An Extension of a Congruence by Kohnen

Abstract

Let $p>3$ be a prime, and let $q_p(2)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base 2. Recently, Z. H. Sun proved that \sum_{k=1}^{p-1}\frac{1}{k\cdot 2^k}\equiv q_p(2)-\frac{p}{2}q_p(2)^2 \pmod{p^2} which is a generalization of a congruence due to W. Kohnen. In this note we give an elementary proof of the above congruence which is based on several combinatorial identities and congruences involving the Fermat quotient $q_p(2)$, harmonic or alternating harmonic sums.