Unique prime

Unique prime
Number of known terms	102
Conjectured number of terms	Infinite
First terms	3, 11, 37, 101
Largest known term	(10²⁷⁰³⁴³-1)/9
OEIS index	A040017

In number theory, a unique 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.^[1] 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. 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 (or, more generally, an integer) $p$ in the numeral base $b$ is periodic of period $n$ if

{\frac {1}{p}}=\sum _{i=1}^{\infty }{\frac {q}{(b^{n})^{i}}},

where $q$ is a positive integer smaller than $b^{n}.$ According to the summation formula of geometric series, this may be rewritten as

{\frac {1}{p}}={\frac {q}{b^{n}-1}}.

In other words, $n$ is a period of the representation of $1/ p$ if and only if $p$ is a divisor of $b^{n}-1.$ Euler's theorem asserts that, if an integer $b$ is coprime with $p$ , then $p$ is a divisor of $p^{\varphi (n)}-1$ where $\varphi$ is Euler's totient function. This proves that, for every integer $p$ coprime with $b$ , the representation of the reciprocal of $p$ is periodic in base $b$ .

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 $p$ in base $b$ the shortest period of the representation of $1/ p$ in base $b$ . Therefore, the period of $p$ in base $b$ is the smallest positive integer $n$ such that that $p$ is a divisor of $b^{n}-1.$ In other words, the period of a prime $p$ in base $b$ is the multiplicative order of $b$ modulo $p$ .

According to Zsigmondy's theorem, every positive integer is a period of some prime in base $b$ except in the following cases:

$b = 2$ and $n = 1$ or $6$
$n = 2$ and $b = 2 k - 1$ for some integer $k > 1$

x^{n}-1=\prod _{i\mid n}\Phi _{n}(x),

where $\Phi _{n}$ is the $n$ th cyclotomic polynomial, the primes of period $n$ in base $b$ are prime divisors of $\Phi _{n}(b).$ More precisely, the primes of period $n$ are exactly the prime divisors of $\Phi _{n}(b)$ that do not divide $n$ (see below for a proof of this result and of the following ones).

If $b$ is even (this includes the binary and the decimal cases), the prime divisors of $\Phi _{n}(b)$ that do not divide $n$ are exactly the prime divisors of

R_{n}(b)={\frac {\Phi _{n}(b)}{\gcd(\Phi _{n}(b),n)}}.

This is wrong if $b$ is odd: if $n = 2$ and $b = 4 k - 1$ , where $k$ is a positive integer, then

R_{2}(b)={\frac {\Phi _{2}(b)}{\gcd(\Phi _{2}(b),2)}}={\frac {b+1}{2}}=2k,

although 2 divides both $n = 2$ and $\Phi _{n}(b).$

If $b$ is odd, the primes of period $n$ are exactly, if $n = 1$ , the prime divisors of $R_{1}(b)=b-1$ , or, if $n > 1$ , the odd prime divisors of $R n (b)$ .

Sketch of the proof of the characterization of primes of period $n$
As the period of every prime $p$ divides $p - 1$ (Fermat's little theorem), if $p$ divides $n$ , then its period is smaller than $n$ . Conversely, if $p$ divides $\Phi _{n}(b)$ and has a period $k$ smaller than $n$ , then it is a common divisor of $\Phi _{n}(b)$ and $\Phi _{k}(b).$ As the resultant of two polynomials is a linear combination of these polynomials, $p$ divides the resultant of $\Phi _{n}(x)$ and $\Phi _{k}(x).$ As these two polynomials are coprime and divide $x^{n}-1,$ $p$ divides also the discriminant $n^n$ of $x^{n}-1.$ Thus, a prime divisor of $\Phi _{n}(b)$ , that has a period smaller than $n$ , is also a divisor of $n$ . Now, we have to prove that, if a prime $p > 2$ divides $n$ and $\Phi _{n}(b),$ then it does not divide $\Phi _{n}(b)/p.$ In fact, this implies immediately that $p$ does not divide $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ If $b$ is even, 2 cannot divide $\Phi _{n}(b),$ (which is odd), and the condition $p > 2$ is not restrictive. Thus, let $n = pm$ . It suffices to prove that $p^{2}$ does not divides $S (b)$ for some polynomial $S (x)$ , which is a multiple of $\Phi _{n}(x).$ We take $S(x)={\frac {x^{n}-1}{x^{m}-1}}=1+x^{m}+x^{2m}+\cdots +x^{(p-1)m}.$ By Fermat's little theorem, we have $b^{p-1}\equiv 1{\pmod {p}}.$ As $p$ divides $\Phi _{n}(b)$ , we have also $b^{n}\equiv 1{\pmod {p}}.$ Thus the multiplicative order of $b$ modulo $p$ divides $gcd(n, p - 1)$ , which is a divisor of $m = n / p$ . Thus $c = b m - 1$ is a multiple of $p$ . Now, $S(b)={\frac {(1+c)^{p}-1}{c}}=p+{\binom {p}{2}}c+\cdots +{\binom {p}{p}}c^{p-1}.$ As $p$ is prime and greater than 2, all the terms but the first one are multiple of $p^{2}.$ This proves that $p^{2}$ does not divides $\Phi _{n}(b).$

