On the prime power factorization of \(n!\). II
type: Item
instance of: scholarly article
title: On the prime power factorization of \(n!\). II
zbMATH Open document ID: 1182.11040
author: Wei Liu
author: Yonggao Chen
published in: Journal of Number Theory
publication date: 2009-03-09T00:00:00Z
review text: For a positive integer \(n\) and a prime \(p\), let \(e_p(n)\) be the power of \(p\) in the prime number factorization of \(n\), i.e. \(p^{e_p(n)}|n\) and \(p^{e_p(n)+1}\nmid n\). The present paper deals with the distribution of \(e_p(n!)\) in residue classes. The following results are proved. Theorem 1. For any positive integer \(m\), any prime \(p\) and any \(\varepsilon \in \mathbb{Z}_m\), there are infinitely many positive integers \(n\) such that \(e_p(n!)\equiv \varepsilon\) (mod \(m\)). Theorem 2. For any positive integer \(m\), there exists a constant \(D(m)\) such that, if \(\varepsilon,\delta\in \mathbb{Z}_m\) and \(p,q\) are two distinct primes with max\(\{p,q\}\geq D(m)\), then there exist infinitely many positive integers \(n\) such that \(e_p(n!)\equiv \varepsilon\) (mod \(m\)) and \(e_q(n!)\equiv \delta\) (mod \(m\)). If \(p\nmid m\) and \(q\nmid m\), these results follow from a theorem of \textit{F. Luca} and \textit{P. Stănică} [J. Number Theory 102, No. 2, 298--305 (2003; Zbl 1049.11092)]. However, their method does not extend to the (more difficult) complementary case which is covered by the above Theorems. The paper concludes with several interesting related open problems.
reviewed by: Stephan Baier
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11B50
zbMATH DE Number: 5526758
zbMATH Keywords: Erdős problems
zbMATH Keywords: prime factorization
zbMATH Keywords: factorials
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.05.016
OpenAlex ID: W2082598440
cites work: On the parity of exponents in the factorization of \(n\)!
cites work: On the parity of exponents in the standard factorization of \(n!\)
cites work: On the prime power factorization of \(n!\)
cites work: Q3872528
cites work: On the prime power factorization of \(n\)!
cites work: On the parity of exponents in the prime factorization of factorials
DOI: 10.1016/J.JNT.2006.05.016
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description: scientific article; zbMATH DE number 5526758