id	sid	tid	token	lemma	pos
ejpam-245	1	1	4_245_jaroma.dvi	4_245_jaroma.dvi	NUM
ejpam-245	1	2	european	european	PROPN
ejpam-245	1	3	journal	journal	PROPN
ejpam-245	1	4	of	of	ADP
ejpam-245	1	5	pure	pure	ADJ
ejpam-245	1	6	and	and	CCONJ
ejpam-245	1	7	applied	apply	VERB
ejpam-245	1	8	mathematics	mathematic	NOUN
ejpam-245	1	9	vol	vol	NOUN
ejpam-245	1	10	.	.	PROPN
ejpam-245	1	11	2	2	NUM
ejpam-245	1	12	,	,	PUNCT
ejpam-245	1	13	no	no	INTJ
ejpam-245	1	14	.	.	NOUN
ejpam-245	1	15	3	3	NUM
ejpam-245	1	16	,	,	PUNCT
ejpam-245	1	17	2009	2009	NUM
ejpam-245	1	18	,	,	PUNCT
ejpam-245	1	19	(	(	PUNCT
ejpam-245	1	20	352	352	NUM
ejpam-245	1	21	-	-	SYM
ejpam-245	1	22	360	360	NUM
ejpam-245	1	23	)	)	PUNCT
ejpam-245	1	24	issn	issn	PROPN
ejpam-245	1	25	1307	1307	NUM
ejpam-245	1	26	-	-	SYM
ejpam-245	1	27	5543	5543	NUM
ejpam-245	1	28	–	–	PUNCT
ejpam-245	1	29	www.ejpam.com	www.ejpam.com	X
ejpam-245	1	30	equivalence	equivalence	NOUN
ejpam-245	1	31	of	of	ADP
ejpam-245	1	32	pepin	pepin	PROPN
ejpam-245	1	33	’s	’s	X
ejpam-245	1	34	and	and	CCONJ
ejpam-245	1	35	the	the	DET
ejpam-245	1	36	lucas	lucas	NOUN
ejpam-245	1	37	-	-	PUNCT
ejpam-245	1	38	lehmer	lehmer	NOUN
ejpam-245	1	39	tests	test	NOUN
ejpam-245	1	40	john	john	PROPN
ejpam-245	1	41	h.	h.	PROPN
ejpam-245	1	42	jaroma	jaroma	PROPN
ejpam-245	1	43	department	department	PROPN
ejpam-245	1	44	of	of	ADP
ejpam-245	1	45	mathematics	mathematics	PROPN
ejpam-245	1	46	&	&	CCONJ
ejpam-245	1	47	physics	physics	PROPN
ejpam-245	1	48	,	,	PUNCT
ejpam-245	1	49	ave	ave	PROPN
ejpam-245	1	50	maria	maria	PROPN
ejpam-245	1	51	university	university	PROPN
ejpam-245	1	52	,	,	PUNCT
ejpam-245	1	53	ave	ave	PROPN
ejpam-245	1	54	maria	maria	PROPN
ejpam-245	1	55	,	,	PUNCT
ejpam-245	1	56	florida	florida	PROPN
ejpam-245	1	57	,	,	PUNCT
ejpam-245	1	58	34142	34142	NUM
ejpam-245	1	59	,	,	PUNCT
ejpam-245	1	60	united	united	PROPN
ejpam-245	1	61	states	states	PROPN
ejpam-245	1	62	abstract	abstract	PROPN
ejpam-245	1	63	.	.	PUNCT
ejpam-245	2	1	pepin	pepin	PROPN
ejpam-245	2	2	’s	’s	PART
ejpam-245	2	3	test	test	NOUN
ejpam-245	2	4	provides	provide	VERB
ejpam-245	2	5	a	a	DET
ejpam-245	2	6	necessary	necessary	ADJ
ejpam-245	2	7	and	and	CCONJ
ejpam-245	2	8	sufficient	sufficient	ADJ
ejpam-245	2	9	condition	condition	NOUN
ejpam-245	2	10	for	for	ADP
ejpam-245	2	11	a	a	DET
ejpam-245	2	12	fermat	fermat	ADJ
ejpam-245	2	13	number	number	NOUN
ejpam-245	2	14	to	to	PART
ejpam-245	2	15	be	be	AUX
ejpam-245	2	16	prime	prime	ADJ
ejpam-245	2	17	.	.	PUNCT
ejpam-245	3	1	the	the	DET
ejpam-245	3	2	lucas	lucas	PROPN
ejpam-245	3	3	-	-	PUNCT
ejpam-245	3	4	lehmer	lehmer	NOUN
ejpam-245	3	5	test	test	NOUN
ejpam-245	3	6	does	do	VERB
ejpam-245	3	7	similarly	similarly	ADV
ejpam-245	3	8	for	for	ADP
ejpam-245	3	9	a	a	DET
ejpam-245	3	10	mersenne	mersenne	NOUN
ejpam-245	3	11	number	number	NOUN
ejpam-245	3	12	.	.	PUNCT
ejpam-245	4	1	these	these	DET
ejpam-245	4	2	tests	test	NOUN
ejpam-245	4	3	share	share	VERB
ejpam-245	4	4	a	a	DET
ejpam-245	4	5	common	common	ADJ
ejpam-245	4	6	nature	nature	NOUN
ejpam-245	4	7	.	.	PUNCT
ejpam-245	5	1	however	however	ADV
ejpam-245	5	2	,	,	PUNCT
ejpam-245	5	3	this	this	PRON
ejpam-245	5	4	is	be	AUX
ejpam-245	5	5	evident	evident	ADJ
ejpam-245	5	6	neither	neither	CCONJ
ejpam-245	5	7	by	by	ADP
ejpam-245	5	8	their	their	PRON
ejpam-245	5	9	usual	usual	ADJ
ejpam-245	5	10	statements	statement	NOUN
ejpam-245	5	11	nor	nor	CCONJ
ejpam-245	5	12	their	their	PRON
ejpam-245	5	13	usual	usual	ADJ
ejpam-245	5	14	treatment	treatment	NOUN
ejpam-245	5	15	in	in	ADP
ejpam-245	5	16	the	the	DET
ejpam-245	5	17	literature	literature	NOUN
ejpam-245	5	18	.	.	PUNCT
ejpam-245	6	1	furthermore	furthermore	ADV
ejpam-245	6	2	,	,	PUNCT
ejpam-245	6	3	it	it	PRON
ejpam-245	6	4	is	be	AUX
ejpam-245	6	5	unusual	unusual	ADJ
ejpam-245	6	6	to	to	PART
ejpam-245	6	7	even	even	ADV
ejpam-245	6	8	find	find	VERB
ejpam-245	6	9	a	a	DET
ejpam-245	6	10	proof	proof	NOUN
ejpam-245	6	11	of	of	ADP
ejpam-245	6	12	the	the	DET
ejpam-245	6	13	latter	latter	ADJ
ejpam-245	6	14	result	result	NOUN
ejpam-245	6	15	in	in	ADP
ejpam-245	6	16	elementary	elementary	ADJ
ejpam-245	6	17	textbooks	textbook	NOUN
ejpam-245	6	18	.	.	PUNCT
ejpam-245	7	1	the	the	DET
ejpam-245	7	2	intent	intent	NOUN
ejpam-245	7	3	of	of	ADP
ejpam-245	7	4	this	this	DET
ejpam-245	7	5	paper	paper	NOUN
ejpam-245	7	6	is	be	AUX
ejpam-245	7	7	to	to	PART
ejpam-245	7	8	bring	bring	VERB
ejpam-245	7	9	to	to	PART
ejpam-245	7	10	light	light	VERB
ejpam-245	7	11	the	the	DET
ejpam-245	7	12	equivalent	equivalent	ADJ
ejpam-245	7	13	structure	structure	NOUN
ejpam-245	7	14	of	of	ADP
ejpam-245	7	15	these	these	DET
ejpam-245	7	16	two	two	NUM
ejpam-245	7	17	primality	primality	NOUN
ejpam-245	7	18	tests	test	NOUN
ejpam-245	7	19	.	.	PUNCT
ejpam-245	8	1	2000	2000	NUM
ejpam-245	8	2	mathematics	mathematic	NOUN
ejpam-245	8	3	subject	subject	NOUN
ejpam-245	8	4	classifications	classification	NOUN
ejpam-245	8	5	:	:	PUNCT
ejpam-245	8	6	11a41	11a41	NUM
ejpam-245	8	7	,	,	PUNCT
ejpam-245	8	8	11a51	11a51	NUM
ejpam-245	8	9	,	,	PUNCT
ejpam-245	8	10	11b39	11b39	NUM
ejpam-245	8	11	key	key	ADJ
ejpam-245	8	12	words	word	NOUN
ejpam-245	8	13	and	and	CCONJ
ejpam-245	8	14	phrases	phrase	NOUN
ejpam-245	8	15	:	:	PUNCT
ejpam-245	8	16	primes	prime	NOUN
ejpam-245	8	17	,	,	PUNCT
ejpam-245	8	18	primality	primality	NOUN
ejpam-245	8	19	test	test	NOUN
ejpam-245	8	20	,	,	PUNCT
ejpam-245	8	21	lehmer	lehmer	NOUN
ejpam-245	8	22	sequence	sequence	NOUN
ejpam-245	8	23	,	,	PUNCT
ejpam-245	8	24	pepin	pepin	PROPN
ejpam-245	8	25	’s	’s	PART
ejpam-245	8	26	test	test	NOUN
ejpam-245	8	27	,	,	PUNCT
ejpam-245	8	28	lucas	lucas	NOUN
ejpam-245	8	29	-	-	PUNCT
ejpam-245	8	30	lehmer	lehmer	NOUN
ejpam-245	8	31	test	test	NOUN
ejpam-245	8	32	.	.	PUNCT
ejpam-245	9	1	1	1	X
ejpam-245	9	2	.	.	X
ejpam-245	9	3	introduction	introduction	NOUN
ejpam-245	9	4	a	a	DET
ejpam-245	9	5	fermat	fermat	ADJ
ejpam-245	9	6	number	number	NOUN
ejpam-245	9	7	is	be	AUX
ejpam-245	9	8	any	any	DET
ejpam-245	9	9	integer	integer	NOUN
ejpam-245	9	10	of	of	ADP
ejpam-245	9	11	the	the	DET
ejpam-245	9	12	form	form	NOUN
ejpam-245	9	13	fn	fn	NOUN
ejpam-245	9	14	=	=	NOUN
ejpam-245	9	15	22n	22n	X
ejpam-245	9	16	+	+	SYM
ejpam-245	9	17	1	1	NUM
ejpam-245	9	18	,	,	PUNCT
ejpam-245	9	19	where	where	SCONJ
ejpam-245	9	20	n	n	PRON
ejpam-245	9	21	≥	≥	NOUN
ejpam-245	9	22	0	0	NUM
ejpam-245	9	23	.	.	PUNCT
ejpam-245	10	1	they	they	PRON
ejpam-245	10	2	are	be	AUX
ejpam-245	10	3	named	name	VERB
ejpam-245	10	4	in	in	ADP
ejpam-245	10	5	honor	honor	NOUN
ejpam-245	10	6	of	of	ADP
ejpam-245	10	7	pierre	pierre	PROPN
ejpam-245	10	8	de	de	X
ejpam-245	10	9	fermat	fermat	PROPN
ejpam-245	10	10	(	(	PUNCT
ejpam-245	10	11	1601–1665	1601–1665	NUM
ejpam-245	10	12	)	)	PUNCT
ejpam-245	10	13	who	who	PRON
ejpam-245	10	14	had	have	AUX
ejpam-245	10	15	expressed	express	VERB
ejpam-245	10	16	a	a	DET
ejpam-245	10	17	belief	belief	NOUN
ejpam-245	10	18	that	that	SCONJ
ejpam-245	10	19	such	such	ADJ
ejpam-245	10	20	numbers	number	NOUN
ejpam-245	10	21	are	be	AUX
ejpam-245	10	22	always	always	ADV
ejpam-245	10	23	prime	prime	ADJ
ejpam-245	10	24	.	.	PUNCT
ejpam-245	11	1	in	in	ADP
ejpam-245	11	2	1732	1732	NUM
ejpam-245	11	3	,	,	PUNCT
ejpam-245	11	4	leonhard	leonhard	PROPN
ejpam-245	11	5	euler	euler	PROPN
ejpam-245	11	6	negatively	negatively	ADV
ejpam-245	11	7	resolved	resolve	VERB
ejpam-245	11	8	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-245	12	1	352	352	NUM
ejpam-245	12	2	c	c	AUX
ejpam-245	12	3	©	©	PROPN
ejpam-245	12	4	2009	2009	NUM
ejpam-245	12	5	ejpam	ejpam	NOUN
ejpam-245	12	6	all	all	DET
ejpam-245	12	7	rights	right	NOUN
ejpam-245	12	8	reserved	reserve	VERB
ejpam-245	12	9	.	.	PUNCT
ejpam-245	13	1	john	john	PROPN
ejpam-245	13	2	h.	h.	PROPN
ejpam-245	13	3	jaroma	jaroma	PROPN
ejpam-245	13	4	/	/	SYM
ejpam-245	13	5	eur	eur	PROPN
ejpam-245	13	6	.	.	PUNCT
ejpam-245	14	1	j.	j.	PROPN
ejpam-245	14	2	pure	pure	PROPN
ejpam-245	14	3	appl	appl	PROPN
ejpam-245	14	4	.	.	PROPN
ejpam-245	14	5	math	math	PROPN
ejpam-245	14	6	,	,	PUNCT
ejpam-245	14	7	2	2	NUM
ejpam-245	14	8	(	(	PUNCT
ejpam-245	14	9	2009	2009	NUM
ejpam-245	14	10	)	)	PUNCT
ejpam-245	14	11	,	,	PUNCT
ejpam-245	14	12	(	(	PUNCT
ejpam-245	14	13	352	352	NUM
ejpam-245	14	14	-	-	SYM
ejpam-245	14	15	360	360	NUM
ejpam-245	14	16	)	)	PUNCT
ejpam-245	14	17	353	353	NUM
ejpam-245	14	18	fermat	fermat	PROPN
ejpam-245	14	19	’s	’s	PART
ejpam-245	14	20	assertion	assertion	NOUN
ejpam-245	14	21	by	by	ADP
ejpam-245	14	22	factoring	factor	VERB
ejpam-245	14	23	f5	f5	NOUN
ejpam-245	14	24	.	.	PUNCT
ejpam-245	15	1	today	today	NOUN
ejpam-245	15	2	,	,	PUNCT
ejpam-245	15	3	the	the	DET
ejpam-245	15	4	prevailing	prevail	VERB
ejpam-245	15	5	conjecture	conjecture	NOUN
ejpam-245	15	6	appears	appear	VERB
ejpam-245	15	7	to	to	PART
ejpam-245	15	8	be	be	AUX
ejpam-245	15	9	that	that	SCONJ
ejpam-245	15	10	no	no	DET
ejpam-245	15	11	fermat	fermat	PROPN
ejpam-245	15	12	primes	prime	VERB
ejpam-245	15	13	beyond	beyond	ADP
ejpam-245	15	14	n	n	NOUN
ejpam-245	15	15	=	=	SYM
ejpam-245	15	16	4	4	NUM
ejpam-245	15	17	exist	exist	VERB
ejpam-245	15	18	.	.	PUNCT
ejpam-245	16	1	a	a	DET
ejpam-245	16	2	necessary	necessary	ADJ
ejpam-245	16	3	and	and	CCONJ
ejpam-245	16	4	sufficient	sufficient	ADJ
ejpam-245	16	5	condition	condition	NOUN
ejpam-245	16	6	for	for	ADP
ejpam-245	16	7	the	the	DET
ejpam-245	16	8	primality	primality	NOUN
ejpam-245	16	9	of	of	ADP
ejpam-245	16	10	a	a	DET
ejpam-245	16	11	fermat	fermat	ADJ
ejpam-245	16	12	number	number	NOUN
ejpam-245	16	13	is	be	AUX
ejpam-245	16	14	provided	provide	VERB
ejpam-245	16	15	by	by	ADP
ejpam-245	16	16	pepin	pepin	PROPN
ejpam-245	16	17	’s	’s	PART
ejpam-245	16	18	test	test	NOUN
ejpam-245	16	19	.	.	PUNCT
ejpam-245	17	1	it	it	PRON
ejpam-245	17	2	is	be	AUX
ejpam-245	17	3	named	name	VERB
ejpam-245	17	4	after	after	ADP
ejpam-245	17	5	fr	fr	PROPN
ejpam-245	17	6	.	.	PUNCT
ejpam-245	18	1	théophile	théophile	PROPN
ejpam-245	18	2	pepin	pepin	PROPN
ejpam-245	18	3	(	(	PUNCT
ejpam-245	18	4	1826–1904	1826–1904	NUM
ejpam-245	18	5	)	)	PUNCT
ejpam-245	18	6	and	and	CCONJ
ejpam-245	18	7	is	be	AUX
ejpam-245	18	8	found	find	VERB
ejpam-245	18	9	in	in	ADP
ejpam-245	18	10	textbooks	textbook	NOUN
ejpam-245	18	11	often	often	ADV
ejpam-245	18	12	stated	state	VERB
ejpam-245	18	13	along	along	ADP
ejpam-245	18	14	the	the	DET
ejpam-245	18	15	lines	line	NOUN
ejpam-245	18	16	of	of	ADP
ejpam-245	18	17	fn	fn	NOUN
ejpam-245	18	18	is	be	AUX
ejpam-245	18	19	prime	prime	ADJ
ejpam-245	18	20	if	if	SCONJ
ejpam-245	18	21	and	and	CCONJ
ejpam-245	18	22	only	only	ADV
ejpam-245	18	23	if	if	SCONJ
ejpam-245	18	24	3	3	NUM
ejpam-245	18	25	fn−1	fn−1	ADJ
ejpam-245	18	26	2	2	NUM
ejpam-245	18	27	≡	≡	PROPN
ejpam-245	18	28	−1	−1	NOUN
ejpam-245	18	29	(	(	PUNCT
ejpam-245	18	30	mod	mod	PROPN
ejpam-245	18	31	fn	fn	NOUN
ejpam-245	18	32	)	)	PUNCT
ejpam-245	19	1	[	[	X
ejpam-245	19	2	1	1	NUM
ejpam-245	19	3	]	]	PUNCT
ejpam-245	19	4	,	,	PUNCT
ejpam-245	19	5	[	[	X
ejpam-245	19	6	11	11	NUM
ejpam-245	19	7	]	]	PUNCT
ejpam-245	19	8	,	,	PUNCT
ejpam-245	19	9	or	or	CCONJ
ejpam-245	19	10	[	[	X
ejpam-245	19	11	15	15	NUM
ejpam-245	19	12	]	]	PUNCT
ejpam-245	19	13	.	.	PUNCT
ejpam-245	20	1	a	a	DET
ejpam-245	20	2	mersenne	mersenne	NOUN
ejpam-245	20	3	number	number	NOUN
ejpam-245	20	4	is	be	AUX
ejpam-245	20	5	any	any	DET
ejpam-245	20	6	integer	integer	NOUN
ejpam-245	20	7	given	give	VERB
ejpam-245	20	8	by	by	ADP
ejpam-245	20	9	mn	mn	PROPN
ejpam-245	20	10	=	=	SYM
ejpam-245	20	11	2n−1	2n−1	PROPN
ejpam-245	20	12	,	,	PUNCT
ejpam-245	20	13	where	where	SCONJ
ejpam-245	20	14	n	n	PRON
ejpam-245	20	15	≥	≥	NOUN
ejpam-245	20	16	1	1	NUM
ejpam-245	20	17	,	,	PUNCT
ejpam-245	20	18	and	and	CCONJ
ejpam-245	20	19	so	so	ADV
ejpam-245	20	20	called	call	VERB
ejpam-245	20	21	because	because	SCONJ
ejpam-245	20	22	of	of	ADP
ejpam-245	20	23	a	a	DET
ejpam-245	20	24	rather	rather	ADV
ejpam-245	20	25	accurate	accurate	ADJ
ejpam-245	20	26	conjecture	conjecture	NOUN
ejpam-245	20	27	made	make	VERB
ejpam-245	20	28	by	by	ADP
ejpam-245	20	29	fr	fr	PROPN
ejpam-245	20	30	.	.	PUNCT
ejpam-245	21	1	marin	marin	PROPN
ejpam-245	21	2	mersenne	mersenne	PROPN
ejpam-245	21	3	(	(	PUNCT
ejpam-245	21	4	1588–1648	1588–1648	NUM
ejpam-245	21	5	)	)	PUNCT
ejpam-245	21	6	who	who	PRON
ejpam-245	21	7	asserted	assert	VERB
ejpam-245	21	8	that	that	SCONJ
ejpam-245	21	9	such	such	ADJ
ejpam-245	21	10	numbers	number	NOUN
ejpam-245	21	11	are	be	AUX
ejpam-245	21	12	prime	prime	ADJ
ejpam-245	21	13	for	for	ADP
ejpam-245	21	14	n	n	PRON
ejpam-245	21	15	∈	∈	NOUN
ejpam-245	21	16	{	{	PUNCT
ejpam-245	21	17	2	2	NUM
ejpam-245	21	18	,	,	PUNCT
ejpam-245	21	19	3	3	NUM
ejpam-245	21	20	,	,	PUNCT
ejpam-245	21	21	5	5	NUM
ejpam-245	21	22	,	,	PUNCT
ejpam-245	21	23	7	7	NUM
ejpam-245	21	24	,	,	PUNCT
ejpam-245	21	25	13	13	NUM
ejpam-245	21	26	,	,	PUNCT
ejpam-245	21	27	17	17	NUM
ejpam-245	21	28	,	,	PUNCT
ejpam-245	21	29	19	19	NUM
ejpam-245	21	30	,	,	PUNCT
ejpam-245	21	31	31	31	NUM
ejpam-245	21	32	,	,	PUNCT
ejpam-245	21	33	67	67	NUM
ejpam-245	21	34	,	,	PUNCT
ejpam-245	21	35	127	127	NUM
ejpam-245	21	36	,	,	PUNCT
ejpam-245	21	37	257	257	NUM
ejpam-245	21	38	}	}	PUNCT
ejpam-245	21	39	and	and	CCONJ
ejpam-245	21	40	composite	composite	VERB
ejpam-245	21	41	for	for	ADP
ejpam-245	21	42	all	all	DET
ejpam-245	21	43	other	other	ADJ
ejpam-245	21	44	values	value	NOUN
ejpam-245	21	45	of	of	ADP
ejpam-245	21	46	n	n	PRON
ejpam-245	21	47	≤	≤	NOUN
ejpam-245	21	48	257	257	NUM
ejpam-245	21	49	.	.	PUNCT
ejpam-245	22	1	it	it	PRON
ejpam-245	22	2	took	take	VERB
ejpam-245	22	3	mathematicians	mathematician	NOUN
ejpam-245	22	4	more	more	ADJ
ejpam-245	22	5	than	than	ADP
ejpam-245	22	6	300	300	NUM
ejpam-245	22	7	years	year	NOUN
