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Math Help - A near-integer expression

  1. #1
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    A near-integer expression

    Computation shows that

    f(n) = \left( 1 + 2 \cos(\frac{\pi}{9}) \right) ^ n

    is, for "large n" (10 or greater, say), just a little less than an integer. For example,

    f(10) \approx 39173.98
    f(20) \approx 1534601033.9998
    f(30) \approx 60116436578864.999997

    Why is this? (I do not know the answer.)

    A related thread is here:

    http://www.mathhelpforum.com/math-he...r-divisib.html
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  2. #2
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    If the integer part of x is denoted by [tex][x][/Math] fractional part of x is denoted by [LaTeX ERROR: Convert failed] , then you can see the following:

    [LaTeX ERROR: Convert failed]

    [LaTeX ERROR: Convert failed]

    [LaTeX ERROR: Convert failed]

    [LaTeX ERROR: Convert failed]

    [LaTeX ERROR: Convert failed]

    You can formulate it as a statement: f(n) gets arbitrarily close to an integer as n goes to infinity. Then maybe you can prove it. This is better than what you state, because we not only see that the function is close to an integer for big n, but it's getting closer and closer to an integer.


    [LaTeX ERROR: Convert failed]
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  3. #3
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    Here's an outline of a proof. No time for details because I'm about to go away for a week's holiday (again that's the advantage of being retired).

    The number \alpha = 1+2\cos(2\pi/9) satisfies the equation z^3-3z^2+1=0 (see here). The other two roots of this equation, call them \beta and \gamma, satisfy |\beta|<1 and |\gamma}<1.

    Define a sequence of integers by x_1=3,\;x_2=9,\;x_3=24 and x_n = 3x_{n-1}-x_{n-3} for n>3.

    Then the theory of difference equations shows that x_n = \alpha^n + \beta^n + \gamma^n. Since |\beta^n|\to0 and |\gamma^n|\to0 as n gets large, it follows that \alpha^n is only slightly different from an integer.
    Last edited by Opalg; August 1st 2010 at 02:51 PM.
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  4. #4
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    Wow, that's so neat! Is the following true?

    Let the solutions to a polynomial equation be [LaTeX ERROR: Convert failed] for k from 1 to n. Then [LaTeX ERROR: Convert failed] .

    EDIT: Never mind. I found some easy counterexamples.
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  5. #5
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    I found something very interesting. (1+2cos(pi/9)) is what's called a PV number. Pisot . Check it out!

    PV numbers are such numbers whose powers reach integers very fast, like the one we have here.
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