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

Isikay S., PEKİN A.

INTERNATIONAL JOURNAL OF NUMBER THEORY, vol.18, no.03, pp.473-484, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 18 Issue: 03
  • Publication Date: 2022
  • Doi Number: 10.1142/s1793042122500270
  • Journal Name: INTERNATIONAL JOURNAL OF NUMBER THEORY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.473-484
  • Istanbul University Affiliated: Yes

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.