Extended Congruences for Harmonic Numbers

TL;DR

This paper derives elementary $p$-adic expansions for generalized harmonic numbers involving Bernoulli numbers and Fermat quotients, extending classical congruences and providing stronger modular relations.

Contribution

It presents new elementary derivations of $p$-adic expansions for harmonic numbers and generalizes Eisenstein's classical congruence to higher prime powers.

Findings

Derived $p$-adic expansions involving Bernoulli numbers and Fermat quotients.

Established a stronger congruence modulo prime powers extending Eisenstein's result.

Provided elementary proofs avoiding $p$-adic L-functions theory.

Abstract

We derive $p$-adic expansions for the generalized Harmonic numbers $H^{(j)}_{p-1}$ and $H^{(j)}_{\frac{p-1}{2}}$ involving the Bernoulli numbers $B_j$ and the the base-2 Fermat quotient $q_p$. While most of our results are not new, we obtain them elementarily, without resorting to the theory of $p$-adic L-functions as was the case previously. Moreover, we show that \begin{equation*}\sum_{j=0}^{n-1}\left(\frac{(2^{j+1}-1)}{(j+1)}\frac{(2^{j+2}-1)}{(j+2)}\frac{B_{j+2}}{2^{j}}H^{(j+1)}_{\frac{p-1}{2}}+2(-1)^j\frac{q_p^{j+1}}{j+1}\right)p^j\equiv 0 \pmod {p^n} \end{equation*} holds under the condition that $p >\frac{n+1}{2}$. This is another generalization, modulo any prime power, of the old $p$-congruence $H_{\frac{p-1}{2}}+2q_p \equiv 0 \bmod p$ attributed to Eisenstein, which is stronger than the one which has been published recently.