Rational $D(q)$-quintuples

TL;DR

This paper studies rational $D(q)$-quintuples, sets of five distinct nonzero rationals with pairwise properties related to squares, and explores the density of such $q$ values assuming the Parity Conjecture.

Contribution

It establishes a high lower bound on the density of $q$ values for which infinitely many rational $D(q)$-quintuples exist, assuming the Parity Conjecture.

Findings

Density of such $q$ is at least approximately 99.5%.

Conditional on the Parity Conjecture, infinitely many $D(q)$-quintuples exist for most $q$.

The study links rational $D(q)$-quintuples to elliptic curve twists.

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 $a_ia_j+q$ is a square for all $1 \leqslant i < j \leqslant n$. 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 $295026/296010\approx 99.5\%$.