id	sid	tid	token	lemma	pos
ejpam-5372	1	1	european	european	PROPN
ejpam-5372	1	2	journal	journal	PROPN
ejpam-5372	1	3	of	of	ADP
ejpam-5372	1	4	pure	pure	ADJ
ejpam-5372	1	5	and	and	CCONJ
ejpam-5372	1	6	applied	apply	VERB
ejpam-5372	1	7	mathematics	mathematic	NOUN
ejpam-5372	1	8	vol	vol	NOUN
ejpam-5372	1	9	.	.	PROPN
ejpam-5372	2	1	17	17	NUM
ejpam-5372	2	2	,	,	PUNCT
ejpam-5372	2	3	no	no	INTJ
ejpam-5372	2	4	.	.	NOUN
ejpam-5372	2	5	4	4	NUM
ejpam-5372	2	6	,	,	PUNCT
ejpam-5372	2	7	2024	2024	NUM
ejpam-5372	2	8	,	,	PUNCT
ejpam-5372	2	9	2431	2431	NUM
ejpam-5372	2	10	-	-	SYM
ejpam-5372	2	11	2447	2447	NUM
ejpam-5372	2	12	issn	issn	PROPN
ejpam-5372	2	13	1307	1307	NUM
ejpam-5372	2	14	-	-	SYM
ejpam-5372	2	15	5543	5543	NUM
ejpam-5372	2	16	–	–	PUNCT
ejpam-5372	2	17	ejpam.com	ejpam.com	X
ejpam-5372	2	18	published	publish	VERB
ejpam-5372	2	19	by	by	ADP
ejpam-5372	2	20	new	new	PROPN
ejpam-5372	2	21	york	york	PROPN
ejpam-5372	2	22	business	business	PROPN
ejpam-5372	2	23	global	global	ADJ
ejpam-5372	2	24	semi	semi	ADJ
ejpam-5372	2	25	-	-	ADJ
ejpam-5372	2	26	primitive	primitive	ADJ
ejpam-5372	2	27	roots	root	NOUN
ejpam-5372	2	28	and	and	CCONJ
ejpam-5372	2	29	irreducible	irreducible	ADJ
ejpam-5372	2	30	quadratic	quadratic	ADJ
ejpam-5372	2	31	forms	form	NOUN
ejpam-5372	2	32	marc	marc	PROPN
ejpam-5372	2	33	wolf1,∗	wolf1,∗	NOUN
ejpam-5372	2	34	,	,	PUNCT
ejpam-5372	2	35	françois	françois	X
ejpam-5372	2	36	wolf1	wolf1	NOUN
ejpam-5372	2	37	1	1	NUM
ejpam-5372	2	38	department	department	NOUN
ejpam-5372	2	39	of	of	ADP
ejpam-5372	2	40	mathematical	mathematical	ADJ
ejpam-5372	2	41	sciences	science	NOUN
ejpam-5372	2	42	at	at	ADP
ejpam-5372	2	43	tsoftemail	tsoftemail	NOUN
ejpam-5372	2	44	,	,	PUNCT
ejpam-5372	2	45	france	france	PROPN
ejpam-5372	2	46	abstract	abstract	NOUN
ejpam-5372	2	47	.	.	PUNCT
ejpam-5372	3	1	modulo	modulo	VERB
ejpam-5372	3	2	a	a	DET
ejpam-5372	3	3	prime	prime	ADJ
ejpam-5372	3	4	number	number	NOUN
ejpam-5372	3	5	,	,	PUNCT
ejpam-5372	3	6	we	we	PRON
ejpam-5372	3	7	define	define	VERB
ejpam-5372	3	8	semi	semi	ADJ
ejpam-5372	3	9	-	-	ADJ
ejpam-5372	3	10	primitive	primitive	ADJ
ejpam-5372	3	11	roots	root	NOUN
ejpam-5372	3	12	as	as	ADP
ejpam-5372	3	13	the	the	DET
ejpam-5372	3	14	square	square	NOUN
ejpam-5372	3	15	of	of	ADP
ejpam-5372	3	16	primitive	primitive	ADJ
ejpam-5372	3	17	roots	root	NOUN
ejpam-5372	3	18	.	.	PUNCT
ejpam-5372	4	1	we	we	PRON
ejpam-5372	4	2	present	present	VERB
ejpam-5372	4	3	a	a	DET
ejpam-5372	4	4	method	method	NOUN
ejpam-5372	4	5	for	for	ADP
ejpam-5372	4	6	calculating	calculate	VERB
ejpam-5372	4	7	primitive	primitive	ADJ
ejpam-5372	4	8	roots	root	NOUN
ejpam-5372	4	9	from	from	ADP
ejpam-5372	4	10	quadratic	quadratic	ADJ
ejpam-5372	4	11	residues	residue	NOUN
ejpam-5372	4	12	,	,	PUNCT
ejpam-5372	4	13	including	include	VERB
ejpam-5372	4	14	semiprimitive	semiprimitive	ADJ
ejpam-5372	4	15	roots	root	NOUN
ejpam-5372	4	16	.	.	PUNCT
ejpam-5372	5	1	we	we	PRON
ejpam-5372	5	2	then	then	ADV
ejpam-5372	5	3	present	present	VERB
ejpam-5372	5	4	progressions	progression	NOUN
ejpam-5372	5	5	that	that	PRON
ejpam-5372	5	6	generate	generate	VERB
ejpam-5372	5	7	primitive	primitive	ADJ
ejpam-5372	5	8	and	and	CCONJ
ejpam-5372	5	9	semi	semi	ADJ
ejpam-5372	5	10	-	-	ADJ
ejpam-5372	5	11	primitive	primitive	ADJ
ejpam-5372	5	12	roots	root	NOUN
ejpam-5372	5	13	,	,	PUNCT
ejpam-5372	5	14	and	and	CCONJ
ejpam-5372	5	15	deduce	deduce	VERB
ejpam-5372	5	16	an	an	DET
ejpam-5372	5	17	algorithm	algorithm	NOUN
ejpam-5372	5	18	to	to	PART
ejpam-5372	5	19	obtain	obtain	VERB
ejpam-5372	5	20	the	the	DET
ejpam-5372	5	21	full	full	ADJ
ejpam-5372	5	22	set	set	NOUN
ejpam-5372	5	23	of	of	ADP
ejpam-5372	5	24	primitive	primitive	ADJ
ejpam-5372	5	25	roots	root	NOUN
ejpam-5372	5	26	without	without	ADP
ejpam-5372	5	27	any	any	DET
ejpam-5372	5	28	gcd	gcd	NOUN
ejpam-5372	5	29	calculation	calculation	NOUN
ejpam-5372	5	30	.	.	PUNCT
ejpam-5372	6	1	next	next	ADV
ejpam-5372	6	2	,	,	PUNCT
ejpam-5372	6	3	we	we	PRON
ejpam-5372	6	4	present	present	VERB
ejpam-5372	6	5	a	a	DET
ejpam-5372	6	6	method	method	NOUN
ejpam-5372	6	7	for	for	ADP
ejpam-5372	6	8	determining	determine	VERB
ejpam-5372	6	9	irreducible	irreducible	ADJ
ejpam-5372	6	10	quadratic	quadratic	ADJ
ejpam-5372	6	11	forms	form	NOUN
ejpam-5372	6	12	with	with	ADP
ejpam-5372	6	13	arbitrarily	arbitrarily	ADV
ejpam-5372	6	14	large	large	ADJ
ejpam-5372	6	15	conjectured	conjecture	VERB
ejpam-5372	6	16	asymptotic	asymptotic	ADJ
ejpam-5372	6	17	density	density	NOUN
ejpam-5372	6	18	of	of	ADP
ejpam-5372	6	19	primes	prime	NOUN
ejpam-5372	6	20	(	(	PUNCT
ejpam-5372	6	21	after	after	ADP
ejpam-5372	6	22	shanks	shank	NOUN
ejpam-5372	6	23	,	,	PUNCT
ejpam-5372	6	24	[	[	X
ejpam-5372	6	25	1][2	1][2	NUM
ejpam-5372	6	26	]	]	PUNCT
ejpam-5372	6	27	)	)	PUNCT
ejpam-5372	6	28	.	.	PUNCT
ejpam-5372	7	1	to	to	ADP
ejpam-5372	7	2	this	this	DET
ejpam-5372	7	3	end	end	NOUN
ejpam-5372	7	4	,	,	PUNCT
ejpam-5372	7	5	we	we	PRON
ejpam-5372	7	6	propose	propose	VERB
ejpam-5372	7	7	an	an	DET
ejpam-5372	7	8	algorithm	algorithm	NOUN
ejpam-5372	7	9	for	for	ADP
ejpam-5372	7	10	calculating	calculate	VERB
ejpam-5372	7	11	the	the	DET
ejpam-5372	7	12	square	square	ADJ
ejpam-5372	7	13	root	root	NOUN
ejpam-5372	7	14	modulo	modulo	VERB
ejpam-5372	7	15	p	p	X
ejpam-5372	7	16	,	,	PUNCT
ejpam-5372	7	17	based	base	VERB
ejpam-5372	7	18	on	on	ADP
ejpam-5372	7	19	the	the	DET
ejpam-5372	7	20	tonelli	tonelli	NOUN
ejpam-5372	7	21	-	-	PUNCT
ejpam-5372	7	22	shanks	shank	NOUN
ejpam-5372	7	23	algorithm	algorithm	NOUN
ejpam-5372	7	24	[	[	X
ejpam-5372	7	25	3	3	NUM
ejpam-5372	7	26	]	]	PUNCT
ejpam-5372	7	27	.	.	PUNCT
ejpam-5372	8	1	2020	2020	NUM
ejpam-5372	8	2	mathematics	mathematic	NOUN
ejpam-5372	8	3	subject	subject	NOUN
ejpam-5372	8	4	classifications	classification	NOUN
ejpam-5372	8	5	:	:	PUNCT
ejpam-5372	8	6	11a07	11a07	NUM
ejpam-5372	8	7	,	,	PUNCT
ejpam-5372	8	8	68p05	68p05	NUM
ejpam-5372	8	9	,	,	PUNCT
ejpam-5372	8	10	11b05	11b05	NUM
ejpam-5372	8	11	key	key	ADJ
ejpam-5372	8	12	words	word	NOUN
ejpam-5372	8	13	and	and	CCONJ
ejpam-5372	8	14	phrases	phrase	NOUN
ejpam-5372	8	15	:	:	PUNCT
ejpam-5372	8	16	primitive	primitive	ADJ
ejpam-5372	8	17	roots	root	NOUN
ejpam-5372	8	18	,	,	PUNCT
ejpam-5372	8	19	semi	semi	ADJ
ejpam-5372	8	20	-	-	ADJ
ejpam-5372	8	21	primitive	primitive	ADJ
ejpam-5372	8	22	roots	root	NOUN
ejpam-5372	8	23	,	,	PUNCT
ejpam-5372	8	24	irreducible	irreducible	ADJ
ejpam-5372	8	25	quadratic	quadratic	ADJ
ejpam-5372	8	26	forms	form	NOUN
ejpam-5372	8	27	,	,	PUNCT
ejpam-5372	8	28	asymptotic	asymptotic	ADJ
ejpam-5372	8	29	density	density	NOUN
ejpam-5372	8	30	,	,	PUNCT
ejpam-5372	8	31	fermat	fermat	PROPN
ejpam-5372	8	32	’s	’s	PART
ejpam-5372	8	33	theorem	theorem	NOUN
ejpam-5372	8	34	on	on	ADP
ejpam-5372	8	35	sums	sum	NOUN
ejpam-5372	8	36	of	of	ADP
ejpam-5372	8	37	two	two	NUM
ejpam-5372	8	38	squares	square	NOUN
ejpam-5372	8	39	1	1	NUM
ejpam-5372	8	40	.	.	PUNCT
ejpam-5372	9	1	introduction	introduction	NOUN
ejpam-5372	9	2	1.1	1.1	NUM
ejpam-5372	9	3	.	.	PUNCT
ejpam-5372	10	1	primitive	primitive	ADJ
ejpam-5372	10	2	root	root	NOUN
ejpam-5372	10	3	modulo	modulo	NOUN
ejpam-5372	10	4	n	n	PRON
ejpam-5372	10	5	quadratic	quadratic	ADJ
ejpam-5372	10	6	residues	residue	NOUN
ejpam-5372	10	7	and	and	CCONJ
ejpam-5372	10	8	non	non	NOUN
ejpam-5372	10	9	-	-	NOUN
ejpam-5372	10	10	residues	residue	NOUN
ejpam-5372	10	11	,	,	PUNCT
ejpam-5372	10	12	as	as	ADV
ejpam-5372	10	13	well	well	ADV
ejpam-5372	10	14	as	as	ADP
ejpam-5372	10	15	primitive	primitive	ADJ
ejpam-5372	10	16	roots	root	NOUN
ejpam-5372	10	17	,	,	PUNCT
ejpam-5372	10	18	have	have	AUX
ejpam-5372	10	19	given	give	VERB
ejpam-5372	10	20	rise	rise	NOUN
ejpam-5372	10	21	to	to	ADP
ejpam-5372	10	22	an	an	DET
ejpam-5372	10	23	abundant	abundant	ADJ
ejpam-5372	10	24	literature	literature	NOUN
ejpam-5372	10	25	in	in	ADP
ejpam-5372	10	26	mathematics	mathematics	PROPN
ejpam-5372	10	27	(	(	PUNCT
ejpam-5372	10	28	group	group	NOUN
ejpam-5372	10	29	theory	theory	NOUN
ejpam-5372	10	30	with	with	ADP
ejpam-5372	10	31	lagrange	lagrange	PROPN
ejpam-5372	10	32	’s	’s	PART
ejpam-5372	10	33	and	and	CCONJ
ejpam-5372	10	34	fermat	fermat	PROPN
ejpam-5372	10	35	’s	’s	PART
ejpam-5372	10	36	theorems	theorem	NOUN
ejpam-5372	10	37	,	,	PUNCT
ejpam-5372	10	38	ring	ring	NOUN
ejpam-5372	10	39	and	and	CCONJ
ejpam-5372	10	40	field	field	NOUN
ejpam-5372	10	41	theory	theory	NOUN
ejpam-5372	10	42	,	,	PUNCT
ejpam-5372	10	43	etc	etc	X
ejpam-5372	10	44	.	.	X
ejpam-5372	10	45	)	)	PUNCT
ejpam-5372	10	46	.	.	PUNCT
ejpam-5372	11	1	they	they	PRON
ejpam-5372	11	2	have	have	VERB
ejpam-5372	11	3	many	many	ADJ
ejpam-5372	11	4	applications	application	NOUN
ejpam-5372	11	5	,	,	PUNCT
ejpam-5372	11	6	including	include	VERB
ejpam-5372	11	7	cryptography	cryptography	NOUN
ejpam-5372	11	8	,	,	PUNCT
ejpam-5372	11	9	primality	primality	NOUN
ejpam-5372	11	10	tests	test	NOUN
ejpam-5372	11	11	and	and	CCONJ
ejpam-5372	11	12	integer	integer	NOUN
ejpam-5372	11	13	factorisation	factorisation	NOUN
ejpam-5372	11	14	.	.	PUNCT
ejpam-5372	12	1	we	we	PRON
ejpam-5372	12	2	recall	recall	VERB
ejpam-5372	12	3	that	that	SCONJ
ejpam-5372	12	4	the	the	DET
ejpam-5372	12	5	group	group	NOUN
ejpam-5372	12	6	of	of	ADP
ejpam-5372	12	7	units	unit	NOUN
ejpam-5372	12	8	of	of	ADP
ejpam-5372	12	9	the	the	DET
ejpam-5372	12	10	ring	ring	NOUN
ejpam-5372	12	11	z	z	PROPN
ejpam-5372	12	12	/	/	SYM
ejpam-5372	12	13	nz	nz	PROPN
ejpam-5372	12	14	,	,	PUNCT
ejpam-5372	12	15	of	of	ADP
ejpam-5372	12	16	order	order	NOUN
ejpam-5372	12	17	φ	φ	X
ejpam-5372	12	18	(	(	PUNCT
ejpam-5372	12	19	n	n	CCONJ
ejpam-5372	12	20	)	)	PUNCT
ejpam-5372	12	21	(	(	PUNCT
ejpam-5372	12	22	euler	euler	VERB
ejpam-5372	12	23	’s	’s	PART
ejpam-5372	12	24	totient	totient	NOUN
ejpam-5372	12	25	of	of	ADP
ejpam-5372	12	26	n	n	CCONJ
ejpam-5372	12	27	)	)	PUNCT
ejpam-5372	12	28	,	,	PUNCT
ejpam-5372	12	29	is	be	AUX
ejpam-5372	12	30	a	a	DET
ejpam-5372	12	31	cyclic	cyclic	ADJ
ejpam-5372	12	32	group	group	NOUN
ejpam-5372	12	33	if	if	SCONJ
ejpam-5372	12	34	and	and	CCONJ
ejpam-5372	12	35	only	only	ADV
ejpam-5372	12	36	if	if	SCONJ
ejpam-5372	12	37	n	n	NOUN
ejpam-5372	12	38	=	=	SYM
ejpam-5372	12	39	2	2	NUM
ejpam-5372	12	40	,	,	PUNCT
ejpam-5372	12	41	n	n	NOUN
ejpam-5372	12	42	=	=	SYM
ejpam-5372	12	43	4	4	NUM
ejpam-5372	12	44	,	,	PUNCT
ejpam-5372	12	45	n	n	NOUN
ejpam-5372	12	46	=	=	SYM
ejpam-5372	12	47	pk	pk	NOUN
ejpam-5372	12	48	or	or	CCONJ
ejpam-5372	12	49	n	n	CCONJ
ejpam-5372	12	50	=	=	SYM
ejpam-5372	12	51	2pk	2pk	NOUN
ejpam-5372	12	52	with	with	ADP
ejpam-5372	12	53	p	p	PROPN
ejpam-5372	12	54	∈	∈	PROPN
ejpam-5372	12	55	p\	p\	NOUN
ejpam-5372	12	56	{	{	PUNCT
ejpam-5372	12	57	2	2	NUM
ejpam-5372	12	58	}	}	PUNCT
ejpam-5372	12	59	and	and	CCONJ
ejpam-5372	12	60	k	k	PROPN
ejpam-5372	12	61	∈	∈	PROPN
ejpam-5372	12	62	n∗.	n∗.	PROPN
ejpam-5372	12	63	for	for	ADP
ejpam-5372	12	64	such	such	ADJ
ejpam-5372	12	65	n	n	NOUN
ejpam-5372	12	66	,	,	PUNCT
ejpam-5372	12	67	a	a	DET
ejpam-5372	12	68	primitive	primitive	ADJ
ejpam-5372	12	69	root	root	NOUN
ejpam-5372	12	70	modulo	modulo	NOUN
ejpam-5372	12	71	n	n	PRON
ejpam-5372	12	72	is	be	AUX
ejpam-5372	12	73	a	a	DET
ejpam-5372	12	74	generator	generator	NOUN
ejpam-5372	12	75	of	of	ADP
ejpam-5372	12	76	this	this	DET
ejpam-5372	12	77	group	group	NOUN
ejpam-5372	12	78	.	.	PUNCT
ejpam-5372	13	1	there	there	PRON
ejpam-5372	13	2	exist	exist	VERB
ejpam-5372	13	3	φ	φ	PROPN
ejpam-5372	13	4	(	(	PUNCT
ejpam-5372	13	5	φ	φ	PROPN
ejpam-5372	13	6	(	(	PUNCT
ejpam-5372	13	7	n	n	CCONJ
ejpam-5372	13	8	)	)	PUNCT
ejpam-5372	13	9	)	)	PUNCT
ejpam-5372	14	1	such	such	ADJ
ejpam-5372	14	2	integers	integer	NOUN
ejpam-5372	14	3	modulo	modulo	VERB
ejpam-5372	14	4	n	n	X
ejpam-5372	14	5	,	,	PUNCT
ejpam-5372	14	6	whose	whose	DET
ejpam-5372	14	7	powers	power	NOUN
ejpam-5372	14	8	generate	generate	VERB
ejpam-5372	14	9	every	every	DET
ejpam-5372	14	10	element	element	NOUN
ejpam-5372	14	11	of	of	ADP
ejpam-5372	14	12	(	(	PUNCT
ejpam-5372	14	13	z	z	NOUN
ejpam-5372	14	14	/	/	SYM
ejpam-5372	14	15	nz)×.	nz)×.	DET
ejpam-5372	14	16	the	the	DET
ejpam-5372	14	17	usual	usual	ADJ
ejpam-5372	14	18	method	method	NOUN
ejpam-5372	14	19	to	to	PART
ejpam-5372	14	20	compute	compute	VERB
ejpam-5372	14	21	one	one	NUM
ejpam-5372	14	22	or	or	CCONJ
ejpam-5372	14	23	more	more	ADV
ejpam-5372	14	24	primitive	primitive	ADJ
ejpam-5372	14	25	roots	root	NOUN
ejpam-5372	14	26	requires	require	VERB
ejpam-5372	14	27	the	the	DET
ejpam-5372	14	28	prime	prime	ADJ
ejpam-5372	14	29	decomposition	decomposition	NOUN
ejpam-5372	14	30	of	of	ADP
ejpam-5372	14	31	φ	φ	PROPN
ejpam-5372	14	32	(	(	PUNCT
ejpam-5372	14	33	n	n	CCONJ
ejpam-5372	14	34	)	)	PUNCT
ejpam-5372	14	35	.	.	PUNCT
ejpam-5372	15	1	in	in	ADP
ejpam-5372	15	2	this	this	DET
ejpam-5372	15	3	article	article	NOUN
ejpam-5372	15	4	,	,	PUNCT
ejpam-5372	15	5	we	we	PRON
ejpam-5372	15	6	will	will	AUX
ejpam-5372	15	7	focus	focus	VERB
ejpam-5372	15	8	on	on	ADP
ejpam-5372	15	9	the	the	DET
ejpam-5372	15	10	case	case	NOUN
ejpam-5372	15	11	n	n	NOUN
ejpam-5372	15	12	=	=	SYM
ejpam-5372	15	13	p	p	PROPN
ejpam-5372	15	14	∈	∈	PROPN
ejpam-5372	15	15	p⧹	p⧹	VERB
ejpam-5372	15	16	{	{	PUNCT
ejpam-5372	15	17	2	2	NUM
ejpam-5372	15	18	}	}	PUNCT
ejpam-5372	15	19	.	.	PUNCT
ejpam-5372	16	1	in	in	ADP
ejpam-5372	16	2	the	the	DET
ejpam-5372	16	3	first	first	ADJ
ejpam-5372	16	4	section	section	NOUN
ejpam-5372	16	5	,	,	PUNCT
ejpam-5372	16	6	we	we	PRON
ejpam-5372	16	7	define	define	VERB
ejpam-5372	16	8	semi	semi	ADJ
ejpam-5372	16	9	-	-	ADJ
ejpam-5372	16	10	primitive	primitive	ADJ
ejpam-5372	16	11	roots	root	NOUN
ejpam-5372	16	12	and	and	CCONJ
ejpam-5372	16	13	study	study	VERB
ejpam-5372	16	14	their	their	PRON
ejpam-5372	16	15	relationship	relationship	NOUN
ejpam-5372	16	16	with	with	ADP
ejpam-5372	16	17	primitive	primitive	ADJ
ejpam-5372	16	18	roots	root	NOUN
ejpam-5372	16	19	modulo	modulo	PART
ejpam-5372	16	20	p.	p.	NOUN
ejpam-5372	16	21	using	use	VERB
ejpam-5372	16	22	these	these	DET
ejpam-5372	16	23	results	result	NOUN
ejpam-5372	16	24	,	,	PUNCT
ejpam-5372	16	25	we	we	PRON
ejpam-5372	16	26	present	present	VERB
ejpam-5372	16	27	a	a	DET
ejpam-5372	16	28	test	test	NOUN
ejpam-5372	16	29	on	on	ADP
ejpam-5372	16	30	quadratic	quadratic	ADJ
ejpam-5372	16	31	residues	residue	NOUN
ejpam-5372	16	32	to	to	PART
ejpam-5372	16	33	determine	determine	VERB
ejpam-5372	16	34	primitive	primitive	ADJ
ejpam-5372	16	35	roots	root	NOUN
ejpam-5372	16	36	.	.	PUNCT
ejpam-5372	17	1	then	then	ADV
ejpam-5372	17	2	,	,	PUNCT
ejpam-5372	17	3	using	use	VERB
ejpam-5372	17	4	recursive	recursive	ADJ
ejpam-5372	17	5	sequences	sequence	NOUN
ejpam-5372	17	6	of	of	ADP
ejpam-5372	17	7	primitive	primitive	ADJ
ejpam-5372	17	8	and	and	CCONJ
ejpam-5372	17	9	semi	semi	ADJ
ejpam-5372	17	10	-	-	ADJ
ejpam-5372	17	11	primitive	primitive	ADJ
ejpam-5372	17	12	∗corresponding	∗corresponding	NOUN
ejpam-5372	17	13	author	author	NOUN
ejpam-5372	17	14	.	.	PUNCT
ejpam-5372	18	1	doi	doi	NOUN
ejpam-5372	18	2	:	:	PUNCT
ejpam-5372	18	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5372	https://doi.org/10.29020/nybg.ejpam.v17i4.5372	ADJ
ejpam-5372	18	4	email	email	NOUN
ejpam-5372	18	5	addresses	address	NOUN
ejpam-5372	18	6	:	:	PUNCT
ejpam-5372	18	7	marc.wolf@tsoftemail.com	marc.wolf@tsoftemail.com	X
ejpam-5372	18	8	(	(	PUNCT
ejpam-5372	18	9	m.	m.	NOUN
ejpam-5372	18	10	wolf	wolf	PROPN
ejpam-5372	18	11	)	)	PUNCT
ejpam-5372	18	12	,	,	PUNCT
ejpam-5372	18	13	francois.wolf@dbmail.com	francois.wolf@dbmail.com	X
ejpam-5372	18	14	(	(	PUNCT
ejpam-5372	18	15	f.	f.	PROPN
ejpam-5372	18	16	wolf	wolf	PROPN
ejpam-5372	18	17	)	)	PUNCT
ejpam-5372	18	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5372	18	19	2431	2431	NUM
ejpam-5372	18	20	copyright	copyright	NOUN
ejpam-5372	18	21	:	:	PUNCT
ejpam-5372	18	22	©	©	PROPN
ejpam-5372	18	23	2024	2024	NUM
ejpam-5372	18	24	the	the	DET
ejpam-5372	18	25	author(s	author(s	NOUN
ejpam-5372	18	26	)	)	PUNCT
ejpam-5372	18	27	.	.	PUNCT
ejpam-5372	19	1	(	(	PUNCT
ejpam-5372	19	2	cc	cc	NOUN
ejpam-5372	19	3	by	by	ADP
ejpam-5372	19	4	-	-	PUNCT
ejpam-5372	19	5	nc	nc	PROPN
ejpam-5372	19	6	4.0	4.0	NUM
ejpam-5372	19	7	)	)	PUNCT
ejpam-5372	19	8	m.	m.	NOUN
ejpam-5372	19	9	wolf	wolf	PROPN
ejpam-5372	19	10	,	,	PUNCT
ejpam-5372	19	11	f.	f.	PROPN
ejpam-5372	19	12	wolf	wolf	PROPN
ejpam-5372	19	13	/	/	SYM
ejpam-5372	19	14	eur	eur	PROPN
ejpam-5372	19	15	.	.	PUNCT
ejpam-5372	20	1	j.	j.	PROPN
ejpam-5372	20	2	pure	pure	PROPN
ejpam-5372	20	3	appl	appl	PROPN
ejpam-5372	20	4	.	.	PROPN
ejpam-5372	20	5	math	math	PROPN
ejpam-5372	20	6	,	,	PUNCT
ejpam-5372	20	7	17	17	NUM
ejpam-5372	20	8	(	(	PUNCT
ejpam-5372	20	9	4	4	NUM
ejpam-5372	20	10	)	)	PUNCT
ejpam-5372	20	11	(	(	PUNCT
ejpam-5372	20	12	2024	2024	NUM
ejpam-5372	20	13	)	)	PUNCT
ejpam-5372	20	14	,	,	PUNCT
ejpam-5372	20	15	2431	2431	NUM
ejpam-5372	20	16	-	-	SYM
ejpam-5372	20	17	2447	2447	NUM
ejpam-5372	20	18	2432	2432	NUM
ejpam-5372	20	19	roots	root	NOUN
ejpam-5372	21	1	,	,	PUNCT
ejpam-5372	21	2	we	we	PRON
ejpam-5372	21	3	describe	describe	VERB
ejpam-5372	21	4	an	an	DET
ejpam-5372	21	5	algorithm	algorithm	NOUN
ejpam-5372	21	6	determining	determine	VERB
ejpam-5372	21	7	all	all	DET
ejpam-5372	21	8	primitive	primitive	ADJ
ejpam-5372	21	9	roots	root	NOUN
ejpam-5372	21	10	without	without	ADP
ejpam-5372	21	11	any	any	DET
ejpam-5372	21	12	gcd	gcd	NOUN
ejpam-5372	21	13	computation	computation	NOUN
ejpam-5372	21	14	.	.	PUNCT
ejpam-5372	22	1	in	in	ADP
ejpam-5372	22	2	[	[	X
ejpam-5372	22	3	2	2	NUM
ejpam-5372	22	4	]	]	PUNCT
ejpam-5372	22	5	,	,	PUNCT
ejpam-5372	22	6	we	we	PRON
ejpam-5372	22	7	studied	study	VERB
ejpam-5372	22	8	the	the	DET
ejpam-5372	22	9	density	density	NOUN
ejpam-5372	22	10	of	of	ADP
ejpam-5372	22	11	primes	prime	NOUN
ejpam-5372	22	12	of	of	ADP
ejpam-5372	22	13	the	the	DET
ejpam-5372	22	14	form	form	NOUN
ejpam-5372	22	15	x2	x2	PROPN
ejpam-5372	22	16	+	+	CCONJ
ejpam-5372	22	17	c	c	X
ejpam-5372	22	18	,	,	PUNCT
ejpam-5372	22	19	for	for	ADP
ejpam-5372	22	20	a	a	DET
ejpam-5372	22	21	fixed	fix	VERB
ejpam-5372	22	22	c	c	PROPN
ejpam-5372	22	23	∈	∈	PROPN
ejpam-5372	22	24	n∗	n∗	PROPN
ejpam-5372	22	25	,	,	PUNCT
ejpam-5372	22	26	based	base	VERB
ejpam-5372	22	27	on	on	ADP
ejpam-5372	22	28	shanks	shank	NOUN
ejpam-5372	22	29	’	'	PUNCT
ejpam-5372	22	30	conjecture	conjecture	NOUN
ejpam-5372	22	31	[	[	X
ejpam-5372	22	32	1	1	NUM
ejpam-5372	22	33	]	]	PUNCT
ejpam-5372	22	34	,	,	PUNCT
ejpam-5372	22	35	which	which	PRON
ejpam-5372	22	36	we	we	PRON
ejpam-5372	22	37	empirically	empirically	ADV
ejpam-5372	22	38	corroborated	corroborate	VERB
ejpam-5372	22	39	.	.	PUNCT
ejpam-5372	23	1	in	in	ADP
ejpam-5372	23	2	the	the	DET
ejpam-5372	23	3	second	second	ADJ
ejpam-5372	23	4	section	section	NOUN
ejpam-5372	23	5	of	of	ADP
ejpam-5372	23	6	this	this	DET
ejpam-5372	23	7	article	article	NOUN
ejpam-5372	23	8	,	,	PUNCT
ejpam-5372	23	9	we	we	PRON
ejpam-5372	23	10	will	will	AUX
ejpam-5372	23	11	continue	continue	VERB
ejpam-5372	23	12	this	this	DET
ejpam-5372	23	13	study	study	NOUN
ejpam-5372	23	14	and	and	CCONJ
ejpam-5372	23	15	propose	propose	VERB
ejpam-5372	23	16	new	new	ADJ
ejpam-5372	23	17	results	result	NOUN
ejpam-5372	23	18	generalised	generalise	VERB
ejpam-5372	23	19	to	to	ADP
ejpam-5372	23	20	irreducible	irreducible	ADJ
ejpam-5372	23	21	quadratic	quadratic	ADJ
ejpam-5372	23	22	forms	form	NOUN
ejpam-5372	23	23	.	.	PUNCT
ejpam-5372	24	1	1.2	1.2	NUM
ejpam-5372	24	2	.	.	PUNCT
ejpam-5372	24	3	basic	basic	ADJ
ejpam-5372	24	4	properties	property	NOUN
ejpam-5372	24	5	,	,	PUNCT
ejpam-5372	24	6	definitions	definition	NOUN
ejpam-5372	24	7	and	and	CCONJ
ejpam-5372	24	8	notations	notation	NOUN
ejpam-5372	24	9	definition	definition	NOUN
ejpam-5372	24	10	1	1	NUM
ejpam-5372	24	11	.	.	PUNCT
ejpam-5372	25	1	(	(	PUNCT
ejpam-5372	25	2	i	i	NOUN
ejpam-5372	25	3	)	)	PUNCT
ejpam-5372	25	4	we	we	PRON
ejpam-5372	25	5	let	let	VERB
ejpam-5372	25	6	d	d	NOUN
ejpam-5372	25	7	(	(	PUNCT
ejpam-5372	25	8	x	x	NOUN
ejpam-5372	25	9	)	)	PUNCT
ejpam-5372	25	10	:	:	PUNCT
ejpam-5372	26	1	=	=	SYM
ejpam-5372	26	2	{	{	PUNCT
ejpam-5372	26	3	d	d	PUNCT
ejpam-5372	26	4	∈	∈	PROPN
ejpam-5372	26	5	n∗	n∗	PROPN
ejpam-5372	26	6	|	|	ADV
ejpam-5372	26	7	d|x	d|x	VERB
ejpam-5372	26	8	}	}	PUNCT
ejpam-5372	26	9	the	the	DET
ejpam-5372	26	10	set	set	NOUN
ejpam-5372	26	11	of	of	ADP
ejpam-5372	26	12	divisors	divisor	NOUN
ejpam-5372	26	13	of	of	ADP
ejpam-5372	26	14	an	an	DET
ejpam-5372	26	15	integer	integer	NOUN
ejpam-5372	26	16	x	x	NOUN
ejpam-5372	26	17	,	,	PUNCT
ejpam-5372	26	18	for	for	ADP
ejpam-5372	26	19	e	e	PROPN
ejpam-5372	26	20	⊂	⊂	PROPN
ejpam-5372	26	21	z	z	VERB
ejpam-5372	27	1	we	we	PRON
ejpam-5372	27	2	let	let	VERB
ejpam-5372	27	3	d	d	X
ejpam-5372	27	4	(	(	PUNCT
ejpam-5372	27	5	e	e	NOUN
ejpam-5372	27	6	)	)	PUNCT
ejpam-5372	27	7	:	:	PUNCT
ejpam-5372	28	1	=	=	SYM
ejpam-5372	28	2	⋃	⋃	NOUN
ejpam-5372	28	3	x∈e	x∈e	NOUN
ejpam-5372	28	4	d	d	X
ejpam-5372	28	5	(	(	PUNCT
ejpam-5372	28	6	x	x	X
ejpam-5372	28	7	)	)	PUNCT
ejpam-5372	28	8	the	the	DET
ejpam-5372	28	9	set	set	NOUN
ejpam-5372	28	10	of	of	ADP
ejpam-5372	28	11	divisors	divisor	NOUN
ejpam-5372	28	12	of	of	ADP
ejpam-5372	28	13	at	at	ADV
ejpam-5372	28	14	least	least	ADV
ejpam-5372	28	15	one	one	NUM
ejpam-5372	28	16	element	element	NOUN
ejpam-5372	28	17	of	of	ADP
ejpam-5372	28	18	e.	e.	PROPN
ejpam-5372	28	19	we	we	PRON
ejpam-5372	28	20	also	also	ADV
ejpam-5372	28	21	define	define	VERB
ejpam-5372	28	22	dp	dp	NOUN
ejpam-5372	28	23	(	(	PUNCT
ejpam-5372	28	24	e	e	NOUN
ejpam-5372	28	25	)	)	PUNCT
ejpam-5372	28	26	=	=	SYM
ejpam-5372	28	27	d	d	X
ejpam-5372	28	28	(	(	PUNCT
ejpam-5372	28	29	e	e	NOUN
ejpam-5372	28	30	)	)	PUNCT
ejpam-5372	28	31	∩	∩	NOUN
ejpam-5372	28	32	p	p	X
ejpam-5372	28	33	the	the	DET
ejpam-5372	28	34	set	set	NOUN
ejpam-5372	28	35	of	of	ADP
ejpam-5372	28	36	prime	prime	ADJ
ejpam-5372	28	37	divisors	divisor	NOUN
ejpam-5372	28	38	of	of	ADP
ejpam-5372	28	39	e.	e.	PROPN
ejpam-5372	28	40	(	(	PUNCT
ejpam-5372	28	41	ii	ii	PROPN
ejpam-5372	28	42	)	)	PUNCT
ejpam-5372	28	43	for	for	ADP
ejpam-5372	28	44	(	(	PUNCT
ejpam-5372	28	45	x	x	NOUN
ejpam-5372	28	46	,	,	PUNCT
ejpam-5372	28	47	y	y	NOUN
ejpam-5372	28	48	)	)	PUNCT
ejpam-5372	28	49	∈	∈	PROPN
ejpam-5372	28	50	z∗	z∗	PROPN
ejpam-5372	28	51	×	×	NOUN
ejpam-5372	28	52	n∗	n∗	NOUN
ejpam-5372	28	53	we	we	PRON
ejpam-5372	28	54	let	let	VERB
ejpam-5372	28	55	px	px	NOUN
ejpam-5372	28	56	,	,	PUNCT
ejpam-5372	28	57	y	y	NOUN
ejpam-5372	28	58	:	:	PUNCT
ejpam-5372	28	59	=	=	SYM
ejpam-5372	28	60	{	{	PUNCT
ejpam-5372	28	61	p	p	X
ejpam-5372	28	62	∈	∈	PROPN
ejpam-5372	29	1	p	p	NOUN
ejpam-5372	29	2	|	|	ADV
ejpam-5372	29	3	p	p	X
ejpam-5372	29	4	≡	≡	PROPN
ejpam-5372	29	5	x	x	PUNCT
ejpam-5372	30	1	[	[	X
ejpam-5372	30	2	y	y	X
ejpam-5372	30	3	]	]	X
ejpam-5372	30	4	}	}	PUNCT
ejpam-5372	30	5	.	.	PUNCT
ejpam-5372	31	1	(	(	PUNCT
ejpam-5372	31	2	iii	iii	X
ejpam-5372	31	3	)	)	PUNCT
ejpam-5372	31	4	let	let	VERB
ejpam-5372	31	5	p	p	PROPN
ejpam-5372	31	6	∈	∈	PROPN
ejpam-5372	31	7	p⧹	p⧹	VERB
ejpam-5372	31	8	{	{	PUNCT
ejpam-5372	31	9	2	2	NUM
ejpam-5372	31	10	}	}	PUNCT
ejpam-5372	31	11	.	.	PUNCT
ejpam-5372	32	1	a	a	DET
ejpam-5372	32	2	semi	semi	ADJ
ejpam-5372	32	3	-	-	ADJ
ejpam-5372	32	4	primitive	primitive	ADJ
ejpam-5372	32	5	root	root	NOUN
ejpam-5372	32	6	modulo	modulo	NOUN
ejpam-5372	32	7	p	p	NOUN
ejpam-5372	32	8	is	be	AUX
ejpam-5372	32	9	defined	define	VERB
ejpam-5372	32	10	as	as	ADP
ejpam-5372	32	11	the	the	DET
ejpam-5372	32	12	square	square	NOUN
ejpam-5372	32	13	of	of	ADP
ejpam-5372	32	14	a	a	DET
ejpam-5372	32	15	primitive	primitive	ADJ
ejpam-5372	32	16	root	root	NOUN
ejpam-5372	32	17	in	in	ADP
ejpam-5372	32	18	(	(	PUNCT
ejpam-5372	32	19	z	z	NOUN
ejpam-5372	32	20	/	/	SYM
ejpam-5372	32	21	pz)×	pz)×	X
ejpam-5372	32	22	=	=	SYM
ejpam-5372	32	23	f∗	f∗	NOUN
ejpam-5372	32	24	p.	p.	NOUN
ejpam-5372	32	25	proposition	proposition	NOUN
ejpam-5372	32	26	1	1	NUM
ejpam-5372	32	27	.	.	PUNCT
ejpam-5372	32	28	g	g	PROPN
ejpam-5372	32	29	∈	∈	PROPN
ejpam-5372	32	30	f∗	f∗	NOUN
ejpam-5372	32	31	p	p	NOUN
ejpam-5372	32	32	is	be	AUX
ejpam-5372	32	33	a	a	DET
ejpam-5372	32	34	semi	semi	ADJ
ejpam-5372	32	35	-	-	ADJ
ejpam-5372	32	36	primitive	primitive	ADJ
ejpam-5372	32	37	root	root	NOUN
ejpam-5372	32	38	if	if	SCONJ
ejpam-5372	32	39	and	and	CCONJ
ejpam-5372	32	40	only	only	ADV
ejpam-5372	32	41	if	if	SCONJ
ejpam-5372	32	42	the	the	DET
ejpam-5372	32	43	order	order	NOUN
ejpam-5372	32	44	of	of	ADP
ejpam-5372	32	45	⟨g⟩	⟨g⟩	PROPN
ejpam-5372	32	46	=	=	PROPN
ejpam-5372	32	47	{	{	PUNCT
ejpam-5372	32	48	gk	gk	PROPN
ejpam-5372	32	49	,	,	PUNCT
ejpam-5372	32	50	k	k	PROPN
ejpam-5372	32	51	∈	∈	PROPN
ejpam-5372	32	52	z	z	PROPN
ejpam-5372	32	53	}	}	PUNCT
ejpam-5372	32	54	is	be	AUX
ejpam-5372	32	55	p−1	p−1	PROPN
ejpam-5372	32	56	2	2	NUM
ejpam-5372	32	57	.	.	PUNCT
ejpam-5372	33	1	proof	proof	NOUN
ejpam-5372	33	2	.	.	PUNCT
ejpam-5372	34	1	if	if	SCONJ
ejpam-5372	34	2	g	g	PROPN
ejpam-5372	34	3	is	be	AUX
ejpam-5372	34	4	a	a	DET
ejpam-5372	34	5	semi	semi	ADJ
ejpam-5372	34	6	-	-	ADJ
ejpam-5372	34	7	primitive	primitive	ADJ
ejpam-5372	34	8	root	root	NOUN
ejpam-5372	34	9	,	,	PUNCT
ejpam-5372	34	10	there	there	PRON
ejpam-5372	34	11	exists	exist	VERB
ejpam-5372	34	12	a	a	DET
ejpam-5372	34	13	primitive	primitive	ADJ
ejpam-5372	34	14	root	root	NOUN
ejpam-5372	34	15	h	h	NOUN
ejpam-5372	34	16	such	such	ADJ
ejpam-5372	34	17	as	as	ADP
ejpam-5372	34	18	g	g	PROPN
ejpam-5372	34	19	=	=	PROPN
ejpam-5372	34	20	h2	h2	PROPN
ejpam-5372	34	21	.	.	PUNCT
ejpam-5372	35	1	since	since	SCONJ
ejpam-5372	35	2	p	p	NOUN
ejpam-5372	35	3	−	−	PROPN
ejpam-5372	35	4	1	1	NUM
ejpam-5372	35	5	is	be	AUX
ejpam-5372	35	6	even	even	ADV
ejpam-5372	35	7	and	and	CCONJ
ejpam-5372	35	8	the	the	DET
ejpam-5372	35	9	order	order	NOUN
ejpam-5372	35	10	of	of	ADP
ejpam-5372	35	11	⟨h⟩	⟨h⟩	PROPN
ejpam-5372	35	12	is	be	AUX
ejpam-5372	35	13	p	p	NOUN
ejpam-5372	35	14	−	−	PROPN
ejpam-5372	35	15	1	1	NUM
ejpam-5372	35	16	,	,	PUNCT
ejpam-5372	35	17	we	we	PRON
ejpam-5372	35	18	deduce	deduce	VERB
ejpam-5372	35	19	that	that	SCONJ
ejpam-5372	35	20	the	the	DET
ejpam-5372	35	21	order	order	NOUN
ejpam-5372	35	22	of	of	ADP
ejpam-5372	35	23	⟨g⟩	⟨g⟩	PROPN
ejpam-5372	35	24	is	be	AUX
ejpam-5372	35	25	p−1	p−1	PROPN
ejpam-5372	35	26	2	2	NUM
ejpam-5372	35	27	.	.	PUNCT
ejpam-5372	36	1	conversely	conversely	ADV
ejpam-5372	36	2	,	,	PUNCT
ejpam-5372	36	3	if	if	SCONJ
ejpam-5372	36	4	the	the	DET
ejpam-5372	36	5	order	order	NOUN
ejpam-5372	36	6	of	of	ADP
ejpam-5372	36	7	⟨g⟩	⟨g⟩	PROPN
ejpam-5372	36	8	is	be	AUX
ejpam-5372	36	9	p−1	p−1	PROPN
ejpam-5372	36	10	2	2	NUM
ejpam-5372	36	11	and	and	CCONJ
ejpam-5372	36	12	if	if	SCONJ
ejpam-5372	36	13	h	h	NOUN
ejpam-5372	36	14	is	be	AUX
ejpam-5372	36	15	a	a	DET
ejpam-5372	36	16	primitive	primitive	ADJ
ejpam-5372	36	17	root	root	NOUN
ejpam-5372	36	18	,	,	PUNCT
ejpam-5372	36	19	let	let	VERB
ejpam-5372	36	20	k	k	NOUN
ejpam-5372	36	21	such	such	ADJ
ejpam-5372	36	22	that	that	SCONJ
ejpam-5372	36	23	g	g	PROPN
ejpam-5372	36	24	=	=	SYM
ejpam-5372	36	25	hk	hk	PROPN
ejpam-5372	36	26	.	.	PUNCT
ejpam-5372	37	1	we	we	PRON
ejpam-5372	37	2	know	know	VERB
ejpam-5372	37	3	that	that	SCONJ
ejpam-5372	37	4	the	the	DET
ejpam-5372	37	5	order	order	NOUN
ejpam-5372	37	6	of	of	ADP
ejpam-5372	37	7	⟨g⟩	⟨g⟩	PROPN
ejpam-5372	37	8	equal	equal	ADJ
ejpam-5372	37	9	to	to	ADP
ejpam-5372	37	10	p−1	p−1	PROPN
ejpam-5372	37	11	gcd(k	gcd(k	PROPN
ejpam-5372	37	12	,	,	PUNCT
ejpam-5372	37	13	p−1	p−1	PROPN
ejpam-5372	37	14	)	)	PUNCT
ejpam-5372	37	15	thus	thus	ADV
ejpam-5372	37	16	gcd	gcd	VERB
ejpam-5372	37	17	(	(	PUNCT
ejpam-5372	37	18	k	k	NOUN
ejpam-5372	37	19	,	,	PUNCT
ejpam-5372	37	20	p−	p−	NOUN
ejpam-5372	37	21	1	1	NUM
ejpam-5372	37	22	)	)	PUNCT
ejpam-5372	37	23	=	=	SYM
ejpam-5372	37	24	2	2	NUM
ejpam-5372	37	25	,	,	PUNCT
ejpam-5372	37	26	i.e.	i.e.	X
ejpam-5372	37	27	we	we	PRON
ejpam-5372	37	28	can	can	AUX
ejpam-5372	37	29	write	write	VERB
ejpam-5372	37	30	k	k	PROPN
ejpam-5372	38	1	=	=	PUNCT
ejpam-5372	38	2	2u	2u	ADJ
ejpam-5372	38	3	with	with	ADP
ejpam-5372	38	4	u	u	NOUN
ejpam-5372	38	5	and	and	CCONJ
ejpam-5372	38	6	p	p	NOUN
ejpam-5372	38	7	−	−	PROPN
ejpam-5372	38	8	1	1	NUM
ejpam-5372	38	9	coprime	coprime	NOUN
ejpam-5372	38	10	.	.	PUNCT
ejpam-5372	39	1	we	we	PRON
ejpam-5372	39	2	conclude	conclude	VERB
ejpam-5372	39	3	that	that	SCONJ
ejpam-5372	39	4	hu	hu	PROPN
ejpam-5372	39	5	is	be	AUX
ejpam-5372	39	6	a	a	DET
ejpam-5372	39	7	primitive	primitive	ADJ
ejpam-5372	39	8	root	root	NOUN
ejpam-5372	39	9	and	and	CCONJ
ejpam-5372	39	10	g	g	NOUN
ejpam-5372	39	11	=	=	PUNCT
ejpam-5372	39	12	(	(	PUNCT
ejpam-5372	39	13	hu)2	hu)2	PROPN
ejpam-5372	39	14	.	.	PUNCT
ejpam-5372	40	1	definition	definition	NOUN
ejpam-5372	40	2	2	2	NUM
ejpam-5372	40	3	.	.	PUNCT
ejpam-5372	41	1	let	let	VERB
ejpam-5372	41	2	p	p	PROPN
ejpam-5372	41	3	∈	∈	PROPN
ejpam-5372	41	4	p⧹	p⧹	VERB
ejpam-5372	41	5	{	{	PUNCT
ejpam-5372	41	6	2	2	NUM
ejpam-5372	41	7	}	}	PUNCT
ejpam-5372	41	8	.	.	PUNCT
ejpam-5372	42	1	gz	gz	PROPN
ejpam-5372	42	2	,	,	PUNCT
ejpam-5372	42	3	p	p	X
ejpam-5372	42	4	(	(	PUNCT
ejpam-5372	42	5	respectively	respectively	ADV
ejpam-5372	42	6	gs	gs	NOUN
ejpam-5372	42	7	,	,	PUNCT
ejpam-5372	42	8	p	p	X
ejpam-5372	42	9	,	,	PUNCT
ejpam-5372	42	10	gqr	gqr	PROPN
ejpam-5372	42	11	,	,	PUNCT
ejpam-5372	42	12	p	p	NOUN
ejpam-5372	42	13	,	,	PUNCT
ejpam-5372	42	14	gqnr	gqnr	NOUN
ejpam-5372	42	15	,	,	PUNCT
ejpam-5372	42	16	p	p	NOUN
ejpam-5372	42	17	)	)	PUNCT
ejpam-5372	42	18	is	be	AUX
ejpam-5372	42	19	the	the	DET
ejpam-5372	42	20	set	set	NOUN
ejpam-5372	42	21	of	of	ADP
ejpam-5372	42	22	primitive	primitive	ADJ
ejpam-5372	42	23	roots	root	NOUN
ejpam-5372	42	24	(	(	PUNCT
ejpam-5372	42	25	respectively	respectively	ADV
ejpam-5372	42	26	semi	semi	ADJ
ejpam-5372	42	27	-	-	ADJ
ejpam-5372	42	28	primitive	primitive	ADJ
ejpam-5372	42	29	roots	root	NOUN
ejpam-5372	42	30	,	,	PUNCT
ejpam-5372	42	31	quadratic	quadratic	ADJ
ejpam-5372	42	32	residues	residue	NOUN
ejpam-5372	42	33	,	,	PUNCT
ejpam-5372	42	34	quadratic	quadratic	ADJ
ejpam-5372	42	35	non	non	ADJ
ejpam-5372	42	36	-	-	NOUN
ejpam-5372	42	37	residues	residue	NOUN
ejpam-5372	42	38	)	)	PUNCT
ejpam-5372	42	39	modulo	modulo	NOUN
ejpam-5372	42	40	p.	p.	NOUN
ejpam-5372	42	41	when	when	SCONJ
ejpam-5372	42	42	the	the	DET
ejpam-5372	42	43	value	value	NOUN
ejpam-5372	42	44	of	of	ADP
ejpam-5372	42	45	p	p	NOUN
ejpam-5372	42	46	is	be	AUX
ejpam-5372	42	47	unambiguous	unambiguous	ADJ
ejpam-5372	42	48	,	,	PUNCT
ejpam-5372	42	49	this	this	DET
ejpam-5372	42	50	set	set	NOUN
ejpam-5372	42	51	is	be	AUX
ejpam-5372	42	52	simply	simply	ADV
ejpam-5372	42	53	noted	note	VERB
ejpam-5372	42	54	gz	gz	PROPN
ejpam-5372	42	55	(	(	PUNCT
ejpam-5372	42	56	respectively	respectively	ADV
ejpam-5372	42	57	gs	gs	PROPN
ejpam-5372	42	58	,	,	PUNCT
ejpam-5372	42	59	gqr	gqr	PROPN
ejpam-5372	42	60	,	,	PUNCT
ejpam-5372	42	61	gqnr	gqnr	NOUN
ejpam-5372	42	62	)	)	PUNCT
ejpam-5372	42	63	.	.	PUNCT
ejpam-5372	43	1	example	example	NOUN
ejpam-5372	44	1	1	1	NUM
ejpam-5372	44	2	.	.	X
ejpam-5372	45	1	for	for	ADP
ejpam-5372	45	2	p	p	NOUN
ejpam-5372	45	3	=	=	SYM
ejpam-5372	45	4	3	3	NUM
ejpam-5372	45	5	,	,	PUNCT
ejpam-5372	45	6	we	we	PRON
ejpam-5372	45	7	have	have	VERB
ejpam-5372	45	8	gs	gs	NOUN
ejpam-5372	45	9	=	=	PUNCT
ejpam-5372	45	10	gqr	gqr	PROPN
ejpam-5372	45	11	=	=	PUNCT
ejpam-5372	45	12	{	{	PUNCT
ejpam-5372	45	13	1	1	NUM
ejpam-5372	45	14	}	}	PUNCT
ejpam-5372	45	15	and	and	CCONJ
ejpam-5372	45	16	gz	gz	X
ejpam-5372	45	17	=	=	NOUN
ejpam-5372	45	18	gqnr	gqnr	NOUN
ejpam-5372	45	19	=	=	PUNCT
ejpam-5372	45	20	{	{	PUNCT
ejpam-5372	45	21	2	2	NUM
ejpam-5372	45	22	}	}	PUNCT
ejpam-5372	45	23	.	.	PUNCT
ejpam-5372	46	1	proposition	proposition	NOUN
ejpam-5372	46	2	2	2	NUM
ejpam-5372	46	3	.	.	PUNCT
ejpam-5372	47	1	let	let	VERB
ejpam-5372	47	2	p	p	PROPN
ejpam-5372	47	3	∈	∈	PROPN
ejpam-5372	47	4	p⧹	p⧹	VERB
ejpam-5372	47	5	{	{	PUNCT
ejpam-5372	47	6	2	2	NUM
ejpam-5372	47	7	}	}	PUNCT
ejpam-5372	47	8	.	.	PUNCT
ejpam-5372	48	1	we	we	PRON
ejpam-5372	48	2	have	have	AUX
ejpam-5372	48	3	gz	gz	NOUN
ejpam-5372	48	4	⊂	⊂	PROPN
ejpam-5372	48	5	gqnr	gqnr	PROPN
ejpam-5372	48	6	and	and	CCONJ
ejpam-5372	48	7	gs	gs	PROPN
ejpam-5372	48	8	⊂	⊂	PROPN
ejpam-5372	48	9	gqr	gqr	PROPN
ejpam-5372	48	10	.	.	PUNCT
ejpam-5372	49	1	proof	proof	NOUN
ejpam-5372	49	2	.	.	PUNCT
ejpam-5372	50	1	the	the	DET
ejpam-5372	50	2	second	second	ADJ
ejpam-5372	50	3	inclusion	inclusion	NOUN
ejpam-5372	50	4	is	be	AUX
ejpam-5372	50	5	immediate	immediate	ADJ
ejpam-5372	50	6	,	,	PUNCT
ejpam-5372	50	7	since	since	SCONJ
ejpam-5372	50	8	a	a	DET
ejpam-5372	50	9	quadratic	quadratic	ADJ
ejpam-5372	50	10	residue	residue	NOUN
ejpam-5372	50	11	is	be	AUX
ejpam-5372	50	12	defined	define	VERB
ejpam-5372	50	13	as	as	ADP
ejpam-5372	50	14	a	a	DET
ejpam-5372	50	15	square	square	ADJ
ejpam-5372	50	16	modulo	modulo	NOUN
ejpam-5372	50	17	p	p	X
ejpam-5372	50	18	,	,	PUNCT
ejpam-5372	50	19	and	and	CCONJ
ejpam-5372	50	20	a	a	DET
ejpam-5372	50	21	semi	semi	ADJ
ejpam-5372	50	22	-	-	ADJ
ejpam-5372	50	23	primitive	primitive	ADJ
ejpam-5372	50	24	root	root	NOUN
ejpam-5372	50	25	is	be	AUX
ejpam-5372	50	26	the	the	DET
ejpam-5372	50	27	square	square	NOUN
ejpam-5372	50	28	of	of	ADP
ejpam-5372	50	29	a	a	DET
ejpam-5372	50	30	primitive	primitive	ADJ
ejpam-5372	50	31	root	root	NOUN
ejpam-5372	50	32	.	.	PUNCT
ejpam-5372	51	1	moreover	moreover	ADV
ejpam-5372	51	2	,	,	PUNCT
ejpam-5372	51	3	g	g	PROPN
ejpam-5372	51	4	∈	∈	PROPN
ejpam-5372	51	5	f∗	f∗	NOUN
ejpam-5372	51	6	p	p	NOUN
ejpam-5372	51	7	is	be	AUX
ejpam-5372	51	8	either	either	CCONJ
ejpam-5372	51	9	a	a	DET
ejpam-5372	51	10	quadratic	quadratic	ADJ
ejpam-5372	51	11	non	non	ADJ
ejpam-5372	51	12	-	-	ADJ
ejpam-5372	51	13	residue	residue	NOUN
ejpam-5372	51	14	or	or	CCONJ
ejpam-5372	51	15	a	a	DET
ejpam-5372	51	16	quadratic	quadratic	ADJ
ejpam-5372	51	17	residue	residue	NOUN
ejpam-5372	51	18	,	,	PUNCT
ejpam-5372	51	19	i.e.	i.e.	X
ejpam-5372	51	20	if	if	SCONJ
ejpam-5372	51	21	h	h	NOUN
ejpam-5372	51	22	is	be	AUX
ejpam-5372	51	23	not	not	PART
ejpam-5372	51	24	a	a	DET
ejpam-5372	51	25	quadratic	quadratic	ADJ
ejpam-5372	51	26	non	non	ADJ
ejpam-5372	51	27	-	-	ADJ
ejpam-5372	51	28	residue	residue	ADJ
ejpam-5372	51	29	,	,	PUNCT
ejpam-5372	51	30	there	there	PRON
ejpam-5372	51	31	exists	exist	VERB
ejpam-5372	51	32	k	k	X
ejpam-5372	51	33	such	such	ADJ
ejpam-5372	51	34	that	that	SCONJ
ejpam-5372	51	35	g	g	PROPN
ejpam-5372	51	36	=	=	SYM
ejpam-5372	51	37	h2k	h2k	PROPN
ejpam-5372	51	38	.	.	PUNCT
ejpam-5372	52	1	thus	thus	ADV
ejpam-5372	52	2	,	,	PUNCT
ejpam-5372	52	3	the	the	DET
ejpam-5372	52	4	order	order	NOUN
ejpam-5372	52	5	of	of	ADP
ejpam-5372	52	6	⟨g⟩	⟨g⟩	PROPN
ejpam-5372	52	7	divides	divide	VERB
ejpam-5372	52	8	p−1	p−1	PROPN
ejpam-5372	52	9	2	2	NUM
ejpam-5372	52	10	and	and	CCONJ
ejpam-5372	52	11	g	g	NOUN
ejpam-5372	52	12	is	be	AUX
ejpam-5372	52	13	therefore	therefore	ADV
ejpam-5372	52	14	not	not	PART
ejpam-5372	52	15	a	a	DET
ejpam-5372	52	16	primitive	primitive	ADJ
ejpam-5372	52	17	root	root	NOUN
ejpam-5372	52	18	,	,	PUNCT
ejpam-5372	52	19	hence	hence	ADV
ejpam-5372	52	20	gz	gz	PROPN
ejpam-5372	52	21	⊂	⊂	PROPN
ejpam-5372	52	22	gqnr	gqnr	PROPN
ejpam-5372	52	23	.	.	PUNCT
ejpam-5372	53	1	definition	definition	NOUN
ejpam-5372	53	2	3	3	NUM
ejpam-5372	53	3	.	.	PUNCT
ejpam-5372	54	1	let	let	VERB
ejpam-5372	54	2	p	p	PROPN
ejpam-5372	54	3	∈	∈	PROPN
ejpam-5372	54	4	p⧹	p⧹	VERB
ejpam-5372	54	5	{	{	PUNCT
ejpam-5372	54	6	2	2	NUM
ejpam-5372	54	7	}	}	PUNCT
ejpam-5372	54	8	and	and	CCONJ
ejpam-5372	54	9	m	m	PROPN
ejpam-5372	54	10	∈	∈	PROPN
ejpam-5372	54	11	z	z	PROPN
ejpam-5372	54	12	/	/	SYM
ejpam-5372	54	13	pz	pz	NOUN
ejpam-5372	54	14	.	.	PUNCT
ejpam-5372	55	1	we	we	PRON
ejpam-5372	55	2	denote	denote	VERB
ejpam-5372	55	3	by	by	ADP
ejpam-5372	55	4	rp	rp	NOUN
ejpam-5372	55	5	(	(	PUNCT
ejpam-5372	55	6	m	m	PROPN
ejpam-5372	55	7	)	)	PUNCT
ejpam-5372	55	8	the	the	DET
ejpam-5372	55	9	set	set	NOUN
ejpam-5372	55	10	of	of	ADP
ejpam-5372	55	11	square	square	ADJ
ejpam-5372	55	12	roots	root	NOUN
ejpam-5372	55	13	of	of	ADP
ejpam-5372	55	14	m	m	PRON
ejpam-5372	55	15	i.e.	i.e.	X
ejpam-5372	55	16	:	:	PUNCT
ejpam-5372	55	17	rp	rp	NOUN
ejpam-5372	55	18	(	(	PUNCT
ejpam-5372	55	19	m	m	NOUN
ejpam-5372	55	20	)	)	PUNCT
ejpam-5372	55	21	=	=	PRON
ejpam-5372	55	22	{	{	PUNCT
ejpam-5372	55	23	g	g	PROPN
ejpam-5372	55	24	∈	∈	PROPN
ejpam-5372	55	25	z	z	PROPN
ejpam-5372	55	26	/	/	SYM
ejpam-5372	55	27	pz	pz	PROPN
ejpam-5372	56	1	∣∣	∣∣	PROPN
ejpam-5372	56	2	g2	g2	PROPN
ejpam-5372	56	3	≡	≡	PROPN
ejpam-5372	56	4	m	m	VERB
ejpam-5372	56	5	}	}	PUNCT
ejpam-5372	56	6	.	.	PUNCT
ejpam-5372	57	1	m.	m.	NOUN
ejpam-5372	57	2	wolf	wolf	PROPN
ejpam-5372	57	3	,	,	PUNCT
ejpam-5372	57	4	f.	f.	PROPN
ejpam-5372	57	5	wolf	wolf	PROPN
ejpam-5372	57	6	/	/	SYM
ejpam-5372	57	7	eur	eur	PROPN
ejpam-5372	57	8	.	.	PUNCT
ejpam-5372	58	1	j.	j.	PROPN
ejpam-5372	58	2	pure	pure	PROPN
ejpam-5372	58	3	appl	appl	PROPN
ejpam-5372	58	4	.	.	PROPN
ejpam-5372	58	5	math	math	PROPN
ejpam-5372	58	6	,	,	PUNCT
ejpam-5372	58	7	17	17	NUM
ejpam-5372	58	8	(	(	PUNCT
ejpam-5372	58	9	4	4	NUM
ejpam-5372	58	10	)	)	PUNCT
ejpam-5372	58	11	(	(	PUNCT
ejpam-5372	58	12	2024	2024	NUM
ejpam-5372	58	13	)	)	PUNCT
ejpam-5372	58	14	,	,	PUNCT
ejpam-5372	58	15	2431	2431	NUM
ejpam-5372	58	16	-	-	SYM
ejpam-5372	58	17	2447	2447	NUM
ejpam-5372	58	18	2433	2433	NUM
ejpam-5372	58	19	more	more	ADV
ejpam-5372	58	20	generally	generally	ADV
ejpam-5372	58	21	,	,	PUNCT
ejpam-5372	58	22	we	we	PRON
ejpam-5372	58	23	recursively	recursively	ADV
ejpam-5372	58	24	define	define	VERB
ejpam-5372	58	25	the	the	DET
ejpam-5372	58	26	sequence	sequence	NOUN
ejpam-5372	58	27	(	(	PUNCT
ejpam-5372	58	28	rk	rk	NOUN
ejpam-5372	58	29	p	p	X
ejpam-5372	58	30	(	(	PUNCT
ejpam-5372	58	31	m	m	NOUN
ejpam-5372	58	32	)	)	PUNCT
ejpam-5372	58	33	)	)	PUNCT
ejpam-5372	59	1	k≥0	k≥0	PROPN
ejpam-5372	59	2	by	by	ADP
ejpam-5372	59	3	:	:	PUNCT
ejpam-5372	59	4	r0	r0	NOUN
ejpam-5372	59	5	p	p	NOUN
ejpam-5372	59	6	(	(	PUNCT
ejpam-5372	59	7	m	m	NOUN
ejpam-5372	59	8	)	)	PUNCT
ejpam-5372	59	9	=	=	PRON
ejpam-5372	59	10	{	{	PUNCT
ejpam-5372	59	11	m	m	X
ejpam-5372	59	12	}	}	PUNCT
ejpam-5372	59	13	,	,	PUNCT
ejpam-5372	59	14	rk+1	rk+1	ADP
ejpam-5372	59	15	p	p	PROPN
ejpam-5372	59	16	(	(	PUNCT
ejpam-5372	59	17	m	m	NOUN
ejpam-5372	59	18	)	)	PUNCT
ejpam-5372	59	19	=	=	SYM
ejpam-5372	60	1	⋃	⋃	ADP
ejpam-5372	60	2	g∈rk	g∈rk	NOUN
ejpam-5372	60	3	p(m)rp	p(m)rp	X
ejpam-5372	60	4	(	(	PUNCT
ejpam-5372	60	5	g	g	NOUN
ejpam-5372	60	6	)	)	PUNCT
ejpam-5372	60	7	so	so	SCONJ
ejpam-5372	60	8	that	that	PRON
ejpam-5372	60	9	rp	rp	NOUN
ejpam-5372	60	10	(	(	PUNCT
ejpam-5372	60	11	m	m	NOUN
ejpam-5372	60	12	)	)	PUNCT
ejpam-5372	60	13	=	=	SYM
ejpam-5372	60	14	r1	r1	PROPN
ejpam-5372	60	15	p	p	PROPN
ejpam-5372	60	16	(	(	PUNCT
ejpam-5372	60	17	m	m	PROPN
ejpam-5372	60	18	)	)	PUNCT
ejpam-5372	60	19	.	.	PUNCT
ejpam-5372	61	1	we	we	PRON
ejpam-5372	61	2	will	will	AUX
ejpam-5372	61	3	simply	simply	ADV
ejpam-5372	61	4	note	note	VERB
ejpam-5372	61	5	r	r	NOUN
ejpam-5372	61	6	(	(	PUNCT
ejpam-5372	61	7	m	m	NOUN
ejpam-5372	61	8	)	)	PUNCT
ejpam-5372	61	9	and	and	CCONJ
ejpam-5372	61	10	rk	rk	PROPN
ejpam-5372	61	11	(	(	PUNCT
ejpam-5372	61	12	m	m	PROPN
ejpam-5372	61	13	)	)	PUNCT
ejpam-5372	61	14	when	when	SCONJ
ejpam-5372	61	15	the	the	DET
ejpam-5372	61	16	value	value	NOUN
ejpam-5372	61	17	of	of	ADP
ejpam-5372	61	18	p	p	NOUN
ejpam-5372	61	19	is	be	AUX
ejpam-5372	61	20	unambiguous	unambiguous	ADJ
ejpam-5372	61	21	.	.	PUNCT
ejpam-5372	62	1	we	we	PRON
ejpam-5372	62	2	will	will	AUX
ejpam-5372	62	3	also	also	ADV
ejpam-5372	62	4	note	note	VERB
ejpam-5372	62	5	rk	rk	NOUN
ejpam-5372	62	6	for	for	ADP
ejpam-5372	62	7	rk	rk	PROPN
ejpam-5372	62	8	(	(	PUNCT
ejpam-5372	62	9	1	1	NUM
ejpam-5372	62	10	)	)	PUNCT
ejpam-5372	62	11	.	.	PUNCT
ejpam-5372	63	1	proposition	proposition	NOUN
ejpam-5372	63	2	3	3	NUM
ejpam-5372	63	3	.	.	PUNCT
ejpam-5372	64	1	rk	rk	NOUN
ejpam-5372	64	2	is	be	AUX
ejpam-5372	64	3	a	a	DET
ejpam-5372	64	4	subgroup	subgroup	NOUN
ejpam-5372	64	5	of	of	ADP
ejpam-5372	64	6	f∗	f∗	NOUN
ejpam-5372	64	7	p.	p.	NOUN
