Bounds on the $p$-adic valuation of the factorial, hyperfactorial and superfactorial

TL;DR

This paper derives simple bounds for the p-adic valuation of factorial-related quantities like factorial, hyperfactorial, and superfactorial, with potential applications in cryptography and number theory.

Contribution

It provides new upper and lower bounds for the p-adic valuation of these quantities using the Legendre-de Polignac formula, extending to related iterated functions.

Findings

Established bounds for $ u_p(n!)$, $H(n)$, and $ ext{sf}(n)$

Extended bounds to Stirling numbers and Catalan numbers

Discussed applications in cryptography and large number analysis

Abstract

In this article, we investigate the $p$-adic valuation $\nu_p$ of quantities such as the factorial $n!$, the hyperfactorial $H(n)$ or the superfactorial $\mathrm{sf}(n)$. In particular, we obtain simple bounds (both upper and lower) for $\nu_p$, using the Legendre-de Polignac formula. Other iterated quantities such as the Berezin function, are also considered. Beyond their recreational character, such quantities, often related to very large numbers, may find applications for cryptography purposes. Finally, lower and upper bounds for the $p$-adic valuation of Stirling numbers of the first kind and Catalan numbers are briefly discussed.