On the special harmonic numbers $H_{\lfloor p/9 \rfloor}$ and $H_{\lfloor p/18 \rfloor}$ modulo $p$

TL;DR

This paper establishes new congruences for specific harmonic numbers modulo primes, expanding understanding of their arithmetic properties and applications to Fermat's Last Theorem.

Contribution

It provides explicit congruences for harmonic numbers involving floors of p/9 and p/18, and fully determines related reciprocal sum families modulo p.

Findings

Derived congruences with three and four arithmetic components

Complete determination of reciprocal sum families modulo p

Applications to Fermat's Last Theorem

Abstract

Building on work of Zhi-Hong Sun, we establish congruences for the special harmonic numbers $H_\lfloor p/9 \rfloor$ and $H_{\lfloor p/18 \rfloor}$ modulo $p$, which contain respectively three and four distinct arithmetic components. We also obtain a complete determination modulo $p$ of the corresponding families of sums of reciprocals of the type studied by Dilcher and Skula. Applications to the first case of Fermat's Last Theorem are considered.