Generalized Unique

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.

This page is about one of those forms.

Definitions and Notes

The reciprocal of every prime p (other than two and five) has a period, that is the decimal expansion of 1/p repeats in blocks of some set length (see the period of a decimal expansion). This is called the period of the prime p. Samuel Yates defined a unique prime (or unique period prime) to be a prime which has a period that it shares with no other prime. For example: 3, 11, 37, and 101 are the only primes with periods one, two, three, and four respectively--so they are unique primes. But 41 and 271 both have period five, 7 and 13 both have period six, 239 and 4649 both have period seven, and each of 353, 449, 641, 1409, and 69857 have period thirty-two, showing that these primes are not unique primes.

As we would expect from any object labeled "unique," unique primes are extremely rare.  For example, even though there are over 10⁴⁷ primes below 10⁵⁰, only eighteen of these primes are unique primes. We can find the unique primes using the following theorem.

Theorem.

The prime p is a unique prime of period n if and only if

is a power of p where is the nth cyclotomic polynomial.

It is possible to generalize this to other bases: A prime p is a unique prime in base b, if and only if, for some integer n, it is the only prime divisor of Φ_n(b) which does not divide n (and it may occur as a prime power). For example, the Gaussian Mersenne norms (other than 5) all have the form

2ⁿ ± 2^(n+1)/2 +1.

Since Φ₄(x) = x²+1, these can be written as Φ₄(2^(n+1)/2± 1)/2, showing these are all generalized unique primes.

Record Primes of this Type

rank prime digits who when comment 1 5166932097152 - 5166931048576 + 1 11981518 L4561 Oct 2023 Generalized unique 2 4658592097152 - 4658591048576 + 1 11887192 L4561 May 2023 Generalized unique 3 1234471048576 - 123447524288 + 1 5338805 L4561 Feb 2017 Generalized unique 4 215317227 + 27658614 + 1 4610945 L5123 Jul 2020 Gaussian Mersenne norm 41?, generalized unique 5 143332786432 - 143332393216 + 1 4055114 L4506 Jan 2017 Generalized unique 6 36608603 - 33304302 + 1 3153105 L5123 Jun 2023 Generalized unique 7 844833524288 - 844833262144 + 1 3107335 L4506 Jan 2017 Generalized unique 8 712012524288 - 712012262144 + 1 3068389 L4506 Jan 2017 Generalized unique 9 558640393216 - 558640196608 + 1 2259865 L4506 Jan 2017 Generalized unique 10 237804393216 - 237804196608 + 1 2114016 L4506 Jan 2017 Generalized unique 11 34043119 + 32021560 + 1 1929059 L5123 Jun 2023 Generalized unique 12 93606354294 + 93606177147 + 1 1761304 p437 Nov 2023 Generalized unique 13 55599354294 + 55599177147 + 1 1681149 p437 Nov 2023 Generalized unique 14 1082083262144 - 1082083131072 + 1 1581846 L4506 Jan 2017 Generalized unique 15 843575262144 - 843575131072 + 1 1553498 L4506 Jan 2017 Generalized unique 16 362978262144 - 362978131072 + 1 1457490 p379 May 2015 Generalized unique 17 24792057 - 22396029 + 1 1442553 L3839 Apr 2014 Gaussian Mersenne norm 40, generalized unique 18 192098262144 - 192098131072 + 1 1385044 p379 Feb 2015 Generalized unique 19 32888387 - 31444194 + 1 1378111 L5123 Jun 2023 Generalized unique 20 1202113196608 - 120211398304 + 1 1195366 L4506 Dec 2016 Generalized unique

References

Caldwell97

C. Caldwell, "Unique (period) primes and the factorization of cyclotomic polynomial minus one," Mathematica Japonica, 46:1 (1997) 189--195.  MR 99b:11139 (Abstract available)

CD1998

C. Caldwell and H. Dubner, "Unique period primes," J. Recreational Math., 29:1 (1998) 43--48.

Yates1980

S. Yates, "Periods of unique primes," Math. Mag., 53:5 (1980) 314.