Nontrivial lower bounds for the $p$-adic valuations of some type of rational numbers

TL;DR

This paper establishes significant lower bounds for the p-adic valuations of certain rational numbers, extending previous results and demonstrating that these valuations can be arbitrarily large through multiple analytical approaches.

Contribution

It generalizes recent findings on p-adic valuations from the case p=2 to arbitrary primes, providing new lower bounds and multiple proof techniques.

Findings

p-adic valuations of certain rational numbers can be arbitrarily large

The paper introduces three different methods to establish unbounded p-adic valuations

Extends previous results from p=2 to general prime p

Abstract

In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both related to the special case $p = 2$. The crucial point for obtaining our main result is the fact that the $p$-adic valuation of the rational numbers in question is unbounded from above. We will confirm this fact by three different methods; the first two are elementary while the third one leans on the $p$-adic analysis.