On a family of Diophantine triples \(\{k,A^2k+2A,(A+1)^2k+2(A+1)\}\) with two parameters
type: Item
instance of: scholarly article
title: On a family of Diophantine triples \(\{k,A^2k+2A,(A+1)^2k+2(A+1)\}\) with two parameters
zbMATH Open document ID: 1199.11074
author: Bo He
author: Alain Togbé
published in: Acta Mathematica Hungarica
publication date: 2009-12-29T00:00:00Z
review text: Let \(A\), \(k\) and \(d\) be positive integers, and put \(H:=\{k,A^2k+2A,(A+1)^2k+2(A+1),d\}\). The authors prove that if \(H\) is a Diophantine quadruple, that is the product of any two distinct elements of \(H\) increased by \(1\) is a perfect square, then in case of \(3\leq A\leq 10\) \[ d=(4A^4+8A^3+4A^2)k^3+(16A^3+24A^2+8A)k^2+(20A^2+20A+4)k+(8A+4) \] holds. This result is an extension of a theorem of \textit{A. Dujella} [Publ. Math. 51, No. 3--4, 311--322 (1997; Zbl 0903.11010)] with \(A=1\), and also of a classical theorem of Baker and Davenport with \(A=k=1\).
reviewed by: Lajos Hajdu
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 5652037
zbMATH Keywords: Diophantine \(m\)-tuple
zbMATH Keywords: Pell equation
zbMATH Keywords: Baker's method
MaRDI profile type: Publication
full work available at URL: s10474-009-8155-5
OpenAlex ID: W2137672922
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DOI: 10.1007/S10474-009-8155-5
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description: scientific article; zbMATH DE number 5652037