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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

where Φn(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 (Sloane's A040017) and their periods (Sloane's A051627):

Period lengthPrime
13
211
337
4101
109,091
129,901
9333,667
14909,091
2499,990,001
36999,999,000,001
489,999,999,900,000,001
38909,090,909,090,909,091
191,111,111,111,111,111,111
2311,111,111,111,111,111,111,111
39900,900,900,900,990,990,990,991
62909,090,909,090,909,090,909,090,909,091
120100,009,999,999,899,989,999,000,000,010,001
15010,000,099,999,999,989,999,899,999,000,000,000,100,001

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