1	now	now	ADV
ejpam-4961	67	2	,	,	PUNCT
ejpam-4961	67	3	we	we	PRON
ejpam-4961	67	4	fix	fix	VERB
ejpam-4961	67	5	p	p	NOUN
ejpam-4961	67	6	=	=	NOUN
ejpam-4961	67	7	5	5	NUM
ejpam-4961	67	8	and	and	CCONJ
ejpam-4961	67	9	vary	vary	VERB
ejpam-4961	67	10	q	q	NOUN
ejpam-4961	67	11	from	from	ADP
ejpam-4961	67	12	3	3	NUM
ejpam-4961	67	13	,	,	PUNCT
ejpam-4961	67	14	5	5	NUM
ejpam-4961	67	15	,	,	PUNCT
ejpam-4961	67	16	7	7	NUM
ejpam-4961	67	17	,	,	PUNCT
ejpam-4961	67	18	...	...	PUNCT
ejpam-4961	67	19	up	up	ADP
ejpam-4961	67	20	to	to	ADP
ejpam-4961	67	21	the	the	DET
ejpam-4961	67	22	prime	prime	ADJ
ejpam-4961	67	23	≤	≤	NOUN
ejpam-4961	67	24	n	n	PRON
ejpam-4961	67	25	5	5	NUM
ejpam-4961	67	26	.	.	PUNCT
ejpam-4961	68	1	then	then	ADV
ejpam-4961	68	2	the	the	DET
ejpam-4961	68	3	number	number	NOUN
ejpam-4961	68	4	of	of	ADP
ejpam-4961	68	5	odd	odd	ADJ
ejpam-4961	68	6	2	2	NUM
ejpam-4961	68	7	-	-	PUNCT
ejpam-4961	68	8	almost	almost	ADV
ejpam-4961	68	9	primes	prime	NOUN
ejpam-4961	68	10	whose	whose	DET
ejpam-4961	68	11	one	one	NUM
ejpam-4961	68	12	factor	factor	NOUN
ejpam-4961	68	13	is	be	AUX
ejpam-4961	68	14	5	5	NUM
ejpam-4961	68	15	is	be	AUX
ejpam-4961	68	16	given	give	VERB
ejpam-4961	68	17	by	by	ADP
ejpam-4961	68	18	π(n5	π(n5	NOUN
ejpam-4961	68	19	)	)	PUNCT
ejpam-4961	68	20	.	.	PUNCT
ejpam-4961	69	1	continue	continue	VERB
ejpam-4961	69	2	the	the	DET
ejpam-4961	69	3	procedure	procedure	NOUN
ejpam-4961	69	4	until	until	ADP
ejpam-4961	69	5	the	the	DET
ejpam-4961	69	6	prime	prime	ADJ
ejpam-4961	69	7	pi	pi	NOUN
ejpam-4961	69	8	such	such	ADJ
ejpam-4961	69	9	that	that	SCONJ
ejpam-4961	69	10	n	n	NUM
ejpam-4961	69	11	pi+1	pi+1	X
ejpam-4961	69	12	<	<	X
ejpam-4961	69	13	3	3	NUM
ejpam-4961	69	14	.	.	PUNCT
ejpam-4961	70	1	following	follow	VERB
ejpam-4961	70	2	the	the	DET
ejpam-4961	70	3	addition	addition	NOUN
ejpam-4961	70	4	of	of	ADP
ejpam-4961	70	5	all	all	DET
ejpam-4961	70	6	the	the	DET
ejpam-4961	70	7	odd	odd	ADJ
ejpam-4961	70	8	2	2	NUM
ejpam-4961	70	9	-	-	PUNCT
ejpam-4961	70	10	almost	almost	ADV
ejpam-4961	70	11	primes	prime	NOUN
ejpam-4961	70	12	so	so	ADV
ejpam-4961	70	13	obtained	obtain	VERB
ejpam-4961	70	14	and	and	CCONJ
ejpam-4961	70	15	the	the	DET
ejpam-4961	70	16	subtraction	subtraction	NOUN
ejpam-4961	70	17	of	of	ADP
ejpam-4961	70	18	the	the	DET
ejpam-4961	70	19	quantity	quantity	NOUN
ejpam-4961	70	20	of	of	ADP
ejpam-4961	70	21	repeated	repeat	VERB
ejpam-4961	70	22	m.	m.	NOUN
ejpam-4961	70	23	m.	m.	NOUN
ejpam-4961	70	24	m.	m.	PROPN
ejpam-4961	70	25	jaradat	jaradat	PROPN
ejpam-4961	70	26	et	et	PROPN
ejpam-4961	70	27	al	al	PROPN
ejpam-4961	70	28	.	.	PUNCT
ejpam-4961	70	29	/	/	SYM
ejpam-4961	70	30	eur	eur	PROPN
ejpam-4961	70	31	.	.	PUNCT
ejpam-4961	71	1	j.	j.	PROPN
ejpam-4961	71	2	pure	pure	PROPN
ejpam-4961	71	3	appl	appl	PROPN
ejpam-4961	71	4	.	.	PROPN
ejpam-4961	71	5	math	math	PROPN
ejpam-4961	71	6	,	,	PUNCT
ejpam-4961	71	7	17	17	NUM
ejpam-4961	71	8	(	(	PUNCT
ejpam-4961	71	9	2	2	NUM
ejpam-4961	71	10	)	)	PUNCT
ejpam-4961	71	11	(	(	PUNCT
ejpam-4961	71	12	2024	2024	NUM
ejpam-4961	71	13	)	)	PUNCT
ejpam-4961	71	14	,	,	PUNCT
ejpam-4961	71	15	1146	1146	NUM
ejpam-4961	71	16	-	-	SYM
ejpam-4961	71	17	1154	1154	NUM
ejpam-4961	71	18	1149	1149	NUM
ejpam-4961	71	19	ones	one	NOUN
ejpam-4961	71	20	,	,	PUNCT
ejpam-4961	71	21	we	we	PRON
ejpam-4961	71	22	arrive	arrive	VERB
ejpam-4961	71	23	at	at	ADP
ejpam-4961	71	24	the	the	DET
ejpam-4961	71	25	formula	formula	NOUN
ejpam-4961	71	26	(	(	PUNCT
ejpam-4961	71	27	5	5	NUM
ejpam-4961	71	28	)	)	PUNCT
ejpam-4961	71	29	.	.	PUNCT
ejpam-4961	72	1	it	it	PRON
ejpam-4961	72	2	should	should	AUX
ejpam-4961	72	3	be	be	AUX
ejpam-4961	72	4	noted	note	VERB
ejpam-4961	72	5	that	that	SCONJ
ejpam-4961	72	6	using	use	VERB
ejpam-4961	72	7	the	the	DET
ejpam-4961	72	8	method	method	NOUN
ejpam-4961	72	9	described	describe	VERB
ejpam-4961	72	10	above	above	ADV
ejpam-4961	72	11	,	,	PUNCT
ejpam-4961	72	12	it	it	PRON
ejpam-4961	72	13	is	be	AUX
ejpam-4961	72	14	simple	simple	ADJ
ejpam-4961	72	15	to	to	PART
ejpam-4961	72	16	infer	infer	VERB
ejpam-4961	72	17	that	that	SCONJ
ejpam-4961	72	18	for	for	ADP
ejpam-4961	72	19	any	any	DET
ejpam-4961	72	20	natural	natural	ADJ
ejpam-4961	72	21	integer	integer	NOUN
ejpam-4961	72	22	n	n	CCONJ
ejpam-4961	72	23	,	,	PUNCT
ejpam-4961	72	24	π(2)(n	π(2)(n	X
ejpam-4961	72	25	)	)	PUNCT
ejpam-4961	72	26	=	=	NOUN
ejpam-4961	72	27	π(2)(n)−	π(2)(n)−	NUM
ejpam-4961	72	28	π	π	X
ejpam-4961	72	29	(	(	PUNCT
ejpam-4961	72	30	n	n	ADV
ejpam-4961	72	31	2	2	NUM
ejpam-4961	72	32	)	)	PUNCT
ejpam-4961	72	33	.	.	PUNCT
ejpam-4961	73	1	(	(	PUNCT
ejpam-4961	73	2	6	6	X
ejpam-4961	73	3	)	)	PUNCT
ejpam-4961	73	4	the	the	DET
ejpam-4961	73	5	second	second	ADJ
ejpam-4961	73	6	proof	proof	ADJ
ejpam-4961	73	7	approach	approach	NOUN
ejpam-4961	73	8	is	be	AUX
ejpam-4961	73	9	as	as	SCONJ
ejpam-4961	73	10	follows	follow	VERB
ejpam-4961	73	11	.	.	PUNCT
ejpam-4961	74	1	using	use	VERB
ejpam-4961	74	2	(	(	PUNCT
ejpam-4961	74	3	1	1	NUM
ejpam-4961	74	4	)	)	PUNCT
ejpam-4961	74	5	.	.	PUNCT
ejpam-4961	75	1	π(2)(n	π(2)(n	X
ejpam-4961	75	2	)	)	PUNCT
ejpam-4961	75	3	=	=	SYM
ejpam-4961	76	1	π(n	π(n	PROPN
ejpam-4961	76	2	1	1	NUM
ejpam-4961	76	3	2	2	NUM
ejpam-4961	76	4	)	)	PUNCT
ejpam-4961	76	5	∑	∑	ADP
ejpam-4961	76	6	i=1	i=1	PROPN
ejpam-4961	76	7	[	[	PUNCT
ejpam-4961	76	8	π	π	PROPN
ejpam-4961	76	9	(	(	PUNCT
ejpam-4961	76	10	n	n	NUM
ejpam-4961	76	11	pi	pi	NOUN
ejpam-4961	76	12	)	)	PUNCT
ejpam-4961	76	13	−	−	NOUN
ejpam-4961	76	14	i+	i+	NUM
ejpam-4961	76	15	1	1	X
ejpam-4961	76	16	]	]	PUNCT
ejpam-4961	77	1	=	=	PUNCT
ejpam-4961	77	2	r∑	r∑	NOUN
ejpam-4961	77	3	i=1	i=1	X
ejpam-4961	78	1	[	[	PUNCT
ejpam-4961	78	2	π	π	PROPN
ejpam-4961	78	3	(	(	PUNCT
ejpam-4961	78	4	n	n	NUM
ejpam-4961	78	5	pi	pi	NOUN
ejpam-4961	78	6	)	)	PUNCT
ejpam-4961	78	7	−	−	NOUN
ejpam-4961	78	8	i+	i+	NUM
ejpam-4961	78	9	1	1	NUM
ejpam-4961	78	10	]	]	PUNCT
ejpam-4961	78	11	,	,	PUNCT
ejpam-4961	78	12	where	where	SCONJ
ejpam-4961	78	13	r	r	NOUN
ejpam-4961	78	14	=	=	SYM
ejpam-4961	78	15	π(n	π(n	PROPN
ejpam-4961	78	16	1	1	NUM
ejpam-4961	78	17	2	2	NUM
ejpam-4961	78	18	)	)	PUNCT
ejpam-4961	78	19	=	=	PUNCT
ejpam-4961	79	1	r∑	r∑	ADP
ejpam-4961	79	2	i=1	i=1	PROPN
ejpam-4961	79	3	π	π	PROPN
ejpam-4961	79	4	(	(	PUNCT
ejpam-4961	79	5	n	n	NUM
ejpam-4961	79	6	pi	pi	NOUN
ejpam-4961	79	7	)	)	PUNCT
ejpam-4961	79	8	−	−	PROPN
ejpam-4961	80	1	r∑	r∑	NOUN
ejpam-4961	80	2	i=1	i=1	PRON
ejpam-4961	80	3	i+	i+	PUNCT
ejpam-4961	80	4	r∑	r∑	NOUN
ejpam-4961	80	5	i=1	i=1	PROPN
ejpam-4961	80	6	1	1	NUM
ejpam-4961	80	7	=	=	NOUN
ejpam-4961	80	8	r∑	r∑	NOUN
ejpam-4961	80	9	i=1	i=1	PROPN
ejpam-4961	80	10	π	π	PROPN
ejpam-4961	80	11	(	(	PUNCT
ejpam-4961	80	12	n	n	NUM
ejpam-4961	80	13	pi	pi	NOUN
ejpam-4961	80	14	)	)	PUNCT
ejpam-4961	80	15	−	−	PROPN
ejpam-4961	81	1	r(r	r(r	NOUN
ejpam-4961	81	2	−	−	PROPN
ejpam-4961	81	3	1	1	NUM
ejpam-4961	81	4	)	)	PUNCT
ejpam-4961	81	5	2	2	NUM
ejpam-4961	81	6	.	.	PUNCT
ejpam-4961	82	1	now	now	ADV
ejpam-4961	82	2	,	,	PUNCT
ejpam-4961	82	3	let	let	VERB
ejpam-4961	82	4	π(n	π(n	PROPN
ejpam-4961	82	5	1	1	NUM
ejpam-4961	82	6	2	2	NUM
ejpam-4961	82	7	)	)	PUNCT
ejpam-4961	82	8	=	=	SYM
ejpam-4961	82	9	t	t	PROPN
ejpam-4961	82	10	,	,	PUNCT
ejpam-4961	82	11	then	then	ADV
ejpam-4961	82	12	clearly	clearly	ADV
ejpam-4961	82	13	by	by	ADP
ejpam-4961	82	14	(	(	PUNCT
ejpam-4961	82	15	4	4	NUM
ejpam-4961	82	16	)	)	PUNCT
ejpam-4961	82	17	,	,	PUNCT
ejpam-4961	83	1	r	r	NOUN
ejpam-4961	83	2	=	=	PUNCT
ejpam-4961	83	3	t+	t+	NOUN
ejpam-4961	83	4	1	1	NUM
ejpam-4961	83	5	.	.	PUNCT
ejpam-4961	84	1	using	use	VERB
ejpam-4961	84	2	this	this	DET
ejpam-4961	84	3	value	value	NOUN
ejpam-4961	84	4	in	in	ADP
ejpam-4961	84	5	above	above	ADP
ejpam-4961	84	6	equation	equation	NOUN
ejpam-4961	84	7	we	we	PRON
ejpam-4961	84	8	get	get	VERB
ejpam-4961	84	9	,	,	PUNCT
ejpam-4961	84	10	π(2)(n	π(2)(n	X
ejpam-4961	84	11	)	)	PUNCT
ejpam-4961	85	1	=	=	SYM
ejpam-4961	85	2	t+1∑	t+1∑	PROPN
ejpam-4961	85	3	i=1	i=1	PROPN
ejpam-4961	86	1	π	π	PROPN
ejpam-4961	86	2	(	(	PUNCT
ejpam-4961	86	3	n	n	NUM
ejpam-4961	86	4	pi	pi	NOUN
ejpam-4961	86	5	)	)	PUNCT
ejpam-4961	86	6	−	−	PROPN
ejpam-4961	87	1	t(t+	t(t+	NUM
ejpam-4961	87	2	1	1	NUM
ejpam-4961	87	3	)	)	SYM
ejpam-4961	87	4	2	2	NUM
ejpam-4961	87	5	=	=	SYM
ejpam-4961	87	6	t+1∑	t+1∑	PROPN
ejpam-4961	87	7	i=1	i=1	X
ejpam-4961	88	1	[	[	X
ejpam-4961	88	2	π	π	X
