Continued fractions in $2$-stage Euclidean quadratic fields
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- by Xavier Guitart and Marc Masdeu PDF
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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.References
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Additional Information
- Xavier Guitart
- Affiliation: Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
- MR Author ID: 887813
- Email: xevi.guitart@gmail.com
- Marc Masdeu
- Affiliation: Department of Mathematics, Columbia University, New York, New York 10001
- MR Author ID: 772165
- Email: masdeu@math.columbia.edu
- 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.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 82 (2013), 1223-1233
- MSC (2010): Primary 13F07, 11A55
- DOI: https://doi.org/10.1090/S0025-5718-2012-02620-2
- MathSciNet review: 3008856