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ON THE OPTIMAL CONTINUED FRACTION EXPANSION OF A QUADRATIC SURD

Part of: Elementary number theory Computational number theory

Published online by Cambridge University Press:  07 February 2013

KEITH R. MATTHEWS
KEITH R. MATTHEWS*
Affiliation:
Department of Mathematics, University of Queensland, Brisbane 4072, Australia email keithmatt@gmail.com Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia email keithmatt@gmail.com
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Abstract

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We describe the period structure of the optimal continued fraction expansion of a quadratic surd, in terms of the period of its nearest square continued fraction expansion. The analysis results in a faster algorithm for determining the optimal continued fraction expansion of a quadratic surd.

Keywords

optimal continued fractionnearest square continued fractionquadratic surdperiod length

MSC classification

Secondary: 11A55: Continued fractions 11Y65: Continued fraction calculations

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 93 , Issue 1-2 , October 2012 , pp. 133 - 156
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc.

References

Ayyangar, A. A. K., ‘Theory of the nearest square continued fraction’, J. Mysore Univ. Sect. A 1 (1941), 97–117.Google Scholar
Bosma, W., ‘Optimal continued fractions’, Indag. Math. 49 (1987), 353–379.CrossRefGoogle Scholar
Bosma, W. and Kraaikamp, C., ‘Optimal approximation by continued fractions’, J. Aust. Math. Soc. Ser. A 50 (1991), 481–504.CrossRefGoogle Scholar
Koksma, J. F., Diophantische Approximation (Springer, Berlin, 1936).Google Scholar
Matthews, K. R., ‘On the nearest square continued fraction expansion of $(p+ q+ \sqrt{{p}^{2} + {q}^{2} } )/ p, p\gt 2q\gt 0$’, see http://www.numbertheory.org/continued_fractions.html.Google Scholar
Matthews, K. R. and Robertson, J. P., ‘Period length equality for the NICF and NSCF expansions of a quadratic surd’, Mat. Glasnik 46 (2011), 269–282.CrossRefGoogle Scholar
Selenius, C.-O., ‘Konstruktion und Theorie halbregelmässiger Kettenbrüche mit idealer relativer approximation’, Acta Acad. Abo. Math. Phys. 22 (1960), 3–77.Google Scholar