A counterexample to the Conjecture of Ankeny, Artin and Chowla

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