Sketch of the proof of the characterization of primes of period

n

As the period of every prime $p$ divides $p - 1$ (Fermat's little theorem), if $p$ divides $n$ , then its period is smaller than $n$ . Conversely, if $p$ divides $\Phi _{n}(b)$ and has a period $k$ smaller than $n$ , then it is a common divisor of $\Phi _{n}(b)$ and $\Phi _{k}(b).$ As the resultant of two polynomials is a linear combination of these polynomials, $p$ divides the resultant of $\Phi _{n}(x)$ and $\Phi _{k}(x).$ As these two polynomials are coprime and divide $x^{n}-1,$ $p$ divides also the discriminant $n^n$ of $x^{n}-1.$ Thus, a prime divisor of $\Phi _{n}(b)$ , that has a period smaller than $n$ , is also a divisor of $n$ .

Now, we have to prove that, if a prime $p > 2$ divides $n$ and $\Phi _{n}(b),$ then it does not divide $\Phi _{n}(b)/p.$ In fact, this implies immediately that $p$ does not divide $\Phi _{n}(b)/\gcd(\Phi _{n}(b),n).$ If $b$ is even, 2 cannot divide $\Phi _{n}(b),$ (which is odd), and the condition $p > 2$ is not restrictive.

Thus, let $n = pm$ . It suffices to prove that $p^{2}$ does not divides $S (b)$ for some polynomial $S (x)$ , which is a multiple of $\Phi _{n}(x).$ We take

S(x)={\frac {x^{n}-1}{x^{m}-1}}=1+x^{m}+x^{2m}+\cdots +x^{(p-1)m}.

By Fermat's little theorem, we have $b^{p-1}\equiv 1{\pmod {p}}.$ As $p$ divides $\Phi _{n}(b)$ , we have also $b^{n}\equiv 1{\pmod {p}}.$ Thus the multiplicative order of $b$ modulo $p$ divides $gcd(n, p - 1)$ , which is a divisor of $m = n / p$ . Thus $c = b m - 1$ is a multiple of $p$ . Now,

S(b)={\frac {(1+c)^{p}-1}{c}}=p+{\binom {p}{2}}c+\cdots +{\binom {p}{p}}c^{p-1}.

As $p$ is prime and greater than 2, all the terms but the first one are multiple of $p^{2}.$ This proves that $p^{2}$ does not divides $\Phi _{n}(b).$

A prime $p$ is a unique prime in base $b$ , if and only if, for some $n$ , it is the unique prime divisor of $\Phi _{n}(b)$ that does not divide $n$ . If $b$ is even (which includes the binary and the decimal cases) this means that

R_{n}(b)={\frac {\Phi _{n}(b)}{\gcd(\Phi _{n}(b),n)}}=p^{c}.

for some positive integer $c$ .

If $b$ is odd, this means that

R_{n}(b)={\frac {\Phi _{n}(b)}{\gcd(\Phi _{n}(b),n)}}=p^{c}2^{d}.

for some integers $c > 0$ and $d \geq 0$ . This provides an efficient method for computing the unique primes and the primes of a given period.

Note that a prime divisor of $b$ is coprime with $b^{n}-1$ , and thus also with its divisor $\Phi _{n}(b).$ Such a prime has no period length, as the representation in base $b$ 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 $b$ . For example, 2 is not considered as a unique prime in binary, although it is the only prime with finite reciprocal in binary.

