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

Isikay S., PEKİN A.

INTERNATIONAL JOURNAL OF NUMBER THEORY, cilt.18, sa.03, ss.473-484, 2022 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 18 Sayı: 03
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1142/s1793042122500270
  • Dergi Adı: INTERNATIONAL JOURNAL OF NUMBER THEORY
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.473-484
  • İstanbul Üniversitesi Adresli: Evet

Özet

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.