Continued fractions in 2-stage euclidean quadratic fields

TL;DR

This paper presents an algorithm to verify whether real quadratic fields of class number 1 are 2-stage euclidean, enabling continued fraction computations and expanding the list of known such fields, especially those with discriminant less than 8000.

Contribution

The paper introduces an algorithm that confirms the 2-stage euclidean property for real quadratic fields of class number 1 and provides a method to compute continued fractions in these fields.

Findings

All real quadratic fields of class number 1 with discriminant less than 8000 are 2-stage euclidean.

The algorithm can verify the 2-stage euclidean property for given fields.

Efficient computation of continued fraction expansions is possible in these 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.