Irregular Diophantine $m$-tuples and elliptic curves of high rank

Abstract

A rational Diophantine $m$-tuple is a set of $m$ nonzero rationals such that the product of any two of them is one less than a perfect square. In this paper we characterize the notions of regular Diophantine quadruples and quintuples, introduced by Gibbs, by means of elliptic curves. Motivated by these characterizations, we find examples of elliptic curves over $\mathbf{Q}$ with torsion group $\mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/2\mathbf{Z}$ and with rank equal 8.