Table of the periods of primes up to 139 in bases up to 24

The mention "terminate" means that the prime divides the base, and thus that the representation of its reciprocal is finite.

prime\base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
2	terminate	1	terminate	1	terminate	1	terminate	1	terminate	1	terminate	1	terminate	1	terminate	1	terminate	1	terminate	1	terminate	1	terminate
3	2	terminate	1	2	terminate	1	2	terminate	1	2	terminate	1	2	terminate	1	2	terminate	1	2	terminate	1	2	terminate
5	4	4	2	terminate	1	4	4	2	terminate	1	4	4	2	terminate	1	4	4	2	terminate	1	4	4	2
7	3	6	3	6	2	terminate	1	3	6	3	6	2	terminate	1	3	6	3	6	2	terminate	1	3	6
11	10	5	5	5	10	10	10	5	2	terminate	1	10	5	5	5	10	10	10	5	2	terminate	1	10
13	12	3	6	4	12	12	4	3	6	12	2	terminate	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	terminate	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	terminate	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	terminate	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

Table of primes of a given period (up to 24) in bases up to 24

Bold for unique primes.

period length\base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
1	(none)	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	(none)	5	3	7	(none)	3	5	11	3	13	7	3, 5	(none)	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	(none)	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 gives an overview of all 23 unique primes below 10¹⁰⁰ (sequence A040017 (sorted) and A007615 (ordered by period length) in OEIS) and their periods (sequence A051627 (ordered by corresponding primes) and A007498 (sorted) in OEIS)

Period length	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 (0.142857142857142857...)

Just after the table, the twenty-fourth unique prime has 128 digits and period length 320. It can be written as (9₃₂0₃₂)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. (Any repunit prime is unique.)

As of 2010 the repunit (10²⁷⁰³⁴³-1)/9 is the largest known probable unique prime.^[2]

In 1996 the largest proven unique prime was (10¹¹³² + 1)/10001 or, using the notation above, (99990000)₁₄₁+ 1. It has 1129 digits. The record has been improved many times since then. As of 2014 the largest proven unique prime is $\Phi _{{47498}}(10)$ , it has 20160 digits.^[3]

Binary unique primes

The first unique primes in binary (base 2) are:

3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, ... (sequence A144755 (sorted) and A161509 (ordered by period length) in OEIS)

The period length of them are:

2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, ... (sequence A247071 (ordered by corresponding primes) and A161508 (sorted) in OEIS)

They include Fermat primes (the period length is a power of 2), Mersenne primes (the period length is a prime) and Wagstaff primes (the period length is twice an odd prime).

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 (mod 8) and n > 20, then at least two primes have period n in base 2, (Thus, n is not a unique period in base 2) because of the Aurifeuillean factorization, for example, 113 (= $\Phi _{{28L}}(2)$ ) and 29 (= $\Phi _{{28M}}(2)$ ) both have period 28 in base 2, 37 (= $\Phi _{{36L}}(2)$ ) and 109 (= $\Phi _{{36M}}(2)$ ) both have period 36 in base 2, and that 397 (= $\Phi _{{44L}}(2)$ ) and 2113 (= $\Phi _{{44M}}(2)$ ) 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

{\frac {\Phi _{n}(2)}{\gcd(\Phi _{n}(2),n)}}=p^{c}.

The only known values of n such that $\Phi_n(2)$ is composite but $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ is prime are 18, 20, 21, 54, 147, 342, 602, and 889 (in these case, $\Phi_n(2)$ has a small factor which divides n). It is a conjecture that there is no other n with this property. All other known base 2 unique primes are of the form $\Phi_n(2)$ .

In fact, no prime with c > 1 (that is $\Phi _{n}(2)/\gcd(\Phi _{n}(2),n)$ is a true power of p) have been discovered, and all known unique primes p have c = 1. It is conjectured that all unique primes have c = 1 (that is, all base 2 unique primes are not Wieferich primes).

The largest known base 2 unique prime is 2^74207281-1, it is also the largest known prime. With an exception of Mersenne primes, the largest known probable base 2 unique prime is ${\frac {2^{{13372531}}+1}{3}}$ ,^[4] and the largest proved base 2 unique prime is $\frac{2^{83339}+1}{3}$ . Besides, the largest known probable base 2 unique prime which is not Mersenne prime or Wagstaff prime is $\frac{2^{4101572}+1}{17}$ .

