On the possible quantities of Fibonacci numbers that occur in some type of intervals

Abstract

In this paper, we show that for any integer $a\ge 2$, each of the intervals $[a^k,a^{k+1})$ ($k\in \mathbb{N}$) contains either $\lfloor \frac{\log a}{\log \Phi }\rfloor$ or $\lceil \frac{\log a}{\log \Phi }\rceil$ Fibonacci numbers. In addition, the density (in $\mathbb{N}$) of the set of the all natural numbers $k$ for which the interval $[a^k,a^{k+1})$ contains exactly $\lfloor \frac{\log a}{\log \Phi }\rfloor$ Fibonacci numbers is equal to $\left(1-⟨\frac{\log a}{\log \Phi }⟩\right)$ and the density of the set of the all natural numbers $k$ for which the interval $[a^k,a^{k+1})$ contains exactly $\lceil \frac{\log a}{\log \Phi }\rceil$ Fibonacci numbers is equal to $⟨\frac{\log a}{\log \Phi }⟩$.

arXiv comment10 pages