id	sid	tid	token	lemma	pos
ejpam-3182	1	1	european	european	PROPN
ejpam-3182	1	2	journal	journal	PROPN
ejpam-3182	1	3	of	of	ADP
ejpam-3182	1	4	pure	pure	ADJ
ejpam-3182	1	5	and	and	CCONJ
ejpam-3182	1	6	applied	apply	VERB
ejpam-3182	1	7	mathematics	mathematic	NOUN
ejpam-3182	1	8	vol	vol	NOUN
ejpam-3182	1	9	.	.	PUNCT
ejpam-3182	2	1	11	11	NUM
ejpam-3182	2	2	,	,	PUNCT
ejpam-3182	2	3	no	no	INTJ
ejpam-3182	2	4	.	.	NOUN
ejpam-3182	2	5	1	1	NUM
ejpam-3182	2	6	,	,	PUNCT
ejpam-3182	2	7	2018	2018	NUM
ejpam-3182	2	8	,	,	PUNCT
ejpam-3182	2	9	23	23	NUM
ejpam-3182	2	10	-	-	SYM
ejpam-3182	2	11	34	34	NUM
ejpam-3182	2	12	issn	issn	PROPN
ejpam-3182	2	13	1307	1307	NUM
ejpam-3182	2	14	-	-	SYM
ejpam-3182	2	15	5543	5543	NUM
ejpam-3182	2	16	–	–	PUNCT
ejpam-3182	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3182	2	18	published	publish	VERB
ejpam-3182	2	19	by	by	ADP
ejpam-3182	2	20	new	new	PROPN
ejpam-3182	2	21	york	york	PROPN
ejpam-3182	2	22	business	business	PROPN
ejpam-3182	2	23	global	global	PROPN
ejpam-3182	2	24	the	the	DET
ejpam-3182	2	25	generalized	generalize	VERB
ejpam-3182	2	26	artin	artin	PROPN
ejpam-3182	2	27	primitive	primitive	ADJ
ejpam-3182	2	28	root	root	NOUN
ejpam-3182	2	29	conjecture	conjecture	NOUN
ejpam-3182	2	30	n.	n.	PROPN
ejpam-3182	2	31	a.	a.	PROPN
ejpam-3182	2	32	carella	carella	PROPN
ejpam-3182	2	33	department	department	PROPN
ejpam-3182	2	34	of	of	ADP
ejpam-3182	2	35	mathematics	mathematics	PROPN
ejpam-3182	2	36	,	,	PUNCT
ejpam-3182	2	37	york	york	PROPN
ejpam-3182	2	38	college	college	PROPN
ejpam-3182	2	39	,	,	PUNCT
ejpam-3182	2	40	jamaica	jamaica	PROPN
ejpam-3182	2	41	,	,	PUNCT
ejpam-3182	2	42	new	new	PROPN
ejpam-3182	2	43	york	york	PROPN
ejpam-3182	2	44	,	,	PUNCT
ejpam-3182	2	45	10451	10451	NUM
ejpam-3182	2	46	abstract	abstract	NOUN
ejpam-3182	2	47	.	.	PUNCT
ejpam-3182	3	1	asymptotic	asymptotic	ADJ
ejpam-3182	3	2	formulas	formula	NOUN
ejpam-3182	3	3	for	for	ADP
ejpam-3182	3	4	the	the	DET
ejpam-3182	3	5	number	number	NOUN
ejpam-3182	3	6	of	of	ADP
ejpam-3182	3	7	integers	integer	NOUN
ejpam-3182	3	8	with	with	ADP
ejpam-3182	3	9	the	the	DET
ejpam-3182	3	10	primitive	primitive	ADJ
ejpam-3182	3	11	root	root	NOUN
ejpam-3182	3	12	2	2	NUM
ejpam-3182	3	13	,	,	PUNCT
ejpam-3182	3	14	and	and	CCONJ
ejpam-3182	3	15	the	the	DET
ejpam-3182	3	16	generalized	generalized	ADJ
ejpam-3182	3	17	artin	artin	PROPN
ejpam-3182	3	18	conjecture	conjecture	NOUN
ejpam-3182	3	19	for	for	ADP
ejpam-3182	3	20	subsets	subset	NOUN
ejpam-3182	3	21	of	of	ADP
ejpam-3182	3	22	composite	composite	ADJ
ejpam-3182	3	23	integers	integer	NOUN
ejpam-3182	3	24	with	with	ADP
ejpam-3182	3	25	fixed	fix	VERB
ejpam-3182	3	26	admissible	admissible	ADJ
ejpam-3182	3	27	primitive	primitive	ADJ
ejpam-3182	3	28	roots	root	NOUN
ejpam-3182	3	29	u	u	PROPN
ejpam-3182	3	30	6=	6=	PRON
ejpam-3182	3	31	±1	±1	VERB
ejpam-3182	3	32	,	,	PUNCT
ejpam-3182	3	33	v2	v2	PROPN
ejpam-3182	3	34	,	,	PUNCT
ejpam-3182	3	35	are	be	AUX
ejpam-3182	3	36	presented	present	VERB
ejpam-3182	3	37	here	here	ADV
ejpam-3182	3	38	.	.	PUNCT
ejpam-3182	4	1	2010	2010	NUM
ejpam-3182	4	2	mathematics	mathematic	NOUN
ejpam-3182	4	3	subject	subject	NOUN
ejpam-3182	4	4	classifications	classification	NOUN
ejpam-3182	4	5	:	:	PUNCT
ejpam-3182	4	6	11a07	11a07	NUM
ejpam-3182	4	7	,	,	PUNCT
ejpam-3182	4	8	11n05	11n05	NUM
ejpam-3182	4	9	.	.	PUNCT
ejpam-3182	5	1	key	key	ADJ
ejpam-3182	5	2	words	word	NOUN
ejpam-3182	5	3	and	and	CCONJ
ejpam-3182	5	4	phrases	phrase	NOUN
ejpam-3182	5	5	:	:	PUNCT
ejpam-3182	5	6	primitive	primitive	ADJ
ejpam-3182	5	7	root	root	NOUN
ejpam-3182	5	8	,	,	PUNCT
ejpam-3182	5	9	generalized	generalize	VERB
ejpam-3182	5	10	primitive	primitive	ADJ
ejpam-3182	5	11	root	root	NOUN
ejpam-3182	5	12	1	1	NUM
ejpam-3182	5	13	.	.	X
ejpam-3182	5	14	introduction	introduction	NOUN
ejpam-3182	5	15	a	a	DET
ejpam-3182	5	16	generalized	generalized	ADJ
ejpam-3182	5	17	artin	artin	NOUN
ejpam-3182	5	18	conjecture	conjecture	NOUN
ejpam-3182	5	19	for	for	ADP
ejpam-3182	5	20	subsets	subset	NOUN
ejpam-3182	5	21	of	of	ADP
ejpam-3182	5	22	composite	composite	ADJ
ejpam-3182	5	23	integers	integer	NOUN
ejpam-3182	5	24	with	with	ADP
ejpam-3182	5	25	fixed	fix	VERB
ejpam-3182	5	26	primitive	primitive	ADJ
ejpam-3182	5	27	roots	root	NOUN
ejpam-3182	5	28	is	be	AUX
ejpam-3182	5	29	presented	present	VERB
ejpam-3182	5	30	here	here	ADV
ejpam-3182	5	31	.	.	PUNCT
ejpam-3182	6	1	the	the	DET
ejpam-3182	6	2	focus	focus	NOUN
ejpam-3182	6	3	is	be	AUX
ejpam-3182	6	4	on	on	ADP
ejpam-3182	6	5	developing	develop	VERB
ejpam-3182	6	6	an	an	DET
ejpam-3182	6	7	asymptotic	asymptotic	ADJ
ejpam-3182	6	8	formula	formula	NOUN
ejpam-3182	6	9	for	for	ADP
ejpam-3182	6	10	the	the	DET
ejpam-3182	6	11	number	number	NOUN
ejpam-3182	6	12	of	of	ADP
ejpam-3182	6	13	integers	integer	NOUN
ejpam-3182	6	14	with	with	ADP
ejpam-3182	6	15	the	the	DET
ejpam-3182	6	16	primitive	primitive	ADJ
ejpam-3182	6	17	root	root	NOUN
ejpam-3182	6	18	2	2	NUM
ejpam-3182	6	19	,	,	PUNCT
ejpam-3182	6	20	modulo	modulo	VERB
ejpam-3182	6	21	the	the	DET
ejpam-3182	6	22	generalized	generalized	ADJ
ejpam-3182	6	23	riemann	riemann	PROPN
ejpam-3182	6	24	hypothesis	hypothesis	NOUN
ejpam-3182	6	25	,	,	PUNCT
ejpam-3182	6	26	but	but	CCONJ
ejpam-3182	6	27	the	the	DET
ejpam-3182	6	28	analysis	analysis	NOUN
ejpam-3182	6	29	easily	easily	ADV
ejpam-3182	6	30	extends	extend	VERB
ejpam-3182	6	31	to	to	ADP
ejpam-3182	6	32	all	all	DET
ejpam-3182	6	33	the	the	DET
ejpam-3182	6	34	admissible	admissible	ADJ
ejpam-3182	6	35	primitive	primitive	ADJ
ejpam-3182	6	36	roots	root	NOUN
ejpam-3182	6	37	u	u	PROPN
ejpam-3182	6	38	6=	6=	PRON
ejpam-3182	6	39	±1	±1	VERB
ejpam-3182	6	40	,	,	PUNCT
ejpam-3182	6	41	v2	v2	PROPN
ejpam-3182	6	42	.	.	PUNCT
ejpam-3182	7	1	this	this	DET
ejpam-3182	7	2	analysis	analysis	NOUN
ejpam-3182	7	3	spawn	spawn	VERB
ejpam-3182	7	4	new	new	ADJ
ejpam-3182	7	5	questions	question	NOUN
ejpam-3182	7	6	about	about	ADP
ejpam-3182	7	7	the	the	DET
ejpam-3182	7	8	structure	structure	NOUN
ejpam-3182	7	9	of	of	ADP
ejpam-3182	7	10	an	an	DET
ejpam-3182	7	11	l	l	NOUN
ejpam-3182	7	12	-	-	PUNCT
ejpam-3182	7	13	series	series	NOUN
ejpam-3182	7	14	associated	associate	VERB
ejpam-3182	7	15	with	with	ADP
ejpam-3182	7	16	the	the	DET
ejpam-3182	7	17	multiplicative	multiplicative	ADJ
ejpam-3182	7	18	subset	subset	NOUN
ejpam-3182	7	19	of	of	ADP
ejpam-3182	7	20	integers	integer	NOUN
ejpam-3182	7	21	with	with	ADP
ejpam-3182	7	22	a	a	DET
ejpam-3182	7	23	fixed	fix	VERB
ejpam-3182	7	24	primitive	primitive	ADJ
ejpam-3182	7	25	root	root	NOUN
ejpam-3182	7	26	2	2	NUM
ejpam-3182	7	27	,	,	PUNCT
ejpam-3182	7	28	and	and	CCONJ
ejpam-3182	7	29	related	related	ADJ
ejpam-3182	7	30	ideas	idea	NOUN
ejpam-3182	7	31	.	.	PUNCT
ejpam-3182	8	1	1.1	1.1	NUM
ejpam-3182	8	2	.	.	PUNCT
ejpam-3182	8	3	subset	subset	NOUN
ejpam-3182	8	4	of	of	ADP
ejpam-3182	8	5	integers	integer	NOUN
ejpam-3182	8	6	with	with	ADP
ejpam-3182	8	7	fixed	fix	VERB
ejpam-3182	8	8	primitive	primitive	ADJ
ejpam-3182	8	9	root	root	NOUN
ejpam-3182	8	10	2	2	NUM
ejpam-3182	8	11	the	the	DET
ejpam-3182	8	12	artin	artin	PROPN
ejpam-3182	8	13	primitive	primitive	ADJ
ejpam-3182	8	14	root	root	NOUN
ejpam-3182	8	15	conjecture	conjecture	NOUN
ejpam-3182	8	16	states	state	VERB
ejpam-3182	8	17	that	that	SCONJ
ejpam-3182	8	18	the	the	DET
ejpam-3182	8	19	integer	integer	NOUN
ejpam-3182	8	20	2	2	NUM
ejpam-3182	8	21	is	be	AUX
ejpam-3182	8	22	a	a	DET
ejpam-3182	8	23	primitive	primitive	ADJ
ejpam-3182	8	24	root	root	NOUN
ejpam-3182	8	25	mod	mod	PROPN
ejpam-3182	8	26	p	p	NOUN
ejpam-3182	8	27	for	for	ADP
ejpam-3182	8	28	infinitely	infinitely	ADV
ejpam-3182	8	29	many	many	ADJ
ejpam-3182	8	30	primes	prime	NOUN
ejpam-3182	8	31	.	.	PUNCT
ejpam-3182	9	1	i	i	PROPN
ejpam-3182	9	2	d	d	PROPN
ejpam-3182	9	3	est	est	X
ejpam-3182	9	4	,	,	PUNCT
ejpam-3182	9	5	p2	p2	PROPN
ejpam-3182	9	6	=	=	SYM
ejpam-3182	9	7	{	{	PUNCT
ejpam-3182	9	8	3	3	NUM
ejpam-3182	9	9	,	,	PUNCT
ejpam-3182	9	10	5	5	NUM
ejpam-3182	9	11	,	,	PUNCT
ejpam-3182	9	12	11	11	NUM
ejpam-3182	9	13	,	,	PUNCT
ejpam-3182	9	14	13	13	NUM
ejpam-3182	9	15	,	,	PUNCT
ejpam-3182	9	16	19	19	NUM
ejpam-3182	9	17	,	,	PUNCT
ejpam-3182	9	18	29	29	NUM
ejpam-3182	9	19	,	,	PUNCT
ejpam-3182	9	20	37	37	NUM
ejpam-3182	9	21	,	,	PUNCT
ejpam-3182	9	22	53	53	NUM
ejpam-3182	9	23	,	,	PUNCT
ejpam-3182	9	24	59	59	NUM
ejpam-3182	9	25	,	,	PUNCT
ejpam-3182	9	26	61	61	NUM
ejpam-3182	9	27	,	,	PUNCT
ejpam-3182	9	28	67	67	NUM
ejpam-3182	9	29	,	,	PUNCT
ejpam-3182	9	30	83	83	NUM
ejpam-3182	9	31	,	,	PUNCT
ejpam-3182	9	32	101	101	NUM
ejpam-3182	9	33	,	,	PUNCT
ejpam-3182	9	34	107	107	NUM
ejpam-3182	9	35	,	,	PUNCT
ejpam-3182	9	36	107	107	NUM
ejpam-3182	9	37	,	,	PUNCT
ejpam-3182	9	38	131	131	NUM
ejpam-3182	9	39	,	,	PUNCT
ejpam-3182	9	40	139	139	NUM
ejpam-3182	9	41	,	,	PUNCT
ejpam-3182	9	42	...	...	PUNCT
ejpam-3182	9	43	}	}	PUNCT
ejpam-3182	9	44	.	.	PUNCT
ejpam-3182	10	1	(	(	PUNCT
ejpam-3182	10	2	1	1	X
ejpam-3182	10	3	)	)	PUNCT
ejpam-3182	10	4	moreover	moreover	ADV
ejpam-3182	10	5	,	,	PUNCT
ejpam-3182	10	6	it	it	PRON
ejpam-3182	10	7	has	have	VERB
ejpam-3182	10	8	the	the	DET
ejpam-3182	10	9	counting	counting	NOUN
ejpam-3182	10	10	function	function	NOUN
ejpam-3182	10	11	π2(x	π2(x	NUM
ejpam-3182	10	12	)	)	PUNCT
ejpam-3182	10	13	=	=	SYM
ejpam-3182	10	14	#	#	NOUN
ejpam-3182	10	15	{	{	PUNCT
ejpam-3182	10	16	p	p	NOUN
ejpam-3182	10	17	≤	≤	PROPN
ejpam-3182	10	18	x	x	SYM
ejpam-3182	10	19	:	:	PUNCT
ejpam-3182	10	20	ordp(2	ordp(2	ADJ
ejpam-3182	10	21	)	)	PUNCT
ejpam-3182	10	22	=	=	PUNCT
ejpam-3182	11	1	p−	p−	NOUN
ejpam-3182	11	2	1	1	X
ejpam-3182	11	3	}	}	PUNCT
ejpam-3182	11	4	=	=	SYM
ejpam-3182	11	5	α2π(x	α2π(x	PROPN
ejpam-3182	11	6	)	)	PUNCT
ejpam-3182	11	7	.	.	PUNCT
ejpam-3182	12	1	here	here	ADV
ejpam-3182	12	2	π(x	π(x	ADP
ejpam-3182	12	3	)	)	PUNCT
ejpam-3182	13	1	=	=	SYM
ejpam-3182	13	2	#	#	NOUN
ejpam-3182	13	3	{	{	PUNCT
ejpam-3182	13	4	p	p	NOUN
ejpam-3182	13	5	≤	≤	PROPN
ejpam-3182	13	6	x	x	X
ejpam-3182	13	7	}	}	PUNCT
ejpam-3182	13	8	is	be	AUX
ejpam-3182	13	9	the	the	DET
ejpam-3182	13	10	primes	prime	NOUN
ejpam-3182	13	11	counting	count	VERB
ejpam-3182	13	12	function	function	NOUN
ejpam-3182	13	13	.	.	PUNCT
ejpam-3182	14	1	conditional	conditional	ADJ
ejpam-3182	14	2	on	on	ADP
ejpam-3182	14	3	the	the	DET
ejpam-3182	14	4	generalized	generalized	ADJ
ejpam-3182	14	5	riemann	riemann	PROPN
ejpam-3182	14	6	hypothesis	hypothesis	NOUN
ejpam-3182	14	7	,	,	PUNCT
ejpam-3182	14	8	hooley	hooley	PROPN
ejpam-3182	14	9	proved	prove	VERB
ejpam-3182	14	10	that	that	SCONJ
ejpam-3182	14	11	the	the	DET
ejpam-3182	14	12	subset	subset	NOUN
ejpam-3182	14	13	of	of	ADP
ejpam-3182	14	14	primes	prime	NOUN
ejpam-3182	14	15	p2	p2	X
ejpam-3182	14	16	=	=	SYM
ejpam-3182	14	17	{	{	PUNCT
ejpam-3182	14	18	p	p	NOUN
ejpam-3182	14	19	∈	∈	PROPN
ejpam-3182	14	20	p	p	NOUN
ejpam-3182	14	21	:	:	PUNCT
ejpam-3182	14	22	ordp(2	ordp(2	ADJ
ejpam-3182	14	23	)	)	PUNCT
ejpam-3182	14	24	=	=	PUNCT
ejpam-3182	15	1	p−	p−	NOUN
ejpam-3182	15	2	1	1	X
ejpam-3182	15	3	}	}	PUNCT
ejpam-3182	15	4	⊂	⊂	PRON
ejpam-3182	15	5	p	p	X
ejpam-3182	15	6	(	(	PUNCT
ejpam-3182	15	7	2	2	NUM
ejpam-3182	15	8	)	)	PUNCT
ejpam-3182	15	9	has	have	VERB
ejpam-3182	15	10	nonzero	nonzero	PROPN
ejpam-3182	15	11	density	density	NOUN
ejpam-3182	15	12	α2	α2	PROPN
ejpam-3182	15	13	=	=	SYM
ejpam-3182	15	14	δ	δ	PROPN
ejpam-3182	15	15	(	(	PUNCT
ejpam-3182	15	16	p2	p2	PROPN
ejpam-3182	15	17	)	)	PUNCT
ejpam-3182	15	18	>	>	X
ejpam-3182	15	19	0	0	NUM
ejpam-3182	15	20	,	,	PUNCT
ejpam-3182	15	21	see	see	VERB
ejpam-3182	15	22	theorem	theorem	VERB
ejpam-3182	15	23	7.1	7.1	NUM
ejpam-3182	15	24	or	or	CCONJ
ejpam-3182	15	25	[	[	X
ejpam-3182	15	26	13	13	NUM
ejpam-3182	15	27	]	]	PUNCT
ejpam-3182	15	28	.	.	PUNCT
ejpam-3182	16	1	partial	partial	ADJ
ejpam-3182	16	2	unconditional	unconditional	ADJ
ejpam-3182	16	3	results	result	NOUN
ejpam-3182	16	4	on	on	ADP
ejpam-3182	16	5	the	the	DET
ejpam-3182	16	6	artin	artin	PROPN
ejpam-3182	16	7	primitive	primitive	ADJ
ejpam-3182	16	8	root	root	NOUN
ejpam-3182	16	9	conjecture	conjecture	NOUN
ejpam-3182	16	10	are	be	AUX
ejpam-3182	16	11	also	also	ADV
ejpam-3182	16	12	available	available	ADJ
ejpam-3182	16	13	in	in	ADP
ejpam-3182	16	14	[	[	X
ejpam-3182	16	15	10	10	NUM
ejpam-3182	16	16	]	]	PUNCT
ejpam-3182	16	17	,	,	PUNCT
ejpam-3182	16	18	et	et	PROPN
ejpam-3182	16	19	alii	alii	PROPN
ejpam-3182	16	20	.	.	PUNCT
ejpam-3182	17	1	the	the	DET
ejpam-3182	17	2	subset	subset	NOUN
ejpam-3182	17	3	p2	p2	PROPN
ejpam-3182	17	4	is	be	AUX
ejpam-3182	17	5	utilized	utilize	VERB
ejpam-3182	17	6	here	here	ADV
ejpam-3182	17	7	to	to	PART
ejpam-3182	17	8	generate	generate	VERB
ejpam-3182	17	9	the	the	DET
ejpam-3182	17	10	subset	subset	NOUN
ejpam-3182	17	11	of	of	ADP
ejpam-3182	17	12	composite	composite	ADJ
ejpam-3182	17	13	integers	integer	NOUN
ejpam-3182	17	14	n2	n2	NOUN
ejpam-3182	17	15	=	=	PUNCT
ejpam-3182	17	16	{	{	PUNCT
ejpam-3182	17	17	3	3	NUM
ejpam-3182	17	18	,	,	PUNCT
ejpam-3182	17	19	5	5	NUM
ejpam-3182	17	20	,	,	PUNCT
ejpam-3182	17	21	32	32	NUM
ejpam-3182	17	22	,	,	PUNCT
ejpam-3182	17	23	11	11	NUM
ejpam-3182	17	24	,	,	PUNCT
ejpam-3182	17	25	3	3	NUM
ejpam-3182	17	26	·	·	SYM
ejpam-3182	17	27	5	5	NUM
ejpam-3182	17	28	,	,	PUNCT
ejpam-3182	17	29	19	19	NUM
ejpam-3182	17	30	,	,	PUNCT
ejpam-3182	17	31	52	52	NUM
ejpam-3182	17	32	,	,	PUNCT
ejpam-3182	17	33	33	33	NUM
ejpam-3182	17	34	,	,	PUNCT
ejpam-3182	17	35	29	29	NUM
ejpam-3182	17	36	,	,	PUNCT
ejpam-3182	17	37	3	3	NUM
ejpam-3182	17	38	·	·	SYM
ejpam-3182	17	39	11	11	NUM
ejpam-3182	17	40	,	,	PUNCT
ejpam-3182	17	41	37	37	NUM
ejpam-3182	17	42	,	,	PUNCT
ejpam-3182	17	43	45	45	NUM
ejpam-3182	17	44	,	,	PUNCT
ejpam-3182	17	45	53	53	NUM
ejpam-3182	17	46	,	,	PUNCT
ejpam-3182	17	47	55	55	NUM
ejpam-3182	17	48	,	,	PUNCT
ejpam-3182	17	49	3	3	NUM
ejpam-3182	17	50	·	·	SYM
ejpam-3182	17	51	19	19	NUM
ejpam-3182	17	52	,	,	PUNCT
ejpam-3182	17	53	61	61	NUM
ejpam-3182	17	54	,	,	PUNCT
ejpam-3182	17	55	...	...	PUNCT
ejpam-3182	17	56	}	}	PUNCT
ejpam-3182	17	57	(	(	PUNCT
ejpam-3182	17	58	3	3	X
ejpam-3182	17	59	)	)	PUNCT
ejpam-3182	17	60	email	email	NOUN
ejpam-3182	17	61	address	address	NOUN
ejpam-3182	17	62	:	:	PUNCT
ejpam-3182	17	63	pobox505@gmail.com	pobox505@gmail.com	X
ejpam-3182	17	64	(	(	PUNCT
ejpam-3182	17	65	n.	n.	NOUN
ejpam-3182	17	66	a.	a.	NOUN
ejpam-3182	17	67	carella	carella	PROPN
ejpam-3182	17	68	)	)	PUNCT
ejpam-3182	17	69	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3182	18	1	23	23	NUM
ejpam-3182	19	1	c	c	X
ejpam-3182	19	2	©	©	PROPN
ejpam-3182	19	3	2018	2018	NUM
ejpam-3182	19	4	ejpam	ejpam	VERB
ejpam-3182	19	5	all	all	DET
ejpam-3182	19	6	rights	right	NOUN
ejpam-3182	19	7	reserved	reserve	VERB
ejpam-3182	19	8	.	.	PUNCT
ejpam-3182	20	1	n.	n.	NOUN
ejpam-3182	20	2	a.	a.	PROPN
ejpam-3182	20	3	carella	carella	PROPN
ejpam-3182	20	4	/	/	SYM
ejpam-3182	20	5	eur	eur	PROPN
ejpam-3182	20	6	.	.	PUNCT
ejpam-3182	21	1	j.	j.	PROPN
ejpam-3182	21	2	pure	pure	PROPN
ejpam-3182	21	3	appl	appl	PROPN
ejpam-3182	21	4	.	.	PROPN
ejpam-3182	21	5	math	math	PROPN
ejpam-3182	21	6	,	,	PUNCT
ejpam-3182	21	7	11	11	NUM
ejpam-3182	21	8	(	(	PUNCT
ejpam-3182	21	9	1	1	NUM
ejpam-3182	21	10	)	)	PUNCT
ejpam-3182	21	11	(	(	PUNCT
ejpam-3182	21	12	2018	2018	NUM
ejpam-3182	21	13	)	)	PUNCT
ejpam-3182	21	14	,	,	PUNCT
ejpam-3182	21	15	23	23	NUM
ejpam-3182	21	16	-	-	SYM
ejpam-3182	21	17	34	34	NUM
ejpam-3182	21	18	24	24	NUM
ejpam-3182	21	19	which	which	PRON
ejpam-3182	21	20	have	have	VERB
ejpam-3182	21	21	u	u	NOUN
ejpam-3182	21	22	=	=	NOUN
ejpam-3182	21	23	2	2	NUM
ejpam-3182	21	24	as	as	ADP
ejpam-3182	21	25	a	a	DET
ejpam-3182	21	26	primitive	primitive	ADJ
ejpam-3182	21	27	root	root	NOUN
ejpam-3182	21	28	.	.	PUNCT
ejpam-3182	22	1	the	the	DET
ejpam-3182	22	2	underlining	underline	VERB
ejpam-3182	22	3	structure	structure	NOUN
ejpam-3182	22	4	of	of	ADP
ejpam-3182	22	5	an	an	DET
ejpam-3182	22	6	asymptotic	asymptotic	ADJ
ejpam-3182	22	7	counting	counting	NOUN
ejpam-3182	22	8	formula	formula	NOUN
ejpam-3182	22	9	n2(x	n2(x	NOUN
ejpam-3182	22	10	)	)	PUNCT
ejpam-3182	22	11	=	=	SYM
ejpam-3182	22	12	#	#	NOUN
ejpam-3182	22	13	{	{	PUNCT
ejpam-3182	22	14	n	n	CCONJ
ejpam-3182	22	15	≤	≤	NOUN
ejpam-3182	22	16	x	x	PUNCT
ejpam-3182	22	17	:	:	PUNCT
ejpam-3182	22	18	ordn(2	ordn(2	ADJ
ejpam-3182	22	19	)	)	PUNCT
ejpam-3182	22	20	=	=	SYM
ejpam-3182	22	21	λ(n	λ(n	PROPN
ejpam-3182	22	22	)	)	PUNCT
ejpam-3182	22	23	}	}	PUNCT
ejpam-3182	22	24	for	for	ADP
ejpam-3182	22	25	the	the	DET
ejpam-3182	22	26	subset	subset	NOUN
ejpam-3182	22	27	of	of	ADP
ejpam-3182	22	28	integers	integer	NOUN
ejpam-3182	22	29	n2	n2	NOUN
ejpam-3182	22	30	=	=	PUNCT
ejpam-3182	22	31	{	{	PUNCT
ejpam-3182	22	32	n	n	NOUN
ejpam-3182	22	33	∈	∈	NOUN
ejpam-3182	22	34	n	n	CCONJ
ejpam-3182	22	35	:	:	PUNCT
ejpam-3182	22	36	ordn(2	ordn(2	ADJ
ejpam-3182	22	37	)	)	PUNCT
ejpam-3182	22	38	=	=	SYM
ejpam-3182	22	39	λ(n	λ(n	PROPN
ejpam-3182	22	40	)	)	PUNCT
ejpam-3182	22	41	}	}	PUNCT
ejpam-3182	22	42	will	will	AUX
ejpam-3182	22	43	be	be	AUX
ejpam-3182	22	44	demonstrated	demonstrate	VERB
ejpam-3182	22	45	here	here	ADV
ejpam-3182	22	46	.	.	PUNCT
ejpam-3182	23	1	this	this	DET
ejpam-3182	23	2	result	result	NOUN
ejpam-3182	23	3	is	be	AUX
ejpam-3182	23	4	consistent	consistent	ADJ
ejpam-3182	23	5	with	with	ADP
ejpam-3182	23	6	the	the	DET
ejpam-3182	23	7	heuristic	heuristic	NOUN
ejpam-3182	23	8	explained	explain	VERB
ejpam-3182	23	9	in	in	ADP
ejpam-3182	23	10	[	[	X
ejpam-3182	23	11	14	14	NUM
ejpam-3182	23	12	,	,	PUNCT
ejpam-3182	23	13	p.	p.	NOUN
ejpam-3182	23	14	10	10	NUM
ejpam-3182	23	15	]	]	PUNCT
ejpam-3182	23	16	,	,	PUNCT
ejpam-3182	23	17	and	and	CCONJ
ejpam-3182	23	18	has	have	VERB
ejpam-3182	23	19	the	the	DET
ejpam-3182	23	20	expected	expect	VERB
ejpam-3182	23	21	asymptotic	asymptotic	ADJ
ejpam-3182	23	22	order	order	NOUN
ejpam-3182	23	23	n2(x	n2(x	NOUN
ejpam-3182	23	24	)	)	PUNCT
ejpam-3182	23	25	=	=	SYM
ejpam-3182	23	26	o(x	o(x	PROPN
ejpam-3182	23	27	)	)	PUNCT
ejpam-3182	23	28	.	.	PUNCT
ejpam-3182	24	1	theorem	theorem	VERB
ejpam-3182	24	2	1.1	1.1	NUM
ejpam-3182	24	3	.	.	PUNCT
ejpam-3182	25	1	assuming	assume	VERB
ejpam-3182	25	2	the	the	DET
ejpam-3182	25	3	generalized	generalized	ADJ
ejpam-3182	25	4	riemann	riemann	PROPN
ejpam-3182	25	5	hypothesis	hypothesis	NOUN
ejpam-3182	25	6	,	,	PUNCT
ejpam-3182	25	7	the	the	DET
ejpam-3182	25	8	integer	integer	NOUN
ejpam-3182	25	9	2	2	NUM
ejpam-3182	25	10	is	be	AUX
ejpam-3182	25	11	a	a	DET
ejpam-3182	25	12	primitive	primitive	ADJ
ejpam-3182	25	13	root	root	NOUN
ejpam-3182	25	14	mod	mod	NOUN
ejpam-3182	25	15	n	n	PROPN
ejpam-3182	25	16	for	for	ADP
ejpam-3182	25	17	infinitely	infinitely	ADV
ejpam-3182	25	18	many	many	ADJ
ejpam-3182	25	19	composite	composite	ADJ
ejpam-3182	25	20	integers	integer	NOUN
ejpam-3182	25	21	n	n	PRON
ejpam-3182	25	22	≥	≥	NOUN
ejpam-3182	25	23	1	1	NUM
ejpam-3182	25	24	.	.	PUNCT
ejpam-3182	26	1	moreover	moreover	ADV
ejpam-3182	26	2	,	,	PUNCT
ejpam-3182	26	3	the	the	DET
ejpam-3182	26	4	number	number	NOUN
ejpam-3182	26	5	of	of	ADP
ejpam-3182	26	6	integers	integer	NOUN
ejpam-3182	26	7	n	n	PRON
ejpam-3182	26	8	≤	≤	NUM
ejpam-3182	26	9	x	x	PUNCT
ejpam-3182	26	10	such	such	ADJ
ejpam-3182	26	11	that	that	SCONJ
ejpam-3182	26	12	2	2	NUM
ejpam-3182	26	13	is	be	AUX
ejpam-3182	26	14	a	a	DET
ejpam-3182	26	15	primitive	primitive	ADJ
ejpam-3182	26	16	root	root	NOUN
ejpam-3182	26	17	mod	mod	NOUN
ejpam-3182	26	18	n	n	PRON
ejpam-3182	26	19	has	have	VERB
ejpam-3182	26	20	the	the	DET
ejpam-3182	26	21	asymptotic	asymptotic	ADJ
ejpam-3182	26	22	formula	formula	NOUN
ejpam-3182	26	23	n2(x	n2(x	NOUN
ejpam-3182	26	24	)	)	PUNCT
ejpam-3182	26	25	=	=	SYM
ejpam-3182	27	1	(	(	PUNCT
ejpam-3182	27	2	eγ2−γα2	eγ2−γα2	ADV
ejpam-3182	27	3	γ	γ	X
ejpam-3182	27	4	(	(	PUNCT
ejpam-3182	27	5	α2	α2	PROPN
ejpam-3182	27	6	)	)	PUNCT
ejpam-3182	27	7	+	+	NUM
ejpam-3182	27	8	o(1	o(1	NOUN
ejpam-3182	27	9	)	)	PUNCT
ejpam-3182	27	10	)	)	PUNCT
ejpam-3182	28	1	x	x	X
ejpam-3182	28	2	(	(	PUNCT
ejpam-3182	28	3	log	log	VERB
ejpam-3182	28	4	x)1−α2	x)1−α2	PROPN
ejpam-3182	28	5	∏	∏	NUM
ejpam-3182	28	6	p∈w	p∈w	NOUN
ejpam-3182	28	7	(	(	PUNCT
ejpam-3182	28	8	1−	1−	NUM
ejpam-3182	28	9	1	1	NUM
ejpam-3182	28	10	p2	p2	PROPN
ejpam-3182	28	11	)	)	PUNCT
ejpam-3182	28	12	,	,	PUNCT
ejpam-3182	28	13	(	(	PUNCT
ejpam-3182	28	14	4	4	X
ejpam-3182	28	15	)	)	PUNCT
ejpam-3182	28	16	where	where	SCONJ
ejpam-3182	28	17	α2	α2	ADV
ejpam-3182	28	18	>	>	X
ejpam-3182	28	19	0	0	PUNCT
ejpam-3182	28	20	is	be	AUX
ejpam-3182	28	21	artin	artin	PROPN
ejpam-3182	28	22	constant	constant	ADJ
ejpam-3182	28	23	,	,	PUNCT
ejpam-3182	28	24	γ2	γ2	PROPN
ejpam-3182	28	25	is	be	AUX
ejpam-3182	28	26	a	a	DET
ejpam-3182	28	27	generalized	generalize	VERB
ejpam-3182	28	28	euler	euler	NOUN
ejpam-3182	28	29	constant	constant	PROPN
ejpam-3182	28	30	,	,	PUNCT
ejpam-3182	28	31	the	the	DET
ejpam-3182	28	32	gamma	gamma	NOUN
ejpam-3182	28	33	function	function	NOUN
ejpam-3182	28	34	is	be	AUX
ejpam-3182	28	35	defined	define	VERB
ejpam-3182	28	36	by	by	ADP
ejpam-3182	28	37	γ(s	γ(	NOUN
ejpam-3182	28	38	)	)	PUNCT
ejpam-3182	29	1	=	=	PUNCT
ejpam-3182	29	2	∫∞	∫∞	NOUN
ejpam-3182	29	3	0	0	NUM
ejpam-3182	30	1	xs−1exdx	xs−1exdx	PROPN
ejpam-3182	30	2	,	,	PUNCT
ejpam-3182	30	3	where	where	SCONJ
ejpam-3182	30	4	s	s	VERB
ejpam-3182	30	5	∈	∈	PROPN
ejpam-3182	30	6	c	c	NOUN
ejpam-3182	30	7	is	be	AUX
ejpam-3182	30	8	a	a	DET
ejpam-3182	30	9	complex	complex	ADJ
ejpam-3182	30	10	number	number	NOUN
ejpam-3182	30	11	,	,	PUNCT
ejpam-3182	30	12	and	and	CCONJ
ejpam-3182	30	13	w	w	X
ejpam-3182	30	14	=	=	PUNCT
ejpam-3182	30	15	{	{	PUNCT
ejpam-3182	30	16	p	p	NOUN
ejpam-3182	30	17	∈	∈	PROPN
ejpam-3182	30	18	p	p	NOUN
ejpam-3182	30	19	:	:	PUNCT
ejpam-3182	30	20	2p−1	2p−1	NUM
ejpam-3182	30	21	−	−	NOUN
ejpam-3182	30	22	1	1	NUM
ejpam-3182	30	23	≡	≡	PROPN
ejpam-3182	30	24	0	0	NUM
ejpam-3182	30	25	mod	mod	PROPN
ejpam-3182	30	26	p	p	PROPN
ejpam-3182	30	27	and	and	CCONJ
ejpam-3182	30	28	2p−1	2p−1	NUM
ejpam-3182	30	29	−	−	NOUN
ejpam-3182	30	30	1	1	NUM
ejpam-3182	30	31	≡	≡	PROPN
ejpam-3182	30	32	0	0	NUM
ejpam-3182	30	33	mod	mod	PROPN
ejpam-3182	30	34	p2	p2	PROPN
ejpam-3182	30	35	}	}	PUNCT
ejpam-3182	30	36	(	(	PUNCT
ejpam-3182	30	37	5	5	NUM
ejpam-3182	30	38	)	)	PUNCT
ejpam-3182	30	39	is	be	AUX
ejpam-3182	30	40	the	the	DET
ejpam-3182	30	41	subset	subset	NOUN
ejpam-3182	30	42	of	of	ADP
ejpam-3182	30	43	wieferich	wieferich	NOUN
ejpam-3182	30	44	primes	prime	VERB
ejpam-3182	30	45	for	for	ADP
ejpam-3182	30	46	base	base	NOUN
ejpam-3182	30	47	2	2	NUM
ejpam-3182	30	48	,	,	PUNCT
ejpam-3182	30	49	for	for	ADP
ejpam-3182	30	50	all	all	DET
ejpam-3182	30	51	large	large	ADJ
ejpam-3182	30	52	numbers	number	NOUN
ejpam-3182	30	53	x	x	X
ejpam-3182	30	54	≥	≥	NUM
ejpam-3182	30	55	1	1	NUM
ejpam-3182	30	56	.	.	PUNCT
ejpam-3182	31	1	the	the	DET
ejpam-3182	31	2	average	average	ADJ
ejpam-3182	31	3	order	order	NOUN
ejpam-3182	31	4	of	of	ADP
ejpam-3182	31	5	the	the	DET
ejpam-3182	31	6	counting	counting	NOUN
