Diophantine \(D(n)\)-quadruples in \(Z[√4k + 2]\)

Let \(d\) be a square-free integer and \(Z[√d]\) a quadratic ring of integers. For a given \(n[√d]\), a set of \(m\) non-zero distinct elements in \(Z[√d]\) is called a Diophantine \(D(n)\)-\(m\)-tuple (or simply \(D(n)\)-\(m\)-tuple) in \(Z[√d]\) if product of any two of them plus \(n\) is a square in \(Z[√d]\). Assume that \(d ≡ 2 4\) 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 \(Z[√d]\) for \(n = 4m + 4k√d\) with \(m, k ∈ Z\) satisfying \(m ≡ 5 {6}\) and \(k ≡ 3 {6}\). Moreover, we prove the same for \(n = (4m + 2) + 4k√d\) when either \(m ≡ 9 {12}\) and \(k ≡ 3 {6}\), or \(m ≡ 0 {12}\) and \(k ≡ 0 {6}\). At the end, some examples supporting the existence of quadruples in \(Z[√d]\) with the property \(D(n)\) for the above exceptional \(n\)'s are provided for \(d = 10\).