Unique prime

In recreational number theory, a unique prime or unique period prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equal to the period length of the reciprocal of q, 1 / q. For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. In contrast, 41 and 271 both have period 5; 7 and 13 both have period 6; 239 and 4649 both have period 7; 73 and 137 both have period 8; 21649 and 513239 both have period 11; 53, 79 and 265371653 all have period 13; 31 and 2906161 both have period 15; 17 and 5882353 both have period 16; 2071723 and 5363222357 both have period 17; 19 and 52579 both have period 18; 3541 and 27961 both have period 20. Therefore, none of these is a unique prime. Unique primes were first described by Samuel Yates in 1980.
The above definition is related to the decimal representation of integers. Unique primes may be defined and have been studied in any numeral base.

Period of a prime in base ''b''

The representation of the reciprocal of a prime number in the numeral base is periodic of period if
where is a positive integer smaller than According to the summation formula of geometric series, this may be rewritten as
In other words, is a period of the representation of if and only if is a divisor of
Euler's theorem asserts that, if an integer is coprime with, then is a divisor of where is Euler's totient function. This proves that, for every integer coprime with, the representation of the reciprocal of is periodic in base.
All the periods of a periodic function are multiples of a shortest period generally called the fundamental period. In this article, we call period of in base the shortest period of the representation of in base. Therefore, the period of in base is the smallest positive integer such that that is a divisor of
In other words, the period of a prime in base is the multiplicative order of modulo.
According to Zsigmondy's theorem, every positive integer is a period of some prime in base except in the following cases:

and or
and for some integer

As
where is the th cyclotomic polynomial, the primes of period in base are prime divisors of More precisely, the primes of period are exactly the prime divisors of that do not divide .
If is even, the prime divisors of that do not divide are exactly the prime divisors of
This is wrong if is odd: if and , where is a positive integer, then
although 2 divides both and
If is odd, the primes of period are exactly, if, the prime divisors of, or, if, the odd prime divisors of.
As the period of every prime divides , if divides, then its period is smaller than. Conversely, if divides and has a period smaller than, then it is a common divisor of and As the resultant of two polynomials is a linear combination of these polynomials, divides the resultant of and As these two polynomials are coprime and divide divides also the discriminant of Thus, a prime divisor of, that has a period smaller than, is also a divisor of.
Now, we have to prove that, if a prime divides and then it does not divide In fact, this implies immediately that does not divide If is even, 2 cannot divide , and the condition is not restrictive.
Thus, let. It suffices to prove that does not divides for some polynomial, which is a multiple of We take
By Fermat's little theorem, we have As divides, we have also Thus the multiplicative order of modulo divides, which is a divisor of. Thus is a multiple of. Now,
As is prime and greater than 2, all the terms but the first one are multiple of This proves that does not divides
A prime is a unique prime in base, if and only if, for some, it is the unique prime divisor of that does not divide. If is even this means that
for some positive integer .
If is odd, this means that
for some integers and. This provides an efficient method for computing the unique primes and the primes of a given period.
Note that a prime divisor of is coprime with, and thus also with its divisor Such a prime has no period length, as the representation in base of its reciprocal is finite instead of being periodic. Thus, such a prime is never considered as a unique prime, even if it is the unique prime that has a finite reciprocal in base. For example, 2 is not considered as a unique prime in binary, although it is the only prime with finite reciprocal in binary.
The mention "terminated" means that the prime divides the base, and thus that the representation of its reciprocal is finite.

	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
2		1		1		1		1		1		1		1		1		1		1		1
3	2		1	2		1	2		1	2		1	2		1	2		1	2		1	2
5	4	4	2		1	4	4	2		1	4	4	2		1	4	4	2		1	4	4	2
7	3	6	3	6	2		1	3	6	3	6	2		1	3	6	3	6	2		1	3	6
11	10	5	5	5	10	10	10	5	2		1	10	5	5	5	10	10	10	5	2		1	10
13	12	3	6	4	12	12	4	3	6	12	2		1	12	3	6	4	12	12	4	3	6	12
17	8	16	4	16	16	16	8	8	16	16	16	4	16	8	2		1	8	16	4	16	16	16
19	18	18	9	9	9	3	6	9	18	3	6	18	18	18	9	9	2		1	18	18	9	9
23	11	11	11	22	11	22	11	11	22	22	11	11	22	22	11	22	11	22	22	22	2		1
29	28	28	14	14	14	7	28	14	28	28	4	14	28	28	7	4	28	28	7	28	14	7	7
31	5	30	5	3	6	15	5	15	15	30	30	30	15	10	5	30	15	15	15	30	30	10	30
37	36	18	18	36	4	9	12	9	3	6	9	36	12	36	9	36	36	36	36	18	36	12	36
41	20	8	10	20	40	40	20	4	5	40	40	40	8	40	5	40	5	40	20	20	40	10	40
43	14	42	7	42	3	6	14	21	21	7	42	21	21	21	7	21	42	42	42	7	14	21	21
47	23	23	23	46	23	23	23	23	46	46	23	46	23	46	23	23	23	46	46	23	46	46	23
53	52	52	26	52	26	26	52	26	13	26	52	13	52	13	13	26	52	52	52	52	52	4	13
59	58	29	29	29	58	29	58	29	58	58	29	58	58	29	29	29	58	29	29	29	29	58	58
61	60	10	30	30	60	60	20	5	60	4	15	3	6	15	15	60	60	30	5	12	15	20	20
67	66	22	33	22	33	66	22	11	33	66	66	66	11	11	33	33	66	33	66	33	11	33	11
71	35	35	35	5	35	70	35	35	35	70	35	70	10	35	35	10	35	35	7	70	70	14	35
73	9	12	9	72	36	24	3	6	8	72	36	72	72	72	9	24	18	36	72	24	8	36	12
79	39	78	39	39	78	78	13	39	13	39	26	39	26	26	39	26	13	39	39	13	13	3	6
83	82	41	41	82	82	41	82	41	41	41	41	82	82	82	41	41	82	82	82	41	82	41	82
89	11	88	11	44	88	88	11	44	44	22	8	88	88	88	11	44	44	88	44	44	22	88	88
97	48	48	24	96	12	96	16	24	96	48	16	96	96	96	12	96	16	32	32	96	4	96	24
101	100	100	50	25	10	100	100	50	4	100	100	50	10	100	25	10	100	25	50	50	50	50	25
103	51	34	51	102	102	51	17	17	34	102	102	17	17	51	51	51	51	51	102	102	34	17	34
107	106	53	53	106	106	106	106	53	53	53	53	53	53	106	53	106	106	53	106	106	106	53	106
109	36	27	18	27	108	27	12	27	108	108	54	108	108	27	9	36	108	36	54	27	27	36	108
113	28	112	14	112	112	14	28	56	112	56	112	56	28	4	7	112	8	112	112	112	56	112	112
127	7	126	7	42	126	126	7	63	42	63	126	63	126	63	7	63	63	3	6	63	9	126	18
131	130	65	65	65	130	65	130	65	130	65	65	65	130	65	65	130	26	26	65	65	130	130	26
137	68	136	34	136	136	68	68	68	8	68	136	136	34	34	17	68	34	68	136	136	34	136	136
139	138	138	69	69	23	69	46	69	46	69	138	69	46	138	69	138	138	138	69	138	138	46	69

Bold for unique primes.

