On a conjecture of Franu\v sić and Jadrijevi\' c: Counter-examples

Let $d\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$ be a square-free integer such that $x^2-d{y}^2=-1$ and $x^2-d{y}^2=6$ are solvable in integers. We prove the existence of infinitely many quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(n)$ when $n\in \{(4m+1)+4k\sqrt{d},(4m+1)+(4k+2)\sqrt{d},(4m+3)+4k\sqrt{d},(4m+3)+(4k+2)\sqrt{d},(4m+2)+(4k+2)\sqrt{d}\}$ for $m,k\in \mathbb{Z}$. As a consequence, we provide few counter examples to a conjecture of Franu\v sić and Jadrijevi' c (see Conjecture 1.1).