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Math Help - Relation between Fibonacci Numbers and Pythagorean Triples

  1. #1
    Junior Member Sarasij's Avatar
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    Relation between Fibonacci Numbers and Pythagorean Triples

    The problem is as follows:
    ------------------------------------

    We all know that this is the Fibonacci recusion.

    Fn = Fn-1 + Fn-2 , F0 = 1 , F1​ = 1

    How to prove that the following triple (x,y,z) follow Pythagorean Theorem ?

    x = (Fi) x (Fi+3)

    y = 2 x (Fi+1) x (Fi+2)

    z = (Fi+1)2 + (Fi+2)2
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  2. #2
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    Re: Relation between Fibonacci Numbers and Pythagorean Triples

    Putting everything in terms of just two of the numbers and expanding both sides (of x^{2}+y^{2}=z^{2}) works.

    F_{i+3}=F_{i+2}+F_{i+1}= F_{i+2}+(F_{i+2}-F_{i})=2F_{i+2}-F_{i}

    and

    F_{i+1}=F_{i+2}-F_{i}.

    Substitute and expand both sides, they turn out to be identical.
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  3. #3
    Junior Member Sarasij's Avatar
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    Re: Relation between Fibonacci Numbers and Pythagorean Triples

    Thanks but I've found another simple proof for this...

    Fi+3 = Fi+2 + Fi+1

    and

    x = Fi x Fi+3 = (Fi+2 - Fi+1) x (Fi+2 + Fi+1) = b2 - a2 where

    b = Fi+2 and a = Fi+1.

    Now (b2 - a2)2 + (2ab)2 = (b2 + a2)2

    Hence proved...

    Anyways thanks for your proof too...
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