ejpam-5372	64	8	proposition	proposition	NOUN
ejpam-5372	64	9	4	4	NUM
ejpam-5372	64	10	.	.	PUNCT
ejpam-5372	65	1	let	let	VERB
ejpam-5372	65	2	p	p	PROPN
ejpam-5372	65	3	∈	∈	PROPN
ejpam-5372	65	4	p⧹	p⧹	VERB
ejpam-5372	65	5	{	{	PUNCT
ejpam-5372	65	6	2	2	NUM
ejpam-5372	65	7	}	}	PUNCT
ejpam-5372	65	8	and	and	CCONJ
ejpam-5372	65	9	n	n	DET
ejpam-5372	65	10	the	the	DET
ejpam-5372	65	11	highest	high	ADJ
ejpam-5372	65	12	power	power	NOUN
ejpam-5372	65	13	of	of	ADP
ejpam-5372	65	14	2	2	NUM
ejpam-5372	65	15	dividing	divide	VERB
ejpam-5372	65	16	p	p	NOUN
ejpam-5372	65	17	−	−	PROPN
ejpam-5372	65	18	1	1	NUM
ejpam-5372	65	19	.	.	PUNCT
ejpam-5372	66	1	for	for	ADP
ejpam-5372	66	2	any	any	DET
ejpam-5372	66	3	0	0	NUM
ejpam-5372	66	4	≤	≤	NUM
ejpam-5372	66	5	k	k	NOUN
ejpam-5372	66	6	≤	≤	PROPN
ejpam-5372	66	7	n	n	CCONJ
ejpam-5372	66	8	,	,	PUNCT
ejpam-5372	66	9	we	we	PRON
ejpam-5372	66	10	have	have	VERB
ejpam-5372	66	11	:	:	PUNCT
ejpam-5372	66	12	∣∣rk	∣∣rk	PROPN
ejpam-5372	66	13	∣∣	∣∣	NUM
ejpam-5372	66	14	=	=	SYM
ejpam-5372	66	15	2k	2k	PROPN
ejpam-5372	66	16	and	and	CCONJ
ejpam-5372	66	17	for	for	ADP
ejpam-5372	66	18	any	any	DET
ejpam-5372	66	19	1	1	NUM
ejpam-5372	66	20	≤	≤	NUM
ejpam-5372	66	21	k	k	NOUN
ejpam-5372	66	22	≤	≤	PROPN
ejpam-5372	67	1	n	n	CCONJ
ejpam-5372	67	2	:	:	PUNCT
ejpam-5372	67	3	rk−1	rk−1	VERB
ejpam-5372	67	4	⊂	⊂	ADJ
ejpam-5372	67	5	rk	rk	NOUN
ejpam-5372	67	6	,	,	PUNCT
ejpam-5372	67	7	∣∣rk⧹rk−1	∣∣rk⧹rk−1	X
ejpam-5372	67	8	∣∣	∣∣	NUM
ejpam-5372	67	9	=	=	SYM
ejpam-5372	67	10	2k−1	2k−1	NUM
ejpam-5372	67	11	for	for	ADP
ejpam-5372	67	12	k	k	PROPN
ejpam-5372	67	13	>	>	PUNCT
ejpam-5372	67	14	n	n	CCONJ
ejpam-5372	67	15	,	,	PUNCT
ejpam-5372	67	16	we	we	PRON
ejpam-5372	67	17	have	have	VERB
ejpam-5372	67	18	rk	rk	NOUN
ejpam-5372	67	19	=	=	SYM
ejpam-5372	67	20	rn	rn	PROPN
ejpam-5372	67	21	.	.	PROPN
ejpam-5372	68	1	proof	proof	NOUN
ejpam-5372	68	2	.	.	PUNCT
ejpam-5372	69	1	if	if	SCONJ
ejpam-5372	69	2	h	h	NOUN
ejpam-5372	69	3	is	be	AUX
ejpam-5372	69	4	a	a	DET
ejpam-5372	69	5	primitive	primitive	ADJ
ejpam-5372	69	6	root	root	NOUN
ejpam-5372	69	7	,	,	PUNCT
ejpam-5372	69	8	we	we	PRON
ejpam-5372	69	9	note	note	VERB
ejpam-5372	69	10	that	that	SCONJ
ejpam-5372	69	11	rk	rk	NOUN
ejpam-5372	69	12	=	=	PUNCT
ejpam-5372	69	13	{	{	PUNCT
ejpam-5372	69	14	hk	hk	PROPN
ejpam-5372	69	15	,	,	PUNCT
ejpam-5372	69	16	k	k	PROPN
ejpam-5372	69	17	∈	∈	PROPN
ejpam-5372	69	18	p−1	p−1	PROPN
ejpam-5372	69	19	2k	2k	PROPN
ejpam-5372	69	20	z	z	NOUN
ejpam-5372	69	21	}	}	PUNCT
ejpam-5372	69	22	.	.	PUNCT
ejpam-5372	70	1	the	the	DET
ejpam-5372	70	2	first	first	ADJ
ejpam-5372	70	3	two	two	NUM
ejpam-5372	70	4	points	point	NOUN
ejpam-5372	70	5	follow	follow	VERB
ejpam-5372	70	6	from	from	ADP
ejpam-5372	70	7	this	this	PRON
ejpam-5372	70	8	.	.	PUNCT
ejpam-5372	71	1	furthermore	furthermore	ADV
ejpam-5372	71	2	,	,	PUNCT
ejpam-5372	71	3	for	for	ADP
ejpam-5372	71	4	k	k	PROPN
ejpam-5372	71	5	>	>	PUNCT
ejpam-5372	71	6	n	n	CCONJ
ejpam-5372	71	7	,	,	PUNCT
ejpam-5372	71	8	f∗	f∗	PROPN
ejpam-5372	71	9	p	p	NOUN
ejpam-5372	71	10	contains	contain	VERB
ejpam-5372	71	11	no	no	DET
ejpam-5372	71	12	element	element	NOUN
ejpam-5372	71	13	of	of	ADP
ejpam-5372	71	14	order	order	NOUN
ejpam-5372	71	15	2k	2k	NUM
ejpam-5372	71	16	,	,	PUNCT
ejpam-5372	71	17	thus	thus	ADV
ejpam-5372	71	18	rk	rk	PROPN
ejpam-5372	71	19	=	=	SYM
ejpam-5372	71	20	rn	rn	PROPN
ejpam-5372	71	21	.	.	PROPN
ejpam-5372	71	22	2	2	NUM
ejpam-5372	71	23	.	.	NUM
ejpam-5372	71	24	generators	generator	NOUN
ejpam-5372	71	25	and	and	CCONJ
ejpam-5372	71	26	semi	semi	NOUN
ejpam-5372	71	27	-	-	NOUN
ejpam-5372	71	28	generators	generator	NOUN
ejpam-5372	71	29	of	of	ADP
ejpam-5372	71	30	f∗	f∗	NOUN
ejpam-5372	71	31	p	p	NOUN
ejpam-5372	71	32	in	in	ADP
ejpam-5372	71	33	this	this	DET
ejpam-5372	71	34	section	section	NOUN
ejpam-5372	71	35	,	,	PUNCT
ejpam-5372	71	36	we	we	PRON
ejpam-5372	71	37	show	show	VERB
ejpam-5372	71	38	that	that	SCONJ
ejpam-5372	71	39	gz	gz	NOUN
ejpam-5372	71	40	can	can	AUX
ejpam-5372	71	41	be	be	AUX
ejpam-5372	71	42	obtained	obtain	VERB
ejpam-5372	71	43	from	from	ADP
ejpam-5372	71	44	quadratic	quadratic	ADJ
ejpam-5372	71	45	residues	residue	NOUN
ejpam-5372	71	46	.	.	PUNCT
ejpam-5372	72	1	we	we	PRON
ejpam-5372	72	2	give	give	VERB
ejpam-5372	72	3	an	an	DET
ejpam-5372	72	4	algorithm	algorithm	NOUN
ejpam-5372	72	5	that	that	PRON
ejpam-5372	72	6	generates	generate	VERB
ejpam-5372	72	7	gz	gz	NOUN
ejpam-5372	72	8	without	without	ADP
ejpam-5372	72	9	computing	compute	VERB
ejpam-5372	72	10	any	any	DET
ejpam-5372	72	11	gcd	gcd	NOUN
ejpam-5372	72	12	.	.	PUNCT
ejpam-5372	73	1	2.1	2.1	NUM
ejpam-5372	73	2	.	.	PUNCT
ejpam-5372	73	3	determining	determine	VERB
ejpam-5372	73	4	primitives	primitive	NOUN
ejpam-5372	73	5	let	let	VERB
ejpam-5372	73	6	p	p	X
ejpam-5372	73	7	∈	∈	PROPN
ejpam-5372	73	8	p⧹	p⧹	VERB
ejpam-5372	73	9	{	{	PUNCT
ejpam-5372	73	10	2	2	NUM
ejpam-5372	73	11	}	}	PUNCT
ejpam-5372	73	12	.	.	PUNCT
ejpam-5372	74	1	the	the	DET
ejpam-5372	74	2	number	number	NOUN
ejpam-5372	74	3	of	of	ADP
ejpam-5372	74	4	generators	generator	NOUN
ejpam-5372	74	5	of	of	ADP
ejpam-5372	74	6	the	the	DET
ejpam-5372	74	7	cyclic	cyclic	ADJ
ejpam-5372	74	8	group	group	NOUN
ejpam-5372	74	9	f∗	f∗	NOUN
ejpam-5372	74	10	p	p	NOUN
ejpam-5372	74	11	is	be	AUX
ejpam-5372	74	12	equal	equal	ADJ
ejpam-5372	74	13	to	to	ADP
ejpam-5372	74	14	φ	φ	PROPN
ejpam-5372	74	15	(	(	PUNCT
ejpam-5372	74	16	φ	φ	PROPN
ejpam-5372	74	17	(	(	PUNCT
ejpam-5372	74	18	p	p	NOUN
ejpam-5372	74	19	)	)	PUNCT
ejpam-5372	74	20	)	)	PUNCT
ejpam-5372	75	1	=	=	SYM
ejpam-5372	75	2	φ	φ	PROPN
ejpam-5372	75	3	(	(	PUNCT
ejpam-5372	75	4	p−	p−	NOUN
ejpam-5372	75	5	1	1	NUM
ejpam-5372	75	6	)	)	PUNCT
ejpam-5372	75	7	.	.	PUNCT
ejpam-5372	76	1	the	the	DET
ejpam-5372	76	2	following	follow	VERB
ejpam-5372	76	3	property	property	NOUN
ejpam-5372	76	4	gives	give	VERB
ejpam-5372	76	5	a	a	DET
ejpam-5372	76	6	constructive	constructive	ADJ
ejpam-5372	76	7	method	method	NOUN
ejpam-5372	76	8	for	for	ADP
ejpam-5372	76	9	determining	determine	VERB
ejpam-5372	76	10	primitive	primitive	ADJ
ejpam-5372	76	11	or	or	CCONJ
ejpam-5372	76	12	semi	semi	ADJ
ejpam-5372	76	13	-	-	ADJ
ejpam-5372	76	14	primitive	primitive	ADJ
ejpam-5372	76	15	roots	root	NOUN
ejpam-5372	76	16	.	.	PUNCT
ejpam-5372	77	1	proposition	proposition	NOUN
ejpam-5372	77	2	5	5	NUM
ejpam-5372	77	3	.	.	PUNCT
ejpam-5372	78	1	we	we	PRON
ejpam-5372	78	2	decompose	decompose	VERB
ejpam-5372	78	3	into	into	ADP
ejpam-5372	78	4	prime	prime	ADJ
ejpam-5372	78	5	factors	factor	NOUN
ejpam-5372	78	6	p	p	NOUN
ejpam-5372	78	7	−	−	PROPN
ejpam-5372	78	8	1	1	NUM
ejpam-5372	78	9	=	=	SYM
ejpam-5372	78	10	∏	∏	NUM
ejpam-5372	78	11	q∈p	q∈p	NOUN
ejpam-5372	78	12	q	q	NOUN
ejpam-5372	78	13	αq	αq	INTJ
ejpam-5372	78	14	.	.	PUNCT
ejpam-5372	79	1	then	then	ADV
ejpam-5372	79	2	g	g	PROPN
ejpam-5372	79	3	∈	∈	PROPN
ejpam-5372	79	4	f∗	f∗	NOUN
ejpam-5372	79	5	p	p	NOUN
ejpam-5372	79	6	is	be	AUX
ejpam-5372	79	7	a	a	DET
ejpam-5372	79	8	primitive	primitive	ADJ
ejpam-5372	79	9	root	root	NOUN
ejpam-5372	79	10	if	if	SCONJ
ejpam-5372	80	1	and	and	CCONJ
ejpam-5372	80	2	only	only	ADV
ejpam-5372	80	3	if	if	SCONJ
ejpam-5372	80	4	:	:	PUNCT
ejpam-5372	80	5	∀q	∀q	PROPN
ejpam-5372	80	6	∈	∈	PROPN
ejpam-5372	80	7	p	p	X
ejpam-5372	80	8	(	(	PUNCT
ejpam-5372	80	9	αq	αq	ADP
ejpam-5372	80	10	≥	≥	NOUN
ejpam-5372	80	11	1	1	NUM
ejpam-5372	80	12	⇒	⇒	NOUN
ejpam-5372	80	13	g	g	PROPN
ejpam-5372	80	14	p−1	p−1	PROPN
ejpam-5372	80	15	q	q	PROPN
ejpam-5372	80	16	̸≡	̸≡	PROPN
ejpam-5372	80	17	1	1	NUM
ejpam-5372	80	18	)	)	PUNCT
ejpam-5372	80	19	in	in	ADP
ejpam-5372	80	20	this	this	DET
ejpam-5372	80	21	case	case	NOUN
ejpam-5372	80	22	we	we	PRON
ejpam-5372	80	23	have	have	AUX
ejpam-5372	80	24	:	:	PUNCT
ejpam-5372	80	25	gz	gz	VERB
ejpam-5372	80	26	=	=	PRON
ejpam-5372	80	27	{	{	PUNCT
ejpam-5372	80	28	gk	gk	PROPN
ejpam-5372	80	29	,	,	PUNCT
ejpam-5372	80	30	1	1	NUM
ejpam-5372	80	31	≤	≤	NUM
ejpam-5372	80	32	k	k	X
ejpam-5372	80	33	≤	≤	NUM
ejpam-5372	80	34	p−	p−	NOUN
ejpam-5372	80	35	2	2	NUM
ejpam-5372	80	36	,	,	PUNCT
ejpam-5372	80	37	gcd	gcd	X
ejpam-5372	80	38	(	(	PUNCT
ejpam-5372	80	39	k	k	NOUN
ejpam-5372	80	40	,	,	PUNCT
ejpam-5372	80	41	p−	p−	NOUN
ejpam-5372	80	42	1	1	NUM
ejpam-5372	80	43	)	)	PUNCT
ejpam-5372	80	44	=	=	SYM
ejpam-5372	80	45	1	1	X
ejpam-5372	80	46	}	}	PUNCT
ejpam-5372	80	47	moreover	moreover	ADV
ejpam-5372	80	48	if	if	SCONJ
ejpam-5372	80	49	p	p	PROPN
ejpam-5372	80	50	∈	∈	PROPN
ejpam-5372	80	51	p1,4	p1,4	PROPN
ejpam-5372	80	52	:	:	PUNCT
ejpam-5372	80	53	g	g	PROPN
ejpam-5372	80	54	∈	∈	PROPN
ejpam-5372	80	55	gs	gs	PROPN
ejpam-5372	80	56	⇔	⇔	PROPN
ejpam-5372	80	57	g	g	PROPN
ejpam-5372	80	58	∈	∈	PROPN
ejpam-5372	80	59	gqr	gqr	PROPN
ejpam-5372	80	60	and	and	CCONJ
ejpam-5372	80	61	∀q	∀q	PROPN
ejpam-5372	80	62	∈	∈	PROPN
ejpam-5372	80	63	p	p	X
ejpam-5372	80	64	(	(	PUNCT
ejpam-5372	80	65	αq	αq	ADP
ejpam-5372	80	66	≥	≥	NOUN
ejpam-5372	80	67	1	1	NUM
ejpam-5372	80	68	⇒	⇒	NOUN
ejpam-5372	80	69	g	g	PROPN
ejpam-5372	80	70	p−1	p−1	PROPN
ejpam-5372	80	71	2q	2q	NUM
ejpam-5372	80	72	̸≡	̸≡	X
ejpam-5372	80	73	1	1	X
ejpam-5372	80	74	)	)	PUNCT
ejpam-5372	80	75	⇔	⇔	NOUN
ejpam-5372	80	76	∅	∅	NOUN
ejpam-5372	80	77	⊊	⊊	VERB
ejpam-5372	80	78	rp	rp	NOUN
ejpam-5372	80	79	(	(	PUNCT
ejpam-5372	80	80	g	g	NOUN
ejpam-5372	80	81	)	)	PUNCT
ejpam-5372	80	82	⊂	⊂	PROPN
ejpam-5372	80	83	gz	gz	PROPN
ejpam-5372	80	84	.	.	PUNCT
ejpam-5372	81	1	if	if	SCONJ
ejpam-5372	81	2	p	p	PROPN
ejpam-5372	81	3	∈	∈	PROPN
ejpam-5372	81	4	p3,4	p3,4	VERB
ejpam-5372	81	5	:	:	PUNCT
ejpam-5372	81	6	g	g	PROPN
ejpam-5372	81	7	∈	∈	PROPN
ejpam-5372	81	8	gs	gs	PROPN
ejpam-5372	81	9	⇔	⇔	PROPN
ejpam-5372	81	10	g	g	PROPN
ejpam-5372	81	11	p−1	p−1	PROPN
ejpam-5372	81	12	2	2	NUM
ejpam-5372	81	13	≡	≡	PROPN
ejpam-5372	81	14	1	1	NUM
ejpam-5372	81	15	and	and	CCONJ
ejpam-5372	81	16	∀q	∀q	PROPN
ejpam-5372	81	17	∈	∈	PROPN
ejpam-5372	81	18	p⧹	p⧹	VERB
ejpam-5372	81	19	{	{	PUNCT
ejpam-5372	81	20	2	2	NUM
ejpam-5372	81	21	}	}	PUNCT
ejpam-5372	81	22	(	(	PUNCT
ejpam-5372	81	23	αq	αq	ADP
ejpam-5372	81	24	≥	≥	NOUN
ejpam-5372	81	25	1	1	NUM
ejpam-5372	81	26	⇒	⇒	NOUN
ejpam-5372	81	27	g	g	PROPN
ejpam-5372	81	28	p−1	p−1	PROPN
ejpam-5372	81	29	2q	2q	NUM
ejpam-5372	81	30	̸≡	̸≡	X
ejpam-5372	81	31	1	1	NUM
ejpam-5372	81	32	)	)	PUNCT
ejpam-5372	81	33	.	.	PUNCT
ejpam-5372	82	1	m.	m.	PROPN
ejpam-5372	82	2	wolf	wolf	PROPN
ejpam-5372	82	3	,	,	PUNCT
ejpam-5372	82	4	f.	f.	PROPN
ejpam-5372	82	5	wolf	wolf	PROPN
ejpam-5372	82	6	/	/	SYM
ejpam-5372	82	7	eur	eur	PROPN
ejpam-5372	82	8	.	.	PUNCT
ejpam-5372	83	1	j.	j.	PROPN
ejpam-5372	83	2	pure	pure	PROPN
ejpam-5372	83	3	appl	appl	PROPN
ejpam-5372	83	4	.	.	PROPN
ejpam-5372	83	5	math	math	PROPN
ejpam-5372	83	6	,	,	PUNCT
ejpam-5372	83	7	17	17	NUM
ejpam-5372	83	8	(	(	PUNCT
ejpam-5372	83	9	4	4	NUM
ejpam-5372	83	10	)	)	PUNCT
ejpam-5372	83	11	(	(	PUNCT
ejpam-5372	83	12	2024	2024	NUM
ejpam-5372	83	13	)	)	PUNCT
ejpam-5372	83	14	,	,	PUNCT
ejpam-5372	83	15	2431	2431	NUM
ejpam-5372	83	16	-	-	SYM
ejpam-5372	83	17	2447	2447	NUM
ejpam-5372	83	18	2434	2434	NUM
ejpam-5372	83	19	in	in	ADP
ejpam-5372	83	20	this	this	DET
ejpam-5372	83	21	second	second	ADJ
ejpam-5372	83	22	case	case	NOUN
ejpam-5372	83	23	,	,	PUNCT
ejpam-5372	83	24	only	only	ADV
ejpam-5372	83	25	one	one	NUM
ejpam-5372	83	26	of	of	ADP
ejpam-5372	83	27	the	the	DET
ejpam-5372	83	28	two	two	NUM
ejpam-5372	83	29	square	square	ADJ
ejpam-5372	83	30	roots	root	NOUN
ejpam-5372	83	31	of	of	ADP
ejpam-5372	83	32	any	any	DET
ejpam-5372	83	33	g	g	PROPN
ejpam-5372	83	34	∈	∈	PROPN
ejpam-5372	83	35	gs	gs	NOUN
ejpam-5372	83	36	is	be	AUX
ejpam-5372	83	37	a	a	DET
ejpam-5372	83	38	primitive	primitive	ADJ
ejpam-5372	83	39	root	root	NOUN
ejpam-5372	83	40	.	.	PUNCT
ejpam-5372	84	1	proof	proof	NOUN
ejpam-5372	84	2	.	.	PUNCT
ejpam-5372	85	1	let	let	VERB
ejpam-5372	85	2	us	we	PRON
ejpam-5372	85	3	take	take	VERB
ejpam-5372	85	4	g	g	PROPN
ejpam-5372	85	5	∈	∈	PROPN
ejpam-5372	85	6	gz	gz	NOUN
ejpam-5372	85	7	.	.	PUNCT
ejpam-5372	86	1	we	we	PRON
ejpam-5372	86	2	know	know	VERB
ejpam-5372	86	3	that	that	SCONJ
ejpam-5372	86	4	the	the	DET
ejpam-5372	86	5	order	order	NOUN
ejpam-5372	86	6	of	of	ADP
ejpam-5372	86	7	gk	gk	PROPN
ejpam-5372	86	8	is	be	AUX
ejpam-5372	86	9	p−1	p−1	PROPN
ejpam-5372	86	10	gcd(k	gcd(k	PROPN
ejpam-5372	86	11	,	,	PUNCT
ejpam-5372	86	12	p−1	p−1	PROPN
ejpam-5372	86	13	)	)	PUNCT
ejpam-5372	86	14	.	.	PUNCT
ejpam-5372	87	1	more	more	ADV
ejpam-5372	87	2	specifically	specifically	ADV
ejpam-5372	87	3	,	,	PUNCT
ejpam-5372	87	4	for	for	ADP
ejpam-5372	87	5	any	any	DET
ejpam-5372	87	6	q	q	NOUN
ejpam-5372	87	7	∈	∈	PROPN
ejpam-5372	87	8	p	p	NOUN
ejpam-5372	87	9	such	such	ADJ
ejpam-5372	87	10	that	that	SCONJ
ejpam-5372	87	11	αq	αq	INTJ
ejpam-5372	87	12	≥	≥	NUM
ejpam-5372	87	13	1	1	NUM
ejpam-5372	87	14	(	(	PUNCT
ejpam-5372	87	15	i.e.	i.e.	X
ejpam-5372	87	16	q	q	X
ejpam-5372	87	17	dividing	dividing	NOUN
ejpam-5372	87	18	p	p	NOUN
ejpam-5372	87	19	−	−	PROPN
ejpam-5372	87	20	1	1	NUM
ejpam-5372	87	21	)	)	PUNCT
ejpam-5372	87	22	the	the	DET
ejpam-5372	87	23	order	order	NOUN
ejpam-5372	87	24	of	of	ADP
ejpam-5372	87	25	g	g	PROPN
ejpam-5372	87	26	p−1	p−1	PROPN
ejpam-5372	87	27	q	q	PROPN
ejpam-5372	87	28	is	be	AUX
ejpam-5372	87	29	p−1	p−1	PROPN
ejpam-5372	87	30	gcd	gcd	NOUN
ejpam-5372	87	31	(	(	PUNCT
ejpam-5372	87	32	p−1	p−1	PROPN
ejpam-5372	87	33	q	q	PROPN
ejpam-5372	87	34	,	,	PUNCT
ejpam-5372	87	35	p−1	p−1	PROPN
ejpam-5372	87	36	)	)	PUNCT
ejpam-5372	88	1	=	=	PUNCT
ejpam-5372	88	2	q	q	PUNCT
ejpam-5372	89	1	so	so	ADV
ejpam-5372	89	2	g	g	PROPN
ejpam-5372	89	3	p−1	p−1	PROPN
ejpam-5372	89	4	q	q	PROPN
ejpam-5372	89	5	̸≡	̸≡	PROPN
ejpam-5372	89	6	1	1	X
ejpam-5372	89	7	.	.	PUNCT
ejpam-5372	90	1	furthermore	furthermore	ADV
ejpam-5372	90	2	,	,	PUNCT
ejpam-5372	90	3	h	h	NOUN
ejpam-5372	90	4	∈	∈	PROPN
ejpam-5372	90	5	gz	gz	VERB
ejpam-5372	90	6	if	if	SCONJ
ejpam-5372	90	7	and	and	CCONJ
ejpam-5372	90	8	only	only	ADV
ejpam-5372	90	9	if	if	SCONJ
ejpam-5372	90	10	there	there	PRON
ejpam-5372	90	11	is	be	VERB
ejpam-5372	90	12	k	k	PROPN
ejpam-5372	90	13	coprime	coprime	NOUN
ejpam-5372	90	14	with	with	ADP
ejpam-5372	90	15	p−	p−	NOUN
ejpam-5372	90	16	1	1	NUM
ejpam-5372	90	17	such	such	ADJ
ejpam-5372	90	18	that	that	DET
ejpam-5372	90	19	h	h	NOUN
ejpam-5372	90	20	=	=	SYM
ejpam-5372	90	21	gk	gk	PROPN
ejpam-5372	90	22	.	.	PUNCT
ejpam-5372	91	1	conversely	conversely	ADV
ejpam-5372	91	2	,	,	PUNCT
ejpam-5372	91	3	let	let	VERB
ejpam-5372	91	4	g	g	PROPN
ejpam-5372	91	5	∈	∈	PROPN
ejpam-5372	91	6	f∗	f∗	NOUN
ejpam-5372	91	7	p	p	NOUN
ejpam-5372	91	8	such	such	ADJ
ejpam-5372	91	9	that	that	PRON
ejpam-5372	91	10	for	for	ADP
ejpam-5372	91	11	any	any	DET
ejpam-5372	91	12	prime	prime	NOUN
ejpam-5372	91	13	q	q	NOUN
ejpam-5372	91	14	dividing	dividing	NOUN
ejpam-5372	91	15	p−	p−	NOUN
ejpam-5372	91	16	1	1	NUM
ejpam-5372	91	17	,	,	PUNCT
ejpam-5372	91	18	g	g	PROPN
ejpam-5372	91	19	p−1	p−1	PROPN
ejpam-5372	91	20	q	q	PROPN
ejpam-5372	91	21	̸≡	̸≡	PROPN
ejpam-5372	91	22	1	1	NUM
ejpam-5372	91	23	.	.	PUNCT
ejpam-5372	92	1	the	the	DET
ejpam-5372	92	2	order	order	NOUN
ejpam-5372	92	3	of	of	ADP
ejpam-5372	92	4	g	g	PROPN
ejpam-5372	92	5	divides	divide	VERB
ejpam-5372	92	6	p	p	NOUN
ejpam-5372	92	7	−	−	PROPN
ejpam-5372	92	8	1	1	NUM
ejpam-5372	92	9	.	.	PUNCT
ejpam-5372	93	1	however	however	ADV
ejpam-5372	93	2	,	,	PUNCT
ejpam-5372	93	3	the	the	DET
ejpam-5372	93	4	hypothesis	hypothesis	NOUN
ejpam-5372	93	5	implies	imply	VERB
ejpam-5372	93	6	that	that	SCONJ
ejpam-5372	93	7	it	it	PRON
ejpam-5372	93	8	can	can	AUX
ejpam-5372	93	9	not	not	PART
ejpam-5372	93	10	be	be	AUX
ejpam-5372	93	11	a	a	DET
ejpam-5372	93	12	strict	strict	ADJ
ejpam-5372	93	13	divisor	divisor	NOUN
ejpam-5372	93	14	of	of	ADP
ejpam-5372	93	15	p	p	NOUN
ejpam-5372	93	16	−	−	PROPN
ejpam-5372	93	17	1	1	NUM
ejpam-5372	93	18	,	,	PUNCT
ejpam-5372	93	19	thus	thus	ADV
ejpam-5372	93	20	g	g	PROPN
ejpam-5372	93	21	is	be	AUX
ejpam-5372	93	22	a	a	DET
ejpam-5372	93	23	primitive	primitive	ADJ
ejpam-5372	93	24	root	root	NOUN
ejpam-5372	93	25	.	.	PUNCT
ejpam-5372	94	1	we	we	PRON
ejpam-5372	94	2	now	now	ADV
ejpam-5372	94	3	assume	assume	VERB
ejpam-5372	94	4	that	that	SCONJ
ejpam-5372	94	5	p	p	PROPN
ejpam-5372	94	6	∈	∈	PROPN
ejpam-5372	94	7	p1,4	p1,4	NOUN
ejpam-5372	94	8	.	.	PUNCT
ejpam-5372	95	1	by	by	ADP
ejpam-5372	95	2	definition	definition	NOUN
ejpam-5372	95	3	,	,	PUNCT
ejpam-5372	95	4	g	g	PROPN
ejpam-5372	95	5	∈	∈	PROPN
ejpam-5372	95	6	gs	gs	PROPN
ejpam-5372	95	7	⇔	⇔	PROPN
ejpam-5372	95	8	∃h	∃h	PROPN
ejpam-5372	95	9	∈	∈	PROPN
ejpam-5372	95	10	gz	gz	NOUN
ejpam-5372	95	11	g	g	PROPN
ejpam-5372	95	12	=	=	PROPN
ejpam-5372	95	13	h2	h2	PROPN
ejpam-5372	95	14	,	,	PUNCT
ejpam-5372	95	15	i.e.	i.e.	X
ejpam-5372	95	16	g	g	PROPN
ejpam-5372	95	17	is	be	AUX
ejpam-5372	95	18	in	in	ADP
ejpam-5372	95	19	gs	gs	INTJ
ejpam-5372	95	20	if	if	SCONJ
ejpam-5372	95	21	and	and	CCONJ
ejpam-5372	95	22	only	only	ADV
ejpam-5372	95	23	if	if	SCONJ
ejpam-5372	95	24	there	there	PRON
ejpam-5372	95	25	exists	exist	VERB
ejpam-5372	95	26	h	h	NOUN
ejpam-5372	95	27	such	such	ADJ
ejpam-5372	95	28	that	that	SCONJ
ejpam-5372	95	29	g	g	PROPN
ejpam-5372	95	30	=	=	SYM
ejpam-5372	95	31	h2	h2	PROPN
ejpam-5372	95	32	and	and	CCONJ
ejpam-5372	95	33	∀q	∀q	PROPN
ejpam-5372	95	34	∈	∈	PROPN
ejpam-5372	95	35	p	p	X
ejpam-5372	95	36	(	(	PUNCT
ejpam-5372	95	37	αq	αq	ADP
ejpam-5372	95	38	≥	≥	NOUN
ejpam-5372	95	39	1	1	NUM
ejpam-5372	95	40	⇒	⇒	NOUN
ejpam-5372	95	41	h	h	NOUN
ejpam-5372	96	1	p−1	p−1	PROPN
ejpam-5372	96	2	q	q	PROPN
ejpam-5372	96	3	̸≡	̸≡	PROPN
ejpam-5372	96	4	1	1	NUM
ejpam-5372	96	5	)	)	PUNCT
ejpam-5372	96	6	.	.	PUNCT
ejpam-5372	97	1	since	since	SCONJ
ejpam-5372	97	2	p−1	p−1	PROPN
ejpam-5372	97	3	q	q	PROPN
ejpam-5372	97	4	is	be	AUX
ejpam-5372	97	5	always	always	ADV
ejpam-5372	97	6	even	even	ADV
ejpam-5372	97	7	(	(	PUNCT
ejpam-5372	97	8	including	include	VERB
ejpam-5372	97	9	when	when	SCONJ
ejpam-5372	97	10	q	q	NOUN
ejpam-5372	97	11	=	=	SYM
ejpam-5372	97	12	2	2	NUM
ejpam-5372	97	13	because	because	SCONJ
ejpam-5372	97	14	p−	p−	NOUN
ejpam-5372	97	15	1	1	NUM
ejpam-5372	97	16	is	be	AUX
ejpam-5372	97	17	a	a	DET
ejpam-5372	97	18	multiple	multiple	NOUN
ejpam-5372	97	19	of	of	ADP
ejpam-5372	97	20	4	4	NUM
ejpam-5372	97	21	)	)	PUNCT
ejpam-5372	97	22	,	,	PUNCT
ejpam-5372	97	23	we	we	PRON
ejpam-5372	97	24	have	have	VERB
ejpam-5372	97	25	h	h	NOUN
ejpam-5372	97	26	p−1	p−1	PROPN
ejpam-5372	97	27	q	q	PROPN
ejpam-5372	98	1	=	=	PUNCT
ejpam-5372	98	2	g	g	PROPN
ejpam-5372	98	3	p−1	p−1	PROPN
ejpam-5372	98	4	2q	2q	NUM
ejpam-5372	98	5	hence	hence	ADV
ejpam-5372	98	6	:	:	PUNCT
ejpam-5372	98	7	g	g	PROPN
ejpam-5372	98	8	∈	∈	PROPN
ejpam-5372	98	9	gs	gs	PROPN
ejpam-5372	98	10	⇔	⇔	PROPN
ejpam-5372	98	11	g	g	PROPN
ejpam-5372	98	12	∈	∈	PROPN
ejpam-5372	98	13	gqr	gqr	PROPN
ejpam-5372	98	14	and	and	CCONJ
ejpam-5372	98	15	∀q	∀q	PROPN
ejpam-5372	98	16	∈	∈	PROPN
ejpam-5372	99	1	p	p	X
ejpam-5372	99	2	(	(	PUNCT
ejpam-5372	99	3	αq	αq	ADP
ejpam-5372	99	4	≥	≥	NOUN
ejpam-5372	99	5	1	1	NUM
ejpam-5372	99	6	⇒	⇒	NOUN
ejpam-5372	99	7	g	g	PROPN
ejpam-5372	99	8	p−1	p−1	PROPN
ejpam-5372	99	9	2q	2q	NUM
ejpam-5372	99	10	̸≡	̸≡	X
ejpam-5372	99	11	1	1	NUM
ejpam-5372	99	12	)	)	PUNCT
ejpam-5372	99	13	.	.	PUNCT
ejpam-5372	100	1	finally	finally	ADV
ejpam-5372	100	2	,	,	PUNCT
ejpam-5372	100	3	g	g	PROPN
ejpam-5372	100	4	∈	∈	PROPN
ejpam-5372	100	5	gs	gs	INTJ
ejpam-5372	101	1	if	if	SCONJ
ejpam-5372	101	2	and	and	CCONJ
ejpam-5372	101	3	only	only	ADV
ejpam-5372	101	4	it	it	PRON
ejpam-5372	101	5	has	have	AUX
ejpam-5372	101	6	two	two	NUM
ejpam-5372	101	7	square	square	ADJ
ejpam-5372	101	8	roots	root	NOUN
ejpam-5372	101	9	h	h	NOUN
ejpam-5372	101	10	and	and	CCONJ
ejpam-5372	101	11	−h	−h	ADJ
ejpam-5372	101	12	,	,	PUNCT
ejpam-5372	101	13	one	one	NUM
ejpam-5372	101	14	of	of	ADP
ejpam-5372	101	15	which	which	PRON
ejpam-5372	101	16	at	at	ADP
ejpam-5372	101	17	least	least	ADJ
ejpam-5372	101	18	(	(	PUNCT
ejpam-5372	101	19	say	say	INTJ
ejpam-5372	101	20	h	h	NOUN
ejpam-5372	101	21	)	)	PUNCT
ejpam-5372	101	22	is	be	AUX
ejpam-5372	101	23	an	an	DET
ejpam-5372	101	24	element	element	NOUN
ejpam-5372	101	25	of	of	ADP
ejpam-5372	101	26	gz	gz	PROPN
ejpam-5372	101	27	.	.	PUNCT
ejpam-5372	102	1	but	but	CCONJ
ejpam-5372	102	2	then	then	ADV
ejpam-5372	102	3	as	as	ADP
ejpam-5372	102	4	for	for	ADP
ejpam-5372	102	5	any	any	DET
ejpam-5372	102	6	prime	prime	NOUN
ejpam-5372	102	7	q	q	NOUN
ejpam-5372	102	8	dividing	dividing	NOUN
ejpam-5372	102	9	p−	p−	NOUN
ejpam-5372	102	10	1	1	NUM
ejpam-5372	102	11	,	,	PUNCT
ejpam-5372	102	12	p−1	p−1	PROPN
ejpam-5372	102	13	q	q	PROPN
ejpam-5372	102	14	is	be	AUX
ejpam-5372	102	15	even	even	ADV
ejpam-5372	102	16	,	,	PUNCT
ejpam-5372	102	17	we	we	PRON
ejpam-5372	102	18	have	have	AUX
ejpam-5372	102	19	(	(	PUNCT
ejpam-5372	102	20	−h	−h	ADV
ejpam-5372	102	21	)	)	PUNCT
ejpam-5372	102	22	p−1	p−1	PROPN
ejpam-5372	102	23	q	q	PROPN
ejpam-5372	103	1	=	=	PUNCT
ejpam-5372	103	2	h	h	NOUN
ejpam-5372	103	3	p−1	p−1	PROPN
ejpam-5372	103	4	q	q	PROPN
ejpam-5372	103	5	̸≡	̸≡	PROPN
ejpam-5372	103	6	1	1	NUM
ejpam-5372	103	7	,	,	PUNCT
ejpam-5372	103	8	hence	hence	ADV
ejpam-5372	103	9	−h	−h	VERB
ejpam-5372	103	10	∈	∈	PROPN
ejpam-5372	103	11	gz	gz	VERB
ejpam-5372	103	12	too	too	ADV
ejpam-5372	103	13	.	.	PUNCT
ejpam-5372	104	1	when	when	SCONJ
ejpam-5372	104	2	p	p	PROPN
ejpam-5372	104	3	∈	∈	PROPN
ejpam-5372	104	4	p3,4	p3,4	VERB
ejpam-5372	104	5	,	,	PUNCT
ejpam-5372	104	6	p−1	p−1	PROPN
ejpam-5372	104	7	2	2	NUM
ejpam-5372	104	8	is	be	AUX
ejpam-5372	104	9	odd	odd	ADJ
ejpam-5372	104	10	and	and	CCONJ
ejpam-5372	104	11	−1	−1	NOUN
ejpam-5372	104	12	is	be	AUX
ejpam-5372	104	13	not	not	PART
ejpam-5372	104	14	a	a	DET
ejpam-5372	104	15	quadratic	quadratic	ADJ
ejpam-5372	104	16	residue	residue	NOUN
ejpam-5372	104	17	.	.	PUNCT
ejpam-5372	105	1	if	if	SCONJ
ejpam-5372	105	2	g	g	PROPN
ejpam-5372	105	3	∈	∈	PROPN
ejpam-5372	105	4	gs	gs	INTJ
ejpam-5372	105	5	then	then	ADV
ejpam-5372	105	6	as	as	ADP
ejpam-5372	105	7	in	in	ADP
ejpam-5372	105	8	the	the	DET
ejpam-5372	105	9	previous	previous	ADJ
ejpam-5372	105	10	paragraph	paragraph	NOUN
ejpam-5372	105	11	,	,	PUNCT
ejpam-5372	105	12	g	g	PROPN
ejpam-5372	105	13	∈	∈	PROPN
ejpam-5372	105	14	gqr	gqr	PROPN
ejpam-5372	105	15	and	and	CCONJ
ejpam-5372	105	16	∀q	∀q	PROPN
ejpam-5372	105	17	∈	∈	PROPN
ejpam-5372	105	18	p⧹	p⧹	VERB
ejpam-5372	105	19	{	{	PUNCT
ejpam-5372	105	20	2	2	NUM
ejpam-5372	105	21	}	}	PUNCT
ejpam-5372	105	22	(	(	PUNCT
ejpam-5372	105	23	αq	αq	ADP
ejpam-5372	105	24	≥	≥	NOUN
ejpam-5372	105	25	1	1	NUM
ejpam-5372	105	26	⇒	⇒	NOUN
ejpam-5372	105	27	g	g	PROPN
ejpam-5372	105	28	p−1	p−1	PROPN
ejpam-5372	105	29	2q	2q	NUM
ejpam-5372	105	30	̸≡	̸≡	X
ejpam-5372	105	31	1	1	NUM
ejpam-5372	105	32	)	)	PUNCT
ejpam-5372	105	33	.	.	PUNCT
ejpam-5372	106	1	conversely	conversely	ADV
ejpam-5372	106	2	,	,	PUNCT
ejpam-5372	106	3	assume	assume	VERB
ejpam-5372	106	4	g	g	PROPN
ejpam-5372	106	5	=	=	PUNCT
ejpam-5372	106	6	h2	h2	PROPN
ejpam-5372	106	7	and	and	CCONJ
ejpam-5372	106	8	∀q	∀q	PROPN
ejpam-5372	106	9	∈	∈	PROPN
ejpam-5372	106	10	p⧹	p⧹	VERB
ejpam-5372	106	11	{	{	PUNCT
ejpam-5372	106	12	2	2	NUM
ejpam-5372	106	13	}	}	PUNCT
ejpam-5372	106	14	(	(	PUNCT
ejpam-5372	106	15	αq	αq	ADP
ejpam-5372	106	16	≥	≥	NOUN
ejpam-5372	106	17	1	1	NUM
ejpam-5372	106	18	⇒	⇒	NOUN
ejpam-5372	106	19	g	g	PROPN
ejpam-5372	106	20	p−1	p−1	PROPN
ejpam-5372	106	21	2q	2q	NUM
ejpam-5372	106	22	̸≡	̸≡	X
ejpam-5372	106	23	1	1	NUM
ejpam-5372	106	24	)	)	PUNCT
ejpam-5372	106	25	.	.	PUNCT
ejpam-5372	107	1	then	then	ADV
ejpam-5372	107	2	∀q	∀q	PROPN
ejpam-5372	107	3	∈	∈	PROPN
ejpam-5372	107	4	p⧹	p⧹	VERB
ejpam-5372	107	5	{	{	PUNCT
ejpam-5372	107	6	2	2	NUM
ejpam-5372	107	7	}	}	PUNCT
ejpam-5372	107	8	(	(	PUNCT
ejpam-5372	107	9	αq	αq	ADP
ejpam-5372	107	10	≥	≥	NOUN
ejpam-5372	107	11	1	1	NUM
ejpam-5372	107	12	⇒	⇒	NOUN
ejpam-5372	107	13	(	(	PUNCT
ejpam-5372	107	14	±h	±h	PROPN
ejpam-5372	107	15	)	)	PUNCT
ejpam-5372	107	16	p−1	p−1	PROPN
ejpam-5372	107	17	q	q	PROPN
ejpam-5372	107	18	̸≡	̸≡	PROPN
ejpam-5372	107	19	1	1	NUM
ejpam-5372	107	20	)	)	PUNCT
ejpam-5372	107	21	and	and	CCONJ
ejpam-5372	107	22	h	h	NOUN
ejpam-5372	107	23	p−1	p−1	PROPN
ejpam-5372	107	24	2	2	NUM
ejpam-5372	107	25	∈	∈	PROPN
ejpam-5372	107	26	{	{	PUNCT
ejpam-5372	107	27	±1	±1	NOUN
ejpam-5372	107	28	}	}	PUNCT
ejpam-5372	107	29	thus	thus	ADV
ejpam-5372	107	30	h	h	NOUN
ejpam-5372	107	31	p−1	p−1	NOUN
ejpam-5372	107	32	2	2	NUM
ejpam-5372	107	33	=	=	SYM
ejpam-5372	107	34	−1	−1	NOUN
ejpam-5372	107	35	or	or	CCONJ
ejpam-5372	107	36	(	(	PUNCT
ejpam-5372	107	37	−h	−h	ADJ
ejpam-5372	107	38	)	)	PUNCT
ejpam-5372	107	39	p−1	p−1	PROPN
ejpam-5372	107	40	2	2	NUM
ejpam-5372	107	41	=	=	SYM
ejpam-5372	107	42	−1	−1	NOUN
ejpam-5372	107	43	.	.	PUNCT
ejpam-5372	108	1	in	in	ADP
ejpam-5372	108	2	all	all	DET
ejpam-5372	108	3	cases	case	NOUN
ejpam-5372	108	4	,	,	PUNCT
ejpam-5372	108	5	either	either	CCONJ
ejpam-5372	108	6	h	h	NOUN
ejpam-5372	108	7	or	or	CCONJ
ejpam-5372	108	8	−h	−h	ADJ
ejpam-5372	108	9	is	be	AUX
ejpam-5372	108	10	a	a	DET
ejpam-5372	108	11	primitive	primitive	ADJ
ejpam-5372	108	12	root	root	NOUN
ejpam-5372	108	13	,	,	PUNCT
ejpam-5372	108	14	but	but	CCONJ
ejpam-5372	108	15	never	never	ADV
ejpam-5372	108	16	both	both	PRON
ejpam-5372	108	17	.	.	PUNCT
ejpam-5372	109	1	2.2	2.2	NUM
ejpam-5372	109	2	.	.	PUNCT
ejpam-5372	109	3	relationships	relationship	NOUN
ejpam-5372	109	4	between	between	ADP
ejpam-5372	109	5	primitive	primitive	ADJ
ejpam-5372	109	6	and	and	CCONJ
ejpam-5372	109	7	semi	semi	ADJ
ejpam-5372	109	8	-	-	ADJ
ejpam-5372	109	9	primitive	primitive	ADJ
ejpam-5372	109	10	roots	root	NOUN
ejpam-5372	109	11	2.2.1	2.2.1	NUM
ejpam-5372	109	12	.	.	PUNCT
ejpam-5372	110	1	primitive	primitive	ADJ
ejpam-5372	110	2	roots	root	NOUN
ejpam-5372	110	3	obtained	obtain	VERB
ejpam-5372	110	4	from	from	ADP
ejpam-5372	110	5	residues	residue	NOUN
ejpam-5372	110	6	definition	definition	NOUN
ejpam-5372	110	7	4	4	NUM
ejpam-5372	110	8	.	.	PUNCT
ejpam-5372	111	1	if	if	SCONJ
ejpam-5372	111	2	h	h	NOUN
ejpam-5372	111	3	is	be	AUX
ejpam-5372	111	4	an	an	DET
ejpam-5372	111	5	element	element	NOUN
ejpam-5372	111	6	of	of	ADP
ejpam-5372	111	7	order	order	NOUN
ejpam-5372	111	8	d	d	NOUN
ejpam-5372	111	9	in	in	ADP
ejpam-5372	111	10	a	a	DET
ejpam-5372	111	11	group	group	NOUN
ejpam-5372	111	12	and	and	CCONJ
ejpam-5372	111	13	0	0	NUM
ejpam-5372	111	14	≤	≤	NUM
ejpam-5372	111	15	x	x	X
ejpam-5372	111	16	≤	≤	NUM
ejpam-5372	111	17	d	d	NOUN
ejpam-5372	111	18	we	we	PRON
ejpam-5372	111	19	note	note	VERB
ejpam-5372	111	20	⟨h⟩(x	⟨h⟩(x	PROPN
ejpam-5372	111	21	)	)	PUNCT
ejpam-5372	112	1	=	=	PRON
ejpam-5372	112	2	{	{	PUNCT
ejpam-5372	112	3	h	h	NOUN
ejpam-5372	112	4	,	,	PUNCT
ejpam-5372	112	5	.	.	PUNCT
ejpam-5372	112	6	.	.	PUNCT
ejpam-5372	112	7	.	.	PUNCT
ejpam-5372	113	1	,	,	PUNCT
ejpam-5372	113	2	hx	hx	PROPN
ejpam-5372	113	3	}	}	PUNCT
ejpam-5372	113	4	.	.	PUNCT
ejpam-5372	114	1	it	it	PRON
ejpam-5372	114	2	is	be	AUX
ejpam-5372	114	3	a	a	DET
ejpam-5372	114	4	subset	subset	NOUN
ejpam-5372	114	5	of	of	ADP
ejpam-5372	114	6	the	the	DET
ejpam-5372	114	7	subgroup	subgroup	NOUN
ejpam-5372	114	8	generated	generate	VERB
ejpam-5372	114	9	by	by	ADP
ejpam-5372	114	10	h	h	NOUN
ejpam-5372	114	11	of	of	ADP
ejpam-5372	114	12	cardinal	cardinal	ADJ
ejpam-5372	114	13	x.	x.	NOUN
ejpam-5372	114	14	in	in	ADP
ejpam-5372	114	15	this	this	DET
ejpam-5372	114	16	whole	whole	ADJ
ejpam-5372	114	17	section	section	NOUN
ejpam-5372	114	18	,	,	PUNCT
ejpam-5372	114	19	we	we	PRON
ejpam-5372	114	20	let	let	VERB
ejpam-5372	114	21	p	p	X
ejpam-5372	114	22	∈	∈	PROPN
ejpam-5372	114	23	p⧹	p⧹	VERB
ejpam-5372	114	24	{	{	PUNCT
ejpam-5372	114	25	2	2	NUM
ejpam-5372	114	26	}	}	PUNCT
ejpam-5372	114	27	and	and	CCONJ
ejpam-5372	114	28	we	we	PRON
ejpam-5372	114	29	write	write	VERB
ejpam-5372	114	30	p−	p−	NOUN
ejpam-5372	114	31	1	1	NUM
ejpam-5372	114	32	=	=	SYM
ejpam-5372	114	33	2nz	2nz	NOUN
ejpam-5372	114	34	with	with	ADP
ejpam-5372	114	35	z	z	NOUN
ejpam-5372	114	36	odd	odd	ADJ
ejpam-5372	114	37	.	.	PUNCT
ejpam-5372	115	1	proposition	proposition	NOUN
ejpam-5372	115	2	6	6	NUM
ejpam-5372	115	3	.	.	PUNCT
ejpam-5372	116	1	let	let	VERB
ejpam-5372	116	2	g	g	PROPN
ejpam-5372	116	3	∈	∈	PROPN
ejpam-5372	116	4	gqnr	gqnr	NOUN
ejpam-5372	116	5	.	.	PUNCT
ejpam-5372	117	1	for	for	ADP
ejpam-5372	117	2	any	any	DET
ejpam-5372	117	3	0	0	NUM
ejpam-5372	117	4	≤	≤	NUM
ejpam-5372	117	5	t	t	NOUN
ejpam-5372	117	6	≤	≤	NOUN
ejpam-5372	117	7	n	n	CCONJ
ejpam-5372	117	8	,	,	PUNCT
ejpam-5372	117	9	the	the	DET
ejpam-5372	117	10	multiplicative	multiplicative	ADJ
ejpam-5372	117	11	order	order	NOUN
ejpam-5372	117	12	of	of	ADP
ejpam-5372	117	13	g2	g2	PROPN
ejpam-5372	117	14	tz	tz	PROPN
ejpam-5372	117	15	is	be	AUX
ejpam-5372	117	16	2n−t	2n−t	ADJ
ejpam-5372	117	17	.	.	PUNCT
ejpam-5372	118	1	in	in	ADP
ejpam-5372	118	2	other	other	ADJ
ejpam-5372	118	3	words	word	NOUN
ejpam-5372	118	4	,	,	PUNCT
ejpam-5372	118	5	〈	〈	PROPN
ejpam-5372	118	6	g2	g2	PROPN
ejpam-5372	118	7	tz	tz	PROPN
ejpam-5372	118	8	〉	〉	NOUN
ejpam-5372	118	9	=	=	PUNCT
ejpam-5372	118	10	r(n−t	r(n−t	NOUN
ejpam-5372	118	11	)	)	PUNCT
ejpam-5372	118	12	.	.	PUNCT
ejpam-5372	119	1	moreover	moreover	ADV
ejpam-5372	119	2	,	,	PUNCT
ejpam-5372	119	3	if	if	SCONJ
ejpam-5372	119	4	we	we	PRON
ejpam-5372	119	5	note	note	VERB
ejpam-5372	119	6	b	b	PROPN
ejpam-5372	119	7	=	=	PROPN
ejpam-5372	119	8	g2	g2	PROPN
ejpam-5372	119	9	tz	tz	NOUN
ejpam-5372	119	10	,	,	PUNCT
ejpam-5372	119	11	with	with	ADP
ejpam-5372	119	12	0	0	NUM
ejpam-5372	119	13	<	<	X
ejpam-5372	119	14	t	t	PROPN
ejpam-5372	119	15	≤	≤	NOUN
ejpam-5372	119	16	n	n	CCONJ
ejpam-5372	119	17	,	,	PUNCT
ejpam-5372	119	18	for	for	ADP
ejpam-5372	119	19	any	any	DET
ejpam-5372	119	20	integer	integer	NOUN
ejpam-5372	119	21	x	x	NOUN
ejpam-5372	119	22	,	,	PUNCT
ejpam-5372	119	23	b2x	b2x	ADP
ejpam-5372	119	24	∈	∈	PROPN
ejpam-5372	119	25	r(n−t−1	r(n−t−1	NOUN
ejpam-5372	119	26	)	)	PUNCT
ejpam-5372	119	27	and	and	CCONJ
ejpam-5372	119	28	b2x+1	b2x+1	PROPN
ejpam-5372	119	29	∈	∈	PROPN
ejpam-5372	119	30	r(n−t)⧹r(n−t−1	r(n−t)⧹r(n−t−1	NOUN
ejpam-5372	119	31	)	)	PUNCT
ejpam-5372	119	32	.	.	PUNCT
ejpam-5372	120	1	in	in	ADP
ejpam-5372	120	2	particular	particular	ADJ
ejpam-5372	120	3	,	,	PUNCT
ejpam-5372	120	4	when	when	SCONJ
ejpam-5372	120	5	t	t	PROPN
ejpam-5372	120	6	=	=	SYM
ejpam-5372	120	7	0	0	PROPN
ejpam-5372	120	8	,	,	PUNCT
ejpam-5372	120	9	the	the	DET
ejpam-5372	120	10	odd	odd	ADJ
ejpam-5372	120	11	powers	power	NOUN
ejpam-5372	120	12	of	of	ADP
ejpam-5372	120	13	b	b	NOUN
ejpam-5372	120	14	are	be	AUX
ejpam-5372	120	15	in	in	ADP
ejpam-5372	120	16	gqnr	gqnr	NOUN
ejpam-5372	120	17	and	and	CCONJ
ejpam-5372	120	18	the	the	DET
ejpam-5372	120	19	even	even	ADJ
ejpam-5372	120	20	powers	power	NOUN
ejpam-5372	120	21	in	in	ADP
ejpam-5372	120	22	gqr	gqr	PROPN
ejpam-5372	120	23	.	.	PUNCT
ejpam-5372	121	1	m.	m.	PROPN
ejpam-5372	121	2	wolf	wolf	PROPN
ejpam-5372	121	3	,	,	PUNCT
ejpam-5372	121	4	f.	f.	PROPN
ejpam-5372	121	5	wolf	wolf	PROPN
ejpam-5372	121	6	/	/	SYM
ejpam-5372	121	7	eur	eur	PROPN
ejpam-5372	121	8	.	.	PUNCT
ejpam-5372	122	1	j.	j.	PROPN
ejpam-5372	122	2	pure	pure	PROPN
ejpam-5372	122	3	appl	appl	PROPN
ejpam-5372	122	4	.	.	PROPN
ejpam-5372	122	5	math	math	PROPN
ejpam-5372	122	6	,	,	PUNCT
ejpam-5372	122	7	17	17	NUM
ejpam-5372	122	8	(	(	PUNCT
ejpam-5372	122	9	4	4	NUM
ejpam-5372	122	10	)	)	PUNCT
ejpam-5372	122	11	(	(	PUNCT
ejpam-5372	122	12	2024	2024	NUM
ejpam-5372	122	13	)	)	PUNCT
ejpam-5372	122	14	,	,	PUNCT
ejpam-5372	122	15	2431	2431	NUM
ejpam-5372	122	16	-	-	SYM
ejpam-5372	122	17	2447	2447	NUM
ejpam-5372	122	18	2435	2435	NUM
ejpam-5372	122	19	proof	proof	NOUN
ejpam-5372	122	20	.	.	PUNCT
ejpam-5372	123	1	the	the	DET
ejpam-5372	123	2	order	order	NOUN
ejpam-5372	123	3	of	of	ADP
ejpam-5372	123	4	gz	gz	NOUN
ejpam-5372	123	5	divides	divide	NOUN
ejpam-5372	123	6	2n	2n	NUM
ejpam-5372	123	7	.	.	PUNCT
ejpam-5372	124	1	by	by	ADP
ejpam-5372	124	2	euler	euler	PROPN
ejpam-5372	124	3	’s	’s	PART
ejpam-5372	124	4	criterion	criterion	NOUN
ejpam-5372	124	5	,	,	PUNCT
ejpam-5372	124	6	(	(	PUNCT
ejpam-5372	124	7	gz	gz	NOUN
ejpam-5372	124	8	)	)	PUNCT
ejpam-5372	124	9	p−1	p−1	PROPN
ejpam-5372	124	10	2	2	NUM
ejpam-5372	124	11	≡	≡	PROPN
ejpam-5372	124	12	(	(	PUNCT
ejpam-5372	124	13	−1)z	−1)z	NOUN
ejpam-5372	124	14	=	=	SYM
ejpam-5372	124	15	−1	−1	NOUN
ejpam-5372	124	16	,	,	PUNCT
ejpam-5372	124	17	thus	thus	ADV
ejpam-5372	124	18	gz	gz	NOUN
ejpam-5372	124	19	is	be	AUX
ejpam-5372	124	20	in	in	ADP
ejpam-5372	124	21	gqnr	gqnr	NOUN
ejpam-5372	124	22	and	and	CCONJ
ejpam-5372	124	23	a	a	DET
ejpam-5372	124	24	generator	generator	NOUN
ejpam-5372	124	25	of	of	ADP
ejpam-5372	124	26	r(n	r(n	PROPN
ejpam-5372	124	27	)	)	PUNCT
ejpam-5372	124	28	.	.	PUNCT
ejpam-5372	125	1	hence	hence	ADV
ejpam-5372	125	2	,	,	PUNCT
ejpam-5372	125	3	g2	g2	PROPN
ejpam-5372	125	4	tz	tz	PROPN
ejpam-5372	125	5	is	be	AUX
ejpam-5372	125	6	a	a	DET
ejpam-5372	125	7	generator	generator	NOUN
ejpam-5372	125	8	of	of	ADP
ejpam-5372	125	9	r(n−t	r(n−t	NOUN
ejpam-5372	125	10	)	)	PUNCT
ejpam-5372	125	11	.	.	PUNCT
ejpam-5372	126	1	the	the	DET
ejpam-5372	126	2	other	other	ADJ
ejpam-5372	126	3	assertions	assertion	NOUN
ejpam-5372	126	4	are	be	AUX
ejpam-5372	126	5	immediate	immediate	ADJ
ejpam-5372	126	6	.	.	PUNCT
ejpam-5372	127	1	corollary	corollary	ADJ
ejpam-5372	127	2	1	1	NUM
ejpam-5372	127	3	.	.	PUNCT
ejpam-5372	128	1	let	let	VERB
ejpam-5372	128	2	m	m	PRON
ejpam-5372	128	3	be	be	AUX
ejpam-5372	128	4	a	a	DET
ejpam-5372	128	5	semi	semi	ADJ
ejpam-5372	128	6	-	-	ADJ
ejpam-5372	128	7	primitive	primitive	ADJ
ejpam-5372	128	8	root	root	NOUN
ejpam-5372	128	9	.	.	PUNCT
ejpam-5372	129	1	then	then	ADV
ejpam-5372	129	2	:	:	PUNCT
ejpam-5372	129	3	gqr	gqr	X
ejpam-5372	129	4	=	=	PUNCT
ejpam-5372	129	5	⟨m⟩	⟨m⟩	X
ejpam-5372	129	6	=	=	SYM
ejpam-5372	129	7	⟨m⟩(z)r(n−1	⟨m⟩(z)r(n−1	PROPN
ejpam-5372	129	8	)	)	PUNCT
ejpam-5372	129	9	.	.	PUNCT
ejpam-5372	130	1	more	more	ADV
ejpam-5372	130	2	generally	generally	ADV
ejpam-5372	130	3	,	,	PUNCT
ejpam-5372	130	4	generators	generator	NOUN
ejpam-5372	130	5	of	of	ADP
ejpam-5372	130	6	gqr	gqr	PROPN
ejpam-5372	130	7	coincide	coincide	NOUN
ejpam-5372	130	8	with	with	ADP
ejpam-5372	130	9	semi	semi	ADJ
ejpam-5372	130	10	-	-	ADJ
ejpam-5372	130	11	primitive	primitive	ADJ
ejpam-5372	130	12	roots	root	NOUN
ejpam-5372	130	13	.	.	PUNCT
ejpam-5372	131	1	proof	proof	NOUN
ejpam-5372	131	2	.	.	PUNCT
ejpam-5372	132	1	we	we	PRON
ejpam-5372	132	2	know	know	VERB
ejpam-5372	132	3	that	that	SCONJ
ejpam-5372	132	4	gqr	gqr	PROPN
ejpam-5372	132	5	is	be	AUX
ejpam-5372	132	6	the	the	DET
ejpam-5372	132	7	subgroup	subgroup	NOUN
ejpam-5372	132	8	made	make	VERB
ejpam-5372	132	9	of	of	ADP
ejpam-5372	132	10	the	the	DET
ejpam-5372	132	11	roots	root	NOUN
ejpam-5372	132	12	of	of	ADP
ejpam-5372	132	13	the	the	DET
ejpam-5372	132	14	polynomial	polynomial	ADJ
ejpam-5372	132	15	(	(	PUNCT
ejpam-5372	132	16	x	x	SYM
ejpam-5372	132	17	p−1	p−1	PROPN
ejpam-5372	132	18	2	2	NUM
ejpam-5372	132	19	−	−	NOUN
ejpam-5372	132	20	1	1	NUM
ejpam-5372	132	21	)	)	PUNCT
ejpam-5372	132	22	.	.	PUNCT
ejpam-5372	133	1	by	by	ADP
ejpam-5372	133	2	definition	definition	NOUN
ejpam-5372	133	3	,	,	PUNCT
ejpam-5372	133	4	a	a	DET
ejpam-5372	133	5	semi	semi	ADJ
ejpam-5372	133	6	-	-	ADJ
ejpam-5372	133	7	primitive	primitive	ADJ
ejpam-5372	133	8	root	root	NOUN
ejpam-5372	133	9	is	be	AUX
ejpam-5372	133	10	of	of	ADP
ejpam-5372	133	11	order	order	NOUN
ejpam-5372	133	12	equal	equal	ADJ
ejpam-5372	133	13	to	to	ADP
ejpam-5372	133	14	p−1	p−1	PROPN
ejpam-5372	133	15	2	2	NUM
ejpam-5372	133	16	and	and	CCONJ
ejpam-5372	133	17	thus	thus	ADV
ejpam-5372	133	18	a	a	DET
ejpam-5372	133	19	generator	generator	NOUN
ejpam-5372	133	20	of	of	ADP
ejpam-5372	133	21	this	this	DET
ejpam-5372	133	22	subgroup	subgroup	NOUN
ejpam-5372	133	23	,	,	PUNCT
ejpam-5372	133	24	and	and	CCONJ
ejpam-5372	133	25	conversely	conversely	ADV
ejpam-5372	133	26	.	.	PUNCT
ejpam-5372	134	1	furthermore	furthermore	ADV
ejpam-5372	134	2	,	,	PUNCT
ejpam-5372	134	3	we	we	PRON
ejpam-5372	134	4	have	have	VERB
ejpam-5372	134	5	⟨m⟩(z	⟨m⟩(z	PROPN
ejpam-5372	134	6	)	)	PUNCT
ejpam-5372	135	1	⊂	⊂	PROPN
ejpam-5372	135	2	gqr	gqr	PROPN
ejpam-5372	135	3	and	and	CCONJ
ejpam-5372	135	4	r(n−1	r(n−1	PRON
ejpam-5372	135	5	)	)	PUNCT
ejpam-5372	135	6	⊂	⊂	PROPN
ejpam-5372	135	7	gqr	gqr	AUX
ejpam-5372	135	8	hence	hence	ADV
ejpam-5372	135	9	⟨m⟩(z)r(n−1	⟨m⟩(z)r(n−1	ADJ
ejpam-5372	135	10	)	)	PUNCT
ejpam-5372	136	1	⊂	⊂	PROPN
ejpam-5372	136	2	gqr	gqr	PROPN
ejpam-5372	136	3	.	.	PUNCT
ejpam-5372	137	1	moreover	moreover	ADV
ejpam-5372	137	2	,	,	PUNCT
ejpam-5372	137	3	∣∣∣⟨m⟩(z	∣∣∣⟨m⟩(z	ADJ
ejpam-5372	137	4	)	)	PUNCT
ejpam-5372	137	5	∣∣∣	∣∣∣	NOUN
ejpam-5372	137	6	=	=	SYM
ejpam-5372	137	7	z	z	NOUN
ejpam-5372	137	8	,	,	PUNCT
ejpam-5372	137	9	∣∣r(n−1	∣∣r(n−1	ADJ
ejpam-5372	137	10	)	)	PUNCT
ejpam-5372	137	11	∣∣	∣∣	X
ejpam-5372	138	1	=	=	SYM
ejpam-5372	138	2	2n−1	2n−1	NUM
ejpam-5372	138	3	.	.	PUNCT
ejpam-5372	139	1	if	if	SCONJ
ejpam-5372	139	2	mkr	mkr	PROPN
ejpam-5372	139	3	≡	≡	PROPN
ejpam-5372	139	4	mk	mk	PROPN
ejpam-5372	139	5	′	′	NUM
ejpam-5372	140	1	r	r	NOUN
ejpam-5372	140	2	′	′	NOUN
ejpam-5372	140	3	with	with	ADP
ejpam-5372	140	4	1	1	NUM
ejpam-5372	140	5	≤	≤	NUM
ejpam-5372	140	6	k	k	NOUN
ejpam-5372	140	7	≤	≤	NUM
ejpam-5372	141	1	k	k	PROPN
ejpam-5372	141	2	′	′	NUM
ejpam-5372	141	3	≤	≤	NUM
ejpam-5372	142	1	z	z	NOUN
ejpam-5372	142	2	and	and	CCONJ
ejpam-5372	142	3	r	r	NOUN
ejpam-5372	142	4	,	,	PUNCT
ejpam-5372	142	5	r	r	NOUN
ejpam-5372	142	6	′	′	NOUN
ejpam-5372	142	7	∈	∈	NOUN
ejpam-5372	142	8	r(n−1	r(n−1	NOUN
ejpam-5372	142	9	)	)	PUNCT
ejpam-5372	142	10	thus	thus	ADV
ejpam-5372	142	11	mk	mk	NOUN
ejpam-5372	143	1	′−kr	′−kr	NOUN
ejpam-5372	143	2	′	′	NUM
ejpam-5372	144	1	≡	≡	PROPN
ejpam-5372	144	2	r	r	NOUN
ejpam-5372	144	3	hence	hence	ADV
ejpam-5372	144	4	m	m	VERB
ejpam-5372	144	5	2n−1	2n−1	NUM
ejpam-5372	144	6	(	(	PUNCT
ejpam-5372	144	7	k	k	PROPN
ejpam-5372	144	8	′−k	′−k	PROPN
ejpam-5372	144	9	)	)	PUNCT
ejpam-5372	145	1	≡	≡	PROPN
ejpam-5372	145	2	1	1	NUM
ejpam-5372	145	3	.	.	PUNCT
ejpam-5372	146	1	this	this	PRON
ejpam-5372	146	2	implies	imply	VERB
ejpam-5372	146	3	k	k	PROPN
ejpam-5372	146	4	=	=	PUNCT
ejpam-5372	146	5	k	k	PROPN
ejpam-5372	147	1	′	′	NOUN
ejpam-5372	148	1	and	and	CCONJ
ejpam-5372	148	2	r	r	NOUN
ejpam-5372	148	3	≡	≡	PROPN
ejpam-5372	148	4	r	r	NOUN
ejpam-5372	148	5	′	′	NOUN
ejpam-5372	148	6	follows	follow	VERB
ejpam-5372	148	7	.	.	PUNCT
ejpam-5372	149	1	thus	thus	ADV
ejpam-5372	149	2	,	,	PUNCT
ejpam-5372	149	3	we	we	PRON
ejpam-5372	149	4	have	have	VERB
ejpam-5372	149	5	∣∣∣⟨m⟩(z	∣∣∣⟨m⟩(z	NOUN
ejpam-5372	149	6	)	)	PUNCT
ejpam-5372	149	7	∣∣∣	∣∣∣	NOUN
ejpam-5372	149	8	.	.	PUNCT
ejpam-5372	150	1	∣∣r(n−1	∣∣r(n−1	ADJ
ejpam-5372	150	2	)	)	PUNCT
ejpam-5372	150	3	∣∣	∣∣	X
ejpam-5372	151	1	=	=	PUNCT
ejpam-5372	151	2	2n−1z	2n−1z	NUM
ejpam-5372	151	3	=	=	SYM
ejpam-5372	151	4	p−1	p−1	PROPN
ejpam-5372	151	5	2	2	NUM
ejpam-5372	151	6	distinct	distinct	ADJ
ejpam-5372	151	7	residues	residue	NOUN
ejpam-5372	151	8	in	in	ADP
ejpam-5372	151	9	⟨m⟩(z)r(n−1	⟨m⟩(z)r(n−1	ADJ
ejpam-5372	151	10	)	)	PUNCT
ejpam-5372	151	11	hence	hence	ADV
ejpam-5372	151	12	⟨m⟩(z)r(n−1	⟨m⟩(z)r(n−1	ADJ
ejpam-5372	151	13	)	)	PUNCT
ejpam-5372	151	14	=	=	SYM
ejpam-5372	152	1	gqr	gqr	PROPN
ejpam-5372	152	2	.	.	PUNCT
ejpam-5372	153	1	proposition	proposition	NOUN
ejpam-5372	153	2	7	7	NUM
ejpam-5372	153	3	.	.	PUNCT
ejpam-5372	153	4	for	for	ADP
ejpam-5372	153	5	m	m	PROPN
ejpam-5372	153	6	a	a	DET
ejpam-5372	153	7	semi	semi	ADJ
ejpam-5372	153	8	-	-	ADJ
ejpam-5372	153	9	primitive	primitive	ADJ
ejpam-5372	153	10	root	root	NOUN
ejpam-5372	153	11	,	,	PUNCT
ejpam-5372	153	12	we	we	PRON
ejpam-5372	153	13	have	have	AUX
ejpam-5372	153	14	:	:	PUNCT
ejpam-5372	153	15	m	m	AUX
ejpam-5372	153	16	(	(	PUNCT
ejpam-5372	153	17	r(n)⧹r(n−1	r(n)⧹r(n−1	PROPN
ejpam-5372	153	18	)	)	PUNCT
ejpam-5372	153	19	)	)	PUNCT
ejpam-5372	154	1	⊂	⊂	PROPN
ejpam-5372	154	2	gz	gz	PROPN
ejpam-5372	154	3	.	.	PUNCT
ejpam-5372	155	1	proof	proof	NOUN
ejpam-5372	155	2	.	.	PUNCT
