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

Authors

• Claire Levaillant, LIPN Universite Sorbonne Paris NordFollow

Abstract

We prove a congruence on the residue of the Fermat quotient in base a which arises from a generalization of the Voronoi congruences and other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we recover a congruence for the generalized harmonic number H_2,(p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known for the harmonic number H_(p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruences that is different from the one of Johnson and more computationally efficient. We prove an additional related congruence, which specialized to base 2, allows to retrieve several congruences of E. Lehmer.

Author ORCID Identifier

0000-0001-6233-6691

Recommended Citation

Levaillant, Claire (2026) "A note on the Voronoi congruences and the residue of the Fermat quotient," Communications on Number Theory and Combinatorial Theory: Vol. 7, Article 3.
Available at: https://research.library.kutztown.edu/contact/vol7/iss1/3