On the \(D(- 1)\)-triple \(\{ 1,k^{2}+1,k^{2}+2k+2\}\) and its unique \(D(1)\)-extension
type: Item
instance of: scholarly article
title: On the \(D(- 1)\)-triple \(\{ 1,k^{2}+1,k^{2}+2k+2\}\) and its unique \(D(1)\)-extension
zbMATH Open document ID: 1223.11036
author: Bo He
author: Alain Togbé
published in: Journal of Number Theory
publication date: 2010-11-19T00:00:00Z
review text: A set of positive integers \(\{a_1,\ldots,a_m\}\) is called a Diophantine \(m\)-tuple of property \(D(n)\) if any \(a_ia_j+n\) is a perfect square. An interesting problem is wether it is possible to extend a Diophantine \(m\)-tuple. The authors consider the \(D(-1)\)-triple \(\{1,k^2+1,(k+1)^2+1\}\). They prove that this triple can not be extended to a Diophantine \(D(-1)\)-quadruple. A \(D(1)\) extension of this Diophantine \(D(-1)\)-triple is found. The proof uses systems of Pellian equations and Baker's method.
reviewed by: István Gaál
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 5817569
zbMATH Keywords: extension of Diophantine tuples
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.07.006
OpenAlex ID: W1966510750
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DOI: 10.1016/J.JNT.2010.07.006
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description: scientific article; zbMATH DE number 5817569