On the \(D(- 1)\)-triple \(\{ 1,k^{2}+1,k^{2}+2k+2\}\) and its unique \(D(1)\)-extension (Q607043)

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.