# 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.

### 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 Φn(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

rankprime digitswhowhencomment
15166932097152 - 5166931048576 + 1 11981518 L4561 Oct 2023 Generalized unique
24658592097152 - 4658591048576 + 1 11887192 L4561 May 2023 Generalized unique
31234471048576 - 123447524288 + 1 5338805 L4561 Feb 2017 Generalized unique
4215317227 + 27658614 + 1 4610945 L5123 Jul 2020 Gaussian Mersenne norm 41?, generalized unique
5143332786432 - 143332393216 + 1 4055114 L4506 Jan 2017 Generalized unique
636608603 - 33304302 + 1 3153105 L5123 Jun 2023 Generalized unique
7844833524288 - 844833262144 + 1 3107335 L4506 Jan 2017 Generalized unique
8712012524288 - 712012262144 + 1 3068389 L4506 Jan 2017 Generalized unique
9558640393216 - 558640196608 + 1 2259865 L4506 Jan 2017 Generalized unique
10237804393216 - 237804196608 + 1 2114016 L4506 Jan 2017 Generalized unique
1134043119 + 32021560 + 1 1929059 L5123 Jun 2023 Generalized unique
1293606354294 + 93606177147 + 1 1761304 p437 Nov 2023 Generalized unique
1355599354294 + 55599177147 + 1 1681149 p437 Nov 2023 Generalized unique
141082083262144 - 1082083131072 + 1 1581846 L4506 Jan 2017 Generalized unique
15843575262144 - 843575131072 + 1 1553498 L4506 Jan 2017 Generalized unique
16362978262144 - 362978131072 + 1 1457490 p379 May 2015 Generalized unique
1724792057 - 22396029 + 1 1442553 L3839 Apr 2014 Gaussian Mersenne norm 40, generalized unique
18192098262144 - 192098131072 + 1 1385044 p379 Feb 2015 Generalized unique
1932888387 - 31444194 + 1 1378111 L5123 Jun 2023 Generalized unique
201202113196608 - 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.