Fermat quotient

In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as^[1][2][3][4]

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

or

${\displaystyle \delta _{p}(a)={\frac {a-a^{p}}{p}}}$.

This article is about the former; for the latter see p-derivation. The quotient is named after Pierre de Fermat.

If the base a is coprime to the exponent p then Fermat's little theorem says that q_p(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then q_p(a) will be a cyclic number, and p will be a full reptend prime.