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

2024/08/01 by Jean‐Christophe Pain, Pain, Jean-Christophe
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2408.00353

openalex publication_date 2024/08/01 · openalex created_date 2024/08/04 · openalex updated_date 2026/07/28

Abstract

In this article, we investigate the p-adic valuation ν_p of quantities such as the factorial n!, the hyperfactorial H(n) or the superfactorial sf(n). In particular, we obtain simple bounds (both upper and lower) for ν_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.

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