Full reptend prime

In number theory, a full reptend prime, full repetend prime, proper prime^[1]: 166 or long prime in base b is an odd prime number p such that the Fermat quotient

${\displaystyle q_{p}(b)={\frac {b^{p-1}-1}{p}}}$

(where p does not divide b) gives a cyclic number. Therefore, the base b expansion of ${\displaystyle 1/p}$ repeats the digits of the corresponding cyclic number infinitely, as does that of ${\displaystyle a/p}$ with rotation of the digits for any a between 1 and p − 1. The cyclic number corresponding to prime p will possess p − 1 digits if and only if p is a full reptend prime. That is, the multiplicative order ord_p b = p − 1, which is equivalent to b being a primitive root modulo p.

The term "long prime" was used by John Conway and Richard Guy in their Book of Numbers. Confusingly, Sloane's OEIS refers to these primes as "cyclic numbers".