Following Yates (1980), a prime such that is a repeating decimal with decimal period shared with no other prime is called a unique prime. For example, 3, 11, 37, and 101 are unique primes, since they are the only primes with periods one (), two (), three (), and four () respectively. On the other hand, 41 and 271 both have period five, so neither is a unique prime.

The unique primes are the primes such that

where is a cyclotomic polynomial, is the period of the unique prime, is the greatest common divisor, and is a positive integer.

The first few unique primes are 3, 11, 37, 101, 9091, 9901, 333667, ... (Sloane's A040017), which have periods 1, 2, 3, 4, 10, 12, 9, 14, 24, ... (Sloane's A051627), respectively.