For a large number x>0, how many Fibonacci numbers are there in the interval [1,x]? I have saw the corresponding results in certain places but I have forgotten now. Can anyone help me? Thanks!
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
closed as too localized by Gjergji Zaimi, Yemon Choi, Wadim Zudilin, Felipe Voloch, S. Carnahan♦ Oct 11 '10 at 11:22This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question. 


Let $G:=(1+\sqrt{5})/2;g:=(1\sqrt{5})/2$, then the $n$th Fibonacci number is $\frac{1}{\sqrt{5}}(G^ng^n)$. Note that $g<1$. Hence the number of Fibonacci numbers $\le x$ is $\frac{\log \sqrt{5}x}{\log G}$ plus or minus 1 (and it is easy to see when you need to add or subtract 1). 

