id	sid	tid	token	lemma	pos
elements-9303	1	1	in	in	ADP
elements-9303	1	2	this	this	DET
elements-9303	1	3	paper	paper	NOUN
elements-9303	1	4	,	,	PUNCT
elements-9303	1	5	we	we	PRON
elements-9303	1	6	discuss	discuss	VERB
elements-9303	1	7	strings	string	NOUN
elements-9303	1	8	of	of	ADP
elements-9303	1	9	3	3	NUM
elements-9303	1	10	’s	’s	NOUN
elements-9303	1	11	and	and	CCONJ
elements-9303	1	12	7	7	NUM
elements-9303	1	13	’s	’s	NOUN
elements-9303	1	14	,	,	PUNCT
elements-9303	1	15	hereby	hereby	ADV
elements-9303	1	16	dubbed	dub	VERB
elements-9303	1	17	“	"	PUNCT
elements-9303	1	18	dreibens	dreiben	NOUN
elements-9303	1	19	.	.	PUNCT
elements-9303	1	20	”	"	PUNCT
elements-9303	2	1	as	as	ADP
elements-9303	2	2	a	a	DET
elements-9303	2	3	first	first	ADJ
elements-9303	2	4	step	step	NOUN
elements-9303	2	5	towards	towards	ADP
elements-9303	2	6	determining	determine	VERB
elements-9303	2	7	whether	whether	SCONJ
elements-9303	2	8	the	the	DET
elements-9303	2	9	set	set	NOUN
elements-9303	2	10	of	of	ADP
elements-9303	2	11	prime	prime	ADJ
elements-9303	2	12	dreibens	dreiben	NOUN
elements-9303	2	13	is	be	AUX
elements-9303	2	14	infinite	infinite	ADJ
elements-9303	2	15	,	,	PUNCT
elements-9303	2	16	we	we	PRON
elements-9303	2	17	examine	examine	VERB
elements-9303	2	18	the	the	DET
elements-9303	2	19	properties	property	NOUN
elements-9303	2	20	of	of	ADP
elements-9303	2	21	dreibens	dreiben	NOUN
elements-9303	2	22	when	when	SCONJ
elements-9303	2	23	divided	divide	VERB
elements-9303	2	24	by	by	ADP
elements-9303	2	25	7	7	NUM
elements-9303	2	26	.	.	PUNCT
elements-9303	2	27	by	by	ADP
elements-9303	2	28	determining	determine	VERB
elements-9303	2	29	the	the	DET
elements-9303	2	30	divisibility	divisibility	NOUN
elements-9303	2	31	of	of	ADP
elements-9303	2	32	a	a	DET
elements-9303	2	33	dreiben	dreiben	NOUN
elements-9303	2	34	by	by	ADP
elements-9303	2	35	7	7	NUM
elements-9303	2	36	,	,	PUNCT
elements-9303	2	37	we	we	PRON
elements-9303	2	38	can	can	AUX
elements-9303	2	39	rule	rule	VERB
elements-9303	2	40	out	out	ADP
elements-9303	2	41	some	some	DET
elements-9303	2	42	composite	composite	ADJ
elements-9303	2	43	dreibens	dreiben	NOUN
elements-9303	2	44	in	in	ADP
elements-9303	2	45	the	the	DET
elements-9303	2	46	search	search	NOUN
elements-9303	2	47	for	for	ADP
elements-9303	2	48	prime	prime	ADJ
elements-9303	2	49	dreibens	dreiben	NOUN
elements-9303	2	50	.	.	PUNCT
elements-9303	3	1	we	we	PRON
elements-9303	3	2	are	be	AUX
elements-9303	3	3	concerned	concerned	ADJ
elements-9303	3	4	with	with	ADP
elements-9303	3	5	the	the	DET
elements-9303	3	6	number	number	NOUN
elements-9303	3	7	of	of	ADP
elements-9303	3	8	dreibens	dreiben	NOUN
elements-9303	3	9	of	of	ADP
elements-9303	3	10	length	length	NOUN
elements-9303	3	11	n	n	PROPN
elements-9303	3	12	that	that	PRON
elements-9303	3	13	leave	leave	VERB
elements-9303	3	14	a	a	DET
elements-9303	3	15	remainder	remainder	NOUN
elements-9303	3	16	i	i	PRON
elements-9303	3	17	when	when	SCONJ
elements-9303	3	18	divided	divide	VERB
elements-9303	3	19	by	by	ADP
elements-9303	3	20	7	7	NUM
elements-9303	3	21	.	.	PUNCT
elements-9303	3	22	by	by	ADP
elements-9303	3	23	using	use	VERB
elements-9303	3	24	number	number	NOUN
elements-9303	3	25	theory	theory	NOUN
elements-9303	3	26	,	,	PUNCT
elements-9303	3	27	linear	linear	PROPN
elements-9303	3	28	algebra	algebra	NOUN
elements-9303	3	29	,	,	PUNCT
elements-9303	3	30	and	and	CCONJ
elements-9303	3	31	abstract	abstract	ADJ
elements-9303	3	32	algebra	algebra	NOUN
elements-9303	3	33	,	,	PUNCT
elements-9303	3	34	we	we	PRON
elements-9303	3	35	arrive	arrive	VERB
elements-9303	3	36	at	at	ADP
elements-9303	3	37	a	a	DET
elements-9303	3	38	formula	formula	NOUN
elements-9303	3	39	that	that	PRON
elements-9303	3	40	tells	tell	VERB
elements-9303	3	41	us	we	PRON
elements-9303	3	42	how	how	SCONJ
elements-9303	3	43	many	many	ADJ
elements-9303	3	44	dreibens	dreiben	NOUN
elements-9303	3	45	of	of	ADP
elements-9303	3	46	length	length	NOUN
elements-9303	3	47	n	n	NUM
elements-9303	3	48	are	be	AUX
elements-9303	3	49	divisible	divisible	ADJ
elements-9303	3	50	by	by	ADP
elements-9303	3	51	7	7	NUM
elements-9303	3	52	.	.	PUNCT
elements-9303	4	1	we	we	PRON
elements-9303	4	2	also	also	ADV
elements-9303	4	3	find	find	VERB
elements-9303	4	4	a	a	DET
elements-9303	4	5	way	way	NOUN
elements-9303	4	6	to	to	PART
elements-9303	4	7	determine	determine	VERB
elements-9303	4	8	the	the	DET
elements-9303	4	9	number	number	NOUN
elements-9303	4	10	of	of	ADP
elements-9303	4	11	dreibens	dreiben	NOUN
elements-9303	4	12	of	of	ADP
elements-9303	4	13	length	length	NOUN
elements-9303	4	14	n	n	PROPN
elements-9303	4	15	that	that	PRON
elements-9303	4	16	leave	leave	VERB
elements-9303	4	17	a	a	DET
elements-9303	4	18	remainder	remainder	NOUN
elements-9303	4	19	i	i	PRON
elements-9303	4	20	when	when	SCONJ
elements-9303	4	21	divided	divide	VERB
elements-9303	4	22	by	by	ADP
elements-9303	4	23	7	7	NUM
elements-9303	4	24	.	.	PUNCT
elements-9303	4	25	further	further	ADJ
elements-9303	4	26	investigation	investigation	NOUN
elements-9303	4	27	from	from	ADP
elements-9303	4	28	a	a	DET
elements-9303	4	29	combinatorial	combinatorial	ADJ
elements-9303	4	30	perspective	perspective	NOUN
elements-9303	4	31	provides	provide	VERB
elements-9303	4	32	additional	additional	ADJ
elements-9303	4	33	insight	insight	NOUN
elements-9303	4	34	into	into	ADP
elements-9303	4	35	the	the	DET
elements-9303	4	36	properties	property	NOUN
elements-9303	4	37	of	of	ADP
elements-9303	4	38	dreibens	dreiben	NOUN
elements-9303	4	39	when	when	SCONJ
elements-9303	4	40	divided	divide	VERB
elements-9303	4	41	by	by	ADP
elements-9303	4	42	7	7	NUM
elements-9303	4	43	.	.	PUNCT
elements-9303	4	44	overall	overall	ADJ
elements-9303	4	45	,	,	PUNCT
elements-9303	4	46	this	this	DET
elements-9303	4	47	paper	paper	NOUN
elements-9303	4	48	helps	help	VERB
elements-9303	4	49	characterize	characterize	VERB
elements-9303	4	50	dreibens	dreiben	NOUN
elements-9303	4	51	,	,	PUNCT
elements-9303	4	52	opens	open	VERB
elements-9303	4	53	up	up	ADP
elements-9303	4	54	more	more	ADJ
elements-9303	4	55	paths	path	NOUN
elements-9303	4	56	of	of	ADP
elements-9303	4	57	inquiry	inquiry	NOUN
elements-9303	4	58	into	into	ADP
elements-9303	4	59	the	the	DET
elements-9303	4	60	nature	nature	NOUN
elements-9303	4	61	of	of	ADP
elements-9303	4	62	dreibens	dreiben	NOUN
elements-9303	4	63	,	,	PUNCT
elements-9303	4	64	and	and	CCONJ
elements-9303	4	65	rules	rule	VERB
elements-9303	4	66	out	out	ADP
elements-9303	4	67	some	some	DET
elements-9303	4	68	composite	composite	ADJ
elements-9303	4	69	dreibens	dreiben	NOUN
elements-9303	4	70	from	from	ADP
elements-9303	4	71	a	a	DET
elements-9303	4	72	prime	prime	ADJ
elements-9303	4	73	dreiben	dreiben	NOUN
elements-9303	4	74	search	search	NOUN
elements-9303	4	75	.	.	PUNCT
elements-9303	5	1	dreibens	dreiben	NOUN
elements-9303	5	2	modulo	modulo	VERB
elements-9303	5	3	7	7	NUM
elements-9303	5	4	a	a	DET
elements-9303	5	5	new	new	ADJ
elements-9303	5	6	formula	formula	NOUN
elements-9303	5	7	for	for	ADP
elements-9303	5	8	primality	primality	NOUN
elements-9303	5	9	testing	testing	NOUN
elements-9303	5	10	arthur	arthur	PROPN
elements-9303	5	11	diep	diep	PROPN
elements-9303	5	12	-	-	PUNCT
elements-9303	5	13	nguyen	nguyen	PROPN
elements-9303	5	14	introduction	introduction	NOUN
elements-9303	5	15	to	to	ADP
elements-9303	5	16	dreibens	dreiben	NOUN
elements-9303	5	17	and	and	CCONJ
elements-9303	5	18	the	the	DET
elements-9303	5	19	search	search	NOUN
elements-9303	5	20	for	for	ADP
elements-9303	5	21	infinitely	infinitely	ADV
elements-9303	5	22	many	many	ADJ
elements-9303	5	23	prime	prime	ADJ
elements-9303	5	24	dreibens	dreiben	NOUN
elements-9303	5	25	the	the	DET
elements-9303	5	26	existence	existence	NOUN
elements-9303	5	27	of	of	ADP
elements-9303	5	28	infinitely	infinitely	ADV
elements-9303	5	29	many	many	ADJ
elements-9303	5	30	prime	prime	ADJ
elements-9303	5	31	numbers	number	NOUN
elements-9303	5	32	is	be	AUX
elements-9303	5	33	an	an	DET
elements-9303	5	34	elementary	elementary	ADJ
elements-9303	5	35	fact	fact	NOUN
elements-9303	5	36	of	of	ADP
elements-9303	5	37	number	number	NOUN
elements-9303	5	38	theory	theory	NOUN
elements-9303	5	39	.	.	PUNCT
elements-9303	6	1	however	however	ADV
elements-9303	6	2	,	,	PUNCT
elements-9303	6	3	mathematicians	mathematician	NOUN
elements-9303	6	4	continue	continue	VERB
elements-9303	6	5	to	to	PART
elements-9303	6	6	ponder	ponder	VERB
elements-9303	6	7	over	over	ADP
elements-9303	6	8	whether	whether	SCONJ
elements-9303	6	9	there	there	PRON
elements-9303	6	10	are	be	VERB
elements-9303	6	11	infinitely	infinitely	ADV
elements-9303	6	12	many	many	ADJ
elements-9303	6	13	primes	prime	NOUN
elements-9303	6	14	of	of	ADP
elements-9303	6	15	a	a	DET
elements-9303	6	16	certain	certain	ADJ
elements-9303	6	17	kind	kind	NOUN
elements-9303	6	18	.	.	PUNCT
elements-9303	7	1	for	for	ADP
elements-9303	7	2	instance	instance	NOUN
elements-9303	7	3	,	,	PUNCT
elements-9303	7	4	it	it	PRON
elements-9303	7	5	is	be	AUX
elements-9303	7	6	not	not	PART
elements-9303	7	7	yet	yet	ADV
elements-9303	7	8	known	know	VERB
elements-9303	7	9	whether	whether	SCONJ
elements-9303	7	10	or	or	CCONJ
elements-9303	7	11	not	not	PART
elements-9303	7	12	the	the	DET
elements-9303	7	13	set	set	NOUN
elements-9303	7	14	of	of	ADP
elements-9303	7	15	mersenne	mersenne	NOUN
elements-9303	7	16	primes	prime	NOUN
elements-9303	7	17	is	be	AUX
elements-9303	7	18	finite	finite	ADJ
elements-9303	7	19	or	or	CCONJ
elements-9303	7	20	infinite	infinite	NOUN
elements-9303	7	21	.	.	PUNCT
elements-9303	8	1	evidently	evidently	ADV
elements-9303	8	2	,	,	PUNCT
elements-9303	8	3	questions	question	NOUN
elements-9303	8	4	about	about	ADP
elements-9303	8	5	primes	prime	NOUN
elements-9303	8	6	remain	remain	VERB
elements-9303	8	7	at	at	ADP
elements-9303	8	8	the	the	DET
elements-9303	8	9	forefront	forefront	NOUN
elements-9303	8	10	of	of	ADP
elements-9303	8	11	number	number	NOUN
elements-9303	8	12	theory	theory	NOUN
elements-9303	8	13	.	.	PUNCT
elements-9303	9	1	our	our	PRON
elements-9303	9	2	current	current	ADJ
elements-9303	9	3	research	research	NOUN
elements-9303	9	4	focuses	focus	VERB
elements-9303	9	5	on	on	ADP
elements-9303	9	6	a	a	DET
elements-9303	9	7	type	type	NOUN
elements-9303	9	8	of	of	ADP
elements-9303	9	9	number	number	NOUN
elements-9303	9	10	that	that	PRON
elements-9303	9	11	we	we	PRON
elements-9303	9	12	dubbed	dub	VERB
elements-9303	9	13	dreibens	dreiben	NOUN
elements-9303	9	14	.	.	PUNCT
elements-9303	10	1	dreibens	dreiben	NOUN
elements-9303	10	2	are	be	AUX
elements-9303	10	3	merely	merely	ADV
elements-9303	10	4	strings	string	NOUN
elements-9303	10	5	of	of	ADP
elements-9303	10	6	3s	3s	NUM
elements-9303	10	7	and	and	CCONJ
elements-9303	10	8	7s	7	NOUN
elements-9303	10	9	(	(	PUNCT
elements-9303	10	10	e.g.	e.g.	ADV
elements-9303	10	11	333	333	NUM
elements-9303	10	12	,	,	PUNCT
elements-9303	10	13	777	777	NUM
elements-9303	10	14	,	,	PUNCT
elements-9303	10	15	3377373737	3377373737	NUM
elements-9303	10	16	,	,	PUNCT
elements-9303	10	17	etc	etc	X
elements-9303	10	18	.	.	X
elements-9303	10	19	)	)	PUNCT
elements-9303	10	20	,	,	PUNCT
elements-9303	10	21	yet	yet	CCONJ
elements-9303	10	22	these	these	DET
elements-9303	10	23	simple	simple	ADJ
elements-9303	10	24	strings	string	NOUN
elements-9303	10	25	provide	provide	VERB
elements-9303	10	26	a	a	DET
elements-9303	10	27	foundation	foundation	NOUN
elements-9303	10	28	for	for	ADP
elements-9303	10	29	a	a	DET
elements-9303	10	30	multitude	multitude	NOUN
elements-9303	10	31	of	of	ADP
elements-9303	10	32	questions	question	NOUN
elements-9303	10	33	.	.	PUNCT
elements-9303	11	1	our	our	PRON
elements-9303	11	2	ultimate	ultimate	ADJ
elements-9303	11	3	goal	goal	NOUN
elements-9303	11	4	is	be	AUX
elements-9303	11	5	to	to	PART
elements-9303	11	6	prove	prove	VERB
elements-9303	11	7	whether	whether	SCONJ
elements-9303	11	8	or	or	CCONJ
elements-9303	11	9	not	not	PART
elements-9303	11	10	infinitely	infinitely	ADV
elements-9303	11	11	many	many	ADJ
elements-9303	11	12	prime	prime	ADJ
elements-9303	11	13	dreibens	dreiben	NOUN
elements-9303	11	14	exist	exist	VERB
elements-9303	11	15	.	.	PUNCT
elements-9303	12	1	while	while	SCONJ
elements-9303	12	2	that	that	DET
elements-9303	12	3	question	question	NOUN
elements-9303	12	4	is	be	AUX
elements-9303	12	5	beyond	beyond	ADP
elements-9303	12	6	the	the	DET
elements-9303	12	7	scope	scope	NOUN
elements-9303	12	8	of	of	ADP
elements-9303	12	9	this	this	DET
elements-9303	12	10	paper	paper	NOUN
elements-9303	12	11	,	,	PUNCT
elements-9303	12	12	this	this	DET
elements-9303	12	13	paper	paper	NOUN
elements-9303	12	14	con	con	PROPN
elements-9303	12	15	tributes	tribute	NOUN
elements-9303	12	16	to	to	ADP
elements-9303	12	17	that	that	DET
elements-9303	12	18	goal	goal	NOUN
elements-9303	12	19	by	by	ADP
elements-9303	12	20	ruling	rule	VERB
elements-9303	12	21	out	out	ADP
elements-9303	12	22	some	some	DET
elements-9303	12	23	composite	composite	ADJ
elements-9303	12	24	dreibens	dreiben	NOUN
elements-9303	12	25	.	.	PUNCT
elements-9303	13	1	in	in	ADP
elements-9303	13	2	fact	fact	NOUN
elements-9303	13	3	,	,	PUNCT
elements-9303	13	4	this	this	DET
elements-9303	13	5	paper	paper	NOUN
elements-9303	13	6	concerns	concern	VERB
elements-9303	13	7	itself	itself	PRON
elements-9303	13	8	with	with	ADP
elements-9303	13	9	dreibens	dreiben	NOUN
elements-9303	13	10	divisible	divisible	ADJ
elements-9303	13	11	by	by	ADP
elements-9303	13	12	7	7	NUM
elements-9303	13	13	;	;	PUNCT
elements-9303	13	14	by	by	ADP
elements-9303	13	15	understanding	understand	VERB
elements-9303	13	16	dreibens	dreiben	NOUN
elements-9303	13	17	modulo	modulo	NOUN
elements-9303	13	18	7	7	NUM
elements-9303	13	19	,	,	PUNCT
elements-9303	13	20	we	we	PRON
elements-9303	13	21	can	can	AUX
elements-9303	13	22	rule	rule	VERB
elements-9303	13	23	out	out	ADP
elements-9303	13	24	some	some	DET
elements-9303	13	25	dreibens	dreiben	NOUN
elements-9303	13	26	that	that	PRON
elements-9303	13	27	are	be	AUX
elements-9303	13	28	certainly	certainly	ADV
elements-9303	13	29	not	not	PART
elements-9303	13	30	prime	prime	ADJ
elements-9303	13	31	.	.	PUNCT
elements-9303	14	1	this	this	DET
elements-9303	14	2	paper	paper	NOUN
elements-9303	14	3	uses	use	VERB
elements-9303	14	4	a	a	DET
elements-9303	14	5	variety	variety	NOUN
elements-9303	14	6	of	of	ADP
elements-9303	14	7	approaches	approach	NOUN
elements-9303	14	8	to	to	PART
elements-9303	14	9	learn	learn	VERB
elements-9303	14	10	about	about	ADP
elements-9303	14	11	dreibens	dreiben	NOUN
elements-9303	14	12	modulo	modulo	NOUN
elements-9303	14	13	7	7	NUM
elements-9303	14	14	,	,	PUNCT
elements-9303	14	15	including	include	VERB
elements-9303	14	16	techniques	technique	NOUN
elements-9303	14	17	from	from	ADP
elements-9303	14	18	number	number	NOUN
elements-9303	14	19	theory	theory	NOUN
elements-9303	14	20	,	,	PUNCT
elements-9303	14	21	abstract	abstract	ADJ
elements-9303	14	22	algebra	algebra	NOUN
elements-9303	14	23	,	,	PUNCT
elements-9303	14	24	linear	linear	PROPN
elements-9303	14	25	algebra	algebra	NOUN
elements-9303	14	26	,	,	PUNCT
elements-9303	14	27	and	and	CCONJ
elements-9303	14	28	combinatorics	combinatoric	NOUN
elements-9303	14	29	.	.	PUNCT
elements-9303	15	1	with	with	ADP
elements-9303	15	2	time	time	NOUN
elements-9303	15	3	,	,	PUNCT
elements-9303	15	4	we	we	PRON
elements-9303	15	5	hope	hope	VERB
elements-9303	15	6	dreibens	dreiben	NOUN
elements-9303	15	7	to	to	ADP
elements-9303	15	8	one	one	NUM
elements-9303	15	9	day	day	NOUN
elements-9303	15	10	become	become	VERB
elements-9303	15	11	a	a	DET
elements-9303	15	12	very	very	ADV
elements-9303	15	13	well	well	ADV
elements-9303	15	14	characterized	characterized	ADJ
elements-9303	15	15	type	type	NOUN
elements-9303	15	16	of	of	ADP
elements-9303	15	17	number	number	NOUN
elements-9303	15	18	.	.	PUNCT
elements-9303	16	1	learning	learn	VERB
elements-9303	16	2	about	about	ADP
elements-9303	16	3	dreibens	dreiben	NOUN
elements-9303	16	4	modulo	modulo	NOUN
elements-9303	16	5	7	7	NUM
elements-9303	16	6	by	by	ADP
elements-9303	16	7	exploring	explore	VERB
elements-9303	16	8	ai	ai	PROPN
elements-9303	16	9	7(n	7(n	NUM
elements-9303	16	10	)	)	PUNCT
elements-9303	17	1	we	we	PRON
elements-9303	17	2	would	would	AUX
elements-9303	17	3	best	well	ADV
elements-9303	17	4	approach	approach	VERB
elements-9303	17	5	dreibens	dreiben	NOUN
elements-9303	17	6	by	by	ADP
elements-9303	17	7	attempting	attempt	VERB
