An Extension of a Congruence by Tauraso

TL;DR

This paper proves new congruences involving harmonic numbers and sums modulo prime squares, extending a problem proposed by Tauraso and using classical number theory techniques.

Contribution

It provides an elementary proof of a congruence involving harmonic numbers, extending Tauraso's congruence and introducing new identities and methods.

Findings

Established congruences for sums involving harmonic numbers modulo p^2.

Provided an elementary proof using classical congruences and combinatorial identities.

Extended Tauraso's problem with new proven congruences.

Abstract

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/n$ be the $n$th harmonic number. In this note we prove that for any prime $p\ge 7$, $$ \sum_{k=1}^{p-1}\frac{H_k}{k^2}\equiv \sum_{k=1}^{p-1}\frac{H_k^2}{k} \equiv\frac{3}{2p}\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}. $$ Notice that the first part of this congruence is recently proposed by R. Tauraso as a problem in Amer. Math. Monthly. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers and a combinatorial identity due to V. Hern\'{a}ndez.