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Periodicity of certain generalized continued fractions

  • Steven H. WeintraubEmail author
Article
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Abstract

We have previously considered continued fractions with “numerator” a positive integer N, which we refer to as \({\text {cf}}_N\) expansions. In particular, let E be a positive integer that is not a perfect square. For \(N > 1\), \(\sqrt{E}\) has infinitely many \({\text {cf}}_N\) expansions. There is a natural notion of the “best” \({\text {cf}}_N\) expansion of \(\sqrt{E}\). We have conjectured, based on extensive numerical evidence, that such a best expansion is not always periodic. From this evidence, it is difficult to predict for which N this expansion will be periodic. We show here that for any such E, there are infinitely many values of N for which this expansion is indeed periodic, more precisely, periodic of period 1 or 2, and we obtain formulas for a subset of these expansions in terms of solutions to Pell’s equation \(x^2 - Ey^2 = 1\).

Keywords

Continued fractions Periodicity Pell’s equation 

Mathematics Subject Classification

11A55 

Notes

Reference

  1. 1.
    Anselm, M., Weintraub, S.H.: A generalization of continued fractions. J. Number Theory 131, 2442–2460 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsLehigh UniversityBethlehemUSA

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