ejpam-3182	31	7	function	function	NOUN
ejpam-3182	31	8	n2(x	n2(x	NOUN
ejpam-3182	31	9	)	)	PUNCT
ejpam-3182	31	10	has	have	VERB
ejpam-3182	31	11	the	the	DET
ejpam-3182	31	12	same	same	ADJ
ejpam-3182	31	13	average	average	ADJ
ejpam-3182	31	14	order	order	NOUN
ejpam-3182	31	15	as	as	SCONJ
ejpam-3182	31	16	the	the	DET
ejpam-3182	31	17	counting	counting	NOUN
ejpam-3182	31	18	function	function	NOUN
ejpam-3182	31	19	l(x	l(x	PROPN
ejpam-3182	31	20	)	)	PUNCT
ejpam-3182	31	21	=	=	PUNCT
ejpam-3182	31	22	#	#	SYM
ejpam-3182	31	23	{	{	PUNCT
ejpam-3182	31	24	n	n	ADV
ejpam-3182	31	25	≤	≤	NOUN
ejpam-3182	31	26	x	x	SYM
ejpam-3182	31	27	:	:	PUNCT
ejpam-3182	31	28	µ(λ(n	µ(λ(n	NUM
ejpam-3182	31	29	)	)	PUNCT
ejpam-3182	31	30	)	)	PUNCT
ejpam-3182	32	1	=	=	PRON
ejpam-3182	32	2	±1	±1	VERB
ejpam-3182	32	3	}	}	PUNCT
ejpam-3182	32	4	=	=	SYM
ejpam-3182	32	5	(	(	PUNCT
ejpam-3182	32	6	κ	κ	NOUN
ejpam-3182	32	7	+	+	X
ejpam-3182	32	8	o(1))x(log	o(1))x(log	NOUN
ejpam-3182	32	9	x)α2−1	x)α2−1	NOUN
ejpam-3182	32	10	for	for	ADP
ejpam-3182	32	11	the	the	DET
ejpam-3182	32	12	number	number	NOUN
ejpam-3182	32	13	of	of	ADP
ejpam-3182	32	14	squarefree	squarefree	NOUN
ejpam-3182	32	15	values	value	NOUN
ejpam-3182	32	16	of	of	ADP
ejpam-3182	32	17	the	the	DET
ejpam-3182	32	18	carmichael	carmichael	PROPN
ejpam-3182	32	19	function	function	PROPN
ejpam-3182	32	20	λ	λ	PROPN
ejpam-3182	32	21	,	,	PUNCT
ejpam-3182	32	22	but	but	CCONJ
ejpam-3182	32	23	it	it	PRON
ejpam-3182	32	24	has	have	VERB
ejpam-3182	32	25	a	a	DET
ejpam-3182	32	26	different	different	ADJ
ejpam-3182	32	27	density	density	NOUN
ejpam-3182	32	28	κ	κ	NOUN
ejpam-3182	32	29	6=	6=	NOUN
ejpam-3182	32	30	eγ2−γα2	eγ2−γα2	ADJ
ejpam-3182	32	31	/	/	SYM
ejpam-3182	32	32	γ	γ	X
ejpam-3182	32	33	(	(	PUNCT
ejpam-3182	32	34	α2	α2	PROPN
ejpam-3182	32	35	)	)	PUNCT
ejpam-3182	32	36	,	,	PUNCT
ejpam-3182	32	37	see	see	VERB
ejpam-3182	32	38	[	[	X
ejpam-3182	32	39	25	25	NUM
ejpam-3182	32	40	]	]	PUNCT
ejpam-3182	32	41	.	.	PUNCT
ejpam-3182	33	1	this	this	PRON
ejpam-3182	33	2	should	should	AUX
ejpam-3182	33	3	be	be	AUX
ejpam-3182	33	4	compared	compare	VERB
ejpam-3182	33	5	to	to	ADP
ejpam-3182	33	6	the	the	DET
ejpam-3182	33	7	counting	counting	NOUN
ejpam-3182	33	8	function	function	NOUN
ejpam-3182	33	9	π2(x	π2(x	NUM
ejpam-3182	33	10	)	)	PUNCT
ejpam-3182	33	11	=	=	SYM
ejpam-3182	33	12	#	#	NOUN
ejpam-3182	33	13	{	{	PUNCT
ejpam-3182	33	14	p	p	NOUN
ejpam-3182	33	15	≤	≤	PROPN
ejpam-3182	33	16	x	x	SYM
ejpam-3182	33	17	:	:	PUNCT
ejpam-3182	33	18	ordp(2	ordp(2	ADJ
ejpam-3182	33	19	)	)	PUNCT
ejpam-3182	33	20	=	=	PUNCT
ejpam-3182	34	1	p−	p−	NOUN
ejpam-3182	34	2	1	1	X
ejpam-3182	34	3	}	}	PUNCT
ejpam-3182	34	4	=	=	SYM
ejpam-3182	34	5	α2π(x	α2π(x	PROPN
ejpam-3182	34	6	)	)	PUNCT
ejpam-3182	34	7	for	for	ADP
ejpam-3182	34	8	the	the	DET
ejpam-3182	34	9	set	set	NOUN
ejpam-3182	34	10	of	of	ADP
ejpam-3182	34	11	primes	prime	NOUN
ejpam-3182	34	12	having	have	VERB
ejpam-3182	34	13	2	2	NUM
ejpam-3182	34	14	as	as	ADP
ejpam-3182	34	15	a	a	DET
ejpam-3182	34	16	primitive	primitive	ADJ
ejpam-3182	34	17	root	root	NOUN
ejpam-3182	34	18	,	,	PUNCT
ejpam-3182	34	19	and	and	CCONJ
ejpam-3182	34	20	the	the	DET
ejpam-3182	34	21	counting	counting	NOUN
ejpam-3182	34	22	function	function	NOUN
ejpam-3182	34	23	t	t	PROPN
ejpam-3182	34	24	(	(	PUNCT
ejpam-3182	34	25	x	x	X
ejpam-3182	34	26	)	)	PUNCT
ejpam-3182	34	27	=	=	SYM
ejpam-3182	34	28	#	#	SYM
ejpam-3182	34	29	{	{	PUNCT
ejpam-3182	34	30	p	p	NOUN
ejpam-3182	34	31	≤	≤	PROPN
ejpam-3182	34	32	x	x	SYM
ejpam-3182	34	33	:	:	PUNCT
ejpam-3182	34	34	µ(ϕ(n	µ(ϕ(n	NOUN
ejpam-3182	34	35	)	)	PUNCT
ejpam-3182	34	36	)	)	PUNCT
ejpam-3182	35	1	=	=	PRON
ejpam-3182	35	2	±1	±1	VERB
ejpam-3182	35	3	}	}	PUNCT
ejpam-3182	35	4	=	=	SYM
ejpam-3182	35	5	α2π(x	α2π(x	PROPN
ejpam-3182	35	6	)	)	PUNCT
ejpam-3182	35	7	of	of	ADP
ejpam-3182	35	8	squarefree	squarefree	NOUN
ejpam-3182	35	9	values	value	NOUN
ejpam-3182	35	10	of	of	ADP
ejpam-3182	35	11	the	the	DET
ejpam-3182	35	12	euler	euler	PROPN
ejpam-3182	35	13	totient	totient	PROPN
ejpam-3182	35	14	function	function	PROPN
ejpam-3182	35	15	ϕ.	ϕ.	NOUN
ejpam-3182	35	16	all	all	DET
ejpam-3182	35	17	these	these	DET
ejpam-3182	35	18	asymptotic	asymptotic	ADJ
ejpam-3182	35	19	formulae	formulae	NOUN
ejpam-3182	35	20	are	be	AUX
ejpam-3182	35	21	closely	closely	ADV
ejpam-3182	35	22	related	relate	VERB
ejpam-3182	35	23	and	and	CCONJ
ejpam-3182	35	24	scaled	scale	VERB
ejpam-3182	35	25	by	by	ADP
ejpam-3182	35	26	the	the	DET
ejpam-3182	35	27	constant	constant	ADJ
ejpam-3182	35	28	α2	α2	NOUN
ejpam-3182	35	29	.	.	PUNCT
ejpam-3182	36	1	the	the	DET
ejpam-3182	36	2	general	general	ADJ
ejpam-3182	36	3	asymptotic	asymptotic	ADJ
ejpam-3182	36	4	formula	formula	NOUN
ejpam-3182	36	5	for	for	ADP
ejpam-3182	36	6	the	the	DET
ejpam-3182	36	7	number	number	NOUN
ejpam-3182	36	8	of	of	ADP
ejpam-3182	36	9	integers	integer	NOUN
ejpam-3182	36	10	n	n	PRON
ejpam-3182	36	11	≤	≤	NUM
ejpam-3182	36	12	x	x	PUNCT
ejpam-3182	36	13	such	such	ADJ
ejpam-3182	36	14	that	that	DET
ejpam-3182	36	15	u	u	PROPN
ejpam-3182	36	16	6=	6=	ADP
ejpam-3182	36	17	±1	±1	VERB
ejpam-3182	36	18	,	,	PUNCT
ejpam-3182	36	19	v2	v2	PROPN
ejpam-3182	36	20	is	be	AUX
ejpam-3182	36	21	a	a	DET
ejpam-3182	36	22	primitive	primitive	ADJ
ejpam-3182	36	23	root	root	NOUN
ejpam-3182	36	24	mod	mod	NOUN
ejpam-3182	36	25	n	n	PRON
ejpam-3182	36	26	has	have	VERB
ejpam-3182	36	27	the	the	DET
ejpam-3182	36	28	form	form	NOUN
ejpam-3182	36	29	nu(x	nu(x	NOUN
ejpam-3182	36	30	)	)	PUNCT
ejpam-3182	36	31	=	=	SYM
ejpam-3182	36	32	(	(	PUNCT
ejpam-3182	36	33	eγu−γαu	eγu−γαu	X
ejpam-3182	36	34	γ	γ	X
ejpam-3182	36	35	(	(	PUNCT
ejpam-3182	36	36	αu	αu	NOUN
ejpam-3182	36	37	)	)	PUNCT
ejpam-3182	36	38	+	+	NUM
ejpam-3182	36	39	o(1	o(1	NOUN
ejpam-3182	36	40	)	)	PUNCT
ejpam-3182	36	41	)	)	PUNCT
ejpam-3182	37	1	x	x	X
ejpam-3182	37	2	(	(	PUNCT
ejpam-3182	37	3	log	log	NOUN
ejpam-3182	37	4	x)1−αu	x)1−αu	PUNCT
ejpam-3182	37	5	∏	∏	NUM
ejpam-3182	37	6	p∈w	p∈w	NOUN
ejpam-3182	37	7	(	(	PUNCT
ejpam-3182	37	8	1−	1−	NUM
ejpam-3182	37	9	1	1	NUM
ejpam-3182	37	10	p2	p2	PROPN
ejpam-3182	37	11	)	)	PUNCT
ejpam-3182	37	12	,	,	PUNCT
ejpam-3182	37	13	(	(	PUNCT
ejpam-3182	37	14	6	6	X
ejpam-3182	37	15	)	)	PUNCT
ejpam-3182	37	16	where	where	SCONJ
ejpam-3182	37	17	αu	αu	ADV
ejpam-3182	37	18	>	>	X
ejpam-3182	37	19	0	0	NUM
ejpam-3182	37	20	is	be	AUX
ejpam-3182	37	21	artin	artin	PROPN
ejpam-3182	37	22	constant	constant	ADJ
ejpam-3182	37	23	,	,	PUNCT
ejpam-3182	37	24	γu	γu	PRON
ejpam-3182	37	25	is	be	AUX
ejpam-3182	37	26	a	a	DET
ejpam-3182	37	27	generalized	generalize	VERB
ejpam-3182	37	28	euler	euler	NOUN
ejpam-3182	37	29	constant	constant	ADJ
ejpam-3182	37	30	,	,	PUNCT
ejpam-3182	37	31	and	and	CCONJ
ejpam-3182	37	32	w	w	X
ejpam-3182	37	33	=	=	PUNCT
ejpam-3182	37	34	{	{	PUNCT
ejpam-3182	37	35	p	p	NOUN
ejpam-3182	37	36	∈	∈	PROPN
ejpam-3182	37	37	p	p	NOUN
ejpam-3182	37	38	:	:	PUNCT
ejpam-3182	37	39	up−1	up−1	PROPN
ejpam-3182	38	1	−	−	PROPN
ejpam-3182	38	2	1	1	NUM
ejpam-3182	38	3	≡	≡	PROPN
ejpam-3182	38	4	0	0	NUM
ejpam-3182	38	5	mod	mod	PROPN
ejpam-3182	38	6	p	p	PROPN
ejpam-3182	38	7	and	and	CCONJ
ejpam-3182	38	8	up−1	up−1	PROPN
ejpam-3182	38	9	−	−	PROPN
ejpam-3182	38	10	1	1	NUM
ejpam-3182	38	11	≡	≡	PROPN
ejpam-3182	38	12	0	0	NUM
ejpam-3182	38	13	mod	mod	PROPN
ejpam-3182	38	14	p2	p2	PROPN
ejpam-3182	38	15	}	}	PUNCT
ejpam-3182	38	16	(	(	PUNCT
ejpam-3182	38	17	7	7	X
ejpam-3182	38	18	)	)	PUNCT
ejpam-3182	38	19	is	be	AUX
ejpam-3182	38	20	the	the	DET
ejpam-3182	38	21	subset	subset	NOUN
ejpam-3182	38	22	of	of	ADP
ejpam-3182	38	23	abel	abel	PROPN
ejpam-3182	38	24	-	-	PUNCT
ejpam-3182	38	25	wieferich	wieferich	NOUN
ejpam-3182	38	26	primes	prime	VERB
ejpam-3182	38	27	for	for	ADP
ejpam-3182	38	28	base	base	NOUN
ejpam-3182	38	29	u	u	PROPN
ejpam-3182	38	30	≥	≥	NUM
ejpam-3182	38	31	2	2	NUM
ejpam-3182	38	32	,	,	PUNCT
ejpam-3182	38	33	for	for	ADP
ejpam-3182	38	34	all	all	DET
ejpam-3182	38	35	large	large	ADJ
ejpam-3182	38	36	numbers	number	NOUN
ejpam-3182	38	37	x	x	X
ejpam-3182	38	38	≥	≥	NUM
ejpam-3182	38	39	1	1	NUM
ejpam-3182	38	40	.	.	PUNCT
ejpam-3182	39	1	the	the	DET
ejpam-3182	39	2	proof	proof	NOUN
ejpam-3182	39	3	is	be	AUX
ejpam-3182	39	4	the	the	DET
ejpam-3182	39	5	same	same	ADJ
ejpam-3182	39	6	as	as	ADP
ejpam-3182	39	7	the	the	DET
ejpam-3182	39	8	proof	proof	NOUN
ejpam-3182	39	9	of	of	ADP
ejpam-3182	39	10	theorem	theorem	ADJ
ejpam-3182	39	11	1.1	1.1	NUM
ejpam-3182	39	12	mutatis	mutatis	NOUN
ejpam-3182	39	13	mutandis	mutandi	NOUN
ejpam-3182	39	14	.	.	PUNCT
ejpam-3182	40	1	sections	section	NOUN
ejpam-3182	40	2	2	2	NUM
ejpam-3182	40	3	to	to	PART
ejpam-3182	40	4	6	6	NUM
ejpam-3182	40	5	provide	provide	VERB
ejpam-3182	40	6	some	some	DET
ejpam-3182	40	7	essential	essential	ADJ
ejpam-3182	40	8	background	background	NOUN
ejpam-3182	40	9	results	result	NOUN
ejpam-3182	40	10	.	.	PUNCT
ejpam-3182	41	1	the	the	DET
ejpam-3182	41	2	proof	proof	NOUN
ejpam-3182	41	3	of	of	ADP
ejpam-3182	41	4	theorem	theorem	ADJ
ejpam-3182	41	5	1.1	1.1	NUM
ejpam-3182	41	6	is	be	AUX
ejpam-3182	41	7	settled	settle	VERB
ejpam-3182	41	8	in	in	ADP
ejpam-3182	41	9	section	section	NOUN
ejpam-3182	41	10	7	7	NUM
ejpam-3182	41	11	.	.	PUNCT
ejpam-3182	41	12	n.	n.	PROPN
ejpam-3182	41	13	a.	a.	PROPN
ejpam-3182	41	14	carella	carella	PROPN
ejpam-3182	41	15	/	/	SYM
ejpam-3182	41	16	eur	eur	PROPN
ejpam-3182	41	17	.	.	PUNCT
ejpam-3182	42	1	j.	j.	PROPN
ejpam-3182	42	2	pure	pure	PROPN
ejpam-3182	42	3	appl	appl	PROPN
ejpam-3182	42	4	.	.	PROPN
ejpam-3182	42	5	math	math	PROPN
ejpam-3182	42	6	,	,	PUNCT
ejpam-3182	42	7	11	11	NUM
ejpam-3182	42	8	(	(	PUNCT
ejpam-3182	42	9	1	1	NUM
ejpam-3182	42	10	)	)	PUNCT
ejpam-3182	42	11	(	(	PUNCT
ejpam-3182	42	12	2018	2018	NUM
ejpam-3182	42	13	)	)	PUNCT
ejpam-3182	42	14	,	,	PUNCT
ejpam-3182	42	15	23	23	NUM
ejpam-3182	42	16	-	-	SYM
ejpam-3182	42	17	34	34	NUM
ejpam-3182	42	18	25	25	NUM
ejpam-3182	42	19	2	2	NUM
ejpam-3182	42	20	.	.	PUNCT
ejpam-3182	43	1	some	some	DET
ejpam-3182	43	2	arithmetic	arithmetic	ADJ
ejpam-3182	43	3	functions	function	NOUN
ejpam-3182	43	4	the	the	DET
ejpam-3182	43	5	euler	euler	PROPN
ejpam-3182	43	6	totient	totient	PROPN
ejpam-3182	43	7	function	function	NOUN
ejpam-3182	43	8	counts	count	VERB
ejpam-3182	43	9	the	the	DET
ejpam-3182	43	10	number	number	NOUN
ejpam-3182	43	11	of	of	ADP
ejpam-3182	43	12	relatively	relatively	ADV
ejpam-3182	43	13	prime	prime	ADJ
ejpam-3182	43	14	integers	integer	NOUN
ejpam-3182	43	15	ϕ(n	ϕ(n	X
ejpam-3182	43	16	)	)	PUNCT
ejpam-3182	44	1	=	=	PUNCT
ejpam-3182	44	2	#	#	SYM
ejpam-3182	44	3	{	{	PUNCT
ejpam-3182	44	4	k	k	NOUN
ejpam-3182	44	5	:	:	PUNCT
ejpam-3182	44	6	gcd(k	gcd(k	PROPN
ejpam-3182	44	7	,	,	PUNCT
ejpam-3182	44	8	n	n	CCONJ
ejpam-3182	44	9	)	)	PUNCT
ejpam-3182	44	10	=	=	SYM
ejpam-3182	44	11	1	1	NUM
ejpam-3182	44	12	}	}	PUNCT
ejpam-3182	44	13	.	.	PUNCT
ejpam-3182	45	1	this	this	DET
ejpam-3182	45	2	counting	counting	NOUN
ejpam-3182	45	3	function	function	NOUN
ejpam-3182	45	4	is	be	AUX
ejpam-3182	45	5	compactly	compactly	ADV
ejpam-3182	45	6	expressed	express	VERB
ejpam-3182	45	7	by	by	ADP
ejpam-3182	45	8	the	the	DET
ejpam-3182	45	9	analytic	analytic	ADJ
ejpam-3182	45	10	formula	formula	NOUN
ejpam-3182	45	11	ϕ(n	ϕ(n	X
ejpam-3182	45	12	)	)	PUNCT
ejpam-3182	46	1	=	=	SYM
ejpam-3182	46	2	n	n	CCONJ
ejpam-3182	46	3	∏	∏	PROPN
ejpam-3182	46	4	p|n(1−	p|n(1−	ADJ
ejpam-3182	46	5	1	1	NUM
ejpam-3182	46	6	/	/	SYM
ejpam-3182	46	7	p	p	NOUN
ejpam-3182	46	8	)	)	PUNCT
ejpam-3182	46	9	,	,	PUNCT
ejpam-3182	46	10	n	n	PROPN
ejpam-3182	46	11	∈	∈	PROPN
ejpam-3182	46	12	n.	n.	NOUN
ejpam-3182	46	13	lemma	lemma	PROPN
ejpam-3182	46	14	2.1	2.1	NUM
ejpam-3182	46	15	.	.	PUNCT
ejpam-3182	47	1	(	(	PUNCT
ejpam-3182	47	2	fermat	fermat	PROPN
ejpam-3182	47	3	-	-	PUNCT
ejpam-3182	47	4	euler	euler	NOUN
ejpam-3182	47	5	)	)	PUNCT
ejpam-3182	47	6	if	if	SCONJ
ejpam-3182	47	7	a	a	DET
ejpam-3182	47	8	∈	∈	PROPN
ejpam-3182	47	9	z	z	NOUN
ejpam-3182	47	10	is	be	AUX
ejpam-3182	47	11	an	an	DET
ejpam-3182	47	12	integer	integer	NOUN
ejpam-3182	47	13	such	such	ADJ
ejpam-3182	47	14	that	that	SCONJ
ejpam-3182	47	15	gcd(a	gcd(a	PROPN
ejpam-3182	47	16	,	,	PUNCT
ejpam-3182	47	17	n	n	CCONJ
ejpam-3182	47	18	)	)	PUNCT
ejpam-3182	47	19	=	=	SYM
ejpam-3182	47	20	1	1	NUM
ejpam-3182	47	21	,	,	PUNCT
ejpam-3182	47	22	then	then	ADV
ejpam-3182	47	23	aϕ(n	aϕ(n	PUNCT
ejpam-3182	47	24	)	)	PUNCT
ejpam-3182	47	25	≡	≡	PROPN
ejpam-3182	47	26	1	1	NUM
ejpam-3182	47	27	mod	mod	PROPN
ejpam-3182	47	28	n.	n.	PROPN
ejpam-3182	47	29	the	the	DET
ejpam-3182	47	30	carmichael	carmichael	PROPN
ejpam-3182	47	31	function	function	NOUN
ejpam-3182	47	32	is	be	AUX
ejpam-3182	47	33	basically	basically	ADV
ejpam-3182	47	34	a	a	DET
ejpam-3182	47	35	refinement	refinement	NOUN
ejpam-3182	47	36	of	of	ADP
ejpam-3182	47	37	the	the	DET
ejpam-3182	47	38	euler	euler	PROPN
ejpam-3182	47	39	totient	totient	PROPN
ejpam-3182	47	40	function	function	NOUN
ejpam-3182	47	41	to	to	ADP
ejpam-3182	47	42	the	the	DET
ejpam-3182	47	43	finite	finite	PROPN
ejpam-3182	47	44	ring	ring	PROPN
ejpam-3182	47	45	z	z	PROPN
ejpam-3182	47	46	/	/	SYM
ejpam-3182	47	47	nz	nz	PROPN
ejpam-3182	47	48	.	.	PROPN
ejpam-3182	47	49	given	give	VERB
ejpam-3182	47	50	an	an	DET
ejpam-3182	47	51	integer	integer	NOUN
ejpam-3182	47	52	n	n	NOUN
ejpam-3182	47	53	=	=	PUNCT
ejpam-3182	48	1	pv11	pv11	PROPN
ejpam-3182	48	2	p	p	PROPN
ejpam-3182	48	3	v2	v2	PROPN
ejpam-3182	48	4	2	2	NUM
ejpam-3182	48	5	·	·	PUNCT
ejpam-3182	48	6	·	·	PUNCT
ejpam-3182	48	7	·	·	PUNCT
ejpam-3182	49	1	p	p	NOUN
ejpam-3182	49	2	vt	vt	PROPN
ejpam-3182	49	3	t	t	PROPN
ejpam-3182	49	4	,	,	PUNCT
ejpam-3182	49	5	the	the	DET
ejpam-3182	49	6	carmichael	carmichael	PROPN
ejpam-3182	49	7	function	function	NOUN
ejpam-3182	49	8	is	be	AUX
ejpam-3182	49	9	defined	define	VERB
ejpam-3182	49	10	by	by	ADP
ejpam-3182	49	11	λ(n	λ(n	PROPN
ejpam-3182	49	12	)	)	PUNCT
ejpam-3182	49	13	=	=	NOUN
ejpam-3182	49	14	lcm	lcm	NOUN
ejpam-3182	49	15	(	(	PUNCT
ejpam-3182	49	16	λ	λ	PROPN
ejpam-3182	49	17	(	(	PUNCT
ejpam-3182	49	18	pv11	pv11	PROPN
ejpam-3182	49	19	)	)	PUNCT
ejpam-3182	49	20	,	,	PUNCT
ejpam-3182	49	21	λ	λ	X
ejpam-3182	49	22	(	(	PUNCT
ejpam-3182	49	23	pv22	pv22	PROPN
ejpam-3182	49	24	)	)	PUNCT
ejpam-3182	49	25	·	·	PUNCT
ejpam-3182	49	26	·	·	PUNCT
ejpam-3182	50	1	·	·	PUNCT
ejpam-3182	50	2	λ	λ	X
ejpam-3182	50	3	(	(	PUNCT
ejpam-3182	50	4	pvtt	pvtt	PROPN
ejpam-3182	50	5	)	)	PUNCT
ejpam-3182	50	6	)	)	PUNCT
ejpam-3182	51	1	=	=	SYM
ejpam-3182	51	2	∏	∏	PROPN
ejpam-3182	51	3	pv	pv	X
ejpam-3182	51	4	||λ(n	||λ(n	ADJ
ejpam-3182	51	5	)	)	PUNCT
ejpam-3182	51	6	pv	pv	INTJ
ejpam-3182	51	7	,	,	PUNCT
ejpam-3182	51	8	(	(	PUNCT
ejpam-3182	51	9	8)	8)	NUM
ejpam-3182	51	10	where	where	SCONJ
ejpam-3182	51	11	the	the	DET
ejpam-3182	51	12	symbol	symbol	NOUN
ejpam-3182	51	13	pv	pv	INTJ
ejpam-3182	51	14	||	||	NOUN
ejpam-3182	52	1	n	n	CCONJ
ejpam-3182	52	2	,	,	PUNCT
ejpam-3182	52	3	ν	ν	X
ejpam-3182	52	4	≥	≥	NOUN
ejpam-3182	52	5	0	0	NUM
ejpam-3182	52	6	,	,	PUNCT
ejpam-3182	52	7	denotes	denote	VERB
ejpam-3182	52	8	the	the	DET
ejpam-3182	52	9	maximal	maximal	ADJ
ejpam-3182	52	10	prime	prime	ADJ
ejpam-3182	52	11	power	power	NOUN
ejpam-3182	52	12	divisor	divisor	NOUN
ejpam-3182	52	13	of	of	ADP
ejpam-3182	52	14	n	n	PRON
ejpam-3182	52	15	≥	≥	NUM
ejpam-3182	52	16	1	1	NUM
ejpam-3182	52	17	,	,	PUNCT
ejpam-3182	52	18	and	and	CCONJ
ejpam-3182	52	19	λ	λ	X
ejpam-3182	52	20	(	(	PUNCT
ejpam-3182	52	21	pv	pv	NOUN
ejpam-3182	52	22	)	)	PUNCT
ejpam-3182	52	23	=	=	PRON
ejpam-3182	52	24	{	{	PUNCT
ejpam-3182	52	25	ϕ	ϕ	NOUN
ejpam-3182	52	26	(	(	PUNCT
ejpam-3182	52	27	pv	pv	NOUN
ejpam-3182	52	28	)	)	PUNCT
ejpam-3182	52	29	if	if	SCONJ
ejpam-3182	52	30	p	p	PROPN
ejpam-3182	52	31	≥	≥	PUNCT
ejpam-3182	52	32	3	3	NUM
ejpam-3182	52	33	or	or	CCONJ
ejpam-3182	52	34	v	v	ADP
ejpam-3182	52	35	≤	≤	NUM
ejpam-3182	52	36	2	2	NUM
ejpam-3182	52	37	,	,	PUNCT
ejpam-3182	52	38	2v−2	2v−2	NUM
ejpam-3182	52	39	if	if	SCONJ
ejpam-3182	52	40	p	p	NOUN
ejpam-3182	52	41	=	=	NOUN
ejpam-3182	52	42	2	2	NUM
ejpam-3182	52	43	and	and	CCONJ
ejpam-3182	52	44	v	v	ADP
ejpam-3182	52	45	≥	≥	NOUN
ejpam-3182	52	46	3	3	NUM
ejpam-3182	52	47	.	.	PUNCT
ejpam-3182	53	1	(	(	PUNCT
ejpam-3182	53	2	9	9	X
ejpam-3182	53	3	)	)	PUNCT
ejpam-3182	53	4	the	the	DET
ejpam-3182	53	5	two	two	NUM
ejpam-3182	53	6	functions	function	NOUN
ejpam-3182	53	7	coincide	coincide	NOUN
ejpam-3182	53	8	,	,	PUNCT
ejpam-3182	53	9	that	that	ADV
ejpam-3182	53	10	is	is	ADV
ejpam-3182	53	11	,	,	PUNCT
ejpam-3182	53	12	ϕ(n	ϕ(n	X
ejpam-3182	53	13	)	)	PUNCT
ejpam-3182	53	14	=	=	SYM
ejpam-3182	54	1	λ(n	λ(n	PROPN
ejpam-3182	54	2	)	)	PUNCT
ejpam-3182	54	3	if	if	SCONJ
ejpam-3182	54	4	n	n	NOUN
ejpam-3182	54	5	=	=	SYM
ejpam-3182	54	6	2	2	NUM
ejpam-3182	54	7	,	,	PUNCT
ejpam-3182	54	8	4	4	NUM
ejpam-3182	54	9	,	,	PUNCT
ejpam-3182	54	10	pm	pm	NOUN
ejpam-3182	54	11	,	,	PUNCT
ejpam-3182	54	12	or	or	CCONJ
ejpam-3182	54	13	2pm	2pm	NOUN
ejpam-3182	54	14	,	,	PUNCT
ejpam-3182	54	15	m	m	VERB
ejpam-3182	54	16	≥	≥	NOUN
ejpam-3182	54	17	1	1	NUM
ejpam-3182	54	18	.	.	PUNCT
ejpam-3182	54	19	and	and	CCONJ
ejpam-3182	54	20	ϕ	ϕ	X
ejpam-3182	54	21	(	(	PUNCT
ejpam-3182	54	22	2	2	NUM
ejpam-3182	54	23	m	m	NOUN
ejpam-3182	54	24	)	)	PUNCT
ejpam-3182	54	25	=	=	SYM
ejpam-3182	54	26	2λ	2λ	NOUN
ejpam-3182	54	27	(	(	PUNCT
ejpam-3182	54	28	2	2	NUM
ejpam-3182	54	29	m	m	NOUN
ejpam-3182	54	30	)	)	PUNCT
ejpam-3182	54	31	.	.	PUNCT
ejpam-3182	55	1	in	in	ADP
ejpam-3182	55	2	a	a	DET
ejpam-3182	55	3	few	few	ADJ
ejpam-3182	55	4	other	other	ADJ
ejpam-3182	55	5	cases	case	NOUN
ejpam-3182	55	6	,	,	PUNCT
ejpam-3182	55	7	there	there	PRON
ejpam-3182	55	8	are	be	VERB
ejpam-3182	55	9	some	some	DET
ejpam-3182	55	10	simple	simple	ADJ
ejpam-3182	55	11	relationships	relationship	NOUN
ejpam-3182	55	12	between	between	ADP
ejpam-3182	55	13	ϕ(n	ϕ(n	PROPN
ejpam-3182	55	14	)	)	PUNCT
ejpam-3182	55	15	and	and	CCONJ
ejpam-3182	55	16	λ(n	λ(n	PROPN
ejpam-3182	55	17	)	)	PUNCT
ejpam-3182	55	18	.	.	PUNCT
ejpam-3182	56	1	in	in	ADP
ejpam-3182	56	2	fact	fact	NOUN
ejpam-3182	56	3	,	,	PUNCT
ejpam-3182	56	4	it	it	PRON
ejpam-3182	56	5	seamlessly	seamlessly	ADV
ejpam-3182	56	6	improves	improve	VERB
ejpam-3182	56	7	the	the	DET
ejpam-3182	56	8	fermat	fermat	PROPN
ejpam-3182	56	9	-	-	PUNCT
ejpam-3182	56	10	euler	euler	NOUN
ejpam-3182	56	11	theorem	theorem	PROPN
ejpam-3182	56	12	:	:	PUNCT
ejpam-3182	56	13	the	the	DET
ejpam-3182	56	14	improvement	improvement	NOUN
ejpam-3182	56	15	provides	provide	VERB
ejpam-3182	56	16	the	the	DET
ejpam-3182	56	17	least	least	ADJ
ejpam-3182	56	18	exponent	exponent	NOUN
ejpam-3182	56	19	λ(n	λ(n	PROPN
ejpam-3182	56	20	)	)	PUNCT
ejpam-3182	56	21	|	|	ADV
ejpam-3182	56	22	ϕ(n	ϕ(n	X
ejpam-3182	56	23	)	)	PUNCT
ejpam-3182	56	24	such	such	ADJ
ejpam-3182	56	25	that	that	PRON
ejpam-3182	56	26	aλ(n	aλ(n	PUNCT
ejpam-3182	56	27	)	)	PUNCT
ejpam-3182	56	28	≡	≡	PROPN
ejpam-3182	56	29	1	1	NUM
ejpam-3182	56	30	mod	mod	PROPN
ejpam-3182	56	31	n.	n.	PROPN
ejpam-3182	56	32	lemma	lemma	PROPN
ejpam-3182	56	33	2.2	2.2	NUM
ejpam-3182	56	34	.	.	PUNCT
ejpam-3182	57	1	(	(	PUNCT
ejpam-3182	57	2	[	[	X
ejpam-3182	57	3	4	4	NUM
ejpam-3182	57	4	]	]	PUNCT
ejpam-3182	57	5	)	)	PUNCT
ejpam-3182	57	6	let	let	VERB
ejpam-3182	57	7	n	n	PRON
ejpam-3182	57	8	∈	∈	PROPN
ejpam-3182	57	9	n	n	AUX
ejpam-3182	57	10	be	be	AUX
ejpam-3182	57	11	any	any	DET
ejpam-3182	57	12	given	give	VERB
ejpam-3182	57	13	integer	integer	NOUN
ejpam-3182	57	14	.	.	PUNCT
ejpam-3182	58	1	then	then	ADV
ejpam-3182	58	2	(	(	PUNCT
ejpam-3182	58	3	i	i	NOUN
ejpam-3182	58	4	)	)	PUNCT
ejpam-3182	58	5	the	the	DET
ejpam-3182	58	6	congruence	congruence	NOUN
ejpam-3182	58	7	aλ(n	aλ(n	X
ejpam-3182	58	8	)	)	PUNCT
ejpam-3182	58	9	≡	≡	PROPN
ejpam-3182	58	10	1	1	NUM
ejpam-3182	58	11	mod	mod	NOUN
ejpam-3182	58	12	n	n	PROPN
ejpam-3182	58	13	is	be	AUX
ejpam-3182	58	14	satisfied	satisfied	ADJ
ejpam-3182	58	15	by	by	ADP
ejpam-3182	58	16	every	every	DET
ejpam-3182	58	17	integer	integer	NOUN
ejpam-3182	58	18	a	a	DET
ejpam-3182	58	19	≥	≥	NUM
ejpam-3182	58	20	1	1	NUM
ejpam-3182	58	21	relatively	relatively	ADV
ejpam-3182	58	22	prime	prime	ADJ
ejpam-3182	58	23	to	to	ADP
ejpam-3182	58	24	n	n	CCONJ
ejpam-3182	58	25	,	,	PUNCT
ejpam-3182	58	26	that	that	PRON
ejpam-3182	58	27	is	be	AUX
ejpam-3182	58	28	gcd(a	gcd(a	PROPN
ejpam-3182	58	29	,	,	PUNCT
ejpam-3182	58	30	n	n	CCONJ
ejpam-3182	58	31	)	)	PUNCT
ejpam-3182	59	1	=	=	SYM
ejpam-3182	59	2	1	1	X
ejpam-3182	59	3	.	.	PUNCT
ejpam-3182	59	4	(	(	PUNCT
ejpam-3182	59	5	ii	ii	NOUN
ejpam-3182	59	6	)	)	PUNCT
ejpam-3182	59	7	in	in	ADP
ejpam-3182	59	8	every	every	DET
ejpam-3182	59	9	congruence	congruence	NOUN
ejpam-3182	59	10	xλ(n	xλ(n	PRON
ejpam-3182	59	11	)	)	PUNCT
ejpam-3182	59	12	≡	≡	PROPN
ejpam-3182	59	13	1	1	NUM
ejpam-3182	59	14	mod	mod	PROPN
ejpam-3182	59	15	n	n	CCONJ
ejpam-3182	59	16	,	,	PUNCT
ejpam-3182	59	17	a	a	DET
ejpam-3182	59	18	solution	solution	NOUN
ejpam-3182	59	19	x	x	PUNCT
ejpam-3182	59	20	=	=	SYM
ejpam-3182	59	21	u	u	NOUN
ejpam-3182	59	22	exists	exist	VERB
ejpam-3182	59	23	which	which	PRON
ejpam-3182	59	24	is	be	AUX
ejpam-3182	59	25	a	a	DET
ejpam-3182	59	26	primitive	primitive	ADJ
ejpam-3182	59	27	root	root	NOUN
ejpam-3182	59	28	mod	mod	NOUN
ejpam-3182	59	29	n	n	CCONJ
ejpam-3182	59	30	,	,	PUNCT
ejpam-3182	59	31	and	and	CCONJ
ejpam-3182	59	32	for	for	ADP
ejpam-3182	59	33	any	any	DET
ejpam-3182	59	34	such	such	ADJ
ejpam-3182	59	35	solution	solution	NOUN
ejpam-3182	59	36	u	u	NOUN
ejpam-3182	59	37	,	,	PUNCT
ejpam-3182	59	38	there	there	PRON
ejpam-3182	59	39	are	be	VERB
ejpam-3182	59	40	ϕ(λ(n	ϕ(λ(n	PROPN
ejpam-3182	59	41	)	)	PUNCT
ejpam-3182	59	42	)	)	PUNCT
ejpam-3182	59	43	primitive	primitive	ADJ
ejpam-3182	59	44	roots	root	NOUN
ejpam-3182	59	45	congruent	congruent	ADJ
ejpam-3182	59	46	to	to	ADP
ejpam-3182	59	47	powers	power	NOUN
ejpam-3182	59	48	of	of	ADP
ejpam-3182	59	49	u.	u.	NOUN
ejpam-3182	59	50	proof	proof	NOUN
ejpam-3182	59	51	.	.	PUNCT
ejpam-3182	60	1	(	(	PUNCT
ejpam-3182	60	2	i	i	NOUN
ejpam-3182	60	3	)	)	PUNCT
ejpam-3182	60	4	the	the	DET
ejpam-3182	60	5	number	number	NOUN
ejpam-3182	60	6	λ(n	λ(n	NOUN
ejpam-3182	60	7	)	)	PUNCT
ejpam-3182	60	8	is	be	AUX
ejpam-3182	60	9	a	a	DET
ejpam-3182	60	10	multiple	multiple	NOUN
ejpam-3182	60	11	of	of	ADP
ejpam-3182	60	12	every	every	DET
ejpam-3182	60	13	λ	λ	PROPN
ejpam-3182	60	14	(	(	PUNCT
ejpam-3182	60	15	pv	pv	NOUN
ejpam-3182	60	16	)	)	PUNCT
ejpam-3182	61	1	=	=	SYM
ejpam-3182	61	2	ϕ	ϕ	X
ejpam-3182	61	3	(	(	PUNCT
ejpam-3182	61	4	pv	pv	NOUN
ejpam-3182	61	5	)	)	PUNCT
ejpam-3182	61	6	such	such	ADJ
ejpam-3182	61	7	that	that	DET
ejpam-3182	61	8	pv	pv	INTJ
ejpam-3182	61	9	|	|	ADV
ejpam-3182	61	10	n.	n.	NOUN
ejpam-3182	61	11	ergo	ergo	NOUN
ejpam-3182	61	12	,	,	PUNCT
ejpam-3182	61	13	for	for	ADP
ejpam-3182	61	14	any	any	DET
ejpam-3182	61	15	relatively	relatively	ADV
ejpam-3182	61	16	prime	prime	ADJ
ejpam-3182	61	17	integer	integer	NOUN
ejpam-3182	61	18	a	a	DET
ejpam-3182	61	19	≥	≥	NOUN
ejpam-3182	61	20	2	2	NUM
ejpam-3182	61	21	,	,	PUNCT
