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

Abstract

In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$ 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.

31 pages. We provide a historical survey of Wolstenholme's type congruences (1862-2012) including more than 70 related results and 106 references. This is in fact version 2 of the paper extended with congruences (12) and (13)

References in corpus (4)

• A generalization of Carlitz's congruence
• On Apery numbers and generalized central trinomial coefficients
• A q-analog of Ljunggren's binomial congruence
• Congruences for Wolstenholme primes

Cited by in corpus (19)

• Congruences for $q$-binomial coefficients
• Some divisibility properties of binomial and q-binomial coefficients
• Multiple harmonic sums and Wolstenholme's theorem
• Lucas' theorem: its generalizations, extensions and applications (1878--2014)
• A congruence involving harmonic sums modulo $p^{\alpha }{q}^{\beta }$
• Congruence properties for the trinomial coefficients
• Wilson's theorem modulo p^2 derived from Faulhaber polynomials
• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
• Several generalizations and variations of Chu-Vandermonde identity
• The $p$-adic Order of Power Sums, the Erdös-Moser Equation, and Bernoulli Numbers
• Variations of Lucas' Theorem Modulo Prime Powers
• On the Congruence of Finite Generalized Harmonic Numbers Sums Modulo $p^2$
• Some divisibility properties of binomial coefficients
• $p$-adic L-functions and Classical Congruences
• Wolstenholme Type Congruences and Framing of Rational 2-Functions
• Généralisation des congruences de Wolstenholme et de Morley
• Values of symmetric polynomials and a truncated analogue of the Riemann zeta function
• Supercongruences for the $(p-1)$th Apéry number
• A variation of $q$-Wolstenholme's theorem