id	sid	tid	token	lemma	pos
ejpam-109	1	1	european	european	PROPN
ejpam-109	1	2	journal	journal	PROPN
ejpam-109	1	3	of	of	ADP
ejpam-109	1	4	pure	pure	ADJ
ejpam-109	1	5	and	and	CCONJ
ejpam-109	1	6	applied	apply	VERB
ejpam-109	1	7	mathematics	mathematic	NOUN
ejpam-109	1	8	vol	vol	NOUN
ejpam-109	1	9	.	.	PROPN
ejpam-109	2	1	1	1	NUM
ejpam-109	2	2	,	,	PUNCT
ejpam-109	2	3	no	no	INTJ
ejpam-109	2	4	.	.	NOUN
ejpam-109	2	5	4	4	NUM
ejpam-109	2	6	,	,	PUNCT
ejpam-109	2	7	2008	2008	NUM
ejpam-109	2	8	,	,	PUNCT
ejpam-109	2	9	(	(	PUNCT
ejpam-109	2	10	3	3	NUM
ejpam-109	2	11	-	-	SYM
ejpam-109	2	12	21	21	NUM
ejpam-109	2	13	)	)	PUNCT
ejpam-109	2	14	issn	issn	PROPN
ejpam-109	2	15	1307	1307	NUM
ejpam-109	2	16	-	-	SYM
ejpam-109	2	17	5543	5543	NUM
ejpam-109	2	18	–	–	PUNCT
ejpam-109	2	19	www.ejpam.com	www.ejpam.com	X
ejpam-109	2	20	voronoï	voronoï	ADJ
ejpam-109	2	21	type	type	NOUN
ejpam-109	2	22	congruences	congruence	NOUN
ejpam-109	2	23	and	and	CCONJ
ejpam-109	2	24	its	its	PRON
ejpam-109	2	25	applications	application	NOUN
ejpam-109	2	26	takashi	takashi	PROPN
ejpam-109	2	27	agoh∗,†	agoh∗,†	PROPN
ejpam-109	2	28	department	department	PROPN
ejpam-109	2	29	of	of	ADP
ejpam-109	2	30	mathematics	mathematics	PROPN
ejpam-109	2	31	,	,	PUNCT
ejpam-109	2	32	tokyo	tokyo	PROPN
ejpam-109	2	33	university	university	PROPN
ejpam-109	2	34	of	of	ADP
ejpam-109	2	35	science	science	PROPN
ejpam-109	2	36	,	,	PUNCT
ejpam-109	2	37	noda	noda	PROPN
ejpam-109	2	38	,	,	PUNCT
ejpam-109	2	39	chiba	chiba	PROPN
ejpam-109	2	40	278	278	NUM
ejpam-109	2	41	-	-	SYM
ejpam-109	2	42	8510	8510	NUM
ejpam-109	2	43	,	,	PUNCT
ejpam-109	2	44	japan	japan	PROPN
ejpam-109	2	45	abstract	abstract	PROPN
ejpam-109	2	46	.	.	PUNCT
ejpam-109	3	1	in	in	ADP
ejpam-109	3	2	this	this	DET
ejpam-109	3	3	paper	paper	NOUN
ejpam-109	3	4	,	,	PUNCT
ejpam-109	3	5	we	we	PRON
ejpam-109	3	6	will	will	AUX
ejpam-109	3	7	first	first	ADV
ejpam-109	3	8	deduce	deduce	VERB
ejpam-109	3	9	voronoï	voronoï	ADJ
ejpam-109	3	10	type	type	NOUN
ejpam-109	3	11	congruences	congruence	NOUN
ejpam-109	3	12	for	for	ADP
ejpam-109	3	13	bernoulli	bernoulli	NOUN
ejpam-109	3	14	numbers	number	NOUN
ejpam-109	3	15	in	in	ADP
ejpam-109	3	16	the	the	DET
ejpam-109	3	17	even	even	ADV
ejpam-109	3	18	suffix	suffix	ADJ
ejpam-109	3	19	notation	notation	NOUN
ejpam-109	3	20	.	.	PUNCT
ejpam-109	4	1	continuously	continuously	ADV
ejpam-109	4	2	,	,	PUNCT
ejpam-109	4	3	we	we	PRON
ejpam-109	4	4	will	will	AUX
ejpam-109	4	5	apply	apply	VERB
ejpam-109	4	6	them	they	PRON
ejpam-109	4	7	to	to	PART
ejpam-109	4	8	extend	extend	VERB
ejpam-109	4	9	very	very	ADV
ejpam-109	4	10	important	important	ADJ
ejpam-109	4	11	arithmetic	arithmetic	ADJ
ejpam-109	4	12	properties	property	NOUN
ejpam-109	4	13	(	(	PUNCT
ejpam-109	4	14	such	such	ADJ
ejpam-109	4	15	as	as	ADP
ejpam-109	4	16	von	von	PROPN
ejpam-109	4	17	staudt	staudt	PROPN
ejpam-109	4	18	-	-	PUNCT
ejpam-109	4	19	clausen	clausen	PROPN
ejpam-109	4	20	’s	’s	PART
ejpam-109	4	21	and	and	CCONJ
ejpam-109	4	22	kummer	kummer	PROPN
ejpam-109	4	23	’s	’s	PART
ejpam-109	4	24	congruences	congruence	NOUN
ejpam-109	4	25	)	)	PUNCT
ejpam-109	4	26	of	of	ADP
ejpam-109	4	27	these	these	DET
ejpam-109	4	28	numbers	number	NOUN
ejpam-109	4	29	to	to	ADP
ejpam-109	4	30	more	more	ADV
ejpam-109	4	31	general	general	ADJ
ejpam-109	4	32	situation	situation	NOUN
ejpam-109	4	33	.	.	PUNCT
ejpam-109	5	1	ams	am	NOUN
ejpam-109	5	2	subject	subject	ADJ
ejpam-109	5	3	classifications	classification	NOUN
ejpam-109	5	4	:	:	PUNCT
ejpam-109	5	5	11a07	11a07	NUM
ejpam-109	5	6	,	,	PUNCT
ejpam-109	5	7	11b68	11b68	NUM
ejpam-109	5	8	,	,	PUNCT
ejpam-109	5	9	11y1	11y1	NUM
ejpam-109	5	10	key	key	ADJ
ejpam-109	5	11	words	word	NOUN
ejpam-109	5	12	:	:	PUNCT
ejpam-109	5	13	bernoulli	bernoulli	NOUN
ejpam-109	5	14	numbers	number	NOUN
ejpam-109	5	15	,	,	PUNCT
ejpam-109	5	16	von	von	PROPN
ejpam-109	5	17	staudt	staudt	PROPN
ejpam-109	5	18	-	-	PUNCT
ejpam-109	5	19	clausen	clausen	PROPN
ejpam-109	5	20	’s	’s	PART
ejpam-109	5	21	theorem	theorem	PROPN
ejpam-109	5	22	,	,	PUNCT
ejpam-109	5	23	voronoï	voronoï	PROPN
ejpam-109	5	24	’s	’s	PART
ejpam-109	5	25	congruence	congruence	NOUN
ejpam-109	5	26	,	,	PUNCT
ejpam-109	5	27	kummer	kummer	PROPN
ejpam-109	5	28	’s	’s	PART
ejpam-109	5	29	congruence	congruence	NOUN
ejpam-109	5	30	,	,	PUNCT
ejpam-109	5	31	irregular	irregular	ADJ
ejpam-109	5	32	primes	prime	NOUN
ejpam-109	5	33	1	1	NUM
ejpam-109	5	34	.	.	X
ejpam-109	6	1	introduction	introduction	NOUN
ejpam-109	6	2	the	the	DET
ejpam-109	6	3	bernoulli	bernoulli	PROPN
ejpam-109	6	4	numbers	number	NOUN
ejpam-109	6	5	bm	bm	PROPN
ejpam-109	6	6	(	(	PUNCT
ejpam-109	6	7	m≥	m≥	NOUN
ejpam-109	6	8	0	0	NUM
ejpam-109	6	9	)	)	PUNCT
ejpam-109	6	10	are	be	AUX
ejpam-109	6	11	defined	define	VERB
ejpam-109	6	12	by	by	ADP
ejpam-109	6	13	the	the	DET
ejpam-109	6	14	taylor	taylor	PROPN
ejpam-109	6	15	expansion	expansion	PROPN
ejpam-109	6	16	t	t	X
ejpam-109	6	17	et	et	NOUN
ejpam-109	7	1	−	−	NOUN
ejpam-109	7	2	1	1	NUM
ejpam-109	7	3	=	=	SYM
ejpam-109	7	4	∞	∞	NUM
ejpam-109	7	5	∑	∑	PROPN
ejpam-109	7	6	m=0	m=0	PROPN
ejpam-109	7	7	bm	bm	PROPN
ejpam-109	7	8	m	m	PROPN
ejpam-109	7	9	!	!	PUNCT
ejpam-109	8	1	tm	tm	PROPN
ejpam-109	8	2	,	,	PUNCT
ejpam-109	8	3	|t|	|t|	VERB
ejpam-109	8	4	<	<	X
ejpam-109	8	5	2π	2π	NOUN
ejpam-109	8	6	.	.	PUNCT
ejpam-109	9	1	these	these	DET
ejpam-109	9	2	numbers	number	NOUN
ejpam-109	9	3	may	may	AUX
ejpam-109	9	4	be	be	AUX
ejpam-109	9	5	also	also	ADV
ejpam-109	9	6	defined	define	VERB
ejpam-109	9	7	by	by	ADP
ejpam-109	9	8	the	the	DET
ejpam-109	9	9	recurrence	recurrence	NOUN
ejpam-109	9	10	relation	relation	PROPN
ejpam-109	9	11	bm	bm	PROPN
ejpam-109	10	1	=	=	NOUN
ejpam-109	10	2	−	−	PROPN
ejpam-109	10	3	1	1	NUM
ejpam-109	10	4	m+	m+	NOUN
ejpam-109	10	5	1	1	NUM
ejpam-109	10	6	m−1	m−1	PROPN
ejpam-109	10	7	∑	∑	PUNCT
ejpam-109	10	8	i=0	i=0	PROPN
ejpam-109	10	9	�	�	PROPN
ejpam-109	11	1	m+	m+	NUM
ejpam-109	11	2	1	1	NUM
ejpam-109	12	1	i	i	PRON
ejpam-109	12	2	�	�	PROPN
ejpam-109	12	3	bi	bi	NOUN
ejpam-109	12	4	,	,	PUNCT
ejpam-109	12	5	b0	b0	NOUN
ejpam-109	12	6	=	=	SYM
ejpam-109	12	7	1	1	X
ejpam-109	12	8	.	.	PUNCT
ejpam-109	13	1	it	it	PRON
ejpam-109	13	2	is	be	AUX
ejpam-109	13	3	easily	easily	ADV
ejpam-109	13	4	seen	see	VERB
ejpam-109	13	5	that	that	SCONJ
ejpam-109	13	6	b2m+1	b2m+1	PROPN
ejpam-109	13	7	=	=	PUNCT
ejpam-109	13	8	0	0	PUNCT
ejpam-109	13	9	and	and	CCONJ
ejpam-109	13	10	(	(	PUNCT
ejpam-109	13	11	−1)m−1b2	−1)m−1b2	PROPN
ejpam-109	13	12	m	m	PROPN
ejpam-109	13	13	>	>	X
ejpam-109	13	14	0	0	PUNCT
ejpam-109	13	15	for	for	ADP
ejpam-109	13	16	all	all	DET
ejpam-109	13	17	m≥	m≥	ADJ
ejpam-109	13	18	1	1	NUM
ejpam-109	13	19	.	.	PUNCT
ejpam-109	13	20	using	use	VERB
ejpam-109	13	21	the	the	DET
ejpam-109	13	22	stirling	stirling	NOUN
ejpam-109	13	23	formula	formula	NOUN
ejpam-109	13	24	n	n	CCONJ
ejpam-109	13	25	!	!	PUNCT
ejpam-109	13	26	∼	∼	NOUN
ejpam-109	13	27	(	(	PUNCT
ejpam-109	13	28	n	n	CCONJ
ejpam-109	13	29	/	/	SYM
ejpam-109	13	30	e)n	e)n	NOUN
ejpam-109	13	31	p	p	NOUN
ejpam-109	13	32	2πn	2πn	NOUN
ejpam-109	13	33	,	,	PUNCT
ejpam-109	13	34	we	we	PRON
ejpam-109	13	35	see	see	VERB
ejpam-109	13	36	asymptotically	asymptotically	ADV
ejpam-109	13	37	|b2m|	|b2m|	NOUN
ejpam-109	13	38	∼	∼	NOUN
ejpam-109	13	39	4	4	NUM
ejpam-109	13	40	p	p	NOUN
ejpam-109	13	41	πm(m	πm(m	NOUN
ejpam-109	13	42	/	/	SYM
ejpam-109	13	43	πe)2	πe)2	NOUN
ejpam-109	13	44	m.	m.	NOUN
ejpam-109	13	45	further	far	ADV
ejpam-109	13	46	,	,	PUNCT
ejpam-109	13	47	if	if	SCONJ
ejpam-109	13	48	3	3	NUM
ejpam-109	13	49	≤	≤	NUM
ejpam-109	13	50	m	m	VERB
ejpam-109	13	51	<	<	NOUN
ejpam-109	13	52	n	n	CCONJ
ejpam-109	13	53	,	,	PUNCT
ejpam-109	13	54	then	then	ADV
ejpam-109	13	55	|b2m/2m|	|b2m/2m|	VERB
ejpam-109	13	56	<	<	X
ejpam-109	13	57	|b2n/2n|	|b2n/2n|	NOUN
ejpam-109	13	58	,	,	PUNCT
ejpam-109	13	59	and	and	CCONJ
ejpam-109	13	60	also	also	ADV
ejpam-109	13	61	|b2m|/(2m)λ→∞	|b2m|/(2m)λ→∞	PROPN
ejpam-109	13	62	as	as	ADP
ejpam-109	13	63	m→∞	m→∞	NOUN
ejpam-109	13	64	for	for	ADP
ejpam-109	13	65	every	every	DET
ejpam-109	13	66	λ≥	λ≥	PROPN
ejpam-109	13	67	1	1	NUM
ejpam-109	13	68	.	.	PUNCT
ejpam-109	14	1	let	let	VERB
ejpam-109	14	2	m	m	PRON
ejpam-109	14	3	≥	≥	NOUN
ejpam-109	14	4	2	2	NUM
ejpam-109	14	5	be	be	AUX
ejpam-109	14	6	even	even	ADV
ejpam-109	14	7	and	and	CCONJ
ejpam-109	14	8	n	n	PRON
ejpam-109	14	9	≥	≥	NOUN
ejpam-109	14	10	1	1	NUM
ejpam-109	14	11	.	.	PUNCT
ejpam-109	15	1	if	if	SCONJ
ejpam-109	15	2	we	we	PRON
ejpam-109	15	3	write	write	VERB
ejpam-109	15	4	bm	bm	PROPN
ejpam-109	15	5	=	=	PUNCT
ejpam-109	15	6	nm	nm	PROPN
ejpam-109	15	7	/	/	SYM
ejpam-109	15	8	dm	dm	PROPN
ejpam-109	15	9	(	(	PUNCT
ejpam-109	15	10	nm	nm	PROPN
ejpam-109	15	11	,	,	PUNCT
ejpam-109	15	12	dm	dm	PROPN
ejpam-109	15	13	∈	∈	PROPN
ejpam-109	15	14	z	z	PROPN
ejpam-109	15	15	,	,	PUNCT
ejpam-109	15	16	dm	dm	INTJ
ejpam-109	15	17	>	>	X
ejpam-109	15	18	0	0	NUM
ejpam-109	15	19	)	)	PUNCT
ejpam-109	15	20	in	in	ADP
ejpam-109	15	21	lowest	low	ADJ
ejpam-109	15	22	terms	term	NOUN
ejpam-109	15	23	,	,	PUNCT
ejpam-109	15	24	then	then	ADV
ejpam-109	15	25	the	the	DET
ejpam-109	15	26	voronoï	voronoï	ADJ
ejpam-109	15	27	congruence	congruence	NOUN
ejpam-109	15	28	can	can	AUX
ejpam-109	15	29	be	be	AUX
ejpam-109	15	30	stated	state	VERB
ejpam-109	15	31	as	as	ADP
ejpam-109	15	32	(	(	PUNCT
ejpam-109	15	33	am−	am−	NUM
ejpam-109	15	34	1)nm	1)nm	PROPN
ejpam-109	15	35	≡	≡	PROPN
ejpam-109	15	36	mdm	mdm	PROPN
ejpam-109	15	37	n−1	n−1	PROPN
ejpam-109	15	38	∑	∑	PUNCT
ejpam-109	15	39	j=1	j=1	PROPN
ejpam-109	15	40	(	(	PUNCT
ejpam-109	15	41	a	a	DET
ejpam-109	15	42	j)m−1	j)m−1	PROPN
ejpam-109	15	43	�	�	PROPN
ejpam-109	15	44	a	a	DET
ejpam-109	15	45	j	j	PROPN
ejpam-109	15	46	n	n	PRON
ejpam-109	15	47	�	�	PROPN
ejpam-109	15	48	(	(	PUNCT
ejpam-109	15	49	mod	mod	PROPN
ejpam-109	15	50	n	n	CCONJ
ejpam-109	15	51	)	)	PUNCT
ejpam-109	15	52	,	,	PUNCT
ejpam-109	15	53	(	(	PUNCT
ejpam-109	15	54	1.1	1.1	NUM
ejpam-109	15	55	)	)	PUNCT
ejpam-109	15	56	∗email	∗email	NOUN
ejpam-109	15	57	address	address	NOUN
ejpam-109	15	58	:	:	PUNCT
ejpam-109	16	1	agoh_takashi@ma.noda.sut.ac.jp	agoh_takashi@ma.noda.sut.ac.jp	PROPN
ejpam-109	16	2	(	(	PUNCT
ejpam-109	16	3	t.	t.	NOUN
ejpam-109	16	4	agoh	agoh	PROPN
ejpam-109	16	5	)	)	PUNCT
ejpam-109	16	6	†the	†the	DET
ejpam-109	16	7	author	author	NOUN
ejpam-109	16	8	was	be	AUX
ejpam-109	16	9	supported	support	VERB
ejpam-109	16	10	in	in	ADP
ejpam-109	16	11	part	part	NOUN
ejpam-109	16	12	by	by	ADP
ejpam-109	16	13	a	a	DET
ejpam-109	16	14	grant	grant	NOUN
ejpam-109	16	15	of	of	ADP
ejpam-109	16	16	the	the	DET
ejpam-109	16	17	ministry	ministry	PROPN
ejpam-109	16	18	of	of	ADP
ejpam-109	16	19	education	education	PROPN
ejpam-109	16	20	,	,	PUNCT
ejpam-109	16	21	science	science	NOUN
ejpam-109	16	22	and	and	CCONJ
ejpam-109	16	23	culture	culture	NOUN
ejpam-109	16	24	of	of	ADP
ejpam-109	16	25	japan	japan	PROPN
ejpam-109	16	26	.	.	PUNCT
ejpam-109	17	1	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-109	18	1	3	3	NUM
ejpam-109	18	2	c	c	X
ejpam-109	18	3	©	©	PROPN
ejpam-109	18	4	2008	2008	NUM
ejpam-109	18	5	ejpam	ejpam	VERB
ejpam-109	18	6	all	all	DET
ejpam-109	18	7	rights	right	NOUN
ejpam-109	18	8	reserved	reserve	VERB
ejpam-109	18	9	.	.	PUNCT
ejpam-109	19	1	t.	t.	PROPN
ejpam-109	19	2	agoh	agoh	PROPN
ejpam-109	19	3	/	/	SYM
ejpam-109	19	4	eur	eur	PROPN
ejpam-109	19	5	.	.	PUNCT
ejpam-109	20	1	j.	j.	PROPN
ejpam-109	20	2	pure	pure	PROPN
ejpam-109	20	3	appl	appl	PROPN
ejpam-109	20	4	.	.	PROPN
ejpam-109	20	5	math	math	PROPN
ejpam-109	20	6	,	,	PUNCT
ejpam-109	20	7	1	1	NUM
ejpam-109	20	8	(	(	PUNCT
ejpam-109	20	9	2008	2008	NUM
ejpam-109	20	10	)	)	PUNCT
ejpam-109	20	11	,	,	PUNCT
ejpam-109	20	12	(	(	PUNCT
ejpam-109	20	13	3	3	NUM
ejpam-109	20	14	-	-	SYM
ejpam-109	20	15	21	21	NUM
ejpam-109	20	16	)	)	PUNCT
ejpam-109	20	17	4	4	NUM
ejpam-109	20	18	where	where	SCONJ
ejpam-109	20	19	a	a	PRON
ejpam-109	20	20	is	be	AUX
ejpam-109	20	21	a	a	DET
ejpam-109	20	22	positive	positive	ADJ
ejpam-109	20	23	integer	integer	NOUN
ejpam-109	20	24	with	with	ADP
ejpam-109	20	25	(	(	PUNCT
ejpam-109	20	26	a	a	PRON
ejpam-109	20	27	,	,	PUNCT
ejpam-109	20	28	n	n	CCONJ
ejpam-109	20	29	)	)	PUNCT
ejpam-109	21	1	=	=	SYM
ejpam-109	21	2	1	1	NUM
ejpam-109	21	3	and	and	CCONJ
ejpam-109	21	4	[	[	X
ejpam-109	21	5	a	a	DET
ejpam-109	21	6	j	j	PROPN
ejpam-109	21	7	/	/	SYM
ejpam-109	21	8	n	n	CCONJ
ejpam-109	21	9	]	]	PUNCT
ejpam-109	21	10	is	be	AUX
ejpam-109	21	11	the	the	DET
ejpam-109	21	12	greatest	great	ADJ
ejpam-109	21	13	integer	integer	NOUN
ejpam-109	21	14	≤	≤	NUM
ejpam-109	21	15	a	a	DET
ejpam-109	21	16	j	j	PROPN
ejpam-109	21	17	/	/	SYM
ejpam-109	21	18	n.	n.	PROPN
ejpam-109	21	19	applying	apply	VERB
ejpam-109	21	20	the	the	DET
ejpam-109	21	21	voronoï	voronoï	NOUN
ejpam-109	21	22	and	and	CCONJ
ejpam-109	21	23	his	his	PRON
ejpam-109	21	24	type	type	NOUN
ejpam-109	21	25	congruences	congruence	NOUN
ejpam-109	21	26	,	,	PUNCT
ejpam-109	21	27	we	we	PRON
ejpam-109	21	28	are	be	AUX
ejpam-109	21	29	able	able	ADJ
ejpam-109	21	30	to	to	PART
ejpam-109	21	31	deduce	deduce	VERB
ejpam-109	21	32	various	various	ADJ
ejpam-109	21	33	arithmetical	arithmetical	ADJ
ejpam-109	21	34	properties	property	NOUN
ejpam-109	21	35	of	of	ADP
ejpam-109	21	36	bernoulli	bernoulli	NOUN
ejpam-109	21	37	numbers	number	NOUN
ejpam-109	21	38	.	.	PUNCT
ejpam-109	22	1	for	for	ADP
ejpam-109	22	2	surrounding	surround	VERB
ejpam-109	22	3	landscape	landscape	NOUN
ejpam-109	22	4	on	on	ADP
ejpam-109	22	5	these	these	DET
ejpam-109	22	6	congruences	congruence	NOUN
ejpam-109	22	7	,	,	PUNCT
ejpam-109	22	8	see	see	VERB
ejpam-109	22	9	porubský	porubský	ADJ
ejpam-109	22	10	’s	’s	PART
ejpam-109	22	11	expository	expository	ADJ
ejpam-109	22	12	article	article	NOUN
ejpam-109	22	13	[	[	X
ejpam-109	22	14	14	14	NUM
ejpam-109	22	15	]	]	PUNCT
ejpam-109	22	16	.	.	PUNCT
ejpam-109	23	1	the	the	DET
ejpam-109	23	2	prime	prime	ADJ
ejpam-109	23	3	factorization	factorization	NOUN
ejpam-109	23	4	of	of	ADP
ejpam-109	23	5	dm	dm	PRON
ejpam-109	23	6	can	can	AUX
ejpam-109	23	7	be	be	AUX
ejpam-109	23	8	explicitly	explicitly	ADV
ejpam-109	23	9	stated	state	VERB
ejpam-109	23	10	by	by	ADP
ejpam-109	23	11	the	the	DET
ejpam-109	23	12	von	von	PROPN
ejpam-109	23	13	staudt	staudt	PROPN
ejpam-109	23	14	-	-	PUNCT
ejpam-109	23	15	clausen	clausen	PROPN
ejpam-109	23	16	theorem	theorem	NOUN
ejpam-109	23	17	which	which	PRON
ejpam-109	23	18	asserts	assert	VERB
ejpam-109	23	19	that	that	SCONJ
ejpam-109	23	20	p−	p−	NOUN
ejpam-109	23	21	1	1	NUM
ejpam-109	24	1	|	|	ADV
ejpam-109	24	2	m	m	VERB
ejpam-109	24	3	if	if	SCONJ
ejpam-109	25	1	and	and	CCONJ
ejpam-109	25	2	only	only	ADV
ejpam-109	25	3	if	if	SCONJ
ejpam-109	25	4	p	p	PRON
ejpam-109	25	5	|	|	ADV
ejpam-109	25	6	dm	dm	X
ejpam-109	25	7	for	for	ADP
ejpam-109	25	8	a	a	DET
ejpam-109	25	9	prime	prime	ADJ
ejpam-109	25	10	p.	p.	NOUN
ejpam-109	25	11	further	far	ADV
ejpam-109	25	12	,	,	PUNCT
ejpam-109	25	13	we	we	PRON
ejpam-109	25	14	know	know	VERB
ejpam-109	25	15	that	that	SCONJ
ejpam-109	25	16	dm	dm	PROPN
ejpam-109	25	17	is	be	AUX
ejpam-109	25	18	square	square	ADV
ejpam-109	25	19	-	-	PUNCT
ejpam-109	25	20	free	free	ADJ
ejpam-109	25	21	.	.	PUNCT
ejpam-109	26	1	in	in	ADP
ejpam-109	26	2	contrast	contrast	NOUN
ejpam-109	26	3	with	with	ADP
ejpam-109	26	4	the	the	DET
ejpam-109	26	5	denominator	denominator	NOUN
ejpam-109	26	6	,	,	PUNCT
ejpam-109	26	7	much	much	ADV
ejpam-109	26	8	less	less	ADJ
ejpam-109	26	9	is	be	AUX
ejpam-109	26	10	known	know	VERB
ejpam-109	26	11	about	about	ADP
ejpam-109	26	12	prime	prime	ADJ
ejpam-109	26	13	divisors	divisor	NOUN
ejpam-109	26	14	of	of	ADP
ejpam-109	26	15	nm	nm	NOUN
ejpam-109	26	16	.	.	PUNCT
ejpam-109	27	1	one	one	NUM
ejpam-109	27	2	of	of	ADP
ejpam-109	27	3	conspicuous	conspicuous	ADJ
ejpam-109	27	4	facts	fact	NOUN
ejpam-109	27	5	is	be	AUX
ejpam-109	27	6	von	von	PROPN
ejpam-109	27	7	staudt	staudt	PROPN
ejpam-109	27	8	’s	’s	PART
ejpam-109	27	9	theorem	theorem	NOUN
ejpam-109	27	10	(	(	PUNCT
ejpam-109	27	11	although	although	SCONJ
ejpam-109	27	12	this	this	PRON
ejpam-109	27	13	is	be	AUX
ejpam-109	27	14	commonly	commonly	ADV
ejpam-109	27	15	called	call	VERB
ejpam-109	27	16	adams	adam	NOUN
ejpam-109	27	17	’	'	PUNCT
ejpam-109	27	18	theorem	theorem	NOUN
ejpam-109	27	19	)	)	PUNCT
ejpam-109	27	20	which	which	PRON
ejpam-109	27	21	mentions	mention	VERB
ejpam-109	27	22	that	that	SCONJ
ejpam-109	27	23	,	,	PUNCT
ejpam-109	27	24	for	for	ADP
ejpam-109	27	25	an	an	DET
ejpam-109	27	26	odd	odd	ADJ
ejpam-109	27	27	prime	prime	NOUN
ejpam-109	27	28	p	p	NOUN
ejpam-109	27	29	and	and	CCONJ
ejpam-109	27	30	an	an	DET
ejpam-109	27	31	even	even	ADV
ejpam-109	27	32	integer	integer	PROPN
ejpam-109	27	33	m	m	PROPN
ejpam-109	27	34	≥	≥	NOUN
ejpam-109	27	35	2	2	NUM
ejpam-109	27	36	,	,	PUNCT
ejpam-109	27	37	if	if	SCONJ
ejpam-109	27	38	p	p	NOUN
ejpam-109	27	39	−	−	PROPN
ejpam-109	27	40	1	1	NUM
ejpam-109	27	41	m	m	NOUN
ejpam-109	28	1	and	and	CCONJ
ejpam-109	28	2	pe	pe	INTJ
ejpam-109	29	1	|	|	ADV
ejpam-109	29	2	m	m	VERB
ejpam-109	29	3	(	(	PUNCT
ejpam-109	29	4	e	e	X
ejpam-109	29	5	≥	≥	NUM
ejpam-109	29	6	1	1	NUM
ejpam-109	29	7	)	)	PUNCT
ejpam-109	29	8	,	,	PUNCT
ejpam-109	29	9	then	then	ADV
ejpam-109	29	10	pe	pe	PROPN
ejpam-109	29	11	|	|	ADV
ejpam-109	29	12	nm	nm	VERB
ejpam-109	29	13	.	.	PUNCT
ejpam-109	30	1	if	if	SCONJ
ejpam-109	30	2	we	we	PRON
ejpam-109	30	3	take	take	VERB
ejpam-109	30	4	account	account	NOUN
ejpam-109	30	5	of	of	ADP
ejpam-109	30	6	this	this	DET
ejpam-109	30	7	result	result	NOUN
ejpam-109	30	8	,	,	PUNCT
ejpam-109	30	9	then	then	ADV
ejpam-109	30	10	the	the	DET
ejpam-109	30	11	question	question	NOUN
ejpam-109	30	12	how	how	SCONJ
ejpam-109	30	13	to	to	PART
ejpam-109	30	14	find	find	VERB
ejpam-109	30	15	prime	prime	ADJ
ejpam-109	30	16	divisors	divisor	NOUN
ejpam-109	30	17	of	of	ADP
ejpam-109	30	18	the	the	DET
ejpam-109	30	19	numerator	numerator	NOUN
ejpam-109	30	20	of	of	ADP
ejpam-109	30	21	bm	bm	PROPN
ejpam-109	30	22	/	/	SYM
ejpam-109	30	23	m	m	VERB
ejpam-109	30	24	necessarily	necessarily	ADV
ejpam-109	30	25	arises	arise	VERB
ejpam-109	30	26	.	.	PUNCT
ejpam-109	31	1	the	the	DET
ejpam-109	31	2	famous	famous	ADJ
ejpam-109	31	3	congruence	congruence	PROPN
ejpam-109	31	4	bm	bm	PROPN
ejpam-109	31	5	/	/	SYM
ejpam-109	31	6	m≡	m≡	NOUN
ejpam-109	31	7	bl	bl	PROPN
ejpam-109	31	8	/	/	SYM
ejpam-109	31	9	l	l	NOUN
ejpam-109	31	10	(	(	PUNCT
ejpam-109	31	11	mod	mod	PROPN
ejpam-109	31	12	p	p	X
ejpam-109	31	13	)	)	PUNCT
ejpam-109	31	14	(	(	PUNCT
ejpam-109	31	15	p	p	X
ejpam-109	31	16	a	a	DET
ejpam-109	31	17	prime	prime	ADJ
ejpam-109	31	18	≥	≥	NOUN
ejpam-109	31	19	5	5	NUM
ejpam-109	31	20	)	)	PUNCT
ejpam-109	31	21	for	for	ADP
ejpam-109	31	22	even	even	ADV
ejpam-109	31	23	integers	integer	NOUN
ejpam-109	31	24	m	m	PRON
ejpam-109	31	25	,	,	PUNCT
ejpam-109	31	26	l	l	PROPN
ejpam-109	31	27	≥	≥	NUM
ejpam-109	31	28	2	2	NUM
ejpam-109	31	29	such	such	ADJ
ejpam-109	31	30	that	that	SCONJ
ejpam-109	31	31	m	m	PROPN
ejpam-109	31	32	≡	≡	PROPN
ejpam-109	31	33	l	l	PROPN
ejpam-109	32	1	(	(	PUNCT
ejpam-109	32	2	mod	mod	PROPN
ejpam-109	32	3	p	p	NOUN
ejpam-109	32	4	−	−	PROPN
ejpam-109	32	5	1	1	NUM
ejpam-109	32	6	)	)	PUNCT
ejpam-109	32	7	and	and	CCONJ
ejpam-109	32	8	p	p	NOUN
ejpam-109	32	9	−	−	PROPN
ejpam-109	32	10	1	1	NUM
ejpam-109	32	11	m	m	NOUN
ejpam-109	32	12	is	be	AUX
ejpam-109	32	13	due	due	ADJ
ejpam-109	32	14	to	to	ADP
ejpam-109	32	15	von	von	PROPN
ejpam-109	32	16	staudt	staudt	PROPN
ejpam-109	32	17	and	and	CCONJ
ejpam-109	32	18	kummer	kummer	NOUN
ejpam-109	32	19	,	,	PUNCT
ejpam-109	32	20	and	and	CCONJ
ejpam-109	32	21	it	it	PRON
ejpam-109	32	22	shows	show	VERB
ejpam-109	32	23	that	that	SCONJ
ejpam-109	32	24	bm	bm	PROPN
ejpam-109	32	25	/	/	SYM
ejpam-109	32	26	m	m	PROPN
ejpam-109	32	27	has	have	VERB
ejpam-109	32	28	period	period	NOUN
ejpam-109	32	29	p	p	NOUN
ejpam-109	32	30	−	−	PROPN
ejpam-109	32	31	1	1	X
ejpam-109	32	32	.	.	PUNCT
ejpam-109	33	1	we	we	PRON
ejpam-109	33	2	can	can	AUX
ejpam-109	33	3	further	far	ADV
ejpam-109	33	4	state	state	VERB
ejpam-109	33	5	that	that	SCONJ
ejpam-109	33	6	if	if	SCONJ
ejpam-109	33	7	p	p	NOUN
ejpam-109	33	8	−	−	PROPN
ejpam-109	33	9	1	1	NUM
ejpam-109	33	10	m	m	NOUN
ejpam-109	33	11	and	and	CCONJ
ejpam-109	33	12	m	m	PROPN
ejpam-109	33	13	≡	≡	PROPN
ejpam-109	33	14	l	l	PROPN
ejpam-109	33	15	(	(	PUNCT
ejpam-109	33	16	mod	mod	PROPN
ejpam-109	33	17	ϕ(ps	ϕ(ps	PROPN
ejpam-109	33	18	)	)	PUNCT
ejpam-109	33	19	)	)	PUNCT
ejpam-109	33	20	(	(	PUNCT
ejpam-109	33	21	s	s	X
ejpam-109	33	22	≥	≥	NOUN
ejpam-109	33	23	1	1	NUM
ejpam-109	33	24	,	,	PUNCT
ejpam-109	33	25	ϕ	ϕ	X
ejpam-109	33	26	the	the	DET
ejpam-109	33	27	euler	euler	PROPN
ejpam-109	33	28	totient	totient	PROPN
ejpam-109	33	29	function	function	PROPN
ejpam-109	33	30	)	)	PUNCT
ejpam-109	33	31	,	,	PUNCT
ejpam-109	33	32	then	then	ADV
ejpam-109	33	33	(	(	PUNCT
ejpam-109	33	34	1−	1−	NUM
ejpam-109	33	35	pm−1	pm−1	NOUN
ejpam-109	33	36	)	)	PUNCT
ejpam-109	33	37	bm	bm	PROPN
ejpam-109	33	38	m	m	PROPN
ejpam-109	33	39	≡	≡	PROPN
ejpam-109	33	40	(	(	PUNCT
ejpam-109	33	41	1−	1−	NUM
ejpam-109	33	42	pl−1	pl−1	NOUN
ejpam-109	33	43	)	)	PUNCT
ejpam-109	33	44	bl	bl	ADP
ejpam-109	33	45	l	l	NOUN
ejpam-109	33	46	(	(	PUNCT
ejpam-109	33	47	mod	mod	PROPN
ejpam-109	33	48	ps	ps	PROPN
ejpam-109	33	49	)	)	PUNCT
ejpam-109	33	50	.	.	PUNCT
ejpam-109	34	1	(	(	PUNCT
ejpam-109	34	2	1.2	1.2	NUM
ejpam-109	34	3	)	)	PUNCT
ejpam-109	34	4	this	this	DET
ejpam-109	34	5	congruence	congruence	NOUN
ejpam-109	34	6	is	be	AUX
ejpam-109	34	7	nothing	nothing	PRON
ejpam-109	34	8	but	but	SCONJ
ejpam-109	34	9	a	a	DET
ejpam-109	34	10	special	special	ADJ
ejpam-109	34	11	case	case	NOUN
ejpam-109	34	12	of	of	ADP
ejpam-109	34	13	more	more	ADJ
ejpam-109	34	14	general	general	ADJ
ejpam-109	34	15	formulas	formula	NOUN
ejpam-109	34	16	given	give	VERB
ejpam-109	34	17	in	in	ADP
ejpam-109	34	18	theorem	theorem	ADJ
ejpam-109	34	19	4.1	4.1	NUM
ejpam-109	34	20	below	below	ADV
ejpam-109	34	21	,	,	PUNCT
ejpam-109	34	22	however	however	ADV
ejpam-109	34	23	it	it	PRON
ejpam-109	34	24	is	be	AUX
ejpam-109	34	25	needless	needless	ADJ
ejpam-109	34	26	to	to	PART
ejpam-109	34	27	say	say	VERB
ejpam-109	34	28	that	that	SCONJ
ejpam-109	34	29	this	this	DET
ejpam-109	34	30	deeply	deeply	ADV
ejpam-109	34	31	concerns	concern	NOUN
ejpam-109	34	32	with	with	ADP
ejpam-109	34	33	the	the	DET
ejpam-109	34	34	construction	construction	NOUN
ejpam-109	34	35	of	of	ADP
ejpam-109	34	36	p	p	NOUN
ejpam-109	34	37	-	-	PUNCT
ejpam-109	34	38	adic	adic	ADJ
ejpam-109	34	39	l	l	NOUN
ejpam-109	34	40	function	function	NOUN
ejpam-109	34	41	.	.	PUNCT
ejpam-109	35	1	one	one	PRON
ejpam-109	35	2	can	can	AUX
ejpam-109	35	3	consult	consult	VERB
ejpam-109	35	4	more	more	ADJ
ejpam-109	35	5	details	detail	NOUN
ejpam-109	35	6	with	with	ADP
ejpam-109	35	7	the	the	DET
ejpam-109	35	8	beautiful	beautiful	ADJ
ejpam-109	35	9	books	book	NOUN
ejpam-109	35	10	by	by	ADP
ejpam-109	35	11	iwasawa	iwasawa	NOUN
ejpam-109	35	12	[	[	X
ejpam-109	35	13	9	9	NUM
ejpam-109	35	14	]	]	PUNCT
ejpam-109	35	15	and	and	CCONJ
ejpam-109	35	16	washington	washington	PROPN
ejpam-109	35	17	[	[	X
ejpam-109	35	18	17	17	NUM
ejpam-109	35	19	]	]	PUNCT
ejpam-109	35	20	.	.	PUNCT
ejpam-109	36	1	an	an	DET
ejpam-109	36	2	odd	odd	ADJ
ejpam-109	36	3	prime	prime	NOUN
ejpam-109	36	4	p	p	NOUN
ejpam-109	36	5	is	be	AUX
ejpam-109	36	6	said	say	VERB
ejpam-109	36	7	to	to	PART
ejpam-109	36	8	be	be	AUX
ejpam-109	36	9	irregular	irregular	ADJ
ejpam-109	36	10	if	if	SCONJ
ejpam-109	36	11	p	p	NOUN
ejpam-109	36	12	divides	divide	VERB
ejpam-109	36	13	the	the	DET
ejpam-109	36	14	class	class	NOUN
ejpam-109	36	15	number	number	NOUN
ejpam-109	36	16	hp	hp	NOUN
ejpam-109	36	17	of	of	ADP
ejpam-109	36	18	the	the	DET
ejpam-109	36	19	cyclotomic	cyclotomic	ADJ
ejpam-109	36	20	field	field	NOUN
ejpam-109	36	21	q(ζp	q(ζp	NOUN
ejpam-109	36	22	)	)	PUNCT
ejpam-109	36	23	defined	define	VERB
ejpam-109	36	24	by	by	ADP
ejpam-109	36	25	ζp	ζp	DET
ejpam-109	36	26	a	a	DET
ejpam-109	36	27	primitive	primitive	ADJ
ejpam-109	36	28	p	p	NOUN
ejpam-109	36	29	th	th	X
ejpam-109	36	30	root	root	NOUN
ejpam-109	36	31	of	of	ADP
ejpam-109	36	32	unity	unity	NOUN
ejpam-109	36	33	.	.	PUNCT
ejpam-109	37	1	otherwise	otherwise	ADV
ejpam-109	37	2	,	,	PUNCT
ejpam-109	37	3	it	it	PRON
ejpam-109	37	4	is	be	AUX
ejpam-109	37	5	said	say	VERB
ejpam-109	37	6	to	to	PART
ejpam-109	37	7	be	be	AUX
ejpam-109	37	8	regular	regular	ADJ
ejpam-109	37	9	.	.	PUNCT
ejpam-109	38	1	these	these	DET
ejpam-109	38	2	notions	notion	NOUN
ejpam-109	38	3	were	be	AUX
ejpam-109	38	4	first	first	ADV
ejpam-109	38	5	brought	bring	VERB
ejpam-109	38	6	by	by	ADP
ejpam-109	38	7	kummer	kummer	NOUN
ejpam-109	38	8	in	in	ADP
ejpam-109	38	9	his	his	PRON
ejpam-109	38	10	work	work	NOUN
ejpam-109	38	11	on	on	ADP
ejpam-109	38	12	fermat	fermat	PROPN
ejpam-109	38	13	’s	’s	PART
ejpam-109	38	14	last	last	ADJ
ejpam-109	38	15	theorem	theorem	NOUN
ejpam-109	38	16	.	.	PUNCT
ejpam-109	39	1	as	as	ADP
ejpam-109	39	2	an	an	DET
ejpam-109	39	3	irregularity	irregularity	NOUN
ejpam-109	39	4	criterion	criterion	NOUN
ejpam-109	39	5	,	,	PUNCT
ejpam-109	39	6	kummer	kummer	PROPN
ejpam-109	39	7	showed	show	VERB
ejpam-109	39	8	that	that	SCONJ
ejpam-109	39	9	p	p	NOUN
ejpam-109	39	10	is	be	AUX
ejpam-109	39	11	irregular	irregular	ADJ
ejpam-109	39	12	if	if	SCONJ
ejpam-109	39	13	and	and	CCONJ
ejpam-109	39	14	only	only	ADV
ejpam-109	39	15	if	if	SCONJ
ejpam-109	39	16	p	p	NOUN
ejpam-109	39	17	divides	divide	VERB
ejpam-109	39	18	at	at	ADV
ejpam-109	39	19	least	least	ADJ
ejpam-109	39	20	one	one	NUM
ejpam-109	39	21	of	of	ADP
ejpam-109	39	22	numerators	numerator	NOUN
ejpam-109	39	23	of	of	ADP
ejpam-109	39	24	the	the	DET
ejpam-109	39	25	(	(	PUNCT
ejpam-109	39	26	p	p	NOUN
ejpam-109	39	27	−	−	PROPN
ejpam-109	39	28	3)/2	3)/2	NUM
ejpam-109	39	29	bernoulli	bernoulli	PROPN
ejpam-109	39	30	numbers	number	NOUN
ejpam-109	39	31	b2	b2	PROPN
ejpam-109	39	32	,	,	PUNCT
ejpam-109	39	33	b4	b4	NOUN
ejpam-109	39	34	,	,	PUNCT
ejpam-109	39	35	...	...	PUNCT
ejpam-109	39	36	,	,	PUNCT
ejpam-109	39	37	bp−3	bp−3	PROPN
ejpam-109	39	38	.	.	PUNCT
ejpam-109	40	1	it	it	PRON
ejpam-109	40	2	therefore	therefore	ADV
ejpam-109	40	3	requires	require	VERB
ejpam-109	40	4	to	to	PART
ejpam-109	40	5	obtain	obtain	VERB
ejpam-109	40	6	appropriate	appropriate	ADJ
ejpam-109	40	7	congruences	congruence	NOUN
ejpam-109	40	8	for	for	ADP
ejpam-109	40	9	getting	get	VERB
ejpam-109	40	10	residues	residue	NOUN
ejpam-109	40	11	modulo	modulo	ADJ
ejpam-109	40	12	p	p	NOUN
ejpam-109	40	13	of	of	ADP
ejpam-109	40	14	b2k	b2k	PROPN
ejpam-109	40	15	(	(	PUNCT
ejpam-109	40	16	1≤	1≤	NUM
ejpam-109	40	17	k	k	PROPN
ejpam-109	40	18	≤	≤	PROPN
ejpam-109	40	19	(	(	PUNCT
ejpam-109	40	20	p−	p−	NOUN
ejpam-109	40	21	3)/2	3)/2	NUM
ejpam-109	40	22	)	)	PUNCT
ejpam-109	40	23	.	.	PUNCT
ejpam-109	41	1	amazingly	amazingly	ADV
ejpam-109	41	2	,	,	PUNCT
ejpam-109	41	3	jensen	jensen	PROPN
ejpam-109	42	1	[	[	X
ejpam-109	42	2	10	10	NUM
ejpam-109	42	3	]	]	PUNCT
ejpam-109	42	4	proved	prove	VERB
ejpam-109	42	5	in	in	ADP
ejpam-109	42	6	1915	1915	NUM
ejpam-109	42	7	that	that	SCONJ
ejpam-109	42	8	there	there	PRON
ejpam-109	42	9	are	be	VERB
ejpam-109	42	10	infinitely	infinitely	ADV
ejpam-109	42	11	many	many	ADJ
ejpam-109	42	12	irregular	irregular	ADJ
ejpam-109	42	13	primes	prime	NOUN
ejpam-109	42	14	of	of	ADP
ejpam-109	42	15	the	the	DET
ejpam-109	42	16	form	form	NOUN
ejpam-109	42	17	p	p	X
ejpam-109	42	18	≡	≡	PROPN
ejpam-109	42	19	3	3	NUM
ejpam-109	42	20	(	(	PUNCT
ejpam-109	42	21	mod	mod	NOUN
ejpam-109	42	22	4	4	NUM
ejpam-109	42	23	)	)	PUNCT
ejpam-109	42	24	.	.	PUNCT
ejpam-109	43	1	the	the	DET
ejpam-109	43	2	proof	proof	NOUN
ejpam-109	43	3	was	be	AUX
ejpam-109	43	4	rather	rather	ADV
ejpam-109	43	5	easy	easy	ADJ
ejpam-109	43	6	and	and	CCONJ
ejpam-109	43	7	,	,	PUNCT
ejpam-109	43	8	in	in	ADP
ejpam-109	43	9	fact	fact	NOUN
ejpam-109	43	10	,	,	PUNCT
ejpam-109	43	11	only	only	ADV
ejpam-109	43	12	some	some	DET
ejpam-109	43	13	basic	basic	ADJ
ejpam-109	43	14	arithmetic	arithmetic	ADJ
ejpam-109	43	15	properties	property	NOUN
ejpam-109	43	16	of	of	ADP
ejpam-109	43	17	bernoulli	bernoulli	NOUN
ejpam-109	43	18	numbers	number	NOUN
ejpam-109	43	19	were	be	AUX
ejpam-109	43	20	used	use	VERB
ejpam-109	43	21	.	.	PUNCT
ejpam-109	44	1	however	however	ADV
ejpam-109	44	2	,	,	PUNCT
ejpam-109	44	3	in	in	ADP
ejpam-109	44	4	spite	spite	NOUN
ejpam-109	44	5	of	of	ADP
ejpam-109	44	6	various	various	ADJ
ejpam-109	44	7	efforts	effort	NOUN
ejpam-109	44	8	by	by	ADP
ejpam-109	44	9	many	many	ADJ
ejpam-109	44	10	mathematicians	mathematician	NOUN
ejpam-109	44	11	,	,	PUNCT
ejpam-109	44	12	it	it	PRON
ejpam-109	44	13	has	have	AUX
ejpam-109	44	14	not	not	PART
ejpam-109	44	15	yet	yet	ADV
ejpam-109	44	16	been	be	AUX
ejpam-109	44	17	shown	show	VERB
ejpam-109	44	18	whether	whether	SCONJ
ejpam-109	44	19	there	there	PRON
ejpam-109	44	20	are	be	VERB
ejpam-109	44	21	infinitely	infinitely	ADV
ejpam-109	44	22	many	many	ADJ
ejpam-109	44	23	regular	regular	ADJ
ejpam-109	44	24	primes	prime	NOUN
ejpam-109	44	25	.	.	PUNCT
ejpam-109	45	1	here	here	ADV
ejpam-109	45	2	we	we	PRON
ejpam-109	45	3	should	should	AUX
ejpam-109	45	4	note	note	VERB
ejpam-109	45	5	that	that	PRON
ejpam-109	45	6	siegel	siegel	NOUN
ejpam-109	45	7	(	(	PUNCT
ejpam-109	45	8	1964	1964	NUM
ejpam-109	45	9	)	)	PUNCT
ejpam-109	45	10	proved	prove	VERB
ejpam-109	45	11	that	that	SCONJ
ejpam-109	45	12	,	,	PUNCT
ejpam-109	45	13	under	under	ADP
ejpam-109	45	14	heuristic	heuristic	ADJ
ejpam-109	45	15	assumptions	assumption	NOUN
ejpam-109	45	16	,	,	PUNCT
ejpam-109	45	17	the	the	DET
ejpam-109	45	18	density	density	NOUN
ejpam-109	45	19	of	of	ADP
ejpam-109	45	20	regular	regular	ADJ
ejpam-109	45	21	primes	prime	NOUN
ejpam-109	45	22	among	among	ADP
ejpam-109	45	23	the	the	DET
ejpam-109	45	24	set	set	NOUN
ejpam-109	45	25	of	of	ADP
ejpam-109	45	26	all	all	DET
ejpam-109	45	27	primes	prime	NOUN
ejpam-109	45	28	is	be	AUX
ejpam-109	45	29	1/	1/	NUM
ejpam-109	45	30	p	p	X
ejpam-109	45	31	e	e	PROPN
ejpam-109	45	32	≈	≈	PROPN
ejpam-109	45	33	0.6065	0.6065	NUM
ejpam-109	45	34	.	.	PUNCT
ejpam-109	46	1	in	in	ADP
ejpam-109	46	2	this	this	DET
ejpam-109	46	3	paper	paper	NOUN
ejpam-109	46	4	,	,	PUNCT
ejpam-109	46	5	we	we	PRON
ejpam-109	46	6	will	will	AUX
ejpam-109	46	7	first	first	ADV
ejpam-109	46	8	deduce	deduce	VERB
ejpam-109	46	9	various	various	ADJ
ejpam-109	46	10	voronoï	voronoï	ADJ
ejpam-109	46	11	type	type	NOUN
ejpam-109	46	12	congruences	congruence	NOUN
ejpam-109	46	13	by	by	ADP
ejpam-109	46	14	generalizing	generalize	VERB
ejpam-109	46	15	his	his	PRON
ejpam-109	46	16	original	original	ADJ
ejpam-109	46	17	(	(	PUNCT
ejpam-109	46	18	1.1	1.1	NUM
ejpam-109	46	19	)	)	PUNCT
ejpam-109	46	20	.	.	PUNCT
ejpam-109	47	1	applying	apply	VERB
ejpam-109	47	2	these	these	PRON
ejpam-109	47	3	,	,	PUNCT
ejpam-109	47	4	we	we	PRON
ejpam-109	47	5	will	will	AUX
ejpam-109	47	6	extend	extend	VERB
ejpam-109	47	7	important	important	ADJ
ejpam-109	47	8	arithmetic	arithmetic	ADJ
ejpam-109	47	9	properties	property	NOUN
ejpam-109	47	10	(	(	PUNCT
ejpam-109	47	11	including	include	VERB
ejpam-109	47	12	von	von	PROPN
ejpam-109	47	13	studt	studt	PROPN
ejpam-109	47	14	-	-	PUNCT
ejpam-109	47	15	clausen	clausen	PROPN
ejpam-109	47	16	’s	’s	PART
ejpam-109	47	17	congruence	congruence	NOUN
ejpam-109	47	18	)	)	PUNCT
ejpam-109	47	19	on	on	ADP
ejpam-109	47	20	bernoulli	bernoulli	NOUN
ejpam-109	47	21	numbers	number	NOUN
ejpam-109	47	22	to	to	ADP
ejpam-109	47	23	more	more	ADV
ejpam-109	47	24	general	general	ADJ
ejpam-109	47	25	situation	situation	NOUN
ejpam-109	47	26	.	.	PUNCT
ejpam-109	48	1	further	far	ADV
ejpam-109	48	2	we	we	PRON
ejpam-109	48	3	will	will	AUX
ejpam-109	48	4	discuss	discuss	VERB
ejpam-109	48	5	specific	specific	ADJ
ejpam-109	48	6	properties	property	NOUN
ejpam-109	48	7	of	of	ADP
ejpam-109	48	8	the	the	DET
ejpam-109	48	9	numerator	numerator	NOUN
ejpam-109	48	10	of	of	ADP
ejpam-109	48	11	bm	bm	PROPN
ejpam-109	48	12	/	/	SYM
ejpam-109	48	13	m	m	PROPN
ejpam-109	48	14	,	,	PUNCT
ejpam-109	48	15	whose	whose	DET
ejpam-109	48	16	prime	prime	ADJ
ejpam-109	48	17	divisors	divisor	NOUN
ejpam-109	48	18	are	be	AUX
ejpam-109	48	19	all	all	ADV
ejpam-109	48	20	irregular	irregular	ADJ
ejpam-109	48	21	.	.	PUNCT
ejpam-109	49	1	2	2	X
ejpam-109	49	2	.	.	X
ejpam-109	49	3	general	general	ADJ
ejpam-109	49	4	discussion	discussion	NOUN
ejpam-109	49	5	on	on	ADP
ejpam-109	49	6	bernoulli	bernoulli	NOUN
ejpam-109	49	7	numbers	number	NOUN
ejpam-109	49	8	in	in	ADP
ejpam-109	49	9	this	this	DET
ejpam-109	49	10	section	section	NOUN
ejpam-109	49	11	,	,	PUNCT
ejpam-109	49	12	we	we	PRON
ejpam-109	49	13	shall	shall	AUX
ejpam-109	49	14	introduce	introduce	VERB
ejpam-109	49	15	fundamental	fundamental	ADJ
ejpam-109	49	16	theorems	theorem	NOUN
ejpam-109	49	17	which	which	PRON
ejpam-109	49	18	are	be	AUX
ejpam-109	49	19	narrating	narrate	VERB
ejpam-109	49	20	to	to	ADP
ejpam-109	49	21	us	we	PRON
ejpam-109	49	22	very	very	ADV
ejpam-109	49	23	important	important	ADJ
ejpam-109	49	24	arithmetical	arithmetical	ADJ
ejpam-109	49	25	properties	property	NOUN
ejpam-109	49	26	of	of	ADP
ejpam-109	49	27	bernoulli	bernoulli	NOUN
ejpam-109	49	28	numbers	number	NOUN
ejpam-109	49	29	.	.	PUNCT
ejpam-109	50	1	t.	t.	PROPN
ejpam-109	50	2	agoh	agoh	PROPN
ejpam-109	50	3	/	/	SYM
ejpam-109	50	4	eur	eur	PROPN
ejpam-109	50	5	.	.	PUNCT
ejpam-109	51	1	j.	j.	PROPN
ejpam-109	51	2	pure	pure	PROPN
ejpam-109	51	3	appl	appl	PROPN
ejpam-109	51	4	.	.	PROPN
ejpam-109	51	5	math	math	PROPN
ejpam-109	51	6	,	,	PUNCT
ejpam-109	51	7	1	1	NUM
ejpam-109	51	8	(	(	PUNCT
ejpam-109	51	9	2008	2008	NUM
ejpam-109	51	10	)	)	PUNCT
ejpam-109	51	11	,	,	PUNCT
ejpam-109	51	12	(	(	PUNCT
ejpam-109	51	13	3	3	NUM
ejpam-109	51	14	-	-	SYM
ejpam-109	51	15	21	21	NUM
ejpam-109	51	16	)	)	PUNCT
ejpam-109	51	17	5	5	NUM
ejpam-109	51	18	in	in	ADP
ejpam-109	51	19	what	what	PRON
ejpam-109	51	20	follows	follow	VERB
ejpam-109	51	21	,	,	PUNCT
ejpam-109	51	22	we	we	PRON
ejpam-109	51	23	write	write	VERB
ejpam-109	51	24	bm	bm	PROPN
ejpam-109	51	25	and	and	CCONJ
ejpam-109	51	26	βm	βm	VERB
ejpam-109	51	27	=	=	PROPN
ejpam-109	51	28	bm	bm	PROPN
ejpam-109	51	29	/	/	SYM
ejpam-109	51	30	m	m	PROPN
ejpam-109	51	31	(	(	PUNCT
ejpam-109	51	32	m≥	m≥	NOUN
ejpam-109	51	33	2	2	NUM
ejpam-109	51	34	is	be	AUX
ejpam-109	51	35	even	even	ADV
ejpam-109	51	36	)	)	PUNCT
ejpam-109	51	37	in	in	ADP
ejpam-109	51	38	lowest	low	ADJ
ejpam-109	51	39	terms	term	NOUN
ejpam-109	51	40	as	as	SCONJ
ejpam-109	51	41	follows	follow	VERB
ejpam-109	51	42	:	:	PUNCT
ejpam-109	51	43	bm	bm	PROPN
ejpam-109	51	44	=	=	PUNCT
ejpam-109	51	45	nm	nm	INTJ
ejpam-109	51	46	dm	dm	VERB
ejpam-109	51	47	and	and	CCONJ
ejpam-109	51	48	βm	βm	VERB
ejpam-109	51	49	=	=	SYM
ejpam-109	51	50	n	n	PRON
ejpam-109	51	51	′m	′m	PROPN
ejpam-109	51	52	d′m	d′m	PROPN
ejpam-109	51	53	,	,	PUNCT
ejpam-109	51	54	where	where	SCONJ
ejpam-109	51	55	(	(	PUNCT
ejpam-109	51	56	nm	nm	NOUN
ejpam-109	51	57	,	,	PUNCT
ejpam-109	51	58	dm	dm	NOUN
ejpam-109	51	59	)	)	PUNCT
ejpam-109	51	60	=	=	SYM
ejpam-109	52	1	(	(	PUNCT
ejpam-109	52	2	n	n	X
ejpam-109	52	3	′m	′m	PROPN
ejpam-109	52	4	,	,	PUNCT
ejpam-109	52	5	d′m	d′m	NOUN
ejpam-109	52	6	)	)	PUNCT
ejpam-109	52	7	=	=	SYM
ejpam-109	52	8	1	1	NUM
ejpam-109	52	9	and	and	CCONJ
ejpam-109	52	10	dm	dm	PROPN
ejpam-109	52	11	,	,	PUNCT
ejpam-109	52	12	d′m	d′m	VERB
ejpam-109	52	13	>	>	X
ejpam-109	52	14	0	0	X
ejpam-109	52	15	.	.	PUNCT
ejpam-109	53	1	concerning	concern	VERB
ejpam-109	53	2	divisibility	divisibility	NOUN
ejpam-109	53	3	properties	property	NOUN
ejpam-109	53	4	of	of	ADP
ejpam-109	53	5	the	the	DET
ejpam-109	53	6	numerator	numerator	NOUN
ejpam-109	53	7	nm	nm	PROPN
ejpam-109	53	8	of	of	ADP
ejpam-109	53	9	bm	bm	PROPN
ejpam-109	53	10	,	,	PUNCT
ejpam-109	53	11	we	we	PRON
ejpam-109	53	12	have	have	VERB
ejpam-109	53	13	the	the	DET
ejpam-109	53	14	following	following	ADJ
ejpam-109	53	15	result	result	NOUN
ejpam-109	53	16	which	which	PRON
ejpam-109	53	17	is	be	AUX
ejpam-109	53	18	widely	widely	ADV
ejpam-109	53	19	known	know	VERB
ejpam-109	53	20	as	as	ADP
ejpam-109	53	21	adams	adam	NOUN
ejpam-109	53	22	’	'	PUNCT
ejpam-109	53	23	theorem	theorem	PROPN
ejpam-109	53	24	.	.	PUNCT
ejpam-109	54	1	however	however	ADV
ejpam-109	54	2	,	,	PUNCT
ejpam-109	54	3	it	it	PRON
ejpam-109	54	4	would	would	AUX
ejpam-109	54	5	be	be	AUX
ejpam-109	54	6	better	well	ADJ
ejpam-109	54	7	to	to	PART
ejpam-109	54	8	call	call	VERB
ejpam-109	54	9	von	von	PROPN
ejpam-109	54	10	staudt	staudt	PROPN
ejpam-109	54	11	’s	’s	PART
ejpam-109	54	12	theorem	theorem	NOUN
ejpam-109	54	13	according	accord	VERB
ejpam-109	54	14	to	to	ADP
ejpam-109	54	15	slavutskii	slavutskii	PROPN
ejpam-109	54	16	’s	’s	PART
ejpam-109	54	17	suggestion	suggestion	NOUN
ejpam-109	54	18	written	write	VERB
ejpam-109	54	19	in	in	ADP
ejpam-109	54	20	his	his	PRON
ejpam-109	54	21	historical	historical	ADJ
ejpam-109	54	22	observation	observation	NOUN
ejpam-109	54	23	[	[	X
ejpam-109	54	24	15	15	NUM
ejpam-109	54	25	]	]	X
ejpam-109	54	26	on	on	ADP
ejpam-109	54	27	von	von	PROPN
ejpam-109	54	28	staudt	staudt	PROPN
ejpam-109	54	29	’s	’s	PART
ejpam-109	54	30	achievements	achievement	NOUN
ejpam-109	54	31	,	,	PUNCT
ejpam-109	54	32	because	because	SCONJ
ejpam-109	54	33	this	this	DET
ejpam-109	54	34	result	result	NOUN
ejpam-109	54	35	was	be	AUX
ejpam-109	54	36	first	first	ADV
ejpam-109	54	37	attributed	attribute	VERB
ejpam-109	54	38	to	to	ADP
ejpam-109	54	39	von	von	PROPN
ejpam-109	54	40	staudt	staudt	NOUN
ejpam-109	54	41	in	in	ADP
ejpam-109	54	42	1845	1845	NUM
ejpam-109	54	43	before	before	ADP
ejpam-109	54	44	adams	adams	PROPN
ejpam-109	54	45	(	(	PUNCT
ejpam-109	54	46	1878	1878	NUM
ejpam-109	54	47	)	)	PUNCT
ejpam-109	54	48	.	.	PUNCT
ejpam-109	55	1	theorem	theorem	VERB
ejpam-109	55	2	2.1	2.1	NUM
ejpam-109	55	3	.	.	PUNCT
ejpam-109	56	1	let	let	VERB
ejpam-109	56	2	m	m	PRON
ejpam-109	56	3	≥	≥	NOUN
ejpam-109	56	4	2	2	NUM
ejpam-109	56	5	be	be	AUX
ejpam-109	56	6	an	an	DET
ejpam-109	56	7	even	even	ADV
ejpam-109	56	8	integer	integer	NOUN
ejpam-109	56	9	.	.	PUNCT
ejpam-109	57	1	if	if	SCONJ
ejpam-109	57	2	m	m	NOUN
ejpam-109	57	3	=	=	VERB
ejpam-109	57	4	pem0	pem0	NOUN
ejpam-109	57	5	(	(	PUNCT
ejpam-109	57	6	e	e	X
ejpam-109	57	7	≥	≥	NUM
ejpam-109	57	8	1	1	NUM
ejpam-109	57	9	,	,	PUNCT
ejpam-109	57	10	p	p	X