ejpam-245	22	8	to	to	PART
ejpam-245	22	9	completely	completely	ADV
ejpam-245	22	10	resolve	resolve	VERB
ejpam-245	22	11	the	the	DET
ejpam-245	22	12	conjecture	conjecture	NOUN
ejpam-245	22	13	.	.	PUNCT
ejpam-245	23	1	upon	upon	SCONJ
ejpam-245	23	2	having	having	AUX
ejpam-245	23	3	done	do	VERB
ejpam-245	23	4	so	so	ADV
ejpam-245	23	5	,	,	PUNCT
ejpam-245	23	6	we	we	PRON
ejpam-245	23	7	learned	learn	VERB
ejpam-245	23	8	that	that	SCONJ
ejpam-245	23	9	mersenne	mersenne	PROPN
ejpam-245	23	10	had	have	AUX
ejpam-245	23	11	made	make	VERB
ejpam-245	23	12	only	only	ADV
ejpam-245	23	13	five	five	NUM
ejpam-245	23	14	mistakes	mistake	NOUN
ejpam-245	23	15	.	.	PUNCT
ejpam-245	24	1	the	the	DET
ejpam-245	24	2	lucas	lucas	PROPN
ejpam-245	24	3	-	-	PUNCT
ejpam-245	24	4	lehmer	lehmer	NOUN
ejpam-245	24	5	test	test	NOUN
ejpam-245	24	6	provides	provide	VERB
ejpam-245	24	7	a	a	DET
ejpam-245	24	8	necessary	necessary	ADJ
ejpam-245	24	9	and	and	CCONJ
ejpam-245	24	10	sufficient	sufficient	ADJ
ejpam-245	24	11	condition	condition	NOUN
ejpam-245	24	12	for	for	ADP
ejpam-245	24	13	a	a	DET
ejpam-245	24	14	mersenne	mersenne	NOUN
ejpam-245	24	15	number	number	NOUN
ejpam-245	24	16	to	to	PART
ejpam-245	24	17	be	be	AUX
ejpam-245	24	18	prime	prime	ADJ
ejpam-245	24	19	.	.	PUNCT
ejpam-245	25	1	letting	let	VERB
ejpam-245	25	2	p	p	PRON
ejpam-245	25	3	denote	denote	VERB
ejpam-245	25	4	a	a	DET
ejpam-245	25	5	prime	prime	NOUN
ejpam-245	25	6	,	,	PUNCT
ejpam-245	25	7	the	the	DET
ejpam-245	25	8	test	test	NOUN
ejpam-245	25	9	is	be	AUX
ejpam-245	25	10	often	often	ADV
ejpam-245	25	11	described	describe	VERB
ejpam-245	25	12	as	as	SCONJ
ejpam-245	25	13	mp	mp	PROPN
ejpam-245	25	14	is	be	AUX
ejpam-245	25	15	prime	prime	ADJ
ejpam-245	25	16	if	if	SCONJ
ejpam-245	25	17	and	and	CCONJ
ejpam-245	25	18	only	only	ADV
ejpam-245	26	1	if	if	SCONJ
ejpam-245	26	2	rp−1	rp−1	PROPN
ejpam-245	26	3	≡	≡	PROPN
ejpam-245	26	4	0	0	PUNCT
ejpam-245	26	5	(	(	PUNCT
ejpam-245	26	6	mod	mod	PROPN
ejpam-245	26	7	mp	mp	PROPN
ejpam-245	26	8	)	)	PUNCT
ejpam-245	26	9	,	,	PUNCT
ejpam-245	26	10	where	where	SCONJ
ejpam-245	26	11	r1	r1	NOUN
ejpam-245	26	12	=	=	SYM
ejpam-245	26	13	4	4	NUM
ejpam-245	26	14	,	,	PUNCT
ejpam-245	26	15	and	and	CCONJ
ejpam-245	26	16	for	for	ADP
ejpam-245	26	17	k	k	PROPN
ejpam-245	26	18	≥	≥	NUM
ejpam-245	26	19	2	2	NUM
ejpam-245	26	20	,	,	PUNCT
ejpam-245	26	21	rk	rk	NOUN
ejpam-245	26	22	=	=	PUNCT
ejpam-245	26	23	r2	r2	PROPN
ejpam-245	26	24	k−1	k−1	PROPN
ejpam-245	26	25	−	−	PROPN
ejpam-245	26	26	2	2	NUM
ejpam-245	26	27	(	(	PUNCT
ejpam-245	26	28	mod	mod	PROPN
ejpam-245	26	29	mp	mp	PROPN
ejpam-245	26	30	)	)	PUNCT
ejpam-245	26	31	,	,	PUNCT
ejpam-245	26	32	0	0	NUM
ejpam-245	26	33	≤	≤	NUM
ejpam-245	26	34	rk	rk	NOUN
ejpam-245	26	35	<	<	X
ejpam-245	26	36	mp	mp	PROPN
ejpam-245	26	37	.	.	PROPN
ejpam-245	27	1	for	for	ADP
ejpam-245	27	2	instance	instance	NOUN
ejpam-245	27	3	[	[	X
ejpam-245	27	4	1	1	NUM
ejpam-245	27	5	]	]	PUNCT
ejpam-245	27	6	,	,	PUNCT
ejpam-245	27	7	[	[	X
ejpam-245	27	8	13	13	NUM
ejpam-245	27	9	]	]	PUNCT
ejpam-245	27	10	or	or	CCONJ
ejpam-245	27	11	[	[	X
ejpam-245	27	12	14	14	NUM
ejpam-245	27	13	]	]	PUNCT
ejpam-245	27	14	.	.	PUNCT
ejpam-245	28	1	although	although	SCONJ
ejpam-245	28	2	not	not	PART
ejpam-245	28	3	evident	evident	ADJ
ejpam-245	28	4	by	by	ADP
ejpam-245	28	5	their	their	PRON
ejpam-245	28	6	usual	usual	ADJ
ejpam-245	28	7	statements	statement	NOUN
ejpam-245	28	8	alone	alone	ADV
ejpam-245	28	9	,	,	PUNCT
ejpam-245	28	10	both	both	DET
ejpam-245	28	11	pepin	pepin	PROPN
ejpam-245	28	12	’s	’s	PART
ejpam-245	28	13	test	test	NOUN
ejpam-245	28	14	and	and	CCONJ
ejpam-245	28	15	the	the	DET
ejpam-245	28	16	lucas	lucas	NOUN
ejpam-245	28	17	-	-	PUNCT
ejpam-245	28	18	lehmer	lehmer	NOUN
ejpam-245	28	19	test	test	NOUN
ejpam-245	28	20	are	be	AUX
ejpam-245	28	21	inherently	inherently	ADV
ejpam-245	28	22	derived	derive	VERB
ejpam-245	28	23	from	from	ADP
ejpam-245	28	24	the	the	DET
ejpam-245	28	25	properties	property	NOUN
ejpam-245	28	26	of	of	ADP
ejpam-245	28	27	the	the	DET
ejpam-245	28	28	lehmer	lehmer	NOUN
ejpam-245	28	29	sequences	sequence	NOUN
ejpam-245	28	30	with	with	ADP
ejpam-245	28	31	both	both	DET
ejpam-245	28	32	tests	test	NOUN
ejpam-245	28	33	being	be	AUX
ejpam-245	28	34	demonstrable	demonstrable	ADJ
ejpam-245	28	35	by	by	ADP
ejpam-245	28	36	similar	similar	ADJ
ejpam-245	28	37	arguments	argument	NOUN
ejpam-245	28	38	.	.	PUNCT
ejpam-245	29	1	the	the	DET
ejpam-245	29	2	intent	intent	NOUN
ejpam-245	29	3	of	of	ADP
ejpam-245	29	4	this	this	DET
ejpam-245	29	5	note	note	NOUN
ejpam-245	29	6	to	to	PART
ejpam-245	29	7	make	make	VERB
ejpam-245	29	8	the	the	DET
ejpam-245	29	9	common	common	ADJ
ejpam-245	29	10	structure	structure	NOUN
ejpam-245	29	11	of	of	ADP
ejpam-245	29	12	these	these	DET
ejpam-245	29	13	two	two	NUM
ejpam-245	29	14	tests	test	NOUN
ejpam-245	29	15	more	more	ADV
ejpam-245	29	16	widely	widely	ADV
ejpam-245	29	17	known	know	VERB
ejpam-245	29	18	.	.	PUNCT
ejpam-245	30	1	the	the	DET
ejpam-245	30	2	similarity	similarity	NOUN
ejpam-245	30	3	between	between	ADP
ejpam-245	30	4	the	the	DET
ejpam-245	30	5	two	two	NUM
ejpam-245	30	6	primality	primality	NOUN
ejpam-245	30	7	tests	test	NOUN
ejpam-245	30	8	appears	appear	VERB
ejpam-245	30	9	to	to	PART
ejpam-245	30	10	have	have	AUX
ejpam-245	30	11	been	be	AUX
ejpam-245	30	12	overlooked	overlook	VERB
ejpam-245	30	13	.	.	PUNCT
ejpam-245	31	1	for	for	ADP
ejpam-245	31	2	example	example	NOUN
ejpam-245	31	3	,	,	PUNCT
ejpam-245	31	4	in	in	ADP
ejpam-245	31	5	[	[	PUNCT
ejpam-245	31	6	10	10	NUM
ejpam-245	31	7	]	]	PUNCT
ejpam-245	31	8	,	,	PUNCT
ejpam-245	31	9	pepin	pepin	PROPN
ejpam-245	31	10	’s	’s	PART
ejpam-245	31	11	test	test	NOUN
ejpam-245	31	12	is	be	AUX
ejpam-245	31	13	discussed	discuss	VERB
ejpam-245	31	14	after	after	ADP
ejpam-245	31	15	the	the	DET
ejpam-245	31	16	section	section	NOUN
ejpam-245	31	17	primality	primality	NOUN
ejpam-245	31	18	tests	test	NOUN
ejpam-245	31	19	based	base	VERB
ejpam-245	31	20	on	on	ADP
ejpam-245	31	21	the	the	DET
ejpam-245	31	22	lucas	lucas	PROPN
ejpam-245	31	23	sequences.∗	sequences.∗	PROPN
ejpam-245	31	24	in	in	ADP
ejpam-245	31	25	addition	addition	NOUN
ejpam-245	31	26	,	,	PUNCT
ejpam-245	31	27	in	in	ADP
ejpam-245	31	28	[	[	PUNCT
ejpam-245	31	29	3	3	NUM
ejpam-245	31	30	]	]	PUNCT
ejpam-245	31	31	,	,	PUNCT
ejpam-245	31	32	derrick	derrick	PROPN
ejpam-245	31	33	lehmer	lehmer	PROPN
ejpam-245	31	34	opts	opt	VERB
ejpam-245	31	35	not	not	PART
ejpam-245	31	36	to	to	PART
ejpam-245	31	37	illustrate	illustrate	VERB
ejpam-245	31	38	a	a	DET
ejpam-245	31	39	test	test	NOUN
ejpam-245	31	40	for	for	ADP
ejpam-245	31	41	the	the	DET
ejpam-245	31	42	primality	primality	NOUN
ejpam-245	31	43	of	of	ADP
ejpam-245	31	44	the	the	DET
ejpam-245	31	45	fermat	fermat	PROPN
ejpam-245	31	46	numbers	number	NOUN
ejpam-245	31	47	but	but	CCONJ
ejpam-245	31	48	instead	instead	ADV
ejpam-245	31	49	remarks	remark	NOUN
ejpam-245	31	50	in	in	ADP
ejpam-245	31	51	particular	particular	ADJ
ejpam-245	31	52	we	we	PRON
ejpam-245	31	53	could	could	AUX
ejpam-245	31	54	give	give	VERB
ejpam-245	31	55	new	new	ADJ
ejpam-245	31	56	tests	test	NOUN
ejpam-245	31	57	for	for	ADP
ejpam-245	31	58	the	the	DET
ejpam-245	31	59	primality	primality	NOUN
ejpam-245	31	60	of	of	ADP
ejpam-245	31	61	22n	22n	NOUN
ejpam-245	31	62	+	+	CCONJ
ejpam-245	31	63	1	1	NUM
ejpam-245	31	64	,	,	PUNCT
ejpam-245	31	65	but	but	CCONJ
ejpam-245	31	66	those	those	DET
ejpam-245	31	67	fermat	fermat	PROPN
ejpam-245	31	68	numbers	number	NOUN
ejpam-245	31	69	which	which	PRON
ejpam-245	31	70	have	have	AUX
ejpam-245	31	71	not	not	PART
ejpam-245	31	72	already	already	ADV
ejpam-245	31	73	been	be	AUX
ejpam-245	31	74	tested	test	VERB
ejpam-245	31	75	are	be	AUX
ejpam-245	31	76	too	too	ADV
ejpam-245	31	77	large	large	ADJ
ejpam-245	31	78	for	for	ADP
ejpam-245	31	79	the	the	DET
ejpam-245	31	80	application	application	NOUN
ejpam-245	31	81	of	of	ADP
ejpam-245	31	82	any	any	DET
ejpam-245	31	83	known	know	VERB
ejpam-245	31	84	test	test	NOUN
ejpam-245	31	85	.	.	PUNCT
ejpam-245	32	1	having	having	AUX
ejpam-245	32	2	said	say	VERB
ejpam-245	32	3	this	this	PRON
ejpam-245	32	4	,	,	PUNCT
ejpam-245	32	5	pepin	pepin	PROPN
ejpam-245	32	6	’s	’s	PART
ejpam-245	32	7	test	test	NOUN
ejpam-245	32	8	is	be	AUX
ejpam-245	32	9	never	never	ADV
ejpam-245	32	10	explicitly	explicitly	ADV
ejpam-245	32	11	mentioned	mention	VERB
ejpam-245	32	12	in	in	ADP
ejpam-245	32	13	lehmer	lehmer	NOUN
ejpam-245	32	14	’s	’s	PART
ejpam-245	32	15	paper	paper	NOUN
ejpam-245	32	16	.	.	PUNCT
ejpam-245	33	1	lastly	lastly	ADV
ejpam-245	33	2	,	,	PUNCT
ejpam-245	33	3	in	in	ADP
ejpam-245	33	4	[	[	X
ejpam-245	33	5	16	16	NUM
ejpam-245	33	6	]	]	PUNCT
ejpam-245	33	7	,	,	PUNCT
ejpam-245	33	8	williams	williams	PROPN
ejpam-245	33	9	∗the	∗the	DET
ejpam-245	33	10	lucas	lucas	PROPN
ejpam-245	33	11	sequences	sequence	NOUN
ejpam-245	33	12	are	be	AUX
ejpam-245	33	13	special	special	ADJ
ejpam-245	33	14	cases	case	NOUN
ejpam-245	33	15	of	of	ADP
ejpam-245	33	16	the	the	DET
ejpam-245	33	17	lehmer	lehmer	NOUN
ejpam-245	33	18	sequences	sequence	NOUN
ejpam-245	33	19	,	,	PUNCT
ejpam-245	33	20	where	where	SCONJ
ejpam-245	33	21	r	r	NOUN
ejpam-245	33	22	is	be	AUX
ejpam-245	33	23	a	a	DET
ejpam-245	33	24	perfect	perfect	ADJ
ejpam-245	33	25	square	square	NOUN
ejpam-245	33	26	.	.	PUNCT
ejpam-245	34	1	john	john	PROPN
ejpam-245	34	2	h.	h.	PROPN
ejpam-245	34	3	jaroma	jaroma	PROPN
ejpam-245	34	4	/	/	SYM
ejpam-245	34	5	eur	eur	PROPN
ejpam-245	34	6	.	.	PUNCT
ejpam-245	35	1	j.	j.	PROPN
ejpam-245	35	2	pure	pure	PROPN
ejpam-245	35	3	appl	appl	PROPN
ejpam-245	35	4	.	.	PROPN
ejpam-245	35	5	math	math	PROPN
ejpam-245	35	6	,	,	PUNCT
ejpam-245	35	7	2	2	NUM
ejpam-245	35	8	(	(	PUNCT
ejpam-245	35	9	2009	2009	NUM
ejpam-245	35	10	)	)	PUNCT
ejpam-245	35	11	,	,	PUNCT
ejpam-245	35	12	(	(	PUNCT
ejpam-245	35	13	352	352	NUM
ejpam-245	35	14	-	-	SYM
ejpam-245	35	15	360	360	NUM
ejpam-245	35	16	)	)	PUNCT
ejpam-245	35	17	354	354	NUM
ejpam-245	35	18	cites	cite	VERB
ejpam-245	35	19	that	that	SCONJ
ejpam-245	35	20	pepin	pepin	PROPN
ejpam-245	35	21	had	have	AUX
ejpam-245	35	22	been	be	AUX
ejpam-245	35	23	aware	aware	ADJ
ejpam-245	35	24	that	that	SCONJ
ejpam-245	35	25	the	the	DET
ejpam-245	35	26	earlier	early	ADJ
ejpam-245	35	27	version	version	NOUN
ejpam-245	35	28	of	of	ADP
ejpam-245	35	29	his	his	PRON
ejpam-245	35	30	test	test	NOUN
ejpam-245	35	31	,	,	PUNCT
ejpam-245	35	32	n	n	PROPN
ejpam-245	35	33	=	=	SYM
ejpam-245	35	34	2r	2r	NUM
ejpam-245	35	35	+	+	CCONJ
ejpam-245	35	36	1	1	NUM
ejpam-245	35	37	is	be	AUX
ejpam-245	35	38	prime	prime	ADJ
ejpam-245	35	39	if	if	SCONJ
ejpam-245	35	40	and	and	CCONJ
ejpam-245	35	41	only	only	ADV
ejpam-245	35	42	if	if	SCONJ
ejpam-245	35	43	5	5	NUM
ejpam-245	35	44	n−1	n−1	PROPN
ejpam-245	35	45	2	2	NUM
ejpam-245	35	46	≡	≡	PROPN
ejpam-245	35	47	−1	−1	NOUN
ejpam-245	35	48	(	(	PUNCT
ejpam-245	35	49	mod	mod	NOUN
ejpam-245	35	50	n	n	CCONJ
ejpam-245	35	51	)	)	PUNCT
ejpam-245	35	52	,	,	PUNCT
ejpam-245	35	53	could	could	AUX
ejpam-245	35	54	be	be	AUX
ejpam-245	35	55	made	make	VERB
ejpam-245	35	56	into	into	ADP
ejpam-245	35	57	a	a	DET
ejpam-245	35	58	simple	simple	ADJ
ejpam-245	35	59	lucas	lucas	NOUN
ejpam-245	35	60	-	-	PUNCT
ejpam-245	35	61	like	like	ADJ
ejpam-245	35	62	test	test	NOUN
ejpam-245	35	63	by	by	ADP
ejpam-245	35	64	defining	define	VERB
ejpam-245	35	65	t1	t1	NOUN
ejpam-245	35	66	=	=	PUNCT
ejpam-245	35	67	52	52	NUM
ejpam-245	35	68	and	and	CCONJ
ejpam-245	35	69	ti+1	ti+1	NOUN
ejpam-245	36	1	=	=	SYM
ejpam-245	36	2	t	t	PROPN
ejpam-245	36	3	2	2	NUM
ejpam-245	37	1	i	i	NOUN
ejpam-245	37	2	,	,	PUNCT
ejpam-245	37	3	where	where	SCONJ
ejpam-245	37	4	i	i	PRON
ejpam-245	37	5	is	be	AUX
ejpam-245	37	6	a	a	DET
ejpam-245	37	7	positive	positive	ADJ
ejpam-245	37	8	integer	integer	NOUN
ejpam-245	37	9	.	.	PUNCT
ejpam-245	38	1	this	this	PRON
ejpam-245	38	2	leads	lead	VERB
ejpam-245	38	3	to	to	ADP
ejpam-245	38	4	the	the	DET
ejpam-245	38	5	result	result	NOUN
ejpam-245	38	6	that	that	SCONJ
ejpam-245	38	7	fn	fn	NOUN
ejpam-245	38	8	is	be	AUX
ejpam-245	38	9	prime	prime	ADJ
ejpam-245	38	10	if	if	SCONJ
ejpam-245	38	11	and	and	CCONJ
ejpam-245	38	12	only	only	ADV
ejpam-245	38	13	if	if	SCONJ
ejpam-245	38	14	fn	fn	PROPN
ejpam-245	38	15	|	|	ADV
ejpam-245	38	16	tr−1	tr−1	PROPN
ejpam-245	38	17	+	+	NOUN
ejpam-245	38	18	1	1	X
ejpam-245	38	19	.	.	PUNCT
ejpam-245	39	1	however	however	ADV
ejpam-245	39	2	,	,	PUNCT
ejpam-245	39	3	a	a	DET
ejpam-245	39	4	direct	direct	ADJ
ejpam-245	39	5	correlation	correlation	NOUN
ejpam-245	39	6	to	to	ADP
ejpam-245	39	7	the	the	DET
ejpam-245	39	8	lucas	lucas	NOUN
ejpam-245	39	9	-	-	PUNCT
ejpam-245	39	10	lehmer	lehmer	NOUN
ejpam-245	39	11	result	result	NOUN
ejpam-245	39	12	does	do	AUX
ejpam-245	39	13	not	not	PART
ejpam-245	39	14	appear	appear	VERB
ejpam-245	39	15	to	to	PART
ejpam-245	39	16	be	be	AUX
ejpam-245	39	17	given	give	VERB
ejpam-245	39	18	in	in	ADP
ejpam-245	39	19	the	the	DET
ejpam-245	39	20	book	book	NOUN
ejpam-245	39	21	.	.	PUNCT
ejpam-245	40	1	2	2	X
ejpam-245	40	2	.	.	X
ejpam-245	40	3	the	the	DET
ejpam-245	40	4	lehmer	lehmer	NOUN
ejpam-245	40	5	sequences	sequence	NOUN
ejpam-245	40	6	let	let	VERB
ejpam-245	40	7	r	r	NOUN
ejpam-245	40	8	and	and	CCONJ
ejpam-245	40	9	q	q	AUX
ejpam-245	40	10	be	be	AUX
ejpam-245	40	11	relatively	relatively	ADV
ejpam-245	40	12	prime	prime	ADJ
ejpam-245	40	13	integers	integer	NOUN
ejpam-245	40	14	.	.	PUNCT
ejpam-245	41	1	the	the	DET
ejpam-245	41	2	lehmer	lehmer	NOUN
ejpam-245	41	3	sequences	sequence	NOUN
ejpam-245	41	4	{	{	PUNCT
ejpam-245	41	5	un	un	PROPN
ejpam-245	41	6	(	(	PUNCT
ejpam-245	41	7	p	p	NOUN
ejpam-245	41	8	r	r	NOUN
ejpam-245	41	9	,	,	PUNCT
