On the prime power factorization of \(n!\)
type: Item
instance of: scholarly article
title: On the prime power factorization of \(n!\)
zbMATH Open document ID: 0999.11015
author: Yaochen Zhu
author: Yonggao Chen
published in: Journal of Number Theory
publication date: 2002-09-17T00:00:00Z
review text: Let \(1=p_1<p_2<\dots\) be the sequence of primes. Let us denote by \(e_i(m)\) the integers satisfying \(p_{i}^{e_i(m)}\mid m\) and \(p_{i}^{e_i(m)+1}\nmid m\) (\(m\) is natural number). Let \(H_i(n)=e_i(n!).\) Motivated by a question of Erdős and Graham, \textit{D. Berend} showed, among other results, [J. Number Theory 64, 13-19 (1997; Zbl 0874.11025)] that for every \(k\geq 1\) there exists an infinite sequence of integers \(n_1<n_2<\dots \) with bounded gaps for which \(H_i(n_m)\equiv 0\pmod 2\), \(i=1,2,\dots ,k.\) In the present paper the authors generalize a result of Berend. They define the sequence \[ S(n)=H_i(u_jn+v_j)\pmod d_{1\leq i\leq k; 1\leq j\leq t}, \] where \(0<u_i\leq u; \;0\leq v_j\leq v\). (For \(v_j=0, u_j=j\) the case was investigated by Berend). For integers \(1\leq n<m\) if \(S(n)=S(m)\) then \(m\) is said to be a successor of \(n\) and \(n\) is a predecessor of \(m\). The authors prove that for every natural number \(n\) there exists at least one successor of \(n\) (and therefore infinitely many). This sequence has bounded gaps and this bound depends only on \(u,v,k,t,p_1,\dots ,p_k.\) As a consequence they prove given a positive \(k\) and \(\varepsilon_1,\varepsilon_2,\dots \varepsilon_k\in \{0,1\}\), if there exists a natural number \(n\) for which \(H_i(N)\equiv \varepsilon_i \pmod 2\), \(i=1,2,\dots ,k,\) then every block of \(m\) consecutive natural numbers contains an integer with the same property where \(m=m(p_1,p_2,\dots p_k).\)
reviewed by: Norbert Hegyvári
Mathematics Subject Classification ID: 11B50
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 1538250
MaRDI profile type: Publication
full work available at URL: jnth.1999.2477
OpenAlex ID: W4213038763
cites work: On the parity of exponents in the factorization of \(n\)!
cites work: Q3872528
cites work: The product of consecutive integers is never a power
cites work: A combinatorial problem on finite Abelian groups. I
DOI: 10.1006/JNTH.1999.2477
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description: scientific article; zbMATH DE number 1538250