A note on the Voronoi congruences and the residue of the Fermat quotient

TL;DR

This paper establishes new congruences related to the Fermat quotient and Voronoi congruences, generalizing classical results and improving computational efficiency, with applications to harmonic numbers and Bernoulli numbers.

Contribution

It introduces a novel congruence for the p-residue of the Fermat quotient based on generalized Voronoi congruences, extending previous results and enhancing computational methods.

Findings

Derived a congruence for the Fermat quotient residue in base a.

Generalized Voronoi congruences with improved computational efficiency.

Connected congruences for harmonic numbers and Bernoulli numbers.

Abstract

Given an odd prime p, we prove a congruence on the p-residue of the Fermat quotient q_p(a) in base a with 0<a<p, which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the first generalized harmonic number of order (p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known since long for the harmonic number of order (p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruences which is different from the one of W. Johnson and which is more computationally efficient. We prove an additional related congruence, which specialized to base 2 allows to retrieve several congruences that were originally due to E. Lehmer.