Indefinite quadratic forms and pell equations involving quadratic ideals

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.