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CLASSIFICATION AND SYMMETRIES OF A FAMILY OF CONTINUED FRACTIONS WITH BOUNDED PERIOD LENGTH

Part of: Elementary number theory

Published online by Cambridge University Press:  03 May 2013

K. H. F. CHENG ,
R. K. GUY ,
R. SCHEIDLER  and
H. C. WILLIAMS
K. H. F. CHENG*
Affiliation:
Department of Mathematics and Information Technology, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong
R. K. GUY
Affiliation:
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 email rkg@cpsc.ucalgary.ca
R. SCHEIDLER
Affiliation:
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 email rscheidl@ucalgary.ca
H. C. WILLIAMS
Affiliation:
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 email hwilliam@ucalgary.ca
*
✉khfcheng@ied.edu.hk
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Abstract

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It is well known that the regular continued fraction expansion of a quadratic irrational is symmetric about its centre; we refer to this symmetry as horizontal. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a family of quadratics known as Schinzel sleepers. This paper provides a method for generating every Schinzel sleeper and investigates their period lengths as well as both their horizontal and vertical symmetries.

Keywords

Schinzel sleepercontinued fraction expansionperiod length

MSC classification

Secondary: 11A55: Continued fractions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 93 , Issue 1-2 , October 2012 , pp. 53 - 76
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc.

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