	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
1		2	3	2	5	2, 3	7	2	3	2, 5	11	2, 3	13	2, 7	3, 5	2	17	2, 3	19	2, 5	3, 7	2, 11	23
2	3		5	3	7		3	5	11	3	13	7	3, 5		17	3	19	5	3, 7	11	23	3	5
3	7	13	7	31	43	19	73	7, 13	37	7, 19	157	61	211	241	7, 13	307	7	127	421	463	13	7, 79	601
4	5	5	17	13	37	5	5, 13	41	101	61	5, 29	5, 17	197	113	257	5, 29	5, 13	181	401	13, 17	5, 97	5, 53	577
5	31	11	11, 31	11, 71	311	2801	31, 151	11, 61	41, 271	3221	22621	30941	11, 3761	11, 4931	11, 31, 41	88741	41, 2711	151, 911	11, 61, 251	40841	245411	292561	346201
6		7	13	7	31	43	19	73	7, 13	37	7, 19	157	61	211	241	7, 13	307	7	127	421	463	13	7, 79
7	127	1093	43, 127	19531	55987	29, 4733	127, 337	547, 1093	239, 4649	43, 45319	659, 4943	5229043	8108731	1743463	29, 43, 113, 127	25646167	449, 80207	701, 70841	29, 71, 32719	43, 631, 3319	16968421	29, 5336717	29, 239, 28771
8	17	41	257	313	1297	1201	17, 241	17, 193	73, 137	7321	89, 233	14281	41, 937	17, 1489	65537	41761	113, 929	17, 3833	160001	97241	73, 3209	139921	331777
9	73	757	19, 73	19, 829	19, 2467	37, 1063	262657	19, 37, 757	333667	1772893	37, 80749	1609669	397, 18973	541, 21061	19, 37, 73, 109	19, 1270657	991, 34327	523, 29989	64008001	85775383	127, 297613	19, 7792003	19, 2017, 4987
10	11	61	41	521	11, 101	11, 191	11, 331	1181	9091	13421	19141	11, 2411	71, 101	31, 1531	61681	11, 71, 101	11, 9041	11, 2251	152381	185641	224071	31, 41, 211	11, 5791
11	23, 89	23, 3851	23, 89, 683	12207031	23, 3154757	1123, 293459	23, 89, 599479	23, 67, 661, 3851	21649, 513239	15797, 1806113	23, 266981089	23, 419, 859, 18041	67, 4027, 1154539	67, 463, 2333, 8537	23, 89, 397, 683, 2113	2141993519227	23, 199, 16127, 51217	104281, 62060021	10778947368421	17513875027111	67, 353, 1176469537	3937230404603	67, 7349, 134367047
12	13	73	241	601	13, 97	13, 181	37, 109	6481	9901	13, 1117	20593	28393	37, 1033	13, 3877	97, 673	83233	229, 457	13, 769	13, 12277	61, 3181	157, 1489	37, 7549	13, 73, 349
13	8191	797161	2731, 8191	305175781	3433, 760891	16148168401	79, 8191, 121369	398581, 797161	53, 79, 265371653	1093, 3158528101	477517, 20369233	53, 264031, 1803647	157, 29914249171	53, 157483, 16655159	53, 157, 1613, 2731, 8191	212057, 2919196853	79, 521, 29759719289	599, 29251, 133338869	3121, 142559, 9690539	79, 189437, 516094151	79, 2003, 85107437663	47691619, 480393499	53, 6553, 15913, 6895253
14	43	547	29, 113	29, 449	29, 197	113, 911	43, 5419	29, 16493	909091	1623931	211, 13063	29, 22079	7027567	10678711	15790321	22796593	32222107	197, 226871	827, 10529	81867661	29, 43, 86969	71, 673, 2969	183458857
15	151	4561	151, 331	181, 1741	1171, 1201	31, 159871	631, 23311	31, 271, 4561	31, 2906161	195019441	61, 661, 9781	4651, 161971	31, 2851, 15511	61, 39225301	61, 151, 331, 1321	6566760001	31, 601, 558721	31, 211, 2460181	31, 3001, 261451	211, 9391, 18181	61, 858794191	74912328481	241, 17881, 24481
16	257	17, 193	65537	17, 11489	17, 98801	17, 169553	97, 257, 673	21523361	17, 5882353	17, 6304673	17, 97, 260753	407865361	17, 5393, 16097	7121, 179953	641, 6700417	18913, 184417	97, 113607841	15073, 563377	17, 1505882353	62897, 300673	17, 3227992561	17, 3697, 623009	17, 2801, 2311681
17	131071	1871, 34511	43691, 131071	409, 466344409	239, 409, 1123, 30839	14009, 2767631689	103, 2143, 11119, 131071	103, 307, 1021, 1871, 34511	2071723, 5363222357	50544702849929377	2693651, 74876782031	103, 443, 15798461357509	103, 22771730193675277	1045002649, 6734509609	137, 953, 26317, 43691, 131071	10949, 1749233, 2699538733	7563707819165039903	3044803, 99995282631947	689852631578947368421	1502097124754084594737	239, 74729519, 176634767651	103, 62246266355102810647	307, 120574031, 341563234253
18	19	19, 37	37, 109	5167	46441	117307	87211	530713	19, 52579	590077	1657, 1801	19, 271, 937	19, 132049	19, 739, 811	433, 38737	1423, 5653	73, 465841	199, 236377	307, 69481	19, 37, 199, 613	19, 5966803	163, 271, 1117	127, 199, 7561
19	524287	1597, 363889	174763, 524287	191, 6271, 3981071	191, 638073026189	419, 4534166740403	32377, 524287, 1212847	1597, 2851, 101917, 363889	1111111111111111111	6115909044841454629	29043636306420266077	12865927, 9468940004449	459715689149916492091	4272113, 370649274902657	229, 457, 174763, 524287, 525313	229, 1103, 202607147, 291973723	6841, 6089884909802812423	109912203092239643840221	75368484119, 192696104561	12061389013, 54921106624003	45943, 341203, 97404596002423	2129, 63877469, 24939218613613	7282588256957615350925401
20	41	1181	61681	41, 9161	241, 6781	281, 4021	41, 61, 1321	42521761	3541, 27961	212601841	85403261	421, 601, 641	1061, 1383881	19421, 131381	4278255361	21881, 63541	15101, 145501	16936647121	41, 2801, 222361	41, 920421641	181, 401, 150901	61, 941, 272341	61, 1801385941
21	337	368089	337, 5419	379, 519499	1822428931	11898664849	92737, 649657	43, 2269, 368089	43, 1933, 10838689	1723, 8527, 27763	8177824843189	43, 337, 547, 2714377	43, 547, 2239000891	43, 2817034275427	337, 1429, 5419, 14449	43, 13567, 940143709	156107192084257	30640261, 68443621	460951, 8442733531	4789, 6427, 227633407	12271836836138419	43, 170689, 408030421	43, 10426753, 78066619
22	683	67, 661	397, 2113	23, 67, 5281	51828151	23, 10746341	67, 683, 20857	5501, 570461	23, 4093, 8779	23, 89, 199, 58367	57154490053	128011456717	23, 11737870057	23, 23504771357	353, 2931542417	23, 947, 87415373	536801, 6301307	23, 253239693257	23, 424016563147	23, 6073, 10362529	89, 285451051007	39700406579747	60867245726761
23	47, 178481	47, 1001523179	47, 178481, 2796203	8971, 332207361361	47, 139, 3221, 7505944891	47, 3083, 31479823396757	47, 178481, 10052678938039	47, 1001523179, 23535794707	11111111111111111111111	829, 28878847, 3740221981231	47, 39891250417, 321218438243	1381, 2519545342349331183143	47, 461, 2347, 10627, 2249861, 14525237	829, 31741, 3046462151831565769	47, 277, 1013, 1657, 30269, 178481, 2796203	47, 26552618219228090162977481	47, 599, 7468009, 20801237997245359	277, 2347, 16497763013, 1335495402823	691, 1381, 46266279097921483078651	47, 19597, 139870566115103282847737	4463, 1323064018651, 60575166785239	461, 1289, 831603031789, 1920647391913	47, 124799, 304751, 58769065453824529
24	241	6481	97, 673	390001	1678321	73, 193, 409	433, 38737	97, 577, 769	99990001	10657, 20113	193, 2227777	815702161	1475750641	2562840001	193, 22253377	73, 1321, 72337	11019855601	4297, 3952393	31177, 821113	73, 518118697	191353, 286777	937, 83575993	97, 1134793633