ejpam-4961	88	3	(	(	PUNCT
ejpam-4961	88	4	n	n	NUM
ejpam-4961	88	5	pi	pi	NOUN
ejpam-4961	88	6	)	)	PUNCT
ejpam-4961	88	7	+	+	CCONJ
ejpam-4961	88	8	1]−	1]−	NUM
ejpam-4961	88	9	t(t+	t(t+	NUM
ejpam-4961	88	10	1	1	NUM
ejpam-4961	88	11	)	)	SYM
ejpam-4961	88	12	2	2	NUM
ejpam-4961	88	13	,	,	PUNCT
ejpam-4961	88	14	∵	∵	X
ejpam-4961	88	15	(	(	PUNCT
ejpam-4961	88	16	4	4	NUM
ejpam-4961	88	17	)	)	PUNCT
ejpam-4961	88	18	=	=	SYM
ejpam-4961	89	1	t+1∑	t+1∑	PROPN
ejpam-4961	89	2	i=1	i=1	PROPN
ejpam-4961	90	1	π	π	PROPN
ejpam-4961	90	2	(	(	PUNCT
ejpam-4961	90	3	n	n	NUM
ejpam-4961	90	4	pi	pi	NOUN
ejpam-4961	90	5	)	)	PUNCT
ejpam-4961	91	1	+	+	CCONJ
ejpam-4961	91	2	(	(	PUNCT
ejpam-4961	91	3	t+	t+	NOUN
ejpam-4961	91	4	1)−	1)−	NUM
ejpam-4961	91	5	t(t+	t(t+	NUM
ejpam-4961	91	6	1	1	NUM
ejpam-4961	91	7	)	)	PUNCT
ejpam-4961	91	8	2	2	NUM
ejpam-4961	91	9	,	,	PUNCT
ejpam-4961	91	10	=	=	SYM
ejpam-4961	91	11	t+1∑	t+1∑	PROPN
ejpam-4961	91	12	i=1	i=1	PROPN
ejpam-4961	92	1	π	π	PROPN
ejpam-4961	92	2	(	(	PUNCT
ejpam-4961	92	3	n	n	NUM
ejpam-4961	92	4	pi	pi	NOUN
ejpam-4961	92	5	)	)	PUNCT
ejpam-4961	93	1	+	+	CCONJ
ejpam-4961	93	2	(	(	PUNCT
ejpam-4961	93	3	t+	t+	PUNCT
ejpam-4961	93	4	1)(2−	1)(2−	PROPN
ejpam-4961	93	5	t	t	PROPN
ejpam-4961	93	6	)	)	PUNCT
ejpam-4961	93	7	2	2	NUM
ejpam-4961	93	8	(	(	PUNCT
ejpam-4961	93	9	7	7	X
ejpam-4961	93	10	)	)	PUNCT
ejpam-4961	93	11	using	use	VERB
ejpam-4961	93	12	(	(	PUNCT
ejpam-4961	93	13	7	7	NUM
ejpam-4961	93	14	)	)	PUNCT
ejpam-4961	93	15	in	in	ADP
ejpam-4961	93	16	(	(	PUNCT
ejpam-4961	93	17	6	6	NUM
ejpam-4961	93	18	)	)	PUNCT
ejpam-4961	93	19	,	,	PUNCT
ejpam-4961	93	20	we	we	PRON
ejpam-4961	93	21	get	get	VERB
ejpam-4961	93	22	π(2)(n	π(2)(n	NOUN
ejpam-4961	93	23	)	)	PUNCT
ejpam-4961	94	1	=	=	SYM
ejpam-4961	94	2	t+1∑	t+1∑	PROPN
ejpam-4961	94	3	i=1	i=1	PROPN
ejpam-4961	95	1	π	π	PROPN
ejpam-4961	95	2	(	(	PUNCT
ejpam-4961	95	3	n	n	NUM
ejpam-4961	95	4	pi	pi	NOUN
ejpam-4961	95	5	)	)	PUNCT
ejpam-4961	96	1	+	+	CCONJ
ejpam-4961	96	2	(	(	PUNCT
ejpam-4961	96	3	t+	t+	PUNCT
ejpam-4961	96	4	1)(2−	1)(2−	PROPN
ejpam-4961	96	5	t	t	PROPN
ejpam-4961	96	6	)	)	PUNCT
ejpam-4961	96	7	2	2	NUM
ejpam-4961	96	8	−	−	PROPN
ejpam-4961	96	9	π	π	PROPN
ejpam-4961	96	10	(	(	PUNCT
ejpam-4961	96	11	n	n	ADV
ejpam-4961	96	12	2	2	NUM
ejpam-4961	96	13	)	)	PUNCT
ejpam-4961	96	14	,	,	PUNCT
ejpam-4961	96	15	=	=	SYM
ejpam-4961	96	16	π	π	PROPN
ejpam-4961	96	17	(	(	PUNCT
ejpam-4961	96	18	n	n	X
ejpam-4961	96	19	p1	p1	NOUN
ejpam-4961	96	20	)	)	PUNCT
ejpam-4961	97	1	+	+	CCONJ
ejpam-4961	97	2	t+1∑	t+1∑	PROPN
ejpam-4961	97	3	i=2	i=2	PROPN
ejpam-4961	97	4	π	π	PROPN
ejpam-4961	97	5	(	(	PUNCT
ejpam-4961	97	6	n	n	NOUN
ejpam-4961	97	7	pi	pi	NOUN
ejpam-4961	97	8	)	)	PUNCT
ejpam-4961	98	1	+	+	CCONJ
ejpam-4961	98	2	(	(	PUNCT
ejpam-4961	98	3	t+	t+	PUNCT
ejpam-4961	98	4	1)(2−	1)(2−	PROPN
ejpam-4961	98	5	t	t	PROPN
ejpam-4961	98	6	)	)	PUNCT
ejpam-4961	98	7	2	2	NUM
ejpam-4961	98	8	−	−	PROPN
ejpam-4961	98	9	π	π	PROPN
ejpam-4961	98	10	(	(	PUNCT
ejpam-4961	98	11	n	n	ADV
ejpam-4961	98	12	2	2	NUM
ejpam-4961	98	13	)	)	PUNCT
ejpam-4961	98	14	,	,	PUNCT
ejpam-4961	98	15	m.	m.	NOUN
ejpam-4961	98	16	m.	m.	PROPN
ejpam-4961	98	17	m.	m.	PROPN
ejpam-4961	98	18	jaradat	jaradat	PROPN
ejpam-4961	98	19	et	et	PROPN
ejpam-4961	98	20	al	al	PROPN
ejpam-4961	98	21	.	.	PUNCT
ejpam-4961	98	22	/	/	SYM
ejpam-4961	98	23	eur	eur	PROPN
ejpam-4961	98	24	.	.	PUNCT
ejpam-4961	99	1	j.	j.	PROPN
ejpam-4961	99	2	pure	pure	PROPN
ejpam-4961	99	3	appl	appl	PROPN
ejpam-4961	99	4	.	.	PROPN
ejpam-4961	99	5	math	math	PROPN
ejpam-4961	99	6	,	,	PUNCT
ejpam-4961	99	7	17	17	NUM
ejpam-4961	99	8	(	(	PUNCT
ejpam-4961	99	9	2	2	NUM
ejpam-4961	99	10	)	)	PUNCT
ejpam-4961	99	11	(	(	PUNCT
ejpam-4961	99	12	2024	2024	NUM
ejpam-4961	99	13	)	)	PUNCT
ejpam-4961	99	14	,	,	PUNCT
ejpam-4961	99	15	1146	1146	NUM
ejpam-4961	99	16	-	-	SYM
ejpam-4961	99	17	1154	1154	NUM
ejpam-4961	99	18	1150	1150	NUM
ejpam-4961	99	19	=	=	SYM
ejpam-4961	99	20	π	π	PROPN
ejpam-4961	99	21	(	(	PUNCT
ejpam-4961	99	22	n	n	X
ejpam-4961	99	23	p1	p1	NOUN
ejpam-4961	99	24	)	)	PUNCT
ejpam-4961	99	25	−	−	PROPN
ejpam-4961	99	26	1	1	NUM
ejpam-4961	100	1	+	+	CCONJ
ejpam-4961	100	2	t+1∑	t+1∑	PROPN
ejpam-4961	100	3	i=2	i=2	PROPN
ejpam-4961	100	4	π	π	PROPN
ejpam-4961	100	5	(	(	PUNCT
ejpam-4961	100	6	n	n	NOUN
ejpam-4961	100	7	pi	pi	NOUN
ejpam-4961	100	8	)	)	PUNCT
ejpam-4961	101	1	+	+	CCONJ
ejpam-4961	101	2	(	(	PUNCT
ejpam-4961	101	3	t+	t+	PUNCT
ejpam-4961	101	4	1)(2−	1)(2−	PROPN
ejpam-4961	101	5	t	t	PROPN
ejpam-4961	101	6	)	)	PUNCT
ejpam-4961	101	7	2	2	NUM
ejpam-4961	101	8	−	−	PROPN
ejpam-4961	101	9	π	π	PROPN
ejpam-4961	101	10	(	(	PUNCT
ejpam-4961	101	11	n	n	ADV
ejpam-4961	101	12	2	2	NUM
ejpam-4961	101	13	)	)	PUNCT
ejpam-4961	101	14	,	,	PUNCT
ejpam-4961	101	15	∵	∵	X
ejpam-4961	101	16	(	(	PUNCT
ejpam-4961	101	17	4	4	NUM
ejpam-4961	101	18	)	)	PUNCT
ejpam-4961	101	19	=	=	SYM
ejpam-4961	101	20	t+1∑	t+1∑	PROPN
ejpam-4961	101	21	i=2	i=2	PROPN
ejpam-4961	101	22	π	π	PROPN
ejpam-4961	101	23	(	(	PUNCT
ejpam-4961	101	24	n	n	NOUN
ejpam-4961	101	25	pi	pi	NOUN
ejpam-4961	101	26	)	)	PUNCT
ejpam-4961	102	1	+	+	CCONJ
ejpam-4961	102	2	(	(	PUNCT
ejpam-4961	102	3	t+	t+	PUNCT
ejpam-4961	102	4	1)(2−	1)(2−	PROPN
ejpam-4961	102	5	t	t	PROPN
ejpam-4961	102	6	)	)	PUNCT
ejpam-4961	102	7	2	2	NUM
ejpam-4961	102	8	−	−	NOUN
ejpam-4961	102	9	1	1	NUM
ejpam-4961	102	10	,	,	PUNCT
ejpam-4961	102	11	∵	∵	NOUN
ejpam-4961	102	12	p1	p1	NOUN
ejpam-4961	102	13	=	=	SYM
ejpam-4961	102	14	2	2	NUM
ejpam-4961	102	15	=	=	SYM
ejpam-4961	102	16	t+1∑	t+1∑	PROPN
ejpam-4961	102	17	i=2	i=2	PROPN
ejpam-4961	102	18	π	π	PROPN
ejpam-4961	102	19	(	(	PUNCT
ejpam-4961	102	20	n	n	NUM
ejpam-4961	102	21	pi	pi	NOUN
ejpam-4961	102	22	)	)	PUNCT
ejpam-4961	102	23	−	−	PROPN
ejpam-4961	102	24	t(t−	t(t−	NOUN
ejpam-4961	102	25	1	1	NUM
ejpam-4961	102	26	)	)	PUNCT
ejpam-4961	102	27	2	2	NUM
ejpam-4961	102	28	,	,	PUNCT
ejpam-4961	102	29	=	=	PUNCT
ejpam-4961	102	30	t∑	t∑	PROPN
ejpam-4961	102	31	i=1	i=1	PROPN
ejpam-4961	102	32	π	π	PROPN
ejpam-4961	102	33	(	(	PUNCT
ejpam-4961	102	34	n	n	ADV
ejpam-4961	102	35	pi+1	pi+1	NOUN
ejpam-4961	102	36	)	)	PUNCT
ejpam-4961	102	37	−	−	PROPN
ejpam-4961	102	38	t(t−	t(t−	NOUN
ejpam-4961	102	39	1	1	NUM
ejpam-4961	102	40	)	)	PUNCT
ejpam-4961	102	41	2	2	NUM
ejpam-4961	102	42	.	.	PUNCT
ejpam-4961	103	1	it	it	PRON
ejpam-4961	103	2	proves	prove	VERB
ejpam-4961	103	3	the	the	DET
ejpam-4961	103	4	result	result	NOUN
ejpam-4961	103	5	.	.	PUNCT
ejpam-4961	104	1	note	note	VERB
ejpam-4961	104	2	:	:	PUNCT
ejpam-4961	104	3	it	it	PRON
ejpam-4961	104	4	is	be	AUX
ejpam-4961	104	5	important	important	ADJ
ejpam-4961	104	6	to	to	PART
ejpam-4961	104	7	note	note	VERB
ejpam-4961	104	8	that	that	DET
ejpam-4961	104	9	formula	formula	NOUN
ejpam-4961	104	10	(	(	PUNCT
ejpam-4961	104	11	5	5	NUM
ejpam-4961	104	12	)	)	PUNCT
ejpam-4961	104	13	is	be	AUX
ejpam-4961	104	14	superior	superior	ADJ
ejpam-4961	104	15	to	to	PART
ejpam-4961	104	16	formula	formula	VERB
ejpam-4961	104	17	(	(	PUNCT
ejpam-4961	104	18	1	1	NUM
ejpam-4961	104	19	)	)	PUNCT
ejpam-4961	104	20	in	in	ADP
ejpam-4961	104	21	two	two	NUM
ejpam-4961	104	22	ways	way	NOUN
ejpam-4961	104	23	:	:	PUNCT
ejpam-4961	104	24	first	first	ADV
ejpam-4961	104	25	,	,	PUNCT
ejpam-4961	104	26	because	because	SCONJ
ejpam-4961	104	27	(	(	PUNCT
ejpam-4961	104	28	5	5	NUM
ejpam-4961	104	29	)	)	PUNCT
ejpam-4961	104	30	only	only	ADV
ejpam-4961	104	31	produces	produce	VERB
ejpam-4961	104	32	odd	odd	ADJ
ejpam-4961	104	33	2	2	NUM
ejpam-4961	104	34	-	-	PUNCT
ejpam-4961	104	35	almost	almost	ADV
ejpam-4961	104	36	primes	prime	NOUN
ejpam-4961	104	37	,	,	PUNCT
ejpam-4961	104	38	and	and	CCONJ
ejpam-4961	104	39	second	second	ADJ
ejpam-4961	104	40	,	,	PUNCT
ejpam-4961	104	41	because	because	SCONJ
ejpam-4961	104	42	(	(	PUNCT
ejpam-4961	104	43	1	1	X
ejpam-4961	104	44	)	)	PUNCT
ejpam-4961	104	45	requires	require	VERB
ejpam-4961	104	46	the	the	DET
ejpam-4961	104	47	list	list	NOUN
ejpam-4961	104	48	of	of	ADP
ejpam-4961	104	49	primes	prime	NOUN
ejpam-4961	104	50	up	up	ADP
ejpam-4961	104	51	to	to	ADP
ejpam-4961	104	52	n	n	DET
ejpam-4961	104	53	2	2	NUM
ejpam-4961	104	54	,	,	PUNCT
ejpam-4961	104	55	whereas	whereas	SCONJ
ejpam-4961	104	56	(	(	PUNCT
ejpam-4961	104	57	5	5	NUM
ejpam-4961	104	58	)	)	PUNCT
ejpam-4961	104	59	only	only	ADV
ejpam-4961	104	60	needs	need	VERB
ejpam-4961	104	61	the	the	DET
