Unique prime number

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In recreational number theory, a unique prime or unique period prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equal to the period length of the reciprocal of q, 1 / q.^[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; 21649 and 513239 both have period 11; 53, 79 and 265371653 all have period 13; 31 and 2906161 both have period 15; 17 and 5882353 both have period 16; 2071723 and 5363222357 both have period 17; 19 and 52579 both have period 18; 3541 and 27961 both have period 20. Therefore, none of these is a unique prime. Unique primes were first described by Samuel Yates in 1980.

The above definition is related to the decimal representation of integers. Unique primes may be defined and have been studied in any numeral base.

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

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

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

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

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

As

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

where ${\displaystyle \Phi _{n}}$ is the nth cyclotomic polynomial, the primes of period n in base b are prime divisors of ${\displaystyle \Phi _{n}(b).}$ More precisely, the primes of period n are exactly the prime divisors of ${\displaystyle \Phi _{n}(b)}$ 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 ${\displaystyle \Phi _{n}(b)}$ that do not divide n are exactly the prime divisors of^{[citation needed]}

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

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

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

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

If b is odd, the primes of period n are exactly, if n = 1, the prime divisors of ${\displaystyle R_{1}(b)=b-1}$, or, if n > 1, the odd prime divisors of R_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 ${\displaystyle \Phi _{n}(b)}$ that does not divide n. If b is even (which includes the binary and the decimal cases) this means that

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

for some positive integer c .

If b is odd, this means that

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

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 ${\displaystyle b^{n}-1}$, and thus also with its divisor ${\displaystyle \Phi _{n}(b).}$ 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.

There are four situations of unique prime in base b:

1. ${\displaystyle \Phi _{n}(b)=p}$ and p is prime.
2. ${\displaystyle {\frac {\Phi _{n}(b)}{\gcd(\Phi _{n}(b),n)}}=p}$ and p is prime and ${\displaystyle \gcd(\Phi _{n}(b),n)>1}$.
3. ${\displaystyle \Phi _{n}(b)=p^{c}}$ and p is prime and ${\displaystyle c>1}$.
4. ${\displaystyle {\frac {\Phi _{n}(b)}{\gcd(\Phi _{n}(b),n)}}=p^{c}}$ and p is prime and ${\displaystyle \gcd(\Phi _{n}(b),n)>1}$ and ${\displaystyle c>1}$.

Conjectures:

• For all positive integers n, there are infinitely many bases b such that there is a unique prime with period length n. (this conjecture is true if Bunyakovsky conjecture is true, since cyclotomic polynomials are irreducible)

• For all bases b, there are infinitely many unique primes.

• For all positive integers n, there are infinitely many bases b such that there is a unique prime with period length n which is the 1st situation. (i.e. for all n≥1, there are infinitely many b≥2 such that ${\displaystyle \Phi _{n}(b)}$ is prime, where ${\displaystyle \Phi }$ is the cyclotomic polynomial) (this conjecture is true if Bunyakovsky conjecture is true, since cyclotomic polynomials are irreducible)

• For all bases b, there are infinitely many unique primes which are the 1st situation. (i.e. for all b≥2, there are infinitely many n≥1 such that ${\displaystyle \Phi _{n}(b)}$ is prime, where ${\displaystyle \Phi }$ is the cyclotomic polynomial)

• For all positive integers n not in (sequence A253235 in the OEIS), there are infinitely many bases b such that there is a unique prime with period length n which is the 2nd situation. (periods lengths n in (sequence A253235 in the OEIS) cannot have any bases which is the 2nd or the 4th situation) (this conjecture is true if Bunyakovsky conjecture is true, since cyclotomic polynomials are irreducible)

• For all bases b, there are only finitely many unique primes (or no unique primes) which are the 2nd situation, specially, in base 2, the only known periods which are the 2nd situation are 18, 20, 21, 54, 147, 342, 602, 889, 258121, and in base 10, the only known periods which are the 2nd situation is 3, 9, 294.

• All (period n, base b) combos which is the 3rd situation are

• • n = 1, b = p^c+1 with p prime and c ≥ 2
  • n = 2, b = p^c−1 with p odd prime and c ≥ 2
  • n = 3, b = 18
  • n = 5, b = 3
  • n = 6, b = 19

• All (period n, base b) combos which is the 4th situation are

• • n = 2, b = 2^rp^c−1 with p odd prime and r ≥ 1 and c ≥ 2
  • n = 3, b in (sequence A028231 in the OEIS) in the case of (b²+b+1)/3 is square of prime (it is conjectured that powers with exponent > 2 cannot exist) (the first such bases b are 22, 313, 4366, 847078, 11798281, 2288805793, 31878952246, ...) (corresponding primes are in (sequence A001570 in the OEIS))
  • n = 4, b in (sequence A002315 in the OEIS) in the case of (b²+1)/2 is square of prime or 28561=13⁴ (it is conjectured that powers with exponent > 2 cannot exist, except (239²+1)/2 = 28561 = 13⁴) (the first such bases b are 7, 41, 239, 8119, 47321, 63018038201, ...) (corresponding primes are in (sequence A001653 in the OEIS))
  • n = 6, b−1 in (sequence A028231 in the OEIS) in the case of (b²−b+1)/3 is square of prime (it is conjectured that powers with exponent > 2 cannot exist) (the first such bases b are 23, 314, 4367, 847079, 11798282, 2288805794, 31878952247, ...) (corresponding primes are in (sequence A001570 in the OEIS))

• Specially, no (period n, base b) combo which is the 3rd or the 4th situation exist for n > 6

• “Almost all” (period n, base b) combos are the 1st situation

(For all (period n, base b) combo which is the 3rd or the 4th situation, the corresponding prime p must be generalized Wieferich prime to base b)

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

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. Following afterward is the repunit prime R₃₁₇ as the 29^th, and at one less than twice the distance from the last unique prime on the list, R₁₀₃₁, as the 46^th.

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 2025^[update] the repunit (10^8177207 – 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 1128 digits. The record has been improved many times since then. As of 2025^[update] the largest proven unique prime is ${\displaystyle \Phi _{11589}(-10000)}$, it has 30897 digits.^[3]

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,

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

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

Empirically, it appears that most cases correspond to the denominator ${\displaystyle \gcd(\Phi _{n}(2),n)}$ being 1, so we have a prime (or prime power) of form ${\displaystyle \Phi _{n}(2)}$. However, for n equal to 18, 20, 21, 54, 147, 342, 602, 889, 258121, ... (sequence A333973 in the OEIS), we get a binary unique prime even though ${\displaystyle \Phi _{n}(2)}$ has a small divisor (which is shared with n).

In fact, no prime with c > 1 (that is ${\displaystyle \Phi _{n}(2)/\gcd(\Phi _{n}(2),n)}$ 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).

As of March 2025, the largest known base 2 unique prime is 2^136279841-1, 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 proven base 2 unique prime is ${\displaystyle {\frac {2^{138937}+1}{3}}}$. 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 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 ${\displaystyle {\frac {\Phi _{n}(2)}{\gcd(\Phi _{n}(2),n)}}}$, it is an overpseudoprime to base 2.^{[clarification needed]}

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

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.

• Chris K. Caldwell, Harvey Dubner, "Unique-period primes", Journal of Recreational Mathematics 29:1:43-48 (1998) preprint

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

• Prime Top Tens
• Unique Period Primes
• Factorization of repunit numbers Archived 2017-01-20 at the Wayback Machine