Wolstenholme's theorem

 Joseph Wolstenholme

Quick Info

Born: 30 September 1829, Eccles near salford, Lancashire, England.

Died: 18 November 1891.

Known for: Wolstenholme primes

                       Wolstenholme's theorem

                       Wolstenholme numbers

Introduction

 In 1862, J. Wolstenholme proved that for any prime p>3 the numerator of the fraction

\begin{equation} \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1} \end{equation}

written in reduced form is divisible by  p^{2}  and that the numerator of the fraction

\begin{equation} \frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+....+\frac{1}{\left ( p-1 \right )^{2}} \end{equation}

written in reduced form is divisible by p.

The first of the above congruences, the so-called Wolstenholme’s theorem, is a fundamental congruence in Combinatorial Number Theory.

Wolstenholme numbers is the numerator of sum

\begin{equation} \sum_{k=1}^{n}\frac{1}{k^{2}} \end{equation}

The first few terms of the Wolstenholme's numbers: 1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141, 1968329,... (See A007406).

References

1   Granville, Andrew (1997), "Binomial coefficients modulo prime powers" , Canadian Mathematical Society Conference Proceedings, 20: 253–275

2   McIntosh, R. J.; Roettger, E. L. (2007), "A search for Fibonacci−Wieferich and Wolstenholme primes", Mathematics of Computation, 76 (260)