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

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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.

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- Mestrovic2011
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- article
- date_added
- 2013-11-21
- date_published
- 2011-11-01

### BibTeX entry

@article{Mestrovic2011, key = {Mestrovic2011}, type = {article}, title = {Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)}, author = {Mestrovic, Romeo}, 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.}, comment = {}, date_added = {2013-11-21}, date_published = {2011-11-01}, urls = {http://arxiv.org/abs/1111.3057,http://arxiv.org/pdf/1111.3057v2}, collections = {History,Integerology}, month = {nov}, url = {http://arxiv.org/abs/1111.3057 http://arxiv.org/pdf/1111.3057v2}, year = 2011, archivePrefix = {arXiv}, eprint = {1111.3057}, primaryClass = {math.NT}, urldate = {2013-11-21} }