Decimal unique primes

At present, more than fifty unique primes or probable primes are known. However, there are only twenty-three unique primes below 10¹⁰⁰. The following table lists all 23 unique primes below 10¹⁰⁰ and and their periods and

	Prime
1	3
2	11
3	37
4	101
10	9,091
12	9,901
9	333,667
14	909,091
24	99,990,001
36	999,999,000,001
48	9,999,999,900,000,001
38	909,090,909,090,909,091
19	1,111,111,111,111,111,111
23	11,111,111,111,111,111,111,111
39	900,900,900,900,990,990,990,991
62	909,090,909,090,909,090,909,090,909,091
120	100,009,999,999,899,989,999,000,000,010,001
150	10,000,099,999,999,989,999,899,999,000,000,000,100,001
106	9,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
93	900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991
134	909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
294	142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143
196	999,999,999,999,990,000,000,000,000,099,999,999,999,999,000,000,000,000,009,999,999,999,999,900,000,000,000,001

The prime with period length 294 is similar to the reciprocal of 7
Just after the table, the twenty-fourth unique prime has 128 digits and period length 320. It can be written as ₂ + 1, where a subscript number n indicates n consecutive copies of the digit or group of digits before the subscript.
Though they are rare, based on the occurrence of repunit primes and probable primes, it is conjectured strongly that there are infinitely many unique primes.
the repunit /9 is the largest known probable unique prime.
In 1996 the largest proven unique prime was /10001 or, using the notation above, ₁₄₁ + 1. It has 1128 digits. The record has been improved many times since then. the largest proven unique prime is, it has 20160 digits.

Binary unique primes

The first unique primes in binary are:
The period length of them are:
They include Fermat primes, Mersenne primes and Wagstaff primes.
Additionally, if n is a natural number which is not equal to 1 or 6, than at least one prime have period n in base 2, because of the Zsigmondy theorem. Besides, if n is congruent to 4 and n > 20, then at least two primes have period n in base 2, because of the Aurifeuillean factorization, for example, 113 and 29 both have period 28 in base 2, 37 and 109 both have period 36 in base 2, and that 397 and 2113 both have period 44 in base 2,
As shown above, a prime p is a unique prime of period n in base 2 if and only if there exists a natural number c such that
The only known values of n such that is composite but is prime are 18, 20, 21, 54, 147, 342, 602, and 889. It is a conjecture that there is no other n with this property. All other known base 2 unique primes are of the form.
In fact, no prime with c > 1 have been discovered, and all known unique primes p have c = 1. It is conjectured that all unique primes have c = 1.
As of September 2019, the largest known base 2 unique prime is 2^82589933-1, it is also the largest known prime. With an exception of Mersenne primes, the largest known probable base 2 unique prime is, and the largest proven base 2 unique prime is. Besides, the largest known probable base 2 unique prime which is not Mersenne prime or Wagstaff prime is.
Similar to base 10, though they are rare, it is conjectured that there are infinitely many base 2 unique primes, because all Mersenne primes are unique in base 2, and it is conjectured they there are infinitely many Mersenne primes.
They divide none of overpseudoprimes to base 2, but every other odd prime number divide one overpseudoprime to base 2, because if and only if a composite number can be written as, it is an overpseudoprime to base 2.
There are 52 unique primes in base 2 below 2⁶⁴, they are:

	Prime	Prime
2	3	11
4	5	101
3	7	111
10	11	1011
12	13	1101
8	17	1 0001
18	19	1 0011
5	31	1 1111
20	41	10 1001
14	43	10 1011
9	73	100 1001
7	127	111 1111
15	151	1001 0111
24	241	1111 0001
16	257	1 0000 0001
30	331	1 0100 1011
21	337	1 0101 0001
22	683	10 1010 1011
26	2,731	1010 1010 1011
42	5,419	1 0101 0010 1011
13	8,191	1 1111 1111 1111
34	43,691	1010 1010 1010 1011
40	61,681	1111 0000 1111 0001
32	65,537	1 0000 0000 0000 0001
54	87,211	1 0101 0100 1010 1011
17	131,071	1 1111 1111 1111 1111
38	174,763	10 1010 1010 1010 1011
27	262,657	100 0000 0010 0000 0001
19	524,287	111 1111 1111 1111 1111
33	599,479	1001 0010 0101 1011 0111
46	2,796,203	10 1010 1010 1010 1010 1011
56	15,790,321	1111 0000 1111 0000 1111 0001
90	18,837,001	1 0001 1111 0110 1110 0000 1001
78	22,366,891	1 0101 0101 0100 1010 1010 1011
62	715,827,883	10 1010 1010 1010 1010 1010 1010 1011
31	2,147,483,647	111 1111 1111 1111 1111 1111 1111 1111
80	4,278,255,361	1111 1111 0000 0000 1111 1111 0000 0001
120	4,562,284,561	1 0000 1111 1110 1110 1111 0000 0001 0001
126	77,158,673,929	1 0001 1111 0111 0000 0011 1110 1110 0000 1001
150	1,133,836,730,401	1 0000 0111 1111 1101 1110 1111 1000 0000 0010 0001
86	2,932,031,007,403	10 1010 1010 1010 1010 1010 1010 1010 1010 1010 1011
98	4,363,953,127,297	11 1111 1000 0000 1111 1110 0000 0011 1111 1000 0001
49	4,432,676,798,593	100 0000 1000 0001 0000 0010 0000 0100 0000 1000 0001
69	10,052,678,938,039	1001 0010 0100 1001 0010 0101 1011 0110 1101 1011 0111
65	145,295,143,558,111	1000 0100 0010 0101 0010 1001 0110 1011 0101 1011 1101 1111
174	96,076,791,871,613,611	1 0101 0101 0101 0101 0101 0101 0100 1010 1010 1010 1010 1010 1010 1011
77	581,283,643,249,112,959	1000 0001 0001 0010 0010 0110 0100 1100 1101 1001 1011 1011 0111 0111 1111
93	658,812,288,653,553,079	1001 0010 0100 1001 0010 0100 1001 0011 0110 1101 1011 0110 1101 1011 0111
122	768,614,336,404,564,651	1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1011
61	2,305,843,009,213,693,951	1 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111
85	9,520,972,806,333,758,431	1000 0100 0010 0001 0100 1010 0101 0010 1011 0101 1010 1101 0111 1011 1101 1111
192	18,446,744,069,414,584,321	1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0001

After the table, the next 10 binary unique prime have period length 170, 234, 158, 165, 147, 129, 184, 89, 208, and 312. Besides, the bits of them are 65, 73, 78, 81, 82, 84, 88, 89, 96, and 97.

Bi-unique primes

Bi-unique primes are a pairs of primes having a period length shared by no other primes. For example, in binary, the bi-unique primes with at least one prime less than 10000 are:

