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

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 \(d = 10\).