ejpam-245	41	10	q	q	NOUN
ejpam-245	41	11	)	)	PUNCT
ejpam-245	41	12	}	}	PUNCT
ejpam-245	41	13	and	and	CCONJ
ejpam-245	41	14	the	the	DET
ejpam-245	41	15	companion	companion	NOUN
ejpam-245	41	16	lehmer	lehmer	NOUN
ejpam-245	41	17	sequences	sequence	NOUN
ejpam-245	41	18	{	{	PUNCT
ejpam-245	41	19	vn	vn	PROPN
ejpam-245	41	20	(	(	PUNCT
ejpam-245	41	21	p	p	NOUN
ejpam-245	41	22	r	r	NOUN
ejpam-245	41	23	,	,	PUNCT
ejpam-245	41	24	q	q	NOUN
ejpam-245	41	25	)	)	PUNCT
ejpam-245	41	26	}	}	PUNCT
ejpam-245	41	27	are	be	AUX
ejpam-245	41	28	defined	define	VERB
ejpam-245	41	29	respectively	respectively	ADV
ejpam-245	41	30	,	,	PUNCT
ejpam-245	41	31	by	by	ADP
ejpam-245	41	32	un+2	un+2	PROPN
ejpam-245	41	33	(	(	PUNCT
ejpam-245	41	34	p	p	NOUN
ejpam-245	41	35	r	r	NOUN
ejpam-245	41	36	,	,	PUNCT
ejpam-245	41	37	q	q	NOUN
ejpam-245	41	38	)	)	PUNCT
ejpam-245	41	39	=	=	SYM
ejpam-245	42	1	p	p	NOUN
ejpam-245	42	2	run+1	run+1	X
ejpam-245	42	3	−qun	−qun	NUM
ejpam-245	42	4	,	,	PUNCT
ejpam-245	42	5	u0	u0	ADJ
ejpam-245	42	6	=	=	SYM
ejpam-245	42	7	0	0	NUM
ejpam-245	42	8	,	,	PUNCT
ejpam-245	42	9	u1	u1	NOUN
ejpam-245	42	10	=	=	SYM
ejpam-245	42	11	1	1	NUM
ejpam-245	42	12	,	,	PUNCT
ejpam-245	42	13	n	n	PRON
ejpam-245	42	14	∈	∈	PROPN
ejpam-245	42	15	{	{	PUNCT
ejpam-245	42	16	0	0	NUM
ejpam-245	42	17	,	,	PUNCT
ejpam-245	42	18	1	1	NUM
ejpam-245	42	19	,	,	PUNCT
ejpam-245	42	20	.	.	PUNCT
ejpam-245	42	21	.	.	PUNCT
ejpam-245	42	22	.	.	PUNCT
ejpam-245	42	23	}	}	PUNCT
ejpam-245	43	1	(	(	PUNCT
ejpam-245	43	2	2.1	2.1	NUM
ejpam-245	43	3	)	)	PUNCT
ejpam-245	43	4	vn+2	vn+2	PROPN
ejpam-245	43	5	(	(	PUNCT
ejpam-245	43	6	p	p	NOUN
ejpam-245	43	7	r	r	NOUN
ejpam-245	43	8	,	,	PUNCT
ejpam-245	43	9	q	q	NOUN
ejpam-245	43	10	)	)	PUNCT
ejpam-245	43	11	=	=	SYM
ejpam-245	44	1	p	p	NOUN
ejpam-245	44	2	rvn+1	rvn+1	NOUN
ejpam-245	44	3	−qvn	−qvn	NOUN
ejpam-245	44	4	,	,	PUNCT
ejpam-245	44	5	v0	v0	NOUN
ejpam-245	44	6	=	=	SYM
ejpam-245	44	7	2	2	NUM
ejpam-245	44	8	,	,	PUNCT
ejpam-245	44	9	v1	v1	NOUN
ejpam-245	44	10	=	=	SYM
ejpam-245	44	11	p	p	NOUN
ejpam-245	44	12	r	r	NOUN
ejpam-245	44	13	,	,	PUNCT
ejpam-245	44	14	n	n	PRON
ejpam-245	44	15	∈	∈	NOUN
ejpam-245	44	16	{	{	PUNCT
ejpam-245	44	17	0	0	NUM
ejpam-245	44	18	,	,	PUNCT
ejpam-245	44	19	1	1	NUM
ejpam-245	44	20	,	,	PUNCT
ejpam-245	44	21	.	.	PUNCT
ejpam-245	44	22	.	.	PUNCT
ejpam-245	44	23	.	.	PUNCT
ejpam-245	44	24	}	}	PUNCT
ejpam-245	44	25	.	.	PUNCT
ejpam-245	45	1	(	(	PUNCT
ejpam-245	45	2	2.2	2.2	NUM
ejpam-245	45	3	)	)	PUNCT
ejpam-245	45	4	furthermore	furthermore	ADV
ejpam-245	45	5	,	,	PUNCT
ejpam-245	45	6	since	since	SCONJ
ejpam-245	45	7	(	(	PUNCT
ejpam-245	45	8	2.1	2.1	NUM
ejpam-245	45	9	)	)	PUNCT
ejpam-245	45	10	and	and	CCONJ
ejpam-245	45	11	(	(	PUNCT
ejpam-245	45	12	2.2	2.2	NUM
ejpam-245	45	13	)	)	PUNCT
ejpam-245	45	14	are	be	AUX
ejpam-245	45	15	linear	linear	ADJ
ejpam-245	45	16	,	,	PUNCT
ejpam-245	45	17	they	they	PRON
ejpam-245	45	18	are	be	AUX
ejpam-245	45	19	solvable	solvable	ADJ
ejpam-245	45	20	and	and	CCONJ
ejpam-245	45	21	given	give	VERB
ejpam-245	45	22	explicitly	explicitly	ADV
ejpam-245	45	23	by	by	ADP
ejpam-245	45	24	(	(	PUNCT
ejpam-245	45	25	2.3	2.3	NUM
ejpam-245	45	26	)	)	PUNCT
ejpam-245	45	27	and	and	CCONJ
ejpam-245	45	28	(	(	PUNCT
ejpam-245	45	29	2.4	2.4	NUM
ejpam-245	45	30	)	)	PUNCT
ejpam-245	45	31	,	,	PUNCT
ejpam-245	45	32	respectively	respectively	ADV
ejpam-245	45	33	.	.	PUNCT
ejpam-245	46	1	un	un	PROPN
ejpam-245	46	2	(	(	PUNCT
ejpam-245	46	3	p	p	NOUN
ejpam-245	46	4	r	r	NOUN
ejpam-245	46	5	,	,	PUNCT
ejpam-245	46	6	q	q	NOUN
ejpam-245	46	7	)	)	PUNCT
ejpam-245	46	8	=	=	SYM
ejpam-245	46	9	θ	θ	NOUN
ejpam-245	46	10	n−φn	n−φn	VERB
ejpam-245	46	11	θ	θ	PROPN
ejpam-245	46	12	−φ	−φ	NOUN
ejpam-245	46	13	,	,	PUNCT
ejpam-245	46	14	n	n	X
ejpam-245	46	15	∈	∈	PROPN
ejpam-245	46	16	{	{	PUNCT
ejpam-245	46	17	0	0	NUM
ejpam-245	46	18	,	,	PUNCT
ejpam-245	46	19	1	1	NUM
ejpam-245	46	20	,	,	PUNCT
ejpam-245	46	21	.	.	PUNCT
ejpam-245	46	22	.	.	PUNCT
ejpam-245	47	1	.	.	PUNCT
ejpam-245	47	2	}	}	PUNCT
ejpam-245	48	1	(	(	PUNCT
ejpam-245	48	2	2.3	2.3	NUM
ejpam-245	48	3	)	)	PUNCT
ejpam-245	48	4	vn	vn	NOUN
ejpam-245	48	5	(	(	PUNCT
ejpam-245	48	6	p	p	NOUN
ejpam-245	48	7	r	r	NOUN
ejpam-245	48	8	,	,	PUNCT
ejpam-245	48	9	q	q	NOUN
ejpam-245	48	10	)	)	PUNCT
ejpam-245	48	11	=	=	SYM
ejpam-245	48	12	θ	θ	NOUN
ejpam-245	48	13	n+φn	n+φn	NOUN
ejpam-245	48	14	,	,	PUNCT
ejpam-245	48	15	n	n	X
ejpam-245	48	16	∈	∈	PROPN
ejpam-245	48	17	{	{	PUNCT
ejpam-245	48	18	0	0	NUM
ejpam-245	48	19	,	,	PUNCT
ejpam-245	48	20	1	1	NUM
ejpam-245	48	21	,	,	PUNCT
ejpam-245	48	22	.	.	PUNCT
ejpam-245	48	23	.	.	PUNCT
ejpam-245	49	1	.	.	PUNCT
ejpam-245	49	2	}	}	PUNCT
ejpam-245	50	1	(	(	PUNCT
ejpam-245	50	2	2.4	2.4	NUM
ejpam-245	50	3	)	)	PUNCT
ejpam-245	50	4	where	where	SCONJ
ejpam-245	50	5	,	,	PUNCT
ejpam-245	50	6	θ	θ	PROPN
ejpam-245	50	7	=	=	PUNCT
ejpam-245	50	8	p	p	PROPN
ejpam-245	50	9	r+	r+	PUNCT
ejpam-245	50	10	p	p	PROPN
ejpam-245	50	11	r−4q	r−4q	ADJ
ejpam-245	50	12	2	2	NUM
ejpam-245	50	13	and	and	CCONJ
ejpam-245	50	14	φ	φ	NUM
ejpam-245	50	15	=	=	SYM
ejpam-245	50	16	p	p	PROPN
ejpam-245	50	17	r−	r−	PROPN
ejpam-245	50	18	p	p	NOUN
ejpam-245	50	19	r−4q	r−4q	VERB
ejpam-245	50	20	2	2	NUM
ejpam-245	50	21	.	.	PUNCT
ejpam-245	51	1	we	we	PRON
ejpam-245	51	2	say	say	VERB
ejpam-245	51	3	that	that	SCONJ
ejpam-245	51	4	the	the	DET
ejpam-245	51	5	rank	rank	NOUN
ejpam-245	51	6	of	of	ADP
ejpam-245	51	7	apparition	apparition	NOUN
ejpam-245	51	8	of	of	ADP
ejpam-245	51	9	a	a	DET
ejpam-245	51	10	number	number	NOUN
ejpam-245	51	11	n	n	NOUN
ejpam-245	51	12	in	in	ADP
ejpam-245	51	13	a	a	DET
ejpam-245	51	14	sequence	sequence	NOUN
ejpam-245	51	15	is	be	AUX
ejpam-245	51	16	the	the	DET
ejpam-245	51	17	index	index	NOUN
ejpam-245	51	18	of	of	ADP
ejpam-245	51	19	the	the	DET
ejpam-245	51	20	first	first	ADJ
ejpam-245	51	21	term	term	NOUN
ejpam-245	51	22	in	in	ADP
ejpam-245	51	23	that	that	DET
ejpam-245	51	24	sequence	sequence	NOUN
ejpam-245	51	25	that	that	PRON
ejpam-245	51	26	contains	contain	VERB
ejpam-245	51	27	n	n	PRON
ejpam-245	51	28	as	as	ADP
ejpam-245	51	29	a	a	DET
ejpam-245	51	30	divisor	divisor	NOUN
ejpam-245	51	31	.	.	PUNCT
ejpam-245	52	1	n	n	PRON
ejpam-245	52	2	is	be	AUX
ejpam-245	52	3	said	say	VERB
ejpam-245	52	4	to	to	PART
ejpam-245	52	5	have	have	VERB
ejpam-245	52	6	maximal	maximal	ADJ
ejpam-245	52	7	rank	rank	NOUN
ejpam-245	52	8	of	of	ADP
ejpam-245	52	9	apparition	apparition	NOUN
ejpam-245	52	10	provided	provide	VERB
ejpam-245	52	11	that	that	SCONJ
ejpam-245	52	12	its	its	PRON
ejpam-245	52	13	rank	rank	NOUN
ejpam-245	52	14	of	of	ADP
ejpam-245	52	15	apparition	apparition	NOUN
ejpam-245	52	16	is	be	AUX
ejpam-245	52	17	either	either	CCONJ
ejpam-245	52	18	n	n	PRON
ejpam-245	52	19	±	±	NUM
ejpam-245	52	20	1	1	NUM
ejpam-245	52	21	.	.	X
ejpam-245	53	1	3	3	X
ejpam-245	53	2	.	.	X
ejpam-245	53	3	properties	property	NOUN
ejpam-245	53	4	of	of	ADP
ejpam-245	53	5	the	the	DET
ejpam-245	53	6	lehmer	lehmer	NOUN
ejpam-245	53	7	sequences	sequence	NOUN
ejpam-245	53	8	let	let	VERB
ejpam-245	53	9	p	p	PRON
ejpam-245	53	10	denote	denote	VERB
ejpam-245	53	11	an	an	DET
ejpam-245	53	12	arbitrary	arbitrary	ADJ
ejpam-245	53	13	odd	odd	ADJ
ejpam-245	53	14	prime	prime	NOUN
ejpam-245	53	15	such	such	ADJ
ejpam-245	53	16	that	that	SCONJ
ejpam-245	53	17	p	p	PROPN
ejpam-245	53	18	∤	∤	PROPN
ejpam-245	53	19	rq	rq	VERB
ejpam-245	53	20	.	.	PUNCT
ejpam-245	54	1	the	the	DET
ejpam-245	54	2	following	follow	VERB
ejpam-245	54	3	propositions	proposition	NOUN
ejpam-245	54	4	are	be	AUX
ejpam-245	54	5	divisibility	divisibility	NOUN
ejpam-245	54	6	properties	property	NOUN
ejpam-245	54	7	associated	associate	VERB
ejpam-245	54	8	with	with	ADP
ejpam-245	54	9	the	the	DET
ejpam-245	54	10	lehmer	lehmer	NOUN
ejpam-245	54	11	sequences	sequence	NOUN
ejpam-245	54	12	found	find	VERB
ejpam-245	54	13	in	in	ADP
ejpam-245	54	14	[	[	X
ejpam-245	54	15	3	3	NUM
ejpam-245	54	16	]	]	PUNCT
ejpam-245	54	17	.	.	PUNCT
ejpam-245	55	1	john	john	PROPN
ejpam-245	55	2	h.	h.	PROPN
ejpam-245	55	3	jaroma	jaroma	PROPN
ejpam-245	55	4	/	/	SYM
ejpam-245	55	5	eur	eur	PROPN
ejpam-245	55	6	.	.	PUNCT
ejpam-245	56	1	j.	j.	PROPN
ejpam-245	56	2	pure	pure	PROPN
ejpam-245	56	3	appl	appl	PROPN
ejpam-245	56	4	.	.	PROPN
ejpam-245	56	5	math	math	PROPN
ejpam-245	56	6	,	,	PUNCT
ejpam-245	56	7	2	2	NUM
ejpam-245	56	8	(	(	PUNCT
ejpam-245	56	9	2009	2009	NUM
ejpam-245	56	10	)	)	PUNCT
ejpam-245	56	11	,	,	PUNCT
ejpam-245	56	12	(	(	PUNCT
ejpam-245	56	13	352	352	NUM
ejpam-245	56	14	-	-	SYM
ejpam-245	56	15	360	360	NUM
ejpam-245	56	16	)	)	PUNCT
ejpam-245	56	17	355	355	NUM
ejpam-245	56	18	lemma	lemma	PROPN
ejpam-245	56	19	3.1	3.1	NUM
ejpam-245	56	20	.	.	PUNCT
ejpam-245	57	1	the	the	DET
ejpam-245	57	2	greatest	great	ADJ
ejpam-245	57	3	common	common	ADJ
ejpam-245	57	4	factor	factor	NOUN
ejpam-245	57	5	of	of	ADP
ejpam-245	57	6	un	un	PROPN
ejpam-245	57	7	(	(	PUNCT
ejpam-245	57	8	p	p	NOUN
ejpam-245	57	9	r	r	NOUN
ejpam-245	57	10	,	,	PUNCT
ejpam-245	57	11	q	q	NOUN
ejpam-245	57	12	)	)	PUNCT
ejpam-245	57	13	and	and	CCONJ
ejpam-245	57	14	vn	vn	X
ejpam-245	57	15	(	(	PUNCT
ejpam-245	57	16	p	p	NOUN
ejpam-245	57	17	r	r	NOUN
ejpam-245	57	18	,	,	PUNCT
ejpam-245	57	19	q	q	NOUN
ejpam-245	57	20	)	)	PUNCT
ejpam-245	57	21	is	be	AUX
ejpam-245	57	22	1	1	NUM
ejpam-245	57	23	or	or	CCONJ
ejpam-245	57	24	2	2	NUM
ejpam-245	57	25	.	.	PUNCT
ejpam-245	58	1	let	let	VERB
ejpam-245	58	2	p	p	PRON
ejpam-245	58	3	be	be	AUX
ejpam-245	58	4	an	an	DET
ejpam-245	58	5	odd	odd	ADJ
ejpam-245	58	6	prime	prime	NOUN
ejpam-245	58	7	and	and	CCONJ
ejpam-245	58	8	a	a	DET
ejpam-245	58	9	any	any	DET
ejpam-245	58	10	integer	integer	NOUN
ejpam-245	58	11	not	not	PART
ejpam-245	58	12	divisible	divisible	ADJ
ejpam-245	58	13	by	by	ADP
ejpam-245	58	14	p.	p.	NOUN
ejpam-245	58	15	then	then	ADV
ejpam-245	58	16	,	,	PUNCT
ejpam-245	58	17	the	the	DET
ejpam-245	58	18	legendre	legendre	PROPN
ejpam-245	58	19	symbol	symbol	NOUN
ejpam-245	58	20	(	(	PUNCT
ejpam-245	58	21	a	a	X
ejpam-245	58	22	/	/	SYM
ejpam-245	58	23	p	p	NOUN
ejpam-245	58	24	)	)	PUNCT
ejpam-245	58	25	is	be	AUX
ejpam-245	58	26	defined	define	VERB
ejpam-245	58	27	to	to	PART
ejpam-245	58	28	be	be	AUX
ejpam-245	58	29	1	1	NUM
ejpam-245	58	30	provided	provide	VERB
ejpam-245	58	31	that	that	SCONJ
ejpam-245	58	32	a	a	PRON
ejpam-245	58	33	is	be	AUX
ejpam-245	58	34	a	a	DET
ejpam-245	58	35	quadratic	quadratic	ADJ
ejpam-245	58	36	residue	residue	NOUN
ejpam-245	58	37	modulo	modulo	NOUN
ejpam-245	58	38	p	p	NOUN
ejpam-245	59	1	and	and	CCONJ
ejpam-245	59	2	-1	-1	X
ejpam-245	59	3	if	if	SCONJ
ejpam-245	59	4	a	a	PRON
ejpam-245	59	5	is	be	AUX
ejpam-245	59	6	a	a	DET
ejpam-245	59	7	quadratic	quadratic	ADJ
ejpam-245	59	8	nonresidue	nonresidue	NOUN
ejpam-245	59	9	modulo	modulo	NOUN
ejpam-245	59	10	p.	p.	NOUN
ejpam-245	59	11	for	for	ADP
ejpam-245	59	12	all	all	DET
ejpam-245	59	13	a	a	DET
ejpam-245	59	14	such	such	ADJ
ejpam-245	59	15	that	that	SCONJ
ejpam-245	59	16	(	(	PUNCT
ejpam-245	59	17	a	a	PRON
ejpam-245	59	18	,	,	PUNCT
ejpam-245	59	19	p	p	NOUN
ejpam-245	59	20	)	)	PUNCT
ejpam-245	59	21	=	=	SYM
ejpam-245	59	22	1	1	NUM
ejpam-245	59	23	,	,	PUNCT
ejpam-245	59	24	a	a	PRON
ejpam-245	59	25	is	be	AUX
ejpam-245	59	26	called	call	VERB
ejpam-245	59	27	a	a	DET
ejpam-245	59	28	quadratic	quadratic	ADJ
ejpam-245	59	29	residue	residue	NOUN
ejpam-245	59	30	modulo	modulo	NOUN
ejpam-245	59	31	p	p	NOUN
ejpam-245	59	32	if	if	SCONJ
ejpam-245	59	33	the	the	DET
ejpam-245	59	34	congruence	congruence	NOUN
ejpam-245	59	35	x2	x2	PROPN
ejpam-245	59	36	≡	≡	PROPN
ejpam-245	59	37	a	a	DET
ejpam-245	59	38	(	(	PUNCT
ejpam-245	59	39	mod	mod	PROPN
ejpam-245	59	40	p	p	NOUN
ejpam-245	59	41	)	)	PUNCT
ejpam-245	59	42	has	have	VERB
ejpam-245	59	43	a	a	DET
ejpam-245	59	44	solution	solution	NOUN
ejpam-245	59	45	.	.	PUNCT
ejpam-245	60	1	otherwise	otherwise	ADV
ejpam-245	60	2	,	,	PUNCT
ejpam-245	60	3	a	a	PRON
ejpam-245	60	4	is	be	AUX
ejpam-245	60	5	called	call	VERB
ejpam-245	60	6	a	a	DET
ejpam-245	60	7	quadratic	quadratic	ADJ
ejpam-245	60	8	nonresidue	nonresidue	NOUN
ejpam-245	60	9	modulo	modulo	NOUN
ejpam-245	60	10	p.	p.	NOUN
ejpam-245	61	1	if	if	SCONJ
ejpam-245	61	2	p	p	PROPN
ejpam-245	61	3	|	|	ADV
ejpam-245	61	4	a	a	PRON
ejpam-245	61	5	then	then	ADV
ejpam-245	61	6	(	(	PUNCT
ejpam-245	61	7	a	a	NOUN
ejpam-245	61	8	/	/	SYM
ejpam-245	61	9	p	p	NOUN
ejpam-245	61	10	)	)	PUNCT
ejpam-245	61	11	=	=	SYM
ejpam-245	61	12	0	0	X
ejpam-245	61	13	.	.	PUNCT
ejpam-245	62	1	consider	consider	VERB
ejpam-245	62	2	the	the	DET
ejpam-245	62	3	legendre	legendre	PROPN
ejpam-245	62	4	symbols	symbols	PROPN
ejpam-245	62	5	σ	σ	PROPN
ejpam-245	62	6	=	=	PUNCT
ejpam-245	62	7	(	(	PUNCT
ejpam-245	62	8	r	r	NOUN
ejpam-245	62	9	/	/	SYM
ejpam-245	62	10	p	p	NOUN
ejpam-245	62	11	)	)	PUNCT
ejpam-245	62	12	and	and	CCONJ
ejpam-245	62	13	ε	ε	PROPN
ejpam-245	62	14	=	=	SYM
ejpam-245	62	15	(	(	PUNCT
ejpam-245	62	16	∆/p	∆/p	NOUN
ejpam-245	62	17	)	)	PUNCT
ejpam-245	62	18	,	,	PUNCT
ejpam-245	62	19	where	where	SCONJ
ejpam-245	62	20	∆	∆	PROPN
ejpam-245	62	21	=	=	SYM
ejpam-245	62	22	r−	r−	PROPN
ejpam-245	62	23	4q	4q	NOUN
ejpam-245	62	24	is	be	AUX
ejpam-245	62	25	the	the	DET
ejpam-245	62	26	discriminant	discriminant	NOUN
ejpam-245	62	27	of	of	ADP
ejpam-245	62	28	the	the	DET
ejpam-245	62	29	characteristic	characteristic	ADJ
ejpam-245	62	30	equation	equation	NOUN
ejpam-245	62	31	of	of	ADP
ejpam-245	62	32	(	(	PUNCT
ejpam-245	62	33	2.1	2.1	NUM
ejpam-245	62	34	)	)	PUNCT
ejpam-245	62	35	and	and	CCONJ
ejpam-245	62	36	(	(	PUNCT
ejpam-245	62	37	2.2	2.2	NUM
ejpam-245	62	38	)	)	PUNCT
ejpam-245	62	39	.	.	PUNCT
ejpam-245	63	1	lemma	lemma	PROPN
ejpam-245	63	2	3.2	3.2	NUM
ejpam-245	63	3	.	.	PUNCT
