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An elementary proof of a congruence by Skula and Granville

Romeo Meštrović (2012)

Archivum Mathematicum

Let p 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 - k = 1 p - 1 2 k k 2 ( mod p ) . In this note we establish the above congruence by entirely elementary number theory arguments.

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