On the equation \(1!^k+2!^k+\cdots+n!^k=x^2\)
type: Item
instance of: scholarly article
title: On the equation \(1!^k+2!^k+\cdots+n!^k=x^2\)
zbMATH Open document ID: 1017.11015
DOI: 10.1023/A:1019648631211
author: Florian Luca
published in: Periodica Mathematica Hungarica
publication date: 2003-08-06T00:00:00Z
review text: Consider the equation given in the title of this paper, where parameters \(n\), \(k\), and \(x\) are positive integers, and ignore the trivial case \(n=1\). One readily verifies that triples \((n,k,x)\) given by \((2,3,3)\), \((3,1,3)\), and \((3,3,15)\) are solutions. The author conjectures that these are the only nontrivial solutions. Indeed, the author shows that no further solutions with \(n=2\) or \(n\geq 4\) are to be found, leaving only the equation \(1+2^k+6^k=x^2\). Moreover, it is shown that the number of solutions to this equation is finite, with \((k,x)=(1,3)\) and \((3,15)\) being two of them, and that if any further solutions exist, they must have \(k\equiv 3\pmod 4\).
reviewed by: Gennady Bachman
Mathematics Subject Classification ID: 11D99
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 1960293
zbMATH Keywords: diophantine equation
MaRDI profile type: Publication
full work available at URL: a:1019648631211
OpenAlex ID: W1773562321
description: scientific article; zbMATH DE number 1960293