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Continued fractions in 2-stage Euclidean quadratic fields
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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.
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GUITART MORALES, Xavier, MASDEU, Marc. Continued fractions in 2-stage Euclidean quadratic fields. _Mathematics of Computation_. 2013. Vol. 82, núm. 282, pàgs. 1223-1233. [consulta: 25 de febrer de 2026]. ISSN: 0025-5718. [Disponible a: https://hdl.handle.net/2445/193371]