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Continued fractionsin $ 2$-stage Euclidean quadratic fields


Authors: Xavier Guitart and Marc Masdeu
Journal: Math. Comp. 82 (2013), 1223-1233
MSC (2010): Primary 13F07, 11A55
Published electronically: October 15, 2012
MathSciNet review: 3008856
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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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Additional Information

Xavier Guitart
Affiliation: Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
Email: xevi.guitart@gmail.com

Marc Masdeu
Affiliation: Department of Mathematics, Columbia University, New York, New York 10001
Email: masdeu@math.columbia.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02620-2
Received by editor(s): June 4, 2011
Received by editor(s) in revised form: September 6, 2011
Published electronically: October 15, 2012
Additional Notes: This work was partially supported by Grants MTM2009-13060-C02-01 and 2009 SGR 1220.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.