On Lerch's formula for the Fermat quotient

TL;DR

This paper investigates Lerch's 1905 formula for the Fermat quotient, generalizes it, and provides new proofs and criteria related to special primes, enhancing understanding of Fermat quotients and related prime classifications.

Contribution

It generalizes Lerch's formula, offers a new proof of Skula's result, and refines Lehmer's criteria for Wieferich and Mirimanoff primes.

Findings

Generalized Lerch's formula for Fermat quotient

Provided a new proof of Skula's sharpened result

Sharpened criteria for Wieferich primes to be Mirimanoff primes

Abstract

This paper explores some previously-unrecognized consequences of Lerch's 1905 formula for the Fermat quotient, with special attention to the sums which he introduced in this context. A generalization of his result is proved, and a new proof given of a sharpened result by Skula (2008). We also sharpen the criteria given by Emma Lehmer in 1938 for a Wieferich prime to be simultaneously a Mirimanoff prime.