On Yokoi's Invariants and the Ankeny-Artin-Chowla conjecture

Abstract

Let d be a positive square-free integer and epsilon(d) = (T-d + U-d root d)/2 > 1 be the fundamental unit of the real quadratic field Q(root d). The Ankeny-Artin-Chowla (AAC) conjecture asserts that Up not equivalent to 0 (mod p) for primes p equivalent to 1 (mod 4), which still remains unsolved. In this paper, sufficient conditions for U-d < d have been given in terms of Yokoi's invariants n(d) and m(d), and it has been shown that the AAC conjecture is true in some special cases.