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Continued fractions for (ir)rationals

  1. May 29, 2007 #1

    CRGreathouse

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    From http://web.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub049.html :

    "Using one of the algorithms, which is based on an identity involving Bessel functions, gamma has been computed to 30,100 decimal places. By computing their regular continued fractions, we show that, if gamma or exp(gamma) is of the form P/Q for integers P and Q, then [itex]|Q|>10^{15000}[/itex]."

    The method of getting to this result is not mentioned in this paper, but an earlier paper by one of its authors says this:

    "Let [itex]x=\gamma\textrm{ or }\exp(\gamma)[/itex]. From Theorem 17 of [15], [itex]|Q_nx-P_n|\le|Qx-P|[/itex] for all integers P and Q with [itex]0<|Q|\le Q_n[/itex]. Using [itex]q_1,\ldots,q_{20000}[/itex], we find [itex]Q_{20000}(\gamma)=5.6\ldots\times10^{10328}[/itex] and [itex]Q_{20000}(\exp(\gamma)=3.3\ldots\times10^{10293}[/itex]. Hence, we have the following result, which makes it highly unlikely that [itex]x=\gamma\textrm{ or }\exp(\gamma)[/itex] is rational.
    THEOREM. If [itex]x=\gamma\textrm{ or }\exp(\gamma)[/itex] = P/Q for integers P and Q, then [itex]|Q|>10^{10000}[/itex]."

    The paper referenced is "A. Ya. KHINTCHINE (A. Ja. HINČIN), Continued Fractions, 3rd ed., (English transl. by P. Wynn), Noordhoff, Groningen, 1963. MR 28 #5038.". The author's name is now usually spelled Khinchin (of Khinchin's Constant fame).


    I'm trying to find that result (possibly even with a proof) which I take to be a basic result. Can anyone state it or give a common name for it?
     
  2. jcsd
  3. May 29, 2007 #2

    Chris Hillman

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    That's not a paper, its a book:

    A. Ya. Khinchin, Continued Fractions, Dover reprint, 1997

    You can purchase it for a five dollars American or something like that. I recommend that you do just that because its a beautiful book. Make sure to look for the wonderful connections with ergodic theory!
     
    Last edited: May 29, 2007
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