Diophantine triples with three parameters (Q6543107)

A \(D(n)\)-\(m\)-tuple is a set of \(m\) positive integers \(\{a_1,\ldots,a_m\}\) such that \(a_ia_j+n\) is a perfect square for all \(1\leq i<j\leq m\). The strong Diophantine quintuple conjecture asserts that a \(D(1)\)-triple (resp. \(D(4)\)-triple) can be extended by a larger element in a unique way to a \(D(1)\)-quadruple (resp. \(D(4)\)-quadruple). This paper confirms this conjecture for a three-parameter family defined as follows. Let \(\epsilon\in\{\pm 1,\pm 2\}\). Consider the sets \(\{f_\nu,f_{\nu+1},f_{\nu+2}\}\) with \(f_\nu=(F_\nu A+F_{\nu-1})((F_\nu A+F_{\nu-1})K+2\epsilon F_\nu)\), where \(A,K\) are positive integers and where \(F_\nu\) is the \(\nu\)-th Fibonacci number. These sets are \(D(\epsilon^2)\)-triples, if \(f_\nu>0\). The strong quintuple conjecture holds for these triples if\N\begin{itemize}\N\item \(K\geq K_0F_\nu A\), where \(K_0=165\), if \(\epsilon^2=1\), and \(K_0=416\), if \(\epsilon^2=4\),\N\item \(1\leq \nu \leq 10\) and \(1\leq A\leq 10\),\N\item \(\nu=1\),\N\item \(A=1\).\N\end{itemize}\NThe proof uses in principle the standard method from the theory of Diophantine tuples. The additional difficulties and the new ways to overcome them, e.g. using other moduls in the congruence method or a new version of the Baker-Davenport reduction lemma, are very clearly described in the well-written paper.