Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jean-Christophe Pain

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.

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 and provide a precise extraction of the 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.

Paper Structure

This paper contains 19 sections, 1 theorem, 70 equations.

Table of Contents

  1. Introduction
  2. Integral representation
  3. Exponential change of variable
  4. Series expansions
  5. Logarithmic expansion
  6. Expansion of $A(t)^p$
  7. Inversion
  8. Expansion of the numerator
  9. Product expansion
  10. $p$-adic justification
  11. Structure of the expansion and valuation estimate
  12. Degree constraint
  13. Link with harmonic numbers
  14. Refinement and explicit extraction of the $B_{p-3}$ term
  15. Interpretation and connection to Wolstenholme primes
  16. ...and 4 more sections

Key Result

Theorem 9.1

For any prime number $p \ge 5$, the following congruence holds: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (2)

  • Theorem 9.1: Wolstenholme, 1862
  • proof