ejpam-5372	156	1	let	let	VERB
ejpam-5372	156	2	r	r	PRON
ejpam-5372	156	3	∈	∈	PROPN
ejpam-5372	156	4	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	156	5	)	)	PUNCT
ejpam-5372	156	6	,	,	PUNCT
ejpam-5372	156	7	and	and	CCONJ
ejpam-5372	156	8	let	let	VERB
ejpam-5372	156	9	us	we	PRON
ejpam-5372	156	10	show	show	VERB
ejpam-5372	156	11	that	that	SCONJ
ejpam-5372	156	12	mr	mr	PROPN
ejpam-5372	156	13	∈	∈	PROPN
ejpam-5372	156	14	gz	gz	PROPN
ejpam-5372	156	15	.	.	PUNCT
ejpam-5372	157	1	let	let	VERB
ejpam-5372	157	2	q	q	PRON
ejpam-5372	157	3	be	be	AUX
ejpam-5372	157	4	a	a	DET
ejpam-5372	157	5	prime	prime	ADJ
ejpam-5372	157	6	factor	factor	NOUN
ejpam-5372	157	7	of	of	ADP
ejpam-5372	157	8	p−	p−	NOUN
ejpam-5372	157	9	1	1	NUM
ejpam-5372	157	10	=	=	NOUN
ejpam-5372	157	11	2nz	2nz	NOUN
ejpam-5372	157	12	.	.	PUNCT
ejpam-5372	158	1	if	if	SCONJ
ejpam-5372	158	2	q	q	NOUN
ejpam-5372	158	3	=	=	NOUN
ejpam-5372	158	4	2	2	NUM
ejpam-5372	158	5	,	,	PUNCT
ejpam-5372	158	6	we	we	PRON
ejpam-5372	158	7	have	have	VERB
ejpam-5372	158	8	(	(	PUNCT
ejpam-5372	158	9	mr	mr	PROPN
ejpam-5372	158	10	)	)	PUNCT
ejpam-5372	158	11	p−1	p−1	PROPN
ejpam-5372	158	12	q	q	PROPN
ejpam-5372	158	13	≡	≡	PROPN
ejpam-5372	158	14	m2n−1zr2	m2n−1zr2	PROPN
ejpam-5372	158	15	n−1z	n−1z	PROPN
ejpam-5372	158	16	≡	≡	PROPN
ejpam-5372	158	17	r2	r2	PROPN
ejpam-5372	158	18	n−1z	n−1z	PROPN
ejpam-5372	158	19	̸≡	̸≡	PROPN
ejpam-5372	158	20	1	1	NUM
ejpam-5372	158	21	because	because	SCONJ
ejpam-5372	158	22	rz	rz	NOUN
ejpam-5372	158	23	∈	∈	PROPN
ejpam-5372	158	24	r(n)⧹r(n−1	r(n)⧹r(n−1	PROPN
ejpam-5372	158	25	)	)	PUNCT
ejpam-5372	158	26	since	since	SCONJ
ejpam-5372	158	27	z	z	PROPN
ejpam-5372	158	28	is	be	AUX
ejpam-5372	158	29	odd	odd	ADJ
ejpam-5372	158	30	.	.	PUNCT
ejpam-5372	159	1	if	if	SCONJ
ejpam-5372	159	2	q	q	PROPN
ejpam-5372	159	3	̸=	̸=	PROPN
ejpam-5372	159	4	2	2	NUM
ejpam-5372	159	5	,	,	PUNCT
ejpam-5372	159	6	we	we	PRON
ejpam-5372	159	7	have	have	VERB
ejpam-5372	159	8	(	(	PUNCT
ejpam-5372	159	9	mr	mr	PROPN
ejpam-5372	159	10	)	)	PUNCT
ejpam-5372	159	11	p−1	p−1	PROPN
ejpam-5372	159	12	q	q	PROPN
ejpam-5372	159	13	≡	≡	PROPN
ejpam-5372	159	14	m	m	PROPN
ejpam-5372	159	15	2nz	2nz	NOUN
ejpam-5372	159	16	q	q	NOUN
ejpam-5372	159	17	r	r	NOUN
ejpam-5372	159	18	2nz	2nz	NOUN
ejpam-5372	159	19	q	q	X
ejpam-5372	159	20	≡	≡	PROPN
ejpam-5372	159	21	m	m	PROPN
ejpam-5372	159	22	2nz	2nz	NOUN
ejpam-5372	159	23	q	q	X
ejpam-5372	159	24	̸≡	̸≡	NOUN
ejpam-5372	159	25	1	1	NUM
ejpam-5372	159	26	because	because	SCONJ
ejpam-5372	159	27	2n−1z	2n−1z	NUM
ejpam-5372	159	28	does	do	AUX
ejpam-5372	159	29	not	not	PART
ejpam-5372	159	30	divide	divide	VERB
ejpam-5372	159	31	2nz	2nz	NOUN
ejpam-5372	159	32	q	q	X
ejpam-5372	159	33	.	.	PUNCT
ejpam-5372	160	1	thus	thus	ADV
ejpam-5372	160	2	,	,	PUNCT
ejpam-5372	160	3	using	use	VERB
ejpam-5372	160	4	proposition	proposition	NOUN
ejpam-5372	160	5	5	5	NUM
ejpam-5372	160	6	,	,	PUNCT
ejpam-5372	160	7	we	we	PRON
ejpam-5372	160	8	deduce	deduce	VERB
ejpam-5372	160	9	that	that	SCONJ
ejpam-5372	160	10	mr	mr	PROPN
ejpam-5372	160	11	∈	∈	PROPN
ejpam-5372	160	12	gz	gz	PROPN
ejpam-5372	160	13	.	.	PUNCT
ejpam-5372	161	1	proposition	proposition	NOUN
ejpam-5372	161	2	8	8	NUM
ejpam-5372	161	3	.	.	PUNCT
ejpam-5372	162	1	if	if	SCONJ
ejpam-5372	162	2	p	p	PROPN
ejpam-5372	162	3	∈	∈	PROPN
ejpam-5372	162	4	p3,4	p3,4	VERB
ejpam-5372	162	5	,	,	PUNCT
ejpam-5372	162	6	there	there	PRON
ejpam-5372	162	7	are	be	VERB
ejpam-5372	162	8	as	as	ADV
ejpam-5372	162	9	many	many	ADJ
ejpam-5372	162	10	semi	semi	ADJ
ejpam-5372	162	11	-	-	ADJ
ejpam-5372	162	12	primitive	primitive	ADJ
ejpam-5372	162	13	roots	root	NOUN
ejpam-5372	162	14	as	as	ADP
ejpam-5372	162	15	primitive	primitive	ADJ
ejpam-5372	162	16	roots	root	NOUN
ejpam-5372	162	17	,	,	PUNCT
ejpam-5372	162	18	and	and	CCONJ
ejpam-5372	162	19	if	if	SCONJ
ejpam-5372	162	20	p	p	PROPN
ejpam-5372	162	21	∈	∈	PROPN
ejpam-5372	162	22	p1,4	p1,4	ADV
ejpam-5372	162	23	there	there	PRON
ejpam-5372	162	24	are	be	VERB
ejpam-5372	162	25	half	half	ADV
ejpam-5372	162	26	as	as	ADV
ejpam-5372	162	27	many	many	ADJ
ejpam-5372	162	28	.	.	PUNCT
ejpam-5372	163	1	proof	proof	NOUN
ejpam-5372	163	2	.	.	PUNCT
ejpam-5372	164	1	we	we	PRON
ejpam-5372	164	2	know	know	VERB
ejpam-5372	164	3	that	that	SCONJ
ejpam-5372	164	4	there	there	PRON
ejpam-5372	164	5	are	be	VERB
ejpam-5372	164	6	φ	φ	NOUN
ejpam-5372	164	7	(	(	PUNCT
ejpam-5372	164	8	p−	p−	NOUN
ejpam-5372	164	9	1	1	NUM
ejpam-5372	164	10	)	)	PUNCT
ejpam-5372	164	11	primitive	primitive	ADJ
ejpam-5372	164	12	roots	root	NOUN
ejpam-5372	164	13	and	and	CCONJ
ejpam-5372	164	14	,	,	PUNCT
ejpam-5372	164	15	by	by	ADP
ejpam-5372	164	16	corollary	corollary	ADJ
ejpam-5372	164	17	1	1	NUM
ejpam-5372	164	18	,	,	PUNCT
ejpam-5372	164	19	φ	φ	PROPN
ejpam-5372	164	20	(	(	PUNCT
ejpam-5372	164	21	p−1	p−1	PROPN
ejpam-5372	164	22	2	2	NUM
ejpam-5372	164	23	)	)	PUNCT
ejpam-5372	164	24	semi	semi	ADJ
ejpam-5372	164	25	-	-	ADJ
ejpam-5372	164	26	primitive	primitive	ADJ
ejpam-5372	164	27	roots	root	NOUN
ejpam-5372	164	28	modulo	modulo	VERB
ejpam-5372	164	29	p.	p.	NOUN
ejpam-5372	164	30	we	we	PRON
ejpam-5372	164	31	also	also	ADV
ejpam-5372	164	32	know	know	VERB
ejpam-5372	164	33	that	that	SCONJ
ejpam-5372	164	34	φ	φ	PROPN
ejpam-5372	164	35	(	(	PUNCT
ejpam-5372	164	36	ab	ab	PROPN
ejpam-5372	164	37	)	)	PUNCT
ejpam-5372	164	38	=	=	SYM
ejpam-5372	164	39	φ	φ	PROPN
ejpam-5372	164	40	(	(	PUNCT
ejpam-5372	164	41	a)φ	a)φ	X
ejpam-5372	164	42	(	(	PUNCT
ejpam-5372	164	43	b	b	X
ejpam-5372	164	44	)	)	PUNCT
ejpam-5372	164	45	if	if	SCONJ
ejpam-5372	164	46	a	a	PRON
ejpam-5372	164	47	,	,	PUNCT
ejpam-5372	164	48	b	b	NOUN
ejpam-5372	164	49	are	be	AUX
ejpam-5372	164	50	coprime	coprime	ADJ
ejpam-5372	164	51	and	and	CCONJ
ejpam-5372	164	52	φ	φ	PROPN
ejpam-5372	164	53	(	(	PUNCT
ejpam-5372	164	54	2k	2k	NUM
ejpam-5372	164	55	)	)	PUNCT
ejpam-5372	165	1	=	=	SYM
ejpam-5372	165	2	2k−1	2k−1	NUM
ejpam-5372	165	3	.	.	PUNCT
ejpam-5372	166	1	if	if	SCONJ
ejpam-5372	166	2	p	p	DET
ejpam-5372	166	3	≡	≡	PROPN
ejpam-5372	166	4	3	3	NUM
ejpam-5372	166	5	[	[	X
ejpam-5372	166	6	4	4	NUM
ejpam-5372	166	7	]	]	PUNCT
ejpam-5372	166	8	then	then	ADV
ejpam-5372	166	9	n	n	CCONJ
ejpam-5372	166	10	=	=	SYM
ejpam-5372	166	11	1	1	NUM
ejpam-5372	166	12	i.e.	i.e.	X
ejpam-5372	166	13	p−1	p−1	PROPN
ejpam-5372	166	14	2	2	NUM
ejpam-5372	166	15	is	be	AUX
ejpam-5372	166	16	odd	odd	ADJ
ejpam-5372	166	17	and	and	CCONJ
ejpam-5372	166	18	φ	φ	PROPN
ejpam-5372	166	19	(	(	PUNCT
ejpam-5372	166	20	p−	p−	NOUN
ejpam-5372	166	21	1	1	NUM
ejpam-5372	166	22	)	)	PUNCT
ejpam-5372	166	23	=	=	SYM
ejpam-5372	166	24	φ	φ	PROPN
ejpam-5372	166	25	(	(	PUNCT
ejpam-5372	166	26	2)φ	2)φ	NUM
ejpam-5372	166	27	(	(	PUNCT
ejpam-5372	166	28	p−1	p−1	PROPN
ejpam-5372	166	29	2	2	NUM
ejpam-5372	166	30	)	)	PUNCT
ejpam-5372	166	31	=	=	SYM
ejpam-5372	167	1	φ	φ	PROPN
ejpam-5372	167	2	(	(	PUNCT
ejpam-5372	167	3	p−1	p−1	PROPN
ejpam-5372	167	4	2	2	NUM
ejpam-5372	167	5	)	)	PUNCT
ejpam-5372	167	6	.	.	PUNCT
ejpam-5372	168	1	if	if	SCONJ
ejpam-5372	168	2	p	p	PRON
ejpam-5372	168	3	≡	≡	PROPN
ejpam-5372	168	4	1	1	NUM
ejpam-5372	169	1	[	[	X
ejpam-5372	169	2	4	4	X
ejpam-5372	169	3	]	]	PUNCT
ejpam-5372	169	4	then	then	ADV
ejpam-5372	169	5	n	n	CCONJ
ejpam-5372	169	6	≥	≥	NOUN
ejpam-5372	169	7	2	2	NUM
ejpam-5372	169	8	and	and	CCONJ
ejpam-5372	169	9	φ	φ	NUM
ejpam-5372	169	10	(	(	PUNCT
ejpam-5372	169	11	p−	p−	NOUN
ejpam-5372	169	12	1	1	NUM
ejpam-5372	169	13	)	)	PUNCT
ejpam-5372	169	14	=	=	SYM
ejpam-5372	169	15	φ	φ	PROPN
ejpam-5372	169	16	(	(	PUNCT
ejpam-5372	169	17	2n)φ	2n)φ	PROPN
ejpam-5372	169	18	(	(	PUNCT
ejpam-5372	169	19	z	z	NOUN
ejpam-5372	169	20	)	)	PUNCT
ejpam-5372	169	21	=	=	SYM
ejpam-5372	170	1	2n−1φ	2n−1φ	NUM
ejpam-5372	170	2	(	(	PUNCT
ejpam-5372	170	3	z	z	NOUN
ejpam-5372	170	4	)	)	PUNCT
ejpam-5372	170	5	whereas	whereas	SCONJ
ejpam-5372	170	6	φ	φ	PROPN
ejpam-5372	170	7	(	(	PUNCT
ejpam-5372	170	8	p−1	p−1	PROPN
ejpam-5372	170	9	2	2	NUM
ejpam-5372	170	10	)	)	PUNCT
ejpam-5372	170	11	=	=	SYM
ejpam-5372	170	12	φ	φ	PROPN
ejpam-5372	170	13	(	(	PUNCT
ejpam-5372	170	14	2n−1	2n−1	NUM
ejpam-5372	170	15	)	)	PUNCT
ejpam-5372	170	16	φ	φ	PROPN
ejpam-5372	170	17	(	(	PUNCT
ejpam-5372	170	18	z	z	NOUN
ejpam-5372	170	19	)	)	PUNCT
ejpam-5372	170	20	=	=	SYM
ejpam-5372	170	21	2n−2φ	2n−2φ	NUM
ejpam-5372	170	22	(	(	PUNCT
ejpam-5372	170	23	z	z	NOUN
ejpam-5372	170	24	)	)	PUNCT
ejpam-5372	170	25	.	.	PUNCT
ejpam-5372	171	1	proposition	proposition	NOUN
ejpam-5372	171	2	9	9	NUM
ejpam-5372	171	3	.	.	PUNCT
ejpam-5372	172	1	we	we	PRON
ejpam-5372	172	2	assume	assume	VERB
ejpam-5372	172	3	p	p	X
ejpam-5372	172	4	∈	∈	PROPN
ejpam-5372	172	5	p1,4	p1,4	PROPN
ejpam-5372	172	6	.	.	PUNCT
ejpam-5372	173	1	let	let	VERB
ejpam-5372	173	2	m	m	PRON
ejpam-5372	173	3	be	be	AUX
ejpam-5372	173	4	a	a	DET
ejpam-5372	173	5	semi	semi	ADJ
ejpam-5372	173	6	-	-	ADJ
ejpam-5372	173	7	primitive	primitive	ADJ
ejpam-5372	173	8	root	root	NOUN
ejpam-5372	173	9	.	.	PUNCT
ejpam-5372	174	1	then	then	ADV
ejpam-5372	174	2	:	:	PUNCT
ejpam-5372	174	3	−m	−m	INTJ
ejpam-5372	174	4	(	(	PUNCT
ejpam-5372	174	5	r(n)⧹r(n−1	r(n)⧹r(n−1	PROPN
ejpam-5372	174	6	)	)	PUNCT
ejpam-5372	174	7	)	)	PUNCT
ejpam-5372	175	1	⊂	⊂	PROPN
ejpam-5372	175	2	gz	gz	VERB
ejpam-5372	175	3	moreover	moreover	ADV
ejpam-5372	175	4	,	,	PUNCT
ejpam-5372	175	5	−m	−m	NOUN
ejpam-5372	175	6	∈	∈	INTJ
ejpam-5372	175	7	gs	gs	INTJ
ejpam-5372	176	1	if	if	SCONJ
ejpam-5372	176	2	and	and	CCONJ
ejpam-5372	176	3	only	only	ADV
ejpam-5372	176	4	if	if	SCONJ
ejpam-5372	176	5	p	p	PROPN
ejpam-5372	176	6	∈	∈	PROPN
ejpam-5372	176	7	p1,8	p1,8	PROPN
ejpam-5372	176	8	.	.	PROPN
ejpam-5372	176	9	m.	m.	PROPN
ejpam-5372	176	10	wolf	wolf	PROPN
ejpam-5372	176	11	,	,	PUNCT
ejpam-5372	176	12	f.	f.	PROPN
ejpam-5372	176	13	wolf	wolf	PROPN
ejpam-5372	176	14	/	/	SYM
ejpam-5372	176	15	eur	eur	PROPN
ejpam-5372	176	16	.	.	PUNCT
ejpam-5372	177	1	j.	j.	PROPN
ejpam-5372	177	2	pure	pure	PROPN
ejpam-5372	177	3	appl	appl	PROPN
ejpam-5372	177	4	.	.	PROPN
ejpam-5372	177	5	math	math	PROPN
ejpam-5372	177	6	,	,	PUNCT
ejpam-5372	177	7	17	17	NUM
ejpam-5372	177	8	(	(	PUNCT
ejpam-5372	177	9	4	4	NUM
ejpam-5372	177	10	)	)	PUNCT
ejpam-5372	177	11	(	(	PUNCT
ejpam-5372	177	12	2024	2024	NUM
ejpam-5372	177	13	)	)	PUNCT
ejpam-5372	177	14	,	,	PUNCT
ejpam-5372	177	15	2431	2431	NUM
ejpam-5372	177	16	-	-	SYM
ejpam-5372	177	17	2447	2447	NUM
ejpam-5372	177	18	2436	2436	NUM
ejpam-5372	177	19	proof	proof	NOUN
ejpam-5372	177	20	.	.	PUNCT
ejpam-5372	178	1	let	let	VERB
ejpam-5372	178	2	r	r	PRON
ejpam-5372	178	3	∈	∈	PROPN
ejpam-5372	178	4	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	178	5	)	)	PUNCT
ejpam-5372	178	6	,	,	PUNCT
ejpam-5372	178	7	and	and	CCONJ
ejpam-5372	178	8	let	let	VERB
ejpam-5372	178	9	us	we	PRON
ejpam-5372	178	10	show	show	VERB
ejpam-5372	178	11	that	that	SCONJ
ejpam-5372	178	12	−mr	−mr	PRON
ejpam-5372	178	13	∈	∈	PROPN
ejpam-5372	178	14	gz	gz	NOUN
ejpam-5372	178	15	like	like	INTJ
ejpam-5372	178	16	in	in	ADP
ejpam-5372	178	17	proposition	proposition	NOUN
ejpam-5372	178	18	7	7	NUM
ejpam-5372	178	19	let	let	VERB
ejpam-5372	178	20	q	q	NOUN
ejpam-5372	178	21	be	be	AUX
ejpam-5372	178	22	a	a	DET
ejpam-5372	178	23	prime	prime	ADJ
ejpam-5372	178	24	factor	factor	NOUN
ejpam-5372	178	25	of	of	ADP
ejpam-5372	178	26	p	p	NOUN
ejpam-5372	178	27	−	−	PROPN
ejpam-5372	178	28	1	1	NUM
ejpam-5372	178	29	=	=	SYM
ejpam-5372	178	30	2nz	2nz	NOUN
ejpam-5372	178	31	.	.	PUNCT
ejpam-5372	179	1	as	as	SCONJ
ejpam-5372	179	2	p	p	PROPN
ejpam-5372	179	3	∈	∈	PROPN
ejpam-5372	179	4	p1,4	p1,4	PROPN
ejpam-5372	179	5	,	,	PUNCT
ejpam-5372	179	6	we	we	PRON
ejpam-5372	179	7	have	have	VERB
ejpam-5372	179	8	n	n	NUM
ejpam-5372	179	9	≥	≥	NUM
ejpam-5372	179	10	2	2	NUM
ejpam-5372	179	11	so	so	ADV
ejpam-5372	179	12	in	in	ADP
ejpam-5372	179	13	all	all	DET
ejpam-5372	179	14	cases	case	NOUN
ejpam-5372	179	15	(	(	PUNCT
ejpam-5372	179	16	−mr	−mr	NOUN
ejpam-5372	179	17	)	)	PUNCT
ejpam-5372	179	18	p−1	p−1	PROPN
ejpam-5372	179	19	q	q	PROPN
ejpam-5372	179	20	≡	≡	PROPN
ejpam-5372	179	21	(	(	PUNCT
ejpam-5372	179	22	mr	mr	PROPN
ejpam-5372	179	23	)	)	PUNCT
ejpam-5372	179	24	p−1	p−1	PROPN
ejpam-5372	179	25	q	q	PROPN
ejpam-5372	179	26	̸≡	̸≡	PROPN
ejpam-5372	179	27	1	1	NUM
ejpam-5372	179	28	,	,	PUNCT
ejpam-5372	179	29	because	because	SCONJ
ejpam-5372	179	30	mr	mr	PROPN
ejpam-5372	179	31	∈	∈	PROPN
ejpam-5372	179	32	gz	gz	PROPN
ejpam-5372	179	33	.	.	PUNCT
ejpam-5372	180	1	therefore	therefore	ADV
ejpam-5372	180	2	−mr	−mr	PRON
ejpam-5372	180	3	∈	∈	PROPN
ejpam-5372	180	4	gz	gz	NOUN
ejpam-5372	180	5	.	.	PUNCT
ejpam-5372	181	1	moreover	moreover	ADV
ejpam-5372	181	2	−m	−m	PROPN
ejpam-5372	181	3	∈	∈	PROPN
ejpam-5372	181	4	gs	gs	INTJ
ejpam-5372	182	1	if	if	SCONJ
ejpam-5372	182	2	and	and	CCONJ
ejpam-5372	182	3	only	only	ADV
ejpam-5372	182	4	if	if	SCONJ
ejpam-5372	182	5	(	(	PUNCT
ejpam-5372	182	6	−m	−m	NOUN
ejpam-5372	182	7	)	)	PUNCT
ejpam-5372	182	8	p−1	p−1	PROPN
ejpam-5372	182	9	2q	2q	NOUN
ejpam-5372	182	10	̸≡	̸≡	VERB
ejpam-5372	182	11	1	1	NUM
ejpam-5372	182	12	for	for	ADP
ejpam-5372	182	13	any	any	DET
ejpam-5372	182	14	prime	prime	ADJ
ejpam-5372	182	15	factor	factor	NOUN
ejpam-5372	182	16	q	q	PROPN
ejpam-5372	182	17	of	of	ADP
ejpam-5372	182	18	p−1	p−1	PROPN
ejpam-5372	182	19	,	,	PUNCT
ejpam-5372	182	20	which	which	PRON
ejpam-5372	182	21	is	be	AUX
ejpam-5372	182	22	true	true	ADJ
ejpam-5372	182	23	when	when	SCONJ
ejpam-5372	182	24	n	n	X
ejpam-5372	182	25	≥	≥	NOUN
ejpam-5372	182	26	3	3	NUM
ejpam-5372	182	27	,	,	PUNCT
ejpam-5372	182	28	because	because	SCONJ
ejpam-5372	182	29	then	then	ADV
ejpam-5372	182	30	(	(	PUNCT
ejpam-5372	182	31	−m	−m	NOUN
ejpam-5372	182	32	)	)	PUNCT
ejpam-5372	182	33	p−1	p−1	PROPN
ejpam-5372	182	34	2q	2q	NUM
ejpam-5372	183	1	≡	≡	PROPN
ejpam-5372	183	2	m	m	VERB
ejpam-5372	183	3	p−1	p−1	NOUN
ejpam-5372	183	4	2q	2q	NUM
ejpam-5372	183	5	.	.	PUNCT
ejpam-5372	184	1	otherwise	otherwise	ADV
ejpam-5372	184	2	m	m	VERB
ejpam-5372	184	3	p−1	p−1	PROPN
ejpam-5372	184	4	4	4	NUM
ejpam-5372	184	5	≡	≡	PROPN
ejpam-5372	184	6	−1	−1	NOUN
ejpam-5372	184	7	thus	thus	ADV
ejpam-5372	184	8	(	(	PUNCT
ejpam-5372	184	9	−m	−m	NOUN
ejpam-5372	184	10	)	)	PUNCT
ejpam-5372	184	11	p−1	p−1	PROPN
ejpam-5372	184	12	4	4	NUM
ejpam-5372	184	13	≡	≡	PROPN
ejpam-5372	184	14	1	1	NUM
ejpam-5372	184	15	i.e.	i.e.	X
ejpam-5372	184	16	−m	−m	ADJ
ejpam-5372	184	17	/∈	/∈	PUNCT
ejpam-5372	185	1	gs	gs	INTJ
ejpam-5372	185	2	.	.	PUNCT
ejpam-5372	186	1	we	we	PRON
ejpam-5372	186	2	hence	hence	ADV
ejpam-5372	186	3	proved	prove	VERB
ejpam-5372	186	4	that	that	SCONJ
ejpam-5372	186	5	−m	−m	NOUN
ejpam-5372	186	6	∈	∈	PROPN
ejpam-5372	186	7	gs	gs	INTJ
ejpam-5372	187	1	if	if	SCONJ
ejpam-5372	187	2	and	and	CCONJ
ejpam-5372	187	3	only	only	ADV
ejpam-5372	187	4	if	if	SCONJ
ejpam-5372	187	5	p	p	PROPN
ejpam-5372	187	6	∈	∈	PROPN
ejpam-5372	187	7	p1,8	p1,8	PROPN
ejpam-5372	187	8	.	.	PUNCT
ejpam-5372	187	9	proposition	proposition	NOUN
ejpam-5372	187	10	10	10	NUM
ejpam-5372	187	11	.	.	PUNCT
ejpam-5372	188	1	let	let	VERB
ejpam-5372	188	2	m	m	PRON
ejpam-5372	188	3	be	be	AUX
ejpam-5372	188	4	a	a	DET
ejpam-5372	188	5	semi	semi	ADJ
ejpam-5372	188	6	-	-	ADJ
ejpam-5372	188	7	primitive	primitive	ADJ
ejpam-5372	188	8	root	root	NOUN
ejpam-5372	188	9	.	.	PUNCT
ejpam-5372	189	1	we	we	PRON
ejpam-5372	189	2	first	first	ADV
ejpam-5372	189	3	assume	assume	VERB
ejpam-5372	189	4	p	p	X
ejpam-5372	189	5	∈	∈	PROPN
ejpam-5372	189	6	p3,4	p3,4	NOUN
ejpam-5372	189	7	.	.	PUNCT
ejpam-5372	190	1	then	then	ADV
ejpam-5372	190	2	m2	m2	PROPN
ejpam-5372	190	3	∈	∈	PROPN
ejpam-5372	190	4	gs	gs	PROPN
ejpam-5372	190	5	and	and	CCONJ
ejpam-5372	190	6	in	in	ADP
ejpam-5372	190	7	particular	particular	ADJ
ejpam-5372	190	8	:	:	PUNCT
ejpam-5372	190	9	m2	m2	PROPN
ejpam-5372	190	10	(	(	PUNCT
ejpam-5372	190	11	r(n)⧹r(n−1	r(n)⧹r(n−1	PROPN
ejpam-5372	190	12	)	)	PUNCT
ejpam-5372	190	13	)	)	PUNCT
ejpam-5372	191	1	⊂	⊂	PROPN
ejpam-5372	191	2	gz	gz	VERB
ejpam-5372	191	3	we	we	PRON
ejpam-5372	191	4	now	now	ADV
ejpam-5372	191	5	assume	assume	VERB
ejpam-5372	191	6	p	p	X
ejpam-5372	191	7	∈	∈	PROPN
ejpam-5372	191	8	p1,4	p1,4	PROPN
ejpam-5372	191	9	.	.	PUNCT
ejpam-5372	192	1	then	then	ADV
ejpam-5372	192	2	:	:	PUNCT
ejpam-5372	192	3	±m2	±m2	X
ejpam-5372	192	4	(	(	PUNCT
ejpam-5372	192	5	r(n)⧹r(n−1	r(n)⧹r(n−1	NUM
ejpam-5372	192	6	)	)	PUNCT
ejpam-5372	192	7	)	)	PUNCT
ejpam-5372	193	1	⊂	⊂	PROPN
ejpam-5372	193	2	gz	gz	VERB
ejpam-5372	193	3	if	if	SCONJ
ejpam-5372	193	4	p	p	PROPN
ejpam-5372	193	5	∈	∈	PROPN
ejpam-5372	193	6	p1,8	p1,8	NOUN
ejpam-5372	193	7	then	then	ADV
ejpam-5372	193	8	±m2	±m2	NOUN
ejpam-5372	193	9	/∈	/∈	PUNCT
ejpam-5372	194	1	gs	gs	INTJ
ejpam-5372	194	2	.	.	PUNCT
ejpam-5372	195	1	if	if	SCONJ
ejpam-5372	195	2	p	p	PROPN
ejpam-5372	195	3	∈	∈	PROPN
ejpam-5372	195	4	p5,8	p5,8	PROPN
ejpam-5372	195	5	then	then	ADV
ejpam-5372	195	6	m2	m2	PROPN
ejpam-5372	195	7	/∈	/∈	PROPN
ejpam-5372	196	1	gs	gs	PROPN
ejpam-5372	196	2	and	and	CCONJ
ejpam-5372	196	3	−m2	−m2	NOUN
ejpam-5372	196	4	∈	∈	PROPN
ejpam-5372	196	5	gs	gs	NOUN
ejpam-5372	196	6	.	.	PUNCT
ejpam-5372	197	1	in	in	ADP
ejpam-5372	197	2	the	the	DET
ejpam-5372	197	3	last	last	ADJ
ejpam-5372	197	4	case	case	NOUN
ejpam-5372	197	5	,	,	PUNCT
ejpam-5372	197	6	we	we	PRON
ejpam-5372	197	7	identify	identify	VERB
ejpam-5372	197	8	−gs	−gs	NOUN
ejpam-5372	197	9	to	to	ADP
ejpam-5372	197	10	the	the	DET
ejpam-5372	197	11	set	set	NOUN
ejpam-5372	197	12	of	of	ADP
ejpam-5372	197	13	squares	square	NOUN
ejpam-5372	197	14	of	of	ADP
ejpam-5372	197	15	gs	gs	PROPN
ejpam-5372	197	16	.	.	PUNCT
ejpam-5372	197	17	proof	proof	NOUN
ejpam-5372	197	18	.	.	PUNCT
ejpam-5372	198	1	when	when	SCONJ
ejpam-5372	198	2	p	p	PROPN
ejpam-5372	198	3	∈	∈	PROPN
ejpam-5372	198	4	p3,4	p3,4	VERB
ejpam-5372	198	5	,	,	PUNCT
ejpam-5372	198	6	n	n	NOUN
ejpam-5372	198	7	=	=	SYM
ejpam-5372	198	8	1	1	NUM
ejpam-5372	198	9	so	so	ADV
ejpam-5372	198	10	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	198	11	)	)	PUNCT
ejpam-5372	198	12	=	=	PRON
ejpam-5372	198	13	{	{	PUNCT
ejpam-5372	198	14	−1	−1	NOUN
ejpam-5372	198	15	}	}	PUNCT
ejpam-5372	198	16	.	.	PUNCT
ejpam-5372	199	1	by	by	ADP
ejpam-5372	199	2	proposition	proposition	NOUN
ejpam-5372	199	3	6	6	NUM
ejpam-5372	199	4	we	we	PRON
ejpam-5372	199	5	have	have	VERB
ejpam-5372	199	6	−m	−m	NOUN
ejpam-5372	199	7	∈	∈	ADJ
ejpam-5372	199	8	gz	gz	NOUN
ejpam-5372	199	9	hence	hence	ADV
ejpam-5372	199	10	m2	m2	PROPN
ejpam-5372	199	11	∈	∈	PROPN
ejpam-5372	199	12	gs	gs	PROPN
ejpam-5372	199	13	.	.	PUNCT
ejpam-5372	200	1	when	when	SCONJ
ejpam-5372	200	2	p	p	PROPN
ejpam-5372	200	3	∈	∈	PROPN
ejpam-5372	200	4	p1,4	p1,4	ADV
ejpam-5372	200	5	i.e.	i.e.	X
ejpam-5372	200	6	n	n	PRON
ejpam-5372	200	7	≥	≥	NOUN
ejpam-5372	200	8	2	2	NUM
ejpam-5372	200	9	,	,	PUNCT
ejpam-5372	200	10	let	let	VERB
ejpam-5372	200	11	us	we	PRON
ejpam-5372	200	12	take	take	VERB
ejpam-5372	200	13	r	r	NOUN
ejpam-5372	200	14	∈	∈	NOUN
ejpam-5372	200	15	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	200	16	)	)	PUNCT
ejpam-5372	200	17	and	and	CCONJ
ejpam-5372	200	18	focus	focus	VERB
ejpam-5372	200	19	on	on	ADP
ejpam-5372	200	20	(	(	PUNCT
ejpam-5372	200	21	±m2r	±m2r	PROPN
ejpam-5372	200	22	)	)	PUNCT
ejpam-5372	201	1	p−1	p−1	PROPN
ejpam-5372	201	2	q	q	PROPN
ejpam-5372	201	3	=(	=(	NOUN
ejpam-5372	201	4	m2r	m2r	PROPN
ejpam-5372	201	5	)	)	PUNCT
ejpam-5372	202	1	p−1	p−1	PROPN
ejpam-5372	202	2	q	q	PROPN
ejpam-5372	202	3	with	with	ADP
ejpam-5372	202	4	q	q	NOUN
ejpam-5372	202	5	dividing	divide	VERB
ejpam-5372	202	6	p−	p−	NOUN
ejpam-5372	202	7	1	1	NUM
ejpam-5372	202	8	.	.	PUNCT
ejpam-5372	203	1	if	if	SCONJ
ejpam-5372	203	2	q	q	NOUN
ejpam-5372	203	3	=	=	NOUN
ejpam-5372	203	4	2	2	NUM
ejpam-5372	203	5	we	we	PRON
ejpam-5372	203	6	have	have	VERB
ejpam-5372	203	7	(	(	PUNCT
ejpam-5372	203	8	m2r	m2r	NOUN
ejpam-5372	203	9	)	)	PUNCT
ejpam-5372	204	1	p−1	p−1	PROPN
ejpam-5372	204	2	2	2	NUM
ejpam-5372	204	3	=	=	SYM
ejpam-5372	204	4	r2	r2	PROPN
ejpam-5372	204	5	n−1z	n−1z	NOUN
ejpam-5372	204	6	̸≡	̸≡	PROPN
ejpam-5372	204	7	1	1	NUM
ejpam-5372	204	8	because	because	SCONJ
ejpam-5372	204	9	rz	rz	NOUN
ejpam-5372	204	10	∈	∈	PROPN
ejpam-5372	204	11	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	204	12	)	)	PUNCT
ejpam-5372	204	13	.	.	PUNCT
ejpam-5372	205	1	if	if	SCONJ
ejpam-5372	205	2	q	q	PROPN
ejpam-5372	205	3	̸=	̸=	PROPN
ejpam-5372	205	4	2	2	NUM
ejpam-5372	205	5	we	we	PRON
ejpam-5372	205	6	have	have	VERB
ejpam-5372	205	7	(	(	PUNCT
ejpam-5372	205	8	m2r	m2r	NOUN
ejpam-5372	205	9	)	)	PUNCT
ejpam-5372	205	10	p−1	p−1	PROPN
ejpam-5372	205	11	q	q	PROPN
ejpam-5372	206	1	=	=	PUNCT
ejpam-5372	206	2	m	m	NOUN
ejpam-5372	206	3	2(p−1	2(p−1	ADJ
ejpam-5372	206	4	)	)	PUNCT
ejpam-5372	207	1	q	q	NOUN
ejpam-5372	207	2	̸≡	̸≡	NOUN
ejpam-5372	207	3	1	1	NUM
ejpam-5372	207	4	because	because	SCONJ
ejpam-5372	207	5	p−1	p−1	PROPN
ejpam-5372	207	6	2	2	NUM
ejpam-5372	207	7	does	do	AUX
ejpam-5372	207	8	not	not	PART
ejpam-5372	207	9	divide	divide	VERB
ejpam-5372	207	10	2(p−1	2(p−1	NOUN
ejpam-5372	207	11	)	)	PUNCT
ejpam-5372	207	12	q	q	NOUN
ejpam-5372	207	13	.	.	PUNCT
ejpam-5372	208	1	we	we	PRON
ejpam-5372	208	2	thus	thus	ADV
ejpam-5372	208	3	proved	prove	VERB
ejpam-5372	208	4	that	that	SCONJ
ejpam-5372	208	5	±m2	±m2	NOUN
ejpam-5372	208	6	(	(	PUNCT
ejpam-5372	208	7	r(n)⧹r(n−1	r(n)⧹r(n−1	NUM
ejpam-5372	208	8	)	)	PUNCT
ejpam-5372	208	9	)	)	PUNCT
ejpam-5372	209	1	⊂	⊂	PROPN
ejpam-5372	209	2	gz	gz	PROPN
ejpam-5372	209	3	.	.	PUNCT
ejpam-5372	210	1	if	if	SCONJ
ejpam-5372	210	2	p	p	PROPN
ejpam-5372	210	3	∈	∈	PROPN
ejpam-5372	210	4	p1,8	p1,8	NOUN
ejpam-5372	210	5	then	then	ADV
ejpam-5372	210	6	(	(	PUNCT
ejpam-5372	210	7	±m2	±m2	NOUN
ejpam-5372	210	8	)	)	PUNCT
ejpam-5372	210	9	p−1	p−1	PROPN
ejpam-5372	210	10	4	4	NUM
ejpam-5372	210	11	≡	≡	PROPN
ejpam-5372	210	12	1	1	NUM
ejpam-5372	210	13	and	and	CCONJ
ejpam-5372	210	14	±m2	±m2	PROPN
ejpam-5372	210	15	/∈	/∈	PUNCT
ejpam-5372	211	1	gs	gs	INTJ
ejpam-5372	211	2	.	.	PUNCT
ejpam-5372	212	1	if	if	SCONJ
ejpam-5372	212	2	p	p	PROPN
ejpam-5372	212	3	∈	∈	PROPN
ejpam-5372	212	4	p5,8	p5,8	NOUN
ejpam-5372	212	5	then	then	ADV
ejpam-5372	212	6	n	n	NOUN
ejpam-5372	212	7	=	=	SYM
ejpam-5372	212	8	2	2	NUM
ejpam-5372	212	9	and	and	CCONJ
ejpam-5372	212	10	(	(	PUNCT
ejpam-5372	212	11	m2	m2	PROPN
ejpam-5372	212	12	)	)	PUNCT
ejpam-5372	212	13	p−1	p−1	PROPN
ejpam-5372	212	14	4	4	NUM
ejpam-5372	212	15	≡	≡	PROPN
ejpam-5372	212	16	1	1	NUM
ejpam-5372	212	17	,	,	PUNCT
ejpam-5372	212	18	(	(	PUNCT
ejpam-5372	212	19	−m2	−m2	NOUN
ejpam-5372	212	20	)	)	PUNCT
ejpam-5372	213	1	p−1	p−1	PROPN
ejpam-5372	213	2	4	4	NUM
ejpam-5372	213	3	≡	≡	PROPN
ejpam-5372	213	4	(	(	PUNCT
ejpam-5372	213	5	−1)z	−1)z	PROPN
ejpam-5372	213	6	(	(	PUNCT
ejpam-5372	213	7	m2	m2	PROPN
ejpam-5372	213	8	)	)	PUNCT
ejpam-5372	213	9	p−1	p−1	PROPN
ejpam-5372	213	10	4	4	NUM
ejpam-5372	213	11	≡	≡	PROPN
ejpam-5372	213	12	−1	−1	NOUN
ejpam-5372	213	13	thus	thus	ADV
ejpam-5372	213	14	m2	m2	PROPN
ejpam-5372	213	15	/∈	/∈	PROPN
ejpam-5372	213	16	gs	gs	PROPN
ejpam-5372	213	17	and	and	CCONJ
ejpam-5372	213	18	−m2	−m2	NOUN
ejpam-5372	213	19	∈	∈	PROPN
ejpam-5372	213	20	gs	gs	X
ejpam-5372	213	21	(	(	PUNCT
ejpam-5372	213	22	checking	check	VERB
ejpam-5372	213	23	the	the	DET
ejpam-5372	213	24	other	other	ADJ
ejpam-5372	213	25	powers	power	NOUN
ejpam-5372	213	26	is	be	AUX
ejpam-5372	213	27	immediate	immediate	ADJ
ejpam-5372	213	28	)	)	PUNCT
ejpam-5372	213	29	.	.	PUNCT
ejpam-5372	214	1	similarly	similarly	ADV
ejpam-5372	214	2	,	,	PUNCT
ejpam-5372	214	3	if	if	SCONJ
ejpam-5372	214	4	m	m	VERB
ejpam-5372	214	5	∈	∈	ADJ
ejpam-5372	214	6	gs	gs	NOUN
ejpam-5372	214	7	,	,	PUNCT
ejpam-5372	214	8	−m	−m	PROPN
ejpam-5372	214	9	/∈	/∈	PUNCT
ejpam-5372	214	10	gs	gs	INTJ
ejpam-5372	215	1	thus	thus	ADV
ejpam-5372	215	2	the	the	DET
ejpam-5372	215	3	square	square	ADJ
ejpam-5372	215	4	function	function	NOUN
ejpam-5372	215	5	is	be	AUX
ejpam-5372	215	6	injective	injective	ADJ
ejpam-5372	215	7	on	on	ADP
ejpam-5372	215	8	gs	gs	X
ejpam-5372	215	9	,	,	PUNCT
ejpam-5372	215	10	which	which	PRON
ejpam-5372	215	11	shows	show	VERB
ejpam-5372	215	12	that	that	SCONJ
ejpam-5372	215	13	−gs	−gs	NOUN
ejpam-5372	215	14	is	be	AUX
ejpam-5372	215	15	equal	equal	ADJ
ejpam-5372	215	16	to	to	ADP
ejpam-5372	215	17	the	the	DET
ejpam-5372	215	18	set	set	NOUN
ejpam-5372	215	19	of	of	ADP
ejpam-5372	215	20	squares	square	NOUN
ejpam-5372	215	21	of	of	ADP
ejpam-5372	215	22	gs	gs	PROPN
ejpam-5372	215	23	.	.	PUNCT
ejpam-5372	216	1	propositions	proposition	NOUN
ejpam-5372	216	2	9	9	NUM
ejpam-5372	216	3	and	and	CCONJ
ejpam-5372	216	4	10	10	NUM
ejpam-5372	216	5	allow	allow	VERB
ejpam-5372	216	6	to	to	PART
ejpam-5372	216	7	build	build	VERB
ejpam-5372	216	8	primitive	primitive	ADJ
ejpam-5372	216	9	roots	root	NOUN
ejpam-5372	216	10	from	from	ADP
ejpam-5372	216	11	semi	semi	ADJ
ejpam-5372	216	12	-	-	ADJ
ejpam-5372	216	13	primitive	primitive	ADJ
ejpam-5372	216	14	roots	root	NOUN
ejpam-5372	216	15	and	and	CCONJ
ejpam-5372	216	16	elements	element	NOUN
ejpam-5372	216	17	of	of	ADP
ejpam-5372	216	18	r(n)⧹r(n−1	r(n)⧹r(n−1	NUM
ejpam-5372	216	19	)	)	PUNCT
ejpam-5372	216	20	.	.	PUNCT
ejpam-5372	217	1	definition	definition	NOUN
ejpam-5372	217	2	5	5	NUM
ejpam-5372	217	3	.	.	PUNCT
ejpam-5372	218	1	for	for	ADP
ejpam-5372	218	2	some	some	DET
ejpam-5372	218	3	r	r	NOUN
ejpam-5372	218	4	∈	∈	NOUN
ejpam-5372	218	5	r(n)⧹r(n−1	r(n)⧹r(n−1	NUM
ejpam-5372	218	6	)	)	PUNCT
ejpam-5372	218	7	,	,	PUNCT
ejpam-5372	218	8	we	we	PRON
ejpam-5372	218	9	define	define	VERB
ejpam-5372	218	10	the	the	DET
ejpam-5372	218	11	set	set	NOUN
ejpam-5372	218	12	g′	g′	NOUN
ejpam-5372	218	13	z	z	NOUN
ejpam-5372	219	1	:	:	PUNCT
ejpam-5372	219	2	=	=	SYM
ejpam-5372	219	3	rgz	rgz	ADJ
ejpam-5372	219	4	.	.	PUNCT
ejpam-5372	220	1	proposition	proposition	NOUN
ejpam-5372	220	2	11	11	NUM
ejpam-5372	220	3	.	.	PUNCT
ejpam-5372	221	1	g′	g′	NOUN
ejpam-5372	221	2	z	z	PROPN
ejpam-5372	221	3	is	be	AUX
ejpam-5372	221	4	independent	independent	ADJ
ejpam-5372	221	5	of	of	ADP
ejpam-5372	221	6	the	the	DET
ejpam-5372	221	7	choice	choice	NOUN
ejpam-5372	221	8	of	of	ADP
ejpam-5372	221	9	r	r	NOUN
ejpam-5372	221	10	and	and	CCONJ
ejpam-5372	221	11	contains	contain	VERB
ejpam-5372	221	12	gs	gs	INTJ
ejpam-5372	221	13	.	.	PUNCT
ejpam-5372	222	1	it	it	PRON
ejpam-5372	222	2	has	have	VERB
ejpam-5372	222	3	same	same	ADJ
ejpam-5372	222	4	size	size	NOUN
ejpam-5372	222	5	as	as	ADP
ejpam-5372	222	6	gz	gz	PROPN
ejpam-5372	222	7	and	and	CCONJ
ejpam-5372	222	8	we	we	PRON
ejpam-5372	222	9	have	have	AUX
ejpam-5372	222	10	gz	gz	NOUN
ejpam-5372	223	1	=	=	SYM
ejpam-5372	223	2	rg′	rg′	PROPN
ejpam-5372	223	3	z	z	NOUN
ejpam-5372	223	4	for	for	ADP
ejpam-5372	223	5	any	any	DET
ejpam-5372	223	6	r	r	NOUN
ejpam-5372	223	7	∈	∈	NOUN
ejpam-5372	223	8	r(n)⧹r(n−1	r(n)⧹r(n−1	NUM
ejpam-5372	223	9	)	)	PUNCT
ejpam-5372	223	10	.	.	PUNCT
ejpam-5372	224	1	moreover	moreover	ADV
ejpam-5372	224	2	,	,	PUNCT
ejpam-5372	224	3	if	if	SCONJ
ejpam-5372	224	4	for	for	ADP
ejpam-5372	224	5	k	k	PROPN
ejpam-5372	224	6	∈	∈	PROPN
ejpam-5372	224	7	n∗	n∗	NOUN
ejpam-5372	224	8	we	we	PRON
ejpam-5372	224	9	define	define	VERB
ejpam-5372	224	10	gs	gs	PROPN
ejpam-5372	224	11	(	(	PUNCT
ejpam-5372	224	12	k	k	NOUN
ejpam-5372	224	13	)	)	PUNCT
ejpam-5372	224	14	=	=	SYM
ejpam-5372	224	15	{	{	PUNCT
ejpam-5372	224	16	mk	mk	PROPN
ejpam-5372	224	17	,	,	PUNCT
ejpam-5372	224	18	m	m	PROPN
ejpam-5372	224	19	∈	∈	ADJ
ejpam-5372	224	20	gs	gs	NOUN
ejpam-5372	224	21	}	}	PUNCT
ejpam-5372	224	22	:	:	PUNCT
ejpam-5372	224	23	•	•	X
ejpam-5372	224	24	when	when	SCONJ
ejpam-5372	224	25	p	p	PROPN
ejpam-5372	224	26	∈	∈	PROPN
ejpam-5372	224	27	p3,4	p3,4	VERB
ejpam-5372	224	28	,	,	PUNCT
ejpam-5372	224	29	g	g	NOUN
ejpam-5372	224	30	′	′	NUM
ejpam-5372	224	31	z	z	NOUN
ejpam-5372	225	1	=	=	PUNCT
ejpam-5372	225	2	gs	gs	NOUN
ejpam-5372	225	3	=	=	PUNCT
ejpam-5372	225	4	gs	gs	PROPN
ejpam-5372	225	5	(	(	PUNCT
ejpam-5372	225	6	2	2	NUM
ejpam-5372	225	7	)	)	PUNCT
ejpam-5372	225	8	whereas	whereas	SCONJ
ejpam-5372	225	9	gz	gz	VERB
ejpam-5372	225	10	=	=	PUNCT
ejpam-5372	225	11	−g′	−g′	PROPN
ejpam-5372	225	12	z	z	NOUN
ejpam-5372	225	13	.	.	PUNCT
ejpam-5372	226	1	m.	m.	PROPN
ejpam-5372	226	2	wolf	wolf	PROPN
ejpam-5372	226	3	,	,	PUNCT
ejpam-5372	226	4	f.	f.	PROPN
ejpam-5372	226	5	wolf	wolf	PROPN
ejpam-5372	226	6	/	/	SYM
ejpam-5372	226	7	eur	eur	PROPN
ejpam-5372	226	8	.	.	PUNCT
ejpam-5372	227	1	j.	j.	PROPN
ejpam-5372	227	2	pure	pure	PROPN
ejpam-5372	227	3	appl	appl	PROPN
ejpam-5372	227	4	.	.	PROPN
ejpam-5372	227	5	math	math	PROPN
ejpam-5372	227	6	,	,	PUNCT
ejpam-5372	227	7	17	17	NUM
ejpam-5372	227	8	(	(	PUNCT
ejpam-5372	227	9	4	4	NUM
ejpam-5372	227	10	)	)	PUNCT
ejpam-5372	227	11	(	(	PUNCT
ejpam-5372	227	12	2024	2024	NUM
ejpam-5372	227	13	)	)	PUNCT
ejpam-5372	227	14	,	,	PUNCT
ejpam-5372	227	15	2431	2431	NUM
ejpam-5372	227	16	-	-	SYM
ejpam-5372	227	17	2447	2447	NUM
ejpam-5372	227	18	2437	2437	NUM
ejpam-5372	227	19	•	•	NOUN
ejpam-5372	227	20	when	when	SCONJ
ejpam-5372	227	21	p	p	PROPN
ejpam-5372	227	22	∈	∈	PROPN
ejpam-5372	227	23	p5,8	p5,8	NOUN
ejpam-5372	227	24	,	,	PUNCT
ejpam-5372	227	25	g	g	NOUN
ejpam-5372	227	26	′	′	NUM
ejpam-5372	227	27	z	z	NOUN
ejpam-5372	228	1	=	=	SYM
ejpam-5372	228	2	gs	gs	INTJ
ejpam-5372	229	1	⊔	⊔	NUM
ejpam-5372	229	2	−gs	−gs	NOUN
ejpam-5372	229	3	=	=	NOUN
ejpam-5372	230	1	gs	gs	INTJ
ejpam-5372	230	2	⊔	⊔	INTJ
ejpam-5372	230	3	gs	gs	INTJ
ejpam-5372	230	4	(	(	PUNCT
ejpam-5372	230	5	2	2	NUM
ejpam-5372	230	6	)	)	PUNCT
ejpam-5372	230	7	.	.	PUNCT
ejpam-5372	231	1	•	•	NUM
ejpam-5372	231	2	when	when	SCONJ
ejpam-5372	231	3	p	p	PROPN
ejpam-5372	231	4	∈	∈	PROPN
ejpam-5372	231	5	p9,16	p9,16	NOUN
ejpam-5372	231	6	,	,	PUNCT
ejpam-5372	231	7	g	g	PROPN
ejpam-5372	231	8	′	′	NUM
ejpam-5372	231	9	z	z	NOUN
ejpam-5372	232	1	=	=	SYM
ejpam-5372	232	2	gs	gs	INTJ
ejpam-5372	232	3	⊔	⊔	INTJ
ejpam-5372	232	4	(	(	PUNCT
ejpam-5372	232	5	gs	gs	INTJ
ejpam-5372	232	6	(	(	PUNCT
ejpam-5372	232	7	2	2	NUM
ejpam-5372	232	8	)	)	PUNCT
ejpam-5372	232	9	⊔	⊔	NUM
ejpam-5372	232	10	−gs	−gs	NOUN
ejpam-5372	232	11	(	(	PUNCT
ejpam-5372	232	12	2	2	NUM
ejpam-5372	232	13	)	)	PUNCT
ejpam-5372	232	14	)	)	PUNCT
ejpam-5372	232	15	•	•	ADP
ejpam-5372	232	16	for	for	ADP
ejpam-5372	232	17	any	any	DET
ejpam-5372	232	18	p	p	NOUN
ejpam-5372	232	19	,	,	PUNCT
ejpam-5372	232	20	we	we	PRON
ejpam-5372	232	21	have	have	VERB
ejpam-5372	232	22	:	:	PUNCT
ejpam-5372	232	23	g′	g′	NOUN
ejpam-5372	232	24	z	z	NOUN
ejpam-5372	232	25	=	=	SYM
ejpam-5372	232	26	⋃n−1	⋃n−1	PROPN
ejpam-5372	232	27	t=0	t=0	PUNCT
ejpam-5372	232	28	gs	gs	PROPN
ejpam-5372	232	29	(	(	PUNCT
ejpam-5372	232	30	2	2	NUM
ejpam-5372	232	31	t	t	NOUN
ejpam-5372	232	32	)	)	PUNCT
ejpam-5372	232	33	=	=	PRON
ejpam-5372	232	34	{	{	PUNCT
ejpam-5372	232	35	g	g	PROPN
ejpam-5372	232	36	∈	∈	PROPN
ejpam-5372	232	37	f∗	f∗	NOUN
ejpam-5372	232	38	p	p	NOUN
ejpam-5372	232	39	∣∣∣	∣∣∣	NOUN
ejpam-5372	232	40	∀q	∀q	PROPN
ejpam-5372	232	41	∈	∈	PROPN
ejpam-5372	232	42	p⧹	p⧹	VERB
ejpam-5372	232	43	{	{	PUNCT
ejpam-5372	232	44	2	2	NUM
ejpam-5372	232	45	}	}	PUNCT
ejpam-5372	232	46	(	(	PUNCT
ejpam-5372	232	47	q|p−	q|p−	PROPN
ejpam-5372	232	48	1	1	NUM
ejpam-5372	232	49	⇒	⇒	NOUN
ejpam-5372	232	50	g	g	PROPN
ejpam-5372	232	51	p−1	p−1	PROPN
ejpam-5372	232	52	2q	2q	NUM
ejpam-5372	232	53	̸≡	̸≡	X
ejpam-5372	232	54	1	1	NUM
ejpam-5372	232	55	)	)	PUNCT
ejpam-5372	232	56	}	}	PUNCT
ejpam-5372	232	57	.	.	PUNCT
ejpam-5372	233	1	proof	proof	NOUN
ejpam-5372	233	2	.	.	PUNCT
ejpam-5372	234	1	to	to	PART
ejpam-5372	234	2	prove	prove	VERB
ejpam-5372	234	3	that	that	SCONJ
ejpam-5372	234	4	g′	g′	NOUN
ejpam-5372	234	5	z	z	NOUN
ejpam-5372	234	6	does	do	AUX
ejpam-5372	234	7	not	not	PART
ejpam-5372	234	8	depend	depend	VERB
ejpam-5372	234	9	on	on	ADP
ejpam-5372	234	10	r	r	NOUN
ejpam-5372	234	11	,	,	PUNCT
ejpam-5372	234	12	it	it	PRON
ejpam-5372	234	13	is	be	AUX
ejpam-5372	234	14	enough	enough	ADJ
ejpam-5372	234	15	to	to	PART
ejpam-5372	234	16	observe	observe	VERB
ejpam-5372	234	17	that	that	SCONJ
ejpam-5372	234	18	if	if	SCONJ
ejpam-5372	234	19	r	r	NOUN
ejpam-5372	234	20	,	,	PUNCT
ejpam-5372	234	21	r	r	NOUN
ejpam-5372	234	22	′	′	NUM
ejpam-5372	234	23	∈	∈	NOUN
ejpam-5372	234	24	r(n)⧹r(n−1	r(n)⧹r(n−1	NUM
ejpam-5372	234	25	)	)	PUNCT
ejpam-5372	234	26	and	and	CCONJ
ejpam-5372	234	27	g	g	PROPN
ejpam-5372	234	28	∈	∈	PROPN
ejpam-5372	234	29	gz	gz	NOUN
ejpam-5372	234	30	,	,	PUNCT
ejpam-5372	234	31	we	we	PRON
ejpam-5372	234	32	have	have	VERB
ejpam-5372	234	33	rr	rr	NUM
ejpam-5372	234	34	′	′	NUM
ejpam-5372	234	35	g	g	PROPN
ejpam-5372	234	36	∈	∈	PROPN
ejpam-5372	234	37	gz	gz	NOUN
ejpam-5372	234	38	.	.	PUNCT
ejpam-5372	235	1	this	this	PRON
ejpam-5372	235	2	can	can	AUX
ejpam-5372	235	3	be	be	AUX
ejpam-5372	235	4	done	do	VERB
ejpam-5372	235	5	using	use	VERB
ejpam-5372	235	6	proposition	proposition	NOUN
ejpam-5372	235	7	5	5	NUM
ejpam-5372	235	8	:	:	PUNCT
ejpam-5372	235	9	we	we	PRON
ejpam-5372	235	10	know	know	VERB
ejpam-5372	235	11	that	that	SCONJ
ejpam-5372	235	12	r	r	NOUN
ejpam-5372	235	13	p−1	p−1	PROPN
ejpam-5372	235	14	2	2	NUM
ejpam-5372	235	15	≡	≡	PROPN
ejpam-5372	235	16	r	r	NOUN
ejpam-5372	235	17	′	′	NUM
ejpam-5372	235	18	p−1	p−1	NOUN
ejpam-5372	235	19	2	2	NUM
ejpam-5372	235	20	≡	≡	PROPN
ejpam-5372	235	21	−1	−1	NOUN
ejpam-5372	235	22	whereas	whereas	SCONJ
ejpam-5372	235	23	if	if	SCONJ
ejpam-5372	235	24	q	q	NOUN
ejpam-5372	235	25	is	be	AUX
ejpam-5372	235	26	an	an	DET
ejpam-5372	235	27	odd	odd	ADJ
ejpam-5372	235	28	prime	prime	ADJ
ejpam-5372	235	29	factor	factor	NOUN
ejpam-5372	235	30	of	of	ADP
ejpam-5372	235	31	p−1	p−1	PROPN
ejpam-5372	235	32	,	,	PUNCT
ejpam-5372	236	1	r	r	PROPN
ejpam-5372	236	2	p−1	p−1	PROPN
ejpam-5372	236	3	q	q	PROPN
ejpam-5372	236	4	≡	≡	PROPN
ejpam-5372	236	5	r	r	NOUN
ejpam-5372	236	6	′	′	NUM
ejpam-5372	237	1	p−1	p−1	PROPN
ejpam-5372	237	2	q	q	PROPN
ejpam-5372	237	3	≡	≡	PROPN
ejpam-5372	237	4	1	1	NUM
ejpam-5372	237	5	.	.	PUNCT
ejpam-5372	237	6	proposition	proposition	NOUN
ejpam-5372	237	7	7	7	NUM
ejpam-5372	237	8	ensures	ensure	VERB
ejpam-5372	237	9	that	that	SCONJ
ejpam-5372	237	10	gs	gs	PROPN
ejpam-5372	237	11	⊂	⊂	PROPN
ejpam-5372	237	12	g′	g′	PROPN
ejpam-5372	238	1	z	z	PROPN
ejpam-5372	238	2	.	.	PUNCT
ejpam-5372	239	1	it	it	PRON
ejpam-5372	239	2	is	be	AUX
ejpam-5372	239	3	also	also	ADV
ejpam-5372	239	4	clear	clear	ADJ
ejpam-5372	239	5	that	that	SCONJ
ejpam-5372	239	6	g′	g′	NOUN
ejpam-5372	239	7	z	z	PROPN
ejpam-5372	239	8	and	and	CCONJ
ejpam-5372	239	9	gz	gz	AUX
ejpam-5372	239	10	have	have	VERB
ejpam-5372	239	11	the	the	DET
ejpam-5372	239	12	same	same	ADJ
ejpam-5372	239	13	size	size	NOUN
ejpam-5372	239	14	and	and	CCONJ
ejpam-5372	239	15	that	that	SCONJ
ejpam-5372	239	16	gz	gz	NOUN
ejpam-5372	240	1	=	=	SYM
ejpam-5372	240	2	rg′	rg′	PROPN
ejpam-5372	240	3	z	z	NOUN
ejpam-5372	240	4	for	for	ADP
ejpam-5372	240	5	any	any	DET
ejpam-5372	240	6	r	r	NOUN
ejpam-5372	240	7	∈	∈	NOUN
ejpam-5372	240	8	r(n)⧹r(n−1	r(n)⧹r(n−1	NUM
ejpam-5372	240	9	)	)	PUNCT
ejpam-5372	240	10	.	.	PUNCT
ejpam-5372	241	1	finally	finally	ADV
ejpam-5372	241	2	:	:	PUNCT
ejpam-5372	241	3	•	•	INTJ
ejpam-5372	241	4	if	if	SCONJ
ejpam-5372	241	5	p	p	PROPN
ejpam-5372	241	6	∈	∈	PROPN
ejpam-5372	241	7	p3,4	p3,4	VERB
ejpam-5372	241	8	,	,	PUNCT
ejpam-5372	241	9	since	since	SCONJ
ejpam-5372	241	10	from	from	ADP
ejpam-5372	241	11	proposition	proposition	NOUN
ejpam-5372	241	12	11	11	NUM
ejpam-5372	241	13	the	the	DET
ejpam-5372	241	14	two	two	NUM
ejpam-5372	241	15	sets	set	NOUN
ejpam-5372	241	16	have	have	VERB
ejpam-5372	241	17	the	the	DET
ejpam-5372	241	18	same	same	ADJ
ejpam-5372	241	19	size	size	NOUN
ejpam-5372	241	20	,	,	PUNCT
ejpam-5372	242	1	g′	g′	NOUN
ejpam-5372	242	2	z	z	NOUN
ejpam-5372	242	3	=	=	SYM
ejpam-5372	242	4	gs	gs	PROPN
ejpam-5372	242	5	.	.	PUNCT
ejpam-5372	242	6	proposition	proposition	PROPN
ejpam-5372	242	7	10	10	NUM
ejpam-5372	242	8	also	also	ADV
ejpam-5372	242	9	ensures	ensure	VERB
ejpam-5372	242	10	that	that	SCONJ
ejpam-5372	242	11	gs	gs	INTJ
ejpam-5372	242	12	(	(	PUNCT
ejpam-5372	242	13	2	2	NUM
ejpam-5372	242	14	)	)	PUNCT
ejpam-5372	242	15	⊂	⊂	PROPN
ejpam-5372	242	16	gs	gs	INTJ
ejpam-5372	243	1	and	and	CCONJ
ejpam-5372	243	2	since	since	SCONJ
ejpam-5372	243	3	−1	−1	NOUN
ejpam-5372	243	4	/∈	/∈	PUNCT
ejpam-5372	244	1	gs	gs	INTJ
ejpam-5372	244	2	the	the	DET
ejpam-5372	244	3	square	square	ADJ
ejpam-5372	244	4	function	function	NOUN
ejpam-5372	244	5	is	be	AUX
ejpam-5372	244	6	injective	injective	ADJ
ejpam-5372	244	7	from	from	ADP
ejpam-5372	244	8	gs	gs	PROPN
ejpam-5372	244	9	to	to	ADP
ejpam-5372	244	10	gs	gs	PROPN
ejpam-5372	244	11	(	(	PUNCT
ejpam-5372	244	12	2	2	NUM
ejpam-5372	244	13	)	)	PUNCT
ejpam-5372	245	1	thus	thus	ADV
ejpam-5372	245	2	gs	gs	X
ejpam-5372	245	3	(	(	PUNCT
ejpam-5372	245	4	2	2	NUM
ejpam-5372	245	5	)	)	PUNCT
ejpam-5372	245	6	=	=	SYM
ejpam-5372	245	7	gs	gs	PROPN
ejpam-5372	245	8	.	.	PUNCT
ejpam-5372	246	1	in	in	ADP
ejpam-5372	246	2	this	this	DET
ejpam-5372	246	3	case	case	NOUN
ejpam-5372	246	4	we	we	PRON
ejpam-5372	246	5	know	know	VERB
ejpam-5372	246	6	that	that	SCONJ
ejpam-5372	246	7	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	246	8	)	)	PUNCT
ejpam-5372	246	9	=	=	PRON
ejpam-5372	246	10	{	{	PUNCT
ejpam-5372	246	11	−1	−1	NOUN
ejpam-5372	246	12	}	}	PUNCT
ejpam-5372	246	13	,	,	PUNCT
ejpam-5372	246	14	proposition	proposition	NOUN
ejpam-5372	246	15	7	7	NUM
ejpam-5372	246	16	ensures	ensure	NOUN
ejpam-5372	246	17	gz	gz	NOUN
ejpam-5372	246	18	=	=	PUNCT
ejpam-5372	246	19	−g′	−g′	PROPN
ejpam-5372	246	20	z	z	NOUN
ejpam-5372	246	21	.	.	PUNCT
ejpam-5372	247	1	•	•	INTJ
ejpam-5372	247	2	if	if	SCONJ
ejpam-5372	247	3	p	p	PROPN
ejpam-5372	247	4	∈	∈	PROPN
ejpam-5372	247	5	p5,8	p5,8	NOUN
ejpam-5372	247	6	,	,	PUNCT
ejpam-5372	247	7	proposition	proposition	NOUN
ejpam-5372	247	8	10	10	NUM
ejpam-5372	247	9	ensures	ensure	VERB
ejpam-5372	247	10	that	that	SCONJ
ejpam-5372	247	11	−gs	−gs	NOUN
ejpam-5372	248	1	=	=	NOUN
ejpam-5372	248	2	gs	gs	X
ejpam-5372	248	3	(	(	PUNCT
ejpam-5372	248	4	2	2	NUM
ejpam-5372	248	5	)	)	PUNCT
ejpam-5372	248	6	is	be	AUX
ejpam-5372	248	7	disjoint	disjoint	NOUN
ejpam-5372	248	8	of	of	ADP
ejpam-5372	248	9	gs	gs	PROPN
ejpam-5372	248	10	and	and	CCONJ
ejpam-5372	248	11	included	include	VERB
ejpam-5372	248	12	in	in	ADP
ejpam-5372	248	13	g′	g′	NOUN
ejpam-5372	248	14	z	z	PROPN
ejpam-5372	248	15	hence	hence	ADV
ejpam-5372	248	16	a	a	DET
ejpam-5372	248	17	cardinality	cardinality	NOUN
ejpam-5372	248	18	argument	argument	NOUN
ejpam-5372	248	19	ensures	ensure	VERB
ejpam-5372	248	20	g′	g′	NOUN
ejpam-5372	248	21	z	z	NOUN
ejpam-5372	249	1	=	=	SYM
ejpam-5372	249	2	gs	gs	INTJ
ejpam-5372	249	3	⊔	⊔	NUM
ejpam-5372	249	4	−gs	−gs	NOUN
ejpam-5372	249	5	=	=	NOUN
ejpam-5372	250	1	gs	gs	INTJ
ejpam-5372	250	2	⊔	⊔	INTJ
ejpam-5372	250	3	gs	gs	INTJ
ejpam-5372	250	4	(	(	PUNCT
ejpam-5372	250	5	2	2	NUM
ejpam-5372	250	6	)	)	PUNCT
ejpam-5372	250	7	.	.	PUNCT
ejpam-5372	251	1	•	•	INTJ
ejpam-5372	251	2	if	if	SCONJ
ejpam-5372	251	3	p	p	PROPN
ejpam-5372	251	4	∈	∈	PROPN
ejpam-5372	251	5	p1,8	p1,8	NOUN
ejpam-5372	251	6	,	,	PUNCT
ejpam-5372	251	7	proposition	proposition	NOUN
ejpam-5372	251	8	5	5	NUM
ejpam-5372	251	9	shows	show	VERB
ejpam-5372	251	10	that	that	SCONJ
ejpam-5372	251	11	gs	gs	INTJ
ejpam-5372	251	12	=	=	PUNCT
ejpam-5372	251	13	−gs	−gs	NOUN
ejpam-5372	251	14	thus	thus	ADV
ejpam-5372	251	15	the	the	DET
ejpam-5372	251	16	size	size	NOUN
ejpam-5372	251	17	of	of	ADP
ejpam-5372	251	18	gs	gs	PROPN
ejpam-5372	251	19	(	(	PUNCT
ejpam-5372	251	20	2	2	NUM
ejpam-5372	251	21	)	)	PUNCT
ejpam-5372	251	22	is	be	AUX
ejpam-5372	251	23	half	half	DET
ejpam-5372	251	24	that	that	PRON
ejpam-5372	251	25	of	of	ADP
ejpam-5372	251	26	gs	gs	PROPN
ejpam-5372	251	27	.	.	PUNCT
ejpam-5372	251	28	proposition	proposition	NOUN
ejpam-5372	251	29	10	10	NUM
ejpam-5372	251	30	ensures	ensure	VERB
ejpam-5372	251	31	gs	gs	X
ejpam-5372	251	32	⊔	⊔	INTJ
ejpam-5372	251	33	(	(	PUNCT
ejpam-5372	251	34	gs	gs	INTJ
ejpam-5372	251	35	(	(	PUNCT
ejpam-5372	251	36	2	2	NUM
ejpam-5372	251	37	)	)	PUNCT
ejpam-5372	251	38	∪	∪	NOUN
ejpam-5372	251	39	−gs	−gs	NOUN
ejpam-5372	251	40	(	(	PUNCT
