Modulus p^2 congruences involving harmonic numbers

TL;DR

This paper investigates congruences involving harmonic numbers modulo p^2 for prime p>3, extending understanding of their properties in number theory.

Contribution

It establishes new congruences of the form sum_{k=1}^{p-1} k^m H_k^n modulo p^2 for various exponents m and n.

Findings

Derived congruences for sums involving harmonic numbers and powers of k

Extended known results to higher powers of harmonic numbers

Enhanced understanding of harmonic number properties modulo prime squares

Abstract

The harmonic number $H_k=\sum_{j=1}^k1/j(k=1,2,3\cdots)$ play an important role in mathematics. Let $p>3$ be a prime. In this paper, we establish a number of congruences with the form $\sum_{k=1}^{p-1}k^mH_k^n(\mod p^2)$ for $m=1,2\cdots,p-2$ and $n=1,2,3$.