Benford's Law in the ring $\mathbb{Z}(\sqrt{D})$

For $D$ a natural number that is not a perfect square and for $k$ a non-zero integer, consider the subset ${\mathbb{Z}}_k(\sqrt{D})$ of the quadratic integer ring $\mathbb{Z}(\sqrt{D})$ consisting of elements $x+y\sqrt{D}$ for which $x^2-D{y}^2=k$ . For each $k$ such that the set ${\mathbb{Z}}_k(\sqrt{D})$ is nonempty, ${\mathbb{Z}}_k(\sqrt{D})$ has a natural arrangement into a sequence for which the corresponding sequence of integers $x$, as well as the corresponding sequence of integers $y$, are strong Benford sequences.