ejpam-5372	251	41	2	2	NUM
ejpam-5372	251	42	)	)	PUNCT
ejpam-5372	251	43	)	)	PUNCT
ejpam-5372	252	1	⊂	⊂	PROPN
ejpam-5372	252	2	g′	g′	PROPN
ejpam-5372	252	3	z	z	PROPN
ejpam-5372	252	4	.	.	PUNCT
ejpam-5372	253	1	remains	remain	VERB
ejpam-5372	253	2	to	to	PART
ejpam-5372	253	3	show	show	VERB
ejpam-5372	253	4	that	that	SCONJ
ejpam-5372	253	5	gs	gs	INTJ
ejpam-5372	253	6	(	(	PUNCT
ejpam-5372	253	7	2	2	NUM
ejpam-5372	253	8	)	)	PUNCT
ejpam-5372	253	9	is	be	AUX
ejpam-5372	253	10	disjoint	disjoint	NOUN
ejpam-5372	253	11	of	of	ADP
ejpam-5372	253	12	−gs	−gs	NOUN
ejpam-5372	253	13	(	(	PUNCT
ejpam-5372	253	14	2	2	NUM
ejpam-5372	253	15	)	)	PUNCT
ejpam-5372	254	1	if	if	SCONJ
ejpam-5372	254	2	and	and	CCONJ
ejpam-5372	254	3	only	only	ADV
ejpam-5372	254	4	if	if	SCONJ
ejpam-5372	254	5	p	p	PROPN
ejpam-5372	254	6	∈	∈	PROPN
ejpam-5372	254	7	p9,16	p9,16	NUM
ejpam-5372	254	8	i.e.	i.e.	X
ejpam-5372	254	9	n	n	X
ejpam-5372	254	10	=	=	SYM
ejpam-5372	254	11	3	3	X
ejpam-5372	254	12	.	.	X
ejpam-5372	255	1	if	if	SCONJ
ejpam-5372	255	2	gs	gs	PROPN
ejpam-5372	255	3	(	(	PUNCT
ejpam-5372	255	4	2	2	NUM
ejpam-5372	255	5	)	)	PUNCT
ejpam-5372	255	6	is	be	AUX
ejpam-5372	255	7	not	not	PART
ejpam-5372	255	8	disjoint	disjoint	NOUN
ejpam-5372	255	9	of	of	ADP
ejpam-5372	255	10	−gs	−gs	NOUN
ejpam-5372	255	11	(	(	PUNCT
ejpam-5372	255	12	2	2	NUM
ejpam-5372	255	13	)	)	PUNCT
ejpam-5372	255	14	,	,	PUNCT
ejpam-5372	255	15	there	there	PRON
ejpam-5372	255	16	exists	exist	VERB
ejpam-5372	255	17	g	g	PROPN
ejpam-5372	255	18	∈	∈	PROPN
ejpam-5372	255	19	gz	gz	PROPN
ejpam-5372	255	20	and	and	CCONJ
ejpam-5372	255	21	k	k	PROPN
ejpam-5372	255	22	coprime	coprime	NOUN
ejpam-5372	255	23	to	to	ADP
ejpam-5372	255	24	p	p	NOUN
ejpam-5372	255	25	−	−	PROPN
ejpam-5372	255	26	1	1	NUM
ejpam-5372	255	27	such	such	ADJ
ejpam-5372	255	28	that	that	DET
ejpam-5372	255	29	g4k	g4k	PROPN
ejpam-5372	255	30	≡	≡	PROPN
ejpam-5372	255	31	−g4	−g4	PROPN
ejpam-5372	255	32	,	,	PUNCT
ejpam-5372	255	33	hence	hence	ADV
ejpam-5372	255	34	,	,	PUNCT
ejpam-5372	255	35	by	by	ADP
ejpam-5372	255	36	taking	take	VERB
ejpam-5372	255	37	odd	odd	ADJ
ejpam-5372	255	38	powers	power	NOUN
ejpam-5372	255	39	coprime	coprime	ADV
ejpam-5372	255	40	to	to	ADP
ejpam-5372	255	41	p	p	NOUN
ejpam-5372	255	42	−	−	PROPN
ejpam-5372	255	43	1	1	NUM
ejpam-5372	255	44	,	,	PUNCT
ejpam-5372	255	45	gs	gs	X
ejpam-5372	255	46	(	(	PUNCT
ejpam-5372	255	47	2	2	NUM
ejpam-5372	255	48	)	)	PUNCT
ejpam-5372	255	49	=	=	SYM
ejpam-5372	255	50	−gs	−gs	NOUN
ejpam-5372	255	51	(	(	PUNCT
ejpam-5372	255	52	2	2	NUM
ejpam-5372	255	53	)	)	PUNCT
ejpam-5372	255	54	while	while	SCONJ
ejpam-5372	255	55	on	on	ADP
ejpam-5372	255	56	the	the	DET
ejpam-5372	255	57	other	other	ADJ
ejpam-5372	255	58	hand	hand	NOUN
ejpam-5372	255	59	(	(	PUNCT
ejpam-5372	255	60	g	g	PROPN
ejpam-5372	255	61	k−1	k−1	PROPN
ejpam-5372	255	62	2	2	NUM
ejpam-5372	255	63	)	)	SYM
ejpam-5372	255	64	8	8	NUM
ejpam-5372	255	65	≡	≡	PROPN
ejpam-5372	255	66	−1	−1	NOUN
ejpam-5372	255	67	≡	≡	PROPN
ejpam-5372	255	68	(	(	PUNCT
ejpam-5372	255	69	gz)2	gz)2	PROPN
ejpam-5372	255	70	n−1	n−1	PROPN
ejpam-5372	255	71	which	which	PRON
ejpam-5372	255	72	implies	imply	VERB
ejpam-5372	255	73	n	n	NUM
ejpam-5372	255	74	≥	≥	NUM
ejpam-5372	255	75	4	4	NUM
ejpam-5372	255	76	.	.	PUNCT
ejpam-5372	256	1	conversely	conversely	ADV
ejpam-5372	256	2	if	if	SCONJ
ejpam-5372	256	3	n	n	NUM
ejpam-5372	256	4	≥	≥	NOUN
ejpam-5372	256	5	4	4	NUM
ejpam-5372	256	6	,	,	PUNCT
ejpam-5372	256	7	then	then	ADV
ejpam-5372	256	8	−1	−1	NOUN
ejpam-5372	256	9	has	have	VERB
ejpam-5372	256	10	a	a	DET
ejpam-5372	256	11	square	square	ADJ
ejpam-5372	256	12	root	root	NOUN
ejpam-5372	256	13	which	which	PRON
ejpam-5372	256	14	leaves	leave	VERB
ejpam-5372	256	15	gs	gs	PRON
ejpam-5372	256	16	stable	stable	ADJ
ejpam-5372	256	17	,	,	PUNCT
ejpam-5372	256	18	thus	thus	ADV
ejpam-5372	256	19	gs	gs	X
ejpam-5372	256	20	(	(	PUNCT
ejpam-5372	256	21	2	2	NUM
ejpam-5372	256	22	)	)	PUNCT
ejpam-5372	256	23	=	=	SYM
ejpam-5372	256	24	−gs	−gs	NOUN
ejpam-5372	256	25	(	(	PUNCT
ejpam-5372	256	26	2	2	NUM
ejpam-5372	256	27	)	)	PUNCT
ejpam-5372	256	28	.	.	PUNCT
ejpam-5372	257	1	•	•	NOUN
ejpam-5372	257	2	in	in	ADP
ejpam-5372	257	3	general	general	ADJ
ejpam-5372	257	4	,	,	PUNCT
ejpam-5372	257	5	similarly	similarly	ADV
ejpam-5372	257	6	to	to	ADP
ejpam-5372	257	7	previous	previous	ADJ
ejpam-5372	257	8	items	item	NOUN
ejpam-5372	257	9	,	,	PUNCT
ejpam-5372	257	10	for	for	ADP
ejpam-5372	257	11	any	any	DET
ejpam-5372	257	12	t	t	NOUN
ejpam-5372	257	13	≤	≤	PROPN
ejpam-5372	257	14	n−1	n−1	PROPN
ejpam-5372	257	15	,	,	PUNCT
ejpam-5372	257	16	gs	gs	X
ejpam-5372	257	17	(	(	PUNCT
ejpam-5372	257	18	2	2	NUM
ejpam-5372	257	19	t	t	NOUN
ejpam-5372	257	20	)	)	PUNCT
ejpam-5372	257	21	is	be	AUX
ejpam-5372	257	22	the	the	DET
ejpam-5372	257	23	set	set	NOUN
ejpam-5372	257	24	of	of	ADP
ejpam-5372	257	25	elements	element	NOUN
ejpam-5372	257	26	of	of	ADP
ejpam-5372	257	27	order	order	NOUN
ejpam-5372	257	28	p−1	p−1	PROPN
ejpam-5372	257	29	2t+1	2t+1	PROPN
ejpam-5372	257	30	,	,	PUNCT
ejpam-5372	257	31	and	and	CCONJ
ejpam-5372	257	32	if	if	SCONJ
ejpam-5372	257	33	moreover	moreover	ADV
ejpam-5372	257	34	t	t	X
ejpam-5372	257	35	̸=	̸=	PROPN
ejpam-5372	257	36	n	n	CCONJ
ejpam-5372	257	37	−	−	PROPN
ejpam-5372	257	38	1	1	NUM
ejpam-5372	258	1	then	then	ADV
ejpam-5372	258	2	gs	gs	INTJ
ejpam-5372	258	3	(	(	PUNCT
ejpam-5372	258	4	2	2	NUM
ejpam-5372	258	5	t	t	NOUN
ejpam-5372	258	6	)	)	PUNCT
ejpam-5372	258	7	=	=	SYM
ejpam-5372	259	1	−gs	−gs	NOUN
ejpam-5372	259	2	(	(	PUNCT
ejpam-5372	259	3	2	2	NUM
ejpam-5372	259	4	t	t	NOUN
ejpam-5372	259	5	)	)	PUNCT
ejpam-5372	259	6	thus	thus	ADV
ejpam-5372	259	7	by	by	ADP
ejpam-5372	259	8	induction∣∣∣gs	induction∣∣∣gs	PROPN
ejpam-5372	259	9	(	(	PUNCT
ejpam-5372	259	10	2	2	NUM
ejpam-5372	259	11	t	t	NOUN
ejpam-5372	259	12	)	)	PUNCT
ejpam-5372	259	13	∣∣∣	∣∣∣	NOUN
ejpam-5372	260	1	=	=	SYM
ejpam-5372	260	2	1	1	NUM
ejpam-5372	260	3	2	2	NUM
ejpam-5372	260	4	t	t	NOUN
ejpam-5372	260	5	|gs	|gs	PUNCT
ejpam-5372	260	6	|	|	NOUN
ejpam-5372	260	7	=	=	SYM
ejpam-5372	260	8	1	1	NUM
ejpam-5372	260	9	2t+1	2t+1	NUM
ejpam-5372	260	10	|gz	|gz	NUM
ejpam-5372	261	1	|	|	ADV
ejpam-5372	261	2	.	.	PUNCT
ejpam-5372	262	1	for	for	ADP
ejpam-5372	262	2	t	t	NOUN
ejpam-5372	262	3	=	=	SYM
ejpam-5372	262	4	n	n	CCONJ
ejpam-5372	262	5	−	−	NUM
ejpam-5372	262	6	1	1	NUM
ejpam-5372	262	7	we	we	PRON
ejpam-5372	262	8	have	have	VERB
ejpam-5372	262	9	−gs	−gs	NOUN
ejpam-5372	262	10	(	(	PUNCT
ejpam-5372	262	11	2n−1	2n−1	NUM
ejpam-5372	262	12	)	)	PUNCT
ejpam-5372	263	1	∩	∩	ADJ
ejpam-5372	263	2	gs	gs	X
ejpam-5372	263	3	(	(	PUNCT
ejpam-5372	263	4	2n−1	2n−1	NUM
ejpam-5372	263	5	)	)	PUNCT
ejpam-5372	263	6	=	=	NOUN
ejpam-5372	263	7	∅	∅	NOUN
ejpam-5372	263	8	hence	hence	ADV
ejpam-5372	263	9	∣∣∣gs	∣∣∣gs	X
ejpam-5372	263	10	(	(	PUNCT
ejpam-5372	263	11	2n−1	2n−1	NUM
ejpam-5372	263	12	)	)	PUNCT
ejpam-5372	263	13	∣∣∣	∣∣∣	NOUN
ejpam-5372	263	14	=	=	SYM
ejpam-5372	263	15	1	1	NUM
ejpam-5372	263	16	2n−1	2n−1	NUM
ejpam-5372	263	17	|gz	|gz	X
ejpam-5372	263	18	|	|	ADV
ejpam-5372	263	19	.	.	PUNCT
ejpam-5372	264	1	we	we	PRON
ejpam-5372	264	2	deduce	deduce	VERB
ejpam-5372	264	3	that	that	SCONJ
ejpam-5372	264	4	the	the	DET
ejpam-5372	264	5	sets	set	NOUN
ejpam-5372	264	6	(	(	PUNCT
ejpam-5372	264	7	gs	gs	INTJ
ejpam-5372	264	8	(	(	PUNCT
ejpam-5372	264	9	2	2	NUM
ejpam-5372	264	10	t	t	NOUN
ejpam-5372	264	11	)	)	PUNCT
ejpam-5372	264	12	)	)	PUNCT
ejpam-5372	264	13	0≤t≤n−1	0≤t≤n−1	NUM
ejpam-5372	264	14	are	be	AUX
ejpam-5372	264	15	pairwise	pairwise	NOUN
ejpam-5372	264	16	disjoint	disjoint	NOUN
ejpam-5372	264	17	and	and	CCONJ
ejpam-5372	264	18	their	their	PRON
ejpam-5372	264	19	union	union	NOUN
ejpam-5372	264	20	has	have	VERB
ejpam-5372	264	21	the	the	DET
ejpam-5372	264	22	same	same	ADJ
ejpam-5372	264	23	cardinal	cardinal	NOUN
ejpam-5372	264	24	as	as	ADP
ejpam-5372	264	25	gz	gz	NOUN
ejpam-5372	264	26	.	.	PUNCT
ejpam-5372	265	1	proposition	proposition	NOUN
ejpam-5372	265	2	5	5	NUM
ejpam-5372	265	3	shows	show	VERB
ejpam-5372	265	4	that	that	SCONJ
ejpam-5372	265	5	if	if	SCONJ
ejpam-5372	265	6	r	r	NOUN
ejpam-5372	265	7	∈	∈	NOUN
ejpam-5372	265	8	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	265	9	)	)	PUNCT
ejpam-5372	265	10	,	,	PUNCT
ejpam-5372	265	11	rgs	rgs	PROPN
ejpam-5372	265	12	(	(	PUNCT
ejpam-5372	265	13	2	2	NUM
ejpam-5372	265	14	t	t	NOUN
ejpam-5372	265	15	)	)	PUNCT
ejpam-5372	265	16	⊂	⊂	PROPN
ejpam-5372	265	17	gz	gz	PROPN
ejpam-5372	265	18	.	.	PUNCT
ejpam-5372	266	1	we	we	PRON
ejpam-5372	266	2	must	must	AUX
ejpam-5372	266	3	then	then	ADV
ejpam-5372	266	4	have	have	VERB
ejpam-5372	266	5	g′	g′	NOUN
ejpam-5372	266	6	z	z	NOUN
ejpam-5372	266	7	=	=	SYM
ejpam-5372	266	8	⋃n−1	⋃n−1	NOUN
ejpam-5372	266	9	t=0	t=0	PUNCT
ejpam-5372	266	10	gs	gs	PROPN
ejpam-5372	266	11	(	(	PUNCT
ejpam-5372	266	12	2	2	NUM
ejpam-5372	266	13	t	t	NOUN
ejpam-5372	266	14	)	)	PUNCT
ejpam-5372	266	15	.	.	PUNCT
ejpam-5372	267	1	proposition	proposition	NOUN
ejpam-5372	267	2	5	5	NUM
ejpam-5372	267	3	also	also	ADV
ejpam-5372	267	4	shows	show	VERB
ejpam-5372	267	5	that	that	SCONJ
ejpam-5372	267	6	g′	g′	NOUN
ejpam-5372	267	7	z	z	NOUN
ejpam-5372	267	8	=	=	PUNCT
ejpam-5372	267	9	{	{	PUNCT
ejpam-5372	267	10	g	g	PROPN
ejpam-5372	267	11	∈	∈	PROPN
ejpam-5372	267	12	f∗	f∗	NOUN
ejpam-5372	267	13	p	p	NOUN
ejpam-5372	267	14	∣∣∣	∣∣∣	NOUN
ejpam-5372	267	15	∀q	∀q	PROPN
ejpam-5372	267	16	∈	∈	PROPN
ejpam-5372	267	17	p⧹	p⧹	VERB
ejpam-5372	267	18	{	{	PUNCT
ejpam-5372	267	19	2	2	NUM
ejpam-5372	267	20	}	}	PUNCT
ejpam-5372	267	21	(	(	PUNCT
ejpam-5372	267	22	q|p−	q|p−	PROPN
ejpam-5372	267	23	1	1	NUM
ejpam-5372	267	24	⇒	⇒	NOUN
ejpam-5372	267	25	g	g	PROPN
ejpam-5372	267	26	p−1	p−1	PROPN
ejpam-5372	267	27	2q	2q	NUM
ejpam-5372	267	28	̸≡	̸≡	X
ejpam-5372	267	29	1	1	NUM
ejpam-5372	267	30	)	)	PUNCT
ejpam-5372	267	31	}	}	PUNCT
ejpam-5372	267	32	.	.	PUNCT
ejpam-5372	268	1	m.	m.	NOUN
ejpam-5372	268	2	wolf	wolf	PROPN
ejpam-5372	268	3	,	,	PUNCT
ejpam-5372	268	4	f.	f.	PROPN
ejpam-5372	268	5	wolf	wolf	PROPN
ejpam-5372	268	6	/	/	SYM
ejpam-5372	268	7	eur	eur	PROPN
ejpam-5372	268	8	.	.	PUNCT
ejpam-5372	269	1	j.	j.	PROPN
ejpam-5372	269	2	pure	pure	PROPN
ejpam-5372	269	3	appl	appl	PROPN
ejpam-5372	269	4	.	.	PROPN
ejpam-5372	269	5	math	math	PROPN
ejpam-5372	269	6	,	,	PUNCT
ejpam-5372	269	7	17	17	NUM
ejpam-5372	269	8	(	(	PUNCT
ejpam-5372	269	9	4	4	NUM
ejpam-5372	269	10	)	)	PUNCT
ejpam-5372	269	11	(	(	PUNCT
ejpam-5372	269	12	2024	2024	NUM
ejpam-5372	269	13	)	)	PUNCT
ejpam-5372	269	14	,	,	PUNCT
ejpam-5372	269	15	2431	2431	NUM
ejpam-5372	269	16	-	-	SYM
ejpam-5372	269	17	2447	2447	NUM
ejpam-5372	269	18	2438	2438	NUM
ejpam-5372	269	19	theorem	theorem	NOUN
ejpam-5372	269	20	1	1	NUM
ejpam-5372	269	21	.	.	PUNCT
ejpam-5372	270	1	we	we	PRON
ejpam-5372	270	2	have	have	VERB
ejpam-5372	270	3	:	:	PUNCT
ejpam-5372	270	4	•	•	NUM
ejpam-5372	270	5	g	g	PROPN
ejpam-5372	270	6	∈	∈	PROPN
ejpam-5372	270	7	gz	gz	NOUN
ejpam-5372	270	8	if	if	SCONJ
ejpam-5372	270	9	and	and	CCONJ
ejpam-5372	270	10	only	only	ADV
ejpam-5372	270	11	if	if	SCONJ
ejpam-5372	270	12	there	there	PRON
ejpam-5372	270	13	exists	exist	VERB
ejpam-5372	270	14	m	m	VERB
ejpam-5372	270	15	∈	∈	NOUN
ejpam-5372	270	16	gqr	gqr	VERB
ejpam-5372	271	1	such	such	DET
ejpam-5372	271	2	that	that	SCONJ
ejpam-5372	271	3	∀q	∀q	PROPN
ejpam-5372	271	4	∈	∈	PROPN
ejpam-5372	271	5	p⧹	p⧹	VERB
ejpam-5372	271	6	{	{	PUNCT
ejpam-5372	271	7	2	2	NUM
ejpam-5372	271	8	}	}	PUNCT
ejpam-5372	271	9	q|p−1	q|p−1	NOUN
ejpam-5372	271	10	⇒	⇒	NOUN
ejpam-5372	271	11	m	m	VERB
ejpam-5372	271	12	p−1	p−1	ADJ
ejpam-5372	271	13	2q	2q	NUM
ejpam-5372	271	14	̸≡	̸≡	NOUN
ejpam-5372	271	15	1	1	NUM
ejpam-5372	271	16	and	and	CCONJ
ejpam-5372	271	17	r	r	PROPN
ejpam-5372	271	18	∈	∈	PROPN
ejpam-5372	271	19	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	271	20	)	)	PUNCT
ejpam-5372	271	21	such	such	ADJ
ejpam-5372	271	22	that	that	SCONJ
ejpam-5372	271	23	g	g	PROPN
ejpam-5372	271	24	=	=	SYM
ejpam-5372	271	25	mr	mr	PROPN
ejpam-5372	271	26	.	.	PROPN
ejpam-5372	271	27	•	•	NUM
ejpam-5372	271	28	g	g	PROPN
ejpam-5372	271	29	∈	∈	PROPN
ejpam-5372	271	30	gz	gz	NOUN
ejpam-5372	271	31	if	if	SCONJ
ejpam-5372	271	32	and	and	CCONJ
ejpam-5372	271	33	only	only	ADV
ejpam-5372	271	34	if	if	SCONJ
ejpam-5372	271	35	g	g	PROPN
ejpam-5372	271	36	∈	∈	PROPN
ejpam-5372	271	37	gqnr	gqnr	NOUN
ejpam-5372	271	38	and	and	CCONJ
ejpam-5372	271	39	∀q	∀q	PROPN
ejpam-5372	271	40	∈	∈	PROPN
ejpam-5372	271	41	p⧹	p⧹	VERB
ejpam-5372	271	42	{	{	PUNCT
ejpam-5372	271	43	2	2	NUM
ejpam-5372	271	44	}	}	PUNCT
ejpam-5372	271	45	q|p−	q|p−	VERB
ejpam-5372	271	46	1	1	NUM
ejpam-5372	271	47	⇒	⇒	NOUN
ejpam-5372	271	48	g	g	PROPN
ejpam-5372	271	49	p−1	p−1	PROPN
ejpam-5372	271	50	q	q	PROPN
ejpam-5372	271	51	̸≡	̸≡	PROPN
ejpam-5372	271	52	1	1	NUM
ejpam-5372	271	53	.	.	PUNCT
ejpam-5372	272	1	proof	proof	NOUN
ejpam-5372	272	2	.	.	PUNCT
ejpam-5372	273	1	we	we	PRON
ejpam-5372	273	2	just	just	ADV
ejpam-5372	273	3	need	need	VERB
ejpam-5372	273	4	to	to	PART
ejpam-5372	273	5	prove	prove	VERB
ejpam-5372	273	6	the	the	DET
ejpam-5372	273	7	first	first	ADJ
ejpam-5372	273	8	item	item	NOUN
ejpam-5372	273	9	,	,	PUNCT
ejpam-5372	273	10	the	the	DET
ejpam-5372	273	11	second	second	ADJ
ejpam-5372	273	12	one	one	NOUN
ejpam-5372	273	13	has	have	AUX
ejpam-5372	273	14	already	already	ADV
ejpam-5372	273	15	been	be	AUX
ejpam-5372	273	16	established	establish	VERB
ejpam-5372	273	17	in	in	ADP
ejpam-5372	273	18	proposition	proposition	NOUN
ejpam-5372	273	19	5	5	NUM
ejpam-5372	273	20	.	.	PUNCT
ejpam-5372	274	1	the	the	DET
ejpam-5372	274	2	chinese	chinese	PROPN
ejpam-5372	274	3	theorem	theorem	NOUN
ejpam-5372	274	4	identifies	identify	VERB
ejpam-5372	274	5	f∗	f∗	NOUN
ejpam-5372	274	6	p	p	X
ejpam-5372	274	7	to	to	ADP
ejpam-5372	274	8	(	(	PUNCT
ejpam-5372	274	9	z/2nz	z/2nz	NOUN
ejpam-5372	274	10	)	)	PUNCT
ejpam-5372	274	11	×	×	NOUN
ejpam-5372	274	12	(	(	PUNCT
ejpam-5372	274	13	z	z	PROPN
ejpam-5372	274	14	/	/	SYM
ejpam-5372	274	15	zz	zz	PROPN
ejpam-5372	274	16	)	)	PUNCT
ejpam-5372	274	17	.	.	PUNCT
ejpam-5372	275	1	a	a	DET
ejpam-5372	275	2	primitive	primitive	ADJ
ejpam-5372	275	3	root	root	NOUN
ejpam-5372	275	4	g	g	NOUN
ejpam-5372	275	5	corresponds	correspond	VERB
ejpam-5372	275	6	to	to	ADP
ejpam-5372	275	7	a	a	DET
ejpam-5372	275	8	pair	pair	NOUN
ejpam-5372	275	9	(	(	PUNCT
ejpam-5372	275	10	x	x	NOUN
ejpam-5372	275	11	,	,	PUNCT
ejpam-5372	275	12	y	y	NOUN
ejpam-5372	275	13	)	)	PUNCT
ejpam-5372	275	14	with	with	ADP
ejpam-5372	275	15	x	x	SYM
ejpam-5372	275	16	odd	odd	ADJ
ejpam-5372	275	17	and	and	CCONJ
ejpam-5372	275	18	y	y	PROPN
ejpam-5372	275	19	coprime	coprime	NOUN
ejpam-5372	275	20	to	to	PART
ejpam-5372	275	21	z.	z.	PROPN
ejpam-5372	275	22	let	let	VERB
ejpam-5372	275	23	m	m	PRON
ejpam-5372	275	24	be	be	AUX
ejpam-5372	275	25	the	the	DET
ejpam-5372	275	26	element	element	NOUN
ejpam-5372	275	27	corresponding	correspond	VERB
ejpam-5372	275	28	to	to	ADP
ejpam-5372	275	29	(	(	PUNCT
ejpam-5372	275	30	2x	2x	NUM
ejpam-5372	275	31	,	,	PUNCT
ejpam-5372	275	32	y	y	NOUN
ejpam-5372	275	33	)	)	PUNCT
ejpam-5372	275	34	and	and	CCONJ
ejpam-5372	275	35	r	r	X
ejpam-5372	275	36	the	the	DET
ejpam-5372	275	37	element	element	NOUN
ejpam-5372	275	38	corresponding	correspond	VERB
ejpam-5372	275	39	to	to	ADP
ejpam-5372	275	40	(	(	PUNCT
ejpam-5372	275	41	−x	−x	NOUN
ejpam-5372	275	42	,	,	PUNCT
ejpam-5372	275	43	0	0	NUM
ejpam-5372	275	44	)	)	PUNCT
ejpam-5372	275	45	.	.	PUNCT
ejpam-5372	276	1	then	then	ADV
ejpam-5372	276	2	it	it	PRON
ejpam-5372	276	3	is	be	AUX
ejpam-5372	276	4	clear	clear	ADJ
ejpam-5372	276	5	that	that	SCONJ
ejpam-5372	276	6	g	g	PROPN
ejpam-5372	276	7	=	=	SYM
ejpam-5372	276	8	mr	mr	PROPN
ejpam-5372	276	9	but	but	CCONJ
ejpam-5372	276	10	r	r	NOUN
ejpam-5372	276	11	is	be	AUX
ejpam-5372	276	12	of	of	ADP
ejpam-5372	276	13	order	order	NOUN
ejpam-5372	276	14	2n	2n	NUM
ejpam-5372	276	15	so	so	CCONJ
ejpam-5372	276	16	in	in	ADP
ejpam-5372	276	17	(	(	PUNCT
ejpam-5372	276	18	r(n)⧹r(n−1	r(n)⧹r(n−1	NUM
ejpam-5372	276	19	)	)	PUNCT
ejpam-5372	276	20	)	)	PUNCT
ejpam-5372	276	21	,	,	PUNCT
ejpam-5372	276	22	while	while	SCONJ
ejpam-5372	276	23	the	the	DET
ejpam-5372	276	24	order	order	NOUN
ejpam-5372	276	25	of	of	ADP
ejpam-5372	276	26	m	m	PROPN
ejpam-5372	276	27	is	be	AUX
ejpam-5372	276	28	2n−1z	2n−1z	NUM
ejpam-5372	276	29	i.e.	i.e.	X
ejpam-5372	276	30	m	m	VERB
ejpam-5372	276	31	is	be	AUX
ejpam-5372	276	32	a	a	DET
ejpam-5372	276	33	semi	semi	ADJ
ejpam-5372	276	34	-	-	ADJ
ejpam-5372	276	35	primitive	primitive	ADJ
ejpam-5372	276	36	root	root	NOUN
ejpam-5372	276	37	and	and	CCONJ
ejpam-5372	276	38	in	in	ADP
ejpam-5372	276	39	particular	particular	ADJ
ejpam-5372	276	40	m	m	PROPN
ejpam-5372	276	41	∈	∈	NOUN
ejpam-5372	276	42	gqr	gqr	VERB
ejpam-5372	276	43	hence	hence	ADV
ejpam-5372	276	44	∀q	∀q	PROPN
ejpam-5372	276	45	∈	∈	PROPN
ejpam-5372	276	46	p⧹	p⧹	VERB
ejpam-5372	276	47	{	{	PUNCT
ejpam-5372	276	48	2	2	NUM
ejpam-5372	276	49	}	}	PUNCT
ejpam-5372	276	50	q|p−	q|p−	VERB
ejpam-5372	276	51	1	1	NUM
ejpam-5372	276	52	⇒	⇒	NOUN
ejpam-5372	276	53	m	m	VERB
ejpam-5372	276	54	p−1	p−1	NOUN
ejpam-5372	276	55	2q	2q	NUM
ejpam-5372	276	56	̸≡	̸≡	VERB
ejpam-5372	276	57	1	1	X
ejpam-5372	276	58	.	.	PUNCT
ejpam-5372	276	59	conversely	conversely	ADV
ejpam-5372	276	60	,	,	PUNCT
ejpam-5372	276	61	using	use	VERB
ejpam-5372	276	62	this	this	DET
ejpam-5372	276	63	identification	identification	NOUN
ejpam-5372	276	64	,	,	PUNCT
ejpam-5372	276	65	m	m	VERB
ejpam-5372	276	66	corresponds	correspond	VERB
ejpam-5372	276	67	to	to	ADP
ejpam-5372	276	68	(	(	PUNCT
ejpam-5372	276	69	x	x	NOUN
ejpam-5372	276	70	,	,	PUNCT
ejpam-5372	276	71	y	y	NOUN
ejpam-5372	276	72	)	)	PUNCT
ejpam-5372	276	73	with	with	ADP
ejpam-5372	276	74	x	x	SYM
ejpam-5372	276	75	even	even	ADV
ejpam-5372	276	76	and	and	CCONJ
ejpam-5372	276	77	y	y	PROPN
ejpam-5372	276	78	coprime	coprime	NOUN
ejpam-5372	276	79	to	to	ADP
ejpam-5372	276	80	z	z	NOUN
ejpam-5372	276	81	,	,	PUNCT
ejpam-5372	276	82	and	and	CCONJ
ejpam-5372	276	83	r	r	NOUN
ejpam-5372	276	84	corresponds	correspond	VERB
ejpam-5372	276	85	to	to	ADP
ejpam-5372	276	86	(	(	PUNCT
ejpam-5372	276	87	t	t	PROPN
ejpam-5372	276	88	,	,	PUNCT
ejpam-5372	276	89	0	0	NUM
ejpam-5372	276	90	)	)	PUNCT
ejpam-5372	276	91	with	with	ADP
ejpam-5372	276	92	t	t	PROPN
ejpam-5372	276	93	odd	odd	ADJ
ejpam-5372	276	94	.	.	PUNCT
ejpam-5372	277	1	thus	thus	ADV
ejpam-5372	277	2	mr	mr	PROPN
ejpam-5372	277	3	corresponds	correspond	VERB
ejpam-5372	277	4	to	to	ADP
ejpam-5372	277	5	(	(	PUNCT
ejpam-5372	277	6	x+	x+	PROPN
ejpam-5372	277	7	t	t	PROPN
ejpam-5372	277	8	,	,	PUNCT
ejpam-5372	277	9	y	y	PROPN
ejpam-5372	277	10	)	)	PUNCT
ejpam-5372	277	11	which	which	PRON
ejpam-5372	277	12	is	be	AUX
ejpam-5372	277	13	indeed	indeed	ADV
ejpam-5372	277	14	a	a	DET
ejpam-5372	277	15	generator	generator	NOUN
ejpam-5372	277	16	of	of	ADP
ejpam-5372	277	17	(	(	PUNCT
ejpam-5372	277	18	z/2nz)×	z/2nz)×	X
ejpam-5372	277	19	(	(	PUNCT
ejpam-5372	277	20	z	z	NOUN
ejpam-5372	277	21	/	/	SYM
ejpam-5372	277	22	zz	zz	PROPN
ejpam-5372	277	23	)	)	PUNCT
ejpam-5372	277	24	because	because	SCONJ
ejpam-5372	277	25	x+	x+	PROPN
ejpam-5372	277	26	t	t	PROPN
ejpam-5372	277	27	is	be	AUX
ejpam-5372	277	28	odd	odd	ADJ
ejpam-5372	277	29	and	and	CCONJ
ejpam-5372	277	30	y	y	PROPN
ejpam-5372	277	31	coprime	coprime	NOUN
ejpam-5372	277	32	to	to	ADP
ejpam-5372	277	33	z.	z.	PROPN
ejpam-5372	277	34	2.2.2	2.2.2	NUM
ejpam-5372	277	35	.	.	PUNCT
ejpam-5372	278	1	a	a	DET
ejpam-5372	278	2	sequence	sequence	NOUN
ejpam-5372	278	3	of	of	ADP
ejpam-5372	278	4	primitive	primitive	ADJ
ejpam-5372	278	5	and	and	CCONJ
ejpam-5372	278	6	semi	semi	ADJ
ejpam-5372	278	7	-	-	ADJ
ejpam-5372	278	8	primitive	primitive	ADJ
ejpam-5372	278	9	roots	root	NOUN
ejpam-5372	278	10	knowing	know	VERB
ejpam-5372	278	11	a	a	DET
ejpam-5372	278	12	single	single	ADJ
ejpam-5372	278	13	primitive	primitive	ADJ
ejpam-5372	278	14	root	root	NOUN
ejpam-5372	278	15	allows	allow	VERB
ejpam-5372	278	16	to	to	PART
ejpam-5372	278	17	obtain	obtain	VERB
ejpam-5372	278	18	every	every	DET
ejpam-5372	278	19	element	element	NOUN
ejpam-5372	278	20	of	of	ADP
ejpam-5372	278	21	gz	gz	NOUN
ejpam-5372	278	22	by	by	ADP
ejpam-5372	278	23	exponentiation	exponentiation	NOUN
ejpam-5372	278	24	.	.	PUNCT
ejpam-5372	279	1	knowing	know	VERB
ejpam-5372	279	2	a	a	DET
ejpam-5372	279	3	semi	semi	ADJ
ejpam-5372	279	4	-	-	ADJ
ejpam-5372	279	5	primitive	primitive	ADJ
ejpam-5372	279	6	root	root	NOUN
ejpam-5372	279	7	and	and	CCONJ
ejpam-5372	279	8	an	an	DET
ejpam-5372	279	9	element	element	NOUN
ejpam-5372	279	10	of	of	ADP
ejpam-5372	279	11	r(n)⧹r(n−1	r(n)⧹r(n−1	PROPN
ejpam-5372	279	12	)	)	PUNCT
ejpam-5372	279	13	also	also	ADV
ejpam-5372	279	14	allows	allow	VERB
ejpam-5372	279	15	to	to	PART
ejpam-5372	279	16	find	find	VERB
ejpam-5372	279	17	one	one	NUM
ejpam-5372	279	18	thus	thus	ADV
ejpam-5372	279	19	every	every	DET
ejpam-5372	279	20	primitive	primitive	ADJ
ejpam-5372	279	21	root	root	NOUN
ejpam-5372	279	22	.	.	PUNCT
ejpam-5372	280	1	we	we	PRON
ejpam-5372	280	2	describe	describe	VERB
ejpam-5372	280	3	here	here	ADV
ejpam-5372	280	4	an	an	DET
ejpam-5372	280	5	alternative	alternative	ADJ
ejpam-5372	280	6	method	method	NOUN
ejpam-5372	280	7	for	for	ADP
ejpam-5372	280	8	generating	generate	VERB
ejpam-5372	280	9	primitive	primitive	ADJ
ejpam-5372	280	10	roots	root	NOUN
ejpam-5372	280	11	.	.	PUNCT
ejpam-5372	281	1	corollary	corollary	ADJ
ejpam-5372	281	2	2	2	NUM
ejpam-5372	281	3	.	.	PUNCT
ejpam-5372	282	1	let	let	VERB
ejpam-5372	282	2	m	m	PRON
ejpam-5372	282	3	∈	∈	VERB
ejpam-5372	282	4	gs	gs	NOUN
ejpam-5372	283	1	and	and	CCONJ
ejpam-5372	283	2	r	r	PROPN
ejpam-5372	283	3	∈	∈	PROPN
ejpam-5372	283	4	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	283	5	)	)	PUNCT
ejpam-5372	283	6	fixed	fix	VERB
ejpam-5372	283	7	,	,	PUNCT
ejpam-5372	283	8	then	then	ADV
ejpam-5372	283	9	:	:	PUNCT
ejpam-5372	283	10	∀k	∀k	NOUN
ejpam-5372	283	11	∈	∈	PROPN
ejpam-5372	283	12	n∗	n∗	VERB
ejpam-5372	283	13	∀a	∀a	X
ejpam-5372	283	14	∈	∈	PROPN
ejpam-5372	283	15	p⧹dp	p⧹dp	PROPN
ejpam-5372	283	16	(	(	PUNCT
ejpam-5372	283	17	z	z	X
ejpam-5372	283	18	)	)	PUNCT
ejpam-5372	283	19	makr	makr	PROPN
ejpam-5372	283	20	∈	∈	PROPN
ejpam-5372	283	21	gz	gz	PROPN
ejpam-5372	283	22	.	.	PUNCT
ejpam-5372	283	23	proof	proof	NOUN
ejpam-5372	283	24	.	.	PUNCT
ejpam-5372	284	1	we	we	PRON
ejpam-5372	284	2	apply	apply	VERB
ejpam-5372	284	3	the	the	DET
ejpam-5372	284	4	first	first	ADJ
ejpam-5372	284	5	item	item	NOUN
ejpam-5372	284	6	of	of	ADP
ejpam-5372	284	7	theorem	theorem	NOUN
ejpam-5372	284	8	1	1	NUM
ejpam-5372	284	9	.	.	PUNCT
ejpam-5372	285	1	if	if	SCONJ
ejpam-5372	285	2	q	q	PROPN
ejpam-5372	285	3	∈	∈	PROPN
ejpam-5372	285	4	p⧹	p⧹	VERB
ejpam-5372	285	5	{	{	PUNCT
ejpam-5372	285	6	2	2	NUM
ejpam-5372	285	7	}	}	PUNCT
ejpam-5372	285	8	divides	divide	VERB
ejpam-5372	285	9	p	p	NOUN
ejpam-5372	285	10	−	−	PROPN
ejpam-5372	285	11	1	1	NUM
ejpam-5372	285	12	,	,	PUNCT
ejpam-5372	285	13	then	then	ADV
ejpam-5372	285	14	m	m	VERB
ejpam-5372	285	15	ak(p−1	ak(p−1	ADJ
ejpam-5372	285	16	)	)	PUNCT
ejpam-5372	285	17	2q	2q	NOUN
ejpam-5372	285	18	̸≡	̸≡	NOUN
ejpam-5372	286	1	1	1	NUM
ejpam-5372	286	2	otherwise	otherwise	ADV
ejpam-5372	286	3	p−1	p−1	PROPN
ejpam-5372	286	4	2	2	NUM
ejpam-5372	286	5	would	would	AUX
ejpam-5372	286	6	divide	divide	VERB
ejpam-5372	286	7	ak(p−1	ak(p−1	NOUN
ejpam-5372	286	8	)	)	PUNCT
ejpam-5372	286	9	2q	2q	NOUN
ejpam-5372	286	10	,	,	PUNCT
ejpam-5372	286	11	hence	hence	ADV
ejpam-5372	286	12	by	by	ADP
ejpam-5372	286	13	gauss	gauss	PROPN
ejpam-5372	286	14	’s	’s	PART
ejpam-5372	286	15	theorem	theorem	NOUN
ejpam-5372	286	16	p−1	p−1	PROPN
ejpam-5372	286	17	2	2	NUM
ejpam-5372	286	18	would	would	AUX
ejpam-5372	286	19	divide	divide	VERB
ejpam-5372	286	20	p−1	p−1	PROPN
ejpam-5372	286	21	2q	2q	NUM
ejpam-5372	286	22	which	which	PRON
ejpam-5372	286	23	is	be	AUX
ejpam-5372	286	24	impossible	impossible	ADJ
ejpam-5372	286	25	.	.	PUNCT
ejpam-5372	287	1	definition	definition	NOUN
ejpam-5372	287	2	6	6	NUM
ejpam-5372	287	3	.	.	PUNCT
ejpam-5372	288	1	let	let	VERB
ejpam-5372	288	2	a	a	DET
ejpam-5372	288	3	∈	∈	NOUN
ejpam-5372	288	4	p	p	NOUN
ejpam-5372	288	5	coprime	coprime	NOUN
ejpam-5372	288	6	to	to	ADP
ejpam-5372	288	7	z	z	PROPN
ejpam-5372	288	8	,	,	PUNCT
ejpam-5372	288	9	the	the	DET
ejpam-5372	288	10	discrete	discrete	ADJ
ejpam-5372	288	11	logarithm	logarithm	NOUN
ejpam-5372	288	12	of	of	ADP
ejpam-5372	288	13	1	1	NUM
ejpam-5372	288	14	to	to	ADP
ejpam-5372	288	15	the	the	DET
ejpam-5372	288	16	base	base	NOUN
ejpam-5372	288	17	a	a	DET
ejpam-5372	288	18	modulo	modulo	NOUN
ejpam-5372	288	19	z	z	NOUN
ejpam-5372	288	20	,	,	PUNCT
ejpam-5372	288	21	denoted	denote	VERB
ejpam-5372	288	22	by	by	ADP
ejpam-5372	288	23	logda	logda	NOUN
ejpam-5372	288	24	(	(	PUNCT
ejpam-5372	288	25	1	1	NUM
ejpam-5372	288	26	,	,	PUNCT
ejpam-5372	288	27	z	z	NOUN
ejpam-5372	288	28	)	)	PUNCT
ejpam-5372	288	29	,	,	PUNCT
ejpam-5372	288	30	is	be	AUX
ejpam-5372	288	31	the	the	DET
ejpam-5372	288	32	multiplicative	multiplicative	ADJ
ejpam-5372	288	33	order	order	NOUN
ejpam-5372	288	34	of	of	ADP
ejpam-5372	288	35	a	a	DET
ejpam-5372	288	36	modulo	modulo	NOUN
ejpam-5372	288	37	z	z	NOUN
ejpam-5372	288	38	,	,	PUNCT
ejpam-5372	288	39	i.e.	i.e.	X
ejpam-5372	288	40	the	the	DET
ejpam-5372	288	41	smallest	small	ADJ
ejpam-5372	288	42	integer	integer	NOUN
ejpam-5372	288	43	k	k	PROPN
ejpam-5372	288	44	such	such	ADJ
ejpam-5372	288	45	that	that	SCONJ
ejpam-5372	288	46	ak	ak	PROPN
ejpam-5372	288	47	≡	≡	PROPN
ejpam-5372	288	48	1	1	NUM
ejpam-5372	289	1	[	[	X
ejpam-5372	289	2	z	z	X
ejpam-5372	289	3	]	]	PUNCT
ejpam-5372	289	4	.	.	PUNCT
ejpam-5372	290	1	proposition	proposition	NOUN
ejpam-5372	290	2	12	12	NUM
ejpam-5372	290	3	.	.	PUNCT
ejpam-5372	291	1	let	let	VERB
ejpam-5372	291	2	m	m	PRON
ejpam-5372	291	3	∈	∈	PROPN
ejpam-5372	291	4	gs	gs	PROPN
ejpam-5372	291	5	,	,	PUNCT
ejpam-5372	291	6	b	b	X
ejpam-5372	291	7	=	=	SYM
ejpam-5372	291	8	m2n−1	m2n−1	PROPN
ejpam-5372	291	9	and	and	CCONJ
ejpam-5372	291	10	k	k	NOUN
ejpam-5372	291	11	=	=	SYM
ejpam-5372	292	1	logd2	logd2	NOUN
ejpam-5372	292	2	(	(	PUNCT
ejpam-5372	292	3	1	1	NUM
ejpam-5372	292	4	,	,	PUNCT
ejpam-5372	292	5	z	z	NOUN
ejpam-5372	292	6	)	)	PUNCT
ejpam-5372	292	7	.	.	PUNCT
ejpam-5372	293	1	then	then	ADV
ejpam-5372	293	2	:	:	PUNCT
ejpam-5372	293	3	b2	b2	PROPN
ejpam-5372	293	4	k	k	PROPN
ejpam-5372	293	5	≡	≡	PROPN
ejpam-5372	293	6	b	b	PROPN
ejpam-5372	294	1	[	[	X
ejpam-5372	294	2	p	p	X
ejpam-5372	294	3	]	]	PUNCT
ejpam-5372	294	4	.	.	PUNCT
ejpam-5372	295	1	proof	proof	NOUN
ejpam-5372	295	2	.	.	PUNCT
ejpam-5372	296	1	by	by	ADP
ejpam-5372	296	2	definition	definition	NOUN
ejpam-5372	296	3	2k	2k	NUM
ejpam-5372	296	4	≡	≡	PROPN
ejpam-5372	296	5	1	1	NUM
ejpam-5372	297	1	[	[	X
ejpam-5372	297	2	z	z	X
ejpam-5372	297	3	]	]	X
ejpam-5372	297	4	.	.	PUNCT
ejpam-5372	298	1	we	we	PRON
ejpam-5372	298	2	also	also	ADV
ejpam-5372	298	3	know	know	VERB
ejpam-5372	298	4	that	that	PRON
ejpam-5372	298	5	b	b	X
ejpam-5372	298	6	=	=	SYM
ejpam-5372	298	7	m2n−1	m2n−1	PROPN
ejpam-5372	298	8	is	be	AUX
ejpam-5372	298	9	of	of	ADP
ejpam-5372	298	10	order	order	NOUN
ejpam-5372	298	11	z	z	NOUN
ejpam-5372	298	12	modulo	modulo	VERB
ejpam-5372	298	13	p	p	X
ejpam-5372	298	14	,	,	PUNCT
ejpam-5372	298	15	so	so	ADV
ejpam-5372	298	16	b2	b2	PROPN
ejpam-5372	298	17	k	k	PROPN
ejpam-5372	298	18	≡	≡	PROPN
ejpam-5372	298	19	b1	b1	PROPN
ejpam-5372	298	20	≡	≡	PROPN
ejpam-5372	298	21	b.	b.	PROPN
ejpam-5372	298	22	m.	m.	PROPN
ejpam-5372	298	23	wolf	wolf	PROPN
ejpam-5372	298	24	,	,	PUNCT
ejpam-5372	298	25	f.	f.	PROPN
ejpam-5372	298	26	wolf	wolf	PROPN
ejpam-5372	298	27	/	/	SYM
ejpam-5372	298	28	eur	eur	PROPN
ejpam-5372	298	29	.	.	PUNCT
ejpam-5372	299	1	j.	j.	PROPN
ejpam-5372	299	2	pure	pure	PROPN
ejpam-5372	299	3	appl	appl	PROPN
ejpam-5372	299	4	.	.	PROPN
ejpam-5372	299	5	math	math	PROPN
ejpam-5372	299	6	,	,	PUNCT
ejpam-5372	299	7	17	17	NUM
ejpam-5372	299	8	(	(	PUNCT
ejpam-5372	299	9	4	4	NUM
ejpam-5372	299	10	)	)	PUNCT
ejpam-5372	299	11	(	(	PUNCT
ejpam-5372	299	12	2024	2024	NUM
ejpam-5372	299	13	)	)	PUNCT
ejpam-5372	299	14	,	,	PUNCT
ejpam-5372	299	15	2431	2431	NUM
ejpam-5372	299	16	-	-	SYM
ejpam-5372	299	17	2447	2447	NUM
ejpam-5372	299	18	2439	2439	NUM
ejpam-5372	299	19	remark	remark	NOUN
ejpam-5372	299	20	1	1	NUM
ejpam-5372	299	21	.	.	NOUN
ejpam-5372	300	1	•	•	NOUN
ejpam-5372	301	1	there	there	PRON
ejpam-5372	301	2	are	be	VERB
ejpam-5372	301	3	several	several	ADJ
ejpam-5372	301	4	algorithms	algorithm	NOUN
ejpam-5372	301	5	to	to	PART
ejpam-5372	301	6	calculate	calculate	VERB
ejpam-5372	301	7	the	the	DET
ejpam-5372	301	8	discrete	discrete	ADJ
ejpam-5372	301	9	logarithm	logarithm	NOUN
ejpam-5372	301	10	,	,	PUNCT
ejpam-5372	301	11	all	all	PRON
ejpam-5372	301	12	of	of	ADP
ejpam-5372	301	13	which	which	PRON
ejpam-5372	301	14	take	take	VERB
ejpam-5372	301	15	subexponential	subexponential	ADJ
ejpam-5372	301	16	time	time	NOUN
ejpam-5372	301	17	.	.	PUNCT
ejpam-5372	302	1	see	see	VERB
ejpam-5372	302	2	for	for	ADP
ejpam-5372	302	3	example	example	NOUN
ejpam-5372	302	4	pollard	pollard	PROPN
ejpam-5372	302	5	’s	’s	PART
ejpam-5372	302	6	rho	rho	ADJ
ejpam-5372	302	7	algorithm	algorithm	NOUN
ejpam-5372	302	8	,	,	PUNCT
ejpam-5372	302	9	pohlig	pohlig	ADJ
ejpam-5372	302	10	-	-	PUNCT
ejpam-5372	302	11	hellman	hellman	NOUN
ejpam-5372	302	12	algorithm	algorithm	NOUN
ejpam-5372	302	13	or	or	CCONJ
ejpam-5372	302	14	the	the	DET
ejpam-5372	302	15	general	general	ADJ
ejpam-5372	302	16	number	number	NOUN
ejpam-5372	302	17	field	field	NOUN
ejpam-5372	302	18	sieve	sieve	NOUN
ejpam-5372	302	19	(	(	PUNCT
ejpam-5372	302	20	gnfs	gnfs	NOUN
ejpam-5372	302	21	)	)	PUNCT
ejpam-5372	302	22	.	.	PUNCT
ejpam-5372	303	1	•	•	NUM
ejpam-5372	303	2	searching	search	VERB
ejpam-5372	303	3	for	for	ADP
ejpam-5372	303	4	the	the	DET
ejpam-5372	303	5	discrete	discrete	ADJ
ejpam-5372	303	6	logarithm	logarithm	NOUN
ejpam-5372	303	7	to	to	ADP
ejpam-5372	303	8	the	the	DET
ejpam-5372	303	9	base	base	NOUN
ejpam-5372	303	10	2	2	NUM
ejpam-5372	303	11	is	be	AUX
ejpam-5372	303	12	adapted	adapt	VERB
ejpam-5372	303	13	to	to	ADP
ejpam-5372	303	14	the	the	DET
ejpam-5372	303	15	binary	binary	ADJ
ejpam-5372	303	16	system	system	NOUN
ejpam-5372	303	17	used	use	VERB
ejpam-5372	303	18	by	by	ADP
ejpam-5372	303	19	the	the	DET
ejpam-5372	303	20	computer	computer	NOUN
ejpam-5372	303	21	.	.	PUNCT
ejpam-5372	304	1	proposition	proposition	NOUN
ejpam-5372	304	2	13	13	NUM
ejpam-5372	304	3	.	.	PUNCT
ejpam-5372	305	1	let	let	VERB
ejpam-5372	305	2	a	a	DET
ejpam-5372	305	3	be	be	AUX
ejpam-5372	305	4	coprime	coprime	ADJ
ejpam-5372	305	5	to	to	ADP
ejpam-5372	305	6	z.	z.	PROPN
ejpam-5372	305	7	then	then	ADV
ejpam-5372	305	8	we	we	PRON
ejpam-5372	305	9	have	have	VERB
ejpam-5372	305	10	:	:	PUNCT
ejpam-5372	305	11	logda	logda	NOUN
ejpam-5372	305	12	(	(	PUNCT
ejpam-5372	305	13	1	1	NUM
ejpam-5372	305	14	,	,	PUNCT
ejpam-5372	305	15	z	z	NOUN
ejpam-5372	305	16	)	)	PUNCT
ejpam-5372	305	17	≤	≤	NOUN
ejpam-5372	305	18	lcm	lcm	NOUN
ejpam-5372	305	19	q∈dp(p−1	q∈dp(p−1	NOUN
ejpam-5372	305	20	)	)	PUNCT
ejpam-5372	306	1	qαq−1	qαq−1	NOUN
ejpam-5372	306	2	(	(	PUNCT
ejpam-5372	306	3	q	q	PROPN
ejpam-5372	306	4	−	−	PROPN
ejpam-5372	306	5	1	1	NUM
ejpam-5372	306	6	)	)	PUNCT
ejpam-5372	306	7	.	.	PUNCT
ejpam-5372	307	1	proof	proof	NOUN
ejpam-5372	307	2	.	.	PUNCT
ejpam-5372	308	1	the	the	DET
ejpam-5372	308	2	chinese	chinese	PROPN
ejpam-5372	308	3	theorem	theorem	NOUN
ejpam-5372	308	4	gives	give	VERB
ejpam-5372	308	5	z	z	PROPN
ejpam-5372	308	6	/	/	SYM
ejpam-5372	308	7	zz	zz	PROPN
ejpam-5372	308	8	≃	≃	PROPN
ejpam-5372	308	9	∏	∏	PROPN
ejpam-5372	308	10	q∈dp(p−1	q∈dp(p−1	NOUN
ejpam-5372	308	11	)	)	PUNCT
ejpam-5372	308	12	z	z	NOUN
ejpam-5372	308	13	/	/	SYM
ejpam-5372	308	14	qαqz	qαqz	NOUN
ejpam-5372	308	15	.	.	PUNCT
ejpam-5372	309	1	therefore	therefore	ADV
ejpam-5372	309	2	the	the	DET
ejpam-5372	309	3	unit	unit	NOUN
ejpam-5372	309	4	group	group	NOUN
ejpam-5372	309	5	(	(	PUNCT
ejpam-5372	309	6	z	z	NOUN
ejpam-5372	309	7	/	/	SYM
ejpam-5372	309	8	zz)×	zz)×	PROPN
ejpam-5372	309	9	can	can	AUX
ejpam-5372	309	10	be	be	AUX
ejpam-5372	309	11	identified	identify	VERB
ejpam-5372	309	12	to	to	ADP
ejpam-5372	309	13	∏	∏	NUM
ejpam-5372	309	14	q∈dp(p−1	q∈dp(p−1	NOUN
ejpam-5372	309	15	)	)	PUNCT
ejpam-5372	309	16	(	(	PUNCT
ejpam-5372	309	17	z	z	X
ejpam-5372	309	18	/	/	SYM
ejpam-5372	309	19	qαqz)×	qαqz)×	NOUN
ejpam-5372	309	20	and	and	CCONJ
ejpam-5372	309	21	any	any	DET
ejpam-5372	309	22	element	element	NOUN
ejpam-5372	309	23	of	of	ADP
ejpam-5372	309	24	this	this	DET
ejpam-5372	309	25	group	group	NOUN
ejpam-5372	309	26	is	be	AUX
ejpam-5372	309	27	of	of	ADP
ejpam-5372	309	28	order	order	NOUN
ejpam-5372	309	29	at	at	ADP
ejpam-5372	309	30	most	most	ADJ
ejpam-5372	309	31	lcmq∈dp(p−1	lcmq∈dp(p−1	NOUN
ejpam-5372	309	32	)	)	PUNCT
ejpam-5372	309	33	∣∣(z	∣∣(z	X
ejpam-5372	309	34	/	/	SYM
ejpam-5372	309	35	qαqz)×	qαqz)×	X
ejpam-5372	309	36	∣∣	∣∣	X
ejpam-5372	309	37	=	=	SYM
ejpam-5372	309	38	lcmq∈dp(p−1	lcmq∈dp(p−1	NOUN
ejpam-5372	309	39	)	)	PUNCT
ejpam-5372	310	1	q	q	PROPN
ejpam-5372	311	1	αq−1	αq−1	INTJ
ejpam-5372	311	2	(	(	PUNCT
ejpam-5372	311	3	q	q	NOUN
ejpam-5372	311	4	−	−	PROPN
ejpam-5372	311	5	1	1	NUM
ejpam-5372	311	6	)	)	PUNCT
ejpam-5372	311	7	.	.	PUNCT
ejpam-5372	312	1	the	the	DET
ejpam-5372	312	2	same	same	ADJ
ejpam-5372	312	3	applies	apply	VERB
ejpam-5372	312	4	to	to	ADP
ejpam-5372	312	5	logda	logda	NOUN
ejpam-5372	312	6	(	(	PUNCT
ejpam-5372	312	7	1	1	NUM
ejpam-5372	312	8	,	,	PUNCT
ejpam-5372	312	9	z	z	NOUN
ejpam-5372	312	10	)	)	PUNCT
ejpam-5372	312	11	which	which	PRON
ejpam-5372	312	12	is	be	AUX
ejpam-5372	312	13	the	the	DET
ejpam-5372	312	14	multiplicative	multiplicative	ADJ
ejpam-5372	312	15	order	order	NOUN
ejpam-5372	312	16	of	of	ADP
ejpam-5372	312	17	a	a	PRON
ejpam-5372	312	18	in	in	ADP
ejpam-5372	312	19	z	z	PROPN
ejpam-5372	312	20	/	/	SYM
ejpam-5372	312	21	zz	zz	PROPN
ejpam-5372	312	22	.	.	PUNCT
ejpam-5372	312	23	proposition	proposition	NOUN
ejpam-5372	312	24	14	14	NUM
ejpam-5372	312	25	.	.	PUNCT
ejpam-5372	313	1	if	if	SCONJ
ejpam-5372	313	2	m	m	PROPN
ejpam-5372	313	3	∈	∈	PROPN
ejpam-5372	313	4	gz	gz	NOUN
ejpam-5372	313	5	(	(	PUNCT
ejpam-5372	313	6	respectively	respectively	ADV
ejpam-5372	313	7	gs	gs	PROPN
ejpam-5372	313	8	,	,	PUNCT
ejpam-5372	313	9	gqr	gqr	PROPN
ejpam-5372	313	10	,	,	PUNCT
ejpam-5372	313	11	gnqr	gnqr	NOUN
ejpam-5372	313	12	)	)	PUNCT
ejpam-5372	313	13	then	then	ADV
ejpam-5372	313	14	mp−2	mp−2	NOUN
ejpam-5372	313	15	∈	∈	PROPN
ejpam-5372	313	16	gz	gz	PROPN
ejpam-5372	313	17	(	(	PUNCT
ejpam-5372	313	18	respectively	respectively	ADV
ejpam-5372	313	19	gs	gs	PROPN
ejpam-5372	313	20	,	,	PUNCT
ejpam-5372	313	21	gqr	gqr	PROPN
ejpam-5372	313	22	,	,	PUNCT
ejpam-5372	313	23	gnqr	gnqr	NOUN
ejpam-5372	313	24	)	)	PUNCT
ejpam-5372	313	25	.	.	PUNCT
ejpam-5372	314	1	proof	proof	NOUN
ejpam-5372	314	2	.	.	PUNCT
ejpam-5372	315	1	mp−2	mp−2	NOUN
ejpam-5372	315	2	is	be	AUX
ejpam-5372	315	3	the	the	DET
ejpam-5372	315	4	multiplicative	multiplicative	ADJ
ejpam-5372	315	5	inverse	inverse	NOUN
ejpam-5372	315	6	of	of	ADP
ejpam-5372	315	7	m	m	PROPN
ejpam-5372	315	8	in	in	ADP
ejpam-5372	315	9	f∗	f∗	NOUN
ejpam-5372	315	10	p.	p.	NOUN
ejpam-5372	315	11	as	as	ADP
ejpam-5372	315	12	the	the	DET
ejpam-5372	315	13	inverse	inverse	NOUN
ejpam-5372	315	14	of	of	ADP
ejpam-5372	315	15	a	a	DET
ejpam-5372	315	16	generator	generator	NOUN
ejpam-5372	315	17	is	be	AUX
ejpam-5372	315	18	a	a	DET
ejpam-5372	315	19	generator	generator	NOUN
ejpam-5372	315	20	,	,	PUNCT
ejpam-5372	315	21	gz	gz	NOUN
ejpam-5372	315	22	and	and	CCONJ
ejpam-5372	315	23	gs	gs	PROPN
ejpam-5372	315	24	are	be	AUX
ejpam-5372	315	25	stable	stable	ADJ
ejpam-5372	315	26	by	by	ADP
ejpam-5372	315	27	conversion	conversion	NOUN
ejpam-5372	315	28	to	to	ADP
ejpam-5372	315	29	the	the	DET
ejpam-5372	315	30	inverse	inverse	NOUN
ejpam-5372	315	31	.	.	PUNCT
ejpam-5372	316	1	moreover	moreover	ADV
ejpam-5372	316	2	,	,	PUNCT
ejpam-5372	316	3	gqr	gqr	PROPN
ejpam-5372	316	4	is	be	AUX
ejpam-5372	316	5	a	a	DET
ejpam-5372	316	6	multiplicative	multiplicative	ADJ
ejpam-5372	316	7	subgroup	subgroup	NOUN
ejpam-5372	316	8	and	and	CCONJ
ejpam-5372	316	9	gnqr	gnqr	NOUN
ejpam-5372	316	10	its	its	PRON
ejpam-5372	316	11	complementary	complementary	ADJ
ejpam-5372	316	12	,	,	PUNCT
ejpam-5372	316	13	so	so	SCONJ
ejpam-5372	316	14	it	it	PRON
ejpam-5372	316	15	’s	’	VERB
ejpam-5372	316	16	the	the	DET
ejpam-5372	316	17	same	same	ADJ
ejpam-5372	316	18	for	for	ADP
ejpam-5372	316	19	these	these	DET
ejpam-5372	316	20	two	two	NUM
ejpam-5372	316	21	sets	set	NOUN
ejpam-5372	316	22	.	.	PUNCT
ejpam-5372	317	1	corollary	corollary	ADJ
ejpam-5372	317	2	3	3	NUM
ejpam-5372	317	3	.	.	PUNCT
ejpam-5372	318	1	for	for	ADP
ejpam-5372	318	2	t	t	PROPN
ejpam-5372	318	3	∈	∈	PROPN
ejpam-5372	318	4	{	{	PUNCT
ejpam-5372	318	5	0	0	NUM
ejpam-5372	318	6	,	,	PUNCT
ejpam-5372	318	7	..	..	PUNCT
ejpam-5372	318	8	,	,	PUNCT
ejpam-5372	318	9	n−	n−	NOUN
ejpam-5372	318	10	1	1	NUM
ejpam-5372	318	11	}	}	PUNCT
ejpam-5372	318	12	,	,	PUNCT
ejpam-5372	318	13	we	we	PRON
ejpam-5372	318	14	let	let	VERB
ejpam-5372	318	15	at	at	ADP
ejpam-5372	318	16	=	=	NOUN
ejpam-5372	318	17	2n−tz	2n−tz	NUM
ejpam-5372	318	18	−	−	NUM
ejpam-5372	318	19	1	1	NUM
ejpam-5372	318	20	.	.	X
ejpam-5372	319	1	for	for	ADP
ejpam-5372	319	2	any	any	DET
ejpam-5372	319	3	m	m	NOUN
ejpam-5372	319	4	∈	∈	PROPN
ejpam-5372	319	5	gz	gz	NOUN
ejpam-5372	319	6	,	,	PUNCT
ejpam-5372	319	7	m2a0	m2a0	PROPN
ejpam-5372	319	8	≡	≡	PROPN
ejpam-5372	319	9	m2a1	m2a1	PROPN
ejpam-5372	319	10	,	,	PUNCT
ejpam-5372	319	11	m2a2	m2a2	INTJ
ejpam-5372	319	12	,	,	PUNCT
ejpam-5372	319	13	.	.	PUNCT
ejpam-5372	319	14	.	.	PUNCT
ejpam-5372	319	15	.	.	PUNCT
ejpam-5372	320	1	,	,	PUNCT
ejpam-5372	320	2	m2an−1	m2an−1	PROPN
ejpam-5372	320	3	are	be	AUX
ejpam-5372	320	4	n−	n−	NUM
ejpam-5372	320	5	1	1	NUM
ejpam-5372	320	6	distinct	distinct	ADJ
ejpam-5372	320	7	semi	semi	ADJ
ejpam-5372	320	8	-	-	ADJ
ejpam-5372	320	9	primitives	primitives	ADJ
ejpam-5372	320	10	roots	root	NOUN
ejpam-5372	320	11	.	.	PUNCT
ejpam-5372	321	1	proof	proof	NOUN
ejpam-5372	321	2	.	.	PUNCT
ejpam-5372	322	1	clearly	clearly	ADV
ejpam-5372	322	2	at	at	ADP
ejpam-5372	322	3	is	be	AUX
ejpam-5372	322	4	odd	odd	ADJ
ejpam-5372	322	5	and	and	CCONJ
ejpam-5372	322	6	coprime	coprime	ADJ
ejpam-5372	322	7	to	to	ADP
ejpam-5372	322	8	z	z	NOUN
ejpam-5372	322	9	,	,	PUNCT
ejpam-5372	322	10	thus	thus	ADV
ejpam-5372	322	11	also	also	ADV
ejpam-5372	322	12	to	to	ADP
ejpam-5372	322	13	p−1	p−1	PROPN
ejpam-5372	322	14	.	.	PUNCT
ejpam-5372	323	1	we	we	PRON
ejpam-5372	323	2	deduce	deduce	VERB
ejpam-5372	323	3	that	that	DET
ejpam-5372	323	4	mat	mat	NOUN
ejpam-5372	323	5	∈	∈	PROPN
ejpam-5372	323	6	gz	gz	NOUN
ejpam-5372	323	7	hence	hence	ADV
ejpam-5372	323	8	m2at	m2at	PROPN
ejpam-5372	323	9	∈	∈	PROPN
ejpam-5372	323	10	gs	gs	PROPN
ejpam-5372	323	11	.	.	PUNCT
ejpam-5372	324	1	moreover	moreover	ADV
ejpam-5372	324	2	,	,	PUNCT
ejpam-5372	324	3	2	2	NUM
ejpam-5372	324	4	≤	≤	NUM
ejpam-5372	324	5	2	2	NUM
ejpam-5372	324	6	×	×	NOUN
ejpam-5372	324	7	(	(	PUNCT
ejpam-5372	324	8	2z	2z	NUM
ejpam-5372	324	9	−	−	NOUN
ejpam-5372	324	10	1	1	X
ejpam-5372	324	11	)	)	PUNCT
ejpam-5372	324	12	=	=	SYM
ejpam-5372	324	13	2an−1	2an−1	NUM
ejpam-5372	324	14	<	<	X
ejpam-5372	324	15	·	·	PUNCT
ejpam-5372	324	16	·	·	PUNCT
ejpam-5372	324	17	·	·	PUNCT
ejpam-5372	325	1	<	<	X
ejpam-5372	325	2	2a1	2a1	NUM
ejpam-5372	325	3	=	=	SYM
ejpam-5372	325	4	2	2	NUM
ejpam-5372	325	5	×	×	NOUN
ejpam-5372	325	6	(	(	PUNCT
ejpam-5372	325	7	2n−1z	2n−1z	NUM
ejpam-5372	325	8	−	−	NOUN
ejpam-5372	325	9	1	1	NUM
ejpam-5372	325	10	)	)	PUNCT
ejpam-5372	325	11	=	=	PUNCT
ejpam-5372	326	1	p−	p−	NOUN
ejpam-5372	326	2	3	3	NUM
ejpam-5372	326	3	so	so	ADV
ejpam-5372	326	4	m2a1	m2a1	NOUN
ejpam-5372	326	5	.	.	PUNCT
ejpam-5372	326	6	.	.	PUNCT
ejpam-5372	327	1	.m2an−1	.m2an−1	PROPN
ejpam-5372	327	2	are	be	AUX
ejpam-5372	327	3	pairwise	pairwise	NOUN
ejpam-5372	327	4	distinct	distinct	ADJ
