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

Christine Patterson, Marion Scheepers

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.

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

Abstract

For a natural number that is not a perfect square and for a non-zero integer, consider the subset of the quadratic integer ring consisting of elements for which . For each such that the set is nonempty, has a natural arrangement into a sequence for which the corresponding sequence of integers , as well as the corresponding sequence of integers , are strong Benford sequences.
Paper Structure (5 sections, 14 theorems, 20 equations, 2 figures)

This paper contains 5 sections, 14 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Canonical orbit sequences for $\mathbb{Z}(\sqrt{D})$
  3. Benford's Law
  4. Benford's Law and canonical orbit sequences.
  5. The interleaving theorem for Benford Sequences

Key Result

Lemma 2

For nonzero integer $k$, if $x+y\sqrt{D} \in \mathbb{Z}_k(\sqrt{D})$, then $Orbit(x+y\sqrt{D})\subseteq \mathbb{Z}_k(\sqrt{D})$.

Figures (2)

  • Figure 1: Newcomb's Table, Benford's Table IV
  • Figure 2: Probability of occurrence of first digit

Theorems & Definitions (27)

  • Definition 1
  • Lemma 2
  • Theorem 3: Nagell, p. 205
  • Definition 4
  • Lemma 5
  • Theorem 6
  • Theorem 7
  • Theorem 8: Classical
  • proof
  • Corollary 9
  • ...and 17 more