Similar to base 10, though they are rare (but more than the case to base 10), 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 ${\frac {\Phi _{n}(2)}{\gcd(\Phi _{n}(2),n)}}$ , it is an overpseudoprime to base 2.

There are 52 unique primes in base 2 below 2⁶⁴, they are:

Period length	Prime (written in decimal)	Prime (written in binary)
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 (digits in binary) 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	197090146431155602193972646715771255052640329744283764892370019904357741 89483906244488746953221813209	519
1291	838618178719251837397922064707038627665630534568678134599691846785465476 94793573468589875745315081	1290
1321	61	60
1327	2365454398418399772605086209214363458552839866247069233	221
1429	14449	84
1471	252359902034571016856214298851708529738525821631	245
1543	496539503006854813427424312497207522543444711437548129903659344272632683 272793440342430995510216284165634152472564121316399840870066338255288866 0520657	771
1697	993352058006638682153966409645670956670946653461410132943205873654433847 19802857319737050495099341955640963272958071602273	848
1753	1795918038741070627	146
1777	25781083	74
1801	601	25
2113	397	44
2281	3011347479614249131	190
2801	111451321936715706754281360936130695725789053113477532787506703859448139 3220804051366788787128409731513666376851495151281817670381468528387601	1400
2971	48912491	110
3011	631215008947706187342830494125660733360092019659681922883823392015121754 384870744044074337887482936870852519582960673945561810148710850934449712 549090934572292098088972061029650939105592263256293676274598529593937386 833315889748213948490958132757432166701901197169972066727635929332437543 971934775961	3010
3259	960843850986532976532466235773483492840618819232206145010143480044702708 779967241439519037158800917230289	1086
3361	88959882481	168
3967	329681082333182744401440483194355858863180343505040423704248576571448633 750584301174148722553932147927597631742347411485337632138078290650210675 876678393486695212411724048483933266891456680698860293140211741652395532 942356085633482633317695457529455010426340441436876126207958684254258686 978025484227726178132865763699306489773212771136387042695385253682824229 1991249685206783121190349820804553	1983
4051	251	50
4129	337707341682536518003709893757969948253892963180186010484820055311728562 60013942500368975908606689	688
4177	9857737155463	87
4523	106788290443848295284382097033	266
4561	510499030505981560130624776542416406578290250029762044510602610086894781 587157297451609248604675303096573768271042333081577723501646221586511876 94109112727796663977157921	2280
4871	820332199631383710976892723082581168416794420573016438739421249911820124 34598644913857356023840478815121709542915222280972560231358838127531337	487
5153	54410972897	112
5281	860573414369008969457638101533827364704684164286188824383996871471626764 468219429432850649798234488791036977295952185305281368586922360240161830 570497885830241813162641447900350702795124321	2640
5347	242099935645987	198
5431	949679298897339527983480966156925132054401430562953533904117600399380467 648519947014606176671467424585748469203514676989673526309460974664365545 499030774032479387078921118363930265760200620672799835715535158837490492 688267807463325287777244213493871271921642282591930541564696983844495622 955980550205990622869155128415290804996456351640552412702829488471650827 3187514617905113012642316992618568629664046915514871575875088457784721	2715
5881	236182562448406188572125221558517145982594227534969066416811777487104605 15038403366198473773770441	1470