ejpam-245	64	1	let	let	VERB
ejpam-245	64	2	p	p	PROPN
ejpam-245	64	3	∤	∤	PROPN
ejpam-245	64	4	rq	rq	VERB
ejpam-245	64	5	.	.	PUNCT
ejpam-245	65	1	then	then	ADV
ejpam-245	65	2	,	,	PUNCT
ejpam-245	65	3	up−σε	up−σε	PROPN
ejpam-245	65	4	(	(	PUNCT
ejpam-245	65	5	p	p	NOUN
ejpam-245	65	6	r	r	NOUN
ejpam-245	65	7	,	,	PUNCT
ejpam-245	65	8	q)≡	q)≡	NOUN
ejpam-245	65	9	0	0	PUNCT
ejpam-245	65	10	(	(	PUNCT
ejpam-245	65	11	mod	mod	PROPN
ejpam-245	65	12	p	p	X
ejpam-245	65	13	)	)	PUNCT
ejpam-245	65	14	.	.	PUNCT
ejpam-245	66	1	let	let	VERB
ejpam-245	66	2	ω(p	ω(p	NOUN
ejpam-245	66	3	)	)	PUNCT
ejpam-245	66	4	denote	denote	VERB
ejpam-245	66	5	the	the	DET
ejpam-245	66	6	rank	rank	NOUN
ejpam-245	66	7	of	of	ADP
ejpam-245	66	8	apparition	apparition	NOUN
ejpam-245	66	9	of	of	ADP
ejpam-245	66	10	p	p	PROPN
ejpam-245	66	11	in	in	ADP
ejpam-245	66	12	{	{	PUNCT
ejpam-245	66	13	un	un	PROPN
ejpam-245	66	14	(	(	PUNCT
ejpam-245	66	15	p	p	NOUN
ejpam-245	66	16	r	r	NOUN
ejpam-245	66	17	,	,	PUNCT
ejpam-245	66	18	q	q	NOUN
ejpam-245	66	19	)	)	PUNCT
ejpam-245	66	20	}	}	PUNCT
ejpam-245	66	21	.	.	PUNCT
ejpam-245	67	1	the	the	DET
ejpam-245	67	2	next	next	ADJ
ejpam-245	67	3	result	result	NOUN
ejpam-245	67	4	tells	tell	VERB
ejpam-245	67	5	us	we	PRON
ejpam-245	67	6	that	that	SCONJ
ejpam-245	67	7	every	every	DET
ejpam-245	67	8	term	term	NOUN
ejpam-245	67	9	with	with	ADP
ejpam-245	67	10	index	index	NOUN
ejpam-245	67	11	equal	equal	ADJ
ejpam-245	67	12	to	to	ADP
ejpam-245	67	13	a	a	DET
ejpam-245	67	14	multiple	multiple	NOUN
ejpam-245	67	15	of	of	ADP
ejpam-245	67	16	ω	ω	PROPN
ejpam-245	67	17	must	must	AUX
ejpam-245	67	18	also	also	ADV
ejpam-245	67	19	contain	contain	VERB
ejpam-245	67	20	p	p	NOUN
ejpam-245	67	21	as	as	ADP
ejpam-245	67	22	a	a	DET
ejpam-245	67	23	factor	factor	NOUN
ejpam-245	67	24	.	.	PUNCT
ejpam-245	68	1	lemma	lemma	PROPN
ejpam-245	68	2	3.3	3.3	NUM
ejpam-245	68	3	.	.	PUNCT
ejpam-245	69	1	let	let	VERB
ejpam-245	69	2	ω	ω	NOUN
ejpam-245	69	3	denote	denote	VERB
ejpam-245	69	4	the	the	DET
ejpam-245	69	5	rank	rank	NOUN
ejpam-245	69	6	of	of	ADP
ejpam-245	69	7	apparition	apparition	NOUN
ejpam-245	69	8	of	of	ADP
ejpam-245	69	9	p	p	NOUN
ejpam-245	69	10	in	in	ADP
ejpam-245	69	11	the	the	DET
ejpam-245	69	12	sequence	sequence	NOUN
ejpam-245	69	13	{	{	PUNCT
ejpam-245	69	14	un	un	PROPN
ejpam-245	69	15	(	(	PUNCT
ejpam-245	69	16	p	p	NOUN
ejpam-245	69	17	r	r	NOUN
ejpam-245	69	18	,	,	PUNCT
ejpam-245	69	19	q	q	NOUN
ejpam-245	69	20	)	)	PUNCT
ejpam-245	69	21	}	}	PUNCT
ejpam-245	69	22	.	.	PUNCT
ejpam-245	70	1	then	then	ADV
ejpam-245	70	2	,	,	PUNCT
ejpam-245	70	3	p	p	PROPN
ejpam-245	70	4	|	|	NOUN
ejpam-245	70	5	un	un	VERB
ejpam-245	70	6	(	(	PUNCT
ejpam-245	70	7	p	p	NOUN
ejpam-245	70	8	r	r	NOUN
ejpam-245	70	9	,	,	PUNCT
ejpam-245	70	10	q	q	NOUN
ejpam-245	70	11	)	)	PUNCT
ejpam-245	70	12	if	if	SCONJ
ejpam-245	70	13	and	and	CCONJ
ejpam-245	70	14	only	only	ADV
ejpam-245	70	15	if	if	SCONJ
ejpam-245	70	16	n	n	PROPN
ejpam-245	70	17	=	=	SYM
ejpam-245	70	18	kω	kω	PROPN
ejpam-245	70	19	,	,	PUNCT
ejpam-245	70	20	where	where	SCONJ
ejpam-245	70	21	k	k	PROPN
ejpam-245	70	22	∈	∈	PROPN
ejpam-245	70	23	{	{	PUNCT
ejpam-245	70	24	1	1	NUM
ejpam-245	70	25	,	,	PUNCT
ejpam-245	70	26	2	2	NUM
ejpam-245	70	27	,	,	PUNCT
ejpam-245	70	28	.	.	PUNCT
ejpam-245	70	29	.	.	PUNCT
ejpam-245	70	30	.	.	PUNCT
ejpam-245	70	31	}	}	PUNCT
ejpam-245	70	32	.	.	PUNCT
ejpam-245	71	1	so	so	ADV
ejpam-245	71	2	far	far	ADV
ejpam-245	71	3	,	,	PUNCT
ejpam-245	71	4	we	we	PRON
ejpam-245	71	5	have	have	AUX
ejpam-245	71	6	seen	see	VERB
ejpam-245	71	7	that	that	SCONJ
ejpam-245	71	8	almost	almost	ADV
ejpam-245	71	9	any	any	DET
ejpam-245	71	10	odd	odd	ADJ
ejpam-245	71	11	prime	prime	NOUN
ejpam-245	71	12	will	will	AUX
ejpam-245	71	13	divide	divide	VERB
ejpam-245	71	14	infinitely	infinitely	ADV
ejpam-245	71	15	many	many	ADJ
ejpam-245	71	16	terms	term	NOUN
ejpam-245	71	17	of	of	ADP
ejpam-245	71	18	{	{	PUNCT
ejpam-245	71	19	un	un	PROPN
ejpam-245	71	20	(	(	PUNCT
ejpam-245	71	21	p	p	NOUN
ejpam-245	71	22	r	r	NOUN
ejpam-245	71	23	,	,	PUNCT
ejpam-245	71	24	q	q	NOUN
ejpam-245	71	25	)	)	PUNCT
ejpam-245	71	26	}	}	PUNCT
ejpam-245	71	27	.	.	PUNCT
ejpam-245	72	1	however	however	ADV
ejpam-245	72	2	,	,	PUNCT
ejpam-245	72	3	there	there	PRON
ejpam-245	72	4	are	be	VERB
ejpam-245	72	5	infinitely	infinitely	ADV
ejpam-245	72	6	many	many	ADJ
ejpam-245	72	7	p	p	NOUN
ejpam-245	72	8	which	which	PRON
ejpam-245	72	9	do	do	AUX
ejpam-245	72	10	not	not	PART
ejpam-245	72	11	divide	divide	VERB
ejpam-245	72	12	{	{	PUNCT
ejpam-245	72	13	vn	vn	NOUN
ejpam-245	72	14	(	(	PUNCT
ejpam-245	72	15	p	p	NOUN
ejpam-245	72	16	r	r	NOUN
ejpam-245	72	17	,	,	PUNCT
ejpam-245	72	18	q	q	NOUN
ejpam-245	72	19	)	)	PUNCT
ejpam-245	72	20	}	}	PUNCT
ejpam-245	72	21	.	.	PUNCT
ejpam-245	73	1	lemma	lemma	PROPN
ejpam-245	73	2	3.4	3.4	NUM
ejpam-245	73	3	.	.	PUNCT
ejpam-245	73	4	suppose	suppose	VERB
ejpam-245	73	5	that	that	SCONJ
ejpam-245	73	6	ω	ω	PROPN
ejpam-245	73	7	is	be	AUX
ejpam-245	73	8	odd	odd	ADJ
ejpam-245	73	9	.	.	PUNCT
ejpam-245	74	1	then	then	ADV
ejpam-245	74	2	vn	vn	INTJ
ejpam-245	74	3	(	(	PUNCT
ejpam-245	74	4	p	p	NOUN
ejpam-245	74	5	r	r	NOUN
ejpam-245	74	6	,	,	PUNCT
ejpam-245	74	7	q	q	NOUN
ejpam-245	74	8	)	)	PUNCT
ejpam-245	74	9	is	be	AUX
ejpam-245	74	10	not	not	PART
ejpam-245	74	11	divisible	divisible	ADJ
ejpam-245	74	12	by	by	ADP
ejpam-245	74	13	p	p	NOUN
ejpam-245	74	14	for	for	ADP
ejpam-245	74	15	any	any	DET
ejpam-245	74	16	value	value	NOUN
ejpam-245	74	17	of	of	ADP
ejpam-245	74	18	n.	n.	NOUN
ejpam-245	74	19	on	on	ADP
ejpam-245	74	20	the	the	DET
ejpam-245	74	21	other	other	ADJ
ejpam-245	74	22	hand	hand	NOUN
ejpam-245	74	23	,	,	PUNCT
ejpam-245	74	24	if	if	SCONJ
ejpam-245	74	25	n	n	PRON
ejpam-245	74	26	is	be	AUX
ejpam-245	74	27	even	even	ADV
ejpam-245	74	28	,	,	PUNCT
ejpam-245	74	29	say	say	VERB
ejpam-245	74	30	2k	2k	NUM
ejpam-245	74	31	,	,	PUNCT
ejpam-245	74	32	then	then	ADV
ejpam-245	74	33	v(2n+1)k	v(2n+1)k	NUM
ejpam-245	74	34	(	(	PUNCT
ejpam-245	74	35	p	p	NOUN
ejpam-245	74	36	r	r	NOUN
ejpam-245	74	37	,	,	PUNCT
ejpam-245	74	38	q	q	NOUN
ejpam-245	74	39	)	)	PUNCT
ejpam-245	74	40	is	be	AUX
ejpam-245	74	41	divisible	divisible	ADJ
ejpam-245	74	42	by	by	ADP
ejpam-245	74	43	p	p	NOUN
ejpam-245	74	44	for	for	ADP
ejpam-245	74	45	every	every	DET
ejpam-245	74	46	n	n	NOUN
ejpam-245	74	47	but	but	CCONJ
ejpam-245	74	48	no	no	DET
ejpam-245	74	49	other	other	ADJ
ejpam-245	74	50	terms	term	NOUN
ejpam-245	74	51	of	of	ADP
ejpam-245	74	52	the	the	DET
ejpam-245	74	53	sequence	sequence	NOUN
ejpam-245	74	54	contain	contain	VERB
ejpam-245	74	55	p	p	NOUN
ejpam-245	74	56	as	as	ADP
ejpam-245	74	57	a	a	DET
ejpam-245	74	58	factor	factor	NOUN
ejpam-245	74	59	.	.	PUNCT
ejpam-245	75	1	let	let	VERB
ejpam-245	75	2	λ(p	λ(p	PROPN
ejpam-245	75	3	)	)	PUNCT
ejpam-245	75	4	denote	denote	VERB
ejpam-245	75	5	the	the	DET
ejpam-245	75	6	rank	rank	NOUN
ejpam-245	75	7	of	of	ADP
ejpam-245	75	8	apparition	apparition	NOUN
ejpam-245	75	9	of	of	ADP
ejpam-245	75	10	p	p	PROPN
ejpam-245	75	11	in	in	ADP
ejpam-245	75	12	{	{	PUNCT
ejpam-245	75	13	vn	vn	X
ejpam-245	75	14	(	(	PUNCT
ejpam-245	75	15	p	p	NOUN
ejpam-245	75	16	r	r	NOUN
ejpam-245	75	17	,	,	PUNCT
ejpam-245	75	18	q	q	NOUN
ejpam-245	75	19	)	)	PUNCT
ejpam-245	75	20	}	}	PUNCT
ejpam-245	75	21	.	.	PUNCT
ejpam-245	76	1	lemma	lemma	PROPN
ejpam-245	76	2	3.5	3.5	NUM
ejpam-245	76	3	.	.	PUNCT
ejpam-245	77	1	let	let	VERB
ejpam-245	77	2	p	p	PROPN
ejpam-245	77	3	∤	∤	PROPN
ejpam-245	77	4	rq∆.	rq∆.	PROPN
ejpam-245	77	5	then	then	ADV
ejpam-245	77	6	,	,	PUNCT
ejpam-245	77	7	u	u	NOUN
ejpam-245	77	8	p−σε	p−σε	PROPN
ejpam-245	77	9	2	2	NUM
ejpam-245	77	10	(	(	PUNCT
ejpam-245	77	11	p	p	NOUN
ejpam-245	77	12	r	r	NOUN
ejpam-245	77	13	,	,	PUNCT
ejpam-245	77	14	q)≡	q)≡	NOUN
ejpam-245	77	15	0	0	PUNCT
ejpam-245	77	16	(	(	PUNCT
ejpam-245	77	17	mod	mod	PROPN
ejpam-245	77	18	p	p	X
ejpam-245	77	19	)	)	PUNCT
ejpam-245	77	20	if	if	SCONJ
ejpam-245	77	21	and	and	CCONJ
ejpam-245	77	22	only	only	ADV
ejpam-245	77	23	if	if	SCONJ
ejpam-245	77	24	σ	σ	PROPN
ejpam-245	77	25	=	=	SYM
ejpam-245	77	26	τ	τ	PROPN
ejpam-245	77	27	,	,	PUNCT
ejpam-245	77	28	where	where	SCONJ
ejpam-245	77	29	τ=	τ=	PRON
ejpam-245	77	30	(	(	PUNCT
ejpam-245	77	31	q	q	NOUN
ejpam-245	77	32	/	/	SYM
ejpam-245	77	33	p	p	NOUN
ejpam-245	77	34	)	)	PUNCT
ejpam-245	77	35	.	.	PUNCT
ejpam-245	78	1	lemma	lemma	PROPN
ejpam-245	78	2	3.6	3.6	NUM
ejpam-245	78	3	.	.	PUNCT
ejpam-245	79	1	let	let	VERB
ejpam-245	79	2	p	p	NOUN
ejpam-245	79	3	∤	∤	PROPN
ejpam-245	79	4	2rq∆.	2rq∆.	PROPN
ejpam-245	80	1	if	if	SCONJ
ejpam-245	80	2	n	n	NOUN
ejpam-245	80	3	±	±	NOUN
ejpam-245	80	4	1	1	NUM
ejpam-245	80	5	is	be	AUX
ejpam-245	80	6	the	the	DET
ejpam-245	80	7	rank	rank	NOUN
ejpam-245	80	8	of	of	ADP
ejpam-245	80	9	apparition	apparition	NOUN
ejpam-245	80	10	of	of	ADP
ejpam-245	80	11	n	n	PROPN
ejpam-245	80	12	then	then	ADV
ejpam-245	80	13	n	n	PROPN
ejpam-245	80	14	is	be	AUX
ejpam-245	80	15	prime	prime	ADJ
ejpam-245	80	16	.	.	PUNCT
ejpam-245	81	1	john	john	PROPN
ejpam-245	81	2	h.	h.	PROPN
ejpam-245	81	3	jaroma	jaroma	PROPN
ejpam-245	81	4	/	/	SYM
ejpam-245	81	5	eur	eur	PROPN
ejpam-245	81	6	.	.	PUNCT
ejpam-245	82	1	j.	j.	PROPN
ejpam-245	82	2	pure	pure	PROPN
ejpam-245	82	3	appl	appl	PROPN
ejpam-245	82	4	.	.	PROPN
ejpam-245	82	5	math	math	PROPN
ejpam-245	82	6	,	,	PUNCT
ejpam-245	82	7	2	2	NUM
ejpam-245	82	8	(	(	PUNCT
ejpam-245	82	9	2009	2009	NUM
ejpam-245	82	10	)	)	PUNCT
ejpam-245	82	11	,	,	PUNCT
ejpam-245	82	12	(	(	PUNCT
ejpam-245	82	13	352	352	NUM
ejpam-245	82	14	-	-	SYM
ejpam-245	82	15	360	360	NUM
ejpam-245	82	16	)	)	PUNCT
ejpam-245	82	17	356	356	NUM
ejpam-245	82	18	4	4	NUM
ejpam-245	82	19	.	.	PUNCT
ejpam-245	82	20	historical	historical	ADJ
ejpam-245	82	21	background	background	NOUN
ejpam-245	82	22	in	in	ADP
ejpam-245	82	23	order	order	NOUN
ejpam-245	82	24	to	to	PART
ejpam-245	82	25	acquire	acquire	VERB
ejpam-245	82	26	a	a	DET
ejpam-245	82	27	more	more	ADV
ejpam-245	82	28	complete	complete	ADJ
ejpam-245	82	29	understanding	understanding	NOUN
ejpam-245	82	30	of	of	ADP
ejpam-245	82	31	these	these	DET
ejpam-245	82	32	two	two	NUM
ejpam-245	82	33	tests	test	NOUN
ejpam-245	82	34	,	,	PUNCT
ejpam-245	82	35	we	we	PRON
ejpam-245	82	36	consider	consider	VERB
ejpam-245	82	37	them	they	PRON
ejpam-245	82	38	first	first	ADV
ejpam-245	82	39	in	in	ADP
ejpam-245	82	40	their	their	PRON
ejpam-245	82	41	historical	historical	ADJ
ejpam-245	82	42	contexts	contexts	NOUN
ejpam-245	82	43	.	.	PUNCT
ejpam-245	83	1	in	in	ADP
ejpam-245	83	2	1877	1877	NUM
ejpam-245	83	3	,	,	PUNCT
ejpam-245	83	4	fr	fr	PROPN
ejpam-245	83	5	.	.	PUNCT
ejpam-245	84	1	pepin	pepin	PROPN
ejpam-245	84	2	formulated	formulate	VERB
ejpam-245	84	3	the	the	DET
ejpam-245	84	4	following	follow	VERB
ejpam-245	84	5	theorem	theorem	NOUN
ejpam-245	84	6	[	[	X
ejpam-245	84	7	7	7	NUM
ejpam-245	84	8	]	]	NUM
ejpam-245	84	9	:	:	PUNCT
ejpam-245	84	10	theorem	theorem	VERB
ejpam-245	84	11	4.1	4.1	NUM
ejpam-245	84	12	.	.	PUNCT
ejpam-245	85	1	pepin	pepin	PROPN
ejpam-245	85	2	’s	’s	PART
ejpam-245	85	3	test	test	NOUN
ejpam-245	85	4	(	(	PUNCT
ejpam-245	85	5	original	original	ADJ
ejpam-245	85	6	version	version	NOUN
ejpam-245	85	7	)	)	PUNCT
ejpam-245	85	8	the	the	DET
ejpam-245	85	9	fermat	fermat	PROPN
ejpam-245	85	10	number	number	NOUN
ejpam-245	85	11	,	,	PUNCT
ejpam-245	85	12	fn	fn	NOUN
ejpam-245	85	13	=	=	SYM
ejpam-245	85	14	22n	22n	NUM
ejpam-245	85	15	+1	+1	PROPN
ejpam-245	85	16	,	,	PUNCT
ejpam-245	85	17	where	where	SCONJ
ejpam-245	85	18	n	n	X
ejpam-245	85	19	>	>	X
ejpam-245	85	20	1	1	NUM
ejpam-245	85	21	is	be	AUX
ejpam-245	85	22	prime	prime	ADJ
ejpam-245	85	23	if	if	SCONJ
ejpam-245	85	24	and	and	CCONJ
ejpam-245	85	25	only	only	ADV
ejpam-245	85	26	if	if	SCONJ
ejpam-245	85	27	5	5	NUM
ejpam-245	85	28	fn−1	fn−1	ADJ
ejpam-245	85	29	2	2	NUM
ejpam-245	85	30	≡−1	≡−1	NOUN
ejpam-245	85	31	(	(	PUNCT
ejpam-245	85	32	mod	mod	PROPN
ejpam-245	85	33	fn	fn	PROPN
ejpam-245	85	34	)	)	PUNCT
ejpam-245	85	35	.	.	PUNCT
ejpam-245	86	1	pepin	pepin	PROPN
ejpam-245	86	2	had	have	AUX
ejpam-245	86	3	noted	note	VERB
ejpam-245	86	4	in	in	ADP
ejpam-245	86	5	[	[	X
ejpam-245	86	6	7	7	X
ejpam-245	86	7	]	]	PUNCT
ejpam-245	86	8	that	that	SCONJ
ejpam-245	86	9	the	the	DET
ejpam-245	86	10	number	number	NOUN
ejpam-245	86	11	10	10	NUM
ejpam-245	86	12	could	could	AUX
ejpam-245	86	13	be	be	AUX
ejpam-245	86	14	used	use	VERB
ejpam-245	86	15	in	in	ADP
ejpam-245	86	16	place	place	NOUN
ejpam-245	86	17	of	of	ADP
ejpam-245	86	18	5	5	NUM
ejpam-245	86	19	.	.	PUNCT
ejpam-245	86	20	prior	prior	ADV
ejpam-245	86	21	to	to	ADP
ejpam-245	86	22	pepin	pepin	PROPN
ejpam-245	86	23	’s	’s	PART
ejpam-245	86	24	remark	remark	NOUN
ejpam-245	86	25	,	,	PUNCT
ejpam-245	86	26	françois	françois	PROPN
ejpam-245	86	27	proth	proth	NOUN
ejpam-245	86	28	(	(	PUNCT
ejpam-245	86	29	1852	1852	NUM
ejpam-245	86	30	–	–	PUNCT
ejpam-245	86	31	1879	1879	NUM
ejpam-245	86	32	)	)	PUNCT
ejpam-245	86	33	noted	note	VERB
ejpam-245	86	34	in	in	ADP
ejpam-245	86	35	1876	1876	NUM
ejpam-245	86	36	and	and	CCONJ
ejpam-245	86	37	then	then	ADV
ejpam-245	86	38	again	again	ADV
ejpam-245	86	39	in	in	ADP
ejpam-245	86	40	1878	1878	NUM
ejpam-245	86	41	that	that	SCONJ
ejpam-245	86	42	one	one	PRON
ejpam-245	86	43	may	may	AUX
ejpam-245	86	44	use	use	VERB
ejpam-245	86	45	the	the	DET
ejpam-245	86	46	number	number	NOUN
ejpam-245	86	47	3	3	NUM
ejpam-245	86	48	in	in	ADP
ejpam-245	86	49	lieu	lieu	NOUN
ejpam-245	86	50	of	of	ADP
ejpam-245	86	51	5	5	NUM
ejpam-245	86	52	for	for	ADP
ejpam-245	86	53	our	our	PRON
ejpam-245	86	54	theorem	theorem	NOUN
ejpam-245	86	55	4.1	4.1	NUM
ejpam-245	87	1	[	[	NOUN
ejpam-245	87	2	8	8	NUM
ejpam-245	87	3	]	]	PUNCT
ejpam-245	87	4	,	,	PUNCT
