Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771

The Fermat quotient $q_p(a):=(a^{p-1}-1)\mathrm{/}p$, for prime $p\nmid a$, and the Wilson quotient $w_p:=((p-1)!+1)\mathrm{/}p$ are integers. If $p\mid w_p,$ then $p$ is a Wilson prime. For odd $p,$ Lerch proved that $(\sum_{a=1}^{p-1} q_p(a) - w_p)/p$isalsoaninteger;wecallittheLerchquotient$\ell_p.$If$p\mid\ell_p$wesay$p$ is a Lerch prime. A simple Bernoulli-number test for Lerch primes is proven. There are four Lerch primes 3, 103, 839, 2237 up to $3\times {10}^6$; we 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 $p\mathrm{.}$ The GCD of all $q_p(w_p)$ is shown to be 24. If $p\mid q_p(a),$ then $p$ is a Wieferich prime base $a$; we give a survey of them. Taking $a=w_p,$ if $p\mid q_p(w_p)$wesay$p$isaWieferich-non-Wilsonprime\mathrm{.}Therearethreeupto$10^7$, namely, 2, 3, 14771. Several open problems are discussed.