# Relation between Fibonacci Numbers and Pythagorean Triples

• Jun 6th 2012, 02:57 AM
Sarasij
Relation between Fibonacci Numbers and Pythagorean Triples
The problem is as follows:
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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
• Jun 6th 2012, 07:36 AM
BobP
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.
• Jun 6th 2012, 08:02 AM
Sarasij
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... (Clapping)