ejpam-245	88	1	[	[	X
ejpam-245	88	2	9	9	NUM
ejpam-245	88	3	]	]	PUNCT
ejpam-245	88	4	)	)	PUNCT
ejpam-245	88	5	but	but	CCONJ
ejpam-245	88	6	offered	offer	VERB
ejpam-245	88	7	no	no	DET
ejpam-245	88	8	proof	proof	NOUN
ejpam-245	88	9	of	of	ADP
ejpam-245	88	10	his	his	PRON
ejpam-245	88	11	assertion	assertion	NOUN
ejpam-245	88	12	.	.	PUNCT
ejpam-245	89	1	then	then	ADV
ejpam-245	89	2	,	,	PUNCT
ejpam-245	89	3	édouard	édouard	PROPN
ejpam-245	89	4	lucas	lucas	PROPN
ejpam-245	89	5	(	(	PUNCT
ejpam-245	89	6	1842	1842	NUM
ejpam-245	89	7	–	–	PUNCT
ejpam-245	89	8	1891	1891	NUM
ejpam-245	89	9	)	)	PUNCT
ejpam-245	89	10	commented	comment	VERB
ejpam-245	89	11	that	that	SCONJ
ejpam-245	89	12	an	an	DET
ejpam-245	89	13	arbitrary	arbitrary	ADJ
ejpam-245	89	14	integer	integer	NOUN
ejpam-245	89	15	a	a	PRON
ejpam-245	89	16	could	could	AUX
ejpam-245	89	17	be	be	AUX
ejpam-245	89	18	used	use	VERB
ejpam-245	89	19	in	in	ADP
ejpam-245	89	20	place	place	NOUN
ejpam-245	89	21	of	of	ADP
ejpam-245	89	22	5	5	NUM
ejpam-245	89	23	provided	provide	VERB
ejpam-245	89	24	that	that	SCONJ
ejpam-245	89	25	the	the	DET
ejpam-245	89	26	jacobi	jacobi	PROPN
ejpam-245	89	27	symbol	symbol	NOUN
ejpam-245	89	28	(	(	PUNCT
ejpam-245	89	29	a	a	X
ejpam-245	89	30	/	/	SYM
ejpam-245	89	31	fn	fn	NOUN
ejpam-245	89	32	)	)	PUNCT
ejpam-245	89	33	has	have	AUX
ejpam-245	89	34	a	a	DET
ejpam-245	89	35	value	value	NOUN
ejpam-245	89	36	equal	equal	ADJ
ejpam-245	89	37	to	to	ADP
ejpam-245	89	38	-1	-1	NOUN
ejpam-245	90	1	[	[	X
ejpam-245	90	2	5	5	NUM
ejpam-245	90	3	]	]	PUNCT
ejpam-245	90	4	and	and	CCONJ
ejpam-245	90	5	in	in	ADP
ejpam-245	90	6	1879	1879	NUM
ejpam-245	90	7	offered	offer	VERB
ejpam-245	90	8	a	a	DET
ejpam-245	90	9	proof	proof	NOUN
ejpam-245	90	10	[	[	X
ejpam-245	90	11	6	6	NUM
ejpam-245	90	12	]	]	PUNCT
ejpam-245	90	13	.	.	PUNCT
ejpam-245	91	1	thus	thus	ADV
ejpam-245	91	2	,	,	PUNCT
ejpam-245	91	3	the	the	DET
ejpam-245	91	4	result	result	NOUN
ejpam-245	91	5	we	we	PRON
ejpam-245	91	6	now	now	ADV
ejpam-245	91	7	call	call	VERB
ejpam-245	91	8	pepin	pepin	PROPN
ejpam-245	91	9	’s	’s	PART
ejpam-245	91	10	test	test	NOUN
ejpam-245	91	11	is	be	AUX
ejpam-245	91	12	actually	actually	ADV
ejpam-245	91	13	a	a	DET
ejpam-245	91	14	theorem	theorem	NOUN
ejpam-245	91	15	suggested	suggest	VERB
ejpam-245	91	16	by	by	ADP
ejpam-245	91	17	proth	proth	NOUN
ejpam-245	91	18	and	and	CCONJ
ejpam-245	91	19	proved	prove	VERB
ejpam-245	91	20	by	by	ADP
ejpam-245	91	21	lucas	lucas	PROPN
ejpam-245	91	22	.	.	PUNCT
ejpam-245	92	1	for	for	ADP
ejpam-245	92	2	a	a	DET
ejpam-245	92	3	more	more	ADV
ejpam-245	92	4	detailed	detailed	ADJ
ejpam-245	92	5	summary	summary	NOUN
ejpam-245	92	6	of	of	ADP
ejpam-245	92	7	these	these	DET
ejpam-245	92	8	events	event	NOUN
ejpam-245	92	9	,	,	PUNCT
ejpam-245	92	10	the	the	DET
ejpam-245	92	11	reader	reader	NOUN
ejpam-245	92	12	is	be	AUX
ejpam-245	92	13	directed	direct	VERB
ejpam-245	92	14	to	to	ADP
ejpam-245	92	15	[	[	X
ejpam-245	92	16	16	16	NUM
ejpam-245	92	17	]	]	PUNCT
ejpam-245	92	18	,	,	PUNCT
ejpam-245	92	19	where	where	SCONJ
ejpam-245	92	20	a	a	DET
ejpam-245	92	21	more	more	ADV
ejpam-245	92	22	detailed	detailed	ADJ
ejpam-245	92	23	account	account	NOUN
ejpam-245	92	24	is	be	AUX
ejpam-245	92	25	found	find	VERB
ejpam-245	92	26	.	.	PUNCT
ejpam-245	93	1	a	a	DET
ejpam-245	93	2	modern	modern	ADJ
ejpam-245	93	3	version	version	NOUN
ejpam-245	93	4	of	of	ADP
ejpam-245	93	5	pepin	pepin	PROPN
ejpam-245	93	6	’s	’s	PART
ejpam-245	93	7	test	test	NOUN
ejpam-245	93	8	is	be	AUX
ejpam-245	93	9	given	give	VERB
ejpam-245	93	10	by	by	ADP
ejpam-245	93	11	:	:	PUNCT
ejpam-245	93	12	theorem	theorem	NOUN
ejpam-245	93	13	4.2	4.2	NUM
ejpam-245	93	14	.	.	PUNCT
ejpam-245	94	1	pepin	pepin	PROPN
ejpam-245	94	2	’s	’s	PART
ejpam-245	94	3	test	test	NOUN
ejpam-245	94	4	(	(	PUNCT
ejpam-245	94	5	modern	modern	ADJ
ejpam-245	94	6	version	version	NOUN
ejpam-245	94	7	)	)	PUNCT
ejpam-245	94	8	the	the	DET
ejpam-245	94	9	fermat	fermat	PROPN
ejpam-245	94	10	number	number	NOUN
ejpam-245	94	11	,	,	PUNCT
ejpam-245	94	12	fn	fn	NOUN
ejpam-245	94	13	=	=	SYM
ejpam-245	94	14	22n	22n	NUM
ejpam-245	94	15	+1	+1	PROPN
ejpam-245	94	16	,	,	PUNCT
ejpam-245	94	17	where	where	SCONJ
ejpam-245	94	18	n	n	DET
ejpam-245	94	19	≥	≥	NOUN
ejpam-245	94	20	1	1	NUM
ejpam-245	94	21	is	be	AUX
ejpam-245	94	22	prime	prime	ADJ
ejpam-245	94	23	if	if	SCONJ
ejpam-245	94	24	and	and	CCONJ
ejpam-245	94	25	only	only	ADV
ejpam-245	94	26	if	if	SCONJ
ejpam-245	94	27	3	3	NUM
ejpam-245	94	28	fn−1	fn−1	ADJ
ejpam-245	94	29	2	2	NUM
ejpam-245	94	30	≡−1	≡−1	NOUN
ejpam-245	94	31	(	(	PUNCT
ejpam-245	94	32	mod	mod	PROPN
ejpam-245	94	33	fn	fn	PROPN
ejpam-245	94	34	)	)	PUNCT
ejpam-245	94	35	.	.	PUNCT
ejpam-245	95	1	testing	test	VERB
ejpam-245	95	2	primality	primality	NOUN
ejpam-245	95	3	of	of	ADP
ejpam-245	95	4	the	the	DET
ejpam-245	95	5	mersenne	mersenne	NOUN
ejpam-245	95	6	numbers	number	NOUN
ejpam-245	95	7	was	be	AUX
ejpam-245	95	8	initiated	initiate	VERB
ejpam-245	95	9	by	by	ADP
ejpam-245	95	10	lucas	lucas	PROPN
ejpam-245	95	11	in	in	ADP
ejpam-245	95	12	1878	1878	NUM
ejpam-245	95	13	.	.	PUNCT
ejpam-245	96	1	lucas	lucas	PROPN
ejpam-245	96	2	proposed	propose	VERB
ejpam-245	96	3	two	two	NUM
ejpam-245	96	4	tests	test	NOUN
ejpam-245	96	5	for	for	ADP
ejpam-245	96	6	determining	determine	VERB
ejpam-245	96	7	if	if	SCONJ
ejpam-245	96	8	2n	2n	NUM
ejpam-245	96	9	−	−	NOUN
ejpam-245	96	10	1	1	NUM
ejpam-245	96	11	is	be	AUX
ejpam-245	96	12	prime	prime	ADJ
ejpam-245	96	13	[	[	X
ejpam-245	96	14	5	5	NUM
ejpam-245	96	15	]	]	PUNCT
ejpam-245	96	16	.	.	PUNCT
ejpam-245	97	1	however	however	ADV
ejpam-245	97	2	,	,	PUNCT
ejpam-245	97	3	neither	neither	CCONJ
ejpam-245	97	4	theorem	theorem	VERB
ejpam-245	97	5	john	john	PROPN
ejpam-245	97	6	h.	h.	PROPN
ejpam-245	97	7	jaroma	jaroma	PROPN
ejpam-245	97	8	/	/	SYM
ejpam-245	97	9	eur	eur	PROPN
ejpam-245	97	10	.	.	PUNCT
ejpam-245	98	1	j.	j.	PROPN
ejpam-245	98	2	pure	pure	PROPN
ejpam-245	98	3	appl	appl	PROPN
ejpam-245	98	4	.	.	PROPN
ejpam-245	98	5	math	math	PROPN
ejpam-245	98	6	,	,	PUNCT
ejpam-245	98	7	2	2	NUM
ejpam-245	98	8	(	(	PUNCT
ejpam-245	98	9	2009	2009	NUM
ejpam-245	98	10	)	)	PUNCT
ejpam-245	98	11	,	,	PUNCT
ejpam-245	98	12	(	(	PUNCT
ejpam-245	98	13	352	352	NUM
ejpam-245	98	14	-	-	SYM
ejpam-245	98	15	360	360	NUM
ejpam-245	98	16	)	)	PUNCT
ejpam-245	98	17	357	357	NUM
ejpam-245	98	18	was	be	AUX
ejpam-245	98	19	given	give	VERB
ejpam-245	98	20	as	as	ADV
ejpam-245	98	21	necessary	necessary	ADJ
ejpam-245	98	22	and	and	CCONJ
ejpam-245	98	23	sufficient	sufficient	ADJ
ejpam-245	98	24	.	.	PUNCT
ejpam-245	99	1	in	in	ADP
ejpam-245	99	2	1930	1930	NUM
ejpam-245	99	3	,	,	PUNCT
ejpam-245	99	4	d.	d.	PROPN
ejpam-245	99	5	h.	h.	PROPN
ejpam-245	99	6	lehmer	lehmer	PROPN
ejpam-245	99	7	wrote	write	VERB
ejpam-245	99	8	,	,	PUNCT
ejpam-245	99	9	his	his	PRON
ejpam-245	99	10	[	[	X
ejpam-245	99	11	lucas	lucas	NOUN
ejpam-245	99	12	’s	’s	PART
ejpam-245	99	13	]	]	PUNCT
ejpam-245	99	14	conditions	condition	NOUN
ejpam-245	99	15	for	for	ADP
ejpam-245	99	16	primality	primality	NOUN
ejpam-245	99	17	are	be	AUX
ejpam-245	99	18	sufficient	sufficient	ADJ
ejpam-245	99	19	but	but	CCONJ
ejpam-245	99	20	not	not	PART
ejpam-245	99	21	necessary.†	necessary.†	NOUN
ejpam-245	99	22	one	one	NOUN
ejpam-245	99	23	is	be	AUX
ejpam-245	99	24	uncertain	uncertain	ADJ
ejpam-245	99	25	whether	whether	SCONJ
ejpam-245	99	26	lucas	luca	NOUN
ejpam-245	99	27	’	'	PUNCT
ejpam-245	99	28	tests	test	NOUN
ejpam-245	99	29	will	will	AUX
ejpam-245	99	30	reveal	reveal	VERB
ejpam-245	99	31	the	the	DET
ejpam-245	99	32	character	character	NOUN
ejpam-245	99	33	of	of	ADP
ejpam-245	99	34	a	a	DET
ejpam-245	99	35	number	number	NOUN
ejpam-245	99	36	which	which	PRON
ejpam-245	99	37	is	be	AUX
ejpam-245	99	38	actually	actually	ADV
ejpam-245	99	39	a	a	DET
ejpam-245	99	40	prime	prime	NOUN
ejpam-245	99	41	[	[	X
ejpam-245	99	42	3	3	NUM
ejpam-245	99	43	]	]	PUNCT
ejpam-245	99	44	.	.	PUNCT
ejpam-245	100	1	lehmer	lehmer	NOUN
ejpam-245	100	2	furthermore	furthermore	ADV
ejpam-245	100	3	noted	note	VERB
ejpam-245	100	4	that	that	SCONJ
ejpam-245	100	5	r.	r.	PROPN
ejpam-245	100	6	d.	d.	PROPN
ejpam-245	100	7	carmichael	carmichael	PROPN
ejpam-245	100	8	in	in	ADP
ejpam-245	100	9	[	[	X
ejpam-245	100	10	2	2	NUM
ejpam-245	100	11	]	]	PUNCT
ejpam-245	100	12	had	have	AUX
ejpam-245	100	13	provided	provide	VERB
ejpam-245	100	14	a	a	DET
ejpam-245	100	15	set	set	NOUN
ejpam-245	100	16	of	of	ADP
ejpam-245	100	17	necessary	necessary	ADJ
ejpam-245	100	18	and	and	CCONJ
ejpam-245	100	19	sufficient	sufficient	ADJ
ejpam-245	100	20	conditions	condition	NOUN
ejpam-245	100	21	for	for	ADP
ejpam-245	100	22	the	the	DET
ejpam-245	100	23	primality	primality	NOUN
ejpam-245	100	24	of	of	ADP
ejpam-245	100	25	such	such	ADJ
ejpam-245	100	26	numbers	number	NOUN
ejpam-245	100	27	.	.	PUNCT
ejpam-245	101	1	however	however	ADV
ejpam-245	101	2	,	,	PUNCT
ejpam-245	101	3	with	with	ADP
ejpam-245	101	4	the	the	DET
ejpam-245	101	5	exception	exception	NOUN
ejpam-245	101	6	of	of	ADP
ejpam-245	101	7	two	two	NUM
ejpam-245	101	8	cases	case	NOUN
ejpam-245	101	9	,	,	PUNCT
ejpam-245	101	10	they	they	PRON
ejpam-245	101	11	depended	depend	VERB
ejpam-245	101	12	on	on	ADP
ejpam-245	101	13	the	the	DET
ejpam-245	101	14	existence	existence	NOUN
ejpam-245	101	15	of	of	ADP
ejpam-245	101	16	an	an	DET
ejpam-245	101	17	auxiliary	auxiliary	ADJ
ejpam-245	101	18	pair	pair	NOUN
ejpam-245	101	19	of	of	ADP
ejpam-245	101	20	numbers	number	NOUN
ejpam-245	101	21	to	to	PART
ejpam-245	101	22	be	be	AUX
ejpam-245	101	23	used	use	VERB
ejpam-245	101	24	in	in	ADP
ejpam-245	101	25	testing	test	VERB
ejpam-245	101	26	a	a	DET
ejpam-245	101	27	given	give	VERB
ejpam-245	101	28	integer	integer	NOUN
ejpam-245	101	29	.	.	PUNCT
ejpam-245	102	1	thus	thus	ADV
ejpam-245	102	2	,	,	PUNCT
ejpam-245	102	3	according	accord	VERB
ejpam-245	102	4	to	to	ADP
ejpam-245	102	5	lehmer	lehmer	NOUN
ejpam-245	102	6	,	,	PUNCT
ejpam-245	102	7	from	from	ADP
ejpam-245	102	8	a	a	DET
ejpam-245	102	9	practical	practical	ADJ
ejpam-245	102	10	point	point	NOUN
ejpam-245	102	11	of	of	ADP
ejpam-245	102	12	view	view	NOUN
ejpam-245	102	13	these	these	DET
ejpam-245	102	14	tests	test	NOUN
ejpam-245	102	15	are	be	AUX
ejpam-245	102	16	not	not	PART
ejpam-245	102	17	applicable	applicable	ADJ
ejpam-245	102	18	since	since	SCONJ
ejpam-245	102	19	no	no	DET
ejpam-245	102	20	method	method	NOUN
ejpam-245	102	21	is	be	AUX
ejpam-245	102	22	given	give	VERB
ejpam-245	102	23	for	for	ADP
ejpam-245	102	24	determining	determine	VERB
ejpam-245	102	25	in	in	ADP
ejpam-245	102	26	advance	advance	NOUN
ejpam-245	102	27	an	an	DET
ejpam-245	102	28	appropriate	appropriate	ADJ
ejpam-245	102	29	number	number	NOUN
ejpam-245	102	30	pair	pair	NOUN
ejpam-245	102	31	.	.	PUNCT
ejpam-245	103	1	finally	finally	ADV
ejpam-245	103	2	,	,	PUNCT
ejpam-245	103	3	in	in	ADP
ejpam-245	103	4	response	response	NOUN
ejpam-245	103	5	to	to	ADP
ejpam-245	103	6	his	his	PRON
ejpam-245	103	7	own	own	ADJ
ejpam-245	103	8	observation	observation	NOUN
ejpam-245	103	9	lehmer	lehmer	NOUN
ejpam-245	103	10	produced	produce	VERB
ejpam-245	103	11	an	an	DET
ejpam-245	103	12	explicit	explicit	ADJ
ejpam-245	103	13	necessary	necessary	ADJ
ejpam-245	103	14	and	and	CCONJ
ejpam-245	103	15	sufficient	sufficient	ADJ
ejpam-245	103	16	condition	condition	NOUN
ejpam-245	103	17	for	for	ADP
ejpam-245	103	18	a	a	DET
ejpam-245	103	19	mersenne	mersenne	NOUN
ejpam-245	103	20	number	number	NOUN
ejpam-245	103	21	to	to	PART
ejpam-245	103	22	be	be	AUX
ejpam-245	103	23	prime	prime	ADJ
ejpam-245	103	24	[	[	X
ejpam-245	103	25	3	3	NUM
ejpam-245	103	26	]	]	PUNCT
ejpam-245	103	27	.	.	PUNCT
ejpam-245	104	1	this	this	DET
ejpam-245	104	2	result	result	NOUN
ejpam-245	104	3	today	today	NOUN
ejpam-245	104	4	is	be	AUX
ejpam-245	104	5	commonly	commonly	ADV
ejpam-245	104	6	known	know	VERB
ejpam-245	104	7	as	as	ADP
ejpam-245	104	8	the	the	DET
ejpam-245	104	9	lucas	lucas	NOUN
ejpam-245	104	10	-	-	PUNCT
ejpam-245	104	11	lehmer	lehmer	NOUN
ejpam-245	104	12	test	test	NOUN
ejpam-245	104	13	.	.	PUNCT
ejpam-245	105	1	the	the	DET
ejpam-245	105	2	original	original	ADJ
ejpam-245	105	3	statement	statement	NOUN
ejpam-245	105	4	of	of	ADP
ejpam-245	105	5	this	this	DET
ejpam-245	105	6	celebrated	celebrate	VERB
ejpam-245	105	7	result	result	NOUN
ejpam-245	105	8	is	be	AUX
ejpam-245	105	9	found	find	VERB
ejpam-245	105	10	in	in	ADP
ejpam-245	105	11	[	[	X
ejpam-245	105	12	3	3	NUM
ejpam-245	105	13	]	]	PUNCT
ejpam-245	105	14	,	,	PUNCT
ejpam-245	105	15	although	although	SCONJ
ejpam-245	105	16	it	it	PRON
ejpam-245	105	17	had	have	AUX
ejpam-245	105	18	been	be	AUX
ejpam-245	105	19	mistakenly	mistakenly	ADV
ejpam-245	105	20	noted	note	VERB
ejpam-245	105	21	in	in	ADP
ejpam-245	105	22	[	[	X
ejpam-245	105	23	13	13	NUM
ejpam-245	105	24	]	]	PUNCT
ejpam-245	105	25	that	that	PRON
ejpam-245	105	26	lehmer	lehmer	NOUN
ejpam-245	105	27	’s	’s	PART
ejpam-245	105	28	original	original	ADJ
ejpam-245	105	29	proof	proof	NOUN
ejpam-245	105	30	of	of	ADP
ejpam-245	105	31	the	the	DET
ejpam-245	105	32	lucas	lucas	NOUN
ejpam-245	105	33	-	-	PUNCT
ejpam-245	105	34	lehmer	lehmer	NOUN
ejpam-245	105	35	test	test	NOUN
ejpam-245	105	36	is	be	AUX
ejpam-245	105	37	given	give	VERB
ejpam-245	105	38	in	in	ADP
ejpam-245	105	39	[	[	X
ejpam-245	105	40	4	4	NUM
ejpam-245	105	41	]	]	PUNCT
ejpam-245	105	42	.	.	PUNCT
ejpam-245	106	1	theorem	theorem	VERB
ejpam-245	106	2	4.3	4.3	NUM
ejpam-245	106	3	.	.	PUNCT
ejpam-245	106	4	lucas	lucas	PROPN
ejpam-245	106	5	-	-	PUNCT
ejpam-245	106	6	lehmer	lehmer	NOUN
ejpam-245	106	7	test	test	NOUN
ejpam-245	106	8	(	(	PUNCT
ejpam-245	106	9	original	original	ADJ
ejpam-245	106	10	version	version	NOUN
