When Fibonacci meets Pythagoras

There is a cool identity relating Fibonacci numbers and Pythagorean triples, which states that given four consecutive Fibonacci numbers Fn,Fn+1,Fn+2,Fn+3{F}_{n},{F}_{n+1},{F}_{n+2},{F}_{n+3}, the triple (FnFn+3,2Fn+1Fn+2,Fn+12+Fn+22)({F}_{n}{F}_{n+3}, 2{F}_{n+1}{F}_{n+2}, {{F}_{n+1}}^{2}+{{F}_{n+2}}^{2}) is a Pythagorean triple.

To prove this, let Fn+1=a{F}_{n+1} = a and Fn+2=b{F}_{n+2} = b. Then Fn=ba{F}_{n} = b-a and Fn+3=b+a{F}_{n+3} = b+a. Substituting the abstract values to the equation (FnFn+3)2+(2Fn+1Fn+2)2=(Fn+12+Fn+22)2{({F}_{n} {F}_{n+3})}^{2} +{(2{F}_{n+1} {F}_{n+2})}^{2} = {({{F}_{n+1}}^{2}+{ {F}_{n+2}}^{2})}^{2} will give ((ba)(b+a))2+(2ab)2=(a2+b2)2.{((b-a)(b+a))}^{2} +{(2ab)}^{2}={({a}^{2}+{b}^{2})}^{2}.

With a little simplification, we arrive at Euclid's formula for Pythagorean triples: (b2a2)2+(2ab)2=(a2+b2)2.{({b}^{2}-{a}^{2})}^{2} +{(2ab)}^{2}={({a}^{2}+{b}^{2})}^{2}.

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
4 years, 9 months ago

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Nice post

Is there any generalized formula to generate pythagoream triples

Aman Sharma - 4 years, 9 months ago

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Euclid's Formula. The link above should bring you to a proof.

Steven Zheng - 4 years, 9 months ago

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How we can solve problems in which product of three pythagorean triples is given and we have to find the triple

Aman Sharma - 4 years, 9 months ago

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@Aman Sharma I'm not sure if that is solvable without more given conditions. You might also need the sum of sides or maybe the radius of an incircle also. Can you come up with an example of your question?

Steven Zheng - 4 years, 9 months ago

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@Steven Zheng Find out the pythagorean triple (x,y,z)(x,y,z) such that xyz=16320xyz=16320

Aman Sharma - 4 years, 9 months ago

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@Aman Sharma Number of solutions a,ba,b such that 2ab(a2+b2)(a2b2)=163202ab({a}^{2}+{b}^{2})({a}^{2}-{b}^{2}) = 16320.

Steven Zheng - 4 years, 9 months ago

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@Steven Zheng Does there exist more the one triples.....also i read on a website that euclid's formula can't generate all pythagorean triples

Aman Sharma - 4 years, 9 months ago

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@Aman Sharma Euclid's formula deal with generating Pythagorean triples that are integers.

Steven Zheng - 4 years, 9 months ago

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Good

Ayanlaja Adebola - 4 years, 9 months ago

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Nice post Is there any generalized formula to generate pythagoream triples.(?) There are several. There is another as yet unpublished formula which will bring an additional as yet unconsidered aspect of Pythagoras.

mike bridger - 1 year, 10 months ago

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