Following Yates, a prime p such that 1/p 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 (1/3 = 0.3^_), two (1/11 = 0.9^_), three (1/37 = 0.27^_), and four (1/101 = 0.99^_) respectively. On the other hand, 41 and 271 both have period five, so neither is a unique prime. The unique primes are the primes p such that (Φ_n(10))/(GCD(Φ_n(10), n)) = p^α, where Φ_n(x) is a cyclotomic polynomial, n is the period of the unique prime, GCD(a, b) is the greatest common divisor, and α is a positive integer.

cyclic number | decimal expansion | full reptend prime