ejpam-245	106	11	)	)	PUNCT
ejpam-245	106	12	the	the	DET
ejpam-245	106	13	number	number	NOUN
ejpam-245	106	14	,	,	PUNCT
ejpam-245	106	15	mn	mn	PROPN
ejpam-245	106	16	=	=	SYM
ejpam-245	106	17	2n	2n	NUM
ejpam-245	107	1	−	−	NOUN
ejpam-245	107	2	1	1	NUM
ejpam-245	107	3	,	,	PUNCT
ejpam-245	107	4	is	be	AUX
ejpam-245	107	5	prime	prime	ADJ
ejpam-245	107	6	if	if	SCONJ
ejpam-245	107	7	and	and	CCONJ
ejpam-245	107	8	only	only	ADV
ejpam-245	107	9	if	if	SCONJ
ejpam-245	107	10	it	it	PRON
ejpam-245	107	11	divides	divide	VERB
ejpam-245	107	12	the	the	DET
ejpam-245	107	13	(	(	PUNCT
ejpam-245	107	14	n−	n−	NOUN
ejpam-245	107	15	1)st	1)st	NUM
ejpam-245	107	16	term	term	NOUN
ejpam-245	107	17	of	of	ADP
ejpam-245	107	18	the	the	DET
ejpam-245	107	19	sequence	sequence	NOUN
ejpam-245	107	20	4	4	NUM
ejpam-245	107	21	,	,	PUNCT
ejpam-245	107	22	14	14	NUM
ejpam-245	107	23	,	,	PUNCT
ejpam-245	107	24	194	194	NUM
ejpam-245	107	25	,	,	PUNCT
ejpam-245	107	26	37634	37634	NUM
ejpam-245	107	27	,	,	PUNCT
ejpam-245	107	28	1416317954	1416317954	NUM
ejpam-245	107	29	,	,	PUNCT
ejpam-245	107	30	.	.	PUNCT
ejpam-245	107	31	.	.	PUNCT
ejpam-245	107	32	.	.	PUNCT
ejpam-245	108	1	sk	sk	INTJ
ejpam-245	108	2	,	,	PUNCT
ejpam-245	108	3	.	.	PUNCT
ejpam-245	108	4	.	.	PUNCT
ejpam-245	108	5	.	.	PUNCT
ejpam-245	109	1	where	where	SCONJ
ejpam-245	109	2	,	,	PUNCT
ejpam-245	109	3	sk	sk	PROPN
ejpam-245	109	4	=	=	PROPN
ejpam-245	109	5	s2	s2	NOUN
ejpam-245	109	6	k−1−	k−1−	PROPN
ejpam-245	109	7	2	2	NUM
ejpam-245	109	8	.	.	NOUN
ejpam-245	109	9	5	5	NUM
ejpam-245	109	10	.	.	NOUN
ejpam-245	109	11	equivalence	equivalence	NOUN
ejpam-245	109	12	of	of	ADP
ejpam-245	109	13	pepin	pepin	PROPN
ejpam-245	109	14	’s	’s	X
ejpam-245	109	15	and	and	CCONJ
ejpam-245	109	16	the	the	DET
ejpam-245	109	17	lucas	lucas	NOUN
ejpam-245	109	18	-	-	PUNCT
ejpam-245	109	19	lehmer	lehmer	NOUN
ejpam-245	109	20	tests	test	NOUN
ejpam-245	109	21	we	we	PRON
ejpam-245	109	22	are	be	AUX
ejpam-245	109	23	ready	ready	ADJ
ejpam-245	109	24	to	to	PART
ejpam-245	109	25	show	show	VERB
ejpam-245	109	26	that	that	SCONJ
ejpam-245	109	27	pepin	pepin	PROPN
ejpam-245	109	28	’s	’s	X
ejpam-245	109	29	and	and	CCONJ
ejpam-245	109	30	the	the	DET
ejpam-245	109	31	lucas	lucas	NOUN
ejpam-245	109	32	-	-	PUNCT
ejpam-245	109	33	lehmer	lehmer	NOUN
ejpam-245	109	34	tests	test	NOUN
ejpam-245	109	35	share	share	VERB
ejpam-245	109	36	a	a	DET
ejpam-245	109	37	similar	similar	ADJ
ejpam-245	109	38	structure	structure	NOUN
ejpam-245	109	39	.	.	PUNCT
ejpam-245	110	1	to	to	ADP
ejpam-245	110	2	this	this	DET
ejpam-245	110	3	end	end	NOUN
ejpam-245	110	4	,	,	PUNCT
ejpam-245	110	5	we	we	PRON
ejpam-245	110	6	make	make	VERB
ejpam-245	110	7	the	the	DET
ejpam-245	110	8	observation	observation	NOUN
ejpam-245	110	9	that	that	SCONJ
ejpam-245	110	10	{	{	PUNCT
ejpam-245	110	11	vn(4	vn(4	PROPN
ejpam-245	110	12	,	,	PUNCT
ejpam-245	110	13	3	3	NUM
ejpam-245	110	14	)	)	PUNCT
ejpam-245	110	15	}	}	PUNCT
ejpam-245	111	1	=	=	PUNCT
ejpam-245	111	2	3n+	3n+	NUM
ejpam-245	111	3	1	1	NUM
ejpam-245	111	4	.	.	PUNCT
ejpam-245	112	1	this	this	PRON
ejpam-245	112	2	follows	follow	VERB
ejpam-245	112	3	subsequently	subsequently	ADV
ejpam-245	112	4	from	from	ADP
ejpam-245	112	5	theorem	theorem	ADJ
ejpam-245	112	6	5.1	5.1	NUM
ejpam-245	112	7	.	.	PUNCT
ejpam-245	112	8	†although	†although	PROPN
ejpam-245	112	9	lucas	lucas	PROPN
ejpam-245	112	10	did	do	AUX
ejpam-245	112	11	not	not	PART
ejpam-245	112	12	attempt	attempt	VERB
ejpam-245	112	13	to	to	PART
ejpam-245	112	14	give	give	VERB
ejpam-245	112	15	necessary	necessary	ADJ
ejpam-245	112	16	tests	test	NOUN
ejpam-245	112	17	,	,	PUNCT
ejpam-245	112	18	one	one	NUM
ejpam-245	112	19	of	of	ADP
ejpam-245	112	20	them	they	PRON
ejpam-245	112	21	is	be	AUX
ejpam-245	112	22	in	in	ADP
ejpam-245	112	23	fact	fact	NOUN
ejpam-245	112	24	necessary	necessary	ADJ
ejpam-245	112	25	.	.	PUNCT
ejpam-245	113	1	john	john	PROPN
ejpam-245	113	2	h.	h.	PROPN
ejpam-245	113	3	jaroma	jaroma	PROPN
ejpam-245	113	4	/	/	SYM
ejpam-245	113	5	eur	eur	PROPN
ejpam-245	113	6	.	.	PUNCT
ejpam-245	114	1	j.	j.	PROPN
ejpam-245	114	2	pure	pure	PROPN
ejpam-245	114	3	appl	appl	PROPN
ejpam-245	114	4	.	.	PROPN
ejpam-245	114	5	math	math	PROPN
ejpam-245	114	6	,	,	PUNCT
ejpam-245	114	7	2	2	NUM
ejpam-245	114	8	(	(	PUNCT
ejpam-245	114	9	2009	2009	NUM
ejpam-245	114	10	)	)	PUNCT
ejpam-245	114	11	,	,	PUNCT
ejpam-245	114	12	(	(	PUNCT
ejpam-245	114	13	352	352	NUM
ejpam-245	114	14	-	-	SYM
ejpam-245	114	15	360	360	NUM
ejpam-245	114	16	)	)	PUNCT
ejpam-245	114	17	358	358	NUM
ejpam-245	114	18	theorem	theorem	VERB
ejpam-245	114	19	5.1	5.1	NUM
ejpam-245	114	20	.	.	PUNCT
ejpam-245	115	1	let	let	VERB
ejpam-245	115	2	a	a	DET
ejpam-245	115	3	be	be	AUX
ejpam-245	115	4	any	any	DET
ejpam-245	115	5	integer	integer	NOUN
ejpam-245	115	6	.	.	PUNCT
ejpam-245	116	1	then	then	ADV
ejpam-245	116	2	the	the	DET
ejpam-245	116	3	terms	term	NOUN
ejpam-245	116	4	of	of	ADP
ejpam-245	116	5	the	the	DET
ejpam-245	116	6	companion	companion	NOUN
ejpam-245	116	7	lehmer	lehmer	NOUN
ejpam-245	116	8	sequence	sequence	NOUN
ejpam-245	116	9	{	{	PUNCT
ejpam-245	116	10	vn	vn	PROPN
ejpam-245	116	11	(	(	PUNCT
ejpam-245	116	12	p	p	NOUN
ejpam-245	116	13	r	r	NOUN
ejpam-245	116	14	,	,	PUNCT
ejpam-245	116	15	q	q	NOUN
ejpam-245	116	16	)	)	PUNCT
ejpam-245	116	17	}	}	PUNCT
ejpam-245	116	18	=	=	PRON
ejpam-245	116	19	{	{	PUNCT
ejpam-245	116	20	vn(a+	vn(a+	PROPN
ejpam-245	116	21	1	1	NUM
ejpam-245	116	22	,	,	PUNCT
ejpam-245	116	23	a	a	PRON
ejpam-245	116	24	)	)	PUNCT
ejpam-245	116	25	}	}	PUNCT
ejpam-245	116	26	are	be	AUX
ejpam-245	116	27	of	of	ADP
ejpam-245	116	28	the	the	DET
ejpam-245	116	29	form	form	NOUN
ejpam-245	116	30	an	an	DET
ejpam-245	116	31	+	+	NOUN
ejpam-245	116	32	1	1	NUM
ejpam-245	116	33	.	.	PUNCT
ejpam-245	117	1	the	the	DET
ejpam-245	117	2	proof	proof	NOUN
ejpam-245	117	3	is	be	AUX
ejpam-245	117	4	straightforward	straightforward	ADJ
ejpam-245	117	5	in	in	ADP
ejpam-245	117	6	light	light	NOUN
ejpam-245	117	7	of	of	ADP
ejpam-245	117	8	(	(	PUNCT
ejpam-245	117	9	2.4	2.4	NUM
ejpam-245	117	10	)	)	PUNCT
ejpam-245	117	11	by	by	ADP
ejpam-245	117	12	letting	let	VERB
ejpam-245	117	13	p	p	PRON
ejpam-245	117	14	r	r	NOUN
ejpam-245	117	15	=	=	PUNCT
ejpam-245	117	16	a	a	DET
ejpam-245	117	17	+	+	NUM
ejpam-245	117	18	1	1	NUM
ejpam-245	117	19	and	and	CCONJ
ejpam-245	117	20	q	q	NOUN
ejpam-245	117	21	=	=	X
ejpam-245	117	22	a	a	NOUN
ejpam-245	117	23	,	,	PUNCT
ejpam-245	117	24	and	and	CCONJ
ejpam-245	117	25	is	be	AUX
ejpam-245	117	26	omitted	omit	VERB
ejpam-245	117	27	.	.	PUNCT
ejpam-245	118	1	hence	hence	ADV
ejpam-245	118	2	,	,	PUNCT
ejpam-245	118	3	the	the	DET
ejpam-245	118	4	terms	term	NOUN
ejpam-245	118	5	of	of	ADP
ejpam-245	118	6	{	{	PUNCT
ejpam-245	118	7	vn(4	vn(4	PROPN
ejpam-245	118	8	,	,	PUNCT
ejpam-245	118	9	3	3	NUM
ejpam-245	118	10	)	)	PUNCT
ejpam-245	118	11	}	}	PUNCT
ejpam-245	118	12	are	be	AUX
ejpam-245	118	13	vn	vn	NOUN
ejpam-245	118	14	=	=	NOUN
ejpam-245	118	15	3n	3n	NOUN
ejpam-245	118	16	+	+	CCONJ
ejpam-245	118	17	1	1	X
ejpam-245	118	18	.	.	PUNCT
ejpam-245	119	1	so	so	ADV
ejpam-245	119	2	,	,	PUNCT
ejpam-245	119	3	vfn−1	vfn−1	PROPN
ejpam-245	119	4	2	2	NUM
ejpam-245	119	5	=	=	SYM
ejpam-245	119	6	3	3	NUM
ejpam-245	119	7	fn−1	fn−1	ADJ
ejpam-245	119	8	2	2	NUM
ejpam-245	119	9	.	.	PUNCT
ejpam-245	120	1	thus	thus	ADV
ejpam-245	120	2	,	,	PUNCT
ejpam-245	120	3	theorem	theorem	VERB
ejpam-245	120	4	4.2	4.2	NUM
ejpam-245	120	5	is	be	AUX
ejpam-245	120	6	revised	revise	VERB
ejpam-245	120	7	equivalently	equivalently	ADV
ejpam-245	120	8	by	by	ADP
ejpam-245	120	9	the	the	DET
ejpam-245	120	10	statement	statement	NOUN
ejpam-245	120	11	of	of	ADP
ejpam-245	120	12	theorem	theorem	ADJ
ejpam-245	120	13	5.2	5.2	NUM
ejpam-245	120	14	.	.	PUNCT
ejpam-245	121	1	its	its	PRON
ejpam-245	121	2	proof	proof	NOUN
ejpam-245	121	3	requires	require	VERB
ejpam-245	121	4	the	the	DET
ejpam-245	121	5	following	follow	VERB
ejpam-245	121	6	identity	identity	NOUN
ejpam-245	121	7	found	find	VERB
ejpam-245	121	8	in	in	ADP
ejpam-245	121	9	[	[	X
ejpam-245	121	10	3	3	NUM
ejpam-245	121	11	]	]	PUNCT
ejpam-245	121	12	.	.	PUNCT
ejpam-245	122	1	u2n	u2n	PUNCT
ejpam-245	123	1	=	=	PRON
ejpam-245	123	2	unvn	unvn	ADJ
ejpam-245	123	3	(	(	PUNCT
ejpam-245	123	4	5.1	5.1	NUM
ejpam-245	123	5	)	)	PUNCT
ejpam-245	123	6	theorem	theorem	VERB
ejpam-245	123	7	5.2	5.2	NUM
ejpam-245	123	8	.	.	PUNCT
ejpam-245	124	1	pepin	pepin	PROPN
ejpam-245	124	2	’s	’s	PART
ejpam-245	124	3	test	test	NOUN
ejpam-245	124	4	the	the	DET
ejpam-245	124	5	fermat	fermat	ADJ
ejpam-245	124	6	number	number	NOUN
ejpam-245	124	7	fn	fn	NOUN
ejpam-245	124	8	=	=	NOUN
ejpam-245	124	9	22n	22n	X
ejpam-245	124	10	+	+	X
ejpam-245	124	11	1	1	NUM
ejpam-245	124	12	where	where	SCONJ
ejpam-245	124	13	,	,	PUNCT
ejpam-245	124	14	n	n	PRON
ejpam-245	124	15	≥	≥	NOUN
ejpam-245	124	16	1	1	NUM
ejpam-245	124	17	is	be	AUX
ejpam-245	124	18	prime	prime	ADJ
ejpam-245	124	19	if	if	SCONJ
ejpam-245	125	1	and	and	CCONJ
ejpam-245	125	2	only	only	ADV
ejpam-245	125	3	if	if	SCONJ
ejpam-245	125	4	fn	fn	PROPN
ejpam-245	125	5	|	|	ADV
ejpam-245	125	6	vfn−1	vfn−1	PROPN
ejpam-245	125	7	2	2	NUM
ejpam-245	125	8	(	(	PUNCT
ejpam-245	125	9	4	4	NUM
ejpam-245	125	10	,	,	PUNCT
ejpam-245	125	11	3	3	NUM
ejpam-245	125	12	)	)	PUNCT
ejpam-245	125	13	.	.	PUNCT
ejpam-245	126	1	proof	proof	NOUN
ejpam-245	126	2	.	.	PUNCT
ejpam-245	127	1	consider	consider	VERB
ejpam-245	127	2	{	{	PUNCT
ejpam-245	127	3	vn(4	vn(4	PROPN
ejpam-245	127	4	,	,	PUNCT
ejpam-245	127	5	3	3	NUM
ejpam-245	127	6	)	)	PUNCT
ejpam-245	127	7	}	}	PUNCT
ejpam-245	127	8	.	.	PUNCT
ejpam-245	128	1	then	then	ADV
ejpam-245	128	2	,	,	PUNCT
ejpam-245	128	3	∆	∆	PROPN
ejpam-245	128	4	=	=	SYM
ejpam-245	128	5	r−	r−	PROPN
ejpam-245	128	6	4q	4q	NOUN
ejpam-245	128	7	=	=	SYM
ejpam-245	128	8	16−	16−	NUM
ejpam-245	128	9	12	12	NUM
ejpam-245	128	10	=	=	SYM
ejpam-245	128	11	4	4	X
ejpam-245	128	12	.	.	X
ejpam-245	128	13	letting	let	VERB
ejpam-245	128	14	fn	fn	NOUN
ejpam-245	128	15	=	=	NOUN
ejpam-245	128	16	22n	22n	X
ejpam-245	128	17	+	+	SYM
ejpam-245	128	18	1	1	NUM
ejpam-245	128	19	be	be	NOUN
ejpam-245	128	20	prime	prime	ADJ
ejpam-245	128	21	,	,	PUNCT
ejpam-245	128	22	it	it	PRON
ejpam-245	128	23	follows	follow	VERB
ejpam-245	128	24	that	that	PRON
ejpam-245	128	25	ε	ε	PROPN
ejpam-245	128	26	=	=	SYM
ejpam-245	128	27	�	�	PROPN
ejpam-245	128	28	∆	∆	PROPN
ejpam-245	128	29	fn	fn	PROPN
ejpam-245	128	30	�	�	PROPN
ejpam-245	128	31	=	=	SYM
ejpam-245	128	32	�	�	PROPN
ejpam-245	128	33	4	4	NUM
ejpam-245	128	34	fn	fn	NOUN
ejpam-245	128	35	�	�	PROPN
ejpam-245	128	36	=	=	SYM
ejpam-245	128	37	�	�	PROPN
ejpam-245	128	38	2	2	NUM
ejpam-245	128	39	fn	fn	PROPN
ejpam-245	128	40	�	�	PROPN
ejpam-245	128	41	�	�	PROPN
ejpam-245	128	42	2	2	NUM
ejpam-245	128	43	fn	fn	NOUN
ejpam-245	128	44	�	�	PROPN
ejpam-245	128	45	=	=	SYM
ejpam-245	128	46	1	1	NUM
ejpam-245	128	47	and	and	CCONJ
ejpam-245	128	48	σ	σ	PROPN
ejpam-245	128	49	=	=	SYM
ejpam-245	128	50	�	�	PROPN
ejpam-245	128	51	r	r	NOUN
ejpam-245	128	52	fn	fn	PROPN
ejpam-245	128	53	�	�	PROPN
ejpam-245	128	54	=	=	SYM
ejpam-245	128	55	�	�	PROPN
ejpam-245	128	56	16	16	NUM
ejpam-245	128	57	fn	fn	PROPN
ejpam-245	128	58	�	�	PROPN
ejpam-245	128	59	=	=	SYM
ejpam-245	128	60	�	�	PROPN
ejpam-245	128	61	4	4	NUM
ejpam-245	128	62	fn	fn	PROPN
ejpam-245	128	63	�	�	PROPN
ejpam-245	128	64	�	�	PROPN
ejpam-245	128	65	4	4	NUM
ejpam-245	128	66	fn	fn	NOUN
ejpam-245	128	67	�	�	PROPN
ejpam-245	128	68	=	=	SYM
ejpam-245	128	69	1	1	X
ejpam-245	128	70	.	.	PUNCT
ejpam-245	129	1	furthermore	furthermore	ADV
ejpam-245	129	2	,	,	PUNCT
ejpam-245	129	3	since	since	SCONJ
ejpam-245	129	4	n	n	PROPN
ejpam-245	129	5	>	>	X
ejpam-245	129	6	1	1	NUM
ejpam-245	129	7	,	,	PUNCT
ejpam-245	129	8	by	by	ADP
ejpam-245	129	9	gauss	gauss	PROPN
ejpam-245	129	10	’s	’s	PART
ejpam-245	129	11	reciprocity	reciprocity	NOUN
ejpam-245	129	12	law	law	NOUN
ejpam-245	129	13	,	,	PUNCT
ejpam-245	129	14	�	�	PROPN
ejpam-245	129	15	3	3	NUM
ejpam-245	129	16	fn	fn	PROPN
ejpam-245	129	17	�	�	PROPN
ejpam-245	129	18	�	�	PROPN
ejpam-245	129	19	fn	fn	NOUN
ejpam-245	129	20	3	3	NUM
ejpam-245	129	21	�	�	PROPN
ejpam-245	129	22	=	=	SYM
ejpam-245	129	23	�	�	PROPN
ejpam-245	129	24	3	3	NUM
ejpam-245	129	25	22n	22n	NUM
ejpam-245	129	26	+1	+1	PROPN
ejpam-245	129	27	�	�	PROPN
ejpam-245	129	28	�	�	PROPN
ejpam-245	129	29	22n	22n	NUM
ejpam-245	129	30	+1	+1	PROPN
ejpam-245	129	31	3	3	NUM
ejpam-245	129	32	�	�	NOUN
ejpam-245	129	33	=	=	SYM
ejpam-245	129	34	(	(	PUNCT
ejpam-245	129	35	−1	−1	NOUN
ejpam-245	129	36	)	)	PUNCT
ejpam-245	129	37	3−1	3−1	NUM
ejpam-245	129	38	2	2	NUM
ejpam-245	129	39	·	·	PUNCT
ejpam-245	129	40	22n	22n	NOUN
ejpam-245	129	41	+1−1	+1−1	ADV
ejpam-245	129	42	2	2	NUM
ejpam-245	129	43	=	=	SYM
ejpam-245	129	44	(	(	PUNCT
ejpam-245	129	45	−1)2	−1)2	X
ejpam-245	129	46	n−1	n−1	NOUN
ejpam-245	129	47	=	=	NOUN
ejpam-245	129	48	1	1	X
ejpam-245	129	49	.	.	PUNCT
ejpam-245	130	1	hence	hence	ADV
ejpam-245	130	2	,	,	PUNCT
ejpam-245	130	3	�	�	PROPN
ejpam-245	130	4	3	3	NUM
ejpam-245	130	5	fn	fn	NOUN
ejpam-245	130	6	�	�	PROPN
ejpam-245	130	7	=	=	SYM
ejpam-245	130	8	�	�	PROPN
ejpam-245	130	9	fn	fn	PROPN
ejpam-245	130	10	3	3	NUM
ejpam-245	130	11	�	�	PROPN
ejpam-245	130	12	.	.	PUNCT
ejpam-245	131	1	thus	thus	ADV
ejpam-245	131	2	,	,	PUNCT
ejpam-245	131	3	τ	τ	PROPN
ejpam-245	131	4	=	=	SYM
ejpam-245	131	5	�	�	PROPN
ejpam-245	131	6	q	q	PROPN
ejpam-245	131	7	fn	fn	PROPN
ejpam-245	131	8	�	�	PROPN
ejpam-245	131	9	=	=	SYM
ejpam-245	131	10	�	�	PROPN
ejpam-245	131	11	3	3	NUM
ejpam-245	131	12	fn	fn	NOUN
ejpam-245	131	13	�	�	PROPN
ejpam-245	131	14	=	=	SYM
ejpam-245	131	15	�	�	PROPN
ejpam-245	131	16	fn	fn	NOUN
