Results 1 to 1 of 1

Thread: Proof of Divisibilty propery of Fibonaccia numbers.

  1. #1
    Member
    Joined
    May 2009
    Posts
    109

    Proof of Divisibilty propery of Fibonaccia numbers.

    I'm struggling with understanding part of this proof.

    It makes use of this relationship.

    $\displaystyle F_{m + n} = F_{m-1}F_n + F_mF_{n + 1}$

    We are supposing that $\displaystyle F_m$ is dvisible by $\displaystyle F_n$ and we have just shown that $\displaystyle F_{nq}$ is divisible by $\displaystyle F_n$. We are trying to proove that m is divisible by n.

    So suppose $\displaystyle m = nq + r$

    We have $\displaystyle F_m = F_{nq+r} = F_{nq-1}F_{r} + F_{nq}F_{r+1}$

    As $\displaystyle F_{nq} $ is divisible by$\displaystyle F_n$ and as $\displaystyle F_m$ is divisible by $\displaystyle F_n$, it follows that $\displaystyle F_n$ divides $\displaystyle F_{nq-1}F_r$

    It does? Doesn't the fact that $\displaystyle F_{nq}$ alone mean that $\displaystyle F_n $ divides $\displaystyle F_{nq-1}F_r$? Why do we have to bother with $\displaystyle F_m$?

    Then we have

    if $\displaystyle gcd(F_{nq-1}, F_n) = 1$ we know by Euclid's Lemma that$\displaystyle F_n$ divides$\displaystyle F_r$. I thought this only work with a prime number? How do we know one of them is prime?
    Last edited by alyosha2; Dec 2nd 2013 at 06:31 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: Nov 1st 2012, 02:29 PM
  2. proof of fermat's divisibilty examination problem?
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: Nov 27th 2011, 05:22 AM
  3. Proof: All rational numbers are algebraic numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Sep 5th 2010, 10:26 AM
  4. Replies: 4
    Last Post: Feb 27th 2010, 02:44 PM
  5. Divisibilty proofs?
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Oct 27th 2008, 05:31 PM

Search Tags


/mathhelpforum @mathhelpforum