On Yokoi’s Invariants and the Ankeny–Artin–Chowla conjecture

Abstract

Let [Formula: see text] be a positive square-free integer and [Formula: see text] be the fundamental unit of the real quadratic field [Formula: see text]. The Ankeny–Artin–Chowla (AAC) conjecture asserts that [Formula: see text] for primes [Formula: see text], which still remains unsolved. In this paper, sufficient conditions for [Formula: see text] have been given in terms of Yokoi’s invariants [Formula: see text] and [Formula: see text], and it has been shown that the AAC conjecture is true in some special cases.