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

Abstract

The Fermat quotient \(q_{p}(a):= (a^{p-1} - 1)/p\), for prime p∤ a, and the Wilson quotient \(w_{p}:= ((p - 1)! + 1)/p\) are integers. If p∣w _p , then p is a Wilson prime. For odd p, Lerch proved that \((\sum\nolimits_{a=1}^{p-1}q_{p}(a) - w_{p})/p\) is also an integer; we call it the Lerch quotient ℓ _p . If p∣ℓ _p we say 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 × 10⁶; 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. 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 \(p\mid q_{p}(w_{p})\) we say p is a Wieferich-non-Wilson prime. There are three up to 10⁷, namely, 2, 3, 14771. Several open problems are discussed.