perplexus.info :: Sequences : Fibonacci yields Pythagoras

Fibonacci yields Pythagoras (Posted on 2016-07-05)

Take four consecutive Fibonacci numbers: a,b,c,d.

Show that a*d and 2*b*c are the two legs of a Pythagorean Triple.

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Solution

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c=a+b and d=a+2b

Then a*d=a^2+2ab and 2*b*c=2ab+2b^2

Then (a^2+2ab)^2 + (2ab+2b^2)^2 = a^4+4a^3b+8a^2b^2+8ab^3+2b^4 = (a^2+2ab+2b^2)^2

Then the hypotenuse can be expressed as c*d-a*b.

Therefore {a*d, 2*b*c, c*d-a*b} is a Pythagorean Triple

Posted by Brian Smith on 2016-07-05 12:58:14

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