Squares and factorials in products of factorials
type: Item
instance of: scholarly article
title: Squares and factorials in products of factorials
zbMATH Open document ID: 1373.11029
author: Florian Luca
author: N. Saradha
author: T. N. Shorey
published in: Monatshefte für Mathematik
publication date: 2014-11-14T00:00:00Z
review text: The authors prove several results in connection with a problem of Erdős and Graham concerning products of factorials yielding a factorial or a perfect square. They considerably improve upon an estimate of Erdős and Graham concerning the number of those positive integers \(n\) up to \(X\), which are not squares, but \(n!a!b!\) is a square with some integers \(a,b\) with \(n\geq a\geq b\). On their way, they also improve upon a related estimate of Erdős and Graham concerning those integers \(n\) for which the exponent of the largest prime factor in the product of a block of consecutive integers starting with \(n\), is at least two. Finally, they derive upper bounds for the solutions of the equation \(a_2!\cdots a_t!=m(m+1)\cdots (m+k-1)\), both under the abc-conjecture and unconditionally. To prove their theorems, the authors combine several results of Erdős and Graham with new ideas and tools from prime number theory in a skillful and sophisticated way.
reviewed by: Lajos Hajdu
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11D72
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 6369395
zbMATH Keywords: factorials
zbMATH Keywords: squares
zbMATH Keywords: applications of sieve methods
zbMATH Keywords: consequences of the abc conjecture
MaRDI profile type: Publication
full work available at URL: s00605-014-0641-3
OpenAlex ID: W2000376795
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DOI: 10.1007/S00605-014-0641-3
description: scientific article; zbMATH DE number 6369395