Geodesic
An elementary proof of a congruence by Skula and Granville
Archivum mathematicum, Tome 48 (2012) no. 2, pp. 113-120.

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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 \[ q_p(2)^2\equiv -\sum _{k=1}^{p-1}\frac{2^k}{k^2}\quad(\operatorname{mod} p)\,. \] In this note we establish the above congruence by entirely elementary number theory arguments.
DOI : 10.5817/AM2012-2-113
Classification : 05A10, 11A07, 11B65
Keywords: congruence; Fermat quotient; harmonic numbers 
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Meštrović, Romeo. An elementary proof of a congruence by Skula and Granville. Archivum mathematicum, Tome 48 (2012) no. 2, pp. 113-120. doi : 10.5817/AM2012-2-113. http://geodesic.mathdoc.fr/articles/10.5817/AM2012-2-113/

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