ejpam-245	131	17	3	3	NUM
ejpam-245	131	18	�	�	PROPN
ejpam-245	131	19	≡	≡	PROPN
ejpam-245	131	20	�	�	PROPN
ejpam-245	131	21	22n	22n	X
ejpam-245	131	22	+	+	CCONJ
ejpam-245	131	23	1	1	NUM
ejpam-245	131	24	�	�	PROPN
ejpam-245	131	25	3−1	3−1	NUM
ejpam-245	131	26	2	2	NUM
ejpam-245	131	27	≡	≡	PROPN
ejpam-245	131	28	−1	−1	NOUN
ejpam-245	131	29	(	(	PUNCT
ejpam-245	131	30	mod	mod	PROPN
ejpam-245	131	31	3	3	NUM
ejpam-245	131	32	)	)	PUNCT
ejpam-245	131	33	.	.	PUNCT
ejpam-245	132	1	since	since	SCONJ
ejpam-245	132	2	σε	σε	PROPN
ejpam-245	132	3	=	=	NOUN
ejpam-245	132	4	1	1	NUM
ejpam-245	132	5	,	,	PUNCT
ejpam-245	132	6	then	then	ADV
ejpam-245	132	7	by	by	ADP
ejpam-245	132	8	lemma	lemma	PROPN
ejpam-245	132	9	3.2	3.2	NUM
ejpam-245	132	10	,	,	PUNCT
ejpam-245	132	11	fn	fn	NOUN
ejpam-245	132	12	|	|	ADV
ejpam-245	132	13	u22n	u22n	NOUN
ejpam-245	132	14	.	.	PUNCT
ejpam-245	133	1	because	because	SCONJ
ejpam-245	133	2	τ	τ	PROPN
ejpam-245	133	3	6=	6=	PROPN
ejpam-245	133	4	σ	σ	PROPN
ejpam-245	133	5	,	,	PUNCT
ejpam-245	133	6	it	it	PRON
ejpam-245	133	7	follows	follow	VERB
ejpam-245	133	8	by	by	ADP
ejpam-245	133	9	lemma	lemma	PROPN
ejpam-245	133	10	3.5	3.5	NUM
ejpam-245	133	11	that	that	PRON
ejpam-245	133	12	fn	fn	PROPN
ejpam-245	133	13	∤	∤	PROPN
ejpam-245	133	14	u22n−1	u22n−1	PUNCT
ejpam-245	133	15	.	.	PUNCT
ejpam-245	134	1	thus	thus	ADV
ejpam-245	134	2	,	,	PUNCT
ejpam-245	134	3	the	the	DET
ejpam-245	134	4	rank	rank	NOUN
ejpam-245	134	5	of	of	ADP
ejpam-245	134	6	apparition	apparition	NOUN
ejpam-245	134	7	of	of	ADP
ejpam-245	134	8	fn	fn	PROPN
ejpam-245	134	9	in	in	ADP
ejpam-245	134	10	{	{	PUNCT
ejpam-245	134	11	un(4	un(4	PROPN
ejpam-245	134	12	,	,	PUNCT
ejpam-245	134	13	3	3	NUM
ejpam-245	134	14	)	)	PUNCT
ejpam-245	134	15	}	}	PUNCT
ejpam-245	134	16	is	be	AUX
ejpam-245	134	17	22n	22n	NOUN
ejpam-245	134	18	;	;	PUNCT
ejpam-245	134	19	that	that	PRON
ejpam-245	134	20	is	is	ADV
ejpam-245	134	21	,	,	PUNCT
ejpam-245	134	22	fn−	fn−	PROPN
ejpam-245	134	23	1	1	NUM
ejpam-245	134	24	.	.	PUNCT
ejpam-245	135	1	(	(	PUNCT
ejpam-245	135	2	otherwise	otherwise	ADV
ejpam-245	135	3	,	,	PUNCT
ejpam-245	135	4	by	by	ADP
ejpam-245	135	5	lemma	lemma	PROPN
ejpam-245	135	6	3.3	3.3	NUM
ejpam-245	135	7	the	the	DET
ejpam-245	135	8	rank	rank	NOUN
ejpam-245	135	9	of	of	ADP
ejpam-245	135	10	apparition	apparition	NOUN
ejpam-245	135	11	is	be	AUX
ejpam-245	135	12	22n−r	22n−r	PROPN
ejpam-245	135	13	,	,	PUNCT
ejpam-245	135	14	for	for	ADP
ejpam-245	135	15	some	some	DET
ejpam-245	135	16	positive	positive	ADJ
ejpam-245	135	17	integer	integer	NOUN
ejpam-245	135	18	r.	r.	PROPN
ejpam-245	135	19	also	also	ADV
ejpam-245	135	20	by	by	ADP
ejpam-245	135	21	lemma	lemma	PROPN
ejpam-245	135	22	3.3	3.3	NUM
ejpam-245	135	23	,	,	PUNCT
ejpam-245	135	24	fn	fn	NOUN
ejpam-245	135	25	|	|	ADV
ejpam-245	135	26	u2n−1	u2n−1	PROPN
ejpam-245	135	27	.	.	PUNCT
ejpam-245	135	28	)	)	PUNCT
ejpam-245	136	1	therefore	therefore	ADV
ejpam-245	136	2	,	,	PUNCT
ejpam-245	136	3	by	by	ADP
ejpam-245	136	4	lemma	lemma	PROPN
ejpam-245	136	5	3.4	3.4	NUM
ejpam-245	136	6	,	,	PUNCT
ejpam-245	136	7	fn	fn	NOUN
ejpam-245	136	8	|	|	ADV
ejpam-245	136	9	vfn−1	vfn−1	PROPN
ejpam-245	136	10	2	2	NUM
ejpam-245	136	11	.	.	PUNCT
ejpam-245	137	1	conversely	conversely	ADV
ejpam-245	137	2	,	,	PUNCT
ejpam-245	137	3	let	let	VERB
ejpam-245	137	4	’s	’s	NOUN
ejpam-245	137	5	suppose	suppose	VERB
ejpam-245	137	6	fn	fn	VERB
ejpam-245	137	7	|	|	ADV
ejpam-245	137	8	vfn−1	vfn−1	PROPN
ejpam-245	137	9	2	2	NUM
ejpam-245	137	10	.	.	PUNCT
ejpam-245	138	1	then	then	ADV
ejpam-245	138	2	by	by	ADP
ejpam-245	138	3	(	(	PUNCT
ejpam-245	138	4	5.1	5.1	NUM
ejpam-245	138	5	)	)	PUNCT
ejpam-245	138	6	,	,	PUNCT
ejpam-245	138	7	fn	fn	VERB
ejpam-245	138	8	|	|	ADV
ejpam-245	138	9	ufn−1	ufn−1	PROPN
ejpam-245	138	10	.	.	PUNCT
ejpam-245	139	1	specifically	specifically	ADV
ejpam-245	139	2	,	,	PUNCT
ejpam-245	139	3	fn	fn	ADV
ejpam-245	139	4	|	|	ADV
ejpam-245	139	5	u22n	u22n	NOUN
ejpam-245	139	6	.	.	PUNCT
ejpam-245	140	1	by	by	ADP
ejpam-245	140	2	lemma	lemma	PROPN
ejpam-245	140	3	3.3	3.3	NUM
ejpam-245	140	4	,	,	PUNCT
ejpam-245	140	5	ω(fn	ω(fn	NUM
ejpam-245	140	6	)	)	PUNCT
ejpam-245	140	7	must	must	AUX
ejpam-245	140	8	be	be	AUX
ejpam-245	140	9	a	a	DET
ejpam-245	140	10	divisor	divisor	NOUN
ejpam-245	140	11	of	of	ADP
ejpam-245	140	12	22n	22n	NOUN
ejpam-245	140	13	.	.	PUNCT
ejpam-245	141	1	but	but	CCONJ
ejpam-245	141	2	by	by	ADP
ejpam-245	141	3	lemma	lemma	PROPN
ejpam-245	141	4	3.1	3.1	NUM
ejpam-245	141	5	,	,	PUNCT
ejpam-245	141	6	u22n	u22n	PRON
ejpam-245	141	7	is	be	AUX
ejpam-245	141	8	relatively	relatively	ADV
ejpam-245	141	9	prime	prime	ADJ
ejpam-245	141	10	to	to	ADP
ejpam-245	141	11	u22n−1	u22n−1	PUNCT
ejpam-245	141	12	.	.	PUNCT
ejpam-245	142	1	thus	thus	ADV
ejpam-245	142	2	,	,	PUNCT
ejpam-245	142	3	ω(fn	ω(fn	NUM
ejpam-245	142	4	)	)	PUNCT
ejpam-245	142	5	=	=	SYM
ejpam-245	142	6	22n	22n	NOUN
ejpam-245	142	7	=	=	SYM
ejpam-245	142	8	fn−	fn−	NUM
ejpam-245	142	9	1	1	X
ejpam-245	142	10	.	.	PUNCT
ejpam-245	142	11	therefore	therefore	ADV
ejpam-245	142	12	,	,	PUNCT
ejpam-245	142	13	by	by	ADP
ejpam-245	142	14	lemma	lemma	PROPN
ejpam-245	142	15	3.6	3.6	NUM
ejpam-245	142	16	,	,	PUNCT
ejpam-245	142	17	fn	fn	NOUN
ejpam-245	142	18	is	be	AUX
ejpam-245	142	19	prime	prime	ADJ
ejpam-245	142	20	.	.	PUNCT
ejpam-245	143	1	next	next	ADV
ejpam-245	143	2	,	,	PUNCT
ejpam-245	143	3	we	we	PRON
ejpam-245	143	4	show	show	VERB
ejpam-245	143	5	that	that	SCONJ
ejpam-245	143	6	the	the	DET
ejpam-245	143	7	sequence	sequence	NOUN
ejpam-245	143	8	of	of	ADP
ejpam-245	143	9	numbers	number	NOUN
ejpam-245	143	10	4	4	NUM
ejpam-245	143	11	,	,	PUNCT
ejpam-245	143	12	14	14	NUM
ejpam-245	143	13	,	,	PUNCT
ejpam-245	143	14	194	194	NUM
ejpam-245	143	15	,	,	PUNCT
ejpam-245	143	16	37634	37634	NUM
ejpam-245	143	17	,	,	PUNCT
ejpam-245	143	18	1416317954	1416317954	NUM
ejpam-245	143	19	,	,	PUNCT
ejpam-245	143	20	.	.	PUNCT
ejpam-245	143	21	.	.	PUNCT
ejpam-245	143	22	.	.	PUNCT
ejpam-245	144	1	given	give	VERB
ejpam-245	144	2	in	in	ADP
ejpam-245	144	3	theorem	theorem	ADJ
ejpam-245	144	4	4.3	4.3	NUM
ejpam-245	144	5	are	be	AUX
ejpam-245	144	6	the	the	DET
ejpam-245	144	7	terms	term	NOUN
ejpam-245	144	8	of	of	ADP
ejpam-245	144	9	the	the	DET
ejpam-245	144	10	companion	companion	NOUN
ejpam-245	144	11	lehmer	lehmer	NOUN
ejpam-245	144	12	sequence	sequence	NOUN
ejpam-245	144	13	{	{	PUNCT
ejpam-245	144	14	vn	vn	PROPN
ejpam-245	144	15	(	(	PUNCT
ejpam-245	144	16	p	p	PROPN
ejpam-245	144	17	2,−1	2,−1	NUM
ejpam-245	144	18	)	)	PUNCT
ejpam-245	144	19	}	}	PUNCT
ejpam-245	144	20	john	john	PROPN
ejpam-245	144	21	h.	h.	PROPN
ejpam-245	144	22	jaroma	jaroma	PROPN
ejpam-245	144	23	/	/	SYM
ejpam-245	144	24	eur	eur	PROPN
ejpam-245	144	25	.	.	PUNCT
ejpam-245	145	1	j.	j.	PROPN
ejpam-245	145	2	pure	pure	PROPN
ejpam-245	145	3	appl	appl	PROPN
ejpam-245	145	4	.	.	PROPN
ejpam-245	145	5	math	math	PROPN
ejpam-245	145	6	,	,	PUNCT
ejpam-245	145	7	2	2	NUM
ejpam-245	145	8	(	(	PUNCT
ejpam-245	145	9	2009	2009	NUM
ejpam-245	145	10	)	)	PUNCT
ejpam-245	145	11	,	,	PUNCT
ejpam-245	145	12	(	(	PUNCT
ejpam-245	145	13	352	352	NUM
ejpam-245	145	14	-	-	SYM
ejpam-245	145	15	360	360	NUM
ejpam-245	145	16	)	)	PUNCT
ejpam-245	145	17	359	359	NUM
ejpam-245	145	18	whose	whose	DET
ejpam-245	145	19	indices	index	NOUN
ejpam-245	145	20	are	be	AUX
ejpam-245	145	21	powers	power	NOUN
ejpam-245	145	22	of	of	ADP
ejpam-245	145	23	2	2	NUM
ejpam-245	145	24	.	.	PUNCT
ejpam-245	146	1	but	but	CCONJ
ejpam-245	146	2	first	first	ADV
ejpam-245	146	3	,	,	PUNCT
ejpam-245	146	4	the	the	DET
ejpam-245	146	5	following	follow	VERB
ejpam-245	146	6	identity	identity	NOUN
ejpam-245	146	7	found	find	VERB
ejpam-245	146	8	in	in	ADP
ejpam-245	146	9	[	[	X
ejpam-245	146	10	3	3	X
ejpam-245	146	11	]	]	PUNCT
ejpam-245	146	12	is	be	AUX
ejpam-245	146	13	needed	need	VERB
ejpam-245	146	14	.	.	PUNCT
ejpam-245	147	1	v2n	v2n	NOUN
ejpam-245	147	2	=	=	SYM
ejpam-245	147	3	v	v	ADP
ejpam-245	147	4	2	2	NUM
ejpam-245	147	5	n	n	CCONJ
ejpam-245	147	6	−	−	NOUN
ejpam-245	147	7	2qn	2qn	NOUN
ejpam-245	147	8	.	.	PUNCT
ejpam-245	148	1	(	(	PUNCT
ejpam-245	148	2	5.2	5.2	NUM
ejpam-245	148	3	)	)	PUNCT
ejpam-245	148	4	now	now	ADV
ejpam-245	148	5	,	,	PUNCT
ejpam-245	148	6	consider	consider	VERB
ejpam-245	148	7	{	{	PUNCT
ejpam-245	148	8	vn	vn	X
ejpam-245	148	9	(	(	PUNCT
ejpam-245	148	10	p	p	PROPN
ejpam-245	148	11	2,−1	2,−1	NUM
ejpam-245	148	12	)	)	PUNCT
ejpam-245	148	13	}	}	PUNCT
ejpam-245	148	14	.	.	PUNCT
ejpam-245	149	1	so	so	ADV
ejpam-245	149	2	,	,	PUNCT
ejpam-245	149	3	v2	v2	PROPN
ejpam-245	149	4	=	=	SYM
ejpam-245	149	5	4	4	X
ejpam-245	149	6	.	.	PUNCT
ejpam-245	150	1	furthermore	furthermore	ADV
ejpam-245	150	2	,	,	PUNCT
ejpam-245	150	3	from	from	ADP
ejpam-245	150	4	theorem	theorem	ADJ
ejpam-245	150	5	4.3	4.3	NUM
ejpam-245	150	6	sk	sk	NOUN
ejpam-245	150	7	=	=	PROPN
ejpam-245	150	8	s2	s2	NOUN
ejpam-245	150	9	k−1	k−1	PROPN
ejpam-245	150	10	−	−	PROPN
ejpam-245	150	11	2	2	NUM
ejpam-245	150	12	,	,	PUNCT
ejpam-245	150	13	where	where	SCONJ
ejpam-245	150	14	s1	s1	NOUN
ejpam-245	150	15	=	=	SYM
ejpam-245	150	16	4	4	NUM
ejpam-245	150	17	=	=	SYM
ejpam-245	150	18	v2	v2	PROPN
ejpam-245	150	19	.	.	PUNCT
ejpam-245	151	1	since	since	SCONJ
ejpam-245	151	2	q	q	NOUN
ejpam-245	151	3	=	=	SYM
ejpam-245	151	4	−1	−1	NOUN
ejpam-245	151	5	,	,	PUNCT
ejpam-245	151	6	by	by	ADP
ejpam-245	151	7	(	(	PUNCT
ejpam-245	151	8	5.2	5.2	NUM
ejpam-245	151	9	)	)	PUNCT
ejpam-245	151	10	it	it	PRON
ejpam-245	151	11	follows	follow	VERB
ejpam-245	151	12	that	that	SCONJ
ejpam-245	151	13	sk	sk	VERB
ejpam-245	151	14	=	=	PUNCT
ejpam-245	151	15	v2k	v2k	NOUN
ejpam-245	151	16	for	for	ADP
ejpam-245	151	17	k	k	PROPN
ejpam-245	151	18	∈	∈	PROPN
ejpam-245	151	19	{	{	PUNCT
ejpam-245	151	20	1	1	NUM
ejpam-245	151	21	,	,	PUNCT
ejpam-245	151	22	2	2	NUM
ejpam-245	151	23	,	,	PUNCT
ejpam-245	151	24	.	.	PUNCT
ejpam-245	151	25	.	.	PUNCT
ejpam-245	152	1	.	.	PUNCT
ejpam-245	152	2	}	}	PUNCT
ejpam-245	152	3	.	.	PUNCT
ejpam-245	153	1	hence	hence	ADV
ejpam-245	153	2	,	,	PUNCT
ejpam-245	153	3	the	the	DET
ejpam-245	153	4	statement	statement	NOUN
ejpam-245	153	5	given	give	VERB
ejpam-245	153	6	in	in	ADP
ejpam-245	153	7	the	the	DET
ejpam-245	153	8	lucas	lucas	NOUN
ejpam-245	153	9	-	-	PUNCT
ejpam-245	153	10	lehmer	lehmer	NOUN
ejpam-245	153	11	test	test	NOUN
ejpam-245	153	12	asserting	assert	VERB
ejpam-245	153	13	mn	mn	PROPN
ejpam-245	153	14	divides	divide	VERB
ejpam-245	153	15	the	the	DET
ejpam-245	153	16	(	(	PUNCT
ejpam-245	153	17	n−	n−	NOUN
ejpam-245	153	18	1)st	1)st	NUM
ejpam-245	153	19	term	term	NOUN
ejpam-245	153	20	of	of	ADP
ejpam-245	153	21	4	4	NUM
ejpam-245	153	22	,	,	PUNCT
ejpam-245	153	23	14	14	NUM
ejpam-245	153	24	,	,	PUNCT
ejpam-245	153	25	194	194	NUM
ejpam-245	153	26	,	,	PUNCT
ejpam-245	153	27	37634	37634	NUM
ejpam-245	153	28	,	,	PUNCT
ejpam-245	153	29	1416317954	1416317954	NUM
ejpam-245	153	30	,	,	PUNCT
ejpam-245	153	31	.	.	PUNCT
ejpam-245	153	32	.	.	PUNCT
ejpam-245	153	33	.	.	PUNCT
ejpam-245	154	1	then	then	ADV
ejpam-245	154	2	mn	mn	PROPN
ejpam-245	154	3	is	be	AUX
ejpam-245	154	4	a	a	DET
ejpam-245	154	5	factor	factor	NOUN
ejpam-245	154	6	of	of	ADP
ejpam-245	154	7	v2n−1	v2n−1	PROPN
ejpam-245	154	8	is	be	AUX
ejpam-245	154	9	equivalent	equivalent	ADJ
ejpam-245	154	10	to	to	ADP
ejpam-245	154	11	saying	say	VERB
ejpam-245	154	12	that	that	SCONJ
ejpam-245	154	13	mn	mn	PROPN
ejpam-245	154	14	|	|	PROPN
ejpam-245	154	15	vmn+1	vmn+1	PROPN
ejpam-245	154	16	2	2	NUM
ejpam-245	154	17	.	.	PUNCT
ejpam-245	155	1	thus	thus	ADV
ejpam-245	155	2	,	,	PUNCT
ejpam-245	155	3	we	we	PRON
ejpam-245	155	4	now	now	ADV
ejpam-245	155	5	state	state	VERB
ejpam-245	155	6	the	the	DET
ejpam-245	155	7	following	follow	VERB
ejpam-245	155	8	equivalent	equivalent	ADJ
ejpam-245	155	9	form	form	NOUN
ejpam-245	155	10	of	of	ADP
ejpam-245	155	11	theorem	theorem	ADJ
ejpam-245	155	12	4.3	4.3	NUM
ejpam-245	155	13	.	.	PUNCT
ejpam-245	155	14	theorem	theorem	VERB
ejpam-245	155	15	5.3	5.3	NUM
ejpam-245	155	16	.	.	PUNCT
ejpam-245	156	1	lucas	lucas	PROPN
ejpam-245	156	2	-	-	PUNCT
ejpam-245	156	3	lehmer	lehmer	NOUN
ejpam-245	156	4	test	test	NOUN
ejpam-245	156	5	the	the	DET
ejpam-245	156	6	number	number	NOUN
ejpam-245	156	7	mn	mn	NOUN
ejpam-245	156	8	=	=	SYM
ejpam-245	156	9	2n	2n	NUM
ejpam-245	157	1	−	−	NOUN
ejpam-245	157	2	1	1	NUM
ejpam-245	157	3	,	,	PUNCT
ejpam-245	157	4	where	where	SCONJ
ejpam-245	157	5	n	n	X
ejpam-245	157	6	>	>	X
ejpam-245	157	7	2	2	NUM
ejpam-245	157	8	is	be	AUX
ejpam-245	157	9	prime	prime	ADJ
ejpam-245	157	10	if	if	SCONJ
ejpam-245	157	11	and	and	CCONJ
ejpam-245	157	12	only	only	ADV
ejpam-245	157	13	if	if	SCONJ
ejpam-245	157	14	mn	mn	PROPN
ejpam-245	157	15	|	|	NOUN
ejpam-245	157	16	vmn+1	vmn+1	PROPN
ejpam-245	157	17	2	2	NUM
ejpam-245	157	18	(	(	PUNCT
ejpam-245	157	19	p	p	NOUN
ejpam-245	157	20	2,−1	2,−1	NUM
ejpam-245	157	21	)	)	PUNCT
ejpam-245	157	22	.	.	PUNCT
ejpam-245	158	1	proof	proof	NOUN
ejpam-245	158	2	.	.	PUNCT
ejpam-245	159	1	as	as	ADP
ejpam-245	159	2	r	r	NOUN
ejpam-245	159	3	=	=	SYM
ejpam-245	159	4	2	2	NUM
ejpam-245	159	5	,	,	PUNCT
ejpam-245	159	6	q	q	NOUN
ejpam-245	159	7	=	=	SYM
ejpam-245	159	8	−1	−1	NOUN
ejpam-245	159	9	,	,	PUNCT
ejpam-245	159	10	and	and	CCONJ
ejpam-245	159	11	∆	∆	X
ejpam-245	159	12	=	=	SYM
ejpam-245	159	13	6	6	NUM
ejpam-245	159	14	,	,	PUNCT
ejpam-245	159	15	it	it	PRON
ejpam-245	159	16	follows	follow	VERB
ejpam-245	159	17	that	that	PRON
ejpam-245	159	18	ε	ε	PROPN
ejpam-245	159	19	=	=	SYM
ejpam-245	159	20	−1	−1	PROPN
ejpam-245	159	21	,	,	PUNCT
ejpam-245	159	22	σ	σ	PROPN
ejpam-245	159	23	=	=	SYM
ejpam-245	159	24	1	1	NUM
ejpam-245	159	25	,	,	PUNCT
ejpam-245	159	26	and	and	CCONJ
