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

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.