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

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.