Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun

TL;DR

This paper proves a conjectured congruence involving harmonic numbers and Bernoulli numbers for primes greater than or equal to 7, confirming a recent conjecture by Z.-W. Sun.

Contribution

It establishes a new congruence relation for harmonic numbers modulo prime squares, confirming Sun's conjecture and providing additional similar results.

Findings

Confirmed Sun's conjecture for primes ≥ 7

Derived congruences involving harmonic and Bernoulli numbers

Extended results to related harmonic number sums

Abstract

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/k$ be the $n$th harmonic number. In this note we prove that for any prime $p\ge 7$, $$ \sum_{k=1}^{p-1}\frac{H_k^2}{k^2} \equiv4/5pB_{p-5}\pmod{p^2}, $$ which confirms the conjecture recently proposed by Z. W. Sun. Furthermore, we also prove two similar congruences modulo $p^2$.