Fermat Quotients and the Ankeny–Artin–Chowla Conjecture

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 \(p\equiv 1 \left( \textrm{mod}\ 4\right) \) while providing a generalization of their conjecture. Using our generalization, we relate Fermat quotients of quadratic non-residues \(\left( \textrm{mod}\ p\right) \) to sums of harmonic numbers.