ejpam-3182	61	22	the	the	DET
ejpam-3182	61	23	system	system	NOUN
ejpam-3182	61	24	of	of	ADP
ejpam-3182	61	25	congruences	congruence	NOUN
ejpam-3182	61	26	aλ(n	aλ(n	PUNCT
ejpam-3182	61	27	)	)	PUNCT
ejpam-3182	61	28	≡	≡	PROPN
ejpam-3182	61	29	1	1	NUM
ejpam-3182	61	30	mod	mod	PROPN
ejpam-3182	61	31	pv11	pv11	PROPN
ejpam-3182	61	32	,	,	PUNCT
ejpam-3182	61	33	aλ(n	aλ(n	NUM
ejpam-3182	61	34	)	)	PUNCT
ejpam-3182	61	35	≡	≡	PROPN
ejpam-3182	61	36	1	1	NUM
ejpam-3182	61	37	mod	mod	PROPN
ejpam-3182	61	38	pv22	pv22	PROPN
ejpam-3182	61	39	,	,	PUNCT
ejpam-3182	61	40	.	.	PUNCT
ejpam-3182	61	41	.	.	PUNCT
ejpam-3182	61	42	.	.	PUNCT
ejpam-3182	62	1	,	,	PUNCT
ejpam-3182	62	2	aλ(n	aλ(n	PUNCT
ejpam-3182	62	3	)	)	PUNCT
ejpam-3182	62	4	≡	≡	PROPN
ejpam-3182	62	5	1	1	NUM
ejpam-3182	62	6	mod	mod	PROPN
ejpam-3182	62	7	pvtt	pvtt	PROPN
ejpam-3182	62	8	,	,	PUNCT
ejpam-3182	62	9	(	(	PUNCT
ejpam-3182	62	10	10	10	NUM
ejpam-3182	62	11	)	)	PUNCT
ejpam-3182	62	12	where	where	SCONJ
ejpam-3182	62	13	t	t	NOUN
ejpam-3182	62	14	=	=	SYM
ejpam-3182	62	15	ω(n	ω(n	NUM
ejpam-3182	62	16	)	)	PUNCT
ejpam-3182	62	17	is	be	AUX
ejpam-3182	62	18	the	the	DET
ejpam-3182	62	19	number	number	NOUN
ejpam-3182	62	20	of	of	ADP
ejpam-3182	62	21	prime	prime	ADJ
ejpam-3182	62	22	divisors	divisor	NOUN
ejpam-3182	62	23	in	in	ADP
ejpam-3182	62	24	n	n	CCONJ
ejpam-3182	62	25	,	,	PUNCT
ejpam-3182	62	26	is	be	AUX
ejpam-3182	62	27	valid	valid	ADJ
ejpam-3182	62	28	.	.	PUNCT
ejpam-3182	63	1	definition	definition	NOUN
ejpam-3182	63	2	2.1	2.1	NUM
ejpam-3182	63	3	.	.	PUNCT
ejpam-3182	64	1	an	an	DET
ejpam-3182	64	2	integer	integer	NOUN
ejpam-3182	64	3	u	u	PROPN
ejpam-3182	64	4	∈	∈	PROPN
ejpam-3182	64	5	z	z	PROPN
ejpam-3182	64	6	is	be	AUX
ejpam-3182	64	7	called	call	VERB
ejpam-3182	64	8	a	a	DET
ejpam-3182	64	9	primitive	primitive	ADJ
ejpam-3182	64	10	root	root	NOUN
ejpam-3182	64	11	mod	mod	NOUN
ejpam-3182	64	12	n	n	PROPN
ejpam-3182	64	13	if	if	SCONJ
ejpam-3182	64	14	the	the	DET
ejpam-3182	64	15	least	least	ADJ
ejpam-3182	64	16	exponent	exponent	ADJ
ejpam-3182	64	17	min	min	PROPN
ejpam-3182	64	18	{	{	PUNCT
ejpam-3182	64	19	m	m	PROPN
ejpam-3182	64	20	∈	∈	PROPN
ejpam-3182	64	21	n	n	NOUN
ejpam-3182	64	22	:	:	PUNCT
ejpam-3182	64	23	um	um	INTJ
ejpam-3182	64	24	≡	≡	PROPN
ejpam-3182	64	25	1	1	NUM
ejpam-3182	64	26	mod	mod	PROPN
ejpam-3182	64	27	n	n	CCONJ
ejpam-3182	64	28	}	}	PUNCT
ejpam-3182	64	29	=	=	SYM
ejpam-3182	64	30	λ(n	λ(n	PROPN
ejpam-3182	64	31	)	)	PUNCT
ejpam-3182	64	32	.	.	PUNCT
ejpam-3182	65	1	lemma	lemma	PROPN
ejpam-3182	65	2	2.3	2.3	NUM
ejpam-3182	65	3	.	.	PUNCT
ejpam-3182	66	1	let	let	VERB
ejpam-3182	66	2	n	n	PRON
ejpam-3182	66	3	,	,	PUNCT
ejpam-3182	66	4	u	u	PROPN
ejpam-3182	66	5	∈	∈	PROPN
ejpam-3182	66	6	n	n	CCONJ
ejpam-3182	66	7	,	,	PUNCT
ejpam-3182	66	8	gcd(u	gcd(u	PROPN
ejpam-3182	66	9	,	,	PUNCT
ejpam-3182	66	10	n	n	CCONJ
ejpam-3182	66	11	)	)	PUNCT
ejpam-3182	66	12	=	=	SYM
ejpam-3182	67	1	1	1	X
ejpam-3182	67	2	.	.	PUNCT
ejpam-3182	68	1	the	the	DET
ejpam-3182	68	2	integer	integer	PROPN
ejpam-3182	68	3	u	u	PROPN
ejpam-3182	68	4	6=	6=	ADP
ejpam-3182	68	5	±1	±1	VERB
ejpam-3182	68	6	,	,	PUNCT
ejpam-3182	68	7	v2	v2	PROPN
ejpam-3182	68	8	is	be	AUX
ejpam-3182	68	9	a	a	DET
ejpam-3182	68	10	primitive	primitive	ADJ
ejpam-3182	68	11	root	root	NOUN
ejpam-3182	68	12	modulo	modulo	NOUN
ejpam-3182	68	13	n	n	CCONJ
ejpam-3182	68	14	if	if	SCONJ
ejpam-3182	69	1	and	and	CCONJ
ejpam-3182	69	2	only	only	ADV
ejpam-3182	69	3	if	if	SCONJ
ejpam-3182	69	4	u	u	NOUN
ejpam-3182	69	5	is	be	AUX
ejpam-3182	69	6	a	a	DET
ejpam-3182	69	7	primitive	primitive	ADJ
ejpam-3182	69	8	root	root	NOUN
ejpam-3182	69	9	modulo	modulo	NOUN
ejpam-3182	69	10	pk	pk	NOUN
ejpam-3182	69	11	for	for	ADP
ejpam-3182	69	12	each	each	DET
ejpam-3182	69	13	prime	prime	ADJ
ejpam-3182	69	14	power	power	NOUN
ejpam-3182	69	15	divisor	divisor	NOUN
ejpam-3182	69	16	pk	pk	NOUN
ejpam-3182	69	17	|	|	ADV
ejpam-3182	69	18	n.	n.	PROPN
ejpam-3182	69	19	n.	n.	PROPN
ejpam-3182	69	20	a.	a.	PROPN
ejpam-3182	69	21	carella	carella	PROPN
ejpam-3182	69	22	/	/	SYM
ejpam-3182	69	23	eur	eur	PROPN
ejpam-3182	69	24	.	.	PUNCT
ejpam-3182	70	1	j.	j.	PROPN
ejpam-3182	70	2	pure	pure	PROPN
ejpam-3182	70	3	appl	appl	PROPN
ejpam-3182	70	4	.	.	PROPN
ejpam-3182	70	5	math	math	PROPN
ejpam-3182	70	6	,	,	PUNCT
ejpam-3182	70	7	11	11	NUM
ejpam-3182	70	8	(	(	PUNCT
ejpam-3182	70	9	1	1	NUM
ejpam-3182	70	10	)	)	PUNCT
ejpam-3182	70	11	(	(	PUNCT
ejpam-3182	70	12	2018	2018	NUM
ejpam-3182	70	13	)	)	PUNCT
ejpam-3182	70	14	,	,	PUNCT
ejpam-3182	70	15	23	23	NUM
ejpam-3182	70	16	-	-	SYM
ejpam-3182	70	17	34	34	NUM
ejpam-3182	70	18	26	26	NUM
ejpam-3182	70	19	proof	proof	NOUN
ejpam-3182	70	20	.	.	PUNCT
ejpam-3182	71	1	without	without	ADP
ejpam-3182	71	2	loss	loss	NOUN
ejpam-3182	71	3	in	in	ADP
ejpam-3182	71	4	generality	generality	NOUN
ejpam-3182	71	5	,	,	PUNCT
ejpam-3182	71	6	let	let	VERB
ejpam-3182	71	7	n	n	PRON
ejpam-3182	71	8	=	=	SYM
ejpam-3182	71	9	pq	pq	PROPN
ejpam-3182	71	10	with	with	ADP
ejpam-3182	71	11	p	p	X
ejpam-3182	71	12	,	,	PUNCT
ejpam-3182	71	13	q	q	ADJ
ejpam-3182	71	14	primes	prime	NOUN
ejpam-3182	71	15	.	.	PUNCT
ejpam-3182	71	16	suppose	suppose	VERB
ejpam-3182	71	17	that	that	SCONJ
ejpam-3182	71	18	u	u	PROPN
ejpam-3182	71	19	is	be	AUX
ejpam-3182	71	20	a	a	DET
ejpam-3182	71	21	primitive	primitive	ADJ
ejpam-3182	71	22	root	root	NOUN
ejpam-3182	71	23	modulo	modulo	NOUN
ejpam-3182	71	24	p	p	NOUN
ejpam-3182	71	25	and	and	CCONJ
ejpam-3182	71	26	modulo	modulo	PROPN
ejpam-3182	71	27	q	q	NOUN
ejpam-3182	71	28	,	,	PUNCT
ejpam-3182	71	29	but	but	CCONJ
ejpam-3182	71	30	it	it	PRON
ejpam-3182	71	31	is	be	AUX
ejpam-3182	71	32	not	not	PART
ejpam-3182	71	33	a	a	DET
ejpam-3182	71	34	primitive	primitive	ADJ
ejpam-3182	71	35	root	root	NOUN
ejpam-3182	71	36	modulo	modulo	NOUN
ejpam-3182	71	37	n.	n.	NOUN
ejpam-3182	71	38	then	then	ADV
ejpam-3182	71	39	uλ(n)/r	uλ(n)/r	PRON
ejpam-3182	71	40	≡	≡	PROPN
ejpam-3182	71	41	1	1	NUM
ejpam-3182	71	42	mod	mod	PROPN
ejpam-3182	71	43	n	n	CCONJ
ejpam-3182	71	44	⇐	⇐	ADJ
ejpam-3182	71	45	⇒	⇒	NOUN
ejpam-3182	71	46	uλ(n)/r	uλ(n)/r	PROPN
ejpam-3182	71	47	=	=	SYM
ejpam-3182	72	1	1	1	NUM
ejpam-3182	72	2	+	+	CCONJ
ejpam-3182	72	3	an	an	X
ejpam-3182	72	4	,	,	PUNCT
ejpam-3182	72	5	(	(	PUNCT
ejpam-3182	72	6	11	11	NUM
ejpam-3182	72	7	)	)	PUNCT
ejpam-3182	72	8	for	for	ADP
ejpam-3182	72	9	some	some	DET
ejpam-3182	72	10	prime	prime	ADJ
ejpam-3182	72	11	r	r	NOUN
ejpam-3182	72	12	|	|	ADV
ejpam-3182	72	13	λ(n	λ(n	NOUN
ejpam-3182	72	14	)	)	PUNCT
ejpam-3182	72	15	,	,	PUNCT
ejpam-3182	72	16	and	and	CCONJ
ejpam-3182	72	17	a	a	DET
ejpam-3182	72	18	∈	∈	PROPN
ejpam-3182	72	19	z.	z.	X
ejpam-3182	73	1	this	this	PRON
ejpam-3182	73	2	in	in	ADP
ejpam-3182	73	3	turns	turn	NOUN
ejpam-3182	73	4	implies	imply	VERB
ejpam-3182	73	5	that	that	SCONJ
ejpam-3182	73	6	both	both	CCONJ
ejpam-3182	73	7	uλ(n)/r	uλ(n)/r	PRON
ejpam-3182	73	8	≡	≡	PROPN
ejpam-3182	73	9	1	1	NUM
ejpam-3182	73	10	mod	mod	PROPN
ejpam-3182	73	11	p	p	PROPN
ejpam-3182	73	12	and	and	CCONJ
ejpam-3182	73	13	uλ(n)/r	uλ(n)/r	PRON
ejpam-3182	73	14	≡	≡	PROPN
ejpam-3182	73	15	1	1	NUM
ejpam-3182	73	16	mod	mod	PROPN
ejpam-3182	73	17	q.	q.	PROPN
ejpam-3182	73	18	this	this	PRON
ejpam-3182	73	19	contradicts	contradict	VERB
ejpam-3182	73	20	the	the	DET
ejpam-3182	73	21	hypothesis	hypothesis	NOUN
ejpam-3182	73	22	that	that	PRON
ejpam-3182	73	23	both	both	CCONJ
ejpam-3182	73	24	uλ(n)/r	uλ(n)/r	NOUN
ejpam-3182	73	25	6≡	6≡	NUM
ejpam-3182	73	26	1	1	NUM
ejpam-3182	73	27	mod	mod	NOUN
ejpam-3182	73	28	p	p	NOUN
ejpam-3182	73	29	and	and	CCONJ
ejpam-3182	73	30	uλ(n)/r	uλ(n)/r	ADJ
ejpam-3182	73	31	6≡	6≡	NUM
ejpam-3182	73	32	1	1	NUM
ejpam-3182	73	33	mod	mod	PROPN
ejpam-3182	73	34	q.	q.	PROPN
ejpam-3182	73	35	conversely	conversely	ADV
ejpam-3182	73	36	,	,	PUNCT
ejpam-3182	73	37	suppose	suppose	VERB
ejpam-3182	73	38	that	that	SCONJ
ejpam-3182	73	39	u	u	PROPN
ejpam-3182	73	40	is	be	AUX
ejpam-3182	73	41	a	a	DET
ejpam-3182	73	42	primitive	primitive	ADJ
ejpam-3182	73	43	root	root	NOUN
ejpam-3182	73	44	modulo	modulo	NOUN
ejpam-3182	73	45	n	n	CCONJ
ejpam-3182	73	46	,	,	PUNCT
ejpam-3182	73	47	but	but	CCONJ
ejpam-3182	73	48	u	u	NOUN
ejpam-3182	73	49	is	be	AUX
ejpam-3182	73	50	not	not	PART
ejpam-3182	73	51	a	a	DET
ejpam-3182	73	52	primitive	primitive	ADJ
ejpam-3182	73	53	root	root	NOUN
ejpam-3182	73	54	either	either	CCONJ
ejpam-3182	73	55	modulo	modulo	NOUN
ejpam-3182	73	56	p	p	NOUN
ejpam-3182	73	57	or	or	CCONJ
ejpam-3182	73	58	modulo	modulo	PROPN
ejpam-3182	73	59	q.	q.	NOUN
ejpam-3182	73	60	write	write	VERB
ejpam-3182	73	61	uλ(n)/r	uλ(n)/r	PROPN
ejpam-3182	73	62	6≡	6≡	NUM
ejpam-3182	73	63	1	1	NUM
ejpam-3182	73	64	mod	mod	NOUN
ejpam-3182	73	65	n	n	CCONJ
ejpam-3182	73	66	,	,	PUNCT
ejpam-3182	73	67	with	with	ADP
ejpam-3182	73	68	r	r	NOUN
ejpam-3182	73	69	≥	≥	NUM
ejpam-3182	73	70	2	2	NUM
ejpam-3182	73	71	prime	prime	NOUN
ejpam-3182	73	72	,	,	PUNCT
ejpam-3182	73	73	and	and	CCONJ
ejpam-3182	73	74	proceed	proceed	VERB
ejpam-3182	73	75	as	as	ADP
ejpam-3182	73	76	before	before	ADV
ejpam-3182	73	77	to	to	PART
ejpam-3182	73	78	derive	derive	VERB
ejpam-3182	73	79	a	a	DET
ejpam-3182	73	80	contradiction	contradiction	NOUN
ejpam-3182	73	81	:	:	PUNCT
ejpam-3182	73	82	uλ(n)/r	uλ(n)/r	NOUN
ejpam-3182	73	83	6≡	6≡	NUM
ejpam-3182	73	84	1	1	NUM
ejpam-3182	73	85	mod	mod	ADJ
ejpam-3182	73	86	n	n	CCONJ
ejpam-3182	73	87	⇐	⇐	ADJ
ejpam-3182	73	88	⇒	⇒	NOUN
ejpam-3182	73	89	uλ(n)/r	uλ(n)/r	PROPN
ejpam-3182	73	90	6=	6=	ADP
ejpam-3182	73	91	1	1	NUM
ejpam-3182	73	92	+	+	NOUN
ejpam-3182	73	93	an	an	X
ejpam-3182	73	94	.	.	PUNCT
ejpam-3182	74	1	(	(	PUNCT
ejpam-3182	74	2	12	12	NUM
ejpam-3182	74	3	)	)	PUNCT
ejpam-3182	75	1	this	this	PRON
ejpam-3182	75	2	in	in	ADP
ejpam-3182	75	3	turns	turn	NOUN
ejpam-3182	75	4	implies	imply	VERB
ejpam-3182	75	5	that	that	SCONJ
ejpam-3182	75	6	either	either	CCONJ
ejpam-3182	75	7	uλ(n)/r	uλ(n)/r	DET
ejpam-3182	75	8	6≡	6≡	NUM
ejpam-3182	75	9	1	1	NUM
ejpam-3182	75	10	mod	mod	NOUN
ejpam-3182	75	11	p	p	NOUN
ejpam-3182	75	12	or	or	CCONJ
ejpam-3182	75	13	uλ(n)/r	uλ(n)/r	ADJ
ejpam-3182	75	14	6≡	6≡	NUM
ejpam-3182	75	15	1	1	NUM
ejpam-3182	75	16	mod	mod	PROPN
ejpam-3182	75	17	q.	q.	PROPN
ejpam-3182	75	18	but	but	CCONJ
ejpam-3182	75	19	,	,	PUNCT
ejpam-3182	75	20	this	this	PRON
ejpam-3182	75	21	contradicts	contradict	VERB
ejpam-3182	75	22	the	the	DET
ejpam-3182	75	23	hypothesis	hypothesis	NOUN
ejpam-3182	75	24	that	that	PRON
ejpam-3182	75	25	u	u	NOUN
ejpam-3182	75	26	is	be	AUX
ejpam-3182	75	27	not	not	PART
ejpam-3182	75	28	a	a	DET
ejpam-3182	75	29	primitive	primitive	ADJ
ejpam-3182	75	30	root	root	NOUN
ejpam-3182	75	31	either	either	CCONJ
ejpam-3182	75	32	modulo	modulo	NOUN
ejpam-3182	75	33	p	p	NOUN
ejpam-3182	75	34	or	or	CCONJ
ejpam-3182	75	35	q.	q.	PROPN
ejpam-3182	75	36	3	3	NUM
ejpam-3182	75	37	.	.	PUNCT
ejpam-3182	75	38	characteristic	characteristic	ADJ
ejpam-3182	75	39	function	function	NOUN
ejpam-3182	75	40	in	in	ADP
ejpam-3182	75	41	finite	finite	ADJ
ejpam-3182	75	42	rings	ring	NOUN
ejpam-3182	75	43	the	the	DET
ejpam-3182	75	44	symbol	symbol	NOUN
ejpam-3182	75	45	ordpk(u	ordpk(u	NOUN
ejpam-3182	75	46	)	)	PUNCT
ejpam-3182	75	47	denotes	denote	VERB
ejpam-3182	75	48	the	the	DET
ejpam-3182	75	49	order	order	NOUN
ejpam-3182	75	50	of	of	ADP
ejpam-3182	75	51	an	an	DET
ejpam-3182	75	52	element	element	NOUN
ejpam-3182	75	53	u	u	NOUN
ejpam-3182	75	54	∈	∈	PROPN
ejpam-3182	75	55	(	(	PUNCT
ejpam-3182	75	56	z	z	NOUN
ejpam-3182	75	57	/	/	SYM
ejpam-3182	75	58	pk	pk	NOUN
ejpam-3182	75	59	z	z	NOUN
ejpam-3182	75	60	)	)	PUNCT
ejpam-3182	75	61	×	×	NOUN
ejpam-3182	75	62	in	in	ADP
ejpam-3182	75	63	the	the	DET
ejpam-3182	75	64	multiplicative	multiplicative	ADJ
ejpam-3182	75	65	group	group	NOUN
ejpam-3182	75	66	of	of	ADP
ejpam-3182	75	67	the	the	DET
ejpam-3182	75	68	integers	integer	NOUN
ejpam-3182	75	69	modulo	modulo	PROPN
ejpam-3182	75	70	pk	pk	PROPN
ejpam-3182	75	71	.	.	PUNCT
ejpam-3182	76	1	the	the	DET
ejpam-3182	76	2	order	order	NOUN
ejpam-3182	76	3	satisfies	satisfy	VERB
ejpam-3182	76	4	the	the	DET
ejpam-3182	76	5	divisibility	divisibility	NOUN
ejpam-3182	76	6	condition	condition	NOUN
ejpam-3182	76	7	ordpk(u	ordpk(u	NOUN
ejpam-3182	76	8	)	)	PUNCT
ejpam-3182	76	9	|	|	ADV
ejpam-3182	76	10	λ(n	λ(n	PROPN
ejpam-3182	76	11	)	)	PUNCT
ejpam-3182	76	12	,	,	PUNCT
ejpam-3182	76	13	and	and	CCONJ
ejpam-3182	76	14	primitive	primitive	ADJ
ejpam-3182	76	15	roots	root	NOUN
ejpam-3182	76	16	have	have	VERB
ejpam-3182	76	17	maximal	maximal	ADJ
ejpam-3182	76	18	orders	order	NOUN
ejpam-3182	76	19	ordpk(u	ordpk(u	NOUN
ejpam-3182	76	20	)	)	PUNCT
ejpam-3182	76	21	=	=	SYM
ejpam-3182	76	22	λ(n	λ(n	PROPN
ejpam-3182	76	23	)	)	PUNCT
ejpam-3182	76	24	.	.	PUNCT
ejpam-3182	77	1	the	the	DET
ejpam-3182	77	2	basic	basic	ADJ
ejpam-3182	77	3	properties	property	NOUN
ejpam-3182	77	4	of	of	ADP
ejpam-3182	77	5	primitive	primitive	ADJ
ejpam-3182	77	6	root	root	NOUN
ejpam-3182	77	7	are	be	AUX
ejpam-3182	77	8	explicated	explicate	VERB
ejpam-3182	77	9	in	in	ADP
ejpam-3182	77	10	[	[	X
ejpam-3182	77	11	1	1	NUM
ejpam-3182	77	12	]	]	PUNCT
ejpam-3182	77	13	,	,	PUNCT
ejpam-3182	77	14	[	[	X
ejpam-3182	77	15	30	30	NUM
ejpam-3182	77	16	]	]	PUNCT
ejpam-3182	77	17	,	,	PUNCT
ejpam-3182	77	18	et	et	PROPN
ejpam-3182	77	19	cetera	cetera	NOUN
ejpam-3182	77	20	.	.	PUNCT
ejpam-3182	78	1	the	the	DET
ejpam-3182	78	2	characteristic	characteristic	ADJ
ejpam-3182	78	3	function	function	NOUN
ejpam-3182	78	4	f	f	NOUN
ejpam-3182	78	5	:	:	PUNCT
ejpam-3182	78	6	n	n	CCONJ
ejpam-3182	78	7	−→	−→	NOUN
ejpam-3182	78	8	{	{	PUNCT
ejpam-3182	78	9	0	0	NUM
ejpam-3182	78	10	,	,	PUNCT
ejpam-3182	78	11	1	1	NUM
ejpam-3182	78	12	}	}	PUNCT
ejpam-3182	78	13	of	of	ADP
ejpam-3182	78	14	a	a	DET
ejpam-3182	78	15	fixed	fix	VERB
ejpam-3182	78	16	primitive	primitive	ADJ
ejpam-3182	78	17	root	root	NOUN
ejpam-3182	78	18	u	u	NOUN
ejpam-3182	78	19	in	in	ADP
ejpam-3182	78	20	the	the	DET
ejpam-3182	78	21	finite	finite	NOUN
ejpam-3182	78	22	ring	ring	NOUN
ejpam-3182	78	23	z	z	PROPN
ejpam-3182	78	24	/	/	SYM
ejpam-3182	78	25	pk	pk	PROPN
ejpam-3182	78	26	z	z	PROPN
ejpam-3182	78	27	,	,	PUNCT
ejpam-3182	78	28	the	the	DET
ejpam-3182	78	29	integers	integer	NOUN
ejpam-3182	78	30	modulo	modulo	PROPN
ejpam-3182	78	31	pk	pk	NOUN
ejpam-3182	78	32	,	,	PUNCT
ejpam-3182	78	33	is	be	AUX
ejpam-3182	78	34	determined	determine	VERB
ejpam-3182	78	35	here	here	ADV
ejpam-3182	78	36	.	.	PUNCT
ejpam-3182	79	1	lemma	lemma	PROPN
ejpam-3182	79	2	3.1	3.1	NUM
ejpam-3182	79	3	.	.	PUNCT
ejpam-3182	80	1	let	let	VERB
ejpam-3182	80	2	pk	pk	NOUN
ejpam-3182	80	3	,	,	PUNCT
ejpam-3182	80	4	k	k	PROPN
ejpam-3182	80	5	≥	≥	NUM
ejpam-3182	80	6	1	1	NUM
ejpam-3182	80	7	,	,	PUNCT
ejpam-3182	80	8	be	be	AUX
ejpam-3182	80	9	a	a	DET
ejpam-3182	80	10	prime	prime	ADJ
ejpam-3182	80	11	power	power	NOUN
ejpam-3182	80	12	,	,	PUNCT
ejpam-3182	80	13	and	and	CCONJ
ejpam-3182	80	14	let	let	VERB
ejpam-3182	80	15	u	u	PRON
ejpam-3182	80	16	∈	∈	PROPN
ejpam-3182	80	17	z	z	AUX
ejpam-3182	80	18	be	be	AUX
ejpam-3182	80	19	an	an	DET
ejpam-3182	80	20	integer	integer	NOUN
ejpam-3182	80	21	such	such	ADJ
ejpam-3182	80	22	that	that	DET
ejpam-3182	80	23	gcd	gcd	PROPN
ejpam-3182	80	24	(	(	PUNCT
ejpam-3182	80	25	u	u	NOUN
ejpam-3182	80	26	,	,	PUNCT
ejpam-3182	80	27	pk	pk	NOUN
ejpam-3182	80	28	)	)	PUNCT
ejpam-3182	81	1	=	=	SYM
ejpam-3182	82	1	1	1	X
ejpam-3182	82	2	.	.	PUNCT
ejpam-3182	83	1	then	then	ADV
ejpam-3182	83	2	(	(	PUNCT
ejpam-3182	83	3	i	i	NOUN
ejpam-3182	83	4	)	)	PUNCT
ejpam-3182	83	5	the	the	DET
ejpam-3182	83	6	characteristic	characteristic	ADJ
ejpam-3182	83	7	f	f	PROPN
ejpam-3182	83	8	function	function	NOUN
ejpam-3182	83	9	of	of	ADP
ejpam-3182	83	10	the	the	DET
ejpam-3182	83	11	primitive	primitive	ADJ
ejpam-3182	83	12	root	root	NOUN
ejpam-3182	83	13	u	u	PROPN
ejpam-3182	83	14	mod	mod	PROPN
ejpam-3182	83	15	pk	pk	PROPN
ejpam-3182	83	16	is	be	AUX
ejpam-3182	83	17	given	give	VERB
ejpam-3182	83	18	by	by	ADP
ejpam-3182	83	19	f	f	PROPN
ejpam-3182	83	20	(	(	PUNCT
ejpam-3182	83	21	pk	pk	NOUN
ejpam-3182	83	22	)	)	PUNCT
ejpam-3182	83	23	=	=	VERB
ejpam-3182	83	24			NUM
ejpam-3182	83	25	1	1	NUM
ejpam-3182	83	26	if	if	SCONJ
ejpam-3182	83	27	pk	pk	NOUN
ejpam-3182	83	28	=	=	SYM
ejpam-3182	83	29	2k	2k	NUM
ejpam-3182	83	30	,	,	PUNCT
ejpam-3182	83	31	k	k	PROPN
ejpam-3182	83	32	≤	≤	PROPN
ejpam-3182	83	33	2	2	NUM
ejpam-3182	83	34	,	,	PUNCT
ejpam-3182	83	35	0	0	NUM
ejpam-3182	84	1	if	if	SCONJ
ejpam-3182	84	2	pk	pk	NOUN
ejpam-3182	84	3	=	=	SYM
ejpam-3182	84	4	2k	2k	NUM
ejpam-3182	84	5	,	,	PUNCT
ejpam-3182	84	6	k	k	PROPN
ejpam-3182	84	7	>	>	X
ejpam-3182	84	8	2	2	NUM
ejpam-3182	84	9	,	,	PUNCT
ejpam-3182	84	10	1	1	NUM
ejpam-3182	84	11	if	if	SCONJ
ejpam-3182	84	12	ordpk(u	ordpk(u	X
ejpam-3182	84	13	)	)	PUNCT
ejpam-3182	84	14	=	=	SYM
ejpam-3182	85	1	pk−1(p−	pk−1(p−	PROPN
ejpam-3182	85	2	1	1	NUM
ejpam-3182	85	3	)	)	PUNCT
ejpam-3182	85	4	,	,	PUNCT
ejpam-3182	85	5	p	p	X
ejpam-3182	85	6	>	>	X
ejpam-3182	85	7	2	2	NUM
ejpam-3182	85	8	,	,	PUNCT
ejpam-3182	85	9	for	for	ADP
ejpam-3182	85	10	any	any	DET
ejpam-3182	85	11	k	k	PROPN
ejpam-3182	85	12	≥	≥	NUM
ejpam-3182	85	13	1	1	NUM
ejpam-3182	85	14	,	,	PUNCT
ejpam-3182	85	15	0	0	NUM
ejpam-3182	85	16	if	if	SCONJ
ejpam-3182	85	17	ordpk(u	ordpk(u	X
ejpam-3182	85	18	)	)	PUNCT
ejpam-3182	85	19	6=	6=	NUM
ejpam-3182	85	20	pk−1(p−	pk−1(p−	PROPN
ejpam-3182	85	21	1	1	NUM
ejpam-3182	85	22	)	)	PUNCT
ejpam-3182	85	23	,	,	PUNCT
ejpam-3182	85	24	and	and	CCONJ
ejpam-3182	85	25	p	p	X
ejpam-3182	85	26	>	>	X
ejpam-3182	85	27	2	2	NUM
ejpam-3182	85	28	,	,	PUNCT
ejpam-3182	85	29	k	k	X
ejpam-3182	85	30	≥	≥	NUM
ejpam-3182	85	31	1	1	NUM
ejpam-3182	85	32	.	.	PUNCT
ejpam-3182	86	1	(	(	PUNCT
ejpam-3182	86	2	13	13	NUM
ejpam-3182	86	3	)	)	PUNCT
ejpam-3182	86	4	(	(	PUNCT
ejpam-3182	86	5	ii	ii	NOUN
ejpam-3182	86	6	)	)	PUNCT
ejpam-3182	86	7	the	the	DET
ejpam-3182	86	8	function	function	NOUN
ejpam-3182	86	9	f	f	PROPN
ejpam-3182	86	10	is	be	AUX
ejpam-3182	86	11	multiplicative	multiplicative	ADJ
ejpam-3182	86	12	,	,	PUNCT
ejpam-3182	86	13	but	but	CCONJ
ejpam-3182	86	14	not	not	PART
ejpam-3182	86	15	completely	completely	ADV
ejpam-3182	86	16	multiplicative	multiplicative	ADJ
ejpam-3182	86	17	since	since	SCONJ
ejpam-3182	86	18	(	(	PUNCT
ejpam-3182	86	19	iii	iii	NOUN
ejpam-3182	86	20	)	)	PUNCT
ejpam-3182	86	21	f(pq	f(pq	PROPN
ejpam-3182	86	22	)	)	PUNCT
ejpam-3182	86	23	=	=	SYM
ejpam-3182	86	24	f(p)f(q	f(p)f(q	NOUN
ejpam-3182	86	25	)	)	PUNCT
ejpam-3182	86	26	,	,	PUNCT
ejpam-3182	86	27	gcd(p	gcd(p	PROPN
ejpam-3182	86	28	,	,	PUNCT
ejpam-3182	86	29	q	q	NOUN
ejpam-3182	86	30	)	)	PUNCT
ejpam-3182	86	31	=	=	SYM
ejpam-3182	86	32	1	1	NUM
ejpam-3182	86	33	,	,	PUNCT
ejpam-3182	86	34	(	(	PUNCT
ejpam-3182	86	35	iv	iv	X
ejpam-3182	86	36	)	)	PUNCT
ejpam-3182	86	37	f	f	PROPN
ejpam-3182	86	38	(	(	PUNCT
ejpam-3182	86	39	p2	p2	PROPN
ejpam-3182	86	40	)	)	PUNCT
ejpam-3182	86	41	6=	6=	ADP
ejpam-3182	86	42	f(p)f(p	f(p)f(p	NOUN
ejpam-3182	86	43	)	)	PUNCT
ejpam-3182	86	44	,	,	PUNCT
ejpam-3182	86	45	if	if	SCONJ
ejpam-3182	86	46	ordp2(u	ordp2(u	NOUN
ejpam-3182	86	47	)	)	PUNCT
ejpam-3182	87	1	6=	6=	NUM
ejpam-3182	87	2	p(p−	p(p−	VERB
ejpam-3182	87	3	1	1	NUM
ejpam-3182	87	4	)	)	PUNCT
ejpam-3182	87	5	.	.	PUNCT
ejpam-3182	88	1	proof	proof	NOUN
ejpam-3182	88	2	.	.	PUNCT
ejpam-3182	89	1	the	the	DET
ejpam-3182	89	2	function	function	NOUN
ejpam-3182	89	3	has	have	VERB
ejpam-3182	89	4	the	the	DET
ejpam-3182	89	5	value	value	NOUN
ejpam-3182	89	6	f	f	X
ejpam-3182	89	7	(	(	PUNCT
ejpam-3182	89	8	pk	pk	NOUN
ejpam-3182	89	9	)	)	PUNCT
ejpam-3182	89	10	=	=	SYM
ejpam-3182	89	11	1	1	NUM
ejpam-3182	89	12	if	if	SCONJ
ejpam-3182	89	13	and	and	CCONJ
ejpam-3182	89	14	only	only	ADV
ejpam-3182	89	15	if	if	SCONJ
ejpam-3182	89	16	the	the	DET
ejpam-3182	89	17	element	element	NOUN
ejpam-3182	89	18	u	u	NOUN
ejpam-3182	89	19	∈	∈	PROPN
ejpam-3182	89	20	(	(	PUNCT
ejpam-3182	89	21	z	z	NOUN
ejpam-3182	89	22	/	/	SYM
ejpam-3182	89	23	pk	pk	NOUN
ejpam-3182	89	24	z	z	NOUN
ejpam-3182	89	25	)	)	PUNCT
ejpam-3182	89	26	×	×	NOUN
ejpam-3182	89	27	is	be	AUX
ejpam-3182	89	28	a	a	DET
ejpam-3182	89	29	primitive	primitive	ADJ
ejpam-3182	89	30	root	root	NOUN
ejpam-3182	89	31	modulo	modulo	PROPN
ejpam-3182	89	32	pk	pk	PROPN
ejpam-3182	89	33	.	.	PROPN
ejpam-3182	89	34	otherwise	otherwise	ADV
ejpam-3182	89	35	,	,	PUNCT
ejpam-3182	89	36	it	it	PRON
ejpam-3182	89	37	vanishes	vanish	VERB
ejpam-3182	89	38	:	:	PUNCT
ejpam-3182	89	39	f	f	PROPN
ejpam-3182	89	40	(	(	PUNCT
ejpam-3182	89	41	pk	pk	NOUN
ejpam-3182	89	42	)	)	PUNCT
ejpam-3182	89	43	=	=	PUNCT
ejpam-3182	90	1	0	0	X
ejpam-3182	90	2	.	.	PUNCT
ejpam-3182	91	1	the	the	DET
ejpam-3182	91	2	completely	completely	ADV
ejpam-3182	91	3	multiplicative	multiplicative	ADJ
ejpam-3182	91	4	property	property	NOUN
ejpam-3182	91	5	fails	fail	VERB
ejpam-3182	91	6	because	because	SCONJ
ejpam-3182	91	7	of	of	ADP
ejpam-3182	91	8	the	the	DET
ejpam-3182	91	9	existence	existence	NOUN
ejpam-3182	91	10	of	of	ADP
ejpam-3182	91	11	wieferich	wieferich	NOUN
ejpam-3182	91	12	primes	prime	NOUN
ejpam-3182	91	13	,	,	PUNCT
ejpam-3182	91	14	exempli	exempli	PROPN
ejpam-3182	91	15	gratia	gratia	PROPN
ejpam-3182	91	16	,	,	PUNCT
ejpam-3182	91	17	n.	n.	PROPN
ejpam-3182	91	18	a.	a.	NOUN
ejpam-3182	91	19	carella	carella	PROPN
ejpam-3182	91	20	/	/	SYM
ejpam-3182	91	21	eur	eur	PROPN
ejpam-3182	91	22	.	.	PUNCT
ejpam-3182	92	1	j.	j.	PROPN
ejpam-3182	92	2	pure	pure	PROPN
ejpam-3182	92	3	appl	appl	PROPN
ejpam-3182	92	4	.	.	PROPN
ejpam-3182	92	5	math	math	PROPN
ejpam-3182	92	6	,	,	PUNCT
ejpam-3182	92	7	11	11	NUM
ejpam-3182	92	8	(	(	PUNCT
ejpam-3182	92	9	1	1	NUM
ejpam-3182	92	10	)	)	PUNCT
ejpam-3182	92	11	(	(	PUNCT
ejpam-3182	92	12	2018	2018	NUM
ejpam-3182	92	13	)	)	PUNCT
ejpam-3182	92	14	,	,	PUNCT
ejpam-3182	92	15	23	23	NUM
ejpam-3182	92	16	-	-	SYM
ejpam-3182	92	17	34	34	NUM
ejpam-3182	92	18	27	27	NUM
ejpam-3182	92	19	0	0	NUM
ejpam-3182	93	1	=	=	SYM
ejpam-3182	93	2	f	f	PROPN
ejpam-3182	93	3	(	(	PUNCT
ejpam-3182	93	4	404872	404872	NUM
ejpam-3182	93	5	)	)	PUNCT
ejpam-3182	93	6	6=	6=	ADP
ejpam-3182	93	7	f(40487)f(40487	f(40487)f(40487	PROPN
ejpam-3182	93	8	)	)	PUNCT
ejpam-3182	93	9	=	=	SYM
ejpam-3182	93	10	1	1	NUM
ejpam-3182	93	11	,	,	PUNCT
ejpam-3182	93	12	see	see	VERB
ejpam-3182	93	13	[	[	X
ejpam-3182	93	14	28	28	NUM
ejpam-3182	93	15	]	]	PUNCT
ejpam-3182	93	16	.	.	PUNCT
ejpam-3182	94	1	otherwise	otherwise	ADV
ejpam-3182	94	2	,	,	PUNCT
ejpam-3182	94	3	it	it	PRON
ejpam-3182	94	4	is	be	AUX
ejpam-3182	94	5	completely	completely	ADV
ejpam-3182	94	6	multiplicative	multiplicative	ADJ
ejpam-3182	94	7	,	,	PUNCT
ejpam-3182	94	8	that	that	ADV
ejpam-3182	94	9	is	is	ADV
ejpam-3182	94	10	,	,	PUNCT
ejpam-3182	94	11	f	f	PROPN
ejpam-3182	94	12	(	(	PUNCT
ejpam-3182	94	13	p2	p2	PROPN
ejpam-3182	94	14	)	)	PUNCT
ejpam-3182	94	15	=	=	SYM
ejpam-3182	94	16	f(p)f(p	f(p)f(p	NOUN
ejpam-3182	94	17	)	)	PUNCT
ejpam-3182	94	18	=	=	SYM
ejpam-3182	94	19	1	1	NUM
ejpam-3182	94	20	for	for	SCONJ
ejpam-3182	94	21	any	any	DET
