Unique prime

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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[edit]

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

where q is a positive integer smaller than According to the summation formula of geometric series, this may be rewritten as

In other words, n is a period of the representation of 1/p if and only if p is a divisor of Euler's theorem asserts that, if an integer b is coprime with p, then p is a divisor of where 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 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= 2k − 1 for some integer k > 1

As

where is the nth cyclotomic polynomial, the primes of period n in base b are prime divisors of More precisely, the primes of period n are exactly the prime divisors of 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 that do not divide n are exactly the prime divisors of[citation needed]

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

although 2 divides both n = 2 and

If b is odd, the primes of period n are exactly, if n = 1, the prime divisors of , or, if n > 1, the odd prime divisors of Rn(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 and has a period k smaller than n, then it is a common divisor of and As the resultant of two polynomials is a linear combination of these polynomials, p divides the resultant of and As these two polynomials are coprime and divide p divides also the discriminant of Thus, a prime divisor of , 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 then it does not divide In fact, this implies immediately that p does not divide If b is even, 2 cannot divide (which is odd), and the condition p > 2 is not restrictive.

Thus, let n = pm. It suffices to prove that does not divides S(b) for some polynomial S(x), which is a multiple of We take

By Fermat's little theorem, we have As p divides , we have also Thus the multiplicative order of b modulo p divides gcd(n, p − 1), which is a divisor of m = n/p. Thus c = bm − 1 is a multiple of p. Now,

As p is prime and greater than 2, all the terms but the first one are multiple of This proves that does not divides

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

for some positive integer c .

If b is odd, this means that

for some integers c > 0 and d ≥ 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 , and thus also with its divisor 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[edit]

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[clarification needed] 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 , it has 20160 digits.[3]

Binary unique primes[edit]

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 (=) 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 (in these case, has a small factor which divides n). It is a conjecture that there is no other n with this property.[citation needed] All other known base 2 unique primes are of the form .

In fact, no prime with c > 1 (that is 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 274207281-1, it is also the largest known prime. With an exception of Mersenne primes, the largest known probable base 2 unique prime is ,[4] and the largest proved 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 (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.[citation needed]

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.[clarification needed]

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[edit]

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 , [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[edit]

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[edit]

  1. ^ Caldwell, Chris. "Unique prime". The Prime Pages. Retrieved 11 April 2014. 
  2. ^ PRP Records: Probable Primes Top 10000
  3. ^ The Top Twenty Unique; Chris Caldwell
  4. ^ PRP records
  5. ^ The Cunningham Project
  6. ^ PRP records
  • Yates, Samuel (1980). "Periods of unique primes". Math. Mag. 53: 314. Zbl 0445.10009. 

External links[edit]