ejpam-4961	104	62	list	list	NOUN
ejpam-4961	104	63	up	up	ADP
ejpam-4961	104	64	to	to	ADP
ejpam-4961	104	65	n	n	PRON
ejpam-4961	104	66	3	3	NUM
ejpam-4961	104	67	,	,	PUNCT
ejpam-4961	104	68	which	which	PRON
ejpam-4961	104	69	is	be	AUX
ejpam-4961	104	70	a	a	DET
ejpam-4961	104	71	very	very	ADV
ejpam-4961	104	72	short	short	ADJ
ejpam-4961	104	73	interval	interval	NOUN
ejpam-4961	104	74	.	.	PUNCT
ejpam-4961	105	1	we	we	PRON
ejpam-4961	105	2	offer	offer	VERB
ejpam-4961	105	3	a	a	DET
ejpam-4961	105	4	few	few	ADJ
ejpam-4961	105	5	straightforward	straightforward	ADJ
ejpam-4961	105	6	examples	example	NOUN
ejpam-4961	105	7	to	to	PART
ejpam-4961	105	8	demonstrate	demonstrate	VERB
ejpam-4961	105	9	our	our	PRON
ejpam-4961	105	10	methodology	methodology	NOUN
ejpam-4961	105	11	.	.	PUNCT
ejpam-4961	106	1	example	example	NOUN
ejpam-4961	107	1	1	1	NUM
ejpam-4961	107	2	.	.	PUNCT
ejpam-4961	107	3	find	find	VERB
ejpam-4961	107	4	the	the	DET
ejpam-4961	107	5	number	number	NOUN
ejpam-4961	107	6	of	of	ADP
ejpam-4961	107	7	odd	odd	ADJ
ejpam-4961	107	8	2	2	NUM
ejpam-4961	107	9	-	-	PUNCT
ejpam-4961	107	10	almost	almost	ADV
ejpam-4961	107	11	primes	prime	NOUN
ejpam-4961	107	12	upto	upto	PROPN
ejpam-4961	107	13	100	100	NUM
ejpam-4961	107	14	,	,	PUNCT
ejpam-4961	107	15	i.e.	i.e.	X
ejpam-4961	107	16	,	,	PUNCT
ejpam-4961	107	17	π(2)(100	π(2)(100	NOUN
ejpam-4961	107	18	)	)	PUNCT
ejpam-4961	107	19	.	.	PUNCT
ejpam-4961	108	1	we	we	PRON
ejpam-4961	108	2	know	know	VERB
ejpam-4961	108	3	that	that	PRON
ejpam-4961	108	4	π(2)(n	π(2)(n	VERB
ejpam-4961	108	5	)	)	PUNCT
ejpam-4961	108	6	=	=	PUNCT
ejpam-4961	109	1	t∑	t∑	PROPN
ejpam-4961	109	2	i=1	i=1	PROPN
ejpam-4961	109	3	π	π	PROPN
ejpam-4961	109	4	(	(	PUNCT
ejpam-4961	109	5	n	n	ADV
ejpam-4961	109	6	pi+1	pi+1	NOUN
ejpam-4961	109	7	)	)	PUNCT
ejpam-4961	109	8	−	−	PROPN
ejpam-4961	109	9	t(t−	t(t−	NOUN
ejpam-4961	109	10	1	1	NUM
ejpam-4961	109	11	)	)	PUNCT
ejpam-4961	109	12	2	2	NUM
ejpam-4961	109	13	.	.	PUNCT
ejpam-4961	110	1	here	here	ADV
ejpam-4961	110	2	,	,	PUNCT
ejpam-4961	110	3	n	n	PROPN
ejpam-4961	110	4	=	=	SYM
ejpam-4961	110	5	100	100	NUM
ejpam-4961	110	6	and	and	CCONJ
ejpam-4961	110	7	t	t	NOUN
ejpam-4961	110	8	=	=	SYM
ejpam-4961	110	9	π(100	π(100	NOUN
ejpam-4961	110	10	1	1	NUM
ejpam-4961	110	11	2	2	NUM
ejpam-4961	110	12	)	)	PUNCT
ejpam-4961	110	13	=	=	SYM
ejpam-4961	111	1	π(10	π(10	X
ejpam-4961	111	2	)	)	PUNCT
ejpam-4961	111	3	=	=	SYM
ejpam-4961	111	4	3	3	NUM
ejpam-4961	111	5	now	now	ADV
ejpam-4961	111	6	,	,	PUNCT
ejpam-4961	111	7	π	π	PROPN
ejpam-4961	111	8	(	(	PUNCT
ejpam-4961	111	9	n	n	ADV
ejpam-4961	111	10	p2	p2	X
ejpam-4961	111	11	)	)	PUNCT
ejpam-4961	112	1	=	=	SYM
ejpam-4961	112	2	π(1003	π(1003	X
ejpam-4961	112	3	)	)	PUNCT
ejpam-4961	112	4	=	=	SYM
ejpam-4961	112	5	π(33.3	π(33.3	X
ejpam-4961	112	6	)	)	PUNCT
ejpam-4961	112	7	=	=	SYM
ejpam-4961	112	8	10	10	NUM
ejpam-4961	112	9	.	.	PUNCT
ejpam-4961	113	1	π	π	PROPN
ejpam-4961	113	2	(	(	PUNCT
ejpam-4961	113	3	n	n	X
ejpam-4961	113	4	p3	p3	NOUN
ejpam-4961	113	5	)	)	PUNCT
ejpam-4961	113	6	=	=	SYM
ejpam-4961	113	7	π(1005	π(1005	ADP
ejpam-4961	113	8	)	)	PUNCT
ejpam-4961	113	9	=	=	SYM
ejpam-4961	113	10	π(20	π(20	X
ejpam-4961	113	11	)	)	PUNCT
ejpam-4961	113	12	=	=	SYM
ejpam-4961	114	1	7	7	X
ejpam-4961	114	2	.	.	X
ejpam-4961	115	1	π	π	PROPN
ejpam-4961	115	2	(	(	PUNCT
ejpam-4961	115	3	n	n	CCONJ
ejpam-4961	115	4	p4	p4	ADJ
ejpam-4961	115	5	)	)	PUNCT
ejpam-4961	115	6	=	=	SYM
ejpam-4961	115	7	π(1007	π(1007	PROPN
ejpam-4961	115	8	)	)	PUNCT
ejpam-4961	115	9	=	=	PUNCT
ejpam-4961	115	10	π(14.2	π(14.2	NOUN
ejpam-4961	115	11	)	)	PUNCT
ejpam-4961	115	12	=	=	SYM
ejpam-4961	115	13	5	5	X
ejpam-4961	115	14	.	.	X
ejpam-4961	115	15	substitute	substitute	VERB
ejpam-4961	115	16	the	the	DET
ejpam-4961	115	17	value	value	NOUN
ejpam-4961	115	18	’s	’	VERB
ejpam-4961	115	19	in	in	ADP
ejpam-4961	115	20	above	above	ADP
ejpam-4961	115	21	equation	equation	NOUN
ejpam-4961	115	22	to	to	PART
ejpam-4961	115	23	get	get	VERB
ejpam-4961	115	24	π(2)(100	π(2)(100	NOUN
ejpam-4961	115	25	)	)	PUNCT
ejpam-4961	115	26	=	=	SYM
ejpam-4961	116	1	10	10	NUM
ejpam-4961	116	2	+	+	NUM
ejpam-4961	116	3	7	7	NUM
ejpam-4961	117	1	+	+	CCONJ
ejpam-4961	117	2	5−	5−	NUM
ejpam-4961	117	3	3	3	NUM
ejpam-4961	117	4	=	=	SYM
ejpam-4961	117	5	19	19	NUM
ejpam-4961	117	6	.	.	NOUN
ejpam-4961	117	7	example	example	NOUN
ejpam-4961	117	8	2	2	NUM
ejpam-4961	117	9	.	.	PUNCT
ejpam-4961	117	10	find	find	VERB
ejpam-4961	117	11	the	the	DET
ejpam-4961	117	12	number	number	NOUN
ejpam-4961	117	13	of	of	ADP
ejpam-4961	117	14	odd	odd	ADJ
ejpam-4961	117	15	2	2	NUM
ejpam-4961	117	16	-	-	PUNCT
ejpam-4961	117	17	almost	almost	ADV
ejpam-4961	117	18	primes	prime	NOUN
ejpam-4961	117	19	upto	upto	ADJ
ejpam-4961	117	20	200	200	NUM
ejpam-4961	117	21	,	,	PUNCT
ejpam-4961	117	22	i.e.	i.e.	X
ejpam-4961	117	23	,	,	PUNCT
ejpam-4961	117	24	π(2)(200	π(2)(200	NUM
ejpam-4961	117	25	)	)	PUNCT
ejpam-4961	117	26	.	.	PUNCT
ejpam-4961	118	1	we	we	PRON
ejpam-4961	118	2	know	know	VERB
ejpam-4961	118	3	that	that	PRON
ejpam-4961	118	4	π(2)(n	π(2)(n	VERB
ejpam-4961	118	5	)	)	PUNCT
ejpam-4961	118	6	=	=	PUNCT
ejpam-4961	119	1	t∑	t∑	PROPN
ejpam-4961	119	2	i=1	i=1	PROPN
ejpam-4961	119	3	π	π	PROPN
ejpam-4961	119	4	(	(	PUNCT
ejpam-4961	119	5	n	n	ADV
ejpam-4961	119	6	pi+1	pi+1	NOUN
ejpam-4961	119	7	)	)	PUNCT
ejpam-4961	119	8	−	−	PROPN
ejpam-4961	119	9	t(t−	t(t−	NOUN
ejpam-4961	119	10	1	1	NUM
ejpam-4961	119	11	)	)	SYM
ejpam-4961	119	12	2	2	NUM
ejpam-4961	119	13	.	.	PUNCT
ejpam-4961	119	14	m.	m.	NOUN
ejpam-4961	119	15	m.	m.	PROPN
ejpam-4961	119	16	m.	m.	PROPN
ejpam-4961	119	17	jaradat	jaradat	PROPN
ejpam-4961	119	18	et	et	PROPN
ejpam-4961	119	19	al	al	PROPN
ejpam-4961	119	20	.	.	PUNCT
ejpam-4961	119	21	/	/	SYM
ejpam-4961	119	22	eur	eur	PROPN
ejpam-4961	119	23	.	.	PUNCT
ejpam-4961	120	1	j.	j.	PROPN
ejpam-4961	120	2	pure	pure	PROPN
ejpam-4961	120	3	appl	appl	PROPN
ejpam-4961	120	4	.	.	PROPN
ejpam-4961	120	5	math	math	PROPN
ejpam-4961	120	6	,	,	PUNCT
ejpam-4961	120	7	17	17	NUM
ejpam-4961	120	8	(	(	PUNCT
ejpam-4961	120	9	2	2	NUM
ejpam-4961	120	10	)	)	PUNCT
ejpam-4961	120	11	(	(	PUNCT
ejpam-4961	120	12	2024	2024	NUM
ejpam-4961	120	13	)	)	PUNCT
ejpam-4961	120	14	,	,	PUNCT
ejpam-4961	120	15	1146	1146	NUM
ejpam-4961	120	16	-	-	SYM
ejpam-4961	120	17	1154	1154	NUM
ejpam-4961	120	18	1151	1151	NUM
ejpam-4961	120	19	here	here	ADV
ejpam-4961	120	20	,	,	PUNCT
ejpam-4961	120	21	n	n	PROPN
ejpam-4961	120	22	=	=	NUM
ejpam-4961	120	23	200	200	NUM
ejpam-4961	120	24	and	and	CCONJ
ejpam-4961	120	25	t	t	NOUN
ejpam-4961	120	26	=	=	PUNCT
ejpam-4961	120	27	π(200	π(200	NOUN
ejpam-4961	120	28	1	1	NUM
ejpam-4961	120	29	2	2	NUM
ejpam-4961	120	30	)	)	PUNCT
ejpam-4961	120	31	=	=	SYM
ejpam-4961	120	32	π(14.14	π(14.14	PROPN
ejpam-4961	120	33	)	)	PUNCT
ejpam-4961	120	34	=	=	SYM
ejpam-4961	120	35	5	5	NUM
ejpam-4961	120	36	∴	∴	NOUN
ejpam-4961	120	37	π(2)(200	π(2)(200	NUM
ejpam-4961	120	38	)	)	PUNCT
ejpam-4961	120	39	=	=	PUNCT
ejpam-4961	121	1	5∑	5∑	NUM
ejpam-4961	121	2	i=1	i=1	PROPN
ejpam-4961	121	3	π	π	PROPN
ejpam-4961	121	4	(	(	PUNCT
ejpam-4961	121	5	200	200	NUM
ejpam-4961	121	6	pi+1	pi+1	NOUN
ejpam-4961	121	7	)	)	PUNCT
ejpam-4961	121	8	−	−	PROPN
ejpam-4961	121	9	10	10	NUM
ejpam-4961	121	10	,	,	PUNCT
ejpam-4961	121	11	=	=	SYM
ejpam-4961	121	12	π	π	PROPN
ejpam-4961	121	13	(	(	PUNCT
ejpam-4961	121	14	200	200	NUM
ejpam-4961	121	15	3	3	NUM
ejpam-4961	121	16	)	)	PUNCT
ejpam-4961	121	17	+	+	CCONJ
ejpam-4961	121	18	π	π	X
ejpam-4961	121	19	(	(	PUNCT
ejpam-4961	121	20	200	200	NUM
ejpam-4961	121	21	5	5	NUM
ejpam-4961	121	22	)	)	PUNCT
ejpam-4961	121	23	+	+	CCONJ
ejpam-4961	121	24	π	π	X
ejpam-4961	121	25	(	(	PUNCT
ejpam-4961	121	26	200	200	NUM
ejpam-4961	121	27	7	7	NUM
ejpam-4961	121	28	)	)	PUNCT
ejpam-4961	121	29	+	+	CCONJ
ejpam-4961	121	30	π	π	X
ejpam-4961	121	31	(	(	PUNCT
ejpam-4961	121	32	200	200	NUM
ejpam-4961	121	33	11	11	NUM
ejpam-4961	121	34	)	)	PUNCT
ejpam-4961	122	1	+	+	CCONJ
ejpam-4961	122	2	π	π	X
ejpam-4961	122	3	(	(	PUNCT
ejpam-4961	122	4	200	200	NUM
ejpam-4961	122	5	13	13	NUM
ejpam-4961	122	6	)	)	PUNCT
ejpam-4961	122	7	−	−	PROPN
ejpam-4961	122	8	10	10	NUM
ejpam-4961	122	9	,	,	PUNCT
ejpam-4961	122	10	=	=	NOUN
ejpam-4961	122	11	17	17	NUM
ejpam-4961	122	12	+	+	NUM
ejpam-4961	122	13	11	11	NUM
ejpam-4961	122	14	+	+	NUM
ejpam-4961	122	15	8	8	NUM