elements-9303	17	8	to	to	ADP
elements-9303	17	9	nd	nd	PRON
elements-9303	17	10	the	the	DET
elements-9303	17	11	number	number	NOUN
elements-9303	17	12	of	of	ADP
elements-9303	17	13	dreibens	dreiben	NOUN
elements-9303	17	14	divisible	divisible	ADJ
elements-9303	17	15	by	by	ADP
elements-9303	17	16	some	some	DET
elements-9303	17	17	number	number	NOUN
elements-9303	17	18	.	.	PUNCT
elements-9303	18	1	for	for	ADP
elements-9303	18	2	this	this	DET
elements-9303	18	3	paper	paper	NOUN
elements-9303	18	4	,	,	PUNCT
elements-9303	18	5	we	we	PRON
elements-9303	18	6	interest	interest	VERB
elements-9303	18	7	ourselves	ourselves	PRON
elements-9303	18	8	in	in	ADP
elements-9303	18	9	the	the	DET
elements-9303	18	10	number	number	NOUN
elements-9303	18	11	of	of	ADP
elements-9303	18	12	dreibens	dreiben	NOUN
elements-9303	18	13	divisible	divisible	ADJ
elements-9303	18	14	by	by	ADP
elements-9303	18	15	7	7	NUM
elements-9303	18	16	.	.	PUNCT
elements-9303	19	1	let	let	VERB
elements-9303	19	2	us	we	PRON
elements-9303	19	3	define	define	VERB
elements-9303	19	4	ai	ai	VERB
elements-9303	19	5	j(n	j(n	PROPN
elements-9303	19	6	)	)	PUNCT
elements-9303	19	7	as	as	ADP
elements-9303	19	8	the	the	DET
elements-9303	19	9	number	number	NOUN
elements-9303	19	10	1	1	NUM
elements-9303	19	11	of	of	ADP
elements-9303	19	12	dreibens	dreiben	NOUN
elements-9303	19	13	of	of	ADP
elements-9303	19	14	length	length	NOUN
elements-9303	19	15	n	n	PROPN
elements-9303	19	16	that	that	PRON
elements-9303	19	17	leave	leave	VERB
elements-9303	19	18	a	a	DET
elements-9303	19	19	remainder	remainder	NOUN
elements-9303	19	20	i	i	PRON
elements-9303	19	21	when	when	SCONJ
elements-9303	19	22	divided	divide	VERB
elements-9303	19	23	by	by	ADP
elements-9303	19	24	j.	j.	PROPN
elements-9303	19	25	in	in	ADP
elements-9303	19	26	this	this	DET
elements-9303	19	27	paper	paper	NOUN
elements-9303	19	28	,	,	PUNCT
elements-9303	19	29	we	we	PRON
elements-9303	19	30	will	will	AUX
elements-9303	19	31	concern	concern	VERB
elements-9303	19	32	ourselves	ourselves	PRON
elements-9303	19	33	with	with	ADP
elements-9303	19	34	ai	ai	PROPN
elements-9303	19	35	7(n	7(n	NUM
elements-9303	19	36	)	)	PUNCT
elements-9303	19	37	.	.	PUNCT
elements-9303	20	1	how	how	SCONJ
elements-9303	20	2	do	do	AUX
elements-9303	20	3	we	we	PRON
elements-9303	20	4	find	find	VERB
elements-9303	20	5	ai	ai	VERB
elements-9303	20	6	7(n	7(n	NUM
elements-9303	20	7	+	+	CCONJ
elements-9303	20	8	1	1	NUM
elements-9303	20	9	)	)	PUNCT
elements-9303	20	10	?	?	PUNCT
elements-9303	21	1	we	we	PRON
elements-9303	21	2	can	can	AUX
elements-9303	21	3	find	find	VERB
elements-9303	21	4	formulas	formula	NOUN
elements-9303	21	5	for	for	ADP
elements-9303	21	6	ai	ai	PROPN
elements-9303	21	7	7(n	7(n	NUM
elements-9303	21	8	+	+	CCONJ
elements-9303	21	9	1	1	X
elements-9303	21	10	)	)	PUNCT
elements-9303	21	11	by	by	ADP
elements-9303	21	12	considering	consider	VERB
elements-9303	21	13	the	the	DET
elements-9303	21	14	first	first	ADJ
elements-9303	21	15	n	n	NOUN
elements-9303	21	16	digits	digit	NOUN
elements-9303	21	17	of	of	ADP
elements-9303	21	18	a	a	DET
elements-9303	21	19	dreiben	dreiben	NOUN
elements-9303	21	20	of	of	ADP
elements-9303	21	21	length	length	NOUN
elements-9303	21	22	n	n	PROPN
elements-9303	21	23	+	+	NUM
elements-9303	21	24	1	1	NUM
elements-9303	21	25	,	,	PUNCT
elements-9303	21	26	then	then	ADV
elements-9303	21	27	seeing	see	VERB
elements-9303	21	28	what	what	PRON
elements-9303	21	29	the	the	DET
elements-9303	21	30	last	last	ADJ
elements-9303	21	31	digit	digit	NOUN
elements-9303	21	32	must	must	AUX
elements-9303	21	33	be	be	AUX
elements-9303	21	34	such	such	ADJ
elements-9303	21	35	that	that	SCONJ
elements-9303	21	36	the	the	DET
elements-9303	21	37	dreiben	dreiben	NOUN
elements-9303	21	38	leaves	leave	VERB
elements-9303	21	39	a	a	DET
elements-9303	21	40	remainder	remainder	NOUN
elements-9303	21	41	i	i	PRON
elements-9303	21	42	when	when	SCONJ
elements-9303	21	43	divided	divide	VERB
elements-9303	21	44	by	by	ADP
elements-9303	21	45	7	7	NUM
elements-9303	21	46	.	.	PUNCT
elements-9303	22	1	let	let	VERB
elements-9303	22	2	us	we	PRON
elements-9303	22	3	consider	consider	VERB
elements-9303	22	4	a	a	DET
elements-9303	22	5	dreiben	dreiben	NOUN
elements-9303	22	6	dn+1	dn+1	ADP
elements-9303	22	7	of	of	ADP
elements-9303	22	8	length	length	NOUN
elements-9303	22	9	n	n	NOUN
elements-9303	22	10	+	+	NUM
elements-9303	22	11	1	1	NUM
elements-9303	22	12	,	,	PUNCT
elements-9303	22	13	which	which	PRON
elements-9303	22	14	could	could	AUX
elements-9303	22	15	leave	leave	VERB
elements-9303	22	16	a	a	DET
elements-9303	22	17	remainder	remainder	NOUN
elements-9303	22	18	of	of	ADP
elements-9303	22	19	0	0	NUM
elements-9303	22	20	,	,	PUNCT
elements-9303	22	21	1	1	NUM
elements-9303	22	22	,	,	PUNCT
elements-9303	22	23	2	2	NUM
elements-9303	22	24	,	,	PUNCT
elements-9303	22	25	3	3	NUM
elements-9303	22	26	,	,	PUNCT
elements-9303	22	27	4	4	NUM
elements-9303	22	28	,	,	PUNCT
elements-9303	22	29	5	5	NUM
elements-9303	22	30	,	,	PUNCT
elements-9303	22	31	or	or	CCONJ
elements-9303	22	32	6	6	NUM
elements-9303	22	33	when	when	SCONJ
elements-9303	22	34	divided	divide	VERB
elements-9303	22	35	by	by	ADP
elements-9303	22	36	7	7	NUM
elements-9303	22	37	.	.	PUNCT
elements-9303	23	1	the	the	DET
elements-9303	23	2	rst	rst	PROPN
elements-9303	23	3	n	n	NOUN
elements-9303	23	4	digits	digit	NOUN
elements-9303	23	5	of	of	ADP
elements-9303	23	6	dn+1	dn+1	ADJ
elements-9303	23	7	form	form	NOUN
elements-9303	23	8	another	another	DET
elements-9303	23	9	dreiben	dreiben	NOUN
elements-9303	23	10	dn	dn	VERB
elements-9303	23	11	that	that	PRON
elements-9303	23	12	could	could	AUX
elements-9303	23	13	leave	leave	VERB
elements-9303	23	14	a	a	DET
elements-9303	23	15	remainder	remainder	NOUN
elements-9303	23	16	of	of	ADP
elements-9303	23	17	0	0	NUM
elements-9303	23	18	,	,	PUNCT
elements-9303	23	19	1	1	NUM
elements-9303	23	20	,	,	PUNCT
elements-9303	23	21	2	2	NUM
elements-9303	23	22	,	,	PUNCT
elements-9303	23	23	3	3	NUM
elements-9303	23	24	,	,	PUNCT
elements-9303	23	25	4	4	NUM
elements-9303	23	26	,	,	PUNCT
elements-9303	23	27	5	5	NUM
elements-9303	23	28	,	,	PUNCT
elements-9303	23	29	or	or	CCONJ
elements-9303	23	30	6	6	NUM
elements-9303	23	31	when	when	SCONJ
elements-9303	23	32	divided	divide	VERB
elements-9303	23	33	by	by	ADP
elements-9303	23	34	7	7	NUM
elements-9303	23	35	.	.	PUNCT
elements-9303	24	1	hence	hence	ADV
elements-9303	24	2	,	,	PUNCT
elements-9303	24	3	we	we	PRON
elements-9303	24	4	have	have	VERB
elements-9303	24	5	seven	seven	NUM
elements-9303	24	6	cases	case	NOUN
elements-9303	24	7	,	,	PUNCT
elements-9303	24	8	each	each	PRON
elements-9303	24	9	with	with	ADP
elements-9303	24	10	seven	seven	NUM
elements-9303	24	11	sub	sub	NOUN
elements-9303	24	12	-	-	NOUN
elements-9303	24	13	cases	case	NOUN
elements-9303	24	14	,	,	PUNCT
elements-9303	24	15	which	which	PRON
elements-9303	24	16	we	we	PRON
elements-9303	24	17	must	must	AUX
elements-9303	24	18	consider	consider	VERB
elements-9303	24	19	.	.	PUNCT
elements-9303	25	1	we	we	PRON
elements-9303	25	2	will	will	AUX
elements-9303	25	3	write	write	VERB
elements-9303	25	4	the	the	DET
elements-9303	25	5	rst	rst	ADJ
elements-9303	25	6	three	three	NUM
elements-9303	25	7	cases	case	NOUN
elements-9303	25	8	below	below	ADV
elements-9303	25	9	.	.	PUNCT
elements-9303	26	1	90	90	NUM
elements-9303	26	2	elements	element	NOUN
elements-9303	26	3	:	:	PUNCT
elements-9303	26	4	:	:	PUNCT
elements-9303	26	5	spring	spring	NOUN
elements-9303	26	6	2016	2016	NUM
elements-9303	26	7	notice	notice	VERB
elements-9303	26	8	that	that	SCONJ
elements-9303	26	9	in	in	ADP
elements-9303	26	10	case	case	NOUN
elements-9303	26	11	1	1	NUM
elements-9303	26	12	,	,	PUNCT
elements-9303	26	13	dn+1	dn+1	NOUN
elements-9303	26	14	covers	cover	VERB
elements-9303	26	15	all	all	DET
elements-9303	26	16	possible	possible	ADJ
elements-9303	26	17	dreibens	dreiben	NOUN
elements-9303	26	18	of	of	ADP
elements-9303	26	19	length	length	NOUN
elements-9303	26	20	n	n	PROPN
elements-9303	26	21	+	+	CCONJ
elements-9303	26	22	1	1	NUM
elements-9303	26	23	divisible	divisible	ADJ
elements-9303	26	24	by	by	ADP
elements-9303	26	25	7	7	NUM
elements-9303	26	26	.	.	PUNCT
elements-9303	27	1	likewise	likewise	ADV
elements-9303	27	2	,	,	PUNCT
elements-9303	27	3	case	case	NOUN
elements-9303	27	4	2	2	NUM
elements-9303	27	5	covers	cover	VERB
elements-9303	27	6	all	all	DET
elements-9303	27	7	possible	possible	ADJ
elements-9303	27	8	dreibens	dreiben	NOUN
elements-9303	27	9	of	of	ADP
elements-9303	27	10	length	length	NOUN
elements-9303	27	11	n	n	PROPN
elements-9303	27	12	+	+	CCONJ
elements-9303	27	13	1	1	NUM
elements-9303	27	14	that	that	PRON
elements-9303	27	15	are	be	AUX
elements-9303	27	16	1	1	NUM
elements-9303	27	17	(	(	PUNCT
elements-9303	27	18	mod	mod	PROPN
elements-9303	27	19	7	7	NUM
elements-9303	27	20	)	)	PUNCT
elements-9303	27	21	,	,	PUNCT
elements-9303	27	22	case	case	NOUN
elements-9303	27	23	3	3	NUM
elements-9303	27	24	covers	cover	VERB
elements-9303	27	25	all	all	DET
elements-9303	27	26	possible	possible	ADJ
elements-9303	27	27	dreibens	dreiben	NOUN
elements-9303	27	28	of	of	ADP
elements-9303	27	29	length	length	NOUN
elements-9303	27	30	n	n	PROPN
elements-9303	27	31	+	+	CCONJ
elements-9303	27	32	1	1	NUM
elements-9303	27	33	that	that	PRON
elements-9303	27	34	are	be	AUX
elements-9303	27	35	2	2	NUM
elements-9303	27	36	(	(	PUNCT
elements-9303	27	37	mod	mod	PROPN
elements-9303	27	38	7	7	NUM
elements-9303	27	39	)	)	PUNCT
elements-9303	27	40	,	,	PUNCT
elements-9303	27	41	and	and	CCONJ
elements-9303	27	42	so	so	ADV
elements-9303	27	43	forth	forth	ADV
elements-9303	27	44	.	.	PUNCT
elements-9303	28	1	similarly	similarly	ADV
elements-9303	28	2	,	,	PUNCT
elements-9303	28	3	each	each	DET
elements-9303	28	4	subcase	subcase	NOUN
elements-9303	28	5	(	(	PUNCT
elements-9303	28	6	a	a	X
elements-9303	28	7	)	)	PUNCT
elements-9303	28	8	covers	cover	VERB
elements-9303	28	9	all	all	DET
elements-9303	28	10	possible	possible	ADJ
elements-9303	28	11	dreibens	dreiben	NOUN
elements-9303	28	12	of	of	ADP
elements-9303	28	13	length	length	NOUN
elements-9303	28	14	n	n	CCONJ
elements-9303	28	15	divisible	divisible	ADJ
elements-9303	28	16	by	by	ADP
elements-9303	28	17	7	7	NUM
elements-9303	28	18	,	,	PUNCT
elements-9303	28	19	each	each	DET
elements-9303	28	20	subcase	subcase	NOUN
elements-9303	28	21	(	(	PUNCT
elements-9303	28	22	b	b	X
elements-9303	28	23	)	)	PUNCT
elements-9303	28	24	covers	cover	VERB
elements-9303	28	25	all	all	DET
elements-9303	28	26	possible	possible	ADJ
elements-9303	28	27	dreibens	dreiben	NOUN
elements-9303	28	28	of	of	ADP
elements-9303	28	29	length	length	NOUN
elements-9303	29	1	n	n	PROPN
elements-9303	29	2	that	that	PRON
elements-9303	29	3	are	be	AUX
elements-9303	29	4	1	1	NUM
elements-9303	29	5	(	(	PUNCT
elements-9303	29	6	mod	mod	PROPN
elements-9303	29	7	7	7	NUM
elements-9303	29	8	)	)	PUNCT
elements-9303	29	9	,	,	PUNCT
elements-9303	29	10	and	and	CCONJ
elements-9303	29	11	so	so	ADV
elements-9303	29	12	forth	forth	ADV
elements-9303	29	13	.	.	PUNCT
elements-9303	30	1	hence	hence	ADV
elements-9303	30	2	,	,	PUNCT
elements-9303	30	3	each	each	DET
elements-9303	30	4	case	case	NOUN
elements-9303	30	5	represents	represent	VERB
elements-9303	30	6	ai	ai	VERB
elements-9303	30	7	7(n	7(n	NUM
elements-9303	30	8	+	+	CCONJ
elements-9303	30	9	1	1	X
elements-9303	30	10	)	)	PUNCT
elements-9303	30	11	for	for	ADP
elements-9303	30	12	some	some	DET
elements-9303	30	13	i	i	PRON
elements-9303	30	14	and	and	CCONJ
elements-9303	30	15	each	each	DET
elements-9303	30	16	sub	sub	NOUN
elements-9303	30	17	-	-	NOUN
elements-9303	30	18	case	case	NOUN
elements-9303	30	19	represents	represents	AUX
elements-9303	30	20	ai	ai	VERB
elements-9303	30	21	7(n	7(n	NUM
elements-9303	30	22	)	)	PUNCT
elements-9303	30	23	for	for	ADP
elements-9303	30	24	some	some	DET
elements-9303	30	25	i.	i.	NOUN
elements-9303	30	26	from	from	ADP
elements-9303	30	27	these	these	DET
elements-9303	30	28	cases	case	NOUN
elements-9303	30	29	,	,	PUNCT
elements-9303	30	30	we	we	PRON
elements-9303	30	31	get	get	VERB
elements-9303	30	32	the	the	DET
elements-9303	30	33	following	follow	VERB
elements-9303	30	34	formulas	formula	NOUN
elements-9303	30	35	for	for	ADP
elements-9303	30	36	ai	ai	PROPN
elements-9303	30	37	7(n	7(n	NUM
elements-9303	30	38	+	+	CCONJ
elements-9303	30	39	1	1	X
elements-9303	30	40	)	)	PUNCT
elements-9303	30	41	in	in	ADP
elements-9303	30	42	terms	term	NOUN
elements-9303	30	43	of	of	ADP
elements-9303	30	44	ai	ai	NOUN
elements-9303	30	45	7(n	7(n	NUM
elements-9303	30	46	):	):	PUNCT
elements-9303	30	47	we	we	PRON
elements-9303	30	48	can	can	AUX
elements-9303	30	49	rewrite	rewrite	VERB
elements-9303	30	50	this	this	DET
elements-9303	30	51	linear	linear	ADJ
elements-9303	30	52	system	system	NOUN
elements-9303	30	53	in	in	ADP
elements-9303	30	54	terms	term	NOUN
elements-9303	30	55	of	of	ADP
elements-9303	30	56	matrices	matrix	NOUN
elements-9303	30	57	.	.	PUNCT
elements-9303	31	1	let	let	VERB
elements-9303	31	2	the	the	DET
elements-9303	31	3	aforementioned	aforementione	VERB
elements-9303	31	4	linear	linear	NOUN
elements-9303	31	5	system	system	NOUN
elements-9303	31	6	is	be	AUX
elements-9303	31	7	the	the	DET
elements-9303	31	8	same	same	ADJ
elements-9303	31	9	as	as	ADP
elements-9303	31	10	by	by	ADP
elements-9303	31	11	induction	induction	NOUN
elements-9303	31	12	,	,	PUNCT
elements-9303	31	13	we	we	PRON
elements-9303	31	14	find	find	VERB
elements-9303	31	15	that	that	SCONJ
elements-9303	31	16	this	this	DET
elements-9303	31	17	linear	linear	ADJ
elements-9303	31	18	system	system	NOUN
elements-9303	31	19	allows	allow	VERB
elements-9303	31	20	us	we	PRON
elements-9303	31	21	to	to	PART
elements-9303	31	22	quickly	quickly	ADV
elements-9303	31	23	find	find	VERB
elements-9303	31	24	the	the	DET
elements-9303	31	25	number	number	NOUN
elements-9303	31	26	of	of	ADP
elements-9303	31	27	dreibens	dreiben	NOUN
elements-9303	31	28	of	of	ADP
elements-9303	31	29	length	length	NOUN
elements-9303	31	30	n	n	PROPN
elements-9303	31	31	that	that	PRON
elements-9303	31	32	leave	leave	VERB
elements-9303	31	33	a	a	DET
elements-9303	31	34	remainder	remainder	NOUN
elements-9303	31	35	i	i	PRON
elements-9303	31	36	when	when	SCONJ
elements-9303	31	37	divided	divide	VERB
elements-9303	31	38	by	by	ADP
elements-9303	31	39	7	7	NUM
elements-9303	31	40	.	.	PUNCT
elements-9303	32	1	we	we	PRON
elements-9303	32	2	need	need	VERB
elements-9303	32	3	only	only	ADV
elements-9303	32	4	calculate	calculate	VERB
elements-9303	32	5	the	the	DET
elements-9303	32	6	following	following	NOUN
elements-9303	32	7	:	:	PUNCT
elements-9303	32	8	let	let	VERB
elements-9303	32	9	us	we	PRON
elements-9303	32	10	investigate	investigate	VERB
elements-9303	32	11	an	an	PRON
elements-9303	32	12	to	to	PART
elements-9303	32	13	see	see	VERB
elements-9303	32	14	if	if	SCONJ
elements-9303	32	15	we	we	PRON
elements-9303	32	16	nd	nd	VERB
elements-9303	32	17	anything	anything	PRON
elements-9303	32	18	interesting	interesting	ADJ
elements-9303	32	19	.	.	PUNCT
elements-9303	33	1	evidently	evidently	ADV
elements-9303	33	2	:	:	PUNCT
elements-9303	33	3	let	let	VERB
elements-9303	33	4	so	so	ADV
elements-9303	33	5	a	a	DET
elements-9303	33	6	=	=	SYM
elements-9303	33	7	b	b	PROPN
elements-9303	33	8	+	+	CCONJ
elements-9303	33	9	c.	c.	NOUN
elements-9303	34	1	"	"	PUNCT
elements-9303	34	2	we	we	PRON
elements-9303	34	3	can	can	AUX
elements-9303	34	4	find	find	VERB
elements-9303	34	5	formulas	formula	NOUN
elements-9303	34	6	for	for	ADP
elements-9303	34	7	ai	ai	PROPN
elements-9303	34	8	7(n	7(n	NUM
elements-9303	34	9	+	+	CCONJ
elements-9303	34	10	1	1	X
elements-9303	34	11	)	)	PUNCT
elements-9303	34	12	by	by	ADP
elements-9303	34	13	considering	consider	VERB
elements-9303	34	14	the	the	DET
elements-9303	34	15	first	first	ADJ
elements-9303	34	16	n	n	NOUN
elements-9303	34	17	digits	digit	NOUN
elements-9303	34	18	of	of	ADP
elements-9303	34	19	a	a	DET
elements-9303	34	20	dreiben	dreiben	NOUN
elements-9303	34	21	of	of	ADP
elements-9303	34	22	length	length	NOUN
elements-9303	34	23	n	n	PROPN
elements-9303	34	24	+	+	NUM
elements-9303	34	25	1	1	NUM
elements-9303	34	26	,	,	PUNCT
elements-9303	34	27	then	then	ADV
elements-9303	34	28	seeing	see	VERB
elements-9303	34	29	what	what	PRON
elements-9303	34	30	the	the	DET
elements-9303	34	31	last	last	ADJ
elements-9303	34	32	digit	digit	NOUN
elements-9303	34	33	must	must	AUX
