Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)

TL;DR

This paper provides a comprehensive historical survey of Wolstenholme's theorem, its numerous generalizations, and related congruences over the past 150 years, including over 70 variations and discussions on Wolstenholme primes.

Contribution

It offers an extensive overview of the developments, variations, and extensions of Wolstenholme's theorem, compiling over 70 generalizations and analyzing their significance in number theory.

Findings

Over 70 variations and generalizations of Wolstenholme's theorem.

Discussion of congruences related to Wolstenholme primes.

Historical survey of 150 years of research in this area.

Abstract

In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$ 1+\frac 12 +\frac 13+...+\frac{1}{p-1} $$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$ 1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2} $$ written in reduced form is divisible by $p$. The first of the above congruences, the so called {\it Wolstenholme's theorem}, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme's type congruences and related problems. Namely, we present and compare several generalizations and extensions of Wolstenholme's theorem obtained in the last hundred and fifty years. In particular, we present more than 70 variations and generalizations of this theorem including congruences for Wolstenholme primes. These congruences are discussed here by 33 remarks. The Bibliography of this article contains 106 references consisting of 13 textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of Integer Sequences. In this article, some results of these references are cited as generalizations of certain Wolstenholme's type congruences, but without the expositions of related congruences. The total number of citations given here is 189.