Benford's Law in the ring $\mathbb{Z}(\sqrt{D})$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910334222073856 |
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| author | Patterson, Christine Scheepers, Marion |
| author_facet | Patterson, Christine Scheepers, Marion |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10864 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Benford's Law in the ring $\mathbb{Z}(\sqrt{D})$ Patterson, Christine Scheepers, Marion Number Theory Primary 11D09, 11B05, 11B99, Secondary 62R01, 11A55 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. |
| title | Benford's Law in the ring $\mathbb{Z}(\sqrt{D})$ |
| topic | Number Theory Primary 11D09, 11B05, 11B99, Secondary 62R01, 11A55 |
| url | https://arxiv.org/abs/2402.10864 |