ejpam-109	57	11	an	an	DET
ejpam-109	57	12	odd	odd	ADJ
ejpam-109	57	13	prime	prime	NOUN
ejpam-109	57	14	,	,	PUNCT
ejpam-109	57	15	p	p	NOUN
ejpam-109	57	16	m0	m0	NOUN
ejpam-109	57	17	)	)	PUNCT
ejpam-109	57	18	and	and	CCONJ
ejpam-109	57	19	p	p	X
ejpam-109	57	20	dm	dm	PROPN
ejpam-109	57	21	,	,	PUNCT
ejpam-109	57	22	then	then	ADV
ejpam-109	57	23	pe	pe	INTJ
ejpam-109	57	24	|	|	ADV
ejpam-109	57	25	nm	nm	VERB
ejpam-109	57	26	.	.	PUNCT
ejpam-109	58	1	we	we	PRON
ejpam-109	58	2	see	see	VERB
ejpam-109	58	3	from	from	ADP
ejpam-109	58	4	this	this	DET
ejpam-109	58	5	theorem	theorem	NOUN
ejpam-109	58	6	that	that	PRON
ejpam-109	58	7	p	p	PROPN
ejpam-109	58	8	dm	dm	X
ejpam-109	59	1	if	if	SCONJ
ejpam-109	60	1	and	and	CCONJ
ejpam-109	60	2	only	only	ADV
ejpam-109	60	3	if	if	SCONJ
ejpam-109	60	4	p	p	PROPN
ejpam-109	60	5	d′m	d′m	NOUN
ejpam-109	60	6	.	.	PUNCT
ejpam-109	61	1	the	the	DET
ejpam-109	61	2	divisor	divisor	NOUN
ejpam-109	61	3	pe	pe	INTJ
ejpam-109	61	4	of	of	ADP
ejpam-109	61	5	nm	nm	PROPN
ejpam-109	61	6	stated	state	VERB
ejpam-109	61	7	above	above	ADV
ejpam-109	61	8	is	be	AUX
ejpam-109	61	9	called	call	VERB
ejpam-109	61	10	an	an	DET
ejpam-109	61	11	improper	improper	ADJ
ejpam-109	61	12	divisor	divisor	NOUN
ejpam-109	61	13	of	of	ADP
ejpam-109	61	14	bm	bm	PROPN
ejpam-109	61	15	.	.	PUNCT
ejpam-109	62	1	if	if	SCONJ
ejpam-109	62	2	d	d	PROPN
ejpam-109	62	3	>	>	X
ejpam-109	62	4	0	0	PUNCT
ejpam-109	62	5	divides	divide	VERB
ejpam-109	62	6	nm	nm	VERB
ejpam-109	62	7	but	but	CCONJ
ejpam-109	62	8	not	not	PART
ejpam-109	62	9	m	m	ADV
ejpam-109	62	10	,	,	PUNCT
ejpam-109	62	11	then	then	ADV
ejpam-109	62	12	it	it	PRON
ejpam-109	62	13	is	be	AUX
ejpam-109	62	14	called	call	VERB
ejpam-109	62	15	a	a	DET
ejpam-109	62	16	proper	proper	ADJ
ejpam-109	62	17	divisor	divisor	NOUN
ejpam-109	62	18	of	of	ADP
ejpam-109	62	19	bm	bm	PROPN
ejpam-109	62	20	.	.	PUNCT
ejpam-109	63	1	we	we	PRON
ejpam-109	63	2	can	can	AUX
ejpam-109	63	3	easily	easily	ADV
ejpam-109	63	4	show	show	VERB
ejpam-109	63	5	that	that	SCONJ
ejpam-109	63	6	p	p	NOUN
ejpam-109	63	7	is	be	AUX
ejpam-109	63	8	irregular	irregular	ADJ
ejpam-109	63	9	if	if	SCONJ
ejpam-109	63	10	and	and	CCONJ
ejpam-109	63	11	only	only	ADV
ejpam-109	63	12	if	if	SCONJ
ejpam-109	63	13	p	p	NOUN
ejpam-109	63	14	is	be	AUX
ejpam-109	63	15	a	a	DET
ejpam-109	63	16	proper	proper	ADJ
ejpam-109	63	17	divisor	divisor	NOUN
ejpam-109	63	18	of	of	ADP
ejpam-109	63	19	some	some	DET
ejpam-109	63	20	bernoulli	bernoulli	NOUN
ejpam-109	63	21	number	number	NOUN
ejpam-109	63	22	.	.	PUNCT
ejpam-109	64	1	it	it	PRON
ejpam-109	64	2	is	be	AUX
ejpam-109	64	3	unknown	unknown	ADJ
ejpam-109	64	4	whether	whether	SCONJ
ejpam-109	64	5	there	there	PRON
ejpam-109	64	6	exists	exist	VERB
ejpam-109	64	7	an	an	DET
ejpam-109	64	8	odd	odd	ADJ
ejpam-109	64	9	prime	prime	NOUN
ejpam-109	64	10	p	p	NOUN
ejpam-109	64	11	such	such	ADJ
ejpam-109	64	12	that	that	SCONJ
ejpam-109	64	13	bm	bm	PROPN
ejpam-109	64	14	≡	≡	PROPN
ejpam-109	64	15	0	0	PUNCT
ejpam-109	65	1	(	(	PUNCT
ejpam-109	65	2	mod	mod	NOUN
ejpam-109	65	3	p2	p2	PROPN
ejpam-109	65	4	)	)	PUNCT
ejpam-109	65	5	for	for	ADP
ejpam-109	65	6	an	an	DET
ejpam-109	65	7	even	even	ADV
ejpam-109	65	8	m	m	PROPN
ejpam-109	65	9	,	,	PUNCT
ejpam-109	65	10	2≤	2≤	NUM
ejpam-109	65	11	m≤	m≤	PROPN
ejpam-109	65	12	p−	p−	NOUN
ejpam-109	65	13	3	3	NUM
ejpam-109	65	14	.	.	PUNCT
ejpam-109	66	1	the	the	DET
ejpam-109	66	2	next	next	ADJ
ejpam-109	66	3	theorem	theorem	NOUN
ejpam-109	66	4	is	be	AUX
ejpam-109	66	5	known	know	VERB
ejpam-109	66	6	as	as	ADP
ejpam-109	66	7	the	the	DET
ejpam-109	66	8	euler	euler	PROPN
ejpam-109	66	9	-	-	PUNCT
ejpam-109	66	10	maclaurin	maclaurin	NOUN
ejpam-109	66	11	summation	summation	NOUN
ejpam-109	66	12	formula	formula	NOUN
ejpam-109	66	13	.	.	PUNCT
ejpam-109	67	1	the	the	DET
ejpam-109	67	2	proof	proof	NOUN
ejpam-109	67	3	can	can	AUX
ejpam-109	67	4	be	be	AUX
ejpam-109	67	5	easily	easily	ADV
ejpam-109	67	6	performed	perform	VERB
ejpam-109	67	7	by	by	ADP
ejpam-109	67	8	equating	equate	VERB
ejpam-109	67	9	the	the	DET
ejpam-109	67	10	coefficients	coefficient	NOUN
ejpam-109	67	11	of	of	ADP
ejpam-109	67	12	tm+1	tm+1	PRON
ejpam-109	67	13	in	in	ADP
ejpam-109	67	14	the	the	DET
ejpam-109	67	15	power	power	NOUN
ejpam-109	67	16	series	series	NOUN
ejpam-109	67	17	expansions	expansion	NOUN
ejpam-109	67	18	of	of	ADP
ejpam-109	67	19	both	both	DET
ejpam-109	67	20	sides	side	NOUN
ejpam-109	67	21	of	of	ADP
ejpam-109	67	22	the	the	DET
ejpam-109	67	23	identity	identity	NOUN
ejpam-109	67	24	t	t	PROPN
ejpam-109	67	25	n−1	n−1	PROPN
ejpam-109	67	26	∑	∑	PUNCT
ejpam-109	68	1	i=0	i=0	PROPN
ejpam-109	68	2	ei	ei	X
ejpam-109	68	3	t	t	PROPN
ejpam-109	68	4	=	=	PUNCT
ejpam-109	68	5	t	t	PROPN
ejpam-109	68	6	et	et	NOUN
ejpam-109	68	7	−	−	NOUN
ejpam-109	68	8	1	1	NUM
ejpam-109	68	9	(	(	PUNCT
ejpam-109	68	10	ent	ent	NOUN
ejpam-109	68	11	−	−	PROPN
ejpam-109	68	12	1	1	NUM
ejpam-109	68	13	)	)	PUNCT
ejpam-109	68	14	.	.	PUNCT
ejpam-109	69	1	theorem	theorem	VERB
ejpam-109	69	2	2.2	2.2	NUM
ejpam-109	69	3	.	.	PUNCT
ejpam-109	70	1	let	let	VERB
ejpam-109	70	2	m	m	PRON
ejpam-109	70	3	,	,	PUNCT
ejpam-109	70	4	n≥	n≥	X
ejpam-109	70	5	1	1	NUM
ejpam-109	70	6	and	and	CCONJ
ejpam-109	70	7	sm(n	sm(n	NOUN
ejpam-109	70	8	)	)	PUNCT
ejpam-109	71	1	=	=	SYM
ejpam-109	71	2	1m+	1m+	NUM
ejpam-109	71	3	2m+	2m+	NUM
ejpam-109	71	4	·	·	PUNCT
ejpam-109	71	5	·	·	PUNCT
ejpam-109	71	6	·	·	PUNCT
ejpam-109	71	7	+	+	CCONJ
ejpam-109	71	8	(	(	PUNCT
ejpam-109	71	9	n−	n−	NOUN
ejpam-109	71	10	1)m	1)m	NUM
ejpam-109	71	11	.	.	PUNCT
ejpam-109	71	12	then	then	ADV
ejpam-109	71	13	sm(n	sm(n	X
ejpam-109	71	14	)	)	PUNCT
ejpam-109	71	15	=	=	SYM
ejpam-109	72	1	m+1	m+1	NUM
ejpam-109	72	2	∑	∑	PUNCT
ejpam-109	72	3	j=1	j=1	PROPN
ejpam-109	72	4	1	1	NUM
ejpam-109	72	5	j	j	PROPN
ejpam-109	72	6	�	�	PROPN
ejpam-109	72	7	m	m	VERB
ejpam-109	72	8	j−	j−	PROPN
ejpam-109	72	9	1	1	NUM
ejpam-109	72	10	�	�	PROPN
ejpam-109	72	11	bm+1−	bm+1−	PROPN
ejpam-109	72	12	jn	jn	PROPN
ejpam-109	72	13	j	j	PROPN
ejpam-109	72	14	.	.	PUNCT
ejpam-109	72	15	suppose	suppose	VERB
ejpam-109	72	16	that	that	SCONJ
ejpam-109	72	17	pα	pα	VERB
ejpam-109	72	18	‖	‖	PROPN
ejpam-109	72	19	2i	2i	NOUN
ejpam-109	72	20	+	+	CCONJ
ejpam-109	72	21	1	1	NUM
ejpam-109	72	22	for	for	ADP
ejpam-109	72	23	p	p	PRON
ejpam-109	72	24	≥	≥	NUM
ejpam-109	72	25	5	5	NUM
ejpam-109	72	26	and	and	CCONJ
ejpam-109	72	27	i	i	PRON
ejpam-109	72	28	≥	≥	VERB
ejpam-109	72	29	2	2	NUM
ejpam-109	72	30	.	.	PUNCT
ejpam-109	73	1	thus	thus	ADV
ejpam-109	73	2	,	,	PUNCT
ejpam-109	73	3	we	we	PRON
ejpam-109	73	4	can	can	AUX
ejpam-109	73	5	write	write	VERB
ejpam-109	73	6	as	as	ADP
ejpam-109	73	7	2i	2i	NUM
ejpam-109	73	8	+	+	CCONJ
ejpam-109	73	9	1	1	NUM
ejpam-109	73	10	=	=	NOUN
ejpam-109	73	11	pαx	pαx	NOUN
ejpam-109	73	12	(	(	PUNCT
ejpam-109	73	13	α	α	PRON
ejpam-109	73	14	≥	≥	NOUN
ejpam-109	73	15	1	1	NUM
ejpam-109	73	16	,	,	PUNCT
ejpam-109	73	17	p	p	NOUN
ejpam-109	73	18	x	x	NOUN
ejpam-109	73	19	)	)	PUNCT
ejpam-109	73	20	.	.	PUNCT
ejpam-109	74	1	since	since	SCONJ
ejpam-109	74	2	pαx	pαx	NOUN
ejpam-109	74	3	−α≥	−α≥	NOUN
ejpam-109	74	4	5α−α≥	5α−α≥	PROPN
ejpam-109	74	5	4	4	NUM
ejpam-109	74	6	,	,	PUNCT
ejpam-109	74	7	1	1	NUM
ejpam-109	74	8	2i+	2i+	NUM
ejpam-109	74	9	1	1	NUM
ejpam-109	75	1	p2i+1	p2i+1	NOUN
ejpam-109	75	2	=	=	NOUN
ejpam-109	75	3	1	1	NUM
ejpam-109	75	4	x	x	SYM
ejpam-109	75	5	ppαx−α	ppαx−α	NUM
ejpam-109	75	6	≡	≡	PROPN
ejpam-109	75	7	0	0	PUNCT
ejpam-109	75	8	(	(	PUNCT
ejpam-109	75	9	mod	mod	PROPN
ejpam-109	75	10	p4	p4	PROPN
ejpam-109	75	11	)	)	PUNCT
ejpam-109	75	12	.	.	PUNCT
ejpam-109	76	1	also	also	ADV
ejpam-109	76	2	,	,	PUNCT
ejpam-109	76	3	since	since	SCONJ
ejpam-109	76	4	ord3(dm	ord3(dm	NUM
ejpam-109	76	5	)	)	PUNCT
ejpam-109	76	6	=	=	SYM
ejpam-109	76	7	1	1	NUM
ejpam-109	76	8	for	for	ADP
ejpam-109	76	9	all	all	PRON
ejpam-109	76	10	even	even	ADV
ejpam-109	76	11	m	m	PRON
ejpam-109	76	12	≥	≥	NOUN
ejpam-109	76	13	2	2	NUM
ejpam-109	76	14	,	,	PUNCT
ejpam-109	76	15	we	we	PRON
ejpam-109	76	16	know	know	VERB
ejpam-109	76	17	�	�	PROPN
ejpam-109	76	18	32bm−2	32bm−2	NUM
ejpam-109	76	19	�	�	PROPN
ejpam-109	76	20	dm	dm	PROPN
ejpam-109	76	21	≡	≡	PROPN
ejpam-109	76	22	0	0	PUNCT
ejpam-109	77	1	(	(	PUNCT
ejpam-109	77	2	mod	mod	PROPN
ejpam-109	77	3	32	32	NUM
ejpam-109	77	4	)	)	PUNCT
ejpam-109	77	5	,	,	PUNCT
ejpam-109	77	6	which	which	PRON
ejpam-109	77	7	implies	imply	VERB
ejpam-109	77	8	from	from	ADP
ejpam-109	77	9	theorem	theorem	ADJ
ejpam-109	77	10	2.2	2.2	NUM
ejpam-109	77	11	the	the	DET
ejpam-109	77	12	following	follow	VERB
ejpam-109	77	13	theorem	theorem	VERB
ejpam-109	77	14	2.3	2.3	NUM
ejpam-109	77	15	.	.	PUNCT
ejpam-109	78	1	if	if	SCONJ
ejpam-109	78	2	m≥	m≥	PROPN
ejpam-109	78	3	2	2	NUM
ejpam-109	78	4	is	be	AUX
ejpam-109	78	5	even	even	ADV
ejpam-109	78	6	and	and	CCONJ
ejpam-109	78	7	n≥	n≥	ADJ
ejpam-109	78	8	1	1	NUM
ejpam-109	78	9	,	,	PUNCT
ejpam-109	78	10	then	then	ADV
ejpam-109	78	11	dmsm(n)≡	dmsm(n)≡	PROPN
ejpam-109	78	12	nmn	nmn	PROPN
ejpam-109	78	13	(	(	PUNCT
ejpam-109	78	14	mod	mod	PROPN
ejpam-109	78	15	n2	n2	PROPN
ejpam-109	78	16	)	)	PUNCT
ejpam-109	78	17	.	.	PUNCT
ejpam-109	79	1	t.	t.	PROPN
ejpam-109	79	2	agoh	agoh	PROPN
ejpam-109	79	3	/	/	SYM
ejpam-109	79	4	eur	eur	PROPN
ejpam-109	79	5	.	.	PUNCT
ejpam-109	80	1	j.	j.	PROPN
ejpam-109	80	2	pure	pure	PROPN
ejpam-109	80	3	appl	appl	PROPN
ejpam-109	80	4	.	.	PROPN
ejpam-109	80	5	math	math	PROPN
ejpam-109	80	6	,	,	PUNCT
ejpam-109	80	7	1	1	NUM
ejpam-109	80	8	(	(	PUNCT
ejpam-109	80	9	2008	2008	NUM
ejpam-109	80	10	)	)	PUNCT
ejpam-109	80	11	,	,	PUNCT
ejpam-109	80	12	(	(	PUNCT
ejpam-109	80	13	3	3	NUM
ejpam-109	80	14	-	-	SYM
ejpam-109	80	15	21	21	NUM
ejpam-109	80	16	)	)	PUNCT
ejpam-109	80	17	6	6	NUM
ejpam-109	80	18	we	we	PRON
ejpam-109	80	19	now	now	ADV
ejpam-109	80	20	introduce	introduce	VERB
ejpam-109	80	21	the	the	DET
ejpam-109	80	22	voronoï	voronoï	ADJ
ejpam-109	80	23	congruence	congruence	NOUN
ejpam-109	80	24	which	which	PRON
ejpam-109	80	25	brings	bring	VERB
ejpam-109	80	26	various	various	ADJ
ejpam-109	80	27	important	important	ADJ
ejpam-109	80	28	properties	property	NOUN
ejpam-109	80	29	of	of	ADP
ejpam-109	80	30	bernoulli	bernoulli	NOUN
ejpam-109	80	31	numbers	number	NOUN
ejpam-109	80	32	in	in	ADP
ejpam-109	80	33	relief	relief	NOUN
ejpam-109	80	34	.	.	PUNCT
ejpam-109	81	1	let	let	VERB
ejpam-109	81	2	a	a	PRON
ejpam-109	81	3	and	and	CCONJ
ejpam-109	81	4	n	n	ADV
ejpam-109	81	5	be	be	AUX
ejpam-109	81	6	positive	positive	ADJ
ejpam-109	81	7	integers	integer	NOUN
ejpam-109	81	8	prime	prime	ADJ
ejpam-109	81	9	to	to	ADP
ejpam-109	81	10	each	each	DET
ejpam-109	81	11	other	other	ADJ
ejpam-109	81	12	.	.	PUNCT
ejpam-109	82	1	suppose	suppose	VERB
ejpam-109	82	2	that	that	SCONJ
ejpam-109	82	3	q	q	PROPN
ejpam-109	82	4	j	j	PROPN
ejpam-109	82	5	and	and	CCONJ
ejpam-109	82	6	r	r	PROPN
ejpam-109	82	7	j	j	PROPN
ejpam-109	82	8	,	,	PUNCT
ejpam-109	82	9	j	j	PROPN
ejpam-109	82	10	=	=	SYM
ejpam-109	82	11	1,2	1,2	NUM
ejpam-109	82	12	,	,	PUNCT
ejpam-109	82	13	...	...	PUNCT
ejpam-109	82	14	,	,	PUNCT
ejpam-109	82	15	n−	n−	NOUN
ejpam-109	82	16	1	1	NUM
ejpam-109	82	17	,	,	PUNCT
ejpam-109	82	18	are	be	AUX
ejpam-109	82	19	positive	positive	ADJ
ejpam-109	82	20	integers	integer	NOUN
ejpam-109	82	21	satisfying	satisfy	VERB
ejpam-109	82	22	a	a	DET
ejpam-109	82	23	j	j	NOUN
ejpam-109	82	24	=	=	PUNCT
ejpam-109	82	25	q	q	X
ejpam-109	82	26	jn+	jn+	PROPN
ejpam-109	82	27	r	r	PROPN
ejpam-109	82	28	j	j	PROPN
ejpam-109	82	29	,	,	PUNCT
ejpam-109	82	30	0	0	PUNCT
ejpam-109	82	31	<	<	X
ejpam-109	82	32	r	r	X
ejpam-109	82	33	j	j	X
ejpam-109	82	34	<	<	X
ejpam-109	82	35	n.	n.	PROPN
ejpam-109	82	36	here	here	ADV
ejpam-109	82	37	q	q	PROPN
ejpam-109	83	1	j	j	PROPN
ejpam-109	83	2	=	=	PUNCT
ejpam-109	84	1	[	[	X
ejpam-109	84	2	a	a	DET
ejpam-109	84	3	j	j	PROPN
ejpam-109	84	4	/	/	SYM
ejpam-109	84	5	n	n	CCONJ
ejpam-109	84	6	]	]	PUNCT
ejpam-109	84	7	.	.	PUNCT
ejpam-109	85	1	by	by	ADP
ejpam-109	85	2	direct	direct	ADJ
ejpam-109	85	3	calculation	calculation	NOUN
ejpam-109	85	4	of	of	ADP
ejpam-109	85	5	(	(	PUNCT
ejpam-109	85	6	a	a	DET
ejpam-109	85	7	j−	j−	NOUN
ejpam-109	85	8	q	q	NOUN
ejpam-109	85	9	jn)m	jn)m	PROPN
ejpam-109	85	10	=	=	SYM
ejpam-109	85	11	rm	rm	PROPN
ejpam-109	86	1	j	j	PROPN
ejpam-109	87	1	we	we	PRON
ejpam-109	87	2	obtain	obtain	VERB
ejpam-109	87	3	(	(	PUNCT
ejpam-109	87	4	a	a	DET
ejpam-109	87	5	j)m−	j)m−	PROPN
ejpam-109	87	6	rm	rm	NOUN
ejpam-109	87	7	j	j	PROPN
ejpam-109	88	1	=	=	PUNCT
ejpam-109	88	2	m	m	PROPN
ejpam-109	88	3	∑	∑	PROPN
ejpam-109	88	4	i=1	i=1	PROPN
ejpam-109	88	5	(	(	PUNCT
ejpam-109	88	6	−1)i−1	−1)i−1	PROPN
ejpam-109	88	7	�	�	PROPN
ejpam-109	88	8	m	m	VERB
ejpam-109	88	9	i	i	PROPN
ejpam-109	88	10	�	�	PROPN
ejpam-109	88	11	(	(	PUNCT
ejpam-109	88	12	a	a	DET
ejpam-109	88	13	j)m−ini	j)m−ini	PROPN
ejpam-109	88	14	�	�	PROPN
ejpam-109	88	15	a	a	DET
ejpam-109	88	16	j	j	PROPN
ejpam-109	88	17	n	n	PROPN
ejpam-109	88	18	�	�	PROPN
ejpam-109	88	19	i	i	PRON
ejpam-109	88	20	.	.	PUNCT
ejpam-109	89	1	noting	note	VERB
ejpam-109	89	2	that	that	SCONJ
ejpam-109	89	3	{	{	PUNCT
ejpam-109	89	4	r	r	NOUN
ejpam-109	89	5	j	j	PROPN
ejpam-109	89	6	|	|	ADV
ejpam-109	89	7	j	j	PROPN
ejpam-109	89	8	=	=	SYM
ejpam-109	89	9	1	1	NUM
ejpam-109	89	10	,	,	PUNCT
ejpam-109	89	11	2	2	NUM
ejpam-109	89	12	,	,	PUNCT
ejpam-109	89	13	...	...	PUNCT
ejpam-109	89	14	,	,	PUNCT
ejpam-109	89	15	n−	n−	PROPN
ejpam-109	89	16	1}=	1}=	NUM
ejpam-109	89	17	{	{	PUNCT
ejpam-109	89	18	1,2	1,2	NUM
ejpam-109	89	19	,	,	PUNCT
ejpam-109	89	20	...	...	PUNCT
ejpam-109	89	21	,	,	PUNCT
ejpam-109	89	22	n−	n−	NOUN
ejpam-109	89	23	1	1	NUM
ejpam-109	89	24	}	}	PUNCT
ejpam-109	89	25	,	,	PUNCT
ejpam-109	89	26	if	if	SCONJ
ejpam-109	89	27	we	we	PRON
ejpam-109	89	28	sum	sum	VERB
ejpam-109	89	29	over	over	ADP
ejpam-109	89	30	j	j	PROPN
ejpam-109	89	31	=	=	SYM
ejpam-109	89	32	1,2	1,2	NUM
ejpam-109	89	33	,	,	PUNCT
ejpam-109	89	34	...	...	PUNCT
ejpam-109	89	35	,	,	PUNCT
ejpam-109	89	36	n−	n−	NOUN
ejpam-109	89	37	1	1	NUM
ejpam-109	89	38	,	,	PUNCT
ejpam-109	89	39	then	then	ADV
ejpam-109	89	40	(	(	PUNCT
ejpam-109	89	41	am−	am−	NUM
ejpam-109	89	42	1)sm(n	1)sm(n	NUM
ejpam-109	89	43	)	)	PUNCT
ejpam-109	89	44	=	=	PUNCT
ejpam-109	89	45	m	m	VERB
ejpam-109	89	46	∑	∑	VERB
ejpam-109	89	47	i=1	i=1	PROPN
ejpam-109	89	48	(	(	PUNCT
ejpam-109	89	49	−1)i−1ni	−1)i−1ni	PROPN
ejpam-109	89	50	�	�	PROPN
ejpam-109	89	51	m	m	VERB
ejpam-109	89	52	i	i	NOUN
ejpam-109	89	53	�	�	PROPN
ejpam-109	89	54	n	n	CCONJ
ejpam-109	89	55	∑	∑	PROPN
ejpam-109	89	56	j=1	j=1	PROPN
ejpam-109	89	57	(	(	PUNCT
ejpam-109	89	58	a	a	DET
ejpam-109	89	59	j)m−i	j)m−i	PROPN
ejpam-109	89	60	�	�	PROPN
ejpam-109	89	61	a	a	DET
ejpam-109	89	62	j	j	PROPN
ejpam-109	89	63	n	n	PROPN
ejpam-109	89	64	�	�	PROPN
ejpam-109	89	65	i	i	PRON
ejpam-109	89	66	,	,	PUNCT
ejpam-109	89	67	(	(	PUNCT
ejpam-109	89	68	2.1	2.1	NUM
ejpam-109	89	69	)	)	PUNCT
ejpam-109	90	1	so	so	SCONJ
ejpam-109	90	2	that	that	SCONJ
ejpam-109	90	3	(	(	PUNCT
ejpam-109	90	4	am−	am−	NUM
ejpam-109	90	5	1)sm(n)≡	1)sm(n)≡	NUM
ejpam-109	90	6	mn	mn	PROPN
ejpam-109	90	7	n−1	n−1	PROPN
ejpam-109	90	8	∑	∑	PUNCT
ejpam-109	90	9	j=1	j=1	PROPN
ejpam-109	90	10	(	(	PUNCT
ejpam-109	90	11	a	a	DET
ejpam-109	90	12	j)m−1	j)m−1	PROPN
ejpam-109	90	13	�	�	PROPN
ejpam-109	90	14	a	a	DET
ejpam-109	90	15	j	j	PROPN
ejpam-109	90	16	n	n	PRON
ejpam-109	90	17	�	�	PROPN
ejpam-109	90	18	(	(	PUNCT
ejpam-109	90	19	mod	mod	PROPN
ejpam-109	90	20	n2	n2	PROPN
ejpam-109	90	21	)	)	PUNCT
ejpam-109	90	22	.	.	PUNCT
ejpam-109	91	1	ultimately	ultimately	ADV
ejpam-109	91	2	,	,	PUNCT
ejpam-109	91	3	from	from	ADP
ejpam-109	91	4	theorem	theorem	ADJ
ejpam-109	91	5	2.3	2.3	NUM
ejpam-109	91	6	we	we	PRON
ejpam-109	91	7	get	get	VERB
ejpam-109	91	8	the	the	DET
ejpam-109	91	9	following	follow	VERB
ejpam-109	91	10	voronoï	voronoï	ADJ
ejpam-109	91	11	congruence	congruence	NOUN
ejpam-109	91	12	:	:	PUNCT
ejpam-109	91	13	theorem	theorem	VERB
ejpam-109	91	14	2.4	2.4	NUM
ejpam-109	91	15	.	.	PUNCT
ejpam-109	92	1	let	let	VERB
ejpam-109	92	2	m≥	m≥	NOUN
ejpam-109	92	3	2	2	NUM
ejpam-109	92	4	be	be	AUX
ejpam-109	92	5	even	even	ADV
ejpam-109	92	6	and	and	CCONJ
ejpam-109	92	7	n≥	n≥	ADJ
ejpam-109	92	8	1	1	NUM
ejpam-109	92	9	.	.	PUNCT
ejpam-109	93	1	if	if	SCONJ
ejpam-109	93	2	a	a	DET
ejpam-109	93	3	≥	≥	NOUN
ejpam-109	93	4	1	1	NUM
ejpam-109	93	5	satisfies	satisfie	NOUN
ejpam-109	93	6	(	(	PUNCT
ejpam-109	93	7	a	a	PRON
ejpam-109	93	8	,	,	PUNCT
ejpam-109	93	9	n	n	CCONJ
ejpam-109	93	10	)	)	PUNCT
ejpam-109	93	11	=	=	SYM
ejpam-109	93	12	1	1	NUM
ejpam-109	93	13	,	,	PUNCT
ejpam-109	93	14	then	then	ADV
ejpam-109	93	15	(	(	PUNCT
ejpam-109	93	16	am−	am−	NUM
ejpam-109	93	17	1)nm	1)nm	PROPN
ejpam-109	93	18	≡	≡	PROPN
ejpam-109	93	19	mdm	mdm	PROPN
ejpam-109	93	20	n−1	n−1	PROPN
ejpam-109	93	21	∑	∑	PUNCT
ejpam-109	93	22	j=1	j=1	PROPN
ejpam-109	93	23	(	(	PUNCT
ejpam-109	93	24	a	a	DET
ejpam-109	93	25	j)m−1	j)m−1	PROPN
ejpam-109	93	26	�	�	PROPN
ejpam-109	93	27	a	a	DET
ejpam-109	93	28	j	j	PROPN
ejpam-109	93	29	n	n	PRON
ejpam-109	93	30	�	�	PROPN
ejpam-109	93	31	(	(	PUNCT
ejpam-109	93	32	mod	mod	PROPN
ejpam-109	93	33	n	n	CCONJ
ejpam-109	93	34	)	)	PUNCT
ejpam-109	93	35	.	.	PUNCT
ejpam-109	94	1	the	the	DET
ejpam-109	94	2	next	next	ADJ
ejpam-109	94	3	one	one	NOUN
ejpam-109	94	4	is	be	AUX
ejpam-109	94	5	known	know	VERB
ejpam-109	94	6	as	as	ADP
ejpam-109	94	7	the	the	DET
ejpam-109	94	8	famous	famous	ADJ
ejpam-109	94	9	von	von	PROPN
ejpam-109	94	10	staudt	staudt	PROPN
ejpam-109	94	11	-	-	PUNCT
ejpam-109	94	12	clausen	clausen	PROPN
ejpam-109	94	13	theorem	theorem	PROPN
ejpam-109	94	14	.	.	PUNCT
ejpam-109	94	15	theorem	theorem	VERB
ejpam-109	94	16	2.5	2.5	NUM
ejpam-109	94	17	.	.	PUNCT
ejpam-109	95	1	let	let	VERB
ejpam-109	95	2	p	p	PRON
ejpam-109	95	3	be	be	AUX
ejpam-109	95	4	a	a	DET
ejpam-109	95	5	prime	prime	NOUN
ejpam-109	95	6	,	,	PUNCT
ejpam-109	95	7	m	m	VERB
ejpam-109	95	8	an	an	DET
ejpam-109	95	9	even	even	ADV
ejpam-109	95	10	integer	integer	ADJ
ejpam-109	95	11	≥	≥	NUM
ejpam-109	95	12	2	2	NUM
ejpam-109	95	13	and	and	CCONJ
ejpam-109	95	14	zp	zp	NOUN
ejpam-109	95	15	the	the	DET
ejpam-109	95	16	ring	ring	NOUN
ejpam-109	95	17	of	of	ADP
ejpam-109	95	18	p	p	NOUN
ejpam-109	95	19	-	-	PUNCT
ejpam-109	95	20	adic	adic	ADJ
ejpam-109	95	21	integers	integer	NOUN
ejpam-109	95	22	.	.	PUNCT
ejpam-109	96	1	if	if	SCONJ
ejpam-109	96	2	p−	p−	NOUN
ejpam-109	96	3	1	1	NUM
ejpam-109	96	4	m	m	NOUN
ejpam-109	96	5	,	,	PUNCT
ejpam-109	96	6	then	then	ADV
ejpam-109	96	7	bm	bm	PROPN
ejpam-109	96	8	∈	∈	PROPN
ejpam-109	96	9	zp	zp	PROPN
ejpam-109	96	10	.	.	PUNCT
ejpam-109	97	1	if	if	SCONJ
ejpam-109	97	2	p−	p−	NOUN
ejpam-109	97	3	1	1	NUM
ejpam-109	97	4	|	|	ADV
ejpam-109	97	5	m	m	VERB
ejpam-109	97	6	,	,	PUNCT
ejpam-109	97	7	then	then	ADV
ejpam-109	97	8	pbm	pbm	NOUN
ejpam-109	97	9	∈	∈	PROPN
ejpam-109	97	10	zp	zp	NOUN
ejpam-109	97	11	,	,	PUNCT
ejpam-109	97	12	precisely	precisely	ADV
ejpam-109	97	13	,	,	PUNCT
ejpam-109	97	14	pbm	pbm	NOUN
ejpam-109	97	15	≡	≡	PROPN
ejpam-109	97	16	p−	p−	NOUN
ejpam-109	97	17	1	1	NUM
ejpam-109	97	18	(	(	PUNCT
ejpam-109	97	19	mod	mod	NOUN
ejpam-109	97	20	p	p	NOUN
ejpam-109	97	21	)	)	PUNCT
ejpam-109	97	22	.	.	PUNCT
ejpam-109	98	1	(	(	PUNCT
ejpam-109	98	2	i	i	NOUN
ejpam-109	98	3	)	)	PUNCT
ejpam-109	98	4	more	more	ADV
ejpam-109	98	5	generally	generally	ADV
ejpam-109	98	6	,	,	PUNCT
ejpam-109	98	7	if	if	SCONJ
ejpam-109	98	8	p	p	NOUN
ejpam-109	98	9	is	be	AUX
ejpam-109	98	10	an	an	DET
ejpam-109	98	11	odd	odd	ADJ
ejpam-109	98	12	prime	prime	NOUN
ejpam-109	98	13	and	and	CCONJ
ejpam-109	98	14	m=	m=	VERB
ejpam-109	98	15	kϕ(ps	kϕ(ps	NOUN
ejpam-109	98	16	)	)	PUNCT
ejpam-109	98	17	(	(	PUNCT
ejpam-109	98	18	k	k	X
ejpam-109	98	19	,	,	PUNCT
ejpam-109	98	20	s	s	X
ejpam-109	98	21	≥	≥	NOUN
ejpam-109	98	22	1	1	NUM
ejpam-109	98	23	)	)	PUNCT
ejpam-109	98	24	,	,	PUNCT
ejpam-109	98	25	then	then	ADV
ejpam-109	98	26	pbm	pbm	NOUN
ejpam-109	98	27	≡	≡	PROPN
ejpam-109	98	28	p−	p−	NOUN
ejpam-109	98	29	1	1	NUM
ejpam-109	98	30	(	(	PUNCT
ejpam-109	98	31	mod	mod	PROPN
ejpam-109	98	32	ps	ps	PROPN
ejpam-109	98	33	)	)	PUNCT
ejpam-109	98	34	.	.	PUNCT
ejpam-109	99	1	(	(	PUNCT
ejpam-109	99	2	ii	ii	NOUN
ejpam-109	99	3	)	)	PUNCT
ejpam-109	99	4	above	above	ADP
ejpam-109	99	5	congruence	congruence	NOUN
ejpam-109	99	6	(	(	PUNCT
ejpam-109	99	7	i	i	NOUN
ejpam-109	99	8	)	)	PUNCT
ejpam-109	99	9	describes	describe	VERB
ejpam-109	99	10	an	an	DET
ejpam-109	99	11	explicit	explicit	ADJ
ejpam-109	99	12	shape	shape	NOUN
ejpam-109	99	13	of	of	ADP
ejpam-109	99	14	the	the	DET
ejpam-109	99	15	denominator	denominator	NOUN
ejpam-109	99	16	of	of	ADP
ejpam-109	99	17	bm	bm	PROPN
ejpam-109	99	18	for	for	ADP
ejpam-109	99	19	an	an	DET
ejpam-109	99	20	even	even	ADV
ejpam-109	99	21	integer	integer	NOUN
ejpam-109	99	22	m≥	m≥	PROPN
ejpam-109	99	23	2	2	NUM
ejpam-109	99	24	.	.	PUNCT
ejpam-109	99	25	to	to	PART
ejpam-109	99	26	be	be	AUX
ejpam-109	99	27	exact	exact	ADJ
ejpam-109	99	28	,	,	PUNCT
ejpam-109	99	29	bm	bm	PROPN
ejpam-109	99	30	is	be	AUX
ejpam-109	99	31	expressed	express	VERB
ejpam-109	99	32	in	in	ADP
ejpam-109	99	33	the	the	DET
ejpam-109	99	34	form	form	NOUN
ejpam-109	99	35	bm	bm	PROPN
ejpam-109	99	36	=	=	AUX
ejpam-109	99	37	ω(m)−	ω(m)−	ADJ
ejpam-109	99	38	∑	∑	PUNCT
ejpam-109	99	39	p−1|m	p−1|m	NOUN
ejpam-109	99	40	1	1	NUM
ejpam-109	99	41	p	p	NOUN
ejpam-109	99	42	,	,	PUNCT
ejpam-109	99	43	where	where	SCONJ
ejpam-109	99	44	ω(m	ω(m	NOUN
ejpam-109	99	45	)	)	PUNCT
ejpam-109	99	46	is	be	AUX
ejpam-109	99	47	an	an	DET
ejpam-109	99	48	integer	integer	NOUN
ejpam-109	99	49	uniquely	uniquely	ADV
ejpam-109	99	50	determined	determine	VERB
ejpam-109	99	51	depending	depend	VERB
ejpam-109	99	52	on	on	ADP
ejpam-109	99	53	m	m	PRON
ejpam-109	99	54	and	and	CCONJ
ejpam-109	99	55	the	the	DET
ejpam-109	99	56	sum	sum	NOUN
ejpam-109	99	57	runs	run	VERB
ejpam-109	99	58	over	over	ADP
ejpam-109	99	59	all	all	DET
ejpam-109	99	60	the	the	DET
ejpam-109	99	61	primes	prime	NOUN
ejpam-109	99	62	p	p	NOUN
ejpam-109	99	63	such	such	ADJ
ejpam-109	100	1	that	that	SCONJ
ejpam-109	100	2	p	p	NOUN
ejpam-109	100	3	−	−	PROPN
ejpam-109	100	4	1	1	NUM
ejpam-109	100	5	|	|	ADV
ejpam-109	100	6	m.	m.	NOUN
ejpam-109	100	7	on	on	ADP
ejpam-109	100	8	the	the	DET
ejpam-109	100	9	one	one	NUM
ejpam-109	100	10	hand	hand	NOUN
ejpam-109	100	11	,	,	PUNCT
ejpam-109	100	12	congruence	congruence	PROPN
ejpam-109	100	13	(	(	PUNCT
ejpam-109	100	14	ii	ii	NOUN
ejpam-109	100	15	)	)	PUNCT
ejpam-109	100	16	asserts	assert	VERB
ejpam-109	100	17	that	that	SCONJ
ejpam-109	100	18	if	if	SCONJ
ejpam-109	100	19	m	m	NOUN
ejpam-109	100	20	=	=	NOUN
ejpam-109	100	21	kϕ(ps	kϕ(ps	NOUN
ejpam-109	100	22	)	)	PUNCT
ejpam-109	100	23	(	(	PUNCT
ejpam-109	100	24	k	k	X
ejpam-109	100	25	,	,	PUNCT
ejpam-109	100	26	s	s	X
ejpam-109	100	27	≥	≥	NOUN
ejpam-109	100	28	1	1	NUM
ejpam-109	100	29	)	)	PUNCT
ejpam-109	100	30	for	for	ADP
ejpam-109	100	31	an	an	DET
ejpam-109	100	32	odd	odd	ADJ
ejpam-109	100	33	prime	prime	NOUN
ejpam-109	100	34	p	p	NOUN
ejpam-109	100	35	,	,	PUNCT
ejpam-109	100	36	then	then	ADV
ejpam-109	100	37	the	the	DET
ejpam-109	100	38	above	above	ADP
ejpam-109	100	39	ω(m	ω(m	NOUN
ejpam-109	100	40	)	)	PUNCT
ejpam-109	100	41	satisfies	satisfie	NOUN
ejpam-109	100	42	ω(m)≡	ω(m)≡	ADP
ejpam-109	100	43	1	1	NUM
ejpam-109	100	44	+	+	NUM
ejpam-109	100	45	∑	∑	ADV
ejpam-109	100	46	q−1|m	q−1|m	ADJ
ejpam-109	100	47	q	q	NOUN
ejpam-109	100	48	6	6	NUM
ejpam-109	100	49	=	=	NOUN
ejpam-109	100	50	p	p	NOUN
ejpam-109	100	51	1	1	NUM
ejpam-109	100	52	q	q	NOUN
ejpam-109	100	53	(	(	PUNCT
ejpam-109	100	54	mod	mod	PROPN
ejpam-109	100	55	ps−1	ps−1	PROPN
ejpam-109	100	56	)	)	PUNCT
ejpam-109	100	57	t.	t.	NOUN
ejpam-109	100	58	agoh	agoh	PROPN
ejpam-109	100	59	/	/	SYM
ejpam-109	100	60	eur	eur	PROPN
ejpam-109	100	61	.	.	PUNCT
ejpam-109	101	1	j.	j.	PROPN
ejpam-109	101	2	pure	pure	PROPN
ejpam-109	101	3	appl	appl	PROPN
ejpam-109	101	4	.	.	PROPN
ejpam-109	101	5	math	math	PROPN
ejpam-109	101	6	,	,	PUNCT
ejpam-109	101	7	1	1	NUM
ejpam-109	101	8	(	(	PUNCT
ejpam-109	101	9	2008	2008	NUM
ejpam-109	101	10	)	)	PUNCT
ejpam-109	101	11	,	,	PUNCT
ejpam-109	101	12	(	(	PUNCT
ejpam-109	101	13	3	3	NUM
ejpam-109	101	14	-	-	SYM
ejpam-109	101	15	21	21	NUM
ejpam-109	101	16	)	)	PUNCT
ejpam-109	101	17	7	7	NUM
ejpam-109	101	18	with	with	ADP
ejpam-109	101	19	the	the	DET
ejpam-109	101	20	sum	sum	NOUN
ejpam-109	101	21	taken	take	VERB
ejpam-109	101	22	over	over	ADP
ejpam-109	101	23	all	all	DET
ejpam-109	101	24	the	the	DET
ejpam-109	101	25	primes	prime	NOUN
ejpam-109	101	26	q	q	NOUN
ejpam-109	102	1	such	such	ADJ
ejpam-109	102	2	that	that	SCONJ
ejpam-109	102	3	q−	q−	PROPN
ejpam-109	102	4	1	1	NUM
ejpam-109	102	5	|	|	ADV
ejpam-109	102	6	m	m	VERB
ejpam-109	102	7	and	and	CCONJ
ejpam-109	102	8	q	q	ADJ
ejpam-109	102	9	6=	6=	NUM
ejpam-109	103	1	p.	p.	NOUN
ejpam-109	103	2	the	the	DET
ejpam-109	103	3	congruences	congruence	NOUN
ejpam-109	103	4	mentioned	mention	VERB
ejpam-109	103	5	below	below	ADV
ejpam-109	103	6	are	be	AUX
ejpam-109	103	7	widely	widely	ADV
ejpam-109	103	8	called	call	VERB
ejpam-109	103	9	kummer	kummer	NOUN
ejpam-109	103	10	’s	’s	PART
ejpam-109	103	11	congruences	congruence	NOUN
ejpam-109	103	12	for	for	ADP
ejpam-109	103	13	bernoulli	bernoulli	NOUN
ejpam-109	103	14	numbers	number	NOUN
ejpam-109	103	15	.	.	PUNCT
ejpam-109	104	1	however	however	ADV
ejpam-109	104	2	,	,	PUNCT
ejpam-109	104	3	if	if	SCONJ
ejpam-109	104	4	we	we	PRON
ejpam-109	104	5	comply	comply	VERB
ejpam-109	104	6	slavutskii	slavutskii	PROPN
ejpam-109	104	7	’s	’s	PART
ejpam-109	104	8	request	request	NOUN
ejpam-109	104	9	written	write	VERB
ejpam-109	104	10	in	in	ADP
ejpam-109	104	11	[	[	X
ejpam-109	104	12	15	15	NUM
ejpam-109	104	13	]	]	PUNCT
ejpam-109	104	14	,	,	PUNCT
ejpam-109	104	15	it	it	PRON
ejpam-109	104	16	should	should	AUX
ejpam-109	104	17	be	be	AUX
ejpam-109	104	18	called	call	VERB
ejpam-109	104	19	von	von	PROPN
ejpam-109	104	20	staudt	staudt	PROPN
ejpam-109	104	21	-	-	PUNCT
ejpam-109	104	22	kummer	kummer	NOUN
ejpam-109	104	23	’s	’s	PART
ejpam-109	104	24	congruences	congruence	NOUN
ejpam-109	104	25	,	,	PUNCT
ejpam-109	104	26	because	because	SCONJ
ejpam-109	104	27	von	von	PROPN
ejpam-109	104	28	staudt	staudt	PROPN
ejpam-109	104	29	was	be	AUX
ejpam-109	104	30	the	the	DET
ejpam-109	104	31	first	first	ADJ
ejpam-109	104	32	contributor	contributor	NOUN
ejpam-109	104	33	who	who	PRON
ejpam-109	104	34	discovered	discover	VERB
ejpam-109	104	35	and	and	CCONJ
ejpam-109	104	36	investigated	investigate	VERB
ejpam-109	104	37	in	in	ADP
ejpam-109	104	38	details	detail	NOUN
ejpam-109	104	39	before	before	ADP
ejpam-109	104	40	kummer	kummer	NOUN
ejpam-109	104	41	.	.	PUNCT
ejpam-109	105	1	theorem	theorem	VERB
ejpam-109	105	2	2.6	2.6	NUM
ejpam-109	105	3	.	.	PUNCT
ejpam-109	106	1	let	let	VERB
ejpam-109	106	2	p	p	PRON
ejpam-109	106	3	be	be	AUX
ejpam-109	106	4	an	an	DET
ejpam-109	106	5	odd	odd	ADJ
ejpam-109	106	6	prime	prime	NOUN
ejpam-109	106	7	,	,	PUNCT
ejpam-109	106	8	m	m	VERB
ejpam-109	106	9	an	an	DET
ejpam-109	106	10	even	even	ADV
ejpam-109	106	11	integer≥	integer≥	NOUN
ejpam-109	106	12	2	2	NUM
ejpam-109	106	13	and	and	CCONJ
ejpam-109	106	14	s	s	VERB
ejpam-109	106	15	an	an	DET
ejpam-109	106	16	integer	integer	NOUN
ejpam-109	106	17	with	with	ADP
ejpam-109	106	18	1≤	1≤	NUM
ejpam-109	106	19	s	s	PART
ejpam-109	106	20	≤	≤	NOUN
ejpam-109	106	21	m−1	m−1	PROPN
ejpam-109	106	22	.	.	PUNCT
ejpam-109	107	1	if	if	SCONJ
ejpam-109	107	2	βi(a	βi(a	PRON
ejpam-109	107	3	)	)	PUNCT
ejpam-109	107	4	=	=	PUNCT
ejpam-109	107	5	(	(	PUNCT
ejpam-109	107	6	ai	ai	VERB
ejpam-109	107	7	−	−	PROPN
ejpam-109	108	1	1)βi	1)βi	PROPN
ejpam-109	108	2	(	(	PUNCT
ejpam-109	108	3	i	i	PRON
ejpam-109	108	4	≥	≥	VERB
ejpam-109	108	5	1	1	NUM
ejpam-109	108	6	)	)	PUNCT
ejpam-109	108	7	for	for	ADP
ejpam-109	108	8	a	a	DET
ejpam-109	108	9	positive	positive	ADJ
ejpam-109	108	10	integer	integer	NOUN
ejpam-109	108	11	a	a	PRON
ejpam-109	108	12	with	with	ADP
ejpam-109	108	13	p	p	PRON
ejpam-109	108	14	a	a	DET
ejpam-109	108	15	,	,	PUNCT
ejpam-109	108	16	then	then	ADV
ejpam-109	108	17	βm(a	βm(a	NOUN
ejpam-109	108	18	)	)	PUNCT
ejpam-109	108	19	�	�	PROPN
ejpam-109	109	1	β	β	X
ejpam-109	109	2	p−1(a)−	p−1(a)−	NOUN
ejpam-109	109	3	1	1	NUM
ejpam-109	109	4	�	�	PROPN
ejpam-109	109	5	s	s	PART
ejpam-109	109	6	≡	≡	PROPN
ejpam-109	109	7	0	0	PUNCT
ejpam-109	110	1	(	(	PUNCT
ejpam-109	110	2	mod	mod	PROPN
ejpam-109	110	3	ps	ps	PROPN
ejpam-109	110	4	)	)	PUNCT
ejpam-109	110	5	.	.	PUNCT
ejpam-109	111	1	(	(	PUNCT
ejpam-109	111	2	i	i	NOUN
ejpam-109	111	3	)	)	PUNCT
ejpam-109	111	4	in	in	ADP
ejpam-109	111	5	particular	particular	ADJ
ejpam-109	111	6	,	,	PUNCT
ejpam-109	111	7	if	if	SCONJ
ejpam-109	111	8	p−	p−	NOUN
ejpam-109	111	9	1	1	NUM
ejpam-109	111	10	m	m	NOUN
ejpam-109	111	11	,	,	PUNCT
ejpam-109	111	12	then	then	ADV
ejpam-109	111	13	βm	βm	VERB
ejpam-109	111	14	�	�	PROPN
ejpam-109	111	15	β	β	X
ejpam-109	111	16	p−1−	p−1−	PROPN
ejpam-109	111	17	1	1	NUM
ejpam-109	111	18	�	�	PROPN
ejpam-109	111	19	s	s	PART
ejpam-109	111	20	≡	≡	PROPN
ejpam-109	111	21	0	0	PUNCT
ejpam-109	112	1	(	(	PUNCT
ejpam-109	112	2	mod	mod	PROPN
ejpam-109	112	3	ps	ps	PROPN
ejpam-109	112	4	)	)	PUNCT
ejpam-109	112	5	.	.	PUNCT
ejpam-109	113	1	(	(	PUNCT
ejpam-109	113	2	ii	ii	X
ejpam-109	113	3	)	)	PUNCT
ejpam-109	113	4	the	the	DET
ejpam-109	113	5	symbolic	symbolic	ADJ
ejpam-109	113	6	notation	notation	NOUN
ejpam-109	113	7	used	use	VERB
ejpam-109	113	8	in	in	ADP
ejpam-109	113	9	above	above	ADP
ejpam-109	113	10	congruences	congruence	NOUN
ejpam-109	113	11	should	should	AUX
ejpam-109	113	12	be	be	AUX
ejpam-109	113	13	understood	understand	VERB
ejpam-109	113	14	that	that	SCONJ
ejpam-109	113	15	we	we	PRON
ejpam-109	113	16	expand	expand	VERB
ejpam-109	113	17	in	in	ADP
ejpam-109	113	18	full	full	ADJ
ejpam-109	113	19	the	the	DET
ejpam-109	113	20	left	left	ADJ
ejpam-109	113	21	-	-	PUNCT
ejpam-109	113	22	hand	hand	NOUN
ejpam-109	113	23	side	side	NOUN
ejpam-109	113	24	of	of	ADP
ejpam-109	113	25	(	(	PUNCT
ejpam-109	113	26	i	i	NOUN
ejpam-109	113	27	)	)	PUNCT
ejpam-109	113	28	(	(	PUNCT
ejpam-109	113	29	resp	resp	NOUN
ejpam-109	113	30	.	.	PUNCT
ejpam-109	114	1	(	(	PUNCT
ejpam-109	114	2	ii	ii	NOUN
ejpam-109	114	3	)	)	PUNCT
ejpam-109	114	4	)	)	PUNCT
ejpam-109	115	1	using	use	VERB
ejpam-109	115	2	the	the	DET
ejpam-109	115	3	binomial	binomial	ADJ
ejpam-109	115	4	theorem	theorem	NOUN
ejpam-109	115	5	and	and	CCONJ
ejpam-109	115	6	β	β	X
ejpam-109	115	7	i(a	i(a	PROPN
ejpam-109	115	8	)	)	PUNCT
ejpam-109	115	9	(	(	PUNCT
ejpam-109	115	10	resp.β	resp.β	PROPN
ejpam-109	115	11	i	i	NOUN
ejpam-109	115	12	)	)	PUNCT
ejpam-109	115	13	is	be	AUX
ejpam-109	115	14	to	to	PART
ejpam-109	115	15	be	be	AUX
ejpam-109	115	16	replaced	replace	VERB
ejpam-109	115	17	by	by	ADP
ejpam-109	115	18	βi(a	βi(a	NOUN
ejpam-109	115	19	)	)	PUNCT
ejpam-109	115	20	(	(	PUNCT
ejpam-109	115	21	resp.βi	resp.βi	NUM
ejpam-109	115	22	)	)	PUNCT
ejpam-109	115	23	for	for	ADP
ejpam-109	115	24	all	all	PRON
ejpam-109	116	1	i	i	PRON
ejpam-109	116	2	=	=	PUNCT
ejpam-109	116	3	m+	m+	NUM
ejpam-109	116	4	c(p−	c(p−	PROPN
ejpam-109	116	5	1	1	NUM
ejpam-109	116	6	)	)	PUNCT
ejpam-109	116	7	,	,	PUNCT
ejpam-109	116	8	c	c	NOUN
ejpam-109	116	9	=	=	SYM
ejpam-109	116	10	0	0	NUM
ejpam-109	116	11	,	,	PUNCT
ejpam-109	116	12	1	1	NUM
ejpam-109	116	13	,	,	PUNCT
ejpam-109	116	14	...	...	PUNCT
ejpam-109	116	15	,	,	PUNCT
ejpam-109	116	16	s.	s.	PROPN
ejpam-109	116	17	that	that	PRON
ejpam-109	116	18	is	be	AUX
ejpam-109	116	19	,	,	PUNCT
ejpam-109	116	20	above	above	ADP
ejpam-109	116	21	(	(	PUNCT
ejpam-109	116	22	i	i	NOUN
ejpam-109	116	23	)	)	PUNCT
ejpam-109	116	24	is	be	AUX
ejpam-109	116	25	exactly	exactly	ADV
ejpam-109	116	26	the	the	DET
ejpam-109	116	27	same	same	ADJ
ejpam-109	116	28	as	as	ADP
ejpam-109	116	29	the	the	DET
ejpam-109	116	30	congruence	congruence	NOUN
ejpam-109	116	31	s	s	X
ejpam-109	116	32	∑	∑	PUNCT
ejpam-109	116	33	c=0	c=0	X
ejpam-109	116	34	(	(	PUNCT
ejpam-109	116	35	−1)c	−1)c	NUM
ejpam-109	116	36	�	�	PROPN
ejpam-109	116	37	s	s	PART
ejpam-109	116	38	c	c	X
ejpam-109	116	39	�	�	X
ejpam-109	116	40	βm+c(p−1)(a)≡	βm+c(p−1)(a)≡	PUNCT
ejpam-109	116	41	0	0	NUM
ejpam-109	116	42	(	(	PUNCT
ejpam-109	116	43	mod	mod	PROPN
ejpam-109	116	44	ps	ps	PROPN
ejpam-109	116	45	)	)	PUNCT
ejpam-109	116	46	,	,	PUNCT
ejpam-109	116	47	and	and	CCONJ
ejpam-109	116	48	also	also	ADV
ejpam-109	116	49	(	(	PUNCT
ejpam-109	116	50	ii	ii	NOUN
ejpam-109	116	51	)	)	PUNCT
ejpam-109	116	52	is	be	AUX
ejpam-109	116	53	the	the	DET
ejpam-109	116	54	congruence	congruence	NOUN
ejpam-109	116	55	exchanged	exchange	VERB
ejpam-109	116	56	here	here	ADV
ejpam-109	116	57	formally	formally	ADV
ejpam-109	116	58	βm+c(p−1)(a	βm+c(p−1)(a	ADJ
ejpam-109	116	59	)	)	PUNCT
ejpam-109	116	60	for	for	ADP
ejpam-109	116	61	βm+c(p−1	βm+c(p−1	NOUN
ejpam-109	116	62	)	)	PUNCT
ejpam-109	116	63	.	.	PUNCT
ejpam-109	117	1	note	note	VERB
ejpam-109	117	2	that	that	SCONJ
ejpam-109	117	3	congruence	congruence	PROPN
ejpam-109	117	4	(	(	PUNCT
ejpam-109	117	5	ii	ii	NOUN
ejpam-109	117	6	)	)	PUNCT
ejpam-109	117	7	is	be	AUX
ejpam-109	117	8	nothing	nothing	PRON
ejpam-109	117	9	but	but	SCONJ
ejpam-109	117	10	a	a	DET
ejpam-109	117	11	special	special	ADJ
ejpam-109	117	12	case	case	NOUN
ejpam-109	117	13	of	of	ADP
ejpam-109	117	14	(	(	PUNCT
ejpam-109	117	15	i	i	NOUN
ejpam-109	117	16	)	)	PUNCT
ejpam-109	117	17	.	.	PUNCT
ejpam-109	118	1	in	in	ADP
ejpam-109	118	2	fact	fact	NOUN
ejpam-109	118	3	,	,	PUNCT
ejpam-109	118	4	taking	take	VERB
ejpam-109	118	5	a	a	DET
ejpam-109	118	6	primitive	primitive	ADJ
ejpam-109	118	7	root	root	NOUN
ejpam-109	118	8	γ	γ	NOUN
ejpam-109	118	9	of	of	ADP
ejpam-109	118	10	p	p	NOUN
ejpam-109	118	11	and	and	CCONJ
ejpam-109	118	12	choosing	choose	VERB
ejpam-109	118	13	an	an	DET
ejpam-109	118	14	integer	integer	NOUN
ejpam-109	118	15	a	a	DET
ejpam-109	118	16	≥	≥	NUM
ejpam-109	118	17	1	1	NUM
ejpam-109	118	18	such	such	ADJ
ejpam-109	118	19	that	that	SCONJ
ejpam-109	118	20	a	a	DET
ejpam-109	118	21	≡	≡	PROPN
ejpam-109	118	22	γps−1	γps−1	PROPN
ejpam-109	118	23	(	(	PUNCT
ejpam-109	118	24	mod	mod	PROPN
ejpam-109	118	25	ps	ps	PROPN
ejpam-109	118	26	)	)	PUNCT
ejpam-109	118	27	,	,	PUNCT
ejpam-109	118	28	we	we	PRON
ejpam-109	118	29	have	have	VERB
ejpam-109	118	30	ap−1	ap−1	PROPN
ejpam-109	118	31	≡	≡	PROPN
ejpam-109	118	32	γϕ(p	γϕ(p	PRON
ejpam-109	118	33	s	s	X
ejpam-109	118	34	)	)	PUNCT
ejpam-109	118	35	≡	≡	PROPN
ejpam-109	118	36	1	1	NUM
ejpam-109	118	37	(	(	PUNCT
ejpam-109	118	38	mod	mod	PROPN
ejpam-109	118	39	ps	ps	PROPN
ejpam-109	118	40	)	)	PUNCT
ejpam-109	118	41	by	by	ADP
ejpam-109	118	42	euler	euler	PROPN
ejpam-109	118	43	’s	’s	PART
ejpam-109	118	44	theorem	theorem	PROPN
ejpam-109	118	45	.	.	PUNCT
ejpam-109	119	1	since	since	SCONJ
ejpam-109	119	2	p	p	PROPN
ejpam-109	119	3	a	a	PROPN
ejpam-109	119	4	and	and	CCONJ
ejpam-109	119	5	p−1	p−1	PROPN
ejpam-109	119	6	m	m	PROPN
ejpam-109	119	7	,	,	PUNCT
ejpam-109	119	8	we	we	PRON
ejpam-109	119	9	also	also	ADV
ejpam-109	119	10	have	have	AUX
ejpam-109	119	11	am	be	AUX
ejpam-109	119	12	≡	≡	PROPN
ejpam-109	119	13	am+c(p−1	am+c(p−1	PROPN
ejpam-109	119	14	)	)	PUNCT
ejpam-109	119	15	(	(	PUNCT
ejpam-109	119	16	mod	mod	PROPN
ejpam-109	119	17	ps	ps	PROPN
ejpam-109	119	18	)	)	PUNCT
ejpam-109	119	19	for	for	ADP
ejpam-109	119	20	each	each	DET
ejpam-109	119	21	c	c	NOUN
ejpam-109	119	22	and	and	CCONJ
ejpam-109	119	23	am	be	AUX
ejpam-109	119	24	6≡	6≡	NUM
ejpam-109	119	25	1	1	NUM
ejpam-109	119	26	(	(	PUNCT
ejpam-109	119	27	mod	mod	NOUN
ejpam-109	119	28	p	p	NOUN
ejpam-109	119	29	)	)	PUNCT
ejpam-109	119	30	.	.	PUNCT
ejpam-109	120	1	so	so	ADV
ejpam-109	120	2	,	,	PUNCT
ejpam-109	120	3	dividing	dividing	NOUN
ejpam-109	120	4	(	(	PUNCT
ejpam-109	120	5	i	i	NOUN
ejpam-109	120	6	)	)	PUNCT
ejpam-109	120	7	by	by	ADP
ejpam-109	120	8	am−	am−	NUM
ejpam-109	120	9	1	1	NUM
ejpam-109	120	10	,	,	PUNCT
ejpam-109	120	11	we	we	PRON
ejpam-109	120	12	can	can	AUX
ejpam-109	120	13	immediately	immediately	ADV
ejpam-109	120	14	deduce	deduce	VERB
ejpam-109	120	15	(	(	PUNCT
ejpam-109	120	16	ii	ii	NOUN
ejpam-109	120	17	)	)	PUNCT
ejpam-109	120	18	.	.	PUNCT
ejpam-109	121	1	here	here	ADV
ejpam-109	121	2	,	,	PUNCT
ejpam-109	121	3	we	we	PRON
ejpam-109	121	4	would	would	AUX
ejpam-109	121	5	like	like	VERB
ejpam-109	121	6	to	to	PART
ejpam-109	121	7	comment	comment	VERB
ejpam-109	121	8	that	that	SCONJ
ejpam-109	121	9	a	a	DET
ejpam-109	121	10	similar	similar	ADJ
ejpam-109	121	11	type	type	NOUN
ejpam-109	121	12	congruence	congruence	NOUN
ejpam-109	121	13	to	to	ADP
ejpam-109	121	14	above	above	ADP
ejpam-109	121	15	(	(	PUNCT
ejpam-109	121	16	ii	ii	NOUN
ejpam-109	121	17	)	)	PUNCT
ejpam-109	121	18	without	without	ADP
ejpam-109	121	19	any	any	DET
ejpam-109	121	20	restriction	restriction	NOUN
ejpam-109	121	21	on	on	ADP
ejpam-109	121	22	m	m	PROPN
ejpam-109	121	23	has	have	AUX
ejpam-109	121	24	been	be	AUX
ejpam-109	121	25	obtained	obtain	VERB
ejpam-109	121	26	by	by	ADP
ejpam-109	121	27	von	von	PROPN
ejpam-109	121	28	staudt	staudt	PROPN
ejpam-109	121	29	.	.	PUNCT
ejpam-109	122	1	for	for	ADP
ejpam-109	122	2	its	its	PRON
ejpam-109	122	3	explicit	explicit	ADJ
ejpam-109	122	4	formula	formula	NOUN
ejpam-109	122	5	,	,	PUNCT
ejpam-109	122	6	see	see	VERB
ejpam-109	122	7	slavutskii	slavutskii	PROPN
ejpam-109	122	8	’s	’s	PART
ejpam-109	122	9	article	article	NOUN
ejpam-109	122	10	(	(	PUNCT
ejpam-109	122	11	(	(	PUNCT
ejpam-109	122	12	4	4	NUM
ejpam-109	122	13	)	)	PUNCT
ejpam-109	122	14	in	in	ADP
ejpam-109	122	15	[	[	X
ejpam-109	122	16	15	15	NUM
ejpam-109	122	17	]	]	NUM
ejpam-109	122	18	)	)	PUNCT
ejpam-109	122	19	.	.	PUNCT
ejpam-109	123	1	further	far	ADV
ejpam-109	123	2	,	,	PUNCT
ejpam-109	123	3	it	it	PRON
ejpam-109	123	4	should	should	AUX
ejpam-109	123	5	be	be	AUX
ejpam-109	123	6	noted	note	VERB
ejpam-109	123	7	that	that	SCONJ
ejpam-109	123	8	a	a	DET
ejpam-109	123	9	sequence	sequence	NOUN
ejpam-109	123	10	βi(a	βi(a	PUNCT
ejpam-109	123	11	)	)	PUNCT
ejpam-109	123	12	(	(	PUNCT
ejpam-109	123	13	i	i	PRON
ejpam-109	123	14	≥	≥	VERB
ejpam-109	123	15	1	1	NUM
ejpam-109	123	16	)	)	PUNCT
ejpam-109	123	17	defined	define	VERB
ejpam-109	123	18	above	above	ADV
ejpam-109	123	19	is	be	AUX
ejpam-109	123	20	well	well	ADV
ejpam-109	123	21	-	-	PUNCT
ejpam-109	123	22	known	know	VERB
ejpam-109	123	23	to	to	PART
ejpam-109	123	24	be	be	AUX
ejpam-109	123	25	the	the	DET