ejpam-5372	327	5	.	.	PUNCT
ejpam-5372	328	1	finally	finally	ADV
ejpam-5372	328	2	,	,	PUNCT
ejpam-5372	328	3	2a0	2a0	NUM
ejpam-5372	328	4	=	=	PUNCT
ejpam-5372	329	1	2p−	2p−	NUM
ejpam-5372	329	2	4	4	NUM
ejpam-5372	329	3	=	=	NOUN
ejpam-5372	329	4	p−	p−	NOUN
ejpam-5372	329	5	1	1	NUM
ejpam-5372	329	6	+	+	NUM
ejpam-5372	329	7	p−	p−	NOUN
ejpam-5372	329	8	3	3	NUM
ejpam-5372	329	9	and	and	CCONJ
ejpam-5372	329	10	mp−1	mp−1	ADJ
ejpam-5372	329	11	≡	≡	PROPN
ejpam-5372	329	12	1	1	NUM
ejpam-5372	329	13	therefore	therefore	ADV
ejpam-5372	329	14	we	we	PRON
ejpam-5372	329	15	have	have	VERB
ejpam-5372	329	16	m2a0	m2a0	PROPN
ejpam-5372	329	17	≡	≡	PROPN
ejpam-5372	329	18	m2a1	m2a1	PROPN
ejpam-5372	329	19	.	.	PUNCT
ejpam-5372	330	1	the	the	DET
ejpam-5372	330	2	following	follow	VERB
ejpam-5372	330	3	proposition	proposition	NOUN
ejpam-5372	330	4	gives	give	VERB
ejpam-5372	330	5	a	a	DET
ejpam-5372	330	6	family	family	NOUN
ejpam-5372	330	7	of	of	ADP
ejpam-5372	330	8	sequences	sequence	NOUN
ejpam-5372	330	9	of	of	ADP
ejpam-5372	330	10	semi	semi	NOUN
ejpam-5372	330	11	-	-	NOUN
ejpam-5372	330	12	generators	generator	NOUN
ejpam-5372	330	13	and	and	CCONJ
ejpam-5372	330	14	generators	generator	NOUN
ejpam-5372	330	15	,	,	PUNCT
ejpam-5372	330	16	as	as	ADV
ejpam-5372	330	17	well	well	ADV
ejpam-5372	330	18	as	as	ADP
ejpam-5372	330	19	a	a	DET
ejpam-5372	330	20	method	method	NOUN
ejpam-5372	330	21	to	to	PART
ejpam-5372	330	22	obtain	obtain	VERB
ejpam-5372	330	23	each	each	DET
ejpam-5372	330	24	element	element	NOUN
ejpam-5372	330	25	of	of	ADP
ejpam-5372	330	26	the	the	DET
ejpam-5372	330	27	sets	set	NOUN
ejpam-5372	330	28	gs	gs	PROPN
ejpam-5372	330	29	and	and	CCONJ
ejpam-5372	330	30	gz	gz	VERB
ejpam-5372	330	31	.	.	PUNCT
ejpam-5372	331	1	proposition	proposition	NOUN
ejpam-5372	331	2	15	15	NUM
ejpam-5372	331	3	.	.	PUNCT
ejpam-5372	332	1	let	let	VERB
ejpam-5372	332	2	g	g	PROPN
ejpam-5372	332	3	∈	∈	PROPN
ejpam-5372	332	4	gs	gs	PROPN
ejpam-5372	332	5	and	and	CCONJ
ejpam-5372	332	6	m	m	VERB
ejpam-5372	332	7	′	′	NUM
ejpam-5372	332	8	∈	∈	PROPN
ejpam-5372	332	9	r(n)⧹r(n−1	r(n)⧹r(n−1	PRON
ejpam-5372	332	10	)	)	PUNCT
ejpam-5372	332	11	fixed	fix	VERB
ejpam-5372	332	12	such	such	ADJ
ejpam-5372	332	13	that	that	DET
ejpam-5372	332	14	gzm	gzm	NOUN
ejpam-5372	332	15	′2z	′2z	PUNCT
ejpam-5372	332	16	≡	≡	PROPN
ejpam-5372	332	17	1	1	X
ejpam-5372	332	18	.	.	PUNCT
ejpam-5372	333	1	we	we	PRON
ejpam-5372	333	2	define	define	VERB
ejpam-5372	333	3	the	the	DET
ejpam-5372	333	4	sequence	sequence	NOUN
ejpam-5372	333	5	(	(	PUNCT
ejpam-5372	333	6	ux	ux	NOUN
ejpam-5372	333	7	)	)	PUNCT
ejpam-5372	333	8	by	by	ADP
ejpam-5372	333	9	:	:	PUNCT
ejpam-5372	333	10	u0	u0	ADJ
ejpam-5372	333	11	=	=	PROPN
ejpam-5372	333	12	g	g	NOUN
ejpam-5372	333	13	,	,	PUNCT
ejpam-5372	333	14	ux+1	ux+1	PROPN
ejpam-5372	333	15	=	=	SYM
ejpam-5372	333	16	(	(	PUNCT
ejpam-5372	333	17	m	m	VERB
ejpam-5372	333	18	′	′	NUM
ejpam-5372	333	19	ux	ux	PROPN
ejpam-5372	333	20	)	)	PUNCT
ejpam-5372	333	21	2	2	NUM
ejpam-5372	333	22	for	for	ADP
ejpam-5372	333	23	all	all	DET
ejpam-5372	333	24	x	x	SYM
ejpam-5372	333	25	∈	∈	PROPN
ejpam-5372	333	26	n	n	CCONJ
ejpam-5372	333	27	,	,	PUNCT
ejpam-5372	333	28	ux	ux	PROPN
ejpam-5372	333	29	∈	∈	PROPN
ejpam-5372	333	30	gs	gs	INTJ
ejpam-5372	333	31	and	and	CCONJ
ejpam-5372	333	32	m	m	VERB
ejpam-5372	333	33	′	′	NUM
ejpam-5372	334	1	ux	ux	INTJ
ejpam-5372	334	2	∈	∈	PROPN
ejpam-5372	334	3	gz	gz	NOUN
ejpam-5372	334	4	.	.	PUNCT
ejpam-5372	335	1	moreover	moreover	ADV
ejpam-5372	335	2	if	if	SCONJ
ejpam-5372	335	3	0	0	NUM
ejpam-5372	335	4	≤	≤	NUM
ejpam-5372	335	5	x	x	X
ejpam-5372	335	6	<	<	X
ejpam-5372	335	7	y	y	X
ejpam-5372	335	8	<	<	X
ejpam-5372	335	9	logd2	logd2	X
ejpam-5372	335	10	(	(	PUNCT
ejpam-5372	335	11	1	1	NUM
ejpam-5372	335	12	,	,	PUNCT
ejpam-5372	335	13	z	z	NOUN
ejpam-5372	335	14	)	)	PUNCT
ejpam-5372	335	15	,	,	PUNCT
ejpam-5372	335	16	ux	ux	PROPN
ejpam-5372	335	17	̸≡	̸≡	PROPN
ejpam-5372	335	18	uy	uy	INTJ
ejpam-5372	336	1	and	and	CCONJ
ejpam-5372	336	2	(	(	PUNCT
ejpam-5372	336	3	ux	ux	NOUN
ejpam-5372	336	4	)	)	PUNCT
ejpam-5372	336	5	is	be	AUX
ejpam-5372	336	6	periodic	periodic	ADJ
ejpam-5372	336	7	of	of	ADP
ejpam-5372	336	8	period	period	NOUN
ejpam-5372	336	9	logd2	logd2	NOUN
ejpam-5372	336	10	(	(	PUNCT
ejpam-5372	336	11	1	1	NUM
ejpam-5372	336	12	,	,	PUNCT
ejpam-5372	336	13	z	z	NOUN
ejpam-5372	336	14	)	)	PUNCT
ejpam-5372	336	15	.	.	PUNCT
ejpam-5372	337	1	proof	proof	NOUN
ejpam-5372	337	2	.	.	PUNCT
ejpam-5372	338	1	in	in	ADP
ejpam-5372	338	2	the	the	DET
ejpam-5372	338	3	identification	identification	NOUN
ejpam-5372	338	4	of	of	ADP
ejpam-5372	338	5	f∗	f∗	NOUN
ejpam-5372	338	6	p	p	X
ejpam-5372	338	7	to	to	ADP
ejpam-5372	338	8	(	(	PUNCT
ejpam-5372	338	9	z/2nz)×	z/2nz)×	NUM
ejpam-5372	338	10	(	(	PUNCT
ejpam-5372	338	11	z	z	NOUN
ejpam-5372	338	12	/	/	SYM
ejpam-5372	338	13	zz	zz	PROPN
ejpam-5372	338	14	)	)	PUNCT
ejpam-5372	338	15	,	,	PUNCT
ejpam-5372	338	16	any	any	DET
ejpam-5372	338	17	element	element	NOUN
ejpam-5372	338	18	of	of	ADP
ejpam-5372	338	19	gs	gs	PROPN
ejpam-5372	338	20	corresponds	correspond	NOUN
ejpam-5372	338	21	to	to	ADP
ejpam-5372	338	22	a	a	DET
ejpam-5372	338	23	pair	pair	NOUN
ejpam-5372	338	24	(	(	PUNCT
ejpam-5372	338	25	2k	2k	NUM
ejpam-5372	338	26	,	,	PUNCT
ejpam-5372	338	27	l	l	NOUN
ejpam-5372	338	28	)	)	PUNCT
ejpam-5372	338	29	with	with	ADP
ejpam-5372	338	30	k	k	PROPN
ejpam-5372	338	31	odd	odd	ADJ
ejpam-5372	338	32	and	and	CCONJ
ejpam-5372	338	33	l	l	NOUN
ejpam-5372	338	34	coprime	coprime	NOUN
ejpam-5372	338	35	to	to	ADP
ejpam-5372	338	36	z	z	NOUN
ejpam-5372	338	37	while	while	SCONJ
ejpam-5372	338	38	any	any	DET
ejpam-5372	338	39	elements	element	NOUN
ejpam-5372	338	40	of	of	ADP
ejpam-5372	338	41	r(n)⧹r(n−1	r(n)⧹r(n−1	X
ejpam-5372	338	42	)	)	PUNCT
ejpam-5372	338	43	correspond	correspond	VERB
ejpam-5372	338	44	to	to	ADP
ejpam-5372	338	45	a	a	DET
ejpam-5372	338	46	pair	pair	NOUN
ejpam-5372	338	47	(	(	PUNCT
ejpam-5372	338	48	i	i	NOUN
ejpam-5372	338	49	,	,	PUNCT
ejpam-5372	338	50	0	0	NUM
ejpam-5372	338	51	)	)	PUNCT
ejpam-5372	338	52	with	with	SCONJ
ejpam-5372	338	53	i	i	PRON
ejpam-5372	338	54	odd	odd	ADJ
ejpam-5372	338	55	.	.	PUNCT
ejpam-5372	339	1	from	from	ADP
ejpam-5372	339	2	this	this	PRON
ejpam-5372	339	3	and	and	CCONJ
ejpam-5372	339	4	the	the	DET
ejpam-5372	339	5	equation	equation	NOUN
ejpam-5372	339	6	gzm	gzm	NOUN
ejpam-5372	339	7	′2z	′2z	PUNCT
ejpam-5372	339	8	≡	≡	PROPN
ejpam-5372	339	9	1	1	NUM
ejpam-5372	339	10	,	,	PUNCT
ejpam-5372	339	11	we	we	PRON
ejpam-5372	339	12	deduce	deduce	VERB
ejpam-5372	339	13	that	that	SCONJ
ejpam-5372	339	14	there	there	PRON
ejpam-5372	339	15	is	be	VERB
ejpam-5372	339	16	k0	k0	PROPN
ejpam-5372	339	17	odd	odd	ADJ
ejpam-5372	339	18	and	and	CCONJ
ejpam-5372	339	19	l0	l0	NOUN
ejpam-5372	339	20	coprime	coprime	NOUN
ejpam-5372	339	21	to	to	ADP
ejpam-5372	339	22	z	z	NOUN
ejpam-5372	339	23	such	such	ADJ
ejpam-5372	339	24	that	that	SCONJ
ejpam-5372	339	25	g	g	PROPN
ejpam-5372	339	26	corresponds	correspond	VERB
ejpam-5372	339	27	to	to	ADP
ejpam-5372	339	28	(	(	PUNCT
ejpam-5372	339	29	2k0	2k0	NUM
ejpam-5372	339	30	,	,	PUNCT
ejpam-5372	339	31	l0	l0	PROPN
ejpam-5372	339	32	)	)	PUNCT
ejpam-5372	339	33	and	and	CCONJ
ejpam-5372	339	34	m	m	AUX
ejpam-5372	339	35	′	′	VERB
ejpam-5372	339	36	to	to	ADP
ejpam-5372	339	37	(	(	PUNCT
ejpam-5372	339	38	−k0	−k0	PROPN
ejpam-5372	339	39	,	,	PUNCT
ejpam-5372	339	40	0	0	NUM
ejpam-5372	339	41	)	)	PUNCT
ejpam-5372	339	42	.	.	PUNCT
ejpam-5372	340	1	m.	m.	PROPN
ejpam-5372	340	2	wolf	wolf	PROPN
ejpam-5372	340	3	,	,	PUNCT
ejpam-5372	340	4	f.	f.	PROPN
ejpam-5372	340	5	wolf	wolf	PROPN
ejpam-5372	340	6	/	/	SYM
ejpam-5372	340	7	eur	eur	PROPN
ejpam-5372	340	8	.	.	PUNCT
ejpam-5372	341	1	j.	j.	PROPN
ejpam-5372	341	2	pure	pure	PROPN
ejpam-5372	341	3	appl	appl	PROPN
ejpam-5372	341	4	.	.	PROPN
ejpam-5372	341	5	math	math	PROPN
ejpam-5372	341	6	,	,	PUNCT
ejpam-5372	341	7	17	17	NUM
ejpam-5372	341	8	(	(	PUNCT
ejpam-5372	341	9	4	4	NUM
ejpam-5372	341	10	)	)	PUNCT
ejpam-5372	341	11	(	(	PUNCT
ejpam-5372	341	12	2024	2024	NUM
ejpam-5372	341	13	)	)	PUNCT
ejpam-5372	341	14	,	,	PUNCT
ejpam-5372	341	15	2431	2431	NUM
ejpam-5372	341	16	-	-	SYM
ejpam-5372	341	17	2447	2447	NUM
ejpam-5372	341	18	2440	2440	NUM
ejpam-5372	341	19	thus	thus	ADV
ejpam-5372	341	20	,	,	PUNCT
ejpam-5372	341	21	by	by	ADP
ejpam-5372	341	22	recurrence	recurrence	NOUN
ejpam-5372	341	23	,	,	PUNCT
ejpam-5372	341	24	we	we	PRON
ejpam-5372	341	25	note	note	VERB
ejpam-5372	341	26	that	that	SCONJ
ejpam-5372	341	27	ux	ux	PROPN
ejpam-5372	341	28	corresponds	correspond	VERB
ejpam-5372	341	29	to	to	ADP
ejpam-5372	341	30	(	(	PUNCT
ejpam-5372	341	31	2kx	2kx	ADJ
ejpam-5372	341	32	,	,	PUNCT
ejpam-5372	341	33	lx	lx	NOUN
ejpam-5372	341	34	)	)	PUNCT
ejpam-5372	341	35	=	=	SYM
ejpam-5372	341	36	(	(	PUNCT
ejpam-5372	341	37	2k0	2k0	NUM
ejpam-5372	341	38	,	,	PUNCT
ejpam-5372	341	39	2	2	NUM
ejpam-5372	341	40	xl0	xl0	NOUN
ejpam-5372	341	41	)	)	PUNCT
ejpam-5372	341	42	where	where	SCONJ
ejpam-5372	341	43	2xl0	2xl0	NUM
ejpam-5372	341	44	is	be	AUX
ejpam-5372	341	45	still	still	ADV
ejpam-5372	341	46	coprime	coprime	ADJ
ejpam-5372	341	47	to	to	ADP
ejpam-5372	341	48	z	z	NOUN
ejpam-5372	341	49	,	,	PUNCT
ejpam-5372	341	50	so	so	ADV
ejpam-5372	341	51	ux	ux	PROPN
ejpam-5372	341	52	∈	∈	PROPN
ejpam-5372	341	53	gs	gs	PROPN
ejpam-5372	341	54	.	.	PUNCT
ejpam-5372	342	1	similarly	similarly	ADV
ejpam-5372	342	2	,	,	PUNCT
ejpam-5372	342	3	m	m	AUX
ejpam-5372	342	4	′	′	NUM
ejpam-5372	342	5	ux	ux	INTJ
ejpam-5372	342	6	corresponds	correspond	VERB
ejpam-5372	342	7	to	to	ADP
ejpam-5372	342	8	(	(	PUNCT
ejpam-5372	342	9	k0	k0	PROPN
ejpam-5372	342	10	,	,	PUNCT
ejpam-5372	342	11	2	2	NUM
ejpam-5372	342	12	xl0	xl0	NOUN
ejpam-5372	342	13	)	)	PUNCT
ejpam-5372	342	14	and	and	CCONJ
ejpam-5372	342	15	we	we	PRON
ejpam-5372	342	16	have	have	VERB
ejpam-5372	342	17	m	m	AUX
ejpam-5372	342	18	′	′	NUM
ejpam-5372	342	19	ux	ux	INTJ
ejpam-5372	342	20	∈	∈	PROPN
ejpam-5372	342	21	gz	gz	NOUN
ejpam-5372	342	22	.	.	PUNCT
ejpam-5372	343	1	moreover	moreover	ADV
ejpam-5372	343	2	,	,	PUNCT
ejpam-5372	343	3	by	by	ADP
ejpam-5372	343	4	definition	definition	NOUN
ejpam-5372	343	5	of	of	ADP
ejpam-5372	343	6	logd2	logd2	NOUN
ejpam-5372	343	7	(	(	PUNCT
ejpam-5372	343	8	1	1	NUM
ejpam-5372	343	9	,	,	PUNCT
ejpam-5372	343	10	z	z	NOUN
ejpam-5372	343	11	)	)	PUNCT
ejpam-5372	343	12	,	,	PUNCT
ejpam-5372	343	13	the	the	DET
ejpam-5372	343	14	elements	element	NOUN
ejpam-5372	343	15	of	of	ADP
ejpam-5372	343	16	ux	ux	PROPN
ejpam-5372	343	17	are	be	AUX
ejpam-5372	343	18	pairwise	pairwise	NOUN
ejpam-5372	343	19	distinct	distinct	ADJ
ejpam-5372	343	20	for	for	ADP
ejpam-5372	343	21	0	0	NUM
ejpam-5372	343	22	≤	≤	NUM
ejpam-5372	343	23	x	x	X
ejpam-5372	343	24	<	<	X
ejpam-5372	343	25	logd2	logd2	X
ejpam-5372	343	26	(	(	PUNCT
ejpam-5372	343	27	1	1	NUM
ejpam-5372	343	28	,	,	PUNCT
ejpam-5372	343	29	z	z	NOUN
ejpam-5372	343	30	)	)	PUNCT
ejpam-5372	343	31	and	and	CCONJ
ejpam-5372	343	32	ulogd2(1,z	ulogd2(1,z	NOUN
ejpam-5372	343	33	)	)	PUNCT
ejpam-5372	343	34	=	=	PUNCT
ejpam-5372	344	1	u0	u0	ADJ
ejpam-5372	344	2	which	which	PRON
ejpam-5372	344	3	implies	imply	VERB
ejpam-5372	344	4	the	the	DET
ejpam-5372	344	5	periodicity	periodicity	NOUN
ejpam-5372	344	6	of	of	ADP
ejpam-5372	344	7	(	(	PUNCT
ejpam-5372	344	8	ux	ux	PROPN
ejpam-5372	344	9	)	)	PUNCT
ejpam-5372	344	10	.	.	PUNCT
ejpam-5372	345	1	remark	remark	NOUN
ejpam-5372	345	2	2	2	NUM
ejpam-5372	345	3	.	.	PUNCT
ejpam-5372	346	1	we	we	PRON
ejpam-5372	346	2	verify	verify	VERB
ejpam-5372	346	3	that	that	PRON
ejpam-5372	346	4	ux	ux	NOUN
ejpam-5372	346	5	=	=	SYM
ejpam-5372	346	6	g2	g2	PROPN
ejpam-5372	346	7	x	x	PUNCT
ejpam-5372	346	8	m	m	NOUN
ejpam-5372	346	9	′2(2x−1	′2(2x−1	NUM
ejpam-5372	346	10	)	)	PUNCT
ejpam-5372	346	11	.	.	PUNCT
ejpam-5372	347	1	in	in	ADP
ejpam-5372	347	2	general	general	ADJ
ejpam-5372	347	3	,	,	PUNCT
ejpam-5372	347	4	the	the	DET
ejpam-5372	347	5	elements	element	NOUN
ejpam-5372	347	6	(	(	PUNCT
ejpam-5372	347	7	m	m	VERB
ejpam-5372	347	8	′	′	NUM
ejpam-5372	347	9	ux	ux	ADV
ejpam-5372	347	10	)	)	PUNCT
ejpam-5372	347	11	form	form	VERB
ejpam-5372	347	12	a	a	DET
ejpam-5372	347	13	strict	strict	ADJ
ejpam-5372	347	14	subset	subset	NOUN
ejpam-5372	347	15	of	of	ADP
ejpam-5372	347	16	gz	gz	PROPN
ejpam-5372	347	17	.	.	PUNCT
ejpam-5372	348	1	however	however	ADV
ejpam-5372	348	2	,	,	PUNCT
ejpam-5372	348	3	for	for	ADP
ejpam-5372	348	4	any	any	DET
ejpam-5372	348	5	generator	generator	NOUN
ejpam-5372	348	6	h	h	NOUN
ejpam-5372	348	7	∈	∈	PROPN
ejpam-5372	348	8	gz	gz	NOUN
ejpam-5372	348	9	there	there	PRON
ejpam-5372	348	10	exists	exist	VERB
ejpam-5372	348	11	a	a	DET
ejpam-5372	348	12	coprime	coprime	NOUN
ejpam-5372	348	13	to	to	ADP
ejpam-5372	348	14	p	p	NOUN
ejpam-5372	348	15	−	−	PROPN
ejpam-5372	348	16	1	1	NUM
ejpam-5372	348	17	such	such	ADJ
ejpam-5372	348	18	that	that	DET
ejpam-5372	348	19	h	h	NOUN
ejpam-5372	348	20	=	=	PUNCT
ejpam-5372	349	1	(	(	PUNCT
ejpam-5372	349	2	m	m	VERB
ejpam-5372	349	3	′	′	NUM
ejpam-5372	349	4	g	g	NOUN
ejpam-5372	349	5	)	)	PUNCT
ejpam-5372	349	6	a	a	PRON
ejpam-5372	349	7	.	.	PUNCT
ejpam-5372	350	1	we	we	PRON
ejpam-5372	350	2	can	can	AUX
ejpam-5372	350	3	also	also	ADV
ejpam-5372	350	4	choose	choose	VERB
ejpam-5372	350	5	a	a	PRON
ejpam-5372	350	6	to	to	PART
ejpam-5372	350	7	be	be	AUX
ejpam-5372	350	8	prime	prime	ADJ
ejpam-5372	350	9	.	.	PUNCT
ejpam-5372	351	1	thus	thus	ADV
ejpam-5372	351	2	,	,	PUNCT
ejpam-5372	351	3	if	if	SCONJ
ejpam-5372	351	4	we	we	PRON
ejpam-5372	351	5	take	take	VERB
ejpam-5372	351	6	a	a	DET
ejpam-5372	351	7	∈	∈	ADJ
ejpam-5372	351	8	p⧹d	p⧹d	NOUN
ejpam-5372	351	9	(	(	PUNCT
ejpam-5372	351	10	p−	p−	NOUN
ejpam-5372	351	11	1	1	NUM
ejpam-5372	351	12	)	)	PUNCT
ejpam-5372	351	13	such	such	ADJ
ejpam-5372	351	14	that	that	SCONJ
ejpam-5372	351	15	gam	gam	NOUN
ejpam-5372	351	16	′a	′a	NOUN
ejpam-5372	351	17	has	have	AUX
ejpam-5372	351	18	not	not	PART
ejpam-5372	351	19	already	already	ADV
ejpam-5372	351	20	been	be	AUX
ejpam-5372	351	21	generated	generate	VERB
ejpam-5372	351	22	then	then	ADV
ejpam-5372	351	23	the	the	DET
ejpam-5372	351	24	logd2	logd2	ADJ
ejpam-5372	351	25	(	(	PUNCT
ejpam-5372	351	26	1	1	NUM
ejpam-5372	351	27	,	,	PUNCT
ejpam-5372	351	28	z	z	NOUN
ejpam-5372	351	29	)	)	PUNCT
ejpam-5372	351	30	first	first	ADJ
ejpam-5372	351	31	terms	term	NOUN
ejpam-5372	351	32	u	u	PROPN
ejpam-5372	351	33	(	(	PUNCT
ejpam-5372	351	34	a	a	NOUN
ejpam-5372	351	35	)	)	PUNCT
ejpam-5372	351	36	x	x	SYM
ejpam-5372	351	37	:	:	PUNCT
ejpam-5372	351	38	=	=	PROPN
ejpam-5372	351	39	g2	g2	PROPN
ejpam-5372	351	40	xam	xam	PROPN
ejpam-5372	351	41	′2a(2x−1	′2a(2x−1	PROPN
ejpam-5372	351	42	)	)	PUNCT
ejpam-5372	351	43	will	will	AUX
ejpam-5372	351	44	all	all	PRON
ejpam-5372	351	45	be	be	AUX
ejpam-5372	351	46	new	new	ADJ
ejpam-5372	351	47	elements	element	NOUN
ejpam-5372	351	48	of	of	ADP
ejpam-5372	351	49	gz	gz	PROPN
ejpam-5372	351	50	.	.	PUNCT
ejpam-5372	352	1	we	we	PRON
ejpam-5372	352	2	deduce	deduce	VERB
ejpam-5372	352	3	that	that	SCONJ
ejpam-5372	352	4	there	there	PRON
ejpam-5372	352	5	exists	exist	VERB
ejpam-5372	352	6	a	a	DET
ejpam-5372	352	7	set	set	NOUN
ejpam-5372	352	8	a	a	PRON
ejpam-5372	352	9	of	of	ADP
ejpam-5372	352	10	φ(p−1	φ(p−1	NOUN
ejpam-5372	352	11	)	)	PUNCT
ejpam-5372	352	12	logd2(1,z	logd2(1,z	PROPN
ejpam-5372	352	13	)	)	PUNCT
ejpam-5372	352	14	primes	prime	NOUN
ejpam-5372	352	15	,	,	PUNCT
ejpam-5372	352	16	odd	odd	ADJ
ejpam-5372	352	17	and	and	CCONJ
ejpam-5372	352	18	coprime	coprime	NOUN
ejpam-5372	352	19	to	to	ADP
ejpam-5372	352	20	z	z	NOUN
ejpam-5372	352	21	,	,	PUNCT
ejpam-5372	352	22	and	and	CCONJ
ejpam-5372	352	23	such	such	ADJ
ejpam-5372	352	24	that	that	PRON
ejpam-5372	352	25	:	:	PUNCT
ejpam-5372	352	26	gz	gz	NOUN
ejpam-5372	352	27	=	=	PRON
ejpam-5372	352	28	{	{	PUNCT
ejpam-5372	352	29	g2	g2	PROPN
ejpam-5372	352	30	xam	xam	PROPN
ejpam-5372	352	31	′(2x+1−1)a	′(2x+1−1)a	PROPN
ejpam-5372	352	32	,	,	PUNCT
ejpam-5372	352	33	(	(	PUNCT
ejpam-5372	352	34	a	a	PRON
ejpam-5372	352	35	,	,	PUNCT
ejpam-5372	352	36	x	x	NOUN
ejpam-5372	352	37	)	)	PUNCT
ejpam-5372	352	38	∈	∈	PROPN
ejpam-5372	352	39	(	(	PUNCT
ejpam-5372	352	40	{	{	PUNCT
ejpam-5372	352	41	1	1	NUM
ejpam-5372	352	42	}	}	PUNCT
ejpam-5372	352	43	∪a)×	∪a)×	NOUN
ejpam-5372	352	44	[	[	X
ejpam-5372	352	45	[	[	X
ejpam-5372	352	46	0	0	NUM
ejpam-5372	352	47	,	,	PUNCT
ejpam-5372	352	48	logd2	logd2	NOUN
ejpam-5372	352	49	(	(	PUNCT
ejpam-5372	352	50	1	1	NUM
ejpam-5372	352	51	,	,	PUNCT
ejpam-5372	352	52	z	z	NOUN
ejpam-5372	352	53	)	)	PUNCT
ejpam-5372	352	54	−	−	PROPN
ejpam-5372	352	55	1	1	NUM
ejpam-5372	352	56	]	]	X
ejpam-5372	352	57	]	]	PUNCT
ejpam-5372	352	58	}	}	PUNCT
ejpam-5372	352	59	proof	proof	NOUN
ejpam-5372	352	60	.	.	PUNCT
ejpam-5372	353	1	by	by	ADP
ejpam-5372	353	2	construction	construction	NOUN
ejpam-5372	353	3	we	we	PRON
ejpam-5372	353	4	have	have	VERB
ejpam-5372	353	5	ux	ux	ADJ
ejpam-5372	353	6	=	=	SYM
ejpam-5372	353	7	g2	g2	PROPN
ejpam-5372	353	8	x	x	PUNCT
ejpam-5372	353	9	m	m	PROPN
ejpam-5372	353	10	′2+···+2x	′2+···+2x	NOUN
ejpam-5372	353	11	=	=	SYM
ejpam-5372	353	12	g2	g2	PROPN
ejpam-5372	353	13	x	x	PUNCT
ejpam-5372	353	14	m	m	NOUN
ejpam-5372	353	15	′2(2x−1	′2(2x−1	NUM
ejpam-5372	353	16	)	)	PUNCT
ejpam-5372	353	17	.	.	PUNCT
ejpam-5372	354	1	if	if	SCONJ
ejpam-5372	354	2	h	h	PROPN
ejpam-5372	354	3	∈	∈	PROPN
ejpam-5372	354	4	gz	gz	NOUN
ejpam-5372	354	5	,	,	PUNCT
ejpam-5372	354	6	we	we	PRON
ejpam-5372	354	7	know	know	VERB
ejpam-5372	354	8	that	that	SCONJ
ejpam-5372	354	9	m	m	VERB
ejpam-5372	354	10	′	′	NUM
ejpam-5372	354	11	g	g	PROPN
ejpam-5372	354	12	∈	∈	PROPN
ejpam-5372	354	13	gz	gz	NOUN
ejpam-5372	355	1	so	so	ADV
ejpam-5372	355	2	there	there	PRON
ejpam-5372	355	3	exists	exist	VERB
ejpam-5372	355	4	a	a	DET
ejpam-5372	355	5	coprime	coprime	NOUN
ejpam-5372	355	6	to	to	ADP
ejpam-5372	355	7	p	p	NOUN
ejpam-5372	355	8	−	−	PROPN
ejpam-5372	355	9	1	1	NUM
ejpam-5372	355	10	(	(	PUNCT
ejpam-5372	355	11	i.e.	i.e.	X
ejpam-5372	355	12	odd	odd	ADJ
ejpam-5372	355	13	and	and	CCONJ
ejpam-5372	355	14	coprime	coprime	NOUN
ejpam-5372	355	15	to	to	ADP
ejpam-5372	355	16	z	z	NOUN
ejpam-5372	355	17	)	)	PUNCT
ejpam-5372	355	18	such	such	ADJ
ejpam-5372	355	19	that	that	DET
ejpam-5372	355	20	h	h	NOUN
ejpam-5372	355	21	=	=	PUNCT
ejpam-5372	356	1	(	(	PUNCT
ejpam-5372	356	2	m	m	VERB
ejpam-5372	356	3	′	′	NUM
ejpam-5372	356	4	g	g	NOUN
ejpam-5372	356	5	)	)	PUNCT
ejpam-5372	356	6	a	a	PRON
ejpam-5372	356	7	.	.	PUNCT
ejpam-5372	357	1	dirichlet	dirichlet	PROPN
ejpam-5372	357	2	’s	’s	PART
ejpam-5372	357	3	theorem	theorem	ADJ
ejpam-5372	357	4	ensures	ensure	VERB
ejpam-5372	357	5	that	that	SCONJ
ejpam-5372	357	6	we	we	PRON
ejpam-5372	357	7	can	can	AUX
ejpam-5372	357	8	choose	choose	VERB
ejpam-5372	357	9	a	a	DET
ejpam-5372	357	10	to	to	PART
ejpam-5372	357	11	be	be	AUX
ejpam-5372	357	12	prime	prime	ADJ
ejpam-5372	357	13	,	,	PUNCT
ejpam-5372	357	14	even	even	ADV
ejpam-5372	357	15	if	if	SCONJ
ejpam-5372	357	16	it	it	PRON
ejpam-5372	357	17	means	mean	VERB
ejpam-5372	357	18	adding	add	VERB
ejpam-5372	357	19	it	it	PRON
ejpam-5372	357	20	a	a	DET
ejpam-5372	357	21	multiple	multiple	NOUN
ejpam-5372	357	22	of	of	ADP
ejpam-5372	357	23	p−	p−	NOUN
ejpam-5372	357	24	1	1	NUM
ejpam-5372	357	25	.	.	X
ejpam-5372	357	26	identifying	identify	VERB
ejpam-5372	357	27	f∗	f∗	NOUN
ejpam-5372	357	28	p	p	X
ejpam-5372	357	29	to	to	ADP
ejpam-5372	357	30	(	(	PUNCT
ejpam-5372	357	31	z/2nz	z/2nz	NOUN
ejpam-5372	357	32	)	)	PUNCT
ejpam-5372	357	33	×	×	NOUN
ejpam-5372	357	34	(	(	PUNCT
ejpam-5372	357	35	z	z	PROPN
ejpam-5372	357	36	/	/	SYM
ejpam-5372	357	37	zz	zz	NOUN
ejpam-5372	357	38	)	)	PUNCT
ejpam-5372	357	39	as	as	ADP
ejpam-5372	357	40	in	in	ADP
ejpam-5372	357	41	the	the	DET
ejpam-5372	357	42	proof	proof	NOUN
ejpam-5372	357	43	of	of	ADP
ejpam-5372	357	44	the	the	DET
ejpam-5372	357	45	previous	previous	ADJ
ejpam-5372	357	46	results	result	NOUN
ejpam-5372	357	47	,	,	PUNCT
ejpam-5372	357	48	m	m	VERB
ejpam-5372	357	49	′a	′a	ADP
ejpam-5372	357	50	u	u	NOUN
ejpam-5372	357	51	(	(	PUNCT
ejpam-5372	357	52	a	a	NOUN
ejpam-5372	357	53	)	)	PUNCT
ejpam-5372	357	54	x	x	PUNCT
ejpam-5372	357	55	correspond	correspond	VERB
ejpam-5372	357	56	to	to	PART
ejpam-5372	357	57	(	(	PUNCT
ejpam-5372	357	58	ak0	ak0	NOUN
ejpam-5372	357	59	,	,	PUNCT
ejpam-5372	357	60	2	2	NUM
ejpam-5372	357	61	xal0	xal0	NOUN
ejpam-5372	357	62	)	)	PUNCT
ejpam-5372	357	63	.	.	PUNCT
ejpam-5372	358	1	let	let	VERB
ejpam-5372	358	2	a	a	DET
ejpam-5372	358	3	,	,	PUNCT
ejpam-5372	358	4	b	b	NOUN
ejpam-5372	358	5	,	,	PUNCT
ejpam-5372	358	6	x	x	PRON
ejpam-5372	358	7	,	,	PUNCT
ejpam-5372	358	8	y	y	PROPN
ejpam-5372	358	9	be	be	VERB
ejpam-5372	358	10	such	such	ADJ
ejpam-5372	358	11	that	that	SCONJ
ejpam-5372	358	12	(	(	PUNCT
ejpam-5372	358	13	ak0	ak0	NOUN
ejpam-5372	358	14	,	,	PUNCT
ejpam-5372	358	15	2	2	NUM
ejpam-5372	358	16	xal0	xal0	NOUN
ejpam-5372	358	17	)	)	PUNCT
ejpam-5372	359	1	=	=	SYM
ejpam-5372	359	2	(	(	PUNCT
ejpam-5372	359	3	bk0	bk0	NOUN
ejpam-5372	359	4	,	,	PUNCT
ejpam-5372	359	5	2	2	NUM
ejpam-5372	359	6	ybl0	ybl0	PROPN
ejpam-5372	359	7	)	)	PUNCT
ejpam-5372	359	8	.	.	PUNCT
ejpam-5372	360	1	in	in	ADP
ejpam-5372	360	2	particular	particular	ADJ
ejpam-5372	360	3	a.	a.	NOUN
ejpam-5372	360	4	(	(	PUNCT
ejpam-5372	360	5	−k0	−k0	PROPN
ejpam-5372	360	6	,	,	PUNCT
ejpam-5372	360	7	0	0	NUM
ejpam-5372	360	8	)	)	PUNCT
ejpam-5372	360	9	=	=	SYM
ejpam-5372	360	10	b.	b.	PROPN
ejpam-5372	360	11	(	(	PUNCT
ejpam-5372	360	12	−k0	−k0	PROPN
ejpam-5372	360	13	,	,	PUNCT
ejpam-5372	360	14	0	0	NUM
ejpam-5372	360	15	)	)	PUNCT
ejpam-5372	360	16	thus	thus	ADV
ejpam-5372	360	17	m	m	VERB
ejpam-5372	360	18	′a	′a	ADP
ejpam-5372	360	19	≡	≡	PROPN
ejpam-5372	360	20	m	m	VERB
ejpam-5372	360	21	′b	′b	PROPN
ejpam-5372	360	22	hence	hence	ADV
ejpam-5372	360	23	for	for	ADP
ejpam-5372	360	24	any	any	DET
ejpam-5372	360	25	k	k	PROPN
ejpam-5372	360	26	∈	∈	PROPN
ejpam-5372	360	27	n	n	CCONJ
ejpam-5372	360	28	,	,	PUNCT
ejpam-5372	360	29	u	u	NOUN
ejpam-5372	360	30	(	(	PUNCT
ejpam-5372	360	31	a	a	NOUN
ejpam-5372	360	32	)	)	PUNCT
ejpam-5372	360	33	x+k	x+k	PUNCT
ejpam-5372	361	1	=	=	SYM
ejpam-5372	361	2	u	u	PROPN
ejpam-5372	361	3	(	(	PUNCT
ejpam-5372	361	4	b	b	NOUN
ejpam-5372	361	5	)	)	PUNCT
ejpam-5372	361	6	y+k	y+k	NUM
ejpam-5372	361	7	from	from	ADP
ejpam-5372	361	8	which	which	PRON
ejpam-5372	361	9	it	it	PRON
ejpam-5372	361	10	follows	follow	VERB
ejpam-5372	361	11	by	by	ADP
ejpam-5372	361	12	periodicity	periodicity	NOUN
ejpam-5372	361	13	of	of	ADP
ejpam-5372	361	14	these	these	DET
ejpam-5372	361	15	sequences	sequence	NOUN
ejpam-5372	361	16	that	that	PRON
ejpam-5372	361	17	m	m	VERB
ejpam-5372	361	18	′b	′b	PROPN
ejpam-5372	361	19	u	u	X
ejpam-5372	361	20	(	(	PUNCT
ejpam-5372	361	21	b	b	NOUN
ejpam-5372	361	22	)	)	PUNCT
ejpam-5372	361	23	0	0	NUM
ejpam-5372	361	24	is	be	AUX
ejpam-5372	361	25	equal	equal	ADJ
ejpam-5372	361	26	to	to	ADP
ejpam-5372	361	27	one	one	NUM
ejpam-5372	361	28	of	of	ADP
ejpam-5372	361	29	the	the	DET
ejpam-5372	361	30	terms	term	NOUN
ejpam-5372	361	31	of	of	ADP
ejpam-5372	361	32	(	(	PUNCT
ejpam-5372	361	33	m	m	VERB
ejpam-5372	361	34	′a	′a	ADP
ejpam-5372	361	35	u	u	NOUN
ejpam-5372	361	36	(	(	PUNCT
ejpam-5372	361	37	a	a	NOUN
ejpam-5372	361	38	)	)	PUNCT
ejpam-5372	361	39	x	x	SYM
ejpam-5372	361	40	)	)	PUNCT
ejpam-5372	361	41	.	.	PUNCT
ejpam-5372	362	1	therefore	therefore	ADV
ejpam-5372	362	2	,	,	PUNCT
ejpam-5372	362	3	if	if	SCONJ
ejpam-5372	362	4	we	we	PRON
ejpam-5372	362	5	construct	construct	VERB
ejpam-5372	362	6	a	a	DET
ejpam-5372	362	7	recursively	recursively	NOUN
ejpam-5372	362	8	in	in	ADP
ejpam-5372	362	9	such	such	DET
ejpam-5372	362	10	a	a	DET
ejpam-5372	362	11	way	way	NOUN
ejpam-5372	362	12	that	that	PRON
ejpam-5372	362	13	m	m	VERB
ejpam-5372	362	14	′a	′a	ADP
ejpam-5372	362	15	u	u	NOUN
ejpam-5372	362	16	(	(	PUNCT
ejpam-5372	362	17	a	a	X
ejpam-5372	362	18	)	)	PUNCT
ejpam-5372	362	19	0	0	NUM
ejpam-5372	362	20	is	be	AUX
ejpam-5372	362	21	not	not	PART
ejpam-5372	362	22	among	among	ADP
ejpam-5372	362	23	the	the	DET
ejpam-5372	362	24	generators	generator	NOUN
ejpam-5372	362	25	already	already	ADV
ejpam-5372	362	26	constructed	construct	VERB
ejpam-5372	362	27	,	,	PUNCT
ejpam-5372	362	28	we	we	PRON
ejpam-5372	362	29	keep	keep	VERB
ejpam-5372	362	30	adding	add	VERB
ejpam-5372	362	31	chunks	chunk	NOUN
ejpam-5372	362	32	of	of	ADP
ejpam-5372	362	33	logd2	logd2	NOUN
ejpam-5372	362	34	(	(	PUNCT
ejpam-5372	362	35	1	1	NUM
ejpam-5372	362	36	,	,	PUNCT
ejpam-5372	362	37	z	z	NOUN
ejpam-5372	362	38	)	)	PUNCT
ejpam-5372	362	39	generators	generator	NOUN
ejpam-5372	362	40	,	,	PUNCT
ejpam-5372	362	41	which	which	PRON
ejpam-5372	362	42	will	will	AUX
ejpam-5372	362	43	eventually	eventually	ADV
ejpam-5372	362	44	cover	cover	VERB
ejpam-5372	362	45	the	the	DET
ejpam-5372	362	46	entire	entire	ADJ
ejpam-5372	362	47	set	set	NOUN
ejpam-5372	362	48	gz	gz	NOUN
ejpam-5372	362	49	.	.	PUNCT
ejpam-5372	363	1	remark	remark	PROPN
ejpam-5372	363	2	3	3	NUM
ejpam-5372	363	3	.	.	PUNCT
ejpam-5372	364	1	for	for	ADP
ejpam-5372	364	2	each	each	DET
ejpam-5372	364	3	value	value	NOUN
ejpam-5372	364	4	of	of	ADP
ejpam-5372	364	5	a	a	DET
ejpam-5372	364	6	∈	∈	NOUN
ejpam-5372	364	7	{	{	PUNCT
ejpam-5372	364	8	1	1	NUM
ejpam-5372	364	9	}	}	PUNCT
ejpam-5372	364	10	∪	∪	NOUN
ejpam-5372	364	11	a	a	PRON
ejpam-5372	364	12	,	,	PUNCT
ejpam-5372	364	13	we	we	PRON
ejpam-5372	364	14	obtain	obtain	VERB
ejpam-5372	364	15	logd2	logd2	NOUN
ejpam-5372	364	16	(	(	PUNCT
ejpam-5372	364	17	1	1	NUM
ejpam-5372	364	18	,	,	PUNCT
ejpam-5372	364	19	z	z	NOUN
ejpam-5372	364	20	)	)	PUNCT
ejpam-5372	364	21	pairwise	pairwise	VERB
ejpam-5372	364	22	distinct	distinct	ADJ
ejpam-5372	364	23	generators	generator	NOUN
ejpam-5372	364	24	.	.	PUNCT
ejpam-5372	365	1	we	we	PRON
ejpam-5372	365	2	deduce	deduce	VERB
ejpam-5372	365	3	that	that	DET
ejpam-5372	365	4	logda	logda	NOUN
ejpam-5372	365	5	(	(	PUNCT
ejpam-5372	365	6	1	1	NUM
ejpam-5372	365	7	,	,	PUNCT
ejpam-5372	365	8	z	z	NOUN
ejpam-5372	365	9	)	)	PUNCT
ejpam-5372	365	10	divides	divide	VERB
ejpam-5372	365	11	|gz	|gz	NUM
ejpam-5372	366	1	|	|	ADV
ejpam-5372	366	2	=	=	SYM
ejpam-5372	366	3	φ	φ	PROPN
ejpam-5372	366	4	(	(	PUNCT
ejpam-5372	366	5	p−	p−	NOUN
ejpam-5372	366	6	1	1	NUM
ejpam-5372	366	7	)	)	PUNCT
ejpam-5372	366	8	=	=	SYM
ejpam-5372	366	9	φ	φ	PROPN
ejpam-5372	366	10	(	(	PUNCT
ejpam-5372	366	11	2n)φ	2n)φ	PROPN
ejpam-5372	366	12	(	(	PUNCT
ejpam-5372	366	13	z	z	NOUN
ejpam-5372	366	14	)	)	PUNCT
ejpam-5372	366	15	.	.	PUNCT
ejpam-5372	367	1	in	in	ADP
ejpam-5372	367	2	fact	fact	NOUN
ejpam-5372	367	3	,	,	PUNCT
ejpam-5372	367	4	since	since	SCONJ
ejpam-5372	367	5	logd2	logd2	NOUN
ejpam-5372	367	6	(	(	PUNCT
ejpam-5372	367	7	1	1	NUM
ejpam-5372	367	8	,	,	PUNCT
ejpam-5372	367	9	z	z	NOUN
ejpam-5372	367	10	)	)	PUNCT
ejpam-5372	367	11	is	be	AUX
ejpam-5372	367	12	the	the	DET
ejpam-5372	367	13	(	(	PUNCT
ejpam-5372	367	14	multiplicative	multiplicative	ADJ
ejpam-5372	367	15	)	)	PUNCT
ejpam-5372	367	16	order	order	NOUN
ejpam-5372	367	17	of	of	ADP
ejpam-5372	367	18	2	2	NUM
ejpam-5372	367	19	modulo	modulo	NOUN
ejpam-5372	367	20	z	z	NOUN
ejpam-5372	367	21	,	,	PUNCT
ejpam-5372	367	22	we	we	PRON
ejpam-5372	367	23	know	know	VERB
ejpam-5372	367	24	that	that	SCONJ
ejpam-5372	367	25	it	it	PRON
ejpam-5372	367	26	divides	divide	VERB
ejpam-5372	367	27	φ	φ	PROPN
ejpam-5372	367	28	(	(	PUNCT
ejpam-5372	367	29	z	z	NOUN
ejpam-5372	367	30	)	)	PUNCT
ejpam-5372	367	31	,	,	PUNCT
ejpam-5372	367	32	the	the	DET
ejpam-5372	367	33	order	order	NOUN
ejpam-5372	367	34	of	of	ADP
ejpam-5372	367	35	the	the	DET
ejpam-5372	367	36	group	group	NOUN
ejpam-5372	367	37	(	(	PUNCT
ejpam-5372	367	38	z	z	NOUN
ejpam-5372	367	39	/	/	SYM
ejpam-5372	367	40	zz)×.	zz)×.	PROPN
ejpam-5372	367	41	remark	remark	NOUN
ejpam-5372	367	42	also	also	ADV
ejpam-5372	367	43	that	that	SCONJ
ejpam-5372	367	44	logd2	logd2	NOUN
ejpam-5372	367	45	(	(	PUNCT
ejpam-5372	367	46	1	1	NUM
ejpam-5372	367	47	,	,	PUNCT
ejpam-5372	367	48	z	z	NOUN
ejpam-5372	367	49	)	)	PUNCT
ejpam-5372	367	50	>	>	X
ejpam-5372	368	1	log2	log2	PROPN
ejpam-5372	368	2	(	(	PUNCT
ejpam-5372	368	3	z	z	NOUN
ejpam-5372	368	4	)	)	PUNCT
ejpam-5372	368	5	since	since	SCONJ
ejpam-5372	368	6	necessarily	necessarily	ADV
ejpam-5372	368	7	2logd2(1,z	2logd2(1,z	NUM
ejpam-5372	368	8	)	)	PUNCT
ejpam-5372	368	9	>	>	PUNCT
ejpam-5372	368	10	z.	z.	PROPN
ejpam-5372	368	11	corollary	corollary	NOUN
ejpam-5372	368	12	4	4	NUM
ejpam-5372	368	13	.	.	PUNCT
ejpam-5372	369	1	we	we	PRON
ejpam-5372	369	2	have	have	VERB
ejpam-5372	369	3	logd2	logd2	NOUN
ejpam-5372	369	4	(	(	PUNCT
ejpam-5372	369	5	1	1	NUM
ejpam-5372	369	6	,	,	PUNCT
ejpam-5372	369	7	z	z	NOUN
ejpam-5372	369	8	)	)	PUNCT
ejpam-5372	369	9	≤	≤	NOUN
ejpam-5372	370	1	φ	φ	PROPN
ejpam-5372	370	2	(	(	PUNCT
ejpam-5372	370	3	z	z	NOUN
ejpam-5372	370	4	)	)	PUNCT
ejpam-5372	370	5	≤	≤	PUNCT
ejpam-5372	371	1	⌊	⌊	PROPN
ejpam-5372	371	2	z	z	NOUN
ejpam-5372	371	3	logd2(1,z	logd2(1,z	PROPN
ejpam-5372	371	4	)	)	PUNCT
ejpam-5372	371	5	⌋	⌋	NOUN
ejpam-5372	371	6	logd2	logd2	NOUN
ejpam-5372	371	7	(	(	PUNCT
ejpam-5372	371	8	1	1	NUM
ejpam-5372	371	9	,	,	PUNCT
ejpam-5372	371	10	z	z	NOUN
ejpam-5372	371	11	)	)	PUNCT
ejpam-5372	371	12	.	.	PUNCT
ejpam-5372	372	1	proof	proof	NOUN
ejpam-5372	372	2	.	.	PUNCT
ejpam-5372	373	1	as	as	SCONJ
ejpam-5372	373	2	stated	state	VERB
ejpam-5372	373	3	in	in	ADP
ejpam-5372	373	4	the	the	DET
ejpam-5372	373	5	remark	remark	NOUN
ejpam-5372	373	6	3	3	NUM
ejpam-5372	373	7	,	,	PUNCT
ejpam-5372	373	8	we	we	PRON
ejpam-5372	373	9	know	know	VERB
ejpam-5372	373	10	that	that	SCONJ
ejpam-5372	373	11	logd2	logd2	NOUN
ejpam-5372	373	12	(	(	PUNCT
ejpam-5372	373	13	1	1	NUM
ejpam-5372	373	14	,	,	PUNCT
ejpam-5372	373	15	z	z	NOUN
ejpam-5372	373	16	)	)	PUNCT
ejpam-5372	373	17	divides	divide	VERB
ejpam-5372	373	18	φ	φ	PROPN
ejpam-5372	373	19	(	(	PUNCT
ejpam-5372	373	20	z	z	NOUN
ejpam-5372	373	21	)	)	PUNCT
ejpam-5372	373	22	.	.	PUNCT
ejpam-5372	374	1	therefore	therefore	ADV
ejpam-5372	374	2	logd2	logd2	NOUN
ejpam-5372	374	3	(	(	PUNCT
ejpam-5372	374	4	1	1	NUM
ejpam-5372	374	5	,	,	PUNCT
ejpam-5372	374	6	z	z	NOUN
ejpam-5372	374	7	)	)	PUNCT
ejpam-5372	374	8	≤	≤	NOUN
ejpam-5372	374	9	φ	φ	PROPN
ejpam-5372	374	10	(	(	PUNCT
ejpam-5372	374	11	z	z	NOUN
ejpam-5372	374	12	)	)	PUNCT
ejpam-5372	374	13	and	and	CCONJ
ejpam-5372	374	14	φ	φ	PROPN
ejpam-5372	374	15	(	(	PUNCT
ejpam-5372	374	16	z	z	NOUN
ejpam-5372	374	17	)	)	PUNCT
ejpam-5372	374	18	=	=	SYM
ejpam-5372	374	19	φ(z	φ(z	ADJ
ejpam-5372	374	20	)	)	PUNCT
ejpam-5372	374	21	logd2(1,z	logd2(1,z	PROPN
ejpam-5372	374	22	)	)	PUNCT
ejpam-5372	374	23	logd2	logd2	NOUN
ejpam-5372	374	24	(	(	PUNCT
ejpam-5372	374	25	1	1	NUM
ejpam-5372	374	26	,	,	PUNCT
ejpam-5372	374	27	z	z	NOUN
ejpam-5372	374	28	)	)	PUNCT
ejpam-5372	374	29	≤	≤	PUNCT
ejpam-5372	374	30	⌊	⌊	PROPN
ejpam-5372	374	31	z	z	NOUN
ejpam-5372	374	32	logd2(1,z	logd2(1,z	PROPN
ejpam-5372	374	33	)	)	PUNCT
ejpam-5372	374	34	⌋	⌋	NOUN
ejpam-5372	375	1	logd2	logd2	NOUN
ejpam-5372	376	1	(	(	PUNCT
ejpam-5372	376	2	1	1	NUM
ejpam-5372	376	3	,	,	PUNCT
ejpam-5372	376	4	z	z	NOUN
ejpam-5372	376	5	)	)	PUNCT
ejpam-5372	376	6	.	.	PUNCT
ejpam-5372	377	1	m.	m.	NOUN
ejpam-5372	377	2	wolf	wolf	PROPN
ejpam-5372	377	3	,	,	PUNCT
ejpam-5372	377	4	f.	f.	PROPN
ejpam-5372	377	5	wolf	wolf	PROPN
ejpam-5372	377	6	/	/	SYM
ejpam-5372	377	7	eur	eur	PROPN
ejpam-5372	377	8	.	.	PUNCT
ejpam-5372	378	1	j.	j.	PROPN
ejpam-5372	378	2	pure	pure	PROPN
ejpam-5372	378	3	appl	appl	PROPN
ejpam-5372	378	4	.	.	PROPN
ejpam-5372	378	5	math	math	PROPN
ejpam-5372	378	6	,	,	PUNCT
ejpam-5372	378	7	17	17	NUM
ejpam-5372	378	8	(	(	PUNCT
ejpam-5372	378	9	4	4	NUM
ejpam-5372	378	10	)	)	PUNCT
ejpam-5372	378	11	(	(	PUNCT
ejpam-5372	378	12	2024	2024	NUM
ejpam-5372	378	13	)	)	PUNCT
ejpam-5372	378	14	,	,	PUNCT
ejpam-5372	378	15	2431	2431	NUM
ejpam-5372	378	16	-	-	SYM
ejpam-5372	378	17	2447	2447	NUM
ejpam-5372	378	18	2441	2441	NUM
ejpam-5372	378	19	proposition	proposition	NOUN
ejpam-5372	378	20	16	16	NUM
ejpam-5372	378	21	.	.	PUNCT
ejpam-5372	379	1	let	let	VERB
ejpam-5372	379	2	g	g	NOUN
ejpam-5372	379	3	and	and	CCONJ
ejpam-5372	379	4	m	m	VERB
ejpam-5372	379	5	′	′	NOUN
ejpam-5372	379	6	be	be	VERB
ejpam-5372	379	7	as	as	ADP
ejpam-5372	379	8	in	in	ADP
ejpam-5372	379	9	proposition	proposition	NOUN
ejpam-5372	379	10	15	15	NUM
ejpam-5372	379	11	.	.	PUNCT
ejpam-5372	380	1	there	there	PRON
ejpam-5372	380	2	exists	exist	VERB
ejpam-5372	380	3	a	a	DET
ejpam-5372	380	4	set	set	NOUN
ejpam-5372	380	5	a	a	PRON
ejpam-5372	380	6	of	of	ADP
ejpam-5372	380	7	φ	φ	PROPN
ejpam-5372	380	8	(	(	PUNCT
ejpam-5372	380	9	p−1	p−1	PROPN
ejpam-5372	380	10	2	2	NUM
ejpam-5372	380	11	)	)	PUNCT
ejpam-5372	380	12	logd2(1,z	logd2(1,z	NOUN
ejpam-5372	380	13	)	)	PUNCT
ejpam-5372	380	14	primes	prime	NOUN
ejpam-5372	380	15	,	,	PUNCT
ejpam-5372	380	16	odd	odd	ADJ
ejpam-5372	380	17	and	and	CCONJ
ejpam-5372	380	18	coprime	coprime	NOUN
ejpam-5372	380	19	to	to	ADP
ejpam-5372	380	20	z	z	NOUN
ejpam-5372	380	21	,	,	PUNCT
ejpam-5372	380	22	such	such	ADJ
ejpam-5372	380	23	that	that	SCONJ
ejpam-5372	380	24	:	:	PUNCT
ejpam-5372	380	25	gs	gs	X
ejpam-5372	380	26	=	=	PUNCT
ejpam-5372	380	27	{	{	PUNCT
ejpam-5372	380	28	g2	g2	PROPN
ejpam-5372	380	29	xam	xam	PROPN
ejpam-5372	380	30	′(2x+1−2)a	′(2x+1−2)a	PROPN
ejpam-5372	380	31	,	,	PUNCT
ejpam-5372	380	32	(	(	PUNCT
ejpam-5372	380	33	a	a	PRON
ejpam-5372	380	34	,	,	PUNCT
ejpam-5372	380	35	x	x	NOUN
ejpam-5372	380	36	)	)	PUNCT
ejpam-5372	380	37	∈	∈	PROPN
ejpam-5372	380	38	(	(	PUNCT
ejpam-5372	380	39	{	{	PUNCT
ejpam-5372	380	40	1	1	NUM
ejpam-5372	380	41	}	}	PUNCT
ejpam-5372	380	42	∪a)×	∪a)×	NOUN
ejpam-5372	381	1	[	[	X
ejpam-5372	381	2	[	[	X
ejpam-5372	381	3	0	0	NUM
ejpam-5372	381	4	,	,	PUNCT
ejpam-5372	381	5	logd2	logd2	NOUN
ejpam-5372	381	6	(	(	PUNCT
ejpam-5372	381	7	1	1	NUM
ejpam-5372	381	8	,	,	PUNCT
ejpam-5372	381	9	z	z	NOUN
ejpam-5372	381	10	)	)	PUNCT
ejpam-5372	381	11	−	−	PROPN
ejpam-5372	381	12	1	1	NUM
ejpam-5372	381	13	]	]	X
ejpam-5372	381	14	]	]	PUNCT
ejpam-5372	381	15	}	}	PUNCT
ejpam-5372	381	16	if	if	SCONJ
ejpam-5372	381	17	p	p	PROPN
ejpam-5372	381	18	∈	∈	PROPN
ejpam-5372	381	19	p3,4	p3,4	VERB
ejpam-5372	381	20	,	,	PUNCT
ejpam-5372	381	21	gz	gz	NOUN
ejpam-5372	381	22	=	=	SYM
ejpam-5372	381	23	{	{	PUNCT
ejpam-5372	381	24	g2	g2	PROPN
ejpam-5372	381	25	xam	xam	PROPN
ejpam-5372	381	26	′(2x+1−1)a	′(2x+1−1)a	PROPN
ejpam-5372	381	27	,	,	PUNCT
ejpam-5372	382	1	(	(	PUNCT
ejpam-5372	382	2	a	a	PRON
ejpam-5372	382	3	,	,	PUNCT
ejpam-5372	382	4	x	x	NOUN
ejpam-5372	382	5	)	)	PUNCT
ejpam-5372	382	6	∈	∈	PROPN
ejpam-5372	382	7	(	(	PUNCT
ejpam-5372	382	8	{	{	PUNCT
ejpam-5372	382	9	1	1	NUM
ejpam-5372	382	10	}	}	PUNCT
ejpam-5372	382	11	∪a)×	∪a)×	NOUN
ejpam-5372	382	12	[	[	X
ejpam-5372	382	13	[	[	X
ejpam-5372	382	14	0	0	NUM
ejpam-5372	382	15	,	,	PUNCT
ejpam-5372	382	16	logd2	logd2	NOUN
ejpam-5372	382	17	(	(	PUNCT
ejpam-5372	382	18	1	1	NUM
ejpam-5372	382	19	,	,	PUNCT
ejpam-5372	382	20	z	z	NOUN
ejpam-5372	382	21	)	)	PUNCT
ejpam-5372	382	22	−	−	PROPN
ejpam-5372	382	23	1	1	NUM
ejpam-5372	382	24	]	]	PUNCT
ejpam-5372	382	25	]	]	PUNCT
ejpam-5372	382	26	}	}	PUNCT
ejpam-5372	382	27	.	.	PUNCT
ejpam-5372	383	1	if	if	SCONJ
ejpam-5372	383	2	p	p	PROPN
ejpam-5372	383	3	∈	∈	PROPN
ejpam-5372	383	4	p1,4	p1,4	PROPN
ejpam-5372	383	5	,	,	PUNCT
ejpam-5372	383	6	gz	gz	NOUN
ejpam-5372	383	7	=	=	SYM
ejpam-5372	383	8	{	{	PUNCT
ejpam-5372	383	9	±g2	±g2	PROPN
ejpam-5372	383	10	xam	xam	PROPN
ejpam-5372	383	11	′(2x+1−1)a	′(2x+1−1)a	PROPN
ejpam-5372	383	12	,	,	PUNCT
ejpam-5372	383	13	(	(	PUNCT
ejpam-5372	383	14	a	a	PRON
ejpam-5372	383	15	,	,	PUNCT
ejpam-5372	383	16	x	x	NOUN
ejpam-5372	383	17	)	)	PUNCT
ejpam-5372	383	18	∈	∈	PROPN
ejpam-5372	383	19	(	(	PUNCT
ejpam-5372	383	20	{	{	PUNCT
ejpam-5372	383	21	1	1	NUM
ejpam-5372	383	22	}	}	PUNCT
ejpam-5372	383	23	∪a)×	∪a)×	NOUN
ejpam-5372	384	1	[	[	X
ejpam-5372	384	2	[	[	X
ejpam-5372	384	3	0	0	NUM
ejpam-5372	384	4	,	,	PUNCT
ejpam-5372	384	5	logd2	logd2	NOUN
ejpam-5372	384	6	(	(	PUNCT
ejpam-5372	384	7	1	1	NUM
ejpam-5372	384	8	,	,	PUNCT
ejpam-5372	384	9	z	z	NOUN
ejpam-5372	384	10	)	)	PUNCT
ejpam-5372	384	11	−	−	PROPN
ejpam-5372	384	12	1	1	NUM
ejpam-5372	384	13	]	]	PUNCT
ejpam-5372	384	14	]	]	PUNCT
ejpam-5372	384	15	}	}	PUNCT
ejpam-5372	384	16	.	.	PUNCT
ejpam-5372	385	1	proof	proof	NOUN
ejpam-5372	385	2	.	.	PUNCT
ejpam-5372	386	1	a	a	PRON
ejpam-5372	386	2	can	can	AUX
ejpam-5372	386	3	be	be	AUX
ejpam-5372	386	4	constructed	construct	VERB
ejpam-5372	386	5	by	by	ADP
ejpam-5372	386	6	induction	induction	NOUN
ejpam-5372	386	7	as	as	ADP
ejpam-5372	386	8	in	in	ADP
ejpam-5372	386	9	remark	remark	NOUN
ejpam-5372	386	10	2	2	NUM
ejpam-5372	386	11	,	,	PUNCT
ejpam-5372	386	12	by	by	ADP
ejpam-5372	386	13	ensuring	ensure	VERB
ejpam-5372	386	14	that	that	SCONJ
ejpam-5372	386	15	u	u	PROPN
ejpam-5372	386	16	(	(	PUNCT
ejpam-5372	386	17	a	a	X
ejpam-5372	386	18	)	)	PUNCT
ejpam-5372	386	19	0	0	NUM
ejpam-5372	386	20	is	be	AUX
ejpam-5372	386	21	a	a	DET
ejpam-5372	386	22	new	new	ADJ
ejpam-5372	386	23	element	element	NOUN
ejpam-5372	386	24	of	of	ADP
ejpam-5372	386	25	gs	gs	PROPN
ejpam-5372	386	26	.	.	PUNCT
ejpam-5372	387	1	using	use	VERB
ejpam-5372	387	2	the	the	DET
ejpam-5372	387	3	identification	identification	NOUN
ejpam-5372	387	4	of	of	ADP
ejpam-5372	387	5	f∗	f∗	NOUN
ejpam-5372	387	6	p	p	X
ejpam-5372	387	7	to	to	ADP
ejpam-5372	387	8	(	(	PUNCT
ejpam-5372	387	9	z/2nz	z/2nz	NOUN
ejpam-5372	387	10	)	)	PUNCT
ejpam-5372	387	11	×	×	NOUN
ejpam-5372	387	12	(	(	PUNCT
ejpam-5372	387	13	z	z	PROPN
ejpam-5372	387	14	/	/	SYM
ejpam-5372	387	15	zz	zz	PROPN
ejpam-5372	387	16	)	)	PUNCT
ejpam-5372	387	17	and	and	CCONJ
ejpam-5372	387	18	a	a	DET
ejpam-5372	387	19	,	,	PUNCT
ejpam-5372	387	20	b	b	X
ejpam-5372	387	21	∈	∈	PROPN
ejpam-5372	387	22	(	(	PUNCT
ejpam-5372	387	23	{	{	PUNCT
ejpam-5372	387	24	1	1	NUM
ejpam-5372	387	25	}	}	PUNCT
ejpam-5372	387	26	∪a	∪a	NUM
ejpam-5372	387	27	)	)	PUNCT
ejpam-5372	387	28	distinct	distinct	ADJ
ejpam-5372	387	29	,	,	PUNCT
ejpam-5372	387	30	suppose	suppose	VERB
ejpam-5372	387	31	(	(	PUNCT
ejpam-5372	387	32	2ak0	2ak0	NUM
ejpam-5372	387	33	,	,	PUNCT
ejpam-5372	387	34	2	2	NUM
ejpam-5372	387	35	xal0	xal0	NOUN
ejpam-5372	387	36	)	)	PUNCT
ejpam-5372	388	1	=	=	PUNCT
ejpam-5372	388	2	(	(	PUNCT
ejpam-5372	388	3	2bk0	2bk0	NUM
ejpam-5372	388	4	,	,	PUNCT
ejpam-5372	388	5	2	2	NUM
ejpam-5372	388	6	ybl0	ybl0	PROPN
ejpam-5372	388	7	)	)	PUNCT
ejpam-5372	388	8	.	.	PUNCT
ejpam-5372	389	1	this	this	PRON
ejpam-5372	389	2	means	mean	VERB
ejpam-5372	389	3	that	that	SCONJ
ejpam-5372	389	4	2xal0	2xal0	NUM
ejpam-5372	389	5	≡	≡	PROPN
ejpam-5372	389	6	2ybl0	2ybl0	PUNCT
ejpam-5372	390	1	[	[	X
ejpam-5372	390	2	z	z	X
ejpam-5372	390	3	]	]	X
ejpam-5372	390	4	thus	thus	ADV
ejpam-5372	390	5	,	,	PUNCT
ejpam-5372	390	6	multiplying	multiply	VERB
ejpam-5372	390	7	by	by	ADP
ejpam-5372	390	8	2logd2(1,z	2logd2(1,z	NUM
ejpam-5372	390	9	)	)	PUNCT
ejpam-5372	390	10	−y	−y	NOUN
ejpam-5372	390	11	,	,	PUNCT
ejpam-5372	390	12	u	u	NOUN
ejpam-5372	390	13	(	(	PUNCT
ejpam-5372	390	14	b	b	NOUN
ejpam-5372	390	15	)	)	PUNCT
ejpam-5372	390	16	0	0	NUM
ejpam-5372	391	1	is	be	AUX
ejpam-5372	391	2	one	one	NUM
ejpam-5372	391	3	of	of	ADP
ejpam-5372	391	4	the	the	DET
ejpam-5372	391	5	(	(	PUNCT
ejpam-5372	391	6	u	u	NOUN
ejpam-5372	391	7	(	(	PUNCT
ejpam-5372	391	8	a	a	NOUN
ejpam-5372	391	9	)	)	PUNCT
ejpam-5372	391	10	x	x	SYM
ejpam-5372	391	11	)	)	PUNCT
ejpam-5372	391	12	,	,	PUNCT
ejpam-5372	391	13	which	which	PRON
ejpam-5372	391	14	is	be	AUX
ejpam-5372	391	15	impossible	impossible	ADJ
ejpam-5372	391	16	.	.	PUNCT
ejpam-5372	392	1	if	if	SCONJ
ejpam-5372	392	2	p	p	PROPN
ejpam-5372	392	3	∈	∈	PROPN
ejpam-5372	392	4	p3,4	p3,4	VERB
ejpam-5372	392	5	,	,	PUNCT
ejpam-5372	392	6	remember	remember	VERB
ejpam-5372	392	7	that	that	SCONJ
ejpam-5372	392	8	in	in	ADP
ejpam-5372	392	9	this	this	DET
ejpam-5372	392	10	case	case	NOUN
ejpam-5372	392	11	gs	gs	NOUN
ejpam-5372	392	12	and	and	CCONJ
ejpam-5372	392	13	gz	gz	AUX
