INDEFINITE QUADRATIC FORMS AND PELL EQUATIONS INVOLVING QUADRATIC IDEALS

TEKCAN A.

MATHEMATICAL REPORTS, cilt.19, sa.2, ss.263-279, 2017 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 19 Sayı: 2
  • Basım Tarihi: 2017
  • Dergi Adı: MATHEMATICAL REPORTS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.263-279
  • Bursa Uludağ Üniversitesi Adresli: Evet

Özet

Let p equivalent to 1(mod 4) be a prime number, let gamma = P+root p/Q be a quadratic irrational, let I-gamma = [Q, P + root p] be a quadratic ideal and let F-gamma = (Q, 2P, -Q) be an indefinite quadratic form of discriminant Delta = 4p, where P and Q are positive integers depending on p. In this work, we first determined the cycle of I, and then proved that the right and left neighbors of F-gamma can be obtained from the cycle of I-gamma. Later we determined the continued fraction expansion of gamma, and then we showed that the continued fraction expansion of root P, the set of proper automorphisms of F-gamma, the fundamental solution of the Pell equation x(2) - py(2) = +/- 1 and the set of all positive integer solutions of the equation x(2) - py(2) = +/- p can be obtained from the continued fraction expansion of gamma.