G\'en\'eralisation des congruences de Wolstenholme et de Morley

TL;DR

This paper generalizes classical congruences related to binomial coefficients for primes, extending results by Wolstenholme, Morley, and others, with precise modular conditions and broader applicability.

Contribution

It introduces a unified congruence formula for binomial coefficients involving p-integers, generalizing multiple known prime-related congruences.

Findings

Generalized congruence for binomial coefficients modulo p^m

Unified framework encompassing Wolstenholme, Morley, Glaisher, Carlitz, and others

Simplified derivation of classical congruences

Abstract

In this paper, we prove that for any odd prime $p$ and for any $p$-integer $\alpha $,we have $ \binom{\alpha p-1}{p-1}\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\leq i<j\leq p-1}\frac{1}{ij} \ \pmod{p^{m}}$, where $m=7$ if $p\neq 7$ and $m=6$ if $p=7$. this congruence generalizes the congruences of Wolstenholme, Morley, Glaisher, Carlitz, McIntosh, Tauraso and Me\v{s}trovi\'c. It allows one to rediscover the congruences of Glaisher, Carlitz and Zhao in a simple way