Fermat quotient

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

$$q_p(a):=\frac{a^{p-1}-1}{p}$$

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

$$\sum _{a=1}^{p-1}{q}_p(a)\equiv w_p(modp)$$

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

$$q_p(w_p)=\frac{w_p^{p-1}-1}{p}$$

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