On a Problem of Diophantus for Rationals

Let q be a nonzero rational number. We investigate for which q there are in nitely many sets consisting of ve nonzero rational numbers such that the product of any two of them plus q is a square of a rational number. We show that there are in nitely many square-free such q and on assuming the Parity Conjecture for the twists of an explicitly given elliptic curve we derive that the density of such q is at least one half. For the proof we consider a related question for polynomials with integral coe cients. We prove that, up to certain admissible transformations, there is precisely one set of non-constant linear polynomials such that the product of any two of them except one combination, plus a given linear polynomial is a perfect square.