Irregular Diophantine \(m\)-tuples and elliptic curves of high rank. (English) Zbl 0964.11027

A set of \(m\) non-zero rational numbers \(\{a_1,\ldots,a_m\}\) is called a 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 \(({\mathbb Z}/2{\mathbb Z})^2\).


11G05 Elliptic curves over global fields
11B99 Sequences and sets
14H52 Elliptic curves


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