On the parity of exponents in the standard factorization of \(n!\)
type: Item
instance of: scholarly article
title: On the parity of exponents in the standard factorization of \(n!\)
zbMATH Open document ID: 1027.11014
DOI: 10.1016/S0022-314X(03)00013-1
published in: Journal of Number Theory
publication date: 2003-06-25T00:00:00Z
review text: Let \(2=p_1<p_2<\dots<p_i<\dots\) be the sequence of primes and denote by \(e_i(m)\) the exponent of the prime \(p_i\) in the standard factorization of the positive integer \(m\). \textit{D. Berend} [J. Number Theory 64, 13--19 (1997; Zbl 0874.11025)] proved a conjecture of Erdős and Graham proving that for every \(k\) there are infinitely many integers \(n\) for which \(e_i(n!)\) an even number for every \(i=1,2,\dots k\). In the present paper the author proves a generalization of this theorem showing: for every integer \(k\) and the sequence \(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_k\) of 0s and 1s (i.e. \(\varepsilon_i\in\{0,1\}\)) there are infinitely many integers \(n\) for which \(e_i(n!)\equiv \varepsilon_i\pmod 2\), \(i=1,2,\dots, k\).
reviewed by: Norbert Hegyvári
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 1934772
zbMATH Keywords: prime factorization
zbMATH Keywords: parity
zbMATH Keywords: factorials
author: Yonggao Chen
MaRDI profile type: Publication
cites work: On the parity of exponents in the factorization of \(n\)!
cites work: On the prime power factorization of \(n!\)
cites work: Q3872528
cites work: On the parity of exponents in the prime factorization of factorials
description: scientific article; zbMATH DE number 1934772