ejpam-3182	94	22	nonwieferich	nonwieferich	NOUN
ejpam-3182	94	23	primes	prime	VERB
ejpam-3182	94	24	p	p	NOUN
ejpam-3182	94	25	≥	≥	NUM
ejpam-3182	94	26	2	2	NUM
ejpam-3182	94	27	.	.	X
ejpam-3182	94	28	observe	observe	VERB
ejpam-3182	94	29	that	that	SCONJ
ejpam-3182	94	30	the	the	DET
ejpam-3182	94	31	conditions	condition	NOUN
ejpam-3182	94	32	ordp(u	ordp(u	NOUN
ejpam-3182	94	33	)	)	PUNCT
ejpam-3182	94	34	=	=	PUNCT
ejpam-3182	95	1	p	p	NOUN
ejpam-3182	95	2	−	−	PROPN
ejpam-3182	95	3	1	1	NUM
ejpam-3182	95	4	and	and	CCONJ
ejpam-3182	95	5	ordp2(u	ordp2(u	NUM
ejpam-3182	95	6	)	)	PUNCT
ejpam-3182	95	7	6=	6=	ADP
ejpam-3182	96	1	p(p	p(p	ADV
ejpam-3182	96	2	−	−	NOUN
ejpam-3182	96	3	1	1	X
ejpam-3182	96	4	)	)	PUNCT
ejpam-3182	96	5	imply	imply	VERB
ejpam-3182	96	6	that	that	SCONJ
ejpam-3182	96	7	the	the	DET
ejpam-3182	96	8	integer	integer	PROPN
ejpam-3182	96	9	u	u	PROPN
ejpam-3182	96	10	6=	6=	ADP
ejpam-3182	96	11	±1	±1	VERB
ejpam-3182	96	12	,	,	PUNCT
ejpam-3182	96	13	v2	v2	PROPN
ejpam-3182	96	14	can	can	AUX
ejpam-3182	96	15	not	not	PART
ejpam-3182	96	16	be	be	AUX
ejpam-3182	96	17	extended	extend	VERB
ejpam-3182	96	18	to	to	ADP
ejpam-3182	96	19	a	a	DET
ejpam-3182	96	20	primitive	primitive	ADJ
ejpam-3182	96	21	root	root	NOUN
ejpam-3182	96	22	u	u	PROPN
ejpam-3182	96	23	mod	mod	PROPN
ejpam-3182	96	24	pk	pk	PROPN
ejpam-3182	96	25	,	,	PUNCT
ejpam-3182	96	26	with	with	ADP
ejpam-3182	96	27	k	k	PROPN
ejpam-3182	96	28	≥	≥	NUM
ejpam-3182	96	29	2	2	NUM
ejpam-3182	96	30	.	.	PUNCT
ejpam-3182	97	1	but	but	CCONJ
ejpam-3182	97	2	that	that	SCONJ
ejpam-3182	97	3	the	the	DET
ejpam-3182	97	4	condition	condition	NOUN
ejpam-3182	97	5	ordp2(u	ordp2(u	NOUN
ejpam-3182	97	6	)	)	PUNCT
ejpam-3182	97	7	=	=	SYM
ejpam-3182	97	8	p(p−1	p(p−1	PROPN
ejpam-3182	97	9	)	)	PUNCT
ejpam-3182	97	10	implies	imply	VERB
ejpam-3182	97	11	that	that	SCONJ
ejpam-3182	97	12	the	the	DET
ejpam-3182	97	13	integer	integer	NOUN
ejpam-3182	97	14	u	u	NOUN
ejpam-3182	97	15	can	can	AUX
ejpam-3182	97	16	be	be	AUX
ejpam-3182	97	17	extended	extend	VERB
ejpam-3182	97	18	to	to	ADP
ejpam-3182	97	19	a	a	DET
ejpam-3182	97	20	primitive	primitive	ADJ
ejpam-3182	97	21	root	root	NOUN
ejpam-3182	97	22	u	u	PROPN
ejpam-3182	97	23	mod	mod	PROPN
ejpam-3182	97	24	pk	pk	PROPN
ejpam-3182	97	25	,	,	PUNCT
ejpam-3182	97	26	with	with	ADP
ejpam-3182	97	27	k	k	PROPN
ejpam-3182	97	28	≥	≥	NUM
ejpam-3182	97	29	2	2	NUM
ejpam-3182	97	30	.	.	NOUN
ejpam-3182	97	31	4	4	NUM
ejpam-3182	97	32	.	.	X
ejpam-3182	97	33	wirsing	wirsing	NOUN
ejpam-3182	97	34	formula	formula	NOUN
ejpam-3182	97	35	this	this	DET
ejpam-3182	97	36	formula	formula	NOUN
ejpam-3182	97	37	provides	provide	VERB
ejpam-3182	97	38	decompositions	decomposition	NOUN
ejpam-3182	97	39	of	of	ADP
ejpam-3182	97	40	some	some	DET
ejpam-3182	97	41	summatory	summatory	ADJ
ejpam-3182	97	42	multiplicative	multiplicative	ADJ
ejpam-3182	97	43	functions	function	NOUN
ejpam-3182	97	44	as	as	ADP
ejpam-3182	97	45	products	product	NOUN
ejpam-3182	97	46	over	over	ADP
ejpam-3182	97	47	the	the	DET
ejpam-3182	97	48	primes	prime	NOUN
ejpam-3182	97	49	supports	support	NOUN
ejpam-3182	97	50	of	of	ADP
ejpam-3182	97	51	the	the	DET
ejpam-3182	97	52	functions	function	NOUN
ejpam-3182	97	53	.	.	PUNCT
ejpam-3182	98	1	this	this	DET
ejpam-3182	98	2	technique	technique	NOUN
ejpam-3182	98	3	works	work	VERB
ejpam-3182	98	4	well	well	ADV
ejpam-3182	98	5	with	with	ADP
ejpam-3182	98	6	certain	certain	ADJ
ejpam-3182	98	7	multiplicative	multiplicative	ADJ
ejpam-3182	98	8	functions	function	NOUN
ejpam-3182	98	9	,	,	PUNCT
ejpam-3182	98	10	which	which	PRON
ejpam-3182	98	11	have	have	VERB
ejpam-3182	98	12	supports	support	NOUN
ejpam-3182	98	13	on	on	ADP
ejpam-3182	98	14	subsets	subset	NOUN
ejpam-3182	98	15	of	of	ADP
ejpam-3182	98	16	primes	prime	NOUN
ejpam-3182	98	17	numbers	number	NOUN
ejpam-3182	98	18	of	of	ADP
ejpam-3182	98	19	nonzero	nonzero	PROPN
ejpam-3182	98	20	densities	density	NOUN
ejpam-3182	98	21	.	.	PUNCT
ejpam-3182	99	1	lemma	lemma	PROPN
ejpam-3182	99	2	4.1	4.1	NUM
ejpam-3182	99	3	.	.	PUNCT
ejpam-3182	100	1	(	(	PUNCT
ejpam-3182	100	2	[	[	X
ejpam-3182	100	3	36	36	NUM
ejpam-3182	100	4	,	,	PUNCT
ejpam-3182	100	5	p.	p.	NOUN
ejpam-3182	100	6	71	71	NUM
ejpam-3182	100	7	]	]	PUNCT
ejpam-3182	100	8	)	)	PUNCT
ejpam-3182	100	9	suppose	suppose	VERB
ejpam-3182	100	10	that	that	SCONJ
ejpam-3182	100	11	f	f	X
ejpam-3182	100	12	:	:	PUNCT
ejpam-3182	100	13	n	n	CCONJ
ejpam-3182	100	14	−→	−→	NOUN
ejpam-3182	100	15	c	c	NOUN
ejpam-3182	100	16	is	be	AUX
ejpam-3182	100	17	a	a	DET
ejpam-3182	100	18	multiplicative	multiplicative	ADJ
ejpam-3182	100	19	function	function	NOUN
ejpam-3182	100	20	with	with	ADP
ejpam-3182	100	21	the	the	DET
ejpam-3182	100	22	following	follow	VERB
ejpam-3182	100	23	properties	property	NOUN
ejpam-3182	100	24	.	.	PUNCT
ejpam-3182	101	1	(	(	PUNCT
ejpam-3182	101	2	i	i	NOUN
ejpam-3182	101	3	)	)	PUNCT
ejpam-3182	101	4	f(n	f(n	PROPN
ejpam-3182	101	5	)	)	PUNCT
ejpam-3182	101	6	≥	≥	NOUN
ejpam-3182	101	7	0	0	NUM
ejpam-3182	101	8	for	for	ADP
ejpam-3182	101	9	all	all	DET
ejpam-3182	101	10	integers	integer	NOUN
ejpam-3182	101	11	n	n	PRON
ejpam-3182	101	12	∈	∈	PROPN
ejpam-3182	101	13	n.	n.	NOUN
ejpam-3182	101	14	(	(	PUNCT
ejpam-3182	101	15	ii	ii	PROPN
ejpam-3182	101	16	)	)	PUNCT
ejpam-3182	101	17	f	f	PROPN
ejpam-3182	101	18	(	(	PUNCT
ejpam-3182	101	19	pk	pk	NOUN
ejpam-3182	101	20	)	)	PUNCT
ejpam-3182	101	21	≤	≤	NOUN
ejpam-3182	101	22	ck	ck	INTJ
ejpam-3182	101	23	for	for	ADP
ejpam-3182	101	24	all	all	DET
ejpam-3182	101	25	integers	integer	NOUN
ejpam-3182	101	26	k	k	PROPN
ejpam-3182	101	27	∈	∈	PROPN
ejpam-3182	101	28	n	n	CCONJ
ejpam-3182	101	29	,	,	PUNCT
ejpam-3182	101	30	and	and	CCONJ
ejpam-3182	101	31	c	c	NOUN
ejpam-3182	101	32	<	<	X
ejpam-3182	101	33	2	2	NUM
ejpam-3182	101	34	constant	constant	ADJ
ejpam-3182	101	35	.	.	PUNCT
ejpam-3182	102	1	(	(	PUNCT
ejpam-3182	102	2	iii	iii	X
ejpam-3182	102	3	)	)	PUNCT
ejpam-3182	102	4	there	there	PRON
ejpam-3182	102	5	is	be	VERB
ejpam-3182	102	6	a	a	DET
ejpam-3182	102	7	constant	constant	ADJ
ejpam-3182	102	8	τ	τ	X
ejpam-3182	102	9	>	>	X
ejpam-3182	102	10	0	0	NUM
ejpam-3182	102	11	such∑	such∑	PROPN
ejpam-3182	102	12	p≤x	p≤x	PROPN
ejpam-3182	102	13	f(p	f(p	PROPN
ejpam-3182	102	14	)	)	PUNCT
ejpam-3182	103	1	=	=	PUNCT
ejpam-3182	103	2	(	(	PUNCT
ejpam-3182	103	3	τ	τ	X
ejpam-3182	103	4	+	+	CCONJ
ejpam-3182	103	5	o(1))x/	o(1))x/	NUM
ejpam-3182	103	6	log	log	NOUN
ejpam-3182	103	7	x	x	SYM
ejpam-3182	103	8	(	(	PUNCT
ejpam-3182	103	9	14	14	NUM
ejpam-3182	103	10	)	)	PUNCT
ejpam-3182	103	11	as	as	ADP
ejpam-3182	103	12	x	x	X
ejpam-3182	103	13	−→∞.	−→∞.	PUNCT
ejpam-3182	103	14	then	then	ADV
ejpam-3182	103	15	∑	∑	PROPN
ejpam-3182	103	16	n≤x	n≤x	PROPN
ejpam-3182	103	17	f(n	f(n	PROPN
ejpam-3182	103	18	)	)	PUNCT
ejpam-3182	103	19	=	=	PRON
ejpam-3182	104	1	(	(	PUNCT
ejpam-3182	104	2	1	1	NUM
ejpam-3182	104	3	eγτγ(τ	eγτγ(τ	NOUN
ejpam-3182	104	4	)	)	PUNCT
ejpam-3182	104	5	+	+	NUM
ejpam-3182	104	6	o(1	o(1	NOUN
ejpam-3182	104	7	)	)	PUNCT
ejpam-3182	104	8	)	)	PUNCT
ejpam-3182	105	1	x	x	SYM
ejpam-3182	105	2	log	log	VERB
ejpam-3182	105	3	x	x	SYM
ejpam-3182	105	4	∏	∏	PROPN
ejpam-3182	105	5	p≤x	p≤x	PROPN
ejpam-3182	105	6	(	(	PUNCT
ejpam-3182	105	7	1	1	NUM
ejpam-3182	105	8	+	+	NUM
ejpam-3182	105	9	f(p	f(p	NOUN
ejpam-3182	105	10	)	)	PUNCT
ejpam-3182	106	1	p	p	NOUN
ejpam-3182	107	1	+	+	NUM
ejpam-3182	107	2	f	f	X
ejpam-3182	107	3	(	(	PUNCT
ejpam-3182	107	4	p2	p2	PROPN
ejpam-3182	107	5	)	)	PUNCT
ejpam-3182	107	6	p2	p2	PROPN
ejpam-3182	107	7	+	+	X
ejpam-3182	107	8	·	·	PUNCT
ejpam-3182	107	9	·	·	PUNCT
ejpam-3182	107	10	·	·	PUNCT
ejpam-3182	107	11	)	)	PUNCT
ejpam-3182	107	12	.	.	PUNCT
ejpam-3182	108	1	(	(	PUNCT
ejpam-3182	108	2	15	15	X
ejpam-3182	108	3	)	)	PUNCT
ejpam-3182	108	4	the	the	DET
ejpam-3182	108	5	gamma	gamma	NOUN
ejpam-3182	108	6	function	function	NOUN
ejpam-3182	108	7	is	be	AUX
ejpam-3182	108	8	defined	define	VERB
ejpam-3182	108	9	by	by	ADP
ejpam-3182	108	10	γ(s	γ(	NOUN
ejpam-3182	108	11	)	)	PUNCT
ejpam-3182	109	1	=	=	PUNCT
ejpam-3182	109	2	∫∞	∫∞	NOUN
ejpam-3182	109	3	0	0	PUNCT
ejpam-3182	110	1	ts−1e−stdt	ts−1e−stdt	PROPN
ejpam-3182	110	2	,	,	PUNCT
ejpam-3182	110	3	where	where	SCONJ
ejpam-3182	110	4	s	s	VERB
ejpam-3182	110	5	∈	∈	PROPN
ejpam-3182	110	6	c	c	NOUN
ejpam-3182	110	7	is	be	AUX
ejpam-3182	110	8	a	a	DET
ejpam-3182	110	9	complex	complex	ADJ
ejpam-3182	110	10	number	number	NOUN
ejpam-3182	110	11	.	.	PUNCT
ejpam-3182	111	1	the	the	DET
ejpam-3182	111	2	intricate	intricate	ADJ
ejpam-3182	111	3	proof	proof	NOUN
ejpam-3182	111	4	of	of	ADP
ejpam-3182	111	5	wirsing	wirsing	NOUN
ejpam-3182	111	6	formula	formula	NOUN
ejpam-3182	111	7	appears	appear	VERB
ejpam-3182	111	8	in	in	ADP
ejpam-3182	111	9	[	[	X
ejpam-3182	111	10	36	36	NUM
ejpam-3182	111	11	]	]	PUNCT
ejpam-3182	111	12	.	.	PUNCT
ejpam-3182	112	1	it	it	PRON
ejpam-3182	112	2	is	be	AUX
ejpam-3182	112	3	also	also	ADV
ejpam-3182	112	4	assembled	assemble	VERB
ejpam-3182	112	5	in	in	ADP
ejpam-3182	112	6	various	various	ADJ
ejpam-3182	112	7	papers	paper	NOUN
ejpam-3182	112	8	,	,	PUNCT
ejpam-3182	112	9	such	such	ADJ
ejpam-3182	112	10	as	as	ADP
ejpam-3182	112	11	[	[	X
ejpam-3182	112	12	12	12	NUM
ejpam-3182	112	13	]	]	PUNCT
ejpam-3182	112	14	,	,	PUNCT
ejpam-3182	112	15	[	[	X
ejpam-3182	112	16	27	27	NUM
ejpam-3182	112	17	,	,	PUNCT
ejpam-3182	112	18	p.	p.	NOUN
ejpam-3182	112	19	195	195	NUM
ejpam-3182	112	20	]	]	PUNCT
ejpam-3182	112	21	,	,	PUNCT
ejpam-3182	112	22	and	and	CCONJ
ejpam-3182	112	23	discussed	discuss	VERB
ejpam-3182	112	24	in	in	ADP
ejpam-3182	112	25	[	[	X
ejpam-3182	112	26	23	23	NUM
ejpam-3182	112	27	,	,	PUNCT
ejpam-3182	112	28	p.	p.	NOUN
ejpam-3182	112	29	70	70	NUM
ejpam-3182	112	30	]	]	PUNCT
ejpam-3182	112	31	,	,	PUNCT
ejpam-3182	112	32	[	[	X
ejpam-3182	112	33	33	33	NUM
ejpam-3182	112	34	,	,	PUNCT
ejpam-3182	112	35	p.	p.	NOUN
ejpam-3182	112	36	308	308	NUM
ejpam-3182	112	37	]	]	PUNCT
ejpam-3182	112	38	.	.	PUNCT
ejpam-3182	113	1	various	various	ADJ
ejpam-3182	113	2	applications	application	NOUN
ejpam-3182	113	3	are	be	AUX
ejpam-3182	113	4	provided	provide	VERB
ejpam-3182	113	5	in	in	ADP
ejpam-3182	113	6	[	[	X
ejpam-3182	113	7	21	21	NUM
ejpam-3182	113	8	]	]	PUNCT
ejpam-3182	113	9	,	,	PUNCT
ejpam-3182	113	10	[	[	X
ejpam-3182	113	11	25	25	NUM
ejpam-3182	113	12	]	]	PUNCT
ejpam-3182	113	13	,	,	PUNCT
ejpam-3182	113	14	[	[	X
ejpam-3182	113	15	37	37	NUM
ejpam-3182	113	16	]	]	PUNCT
ejpam-3182	113	17	,	,	PUNCT
ejpam-3182	113	18	et	et	PROPN
ejpam-3182	113	19	alii	alii	PROPN
ejpam-3182	113	20	.	.	PUNCT
ejpam-3182	114	1	5	5	X
ejpam-3182	114	2	.	.	X
ejpam-3182	114	3	harmonic	harmonic	ADJ
ejpam-3182	114	4	sums	sum	NOUN
ejpam-3182	114	5	and	and	CCONJ
ejpam-3182	114	6	products	product	NOUN
ejpam-3182	114	7	over	over	ADP
ejpam-3182	114	8	primes	prime	NOUN
ejpam-3182	114	9	with	with	ADP
ejpam-3182	114	10	fixed	fix	VERB
ejpam-3182	114	11	primitive	primitive	ADJ
ejpam-3182	114	12	roots	root	NOUN
ejpam-3182	114	13	the	the	DET
ejpam-3182	114	14	subset	subset	NOUN
ejpam-3182	114	15	of	of	ADP
ejpam-3182	114	16	primes	prime	NOUN
ejpam-3182	114	17	pu	pu	PROPN
ejpam-3182	114	18	=	=	PUNCT
ejpam-3182	114	19	{	{	PUNCT
ejpam-3182	114	20	p	p	NOUN
ejpam-3182	114	21	∈	∈	PROPN
ejpam-3182	114	22	p	p	X
ejpam-3182	114	23	:	:	PUNCT
ejpam-3182	114	24	ordp(u	ordp(u	NOUN
ejpam-3182	114	25	)	)	PUNCT
ejpam-3182	114	26	=	=	PUNCT
ejpam-3182	115	1	p−	p−	NOUN
ejpam-3182	115	2	1	1	X
ejpam-3182	115	3	}	}	PUNCT
ejpam-3182	115	4	⊂	⊂	PRON
ejpam-3182	115	5	p	p	NOUN
ejpam-3182	115	6	consists	consist	VERB
ejpam-3182	115	7	of	of	ADP
ejpam-3182	115	8	all	all	DET
ejpam-3182	115	9	the	the	DET
ejpam-3182	115	10	primes	prime	NOUN
ejpam-3182	115	11	with	with	ADP
ejpam-3182	115	12	a	a	DET
ejpam-3182	115	13	fixed	fix	VERB
ejpam-3182	115	14	primitive	primitive	ADJ
ejpam-3182	115	15	root	root	NOUN
ejpam-3182	115	16	u	u	PROPN
ejpam-3182	115	17	∈	∈	PROPN
ejpam-3182	115	18	z.	z.	PROPN
ejpam-3182	115	19	by	by	ADP
ejpam-3182	115	20	hooley	hooley	PROPN
ejpam-3182	115	21	theorem	theorem	PROPN
ejpam-3182	115	22	,	,	PUNCT
ejpam-3182	115	23	which	which	PRON
ejpam-3182	115	24	is	be	AUX
ejpam-3182	115	25	conditional	conditional	ADJ
ejpam-3182	115	26	on	on	ADP
ejpam-3182	115	27	the	the	DET
ejpam-3182	115	28	generalized	generalized	ADJ
ejpam-3182	115	29	riemann	riemann	PROPN
ejpam-3182	115	30	hypothesis	hypothesis	NOUN
ejpam-3182	115	31	,	,	PUNCT
ejpam-3182	115	32	it	it	PRON
ejpam-3182	115	33	has	have	VERB
ejpam-3182	115	34	nonzero	nonzero	PROPN
ejpam-3182	115	35	density	density	NOUN
ejpam-3182	115	36	αu	αu	NOUN
ejpam-3182	115	37	=	=	SYM
ejpam-3182	115	38	δ	δ	PROPN
ejpam-3182	115	39	(	(	PUNCT
ejpam-3182	115	40	pu	pu	PROPN
ejpam-3182	115	41	)	)	PUNCT
ejpam-3182	115	42	>	>	X
ejpam-3182	116	1	0	0	X
ejpam-3182	116	2	.	.	PUNCT
ejpam-3182	117	1	the	the	DET
ejpam-3182	117	2	real	real	ADJ
ejpam-3182	117	3	number	number	NOUN
ejpam-3182	117	4	n.	n.	NOUN
ejpam-3182	117	5	a.	a.	NOUN
ejpam-3182	117	6	carella	carella	PROPN
ejpam-3182	117	7	/	/	SYM
ejpam-3182	117	8	eur	eur	PROPN
ejpam-3182	117	9	.	.	PUNCT
ejpam-3182	118	1	j.	j.	PROPN
ejpam-3182	118	2	pure	pure	PROPN
ejpam-3182	118	3	appl	appl	PROPN
ejpam-3182	118	4	.	.	PROPN
ejpam-3182	118	5	math	math	PROPN
ejpam-3182	118	6	,	,	PUNCT
ejpam-3182	118	7	11	11	NUM
ejpam-3182	118	8	(	(	PUNCT
ejpam-3182	118	9	1	1	NUM
ejpam-3182	118	10	)	)	PUNCT
ejpam-3182	118	11	(	(	PUNCT
ejpam-3182	118	12	2018	2018	NUM
ejpam-3182	118	13	)	)	PUNCT
ejpam-3182	118	14	,	,	PUNCT
ejpam-3182	118	15	23	23	NUM
ejpam-3182	118	16	-	-	SYM
ejpam-3182	118	17	34	34	NUM
ejpam-3182	118	18	28	28	NUM
ejpam-3182	118	19	αu	αu	PROPN
ejpam-3182	118	20	>	>	X
ejpam-3182	118	21	0	0	NUM
ejpam-3182	118	22	coincides	coincide	VERB
ejpam-3182	118	23	with	with	ADP
ejpam-3182	118	24	the	the	DET
ejpam-3182	118	25	corresponding	corresponding	ADJ
ejpam-3182	118	26	artin	artin	PROPN
ejpam-3182	118	27	constant	constant	ADJ
ejpam-3182	118	28	,	,	PUNCT
ejpam-3182	118	29	see	see	VERB
ejpam-3182	118	30	[	[	X
ejpam-3182	118	31	13	13	NUM
ejpam-3182	118	32	,	,	PUNCT
ejpam-3182	118	33	p.	p.	NOUN
ejpam-3182	118	34	220	220	NUM
ejpam-3182	118	35	]	]	PUNCT
ejpam-3182	118	36	,	,	PUNCT
ejpam-3182	118	37	for	for	ADP
ejpam-3182	118	38	the	the	DET
ejpam-3182	118	39	formula	formula	NOUN
ejpam-3182	118	40	.	.	PUNCT
ejpam-3182	119	1	the	the	DET
ejpam-3182	119	2	proof	proof	NOUN
ejpam-3182	119	3	of	of	ADP
ejpam-3182	119	4	the	the	DET
ejpam-3182	119	5	next	next	ADJ
ejpam-3182	119	6	result	result	NOUN
ejpam-3182	119	7	is	be	AUX
ejpam-3182	119	8	based	base	VERB
ejpam-3182	119	9	on	on	ADP
ejpam-3182	119	10	standard	standard	ADJ
ejpam-3182	119	11	analytic	analytic	ADJ
ejpam-3182	119	12	number	number	NOUN
ejpam-3182	119	13	theory	theory	NOUN
ejpam-3182	119	14	methods	method	NOUN
ejpam-3182	119	15	in	in	ADP
ejpam-3182	119	16	the	the	DET
ejpam-3182	119	17	literature	literature	NOUN
ejpam-3182	119	18	,	,	PUNCT
ejpam-3182	119	19	refer	refer	VERB
ejpam-3182	119	20	to	to	ADP
ejpam-3182	119	21	[	[	X
ejpam-3182	119	22	25	25	NUM
ejpam-3182	119	23	,	,	PUNCT
ejpam-3182	119	24	lemma	lemma	PROPN
ejpam-3182	119	25	4	4	NUM
ejpam-3182	119	26	]	]	PUNCT
ejpam-3182	119	27	.	.	PUNCT
ejpam-3182	120	1	lemma	lemma	PROPN
ejpam-3182	120	2	5.1	5.1	NUM
ejpam-3182	120	3	.	.	PUNCT
ejpam-3182	121	1	assume	assume	VERB
ejpam-3182	121	2	the	the	DET
ejpam-3182	121	3	generalized	generalized	ADJ
ejpam-3182	121	4	riemann	riemann	PROPN
ejpam-3182	121	5	hypothesis	hypothesis	NOUN
ejpam-3182	121	6	,	,	PUNCT
ejpam-3182	121	7	and	and	CCONJ
ejpam-3182	121	8	let	let	VERB
ejpam-3182	121	9	x	x	PRON
ejpam-3182	121	10	≥	≥	PRON
ejpam-3182	121	11	1	1	NUM
ejpam-3182	121	12	be	be	AUX
ejpam-3182	121	13	a	a	DET
ejpam-3182	121	14	large	large	ADJ
ejpam-3182	121	15	number	number	NOUN
ejpam-3182	121	16	.	.	PUNCT
ejpam-3182	122	1	then	then	ADV
ejpam-3182	122	2	,	,	PUNCT
ejpam-3182	122	3	there	there	PRON
ejpam-3182	122	4	exists	exist	VERB
ejpam-3182	122	5	a	a	DET
ejpam-3182	122	6	pair	pair	NOUN
ejpam-3182	122	7	of	of	ADP
ejpam-3182	122	8	constants	constant	NOUN
ejpam-3182	122	9	βu	βu	INTJ
ejpam-3182	122	10	>	>	X
ejpam-3182	122	11	0	0	NUM
ejpam-3182	122	12	,	,	PUNCT
ejpam-3182	122	13	and	and	CCONJ
ejpam-3182	122	14	γu	γu	INTJ
ejpam-3182	122	15	>	>	X
ejpam-3182	122	16	0	0	NUM
ejpam-3182	122	17	such	such	ADJ
ejpam-3182	122	18	that	that	SCONJ
ejpam-3182	122	19	(	(	PUNCT
ejpam-3182	122	20	i	i	NOUN
ejpam-3182	122	21	)	)	PUNCT
ejpam-3182	122	22	∑	∑	PROPN
ejpam-3182	122	23	p≤x	p≤x	PROPN
ejpam-3182	122	24	,	,	PUNCT
ejpam-3182	122	25	p∈pu	p∈pu	NOUN
ejpam-3182	122	26	1	1	NUM
ejpam-3182	122	27	p	p	NOUN
ejpam-3182	122	28	=	=	NOUN
ejpam-3182	122	29	αu	αu	NOUN
ejpam-3182	122	30	log	log	NOUN
ejpam-3182	122	31	log	log	NOUN
ejpam-3182	122	32	x+	x+	PUNCT
ejpam-3182	122	33	βu	βu	PUNCT
ejpam-3182	123	1	+	+	NOUN
ejpam-3182	123	2	o	o	X
ejpam-3182	123	3	(	(	PUNCT
ejpam-3182	123	4	log	log	NOUN
ejpam-3182	123	5	log	log	NOUN
ejpam-3182	123	6	x	x	PUNCT
ejpam-3182	123	7	log	log	NOUN
ejpam-3182	123	8	x	x	PUNCT
ejpam-3182	123	9	)	)	PUNCT
ejpam-3182	123	10	.	.	PUNCT
ejpam-3182	124	1	(	(	PUNCT
ejpam-3182	124	2	ii	ii	NOUN
ejpam-3182	124	3	)	)	PUNCT
ejpam-3182	124	4	∑	∑	PROPN
ejpam-3182	124	5	p≤x	p≤x	PROPN
ejpam-3182	124	6	,	,	PUNCT
ejpam-3182	124	7	p∈pu	p∈pu	PROPN
ejpam-3182	124	8	log	log	VERB
ejpam-3182	124	9	p	p	NOUN
ejpam-3182	124	10	p−	p−	NOUN
ejpam-3182	124	11	1	1	NUM
ejpam-3182	124	12	=	=	SYM
ejpam-3182	124	13	αu	αu	NOUN
ejpam-3182	124	14	log	log	VERB
ejpam-3182	124	15	x−	x−	PROPN
ejpam-3182	124	16	γu	γu	PROPN
ejpam-3182	125	1	+	+	PROPN
ejpam-3182	125	2	o	o	X
ejpam-3182	125	3	(	(	PUNCT
ejpam-3182	125	4	log	log	NOUN
ejpam-3182	125	5	log	log	NOUN
ejpam-3182	125	6	x	x	PUNCT
ejpam-3182	125	7	log	log	NOUN
ejpam-3182	125	8	x	x	PUNCT
ejpam-3182	125	9	)	)	PUNCT
ejpam-3182	125	10	.	.	PUNCT
ejpam-3182	126	1	proof	proof	NOUN
ejpam-3182	126	2	.	.	PUNCT
ejpam-3182	127	1	(	(	PUNCT
ejpam-3182	127	2	i	i	NOUN
ejpam-3182	127	3	)	)	PUNCT
ejpam-3182	127	4	.	.	PUNCT
ejpam-3182	128	1	let	let	VERB
ejpam-3182	128	2	πu(x	πu(x	NOUN
ejpam-3182	128	3	)	)	PUNCT
ejpam-3182	128	4	=	=	SYM
ejpam-3182	128	5	#	#	NOUN
ejpam-3182	128	6	{	{	PUNCT
ejpam-3182	128	7	p	p	NOUN
ejpam-3182	128	8	≤	≤	PROPN
ejpam-3182	128	9	x	x	X
ejpam-3182	128	10	:	:	PUNCT
ejpam-3182	128	11	ordp(u	ordp(u	NOUN
ejpam-3182	128	12	)	)	PUNCT
ejpam-3182	128	13	=	=	PUNCT
ejpam-3182	129	1	p−	p−	NOUN
ejpam-3182	129	2	1	1	X
ejpam-3182	129	3	}	}	PUNCT
ejpam-3182	129	4	=	=	SYM
ejpam-3182	129	5	αuπ(x	αuπ(x	PROPN
ejpam-3182	129	6	)	)	PUNCT
ejpam-3182	129	7	be	be	VERB
ejpam-3182	129	8	the	the	DET
ejpam-3182	129	9	counting	counting	NOUN
ejpam-3182	129	10	measure	measure	NOUN
ejpam-3182	129	11	of	of	ADP
ejpam-3182	129	12	the	the	DET
ejpam-3182	129	13	corresponding	corresponding	ADJ
ejpam-3182	129	14	subset	subset	NOUN
ejpam-3182	129	15	of	of	ADP
ejpam-3182	129	16	primes	prime	NOUN
ejpam-3182	129	17	pu	pu	PROPN
ejpam-3182	129	18	.	.	PUNCT
ejpam-3182	130	1	to	to	PART
ejpam-3182	130	2	estimate	estimate	VERB
ejpam-3182	130	3	the	the	DET
ejpam-3182	130	4	asymptotic	asymptotic	ADJ
ejpam-3182	130	5	order	order	NOUN
ejpam-3182	130	6	of	of	ADP
ejpam-3182	130	7	the	the	DET
ejpam-3182	130	8	prime	prime	ADJ
ejpam-3182	130	9	harmonic	harmonic	ADJ
ejpam-3182	130	10	sum	sum	NOUN
ejpam-3182	130	11	,	,	PUNCT
ejpam-3182	130	12	use	use	VERB
ejpam-3182	130	13	the	the	DET
ejpam-3182	130	14	stieltjes	stieltjes	NOUN
ejpam-3182	130	15	integral	integral	ADJ
ejpam-3182	130	16	representation:∑	representation:∑	PROPN
ejpam-3182	130	17	p≤x	p≤x	PROPN
ejpam-3182	130	18	,	,	PUNCT
ejpam-3182	130	19	p∈pu	p∈pu	NOUN
ejpam-3182	130	20	1	1	NUM
ejpam-3182	130	21	p	p	NOUN
ejpam-3182	130	22	=	=	PUNCT
ejpam-3182	130	23	∫	∫	PROPN
ejpam-3182	131	1	x	x	SYM
ejpam-3182	131	2	x0	x0	PROPN
ejpam-3182	131	3	1	1	NUM
ejpam-3182	131	4	t	t	NOUN
ejpam-3182	131	5	dπu(t	dπu(t	PROPN
ejpam-3182	131	6	)	)	PUNCT
ejpam-3182	131	7	=	=	PUNCT
ejpam-3182	131	8	πu(x	πu(x	X
ejpam-3182	131	9	)	)	PUNCT
ejpam-3182	131	10	x	x	SYM
ejpam-3182	132	1	+	+	NUM
ejpam-3182	132	2	c(x0	c(x0	NOUN
ejpam-3182	132	3	)	)	PUNCT
ejpam-3182	133	1	+	+	CCONJ
ejpam-3182	133	2	∫	∫	PROPN
ejpam-3182	133	3	x	x	SYM
ejpam-3182	133	4	x0	x0	PROPN
ejpam-3182	133	5	πu(t	πu(t	PRON
ejpam-3182	133	6	)	)	PUNCT
ejpam-3182	133	7	t2	t2	NOUN
ejpam-3182	133	8	dt	dt	PROPN
ejpam-3182	133	9	,	,	PUNCT
ejpam-3182	133	10	(	(	PUNCT
ejpam-3182	133	11	16	16	NUM
ejpam-3182	133	12	)	)	PUNCT
ejpam-3182	133	13	where	where	SCONJ
ejpam-3182	133	14	x0	x0	PROPN
ejpam-3182	133	15	>	>	X
ejpam-3182	133	16	0	0	PUNCT
ejpam-3182	133	17	is	be	AUX
ejpam-3182	133	18	a	a	DET
ejpam-3182	133	19	constant	constant	ADJ
ejpam-3182	133	20	.	.	PUNCT
ejpam-3182	134	1	applying	apply	VERB
ejpam-3182	134	2	theorem	theorem	VERB
ejpam-3182	134	3	7.1	7.1	NUM
ejpam-3182	134	4	yields∫	yields∫	NOUN
ejpam-3182	134	5	x	x	SYM
ejpam-3182	134	6	x0	x0	PROPN
ejpam-3182	134	7	1	1	NUM
ejpam-3182	134	8	t	t	NOUN
ejpam-3182	134	9	dπu(t	dπu(t	PROPN
ejpam-3182	134	10	)	)	PUNCT
ejpam-3182	134	11	=	=	PUNCT
ejpam-3182	135	1	αu	αu	NOUN
ejpam-3182	135	2	log	log	VERB
ejpam-3182	135	3	x	x	PUNCT
ejpam-3182	136	1	+	+	NOUN
ejpam-3182	136	2	o	o	X
ejpam-3182	136	3	(	(	PUNCT
ejpam-3182	136	4	log	log	NOUN
ejpam-3182	136	5	log	log	NOUN
ejpam-3182	136	6	x	x	PUNCT
ejpam-3182	136	7	log2(x	log2(x	NOUN
ejpam-3182	136	8	)	)	PUNCT
ejpam-3182	136	9	)	)	PUNCT
ejpam-3182	137	1	+	+	CCONJ
ejpam-3182	137	2	c0(x0	c0(x0	X
ejpam-3182	137	3	)	)	PUNCT
ejpam-3182	137	4	+	+	NOUN
ejpam-3182	137	5	αu	αu	NOUN
ejpam-3182	137	6	∫	∫	PROPN
ejpam-3182	137	7	x	x	X
ejpam-3182	137	8	x0	x0	PROPN
ejpam-3182	137	9	(	(	PUNCT
ejpam-3182	137	10	1	1	NUM
ejpam-3182	137	11	t	t	NOUN
ejpam-3182	137	12	log	log	NOUN
ejpam-3182	137	13	t	t	PROPN
ejpam-3182	138	1	+	+	NOUN
ejpam-3182	138	2	o	o	X
ejpam-3182	138	3	(	(	PUNCT
ejpam-3182	138	4	log	log	PROPN
ejpam-3182	138	5	log	log	NOUN
ejpam-3182	138	6	t	t	PROPN
ejpam-3182	138	7	t	t	PROPN
ejpam-3182	138	8	log2(t	log2(t	NUM
ejpam-3182	138	9	)	)	PUNCT
ejpam-3182	138	10	)	)	PUNCT
ejpam-3182	138	11	)	)	PUNCT
ejpam-3182	139	1	dt	dt	X
ejpam-3182	139	2	(	(	PUNCT
ejpam-3182	139	3	17	17	NUM
ejpam-3182	139	4	)	)	PUNCT
ejpam-3182	139	5	=	=	NOUN
ejpam-3182	140	1	αu	αu	NOUN
ejpam-3182	140	2	log	log	NOUN
ejpam-3182	140	3	log	log	NOUN
ejpam-3182	140	4	x−	x−	PROPN
ejpam-3182	140	5	log	log	PROPN
ejpam-3182	140	6	log	log	NOUN
ejpam-3182	140	7	x0	x0	PROPN
ejpam-3182	140	8	+	+	CCONJ
ejpam-3182	140	9	c0	c0	PROPN
ejpam-3182	140	10	(	(	PUNCT
ejpam-3182	140	11	x0	x0	PROPN
ejpam-3182	140	12	)	)	PUNCT
ejpam-3182	141	1	+	+	NOUN
ejpam-3182	141	2	o	o	X
ejpam-3182	141	3	(	(	PUNCT
ejpam-3182	141	4	log	log	NOUN
ejpam-3182	141	5	log	log	NOUN
ejpam-3182	141	6	x	x	PUNCT
ejpam-3182	141	7	log	log	NOUN
ejpam-3182	141	8	x	x	PUNCT
ejpam-3182	141	9	)	)	PUNCT
ejpam-3182	141	10	,	,	PUNCT
ejpam-3182	141	11	where	where	SCONJ
ejpam-3182	141	12	βu	βu	ADP
ejpam-3182	141	13	=	=	SYM
ejpam-3182	142	1	−	−	PROPN
ejpam-3182	142	2	log	log	NOUN
ejpam-3182	142	3	log	log	NOUN
