Irregular Diophantine \(m\)-tuples and elliptic curves of high rank
A set of \(m\) non-zero rational numbers \(\{a_1,\ldots,a_m\}\) is called a \textit{Diophantine \(m\)-tuple} if for all \(1\leq i<j\leq m\) the rational number \(a_ia_j+1\) is a perfect square. In this very short note the author interprets certain properties of Diophantine \(4\)-tuples and \(5\)-tuples in terms of the existence of some relations connecting rational points of elliptic curves defined over \({\mathbb Q}\). As an application some elliptic curves over \({\mathbb Q}\) are found, whose Mordell-Weil group has rank \(8\) and torsion \(({\mathbb Z}/2{\mathbb Z})^2\).
scientific article; zbMATH DE number 1538524