Généralisation des congruences de Wolstenholme et de Morley

Abstract

In this paper, we prove that for any odd prime $p$ and for any $p$-integer $\alpha$,we have $\left(\genfrac{}{}{0ex}{}{\alpha p-1}{p-1}\right)\equiv 1-\alpha (\alpha -1)({\alpha }^2-\alpha -1)p{\sum }_{k=1}^{p-1}\frac{1}{k}+{\alpha }^2(\alpha -1{)}^2{p}^2{\sum }_{1\le i<j\le p-1}\frac{1}{ij}(\mathrm{mod}{p}^m)$, where $m=7$ if $p\ne 7$ and $m=6$ if $p=7$. this congruence generalizes the congruences of Wolstenholme, Morley, Glaisher, Carlitz, McIntosh, Tauraso and Meštrović. It allows one to rediscover the congruences of Glaisher, Carlitz and Zhao in a simple way

10 pages, in French

References in corpus (2)

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