ejpam-109	123	26	sequence	sequence	NOUN
ejpam-109	123	27	of	of	ADP
ejpam-109	123	28	moments	moment	NOUN
ejpam-109	123	29	of	of	ADP
ejpam-109	123	30	a	a	DET
ejpam-109	123	31	p	p	NOUN
ejpam-109	123	32	-	-	PUNCT
ejpam-109	123	33	adic	adic	ADJ
ejpam-109	123	34	measure	measure	NOUN
ejpam-109	123	35	on	on	ADP
ejpam-109	123	36	the	the	DET
ejpam-109	123	37	ring	ring	NOUN
ejpam-109	123	38	zp	zp	PROPN
ejpam-109	123	39	.	.	PUNCT
ejpam-109	124	1	the	the	DET
ejpam-109	124	2	following	follow	VERB
ejpam-109	124	3	theorem	theorem	NOUN
ejpam-109	124	4	discovered	discover	VERB
ejpam-109	124	5	by	by	ADP
ejpam-109	124	6	e.	e.	PROPN
ejpam-109	124	7	lehmer	lehmer	PROPN
ejpam-109	124	8	[	[	X
ejpam-109	124	9	13	13	NUM
ejpam-109	124	10	]	]	PUNCT
ejpam-109	124	11	in	in	ADP
ejpam-109	124	12	1939	1939	NUM
ejpam-109	124	13	was	be	AUX
ejpam-109	124	14	very	very	ADV
ejpam-109	124	15	important	important	ADJ
ejpam-109	124	16	for	for	ADP
ejpam-109	124	17	the	the	DET
ejpam-109	124	18	irregularity	irregularity	NOUN
ejpam-109	124	19	testing	testing	NOUN
ejpam-109	124	20	of	of	ADP
ejpam-109	124	21	primes	prime	NOUN
ejpam-109	124	22	for	for	ADP
ejpam-109	124	23	many	many	ADJ
ejpam-109	124	24	years	year	NOUN
ejpam-109	124	25	.	.	PUNCT
ejpam-109	125	1	theorem	theorem	ADJ
ejpam-109	125	2	2.7	2.7	NUM
ejpam-109	125	3	.	.	PUNCT
ejpam-109	126	1	let	let	VERB
ejpam-109	126	2	p	p	PRON
ejpam-109	126	3	be	be	AUX
ejpam-109	126	4	an	an	DET
ejpam-109	126	5	odd	odd	ADJ
ejpam-109	126	6	prime	prime	NOUN
ejpam-109	126	7	and	and	CCONJ
ejpam-109	126	8	m	m	AUX
ejpam-109	126	9	be	be	AUX
ejpam-109	126	10	an	an	DET
ejpam-109	126	11	even	even	ADV
ejpam-109	126	12	integer	integer	NOUN
ejpam-109	126	13	≥	≥	NOUN
ejpam-109	126	14	2	2	NUM
ejpam-109	126	15	.	.	PUNCT
ejpam-109	126	16	also	also	ADV
ejpam-109	126	17	put	put	VERB
ejpam-109	126	18	q2(m	q2(m	PRON
ejpam-109	126	19	)	)	PUNCT
ejpam-109	126	20	=	=	PUNCT
ejpam-109	126	21	2	2	NUM
ejpam-109	126	22	m	m	NOUN
ejpam-109	126	23	−	−	NOUN
ejpam-109	126	24	1,q3(m	1,q3(m	NUM
ejpam-109	126	25	)	)	PUNCT
ejpam-109	127	1	=	=	SYM
ejpam-109	127	2	1	1	NUM
ejpam-109	127	3	2	2	NUM
ejpam-109	127	4	(	(	PUNCT
ejpam-109	127	5	3m−1),q4(m	3m−1),q4(m	NUM
ejpam-109	127	6	)	)	PUNCT
ejpam-109	127	7	=	=	SYM
ejpam-109	127	8	1	1	NUM
ejpam-109	127	9	2	2	NUM
ejpam-109	127	10	(	(	PUNCT
ejpam-109	127	11	2m−1)(2m−1−1	2m−1)(2m−1−1	NUM
ejpam-109	127	12	)	)	PUNCT
ejpam-109	127	13	and	and	CCONJ
ejpam-109	127	14	q6(m	q6(m	NOUN
ejpam-109	127	15	)	)	PUNCT
ejpam-109	127	16	=	=	SYM
ejpam-109	127	17	1	1	NUM
ejpam-109	127	18	2	2	NUM
ejpam-109	127	19	(	(	PUNCT
ejpam-109	127	20	6m−1	6m−1	NUM
ejpam-109	127	21	+	+	NOUN
ejpam-109	127	22	3m−1	3m−1	PROPN
ejpam-109	127	23	+	+	ADJ
ejpam-109	127	24	2m−1−1	2m−1−1	NUM
ejpam-109	127	25	)	)	PUNCT
ejpam-109	127	26	.	.	PUNCT
ejpam-109	128	1	if	if	SCONJ
ejpam-109	128	2	p−	p−	NOUN
ejpam-109	128	3	1	1	NUM
ejpam-109	128	4	m−	m−	PROPN
ejpam-109	128	5	2	2	NUM
ejpam-109	128	6	,	,	PUNCT
ejpam-109	128	7	then	then	ADV
ejpam-109	128	8	qk(m)βm	qk(m)βm	NOUN
ejpam-109	128	9	≡	≡	PROPN
ejpam-109	128	10	∑	∑	PUNCT
ejpam-109	128	11	0	0	PUNCT
ejpam-109	128	12	<	<	X
ejpam-109	128	13	i	i	X
ejpam-109	128	14	<	<	X
ejpam-109	128	15	p	p	PROPN
ejpam-109	128	16	/	/	SYM
ejpam-109	128	17	k	k	PROPN
ejpam-109	128	18	(	(	PUNCT
ejpam-109	128	19	p−	p−	INTJ
ejpam-109	128	20	ik)m−1	ik)m−1	NOUN
ejpam-109	128	21	(	(	PUNCT
ejpam-109	128	22	mod	mod	ADJ
ejpam-109	128	23	p2	p2	PROPN
ejpam-109	128	24	)	)	PUNCT
ejpam-109	128	25	,	,	PUNCT
ejpam-109	128	26	k	k	X
ejpam-109	128	27	=	=	SYM
ejpam-109	128	28	2,3	2,3	NUM
ejpam-109	128	29	,	,	PUNCT
ejpam-109	128	30	4,6	4,6	NUM
ejpam-109	128	31	,	,	PUNCT
ejpam-109	128	32	provided	provide	VERB
ejpam-109	128	33	that	that	SCONJ
ejpam-109	128	34	p	p	NOUN
ejpam-109	128	35	≥	≥	NUM
ejpam-109	128	36	7	7	NUM
ejpam-109	128	37	for	for	ADP
ejpam-109	128	38	k	k	PROPN
ejpam-109	128	39	=	=	SYM
ejpam-109	128	40	6	6	NUM
ejpam-109	128	41	.	.	PUNCT
ejpam-109	128	42	applying	apply	VERB
ejpam-109	128	43	the	the	DET
ejpam-109	128	44	above	above	ADJ
ejpam-109	128	45	congruence	congruence	NOUN
ejpam-109	128	46	for	for	ADP
ejpam-109	128	47	k	k	PROPN
ejpam-109	128	48	=	=	SYM
ejpam-109	128	49	2	2	NUM
ejpam-109	128	50	,	,	PUNCT
ejpam-109	128	51	lehmer	lehmer	NOUN
ejpam-109	128	52	showed	show	VERB
ejpam-109	128	53	that	that	SCONJ
ejpam-109	128	54	if	if	SCONJ
ejpam-109	128	55	p−	p−	NOUN
ejpam-109	128	56	1	1	NUM
ejpam-109	128	57	m−	m−	PROPN
ejpam-109	128	58	2	2	NUM
ejpam-109	128	59	,	,	PUNCT
ejpam-109	128	60	then	then	ADV
ejpam-109	128	61	2m−1pβm	2m−1pβm	NUM
ejpam-109	128	62	≡	≡	PROPN
ejpam-109	128	63	1	1	NUM
ejpam-109	128	64	m	m	NOUN
ejpam-109	128	65	∑	∑	PROPN
ejpam-109	128	66	0	0	PUNCT
ejpam-109	128	67	<	<	X
ejpam-109	128	68	i	i	X
ejpam-109	128	69	<	<	X
ejpam-109	128	70	p/2	p/2	X
ejpam-109	128	71	(	(	PUNCT
ejpam-109	128	72	p−	p−	NOUN
ejpam-109	128	73	2i)m	2i)m	NUM
ejpam-109	128	74	(	(	PUNCT
ejpam-109	128	75	mod	mod	PROPN
ejpam-109	128	76	p3	p3	PROPN
ejpam-109	128	77	)	)	PUNCT
ejpam-109	128	78	.	.	PUNCT
ejpam-109	129	1	t.	t.	PROPN
ejpam-109	129	2	agoh	agoh	PROPN
ejpam-109	129	3	/	/	SYM
ejpam-109	129	4	eur	eur	PROPN
ejpam-109	129	5	.	.	PUNCT
ejpam-109	130	1	j.	j.	PROPN
ejpam-109	130	2	pure	pure	PROPN
ejpam-109	130	3	appl	appl	PROPN
ejpam-109	130	4	.	.	PROPN
ejpam-109	130	5	math	math	PROPN
ejpam-109	130	6	,	,	PUNCT
ejpam-109	130	7	1	1	NUM
ejpam-109	130	8	(	(	PUNCT
ejpam-109	130	9	2008	2008	NUM
ejpam-109	130	10	)	)	PUNCT
ejpam-109	130	11	,	,	PUNCT
ejpam-109	130	12	(	(	PUNCT
ejpam-109	130	13	3	3	NUM
ejpam-109	130	14	-	-	SYM
ejpam-109	130	15	21	21	NUM
ejpam-109	130	16	)	)	PUNCT
ejpam-109	130	17	8	8	NUM
ejpam-109	130	18	we	we	PRON
ejpam-109	130	19	note	note	VERB
ejpam-109	130	20	that	that	SCONJ
ejpam-109	130	21	very	very	ADV
ejpam-109	130	22	simple	simple	ADJ
ejpam-109	130	23	and	and	CCONJ
ejpam-109	130	24	elegant	elegant	ADJ
ejpam-109	130	25	p	p	NOUN
ejpam-109	130	26	-	-	PUNCT
ejpam-109	130	27	adic	adic	ADJ
ejpam-109	130	28	proofs	proof	NOUN
ejpam-109	130	29	of	of	ADP
ejpam-109	130	30	above	above	ADP
ejpam-109	130	31	lehmer	lehmer	NOUN
ejpam-109	130	32	’s	’s	PART
ejpam-109	130	33	congruences	congruence	NOUN
ejpam-109	130	34	were	be	AUX
ejpam-109	130	35	given	give	VERB
ejpam-109	130	36	by	by	ADP
ejpam-109	130	37	johnson	johnson	PROPN
ejpam-109	131	1	[	[	X
ejpam-109	131	2	11	11	NUM
ejpam-109	131	3	]	]	PUNCT
ejpam-109	131	4	.	.	PUNCT
ejpam-109	132	1	as	as	SCONJ
ejpam-109	132	2	mentioned	mention	VERB
ejpam-109	132	3	in	in	ADP
ejpam-109	132	4	the	the	DET
ejpam-109	132	5	introduction	introduction	NOUN
ejpam-109	132	6	,	,	PUNCT
ejpam-109	132	7	jensen	jensen	PROPN
ejpam-109	133	1	[	[	X
ejpam-109	133	2	10	10	NUM
ejpam-109	133	3	]	]	PUNCT
ejpam-109	133	4	investigated	investigate	VERB
ejpam-109	133	5	the	the	DET
ejpam-109	133	6	distribution	distribution	NOUN
ejpam-109	133	7	of	of	ADP
ejpam-109	133	8	irregular	irregular	ADJ
ejpam-109	133	9	primes	prime	NOUN
ejpam-109	133	10	and	and	CCONJ
ejpam-109	133	11	proved	prove	VERB
ejpam-109	133	12	in	in	ADP
ejpam-109	133	13	1915	1915	NUM
ejpam-109	133	14	the	the	DET
ejpam-109	133	15	following	follow	VERB
ejpam-109	133	16	remarkable	remarkable	ADJ
ejpam-109	133	17	theorem	theorem	NOUN
ejpam-109	133	18	.	.	PUNCT
ejpam-109	133	19	theorem	theorem	NOUN
ejpam-109	133	20	2.8	2.8	NUM
ejpam-109	133	21	.	.	PUNCT
ejpam-109	134	1	there	there	PRON
ejpam-109	134	2	are	be	VERB
ejpam-109	134	3	infinitely	infinitely	ADV
ejpam-109	134	4	many	many	ADJ
ejpam-109	134	5	irregular	irregular	ADJ
ejpam-109	134	6	primes	prime	NOUN
ejpam-109	134	7	p	p	NOUN
ejpam-109	134	8	such	such	ADJ
ejpam-109	134	9	that	that	SCONJ
ejpam-109	134	10	p	p	PROPN
ejpam-109	134	11	≡	≡	PROPN
ejpam-109	134	12	3	3	NUM
ejpam-109	134	13	(	(	PUNCT
ejpam-109	134	14	mod	mod	NOUN
ejpam-109	134	15	4	4	NUM
ejpam-109	134	16	)	)	PUNCT
ejpam-109	134	17	.	.	PUNCT
ejpam-109	135	1	a	a	DET
ejpam-109	135	2	simple	simple	ADJ
ejpam-109	135	3	proof	proof	NOUN
ejpam-109	135	4	of	of	ADP
ejpam-109	135	5	the	the	DET
ejpam-109	135	6	weaker	weak	ADJ
ejpam-109	135	7	theorem	theorem	NOUN
ejpam-109	135	8	“	"	PUNCT
ejpam-109	135	9	there	there	PRON
ejpam-109	135	10	are	be	VERB
ejpam-109	135	11	infinitely	infinitely	ADV
ejpam-109	135	12	many	many	ADJ
ejpam-109	135	13	irregular	irregular	ADJ
ejpam-109	135	14	primes	prime	NOUN
ejpam-109	135	15	”	"	PUNCT
ejpam-109	135	16	was	be	AUX
ejpam-109	135	17	given	give	VERB
ejpam-109	135	18	by	by	ADP
ejpam-109	135	19	carlitz	carlitz	NOUN
ejpam-109	135	20	[	[	X
ejpam-109	135	21	6	6	NUM
ejpam-109	135	22	]	]	PUNCT
ejpam-109	135	23	.	.	PUNCT
ejpam-109	136	1	recently	recently	ADV
ejpam-109	136	2	,	,	PUNCT
ejpam-109	136	3	using	use	VERB
ejpam-109	136	4	multisectioning	multisectioning	NOUN
ejpam-109	136	5	and	and	CCONJ
ejpam-109	136	6	convolution	convolution	NOUN
ejpam-109	136	7	methods	method	NOUN
ejpam-109	136	8	,	,	PUNCT
ejpam-109	136	9	buhler	buhler	PROPN
ejpam-109	136	10	et	et	PROPN
ejpam-109	136	11	al	al	PROPN
ejpam-109	136	12	.	.	PUNCT
ejpam-109	137	1	[	[	X
ejpam-109	137	2	5	5	NUM
ejpam-109	137	3	]	]	PUNCT
ejpam-109	137	4	determined	determine	VERB
ejpam-109	137	5	all	all	DET
ejpam-109	137	6	the	the	DET
ejpam-109	137	7	irregular	irregular	ADJ
ejpam-109	137	8	primes	prime	NOUN
ejpam-109	137	9	less	less	ADJ
ejpam-109	137	10	than	than	ADP
ejpam-109	137	11	12	12	NUM
ejpam-109	137	12	×	×	NOUN
ejpam-109	137	13	106	106	NUM
ejpam-109	137	14	and	and	CCONJ
ejpam-109	137	15	their	their	PRON
ejpam-109	137	16	irregular	irregular	ADJ
ejpam-109	137	17	indices	index	NOUN
ejpam-109	137	18	i(p	i(p	NOUN
ejpam-109	137	19	)	)	PUNCT
ejpam-109	137	20	=	=	PUNCT
ejpam-109	137	21	#	#	SYM
ejpam-109	137	22	{	{	PUNCT
ejpam-109	137	23	m	m	PROPN
ejpam-109	137	24	|	|	NOUN
ejpam-109	137	25	b2	b2	NOUN
ejpam-109	137	26	m	m	NOUN
ejpam-109	137	27	≡	≡	PROPN
ejpam-109	137	28	0	0	PUNCT
ejpam-109	138	1	(	(	PUNCT
ejpam-109	138	2	mod	mod	PROPN
ejpam-109	138	3	p	p	X
ejpam-109	138	4	)	)	PUNCT
ejpam-109	138	5	,	,	PUNCT
ejpam-109	138	6	1≤	1≤	NUM
ejpam-109	138	7	m≤	m≤	NOUN
ejpam-109	138	8	(	(	PUNCT
ejpam-109	138	9	p−	p−	NOUN
ejpam-109	138	10	3)/2	3)/2	NUM
ejpam-109	138	11	}	}	PUNCT
ejpam-109	138	12	.	.	PUNCT
ejpam-109	139	1	3	3	X
ejpam-109	139	2	.	.	X
ejpam-109	139	3	voronoï	voronoï	ADJ
ejpam-109	139	4	type	type	NOUN
ejpam-109	139	5	congruences	congruence	NOUN
ejpam-109	139	6	in	in	ADP
ejpam-109	139	7	this	this	DET
ejpam-109	139	8	section	section	NOUN
ejpam-109	139	9	,	,	PUNCT
ejpam-109	139	10	we	we	PRON
ejpam-109	139	11	will	will	AUX
ejpam-109	139	12	first	first	ADV
ejpam-109	139	13	introduce	introduce	VERB
ejpam-109	139	14	some	some	DET
ejpam-109	139	15	generalized	generalized	ADJ
ejpam-109	139	16	voronoï	voronoï	ADJ
ejpam-109	139	17	type	type	NOUN
ejpam-109	139	18	congruences	congruence	NOUN
ejpam-109	139	19	.	.	PUNCT
ejpam-109	140	1	as	as	ADP
ejpam-109	140	2	one	one	NUM
ejpam-109	140	3	of	of	ADP
ejpam-109	140	4	applications	application	NOUN
ejpam-109	140	5	of	of	ADP
ejpam-109	140	6	these	these	PRON
ejpam-109	140	7	,	,	PUNCT
ejpam-109	140	8	we	we	PRON
ejpam-109	140	9	extend	extend	VERB
ejpam-109	140	10	the	the	DET
ejpam-109	140	11	von	von	PROPN
ejpam-109	140	12	staudt	staudt	PROPN
ejpam-109	140	13	-	-	PUNCT
ejpam-109	140	14	clausen	clausen	PROPN
ejpam-109	140	15	congruences	congruence	VERB
ejpam-109	140	16	to	to	ADP
ejpam-109	140	17	more	more	ADV
ejpam-109	140	18	general	general	ADJ
ejpam-109	140	19	situation	situation	NOUN
ejpam-109	140	20	.	.	PUNCT
ejpam-109	141	1	in	in	ADP
ejpam-109	141	2	addition	addition	NOUN
ejpam-109	141	3	,	,	PUNCT
ejpam-109	141	4	we	we	PRON
ejpam-109	141	5	study	study	VERB
ejpam-109	141	6	giuga	giuga	PROPN
ejpam-109	141	7	’s	’s	PART
ejpam-109	141	8	conjecture	conjecture	NOUN
ejpam-109	141	9	by	by	ADP
ejpam-109	141	10	means	mean	NOUN
ejpam-109	141	11	of	of	ADP
ejpam-109	141	12	bernoulli	bernoulli	NOUN
ejpam-109	141	13	numbers	number	NOUN
ejpam-109	141	14	.	.	PUNCT
ejpam-109	142	1	for	for	ADP
ejpam-109	142	2	a	a	DET
ejpam-109	142	3	positive	positive	ADJ
ejpam-109	142	4	integer	integer	NOUN
ejpam-109	142	5	n	n	PROPN
ejpam-109	142	6	and	and	CCONJ
ejpam-109	142	7	an	an	DET
ejpam-109	142	8	even	even	ADV
ejpam-109	142	9	integer	integer	NOUN
ejpam-109	142	10	m≥	m≥	PROPN
ejpam-109	142	11	2	2	NUM
ejpam-109	142	12	,	,	PUNCT
ejpam-109	142	13	we	we	PRON
ejpam-109	142	14	define	define	VERB
ejpam-109	142	15	δ	δ	PROPN
ejpam-109	142	16	=	=	SYM
ejpam-109	142	17	δ(m	δ(m	PROPN
ejpam-109	142	18	,	,	PUNCT
ejpam-109	142	19	n	n	CCONJ
ejpam-109	142	20	)	)	PUNCT
ejpam-109	142	21	=	=	SYM
ejpam-109	142	22	∏	∏	PROPN
ejpam-109	142	23	p|n	p|n	NOUN
ejpam-109	142	24	pup	pup	VERB
ejpam-109	142	25	�	�	PROPN
ejpam-109	142	26	where	where	SCONJ
ejpam-109	142	27	up	up	ADV
ejpam-109	142	28	=	=	SYM
ejpam-109	142	29	ordp(m	ordp(m	NUM
ejpam-109	142	30	)	)	PUNCT
ejpam-109	142	31	�	�	PROPN
ejpam-109	142	32	,	,	PUNCT
ejpam-109	142	33	ν	ν	X
ejpam-109	142	34	=	=	SYM
ejpam-109	142	35	ν(m	ν(m	NOUN
ejpam-109	142	36	,	,	PUNCT
ejpam-109	142	37	n	n	CCONJ
ejpam-109	142	38	)	)	PUNCT
ejpam-109	142	39	=	=	SYM
ejpam-109	142	40	∏	∏	PROPN
ejpam-109	142	41	p|n	p|n	NOUN
ejpam-109	142	42	pvp	pvp	NOUN
ejpam-109	142	43	�	�	PROPN
ejpam-109	142	44	where	where	SCONJ
ejpam-109	142	45	vp	vp	PROPN
ejpam-109	142	46	=	=	SYM
ejpam-109	142	47	ordp(dm	ordp(dm	PROPN
ejpam-109	142	48	)	)	PUNCT
ejpam-109	142	49	�	�	PROPN
ejpam-109	142	50	.	.	PUNCT
ejpam-109	143	1	particularly	particularly	ADV
ejpam-109	143	2	,	,	PUNCT
ejpam-109	143	3	if	if	SCONJ
ejpam-109	143	4	n=	n=	ADJ
ejpam-109	143	5	1	1	NUM
ejpam-109	143	6	,	,	PUNCT
ejpam-109	143	7	then	then	ADV
ejpam-109	143	8	we	we	PRON
ejpam-109	143	9	put	put	VERB
ejpam-109	143	10	δ	δ	X
ejpam-109	143	11	=	=	PUNCT
ejpam-109	143	12	ν	ν	X
ejpam-109	143	13	=	=	SYM
ejpam-109	143	14	1	1	NUM
ejpam-109	143	15	by	by	ADP
ejpam-109	143	16	convention	convention	NOUN
ejpam-109	143	17	.	.	PUNCT
ejpam-109	144	1	further	far	ADV
ejpam-109	144	2	,	,	PUNCT
ejpam-109	144	3	letting	let	VERB
ejpam-109	144	4	εm(n	εm(n	NOUN
ejpam-109	144	5	)	)	PUNCT
ejpam-109	144	6	=	=	PUNCT
ejpam-109	144	7			PROPN
ejpam-109	144	8			X
ejpam-109	144	9			ADJ
ejpam-109	144	10	1	1	NUM
ejpam-109	144	11	if	if	SCONJ
ejpam-109	144	12	n=	n=	ADJ
ejpam-109	144	13	1	1	NUM
ejpam-109	144	14	,	,	PUNCT
ejpam-109	144	15	∏	∏	NUM
ejpam-109	144	16	p|n	p|n	X
ejpam-109	144	17	(	(	PUNCT
ejpam-109	144	18	1−	1−	NUM
ejpam-109	144	19	pm−1	pm−1	NOUN
ejpam-109	144	20	)	)	PUNCT
ejpam-109	144	21	otherwise	otherwise	ADV
ejpam-109	144	22	,	,	PUNCT
ejpam-109	144	23	we	we	PRON
ejpam-109	144	24	define	define	VERB
ejpam-109	144	25	,	,	PUNCT
ejpam-109	144	26	for	for	ADP
ejpam-109	144	27	an	an	DET
ejpam-109	144	28	integer	integer	NOUN
ejpam-109	144	29	a	a	PRON
ejpam-109	144	30	>	>	X
ejpam-109	144	31	0	0	NUM
ejpam-109	144	32	,	,	PUNCT
ejpam-109	144	33	hm(n	hm(n	X
ejpam-109	144	34	)	)	PUNCT
ejpam-109	145	1	=	=	SYM
ejpam-109	145	2	εm(n)βm	εm(n)βm	NOUN
ejpam-109	145	3	,	,	PUNCT
ejpam-109	145	4	km(n	km(n	NOUN
ejpam-109	145	5	;	;	PUNCT
ejpam-109	145	6	a	a	X
ejpam-109	145	7	)	)	PUNCT
ejpam-109	145	8	=(	=(	NOUN
ejpam-109	145	9	am−	am−	NUM
ejpam-109	145	10	1)hm(n	1)hm(n	NUM
ejpam-109	145	11	)	)	PUNCT
ejpam-109	145	12	=	=	SYM
ejpam-109	145	13	εm(n)βm(a	εm(n)βm(a	PROPN
ejpam-109	145	14	)	)	PUNCT
ejpam-109	145	15	,	,	PUNCT
ejpam-109	145	16	h	h	NOUN
ejpam-109	145	17	′m(n	′m(n	PROPN
ejpam-109	145	18	)	)	PUNCT
ejpam-109	146	1	=	=	SYM
ejpam-109	146	2	mhm(n	mhm(n	X
ejpam-109	146	3	)	)	PUNCT
ejpam-109	146	4	=	=	PUNCT
ejpam-109	146	5	εm(n)bm	εm(n)bm	PROPN
ejpam-109	146	6	,	,	PUNCT
ejpam-109	146	7	k	k	PROPN
ejpam-109	146	8	′m(n	′m(n	PROPN
ejpam-109	146	9	;	;	PUNCT
ejpam-109	146	10	a	a	DET
ejpam-109	146	11	)	)	PUNCT
ejpam-109	146	12	=(	=(	NOUN
ejpam-109	146	13	am−	am−	NUM
ejpam-109	146	14	1)h	1)h	NUM
ejpam-109	146	15	′m(n	′m(n	NOUN
ejpam-109	146	16	)	)	PUNCT
ejpam-109	146	17	=	=	PUNCT
ejpam-109	146	18	εm(n)(a	εm(n)(a	NOUN
ejpam-109	146	19	m−	m−	PROPN
ejpam-109	146	20	1)bm	1)bm	NUM
ejpam-109	146	21	.	.	PUNCT
ejpam-109	147	1	to	to	PART
ejpam-109	147	2	deduce	deduce	VERB
ejpam-109	147	3	various	various	ADJ
ejpam-109	147	4	important	important	ADJ
ejpam-109	147	5	congruences	congruence	NOUN
ejpam-109	147	6	for	for	ADP
ejpam-109	147	7	composite	composite	ADJ
ejpam-109	147	8	moduli	modulus	NOUN
ejpam-109	147	9	,	,	PUNCT
ejpam-109	147	10	we	we	PRON
ejpam-109	147	11	introduce	introduce	VERB
ejpam-109	147	12	generalized	generalized	ADJ
ejpam-109	147	13	voronoï	voronoï	ADJ
ejpam-109	147	14	type	type	NOUN
ejpam-109	147	15	congruences	congruence	NOUN
ejpam-109	147	16	as	as	SCONJ
ejpam-109	147	17	follows	follow	VERB
ejpam-109	147	18	:	:	PUNCT
ejpam-109	147	19	t.	t.	NOUN
ejpam-109	147	20	agoh	agoh	PROPN
ejpam-109	147	21	/	/	SYM
ejpam-109	147	22	eur	eur	PROPN
ejpam-109	147	23	.	.	PUNCT
ejpam-109	148	1	j.	j.	PROPN
ejpam-109	148	2	pure	pure	PROPN
ejpam-109	148	3	appl	appl	PROPN
ejpam-109	148	4	.	.	PROPN
ejpam-109	148	5	math	math	PROPN
ejpam-109	148	6	,	,	PUNCT
ejpam-109	148	7	1	1	NUM
ejpam-109	148	8	(	(	PUNCT
ejpam-109	148	9	2008	2008	NUM
ejpam-109	148	10	)	)	PUNCT
ejpam-109	148	11	,	,	PUNCT
ejpam-109	148	12	(	(	PUNCT
ejpam-109	148	13	3	3	NUM
ejpam-109	148	14	-	-	SYM
ejpam-109	148	15	21	21	NUM
ejpam-109	148	16	)	)	PUNCT
ejpam-109	148	17	9	9	NUM
ejpam-109	148	18	theorem	theorem	VERB
ejpam-109	148	19	3.1	3.1	NUM
ejpam-109	148	20	.	.	PUNCT
ejpam-109	149	1	let	let	VERB
ejpam-109	149	2	m	m	PRON
ejpam-109	149	3	≥	≥	NOUN
ejpam-109	149	4	2	2	NUM
ejpam-109	149	5	be	be	AUX
ejpam-109	149	6	even	even	ADV
ejpam-109	149	7	and	and	CCONJ
ejpam-109	149	8	n	n	PRON
ejpam-109	149	9	≥	≥	NOUN
ejpam-109	149	10	1	1	NUM
ejpam-109	149	11	.	.	PUNCT
ejpam-109	150	1	also	also	ADV
ejpam-109	150	2	,	,	PUNCT
ejpam-109	150	3	let	let	VERB
ejpam-109	150	4	w	w	NOUN
ejpam-109	150	5	and	and	CCONJ
ejpam-109	150	6	w′	w′	PROPN
ejpam-109	150	7	be	be	VERB
ejpam-109	150	8	arbitrary	arbitrary	ADJ
ejpam-109	150	9	positive	positive	ADJ
ejpam-109	150	10	multiples	multiple	NOUN
ejpam-109	150	11	of	of	ADP
ejpam-109	150	12	nδν	nδν	NOUN
ejpam-109	150	13	and	and	CCONJ
ejpam-109	150	14	nν	nν	ADV
ejpam-109	150	15	,	,	PUNCT
ejpam-109	150	16	respectively	respectively	ADV
ejpam-109	150	17	.	.	PUNCT
ejpam-109	151	1	for	for	ADP
ejpam-109	151	2	a	a	DET
ejpam-109	151	3	positive	positive	ADJ
ejpam-109	151	4	integer	integer	NOUN
ejpam-109	151	5	a	a	DET
ejpam-109	151	6	with	with	ADP
ejpam-109	151	7	(	(	PUNCT
ejpam-109	151	8	a	a	DET
ejpam-109	151	9	,	,	PUNCT
ejpam-109	151	10	w	w	NOUN
ejpam-109	151	11	)	)	PUNCT
ejpam-109	151	12	=	=	SYM
ejpam-109	151	13	(	(	PUNCT
ejpam-109	151	14	a	a	PRON
ejpam-109	151	15	,	,	PUNCT
ejpam-109	151	16	w′	w′	NOUN
ejpam-109	151	17	)	)	PUNCT
ejpam-109	151	18	=	=	SYM
ejpam-109	151	19	1	1	NUM
ejpam-109	151	20	,	,	PUNCT
ejpam-109	151	21	we	we	PRON
ejpam-109	151	22	have	have	VERB
ejpam-109	151	23	km(n	km(n	NOUN
ejpam-109	151	24	;	;	PUNCT
ejpam-109	152	1	a)≡	a)≡	PROPN
ejpam-109	152	2	w−1	w−1	PROPN
ejpam-109	152	3	∑	∑	PUNCT
ejpam-109	152	4	j=1	j=1	PROPN
ejpam-109	152	5	(	(	PUNCT
ejpam-109	152	6	j	j	PROPN
ejpam-109	152	7	,	,	PUNCT
ejpam-109	152	8	n)=1	n)=1	PROPN
ejpam-109	152	9	(	(	PUNCT
ejpam-109	152	10	a	a	DET
ejpam-109	152	11	j)m−1	j)m−1	PROPN
ejpam-109	152	12	�	�	PROPN
ejpam-109	152	13	a	a	DET
ejpam-109	152	14	j	j	PROPN
ejpam-109	152	15	w	w	PROPN
ejpam-109	152	16	�	�	PROPN
ejpam-109	152	17	(	(	PUNCT
ejpam-109	152	18	mod	mod	PROPN
ejpam-109	152	19	n	n	CCONJ
ejpam-109	152	20	)	)	PUNCT
ejpam-109	152	21	,	,	PUNCT
ejpam-109	152	22	(	(	PUNCT
ejpam-109	152	23	i	i	NOUN
ejpam-109	152	24	)	)	PUNCT
ejpam-109	152	25	k	k	PROPN
ejpam-109	152	26	′m(n	′m(n	PROPN
ejpam-109	152	27	;	;	PUNCT
ejpam-109	152	28	a)≡m	a)≡m	PRON
ejpam-109	152	29	w′−1	w′−1	VERB
ejpam-109	152	30	∑	∑	PUNCT
ejpam-109	152	31	j=1	j=1	PROPN
ejpam-109	152	32	(	(	PUNCT
ejpam-109	152	33	j	j	PROPN
ejpam-109	152	34	,	,	PUNCT
ejpam-109	152	35	n)=1	n)=1	PROPN
ejpam-109	152	36	(	(	PUNCT
ejpam-109	152	37	a	a	DET
ejpam-109	152	38	j)m−1	j)m−1	PROPN
ejpam-109	152	39	�	�	PROPN
ejpam-109	152	40	a	a	DET
ejpam-109	152	41	j	j	PROPN
ejpam-109	152	42	w′	w′	PROPN
ejpam-109	152	43	�	�	PROPN
ejpam-109	152	44	(	(	PUNCT
ejpam-109	152	45	mod	mod	PROPN
ejpam-109	152	46	n	n	CCONJ
ejpam-109	152	47	)	)	PUNCT
ejpam-109	152	48	.	.	PUNCT
ejpam-109	153	1	(	(	PUNCT
ejpam-109	153	2	ii	ii	NOUN
ejpam-109	153	3	)	)	PUNCT
ejpam-109	153	4	proof	proof	NOUN
ejpam-109	153	5	.	.	PUNCT
ejpam-109	154	1	since	since	SCONJ
ejpam-109	154	2	the	the	DET
ejpam-109	154	3	proofs	proof	NOUN
ejpam-109	154	4	of	of	ADP
ejpam-109	154	5	(	(	PUNCT
ejpam-109	154	6	i	i	NOUN
ejpam-109	154	7	)	)	PUNCT
ejpam-109	154	8	and	and	CCONJ
ejpam-109	154	9	(	(	PUNCT
ejpam-109	154	10	ii	ii	NOUN
ejpam-109	154	11	)	)	PUNCT
ejpam-109	154	12	are	be	AUX
ejpam-109	154	13	almost	almost	ADV
ejpam-109	154	14	the	the	DET
ejpam-109	154	15	same	same	ADJ
ejpam-109	154	16	,	,	PUNCT
ejpam-109	154	17	we	we	PRON
ejpam-109	154	18	shall	shall	AUX
ejpam-109	154	19	give	give	VERB
ejpam-109	154	20	below	below	ADP
ejpam-109	154	21	only	only	ADV
ejpam-109	154	22	the	the	DET
ejpam-109	154	23	proof	proof	NOUN
ejpam-109	154	24	of	of	ADP
ejpam-109	154	25	(	(	PUNCT
ejpam-109	154	26	i	i	PROPN
ejpam-109	154	27	)	)	PUNCT
ejpam-109	154	28	.	.	PUNCT
ejpam-109	155	1	first	first	ADV
ejpam-109	155	2	,	,	PUNCT
ejpam-109	155	3	consider	consider	VERB
ejpam-109	155	4	the	the	DET
ejpam-109	155	5	voronoï	voronoï	ADJ
ejpam-109	155	6	congruence	congruence	NOUN
ejpam-109	155	7	in	in	ADP
ejpam-109	155	8	theorem	theorem	ADJ
ejpam-109	155	9	2.4	2.4	NUM
ejpam-109	155	10	replaced	replace	VERB
ejpam-109	155	11	n	n	INTJ
ejpam-109	155	12	by	by	ADP
ejpam-109	155	13	w	w	PROPN
ejpam-109	155	14	:	:	PUNCT
ejpam-109	155	15	(	(	PUNCT
ejpam-109	155	16	am−	am−	NUM
ejpam-109	155	17	1)nm	1)nm	PROPN
ejpam-109	155	18	≡	≡	PROPN
ejpam-109	155	19	mdm	mdm	PROPN
ejpam-109	156	1	w−1	w−1	PROPN
ejpam-109	156	2	∑	∑	PROPN
ejpam-109	157	1	j=1	j=1	PROPN
ejpam-109	157	2	(	(	PUNCT
ejpam-109	157	3	a	a	DET
ejpam-109	157	4	j)m−1	j)m−1	PROPN
ejpam-109	157	5	�	�	PROPN
ejpam-109	157	6	a	a	DET
ejpam-109	157	7	j	j	PROPN
ejpam-109	157	8	w	w	PROPN
ejpam-109	157	9	�	�	PROPN
ejpam-109	157	10	(	(	PUNCT
ejpam-109	157	11	mod	mod	PROPN
ejpam-109	157	12	w	w	PROPN
ejpam-109	157	13	)	)	PUNCT
ejpam-109	157	14	.	.	PUNCT
ejpam-109	158	1	since	since	SCONJ
ejpam-109	158	2	(	(	PUNCT
ejpam-109	158	3	mdm	mdm	PROPN
ejpam-109	158	4	/	/	SYM
ejpam-109	158	5	δν	δν	PROPN
ejpam-109	158	6	,	,	PUNCT
ejpam-109	158	7	n	n	CCONJ
ejpam-109	158	8	)	)	PUNCT
ejpam-109	158	9	=	=	SYM
ejpam-109	158	10	1	1	NUM
ejpam-109	158	11	and	and	CCONJ
ejpam-109	158	12	w	w	PROPN
ejpam-109	158	13	/	/	SYM
ejpam-109	158	14	δν	δν	NOUN
ejpam-109	158	15	≡	≡	PROPN
ejpam-109	158	16	0	0	PUNCT
ejpam-109	158	17	(	(	PUNCT
ejpam-109	158	18	mod	mod	NOUN
ejpam-109	158	19	n	n	CCONJ
ejpam-109	158	20	)	)	PUNCT
ejpam-109	158	21	,	,	PUNCT
ejpam-109	158	22	dividing	divide	VERB
ejpam-109	158	23	this	this	PRON
ejpam-109	158	24	by	by	ADP
ejpam-109	158	25	mdm	mdm	PROPN
ejpam-109	158	26	we	we	PRON
ejpam-109	158	27	get	get	VERB
ejpam-109	158	28	(	(	PUNCT
ejpam-109	158	29	am−	am−	NUM
ejpam-109	158	30	1)βm	1)βm	NUM
ejpam-109	158	31	≡	≡	PROPN
ejpam-109	159	1	w−1	w−1	INTJ
ejpam-109	159	2	∑	∑	PUNCT
ejpam-109	159	3	j=1	j=1	PROPN
ejpam-109	159	4	(	(	PUNCT
ejpam-109	159	5	a	a	DET
ejpam-109	159	6	j)m−1	j)m−1	PROPN
ejpam-109	159	7	�	�	PROPN
ejpam-109	159	8	a	a	DET
ejpam-109	159	9	j	j	PROPN
ejpam-109	159	10	w	w	PROPN
ejpam-109	159	11	�	�	PROPN
ejpam-109	159	12	(	(	PUNCT
ejpam-109	159	13	mod	mod	PROPN
ejpam-109	159	14	n	n	CCONJ
ejpam-109	159	15	)	)	PUNCT
ejpam-109	159	16	.	.	PUNCT
ejpam-109	160	1	(	(	PUNCT
ejpam-109	160	2	3.1	3.1	NUM
ejpam-109	160	3	)	)	PUNCT
ejpam-109	160	4	here	here	ADV
ejpam-109	160	5	the	the	DET
ejpam-109	160	6	sum	sum	NOUN
ejpam-109	160	7	on	on	ADP
ejpam-109	160	8	the	the	DET
ejpam-109	160	9	right	right	ADJ
ejpam-109	160	10	-	-	PUNCT
ejpam-109	160	11	hand	hand	NOUN
ejpam-109	160	12	side	side	NOUN
ejpam-109	160	13	can	can	AUX
ejpam-109	160	14	be	be	AUX
ejpam-109	160	15	expressed	express	VERB
ejpam-109	160	16	as	as	ADP
ejpam-109	160	17	w−1	w−1	PROPN
ejpam-109	160	18	∑	∑	PROPN
ejpam-109	160	19	j=1	j=1	PROPN
ejpam-109	160	20	(	(	PUNCT
ejpam-109	160	21	a	a	DET
ejpam-109	160	22	j)m−1	j)m−1	PROPN
ejpam-109	160	23	�	�	PROPN
ejpam-109	161	1	a	a	DET
ejpam-109	161	2	j	j	PROPN
ejpam-109	161	3	w	w	PROPN
ejpam-109	161	4	�	�	PROPN
ejpam-109	161	5	=	=	SYM
ejpam-109	161	6	w−1	w−1	PROPN
ejpam-109	161	7	∑	∑	PUNCT
ejpam-109	161	8	j=1	j=1	PROPN
ejpam-109	161	9	(	(	PUNCT
ejpam-109	161	10	j	j	PROPN
ejpam-109	161	11	,	,	PUNCT
ejpam-109	161	12	n)=1	n)=1	PROPN
ejpam-109	161	13	(	(	PUNCT
ejpam-109	161	14	a	a	DET
ejpam-109	161	15	j)m−1	j)m−1	PROPN
ejpam-109	161	16	�	�	PROPN
ejpam-109	161	17	a	a	DET
ejpam-109	161	18	j	j	PROPN
ejpam-109	161	19	w	w	PROPN
ejpam-109	161	20	�	�	PROPN
ejpam-109	161	21	+	+	CCONJ
ejpam-109	161	22	w−1	w−1	PROPN
ejpam-109	161	23	∑	∑	PROPN
ejpam-109	161	24	j=1	j=1	PROPN
ejpam-109	161	25	(	(	PUNCT
ejpam-109	161	26	j	j	PROPN
ejpam-109	161	27	,	,	PUNCT
ejpam-109	161	28	n)6=1	n)6=1	NOUN
ejpam-109	161	29	(	(	PUNCT
ejpam-109	161	30	a	a	DET
ejpam-109	161	31	j)m−1	j)m−1	PROPN
ejpam-109	161	32	�	�	PROPN
ejpam-109	161	33	a	a	DET
ejpam-109	161	34	j	j	PROPN
ejpam-109	161	35	w	w	PROPN
ejpam-109	161	36	�	�	PROPN
ejpam-109	161	37	=	=	SYM
ejpam-109	161	38	w−1	w−1	PROPN
ejpam-109	161	39	∑	∑	PUNCT
ejpam-109	161	40	j=1	j=1	PROPN
ejpam-109	161	41	(	(	PUNCT
ejpam-109	161	42	j	j	PROPN
ejpam-109	161	43	,	,	PUNCT
ejpam-109	161	44	n)=1	n)=1	PROPN
ejpam-109	161	45	(	(	PUNCT
ejpam-109	161	46	a	a	DET
ejpam-109	161	47	j)m−1	j)m−1	PROPN
ejpam-109	161	48	�	�	PROPN
ejpam-109	161	49	a	a	DET
ejpam-109	161	50	j	j	PROPN
ejpam-109	161	51	w	w	PROPN
ejpam-109	161	52	�	�	PROPN
ejpam-109	161	53	−	−	PROPN
ejpam-109	161	54	∑	∑	PUNCT
ejpam-109	161	55	d|n	d|n	PROPN
ejpam-109	161	56	d>1	d>1	NOUN
ejpam-109	161	57	µ(d)dm−1	µ(d)dm−1	NOUN
ejpam-109	161	58			NOUN
ejpam-109	161	59			NOUN
ejpam-109	161	60			PROPN
ejpam-109	161	61	w	w	VERB
ejpam-109	161	62	/	/	SYM
ejpam-109	161	63	d−1	d−1	PROPN
ejpam-109	161	64	∑	∑	PROPN
ejpam-109	161	65	j=1	j=1	PROPN
ejpam-109	161	66	(	(	PUNCT
ejpam-109	161	67	a	a	DET
ejpam-109	161	68	j)m−1	j)m−1	PROPN
ejpam-109	161	69	�	�	PROPN
ejpam-109	161	70	a	a	DET
ejpam-109	161	71	j	j	PROPN
ejpam-109	161	72	w	w	PROPN
ejpam-109	161	73	/	/	SYM
ejpam-109	161	74	d	d	PROPN
ejpam-109	161	75	�	�	PROPN
ejpam-109	161	76			PROPN
ejpam-109	161	77			VERB
ejpam-109	161	78			PUNCT
ejpam-109	161	79	,	,	PUNCT
ejpam-109	161	80	(	(	PUNCT
ejpam-109	161	81	3.2	3.2	NUM
ejpam-109	161	82	)	)	PUNCT
ejpam-109	161	83	where	where	SCONJ
ejpam-109	161	84	µ	µ	NOUN
ejpam-109	161	85	is	be	AUX
ejpam-109	161	86	the	the	DET
ejpam-109	161	87	möbius	möbius	NOUN
ejpam-109	161	88	function	function	NOUN
ejpam-109	161	89	.	.	PUNCT
ejpam-109	162	1	now	now	ADV
ejpam-109	162	2	consider	consider	VERB
ejpam-109	162	3	congruence	congruence	PRON
ejpam-109	162	4	(	(	PUNCT
ejpam-109	162	5	3.1	3.1	NUM
ejpam-109	162	6	)	)	PUNCT
ejpam-109	162	7	replaced	replace	VERB
ejpam-109	162	8	w	w	NOUN
ejpam-109	162	9	by	by	ADP
ejpam-109	162	10	w	w	PROPN
ejpam-109	162	11	/	/	SYM
ejpam-109	162	12	d	d	NOUN
ejpam-109	162	13	,	,	PUNCT
ejpam-109	162	14	and	and	CCONJ
ejpam-109	162	15	multiply	multiply	VERB
ejpam-109	162	16	it	it	PRON
ejpam-109	162	17	by	by	ADP
ejpam-109	162	18	µ(d)dm−1	µ(d)dm−1	PROPN
ejpam-109	162	19	.	.	PUNCT
ejpam-109	163	1	then	then	ADV
ejpam-109	163	2	we	we	PRON
ejpam-109	163	3	have	have	VERB
ejpam-109	163	4	µ(d)dm−1βm(a)≡	µ(d)dm−1βm(a)≡	NOUN
ejpam-109	163	5	µ(d)dm−1	µ(d)dm−1	PROPN
ejpam-109	163	6	w	w	PROPN
ejpam-109	163	7	/	/	SYM
ejpam-109	163	8	d−1	d−1	PROPN
ejpam-109	163	9	∑	∑	PROPN
ejpam-109	163	10	j=1	j=1	PROPN
ejpam-109	163	11	(	(	PUNCT
ejpam-109	163	12	a	a	DET
ejpam-109	163	13	j)m−1	j)m−1	PROPN
ejpam-109	163	14	�	�	PROPN
ejpam-109	163	15	a	a	DET
ejpam-109	163	16	j	j	PROPN
ejpam-109	163	17	w	w	PROPN
ejpam-109	163	18	/	/	SYM
ejpam-109	163	19	d	d	PROPN
ejpam-109	163	20	�	�	PROPN
ejpam-109	163	21	(	(	PUNCT
ejpam-109	163	22	mod	mod	PROPN
ejpam-109	163	23	n	n	CCONJ
ejpam-109	163	24	)	)	PUNCT
ejpam-109	163	25	.	.	PUNCT
ejpam-109	164	1	noting	note	VERB
ejpam-109	164	2	that	that	SCONJ
ejpam-109	164	3	εm(n	εm(n	VERB
ejpam-109	164	4	)	)	PUNCT
ejpam-109	164	5	=	=	SYM
ejpam-109	165	1	∑	∑	PUNCT
ejpam-109	165	2	d|nµ(d)d	d|nµ(d)d	PROPN
ejpam-109	165	3	m−1	m−1	PROPN
ejpam-109	165	4	(	(	PUNCT
ejpam-109	165	5	where	where	SCONJ
ejpam-109	165	6	µ(1	µ(1	PROPN
ejpam-109	165	7	)	)	PUNCT
ejpam-109	165	8	=	=	NOUN
ejpam-109	165	9	1	1	NUM
ejpam-109	165	10	)	)	PUNCT
ejpam-109	165	11	,	,	PUNCT
ejpam-109	165	12	from	from	ADP
ejpam-109	165	13	(	(	PUNCT
ejpam-109	165	14	3.1	3.1	NUM
ejpam-109	165	15	)	)	PUNCT
ejpam-109	165	16	and	and	CCONJ
ejpam-109	165	17	(	(	PUNCT
ejpam-109	165	18	3.2	3.2	NUM
ejpam-109	165	19	)	)	PUNCT
ejpam-109	165	20	km(n	km(n	NOUN
ejpam-109	165	21	;	;	PUNCT
ejpam-109	165	22	a	a	X
ejpam-109	165	23	)	)	PUNCT
ejpam-109	165	24	=	=	SYM
ejpam-109	165	25	�	�	PROPN
ejpam-109	165	26	1	1	NUM
ejpam-109	165	27	+	+	NUM
ejpam-109	165	28	∑	∑	PUNCT
ejpam-109	165	29	d|n	d|n	PROPN
ejpam-109	165	30	d>1	d>1	NOUN
ejpam-109	165	31	µ(d)dm−1	µ(d)dm−1	PROPN
ejpam-109	165	32	�	�	NOUN
ejpam-109	165	33	βm(a)≡	βm(a)≡	PUNCT
ejpam-109	165	34	w−1	w−1	PROPN
ejpam-109	165	35	∑	∑	PUNCT
ejpam-109	165	36	j=1	j=1	PROPN
ejpam-109	165	37	(	(	PUNCT
ejpam-109	165	38	j	j	PROPN
ejpam-109	165	39	,	,	PUNCT
ejpam-109	165	40	n)=1	n)=1	PROPN
ejpam-109	165	41	(	(	PUNCT
ejpam-109	165	42	a	a	DET
ejpam-109	165	43	j)m−1	j)m−1	PROPN
ejpam-109	165	44	�	�	PROPN
ejpam-109	165	45	a	a	DET
ejpam-109	165	46	j	j	PROPN
ejpam-109	165	47	w	w	PROPN
ejpam-109	165	48	�	�	PROPN
ejpam-109	165	49	(	(	PUNCT
ejpam-109	165	50	mod	mod	PROPN
ejpam-109	165	51	n	n	CCONJ
ejpam-109	165	52	)	)	PUNCT
ejpam-109	165	53	,	,	PUNCT
ejpam-109	165	54	which	which	PRON
ejpam-109	165	55	is	be	AUX
ejpam-109	165	56	exactly	exactly	ADV
ejpam-109	165	57	congruence	congruence	ADJ
ejpam-109	165	58	(	(	PUNCT
ejpam-109	165	59	i	i	NOUN
ejpam-109	165	60	)	)	PUNCT
ejpam-109	165	61	,	,	PUNCT
ejpam-109	165	62	as	as	SCONJ
ejpam-109	165	63	desired	desire	VERB
ejpam-109	165	64	.	.	PUNCT
ejpam-109	166	1	t.	t.	PROPN
ejpam-109	166	2	agoh	agoh	PROPN
ejpam-109	166	3	/	/	SYM
ejpam-109	166	4	eur	eur	PROPN
ejpam-109	166	5	.	.	PUNCT
ejpam-109	167	1	j.	j.	PROPN
ejpam-109	167	2	pure	pure	PROPN
ejpam-109	167	3	appl	appl	PROPN
ejpam-109	167	4	.	.	PROPN
ejpam-109	167	5	math	math	PROPN
ejpam-109	167	6	,	,	PUNCT
ejpam-109	167	7	1	1	NUM
ejpam-109	167	8	(	(	PUNCT
ejpam-109	167	9	2008	2008	NUM
ejpam-109	167	10	)	)	PUNCT
ejpam-109	167	11	,	,	PUNCT
ejpam-109	167	12	(	(	PUNCT
ejpam-109	167	13	3	3	NUM
ejpam-109	167	14	-	-	SYM
ejpam-109	167	15	21	21	NUM
ejpam-109	167	16	)	)	PUNCT
ejpam-109	167	17	10	10	NUM
ejpam-109	167	18	theorem	theorem	VERB
ejpam-109	167	19	3.2	3.2	NUM
ejpam-109	167	20	.	.	PUNCT
ejpam-109	168	1	let	let	VERB
ejpam-109	168	2	m≥	m≥	NOUN
ejpam-109	168	3	2	2	NUM
ejpam-109	168	4	be	be	AUX
ejpam-109	168	5	even	even	ADV
ejpam-109	168	6	and	and	CCONJ
ejpam-109	168	7	n≥	n≥	ADJ
ejpam-109	168	8	1	1	NUM
ejpam-109	168	9	.	.	PUNCT
ejpam-109	169	1	then	then	ADV
ejpam-109	169	2	nh	nh	PROPN
ejpam-109	169	3	′m(n)≡	′m(n)≡	NUM
ejpam-109	169	4	n−1	n−1	PROPN
ejpam-109	169	5	∑	∑	PUNCT
ejpam-109	169	6	j=1	j=1	PROPN
ejpam-109	169	7	(	(	PUNCT
ejpam-109	169	8	j	j	PROPN
ejpam-109	169	9	,	,	PUNCT
ejpam-109	169	10	n)=1	n)=1	PROPN
ejpam-109	169	11	jm	jm	PROPN
ejpam-109	169	12	(	(	PUNCT
ejpam-109	169	13	mod	mod	PROPN
ejpam-109	169	14	n	n	CCONJ
ejpam-109	169	15	)	)	PUNCT
ejpam-109	169	16	.	.	PUNCT
ejpam-109	170	1	proof	proof	NOUN
ejpam-109	170	2	.	.	PUNCT
ejpam-109	171	1	using	use	VERB
ejpam-109	171	2	the	the	DET
ejpam-109	171	3	similar	similar	ADJ
ejpam-109	171	4	method	method	NOUN
ejpam-109	171	5	to	to	ADP
ejpam-109	171	6	that	that	PRON
ejpam-109	171	7	stated	state	VERB
ejpam-109	171	8	in	in	ADP
ejpam-109	171	9	the	the	DET
ejpam-109	171	10	proof	proof	NOUN
ejpam-109	171	11	of	of	ADP
ejpam-109	171	12	theorem	theorem	ADJ
ejpam-109	171	13	3.1	3.1	NUM
ejpam-109	171	14	,	,	PUNCT
ejpam-109	171	15	we	we	PRON
ejpam-109	171	16	express	express	VERB
ejpam-109	171	17	sm(n	sm(n	NOUN
ejpam-109	171	18	)	)	PUNCT
ejpam-109	171	19	as	as	ADP
ejpam-109	171	20	sm(n	sm(n	NOUN
ejpam-109	171	21	)	)	PUNCT
ejpam-109	172	1	=	=	SYM
ejpam-109	172	2	n−1	n−1	PROPN
ejpam-109	172	3	∑	∑	PUNCT
ejpam-109	172	4	j=1	j=1	PROPN
ejpam-109	172	5	(	(	PUNCT
ejpam-109	172	6	j	j	PROPN
ejpam-109	172	7	,	,	PUNCT
ejpam-109	172	8	n)=1	n)=1	PROPN
ejpam-109	172	9	jm+	jm+	NOUN
ejpam-109	172	10	n−1	n−1	PROPN
ejpam-109	172	11	∑	∑	PUNCT
ejpam-109	172	12	j=1	j=1	PROPN
ejpam-109	172	13	(	(	PUNCT
ejpam-109	172	14	j	j	PROPN
ejpam-109	172	15	,	,	PUNCT
ejpam-109	172	16	n)6=1	n)6=1	NOUN
ejpam-109	172	17	jm	jm	PROPN
ejpam-109	172	18	=	=	SYM
ejpam-109	172	19	n−1	n−1	PROPN
ejpam-109	172	20	∑	∑	PUNCT
ejpam-109	172	21	j=1	j=1	PROPN
ejpam-109	172	22	(	(	PUNCT
ejpam-109	172	23	j	j	PROPN
ejpam-109	172	24	,	,	PUNCT
ejpam-109	172	25	n)=1	n)=1	PROPN
ejpam-109	172	26	jm−	jm−	PROPN
ejpam-109	172	27	∑	∑	PUNCT
ejpam-109	172	28	d|n	d|n	PROPN
ejpam-109	172	29	d>1	d>1	NOUN
ejpam-109	172	30	µ(d)dm	µ(d)dm	DET
ejpam-109	172	31	�	�	PROPN
ejpam-109	172	32	n	n	CCONJ
ejpam-109	172	33	/	/	SYM
ejpam-109	172	34	d−1	d−1	PROPN
ejpam-109	172	35	∑	∑	PROPN
ejpam-109	172	36	j=1	j=1	PROPN
ejpam-109	172	37	jm	jm	PROPN
ejpam-109	172	38	�	�	PROPN
ejpam-109	172	39	.	.	PUNCT
ejpam-109	173	1	by	by	ADP
ejpam-109	173	2	theorem	theorem	VERB
ejpam-109	173	3	2.5	2.5	NUM
ejpam-109	173	4	we	we	PRON
ejpam-109	173	5	see	see	VERB
ejpam-109	173	6	ordp(dm	ordp(dm	ADJ
ejpam-109	173	7	)	)	PUNCT
ejpam-109	173	8	∈	∈	PROPN
ejpam-109	173	9	{	{	PUNCT
ejpam-109	173	10	0,1	0,1	NOUN
ejpam-109	173	11	}	}	PUNCT
ejpam-109	173	12	for	for	ADP
ejpam-109	173	13	all	all	DET
ejpam-109	173	14	prime	prime	ADJ
ejpam-109	173	15	divisors	divisor	NOUN
ejpam-109	173	16	p	p	NOUN
ejpam-109	173	17	of	of	ADP
ejpam-109	173	18	n	n	CCONJ
ejpam-109	173	19	,	,	PUNCT
ejpam-109	173	20	hence	hence	ADV
ejpam-109	173	21	from	from	ADP
ejpam-109	173	22	theorem	theorem	ADJ
ejpam-109	173	23	2.2	2.2	NUM
ejpam-109	173	24	sm(n	sm(n	NOUN
ejpam-109	173	25	)	)	PUNCT
ejpam-109	173	26	≡	≡	PROPN
ejpam-109	173	27	nbm	nbm	PROPN
ejpam-109	173	28	(	(	PUNCT
ejpam-109	173	29	mod	mod	PROPN
ejpam-109	173	30	n	n	CCONJ
ejpam-109	173	31	)	)	PUNCT
ejpam-109	173	32	and	and	CCONJ
ejpam-109	173	33	sm(n	sm(n	NOUN
ejpam-109	173	34	/	/	SYM
ejpam-109	173	35	d	d	NOUN
ejpam-109	173	36	)	)	PUNCT
ejpam-109	173	37	≡	≡	PROPN
ejpam-109	173	38	(	(	PUNCT
ejpam-109	173	39	n	n	CCONJ
ejpam-109	173	40	/	/	SYM
ejpam-109	174	1	d)bm	d)bm	ADJ
ejpam-109	174	2	(	(	PUNCT
ejpam-109	174	3	mod	mod	PROPN
ejpam-109	174	4	n	n	CCONJ
ejpam-109	174	5	/	/	SYM
ejpam-109	174	6	d	d	NOUN
ejpam-109	174	7	)	)	PUNCT
ejpam-109	174	8	for	for	ADP
ejpam-109	174	9	any	any	DET
ejpam-109	174	10	positive	positive	ADJ
ejpam-109	174	11	divisor	divisor	NOUN
ejpam-109	174	12	d	d	PROPN
ejpam-109	174	13	of	of	ADP
ejpam-109	174	14	n.	n.	PROPN
ejpam-109	174	15	multiplying	multiply	VERB
ejpam-109	174	16	the	the	DET
ejpam-109	174	17	latter	latter	ADJ
ejpam-109	174	18	one	one	NUM
ejpam-109	174	19	by	by	ADP
ejpam-109	174	20	µ(d)dm	µ(d)dm	ADP
ejpam-109	174	21	,	,	PUNCT
ejpam-109	174	22	we	we	PRON
ejpam-109	174	23	have	have	VERB
ejpam-109	174	24	µ(d)dmsm	µ(d)dmsm	NOUN
ejpam-109	174	25	(	(	PUNCT
ejpam-109	174	26	n	n	CCONJ
ejpam-109	174	27	/	/	SYM
ejpam-109	174	28	d)≡	d)≡	NOUN
ejpam-109	174	29	µ(d)dm−1nbm	µ(d)dm−1nbm	ADV
ejpam-109	174	30	(	(	PUNCT
ejpam-109	174	31	mod	mod	PROPN
ejpam-109	174	32	n	n	CCONJ
ejpam-109	174	33	)	)	PUNCT
ejpam-109	174	34	.	.	PUNCT
ejpam-109	175	1	substituting	substitute	VERB
ejpam-109	175	2	these	these	DET
ejpam-109	175	3	congruences	congruence	NOUN
ejpam-109	175	4	for	for	ADP
ejpam-109	175	5	every	every	DET
ejpam-109	175	6	d	d	NOUN
ejpam-109	175	7	into	into	ADP
ejpam-109	175	8	the	the	DET
ejpam-109	175	9	above	above	NOUN
ejpam-109	175	10	,	,	PUNCT
ejpam-109	175	11	it	it	PRON
ejpam-109	175	12	follows	follow	VERB
ejpam-109	175	13	that	that	SCONJ
ejpam-109	175	14	nh	nh	PROPN
ejpam-109	175	15	′m(n	′m(n	PROPN
ejpam-109	175	16	)	)	PUNCT
ejpam-109	175	17	=	=	SYM
ejpam-109	175	18	�	�	PROPN
ejpam-109	175	19	1	1	NUM
ejpam-109	175	20	+	+	NUM
ejpam-109	175	21	∑	∑	PUNCT
ejpam-109	175	22	d|n	d|n	PROPN
ejpam-109	175	23	d>1	d>1	NOUN
ejpam-109	175	24	µ(d)dm−1	µ(d)dm−1	PROPN
ejpam-109	175	25	�	�	PROPN
ejpam-109	175	26	nbm	nbm	PROPN
ejpam-109	175	27	≡	≡	PROPN
ejpam-109	175	28	n−1	n−1	PROPN
ejpam-109	176	1	∑	∑	PUNCT
ejpam-109	176	2	j=1	j=1	PROPN
ejpam-109	176	3	(	(	PUNCT
ejpam-109	176	4	j	j	PROPN
ejpam-109	176	5	,	,	PUNCT
ejpam-109	176	6	n)=1	n)=1	PROPN
ejpam-109	176	7	jm	jm	PROPN
ejpam-109	176	8	(	(	PUNCT
ejpam-109	176	9	mod	mod	PROPN
ejpam-109	176	10	n	n	CCONJ
ejpam-109	176	11	)	)	PUNCT
ejpam-109	176	12	,	,	PUNCT
ejpam-109	176	13	which	which	PRON
ejpam-109	176	14	is	be	AUX
ejpam-109	176	15	precisely	precisely	ADV
ejpam-109	176	16	the	the	DET
ejpam-109	176	17	congruence	congruence	NOUN
ejpam-109	176	18	indicated	indicate	VERB
ejpam-109	176	19	.	.	PUNCT
ejpam-109	177	1	with	with	ADP
ejpam-109	177	2	above	above	ADP
ejpam-109	177	3	notations	notation	NOUN
ejpam-109	177	4	,	,	PUNCT
ejpam-109	177	5	one	one	PRON
ejpam-109	177	6	can	can	AUX
ejpam-109	177	7	state	state	VERB
ejpam-109	177	8	corollary	corollary	ADJ
ejpam-109	177	9	3.3	3.3	NUM
ejpam-109	177	10	.	.	PUNCT
ejpam-109	178	1	if	if	SCONJ
ejpam-109	178	2	ϕ(n	ϕ(n	X
ejpam-109	178	3	)	)	PUNCT
ejpam-109	178	4	|	|	ADV
ejpam-109	178	5	m	m	ADV
ejpam-109	178	6	,	,	PUNCT
ejpam-109	178	7	then	then	ADV
ejpam-109	178	8	nh	nh	PROPN
ejpam-109	178	9	′m(n)≡	′m(n)≡	NUM
ejpam-109	178	10	ϕ(n	ϕ(n	X
ejpam-109	178	11	)	)	PUNCT
ejpam-109	178	12	(	(	PUNCT
ejpam-109	178	13	mod	mod	NOUN
ejpam-109	178	14	n	n	CCONJ
ejpam-109	178	15	)	)	PUNCT
ejpam-109	178	16	.	.	PUNCT
ejpam-109	179	1	proof	proof	NOUN
ejpam-109	179	2	.	.	PUNCT
ejpam-109	180	1	if	if	SCONJ
ejpam-109	180	2	(	(	PUNCT
ejpam-109	180	3	j	j	NOUN
ejpam-109	180	4	,	,	PUNCT
ejpam-109	180	5	n	n	CCONJ
ejpam-109	180	6	)	)	PUNCT
ejpam-109	180	7	=	=	SYM
ejpam-109	180	8	1	1	NUM
ejpam-109	180	9	and	and	CCONJ
ejpam-109	180	10	ϕ(n	ϕ(n	NUM
ejpam-109	180	11	)	)	PUNCT
ejpam-109	181	1	|	|	ADV
ejpam-109	181	2	m	m	ADV