prime p	the only other prime having the same period as p	period length
23	89	11
29	113	28
37	109	36
47	178481	23
59	3033169	58
61	1321	60
67	20857	66
71	122921	35
79	121369	39
83	8831418697	82
89	23	11
97	673	48
107	28059810762433	106
109	37	36
113	29	28
139	168749965921	138
167	57912614113275649087721	83
193	22253377	96
223	616318177	37
251	4051	50
263	10350794431055162386718619237468234569	131
281	86171	70
283	165768537521	94
353	2931542417	88
397	2113	44
433	38737	72
463	4982397651178256151338302204762057	231
571	160465489	114
577	487824887233	144
601	1801	25
607	1512768222413735255864403005264105839324374778520631853993	303
631	23311	45
641	6700417	64
643	84115747449047881488635567801	214
673	97	48
727	1786393878363164227858270210279	121
751	2139731020464054092520609592459940706818275139793055476751	375
769	442499826945303593556473164314770689	384
919	75582488424179347083438319	153
1039	19709014643115560219397264671577125505264032974428376489237001990435774189483906244488746953221813209	519
1291	83861817871925183739792206470703862766563053456867813459969184678546547694793573468589875745315081	1290
1321	61	60
1327	2365454398418399772605086209214363458552839866247069233	221
1429	14449	84
1471	252359902034571016856214298851708529738525821631	245
1543	4965395030068548134274243124972075225434447114375481299036593442726326832727934403424309955102162841656341524725641213163998408700663382552888660520657	771
1697	99335205800663868215396640964567095667094665346141013294320587365443384719802857319737050495099341955640963272958071602273	848
1753	1795918038741070627	146
1777	25781083	74
1801	601	25
2113	397	44
2281	3011347479614249131	190
2801	1114513219367157067542813609361306957257890531134775327875067038594481393220804051366788787128409731513666376851495151281817670381468528387601	1400
2971	48912491	110
3011	631215008947706187342830494125660733360092019659681922883823392015121754384870744044074337887482936870852519582960673945561810148710850934449712549090934572292098088972061029650939105592263256293676274598529593937386833315889748213948490958132757432166701901197169972066727635929332437543971934775961	3010
3259	960843850986532976532466235773483492840618819232206145010143480044702708779967241439519037158800917230289	1086
3361	88959882481	168
3967	3296810823331827444014404831943558588631803435050404237042485765714486337505843011741487225539321479275976317423474114853376321380782906502106758766783934866952124117240484839332668914566806988602931402117416523955329423560856334826333176954575294550104263404414368761262079586842542586869780254842277261781328657636993064897732127711363870426953852536828242291991249685206783121190349820804553	1983
4051	251	50
4129	33770734168253651800370989375796994825389296318018601048482005531172856260013942500368975908606689	688
4177	9857737155463	87
4523	106788290443848295284382097033	266
4561	51049903050598156013062477654241640657829025002976204451060261008689478158715729745160924860467530309657376827104233308157772350164622158651187694109112727796663977157921	2280
4871	82033219963138371097689272308258116841679442057301643873942124991182012434598644913857356023840478815121709542915222280972560231358838127531337	487
5153	54410972897	112
5281	860573414369008969457638101533827364704684164286188824383996871471626764468219429432850649798234488791036977295952185305281368586922360240161830570497885830241813162641447900350702795124321	2640
5347	242099935645987	198
5431	9496792988973395279834809661569251320544014305629535339041176003993804676485199470146061766714674245857484692035146769896735263094609746643655454990307740324793870789211183639302657602006206727998357155351588374904926882678074633252877772442134938712719216422825919305415646969838444956229559805502059906228691551284152908049964563516405524127028294884716508273187514617905113012642316992618568629664046915514871575875088457784721	2715
5881	23618256244840618857212522155851714598259422753496906641681177748710460515038403366198473773770441	1470
6043	4475130366518102084427698737	318
6659	67348091890626757137914773048080151982788009808953349522971703676209072447919253736713943719839321930921001920443146832964494535806286153336758808764827052922284527987735369082658226155752758837734585788583920437031679967832798697874531506375848295306002069974292647012532975382657869395896549536084026643086849168707638035984652951756449232381922841508279043449553683642165895518852273616853752192626547082225232322662203349617421624450106130361133033050996977517456780759336980017504035553443204836636663583950658161718264375035034007950563418777684540446570742277682688819816930529249402141227674467861784042184664703273065772145626307008333021727910295689307897812592176340797189662547498298629041419687123412984210580325376481846316396566413701109848755348878790562296275295266190788880122151835567771665815565611863261470167857288685014242115505182159685753576612239477286620238583071292970734389580521730540789853959607322402465845627773409459421340250476125665859926003121138412497735360569	6658
6719	215006106257113223254503015023149432126193150293791416185445173578281597218315377296589584591228602041183907532584815068471747291177386898925622477208530115714962355294842135137890474394949339249259335407710018584480055157825387089416912233252714054247018216597994795059161567922302450277281351135838393171424038832688432240078361264161523904355539085927738753968157018550258476163852090826756157915705283413226000816151712543838581066281600650278690534719371112997393190721068136840596790525950480851370510277560248341182341805553054000587378384785994695875587394905226703149605689830257768229987714770949192483302583569799141079867597051190134078718011730508482542567284418838119000563443985593691221203060137047648713095502877775290062907208508727269017130916691676838817452529964938878349395785642430571852241837461604136374448443730175081889502056290512497717177577492736555784081731998565765598518104822516520340301701034123926767472784665776779480628821628279687651736198541330802238405154786248043073	3359
7487	26828803997912886929710867041891989490486893845712448833	197
8929	197107422273014301919781414466039325387889623676342705850752210599969	496
8969	10508537584872980049787749414505440238543661684506416445249892188329191267897669657242625405655025902294996965713681247700894953567276596965114308183649957469931262029470372188492494505614207827774171575432114297123003373257035070542940532411186322417809411123684246738342720455933424175399671044286557638075591	1121
9547	1621441292160739312484402643488810210953460916758334047593952342310982348899125523375207637304333778211869062392988099059802528019593682234941755422758885656395068722385037980657466257618112582188770921312100125511337836412531718154395821529210922389443733616354268820219577863577759459082447218927273695668223251258943006743614909639761127161704816862626236353032622115795192245125083091261029053988053316433377173895018793740052548266015018756763150731725385456332232982433576015547722563978072554378000015707071821371450842910648052930276764535303424167478747579771592484270800978561959411183367133498969236434846108865206764889977963554070295936092795484663326925724277620386077381551473009733178990983013487085676185144378484902884955972104873606558173086267499547566081780818064857671567480196001693236835136811036110768546793929610732909274227296407079545788520811837495181586420117807667033593394473	9546

Although there are 1228 odd primes below 10000, only 21 of them are unique and 76 of them are bi-unique in binary.
A classic example of binary bi-unique primes are
and
they are the two prime factors of the Mersenne number 2¹⁰⁶¹−1. Thus, the period
length of them is 1061.
As of October 2016, the largest known probable binary bi-unique prime is, it has a period
length of 5240707 shares with only the prime 75392810903.
Similarly, we can define "tri-unique primes" as a triple of primes having a period
length shared by no other primes. The first few tri-unique primes are:

prime p	the only two other primes having the same period as p	period length
53	157, 1613	52
101	8101, 268501	100
103	2143, 11119	51
131	409891, 7623851	130
137	953, 26317	68
157	53, 1613	52
163	135433, 272010961	162
179	62020897, 18584774046020617	178
181	54001, 29247661	180
191	420778751, 30327152671	95
197	19707683773, 4981857697937	196
199	153649, 33057806959	99
211	664441, 1564921	210
229	457, 525313	76
233	1103, 2089	29
271	348031, 49971617830801	135
307	2857, 6529	102
317	381364611866507317969, 604462909806215075725313	316
359	1433, 1489459109360039866456940197095433721664951999121	179
367	55633, 37201708625305146303973352041	183
373	951088215727633, 4611545283086450689	372
419	3410623284654639440707, 1607792018780394024095514317003	418
421	146919792181, 1041815865690181	420
431	9719, 2099863	43
439	2298041, 9361973132609	73
443	4714692062809, 4507513575406446515845401458366741487526913	442
457	229, 525313	76
467	27961, 352369374013660139472574531568890678155040563007620742839120913	466
491	15162868758218274451, 50647282035796125885000330641	490

In binary, the smallest n-unique prime are
In binary, the period length of odd primes are:

prime	period length	prime	period length	prime	period length	prime	period length	prime	period length	prime	period length	prime	period length
3	2	79	39	181	180	293	292	421	420	557	556	673	48
5	4	83	82	191	95	307	102	431	43	563	562	677	676
7	3	89	11	193	96	311	155	433	72	569	284	683	22
11	10	97	48	197	196	313	156	439	73	571	114	691	230
13	12	101	100	199	99	317	316	443	442	577	144	701	700
17	8	103	51	211	210	331	30	449	224	587	586	709	708
19	18	107	106	223	37	337	21	457	76	593	148	719	359
23	11	109	36	227	226	347	346	461	460	599	299	727	121
29	28	113	28	229	76	349	348	463	231	601	25	733	244
31	5	127	7	233	29	353	88	467	466	607	303	739	246
37	36	131	130	239	119	359	179	479	239	613	612	743	371
41	20	137	68	241	24	367	183	487	243	617	154	751	375
43	14	139	138	251	50	373	372	491	490	619	618	757	756
47	23	149	148	257	16	379	378	499	166	631	45	761	380
53	52	151	15	263	131	383	191	503	251	641	64	769	384
59	58	157	52	269	268	389	388	509	508	643	214	773	772
61	60	163	162	271	135	397	44	521	260	647	323	787	786
67	66	167	83	277	92	401	200	523	522	653	652	797	796
71	35	173	172	281	70	409	204	541	540	659	658	809	404
73	9	179	178	283	94	419	418	547	546	661	660	811	270

