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 $d\equiv 2(\mathrm{mod}4)$ is a positive integer such that $x^2-d{y}^2=-1$ and $x^2-d{y}^2=6$ 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 $m\not\equiv 5(\mathrm{mod}6)$ and $k\not\equiv 3(\mathrm{mod}6)$. Moreover, we prove the same for $n=(4m+2)+4k\sqrt{d}$ when either $m\not\equiv 9(\mathrm{mod}12)$ and $k\not\equiv 3(\mathrm{mod}6)$, or $m\not\equiv 0(\mathrm{mod}12)$ and $k\not\equiv 0(\mathrm{mod}6)$. 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 $d=10$.

18 pages. To appear in `Glasnik matematički'