ejpam-109	181	3	,	,	PUNCT
ejpam-109	181	4	then	then	ADV
ejpam-109	181	5	jm	jm	PROPN
ejpam-109	181	6	≡	≡	PROPN
ejpam-109	181	7	1	1	NUM
ejpam-109	181	8	(	(	PUNCT
ejpam-109	181	9	mod	mod	NOUN
ejpam-109	181	10	n	n	CCONJ
ejpam-109	181	11	)	)	PUNCT
ejpam-109	181	12	.	.	PUNCT
ejpam-109	182	1	so	so	ADV
ejpam-109	182	2	the	the	DET
ejpam-109	182	3	result	result	NOUN
ejpam-109	182	4	follows	follow	VERB
ejpam-109	182	5	immediately	immediately	ADV
ejpam-109	182	6	from	from	ADP
ejpam-109	182	7	theorem	theorem	ADJ
ejpam-109	182	8	3.2	3.2	NUM
ejpam-109	182	9	.	.	PUNCT
ejpam-109	183	1	note	note	VERB
ejpam-109	183	2	that	that	SCONJ
ejpam-109	183	3	congruence	congruence	NOUN
ejpam-109	183	4	(	(	PUNCT
ejpam-109	183	5	i	i	NOUN
ejpam-109	183	6	)	)	PUNCT
ejpam-109	183	7	in	in	ADP
ejpam-109	183	8	theorem	theorem	ADJ
ejpam-109	183	9	2.5	2.5	NUM
ejpam-109	183	10	is	be	AUX
ejpam-109	183	11	given	give	VERB
ejpam-109	183	12	as	as	ADP
ejpam-109	183	13	a	a	DET
ejpam-109	183	14	special	special	ADJ
ejpam-109	183	15	case	case	NOUN
ejpam-109	183	16	of	of	ADP
ejpam-109	183	17	corollary	corollary	ADJ
ejpam-109	183	18	3.3	3.3	NUM
ejpam-109	183	19	for	for	ADP
ejpam-109	183	20	the	the	DET
ejpam-109	183	21	case	case	NOUN
ejpam-109	183	22	n=	n=	ADJ
ejpam-109	183	23	p.	p.	NOUN
ejpam-109	183	24	let	let	VERB
ejpam-109	183	25	zn	zn	PROPN
ejpam-109	183	26	=	=	SYM
ejpam-109	183	27	∩p|nzp	∩p|nzp	X
ejpam-109	184	1	(	(	PUNCT
ejpam-109	184	2	n	n	X
ejpam-109	184	3	≥	≥	NOUN
ejpam-109	184	4	2	2	NUM
ejpam-109	184	5	)	)	PUNCT
ejpam-109	184	6	be	be	AUX
ejpam-109	184	7	the	the	DET
ejpam-109	184	8	ring	ring	NOUN
ejpam-109	184	9	of	of	ADP
ejpam-109	184	10	rational	rational	ADJ
ejpam-109	184	11	numbers	number	NOUN
ejpam-109	184	12	which	which	PRON
ejpam-109	184	13	are	be	AUX
ejpam-109	184	14	n	n	PRON
ejpam-109	184	15	-	-	PUNCT
ejpam-109	184	16	integral	integral	ADJ
ejpam-109	184	17	.	.	PUNCT
ejpam-109	185	1	suppose	suppose	VERB
ejpam-109	185	2	that	that	SCONJ
ejpam-109	185	3	m≥	m≥	PROPN
ejpam-109	185	4	2	2	NUM
ejpam-109	185	5	is	be	AUX
ejpam-109	185	6	even	even	ADV
ejpam-109	185	7	and	and	CCONJ
ejpam-109	185	8	ϕ(n	ϕ(n	NUM
ejpam-109	185	9	)	)	PUNCT
ejpam-109	185	10	|	|	ADV
ejpam-109	185	11	m.	m.	NOUN
ejpam-109	185	12	writing	write	VERB
ejpam-109	185	13	as	as	ADP
ejpam-109	185	14	bm	bm	PROPN
ejpam-109	185	15	=	=	PROPN
ejpam-109	185	16	ω(m)+	ω(m)+	NOUN
ejpam-109	185	17	xm+	xm+	PROPN
ejpam-109	185	18	ym	ym	PROPN
ejpam-109	185	19	,	,	PUNCT
ejpam-109	185	20	where	where	SCONJ
ejpam-109	185	21	ω(m	ω(m	NOUN
ejpam-109	185	22	)	)	PUNCT
ejpam-109	185	23	is	be	AUX
ejpam-109	185	24	the	the	DET
ejpam-109	185	25	integer	integer	NOUN
ejpam-109	185	26	mentioned	mention	VERB
ejpam-109	185	27	in	in	ADP
ejpam-109	185	28	section	section	NOUN
ejpam-109	185	29	2	2	NUM
ejpam-109	185	30	,	,	PUNCT
ejpam-109	185	31	xm	xm	PROPN
ejpam-109	185	32	∈	∈	PROPN
ejpam-109	185	33	zn	zn	PROPN
ejpam-109	185	34	and	and	CCONJ
ejpam-109	185	35	ym	ym	PROPN
ejpam-109	185	36	/∈	/∈	PUNCT
ejpam-109	186	1	zn	zn	X
ejpam-109	186	2	,	,	PUNCT
ejpam-109	186	3	we	we	PRON
ejpam-109	186	4	obviously	obviously	ADV
ejpam-109	186	5	see	see	VERB
ejpam-109	186	6	ym	ym	NOUN
ejpam-109	186	7	=	=	SYM
ejpam-109	187	1	−	−	PROPN
ejpam-109	187	2	∑	∑	PUNCT
ejpam-109	187	3	p|n	p|n	NOUN
ejpam-109	187	4	1	1	NUM
ejpam-109	187	5	/	/	SYM
ejpam-109	187	6	p.	p.	NOUN
ejpam-109	187	7	on	on	ADP
ejpam-109	187	8	the	the	DET
ejpam-109	187	9	one	one	NUM
ejpam-109	187	10	hand	hand	NOUN
ejpam-109	188	1	,	,	PUNCT
ejpam-109	188	2	the	the	DET
ejpam-109	188	3	right	right	ADJ
ejpam-109	188	4	-	-	PUNCT
ejpam-109	188	5	hand	hand	NOUN
ejpam-109	188	6	side	side	NOUN
ejpam-109	188	7	of	of	ADP
ejpam-109	188	8	the	the	DET
ejpam-109	188	9	congruence	congruence	NOUN
ejpam-109	188	10	in	in	ADP
ejpam-109	188	11	theorem	theorem	ADJ
ejpam-109	188	12	3.2	3.2	NUM
ejpam-109	188	13	is	be	AUX
ejpam-109	188	14	unchanged	unchanged	ADJ
ejpam-109	188	15	modulo	modulo	NOUN
ejpam-109	188	16	n	n	CCONJ
ejpam-109	188	17	even	even	ADV
ejpam-109	188	18	if	if	SCONJ
ejpam-109	188	19	we	we	PRON
ejpam-109	188	20	replace	replace	VERB
ejpam-109	188	21	m	m	PRON
ejpam-109	188	22	by	by	ADP
ejpam-109	188	23	l	l	PROPN
ejpam-109	188	24	≥	≥	NUM
ejpam-109	188	25	2	2	NUM
ejpam-109	188	26	satisfying	satisfy	VERB
ejpam-109	188	27	m	m	NOUN
ejpam-109	188	28	≡	≡	PROPN
ejpam-109	188	29	l	l	PROPN
ejpam-109	188	30	(	(	PUNCT
ejpam-109	188	31	mod	mod	PROPN
ejpam-109	188	32	ϕ(n	ϕ(n	PROPN
ejpam-109	188	33	)	)	PUNCT
ejpam-109	188	34	)	)	PUNCT
ejpam-109	188	35	.	.	PUNCT
ejpam-109	189	1	therefore	therefore	ADV
ejpam-109	189	2	,	,	PUNCT
ejpam-109	189	3	if	if	SCONJ
ejpam-109	189	4	m	m	VERB
ejpam-109	189	5	≡	≡	PROPN
ejpam-109	189	6	l	l	PROPN
ejpam-109	189	7	(	(	PUNCT
ejpam-109	189	8	mod	mod	PROPN
ejpam-109	189	9	ϕ(n	ϕ(n	PROPN
ejpam-109	189	10	)	)	PUNCT
ejpam-109	189	11	)	)	PUNCT
ejpam-109	189	12	for	for	ADP
ejpam-109	189	13	n≥	n≥	PROPN
ejpam-109	189	14	3	3	NUM
ejpam-109	189	15	,	,	PUNCT
ejpam-109	189	16	then	then	ADV
ejpam-109	189	17	we	we	PRON
ejpam-109	189	18	have	have	VERB
ejpam-109	189	19	nh	nh	PROPN
ejpam-109	189	20	′m(n)≡	′m(n)≡	NUM
ejpam-109	189	21	nh	nh	PROPN
ejpam-109	189	22	′l	′l	PROPN
ejpam-109	189	23	(	(	PUNCT
ejpam-109	189	24	n	n	CCONJ
ejpam-109	189	25	)	)	PUNCT
ejpam-109	189	26	(	(	PUNCT
ejpam-109	189	27	mod	mod	PROPN
ejpam-109	189	28	n	n	CCONJ
ejpam-109	189	29	)	)	PUNCT
ejpam-109	189	30	.	.	PUNCT
ejpam-109	190	1	however	however	ADV
ejpam-109	190	2	,	,	PUNCT
ejpam-109	190	3	since	since	SCONJ
ejpam-109	190	4	bm	bm	PROPN
ejpam-109	190	5	∈	∈	PROPN
ejpam-109	190	6	zn	zn	PROPN
ejpam-109	190	7	if	if	SCONJ
ejpam-109	190	8	and	and	CCONJ
ejpam-109	190	9	only	only	ADV
ejpam-109	190	10	if	if	SCONJ
ejpam-109	190	11	bl	bl	PROPN
ejpam-109	190	12	∈	∈	PROPN
ejpam-109	190	13	zn	zn	PROPN
ejpam-109	190	14	,	,	PUNCT
ejpam-109	190	15	this	this	DET
ejpam-109	190	16	congruence	congruence	NOUN
ejpam-109	190	17	is	be	AUX
ejpam-109	190	18	interesting	interesting	ADJ
ejpam-109	190	19	only	only	ADV
ejpam-109	190	20	in	in	ADP
ejpam-109	190	21	the	the	DET
ejpam-109	190	22	case	case	NOUN
ejpam-109	190	23	bm	bm	PROPN
ejpam-109	190	24	/∈	/∈	PUNCT
ejpam-109	191	1	zn	zn	X
ejpam-109	191	2	.	.	PUNCT
ejpam-109	192	1	as	as	ADP
ejpam-109	192	2	recurrence	recurrence	NOUN
ejpam-109	192	3	relations	relation	NOUN
ejpam-109	192	4	for	for	ADP
ejpam-109	192	5	h	h	NOUN
ejpam-109	192	6	′m(n	′m(n	NOUN
ejpam-109	192	7	)	)	PUNCT
ejpam-109	192	8	,	,	PUNCT
ejpam-109	192	9	we	we	PRON
ejpam-109	192	10	can	can	AUX
ejpam-109	192	11	give	give	VERB
ejpam-109	192	12	the	the	DET
ejpam-109	192	13	following	follow	VERB
ejpam-109	192	14	t.	t.	NOUN
ejpam-109	192	15	agoh	agoh	PROPN
ejpam-109	192	16	/	/	SYM
ejpam-109	192	17	eur	eur	PROPN
ejpam-109	192	18	.	.	PUNCT
ejpam-109	193	1	j.	j.	PROPN
ejpam-109	193	2	pure	pure	PROPN
ejpam-109	193	3	appl	appl	PROPN
ejpam-109	193	4	.	.	PROPN
ejpam-109	193	5	math	math	PROPN
ejpam-109	193	6	,	,	PUNCT
ejpam-109	193	7	1	1	NUM
ejpam-109	193	8	(	(	PUNCT
ejpam-109	193	9	2008	2008	NUM
ejpam-109	193	10	)	)	PUNCT
ejpam-109	193	11	,	,	PUNCT
ejpam-109	193	12	(	(	PUNCT
ejpam-109	193	13	3	3	NUM
ejpam-109	193	14	-	-	SYM
ejpam-109	193	15	21	21	NUM
ejpam-109	193	16	)	)	PUNCT
ejpam-109	193	17	11	11	NUM
ejpam-109	193	18	theorem	theorem	VERB
ejpam-109	193	19	3.4	3.4	NUM
ejpam-109	193	20	.	.	PUNCT
ejpam-109	194	1	let	let	VERB
ejpam-109	194	2	m≥	m≥	NOUN
ejpam-109	194	3	1	1	NUM
ejpam-109	194	4	,	,	PUNCT
ejpam-109	194	5	n≥	n≥	PROPN
ejpam-109	194	6	2	2	NUM
ejpam-109	194	7	and	and	CCONJ
ejpam-109	194	8	a	a	DET
ejpam-109	194	9	≥	≥	NOUN
ejpam-109	194	10	1	1	NUM
ejpam-109	194	11	be	be	AUX
ejpam-109	194	12	integers	integer	NOUN
ejpam-109	194	13	with	with	ADP
ejpam-109	194	14	(	(	PUNCT
ejpam-109	194	15	a	a	PRON
ejpam-109	194	16	,	,	PUNCT
ejpam-109	194	17	n	n	CCONJ
ejpam-109	194	18	)	)	PUNCT
ejpam-109	194	19	=	=	SYM
ejpam-109	195	1	1	1	X
ejpam-109	195	2	.	.	X
ejpam-109	195	3	then	then	ADV
ejpam-109	195	4	�	�	PROPN
ejpam-109	195	5	h	h	NOUN
ejpam-109	195	6	′(n	′(n	X
ejpam-109	195	7	)	)	PUNCT
ejpam-109	196	1	+	+	CCONJ
ejpam-109	196	2	n	n	PRON
ejpam-109	196	3	�	�	NOUN
ejpam-109	196	4	m+1−h	m+1−h	NOUN
ejpam-109	196	5	′m+1(n	′m+1(n	PROPN
ejpam-109	196	6	)	)	PUNCT
ejpam-109	197	1	=	=	SYM
ejpam-109	197	2	(	(	PUNCT
ejpam-109	197	3	m+	m+	NOUN
ejpam-109	197	4	1	1	NUM
ejpam-109	197	5	)	)	PUNCT
ejpam-109	197	6	n−1	n−1	PROPN
ejpam-109	197	7	∑	∑	PUNCT
ejpam-109	197	8	j=1	j=1	PROPN
ejpam-109	197	9	(	(	PUNCT
ejpam-109	197	10	j	j	PROPN
ejpam-109	197	11	,	,	PUNCT
ejpam-109	197	12	n)=1	n)=1	PROPN
ejpam-109	197	13	jm	jm	PROPN
ejpam-109	197	14	,	,	PUNCT
ejpam-109	197	15	(	(	PUNCT
ejpam-109	197	16	i	i	NOUN
ejpam-109	197	17	)	)	PUNCT
ejpam-109	197	18	(	(	PUNCT
ejpam-109	197	19	am−	am−	NUM
ejpam-109	197	20	1	1	NUM
ejpam-109	197	21	)	)	PUNCT
ejpam-109	197	22	n	n	PRON
ejpam-109	197	23	�	�	PROPN
ejpam-109	197	24	h	h	NOUN
ejpam-109	197	25	′(n	′(n	X
ejpam-109	197	26	)	)	PUNCT
ejpam-109	197	27	+	+	CCONJ
ejpam-109	197	28	n	n	PRON
ejpam-109	197	29	�	�	NOUN
ejpam-109	197	30	m+1−h	m+1−h	NOUN
ejpam-109	197	31	′m+1(n	′m+1(n	PROPN
ejpam-109	197	32	)	)	PUNCT
ejpam-109	197	33	o	o	PROPN
ejpam-109	197	34	(	(	PUNCT
ejpam-109	197	35	ii	ii	NOUN
ejpam-109	197	36	)	)	PUNCT
ejpam-109	197	37	=(	=(	NOUN
ejpam-109	197	38	m+	m+	NUM
ejpam-109	197	39	1	1	NUM
ejpam-109	197	40	)	)	PUNCT
ejpam-109	197	41	m	m	VERB
ejpam-109	197	42	∑	∑	PROPN
ejpam-109	197	43	i=1	i=1	PROPN
ejpam-109	197	44	(	(	PUNCT
ejpam-109	197	45	−1)i+1	−1)i+1	NOUN
ejpam-109	197	46	�	�	PROPN
ejpam-109	197	47	m	m	VERB
ejpam-109	197	48	i	i	NOUN
ejpam-109	197	49	�	�	PROPN
ejpam-109	197	50	ni	ni	PROPN
ejpam-109	197	51	n−1	n−1	PROPN
ejpam-109	197	52	∑	∑	PUNCT
ejpam-109	197	53	j=1	j=1	PROPN
ejpam-109	197	54	(	(	PUNCT
ejpam-109	197	55	j	j	PROPN
ejpam-109	197	56	,	,	PUNCT
ejpam-109	197	57	n)=1	n)=1	PROPN
ejpam-109	197	58	(	(	PUNCT
ejpam-109	197	59	a	a	DET
ejpam-109	197	60	j)m−i	j)m−i	X
ejpam-109	197	61	�	�	PROPN
ejpam-109	197	62	a	a	DET
ejpam-109	197	63	j	j	PROPN
ejpam-109	197	64	n	n	PROPN
ejpam-109	197	65	�	�	PROPN
ejpam-109	197	66	i	i	PRON
ejpam-109	197	67	.	.	PUNCT
ejpam-109	198	1	we	we	PRON
ejpam-109	198	2	do	do	AUX
ejpam-109	198	3	not	not	PART
ejpam-109	198	4	give	give	VERB
ejpam-109	198	5	the	the	DET
ejpam-109	198	6	proof	proof	NOUN
ejpam-109	198	7	of	of	ADP
ejpam-109	198	8	this	this	DET
ejpam-109	198	9	theorem	theorem	NOUN
ejpam-109	198	10	,	,	PUNCT
ejpam-109	198	11	because	because	SCONJ
ejpam-109	198	12	both	both	DET
ejpam-109	198	13	congruences	congruence	NOUN
ejpam-109	198	14	(	(	PUNCT
ejpam-109	198	15	i	i	NOUN
ejpam-109	198	16	)	)	PUNCT
ejpam-109	198	17	and	and	CCONJ
ejpam-109	198	18	(	(	PUNCT
ejpam-109	198	19	ii	ii	NOUN
ejpam-109	198	20	)	)	PUNCT
ejpam-109	198	21	are	be	AUX
ejpam-109	198	22	nothing	nothing	PRON
ejpam-109	198	23	but	but	SCONJ
ejpam-109	198	24	special	special	ADJ
ejpam-109	198	25	cases	case	NOUN
ejpam-109	198	26	of	of	ADP
ejpam-109	198	27	known	know	VERB
ejpam-109	198	28	formulas	formula	NOUN
ejpam-109	198	29	for	for	ADP
ejpam-109	198	30	generalized	generalized	ADJ
ejpam-109	198	31	bernoulli	bernoulli	NOUN
ejpam-109	198	32	numbers	number	NOUN
ejpam-109	198	33	(	(	PUNCT
ejpam-109	198	34	see	see	VERB
ejpam-109	198	35	theorems	theorem	NOUN
ejpam-109	198	36	1	1	NUM
ejpam-109	198	37	and	and	CCONJ
ejpam-109	198	38	3	3	NUM
ejpam-109	198	39	in	in	ADP
ejpam-109	198	40	agoh	agoh	NOUN
ejpam-109	198	41	[	[	X
ejpam-109	198	42	1	1	NUM
ejpam-109	198	43	]	]	PUNCT
ejpam-109	198	44	)	)	PUNCT
ejpam-109	198	45	.	.	PUNCT
ejpam-109	199	1	in	in	ADP
ejpam-109	199	2	addition	addition	NOUN
ejpam-109	199	3	,	,	PUNCT
ejpam-109	199	4	we	we	PRON
ejpam-109	199	5	note	note	VERB
ejpam-109	199	6	that	that	SCONJ
ejpam-109	199	7	theorem	theorem	VERB
ejpam-109	199	8	3.2	3.2	NUM
ejpam-109	199	9	can	can	AUX
ejpam-109	199	10	be	be	AUX
ejpam-109	199	11	also	also	ADV
ejpam-109	199	12	derived	derive	VERB
ejpam-109	199	13	from	from	ADP
ejpam-109	199	14	above	above	ADP
ejpam-109	199	15	(	(	PUNCT
ejpam-109	199	16	i	i	NOUN
ejpam-109	199	17	)	)	PUNCT
ejpam-109	199	18	.	.	PUNCT
ejpam-109	200	1	now	now	ADV
ejpam-109	200	2	,	,	PUNCT
ejpam-109	200	3	we	we	PRON
ejpam-109	200	4	would	would	AUX
ejpam-109	200	5	like	like	VERB
ejpam-109	200	6	to	to	PART
ejpam-109	200	7	introduce	introduce	VERB
ejpam-109	200	8	an	an	DET
ejpam-109	200	9	interesting	interesting	ADJ
ejpam-109	200	10	problem	problem	NOUN
ejpam-109	200	11	related	relate	VERB
ejpam-109	200	12	to	to	ADP
ejpam-109	200	13	a	a	DET
ejpam-109	200	14	characterization	characterization	NOUN
ejpam-109	200	15	of	of	ADP
ejpam-109	200	16	primes	prime	NOUN
ejpam-109	200	17	,	,	PUNCT
ejpam-109	200	18	which	which	PRON
ejpam-109	200	19	is	be	AUX
ejpam-109	200	20	deeply	deeply	ADV
ejpam-109	200	21	related	relate	VERB
ejpam-109	200	22	to	to	ADP
ejpam-109	200	23	the	the	DET
ejpam-109	200	24	von	von	PROPN
ejpam-109	200	25	staudt	staudt	PROPN
ejpam-109	200	26	-	-	PUNCT
ejpam-109	200	27	clausen	clausen	PROPN
ejpam-109	200	28	congruence	congruence	PROPN
ejpam-109	200	29	.	.	PUNCT
ejpam-109	201	1	put	put	VERB
ejpam-109	201	2	g	g	NOUN
ejpam-109	201	3	=	=	PUNCT
ejpam-109	201	4	{	{	PUNCT
ejpam-109	201	5	n≥	n≥	NOUN
ejpam-109	201	6	2	2	NUM
ejpam-109	201	7	|	|	ADV
ejpam-109	201	8	sn−1(n)≡	sn−1(n)≡	NUM
ejpam-109	201	9	n−	n−	NOUN
ejpam-109	201	10	1	1	NUM
ejpam-109	201	11	(	(	PUNCT
ejpam-109	201	12	mod	mod	NOUN
ejpam-109	201	13	n	n	CCONJ
ejpam-109	201	14	)	)	PUNCT
ejpam-109	201	15	}	}	PUNCT
ejpam-109	201	16	.	.	PUNCT
ejpam-109	202	1	by	by	ADP
ejpam-109	202	2	fermat	fermat	PROPN
ejpam-109	202	3	’s	’s	PART
ejpam-109	202	4	little	little	ADJ
ejpam-109	202	5	theorem	theorem	ADJ
ejpam-109	202	6	,	,	PUNCT
ejpam-109	202	7	it	it	PRON
ejpam-109	202	8	is	be	AUX
ejpam-109	202	9	clear	clear	ADJ
ejpam-109	202	10	that	that	SCONJ
ejpam-109	202	11	the	the	DET
ejpam-109	202	12	set	set	NOUN
ejpam-109	202	13	g	g	NOUN
ejpam-109	202	14	contains	contain	VERB
ejpam-109	202	15	any	any	DET
ejpam-109	202	16	primes	prime	NOUN
ejpam-109	202	17	.	.	PUNCT
ejpam-109	203	1	in	in	ADP
ejpam-109	203	2	1950	1950	NUM
ejpam-109	203	3	,	,	PUNCT
ejpam-109	203	4	giuga	giuga	NOUN
ejpam-109	203	5	[	[	X
ejpam-109	203	6	8	8	NUM
ejpam-109	203	7	]	]	PUNCT
ejpam-109	203	8	conjectured	conjecture	VERB
ejpam-109	203	9	that	that	SCONJ
ejpam-109	203	10	g	g	PROPN
ejpam-109	203	11	is	be	AUX
ejpam-109	203	12	precisely	precisely	ADV
ejpam-109	203	13	the	the	DET
ejpam-109	203	14	same	same	ADJ
ejpam-109	203	15	as	as	ADP
ejpam-109	203	16	the	the	DET
ejpam-109	203	17	set	set	NOUN
ejpam-109	203	18	of	of	ADP
ejpam-109	203	19	all	all	DET
ejpam-109	203	20	the	the	DET
ejpam-109	203	21	primes	prime	NOUN
ejpam-109	203	22	.	.	PUNCT
ejpam-109	204	1	by	by	ADP
ejpam-109	204	2	elementary	elementary	ADJ
ejpam-109	204	3	observation	observation	NOUN
ejpam-109	204	4	,	,	PUNCT
ejpam-109	204	5	we	we	PRON
ejpam-109	204	6	see	see	VERB
ejpam-109	204	7	that	that	SCONJ
ejpam-109	204	8	the	the	DET
ejpam-109	204	9	following	follow	VERB
ejpam-109	204	10	statements	statement	NOUN
ejpam-109	204	11	are	be	AUX
ejpam-109	204	12	all	all	ADV
ejpam-109	204	13	equivalent	equivalent	ADJ
ejpam-109	204	14	(	(	PUNCT
ejpam-109	204	15	for	for	ADP
ejpam-109	204	16	details	detail	NOUN
ejpam-109	204	17	,	,	PUNCT
ejpam-109	204	18	see	see	VERB
ejpam-109	204	19	agoh	agoh	NOUN
ejpam-109	205	1	[	[	X
ejpam-109	205	2	2	2	NUM
ejpam-109	205	3	]	]	PUNCT
ejpam-109	205	4	,	,	PUNCT
ejpam-109	205	5	borwein	borwein	NOUN
ejpam-109	205	6	et	et	PROPN
ejpam-109	205	7	al	al	PROPN
ejpam-109	205	8	.	.	PUNCT
ejpam-109	206	1	[	[	X
ejpam-109	206	2	4	4	X
ejpam-109	206	3	]	]	PUNCT
ejpam-109	206	4	and	and	CCONJ
ejpam-109	206	5	kellner	kellner	PROPN
ejpam-109	207	1	[	[	X
ejpam-109	207	2	12	12	NUM
ejpam-109	207	3	]	]	PUNCT
ejpam-109	207	4	):	):	PUNCT
ejpam-109	207	5	(	(	PUNCT
ejpam-109	207	6	a	a	NOUN
ejpam-109	207	7	)	)	PUNCT
ejpam-109	207	8	n	n	NOUN
ejpam-109	207	9	∈	∈	PROPN
ejpam-109	207	10	g	g	NOUN
ejpam-109	207	11	,	,	PUNCT
ejpam-109	207	12	(	(	PUNCT
ejpam-109	207	13	b	b	NOUN
ejpam-109	207	14	)	)	PUNCT
ejpam-109	207	15	p2(p−	p2(p−	X
ejpam-109	207	16	1	1	X
ejpam-109	207	17	)	)	PUNCT
ejpam-109	208	1	|	|	ADV
ejpam-109	208	2	n−	n−	NOUN
ejpam-109	208	3	p	p	NOUN
ejpam-109	208	4	for	for	ADP
ejpam-109	208	5	any	any	DET
ejpam-109	208	6	prime	prime	ADJ
ejpam-109	208	7	divisor	divisor	NOUN
ejpam-109	208	8	p	p	NOUN
ejpam-109	208	9	of	of	ADP
ejpam-109	208	10	n	n	CCONJ
ejpam-109	208	11	,	,	PUNCT
ejpam-109	208	12	(	(	PUNCT
ejpam-109	208	13	c	c	NOUN
ejpam-109	208	14	)	)	PUNCT
ejpam-109	208	15	nbn−1	nbn−1	PROPN
ejpam-109	208	16	≡	≡	PROPN
ejpam-109	208	17	n−	n−	PROPN
ejpam-109	208	18	1	1	NUM
ejpam-109	208	19	(	(	PUNCT
ejpam-109	208	20	mod	mod	NOUN
ejpam-109	208	21	n	n	CCONJ
ejpam-109	208	22	)	)	PUNCT
ejpam-109	208	23	.	.	PUNCT
ejpam-109	209	1	therefore	therefore	ADV
ejpam-109	209	2	,	,	PUNCT
ejpam-109	209	3	we	we	PRON
ejpam-109	209	4	can	can	AUX
ejpam-109	209	5	restate	restate	VERB
ejpam-109	209	6	giuga	giuga	NOUN
ejpam-109	209	7	’s	’s	PART
ejpam-109	209	8	conjecture	conjecture	NOUN
ejpam-109	209	9	as	as	SCONJ
ejpam-109	209	10	follows	follow	VERB
ejpam-109	209	11	:	:	PUNCT
ejpam-109	209	12	conjecture	conjecture	NOUN
ejpam-109	209	13	3.5	3.5	NUM
ejpam-109	209	14	.	.	PUNCT
ejpam-109	210	1	let	let	VERB
ejpam-109	210	2	n≥	n≥	NOUN
ejpam-109	210	3	2	2	NUM
ejpam-109	210	4	.	.	PUNCT
ejpam-109	211	1	then	then	ADV
ejpam-109	211	2	nbn−1	nbn−1	PROPN
ejpam-109	211	3	≡	≡	PROPN
ejpam-109	211	4	n−	n−	PROPN
ejpam-109	211	5	1	1	NUM
ejpam-109	211	6	(	(	PUNCT
ejpam-109	211	7	mod	mod	NOUN
ejpam-109	211	8	n	n	CCONJ
ejpam-109	211	9	)	)	PUNCT
ejpam-109	211	10	if	if	SCONJ
ejpam-109	211	11	and	and	CCONJ
ejpam-109	211	12	only	only	ADV
ejpam-109	211	13	if	if	SCONJ
ejpam-109	211	14	n	n	PRON
ejpam-109	211	15	is	be	AUX
ejpam-109	211	16	a	a	DET
ejpam-109	211	17	prime	prime	NOUN
ejpam-109	211	18	.	.	PUNCT
ejpam-109	212	1	a	a	DET
ejpam-109	212	2	composite	composite	ADJ
ejpam-109	212	3	n	n	NOUN
ejpam-109	212	4	is	be	AUX
ejpam-109	212	5	called	call	VERB
ejpam-109	212	6	a	a	DET
ejpam-109	212	7	carmichael	carmichael	PROPN
ejpam-109	212	8	number	number	NOUN
ejpam-109	212	9	if	if	SCONJ
ejpam-109	212	10	an−1	an−1	PROPN
ejpam-109	212	11	≡	≡	PROPN
ejpam-109	212	12	1	1	NUM
ejpam-109	212	13	(	(	PUNCT
ejpam-109	212	14	mod	mod	NOUN
ejpam-109	212	15	n	n	CCONJ
ejpam-109	212	16	)	)	PUNCT
ejpam-109	212	17	for	for	ADP
ejpam-109	212	18	every	every	DET
ejpam-109	212	19	positive	positive	ADJ
ejpam-109	212	20	integer	integer	NOUN
ejpam-109	212	21	a	a	PRON
ejpam-109	212	22	with	with	ADP
ejpam-109	212	23	(	(	PUNCT
ejpam-109	212	24	a	a	PRON
ejpam-109	212	25	,	,	PUNCT
ejpam-109	212	26	n	n	CCONJ
ejpam-109	212	27	)	)	PUNCT
ejpam-109	212	28	=	=	SYM
ejpam-109	213	1	1	1	X
ejpam-109	213	2	.	.	X
ejpam-109	213	3	such	such	DET
ejpam-109	213	4	the	the	DET
ejpam-109	213	5	smallest	small	ADJ
ejpam-109	213	6	number	number	NOUN
ejpam-109	213	7	is	be	AUX
ejpam-109	213	8	561	561	NUM
ejpam-109	213	9	=	=	SYM
ejpam-109	213	10	3	3	NUM
ejpam-109	213	11	·	·	SYM
ejpam-109	213	12	11	11	NUM
ejpam-109	213	13	·	·	SYM
ejpam-109	213	14	17	17	NUM
ejpam-109	213	15	.	.	PUNCT
ejpam-109	214	1	it	it	PRON
ejpam-109	214	2	is	be	AUX
ejpam-109	214	3	easily	easily	ADV
ejpam-109	214	4	shown	show	VERB
ejpam-109	214	5	that	that	SCONJ
ejpam-109	214	6	n	n	PRON
ejpam-109	214	7	is	be	AUX
ejpam-109	214	8	a	a	DET
ejpam-109	214	9	carmichael	carmichael	PROPN
ejpam-109	214	10	number	number	NOUN
ejpam-109	214	11	if	if	SCONJ
ejpam-109	214	12	and	and	CCONJ
ejpam-109	214	13	only	only	ADV
ejpam-109	214	14	if	if	SCONJ
ejpam-109	214	15	n	n	NOUN
ejpam-109	214	16	=	=	X
ejpam-109	214	17	p1p2	p1p2	X
ejpam-109	214	18	·	·	PUNCT
ejpam-109	214	19	·	·	PUNCT
ejpam-109	214	20	·	·	PUNCT
ejpam-109	215	1	pg	pg	PRON
ejpam-109	215	2	(	(	PUNCT
ejpam-109	215	3	the	the	DET
ejpam-109	215	4	product	product	NOUN
ejpam-109	215	5	of	of	ADP
ejpam-109	215	6	some	some	DET
ejpam-109	215	7	distinct	distinct	ADJ
ejpam-109	215	8	odd	odd	ADJ
ejpam-109	215	9	primes	prime	NOUN
ejpam-109	215	10	)	)	PUNCT
ejpam-109	215	11	and	and	CCONJ
ejpam-109	215	12	p(p	p(p	ADV
ejpam-109	215	13	−	−	NOUN
ejpam-109	215	14	1	1	X
ejpam-109	215	15	)	)	PUNCT
ejpam-109	215	16	|	|	ADV
ejpam-109	215	17	n−	n−	VERB
ejpam-109	215	18	p	p	NOUN
ejpam-109	215	19	for	for	ADP
ejpam-109	215	20	each	each	DET
ejpam-109	215	21	prime	prime	ADJ
ejpam-109	215	22	divisor	divisor	NOUN
ejpam-109	215	23	p	p	NOUN
ejpam-109	215	24	of	of	ADP
ejpam-109	215	25	n.	n.	NOUN
ejpam-109	215	26	further	far	ADV
ejpam-109	215	27	,	,	PUNCT
ejpam-109	215	28	this	this	DET
ejpam-109	215	29	number	number	NOUN
ejpam-109	215	30	may	may	AUX
ejpam-109	215	31	be	be	AUX
ejpam-109	215	32	characterized	characterize	VERB
ejpam-109	215	33	by	by	ADP
ejpam-109	215	34	nbn−1	nbn−1	PROPN
ejpam-109	215	35	≡	≡	PROPN
ejpam-109	215	36	−n	−n	NOUN
ejpam-109	215	37	∑	∑	PROPN
ejpam-109	215	38	p|n	p|n	PROPN
ejpam-109	215	39	1	1	NUM
ejpam-109	215	40	/	/	SYM
ejpam-109	215	41	p	p	X
ejpam-109	215	42	(	(	PUNCT
ejpam-109	215	43	mod	mod	PROPN
ejpam-109	215	44	n	n	CCONJ
ejpam-109	215	45	)	)	PUNCT
ejpam-109	215	46	.	.	PUNCT
ejpam-109	216	1	in	in	ADP
ejpam-109	216	2	their	their	PRON
ejpam-109	216	3	outstanding	outstanding	ADJ
ejpam-109	216	4	paper	paper	NOUN
ejpam-109	216	5	[	[	X
ejpam-109	216	6	3	3	NUM
ejpam-109	216	7	]	]	PUNCT
ejpam-109	216	8	,	,	PUNCT
ejpam-109	216	9	alford	alford	PROPN
ejpam-109	216	10	,	,	PUNCT
ejpam-109	216	11	granville	granville	PROPN
ejpam-109	216	12	and	and	CCONJ
ejpam-109	216	13	pomerance	pomerance	NOUN
ejpam-109	216	14	proved	prove	VERB
ejpam-109	216	15	that	that	SCONJ
ejpam-109	216	16	there	there	PRON
ejpam-109	216	17	exist	exist	VERB
ejpam-109	216	18	infinitely	infinitely	ADV
ejpam-109	216	19	many	many	ADJ
ejpam-109	216	20	carmichael	carmichael	PROPN
ejpam-109	216	21	numbers	number	NOUN
ejpam-109	216	22	.	.	PUNCT
ejpam-109	217	1	we	we	PRON
ejpam-109	217	2	can	can	AUX
ejpam-109	217	3	show	show	VERB
ejpam-109	217	4	that	that	SCONJ
ejpam-109	217	5	if	if	SCONJ
ejpam-109	217	6	there	there	PRON
ejpam-109	217	7	is	be	VERB
ejpam-109	217	8	a	a	DET
ejpam-109	217	9	composite	composite	ADJ
ejpam-109	217	10	number	number	NOUN
ejpam-109	217	11	n	n	NUM
ejpam-109	217	12	∈	∈	NOUN
ejpam-109	217	13	g	g	NOUN
ejpam-109	217	14	,	,	PUNCT
ejpam-109	217	15	then	then	ADV
ejpam-109	217	16	n	n	PRON
ejpam-109	217	17	is	be	AUX
ejpam-109	217	18	the	the	DET
ejpam-109	217	19	carmichael	carmichael	PROPN
ejpam-109	217	20	number	number	NOUN
ejpam-109	217	21	and	and	CCONJ
ejpam-109	217	22	the	the	DET
ejpam-109	217	23	number	number	NOUN
ejpam-109	217	24	of	of	ADP
ejpam-109	217	25	its	its	PRON
ejpam-109	217	26	prime	prime	ADJ
ejpam-109	217	27	divisors	divisor	NOUN
ejpam-109	217	28	must	must	AUX
ejpam-109	217	29	be	be	AUX
ejpam-109	217	30	greater	great	ADJ
ejpam-109	217	31	than	than	ADP
ejpam-109	217	32	the	the	DET
ejpam-109	217	33	smallest	small	ADJ
ejpam-109	217	34	prime	prime	ADJ
ejpam-109	217	35	divisor	divisor	NOUN
ejpam-109	217	36	of	of	ADP
ejpam-109	217	37	n.	n.	NOUN
ejpam-109	217	38	also	also	ADV
ejpam-109	217	39	we	we	PRON
ejpam-109	217	40	can	can	AUX
ejpam-109	217	41	say	say	VERB
ejpam-109	217	42	that	that	SCONJ
ejpam-109	217	43	if	if	SCONJ
ejpam-109	217	44	n	n	NUM
ejpam-109	217	45	∈	∈	PROPN
ejpam-109	217	46	g	g	PROPN
ejpam-109	217	47	is	be	AUX
ejpam-109	217	48	composite	composite	ADJ
ejpam-109	217	49	,	,	PUNCT
ejpam-109	217	50	then	then	ADV
ejpam-109	218	1	(	(	PUNCT
ejpam-109	218	2	cf	cf	NOUN
ejpam-109	218	3	.	.	PUNCT
ejpam-109	218	4	agoh	agoh	PROPN
ejpam-109	219	1	[	[	X
ejpam-109	219	2	2	2	NUM
ejpam-109	219	3	]	]	PUNCT
ejpam-109	219	4	)	)	PUNCT
ejpam-109	219	5			PROPN
ejpam-109	219	6			ADP
ejpam-109	219	7			ADJ
ejpam-109	219	8	pbn−p	pbn−p	NOUN
ejpam-109	219	9	≡	≡	PROPN
ejpam-109	219	10	p−	p−	NOUN
ejpam-109	219	11	1	1	NUM
ejpam-109	219	12	(	(	PUNCT
ejpam-109	219	13	mod	mod	PROPN
ejpam-109	219	14	p3	p3	PROPN
ejpam-109	219	15	)	)	PUNCT
ejpam-109	219	16	,	,	PUNCT
ejpam-109	219	17	pb(n	pb(n	PUNCT
ejpam-109	219	18	/	/	SYM
ejpam-109	219	19	p)−1	p)−1	PROPN
ejpam-109	219	20	≡	≡	PROPN
ejpam-109	219	21	p−	p−	NOUN
ejpam-109	219	22	1	1	NUM
ejpam-109	219	23	(	(	PUNCT
ejpam-109	219	24	mod	mod	ADJ
ejpam-109	219	25	p2	p2	PROPN
ejpam-109	219	26	)	)	PUNCT
ejpam-109	219	27	,	,	PUNCT
ejpam-109	219	28	pb((n	pb((n	NOUN
ejpam-109	219	29	/	/	SYM
ejpam-109	219	30	p)−1)/p	p)−1)/p	NOUN
ejpam-109	219	31	≡	≡	PROPN
ejpam-109	219	32	p−	p−	NOUN
ejpam-109	219	33	1	1	NUM
ejpam-109	219	34	(	(	PUNCT
ejpam-109	219	35	mod	mod	NOUN
ejpam-109	219	36	p	p	X
ejpam-109	219	37	)	)	PUNCT
ejpam-109	219	38	t.	t.	NOUN
ejpam-109	219	39	agoh	agoh	PROPN
ejpam-109	219	40	/	/	SYM
ejpam-109	219	41	eur	eur	PROPN
ejpam-109	219	42	.	.	PUNCT
ejpam-109	220	1	j.	j.	PROPN
ejpam-109	220	2	pure	pure	PROPN
ejpam-109	220	3	appl	appl	PROPN
ejpam-109	220	4	.	.	PROPN
ejpam-109	220	5	math	math	PROPN
ejpam-109	220	6	,	,	PUNCT
ejpam-109	220	7	1	1	NUM
ejpam-109	220	8	(	(	PUNCT
ejpam-109	220	9	2008	2008	NUM
ejpam-109	220	10	)	)	PUNCT
ejpam-109	220	11	,	,	PUNCT
ejpam-109	220	12	(	(	PUNCT
ejpam-109	220	13	3	3	NUM
ejpam-109	220	14	-	-	SYM
ejpam-109	220	15	21	21	NUM
ejpam-109	220	16	)	)	PUNCT
ejpam-109	220	17	12	12	NUM
ejpam-109	220	18	for	for	ADP
ejpam-109	220	19	all	all	DET
ejpam-109	220	20	prime	prime	ADJ
ejpam-109	220	21	divisors	divisor	NOUN
ejpam-109	220	22	p	p	NOUN
ejpam-109	220	23	of	of	ADP
ejpam-109	220	24	n.	n.	NOUN
ejpam-109	220	25	it	it	PRON
ejpam-109	220	26	is	be	AUX
ejpam-109	220	27	unknown	unknown	ADJ
ejpam-109	220	28	whether	whether	SCONJ
ejpam-109	220	29	there	there	PRON
ejpam-109	220	30	is	be	VERB
ejpam-109	220	31	a	a	DET
ejpam-109	220	32	composite	composite	ADJ
ejpam-109	220	33	number	number	NOUN
ejpam-109	220	34	n	n	ADP
ejpam-109	220	35	belonging	belong	VERB
ejpam-109	220	36	to	to	ADP
ejpam-109	220	37	g.	g.	PROPN
ejpam-109	220	38	recently	recently	ADV
ejpam-109	220	39	,	,	PUNCT
ejpam-109	220	40	it	it	PRON
ejpam-109	220	41	was	be	AUX
ejpam-109	220	42	shown	show	VERB
ejpam-109	220	43	in	in	ADP
ejpam-109	220	44	borwein	borwein	PROPN
ejpam-109	220	45	et	et	PROPN
ejpam-109	220	46	al	al	PROPN
ejpam-109	220	47	.	.	PUNCT
ejpam-109	221	1	[	[	X
ejpam-109	221	2	4	4	X
ejpam-109	221	3	]	]	PUNCT
ejpam-109	221	4	that	that	SCONJ
ejpam-109	221	5	any	any	DET
ejpam-109	221	6	counter	counter	NOUN
ejpam-109	221	7	example	example	NOUN
ejpam-109	221	8	has	have	VERB
ejpam-109	221	9	at	at	ADV
ejpam-109	221	10	least	least	ADJ
ejpam-109	221	11	12055	12055	NUM
ejpam-109	221	12	digits	digit	NOUN
ejpam-109	221	13	.	.	PUNCT
ejpam-109	222	1	for	for	ADP
ejpam-109	222	2	more	more	ADV
ejpam-109	222	3	extensive	extensive	ADJ
ejpam-109	222	4	computations	computation	NOUN
ejpam-109	222	5	,	,	PUNCT
ejpam-109	222	6	see	see	VERB
ejpam-109	222	7	fee	fee	NOUN
ejpam-109	222	8	and	and	CCONJ
ejpam-109	222	9	plouffe	plouffe	PROPN
ejpam-109	223	1	[	[	X
ejpam-109	223	2	7	7	NUM
ejpam-109	223	3	]	]	PUNCT
ejpam-109	223	4	.	.	PUNCT
ejpam-109	224	1	if	if	SCONJ
ejpam-109	224	2	it	it	PRON
ejpam-109	224	3	is	be	AUX
ejpam-109	224	4	possible	possible	ADJ
ejpam-109	224	5	to	to	PART
ejpam-109	224	6	appreciate	appreciate	VERB
ejpam-109	224	7	the	the	DET
ejpam-109	224	8	justification	justification	NOUN
ejpam-109	224	9	of	of	ADP
ejpam-109	224	10	the	the	DET
ejpam-109	224	11	above	above	ADJ
ejpam-109	224	12	conjecture	conjecture	NOUN
ejpam-109	224	13	,	,	PUNCT
ejpam-109	224	14	then	then	ADV
ejpam-109	224	15	an	an	DET
ejpam-109	224	16	interesting	interesting	ADJ
ejpam-109	224	17	characterization	characterization	NOUN
ejpam-109	224	18	of	of	ADP
ejpam-109	224	19	primes	prime	NOUN
ejpam-109	224	20	can	can	AUX
ejpam-109	224	21	be	be	AUX
ejpam-109	224	22	given	give	VERB
ejpam-109	224	23	by	by	ADP
ejpam-109	224	24	means	mean	NOUN
ejpam-109	224	25	of	of	ADP
ejpam-109	224	26	the	the	DET
ejpam-109	224	27	bernoulli	bernoulli	NOUN
ejpam-109	224	28	number	number	NOUN
ejpam-109	224	29	.	.	PUNCT
ejpam-109	225	1	4	4	X
ejpam-109	225	2	.	.	NUM
ejpam-109	225	3	generalized	generalized	ADJ
ejpam-109	225	4	von	von	PROPN
ejpam-109	225	5	staudt	staudt	PROPN
ejpam-109	225	6	-	-	PUNCT
ejpam-109	225	7	kummer	kummer	NOUN
ejpam-109	225	8	congruences	congruence	NOUN
ejpam-109	225	9	in	in	ADP
ejpam-109	225	10	this	this	DET
ejpam-109	225	11	section	section	NOUN
ejpam-109	225	12	,	,	PUNCT
ejpam-109	225	13	we	we	PRON
ejpam-109	225	14	will	will	AUX
ejpam-109	225	15	generalize	generalize	VERB
ejpam-109	225	16	von	von	PROPN
ejpam-109	225	17	staudt	staudt	PROPN
ejpam-109	225	18	-	-	PUNCT
ejpam-109	225	19	kummer	kummer	NOUN
ejpam-109	225	20	congruences	congruence	NOUN
ejpam-109	225	21	stated	state	VERB
ejpam-109	225	22	in	in	ADP
ejpam-109	225	23	theorem	theorem	ADJ
ejpam-109	225	24	2.6	2.6	NUM
ejpam-109	225	25	.	.	PUNCT
ejpam-109	226	1	in	in	ADP
ejpam-109	226	2	what	what	PRON
ejpam-109	226	3	follows	follow	VERB
ejpam-109	226	4	,	,	PUNCT
ejpam-109	226	5	we	we	PRON
ejpam-109	226	6	assume	assume	VERB
ejpam-109	226	7	that	that	SCONJ
ejpam-109	226	8	n	n	CCONJ
ejpam-109	226	9	,	,	PUNCT
ejpam-109	226	10	s	s	PROPN
ejpam-109	226	11	,	,	PUNCT
ejpam-109	226	12	mr	mr	PROPN
ejpam-109	226	13	,	,	PUNCT
ejpam-109	226	14	ar	ar	PROPN
ejpam-109	226	15	,	,	PUNCT
ejpam-109	226	16	br	br	PROPN
ejpam-109	227	1	(	(	PUNCT
ejpam-109	227	2	r	r	NOUN
ejpam-109	227	3	=	=	SYM
ejpam-109	227	4	1	1	NUM
ejpam-109	227	5	,	,	PUNCT
ejpam-109	227	6	2	2	NUM
ejpam-109	227	7	,	,	PUNCT
ejpam-109	227	8	...	...	PUNCT
ejpam-109	227	9	,	,	PUNCT
ejpam-109	228	1	k	k	X
ejpam-109	228	2	)	)	PUNCT
ejpam-109	228	3	are	be	AUX
ejpam-109	228	4	positive	positive	ADJ
ejpam-109	228	5	integers	integer	NOUN
ejpam-109	228	6	with	with	ADP
ejpam-109	228	7	n≥	n≥	PROPN
ejpam-109	228	8	3	3	NUM
ejpam-109	228	9	and	and	CCONJ
ejpam-109	228	10	(	(	PUNCT
ejpam-109	228	11	ar	ar	PROPN
ejpam-109	228	12	,	,	PUNCT
ejpam-109	228	13	n	n	CCONJ
ejpam-109	228	14	)	)	PUNCT
ejpam-109	228	15	=	=	SYM
ejpam-109	228	16	1	1	X
ejpam-109	228	17	.	.	X
ejpam-109	228	18	theorem	theorem	VERB
ejpam-109	228	19	4.1	4.1	NUM
ejpam-109	228	20	.	.	PUNCT
ejpam-109	229	1	for	for	ADP
ejpam-109	229	2	each	each	DET
ejpam-109	229	3	r	r	NOUN
ejpam-109	229	4	=	=	SYM
ejpam-109	229	5	1	1	NUM
ejpam-109	229	6	,	,	PUNCT
ejpam-109	229	7	2	2	NUM
ejpam-109	229	8	,	,	PUNCT
ejpam-109	229	9	...	...	PUNCT
ejpam-109	229	10	,	,	PUNCT
ejpam-109	229	11	k	k	PROPN
ejpam-109	229	12	+	+	NOUN
ejpam-109	229	13	1	1	NUM
ejpam-109	229	14	,	,	PUNCT
ejpam-109	229	15	we	we	PRON
ejpam-109	229	16	assume	assume	VERB
ejpam-109	229	17	that	that	SCONJ
ejpam-109	229	18	mr	mr	PROPN
ejpam-109	229	19	is	be	AUX
ejpam-109	229	20	even	even	ADV
ejpam-109	229	21	and	and	CCONJ
ejpam-109	229	22	λr	λr	NOUN
ejpam-109	229	23	∈	∈	NOUN
ejpam-109	229	24	zn	zn	NOUN
ejpam-109	229	25	satisfies	satisfie	NOUN
ejpam-109	229	26	∑k+1	∑k+1	VERB
ejpam-109	229	27	i=1	i=1	PROPN
ejpam-109	230	1	λi	λi	INTJ
ejpam-109	230	2	≡	≡	PROPN
ejpam-109	230	3	0(mod	0(mod	NOUN
ejpam-109	230	4	n	n	CCONJ
ejpam-109	230	5	)	)	PUNCT
ejpam-109	230	6	.	.	PUNCT
ejpam-109	231	1	then	then	ADV
ejpam-109	231	2	we	we	PRON
ejpam-109	231	3	have	have	VERB
ejpam-109	231	4	k	k	PROPN
ejpam-109	231	5	∏	∏	PROPN
ejpam-109	231	6	r=1	r=1	NOUN
ejpam-109	231	7	kmr	kmr	NOUN
ejpam-109	231	8	(	(	PUNCT
ejpam-109	231	9	n	n	NUM
ejpam-109	231	10	;	;	PUNCT
ejpam-109	231	11	ar	ar	NOUN
ejpam-109	231	12	)	)	PUNCT
ejpam-109	231	13	k	k	NOUN
ejpam-109	231	14	∑	∑	PUNCT
ejpam-109	231	15	r=1	r=1	NOUN
ejpam-109	231	16	λr	λr	ADP
ejpam-109	231	17	k	k	PROPN
ejpam-109	231	18	brϕ(n)(n	brϕ(n)(n	PROPN
ejpam-109	231	19	;	;	PUNCT
ejpam-109	231	20	ar	ar	NOUN
ejpam-109	231	21	)	)	PUNCT
ejpam-109	231	22	+	+	NOUN
ejpam-109	231	23	λk+1	λk+1	NUM
ejpam-109	231	24	!	!	PUNCT
ejpam-109	231	25	s	s	PART
ejpam-109	232	1	≡	≡	PROPN
ejpam-109	232	2	0	0	PUNCT
ejpam-109	233	1	(	(	PUNCT
ejpam-109	233	2	mod	mod	PROPN
ejpam-109	233	3	ns	ns	NUM
ejpam-109	233	4	)	)	PUNCT
ejpam-109	233	5	.	.	PUNCT
ejpam-109	234	1	(	(	PUNCT
ejpam-109	234	2	i	i	NOUN
ejpam-109	234	3	)	)	PUNCT
ejpam-109	234	4	in	in	ADP
ejpam-109	234	5	particular	particular	ADJ
ejpam-109	234	6	,	,	PUNCT
ejpam-109	234	7	if	if	SCONJ
ejpam-109	234	8	n=	n=	ADJ
ejpam-109	234	9	pα	pα	INTJ
ejpam-109	234	10	(	(	PUNCT
ejpam-109	234	11	α≥	α≥	PROPN
ejpam-109	234	12	1	1	NUM
ejpam-109	234	13	,	,	PUNCT
ejpam-109	234	14	p	p	X
ejpam-109	234	15	an	an	DET
ejpam-109	234	16	odd	odd	ADJ
ejpam-109	234	17	prime	prime	NOUN
ejpam-109	234	18	)	)	PUNCT
ejpam-109	234	19	and	and	CCONJ
ejpam-109	234	20	p−	p−	NOUN
ejpam-109	234	21	1	1	NUM
ejpam-109	234	22	mr	mr	PROPN
ejpam-109	234	23	,	,	PUNCT
ejpam-109	234	24	r	r	NOUN
ejpam-109	234	25	=	=	SYM
ejpam-109	234	26	1,2	1,2	NUM
ejpam-109	234	27	,	,	PUNCT
ejpam-109	234	28	...	...	PUNCT
ejpam-109	234	29	,	,	PUNCT
ejpam-109	234	30	k	k	NOUN
ejpam-109	234	31	,	,	PUNCT
ejpam-109	234	32	then	then	ADV
ejpam-109	234	33	k	k	PROPN
ejpam-109	234	34	∏	∏	PROPN
ejpam-109	234	35	r=1	r=1	NOUN
ejpam-109	234	36	hmr	hmr	PROPN
ejpam-109	234	37	r	r	NOUN
ejpam-109	234	38	(	(	PUNCT
ejpam-109	234	39	n	n	CCONJ
ejpam-109	234	40	)	)	PUNCT
ejpam-109	234	41	k	k	NOUN
ejpam-109	234	42	∑	∑	PUNCT
ejpam-109	234	43	r=1	r=1	NOUN
ejpam-109	234	44	λr	λr	ADP
ejpam-109	234	45	h	h	NOUN
ejpam-109	234	46	brϕ(n	brϕ(n	NOUN
ejpam-109	234	47	)	)	PUNCT
ejpam-109	234	48	r	r	NOUN
ejpam-109	234	49	(	(	PUNCT
ejpam-109	234	50	n	n	CCONJ
ejpam-109	234	51	)	)	PUNCT
ejpam-109	235	1	+	+	ADV
ejpam-109	235	2	λk+1	λk+1	X
ejpam-109	235	3	!	!	PUNCT
ejpam-109	235	4	s	s	PART
ejpam-109	236	1	≡	≡	PROPN
ejpam-109	236	2	0	0	PUNCT
ejpam-109	237	1	(	(	PUNCT
ejpam-109	237	2	mod	mod	PROPN
ejpam-109	237	3	ns	ns	NUM
ejpam-109	237	4	)	)	PUNCT
ejpam-109	237	5	.	.	PUNCT
ejpam-109	238	1	(	(	PUNCT
ejpam-109	238	2	ii	ii	X
ejpam-109	238	3	)	)	PUNCT
ejpam-109	238	4	the	the	DET
ejpam-109	238	5	symbolic	symbolic	ADJ
ejpam-109	238	6	expression	expression	NOUN
ejpam-109	238	7	used	use	VERB
ejpam-109	238	8	in	in	ADP
ejpam-109	238	9	the	the	DET
ejpam-109	238	10	above	above	ADJ
ejpam-109	238	11	statement	statement	NOUN
ejpam-109	238	12	means	mean	VERB
ejpam-109	238	13	that	that	SCONJ
ejpam-109	238	14	we	we	PRON
ejpam-109	238	15	expand	expand	VERB
ejpam-109	238	16	the	the	DET
ejpam-109	238	17	left	left	ADJ
ejpam-109	238	18	-	-	PUNCT
ejpam-109	238	19	hand	hand	NOUN
ejpam-109	238	20	side	side	NOUN
ejpam-109	238	21	of	of	ADP
ejpam-109	238	22	(	(	PUNCT
ejpam-109	238	23	i	i	NOUN
ejpam-109	238	24	)	)	PUNCT
ejpam-109	238	25	(	(	PUNCT
ejpam-109	238	26	resp	resp	NOUN
ejpam-109	238	27	.	.	PUNCT
ejpam-109	239	1	(	(	PUNCT
ejpam-109	239	2	ii	ii	NOUN
ejpam-109	239	3	)	)	PUNCT
ejpam-109	239	4	)	)	PUNCT
ejpam-109	239	5	in	in	ADP
ejpam-109	239	6	full	full	ADJ
ejpam-109	239	7	by	by	ADP
ejpam-109	239	8	making	make	VERB
ejpam-109	239	9	use	use	NOUN
ejpam-109	239	10	of	of	ADP
ejpam-109	239	11	the	the	DET
ejpam-109	239	12	multinomial	multinomial	ADJ
ejpam-109	239	13	theorem	theorem	NOUN
ejpam-109	239	14	regarding	regard	VERB
ejpam-109	239	15	the	the	DET
ejpam-109	239	16	k	k	PROPN
ejpam-109	239	17	’s	’s	PROPN
ejpam-109	239	18	(	(	PUNCT
ejpam-109	239	19	resp	resp	PROPN
ejpam-109	239	20	.	.	PUNCT
ejpam-109	240	1	h	h	PROPN
ejpam-109	240	2	’s	’s	ADV
ejpam-109	240	3	)	)	PUNCT
ejpam-109	240	4	as	as	ADP
ejpam-109	240	5	ordinary	ordinary	ADJ
ejpam-109	240	6	real	real	ADJ
ejpam-109	240	7	numbers	number	NOUN
ejpam-109	240	8	,	,	PUNCT
ejpam-109	240	9	and	and	CCONJ
ejpam-109	240	10	afterwards	afterwards	ADV
ejpam-109	240	11	,	,	PUNCT
ejpam-109	240	12	k	k	PROPN
ejpam-109	240	13	l(n	l(n	PROPN
ejpam-109	240	14	;	;	PUNCT
ejpam-109	240	15	ar	ar	PROPN
ejpam-109	240	16	)	)	PUNCT
ejpam-109	240	17	(	(	PUNCT
ejpam-109	240	18	resp	resp	NOUN
ejpam-109	240	19	.	.	PUNCT
ejpam-109	241	1	h	h	NOUN
ejpam-109	241	2	l	l	NOUN
ejpam-109	241	3	r(n	r(n	PROPN
ejpam-109	241	4	)	)	PUNCT
ejpam-109	241	5	)	)	PUNCT
ejpam-109	241	6	is	be	AUX
ejpam-109	241	7	to	to	PART
ejpam-109	241	8	be	be	AUX
ejpam-109	241	9	replaced	replace	VERB
ejpam-109	241	10	by	by	ADP
ejpam-109	241	11	kl(n	kl(n	NUM
ejpam-109	241	12	;	;	PUNCT
ejpam-109	241	13	ar	ar	NOUN
ejpam-109	241	14	)	)	PUNCT
ejpam-109	241	15	(	(	PUNCT
ejpam-109	241	16	resp	resp	NOUN
ejpam-109	241	17	.	.	PUNCT
ejpam-109	241	18	hl(n	hl(n	PROPN
ejpam-109	241	19	)	)	PUNCT
ejpam-109	241	20	)	)	PUNCT
ejpam-109	241	21	for	for	ADP
ejpam-109	241	22	each	each	DET
ejpam-109	241	23	r.	r.	PROPN
ejpam-109	241	24	proof	proof	NOUN
ejpam-109	241	25	.	.	PUNCT
ejpam-109	242	1	since	since	SCONJ
ejpam-109	242	2	n	n	PROPN
ejpam-109	242	3	≥	≥	NOUN
ejpam-109	242	4	3	3	NUM
ejpam-109	242	5	,	,	PUNCT
ejpam-109	242	6	we	we	PRON
ejpam-109	242	7	know	know	VERB
ejpam-109	242	8	that	that	SCONJ
ejpam-109	242	9	all	all	PRON
ejpam-109	242	10	of	of	ADP
ejpam-109	242	11	mr	mr	PROPN
ejpam-109	242	12	+	+	PROPN
ejpam-109	242	13	cbrϕ(n	cbrϕ(n	NOUN
ejpam-109	242	14	)	)	PUNCT
ejpam-109	242	15	for	for	ADP
ejpam-109	242	16	r	r	NOUN
ejpam-109	242	17	=	=	SYM
ejpam-109	242	18	1	1	NUM
ejpam-109	242	19	,	,	PUNCT
ejpam-109	242	20	2	2	NUM
ejpam-109	242	21	,	,	PUNCT
ejpam-109	242	22	...	...	PUNCT
ejpam-109	242	23	,	,	PUNCT
ejpam-109	242	24	k	k	PROPN
ejpam-109	242	25	and	and	CCONJ
ejpam-109	242	26	c	c	PROPN
ejpam-109	242	27	=	=	SYM
ejpam-109	242	28	0	0	NUM
ejpam-109	242	29	,	,	PUNCT
ejpam-109	242	30	1	1	NUM
ejpam-109	242	31	,	,	PUNCT
ejpam-109	242	32	...	...	PUNCT
ejpam-109	242	33	,	,	PUNCT
ejpam-109	242	34	s	s	VERB
ejpam-109	242	35	are	be	AUX
ejpam-109	242	36	even	even	ADV
ejpam-109	242	37	.	.	PUNCT
ejpam-109	243	1	also	also	ADV
ejpam-109	243	2	,	,	PUNCT
ejpam-109	243	3	it	it	PRON
ejpam-109	243	4	follows	follow	VERB
ejpam-109	243	5	that	that	SCONJ
ejpam-109	243	6	p	p	NOUN
ejpam-109	243	7	−	−	PROPN
ejpam-109	243	8	1	1	NUM
ejpam-109	244	1	|	|	ADV
ejpam-109	244	2	mr	mr	PROPN
ejpam-109	244	3	if	if	SCONJ
ejpam-109	244	4	and	and	CCONJ
ejpam-109	244	5	only	only	ADV
ejpam-109	244	6	if	if	SCONJ
ejpam-109	244	7	p	p	NOUN
ejpam-109	244	8	−	−	PROPN
ejpam-109	244	9	1	1	NUM
ejpam-109	244	10	|	|	ADV
ejpam-109	244	11	mr	mr	PROPN
ejpam-109	244	12	+	+	PROPN
ejpam-109	244	13	cbrϕ(n	cbrϕ(n	NOUN
ejpam-109	244	14	)	)	PUNCT
ejpam-109	244	15	for	for	ADP
ejpam-109	244	16	each	each	DET
ejpam-109	244	17	prime	prime	ADJ
ejpam-109	244	18	divisor	divisor	NOUN
ejpam-109	244	19	p	p	NOUN
ejpam-109	244	20	of	of	ADP
ejpam-109	244	21	n	n	CCONJ
ejpam-109	244	22	,	,	PUNCT
ejpam-109	244	23	hence	hence	ADV
ejpam-109	244	24	ordp(dmr	ordp(dmr	NOUN
