A note on $p$-adic locally analytic functions with application to behavior of the $p$-adic valuations of Stirling numbers

TL;DR

This paper proves conjectures about the $p$-adic valuations of Stirling numbers of the second kind using elementary $p$-adic analysis, advancing understanding of their behavior in number theory.

Contribution

It provides the first proof of conjectures on $p$-adic valuations of Stirling numbers, connecting $p$-adic analysis with combinatorial number theory.

Findings

Confirmed conjectures on $p$-adic valuations of Stirling numbers

Established new properties of $p$-adic valuations using elementary methods

Enhanced understanding of the behavior of Stirling numbers in $p$-adic contexts

Abstract

The aim of this paper is to prove conjectures concerning $p$-adic valuations of Stirling numbers of the second kind $S(n,k)$, $n,k\in\mathbb{N}_+$, stated by Amdeberhan, Manna and Moll and Berrizbeitia et al., where $p$ is a prime number. The proof is based on elementary facts from $p$-adic analysis.