A proof of Wolstenholme's theorem and congruence properties via an Egorychev-type integral

TL;DR

This paper offers a detailed complex analysis-based proof of Wolstenholme's theorem, explicitly connecting harmonic sums and Bernoulli numbers, and extends the classical result to a refinement modulo p^4.

Contribution

It introduces a general complex analysis method for deriving number-theoretic congruences, explicitly linking harmonic sums, Bernoulli numbers, and congruence properties.

Findings

Proof of Wolstenholme's theorem using Egorychev-type integral

Explicit connection between harmonic sums and Bernoulli numbers

Refinement of the theorem modulo p^4

Abstract

We present a detailed proof of Wolstenholme's theorem using an Egorychev-type contour integral and an exponential change of variables. All formal series manipulations are justified, and the connection with harmonic sums and Bernoulli numbers is made completely explicit. We further derive the classical refinement modulo $p^4$ and provide a precise extraction of the $B_{p-3}$ term. Our purpose is not to provide the most concise proofs, but rather to demonstrate, by showing how established results can be recovered, a general method based on complex analysis for deriving congruence properties in number theory.