ejpam-3182	142	4	x0	x0	PROPN
ejpam-3182	143	1	+	+	PROPN
ejpam-3182	143	2	c0	c0	PROPN
ejpam-3182	143	3	(	(	PUNCT
ejpam-3182	143	4	x0	x0	PROPN
ejpam-3182	143	5	)	)	PUNCT
ejpam-3182	143	6	is	be	AUX
ejpam-3182	143	7	the	the	DET
ejpam-3182	143	8	artin	artin	PROPN
ejpam-3182	143	9	-	-	PUNCT
ejpam-3182	143	10	mertens	mertens	PROPN
ejpam-3182	143	11	constant	constant	PROPN
ejpam-3182	143	12	.	.	PUNCT
ejpam-3182	144	1	the	the	DET
ejpam-3182	144	2	statement	statement	NOUN
ejpam-3182	144	3	(	(	PUNCT
ejpam-3182	144	4	ii	ii	NOUN
ejpam-3182	144	5	)	)	PUNCT
ejpam-3182	144	6	follows	follow	VERB
ejpam-3182	144	7	from	from	ADP
ejpam-3182	144	8	statement	statement	NOUN
ejpam-3182	144	9	(	(	PUNCT
ejpam-3182	144	10	i	i	NOUN
ejpam-3182	144	11	)	)	PUNCT
ejpam-3182	144	12	and	and	CCONJ
ejpam-3182	144	13	partial	partial	ADJ
ejpam-3182	144	14	summation	summation	NOUN
ejpam-3182	144	15	.	.	PUNCT
ejpam-3182	145	1	the	the	DET
ejpam-3182	145	2	artin	artin	PROPN
ejpam-3182	145	3	-	-	PUNCT
ejpam-3182	145	4	mertens	mertens	PROPN
ejpam-3182	145	5	constant	constant	ADJ
ejpam-3182	145	6	βu	βu	X
ejpam-3182	145	7	and	and	CCONJ
ejpam-3182	145	8	the	the	DET
ejpam-3182	145	9	artin	artin	PROPN
ejpam-3182	145	10	-	-	PUNCT
ejpam-3182	145	11	euler	euler	PROPN
ejpam-3182	145	12	constant	constant	PROPN
ejpam-3182	145	13	γu	γu	NOUN
ejpam-3182	145	14	have	have	VERB
ejpam-3182	145	15	other	other	ADJ
ejpam-3182	145	16	equivalent	equivalent	ADJ
ejpam-3182	145	17	definitions	definition	NOUN
ejpam-3182	145	18	such	such	ADJ
ejpam-3182	145	19	as	as	ADP
ejpam-3182	145	20	βu	βu	PROPN
ejpam-3182	145	21	=	=	PROPN
ejpam-3182	145	22	lim	lim	PROPN
ejpam-3182	145	23	x→∞	x→∞	X
ejpam-3182	146	1			PROPN
ejpam-3182	146	2	∑	∑	PUNCT
ejpam-3182	146	3	p≤x	p≤x	PROPN
ejpam-3182	146	4	,	,	PUNCT
ejpam-3182	146	5	p∈pu	p∈pu	PROPN
ejpam-3182	146	6	1	1	NUM
ejpam-3182	146	7	p	p	NOUN
ejpam-3182	146	8	−	−	PROPN
ejpam-3182	146	9	αu	αu	NOUN
ejpam-3182	146	10	log	log	NOUN
ejpam-3182	146	11	log	log	NOUN
ejpam-3182	146	12	x	x	PUNCT
ejpam-3182	146	13			PROPN
ejpam-3182	146	14	and	and	CCONJ
ejpam-3182	146	15	βu	βu	NOUN
ejpam-3182	146	16	=	=	NOUN
ejpam-3182	146	17	γu	γu	NOUN
ejpam-3182	146	18	−	−	PROPN
ejpam-3182	146	19	∑	∑	ADV
ejpam-3182	146	20	p∈pu	p∈pu	PROPN
ejpam-3182	146	21	,	,	PUNCT
ejpam-3182	146	22	∑	∑	PUNCT
ejpam-3182	146	23	k≥2	k≥2	PROPN
ejpam-3182	146	24	1	1	NUM
ejpam-3182	146	25	kpk	kpk	PROPN
ejpam-3182	146	26	,	,	PUNCT
ejpam-3182	146	27	(	(	PUNCT
ejpam-3182	146	28	18	18	NUM
ejpam-3182	146	29	)	)	PUNCT
ejpam-3182	146	30	respectively	respectively	ADV
ejpam-3182	146	31	.	.	PUNCT
ejpam-3182	147	1	these	these	DET
ejpam-3182	147	2	constants	constant	NOUN
ejpam-3182	147	3	satisfy	satisfy	VERB
ejpam-3182	147	4	βu	βu	PUNCT
ejpam-3182	148	1	=	=	PUNCT
ejpam-3182	148	2	β1αu	β1αu	PUNCT
ejpam-3182	148	3	and	and	CCONJ
ejpam-3182	148	4	γu	γu	NOUN
ejpam-3182	148	5	=	=	SYM
ejpam-3182	148	6	γαu	γαu	PROPN
ejpam-3182	148	7	.	.	PUNCT
ejpam-3182	149	1	if	if	SCONJ
ejpam-3182	149	2	the	the	DET
ejpam-3182	149	3	density	density	NOUN
ejpam-3182	149	4	αu	αu	NOUN
ejpam-3182	149	5	=	=	SYM
ejpam-3182	149	6	1	1	NUM
ejpam-3182	149	7	,	,	PUNCT
ejpam-3182	149	8	these	these	DET
ejpam-3182	149	9	definitions	definition	NOUN
ejpam-3182	149	10	reduce	reduce	VERB
ejpam-3182	149	11	to	to	ADP
ejpam-3182	149	12	the	the	DET
ejpam-3182	149	13	usual	usual	ADJ
ejpam-3182	149	14	euler	euler	NOUN
ejpam-3182	149	15	constant	constant	ADJ
ejpam-3182	149	16	and	and	CCONJ
ejpam-3182	149	17	the	the	DET
ejpam-3182	149	18	mertens	mertens	PROPN
ejpam-3182	149	19	constant	constant	PROPN
ejpam-3182	149	20	,	,	PUNCT
ejpam-3182	149	21	which	which	PRON
ejpam-3182	149	22	are	be	AUX
ejpam-3182	149	23	defined	define	VERB
ejpam-3182	149	24	by	by	ADP
ejpam-3182	149	25	the	the	DET
ejpam-3182	149	26	limits	limit	NOUN
ejpam-3182	149	27	γ	γ	X
ejpam-3182	149	28	=	=	PROPN
ejpam-3182	149	29	lim	lim	PROPN
ejpam-3182	149	30	x→∞	x→∞	NUM
ejpam-3182	150	1	∑	∑	X
ejpam-3182	150	2	p≤x	p≤x	NOUN
ejpam-3182	150	3	log	log	NOUN
ejpam-3182	150	4	p	p	NOUN
ejpam-3182	150	5	p−	p−	NOUN
ejpam-3182	150	6	1	1	NUM
ejpam-3182	150	7	−	−	NOUN
ejpam-3182	150	8	log	log	NOUN
ejpam-3182	150	9	x	x	PUNCT
ejpam-3182	150	10			PROPN
ejpam-3182	150	11	and	and	CCONJ
ejpam-3182	150	12	β1	β1	PROPN
ejpam-3182	150	13	=	=	SYM
ejpam-3182	150	14	lim	lim	PROPN
ejpam-3182	150	15	x→∞	x→∞	X
ejpam-3182	151	1	∑	∑	X
ejpam-3182	151	2	p≤x	p≤x	PROPN
ejpam-3182	151	3	1	1	NUM
ejpam-3182	152	1	p	p	NOUN
ejpam-3182	152	2	−	−	PROPN
ejpam-3182	152	3	log	log	NOUN
ejpam-3182	152	4	log	log	NOUN
ejpam-3182	152	5	x	x	PUNCT
ejpam-3182	152	6			PROPN
ejpam-3182	152	7	,	,	PUNCT
ejpam-3182	152	8	(	(	PUNCT
ejpam-3182	152	9	19	19	NUM
ejpam-3182	152	10	)	)	PUNCT
ejpam-3182	152	11	n.	n.	NOUN
ejpam-3182	152	12	a.	a.	NOUN
ejpam-3182	152	13	carella	carella	PROPN
ejpam-3182	152	14	/	/	SYM
ejpam-3182	152	15	eur	eur	PROPN
ejpam-3182	152	16	.	.	PUNCT
ejpam-3182	153	1	j.	j.	PROPN
ejpam-3182	153	2	pure	pure	PROPN
ejpam-3182	153	3	appl	appl	PROPN
ejpam-3182	153	4	.	.	PROPN
ejpam-3182	153	5	math	math	PROPN
ejpam-3182	153	6	,	,	PUNCT
ejpam-3182	153	7	11	11	NUM
ejpam-3182	153	8	(	(	PUNCT
ejpam-3182	153	9	1	1	NUM
ejpam-3182	153	10	)	)	PUNCT
ejpam-3182	153	11	(	(	PUNCT
ejpam-3182	153	12	2018	2018	NUM
ejpam-3182	153	13	)	)	PUNCT
ejpam-3182	153	14	,	,	PUNCT
ejpam-3182	153	15	23	23	NUM
ejpam-3182	153	16	-	-	SYM
ejpam-3182	153	17	34	34	NUM
ejpam-3182	153	18	29	29	NUM
ejpam-3182	153	19	or	or	CCONJ
ejpam-3182	153	20	some	some	DET
ejpam-3182	153	21	other	other	ADJ
ejpam-3182	153	22	equivalent	equivalent	ADJ
ejpam-3182	153	23	definitions	definition	NOUN
ejpam-3182	153	24	,	,	PUNCT
ejpam-3182	153	25	respectively	respectively	ADV
ejpam-3182	153	26	.	.	PUNCT
ejpam-3182	154	1	moreover	moreover	ADV
ejpam-3182	154	2	,	,	PUNCT
ejpam-3182	154	3	the	the	DET
ejpam-3182	154	4	linear	linear	ADJ
ejpam-3182	154	5	independence	independence	NOUN
ejpam-3182	154	6	relation	relation	NOUN
ejpam-3182	154	7	in	in	ADP
ejpam-3182	154	8	(	(	PUNCT
ejpam-3182	154	9	18	18	NUM
ejpam-3182	154	10	)	)	PUNCT
ejpam-3182	154	11	becomes	become	VERB
ejpam-3182	154	12	β	β	NOUN
ejpam-3182	154	13	=	=	PUNCT
ejpam-3182	154	14	γ	γ	X
ejpam-3182	154	15	−	−	NOUN
ejpam-3182	154	16	∑	∑	PUNCT
ejpam-3182	154	17	p≥2	p≥2	PROPN
ejpam-3182	154	18	∑	∑	PROPN
ejpam-3182	154	19	k≥2	k≥2	PROPN
ejpam-3182	154	20	(	(	PUNCT
ejpam-3182	154	21	kpk	kpk	PROPN
ejpam-3182	154	22	)	)	PUNCT
ejpam-3182	154	23	−1	−1	NOUN
ejpam-3182	154	24	,	,	PUNCT
ejpam-3182	154	25	see	see	VERB
ejpam-3182	154	26	[	[	X
ejpam-3182	154	27	11	11	NUM
ejpam-3182	154	28	,	,	PUNCT
ejpam-3182	154	29	theorem	theorem	VERB
ejpam-3182	154	30	427	427	NUM
ejpam-3182	154	31	]	]	PUNCT
ejpam-3182	154	32	.	.	PUNCT
ejpam-3182	155	1	a	a	DET
ejpam-3182	155	2	numerical	numerical	ADJ
ejpam-3182	155	3	experiment	experiment	NOUN
ejpam-3182	155	4	for	for	ADP
ejpam-3182	155	5	the	the	DET
ejpam-3182	155	6	primitive	primitive	ADJ
ejpam-3182	155	7	root	root	NOUN
ejpam-3182	155	8	u	u	NOUN
ejpam-3182	155	9	=	=	SYM
ejpam-3182	155	10	2	2	NUM
ejpam-3182	155	11	gives	give	VERB
ejpam-3182	155	12	the	the	DET
ejpam-3182	155	13	approximate	approximate	ADJ
ejpam-3182	155	14	values	value	NOUN
ejpam-3182	155	15	(	(	PUNCT
ejpam-3182	155	16	i	i	NOUN
ejpam-3182	155	17	)	)	PUNCT
ejpam-3182	156	1	α2	α2	PROPN
ejpam-3182	157	1	=	=	SYM
ejpam-3182	158	1	∏	∏	PROPN
ejpam-3182	159	1	p≥2	p≥2	NOUN
ejpam-3182	159	2	(	(	PUNCT
ejpam-3182	159	3	1−	1−	NUM
ejpam-3182	159	4	1	1	NUM
ejpam-3182	159	5	p(p−	p(p−	VERB
ejpam-3182	159	6	1	1	NUM
ejpam-3182	159	7	)	)	PUNCT
ejpam-3182	159	8	)	)	PUNCT
ejpam-3182	160	1	=	=	PUNCT
ejpam-3182	161	1	0.3739558667768911078453786	0.3739558667768911078453786	NUM
ejpam-3182	161	2	.	.	PUNCT
ejpam-3182	161	3	.	.	PUNCT
ejpam-3182	161	4	.	.	PUNCT
ejpam-3182	161	5	.	.	PUNCT
ejpam-3182	162	1	(	(	PUNCT
ejpam-3182	162	2	ii	ii	NOUN
ejpam-3182	162	3	)	)	PUNCT
ejpam-3182	162	4	β2	β2	PROPN
ejpam-3182	162	5	≈	≈	PROPN
ejpam-3182	162	6	∑	∑	PROPN
ejpam-3182	162	7	p≤1000,p∈p2	p≤1000,p∈p2	PROPN
ejpam-3182	162	8	1	1	NUM
ejpam-3182	162	9	p	p	NOUN
ejpam-3182	162	10	−	−	NOUN
ejpam-3182	162	11	α2	α2	ADJ
ejpam-3182	162	12	log	log	NOUN
ejpam-3182	162	13	log	log	NOUN
ejpam-3182	162	14	x	x	PUNCT
ejpam-3182	163	1	=	=	PUNCT
ejpam-3182	163	2	0.328644525584805374999956	0.328644525584805374999956	NUM
ejpam-3182	163	3	...	...	PUNCT
ejpam-3182	163	4	,	,	PUNCT
ejpam-3182	163	5	and	and	CCONJ
ejpam-3182	163	6	(	(	PUNCT
ejpam-3182	163	7	iii	iii	X
ejpam-3182	163	8	)	)	PUNCT
ejpam-3182	163	9	γ2	γ2	PROPN
ejpam-3182	163	10	≈	≈	PROPN
ejpam-3182	163	11	∑	∑	PROPN
ejpam-3182	163	12	p≤1000,p∈p2	p≤1000,p∈p2	PROPN
ejpam-3182	163	13	log	log	VERB
ejpam-3182	163	14	p	p	NOUN
ejpam-3182	163	15	p−	p−	NOUN
ejpam-3182	163	16	1	1	NUM
ejpam-3182	163	17	−	−	NOUN
ejpam-3182	163	18	α2	α2	ADJ
ejpam-3182	163	19	log	log	NOUN
ejpam-3182	163	20	x	x	PUNCT
ejpam-3182	163	21	=	=	NOUN
ejpam-3182	163	22	0.424902273366234745796616	0.424902273366234745796616	NUM
ejpam-3182	163	23	.	.	PUNCT
ejpam-3182	163	24	.	.	PUNCT
ejpam-3182	163	25	.	.	PUNCT
ejpam-3182	163	26	.	.	PUNCT
ejpam-3182	164	1	lemma	lemma	PROPN
ejpam-3182	164	2	5.2	5.2	NUM
ejpam-3182	164	3	.	.	PUNCT
ejpam-3182	165	1	assume	assume	VERB
ejpam-3182	165	2	the	the	DET
ejpam-3182	165	3	generalized	generalized	ADJ
ejpam-3182	165	4	riemann	riemann	PROPN
ejpam-3182	165	5	hypothesis	hypothesis	NOUN
ejpam-3182	165	6	,	,	PUNCT
ejpam-3182	165	7	and	and	CCONJ
ejpam-3182	165	8	let	let	VERB
ejpam-3182	165	9	x	x	PRON
ejpam-3182	165	10	≥	≥	PRON
ejpam-3182	165	11	1	1	NUM
ejpam-3182	165	12	be	be	AUX
ejpam-3182	165	13	a	a	DET
ejpam-3182	165	14	large	large	ADJ
ejpam-3182	165	15	number	number	NOUN
ejpam-3182	165	16	.	.	PUNCT
ejpam-3182	166	1	then	then	ADV
ejpam-3182	166	2	,	,	PUNCT
ejpam-3182	166	3	there	there	PRON
ejpam-3182	166	4	exists	exist	VERB
ejpam-3182	166	5	a	a	DET
ejpam-3182	166	6	pair	pair	NOUN
ejpam-3182	166	7	of	of	ADP
ejpam-3182	166	8	constants	constant	NOUN
ejpam-3182	166	9	γu	γu	INTJ
ejpam-3182	166	10	>	>	X
ejpam-3182	166	11	0	0	PUNCT
ejpam-3182	166	12	and	and	CCONJ
ejpam-3182	166	13	νu	νu	X
ejpam-3182	166	14	>	>	X
ejpam-3182	166	15	0	0	NUM
ejpam-3182	166	16	such	such	ADJ
ejpam-3182	166	17	that	that	SCONJ
ejpam-3182	166	18	(	(	PUNCT
ejpam-3182	166	19	i	i	NOUN
ejpam-3182	166	20	)	)	PUNCT
ejpam-3182	166	21	∏	∏	PROPN
ejpam-3182	166	22	p≤x	p≤x	PROPN
ejpam-3182	166	23	,	,	PUNCT
ejpam-3182	166	24	p∈pu	p∈pu	NOUN
ejpam-3182	166	25	(	(	PUNCT
ejpam-3182	166	26	1−	1−	NUM
ejpam-3182	166	27	1	1	NUM
ejpam-3182	166	28	p	p	NOUN
ejpam-3182	166	29	)	)	PUNCT
ejpam-3182	166	30	−1	−1	NOUN
ejpam-3182	166	31	=	=	NOUN
ejpam-3182	166	32	eγu	eγu	VERB
ejpam-3182	166	33	log(x)αu	log(x)αu	PROPN
ejpam-3182	167	1	+	+	NOUN
ejpam-3182	167	2	o	o	X
ejpam-3182	167	3	(	(	PUNCT
ejpam-3182	167	4	log	log	NOUN
ejpam-3182	167	5	log	log	NOUN
ejpam-3182	167	6	x	x	PUNCT
ejpam-3182	167	7	log	log	NOUN
ejpam-3182	167	8	x	x	PUNCT
ejpam-3182	167	9	)	)	PUNCT
ejpam-3182	167	10	.	.	PUNCT
ejpam-3182	168	1	(	(	PUNCT
ejpam-3182	168	2	ii	ii	X
ejpam-3182	168	3	)	)	PUNCT
ejpam-3182	168	4	∏	∏	PROPN
ejpam-3182	168	5	p≤x	p≤x	PROPN
ejpam-3182	168	6	,	,	PUNCT
ejpam-3182	168	7	p∈pu	p∈pu	NOUN
ejpam-3182	168	8	(	(	PUNCT
ejpam-3182	168	9	1	1	NUM
ejpam-3182	168	10	+	+	SYM
ejpam-3182	168	11	1	1	NUM
ejpam-3182	168	12	p	p	NOUN
ejpam-3182	168	13	)	)	PUNCT
ejpam-3182	169	1	=	=	PRON
ejpam-3182	169	2	eγu	eγu	VERB
ejpam-3182	169	3	∏	∏	PROPN
ejpam-3182	169	4	p∈pu	p∈pu	PROPN
ejpam-3182	169	5	(	(	PUNCT
ejpam-3182	169	6	1−	1−	NUM
ejpam-3182	169	7	p−2	p−2	PROPN
ejpam-3182	169	8	)	)	PUNCT
ejpam-3182	169	9	log(x)αu	log(x)αu	PUNCT
ejpam-3182	170	1	+	+	PUNCT
ejpam-3182	170	2	o	o	X
ejpam-3182	170	3	(	(	PUNCT
ejpam-3182	170	4	log	log	NOUN
ejpam-3182	170	5	log	log	NOUN
ejpam-3182	170	6	x	x	PUNCT
ejpam-3182	170	7	log	log	NOUN
ejpam-3182	170	8	x	x	PUNCT
ejpam-3182	170	9	)	)	PUNCT
ejpam-3182	170	10	.	.	PUNCT
ejpam-3182	171	1	(	(	PUNCT
ejpam-3182	171	2	iii	iii	X
ejpam-3182	171	3	)	)	PUNCT
ejpam-3182	171	4	∏	∏	PROPN
ejpam-3182	171	5	p≤x	p≤x	PROPN
ejpam-3182	171	6	,	,	PUNCT
ejpam-3182	171	7	p∈pu	p∈pu	NOUN
ejpam-3182	171	8	(	(	PUNCT
ejpam-3182	171	9	1−	1−	NUM
ejpam-3182	171	10	log	log	NOUN
ejpam-3182	171	11	p	p	NOUN
ejpam-3182	171	12	p−	p−	NOUN
ejpam-3182	171	13	1	1	NUM
ejpam-3182	171	14	)	)	PUNCT
ejpam-3182	171	15	−1	−1	NOUN
ejpam-3182	171	16	=	=	PUNCT
ejpam-3182	171	17	eνu−γuxαu	eνu−γuxαu	PUNCT
ejpam-3182	172	1	+	+	ADJ
ejpam-3182	172	2	o	o	X
ejpam-3182	172	3	(	(	PUNCT
ejpam-3182	172	4	xαu	xαu	PROPN
ejpam-3182	172	5	log	log	VERB
ejpam-3182	172	6	log	log	NOUN
ejpam-3182	172	7	x	x	PUNCT
ejpam-3182	172	8	log	log	NOUN
ejpam-3182	172	9	x	x	PUNCT
ejpam-3182	172	10	)	)	PUNCT
ejpam-3182	172	11	.	.	PUNCT
ejpam-3182	173	1	proof	proof	NOUN
ejpam-3182	173	2	.	.	PUNCT
ejpam-3182	174	1	(	(	PUNCT
ejpam-3182	174	2	i	i	NOUN
ejpam-3182	174	3	)	)	PUNCT
ejpam-3182	174	4	.	.	PUNCT
ejpam-3182	175	1	express	express	VERB
ejpam-3182	175	2	the	the	DET
ejpam-3182	175	3	logarithm	logarithm	NOUN
ejpam-3182	175	4	of	of	ADP
ejpam-3182	175	5	the	the	DET
ejpam-3182	175	6	product	product	NOUN
ejpam-3182	175	7	as	as	ADP
ejpam-3182	175	8	∑	∑	PROPN
ejpam-3182	175	9	p≤x	p≤x	PROPN
ejpam-3182	175	10	,	,	PUNCT
ejpam-3182	175	11	p∈pu	p∈pu	PROPN
ejpam-3182	175	12	log	log	VERB
ejpam-3182	175	13	(	(	PUNCT
ejpam-3182	175	14	1−	1−	NUM
ejpam-3182	175	15	1	1	NUM
ejpam-3182	175	16	p	p	NOUN
ejpam-3182	175	17	)	)	PUNCT
ejpam-3182	175	18	−1	−1	NOUN
ejpam-3182	175	19	=	=	SYM
ejpam-3182	175	20	∑	∑	PUNCT
ejpam-3182	175	21	p≤x	p≤x	PROPN
ejpam-3182	175	22	,	,	PUNCT
ejpam-3182	175	23	p∈pu	p∈pu	NOUN
ejpam-3182	175	24	,	,	PUNCT
ejpam-3182	175	25	∑	∑	PUNCT
ejpam-3182	175	26	k≥1	k≥1	PROPN
ejpam-3182	175	27	1	1	NUM
ejpam-3182	175	28	kpk	kpk	PROPN
ejpam-3182	175	29	=	=	SYM
ejpam-3182	175	30	∑	∑	PUNCT
ejpam-3182	175	31	p≤x	p≤x	PROPN
ejpam-3182	175	32	,	,	PUNCT
ejpam-3182	175	33	p∈pu	p∈pu	NOUN
ejpam-3182	175	34	1	1	NUM
ejpam-3182	175	35	p	p	NOUN
ejpam-3182	175	36	+	+	PROPN
ejpam-3182	175	37	∑	∑	PROPN
ejpam-3182	175	38	p≤x	p≤x	PROPN
ejpam-3182	175	39	,	,	PUNCT
ejpam-3182	175	40	p∈pu	p∈pu	NOUN
ejpam-3182	175	41	,	,	PUNCT
ejpam-3182	175	42	∑	∑	PUNCT
ejpam-3182	175	43	k≥2	k≥2	PROPN
ejpam-3182	175	44	1	1	NUM
ejpam-3182	175	45	kpk	kpk	PROPN
ejpam-3182	175	46	.	.	PUNCT
ejpam-3182	176	1	(	(	PUNCT
ejpam-3182	176	2	20	20	NUM
ejpam-3182	176	3	)	)	PUNCT
ejpam-3182	176	4	apply	apply	VERB
ejpam-3182	176	5	lemma	lemma	PROPN
ejpam-3182	176	6	5.1	5.1	NUM
ejpam-3182	176	7	to	to	PART
ejpam-3182	176	8	complete	complete	VERB
ejpam-3182	176	9	the	the	DET
ejpam-3182	176	10	verification	verification	NOUN
ejpam-3182	176	11	.	.	PUNCT
ejpam-3182	177	1	for	for	ADP
ejpam-3182	177	2	statements	statement	NOUN
ejpam-3182	177	3	(	(	PUNCT
ejpam-3182	177	4	ii	ii	NOUN
ejpam-3182	177	5	)	)	PUNCT
ejpam-3182	177	6	and	and	CCONJ
ejpam-3182	177	7	(	(	PUNCT
ejpam-3182	177	8	iii	iii	NOUN
ejpam-3182	177	9	)	)	PUNCT
ejpam-3182	177	10	,	,	PUNCT
ejpam-3182	177	11	use	use	VERB
ejpam-3182	177	12	similar	similar	ADJ
ejpam-3182	177	13	methods	method	NOUN
ejpam-3182	177	14	as	as	ADP
ejpam-3182	177	15	in	in	ADP
ejpam-3182	177	16	the	the	DET
ejpam-3182	177	17	first	first	ADJ
ejpam-3182	177	18	one	one	NUM
ejpam-3182	177	19	.	.	PUNCT
ejpam-3182	178	1	the	the	DET
ejpam-3182	178	2	constant	constant	ADJ
ejpam-3182	178	3	νu	νu	X
ejpam-3182	178	4	>	>	X
ejpam-3182	178	5	0	0	NUM
ejpam-3182	178	6	is	be	AUX
ejpam-3182	178	7	defined	define	VERB
ejpam-3182	178	8	by	by	ADP
ejpam-3182	178	9	the	the	DET
ejpam-3182	178	10	double	double	ADJ
ejpam-3182	178	11	power	power	NOUN
ejpam-3182	178	12	series	series	NOUN
ejpam-3182	178	13	(	(	PUNCT
ejpam-3182	178	14	an	an	DET
ejpam-3182	178	15	approximate	approximate	ADJ
ejpam-3182	178	16	numerical	numerical	ADJ
ejpam-3182	178	17	value	value	NOUN
ejpam-3182	178	18	for	for	ADP
ejpam-3182	178	19	set	set	VERB
ejpam-3182	178	20	p2	p2	NOUN
ejpam-3182	178	21	=	=	SYM
ejpam-3182	178	22	{	{	PUNCT
ejpam-3182	178	23	3	3	NUM
ejpam-3182	178	24	,	,	PUNCT
ejpam-3182	178	25	5	5	NUM
ejpam-3182	178	26	,	,	PUNCT
ejpam-3182	178	27	11	11	NUM
ejpam-3182	178	28	,	,	PUNCT
ejpam-3182	178	29	13	13	NUM
ejpam-3182	178	30	,	,	PUNCT
ejpam-3182	178	31	...	...	PUNCT
ejpam-3182	178	32	}	}	PUNCT
ejpam-3182	178	33	is	be	AUX
ejpam-3182	178	34	shown	show	VERB
ejpam-3182	178	35	):	):	PUNCT
ejpam-3182	179	1	ν2	ν2	NOUN
ejpam-3182	179	2	=	=	SYM
ejpam-3182	179	3	∑	∑	PUNCT
ejpam-3182	179	4	p∈p2	p∈p2	NOUN
ejpam-3182	179	5	,	,	PUNCT
ejpam-3182	179	6	∑	∑	PUNCT
ejpam-3182	179	7	k≥2	k≥2	PROPN
ejpam-3182	179	8	1	1	NUM
ejpam-3182	179	9	k	k	NOUN
ejpam-3182	179	10	(	(	PUNCT
ejpam-3182	179	11	log	log	VERB
ejpam-3182	179	12	p	p	NOUN
ejpam-3182	179	13	p−	p−	NOUN
ejpam-3182	179	14	1	1	NUM
ejpam-3182	179	15	)	)	PUNCT
ejpam-3182	179	16	k	k	PROPN
ejpam-3182	180	1	≈	≈	PROPN
ejpam-3182	180	2	0.163507781570971567408003	0.163507781570971567408003	NUM
ejpam-3182	180	3	...	...	PUNCT
ejpam-3182	180	4	.	.	PUNCT
ejpam-3182	181	1	(	(	PUNCT
ejpam-3182	181	2	21	21	NUM
ejpam-3182	181	3	)	)	PUNCT
ejpam-3182	181	4	n.	n.	NOUN
ejpam-3182	181	5	a.	a.	NOUN
ejpam-3182	181	6	carella	carella	PROPN
ejpam-3182	181	7	/	/	SYM
ejpam-3182	181	8	eur	eur	PROPN
ejpam-3182	181	9	.	.	PUNCT
ejpam-3182	182	1	j.	j.	PROPN
ejpam-3182	182	2	pure	pure	PROPN
ejpam-3182	182	3	appl	appl	PROPN
ejpam-3182	182	4	.	.	PROPN
ejpam-3182	182	5	math	math	PROPN
ejpam-3182	182	6	,	,	PUNCT
ejpam-3182	182	7	11	11	NUM
ejpam-3182	182	8	(	(	PUNCT
ejpam-3182	182	9	1	1	NUM
ejpam-3182	182	10	)	)	PUNCT
ejpam-3182	182	11	(	(	PUNCT
ejpam-3182	182	12	2018	2018	NUM
ejpam-3182	182	13	)	)	PUNCT
ejpam-3182	182	14	,	,	PUNCT
ejpam-3182	182	15	23	23	NUM
ejpam-3182	182	16	-	-	SYM
ejpam-3182	182	17	34	34	NUM
ejpam-3182	182	18	30	30	NUM
ejpam-3182	182	19	6	6	NUM
ejpam-3182	182	20	.	.	PUNCT
ejpam-3182	183	1	density	density	NOUN
ejpam-3182	183	2	correction	correction	NOUN
ejpam-3182	183	3	factor	factor	VERB
ejpam-3182	183	4	the	the	DET
ejpam-3182	183	5	sporadic	sporadic	ADJ
ejpam-3182	183	6	subsets	subset	NOUN
ejpam-3182	183	7	of	of	ADP
ejpam-3182	183	8	abel	abel	PROPN
ejpam-3182	183	9	-	-	PUNCT
ejpam-3182	183	10	wieferich	wieferich	NOUN
ejpam-3182	183	11	primes	prime	NOUN
ejpam-3182	183	12	,	,	PUNCT
ejpam-3182	183	13	see	see	VERB
ejpam-3182	183	14	[	[	X
ejpam-3182	183	15	31	31	NUM
ejpam-3182	183	16	,	,	PUNCT
ejpam-3182	183	17	p.	p.	NOUN
ejpam-3182	183	18	333	333	NUM
ejpam-3182	183	19	]	]	PUNCT
ejpam-3182	183	20	for	for	ADP
ejpam-3182	183	21	other	other	ADJ
ejpam-3182	183	22	details	detail	NOUN
ejpam-3182	183	23	,	,	PUNCT
ejpam-3182	183	24	have	have	VERB
ejpam-3182	183	25	roles	role	NOUN
ejpam-3182	183	26	in	in	ADP
ejpam-3182	183	27	the	the	DET
ejpam-3182	183	28	determination	determination	NOUN
ejpam-3182	183	29	of	of	ADP
ejpam-3182	183	30	the	the	DET
ejpam-3182	183	31	densities	density	NOUN
ejpam-3182	183	32	of	of	ADP
ejpam-3182	183	33	the	the	DET
ejpam-3182	183	34	subsets	subset	NOUN
ejpam-3182	183	35	of	of	ADP
ejpam-3182	183	36	integers	integer	NOUN
ejpam-3182	183	37	nu	nu	INTJ
ejpam-3182	183	38	=	=	SYM
ejpam-3182	183	39	{	{	PUNCT
ejpam-3182	183	40	n	n	NOUN
ejpam-3182	183	41	∈	∈	PROPN
ejpam-3182	183	42	n	n	CCONJ
ejpam-3182	183	43	:	:	PUNCT
ejpam-3182	183	44	ordn(u	ordn(u	NOUN
ejpam-3182	183	45	)	)	PUNCT
ejpam-3182	183	46	=	=	SYM
ejpam-3182	183	47	λ(n	λ(n	PROPN
ejpam-3182	183	48	)	)	PUNCT
ejpam-3182	183	49	}	}	PUNCT
ejpam-3182	183	50	(	(	PUNCT
ejpam-3182	183	51	22	22	NUM
ejpam-3182	183	52	)	)	PUNCT
ejpam-3182	183	53	with	with	ADP
ejpam-3182	183	54	fixed	fix	VERB
ejpam-3182	183	55	primitive	primitive	ADJ
ejpam-3182	183	56	roots	root	NOUN
ejpam-3182	183	57	u	u	NOUN
ejpam-3182	183	58	∈	∈	PROPN
ejpam-3182	183	59	z.	z.	X
ejpam-3182	183	60	the	the	DET
ejpam-3182	183	61	prime	prime	ADJ
ejpam-3182	183	62	product	product	NOUN
ejpam-3182	183	63	arising	arise	VERB
ejpam-3182	183	64	from	from	ADP
ejpam-3182	183	65	the	the	DET
ejpam-3182	183	66	sporadic	sporadic	ADJ
ejpam-3182	183	67	existence	existence	NOUN
ejpam-3182	183	68	of	of	ADP
ejpam-3182	183	69	the	the	DET
ejpam-3182	183	70	abel	abel	PROPN
ejpam-3182	183	71	-	-	PUNCT
ejpam-3182	183	72	wieferich	wieferich	NOUN
ejpam-3182	183	73	primes	prime	VERB
ejpam-3182	183	74	p	p	NOUN
ejpam-3182	183	75	≥	≥	NUM
ejpam-3182	183	76	3	3	NUM
ejpam-3182	183	77	is	be	AUX
ejpam-3182	183	78	reformulated	reformulate	VERB
ejpam-3182	183	79	in	in	ADP
ejpam-3182	183	80	the	the	DET
ejpam-3182	183	81	equivalent	equivalent	ADJ
ejpam-3182	183	82	expression	expression	NOUN
ejpam-3182	183	83	p	p	X
ejpam-3182	183	84	(	(	PUNCT
ejpam-3182	183	85	x	x	NOUN
ejpam-3182	183	86	)	)	PUNCT
ejpam-3182	183	87	=	=	SYM
ejpam-3182	183	88	∏	∏	NUM
ejpam-3182	183	89	pk≤x	pk≤x	PROPN
ejpam-3182	183	90	,	,	PUNCT
ejpam-3182	183	91	ordp(u)=p−1	ordp(u)=p−1	NOUN
ejpam-3182	183	92	,	,	PUNCT
ejpam-3182	183	93	ordp2	ordp2	NOUN
ejpam-3182	183	94	(	(	PUNCT
ejpam-3182	183	95	u)6	u)6	PROPN
ejpam-3182	183	96	=	=	SYM
ejpam-3182	183	97	p(p−1	p(p−1	X
ejpam-3182	183	98	)	)	PUNCT
ejpam-3182	183	99	(	(	PUNCT
ejpam-3182	183	100	1	1	NUM
ejpam-3182	183	101	+	+	SYM
ejpam-3182	183	102	1	1	NUM
ejpam-3182	183	103	p	p	NOUN
ejpam-3182	183	104	)	)	PUNCT
ejpam-3182	183	105	∏	∏	PROPN
ejpam-3182	183	106	pk≤x	pk≤x	NOUN
ejpam-3182	183	107	,	,	PUNCT
ejpam-3182	183	108	ordp2	ordp2	NOUN
ejpam-3182	183	109	(	(	PUNCT
ejpam-3182	183	110	u)=p(p−1	u)=p(p−1	ADJ
ejpam-3182	183	111	)	)	PUNCT
ejpam-3182	183	112	(	(	PUNCT
ejpam-3182	183	113	1	1	NUM
ejpam-3182	183	114	+	+	SYM
ejpam-3182	183	115	1	1	NUM
ejpam-3182	183	116	p	p	NOUN
ejpam-3182	183	117	+	+	NOUN
ejpam-3182	183	118	1	1	NUM
ejpam-3182	183	119	p2	p2	NOUN
ejpam-3182	183	120	+	+	X
ejpam-3182	183	121	·	·	PUNCT
ejpam-3182	183	122	·	·	PUNCT
ejpam-3182	183	123	·	·	PUNCT
ejpam-3182	183	124	)	)	PUNCT
ejpam-3182	184	1	=	=	SYM
ejpam-3182	184	2	∏	∏	PROPN
ejpam-3182	184	3	p≤x	p≤x	PROPN
ejpam-3182	184	4	,	,	PUNCT
ejpam-3182	184	5	p∈w	p∈w	NOUN
ejpam-3182	184	6	(	(	PUNCT
ejpam-3182	184	7	1−	1−	NUM
ejpam-3182	184	8	1	1	NUM
ejpam-3182	184	9	p2	p2	PROPN
ejpam-3182	184	10	)	)	PUNCT
ejpam-3182	184	11	∏	∏	PROPN
ejpam-3182	184	12	p≤x	p≤x	PROPN
ejpam-3182	184	13	,	,	PUNCT
ejpam-3182	184	14	p∈pu	p∈pu	NOUN
ejpam-3182	184	15	(	(	PUNCT
ejpam-3182	184	16	1−	1−	NUM
ejpam-3182	184	17	1	1	NUM
ejpam-3182	184	18	p	p	NOUN
ejpam-3182	184	19	)	)	PUNCT
ejpam-3182	184	20	−1	−1	NOUN
ejpam-3182	185	1	+	+	NOUN
ejpam-3182	185	2	o	o	X
ejpam-3182	185	3	(	(	PUNCT
ejpam-3182	185	4	1	1	NUM
ejpam-3182	185	5	x	x	NOUN
ejpam-3182	185	6	)	)	PUNCT
ejpam-3182	185	7	(	(	PUNCT
ejpam-3182	185	8	23	23	NUM
ejpam-3182	185	9	)	)	PUNCT
ejpam-3182	186	1	=	=	SYM
ejpam-3182	186	2	∏	∏	NUM
ejpam-3182	186	3	p∈w	p∈w	NOUN
ejpam-3182	186	4	(	(	PUNCT
ejpam-3182	186	5	1−	1−	NUM
ejpam-3182	186	6	1	1	NUM
ejpam-3182	186	7	p2	p2	PROPN
ejpam-3182	186	8	)	)	PUNCT
ejpam-3182	186	9	∏	∏	PROPN
ejpam-3182	186	10	p≤x	p≤x	PROPN
ejpam-3182	186	11	,	,	PUNCT
ejpam-3182	186	12	p∈pu	p∈pu	NOUN
ejpam-3182	186	13	(	(	PUNCT
ejpam-3182	186	14	1−	1−	NUM
ejpam-3182	186	15	1	1	NUM
ejpam-3182	186	16	p	p	NOUN
