On the prime power factorization of n!

2003/04/19 by Florian Luca, Luca, Florian, Pantelimon Stănică +2
Mathematics · #11B50 #11N25 #Advanced Topology and Set Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras #math.NT #msc:11B50 #msc:11N25

paper · pdf · doi:10.48550/arxiv.math/0304272

7 pages; accepted Journal of Number Theory

arxiv created 2003/04/19 · openalex publication_date 2003/04/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove two results. The first theorem uses a paper of Kim \citeK to show that for fixed primes p₁,...,p_k, and for fixed integers m₁,...,m_k, with p_i\not|m_i, the numbers (e_p1(n),...,e_pk(n)) are uniformly distributed modulo (m₁,...,m_k), where e_p(n) is the order of the prime p in the factorization of n!. That implies one of Sander's conjecture from \citeS, for any set of odd primes. Berend \citeB asks to find the fastest growing function f(x) so that for large x and any given finite sequence ε_i∈ \0,1\, i≤ f(x), there exists n

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