ejpam-5372	392	14	have	have	VERB
ejpam-5372	392	15	the	the	DET
ejpam-5372	392	16	same	same	ADJ
ejpam-5372	392	17	size	size	NOUN
ejpam-5372	392	18	.	.	PUNCT
ejpam-5372	393	1	thus	thus	ADV
ejpam-5372	393	2	,	,	PUNCT
ejpam-5372	393	3	once	once	SCONJ
ejpam-5372	393	4	we	we	PRON
ejpam-5372	393	5	have	have	VERB
ejpam-5372	393	6	gs	gs	INTJ
ejpam-5372	393	7	,	,	PUNCT
ejpam-5372	393	8	we	we	PRON
ejpam-5372	393	9	also	also	ADV
ejpam-5372	393	10	get	get	VERB
ejpam-5372	393	11	gz	gz	NOUN
ejpam-5372	393	12	by	by	ADP
ejpam-5372	393	13	multiplying	multiply	VERB
ejpam-5372	393	14	all	all	DET
ejpam-5372	393	15	elements	element	NOUN
ejpam-5372	393	16	by	by	ADP
ejpam-5372	393	17	the	the	DET
ejpam-5372	393	18	suitable	suitable	ADJ
ejpam-5372	393	19	power	power	NOUN
ejpam-5372	393	20	of	of	ADP
ejpam-5372	393	21	m	m	PROPN
ejpam-5372	393	22	′	′	NUM
ejpam-5372	393	23	,	,	PUNCT
ejpam-5372	393	24	which	which	PRON
ejpam-5372	393	25	is	be	AUX
ejpam-5372	393	26	−1	−1	ADV
ejpam-5372	393	27	anyway	anyway	ADV
ejpam-5372	393	28	.	.	PUNCT
ejpam-5372	394	1	when	when	SCONJ
ejpam-5372	394	2	p	p	PROPN
ejpam-5372	394	3	∈	∈	PROPN
ejpam-5372	394	4	p1,4	p1,4	PROPN
ejpam-5372	394	5	,	,	PUNCT
ejpam-5372	394	6	let	let	VERB
ejpam-5372	394	7	us	we	PRON
ejpam-5372	394	8	show	show	VERB
ejpam-5372	394	9	that	that	SCONJ
ejpam-5372	394	10	elements	element	NOUN
ejpam-5372	394	11	±g2	±g2	PROPN
ejpam-5372	394	12	xam	xam	PROPN
ejpam-5372	394	13	′(2x+1−1)a	′(2x+1−1)a	VERB
ejpam-5372	394	14	are	be	AUX
ejpam-5372	394	15	pairwise	pairwise	NOUN
ejpam-5372	394	16	distinct	distinct	ADJ
ejpam-5372	394	17	.	.	PUNCT
ejpam-5372	394	18	suppose	suppose	VERB
ejpam-5372	394	19	that	that	SCONJ
ejpam-5372	394	20	m	m	VERB
ejpam-5372	394	21	′a	′a	ADP
ejpam-5372	394	22	u	u	NOUN
ejpam-5372	394	23	(	(	PUNCT
ejpam-5372	394	24	a	a	NOUN
ejpam-5372	394	25	)	)	PUNCT
ejpam-5372	394	26	x	x	X
ejpam-5372	394	27	=	=	PUNCT
ejpam-5372	394	28	±m	±m	PROPN
ejpam-5372	394	29	′b	′b	PROPN
ejpam-5372	394	30	u	u	NOUN
ejpam-5372	394	31	(	(	PUNCT
ejpam-5372	394	32	b	b	NOUN
ejpam-5372	394	33	)	)	PUNCT
ejpam-5372	394	34	y	y	PROPN
ejpam-5372	394	35	.	.	PUNCT
ejpam-5372	395	1	then	then	ADV
ejpam-5372	395	2	,	,	PUNCT
ejpam-5372	395	3	taking	take	VERB
ejpam-5372	395	4	the	the	DET
ejpam-5372	395	5	square	square	ADJ
ejpam-5372	395	6	yields	yield	NOUN
ejpam-5372	395	7	that	that	PRON
ejpam-5372	395	8	u	u	PROPN
ejpam-5372	395	9	(	(	PUNCT
ejpam-5372	395	10	a	a	NOUN
ejpam-5372	395	11	)	)	PUNCT
ejpam-5372	395	12	x+1	x+1	PUNCT
ejpam-5372	396	1	=	=	SYM
ejpam-5372	396	2	u	u	PROPN
ejpam-5372	396	3	(	(	PUNCT
ejpam-5372	396	4	b	b	NOUN
ejpam-5372	396	5	)	)	PUNCT
ejpam-5372	396	6	y+1	y+1	PROPN
ejpam-5372	396	7	,	,	PUNCT
ejpam-5372	396	8	which	which	PRON
ejpam-5372	396	9	again	again	ADV
ejpam-5372	396	10	is	be	AUX
ejpam-5372	396	11	impossible	impossible	ADJ
ejpam-5372	396	12	.	.	PUNCT
ejpam-5372	396	13	remark	remark	PROPN
ejpam-5372	396	14	4	4	NUM
ejpam-5372	396	15	.	.	PUNCT
ejpam-5372	396	16	when	when	SCONJ
ejpam-5372	396	17	z	z	NOUN
ejpam-5372	396	18	is	be	AUX
ejpam-5372	396	19	a	a	DET
ejpam-5372	396	20	prime	prime	ADJ
ejpam-5372	396	21	number	number	NOUN
ejpam-5372	396	22	,	,	PUNCT
ejpam-5372	396	23	all	all	DET
ejpam-5372	396	24	non	non	NOUN
ejpam-5372	396	25	-	-	NOUN
ejpam-5372	396	26	residues	residue	NOUN
ejpam-5372	396	27	g	g	NOUN
ejpam-5372	396	28	not	not	PART
ejpam-5372	396	29	in	in	ADP
ejpam-5372	396	30	r(n	r(n	PROPN
ejpam-5372	396	31	)	)	PUNCT
ejpam-5372	396	32	are	be	AUX
ejpam-5372	396	33	generators	generator	NOUN
ejpam-5372	396	34	.	.	PUNCT
ejpam-5372	397	1	if	if	SCONJ
ejpam-5372	397	2	,	,	PUNCT
ejpam-5372	397	3	additionally	additionally	ADV
ejpam-5372	397	4	,	,	PUNCT
ejpam-5372	397	5	p	p	PROPN
ejpam-5372	397	6	∈	∈	PROPN
ejpam-5372	397	7	p3,4	p3,4	ADJ
ejpam-5372	397	8	and	and	CCONJ
ejpam-5372	397	9	logd2	logd2	ADJ
ejpam-5372	397	10	(	(	PUNCT
ejpam-5372	397	11	1	1	NUM
ejpam-5372	397	12	,	,	PUNCT
ejpam-5372	397	13	z	z	NOUN
ejpam-5372	397	14	)	)	PUNCT
ejpam-5372	397	15	=	=	PUNCT
ejpam-5372	398	1	z	z	NOUN
ejpam-5372	398	2	−	−	NOUN
ejpam-5372	398	3	1	1	NUM
ejpam-5372	398	4	=	=	SYM
ejpam-5372	398	5	φ	φ	X
ejpam-5372	398	6	(	(	PUNCT
ejpam-5372	398	7	p−	p−	NOUN
ejpam-5372	398	8	1	1	NUM
ejpam-5372	398	9	)	)	PUNCT
ejpam-5372	398	10	then	then	ADV
ejpam-5372	398	11	a	a	DET
ejpam-5372	398	12	=	=	NOUN
ejpam-5372	398	13	∅	∅	NOUN
ejpam-5372	398	14	for	for	ADP
ejpam-5372	398	15	both	both	PRON
ejpam-5372	398	16	remark	remark	NOUN
ejpam-5372	398	17	2	2	NUM
ejpam-5372	398	18	and	and	CCONJ
ejpam-5372	398	19	proposition	proposition	NOUN
ejpam-5372	398	20	16	16	NUM
ejpam-5372	398	21	.	.	PUNCT
ejpam-5372	398	22	remark	remark	PROPN
ejpam-5372	398	23	5	5	NUM
ejpam-5372	398	24	.	.	PUNCT
ejpam-5372	399	1	when	when	SCONJ
ejpam-5372	399	2	p	p	X
ejpam-5372	399	3	>	>	X
ejpam-5372	399	4	3	3	NUM
ejpam-5372	399	5	and	and	CCONJ
ejpam-5372	399	6	z	z	NOUN
ejpam-5372	399	7	=	=	SYM
ejpam-5372	399	8	1	1	NUM
ejpam-5372	399	9	,	,	PUNCT
ejpam-5372	399	10	any	any	PRON
ejpam-5372	399	11	of	of	ADP
ejpam-5372	399	12	the	the	DET
ejpam-5372	399	13	2n−1	2n−1	NUM
ejpam-5372	399	14	non	non	NOUN
ejpam-5372	399	15	-	-	NOUN
ejpam-5372	399	16	residues	residue	NOUN
ejpam-5372	399	17	is	be	AUX
ejpam-5372	399	18	a	a	DET
ejpam-5372	399	19	generator	generator	NOUN
ejpam-5372	399	20	,	,	PUNCT
ejpam-5372	399	21	and	and	CCONJ
ejpam-5372	399	22	the	the	DET
ejpam-5372	399	23	sequences	sequence	NOUN
ejpam-5372	399	24	(	(	PUNCT
ejpam-5372	399	25	u	u	NOUN
ejpam-5372	399	26	(	(	PUNCT
ejpam-5372	399	27	a	a	NOUN
ejpam-5372	399	28	)	)	PUNCT
ejpam-5372	399	29	x	x	X
ejpam-5372	399	30	)	)	PUNCT
ejpam-5372	399	31	are	be	AUX
ejpam-5372	399	32	constants	constant	NOUN
ejpam-5372	399	33	.	.	PUNCT
ejpam-5372	400	1	therefore	therefore	ADV
ejpam-5372	400	2	we	we	PRON
ejpam-5372	400	3	need	need	VERB
ejpam-5372	400	4	a	a	DET
ejpam-5372	400	5	set	set	NOUN
ejpam-5372	400	6	a	a	PRON
ejpam-5372	400	7	of	of	ADP
ejpam-5372	400	8	size	size	NOUN
ejpam-5372	400	9	2n−2	2n−2	PROPN
ejpam-5372	400	10	−	−	PROPN
ejpam-5372	400	11	1	1	NUM
ejpam-5372	400	12	to	to	PART
ejpam-5372	400	13	enumerate	enumerate	VERB
ejpam-5372	400	14	all	all	DET
ejpam-5372	400	15	the	the	DET
ejpam-5372	400	16	generators	generator	NOUN
ejpam-5372	400	17	using	use	VERB
ejpam-5372	400	18	proposition	proposition	NOUN
ejpam-5372	400	19	16	16	NUM
ejpam-5372	400	20	.	.	PUNCT
ejpam-5372	401	1	for	for	ADP
ejpam-5372	401	2	example	example	NOUN
ejpam-5372	401	3	,	,	PUNCT
ejpam-5372	401	4	for	for	ADP
ejpam-5372	401	5	p	p	NOUN
ejpam-5372	401	6	=	=	NOUN
ejpam-5372	401	7	257	257	NUM
ejpam-5372	401	8	,	,	PUNCT
ejpam-5372	401	9	we	we	PRON
ejpam-5372	401	10	need	need	VERB
ejpam-5372	401	11	63	63	NUM
ejpam-5372	401	12	primes	prime	NOUN
ejpam-5372	401	13	,	,	PUNCT
ejpam-5372	401	14	which	which	PRON
ejpam-5372	401	15	means	mean	VERB
ejpam-5372	401	16	that	that	SCONJ
ejpam-5372	401	17	we	we	PRON
ejpam-5372	401	18	must	must	AUX
ejpam-5372	401	19	choose	choose	VERB
ejpam-5372	401	20	primes	prime	NOUN
ejpam-5372	401	21	greater	great	ADJ
ejpam-5372	401	22	than	than	ADP
ejpam-5372	401	23	p.	p.	NOUN
ejpam-5372	401	24	remark	remark	NOUN
ejpam-5372	401	25	6	6	NUM
ejpam-5372	401	26	.	.	PUNCT
ejpam-5372	402	1	for	for	SCONJ
ejpam-5372	402	2	each	each	DET
ejpam-5372	402	3	generator	generator	NOUN
ejpam-5372	402	4	found	find	VERB
ejpam-5372	402	5	,	,	PUNCT
ejpam-5372	402	6	proposition	proposition	NOUN
ejpam-5372	402	7	14	14	NUM
ejpam-5372	402	8	also	also	ADV
ejpam-5372	402	9	gives	give	VERB
ejpam-5372	402	10	potential	potential	ADJ
ejpam-5372	402	11	new	new	ADJ
ejpam-5372	402	12	generators	generator	NOUN
ejpam-5372	402	13	,	,	PUNCT
ejpam-5372	402	14	which	which	PRON
ejpam-5372	402	15	can	can	AUX
ejpam-5372	402	16	reduce	reduce	VERB
ejpam-5372	402	17	the	the	DET
ejpam-5372	402	18	number	number	NOUN
ejpam-5372	402	19	of	of	ADP
ejpam-5372	402	20	primes	prime	NOUN
ejpam-5372	402	21	needed	need	VERB
ejpam-5372	402	22	in	in	ADP
ejpam-5372	402	23	remark	remark	NOUN
ejpam-5372	402	24	2	2	NUM
ejpam-5372	402	25	or	or	CCONJ
ejpam-5372	402	26	proposition	proposition	NOUN
ejpam-5372	402	27	16	16	NUM
ejpam-5372	402	28	.	.	PUNCT
ejpam-5372	403	1	this	this	DET
ejpam-5372	403	2	approach	approach	NOUN
ejpam-5372	403	3	is	be	AUX
ejpam-5372	403	4	left	leave	VERB
ejpam-5372	403	5	to	to	ADP
ejpam-5372	403	6	the	the	DET
ejpam-5372	403	7	reader	reader	NOUN
ejpam-5372	403	8	as	as	SCONJ
ejpam-5372	403	9	it	it	PRON
ejpam-5372	403	10	seems	seem	VERB
ejpam-5372	403	11	less	less	ADV
ejpam-5372	403	12	predictable	predictable	ADJ
ejpam-5372	403	13	but	but	CCONJ
ejpam-5372	403	14	could	could	AUX
ejpam-5372	403	15	prove	prove	VERB
ejpam-5372	403	16	interesting	interesting	ADJ
ejpam-5372	403	17	for	for	ADP
ejpam-5372	403	18	values	value	NOUN
ejpam-5372	403	19	of	of	ADP
ejpam-5372	403	20	p	p	NOUN
ejpam-5372	403	21	greater	great	ADJ
ejpam-5372	403	22	than	than	ADP
ejpam-5372	403	23	those	those	PRON
ejpam-5372	403	24	we	we	PRON
ejpam-5372	403	25	have	have	AUX
ejpam-5372	403	26	tested	test	VERB
ejpam-5372	403	27	and	and	CCONJ
ejpam-5372	403	28	such	such	ADJ
ejpam-5372	403	29	that	that	DET
ejpam-5372	403	30	logd2	logd2	NOUN
ejpam-5372	403	31	(	(	PUNCT
ejpam-5372	403	32	1	1	NUM
ejpam-5372	403	33	,	,	PUNCT
ejpam-5372	403	34	z	z	NOUN
ejpam-5372	403	35	)	)	PUNCT
ejpam-5372	403	36	is	be	AUX
ejpam-5372	403	37	small	small	ADJ
ejpam-5372	403	38	,	,	PUNCT
ejpam-5372	403	39	since	since	SCONJ
ejpam-5372	403	40	the	the	DET
ejpam-5372	403	41	density	density	NOUN
ejpam-5372	403	42	of	of	ADP
ejpam-5372	403	43	primes	prime	NOUN
ejpam-5372	403	44	goes	go	VERB
ejpam-5372	403	45	asymptotically	asymptotically	ADV
ejpam-5372	403	46	towards	towards	ADP
ejpam-5372	403	47	0	0	NUM
ejpam-5372	403	48	.	.	PUNCT
ejpam-5372	404	1	alternatively	alternatively	ADV
ejpam-5372	404	2	,	,	PUNCT
ejpam-5372	404	3	we	we	PRON
ejpam-5372	404	4	could	could	AUX
ejpam-5372	404	5	“	"	PUNCT
ejpam-5372	404	6	sieve	sieve	VERB
ejpam-5372	404	7	”	"	PUNCT
ejpam-5372	404	8	the	the	DET
ejpam-5372	404	9	set	set	NOUN
ejpam-5372	404	10	{	{	PUNCT
ejpam-5372	404	11	k	k	PROPN
ejpam-5372	404	12	∈	∈	PROPN
ejpam-5372	405	1	[	[	X
ejpam-5372	405	2	[	[	X
ejpam-5372	405	3	1	1	NUM
ejpam-5372	405	4	,	,	PUNCT
ejpam-5372	405	5	p−	p−	NOUN
ejpam-5372	405	6	2	2	NUM
ejpam-5372	405	7	]	]	PUNCT
ejpam-5372	405	8	]	]	PUNCT
ejpam-5372	406	1	|	|	ADV
ejpam-5372	406	2	gcd	gcd	VERB
ejpam-5372	406	3	(	(	PUNCT
ejpam-5372	406	4	k	k	NOUN
ejpam-5372	406	5	,	,	PUNCT
ejpam-5372	406	6	p−	p−	NOUN
ejpam-5372	406	7	1	1	NUM
ejpam-5372	406	8	)	)	PUNCT
ejpam-5372	406	9	=	=	SYM
ejpam-5372	406	10	1	1	X
ejpam-5372	406	11	}	}	PUNCT
ejpam-5372	406	12	by	by	ADP
ejpam-5372	406	13	eliminating	eliminate	VERB
ejpam-5372	406	14	multiples	multiple	NOUN
ejpam-5372	406	15	of	of	ADP
ejpam-5372	406	16	prime	prime	ADJ
ejpam-5372	406	17	factors	factor	NOUN
ejpam-5372	406	18	of	of	ADP
ejpam-5372	406	19	p−	p−	NOUN
ejpam-5372	406	20	1	1	NUM
ejpam-5372	406	21	.	.	X
ejpam-5372	406	22	2.2.3	2.2.3	NUM
ejpam-5372	406	23	.	.	PUNCT
ejpam-5372	407	1	algorithm	algorithm	NOUN
ejpam-5372	407	2	to	to	PART
ejpam-5372	407	3	obtain	obtain	VERB
ejpam-5372	407	4	the	the	DET
ejpam-5372	407	5	set	set	NOUN
ejpam-5372	407	6	of	of	ADP
ejpam-5372	407	7	primitive	primitive	ADJ
ejpam-5372	407	8	roots	root	NOUN
ejpam-5372	407	9	the	the	DET
ejpam-5372	407	10	general	general	ADJ
ejpam-5372	407	11	idea	idea	NOUN
ejpam-5372	407	12	of	of	ADP
ejpam-5372	407	13	the	the	DET
ejpam-5372	407	14	algorithm	algorithm	NOUN
ejpam-5372	407	15	is	be	AUX
ejpam-5372	407	16	to	to	PART
ejpam-5372	407	17	search	search	VERB
ejpam-5372	407	18	for	for	ADP
ejpam-5372	407	19	a	a	DET
ejpam-5372	407	20	semi	semi	ADJ
ejpam-5372	407	21	-	-	ADJ
ejpam-5372	407	22	primitive	primitive	ADJ
ejpam-5372	407	23	root	root	NOUN
ejpam-5372	407	24	g	g	PROPN
ejpam-5372	407	25	∈	∈	PROPN
ejpam-5372	407	26	gs	gs	NOUN
ejpam-5372	407	27	and	and	CCONJ
ejpam-5372	407	28	the	the	DET
ejpam-5372	407	29	associated	associated	ADJ
ejpam-5372	407	30	r	r	NOUN
ejpam-5372	407	31	in	in	ADP
ejpam-5372	407	32	r(n	r(n	PROPN
ejpam-5372	407	33	)	)	PUNCT
ejpam-5372	407	34	p	p	X
ejpam-5372	407	35	\r(n−1	\r(n−1	PROPN
ejpam-5372	407	36	)	)	PUNCT
ejpam-5372	407	37	p	p	NOUN
ejpam-5372	407	38	such	such	ADJ
ejpam-5372	407	39	that	that	DET
ejpam-5372	407	40	gzr2z	gzr2z	PROPN
ejpam-5372	407	41	≡	≡	PROPN
ejpam-5372	407	42	1	1	NUM
ejpam-5372	407	43	.	.	PUNCT
ejpam-5372	408	1	following	follow	VERB
ejpam-5372	408	2	the	the	DET
ejpam-5372	408	3	proof	proof	NOUN
ejpam-5372	408	4	of	of	ADP
ejpam-5372	408	5	proposition	proposition	NOUN
ejpam-5372	408	6	16	16	NUM
ejpam-5372	408	7	,	,	PUNCT
ejpam-5372	408	8	we	we	PRON
ejpam-5372	408	9	then	then	ADV
ejpam-5372	408	10	construct	construct	VERB
ejpam-5372	408	11	a	a	DET
ejpam-5372	408	12	set	set	NOUN
ejpam-5372	408	13	a	a	PRON
ejpam-5372	408	14	of	of	ADP
ejpam-5372	408	15	primes	prime	NOUN
ejpam-5372	408	16	to	to	PART
ejpam-5372	408	17	obtain	obtain	VERB
ejpam-5372	408	18	the	the	DET
ejpam-5372	408	19	full	full	ADJ
ejpam-5372	408	20	set	set	NOUN
ejpam-5372	408	21	gz	gz	NOUN
ejpam-5372	408	22	.	.	PUNCT
ejpam-5372	409	1	let	let	VERB
ejpam-5372	409	2	us	we	PRON
ejpam-5372	409	3	describe	describe	VERB
ejpam-5372	409	4	the	the	DET
ejpam-5372	409	5	algorithm	algorithm	NOUN
ejpam-5372	409	6	a	a	DET
ejpam-5372	409	7	bit	bit	NOUN
ejpam-5372	409	8	more	more	ADV
ejpam-5372	409	9	in	in	ADP
ejpam-5372	409	10	details	detail	NOUN
ejpam-5372	409	11	below	below	ADP
ejpam-5372	409	12	.	.	PUNCT
ejpam-5372	410	1	m.	m.	NOUN
ejpam-5372	410	2	wolf	wolf	PROPN
ejpam-5372	410	3	,	,	PUNCT
ejpam-5372	410	4	f.	f.	PROPN
ejpam-5372	410	5	wolf	wolf	PROPN
ejpam-5372	410	6	/	/	SYM
ejpam-5372	410	7	eur	eur	PROPN
ejpam-5372	410	8	.	.	PUNCT
ejpam-5372	411	1	j.	j.	PROPN
ejpam-5372	411	2	pure	pure	PROPN
ejpam-5372	411	3	appl	appl	PROPN
ejpam-5372	411	4	.	.	PROPN
ejpam-5372	411	5	math	math	PROPN
ejpam-5372	411	6	,	,	PUNCT
ejpam-5372	411	7	17	17	NUM
ejpam-5372	411	8	(	(	PUNCT
ejpam-5372	411	9	4	4	NUM
ejpam-5372	411	10	)	)	PUNCT
ejpam-5372	411	11	(	(	PUNCT
ejpam-5372	411	12	2024	2024	NUM
ejpam-5372	411	13	)	)	PUNCT
ejpam-5372	411	14	,	,	PUNCT
ejpam-5372	411	15	2431	2431	NUM
ejpam-5372	411	16	-	-	SYM
ejpam-5372	411	17	2447	2447	NUM
ejpam-5372	411	18	2442	2442	NUM
ejpam-5372	411	19	algorithm	algorithm	NOUN
ejpam-5372	411	20	1	1	NUM
ejpam-5372	411	21	.	.	PUNCT
ejpam-5372	412	1	description	description	NOUN
ejpam-5372	412	2	:	:	PUNCT
ejpam-5372	412	3	we	we	PRON
ejpam-5372	412	4	still	still	ADV
ejpam-5372	412	5	write	write	VERB
ejpam-5372	412	6	p	p	NOUN
ejpam-5372	412	7	−	−	PROPN
ejpam-5372	412	8	1	1	NUM
ejpam-5372	412	9	=	=	SYM
ejpam-5372	412	10	2nz	2nz	NOUN
ejpam-5372	412	11	with	with	ADP
ejpam-5372	412	12	z	z	NOUN
ejpam-5372	412	13	odd	odd	ADJ
ejpam-5372	412	14	.	.	PUNCT
ejpam-5372	413	1	q	q	PUNCT
ejpam-5372	414	1	:	:	PUNCT
ejpam-5372	414	2	=	=	SYM
ejpam-5372	414	3	dp	dp	NOUN
ejpam-5372	414	4	(	(	PUNCT
ejpam-5372	414	5	{	{	PUNCT
ejpam-5372	414	6	z	z	NOUN
ejpam-5372	414	7	}	}	PUNCT
ejpam-5372	414	8	)	)	PUNCT
ejpam-5372	414	9	,	,	PUNCT
ejpam-5372	414	10	the	the	DET
ejpam-5372	414	11	set	set	NOUN
ejpam-5372	414	12	of	of	ADP
ejpam-5372	414	13	prime	prime	ADJ
ejpam-5372	414	14	divisors	divisor	NOUN
ejpam-5372	414	15	of	of	ADP
ejpam-5372	414	16	z	z	PROPN
ejpam-5372	414	17	,	,	PUNCT
ejpam-5372	414	18	is	be	AUX
ejpam-5372	414	19	also	also	ADV
ejpam-5372	414	20	assumed	assume	VERB
ejpam-5372	414	21	to	to	PART
ejpam-5372	414	22	be	be	AUX
ejpam-5372	414	23	known	know	VERB
ejpam-5372	414	24	,	,	PUNCT
ejpam-5372	414	25	allowing	allow	VERB
ejpam-5372	414	26	to	to	PART
ejpam-5372	414	27	calculate	calculate	VERB
ejpam-5372	414	28	φ	φ	PROPN
ejpam-5372	414	29	(	(	PUNCT
ejpam-5372	414	30	z	z	NOUN
ejpam-5372	414	31	)	)	PUNCT
ejpam-5372	414	32	.	.	PUNCT
ejpam-5372	415	1	(	(	PUNCT
ejpam-5372	415	2	i	i	NOUN
ejpam-5372	415	3	)	)	PUNCT
ejpam-5372	415	4	we	we	PRON
ejpam-5372	415	5	iterate	iterate	VERB
ejpam-5372	415	6	over	over	ADP
ejpam-5372	415	7	the	the	DET
ejpam-5372	415	8	elements	element	NOUN
ejpam-5372	415	9	m	m	VERB
ejpam-5372	415	10	in	in	ADP
ejpam-5372	415	11	[	[	X
ejpam-5372	415	12	[	[	X
ejpam-5372	415	13	2	2	NUM
ejpam-5372	415	14	,	,	PUNCT
ejpam-5372	415	15	p−	p−	NOUN
ejpam-5372	415	16	2	2	NUM
ejpam-5372	415	17	]	]	NOUN
ejpam-5372	415	18	]	]	PUNCT
ejpam-5372	415	19	.	.	PUNCT
ejpam-5372	416	1	for	for	ADP
ejpam-5372	416	2	each	each	DET
ejpam-5372	416	3	iteration	iteration	NOUN
ejpam-5372	416	4	,	,	PUNCT
ejpam-5372	416	5	two	two	NUM
ejpam-5372	416	6	tests	test	NOUN
ejpam-5372	416	7	are	be	AUX
ejpam-5372	416	8	performed	perform	VERB
ejpam-5372	416	9	,	,	PUNCT
ejpam-5372	416	10	and	and	CCONJ
ejpam-5372	416	11	we	we	PRON
ejpam-5372	416	12	go	go	VERB
ejpam-5372	416	13	to	to	ADP
ejpam-5372	416	14	the	the	DET
ejpam-5372	416	15	next	next	ADJ
ejpam-5372	416	16	step	step	NOUN
ejpam-5372	416	17	as	as	ADV
ejpam-5372	416	18	soon	soon	ADV
ejpam-5372	416	19	as	as	SCONJ
ejpam-5372	416	20	one	one	PRON
ejpam-5372	416	21	succeeds	succeed	VERB
ejpam-5372	416	22	:	:	PUNCT
ejpam-5372	416	23	•	•	NOUN
ejpam-5372	416	24	if	if	SCONJ
ejpam-5372	416	25	m	m	VERB
ejpam-5372	416	26	∈	∈	PROPN
ejpam-5372	416	27	gqr	gqr	PROPN
ejpam-5372	416	28	and	and	CCONJ
ejpam-5372	416	29	∀q	∀q	PROPN
ejpam-5372	416	30	∈	∈	PROPN
ejpam-5372	416	31	q⧹	q⧹	NOUN
ejpam-5372	416	32	{	{	PUNCT
ejpam-5372	416	33	2	2	NUM
ejpam-5372	416	34	}	}	PUNCT
ejpam-5372	416	35	m	m	VERB
ejpam-5372	416	36	φ(p	φ(p	NOUN
ejpam-5372	416	37	)	)	PUNCT
ejpam-5372	416	38	2q	2q	NOUN
ejpam-5372	416	39	̸≡	̸≡	NOUN
ejpam-5372	417	1	1	1	NUM
ejpam-5372	417	2	then	then	ADV
ejpam-5372	417	3	(	(	PUNCT
ejpam-5372	417	4	proposition	proposition	NOUN
ejpam-5372	417	5	11	11	NUM
ejpam-5372	417	6	)	)	PUNCT
ejpam-5372	417	7	m	m	PROPN
ejpam-5372	417	8	∈	∈	PROPN
ejpam-5372	417	9	g′	g′	NOUN
ejpam-5372	417	10	z	z	NOUN
ejpam-5372	418	1	and	and	CCONJ
ejpam-5372	418	2	we	we	PRON
ejpam-5372	418	3	get	get	VERB
ejpam-5372	418	4	an	an	DET
ejpam-5372	418	5	element	element	NOUN
ejpam-5372	418	6	r	r	NOUN
ejpam-5372	418	7	∈	∈	NOUN
ejpam-5372	418	8	gqnr	gqnr	NOUN
ejpam-5372	418	9	(	(	PUNCT
ejpam-5372	418	10	possibly	possibly	ADV
ejpam-5372	418	11	a	a	DET
ejpam-5372	418	12	value	value	NOUN
ejpam-5372	418	13	r	r	NOUN
ejpam-5372	418	14	<	<	X
ejpam-5372	418	15	m	m	VERB
ejpam-5372	418	16	on	on	ADP
ejpam-5372	418	17	which	which	PRON
ejpam-5372	418	18	we	we	PRON
ejpam-5372	418	19	have	have	AUX
ejpam-5372	418	20	already	already	ADV
ejpam-5372	418	21	performed	perform	VERB
ejpam-5372	418	22	the	the	DET
ejpam-5372	418	23	test	test	NOUN
ejpam-5372	418	24	)	)	PUNCT
ejpam-5372	418	25	.	.	PUNCT
ejpam-5372	419	1	we	we	PRON
ejpam-5372	419	2	then	then	ADV
ejpam-5372	419	3	set	set	VERB
ejpam-5372	419	4	gz	gz	PROPN
ejpam-5372	419	5	≡	≡	PROPN
ejpam-5372	419	6	m.rz	m.rz	PROPN
ejpam-5372	419	7	and	and	CCONJ
ejpam-5372	419	8	g	g	PROPN
ejpam-5372	419	9	≡	≡	PROPN
ejpam-5372	419	10	gz	gz	PROPN
ejpam-5372	419	11	2	2	NUM
ejpam-5372	419	12	.	.	NOUN
ejpam-5372	420	1	•	•	NOUN
ejpam-5372	420	2	otherwise	otherwise	ADV
ejpam-5372	420	3	m	m	VERB
ejpam-5372	420	4	∈	∈	NOUN
ejpam-5372	420	5	gqnr	gqnr	NOUN
ejpam-5372	420	6	and	and	CCONJ
ejpam-5372	420	7	if	if	SCONJ
ejpam-5372	420	8	,	,	PUNCT
ejpam-5372	420	9	in	in	ADP
ejpam-5372	420	10	addition	addition	NOUN
ejpam-5372	420	11	,	,	PUNCT
ejpam-5372	420	12	∀q	∀q	PROPN
ejpam-5372	420	13	∈	∈	PROPN
ejpam-5372	420	14	q	q	NOUN
ejpam-5372	420	15	m	m	VERB
ejpam-5372	420	16	φ(p	φ(p	PROPN
ejpam-5372	420	17	)	)	PUNCT
ejpam-5372	420	18	q	q	NOUN
ejpam-5372	421	1	̸≡	̸≡	NOUN
ejpam-5372	421	2	1	1	NUM
ejpam-5372	421	3	then	then	ADV
ejpam-5372	421	4	(	(	PUNCT
ejpam-5372	421	5	proposition	proposition	NOUN
ejpam-5372	421	6	5	5	NUM
ejpam-5372	421	7	)	)	PUNCT
ejpam-5372	421	8	m	m	VERB
ejpam-5372	421	9	∈	∈	NOUN
ejpam-5372	421	10	gz	gz	NOUN
ejpam-5372	422	1	and	and	CCONJ
ejpam-5372	422	2	we	we	PRON
ejpam-5372	422	3	set	set	VERB
ejpam-5372	422	4	gz	gz	PROPN
ejpam-5372	422	5	=	=	NOUN
ejpam-5372	422	6	m	m	PROPN
ejpam-5372	422	7	and	and	CCONJ
ejpam-5372	422	8	g	g	NOUN
ejpam-5372	422	9	=	=	SYM
ejpam-5372	422	10	gz	gz	PROPN
ejpam-5372	422	11	2	2	NUM
ejpam-5372	422	12	.	.	PUNCT
ejpam-5372	422	13	(	(	PUNCT
ejpam-5372	422	14	ii	ii	NOUN
ejpam-5372	422	15	)	)	PUNCT
ejpam-5372	422	16	we	we	PRON
ejpam-5372	422	17	iterate	iterate	VERB
ejpam-5372	422	18	over	over	ADP
ejpam-5372	422	19	r(n	r(n	PROPN
ejpam-5372	422	20	)	)	PUNCT
ejpam-5372	422	21	p	p	X
ejpam-5372	422	22	\r(n−1	\r(n−1	PROPN
ejpam-5372	422	23	)	)	PUNCT
ejpam-5372	422	24	p	p	NOUN
ejpam-5372	423	1	=	=	X
ejpam-5372	423	2	{	{	PUNCT
ejpam-5372	423	3	±(gz	±(gz	X
ejpam-5372	423	4	z)2k+1	z)2k+1	PROPN
ejpam-5372	423	5	,	,	PUNCT
ejpam-5372	423	6	0	0	NUM
ejpam-5372	423	7	≤	≤	PUNCT
ejpam-5372	424	1	k	k	X
ejpam-5372	424	2	<	<	X
ejpam-5372	424	3	2n−1	2n−1	NUM
ejpam-5372	424	4	}	}	PUNCT
ejpam-5372	424	5	,	,	PUNCT
ejpam-5372	424	6	searching	search	VERB
ejpam-5372	424	7	for	for	ADP
ejpam-5372	424	8	a	a	DET
ejpam-5372	424	9	value	value	NOUN
ejpam-5372	424	10	m	m	NOUN
ejpam-5372	424	11	′	′	NOUN
ejpam-5372	424	12	of	of	ADP
ejpam-5372	424	13	r(n	r(n	PROPN
ejpam-5372	424	14	)	)	PUNCT
ejpam-5372	424	15	p	p	X
ejpam-5372	424	16	\r(n−1	\r(n−1	PROPN
ejpam-5372	424	17	)	)	PUNCT
ejpam-5372	424	18	p	p	NOUN
ejpam-5372	424	19	such	such	ADJ
ejpam-5372	424	20	that	that	DET
ejpam-5372	424	21	gzm	gzm	NOUN
ejpam-5372	424	22	′2z	′2z	PUNCT
ejpam-5372	424	23	≡	≡	PROPN
ejpam-5372	424	24	1	1	NUM
ejpam-5372	424	25	.	.	PUNCT
ejpam-5372	425	1	(	(	PUNCT
ejpam-5372	425	2	iii	iii	X
ejpam-5372	425	3	)	)	PUNCT
ejpam-5372	425	4	we	we	PRON
ejpam-5372	425	5	determine	determine	VERB
ejpam-5372	425	6	logd2	logd2	NOUN
ejpam-5372	425	7	(	(	PUNCT
ejpam-5372	425	8	1	1	NUM
ejpam-5372	425	9	,	,	PUNCT
ejpam-5372	425	10	z	z	NOUN
ejpam-5372	425	11	)	)	PUNCT
ejpam-5372	425	12	by	by	ADP
ejpam-5372	425	13	calculating	calculate	VERB
ejpam-5372	425	14	the	the	DET
ejpam-5372	425	15	first	first	ADJ
ejpam-5372	425	16	terms	term	NOUN
ejpam-5372	425	17	of	of	ADP
ejpam-5372	425	18	the	the	DET
ejpam-5372	425	19	sequence	sequence	NOUN
ejpam-5372	425	20	(	(	PUNCT
ejpam-5372	425	21	ux	ux	NOUN
ejpam-5372	425	22	)	)	PUNCT
ejpam-5372	425	23	(	(	PUNCT
ejpam-5372	425	24	proposition	proposition	NOUN
ejpam-5372	425	25	11	11	NUM
ejpam-5372	425	26	)	)	PUNCT
ejpam-5372	425	27	.	.	PUNCT
ejpam-5372	426	1	the	the	DET
ejpam-5372	426	2	first	first	ADJ
ejpam-5372	426	3	value	value	NOUN
ejpam-5372	426	4	of	of	ADP
ejpam-5372	426	5	x	x	X
ejpam-5372	426	6	≥	≥	NUM
ejpam-5372	426	7	1	1	NUM
ejpam-5372	426	8	such	such	ADJ
ejpam-5372	426	9	that	that	SCONJ
ejpam-5372	426	10	ux	ux	PROPN
ejpam-5372	426	11	≡	≡	PROPN
ejpam-5372	426	12	g	g	PROPN
ejpam-5372	426	13	is	be	AUX
ejpam-5372	426	14	indeed	indeed	ADV
ejpam-5372	426	15	x	x	NOUN
ejpam-5372	426	16	=	=	SYM
ejpam-5372	426	17	logd2	logd2	NOUN
ejpam-5372	426	18	(	(	PUNCT
ejpam-5372	426	19	1	1	NUM
ejpam-5372	426	20	,	,	PUNCT
ejpam-5372	426	21	z	z	NOUN
ejpam-5372	426	22	)	)	PUNCT
ejpam-5372	426	23	.	.	PUNCT
ejpam-5372	427	1	(	(	PUNCT
ejpam-5372	427	2	iv	iv	X
ejpam-5372	427	3	)	)	PUNCT
ejpam-5372	427	4	we	we	PRON
ejpam-5372	427	5	generate	generate	VERB
ejpam-5372	427	6	odd	odd	ADJ
ejpam-5372	427	7	prime	prime	ADJ
ejpam-5372	427	8	numbers	number	NOUN
ejpam-5372	427	9	a1	a1	NOUN
ejpam-5372	427	10	.	.	PUNCT
ejpam-5372	427	11	.	.	PUNCT
ejpam-5372	427	12	.	.	PUNCT
ejpam-5372	428	1	an	an	DET
ejpam-5372	428	2	/∈	/∈	INTJ
ejpam-5372	428	3	dp	dp	NOUN
ejpam-5372	428	4	(	(	PUNCT
ejpam-5372	428	5	{	{	PUNCT
ejpam-5372	428	6	z	z	NOUN
ejpam-5372	428	7	}	}	PUNCT
ejpam-5372	428	8	)	)	PUNCT
ejpam-5372	429	1	such	such	ADJ
ejpam-5372	429	2	that	that	DET
ejpam-5372	429	3	u	u	PROPN
ejpam-5372	429	4	(	(	PUNCT
ejpam-5372	429	5	ak	ak	PROPN
ejpam-5372	429	6	)	)	PUNCT
ejpam-5372	429	7	0	0	NUM
ejpam-5372	429	8	/∈	/∈	PUNCT
ejpam-5372	429	9	{	{	PUNCT
ejpam-5372	429	10	u	u	NOUN
ejpam-5372	429	11	(	(	PUNCT
ejpam-5372	429	12	ak	ak	PROPN
ejpam-5372	429	13	)	)	PUNCT
ejpam-5372	429	14	x	x	NOUN
ejpam-5372	429	15	,	,	PUNCT
ejpam-5372	429	16	x	x	SYM
ejpam-5372	429	17	∈	∈	PROPN
ejpam-5372	429	18	n	n	CCONJ
ejpam-5372	429	19	,	,	PUNCT
ejpam-5372	429	20	k	k	PROPN
ejpam-5372	429	21	∈	∈	PROPN
ejpam-5372	430	1	[	[	X
ejpam-5372	430	2	[	[	X
ejpam-5372	430	3	0,k	0,k	NUM
ejpam-5372	430	4	−	−	NUM
ejpam-5372	430	5	1	1	NUM
ejpam-5372	430	6	]	]	X
ejpam-5372	430	7	]	]	PUNCT
ejpam-5372	430	8	}	}	PUNCT
ejpam-5372	430	9	(	(	PUNCT
ejpam-5372	430	10	where	where	SCONJ
ejpam-5372	430	11	by	by	ADP
ejpam-5372	430	12	convention	convention	NOUN
ejpam-5372	430	13	u	u	PROPN
ejpam-5372	430	14	(	(	PUNCT
ejpam-5372	430	15	a0	a0	PROPN
ejpam-5372	430	16	)	)	PUNCT
ejpam-5372	430	17	x	x	X
ejpam-5372	430	18	:	:	PUNCT
ejpam-5372	430	19	=	=	SYM
ejpam-5372	430	20	ux	ux	NOUN
ejpam-5372	430	21	)	)	PUNCT
ejpam-5372	430	22	and	and	CCONJ
ejpam-5372	430	23	n	n	CCONJ
ejpam-5372	430	24	=	=	SYM
ejpam-5372	430	25	(	(	PUNCT
ejpam-5372	430	26	φ	φ	PROPN
ejpam-5372	430	27	(	(	PUNCT
ejpam-5372	430	28	p−1	p−1	PROPN
ejpam-5372	430	29	2	2	NUM
ejpam-5372	430	30	)	)	PUNCT
ejpam-5372	430	31	/logd2	/logd2	NOUN
ejpam-5372	430	32	(	(	PUNCT
ejpam-5372	430	33	1	1	NUM
ejpam-5372	430	34	,	,	PUNCT
ejpam-5372	430	35	z	z	NOUN
ejpam-5372	430	36	)	)	PUNCT
ejpam-5372	430	37	)	)	PUNCT
ejpam-5372	430	38	−	−	PROPN
ejpam-5372	431	1	1	1	X
ejpam-5372	431	2	.	.	X
ejpam-5372	431	3	for	for	ADP
ejpam-5372	431	4	example	example	NOUN
ejpam-5372	431	5	,	,	PUNCT
ejpam-5372	431	6	these	these	DET
ejpam-5372	431	7	numbers	number	NOUN
ejpam-5372	431	8	can	can	AUX
ejpam-5372	431	9	be	be	AUX
ejpam-5372	431	10	obtained	obtain	VERB
ejpam-5372	431	11	from	from	ADP
ejpam-5372	431	12	atkin	atkin	PROPN
ejpam-5372	431	13	’s	’s	PART
ejpam-5372	431	14	algorithm	algorithm	NOUN
ejpam-5372	431	15	[	[	X
ejpam-5372	431	16	4	4	X
ejpam-5372	431	17	]	]	PUNCT
ejpam-5372	431	18	by	by	ADP
ejpam-5372	431	19	rejecting	reject	VERB
ejpam-5372	431	20	those	those	PRON
ejpam-5372	431	21	that	that	PRON
ejpam-5372	431	22	are	be	AUX
ejpam-5372	431	23	not	not	PART
ejpam-5372	431	24	suitable	suitable	ADJ
ejpam-5372	431	25	.	.	PUNCT
ejpam-5372	432	1	(	(	PUNCT
ejpam-5372	432	2	v	v	X
ejpam-5372	432	3	)	)	PUNCT
ejpam-5372	432	4	we	we	PRON
ejpam-5372	432	5	have	have	AUX
ejpam-5372	432	6	thus	thus	ADV
ejpam-5372	432	7	recursively	recursively	ADV
ejpam-5372	432	8	constructed	construct	VERB
ejpam-5372	432	9	gs	gs	NOUN
ejpam-5372	432	10	=	=	PUNCT
ejpam-5372	432	11	{	{	PUNCT
ejpam-5372	432	12	u	u	PROPN
ejpam-5372	432	13	(	(	PUNCT
ejpam-5372	432	14	ak	ak	PROPN
ejpam-5372	432	15	)	)	PUNCT
ejpam-5372	432	16	x	x	NOUN
ejpam-5372	432	17	,	,	PUNCT
ejpam-5372	432	18	x	x	SYM
ejpam-5372	432	19	∈	∈	PROPN
ejpam-5372	432	20	n	n	CCONJ
ejpam-5372	432	21	,	,	PUNCT
ejpam-5372	432	22	k	k	PROPN
ejpam-5372	432	23	∈	∈	PROPN
ejpam-5372	433	1	[	[	X
ejpam-5372	433	2	[	[	X
ejpam-5372	433	3	0,k	0,k	NUM
ejpam-5372	433	4	−	−	NUM
ejpam-5372	433	5	1	1	NUM
ejpam-5372	433	6	]	]	X
ejpam-5372	433	7	]	]	PUNCT
ejpam-5372	433	8	}	}	PUNCT
ejpam-5372	433	9	and	and	CCONJ
ejpam-5372	433	10	gz	gz	VERB
ejpam-5372	433	11	=	=	NOUN
ejpam-5372	433	12	{	{	PUNCT
ejpam-5372	433	13	m	m	PROPN
ejpam-5372	433	14	′ak	′ak	PROPN
ejpam-5372	433	15	u	u	PROPN
ejpam-5372	433	16	(	(	PUNCT
ejpam-5372	433	17	ak	ak	PROPN
ejpam-5372	433	18	)	)	PUNCT
ejpam-5372	433	19	x	x	PUNCT
ejpam-5372	433	20	}	}	PUNCT
ejpam-5372	433	21	when	when	SCONJ
ejpam-5372	433	22	p	p	PROPN
ejpam-5372	433	23	∈	∈	PROPN
ejpam-5372	433	24	p1,4	p1,4	PROPN
ejpam-5372	433	25	,	,	PUNCT
ejpam-5372	433	26	gz	gz	NOUN
ejpam-5372	433	27	=	=	SYM
ejpam-5372	433	28	{	{	PUNCT
ejpam-5372	433	29	±m	±m	PROPN
ejpam-5372	433	30	′ak	′ak	NOUN
ejpam-5372	433	31	u	u	NOUN
ejpam-5372	433	32	(	(	PUNCT
ejpam-5372	433	33	ak	ak	PROPN
ejpam-5372	433	34	)	)	PUNCT
ejpam-5372	433	35	x	x	SYM
ejpam-5372	433	36	}	}	PUNCT
ejpam-5372	433	37	otherwise	otherwise	ADV
ejpam-5372	433	38	.	.	PUNCT
ejpam-5372	434	1	3	3	X
ejpam-5372	434	2	.	.	X
ejpam-5372	434	3	irreducible	irreducible	ADJ
ejpam-5372	434	4	quadratic	quadratic	ADJ
ejpam-5372	434	5	forms	form	NOUN
ejpam-5372	434	6	in	in	ADP
ejpam-5372	434	7	[	[	X
ejpam-5372	434	8	2	2	NUM
ejpam-5372	434	9	]	]	PUNCT
ejpam-5372	434	10	,	,	PUNCT
ejpam-5372	434	11	we	we	PRON
ejpam-5372	434	12	studied	study	VERB
ejpam-5372	434	13	the	the	DET
ejpam-5372	434	14	density	density	NOUN
ejpam-5372	434	15	of	of	ADP
ejpam-5372	434	16	primes	prime	NOUN
ejpam-5372	434	17	in	in	ADP
ejpam-5372	434	18	ec	ec	PROPN
ejpam-5372	434	19	=	=	PUNCT
ejpam-5372	434	20	{	{	PUNCT
ejpam-5372	435	1	x2	x2	PROPN
ejpam-5372	436	1	+	+	CCONJ
ejpam-5372	436	2	c	c	X
ejpam-5372	436	3	,	,	PUNCT
ejpam-5372	436	4	x	x	SYM
ejpam-5372	436	5	∈	∈	PROPN
ejpam-5372	436	6	2n+r	2n+r	NUM
ejpam-5372	436	7	}	}	PUNCT
ejpam-5372	436	8	,	,	PUNCT
ejpam-5372	436	9	with	with	ADP
ejpam-5372	436	10	c	c	PROPN
ejpam-5372	436	11	∈	∈	PROPN
ejpam-5372	436	12	n∗	n∗	PROPN
ejpam-5372	436	13	and	and	CCONJ
ejpam-5372	436	14	r	r	NOUN
ejpam-5372	436	15	∈	∈	PROPN
ejpam-5372	436	16	{	{	PUNCT
ejpam-5372	436	17	0	0	NUM
ejpam-5372	436	18	,	,	PUNCT
ejpam-5372	436	19	1	1	NUM
ejpam-5372	436	20	}	}	PUNCT
ejpam-5372	436	21	such	such	ADJ
ejpam-5372	436	22	that	that	SCONJ
ejpam-5372	436	23	r	r	NOUN
ejpam-5372	436	24	≡	≡	PROPN
ejpam-5372	436	25	1−	1−	NUM
ejpam-5372	436	26	c	c	PROPN
ejpam-5372	437	1	[	[	X
ejpam-5372	437	2	2	2	NUM
ejpam-5372	437	3	]	]	PUNCT
ejpam-5372	437	4	.	.	PUNCT
ejpam-5372	438	1	we	we	PRON
ejpam-5372	438	2	empirically	empirically	ADV
ejpam-5372	438	3	corroborated	corroborate	VERB
ejpam-5372	438	4	shanks	shank	NOUN
ejpam-5372	438	5	’	’	PART
ejpam-5372	438	6	conjecture	conjecture	NOUN
ejpam-5372	438	7	,	,	PUNCT
ejpam-5372	438	8	which	which	PRON
ejpam-5372	438	9	gives	give	VERB
ejpam-5372	438	10	the	the	DET
ejpam-5372	438	11	asymptotic	asymptotic	ADJ
ejpam-5372	438	12	density	density	NOUN
ejpam-5372	438	13	of	of	ADP
ejpam-5372	438	14	primes	prime	NOUN
ejpam-5372	438	15	in	in	ADP
ejpam-5372	438	16	ec	ec	PROPN
ejpam-5372	438	17	:	:	PUNCT
ejpam-5372	438	18	dp|ec	dp|ec	X
ejpam-5372	438	19	(	(	PUNCT
ejpam-5372	438	20	x	x	NOUN
ejpam-5372	438	21	)	)	PUNCT
ejpam-5372	438	22	∼	∼	NOUN
ejpam-5372	438	23	hc	hc	NOUN
ejpam-5372	438	24	2ln(x	2ln(x	NUM
ejpam-5372	438	25	)	)	PUNCT
ejpam-5372	438	26	with	with	ADP
ejpam-5372	438	27	hc	hc	PROPN
ejpam-5372	438	28	=	=	SYM
ejpam-5372	438	29	∏	∏	PROPN
ejpam-5372	438	30	p∈p	p∈p	NOUN
ejpam-5372	438	31	p−tp(c	p−tp(c	ADV
ejpam-5372	438	32	)	)	PUNCT
ejpam-5372	438	33	p−1	p−1	PROPN
ejpam-5372	438	34	<	<	X
ejpam-5372	438	35	∞	∞	PROPN
ejpam-5372	438	36	and	and	CCONJ
ejpam-5372	438	37	tp	tp	X
ejpam-5372	438	38	(	(	PUNCT
ejpam-5372	438	39	c	c	NOUN
ejpam-5372	438	40	)	)	PUNCT
ejpam-5372	438	41	=	=	PUNCT
ejpam-5372	439	1			PUNCT
ejpam-5372	439	2	0	0	NUM
ejpam-5372	439	3	if	if	SCONJ
ejpam-5372	439	4	p	p	X
ejpam-5372	439	5	/∈	/∈	INTJ
ejpam-5372	440	1	dp	dp	NOUN
ejpam-5372	440	2	(	(	PUNCT
ejpam-5372	440	3	ec	ec	PROPN
ejpam-5372	440	4	)	)	PUNCT
ejpam-5372	440	5	1	1	NUM
ejpam-5372	440	6	if	if	SCONJ
ejpam-5372	440	7	p|c	p|c	NOUN
ejpam-5372	440	8	2	2	NUM
ejpam-5372	440	9	otherwise	otherwise	ADV
ejpam-5372	440	10	.	.	PUNCT
ejpam-5372	441	1	here	here	ADV
ejpam-5372	441	2	,	,	PUNCT
ejpam-5372	441	3	we	we	PRON
ejpam-5372	441	4	extend	extend	VERB
ejpam-5372	441	5	the	the	DET
ejpam-5372	441	6	definition	definition	NOUN
ejpam-5372	441	7	of	of	ADP
ejpam-5372	441	8	hc	hc	PRON
ejpam-5372	441	9	to	to	PART
ejpam-5372	441	10	integer	integer	VERB
ejpam-5372	441	11	quadratic	quadratic	ADJ
ejpam-5372	441	12	forms	form	NOUN
ejpam-5372	441	13	by	by	ADP
ejpam-5372	441	14	letting	let	VERB
ejpam-5372	441	15	,	,	PUNCT
ejpam-5372	441	16	for	for	ADP
ejpam-5372	441	17	q	q	NOUN
ejpam-5372	441	18	=	=	SYM
ejpam-5372	441	19	ax2	ax2	NOUN
ejpam-5372	441	20	+	+	CCONJ
ejpam-5372	441	21	bx	bx	NOUN
ejpam-5372	442	1	+	+	NOUN
ejpam-5372	442	2	c	c	NOUN
ejpam-5372	442	3	:	:	PUNCT
ejpam-5372	442	4	tp	tp	X
ejpam-5372	442	5	(	(	PUNCT
ejpam-5372	442	6	q	q	X
ejpam-5372	442	7	)	)	PUNCT
ejpam-5372	442	8	=	=	SYM
ejpam-5372	442	9	|{x	|{x	SYM
ejpam-5372	442	10	∈	∈	NOUN
ejpam-5372	443	1	fp	fp	X
ejpam-5372	443	2	|	|	ADV
ejpam-5372	443	3	q	q	X
ejpam-5372	443	4	(	(	PUNCT
ejpam-5372	443	5	x	x	NOUN
ejpam-5372	443	6	)	)	PUNCT
ejpam-5372	443	7	≡	≡	PROPN
ejpam-5372	443	8	0	0	PUNCT
ejpam-5372	444	1	[	[	X
ejpam-5372	444	2	p]}|	p]}|	X
ejpam-5372	444	3	,	,	PUNCT
ejpam-5372	444	4	hq	hq	NOUN
ejpam-5372	444	5	=	=	SYM
ejpam-5372	444	6	∏	∏	PROPN
ejpam-5372	444	7	p∈p⧹{2	p∈p⧹{2	PROPN
ejpam-5372	444	8	}	}	PUNCT
ejpam-5372	444	9	p−tp(q	p−tp(q	NOUN
ejpam-5372	444	10	)	)	PUNCT
ejpam-5372	444	11	p−1	p−1	PROPN
ejpam-5372	444	12	.	.	PUNCT
ejpam-5372	445	1	this	this	DET
ejpam-5372	445	2	definition	definition	NOUN
ejpam-5372	445	3	extends	extend	VERB
ejpam-5372	445	4	that	that	PRON
ejpam-5372	445	5	of	of	ADP
ejpam-5372	445	6	hc	hc	PROPN
ejpam-5372	445	7	i.e.	i.e.	X
ejpam-5372	445	8	hc	hc	X
ejpam-5372	445	9	=	=	PUNCT
ejpam-5372	445	10	h(2x+r)2+c	h(2x+r)2+c	PROPN
ejpam-5372	445	11	.	.	PUNCT
ejpam-5372	446	1	we	we	PRON
ejpam-5372	446	2	continue	continue	VERB
ejpam-5372	446	3	this	this	DET
ejpam-5372	446	4	study	study	NOUN
ejpam-5372	446	5	here	here	ADV
ejpam-5372	446	6	and	and	CCONJ
ejpam-5372	446	7	present	present	VERB
ejpam-5372	446	8	several	several	ADJ
ejpam-5372	446	9	results	result	NOUN
ejpam-5372	446	10	about	about	ADP
ejpam-5372	446	11	hq	hq	NOUN
ejpam-5372	446	12	.	.	PUNCT
ejpam-5372	446	13	m.	m.	PROPN
ejpam-5372	446	14	wolf	wolf	PROPN
ejpam-5372	446	15	,	,	PUNCT
ejpam-5372	446	16	f.	f.	PROPN
ejpam-5372	446	17	wolf	wolf	PROPN
ejpam-5372	446	18	/	/	SYM
ejpam-5372	446	19	eur	eur	PROPN
ejpam-5372	446	20	.	.	PUNCT
ejpam-5372	447	1	j.	j.	PROPN
ejpam-5372	447	2	pure	pure	PROPN
ejpam-5372	447	3	appl	appl	PROPN
ejpam-5372	447	4	.	.	PROPN
ejpam-5372	447	5	math	math	PROPN
ejpam-5372	447	6	,	,	PUNCT
ejpam-5372	447	7	17	17	NUM
ejpam-5372	447	8	(	(	PUNCT
ejpam-5372	447	9	4	4	NUM
ejpam-5372	447	10	)	)	PUNCT
ejpam-5372	447	11	(	(	PUNCT
ejpam-5372	447	12	2024	2024	NUM
ejpam-5372	447	13	)	)	PUNCT
ejpam-5372	447	14	,	,	PUNCT
ejpam-5372	447	15	2431	2431	NUM
ejpam-5372	447	16	-	-	SYM
ejpam-5372	447	17	2447	2447	NUM
ejpam-5372	447	18	2443	2443	NUM
ejpam-5372	447	19	3.1	3.1	NUM
ejpam-5372	447	20	.	.	PUNCT
ejpam-5372	448	1	application	application	NOUN
ejpam-5372	448	2	of	of	ADP
ejpam-5372	448	3	fermat	fermat	PROPN
ejpam-5372	448	4	’s	’s	PART
ejpam-5372	448	5	sum	sum	NOUN
ejpam-5372	448	6	of	of	ADP
ejpam-5372	448	7	two	two	NUM
ejpam-5372	448	8	squares	square	NOUN
ejpam-5372	448	9	theorem	theorem	ADJ
ejpam-5372	448	10	theorem	theorem	NOUN
ejpam-5372	448	11	2	2	X
ejpam-5372	448	12	.	.	PUNCT
ejpam-5372	449	1	let	let	VERB
ejpam-5372	449	2	c	c	NOUN
ejpam-5372	449	3	be	be	AUX
ejpam-5372	449	4	the	the	DET
ejpam-5372	449	5	square	square	NOUN
ejpam-5372	449	6	of	of	ADP
ejpam-5372	449	7	a	a	DET
ejpam-5372	449	8	non	non	ADJ
ejpam-5372	449	9	-	-	ADJ
ejpam-5372	449	10	zero	zero	NUM
ejpam-5372	449	11	integer	integer	NOUN
ejpam-5372	449	12	.	.	PUNCT
ejpam-5372	450	1	we	we	PRON
ejpam-5372	450	2	have	have	VERB
ejpam-5372	450	3	:	:	PUNCT
ejpam-5372	450	4	dp	dp	NOUN
ejpam-5372	450	5	(	(	PUNCT
ejpam-5372	450	6	ec	ec	PROPN
ejpam-5372	450	7	)	)	PUNCT
ejpam-5372	451	1	=	=	PRON
ejpam-5372	451	2	p1,4	p1,4	ADV
ejpam-5372	451	3	∪	∪	VERB
ejpam-5372	451	4	dp	dp	NOUN
ejpam-5372	451	5	(	(	PUNCT
ejpam-5372	451	6	{	{	PUNCT
ejpam-5372	451	7	c	c	NOUN
ejpam-5372	451	8	}	}	PUNCT
ejpam-5372	451	9	)	)	PUNCT
ejpam-5372	452	1	=	=	SYM
ejpam-5372	452	2	p1,4	p1,4	ADV
ejpam-5372	452	3	⊔	⊔	INTJ
ejpam-5372	452	4	(	(	PUNCT
ejpam-5372	452	5	p3,4	p3,4	ADJ
ejpam-5372	452	6	∩	∩	ADJ
ejpam-5372	452	7	dp	dp	NOUN
ejpam-5372	452	8	(	(	PUNCT
ejpam-5372	452	9	{	{	PUNCT
ejpam-5372	452	10	c	c	NOUN
ejpam-5372	452	11	}	}	PUNCT
ejpam-5372	452	12	)	)	PUNCT
ejpam-5372	452	13	)	)	PUNCT
ejpam-5372	452	14	proof	proof	NOUN
ejpam-5372	452	15	.	.	PUNCT
ejpam-5372	453	1	let	let	VERB
ejpam-5372	453	2	us	we	PRON
ejpam-5372	453	3	write	write	VERB
ejpam-5372	453	4	c	c	NOUN
ejpam-5372	453	5	=	=	SYM
ejpam-5372	453	6	y2	y2	PROPN
ejpam-5372	453	7	.	.	PUNCT
ejpam-5372	454	1	let	let	VERB
ejpam-5372	454	2	p	p	PRON
ejpam-5372	454	3	∈	∈	PROPN
ejpam-5372	454	4	dp	dp	NOUN
ejpam-5372	454	5	(	(	PUNCT
ejpam-5372	454	6	ec	ec	PROPN
ejpam-5372	454	7	)	)	PUNCT
ejpam-5372	454	8	,	,	PUNCT
ejpam-5372	454	9	and	and	CCONJ
ejpam-5372	454	10	x	x	X
ejpam-5372	454	11	∈	∈	NOUN
ejpam-5372	454	12	n	n	PRON
ejpam-5372	454	13	such	such	ADJ
ejpam-5372	454	14	that	that	SCONJ
ejpam-5372	454	15	p	p	NOUN
ejpam-5372	454	16	divides	divide	VERB
ejpam-5372	454	17	x2	x2	PROPN
ejpam-5372	455	1	+	+	CCONJ
ejpam-5372	456	1	y2	y2	NOUN
ejpam-5372	456	2	.	.	PUNCT
ejpam-5372	457	1	we	we	PRON
ejpam-5372	457	2	deduce	deduce	VERB
ejpam-5372	457	3	that	that	SCONJ
ejpam-5372	457	4	p	p	NOUN
ejpam-5372	457	5	divides	divide	VERB
ejpam-5372	457	6	y	y	PROPN
ejpam-5372	457	7	or	or	CCONJ
ejpam-5372	457	8	that	that	DET
ejpam-5372	457	9	−1	−1	NOUN
ejpam-5372	457	10	is	be	AUX
ejpam-5372	457	11	a	a	DET
ejpam-5372	457	12	square	square	ADJ
ejpam-5372	457	13	modulo	modulo	NOUN
ejpam-5372	457	14	p.	p.	NOUN
ejpam-5372	457	15	but	but	CCONJ
ejpam-5372	457	16	−1	−1	NOUN
ejpam-5372	457	17	is	be	AUX
ejpam-5372	457	18	a	a	DET
ejpam-5372	457	19	square	square	ADJ
ejpam-5372	457	20	modulo	modulo	NOUN
ejpam-5372	458	1	p	p	NOUN
ejpam-5372	458	2	if	if	SCONJ
ejpam-5372	458	3	and	and	CCONJ
ejpam-5372	458	4	only	only	ADV
ejpam-5372	458	5	if	if	SCONJ
ejpam-5372	458	6	p	p	PROPN
ejpam-5372	458	7	∈	∈	PROPN
ejpam-5372	458	8	p1,4	p1,4	PROPN
ejpam-5372	458	9	,	,	PUNCT
ejpam-5372	458	10	thus	thus	ADV
ejpam-5372	458	11	we	we	PRON
ejpam-5372	458	12	have	have	VERB
ejpam-5372	458	13	p	p	NOUN
ejpam-5372	458	14	∈	∈	PROPN
ejpam-5372	458	15	p1,4	p1,4	ADV
ejpam-5372	458	16	∪	∪	VERB
ejpam-5372	458	17	dp	dp	NOUN
ejpam-5372	458	18	(	(	PUNCT
ejpam-5372	458	19	{	{	PUNCT
ejpam-5372	458	20	y	y	NOUN
ejpam-5372	458	21	}	}	PUNCT
ejpam-5372	458	22	)	)	PUNCT
ejpam-5372	459	1	=	=	SYM
ejpam-5372	459	2	p1,4	p1,4	ADV
ejpam-5372	459	3	⊔	⊔	INTJ
ejpam-5372	459	4	(	(	PUNCT
ejpam-5372	459	5	p3,4	p3,4	ADJ
ejpam-5372	459	6	∩	∩	ADJ
ejpam-5372	459	7	dp	dp	NOUN
ejpam-5372	459	8	(	(	PUNCT
ejpam-5372	459	9	{	{	PUNCT
ejpam-5372	459	10	y	y	NOUN
ejpam-5372	459	11	}	}	PUNCT
ejpam-5372	459	12	)	)	PUNCT
ejpam-5372	459	13	)	)	PUNCT
ejpam-5372	459	14	.	.	PUNCT
ejpam-5372	460	1	conversely	conversely	ADV
ejpam-5372	460	2	,	,	PUNCT
ejpam-5372	460	3	let	let	VERB
ejpam-5372	460	4	p	p	PRON
ejpam-5372	460	5	∈	∈	PROPN
ejpam-5372	460	6	p1,4	p1,4	ADV
ejpam-5372	460	7	∪	∪	VERB
ejpam-5372	460	8	dp	dp	NOUN
ejpam-5372	460	9	(	(	PUNCT
ejpam-5372	460	10	{	{	PUNCT
ejpam-5372	460	11	y	y	NOUN
ejpam-5372	460	12	}	}	PUNCT
ejpam-5372	460	13	)	)	PUNCT
ejpam-5372	460	14	.	.	PUNCT
ejpam-5372	461	1	if	if	SCONJ
ejpam-5372	461	2	p	p	PROPN
ejpam-5372	461	3	∈	∈	PROPN
ejpam-5372	461	4	dp	dp	NOUN
ejpam-5372	461	5	(	(	PUNCT
ejpam-5372	461	6	{	{	PUNCT
ejpam-5372	461	7	y	y	NOUN
ejpam-5372	461	8	}	}	PUNCT
ejpam-5372	461	9	)	)	PUNCT
ejpam-5372	461	10	it	it	PRON
ejpam-5372	461	11	is	be	AUX
ejpam-5372	461	12	clear	clear	ADJ
ejpam-5372	461	13	that	that	SCONJ
ejpam-5372	461	14	p	p	NOUN
ejpam-5372	461	15	divides	divide	VERB
ejpam-5372	461	16	02	02	NUM
ejpam-5372	461	17	+	+	NUM
ejpam-5372	461	18	y2	y2	NOUN
ejpam-5372	461	19	∈	∈	PROPN
ejpam-5372	461	20	ec	ec	PROPN
ejpam-5372	462	1	so	so	ADV
ejpam-5372	462	2	p	p	PROPN
ejpam-5372	462	3	∈	∈	PROPN
ejpam-5372	462	4	ec	ec	PROPN
ejpam-5372	462	5	.	.	PUNCT
ejpam-5372	463	1	now	now	ADV
ejpam-5372	463	2	assume	assume	VERB
ejpam-5372	463	3	p	p	X
ejpam-5372	463	4	∈	∈	PROPN
ejpam-5372	463	5	p1,4	p1,4	PROPN
ejpam-5372	463	6	.	.	PUNCT
ejpam-5372	464	1	fermat	fermat	PROPN
ejpam-5372	464	2	’s	’s	PART
ejpam-5372	464	3	sum	sum	NOUN
ejpam-5372	464	4	of	of	ADP
ejpam-5372	464	5	two	two	NUM
ejpam-5372	464	6	squares	square	NOUN
ejpam-5372	464	7	theorem	theorem	VERB
ejpam-5372	464	8	[	[	X
ejpam-5372	464	9	5	5	NUM
ejpam-5372	464	10	]	]	PUNCT
ejpam-5372	464	11	ensures	ensure	VERB
ejpam-5372	464	12	that	that	SCONJ
ejpam-5372	464	13	there	there	PRON
ejpam-5372	464	14	exist	exist	VERB
