Condition for a number to be a Fibonacci one

vendredi 30 mai 2014

Hi all, I found the following statement on a magazine page and cannot understand it. It is possibly very distant from the little maths I know, but made me very curious.

It is therein said that if a number $$n$$ is a Fibonacci number, then one of the conditions $$ 5n^2 + 4$$ or $$5n^2-4$$ is true.

The conclusion follows from the following relationship, where $$A_n$$ is the n-th number in the Fibonacci sequence.

$$n = log_{golden ratio} \frac{A_n \sqrt{5} + \sqrt{5A_n^2 +-4}}{2}$$.

I am unable to see how the conclusion is drawn, any hint would be most appreciated.

Thanks a lot





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