On the parity of exponents in the factorization of \(n\)!
type: Item
instance of: scholarly article
title: On the parity of exponents in the factorization of \(n\)!
zbMATH Open document ID: 0874.11025
author: Daniel Berend
published in: Journal of Number Theory
publication date: 1997-06-08T00:00:00Z
review text: Let \(2=p_1<p_2<\cdots\) be the sequence of all primes in ascending order. Furthermore, let \(n!= p^{\alpha_1(n)}_1\cdots p^{\alpha_m(n)}_m\). \textit{P. Erdös} and \textit{R. Graham} in their celebrated problem book [Old and new problems and results in combinatorial number theory, Monographie No. 28 de L'Enseign. Math. (1980; Zbl 0434.10001)] raised the following question: Does there exist, for every \(k\), some \(n>1\) with all exponents \(\alpha_1(n),\alpha_2(n),\dots,\alpha_k(n)\) even? In the present paper, the author gives an affirmative answer to this question. His proof is elementary. Furthermore, the author proves that for arbitrary fixed positive integers \(m\), \(k\), \(d\) there exist infinitely many positive integers \(n\) for which all the exponents \(\alpha_1(tn),\alpha_2(tn),\dots,\alpha_k(tn)\), \(1\leq t\leq m\), are divisible by \(d\). The proof of this theorem uses some results of dynamical systems.
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 1017562
zbMATH Keywords: parity
zbMATH Keywords: exponents
zbMATH Keywords: factorials
reviewed by: Norbert Hegyvári
MaRDI profile type: Publication
full work available at URL: jnth.1997.2106
OpenAlex ID: W2012164061
cites work: Q3872528
cites work: The product of consecutive integers is never a power
cites work: Q3909268
cites work: A combinatorial problem on finite Abelian groups. II
DOI: 10.1006/JNTH.1997.2106
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description: scientific article; zbMATH DE number 1017562