Infinitely Many Counter Examples of a Conjecture of Franu\v{s}i\'c and Jadrijevi\'c

Abstract

Let $d$ be a square-free integer such that $d \equiv 15 \pmod{60}$ and the 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 Franu\v{s}i\'c and Jadrijevi\'c (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.