An elementary proof of a congruence [4pt] by Skula and Granville

Romeo Meštrović

Address: Maritime Faculty, University of Montenegro, Dobrota 36, 85330 Kotor, Montenegro


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 \[ q_p(2)^2\equiv -\sum _{k=1}^{p-1}\frac{2^k}{k^2}\hspace{10.0pt}(\@mod \; p)\,. \] In this note we establish the above congruence by entirely elementary number theory arguments.

AMSclassification: primary 11A07; secondary 11B65, 05A10.

Keywords: congruence, Fermat quotient, harmonic numbers.

DOI: 10.5817/AM2012-2-113