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 \geq 2, each of the intervals[a^k; a^{k+1}) (k \in N) contains either ⌊\log a/\log\Phi ⌋ or ⌈\log a/\log\Phi⌉ Fibonacci numbers. In addi-tion, the density (in N) of the set of the all natural numbers k for which the interval[a^k; a^{k+1}) contains exactly ⌊\log a/\log\Phi ⌋ Fibonacci numbers is equal to (1 - \langle\log a/\log\Phi\rangle) and the density of the set of the all natural numbers k for which the interval[a^k; a^{k+1}) contains exactly ⌈\log a/\log\Phi⌉ Fibonacci numbers is equal to \langle\log a/\log\Phi\rangle.