ejpam-5372	464	15	a	a	DET
ejpam-5372	464	16	,	,	PUNCT
ejpam-5372	464	17	b	b	PROPN
ejpam-5372	464	18	∈	∈	PROPN
ejpam-5372	464	19	n∗	n∗	PROPN
ejpam-5372	464	20	coprime	coprime	VERB
ejpam-5372	464	21	such	such	ADJ
ejpam-5372	464	22	that	that	SCONJ
ejpam-5372	464	23	p	p	PROPN
ejpam-5372	464	24	=	=	PROPN
ejpam-5372	464	25	a2	a2	PROPN
ejpam-5372	464	26	+	+	CCONJ
ejpam-5372	464	27	b2	b2	NOUN
ejpam-5372	464	28	.	.	PUNCT
ejpam-5372	465	1	but	but	CCONJ
ejpam-5372	465	2	then	then	ADV
ejpam-5372	465	3	we	we	PRON
ejpam-5372	465	4	note	note	VERB
ejpam-5372	465	5	that	that	SCONJ
ejpam-5372	466	1	p	p	PROPN
ejpam-5372	466	2	(	(	PUNCT
ejpam-5372	466	3	c2	c2	PROPN
ejpam-5372	466	4	+	+	CCONJ
ejpam-5372	466	5	d2	d2	PROPN
ejpam-5372	466	6	)	)	PUNCT
ejpam-5372	466	7	=	=	PUNCT
ejpam-5372	466	8	(	(	PUNCT
ejpam-5372	466	9	ac−	ac−	PROPN
ejpam-5372	466	10	bd)2+(ad+	bd)2+(ad+	NOUN
ejpam-5372	466	11	bc)2	bc)2	NOUN
ejpam-5372	466	12	.	.	PUNCT
ejpam-5372	467	1	since	since	SCONJ
ejpam-5372	467	2	a	a	PRON
ejpam-5372	467	3	and	and	CCONJ
ejpam-5372	467	4	b	b	NOUN
ejpam-5372	467	5	are	be	AUX
ejpam-5372	467	6	coprime	coprime	ADJ
ejpam-5372	467	7	,	,	PUNCT
ejpam-5372	467	8	there	there	PRON
ejpam-5372	467	9	exist	exist	VERB
ejpam-5372	467	10	c	c	NOUN
ejpam-5372	467	11	,	,	PUNCT
ejpam-5372	467	12	d	d	PROPN
ejpam-5372	467	13	∈	∈	PROPN
ejpam-5372	467	14	z	z	NOUN
ejpam-5372	467	15	such	such	ADJ
ejpam-5372	467	16	that	that	SCONJ
ejpam-5372	467	17	ad+bc	ad+bc	PRON
ejpam-5372	467	18	=	=	SYM
ejpam-5372	467	19	y	y	PROPN
ejpam-5372	467	20	,	,	PUNCT
ejpam-5372	467	21	thus	thus	ADV
ejpam-5372	467	22	by	by	ADP
ejpam-5372	467	23	letting	let	VERB
ejpam-5372	467	24	x	x	SYM
ejpam-5372	468	1	=	=	SYM
ejpam-5372	468	2	|ac−	|ac−	PROPN
ejpam-5372	468	3	bd|	bd|	NOUN
ejpam-5372	468	4	we	we	PRON
ejpam-5372	468	5	obtain	obtain	VERB
ejpam-5372	468	6	that	that	PRON
ejpam-5372	468	7	x2	x2	PROPN
ejpam-5372	469	1	+	+	CCONJ
ejpam-5372	469	2	y2	y2	PROPN
ejpam-5372	469	3	is	be	AUX
ejpam-5372	469	4	a	a	DET
ejpam-5372	469	5	multiple	multiple	NOUN
ejpam-5372	469	6	of	of	ADP
ejpam-5372	469	7	p	p	NOUN
ejpam-5372	469	8	,	,	PUNCT
ejpam-5372	469	9	hence	hence	ADV
ejpam-5372	469	10	p	p	PROPN
ejpam-5372	469	11	∈	∈	PROPN
ejpam-5372	469	12	ec	ec	PROPN
ejpam-5372	469	13	.	.	PUNCT
ejpam-5372	470	1	proposition	proposition	NOUN
ejpam-5372	470	2	17	17	NUM
ejpam-5372	470	3	.	.	PUNCT
ejpam-5372	471	1	if	if	SCONJ
ejpam-5372	471	2	c	c	PROPN
ejpam-5372	471	3	is	be	AUX
ejpam-5372	471	4	a	a	DET
ejpam-5372	471	5	non	non	ADJ
ejpam-5372	471	6	-	-	ADJ
ejpam-5372	471	7	zero	zero	NUM
ejpam-5372	471	8	square	square	NOUN
ejpam-5372	471	9	,	,	PUNCT
ejpam-5372	471	10	we	we	PRON
ejpam-5372	471	11	have	have	VERB
ejpam-5372	471	12	:	:	PUNCT
ejpam-5372	471	13	hc	hc	VERB
ejpam-5372	471	14	=	=	PROPN
ejpam-5372	471	15	h1	h1	PROPN
ejpam-5372	471	16			PROPN
ejpam-5372	471	17	∏	∏	PROPN
ejpam-5372	471	18	p∈p3,4∩dp({c	p∈p3,4∩dp({c	NOUN
ejpam-5372	471	19	}	}	PUNCT
ejpam-5372	471	20	)	)	PUNCT
ejpam-5372	472	1	p−	p−	NOUN
ejpam-5372	472	2	1	1	NUM
ejpam-5372	472	3	p	p	NOUN
ejpam-5372	472	4			PUNCT
ejpam-5372	472	5	∏	∏	PROPN
ejpam-5372	472	6	p∈p1,4∩dp({c	p∈p1,4∩dp({c	NOUN
ejpam-5372	472	7	}	}	PUNCT
ejpam-5372	472	8	)	)	PUNCT
ejpam-5372	473	1	p−	p−	NOUN
ejpam-5372	473	2	1	1	NUM
ejpam-5372	473	3	p−	p−	NOUN
ejpam-5372	473	4	2	2	NUM
ejpam-5372	473	5			PROPN
ejpam-5372	473	6	proof	proof	NOUN
ejpam-5372	473	7	.	.	PUNCT
ejpam-5372	474	1	we	we	PRON
ejpam-5372	474	2	recall	recall	VERB
ejpam-5372	474	3	that	that	PRON
ejpam-5372	474	4	hc	hc	PROPN
ejpam-5372	474	5	=	=	SYM
ejpam-5372	474	6	∏	∏	PROPN
ejpam-5372	474	7	p∈p⧹{2	p∈p⧹{2	PROPN
ejpam-5372	474	8	}	}	PUNCT
ejpam-5372	474	9	p−tp(c	p−tp(c	ADJ
ejpam-5372	474	10	)	)	PUNCT
ejpam-5372	474	11	p−1	p−1	PROPN
ejpam-5372	474	12	.	.	PUNCT
ejpam-5372	475	1	thus	thus	ADV
ejpam-5372	475	2	by	by	ADP
ejpam-5372	475	3	theorem	theorem	NOUN
ejpam-5372	475	4	2	2	NUM
ejpam-5372	475	5	,	,	PUNCT
ejpam-5372	475	6	h1	h1	NOUN
ejpam-5372	475	7	=	=	PUNCT
ejpam-5372	475	8	limn→∞	limn→∞	PROPN
ejpam-5372	475	9	∏	∏	VERB
ejpam-5372	475	10	p	p	ADJ
ejpam-5372	475	11	∈	∈	PROPN
ejpam-5372	475	12	p3,4	p3,4	ADJ
ejpam-5372	475	13	p	p	NOUN
ejpam-5372	475	14	≤	≤	NUM
ejpam-5372	475	15	n	n	CCONJ
ejpam-5372	475	16	p	p	X
ejpam-5372	475	17	p−1	p−1	PROPN
ejpam-5372	475	18	×	×	NOUN
ejpam-5372	475	19	∏	∏	VERB
ejpam-5372	475	20	p	p	PROPN
ejpam-5372	475	21	∈	∈	PROPN
ejpam-5372	475	22	p1,4	p1,4	ADV
ejpam-5372	475	23	p	p	NOUN
ejpam-5372	475	24	≤	≤	NOUN
ejpam-5372	475	25	n	n	CCONJ
ejpam-5372	475	26	p−2	p−2	PROPN
ejpam-5372	475	27	p−1	p−1	PROPN
ejpam-5372	475	28			NOUN
ejpam-5372	475	29	and	and	CCONJ
ejpam-5372	475	30	hc	hc	X
ejpam-5372	475	31	=	=	NOUN
ejpam-5372	475	32	limn→∞	limn→∞	PROPN
ejpam-5372	475	33	∏	∏	VERB
ejpam-5372	475	34	p	p	PROPN
ejpam-5372	475	35	∈	∈	ADJ
ejpam-5372	475	36	p3,4\d	p3,4\d	X
ejpam-5372	475	37	(	(	PUNCT
ejpam-5372	475	38	c	c	NOUN
ejpam-5372	475	39	)	)	PUNCT
ejpam-5372	475	40	p	p	NOUN
ejpam-5372	475	41	≤	≤	NOUN
ejpam-5372	475	42	n	n	CCONJ
ejpam-5372	475	43	p	p	X
ejpam-5372	475	44	p−1	p−1	PROPN
ejpam-5372	475	45	×	×	NOUN
ejpam-5372	475	46	∏	∏	VERB
ejpam-5372	475	47	p	p	PROPN
ejpam-5372	475	48	∈	∈	ADJ
ejpam-5372	475	49	p1,4\d	p1,4\d	X
ejpam-5372	475	50	(	(	PUNCT
ejpam-5372	475	51	c	c	NOUN
ejpam-5372	475	52	)	)	PUNCT
ejpam-5372	475	53	p	p	NOUN
ejpam-5372	475	54	≤	≤	NOUN
ejpam-5372	475	55	n	n	CCONJ
ejpam-5372	475	56	p−2	p−2	PROPN
ejpam-5372	475	57	p−1	p−1	PROPN
ejpam-5372	475	58			NOUN
ejpam-5372	475	59	.	.	PUNCT
ejpam-5372	476	1	which	which	PRON
ejpam-5372	476	2	shows	show	VERB
ejpam-5372	476	3	indeed	indeed	ADV
ejpam-5372	476	4	that	that	SCONJ
ejpam-5372	476	5	hc	hc	VERB
ejpam-5372	476	6	h1	h1	PROPN
ejpam-5372	476	7	=	=	SYM
ejpam-5372	476	8	(	(	PUNCT
ejpam-5372	476	9	∏	∏	PROPN
ejpam-5372	476	10	p∈p3,4∩dp({c	p∈p3,4∩dp({c	NOUN
ejpam-5372	476	11	}	}	PUNCT
ejpam-5372	476	12	)	)	PUNCT
ejpam-5372	477	1	p−1	p−1	PROPN
ejpam-5372	477	2	p	p	PROPN
ejpam-5372	477	3	)	)	PUNCT
ejpam-5372	477	4	(	(	PUNCT
ejpam-5372	477	5	∏	∏	PROPN
ejpam-5372	477	6	p∈p1,4∩dp({c	p∈p1,4∩dp({c	NOUN
ejpam-5372	477	7	}	}	PUNCT
ejpam-5372	477	8	)	)	PUNCT
ejpam-5372	478	1	p−1	p−1	PROPN
ejpam-5372	478	2	p−2	p−2	PROPN
ejpam-5372	478	3	)	)	PUNCT
ejpam-5372	478	4	.	.	PUNCT
ejpam-5372	479	1	corollary	corollary	ADJ
ejpam-5372	479	2	5	5	NUM
ejpam-5372	479	3	.	.	PUNCT
ejpam-5372	480	1	let	let	VERB
ejpam-5372	480	2	h	h	NOUN
ejpam-5372	480	3	̸=	̸=	PROPN
ejpam-5372	480	4	h1	h1	PROPN
ejpam-5372	480	5	∈	∈	PROPN
ejpam-5372	480	6	r+	r+	NOUN
ejpam-5372	480	7	.	.	PUNCT
ejpam-5372	481	1	if	if	SCONJ
ejpam-5372	481	2	there	there	PRON
ejpam-5372	481	3	is	be	VERB
ejpam-5372	481	4	a	a	DET
ejpam-5372	481	5	square	square	ADJ
ejpam-5372	481	6	c	c	NOUN
ejpam-5372	481	7	such	such	ADJ
ejpam-5372	481	8	that	that	DET
ejpam-5372	481	9	hc	hc	PROPN
ejpam-5372	481	10	=	=	NOUN
ejpam-5372	481	11	h	h	NOUN
ejpam-5372	481	12	then	then	ADV
ejpam-5372	481	13	there	there	PRON
ejpam-5372	481	14	exists	exist	VERB
ejpam-5372	481	15	an	an	DET
ejpam-5372	481	16	infinity	infinity	NOUN
ejpam-5372	481	17	of	of	ADP
ejpam-5372	481	18	such	such	ADJ
ejpam-5372	481	19	c.	c.	NOUN
ejpam-5372	481	20	proof	proof	NOUN
ejpam-5372	481	21	.	.	PUNCT
ejpam-5372	482	1	since	since	SCONJ
ejpam-5372	482	2	the	the	DET
ejpam-5372	482	3	formula	formula	NOUN
ejpam-5372	482	4	in	in	ADP
ejpam-5372	482	5	proposition	proposition	NOUN
ejpam-5372	482	6	17	17	NUM
ejpam-5372	482	7	depends	depend	VERB
ejpam-5372	482	8	only	only	ADV
ejpam-5372	482	9	on	on	ADP
ejpam-5372	482	10	dp	dp	NOUN
ejpam-5372	482	11	(	(	PUNCT
ejpam-5372	482	12	{	{	PUNCT
ejpam-5372	482	13	c	c	NOUN
ejpam-5372	482	14	}	}	PUNCT
ejpam-5372	482	15	)	)	PUNCT
ejpam-5372	482	16	̸=	̸=	NOUN
ejpam-5372	482	17	∅	∅	NOUN
ejpam-5372	482	18	,	,	PUNCT
ejpam-5372	482	19	it	it	PRON
ejpam-5372	482	20	suffices	suffice	VERB
ejpam-5372	482	21	to	to	PART
ejpam-5372	482	22	observe	observe	VERB
ejpam-5372	482	23	that	that	SCONJ
ejpam-5372	482	24	there	there	PRON
ejpam-5372	482	25	is	be	VERB
ejpam-5372	482	26	an	an	DET
ejpam-5372	482	27	infinity	infinity	NOUN
ejpam-5372	482	28	of	of	ADP
ejpam-5372	482	29	c	c	NOUN
ejpam-5372	482	30	which	which	PRON
ejpam-5372	482	31	share	share	VERB
ejpam-5372	482	32	the	the	DET
ejpam-5372	482	33	same	same	ADJ
ejpam-5372	482	34	set	set	NOUN
ejpam-5372	482	35	of	of	ADP
ejpam-5372	482	36	prime	prime	ADJ
ejpam-5372	482	37	factors	factor	NOUN
ejpam-5372	482	38	.	.	PUNCT
ejpam-5372	483	1	remark	remark	VERB
ejpam-5372	483	2	:	:	PUNCT
ejpam-5372	483	3	if	if	SCONJ
ejpam-5372	483	4	we	we	PRON
ejpam-5372	483	5	choose	choose	VERB
ejpam-5372	483	6	all	all	DET
ejpam-5372	483	7	the	the	DET
ejpam-5372	483	8	prime	prime	ADJ
ejpam-5372	483	9	factors	factor	NOUN
ejpam-5372	483	10	of	of	ADP
ejpam-5372	483	11	c	c	PROPN
ejpam-5372	483	12	in	in	ADP
ejpam-5372	483	13	p1,4	p1,4	PROPN
ejpam-5372	483	14	,	,	PUNCT
ejpam-5372	483	15	then	then	ADV
ejpam-5372	483	16	hc	hc	PROPN
ejpam-5372	483	17	≥	≥	PROPN
ejpam-5372	483	18	h1	h1	PROPN
ejpam-5372	483	19	.	.	PUNCT
ejpam-5372	484	1	proposition	proposition	NOUN
ejpam-5372	484	2	18	18	NUM
ejpam-5372	484	3	.	.	PUNCT
ejpam-5372	485	1	let	let	VERB
ejpam-5372	485	2	a	a	DET
ejpam-5372	485	3	,	,	PUNCT
ejpam-5372	485	4	b	b	X
ejpam-5372	485	5	∈	∈	PROPN
ejpam-5372	485	6	n	n	PRON
ejpam-5372	485	7	such	such	ADJ
ejpam-5372	485	8	that	that	SCONJ
ejpam-5372	485	9	dp	dp	NOUN
ejpam-5372	485	10	(	(	PUNCT
ejpam-5372	485	11	{	{	PUNCT
ejpam-5372	485	12	a	a	NOUN
ejpam-5372	485	13	}	}	PUNCT
ejpam-5372	485	14	)	)	PUNCT
ejpam-5372	486	1	⊂	⊂	PROPN
ejpam-5372	486	2	dp	dp	INTJ
ejpam-5372	486	3	(	(	PUNCT
ejpam-5372	486	4	{	{	PUNCT
ejpam-5372	486	5	c	c	NOUN
ejpam-5372	486	6	}	}	PUNCT
ejpam-5372	486	7	)	)	PUNCT
ejpam-5372	486	8	and	and	CCONJ
ejpam-5372	486	9	b	b	X
ejpam-5372	486	10	≡	≡	PROPN
ejpam-5372	486	11	r	r	NOUN
ejpam-5372	487	1	[	[	X
ejpam-5372	487	2	2	2	NUM
ejpam-5372	487	3	]	]	PUNCT
ejpam-5372	487	4	.	.	PUNCT
ejpam-5372	488	1	let	let	VERB
ejpam-5372	488	2	q	q	NOUN
ejpam-5372	489	1	=	=	PUNCT
ejpam-5372	489	2	(	(	PUNCT
ejpam-5372	489	3	2ax	2ax	ADJ
ejpam-5372	489	4	+	+	CCONJ
ejpam-5372	489	5	b)2	b)2	ADJ
ejpam-5372	489	6	+	+	CCONJ
ejpam-5372	489	7	c	c	NOUN
ejpam-5372	489	8	and	and	CCONJ
ejpam-5372	489	9	eq	eq	NOUN
ejpam-5372	489	10	=	=	NOUN
ejpam-5372	489	11	{	{	PUNCT
ejpam-5372	489	12	q	q	X
ejpam-5372	489	13	(	(	PUNCT
ejpam-5372	489	14	x	x	NOUN
ejpam-5372	489	15	)	)	PUNCT
ejpam-5372	489	16	,	,	PUNCT
ejpam-5372	489	17	x	x	PUNCT
ejpam-5372	489	18	∈	∈	PROPN
ejpam-5372	489	19	n	n	CCONJ
ejpam-5372	489	20	}	}	PUNCT
ejpam-5372	489	21	.	.	PUNCT
ejpam-5372	490	1	we	we	PRON
ejpam-5372	490	2	assume	assume	VERB
ejpam-5372	490	3	that	that	SCONJ
ejpam-5372	490	4	dp	dp	NOUN
ejpam-5372	490	5	(	(	PUNCT
ejpam-5372	490	6	{	{	PUNCT
ejpam-5372	490	7	c})∩dp	c})∩dp	NOUN
ejpam-5372	490	8	(	(	PUNCT
ejpam-5372	490	9	eq	eq	NOUN
ejpam-5372	490	10	)	)	PUNCT
ejpam-5372	490	11	=	=	PUNCT
ejpam-5372	490	12	∅.	∅.	ADP
ejpam-5372	490	13	then	then	ADV
ejpam-5372	490	14	:	:	PUNCT
ejpam-5372	490	15	hq	hq	NOUN
ejpam-5372	490	16	=	=	PUNCT
ejpam-5372	490	17	limn→∞	limn→∞	X
ejpam-5372	490	18	∏	∏	VERB
ejpam-5372	490	19	p	p	PROPN
ejpam-5372	490	20	∈	∈	PROPN
ejpam-5372	490	21	p⧹dp	p⧹dp	PROPN
ejpam-5372	490	22	(	(	PUNCT
ejpam-5372	490	23	eq	eq	NOUN
ejpam-5372	490	24	)	)	PUNCT
ejpam-5372	490	25	p	p	NOUN
ejpam-5372	490	26	≤	≤	NOUN
ejpam-5372	491	1	n	n	CCONJ
ejpam-5372	491	2	p	p	X
ejpam-5372	491	3	p−1	p−1	PROPN
ejpam-5372	491	4	×	×	NOUN
ejpam-5372	491	5	∏	∏	VERB
ejpam-5372	491	6	p	p	PROPN
ejpam-5372	491	7	∈	∈	ADJ
ejpam-5372	491	8	dp	dp	NOUN
ejpam-5372	491	9	(	(	PUNCT
ejpam-5372	491	10	eq	eq	NOUN
ejpam-5372	491	11	)	)	PUNCT
ejpam-5372	491	12	p	p	NOUN
ejpam-5372	491	13	≤	≤	NOUN
ejpam-5372	491	14	n	n	CCONJ
ejpam-5372	491	15	p−2	p−2	PROPN
ejpam-5372	491	16	p−1	p−1	PROPN
ejpam-5372	491	17			NOUN
ejpam-5372	491	18	m.	m.	NOUN
ejpam-5372	491	19	wolf	wolf	PROPN
ejpam-5372	491	20	,	,	PUNCT
ejpam-5372	491	21	f.	f.	PROPN
ejpam-5372	491	22	wolf	wolf	PROPN
ejpam-5372	491	23	/	/	SYM
ejpam-5372	491	24	eur	eur	PROPN
ejpam-5372	491	25	.	.	PUNCT
ejpam-5372	492	1	j.	j.	PROPN
ejpam-5372	492	2	pure	pure	PROPN
ejpam-5372	492	3	appl	appl	PROPN
ejpam-5372	492	4	.	.	PROPN
ejpam-5372	492	5	math	math	PROPN
ejpam-5372	492	6	,	,	PUNCT
ejpam-5372	492	7	17	17	NUM
ejpam-5372	492	8	(	(	PUNCT
ejpam-5372	492	9	4	4	NUM
ejpam-5372	492	10	)	)	PUNCT
ejpam-5372	492	11	(	(	PUNCT
ejpam-5372	492	12	2024	2024	NUM
ejpam-5372	492	13	)	)	PUNCT
ejpam-5372	492	14	,	,	PUNCT
ejpam-5372	492	15	2431	2431	NUM
ejpam-5372	492	16	-	-	SYM
ejpam-5372	492	17	2447	2447	NUM
ejpam-5372	492	18	2444	2444	NUM
ejpam-5372	492	19	proof	proof	NOUN
ejpam-5372	492	20	.	.	PUNCT
ejpam-5372	493	1	we	we	PRON
ejpam-5372	493	2	must	must	AUX
ejpam-5372	493	3	show	show	VERB
ejpam-5372	493	4	that	that	SCONJ
ejpam-5372	493	5	tp	tp	X
ejpam-5372	493	6	(	(	PUNCT
ejpam-5372	493	7	q	q	X
ejpam-5372	493	8	)	)	PUNCT
ejpam-5372	493	9	=	=	SYM
ejpam-5372	493	10	2	2	NUM
ejpam-5372	493	11	when	when	SCONJ
ejpam-5372	493	12	p	p	PROPN
ejpam-5372	493	13	∈	∈	PROPN
ejpam-5372	493	14	dp	dp	NOUN
ejpam-5372	493	15	(	(	PUNCT
ejpam-5372	493	16	eq	eq	NOUN
ejpam-5372	493	17	)	)	PUNCT
ejpam-5372	493	18	.	.	PUNCT
ejpam-5372	494	1	as	as	SCONJ
ejpam-5372	494	2	by	by	ADP
ejpam-5372	494	3	construction	construction	NOUN
ejpam-5372	494	4	eq	eq	ADP
ejpam-5372	494	5	⊂	⊂	PROPN
ejpam-5372	494	6	ec	ec	PROPN
ejpam-5372	494	7	and	and	CCONJ
ejpam-5372	494	8	dp	dp	PROPN
ejpam-5372	494	9	(	(	PUNCT
ejpam-5372	494	10	{	{	PUNCT
ejpam-5372	494	11	c	c	NOUN
ejpam-5372	494	12	}	}	PUNCT
ejpam-5372	494	13	)	)	PUNCT
ejpam-5372	494	14	∩	∩	ADJ
ejpam-5372	494	15	dp	dp	NOUN
ejpam-5372	494	16	(	(	PUNCT
ejpam-5372	494	17	eq	eq	NOUN
ejpam-5372	494	18	)	)	PUNCT
ejpam-5372	494	19	=	=	NOUN
ejpam-5372	494	20	∅	∅	NOUN
ejpam-5372	494	21	,	,	PUNCT
ejpam-5372	494	22	p	p	PROPN
ejpam-5372	494	23	∈	∈	NOUN
ejpam-5372	494	24	dp	dp	NOUN
ejpam-5372	494	25	(	(	PUNCT
ejpam-5372	494	26	eq	eq	NOUN
ejpam-5372	494	27	)	)	PUNCT
ejpam-5372	494	28	implies	imply	VERB
ejpam-5372	494	29	tp	tp	X
ejpam-5372	494	30	(	(	PUNCT
ejpam-5372	494	31	c	c	NOUN
ejpam-5372	494	32	)	)	PUNCT
ejpam-5372	495	1	=	=	SYM
ejpam-5372	495	2	2	2	X
ejpam-5372	495	3	.	.	PUNCT
ejpam-5372	496	1	furthermore	furthermore	ADV
ejpam-5372	496	2	,	,	PUNCT
ejpam-5372	496	3	dp	dp	X
ejpam-5372	496	4	(	(	PUNCT
ejpam-5372	496	5	{	{	PUNCT
ejpam-5372	496	6	a	a	NOUN
ejpam-5372	496	7	}	}	PUNCT
ejpam-5372	496	8	)	)	PUNCT
ejpam-5372	496	9	⊂	⊂	PROPN
ejpam-5372	496	10	dp	dp	INTJ
ejpam-5372	496	11	(	(	PUNCT
ejpam-5372	496	12	{	{	PUNCT
ejpam-5372	496	13	c	c	NOUN
ejpam-5372	496	14	}	}	PUNCT
ejpam-5372	496	15	)	)	PUNCT
ejpam-5372	496	16	implies	imply	VERB
ejpam-5372	496	17	that	that	SCONJ
ejpam-5372	496	18	2a	2a	NUM
ejpam-5372	496	19	is	be	AUX
ejpam-5372	496	20	invertible	invertible	ADJ
ejpam-5372	496	21	modulo	modulo	NOUN
ejpam-5372	496	22	p	p	X
ejpam-5372	496	23	,	,	PUNCT
ejpam-5372	496	24	so	so	SCONJ
ejpam-5372	496	25	tp	tp	X
ejpam-5372	496	26	(	(	PUNCT
ejpam-5372	496	27	q	q	X
ejpam-5372	496	28	)	)	PUNCT
ejpam-5372	496	29	=	=	SYM
ejpam-5372	496	30	2	2	NUM
ejpam-5372	496	31	too	too	ADV
ejpam-5372	496	32	.	.	PUNCT
ejpam-5372	497	1	corollary	corollary	ADJ
ejpam-5372	497	2	6	6	NUM
ejpam-5372	497	3	.	.	PUNCT
ejpam-5372	498	1	let	let	VERB
ejpam-5372	498	2	c	c	NOUN
ejpam-5372	498	3	≥	≥	VERB
ejpam-5372	498	4	2	2	NUM
ejpam-5372	498	5	a	a	DET
ejpam-5372	498	6	square	square	NOUN
ejpam-5372	498	7	such	such	ADJ
ejpam-5372	498	8	that	that	SCONJ
ejpam-5372	498	9	p1,4	p1,4	ADV
ejpam-5372	498	10	∩	∩	ADJ
ejpam-5372	498	11	dp	dp	NOUN
ejpam-5372	498	12	(	(	PUNCT
ejpam-5372	498	13	{	{	PUNCT
ejpam-5372	498	14	c	c	NOUN
ejpam-5372	498	15	}	}	PUNCT
ejpam-5372	498	16	)	)	PUNCT
ejpam-5372	498	17	=	=	VERB
ejpam-5372	498	18	∅.	∅.	AUX
ejpam-5372	498	19	let	let	VERB
ejpam-5372	498	20	f	f	NOUN
ejpam-5372	498	21	=	=	PUNCT
ejpam-5372	498	22	dp	dp	PROPN
ejpam-5372	498	23	(	(	PUNCT
ejpam-5372	498	24	{	{	PUNCT
ejpam-5372	498	25	c	c	NOUN
ejpam-5372	498	26	}	}	PUNCT
ejpam-5372	498	27	)	)	PUNCT
ejpam-5372	498	28	and	and	CCONJ
ejpam-5372	498	29	α	α	PRON
ejpam-5372	498	30	∈	∈	PROPN
ejpam-5372	498	31	(	(	PUNCT
ejpam-5372	498	32	n∗)f	n∗)f	PROPN
ejpam-5372	498	33	.	.	PUNCT
ejpam-5372	499	1	we	we	PRON
ejpam-5372	499	2	set	set	VERB
ejpam-5372	499	3	pαf	pαf	PROPN
ejpam-5372	499	4	=	=	SYM
ejpam-5372	499	5	∏	∏	PROPN
ejpam-5372	499	6	p∈f	p∈f	NOUN
ejpam-5372	499	7	pαp	pαp	NOUN
ejpam-5372	499	8	,	,	PUNCT
ejpam-5372	499	9	then	then	ADV
ejpam-5372	499	10	for	for	ADP
ejpam-5372	499	11	any	any	DET
ejpam-5372	499	12	b	b	PROPN
ejpam-5372	499	13	∈	∈	PROPN
ejpam-5372	500	1	[	[	X
ejpam-5372	500	2	[	[	X
ejpam-5372	500	3	1	1	NUM
ejpam-5372	500	4	,	,	PUNCT
ejpam-5372	500	5	pαf	pαf	PROPN
ejpam-5372	500	6	−	−	PROPN
ejpam-5372	500	7	1	1	NUM
ejpam-5372	500	8	]	]	X
ejpam-5372	500	9	]	]	X
ejpam-5372	500	10	coprime	coprime	NOUN
ejpam-5372	500	11	to	to	ADP
ejpam-5372	500	12	elements	element	NOUN
ejpam-5372	500	13	of	of	ADP
ejpam-5372	500	14	f	f	PROPN
ejpam-5372	500	15	and	and	CCONJ
ejpam-5372	500	16	of	of	ADP
ejpam-5372	500	17	opposite	opposite	ADJ
ejpam-5372	500	18	parity	parity	NOUN
ejpam-5372	500	19	to	to	ADP
ejpam-5372	500	20	c	c	PROPN
ejpam-5372	500	21	,	,	PUNCT
ejpam-5372	500	22	the	the	DET
ejpam-5372	500	23	quadratic	quadratic	ADJ
ejpam-5372	500	24	form	form	NOUN
ejpam-5372	500	25	:	:	PUNCT
ejpam-5372	500	26	qc	qc	PROPN
ejpam-5372	500	27	,	,	PUNCT
ejpam-5372	500	28	α	α	PROPN
ejpam-5372	500	29	,	,	PUNCT
ejpam-5372	500	30	b	b	NOUN
ejpam-5372	500	31	=	=	SYM
ejpam-5372	500	32	(	(	PUNCT
ejpam-5372	500	33	2pαfx	2pαfx	NUM
ejpam-5372	500	34	+	+	CCONJ
ejpam-5372	500	35	b)2	b)2	ADJ
ejpam-5372	500	36	+	+	CCONJ
ejpam-5372	500	37	c	c	NOUN
ejpam-5372	500	38	verifies	verifie	NOUN
ejpam-5372	500	39	hqc	hqc	INTJ
ejpam-5372	500	40	,	,	PUNCT
ejpam-5372	500	41	α	α	NOUN
ejpam-5372	500	42	,	,	PUNCT
ejpam-5372	500	43	b	b	NOUN
ejpam-5372	500	44	=	=	SYM
ejpam-5372	500	45	h1	h1	PROPN
ejpam-5372	500	46	.	.	PUNCT
ejpam-5372	501	1	moreover	moreover	ADV
ejpam-5372	501	2	if	if	SCONJ
ejpam-5372	501	3	b	b	PROPN
ejpam-5372	501	4	̸=	̸=	PROPN
ejpam-5372	501	5	b	b	PROPN
ejpam-5372	502	1	′	′	NOUN
ejpam-5372	503	1	then	then	ADV
ejpam-5372	503	2	the	the	DET
ejpam-5372	503	3	terms	term	NOUN
ejpam-5372	503	4	of	of	ADP
ejpam-5372	503	5	qc	qc	PROPN
ejpam-5372	503	6	,	,	PUNCT
ejpam-5372	503	7	α	α	PROPN
ejpam-5372	503	8	,	,	PUNCT
ejpam-5372	503	9	b	b	PROPN
ejpam-5372	503	10	are	be	AUX
ejpam-5372	503	11	disjoint	disjoint	NOUN
ejpam-5372	503	12	from	from	ADP
ejpam-5372	503	13	those	those	PRON
ejpam-5372	503	14	of	of	ADP
ejpam-5372	503	15	qc	qc	PROPN
ejpam-5372	503	16	,	,	PUNCT
ejpam-5372	503	17	α	α	NOUN
ejpam-5372	503	18	,	,	PUNCT
ejpam-5372	503	19	b′	b′	NOUN
ejpam-5372	503	20	.	.	PUNCT
ejpam-5372	504	1	proof	proof	NOUN
ejpam-5372	504	2	.	.	PUNCT
ejpam-5372	505	1	let	let	VERB
ejpam-5372	505	2	q	q	NOUN
ejpam-5372	505	3	=	=	SYM
ejpam-5372	505	4	qc	qc	PROPN
ejpam-5372	505	5	,	,	PUNCT
ejpam-5372	505	6	α	α	PROPN
ejpam-5372	505	7	,	,	PUNCT
ejpam-5372	505	8	b.	b.	PROPN
ejpam-5372	505	9	proposition	proposition	NOUN
ejpam-5372	505	10	17	17	NUM
ejpam-5372	505	11	and	and	CCONJ
ejpam-5372	505	12	the	the	DET
ejpam-5372	505	13	assumption	assumption	NOUN
ejpam-5372	505	14	p1,4	p1,4	ADV
ejpam-5372	505	15	∩	∩	ADJ
ejpam-5372	505	16	dp	dp	NOUN
ejpam-5372	505	17	(	(	PUNCT
ejpam-5372	505	18	{	{	PUNCT
ejpam-5372	505	19	c	c	NOUN
ejpam-5372	505	20	}	}	PUNCT
ejpam-5372	505	21	)	)	PUNCT
ejpam-5372	506	1	=	=	NOUN
ejpam-5372	506	2	∅	∅	NOUN
ejpam-5372	506	3	yield	yield	NOUN
ejpam-5372	506	4	dp	dp	NOUN
ejpam-5372	506	5	(	(	PUNCT
ejpam-5372	506	6	ec)⧹dp	ec)⧹dp	NOUN
ejpam-5372	506	7	(	(	PUNCT
ejpam-5372	506	8	{	{	PUNCT
ejpam-5372	506	9	c	c	NOUN
ejpam-5372	506	10	}	}	PUNCT
ejpam-5372	506	11	)	)	PUNCT
ejpam-5372	506	12	=	=	SYM
ejpam-5372	507	1	p1,4	p1,4	PROPN
ejpam-5372	507	2	.	.	PUNCT
ejpam-5372	508	1	however	however	ADV
ejpam-5372	508	2	,	,	PUNCT
ejpam-5372	508	3	by	by	ADP
ejpam-5372	508	4	construction	construction	NOUN
ejpam-5372	508	5	we	we	PRON
ejpam-5372	508	6	also	also	ADV
ejpam-5372	508	7	have	have	VERB
ejpam-5372	508	8	dp	dp	NOUN
ejpam-5372	508	9	(	(	PUNCT
ejpam-5372	508	10	ec)⧹dp	ec)⧹dp	NOUN
ejpam-5372	508	11	(	(	PUNCT
ejpam-5372	508	12	{	{	PUNCT
ejpam-5372	508	13	c	c	NOUN
ejpam-5372	508	14	}	}	PUNCT
ejpam-5372	508	15	)	)	PUNCT
ejpam-5372	509	1	=	=	SYM
ejpam-5372	509	2	dp	dp	NOUN
ejpam-5372	509	3	(	(	PUNCT
ejpam-5372	509	4	eq	eq	NOUN
ejpam-5372	509	5	)	)	PUNCT
ejpam-5372	509	6	,	,	PUNCT
ejpam-5372	509	7	thus	thus	ADV
ejpam-5372	509	8	proposition	proposition	VERB
ejpam-5372	509	9	18	18	NUM
ejpam-5372	509	10	yields	yield	NOUN
ejpam-5372	509	11	hq	hq	NOUN
ejpam-5372	510	1	=	=	PUNCT
ejpam-5372	510	2	h1	h1	PROPN
ejpam-5372	510	3	.	.	PUNCT
ejpam-5372	511	1	assume	assume	VERB
ejpam-5372	511	2	that	that	SCONJ
ejpam-5372	511	3	there	there	PRON
ejpam-5372	511	4	exists	exist	VERB
ejpam-5372	511	5	b	b	NOUN
ejpam-5372	511	6	′	′	NUM
ejpam-5372	511	7	∈	∈	PROPN
ejpam-5372	512	1	[	[	X
ejpam-5372	512	2	[	[	X
ejpam-5372	512	3	1	1	NUM
ejpam-5372	512	4	,	,	PUNCT
ejpam-5372	512	5	pαf	pαf	PROPN
ejpam-5372	512	6	−	−	PROPN
ejpam-5372	512	7	1	1	NUM
ejpam-5372	512	8	]	]	PUNCT
ejpam-5372	512	9	]	]	PUNCT
ejpam-5372	512	10	and	and	CCONJ
ejpam-5372	512	11	x	x	X
ejpam-5372	512	12	,	,	PUNCT
ejpam-5372	512	13	y	y	PROPN
ejpam-5372	512	14	∈	∈	PROPN
ejpam-5372	512	15	z	z	NOUN
ejpam-5372	512	16	such	such	ADJ
ejpam-5372	512	17	that	that	DET
ejpam-5372	512	18	q	q	X
ejpam-5372	512	19	(	(	PUNCT
ejpam-5372	512	20	x	x	NOUN
ejpam-5372	512	21	)	)	PUNCT
ejpam-5372	512	22	=	=	SYM
ejpam-5372	512	23	qc	qc	PROPN
ejpam-5372	512	24	,	,	PUNCT
ejpam-5372	512	25	α	α	NOUN
ejpam-5372	512	26	,	,	PUNCT
ejpam-5372	512	27	b′	b′	NUM
ejpam-5372	512	28	(	(	PUNCT
ejpam-5372	512	29	y	y	NOUN
ejpam-5372	512	30	)	)	PUNCT
ejpam-5372	512	31	.	.	PUNCT
ejpam-5372	513	1	we	we	PRON
ejpam-5372	513	2	deduce	deduce	VERB
ejpam-5372	513	3	2pαf	2pαf	PROPN
ejpam-5372	513	4	(	(	PUNCT
ejpam-5372	513	5	x±	x±	PROPN
ejpam-5372	513	6	y	y	X
ejpam-5372	513	7	)	)	PUNCT
ejpam-5372	514	1	=	=	SYM
ejpam-5372	514	2	−	−	PROPN
ejpam-5372	514	3	(	(	PUNCT
ejpam-5372	514	4	b±	b±	PROPN
ejpam-5372	514	5	b	b	NOUN
ejpam-5372	514	6	′	′	NUM
ejpam-5372	514	7	)	)	PUNCT
ejpam-5372	514	8	which	which	PRON
ejpam-5372	514	9	implies	imply	VERB
ejpam-5372	514	10	b	b	NOUN
ejpam-5372	514	11	=	=	SYM
ejpam-5372	514	12	b	b	PROPN
ejpam-5372	514	13	′	′	NUM
ejpam-5372	514	14	and	and	CCONJ
ejpam-5372	514	15	x	x	X
ejpam-5372	515	1	=	=	PUNCT
ejpam-5372	515	2	y.	y.	NOUN
ejpam-5372	515	3	the	the	DET
ejpam-5372	515	4	terms	term	NOUN
ejpam-5372	515	5	of	of	ADP
ejpam-5372	515	6	qc	qc	PROPN
ejpam-5372	515	7	,	,	PUNCT
ejpam-5372	515	8	α	α	PROPN
ejpam-5372	515	9	,	,	PUNCT
ejpam-5372	515	10	b	b	PROPN
ejpam-5372	515	11	and	and	CCONJ
ejpam-5372	515	12	qc	qc	PROPN
ejpam-5372	515	13	,	,	PUNCT
ejpam-5372	515	14	α	α	PROPN
ejpam-5372	515	15	,	,	PUNCT
ejpam-5372	515	16	b′	b′	NUM
ejpam-5372	515	17	are	be	AUX
ejpam-5372	515	18	hence	hence	ADV
ejpam-5372	515	19	disjoint	disjoint	NOUN
ejpam-5372	515	20	.	.	PUNCT
ejpam-5372	516	1	theorem	theorem	NOUN
ejpam-5372	516	2	3	3	X
ejpam-5372	516	3	.	.	X
ejpam-5372	517	1	there	there	PRON
ejpam-5372	517	2	are	be	VERB
ejpam-5372	517	3	infinitely	infinitely	ADV
ejpam-5372	517	4	many	many	ADJ
ejpam-5372	517	5	quadratic	quadratic	ADJ
ejpam-5372	517	6	forms	form	NOUN
ejpam-5372	517	7	q	q	NOUN
ejpam-5372	517	8	with	with	ADP
ejpam-5372	517	9	disjoint	disjoint	NOUN
ejpam-5372	517	10	terms	term	NOUN
ejpam-5372	517	11	and	and	CCONJ
ejpam-5372	517	12	the	the	DET
ejpam-5372	517	13	same	same	ADJ
ejpam-5372	517	14	value	value	NOUN
ejpam-5372	517	15	hq	hq	NOUN
ejpam-5372	517	16	,	,	PUNCT
ejpam-5372	517	17	which	which	PRON
ejpam-5372	517	18	can	can	AUX
ejpam-5372	517	19	also	also	ADV
ejpam-5372	517	20	be	be	AUX
ejpam-5372	517	21	assumed	assume	VERB
ejpam-5372	517	22	to	to	PART
ejpam-5372	517	23	be	be	AUX
ejpam-5372	517	24	arbitrarily	arbitrarily	ADV
ejpam-5372	517	25	close	close	ADJ
ejpam-5372	517	26	to	to	ADP
ejpam-5372	517	27	any	any	DET
ejpam-5372	517	28	value	value	NOUN
ejpam-5372	517	29	h	h	NOUN
ejpam-5372	517	30	≥	≥	NOUN
ejpam-5372	517	31	h1	h1	PROPN
ejpam-5372	517	32	.	.	PUNCT
ejpam-5372	518	1	two	two	NUM
ejpam-5372	518	2	proofs	proof	NOUN
ejpam-5372	518	3	of	of	ADP
ejpam-5372	518	4	this	this	DET
ejpam-5372	518	5	theorem	theorem	NOUN
ejpam-5372	518	6	are	be	AUX
ejpam-5372	518	7	given	give	VERB
ejpam-5372	518	8	.	.	PUNCT
ejpam-5372	519	1	proof	proof	NOUN
ejpam-5372	519	2	.	.	PUNCT
ejpam-5372	520	1	first	first	ADJ
ejpam-5372	520	2	proof	proof	NOUN
ejpam-5372	520	3	:	:	PUNCT
ejpam-5372	520	4	proposition	proposition	NOUN
ejpam-5372	520	5	17	17	NUM
ejpam-5372	520	6	and	and	CCONJ
ejpam-5372	520	7	its	its	PRON
ejpam-5372	520	8	corollary	corollary	ADJ
ejpam-5372	520	9	yield	yield	NOUN
ejpam-5372	520	10	that	that	PRON
ejpam-5372	520	11	,	,	PUNCT
ejpam-5372	520	12	for	for	ADP
ejpam-5372	520	13	any	any	DET
ejpam-5372	520	14	subset	subset	NOUN
ejpam-5372	520	15	f	f	PROPN
ejpam-5372	520	16	of	of	ADP
ejpam-5372	520	17	p1,4	p1,4	PROPN
ejpam-5372	520	18	,	,	PUNCT
ejpam-5372	520	19	there	there	PRON
ejpam-5372	520	20	are	be	VERB
ejpam-5372	520	21	infinitely	infinitely	ADV
ejpam-5372	520	22	many	many	ADJ
ejpam-5372	520	23	odd	odd	ADJ
ejpam-5372	520	24	squares	square	NOUN
ejpam-5372	520	25	such	such	ADJ
ejpam-5372	520	26	that	that	SCONJ
ejpam-5372	520	27	:	:	PUNCT
ejpam-5372	520	28	hc	hc	X
ejpam-5372	520	29	=	=	PUNCT
ejpam-5372	520	30	h1	h1	PROPN
ejpam-5372	520	31	(	(	PUNCT
ejpam-5372	520	32	∏	∏	PROPN
ejpam-5372	520	33	p∈f	p∈f	NOUN
ejpam-5372	520	34	p−1	p−1	PROPN
ejpam-5372	520	35	p−2	p−2	PROPN
ejpam-5372	520	36	)	)	PUNCT
ejpam-5372	521	1	if	if	SCONJ
ejpam-5372	521	2	c	c	NOUN
ejpam-5372	521	3	=	=	SYM
ejpam-5372	521	4	y2	y2	PROPN
ejpam-5372	521	5	,	,	PUNCT
ejpam-5372	521	6	c	c	NOUN
ejpam-5372	521	7	′	′	NOUN
ejpam-5372	521	8	=	=	SYM
ejpam-5372	521	9	y	y	NUM
ejpam-5372	521	10	′2	′2	NOUN
ejpam-5372	521	11	are	be	AUX
ejpam-5372	521	12	such	such	ADJ
ejpam-5372	521	13	perfect	perfect	ADJ
ejpam-5372	521	14	squares	square	NOUN
ejpam-5372	521	15	then	then	ADV
ejpam-5372	521	16	the	the	DET
ejpam-5372	521	17	equality	equality	NOUN
ejpam-5372	521	18	x2	x2	PROPN
ejpam-5372	522	1	+	+	CCONJ
ejpam-5372	523	1	y2	y2	NOUN
ejpam-5372	523	2	=	=	SYM
ejpam-5372	523	3	x	x	PUNCT
ejpam-5372	523	4	′2	′2	NOUN
ejpam-5372	523	5	+	+	CCONJ
ejpam-5372	523	6	y	y	PROPN
ejpam-5372	523	7	′2	′2	NOUN
ejpam-5372	523	8	implies	imply	VERB
ejpam-5372	523	9	(	(	PUNCT
ejpam-5372	523	10	x−	x−	PROPN
ejpam-5372	523	11	x	x	NOUN
ejpam-5372	523	12	′	′	NUM
ejpam-5372	523	13	)	)	PUNCT
ejpam-5372	523	14	(	(	PUNCT
ejpam-5372	523	15	x+	x+	PUNCT
ejpam-5372	523	16	x	x	NOUN
ejpam-5372	523	17	′	′	NUM
ejpam-5372	523	18	)	)	PUNCT
ejpam-5372	524	1	=	=	SYM
ejpam-5372	524	2	y	y	PROPN
ejpam-5372	524	3	′2	′2	NOUN
ejpam-5372	524	4	−	−	NOUN
ejpam-5372	525	1	y2	y2	INTJ
ejpam-5372	525	2	hence	hence	ADV
ejpam-5372	525	3	necessary	necessary	ADJ
ejpam-5372	525	4	x	x	X
ejpam-5372	525	5	and	and	CCONJ
ejpam-5372	525	6	x	x	SYM
ejpam-5372	525	7	′	′	NOUN
ejpam-5372	525	8	are	be	AUX
ejpam-5372	525	9	smaller	small	ADJ
ejpam-5372	525	10	than	than	ADP
ejpam-5372	525	11	y	y	PROPN
ejpam-5372	525	12	′2	′2	NOUN
ejpam-5372	525	13	−	−	PROPN
ejpam-5372	525	14	y2	y2	PROPN
ejpam-5372	525	15	.	.	PUNCT
ejpam-5372	526	1	thus	thus	ADV
ejpam-5372	526	2	we	we	PRON
ejpam-5372	526	3	can	can	AUX
ejpam-5372	526	4	construct	construct	VERB
ejpam-5372	526	5	quadratic	quadratic	ADJ
ejpam-5372	526	6	forms	form	NOUN
ejpam-5372	526	7	qk	qk	NOUN
ejpam-5372	526	8	=	=	SYM
ejpam-5372	526	9	(	(	PUNCT
ejpam-5372	526	10	2x	2x	NUM
ejpam-5372	526	11	+	+	ADJ
ejpam-5372	526	12	mk	mk	NOUN
ejpam-5372	526	13	)	)	PUNCT
ejpam-5372	526	14	2	2	NUM
ejpam-5372	527	1	+	+	CCONJ
ejpam-5372	527	2	ck	ck	PRON
ejpam-5372	527	3	such	such	ADJ
ejpam-5372	527	4	that	that	DET
ejpam-5372	527	5	hqk	hqk	NOUN
ejpam-5372	527	6	=	=	PROPN
ejpam-5372	527	7	hck	hck	PROPN
ejpam-5372	527	8	=	=	PROPN
ejpam-5372	527	9	h1	h1	PROPN
ejpam-5372	527	10	(	(	PUNCT
ejpam-5372	527	11	∏	∏	PROPN
ejpam-5372	527	12	p∈f	p∈f	NOUN
ejpam-5372	527	13	p−1	p−1	PROPN
ejpam-5372	527	14	p−2	p−2	PROPN
ejpam-5372	527	15	)	)	PUNCT
ejpam-5372	527	16	.	.	PUNCT
ejpam-5372	528	1	moreover	moreover	ADV
ejpam-5372	528	2	the	the	DET
ejpam-5372	528	3	series	series	NOUN
ejpam-5372	528	4	∑	∑	PROPN
ejpam-5372	528	5	p∈p1,4	p∈p1,4	PROPN
ejpam-5372	528	6	1	1	NUM
ejpam-5372	528	7	p	p	NOUN
ejpam-5372	528	8	diverge	diverge	NOUN
ejpam-5372	528	9	:	:	PUNCT
ejpam-5372	528	10	indeed	indeed	ADV
ejpam-5372	528	11	chebotarev	chebotarev	X
ejpam-5372	528	12	’s	’s	PART
ejpam-5372	528	13	theorem	theorem	ADJ
ejpam-5372	528	14	states	state	NOUN
ejpam-5372	528	15	that	that	SCONJ
ejpam-5372	528	16	dp|4n+1	dp|4n+1	ADJ
ejpam-5372	528	17	(	(	PUNCT
ejpam-5372	528	18	x	x	X
ejpam-5372	528	19	)	)	PUNCT
ejpam-5372	528	20	∼	∼	NOUN
ejpam-5372	528	21	1	1	NUM
ejpam-5372	528	22	2ln(x	2ln(x	NUM
ejpam-5372	528	23	)	)	PUNCT
ejpam-5372	528	24	,	,	PUNCT
ejpam-5372	528	25	so	so	SCONJ
ejpam-5372	528	26	tn	tn	NOUN
ejpam-5372	528	27	:	:	PUNCT
ejpam-5372	528	28	=	=	PUNCT
ejpam-5372	528	29	∑	∑	PUNCT
ejpam-5372	528	30	p	p	X
ejpam-5372	528	31	∈	∈	PROPN
ejpam-5372	528	32	p1,4	p1,4	ADV
ejpam-5372	528	33	en	en	X
ejpam-5372	528	34	<	<	X
ejpam-5372	528	35	p	p	X
ejpam-5372	528	36	≤	≤	X
ejpam-5372	528	37	en+1	en+1	ADJ
ejpam-5372	528	38	1	1	NUM
ejpam-5372	528	39	p	p	NOUN
ejpam-5372	528	40	≥	≥	NOUN
ejpam-5372	528	41	e−(n+1	e−(n+1	PROPN
ejpam-5372	528	42	)	)	PUNCT
ejpam-5372	529	1	[	[	X
ejpam-5372	529	2	⌊	⌊	X
ejpam-5372	529	3	en+1	en+1	NUM
ejpam-5372	529	4	⌋	⌋	NOUN
ejpam-5372	529	5	dp|4n+1	dp|4n+1	ADJ
ejpam-5372	529	6	(	(	PUNCT
ejpam-5372	529	7	en+1	en+1	PROPN
ejpam-5372	529	8	)	)	PUNCT
ejpam-5372	529	9	−	−	PROPN
ejpam-5372	530	1	⌊en⌋	⌊en⌋	X
ejpam-5372	531	1	dp|4n+1	dp|4n+1	PROPN
ejpam-5372	531	2	(	(	PUNCT
ejpam-5372	531	3	e	e	NOUN
ejpam-5372	531	4	n	n	CCONJ
ejpam-5372	531	5	)	)	PUNCT
ejpam-5372	531	6	]	]	PUNCT
ejpam-5372	532	1	∼	∼	NOUN
ejpam-5372	532	2	e−1	e−1	PROPN
ejpam-5372	532	3	2ne	2ne	NOUN
ejpam-5372	532	4	is	be	AUX
ejpam-5372	532	5	a	a	DET
ejpam-5372	532	6	divergent	divergent	ADJ
ejpam-5372	532	7	series	series	NOUN
ejpam-5372	532	8	,	,	PUNCT
ejpam-5372	532	9	which	which	PRON
ejpam-5372	532	10	implies	imply	VERB
ejpam-5372	532	11	that	that	SCONJ
ejpam-5372	532	12	∑	∑	PUNCT
ejpam-5372	532	13	p∈p1,4	p∈p1,4	PROPN
ejpam-5372	532	14	1	1	NUM
ejpam-5372	532	15	p	p	NOUN
ejpam-5372	532	16	diverges	diverge	VERB
ejpam-5372	532	17	too	too	ADV
ejpam-5372	532	18	.	.	PUNCT
ejpam-5372	533	1	additionally	additionally	ADV
ejpam-5372	533	2	,	,	PUNCT
ejpam-5372	533	3	1	1	NUM
ejpam-5372	533	4	p→p→∞0	p→p→∞0	NOUN
ejpam-5372	533	5	.	.	PUNCT
ejpam-5372	534	1	thus	thus	ADV
ejpam-5372	534	2	,	,	PUNCT
ejpam-5372	534	3	we	we	PRON
ejpam-5372	534	4	can	can	AUX
ejpam-5372	534	5	choose	choose	VERB
ejpam-5372	534	6	f	f	PROPN
ejpam-5372	534	7	such	such	ADJ
ejpam-5372	534	8	that	that	PRON
ejpam-5372	534	9	(	(	PUNCT
ejpam-5372	534	10	∏	∏	PROPN
ejpam-5372	534	11	p∈f	p∈f	NOUN
ejpam-5372	534	12	p−1	p−1	PROPN
ejpam-5372	534	13	p−2	p−2	PROPN
ejpam-5372	534	14	)	)	PUNCT
ejpam-5372	534	15	be	be	AUX
ejpam-5372	534	16	arbitrarily	arbitrarily	ADV
ejpam-5372	534	17	close	close	ADJ
ejpam-5372	534	18	to	to	ADP
ejpam-5372	534	19	h	h	PROPN
ejpam-5372	534	20	≥	≥	NOUN
ejpam-5372	534	21	h1	h1	AUX
ejpam-5372	534	22	fixed	fix	VERB
ejpam-5372	534	23	.	.	PUNCT
ejpam-5372	535	1	remark	remark	NOUN
ejpam-5372	535	2	:	:	PUNCT
ejpam-5372	535	3	similarly	similarly	ADV
ejpam-5372	535	4	,	,	PUNCT
ejpam-5372	535	5	if	if	SCONJ
ejpam-5372	535	6	we	we	PRON
ejpam-5372	535	7	choose	choose	VERB
ejpam-5372	535	8	f	f	PROPN
ejpam-5372	535	9	⊂	⊂	PROPN
ejpam-5372	535	10	p3,4	p3,4	ADJ
ejpam-5372	535	11	we	we	PRON
ejpam-5372	535	12	can	can	AUX
ejpam-5372	535	13	also	also	ADV
ejpam-5372	535	14	make	make	VERB
ejpam-5372	535	15	hq	hq	NOUN
ejpam-5372	535	16	arbitrarily	arbitrarily	ADV
ejpam-5372	535	17	close	close	ADJ
ejpam-5372	535	18	to	to	ADP
ejpam-5372	535	19	any	any	DET
ejpam-5372	535	20	h	h	NOUN
ejpam-5372	535	21	∈	∈	PROPN
ejpam-5372	536	1	[	[	X
ejpam-5372	536	2	0	0	NUM
ejpam-5372	536	3	,	,	PUNCT
ejpam-5372	536	4	h1	h1	NOUN
ejpam-5372	536	5	]	]	PUNCT
ejpam-5372	536	6	.	.	PUNCT
ejpam-5372	537	1	second	second	ADJ
ejpam-5372	537	2	proof	proof	NOUN
ejpam-5372	537	3	:	:	PUNCT
ejpam-5372	537	4	as	as	ADP
ejpam-5372	537	5	in	in	ADP
ejpam-5372	537	6	corollary	corollary	ADJ
ejpam-5372	537	7	6	6	NUM
ejpam-5372	537	8	,	,	PUNCT
ejpam-5372	537	9	for	for	ADP
ejpam-5372	537	10	any	any	DET
ejpam-5372	537	11	subset	subset	NOUN
ejpam-5372	537	12	f	f	PROPN
ejpam-5372	537	13	of	of	ADP
ejpam-5372	537	14	p⧹	p⧹	VERB
ejpam-5372	537	15	{	{	PUNCT
ejpam-5372	537	16	2	2	NUM
ejpam-5372	537	17	}	}	PUNCT
ejpam-5372	537	18	and	and	CCONJ
ejpam-5372	537	19	for	for	ADP
ejpam-5372	537	20	any	any	DET
ejpam-5372	537	21	square	square	NOUN
ejpam-5372	537	22	c	c	NOUN
ejpam-5372	537	23	such	such	ADJ
ejpam-5372	537	24	that	that	DET
ejpam-5372	537	25	dp	dp	NOUN
ejpam-5372	537	26	(	(	PUNCT
ejpam-5372	537	27	{	{	PUNCT
ejpam-5372	537	28	c	c	NOUN
ejpam-5372	537	29	}	}	PUNCT
ejpam-5372	537	30	)	)	PUNCT
ejpam-5372	538	1	⊂	⊂	PROPN
ejpam-5372	538	2	f	f	PROPN
ejpam-5372	538	3	,	,	PUNCT
ejpam-5372	538	4	we	we	PRON
ejpam-5372	538	5	can	can	AUX
ejpam-5372	538	6	get	get	VERB
ejpam-5372	538	7	a	a	DET
ejpam-5372	538	8	,	,	PUNCT
ejpam-5372	538	9	b	b	X
ejpam-5372	538	10	such	such	ADJ
ejpam-5372	538	11	that	that	PRON
ejpam-5372	538	12	:	:	PUNCT
ejpam-5372	538	13	h(2ax+b)2+c	h(2ax+b)2+c	PROPN
ejpam-5372	538	14	=	=	PUNCT
ejpam-5372	538	15	h1	h1	PROPN
ejpam-5372	538	16	(	(	PUNCT
ejpam-5372	538	17	∏	∏	PROPN
ejpam-5372	538	18	p∈f∩p1,4	p∈f∩p1,4	NOUN
ejpam-5372	538	19	p−1	p−1	PROPN
ejpam-5372	538	20	p−2	p−2	PROPN
ejpam-5372	538	21	)	)	PUNCT
ejpam-5372	538	22	.	.	PUNCT
ejpam-5372	539	1	m.	m.	NOUN
ejpam-5372	539	2	wolf	wolf	PROPN
ejpam-5372	539	3	,	,	PUNCT
ejpam-5372	539	4	f.	f.	PROPN
ejpam-5372	539	5	wolf	wolf	PROPN
ejpam-5372	539	6	/	/	SYM
ejpam-5372	539	7	eur	eur	PROPN
ejpam-5372	539	8	.	.	PUNCT
ejpam-5372	540	1	j.	j.	PROPN
ejpam-5372	540	2	pure	pure	PROPN
ejpam-5372	540	3	appl	appl	PROPN
ejpam-5372	540	4	.	.	PROPN
ejpam-5372	540	5	math	math	PROPN
ejpam-5372	540	6	,	,	PUNCT
ejpam-5372	540	7	17	17	NUM
ejpam-5372	540	8	(	(	PUNCT
ejpam-5372	540	9	4	4	NUM
ejpam-5372	540	10	)	)	PUNCT
ejpam-5372	540	11	(	(	PUNCT
ejpam-5372	540	12	2024	2024	NUM
ejpam-5372	540	13	)	)	PUNCT
ejpam-5372	540	14	,	,	PUNCT
ejpam-5372	540	15	2431	2431	NUM
ejpam-5372	540	16	-	-	SYM
ejpam-5372	540	17	2447	2447	NUM
ejpam-5372	540	18	2445	2445	NUM
ejpam-5372	540	19	we	we	PRON
ejpam-5372	540	20	conclude	conclude	VERB
ejpam-5372	540	21	as	as	ADP
ejpam-5372	540	22	in	in	ADP
ejpam-5372	540	23	the	the	DET
ejpam-5372	540	24	first	first	ADJ
ejpam-5372	540	25	proof	proof	NOUN
ejpam-5372	540	26	:	:	PUNCT
ejpam-5372	540	27	we	we	PRON
ejpam-5372	540	28	can	can	AUX
ejpam-5372	540	29	either	either	CCONJ
ejpam-5372	540	30	take	take	VERB
ejpam-5372	540	31	an	an	DET
ejpam-5372	540	32	infinity	infinity	NOUN
ejpam-5372	540	33	of	of	ADP
ejpam-5372	540	34	values	value	NOUN
ejpam-5372	540	35	for	for	ADP
ejpam-5372	540	36	c	c	NOUN
ejpam-5372	540	37	as	as	ADP
ejpam-5372	540	38	in	in	ADP
ejpam-5372	540	39	the	the	DET
ejpam-5372	540	40	first	first	ADJ
ejpam-5372	540	41	proof	proof	NOUN
ejpam-5372	540	42	,	,	PUNCT
ejpam-5372	540	43	or	or	CCONJ
ejpam-5372	540	44	build	build	VERB
ejpam-5372	540	45	a	a	DET
ejpam-5372	540	46	sequence	sequence	NOUN
ejpam-5372	540	47	(	(	PUNCT
ejpam-5372	540	48	ak	ak	PROPN
ejpam-5372	540	49	,	,	PUNCT
ejpam-5372	540	50	bk	bk	PROPN
ejpam-5372	540	51	)	)	PUNCT
ejpam-5372	540	52	such	such	ADJ
ejpam-5372	540	53	that	that	SCONJ
ejpam-5372	540	54	ak+1	ak+1	VERB
ejpam-5372	540	55	is	be	AUX
ejpam-5372	540	56	a	a	DET
ejpam-5372	540	57	multiple	multiple	NOUN
ejpam-5372	540	58	of	of	ADP
ejpam-5372	540	59	ak	ak	PROPN
ejpam-5372	540	60	and	and	CCONJ
ejpam-5372	540	61	for	for	ADP
ejpam-5372	540	62	any	any	DET
ejpam-5372	540	63	l	l	NOUN
ejpam-5372	540	64	≤	≤	NOUN
ejpam-5372	541	1	k	k	X
ejpam-5372	541	2	,	,	PUNCT
ejpam-5372	541	3	bl	bl	PROPN
ejpam-5372	541	4	̸≡	̸≡	PROPN
ejpam-5372	541	5	bk+1	bk+1	PROPN
ejpam-5372	542	1	[	[	X
ejpam-5372	542	2	al	al	X
ejpam-5372	542	3	]	]	PUNCT
ejpam-5372	542	4	.	.	PUNCT
ejpam-5372	543	1	remark	remark	NOUN
ejpam-5372	543	2	:	:	PUNCT
ejpam-5372	543	3	if	if	SCONJ
ejpam-5372	543	4	p	p	PROPN
ejpam-5372	543	5	∈	∈	PROPN
ejpam-5372	543	6	f	f	PROPN
ejpam-5372	543	7	∩	∩	X
ejpam-5372	543	8	p1,4⧹dp	p1,4⧹dp	PRON
ejpam-5372	543	9	(	(	PUNCT
ejpam-5372	543	10	{	{	PUNCT
ejpam-5372	543	11	c	c	NOUN
ejpam-5372	543	12	}	}	PUNCT
ejpam-5372	543	13	)	)	PUNCT
ejpam-5372	543	14	,	,	PUNCT
ejpam-5372	543	15	to	to	PART
ejpam-5372	543	16	”	"	PUNCT
ejpam-5372	543	17	eliminate	eliminate	VERB
ejpam-5372	543	18	”	"	PUNCT
ejpam-5372	543	19	it	it	PRON
ejpam-5372	543	20	,	,	PUNCT
ejpam-5372	543	21	we	we	PRON
ejpam-5372	543	22	need	need	VERB
ejpam-5372	543	23	to	to	PART
ejpam-5372	543	24	determine	determine	VERB
ejpam-5372	543	25	the	the	DET
ejpam-5372	543	26	solutions	solution	NOUN
ejpam-5372	543	27	of	of	ADP
ejpam-5372	543	28	x2	x2	PROPN
ejpam-5372	543	29	≡	≡	PROPN
ejpam-5372	543	30	−c	−c	NOUN
ejpam-5372	544	1	[	[	X
ejpam-5372	544	2	p	p	X
ejpam-5372	544	3	]	]	X
ejpam-5372	544	4	.	.	PUNCT
ejpam-5372	545	1	this	this	DET
ejpam-5372	545	2	equation	equation	NOUN
ejpam-5372	545	3	is	be	AUX
ejpam-5372	545	4	discussed	discuss	VERB
ejpam-5372	545	5	in	in	ADP
ejpam-5372	545	6	the	the	DET
ejpam-5372	545	7	next	next	ADJ
ejpam-5372	545	8	section	section	NOUN
ejpam-5372	545	9	.	.	PUNCT
ejpam-5372	546	1	by	by	ADP
ejpam-5372	546	2	analogy	analogy	NOUN
ejpam-5372	546	3	with	with	ADP
ejpam-5372	546	4	shanks	shank	NOUN
ejpam-5372	546	5	’	'	PUNCT
ejpam-5372	546	6	conjecture	conjecture	NOUN
ejpam-5372	546	7	,	,	PUNCT
ejpam-5372	546	8	hq	hq	PROPN
ejpam-5372	546	9	can	can	AUX
ejpam-5372	546	10	be	be	AUX
ejpam-5372	546	11	conjectured	conjecture	VERB
ejpam-5372	546	12	to	to	PART
ejpam-5372	546	13	be	be	AUX
ejpam-5372	546	14	linked	link	VERB
ejpam-5372	546	15	with	with	ADP
ejpam-5372	546	16	the	the	DET
ejpam-5372	546	17	density	density	NOUN
ejpam-5372	546	18	of	of	ADP
ejpam-5372	546	19	prime	prime	ADJ
ejpam-5372	546	20	numbers	number	NOUN
ejpam-5372	546	21	of	of	ADP
ejpam-5372	546	22	the	the	DET
ejpam-5372	546	23	form	form	NOUN
ejpam-5372	547	1	q	q	X
ejpam-5372	547	2	(	(	PUNCT
ejpam-5372	547	3	x	x	NOUN
ejpam-5372	547	4	)	)	PUNCT
ejpam-5372	547	5	as	as	ADP
ejpam-5372	547	6	in	in	ADP
ejpam-5372	547	7	the	the	DET
ejpam-5372	547	8	following	follow	VERB
ejpam-5372	547	9	generalisation	generalisation	NOUN
ejpam-5372	547	10	:	:	PUNCT
ejpam-5372	547	11	assumption	assumption	NOUN
ejpam-5372	547	12	1	1	X
ejpam-5372	547	13	.	.	PUNCT
ejpam-5372	548	1	let	let	VERB
ejpam-5372	548	2	q	q	PART
ejpam-5372	548	3	be	be	AUX
ejpam-5372	548	4	an	an	DET
ejpam-5372	548	5	irreducible	irreducible	ADJ
ejpam-5372	548	6	quadratic	quadratic	ADJ
ejpam-5372	548	7	form	form	NOUN
ejpam-5372	548	8	and	and	CCONJ
ejpam-5372	548	9	eq	eq	NOUN
ejpam-5372	548	10	=	=	NOUN
ejpam-5372	548	11	{	{	PUNCT
ejpam-5372	548	12	q	q	X
ejpam-5372	548	13	(	(	PUNCT
ejpam-5372	548	14	x	x	NOUN
ejpam-5372	548	15	)	)	PUNCT
ejpam-5372	548	16	,	,	PUNCT
ejpam-5372	548	17	x	x	PUNCT
ejpam-5372	548	18	∈	∈	PROPN
ejpam-5372	548	19	n	n	CCONJ
ejpam-5372	548	20	}	}	PUNCT
ejpam-5372	548	21	.	.	PUNCT
ejpam-5372	549	1	then	then	ADV
ejpam-5372	549	2	:	:	PUNCT
ejpam-5372	549	3	dp|eq	dp|eq	X
ejpam-5372	549	4	(	(	PUNCT
ejpam-5372	549	5	x	x	X
ejpam-5372	549	6	)	)	PUNCT
ejpam-5372	549	7	=	=	SYM
ejpam-5372	549	8	hq	hq	NOUN
ejpam-5372	549	9	ln(x	ln(x	X
ejpam-5372	549	10	)	)	PUNCT
ejpam-5372	550	1	+	+	CCONJ
ejpam-5372	550	2	o	o	X
ejpam-5372	550	3	(	(	PUNCT
ejpam-5372	550	4	1	1	NUM
ejpam-5372	550	5	ln(x	ln(x	X
ejpam-5372	550	6	)	)	PUNCT
ejpam-5372	550	7	)	)	PUNCT
ejpam-5372	550	8	where	where	SCONJ
ejpam-5372	550	9	dp|eq	dp|eq	PROPN
ejpam-5372	550	10	is	be	AUX
ejpam-5372	550	11	the	the	DET