elements-9303	34	34	be	be	AUX
elements-9303	34	35	such	such	ADJ
elements-9303	34	36	that	that	SCONJ
elements-9303	34	37	the	the	DET
elements-9303	34	38	dreiben	dreiben	NOUN
elements-9303	34	39	leaves	leave	VERB
elements-9303	34	40	a	a	DET
elements-9303	34	41	remainder	remainder	NOUN
elements-9303	34	42	i	i	PRON
elements-9303	34	43	when	when	SCONJ
elements-9303	34	44	divided	divide	VERB
elements-9303	34	45	by	by	ADP
elements-9303	34	46	7	7	NUM
elements-9303	34	47	.	.	PUNCT
elements-9303	34	48	"	"	PUNCT
elements-9303	35	1	91	91	NUM
elements-9303	35	2	dreibens	dreiben	NOUN
elements-9303	35	3	modulo	modulo	VERB
elements-9303	35	4	7	7	NUM
elements-9303	35	5	if	if	SCONJ
elements-9303	35	6	we	we	PRON
elements-9303	35	7	look	look	VERB
elements-9303	35	8	at	at	ADP
elements-9303	35	9	the	the	DET
elements-9303	35	10	powers	power	NOUN
elements-9303	35	11	of	of	ADP
elements-9303	35	12	b	b	PROPN
elements-9303	35	13	and	and	CCONJ
elements-9303	35	14	c	c	X
elements-9303	35	15	,	,	PUNCT
elements-9303	35	16	we	we	PRON
elements-9303	35	17	find	find	VERB
elements-9303	35	18	that	that	PRON
elements-9303	35	19	notice	notice	VERB
elements-9303	35	20	how	how	SCONJ
elements-9303	35	21	for	for	ADP
elements-9303	35	22	i	i	PRON
elements-9303	35	23	=	=	SYM
elements-9303	35	24	0	0	PUNCT
elements-9303	36	1	(	(	PUNCT
elements-9303	36	2	mod	mod	PROPN
elements-9303	36	3	6	6	NUM
elements-9303	36	4	)	)	PUNCT
elements-9303	36	5	,	,	PUNCT
elements-9303	36	6	both	both	CCONJ
elements-9303	36	7	bi	bi	NOUN
elements-9303	36	8	and	and	CCONJ
elements-9303	36	9	ci	ci	NOUN
elements-9303	36	10	are	be	AUX
elements-9303	36	11	equal	equal	ADJ
elements-9303	36	12	to	to	ADP
elements-9303	36	13	the	the	DET
elements-9303	36	14	identity	identity	NOUN
elements-9303	36	15	matrix	matrix	NOUN
elements-9303	36	16	i7	i7	NOUN
elements-9303	36	17	.	.	PUNCT
elements-9303	37	1	both	both	CCONJ
elements-9303	37	2	the	the	DET
elements-9303	37	3	powers	power	NOUN
elements-9303	37	4	of	of	ADP
elements-9303	37	5	b	b	PROPN
elements-9303	37	6	and	and	CCONJ
elements-9303	37	7	c	c	PROPN
elements-9303	37	8	are	be	AUX
elements-9303	37	9	cyclic	cyclic	ADJ
elements-9303	37	10	groups	group	NOUN
elements-9303	37	11	of	of	ADP
elements-9303	37	12	order	order	NOUN
elements-9303	37	13	6	6	NUM
elements-9303	37	14	.	.	PUNCT
elements-9303	37	15	before	before	ADP
elements-9303	37	16	examining	examine	VERB
elements-9303	37	17	an	an	PRON
elements-9303	37	18	,	,	PUNCT
elements-9303	37	19	let	let	VERB
elements-9303	37	20	us	we	PRON
elements-9303	37	21	define	define	VERB
elements-9303	37	22	the	the	DET
elements-9303	37	23	following	following	NOUN
elements-9303	37	24	:	:	PUNCT
elements-9303	37	25	these	these	DET
elements-9303	37	26	formulas	formula	NOUN
elements-9303	37	27	will	will	AUX
elements-9303	37	28	be	be	AUX
elements-9303	37	29	useful	useful	ADJ
elements-9303	37	30	for	for	ADP
elements-9303	37	31	condensing	condense	VERB
elements-9303	37	32	our	our	PRON
elements-9303	37	33	notation	notation	NOUN
elements-9303	37	34	.	.	PUNCT
elements-9303	38	1	now	now	ADV
elements-9303	38	2	we	we	PRON
elements-9303	38	3	can	can	AUX
elements-9303	38	4	finally	finally	ADV
elements-9303	38	5	begin	begin	VERB
elements-9303	38	6	examining	examine	VERB
elements-9303	38	7	an	an	PRON
elements-9303	38	8	:	:	PUNCT
elements-9303	38	9	now	now	ADV
elements-9303	38	10	we	we	PRON
elements-9303	38	11	have	have	AUX
elements-9303	38	12	the	the	DET
elements-9303	38	13	problem	problem	NOUN
elements-9303	38	14	with	with	ADP
elements-9303	38	15	the	the	DET
elements-9303	38	16	above	above	ADJ
elements-9303	38	17	equation	equation	NOUN
elements-9303	38	18	is	be	AUX
elements-9303	38	19	that	that	SCONJ
elements-9303	38	20	we	we	PRON
elements-9303	38	21	still	still	ADV
elements-9303	38	22	have	have	VERB
elements-9303	38	23	the	the	DET
elements-9303	38	24	powers	power	NOUN
elements-9303	38	25	of	of	ADP
elements-9303	38	26	b	b	NOUN
elements-9303	38	27	in	in	ADP
elements-9303	38	28	terms	term	NOUN
elements-9303	38	29	of	of	ADP
elements-9303	38	30	n.	n.	NOUN
elements-9303	38	31	however	however	ADV
elements-9303	38	32	,	,	PUNCT
elements-9303	38	33	we	we	PRON
elements-9303	38	34	can	can	AUX
elements-9303	38	35	easily	easily	ADV
elements-9303	38	36	solve	solve	VERB
elements-9303	38	37	this	this	DET
elements-9303	38	38	problem	problem	NOUN
elements-9303	38	39	because	because	SCONJ
elements-9303	38	40	the	the	DET
elements-9303	38	41	powers	power	NOUN
elements-9303	38	42	of	of	ADP
elements-9303	38	43	b	b	PROPN
elements-9303	38	44	form	form	NOUN
elements-9303	38	45	a	a	DET
elements-9303	38	46	cyclic	cyclic	ADJ
elements-9303	38	47	group	group	NOUN
elements-9303	38	48	.	.	PUNCT
elements-9303	39	1	since	since	SCONJ
elements-9303	39	2	this	this	DET
elements-9303	39	3	cyclic	cyclic	ADJ
elements-9303	39	4	group	group	NOUN
elements-9303	39	5	has	have	VERB
elements-9303	39	6	order	order	NOUN
elements-9303	39	7	6	6	NUM
elements-9303	39	8	,	,	PUNCT
elements-9303	39	9	we	we	PRON
elements-9303	39	10	must	must	AUX
elements-9303	39	11	consider	consider	VERB
elements-9303	39	12	six	six	NUM
elements-9303	39	13	cases	case	NOUN
elements-9303	39	14	.	.	PUNCT
elements-9303	40	1	92	92	NUM
elements-9303	40	2	elements	element	NOUN
elements-9303	40	3	:	:	PUNCT
elements-9303	40	4	:	:	PUNCT
elements-9303	40	5	spring	spring	NOUN
elements-9303	40	6	2016	2016	NUM
elements-9303	40	7	evidently	evidently	ADV
elements-9303	40	8	,	,	PUNCT
elements-9303	40	9	no	no	ADV
elements-9303	40	10	matter	matter	ADV
elements-9303	40	11	what	what	PRON
elements-9303	40	12	n	n	X
elements-9303	40	13	is	be	AUX
elements-9303	40	14	,	,	PUNCT
elements-9303	40	15	the	the	DET
elements-9303	40	16	first	first	ADJ
elements-9303	40	17	row	row	NOUN
elements-9303	40	18	of	of	ADP
elements-9303	40	19	the	the	DET
elements-9303	40	20	matrix	matrix	NOUN
elements-9303	40	21	an	an	PRON
elements-9303	40	22	is	be	AUX
elements-9303	40	23	always	always	ADV
elements-9303	40	24	recall	recall	ADJ
elements-9303	40	25	that	that	SCONJ
elements-9303	40	26	if	if	SCONJ
elements-9303	40	27	we	we	PRON
elements-9303	40	28	want	want	VERB
elements-9303	40	29	to	to	PART
elements-9303	40	30	find	find	VERB
elements-9303	40	31	a0	a0	NOUN
elements-9303	40	32	7(n	7(n	NOUN
elements-9303	40	33	+	+	CCONJ
elements-9303	40	34	1	1	NUM
elements-9303	40	35	)	)	PUNCT
elements-9303	40	36	,	,	PUNCT
elements-9303	40	37	i.e.	i.e.	X
elements-9303	40	38	the	the	DET
elements-9303	40	39	first	first	ADJ
elements-9303	40	40	entry	entry	NOUN
elements-9303	40	41	of	of	ADP
elements-9303	40	42	→vn+1	→vn+1	NOUN
elements-9303	40	43	,	,	PUNCT
elements-9303	40	44	we	we	PRON
elements-9303	40	45	need	need	AUX
elements-9303	40	46	only	only	ADV
elements-9303	40	47	consider	consider	VERB
elements-9303	40	48	the	the	DET
elements-9303	40	49	first	first	ADJ
elements-9303	40	50	row	row	NOUN
elements-9303	40	51	of	of	ADP
elements-9303	40	52	an	an	PRON
elements-9303	40	53	:	:	PUNCT
elements-9303	40	54	now	now	ADV
elements-9303	40	55	we	we	PRON
elements-9303	40	56	have	have	VERB
elements-9303	40	57	a	a	DET
elements-9303	40	58	formula	formula	NOUN
elements-9303	40	59	with	with	ADP
elements-9303	40	60	which	which	PRON
elements-9303	40	61	to	to	PART
elements-9303	40	62	find	find	VERB
elements-9303	40	63	the	the	DET
elements-9303	40	64	number	number	NOUN
elements-9303	40	65	of	of	ADP
elements-9303	40	66	dreibens	dreiben	NOUN
elements-9303	40	67	of	of	ADP
elements-9303	40	68	length	length	NOUN
elements-9303	40	69	n	n	PROPN
elements-9303	40	70	that	that	PRON
elements-9303	40	71	are	be	AUX
elements-9303	40	72	divisible	divisible	ADJ
elements-9303	40	73	by	by	ADP
elements-9303	40	74	7	7	NUM
elements-9303	40	75	:	:	PUNCT
elements-9303	40	76	what	what	PRON
elements-9303	40	77	if	if	SCONJ
elements-9303	40	78	we	we	PRON
elements-9303	40	79	want	want	VERB
elements-9303	40	80	to	to	PART
elements-9303	40	81	find	find	VERB
elements-9303	40	82	any	any	DET
elements-9303	40	83	ai	ai	NOUN
elements-9303	40	84	7(n	7(n	NUM
elements-9303	40	85	)	)	PUNCT
elements-9303	40	86	,	,	PUNCT
elements-9303	40	87	i	i	PRON
elements-9303	40	88	=	=	NOUN
elements-9303	40	89	0	0	NUM
elements-9303	40	90	,	,	PUNCT
elements-9303	40	91	1	1	NUM
elements-9303	40	92	,	,	PUNCT
elements-9303	40	93	2	2	NUM
elements-9303	40	94	,	,	PUNCT
elements-9303	40	95	3	3	NUM
elements-9303	40	96	,	,	PUNCT
elements-9303	40	97	4	4	NUM
elements-9303	40	98	,	,	PUNCT
elements-9303	40	99	5	5	NUM
elements-9303	40	100	,	,	PUNCT
elements-9303	40	101	6	6	NUM
elements-9303	40	102	?	?	PUNCT
elements-9303	41	1	we	we	PRON
elements-9303	41	2	know	know	VERB
elements-9303	41	3	that	that	SCONJ
elements-9303	41	4	these	these	PRON
elements-9303	41	5	ai	ai	VERB
elements-9303	41	6	7(n	7(n	NOUN
elements-9303	41	7	)	)	PUNCT
elements-9303	41	8	are	be	AUX
elements-9303	41	9	entries	entry	NOUN
elements-9303	41	10	of	of	ADP
elements-9303	41	11	→vn	→vn	PRON
elements-9303	41	12	.	.	PUNCT
elements-9303	42	1	recall	recall	VERB
elements-9303	42	2	that	that	PRON
elements-9303	42	3	to	to	PART
elements-9303	42	4	solve	solve	VERB
elements-9303	42	5	for	for	ADP
elements-9303	42	6	→vn+1	→vn+1	NOUN
elements-9303	42	7	,	,	PUNCT
elements-9303	42	8	we	we	PRON
elements-9303	42	9	must	must	AUX
elements-9303	42	10	first	first	ADV
elements-9303	42	11	understand	understand	VERB
elements-9303	42	12	an	an	DET
elements-9303	42	13	.	.	NOUN
elements-9303	42	14	recall	recall	VERB
elements-9303	42	15	that	that	PRON
elements-9303	42	16	as	as	SCONJ
elements-9303	42	17	it	it	PRON
elements-9303	42	18	turns	turn	VERB
elements-9303	42	19	out	out	ADP
elements-9303	42	20	,	,	PUNCT
elements-9303	42	21	a	a	PRON
elements-9303	42	22	is	be	AUX
elements-9303	42	23	a	a	DET
elements-9303	42	24	diagonalizable	diagonalizable	ADJ
elements-9303	42	25	matrix	matrix	NOUN
elements-9303	42	26	,	,	PUNCT
elements-9303	42	27	so	so	CCONJ
elements-9303	42	28	an	an	PRON
elements-9303	42	29	is	be	AUX
elements-9303	42	30	also	also	ADV
elements-9303	42	31	diagonalizable	diagonalizable	ADJ
elements-9303	42	32	:	:	PUNCT
elements-9303	42	33	interestingly	interestingly	ADV
elements-9303	42	34	,	,	PUNCT
elements-9303	42	35	some	some	DET
elements-9303	42	36	entries	entry	NOUN
elements-9303	42	37	of	of	ADP
elements-9303	42	38	d	d	NOUN
elements-9303	42	39	are	be	AUX
elements-9303	42	40	6th	6th	ADJ
elements-9303	42	41	roots	root	NOUN
elements-9303	42	42	of	of	ADP
elements-9303	42	43	unity	unity	NOUN
elements-9303	42	44	.	.	PUNCT
elements-9303	43	1	if	if	SCONJ
elements-9303	43	2	we	we	PRON
elements-9303	43	3	let	let	VERB
elements-9303	43	4	be	be	AUX
elements-9303	43	5	the	the	DET
elements-9303	43	6	jth	jth	PROPN
elements-9303	43	7	ith	ith	PROPN
elements-9303	43	8	-	-	PUNCT
elements-9303	43	9	root	root	NOUN
elements-9303	43	10	of	of	ADP
elements-9303	43	11	unity	unity	NOUN
elements-9303	43	12	,	,	PUNCT
elements-9303	43	13	then	then	ADV
elements-9303	43	14	93	93	NUM
elements-9303	43	15	dreibens	dreiben	NOUN
elements-9303	43	16	modulo	modulo	VERB
elements-9303	43	17	7	7	NUM
elements-9303	43	18	recall	recall	NOUN
elements-9303	43	19	that	that	SCONJ
elements-9303	43	20	the	the	DET
elements-9303	43	21	nth	nth	NOUN
elements-9303	43	22	roots	root	NOUN
elements-9303	43	23	of	of	ADP
elements-9303	43	24	unity	unity	NOUN
elements-9303	43	25	form	form	VERB
elements-9303	43	26	a	a	DET
elements-9303	43	27	cyclic	cyclic	ADJ
elements-9303	43	28	group	group	NOUN
elements-9303	43	29	of	of	ADP
elements-9303	43	30	order	order	NOUN
elements-9303	43	31	n	n	CCONJ
elements-9303	43	32	,	,	PUNCT
elements-9303	43	33	as	as	ADP
elements-9303	43	34	in	in	ADP
elements-9303	43	35	table	table	NOUN
elements-9303	43	36	1	1	NUM
elements-9303	43	37	below	below	ADV
elements-9303	43	38	:	:	PUNCT
elements-9303	43	39	based	base	VERB
elements-9303	43	40	on	on	ADP
elements-9303	43	41	the	the	DET
elements-9303	43	42	properties	property	NOUN
elements-9303	43	43	of	of	ADP
elements-9303	43	44	the	the	DET
elements-9303	43	45	6th	6th	ADJ
elements-9303	43	46	roots	root	NOUN
elements-9303	43	47	of	of	ADP
elements-9303	43	48	unity	unity	NOUN
elements-9303	43	49	,	,	PUNCT
elements-9303	43	50	we	we	PRON
elements-9303	43	51	find	find	VERB
elements-9303	43	52	that	that	SCONJ
elements-9303	43	53	now	now	ADV
elements-9303	43	54	that	that	SCONJ
elements-9303	43	55	we	we	PRON
elements-9303	43	56	understand	understand	VERB
elements-9303	43	57	dn	dn	PROPN
elements-9303	43	58	,	,	PUNCT
elements-9303	43	59	we	we	PRON
elements-9303	43	60	are	be	AUX
elements-9303	43	61	much	much	ADV
elements-9303	43	62	closer	close	ADJ
elements-9303	43	63	to	to	ADP
elements-9303	43	64	characterizing	characterize	VERB
elements-9303	43	65	an	an	PRON
elements-9303	43	66	.	.	PUNCT
elements-9303	44	1	all	all	PRON
elements-9303	44	2	we	we	PRON
elements-9303	44	3	need	need	VERB
elements-9303	44	4	to	to	PART
elements-9303	44	5	do	do	VERB
elements-9303	44	6	is	be	AUX
elements-9303	44	7	to	to	PART
elements-9303	44	8	calculate	calculate	VERB
elements-9303	44	9	pdnp-1	pdnp-1	NOUN
elements-9303	44	10	for	for	ADP
elements-9303	44	11	each	each	DET
elements-9303	44	12	case	case	NOUN
elements-9303	44	13	of	of	ADP
elements-9303	44	14	dn	dn	PROPN
elements-9303	44	15	.	.	PROPN
elements-9303	44	16	below	below	ADV
elements-9303	44	17	,	,	PUNCT
elements-9303	44	18	we	we	PRON
elements-9303	44	19	will	will	AUX
elements-9303	44	20	calculate	calculate	VERB
elements-9303	44	21	the	the	DET
elements-9303	44	22	first	first	ADJ
elements-9303	44	23	case	case	NOUN
elements-9303	44	24	,	,	PUNCT
elements-9303	44	25	i.e.	i.e.	X
elements-9303	44	26	where	where	SCONJ
elements-9303	44	27	n	n	X
elements-9303	44	28	≡	≡	PROPN
elements-9303	44	29	0	0	PUNCT
elements-9303	44	30	(	(	PUNCT
elements-9303	44	31	mod	mod	PROPN
elements-9303	44	32	6	6	NUM
elements-9303	44	33	)	)	PUNCT
elements-9303	44	34	.	.	PUNCT
elements-9303	45	1	if	if	SCONJ
elements-9303	45	2	n	n	PRON
elements-9303	45	3	≡	≡	PROPN
elements-9303	45	4	0	0	PUNCT
elements-9303	45	5	(	(	PUNCT
elements-9303	45	6	mod	mod	PROPN
elements-9303	45	7	6	6	NUM
elements-9303	45	8	)	)	PUNCT
elements-9303	45	9	,	,	PUNCT
elements-9303	45	10	where	where	SCONJ
elements-9303	45	11	n	n	PRON
elements-9303	45	12	≡	≡	PROPN
elements-9303	45	13	0	0	PUNCT
elements-9303	45	14	(	(	PUNCT
elements-9303	45	15	mod	mod	PROPN
elements-9303	45	16	6	6	NUM
elements-9303	45	17	)	)	PUNCT
elements-9303	45	18	,	,	PUNCT
elements-9303	45	19	the	the	DET
elements-9303	45	20	entries	entry	NOUN
elements-9303	45	21	of	of	ADP
elements-9303	45	22	an	an	PRON
elements-9303	45	23	are	be	AUX
elements-9303	45	24	all	all	PRON
elements-9303	45	25	either	either	DET
elements-9303	45	26	1/7(2n	1/7(2n	NUM
elements-9303	45	27	–	–	SYM
elements-9303	45	28	1	1	NUM
elements-9303	45	29	)	)	PUNCT
elements-9303	45	30	or	or	CCONJ
elements-9303	45	31	1/7(2n	1/7(2n	NUM
elements-9303	45	32	+	+	CCONJ
elements-9303	45	33	6	6	NUM
elements-9303	45	34	)	)	PUNCT
elements-9303	45	35	.	.	PUNCT
elements-9303	46	1	hence	hence	ADV
elements-9303	46	2	,	,	PUNCT
elements-9303	46	3	an	an	PRON
elements-9303	46	4	can	can	AUX
elements-9303	46	5	be	be	AUX
elements-9303	46	6	much	much	ADV
elements-9303	46	7	more	more	ADV
elements-9303	46	8	easily	easily	ADV
elements-9303	46	9	calculated	calculate	VERB
elements-9303	46	10	;	;	PUNCT
elements-9303	46	11	instead	instead	ADV
elements-9303	46	12	of	of	ADP
elements-9303	46	13	performing	perform	VERB
elements-9303	46	14	lengthy	lengthy	ADJ
elements-9303	46	15	matrix	matrix	NOUN
elements-9303	46	16	power	power	NOUN
elements-9303	46	17	operations	operation	NOUN
elements-9303	46	18	,	,	PUNCT
elements-9303	46	19	one	one	PRON
elements-9303	46	20	can	can	AUX
elements-9303	46	21	merely	merely	ADV
elements-9303	46	22	find	find	VERB
elements-9303	46	23	1/7(2n	1/7(2n	NUM
elements-9303	46	24	1	1	NUM
elements-9303	46	25	)	)	PUNCT
elements-9303	46	26	and	and	CCONJ
elements-9303	46	27	1/7(2n	1/7(2n	NUM
elements-9303	46	28	+	+	CCONJ
elements-9303	46	29	6	6	NUM
elements-9303	46	30	)	)	PUNCT
elements-9303	46	31	to	to	PART
elements-9303	46	32	determine	determine	VERB
elements-9303	46	33	an	an	PRON
elements-9303	46	34	.	.	PUNCT
elements-9303	47	1	since	since	SCONJ
elements-9303	47	2	→vn	→vn	X
elements-9303	47	3	=	=	SYM
elements-9303	47	4	an-1→v1	an-1→v1	NOUN
elements-9303	47	5	,	,	PUNCT
elements-9303	47	6	one	one	PRON
elements-9303	47	7	can	can	AUX
elements-9303	47	8	now	now	ADV
elements-9303	47	9	easily	easily	ADV
elements-9303	47	10	determine	determine	VERB