ejpam-4961	122	16	+	+	NUM
ejpam-4961	122	17	6	6	NUM
ejpam-4961	122	18	+	+	CCONJ
ejpam-4961	122	19	5−	5−	NUM
ejpam-4961	122	20	10	10	NUM
ejpam-4961	122	21	,	,	PUNCT
ejpam-4961	122	22	=	=	NOUN
ejpam-4961	122	23	37	37	NUM
ejpam-4961	122	24	.	.	PUNCT
ejpam-4961	122	25	number	number	NOUN
ejpam-4961	122	26	of	of	ADP
ejpam-4961	122	27	odd	odd	ADJ
ejpam-4961	122	28	3	3	NUM
ejpam-4961	122	29	-	-	PUNCT
ejpam-4961	122	30	almost	almost	ADV
ejpam-4961	122	31	primes	prime	NOUN
ejpam-4961	122	32	≤	≤	NUM
ejpam-4961	122	33	n	n	CCONJ
ejpam-4961	122	34	,	,	PUNCT
ejpam-4961	122	35	i.e.	i.e.	X
ejpam-4961	122	36	,π(3)(n	,π(3)(n	PUNCT
ejpam-4961	122	37	)	)	PUNCT
ejpam-4961	122	38	.	.	PUNCT
ejpam-4961	123	1	in	in	ADP
ejpam-4961	123	2	this	this	DET
ejpam-4961	123	3	section	section	NOUN
ejpam-4961	123	4	,	,	PUNCT
ejpam-4961	123	5	we	we	PRON
ejpam-4961	123	6	give	give	VERB
ejpam-4961	123	7	the	the	DET
ejpam-4961	123	8	formula	formula	NOUN
ejpam-4961	123	9	for	for	ADP
ejpam-4961	123	10	finding	find	VERB
ejpam-4961	123	11	the	the	DET
ejpam-4961	123	12	number	number	NOUN
ejpam-4961	123	13	of	of	ADP
ejpam-4961	123	14	odd	odd	ADJ
ejpam-4961	123	15	3	3	NUM
ejpam-4961	123	16	-	-	PUNCT
ejpam-4961	123	17	almost	almost	ADV
ejpam-4961	123	18	primes	prime	NOUN
ejpam-4961	123	19	≤	≤	NUM
ejpam-4961	123	20	n.	n.	NOUN
ejpam-4961	123	21	theorem	theorem	NOUN
ejpam-4961	123	22	2	2	NUM
ejpam-4961	123	23	.	.	X
ejpam-4961	124	1	for	for	ADP
ejpam-4961	124	2	any	any	DET
ejpam-4961	124	3	natural	natural	ADJ
ejpam-4961	124	4	number	number	NOUN
ejpam-4961	124	5	n	n	CCONJ
ejpam-4961	124	6	,	,	PUNCT
ejpam-4961	124	7	the	the	DET
ejpam-4961	124	8	number	number	NOUN
ejpam-4961	124	9	of	of	ADP
ejpam-4961	124	10	odd	odd	ADJ
ejpam-4961	124	11	3	3	NUM
ejpam-4961	124	12	-	-	PUNCT
ejpam-4961	124	13	almost	almost	ADV
ejpam-4961	124	14	primes	prime	NOUN
ejpam-4961	124	15	is	be	AUX
ejpam-4961	124	16	given	give	VERB
ejpam-4961	124	17	by	by	ADP
ejpam-4961	124	18	π3(n	π3(n	NUM
ejpam-4961	124	19	)	)	PUNCT
ejpam-4961	124	20	=	=	PUNCT
ejpam-4961	125	1	s−1∑	s−1∑	NUM
ejpam-4961	125	2	j=2	j=2	PROPN
ejpam-4961	125	3	[	[	PUNCT
ejpam-4961	125	4	t∑	t∑	PUNCT
ejpam-4961	125	5	i=1	i=1	PROPN
ejpam-4961	125	6	πpj	πpj	NOUN
ejpam-4961	125	7	(	(	PUNCT
ejpam-4961	125	8	n	n	CCONJ
ejpam-4961	125	9	pjpi+j−1	pjpi+j−1	NOUN
ejpam-4961	125	10	)	)	PUNCT
ejpam-4961	125	11	−	−	NOUN
ejpam-4961	125	12	t(t−	t(t−	NOUN
ejpam-4961	125	13	1	1	NUM
ejpam-4961	125	14	)	)	PUNCT
ejpam-4961	125	15	2	2	NUM
ejpam-4961	125	16	]	]	PUNCT
ejpam-4961	125	17	(	(	PUNCT
ejpam-4961	125	18	8)	8)	NUM
ejpam-4961	125	19	where	where	SCONJ
ejpam-4961	125	20	t	t	NOUN
ejpam-4961	125	21	=	=	PUNCT
ejpam-4961	125	22	πpj	πpj	X
ejpam-4961	126	1	[	[	X
ejpam-4961	126	2	(	(	PUNCT
ejpam-4961	126	3	n	n	X
ejpam-4961	126	4	pj	pj	PROPN
ejpam-4961	126	5	)	)	PUNCT
ejpam-4961	126	6	1	1	NUM
ejpam-4961	126	7	2	2	NUM
ejpam-4961	126	8	]	]	PUNCT
ejpam-4961	126	9	,	,	PUNCT
ejpam-4961	126	10	s	s	X
ejpam-4961	126	11	is	be	AUX
ejpam-4961	126	12	the	the	DET
ejpam-4961	126	13	least	least	ADJ
ejpam-4961	126	14	value	value	NOUN
ejpam-4961	126	15	for	for	ADP
ejpam-4961	126	16	which	which	PRON
ejpam-4961	126	17	t	t	NOUN
ejpam-4961	126	18	=	=	SYM
ejpam-4961	126	19	0	0	NUM
ejpam-4961	126	20	,	,	PUNCT
ejpam-4961	126	21	πk(n	πk(n	PUNCT
ejpam-4961	126	22	)	)	PUNCT
ejpam-4961	126	23	is	be	AUX
ejpam-4961	126	24	the	the	DET
ejpam-4961	126	25	number	number	NOUN
ejpam-4961	126	26	of	of	ADP
ejpam-4961	126	27	primes	prime	NOUN
ejpam-4961	126	28	≥	≥	PROPN
ejpam-4961	126	29	k	k	NOUN
ejpam-4961	126	30	and	and	CCONJ
ejpam-4961	126	31	≤	≤	NUM
ejpam-4961	126	32	n	n	PROPN
ejpam-4961	126	33	and	and	CCONJ
ejpam-4961	126	34	pj	pj	PROPN
ejpam-4961	126	35	is	be	AUX
ejpam-4961	126	36	the	the	DET
ejpam-4961	126	37	jth	jth	PROPN
ejpam-4961	126	38	prime	prime	NOUN
ejpam-4961	126	39	.	.	PUNCT
ejpam-4961	127	1	proof	proof	NOUN
ejpam-4961	127	2	.	.	PUNCT
ejpam-4961	128	1	the	the	DET
ejpam-4961	128	2	proof	proof	NOUN
ejpam-4961	128	3	of	of	ADP
ejpam-4961	128	4	(	(	PUNCT
ejpam-4961	128	5	8)	8)	NUM
ejpam-4961	128	6	can	can	AUX
ejpam-4961	128	7	be	be	AUX
ejpam-4961	128	8	easily	easily	ADV
ejpam-4961	128	9	verified	verify	VERB
ejpam-4961	128	10	by	by	ADP
ejpam-4961	128	11	following	follow	VERB
ejpam-4961	128	12	the	the	DET
ejpam-4961	128	13	procedure	procedure	NOUN
ejpam-4961	128	14	in	in	ADP
ejpam-4961	128	15	the	the	DET
ejpam-4961	128	16	proof	proof	NOUN
ejpam-4961	128	17	of	of	ADP
ejpam-4961	128	18	formula	formula	NOUN
ejpam-4961	128	19	(	(	PUNCT
ejpam-4961	128	20	5	5	NUM
ejpam-4961	128	21	)	)	PUNCT
ejpam-4961	128	22	.	.	PUNCT
ejpam-4961	129	1	note	note	VERB
ejpam-4961	129	2	:	:	PUNCT
ejpam-4961	129	3	it	it	PRON
ejpam-4961	129	4	should	should	AUX
ejpam-4961	129	5	be	be	AUX
ejpam-4961	129	6	noted	note	VERB
ejpam-4961	129	7	that	that	SCONJ
ejpam-4961	129	8	formula	formula	NOUN
ejpam-4961	129	9	(	(	PUNCT
ejpam-4961	129	10	8)	8)	NUM
ejpam-4961	129	11	gives	give	VERB
ejpam-4961	129	12	the	the	DET
ejpam-4961	129	13	number	number	NOUN
ejpam-4961	129	14	of	of	ADP
ejpam-4961	129	15	only	only	ADV
ejpam-4961	129	16	odd	odd	ADJ
ejpam-4961	129	17	3	3	NUM
ejpam-4961	129	18	-	-	PUNCT
ejpam-4961	129	19	almost	almost	ADV
ejpam-4961	129	20	primes	prime	NOUN
ejpam-4961	129	21	and	and	CCONJ
ejpam-4961	129	22	requires	require	VERB
ejpam-4961	129	23	list	list	NOUN
ejpam-4961	129	24	of	of	ADP
ejpam-4961	129	25	primes	prime	NOUN
ejpam-4961	129	26	upto	upto	VERB
ejpam-4961	129	27	n	n	CCONJ
ejpam-4961	129	28	9	9	NUM
ejpam-4961	129	29	to	to	PART
ejpam-4961	129	30	find	find	VERB
ejpam-4961	129	31	value	value	NOUN
ejpam-4961	129	32	of	of	ADP
ejpam-4961	129	33	π3(n	π3(n	PROPN
ejpam-4961	129	34	)	)	PUNCT
ejpam-4961	129	35	,	,	PUNCT
ejpam-4961	129	36	which	which	PRON
ejpam-4961	129	37	is	be	AUX
ejpam-4961	129	38	a	a	DET
ejpam-4961	129	39	very	very	ADV
ejpam-4961	129	40	short	short	ADJ
ejpam-4961	129	41	interval	interval	NOUN
ejpam-4961	129	42	.	.	PUNCT
ejpam-4961	130	1	example	example	NOUN
ejpam-4961	131	1	3	3	X
ejpam-4961	131	2	.	.	PUNCT
ejpam-4961	131	3	find	find	VERB
ejpam-4961	131	4	the	the	DET
ejpam-4961	131	5	number	number	NOUN
ejpam-4961	131	6	of	of	ADP
ejpam-4961	131	7	odd	odd	ADJ
ejpam-4961	131	8	3	3	NUM
ejpam-4961	131	9	-	-	PUNCT
ejpam-4961	131	10	almost	almost	ADV
ejpam-4961	131	11	primes	prime	NOUN
ejpam-4961	131	12	up	up	ADP
ejpam-4961	131	13	to	to	ADP
ejpam-4961	131	14	1000	1000	NUM
ejpam-4961	131	15	,	,	PUNCT
ejpam-4961	131	16	i.e.	i.e.	X
ejpam-4961	131	17	,	,	PUNCT
ejpam-4961	131	18	π(3)(1000	π(3)(1000	NUM
ejpam-4961	131	19	)	)	PUNCT
ejpam-4961	131	20	.	.	PUNCT
ejpam-4961	132	1	we	we	PRON
ejpam-4961	132	2	know	know	VERB
ejpam-4961	132	3	that	that	SCONJ
ejpam-4961	132	4	,	,	PUNCT
ejpam-4961	132	5	π3(n	π3(n	X
ejpam-4961	132	6	)	)	PUNCT
ejpam-4961	132	7	=	=	PUNCT
ejpam-4961	133	1	s−1∑	s−1∑	NUM
ejpam-4961	133	2	j=2	j=2	PROPN
ejpam-4961	134	1	[	[	PUNCT
ejpam-4961	134	2	t∑	t∑	PUNCT
ejpam-4961	134	3	i=1	i=1	PROPN
ejpam-4961	134	4	πpj	πpj	NOUN
ejpam-4961	134	5	(	(	PUNCT
ejpam-4961	134	6	n	n	CCONJ
ejpam-4961	134	7	pjpi+j−1	pjpi+j−1	NOUN
ejpam-4961	134	8	)	)	PUNCT
ejpam-4961	134	9	−	−	NOUN
ejpam-4961	134	10	t(t−	t(t−	NOUN
ejpam-4961	134	11	1	1	NUM
ejpam-4961	134	12	)	)	PUNCT
ejpam-4961	134	13	2	2	NUM
ejpam-4961	134	14	]	]	PUNCT
ejpam-4961	134	15	.	.	PUNCT
ejpam-4961	135	1	here	here	ADV
ejpam-4961	135	2	n	n	PROPN
ejpam-4961	135	3	=	=	SYM
ejpam-4961	135	4	1000	1000	NUM
ejpam-4961	135	5	.	.	PUNCT
ejpam-4961	136	1	the	the	DET
ejpam-4961	136	2	prerequisite	prerequisite	NOUN
ejpam-4961	136	3	here	here	ADV
ejpam-4961	136	4	is	be	AUX
ejpam-4961	136	5	only	only	ADV
ejpam-4961	136	6	the	the	DET
ejpam-4961	136	7	list	list	NOUN
ejpam-4961	136	8	of	of	ADP
ejpam-4961	136	9	primes	prime	NOUN
ejpam-4961	136	10	≤	≤	NUM
ejpam-4961	136	11	111	111	NUM
ejpam-4961	136	12	.	.	PUNCT
ejpam-4961	137	1	for	for	ADP
ejpam-4961	137	2	j	j	PROPN
ejpam-4961	137	3	=	=	SYM
ejpam-4961	137	4	2	2	NUM
ejpam-4961	137	5	,	,	PUNCT
ejpam-4961	137	6	t	t	NOUN
ejpam-4961	137	7	=	=	PUNCT
ejpam-4961	137	8	π3	π3	PROPN
ejpam-4961	137	9	[	[	X
ejpam-4961	137	10	(	(	PUNCT
ejpam-4961	137	11	1000	1000	NUM
ejpam-4961	137	12	3	3	NUM
ejpam-4961	137	13	)	)	PUNCT
ejpam-4961	137	14	1	1	NUM
ejpam-4961	137	15	2	2	NUM
ejpam-4961	137	16	]	]	PUNCT
ejpam-4961	137	17	=	=	SYM
ejpam-4961	137	18	π3(18.2	π3(18.2	PROPN
ejpam-4961	137	19	)	)	PUNCT
