Rational $D(q)$-quadruples

Abstract

For a rational number $q$, a rational $D(q)$-$n$-tuple is a set of $n$ distinct nonzero rationals $\{a_1,a_2,\dots ,a_n\}$ such that $a_i{a}_j+q$ is a rational square for all $1⩽i<j⩽n$. For every $q$ we find all rational $m$ such that there exists a $D(q)$-quadruple with product $abcd=m$. We describe all such quadruples using points on a specific elliptic curve depending on $(q,m).$

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• Rational $D(q)$-quintuples