elements-9303	47	11	any	any	DET
elements-9303	47	12	ai	ai	NOUN
elements-9303	47	13	7(n	7(n	NUM
elements-9303	47	14	)	)	PUNCT
elements-9303	47	15	where	where	SCONJ
elements-9303	47	16	n	n	PRON
elements-9303	47	17	≡	≡	PROPN
elements-9303	47	18	0	0	PUNCT
elements-9303	47	19	(	(	PUNCT
elements-9303	47	20	mod	mod	PROPN
elements-9303	47	21	6	6	NUM
elements-9303	47	22	)	)	PUNCT
elements-9303	47	23	.	.	PUNCT
elements-9303	48	1	in	in	ADP
elements-9303	48	2	other	other	ADJ
elements-9303	48	3	words	word	NOUN
elements-9303	48	4	,	,	PUNCT
elements-9303	48	5	one	one	PRON
elements-9303	48	6	can	can	AUX
elements-9303	48	7	easily	easily	ADV
elements-9303	48	8	determine	determine	VERB
elements-9303	48	9	the	the	DET
elements-9303	48	10	number	number	NOUN
elements-9303	48	11	of	of	ADP
elements-9303	48	12	dreibens	dreiben	NOUN
elements-9303	48	13	of	of	ADP
elements-9303	48	14	length	length	NOUN
elements-9303	48	15	n	n	X
elements-9303	48	16	≡	≡	PROPN
elements-9303	48	17	0	0	PUNCT
elements-9303	49	1	(	(	PUNCT
elements-9303	49	2	mod	mod	PROPN
elements-9303	49	3	6	6	NUM
elements-9303	49	4	)	)	PUNCT
elements-9303	49	5	that	that	PRON
elements-9303	49	6	leave	leave	VERB
elements-9303	49	7	remainder	remainder	NOUN
elements-9303	49	8	i	i	PRON
elements-9303	49	9	when	when	SCONJ
elements-9303	49	10	divided	divide	VERB
elements-9303	49	11	by	by	ADP
elements-9303	49	12	7	7	NUM
elements-9303	49	13	.	.	PUNCT
elements-9303	49	14	by	by	ADP
elements-9303	49	15	extension	extension	NOUN
elements-9303	49	16	,	,	PUNCT
elements-9303	49	17	one	one	PRON
elements-9303	49	18	can	can	AUX
elements-9303	49	19	determine	determine	VERB
elements-9303	49	20	the	the	DET
elements-9303	49	21	number	number	NOUN
elements-9303	49	22	of	of	ADP
elements-9303	49	23	dreibens	dreiben	NOUN
elements-9303	49	24	of	of	ADP
elements-9303	49	25	length	length	NOUN
elements-9303	49	26	n	n	X
elements-9303	49	27	≡	≡	PROPN
elements-9303	49	28	0	0	PUNCT
elements-9303	50	1	(	(	PUNCT
elements-9303	50	2	mod	mod	NOUN
elements-9303	50	3	6	6	NUM
elements-9303	50	4	)	)	PUNCT
elements-9303	50	5	divisible	divisible	ADJ
elements-9303	50	6	by	by	ADP
elements-9303	50	7	7	7	NUM
elements-9303	50	8	;	;	PUNCT
elements-9303	50	9	all	all	PRON
elements-9303	50	10	of	of	ADP
elements-9303	50	11	these	these	DET
elements-9303	50	12	dreibens	dreiben	NOUN
elements-9303	50	13	are	be	AUX
elements-9303	50	14	necessarily	necessarily	ADV
elements-9303	50	15	composite	composite	ADJ
elements-9303	50	16	,	,	PUNCT
elements-9303	50	17	so	so	SCONJ
elements-9303	50	18	they	they	PRON
elements-9303	50	19	can	can	AUX
elements-9303	50	20	be	be	AUX
elements-9303	50	21	ruled	rule	VERB
elements-9303	50	22	out	out	ADP
elements-9303	50	23	during	during	ADP
elements-9303	50	24	our	our	PRON
elements-9303	50	25	search	search	NOUN
elements-9303	50	26	for	for	ADP
elements-9303	50	27	infinitely	infinitely	ADV
elements-9303	50	28	many	many	ADJ
elements-9303	50	29	prime	prime	ADJ
elements-9303	50	30	dreibens	dreiben	NOUN
elements-9303	50	31	.	.	PUNCT
elements-9303	51	1	we	we	PRON
elements-9303	51	2	can	can	AUX
elements-9303	51	3	perform	perform	VERB
elements-9303	51	4	similar	similar	ADJ
elements-9303	51	5	calculations	calculation	NOUN
elements-9303	51	6	for	for	ADP
elements-9303	51	7	an	an	DET
elements-9303	51	8	where	where	SCONJ
elements-9303	51	9	n	n	X
elements-9303	51	10	≡	≡	PROPN
elements-9303	51	11	1	1	NUM
elements-9303	51	12	,	,	PUNCT
elements-9303	51	13	2	2	NUM
elements-9303	51	14	,	,	PUNCT
elements-9303	51	15	3	3	NUM
elements-9303	51	16	,	,	PUNCT
elements-9303	51	17	4	4	NUM
elements-9303	51	18	,	,	PUNCT
elements-9303	51	19	5	5	NUM
elements-9303	51	20	(	(	PUNCT
elements-9303	51	21	mod	mod	PROPN
elements-9303	51	22	6	6	NUM
elements-9303	51	23	)	)	PUNCT
elements-9303	51	24	,	,	PUNCT
elements-9303	51	25	thereby	thereby	ADV
elements-9303	51	26	allowing	allow	VERB
elements-9303	51	27	us	we	PRON
elements-9303	51	28	to	to	PART
elements-9303	51	29	quickly	quickly	ADV
elements-9303	51	30	find	find	VERB
elements-9303	51	31	an	an	PRON
elements-9303	51	32	for	for	ADP
elements-9303	51	33	any	any	DET
elements-9303	51	34	n.	n.	NOUN
elements-9303	51	35	hence	hence	ADV
elements-9303	51	36	,	,	PUNCT
elements-9303	51	37	we	we	PRON
elements-9303	51	38	can	can	AUX
elements-9303	51	39	easily	easily	ADV
elements-9303	51	40	determine	determine	VERB
elements-9303	51	41	an	an	PRON
elements-9303	51	42	for	for	ADP
elements-9303	51	43	any	any	DET
elements-9303	51	44	n	n	NOUN
elements-9303	51	45	,	,	PUNCT
elements-9303	51	46	allowing	allow	VERB
elements-9303	51	47	us	we	PRON
elements-9303	51	48	to	to	PART
elements-9303	51	49	rule	rule	VERB
elements-9303	51	50	out	out	ADP
elements-9303	51	51	many	many	ADJ
elements-9303	51	52	composite	composite	ADJ
elements-9303	51	53	dreibens	dreiben	NOUN
elements-9303	51	54	.	.	PUNCT
elements-9303	52	1	combinatorial	combinatorial	ADJ
elements-9303	52	2	approach	approach	NOUN
elements-9303	52	3	to	to	ADP
elements-9303	52	4	dreibens	dreiben	NOUN
elements-9303	52	5	modulo	modulo	PROPN
elements-9303	52	6	7	7	NUM
elements-9303	52	7	let	let	VERB
elements-9303	52	8	us	we	PRON
elements-9303	52	9	take	take	VERB
elements-9303	52	10	a	a	DET
elements-9303	52	11	combinatorial	combinatorial	ADJ
elements-9303	52	12	approach	approach	NOUN
elements-9303	52	13	to	to	ADP
elements-9303	52	14	ai	ai	VERB
elements-9303	52	15	7(n	7(n	NUM
elements-9303	52	16	)	)	PUNCT
elements-9303	52	17	.	.	PUNCT
elements-9303	53	1	for	for	ADP
elements-9303	53	2	now	now	ADV
elements-9303	53	3	,	,	PUNCT
elements-9303	53	4	let	let	VERB
elements-9303	53	5	us	we	PRON
elements-9303	53	6	restrict	restrict	VERB
elements-9303	53	7	ourselves	ourselves	PRON
elements-9303	53	8	to	to	ADP
elements-9303	53	9	a0	a0	PROPN
elements-9303	53	10	7(n	7(n	NUM
elements-9303	53	11	)	)	PUNCT
elements-9303	53	12	;	;	PUNCT
elements-9303	53	13	in	in	ADP
elements-9303	53	14	other	other	ADJ
elements-9303	53	15	words	word	NOUN
elements-9303	53	16	,	,	PUNCT
elements-9303	53	17	we	we	PRON
elements-9303	53	18	are	be	AUX
elements-9303	53	19	looking	look	VERB
elements-9303	53	20	for	for	ADP
elements-9303	53	21	when	when	SCONJ
elements-9303	53	22	a	a	DET
elements-9303	53	23	dreiben	dreiben	NOUN
elements-9303	53	24	of	of	ADP
elements-9303	53	25	length	length	NOUN
elements-9303	53	26	n	n	NUM
elements-9303	53	27	is	be	AUX
elements-9303	53	28	divisible	divisible	ADJ
elements-9303	53	29	by	by	ADP
elements-9303	53	30	7	7	NUM
elements-9303	53	31	.	.	PUNCT
elements-9303	54	1	these	these	DET
elements-9303	54	2	dreibens	dreiben	NOUN
elements-9303	54	3	are	be	AUX
elements-9303	54	4	obviously	obviously	ADV
elements-9303	54	5	composite	composite	ADJ
elements-9303	54	6	,	,	PUNCT
elements-9303	54	7	so	so	ADV
elements-9303	54	8	learning	learn	VERB
elements-9303	54	9	to	to	PART
elements-9303	54	10	identify	identify	VERB
elements-9303	54	11	them	they	PRON
elements-9303	54	12	will	will	AUX
elements-9303	54	13	help	help	VERB
elements-9303	54	14	rule	rule	VERB
elements-9303	54	15	out	out	ADP
elements-9303	54	16	composite	composite	ADJ
elements-9303	54	17	dreibens	dreiben	NOUN
elements-9303	54	18	in	in	ADP
elements-9303	54	19	a	a	DET
elements-9303	54	20	prime	prime	ADJ
elements-9303	54	21	dreiben	dreiben	NOUN
elements-9303	54	22	search	search	NOUN
elements-9303	54	23	.	.	PUNCT
elements-9303	55	1	94	94	NUM
elements-9303	55	2	elements	element	NOUN
elements-9303	55	3	:	:	PUNCT
elements-9303	55	4	:	:	PUNCT
elements-9303	55	5	spring	spring	NOUN
elements-9303	55	6	2016	2016	NUM
elements-9303	55	7	here	here	ADV
elements-9303	55	8	,	,	PUNCT
elements-9303	55	9	we	we	PRON
elements-9303	55	10	will	will	AUX
elements-9303	55	11	use	use	VERB
elements-9303	55	12	the	the	DET
elements-9303	55	13	divisibility	divisibility	NOUN
elements-9303	55	14	rule	rule	NOUN
elements-9303	55	15	for	for	ADP
elements-9303	55	16	7	7	NUM
elements-9303	55	17	to	to	PART
elements-9303	55	18	aid	aid	VERB
elements-9303	55	19	us	we	PRON
elements-9303	55	20	.	.	PUNCT
elements-9303	56	1	recall	recall	VERB
elements-9303	56	2	the	the	DET
elements-9303	56	3	divisibility	divisibility	NOUN
elements-9303	56	4	rule	rule	NOUN
elements-9303	56	5	for	for	ADP
elements-9303	56	6	7	7	NUM
elements-9303	56	7	:	:	PUNCT
elements-9303	56	8	notice	notice	VERB
elements-9303	56	9	that	that	SCONJ
elements-9303	56	10	the	the	DET
elements-9303	56	11	powers	power	NOUN
elements-9303	56	12	of	of	ADP
elements-9303	56	13	10	10	NUM
elements-9303	56	14	(	(	PUNCT
elements-9303	56	15	mod	mod	PROPN
elements-9303	56	16	7	7	NUM
elements-9303	56	17	)	)	PUNCT
elements-9303	56	18	forms	form	VERB
elements-9303	56	19	a	a	DET
elements-9303	56	20	cyclic	cyclic	ADJ
elements-9303	56	21	group	group	NOUN
elements-9303	56	22	of	of	ADP
elements-9303	56	23	order	order	NOUN
elements-9303	56	24	6	6	NUM
elements-9303	56	25	.	.	PUNCT
elements-9303	57	1	let	let	VERB
elements-9303	57	2	us	we	PRON
elements-9303	57	3	use	use	VERB
elements-9303	57	4	the	the	DET
elements-9303	57	5	above	above	ADJ
elements-9303	57	6	divisibility	divisibility	NOUN
elements-9303	57	7	rule	rule	NOUN
elements-9303	57	8	to	to	PART
elements-9303	57	9	determine	determine	VERB
elements-9303	57	10	,	,	PUNCT
elements-9303	57	11	as	as	ADP
elements-9303	57	12	an	an	DET
elements-9303	57	13	example	example	NOUN
elements-9303	57	14	,	,	PUNCT
elements-9303	57	15	9487310294	9487310294	NUM
elements-9303	57	16	(	(	PUNCT
elements-9303	57	17	mod	mod	PROPN
elements-9303	57	18	7	7	NUM
elements-9303	57	19	):	):	PUNCT
elements-9303	57	20	dreibens	dreiben	NOUN
elements-9303	57	21	consist	consist	NOUN
elements-9303	57	22	of	of	ADP
elements-9303	57	23	only	only	ADV
elements-9303	57	24	7	7	NUM
elements-9303	57	25	’s	’s	NOUN
elements-9303	57	26	and	and	CCONJ
elements-9303	57	27	3	3	NUM
elements-9303	57	28	’s	’s	NOUN
elements-9303	57	29	;	;	PUNCT
elements-9303	57	30	hence	hence	ADV
elements-9303	57	31	,	,	PUNCT
elements-9303	57	32	the	the	DET
elements-9303	57	33	divisibility	divisibility	NOUN
elements-9303	57	34	of	of	ADP
elements-9303	57	35	a	a	DET
elements-9303	57	36	dreiben	dreiben	NOUN
elements-9303	57	37	by	by	ADP
elements-9303	57	38	7	7	NUM
elements-9303	57	39	depends	depend	VERB
elements-9303	57	40	on	on	ADP
elements-9303	57	41	both	both	CCONJ
elements-9303	57	42	the	the	DET
elements-9303	57	43	number	number	NOUN
elements-9303	57	44	and	and	CCONJ
elements-9303	57	45	placement	placement	NOUN
elements-9303	57	46	of	of	ADP
elements-9303	57	47	3	3	NUM
elements-9303	57	48	’s	’s	NOUN
elements-9303	57	49	amongst	amongst	ADP
elements-9303	57	50	its	its	PRON
elements-9303	57	51	digits	digit	NOUN
elements-9303	57	52	.	.	PUNCT
elements-9303	58	1	(	(	PUNCT
elements-9303	58	2	neither	neither	CCONJ
elements-9303	58	3	the	the	DET
elements-9303	58	4	number	number	NOUN
elements-9303	58	5	nor	nor	CCONJ
elements-9303	58	6	placement	placement	NOUN
elements-9303	58	7	of	of	ADP
elements-9303	58	8	7	7	NUM
elements-9303	58	9	’s	’s	NOUN
elements-9303	58	10	in	in	ADP
elements-9303	58	11	a	a	DET
elements-9303	58	12	dreiben	dreiben	NOUN
elements-9303	58	13	affect	affect	VERB
elements-9303	58	14	its	its	PRON
elements-9303	58	15	divisibility	divisibility	NOUN
elements-9303	58	16	by	by	ADP
elements-9303	58	17	7	7	NUM
elements-9303	58	18	,	,	PUNCT
elements-9303	58	19	as	as	SCONJ
elements-9303	58	20	we	we	PRON
elements-9303	58	21	can	can	AUX
elements-9303	58	22	tell	tell	VERB
elements-9303	58	23	from	from	ADP
elements-9303	58	24	the	the	DET
elements-9303	58	25	divisibility	divisibility	NOUN
elements-9303	58	26	rule	rule	NOUN
elements-9303	58	27	for	for	ADP
elements-9303	58	28	7	7	NUM
elements-9303	58	29	.	.	PUNCT
elements-9303	58	30	)	)	PUNCT
elements-9303	59	1	let	let	VERB
elements-9303	59	2	dn	dn	PROPN
elements-9303	59	3	j(i	j(i	PROPN
elements-9303	59	4	)	)	PUNCT
elements-9303	59	5	be	be	VERB
elements-9303	59	6	the	the	DET
elements-9303	59	7	number	number	NOUN
elements-9303	59	8	of	of	ADP
elements-9303	59	9	dreibens	dreiben	NOUN
elements-9303	59	10	of	of	ADP
elements-9303	59	11	length	length	NOUN
elements-9303	59	12	n	n	PROPN
elements-9303	59	13	with	with	ADP
elements-9303	59	14	i	i	PRON
elements-9303	59	15	number	number	NOUN
elements-9303	59	16	of	of	ADP
elements-9303	59	17	j	j	PROPN
elements-9303	59	18	’s	’s	X
elements-9303	59	19	in	in	ADP
elements-9303	59	20	it	it	PRON
elements-9303	59	21	,	,	PUNCT
elements-9303	59	22	e.g.	e.g.	ADV
elements-9303	59	23	dn	dn	ADP
elements-9303	59	24	3(2	3(2	NUM
elements-9303	59	25	)	)	PUNCT
elements-9303	59	26	is	be	AUX
elements-9303	59	27	the	the	DET
elements-9303	59	28	number	number	NOUN
elements-9303	59	29	of	of	ADP
elements-9303	59	30	dreibens	dreiben	NOUN
elements-9303	59	31	of	of	ADP
elements-9303	59	32	length	length	NOUN
elements-9303	59	33	n	n	CCONJ
elements-9303	59	34	where	where	SCONJ
elements-9303	59	35	3	3	NUM
elements-9303	59	36	occupies	occupy	VERB
elements-9303	59	37	exactly	exactly	ADV
elements-9303	59	38	2	2	NUM
elements-9303	59	39	digits	digit	NOUN
elements-9303	59	40	of	of	ADP
elements-9303	59	41	each	each	DET
elements-9303	59	42	dreiben	dreiben	NOUN
elements-9303	59	43	.	.	PUNCT
elements-9303	60	1	since	since	SCONJ
elements-9303	60	2	the	the	DET
elements-9303	60	3	divisibility	divisibility	NOUN
elements-9303	60	4	of	of	ADP
elements-9303	60	5	a	a	DET
elements-9303	60	6	dreiben	dreiben	NOUN
elements-9303	60	7	by	by	ADP
elements-9303	60	8	7	7	NUM
elements-9303	60	9	depends	depend	VERB
elements-9303	60	10	on	on	ADP
elements-9303	60	11	the	the	DET
elements-9303	60	12	number	number	NOUN
elements-9303	60	13	and	and	CCONJ
elements-9303	60	14	placement	placement	NOUN
elements-9303	60	15	of	of	ADP
elements-9303	60	16	3	3	NUM
elements-9303	60	17	’s	’s	NOUN
elements-9303	60	18	amongst	amongst	ADP
elements-9303	60	19	that	that	DET
elements-9303	60	20	dreiben	dreiben	PROPN
elements-9303	60	21	’s	’s	PART
elements-9303	60	22	digits	digit	NOUN
elements-9303	60	23	,	,	PUNCT
elements-9303	60	24	we	we	PRON
elements-9303	60	25	would	would	AUX
elements-9303	60	26	stand	stand	VERB
elements-9303	60	27	to	to	PART
elements-9303	60	28	gain	gain	VERB
elements-9303	60	29	from	from	ADP
elements-9303	60	30	investigating	investigate	VERB
elements-9303	60	31	dn	dn	PROPN
elements-9303	60	32	3(i	3(i	NUM
elements-9303	60	33	)	)	PUNCT
elements-9303	60	34	.	.	PUNCT
elements-9303	61	1	as	as	SCONJ
elements-9303	61	2	you	you	PRON
elements-9303	61	3	may	may	AUX
elements-9303	61	4	have	have	AUX
elements-9303	61	5	realized	realize	VERB
elements-9303	61	6	already	already	ADV
elements-9303	61	7	,	,	PUNCT
elements-9303	61	8	evidently	evidently	ADV
elements-9303	61	9	,	,	PUNCT
elements-9303	61	10	by	by	ADP
elements-9303	61	11	examining	examine	VERB
elements-9303	61	12	dn	dn	PROPN
elements-9303	61	13	3(i	3(i	NUM
elements-9303	61	14	)	)	PUNCT
elements-9303	61	15	,	,	PUNCT
elements-9303	61	16	we	we	PRON
elements-9303	61	17	can	can	AUX
elements-9303	61	18	approach	approach	VERB
elements-9303	61	19	the	the	DET
elements-9303	61	20	problem	problem	NOUN
elements-9303	61	21	of	of	ADP
elements-9303	61	22	determining	determine	VERB
elements-9303	61	23	the	the	DET
elements-9303	61	24	number	number	NOUN
elements-9303	61	25	of	of	ADP
elements-9303	61	26	dreibens	dreiben	NOUN
elements-9303	61	27	of	of	ADP
elements-9303	61	28	length	length	NOUN
elements-9303	61	29	n	n	CCONJ
elements-9303	61	30	divisible	divisible	ADJ
elements-9303	61	31	by	by	ADP
elements-9303	61	32	7	7	NUM
elements-9303	61	33	.	.	PUNCT
elements-9303	62	1	our	our	PRON
elements-9303	62	2	approach	approach	NOUN
elements-9303	62	3	will	will	AUX
elements-9303	62	4	involve	involve	VERB
elements-9303	62	5	investigating	investigate	VERB
elements-9303	62	6	dn	dn	ADP
elements-9303	62	7	3(i	3(i	NUM
elements-9303	62	8	)	)	PUNCT
elements-9303	62	9	with	with	ADP
elements-9303	62	10	varying	vary	VERB
elements-9303	62	11	i	i	PRON
elements-9303	62	12	to	to	PART
elements-9303	62	13	see	see	VERB
elements-9303	62	14	if	if	SCONJ
elements-9303	62	15	we	we	PRON
elements-9303	62	16	can	can	AUX
elements-9303	62	17	make	make	VERB
elements-9303	62	18	any	any	DET
elements-9303	62	19	generalizations	generalization	NOUN
elements-9303	62	20	about	about	ADP
elements-9303	62	21	dn	dn	PROPN
elements-9303	62	22	3(i	3(i	NUM
elements-9303	62	23	)	)	PUNCT
elements-9303	62	24	.	.	PUNCT
elements-9303	63	1	first	first	ADV
elements-9303	63	2	,	,	PUNCT
elements-9303	63	3	let	let	VERB
elements-9303	63	4	us	we	PRON
elements-9303	63	5	examine	examine	VERB
