long prime | reptend prime

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 (Conway and Guy 1996, pp. 157-163 and 166-171). 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 congruent 1 (mod p) for k = p - 1 and no k less than this. In other words, the multiplicative 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) congruent (3, 2, 6, 4, 5, 1) (mod 7).

Artin's constant | cyclic number | decimal expansion | Fermat prime | multiplicative order | primitive root | repeating decimal | unique prime