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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.

References

Cheng, K. H. F., ‘Some results concerning periodic continued fractions’, Doctoral Dissertation, University of Calgary, Canada, 2003.Google Scholar
Cheng, K. H. F. and Williams, H. C., ‘Some results concerning certain periodic continued fractions’, Acta Arith. 117 (2005), 247–264.CrossRefGoogle Scholar
Cohen, H. and Lenstra, H. W. Jr, ‘Heuristics on class groups of number fields’, in: Number Theory (Noordwijkerhout 1983), Lecture Notes in Mathematics, 1068 (Springer, Berlin, 1984), pp. 33–62.Google Scholar
Jacobson, M. J. Jr and Williams, H. C., Solving the Pell Equation, CMS Books in Mathematics (Springer, New York, 2009).CrossRefGoogle Scholar
Kaplansky, I., ‘Letter to R. A. Mollin, H. C. Williams and K. S. Williams’, 1998.Google Scholar
Knuth, D., The Art of Computer Programming, vol. 2. Seminumerical algorithms, 2nd edn (Addison-Wesley, Reading, MA, 1981).Google Scholar
Patterson, R. D., ‘Creepers—real quadratic fields with large class number’, Doctoral Dissertation, Macquarie University, Sydney, 2003.Google Scholar
Patterson, R. D., van der Poorten, A. J. and Williams, H. C., ‘Characterization of a generalized Shanks sequence’, Pacific J. Math. 230 (2007), 185–215.CrossRefGoogle Scholar
Ribenboim, P., The Book of Prime Number Records (Springer, New York, 1988).CrossRefGoogle Scholar
Schinzel, A., ‘On some problems of the arithmetical theory of continued fractions’, Acta Arith. 6 (1961), 393–413.CrossRefGoogle Scholar
Schinzel, A., ‘On some problems of the arithmetical theory of continued fractions II’, Acta Arith. 7 (1962), 187–298.CrossRefGoogle Scholar
Schinzel, A., ‘Corrigendum to “On some problems of the arithmetical theory of continued fractions II”’, Acta Arith. 47 (1986), 295.Google Scholar
van der Poorten, A. J., ‘Beer and continued fractions with periodic periods’, in: Number Theory (Ottawa, ON, 1996), CRM Proceedings and Lecture Notes, 19 (American Mathematical Society, Providence, RI, 1999), pp. 309–314.Google Scholar
van der Poorten, A. J. and Williams, H. C., ‘On certain continued fraction expansions of fixed period length’, Acta Arith. 89 (1999), 23–35.CrossRefGoogle Scholar
Williams, H. C., ‘Some generalizations of the ${S}_{n} $ sequence of Shanks’, Acta Arith. 69 (1995), 199–215.CrossRefGoogle Scholar
Williams, H. C., Édouard Lucas and Primality Testing, Canadian Mathematical Society Series of Monographs and Advanced Texts, 22 (Wiley, New York, 1998).Google Scholar