elements-9303	63	6	dn	dn	PROPN
elements-9303	63	7	3(0	3(0	NUM
elements-9303	63	8	)	)	PUNCT
elements-9303	63	9	.	.	PUNCT
elements-9303	64	1	there	there	PRON
elements-9303	64	2	is	be	VERB
elements-9303	64	3	only	only	ADV
elements-9303	64	4	one	one	NUM
elements-9303	64	5	way	way	NOUN
elements-9303	64	6	to	to	PART
elements-9303	64	7	put	put	VERB
elements-9303	64	8	exactly	exactly	ADV
elements-9303	64	9	zero	zero	NUM
elements-9303	64	10	3	3	NUM
elements-9303	64	11	’s	’s	NOUN
elements-9303	64	12	in	in	ADP
elements-9303	64	13	a	a	DET
elements-9303	64	14	dreiben	dreiben	NOUN
elements-9303	64	15	;	;	PUNCT
elements-9303	64	16	every	every	DET
elements-9303	64	17	digit	digit	NOUN
elements-9303	64	18	of	of	ADP
elements-9303	64	19	that	that	DET
elements-9303	64	20	dreiben	dreiben	NOUN
elements-9303	64	21	must	must	AUX
elements-9303	64	22	be	be	AUX
elements-9303	64	23	7	7	NUM
elements-9303	64	24	.	.	PUNCT
elements-9303	65	1	obviously	obviously	ADV
elements-9303	65	2	,	,	PUNCT
elements-9303	65	3	if	if	SCONJ
elements-9303	65	4	a	a	DET
elements-9303	65	5	number	number	NOUN
elements-9303	65	6	’s	’s	PART
elements-9303	65	7	digits	digit	NOUN
elements-9303	65	8	are	be	AUX
elements-9303	65	9	all	all	PRON
elements-9303	65	10	7	7	NUM
elements-9303	65	11	’s	’s	NOUN
elements-9303	65	12	,	,	PUNCT
elements-9303	65	13	then	then	ADV
elements-9303	65	14	the	the	DET
elements-9303	65	15	number	number	NOUN
elements-9303	65	16	is	be	AUX
elements-9303	65	17	divisible	divisible	ADJ
elements-9303	65	18	by	by	ADP
elements-9303	65	19	7	7	NUM
elements-9303	65	20	.	.	PUNCT
elements-9303	66	1	since	since	SCONJ
elements-9303	66	2	all	all	PRON
elements-9303	66	3	of	of	ADP
elements-9303	66	4	the	the	DET
elements-9303	66	5	digits	digit	NOUN
elements-9303	66	6	of	of	ADP
elements-9303	66	7	a	a	DET
elements-9303	66	8	dreiben	dreiben	NOUN
elements-9303	66	9	with	with	ADP
elements-9303	66	10	zero	zero	NUM
elements-9303	66	11	3	3	NUM
elements-9303	66	12	’s	’s	PART
elements-9303	66	13	is	be	AUX
elements-9303	66	14	7	7	NUM
elements-9303	66	15	,	,	PUNCT
elements-9303	66	16	and	and	CCONJ
elements-9303	66	17	since	since	SCONJ
elements-9303	66	18	there	there	PRON
elements-9303	66	19	is	be	VERB
elements-9303	66	20	only	only	ADV
elements-9303	66	21	one	one	NUM
elements-9303	66	22	way	way	NOUN
elements-9303	66	23	to	to	PART
elements-9303	66	24	make	make	VERB
elements-9303	66	25	a	a	DET
elements-9303	66	26	dreiben	dreiben	NOUN
elements-9303	66	27	of	of	ADP
elements-9303	66	28	length	length	NOUN
elements-9303	66	29	n	n	PROPN
elements-9303	66	30	with	with	ADP
elements-9303	66	31	zero	zero	NUM
elements-9303	66	32	3	3	NUM
elements-9303	66	33	’s	’s	ADV
elements-9303	66	34	,	,	PUNCT
elements-9303	66	35	we	we	PRON
elements-9303	66	36	can	can	AUX
elements-9303	66	37	easily	easily	ADV
elements-9303	66	38	see	see	VERB
elements-9303	66	39	that	that	PRON
elements-9303	66	40	dn	dn	PROPN
elements-9303	66	41	3(0	3(0	NUM
elements-9303	66	42	)	)	PUNCT
elements-9303	67	1	=	=	PUNCT
elements-9303	68	1	1	1	X
elements-9303	68	2	.	.	PUNCT
elements-9303	69	1	next	next	ADV
elements-9303	69	2	,	,	PUNCT
elements-9303	69	3	we	we	PRON
elements-9303	69	4	can	can	AUX
elements-9303	69	5	examine	examine	VERB
elements-9303	69	6	dn	dn	NOUN
elements-9303	69	7	3(1	3(1	NUM
elements-9303	69	8	)	)	PUNCT
elements-9303	70	1	=	=	SYM
elements-9303	70	2	0	0	X
elements-9303	70	3	.	.	PUNCT
elements-9303	71	1	there	there	PRON
elements-9303	71	2	are	be	VERB
elements-9303	71	3	no	no	DET
elements-9303	71	4	dreibens	dreiben	NOUN
elements-9303	71	5	with	with	ADP
elements-9303	71	6	only	only	ADV
elements-9303	71	7	one	one	NUM
elements-9303	71	8	3	3	NUM
elements-9303	71	9	,	,	PUNCT
elements-9303	71	10	since	since	SCONJ
elements-9303	71	11	3	3	NUM
elements-9303	71	12	times	time	NOUN
elements-9303	71	13	any	any	DET
elements-9303	71	14	digit	digit	NOUN
elements-9303	71	15	place	place	NOUN
elements-9303	71	16	10x	10x	NUM
elements-9303	71	17	is	be	AUX
elements-9303	71	18	never	never	ADV
elements-9303	71	19	divisible	divisible	ADJ
elements-9303	71	20	by	by	ADP
elements-9303	71	21	7	7	NUM
elements-9303	71	22	,	,	PUNCT
elements-9303	71	23	so	so	ADV
elements-9303	71	24	dn	dn	PROPN
elements-9303	71	25	3(1	3(1	NUM
elements-9303	71	26	)	)	PUNCT
elements-9303	72	1	=	=	SYM
elements-9303	72	2	0	0	X
elements-9303	72	3	.	.	PUNCT
elements-9303	73	1	so	so	ADV
elements-9303	73	2	far	far	ADV
elements-9303	73	3	,	,	PUNCT
elements-9303	73	4	we	we	PRON
elements-9303	73	5	have	have	AUX
elements-9303	73	6	covered	cover	VERB
elements-9303	73	7	two	two	NUM
elements-9303	73	8	very	very	ADV
elements-9303	73	9	simple	simple	ADJ
elements-9303	73	10	instances	instance	NOUN
elements-9303	73	11	of	of	ADP
elements-9303	73	12	dn	dn	NOUN
elements-9303	73	13	3(i	3(i	NUM
elements-9303	73	14	)	)	PUNCT
elements-9303	73	15	.	.	PUNCT
elements-9303	74	1	when	when	SCONJ
elements-9303	74	2	i	i	PRON
elements-9303	74	3	=	=	NOUN
elements-9303	74	4	0	0	NUM
elements-9303	74	5	,	,	PUNCT
elements-9303	74	6	there	there	PRON
elements-9303	74	7	is	be	VERB
elements-9303	74	8	only	only	ADV
elements-9303	74	9	a	a	DET
elements-9303	74	10	single	single	ADJ
elements-9303	74	11	way	way	NOUN
elements-9303	74	12	to	to	PART
elements-9303	74	13	make	make	VERB
elements-9303	74	14	a	a	DET
elements-9303	74	15	dreiben	dreiben	NOUN
elements-9303	74	16	divisible	divisible	ADJ
elements-9303	74	17	by	by	ADP
elements-9303	74	18	7	7	NUM
elements-9303	74	19	.	.	PUNCT
elements-9303	75	1	when	when	SCONJ
elements-9303	75	2	i	i	PRON
elements-9303	75	3	=	=	NOUN
elements-9303	75	4	1	1	NUM
elements-9303	75	5	,	,	PUNCT
elements-9303	75	6	it	it	PRON
elements-9303	75	7	is	be	AUX
elements-9303	75	8	impossible	impossible	ADJ
elements-9303	75	9	to	to	PART
elements-9303	75	10	make	make	VERB
elements-9303	75	11	a	a	DET
elements-9303	75	12	dreiben	dreiben	NOUN
elements-9303	75	13	divisible	divisible	ADJ
elements-9303	75	14	by	by	ADP
elements-9303	75	15	7	7	NUM
elements-9303	75	16	.	.	PUNCT
elements-9303	76	1	however	however	ADV
elements-9303	76	2	,	,	PUNCT
elements-9303	76	3	as	as	SCONJ
elements-9303	76	4	i	i	PRON
elements-9303	76	5	increases	increase	VERB
elements-9303	76	6	,	,	PUNCT
elements-9303	76	7	the	the	DET
elements-9303	76	8	problem	problem	NOUN
elements-9303	76	9	of	of	ADP
elements-9303	76	10	describing	describe	VERB
elements-9303	76	11	dn	dn	PROPN
elements-9303	76	12	3(i	3(i	NUM
elements-9303	76	13	)	)	PUNCT
elements-9303	76	14	becomes	become	VERB
elements-9303	76	15	much	much	ADV
elements-9303	76	16	more	more	ADV
elements-9303	76	17	complicated	complicated	ADJ
elements-9303	76	18	.	.	PUNCT
elements-9303	77	1	the	the	DET
elements-9303	77	2	next	next	ADJ
elements-9303	77	3	question	question	NOUN
elements-9303	77	4	in	in	ADP
elements-9303	77	5	our	our	PRON
elements-9303	77	6	investigation	investigation	NOUN
elements-9303	77	7	of	of	ADP
elements-9303	77	8	dn	dn	PROPN
elements-9303	77	9	3(i	3(i	NUM
elements-9303	77	10	)	)	PUNCT
elements-9303	77	11	involves	involve	VERB
elements-9303	77	12	i	i	NOUN
elements-9303	77	13	=	=	NOUN
elements-9303	77	14	2	2	X
elements-9303	77	15	.	.	PUNCT
elements-9303	78	1	in	in	ADP
elements-9303	78	2	other	other	ADJ
elements-9303	78	3	words	word	NOUN
elements-9303	78	4	,	,	PUNCT
elements-9303	78	5	how	how	SCONJ
elements-9303	78	6	many	many	ADJ
elements-9303	78	7	ways	way	NOUN
elements-9303	78	8	are	be	AUX
elements-9303	78	9	there	there	ADV
elements-9303	78	10	to	to	PART
elements-9303	78	11	put	put	VERB
elements-9303	78	12	exactly	exactly	ADV
elements-9303	78	13	two	two	NUM
elements-9303	78	14	3	3	NUM
elements-9303	78	15	’s	’s	NOUN
elements-9303	78	16	in	in	ADP
elements-9303	78	17	a	a	DET
elements-9303	78	18	dreiben	dreiben	NOUN
elements-9303	78	19	of	of	ADP
elements-9303	78	20	length	length	NOUN
elements-9303	78	21	n	n	CCONJ
elements-9303	78	22	?	?	PUNCT
elements-9303	79	1	putting	put	VERB
elements-9303	79	2	exactly	exactly	ADV
elements-9303	79	3	two	two	NUM
elements-9303	79	4	3	3	NUM
elements-9303	79	5	’s	’s	NOUN
elements-9303	79	6	in	in	ADP
elements-9303	79	7	a	a	DET
elements-9303	79	8	dreiben	dreiben	NOUN
elements-9303	79	9	of	of	ADP
elements-9303	79	10	length	length	NOUN
elements-9303	79	11	n	n	PROPN
elements-9303	79	12	means	mean	VERB
elements-9303	79	13	we	we	PRON
elements-9303	79	14	must	must	AUX
elements-9303	79	15	choose	choose	VERB
elements-9303	79	16	two	two	NUM
elements-9303	79	17	digit	digit	NOUN
elements-9303	79	18	places	place	NOUN
elements-9303	79	19	10x	10x	NUM
elements-9303	79	20	and	and	CCONJ
elements-9303	79	21	10y	10y	PROPN
elements-9303	80	1	where	where	SCONJ
elements-9303	80	2	x	x	SYM
elements-9303	80	3	≠	≠	PROPN
elements-9303	80	4	y	y	PROPN
elements-9303	80	5	and	and	CCONJ
elements-9303	80	6	0	0	NUM
elements-9303	80	7	≤	≤	NUM
elements-9303	80	8	x	x	X
elements-9303	80	9	,	,	PUNCT
elements-9303	80	10	y	y	PROPN
elements-9303	80	11	≤	≤	PROPN
elements-9303	80	12	n	n	PRON
elements-9303	80	13	1	1	NUM
elements-9303	80	14	.	.	PUNCT
elements-9303	80	15	putting	put	VERB
elements-9303	80	16	our	our	PRON
elements-9303	80	17	two	two	NUM
elements-9303	80	18	3	3	NUM
elements-9303	80	19	’s	’s	NOUN
elements-9303	80	20	in	in	ADP
elements-9303	80	21	these	these	DET
elements-9303	80	22	two	two	NUM
elements-9303	80	23	digit	digit	NOUN
elements-9303	80	24	places	place	NOUN
elements-9303	80	25	,	,	PUNCT
elements-9303	80	26	we	we	PRON
elements-9303	80	27	find	find	VERB
elements-9303	80	28	that	that	SCONJ
elements-9303	80	29	the	the	DET
elements-9303	80	30	dreiben	dreiben	NOUN
elements-9303	80	31	is	be	AUX
elements-9303	80	32	congruent	congruent	ADJ
elements-9303	80	33	to	to	ADP
elements-9303	80	34	3(10x	3(10x	NUM
elements-9303	80	35	+	+	NUM
elements-9303	80	36	10y	10y	NUM
elements-9303	80	37	)	)	PUNCT
elements-9303	80	38	(	(	PUNCT
elements-9303	80	39	mod	mod	PROPN
elements-9303	80	40	7	7	NUM
elements-9303	80	41	)	)	PUNCT
elements-9303	80	42	.	.	PUNCT
elements-9303	81	1	clearly	clearly	ADV
elements-9303	81	2	,	,	PUNCT
elements-9303	81	3	the	the	DET
elements-9303	81	4	dreiben	dreiben	NOUN
elements-9303	81	5	is	be	AUX
elements-9303	81	6	divisible	divisible	ADJ
elements-9303	81	7	by	by	ADP
elements-9303	81	8	7	7	NUM
elements-9303	81	9	if	if	SCONJ
elements-9303	81	10	and	and	CCONJ
elements-9303	81	11	only	only	ADV
elements-9303	81	12	if	if	SCONJ
elements-9303	81	13	10x	10x	NOUN
elements-9303	81	14	+	+	CCONJ
elements-9303	81	15	10y	10y	PROPN
elements-9303	81	16	is	be	AUX
elements-9303	81	17	divisible	divisible	ADJ
elements-9303	81	18	by	by	ADP
elements-9303	81	19	7	7	NUM
elements-9303	81	20	,	,	PUNCT
elements-9303	81	21	so	so	SCONJ
elements-9303	81	22	we	we	PRON
elements-9303	81	23	need	need	AUX
elements-9303	81	24	only	only	ADV
elements-9303	81	25	consider	consider	VERB
elements-9303	81	26	10x	10x	NOUN
elements-9303	81	27	+	+	CCONJ
elements-9303	81	28	10y	10y	PROPN
elements-9303	81	29	(	(	PUNCT
elements-9303	81	30	mod	mod	PROPN
elements-9303	81	31	7	7	NUM
elements-9303	81	32	)	)	PUNCT
elements-9303	81	33	.	.	PUNCT
elements-9303	82	1	what	what	PRON
elements-9303	82	2	are	be	AUX
elements-9303	82	3	the	the	DET
elements-9303	82	4	possible	possible	ADJ
elements-9303	82	5	ways	way	NOUN
elements-9303	82	6	to	to	PART
elements-9303	82	7	make	make	VERB
elements-9303	82	8	10x	10x	NOUN
elements-9303	82	9	+	+	CCONJ
elements-9303	82	10	10y	10y	PROPN
elements-9303	82	11	≡	≡	PROPN
elements-9303	82	12	0	0	PUNCT
elements-9303	83	1	(	(	PUNCT
elements-9303	83	2	mod	mod	PROPN
elements-9303	83	3	7	7	NUM
elements-9303	83	4	)	)	PUNCT
elements-9303	83	5	?	?	PUNCT
elements-9303	84	1	the	the	DET
elements-9303	84	2	powers	power	NOUN
elements-9303	84	3	of	of	ADP
elements-9303	84	4	10	10	NUM
elements-9303	84	5	modulo	modulo	NOUN
elements-9303	84	6	7	7	NUM
elements-9303	84	7	form	form	NOUN
elements-9303	84	8	a	a	DET
elements-9303	84	9	cyclic	cyclic	ADJ
elements-9303	84	10	group	group	NOUN
elements-9303	84	11	of	of	ADP
elements-9303	84	12	order	order	NOUN
elements-9303	84	13	6	6	NUM
elements-9303	84	14	;	;	PUNCT
elements-9303	84	15	without	without	ADP
elements-9303	84	16	loss	loss	NOUN
elements-9303	84	17	of	of	ADP
elements-9303	84	18	generality	generality	NOUN
elements-9303	84	19	,	,	PUNCT
elements-9303	84	20	the	the	DET
elements-9303	84	21	possible	possible	ADJ
elements-9303	84	22	solutions	solution	NOUN
elements-9303	84	23	are	be	AUX
elements-9303	84	24	(	(	PUNCT
elements-9303	84	25	a	a	PRON
elements-9303	84	26	)	)	PUNCT
elements-9303	84	27	10x	10x	NUM
elements-9303	84	28	≡	≡	PROPN
elements-9303	84	29	1	1	NUM
elements-9303	84	30	(	(	PUNCT
elements-9303	84	31	mod	mod	PROPN
elements-9303	84	32	7	7	NUM
elements-9303	84	33	)	)	PUNCT
elements-9303	84	34	,	,	PUNCT
elements-9303	84	35	10y	10y	PROPN
elements-9303	84	36	≡	≡	PROPN
elements-9303	84	37	6	6	NUM
elements-9303	84	38	(	(	PUNCT
elements-9303	84	39	mod	mod	PROPN
elements-9303	84	40	7	7	NUM
elements-9303	84	41	)	)	PUNCT
elements-9303	84	42	(	(	PUNCT
elements-9303	84	43	b	b	NOUN
elements-9303	84	44	)	)	PUNCT
elements-9303	84	45	10x	10x	X
elements-9303	85	1	≡	≡	PROPN
elements-9303	85	2	2	2	NUM
elements-9303	85	3	(	(	PUNCT
elements-9303	85	4	mod	mod	PROPN
elements-9303	85	5	7	7	NUM
elements-9303	85	6	)	)	PUNCT
elements-9303	85	7	,	,	PUNCT
elements-9303	85	8	10y	10y	PROPN
elements-9303	85	9	≡	≡	PROPN
elements-9303	85	10	5	5	NUM
elements-9303	85	11	(	(	PUNCT
elements-9303	85	12	mod	mod	NOUN
elements-9303	85	13	7	7	NUM
elements-9303	85	14	)	)	PUNCT
elements-9303	85	15	(	(	PUNCT
elements-9303	85	16	c	c	NOUN
elements-9303	85	17	)	)	PUNCT
elements-9303	85	18	10x	10x	NUM
elements-9303	85	19	≡	≡	PROPN
elements-9303	85	20	3	3	NUM
elements-9303	85	21	(	(	PUNCT
elements-9303	85	22	mod	mod	PROPN
elements-9303	85	23	7	7	NUM
elements-9303	85	24	)	)	PUNCT
elements-9303	85	25	,	,	PUNCT
elements-9303	85	26	10y	10y	PROPN
elements-9303	85	27	≡	≡	PROPN
elements-9303	85	28	4	4	NUM
elements-9303	85	29	(	(	PUNCT
elements-9303	85	30	mod	mod	PROPN
elements-9303	85	31	7	7	NUM
elements-9303	85	32	)	)	PUNCT
elements-9303	85	33	another	another	DET
elements-9303	85	34	way	way	NOUN
elements-9303	85	35	to	to	PART
elements-9303	85	36	write	write	VERB
elements-9303	85	37	the	the	DET
elements-9303	85	38	possible	possible	ADJ
elements-9303	85	39	solutions	solution	NOUN
elements-9303	85	40	is	be	AUX
elements-9303	85	41	(	(	PUNCT
elements-9303	85	42	a	a	X
elements-9303	85	43	)	)	PUNCT
elements-9303	85	44	x	x	SYM
elements-9303	85	45	≡	≡	PROPN
elements-9303	85	46	0	0	PUNCT
elements-9303	86	1	(	(	PUNCT
elements-9303	86	2	mod	mod	PROPN
elements-9303	86	3	6	6	NUM
elements-9303	86	4	)	)	PUNCT
elements-9303	86	5	,	,	PUNCT
elements-9303	86	6	y	y	PROPN
elements-9303	86	7	≡	≡	PROPN
elements-9303	86	8	3	3	NUM
elements-9303	86	9	(	(	PUNCT
elements-9303	86	10	mod	mod	PROPN
elements-9303	86	11	6	6	NUM
elements-9303	86	12	)	)	PUNCT
elements-9303	86	13	(	(	PUNCT
elements-9303	86	14	b	b	X
elements-9303	86	15	)	)	PUNCT
elements-9303	86	16	x	x	SYM
elements-9303	86	17	≡	≡	PROPN
elements-9303	86	18	2	2	NUM
elements-9303	86	19	(	(	PUNCT
elements-9303	86	20	mod	mod	PROPN
elements-9303	86	21	6	6	NUM
elements-9303	86	22	)	)	PUNCT
elements-9303	86	23	,	,	PUNCT
elements-9303	86	24	y	y	PROPN
elements-9303	86	25	≡	≡	PROPN
elements-9303	86	26	5	5	NUM
elements-9303	86	27	(	(	PUNCT
elements-9303	86	28	mod	mod	PROPN
elements-9303	86	29	6	6	NUM
elements-9303	86	30	)	)	PUNCT
elements-9303	86	31	(	(	PUNCT
elements-9303	86	32	c	c	X
elements-9303	86	33	)	)	PUNCT
elements-9303	86	34	x	x	SYM
elements-9303	86	35	≡	≡	PROPN
elements-9303	86	36	1	1	NUM
elements-9303	86	37	(	(	PUNCT
elements-9303	86	38	mod	mod	PROPN
elements-9303	86	39	6	6	NUM
elements-9303	86	40	)	)	PUNCT
elements-9303	86	41	,	,	PUNCT
elements-9303	86	42	y	y	PROPN
elements-9303	86	43	≡	≡	PROPN
elements-9303	86	44	4	4	NUM
elements-9303	86	45	(	(	PUNCT
elements-9303	86	46	mod	mod	PROPN
elements-9303	86	47	6	6	NUM
elements-9303	86	48	)	)	PUNCT
elements-9303	86	49	for	for	ADP
elements-9303	86	50	each	each	DET
elements-9303	86	51	solution	solution	NOUN
elements-9303	86	52	,	,	PUNCT
elements-9303	86	53	we	we	PRON
elements-9303	86	54	choose	choose	VERB
elements-9303	86	55	one	one	NUM
elements-9303	86	56	digit	digit	NOUN
elements-9303	86	57	place	place	NOUN
elements-9303	86	58	each	each	PRON
elements-9303	86	59	from	from	ADP
elements-9303	86	60	two	two	NUM
elements-9303	86	61	sets	set	NOUN
elements-9303	86	62	of	of	ADP
elements-9303	86	63	digit	digit	NOUN
elements-9303	86	64	places	place	NOUN
