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In the search of convergents to 23

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  • Petek, Peter
  • Lakner, Mitja
  • Škapin Rugelj, Marjeta

Abstract

Application of continued fractions in high energy physics is well known, especially via the K.A.M. theorem and mostly for quadratic irrationals. Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow Kuzmin’s probability law. Here a combinatorial approach to the search of convergents is presented. We resort to the adjunction ring Z(23), representing its elements in the irrational basis ρ=1+23+43.

Suggested Citation

  • Petek, Peter & Lakner, Mitja & Škapin Rugelj, Marjeta, 2009. "In the search of convergents to 23," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 811-817.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:2:p:811-817
    DOI: 10.1016/j.chaos.2008.04.004
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    References listed on IDEAS

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    1. Marek Crnjac, L., 2005. "Periodic continued fraction representations of different quark’s mass ratios," Chaos, Solitons & Fractals, Elsevier, vol. 25(4), pages 807-814.
    2. El Naschie, M.S., 2006. "Elementary number theory in superstrings, loop quantum mechanics, twistors and E-infinity high energy physics," Chaos, Solitons & Fractals, Elsevier, vol. 27(2), pages 297-330.
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