In binary, the primes with given period length are:

period length	prime	period length	prime	period length	prime	period length	prime
1		26	2731	51	103, 2143, 11119	76	229, 457, 525313
2	3	27	262657	52	53, 157, 1613	77	581283643249112959
3	7	28	29, 113	53	6361, 69431, 20394401	78	22366891
4	5	29	233, 1103, 2089	54	87211	79	2687, 202029703, 1113491139767
5	31	30	331	55	881, 3191, 201961	80	4278255361
6		31	2147483647	56	15790321	81	2593, 71119, 97685839
7	127	32	65537	57	32377, 1212847	82	83, 8831418697
8	17	33	599479	58	59, 3033169	83	167, 57912614113275649087721
9	73	34	43691	59	179951, 3203431780337	84	1429, 14449
10	11	35	71, 122921	60	61, 1321	85	9520972806333758431
11	23, 89	36	37, 109	61	2305843009213693951	86	2932031007403
12	13	37	223, 616318177	62	715827883	87	4177, 9857737155463
13	8191	38	174763	63	92737, 649657	88	353, 2931542417
14	43	39	79, 121369	64	641, 6700417	89	618970019642690137449562111
15	151	40	61681	65	145295143558111	90	18837001
16	257	41	13367, 164511353	66	67, 20857	91	911, 112901153, 23140471537
17	131071	42	5419	67	193707721, 761838257287	92	277, 1013, 1657, 30269
18	19	43	431, 9719, 2099863	68	137, 953, 26317	93	658812288653553079
19	524287	44	397, 2113	69	10052678938039	94	283, 165768537521
20	41	45	631, 23311	70	281, 86171	95	191, 420778751, 30327152671
21	337	46	2796203	71	228479, 48544121, 212885833	96	193, 22253377
22	683	47	2351, 4513, 13264529	72	433, 38737	97	11447, 13842607235828485645766393
23	47, 178481	48	97, 673	73	439, 2298041, 9361973132609	98	4363953127297
24	241	49	4432676798593	74	1777, 25781083	99	199, 153649, 33057806959
25	601, 1801	50	251, 4051	75	100801, 10567201	100	101, 8101, 268501

Period lengths

Period length	Primes	Period length	Primes	Period length	Primes	Period length	Primes	Period length	Primes
1	3	21	43, 1933, 10838689	41	83, 1231, 538987, 201763709900322803748657942361	61	733, 4637, 329401, 974293, 1360682471, 106007173861643, 7061709990156159479	81	163, 9397, 2462401, 676421558270641, 130654897808007778425046117
2	11	22	23, 4093, 8779	42	127, 2689, 459691	62	909090909090909090909090909091	82	2670502781396266997, 3404193829806058997303
3	37	23	11111111111111111111111	43	173, 1527791, 1963506722254397, 2140992015395526641	63	10837, 23311, 45613, 45121231, 1921436048294281	83	3367147378267, 9512538508624154373682136329, 346895716385857804544741137394505425384477
4	101	24	99990001	44	89, 1052788969, 1056689261	64	19841, 976193, 6187457, 834427406578561	84	226549, 4458192223320340849
5	41, 271	25	21401, 25601, 182521213001	45	238681, 4185502830133110721	65	162503518711, 5538396997364024056286510640780600481	85	262533041, 8119594779271, 4222100119405530170179331190291488789678081
6	7, 13	26	859, 1058313049	46	47, 139, 2531, 549797184491917	66	599144041, 183411838171	86	57009401, 2182600451, 7306116556571817748755241
7	239, 4649	27	757, 440334654777631	47	35121409, 316362908763458525001406154038726382279	67	493121, 79863595778924342083, 28213380943176667001263153660999177245677	87	4003, 72559, 310170251658029759045157793237339498342763245483
8	73, 137	28	29, 281, 121499449	48	9999999900000001	68	28559389, 1491383821, 2324557465671829	88	617, 16205834846012967584927082656402106953
9	333667	29	3191, 16763, 43037, 62003, 77843839397	49	505885997, 1976730144598190963568023014679333	69	277, 203864078068831, 1595352086329224644348978893	89	497867, 103733951, 104984505733, 5078554966026315671444089, 403513310222809053284932818475878953159
10	9091	30	211, 241, 2161	50	251, 5051, 78875943472201	70	4147571, 265212793249617641	90	29611, 3762091, 8985695684401
11	21649, 513239	31	2791, 6943319, 57336415063790604359	51	613, 210631, 52986961, 13168164561429877	71	241573142393627673576957439049, 45994811347886846310221728895223034301839	91	547, 14197, 17837, 4262077, 43442141653, 316877365766624209, 110742186470530054291318013
12	9901	32	353, 449, 641, 1409, 69857	52	521, 1900381976777332243781	72	3169, 98641, 3199044596370769	92	1289, 18371524594609, 4181003300071669867932658901
13	53, 79, 265371653	33	67, 1344628210313298373	53	107, 1659431, 1325815267337711173, 47198858799491425660200071	73	12171337159, 1855193842151350117, 49207341634646326934001739482502131487446637	93	900900900900900900900900900900990990990990990990990990990991
14	909091	34	103, 4013, 21993833369	54	70541929, 14175966169	74	7253, 422650073734453, 296557347313446299	94	6299, 4855067598095567, 297262705009139006771611927
15	31, 2906161	35	71, 123551, 102598800232111471	55	1321, 62921, 83251631, 1300635692678058358830121	75	151, 4201, 15763985553739191709164170940063151	95	191, 59281, 63841, 1289981231950849543985493631, 965194617121640791456070347951751
16	17, 5882353	36	999999000001	56	7841, 127522001020150503761	76	722817036322379041, 1369778187490592461	96	97, 206209, 66554101249, 75118313082913
17	2071723, 5363222357	37	2028119, 247629013, 2212394296770203368013	57	21319, 10749631, 3931123022305129377976519	77	5237, 42043, 29920507, 136614668576002329371496447555915740910181043	97	12004721, 92556179448994367391887834053878562534782033760810527051075248738484727059555245899601591
18	19, 52579	38	909090909090909091	58	59, 154083204930662557781201849	78	157, 6397, 216451, 388847808493	98	197, 5076141624365532994918781726395939035533
19	1111111111111111111	39	900900900900990990990991	59	2559647034361, 4340876285657460212144534289928559826755746751	79	317, 6163, 10271, 307627, 49172195536083790769, 3660574762725521461527140564875080461079917	99	199, 397, 34849, 362853724342990469324766235474268869786311886053883
20	3541, 27961	40	1676321, 5964848081	60	61, 4188901, 39526741	80	5070721, 19721061166646717498359681	100	60101, 7019801, 14103673319201, 1680588011350901

