Rational $D(q)$-quintuples

Abstract

For a nonzero 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 square for all $1⩽i<j⩽n$. We investigate for which $q$ there exist infinitely many rational $D(q)$-quintuples. We show that assuming the Parity Conjecture for the twists of several explicitly given elliptic curves, the density of such $q$ is at least $295026/296010\approx 99.5\%$.

16 pages

References in corpus (5)

• Doubly regular Diophantine quadruples
• Elliptic curves induced by Diophantine triples
• D(n)-quintuples with square elements
• Diophantine quadruples with the properties $D(n_1)$ and $D(n_2)$
• Rational $D(q)$-quadruples