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arXiv · 2410.21864

A counterexample to the Conjecture of Ankeny, Artin and Chowla

Andreas Reinhart
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Abstract

Let $p$ be a prime number with $p\equiv 1\mod 4$, let $\omega=\frac{1+\sqrt{p}}{2}$, let $\varepsilon>1$ be the fundamental unit of $\mathbb{Z}[\omega]$ and let $x$ and $y$ be the unique nonnegative integers with $\varepsilon=x+y\omega$. 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.

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BibTeXRIS

Andreas Reinhart. 2024-10-29. A counterexample to the Conjecture of Ankeny, Artin and Chowla. https://arxiv.org/abs/2410.21864

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