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

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 $p\ge 7$, $$\sum _{k=1}^{p-1}\frac{H_k^2}{k^2}\equiv 4/5p{B}_{p-5}(\mathrm{mod}{p}^2),$$ which confirms the conjecture recently proposed by Z. W. Sun. Furthermore, we also prove two similar congruences modulo $p^2$.

This is a final version accepted for publication in International Journal of Number Theory

References in corpus (3)

• Open Conjectures on Congruences
• New harmonic number identities with applications
• Arithmetic theory of harmonic numbers (II)

Cited by in corpus (1)

• An Extension of a Congruence by Tauraso