ejpam-109	244	25	)	)	PUNCT
ejpam-109	244	26	=	=	SYM
ejpam-109	245	1	ordp(dmr+cbrϕ(n	ordp(dmr+cbrϕ(n	PROPN
ejpam-109	245	2	)	)	PUNCT
ejpam-109	245	3	)	)	PUNCT
ejpam-109	245	4	.	.	PUNCT
ejpam-109	246	1	we	we	PRON
ejpam-109	246	2	now	now	ADV
ejpam-109	246	3	put	put	VERB
ejpam-109	246	4	ξ=	ξ=	NOUN
ejpam-109	246	5	ns	ns	NUM
ejpam-109	246	6	∏	∏	PROPN
ejpam-109	246	7	p|n	p|n	NOUN
ejpam-109	246	8	p	p	X
ejpam-109	246	9	fp+gp	fp+gp	NOUN
ejpam-109	246	10	,	,	PUNCT
ejpam-109	246	11	where	where	SCONJ
ejpam-109	246	12	fp	fp	PROPN
ejpam-109	246	13	=	=	SYM
ejpam-109	246	14	max	max	PROPN
ejpam-109	246	15	1≤r≤k	1≤r≤k	NUM
ejpam-109	246	16	0≤c≤s	0≤c≤s	NUM
ejpam-109	246	17	¦	¦	PROPN
ejpam-109	246	18	ordp(mr	ordp(mr	PROPN
ejpam-109	246	19	+	+	CCONJ
ejpam-109	246	20	cbrϕ(n	cbrϕ(n	NOUN
ejpam-109	246	21	)	)	PUNCT
ejpam-109	246	22	)	)	PUNCT
ejpam-109	246	23	©	©	PROPN
ejpam-109	246	24	and	and	CCONJ
ejpam-109	246	25	gp	gp	NOUN
ejpam-109	246	26	=	=	SYM
ejpam-109	246	27	max	max	PROPN
ejpam-109	246	28	1≤r≤k	1≤r≤k	NUM
ejpam-109	246	29	¦	¦	PROPN
ejpam-109	246	30	ordp(dmr	ordp(dmr	NOUN
ejpam-109	246	31	)	)	PUNCT
ejpam-109	247	1	©	©	PROPN
ejpam-109	247	2	.	.	PUNCT
ejpam-109	248	1	noticing	notice	VERB
ejpam-109	248	2	that	that	SCONJ
ejpam-109	248	3	m	m	VERB
ejpam-109	248	4	and	and	CCONJ
ejpam-109	248	5	n	n	PRON
ejpam-109	248	6	can	can	AUX
ejpam-109	248	7	be	be	AUX
ejpam-109	248	8	chosen	choose	VERB
ejpam-109	248	9	independently	independently	ADV
ejpam-109	248	10	,	,	PUNCT
ejpam-109	248	11	we	we	PRON
ejpam-109	248	12	may	may	AUX
ejpam-109	248	13	consider	consider	VERB
ejpam-109	248	14	congruence	congruence	NOUN
ejpam-109	248	15	(	(	PUNCT
ejpam-109	248	16	i	i	NOUN
ejpam-109	248	17	)	)	PUNCT
ejpam-109	248	18	in	in	ADP
ejpam-109	248	19	theorem	theorem	NOUN
ejpam-109	248	20	3.1	3.1	NUM
ejpam-109	248	21	which	which	PRON
ejpam-109	248	22	is	be	AUX
ejpam-109	248	23	replaced	replace	VERB
ejpam-109	248	24	a	a	DET
ejpam-109	248	25	,	,	PUNCT
ejpam-109	248	26	m	m	PROPN
ejpam-109	248	27	,	,	PUNCT
ejpam-109	248	28	n	n	PROPN
ejpam-109	248	29	and	and	CCONJ
ejpam-109	248	30	w	w	NOUN
ejpam-109	248	31	by	by	ADP
ejpam-109	248	32	ar	ar	PROPN
ejpam-109	248	33	,	,	PUNCT
ejpam-109	248	34	mr	mr	PROPN
ejpam-109	248	35	+	+	PROPN
ejpam-109	248	36	cbrϕ(n	cbrϕ(n	NOUN
ejpam-109	248	37	)	)	PUNCT
ejpam-109	248	38	,	,	PUNCT
ejpam-109	248	39	ns	ns	NUM
ejpam-109	248	40	and	and	CCONJ
ejpam-109	248	41	ξ	ξ	NOUN
ejpam-109	248	42	,	,	PUNCT
ejpam-109	248	43	respectively	respectively	ADV
ejpam-109	248	44	:	:	PUNCT
ejpam-109	249	1	kmr+cbrϕ(n)(n	kmr+cbrϕ(n)(n	PROPN
ejpam-109	249	2	;	;	PUNCT
ejpam-109	249	3	ar)≡	ar)≡	PUNCT
ejpam-109	249	4	ξ−1	ξ−1	NOUN
ejpam-109	249	5	∑	∑	PUNCT
ejpam-109	249	6	jr=1	jr=1	PROPN
ejpam-109	249	7	(	(	PUNCT
ejpam-109	249	8	jr	jr	PROPN
ejpam-109	249	9	,	,	PUNCT
ejpam-109	249	10	n)=1	n)=1	PROPN
ejpam-109	249	11	(	(	PUNCT
ejpam-109	249	12	ar	ar	PROPN
ejpam-109	249	13	jr	jr	PROPN
ejpam-109	249	14	)	)	PUNCT
ejpam-109	249	15	mr+cbrϕ(n)−1	mr+cbrϕ(n)−1	PROPN
ejpam-109	249	16	�	�	PROPN
ejpam-109	249	17	ar	ar	PROPN
ejpam-109	249	18	jr	jr	PROPN
ejpam-109	249	19	ξ	ξ	PROPN
ejpam-109	249	20	�	�	PROPN
ejpam-109	249	21	(	(	PUNCT
ejpam-109	249	22	mod	mod	PROPN
ejpam-109	249	23	ns	ns	NUM
ejpam-109	249	24	)	)	PUNCT
ejpam-109	249	25	,	,	PUNCT
ejpam-109	249	26	t.	t.	NOUN
ejpam-109	249	27	agoh	agoh	PROPN
ejpam-109	249	28	/	/	SYM
ejpam-109	249	29	eur	eur	PROPN
ejpam-109	249	30	.	.	PUNCT
ejpam-109	250	1	j.	j.	PROPN
ejpam-109	250	2	pure	pure	PROPN
ejpam-109	250	3	appl	appl	PROPN
ejpam-109	250	4	.	.	PROPN
ejpam-109	250	5	math	math	PROPN
ejpam-109	250	6	,	,	PUNCT
ejpam-109	250	7	1	1	NUM
ejpam-109	250	8	(	(	PUNCT
ejpam-109	250	9	2008	2008	NUM
ejpam-109	250	10	)	)	PUNCT
ejpam-109	250	11	,	,	PUNCT
ejpam-109	250	12	(	(	PUNCT
ejpam-109	250	13	3	3	NUM
ejpam-109	250	14	-	-	SYM
ejpam-109	250	15	21	21	NUM
ejpam-109	250	16	)	)	PUNCT
ejpam-109	250	17	13	13	NUM
ejpam-109	250	18	which	which	PRON
ejpam-109	250	19	is	be	AUX
ejpam-109	250	20	valid	valid	ADJ
ejpam-109	250	21	for	for	ADP
ejpam-109	250	22	all	all	DET
ejpam-109	250	23	r	r	NOUN
ejpam-109	250	24	=	=	SYM
ejpam-109	250	25	1,2	1,2	NUM
ejpam-109	250	26	,	,	PUNCT
ejpam-109	250	27	...	...	PUNCT
ejpam-109	250	28	,	,	PUNCT
ejpam-109	250	29	k	k	PROPN
ejpam-109	250	30	and	and	CCONJ
ejpam-109	250	31	c	c	NOUN
ejpam-109	250	32	=	=	SYM
ejpam-109	250	33	0,1	0,1	NUM
ejpam-109	250	34	,	,	PUNCT
ejpam-109	250	35	...	...	PUNCT
ejpam-109	250	36	,	,	PUNCT
ejpam-109	250	37	s.	s.	PROPN
ejpam-109	250	38	making	make	VERB
ejpam-109	250	39	use	use	NOUN
ejpam-109	250	40	of	of	ADP
ejpam-109	250	41	this	this	DET
ejpam-109	250	42	congruence	congruence	NOUN
ejpam-109	250	43	,	,	PUNCT
ejpam-109	250	44	we	we	PRON
ejpam-109	250	45	can	can	AUX
ejpam-109	250	46	deduce	deduce	VERB
ejpam-109	250	47	the	the	DET
ejpam-109	250	48	following	follow	VERB
ejpam-109	250	49	k	k	PROPN
ejpam-109	250	50	∏	∏	PROPN
ejpam-109	250	51	r=1	r=1	NOUN
ejpam-109	250	52	kmr	kmr	NOUN
ejpam-109	250	53	(	(	PUNCT
ejpam-109	250	54	n	n	NUM
ejpam-109	250	55	;	;	PUNCT
ejpam-109	250	56	ar	ar	NOUN
ejpam-109	250	57	)	)	PUNCT
ejpam-109	250	58	k	k	NOUN
ejpam-109	250	59	∑	∑	PUNCT
ejpam-109	250	60	r=1	r=1	NOUN
ejpam-109	250	61	λr	λr	ADP
ejpam-109	250	62	k	k	PROPN
ejpam-109	250	63	brϕ(n)(n	brϕ(n)(n	PROPN
ejpam-109	250	64	;	;	PUNCT
ejpam-109	250	65	ar	ar	NOUN
ejpam-109	250	66	)	)	PUNCT
ejpam-109	250	67	+	+	NOUN
ejpam-109	250	68	λk+1	λk+1	X
ejpam-109	250	69	!	!	PUNCT
ejpam-109	250	70	s	s	PART
ejpam-109	251	1	=	=	PUNCT
ejpam-109	251	2	∑	∑	PROPN
ejpam-109	251	3	0≤c1,	0≤c1,	NUM
ejpam-109	251	4	...	...	PUNCT
ejpam-109	251	5	,ck+1≤s	,ck+1≤s	PUNCT
ejpam-109	251	6	c1+···+ck+1	c1+···+ck+1	NOUN
ejpam-109	251	7	=	=	SYM
ejpam-109	251	8	s	s	PART
ejpam-109	251	9	�	�	PROPN
ejpam-109	251	10	s	s	PART
ejpam-109	251	11	c1	c1	PROPN
ejpam-109	251	12	,	,	PUNCT
ejpam-109	251	13	...	...	PUNCT
ejpam-109	251	14	,	,	PUNCT
ejpam-109	251	15	ck+1	ck+1	X
ejpam-109	251	16	�	�	PROPN
ejpam-109	251	17	k	k	PROPN
ejpam-109	251	18	∏	∏	PROPN
ejpam-109	251	19	r=1	r=1	NOUN
ejpam-109	251	20	λcr	λcr	NOUN
ejpam-109	251	21	r	r	NOUN
ejpam-109	251	22	kmr+cr	kmr+cr	PROPN
ejpam-109	251	23	brϕ(n)(n	brϕ(n)(n	PROPN
ejpam-109	251	24	;	;	PUNCT
ejpam-109	251	25	ar	ar	NOUN
ejpam-109	251	26	)	)	PUNCT
ejpam-109	251	27	!	!	PUNCT
ejpam-109	252	1	λ	λ	NOUN
ejpam-109	252	2	ck+1	ck+1	VERB
ejpam-109	252	3	k+1	k+1	X
ejpam-109	252	4	≡	≡	PROPN
ejpam-109	252	5	∑	∑	PROPN
ejpam-109	252	6	0≤c1,	0≤c1,	PROPN
ejpam-109	252	7	...	...	PUNCT
ejpam-109	252	8	,ck+1≤s	,ck+1≤s	PUNCT
ejpam-109	252	9	c1+···+ck+1	c1+···+ck+1	NOUN
ejpam-109	252	10	=	=	SYM
ejpam-109	252	11	s	s	PART
ejpam-109	252	12	�	�	PROPN
ejpam-109	252	13	s	s	PART
ejpam-109	252	14	c1	c1	PROPN
ejpam-109	252	15	,	,	PUNCT
ejpam-109	252	16	...	...	PUNCT
ejpam-109	252	17	,	,	PUNCT
ejpam-109	252	18	ck+1	ck+1	NUM
ejpam-109	252	19	�	�	PROPN
ejpam-109	252	20	�	�	PROPN
ejpam-109	252	21	k	k	PROPN
ejpam-109	252	22	∏	∏	PROPN
ejpam-109	252	23	r=1	r=1	NOUN
ejpam-109	252	24	�	�	PROPN
ejpam-109	252	25	λcr	λcr	NOUN
ejpam-109	252	26	r	r	NOUN
ejpam-109	252	27	ξ−1	ξ−1	PROPN
ejpam-109	252	28	∑	∑	PUNCT
ejpam-109	252	29	jr=1	jr=1	PROPN
ejpam-109	252	30	(	(	PUNCT
ejpam-109	252	31	jr	jr	PROPN
ejpam-109	252	32	,	,	PUNCT
ejpam-109	252	33	n)=1	n)=1	PROPN
ejpam-109	252	34	(	(	PUNCT
ejpam-109	252	35	ar	ar	PROPN
ejpam-109	252	36	jr	jr	PROPN
ejpam-109	252	37	)	)	PUNCT
ejpam-109	252	38	mr+cr	mr+cr	NOUN
ejpam-109	252	39	brϕ(n)−1	brϕ(n)−1	NOUN
ejpam-109	252	40	�	�	PROPN
ejpam-109	252	41	ar	ar	PROPN
ejpam-109	252	42	jr	jr	PROPN
ejpam-109	252	43	ξ	ξ	PROPN
ejpam-109	252	44	�	�	PROPN
ejpam-109	252	45	�	�	PROPN
ejpam-109	252	46	�	�	PROPN
ejpam-109	252	47	λ	λ	PROPN
ejpam-109	252	48	ck+1	ck+1	AUX
ejpam-109	252	49	k+1	k+1	X
ejpam-109	252	50	≡	≡	PROPN
ejpam-109	252	51	ξ−1	ξ−1	PROPN
ejpam-109	252	52	∑	∑	PROPN
ejpam-109	252	53	j1=1	j1=1	PROPN
ejpam-109	252	54	(	(	PUNCT
ejpam-109	252	55	j1,n)=1	j1,n)=1	PROPN
ejpam-109	252	56	·	·	PUNCT
ejpam-109	252	57	·	·	PUNCT
ejpam-109	252	58	·	·	PUNCT
ejpam-109	253	1	ξ−1	ξ−1	NOUN
ejpam-109	253	2	∑	∑	PUNCT
ejpam-109	253	3	jk=1	jk=1	PROPN
ejpam-109	253	4	(	(	PUNCT
ejpam-109	253	5	jk	jk	PROPN
ejpam-109	253	6	,	,	PUNCT
ejpam-109	253	7	n)=1	n)=1	PROPN
ejpam-109	253	8	¨	¨	NOUN
ejpam-109	253	9	∑	∑	PROPN
ejpam-109	253	10	0≤c1,	0≤c1,	NUM
ejpam-109	253	11	...	...	PUNCT
ejpam-109	253	12	,ck+1≤s	,ck+1≤s	PUNCT
ejpam-109	253	13	c1+···+ck+1	c1+···+ck+1	NOUN
ejpam-109	253	14	=	=	SYM
ejpam-109	253	15	s	s	PART
ejpam-109	253	16	�	�	PROPN
ejpam-109	253	17	s	s	PART
ejpam-109	253	18	c1	c1	PROPN
ejpam-109	253	19	,	,	PUNCT
ejpam-109	253	20	...	...	PUNCT
ejpam-109	253	21	,	,	PUNCT
ejpam-109	253	22	ck+1	ck+1	NUM
ejpam-109	253	23	�	�	PROPN
ejpam-109	253	24	×	×	PROPN
ejpam-109	253	25	k	k	PROPN
ejpam-109	253	26	∏	∏	PROPN
ejpam-109	253	27	r=1	r=1	NOUN
ejpam-109	253	28	λcr	λcr	NOUN
ejpam-109	253	29	r	r	NOUN
ejpam-109	253	30	(	(	PUNCT
ejpam-109	253	31	ar	ar	PROPN
ejpam-109	253	32	jr	jr	PROPN
ejpam-109	253	33	)	)	PUNCT
ejpam-109	253	34	mr+cr	mr+cr	NOUN
ejpam-109	253	35	brϕ(n)−1	brϕ(n)−1	NOUN
ejpam-109	253	36	�	�	PROPN
ejpam-109	253	37	ar	ar	PROPN
ejpam-109	253	38	jr	jr	PROPN
ejpam-109	253	39	ξ	ξ	PROPN
ejpam-109	253	40	�	�	PROPN
ejpam-109	253	41	!	!	PUNCT
ejpam-109	254	1	λ	λ	NOUN
ejpam-109	254	2	ck+1	ck+1	VERB
ejpam-109	254	3	k+1	k+1	X
ejpam-109	254	4	«	«	PUNCT
ejpam-109	254	5	≡	≡	PROPN
ejpam-109	254	6	ξ−1	ξ−1	PROPN
ejpam-109	254	7	∑	∑	PUNCT
ejpam-109	254	8	j1=1	j1=1	PROPN
ejpam-109	254	9	(	(	PUNCT
ejpam-109	254	10	j1,n)=1	j1,n)=1	PROPN
ejpam-109	254	11	·	·	PUNCT
ejpam-109	254	12	·	·	PUNCT
ejpam-109	254	13	·	·	PUNCT
ejpam-109	255	1	ξ−1	ξ−1	NOUN
ejpam-109	255	2	∑	∑	PUNCT
ejpam-109	255	3	jk=1	jk=1	PROPN
ejpam-109	255	4	(	(	PUNCT
ejpam-109	255	5	jk	jk	PROPN
ejpam-109	255	6	,	,	PUNCT
ejpam-109	255	7	n)=1	n)=1	PROPN
ejpam-109	255	8			NOUN
ejpam-109	255	9			PRON
ejpam-109	255	10			NOUN
ejpam-109	255	11	k	k	PROPN
ejpam-109	255	12	∑	∑	PUNCT
ejpam-109	255	13	r=1	r=1	PROPN
ejpam-109	255	14	λr(ar	λr(ar	PROPN
ejpam-109	255	15	jr	jr	NOUN
ejpam-109	255	16	)	)	PUNCT
ejpam-109	255	17	brϕ(n)+λk+1	brϕ(n)+λk+1	X
ejpam-109	255	18	!	!	PUNCT
ejpam-109	255	19	s	s	PART
ejpam-109	255	20	k	k	PROPN
ejpam-109	255	21	∏	∏	PROPN
ejpam-109	255	22	r=1	r=1	PROPN
ejpam-109	255	23	(	(	PUNCT
ejpam-109	255	24	ar	ar	PROPN
ejpam-109	255	25	jr	jr	PROPN
ejpam-109	255	26	)	)	PUNCT
ejpam-109	255	27	mr−1	mr−1	PROPN
ejpam-109	255	28	�	�	PROPN
ejpam-109	255	29	ar	ar	PROPN
ejpam-109	255	30	jr	jr	PROPN
ejpam-109	255	31	ξ	ξ	PROPN
ejpam-109	255	32	�	�	PROPN
ejpam-109	255	33			PROPN
ejpam-109	255	34			PROPN
ejpam-109	255	35			NOUN
ejpam-109	255	36	(	(	PUNCT
ejpam-109	255	37	mod	mod	PROPN
ejpam-109	255	38	ns	ns	NUM
ejpam-109	255	39	)	)	PUNCT
ejpam-109	255	40	.	.	PUNCT
ejpam-109	256	1	since	since	SCONJ
ejpam-109	256	2	(	(	PUNCT
ejpam-109	256	3	ar	ar	PROPN
ejpam-109	256	4	jr	jr	PROPN
ejpam-109	256	5	,	,	PUNCT
ejpam-109	256	6	n	n	CCONJ
ejpam-109	256	7	)	)	PUNCT
ejpam-109	256	8	=	=	SYM
ejpam-109	256	9	1	1	NUM
ejpam-109	256	10	for	for	ADP
ejpam-109	256	11	each	each	DET
ejpam-109	256	12	r	r	NOUN
ejpam-109	256	13	,	,	PUNCT
ejpam-109	256	14	under	under	ADP
ejpam-109	256	15	the	the	DET
ejpam-109	256	16	assumption	assumption	NOUN
ejpam-109	256	17	for	for	ADP
ejpam-109	256	18	the	the	DET
ejpam-109	256	19	λ	λ	NOUN
ejpam-109	256	20	’s	’s	X
ejpam-109	256	21	we	we	PRON
ejpam-109	256	22	have	have	VERB
ejpam-109	256	23	k	k	NOUN
ejpam-109	256	24	∑	∑	PART
ejpam-109	256	25	r=1	r=1	PROPN
ejpam-109	256	26	λr(ar	λr(ar	PROPN
ejpam-109	256	27	jr	jr	PROPN
ejpam-109	256	28	)	)	PUNCT
ejpam-109	256	29	brϕ(n)+λk+1	brϕ(n)+λk+1	VERB
ejpam-109	257	1	≡	≡	PROPN
ejpam-109	258	1	k+1	k+1	X
ejpam-109	258	2	∑	∑	PROPN
ejpam-109	258	3	i=1	i=1	PROPN
ejpam-109	258	4	λi	λi	INTJ
ejpam-109	258	5	≡	≡	PROPN
ejpam-109	258	6	0	0	PUNCT
ejpam-109	258	7	(	(	PUNCT
ejpam-109	258	8	mod	mod	NOUN
ejpam-109	258	9	n	n	CCONJ
ejpam-109	258	10	)	)	PUNCT
ejpam-109	258	11	,	,	PUNCT
ejpam-109	258	12	which	which	PRON
ejpam-109	258	13	offers	offer	VERB
ejpam-109	258	14	congruence	congruence	NOUN
ejpam-109	258	15	(	(	PUNCT
ejpam-109	258	16	i	i	NOUN
ejpam-109	258	17	)	)	PUNCT
ejpam-109	258	18	as	as	SCONJ
ejpam-109	258	19	desired	desire	VERB
ejpam-109	258	20	.	.	PUNCT
ejpam-109	259	1	next	next	ADV
ejpam-109	259	2	,	,	PUNCT
ejpam-109	259	3	assume	assume	VERB
ejpam-109	259	4	that	that	SCONJ
ejpam-109	259	5	n	n	PRON
ejpam-109	259	6	=	=	VERB
ejpam-109	259	7	pα	pα	INTJ
ejpam-109	259	8	(	(	PUNCT
ejpam-109	259	9	α	α	PRON
ejpam-109	259	10	≥	≥	NOUN
ejpam-109	259	11	1	1	NUM
ejpam-109	259	12	,	,	PUNCT
ejpam-109	259	13	p	p	X
ejpam-109	259	14	an	an	DET
ejpam-109	259	15	odd	odd	ADJ
ejpam-109	259	16	prime	prime	NOUN
ejpam-109	259	17	)	)	PUNCT
ejpam-109	259	18	and	and	CCONJ
ejpam-109	259	19	p−	p−	NOUN
ejpam-109	259	20	1	1	NUM
ejpam-109	259	21	mr	mr	PROPN
ejpam-109	259	22	(	(	PUNCT
ejpam-109	259	23	r	r	NOUN
ejpam-109	259	24	=	=	SYM
ejpam-109	259	25	1	1	NUM
ejpam-109	259	26	,	,	PUNCT
ejpam-109	259	27	2	2	NUM
ejpam-109	259	28	,	,	PUNCT
ejpam-109	259	29	...	...	PUNCT
ejpam-109	259	30	,	,	PUNCT
ejpam-109	259	31	k	k	NOUN
ejpam-109	259	32	)	)	PUNCT
ejpam-109	259	33	.	.	PUNCT
ejpam-109	260	1	in	in	ADP
ejpam-109	260	2	this	this	DET
ejpam-109	260	3	case	case	NOUN
ejpam-109	260	4	,	,	PUNCT
ejpam-109	260	5	n	n	PRON
ejpam-109	260	6	has	have	VERB
ejpam-109	260	7	a	a	DET
ejpam-109	260	8	primitive	primitive	ADJ
ejpam-109	260	9	root	root	NOUN
ejpam-109	260	10	γ	γ	NOUN
ejpam-109	260	11	,	,	PUNCT
ejpam-109	260	12	so	so	ADV
ejpam-109	260	13	take	take	VERB
ejpam-109	260	14	a	a	DET
ejpam-109	260	15	positive	positive	ADJ
ejpam-109	260	16	integer	integer	NOUN
ejpam-109	260	17	a	a	PRON
ejpam-109	260	18	with	with	ADP
ejpam-109	260	19	a	a	DET
ejpam-109	260	20	≡	≡	PROPN
ejpam-109	260	21	γns−1	γns−1	PROPN
ejpam-109	260	22	(	(	PUNCT
ejpam-109	260	23	mod	mod	PROPN
ejpam-109	260	24	ns	ns	NUM
ejpam-109	260	25	)	)	PUNCT
ejpam-109	260	26	.	.	PUNCT
ejpam-109	261	1	then	then	ADV
ejpam-109	261	2	,	,	PUNCT
ejpam-109	261	3	by	by	ADP
ejpam-109	261	4	euler	euler	PROPN
ejpam-109	261	5	’s	’s	PART
ejpam-109	261	6	theorem	theorem	NOUN
ejpam-109	261	7	we	we	PRON
ejpam-109	261	8	see	see	VERB
ejpam-109	261	9	aϕ(n	aϕ(n	NOUN
ejpam-109	261	10	)	)	PUNCT
ejpam-109	261	11	≡	≡	PROPN
ejpam-109	261	12	γns−1ϕ(n	γns−1ϕ(n	NOUN
ejpam-109	261	13	)	)	PUNCT
ejpam-109	261	14	≡	≡	PROPN
ejpam-109	261	15	γϕ(n	γϕ(n	X
ejpam-109	261	16	s	s	PART
ejpam-109	261	17	)	)	PUNCT
ejpam-109	261	18	≡	≡	PROPN
ejpam-109	261	19	1	1	NUM
ejpam-109	261	20	(	(	PUNCT
ejpam-109	261	21	mod	mod	PROPN
ejpam-109	261	22	ns	ns	NUM
ejpam-109	261	23	)	)	PUNCT
ejpam-109	261	24	.	.	PUNCT
ejpam-109	262	1	now	now	ADV
ejpam-109	262	2	choose	choose	VERB
ejpam-109	262	3	especially	especially	ADV
ejpam-109	262	4	a1	a1	NOUN
ejpam-109	262	5	=	=	SYM
ejpam-109	262	6	a2	a2	PROPN
ejpam-109	262	7	=	=	SYM
ejpam-109	262	8	·	·	PUNCT
ejpam-109	262	9	·	·	PUNCT
ejpam-109	262	10	·	·	PUNCT
ejpam-109	263	1	=	=	SYM
ejpam-109	263	2	ak	ak	PROPN
ejpam-109	263	3	=	=	PROPN
ejpam-109	263	4	a	a	PRON
ejpam-109	263	5	in	in	ADP
ejpam-109	263	6	(	(	PUNCT
ejpam-109	263	7	i	i	NOUN
ejpam-109	263	8	)	)	PUNCT
ejpam-109	263	9	.	.	PUNCT
ejpam-109	264	1	then	then	ADV
ejpam-109	264	2	acϕ(n	acϕ(n	PROPN
ejpam-109	264	3	)	)	PUNCT
ejpam-109	264	4	≡	≡	PROPN
ejpam-109	264	5	1	1	NUM
ejpam-109	264	6	(	(	PUNCT
ejpam-109	264	7	mod	mod	NOUN
ejpam-109	264	8	ns	ns	NUM
ejpam-109	264	9	)	)	PUNCT
ejpam-109	264	10	for	for	ADP
ejpam-109	264	11	any	any	DET
ejpam-109	264	12	c	c	PROPN
ejpam-109	264	13	≥	≥	NOUN
ejpam-109	264	14	0	0	NUM
ejpam-109	264	15	and	and	CCONJ
ejpam-109	264	16	(	(	PUNCT
ejpam-109	264	17	amr	amr	NOUN
ejpam-109	264	18	−	−	PROPN
ejpam-109	264	19	1	1	NUM
ejpam-109	264	20	,	,	PUNCT
ejpam-109	264	21	n	n	CCONJ
ejpam-109	264	22	)	)	PUNCT
ejpam-109	264	23	=	=	SYM
ejpam-109	264	24	1	1	NUM
ejpam-109	264	25	for	for	ADP
ejpam-109	264	26	each	each	DET
ejpam-109	264	27	r.	r.	NOUN
ejpam-109	264	28	dividing	dividing	NOUN
ejpam-109	264	29	(	(	PUNCT
ejpam-109	264	30	i	i	NOUN
ejpam-109	264	31	)	)	PUNCT
ejpam-109	264	32	by	by	ADP
ejpam-109	264	33	∏k	∏k	NOUN
ejpam-109	264	34	r=1	r=1	NOUN
ejpam-109	264	35	(	(	PUNCT
ejpam-109	264	36	a	a	DET
ejpam-109	264	37	mr	mr	PROPN
ejpam-109	264	38	−	−	PROPN
ejpam-109	264	39	1	1	NUM
ejpam-109	264	40	)	)	PUNCT
ejpam-109	264	41	,	,	PUNCT
ejpam-109	264	42	we	we	PRON
ejpam-109	264	43	can	can	AUX
ejpam-109	264	44	deduce	deduce	VERB
ejpam-109	264	45	(	(	PUNCT
ejpam-109	264	46	ii	ii	NOUN
ejpam-109	264	47	)	)	PUNCT
ejpam-109	264	48	immediately	immediately	ADV
ejpam-109	264	49	.	.	PUNCT
ejpam-109	265	1	with	with	ADP
ejpam-109	265	2	above	above	ADJ
ejpam-109	265	3	notations	notation	NOUN
ejpam-109	265	4	,	,	PUNCT
ejpam-109	265	5	we	we	PRON
ejpam-109	265	6	have	have	VERB
ejpam-109	265	7	corollary	corollary	ADJ
ejpam-109	265	8	4.2	4.2	NUM
ejpam-109	265	9	.	.	PUNCT
ejpam-109	266	1	let	let	VERB
ejpam-109	266	2	p	p	PRON
ejpam-109	266	3	be	be	AUX
ejpam-109	266	4	an	an	DET
ejpam-109	266	5	odd	odd	ADJ
ejpam-109	266	6	prime	prime	NOUN
ejpam-109	266	7	,	,	PUNCT
ejpam-109	266	8	mr	mr	PROPN
ejpam-109	266	9	≥	≥	PROPN
ejpam-109	266	10	s+1	s+1	PROPN
ejpam-109	266	11	for	for	ADP
ejpam-109	266	12	r	r	NOUN
ejpam-109	266	13	=	=	SYM
ejpam-109	266	14	1	1	NUM
ejpam-109	266	15	,	,	PUNCT
ejpam-109	266	16	2	2	NUM
ejpam-109	266	17	,	,	PUNCT
ejpam-109	266	18	...	...	PUNCT
ejpam-109	266	19	,	,	PUNCT
ejpam-109	266	20	k	k	PROPN
ejpam-109	266	21	and	and	CCONJ
ejpam-109	266	22	assume	assume	VERB
ejpam-109	266	23	that	that	SCONJ
ejpam-109	266	24	λi	λi	ADP
ejpam-109	266	25	∈	∈	PROPN
ejpam-109	266	26	zp	zp	X
ejpam-109	266	27	(	(	PUNCT
ejpam-109	266	28	i	i	NOUN
ejpam-109	266	29	=	=	SYM
ejpam-109	266	30	1,2	1,2	NUM
ejpam-109	266	31	,	,	PUNCT
ejpam-109	266	32	...	...	PUNCT
ejpam-109	266	33	,	,	PUNCT
ejpam-109	266	34	k+	k+	NOUN
ejpam-109	266	35	1	1	X
ejpam-109	266	36	)	)	PUNCT
ejpam-109	266	37	satisfy	satisfy	NOUN
ejpam-109	266	38	∑k+1	∑k+1	PUNCT
ejpam-109	266	39	i=1	i=1	PROPN
ejpam-109	266	40	λi	λi	INTJ
ejpam-109	266	41	≡	≡	PROPN
ejpam-109	266	42	0(mod	0(mod	NOUN
ejpam-109	267	1	p	p	X
ejpam-109	267	2	)	)	PUNCT
ejpam-109	267	3	.	.	PUNCT
ejpam-109	268	1	then	then	ADV
ejpam-109	268	2	k	k	PROPN
ejpam-109	268	3	∏	∏	PROPN
ejpam-109	268	4	r=1	r=1	NOUN
ejpam-109	268	5	βmr	βmr	NOUN
ejpam-109	268	6	r	r	NOUN
ejpam-109	268	7	(	(	PUNCT
ejpam-109	268	8	ar	ar	NOUN
ejpam-109	268	9	)	)	PUNCT
ejpam-109	268	10	k	k	NOUN
ejpam-109	268	11	∑	∑	PUNCT
ejpam-109	268	12	r=1	r=1	NOUN
ejpam-109	268	13	λrβ	λrβ	ADP
ejpam-109	268	14	br	br	PROPN
ejpam-109	268	15	(	(	PUNCT
ejpam-109	268	16	p−1	p−1	PROPN
ejpam-109	268	17	)	)	PUNCT
ejpam-109	268	18	r	r	NOUN
ejpam-109	268	19	(	(	PUNCT
ejpam-109	268	20	ar	ar	NOUN
ejpam-109	268	21	)	)	PUNCT
ejpam-109	268	22	+	+	NOUN
ejpam-109	268	23	λk+1	λk+1	NUM
ejpam-109	268	24	!	!	PUNCT
ejpam-109	268	25	s	s	PART
ejpam-109	269	1	≡	≡	PROPN
ejpam-109	269	2	0	0	PUNCT
ejpam-109	269	3	(	(	PUNCT
ejpam-109	269	4	mod	mod	PROPN
ejpam-109	269	5	ps	ps	PROPN
ejpam-109	269	6	)	)	PUNCT
ejpam-109	269	7	.	.	PUNCT
ejpam-109	270	1	(	(	PUNCT
ejpam-109	270	2	i	i	NOUN
ejpam-109	270	3	)	)	PUNCT
ejpam-109	270	4	t.	t.	NOUN
ejpam-109	270	5	agoh	agoh	PROPN
ejpam-109	270	6	/	/	SYM
ejpam-109	270	7	eur	eur	PROPN
ejpam-109	270	8	.	.	PUNCT
ejpam-109	271	1	j.	j.	PROPN
ejpam-109	271	2	pure	pure	PROPN
ejpam-109	271	3	appl	appl	PROPN
ejpam-109	271	4	.	.	PROPN
ejpam-109	271	5	math	math	PROPN
ejpam-109	271	6	,	,	PUNCT
ejpam-109	271	7	1	1	NUM
ejpam-109	271	8	(	(	PUNCT
ejpam-109	271	9	2008	2008	NUM
ejpam-109	271	10	)	)	PUNCT
ejpam-109	271	11	,	,	PUNCT
ejpam-109	271	12	(	(	PUNCT
ejpam-109	271	13	3	3	NUM
ejpam-109	271	14	-	-	SYM
ejpam-109	271	15	21	21	NUM
ejpam-109	271	16	)	)	PUNCT
ejpam-109	271	17	14	14	NUM
ejpam-109	271	18	in	in	ADP
ejpam-109	271	19	particular	particular	ADJ
ejpam-109	271	20	,	,	PUNCT
ejpam-109	271	21	if	if	SCONJ
ejpam-109	271	22	p−	p−	NOUN
ejpam-109	271	23	1	1	NUM
ejpam-109	271	24	mr	mr	PROPN
ejpam-109	271	25	(	(	PUNCT
ejpam-109	271	26	r	r	NOUN
ejpam-109	271	27	=	=	SYM
ejpam-109	271	28	1	1	NUM
ejpam-109	271	29	,	,	PUNCT
ejpam-109	271	30	2	2	NUM
ejpam-109	271	31	,	,	PUNCT
ejpam-109	271	32	...	...	PUNCT
ejpam-109	271	33	,	,	PUNCT
ejpam-109	271	34	k	k	PROPN
ejpam-109	271	35	)	)	PUNCT
ejpam-109	271	36	,	,	PUNCT
ejpam-109	271	37	then	then	ADV
ejpam-109	271	38	k	k	PROPN
ejpam-109	271	39	∏	∏	PROPN
ejpam-109	271	40	r=1	r=1	NOUN
ejpam-109	271	41	βmr	βmr	PROPN
ejpam-109	271	42	r	r	NOUN
ejpam-109	271	43	k	k	NOUN
ejpam-109	271	44	∑	∑	PUNCT
ejpam-109	271	45	r=1	r=1	NOUN
ejpam-109	271	46	λrβ	λrβ	ADP
ejpam-109	271	47	br	br	PROPN
ejpam-109	271	48	(	(	PUNCT
ejpam-109	271	49	p−1	p−1	PROPN
ejpam-109	271	50	)	)	PUNCT
ejpam-109	271	51	r	r	NOUN
ejpam-109	271	52	+	+	NOUN
ejpam-109	271	53	λk+1	λk+1	X
ejpam-109	271	54	!	!	PUNCT
ejpam-109	271	55	s	s	PART
ejpam-109	272	1	≡	≡	PROPN
ejpam-109	272	2	0	0	PUNCT
ejpam-109	272	3	(	(	PUNCT
ejpam-109	272	4	mod	mod	PROPN
ejpam-109	272	5	ps	ps	PROPN
ejpam-109	272	6	)	)	PUNCT
ejpam-109	272	7	.	.	PUNCT
ejpam-109	273	1	(	(	PUNCT
ejpam-109	273	2	ii	ii	X
ejpam-109	273	3	)	)	PUNCT
ejpam-109	273	4	it	it	PRON
ejpam-109	273	5	may	may	AUX
ejpam-109	273	6	be	be	AUX
ejpam-109	273	7	unnecessary	unnecessary	ADJ
ejpam-109	273	8	to	to	PART
ejpam-109	273	9	comment	comment	VERB
ejpam-109	273	10	on	on	ADP
ejpam-109	273	11	the	the	DET
ejpam-109	273	12	symbolic	symbolic	ADJ
ejpam-109	273	13	notation	notation	NOUN
ejpam-109	273	14	used	use	VERB
ejpam-109	273	15	in	in	ADP
ejpam-109	273	16	the	the	DET
ejpam-109	273	17	above	above	ADJ
ejpam-109	273	18	corollary	corollary	NOUN
ejpam-109	273	19	,	,	PUNCT
ejpam-109	273	20	but	but	CCONJ
ejpam-109	273	21	we	we	PRON
ejpam-109	273	22	want	want	VERB
ejpam-109	273	23	to	to	PART
ejpam-109	273	24	explain	explain	VERB
ejpam-109	273	25	it	it	PRON
ejpam-109	273	26	once	once	ADV
ejpam-109	273	27	more	more	ADJ
ejpam-109	273	28	to	to	PART
ejpam-109	273	29	make	make	VERB
ejpam-109	273	30	doubly	doubly	ADV
ejpam-109	273	31	sure	sure	ADJ
ejpam-109	273	32	.	.	PUNCT
ejpam-109	274	1	above	above	ADP
ejpam-109	274	2	congruence	congruence	PROPN
ejpam-109	274	3	(	(	PUNCT
ejpam-109	274	4	i	i	NOUN
ejpam-109	274	5	)	)	PUNCT
ejpam-109	274	6	means	mean	VERB
ejpam-109	274	7	that	that	SCONJ
ejpam-109	274	8	we	we	PRON
ejpam-109	274	9	calculate	calculate	VERB
ejpam-109	274	10	the	the	DET
ejpam-109	274	11	left	left	ADJ
ejpam-109	274	12	-	-	PUNCT
ejpam-109	274	13	hand	hand	NOUN
ejpam-109	274	14	side	side	NOUN
ejpam-109	274	15	of	of	ADP
ejpam-109	274	16	(	(	PUNCT
ejpam-109	274	17	i	i	NOUN
ejpam-109	274	18	)	)	PUNCT
ejpam-109	274	19	in	in	ADP
ejpam-109	274	20	full	full	ADJ
ejpam-109	274	21	regarding	regard	VERB
ejpam-109	274	22	βr(ar	βr(ar	NOUN
ejpam-109	274	23	)	)	PUNCT
ejpam-109	275	1	(	(	PUNCT
ejpam-109	275	2	r	r	NOUN
ejpam-109	275	3	=	=	SYM
ejpam-109	275	4	1,2	1,2	NUM
ejpam-109	275	5	,	,	PUNCT
ejpam-109	275	6	...	...	PUNCT
ejpam-109	275	7	,	,	PUNCT
ejpam-109	275	8	k	k	X
ejpam-109	275	9	)	)	PUNCT
ejpam-109	275	10	as	as	ADP
ejpam-109	275	11	ordinary	ordinary	ADJ
ejpam-109	275	12	real	real	ADJ
ejpam-109	275	13	numbers	number	NOUN
ejpam-109	275	14	,	,	PUNCT
ejpam-109	275	15	and	and	CCONJ
ejpam-109	275	16	replace	replace	VERB
ejpam-109	275	17	β	β	X
ejpam-109	275	18	g(r	g(r	NOUN
ejpam-109	275	19	)	)	PUNCT
ejpam-109	275	20	r	r	NOUN
ejpam-109	275	21	(	(	PUNCT
ejpam-109	275	22	ar	ar	NOUN
ejpam-109	275	23	)	)	PUNCT
ejpam-109	275	24	by	by	ADP
ejpam-109	275	25	βg(r)(ar	βg(r)(ar	NOUN
ejpam-109	275	26	)	)	PUNCT
ejpam-109	275	27	for	for	ADP
ejpam-109	275	28	each	each	DET
ejpam-109	275	29	r	r	NOUN
ejpam-109	275	30	and	and	CCONJ
ejpam-109	275	31	various	various	ADJ
ejpam-109	275	32	values	value	NOUN
ejpam-109	275	33	of	of	ADP
ejpam-109	275	34	g(r	g(r	NOUN
ejpam-109	275	35	)	)	PUNCT
ejpam-109	275	36	.	.	PUNCT
ejpam-109	276	1	hence	hence	ADV
ejpam-109	276	2	,	,	PUNCT
ejpam-109	276	3	(	(	PUNCT
ejpam-109	276	4	i	i	NOUN
ejpam-109	276	5	)	)	PUNCT
ejpam-109	276	6	is	be	AUX
ejpam-109	276	7	precisely	precisely	ADV
ejpam-109	276	8	the	the	DET
ejpam-109	276	9	same	same	ADJ
ejpam-109	276	10	as	as	ADP
ejpam-109	276	11	∑	∑	PROPN
ejpam-109	276	12	0≤c1,	0≤c1,	NUM
ejpam-109	276	13	...	...	PUNCT
ejpam-109	276	14	,ck≤s	,ck≤s	PUNCT
ejpam-109	276	15	c1+···+ck+1	c1+···+ck+1	NOUN
ejpam-109	276	16	=	=	SYM
ejpam-109	276	17	s	s	PART
ejpam-109	276	18	�	�	PROPN
ejpam-109	276	19	s	s	PART
ejpam-109	276	20	c1	c1	PROPN
ejpam-109	276	21	,	,	PUNCT
ejpam-109	276	22	...	...	PUNCT
ejpam-109	276	23	,	,	PUNCT
ejpam-109	276	24	ck+1	ck+1	X
ejpam-109	276	25	�	�	PROPN
ejpam-109	276	26	k	k	PROPN
ejpam-109	276	27	∏	∏	PROPN
ejpam-109	276	28	r=1	r=1	NOUN
ejpam-109	276	29	λcr	λcr	NOUN
ejpam-109	276	30	r	r	NOUN
ejpam-109	276	31	βmr+cr	βmr+cr	VERB
ejpam-109	276	32	br	br	NOUN
ejpam-109	276	33	(	(	PUNCT
ejpam-109	276	34	p−1)(ar	p−1)(ar	NOUN
ejpam-109	276	35	)	)	PUNCT
ejpam-109	276	36	!	!	PUNCT
ejpam-109	277	1	λ	λ	NOUN
ejpam-109	277	2	ck+1	ck+1	VERB
ejpam-109	277	3	k+1	k+1	X
ejpam-109	277	4	≡	≡	PROPN
ejpam-109	277	5	0	0	PUNCT
ejpam-109	278	1	(	(	PUNCT
ejpam-109	278	2	mod	mod	PROPN
ejpam-109	278	3	ps	ps	PROPN
ejpam-109	278	4	)	)	PUNCT
ejpam-109	278	5	.	.	PUNCT
ejpam-109	279	1	we	we	PRON
ejpam-109	279	2	do	do	AUX
ejpam-109	279	3	not	not	PART
ejpam-109	279	4	say	say	VERB
ejpam-109	279	5	fully	fully	ADV
ejpam-109	279	6	,	,	PUNCT
ejpam-109	279	7	but	but	CCONJ
ejpam-109	279	8	the	the	DET
ejpam-109	279	9	symbolic	symbolic	ADJ
ejpam-109	279	10	notation	notation	NOUN
ejpam-109	279	11	used	use	VERB
ejpam-109	279	12	in	in	ADP
ejpam-109	279	13	(	(	PUNCT
ejpam-109	279	14	ii	ii	NOUN
ejpam-109	279	15	)	)	PUNCT
ejpam-109	279	16	should	should	AUX
ejpam-109	279	17	be	be	AUX
ejpam-109	279	18	also	also	ADV
ejpam-109	279	19	understood	understand	VERB
ejpam-109	279	20	similarly	similarly	ADV
ejpam-109	279	21	to	to	ADP
ejpam-109	279	22	that	that	PRON
ejpam-109	279	23	used	use	VERB
ejpam-109	279	24	in	in	ADP
ejpam-109	279	25	(	(	PUNCT
ejpam-109	279	26	i	i	NOUN
ejpam-109	279	27	)	)	PUNCT
ejpam-109	279	28	.	.	PUNCT
ejpam-109	280	1	proof	proof	NOUN
ejpam-109	280	2	.	.	PUNCT
ejpam-109	281	1	take	take	VERB
ejpam-109	281	2	n=	n=	ADJ
ejpam-109	281	3	p	p	NOUN
ejpam-109	281	4	in	in	ADP
ejpam-109	281	5	theorem	theorem	NOUN
ejpam-109	281	6	4.1	4.1	NUM
ejpam-109	281	7	.	.	PUNCT
ejpam-109	282	1	since	since	SCONJ
ejpam-109	282	2	mr−1≥	mr−1≥	PROPN
ejpam-109	282	3	s	s	VERB
ejpam-109	282	4	for	for	ADP
ejpam-109	282	5	each	each	DET
ejpam-109	282	6	r	r	NOUN
ejpam-109	282	7	,	,	PUNCT
ejpam-109	282	8	we	we	PRON
ejpam-109	282	9	get	get	AUX
ejpam-109	282	10	εmr	εmr	VERB
ejpam-109	282	11	(	(	PUNCT
ejpam-109	282	12	ps	ps	NOUN
ejpam-109	282	13	)	)	PUNCT
ejpam-109	282	14	=	=	SYM
ejpam-109	283	1	1−pmr−1	1−pmr−1	NUM
ejpam-109	283	2	≡	≡	PROPN
ejpam-109	283	3	1	1	NUM
ejpam-109	283	4	(	(	PUNCT
ejpam-109	283	5	mod	mod	PROPN
ejpam-109	283	6	ps	ps	PROPN
ejpam-109	283	7	)	)	PUNCT
ejpam-109	283	8	,	,	PUNCT
ejpam-109	283	9	so	so	SCONJ
ejpam-109	283	10	that	that	SCONJ
ejpam-109	283	11	hmr	hmr	NOUN
ejpam-109	283	12	(	(	PUNCT
ejpam-109	283	13	ps)(ar)≡	ps)(ar)≡	NOUN
ejpam-109	283	14	βmr	βmr	PROPN
ejpam-109	283	15	(	(	PUNCT
ejpam-109	283	16	ar	ar	NOUN
ejpam-109	283	17	)	)	PUNCT
ejpam-109	283	18	(	(	PUNCT
ejpam-109	283	19	mod	mod	PROPN
ejpam-109	283	20	ps	ps	PROPN
ejpam-109	283	21	)	)	PUNCT
ejpam-109	283	22	.	.	PUNCT
ejpam-109	284	1	additionally	additionally	ADV
ejpam-109	284	2	,	,	PUNCT
ejpam-109	284	3	if	if	SCONJ
ejpam-109	284	4	p−1	p−1	PROPN
ejpam-109	284	5	mr	mr	PROPN
ejpam-109	284	6	(	(	PUNCT
ejpam-109	284	7	r	r	NOUN
ejpam-109	284	8	=	=	SYM
ejpam-109	284	9	1,2	1,2	NUM
ejpam-109	284	10	,	,	PUNCT
ejpam-109	284	11	...	...	PUNCT
ejpam-109	284	12	,	,	PUNCT
ejpam-109	284	13	k	k	PROPN
ejpam-109	284	14	)	)	PUNCT
ejpam-109	284	15	,	,	PUNCT
ejpam-109	284	16	then	then	ADV
ejpam-109	284	17	we	we	PRON
ejpam-109	284	18	can	can	AUX
ejpam-109	284	19	choose	choose	VERB
ejpam-109	284	20	a	a	DET
ejpam-109	284	21	positive	positive	ADJ
ejpam-109	284	22	integer	integer	NOUN
ejpam-109	284	23	a	a	DET
ejpam-109	284	24	with	with	ADP
ejpam-109	284	25	(	(	PUNCT
ejpam-109	284	26	amr	amr	NOUN
ejpam-109	284	27	−	−	PROPN
ejpam-109	284	28	1	1	NUM
ejpam-109	284	29	,	,	PUNCT
ejpam-109	284	30	p	p	NOUN
ejpam-109	284	31	)	)	PUNCT
ejpam-109	284	32	=	=	SYM
ejpam-109	284	33	1	1	NUM
ejpam-109	284	34	for	for	ADP
ejpam-109	284	35	each	each	DET
ejpam-109	284	36	r.	r.	PROPN
ejpam-109	284	37	therefore	therefore	ADV
ejpam-109	284	38	,	,	PUNCT
ejpam-109	284	39	using	use	VERB
ejpam-109	284	40	the	the	DET
ejpam-109	284	41	same	same	ADJ
ejpam-109	284	42	methods	method	NOUN
ejpam-109	284	43	as	as	ADP
ejpam-109	284	44	in	in	ADP
ejpam-109	284	45	the	the	DET
ejpam-109	284	46	proof	proof	NOUN
ejpam-109	284	47	of	of	ADP
ejpam-109	284	48	theorem	theorem	NOUN
ejpam-109	284	49	4.1	4.1	NUM
ejpam-109	284	50	,	,	PUNCT
ejpam-109	284	51	we	we	PRON
ejpam-109	284	52	get	get	VERB
ejpam-109	284	53	the	the	DET
ejpam-109	284	54	indicated	indicate	VERB
ejpam-109	284	55	congruences	congruence	NOUN
ejpam-109	284	56	.	.	PUNCT
ejpam-109	285	1	as	as	ADP
ejpam-109	285	2	a	a	DET
ejpam-109	285	3	special	special	ADJ
ejpam-109	285	4	case	case	NOUN
ejpam-109	285	5	of	of	ADP
ejpam-109	285	6	corollary	corollary	ADJ
ejpam-109	285	7	4.2	4.2	NUM
ejpam-109	285	8	,	,	PUNCT
ejpam-109	285	9	we	we	PRON
ejpam-109	285	10	may	may	AUX
ejpam-109	285	11	state	state	VERB
ejpam-109	285	12	the	the	DET
ejpam-109	285	13	following	follow	VERB
ejpam-109	285	14	corollary	corollary	NOUN
ejpam-109	285	15	4.3	4.3	NUM
ejpam-109	285	16	.	.	PUNCT
ejpam-109	286	1	let	let	VERB
ejpam-109	286	2	p	p	PRON
ejpam-109	286	3	be	be	AUX
ejpam-109	286	4	an	an	DET
ejpam-109	286	5	odd	odd	ADJ
ejpam-109	286	6	prime	prime	NOUN
ejpam-109	286	7	,	,	PUNCT
ejpam-109	286	8	s	s	PROPN
ejpam-109	286	9	,	,	PUNCT
ejpam-109	286	10	b	b	PROPN
ejpam-109	286	11	≥	≥	NUM
ejpam-109	286	12	1	1	NUM
ejpam-109	286	13	,	,	PUNCT
ejpam-109	286	14	m	m	AUX
ejpam-109	286	15	be	be	AUX
ejpam-109	286	16	an	an	DET
ejpam-109	286	17	even	even	ADV
ejpam-109	286	18	integer	integer	NOUN
ejpam-109	286	19	≥	≥	NOUN
ejpam-109	286	20	2	2	NUM
ejpam-109	286	21	,	,	PUNCT
ejpam-109	286	22	m≥	m≥	VERB
ejpam-109	286	23	s+1	s+1	NOUN
ejpam-109	286	24	and	and	CCONJ
ejpam-109	286	25	assume	assume	VERB
ejpam-109	286	26	that	that	SCONJ
ejpam-109	286	27	λ1,λ2	λ1,λ2	PROPN
ejpam-109	286	28	∈	∈	PROPN
ejpam-109	286	29	zp	zp	NOUN
ejpam-109	286	30	satisfy	satisfy	NOUN
ejpam-109	286	31	λ1+λ2	λ1+λ2	PROPN
ejpam-109	286	32	≡	≡	PROPN
ejpam-109	286	33	0	0	PUNCT
ejpam-109	287	1	(	(	PUNCT
ejpam-109	287	2	mod	mod	PROPN
ejpam-109	287	3	p	p	NOUN
ejpam-109	287	4	)	)	PUNCT
ejpam-109	287	5	.	.	PUNCT
ejpam-109	288	1	if	if	SCONJ
ejpam-109	288	2	a	a	PRON
ejpam-109	288	3	is	be	AUX
ejpam-109	288	4	a	a	DET
ejpam-109	288	5	positive	positive	ADJ
ejpam-109	288	6	integer	integer	NOUN
ejpam-109	288	7	with	with	ADP
ejpam-109	288	8	(	(	PUNCT
ejpam-109	288	9	a	a	PRON
ejpam-109	288	10	,	,	PUNCT
ejpam-109	288	11	p	p	NOUN
ejpam-109	288	12	)	)	PUNCT
ejpam-109	288	13	=	=	SYM
ejpam-109	288	14	1	1	NUM
ejpam-109	288	15	,	,	PUNCT
ejpam-109	288	16	then	then	ADV
ejpam-109	288	17	βm(a	βm(a	NOUN
ejpam-109	288	18	)	)	PUNCT
ejpam-109	288	19	�	�	PROPN
ejpam-109	288	20	λ1β	λ1β	PROPN
ejpam-109	288	21	b(p−1)(a	b(p−1)(a	PROPN
ejpam-109	288	22	)	)	PUNCT
ejpam-109	289	1	+	+	ADJ
ejpam-109	289	2	λ2	λ2	PROPN
ejpam-109	289	3	�	�	SYM
ejpam-109	289	4	s	s	PART
ejpam-109	289	5	≡	≡	PROPN
ejpam-109	289	6	0	0	PUNCT
ejpam-109	290	1	(	(	PUNCT
ejpam-109	290	2	mod	mod	PROPN
ejpam-109	290	3	ps	ps	PROPN
ejpam-109	290	4	)	)	PUNCT
ejpam-109	290	5	.	.	PUNCT
ejpam-109	291	1	(	(	PUNCT
ejpam-109	291	2	i	i	NOUN
ejpam-109	291	3	)	)	PUNCT
ejpam-109	291	4	in	in	ADP
ejpam-109	291	5	particular	particular	ADJ
ejpam-109	291	6	,	,	PUNCT
ejpam-109	291	7	if	if	SCONJ
ejpam-109	291	8	p−	p−	NOUN
ejpam-109	291	9	1	1	NUM
ejpam-109	291	10	m	m	NOUN
ejpam-109	291	11	,	,	PUNCT
ejpam-109	291	12	then	then	ADV
ejpam-109	291	13	βm	βm	VERB
ejpam-109	291	14	�	�	PROPN
ejpam-109	291	15	λ1β	λ1β	PROPN
ejpam-109	291	16	b(p−1)+λ2	b(p−1)+λ2	PROPN
ejpam-109	291	17	�	�	PROPN
ejpam-109	291	18	s	s	PART
ejpam-109	291	19	≡	≡	PROPN
ejpam-109	291	20	0	0	PUNCT
ejpam-109	292	1	(	(	PUNCT
ejpam-109	292	2	mod	mod	PROPN
ejpam-109	292	3	ps	ps	PROPN
ejpam-109	292	4	)	)	PUNCT
ejpam-109	292	5	.	.	PUNCT
ejpam-109	293	1	(	(	PUNCT
ejpam-109	293	2	ii	ii	NOUN
ejpam-109	293	3	)	)	PUNCT
ejpam-109	293	4	proof	proof	NOUN
ejpam-109	293	5	.	.	PUNCT
ejpam-109	294	1	take	take	VERB
ejpam-109	294	2	k	k	NOUN
ejpam-109	294	3	=	=	PUNCT
ejpam-109	294	4	1	1	NUM
ejpam-109	294	5	in	in	ADP
ejpam-109	294	6	corollary	corollary	ADJ
ejpam-109	294	7	4.2	4.2	NUM
ejpam-109	294	8	.	.	PUNCT
ejpam-109	295	1	choosing	choose	VERB
ejpam-109	295	2	especially	especially	ADV
ejpam-109	295	3	b	b	NOUN
ejpam-109	295	4	=	=	SYM
ejpam-109	295	5	λ1	λ1	PROPN
ejpam-109	295	6	=	=	SYM
ejpam-109	295	7	1	1	NUM
ejpam-109	295	8	and	and	CCONJ
ejpam-109	295	9	λ2	λ2	NOUN
ejpam-109	295	10	=	=	NOUN
ejpam-109	295	11	−1	−1	NOUN
ejpam-109	295	12	in	in	ADP
ejpam-109	295	13	the	the	DET
ejpam-109	295	14	above	above	ADJ
ejpam-109	295	15	corollary	corollary	NOUN
ejpam-109	295	16	,	,	PUNCT
ejpam-109	295	17	we	we	PRON
ejpam-109	295	18	can	can	AUX
ejpam-109	295	19	derive	derive	VERB
ejpam-109	295	20	readily	readily	ADV
ejpam-109	295	21	theorem	theorem	VERB
ejpam-109	295	22	2.6	2.6	NUM
ejpam-109	295	23	.	.	PUNCT
ejpam-109	296	1	corollary	corollary	NOUN
ejpam-109	296	2	4.4	4.4	NUM
ejpam-109	296	3	.	.	PUNCT
ejpam-109	297	1	let	let	VERB
ejpam-109	297	2	m	m	PRON
ejpam-109	297	3	be	be	AUX
ejpam-109	297	4	an	an	DET
ejpam-109	297	5	even	even	ADV
ejpam-109	297	6	integer	integer	NOUN
ejpam-109	297	7	≥	≥	NOUN
ejpam-109	297	8	2	2	NUM
ejpam-109	297	9	,	,	PUNCT
ejpam-109	297	10	n	n	CCONJ
ejpam-109	297	11	,	,	PUNCT
ejpam-109	297	12	a	a	DET
ejpam-109	297	13	be	be	AUX
ejpam-109	297	14	positive	positive	ADJ
ejpam-109	297	15	integers	integer	NOUN
ejpam-109	297	16	with	with	ADP
ejpam-109	297	17	(	(	PUNCT
ejpam-109	297	18	a	a	PRON
ejpam-109	297	19	,	,	PUNCT
ejpam-109	297	20	n	n	CCONJ
ejpam-109	297	21	)	)	PUNCT
ejpam-109	297	22	=	=	SYM
ejpam-109	297	23	1	1	NUM
ejpam-109	297	24	and	and	CCONJ
ejpam-109	297	25	n≥	n≥	ADJ
ejpam-109	297	26	3	3	X
ejpam-109	297	27	.	.	PUNCT
ejpam-109	298	1	if	if	SCONJ
ejpam-109	298	2	m≡	m≡	NOUN
ejpam-109	298	3	l	l	NOUN
ejpam-109	298	4	(	(	PUNCT
ejpam-109	298	5	mod	mod	PROPN
ejpam-109	298	6	ϕ(n	ϕ(n	PROPN
ejpam-109	298	7	)	)	PUNCT
ejpam-109	298	8	)	)	PUNCT
ejpam-109	299	1	for	for	ADP
ejpam-109	299	2	l	l	PROPN
ejpam-109	299	3	≥	≥	NUM
ejpam-109	299	4	2	2	NUM
ejpam-109	299	5	,	,	PUNCT
ejpam-109	299	6	then	then	ADV
ejpam-109	299	7	km(n	km(n	NOUN
ejpam-109	299	8	;	;	PUNCT
ejpam-109	299	9	a)≡	a)≡	NOUN
ejpam-109	299	10	kl(n	kl(n	X
ejpam-109	299	11	;	;	PUNCT
ejpam-109	299	12	a	a	X
ejpam-109	299	13	)	)	PUNCT
ejpam-109	299	14	(	(	PUNCT
ejpam-109	299	15	mod	mod	NOUN
ejpam-109	299	16	n	n	CCONJ
ejpam-109	299	17	)	)	PUNCT
ejpam-109	299	18	.	.	PUNCT
ejpam-109	300	1	(	(	PUNCT
ejpam-109	300	2	i	i	NOUN
ejpam-109	300	3	)	)	PUNCT
ejpam-109	300	4	in	in	ADP
ejpam-109	300	5	particular	particular	ADJ
ejpam-109	300	6	,	,	PUNCT
ejpam-109	300	7	if	if	SCONJ
ejpam-109	300	8	p	p	NOUN
ejpam-109	300	9	is	be	AUX
ejpam-109	300	10	an	an	DET
ejpam-109	300	11	odd	odd	ADJ
ejpam-109	300	12	prime	prime	NOUN
ejpam-109	300	13	with	with	ADP
ejpam-109	300	14	p−1	p−1	PROPN
ejpam-109	300	15	m	m	PROPN
ejpam-109	300	16	and	and	CCONJ
ejpam-109	300	17	m≡	m≡	NOUN
ejpam-109	300	18	l	l	NOUN
ejpam-109	300	19	(	(	PUNCT
ejpam-109	300	20	mod	mod	NOUN
ejpam-109	300	21	ϕ(pα	ϕ(pα	PROPN
ejpam-109	300	22	)	)	PUNCT
ejpam-109	300	23	)	)	PUNCT
ejpam-109	300	24	(	(	PUNCT
ejpam-109	300	25	α≥	α≥	NOUN
ejpam-109	300	26	1	1	NUM
ejpam-109	300	27	)	)	PUNCT
ejpam-109	300	28	for	for	ADP
ejpam-109	300	29	l	l	PROPN
ejpam-109	300	30	≥	≥	NUM
ejpam-109	300	31	2	2	NUM
ejpam-109	300	32	,	,	PUNCT
ejpam-109	300	33	then	then	ADV
ejpam-109	300	34	hm(p)≡	hm(p)≡	PRON
ejpam-109	300	35	hl(p	hl(p	NOUN
ejpam-109	300	36	)	)	PUNCT
ejpam-109	300	37	(	(	PUNCT
ejpam-109	300	38	mod	mod	PROPN
ejpam-109	300	39	pα	pα	PROPN
ejpam-109	300	40	)	)	PUNCT
ejpam-109	300	41	.	.	PUNCT
ejpam-109	301	1	(	(	PUNCT
ejpam-109	301	2	ii	ii	NOUN
ejpam-109	301	3	)	)	PUNCT
ejpam-109	301	4	t.	t.	NOUN
ejpam-109	301	5	agoh	agoh	PROPN
ejpam-109	301	6	/	/	SYM
ejpam-109	301	7	eur	eur	PROPN
ejpam-109	301	8	.	.	PUNCT
ejpam-109	302	1	j.	j.	PROPN
ejpam-109	302	2	pure	pure	PROPN
ejpam-109	302	3	appl	appl	PROPN
ejpam-109	302	4	.	.	PROPN
ejpam-109	302	5	math	math	PROPN
ejpam-109	302	6	,	,	PUNCT
ejpam-109	302	7	1	1	NUM
ejpam-109	302	8	(	(	PUNCT
ejpam-109	302	9	2008	2008	NUM
ejpam-109	302	10	)	)	PUNCT
ejpam-109	302	11	,	,	PUNCT
ejpam-109	302	12	(	(	PUNCT
ejpam-109	302	13	3	3	NUM
ejpam-109	302	14	-	-	SYM
ejpam-109	302	15	21	21	NUM
ejpam-109	302	16	)	)	PUNCT
ejpam-109	302	17	15	15	NUM
ejpam-109	302	18	proof	proof	NOUN
ejpam-109	302	19	.	.	PUNCT
ejpam-109	303	1	take	take	VERB
ejpam-109	303	2	s	s	PART
ejpam-109	304	1	=	=	X
ejpam-109	304	2	k	k	NOUN
ejpam-109	304	3	=	=	PUNCT
ejpam-109	304	4	λ1	λ1	PROPN
ejpam-109	304	5	=	=	SYM
ejpam-109	304	6	1	1	NUM
ejpam-109	304	7	and	and	CCONJ
ejpam-109	304	8	λ2	λ2	NOUN
ejpam-109	304	9	=	=	NOUN
ejpam-109	304	10	−1	−1	NOUN
ejpam-109	304	11	in	in	ADP
ejpam-109	304	12	theorem	theorem	ADJ
ejpam-109	304	13	4.1	4.1	NUM
ejpam-109	304	14	and	and	CCONJ
ejpam-109	304	15	note	note	VERB
ejpam-109	304	16	that	that	SCONJ
ejpam-109	304	17	hi(pα	hi(pα	ADJ
ejpam-109	304	18	)	)	PUNCT
ejpam-109	304	19	=	=	SYM
ejpam-109	304	20	hi(p	hi(p	NOUN
ejpam-109	304	21	)	)	PUNCT
ejpam-109	304	22	for	for	ADP
ejpam-109	304	23	any	any	DET
ejpam-109	304	24	i	i	PRON
ejpam-109	304	25	≥	≥	NOUN
ejpam-109	304	26	1	1	NUM
ejpam-109	304	27	.	.	PUNCT
ejpam-109	305	1	then	then	ADV
ejpam-109	305	2	,	,	PUNCT
ejpam-109	305	3	congruence	congruence	PROPN
ejpam-109	305	4	(	(	PUNCT
ejpam-109	305	5	i	i	NOUN
ejpam-109	305	6	)	)	PUNCT
ejpam-109	305	7	and	and	CCONJ
ejpam-109	305	8	(	(	PUNCT
ejpam-109	305	9	ii	ii	NOUN
ejpam-109	305	10	)	)	PUNCT
ejpam-109	305	11	are	be	AUX
ejpam-109	305	12	immediate	immediate	ADJ
ejpam-109	305	13	.	.	PUNCT
ejpam-109	306	1	let	let	VERB
ejpam-109	306	2	n	n	CCONJ
ejpam-109	306	3	,	,	PUNCT
ejpam-109	306	4	a	a	DET
ejpam-109	306	5	be	be	AUX
ejpam-109	306	6	positive	positive	ADJ