Unique prime in various bases

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	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
2		1		1		1		1		1		1		1		1		1		1		1
3	2		1	2		1	2		1	2		1	2		1	2		1	2		1	2
5	4	4	2		1	4	4	2		1	4	4	2		1	4	4	2		1	4	4	2
7	3	6	3	6	2		1	3	6	3	6	2		1	3	6	3	6	2		1	3	6
11	10	5	5	5	10	10	10	5	2		1	10	5	5	5	10	10	10	5	2		1	10
13	12	3	6	4	12	12	4	3	6	12	2		1	12	3	6	4	12	12	4	3	6	12
17	8	16	4	16	16	16	8	8	16	16	16	4	16	8	2		1	8	16	4	16	16	16
19	18	18	9	9	9	3	6	9	18	3	6	18	18	18	9	9	2		1	18	18	9	9
23	11	11	11	22	11	22	11	11	22	22	11	11	22	22	11	22	11	22	22	22	2		1
29	28	28	14	14	14	7	28	14	28	28	4	14	28	28	7	4	28	28	7	28	14	7	7
31	5	30	5	3	6	15	5	15	15	30	30	30	15	10	5	30	15	15	15	30	30	10	30
37	36	18	18	36	4	9	12	9	3	6	9	36	12	36	9	36	36	36	36	18	36	12	36
41	20	8	10	20	40	40	20	4	5	40	40	40	8	40	5	40	5	40	20	20	40	10	40
43	14	42	7	42	3	6	14	21	21	7	42	21	21	21	7	21	42	42	42	7	14	21	21
47	23	23	23	46	23	23	23	23	46	46	23	46	23	46	23	23	23	46	46	23	46	46	23
53	52	52	26	52	26	26	52	26	13	26	52	13	52	13	13	26	52	52	52	52	52	4	13
59	58	29	29	29	58	29	58	29	58	58	29	58	58	29	29	29	58	29	29	29	29	58	58
61	60	10	30	30	60	60	20	5	60	4	15	3	6	15	15	60	60	30	5	12	15	20	20
67	66	22	33	22	33	66	22	11	33	66	66	66	11	11	33	33	66	33	66	33	11	33	11
71	35	35	35	5	35	70	35	35	35	70	35	70	10	35	35	10	35	35	7	70	70	14	35
73	9	12	9	72	36	24	3	6	8	72	36	72	72	72	9	24	18	36	72	24	8	36	12
79	39	78	39	39	78	78	13	39	13	39	26	39	26	26	39	26	13	39	39	13	13	3	6
83	82	41	41	82	82	41	82	41	41	41	41	82	82	82	41	41	82	82	82	41	82	41	82
89	11	88	11	44	88	88	11	44	44	22	8	88	88	88	11	44	44	88	44	44	22	88	88
97	48	48	24	96	12	96	16	24	96	48	16	96	96	96	12	96	16	32	32	96	4	96	24
101	100	100	50	25	10	100	100	50	4	100	100	50	10	100	25	10	100	25	50	50	50	50	25
103	51	34	51	102	102	51	17	17	34	102	102	17	17	51	51	51	51	51	102	102	34	17	34
107	106	53	53	106	106	106	106	53	53	53	53	53	53	106	53	106	106	53	106	106	106	53	106
109	36	27	18	27	108	27	12	27	108	108	54	108	108	27	9	36	108	36	54	27	27	36	108
113	28	112	14	112	112	14	28	56	112	56	112	56	28	4	7	112	8	112	112	112	56	112	112
127	7	126	7	42	126	126	7	63	42	63	126	63	126	63	7	63	63	3	6	63	9	126	18
131	130	65	65	65	130	65	130	65	130	65	65	65	130	65	65	130	26	26	65	65	130	130	26
137	68	136	34	136	136	68	68	68	8	68	136	136	34	34	17	68	34	68	136	136	34	136	136
139	138	138	69	69	23	69	46	69	46	69	138	69	46	138	69	138	138	138	69	138	138	46	69

