# Unique prime

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 contrary, 41 and 271 have both period 5, 7 and 13 have both period 6, 239 and 4649 have both period 7, 73 and 137 have both period 8, so they are not unique primes. 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

$\frac{\Phi_n(10)}{\gcd(\Phi_n(10),n)} = p^c$

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 in OEIS) and their periods (sequence A051627 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]

## Base 2 Unique primes

The first base 2 unique primes are:

3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, ... ()

The period length of them are:

2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, ... ()

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).

By the way, if n is a natural number which does not equal to 1 or 6, than there are at least one prime with period n 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

$\frac{\Phi_n(2)}{\gcd(\Phi_n(2),n)} = p^c$

In fact, there are no terms which c > 1 (means it is a true power of p) have been discovered, all known unique primes p have that c = 1. It is a conjecture that all c = 1, and it is very possible, because it's very possible that all $\Phi_n(2)$ are square-free except while n = 364 or n = 1755, if 1093 and 3511 are only two known Wieferich primes, and even if there are other Wieferich prime, they are rare! (If a term which c > 1 exists, the prime p is another Wieferich prime, because pc divides $\Phi_n(2)$, and $\Phi_n(2)$ divides 2n-1, and that n must divide p-1, so pc divides 2p-1-1)

The largest known base 2 unique prime is 257885161-1, it is also the largest known prime. With an exception of Mersenne primes, the largest probable base 2 unique prime is $\frac{2^{13372531}+1}{3}$,[4] and the largest proved base 2 unique prime is $\frac{2^{42737}+1}{3}$. Besides, the largest probable base 2 unique prime which is not Mersenne prime or Wagstaff prime is $\Phi_{164591}(2)$.[5]

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 infinitely.

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 conposite 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 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

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