$p$-adic L-functions and Classical Congruences

TL;DR

This paper extends classical number theory congruences to higher powers of primes using $p$-adic analysis and $p$-adic L-functions, broadening their applicability.

Contribution

It introduces a method to generalize classical congruences to modulo $p^k$ for any $k>0$ using $p$-adic tools, which is a novel approach.

Findings

Classical congruences are extended to higher prime powers.

$p$-adic analysis provides a framework for these generalizations.

New congruences apply to a wide range of classical results.

Abstract

In this paper, using $p$-adic analysis and $p$-adic L-functions, we show how to extend classical congruences (due to Wilson, Gauss, Dirichlet, Jacobi, Wolstenholme, Glaisher, Morley, Lemher and other people) to modulo $p^k$ for any $k>0$.