ejpam-5372	550	12	density	density	NOUN
ejpam-5372	550	13	of	of	ADP
ejpam-5372	550	14	primes	prime	NOUN
ejpam-5372	550	15	in	in	ADP
ejpam-5372	550	16	eq	eq	NOUN
ejpam-5372	550	17	(	(	PUNCT
ejpam-5372	550	18	see	see	VERB
ejpam-5372	550	19	[	[	X
ejpam-5372	550	20	2	2	X
ejpam-5372	550	21	]	]	PUNCT
ejpam-5372	550	22	for	for	ADP
ejpam-5372	550	23	a	a	DET
ejpam-5372	550	24	formal	formal	ADJ
ejpam-5372	550	25	definition	definition	NOUN
ejpam-5372	550	26	)	)	PUNCT
ejpam-5372	550	27	.	.	PUNCT
ejpam-5372	551	1	3.2	3.2	NUM
ejpam-5372	551	2	.	.	PUNCT
ejpam-5372	552	1	tonelli	tonelli	NOUN
ejpam-5372	552	2	-	-	PUNCT
ejpam-5372	552	3	shanks	shank	NOUN
ejpam-5372	552	4	algorithm	algorithm	NOUN
ejpam-5372	552	5	there	there	PRON
ejpam-5372	552	6	are	be	VERB
ejpam-5372	552	7	several	several	ADJ
ejpam-5372	552	8	algorithms	algorithm	NOUN
ejpam-5372	552	9	to	to	PART
ejpam-5372	552	10	calculate	calculate	VERB
ejpam-5372	552	11	the	the	DET
ejpam-5372	552	12	square	square	ADJ
ejpam-5372	552	13	root	root	NOUN
ejpam-5372	552	14	of	of	ADP
ejpam-5372	552	15	an	an	DET
ejpam-5372	552	16	integer	integer	NOUN
ejpam-5372	552	17	modulo	modulo	NOUN
ejpam-5372	553	1	p	p	X
ejpam-5372	553	2	,	,	PUNCT
ejpam-5372	553	3	e.g.	e.g.	ADV
ejpam-5372	553	4	tonelli	tonelli	NOUN
ejpam-5372	553	5	-	-	PUNCT
ejpam-5372	553	6	shanks	shank	NOUN
ejpam-5372	554	1	[	[	X
ejpam-5372	554	2	3][6	3][6	NOUN
ejpam-5372	554	3	]	]	PUNCT
ejpam-5372	554	4	,	,	PUNCT
ejpam-5372	554	5	cipolla	cipolla	X
ejpam-5372	555	1	[	[	X
ejpam-5372	555	2	7	7	NUM
ejpam-5372	555	3	]	]	PUNCT
ejpam-5372	555	4	and	and	CCONJ
ejpam-5372	555	5	daniel	daniel	PROPN
ejpam-5372	555	6	bernstein	bernstein	PROPN
ejpam-5372	556	1	[	[	X
ejpam-5372	556	2	8	8	NUM
ejpam-5372	556	3	]	]	SYM
ejpam-5372	556	4	algorithms	algorithm	NOUN
ejpam-5372	556	5	.	.	PUNCT
ejpam-5372	557	1	in	in	ADP
ejpam-5372	557	2	this	this	DET
ejpam-5372	557	3	section	section	NOUN
ejpam-5372	557	4	,	,	PUNCT
ejpam-5372	557	5	we	we	PRON
ejpam-5372	557	6	rewrite	rewrite	VERB
ejpam-5372	557	7	the	the	DET
ejpam-5372	557	8	tonelli	tonelli	NOUN
ejpam-5372	557	9	-	-	PUNCT
ejpam-5372	557	10	shanks	shank	NOUN
ejpam-5372	557	11	algorithm	algorithm	NOUN
ejpam-5372	558	1	[	[	X
ejpam-5372	558	2	3][6	3][6	NOUN
ejpam-5372	558	3	]	]	PUNCT
ejpam-5372	558	4	to	to	PART
ejpam-5372	558	5	solve	solve	VERB
ejpam-5372	558	6	equation	equation	NOUN
ejpam-5372	558	7	(	(	PUNCT
ejpam-5372	558	8	3.2	3.2	NUM
ejpam-5372	558	9	)	)	PUNCT
ejpam-5372	558	10	x2	x2	NOUN
ejpam-5372	559	1	+	+	CCONJ
ejpam-5372	559	2	c	c	X
ejpam-5372	559	3	≡	≡	PROPN
ejpam-5372	559	4	0	0	PUNCT
ejpam-5372	560	1	[	[	X
ejpam-5372	560	2	p	p	X
ejpam-5372	560	3	]	]	PUNCT
ejpam-5372	560	4	simultaneously	simultaneously	ADV
ejpam-5372	560	5	for	for	ADP
ejpam-5372	560	6	several	several	ADJ
ejpam-5372	560	7	values	value	NOUN
ejpam-5372	560	8	of	of	ADP
ejpam-5372	560	9	c.	c.	NOUN
ejpam-5372	560	10	proposition	proposition	NOUN
ejpam-5372	560	11	19	19	NUM
ejpam-5372	560	12	.	.	PUNCT
ejpam-5372	561	1	as	as	ADP
ejpam-5372	561	2	in	in	ADP
ejpam-5372	561	3	section	section	NOUN
ejpam-5372	561	4	2	2	NUM
ejpam-5372	561	5	,	,	PUNCT
ejpam-5372	561	6	let	let	VERB
ejpam-5372	561	7	us	we	PRON
ejpam-5372	561	8	write	write	VERB
ejpam-5372	561	9	p−	p−	NOUN
ejpam-5372	561	10	1	1	NUM
ejpam-5372	561	11	=	=	SYM
ejpam-5372	561	12	2nz	2nz	NOUN
ejpam-5372	561	13	with	with	ADP
ejpam-5372	561	14	z	z	NOUN
ejpam-5372	561	15	odd	odd	ADJ
ejpam-5372	561	16	.	.	PUNCT
ejpam-5372	562	1	if	if	SCONJ
ejpam-5372	562	2	m	m	NOUN
ejpam-5372	562	3	is	be	AUX
ejpam-5372	562	4	a	a	DET
ejpam-5372	562	5	quadratic	quadratic	ADJ
ejpam-5372	562	6	residue	residue	NOUN
ejpam-5372	562	7	,	,	PUNCT
ejpam-5372	562	8	there	there	PRON
ejpam-5372	562	9	exists	exist	VERB
ejpam-5372	562	10	r	r	NOUN
ejpam-5372	562	11	∈	∈	PROPN
ejpam-5372	562	12	r(n	r(n	PROPN
ejpam-5372	562	13	)	)	PUNCT
ejpam-5372	562	14	such	such	ADJ
ejpam-5372	562	15	that	that	SCONJ
ejpam-5372	562	16	:(	:(	PUNCT
ejpam-5372	562	17	m	m	PRON
ejpam-5372	562	18	p−z	p−z	NOUN
ejpam-5372	562	19	2	2	NUM
ejpam-5372	562	20	r	r	NOUN
ejpam-5372	562	21	)	)	PUNCT
ejpam-5372	562	22	2	2	NUM
ejpam-5372	562	23	≡	≡	PROPN
ejpam-5372	562	24	m	m	VERB
ejpam-5372	562	25	the	the	DET
ejpam-5372	562	26	square	square	ADJ
ejpam-5372	562	27	roots	root	NOUN
ejpam-5372	562	28	of	of	ADP
ejpam-5372	562	29	m	m	NOUN
ejpam-5372	562	30	are	be	AUX
ejpam-5372	562	31	thus	thus	ADV
ejpam-5372	562	32	±m	±m	VERB
ejpam-5372	562	33	p−z	p−z	NOUN
ejpam-5372	562	34	2	2	NUM
ejpam-5372	562	35	r	r	NOUN
ejpam-5372	562	36	and	and	CCONJ
ejpam-5372	562	37	m	m	NOUN
ejpam-5372	562	38	p−z	p−z	NOUN
ejpam-5372	562	39	2	2	NUM
ejpam-5372	562	40	r	r	NOUN
ejpam-5372	562	41	is	be	AUX
ejpam-5372	562	42	a	a	DET
ejpam-5372	562	43	quadratic	quadratic	ADJ
ejpam-5372	562	44	residue	residue	NOUN
ejpam-5372	562	45	if	if	SCONJ
ejpam-5372	562	46	and	and	CCONJ
ejpam-5372	562	47	only	only	ADV
ejpam-5372	562	48	if	if	SCONJ
ejpam-5372	562	49	r	r	NOUN
ejpam-5372	562	50	also	also	ADV
ejpam-5372	562	51	is	be	AUX
ejpam-5372	562	52	.	.	PUNCT
ejpam-5372	563	1	proof	proof	NOUN
ejpam-5372	563	2	.	.	PUNCT
ejpam-5372	564	1	we	we	PRON
ejpam-5372	564	2	still	still	ADV
ejpam-5372	564	3	identify	identify	VERB
ejpam-5372	564	4	f∗	f∗	NOUN
ejpam-5372	564	5	p	p	X
ejpam-5372	564	6	to	to	ADP
ejpam-5372	564	7	(	(	PUNCT
ejpam-5372	564	8	z/2nz)×(z	z/2nz)×(z	NOUN
ejpam-5372	564	9	/	/	SYM
ejpam-5372	564	10	zz	zz	PROPN
ejpam-5372	564	11	)	)	PUNCT
ejpam-5372	564	12	.	.	PUNCT
ejpam-5372	565	1	m	m	PROPN
ejpam-5372	565	2	is	be	AUX
ejpam-5372	565	3	a	a	DET
ejpam-5372	565	4	quadratic	quadratic	ADJ
ejpam-5372	565	5	residue	residue	NOUN
ejpam-5372	565	6	and	and	CCONJ
ejpam-5372	565	7	therefore	therefore	ADV
ejpam-5372	565	8	corresponds	correspond	VERB
ejpam-5372	565	9	to	to	ADP
ejpam-5372	565	10	an	an	DET
ejpam-5372	565	11	element	element	NOUN
ejpam-5372	565	12	in	in	ADP
ejpam-5372	565	13	form	form	NOUN
ejpam-5372	565	14	(	(	PUNCT
ejpam-5372	565	15	2k	2k	NUM
ejpam-5372	565	16	,	,	PUNCT
ejpam-5372	565	17	l	l	NOUN
ejpam-5372	565	18	)	)	PUNCT
ejpam-5372	565	19	.	.	PUNCT
ejpam-5372	566	1	we	we	PRON
ejpam-5372	566	2	know	know	VERB
ejpam-5372	566	3	that	that	SCONJ
ejpam-5372	566	4	p−z	p−z	NOUN
ejpam-5372	566	5	=	=	NOUN
ejpam-5372	566	6	1+(2n	1+(2n	NUM
ejpam-5372	566	7	−	−	NUM
ejpam-5372	566	8	1	1	NUM
ejpam-5372	566	9	)	)	PUNCT
ejpam-5372	566	10	z	z	NOUN
ejpam-5372	566	11	is	be	AUX
ejpam-5372	566	12	even	even	ADV
ejpam-5372	566	13	and	and	CCONJ
ejpam-5372	566	14	p−	p−	NOUN
ejpam-5372	566	15	z	z	PROPN
ejpam-5372	566	16	≡	≡	PROPN
ejpam-5372	566	17	1	1	NUM
ejpam-5372	567	1	[	[	X
ejpam-5372	567	2	z	z	X
ejpam-5372	567	3	]	]	X
ejpam-5372	567	4	hence	hence	ADV
ejpam-5372	567	5	p−z	p−z	VERB
ejpam-5372	567	6	2	2	NUM
ejpam-5372	567	7	×	×	NOUN
ejpam-5372	567	8	2	2	NUM
ejpam-5372	567	9	≡	≡	PROPN
ejpam-5372	567	10	1	1	NUM
ejpam-5372	568	1	[	[	X
ejpam-5372	568	2	z	z	X
ejpam-5372	568	3	]	]	X
ejpam-5372	568	4	.	.	PUNCT
ejpam-5372	569	1	on	on	ADP
ejpam-5372	569	2	the	the	DET
ejpam-5372	569	3	other	other	ADJ
ejpam-5372	569	4	hand	hand	NOUN
ejpam-5372	569	5	,	,	PUNCT
ejpam-5372	569	6	modulo	modulo	PROPN
ejpam-5372	569	7	2n	2n	NUM
ejpam-5372	569	8	,	,	PUNCT
ejpam-5372	569	9	p−z	p−z	NOUN
ejpam-5372	569	10	2	2	NUM
ejpam-5372	569	11	.2k	.2k	NOUN
ejpam-5372	569	12	≡	≡	PROPN
ejpam-5372	569	13	(	(	PUNCT
ejpam-5372	569	14	p−	p−	PROPN
ejpam-5372	569	15	z	z	NOUN
ejpam-5372	569	16	)	)	PUNCT
ejpam-5372	569	17	k	k	PROPN
ejpam-5372	569	18	≡	≡	PROPN
ejpam-5372	569	19	(	(	PUNCT
ejpam-5372	569	20	1−	1−	NUM
ejpam-5372	569	21	z	z	NOUN
ejpam-5372	569	22	)	)	PUNCT
ejpam-5372	569	23	k.	k.	PROPN
ejpam-5372	569	24	let	let	VERB
ejpam-5372	569	25	r	r	NOUN
ejpam-5372	569	26	∈	∈	PROPN
ejpam-5372	569	27	f∗	f∗	NOUN
ejpam-5372	569	28	p	p	X
ejpam-5372	569	29	which	which	PRON
ejpam-5372	569	30	corresponds	correspond	VERB
ejpam-5372	569	31	to	to	ADP
ejpam-5372	569	32	(	(	PUNCT
ejpam-5372	569	33	zk	zk	PROPN
ejpam-5372	569	34	,	,	PUNCT
ejpam-5372	569	35	0	0	NUM
ejpam-5372	569	36	)	)	PUNCT
ejpam-5372	569	37	.	.	PUNCT
ejpam-5372	570	1	then	then	ADV
ejpam-5372	570	2	m	m	VERB
ejpam-5372	570	3	p−z	p−z	NOUN
ejpam-5372	570	4	2	2	NUM
ejpam-5372	570	5	r	r	NOUN
ejpam-5372	570	6	corresponds	correspond	NOUN
ejpam-5372	570	7	to	to	ADP
ejpam-5372	570	8	(	(	PUNCT
ejpam-5372	570	9	k	k	NOUN
ejpam-5372	570	10	,	,	PUNCT
ejpam-5372	570	11	p−z	p−z	NOUN
ejpam-5372	570	12	2	2	NUM
ejpam-5372	570	13	l	l	NOUN
ejpam-5372	570	14	)	)	PUNCT
ejpam-5372	570	15	,	,	PUNCT
ejpam-5372	570	16	so	so	CCONJ
ejpam-5372	570	17	its	its	PRON
ejpam-5372	570	18	square	square	NOUN
ejpam-5372	570	19	is	be	AUX
ejpam-5372	570	20	so	so	ADV
ejpam-5372	570	21	equal	equal	ADJ
ejpam-5372	570	22	to	to	ADP
ejpam-5372	570	23	m.	m.	NOUN
ejpam-5372	570	24	moreover	moreover	ADV
ejpam-5372	570	25	,	,	PUNCT
ejpam-5372	570	26	m	m	VERB
ejpam-5372	570	27	p−z	p−z	NOUN
ejpam-5372	570	28	2	2	NUM
ejpam-5372	570	29	r	r	NOUN
ejpam-5372	570	30	is	be	AUX
ejpam-5372	570	31	a	a	DET
ejpam-5372	570	32	quadratic	quadratic	ADJ
ejpam-5372	570	33	residue	residue	NOUN
ejpam-5372	570	34	if	if	SCONJ
ejpam-5372	570	35	and	and	CCONJ
ejpam-5372	570	36	only	only	ADV
ejpam-5372	570	37	if	if	SCONJ
ejpam-5372	570	38	k	k	PROPN
ejpam-5372	570	39	is	be	AUX
ejpam-5372	570	40	even	even	ADV
ejpam-5372	570	41	,	,	PUNCT
ejpam-5372	570	42	which	which	PRON
ejpam-5372	570	43	is	be	AUX
ejpam-5372	570	44	equivalent	equivalent	ADJ
ejpam-5372	570	45	to	to	ADP
ejpam-5372	570	46	zk	zk	PROPN
ejpam-5372	570	47	being	be	AUX
ejpam-5372	570	48	even	even	ADV
ejpam-5372	570	49	,	,	PUNCT
ejpam-5372	570	50	i.e.	i.e.	X
ejpam-5372	570	51	to	to	ADP
ejpam-5372	570	52	r	r	NOUN
ejpam-5372	570	53	being	be	AUX
ejpam-5372	570	54	quadratic	quadratic	ADJ
ejpam-5372	570	55	residue	residue	NOUN
ejpam-5372	570	56	.	.	PUNCT
ejpam-5372	571	1	remark	remark	NOUN
ejpam-5372	571	2	7	7	NUM
ejpam-5372	571	3	.	.	PUNCT
ejpam-5372	572	1	if	if	SCONJ
ejpam-5372	572	2	p	p	PROPN
ejpam-5372	572	3	∈	∈	PROPN
ejpam-5372	572	4	p1,4	p1,4	PROPN
ejpam-5372	572	5	,	,	PUNCT
ejpam-5372	572	6	we	we	PRON
ejpam-5372	572	7	can	can	AUX
ejpam-5372	572	8	also	also	ADV
ejpam-5372	572	9	write	write	VERB
ejpam-5372	572	10	the	the	DET
ejpam-5372	572	11	square	square	ADJ
ejpam-5372	572	12	root	root	NOUN
ejpam-5372	572	13	of	of	ADP
ejpam-5372	572	14	m	m	PRON
ejpam-5372	572	15	as	as	ADP
ejpam-5372	572	16	m	m	PROPN
ejpam-5372	572	17	p+1−2z	p+1−2z	NOUN
ejpam-5372	572	18	4	4	NUM
ejpam-5372	572	19	r.	r.	NOUN
ejpam-5372	572	20	indeed	indeed	ADV
ejpam-5372	572	21	,	,	PUNCT
ejpam-5372	572	22	on	on	ADP
ejpam-5372	572	23	one	one	NUM
ejpam-5372	572	24	hand	hand	NOUN
ejpam-5372	572	25	p+1	p+1	NOUN
ejpam-5372	572	26	2	2	NUM
ejpam-5372	572	27	and	and	CCONJ
ejpam-5372	572	28	z	z	NOUN
ejpam-5372	572	29	are	be	AUX
ejpam-5372	572	30	both	both	PRON
ejpam-5372	572	31	odd	odd	ADJ
ejpam-5372	572	32	with	with	ADP
ejpam-5372	572	33	z	z	NOUN
ejpam-5372	572	34	≤	≤	NOUN
ejpam-5372	573	1	p−1	p−1	PROPN
ejpam-5372	573	2	4	4	NUM
ejpam-5372	573	3	<	<	NOUN
ejpam-5372	573	4	p+1	p+1	PROPN
ejpam-5372	573	5	2	2	NUM
ejpam-5372	573	6	so	so	ADV
ejpam-5372	573	7	p+1−2z	p+1−2z	NOUN
ejpam-5372	573	8	4	4	NUM
ejpam-5372	573	9	∈	∈	NOUN
ejpam-5372	573	10	n∗	n∗	NOUN
ejpam-5372	573	11	,	,	PUNCT
ejpam-5372	573	12	and	and	CCONJ
ejpam-5372	573	13	on	on	ADP
ejpam-5372	573	14	the	the	DET
ejpam-5372	573	15	other	other	ADJ
ejpam-5372	573	16	hand	hand	NOUN
ejpam-5372	573	17	:	:	PUNCT
ejpam-5372	573	18	(	(	PUNCT
ejpam-5372	573	19	m	m	VERB
ejpam-5372	573	20	p+1−2z	p+1−2z	NOUN
ejpam-5372	573	21	4	4	NUM
ejpam-5372	573	22	r	r	NOUN
ejpam-5372	573	23	)	)	PUNCT
ejpam-5372	573	24	2	2	NUM
ejpam-5372	573	25	≡	≡	PROPN
ejpam-5372	573	26	m	m	NOUN
ejpam-5372	573	27	p−1	p−1	PROPN
ejpam-5372	573	28	2	2	NUM
ejpam-5372	573	29	mp−zr2	mp−zr2	NOUN
ejpam-5372	573	30	≡	≡	PROPN
ejpam-5372	573	31	1×	1×	PROPN
ejpam-5372	573	32	(	(	PUNCT
ejpam-5372	573	33	m	m	PROPN
ejpam-5372	573	34	p−z	p−z	NOUN
ejpam-5372	573	35	2	2	NUM
ejpam-5372	573	36	r	r	NOUN
ejpam-5372	573	37	)	)	PUNCT
ejpam-5372	573	38	2	2	NUM
ejpam-5372	573	39	≡	≡	PROPN
ejpam-5372	573	40	m	m	VERB
ejpam-5372	574	1	[	[	X
ejpam-5372	574	2	p	p	X
ejpam-5372	574	3	]	]	PUNCT
ejpam-5372	574	4	.	.	PUNCT
ejpam-5372	575	1	corollary	corollary	ADJ
ejpam-5372	575	2	7	7	NUM
ejpam-5372	575	3	.	.	PUNCT
ejpam-5372	576	1	if	if	SCONJ
ejpam-5372	576	2	p	p	PROPN
ejpam-5372	576	3	∈	∈	PROPN
ejpam-5372	576	4	p3,4	p3,4	ADJ
ejpam-5372	576	5	and	and	CCONJ
ejpam-5372	576	6	m	m	PROPN
ejpam-5372	576	7	∈	∈	PROPN
ejpam-5372	576	8	gqr	gqr	PROPN
ejpam-5372	576	9	,	,	PUNCT
ejpam-5372	576	10	then	then	ADV
ejpam-5372	576	11	its	its	PRON
ejpam-5372	576	12	square	square	ADJ
ejpam-5372	576	13	roots	root	NOUN
ejpam-5372	576	14	are	be	AUX
ejpam-5372	576	15	±m	±m	PROPN
ejpam-5372	576	16	p+1	p+1	NOUN
ejpam-5372	576	17	4	4	NUM
ejpam-5372	576	18	.	.	PUNCT
ejpam-5372	577	1	proof	proof	NOUN
ejpam-5372	577	2	.	.	PUNCT
ejpam-5372	578	1	this	this	PRON
ejpam-5372	578	2	follows	follow	VERB
ejpam-5372	578	3	directly	directly	ADV
ejpam-5372	578	4	from	from	ADP
ejpam-5372	578	5	proposition	proposition	NOUN
ejpam-5372	578	6	19	19	NUM
ejpam-5372	578	7	,	,	PUNCT
ejpam-5372	578	8	since	since	SCONJ
ejpam-5372	578	9	then	then	ADV
ejpam-5372	578	10	r	r	NOUN
ejpam-5372	578	11	=	=	SYM
ejpam-5372	578	12	±1	±1	VERB
ejpam-5372	578	13	and	and	CCONJ
ejpam-5372	578	14	p	p	NOUN
ejpam-5372	578	15	−	−	PROPN
ejpam-5372	578	16	z	z	NOUN
ejpam-5372	578	17	=	=	PUNCT
ejpam-5372	578	18	p+1	p+1	NOUN
ejpam-5372	578	19	2	2	NUM
ejpam-5372	578	20	.	.	PUNCT
ejpam-5372	579	1	we	we	PRON
ejpam-5372	579	2	can	can	AUX
ejpam-5372	579	3	also	also	ADV
ejpam-5372	579	4	verify	verify	VERB
ejpam-5372	579	5	directly	directly	ADV
ejpam-5372	579	6	m	m	VERB
ejpam-5372	579	7	p+1	p+1	NOUN
ejpam-5372	579	8	2	2	NUM
ejpam-5372	579	9	=	=	SYM
ejpam-5372	579	10	m	m	VERB
ejpam-5372	579	11	p−1	p−1	NOUN
ejpam-5372	579	12	2	2	NUM
ejpam-5372	579	13	+1	+1	NOUN
ejpam-5372	579	14	=	=	PUNCT
ejpam-5372	579	15	m	m	VERB
ejpam-5372	579	16	by	by	ADP
ejpam-5372	579	17	euler	euler	NOUN
ejpam-5372	579	18	’s	’s	PART
ejpam-5372	579	19	criterion	criterion	NOUN
ejpam-5372	579	20	,	,	PUNCT
ejpam-5372	579	21	since	since	SCONJ
ejpam-5372	579	22	m	m	PROPN
ejpam-5372	579	23	∈	∈	PROPN
ejpam-5372	579	24	gqr	gqr	PROPN
ejpam-5372	579	25	.	.	PUNCT
ejpam-5372	579	26	m.	m.	PROPN
ejpam-5372	579	27	wolf	wolf	PROPN
ejpam-5372	579	28	,	,	PUNCT
ejpam-5372	579	29	f.	f.	PROPN
ejpam-5372	579	30	wolf	wolf	PROPN
ejpam-5372	579	31	/	/	SYM
ejpam-5372	579	32	eur	eur	PROPN
ejpam-5372	579	33	.	.	PUNCT
ejpam-5372	580	1	j.	j.	PROPN
ejpam-5372	580	2	pure	pure	PROPN
ejpam-5372	580	3	appl	appl	PROPN
ejpam-5372	580	4	.	.	PROPN
ejpam-5372	580	5	math	math	PROPN
ejpam-5372	580	6	,	,	PUNCT
ejpam-5372	580	7	17	17	NUM
ejpam-5372	580	8	(	(	PUNCT
ejpam-5372	580	9	4	4	NUM
ejpam-5372	580	10	)	)	PUNCT
ejpam-5372	580	11	(	(	PUNCT
ejpam-5372	580	12	2024	2024	NUM
ejpam-5372	580	13	)	)	PUNCT
ejpam-5372	580	14	,	,	PUNCT
ejpam-5372	580	15	2431	2431	NUM
ejpam-5372	580	16	-	-	SYM
ejpam-5372	580	17	2447	2447	NUM
ejpam-5372	580	18	2446	2446	NUM
ejpam-5372	580	19	searching	search	VERB
ejpam-5372	580	20	for	for	ADP
ejpam-5372	580	21	the	the	DET
ejpam-5372	580	22	roots	root	NOUN
ejpam-5372	580	23	of	of	ADP
ejpam-5372	580	24	x2	x2	PROPN
ejpam-5372	581	1	+	+	CCONJ
ejpam-5372	581	2	c	c	NOUN
ejpam-5372	581	3	modulo	modulo	NOUN
ejpam-5372	581	4	p	p	NOUN
ejpam-5372	581	5	allows	allow	VERB
ejpam-5372	581	6	us	we	PRON
ejpam-5372	581	7	to	to	PART
ejpam-5372	581	8	obtain	obtain	VERB
ejpam-5372	581	9	,	,	PUNCT
ejpam-5372	581	10	if	if	SCONJ
ejpam-5372	581	11	it	it	PRON
ejpam-5372	581	12	exists	exist	VERB
ejpam-5372	581	13	,	,	PUNCT
ejpam-5372	581	14	the	the	DET
ejpam-5372	581	15	smallest	small	ADJ
ejpam-5372	581	16	multiple	multiple	NOUN
ejpam-5372	581	17	of	of	ADP
ejpam-5372	581	18	p	p	NOUN
ejpam-5372	581	19	in	in	ADP
ejpam-5372	581	20	ec	ec	PROPN
ejpam-5372	581	21	.	.	PUNCT
ejpam-5372	582	1	we	we	PRON
ejpam-5372	582	2	describe	describe	VERB
ejpam-5372	582	3	such	such	DET
ejpam-5372	582	4	an	an	DET
ejpam-5372	582	5	algorithm	algorithm	NOUN
ejpam-5372	582	6	below	below	ADV
ejpam-5372	582	7	.	.	PUNCT
ejpam-5372	583	1	algorithm	algorithm	NOUN
ejpam-5372	583	2	2	2	NUM
ejpam-5372	583	3	.	.	PUNCT
ejpam-5372	583	4	description	description	NOUN
ejpam-5372	583	5	:	:	PUNCT
ejpam-5372	583	6	search	search	NOUN
ejpam-5372	583	7	for	for	ADP
ejpam-5372	583	8	(	(	PUNCT
ejpam-5372	583	9	x1	x1	PROPN
ejpam-5372	583	10	.	.	PUNCT
ejpam-5372	583	11	.	.	PUNCT
ejpam-5372	583	12	.	.	PUNCT
ejpam-5372	584	1	xn	xn	X
ejpam-5372	584	2	)	)	PUNCT
ejpam-5372	584	3	such	such	ADJ
ejpam-5372	584	4	that	that	SCONJ
ejpam-5372	584	5	xk	xk	PROPN
ejpam-5372	584	6	is	be	AUX
ejpam-5372	584	7	the	the	DET
ejpam-5372	584	8	smallest	small	ADJ
ejpam-5372	584	9	nonnegative	nonnegative	ADJ
ejpam-5372	584	10	integer	integer	NOUN
ejpam-5372	584	11	verifying	verify	VERB
ejpam-5372	584	12	x2k	x2k	NOUN
ejpam-5372	585	1	+	+	CCONJ
ejpam-5372	585	2	ck	ck	PROPN
ejpam-5372	585	3	≡	≡	PROPN
ejpam-5372	585	4	0	0	PUNCT
ejpam-5372	586	1	[	[	X
ejpam-5372	586	2	p	p	X
ejpam-5372	586	3	]	]	X
ejpam-5372	586	4	with	with	ADP
ejpam-5372	586	5	xk	xk	PROPN
ejpam-5372	586	6	and	and	CCONJ
ejpam-5372	586	7	c	c	PROPN
ejpam-5372	586	8	of	of	ADP
ejpam-5372	586	9	opposite	opposite	ADJ
ejpam-5372	586	10	parity	parity	NOUN
ejpam-5372	586	11	(	(	PUNCT
ejpam-5372	586	12	i.e.	i.e.	X
ejpam-5372	586	13	x2k	x2k	NOUN
ejpam-5372	586	14	+	+	CCONJ
ejpam-5372	586	15	ck	ck	X
ejpam-5372	586	16	is	be	AUX
ejpam-5372	586	17	the	the	DET
ejpam-5372	586	18	smallest	small	ADJ
ejpam-5372	586	19	multiple	multiple	NOUN
ejpam-5372	586	20	of	of	ADP
ejpam-5372	586	21	p	p	NOUN
ejpam-5372	586	22	in	in	ADP
ejpam-5372	586	23	eck	eck	PROPN
ejpam-5372	586	24	)	)	PUNCT
ejpam-5372	586	25	,	,	PUNCT
ejpam-5372	586	26	for	for	ADP
ejpam-5372	586	27	a	a	DET
ejpam-5372	586	28	n	n	NOUN
ejpam-5372	586	29	-	-	PUNCT
ejpam-5372	586	30	uplet	uplet	NOUN
ejpam-5372	586	31	(	(	PUNCT
ejpam-5372	586	32	c1	c1	PROPN
ejpam-5372	586	33	.	.	PUNCT
ejpam-5372	586	34	.	.	PUNCT
ejpam-5372	586	35	.	.	PUNCT
ejpam-5372	587	1	cn	cn	X
ejpam-5372	587	2	)	)	PUNCT
ejpam-5372	587	3	∈	∈	PROPN
ejpam-5372	587	4	(	(	PUNCT
ejpam-5372	587	5	n∗)n	n∗)n	PROPN
ejpam-5372	587	6	and	and	CCONJ
ejpam-5372	587	7	p	p	PROPN
ejpam-5372	587	8	∈	∈	PROPN
ejpam-5372	587	9	p⧹	p⧹	VERB
ejpam-5372	587	10	{	{	PUNCT
ejpam-5372	587	11	2	2	NUM
ejpam-5372	587	12	}	}	PUNCT
ejpam-5372	587	13	fixed	fix	VERB
ejpam-5372	587	14	.	.	PUNCT
ejpam-5372	588	1	we	we	PRON
ejpam-5372	588	2	write	write	VERB
ejpam-5372	588	3	p	p	NOUN
ejpam-5372	588	4	−	−	PROPN
ejpam-5372	588	5	1	1	NUM
ejpam-5372	588	6	=	=	SYM
ejpam-5372	588	7	2nz	2nz	NOUN
ejpam-5372	588	8	with	with	ADP
ejpam-5372	588	9	z	z	NOUN
ejpam-5372	588	10	odd	odd	ADJ
ejpam-5372	588	11	.	.	PUNCT
ejpam-5372	589	1	by	by	ADP
ejpam-5372	589	2	convention	convention	NOUN
ejpam-5372	589	3	,	,	PUNCT
ejpam-5372	589	4	we	we	PRON
ejpam-5372	589	5	let	let	VERB
ejpam-5372	589	6	xk	xk	PROPN
ejpam-5372	589	7	=	=	PUNCT
ejpam-5372	589	8	∞	∞	PROPN
ejpam-5372	590	1	if	if	SCONJ
ejpam-5372	590	2	x2	x2	PROPN
ejpam-5372	590	3	+	+	CCONJ
ejpam-5372	590	4	ck	ck	PROPN
ejpam-5372	590	5	has	have	VERB
ejpam-5372	590	6	no	no	DET
ejpam-5372	590	7	root	root	NOUN
ejpam-5372	590	8	modulo	modulo	NOUN
ejpam-5372	591	1	p.	p.	NOUN
ejpam-5372	591	2	(	(	PUNCT
ejpam-5372	591	3	i	i	NOUN
ejpam-5372	591	4	)	)	PUNCT
ejpam-5372	591	5	if	if	SCONJ
ejpam-5372	591	6	p	p	PROPN
ejpam-5372	591	7	∈	∈	PROPN
ejpam-5372	591	8	p3,4	p3,4	VERB
ejpam-5372	591	9	,	,	PUNCT
ejpam-5372	591	10	for	for	ADP
ejpam-5372	591	11	each	each	DET
ejpam-5372	591	12	value	value	NOUN
ejpam-5372	591	13	of	of	ADP
ejpam-5372	591	14	ck	ck	PROPN
ejpam-5372	591	15	,	,	PUNCT
ejpam-5372	591	16	we	we	PRON
ejpam-5372	591	17	compute	compute	VERB
ejpam-5372	591	18	c	c	PROPN
ejpam-5372	591	19	p−1	p−1	PROPN
ejpam-5372	591	20	2	2	NUM
ejpam-5372	591	21	k	k	PROPN
ejpam-5372	591	22	modulo	modulo	NOUN
ejpam-5372	592	1	p.	p.	NOUN
ejpam-5372	592	2	if	if	SCONJ
ejpam-5372	592	3	c	c	PROPN
ejpam-5372	592	4	p−1	p−1	PROPN
ejpam-5372	592	5	2	2	NUM
ejpam-5372	592	6	k	k	PROPN
ejpam-5372	592	7	≡	≡	PROPN
ejpam-5372	592	8	1	1	NUM
ejpam-5372	593	1	[	[	X
ejpam-5372	593	2	p	p	X
ejpam-5372	593	3	]	]	X
ejpam-5372	593	4	,	,	PUNCT
ejpam-5372	593	5	we	we	PRON
ejpam-5372	593	6	set	set	VERB
ejpam-5372	593	7	xk	xk	PROPN
ejpam-5372	593	8	=	=	SYM
ejpam-5372	593	9	∞	∞	PROPN
ejpam-5372	593	10	,	,	PUNCT
ejpam-5372	593	11	otherwise	otherwise	ADV
ejpam-5372	593	12	we	we	PRON
ejpam-5372	593	13	set	set	VERB
ejpam-5372	593	14	xk	xk	PROPN
ejpam-5372	593	15	the	the	DET
ejpam-5372	593	16	smallest	small	ADJ
ejpam-5372	593	17	positive	positive	ADJ
ejpam-5372	593	18	integer	integer	NOUN
ejpam-5372	593	19	with	with	ADP
ejpam-5372	593	20	parity	parity	NOUN
ejpam-5372	593	21	opposite	opposite	NOUN
ejpam-5372	593	22	to	to	ADP
ejpam-5372	593	23	that	that	PRON
ejpam-5372	593	24	of	of	ADP
ejpam-5372	593	25	ck	ck	PROPN
ejpam-5372	593	26	and	and	CCONJ
ejpam-5372	593	27	such	such	ADJ
ejpam-5372	593	28	that	that	SCONJ
ejpam-5372	593	29	xk	xk	PROPN
ejpam-5372	593	30	≡	≡	PROPN
ejpam-5372	593	31	±c	±c	PROPN
ejpam-5372	593	32	p+1	p+1	NOUN
ejpam-5372	593	33	4	4	NUM
ejpam-5372	593	34	k	k	NOUN
ejpam-5372	594	1	[	[	X
ejpam-5372	594	2	p	p	X
ejpam-5372	594	3	]	]	X
ejpam-5372	594	4	.	.	PUNCT
ejpam-5372	595	1	(	(	PUNCT
ejpam-5372	595	2	ii	ii	NOUN
ejpam-5372	595	3	)	)	PUNCT
ejpam-5372	595	4	if	if	SCONJ
ejpam-5372	595	5	p	p	PROPN
ejpam-5372	595	6	∈	∈	PROPN
ejpam-5372	595	7	p1,4	p1,4	PROPN
ejpam-5372	595	8	,	,	PUNCT
ejpam-5372	595	9	we	we	PRON
ejpam-5372	595	10	set	set	VERB
ejpam-5372	595	11	g	g	NOUN
ejpam-5372	595	12	=	=	SYM
ejpam-5372	595	13	2	2	NUM
ejpam-5372	595	14	and	and	CCONJ
ejpam-5372	595	15	increase	increase	VERB
ejpam-5372	595	16	it	it	PRON
ejpam-5372	595	17	until	until	ADP
ejpam-5372	595	18	g	g	PROPN
ejpam-5372	595	19	∈	∈	PROPN
ejpam-5372	595	20	gqnr	gqnr	NOUN
ejpam-5372	595	21	(	(	PUNCT
ejpam-5372	595	22	i.e.	i.e.	X
ejpam-5372	595	23	by	by	ADP
ejpam-5372	595	24	euler	euler	NOUN
ejpam-5372	595	25	criterion	criterion	NOUN
ejpam-5372	595	26	g	g	PROPN
ejpam-5372	595	27	p−1	p−1	PROPN
ejpam-5372	595	28	2	2	NUM
ejpam-5372	595	29	≡	≡	PROPN
ejpam-5372	595	30	−1	−1	NOUN
ejpam-5372	595	31	)	)	PUNCT
ejpam-5372	595	32	then	then	ADV
ejpam-5372	595	33	:	:	PUNCT
ejpam-5372	595	34	•	•	X
ejpam-5372	595	35	we	we	PRON
ejpam-5372	595	36	enumerate	enumerate	VERB
ejpam-5372	595	37	r(n−1	r(n−1	PRON
ejpam-5372	595	38	)	)	PUNCT
ejpam-5372	595	39	p	p	NOUN
ejpam-5372	595	40	=	=	X
ejpam-5372	595	41	{	{	PUNCT
ejpam-5372	595	42	±(gz)2k	±(gz)2k	PROPN
ejpam-5372	595	43	[	[	X
ejpam-5372	595	44	p	p	X
ejpam-5372	595	45	]	]	X
ejpam-5372	595	46	,	,	PUNCT
ejpam-5372	595	47	1	1	NUM
ejpam-5372	595	48	≤	≤	NUM
ejpam-5372	595	49	k	k	PROPN
ejpam-5372	595	50	≤	≤	PROPN
ejpam-5372	595	51	2n−2	2n−2	NUM
ejpam-5372	595	52	}	}	PUNCT
ejpam-5372	595	53	.	.	PUNCT
ejpam-5372	596	1	•	•	NOUN
ejpam-5372	596	2	for	for	SCONJ
ejpam-5372	596	3	each	each	DET
ejpam-5372	596	4	value	value	NOUN
ejpam-5372	596	5	ck	ck	NOUN
ejpam-5372	596	6	:	:	PUNCT
ejpam-5372	596	7	(	(	PUNCT
ejpam-5372	596	8	a	a	X
ejpam-5372	596	9	)	)	PUNCT
ejpam-5372	596	10	if	if	SCONJ
ejpam-5372	596	11	c	c	PROPN
ejpam-5372	596	12	p−1	p−1	PROPN
ejpam-5372	596	13	2	2	NUM
ejpam-5372	596	14	k	k	PROPN
ejpam-5372	596	15	≡	≡	PROPN
ejpam-5372	596	16	−1	−1	NOUN
ejpam-5372	597	1	[	[	X
ejpam-5372	597	2	p	p	X
ejpam-5372	597	3	]	]	X
ejpam-5372	597	4	,	,	PUNCT
ejpam-5372	597	5	we	we	PRON
ejpam-5372	597	6	set	set	VERB
ejpam-5372	597	7	xk	xk	PROPN
ejpam-5372	597	8	=	=	PROPN
ejpam-5372	597	9	∞.	∞.	PROPN
ejpam-5372	597	10	(	(	PUNCT
ejpam-5372	597	11	b	b	NOUN
ejpam-5372	597	12	)	)	PUNCT
ejpam-5372	597	13	otherwise	otherwise	ADV
ejpam-5372	597	14	for	for	ADP
ejpam-5372	597	15	each	each	DET
ejpam-5372	597	16	element	element	NOUN
ejpam-5372	597	17	r	r	NOUN
ejpam-5372	597	18	of	of	ADP
ejpam-5372	597	19	r(n−1	r(n−1	PROPN
ejpam-5372	597	20	)	)	PUNCT
ejpam-5372	597	21	p	p	NOUN
ejpam-5372	597	22	,	,	PUNCT
ejpam-5372	597	23	we	we	PRON
ejpam-5372	597	24	test	test	VERB
ejpam-5372	598	1	if	if	SCONJ
ejpam-5372	598	2	(	(	PUNCT
ejpam-5372	598	3	−c	−c	NOUN
ejpam-5372	598	4	)	)	PUNCT
ejpam-5372	598	5	p+1−2z	p+1−2z	NOUN
ejpam-5372	598	6	2	2	NUM
ejpam-5372	598	7	r2	r2	PROPN
ejpam-5372	598	8	≡	≡	PROPN
ejpam-5372	598	9	−c	−c	NOUN
ejpam-5372	599	1	[	[	X
ejpam-5372	599	2	p	p	X
ejpam-5372	599	3	]	]	X
ejpam-5372	599	4	modulo	modulo	NOUN
ejpam-5372	599	5	p.	p.	NOUN
ejpam-5372	599	6	if	if	SCONJ
ejpam-5372	599	7	true	true	ADJ
ejpam-5372	599	8	,	,	PUNCT
ejpam-5372	599	9	we	we	PRON
ejpam-5372	599	10	set	set	VERB
ejpam-5372	599	11	xk	xk	PROPN
ejpam-5372	599	12	the	the	DET
ejpam-5372	599	13	smallest	small	ADJ
ejpam-5372	599	14	positive	positive	ADJ
ejpam-5372	599	15	integer	integer	NOUN
ejpam-5372	599	16	with	with	ADP
ejpam-5372	599	17	parity	parity	NOUN
ejpam-5372	599	18	opposite	opposite	NOUN
ejpam-5372	599	19	to	to	ADP
ejpam-5372	599	20	that	that	PRON
ejpam-5372	599	21	of	of	ADP
ejpam-5372	599	22	ck	ck	PROPN
ejpam-5372	599	23	and	and	CCONJ
ejpam-5372	599	24	such	such	ADJ
ejpam-5372	599	25	that	that	SCONJ
ejpam-5372	599	26	xk	xk	PROPN
ejpam-5372	599	27	≡	≡	PROPN
ejpam-5372	599	28	±(−ck	±(−ck	PROPN
ejpam-5372	599	29	)	)	PUNCT
ejpam-5372	599	30	p+1−2z	p+1−2z	NOUN
ejpam-5372	599	31	4	4	NUM
ejpam-5372	599	32	r.	r.	PROPN
ejpam-5372	599	33	4	4	NUM
ejpam-5372	599	34	.	.	PUNCT
ejpam-5372	599	35	conclusion	conclusion	NOUN
ejpam-5372	599	36	based	base	VERB
ejpam-5372	599	37	on	on	ADP
ejpam-5372	599	38	the	the	DET
ejpam-5372	599	39	study	study	NOUN
ejpam-5372	599	40	of	of	ADP
ejpam-5372	599	41	primitive	primitive	ADJ
ejpam-5372	599	42	roots	root	NOUN
ejpam-5372	599	43	(	(	PUNCT
ejpam-5372	599	44	gz	gz	NOUN
ejpam-5372	599	45	)	)	PUNCT
ejpam-5372	599	46	and	and	CCONJ
ejpam-5372	599	47	semi	semi	ADJ
ejpam-5372	599	48	-	-	ADJ
ejpam-5372	599	49	primitive	primitive	ADJ
ejpam-5372	599	50	roots	root	NOUN
ejpam-5372	599	51	(	(	PUNCT
ejpam-5372	599	52	gs	gs	NOUN
ejpam-5372	599	53	)	)	PUNCT
ejpam-5372	599	54	modulo	modulo	VERB
ejpam-5372	599	55	p	p	PROPN
ejpam-5372	599	56	∈	∈	PROPN
ejpam-5372	599	57	p⧹	p⧹	VERB
ejpam-5372	599	58	{	{	PUNCT
ejpam-5372	599	59	2	2	NUM
ejpam-5372	599	60	}	}	PUNCT
ejpam-5372	599	61	,	,	PUNCT
ejpam-5372	599	62	and	and	CCONJ
ejpam-5372	599	63	in	in	ADP
ejpam-5372	599	64	particular	particular	ADJ
ejpam-5372	599	65	thanks	thank	NOUN
ejpam-5372	599	66	to	to	ADP
ejpam-5372	599	67	recursive	recursive	ADJ
ejpam-5372	599	68	sequences	sequence	NOUN
ejpam-5372	599	69	of	of	ADP
ejpam-5372	599	70	elements	element	NOUN
ejpam-5372	599	71	of	of	ADP
ejpam-5372	599	72	gz	gz	NOUN
ejpam-5372	599	73	and	and	CCONJ
ejpam-5372	599	74	gs	gs	INTJ
ejpam-5372	599	75	,	,	PUNCT
ejpam-5372	599	76	we	we	PRON
ejpam-5372	599	77	proposed	propose	VERB
ejpam-5372	599	78	an	an	DET
ejpam-5372	599	79	original	original	ADJ
ejpam-5372	599	80	algorithm	algorithm	NOUN
ejpam-5372	599	81	returning	return	VERB
ejpam-5372	599	82	these	these	DET
ejpam-5372	599	83	two	two	NUM
ejpam-5372	599	84	sets	set	NOUN
ejpam-5372	599	85	in	in	ADP
ejpam-5372	599	86	full	full	ADJ
ejpam-5372	599	87	,	,	PUNCT
ejpam-5372	599	88	which	which	PRON
ejpam-5372	599	89	is	be	AUX
ejpam-5372	599	90	based	base	VERB
ejpam-5372	599	91	on	on	ADP
ejpam-5372	599	92	a	a	DET
ejpam-5372	599	93	prime	prime	ADJ
ejpam-5372	599	94	number	number	NOUN
ejpam-5372	599	95	generator	generator	NOUN
ejpam-5372	599	96	algorithm	algorithm	NOUN
ejpam-5372	599	97	rather	rather	ADV
ejpam-5372	599	98	than	than	ADP
ejpam-5372	599	99	on	on	ADP
ejpam-5372	599	100	gcd	gcd	NOUN
ejpam-5372	599	101	calculations	calculation	NOUN
ejpam-5372	599	102	.	.	PUNCT
ejpam-5372	600	1	we	we	PRON
ejpam-5372	600	2	then	then	ADV
ejpam-5372	600	3	came	come	VERB
ejpam-5372	600	4	back	back	ADV
ejpam-5372	600	5	to	to	ADP
ejpam-5372	600	6	shanks	shank	NOUN
ejpam-5372	600	7	’	'	PUNCT
ejpam-5372	600	8	conjecture	conjecture	NOUN
ejpam-5372	600	9	,	,	PUNCT
ejpam-5372	600	10	which	which	PRON
ejpam-5372	600	11	we	we	PRON
ejpam-5372	600	12	generalised	generalise	VERB
ejpam-5372	600	13	to	to	ADP
ejpam-5372	600	14	irreducible	irreducible	ADJ
ejpam-5372	600	15	quadratic	quadratic	ADJ
ejpam-5372	600	16	forms	form	NOUN
ejpam-5372	600	17	,	,	PUNCT
ejpam-5372	600	18	and	and	CCONJ
ejpam-5372	600	19	showed	show	VERB
ejpam-5372	600	20	that	that	SCONJ
ejpam-5372	600	21	the	the	DET
ejpam-5372	600	22	conjectured	conjecture	VERB
ejpam-5372	600	23	asymptotic	asymptotic	ADJ
ejpam-5372	600	24	density	density	NOUN
ejpam-5372	600	25	constant	constant	ADJ
ejpam-5372	600	26	could	could	AUX
ejpam-5372	600	27	be	be	AUX
ejpam-5372	600	28	well	well	ADV
ejpam-5372	600	29	controlled	control	VERB
ejpam-5372	600	30	.	.	PUNCT
ejpam-5372	601	1	to	to	ADP
ejpam-5372	601	2	this	this	DET
ejpam-5372	601	3	end	end	NOUN
ejpam-5372	601	4	,	,	PUNCT
ejpam-5372	601	5	we	we	PRON
ejpam-5372	601	6	rewrote	rewrote	VERB
ejpam-5372	601	7	the	the	DET
ejpam-5372	601	8	tonelli	tonelli	NOUN
ejpam-5372	601	9	-	-	PUNCT
ejpam-5372	601	10	shanks	shank	NOUN
ejpam-5372	601	11	algorithm	algorithm	NOUN
ejpam-5372	601	12	for	for	ADP
ejpam-5372	601	13	solving	solve	VERB
ejpam-5372	601	14	the	the	DET
ejpam-5372	601	15	equation	equation	NOUN
ejpam-5372	601	16	x2	x2	PROPN
ejpam-5372	602	1	+	+	CCONJ
ejpam-5372	602	2	c	c	X
ejpam-5372	602	3	≡	≡	PROPN
ejpam-5372	602	4	0	0	PUNCT
ejpam-5372	603	1	[	[	X
ejpam-5372	603	2	p	p	X
ejpam-5372	603	3	]	]	PUNCT
ejpam-5372	603	4	.	.	PUNCT
ejpam-5372	604	1	acknowledgements	acknowledgement	NOUN
ejpam-5372	604	2	we	we	PRON
ejpam-5372	604	3	would	would	AUX
ejpam-5372	604	4	like	like	VERB
ejpam-5372	604	5	to	to	PART
ejpam-5372	604	6	thank	thank	VERB
ejpam-5372	604	7	françois	françois	NOUN
ejpam-5372	604	8	-	-	PUNCT
ejpam-5372	604	9	xavier	xavier	NOUN
ejpam-5372	604	10	villemin	villemin	PROPN
ejpam-5372	604	11	for	for	ADP
ejpam-5372	604	12	his	his	PRON
ejpam-5372	604	13	attentive	attentive	ADJ
ejpam-5372	604	14	comments	comment	NOUN
ejpam-5372	604	15	and	and	CCONJ
ejpam-5372	604	16	suggestions	suggestion	NOUN
ejpam-5372	604	17	.	.	PUNCT
ejpam-5372	605	1	references	reference	NOUN
ejpam-5372	605	2	2447	2447	NUM
ejpam-5372	605	3	references	reference	NOUN
ejpam-5372	605	4	[	[	X
ejpam-5372	605	5	1	1	NUM
ejpam-5372	605	6	]	]	PUNCT
ejpam-5372	605	7	shanks	shank	NOUN
ejpam-5372	605	8	daniel	daniel	PROPN
ejpam-5372	605	9	.	.	PUNCT
ejpam-5372	606	1	on	on	ADP
ejpam-5372	606	2	the	the	DET
ejpam-5372	606	3	conjecture	conjecture	NOUN
ejpam-5372	606	4	of	of	ADP
ejpam-5372	606	5	hardy	hardy	ADJ
ejpam-5372	606	6	and	and	CCONJ
ejpam-5372	606	7	littlewood	littlewood	NOUN
ejpam-5372	606	8	concerning	concern	VERB
ejpam-5372	606	9	the	the	DET
ejpam-5372	606	10	number	number	NOUN
ejpam-5372	606	11	of	of	ADP
ejpam-5372	606	12	primes	prime	NOUN
ejpam-5372	606	13	of	of	ADP
ejpam-5372	606	14	the	the	DET
ejpam-5372	606	15	form	form	NOUN
ejpam-5372	606	16	n2	n2	NOUN
ejpam-5372	606	17	+	+	CCONJ
ejpam-5372	606	18	a.	a.	NOUN
ejpam-5372	606	19	mathematics	mathematic	NOUN
ejpam-5372	606	20	of	of	ADP
ejpam-5372	606	21	computation	computation	NOUN
ejpam-5372	606	22	,	,	PUNCT
ejpam-5372	606	23	vol	vol	NOUN
ejpam-5372	606	24	.	.	PUNCT
ejpam-5372	607	1	14(no	14(no	NUM
ejpam-5372	607	2	.	.	PUNCT
ejpam-5372	608	1	72):pp	72):pp	NUM
ejpam-5372	608	2	.	.	PUNCT
ejpam-5372	609	1	321–332	321–332	NUM
ejpam-5372	609	2	,	,	PUNCT
ejpam-5372	609	3	1960	1960	NUM
ejpam-5372	609	4	.	.	PUNCT
ejpam-5372	609	5	https://doi.org/10.2307/2003891	https://doi.org/10.2307/2003891	X
ejpam-5372	609	6	.	.	PUNCT
ejpam-5372	610	1	[	[	X
ejpam-5372	610	2	2	2	X
ejpam-5372	610	3	]	]	X
ejpam-5372	610	4	marc	marc	PROPN
ejpam-5372	610	5	wolf	wolf	PROPN
ejpam-5372	610	6	françois	françois	PROPN
ejpam-5372	610	7	wolf	wolf	NOUN
ejpam-5372	610	8	.	.	PUNCT
ejpam-5372	611	1	on	on	ADP
ejpam-5372	611	2	the	the	DET
ejpam-5372	611	3	density	density	NOUN
ejpam-5372	611	4	of	of	ADP
ejpam-5372	611	5	primes	prime	NOUN
ejpam-5372	611	6	of	of	ADP
ejpam-5372	611	7	the	the	DET
ejpam-5372	611	8	form	form	NOUN
ejpam-5372	611	9	x2+c	x2+c	PROPN
ejpam-5372	611	10	.	.	PUNCT
ejpam-5372	611	11	transactions	transaction	NOUN
ejpam-5372	611	12	on	on	ADP
ejpam-5372	611	13	machine	machine	NOUN
ejpam-5372	611	14	learning	learning	NOUN
ejpam-5372	611	15	and	and	CCONJ
ejpam-5372	611	16	artificial	artificial	ADJ
ejpam-5372	611	17	intelligence	intelligence	NOUN
ejpam-5372	611	18	,	,	PUNCT
ejpam-5372	611	19	vol	vol	NOUN
ejpam-5372	611	20	.	.	PUNCT
ejpam-5372	611	21	11(no	11(no	NUM
ejpam-5372	611	22	.	.	PUNCT
ejpam-5372	612	1	6):pp	6):pp	NUM
ejpam-5372	612	2	.	.	PUNCT
ejpam-5372	613	1	80–105	80–105	NUM
ejpam-5372	613	2	,	,	PUNCT
ejpam-5372	613	3	2023	2023	NUM
ejpam-5372	613	4	.	.	PUNCT
ejpam-5372	614	1	https://doi.org/10.14738/tecs.116.15890	https://doi.org/10.14738/tecs.116.15890	VERB
ejpam-5372	614	2	.	.	PUNCT
ejpam-5372	615	1	[	[	X
ejpam-5372	615	2	3	3	X
ejpam-5372	615	3	]	]	X
ejpam-5372	615	4	tonelli	tonelli	PROPN
ejpam-5372	615	5	alberto	alberto	PROPN
ejpam-5372	615	6	.	.	PUNCT
ejpam-5372	615	7	bemerkung	bemerkung	PROPN
ejpam-5372	615	8	über	über	PROPN
ejpam-5372	615	9	die	die	VERB
ejpam-5372	615	10	auflösung	auflösung	ADP
ejpam-5372	615	11	quadratischer	quadratischer	ADJ
ejpam-5372	615	12	congruenzen	congruenzen	NOUN
ejpam-5372	615	13	.	.	PUNCT
ejpam-5372	616	1	nachrichten	nachrichten	PROPN
ejpam-5372	616	2	von	von	PROPN
ejpam-5372	616	3	der	der	PROPN
ejpam-5372	616	4	königl	königl	PROPN
ejpam-5372	616	5	.	.	PUNCT
ejpam-5372	617	1	gesellschaft	gesellschaft	PROPN
ejpam-5372	617	2	der	der	PROPN
ejpam-5372	617	3	wissenschaften	wissenschaften	AUX
ejpam-5372	617	4	und	und	VERB
ejpam-5372	617	5	der	der	ADJ
ejpam-5372	617	6	georg	georg	NOUN
ejpam-5372	617	7	-	-	PUNCT
ejpam-5372	617	8	augustsuniversität	augustsuniversität	PROPN
ejpam-5372	617	9	zu	zu	PROPN
ejpam-5372	617	10	göttingen	göttingen	NOUN
ejpam-5372	617	11	,	,	PUNCT
ejpam-5372	617	12	vol	vol	NOUN
ejpam-5372	617	13	.	.	PUNCT
ejpam-5372	618	1	1891(no	1891(no	NUM
ejpam-5372	618	2	.	.	PUNCT
ejpam-5372	619	1	6):pp	6):pp	NUM
ejpam-5372	619	2	.	.	PUNCT
ejpam-5372	620	1	344–346	344–346	NUM
ejpam-5372	620	2	,	,	PUNCT
ejpam-5372	620	3	1891	1891	NUM
ejpam-5372	620	4	.	.	PUNCT
ejpam-5372	621	1	https://gdz.sub.unigoettingen.de/id/ppn2524570721891?tify	https://gdz.sub.unigoettingen.de/id/ppn2524570721891?tify	PUNCT
ejpam-5372	622	1	=	=	NOUN
ejpam-5372	623	1	[	[	X
ejpam-5372	623	2	4	4	X
ejpam-5372	623	3	]	]	X
ejpam-5372	623	4	marc	marc	PROPN
ejpam-5372	623	5	wolf	wolf	PROPN
ejpam-5372	623	6	françois	françois	PROPN
ejpam-5372	623	7	wolf	wolf	NOUN
ejpam-5372	623	8	.	.	PUNCT
ejpam-5372	624	1	primality	primality	PROPN
ejpam-5372	624	2	test	test	NOUN
ejpam-5372	624	3	and	and	CCONJ
ejpam-5372	624	4	primes	prime	VERB
ejpam-5372	624	5	enumeration	enumeration	NOUN
ejpam-5372	624	6	using	use	VERB
ejpam-5372	624	7	odd	odd	ADJ
ejpam-5372	624	8	numbers	number	NOUN
ejpam-5372	624	9	indexation	indexation	NOUN
ejpam-5372	624	10	.	.	PUNCT
ejpam-5372	625	1	transactions	transaction	NOUN
ejpam-5372	625	2	on	on	ADP
ejpam-5372	625	3	machine	machine	NOUN
ejpam-5372	625	4	learning	learning	NOUN
ejpam-5372	625	5	and	and	CCONJ
ejpam-5372	625	6	artificial	artificial	ADJ
ejpam-5372	625	7	intelligence	intelligence	NOUN
ejpam-5372	625	8	,	,	PUNCT
ejpam-5372	625	9	vol	vol	NOUN
ejpam-5372	625	10	.	.	PUNCT
ejpam-5372	625	11	8(no	8(no	NUM
ejpam-5372	625	12	.	.	PUNCT
ejpam-5372	626	1	2):pp	2):pp	NUM
ejpam-5372	626	2	.	.	PUNCT
ejpam-5372	627	1	11–41	11–41	NUM
ejpam-5372	627	2	,	,	PUNCT
ejpam-5372	627	3	2020	2020	NUM
ejpam-5372	627	4	.	.	PUNCT
ejpam-5372	628	1	https://doi.org/10.14738/tmlai.82.8054	https://doi.org/10.14738/tmlai.82.8054	NOUN
ejpam-5372	628	2	.	.	PUNCT
ejpam-5372	629	1	[	[	X
ejpam-5372	629	2	5	5	X
ejpam-5372	629	3	]	]	PUNCT
ejpam-5372	629	4	herbert	herbert	PROPN
ejpam-5372	629	5	s.	s.	PROPN
ejpam-5372	629	6	zuckerman	zuckerman	PROPN
ejpam-5372	629	7	ivan	ivan	PROPN
ejpam-5372	629	8	niven	niven	PROPN
ejpam-5372	629	9	.	.	PUNCT
ejpam-5372	630	1	improved	improve	VERB
ejpam-5372	630	2	incremental	incremental	ADJ
ejpam-5372	630	3	prime	prime	ADJ
ejpam-5372	630	4	number	number	NOUN
ejpam-5372	630	5	sieves	sieve	NOUN
ejpam-5372	630	6	.	.	PUNCT
ejpam-5372	631	1	algorithmic	algorithmic	ADJ
ejpam-5372	631	2	number	number	NOUN
ejpam-5372	631	3	theory	theory	NOUN
ejpam-5372	631	4	.	.	PUNCT
ejpam-5372	632	1	ants	ant	NOUN
ejpam-5372	632	2	1994	1994	NUM
ejpam-5372	632	3	.	.	PUNCT
ejpam-5372	633	1	lecture	lecture	NOUN
ejpam-5372	633	2	notes	note	NOUN
ejpam-5372	633	3	in	in	ADP
ejpam-5372	633	4	computer	computer	NOUN
ejpam-5372	633	5	science	science	NOUN
ejpam-5372	633	6	,	,	PUNCT
ejpam-5372	633	7	vol	vol	NOUN
ejpam-5372	633	8	.	.	PUNCT
ejpam-5372	634	1	270(no	270(no	NUM
ejpam-5372	634	2	.	.	PUNCT
ejpam-5372	635	1	6):pp	6):pp	NUM
ejpam-5372	635	2	.	.	PROPN
ejpam-5372	635	3	540	540	NUM
ejpam-5372	635	4	,	,	PUNCT
ejpam-5372	635	5	1994	1994	NUM
ejpam-5372	635	6	.	.	PUNCT
ejpam-5372	636	1	https://doi.org/10.1016/0016-0032(60)90676-1	https://doi.org/10.1016/0016-0032(60)90676-1	NOUN
ejpam-5372	636	2	.	.	PUNCT
ejpam-5372	637	1	[	[	X
ejpam-5372	637	2	6	6	NUM
ejpam-5372	637	3	]	]	PUNCT
ejpam-5372	637	4	shanks	shanks	PROPN
ejpam-5372	637	5	daniel	daniel	PROPN
ejpam-5372	637	6	.	.	PUNCT
ejpam-5372	638	1	five	five	NUM
ejpam-5372	638	2	number	number	NOUN
ejpam-5372	638	3	-	-	PUNCT
ejpam-5372	638	4	theoretic	theoretic	NOUN
ejpam-5372	638	5	algorithms	algorithm	NOUN
ejpam-5372	638	6	.	.	PUNCT
ejpam-5372	639	1	proceedings	proceeding	NOUN
ejpam-5372	639	2	of	of	ADP
ejpam-5372	639	3	the	the	DET
ejpam-5372	639	4	second	second	PROPN
ejpam-5372	639	5	manitoba	manitoba	PROPN
ejpam-5372	639	6	conference	conference	PROPN
ejpam-5372	639	7	on	on	ADP
ejpam-5372	639	8	numerical	numerical	ADJ
ejpam-5372	639	9	mathematics	mathematics	PROPN
ejpam-5372	639	10	,	,	PUNCT
ejpam-5372	639	11	congressus	congressus	PROPN
ejpam-5372	639	12	numerantium	numerantium	PROPN
ejpam-5372	639	13	,	,	PUNCT
ejpam-5372	639	14	vol	vol	NOUN
ejpam-5372	639	15	.	.	PUNCT
ejpam-5372	640	1	1(no	1(no	NUM
ejpam-5372	640	2	.	.	PUNCT
ejpam-5372	641	1	vii):pp	vii):pp	PROPN
ejpam-5372	641	2	.	.	PUNCT
ejpam-5372	642	1	51–70	51–70	NUM
ejpam-5372	642	2	,	,	PUNCT
ejpam-5372	642	3	1973	1973	NUM
ejpam-5372	642	4	.	.	PUNCT
ejpam-5372	643	1	[	[	X
ejpam-5372	643	2	7	7	X
ejpam-5372	643	3	]	]	X
ejpam-5372	643	4	michel	michel	PROPN
ejpam-5372	643	5	cipolla	cipolla	PROPN
ejpam-5372	643	6	.	.	PUNCT
ejpam-5372	643	7	un	un	PROPN
ejpam-5372	643	8	metodo	metodo	PROPN
ejpam-5372	643	9	per	per	ADP
ejpam-5372	643	10	la	la	X
ejpam-5372	643	11	risoluzione	risoluzione	PROPN
ejpam-5372	643	12	della	della	PROPN
ejpam-5372	643	13	congruenza	congruenza	PROPN
ejpam-5372	643	14	di	di	PROPN
ejpam-5372	643	15	secondo	secondo	PROPN
ejpam-5372	643	16	grado	grado	PROPN
ejpam-5372	643	17	.	.	PUNCT
ejpam-5372	644	1	napoli	napoli	PROPN
ejpam-5372	644	2	rend	rend	VERB
ejpam-5372	644	3	,	,	PUNCT
ejpam-5372	644	4	vol	vol	NOUN
ejpam-5372	644	5	.	.	PUNCT
ejpam-5372	645	1	9	9	NUM
ejpam-5372	645	2	:	:	SYM
ejpam-5372	645	3	pp	pp	ADJ
ejpam-5372	645	4	.	.	PUNCT
ejpam-5372	646	1	154–163	154–163	NUM
ejpam-5372	646	2	,	,	PUNCT
ejpam-5372	646	3	1903	1903	NUM
ejpam-5372	646	4	.	.	PUNCT
ejpam-5372	647	1	[	[	X
ejpam-5372	647	2	8	8	NUM
ejpam-5372	647	3	]	]	X
ejpam-5372	647	4	daniel	daniel	PROPN
ejpam-5372	647	5	j.	j.	PROPN
ejpam-5372	647	6	bernstein	bernstein	PROPN
ejpam-5372	647	7	.	.	PUNCT
ejpam-5372	648	1	faster	fast	ADJ
ejpam-5372	648	2	square	square	ADJ
ejpam-5372	648	3	roots	root	NOUN
ejpam-5372	648	4	in	in	ADP
ejpam-5372	648	5	annoying	annoying	ADJ
ejpam-5372	648	6	finite	finite	ADJ
ejpam-5372	648	7	fields	field	NOUN
ejpam-5372	648	8	.	.	PUNCT
ejpam-5372	649	1	in	in	ADP
ejpam-5372	649	2	.	.	PUNCT
ejpam-5372	649	3	,	,	PUNCT
ejpam-5372	649	4	2007	2007	NUM
ejpam-5372	649	5	.	.	PUNCT
ejpam-5372	650	1	https://api.semanticscholar.org/corpusid:247863028	https://api.semanticscholar.org/corpusid:247863028	NOUN
ejpam-5372	650	2	.	.	PUNCT