An elementary proof of a congruence by Skula and Granville

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.