A full reptend prime is a prime p for which 1/p has a maximal period decimal expansion of p-1 digits. Full reptend primes are sometimes also called long primes. (There is a surprising connection between full reptend primes and Fermat primes.)

A prime p is full reptend iff 10 is a primitive root modulo p, which means that

10^{k} is congruent to 1 mod p

for k = p-1 and no k less than this. In other words, the modulo order of p(mod 10) is p-1. For example, 7 is a full reptend prime since (10^{1}, 10^{2}, 10^{3}, 10^{4}, 10^{5}, 10^{6}) is congruent to (3, 2, 6, 4, 5, 1) mod 7.

The full reptend primes are 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, … .

However, there is no general method known for finding full reptend primes!