Rational D(q)-quintuples.

For a nonzero rational number q, a rationalD(q)-n-tuple is a set of n distinct nonzero rationals { a 1 , a 2 , ⋯ , 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 ≈ 99.5 % . [ABSTRACT FROM AUTHOR]