ejpam-4961	137	20	=	=	SYM
ejpam-4961	138	1	6	6	X
ejpam-4961	138	2	.	.	PUNCT
ejpam-4961	139	1	j	j	PROPN
ejpam-4961	139	2	=	=	SYM
ejpam-4961	139	3	3	3	NUM
ejpam-4961	139	4	,	,	PUNCT
ejpam-4961	139	5	t	t	NOUN
ejpam-4961	139	6	=	=	SYM
ejpam-4961	139	7	π5	π5	PROPN
ejpam-4961	140	1	[	[	X
ejpam-4961	140	2	(	(	PUNCT
ejpam-4961	140	3	1000	1000	NUM
ejpam-4961	140	4	5	5	NUM
ejpam-4961	140	5	)	)	PUNCT
ejpam-4961	140	6	1	1	NUM
ejpam-4961	140	7	2	2	NUM
ejpam-4961	140	8	]	]	PUNCT
ejpam-4961	140	9	=	=	SYM
ejpam-4961	140	10	π5(14.14	π5(14.14	X
ejpam-4961	140	11	)	)	PUNCT
ejpam-4961	140	12	=	=	SYM
ejpam-4961	141	1	4	4	X
ejpam-4961	141	2	.	.	X
ejpam-4961	141	3	j	j	PROPN
ejpam-4961	141	4	=	=	SYM
ejpam-4961	141	5	4	4	NUM
ejpam-4961	141	6	,	,	PUNCT
ejpam-4961	141	7	t	t	NOUN
ejpam-4961	141	8	=	=	SYM
ejpam-4961	141	9	π7	π7	PROPN
ejpam-4961	142	1	[	[	X
ejpam-4961	142	2	(	(	PUNCT
ejpam-4961	142	3	1000	1000	NUM
ejpam-4961	142	4	7	7	NUM
ejpam-4961	142	5	)	)	PUNCT
ejpam-4961	142	6	1	1	NUM
ejpam-4961	142	7	2	2	NUM
ejpam-4961	142	8	]	]	PUNCT
ejpam-4961	142	9	=	=	PUNCT
ejpam-4961	142	10	π7(11.9	π7(11.9	ADJ
ejpam-4961	142	11	)	)	PUNCT
ejpam-4961	142	12	=	=	SYM
ejpam-4961	142	13	2	2	X
ejpam-4961	142	14	.	.	PUNCT
ejpam-4961	142	15	m.	m.	NOUN
ejpam-4961	142	16	m.	m.	PROPN
ejpam-4961	142	17	m.	m.	PROPN
ejpam-4961	142	18	jaradat	jaradat	PROPN
ejpam-4961	142	19	et	et	PROPN
ejpam-4961	142	20	al	al	PROPN
ejpam-4961	142	21	.	.	PUNCT
ejpam-4961	142	22	/	/	SYM
ejpam-4961	142	23	eur	eur	PROPN
ejpam-4961	142	24	.	.	PUNCT
ejpam-4961	143	1	j.	j.	PROPN
ejpam-4961	143	2	pure	pure	PROPN
ejpam-4961	143	3	appl	appl	PROPN
ejpam-4961	143	4	.	.	PROPN
ejpam-4961	143	5	math	math	PROPN
ejpam-4961	143	6	,	,	PUNCT
ejpam-4961	143	7	17	17	NUM
ejpam-4961	143	8	(	(	PUNCT
ejpam-4961	143	9	2	2	NUM
ejpam-4961	143	10	)	)	PUNCT
ejpam-4961	143	11	(	(	PUNCT
ejpam-4961	143	12	2024	2024	NUM
ejpam-4961	143	13	)	)	PUNCT
ejpam-4961	143	14	,	,	PUNCT
ejpam-4961	143	15	1146	1146	NUM
ejpam-4961	143	16	-	-	SYM
ejpam-4961	143	17	1154	1154	NUM
ejpam-4961	143	18	1152	1152	NUM
ejpam-4961	143	19	j	j	PROPN
ejpam-4961	143	20	=	=	SYM
ejpam-4961	143	21	5	5	NUM
ejpam-4961	143	22	,	,	PUNCT
ejpam-4961	143	23	t	t	NOUN
ejpam-4961	143	24	=	=	SYM
ejpam-4961	143	25	π11	π11	NOUN
ejpam-4961	144	1	[	[	X
ejpam-4961	144	2	(	(	PUNCT
ejpam-4961	144	3	1000	1000	NUM
ejpam-4961	144	4	11	11	NUM
ejpam-4961	144	5	)	)	PUNCT
ejpam-4961	144	6	1	1	NUM
ejpam-4961	144	7	2	2	NUM
ejpam-4961	144	8	]	]	PUNCT
ejpam-4961	144	9	=	=	SYM
ejpam-4961	144	10	π11(9.5	π11(9.5	X
ejpam-4961	144	11	)	)	PUNCT
ejpam-4961	144	12	=	=	SYM
ejpam-4961	144	13	0	0	X
ejpam-4961	144	14	.	.	PUNCT
ejpam-4961	145	1	thus	thus	ADV
ejpam-4961	145	2	,	,	PUNCT
ejpam-4961	145	3	s	s	VERB
ejpam-4961	145	4	=	=	SYM
ejpam-4961	145	5	5	5	X
ejpam-4961	145	6	.	.	PUNCT
ejpam-4961	145	7	∴	∴	PROPN
ejpam-4961	145	8	π(3)(1000	π(3)(1000	NUM
ejpam-4961	145	9	)	)	PUNCT
ejpam-4961	146	1	=	=	SYM
ejpam-4961	146	2	4∑	4∑	NUM
ejpam-4961	146	3	j=2	j=2	NOUN
ejpam-4961	147	1	[	[	PUNCT
ejpam-4961	147	2	t∑	t∑	PUNCT
ejpam-4961	147	3	i=1	i=1	PROPN
ejpam-4961	147	4	πpj	πpj	NOUN
ejpam-4961	147	5	(	(	PUNCT
ejpam-4961	147	6	n	n	CCONJ
ejpam-4961	147	7	pjpi+j−1	pjpi+j−1	NOUN
ejpam-4961	147	8	)	)	PUNCT
ejpam-4961	147	9	−	−	NOUN
ejpam-4961	147	10	t(t−	t(t−	NOUN
ejpam-4961	147	11	1	1	NUM
ejpam-4961	147	12	)	)	PUNCT
ejpam-4961	147	13	2	2	NUM
ejpam-4961	147	14	]	]	PUNCT
ejpam-4961	147	15	=	=	PUNCT
ejpam-4961	148	1	[	[	PUNCT
ejpam-4961	148	2	6∑	6∑	NUM
ejpam-4961	148	3	i=1	i=1	X
ejpam-4961	149	1	πp2	πp2	PROPN
ejpam-4961	149	2	(	(	PUNCT
ejpam-4961	149	3	n	n	CCONJ
ejpam-4961	149	4	p2pi+1	p2pi+1	NOUN
ejpam-4961	149	5	)	)	PUNCT
ejpam-4961	149	6	−	−	PROPN
ejpam-4961	150	1	6.5	6.5	NUM
ejpam-4961	150	2	2	2	NUM
ejpam-4961	150	3	]	]	PUNCT
ejpam-4961	151	1	+	+	CCONJ
ejpam-4961	151	2	[	[	PUNCT
ejpam-4961	151	3	4∑	4∑	NOUN
ejpam-4961	151	4	i=1	i=1	PRON
ejpam-4961	151	5	πp3	πp3	PROPN
ejpam-4961	151	6	(	(	PUNCT
ejpam-4961	151	7	n	n	CCONJ
ejpam-4961	151	8	p3pi+2	p3pi+2	NOUN
ejpam-4961	151	9	)	)	PUNCT
ejpam-4961	151	10	−	−	PROPN
ejpam-4961	151	11	4.3	4.3	NUM
ejpam-4961	151	12	2	2	NUM
ejpam-4961	151	13	]	]	PUNCT
ejpam-4961	152	1	+	+	CCONJ
ejpam-4961	152	2	[	[	PUNCT
ejpam-4961	152	3	2∑	2∑	NUM
ejpam-4961	152	4	i=1	i=1	PROPN
ejpam-4961	152	5	πp4	πp4	PROPN
ejpam-4961	152	6	(	(	PUNCT
ejpam-4961	152	7	n	n	NOUN
ejpam-4961	152	8	p4pi+3	p4pi+3	NOUN
ejpam-4961	152	9	)	)	PUNCT
ejpam-4961	152	10	−	−	ADP
ejpam-4961	152	11	2.1	2.1	NUM
ejpam-4961	152	12	2	2	NUM
ejpam-4961	152	13	]	]	PUNCT
ejpam-4961	152	14	=	=	PUNCT
ejpam-4961	153	1	[	[	PUNCT
ejpam-4961	153	2	6∑	6∑	NUM
ejpam-4961	153	3	i=1	i=1	PROPN
ejpam-4961	153	4	π3	π3	PROPN
ejpam-4961	153	5	(	(	PUNCT
ejpam-4961	153	6	333.3	333.3	NUM
ejpam-4961	153	7	pi+1	pi+1	NOUN
ejpam-4961	153	8	)	)	PUNCT
ejpam-4961	153	9	−	−	PROPN
ejpam-4961	153	10	15	15	NUM
ejpam-4961	153	11	]	]	PUNCT
ejpam-4961	154	1	+	+	CCONJ
ejpam-4961	154	2	[	[	PUNCT
ejpam-4961	154	3	4∑	4∑	NUM
ejpam-4961	154	4	i=1	i=1	PROPN
ejpam-4961	154	5	π5	π5	PROPN
ejpam-4961	154	6	(	(	PUNCT
ejpam-4961	154	7	200	200	NUM
ejpam-4961	154	8	pi+2	pi+2	NOUN
ejpam-4961	154	9	)	)	PUNCT
ejpam-4961	154	10	−	−	PROPN
ejpam-4961	154	11	6	6	NUM
ejpam-4961	154	12	]	]	PUNCT
ejpam-4961	155	1	+	+	CCONJ
ejpam-4961	155	2	[	[	PUNCT
ejpam-4961	155	3	2∑	2∑	NUM
ejpam-4961	155	4	i=1	i=1	PROPN
ejpam-4961	155	5	π7	π7	PROPN
ejpam-4961	155	6	(	(	PUNCT
ejpam-4961	155	7	142.8	142.8	NUM
ejpam-4961	155	8	pi+3	pi+3	NOUN
ejpam-4961	155	9	)	)	PUNCT
ejpam-4961	155	10	−	−	PROPN
ejpam-4961	155	11	1	1	NUM
ejpam-4961	155	12	]	]	PUNCT
ejpam-4961	155	13	=	=	SYM
ejpam-4961	155	14	94	94	NUM
ejpam-4961	155	15	.	.	PUNCT
ejpam-4961	155	16	number	number	NOUN
ejpam-4961	155	17	of	of	ADP
ejpam-4961	155	18	odd	odd	ADJ
ejpam-4961	155	19	4	4	NUM
ejpam-4961	155	20	-	-	PUNCT
ejpam-4961	155	21	almost	almost	ADV
ejpam-4961	155	22	primes	prime	NOUN
ejpam-4961	155	23	≤	≤	NUM
ejpam-4961	155	24	n	n	CCONJ
ejpam-4961	155	25	,	,	PUNCT
ejpam-4961	155	26	i.e.	i.e.	X
ejpam-4961	155	27	,π(4)(n	,π(4)(n	PUNCT
ejpam-4961	155	28	)	)	PUNCT
ejpam-4961	155	29	.	.	PUNCT
ejpam-4961	156	1	in	in	ADP
ejpam-4961	156	2	this	this	DET
ejpam-4961	156	3	section	section	NOUN
ejpam-4961	156	4	,	,	PUNCT
ejpam-4961	156	5	we	we	PRON
ejpam-4961	156	6	give	give	VERB
ejpam-4961	156	7	the	the	DET
ejpam-4961	156	8	formula	formula	NOUN
ejpam-4961	156	9	for	for	ADP
ejpam-4961	156	10	finding	find	VERB
ejpam-4961	156	11	the	the	DET
ejpam-4961	156	12	number	number	NOUN
ejpam-4961	156	13	of	of	ADP
ejpam-4961	156	14	odd	odd	ADJ
ejpam-4961	156	15	4	4	NUM
ejpam-4961	156	16	-	-	PUNCT
ejpam-4961	156	17	almost	almost	ADV
ejpam-4961	156	18	primes	prime	NOUN
ejpam-4961	156	19	≤	≤	NUM
ejpam-4961	156	20	n.	n.	NOUN
ejpam-4961	156	21	theorem	theorem	VERB
ejpam-4961	156	22	3	3	NUM
ejpam-4961	156	23	.	.	X
ejpam-4961	157	1	for	for	ADP
ejpam-4961	157	2	any	any	DET
ejpam-4961	157	3	natural	natural	ADJ
ejpam-4961	157	4	number	number	NOUN
ejpam-4961	157	5	n	n	CCONJ
ejpam-4961	157	6	,	,	PUNCT
ejpam-4961	157	7	the	the	DET
ejpam-4961	157	8	number	number	NOUN
ejpam-4961	157	9	of	of	ADP
ejpam-4961	157	10	odd	odd	ADJ
ejpam-4961	157	11	4	4	NUM
ejpam-4961	157	12	-	-	PUNCT
ejpam-4961	157	13	almost	almost	ADV
ejpam-4961	157	14	primes	prime	NOUN
ejpam-4961	157	15	is	be	AUX
ejpam-4961	157	16	given	give	VERB
ejpam-4961	157	17	by	by	ADP
ejpam-4961	157	18	π4(n	π4(n	PROPN
ejpam-4961	157	19	)	)	PUNCT
ejpam-4961	157	20	=	=	PUNCT
ejpam-4961	158	1	l−1∑	l−1∑	ADJ
ejpam-4961	158	2	k=2	k=2	PROPN
ejpam-4961	159	1	[	[	X
ejpam-4961	159	2	m−1∑	m−1∑	NUM
ejpam-4961	159	3	j	j	PROPN
ejpam-4961	159	4	=	=	PROPN
ejpam-4961	159	5	k	k	X
ejpam-4961	159	6	[	[	PUNCT
ejpam-4961	159	7	t∑	t∑	PUNCT
ejpam-4961	159	8	i=1	i=1	PROPN
ejpam-4961	159	9	πpj	πpj	NOUN
ejpam-4961	159	10	(	(	PUNCT
ejpam-4961	159	11	n	n	CCONJ
ejpam-4961	159	12	pkpjpi+j−1	pkpjpi+j−1	NOUN
ejpam-4961	159	13	)	)	PUNCT
ejpam-4961	159	14	−	−	ADP
ejpam-4961	159	15	t(t−	t(t−	NOUN
ejpam-4961	159	16	1	1	NUM
ejpam-4961	159	17	)	)	PUNCT
ejpam-4961	159	18	2	2	NUM
ejpam-4961	159	19	]	]	PUNCT
ejpam-4961	159	20	]	]	X
ejpam-4961	159	21	(	(	PUNCT
ejpam-4961	159	22	9	9	NUM
ejpam-4961	159	23	)	)	PUNCT
ejpam-4961	159	24	where	where	SCONJ
ejpam-4961	159	25	t	t	NOUN
ejpam-4961	159	26	=	=	PRON
ejpam-4961	159	27	πpj	πpj	X
ejpam-4961	160	1	[	[	X
ejpam-4961	160	2	(	(	PUNCT