ejpam-109	306	7	integers	integer	NOUN
ejpam-109	306	8	with	with	ADP
ejpam-109	306	9	n	n	NUM
ejpam-109	306	10	≥	≥	NOUN
ejpam-109	306	11	3	3	NUM
ejpam-109	306	12	and	and	CCONJ
ejpam-109	306	13	(	(	PUNCT
ejpam-109	306	14	a	a	PRON
ejpam-109	306	15	,	,	PUNCT
ejpam-109	306	16	n	n	CCONJ
ejpam-109	306	17	)	)	PUNCT
ejpam-109	306	18	=	=	SYM
ejpam-109	306	19	1	1	NUM
ejpam-109	306	20	,	,	PUNCT
ejpam-109	306	21	and	and	CCONJ
ejpam-109	306	22	let	let	VERB
ejpam-109	306	23	s	s	PRON
ejpam-109	306	24	,	,	PUNCT
ejpam-109	306	25	b	b	X
ejpam-109	306	26	be	be	AUX
ejpam-109	306	27	positive	positive	ADJ
ejpam-109	306	28	integers	integer	NOUN
ejpam-109	306	29	.	.	PUNCT
ejpam-109	307	1	then	then	ADV
ejpam-109	307	2	we	we	PRON
ejpam-109	307	3	obtain	obtain	VERB
ejpam-109	307	4	theorem	theorem	ADJ
ejpam-109	307	5	4.5	4.5	NUM
ejpam-109	307	6	.	.	PUNCT
ejpam-109	308	1	let	let	VERB
ejpam-109	308	2	m	m	PRON
ejpam-109	308	3	≥	≥	NOUN
ejpam-109	308	4	2	2	NUM
ejpam-109	308	5	be	be	AUX
ejpam-109	308	6	an	an	DET
ejpam-109	308	7	even	even	ADV
ejpam-109	308	8	integer	integer	NOUN
ejpam-109	308	9	and	and	CCONJ
ejpam-109	308	10	assume	assume	VERB
ejpam-109	308	11	that	that	SCONJ
ejpam-109	308	12	λ1,λ2	λ1,λ2	PROPN
ejpam-109	308	13	∈	∈	PROPN
ejpam-109	308	14	zn	zn	PROPN
ejpam-109	308	15	satisfy	satisfy	VERB
ejpam-109	308	16	λ1	λ1	PROPN
ejpam-109	309	1	+	+	CCONJ
ejpam-109	309	2	λ2	λ2	PROPN
ejpam-109	309	3	≡	≡	PROPN
ejpam-109	309	4	0	0	PUNCT
ejpam-109	309	5	(	(	PUNCT
ejpam-109	309	6	mod	mod	NOUN
ejpam-109	309	7	n	n	CCONJ
ejpam-109	309	8	)	)	PUNCT
ejpam-109	309	9	.	.	PUNCT
ejpam-109	310	1	then	then	ADV
ejpam-109	310	2	k	k	PROPN
ejpam-109	310	3	′m(n	′m(n	PROPN
ejpam-109	310	4	;	;	PUNCT
ejpam-109	310	5	a	a	DET
ejpam-109	310	6	)	)	PUNCT
ejpam-109	310	7	�	�	PROPN
ejpam-109	310	8	λ1k	λ1k	NOUN
ejpam-109	310	9	′bϕ(n)(n	′bϕ(n)(n	NOUN
ejpam-109	310	10	;	;	PUNCT
ejpam-109	310	11	a	a	X
ejpam-109	310	12	)	)	PUNCT
ejpam-109	310	13	+	+	ADJ
ejpam-109	310	14	λ2	λ2	PROPN
ejpam-109	310	15	�	�	SYM
ejpam-109	310	16	s	s	PART
ejpam-109	310	17	≡	≡	PROPN
ejpam-109	310	18	0	0	PUNCT
ejpam-109	311	1	(	(	PUNCT
ejpam-109	311	2	mod	mod	PROPN
ejpam-109	311	3	ns−1	ns−1	PROPN
ejpam-109	311	4	)	)	PUNCT
ejpam-109	311	5	.	.	PUNCT
ejpam-109	312	1	(	(	PUNCT
ejpam-109	312	2	i	i	NOUN
ejpam-109	312	3	)	)	PUNCT
ejpam-109	312	4	in	in	ADP
ejpam-109	312	5	particular	particular	ADJ
ejpam-109	312	6	,	,	PUNCT
ejpam-109	312	7	if	if	SCONJ
ejpam-109	312	8	n	n	NOUN
ejpam-109	312	9	=	=	VERB
ejpam-109	312	10	pα	pα	INTJ
ejpam-109	312	11	(	(	PUNCT
ejpam-109	312	12	α	α	PRON
ejpam-109	312	13	≥	≥	NOUN
ejpam-109	312	14	1	1	NUM
ejpam-109	312	15	,	,	PUNCT
ejpam-109	312	16	p	p	X
ejpam-109	312	17	an	an	DET
ejpam-109	312	18	odd	odd	ADJ
ejpam-109	312	19	prime	prime	NOUN
ejpam-109	312	20	)	)	PUNCT
ejpam-109	312	21	,	,	PUNCT
ejpam-109	312	22	p−	p−	NOUN
ejpam-109	312	23	1	1	NUM
ejpam-109	312	24	m	m	NOUN
ejpam-109	312	25	and	and	CCONJ
ejpam-109	312	26	λ1,λ2	λ1,λ2	PROPN
ejpam-109	312	27	∈	∈	PROPN
ejpam-109	312	28	zp	zp	NOUN
ejpam-109	312	29	satisfy	satisfy	NOUN
ejpam-109	313	1	λ1	λ1	PROPN
ejpam-109	314	1	+	+	CCONJ
ejpam-109	314	2	λ2	λ2	PROPN
ejpam-109	314	3	≡	≡	PROPN
ejpam-109	314	4	0	0	PUNCT
ejpam-109	315	1	(	(	PUNCT
ejpam-109	315	2	mod	mod	PROPN
ejpam-109	315	3	p	p	PROPN
ejpam-109	315	4	)	)	PUNCT
ejpam-109	315	5	,	,	PUNCT
ejpam-109	315	6	then	then	ADV
ejpam-109	315	7	h	h	PROPN
ejpam-109	315	8	′m(n	′m(n	PROPN
ejpam-109	315	9	)	)	PUNCT
ejpam-109	315	10	�	�	PROPN
ejpam-109	315	11	λ1h	λ1h	X
ejpam-109	315	12	′bϕ(n)(n	′bϕ(n)(n	NOUN
ejpam-109	315	13	)	)	PUNCT
ejpam-109	316	1	+	+	SYM
ejpam-109	316	2	λ2	λ2	PROPN
ejpam-109	316	3	�	�	SYM
ejpam-109	316	4	s	s	PART
ejpam-109	316	5	≡	≡	PROPN
ejpam-109	316	6	0	0	PUNCT
ejpam-109	317	1	(	(	PUNCT
ejpam-109	317	2	mod	mod	PROPN
ejpam-109	317	3	ns−1	ns−1	PROPN
ejpam-109	317	4	)	)	PUNCT
ejpam-109	317	5	.	.	PUNCT
ejpam-109	318	1	(	(	PUNCT
ejpam-109	318	2	ii	ii	NOUN
ejpam-109	318	3	)	)	PUNCT
ejpam-109	318	4	proof	proof	NOUN
ejpam-109	318	5	.	.	PUNCT
ejpam-109	319	1	as	as	SCONJ
ejpam-109	319	2	defined	define	VERB
ejpam-109	319	3	in	in	ADP
ejpam-109	319	4	section	section	NOUN
ejpam-109	319	5	3	3	NUM
ejpam-109	319	6	,	,	PUNCT
ejpam-109	319	7	let	let	VERB
ejpam-109	319	8	ν	ν	X
ejpam-109	319	9	=	=	SYM
ejpam-109	319	10	∏	∏	PROPN
ejpam-109	319	11	p|n	p|n	NOUN
ejpam-109	319	12	pvp	pvp	NOUN
ejpam-109	319	13	(	(	PUNCT
ejpam-109	319	14	where	where	SCONJ
ejpam-109	319	15	vp	vp	NOUN
ejpam-109	319	16	=	=	SYM
ejpam-109	319	17	ordp(dm	ordp(dm	PROPN
ejpam-109	319	18	)	)	PUNCT
ejpam-109	319	19	)	)	PUNCT
ejpam-109	319	20	and	and	CCONJ
ejpam-109	319	21	put	put	VERB
ejpam-109	319	22	η	η	PROPN
ejpam-109	319	23	=	=	PROPN
ejpam-109	319	24	nsν	nsν	PROPN
ejpam-109	319	25	.	.	PUNCT
ejpam-109	320	1	since	since	SCONJ
ejpam-109	320	2	ordp(dm	ordp(dm	NOUN
ejpam-109	320	3	)	)	PUNCT
ejpam-109	320	4	=	=	PUNCT
ejpam-109	320	5	ordp(dm+cbϕ(n	ordp(dm+cbϕ(n	NUM
ejpam-109	320	6	)	)	PUNCT
ejpam-109	320	7	)	)	PUNCT
ejpam-109	320	8	for	for	ADP
ejpam-109	320	9	each	each	DET
ejpam-109	320	10	prime	prime	ADJ
ejpam-109	320	11	divisor	divisor	NOUN
ejpam-109	320	12	p	p	NOUN
ejpam-109	320	13	of	of	ADP
ejpam-109	320	14	n	n	PROPN
ejpam-109	320	15	and	and	CCONJ
ejpam-109	320	16	c	c	NOUN
ejpam-109	320	17	=	=	SYM
ejpam-109	320	18	0	0	NUM
ejpam-109	320	19	,	,	PUNCT
ejpam-109	320	20	1	1	NUM
ejpam-109	320	21	,	,	PUNCT
ejpam-109	320	22	...	...	PUNCT
ejpam-109	320	23	,	,	PUNCT
ejpam-109	320	24	s	s	AUX
ejpam-109	320	25	,	,	PUNCT
ejpam-109	320	26	we	we	PRON
ejpam-109	320	27	may	may	AUX
ejpam-109	320	28	consider	consider	VERB
ejpam-109	320	29	the	the	DET
ejpam-109	320	30	voronoï	voronoï	ADJ
ejpam-109	320	31	type	type	NOUN
ejpam-109	320	32	congruence	congruence	NOUN
ejpam-109	320	33	(	(	PUNCT
ejpam-109	320	34	ii	ii	NOUN
ejpam-109	320	35	)	)	PUNCT
ejpam-109	320	36	in	in	ADP
ejpam-109	320	37	theorem	theorem	ADJ
ejpam-109	320	38	3.1	3.1	NUM
ejpam-109	320	39	replaced	replace	VERB
ejpam-109	320	40	m	m	PRON
ejpam-109	320	41	,	,	PUNCT
ejpam-109	320	42	n	n	PROPN
ejpam-109	320	43	and	and	CCONJ
ejpam-109	320	44	w′	w′	PROPN
ejpam-109	320	45	by	by	ADP
ejpam-109	320	46	m+	m+	NUM
ejpam-109	320	47	cbϕ(n	cbϕ(n	NOUN
ejpam-109	320	48	)	)	PUNCT
ejpam-109	320	49	,	,	PUNCT
ejpam-109	320	50	ns	ns	NUM
ejpam-109	320	51	and	and	CCONJ
ejpam-109	320	52	η	η	PROPN
ejpam-109	320	53	,	,	PUNCT
ejpam-109	320	54	respectively	respectively	ADV
ejpam-109	320	55	.	.	PUNCT
ejpam-109	321	1	noting	note	VERB
ejpam-109	321	2	that	that	SCONJ
ejpam-109	321	3	εm(ns	εm(ns	NOUN
ejpam-109	321	4	)	)	PUNCT
ejpam-109	321	5	=	=	NOUN
ejpam-109	321	6	εm(n	εm(n	X
ejpam-109	321	7	)	)	PUNCT
ejpam-109	321	8	and	and	CCONJ
ejpam-109	321	9	hence	hence	ADV
ejpam-109	321	10	k	k	PROPN
ejpam-109	321	11	′m(n	′m(n	PROPN
ejpam-109	321	12	s	s	PROPN
ejpam-109	321	13	;	;	PUNCT
ejpam-109	321	14	a	a	X
ejpam-109	321	15	)	)	PUNCT
ejpam-109	321	16	=	=	SYM
ejpam-109	321	17	k	k	PROPN
ejpam-109	321	18	′m(n	′m(n	PROPN
ejpam-109	321	19	;	;	PUNCT
ejpam-109	321	20	a	a	X
ejpam-109	321	21	)	)	PUNCT
ejpam-109	321	22	for	for	ADP
ejpam-109	321	23	any	any	DET
ejpam-109	321	24	m	m	NOUN
ejpam-109	321	25	,	,	PUNCT
ejpam-109	321	26	we	we	PRON
ejpam-109	321	27	have	have	VERB
ejpam-109	321	28	k	k	PROPN
ejpam-109	321	29	′m+cbϕ(n)(n	′m+cbϕ(n)(n	PROPN
ejpam-109	321	30	;	;	PUNCT
ejpam-109	321	31	a)≡	a)≡	X
ejpam-109	321	32	(	(	PUNCT
ejpam-109	321	33	m+	m+	NUM
ejpam-109	321	34	cbϕ(n	cbϕ(n	NOUN
ejpam-109	321	35	)	)	PUNCT
ejpam-109	321	36	)	)	PUNCT
ejpam-109	322	1	η−1	η−1	PROPN
ejpam-109	322	2	∑	∑	PUNCT
ejpam-109	322	3	j=1	j=1	PROPN
ejpam-109	322	4	(	(	PUNCT
ejpam-109	322	5	j	j	PROPN
ejpam-109	322	6	,	,	PUNCT
ejpam-109	322	7	n)=1	n)=1	PROPN
ejpam-109	322	8	(	(	PUNCT
ejpam-109	322	9	a	a	DET
ejpam-109	322	10	j)m+cbϕ(n)−1	j)m+cbϕ(n)−1	NOUN
ejpam-109	322	11	�	�	PROPN
ejpam-109	322	12	a	a	DET
ejpam-109	322	13	j	j	PROPN
ejpam-109	322	14	η	η	PROPN
ejpam-109	322	15	�	�	PROPN
ejpam-109	322	16	(	(	PUNCT
ejpam-109	322	17	mod	mod	PROPN
ejpam-109	322	18	ns	ns	NUM
ejpam-109	322	19	)	)	PUNCT
ejpam-109	322	20	.	.	PUNCT
ejpam-109	323	1	using	use	VERB
ejpam-109	323	2	this	this	DET
ejpam-109	323	3	congruence	congruence	NOUN
ejpam-109	323	4	,	,	PUNCT
ejpam-109	323	5	it	it	PRON
ejpam-109	323	6	follows	follow	VERB
ejpam-109	323	7	that	that	SCONJ
ejpam-109	323	8	k	k	PROPN
ejpam-109	323	9	′m(n	′m(n	PROPN
ejpam-109	323	10	;	;	PUNCT
ejpam-109	323	11	a	a	DET
ejpam-109	323	12	)	)	PUNCT
ejpam-109	323	13	�	�	PROPN
ejpam-109	323	14	λ1k	λ1k	NOUN
ejpam-109	323	15	′bϕ(n)(n	′bϕ(n)(n	NOUN
ejpam-109	323	16	;	;	PUNCT
ejpam-109	323	17	a	a	X
ejpam-109	323	18	)	)	PUNCT
ejpam-109	323	19	+	+	ADJ
ejpam-109	323	20	λ2	λ2	PROPN
ejpam-109	323	21	�	�	NOUN
ejpam-109	323	22	s	s	PART
ejpam-109	323	23	=	=	X
ejpam-109	323	24	s	s	PART
ejpam-109	323	25	∑	∑	PUNCT
ejpam-109	323	26	c=0	c=0	PROPN
ejpam-109	323	27	�	�	PROPN
ejpam-109	323	28	s	s	PART
ejpam-109	323	29	c	c	X
ejpam-109	323	30	�	�	PROPN
ejpam-109	323	31	λc	λc	PROPN
ejpam-109	323	32	1k	1k	PROPN
ejpam-109	323	33	′m+cbϕ(n)(n	′m+cbϕ(n)(n	PROPN
ejpam-109	323	34	;	;	PUNCT
ejpam-109	323	35	a)λs−c	a)λs−c	PROPN
ejpam-109	323	36	2	2	NUM
ejpam-109	323	37	≡	≡	PROPN
ejpam-109	323	38	s	s	PART
ejpam-109	323	39	∑	∑	PROPN
ejpam-109	323	40	c=0	c=0	PROPN
ejpam-109	323	41	�	�	PROPN
ejpam-109	323	42	s	s	PART
ejpam-109	323	43	c	c	NOUN
ejpam-109	323	44	�	�	PROPN
ejpam-109	323	45	λc	λc	PRON
ejpam-109	323	46	1λ	1λ	PROPN
ejpam-109	323	47	s−c	s−c	PROPN
ejpam-109	323	48	2	2	NUM
ejpam-109	323	49	�	�	PROPN
ejpam-109	323	50	(	(	PUNCT
ejpam-109	323	51	m+	m+	NUM
ejpam-109	323	52	cbϕ(n	cbϕ(n	NOUN
ejpam-109	323	53	)	)	PUNCT
ejpam-109	323	54	)	)	PUNCT
ejpam-109	324	1	η−1	η−1	PROPN
ejpam-109	324	2	∑	∑	PUNCT
ejpam-109	324	3	j=1	j=1	PROPN
ejpam-109	324	4	(	(	PUNCT
ejpam-109	324	5	j	j	PROPN
ejpam-109	324	6	,	,	PUNCT
ejpam-109	324	7	n)=1	n)=1	PROPN
ejpam-109	324	8	(	(	PUNCT
ejpam-109	324	9	a	a	DET
ejpam-109	324	10	j)m+cbϕ(n)−1	j)m+cbϕ(n)−1	NOUN
ejpam-109	324	11	�	�	PROPN
ejpam-109	324	12	a	a	DET
ejpam-109	324	13	j	j	PROPN
ejpam-109	324	14	η	η	PROPN
ejpam-109	324	15	�	�	PROPN
ejpam-109	324	16	�	�	PROPN
ejpam-109	324	17	≡	≡	PROPN
ejpam-109	324	18	s	s	PART
ejpam-109	324	19	∑	∑	PROPN
ejpam-109	324	20	c=0	c=0	PROPN
ejpam-109	324	21	�	�	PROPN
ejpam-109	324	22	s	s	PART
ejpam-109	324	23	c	c	NOUN
ejpam-109	324	24	�	�	PROPN
ejpam-109	325	1	λc	λc	PRON
ejpam-109	325	2	1λ	1λ	PROPN
ejpam-109	325	3	s−c	s−c	PROPN
ejpam-109	325	4	2	2	NUM
ejpam-109	325	5	�	�	PROPN
ejpam-109	325	6	m	m	PROPN
ejpam-109	325	7	η−1	η−1	PROPN
ejpam-109	325	8	∑	∑	INTJ
ejpam-109	325	9	j=1	j=1	PROPN
ejpam-109	325	10	(	(	PUNCT
ejpam-109	325	11	j	j	PROPN
ejpam-109	325	12	,	,	PUNCT
ejpam-109	325	13	n)=1	n)=1	PROPN
ejpam-109	325	14	(	(	PUNCT
ejpam-109	325	15	a	a	DET
ejpam-109	325	16	j)m+cbϕ(n)−1	j)m+cbϕ(n)−1	NOUN
ejpam-109	325	17	�	�	PROPN
ejpam-109	325	18	a	a	DET
ejpam-109	325	19	j	j	PROPN
ejpam-109	325	20	η	η	PROPN
ejpam-109	325	21	�	�	PROPN
ejpam-109	325	22	�	�	PROPN
ejpam-109	325	23	+	+	CCONJ
ejpam-109	325	24	s	s	PART
ejpam-109	325	25	∑	∑	PUNCT
ejpam-109	325	26	c=0	c=0	PROPN
ejpam-109	325	27	�	�	PROPN
ejpam-109	325	28	s	s	PART
ejpam-109	325	29	c	c	NOUN
ejpam-109	325	30	�	�	PROPN
ejpam-109	325	31	λc	λc	PRON
ejpam-109	325	32	1λ	1λ	PROPN
ejpam-109	325	33	s−c	s−c	PROPN
ejpam-109	325	34	2	2	NUM
ejpam-109	325	35	�	�	PROPN
ejpam-109	325	36	cbϕ(n	cbϕ(n	PROPN
ejpam-109	325	37	)	)	PUNCT
ejpam-109	325	38	η−1	η−1	PROPN
ejpam-109	325	39	∑	∑	PUNCT
ejpam-109	325	40	j=1	j=1	PROPN
ejpam-109	325	41	(	(	PUNCT
ejpam-109	325	42	j	j	PROPN
ejpam-109	325	43	,	,	PUNCT
ejpam-109	325	44	n)=1	n)=1	PROPN
ejpam-109	325	45	(	(	PUNCT
ejpam-109	325	46	a	a	DET
ejpam-109	325	47	j)m+cbϕ(n)−1	j)m+cbϕ(n)−1	NOUN
ejpam-109	325	48	�	�	PROPN
ejpam-109	325	49	a	a	DET
ejpam-109	325	50	j	j	PROPN
ejpam-109	325	51	η	η	PROPN
ejpam-109	325	52	�	�	PROPN
ejpam-109	325	53	�	�	PROPN
ejpam-109	325	54	≡m	≡m	ADV
ejpam-109	325	55	η−1	η−1	PROPN
ejpam-109	325	56	∑	∑	PROPN
ejpam-109	325	57	j=1	j=1	PROPN
ejpam-109	325	58	(	(	PUNCT
ejpam-109	325	59	j	j	PROPN
ejpam-109	325	60	,	,	PUNCT
ejpam-109	325	61	n)=1	n)=1	PROPN
ejpam-109	325	62	(	(	PUNCT
ejpam-109	325	63	a	a	DET
ejpam-109	325	64	j)m−1	j)m−1	PROPN
ejpam-109	325	65	s	s	PART
ejpam-109	325	66	∑	∑	PROPN
ejpam-109	325	67	c=0	c=0	PROPN
ejpam-109	325	68	�	�	PROPN
ejpam-109	325	69	s	s	PART
ejpam-109	325	70	c	c	NOUN
ejpam-109	325	71	�	�	PROPN
ejpam-109	325	72	λc	λc	PRON
ejpam-109	325	73	1λ	1λ	NUM
ejpam-109	325	74	s−c	s−c	NOUN
ejpam-109	325	75	2	2	NUM
ejpam-109	325	76	(	(	PUNCT
ejpam-109	325	77	a	a	DET
ejpam-109	325	78	j)cbϕ(n	j)cbϕ(n	NOUN
ejpam-109	325	79	)	)	PUNCT
ejpam-109	325	80	!	!	PUNCT
ejpam-109	326	1	�	�	PROPN
ejpam-109	326	2	a	a	DET
ejpam-109	326	3	j	j	PROPN
ejpam-109	326	4	η	η	PROPN
ejpam-109	326	5	�	�	PROPN
ejpam-109	326	6	+	+	CCONJ
ejpam-109	326	7	sλ1	sλ1	PROPN
ejpam-109	326	8	bϕ(n	bϕ(n	X
ejpam-109	326	9	)	)	PUNCT
ejpam-109	326	10	η−1	η−1	PROPN
ejpam-109	327	1	∑	∑	PUNCT
ejpam-109	327	2	j=1	j=1	PROPN
ejpam-109	327	3	(	(	PUNCT
ejpam-109	327	4	j	j	PROPN
ejpam-109	327	5	,	,	PUNCT
ejpam-109	327	6	n)=1	n)=1	PROPN
ejpam-109	327	7	(	(	PUNCT
ejpam-109	327	8	a	a	DET
ejpam-109	327	9	j)m+bϕ(n)−1	j)m+bϕ(n)−1	PROPN
ejpam-109	327	10	s	s	PART
ejpam-109	327	11	∑	∑	PROPN
ejpam-109	327	12	c′=0	c′=0	PROPN
ejpam-109	327	13	�	�	PROPN
ejpam-109	327	14	s−	s−	PROPN
ejpam-109	327	15	1	1	NUM
ejpam-109	327	16	c′	c′	NOUN
ejpam-109	327	17	�	�	NOUN
ejpam-109	327	18	λc′	λc′	ADP
ejpam-109	327	19	1	1	NUM
ejpam-109	327	20	λ	λ	NOUN
ejpam-109	327	21	s−1−c′	s−1−c′	PROPN
ejpam-109	327	22	2	2	NUM
ejpam-109	327	23	(	(	PUNCT
ejpam-109	327	24	a	a	DET
ejpam-109	327	25	j)c	j)c	ADJ
ejpam-109	327	26	′bϕ(n	′bϕ(n	NOUN
ejpam-109	327	27	)	)	PUNCT
ejpam-109	327	28	!	!	PUNCT
ejpam-109	328	1	�	�	PROPN
ejpam-109	328	2	a	a	DET
ejpam-109	328	3	j	j	PROPN
ejpam-109	328	4	η	η	PROPN
ejpam-109	328	5	�	�	PROPN
ejpam-109	328	6	t.	t.	PROPN
ejpam-109	328	7	agoh	agoh	PROPN
ejpam-109	328	8	/	/	SYM
ejpam-109	328	9	eur	eur	PROPN
ejpam-109	328	10	.	.	PUNCT
ejpam-109	329	1	j.	j.	PROPN
ejpam-109	329	2	pure	pure	PROPN
ejpam-109	329	3	appl	appl	PROPN
ejpam-109	329	4	.	.	PROPN
ejpam-109	329	5	math	math	PROPN
ejpam-109	329	6	,	,	PUNCT
ejpam-109	329	7	1	1	NUM
ejpam-109	329	8	(	(	PUNCT
ejpam-109	329	9	2008	2008	NUM
ejpam-109	329	10	)	)	PUNCT
ejpam-109	329	11	,	,	PUNCT
ejpam-109	329	12	(	(	PUNCT
ejpam-109	329	13	3	3	NUM
ejpam-109	329	14	-	-	SYM
ejpam-109	329	15	21	21	NUM
ejpam-109	329	16	)	)	PUNCT
ejpam-109	329	17	16	16	NUM
ejpam-109	329	18	≡m	≡m	ADV
ejpam-109	329	19	η−1	η−1	PROPN
ejpam-109	329	20	∑	∑	PUNCT
ejpam-109	329	21	j=1	j=1	PROPN
ejpam-109	329	22	(	(	PUNCT
ejpam-109	329	23	j	j	PROPN
ejpam-109	329	24	,	,	PUNCT
ejpam-109	329	25	n)=1	n)=1	PROPN
ejpam-109	329	26	(	(	PUNCT
ejpam-109	329	27	a	a	DET
ejpam-109	329	28	j)m−1	j)m−1	PROPN
ejpam-109	329	29	�	�	PROPN
ejpam-109	330	1	λ1(a	λ1(a	PRON
ejpam-109	330	2	j)bϕ(n)+λ2	j)bϕ(n)+λ2	PROPN
ejpam-109	330	3	�	�	PROPN
ejpam-109	330	4	s	s	PART
ejpam-109	330	5	�	�	PROPN
ejpam-109	330	6	a	a	DET
ejpam-109	330	7	j	j	PROPN
ejpam-109	330	8	η	η	PROPN
ejpam-109	330	9	�	�	PROPN
ejpam-109	330	10	+	+	CCONJ
ejpam-109	330	11	sλ1	sλ1	PROPN
ejpam-109	330	12	bϕ(n	bϕ(n	X
ejpam-109	330	13	)	)	PUNCT
ejpam-109	330	14	η−1	η−1	PROPN
ejpam-109	330	15	∑	∑	PUNCT
ejpam-109	330	16	j=1	j=1	PROPN
ejpam-109	330	17	(	(	PUNCT
ejpam-109	330	18	j	j	PROPN
ejpam-109	330	19	,	,	PUNCT
ejpam-109	330	20	n)=1	n)=1	PROPN
ejpam-109	330	21	(	(	PUNCT
ejpam-109	330	22	a	a	DET
ejpam-109	330	23	j)m+bϕ(n)−1	j)m+bϕ(n)−1	PROPN
ejpam-109	330	24	�	�	PROPN
ejpam-109	330	25	λ1(a	λ1(a	SYM
ejpam-109	331	1	j)bϕ(n)+λ2	j)bϕ(n)+λ2	PROPN
ejpam-109	331	2	�	�	PROPN
ejpam-109	331	3	s−1	s−1	PROPN
ejpam-109	331	4	�	�	PROPN
ejpam-109	331	5	a	a	DET
ejpam-109	331	6	j	j	PROPN
ejpam-109	331	7	η	η	PROPN
ejpam-109	331	8	�	�	PROPN
ejpam-109	331	9	(	(	PUNCT
ejpam-109	331	10	mod	mod	PROPN
ejpam-109	331	11	ns	ns	NUM
ejpam-109	331	12	)	)	PUNCT
ejpam-109	331	13	.	.	PUNCT
ejpam-109	332	1	since	since	SCONJ
ejpam-109	332	2	(	(	PUNCT
ejpam-109	332	3	a	a	DET
ejpam-109	332	4	j)bϕ(n	j)bϕ(n	NOUN
ejpam-109	332	5	)	)	PUNCT
ejpam-109	332	6	≡	≡	PROPN
ejpam-109	332	7	1	1	NUM
ejpam-109	332	8	(	(	PUNCT
ejpam-109	332	9	mod	mod	NOUN
ejpam-109	332	10	n	n	CCONJ
ejpam-109	332	11	)	)	PUNCT
ejpam-109	332	12	if	if	SCONJ
ejpam-109	332	13	(	(	PUNCT
ejpam-109	332	14	a	a	DET
ejpam-109	332	15	j	j	PROPN
ejpam-109	332	16	,	,	PUNCT
ejpam-109	332	17	n	n	CCONJ
ejpam-109	332	18	)	)	PUNCT
ejpam-109	332	19	=	=	SYM
ejpam-109	332	20	1	1	NUM
ejpam-109	332	21	,	,	PUNCT
ejpam-109	332	22	we	we	PRON
ejpam-109	332	23	have	have	VERB
ejpam-109	332	24	,	,	PUNCT
ejpam-109	332	25	for	for	ADP
ejpam-109	332	26	κ=	κ=	PROPN
ejpam-109	332	27	s	s	NOUN
ejpam-109	332	28	,	,	PUNCT
ejpam-109	332	29	s−	s−	PROPN
ejpam-109	332	30	1	1	NUM
ejpam-109	332	31	,	,	PUNCT
ejpam-109	332	32	�	�	PROPN
ejpam-109	333	1	λ1(a	λ1(a	PRON
ejpam-109	333	2	j)bϕ(n)+λ2	j)bϕ(n)+λ2	PROPN
ejpam-109	333	3	�	�	PROPN
ejpam-109	333	4	κ	κ	PROPN
ejpam-109	333	5	≡	≡	PROPN
ejpam-109	333	6	�	�	PROPN
ejpam-109	333	7	λ1+λ2	λ1+λ2	PROPN
ejpam-109	333	8	�	�	PROPN
ejpam-109	333	9	κ	κ	PROPN
ejpam-109	333	10	≡	≡	PROPN
ejpam-109	333	11	0	0	PUNCT
ejpam-109	334	1	(	(	PUNCT
ejpam-109	334	2	mod	mod	PROPN
ejpam-109	334	3	ns−1	ns−1	PROPN
ejpam-109	334	4	)	)	PUNCT
ejpam-109	334	5	.	.	PUNCT
ejpam-109	335	1	from	from	ADP
ejpam-109	335	2	this	this	DET
ejpam-109	335	3	congruence	congruence	NOUN
ejpam-109	335	4	we	we	PRON
ejpam-109	335	5	can	can	AUX
ejpam-109	335	6	readily	readily	ADV
ejpam-109	335	7	derive	derive	VERB
ejpam-109	335	8	(	(	PUNCT
ejpam-109	335	9	i	i	NOUN
ejpam-109	335	10	)	)	PUNCT
ejpam-109	335	11	.	.	PUNCT
ejpam-109	336	1	when	when	SCONJ
ejpam-109	336	2	p−	p−	NOUN
ejpam-109	336	3	1	1	NUM
ejpam-109	336	4	m	m	NOUN
ejpam-109	336	5	,	,	PUNCT
ejpam-109	336	6	(	(	PUNCT
ejpam-109	336	7	ii	ii	NOUN
ejpam-109	336	8	)	)	PUNCT
ejpam-109	336	9	can	can	AUX
ejpam-109	336	10	be	be	AUX
ejpam-109	336	11	easily	easily	ADV
ejpam-109	336	12	given	give	VERB
ejpam-109	336	13	from	from	ADP
ejpam-109	336	14	(	(	PUNCT
ejpam-109	336	15	i	i	NOUN
ejpam-109	336	16	)	)	PUNCT
ejpam-109	336	17	by	by	ADP
ejpam-109	336	18	the	the	DET
ejpam-109	336	19	same	same	ADJ
ejpam-109	336	20	arguments	argument	NOUN
ejpam-109	336	21	as	as	SCONJ
ejpam-109	336	22	done	do	VERB
ejpam-109	336	23	for	for	ADP
ejpam-109	336	24	the	the	DET
ejpam-109	336	25	proof	proof	NOUN
ejpam-109	336	26	of	of	ADP
ejpam-109	336	27	(	(	PUNCT
ejpam-109	336	28	ii	ii	NOUN
ejpam-109	336	29	)	)	PUNCT
ejpam-109	336	30	in	in	ADP
ejpam-109	336	31	theorem	theorem	NOUN
ejpam-109	336	32	4.1	4.1	NUM
ejpam-109	336	33	.	.	PUNCT
ejpam-109	336	34	using	use	VERB
ejpam-109	336	35	the	the	DET
ejpam-109	336	36	same	same	ADJ
ejpam-109	336	37	notations	notation	NOUN
ejpam-109	336	38	as	as	ADP
ejpam-109	336	39	in	in	ADP
ejpam-109	336	40	theorem	theorem	ADJ
ejpam-109	336	41	4.5	4.5	NUM
ejpam-109	336	42	,	,	PUNCT
ejpam-109	336	43	we	we	PRON
ejpam-109	336	44	can	can	AUX
ejpam-109	336	45	state	state	VERB
ejpam-109	336	46	corollary	corollary	ADJ
ejpam-109	336	47	4.6	4.6	NUM
ejpam-109	336	48	.	.	PUNCT
ejpam-109	337	1	let	let	VERB
ejpam-109	337	2	p	p	PRON
ejpam-109	337	3	be	be	AUX
ejpam-109	337	4	an	an	DET
ejpam-109	337	5	odd	odd	ADJ
ejpam-109	337	6	prime	prime	NOUN
ejpam-109	337	7	and	and	CCONJ
ejpam-109	337	8	assume	assume	VERB
ejpam-109	337	9	that	that	SCONJ
ejpam-109	337	10	λ1,λ2	λ1,λ2	PROPN
ejpam-109	337	11	∈	∈	PROPN
ejpam-109	337	12	zp	zp	NOUN
ejpam-109	337	13	satisfy	satisfy	NOUN
ejpam-109	337	14	λ1+λ2	λ1+λ2	PROPN
ejpam-109	337	15	≡	≡	PROPN
ejpam-109	337	16	0	0	PUNCT
ejpam-109	338	1	(	(	PUNCT
ejpam-109	338	2	mod	mod	PROPN
ejpam-109	338	3	p	p	NOUN
ejpam-109	338	4	)	)	PUNCT
ejpam-109	338	5	.	.	PUNCT
ejpam-109	339	1	if	if	SCONJ
ejpam-109	339	2	m≥	m≥	PROPN
ejpam-109	339	3	s	s	PROPN
ejpam-109	339	4	and	and	CCONJ
ejpam-109	339	5	β	β	X
ejpam-109	339	6	′i	′i	NOUN
ejpam-109	339	7	(	(	PUNCT
ejpam-109	339	8	a	a	X
ejpam-109	339	9	)	)	PUNCT
ejpam-109	339	10	=	=	SYM
ejpam-109	339	11	iβi(a	iβi(a	ADJ
ejpam-109	339	12	)	)	PUNCT
ejpam-109	339	13	=	=	PUNCT
ejpam-109	339	14	(	(	PUNCT
ejpam-109	339	15	ai	ai	VERB
ejpam-109	339	16	−	−	PROPN
ejpam-109	339	17	1)bi	1)bi	NUM
ejpam-109	340	1	(	(	PUNCT
ejpam-109	340	2	i	i	PRON
ejpam-109	340	3	≥	≥	VERB
ejpam-109	340	4	1	1	NUM
ejpam-109	340	5	)	)	PUNCT
ejpam-109	340	6	,	,	PUNCT
ejpam-109	340	7	then	then	ADV
ejpam-109	340	8	β	β	NOUN
ejpam-109	340	9	′m(a	′m(a	NOUN
ejpam-109	340	10	)	)	PUNCT
ejpam-109	340	11	�	�	NOUN
ejpam-109	341	1	λ1β	λ1β	PROPN
ejpam-109	341	2	′b(p−1)(a	′b(p−1)(a	NOUN
ejpam-109	341	3	)	)	PUNCT
ejpam-109	341	4	+	+	ADJ
ejpam-109	341	5	λ2	λ2	PROPN
ejpam-109	341	6	�	�	SYM
ejpam-109	341	7	s	s	PART
ejpam-109	341	8	≡	≡	PROPN
ejpam-109	341	9	0	0	PUNCT
ejpam-109	342	1	(	(	PUNCT
ejpam-109	342	2	mod	mod	PROPN
ejpam-109	342	3	ps−1	ps−1	PROPN
ejpam-109	342	4	)	)	PUNCT
ejpam-109	342	5	.	.	PUNCT
ejpam-109	343	1	(	(	PUNCT
ejpam-109	343	2	i	i	NOUN
ejpam-109	343	3	)	)	PUNCT
ejpam-109	343	4	in	in	ADP
ejpam-109	343	5	particular	particular	ADJ
ejpam-109	343	6	,	,	PUNCT
ejpam-109	343	7	if	if	SCONJ
ejpam-109	343	8	p−	p−	NOUN
ejpam-109	343	9	1	1	NUM
ejpam-109	343	10	m	m	NOUN
ejpam-109	343	11	,	,	PUNCT
ejpam-109	343	12	then	then	ADV
ejpam-109	343	13	bm	bm	PROPN
ejpam-109	343	14	�	�	PROPN
ejpam-109	343	15	λ1bb(p−1)+λ2	λ1bb(p−1)+λ2	PUNCT
ejpam-109	343	16	�	�	PROPN
ejpam-109	343	17	s	s	PART
ejpam-109	343	18	≡	≡	PROPN
ejpam-109	343	19	0	0	PUNCT
ejpam-109	344	1	(	(	PUNCT
ejpam-109	344	2	mod	mod	PROPN
ejpam-109	344	3	ps−1	ps−1	PROPN
ejpam-109	344	4	)	)	PUNCT
ejpam-109	344	5	.	.	PUNCT
ejpam-109	345	1	(	(	PUNCT
ejpam-109	345	2	ii	ii	NOUN
ejpam-109	345	3	)	)	PUNCT
ejpam-109	345	4	proof	proof	NOUN
ejpam-109	345	5	.	.	PUNCT
ejpam-109	346	1	since	since	SCONJ
ejpam-109	346	2	m≥	m≥	PROPN
ejpam-109	346	3	s	s	PROPN
ejpam-109	346	4	,	,	PUNCT
ejpam-109	346	5	we	we	PRON
ejpam-109	346	6	see	see	VERB
ejpam-109	346	7	εm(p	εm(p	PUNCT
ejpam-109	346	8	)	)	PUNCT
ejpam-109	346	9	=	=	SYM
ejpam-109	346	10	1−	1−	NUM
ejpam-109	346	11	pm−1	pm−1	PROPN
ejpam-109	346	12	≡	≡	PROPN
ejpam-109	346	13	1	1	NUM
ejpam-109	346	14	(	(	PUNCT
ejpam-109	346	15	mod	mod	PROPN
ejpam-109	346	16	ps−1	ps−1	PROPN
ejpam-109	346	17	)	)	PUNCT
ejpam-109	346	18	,	,	PUNCT
ejpam-109	346	19	and	and	CCONJ
ejpam-109	346	20	so	so	ADV
ejpam-109	346	21	the	the	DET
ejpam-109	346	22	corollary	corollary	NOUN
ejpam-109	346	23	follows	follow	VERB
ejpam-109	346	24	from	from	ADP
ejpam-109	346	25	theorem	theorem	ADJ
ejpam-109	346	26	4.5	4.5	NUM
ejpam-109	346	27	.	.	PUNCT
ejpam-109	347	1	to	to	PART
ejpam-109	347	2	obtain	obtain	VERB
ejpam-109	347	3	some	some	DET
ejpam-109	347	4	generalized	generalize	VERB
ejpam-109	347	5	von	von	PROPN
ejpam-109	347	6	staudt	staudt	PROPN
ejpam-109	347	7	-	-	PUNCT
ejpam-109	347	8	kummer	kummer	NOUN
ejpam-109	347	9	congruences	congruence	NOUN
ejpam-109	347	10	in	in	ADP
ejpam-109	347	11	this	this	DET
ejpam-109	347	12	section	section	NOUN
ejpam-109	347	13	,	,	PUNCT
ejpam-109	347	14	we	we	PRON
ejpam-109	347	15	made	make	VERB
ejpam-109	347	16	use	use	NOUN
ejpam-109	347	17	of	of	ADP
ejpam-109	347	18	voronoï	voronoï	ADJ
ejpam-109	347	19	type	type	NOUN
ejpam-109	347	20	congruences	congruence	NOUN
ejpam-109	347	21	stated	state	VERB
ejpam-109	347	22	in	in	ADP
ejpam-109	347	23	section	section	NOUN
ejpam-109	347	24	3	3	NUM
ejpam-109	347	25	.	.	PUNCT
ejpam-109	348	1	however	however	ADV
ejpam-109	348	2	,	,	PUNCT
ejpam-109	348	3	besides	besides	SCONJ
ejpam-109	348	4	our	our	PRON
ejpam-109	348	5	method	method	NOUN
ejpam-109	348	6	,	,	PUNCT
ejpam-109	348	7	there	there	PRON
ejpam-109	348	8	are	be	VERB
ejpam-109	348	9	other	other	ADJ
ejpam-109	348	10	sagacious	sagacious	ADJ
ejpam-109	348	11	ones	one	NOUN
ejpam-109	348	12	to	to	PART
ejpam-109	348	13	accomplish	accomplish	VERB
ejpam-109	348	14	the	the	DET
ejpam-109	348	15	purpose	purpose	NOUN
ejpam-109	348	16	.	.	PUNCT
ejpam-109	349	1	indeed	indeed	ADV
ejpam-109	349	2	,	,	PUNCT
ejpam-109	349	3	two	two	NUM
ejpam-109	349	4	sequences	sequence	NOUN
ejpam-109	349	5	hm(n	hm(n	NOUN
ejpam-109	349	6	)	)	PUNCT
ejpam-109	349	7	and	and	CCONJ
ejpam-109	349	8	km(n	km(n	NOUN
ejpam-109	349	9	;	;	PUNCT
ejpam-109	349	10	a	a	X
ejpam-109	349	11	)	)	PUNCT
ejpam-109	349	12	(	(	PUNCT
ejpam-109	349	13	m≥	m≥	NOUN
ejpam-109	349	14	2	2	NUM
ejpam-109	349	15	,	,	PUNCT
ejpam-109	349	16	even	even	ADV
ejpam-109	349	17	)	)	PUNCT
ejpam-109	349	18	defined	define	VERB
ejpam-109	349	19	in	in	ADP
ejpam-109	349	20	section	section	NOUN
ejpam-109	349	21	3	3	NUM
ejpam-109	349	22	are	be	AUX
ejpam-109	349	23	the	the	DET
ejpam-109	349	24	moments	moment	NOUN
ejpam-109	349	25	of	of	ADP
ejpam-109	349	26	p	p	NOUN
ejpam-109	349	27	-	-	PUNCT
ejpam-109	349	28	adic	adic	ADJ
ejpam-109	349	29	measures	measure	NOUN
ejpam-109	349	30	on	on	ADP
ejpam-109	349	31	the	the	DET
ejpam-109	349	32	unit	unit	NOUN
ejpam-109	349	33	group	group	NOUN
ejpam-109	349	34	z×p	z×p	PROPN
ejpam-109	349	35	of	of	ADP
ejpam-109	349	36	zp	zp	PROPN
ejpam-109	349	37	and	and	CCONJ
ejpam-109	349	38	this	this	DET
ejpam-109	349	39	realization	realization	NOUN
ejpam-109	349	40	brings	bring	VERB
ejpam-109	349	41	us	we	PRON
ejpam-109	349	42	the	the	DET
ejpam-109	349	43	congruences	congruence	NOUN
ejpam-109	349	44	such	such	ADJ
ejpam-109	349	45	as	as	ADP
ejpam-109	349	46	theorem	theorem	VERB
ejpam-109	349	47	4.1	4.1	NUM
ejpam-109	349	48	(	(	PUNCT
ejpam-109	349	49	cf	cf	NOUN
ejpam-109	349	50	.	.	NOUN
ejpam-109	349	51	,	,	PUNCT
ejpam-109	349	52	e.g.	e.g.	ADV
ejpam-109	349	53	,	,	PUNCT
ejpam-109	349	54	young	young	ADJ
ejpam-109	349	55	[	[	X
ejpam-109	349	56	18	18	NUM
ejpam-109	349	57	,	,	PUNCT
ejpam-109	349	58	19	19	NUM
ejpam-109	349	59	]	]	PUNCT
ejpam-109	349	60	)	)	PUNCT
ejpam-109	349	61	.	.	PUNCT
ejpam-109	350	1	for	for	ADP
ejpam-109	350	2	the	the	DET
ejpam-109	350	3	other	other	ADJ
ejpam-109	350	4	method	method	NOUN
ejpam-109	350	5	obtaining	obtain	VERB
ejpam-109	350	6	extended	extend	VERB
ejpam-109	350	7	voronoï	voronoï	ADJ
ejpam-109	350	8	and	and	CCONJ
ejpam-109	350	9	von	von	PROPN
ejpam-109	350	10	staudt	staudt	PROPN
ejpam-109	350	11	-	-	PUNCT
ejpam-109	350	12	kummer	kummer	NOUN
ejpam-109	350	13	congruences	congruence	NOUN
ejpam-109	350	14	,	,	PUNCT
ejpam-109	350	15	see	see	VERB
ejpam-109	350	16	sun	sun	PROPN
ejpam-109	350	17	’s	’s	PART
ejpam-109	350	18	interesting	interesting	ADJ
ejpam-109	350	19	approach	approach	NOUN
ejpam-109	350	20	in	in	ADP
ejpam-109	350	21	[	[	X
ejpam-109	350	22	16	16	NUM
ejpam-109	350	23	]	]	PUNCT
ejpam-109	350	24	.	.	PUNCT
ejpam-109	351	1	5	5	X
ejpam-109	351	2	.	.	X
ejpam-109	351	3	generalization	generalization	NOUN
ejpam-109	351	4	of	of	ADP
ejpam-109	351	5	lehmer	lehmer	NOUN
ejpam-109	351	6	’s	’s	PART
ejpam-109	351	7	congruences	congruence	NOUN
ejpam-109	351	8	in	in	ADP
ejpam-109	351	9	this	this	DET
ejpam-109	351	10	section	section	NOUN
ejpam-109	351	11	,	,	PUNCT
ejpam-109	351	12	we	we	PRON
ejpam-109	351	13	will	will	AUX
ejpam-109	351	14	extend	extend	VERB
ejpam-109	351	15	lehmer	lehmer	NOUN
ejpam-109	351	16	’s	’s	PART
ejpam-109	351	17	congruences	congruence	NOUN
ejpam-109	351	18	in	in	ADP
ejpam-109	351	19	theorem	theorem	ADJ
ejpam-109	351	20	2.7	2.7	NUM
ejpam-109	351	21	to	to	ADP
ejpam-109	351	22	more	more	ADV
ejpam-109	351	23	general	general	ADJ
ejpam-109	351	24	moduli	modulus	NOUN
ejpam-109	351	25	.	.	PUNCT
ejpam-109	352	1	assume	assume	VERB
ejpam-109	352	2	that	that	SCONJ
ejpam-109	352	3	m	m	PROPN
ejpam-109	352	4	≥	≥	NUM
ejpam-109	352	5	2	2	NUM
ejpam-109	352	6	is	be	AUX
ejpam-109	352	7	even	even	ADV
ejpam-109	352	8	and	and	CCONJ
ejpam-109	352	9	p−	p−	NOUN
ejpam-109	352	10	1	1	NUM
ejpam-109	352	11	m−	m−	PROPN
ejpam-109	352	12	2	2	NUM
ejpam-109	352	13	for	for	ADP
ejpam-109	352	14	all	all	DET
ejpam-109	352	15	prime	prime	ADJ
ejpam-109	352	16	divisors	divisor	NOUN
ejpam-109	352	17	p	p	NOUN
ejpam-109	352	18	of	of	ADP
ejpam-109	352	19	n.	n.	NOUN
ejpam-109	352	20	then	then	ADV
ejpam-109	352	21	we	we	PRON
ejpam-109	352	22	have	have	VERB
ejpam-109	352	23	m	m	PROPN
ejpam-109	352	24	6=	6=	NUM
ejpam-109	352	25	2	2	NUM
ejpam-109	352	26	(	(	PUNCT
ejpam-109	352	27	hence	hence	ADV
ejpam-109	352	28	m	m	VERB
ejpam-109	352	29	≥	≥	NOUN
ejpam-109	352	30	4	4	NUM
ejpam-109	352	31	)	)	PUNCT
ejpam-109	352	32	and	and	CCONJ
ejpam-109	352	33	bm−2	bm−2	PROPN
ejpam-109	352	34	∈	∈	PROPN
ejpam-109	352	35	zp	zp	X
ejpam-109	352	36	.	.	PUNCT
ejpam-109	353	1	also	also	ADV
ejpam-109	353	2	it	it	PRON
ejpam-109	353	3	is	be	AUX
ejpam-109	353	4	clear	clear	ADJ
ejpam-109	353	5	that	that	SCONJ
ejpam-109	353	6	2	2	NUM
ejpam-109	353	7	n	n	NOUN
ejpam-109	353	8	and	and	CCONJ
ejpam-109	353	9	3	3	NUM
ejpam-109	353	10	n	n	CCONJ
ejpam-109	353	11	,	,	PUNCT
ejpam-109	353	12	hence	hence	ADV
ejpam-109	353	13	p	p	X
ejpam-109	353	14	≥	≥	NUM
ejpam-109	353	15	5	5	NUM
ejpam-109	353	16	.	.	PUNCT
ejpam-109	353	17	recall	recall	VERB
ejpam-109	353	18	the	the	DET
ejpam-109	353	19	euler	euler	PROPN
ejpam-109	353	20	-	-	PUNCT
ejpam-109	353	21	maclaurin	maclaurin	NOUN
ejpam-109	353	22	summation	summation	NOUN
ejpam-109	353	23	formula	formula	NOUN
ejpam-109	353	24	(	(	PUNCT
ejpam-109	353	25	theorem	theorem	VERB
ejpam-109	353	26	2.2	2.2	NUM
ejpam-109	353	27	)	)	PUNCT
ejpam-109	353	28	and	and	CCONJ
ejpam-109	353	29	observe	observe	VERB
ejpam-109	353	30	each	each	DET
ejpam-109	353	31	term	term	NOUN
ejpam-109	353	32	on	on	ADP
ejpam-109	353	33	the	the	DET
ejpam-109	353	34	right	right	ADJ
ejpam-109	353	35	-	-	PUNCT
ejpam-109	353	36	hand	hand	NOUN
ejpam-109	353	37	side	side	NOUN
ejpam-109	353	38	of	of	ADP
ejpam-109	353	39	this	this	DET
ejpam-109	353	40	formula	formula	NOUN
ejpam-109	353	41	.	.	PUNCT
ejpam-109	354	1	for	for	ADP
ejpam-109	354	2	simplicity	simplicity	NOUN
ejpam-109	354	3	,	,	PUNCT
ejpam-109	354	4	we	we	PRON
ejpam-109	354	5	put	put	VERB
ejpam-109	354	6	x	x	PUNCT
ejpam-109	354	7	j	j	NOUN
ejpam-109	354	8	=	=	SYM
ejpam-109	354	9	1	1	NUM
ejpam-109	354	10	j	j	PROPN
ejpam-109	354	11	�	�	PROPN
ejpam-109	354	12	m	m	VERB
ejpam-109	354	13	j−	j−	PROPN
ejpam-109	354	14	1	1	NUM
ejpam-109	354	15	�	�	PROPN
ejpam-109	354	16	bm+1−	bm+1−	PROPN
ejpam-109	354	17	jn	jn	PROPN
ejpam-109	354	18	j	j	PROPN
ejpam-109	354	19	for	for	ADP
ejpam-109	354	20	j	j	PROPN
ejpam-109	354	21	=	=	SYM
ejpam-109	354	22	1	1	NUM
ejpam-109	354	23	,	,	PUNCT
ejpam-109	354	24	2	2	NUM
ejpam-109	354	25	,	,	PUNCT
ejpam-109	354	26	...	...	PUNCT
ejpam-109	354	27	,	,	PUNCT
ejpam-109	354	28	m+	m+	NOUN
ejpam-109	354	29	1	1	NUM
ejpam-109	354	30	.	.	PUNCT
ejpam-109	355	1	t.	t.	PROPN
ejpam-109	355	2	agoh	agoh	PROPN
ejpam-109	355	3	/	/	SYM
ejpam-109	355	4	eur	eur	PROPN
ejpam-109	355	5	.	.	PUNCT
ejpam-109	356	1	j.	j.	PROPN
ejpam-109	356	2	pure	pure	PROPN
ejpam-109	356	3	appl	appl	PROPN
ejpam-109	356	4	.	.	PROPN
ejpam-109	356	5	math	math	PROPN
ejpam-109	356	6	,	,	PUNCT
ejpam-109	356	7	1	1	NUM
ejpam-109	356	8	(	(	PUNCT
ejpam-109	356	9	2008	2008	NUM
ejpam-109	356	10	)	)	PUNCT
ejpam-109	356	11	,	,	PUNCT
ejpam-109	356	12	(	(	PUNCT
ejpam-109	356	13	3	3	NUM
ejpam-109	356	14	-	-	SYM
ejpam-109	356	15	21	21	NUM
ejpam-109	356	16	)	)	PUNCT
ejpam-109	356	17	17	17	NUM
ejpam-109	356	18	if	if	SCONJ
ejpam-109	356	19	j	j	PROPN
ejpam-109	356	20	is	be	AUX
ejpam-109	356	21	an	an	DET
ejpam-109	356	22	even	even	ADV
ejpam-109	356	23	integer	integer	NOUN
ejpam-109	356	24	with	with	ADP
ejpam-109	356	25	m−	m−	PROPN
ejpam-109	356	26	2	2	NUM
ejpam-109	356	27	≥	≥	NOUN
ejpam-109	356	28	j	j	PROPN
ejpam-109	356	29	≥	≥	NUM
ejpam-109	356	30	2	2	NUM
ejpam-109	356	31	,	,	PUNCT
ejpam-109	356	32	then	then	ADV
ejpam-109	356	33	x	x	X
ejpam-109	356	34	j	j	PROPN
ejpam-109	356	35	=	=	NOUN
ejpam-109	356	36	0	0	PROPN
ejpam-109	356	37	.	.	PUNCT
ejpam-109	357	1	so	so	ADV
ejpam-109	357	2	we	we	PRON
ejpam-109	357	3	observe	observe	VERB
ejpam-109	357	4	here	here	ADV
ejpam-109	357	5	only	only	ADV
ejpam-109	357	6	the	the	DET
ejpam-109	357	7	terms	term	NOUN
ejpam-109	357	8	x	x	X
ejpam-109	357	9	j	j	NOUN
ejpam-109	357	10	with	with	ADP
ejpam-109	357	11	j	j	PROPN
ejpam-109	357	12	=	=	SYM
ejpam-109	358	1	2i+	2i+	NUM
ejpam-109	358	2	1	1	NUM
ejpam-109	358	3	(	(	PUNCT
ejpam-109	358	4	i	i	NOUN
ejpam-109	358	5	=	=	NOUN
ejpam-109	358	6	0,1	0,1	NUM
ejpam-109	358	7	,	,	PUNCT
ejpam-109	358	8	...	...	PUNCT
ejpam-109	358	9	,	,	PUNCT
ejpam-109	358	10	m/2	m/2	NUM
ejpam-109	358	11	)	)	PUNCT
ejpam-109	358	12	and	and	CCONJ
ejpam-109	358	13	j	j	PROPN
ejpam-109	358	14	=	=	SYM
ejpam-109	358	15	m.	m.	NOUN
ejpam-109	358	16	let	let	VERB
ejpam-109	358	17	p	p	PRON
ejpam-109	358	18	be	be	AUX
ejpam-109	358	19	any	any	DET
ejpam-109	358	20	prime	prime	ADJ
ejpam-109	358	21	divisor	divisor	NOUN
ejpam-109	358	22	of	of	ADP
ejpam-109	358	23	n.	n.	NOUN
ejpam-109	358	24	if	if	SCONJ
ejpam-109	358	25	m	m	VERB
ejpam-109	358	26	>	>	X
ejpam-109	358	27	2i	2i	NUM
ejpam-109	358	28	≥	≥	NOUN
ejpam-109	358	29	4	4	NUM
ejpam-109	358	30	,	,	PUNCT
ejpam-109	358	31	then	then	ADV
ejpam-109	358	32	ordp(dm−2i	ordp(dm−2i	X
ejpam-109	358	33	)	)	PUNCT
ejpam-109	358	34	∈	∈	PROPN
ejpam-109	358	35	{	{	PUNCT
ejpam-109	358	36	0,1	0,1	NOUN
ejpam-109	358	37	}	}	PUNCT
ejpam-109	358	38	and	and	CCONJ
ejpam-109	358	39	p2i−3	p2i−3	PROPN
ejpam-109	358	40	≥	≥	NOUN
ejpam-109	358	41	52i−3	52i−3	NUM
ejpam-109	358	42	≥	≥	NOUN
ejpam-109	358	43	2i	2i	NOUN
ejpam-109	358	44	+	+	CCONJ
ejpam-109	358	45	1	1	NUM
ejpam-109	358	46	,	,	PUNCT
ejpam-109	358	47	hence	hence	ADV
ejpam-109	358	48	ordp	ordp	PROPN
ejpam-109	358	49	�	�	PROPN
ejpam-109	358	50	x2i+1	x2i+1	PROPN
ejpam-109	358	51	�	�	PROPN
ejpam-109	358	52	≥	≥	NUM
ejpam-109	358	53	−(2i	−(2i	NUM
ejpam-109	358	54	−	−	NOUN
ejpam-109	358	55	3	3	NUM
ejpam-109	358	56	)	)	PUNCT
ejpam-109	358	57	−	−	NOUN
ejpam-109	358	58	1	1	NUM
ejpam-109	358	59	+	+	CCONJ
ejpam-109	358	60	(	(	PUNCT
ejpam-109	358	61	2i	2i	NUM
ejpam-109	358	62	+	+	CCONJ
ejpam-109	358	63	1	1	X
ejpam-109	358	64	)	)	PUNCT
ejpam-109	358	65	=	=	SYM
ejpam-109	359	1	3	3	X
ejpam-109	359	2	.	.	PUNCT
ejpam-109	360	1	this	this	PRON
ejpam-109	360	2	gives	give	VERB
ejpam-109	360	3	x2i+1	x2i+1	PROPN
ejpam-109	360	4	≡	≡	PROPN
ejpam-109	360	5	0	0	PUNCT
ejpam-109	361	1	(	(	PUNCT
ejpam-109	361	2	mod	mod	PROPN
ejpam-109	361	3	n3	n3	PROPN
ejpam-109	361	4	)	)	PUNCT
ejpam-109	361	5	for	for	ADP
ejpam-109	361	6	all	all	PRON
ejpam-109	361	7	i	i	PRON
ejpam-109	361	8	≥	≥	VERB
ejpam-109	361	9	2	2	NUM
ejpam-109	361	10	.	.	PUNCT
ejpam-109	362	1	also	also	ADV
ejpam-109	362	2	,	,	PUNCT
ejpam-109	362	3	since	since	SCONJ
ejpam-109	362	4	ordp(dm−2	ordp(dm−2	ADV
ejpam-109	362	5	)	)	PUNCT
ejpam-109	362	6	=	=	SYM
ejpam-109	362	7	0	0	NUM
ejpam-109	362	8	by	by	ADP
ejpam-109	362	9	the	the	DET
ejpam-109	362	10	assumption	assumption	NOUN
ejpam-109	362	11	,	,	PUNCT
ejpam-109	362	12	m≥	m≥	NOUN
ejpam-109	362	13	4	4	NUM
ejpam-109	362	14	and	and	CCONJ
ejpam-109	362	15	(	(	PUNCT
ejpam-109	362	16	n	n	CCONJ
ejpam-109	362	17	,	,	PUNCT
ejpam-109	362	18	6	6	NUM
ejpam-109	362	19	)	)	PUNCT
ejpam-109	362	20	=	=	SYM
ejpam-109	362	21	1	1	NUM
ejpam-109	362	22	,	,	PUNCT
ejpam-109	362	23	it	it	PRON
ejpam-109	362	24	follows	follow	VERB
ejpam-109	362	25	that	that	SCONJ
ejpam-109	363	1	x3	x3	ADJ
ejpam-109	363	2	=	=	SYM
ejpam-109	363	3	1	1	NUM
ejpam-109	363	4	3	3	NUM
ejpam-109	363	5	�	�	PROPN
ejpam-109	363	6	m	m	VERB
ejpam-109	363	7	2	2	NUM
ejpam-109	363	8	�	�	PROPN
ejpam-109	363	9	bm−2n3	bm−2n3	NOUN
ejpam-109	363	10	≡	≡	PROPN
ejpam-109	363	11	0	0	PUNCT
ejpam-109	364	1	(	(	PUNCT
ejpam-109	364	2	mod	mod	PROPN
ejpam-109	364	3	n3	n3	PROPN
ejpam-109	364	4	)	)	PUNCT
ejpam-109	364	5	and	and	CCONJ
ejpam-109	364	6	xm	xm	X
ejpam-109	364	7	=	=	SYM
ejpam-109	364	8	1	1	NUM
ejpam-109	364	9	m	m	NOUN
ejpam-109	364	10	�	�	PROPN
ejpam-109	364	11	m	m	NOUN
ejpam-109	364	12	m−1	m−1	PROPN
ejpam-109	364	13	�	�	PROPN
ejpam-109	364	14	b1	b1	PROPN
ejpam-109	364	15	nm	nm	ADJ
ejpam-109	365	1	=	=	NOUN
ejpam-109	365	2	−1	−1	NOUN
ejpam-109	365	3	2	2	NUM
ejpam-109	365	4	nm	nm	NOUN
ejpam-109	365	5	≡	≡	PROPN
ejpam-109	365	6	0	0	PUNCT
ejpam-109	366	1	(	(	PUNCT
ejpam-109	366	2	mod	mod	PROPN
ejpam-109	366	3	n3	n3	PROPN
ejpam-109	366	4	)	)	PUNCT
ejpam-109	366	5	.	.	PUNCT
ejpam-109	367	1	consequently	consequently	ADV
ejpam-109	367	2	,	,	PUNCT
ejpam-109	367	3	we	we	PRON
ejpam-109	367	4	get	get	VERB
ejpam-109	367	5	from	from	ADP
ejpam-109	367	6	theorem	theorem	ADJ
ejpam-109	367	7	2.2	2.2	NUM
ejpam-109	367	8	sm(n)≡	sm(n)≡	NUM
ejpam-109	367	9	bmn	bmn	NOUN
ejpam-109	367	10	(	(	PUNCT
ejpam-109	367	11	mod	mod	PROPN
ejpam-109	367	12	n3	n3	PROPN
ejpam-109	367	13	)	)	PUNCT
ejpam-109	367	14	.	.	PUNCT
ejpam-109	368	1	on	on	ADP
ejpam-109	368	2	the	the	DET
ejpam-109	368	3	other	other	ADJ
ejpam-109	368	4	hand	hand	NOUN
ejpam-109	368	5	,	,	PUNCT
ejpam-109	368	6	we	we	PRON
ejpam-109	368	7	obtain	obtain	VERB
ejpam-109	368	8	from	from	ADP
ejpam-109	368	9	(	(	PUNCT
ejpam-109	368	10	2.1	2.1	NUM
ejpam-109	368	11	)	)	PUNCT
ejpam-109	368	12	that	that	SCONJ
ejpam-109	369	1	if	if	SCONJ
ejpam-109	369	2	(	(	PUNCT
ejpam-109	369	3	a	a	PRON
ejpam-109	369	4	,	,	PUNCT
ejpam-109	369	5	n	n	CCONJ
ejpam-109	369	6	)	)	PUNCT
ejpam-109	369	7	=	=	SYM
ejpam-109	369	8	1	1	NUM
ejpam-109	369	9	,	,	PUNCT
ejpam-109	369	10	a	a	DET
ejpam-109	369	11	≥	≥	NOUN
ejpam-109	369	12	1	1	NUM
ejpam-109	369	13	,	,	PUNCT
ejpam-109	369	14	then	then	ADV
ejpam-109	369	15	(	(	PUNCT
ejpam-109	369	16	am−	am−	NUM
ejpam-109	369	17	1)sm(n)≡	1)sm(n)≡	NUM
ejpam-109	369	18	mn	mn	PROPN
ejpam-109	369	19	n−1	n−1	PROPN
ejpam-109	369	20	∑	∑	PUNCT
ejpam-109	369	21	j=1	j=1	PROPN
ejpam-109	369	22	(	(	PUNCT
ejpam-109	369	23	a	a	DET
ejpam-109	369	24	j)m−1	j)m−1	PROPN
ejpam-109	369	25	�	�	PROPN
ejpam-109	369	26	a	a	DET
ejpam-109	369	27	j	j	PROPN
ejpam-109	369	28	n	n	PRON
ejpam-109	369	29	�	�	PROPN
ejpam-109	369	30	−	−	PROPN
ejpam-109	369	31	m(m−	m(m−	PROPN
ejpam-109	369	32	1	1	NUM
ejpam-109	369	33	)	)	PUNCT
ejpam-109	369	34	2	2	NUM
ejpam-109	369	35	n2	n2	NOUN
ejpam-109	369	36	n−1	n−1	PROPN
ejpam-109	369	37	∑	∑	PUNCT
ejpam-109	369	38	j=1	j=1	PROPN
ejpam-109	369	39	(	(	PUNCT
ejpam-109	369	40	a	a	DET
ejpam-109	369	41	j)m−2	j)m−2	PROPN
ejpam-109	369	42	�	�	PROPN
ejpam-109	369	43	a	a	DET
ejpam-109	369	44	j	j	PROPN
ejpam-109	369	45	n	n	X
ejpam-109	369	46	�	�	PROPN
ejpam-109	369	47	2	2	NUM
ejpam-109	369	48	(	(	PUNCT
ejpam-109	369	49	mod	mod	PROPN
ejpam-109	369	50	n3	n3	PROPN
ejpam-109	369	51	)	)	PUNCT