elements-9303	86	65	.	.	PUNCT
elements-9303	87	1	here	here	ADV
elements-9303	87	2	,	,	PUNCT
elements-9303	87	3	we	we	PRON
elements-9303	87	4	introduce	introduce	VERB
elements-9303	87	5	some	some	DET
elements-9303	87	6	notation	notation	NOUN
elements-9303	87	7	that	that	PRON
elements-9303	87	8	will	will	AUX
elements-9303	87	9	make	make	VERB
elements-9303	87	10	the	the	DET
elements-9303	87	11	later	late	ADJ
elements-9303	87	12	calculations	calculation	NOUN
elements-9303	87	13	more	more	ADV
elements-9303	87	14	concise	concise	ADJ
elements-9303	87	15	:	:	PUNCT
elements-9303	87	16	•	•	NUM
elements-9303	87	17	α	α	NOUN
elements-9303	87	18	is	be	AUX
elements-9303	87	19	the	the	DET
elements-9303	87	20	set	set	NOUN
elements-9303	87	21	of	of	ADP
elements-9303	87	22	digit	digit	NOUN
elements-9303	87	23	places	place	NOUN
elements-9303	87	24	10x	10x	NUM
elements-9303	87	25	≡	≡	PROPN
elements-9303	87	26	1	1	NUM
elements-9303	87	27	(	(	PUNCT
elements-9303	87	28	mod	mod	PROPN
elements-9303	87	29	7	7	NUM
elements-9303	87	30	)	)	PUNCT
elements-9303	87	31	,	,	PUNCT
elements-9303	87	32	i.e.	i.e.	X
elements-9303	87	33	x	x	SYM
elements-9303	87	34	≡	≡	PROPN
elements-9303	87	35	0	0	NUM
elements-9303	87	36	(	(	PUNCT
elements-9303	87	37	mod	mod	PROPN
elements-9303	87	38	6	6	NUM
elements-9303	87	39	)	)	PUNCT
elements-9303	87	40	.	.	PUNCT
elements-9303	88	1	•	•	NUM
elements-9303	88	2	β	β	X
elements-9303	88	3	is	be	AUX
elements-9303	88	4	the	the	DET
elements-9303	88	5	set	set	NOUN
elements-9303	88	6	of	of	ADP
elements-9303	88	7	digit	digit	NOUN
elements-9303	88	8	places	place	NOUN
elements-9303	88	9	10x	10x	NUM
elements-9303	88	10	≡	≡	PROPN
elements-9303	88	11	2	2	NUM
elements-9303	88	12	(	(	PUNCT
elements-9303	88	13	mod	mod	PROPN
elements-9303	88	14	7	7	NUM
elements-9303	88	15	)	)	PUNCT
elements-9303	88	16	,	,	PUNCT
elements-9303	88	17	i.e.	i.e.	X
elements-9303	88	18	x	x	SYM
elements-9303	88	19	≡	≡	ADJ
elements-9303	88	20	2	2	NUM
elements-9303	88	21	(	(	PUNCT
elements-9303	88	22	mod	mod	PROPN
elements-9303	88	23	6	6	NUM
elements-9303	88	24	)	)	PUNCT
elements-9303	88	25	.	.	PUNCT
elements-9303	89	1	•	•	NOUN
elements-9303	89	2	γ	γ	X
elements-9303	89	3	is	be	AUX
elements-9303	89	4	the	the	DET
elements-9303	89	5	set	set	NOUN
elements-9303	89	6	of	of	ADP
elements-9303	89	7	digit	digit	NOUN
elements-9303	89	8	places	place	NOUN
elements-9303	89	9	10x	10x	PROPN
elements-9303	89	10	≡	≡	PROPN
elements-9303	89	11	3	3	NUM
elements-9303	89	12	(	(	PUNCT
elements-9303	89	13	mod	mod	PROPN
elements-9303	89	14	7	7	NUM
elements-9303	89	15	)	)	PUNCT
elements-9303	89	16	,	,	PUNCT
elements-9303	89	17	i.e.	i.e.	X
elements-9303	89	18	x	x	SYM
elements-9303	89	19	≡	≡	ADJ
elements-9303	89	20	1	1	NUM
elements-9303	89	21	(	(	PUNCT
elements-9303	89	22	mod	mod	PROPN
elements-9303	89	23	6	6	NUM
elements-9303	89	24	)	)	PUNCT
elements-9303	89	25	.	.	PUNCT
elements-9303	90	1	•	•	NUM
elements-9303	90	2	δ	δ	PROPN
elements-9303	90	3	is	be	AUX
elements-9303	90	4	the	the	DET
elements-9303	90	5	set	set	NOUN
elements-9303	90	6	of	of	ADP
elements-9303	90	7	digit	digit	NOUN
elements-9303	90	8	places	place	NOUN
elements-9303	90	9	10x	10x	NUM
elements-9303	90	10	≡	≡	PROPN
elements-9303	90	11	4	4	NUM
elements-9303	90	12	(	(	PUNCT
elements-9303	90	13	mod	mod	PROPN
elements-9303	90	14	7	7	NUM
elements-9303	90	15	)	)	PUNCT
elements-9303	90	16	,	,	PUNCT
elements-9303	90	17	i.e.	i.e.	X
elements-9303	90	18	x	x	SYM
elements-9303	90	19	≡	≡	ADJ
elements-9303	90	20	4	4	NUM
elements-9303	90	21	(	(	PUNCT
elements-9303	90	22	mod	mod	PROPN
elements-9303	90	23	6	6	NUM
elements-9303	90	24	)	)	PUNCT
elements-9303	90	25	.	.	PUNCT
elements-9303	91	1	•	•	NUM
elements-9303	91	2	ε	ε	PROPN
elements-9303	91	3	is	be	AUX
elements-9303	91	4	the	the	DET
elements-9303	91	5	set	set	NOUN
elements-9303	91	6	of	of	ADP
elements-9303	91	7	digit	digit	NOUN
elements-9303	91	8	places	place	NOUN
elements-9303	91	9	10x	10x	NUM
elements-9303	91	10	≡	≡	PROPN
elements-9303	91	11	5	5	NUM
elements-9303	91	12	(	(	PUNCT
elements-9303	91	13	mod	mod	PROPN
elements-9303	91	14	7	7	NUM
elements-9303	91	15	)	)	PUNCT
elements-9303	91	16	,	,	PUNCT
elements-9303	91	17	i.e.	i.e.	X
elements-9303	91	18	x	x	SYM
elements-9303	91	19	≡	≡	ADJ
elements-9303	91	20	5	5	NUM
elements-9303	91	21	(	(	PUNCT
elements-9303	91	22	mod	mod	PROPN
elements-9303	91	23	6	6	NUM
elements-9303	91	24	)	)	PUNCT
elements-9303	91	25	.	.	PUNCT
elements-9303	92	1	•	•	NUM
elements-9303	92	2	ζ	ζ	NOUN
elements-9303	92	3	is	be	AUX
elements-9303	92	4	the	the	DET
elements-9303	92	5	set	set	NOUN
elements-9303	92	6	of	of	ADP
elements-9303	92	7	digit	digit	NOUN
elements-9303	92	8	places	place	NOUN
elements-9303	92	9	10x	10x	NUM
elements-9303	92	10	≡	≡	PROPN
elements-9303	92	11	6	6	NUM
elements-9303	92	12	(	(	PUNCT
elements-9303	92	13	mod	mod	PROPN
elements-9303	92	14	7	7	NUM
elements-9303	92	15	)	)	PUNCT
elements-9303	92	16	,	,	PUNCT
elements-9303	92	17	i.e.	i.e.	X
elements-9303	92	18	x	x	SYM
elements-9303	92	19	≡	≡	ADJ
elements-9303	92	20	3	3	NUM
elements-9303	92	21	(	(	PUNCT
elements-9303	92	22	mod	mod	PROPN
elements-9303	92	23	6	6	NUM
elements-9303	92	24	)	)	PUNCT
elements-9303	92	25	.	.	PUNCT
elements-9303	93	1	95	95	NUM
elements-9303	93	2	dreibens	dreiben	NOUN
elements-9303	93	3	modulo	modulo	VERB
elements-9303	93	4	7	7	NUM
elements-9303	93	5	as	as	ADP
elements-9303	93	6	an	an	DET
elements-9303	93	7	example	example	NOUN
elements-9303	93	8	,	,	PUNCT
elements-9303	93	9	one	one	NUM
elements-9303	93	10	possible	possible	ADJ
elements-9303	93	11	way	way	NOUN
elements-9303	93	12	to	to	PART
elements-9303	93	13	put	put	VERB
elements-9303	93	14	two	two	NUM
elements-9303	93	15	3	3	NUM
elements-9303	93	16	’s	’s	NOUN
elements-9303	93	17	into	into	ADP
elements-9303	93	18	a	a	DET
elements-9303	93	19	dreiben	dreiben	NOUN
elements-9303	93	20	is	be	AUX
elements-9303	93	21	to	to	PART
elements-9303	93	22	choose	choose	VERB
elements-9303	93	23	one	one	NUM
elements-9303	93	24	digit	digit	NOUN
elements-9303	93	25	place	place	NOUN
elements-9303	93	26	from	from	ADP
elements-9303	93	27	α	α	NOUN
elements-9303	93	28	and	and	CCONJ
elements-9303	93	29	another	another	DET
elements-9303	93	30	digit	digit	NOUN
elements-9303	93	31	place	place	NOUN
elements-9303	93	32	from	from	ADP
elements-9303	93	33	ζ	ζ	NOUN
elements-9303	93	34	.	.	PUNCT
elements-9303	94	1	the	the	DET
elements-9303	94	2	number	number	NOUN
elements-9303	94	3	of	of	ADP
elements-9303	94	4	possible	possible	ADJ
elements-9303	94	5	combinations	combination	NOUN
elements-9303	94	6	of	of	ADP
elements-9303	94	7	choosing	choose	VERB
elements-9303	94	8	two	two	NUM
elements-9303	94	9	digit	digit	NOUN
elements-9303	94	10	places	place	NOUN
elements-9303	94	11	,	,	PUNCT
elements-9303	94	12	one	one	NUM
elements-9303	94	13	from	from	ADP
elements-9303	94	14	α	α	NOUN
elements-9303	94	15	and	and	CCONJ
elements-9303	94	16	one	one	NUM
elements-9303	94	17	from	from	ADP
elements-9303	94	18	ζ	ζ	NOUN
elements-9303	94	19	,	,	PUNCT
elements-9303	94	20	is	be	AUX
elements-9303	94	21	(	(	PUNCT
elements-9303	94	22	|α|	|α|	PROPN
elements-9303	94	23	1	1	NUM
elements-9303	94	24	)	)	PUNCT
elements-9303	94	25	(	(	PUNCT
elements-9303	94	26	|ζ|	|ζ|	SYM
elements-9303	94	27	1	1	NUM
elements-9303	94	28	)	)	PUNCT
elements-9303	94	29	=	=	SYM
elements-9303	94	30	|α|·|ζ|	|α|·|ζ|	NOUN
elements-9303	94	31	.	.	PUNCT
elements-9303	95	1	applying	apply	VERB
elements-9303	95	2	this	this	PRON
elements-9303	95	3	to	to	ADP
elements-9303	95	4	the	the	DET
elements-9303	95	5	other	other	ADJ
elements-9303	95	6	sets	set	NOUN
elements-9303	95	7	of	of	ADP
elements-9303	95	8	digit	digit	NOUN
elements-9303	95	9	places	place	NOUN
elements-9303	95	10	,	,	PUNCT
elements-9303	95	11	we	we	PRON
elements-9303	95	12	find	find	VERB
elements-9303	95	13	that	that	SCONJ
elements-9303	95	14	as	as	SCONJ
elements-9303	95	15	you	you	PRON
elements-9303	95	16	may	may	AUX
elements-9303	95	17	have	have	AUX
elements-9303	95	18	noticed	notice	VERB
elements-9303	95	19	,	,	PUNCT
elements-9303	95	20	the	the	DET
elements-9303	95	21	cardinality	cardinality	NOUN
elements-9303	95	22	of	of	ADP
elements-9303	95	23	the	the	DET
elements-9303	95	24	six	six	NUM
elements-9303	95	25	sets	set	NOUN
elements-9303	95	26	above	above	ADV
elements-9303	95	27	depends	depend	VERB
elements-9303	95	28	on	on	ADP
elements-9303	95	29	n	n	PROPN
elements-9303	95	30	(	(	PUNCT
elements-9303	95	31	mod	mod	PROPN
elements-9303	95	32	6	6	NUM
elements-9303	95	33	)	)	PUNCT
elements-9303	95	34	.	.	PUNCT
elements-9303	96	1	this	this	PRON
elements-9303	96	2	can	can	AUX
elements-9303	96	3	be	be	AUX
elements-9303	96	4	seen	see	VERB
elements-9303	96	5	in	in	ADP
elements-9303	96	6	table	table	NOUN
elements-9303	96	7	2	2	NUM
elements-9303	96	8	below	below	ADV
elements-9303	96	9	:	:	PUNCT
elements-9303	96	10	hence	hence	ADV
elements-9303	96	11	,	,	PUNCT
elements-9303	96	12	•	•	ADP
elements-9303	96	13	if	if	SCONJ
elements-9303	96	14	n	n	PRON
elements-9303	96	15	≡	≡	PROPN
elements-9303	96	16	0	0	PUNCT
elements-9303	97	1	(	(	PUNCT
elements-9303	97	2	mod	mod	PROPN
elements-9303	97	3	6	6	NUM
elements-9303	97	4	)	)	PUNCT
elements-9303	97	5	,	,	PUNCT
elements-9303	97	6	then	then	ADV
elements-9303	97	7	•	•	INTJ
elements-9303	97	8	if	if	SCONJ
elements-9303	97	9	n	n	PRON
elements-9303	97	10	≡	≡	PROPN
elements-9303	97	11	1	1	NUM
elements-9303	97	12	(	(	PUNCT
elements-9303	97	13	mod	mod	PROPN
elements-9303	97	14	6	6	NUM
elements-9303	97	15	)	)	PUNCT
elements-9303	97	16	,	,	PUNCT
elements-9303	97	17	then	then	ADV
elements-9303	97	18	•	•	INTJ
elements-9303	97	19	if	if	SCONJ
elements-9303	97	20	n	n	PRON
elements-9303	97	21	≡	≡	PROPN
elements-9303	97	22	2	2	NUM
elements-9303	97	23	(	(	PUNCT
elements-9303	97	24	mod	mod	PROPN
elements-9303	97	25	6	6	NUM
elements-9303	97	26	)	)	PUNCT
elements-9303	97	27	,	,	PUNCT
elements-9303	97	28	then	then	ADV
elements-9303	97	29	•	•	INTJ
elements-9303	97	30	if	if	SCONJ
elements-9303	97	31	n	n	PRON
elements-9303	97	32	≡	≡	PROPN
elements-9303	97	33	3	3	NUM
elements-9303	97	34	(	(	PUNCT
elements-9303	97	35	mod	mod	PROPN
elements-9303	97	36	6	6	NUM
elements-9303	97	37	)	)	PUNCT
elements-9303	97	38	,	,	PUNCT
elements-9303	97	39	then	then	ADV
elements-9303	97	40	•	•	INTJ
elements-9303	97	41	if	if	SCONJ
elements-9303	97	42	n	n	PRON
elements-9303	97	43	≡	≡	PROPN
elements-9303	97	44	4	4	NUM
elements-9303	97	45	(	(	PUNCT
elements-9303	97	46	mod	mod	PROPN
elements-9303	97	47	6	6	NUM
elements-9303	97	48	)	)	PUNCT
elements-9303	97	49	,	,	PUNCT
elements-9303	97	50	then	then	ADV
elements-9303	97	51	•	•	INTJ
elements-9303	97	52	if	if	SCONJ
elements-9303	97	53	n	n	PRON
elements-9303	97	54	≡	≡	PROPN
elements-9303	97	55	5	5	NUM
elements-9303	97	56	(	(	PUNCT
elements-9303	97	57	mod	mod	PROPN
elements-9303	97	58	6	6	NUM
elements-9303	97	59	)	)	PUNCT
elements-9303	97	60	,	,	PUNCT
elements-9303	97	61	then	then	ADV
elements-9303	97	62	evidently	evidently	ADV
elements-9303	97	63	,	,	PUNCT
elements-9303	97	64	dn	dn	PROPN
elements-9303	97	65	3(2	3(2	NUM
elements-9303	97	66	)	)	PUNCT
elements-9303	98	1	=	=	NOUN
elements-9303	98	2	[	[	PUNCT
elements-9303	98	3	n2/12	n2/12	X
elements-9303	98	4	]	]	PUNCT
elements-9303	98	5	.	.	PUNCT
elements-9303	99	1	let	let	VERB
elements-9303	99	2	us	we	PRON
elements-9303	99	3	prove	prove	VERB
elements-9303	99	4	this	this	PRON
elements-9303	99	5	.	.	PUNCT
elements-9303	100	1	proof	proof	NOUN
elements-9303	100	2	.	.	PUNCT
elements-9303	101	1	if	if	SCONJ
elements-9303	101	2	n	n	PRON
elements-9303	101	3	≡	≡	PROPN
elements-9303	101	4	0	0	PUNCT
elements-9303	101	5	(	(	PUNCT
elements-9303	101	6	mod	mod	PROPN
elements-9303	101	7	6	6	NUM
elements-9303	101	8	)	)	PUNCT
elements-9303	101	9	,	,	PUNCT
elements-9303	101	10	i.e.	i.e.	X
elements-9303	101	11	n	n	CCONJ
elements-9303	101	12	=	=	SYM
elements-9303	101	13	6	6	NUM
elements-9303	101	14	m	m	NOUN
elements-9303	101	15	where	where	SCONJ
elements-9303	101	16	m	m	NOUN
elements-9303	101	17	is	be	AUX
elements-9303	101	18	an	an	DET
elements-9303	101	19	integer	integer	NOUN
elements-9303	101	20	,	,	PUNCT
elements-9303	101	21	then	then	ADV
elements-9303	101	22	if	if	SCONJ
elements-9303	101	23	n	n	PRON
elements-9303	101	24	≡	≡	PROPN
elements-9303	101	25	1	1	NUM
elements-9303	101	26	(	(	PUNCT
elements-9303	101	27	mod	mod	PROPN
elements-9303	101	28	6	6	NUM
elements-9303	101	29	)	)	PUNCT
elements-9303	101	30	,	,	PUNCT
elements-9303	101	31	i.e.	i.e.	X
elements-9303	101	32	n	n	CCONJ
elements-9303	101	33	=	=	SYM
elements-9303	101	34	6	6	NUM
elements-9303	101	35	m	m	NOUN
elements-9303	101	36	+	+	NOUN
elements-9303	101	37	1	1	NUM
elements-9303	101	38	where	where	SCONJ
elements-9303	101	39	m	m	PRON
elements-9303	101	40	is	be	AUX
elements-9303	101	41	an	an	DET
elements-9303	101	42	integer	integer	NOUN
elements-9303	101	43	,	,	PUNCT
elements-9303	101	44	then	then	ADV
elements-9303	101	45	if	if	SCONJ
elements-9303	101	46	n	n	PRON
elements-9303	101	47	≡	≡	PROPN
elements-9303	101	48	2	2	NUM
elements-9303	101	49	(	(	PUNCT
elements-9303	101	50	mod	mod	PROPN
elements-9303	101	51	6	6	NUM
elements-9303	101	52	)	)	PUNCT
elements-9303	101	53	,	,	PUNCT
elements-9303	101	54	i.e.	i.e.	X
elements-9303	101	55	n	n	CCONJ
elements-9303	101	56	=	=	SYM
elements-9303	101	57	6	6	NUM
elements-9303	101	58	m	m	NOUN
elements-9303	101	59	+	+	NOUN
elements-9303	101	60	2	2	NUM
elements-9303	101	61	where	where	SCONJ
elements-9303	101	62	m	m	PRON
elements-9303	101	63	is	be	AUX
elements-9303	101	64	an	an	DET
elements-9303	101	65	integer	integer	NOUN
elements-9303	101	66	,	,	PUNCT
elements-9303	101	67	then	then	ADV
elements-9303	101	68	if	if	SCONJ
elements-9303	101	69	n	n	PRON
elements-9303	101	70	≡	≡	PROPN
elements-9303	101	71	3	3	NUM
elements-9303	101	72	(	(	PUNCT
elements-9303	101	73	mod	mod	PROPN
elements-9303	101	74	6	6	NUM
elements-9303	101	75	)	)	PUNCT
elements-9303	101	76	,	,	PUNCT
elements-9303	101	77	i.e.	i.e.	X
elements-9303	101	78	n	n	CCONJ
elements-9303	101	79	=	=	SYM
elements-9303	101	80	6	6	NUM
elements-9303	101	81	m	m	NOUN
elements-9303	101	82	+	+	NOUN
elements-9303	101	83	3	3	NUM
elements-9303	101	84	where	where	SCONJ
elements-9303	101	85	m	m	PRON
elements-9303	101	86	is	be	AUX
elements-9303	101	87	an	an	DET
elements-9303	101	88	integer	integer	NOUN
elements-9303	101	89	,	,	PUNCT
elements-9303	101	90	then	then	ADV
elements-9303	101	91	96	96	NUM
elements-9303	101	92	elements	element	NOUN
elements-9303	101	93	:	:	PUNCT
elements-9303	101	94	:	:	PUNCT
elements-9303	101	95	spring	spring	NOUN
elements-9303	101	96	2016	2016	NUM
elements-9303	101	97	if	if	SCONJ
elements-9303	101	98	n	n	PRON
elements-9303	101	99	≡	≡	PROPN
elements-9303	101	100	4	4	NUM
elements-9303	101	101	(	(	PUNCT
elements-9303	101	102	mod	mod	PROPN
elements-9303	101	103	6	6	NUM
elements-9303	101	104	)	)	PUNCT
elements-9303	101	105	,	,	PUNCT
elements-9303	101	106	i.e.	i.e.	X
elements-9303	101	107	n	n	CCONJ
elements-9303	101	108	=	=	SYM
elements-9303	101	109	6	6	NUM
elements-9303	101	110	m	m	NOUN
elements-9303	101	111	+	+	NOUN
elements-9303	101	112	4	4	NUM
elements-9303	101	113	where	where	SCONJ
elements-9303	101	114	m	m	PRON
elements-9303	101	115	is	be	AUX
elements-9303	101	116	an	an	DET
elements-9303	101	117	integer	integer	NOUN
elements-9303	101	118	,	,	PUNCT
elements-9303	101	119	then	then	ADV
elements-9303	101	120	if	if	SCONJ
elements-9303	101	121	n	n	PRON
elements-9303	101	122	≡	≡	PROPN
elements-9303	101	123	5	5	NUM
elements-9303	101	124	(	(	PUNCT
elements-9303	101	125	mod	mod	PROPN
elements-9303	101	126	6	6	NUM
elements-9303	101	127	)	)	PUNCT
elements-9303	101	128	,	,	PUNCT
elements-9303	101	129	i.e.	i.e.	X
elements-9303	101	130	n	n	CCONJ
elements-9303	101	131	=	=	SYM
elements-9303	101	132	6	6	NUM
elements-9303	101	133	m	m	NOUN
elements-9303	101	134	+	+	NOUN
elements-9303	101	135	5	5	NUM
elements-9303	101	136	where	where	SCONJ
elements-9303	101	137	m	m	PRON
elements-9303	101	138	is	be	AUX
elements-9303	101	139	an	an	DET
elements-9303	101	140	integer	integer	NOUN
elements-9303	101	141	,	,	PUNCT
elements-9303	101	142	then	then	ADV
elements-9303	101	143	therefore	therefore	ADV
elements-9303	101	144	,	,	PUNCT
elements-9303	101	145	in	in	ADP
elements-9303	101	146	general	general	ADJ
elements-9303	101	147	,	,	PUNCT
elements-9303	101	148	dn	dn	NOUN
elements-9303	101	149	3(2	3(2	NUM
elements-9303	101	150	)	)	PUNCT
elements-9303	102	1	=	=	NOUN
