Continued fractionsin stage Euclidean quadratic fields
Authors:
Xavier Guitart and Marc Masdeu
Journal:
Math. Comp. 82 (2013), 12231233
MSC (2010):
Primary 13F07, 11A55
Published electronically:
October 15, 2012
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Abstract: We discuss continued fractions on real quadratic number fields of class number . If the field has the property of being 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 are 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 , verifies this conjecture, and produces as byproduct enough data to efficiently compute continued fraction expansions. If the field was not stage euclidean, then the algorithm would not terminate. As an application, we enlarge the list of known stage euclidean fields, by proving that all real quadratic fields of class number and discriminant less than are 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/S002557182012026202
PII:
S 00255718(2012)026202
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 MTM200913060C0201 and 2009 SGR 1220.
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
