Polynomial identities and Fermat quotients

TL;DR

This paper establishes new polynomial identities that lead to simplified congruences modulo p^2 for Fermat quotients, extending classical results and deriving new sum formulas involving harmonic numbers.

Contribution

It introduces novel polynomial identities that yield simpler congruences for Fermat quotients and generalizes classical Eisenstein's congruence.

Findings

Simpler congruences for Fermat quotients modulo p^2

Generalizations of Eisenstein's classical congruence

New formulas for sums of harmonic numbers and their higher order variants

Abstract

We prove some polynomial identities from which we deduce congruences modulo $p^2$ for the Fermat quotient $\frac{2^p-2}{p}$ for any odd prime $p$ (Proposition 1 and Theorem 1). These congruences are simpler than the one obtained by Jothilingam in 1985 which involves listing quadratic residues in some order. On the way, we also observe some more congruences for the Fermat quotient that generalize Eisenstein's classical congruence. Using such polynomial identities, we obtain some sums involving harmonic numbers. We also prove formulae for binomial sums of harmonic numbers of higher order.