ejpam-245	159	27	τ	τ	PROPN
ejpam-245	159	28	=	=	SYM
ejpam-245	159	29	−1	−1	NOUN
ejpam-245	159	30	.	.	PUNCT
ejpam-245	160	1	the	the	DET
ejpam-245	160	2	proof	proof	NOUN
ejpam-245	160	3	then	then	ADV
ejpam-245	160	4	follows	follow	VERB
ejpam-245	160	5	similarly	similarly	ADV
ejpam-245	160	6	to	to	ADP
ejpam-245	160	7	that	that	PRON
ejpam-245	160	8	presented	present	VERB
ejpam-245	160	9	for	for	ADP
ejpam-245	160	10	theorem	theorem	ADJ
ejpam-245	160	11	5.2	5.2	NUM
ejpam-245	160	12	.	.	PUNCT
ejpam-245	161	1	it	it	PRON
ejpam-245	161	2	may	may	AUX
ejpam-245	161	3	be	be	AUX
ejpam-245	161	4	found	find	VERB
ejpam-245	161	5	in	in	ADP
ejpam-245	161	6	[	[	X
ejpam-245	161	7	3	3	NUM
ejpam-245	161	8	]	]	PUNCT
ejpam-245	161	9	.	.	PUNCT
ejpam-245	162	1	references	reference	NOUN
ejpam-245	162	2	[	[	X
ejpam-245	162	3	1	1	X
ejpam-245	162	4	]	]	PUNCT
ejpam-245	162	5	d.	d.	PROPN
ejpam-245	162	6	m.	m.	PROPN
ejpam-245	162	7	burton	burton	PROPN
ejpam-245	162	8	,	,	PUNCT
ejpam-245	162	9	elementary	elementary	ADJ
ejpam-245	162	10	number	number	NOUN
ejpam-245	162	11	theory	theory	NOUN
ejpam-245	162	12	,	,	PUNCT
ejpam-245	162	13	6th	6th	ADJ
ejpam-245	162	14	ed	ed	NOUN
ejpam-245	162	15	.	.	PROPN
ejpam-245	162	16	,	,	PUNCT
ejpam-245	162	17	mcgraw	mcgraw	PROPN
ejpam-245	162	18	-	-	PUNCT
ejpam-245	162	19	hill	hill	PROPN
ejpam-245	162	20	,	,	PUNCT
ejpam-245	162	21	new	new	PROPN
ejpam-245	162	22	york	york	PROPN
ejpam-245	162	23	,	,	PUNCT
ejpam-245	162	24	2005	2005	NUM
ejpam-245	162	25	.	.	PUNCT
ejpam-245	163	1	[	[	X
ejpam-245	163	2	2	2	X
ejpam-245	163	3	]	]	PUNCT
ejpam-245	163	4	r.	r.	PROPN
ejpam-245	163	5	d.	d.	PROPN
ejpam-245	163	6	carmichael	carmichael	PROPN
ejpam-245	163	7	,	,	PUNCT
ejpam-245	163	8	on	on	ADP
ejpam-245	163	9	the	the	DET
ejpam-245	163	10	numerical	numerical	ADJ
ejpam-245	163	11	factors	factor	NOUN
ejpam-245	163	12	of	of	ADP
ejpam-245	163	13	the	the	DET
ejpam-245	163	14	arithmetic	arithmetic	ADJ
ejpam-245	163	15	forms	form	NOUN
ejpam-245	163	16	αn±βn	αn±βn	VERB
ejpam-245	163	17	,	,	PUNCT
ejpam-245	163	18	ann	ann	PROPN
ejpam-245	163	19	.	.	PROPN
ejpam-245	163	20	math	math	PROPN
ejpam-245	163	21	.	.	PUNCT
ejpam-245	164	1	2nd	2nd	ADJ
ejpam-245	164	2	ser	ser	PROPN
ejpam-245	164	3	.	.	PUNCT
ejpam-245	165	1	15	15	NUM
ejpam-245	165	2	:	:	PUNCT
ejpam-245	165	3	30–70	30–70	NUM
ejpam-245	165	4	(	(	PUNCT
ejpam-245	165	5	1913	1913	NUM
ejpam-245	165	6	)	)	PUNCT
ejpam-245	165	7	.	.	PUNCT
ejpam-245	166	1	[	[	X
ejpam-245	166	2	3	3	X
ejpam-245	166	3	]	]	PUNCT
ejpam-245	166	4	d.	d.	PROPN
ejpam-245	166	5	h.	h.	PROPN
ejpam-245	166	6	lehmer	lehmer	PROPN
ejpam-245	166	7	,	,	PUNCT
ejpam-245	166	8	an	an	DET
ejpam-245	166	9	extended	extended	ADJ
ejpam-245	166	10	theory	theory	NOUN
ejpam-245	166	11	of	of	ADP
ejpam-245	166	12	lucas	lucas	PROPN
ejpam-245	166	13	’	'	PUNCT
ejpam-245	166	14	functions	function	NOUN
ejpam-245	166	15	,	,	PUNCT
ejpam-245	166	16	ann	ann	PROPN
ejpam-245	166	17	.	.	PROPN
ejpam-245	166	18	math	math	PROPN
ejpam-245	166	19	.	.	PUNCT
ejpam-245	167	1	31	31	NUM
ejpam-245	167	2	:	:	PUNCT
ejpam-245	168	1	419–448	419–448	NUM
ejpam-245	168	2	(	(	PUNCT
ejpam-245	168	3	1930	1930	NUM
ejpam-245	168	4	)	)	PUNCT
ejpam-245	168	5	.	.	PUNCT
ejpam-245	169	1	[	[	X
ejpam-245	169	2	4	4	X
ejpam-245	169	3	]	]	PUNCT
ejpam-245	169	4	d.	d.	PROPN
ejpam-245	169	5	h.	h.	PROPN
ejpam-245	169	6	lemer	lemer	PROPN
ejpam-245	169	7	,	,	PUNCT
ejpam-245	169	8	on	on	ADP
ejpam-245	169	9	lucas	lucas	NOUN
ejpam-245	169	10	’	'	PUNCT
ejpam-245	169	11	test	test	NOUN
ejpam-245	169	12	for	for	ADP
ejpam-245	169	13	the	the	DET
ejpam-245	169	14	primality	primality	NOUN
ejpam-245	169	15	of	of	ADP
ejpam-245	169	16	mersenne	mersenne	PROPN
ejpam-245	169	17	’s	’s	PART
ejpam-245	169	18	numbers	number	NOUN
ejpam-245	169	19	,	,	PUNCT
ejpam-245	169	20	j.	j.	PROPN
ejpam-245	169	21	lon	lon	PROPN
ejpam-245	169	22	.	.	PROPN
ejpam-245	169	23	math	math	PROPN
ejpam-245	169	24	.	.	PUNCT
ejpam-245	170	1	soc	soc	PROPN
ejpam-245	170	2	.	.	PUNCT
ejpam-245	171	1	10	10	NUM
ejpam-245	171	2	:	:	PUNCT
ejpam-245	171	3	162–165	162–165	NUM
ejpam-245	171	4	(	(	PUNCT
ejpam-245	171	5	1935	1935	NUM
ejpam-245	171	6	)	)	PUNCT
ejpam-245	171	7	.	.	PUNCT
ejpam-245	172	1	[	[	X
ejpam-245	172	2	5	5	NUM
ejpam-245	172	3	]	]	PUNCT
ejpam-245	172	4	é	é	PROPN
ejpam-245	172	5	.	.	PUNCT
ejpam-245	172	6	lucas	lucas	PROPN
ejpam-245	172	7	,	,	PUNCT
ejpam-245	172	8	théorie	théorie	PROPN
ejpam-245	172	9	des	des	PROPN
ejpam-245	172	10	fonctions	fonctions	PROPN
ejpam-245	172	11	numériques	numérique	NOUN
ejpam-245	172	12	simplement	simplement	PROPN
ejpam-245	172	13	périodiques	périodiques	PROPN
ejpam-245	172	14	,	,	PUNCT
ejpam-245	172	15	amer	amer	PROPN
ejpam-245	172	16	.	.	PUNCT
ejpam-245	173	1	j.	j.	PROPN
ejpam-245	173	2	math	math	PROPN
ejpam-245	173	3	.	.	PUNCT
ejpam-245	174	1	1	1	NUM
ejpam-245	174	2	:	:	PUNCT
ejpam-245	174	3	184–240	184–240	NUM
ejpam-245	174	4	,	,	PUNCT
ejpam-245	174	5	289–321	289–321	NUM
ejpam-245	174	6	(	(	PUNCT
ejpam-245	174	7	1878	1878	NUM
ejpam-245	174	8	)	)	PUNCT
ejpam-245	174	9	.	.	PUNCT
ejpam-245	175	1	[	[	X
ejpam-245	175	2	6	6	NUM
ejpam-245	175	3	]	]	PUNCT
ejpam-245	175	4	é	é	PROPN
ejpam-245	175	5	.	.	PUNCT
ejpam-245	175	6	lucas	lucas	PROPN
ejpam-245	175	7	,	,	PUNCT
ejpam-245	175	8	question	question	NOUN
ejpam-245	175	9	453	453	NUM
ejpam-245	175	10	,	,	PUNCT
ejpam-245	175	11	nouv	nouv	PROPN
ejpam-245	175	12	.	.	PUNCT
ejpam-245	176	1	cor	cor	PROPN
ejpam-245	176	2	.	.	PROPN
ejpam-245	176	3	math	math	PROPN
ejpam-245	176	4	.	.	PUNCT
ejpam-245	177	1	5	5	NUM
ejpam-245	177	2	:	:	PUNCT
ejpam-245	177	3	p.137	p.137	NOUN
ejpam-245	177	4	(	(	PUNCT
ejpam-245	177	5	1879	1879	NUM
ejpam-245	177	6	)	)	PUNCT
ejpam-245	177	7	.	.	PUNCT
ejpam-245	178	1	john	john	PROPN
ejpam-245	178	2	h.	h.	PROPN
ejpam-245	178	3	jaroma	jaroma	PROPN
ejpam-245	178	4	/	/	SYM
ejpam-245	178	5	eur	eur	PROPN
ejpam-245	178	6	.	.	PUNCT
ejpam-245	179	1	j.	j.	PROPN
ejpam-245	179	2	pure	pure	PROPN
ejpam-245	179	3	appl	appl	PROPN
ejpam-245	179	4	.	.	PROPN
ejpam-245	179	5	math	math	PROPN
ejpam-245	179	6	,	,	PUNCT
ejpam-245	179	7	2	2	NUM
ejpam-245	179	8	(	(	PUNCT
ejpam-245	179	9	2009	2009	NUM
ejpam-245	179	10	)	)	PUNCT
ejpam-245	179	11	,	,	PUNCT
ejpam-245	179	12	(	(	PUNCT
ejpam-245	179	13	352	352	NUM
ejpam-245	179	14	-	-	SYM
ejpam-245	179	15	360	360	NUM
ejpam-245	179	16	)	)	PUNCT
ejpam-245	179	17	360	360	NUM
ejpam-245	180	1	[	[	X
ejpam-245	180	2	7	7	NUM
ejpam-245	180	3	]	]	PUNCT
ejpam-245	180	4	t.	t.	PROPN
ejpam-245	180	5	pepin	pepin	PROPN
ejpam-245	180	6	,	,	PUNCT
ejpam-245	180	7	sur	sur	PROPN
ejpam-245	180	8	la	la	PROPN
ejpam-245	180	9	formule	formule	PROPN
ejpam-245	180	10	22n	22n	X
ejpam-245	180	11	+	+	CCONJ
ejpam-245	180	12	1	1	NUM
ejpam-245	180	13	,	,	PUNCT
ejpam-245	180	14	comp	comp	NOUN
ejpam-245	180	15	.	.	PUNCT
ejpam-245	181	1	rend	rend	VERB
ejpam-245	181	2	.	.	PUNCT
ejpam-245	182	1	acad	acad	PROPN
ejpam-245	182	2	.	.	PUNCT
ejpam-245	183	1	sci	sci	PROPN
ejpam-245	183	2	.	.	PROPN
ejpam-245	183	3	85	85	NUM
ejpam-245	183	4	:	:	PUNCT
ejpam-245	183	5	329–331	329–331	NUM
ejpam-245	183	6	(	(	PUNCT
ejpam-245	183	7	1877	1877	NUM
ejpam-245	183	8	)	)	PUNCT
ejpam-245	183	9	.	.	PUNCT
ejpam-245	184	1	[	[	X
ejpam-245	184	2	8	8	X
ejpam-245	184	3	]	]	PUNCT
ejpam-245	184	4	f.	f.	PROPN
ejpam-245	184	5	proth	proth	PROPN
ejpam-245	184	6	,	,	PUNCT
ejpam-245	184	7	énoncés	énoncés	PROPN
ejpam-245	184	8	de	de	X
ejpam-245	184	9	divers	divers	PROPN
ejpam-245	184	10	théorèmes	théorèmes	PROPN
ejpam-245	184	11	sur	sur	PROPN
ejpam-245	184	12	les	les	PROPN
ejpam-245	184	13	nombres	nombre	NOUN
ejpam-245	184	14	,	,	PUNCT
ejpam-245	184	15	comp	comp	PROPN
ejpam-245	184	16	.	.	PUNCT
ejpam-245	185	1	rend	rend	VERB
ejpam-245	185	2	.	.	PUNCT
ejpam-245	186	1	acad	acad	PROPN
ejpam-245	186	2	.	.	PUNCT
ejpam-245	187	1	sci	sci	PROPN
ejpam-245	187	2	.	.	PROPN
ejpam-245	188	1	83	83	NUM
ejpam-245	188	2	:	:	SYM
ejpam-245	188	3	1288–1289	1288–1289	NUM
ejpam-245	188	4	(	(	PUNCT
ejpam-245	188	5	1876	1876	NUM
ejpam-245	188	6	)	)	PUNCT
ejpam-245	188	7	.	.	PUNCT
ejpam-245	189	1	[	[	X
ejpam-245	189	2	9	9	NUM
ejpam-245	189	3	]	]	PUNCT
ejpam-245	189	4	f.	f.	NOUN
ejpam-245	189	5	proth	proth	PROPN
ejpam-245	189	6	,	,	PUNCT
ejpam-245	189	7	mémoires	mémoire	NOUN
ejpam-245	189	8	présentés	présentés	NOUN
ejpam-245	189	9	,	,	PUNCT
ejpam-245	189	10	comp	comp	NOUN
ejpam-245	189	11	.	.	PUNCT
ejpam-245	190	1	rend	rend	VERB
ejpam-245	190	2	.	.	PUNCT
ejpam-245	191	1	acad	acad	PROPN
ejpam-245	191	2	.	.	PUNCT
ejpam-245	192	1	sci	sci	PROPN
ejpam-245	192	2	.	.	PROPN
ejpam-245	193	1	87	87	NUM
ejpam-245	193	2	:	:	PUNCT
ejpam-245	193	3	p.374	p.374	NOUN
ejpam-245	193	4	(	(	PUNCT
ejpam-245	193	5	1878	1878	NUM
ejpam-245	193	6	)	)	PUNCT
ejpam-245	193	7	.	.	PUNCT
ejpam-245	194	1	[	[	X
ejpam-245	194	2	10	10	NUM
ejpam-245	194	3	]	]	PUNCT
ejpam-245	194	4	p.	p.	NOUN
ejpam-245	194	5	ribenboim	ribenboim	NOUN
ejpam-245	194	6	,	,	PUNCT
ejpam-245	194	7	the	the	DET
ejpam-245	194	8	new	new	ADJ
ejpam-245	194	9	book	book	NOUN
ejpam-245	194	10	of	of	ADP
ejpam-245	194	11	prime	prime	ADJ
ejpam-245	194	12	number	number	NOUN
ejpam-245	194	13	records	record	NOUN
ejpam-245	194	14	,	,	PUNCT
ejpam-245	194	15	springer	springer	NOUN
ejpam-245	194	16	-	-	PUNCT
ejpam-245	194	17	verlag	verlag	PROPN
ejpam-245	194	18	,	,	PUNCT
ejpam-245	194	19	new	new	PROPN
ejpam-245	194	20	york	york	PROPN
ejpam-245	194	21	,	,	PUNCT
ejpam-245	194	22	1996	1996	NUM
ejpam-245	194	23	.	.	PUNCT
ejpam-245	195	1	[	[	X
ejpam-245	195	2	11	11	NUM
ejpam-245	195	3	]	]	PUNCT
ejpam-245	195	4	n.	n.	NOUN
ejpam-245	195	5	robbins	robbins	PROPN
ejpam-245	195	6	,	,	PUNCT
ejpam-245	195	7	beginning	begin	VERB
ejpam-245	195	8	number	number	NOUN
ejpam-245	195	9	theory	theory	NOUN
ejpam-245	195	10	,	,	PUNCT
ejpam-245	195	11	wm	wm	PROPN
ejpam-245	195	12	.	.	PROPN
ejpam-245	195	13	c.	c.	PROPN
ejpam-245	195	14	brown	brown	PROPN
ejpam-245	195	15	,	,	PUNCT
ejpam-245	195	16	dubuque	dubuque	PROPN
ejpam-245	195	17	,	,	PUNCT
ejpam-245	195	18	1993	1993	NUM
ejpam-245	195	19	.	.	PUNCT
ejpam-245	196	1	[	[	X
ejpam-245	196	2	12	12	NUM
ejpam-245	196	3	]	]	PUNCT
ejpam-245	196	4	k.	k.	PROPN
ejpam-245	196	5	h.	h.	PROPN
ejpam-245	196	6	rosen	rosen	PROPN
ejpam-245	196	7	,	,	PUNCT
ejpam-245	196	8	elementary	elementary	ADJ
ejpam-245	196	9	number	number	NOUN
ejpam-245	196	10	theory	theory	NOUN
ejpam-245	196	11	,	,	PUNCT
ejpam-245	196	12	4th	4th	ADJ
ejpam-245	196	13	ed	ed	NOUN
ejpam-245	196	14	.	.	PROPN
ejpam-245	196	15	,	,	PUNCT
ejpam-245	196	16	addison	addison	PROPN
ejpam-245	196	17	wesley	wesley	PROPN
ejpam-245	196	18	longman	longman	PROPN
ejpam-245	196	19	,	,	PUNCT
ejpam-245	196	20	reading	reading	NOUN
ejpam-245	196	21	,	,	PUNCT
ejpam-245	196	22	2000	2000	NUM
ejpam-245	196	23	.	.	PUNCT
ejpam-245	197	1	[	[	X
ejpam-245	197	2	13	13	NUM
ejpam-245	197	3	]	]	PUNCT
ejpam-245	197	4	m.	m.	PROPN
ejpam-245	197	5	rosen	rosen	PROPN
ejpam-245	197	6	,	,	PUNCT
ejpam-245	197	7	a	a	DET
ejpam-245	197	8	proof	proof	NOUN
ejpam-245	197	9	of	of	ADP
ejpam-245	197	10	the	the	DET
ejpam-245	197	11	lucas	lucas	NOUN
ejpam-245	197	12	-	-	PUNCT
ejpam-245	197	13	lehmer	lehmer	NOUN
ejpam-245	197	14	test	test	NOUN
ejpam-245	197	15	,	,	PUNCT
ejpam-245	197	16	amer	amer	PROPN
ejpam-245	197	17	.	.	PROPN
ejpam-245	197	18	math	math	PROPN
ejpam-245	197	19	.	.	PUNCT
ejpam-245	198	1	mon	mon	PROPN
ejpam-245	198	2	.	.	PUNCT
ejpam-245	199	1	95	95	NUM
ejpam-245	199	2	:	:	PUNCT
ejpam-245	199	3	855–856	855–856	NUM
ejpam-245	199	4	(	(	PUNCT
ejpam-245	199	5	1988	1988	NUM
ejpam-245	199	6	)	)	PUNCT
ejpam-245	199	7	.	.	PUNCT
ejpam-245	200	1	[	[	X
ejpam-245	200	2	14	14	NUM
ejpam-245	200	3	]	]	PUNCT
ejpam-245	200	4	p.	p.	PROPN
ejpam-245	200	5	schumer	schumer	PROPN
ejpam-245	200	6	,	,	PUNCT
ejpam-245	200	7	introduction	introduction	NOUN
ejpam-245	200	8	to	to	ADP
ejpam-245	200	9	number	number	NOUN
ejpam-245	200	10	theory	theory	NOUN
ejpam-245	200	11	,	,	PUNCT
ejpam-245	200	12	pws	pws	NOUN
ejpam-245	200	13	,	,	PUNCT
ejpam-245	200	14	boston	boston	PROPN
ejpam-245	200	15	,	,	PUNCT
ejpam-245	200	16	1996	1996	NUM
ejpam-245	200	17	.	.	PUNCT
ejpam-245	201	1	[	[	X
ejpam-245	201	2	15	15	NUM
ejpam-245	201	3	]	]	PUNCT
ejpam-245	201	4	j.	j.	PROPN
ejpam-245	201	5	j.	j.	PROPN
ejpam-245	201	6	tattersall	tattersall	PROPN
ejpam-245	201	7	,	,	PUNCT
ejpam-245	201	8	elementary	elementary	ADJ
ejpam-245	201	9	number	number	NOUN
ejpam-245	201	10	theory	theory	NOUN
ejpam-245	201	11	in	in	ADP
ejpam-245	201	12	nine	nine	NUM
ejpam-245	201	13	chapters	chapter	NOUN
ejpam-245	201	14	,	,	PUNCT
ejpam-245	201	15	2nd	2nd	ADJ
ejpam-245	201	16	ed	ed	NOUN
ejpam-245	201	17	.	.	PROPN
ejpam-245	201	18	,	,	PUNCT
ejpam-245	201	19	cambridge	cambridge	PROPN
ejpam-245	201	20	univ	univ	PROPN
ejpam-245	201	21	.	.	PUNCT
ejpam-245	202	1	press	press	PROPN
ejpam-245	202	2	,	,	PUNCT
ejpam-245	202	3	cambridge	cambridge	PROPN
ejpam-245	202	4	,	,	PUNCT
ejpam-245	202	5	2005	2005	NUM
ejpam-245	202	6	.	.	PUNCT
ejpam-245	203	1	[	[	X
ejpam-245	203	2	16	16	NUM
ejpam-245	203	3	]	]	PUNCT
ejpam-245	203	4	h.	h.	PROPN
ejpam-245	203	5	c.	c.	PROPN
ejpam-245	203	6	williams	williams	PROPN
ejpam-245	203	7	,	,	PUNCT
ejpam-245	203	8	édouard	édouard	PROPN
ejpam-245	203	9	lucas	lucas	PROPN
ejpam-245	203	10	and	and	CCONJ
ejpam-245	203	11	primality	primality	PROPN
ejpam-245	203	12	testing	testing	NOUN
ejpam-245	203	13	,	,	PUNCT
ejpam-245	203	14	john	john	PROPN
ejpam-245	203	15	wiley	wiley	PROPN
ejpam-245	203	16	&	&	CCONJ
ejpam-245	203	17	sons	son	NOUN
ejpam-245	203	18	,	,	PUNCT
ejpam-245	203	19	new	new	PROPN
ejpam-245	203	20	york	york	PROPN
ejpam-245	203	21	,	,	PUNCT
ejpam-245	203	22	1998	1998	NUM
ejpam-245	203	23	.	.	PUNCT