# Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771

The Fermat quotient $q_p(a):=(a^{p-1}-1)/p$ and the Wilson quotient $w_p:=((p-1)!+1)/p$ are integers. If $p$ divides $w_p$, then $p$ is a Wilson prime. For odd $p$, Lerch proved that $(\sum_{a=1}^{p-1} q_p(a) - w_p)/p$ is also an integer; we call it the Lerch quotient $\ell_p$. If $p$ divides $\ell_p$, we say $p$ is a Lerch prime. A simple Bernoulli-number test for Lerch primes is proven. We find four Lerch primes 3, 103, 839, 2237 up to $10^6$ and relate them to the known Wilson primes 5, 13, 563. Generalizations are suggested. Next, if $p$ is a non-Wilson prime, then $q_p(w_p)$ is an integer that we call the Fermat-Wilson quotient of $p$. The GCD of all $q_p(w_p)$ is shown to be 24. If $p$ divides $q_p(a)$, then $p$ is a Wieferich prime base $a$; we give a survey of them. Taking $a=w_p$, if $p$ divides $q_p(w_p)$, we call $p$ a Wieferich-non-Wilson prime; the first three are 2, 3, 14771. Several open problems are discussed. Here is the preprint.

Jonathan Sondow