INDEFINITE QUADRATIC FORMS AND PELL EQUATIONS INVOLVING QUADRATIC IDEALS

TEKCAN A.

MATHEMATICAL REPORTS, vol.19, no.2, pp.263-279, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 19 Issue: 2
  • Publication Date: 2017
  • Journal Name: MATHEMATICAL REPORTS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.263-279
  • Bursa Uludag University Affiliated: Yes

Abstract

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.