ejpam-109	369	52	.	.	PUNCT
ejpam-109	370	1	consequently	consequently	ADV
ejpam-109	370	2	,	,	PUNCT
ejpam-109	370	3	we	we	PRON
ejpam-109	370	4	have	have	VERB
ejpam-109	370	5	(	(	PUNCT
ejpam-109	370	6	am−	am−	NUM
ejpam-109	370	7	1)bm	1)bm	NUM
ejpam-109	370	8	≡	≡	PROPN
ejpam-109	370	9	m	m	VERB
ejpam-109	370	10	n−1	n−1	ADJ
ejpam-109	370	11	∑	∑	INTJ
ejpam-109	370	12	j=1	j=1	PROPN
ejpam-109	370	13	(	(	PUNCT
ejpam-109	370	14	a	a	DET
ejpam-109	370	15	j)m−1	j)m−1	PROPN
ejpam-109	370	16	�	�	PROPN
ejpam-109	370	17	a	a	DET
ejpam-109	370	18	j	j	PROPN
ejpam-109	370	19	n	n	PRON
ejpam-109	370	20	�	�	PROPN
ejpam-109	370	21	−	−	PROPN
ejpam-109	370	22	m(m−	m(m−	PROPN
ejpam-109	370	23	1	1	NUM
ejpam-109	370	24	)	)	SYM
ejpam-109	370	25	2	2	NUM
ejpam-109	370	26	n	n	NUM
ejpam-109	370	27	n−1	n−1	PROPN
ejpam-109	370	28	∑	∑	PUNCT
ejpam-109	370	29	j=1	j=1	PROPN
ejpam-109	370	30	(	(	PUNCT
ejpam-109	370	31	a	a	DET
ejpam-109	370	32	j)m−2	j)m−2	PROPN
ejpam-109	370	33	�	�	PROPN
ejpam-109	370	34	a	a	DET
ejpam-109	370	35	j	j	PROPN
ejpam-109	370	36	n	n	X
ejpam-109	370	37	�	�	PROPN
ejpam-109	370	38	2	2	NUM
ejpam-109	370	39	(	(	PUNCT
ejpam-109	370	40	mod	mod	PROPN
ejpam-109	370	41	n2	n2	PROPN
ejpam-109	370	42	)	)	PUNCT
ejpam-109	370	43	.	.	PUNCT
ejpam-109	371	1	let	let	VERB
ejpam-109	371	2	δ	δ	PRON
ejpam-109	371	3	be	be	AUX
ejpam-109	371	4	as	as	ADP
ejpam-109	371	5	in	in	ADP
ejpam-109	371	6	section	section	NOUN
ejpam-109	371	7	2	2	NUM
ejpam-109	371	8	and	and	CCONJ
ejpam-109	371	9	set	set	VERB
ejpam-109	371	10	η′	η′	NOUN
ejpam-109	371	11	=	=	SYM
ejpam-109	371	12	nδ	nδ	PROPN
ejpam-109	371	13	.	.	PUNCT
ejpam-109	372	1	since	since	SCONJ
ejpam-109	372	2	(	(	PUNCT
ejpam-109	372	3	a	a	PRON
ejpam-109	372	4	,	,	PUNCT
ejpam-109	372	5	η′	η′	NUM
ejpam-109	372	6	)	)	PUNCT
ejpam-109	372	7	=	=	SYM
ejpam-109	372	8	1	1	NUM
ejpam-109	372	9	,	,	PUNCT
ejpam-109	372	10	we	we	PRON
ejpam-109	372	11	may	may	AUX
ejpam-109	372	12	consider	consider	VERB
ejpam-109	372	13	the	the	DET
ejpam-109	372	14	above	above	ADJ
ejpam-109	372	15	congruence	congruence	NOUN
ejpam-109	372	16	replaced	replace	VERB
ejpam-109	372	17	n	n	INTJ
ejpam-109	372	18	by	by	ADP
ejpam-109	372	19	η′.	η′.	NOUN
ejpam-109	372	20	that	that	PRON
ejpam-109	372	21	is	be	AUX
ejpam-109	372	22	,	,	PUNCT
ejpam-109	372	23	(	(	PUNCT
ejpam-109	372	24	am−	am−	NUM
ejpam-109	372	25	1)bm	1)bm	NUM
ejpam-109	372	26	≡	≡	PROPN
ejpam-109	372	27	m	m	VERB
ejpam-109	372	28	η′−1	η′−1	VERB
ejpam-109	372	29	∑	∑	PUNCT
ejpam-109	372	30	j=1	j=1	PROPN
ejpam-109	372	31	(	(	PUNCT
ejpam-109	372	32	a	a	DET
ejpam-109	372	33	j)m−1	j)m−1	PROPN
ejpam-109	372	34	�	�	PROPN
ejpam-109	372	35	a	a	DET
ejpam-109	372	36	j	j	PROPN
ejpam-109	372	37	η′	η′	PROPN
ejpam-109	372	38	�	�	PROPN
ejpam-109	372	39	−	−	NOUN
ejpam-109	372	40	m(m−	m(m−	PROPN
ejpam-109	372	41	1	1	NUM
ejpam-109	372	42	)	)	SYM
ejpam-109	372	43	2	2	NUM
ejpam-109	372	44	η′	η′	NOUN
ejpam-109	372	45	η′−1	η′−1	VERB
ejpam-109	372	46	∑	∑	PUNCT
ejpam-109	372	47	j=1	j=1	PROPN
ejpam-109	372	48	(	(	PUNCT
ejpam-109	372	49	a	a	DET
ejpam-109	372	50	j)m−2	j)m−2	PROPN
ejpam-109	372	51	�	�	PROPN
ejpam-109	372	52	a	a	DET
ejpam-109	372	53	j	j	PROPN
ejpam-109	372	54	η′	η′	X
ejpam-109	372	55	�	�	PROPN
ejpam-109	372	56	2	2	NUM
ejpam-109	372	57	(	(	PUNCT
ejpam-109	372	58	mod	mod	PROPN
ejpam-109	372	59	η	η	PROPN
ejpam-109	372	60	′2	′2	PROPN
ejpam-109	372	61	)	)	PUNCT
ejpam-109	372	62	,	,	PUNCT
ejpam-109	372	63	which	which	PRON
ejpam-109	372	64	implies	imply	VERB
ejpam-109	372	65	(	(	PUNCT
ejpam-109	372	66	am−	am−	NUM
ejpam-109	372	67	1)βm	1)βm	NUM
ejpam-109	372	68	≡	≡	PROPN
ejpam-109	372	69	η′−1	η′−1	VERB
ejpam-109	372	70	∑	∑	PUNCT
ejpam-109	372	71	j=1	j=1	PROPN
ejpam-109	372	72	(	(	PUNCT
ejpam-109	372	73	a	a	DET
ejpam-109	372	74	j)m−1	j)m−1	PROPN
ejpam-109	372	75	�	�	PROPN
ejpam-109	372	76	a	a	DET
ejpam-109	372	77	j	j	PROPN
ejpam-109	372	78	η′	η′	PROPN
ejpam-109	372	79	�	�	PROPN
ejpam-109	372	80	−	−	PROPN
ejpam-109	372	81	m−	m−	PROPN
ejpam-109	372	82	1	1	NUM
ejpam-109	372	83	2	2	NUM
ejpam-109	372	84	η′	η′	NOUN
ejpam-109	372	85	η′−1	η′−1	VERB
ejpam-109	372	86	∑	∑	PUNCT
ejpam-109	372	87	j=1	j=1	PROPN
ejpam-109	372	88	(	(	PUNCT
ejpam-109	372	89	a	a	DET
ejpam-109	372	90	j)m−2	j)m−2	PROPN
ejpam-109	372	91	�	�	PROPN
ejpam-109	372	92	a	a	DET
ejpam-109	372	93	j	j	PROPN
ejpam-109	372	94	η′	η′	X
ejpam-109	372	95	�	�	PROPN
ejpam-109	372	96	2	2	NUM
ejpam-109	372	97	(	(	PUNCT
ejpam-109	372	98	mod	mod	PROPN
ejpam-109	372	99	n2	n2	PROPN
ejpam-109	372	100	)	)	PUNCT
ejpam-109	372	101	.	.	PUNCT
ejpam-109	373	1	(	(	PUNCT
ejpam-109	373	2	5.1	5.1	NUM
ejpam-109	373	3	)	)	PUNCT
ejpam-109	373	4	putting	putting	NOUN
ejpam-109	373	5	,	,	PUNCT
ejpam-109	373	6	for	for	ADP
ejpam-109	373	7	i	i	PROPN
ejpam-109	373	8	=	=	SYM
ejpam-109	373	9	0	0	NUM
ejpam-109	373	10	,	,	PUNCT
ejpam-109	373	11	1	1	NUM
ejpam-109	373	12	,	,	PUNCT
ejpam-109	373	13	...	...	PUNCT
ejpam-109	373	14	,	,	PUNCT
ejpam-109	373	15	a−	a−	PROPN
ejpam-109	373	16	1	1	NUM
ejpam-109	373	17	,	,	PUNCT
ejpam-109	373	18	g(a)i	g(a)i	PROPN
ejpam-109	373	19	=	=	PUNCT
ejpam-109	373	20	¨	¨	PROPN
ejpam-109	373	21	j	j	PROPN
ejpam-109	373	22	∈	∈	PROPN
ejpam-109	373	23	z	z	NOUN
ejpam-109	373	24	|	|	ADV
ejpam-109	373	25	iη′	iη′	VERB
ejpam-109	373	26	a	a	PRON
ejpam-109	373	27	<	<	X
ejpam-109	373	28	j	j	X
ejpam-109	373	29	<	<	X
ejpam-109	373	30	(	(	PUNCT
ejpam-109	373	31	i+	i+	PROPN
ejpam-109	373	32	1)η′	1)η′	PRON
ejpam-109	373	33	a	a	PRON
ejpam-109	373	34	«	«	PUNCT
ejpam-109	373	35	,	,	PUNCT
ejpam-109	373	36	and	and	CCONJ
ejpam-109	373	37	u	u	NOUN
ejpam-109	373	38	(	(	PUNCT
ejpam-109	373	39	a)i	a)i	X
ejpam-109	373	40	=	=	PUNCT
ejpam-109	373	41	∑	∑	PUNCT
ejpam-109	373	42	j∈g(a)i	j∈g(a)i	PROPN
ejpam-109	373	43	jm−1	jm−1	PROPN
ejpam-109	373	44	,	,	PUNCT
ejpam-109	373	45	v	v	NOUN
ejpam-109	373	46	(	(	PUNCT
ejpam-109	373	47	a)i	a)i	X
ejpam-109	373	48	=	=	PUNCT
ejpam-109	373	49	∑	∑	PUNCT
ejpam-109	373	50	j∈g(a)i	j∈g(a)i	PROPN
ejpam-109	373	51	jm−2	jm−2	PROPN
ejpam-109	373	52	,	,	PUNCT
ejpam-109	373	53	above	above	ADV
ejpam-109	373	54	(	(	PUNCT
ejpam-109	373	55	5.1	5.1	NUM
ejpam-109	373	56	)	)	PUNCT
ejpam-109	373	57	can	can	AUX
ejpam-109	373	58	be	be	AUX
ejpam-109	373	59	expressed	express	VERB
ejpam-109	373	60	as	as	ADP
ejpam-109	373	61	(	(	PUNCT
ejpam-109	373	62	am−	am−	NUM
ejpam-109	373	63	1)βm	1)βm	PROPN
ejpam-109	373	64	≡	≡	PROPN
ejpam-109	373	65	am−1	am−1	PROPN
ejpam-109	373	66	a−1	a−1	PROPN
ejpam-109	373	67	∑	∑	PROPN
ejpam-109	373	68	i=1	i=1	PROPN
ejpam-109	373	69	iu	iu	ADV
ejpam-109	373	70	(	(	PUNCT
ejpam-109	373	71	a)i	a)i	X
ejpam-109	373	72	−	−	PROPN
ejpam-109	373	73	m−	m−	PROPN
ejpam-109	373	74	1	1	NUM
ejpam-109	373	75	2	2	NUM
ejpam-109	373	76	am−2η′	am−2η′	ADP
ejpam-109	373	77	a−1	a−1	NOUN
ejpam-109	373	78	∑	∑	PROPN
ejpam-109	373	79	i=1	i=1	PROPN
ejpam-109	373	80	i2v	i2v	PRON
ejpam-109	373	81	(	(	PUNCT
ejpam-109	373	82	a)i	a)i	X
ejpam-109	373	83	(	(	PUNCT
ejpam-109	373	84	mod	mod	PROPN
ejpam-109	373	85	n2	n2	PROPN
ejpam-109	373	86	)	)	PUNCT
ejpam-109	373	87	.	.	PUNCT
ejpam-109	374	1	(	(	PUNCT
ejpam-109	374	2	5.2	5.2	NUM
ejpam-109	374	3	)	)	PUNCT
ejpam-109	374	4	t.	t.	NOUN
ejpam-109	374	5	agoh	agoh	PROPN
ejpam-109	374	6	/	/	SYM
ejpam-109	374	7	eur	eur	PROPN
ejpam-109	374	8	.	.	PUNCT
ejpam-109	375	1	j.	j.	PROPN
ejpam-109	375	2	pure	pure	PROPN
ejpam-109	375	3	appl	appl	PROPN
ejpam-109	375	4	.	.	PROPN
ejpam-109	375	5	math	math	PROPN
ejpam-109	375	6	,	,	PUNCT
ejpam-109	375	7	1	1	NUM
ejpam-109	375	8	(	(	PUNCT
ejpam-109	375	9	2008	2008	NUM
ejpam-109	375	10	)	)	PUNCT
ejpam-109	375	11	,	,	PUNCT
ejpam-109	375	12	(	(	PUNCT
ejpam-109	375	13	3	3	NUM
ejpam-109	375	14	-	-	SYM
ejpam-109	375	15	21	21	NUM
ejpam-109	375	16	)	)	PUNCT
ejpam-109	375	17	18	18	NUM
ejpam-109	375	18	on	on	ADP
ejpam-109	375	19	the	the	DET
ejpam-109	375	20	other	other	ADJ
ejpam-109	375	21	hand	hand	NOUN
ejpam-109	375	22	,	,	PUNCT
ejpam-109	375	23	since	since	SCONJ
ejpam-109	375	24	m	m	PROPN
ejpam-109	375	25	is	be	AUX
ejpam-109	375	26	even	even	ADV
ejpam-109	375	27	,	,	PUNCT
ejpam-109	375	28	it	it	PRON
ejpam-109	375	29	follows	follow	VERB
ejpam-109	375	30	that	that	SCONJ
ejpam-109	375	31	u	u	PROPN
ejpam-109	375	32	(	(	PUNCT
ejpam-109	375	33	a)i	a)i	X
ejpam-109	375	34	=	=	SYM
ejpam-109	375	35	∑	∑	PUNCT
ejpam-109	375	36	j∈g(a)a−1−i	j∈g(a)a−1−i	X
ejpam-109	375	37	(	(	PUNCT
ejpam-109	375	38	η′−	η′−	NUM
ejpam-109	375	39	j)m−1	j)m−1	PROPN
ejpam-109	375	40	≡−	≡−	PROPN
ejpam-109	375	41	∑	∑	PROPN
ejpam-109	375	42	j∈g(a)a−1−i	j∈g(a)a−1−i	ADP
ejpam-109	375	43	jm−1	jm−1	PROPN
ejpam-109	375	44	+	+	CCONJ
ejpam-109	375	45	(	(	PUNCT
ejpam-109	375	46	m−	m−	PROPN
ejpam-109	375	47	1)η′	1)η′	NUM
ejpam-109	375	48	∑	∑	PUNCT
ejpam-109	375	49	j∈g(a)a−1−i	j∈g(a)a−1−i	PRON
ejpam-109	375	50	jm−2	jm−2	PROPN
ejpam-109	375	51	≡−	≡−	ADP
ejpam-109	375	52	u	u	NOUN
ejpam-109	375	53	(	(	PUNCT
ejpam-109	375	54	a)a−1−i	a)a−1−i	PROPN
ejpam-109	375	55	+	+	X
ejpam-109	375	56	(	(	PUNCT
ejpam-109	375	57	m−	m−	PROPN
ejpam-109	375	58	1)η′v	1)η′v	PROPN
ejpam-109	375	59	(	(	PUNCT
ejpam-109	375	60	a)a−1−i	a)a−1−i	PROPN
ejpam-109	375	61	(	(	PUNCT
ejpam-109	375	62	mod	mod	PROPN
ejpam-109	375	63	n2	n2	PROPN
ejpam-109	375	64	)	)	PUNCT
ejpam-109	375	65	.	.	PUNCT
ejpam-109	376	1	we	we	PRON
ejpam-109	376	2	also	also	ADV
ejpam-109	376	3	have	have	VERB
ejpam-109	376	4	v	v	NOUN
ejpam-109	376	5	(	(	PUNCT
ejpam-109	376	6	a)i	a)i	X
ejpam-109	376	7	=	=	SYM
ejpam-109	376	8	∑	∑	PUNCT
ejpam-109	376	9	j∈g(a)a−1−i	j∈g(a)a−1−i	X
ejpam-109	376	10	(	(	PUNCT
ejpam-109	376	11	η′−	η′−	NOUN
ejpam-109	376	12	j)m−2	j)m−2	PROPN
ejpam-109	376	13	≡	≡	PROPN
ejpam-109	376	14	∑	∑	PROPN
ejpam-109	377	1	j∈g(a)a−1−i	j∈g(a)a−1−i	PROPN
ejpam-109	377	2	jm−2	jm−2	PROPN
ejpam-109	377	3	≡	≡	PROPN
ejpam-109	377	4	v	v	ADP
ejpam-109	377	5	(	(	PUNCT
ejpam-109	377	6	a)a−1−i	a)a−1−i	INTJ
ejpam-109	377	7	(	(	PUNCT
ejpam-109	377	8	mod	mod	NOUN
ejpam-109	377	9	n	n	CCONJ
ejpam-109	377	10	)	)	PUNCT
ejpam-109	377	11	.	.	PUNCT
ejpam-109	378	1	by	by	ADP
ejpam-109	378	2	theorem	theorem	NOUN
ejpam-109	378	3	2.2	2.2	NUM
ejpam-109	378	4	we	we	PRON
ejpam-109	378	5	see	see	VERB
ejpam-109	378	6	,	,	PUNCT
ejpam-109	378	7	since	since	SCONJ
ejpam-109	378	8	bm−2	bm−2	PROPN
ejpam-109	378	9	∈	∈	PROPN
ejpam-109	378	10	zp	zp	X
ejpam-109	378	11	for	for	ADP
ejpam-109	378	12	any	any	DET
ejpam-109	378	13	prime	prime	ADJ
ejpam-109	378	14	divisor	divisor	NOUN
ejpam-109	378	15	p	p	NOUN
ejpam-109	378	16	of	of	ADP
ejpam-109	378	17	n	n	PRON
ejpam-109	378	18	such	such	ADJ
ejpam-109	378	19	that	that	SCONJ
ejpam-109	378	20	p−	p−	NOUN
ejpam-109	378	21	1	1	NUM
ejpam-109	378	22	m−	m−	PROPN
ejpam-109	378	23	2	2	NUM
ejpam-109	378	24	,	,	PUNCT
ejpam-109	378	25	sm−2(η	sm−2(η	NOUN
ejpam-109	378	26	′	′	NOUN
ejpam-109	378	27	)	)	PUNCT
ejpam-109	379	1	=	=	SYM
ejpam-109	379	2	a−1	a−1	PROPN
ejpam-109	379	3	∑	∑	PUNCT
ejpam-109	379	4	i=0	i=0	PROPN
ejpam-109	379	5	v	v	X
ejpam-109	379	6	(	(	PUNCT
ejpam-109	379	7	a)i	a)i	X
ejpam-109	379	8	≡	≡	PROPN
ejpam-109	379	9	0	0	PUNCT
ejpam-109	380	1	(	(	PUNCT
ejpam-109	380	2	mod	mod	NOUN
ejpam-109	380	3	n	n	CCONJ
ejpam-109	380	4	)	)	PUNCT
ejpam-109	380	5	.	.	PUNCT
ejpam-109	381	1	we	we	PRON
ejpam-109	381	2	are	be	AUX
ejpam-109	381	3	now	now	ADV
ejpam-109	381	4	able	able	ADJ
ejpam-109	381	5	to	to	PART
ejpam-109	381	6	prove	prove	VERB
ejpam-109	381	7	the	the	DET
ejpam-109	381	8	following	follow	VERB
ejpam-109	381	9	generalizations	generalization	NOUN
ejpam-109	381	10	of	of	ADP
ejpam-109	381	11	lehmer	lehmer	NOUN
ejpam-109	381	12	’s	’s	PART
ejpam-109	381	13	congruences	congruence	NOUN
ejpam-109	381	14	stated	state	VERB
ejpam-109	381	15	in	in	ADP
ejpam-109	381	16	theorem	theorem	ADJ
ejpam-109	381	17	2.7	2.7	NUM
ejpam-109	381	18	.	.	PUNCT
ejpam-109	382	1	theorem	theorem	VERB
ejpam-109	382	2	5.1	5.1	NUM
ejpam-109	382	3	.	.	PUNCT
ejpam-109	383	1	let	let	VERB
ejpam-109	383	2	n	n	PRON
ejpam-109	383	3	be	be	AUX
ejpam-109	383	4	an	an	DET
ejpam-109	383	5	odd	odd	ADJ
ejpam-109	383	6	integer	integer	NOUN
ejpam-109	383	7	,	,	PUNCT
ejpam-109	383	8	m	m	VERB
ejpam-109	383	9	an	an	PRON
ejpam-109	383	10	even	even	ADV
ejpam-109	383	11	integer	integer	NOUN
ejpam-109	383	12	≥	≥	NOUN
ejpam-109	383	13	2	2	NUM
ejpam-109	383	14	,	,	PUNCT
ejpam-109	383	15	η′	η′	PROPN
ejpam-109	383	16	=	=	SYM
ejpam-109	383	17	nδ	nδ	NOUN
ejpam-109	383	18	,	,	PUNCT
ejpam-109	383	19	and	and	CCONJ
ejpam-109	383	20	let	let	VERB
ejpam-109	383	21	qk(m	qk(m	NUM
ejpam-109	383	22	)	)	PUNCT
ejpam-109	384	1	(	(	PUNCT
ejpam-109	384	2	k	k	NOUN
ejpam-109	384	3	=	=	SYM
ejpam-109	384	4	2,3	2,3	NUM
ejpam-109	384	5	,	,	PUNCT
ejpam-109	384	6	4,6	4,6	NUM
ejpam-109	384	7	)	)	PUNCT
ejpam-109	384	8	be	be	AUX
ejpam-109	384	9	as	as	SCONJ
ejpam-109	384	10	mentioned	mention	VERB
ejpam-109	384	11	in	in	ADP
ejpam-109	384	12	theorem	theorem	NOUN
ejpam-109	384	13	2.7	2.7	NUM
ejpam-109	384	14	.	.	PUNCT
ejpam-109	385	1	if	if	SCONJ
ejpam-109	385	2	p−	p−	NOUN
ejpam-109	385	3	1	1	NUM
ejpam-109	385	4	m−	m−	PROPN
ejpam-109	385	5	2	2	NUM
ejpam-109	385	6	for	for	ADP
ejpam-109	385	7	every	every	DET
ejpam-109	385	8	prime	prime	ADJ
ejpam-109	385	9	divisor	divisor	NOUN
ejpam-109	385	10	p	p	NOUN
ejpam-109	385	11	of	of	ADP
ejpam-109	385	12	n	n	CCONJ
ejpam-109	385	13	,	,	PUNCT
ejpam-109	385	14	then	then	ADV
ejpam-109	385	15	qk(m)βm	qk(m)βm	NOUN
ejpam-109	385	16	≡	≡	PROPN
ejpam-109	385	17	∑	∑	PUNCT
ejpam-109	385	18	0	0	PUNCT
ejpam-109	385	19	<	<	X
ejpam-109	385	20	j	j	X
ejpam-109	385	21	<	<	X
ejpam-109	385	22	η′/k	η′/k	NOUN
ejpam-109	385	23	(	(	PUNCT
ejpam-109	385	24	η′−	η′−	PROPN
ejpam-109	385	25	k	k	PROPN
ejpam-109	385	26	j)m−1	j)m−1	PROPN
ejpam-109	385	27	(	(	PUNCT
ejpam-109	385	28	mod	mod	PROPN
ejpam-109	385	29	n2	n2	PROPN
ejpam-109	385	30	)	)	PUNCT
ejpam-109	385	31	,	,	PUNCT
ejpam-109	386	1	k	k	PROPN
ejpam-109	386	2	=	=	SYM
ejpam-109	386	3	2	2	NUM
ejpam-109	386	4	,	,	PUNCT
ejpam-109	386	5	3,4	3,4	NUM
ejpam-109	386	6	,	,	PUNCT
ejpam-109	386	7	6	6	NUM
ejpam-109	386	8	,	,	PUNCT
ejpam-109	386	9	(	(	PUNCT
ejpam-109	386	10	5.3	5.3	NUM
ejpam-109	386	11	)	)	PUNCT
ejpam-109	386	12	provided	provide	VERB
ejpam-109	386	13	that	that	SCONJ
ejpam-109	386	14	every	every	DET
ejpam-109	386	15	prime	prime	ADJ
ejpam-109	386	16	divisor	divisor	NOUN
ejpam-109	386	17	p	p	NOUN
ejpam-109	386	18	of	of	ADP
ejpam-109	386	19	n	n	PROPN
ejpam-109	386	20	is	be	AUX
ejpam-109	386	21	≥	≥	NOUN
ejpam-109	386	22	7	7	NUM
ejpam-109	386	23	when	when	SCONJ
ejpam-109	386	24	k	k	PROPN
ejpam-109	386	25	=	=	SYM
ejpam-109	386	26	6	6	X
ejpam-109	386	27	.	.	PUNCT
ejpam-109	387	1	proof	proof	NOUN
ejpam-109	387	2	.	.	PUNCT
ejpam-109	388	1	since	since	SCONJ
ejpam-109	388	2	each	each	DET
ejpam-109	388	3	proof	proof	NOUN
ejpam-109	388	4	of	of	ADP
ejpam-109	388	5	(	(	PUNCT
ejpam-109	388	6	5.3	5.3	NUM
ejpam-109	388	7	)	)	PUNCT
ejpam-109	388	8	for	for	ADP
ejpam-109	388	9	k	k	PROPN
ejpam-109	388	10	=	=	SYM
ejpam-109	388	11	2,3	2,3	NUM
ejpam-109	388	12	,	,	PUNCT
ejpam-109	388	13	4,6	4,6	PRON
ejpam-109	388	14	is	be	AUX
ejpam-109	388	15	similar	similar	ADJ
ejpam-109	388	16	,	,	PUNCT
ejpam-109	388	17	we	we	PRON
ejpam-109	388	18	give	give	VERB
ejpam-109	388	19	only	only	ADV
ejpam-109	388	20	the	the	DET
ejpam-109	388	21	proof	proof	NOUN
ejpam-109	388	22	for	for	ADP
ejpam-109	388	23	k	k	PROPN
ejpam-109	388	24	=	=	SYM
ejpam-109	388	25	6	6	NUM
ejpam-109	388	26	,	,	PUNCT
ejpam-109	388	27	under	under	ADP
ejpam-109	388	28	the	the	DET
ejpam-109	388	29	assumption	assumption	NOUN
ejpam-109	388	30	that	that	SCONJ
ejpam-109	388	31	(	(	PUNCT
ejpam-109	388	32	5.3	5.3	NUM
ejpam-109	388	33	)	)	PUNCT
ejpam-109	388	34	holds	hold	VERB
ejpam-109	388	35	for	for	ADP
ejpam-109	388	36	k	k	X
ejpam-109	388	37	=	=	SYM
ejpam-109	388	38	2,3	2,3	NUM
ejpam-109	388	39	.	.	PUNCT
ejpam-109	389	1	in	in	ADV
ejpam-109	389	2	here	here	ADV
ejpam-109	389	3	and	and	CCONJ
ejpam-109	389	4	what	what	PRON
ejpam-109	389	5	follows	follow	VERB
ejpam-109	389	6	,	,	PUNCT
ejpam-109	389	7	we	we	PRON
ejpam-109	389	8	write	write	VERB
ejpam-109	389	9	ui	ui	PROPN
ejpam-109	390	1	=	=	SYM
ejpam-109	390	2	u	u	PROPN
ejpam-109	390	3	(	(	PUNCT
ejpam-109	390	4	6)i	6)i	NOUN
ejpam-109	390	5	and	and	CCONJ
ejpam-109	390	6	vi	vi	NOUN
ejpam-109	390	7	=	=	SYM
ejpam-109	390	8	v	v	PROPN
ejpam-109	390	9	(	(	PUNCT
ejpam-109	390	10	6)i	6)i	NUM
ejpam-109	390	11	(	(	PUNCT
ejpam-109	390	12	i	i	NOUN
ejpam-109	390	13	=	=	NOUN
ejpam-109	390	14	0	0	NUM
ejpam-109	390	15	,	,	PUNCT
ejpam-109	390	16	1	1	NUM
ejpam-109	390	17	,	,	PUNCT
ejpam-109	390	18	...	...	PUNCT
ejpam-109	390	19	,	,	PUNCT
ejpam-109	390	20	5	5	X
ejpam-109	390	21	)	)	PUNCT
ejpam-109	390	22	for	for	ADP
ejpam-109	390	23	simplification	simplification	NOUN
ejpam-109	390	24	.	.	PUNCT
ejpam-109	391	1	from	from	ADP
ejpam-109	391	2	the	the	DET
ejpam-109	391	3	congruences	congruence	NOUN
ejpam-109	391	4	mentioned	mention	VERB
ejpam-109	391	5	above	above	ADV
ejpam-109	391	6	we	we	PRON
ejpam-109	391	7	get	get	VERB
ejpam-109	391	8	ui	ui	ADP
ejpam-109	391	9	≡−u5−i+(m−1)η′v5−i	≡−u5−i+(m−1)η′v5−i	PROPN
ejpam-109	391	10	(	(	PUNCT
ejpam-109	391	11	mod	mod	PROPN
ejpam-109	391	12	n2	n2	PROPN
ejpam-109	391	13	)	)	PUNCT
ejpam-109	391	14	and	and	CCONJ
ejpam-109	391	15	vi	vi	PROPN
ejpam-109	391	16	≡	≡	PROPN
ejpam-109	391	17	v5−i	v5−i	PROPN
ejpam-109	391	18	(	(	PUNCT
ejpam-109	391	19	mod	mod	PROPN
ejpam-109	391	20	n	n	CCONJ
ejpam-109	391	21	)	)	PUNCT
ejpam-109	391	22	for	for	ADP
ejpam-109	391	23	i	i	PRON
ejpam-109	391	24	=	=	NOUN
ejpam-109	391	25	3,4	3,4	NUM
ejpam-109	391	26	,	,	PUNCT
ejpam-109	391	27	5	5	NUM
ejpam-109	391	28	.	.	PUNCT
ejpam-109	392	1	since	since	SCONJ
ejpam-109	392	2	(	(	PUNCT
ejpam-109	392	3	n	n	CCONJ
ejpam-109	392	4	,	,	PUNCT
ejpam-109	392	5	6	6	NUM
ejpam-109	392	6	)	)	PUNCT
ejpam-109	392	7	=	=	SYM
ejpam-109	392	8	1	1	NUM
ejpam-109	392	9	,	,	PUNCT
ejpam-109	392	10	by	by	ADP
ejpam-109	392	11	taking	take	VERB
ejpam-109	392	12	a	a	DET
ejpam-109	392	13	=	=	NOUN
ejpam-109	392	14	6	6	NUM
ejpam-109	392	15	in	in	ADP
ejpam-109	392	16	(	(	PUNCT
ejpam-109	392	17	5.2	5.2	NUM
ejpam-109	392	18	)	)	PUNCT
ejpam-109	392	19	,	,	PUNCT
ejpam-109	392	20	we	we	PRON
ejpam-109	392	21	obtain	obtain	VERB
ejpam-109	392	22	(	(	PUNCT
ejpam-109	392	23	6m−	6m−	PROPN
ejpam-109	392	24	1)βm	1)βm	PROPN
ejpam-109	392	25	≡6m−1	≡6m−1	NOUN
ejpam-109	392	26	5	5	NUM
ejpam-109	392	27	∑	∑	PROPN
ejpam-109	392	28	i=1	i=1	PROPN
ejpam-109	392	29	iui	iui	PROPN
ejpam-109	392	30	−	−	PROPN
ejpam-109	393	1	m−	m−	PROPN
ejpam-109	393	2	1	1	NUM
ejpam-109	393	3	2	2	NUM
ejpam-109	393	4	6m−2η′	6m−2η′	NUM
ejpam-109	393	5	5	5	NUM
ejpam-109	393	6	∑	∑	PROPN
ejpam-109	393	7	i=1	i=1	PROPN
ejpam-109	393	8	i2vi	i2vi	X
ejpam-109	393	9	≡−	≡−	X
ejpam-109	393	10	6m−1(5u0	6m−1(5u0	NOUN
ejpam-109	393	11	+	+	SYM
ejpam-109	393	12	3u1	3u1	NUM
ejpam-109	393	13	+	+	CCONJ
ejpam-109	393	14	u2	u2	NOUN
ejpam-109	393	15	)	)	PUNCT
ejpam-109	393	16	+	+	CCONJ
ejpam-109	393	17	m−	m−	PROPN
ejpam-109	393	18	1	1	NUM
ejpam-109	393	19	2	2	NUM
ejpam-109	393	20	6m−2η′(35v0	6m−2η′(35v0	NUM
ejpam-109	393	21	+	+	NOUN
ejpam-109	393	22	31v1	31v1	NUM
ejpam-109	393	23	+	+	NUM
ejpam-109	393	24	23v2	23v2	NUM
ejpam-109	393	25	)	)	PUNCT
ejpam-109	393	26	(	(	PUNCT
ejpam-109	393	27	mod	mod	PROPN
ejpam-109	393	28	n2	n2	PROPN
ejpam-109	393	29	)	)	PUNCT
ejpam-109	393	30	.	.	PUNCT
ejpam-109	394	1	under	under	ADP
ejpam-109	394	2	the	the	DET
ejpam-109	394	3	given	give	VERB
ejpam-109	394	4	condition	condition	NOUN
ejpam-109	394	5	,	,	PUNCT
ejpam-109	394	6	bm−2	bm−2	PROPN
ejpam-109	394	7	∈	∈	PROPN
ejpam-109	394	8	zp	zp	X
ejpam-109	394	9	for	for	ADP
ejpam-109	394	10	every	every	DET
ejpam-109	394	11	prime	prime	ADJ
ejpam-109	394	12	divisor	divisor	NOUN
ejpam-109	394	13	p	p	NOUN
ejpam-109	394	14	of	of	ADP
ejpam-109	394	15	n	n	CCONJ
ejpam-109	394	16	,	,	PUNCT
ejpam-109	394	17	so	so	SCONJ
ejpam-109	394	18	we	we	PRON
ejpam-109	394	19	get	get	VERB
ejpam-109	394	20	∑5	∑5	PROPN
ejpam-109	394	21	i=0	i=0	PROPN
ejpam-109	394	22	vi	vi	PROPN
ejpam-109	394	23	≡	≡	PROPN
ejpam-109	394	24	2	2	NUM
ejpam-109	394	25	�	�	PROPN
ejpam-109	394	26	v0	v0	PROPN
ejpam-109	394	27	+	+	CCONJ
ejpam-109	394	28	v1	v1	NOUN
ejpam-109	394	29	+	+	CCONJ
ejpam-109	394	30	v2	v2	PROPN
ejpam-109	394	31	�	�	PROPN
ejpam-109	394	32	≡	≡	PROPN
ejpam-109	394	33	0	0	PUNCT
ejpam-109	395	1	(	(	PUNCT
ejpam-109	395	2	mod	mod	NOUN
ejpam-109	395	3	n	n	CCONJ
ejpam-109	395	4	)	)	PUNCT
ejpam-109	395	5	,	,	PUNCT
ejpam-109	395	6	which	which	PRON
ejpam-109	395	7	implies	imply	VERB
ejpam-109	395	8	v2	v2	PROPN
ejpam-109	395	9	≡	≡	PROPN
ejpam-109	395	10	−v0	−v0	PROPN
ejpam-109	395	11	−	−	PROPN
ejpam-109	395	12	v1	v1	PROPN
ejpam-109	395	13	(	(	PUNCT
ejpam-109	395	14	mod	mod	NOUN
ejpam-109	395	15	n	n	CCONJ
ejpam-109	395	16	)	)	PUNCT
ejpam-109	395	17	since	since	SCONJ
ejpam-109	395	18	n	n	PRON
ejpam-109	395	19	is	be	AUX
ejpam-109	395	20	odd	odd	ADJ
ejpam-109	395	21	.	.	PUNCT
ejpam-109	396	1	consequently	consequently	ADV
ejpam-109	396	2	,	,	PUNCT
ejpam-109	396	3	dividing	divide	VERB
ejpam-109	396	4	the	the	DET
ejpam-109	396	5	above	above	ADJ
ejpam-109	396	6	congruence	congruence	NOUN
ejpam-109	396	7	by	by	ADP
ejpam-109	396	8	6m−1	6m−1	NUM
ejpam-109	396	9	we	we	PRON
ejpam-109	396	10	have	have	VERB
ejpam-109	396	11	(	(	PUNCT
ejpam-109	396	12	6−	6−	NUM
ejpam-109	396	13	61−m)βm	61−m)βm	NOUN
ejpam-109	396	14	≡−(5u0	≡−(5u0	PRON
ejpam-109	396	15	+	+	NUM
ejpam-109	396	16	3u1	3u1	NUM
ejpam-109	396	17	+	+	CCONJ
ejpam-109	396	18	u2	u2	NOUN
ejpam-109	396	19	)	)	PUNCT
ejpam-109	396	20	+	+	CCONJ
ejpam-109	397	1	m−	m−	PROPN
ejpam-109	397	2	1	1	NUM
ejpam-109	397	3	3	3	NUM
ejpam-109	397	4	η′(3v0	η′(3v0	PROPN
ejpam-109	397	5	+	+	NUM
ejpam-109	397	6	2v1	2v1	NUM
ejpam-109	397	7	)	)	PUNCT
ejpam-109	397	8	(	(	PUNCT
ejpam-109	397	9	mod	mod	PROPN
ejpam-109	397	10	n2	n2	PROPN
ejpam-109	397	11	)	)	PUNCT
ejpam-109	397	12	.	.	PUNCT
ejpam-109	398	1	t.	t.	PROPN
ejpam-109	398	2	agoh	agoh	PROPN
ejpam-109	398	3	/	/	SYM
ejpam-109	398	4	eur	eur	PROPN
ejpam-109	398	5	.	.	PUNCT
ejpam-109	399	1	j.	j.	PROPN
ejpam-109	399	2	pure	pure	PROPN
ejpam-109	399	3	appl	appl	PROPN
ejpam-109	399	4	.	.	PROPN
ejpam-109	399	5	math	math	PROPN
ejpam-109	399	6	,	,	PUNCT
ejpam-109	399	7	1	1	NUM
ejpam-109	399	8	(	(	PUNCT
ejpam-109	399	9	2008	2008	NUM
ejpam-109	399	10	)	)	PUNCT
ejpam-109	399	11	,	,	PUNCT
ejpam-109	399	12	(	(	PUNCT
ejpam-109	399	13	3	3	NUM
ejpam-109	399	14	-	-	SYM
ejpam-109	399	15	21	21	NUM
ejpam-109	399	16	)	)	PUNCT
ejpam-109	399	17	19	19	NUM
ejpam-109	399	18	assume	assume	VERB
ejpam-109	399	19	that	that	SCONJ
ejpam-109	399	20	(	(	PUNCT
ejpam-109	399	21	5.3	5.3	NUM
ejpam-109	399	22	)	)	PUNCT
ejpam-109	399	23	holds	hold	VERB
ejpam-109	399	24	for	for	ADP
ejpam-109	399	25	k	k	PROPN
ejpam-109	399	26	=	=	SYM
ejpam-109	399	27	2	2	NUM
ejpam-109	399	28	and	and	CCONJ
ejpam-109	399	29	3	3	NUM
ejpam-109	399	30	,	,	PUNCT
ejpam-109	399	31	i.e.	i.e.	X
ejpam-109	399	32	,	,	PUNCT
ejpam-109	399	33	in	in	ADP
ejpam-109	399	34	equivalent	equivalent	ADJ
ejpam-109	399	35	forms	form	NOUN
ejpam-109	399	36	(	(	PUNCT
ejpam-109	399	37	2−	2−	NUM
ejpam-109	399	38	21−m)βm	21−m)βm	PROPN
ejpam-109	399	39	≡−	≡−	X
ejpam-109	399	40	(	(	PUNCT
ejpam-109	399	41	u0	u0	ADJ
ejpam-109	399	42	+	+	NOUN
ejpam-109	399	43	u1	u1	NOUN
ejpam-109	399	44	+	+	CCONJ
ejpam-109	399	45	u2	u2	NOUN
ejpam-109	399	46	)	)	PUNCT
ejpam-109	399	47	+	+	CCONJ
ejpam-109	399	48	m−	m−	PROPN
ejpam-109	399	49	1	1	NUM
ejpam-109	399	50	2	2	NUM
ejpam-109	399	51	η′(v0	η′(v0	NOUN
ejpam-109	399	52	+	+	CCONJ
ejpam-109	399	53	v1	v1	PROPN
ejpam-109	399	54	+	+	CCONJ
ejpam-109	399	55	v2	v2	NOUN
ejpam-109	399	56	)	)	PUNCT
ejpam-109	399	57	≡−	≡−	NOUN
ejpam-109	399	58	(	(	PUNCT
ejpam-109	399	59	u0	u0	ADJ
ejpam-109	399	60	+	+	NOUN
ejpam-109	399	61	u1	u1	NOUN
ejpam-109	399	62	+	+	CCONJ
ejpam-109	399	63	u2	u2	NOUN
ejpam-109	399	64	)	)	PUNCT
ejpam-109	399	65	(	(	PUNCT
ejpam-109	399	66	mod	mod	PROPN
ejpam-109	399	67	n2	n2	PROPN
ejpam-109	399	68	)	)	PUNCT
ejpam-109	399	69	,	,	PUNCT
ejpam-109	399	70	(	(	PUNCT
ejpam-109	399	71	3−	3−	NUM
ejpam-109	399	72	31−m)βm	31−m)βm	PROPN
ejpam-109	399	73	≡−	≡−	ADP
ejpam-109	399	74	2(u0	2(u0	ADJ
ejpam-109	399	75	+	+	SYM
ejpam-109	399	76	u1	u1	NOUN
ejpam-109	399	77	)	)	PUNCT
ejpam-109	400	1	+	+	CCONJ
ejpam-109	400	2	2(m−	2(m−	NUM
ejpam-109	400	3	1	1	NUM
ejpam-109	400	4	)	)	PUNCT
ejpam-109	400	5	3	3	NUM
ejpam-109	400	6	η′(v0	η′(v0	NOUN
ejpam-109	400	7	+	+	CCONJ
ejpam-109	400	8	v1	v1	NOUN
ejpam-109	400	9	)	)	PUNCT
ejpam-109	400	10	(	(	PUNCT
ejpam-109	400	11	mod	mod	PROPN
ejpam-109	400	12	n2	n2	PROPN
ejpam-109	400	13	)	)	PUNCT
ejpam-109	400	14	.	.	PUNCT
ejpam-109	401	1	combining	combine	VERB
ejpam-109	401	2	these	these	DET
ejpam-109	401	3	congruences	congruence	NOUN
ejpam-109	401	4	,	,	PUNCT
ejpam-109	401	5	we	we	PRON
ejpam-109	401	6	deduce	deduce	VERB
ejpam-109	401	7	(	(	PUNCT
ejpam-109	401	8	1	1	NUM
ejpam-109	401	9	+	+	NUM
ejpam-109	401	10	21−m+	21−m+	NUM
ejpam-109	401	11	31−m−	31−m−	NUM
ejpam-109	401	12	61−m)βm	61−m)βm	NOUN
ejpam-109	401	13	=	=	SYM
ejpam-109	402	1	¦	¦	PROPN
ejpam-109	402	2	−(2−	−(2−	PRON
ejpam-109	402	3	21−m)−	21−m)−	NUM
ejpam-109	402	4	(	(	PUNCT
ejpam-109	402	5	3−	3−	NUM
ejpam-109	402	6	31−m	31−m	NUM
ejpam-109	402	7	)	)	PUNCT
ejpam-109	403	1	+	+	CCONJ
ejpam-109	403	2	(	(	PUNCT
ejpam-109	403	3	6−	6−	NUM
ejpam-109	403	4	61−m	61−m	NUM
ejpam-109	403	5	)	)	PUNCT
ejpam-109	404	1	©	©	PROPN
ejpam-109	404	2	βm	βm	NOUN
ejpam-109	404	3	≡−	≡−	PROPN
ejpam-109	404	4	2u0	2u0	NUM
ejpam-109	404	5	+	+	NOUN
ejpam-109	404	6	m−	m−	PROPN
ejpam-109	404	7	1	1	NUM
ejpam-109	404	8	3	3	NUM
ejpam-109	404	9	η′v0	η′v0	NOUN
ejpam-109	404	10	(	(	PUNCT
ejpam-109	404	11	mod	mod	PROPN
ejpam-109	404	12	n2	n2	PROPN
ejpam-109	404	13	)	)	PUNCT
ejpam-109	404	14	,	,	PUNCT
ejpam-109	404	15	that	that	ADV
ejpam-109	404	16	is	is	ADV
ejpam-109	404	17	,	,	PUNCT
ejpam-109	404	18	(	(	PUNCT
ejpam-109	404	19	6m−1	6m−1	NOUN
ejpam-109	404	20	+	+	SYM
ejpam-109	404	21	3m−1	3m−1	NUM
ejpam-109	404	22	+	+	SYM
ejpam-109	404	23	2m−1−	2m−1−	NUM
ejpam-109	404	24	1)βm	1)βm	PROPN
ejpam-109	404	25	≡2	≡2	NUM
ejpam-109	404	26	�	�	PROPN
ejpam-109	404	27	−6m−1u0	−6m−1u0	PROPN
ejpam-109	404	28	+	+	PROPN
ejpam-109	404	29	(	(	PUNCT
ejpam-109	404	30	m−	m−	PROPN
ejpam-109	404	31	1)6m−2η′v0	1)6m−2η′v0	NUM
ejpam-109	404	32	�	�	PROPN
ejpam-109	404	33	≡2	≡2	NUM
ejpam-109	404	34	∑	∑	NOUN
ejpam-109	404	35	j∈g(6)0	j∈g(6)0	NOUN
ejpam-109	404	36	(	(	PUNCT
ejpam-109	404	37	η′−	η′−	NUM
ejpam-109	404	38	6	6	NUM
ejpam-109	404	39	j)m−1	j)m−1	PROPN
ejpam-109	404	40	(	(	PUNCT
ejpam-109	404	41	mod	mod	PROPN
ejpam-109	404	42	n2	n2	PROPN
ejpam-109	404	43	)	)	PUNCT
ejpam-109	404	44	,	,	PUNCT
ejpam-109	404	45	which	which	PRON
ejpam-109	404	46	completes	complete	VERB
ejpam-109	404	47	the	the	DET
ejpam-109	404	48	proof	proof	NOUN
ejpam-109	404	49	of	of	ADP
ejpam-109	404	50	(	(	PUNCT
ejpam-109	404	51	5.3	5.3	NUM
ejpam-109	404	52	)	)	PUNCT
ejpam-109	404	53	for	for	ADP
ejpam-109	404	54	k	k	PROPN
ejpam-109	404	55	=	=	SYM
ejpam-109	404	56	6	6	NUM
ejpam-109	404	57	.	.	NOUN
ejpam-109	404	58	6	6	NUM
ejpam-109	404	59	.	.	PUNCT
ejpam-109	404	60	properties	property	NOUN
ejpam-109	404	61	of	of	ADP
ejpam-109	404	62	the	the	DET
ejpam-109	404	63	numerator	numerator	NOUN
ejpam-109	404	64	of	of	ADP
ejpam-109	404	65	βm	βm	VERB
ejpam-109	404	66	in	in	ADP
ejpam-109	404	67	this	this	DET
ejpam-109	404	68	final	final	ADJ
ejpam-109	404	69	section	section	NOUN
ejpam-109	404	70	,	,	PUNCT
ejpam-109	404	71	we	we	PRON
ejpam-109	404	72	would	would	AUX
ejpam-109	404	73	like	like	VERB
ejpam-109	404	74	to	to	PART
ejpam-109	404	75	discuss	discuss	VERB
ejpam-109	404	76	some	some	DET
ejpam-109	404	77	basic	basic	ADJ
ejpam-109	404	78	properties	property	NOUN
ejpam-109	404	79	of	of	ADP
ejpam-109	404	80	the	the	DET
ejpam-109	404	81	numerator	numerator	NOUN
ejpam-109	404	82	of	of	ADP
ejpam-109	404	83	βm	βm	VERB
ejpam-109	404	84	for	for	ADP
ejpam-109	404	85	an	an	DET
ejpam-109	404	86	even	even	ADV
ejpam-109	404	87	integer	integer	NOUN
ejpam-109	404	88	m≥	m≥	PROPN
ejpam-109	404	89	2	2	NUM
ejpam-109	404	90	.	.	PUNCT
ejpam-109	404	91	as	as	SCONJ
ejpam-109	404	92	defined	define	VERB
ejpam-109	404	93	in	in	ADP
ejpam-109	404	94	section	section	NOUN
ejpam-109	404	95	2	2	NUM
ejpam-109	404	96	,	,	PUNCT
ejpam-109	404	97	let	let	VERB
ejpam-109	404	98	n	n	PRON
ejpam-109	404	99	′m	′m	VERB
ejpam-109	404	100	be	be	AUX
ejpam-109	404	101	the	the	DET
ejpam-109	404	102	numerator	numerator	NOUN
ejpam-109	404	103	of	of	ADP
ejpam-109	404	104	βm	βm	PROPN
ejpam-109	404	105	.	.	PROPN
ejpam-109	404	106	theorem	theorem	VERB
ejpam-109	404	107	6.1	6.1	NUM
ejpam-109	404	108	.	.	PUNCT
ejpam-109	405	1	let	let	VERB
ejpam-109	405	2	n	n	NOUN
ejpam-109	405	3	=	=	PUNCT
ejpam-109	405	4	p1p2	p1p2	X
ejpam-109	405	5	·	·	PUNCT
ejpam-109	405	6	·	·	PUNCT
ejpam-109	405	7	·	·	PUNCT
ejpam-109	406	1	ps	ps	INTJ
ejpam-109	406	2	be	be	AUX
ejpam-109	406	3	the	the	DET
ejpam-109	406	4	product	product	NOUN
ejpam-109	406	5	of	of	ADP
ejpam-109	406	6	some	some	DET
ejpam-109	406	7	distinct	distinct	ADJ
ejpam-109	406	8	irregular	irregular	ADJ
ejpam-109	406	9	primes	prime	NOUN
ejpam-109	406	10	such	such	ADJ
ejpam-109	406	11	that	that	SCONJ
ejpam-109	406	12	(	(	PUNCT
ejpam-109	406	13	1	1	NUM
ejpam-109	406	14	2	2	NUM
ejpam-109	406	15	(	(	PUNCT
ejpam-109	406	16	pi	pi	NOUN
ejpam-109	406	17	−	−	NOUN
ejpam-109	406	18	1	1	NUM
ejpam-109	406	19	)	)	PUNCT
ejpam-109	406	20	,	,	PUNCT
ejpam-109	406	21	1	1	NUM
ejpam-109	406	22	2	2	NUM
ejpam-109	406	23	(	(	PUNCT
ejpam-109	406	24	p	p	X
ejpam-109	406	25	j	j	PROPN
ejpam-109	406	26	−	−	NOUN
ejpam-109	406	27	1	1	NUM
ejpam-109	406	28	)	)	PUNCT
ejpam-109	406	29	)	)	PUNCT
ejpam-109	407	1	=	=	SYM
ejpam-109	407	2	1	1	NUM
ejpam-109	407	3	for	for	ADP
ejpam-109	407	4	every	every	DET
ejpam-109	407	5	i	i	PROPN
ejpam-109	407	6	,	,	PUNCT
ejpam-109	407	7	j	j	PROPN
ejpam-109	407	8	=	=	SYM
ejpam-109	407	9	1,2	1,2	NUM
ejpam-109	407	10	,	,	PUNCT
ejpam-109	407	11	...	...	PUNCT
ejpam-109	407	12	,	,	PUNCT
ejpam-109	407	13	s	s	VERB
ejpam-109	407	14	with	with	ADP
ejpam-109	407	15	i	i	PROPN
ejpam-109	407	16	6=	6=	PROPN
ejpam-109	408	1	j.	j.	PROPN
ejpam-109	408	2	then	then	ADV
ejpam-109	408	3	there	there	PRON
ejpam-109	408	4	exists	exist	VERB
ejpam-109	408	5	an	an	DET
ejpam-109	408	6	even	even	ADV
ejpam-109	408	7	integer	integer	NOUN
ejpam-109	408	8	m≥	m≥	PROPN
ejpam-109	408	9	2	2	NUM
ejpam-109	408	10	such	such	ADJ
ejpam-109	408	11	that	that	SCONJ
ejpam-109	408	12	n	n	X
ejpam-109	408	13	′m	′m	PROPN
ejpam-109	408	14	≡	≡	PROPN
ejpam-109	408	15	0	0	PUNCT
ejpam-109	408	16	(	(	PUNCT
ejpam-109	408	17	mod	mod	NOUN
ejpam-109	408	18	n	n	CCONJ
ejpam-109	408	19	)	)	PUNCT
ejpam-109	408	20	.	.	PUNCT
ejpam-109	409	1	proof	proof	NOUN
ejpam-109	409	2	.	.	PUNCT
ejpam-109	410	1	since	since	SCONJ
ejpam-109	410	2	all	all	DET
ejpam-109	410	3	the	the	DET
ejpam-109	410	4	primes	prime	NOUN
ejpam-109	410	5	pi	pi	NOUN
ejpam-109	410	6	,	,	PUNCT
ejpam-109	410	7	i	i	NOUN
ejpam-109	410	8	=	=	NOUN
ejpam-109	410	9	1,2	1,2	NUM
ejpam-109	410	10	,	,	PUNCT
ejpam-109	410	11	...	...	PUNCT
ejpam-109	410	12	,	,	PUNCT
ejpam-109	410	13	s	s	AUX
ejpam-109	410	14	,	,	PUNCT
ejpam-109	410	15	are	be	AUX
ejpam-109	410	16	irregular	irregular	ADJ
ejpam-109	410	17	,	,	PUNCT
ejpam-109	410	18	there	there	PRON
ejpam-109	410	19	exist	exist	VERB
ejpam-109	410	20	even	even	ADV
ejpam-109	410	21	integers	integer	NOUN
ejpam-109	410	22	2mi	2mi	ADJ
ejpam-109	410	23	,	,	PUNCT
ejpam-109	410	24	1	1	NUM
ejpam-109	410	25	≤	≤	NUM
ejpam-109	410	26	mi	mi	X
ejpam-109	410	27	≤	≤	ADV
ejpam-109	410	28	1	1	NUM
ejpam-109	410	29	2	2	NUM
ejpam-109	410	30	(	(	PUNCT
ejpam-109	410	31	pi	pi	NOUN
ejpam-109	410	32	−	−	NOUN
ejpam-109	410	33	3	3	NUM
ejpam-109	410	34	)	)	PUNCT
ejpam-109	410	35	,	,	PUNCT
ejpam-109	410	36	such	such	ADJ
ejpam-109	410	37	that	that	SCONJ
ejpam-109	410	38	b2mi	b2mi	ADJ
ejpam-109	410	39	≡	≡	PROPN
ejpam-109	410	40	0	0	PUNCT
ejpam-109	410	41	(	(	PUNCT
ejpam-109	410	42	mod	mod	PROPN
ejpam-109	410	43	pi	pi	PROPN
ejpam-109	410	44	)	)	PUNCT
ejpam-109	410	45	.	.	PUNCT
ejpam-109	411	1	since	since	SCONJ
ejpam-109	411	2	(	(	PUNCT
ejpam-109	411	3	1	1	NUM
ejpam-109	411	4	2	2	NUM
ejpam-109	411	5	(	(	PUNCT
ejpam-109	411	6	pi	pi	NOUN
ejpam-109	411	7	−	−	NOUN
ejpam-109	411	8	1	1	NUM
ejpam-109	411	9	)	)	PUNCT
ejpam-109	411	10	,	,	PUNCT
ejpam-109	411	11	1	1	NUM
ejpam-109	411	12	2	2	NUM
ejpam-109	411	13	(	(	PUNCT
ejpam-109	411	14	p	p	X
ejpam-109	411	15	j	j	PROPN
ejpam-109	411	16	−	−	NOUN
ejpam-109	411	17	1	1	NUM
ejpam-109	411	18	)	)	PUNCT
ejpam-109	411	19	)	)	PUNCT
ejpam-109	411	20	=	=	PUNCT
ejpam-109	411	21	1	1	NUM
ejpam-109	411	22	if	if	SCONJ
ejpam-109	411	23	i	i	PRON
ejpam-109	411	24	6=	6=	PROPN
ejpam-109	411	25	j	j	PROPN
ejpam-109	411	26	,	,	PUNCT
ejpam-109	411	27	by	by	ADP
ejpam-109	411	28	the	the	DET
ejpam-109	411	29	chinese	chinese	ADJ
ejpam-109	411	30	remainder	remainder	NOUN
ejpam-109	411	31	theorem	theorem	VERB
ejpam-109	411	32	the	the	DET
ejpam-109	411	33	system	system	NOUN
ejpam-109	411	34	of	of	ADP
ejpam-109	411	35	congruences	congruence	NOUN
ejpam-109	411	36	x	x	PROPN
ejpam-109	411	37	≡	≡	PROPN
ejpam-109	411	38	mi	mi	PROPN
ejpam-109	411	39	(	(	PUNCT
ejpam-109	411	40	mod	mod	PROPN
ejpam-109	411	41	1	1	NUM
ejpam-109	411	42	2	2	NUM
ejpam-109	411	43	(	(	PUNCT
ejpam-109	411	44	pi	pi	NOUN
ejpam-109	411	45	−	−	NOUN
ejpam-109	411	46	1	1	NUM
ejpam-109	411	47	)	)	PUNCT
ejpam-109	411	48	)	)	PUNCT
ejpam-109	411	49	,	,	PUNCT
ejpam-109	411	50	i	i	PRON
ejpam-109	411	51	=	=	NOUN
ejpam-109	411	52	1	1	NUM
ejpam-109	411	53	,	,	PUNCT
ejpam-109	411	54	2	2	NUM
ejpam-109	411	55	,	,	PUNCT
ejpam-109	411	56	...	...	PUNCT
ejpam-109	411	57	,	,	PUNCT
ejpam-109	411	58	s	s	AUX
ejpam-109	411	59	,	,	PUNCT
ejpam-109	411	60	has	have	VERB
ejpam-109	411	61	uniquely	uniquely	ADV
ejpam-109	411	62	the	the	DET
ejpam-109	411	63	solution	solution	NOUN
ejpam-109	411	64	x	x	PUNCT
ejpam-109	411	65	=	=	SYM
ejpam-109	411	66	α	α	X
ejpam-109	411	67	>	>	X
ejpam-109	411	68	0	0	NUM
ejpam-109	411	69	modulo	modulo	NOUN
ejpam-109	411	70	1	1	NUM
ejpam-109	411	71	2s	2s	NUM
ejpam-109	411	72	∏s	∏s	NUM
ejpam-109	411	73	i=1(pi	i=1(pi	NUM
ejpam-109	411	74	−	−	NOUN
ejpam-109	411	75	1	1	NUM
ejpam-109	411	76	)	)	PUNCT
ejpam-109	411	77	in	in	ADP
ejpam-109	411	78	common	common	ADJ
ejpam-109	411	79	.	.	PUNCT
ejpam-109	412	1	if	if	SCONJ
ejpam-109	412	2	we	we	PRON
ejpam-109	412	3	choose	choose	VERB
ejpam-109	412	4	m	m	NOUN
ejpam-109	412	5	=	=	NOUN
ejpam-109	412	6	2α	2α	NOUN
ejpam-109	412	7	,	,	PUNCT
ejpam-109	412	8	then	then	ADV
ejpam-109	412	9	pi	pi	NOUN
ejpam-109	412	10	−	−	PROPN
ejpam-109	412	11	1	1	NUM
ejpam-109	412	12	m	m	NOUN
ejpam-109	412	13	and	and	CCONJ
ejpam-109	412	14	βm	βm	VERB
ejpam-109	412	15	≡	≡	PROPN
ejpam-109	412	16	β2mi	β2mi	PUNCT
ejpam-109	412	17	≡	≡	PROPN
ejpam-109	412	18	0	0	PUNCT
ejpam-109	413	1	(	(	PUNCT
ejpam-109	413	2	mod	mod	PROPN
ejpam-109	413	3	pi	pi	NOUN
ejpam-109	413	4	)	)	PUNCT
ejpam-109	413	5	by	by	ADP
ejpam-109	413	6	theorem	theorem	NOUN
ejpam-109	413	7	2.6	2.6	NUM
ejpam-109	413	8	,	,	PUNCT
ejpam-109	413	9	which	which	PRON
ejpam-109	413	10	are	be	AUX
ejpam-109	413	11	valid	valid	ADJ
ejpam-109	413	12	for	for	ADP
ejpam-109	413	13	all	all	DET
ejpam-109	413	14	i	i	NOUN
ejpam-109	413	15	=	=	SYM
ejpam-109	413	16	1,2	1,2	NUM
ejpam-109	413	17	,	,	PUNCT
ejpam-109	413	18	...	...	PUNCT
ejpam-109	413	19	,	,	PUNCT
ejpam-109	413	20	s.	s.	PROPN
ejpam-109	413	21	this	this	PRON
ejpam-109	413	22	completes	complete	VERB
ejpam-109	413	23	the	the	DET
ejpam-109	413	24	proof	proof	NOUN
ejpam-109	413	25	.	.	PUNCT
ejpam-109	414	1	it	it	PRON
ejpam-109	414	2	is	be	AUX
ejpam-109	414	3	evident	evident	ADJ
ejpam-109	414	4	that	that	SCONJ
ejpam-109	414	5	if	if	SCONJ
ejpam-109	414	6	n	n	PRON
ejpam-109	414	7	is	be	AUX
ejpam-109	414	8	divisible	divisible	ADJ
ejpam-109	414	9	by	by	ADP
ejpam-109	414	10	at	at	ADV
ejpam-109	414	11	least	least	ADV
ejpam-109	414	12	two	two	NUM
ejpam-109	414	13	distinct	distinct	ADJ
ejpam-109	414	14	irregular	irregular	ADJ
ejpam-109	414	15	primes	prime	NOUN
ejpam-109	414	16	p	p	NOUN
ejpam-109	414	17	and	and	CCONJ
ejpam-109	414	18	q	q	NOUN
ejpam-109	414	19	with	with	ADP
ejpam-109	414	20	(	(	PUNCT
ejpam-109	414	21	1	1	NUM
ejpam-109	414	22	2	2	NUM
ejpam-109	414	23	(	(	PUNCT
ejpam-109	414	24	p	p	NOUN
ejpam-109	414	25	−	−	PROPN
ejpam-109	414	26	1	1	NUM
ejpam-109	414	27	)	)	PUNCT
ejpam-109	414	28	,	,	PUNCT
ejpam-109	414	29	1	1	NUM
ejpam-109	414	30	2	2	NUM
ejpam-109	414	31	(	(	PUNCT
ejpam-109	414	32	q	q	NOUN
ejpam-109	414	33	−	−	PROPN
ejpam-109	414	34	1	1	NUM
ejpam-109	414	35	)	)	PUNCT
ejpam-109	414	36	)	)	PUNCT
ejpam-109	415	1	6=	6=	ADP
ejpam-109	415	2	1	1	NUM
ejpam-109	415	3	,	,	PUNCT
ejpam-109	415	4	then	then	ADV
ejpam-109	415	5	there	there	PRON
ejpam-109	415	6	does	do	AUX
ejpam-109	415	7	not	not	PART
ejpam-109	415	8	exist	exist	VERB
ejpam-109	415	9	an	an	DET
ejpam-109	415	10	even	even	ADV
ejpam-109	415	11	integer	integer	PROPN
ejpam-109	415	12	m	m	PROPN
ejpam-109	415	13	≥	≥	NOUN
ejpam-109	415	14	2	2	NUM
ejpam-109	415	15	such	such	ADJ
ejpam-109	415	16	that	that	SCONJ
ejpam-109	415	17	n	n	X
ejpam-109	415	18	′m	′m	PROPN
ejpam-109	415	19	≡	≡	PROPN
ejpam-109	415	20	0	0	PUNCT
ejpam-109	415	21	(	(	PUNCT
ejpam-109	415	22	mod	mod	NOUN
ejpam-109	415	23	n	n	CCONJ
ejpam-109	415	24	)	)	PUNCT
ejpam-109	415	25	.	.	PUNCT
ejpam-109	416	1	we	we	PRON
ejpam-109	416	2	shall	shall	AUX
ejpam-109	416	3	give	give	VERB
ejpam-109	416	4	here	here	ADV
ejpam-109	416	5	an	an	DET
ejpam-109	416	6	example	example	NOUN
ejpam-109	416	7	of	of	ADP
ejpam-109	416	8	theorem	theorem	NOUN
ejpam-109	416	9	6.1	6.1	NUM
ejpam-109	416	10	for	for	ADP
ejpam-109	416	11	n	n	NOUN
ejpam-109	416	12	=	=	SYM
ejpam-109	416	13	37	37	NUM
ejpam-109	416	14	·	·	SYM
ejpam-109	416	15	59	59	NUM
ejpam-109	416	16	·	·	SYM
ejpam-109	416	17	131	131	NUM
ejpam-109	416	18	,	,	PUNCT
ejpam-109	416	19	the	the	DET
ejpam-109	416	20	product	product	NOUN
ejpam-109	416	21	of	of	ADP
ejpam-109	416	22	three	three	NUM
ejpam-109	416	23	irregular	irregular	ADJ
ejpam-109	416	24	primes	prime	NOUN
ejpam-109	416	25	satisfying	satisfy	VERB
ejpam-109	416	26	the	the	DET
ejpam-109	416	27	indicated	indicate	VERB
ejpam-109	416	28	condition	condition	NOUN
ejpam-109	416	29	.	.	PUNCT
ejpam-109	417	1	all	all	DET