	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
2		1		1		1		1		1		1		1		1		1		1		1
3	2		1	2		1	2		1	2		1	2		1	2		1	2		1	2
5	4	4	2		1	4	4	2		1	4	4	2		1	4	4	2		1	4	4	2
7	3	6	3	6	2		1	3	6	3	6	2		1	3	6	3	6	2		1	3	6
11	10	5	5	5	10	10	10	5	2		1	10	5	5	5	10	10	10	5	2		1	10
13	12	3	6	4	12	12	4	3	6	12	2		1	12	3	6	4	12	12	4	3	6	12
17	8	16	4	16	16	16	8	8	16	16	16	4	16	8	2		1	8	16	4	16	16	16
19	18	18	9	9	9	3	6	9	18	3	6	18	18	18	9	9	2		1	18	18	9	9
23	11	11	11	22	11	22	11	11	22	22	11	11	22	22	11	22	11	22	22	22	2		1
29	28	28	14	14	14	7	28	14	28	28	4	14	28	28	7	4	28	28	7	28	14	7	7
31	5	30	5	3	6	15	5	15	15	30	30	30	15	10	5	30	15	15	15	30	30	10	30
37	36	18	18	36	4	9	12	9	3	6	9	36	12	36	9	36	36	36	36	18	36	12	36
41	20	8	10	20	40	40	20	4	5	40	40	40	8	40	5	40	5	40	20	20	40	10	40
43	14	42	7	42	3	6	14	21	21	7	42	21	21	21	7	21	42	42	42	7	14	21	21
47	23	23	23	46	23	23	23	23	46	46	23	46	23	46	23	23	23	46	46	23	46	46	23
53	52	52	26	52	26	26	52	26	13	26	52	13	52	13	13	26	52	52	52	52	52	4	13
59	58	29	29	29	58	29	58	29	58	58	29	58	58	29	29	29	58	29	29	29	29	58	58
61	60	10	30	30	60	60	20	5	60	4	15	3	6	15	15	60	60	30	5	12	15	20	20
67	66	22	33	22	33	66	22	11	33	66	66	66	11	11	33	33	66	33	66	33	11	33	11
71	35	35	35	5	35	70	35	35	35	70	35	70	10	35	35	10	35	35	7	70	70	14	35
73	9	12	9	72	36	24	3	6	8	72	36	72	72	72	9	24	18	36	72	24	8	36	12
79	39	78	39	39	78	78	13	39	13	39	26	39	26	26	39	26	13	39	39	13	13	3	6
83	82	41	41	82	82	41	82	41	41	41	41	82	82	82	41	41	82	82	82	41	82	41	82
89	11	88	11	44	88	88	11	44	44	22	8	88	88	88	11	44	44	88	44	44	22	88	88
97	48	48	24	96	12	96	16	24	96	48	16	96	96	96	12	96	16	32	32	96	4	96	24
101	100	100	50	25	10	100	100	50	4	100	100	50	10	100	25	10	100	25	50	50	50	50	25
103	51	34	51	102	102	51	17	17	34	102	102	17	17	51	51	51	51	51	102	102	34	17	34
107	106	53	53	106	106	106	106	53	53	53	53	53	53	106	53	106	106	53	106	106	106	53	106
109	36	27	18	27	108	27	12	27	108	108	54	108	108	27	9	36	108	36	54	27	27	36	108
113	28	112	14	112	112	14	28	56	112	56	112	56	28	4	7	112	8	112	112	112	56	112	112
127	7	126	7	42	126	126	7	63	42	63	126	63	126	63	7	63	63	3	6	63	9	126	18
131	130	65	65	65	130	65	130	65	130	65	65	65	130	65	65	130	26	26	65	65	130	130	26
137	68	136	34	136	136	68	68	68	8	68	136	136	34	34	17	68	34	68	136	136	34	136	136
139	138	138	69	69	23	69	46	69	46	69	138	69	46	138	69	138	138	138	69	138	138	46	69

	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
2		1		1		1		1		1		1		1		1		1		1		1
3	2		1	2		1	2		1	2		1	2		1	2		1	2		1	2
5	4	4	2		1	4	4	2		1	4	4	2		1	4	4	2		1	4	4	2
7	3	6	3	6	2		1	3	6	3	6	2		1	3	6	3	6	2		1	3	6
11	10	5	5	5	10	10	10	5	2		1	10	5	5	5	10	10	10	5	2		1	10
13	12	3	6	4	12	12	4	3	6	12	2		1	12	3	6	4	12	12	4	3	6	12
17	8	16	4	16	16	16	8	8	16	16	16	4	16	8	2		1	8	16	4	16	16	16
19	18	18	9	9	9	3	6	9	18	3	6	18	18	18	9	9	2		1	18	18	9	9
23	11	11	11	22	11	22	11	11	22	22	11	11	22	22	11	22	11	22	22	22	2		1
29	28	28	14	14	14	7	28	14	28	28	4	14	28	28	7	4	28	28	7	28	14	7	7
31	5	30	5	3	6	15	5	15	15	30	30	30	15	10	5	30	15	15	15	30	30	10	30
37	36	18	18	36	4	9	12	9	3	6	9	36	12	36	9	36	36	36	36	18	36	12	36
41	20	8	10	20	40	40	20	4	5	40	40	40	8	40	5	40	5	40	20	20	40	10	40
43	14	42	7	42	3	6	14	21	21	7	42	21	21	21	7	21	42	42	42	7	14	21	21
47	23	23	23	46	23	23	23	23	46	46	23	46	23	46	23	23	23	46	46	23	46	46	23
53	52	52	26	52	26	26	52	26	13	26	52	13	52	13	13	26	52	52	52	52	52	4	13
59	58	29	29	29	58	29	58	29	58	58	29	58	58	29	29	29	58	29	29	29	29	58	58
61	60	10	30	30	60	60	20	5	60	4	15	3	6	15	15	60	60	30	5	12	15	20	20
67	66	22	33	22	33	66	22	11	33	66	66	66	11	11	33	33	66	33	66	33	11	33	11
71	35	35	35	5	35	70	35	35	35	70	35	70	10	35	35	10	35	35	7	70	70	14	35
73	9	12	9	72	36	24	3	6	8	72	36	72	72	72	9	24	18	36	72	24	8	36	12
79	39	78	39	39	78	78	13	39	13	39	26	39	26	26	39	26	13	39	39	13	13	3	6
83	82	41	41	82	82	41	82	41	41	41	41	82	82	82	41	41	82	82	82	41	82	41	82
89	11	88	11	44	88	88	11	44	44	22	8	88	88	88	11	44	44	88	44	44	22	88	88
97	48	48	24	96	12	96	16	24	96	48	16	96	96	96	12	96	16	32	32	96	4	96	24
101	100	100	50	25	10	100	100	50	4	100	100	50	10	100	25	10	100	25	50	50	50	50	25
103	51	34	51	102	102	51	17	17	34	102	102	17	17	51	51	51	51	51	102	102	34	17	34
107	106	53	53	106	106	106	106	53	53	53	53	53	53	106	53	106	106	53	106	106	106	53	106
109	36	27	18	27	108	27	12	27	108	108	54	108	108	27	9	36	108	36	54	27	27	36	108
113	28	112	14	112	112	14	28	56	112	56	112	56	28	4	7	112	8	112	112	112	56	112	112
127	7	126	7	42	126	126	7	63	42	63	126	63	126	63	7	63	63	3	6	63	9	126	18
131	130	65	65	65	130	65	130	65	130	65	65	65	130	65	65	130	26	26	65	65	130	130	26
137	68	136	34	136	136	68	68	68	8	68	136	136	34	34	17	68	34	68	136	136	34	136	136
139	138	138	69	69	23	69	46	69	46	69	138	69	46	138	69	138	138	138	69	138	138	46	69