Nontrivial lower bounds for the $p$-adic valuations of some type of rational numbers and an application for establishing the integrality of some rational sequences

TL;DR

This paper develops lower bounds for the p-adic valuations of certain rational numbers involving harmonic numbers, and applies these bounds to prove the integrality of specific rational sequences that are not obviously integral.

Contribution

It introduces a novel method based on dilogarithm functional equations to establish p-adic valuation bounds and proves the integrality of sequences previously not known to be integral.

Findings

Established nontrivial lower bounds for p-adic valuations.

Proved the integrality of certain rational sequences.

Demonstrated the application of dilogarithm functional equations in number theory.

Abstract

In this note, basing on a certain functional equation of the dilogarithm function, we establish nontrivial lower bounds for the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers involving harmonic numbers. Then we use our estimate to derive the integrality of some sequences of rational numbers, which cannot be seen directly from their definitions.