6043	4475130366518102084427698737	318
6659	673480918906267571379147730480801519827880098089533495229717036762090724 479192537367139437198393219309210019204431468329644945358062861533367588 087648270529222845279877353690826582261557527588377345857885839204370316 799678327986978745315063758482953060020699742926470125329753826578693958 965495360840266430868491687076380359846529517564492323819228415082790434 495536836421658955188522736168537521926265470822252323226622033496174216 244501061303611330330509969775174567807593369800175040355534432048366366 635839506581617182643750350340079505634187776845404465707422776826888198 169305292494021412276744678617840421846647032730657721456263070083330217 279102956893078978125921763407971896625474982986290414196871234129842105 803253764818463163965664137011098487553488787905622962752952661907888801 221518355677716658155656118632614701678572886850142421155051821596857535 766122394772866202385830712929707343895805217305407898539596073224024658 45627773409459421340250476125665859926003121138412497735360569	6658
6719	215006106257113223254503015023149432126193150293791416185445173578281597 218315377296589584591228602041183907532584815068471747291177386898925622 477208530115714962355294842135137890474394949339249259335407710018584480 055157825387089416912233252714054247018216597994795059161567922302450277 281351135838393171424038832688432240078361264161523904355539085927738753 968157018550258476163852090826756157915705283413226000816151712543838581 066281600650278690534719371112997393190721068136840596790525950480851370 510277560248341182341805553054000587378384785994695875587394905226703149 605689830257768229987714770949192483302583569799141079867597051190134078 718011730508482542567284418838119000563443985593691221203060137047648713 095502877775290062907208508727269017130916691676838817452529964938878349 395785642430571852241837461604136374448443730175081889502056290512497717 177577492736555784081731998565765598518104822516520340301701034123926767 472784665776779480628821628279687651736198541330802238405154786248043073	3359
7487	26828803997912886929710867041891989490486893845712448833	197
8929	197107422273014301919781414466039325387889623676342705850752210599969	496
8969	105085375848729800497877494145054402385436616845064164452498921883291912 678976696572426254056550259022949969657136812477008949535672765969651143 081836499574699312620294703721884924945056142078277741715754321142971230 033732570350705429405324111863224178094111236842467383427204559334241753 99671044286557638075591	1121
9547	162144129216073931248440264348881021095346091675833404759395234231098234 889912552337520763730433377821186906239298809905980252801959368223494175 542275888565639506872238503798065746625761811258218877092131210012551133 783641253171815439582152921092238944373361635426882021957786357775945908 244721892727369566822325125894300674361490963976112716170481686262623635 303262211579519224512508309126102905398805331643337717389501879374005254 826601501875676315073172538545633223298243357601554772256397807255437800 001570707182137145084291064805293027676453530342416747874757977159248427 080097856195941118336713349896923643484610886520676488997796355407029593 609279548466332692572427762038607738155147300973317899098301348708567618 514437848490288495597210487360655817308626749954756608178081806485767156 748019600169323683513681103611076854679392961073290927422729640707954578 8520811837495181586420117807667033593394473	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

