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ON THE OPTIMAL CONTINUED FRACTION EXPANSION OF A QUADRATIC SURD

  • KEITH R. MATTHEWS (a1) (a2)
Abstract
Abstract

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.

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References
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Bosma W., ‘Optimal continued fractions’, Indag. Math. 49 (1987), 353379.
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Koksma J. F., Diophantische Approximation (Springer, Berlin, 1936).
Matthews K. R., ‘On the nearest square continued fraction expansion of inline-graphic$(p+ q+ \sqrt{{p}^{2} + {q}^{2} } )/ p, p\gt 2q\gt 0$’, see http://www.numbertheory.org/continued_fractions.html.
Matthews K. R. and Robertson J. P., ‘Period length equality for the NICF and NSCF expansions of a quadratic surd’, Mat. Glasnik 46 (2011), 269282.
Selenius C.-O., ‘Konstruktion und Theorie halbregelmässiger Kettenbrüche mit idealer relativer approximation’, Acta Acad. Abo. Math. Phys. 22 (1960), 377.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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