A counterexample to the Conjecture of Ankeny, Artin and Chowla

TL;DR

This paper presents a counterexample to the longstanding Ankeny-Artin-Chowla Conjecture, which claims that for primes p ≡ 1 mod 4, p does not divide the integer y associated with the fundamental unit in a quadratic field.

Contribution

The paper provides the first explicit counterexample to the conjecture, challenging previously held assumptions in algebraic number theory.

Findings

Counterexample disproves the conjecture

p divides y in the counterexample case

Implications for related number theory conjectures

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.