elements-9303	102	2	[	[	PUNCT
elements-9303	102	3	n2/12	n2/12	X
elements-9303	102	4	]	]	PUNCT
elements-9303	102	5	.	.	PUNCT
elements-9303	103	1	how	how	SCONJ
elements-9303	103	2	many	many	ADJ
elements-9303	103	3	ways	way	NOUN
elements-9303	103	4	are	be	AUX
elements-9303	103	5	there	there	ADV
elements-9303	103	6	to	to	PART
elements-9303	103	7	put	put	VERB
elements-9303	103	8	exactly	exactly	ADV
elements-9303	103	9	three	three	NUM
elements-9303	103	10	3	3	NUM
elements-9303	103	11	’s	’s	NOUN
elements-9303	103	12	in	in	ADP
elements-9303	103	13	a	a	DET
elements-9303	103	14	dreiben	dreiben	NOUN
elements-9303	103	15	of	of	ADP
elements-9303	103	16	length	length	NOUN
elements-9303	103	17	n	n	CCONJ
elements-9303	103	18	,	,	PUNCT
elements-9303	103	19	i.e.	i.e.	X
elements-9303	103	20	what	what	PRON
elements-9303	103	21	is	be	AUX
elements-9303	103	22	dn	dn	ADP
elements-9303	103	23	3(3	3(3	NUM
elements-9303	103	24	)	)	PUNCT
elements-9303	103	25	?	?	PUNCT
elements-9303	104	1	we	we	PRON
elements-9303	104	2	must	must	AUX
elements-9303	104	3	choose	choose	VERB
elements-9303	104	4	three	three	NUM
elements-9303	104	5	digit	digit	NOUN
elements-9303	104	6	places	place	NOUN
elements-9303	104	7	10x	10x	NUM
elements-9303	104	8	,	,	PUNCT
elements-9303	104	9	10y	10y	NUM
elements-9303	104	10	,	,	PUNCT
elements-9303	104	11	and	and	CCONJ
elements-9303	104	12	10z	10z	NOUN
elements-9303	104	13	where	where	SCONJ
elements-9303	104	14	x	x	PUNCT
elements-9303	104	15	≠	≠	PROPN
elements-9303	104	16	y	y	PROPN
elements-9303	104	17	≠	≠	PROPN
elements-9303	104	18	z	z	NOUN
elements-9303	104	19	and	and	CCONJ
elements-9303	104	20	0	0	NUM
elements-9303	104	21	≤	≤	NUM
elements-9303	104	22	x	x	X
elements-9303	104	23	,	,	PUNCT
elements-9303	104	24	y	y	PROPN
elements-9303	104	25	,	,	PUNCT
elements-9303	104	26	z	z	NOUN
elements-9303	104	27	≤	≤	NOUN
elements-9303	104	28	n	n	DET
elements-9303	104	29	1	1	NUM
elements-9303	104	30	.	.	PUNCT
elements-9303	105	1	now	now	ADV
elements-9303	105	2	the	the	DET
elements-9303	105	3	question	question	NOUN
elements-9303	105	4	is	be	AUX
elements-9303	105	5	,	,	PUNCT
elements-9303	105	6	what	what	PRON
elements-9303	105	7	are	be	AUX
elements-9303	105	8	the	the	DET
elements-9303	105	9	possible	possible	ADJ
elements-9303	105	10	ways	way	NOUN
elements-9303	105	11	to	to	PART
elements-9303	105	12	make	make	VERB
elements-9303	105	13	10x	10x	NOUN
elements-9303	105	14	+	+	SYM
elements-9303	105	15	10y	10y	PROPN
elements-9303	105	16	+	+	SYM
elements-9303	105	17	10z	10z	NOUN
elements-9303	105	18	≡	≡	PROPN
elements-9303	105	19	0	0	PUNCT
elements-9303	106	1	(	(	PUNCT
elements-9303	106	2	mod	mod	PROPN
elements-9303	106	3	7	7	NUM
elements-9303	106	4	)	)	PUNCT
elements-9303	106	5	?	?	PUNCT
elements-9303	107	1	in	in	ADP
elements-9303	107	2	fact	fact	NOUN
elements-9303	107	3	,	,	PUNCT
elements-9303	107	4	there	there	PRON
elements-9303	107	5	are	be	VERB
elements-9303	107	6	eight	eight	NUM
elements-9303	107	7	cases	case	NOUN
elements-9303	107	8	:	:	PUNCT
elements-9303	107	9	without	without	ADP
elements-9303	107	10	loss	loss	NOUN
elements-9303	107	11	of	of	ADP
elements-9303	107	12	generality	generality	NOUN
elements-9303	107	13	,	,	PUNCT
elements-9303	107	14	the	the	DET
elements-9303	107	15	possible	possible	ADJ
elements-9303	107	16	solutions	solution	NOUN
elements-9303	107	17	are	be	AUX
elements-9303	107	18	(	(	PUNCT
elements-9303	107	19	a	a	PRON
elements-9303	107	20	)	)	PUNCT
elements-9303	107	21	10x	10x	NUM
elements-9303	107	22	≡	≡	PROPN
elements-9303	107	23	1	1	NUM
elements-9303	107	24	(	(	PUNCT
elements-9303	107	25	mod	mod	PROPN
elements-9303	107	26	7	7	NUM
elements-9303	107	27	)	)	PUNCT
elements-9303	107	28	,	,	PUNCT
elements-9303	107	29	10y	10y	PROPN
elements-9303	107	30	≡	≡	PROPN
elements-9303	107	31	1	1	NUM
elements-9303	107	32	(	(	PUNCT
elements-9303	107	33	mod	mod	PROPN
elements-9303	107	34	7	7	NUM
elements-9303	107	35	)	)	PUNCT
elements-9303	107	36	,	,	PUNCT
elements-9303	107	37	10z	10z	NOUN
elements-9303	107	38	≡	≡	PROPN
elements-9303	107	39	5	5	NUM
elements-9303	107	40	(	(	PUNCT
elements-9303	107	41	mod	mod	PROPN
elements-9303	107	42	7	7	NUM
elements-9303	107	43	)	)	PUNCT
elements-9303	107	44	(	(	PUNCT
elements-9303	107	45	b	b	NOUN
elements-9303	107	46	)	)	PUNCT
elements-9303	107	47	10x	10x	PUNCT
elements-9303	108	1	≡	≡	PROPN
elements-9303	108	2	1	1	NUM
elements-9303	108	3	(	(	PUNCT
elements-9303	108	4	mod	mod	PROPN
elements-9303	108	5	7	7	NUM
elements-9303	108	6	)	)	PUNCT
elements-9303	108	7	,	,	PUNCT
elements-9303	108	8	10y	10y	PROPN
elements-9303	108	9	≡	≡	PROPN
elements-9303	108	10	2	2	NUM
elements-9303	108	11	(	(	PUNCT
elements-9303	108	12	mod	mod	PROPN
elements-9303	108	13	7	7	NUM
elements-9303	108	14	)	)	PUNCT
elements-9303	108	15	,	,	PUNCT
elements-9303	108	16	10z	10z	NOUN
elements-9303	108	17	≡	≡	PROPN
elements-9303	108	18	4	4	NUM
elements-9303	108	19	(	(	PUNCT
elements-9303	108	20	mod	mod	PROPN
elements-9303	108	21	7	7	NUM
elements-9303	108	22	)	)	PUNCT
elements-9303	108	23	(	(	PUNCT
elements-9303	108	24	c	c	NOUN
elements-9303	108	25	)	)	PUNCT
elements-9303	108	26	10x	10x	X
elements-9303	108	27	≡	≡	PROPN
elements-9303	108	28	1	1	NUM
elements-9303	108	29	(	(	PUNCT
elements-9303	108	30	mod	mod	PROPN
elements-9303	108	31	7	7	NUM
elements-9303	108	32	)	)	PUNCT
elements-9303	108	33	,	,	PUNCT
elements-9303	108	34	10y	10y	PROPN
elements-9303	108	35	≡	≡	PROPN
elements-9303	108	36	3	3	NUM
elements-9303	108	37	(	(	PUNCT
elements-9303	108	38	mod	mod	PROPN
elements-9303	108	39	7	7	NUM
elements-9303	108	40	)	)	PUNCT
elements-9303	108	41	,	,	PUNCT
elements-9303	108	42	10z	10z	NOUN
elements-9303	108	43	≡	≡	PROPN
elements-9303	108	44	3	3	NUM
elements-9303	108	45	(	(	PUNCT
elements-9303	108	46	mod	mod	PROPN
elements-9303	108	47	7	7	NUM
elements-9303	108	48	)	)	PUNCT
elements-9303	108	49	(	(	PUNCT
elements-9303	108	50	d	d	NOUN
elements-9303	108	51	)	)	PUNCT
elements-9303	108	52	10x	10x	X
elements-9303	108	53	≡	≡	PROPN
elements-9303	108	54	2	2	NUM
elements-9303	108	55	(	(	PUNCT
elements-9303	108	56	mod	mod	PROPN
elements-9303	108	57	7	7	NUM
elements-9303	108	58	)	)	PUNCT
elements-9303	108	59	,	,	PUNCT
elements-9303	108	60	10y	10y	PROPN
elements-9303	108	61	≡	≡	PROPN
elements-9303	108	62	2	2	NUM
elements-9303	108	63	(	(	PUNCT
elements-9303	108	64	mod	mod	PROPN
elements-9303	108	65	7	7	NUM
elements-9303	108	66	)	)	PUNCT
elements-9303	108	67	,	,	PUNCT
elements-9303	108	68	10z	10z	NOUN
elements-9303	108	69	≡	≡	PROPN
elements-9303	108	70	3	3	NUM
elements-9303	108	71	(	(	PUNCT
elements-9303	108	72	mod	mod	PROPN
elements-9303	108	73	7	7	NUM
elements-9303	108	74	)	)	PUNCT
elements-9303	108	75	(	(	PUNCT
elements-9303	108	76	e	e	NOUN
elements-9303	108	77	)	)	PUNCT
elements-9303	108	78	10x	10x	X
elements-9303	108	79	≡	≡	PROPN
elements-9303	108	80	2	2	NUM
elements-9303	108	81	(	(	PUNCT
elements-9303	108	82	mod	mod	PROPN
elements-9303	108	83	7	7	NUM
elements-9303	108	84	)	)	PUNCT
elements-9303	108	85	,	,	PUNCT
elements-9303	108	86	10y	10y	PROPN
elements-9303	108	87	≡	≡	PROPN
elements-9303	108	88	6	6	NUM
elements-9303	108	89	(	(	PUNCT
elements-9303	108	90	mod	mod	PROPN
elements-9303	108	91	7	7	NUM
elements-9303	108	92	)	)	PUNCT
elements-9303	108	93	,	,	PUNCT
elements-9303	108	94	10z	10z	NOUN
elements-9303	108	95	≡	≡	PROPN
elements-9303	108	96	6	6	NUM
elements-9303	108	97	(	(	PUNCT
elements-9303	108	98	mod	mod	PROPN
elements-9303	108	99	7	7	NUM
elements-9303	108	100	)	)	PUNCT
elements-9303	108	101	(	(	PUNCT
elements-9303	108	102	f	f	X
elements-9303	108	103	)	)	PUNCT
elements-9303	108	104	10x	10x	PROPN
elements-9303	108	105	≡	≡	PROPN
elements-9303	108	106	3	3	NUM
elements-9303	108	107	(	(	PUNCT
elements-9303	108	108	mod	mod	PROPN
elements-9303	108	109	7	7	NUM
elements-9303	108	110	)	)	PUNCT
elements-9303	108	111	,	,	PUNCT
elements-9303	108	112	10y	10y	PROPN
elements-9303	108	113	≡	≡	PROPN
elements-9303	108	114	5	5	NUM
elements-9303	108	115	(	(	PUNCT
elements-9303	108	116	mod	mod	PROPN
elements-9303	108	117	7	7	NUM
elements-9303	108	118	)	)	PUNCT
elements-9303	108	119	,	,	PUNCT
elements-9303	108	120	10z	10z	NOUN
elements-9303	108	121	≡	≡	PROPN
elements-9303	108	122	5	5	NUM
elements-9303	108	123	(	(	PUNCT
elements-9303	108	124	mod	mod	NOUN
elements-9303	108	125	7	7	NUM
elements-9303	108	126	)	)	PUNCT
elements-9303	108	127	(	(	PUNCT
elements-9303	108	128	g	g	NOUN
elements-9303	108	129	)	)	PUNCT
elements-9303	108	130	10x	10x	PUNCT
elements-9303	108	131	≡	≡	PROPN
elements-9303	108	132	4	4	NUM
elements-9303	108	133	(	(	PUNCT
elements-9303	108	134	mod	mod	PROPN
elements-9303	108	135	7	7	NUM
elements-9303	108	136	)	)	PUNCT
elements-9303	108	137	,	,	PUNCT
elements-9303	108	138	10y	10y	PROPN
elements-9303	108	139	≡	≡	PROPN
elements-9303	108	140	4	4	NUM
elements-9303	108	141	(	(	PUNCT
elements-9303	108	142	mod	mod	PROPN
elements-9303	108	143	7	7	NUM
elements-9303	108	144	)	)	PUNCT
elements-9303	108	145	,	,	PUNCT
elements-9303	108	146	10z	10z	NOUN
elements-9303	108	147	≡	≡	PROPN
elements-9303	108	148	6	6	NUM
elements-9303	108	149	(	(	PUNCT
elements-9303	108	150	mod	mod	PROPN
elements-9303	108	151	7	7	NUM
elements-9303	108	152	)	)	PUNCT
elements-9303	108	153	(	(	PUNCT
elements-9303	108	154	h	h	NOUN
elements-9303	108	155	)	)	PUNCT
elements-9303	108	156	10x	10x	PUNCT
elements-9303	108	157	≡	≡	PROPN
elements-9303	108	158	4	4	NUM
elements-9303	108	159	(	(	PUNCT
elements-9303	108	160	mod	mod	PROPN
elements-9303	108	161	7	7	NUM
elements-9303	108	162	)	)	PUNCT
elements-9303	108	163	,	,	PUNCT
elements-9303	108	164	10y	10y	PROPN
elements-9303	108	165	≡	≡	PROPN
elements-9303	108	166	5	5	NUM
elements-9303	108	167	(	(	PUNCT
elements-9303	108	168	mod	mod	PROPN
elements-9303	108	169	7	7	NUM
elements-9303	108	170	)	)	PUNCT
elements-9303	108	171	,	,	PUNCT
elements-9303	108	172	10z	10z	NOUN
elements-9303	108	173	≡	≡	PROPN
elements-9303	108	174	5	5	NUM
elements-9303	108	175	(	(	PUNCT
elements-9303	108	176	mod	mod	PROPN
elements-9303	108	177	7	7	X
elements-9303	108	178	)	)	PUNCT
elements-9303	108	179	these	these	DET
elements-9303	108	180	cases	case	NOUN
elements-9303	108	181	can	can	AUX
elements-9303	108	182	be	be	AUX
elements-9303	108	183	split	split	VERB
elements-9303	108	184	into	into	ADP
elements-9303	108	185	two	two	NUM
elements-9303	108	186	categories	category	NOUN
elements-9303	108	187	.	.	PUNCT
elements-9303	109	1	cases	case	NOUN
elements-9303	109	2	(	(	PUNCT
elements-9303	109	3	b	b	NOUN
elements-9303	109	4	)	)	PUNCT
elements-9303	109	5	and	and	CCONJ
elements-9303	109	6	(	(	PUNCT
elements-9303	109	7	f	f	X
elements-9303	109	8	)	)	PUNCT
elements-9303	109	9	entail	entail	VERB
elements-9303	109	10	choosing	choose	VERB
elements-9303	109	11	one	one	NUM
elements-9303	109	12	digit	digit	NOUN
elements-9303	109	13	place	place	NOUN
elements-9303	109	14	from	from	ADP
elements-9303	109	15	each	each	PRON
elements-9303	109	16	of	of	ADP
elements-9303	109	17	three	three	NUM
elements-9303	109	18	sets	set	NOUN
elements-9303	109	19	,	,	PUNCT
elements-9303	109	20	whereas	whereas	SCONJ
elements-9303	109	21	the	the	DET
elements-9303	109	22	remaining	remain	VERB
elements-9303	109	23	cases	case	NOUN
elements-9303	109	24	entail	entail	VERB
elements-9303	109	25	choosing	choose	VERB
elements-9303	109	26	two	two	NUM
elements-9303	109	27	digit	digit	NOUN
elements-9303	109	28	places	place	NOUN
elements-9303	109	29	from	from	ADP
elements-9303	109	30	one	one	NUM
elements-9303	109	31	set	set	NOUN
elements-9303	109	32	and	and	CCONJ
elements-9303	109	33	one	one	NUM
elements-9303	109	34	digit	digit	NOUN
elements-9303	109	35	place	place	NOUN
elements-9303	109	36	from	from	ADP
elements-9303	109	37	another	another	PRON
elements-9303	109	38	.	.	PUNCT
elements-9303	110	1	to	to	PART
elements-9303	110	2	describe	describe	VERB
elements-9303	110	3	these	these	DET
elements-9303	110	4	two	two	NUM
elements-9303	110	5	categories	category	NOUN
elements-9303	110	6	,	,	PUNCT
elements-9303	110	7	let	let	VERB
elements-9303	110	8	us	we	PRON
elements-9303	110	9	examine	examine	VERB
elements-9303	110	10	cases	case	NOUN
elements-9303	110	11	(	(	PUNCT
elements-9303	110	12	b	b	NOUN
elements-9303	110	13	)	)	PUNCT
elements-9303	110	14	and	and	CCONJ
elements-9303	110	15	(	(	PUNCT
elements-9303	110	16	c	c	NOUN
elements-9303	110	17	)	)	PUNCT
elements-9303	110	18	.	.	PUNCT
elements-9303	111	1	in	in	ADP
elements-9303	111	2	case	case	NOUN
elements-9303	111	3	(	(	PUNCT
elements-9303	111	4	b	b	NOUN
elements-9303	111	5	)	)	PUNCT
elements-9303	111	6	,	,	PUNCT
elements-9303	111	7	we	we	PRON
elements-9303	111	8	are	be	AUX
elements-9303	111	9	choosing	choose	VERB
elements-9303	111	10	one	one	NUM
elements-9303	111	11	digit	digit	NOUN
elements-9303	111	12	place	place	NOUN
elements-9303	111	13	each	each	PRON
elements-9303	111	14	from	from	ADP
elements-9303	111	15	the	the	DET
elements-9303	111	16	sets	set	NOUN
elements-9303	111	17	α	α	NOUN
elements-9303	111	18	,	,	PUNCT
elements-9303	111	19	β	β	NOUN
elements-9303	111	20	,	,	PUNCT
elements-9303	111	21	and	and	CCONJ
elements-9303	111	22	ζ	ζ	NOUN
elements-9303	111	23	.	.	NOUN
elements-9303	112	1	hence	hence	ADV
elements-9303	112	2	,	,	PUNCT
elements-9303	112	3	the	the	DET
elements-9303	112	4	number	number	NOUN
elements-9303	112	5	of	of	ADP
elements-9303	112	6	possible	possible	ADJ
elements-9303	112	7	combinations	combination	NOUN
elements-9303	112	8	of	of	ADP
elements-9303	112	9	three	three	NUM
elements-9303	112	10	digit	digit	NOUN
elements-9303	112	11	places	place	NOUN
elements-9303	112	12	chosen	choose	VERB
elements-9303	112	13	thusly	thusly	ADV
elements-9303	112	14	is	be	AUX
elements-9303	112	15	(	(	PUNCT
elements-9303	112	16	|α|	|α|	PROPN
elements-9303	112	17	1	1	NUM
elements-9303	112	18	)	)	PUNCT
elements-9303	112	19	(	(	PUNCT
elements-9303	112	20	|β|	|β|	NOUN
elements-9303	112	21	1	1	NUM
elements-9303	112	22	)	)	PUNCT
elements-9303	112	23	(	(	PUNCT
elements-9303	112	24	|ζ|	|ζ|	SYM
elements-9303	112	25	1	1	NUM
elements-9303	112	26	)	)	PUNCT
elements-9303	112	27	=	=	SYM
elements-9303	112	28	|α|·|β|·|ζ|	|α|·|β|·|ζ|	PROPN
elements-9303	112	29	.	.	PUNCT
elements-9303	113	1	in	in	ADP
elements-9303	113	2	case	case	NOUN
elements-9303	113	3	(	(	PUNCT
elements-9303	113	4	c	c	NOUN
elements-9303	113	5	)	)	PUNCT
elements-9303	113	6	,	,	PUNCT
elements-9303	113	7	we	we	PRON
elements-9303	113	8	are	be	AUX
elements-9303	113	9	choosing	choose	VERB
elements-9303	113	10	one	one	NUM
elements-9303	113	11	digit	digit	NOUN
elements-9303	113	12	place	place	NOUN
elements-9303	113	13	from	from	ADP
elements-9303	113	14	α	α	PROPN
elements-9303	113	15	and	and	CCONJ
elements-9303	113	16	two	two	NUM
elements-9303	113	17	digit	digit	NOUN
elements-9303	113	18	places	place	NOUN
elements-9303	113	19	from	from	ADP
elements-9303	113	20	γ	γ	PROPN
elements-9303	113	21	.	.	PROPN
elements-9303	113	22	hence	hence	ADV
elements-9303	113	23	,	,	PUNCT
elements-9303	113	24	the	the	DET
elements-9303	113	25	number	number	NOUN
elements-9303	113	26	of	of	ADP
elements-9303	113	27	possible	possible	ADJ
elements-9303	113	28	combinations	combination	NOUN
elements-9303	113	29	of	of	ADP
elements-9303	113	30	three	three	NUM
elements-9303	113	31	digit	digit	NOUN
elements-9303	113	32	places	place	NOUN
elements-9303	113	33	chosen	choose	VERB
elements-9303	113	34	thusly	thusly	ADV
elements-9303	113	35	is	be	AUX
elements-9303	113	36	(	(	PUNCT
elements-9303	113	37	|α|	|α|	PROPN
elements-9303	113	38	1	1	NUM
elements-9303	113	39	)	)	PUNCT
elements-9303	113	40	(	(	PUNCT
elements-9303	113	41	|γ|	|γ|	ADP
elements-9303	113	42	2	2	NUM
elements-9303	113	43	)	)	PUNCT
elements-9303	113	44	=	=	SYM
elements-9303	114	1	|α|·(|γ|)(|γ|-1)/2	|α|·(|γ|)(|γ|-1)/2	NOUN
elements-9303	114	2	.	.	NOUN
elements-9303	114	3	describing	describe	VERB
elements-9303	114	4	the	the	DET
elements-9303	114	5	other	other	ADJ
elements-9303	114	6	cases	case	NOUN
elements-9303	114	7	similarly	similarly	ADV
elements-9303	114	8	,	,	PUNCT
elements-9303	114	9	we	we	PRON
elements-9303	114	10	find	find	VERB
elements-9303	114	11	that	that	SCONJ
elements-9303	114	12	as	as	ADP
elements-9303	114	13	before	before	ADV
elements-9303	114	14	,	,	PUNCT
elements-9303	114	15	since	since	SCONJ
elements-9303	114	16	the	the	DET
elements-9303	114	17	cardinalities	cardinality	NOUN
elements-9303	114	18	of	of	ADP
elements-9303	114	19	the	the	DET
elements-9303	114	20	sets	set	NOUN
elements-9303	114	21	α	α	NOUN
