Diophantine $D(n)$-quadruples in $\mathbb{Z}[\sqrt{4k + 2}]$

Abstract: Let $d$ be a square-free integer and $\mathbb{Z}[\sqrt{d}]$ a quadratic ring of integers. For a given $n\in\mathbb{Z}[\sqrt{d}]$, a set of $m$ non-zero distinct elements in $\mathbb{Z}[\sqrt{d}]$ is called a Diophantine $D(n)$-$m$-tuple (or simply $D(n)$-$m$-tuple) in $\mathbb{Z}[\sqrt{d}]$ if product of any two of them plus $n$ is a square in $\mathbb{Z}[\sqrt{d}]$. Assume that $d \equiv 2 \pmod 4$ is a positive integer such that $x^2 - dy^2 = -1$ and $x^2 - dy^2 = 6$ are solvable in integers. In this paper, we prove the existence of infinitely many $D(n)$-quadruples in $\mathbb{Z}[\sqrt{d}]$ for $n = 4m + 4k\sqrt{d}$ with $m, k \in \mathbb{Z}$ satisfying $m \not\equiv 5 \pmod{6}$ and $k \not\equiv 3 \pmod{6}$. Moreover, we prove the same for $n = (4m + 2) + 4k\sqrt{d}$ when either $m \not\equiv 9 \pmod{12}$ and $k \not\equiv 3 \pmod{6}$, or $m \not\equiv 0 \pmod{12}$ and $k \not\equiv 0 \pmod{6}$. At the end, some examples supporting the existence of quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(n)$ for the above exceptional $n$'s are provided for $d = 10$.