ejpam-4961	160	3	n	n	ADP
ejpam-4961	160	4	pkpj	pkpj	NOUN
ejpam-4961	160	5	)	)	PUNCT
ejpam-4961	160	6	1	1	NUM
ejpam-4961	160	7	2	2	NUM
ejpam-4961	160	8	]	]	PUNCT
ejpam-4961	160	9	,	,	PUNCT
ejpam-4961	160	10	l	l	NOUN
ejpam-4961	160	11	is	be	AUX
ejpam-4961	160	12	the	the	DET
ejpam-4961	160	13	least	least	ADJ
ejpam-4961	160	14	value	value	NOUN
ejpam-4961	160	15	for	for	ADP
ejpam-4961	160	16	which	which	PRON
ejpam-4961	160	17	πpl	πpl	NOUN
ejpam-4961	160	18	[	[	X
ejpam-4961	160	19	(	(	PUNCT
ejpam-4961	160	20	n	n	NUM
ejpam-4961	160	21	3pl	3pl	ADJ
ejpam-4961	160	22	)	)	PUNCT
ejpam-4961	160	23	1	1	NUM
ejpam-4961	160	24	2	2	NUM
ejpam-4961	160	25	]	]	PUNCT
ejpam-4961	160	26	=	=	SYM
ejpam-4961	160	27	0	0	NUM
ejpam-4961	160	28	,	,	PUNCT
ejpam-4961	160	29	m	m	VERB
ejpam-4961	160	30	is	be	AUX
ejpam-4961	160	31	the	the	DET
ejpam-4961	160	32	least	least	ADJ
ejpam-4961	160	33	value	value	NOUN
ejpam-4961	160	34	for	for	ADP
ejpam-4961	160	35	which	which	DET
ejpam-4961	160	36	πpm	πpm	NOUN
ejpam-4961	160	37	[	[	X
ejpam-4961	160	38	(	(	PUNCT
ejpam-4961	160	39	n	n	CCONJ
ejpam-4961	160	40	pm	pm	NOUN
ejpam-4961	160	41	)	)	PUNCT
ejpam-4961	160	42	1	1	NUM
ejpam-4961	160	43	2	2	NUM
ejpam-4961	160	44	]	]	PUNCT
ejpam-4961	160	45	=	=	SYM
ejpam-4961	160	46	0	0	NUM
ejpam-4961	160	47	,	,	PUNCT
ejpam-4961	160	48	πk(n	πk(n	PUNCT
ejpam-4961	160	49	)	)	PUNCT
ejpam-4961	160	50	is	be	AUX
ejpam-4961	160	51	the	the	DET
ejpam-4961	160	52	number	number	NOUN
ejpam-4961	160	53	of	of	ADP
ejpam-4961	160	54	primes	prime	NOUN
ejpam-4961	160	55	≥	≥	PROPN
ejpam-4961	160	56	k	k	NOUN
ejpam-4961	160	57	and	and	CCONJ
ejpam-4961	160	58	≤	≤	NUM
ejpam-4961	160	59	n	n	PROPN
ejpam-4961	160	60	and	and	CCONJ
ejpam-4961	160	61	pj	pj	PROPN
ejpam-4961	160	62	is	be	AUX
ejpam-4961	160	63	the	the	DET
ejpam-4961	160	64	jth	jth	PROPN
ejpam-4961	160	65	prime	prime	NOUN
ejpam-4961	160	66	.	.	PUNCT
ejpam-4961	161	1	proof	proof	NOUN
ejpam-4961	161	2	.	.	PUNCT
ejpam-4961	162	1	the	the	DET
ejpam-4961	162	2	proof	proof	NOUN
ejpam-4961	162	3	of	of	ADP
ejpam-4961	162	4	(	(	PUNCT
ejpam-4961	162	5	9	9	NUM
ejpam-4961	162	6	)	)	PUNCT
ejpam-4961	162	7	can	can	AUX
ejpam-4961	162	8	be	be	AUX
ejpam-4961	162	9	easily	easily	ADV
ejpam-4961	162	10	verified	verify	VERB
ejpam-4961	162	11	by	by	ADP
ejpam-4961	162	12	following	follow	VERB
ejpam-4961	162	13	the	the	DET
ejpam-4961	162	14	procedure	procedure	NOUN
ejpam-4961	162	15	in	in	ADP
ejpam-4961	162	16	the	the	DET
ejpam-4961	162	17	proof	proof	NOUN
ejpam-4961	162	18	of	of	ADP
ejpam-4961	162	19	formula	formula	NOUN
ejpam-4961	162	20	(	(	PUNCT
ejpam-4961	162	21	5	5	NUM
ejpam-4961	162	22	)	)	PUNCT
ejpam-4961	162	23	and	and	CCONJ
ejpam-4961	162	24	(	(	PUNCT
ejpam-4961	162	25	8)	8)	NUM
ejpam-4961	162	26	.	.	PUNCT
ejpam-4961	163	1	note	note	NOUN
ejpam-4961	163	2	:	:	PUNCT
ejpam-4961	163	3	it	it	PRON
ejpam-4961	163	4	is	be	AUX
ejpam-4961	163	5	important	important	ADJ
ejpam-4961	163	6	noting	note	VERB
ejpam-4961	163	7	that	that	SCONJ
ejpam-4961	163	8	formula	formula	NOUN
ejpam-4961	163	9	(	(	PUNCT
ejpam-4961	163	10	9	9	X
ejpam-4961	163	11	)	)	PUNCT
ejpam-4961	163	12	gives	give	VERB
ejpam-4961	163	13	the	the	DET
ejpam-4961	163	14	number	number	NOUN
ejpam-4961	163	15	of	of	ADP
ejpam-4961	163	16	only	only	ADV
ejpam-4961	163	17	odd	odd	ADJ
ejpam-4961	163	18	4	4	NUM
ejpam-4961	163	19	-	-	PUNCT
ejpam-4961	163	20	almost	almost	ADV
ejpam-4961	163	21	primes	prime	NOUN
ejpam-4961	163	22	and	and	CCONJ
ejpam-4961	163	23	requires	require	VERB
ejpam-4961	163	24	list	list	NOUN
ejpam-4961	163	25	of	of	ADP
ejpam-4961	163	26	primes	prime	NOUN
ejpam-4961	163	27	up	up	ADP
ejpam-4961	163	28	to	to	ADP
ejpam-4961	163	29	n	n	PRON
ejpam-4961	163	30	27	27	NUM
ejpam-4961	163	31	,	,	PUNCT
ejpam-4961	163	32	to	to	PART
ejpam-4961	163	33	find	find	VERB
ejpam-4961	163	34	value	value	NOUN
ejpam-4961	163	35	of	of	ADP
ejpam-4961	163	36	π4(n	π4(n	PROPN
ejpam-4961	163	37	)	)	PUNCT
ejpam-4961	163	38	,	,	PUNCT
ejpam-4961	163	39	which	which	PRON
ejpam-4961	163	40	is	be	AUX
ejpam-4961	163	41	a	a	DET
ejpam-4961	163	42	very	very	ADV
ejpam-4961	163	43	short	short	ADJ
ejpam-4961	163	44	interval	interval	NOUN
ejpam-4961	163	45	.	.	PUNCT
ejpam-4961	164	1	prime	prime	ADJ
ejpam-4961	164	2	counting	counting	NOUN
ejpam-4961	164	3	function	function	NOUN
ejpam-4961	164	4	using	use	VERB
ejpam-4961	164	5	odd	odd	ADJ
ejpam-4961	164	6	k	k	ADJ
ejpam-4961	164	7	-	-	PUNCT
ejpam-4961	164	8	almost	almost	ADV
ejpam-4961	164	9	primes	prime	NOUN
ejpam-4961	164	10	so	so	ADV
ejpam-4961	164	11	far	far	ADV
ejpam-4961	164	12	,	,	PUNCT
ejpam-4961	164	13	we	we	PRON
ejpam-4961	164	14	have	have	AUX
ejpam-4961	164	15	discussed	discuss	VERB
ejpam-4961	164	16	oddk	oddk	NOUN
ejpam-4961	164	17	-	-	PUNCT
ejpam-4961	164	18	almost	almost	ADV
ejpam-4961	164	19	primes	prime	NOUN
ejpam-4961	164	20	and	and	CCONJ
ejpam-4961	164	21	construction	construction	NOUN
ejpam-4961	164	22	of	of	ADP
ejpam-4961	164	23	formula	formula	NOUN
ejpam-4961	164	24	’s	’	VERB
ejpam-4961	164	25	for	for	ADP
ejpam-4961	164	26	getting	get	VERB
ejpam-4961	164	27	their	their	PRON
ejpam-4961	164	28	number	number	NOUN
ejpam-4961	164	29	.	.	PUNCT
ejpam-4961	165	1	using	use	VERB
ejpam-4961	165	2	that	that	PRON
ejpam-4961	165	3	,	,	PUNCT
ejpam-4961	165	4	the	the	DET
ejpam-4961	165	5	prime	prime	ADJ
ejpam-4961	165	6	counting	counting	NOUN
ejpam-4961	165	7	function	function	NOUN
ejpam-4961	165	8	for	for	ADP
ejpam-4961	165	9	any	any	DET
ejpam-4961	165	10	n	n	NOUN
ejpam-4961	165	11	can	can	AUX
ejpam-4961	165	12	be	be	AUX
ejpam-4961	165	13	given	give	VERB
ejpam-4961	165	14	by	by	ADP
ejpam-4961	165	15	π(n	π(n	PROPN
ejpam-4961	165	16	)	)	PUNCT
ejpam-4961	166	1	=	=	SYM
ejpam-4961	166	2	n−	n−	NOUN
ejpam-4961	166	3	(	(	PUNCT
ejpam-4961	166	4	number	number	NOUN
ejpam-4961	166	5	of	of	ADP
ejpam-4961	166	6	even	even	ADV
ejpam-4961	166	7	’s	’s	NOUN
ejpam-4961	166	8	≤	≤	ADJ
ejpam-4961	166	9	n)−	n)−	NOUN
ejpam-4961	166	10	(	(	PUNCT
ejpam-4961	166	11	number	number	NOUN
ejpam-4961	166	12	of	of	ADP
ejpam-4961	166	13	odd	odd	ADJ
ejpam-4961	166	14	k	k	ADJ
ejpam-4961	166	15	-	-	PUNCT
ejpam-4961	166	16	almost	almost	ADV
ejpam-4961	166	17	prime	prime	NOUN
ejpam-4961	166	18	’s	’s	PART
ejpam-4961	166	19	≤	≤	NUM
ejpam-4961	166	20	n.	n.	NOUN
ejpam-4961	166	21	remember	remember	VERB
ejpam-4961	166	22	that	that	SCONJ
ejpam-4961	166	23	2	2	NUM
ejpam-4961	166	24	is	be	AUX
ejpam-4961	166	25	even	even	ADV
ejpam-4961	166	26	but	but	CCONJ
ejpam-4961	166	27	prime	prime	ADJ
ejpam-4961	166	28	,	,	PUNCT
ejpam-4961	166	29	while	while	SCONJ
ejpam-4961	166	30	1	1	NUM
ejpam-4961	166	31	is	be	AUX
ejpam-4961	166	32	odd	odd	ADJ
ejpam-4961	166	33	but	but	CCONJ
ejpam-4961	166	34	not	not	PART
ejpam-4961	166	35	prime	prime	ADJ
ejpam-4961	166	36	.	.	PUNCT
ejpam-4961	167	1	references	reference	NOUN
ejpam-4961	167	2	1153	1153	NUM
ejpam-4961	167	3	example	example	NOUN
ejpam-4961	167	4	4	4	NUM
ejpam-4961	167	5	.	.	PUNCT
ejpam-4961	167	6	find	find	VERB
ejpam-4961	167	7	the	the	DET
ejpam-4961	167	8	number	number	NOUN
ejpam-4961	167	9	of	of	ADP
ejpam-4961	167	10	primes	prime	NOUN
ejpam-4961	167	11	≤	≤	NUM
ejpam-4961	167	12	100	100	NUM
ejpam-4961	167	13	.	.	PUNCT
ejpam-4961	168	1	here	here	ADV
ejpam-4961	168	2	the	the	DET
ejpam-4961	168	3	prerequisite	prerequisite	NOUN
ejpam-4961	168	4	is	be	AUX
ejpam-4961	168	5	only	only	ADV
ejpam-4961	168	6	the	the	DET
ejpam-4961	168	7	list	list	NOUN
ejpam-4961	168	8	of	of	ADP
ejpam-4961	168	9	primes	prime	NOUN
ejpam-4961	168	10	≤	≤	NUM
ejpam-4961	168	11	33	33	NUM
ejpam-4961	168	12	.	.	PUNCT
ejpam-4961	169	1	since	since	SCONJ
ejpam-4961	169	2	up	up	ADP
ejpam-4961	169	3	to	to	PART
ejpam-4961	169	4	100	100	NUM
ejpam-4961	169	5	there	there	PRON
ejpam-4961	169	6	are	be	VERB
ejpam-4961	169	7	50	50	NUM
ejpam-4961	169	8	even	even	ADV
ejpam-4961	169	9	numbers	number	NOUN
ejpam-4961	169	10	,	,	PUNCT
ejpam-4961	169	11	19	19	NUM
ejpam-4961	169	12	odd	odd	ADJ
ejpam-4961	169	13	2	2	NUM
ejpam-4961	169	14	-	-	PUNCT
ejpam-4961	169	15	almost	almost	ADV
ejpam-4961	169	16	primes	prime	NOUN
ejpam-4961	169	17	,	,	PUNCT
ejpam-4961	169	18	5	5	NUM
ejpam-4961	169	19	odd	odd	ADJ
ejpam-4961	169	20	3	3	NUM
ejpam-4961	169	21	-	-	PUNCT
ejpam-4961	169	22	almost	almost	ADV
ejpam-4961	169	23	primes	prime	NOUN
ejpam-4961	169	24	and	and	CCONJ
ejpam-4961	169	25	only	only	ADV
