Continued fractions in 2-stage euclidean quadratic fields

Abstract

We discuss continued fractions on real quadratic number fields of class number 1. If the field has the property of being 2-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions. Although it is conjectured that all real quadratic fields of class number 1 are 2-stage euclidean, this property has been proven for only a few of them. The main result of this paper is an algorithm that, given a real quadratic field of class number 1, verifies this conjecture, and produces as byproduct enough data to efficiently compute continued fraction expansions. If the field was not 2-stage euclidean, then the algorithm would not terminate. As an application, we enlarge the list of known 2-stage euclidean fields, by proving that all real quadratic fields of class number 1 and discriminant less than 8000 are 2-stage euclidean.