# 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. Comments and suggestions requested.

### 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, and the generalized unique primes in base-x (any integer greater than one) are the prime factors of which do not divide x.

### Record Primes of this Type

rankprime digitswhowhencomment
1Phi(5, (1242474561 · 16001#/5 + 1) · (6071 · 16001# - 1)9) 276320 x38 Dec 2013 Generalized unique
2Phi(5, (364740596 · 16001#/5 + 1) · (6071 · 16001# - 1)9) 276318 x38 Dec 2013 Generalized unique
3Phi(5, (1512607666 · 16001#/5 + 1) · (4961 · 16001# - 1)9) 276317 x38 Dec 2013 Generalized unique
4Phi(5, (855408961 · 16001#/5 + 1) · (4961 · 16001# - 1)9) 276316 x38 Nov 2013 Generalized unique
5Phi(5, (207685826 · 16001#/5 + 1) · (3668 · 16001# - 1)9) 276309 x38 Nov 2013 Generalized unique
6Phi(5, (277516296 · 16001#/5 + 1) · (1085 · 16001# - 1)9) 276291 x38 Nov 2013 Generalized unique
7Phi(3, - 3267414 + 1)/3 255178 x28 Nov 2005 Generalized unique
8Phi(5, (3668 · 16001# - 1) · (378266 · 16001#/5 + 1)7) 221071 x34 Oct 2008 Generalized unique
9Phi(3, - 232257316384) 208601 p72 Jan 2008 Generalized unique
10Phi(3, - 231351616384) 208545 p72 Jan 2008 Generalized unique
11Phi(3, - 218252816384) 207716 f7 Feb 2007 Generalized unique
12Phi(3, - 217899616384) 207692 f7 Feb 2007 Generalized unique
13Phi(3, - 211508416384) 207269 f7 Jan 2007 Generalized unique
14Phi(3, - 211019916384) 207236 f7 Jan 2007 Generalized unique
15Phi(3, - 207450716384) 206993 f7 Jan 2007 Generalized unique
16Phi(3, - 202982716384) 206683 f7 Dec 2006 Generalized unique
17Phi(3, - 198980116384) 206400 f7 Dec 2006 Generalized unique
18Phi(3, - 38987712192328192) 206290 f14 Nov 2007 Generalized unique
19Phi(3, - 38249907698008192) 206154 f14 Nov 2007 Generalized unique
20Phi(3, - 38042639113688192) 206116 f14 Oct 2007 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.