ejpam-4961	169	26	1	1	NUM
ejpam-4961	169	27	odd	odd	ADJ
ejpam-4961	169	28	4	4	NUM
ejpam-4961	169	29	-	-	PUNCT
ejpam-4961	169	30	almost	almost	ADV
ejpam-4961	169	31	prime	prime	ADJ
ejpam-4961	169	32	.	.	PUNCT
ejpam-4961	170	1	hence	hence	ADV
ejpam-4961	170	2	,	,	PUNCT
ejpam-4961	170	3	π(100	π(100	NOUN
ejpam-4961	170	4	)	)	PUNCT
ejpam-4961	170	5	=	=	SYM
ejpam-4961	171	1	100−	100−	NUM
ejpam-4961	171	2	50−	50−	NUM
ejpam-4961	171	3	(	(	PUNCT
ejpam-4961	171	4	19	19	NUM
ejpam-4961	171	5	+	+	NUM
ejpam-4961	171	6	5	5	NUM
ejpam-4961	171	7	+	+	CCONJ
ejpam-4961	171	8	1	1	NUM
ejpam-4961	171	9	)	)	PUNCT
ejpam-4961	171	10	=	=	SYM
ejpam-4961	171	11	25	25	NUM
ejpam-4961	171	12	.	.	PUNCT
ejpam-4961	172	1	it	it	PRON
ejpam-4961	172	2	is	be	AUX
ejpam-4961	172	3	the	the	DET
ejpam-4961	172	4	exact	exact	ADJ
ejpam-4961	172	5	number	number	NOUN
ejpam-4961	172	6	of	of	ADP
ejpam-4961	172	7	primes	prime	NOUN
ejpam-4961	172	8	≤	≤	NUM
ejpam-4961	172	9	100	100	NUM
ejpam-4961	172	10	.	.	PUNCT
ejpam-4961	173	1	3	3	X
ejpam-4961	173	2	.	.	X
ejpam-4961	173	3	conclusions	conclusion	NOUN
ejpam-4961	173	4	an	an	DET
ejpam-4961	173	5	extensive	extensive	ADJ
ejpam-4961	173	6	examination	examination	NOUN
ejpam-4961	173	7	of	of	ADP
ejpam-4961	173	8	the	the	DET
ejpam-4961	173	9	idea	idea	NOUN
ejpam-4961	173	10	of	of	ADP
ejpam-4961	173	11	k	k	ADJ
ejpam-4961	173	12	-	-	PUNCT
ejpam-4961	173	13	almost	almost	ADV
ejpam-4961	173	14	prime	prime	ADJ
ejpam-4961	173	15	numbers	number	NOUN
ejpam-4961	173	16	is	be	AUX
ejpam-4961	173	17	provided	provide	VERB
ejpam-4961	173	18	in	in	ADP
ejpam-4961	173	19	this	this	DET
ejpam-4961	173	20	article	article	NOUN
ejpam-4961	173	21	.	.	PUNCT
ejpam-4961	174	1	it	it	PRON
ejpam-4961	174	2	demonstrates	demonstrate	VERB
ejpam-4961	174	3	the	the	DET
ejpam-4961	174	4	essential	essential	ADJ
ejpam-4961	174	5	circumstances	circumstance	NOUN
ejpam-4961	174	6	for	for	ADP
ejpam-4961	174	7	odd	odd	ADJ
ejpam-4961	174	8	k	k	ADJ
ejpam-4961	174	9	-	-	PUNCT
ejpam-4961	174	10	almost	almost	ADV
ejpam-4961	174	11	prime	prime	ADJ
ejpam-4961	174	12	numbers	number	NOUN
ejpam-4961	174	13	to	to	PART
ejpam-4961	174	14	exist	exist	VERB
ejpam-4961	174	15	and	and	CCONJ
ejpam-4961	174	16	creates	create	VERB
ejpam-4961	174	17	a	a	DET
ejpam-4961	174	18	formula	formula	NOUN
ejpam-4961	174	19	for	for	ADP
ejpam-4961	174	20	counting	count	VERB
ejpam-4961	174	21	these	these	DET
ejpam-4961	174	22	numbers	number	NOUN
ejpam-4961	174	23	up	up	ADP
ejpam-4961	174	24	to	to	ADP
ejpam-4961	174	25	a	a	DET
ejpam-4961	174	26	certain	certain	ADJ
ejpam-4961	174	27	limit	limit	NOUN
ejpam-4961	174	28	while	while	SCONJ
ejpam-4961	174	29	taking	take	VERB
ejpam-4961	174	30	different	different	ADJ
ejpam-4961	174	31	values	value	NOUN
ejpam-4961	174	32	of	of	ADP
ejpam-4961	174	33	k	k	PROPN
ejpam-4961	174	34	into	into	ADP
ejpam-4961	174	35	consideration	consideration	NOUN
ejpam-4961	174	36	.	.	PUNCT
ejpam-4961	175	1	the	the	DET
ejpam-4961	175	2	precise	precise	ADJ
ejpam-4961	175	3	determination	determination	NOUN
ejpam-4961	175	4	of	of	ADP
ejpam-4961	175	5	the	the	DET
ejpam-4961	175	6	prime	prime	ADJ
ejpam-4961	175	7	counting	counting	NOUN
ejpam-4961	175	8	function	function	NOUN
ejpam-4961	175	9	,	,	PUNCT
ejpam-4961	175	10	π(n	π(n	PROPN
ejpam-4961	175	11	)	)	PUNCT
ejpam-4961	175	12	,	,	PUNCT
ejpam-4961	175	13	further	far	ADV
ejpam-4961	175	14	illustrates	illustrate	VERB
ejpam-4961	175	15	the	the	DET
ejpam-4961	175	16	practical	practical	ADJ
ejpam-4961	175	17	applicability	applicability	NOUN
ejpam-4961	175	18	of	of	ADP
ejpam-4961	175	19	these	these	DET
ejpam-4961	175	20	procedures	procedure	NOUN
ejpam-4961	175	21	.	.	PUNCT
ejpam-4961	176	1	this	this	DET
ejpam-4961	176	2	paper	paper	NOUN
ejpam-4961	176	3	makes	make	VERB
ejpam-4961	176	4	a	a	DET
ejpam-4961	176	5	significant	significant	ADJ
ejpam-4961	176	6	contribution	contribution	NOUN
ejpam-4961	176	7	to	to	ADP
ejpam-4961	176	8	the	the	DET
ejpam-4961	176	9	subject	subject	NOUN
ejpam-4961	176	10	by	by	ADP
ejpam-4961	176	11	demonstrating	demonstrate	VERB
ejpam-4961	176	12	the	the	DET
ejpam-4961	176	13	viability	viability	NOUN
ejpam-4961	176	14	of	of	ADP
ejpam-4961	176	15	estimating	estimate	VERB
ejpam-4961	176	16	the	the	DET
ejpam-4961	176	17	precise	precise	ADJ
ejpam-4961	176	18	number	number	NOUN
ejpam-4961	176	19	of	of	ADP
ejpam-4961	176	20	primes	prime	NOUN
ejpam-4961	176	21	within	within	ADP
ejpam-4961	176	22	a	a	DET
ejpam-4961	176	23	period	period	NOUN
ejpam-4961	176	24	three	three	NUM
ejpam-4961	176	25	times	time	NOUN
ejpam-4961	176	26	the	the	DET
ejpam-4961	176	27	length	length	NOUN
ejpam-4961	176	28	of	of	ADP
ejpam-4961	176	29	another	another	DET
ejpam-4961	176	30	interval	interval	NOUN
ejpam-4961	176	31	,	,	PUNCT
ejpam-4961	176	32	based	base	VERB
ejpam-4961	176	33	on	on	ADP
ejpam-4961	176	34	a	a	DET
ejpam-4961	176	35	given	give	VERB
ejpam-4961	176	36	set	set	NOUN
ejpam-4961	176	37	of	of	ADP
ejpam-4961	176	38	primes	prime	NOUN
ejpam-4961	176	39	.	.	PUNCT
ejpam-4961	177	1	our	our	PRON
ejpam-4961	177	2	understanding	understanding	NOUN
ejpam-4961	177	3	of	of	ADP
ejpam-4961	177	4	prime	prime	ADJ
ejpam-4961	177	5	numbers	number	NOUN
ejpam-4961	177	6	and	and	CCONJ
ejpam-4961	177	7	their	their	PRON
ejpam-4961	177	8	distribution	distribution	NOUN
ejpam-4961	177	9	is	be	AUX
ejpam-4961	177	10	improved	improve	VERB
ejpam-4961	177	11	by	by	ADP
ejpam-4961	177	12	the	the	DET
ejpam-4961	177	13	conclusions	conclusion	NOUN
ejpam-4961	177	14	reported	report	VERB
ejpam-4961	177	15	here	here	ADV
ejpam-4961	177	16	,	,	PUNCT
ejpam-4961	177	17	which	which	PRON
ejpam-4961	177	18	also	also	ADV
ejpam-4961	177	19	may	may	AUX
ejpam-4961	177	20	have	have	VERB
ejpam-4961	177	21	consequences	consequence	NOUN
ejpam-4961	177	22	for	for	ADP
ejpam-4961	177	23	numerous	numerous	ADJ
ejpam-4961	177	24	mathematics	mathematic	NOUN
ejpam-4961	177	25	and	and	CCONJ
ejpam-4961	177	26	computational	computational	ADJ
ejpam-4961	177	27	references	reference	NOUN
ejpam-4961	177	28	[	[	X
ejpam-4961	177	29	1	1	X
ejpam-4961	177	30	]	]	X
ejpam-4961	177	31	t.w	t.w	PROPN
ejpam-4961	177	32	.	.	PROPN
ejpam-4961	177	33	ching	ching	PROPN
ejpam-4961	177	34	.	.	PUNCT
ejpam-4961	178	1	lagrange	lagrange	PROPN
ejpam-4961	178	2	’s	’s	PART
ejpam-4961	178	3	equation	equation	NOUN
ejpam-4961	178	4	with	with	ADP
ejpam-4961	178	5	one	one	NUM
ejpam-4961	178	6	prime	prime	NOUN
ejpam-4961	178	7	and	and	CCONJ
ejpam-4961	178	8	three	three	NUM
ejpam-4961	178	9	almost	almost	ADV
ejpam-4961	178	10	-	-	PUNCT
ejpam-4961	178	11	primes	prime	NOUN
ejpam-4961	178	12	.	.	PUNCT
ejpam-4961	179	1	journal	journal	NOUN
ejpam-4961	179	2	of	of	ADP
ejpam-4961	179	3	number	number	NOUN
ejpam-4961	179	4	theory	theory	NOUN
ejpam-4961	179	5	,	,	PUNCT
ejpam-4961	179	6	183:442–465	183:442–465	NUM
ejpam-4961	179	7	,	,	PUNCT
ejpam-4961	179	8	2018	2018	NUM
ejpam-4961	179	9	.	.	PUNCT
ejpam-4961	180	1	[	[	X
ejpam-4961	180	2	2	2	X
ejpam-4961	180	3	]	]	X
ejpam-4961	180	4	t.w	t.w	PROPN
ejpam-4961	180	5	.	.	PROPN
ejpam-4961	180	6	ching	ching	PROPN
ejpam-4961	180	7	.	.	PUNCT
ejpam-4961	181	1	lagrange	lagrange	PROPN
ejpam-4961	181	2	’s	’s	PART
ejpam-4961	181	3	equation	equation	NOUN
ejpam-4961	181	4	with	with	ADP
ejpam-4961	181	5	almost	almost	ADV
ejpam-4961	181	6	-	-	PUNCT
ejpam-4961	181	7	prime	prime	ADJ
ejpam-4961	181	8	variables	variable	NOUN
ejpam-4961	181	9	.	.	PUNCT
ejpam-4961	182	1	trans	trans	PROPN
ejpam-4961	182	2	.	.	PUNCT
ejpam-4961	183	1	amer	amer	PROPN
ejpam-4961	183	2	.	.	PUNCT
ejpam-4961	183	3	math	math	PROPN
ejpam-4961	183	4	.	.	PUNCT
ejpam-4961	184	1	soc	soc	PROPN
ejpam-4961	184	2	.	.	PUNCT
ejpam-4961	184	3	,	,	PUNCT
ejpam-4961	184	4	375:7669–7714	375:7669–7714	NUM
ejpam-4961	184	5	,	,	PUNCT
ejpam-4961	184	6	2022	2022	NUM
ejpam-4961	184	7	.	.	PUNCT
ejpam-4961	185	1	[	[	X
ejpam-4961	185	2	3	3	X
ejpam-4961	185	3	]	]	X
ejpam-4961	185	4	e.m	e.m	PROPN
ejpam-4961	185	5	.	.	PROPN
ejpam-4961	185	6	wright	wright	PROPN
ejpam-4961	185	7	g.h	g.h	PROPN
ejpam-4961	185	8	.	.	PROPN
ejpam-4961	185	9	hardy	hardy	PROPN
ejpam-4961	185	10	.	.	PUNCT
ejpam-4961	186	1	an	an	DET
ejpam-4961	186	2	introduction	introduction	NOUN
ejpam-4961	186	3	to	to	ADP
ejpam-4961	186	4	the	the	DET
ejpam-4961	186	5	theory	theory	NOUN
ejpam-4961	186	6	of	of	ADP
ejpam-4961	186	7	numbers	number	NOUN
