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

Abstract

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 $x^2 - Dy^2 = k$ . 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.