Representation of real numbers by means of Fibonacci numbers (Q1072582)

The Fibonacci sequence is defined by the recurrence relation \(F_ n=F_{n-1}+F_{n-2}\) for \(n\geq 3\) with \(F_ 1=F_ 2=1\). Define \(I_ 0=0\) and if \(m\geq 1\) let \(I_ m=\sum^{\infty}_{n=1}1/F_ n^{1/m}.\) It is not difficult to show that \(I_ m\) is a convergent series and that \(I_ 1<I_ 2<I_ 3<..\). with \(\lim_{m\to \infty}I_ m=\infty.\) In the present paper it is proved that if x is a positive real number then a unique m exists such that \(x=\sum^{\infty}_{j=1}1/F_{i_ j}^{1/m},\) but x is not of the form \(\sum^{\infty}_{j=1}1/F_{i_ j}^{1/(m-1)}.\) Here, \(1\leq i_ 1<i_ 2<..\). is a sequence of positive integers. The proof depends on a result due to Kakeya concerning conditions which imply that a given positive real number is ''representable'' by a decreasing sequence of positive numbers with a limit of zero. The paper concludes with a proof, due originally to Landau, that \(\sum^{\infty}_{n=1}1/F_{2n}\quad and\quad \sum^{\infty}_{n=1}1/F_{2n+1}\) can be expressed in terms of Lambert series and Jacobi theta series, respectively.