A proof of a simple conjecture about harmonic numbers

TL;DR

This paper proves that for any prime number greater than or equal to 5, only finitely many natural numbers n exist such that p divides the nth harmonic number.

Contribution

It establishes a new finiteness result regarding the divisibility of harmonic numbers by primes p ≥ 5.

Findings

Finiteness of n such that p divides H_n for p ≥ 5

Supports a conjecture about harmonic number divisibility

Advances understanding of harmonic number properties

Abstract

We prove that, for any prime number $p\geq 5$, the set of natural numbers $n$ such that $p\mid H_n$ is finite.