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

19 pages