If $a$ is an integer not divisible by a positive prime $p$, then Fermat’s little theorem (a.k.a. Fermat’s theorem) guarantees that the difference $a^{{p-1}}\!-\!1$ is divisible by $p$.  The integer

is called the Fermat quotient of $a$ modulo $p$.  Compare it with the Wilson quotient $w_{p}$, which is similarly related to Wilson’s theorem.

Lerch’s formula

for an odd prime $p$ connects the Fermat quotients and the Wilson quotient.

If $p$ is a positive prime but not a Wilson prime, and $w_{p}$ is its Wilson quotient, then the expression

is called the Fermat–Wilson quotient of $p$.  Sondow proves in [1] that the greatest common divisor of all Fermat–Wilson quotients is 24.