Infinitely many counterexamples of a conjecture of Franušić and Jadrijević

Abstract

Let d be a square-free integer such that \(d \equiv 15 \pmod {60}\) and Pell’s equation \(x^2 - dy^2 = -6\) is solvable in rational integers x and y. In this paper, we prove that there exist infinitely many Diophantine quadruples in \(\mathbb {Z}[\sqrt{d}]\) with the property D(n) for certain n’s. As an application of it, we ‘unconditionally’ prove the existence of infinitely many rings \(\mathbb {Z}[\sqrt{d}]\) for which the conjecture of Franušić and Jadrijević (Conjecture 1.1) does ‘not’ hold. This conjecture states a relationship between the existence of a Diophantine quadruple in \(\mathcal {R}\) with the property D(n) and the representability of n as a difference of two squares in \(\mathcal {R}\), where \(\mathcal {R}\) is a commutative ring with unity.