A note on the converse of Wolstenholme's Theorem

TL;DR

This paper explores the conditions under which the converse of Wolstenholme's Theorem holds for specific prime-related products, extending previous results to broader cases involving powers of primes.

Contribution

It generalizes the converse of Wolstenholme's Theorem for products of primes and their powers, building upon and extending Helou and Terjanian's earlier work.

Findings

The converse holds for mp^k under certain inequalities.

Generalization of previous theorems to broader prime power cases.

Provides conditions for the converse to be valid for composite numbers.

Abstract

Given a prime p and a positive integer m satisfying a certain inequality, the converse of Wolstenholme's Theorem is shown to hold for the product mp^k where k is any positive integer, generalizing a result by Helou and Terjanian.