elements-9303	114	22	through	through	ADP
elements-9303	114	23	ζ	ζ	NOUN
elements-9303	114	24	are	be	AUX
elements-9303	114	25	dependent	dependent	ADJ
elements-9303	114	26	on	on	ADP
elements-9303	114	27	n	n	PROPN
elements-9303	114	28	(	(	PUNCT
elements-9303	114	29	mod	mod	PROPN
elements-9303	114	30	6	6	NUM
elements-9303	114	31	)	)	PUNCT
elements-9303	114	32	,	,	PUNCT
elements-9303	114	33	we	we	PRON
elements-9303	114	34	would	would	AUX
elements-9303	114	35	be	be	AUX
elements-9303	114	36	best	well	ADV
elements-9303	114	37	served	serve	VERB
elements-9303	114	38	writing	write	VERB
elements-9303	114	39	out	out	ADP
elements-9303	114	40	each	each	PRON
elements-9303	114	41	of	of	ADP
elements-9303	114	42	the	the	DET
elements-9303	114	43	six	six	NUM
elements-9303	114	44	possible	possible	ADJ
elements-9303	114	45	cases	case	NOUN
elements-9303	114	46	to	to	PART
elements-9303	114	47	see	see	VERB
elements-9303	114	48	if	if	SCONJ
elements-9303	114	49	we	we	PRON
elements-9303	114	50	notice	notice	VERB
elements-9303	114	51	a	a	DET
elements-9303	114	52	pattern	pattern	NOUN
elements-9303	114	53	:	:	PUNCT
elements-9303	114	54	•	•	ADP
elements-9303	114	55	if	if	SCONJ
elements-9303	114	56	n	n	X
elements-9303	114	57	≡	≡	PROPN
elements-9303	114	58	0	0	PUNCT
elements-9303	115	1	(	(	PUNCT
elements-9303	115	2	mod	mod	PROPN
elements-9303	115	3	6	6	NUM
elements-9303	115	4	)	)	PUNCT
elements-9303	115	5	,	,	PUNCT
elements-9303	115	6	then	then	ADV
elements-9303	115	7	•	•	INTJ
elements-9303	115	8	if	if	SCONJ
elements-9303	115	9	n	n	PRON
elements-9303	115	10	≡	≡	PROPN
elements-9303	115	11	1	1	NUM
elements-9303	115	12	(	(	PUNCT
elements-9303	115	13	mod	mod	PROPN
elements-9303	115	14	6	6	NUM
elements-9303	115	15	)	)	PUNCT
elements-9303	115	16	,	,	PUNCT
elements-9303	115	17	then	then	ADV
elements-9303	115	18	•	•	INTJ
elements-9303	115	19	if	if	SCONJ
elements-9303	115	20	n	n	PRON
elements-9303	115	21	≡	≡	PROPN
elements-9303	115	22	2	2	NUM
elements-9303	115	23	(	(	PUNCT
elements-9303	115	24	mod	mod	PROPN
elements-9303	115	25	6	6	NUM
elements-9303	115	26	)	)	PUNCT
elements-9303	115	27	,	,	PUNCT
elements-9303	115	28	then	then	ADV
elements-9303	115	29	97	97	NUM
elements-9303	115	30	dreibens	dreiben	NOUN
elements-9303	115	31	modulo	modulo	VERB
elements-9303	115	32	7	7	NUM
elements-9303	115	33	•	•	NOUN
elements-9303	115	34	if	if	SCONJ
elements-9303	115	35	n	n	PRON
elements-9303	115	36	≡	≡	PROPN
elements-9303	115	37	3	3	NUM
elements-9303	115	38	(	(	PUNCT
elements-9303	115	39	mod	mod	PROPN
elements-9303	115	40	6	6	NUM
elements-9303	115	41	)	)	PUNCT
elements-9303	115	42	,	,	PUNCT
elements-9303	115	43	then	then	ADV
elements-9303	115	44	•	•	INTJ
elements-9303	115	45	if	if	SCONJ
elements-9303	115	46	n	n	PRON
elements-9303	115	47	≡	≡	PROPN
elements-9303	115	48	4	4	NUM
elements-9303	115	49	(	(	PUNCT
elements-9303	115	50	mod	mod	PROPN
elements-9303	115	51	6	6	NUM
elements-9303	115	52	)	)	PUNCT
elements-9303	115	53	,	,	PUNCT
elements-9303	115	54	then	then	ADV
elements-9303	115	55	•	•	INTJ
elements-9303	115	56	if	if	SCONJ
elements-9303	115	57	n	n	PRON
elements-9303	115	58	≡	≡	PROPN
elements-9303	115	59	5	5	NUM
elements-9303	115	60	(	(	PUNCT
elements-9303	115	61	mod	mod	PROPN
elements-9303	115	62	6	6	NUM
elements-9303	115	63	)	)	PUNCT
elements-9303	115	64	,	,	PUNCT
elements-9303	115	65	then	then	ADV
elements-9303	115	66	table	table	VERB
elements-9303	115	67	3	3	NUM
elements-9303	115	68	summarizes	summarize	NOUN
elements-9303	115	69	what	what	PRON
elements-9303	115	70	we	we	PRON
elements-9303	115	71	have	have	AUX
elements-9303	115	72	learned	learn	VERB
elements-9303	115	73	so	so	ADV
elements-9303	115	74	far	far	ADV
elements-9303	115	75	about	about	ADP
elements-9303	115	76	dn	dn	PROPN
elements-9303	115	77	3(2	3(2	NUM
elements-9303	115	78	)	)	PUNCT
elements-9303	115	79	and	and	CCONJ
elements-9303	115	80	dn	dn	ADP
elements-9303	115	81	3(3	3(3	NUM
elements-9303	115	82	):	):	PUNCT
elements-9303	115	83	given	give	VERB
elements-9303	115	84	that	that	PRON
elements-9303	115	85	dn	dn	NOUN
elements-9303	115	86	3(2	3(2	NUM
elements-9303	115	87	)	)	PUNCT
elements-9303	116	1	=	=	NOUN
elements-9303	116	2	,	,	PUNCT
elements-9303	116	3	it	it	PRON
elements-9303	116	4	seems	seem	VERB
elements-9303	116	5	reasonable	reasonable	ADJ
elements-9303	116	6	to	to	PART
elements-9303	116	7	suspect	suspect	VERB
elements-9303	116	8	that	that	SCONJ
elements-9303	116	9	dn	dn	PROPN
elements-9303	116	10	3(3	3(3	NUM
elements-9303	116	11	)	)	PUNCT
elements-9303	116	12	=	=	PUNCT
elements-9303	116	13	.	.	PUNCT
elements-9303	117	1	let	let	VERB
elements-9303	117	2	us	we	PRON
elements-9303	117	3	see	see	VERB
elements-9303	117	4	if	if	SCONJ
elements-9303	117	5	examining	examine	VERB
elements-9303	117	6	some	some	DET
elements-9303	117	7	values	value	NOUN
elements-9303	117	8	of	of	ADP
elements-9303	117	9	dn	dn	NOUN
elements-9303	117	10	3(3	3(3	NUM
elements-9303	117	11	)	)	PUNCT
elements-9303	117	12	and	and	CCONJ
elements-9303	117	13	will	will	AUX
elements-9303	117	14	enlighten	enlighten	VERB
elements-9303	117	15	us	we	PRON
elements-9303	117	16	.	.	PUNCT
elements-9303	118	1	as	as	ADP
elements-9303	118	2	evident	evident	ADJ
elements-9303	118	3	from	from	ADP
elements-9303	118	4	table	table	NOUN
elements-9303	118	5	4	4	NUM
elements-9303	118	6	above	above	ADV
elements-9303	118	7	,	,	PUNCT
elements-9303	118	8	dn	dn	NOUN
elements-9303	118	9	3(3	3(3	NUM
elements-9303	118	10	)	)	PUNCT
elements-9303	118	11	≠	≠	PROPN
elements-9303	118	12	,	,	PUNCT
elements-9303	118	13	but	but	CCONJ
elements-9303	118	14	they	they	PRON
elements-9303	118	15	seem	seem	VERB
elements-9303	118	16	very	very	ADV
elements-9303	118	17	similar	similar	ADJ
elements-9303	118	18	.	.	PUNCT
elements-9303	119	1	determining	determine	VERB
elements-9303	119	2	the	the	DET
elements-9303	119	3	relationship	relationship	NOUN
elements-9303	119	4	between	between	ADP
elements-9303	119	5	dn	dn	PROPN
elements-9303	119	6	3(3	3(3	NUM
elements-9303	119	7	)	)	PUNCT
elements-9303	119	8	and	and	CCONJ
elements-9303	119	9	may	may	AUX
elements-9303	119	10	be	be	AUX
elements-9303	119	11	a	a	DET
elements-9303	119	12	topic	topic	NOUN
elements-9303	119	13	for	for	ADP
elements-9303	119	14	future	future	ADJ
elements-9303	119	15	research	research	NOUN
elements-9303	119	16	,	,	PUNCT
elements-9303	119	17	but	but	CCONJ
elements-9303	119	18	is	be	AUX
elements-9303	119	19	beyond	beyond	ADP
elements-9303	119	20	the	the	DET
elements-9303	119	21	scope	scope	NOUN
elements-9303	119	22	of	of	ADP
elements-9303	119	23	this	this	DET
elements-9303	119	24	paper	paper	NOUN
elements-9303	119	25	.	.	PUNCT
elements-9303	120	1	finding	find	VERB
elements-9303	120	2	dn	dn	ADP
elements-9303	120	3	3(i	3(i	NUM
elements-9303	120	4	)	)	PUNCT
elements-9303	120	5	for	for	ADP
elements-9303	120	6	greater	great	ADJ
elements-9303	120	7	values	value	NOUN
elements-9303	120	8	of	of	ADP
elements-9303	120	9	i	i	PRON
elements-9303	120	10	becomes	become	VERB
elements-9303	120	11	increasingly	increasingly	ADV
elements-9303	120	12	difficult	difficult	ADJ
elements-9303	120	13	,	,	PUNCT
elements-9303	120	14	perhaps	perhaps	ADV
elements-9303	120	15	even	even	ADV
elements-9303	120	16	impractical	impractical	ADJ
elements-9303	120	17	.	.	PUNCT
elements-9303	121	1	it	it	PRON
elements-9303	121	2	is	be	AUX
elements-9303	121	3	beyond	beyond	ADP
elements-9303	121	4	the	the	DET
elements-9303	121	5	scope	scope	NOUN
elements-9303	121	6	of	of	ADP
elements-9303	121	7	this	this	DET
elements-9303	121	8	paper	paper	NOUN
elements-9303	121	9	,	,	PUNCT
elements-9303	121	10	but	but	CCONJ
elements-9303	121	11	should	should	AUX
elements-9303	121	12	a	a	DET
elements-9303	121	13	general	general	ADJ
elements-9303	121	14	form	form	NOUN
elements-9303	121	15	for	for	ADP
elements-9303	121	16	dn	dn	NOUN
elements-9303	121	17	3(i	3(i	NUM
elements-9303	121	18	)	)	PUNCT
elements-9303	121	19	be	be	AUX
elements-9303	121	20	found	find	VERB
elements-9303	121	21	,	,	PUNCT
elements-9303	121	22	one	one	NUM
elements-9303	121	23	could	could	AUX
elements-9303	121	24	,	,	PUNCT
elements-9303	121	25	from	from	ADP
elements-9303	121	26	a	a	DET
elements-9303	121	27	combinatorial	combinatorial	ADJ
elements-9303	121	28	perspective	perspective	NOUN
elements-9303	121	29	,	,	PUNCT
elements-9303	121	30	corroborate	corroborate	VERB
elements-9303	121	31	the	the	DET
elements-9303	121	32	results	result	NOUN
elements-9303	121	33	of	of	ADP
elements-9303	121	34	the	the	DET
elements-9303	121	35	second	second	ADJ
elements-9303	121	36	section	section	NOUN
elements-9303	121	37	,	,	PUNCT
elements-9303	121	38	allowing	allow	VERB
elements-9303	121	39	one	one	PRON
elements-9303	121	40	to	to	PART
elements-9303	121	41	find	find	VERB
elements-9303	121	42	a	a	DET
elements-9303	121	43	different	different	ADJ
elements-9303	121	44	form	form	NOUN
elements-9303	121	45	for	for	ADP
elements-9303	121	46	the	the	DET
elements-9303	121	47	explicit	explicit	ADJ
elements-9303	121	48	equation	equation	NOUN
elements-9303	121	49	of	of	ADP
elements-9303	121	50	ai	ai	ADJ
elements-9303	121	51	7(n	7(n	NUM
elements-9303	121	52	)	)	PUNCT
elements-9303	121	53	.	.	PUNCT
elements-9303	122	1	in	in	ADP
elements-9303	122	2	other	other	ADJ
elements-9303	122	3	words	word	NOUN
elements-9303	122	4	,	,	PUNCT
elements-9303	122	5	one	one	PRON
elements-9303	122	6	would	would	AUX
elements-9303	122	7	have	have	VERB
elements-9303	122	8	yet	yet	ADV
elements-9303	122	9	another	another	PRON
elements-9303	122	10	method	method	NOUN
elements-9303	122	11	to	to	PART
elements-9303	122	12	determine	determine	VERB
elements-9303	122	13	the	the	DET
elements-9303	122	14	number	number	NOUN
elements-9303	122	15	of	of	ADP
elements-9303	122	16	dreibens	dreiben	NOUN
elements-9303	122	17	of	of	ADP
elements-9303	122	18	length	length	NOUN
elements-9303	122	19	n	n	CCONJ
elements-9303	122	20	divisible	divisible	ADJ
elements-9303	122	21	by	by	ADP
elements-9303	122	22	7	7	NUM
elements-9303	122	23	.	.	NOUN
elements-9303	122	24	98	98	NUM
elements-9303	122	25	elements	element	NOUN
elements-9303	122	26	:	:	PUNCT
elements-9303	122	27	:	:	PUNCT
elements-9303	122	28	spring	spring	NOUN
elements-9303	122	29	2016	2016	NUM
elements-9303	122	30	conclusion	conclusion	NOUN
elements-9303	122	31	while	while	SCONJ
elements-9303	122	32	we	we	PRON
elements-9303	122	33	are	be	AUX
elements-9303	122	34	still	still	ADV
elements-9303	122	35	far	far	ADV
elements-9303	122	36	from	from	ADP
elements-9303	122	37	determining	determine	VERB
elements-9303	122	38	whether	whether	SCONJ
elements-9303	122	39	the	the	DET
elements-9303	122	40	set	set	NOUN
elements-9303	122	41	of	of	ADP
elements-9303	122	42	prime	prime	ADJ
elements-9303	122	43	dreibens	dreiben	NOUN
elements-9303	122	44	is	be	AUX
elements-9303	122	45	finite	finite	ADJ
elements-9303	122	46	or	or	CCONJ
elements-9303	122	47	infinite	infinite	VERB
elements-9303	122	48	,	,	PUNCT
elements-9303	122	49	our	our	PRON
elements-9303	122	50	exploration	exploration	NOUN
elements-9303	122	51	of	of	ADP
elements-9303	122	52	dreibens	dreiben	NOUN
elements-9303	122	53	modulo	modulo	NOUN
elements-9303	122	54	7	7	NUM
elements-9303	122	55	has	have	AUX
elements-9303	122	56	cast	cast	VERB
elements-9303	122	57	much	much	ADJ
elements-9303	122	58	light	light	NOUN
elements-9303	122	59	on	on	ADP
elements-9303	122	60	the	the	DET
elements-9303	122	61	problem	problem	NOUN
elements-9303	122	62	.	.	PUNCT
elements-9303	123	1	now	now	ADV
elements-9303	123	2	,	,	PUNCT
elements-9303	123	3	we	we	PRON
elements-9303	123	4	have	have	VERB
elements-9303	123	5	several	several	ADJ
elements-9303	123	6	methods	method	NOUN
elements-9303	123	7	to	to	PART
elements-9303	123	8	determine	determine	VERB
elements-9303	123	9	how	how	SCONJ
elements-9303	123	10	many	many	ADJ
elements-9303	123	11	dreibens	dreiben	NOUN
elements-9303	123	12	of	of	ADP
elements-9303	123	13	length	length	NOUN
elements-9303	123	14	n	n	NUM
elements-9303	123	15	are	be	AUX
elements-9303	123	16	divisible	divisible	ADJ
elements-9303	123	17	by	by	ADP
elements-9303	123	18	7	7	NUM
elements-9303	123	19	,	,	PUNCT
elements-9303	123	20	therefore	therefore	ADV
elements-9303	123	21	allowing	allow	VERB
elements-9303	123	22	these	these	DET
elements-9303	123	23	dreibens	dreiben	NOUN
elements-9303	123	24	to	to	PART
elements-9303	123	25	be	be	AUX
elements-9303	123	26	ruled	rule	VERB
elements-9303	123	27	out	out	ADP
elements-9303	123	28	during	during	ADP
elements-9303	123	29	our	our	PRON
elements-9303	123	30	search	search	NOUN
elements-9303	123	31	for	for	ADP
elements-9303	123	32	prime	prime	ADJ
elements-9303	123	33	dreibens	dreiben	NOUN
elements-9303	123	34	.	.	PUNCT
elements-9303	124	1	we	we	PRON
elements-9303	124	2	have	have	AUX
elements-9303	124	3	also	also	ADV
elements-9303	124	4	opened	open	VERB
elements-9303	124	5	up	up	ADP
elements-9303	124	6	several	several	ADJ
elements-9303	124	7	opportunities	opportunity	NOUN
elements-9303	124	8	for	for	ADP
elements-9303	124	9	inquiry	inquiry	NOUN
elements-9303	124	10	.	.	PUNCT
elements-9303	125	1	for	for	ADP
elements-9303	125	2	instance	instance	NOUN
elements-9303	125	3	,	,	PUNCT
elements-9303	125	4	explicit	explicit	ADJ
elements-9303	125	5	matrices	matrix	NOUN
elements-9303	125	6	for	for	ADP
elements-9303	125	7	an	an	PRON
elements-9303	125	8	where	where	SCONJ
elements-9303	125	9	n	n	X
elements-9303	125	10	≡	≡	PROPN
elements-9303	125	11	1	1	NUM
elements-9303	125	12	,	,	PUNCT
elements-9303	125	13	2	2	NUM
elements-9303	125	14	,	,	PUNCT
elements-9303	125	15	3	3	NUM
elements-9303	125	16	,	,	PUNCT
elements-9303	125	17	4	4	NUM
elements-9303	125	18	,	,	PUNCT
elements-9303	125	19	5	5	NUM
elements-9303	125	20	(	(	PUNCT
elements-9303	125	21	mod	mod	NOUN
elements-9303	125	22	6	6	NUM
elements-9303	125	23	)	)	PUNCT
elements-9303	125	24	must	must	AUX
elements-9303	125	25	still	still	ADV
elements-9303	125	26	be	be	AUX
elements-9303	125	27	determined	determine	VERB
elements-9303	125	28	.	.	PUNCT
elements-9303	126	1	the	the	DET
elements-9303	126	2	relationship	relationship	NOUN
elements-9303	126	3	between	between	ADP
elements-9303	126	4	dn	dn	PROPN
elements-9303	126	5	3(3	3(3	NUM
elements-9303	126	6	)	)	PUNCT
elements-9303	126	7	and	and	CCONJ
elements-9303	126	8	remains	remain	VERB
elements-9303	126	9	a	a	DET
elements-9303	126	10	promising	promising	ADJ
elements-9303	126	11	mystery	mystery	NOUN
elements-9303	126	12	,	,	PUNCT
elements-9303	126	13	and	and	CCONJ
elements-9303	126	14	we	we	PRON
elements-9303	126	15	have	have	VERB
elements-9303	126	16	yet	yet	ADV
elements-9303	126	17	to	to	PART
elements-9303	126	18	find	find	VERB
elements-9303	126	19	a	a	DET
elements-9303	126	20	generalized	generalized	ADJ
elements-9303	126	21	formula	formula	NOUN
elements-9303	126	22	for	for	ADP
elements-9303	126	23	dn	dn	NOUN
elements-9303	126	24	3(i	3(i	NUM
elements-9303	126	25	)	)	PUNCT
elements-9303	126	26	.	.	PUNCT
elements-9303	127	1	although	although	SCONJ
elements-9303	127	2	dreibens	dreiben	NOUN
elements-9303	127	3	remain	remain	VERB
elements-9303	127	4	a	a	DET
elements-9303	127	5	mysterious	mysterious	ADJ
elements-9303	127	6	type	type	NOUN
elements-9303	127	7	of	of	ADP
elements-9303	127	8	number	number	NOUN
elements-9303	127	9	,	,	PUNCT
elements-9303	127	10	current	current	ADJ
elements-9303	127	11	research	research	NOUN
elements-9303	127	12	in	in	ADP
elements-9303	127	13	dreibens	dreiben	NOUN
elements-9303	127	14	show	show	NOUN
elements-9303	127	15	promise	promise	NOUN
elements-9303	127	16	in	in	ADP
elements-9303	127	17	both	both	PRON
elements-9303	127	18	uncovering	uncover	VERB
elements-9303	127	19	new	new	ADJ
elements-9303	127	20	mathematical	mathematical	ADJ
elements-9303	127	21	problems	problem	NOUN
elements-9303	127	22	and	and	CCONJ
elements-9303	127	23	achieving	achieve	VERB
elements-9303	127	24	our	our	PRON
elements-9303	127	25	ultimate	ultimate	ADJ
elements-9303	127	26	goal	goal	NOUN
elements-9303	127	27	of	of	ADP
elements-9303	127	28	finding	find	VERB
elements-9303	127	29	infinitely	infinitely	ADV
elements-9303	127	30	many	many	ADJ
elements-9303	127	31	prime	prime	ADJ
elements-9303	127	32	dreibens	dreiben	NOUN
elements-9303	127	33	,	,	PUNCT
elements-9303	127	34	if	if	SCONJ
elements-9303	127	35	they	they	PRON
elements-9303	127	36	exist	exist	VERB
elements-9303	127	37	.	.	PUNCT
elements-9303	128	1	99	99	NUM
elements-9303	128	2	dreibens	dreiben	NOUN
elements-9303	128	3	modulo	modulo	VERB
elements-9303	128	4	7	7	NUM