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Thread: Fibonacci numbers.

  1. #1
    Senior Member I-Think's Avatar
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    Fibonacci numbers.

    The Fibonacci numbers are defined by $\displaystyle F_0=0$,$\displaystyle F_1=1$ and $\displaystyle F_n=F_{n-1}+F_{n-2}$
    Prove if $\displaystyle 2|F_n$, then $\displaystyle 3|n$

    This is one of those times where it's easy to see intuitively, it's hard to see formally.
    Considering the Fibonacci sequence
    $\displaystyle 0,1,1,2,3,5,8,13,21,36...$

    It falls into a pattern of
    $\displaystyle E,O,O,E,O,O,E,O,O,E,...$

    From, as the first term $\displaystyle F_0$ is even, it's clear to see that every third term beginning from $\displaystyle F_1$ is divisible by 3, as it is the sum of the 2 previous numbers which are odd.

    But this just scaffolding, scrap work.
    Any tips on how to codify this formally?
    Last edited by mr fantastic; Oct 25th 2010 at 05:37 PM. Reason: Re-titled.
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  2. #2
    Pim
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    Given the recursive nature of Fibonacci numbers, a proof by induction seems to be the logical way to go. Have you considered that?
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