Fermat quotients and the Ankeny-Artin-Chowla conjecture

Abstract: In this article, we present streamlined proofs of results of Ankeny, Artin, and Chowla concerning the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{p})$ for primes $p\equiv 1 \bmod{4}$ while providing a generalization of their conjecture. Using our generalization, we relate Fermat quotients of quadratic non-residues $\bmod{p}$ to sums of harmonic numbers.