ejpam-3182	186	17	)	)	PUNCT
ejpam-3182	186	18	−1	−1	NOUN
ejpam-3182	187	1	+	+	NOUN
ejpam-3182	187	2	o	o	X
ejpam-3182	187	3	(	(	PUNCT
ejpam-3182	187	4	1	1	NUM
ejpam-3182	187	5	x	x	NOUN
ejpam-3182	187	6	)	)	PUNCT
ejpam-3182	187	7	.	.	PUNCT
ejpam-3182	188	1	note	note	VERB
ejpam-3182	188	2	that	that	SCONJ
ejpam-3182	188	3	the	the	DET
ejpam-3182	188	4	subset	subset	NOUN
ejpam-3182	188	5	of	of	ADP
ejpam-3182	188	6	primes	prime	NOUN
ejpam-3182	188	7	has	have	VERB
ejpam-3182	188	8	the	the	DET
ejpam-3182	188	9	disjoint	disjoint	ADJ
ejpam-3182	188	10	partition	partition	NOUN
ejpam-3182	188	11	pu	pu	PROPN
ejpam-3182	188	12	=	=	PUNCT
ejpam-3182	188	13	{	{	PUNCT
ejpam-3182	188	14	p	p	NOUN
ejpam-3182	188	15	∈	∈	PROPN
ejpam-3182	188	16	p	p	X
ejpam-3182	188	17	:	:	PUNCT
ejpam-3182	188	18	ordp(u	ordp(u	NOUN
ejpam-3182	188	19	)	)	PUNCT
ejpam-3182	188	20	=	=	PUNCT
ejpam-3182	189	1	p−	p−	NOUN
ejpam-3182	189	2	1	1	X
ejpam-3182	189	3	}	}	PUNCT
ejpam-3182	189	4	=	=	NOUN
ejpam-3182	189	5	w	w	NOUN
ejpam-3182	189	6	∪w	∪w	PROPN
ejpam-3182	189	7	,	,	PUNCT
ejpam-3182	189	8	(	(	PUNCT
ejpam-3182	189	9	24	24	NUM
ejpam-3182	189	10	)	)	PUNCT
ejpam-3182	189	11	where	where	SCONJ
ejpam-3182	189	12	w	w	NOUN
ejpam-3182	189	13	=	=	X
ejpam-3182	189	14	{	{	PUNCT
ejpam-3182	189	15	p	p	NOUN
ejpam-3182	189	16	∈	∈	PROPN
ejpam-3182	189	17	p	p	X
ejpam-3182	189	18	:	:	PUNCT
ejpam-3182	189	19	ordp(u	ordp(u	NOUN
ejpam-3182	189	20	)	)	PUNCT
ejpam-3182	189	21	|	|	ADV
ejpam-3182	189	22	p−	p−	NOUN
ejpam-3182	189	23	1	1	NUM
ejpam-3182	189	24	,	,	PUNCT
ejpam-3182	189	25	ordp2(u	ordp2(u	NOUN
ejpam-3182	189	26	)	)	PUNCT
ejpam-3182	189	27	6=	6=	NUM
ejpam-3182	189	28	p(p−	p(p−	VERB
ejpam-3182	189	29	1	1	NUM
ejpam-3182	189	30	)	)	PUNCT
ejpam-3182	189	31	}	}	PUNCT
ejpam-3182	189	32	(	(	PUNCT
ejpam-3182	189	33	25	25	NUM
ejpam-3182	189	34	)	)	PUNCT
ejpam-3182	189	35	and	and	CCONJ
ejpam-3182	189	36	w	w	X
ejpam-3182	189	37	=	=	X
ejpam-3182	189	38	{	{	PUNCT
ejpam-3182	189	39	p	p	NOUN
ejpam-3182	189	40	∈	∈	PROPN
ejpam-3182	189	41	p	p	NOUN
ejpam-3182	189	42	:	:	PUNCT
ejpam-3182	189	43	ordp2(u	ordp2(u	NUM
ejpam-3182	189	44	)	)	PUNCT
ejpam-3182	189	45	=	=	PUNCT
ejpam-3182	189	46	p(p−	p(p−	VERB
ejpam-3182	189	47	1	1	NUM
ejpam-3182	189	48	)	)	PUNCT
ejpam-3182	189	49	}	}	PUNCT
ejpam-3182	189	50	.	.	PUNCT
ejpam-3182	190	1	(	(	PUNCT
ejpam-3182	190	2	26	26	NUM
ejpam-3182	190	3	)	)	PUNCT
ejpam-3182	190	4	the	the	DET
ejpam-3182	190	5	convergent	convergent	ADJ
ejpam-3182	190	6	partial	partial	ADJ
ejpam-3182	190	7	product	product	NOUN
ejpam-3182	190	8	(	(	PUNCT
ejpam-3182	190	9	23	23	NUM
ejpam-3182	190	10	)	)	PUNCT
ejpam-3182	190	11	is	be	AUX
ejpam-3182	190	12	replaced	replace	VERB
ejpam-3182	190	13	with	with	ADP
ejpam-3182	190	14	the	the	DET
ejpam-3182	190	15	approximation∏	approximation∏	PROPN
ejpam-3182	190	16	p≤x	p≤x	PROPN
ejpam-3182	190	17	,	,	PUNCT
ejpam-3182	190	18	p∈w	p∈w	NOUN
ejpam-3182	190	19	(	(	PUNCT
ejpam-3182	190	20	1−	1−	NUM
ejpam-3182	190	21	1	1	NUM
ejpam-3182	190	22	p2	p2	PROPN
ejpam-3182	190	23	)	)	PUNCT
ejpam-3182	191	1	=	=	SYM
ejpam-3182	191	2	∏	∏	NUM
ejpam-3182	191	3	p∈w	p∈w	NOUN
ejpam-3182	191	4	(	(	PUNCT
ejpam-3182	191	5	1−	1−	NUM
ejpam-3182	191	6	1	1	NUM
ejpam-3182	191	7	p2	p2	PROPN
ejpam-3182	191	8	)	)	PUNCT
ejpam-3182	192	1	+	+	ADP
ejpam-3182	192	2	o	o	NOUN
ejpam-3182	192	3	(	(	PUNCT
ejpam-3182	192	4	1	1	NUM
ejpam-3182	192	5	x	x	NOUN
ejpam-3182	192	6	)	)	PUNCT
ejpam-3182	192	7	.	.	PUNCT
ejpam-3182	193	1	(	(	PUNCT
ejpam-3182	193	2	27	27	NUM
ejpam-3182	193	3	)	)	PUNCT
ejpam-3182	193	4	for	for	ADP
ejpam-3182	193	5	u	u	NOUN
ejpam-3182	193	6	=	=	SYM
ejpam-3182	193	7	2	2	NUM
ejpam-3182	193	8	,	,	PUNCT
ejpam-3182	193	9	the	the	DET
ejpam-3182	193	10	subset	subset	NOUN
ejpam-3182	193	11	of	of	ADP
ejpam-3182	193	12	primes	prime	NOUN
ejpam-3182	193	13	w	w	VERB
ejpam-3182	193	14	is	be	AUX
ejpam-3182	193	15	the	the	DET
ejpam-3182	193	16	subset	subset	NOUN
ejpam-3182	193	17	of	of	ADP
ejpam-3182	193	18	wieferich	wieferich	NOUN
ejpam-3182	193	19	primes	prime	NOUN
ejpam-3182	193	20	.	.	PUNCT
ejpam-3182	194	1	this	this	DET
ejpam-3182	194	2	subset	subset	NOUN
ejpam-3182	194	3	of	of	ADP
ejpam-3182	194	4	primes	prime	NOUN
ejpam-3182	194	5	is	be	AUX
ejpam-3182	194	6	usually	usually	ADV
ejpam-3182	194	7	characterized	characterize	VERB
ejpam-3182	194	8	in	in	ADP
ejpam-3182	194	9	terms	term	NOUN
ejpam-3182	194	10	of	of	ADP
ejpam-3182	194	11	the	the	DET
ejpam-3182	194	12	congruence	congruence	NOUN
ejpam-3182	194	13	{	{	PUNCT
ejpam-3182	194	14	p	p	NOUN
ejpam-3182	194	15	∈	∈	PROPN
ejpam-3182	194	16	p	p	NOUN
ejpam-3182	194	17	:	:	PUNCT
ejpam-3182	194	18	2p−1	2p−1	NUM
ejpam-3182	194	19	≡	≡	PROPN
ejpam-3182	194	20	1	1	NUM
ejpam-3182	194	21	mod	mod	NOUN
ejpam-3182	194	22	p2	p2	PROPN
ejpam-3182	194	23	}	}	PUNCT
ejpam-3182	194	24	=	=	PUNCT
ejpam-3182	194	25	{	{	PUNCT
ejpam-3182	194	26	1093	1093	NUM
ejpam-3182	194	27	,	,	PUNCT
ejpam-3182	194	28	3511	3511	NUM
ejpam-3182	194	29	,	,	PUNCT
ejpam-3182	194	30	....	....	PUNCT
ejpam-3182	194	31	}	}	PUNCT
ejpam-3182	194	32	.	.	PUNCT
ejpam-3182	195	1	(	(	PUNCT
ejpam-3182	195	2	28	28	NUM
ejpam-3182	195	3	)	)	PUNCT
ejpam-3182	195	4	given	give	VERB
ejpam-3182	195	5	a	a	DET
ejpam-3182	195	6	fixed	fix	VERB
ejpam-3182	195	7	u	u	NOUN
ejpam-3182	195	8	6=	6=	ADP
ejpam-3182	195	9	±1	±1	VERB
ejpam-3182	195	10	,	,	PUNCT
ejpam-3182	195	11	v2	v2	PROPN
ejpam-3182	195	12	,	,	PUNCT
ejpam-3182	195	13	the	the	DET
ejpam-3182	195	14	product	product	NOUN
ejpam-3182	195	15	∏	∏	NUM
ejpam-3182	195	16	p∈w	p∈w	NOUN
ejpam-3182	195	17	(	(	PUNCT
ejpam-3182	195	18	1−	1−	NUM
ejpam-3182	195	19	p−2	p−2	NOUN
ejpam-3182	195	20	)	)	PUNCT
ejpam-3182	195	21	reduces	reduce	VERB
ejpam-3182	195	22	the	the	DET
ejpam-3182	195	23	density	density	NOUN
ejpam-3182	195	24	to	to	PART
ejpam-3182	195	25	compensate	compensate	VERB
ejpam-3182	195	26	for	for	ADP
ejpam-3182	195	27	those	those	DET
ejpam-3182	195	28	primes	prime	NOUN
ejpam-3182	195	29	for	for	ADP
ejpam-3182	195	30	which	which	PRON
ejpam-3182	195	31	the	the	DET
ejpam-3182	195	32	primitive	primitive	ADJ
ejpam-3182	195	33	root	root	NOUN
ejpam-3182	195	34	u	u	NOUN
ejpam-3182	195	35	mod	mod	PROPN
ejpam-3182	195	36	p	p	PROPN
ejpam-3182	195	37	can	can	AUX
ejpam-3182	195	38	not	not	PART
ejpam-3182	195	39	be	be	AUX
ejpam-3182	195	40	extended	extend	VERB
ejpam-3182	195	41	to	to	ADP
ejpam-3182	195	42	a	a	DET
ejpam-3182	195	43	primitive	primitive	ADJ
ejpam-3182	195	44	root	root	NOUN
ejpam-3182	195	45	u	u	NOUN
ejpam-3182	195	46	mod	mod	PROPN
ejpam-3182	195	47	p2	p2	PROPN
ejpam-3182	195	48	.	.	PUNCT
ejpam-3182	196	1	this	this	PRON
ejpam-3182	196	2	seems	seem	VERB
ejpam-3182	196	3	to	to	PART
ejpam-3182	196	4	be	be	AUX
ejpam-3182	196	5	a	a	DET
ejpam-3182	196	6	density	density	NOUN
ejpam-3182	196	7	correction	correction	NOUN
ejpam-3182	196	8	factor	factor	NOUN
ejpam-3182	196	9	similar	similar	ADJ
ejpam-3182	196	10	to	to	ADP
ejpam-3182	196	11	the	the	DET
ejpam-3182	196	12	case	case	NOUN
ejpam-3182	196	13	for	for	ADP
ejpam-3182	196	14	primitive	primitive	ADJ
ejpam-3182	196	15	roots	root	NOUN
ejpam-3182	196	16	over	over	ADP
ejpam-3182	196	17	the	the	DET
ejpam-3182	196	18	prime	prime	ADJ
ejpam-3182	196	19	numbers	number	NOUN
ejpam-3182	196	20	.	.	PUNCT
ejpam-3182	197	1	the	the	DET
ejpam-3182	197	2	correction	correction	NOUN
ejpam-3182	197	3	required	require	VERB
ejpam-3182	197	4	for	for	ADP
ejpam-3182	197	5	certain	certain	ADJ
ejpam-3182	197	6	densities	density	NOUN
ejpam-3182	197	7	of	of	ADP
ejpam-3182	197	8	primes	prime	NOUN
ejpam-3182	197	9	with	with	ADP
ejpam-3182	197	10	respect	respect	NOUN
ejpam-3182	197	11	to	to	ADP
ejpam-3182	197	12	fixed	fix	VERB
ejpam-3182	197	13	primitive	primitive	ADJ
ejpam-3182	197	14	roots	root	NOUN
ejpam-3182	197	15	over	over	ADP
ejpam-3182	197	16	the	the	DET
ejpam-3182	197	17	primes	prime	NOUN
ejpam-3182	197	18	was	be	AUX
ejpam-3182	197	19	discovered	discover	VERB
ejpam-3182	197	20	by	by	ADP
ejpam-3182	197	21	the	the	DET
ejpam-3182	197	22	lehmers	lehmer	NOUN
ejpam-3182	197	23	,	,	PUNCT
ejpam-3182	197	24	see	see	VERB
ejpam-3182	197	25	[	[	X
ejpam-3182	197	26	32	32	NUM
ejpam-3182	197	27	]	]	PUNCT
ejpam-3182	197	28	.	.	PUNCT
ejpam-3182	198	1	n.	n.	PROPN
ejpam-3182	198	2	a.	a.	PROPN
ejpam-3182	198	3	carella	carella	PROPN
ejpam-3182	198	4	/	/	SYM
ejpam-3182	198	5	eur	eur	PROPN
ejpam-3182	198	6	.	.	PUNCT
ejpam-3182	199	1	j.	j.	PROPN
ejpam-3182	199	2	pure	pure	PROPN
ejpam-3182	199	3	appl	appl	PROPN
ejpam-3182	199	4	.	.	PROPN
ejpam-3182	199	5	math	math	PROPN
ejpam-3182	199	6	,	,	PUNCT
ejpam-3182	199	7	11	11	NUM
ejpam-3182	199	8	(	(	PUNCT
ejpam-3182	199	9	1	1	NUM
ejpam-3182	199	10	)	)	PUNCT
ejpam-3182	199	11	(	(	PUNCT
ejpam-3182	199	12	2018	2018	NUM
ejpam-3182	199	13	)	)	PUNCT
ejpam-3182	199	14	,	,	PUNCT
ejpam-3182	199	15	23	23	NUM
ejpam-3182	199	16	-	-	SYM
ejpam-3182	199	17	34	34	NUM
ejpam-3182	199	18	31	31	NUM
ejpam-3182	199	19	7	7	NUM
ejpam-3182	199	20	.	.	PUNCT
ejpam-3182	200	1	the	the	DET
ejpam-3182	200	2	proof	proof	NOUN
ejpam-3182	200	3	of	of	ADP
ejpam-3182	200	4	the	the	DET
ejpam-3182	200	5	theorem	theorem	NOUN
ejpam-3182	200	6	the	the	DET
ejpam-3182	200	7	result	result	NOUN
ejpam-3182	200	8	below	below	ADV
ejpam-3182	200	9	has	have	AUX
ejpam-3182	200	10	served	serve	VERB
ejpam-3182	200	11	as	as	ADP
ejpam-3182	200	12	the	the	DET
ejpam-3182	200	13	foundation	foundation	NOUN
ejpam-3182	200	14	for	for	ADP
ejpam-3182	200	15	various	various	ADJ
ejpam-3182	200	16	other	other	ADJ
ejpam-3182	200	17	results	result	NOUN
ejpam-3182	200	18	about	about	ADP
ejpam-3182	200	19	primitive	primitive	ADJ
ejpam-3182	200	20	roots	root	NOUN
ejpam-3182	200	21	.	.	PUNCT
ejpam-3182	201	1	most	most	ADV
ejpam-3182	201	2	recently	recently	ADV
ejpam-3182	201	3	,	,	PUNCT
ejpam-3182	201	4	it	it	PRON
ejpam-3182	201	5	was	be	AUX
ejpam-3182	201	6	used	use	VERB
ejpam-3182	201	7	to	to	PART
ejpam-3182	201	8	prove	prove	VERB
ejpam-3182	201	9	the	the	DET
ejpam-3182	201	10	existence	existence	NOUN
ejpam-3182	201	11	of	of	ADP
ejpam-3182	201	12	infinite	infinite	ADJ
ejpam-3182	201	13	sequences	sequence	NOUN
ejpam-3182	201	14	of	of	ADP
ejpam-3182	201	15	primes	prime	NOUN
ejpam-3182	201	16	with	with	ADP
ejpam-3182	201	17	fixed	fix	VERB
ejpam-3182	201	18	prime	prime	ADJ
ejpam-3182	201	19	roots	root	NOUN
ejpam-3182	201	20	,	,	PUNCT
ejpam-3182	201	21	and	and	CCONJ
ejpam-3182	201	22	bounded	bound	VERB
ejpam-3182	201	23	gaps	gap	NOUN
ejpam-3182	201	24	,	,	PUNCT
ejpam-3182	201	25	confer	confer	NOUN
ejpam-3182	201	26	[	[	X
ejpam-3182	201	27	3	3	NUM
ejpam-3182	201	28	]	]	PUNCT
ejpam-3182	201	29	.	.	PUNCT
ejpam-3182	202	1	theorem	theorem	VERB
ejpam-3182	202	2	7.1	7.1	NUM
ejpam-3182	202	3	.	.	PUNCT
ejpam-3182	203	1	(	(	PUNCT
ejpam-3182	203	2	[	[	X
ejpam-3182	203	3	13	13	NUM
ejpam-3182	203	4	]	]	SYM
ejpam-3182	203	5	)	)	PUNCT
ejpam-3182	203	6	if	if	SCONJ
ejpam-3182	203	7	it	it	PRON
ejpam-3182	203	8	be	be	AUX
ejpam-3182	203	9	assumed	assume	VERB
ejpam-3182	203	10	that	that	SCONJ
ejpam-3182	203	11	the	the	DET
ejpam-3182	203	12	extended	extended	ADJ
ejpam-3182	203	13	riemann	riemann	PROPN
ejpam-3182	203	14	hypothesis	hypothesis	NOUN
ejpam-3182	203	15	hold	hold	NOUN
ejpam-3182	203	16	for	for	SCONJ
ejpam-3182	203	17	the	the	DET
ejpam-3182	203	18	dedekind	dedekind	PROPN
ejpam-3182	203	19	zeta	zeta	NOUN
ejpam-3182	203	20	function	function	VERB
ejpam-3182	203	21	over	over	ADP
ejpam-3182	203	22	galois	galois	PROPN
ejpam-3182	203	23	fields	field	NOUN
ejpam-3182	203	24	of	of	ADP
ejpam-3182	203	25	the	the	DET
ejpam-3182	203	26	type	type	NOUN
ejpam-3182	203	27	q	q	NOUN
ejpam-3182	204	1	(	(	PUNCT
ejpam-3182	204	2	d	d	NOUN
ejpam-3182	204	3	√	√	NUM
ejpam-3182	204	4	u	u	NOUN
ejpam-3182	204	5	,	,	PUNCT
ejpam-3182	204	6	n	n	PROPN
ejpam-3182	204	7	√	√	PROPN
ejpam-3182	204	8	1	1	NUM
ejpam-3182	204	9	)	)	PUNCT
ejpam-3182	204	10	,	,	PUNCT
ejpam-3182	204	11	where	where	SCONJ
ejpam-3182	204	12	n	n	PRON
ejpam-3182	204	13	is	be	AUX
ejpam-3182	204	14	a	a	DET
ejpam-3182	204	15	squarefree	squarefree	NOUN
ejpam-3182	204	16	integer	integer	NOUN
ejpam-3182	204	17	,	,	PUNCT
ejpam-3182	204	18	and	and	CCONJ
ejpam-3182	204	19	d	d	NOUN
ejpam-3182	204	20	|	|	ADV
ejpam-3182	204	21	n.	n.	NOUN
ejpam-3182	204	22	for	for	ADP
ejpam-3182	204	23	a	a	DET
ejpam-3182	204	24	given	give	VERB
ejpam-3182	204	25	nonzero	nonzero	PROPN
ejpam-3182	204	26	integer	integer	PROPN
ejpam-3182	204	27	u	u	PROPN
ejpam-3182	204	28	6=	6=	ADP
ejpam-3182	204	29	±1	±1	PROPN
ejpam-3182	204	30	,	,	PUNCT
ejpam-3182	204	31	v2	v2	PROPN
ejpam-3182	204	32	,	,	PUNCT
ejpam-3182	204	33	let	let	VERB
ejpam-3182	204	34	nu(x	nu(x	NOUN
ejpam-3182	204	35	)	)	PUNCT
ejpam-3182	204	36	be	be	VERB
ejpam-3182	204	37	the	the	DET
ejpam-3182	204	38	number	number	NOUN
ejpam-3182	204	39	of	of	ADP
ejpam-3182	204	40	primes	prime	NOUN
ejpam-3182	204	41	p	p	NOUN
ejpam-3182	204	42	≤	≤	NOUN
ejpam-3182	204	43	x	x	PUNCT
ejpam-3182	204	44	for	for	ADP
ejpam-3182	204	45	which	which	PRON
ejpam-3182	204	46	u	u	NOUN
ejpam-3182	204	47	is	be	AUX
ejpam-3182	204	48	a	a	DET
ejpam-3182	204	49	primitive	primitive	ADJ
ejpam-3182	204	50	root	root	NOUN
ejpam-3182	204	51	modulo	modulo	NOUN
ejpam-3182	204	52	p.	p.	NOUN
ejpam-3182	204	53	let	let	VERB
ejpam-3182	204	54	u	u	NOUN
ejpam-3182	204	55	=	=	PROPN
ejpam-3182	204	56	um1	um1	X
ejpam-3182	204	57	·	·	PUNCT
ejpam-3182	204	58	u22	u22	PROPN
ejpam-3182	204	59	,	,	PUNCT
ejpam-3182	204	60	where	where	SCONJ
ejpam-3182	204	61	u1	u1	NOUN
ejpam-3182	204	62	>	>	X
ejpam-3182	204	63	1	1	NUM
ejpam-3182	204	64	is	be	AUX
ejpam-3182	204	65	squarefree	squarefree	NOUN
ejpam-3182	204	66	,	,	PUNCT
ejpam-3182	204	67	and	and	CCONJ
ejpam-3182	204	68	m	m	PRON
ejpam-3182	204	69	≥	≥	NUM
ejpam-3182	204	70	1	1	NUM
ejpam-3182	204	71	is	be	AUX
ejpam-3182	204	72	odd	odd	ADJ
ejpam-3182	204	73	.	.	PUNCT
ejpam-3182	205	1	then	then	ADV
ejpam-3182	205	2	,	,	PUNCT
ejpam-3182	205	3	there	there	PRON
ejpam-3182	205	4	is	be	VERB
ejpam-3182	205	5	a	a	DET
ejpam-3182	205	6	constant	constant	ADJ
ejpam-3182	205	7	αu	αu	NOUN
ejpam-3182	205	8	≥	≥	NOUN
ejpam-3182	205	9	0	0	NUM
ejpam-3182	205	10	such	such	ADJ
ejpam-3182	205	11	that	that	SCONJ
ejpam-3182	205	12	nu(x	nu(x	NOUN
ejpam-3182	205	13	)	)	PUNCT
ejpam-3182	205	14	=	=	SYM
ejpam-3182	205	15	αu	αu	NOUN
ejpam-3182	205	16	x	x	SYM
ejpam-3182	205	17	log	log	VERB
ejpam-3182	205	18	x	x	PUNCT
ejpam-3182	206	1	+	+	NOUN
ejpam-3182	206	2	o	o	X
ejpam-3182	206	3	(	(	PUNCT
ejpam-3182	206	4	x	x	SYM
ejpam-3182	206	5	log	log	VERB
ejpam-3182	206	6	log	log	NOUN
ejpam-3182	206	7	x	x	PUNCT
ejpam-3182	206	8	log2	log2	PROPN
ejpam-3182	206	9	x	x	X
ejpam-3182	206	10	)	)	PUNCT
ejpam-3182	206	11	as	as	ADP
ejpam-3182	206	12	x	x	X
ejpam-3182	206	13	−→∞.	−→∞.	X
ejpam-3182	206	14	(	(	PUNCT
ejpam-3182	206	15	29	29	NUM
ejpam-3182	206	16	)	)	PUNCT
ejpam-3182	206	17	proof	proof	NOUN
ejpam-3182	206	18	.	.	PUNCT
ejpam-3182	207	1	(	(	PUNCT
ejpam-3182	207	2	theorem	theorem	VERB
ejpam-3182	207	3	1.1	1.1	NUM
ejpam-3182	207	4	)	)	PUNCT
ejpam-3182	207	5	by	by	ADP
ejpam-3182	207	6	the	the	DET
ejpam-3182	207	7	generalized	generalize	VERB
ejpam-3182	207	8	riemann	riemann	PROPN
ejpam-3182	207	9	hypothesis	hypothesis	NOUN
ejpam-3182	207	10	or	or	CCONJ
ejpam-3182	207	11	theorem	theorem	VERB
ejpam-3182	207	12	7.1	7.1	NUM
ejpam-3182	207	13	,	,	PUNCT
ejpam-3182	207	14	the	the	DET
ejpam-3182	207	15	density	density	NOUN
ejpam-3182	207	16	α2	α2	PROPN
ejpam-3182	207	17	=	=	SYM
ejpam-3182	207	18	δ	δ	PROPN
ejpam-3182	207	19	(	(	PUNCT
ejpam-3182	207	20	p2	p2	PROPN
ejpam-3182	207	21	)	)	PUNCT
ejpam-3182	207	22	>	>	X
ejpam-3182	207	23	0	0	NUM
ejpam-3182	207	24	of	of	ADP
ejpam-3182	207	25	the	the	DET
ejpam-3182	207	26	subset	subset	NOUN
ejpam-3182	207	27	of	of	ADP
ejpam-3182	207	28	primes	prime	NOUN
ejpam-3182	207	29	p2	p2	PROPN
ejpam-3182	207	30	is	be	AUX
ejpam-3182	207	31	nonzero	nonzero	NOUN
ejpam-3182	207	32	.	.	PUNCT
ejpam-3182	208	1	put	put	VERB
ejpam-3182	208	2	τ	τ	X
ejpam-3182	208	3	=	=	SYM
ejpam-3182	208	4	α2	α2	ADJ
ejpam-3182	208	5	in	in	ADP
ejpam-3182	208	6	wirsing	wirsing	NOUN
ejpam-3182	208	7	formula	formula	NOUN
ejpam-3182	208	8	,	,	PUNCT
ejpam-3182	208	9	lemma	lemma	PROPN
ejpam-3182	208	10	4.1	4.1	NUM
ejpam-3182	208	11	,	,	PUNCT
ejpam-3182	208	12	and	and	CCONJ
ejpam-3182	208	13	replace	replace	VERB
ejpam-3182	208	14	the	the	DET
ejpam-3182	208	15	characteristic	characteristic	ADJ
ejpam-3182	208	16	function	function	NOUN
ejpam-3182	208	17	f(n	f(n	PROPN
ejpam-3182	208	18	)	)	PUNCT
ejpam-3182	208	19	of	of	ADP
ejpam-3182	208	20	primitive	primitive	ADJ
ejpam-3182	208	21	roots	root	NOUN
ejpam-3182	208	22	in	in	ADP
ejpam-3182	208	23	the	the	DET
ejpam-3182	208	24	finite	finite	NOUN
ejpam-3182	208	25	ring	ring	NOUN
ejpam-3182	208	26	z	z	PROPN
ejpam-3182	208	27	/	/	SYM
ejpam-3182	208	28	pk	pk	PROPN
ejpam-3182	208	29	z	z	PROPN
ejpam-3182	208	30	,	,	PUNCT
ejpam-3182	208	31	k	k	PROPN
ejpam-3182	208	32	≥	≥	NUM
ejpam-3182	208	33	1	1	NUM
ejpam-3182	208	34	,	,	PUNCT
ejpam-3182	208	35	see	see	VERB
ejpam-3182	208	36	lemma	lemma	PROPN
ejpam-3182	208	37	4.1	4.1	NUM
ejpam-3182	208	38	,	,	PUNCT
ejpam-3182	208	39	to	to	PART
ejpam-3182	208	40	produce	produce	VERB
ejpam-3182	208	41	∑	∑	PUNCT
ejpam-3182	208	42	n≤x	n≤x	PROPN
ejpam-3182	208	43	f(n	f(n	PROPN
ejpam-3182	208	44	)	)	PUNCT
ejpam-3182	209	1	=	=	PRON
ejpam-3182	209	2	(	(	PUNCT
ejpam-3182	209	3	1	1	NUM
ejpam-3182	209	4	eγτγ(τ	eγτγ(τ	NOUN
ejpam-3182	209	5	)	)	PUNCT
ejpam-3182	209	6	+	+	NUM
ejpam-3182	209	7	o(1	o(1	NOUN
ejpam-3182	209	8	)	)	PUNCT
ejpam-3182	209	9	)	)	PUNCT
ejpam-3182	210	1	x	x	SYM
ejpam-3182	210	2	log	log	VERB
ejpam-3182	210	3	x	x	SYM
ejpam-3182	210	4	∏	∏	PROPN
ejpam-3182	210	5	pk≤x	pk≤x	PROPN
ejpam-3182	210	6	(	(	PUNCT
ejpam-3182	210	7	1	1	NUM
ejpam-3182	210	8	+	+	NUM
ejpam-3182	210	9	f(p	f(p	NOUN
ejpam-3182	210	10	)	)	PUNCT
ejpam-3182	210	11	p	p	NOUN
ejpam-3182	211	1	+	+	NUM
ejpam-3182	211	2	f	f	X
ejpam-3182	211	3	(	(	PUNCT
ejpam-3182	211	4	p2	p2	PROPN
ejpam-3182	211	5	)	)	PUNCT
ejpam-3182	211	6	p2	p2	PROPN
ejpam-3182	211	7	+	+	X
ejpam-3182	211	8	·	·	PUNCT
ejpam-3182	211	9	·	·	PUNCT
ejpam-3182	211	10	·	·	PUNCT
ejpam-3182	211	11	)	)	PUNCT
ejpam-3182	212	1	=	=	PUNCT
ejpam-3182	212	2	(	(	PUNCT
ejpam-3182	212	3	1	1	NUM
ejpam-3182	212	4	eγα2γ	eγα2γ	NOUN
ejpam-3182	212	5	(	(	PUNCT
ejpam-3182	212	6	α2	α2	ADV
ejpam-3182	212	7	)	)	PUNCT
ejpam-3182	212	8	+	+	NUM
ejpam-3182	212	9	o(1	o(1	NOUN
ejpam-3182	212	10	)	)	PUNCT
ejpam-3182	212	11	)	)	PUNCT
ejpam-3182	213	1	x	x	SYM
ejpam-3182	213	2	log	log	VERB
ejpam-3182	213	3	x	x	SYM
ejpam-3182	213	4	(	(	PUNCT
ejpam-3182	213	5	30	30	NUM
ejpam-3182	213	6	)	)	PUNCT
ejpam-3182	213	7	×	×	NOUN
ejpam-3182	213	8	∏	∏	PROPN
ejpam-3182	213	9	pk≤x	pk≤x	NOUN
ejpam-3182	213	10	,	,	PUNCT
ejpam-3182	213	11	ord(2)=p−1	ord(2)=p−1	NUM
ejpam-3182	213	12	,	,	PUNCT
ejpam-3182	213	13	ord(2)6	ord(2)6	NOUN
ejpam-3182	213	14	=	=	SYM
ejpam-3182	213	15	p(p−1	p(p−1	X
ejpam-3182	213	16	)	)	PUNCT
ejpam-3182	213	17	(	(	PUNCT
ejpam-3182	213	18	1	1	NUM
ejpam-3182	213	19	+	+	SYM
ejpam-3182	213	20	1	1	NUM
ejpam-3182	213	21	p	p	NOUN
ejpam-3182	213	22	)	)	PUNCT
ejpam-3182	213	23	∏	∏	PROPN
ejpam-3182	213	24	pk≤x	pk≤x	NOUN
ejpam-3182	213	25	,	,	PUNCT
ejpam-3182	213	26	ord(2)=p(p−1	ord(2)=p(p−1	NOUN
ejpam-3182	213	27	)	)	PUNCT
ejpam-3182	213	28	(	(	PUNCT
ejpam-3182	213	29	1	1	NUM
ejpam-3182	213	30	+	+	SYM
ejpam-3182	213	31	1	1	NUM
ejpam-3182	213	32	p	p	NOUN
ejpam-3182	213	33	+	+	NOUN
ejpam-3182	213	34	1	1	NUM
ejpam-3182	213	35	p2	p2	NOUN
ejpam-3182	213	36	+	+	X
ejpam-3182	213	37	·	·	PUNCT
ejpam-3182	213	38	·	·	PUNCT
ejpam-3182	213	39	·	·	PUNCT
ejpam-3182	213	40	)	)	PUNCT
ejpam-3182	213	41	.	.	PUNCT
ejpam-3182	214	1	in	in	ADP
ejpam-3182	214	2	equation	equation	NOUN
ejpam-3182	214	3	(	(	PUNCT
ejpam-3182	214	4	30	30	NUM
ejpam-3182	214	5	)	)	PUNCT
ejpam-3182	214	6	,	,	PUNCT
ejpam-3182	214	7	line	line	NOUN
ejpam-3182	214	8	1	1	NUM
ejpam-3182	214	9	,	,	PUNCT
ejpam-3182	214	10	the	the	DET
ejpam-3182	214	11	product	product	NOUN
ejpam-3182	214	12	over	over	ADP
ejpam-3182	214	13	the	the	DET
ejpam-3182	214	14	prime	prime	ADJ
ejpam-3182	214	15	powers	power	NOUN
ejpam-3182	214	16	pk	pk	NOUN
ejpam-3182	214	17	≤	≤	NUM
ejpam-3182	214	18	x	x	PUNCT
ejpam-3182	214	19	is	be	AUX
ejpam-3182	214	20	broken	break	VERB
ejpam-3182	214	21	up	up	ADP
ejpam-3182	214	22	into	into	ADP
ejpam-3182	214	23	two	two	NUM
ejpam-3182	214	24	subproducts	subproduct	NOUN
ejpam-3182	214	25	.	.	PUNCT
ejpam-3182	215	1	in	in	ADP
ejpam-3182	215	2	equation	equation	NOUN
ejpam-3182	215	3	(	(	PUNCT
ejpam-3182	215	4	30	30	NUM
ejpam-3182	215	5	)	)	PUNCT
ejpam-3182	215	6	,	,	PUNCT
ejpam-3182	215	7	line	line	NOUN
ejpam-3182	215	8	2	2	NUM
ejpam-3182	215	9	,	,	PUNCT
ejpam-3182	215	10	the	the	DET
ejpam-3182	215	11	first	first	ADJ
ejpam-3182	215	12	subproduct	subproduct	NOUN
ejpam-3182	215	13	is	be	AUX
ejpam-3182	215	14	restricted	restrict	VERB
ejpam-3182	215	15	to	to	ADP
ejpam-3182	215	16	the	the	DET
ejpam-3182	215	17	subset	subset	NOUN
ejpam-3182	215	18	of	of	ADP
ejpam-3182	215	19	wieferich	wieferich	PROPN
ejpam-3182	215	20	prime	prime	ADJ
ejpam-3182	215	21	powers	power	NOUN
ejpam-3182	215	22	which	which	PRON
ejpam-3182	215	23	do	do	AUX
ejpam-3182	215	24	not	not	PART
ejpam-3182	215	25	satisfy	satisfy	VERB
ejpam-3182	215	26	the	the	DET
ejpam-3182	215	27	completely	completely	ADV
ejpam-3182	215	28	multiplicative	multiplicative	ADJ
ejpam-3182	215	29	property	property	NOUN
ejpam-3182	215	30	f(p2	f(p2	NOUN
ejpam-3182	215	31	)	)	PUNCT
ejpam-3182	215	32	6=	6=	ADP
ejpam-3182	215	33	f(p)f(p	f(p)f(p	NOUN
ejpam-3182	215	34	)	)	PUNCT
ejpam-3182	215	35	of	of	ADP
ejpam-3182	215	36	the	the	DET
ejpam-3182	215	37	characteristic	characteristic	ADJ
ejpam-3182	215	38	function	function	NOUN
ejpam-3182	215	39	;	;	PUNCT
ejpam-3182	215	40	the	the	DET
ejpam-3182	215	41	second	second	ADJ
ejpam-3182	215	42	subproduct	subproduct	NOUN
ejpam-3182	215	43	is	be	AUX
ejpam-3182	215	44	restricted	restrict	VERB
ejpam-3182	215	45	to	to	ADP
ejpam-3182	215	46	the	the	DET
ejpam-3182	215	47	subset	subset	NOUN
ejpam-3182	215	48	of	of	ADP
ejpam-3182	215	49	nonwieferich	nonwieferich	ADV
ejpam-3182	215	50	prime	prime	ADJ
ejpam-3182	215	51	powers	power	NOUN
ejpam-3182	215	52	which	which	PRON
ejpam-3182	215	53	do	do	AUX
ejpam-3182	215	54	satisfy	satisfy	VERB
ejpam-3182	215	55	the	the	DET
ejpam-3182	215	56	completely	completely	ADV
ejpam-3182	215	57	multiplicative	multiplicative	ADJ
ejpam-3182	215	58	property	property	NOUN
ejpam-3182	215	59	f(p2	f(p2	NOUN
ejpam-3182	215	60	)	)	PUNCT
ejpam-3182	215	61	=	=	SYM
ejpam-3182	215	62	f(p)f(p	f(p)f(p	NOUN
ejpam-3182	215	63	)	)	PUNCT
ejpam-3182	215	64	of	of	ADP
ejpam-3182	215	65	the	the	DET
ejpam-3182	215	66	characteristic	characteristic	ADJ
ejpam-3182	215	67	function	function	NOUN
ejpam-3182	215	68	,	,	PUNCT
ejpam-3182	215	69	lemma	lemma	PROPN
ejpam-3182	215	70	3.1	3.1	NUM
ejpam-3182	215	71	.	.	PUNCT
ejpam-3182	216	1	replacing	replace	VERB