ejpam-4961	186	8	.	.	PUNCT
ejpam-4961	187	1	5th	5th	ADJ
ejpam-4961	187	2	ed	ed	NOUN
ejpam-4961	187	3	.	.	PUNCT
ejpam-4961	188	1	oxford	oxford	PROPN
ejpam-4961	188	2	,	,	PUNCT
ejpam-4961	188	3	england	england	PROPN
ejpam-4961	188	4	:	:	PUNCT
ejpam-4961	188	5	clarendon	clarendon	PROPN
ejpam-4961	188	6	press	press	PROPN
ejpam-4961	188	7	,	,	PUNCT
ejpam-4961	188	8	1979	1979	NUM
ejpam-4961	188	9	.	.	PUNCT
ejpam-4961	189	1	[	[	X
ejpam-4961	189	2	4	4	X
ejpam-4961	189	3	]	]	PUNCT
ejpam-4961	189	4	j.	j.	PROPN
ejpam-4961	189	5	havil	havil	PROPN
ejpam-4961	189	6	.	.	PUNCT
ejpam-4961	190	1	gamma	gamma	PROPN
ejpam-4961	190	2	:	:	PUNCT
ejpam-4961	190	3	exploring	explore	VERB
ejpam-4961	190	4	euler	euler	PROPN
ejpam-4961	190	5	’s	’s	PART
ejpam-4961	190	6	constant	constant	ADJ
ejpam-4961	190	7	.	.	PUNCT
ejpam-4961	191	1	princeton	princeton	PROPN
ejpam-4961	191	2	.	.	PUNCT
ejpam-4961	192	1	nj	nj	PROPN
ejpam-4961	192	2	:	:	PUNCT
ejpam-4961	192	3	princeton	princeton	PROPN
ejpam-4961	192	4	university	university	PROPN
ejpam-4961	192	5	press	press	NOUN
ejpam-4961	192	6	,	,	PUNCT
ejpam-4961	192	7	pages	page	NOUN
ejpam-4961	192	8	164–188	164–188	NUM
ejpam-4961	192	9	,	,	PUNCT
ejpam-4961	192	10	2003	2003	NUM
ejpam-4961	192	11	.	.	PUNCT
ejpam-4961	193	1	[	[	X
ejpam-4961	193	2	5	5	NUM
ejpam-4961	193	3	]	]	PUNCT
ejpam-4961	193	4	a.	a.	NOUN
ejpam-4961	193	5	odlyzko	odlyzko	PROPN
ejpam-4961	193	6	j.	j.	PROPN
ejpam-4961	193	7	lagarias	lagarias	PROPN
ejpam-4961	193	8	.	.	PUNCT
ejpam-4961	194	1	computing	compute	VERB
ejpam-4961	194	2	π(x	π(x	PROPN
ejpam-4961	194	3	):	):	PUNCT
ejpam-4961	194	4	an	an	DET
ejpam-4961	194	5	analytic	analytic	ADJ
ejpam-4961	194	6	method	method	NOUN
ejpam-4961	194	7	.	.	PUNCT
ejpam-4961	195	1	j.	j.	PROPN
ejpam-4961	195	2	algorithms	algorithms	PROPN
ejpam-4961	195	3	,	,	PUNCT
ejpam-4961	195	4	8:173	8:173	NUM
ejpam-4961	195	5	–	–	PUNCT
ejpam-4961	195	6	191	191	NUM
ejpam-4961	195	7	,	,	PUNCT
ejpam-4961	195	8	1987	1987	NUM
ejpam-4961	195	9	.	.	PUNCT
ejpam-4961	196	1	[	[	X
ejpam-4961	196	2	6	6	NUM
ejpam-4961	196	3	]	]	PUNCT
ejpam-4961	196	4	a.	a.	NOUN
ejpam-4961	196	5	odlyzko	odlyzko	PROPN
ejpam-4961	196	6	j.	j.	PROPN
ejpam-4961	196	7	lagarias	lagarias	PROPN
ejpam-4961	196	8	,	,	PUNCT
ejpam-4961	196	9	v.s.	v.s.	PROPN
ejpam-4961	196	10	miller	miller	PROPN
ejpam-4961	196	11	.	.	PUNCT
ejpam-4961	197	1	computing	compute	VERB
ejpam-4961	197	2	π(x	π(x	NOUN
ejpam-4961	197	3	):	):	PUNCT
ejpam-4961	197	4	the	the	DET
ejpam-4961	197	5	meissel	meissel	ADJ
ejpam-4961	197	6	-	-	PUNCT
ejpam-4961	197	7	lehmer	lehmer	NOUN
ejpam-4961	197	8	method	method	NOUN
ejpam-4961	197	9	.	.	PUNCT
ejpam-4961	198	1	math	math	NOUN
ejpam-4961	198	2	.	.	PUNCT
ejpam-4961	199	1	comput	comput	NOUN
ejpam-4961	199	2	.	.	PUNCT
ejpam-4961	199	3	,	,	PUNCT
ejpam-4961	200	1	44:537–560	44:537–560	PROPN
ejpam-4961	200	2	,	,	PUNCT
ejpam-4961	200	3	1985	1985	NUM
ejpam-4961	200	4	.	.	PUNCT
ejpam-4961	201	1	[	[	X
ejpam-4961	201	2	7	7	X
ejpam-4961	201	3	]	]	PUNCT
ejpam-4961	201	4	l.	l.	PROPN
ejpam-4961	201	5	schoenfeld	schoenfeld	PROPN
ejpam-4961	201	6	j.b	j.b	PROPN
ejpam-4961	201	7	.	.	PUNCT
ejpam-4961	201	8	rosser	rosser	PROPN
ejpam-4961	201	9	.	.	PUNCT
ejpam-4961	202	1	approximate	approximate	ADJ
ejpam-4961	202	2	formulas	formula	NOUN
ejpam-4961	202	3	for	for	ADP
ejpam-4961	202	4	some	some	DET
ejpam-4961	202	5	functions	function	NOUN
ejpam-4961	202	6	of	of	ADP
ejpam-4961	202	7	prime	prime	ADJ
ejpam-4961	202	8	numbers	number	NOUN
ejpam-4961	202	9	.	.	PUNCT
ejpam-4961	203	1	illinois	illinois	PROPN
ejpam-4961	203	2	j.	j.	PROPN
ejpam-4961	203	3	math	math	PROPN
ejpam-4961	203	4	.	.	PUNCT
ejpam-4961	203	5	,	,	PUNCT
ejpam-4961	203	6	6:64–97	6:64–97	PROPN
ejpam-4961	203	7	,	,	PUNCT
ejpam-4961	203	8	1962	1962	NUM
ejpam-4961	203	9	.	.	PUNCT
ejpam-4961	204	1	[	[	X
ejpam-4961	204	2	8	8	NUM
ejpam-4961	204	3	]	]	X
ejpam-4961	204	4	e.a	e.a	PROPN
ejpam-4961	204	5	.	.	PROPN
ejpam-4961	204	6	o’brien	o’brien	PROPN
ejpam-4961	204	7	j.h	j.h	PROPN
ejpam-4961	204	8	.	.	PROPN
ejpam-4961	204	9	conway	conway	PROPN
ejpam-4961	204	10	,	,	PUNCT
ejpam-4961	204	11	h.	h.	PROPN
ejpam-4961	204	12	dietrich	dietrich	PROPN
ejpam-4961	204	13	.	.	PUNCT
ejpam-4961	205	1	counting	count	VERB
ejpam-4961	205	2	groups	group	NOUN
ejpam-4961	205	3	:	:	PUNCT
ejpam-4961	205	4	gnus	gnus	PROPN
ejpam-4961	205	5	,	,	PUNCT
ejpam-4961	205	6	moas	moas	PROPN
ejpam-4961	205	7	and	and	CCONJ
ejpam-4961	205	8	other	other	ADJ
ejpam-4961	205	9	exotica	exotica	PROPN
ejpam-4961	205	10	.	.	PUNCT
ejpam-4961	206	1	math	math	NOUN
ejpam-4961	206	2	.	.	PUNCT
ejpam-4961	207	1	intell	intell	PROPN
ejpam-4961	207	2	.	.	PUNCT
ejpam-4961	207	3	,	,	PUNCT
ejpam-4961	207	4	30:6–18	30:6–18	NUM
ejpam-4961	207	5	,	,	PUNCT
ejpam-4961	207	6	2008	2008	NUM
ejpam-4961	207	7	.	.	PUNCT
ejpam-4961	208	1	references	reference	NOUN
ejpam-4961	208	2	1154	1154	NUM
ejpam-4961	208	3	[	[	X
ejpam-4961	208	4	9	9	NUM
ejpam-4961	208	5	]	]	X
ejpam-4961	208	6	l.	l.	PROPN
ejpam-4961	208	7	locker	locker	PROPN
ejpam-4961	208	8	-	-	PUNCT
ejpam-4961	208	9	ernst	ernst	PROPN
ejpam-4961	208	10	.	.	PUNCT
ejpam-4961	209	1	bemerkung	bemerkung	PROPN
ejpam-4961	209	2	über	über	PROPN
ejpam-4961	209	3	die	die	PROPN
ejpam-4961	209	4	verteilung	verteilung	PROPN
ejpam-4961	209	5	der	der	NOUN
ejpam-4961	209	6	primzahlen	primzahlen	VERB
ejpam-4961	209	7	.	.	PUNCT
ejpam-4961	210	1	elemente	elemente	PROPN
ejpam-4961	210	2	math	math	PROPN
ejpam-4961	210	3	.	.	PUNCT
ejpam-4961	211	1	,	,	PUNCT
ejpam-4961	211	2	14:1–5	14:1–5	NUM
ejpam-4961	211	3	,	,	PUNCT
ejpam-4961	211	4	1959	1959	NUM
ejpam-4961	211	5	.	.	PUNCT
ejpam-4961	212	1	[	[	X
ejpam-4961	212	2	10	10	NUM
ejpam-4961	212	3	]	]	X
ejpam-4961	212	4	m.p	m.p	AUX
ejpam-4961	212	5	.	.	PROPN
ejpam-4961	212	6	may	may	AUX
ejpam-4961	212	7	.	.	PUNCT
ejpam-4961	213	1	relationship	relationship	NOUN
ejpam-4961	213	2	between	between	ADP
ejpam-4961	213	3	the	the	DET
ejpam-4961	213	4	prime	prime	ADJ
ejpam-4961	213	5	-	-	PUNCT
ejpam-4961	213	6	counting	count	VERB
ejpam-4961	213	7	function	function	NOUN
ejpam-4961	213	8	and	and	CCONJ
ejpam-4961	213	9	a	a	DET
ejpam-4961	213	10	unique	unique	ADJ
ejpam-4961	213	11	prime	prime	ADJ
ejpam-4961	213	12	number	number	NOUN
ejpam-4961	213	13	sequence	sequence	NOUN
ejpam-4961	213	14	.	.	PUNCT
ejpam-4961	214	1	missouri	missouri	PROPN
ejpam-4961	214	2	j.	j.	PROPN
ejpam-4961	214	3	math	math	PROPN
ejpam-4961	214	4	.	.	PUNCT
ejpam-4961	215	1	sci	sci	PROPN
ejpam-4961	215	2	.	.	PROPN
ejpam-4961	215	3	,	,	PUNCT
ejpam-4961	215	4	35(1):105–116	35(1):105–116	PROPN
ejpam-4961	215	5	,	,	PUNCT
ejpam-4961	215	6	2023	2023	NUM
ejpam-4961	215	7	.	.	PUNCT
ejpam-4961	216	1	[	[	X
ejpam-4961	216	2	11	11	NUM
ejpam-4961	216	3	]	]	X
ejpam-4961	216	4	b.	b.	PROPN
ejpam-4961	216	5	kane	kane	PROPN
ejpam-4961	216	6	s.	s.	PROPN
ejpam-4961	216	7	banerjee	banerjee	PROPN
ejpam-4961	216	8	.	.	PUNCT
ejpam-4961	217	1	finiteness	finiteness	PROPN
ejpam-4961	217	2	theorems	theorem	NOUN
ejpam-4961	217	3	for	for	ADP
ejpam-4961	217	4	universal	universal	ADJ
ejpam-4961	217	5	sums	sum	NOUN
ejpam-4961	217	6	of	of	ADP
ejpam-4961	217	7	squares	square	NOUN
ejpam-4961	217	8	of	of	ADP
ejpam-4961	217	9	almost	almost	ADV
ejpam-4961	217	10	primes	prime	NOUN
ejpam-4961	217	11	.	.	PUNCT
ejpam-4961	218	1	journal	journal	NOUN
ejpam-4961	218	2	of	of	ADP
ejpam-4961	218	3	number	number	NOUN
ejpam-4961	218	4	theory	theory	NOUN
ejpam-4961	218	5	,	,	PUNCT
ejpam-4961	218	6	212:233–264	212:233–264	NUM
ejpam-4961	218	7	,	,	PUNCT
ejpam-4961	218	8	2020	2020	NUM
ejpam-4961	218	9	.	.	PUNCT
ejpam-4961	219	1	[	[	X
ejpam-4961	219	2	12	12	NUM
ejpam-4961	219	3	]	]	X
ejpam-4961	219	4	f.f	f.f	PROPN
ejpam-4961	219	5	.	.	PROPN
ejpam-4961	219	6	sharifullina	sharifullina	PROPN
ejpam-4961	219	7	s.t	s.t	PROPN
ejpam-4961	219	8	.	.	PROPN
ejpam-4961	219	9	ishmukhametov	ishmukhametov	PROPN
ejpam-4961	219	10	.	.	PUNCT
ejpam-4961	220	1	on	on	ADP
ejpam-4961	220	2	distribution	distribution	NOUN
ejpam-4961	220	3	of	of	ADP
ejpam-4961	220	4	semiprime	semiprime	NOUN
ejpam-4961	220	5	numbers	number	NOUN
ejpam-4961	220	6	.	.	PUNCT
ejpam-4961	221	1	russian	russian	ADJ
ejpam-4961	221	2	mathematics	mathematic	NOUN
ejpam-4961	221	3	,	,	PUNCT
ejpam-4961	221	4	58(8):43–48	58(8):43–48	NUM
ejpam-4961	221	5	,	,	PUNCT
ejpam-4961	221	6	2014	2014	NUM
ejpam-4961	221	7	.	.	PUNCT