A counterexample to the Conjecture of Ankeny, Artin and Chowla
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911316999929856 |
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| author | Reinhart, Andreas |
| author_facet | Reinhart, Andreas |
| contents | Let $p$ be a prime number with $p\equiv 1\mod 4$, let $ω=\frac{1+\sqrt{p}}{2}$, let $\varepsilon>1$ be the fundamental unit of $\mathbb{Z}[ω]$ and let $x$ and $y$ be the unique nonnegative integers with $\varepsilon=x+yω$. The Ankeny-Artin-Chowla-Conjecture states that $p$ is not a divisor of $y$. In this note, we provide and discuss a counterexample to this conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_21864 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A counterexample to the Conjecture of Ankeny, Artin and Chowla Reinhart, Andreas Number Theory 11R11, 11R27 Let $p$ be a prime number with $p\equiv 1\mod 4$, let $ω=\frac{1+\sqrt{p}}{2}$, let $\varepsilon>1$ be the fundamental unit of $\mathbb{Z}[ω]$ and let $x$ and $y$ be the unique nonnegative integers with $\varepsilon=x+yω$. The Ankeny-Artin-Chowla-Conjecture states that $p$ is not a divisor of $y$. In this note, we provide and discuss a counterexample to this conjecture. |
| title | A counterexample to the Conjecture of Ankeny, Artin and Chowla |
| topic | Number Theory 11R11, 11R27 |
| url | https://arxiv.org/abs/2410.21864 |