An elementary proof of a congruence by Skula and Granville

TL;DR

This paper provides an elementary proof of a conjectured congruence involving Fermat quotients and exponential sums, previously proven by Granville, using only basic number theory techniques.

Contribution

It offers a new, elementary proof of a known congruence relating Fermat quotients and exponential sums, avoiding complex methods.

Findings

Confirmed the congruence using elementary number theory

Provided a simpler proof accessible to a broader audience

Strengthened understanding of Fermat quotients and related sums

Abstract

Let $p\ge 5$ be a prime, and let $q_p(2):=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base 2. The following curious congruence was conjectured by L. Skula and proved by A. Granville $$ q_p(2)^2\equiv -\sum_{k=1}^{p-1}\frac{2^k}{k^2}\pmod{p}. $$ In this note we establish the above congruence by entirely elementary number theory arguments.