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 $$q_p(2{)}^2\equiv -\sum _{k=1}^{p-1}\frac{2^k}{k^2}(\mathrm{mod}p).$$ In this note we establish the above congruence by entirely elementary number theory arguments.

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