ejpam-3182	216	2	the	the	DET
ejpam-3182	216	3	equivalent	equivalent	ADJ
ejpam-3182	216	4	product	product	NOUN
ejpam-3182	216	5	,	,	PUNCT
ejpam-3182	216	6	see	see	VERB
ejpam-3182	216	7	(	(	PUNCT
ejpam-3182	216	8	23	23	NUM
ejpam-3182	216	9	)	)	PUNCT
ejpam-3182	216	10	in	in	ADP
ejpam-3182	216	11	section	section	NOUN
ejpam-3182	216	12	6	6	NUM
ejpam-3182	216	13	,	,	PUNCT
ejpam-3182	216	14	and	and	CCONJ
ejpam-3182	216	15	using	use	VERB
ejpam-3182	216	16	lemma	lemma	PROPN
ejpam-3182	216	17	5.2	5.2	NUM
ejpam-3182	216	18	,	,	PUNCT
ejpam-3182	216	19	yield	yield	VERB
ejpam-3182	216	20	∑	∑	PROPN
ejpam-3182	216	21	n≤x	n≤x	PROPN
ejpam-3182	216	22	f(n	f(n	PROPN
ejpam-3182	216	23	)	)	PUNCT
ejpam-3182	217	1	=	=	PRON
ejpam-3182	217	2	(	(	PUNCT
ejpam-3182	217	3	1	1	NUM
ejpam-3182	217	4	eγα2γ	eγα2γ	NOUN
ejpam-3182	217	5	(	(	PUNCT
ejpam-3182	217	6	α2	α2	ADV
ejpam-3182	217	7	)	)	PUNCT
ejpam-3182	217	8	+	+	NUM
ejpam-3182	217	9	o(1	o(1	NOUN
ejpam-3182	217	10	)	)	PUNCT
ejpam-3182	217	11	)	)	PUNCT
ejpam-3182	218	1	x	x	SYM
ejpam-3182	218	2	log	log	VERB
ejpam-3182	218	3	x	x	SYM
ejpam-3182	218	4	∏	∏	NUM
ejpam-3182	218	5	p∈w	p∈w	NOUN
ejpam-3182	218	6	(	(	PUNCT
ejpam-3182	218	7	1−	1−	NUM
ejpam-3182	218	8	1	1	NUM
ejpam-3182	218	9	p2	p2	PROPN
ejpam-3182	218	10	)	)	PUNCT
ejpam-3182	218	11	∏	∏	PROPN
ejpam-3182	218	12	pk≤x	pk≤x	NOUN
ejpam-3182	218	13	,	,	PUNCT
ejpam-3182	218	14	p∈pu	p∈pu	NOUN
ejpam-3182	218	15	(	(	PUNCT
ejpam-3182	218	16	1−	1−	NUM
ejpam-3182	218	17	1	1	NUM
ejpam-3182	218	18	p	p	NOUN
ejpam-3182	218	19	)	)	PUNCT
ejpam-3182	218	20	−1	−1	NOUN
ejpam-3182	218	21	references	reference	NOUN
ejpam-3182	218	22	32	32	NUM
ejpam-3182	218	23	=	=	SYM
ejpam-3182	218	24	(	(	PUNCT
ejpam-3182	218	25	eγ2−γα2	eγ2−γα2	ADV
ejpam-3182	218	26	γ	γ	X
ejpam-3182	218	27	(	(	PUNCT
ejpam-3182	218	28	α2	α2	PROPN
ejpam-3182	218	29	)	)	PUNCT
ejpam-3182	218	30	+	+	NUM
ejpam-3182	218	31	o(1	o(1	NOUN
ejpam-3182	218	32	)	)	PUNCT
ejpam-3182	218	33	)	)	PUNCT
ejpam-3182	219	1	x	x	X
ejpam-3182	219	2	(	(	PUNCT
ejpam-3182	219	3	log	log	VERB
ejpam-3182	219	4	x)1−α2	x)1−α2	PROPN
ejpam-3182	219	5	∏	∏	NUM
ejpam-3182	219	6	p∈w	p∈w	NOUN
ejpam-3182	219	7	(	(	PUNCT
ejpam-3182	219	8	1−	1−	NUM
ejpam-3182	219	9	1	1	NUM
ejpam-3182	219	10	p2	p2	PROPN
ejpam-3182	219	11	)	)	PUNCT
ejpam-3182	219	12	,	,	PUNCT
ejpam-3182	219	13	(	(	PUNCT
ejpam-3182	219	14	31	31	NUM
ejpam-3182	219	15	)	)	PUNCT
ejpam-3182	219	16	where	where	SCONJ
ejpam-3182	219	17	γ2	γ2	PROPN
ejpam-3182	219	18	is	be	AUX
ejpam-3182	219	19	the	the	DET
ejpam-3182	219	20	artin	artin	PROPN
ejpam-3182	219	21	-	-	PUNCT
ejpam-3182	219	22	euler	euler	NOUN
ejpam-3182	219	23	constant	constant	NOUN
ejpam-3182	219	24	,	,	PUNCT
ejpam-3182	219	25	see	see	VERB
ejpam-3182	219	26	lemmas	lemmas	PROPN
ejpam-3182	219	27	5.1	5.1	NUM
ejpam-3182	219	28	and	and	CCONJ
ejpam-3182	219	29	5.2	5.2	NUM
ejpam-3182	219	30	for	for	ADP
ejpam-3182	219	31	details	detail	NOUN
ejpam-3182	219	32	.	.	PUNCT
ejpam-3182	220	1	lastly	lastly	ADV
ejpam-3182	220	2	,	,	PUNCT
ejpam-3182	220	3	the	the	DET
ejpam-3182	220	4	error	error	NOUN
ejpam-3182	220	5	term	term	NOUN
ejpam-3182	220	6	o	o	NOUN
ejpam-3182	220	7	(	(	PUNCT
ejpam-3182	220	8	x(log	x(log	PROPN
ejpam-3182	220	9	x)α2−1	x)α2−1	NOUN
ejpam-3182	220	10	)	)	PUNCT
ejpam-3182	220	11	absorbs	absorb	VERB
ejpam-3182	220	12	all	all	DET
ejpam-3182	220	13	the	the	DET
ejpam-3182	220	14	errors	error	NOUN
ejpam-3182	220	15	.	.	PUNCT
ejpam-3182	221	1	quod	quod	PROPN
ejpam-3182	221	2	erat	erat	PROPN
ejpam-3182	221	3	demonstrandum	demonstrandum	PROPN
ejpam-3182	221	4	.	.	PROPN
ejpam-3182	222	1	7.1	7.1	NUM
ejpam-3182	222	2	.	.	PUNCT
ejpam-3182	223	1	summary	summary	NOUN
ejpam-3182	223	2	this	this	DET
ejpam-3182	223	3	note	note	NOUN
ejpam-3182	223	4	proves	prove	VERB
ejpam-3182	223	5	a	a	DET
ejpam-3182	223	6	generalization	generalization	NOUN
ejpam-3182	223	7	of	of	ADP
ejpam-3182	223	8	hooley	hooley	PROPN
ejpam-3182	223	9	theorem	theorem	VERB
ejpam-3182	223	10	for	for	ADP
ejpam-3182	223	11	fixed	fix	VERB
ejpam-3182	223	12	primitive	primitive	ADJ
ejpam-3182	223	13	root	root	NOUN
ejpam-3182	223	14	u	u	PROPN
ejpam-3182	223	15	6=	6=	PRON
ejpam-3182	223	16	±1	±1	VERB
ejpam-3182	223	17	,	,	PUNCT
ejpam-3182	223	18	v2	v2	PROPN
ejpam-3182	223	19	modulo	modulo	VERB
ejpam-3182	223	20	the	the	DET
ejpam-3182	223	21	prime	prime	ADJ
ejpam-3182	223	22	numbers	number	NOUN
ejpam-3182	223	23	p	p	PROPN
ejpam-3182	223	24	≥	≥	NUM
ejpam-3182	223	25	2	2	NUM
ejpam-3182	223	26	,	,	PUNCT
ejpam-3182	223	27	see	see	VERB
ejpam-3182	223	28	theorem	theorem	VERB
ejpam-3182	223	29	7.1	7.1	NUM
ejpam-3182	223	30	,	,	PUNCT
ejpam-3182	223	31	to	to	PART
ejpam-3182	223	32	theorem	theorem	VERB
ejpam-3182	223	33	1.1	1.1	NUM
ejpam-3182	223	34	for	for	ADP
ejpam-3182	223	35	fixed	fix	VERB
ejpam-3182	223	36	primitive	primitive	ADJ
ejpam-3182	223	37	root	root	NOUN
ejpam-3182	223	38	u	u	PROPN
ejpam-3182	223	39	6=	6=	NUM
ejpam-3182	223	40	±1	±1	VERB
ejpam-3182	223	41	,	,	PUNCT
ejpam-3182	223	42	v2	v2	PROPN
ejpam-3182	223	43	in	in	ADP
ejpam-3182	223	44	the	the	DET
ejpam-3182	223	45	maximal	maximal	ADJ
ejpam-3182	223	46	cyclic	cyclic	ADJ
ejpam-3182	223	47	groups	group	NOUN
ejpam-3182	223	48	g	g	PROPN
ejpam-3182	223	49	⊂	⊂	PROPN
ejpam-3182	223	50	z	z	PROPN
ejpam-3182	223	51	/	/	SYM
ejpam-3182	223	52	nz	nz	PROPN
ejpam-3182	223	53	modulo	modulo	VERB
ejpam-3182	223	54	the	the	DET
ejpam-3182	223	55	integers	integer	NOUN
ejpam-3182	223	56	n	n	PRON
ejpam-3182	223	57	≥	≥	NOUN
ejpam-3182	223	58	2	2	NUM
ejpam-3182	223	59	.	.	PUNCT
ejpam-3182	224	1	this	this	DET
ejpam-3182	224	2	technique	technique	NOUN
ejpam-3182	224	3	uses	use	VERB
ejpam-3182	224	4	a	a	DET
ejpam-3182	224	5	new	new	ADJ
ejpam-3182	224	6	characteristic	characteristic	ADJ
ejpam-3182	224	7	function	function	NOUN
ejpam-3182	224	8	for	for	ADP
ejpam-3182	224	9	primitive	primitive	ADJ
ejpam-3182	224	10	roots	root	NOUN
ejpam-3182	224	11	in	in	ADP
ejpam-3182	224	12	the	the	DET
ejpam-3182	224	13	rings	ring	NOUN
ejpam-3182	224	14	z	z	PROPN
ejpam-3182	224	15	/	/	SYM
ejpam-3182	224	16	pkz	pkz	NOUN
ejpam-3182	224	17	,	,	PUNCT
ejpam-3182	224	18	see	see	VERB
ejpam-3182	224	19	lemma	lemma	PROPN
ejpam-3182	224	20	3.1	3.1	NUM
ejpam-3182	224	21	,	,	PUNCT
ejpam-3182	224	22	and	and	CCONJ
ejpam-3182	224	23	a	a	DET
ejpam-3182	224	24	new	new	ADJ
ejpam-3182	224	25	application	application	NOUN
ejpam-3182	224	26	of	of	ADP
ejpam-3182	224	27	wirsing	wirsing	NOUN
ejpam-3182	224	28	theorem	theorem	NOUN
ejpam-3182	224	29	,	,	PUNCT
ejpam-3182	224	30	see	see	VERB
ejpam-3182	224	31	lemma	lemma	PROPN
ejpam-3182	224	32	4.1	4.1	NUM
ejpam-3182	224	33	.	.	PUNCT
ejpam-3182	225	1	furthermore	furthermore	ADV
ejpam-3182	225	2	,	,	PUNCT
ejpam-3182	225	3	the	the	DET
ejpam-3182	225	4	corresponding	correspond	VERB
ejpam-3182	225	5	generalizations	generalization	NOUN
ejpam-3182	225	6	for	for	ADP
ejpam-3182	225	7	the	the	DET
ejpam-3182	225	8	euler	euler	PROPN
ejpam-3182	225	9	constant	constant	ADJ
ejpam-3182	225	10	and	and	CCONJ
ejpam-3182	225	11	the	the	DET
ejpam-3182	225	12	mertens	mertens	PROPN
ejpam-3182	225	13	constant	constant	NOUN
ejpam-3182	225	14	are	be	AUX
ejpam-3182	225	15	included	include	VERB
ejpam-3182	225	16	in	in	ADP
ejpam-3182	225	17	lemma	lemma	PROPN
ejpam-3182	225	18	5.1	5.1	NUM
ejpam-3182	225	19	.	.	PUNCT
ejpam-3182	226	1	references	reference	NOUN
ejpam-3182	226	2	[	[	X
ejpam-3182	226	3	1	1	NUM
ejpam-3182	226	4	]	]	PUNCT
ejpam-3182	226	5	apostol	apostol	NOUN
ejpam-3182	226	6	,	,	PUNCT
ejpam-3182	226	7	tom	tom	PROPN
ejpam-3182	226	8	m.	m.	NOUN
ejpam-3182	226	9	introduction	introduction	NOUN
ejpam-3182	226	10	to	to	ADP
ejpam-3182	226	11	analytic	analytic	ADJ
ejpam-3182	226	12	number	number	NOUN
ejpam-3182	226	13	theory	theory	NOUN
ejpam-3182	226	14	.	.	PUNCT
ejpam-3182	227	1	undergraduate	undergraduate	ADJ
ejpam-3182	227	2	texts	text	NOUN
ejpam-3182	227	3	in	in	ADP
ejpam-3182	227	4	mathematics	mathematic	NOUN
ejpam-3182	227	5	.	.	PUNCT
ejpam-3182	228	1	springer	springer	NOUN
ejpam-3182	228	2	-	-	PUNCT
ejpam-3182	228	3	verlag	verlag	PROPN
ejpam-3182	228	4	,	,	PUNCT
ejpam-3182	228	5	new	new	PROPN
ejpam-3182	228	6	york	york	PROPN
ejpam-3182	228	7	-	-	PUNCT
ejpam-3182	228	8	heidelberg	heidelberg	PROPN
ejpam-3182	228	9	,	,	PUNCT
ejpam-3182	228	10	1976	1976	NUM
ejpam-3182	228	11	.	.	PUNCT
ejpam-3182	229	1	[	[	X
ejpam-3182	229	2	2	2	NUM
ejpam-3182	229	3	]	]	PUNCT
ejpam-3182	229	4	balog	balog	NOUN
ejpam-3182	229	5	,	,	PUNCT
ejpam-3182	229	6	antal	antal	PROPN
ejpam-3182	229	7	;	;	PUNCT
ejpam-3182	229	8	cojocaru	cojocaru	PROPN
ejpam-3182	229	9	,	,	PUNCT
ejpam-3182	229	10	alina	alina	NOUN
ejpam-3182	229	11	-	-	PUNCT
ejpam-3182	229	12	carmen	carmen	PROPN
ejpam-3182	229	13	;	;	PUNCT
ejpam-3182	229	14	david	david	PROPN
ejpam-3182	229	15	,	,	PUNCT
ejpam-3182	229	16	chantal	chantal	PROPN
ejpam-3182	229	17	.	.	PUNCT
ejpam-3182	230	1	average	average	ADJ
ejpam-3182	230	2	twin	twin	ADJ
ejpam-3182	230	3	prime	prime	ADJ
ejpam-3182	230	4	conjecture	conjecture	NOUN
ejpam-3182	230	5	for	for	ADP
ejpam-3182	230	6	elliptic	elliptic	ADJ
ejpam-3182	230	7	curves	curve	NOUN
ejpam-3182	230	8	.	.	PUNCT
ejpam-3182	231	1	amer	amer	PROPN
ejpam-3182	231	2	.	.	PUNCT
ejpam-3182	232	1	j.	j.	PROPN
ejpam-3182	232	2	math	math	PROPN
ejpam-3182	232	3	.	.	PUNCT
ejpam-3182	233	1	133,(2011	133,(2011	NUM
ejpam-3182	233	2	)	)	PUNCT
ejpam-3182	233	3	,	,	PUNCT
ejpam-3182	234	1	no	no	INTJ
ejpam-3182	234	2	.	.	NOUN
ejpam-3182	234	3	5	5	NUM
ejpam-3182	234	4	,	,	PUNCT
ejpam-3182	234	5	1179	1179	NUM
ejpam-3182	234	6	-	-	SYM
ejpam-3182	234	7	1229	1229	NUM
ejpam-3182	234	8	.	.	PUNCT
ejpam-3182	235	1	[	[	X
ejpam-3182	235	2	3	3	X
ejpam-3182	235	3	]	]	X
ejpam-3182	235	4	roger	roger	PROPN
ejpam-3182	235	5	c.	c.	PROPN
ejpam-3182	235	6	baker	baker	PROPN
ejpam-3182	235	7	,	,	PUNCT
ejpam-3182	235	8	paul	paul	PROPN
ejpam-3182	235	9	pollack	pollack	PROPN
ejpam-3182	235	10	,	,	PUNCT
ejpam-3182	235	11	bounded	bound	VERB
ejpam-3182	235	12	gaps	gap	NOUN
ejpam-3182	235	13	between	between	ADP
ejpam-3182	235	14	primes	prime	NOUN
ejpam-3182	235	15	with	with	ADP
ejpam-3182	235	16	a	a	DET
ejpam-3182	235	17	given	give	VERB
ejpam-3182	235	18	primitive	primitive	ADJ
ejpam-3182	235	19	root	root	NOUN
ejpam-3182	235	20	,	,	PUNCT
ejpam-3182	235	21	ii	ii	NOUN
ejpam-3182	235	22	,	,	PUNCT
ejpam-3182	235	23	arxiv:1407.7186	arxiv:1407.7186	ADJ
ejpam-3182	235	24	.	.	PUNCT
ejpam-3182	236	1	[	[	X
ejpam-3182	236	2	4	4	NUM
ejpam-3182	236	3	]	]	X
ejpam-3182	236	4	carmichael	carmichael	PROPN
ejpam-3182	236	5	,	,	PUNCT
ejpam-3182	236	6	r.	r.	PROPN
ejpam-3182	236	7	d.	d.	PROPN
ejpam-3182	236	8	note	note	VERB
ejpam-3182	236	9	on	on	ADP
ejpam-3182	236	10	a	a	DET
ejpam-3182	236	11	new	new	ADJ
ejpam-3182	236	12	number	number	NOUN
ejpam-3182	236	13	theory	theory	NOUN
ejpam-3182	236	14	function	function	NOUN
ejpam-3182	236	15	.	.	PUNCT
ejpam-3182	237	1	bull	bull	NOUN
ejpam-3182	237	2	.	.	PUNCT
ejpam-3182	238	1	amer	amer	PROPN
ejpam-3182	238	2	.	.	PUNCT
ejpam-3182	238	3	math	math	PROPN
ejpam-3182	238	4	.	.	PUNCT
ejpam-3182	239	1	soc	soc	PROPN
ejpam-3182	239	2	.	.	PUNCT
ejpam-3182	240	1	16	16	NUM
ejpam-3182	240	2	(	(	PUNCT
ejpam-3182	240	3	1910	1910	NUM
ejpam-3182	240	4	)	)	PUNCT
ejpam-3182	240	5	,	,	PUNCT
ejpam-3182	240	6	no	no	INTJ
ejpam-3182	240	7	.	.	NOUN
ejpam-3182	240	8	5	5	NUM
ejpam-3182	240	9	,	,	PUNCT
ejpam-3182	240	10	232	232	NUM
ejpam-3182	240	11	-	-	SYM
ejpam-3182	240	12	238	238	NUM
ejpam-3182	240	13	.	.	PUNCT
ejpam-3182	241	1	[	[	X
ejpam-3182	241	2	5	5	X
ejpam-3182	241	3	]	]	X
ejpam-3182	241	4	peter	peter	PROPN
ejpam-3182	241	5	j.	j.	PROPN
ejpam-3182	241	6	cameron	cameron	PROPN
ejpam-3182	241	7	and	and	CCONJ
ejpam-3182	241	8	d.	d.	PROPN
ejpam-3182	241	9	a.	a.	PROPN
ejpam-3182	241	10	preece	preece	PROPN
ejpam-3182	241	11	,	,	PUNCT
ejpam-3182	241	12	notes	note	NOUN
ejpam-3182	241	13	on	on	ADP
ejpam-3182	241	14	primitive	primitive	ADJ
ejpam-3182	241	15	lambda	lambda	NOUN
ejpam-3182	241	16	-	-	PUNCT
ejpam-3182	241	17	roots	root	NOUN
ejpam-3182	241	18	,	,	PUNCT
ejpam-3182	241	19	http://www.maths.qmul.ac.uk/˜pjc/csgnotes/lambda.pdf	http://www.maths.qmul.ac.uk/˜pjc/csgnotes/lambda.pdf	NOUN
ejpam-3182	241	20	[	[	X
ejpam-3182	241	21	6	6	NUM
ejpam-3182	241	22	]	]	X
ejpam-3182	241	23	joseph	joseph	PROPN
ejpam-3182	241	24	cohen	cohen	PROPN
ejpam-3182	241	25	,	,	PUNCT
ejpam-3182	241	26	primitive	primitive	ADJ
ejpam-3182	241	27	roots	root	NOUN
ejpam-3182	241	28	in	in	ADP
ejpam-3182	241	29	quadratic	quadratic	ADJ
ejpam-3182	241	30	fields	field	NOUN
ejpam-3182	241	31	,	,	PUNCT
ejpam-3182	241	32	ii	ii	PROPN
ejpam-3182	241	33	,	,	PUNCT
ejpam-3182	241	34	journal	journal	NOUN
ejpam-3182	241	35	of	of	ADP
ejpam-3182	241	36	number	number	NOUN
ejpam-3182	241	37	theory	theory	NOUN
ejpam-3182	241	38	124	124	NUM
ejpam-3182	241	39	(	(	PUNCT
ejpam-3182	241	40	2007	2007	NUM
ejpam-3182	241	41	)	)	PUNCT
ejpam-3182	241	42	429	429	NUM
ejpam-3182	241	43	-	-	SYM
ejpam-3182	241	44	441	441	NUM
ejpam-3182	241	45	.	.	PUNCT
ejpam-3182	242	1	[	[	X
ejpam-3182	242	2	7	7	X
ejpam-3182	242	3	]	]	X
ejpam-3182	242	4	h.	h.	PROPN
ejpam-3182	242	5	davenport	davenport	PROPN
ejpam-3182	242	6	,	,	PUNCT
ejpam-3182	242	7	on	on	ADP
ejpam-3182	242	8	primitive	primitive	ADJ
ejpam-3182	242	9	roots	root	NOUN
ejpam-3182	242	10	in	in	ADP
ejpam-3182	242	11	finite	finite	ADJ
ejpam-3182	242	12	fields	field	NOUN
ejpam-3182	242	13	,	,	PUNCT
ejpam-3182	242	14	quarterly	quarterly	PROPN
ejpam-3182	242	15	j.	j.	PROPN
ejpam-3182	242	16	math	math	PROPN
ejpam-3182	242	17	.	.	PUNCT
ejpam-3182	243	1	1937	1937	NUM
ejpam-3182	243	2	,	,	PUNCT
ejpam-3182	243	3	308312	308312	NUM
ejpam-3182	243	4	.	.	PUNCT
ejpam-3182	244	1	[	[	X
ejpam-3182	244	2	8	8	NUM
ejpam-3182	244	3	]	]	PUNCT
ejpam-3182	244	4	rainer	rainer	NOUN
ejpam-3182	244	5	dietmann	dietmann	PROPN
ejpam-3182	244	6	,	,	PUNCT
ejpam-3182	244	7	christian	christian	ADJ
ejpam-3182	244	8	elsholtz	elsholtz	NOUN
ejpam-3182	244	9	,	,	PUNCT
ejpam-3182	244	10	igor	igor	PROPN
ejpam-3182	244	11	e.	e.	PROPN
ejpam-3182	244	12	shparlinski	shparlinski	PROPN
ejpam-3182	244	13	,	,	PUNCT
ejpam-3182	244	14	on	on	ADP
ejpam-3182	244	15	gaps	gap	NOUN
ejpam-3182	244	16	between	between	ADP
ejpam-3182	244	17	primitive	primitive	ADJ
ejpam-3182	244	18	roots	root	NOUN
ejpam-3182	244	19	in	in	ADP
ejpam-3182	244	20	the	the	DET
ejpam-3182	244	21	hamming	hamming	NOUN
ejpam-3182	244	22	metric	metric	NOUN
ejpam-3182	244	23	,	,	PUNCT
ejpam-3182	244	24	arxiv:1207.0842	arxiv:1207.0842	PROPN
ejpam-3182	244	25	.	.	PUNCT
ejpam-3182	245	1	[	[	X
ejpam-3182	245	2	9	9	NUM
ejpam-3182	245	3	]	]	X
ejpam-3182	245	4	paul	paul	PROPN
ejpam-3182	245	5	erdos	erdos	PROPN
ejpam-3182	245	6	,	,	PUNCT
ejpam-3182	245	7	harold	harold	PROPN
ejpam-3182	245	8	n.	n.	PROPN
ejpam-3182	245	9	shapiro	shapiro	PROPN
ejpam-3182	245	10	,	,	PUNCT
ejpam-3182	245	11	on	on	ADP
ejpam-3182	245	12	the	the	DET
ejpam-3182	245	13	least	least	ADJ
ejpam-3182	245	14	primitive	primitive	ADJ
ejpam-3182	245	15	root	root	NOUN
ejpam-3182	245	16	of	of	ADP
ejpam-3182	245	17	a	a	DET
ejpam-3182	245	18	prime	prime	NOUN
ejpam-3182	245	19	,	,	PUNCT
ejpam-3182	245	20	1957	1957	NUM
ejpam-3182	245	21	,	,	PUNCT
ejpam-3182	245	22	euclidproject.org	euclidproject.org	X
ejpam-3182	245	23	.	.	PUNCT
ejpam-3182	245	24	references	reference	NOUN
ejpam-3182	245	25	33	33	NUM
ejpam-3182	245	26	[	[	SYM
ejpam-3182	245	27	10	10	NUM
ejpam-3182	245	28	]	]	X
ejpam-3182	245	29	gupta	gupta	PROPN
ejpam-3182	245	30	,	,	PUNCT
ejpam-3182	245	31	rajiv	rajiv	PROPN
ejpam-3182	245	32	;	;	PUNCT
ejpam-3182	245	33	murty	murty	NOUN
ejpam-3182	245	34	,	,	PUNCT
ejpam-3182	245	35	m.	m.	NOUN
ejpam-3182	245	36	ram	ram	PROPN
ejpam-3182	245	37	.	.	PUNCT
ejpam-3182	246	1	a	a	DET
ejpam-3182	246	2	remark	remark	NOUN
ejpam-3182	246	3	on	on	ADP
ejpam-3182	246	4	artin	artin	PROPN
ejpam-3182	246	5	’s	’s	PART
ejpam-3182	246	6	conjecture	conjecture	NOUN
ejpam-3182	246	7	.	.	PUNCT
ejpam-3182	247	1	invent	invent	NOUN
ejpam-3182	247	2	.	.	PUNCT
ejpam-3182	248	1	math	math	NOUN
ejpam-3182	248	2	.	.	PUNCT
ejpam-3182	249	1	78	78	NUM
ejpam-3182	249	2	(	(	PUNCT
ejpam-3182	249	3	1984	1984	NUM
ejpam-3182	249	4	)	)	PUNCT
ejpam-3182	249	5	,	,	PUNCT
ejpam-3182	249	6	no	no	INTJ
ejpam-3182	249	7	.	.	NOUN
ejpam-3182	249	8	1	1	NUM
ejpam-3182	249	9	,	,	PUNCT
ejpam-3182	249	10	127	127	NUM
ejpam-3182	249	11	-	-	SYM
ejpam-3182	249	12	130	130	NUM
ejpam-3182	249	13	.	.	PUNCT
ejpam-3182	250	1	[	[	X
ejpam-3182	250	2	11	11	NUM
ejpam-3182	250	3	]	]	X
ejpam-3182	250	4	hardy	hardy	ADJ
ejpam-3182	250	5	,	,	PUNCT
ejpam-3182	250	6	g.	g.	PROPN
ejpam-3182	250	7	h.	h.	PROPN
ejpam-3182	250	8	;	;	PUNCT
ejpam-3182	250	9	wright	wright	PROPN
ejpam-3182	250	10	,	,	PUNCT
ejpam-3182	250	11	e.	e.	PROPN
ejpam-3182	250	12	m.	m.	PROPN
ejpam-3182	250	13	an	an	DET
ejpam-3182	250	14	introduction	introduction	NOUN
ejpam-3182	250	15	to	to	ADP
ejpam-3182	250	16	the	the	DET
ejpam-3182	250	17	theory	theory	NOUN
ejpam-3182	250	18	of	of	ADP
ejpam-3182	250	19	numbers	number	NOUN
ejpam-3182	250	20	.	.	PUNCT
ejpam-3182	251	1	sixth	sixth	ADJ
ejpam-3182	251	2	edition	edition	NOUN
ejpam-3182	251	3	.	.	PUNCT
ejpam-3182	252	1	revised	revise	VERB
ejpam-3182	252	2	by	by	ADP
ejpam-3182	252	3	d.	d.	PROPN
ejpam-3182	252	4	r.	r.	PROPN
ejpam-3182	252	5	heath	heath	PROPN
ejpam-3182	252	6	-	-	PUNCT
ejpam-3182	252	7	brown	brown	PROPN
ejpam-3182	252	8	and	and	CCONJ
ejpam-3182	252	9	j.	j.	PROPN
ejpam-3182	252	10	h.	h.	PROPN
ejpam-3182	252	11	silverman	silverman	PROPN
ejpam-3182	252	12	.	.	PUNCT
ejpam-3182	253	1	with	with	ADP
ejpam-3182	253	2	a	a	DET
ejpam-3182	253	3	foreword	foreword	NOUN
ejpam-3182	253	4	by	by	ADP
ejpam-3182	253	5	andrew	andrew	PROPN
ejpam-3182	253	6	wiles	wile	NOUN
ejpam-3182	253	7	.	.	PUNCT
ejpam-3182	254	1	oxford	oxford	PROPN
ejpam-3182	254	2	university	university	PROPN
ejpam-3182	254	3	press	press	NOUN
ejpam-3182	254	4	,	,	PUNCT
ejpam-3182	254	5	2008	2008	NUM
ejpam-3182	254	6	.	.	PUNCT
ejpam-3182	255	1	[	[	X
ejpam-3182	255	2	12	12	NUM
ejpam-3182	255	3	]	]	X
ejpam-3182	255	4	hildebrand	hildebrand	NOUN
ejpam-3182	255	5	,	,	PUNCT
ejpam-3182	255	6	adolf	adolf	PROPN
ejpam-3182	255	7	.	.	PUNCT
ejpam-3182	256	1	quantitative	quantitative	ADJ
ejpam-3182	256	2	mean	mean	NOUN
ejpam-3182	256	3	value	value	NOUN
ejpam-3182	256	4	theorems	theorem	NOUN
ejpam-3182	256	5	for	for	ADP
ejpam-3182	256	6	nonnegative	nonnegative	ADJ
ejpam-3182	256	7	multiplicative	multiplicative	ADJ
ejpam-3182	256	8	functions	function	NOUN
ejpam-3182	256	9	.	.	PUNCT
ejpam-3182	257	1	ii	ii	PROPN
ejpam-3182	257	2	.	.	PUNCT
ejpam-3182	258	1	acta	acta	PROPN
ejpam-3182	258	2	arith	arith	PROPN
ejpam-3182	258	3	.	.	PUNCT
ejpam-3182	259	1	48	48	NUM
ejpam-3182	259	2	(	(	PUNCT
ejpam-3182	259	3	1987	1987	NUM
ejpam-3182	259	4	)	)	PUNCT
ejpam-3182	259	5	,	,	PUNCT
ejpam-3182	259	6	no	no	INTJ
ejpam-3182	259	7	.	.	NOUN
ejpam-3182	259	8	3	3	NUM
ejpam-3182	259	9	,	,	PUNCT
ejpam-3182	259	10	209	209	NUM
ejpam-3182	259	11	-	-	SYM
ejpam-3182	259	12	260	260	NUM
ejpam-3182	259	13	.	.	PUNCT
ejpam-3182	260	1	[	[	X
ejpam-3182	260	2	13	13	NUM
ejpam-3182	260	3	]	]	X
ejpam-3182	260	4	c.	c.	PROPN
ejpam-3182	260	5	hooley	hooley	PROPN
ejpam-3182	260	6	,	,	PUNCT
ejpam-3182	260	7	on	on	ADP
ejpam-3182	260	8	artins	artin	NOUN
ejpam-3182	260	9	conjecture	conjecture	VERB
ejpam-3182	260	10	,	,	PUNCT
ejpam-3182	260	11	j.	j.	PROPN
ejpam-3182	260	12	reine	reine	PROPN
ejpam-3182	260	13	angew	angew	PROPN
ejpam-3182	260	14	.	.	PUNCT
ejpam-3182	261	1	math	math	NOUN
ejpam-3182	261	2	.	.	PUNCT
ejpam-3182	262	1	225	225	NUM
ejpam-3182	262	2	,	,	PUNCT
ejpam-3182	262	3	209	209	NUM
ejpam-3182	262	4	-	-	SYM
ejpam-3182	262	5	220	220	NUM
ejpam-3182	262	6	,	,	PUNCT
ejpam-3182	262	7	1967	1967	NUM
ejpam-3182	262	8	.	.	PUNCT
ejpam-3182	263	1	[	[	X
ejpam-3182	263	2	14	14	NUM
ejpam-3182	263	3	]	]	SYM
ejpam-3182	263	4	li	li	PROPN
ejpam-3182	263	5	,	,	PUNCT
ejpam-3182	263	6	shuguang	shuguang	PROPN
ejpam-3182	263	7	;	;	PUNCT
ejpam-3182	263	8	pomerance	pomerance	NOUN
ejpam-3182	263	9	,	,	PUNCT
ejpam-3182	263	10	carl	carl	PROPN
ejpam-3182	263	11	.	.	PROPN
ejpam-3182	263	12	primitive	primitive	ADJ
ejpam-3182	263	13	roots	root	NOUN
ejpam-3182	263	14	:	:	PUNCT
ejpam-3182	263	15	a	a	DET
ejpam-3182	263	16	survey	survey	NOUN
ejpam-3182	263	17	.	.	PUNCT
ejpam-3182	264	1	number	number	NOUN
ejpam-3182	264	2	theoretic	theoretic	ADJ
ejpam-3182	264	3	methods	method	NOUN
ejpam-3182	264	4	,	,	PUNCT
ejpam-3182	264	5	iizuka	iizuka	NOUN
ejpam-3182	264	6	,	,	PUNCT
ejpam-3182	264	7	2001	2001	NUM
ejpam-3182	264	8	,	,	PUNCT
ejpam-3182	264	9	219	219	NUM
ejpam-3182	264	10	-	-	SYM
ejpam-3182	264	11	231	231	NUM
ejpam-3182	264	12	,	,	PUNCT
ejpam-3182	264	13	dev	dev	PROPN
ejpam-3182	264	14	.	.	PROPN
ejpam-3182	264	15	math	math	PROPN
ejpam-3182	264	16	.	.	PUNCT
ejpam-3182	264	17	,	,	PUNCT
ejpam-3182	264	18	8	8	NUM
ejpam-3182	264	19	,	,	PUNCT
ejpam-3182	264	20	kluwer	kluwer	NOUN
ejpam-3182	264	21	acad	acad	PROPN
ejpam-3182	264	22	.	.	PUNCT
ejpam-3182	265	1	publ	publ	PROPN
ejpam-3182	265	2	.	.	PUNCT
ejpam-3182	265	3	,	,	PUNCT
ejpam-3182	265	4	dordrecht	dordrecht	PROPN
ejpam-3182	265	5	,	,	PUNCT
ejpam-3182	265	6	2002	2002	NUM
ejpam-3182	265	7	.	.	PUNCT
ejpam-3182	266	1	[	[	X
ejpam-3182	266	2	15	15	NUM
ejpam-3182	266	3	]	]	X
ejpam-3182	266	4	li	li	PROPN
ejpam-3182	266	5	,	,	PUNCT
ejpam-3182	266	6	shuguang	shuguang	PROPN
ejpam-3182	266	7	;	;	PUNCT
ejpam-3182	266	8	pomerance	pomerance	NOUN
ejpam-3182	266	9	,	,	PUNCT
ejpam-3182	266	10	carl	carl	PROPN
ejpam-3182	266	11	.	.	PROPN
ejpam-3182	266	12	on	on	ADP
ejpam-3182	266	13	generalizing	generalize	VERB
ejpam-3182	266	14	artin	artin	PROPN
ejpam-3182	266	15	’s	’s	PART
ejpam-3182	266	16	conjecture	conjecture	NOUN
ejpam-3182	266	17	on	on	ADP
ejpam-3182	266	18	primitive	primitive	ADJ
ejpam-3182	266	19	roots	root	NOUN
ejpam-3182	266	20	to	to	ADP
ejpam-3182	266	21	composite	composite	ADJ
ejpam-3182	266	22	moduli	modulus	NOUN
ejpam-3182	266	23	.	.	PUNCT
ejpam-3182	267	1	j.	j.	PROPN
ejpam-3182	267	2	reine	reine	PROPN
ejpam-3182	267	3	angew	angew	PROPN
ejpam-3182	267	4	.	.	PUNCT
ejpam-3182	268	1	math	math	NOUN
ejpam-3182	268	2	.	.	PUNCT
ejpam-3182	269	1	556	556	NUM
ejpam-3182	269	2	(	(	PUNCT
ejpam-3182	269	3	2003	2003	NUM
ejpam-3182	269	4	)	)	PUNCT
ejpam-3182	269	5	,	,	PUNCT
ejpam-3182	269	6	205	205	NUM
ejpam-3182	269	7	-	-	SYM
ejpam-3182	269	8	224	224	NUM
ejpam-3182	269	9	.	.	PUNCT
ejpam-3182	270	1	[	[	X
ejpam-3182	270	2	16	16	NUM
ejpam-3182	270	3	]	]	X
ejpam-3182	270	4	iwaniec	iwaniec	PROPN
ejpam-3182	270	5	,	,	PUNCT
ejpam-3182	270	6	henryk	henryk	PROPN
ejpam-3182	270	7	;	;	PUNCT
ejpam-3182	270	8	kowalski	kowalski	PROPN
ejpam-3182	270	9	,	,	PUNCT
ejpam-3182	270	10	emmanuel	emmanuel	PROPN
ejpam-3182	270	11	.	.	PUNCT
ejpam-3182	270	12	analytic	analytic	ADJ
ejpam-3182	270	13	number	number	NOUN
ejpam-3182	270	14	theory	theory	NOUN
ejpam-3182	270	15	.	.	PUNCT
ejpam-3182	271	1	ams	am	NOUN
