A counterexample to the Conjecture of Ankeny, Artin and Chowla

Abstract

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