Powers of two weighted sum of the first p divided Bernoulli numbers modulo p

TL;DR

This paper explores the properties of Bernoulli numbers modulo prime p, linking them to permutation counts and providing combinatorial characterizations of special primes like Wieferich primes.

Contribution

It introduces a novel connection between Bernoulli numbers, permutation statistics, and prime characterizations, expanding understanding of prime-related number theory.

Findings

The weighted sum of Bernoulli numbers relates to the Agoh-Giuga quotient.

Provides a combinatorial characterization of Wieferich primes.

Identifies conditions for primes where p^2 divides the Fermat quotient q_p(2).

Abstract

We show that, modulo some odd prime p, the powers of two weighted sum of the first p-2 divided Bernoulli numbers equals the Agoh-Giuga quotient plus twice the number of permutations on p-2 letters with an even number of ascents and distinct from the identity. We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p^2 divides the Fermat quotient q_p(2).