Unique prime

Number of known terms 102 Infinite 3, 11, 37, 101 (10270343-1)/9 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 equivalent 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.

It can be shown that a prime p is of unique period n if and only if there exists a natural number c such that

${\displaystyle {\frac {\Phi _{n}(10)}{\gcd(\Phi _{n}(10),n)}}=p^{c}}$

where Φn(x) is the n-th cyclotomic polynomial. 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 ${\displaystyle \Phi _{47498}(10)}$, 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 (=${\displaystyle \Phi _{28L}(2)}$) and 29 (=${\displaystyle \Phi _{28M}(2)}$) both have period 28 in base 2, 37 (=${\displaystyle \Phi _{36L}(2)}$) and 109 (=${\displaystyle \Phi _{36M}(2)}$) both have period 36 in base 2, and that 397 (=${\displaystyle \Phi _{44L}(2)}$) and 2113 (=${\displaystyle \Phi _{44M}(2)}$) both have period 44 in base 2,

It can be shown that a prime p is of unique period n in base 2 if and only if there exists a natural number c such that

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

where ${\displaystyle \Phi _{n}(2)}$ is the nth cyclotomic polynomial at 2, because if and only if a prime p divides ${\displaystyle {\frac {\Phi _{n}(2)}{\gcd(\Phi _{n}(2),n)}}}$, then the period length of ${\displaystyle {\frac {1}{p}}}$ in base 2 is n.

The only known values of n such that ${\displaystyle \Phi _{n}(2)}$ is composite but ${\displaystyle {\frac {\Phi _{n}(2)}{\gcd(\Phi _{n}(2),n)}}}$ is prime is 18, 20, 21, 54, 147, 342, 602, and 889 (If so, ${\displaystyle \Phi _{n}(2)}$ must have a small factor which is also a factor of n), and might have other terms (However, it is a conjecture that there is no others). Thus, they are also unique period length in base 2, but the corresponding primes of them are not of the form ${\displaystyle \Phi _{n}(2)}$, and all other base 2 unique primes are of the form ${\displaystyle \Phi _{n}(2)}$.

In fact, there are no primes which c > 1 (means it is a true power of p) have been discovered, 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), and it is very possible, because it's very possible that all ${\displaystyle \Phi _{n}(2)}$ are square-free except while n = 364 or n = 1755, if 1093 and 3511 are only two Wieferich primes (1093 and 3511 are only two known Wieferich primes, and neither 1093 nor 3511 is unique in base 2, that is, neither 364 nor 1755 is a unique period in base 2), and even if there are other Wieferich prime, they are rare!

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 ${\displaystyle {\frac {2^{13372531}+1}{3}}}$,[4] and the largest proved base 2 unique prime is ${\displaystyle {\frac {2^{83339}+1}{3}}}$. Besides, the largest known probable base 2 unique prime which is not Mersenne prime or Wagstaff prime is ${\displaystyle {\frac {2^{4101572}+1}{17}}}$.

Similar to base 10, though they are rare (but more than the case to base 10), it is conjectured strongly that there are infinitely many base 2 unique primes, because all Mersenne primes are unique in base 2, and Mersenne primes are conjectured to be infinite.

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 ${\displaystyle {\frac {\Phi _{n}(2)}{\gcd(\Phi _{n}(2),n)}}}$, it is an overpseudoprime to base 2.

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

Period length Prime Base 2
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 base 2 unique prime have period length 170, 234, 158, 165, 147, 129, 184, 89, 208, and 312, and the bits of them are 65, 73, 78, 81, 82, 84, 88, 89, 96, and 97.

The binary period of nth prime are

2, 4, 3, 10, 12, 8, 18, 11, 28, 5, 36, 20, 14, 23, 52, 58, 60, 66, 35, 9, 39, 82, 11, 48, 100, 51, 106, 36, 28, 7, 130, 68, 138, 148, 15, 52, 162, 83, 172, 178, 180, 95, 96, 196, 99, 210, 37, 226, 76, 29, 119, 24, 50, 16, 131, 268, 135, 92, 70, 94, 292, 102, 155, 156, 316, 30, 21, 346, 348, 88, 179, 183, 372, 378, 191, 388, 44, ... (this sequence starts at n = 2, or the prime = 3) (sequence A014664 in the OEIS)

The least prime with binary period n are

1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 47, 241, 601, 2731, 262657, 29, 233, 331, 2147483647, 65537, 599479, 43691, 71, 37, 223, 174763, 79, 61681, 13367, 5419, 431, 397, 631, 2796203, 2351, 97, 4432676798593, 251, ... (sequence A112927 in the OEIS)

The number of primes with binary period n are

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 5, 2, 3, 2, 4, 3, 4, 1, 2, 1, 2, 4, 2, 1, 1, 2, ... (sequence A086251 in the OEIS)

Product of primes with binary period n are (it is the primitive part of 2n - 1)

1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, 2147483647, 65537, 599479, 43691, 8727391, 4033, 137438953471, 174763, 9588151, 61681, ... (sequence A064078 in the OEIS)

The binary period level of nth prime are

1, 1, 2, 1, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 1, 1, 1, 2, 8, 2, 1, 8, 2, 1, 2, 1, 3, 4, 18, 1, 2, 1, 1, 10, 3, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 6, 1, 3, 8, 2, 10, 5, 16, 2, 1, 2, 3, 4, 3, 1, 3, 2, 2, 1, 11, 16, 1, 1, 4, 2, 2, 1, 1, 2, 1, 9, 2, 2, 1, 1, 10, 6, 6, 1, 2, 6, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, ... (sequence A001917 in the OEIS)

The least prime with binary period level n are

3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257, 1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857, 72269, 3391, 683, 2593, 17029, 2687, 42701, 11161, 13099, 1103, 71293, 13121, 17467, 2143, 83077, 25609, 5581, 5153, 26227, 2113, 51941, 2351, ... (sequence A101208 in the OEIS)

Unique prime in vary bases

 Base Unique period 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, ... 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, ... 4 1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, 596, 612, 692, 732, 756, 800, 952, 996, ... 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, ... 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, ... 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, ... 8 1, 2, 3, 6, 9, 18, 30, 42, 78, 87, 114, 138, 189, 303, 318, 330, 408, 462, 504, 561, 1002, 1389, 1746, 1794, 2040, 2418, 2790, ... 9 1, 2, 4, 6, 10, 12, 16, 18, 20, 30, 32, 36, 54, 66, 118, 138, 152, 182, 232, 264, 336, 340, 380, 414, 446, 492, 540, 624, 720, 762, ... 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, 586, 597, 654, 738, 945, ... 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, ... 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, ... 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, 584, 864, 1006, ... 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, 590, 1029, 1194, ... 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, 614, 634, 1120, 1266, ...

References

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