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.

13 pages; This is the same as version 2 with extended Remarks on page 3 concerning a search of Euler numbers arising supercongruences and a new conjecture on page 4

References in corpus (3)

• On congruences related to central binomial coefficients
• On Delannoy numbers and Schröder numbers
• Arithmetic theory of harmonic numbers (II)

Cited by in corpus (2)

• A search for primes $p$ such that Euler number $E_{p-3}$ is divisible by $p$
• Some congruences related to the q-Fermat quotients