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 −

A characterization of Wilson-Lerch primes

“…p−1 a=1 q p (a) ≡ W p (mod p 2 ). The Wilson primes < 2 • 10 13 are 5, 13, and 563 ( [3], [4]), and the Lerch primes < 4, 496, 113 are 3, 103, 839, and 2237 [12], with no overlap between the two sequences in the ranges examined. In this note, we present analogous criteria for a prime to possess both of these properties simultaneously.…”

“…where B p−j vanishes for all even j except j = p − 1. This identity, in which the sum in the right-hand side is really just the usual expansion of 1 p 2 {B p (p) − B p } with the terms reversed, can be used to obtain congruences like (12) to any desired precision, though ( 12) is sufficient for our purpose.…”

“…where the sum of the first two terms in the right-hand side is congruent to W p (mod p), and the sum of the last two terms is a multiple of p. This result, incidentally, establishes that the mod p 2 evaluation of the sum in (12) in terms of a Bernoulli number has no such simple counterpart in terms of the Wilson quotient. It also allows us to state:…”