Unique prime

In mathematics, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique iff there is no other prime q such that the period length of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q. Unique primes were first described by Samuel Yates in 1980.

It can be shown that a prime p is of unique period n iff there exists a natural number c such that

<math>\frac{\Phi_n(10)}{\gcd(\Phi_n(10),n)} = p^c<math>

where Φ_n(x) is the n-th cyclotomic polynomial; until today, 18 unique primes are known, and no others exist below 10⁵⁰. The following table gives an overview of all known unique primes (sequence in OEIS) and their periods (sequence 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