On the Diophantine equation \((x^2+k)(y^2+k)=(z^2+k)^2\)
type: Item
instance of: scholarly article
title: On the Diophantine equation \((x^2+k)(y^2+k)=(z^2+k)^2\)
zbMATH Open document ID: 1238.11039
DOI: 10.1216/RMJ-2008-38-6-2091
author: Maciej Ulas
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-04-07T00:00:00Z
review text: With fixed \(k\), a solution to the equation in the title gives three values of the polynomial \(x^2 +k\) forming a geometric progression. First the author reproves a result known since 1960's that for \(k\) of the shape \(\pm(a^2-2b^2)\) there are infinitely many solutions in integers \(x,y,z\). Then he proves that for infinitely many integers \(k\) there exists a four term geometric progression of integers of the form \(x^2 + k\). Reviewer's remark. A case of special interest is when \(k=-1\) and \(x,y,z\) are odd since then a solution to the equation produces three triangular numbers in a geometric progression. It was proved recently by \textit{Y. G. Chen} and \textit{J. H. Fang} [Integers 7, No. 1, Paper A19, 2 p. (2007; Zbl 1130.11014)] that there are no geometric progressions formed by four triangular numbers.
reviewed by: Kazimierz Szymiczek
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 5541312
zbMATH Keywords: systems of quadratic Diophantine equations
MaRDI profile type: Publication
cites work: On the Diophantine equation \((x^2-1)(y^2-1) = [(\tfrac 12 (y-x))^2-1]^2\)
cites work: On a Diophantine equation
description: scientific article; zbMATH DE number 5541312