ejpam-3182	271	2	colloquium	colloquium	NOUN
ejpam-3182	271	3	publications	publication	NOUN
ejpam-3182	271	4	,	,	PUNCT
ejpam-3182	271	5	53	53	NUM
ejpam-3182	271	6	.	.	PUNCT
ejpam-3182	272	1	american	american	PROPN
ejpam-3182	272	2	mathematical	mathematical	PROPN
ejpam-3182	272	3	society	society	NOUN
ejpam-3182	272	4	,	,	PUNCT
ejpam-3182	272	5	providence	providence	NOUN
ejpam-3182	272	6	,	,	PUNCT
ejpam-3182	272	7	ri	ri	PROPN
ejpam-3182	272	8	,	,	PUNCT
ejpam-3182	272	9	2004	2004	NUM
ejpam-3182	272	10	.	.	PUNCT
ejpam-3182	273	1	[	[	X
ejpam-3182	273	2	17	17	NUM
ejpam-3182	273	3	]	]	PUNCT
ejpam-3182	273	4	konyagin	konyagin	NOUN
ejpam-3182	273	5	,	,	PUNCT
ejpam-3182	273	6	sergei	sergei	PROPN
ejpam-3182	273	7	v.	v.	PROPN
ejpam-3182	273	8	;	;	PUNCT
ejpam-3182	273	9	shparlinski	shparlinski	NOUN
ejpam-3182	273	10	,	,	PUNCT
ejpam-3182	273	11	igor	igor	PROPN
ejpam-3182	273	12	e.	e.	PROPN
ejpam-3182	273	13	on	on	ADP
ejpam-3182	273	14	the	the	DET
ejpam-3182	273	15	consecutive	consecutive	ADJ
ejpam-3182	273	16	powers	power	NOUN
ejpam-3182	273	17	of	of	ADP
ejpam-3182	273	18	a	a	DET
ejpam-3182	273	19	primitive	primitive	ADJ
ejpam-3182	273	20	root	root	NOUN
ejpam-3182	273	21	:	:	PUNCT
ejpam-3182	273	22	gaps	gap	NOUN
ejpam-3182	273	23	and	and	CCONJ
ejpam-3182	273	24	exponential	exponential	ADJ
ejpam-3182	273	25	sums	sum	NOUN
ejpam-3182	273	26	.	.	PUNCT
ejpam-3182	274	1	mathematika	mathematika	NOUN
ejpam-3182	274	2	58	58	NUM
ejpam-3182	274	3	(	(	PUNCT
ejpam-3182	274	4	2012	2012	NUM
ejpam-3182	274	5	)	)	PUNCT
ejpam-3182	274	6	,	,	PUNCT
ejpam-3182	274	7	no	no	INTJ
ejpam-3182	274	8	.	.	NOUN
ejpam-3182	274	9	1	1	NUM
ejpam-3182	274	10	,	,	PUNCT
ejpam-3182	274	11	11	11	NUM
ejpam-3182	274	12	-	-	SYM
ejpam-3182	274	13	20	20	NUM
ejpam-3182	274	14	.	.	PUNCT
ejpam-3182	275	1	[	[	X
ejpam-3182	275	2	18	18	NUM
ejpam-3182	275	3	]	]	X
ejpam-3182	275	4	h.	h.	PROPN
ejpam-3182	275	5	w.	w.	PROPN
ejpam-3182	275	6	lenstra	lenstra	PROPN
ejpam-3182	275	7	jr	jr	PROPN
ejpam-3182	275	8	,	,	PUNCT
ejpam-3182	275	9	p.	p.	NOUN
ejpam-3182	275	10	moree	moree	NOUN
ejpam-3182	275	11	,	,	PUNCT
ejpam-3182	275	12	p.	p.	NOUN
ejpam-3182	275	13	stevenhagen	stevenhagen	PROPN
ejpam-3182	275	14	,	,	PUNCT
ejpam-3182	275	15	character	character	NOUN
ejpam-3182	275	16	sums	sum	NOUN
ejpam-3182	275	17	for	for	ADP
ejpam-3182	275	18	primitive	primitive	ADJ
ejpam-3182	275	19	root	root	NOUN
ejpam-3182	275	20	densities	density	NOUN
ejpam-3182	275	21	,	,	PUNCT
ejpam-3182	275	22	arxiv:1112.4816	arxiv:1112.4816	PROPN
ejpam-3182	275	23	.	.	PUNCT
ejpam-3182	276	1	[	[	X
ejpam-3182	276	2	19	19	NUM
ejpam-3182	276	3	]	]	PUNCT
ejpam-3182	276	4	lenstra	lenstra	NOUN
ejpam-3182	276	5	,	,	PUNCT
ejpam-3182	276	6	h.	h.	PROPN
ejpam-3182	276	7	w.	w.	PROPN
ejpam-3182	276	8	,	,	PUNCT
ejpam-3182	276	9	jr	jr	PROPN
ejpam-3182	276	10	.	.	PROPN
ejpam-3182	276	11	on	on	ADP
ejpam-3182	276	12	artin	artin	PROPN
ejpam-3182	276	13	conjecture	conjecture	NOUN
ejpam-3182	276	14	and	and	CCONJ
ejpam-3182	276	15	euclid	euclid	PROPN
ejpam-3182	276	16	algorithm	algorithm	NOUN
ejpam-3182	276	17	in	in	ADP
ejpam-3182	276	18	global	global	ADJ
ejpam-3182	276	19	fields	field	NOUN
ejpam-3182	276	20	.	.	PUNCT
ejpam-3182	276	21	invent	invent	NOUN
ejpam-3182	276	22	.	.	PUNCT
ejpam-3182	277	1	math	math	NOUN
ejpam-3182	277	2	.	.	PUNCT
ejpam-3182	278	1	42	42	NUM
ejpam-3182	278	2	,	,	PUNCT
ejpam-3182	278	3	(	(	PUNCT
ejpam-3182	278	4	1977	1977	NUM
ejpam-3182	278	5	)	)	PUNCT
ejpam-3182	278	6	,	,	PUNCT
ejpam-3182	278	7	201	201	NUM
ejpam-3182	278	8	-	-	SYM
ejpam-3182	278	9	224	224	NUM
ejpam-3182	278	10	.	.	PUNCT
ejpam-3182	279	1	[	[	X
ejpam-3182	279	2	20	20	NUM
ejpam-3182	279	3	]	]	SYM
ejpam-3182	279	4	lidl	lidl	PROPN
ejpam-3182	279	5	,	,	PUNCT
ejpam-3182	279	6	rudolf	rudolf	PROPN
ejpam-3182	279	7	;	;	PUNCT
ejpam-3182	279	8	niederreiter	niederreiter	PROPN
ejpam-3182	279	9	,	,	PUNCT
ejpam-3182	279	10	harald	harald	PROPN
ejpam-3182	279	11	.	.	PUNCT
ejpam-3182	280	1	finite	finite	PROPN
ejpam-3182	280	2	fields	field	NOUN
ejpam-3182	280	3	.	.	PUNCT
ejpam-3182	281	1	with	with	ADP
ejpam-3182	281	2	a	a	DET
ejpam-3182	281	3	foreword	foreword	NOUN
ejpam-3182	281	4	by	by	ADP
ejpam-3182	281	5	p.	p.	PROPN
ejpam-3182	281	6	m.	m.	PROPN
ejpam-3182	281	7	cohn	cohn	PROPN
ejpam-3182	281	8	.	.	PUNCT
ejpam-3182	281	9	second	second	PROPN
ejpam-3182	281	10	edition	edition	PROPN
ejpam-3182	281	11	.	.	PUNCT
ejpam-3182	282	1	encyclopedia	encyclopedia	NOUN
ejpam-3182	282	2	of	of	ADP
ejpam-3182	282	3	mathematics	mathematic	NOUN
ejpam-3182	282	4	and	and	CCONJ
ejpam-3182	282	5	its	its	PRON
ejpam-3182	282	6	applications	application	NOUN
ejpam-3182	282	7	,	,	PUNCT
ejpam-3182	282	8	20	20	NUM
ejpam-3182	282	9	.	.	PUNCT
ejpam-3182	283	1	cambridge	cambridge	PROPN
ejpam-3182	283	2	university	university	PROPN
ejpam-3182	283	3	press	press	PROPN
ejpam-3182	283	4	,	,	PUNCT
ejpam-3182	283	5	cambridge	cambridge	PROPN
ejpam-3182	283	6	,	,	PUNCT
ejpam-3182	283	7	1997	1997	NUM
ejpam-3182	283	8	.	.	PUNCT
ejpam-3182	284	1	[	[	X
ejpam-3182	284	2	21	21	NUM
ejpam-3182	284	3	]	]	X
ejpam-3182	284	4	moree	moree	NOUN
ejpam-3182	284	5	,	,	PUNCT
ejpam-3182	284	6	pieter	pieter	NOUN
ejpam-3182	284	7	.	.	PUNCT
ejpam-3182	285	1	counting	count	VERB
ejpam-3182	285	2	numbers	number	NOUN
ejpam-3182	285	3	in	in	ADP
ejpam-3182	285	4	multiplicative	multiplicative	ADJ
ejpam-3182	285	5	sets	set	NOUN
ejpam-3182	285	6	:	:	PUNCT
ejpam-3182	285	7	landau	landau	NOUN
ejpam-3182	285	8	versus	versus	ADP
ejpam-3182	285	9	ramanujan	ramanujan	PROPN
ejpam-3182	285	10	.	.	PROPN
ejpam-3182	285	11	math	math	PROPN
ejpam-3182	285	12	.	.	PUNCT
ejpam-3182	286	1	newsl	newsl	PROPN
ejpam-3182	286	2	.	.	PUNCT
ejpam-3182	287	1	21	21	NUM
ejpam-3182	287	2	(	(	PUNCT
ejpam-3182	287	3	2011	2011	NUM
ejpam-3182	287	4	)	)	PUNCT
ejpam-3182	287	5	,	,	PUNCT
ejpam-3182	287	6	no	no	INTJ
ejpam-3182	287	7	.	.	NOUN
ejpam-3182	287	8	3	3	NUM
ejpam-3182	287	9	,	,	PUNCT
ejpam-3182	287	10	73	73	NUM
ejpam-3182	287	11	-	-	SYM
ejpam-3182	287	12	81	81	NUM
ejpam-3182	287	13	.	.	PUNCT
ejpam-3182	288	1	[	[	X
ejpam-3182	288	2	22	22	NUM
ejpam-3182	288	3	]	]	PUNCT
ejpam-3182	288	4	pieter	pieter	NOUN
ejpam-3182	288	5	moree	moree	PROPN
ejpam-3182	288	6	.	.	PUNCT
ejpam-3182	289	1	artin	artin	PROPN
ejpam-3182	289	2	’s	’s	PART
ejpam-3182	289	3	primitive	primitive	ADJ
ejpam-3182	289	4	root	root	NOUN
ejpam-3182	289	5	conjecture	conjecture	NOUN
ejpam-3182	289	6	-a	-a	X
ejpam-3182	289	7	survey	survey	NOUN
ejpam-3182	289	8	.	.	PUNCT
ejpam-3182	290	1	arxiv	arxiv	NOUN
ejpam-3182	290	2	:	:	PUNCT
ejpam-3182	290	3	math/0412262	math/0412262	NOUN
ejpam-3182	290	4	.	.	PUNCT
ejpam-3182	291	1	[	[	X
ejpam-3182	291	2	23	23	NUM
ejpam-3182	291	3	]	]	X
ejpam-3182	291	4	montgomery	montgomery	PROPN
ejpam-3182	291	5	,	,	PUNCT
ejpam-3182	291	6	hugh	hugh	PROPN
ejpam-3182	291	7	l.	l.	PROPN
ejpam-3182	291	8	;	;	PUNCT
ejpam-3182	291	9	vaughan	vaughan	PROPN
ejpam-3182	291	10	,	,	PUNCT
ejpam-3182	291	11	robert	robert	PROPN
ejpam-3182	291	12	c.	c.	PROPN
ejpam-3182	291	13	multiplicative	multiplicative	PROPN
ejpam-3182	291	14	number	number	NOUN
ejpam-3182	291	15	theory	theory	NOUN
ejpam-3182	291	16	.	.	PUNCT
ejpam-3182	292	1	i.	i.	PROPN
ejpam-3182	292	2	classical	classical	PROPN
ejpam-3182	292	3	theory	theory	PROPN
ejpam-3182	292	4	.	.	PUNCT
ejpam-3182	293	1	cambridge	cambridge	PROPN
ejpam-3182	293	2	university	university	PROPN
ejpam-3182	293	3	press	press	PROPN
ejpam-3182	293	4	,	,	PUNCT
ejpam-3182	293	5	cambridge	cambridge	PROPN
ejpam-3182	293	6	,	,	PUNCT
ejpam-3182	293	7	2007	2007	NUM
ejpam-3182	293	8	.	.	PUNCT
ejpam-3182	294	1	[	[	X
ejpam-3182	294	2	24	24	NUM
ejpam-3182	294	3	]	]	X
ejpam-3182	294	4	narkiewicz	narkiewicz	NOUN
ejpam-3182	294	5	,	,	PUNCT
ejpam-3182	294	6	w.	w.	NOUN
ejpam-3182	294	7	the	the	DET
ejpam-3182	294	8	development	development	NOUN
ejpam-3182	294	9	of	of	ADP
ejpam-3182	294	10	prime	prime	ADJ
ejpam-3182	294	11	number	number	NOUN
ejpam-3182	294	12	theory	theory	NOUN
ejpam-3182	294	13	.	.	PUNCT
ejpam-3182	295	1	from	from	ADP
ejpam-3182	295	2	euclid	euclid	PROPN
ejpam-3182	295	3	to	to	ADP
ejpam-3182	295	4	hardy	hardy	VERB
ejpam-3182	295	5	and	and	CCONJ
ejpam-3182	295	6	littlewood	littlewood	PROPN
ejpam-3182	295	7	.	.	PUNCT
ejpam-3182	296	1	springer	springer	NOUN
ejpam-3182	296	2	monographs	monograph	NOUN
ejpam-3182	296	3	in	in	ADP
ejpam-3182	296	4	mathematics	mathematic	NOUN
ejpam-3182	296	5	.	.	PUNCT
ejpam-3182	297	1	springer	springer	NOUN
ejpam-3182	297	2	-	-	PUNCT
ejpam-3182	297	3	verlag	verlag	PROPN
ejpam-3182	297	4	,	,	PUNCT
ejpam-3182	297	5	berlin	berlin	PROPN
ejpam-3182	297	6	,	,	PUNCT
ejpam-3182	297	7	2000	2000	NUM
ejpam-3182	297	8	.	.	PUNCT
ejpam-3182	298	1	[	[	X
ejpam-3182	298	2	25	25	NUM
ejpam-3182	298	3	]	]	X
ejpam-3182	298	4	pappalardi	pappalardi	NOUN
ejpam-3182	298	5	,	,	PUNCT
ejpam-3182	298	6	francesco	francesco	PROPN
ejpam-3182	298	7	;	;	PUNCT
ejpam-3182	298	8	saidak	saidak	PROPN
ejpam-3182	298	9	,	,	PUNCT
ejpam-3182	298	10	filip	filip	PROPN
ejpam-3182	298	11	;	;	PUNCT
ejpam-3182	298	12	shparlinski	shparlinski	NOUN
ejpam-3182	298	13	,	,	PUNCT
ejpam-3182	298	14	igor	igor	PROPN
ejpam-3182	298	15	e.	e.	PROPN
ejpam-3182	298	16	square	square	PROPN
ejpam-3182	298	17	-	-	PUNCT
ejpam-3182	298	18	free	free	ADJ
ejpam-3182	298	19	values	value	NOUN
ejpam-3182	298	20	of	of	ADP
ejpam-3182	298	21	the	the	DET
ejpam-3182	298	22	carmichael	carmichael	PROPN
ejpam-3182	298	23	function	function	PROPN
ejpam-3182	298	24	.	.	PUNCT
ejpam-3182	299	1	j.	j.	PROPN
ejpam-3182	299	2	number	number	PROPN
ejpam-3182	299	3	theory	theory	NOUN
ejpam-3182	299	4	103	103	NUM
ejpam-3182	299	5	(	(	PUNCT
ejpam-3182	299	6	2003	2003	NUM
ejpam-3182	299	7	)	)	PUNCT
ejpam-3182	299	8	,	,	PUNCT
ejpam-3182	299	9	no	no	INTJ
ejpam-3182	299	10	.	.	NOUN
ejpam-3182	299	11	1	1	NUM
ejpam-3182	299	12	,	,	PUNCT
ejpam-3182	299	13	122	122	NUM
ejpam-3182	299	14	-	-	SYM
ejpam-3182	299	15	131	131	NUM
ejpam-3182	299	16	.	.	PUNCT
ejpam-3182	300	1	references	reference	NOUN
ejpam-3182	300	2	34	34	NUM
ejpam-3182	300	3	[	[	X
ejpam-3182	300	4	26	26	NUM
ejpam-3182	300	5	]	]	X
ejpam-3182	300	6	pappalardi	pappalardi	NOUN
ejpam-3182	300	7	,	,	PUNCT
ejpam-3182	300	8	francesco	francesco	PROPN
ejpam-3182	300	9	;	;	PUNCT
ejpam-3182	300	10	susa	susa	PROPN
ejpam-3182	300	11	,	,	PUNCT
ejpam-3182	300	12	andrea	andrea	PROPN
ejpam-3182	300	13	.	.	PUNCT
ejpam-3182	301	1	an	an	DET
ejpam-3182	301	2	analogue	analogue	NOUN
ejpam-3182	301	3	of	of	ADP
ejpam-3182	301	4	artin	artin	PROPN
ejpam-3182	301	5	conjecture	conjecture	NOUN
ejpam-3182	301	6	for	for	ADP
ejpam-3182	301	7	multiplicative	multiplicative	ADJ
ejpam-3182	301	8	subgroups	subgroup	NOUN
ejpam-3182	301	9	of	of	ADP
ejpam-3182	301	10	the	the	DET
ejpam-3182	301	11	rationals	rational	NOUN
ejpam-3182	301	12	.	.	PUNCT
ejpam-3182	302	1	arch	arch	PROPN
ejpam-3182	302	2	.	.	PUNCT
ejpam-3182	303	1	math	math	NOUN
ejpam-3182	303	2	.	.	PUNCT
ejpam-3182	304	1	(	(	PUNCT
ejpam-3182	304	2	basel	basel	PROPN
ejpam-3182	304	3	)	)	PUNCT
ejpam-3182	304	4	101	101	NUM
ejpam-3182	304	5	,	,	PUNCT
ejpam-3182	304	6	(	(	PUNCT
ejpam-3182	304	7	2013	2013	NUM
ejpam-3182	304	8	)	)	PUNCT
ejpam-3182	304	9	,	,	PUNCT
ejpam-3182	304	10	no	no	INTJ
ejpam-3182	304	11	.	.	NOUN
ejpam-3182	304	12	4	4	NUM
ejpam-3182	304	13	,	,	PUNCT
ejpam-3182	304	14	319	319	NUM
ejpam-3182	304	15	-	-	SYM
ejpam-3182	304	16	330	330	NUM
ejpam-3182	304	17	.	.	PUNCT
ejpam-3182	305	1	[	[	X
ejpam-3182	305	2	27	27	NUM
ejpam-3182	305	3	]	]	PUNCT
ejpam-3182	305	4	a.	a.	NOUN
ejpam-3182	305	5	g.	g.	PROPN
ejpam-3182	305	6	postnikov	postnikov	PROPN
ejpam-3182	305	7	,	,	PUNCT
ejpam-3182	305	8	introduction	introduction	NOUN
ejpam-3182	305	9	to	to	ADP
ejpam-3182	305	10	analytic	analytic	ADJ
ejpam-3182	305	11	number	number	NOUN
ejpam-3182	305	12	theory	theory	NOUN
ejpam-3182	305	13	,	,	PUNCT
ejpam-3182	305	14	translations	translation	NOUN
ejpam-3182	305	15	of	of	ADP
ejpam-3182	305	16	mathematical	mathematical	ADJ
ejpam-3182	305	17	monographs	monograph	NOUN
ejpam-3182	305	18	,	,	PUNCT
ejpam-3182	305	19	vol	vol	NOUN
ejpam-3182	305	20	.	.	PROPN
ejpam-3182	305	21	68	68	NUM
ejpam-3182	305	22	,	,	PUNCT
ejpam-3182	305	23	american	american	PROPN
ejpam-3182	305	24	mathematical	mathematical	ADJ
ejpam-3182	305	25	society	society	NOUN
ejpam-3182	305	26	,	,	PUNCT
ejpam-3182	305	27	providence	providence	NOUN
ejpam-3182	305	28	,	,	PUNCT
ejpam-3182	305	29	ri	ri	NOUN
ejpam-3182	305	30	,	,	PUNCT
ejpam-3182	305	31	1988	1988	NUM
ejpam-3182	305	32	.	.	PUNCT
ejpam-3182	306	1	[	[	X
ejpam-3182	306	2	28	28	NUM
ejpam-3182	306	3	]	]	X
ejpam-3182	306	4	paszkiewicz	paszkiewicz	NOUN
ejpam-3182	306	5	,	,	PUNCT
ejpam-3182	306	6	a.	a.	NOUN
ejpam-3182	306	7	a	a	DET
ejpam-3182	306	8	new	new	ADJ
ejpam-3182	306	9	prime	prime	NOUN
ejpam-3182	306	10	p	p	NOUN
ejpam-3182	306	11	for	for	ADP
ejpam-3182	306	12	which	which	PRON
ejpam-3182	306	13	the	the	DET
ejpam-3182	306	14	least	least	ADV
ejpam-3182	306	15	primitive	primitive	ADJ
ejpam-3182	306	16	root	root	NOUN
ejpam-3182	306	17	mod	mod	PROPN
ejpam-3182	306	18	p	p	PROPN
ejpam-3182	306	19	and	and	CCONJ
ejpam-3182	306	20	the	the	DET
ejpam-3182	306	21	least	least	ADJ
ejpam-3182	306	22	primitive	primitive	ADJ
ejpam-3182	306	23	root	root	NOUN
ejpam-3182	306	24	modp2	modp2	ADV
ejpam-3182	306	25	are	be	AUX
ejpam-3182	306	26	not	not	PART
ejpam-3182	306	27	equal	equal	ADJ
ejpam-3182	306	28	.	.	PUNCT
ejpam-3182	307	1	math	math	NOUN
ejpam-3182	307	2	.	.	PUNCT
ejpam-3182	308	1	comp	comp	NOUN
ejpam-3182	308	2	.	.	PUNCT
ejpam-3182	309	1	78	78	NUM
ejpam-3182	309	2	(	(	PUNCT
ejpam-3182	309	3	2009	2009	NUM
ejpam-3182	309	4	)	)	PUNCT
ejpam-3182	309	5	,	,	PUNCT
ejpam-3182	309	6	no	no	INTJ
ejpam-3182	309	7	.	.	NOUN
ejpam-3182	309	8	266	266	NUM
ejpam-3182	309	9	,	,	PUNCT
ejpam-3182	309	10	1193	1193	NUM
ejpam-3182	309	11	-	-	SYM
ejpam-3182	309	12	1195	1195	NUM
ejpam-3182	309	13	.	.	PUNCT
ejpam-3182	310	1	[	[	X
ejpam-3182	310	2	29	29	NUM
ejpam-3182	310	3	]	]	X
ejpam-3182	310	4	roskam	roskam	NOUN
ejpam-3182	310	5	,	,	PUNCT
ejpam-3182	310	6	hans	hans	PROPN
ejpam-3182	310	7	.	.	PUNCT
ejpam-3182	311	1	artin	artin	PROPN
ejpam-3182	311	2	primitive	primitive	ADJ
ejpam-3182	311	3	root	root	NOUN
ejpam-3182	311	4	conjecture	conjecture	NOUN
ejpam-3182	311	5	for	for	ADP
ejpam-3182	311	6	quadratic	quadratic	ADJ
ejpam-3182	311	7	fields	field	NOUN
ejpam-3182	311	8	.	.	PUNCT
ejpam-3182	312	1	j.	j.	PROPN
ejpam-3182	312	2	theory	theory	PROPN
ejpam-3182	312	3	nombres	nombre	NOUN
ejpam-3182	312	4	bordeaux	bordeaux	PROPN
ejpam-3182	312	5	,	,	PUNCT
ejpam-3182	312	6	14	14	NUM
ejpam-3182	312	7	,	,	PUNCT
ejpam-3182	312	8	(	(	PUNCT
ejpam-3182	312	9	2002	2002	NUM
ejpam-3182	312	10	)	)	PUNCT
ejpam-3182	312	11	,	,	PUNCT
ejpam-3182	312	12	no	no	INTJ
ejpam-3182	312	13	.	.	NOUN
ejpam-3182	312	14	1	1	NUM
ejpam-3182	312	15	,	,	PUNCT
ejpam-3182	312	16	287	287	NUM
ejpam-3182	312	17	-	-	SYM
ejpam-3182	312	18	324	324	NUM
ejpam-3182	312	19	.	.	PUNCT
ejpam-3182	313	1	[	[	X
ejpam-3182	313	2	30	30	NUM
ejpam-3182	313	3	]	]	PUNCT
ejpam-3182	313	4	rose	rise	VERB
ejpam-3182	313	5	,	,	PUNCT
ejpam-3182	313	6	h.	h.	PROPN
ejpam-3182	313	7	e.	e.	PROPN
ejpam-3182	314	1	a	a	DET
ejpam-3182	314	2	course	course	NOUN
ejpam-3182	314	3	in	in	ADP
ejpam-3182	314	4	number	number	NOUN
ejpam-3182	314	5	theory	theory	NOUN
ejpam-3182	314	6	.	.	PUNCT
ejpam-3182	315	1	second	second	ADJ
ejpam-3182	315	2	edition	edition	NOUN
ejpam-3182	315	3	.	.	PUNCT
ejpam-3182	316	1	oxford	oxford	PROPN
ejpam-3182	316	2	science	science	PROPN
ejpam-3182	316	3	publications	publication	NOUN
ejpam-3182	316	4	.	.	PUNCT
ejpam-3182	317	1	the	the	DET
ejpam-3182	317	2	clarendon	clarendon	PROPN
ejpam-3182	317	3	press	press	NOUN
ejpam-3182	317	4	,	,	PUNCT
ejpam-3182	317	5	oxford	oxford	PROPN
ejpam-3182	317	6	university	university	PROPN
ejpam-3182	317	7	press	press	NOUN
ejpam-3182	317	8	,	,	PUNCT
ejpam-3182	317	9	new	new	PROPN
ejpam-3182	317	10	york	york	PROPN
ejpam-3182	317	11	,	,	PUNCT
ejpam-3182	317	12	1994	1994	NUM
ejpam-3182	317	13	.	.	PUNCT
ejpam-3182	318	1	[	[	X
ejpam-3182	318	2	31	31	NUM
ejpam-3182	318	3	]	]	PUNCT
ejpam-3182	318	4	ribenboim	ribenboim	NOUN
ejpam-3182	318	5	,	,	PUNCT
ejpam-3182	318	6	paulo	paulo	PROPN
ejpam-3182	318	7	,	,	PUNCT
ejpam-3182	318	8	the	the	DET
ejpam-3182	318	9	new	new	ADJ
ejpam-3182	318	10	book	book	NOUN
ejpam-3182	318	11	of	of	ADP
ejpam-3182	318	12	prime	prime	ADJ
ejpam-3182	318	13	number	number	NOUN
ejpam-3182	318	14	records	record	NOUN
ejpam-3182	318	15	,	,	PUNCT
ejpam-3182	318	16	berlin	berlin	PROPN
ejpam-3182	318	17	,	,	PUNCT
ejpam-3182	318	18	new	new	PROPN
ejpam-3182	318	19	york	york	PROPN
ejpam-3182	318	20	:	:	PUNCT
ejpam-3182	318	21	springer	springer	NOUN
ejpam-3182	318	22	-	-	PUNCT
ejpam-3182	318	23	verlag	verlag	PROPN
ejpam-3182	318	24	,	,	PUNCT
ejpam-3182	318	25	1996	1996	NUM
ejpam-3182	318	26	.	.	PUNCT
ejpam-3182	319	1	[	[	X
ejpam-3182	319	2	32	32	NUM
ejpam-3182	319	3	]	]	SYM
ejpam-3182	319	4	stevenhagen	stevenhagen	PROPN
ejpam-3182	319	5	,	,	PUNCT
ejpam-3182	319	6	peter	peter	PROPN
ejpam-3182	319	7	.	.	PUNCT
ejpam-3182	320	1	the	the	DET
ejpam-3182	320	2	correction	correction	NOUN
ejpam-3182	320	3	factor	factor	NOUN
ejpam-3182	320	4	in	in	ADP
ejpam-3182	320	5	artin	artin	PROPN
ejpam-3182	320	6	’s	’s	PART
ejpam-3182	320	7	primitive	primitive	ADJ
ejpam-3182	320	8	root	root	NOUN
ejpam-3182	320	9	conjecture	conjecture	NOUN
ejpam-3182	320	10	.	.	PUNCT
ejpam-3182	321	1	les	les	PROPN
ejpam-3182	321	2	xxii	xxii	PROPN
ejpam-3182	321	3	emes	emes	PROPN
ejpam-3182	321	4	journees	journees	PROPN
ejpam-3182	321	5	arithmetiques	arithmetiques	PROPN
ejpam-3182	321	6	(	(	PUNCT
ejpam-3182	321	7	lille	lille	PROPN
ejpam-3182	321	8	,	,	PUNCT
ejpam-3182	321	9	2001	2001	NUM
ejpam-3182	321	10	)	)	PUNCT
ejpam-3182	321	11	.	.	PUNCT
ejpam-3182	322	1	j.	j.	PROPN
ejpam-3182	322	2	theor	theor	PROPN
ejpam-3182	322	3	.	.	PUNCT
ejpam-3182	323	1	nombres	nombre	NOUN
ejpam-3182	323	2	bordeaux	bordeaux	PROPN
ejpam-3182	323	3	15	15	NUM
ejpam-3182	323	4	(	(	PUNCT
ejpam-3182	323	5	2003	2003	NUM
ejpam-3182	323	6	)	)	PUNCT
ejpam-3182	323	7	,	,	PUNCT
ejpam-3182	323	8	no	no	INTJ
ejpam-3182	323	9	.	.	NOUN
ejpam-3182	323	10	1	1	NUM
ejpam-3182	323	11	,	,	PUNCT
ejpam-3182	323	12	383	383	NUM
ejpam-3182	323	13	-	-	SYM
ejpam-3182	323	14	391	391	NUM
ejpam-3182	323	15	.	.	PUNCT
ejpam-3182	324	1	[	[	X
ejpam-3182	324	2	33	33	NUM
ejpam-3182	324	3	]	]	PUNCT
ejpam-3182	324	4	g.	g.	PROPN
ejpam-3182	324	5	tenenbaum	tenenbaum	PROPN
ejpam-3182	324	6	,	,	PUNCT
ejpam-3182	324	7	introduction	introduction	NOUN
ejpam-3182	324	8	to	to	ADP
ejpam-3182	324	9	analytic	analytic	ADJ
ejpam-3182	324	10	and	and	CCONJ
ejpam-3182	324	11	probabilistic	probabilistic	ADJ
ejpam-3182	324	12	number	number	NOUN
ejpam-3182	324	13	theory	theory	NOUN
ejpam-3182	324	14	,	,	PUNCT
ejpam-3182	324	15	cambridge	cambridge	PROPN
ejpam-3182	324	16	studies	study	NOUN
ejpam-3182	324	17	in	in	ADP
ejpam-3182	324	18	advanced	advanced	ADJ
ejpam-3182	324	19	mathematics	mathematic	NOUN
ejpam-3182	324	20	46	46	NUM
ejpam-3182	324	21	,	,	PUNCT
ejpam-3182	324	22	cambridge	cambridge	PROPN
ejpam-3182	324	23	university	university	PROPN
ejpam-3182	324	24	press	press	PROPN
ejpam-3182	324	25	,	,	PUNCT
ejpam-3182	324	26	cambridge	cambridge	PROPN
ejpam-3182	324	27	,	,	PUNCT
ejpam-3182	324	28	1995	1995	NUM
ejpam-3182	324	29	.	.	PUNCT
ejpam-3182	325	1	[	[	X
ejpam-3182	325	2	34	34	NUM
ejpam-3182	325	3	]	]	PUNCT
ejpam-3182	325	4	stephens	stephen	NOUN
ejpam-3182	325	5	,	,	PUNCT
ejpam-3182	325	6	p.	p.	PROPN
ejpam-3182	325	7	j.	j.	PROPN
ejpam-3182	326	1	an	an	DET
ejpam-3182	326	2	average	average	ADJ
ejpam-3182	326	3	result	result	NOUN
ejpam-3182	326	4	for	for	ADP
ejpam-3182	326	5	artin	artin	PROPN
ejpam-3182	326	6	conjecture	conjecture	NOUN
ejpam-3182	326	7	.	.	PUNCT
ejpam-3182	327	1	mathematika	mathematika	NOUN
ejpam-3182	327	2	16	16	NUM
ejpam-3182	327	3	,	,	PUNCT
ejpam-3182	327	4	(	(	PUNCT
ejpam-3182	327	5	1969	1969	NUM
ejpam-3182	327	6	)	)	PUNCT
ejpam-3182	327	7	,	,	PUNCT
ejpam-3182	327	8	178	178	NUM
ejpam-3182	327	9	-	-	SYM
ejpam-3182	327	10	188	188	NUM
ejpam-3182	327	11	.	.	PUNCT
ejpam-3182	328	1	[	[	X
ejpam-3182	328	2	35	35	NUM
ejpam-3182	328	3	]	]	X
ejpam-3182	328	4	vaughan	vaughan	PROPN
ejpam-3182	328	5	,	,	PUNCT
ejpam-3182	328	6	r.	r.	PROPN
ejpam-3182	328	7	c.	c.	PROPN
ejpam-3182	328	8	some	some	DET
ejpam-3182	328	9	applications	application	NOUN
ejpam-3182	328	10	of	of	ADP
ejpam-3182	328	11	montgomery	montgomery	PROPN
ejpam-3182	328	12	’s	’s	PART
ejpam-3182	328	13	sieve	sieve	NOUN
ejpam-3182	328	14	.	.	PUNCT
ejpam-3182	329	1	j.	j.	PROPN
ejpam-3182	329	2	number	number	PROPN
ejpam-3182	329	3	theory	theory	NOUN
ejpam-3182	329	4	5	5	NUM
ejpam-3182	329	5	(	(	PUNCT
ejpam-3182	329	6	1973	1973	NUM
ejpam-3182	329	7	)	)	PUNCT
ejpam-3182	329	8	,	,	PUNCT
ejpam-3182	329	9	64	64	NUM
ejpam-3182	329	10	-	-	SYM
ejpam-3182	329	11	79	79	NUM
ejpam-3182	329	12	.	.	PUNCT
ejpam-3182	330	1	[	[	X
ejpam-3182	330	2	36	36	NUM
ejpam-3182	330	3	]	]	X
ejpam-3182	330	4	e.	e.	PROPN
ejpam-3182	330	5	wirsing	wirsing	PROPN
ejpam-3182	330	6	,	,	PUNCT
ejpam-3182	330	7	das	das	PROPN
ejpam-3182	330	8	asymptotische	asymptotische	PROPN
ejpam-3182	330	9	verhalten	verhalten	VERB
ejpam-3182	330	10	von	von	PROPN
ejpam-3182	330	11	summen	summen	PROPN
ejpam-3182	330	12	uber	uber	VERB
ejpam-3182	330	13	multiplikative	multiplikative	ADJ
ejpam-3182	330	14	funktionen	funktionen	NOUN
ejpam-3182	330	15	,	,	PUNCT
ejpam-3182	330	16	math	math	NOUN
ejpam-3182	330	17	.	.	PUNCT
ejpam-3182	331	1	ann	ann	PROPN
ejpam-3182	331	2	.	.	PUNCT
ejpam-3182	332	1	143	143	NUM
ejpam-3182	332	2	(	(	PUNCT
ejpam-3182	332	3	1961	1961	NUM
ejpam-3182	332	4	)	)	PUNCT
ejpam-3182	332	5	75	75	NUM
ejpam-3182	332	6	-	-	SYM
ejpam-3182	332	7	102	102	NUM
ejpam-3182	332	8	.	.	PUNCT
ejpam-3182	333	1	[	[	X
ejpam-3182	333	2	37	37	NUM
ejpam-3182	333	3	]	]	X
ejpam-3182	333	4	williams	williams	PROPN
ejpam-3182	333	5	,	,	PUNCT
ejpam-3182	333	6	kenneth	kenneth	PROPN
ejpam-3182	333	7	s.	s.	PROPN
ejpam-3182	333	8	note	note	PROPN
ejpam-3182	333	9	on	on	ADP
ejpam-3182	333	10	integers	integer	NOUN
ejpam-3182	333	11	representable	representable	ADJ
ejpam-3182	333	12	by	by	ADP
ejpam-3182	333	13	binary	binary	ADJ
ejpam-3182	333	14	quadratic	quadratic	ADJ
ejpam-3182	333	15	forms	form	NOUN
ejpam-3182	333	16	.	.	PUNCT
ejpam-3182	334	1	canad	canad	PROPN
ejpam-3182	334	2	.	.	PUNCT
ejpam-3182	335	1	math	math	NOUN
ejpam-3182	335	2	.	.	PUNCT
ejpam-3182	336	1	bull	bull	NOUN
ejpam-3182	336	2	.	.	PUNCT
ejpam-3182	337	1	18	18	NUM
ejpam-3182	337	2	(	(	PUNCT
ejpam-3182	337	3	1975	1975	NUM
ejpam-3182	337	4	)	)	PUNCT
ejpam-3182	337	5	,	,	PUNCT
ejpam-3182	337	6	no	no	INTJ
ejpam-3182	337	7	.	.	NOUN
ejpam-3182	337	8	1	1	NUM
ejpam-3182	337	9	,	,	PUNCT
ejpam-3182	337	10	123	123	NUM
ejpam-3182	337	11	-	-	SYM
ejpam-3182	337	12	125	125	NUM
ejpam-3182	337	13	.	.	PUNCT