ejpam-109	417	2	the	the	DET
ejpam-109	417	3	irregularity	irregularity	NOUN
ejpam-109	417	4	indices	index	NOUN
ejpam-109	417	5	of	of	ADP
ejpam-109	417	6	37,59	37,59	NUM
ejpam-109	417	7	and	and	CCONJ
ejpam-109	417	8	t.	t.	NOUN
ejpam-109	417	9	agoh	agoh	PROPN
ejpam-109	417	10	/	/	SYM
ejpam-109	417	11	eur	eur	PROPN
ejpam-109	417	12	.	.	PUNCT
ejpam-109	418	1	j.	j.	PROPN
ejpam-109	418	2	pure	pure	PROPN
ejpam-109	418	3	appl	appl	PROPN
ejpam-109	418	4	.	.	PROPN
ejpam-109	418	5	math	math	PROPN
ejpam-109	418	6	,	,	PUNCT
ejpam-109	418	7	1	1	NUM
ejpam-109	418	8	(	(	PUNCT
ejpam-109	418	9	2008	2008	NUM
ejpam-109	418	10	)	)	PUNCT
ejpam-109	418	11	,	,	PUNCT
ejpam-109	418	12	(	(	PUNCT
ejpam-109	418	13	3	3	NUM
ejpam-109	418	14	-	-	SYM
ejpam-109	418	15	21	21	NUM
ejpam-109	418	16	)	)	PUNCT
ejpam-109	418	17	20	20	NUM
ejpam-109	418	18	131	131	NUM
ejpam-109	418	19	are	be	AUX
ejpam-109	418	20	equal	equal	ADJ
ejpam-109	418	21	to	to	ADP
ejpam-109	418	22	1	1	NUM
ejpam-109	418	23	and	and	CCONJ
ejpam-109	418	24	the	the	DET
ejpam-109	418	25	corresponding	corresponding	ADJ
ejpam-109	418	26	irregular	irregular	ADJ
ejpam-109	418	27	pairs	pair	NOUN
ejpam-109	418	28	(	(	PUNCT
ejpam-109	418	29	p	p	X
ejpam-109	418	30	,	,	PUNCT
ejpam-109	418	31	2	2	NUM
ejpam-109	418	32	m	m	NOUN
ejpam-109	418	33	)	)	PUNCT
ejpam-109	418	34	satisfying	satisfy	VERB
ejpam-109	418	35	b2	b2	NOUN
ejpam-109	418	36	m	m	NOUN
ejpam-109	418	37	≡	≡	PROPN
ejpam-109	418	38	0	0	PUNCT
ejpam-109	419	1	(	(	PUNCT
ejpam-109	419	2	mod	mod	PROPN
ejpam-109	419	3	p	p	X
ejpam-109	419	4	)	)	PUNCT
ejpam-109	419	5	(	(	PUNCT
ejpam-109	419	6	1	1	NUM
ejpam-109	419	7	≤	≤	NUM
ejpam-109	419	8	m	m	VERB
ejpam-109	419	9	≤	≤	NOUN
ejpam-109	419	10	(	(	PUNCT
ejpam-109	419	11	p−	p−	NOUN
ejpam-109	419	12	3)/2	3)/2	NUM
ejpam-109	419	13	)	)	PUNCT
ejpam-109	419	14	are	be	AUX
ejpam-109	419	15	(	(	PUNCT
ejpam-109	419	16	37	37	NUM
ejpam-109	419	17	,	,	PUNCT
ejpam-109	419	18	32	32	NUM
ejpam-109	419	19	)	)	PUNCT
ejpam-109	419	20	,	,	PUNCT
ejpam-109	419	21	(	(	PUNCT
ejpam-109	419	22	59,44	59,44	X
ejpam-109	419	23	)	)	PUNCT
ejpam-109	419	24	and	and	CCONJ
ejpam-109	419	25	(	(	PUNCT
ejpam-109	419	26	131,22	131,22	ADV
ejpam-109	419	27	)	)	PUNCT
ejpam-109	419	28	,	,	PUNCT
ejpam-109	419	29	respectively	respectively	ADV
ejpam-109	419	30	.	.	PUNCT
ejpam-109	420	1	so	so	ADV
ejpam-109	420	2	consider	consider	VERB
ejpam-109	420	3	the	the	DET
ejpam-109	420	4	system	system	NOUN
ejpam-109	420	5	of	of	ADP
ejpam-109	420	6	congruences	congruence	NOUN
ejpam-109	420	7	x	x	SYM
ejpam-109	420	8	≡	≡	PROPN
ejpam-109	420	9	16	16	NUM
ejpam-109	420	10	(	(	PUNCT
ejpam-109	420	11	mod	mod	PROPN
ejpam-109	420	12	18	18	NUM
ejpam-109	420	13	)	)	PUNCT
ejpam-109	420	14	,	,	PUNCT
ejpam-109	420	15	x	x	PROPN
ejpam-109	420	16	≡	≡	PROPN
ejpam-109	420	17	22	22	NUM
ejpam-109	420	18	(	(	PUNCT
ejpam-109	420	19	mod	mod	PROPN
ejpam-109	420	20	29	29	NUM
ejpam-109	420	21	)	)	PUNCT
ejpam-109	420	22	,	,	PUNCT
ejpam-109	420	23	x	x	SYM
ejpam-109	420	24	≡	≡	PROPN
ejpam-109	420	25	11	11	NUM
ejpam-109	420	26	(	(	PUNCT
ejpam-109	420	27	mod	mod	PROPN
ejpam-109	420	28	65	65	NUM
ejpam-109	420	29	)	)	PUNCT
ejpam-109	420	30	.	.	PUNCT
ejpam-109	421	1	then	then	ADV
ejpam-109	421	2	we	we	PRON
ejpam-109	421	3	find	find	VERB
ejpam-109	421	4	the	the	DET
ejpam-109	421	5	common	common	ADJ
ejpam-109	421	6	solution	solution	NOUN
ejpam-109	421	7	x	x	SYM
ejpam-109	421	8	≡	≡	PROPN
ejpam-109	421	9	2806	2806	NUM
ejpam-109	421	10	(	(	PUNCT
ejpam-109	421	11	mod	mod	PROPN
ejpam-109	421	12	18	18	NUM
ejpam-109	421	13	·	·	SYM
ejpam-109	421	14	29	29	NUM
ejpam-109	421	15	·	·	SYM
ejpam-109	421	16	65	65	NUM
ejpam-109	421	17	)	)	PUNCT
ejpam-109	421	18	and	and	CCONJ
ejpam-109	421	19	therefore	therefore	ADV
ejpam-109	421	20	,	,	PUNCT
ejpam-109	421	21	letting	let	VERB
ejpam-109	421	22	m	m	VERB
ejpam-109	421	23	=	=	SYM
ejpam-109	421	24	2	2	NUM
ejpam-109	421	25	·	·	PUNCT
ejpam-109	421	26	2806=	2806=	NUM
ejpam-109	421	27	5612	5612	NUM
ejpam-109	421	28	,	,	PUNCT
ejpam-109	421	29	we	we	PRON
ejpam-109	421	30	have	have	VERB
ejpam-109	421	31	n	n	ADV
ejpam-109	421	32	′m	′m	VERB
ejpam-109	421	33	≡	≡	PROPN
ejpam-109	421	34	0	0	PUNCT
ejpam-109	422	1	(	(	PUNCT
ejpam-109	422	2	mod	mod	PROPN
ejpam-109	422	3	37	37	NUM
ejpam-109	422	4	·	·	SYM
ejpam-109	422	5	59	59	NUM
ejpam-109	422	6	·	·	PUNCT
ejpam-109	422	7	131	131	NUM
ejpam-109	422	8	)	)	PUNCT
ejpam-109	422	9	.	.	PUNCT
ejpam-109	423	1	let	let	VERB
ejpam-109	423	2	n	n	PRON
ejpam-109	423	3	≥	≥	X
ejpam-109	423	4	3	3	NUM
ejpam-109	423	5	and	and	CCONJ
ejpam-109	423	6	εm(n	εm(n	NOUN
ejpam-109	423	7	)	)	PUNCT
ejpam-109	423	8	(	(	PUNCT
ejpam-109	423	9	m	m	VERB
ejpam-109	423	10	an	an	DET
ejpam-109	423	11	even	even	ADV
ejpam-109	423	12	integer	integer	NOUN
ejpam-109	423	13	≥	≥	NOUN
ejpam-109	423	14	2	2	NUM
ejpam-109	423	15	)	)	PUNCT
ejpam-109	423	16	be	be	AUX
ejpam-109	423	17	as	as	ADP
ejpam-109	423	18	in	in	ADP
ejpam-109	423	19	section	section	NOUN
ejpam-109	423	20	3	3	NUM
ejpam-109	423	21	.	.	PUNCT
ejpam-109	424	1	if	if	SCONJ
ejpam-109	424	2	m	m	VERB
ejpam-109	424	3	≡	≡	PROPN
ejpam-109	424	4	l	l	PROPN
ejpam-109	424	5	(	(	PUNCT
ejpam-109	424	6	mod	mod	PROPN
ejpam-109	424	7	ϕ(n	ϕ(n	PROPN
ejpam-109	424	8	)	)	PUNCT
ejpam-109	424	9	)	)	PUNCT
ejpam-109	424	10	for	for	ADP
ejpam-109	424	11	m	m	PRON
ejpam-109	424	12	,	,	PUNCT
ejpam-109	424	13	l	l	PROPN
ejpam-109	424	14	≥	≥	NUM
ejpam-109	424	15	2	2	NUM
ejpam-109	424	16	,	,	PUNCT
ejpam-109	424	17	then	then	ADV
ejpam-109	424	18	we	we	PRON
ejpam-109	424	19	know	know	VERB
ejpam-109	424	20	that	that	SCONJ
ejpam-109	424	21	(	(	PUNCT
ejpam-109	424	22	εm(n	εm(n	NOUN
ejpam-109	424	23	)	)	PUNCT
ejpam-109	424	24	,	,	PUNCT
ejpam-109	424	25	n	n	CCONJ
ejpam-109	424	26	)	)	PUNCT
ejpam-109	424	27	=	=	PUNCT
ejpam-109	425	1	d	d	NOUN
ejpam-109	425	2	if	if	SCONJ
ejpam-109	425	3	and	and	CCONJ
ejpam-109	425	4	only	only	ADV
ejpam-109	425	5	if	if	SCONJ
ejpam-109	425	6	(	(	PUNCT
ejpam-109	425	7	εl(n	εl(n	NUM
ejpam-109	425	8	)	)	PUNCT
ejpam-109	425	9	,	,	PUNCT
ejpam-109	425	10	n	n	CCONJ
ejpam-109	425	11	)	)	PUNCT
ejpam-109	425	12	=	=	SYM
ejpam-109	425	13	d.	d.	NOUN
ejpam-109	425	14	if	if	SCONJ
ejpam-109	425	15	p	p	NOUN
ejpam-109	425	16	−	−	PROPN
ejpam-109	425	17	1	1	NUM
ejpam-109	425	18	m	m	NOUN
ejpam-109	425	19	for	for	ADP
ejpam-109	425	20	all	all	DET
ejpam-109	425	21	prime	prime	ADJ
ejpam-109	425	22	divisors	divisor	NOUN
ejpam-109	425	23	p	p	NOUN
ejpam-109	425	24	of	of	ADP
ejpam-109	425	25	n	n	CCONJ
ejpam-109	425	26	,	,	PUNCT
ejpam-109	425	27	then	then	ADV
ejpam-109	425	28	hm(n)≡	hm(n)≡	PROPN
ejpam-109	425	29	hl(n	hl(n	NUM
ejpam-109	425	30	)	)	PUNCT
ejpam-109	425	31	(	(	PUNCT
ejpam-109	425	32	mod	mod	NOUN
ejpam-109	425	33	n	n	CCONJ
ejpam-109	425	34	)	)	PUNCT
ejpam-109	425	35	by	by	ADP
ejpam-109	425	36	corollary	corollary	ADJ
ejpam-109	425	37	4.3	4.3	NUM
ejpam-109	425	38	,	,	PUNCT
ejpam-109	425	39	hence	hence	ADV
ejpam-109	425	40	putting	put	VERB
ejpam-109	425	41	n0	n0	X
ejpam-109	425	42	=	=	PUNCT
ejpam-109	425	43	n	n	CCONJ
ejpam-109	425	44	/	/	SYM
ejpam-109	425	45	d	d	PROPN
ejpam-109	425	46	,	,	PUNCT
ejpam-109	425	47	we	we	PRON
ejpam-109	425	48	have	have	VERB
ejpam-109	425	49	εm(n	εm(n	NOUN
ejpam-109	425	50	)	)	PUNCT
ejpam-109	426	1	d	d	NOUN
ejpam-109	426	2	βm	βm	ADP
ejpam-109	426	3	≡	≡	PROPN
ejpam-109	426	4	εl(n	εl(n	PUNCT
ejpam-109	426	5	)	)	PUNCT
ejpam-109	426	6	d	d	X
ejpam-109	426	7	βl	βl	PROPN
ejpam-109	426	8	(	(	PUNCT
ejpam-109	426	9	mod	mod	PROPN
ejpam-109	426	10	n0	n0	PROPN
ejpam-109	426	11	)	)	PUNCT
ejpam-109	426	12	,	,	PUNCT
ejpam-109	426	13	where	where	SCONJ
ejpam-109	426	14	(	(	PUNCT
ejpam-109	426	15	εm(n)/d	εm(n)/d	NOUN
ejpam-109	426	16	,	,	PUNCT
ejpam-109	426	17	n0	n0	NUM
ejpam-109	426	18	)	)	PUNCT
ejpam-109	426	19	=	=	SYM
ejpam-109	426	20	(	(	PUNCT
ejpam-109	426	21	εl(n)/d	εl(n)/d	PROPN
ejpam-109	426	22	,	,	PUNCT
ejpam-109	426	23	n0	n0	NUM
ejpam-109	426	24	)	)	PUNCT
ejpam-109	426	25	=	=	SYM
ejpam-109	426	26	1	1	X
ejpam-109	426	27	.	.	PUNCT
ejpam-109	427	1	consequently	consequently	ADV
ejpam-109	427	2	,	,	PUNCT
ejpam-109	427	3	if	if	SCONJ
ejpam-109	427	4	m≡	m≡	PRON
ejpam-109	427	5	l	l	NOUN
ejpam-109	427	6	(	(	PUNCT
ejpam-109	427	7	mod	mod	PROPN
ejpam-109	427	8	ϕ(n	ϕ(n	PROPN
ejpam-109	427	9	)	)	PUNCT
ejpam-109	427	10	)	)	PUNCT
ejpam-109	427	11	,	,	PUNCT
ejpam-109	427	12	then	then	ADV
ejpam-109	427	13	(	(	PUNCT
ejpam-109	427	14	n	n	X
ejpam-109	427	15	′m	′m	PROPN
ejpam-109	427	16	,	,	PUNCT
ejpam-109	427	17	n0	n0	NUM
ejpam-109	427	18	)	)	PUNCT
ejpam-109	427	19	=	=	PUNCT
ejpam-109	427	20	(	(	PUNCT
ejpam-109	427	21	n	n	CCONJ
ejpam-109	427	22	′l	′l	PROPN
ejpam-109	427	23	,	,	PUNCT
ejpam-109	427	24	n0	n0	NUM
ejpam-109	427	25	)	)	PUNCT
ejpam-109	427	26	.	.	PUNCT
ejpam-109	428	1	here	here	ADV
ejpam-109	428	2	we	we	PRON
ejpam-109	428	3	see	see	VERB
ejpam-109	428	4	p−1	p−1	PROPN
ejpam-109	428	5	m	m	PROPN
ejpam-109	428	6	for	for	ADP
ejpam-109	428	7	any	any	DET
ejpam-109	428	8	prime	prime	ADJ
ejpam-109	428	9	divisor	divisor	NOUN
ejpam-109	428	10	p	p	NOUN
ejpam-109	428	11	of	of	ADP
ejpam-109	428	12	n	n	PRON
ejpam-109	428	13	′m	′m	PROPN
ejpam-109	428	14	.	.	PUNCT
ejpam-109	429	1	indeed	indeed	ADV
ejpam-109	429	2	,	,	PUNCT
ejpam-109	429	3	if	if	SCONJ
ejpam-109	429	4	p−1	p−1	PROPN
ejpam-109	429	5	|	|	ADV
ejpam-109	429	6	m	m	VERB
ejpam-109	429	7	,	,	PUNCT
ejpam-109	429	8	then	then	ADV
ejpam-109	429	9	bm	bm	PROPN
ejpam-109	429	10	6∈	6∈	PROPN
ejpam-109	429	11	zp	zp	PROPN
ejpam-109	429	12	by	by	ADP
ejpam-109	429	13	theorem	theorem	VERB
ejpam-109	429	14	2.5	2.5	NUM
ejpam-109	429	15	,	,	PUNCT
ejpam-109	429	16	which	which	PRON
ejpam-109	429	17	is	be	AUX
ejpam-109	429	18	contrary	contrary	ADJ
ejpam-109	429	19	to	to	ADP
ejpam-109	429	20	(	(	PUNCT
ejpam-109	429	21	n	n	PRON
ejpam-109	429	22	′m	′m	PROPN
ejpam-109	429	23	,	,	PUNCT
ejpam-109	429	24	d′m	d′m	NOUN
ejpam-109	429	25	)	)	PUNCT
ejpam-109	429	26	=	=	SYM
ejpam-109	429	27	1	1	X
ejpam-109	429	28	.	.	PUNCT
ejpam-109	430	1	in	in	ADP
ejpam-109	430	2	particular	particular	ADJ
ejpam-109	430	3	,	,	PUNCT
ejpam-109	430	4	taking	take	VERB
ejpam-109	430	5	n	n	X
ejpam-109	430	6	=	=	PUNCT
ejpam-109	430	7	|n	|n	NOUN
ejpam-109	430	8	′m|	′m|	PROPN
ejpam-109	430	9	>	>	X
ejpam-109	430	10	3	3	NUM
ejpam-109	430	11	and	and	CCONJ
ejpam-109	430	12	d	d	NOUN
ejpam-109	430	13	=	=	PUNCT
ejpam-109	430	14	(	(	PUNCT
ejpam-109	430	15	εm(n	εm(n	NOUN
ejpam-109	430	16	)	)	PUNCT
ejpam-109	430	17	,	,	PUNCT
ejpam-109	430	18	n	n	PROPN
ejpam-109	430	19	′m	′m	PROPN
ejpam-109	430	20	)	)	PUNCT
ejpam-109	430	21	,	,	PUNCT
ejpam-109	430	22	it	it	PRON
ejpam-109	430	23	is	be	AUX
ejpam-109	430	24	easily	easily	ADV
ejpam-109	430	25	seen	see	VERB
ejpam-109	430	26	that	that	SCONJ
ejpam-109	430	27	if	if	SCONJ
ejpam-109	430	28	m≡	m≡	NOUN
ejpam-109	430	29	l	l	NOUN
ejpam-109	430	30	(	(	PUNCT
ejpam-109	430	31	mod	mod	PROPN
ejpam-109	430	32	ϕ(n	ϕ(n	X
ejpam-109	430	33	′m	′m	PROPN
ejpam-109	430	34	)	)	PUNCT
ejpam-109	430	35	)	)	PUNCT
ejpam-109	430	36	,	,	PUNCT
ejpam-109	430	37	then	then	ADV
ejpam-109	430	38	|n	|n	PRON
ejpam-109	430	39	′m	′m	PROPN
ejpam-109	430	40	/	/	SYM
ejpam-109	430	41	d|=	d|=	PROPN
ejpam-109	430	42	(	(	PUNCT
ejpam-109	430	43	n	n	NOUN
ejpam-109	430	44	′	′	NUM
ejpam-109	430	45	l	l	NOUN
ejpam-109	430	46	,	,	PUNCT
ejpam-109	430	47	n	n	PROPN
ejpam-109	430	48	′m	′m	PROPN
ejpam-109	430	49	/	/	SYM
ejpam-109	430	50	d	d	NOUN
ejpam-109	430	51	)	)	PUNCT
ejpam-109	430	52	.	.	PUNCT
ejpam-109	431	1	the	the	DET
ejpam-109	431	2	next	next	ADJ
ejpam-109	431	3	theorem	theorem	NOUN
ejpam-109	431	4	can	can	AUX
ejpam-109	431	5	be	be	AUX
ejpam-109	431	6	deduced	deduce	VERB
ejpam-109	431	7	immediately	immediately	ADV
ejpam-109	431	8	from	from	ADP
ejpam-109	431	9	theorem	theorem	ADJ
ejpam-109	431	10	2.8	2.8	NUM
ejpam-109	431	11	,	,	PUNCT
ejpam-109	431	12	however	however	ADV
ejpam-109	431	13	we	we	PRON
ejpam-109	431	14	would	would	AUX
ejpam-109	431	15	like	like	VERB
ejpam-109	431	16	to	to	PART
ejpam-109	431	17	give	give	VERB
ejpam-109	431	18	the	the	DET
ejpam-109	431	19	proof	proof	NOUN
ejpam-109	431	20	from	from	ADP
ejpam-109	431	21	a	a	DET
ejpam-109	431	22	different	different	ADJ
ejpam-109	431	23	viewpoint	viewpoint	NOUN
ejpam-109	431	24	without	without	ADP
ejpam-109	431	25	of	of	ADP
ejpam-109	431	26	use	use	NOUN
ejpam-109	431	27	primes	prime	NOUN
ejpam-109	431	28	.	.	PUNCT
ejpam-109	432	1	theorem	theorem	VERB
ejpam-109	432	2	6.2	6.2	NUM
ejpam-109	432	3	.	.	PUNCT
ejpam-109	433	1	the	the	DET
ejpam-109	433	2	set	set	NOUN
ejpam-109	433	3	s	s	PART
ejpam-109	433	4	=	=	X
ejpam-109	433	5	{	{	PUNCT
ejpam-109	433	6	|n	|n	NOUN
ejpam-109	433	7	′2m|	′2m|	NOUN
ejpam-109	433	8	|	|	ADV
ejpam-109	433	9	m	m	NOUN
ejpam-109	433	10	=	=	NOUN
ejpam-109	433	11	1	1	NUM
ejpam-109	433	12	,	,	PUNCT
ejpam-109	433	13	2,3	2,3	NUM
ejpam-109	433	14	,	,	PUNCT
ejpam-109	433	15	...	...	PUNCT
ejpam-109	433	16	}	}	PUNCT
ejpam-109	433	17	contains	contain	VERB
ejpam-109	433	18	infinitely	infinitely	ADV
ejpam-109	433	19	many	many	ADJ
ejpam-109	433	20	elements	element	NOUN
ejpam-109	433	21	that	that	PRON
ejpam-109	433	22	are	be	AUX
ejpam-109	433	23	relatively	relatively	ADV
ejpam-109	433	24	prime	prime	ADJ
ejpam-109	433	25	in	in	ADP
ejpam-109	433	26	pairs	pair	NOUN
ejpam-109	433	27	.	.	PUNCT
ejpam-109	434	1	proof	proof	NOUN
ejpam-109	434	2	.	.	PUNCT
ejpam-109	435	1	assume	assume	VERB
ejpam-109	435	2	that	that	SCONJ
ejpam-109	435	3	|n	|n	NOUN
ejpam-109	435	4	′2m1	′2m1	ADP
ejpam-109	435	5	|	|	ADV
ejpam-109	435	6	,	,	PUNCT
ejpam-109	435	7	|n	|n	NOUN
ejpam-109	435	8	′2m2	′2m2	VERB
ejpam-109	435	9	|	|	ADV
ejpam-109	435	10	,	,	PUNCT
ejpam-109	435	11	...	...	PUNCT
ejpam-109	435	12	,	,	PUNCT
ejpam-109	435	13	|n	|n	PRON
ejpam-109	435	14	′2mk	′2mk	VERB
ejpam-109	435	15	|	|	ADV
ejpam-109	435	16	are	be	AUX
ejpam-109	435	17	all	all	DET
ejpam-109	435	18	the	the	DET
ejpam-109	435	19	elements	element	NOUN
ejpam-109	435	20	in	in	ADP
ejpam-109	435	21	s	s	PRON
ejpam-109	435	22	which	which	PRON
ejpam-109	435	23	are	be	AUX
ejpam-109	435	24	relatively	relatively	ADV
ejpam-109	435	25	prime	prime	ADJ
ejpam-109	435	26	in	in	ADP
ejpam-109	435	27	pairs	pair	NOUN
ejpam-109	435	28	.	.	PUNCT
ejpam-109	436	1	since	since	SCONJ
ejpam-109	436	2	|β2m|	|β2m|	NOUN
ejpam-109	436	3	→	→	SYM
ejpam-109	436	4	∞	∞	PROPN
ejpam-109	436	5	as	as	ADP
ejpam-109	436	6	m→∞	m→∞	NOUN
ejpam-109	436	7	,	,	PUNCT
ejpam-109	436	8	we	we	PRON
ejpam-109	436	9	can	can	AUX
ejpam-109	436	10	choose	choose	VERB
ejpam-109	436	11	x	x	PUNCT
ejpam-109	436	12	≥	≥	NUM
ejpam-109	436	13	1	1	NUM
ejpam-109	436	14	such	such	ADJ
ejpam-109	436	15	that	that	DET
ejpam-109	436	16	ϕ(m1	ϕ(m1	NOUN
ejpam-109	436	17	)	)	PUNCT
ejpam-109	436	18	≥	≥	NOUN
ejpam-109	436	19	6	6	NUM
ejpam-109	436	20	and	and	CCONJ
ejpam-109	436	21	|βϕ(m1)|	|βϕ(m1)|	PROPN
ejpam-109	436	22	>	>	X
ejpam-109	436	23	1	1	NUM
ejpam-109	436	24	for	for	ADP
ejpam-109	436	25	m1	m1	NOUN
ejpam-109	436	26	=	=	PUNCT
ejpam-109	437	1	x	x	X
ejpam-109	437	2	∏k	∏k	X
ejpam-109	437	3	i=1	i=1	X
ejpam-109	437	4	|n	|n	X
ejpam-109	437	5	′	′	NUM
ejpam-109	437	6	2mi	2mi	ADJ
ejpam-109	437	7	|	|	NOUN
ejpam-109	437	8	.	.	PUNCT
ejpam-109	438	1	then	then	ADV
ejpam-109	438	2	we	we	PRON
ejpam-109	438	3	see	see	VERB
ejpam-109	438	4	from	from	ADP
ejpam-109	438	5	theorem	theorem	ADJ
ejpam-109	438	6	2.5	2.5	NUM
ejpam-109	438	7	that	that	PRON
ejpam-109	438	8	∏	∏	NUM
ejpam-109	438	9	p|m1	p|m1	PROPN
ejpam-109	438	10	p	p	NOUN
ejpam-109	438	11	divides	divide	VERB
ejpam-109	438	12	d′	d′	PROPN
ejpam-109	438	13	ϕ(m1	ϕ(m1	NOUN
ejpam-109	438	14	)	)	PUNCT
ejpam-109	438	15	,	,	PUNCT
ejpam-109	438	16	hence	hence	ADV
ejpam-109	438	17	(	(	PUNCT
ejpam-109	438	18	|n	|n	NOUN
ejpam-109	438	19	′	′	NUM
ejpam-109	438	20	ϕ(m1	ϕ(m1	NOUN
ejpam-109	438	21	)	)	PUNCT
ejpam-109	439	1	|	|	ADV
ejpam-109	439	2	,	,	PUNCT
ejpam-109	439	3	m1	m1	NOUN
ejpam-109	439	4	)	)	PUNCT
ejpam-109	439	5	=	=	SYM
ejpam-109	440	1	1	1	X
ejpam-109	440	2	.	.	PUNCT
ejpam-109	441	1	next	next	ADJ
ejpam-109	441	2	put	put	VERB
ejpam-109	441	3	m2	m2	PROPN
ejpam-109	441	4	=	=	PROPN
ejpam-109	441	5	m1|n	m1|n	PROPN
ejpam-109	441	6	′ϕ(m1	′ϕ(m1	NOUN
ejpam-109	441	7	)	)	PUNCT
ejpam-109	441	8	|	|	ADV
ejpam-109	441	9	.	.	PUNCT
ejpam-109	442	1	then	then	ADV
ejpam-109	442	2	1	1	X
ejpam-109	442	3	<	<	X
ejpam-109	442	4	|βϕ(m1)|	|βϕ(m1)|	PROPN
ejpam-109	442	5	<	<	X
ejpam-109	442	6	|βϕ(m2)|	|βϕ(m2)|	PROPN
ejpam-109	442	7	since	since	SCONJ
ejpam-109	442	8	6	6	NUM
ejpam-109	442	9	≤	≤	NUM
ejpam-109	442	10	ϕ(m1	ϕ(m1	NOUN
ejpam-109	442	11	)	)	PUNCT
ejpam-109	442	12	<	<	X
ejpam-109	442	13	ϕ(m2	ϕ(m2	X
ejpam-109	442	14	)	)	PUNCT
ejpam-109	442	15	.	.	PUNCT
ejpam-109	443	1	by	by	ADP
ejpam-109	443	2	the	the	DET
ejpam-109	443	3	same	same	ADJ
ejpam-109	443	4	reason	reason	NOUN
ejpam-109	443	5	as	as	SCONJ
ejpam-109	443	6	mentioned	mention	VERB
ejpam-109	443	7	above	above	ADV
ejpam-109	443	8	for	for	ADP
ejpam-109	443	9	m1	m1	NOUN
ejpam-109	443	10	,	,	PUNCT
ejpam-109	443	11	we	we	PRON
ejpam-109	443	12	see	see	VERB
ejpam-109	443	13	again	again	ADV
ejpam-109	443	14	(	(	PUNCT
ejpam-109	443	15	|n	|n	NOUN
ejpam-109	443	16	′	′	PROPN
ejpam-109	443	17	ϕ(m2	ϕ(m2	NOUN
ejpam-109	443	18	)	)	PUNCT
ejpam-109	443	19	|	|	ADV
ejpam-109	443	20	,	,	PUNCT
ejpam-109	443	21	m2	m2	PROPN
ejpam-109	443	22	)	)	PUNCT
ejpam-109	443	23	=	=	SYM
ejpam-109	444	1	1	1	X
ejpam-109	444	2	.	.	PUNCT
ejpam-109	444	3	repeating	repeat	VERB
ejpam-109	444	4	these	these	DET
ejpam-109	444	5	procedures	procedure	NOUN
ejpam-109	444	6	,	,	PUNCT
ejpam-109	444	7	we	we	PRON
ejpam-109	444	8	are	be	AUX
ejpam-109	444	9	able	able	ADJ
ejpam-109	444	10	to	to	PART
ejpam-109	444	11	produce	produce	VERB
ejpam-109	444	12	an	an	DET
ejpam-109	444	13	infinite	infinite	ADJ
ejpam-109	444	14	sequence	sequence	NOUN
ejpam-109	444	15	|n	|n	NOUN
ejpam-109	444	16	′2m1	′2m1	ADP
ejpam-109	444	17	|	|	ADV
ejpam-109	444	18	,	,	PUNCT
ejpam-109	444	19	...	...	PUNCT
ejpam-109	444	20	,	,	PUNCT
ejpam-109	444	21	|n	|n	PRON
ejpam-109	444	22	′2mk	′2mk	VERB
ejpam-109	444	23	|	|	ADV
ejpam-109	444	24	,	,	PUNCT
ejpam-109	444	25	|n	|n	NOUN
ejpam-109	444	26	′ϕ(m1	′ϕ(m1	NOUN
ejpam-109	444	27	)	)	PUNCT
ejpam-109	444	28	|	|	ADV
ejpam-109	444	29	,	,	PUNCT
ejpam-109	444	30	...	...	PUNCT
ejpam-109	444	31	,	,	PUNCT
ejpam-109	444	32	|n	|n	PROPN
ejpam-109	444	33	′ϕ(mn	′ϕ(mn	NOUN
ejpam-109	444	34	)	)	PUNCT
ejpam-109	444	35	|	|	ADV
ejpam-109	444	36	,	,	PUNCT
ejpam-109	444	37	...	...	PUNCT
ejpam-109	444	38	,	,	PUNCT
ejpam-109	444	39	which	which	PRON
ejpam-109	444	40	consists	consist	VERB
ejpam-109	444	41	of	of	ADP
ejpam-109	444	42	relatively	relatively	ADV
ejpam-109	444	43	prime	prime	ADJ
ejpam-109	444	44	members	member	NOUN
ejpam-109	444	45	in	in	ADP
ejpam-109	444	46	pairs	pair	NOUN
ejpam-109	444	47	.	.	PUNCT
ejpam-109	445	1	this	this	PRON
ejpam-109	445	2	is	be	AUX
ejpam-109	445	3	however	however	ADV
ejpam-109	445	4	contrary	contrary	ADJ
ejpam-109	445	5	to	to	ADP
ejpam-109	445	6	the	the	DET
ejpam-109	445	7	assumption	assumption	NOUN
ejpam-109	445	8	and	and	CCONJ
ejpam-109	445	9	therefore	therefore	ADV
ejpam-109	445	10	the	the	DET
ejpam-109	445	11	assertion	assertion	NOUN
ejpam-109	445	12	follows	follow	VERB
ejpam-109	445	13	.	.	PUNCT
ejpam-109	446	1	it	it	PRON
ejpam-109	446	2	is	be	AUX
ejpam-109	446	3	obvious	obvious	ADJ
ejpam-109	446	4	that	that	SCONJ
ejpam-109	446	5	theorem	theorem	VERB
ejpam-109	446	6	6.2	6.2	NUM
ejpam-109	446	7	contains	contain	VERB
ejpam-109	446	8	the	the	DET
ejpam-109	446	9	statement	statement	NOUN
ejpam-109	446	10	that	that	SCONJ
ejpam-109	446	11	there	there	PRON
ejpam-109	446	12	are	be	VERB
ejpam-109	446	13	infinitely	infinitely	ADV
ejpam-109	446	14	many	many	ADJ
ejpam-109	446	15	irregular	irregular	ADJ
ejpam-109	446	16	primes	prime	NOUN
ejpam-109	446	17	.	.	PUNCT
ejpam-109	447	1	acknowledgment	acknowledgment	NOUN
ejpam-109	447	2	the	the	DET
ejpam-109	447	3	author	author	NOUN
ejpam-109	447	4	is	be	AUX
ejpam-109	447	5	grateful	grateful	ADJ
ejpam-109	447	6	to	to	ADP
ejpam-109	447	7	the	the	DET
ejpam-109	447	8	anonymous	anonymous	ADJ
ejpam-109	447	9	referee	referee	NOUN
ejpam-109	447	10	for	for	ADP
ejpam-109	447	11	his	his	PRON
ejpam-109	447	12	/	/	SYM
ejpam-109	447	13	her	her	PRON
ejpam-109	447	14	useful	useful	ADJ
ejpam-109	447	15	comments	comment	NOUN
ejpam-109	447	16	on	on	ADP
ejpam-109	447	17	the	the	DET
ejpam-109	447	18	first	first	ADJ
ejpam-109	447	19	version	version	NOUN
ejpam-109	447	20	of	of	ADP
ejpam-109	447	21	this	this	DET
ejpam-109	447	22	paper	paper	NOUN
ejpam-109	447	23	.	.	PUNCT
ejpam-109	448	1	references	reference	NOUN
ejpam-109	448	2	21	21	NUM
ejpam-109	448	3	references	reference	NOUN
ejpam-109	448	4	[	[	X
ejpam-109	448	5	1	1	NUM
ejpam-109	448	6	]	]	PUNCT
ejpam-109	448	7	t.	t.	NOUN
ejpam-109	448	8	agoh	agoh	PROPN
ejpam-109	448	9	,	,	PUNCT
ejpam-109	448	10	on	on	ADP
ejpam-109	448	11	bernoulli	bernoulli	PROPN
ejpam-109	448	12	and	and	CCONJ
ejpam-109	448	13	euler	euler	NOUN
ejpam-109	448	14	numbers	number	NOUN
ejpam-109	448	15	,	,	PUNCT
ejpam-109	448	16	manuscripta	manuscripta	NOUN
ejpam-109	448	17	math	math	NOUN
ejpam-109	448	18	.	.	PUNCT
ejpam-109	449	1	61	61	NUM
ejpam-109	449	2	(	(	PUNCT
ejpam-109	449	3	1988	1988	NUM
ejpam-109	449	4	)	)	PUNCT
ejpam-109	449	5	,	,	PUNCT
ejpam-109	449	6	1	1	NUM
ejpam-109	449	7	-	-	SYM
ejpam-109	449	8	10	10	NUM
ejpam-109	449	9	.	.	PUNCT
ejpam-109	450	1	[	[	X
ejpam-109	450	2	2	2	X
ejpam-109	450	3	]	]	PUNCT
ejpam-109	450	4	t.	t.	NOUN
ejpam-109	450	5	agoh	agoh	NOUN
ejpam-109	450	6	,	,	PUNCT
ejpam-109	450	7	on	on	ADP
ejpam-109	450	8	giuga	giuga	PROPN
ejpam-109	450	9	’s	’s	PART
ejpam-109	450	10	conjecture	conjecture	NOUN
ejpam-109	450	11	,	,	PUNCT
ejpam-109	450	12	manuscripta	manuscripta	NOUN
ejpam-109	450	13	math	math	NOUN
ejpam-109	450	14	.	.	PUNCT
ejpam-109	451	1	87	87	NUM
ejpam-109	451	2	(	(	PUNCT
ejpam-109	451	3	1995	1995	NUM
ejpam-109	451	4	)	)	PUNCT
ejpam-109	451	5	,	,	PUNCT
ejpam-109	451	6	501	501	NUM
ejpam-109	451	7	-	-	SYM
ejpam-109	451	8	510	510	NUM
ejpam-109	451	9	.	.	PUNCT
ejpam-109	452	1	[	[	X
ejpam-109	452	2	3	3	X
ejpam-109	452	3	]	]	X
ejpam-109	452	4	w.r	w.r	PROPN
ejpam-109	452	5	.	.	PROPN
ejpam-109	452	6	alford	alford	PROPN
ejpam-109	452	7	,	,	PUNCT
ejpam-109	452	8	a.	a.	NOUN
ejpam-109	452	9	granville	granville	PROPN
ejpam-109	452	10	and	and	CCONJ
ejpam-109	452	11	c.	c.	PROPN
ejpam-109	452	12	pomerance	pomerance	NOUN
ejpam-109	452	13	,	,	PUNCT
ejpam-109	452	14	there	there	PRON
ejpam-109	452	15	are	be	VERB
ejpam-109	452	16	infinitely	infinitely	ADV
ejpam-109	452	17	many	many	ADJ
ejpam-109	452	18	carmichael	carmichael	PROPN
ejpam-109	452	19	numbers	number	NOUN
ejpam-109	452	20	,	,	PUNCT
ejpam-109	452	21	ann	ann	PROPN
ejpam-109	452	22	.	.	PROPN
ejpam-109	452	23	of	of	ADP
ejpam-109	452	24	math	math	NOUN
ejpam-109	452	25	.	.	PUNCT
ejpam-109	453	1	140	140	NUM
ejpam-109	453	2	(	(	PUNCT
ejpam-109	453	3	1994	1994	NUM
ejpam-109	453	4	)	)	PUNCT
ejpam-109	453	5	,	,	PUNCT
ejpam-109	453	6	1	1	NUM
ejpam-109	453	7	-	-	SYM
ejpam-109	453	8	20	20	NUM
ejpam-109	453	9	.	.	PUNCT
ejpam-109	454	1	[	[	X
ejpam-109	454	2	4	4	X
ejpam-109	454	3	]	]	X
ejpam-109	454	4	d.	d.	PROPN
ejpam-109	454	5	borwein	borwein	PROPN
ejpam-109	454	6	,	,	PUNCT
ejpam-109	454	7	j.m	j.m	PROPN
ejpam-109	454	8	.	.	PROPN
ejpam-109	454	9	borwein	borwein	PROPN
ejpam-109	454	10	,	,	PUNCT
ejpam-109	454	11	p.b	p.b	PROPN
ejpam-109	454	12	.	.	PROPN
ejpam-109	454	13	borwein	borwein	PROPN
ejpam-109	454	14	and	and	CCONJ
ejpam-109	454	15	r.	r.	PROPN
ejpam-109	454	16	girgensohn	girgensohn	PROPN
ejpam-109	454	17	,	,	PUNCT
ejpam-109	454	18	giuga	giuga	PROPN
ejpam-109	454	19	’s	’s	PART
ejpam-109	454	20	conjecture	conjecture	NOUN
ejpam-109	454	21	on	on	ADP
ejpam-109	454	22	primality	primality	NOUN
ejpam-109	454	23	,	,	PUNCT
ejpam-109	454	24	amer	amer	PROPN
ejpam-109	454	25	.	.	PROPN
ejpam-109	454	26	math	math	PROPN
ejpam-109	454	27	.	.	PUNCT
ejpam-109	455	1	monthly	monthly	ADJ
ejpam-109	455	2	,	,	PUNCT
ejpam-109	455	3	103	103	NUM
ejpam-109	455	4	(	(	PUNCT
ejpam-109	455	5	1996	1996	NUM
ejpam-109	455	6	)	)	PUNCT
ejpam-109	455	7	,	,	PUNCT
ejpam-109	455	8	40	40	NUM
ejpam-109	455	9	-	-	SYM
ejpam-109	455	10	50	50	NUM
ejpam-109	455	11	.	.	PUNCT
ejpam-109	456	1	[	[	X
ejpam-109	456	2	5	5	X
ejpam-109	456	3	]	]	PUNCT
ejpam-109	456	4	j.	j.	PROPN
ejpam-109	456	5	buhler	buhler	PROPN
ejpam-109	456	6	,	,	PUNCT
ejpam-109	456	7	r.	r.	PROPN
ejpam-109	456	8	crandall	crandall	PROPN
ejpam-109	456	9	,	,	PUNCT
ejpam-109	456	10	r.	r.	PROPN
ejpam-109	456	11	ernvall	ernvall	PROPN
ejpam-109	456	12	,	,	PUNCT
ejpam-109	456	13	t.	t.	NOUN
ejpam-109	456	14	metsankyla	metsankyla	NOUN
ejpam-109	456	15	and	and	CCONJ
ejpam-109	456	16	m.	m.	NOUN
ejpam-109	456	17	shokrollahi	shokrollahi	PROPN
ejpam-109	456	18	,	,	PUNCT
ejpam-109	456	19	irregular	irregular	ADJ
ejpam-109	456	20	primes	prime	NOUN
ejpam-109	456	21	and	and	CCONJ
ejpam-109	456	22	cyclotomic	cyclotomic	ADJ
ejpam-109	456	23	invariants	invariant	NOUN
ejpam-109	456	24	to	to	ADP
ejpam-109	456	25	12	12	NUM
ejpam-109	456	26	million	million	NUM
ejpam-109	456	27	,	,	PUNCT
ejpam-109	456	28	j.	j.	PROPN
ejpam-109	456	29	symbolic	symbolic	PROPN
ejpam-109	456	30	comput	comput	PROPN
ejpam-109	456	31	.	.	PUNCT
ejpam-109	457	1	31(2001	31(2001	NUM
ejpam-109	457	2	)	)	PUNCT
ejpam-109	457	3	,	,	PUNCT
ejpam-109	457	4	89–96	89–96	NUM
ejpam-109	457	5	.	.	PUNCT
ejpam-109	458	1	[	[	X
ejpam-109	458	2	6	6	NUM
ejpam-109	458	3	]	]	PUNCT
ejpam-109	458	4	l.	l.	PROPN
ejpam-109	458	5	carlitz	carlitz	PROPN
ejpam-109	458	6	,	,	PUNCT
ejpam-109	458	7	note	note	VERB
ejpam-109	458	8	on	on	ADP
ejpam-109	458	9	irregular	irregular	ADJ
ejpam-109	458	10	primes	prime	NOUN
ejpam-109	458	11	,	,	PUNCT
ejpam-109	458	12	proc	proc	NOUN
ejpam-109	458	13	.	.	PUNCT
ejpam-109	459	1	amer	amer	PROPN
ejpam-109	459	2	.	.	PUNCT
ejpam-109	459	3	math	math	PROPN
ejpam-109	459	4	.	.	PUNCT
ejpam-109	460	1	soc	soc	PROPN
ejpam-109	460	2	.	.	PUNCT
ejpam-109	461	1	5	5	NUM
ejpam-109	461	2	(	(	PUNCT
ejpam-109	461	3	1954	1954	NUM
ejpam-109	461	4	)	)	PUNCT
ejpam-109	461	5	,	,	PUNCT
ejpam-109	461	6	329	329	NUM
ejpam-109	461	7	-	-	SYM
ejpam-109	461	8	331	331	NUM
ejpam-109	461	9	.	.	PUNCT
ejpam-109	462	1	[	[	X
ejpam-109	462	2	7	7	X
ejpam-109	462	3	]	]	X
ejpam-109	462	4	g.	g.	NOUN
ejpam-109	462	5	fee	fee	NOUN
ejpam-109	462	6	and	and	CCONJ
ejpam-109	462	7	s.	s.	PROPN
ejpam-109	462	8	plouffe	plouffe	PROPN
ejpam-109	462	9	,	,	PUNCT
ejpam-109	462	10	an	an	DET
ejpam-109	462	11	efficient	efficient	ADJ
ejpam-109	462	12	algorithm	algorithm	NOUN
ejpam-109	462	13	for	for	ADP
ejpam-109	462	14	the	the	DET
ejpam-109	462	15	computation	computation	NOUN
ejpam-109	462	16	of	of	ADP
ejpam-109	462	17	bernoulli	bernoulli	NOUN
ejpam-109	462	18	numbers	number	NOUN
ejpam-109	462	19	,	,	PUNCT
ejpam-109	462	20	2007	2007	NUM
ejpam-109	462	21	,	,	PUNCT
ejpam-109	462	22	http://arxiv.org/pdf/math.nt/0702300	http://arxiv.org/pdf/math.nt/0702300	PROPN
ejpam-109	462	23	.	.	PUNCT
ejpam-109	463	1	[	[	X
ejpam-109	463	2	8	8	NUM
ejpam-109	463	3	]	]	X
ejpam-109	463	4	g.	g.	PROPN
ejpam-109	463	5	giuga	giuga	PROPN
ejpam-109	463	6	,	,	PUNCT
ejpam-109	463	7	su	su	PROPN
ejpam-109	463	8	una	una	PROPN
ejpam-109	463	9	presumibile	presumibile	PROPN
ejpam-109	463	10	proprietà	proprietà	PROPN
ejpam-109	463	11	caratteristica	caratteristica	PROPN
ejpam-109	463	12	dei	dei	PROPN
ejpam-109	463	13	numeri	numeri	PROPN
ejpam-109	463	14	primi	primi	PROPN
ejpam-109	463	15	,	,	PUNCT
ejpam-109	463	16	ist	ist	PROPN
ejpam-109	463	17	.	.	PROPN
ejpam-109	464	1	lombardo	lombardo	PROPN
ejpam-109	464	2	sci	sci	PROPN
ejpam-109	464	3	.	.	PROPN
ejpam-109	464	4	lett	lett	PROPN
ejpam-109	464	5	.	.	PUNCT
ejpam-109	465	1	rend	rend	VERB
ejpam-109	465	2	.	.	PUNCT
ejpam-109	466	1	cl	cl	NOUN
ejpam-109	466	2	.	.	PUNCT
ejpam-109	467	1	sci	sci	PROPN
ejpam-109	467	2	.	.	PROPN
ejpam-109	467	3	mat	mat	PROPN
ejpam-109	467	4	.	.	PUNCT
ejpam-109	468	1	nat	nat	PROPN
ejpam-109	468	2	.	.	PUNCT
ejpam-109	469	1	83	83	NUM
ejpam-109	469	2	(	(	PUNCT
ejpam-109	469	3	1950	1950	NUM
ejpam-109	469	4	)	)	PUNCT
ejpam-109	469	5	,	,	PUNCT
ejpam-109	469	6	511	511	NUM
ejpam-109	469	7	-	-	SYM
ejpam-109	469	8	528	528	NUM
ejpam-109	469	9	.	.	PUNCT
ejpam-109	470	1	[	[	X
ejpam-109	470	2	9	9	NUM
ejpam-109	470	3	]	]	PUNCT
ejpam-109	470	4	k.	k.	PROPN
ejpam-109	470	5	iwasawa	iwasawa	PROPN
ejpam-109	470	6	,	,	PUNCT
ejpam-109	470	7	lectures	lecture	VERB
ejpam-109	470	8	on	on	ADP
ejpam-109	470	9	p	p	NOUN
ejpam-109	470	10	-	-	PUNCT
ejpam-109	470	11	adic	adic	ADJ
ejpam-109	470	12	l	l	NOUN
ejpam-109	470	13	functions	function	NOUN
ejpam-109	470	14	,	,	PUNCT
ejpam-109	470	15	annals	annal	NOUN
ejpam-109	470	16	of	of	ADP
ejpam-109	470	17	math	math	NOUN
ejpam-109	470	18	.	.	PUNCT
ejpam-109	471	1	studies	study	NOUN
ejpam-109	471	2	no	no	INTJ
ejpam-109	471	3	.	.	PROPN
ejpam-109	471	4	74	74	NUM
ejpam-109	471	5	,	,	PUNCT
ejpam-109	471	6	princeton	princeton	PROPN
ejpam-109	471	7	univ	univ	PROPN
ejpam-109	471	8	.	.	PUNCT
ejpam-109	472	1	press	press	PROPN
ejpam-109	472	2	,	,	PUNCT
ejpam-109	472	3	princeton	princeton	PROPN
ejpam-109	472	4	,	,	PUNCT
ejpam-109	472	5	1972	1972	NUM
ejpam-109	472	6	.	.	PUNCT
ejpam-109	473	1	[	[	X
ejpam-109	473	2	10	10	NUM
ejpam-109	473	3	]	]	X
ejpam-109	473	4	k.l	k.l	PROPN
ejpam-109	473	5	.	.	PROPN
ejpam-109	473	6	jensen	jensen	PROPN
ejpam-109	473	7	,	,	PUNCT
ejpam-109	473	8	om	om	PROPN
ejpam-109	473	9	talteoretiske	talteoretiske	PROPN
ejpam-109	473	10	egenskaber	egenskaber	PROPN
ejpam-109	473	11	ved	ve	VERB
ejpam-109	473	12	de	de	ADP
ejpam-109	473	13	bernoulliske	bernoulliske	PROPN
ejpam-109	473	14	tal	tal	PROPN
ejpam-109	473	15	,	,	PUNCT
ejpam-109	473	16	nyt	nyt	PROPN
ejpam-109	473	17	tidsskrift	tidsskrift	PROPN
ejpam-109	473	18	f.	f.	PROPN
ejpam-109	473	19	math	math	PROPN
ejpam-109	473	20	.	.	PUNCT
ejpam-109	474	1	,	,	PUNCT
ejpam-109	474	2	b	b	X
ejpam-109	474	3	,	,	PUNCT
ejpam-109	474	4	26	26	NUM
ejpam-109	474	5	(	(	PUNCT
ejpam-109	474	6	1915	1915	NUM
ejpam-109	474	7	)	)	PUNCT
ejpam-109	474	8	,	,	PUNCT
ejpam-109	474	9	73	73	NUM
ejpam-109	474	10	-	-	SYM
ejpam-109	474	11	83	83	NUM
ejpam-109	474	12	.	.	PUNCT
ejpam-109	475	1	[	[	X
ejpam-109	475	2	11	11	NUM
ejpam-109	475	3	]	]	PUNCT
ejpam-109	475	4	w.	w.	PROPN
ejpam-109	475	5	johnson	johnson	PROPN
ejpam-109	475	6	,	,	PUNCT
ejpam-109	475	7	p	p	ADJ
ejpam-109	475	8	-	-	PUNCT
ejpam-109	475	9	adic	adic	ADJ
ejpam-109	475	10	proofs	proof	NOUN
ejpam-109	475	11	of	of	ADP
ejpam-109	475	12	congruences	congruence	NOUN
ejpam-109	475	13	for	for	ADP
ejpam-109	475	14	the	the	DET
ejpam-109	475	15	bernoulli	bernoulli	PROPN
ejpam-109	475	16	numbers	number	NOUN
ejpam-109	475	17	,	,	PUNCT
ejpam-109	475	18	j.	j.	PROPN
ejpam-109	475	19	number	number	PROPN
ejpam-109	475	20	theory	theory	NOUN
ejpam-109	475	21	,	,	PUNCT
ejpam-109	475	22	7	7	NUM
ejpam-109	475	23	(	(	PUNCT
ejpam-109	475	24	1975	1975	NUM
ejpam-109	475	25	)	)	PUNCT
ejpam-109	475	26	,	,	PUNCT
ejpam-109	475	27	251	251	NUM
ejpam-109	475	28	-	-	SYM
ejpam-109	475	29	265	265	NUM
ejpam-109	475	30	.	.	PUNCT
ejpam-109	476	1	[	[	X
ejpam-109	476	2	12	12	NUM
ejpam-109	476	3	]	]	X
ejpam-109	476	4	b.c	b.c	PROPN
ejpam-109	476	5	.	.	PROPN
ejpam-109	476	6	kellner	kellner	PROPN
ejpam-109	476	7	,	,	PUNCT
ejpam-109	476	8	the	the	DET
ejpam-109	476	9	equivalence	equivalence	NOUN
ejpam-109	476	10	of	of	ADP
ejpam-109	476	11	giuga	giuga	PROPN
ejpam-109	476	12	’s	’s	PART
ejpam-109	476	13	and	and	CCONJ
ejpam-109	476	14	agoh	agoh	VERB
ejpam-109	476	15	’s	’s	PART
ejpam-109	476	16	conjectures	conjecture	NOUN
ejpam-109	476	17	,	,	PUNCT
ejpam-109	476	18	2004	2004	NUM
ejpam-109	476	19	,	,	PUNCT
ejpam-109	476	20	http://arxiv.org/abs/math.nt/0409259	http://arxiv.org/abs/math.nt/0409259	NOUN
ejpam-109	476	21	.	.	PUNCT
ejpam-109	477	1	[	[	X
ejpam-109	477	2	13	13	NUM
ejpam-109	477	3	]	]	X
ejpam-109	477	4	e.	e.	PROPN
ejpam-109	477	5	lehmer	lehmer	PROPN
ejpam-109	477	6	,	,	PUNCT
ejpam-109	477	7	on	on	ADP
ejpam-109	477	8	congruences	congruence	NOUN
ejpam-109	477	9	involving	involve	VERB
ejpam-109	477	10	bernoulli	bernoulli	NOUN
ejpam-109	477	11	numbers	number	NOUN
ejpam-109	477	12	and	and	CCONJ
ejpam-109	477	13	the	the	DET
ejpam-109	477	14	quotients	quotient	NOUN
ejpam-109	477	15	of	of	ADP
ejpam-109	477	16	fermat	fermat	PROPN
ejpam-109	477	17	and	and	CCONJ
ejpam-109	477	18	wilson	wilson	PROPN
ejpam-109	477	19	,	,	PUNCT
ejpam-109	477	20	ann	ann	PROPN
ejpam-109	477	21	.	.	PROPN
ejpam-109	477	22	of	of	ADP
ejpam-109	477	23	math	math	NOUN
ejpam-109	477	24	.	.	PUNCT
ejpam-109	478	1	39	39	NUM
ejpam-109	478	2	(	(	PUNCT
ejpam-109	478	3	1938	1938	NUM
ejpam-109	478	4	)	)	PUNCT
ejpam-109	478	5	,	,	PUNCT
ejpam-109	478	6	350	350	NUM
ejpam-109	478	7	-	-	SYM
ejpam-109	478	8	360	360	NUM
ejpam-109	478	9	.	.	PUNCT
ejpam-109	479	1	[	[	X
ejpam-109	479	2	14	14	NUM
ejpam-109	479	3	]	]	X
ejpam-109	479	4	š	š	PROPN
ejpam-109	479	5	.	.	PUNCT
ejpam-109	479	6	porubský	porubský	ADJ
ejpam-109	479	7	,	,	PUNCT
ejpam-109	479	8	voronoï	voronoï	ADJ
ejpam-109	479	9	type	type	NOUN
ejpam-109	479	10	congruences	congruence	NOUN
ejpam-109	479	11	for	for	ADP
ejpam-109	479	12	bernoulli	bernoulli	NOUN
ejpam-109	479	13	numbers	number	NOUN
ejpam-109	479	14	,	,	PUNCT
ejpam-109	479	15	voronoï	voronoï	ADJ
ejpam-109	479	16	’s	’s	PART
ejpam-109	479	17	impact	impact	NOUN
ejpam-109	479	18	on	on	ADP
ejpam-109	479	19	modern	modern	ADJ
ejpam-109	479	20	science	science	NOUN
ejpam-109	479	21	,	,	PUNCT
ejpam-109	479	22	book	book	NOUN
ejpam-109	479	23	1	1	NUM
ejpam-109	479	24	,	,	PUNCT
ejpam-109	479	25	proc	proc	NOUN
ejpam-109	479	26	.	.	PUNCT
ejpam-109	480	1	inst	inst	PROPN
ejpam-109	480	2	.	.	PROPN
ejpam-109	480	3	of	of	ADP
ejpam-109	480	4	math	math	NOUN
ejpam-109	480	5	.	.	PUNCT
ejpam-109	480	6	,	,	PUNCT
ejpam-109	480	7	nat	nat	PROPN
ejpam-109	480	8	.	.	PUNCT
ejpam-109	481	1	acad	acad	PROPN
ejpam-109	481	2	.	.	PUNCT
ejpam-109	482	1	sci	sci	PROPN
ejpam-109	482	2	.	.	PROPN
ejpam-109	482	3	of	of	ADP
ejpam-109	482	4	ukraine	ukraine	PROPN
ejpam-109	482	5	,	,	PUNCT
ejpam-109	482	6	1998	1998	NUM
ejpam-109	482	7	,	,	PUNCT
ejpam-109	482	8	71	71	NUM
ejpam-109	482	9	-	-	SYM
ejpam-109	482	10	98	98	NUM
ejpam-109	482	11	.	.	PUNCT
ejpam-109	483	1	[	[	X
ejpam-109	483	2	15	15	NUM
ejpam-109	483	3	]	]	X
ejpam-109	483	4	i.sh	i.sh	PROPN
ejpam-109	483	5	.	.	PUNCT
ejpam-109	484	1	slavutskii	slavutskii	PROPN
ejpam-109	484	2	,	,	PUNCT
ejpam-109	484	3	staudt	staudt	NOUN
ejpam-109	484	4	and	and	CCONJ
ejpam-109	484	5	arithmetical	arithmetical	ADJ
ejpam-109	484	6	properties	property	NOUN
ejpam-109	484	7	on	on	ADP
ejpam-109	484	8	bernoulli	bernoulli	NOUN
ejpam-109	484	9	numbers	number	NOUN
ejpam-109	484	10	,	,	PUNCT
ejpam-109	484	11	hist	hist	PROPN
ejpam-109	484	12	.	.	PUNCT
ejpam-109	485	1	sci	sci	PROPN
ejpam-109	485	2	.	.	PROPN
ejpam-109	485	3	5	5	NUM
ejpam-109	485	4	(	(	PUNCT
ejpam-109	485	5	1995	1995	NUM
ejpam-109	485	6	)	)	PUNCT
ejpam-109	485	7	,	,	PUNCT
ejpam-109	485	8	70	70	NUM
ejpam-109	485	9	-	-	SYM
ejpam-109	485	10	74	74	NUM
ejpam-109	485	11	.	.	PUNCT
ejpam-109	486	1	[	[	X
ejpam-109	486	2	16	16	NUM
ejpam-109	486	3	]	]	X
ejpam-109	486	4	z.-w	z.-w	PROPN
ejpam-109	486	5	.	.	PUNCT
ejpam-109	486	6	sun	sun	PROPN
ejpam-109	486	7	,	,	PUNCT
ejpam-109	486	8	general	general	ADJ
ejpam-109	486	9	congruences	congruence	VERB
ejpam-109	486	10	for	for	ADP
ejpam-109	486	11	bernoulli	bernoulli	NOUN
ejpam-109	486	12	polynomials	polynomial	NOUN
ejpam-109	486	13	,	,	PUNCT
ejpam-109	486	14	discrete	discrete	ADJ
ejpam-109	486	15	math	math	NOUN
ejpam-109	486	16	.	.	PUNCT
ejpam-109	487	1	262(2003	262(2003	NUM
ejpam-109	487	2	)	)	PUNCT
ejpam-109	487	3	,	,	PUNCT
ejpam-109	487	4	253	253	NUM
ejpam-109	487	5	-	-	SYM
ejpam-109	487	6	276	276	NUM
ejpam-109	487	7	.	.	PUNCT
ejpam-109	488	1	[	[	X
ejpam-109	488	2	17	17	NUM
ejpam-109	488	3	]	]	X
ejpam-109	488	4	l.c	l.c	PROPN
ejpam-109	488	5	.	.	PROPN
ejpam-109	488	6	washington	washington	PROPN
ejpam-109	488	7	,	,	PUNCT
ejpam-109	488	8	introduction	introduction	NOUN
ejpam-109	488	9	to	to	ADP
ejpam-109	488	10	cyclotomic	cyclotomic	ADJ
ejpam-109	488	11	fields	field	NOUN
ejpam-109	488	12	,	,	PUNCT
ejpam-109	488	13	springer	springer	NOUN
ejpam-109	488	14	,	,	PUNCT
ejpam-109	488	15	new	new	PROPN
ejpam-109	488	16	york	york	PROPN
ejpam-109	488	17	,	,	PUNCT
ejpam-109	488	18	1982	1982	NUM
ejpam-109	488	19	.	.	PUNCT
ejpam-109	489	1	[	[	X
ejpam-109	489	2	18	18	NUM
ejpam-109	489	3	]	]	X
ejpam-109	489	4	p.t	p.t	PROPN
ejpam-109	489	5	.	.	PROPN
ejpam-109	489	6	young	young	ADJ
ejpam-109	489	7	,	,	PUNCT
ejpam-109	489	8	congruences	congruence	VERB
ejpam-109	489	9	for	for	ADP
ejpam-109	489	10	bernoulli	bernoulli	PROPN
ejpam-109	489	11	,	,	PUNCT
ejpam-109	489	12	euler	euler	NOUN
ejpam-109	489	13	,	,	PUNCT
ejpam-109	489	14	and	and	CCONJ
ejpam-109	489	15	stirling	stirling	NOUN
ejpam-109	489	16	numbers	number	NOUN
ejpam-109	489	17	,	,	PUNCT
ejpam-109	489	18	j.	j.	PROPN
ejpam-109	489	19	number	number	PROPN
ejpam-109	489	20	theory	theory	NOUN
ejpam-109	489	21	,	,	PUNCT
ejpam-109	489	22	78(1999	78(1999	NUM
ejpam-109	489	23	)	)	PUNCT
ejpam-109	489	24	,	,	PUNCT
ejpam-109	489	25	204	204	NUM
ejpam-109	489	26	-	-	SYM
ejpam-109	489	27	227	227	NUM
ejpam-109	489	28	,	,	PUNCT
ejpam-109	489	29	[	[	X
ejpam-109	489	30	19	19	NUM
ejpam-109	489	31	]	]	X
ejpam-109	489	32	p.t	p.t	PROPN
ejpam-109	489	33	.	.	PROPN
ejpam-109	489	34	young	young	ADJ
ejpam-109	489	35	,	,	PUNCT
ejpam-109	489	36	degenerate	degenerate	ADJ
ejpam-109	489	37	and	and	CCONJ
ejpam-109	489	38	n	n	CCONJ
ejpam-109	489	39	-	-	PUNCT
ejpam-109	489	40	adic	adic	ADJ
ejpam-109	489	41	versions	version	NOUN
ejpam-109	489	42	of	of	ADP
ejpam-109	489	43	kummer	kummer	NOUN
ejpam-109	489	44	’s	’s	PART
ejpam-109	489	45	congruences	congruence	NOUN
ejpam-109	489	46	for	for	ADP
ejpam-109	489	47	values	value	NOUN
ejpam-109	489	48	of	of	ADP
ejpam-109	489	49	bernoulli	bernoulli	NOUN
ejpam-109	489	50	polynomials	polynomial	NOUN
ejpam-109	489	51	,	,	PUNCT
ejpam-109	489	52	discrete	discrete	ADJ
ejpam-109	489	53	math	math	NOUN
ejpam-109	489	54	.	.	PUNCT
ejpam-109	490	1	285(2004	285(2004	NUM
ejpam-109	490	2	)	)	PUNCT
ejpam-109	490	3	,	,	PUNCT
ejpam-109	490	4	289	289	NUM
ejpam-109	490	5	-	-	SYM
ejpam-109	490	6	296	296	NUM
ejpam-109	490	7	.	.	PUNCT