468172263510722656207776706750069723016189792142528328750689763038394004
13682313921168154465151768472420980044715745858522803980473207943564433 (143 digits)

and

527739642811233917558838216073534609312522896254707972010583175760467054
896492872702786549764052643493511382273226052631979775533936351462037464
331880467187717179256707148303247 (177 digits)

they are the two prime factor of the Mersenne number 2¹⁰⁶¹−1. ^[5] Thus, the period length of them is 1061.

As of October 2016, the largest known probable binary bi-unique prime is ${\frac {2^{5240707}-1}{75392810903}}$ , ^[6] 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

3, 23, 53, 149, 269, 461, 619, 389, ...

In binary, the period length of odd primes are: (sequence A014664 in the OEIS)

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: (sequence A108974 in the OEIS)

period length	prime(s)	period length	prime(s)	period length	prime(s)	period length	prime(s)
1	(none)	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	(none)	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

Unique prime in various bases

base	unique period length
2	2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 342, 345, 366, 374, 382, 398, 410, 414, 425, 447, 471, 507, 521, 550, 567, ...
3	1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 20, 21, 24, 26, 32, 33, 36, 40, 46, 60, 63, 64, 70, 71, 72, 86, 103, 108, 128, 130, 132, 143, 145, 154, 161, 236, 255, 261, 276, 279, 287, 304, 364, 430, 464, 513, 528, 541, 562, ...
4	1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, ...
5	1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 18, 24, 28, 47, 48, 49, 56, 57, 88, 90, 92, 108, 110, 116, 120, 127, 134, 141, 149, 161, 171, 181, 198, 202, 206, 236, 248, 288, 357, 384, 420, 458, 500, 530, 536, ...
6	1, 2, 3, 4, 5, 6, 7, 8, 18, 21, 22, 24, 29, 30, 42, 50, 62, 71, 86, 90, 94, 118, 124, 127, 129, 144, 154, 186, 192, 214, 271, 354, 360, 411, 480, 509, 558, 575, ...
7	3, 4, 5, 6, 8, 13, 18, 21, 28, 30, 34, 36, 46, 48, 50, 54, 55, 58, 63, 76, 84, 94, 105, 122, 131, 148, 149, 224, 280, 288, 296, 332, 352, 456, 528, 531, ...
8	1, 2, 3, 6, 9, 18, 30, 42, 78, 87, 114, 138, 189, 303, 318, 330, 408, 462, 504, 561, ...
9	1, 2, 4, 6, 10, 12, 16, 18, 20, 30, 32, 36, 54, 64, 66, 118, 138, 152, 182, 232, 264, 336, 340, 380, 414, 446, 492, 540, ...
10	1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385, ...
11	2, 4, 5, 6, 8, 9, 10, 14, 15, 17, 18, 19, 20, 27, 36, 42, 45, 52, 60, 73, 91, 104, 139, 205, 234, 246, 318, 358, 388, 403, 458, 552, ...
12	1, 2, 3, 5, 10, 12, 19, 20, 21, 22, 56, 60, 63, 70, 80, 84, 92, 97, 109, 111, 123, 164, 189, 218, 276, 317, 353, 364, 386, 405, 456, 511, ...
13	2, 3, 5, 6, 7, 8, 9, 12, 16, 22, 24, 28, 33, 34, 38, 78, 80, 102, 137, 140, 147, 224, 230, 283, 304, 341, 360, 372, 384, 418, 420, 436, 483, 568, 570, ...
14	1, 3, 4, 6, 7, 14, 19, 24, 31, 33, 35, 36, 41, 55, 60, 106, 114, 129, 152, 153, 172, 222, 265, 286, 400, 448, 560, ...
15	3, 4, 6, 7, 14, 24, 43, 54, 58, 73, 85, 93, 102, 184, 220, 221, 228, 232, 247, 291, 305, 486, 487, 505, 551, 552, ...
16	2, 4, 6, 8, 10, 14, 20, 30, 46, 48, 52, 70, 74, 78, 150, 178, 204, 298, 306, 346, 366, 378, 400, 476, 498, 502, ...
17	1, 2, 3, 5, 7, 8, 11, 12, 14, 15, 34, 42, 46, 47, 48, 50, 71, 77, 94, 110, 114, 147, 154, 176, 228, 235, 258, 275, 338, 350, 419, 450, 480, 515, ...
18	1, 2, 3, 6, 14, 17, 21, 24, 30, 33, 38, 45, 46, 72, 78, 114, 146, 168, 288, 414, 440, 448, ...
19	2, 3, 4, 6, 19, 20, 31, 34, 47, 56, 59, 61, 70, 74, 91, 92, 96, 98, 107, 120, 145, 156, 168, 242, 276, 314, 326, 337, 387, 565, ...
20	1, 3, 4, 6, 8, 9, 10, 11, 17, 30, 98, 100, 110, 126, 154, 158, 160, 168, 178, 182, 228, 266, 270, 280, 340, 416, 480, 574, ...
21	2, 3, 5, 6, 8, 9, 10, 11, 14, 17, 26, 43, 64, 74, 81, 104, 192, 271, 321, 335, 348, 404, 437, 445, 516, ...
22	2, 5, 6, 7, 10, 21, 25, 26, 69, 79, 86, 93, 100, 101, 154, 158, 161, 171, 202, 214, 294, 354, 359, 424, 454, ...
23	2, 5, 8, 11, 15, 22, 26, 39, 42, 45, 54, 56, 132, 134, 145, 147, 196, 212, 218, 252, 343, ...
24	1, 2, 3, 4, 5, 8, 14, 19, 22, 38, 45, 53, 54, 70, 71, 117, 140, 144, 169, 186, 192, 195, 196, 430, ...

References

↑ Caldwell, Chris. "Unique prime". The Prime Pages. Retrieved 11 April 2014.
↑ PRP Records: Probable Primes Top 10000
↑ The Top Twenty Unique; Chris Caldwell
↑ PRP records
↑ The Cunningham Project
↑ PRP records

Yates, Samuel (1980). "Periods of unique primes". Math. Mag. 53: 314. Zbl 0445.10009.

External links

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 50 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229

List of prime numbers

This article is issued from Wikipedia - version of the 11/17/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

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

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

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