×

A new characterization of the Fibonacci-free partition. (English) Zbl 0729.11048

The reviewer along with P. Erdős and V. E. Hoggatt, jun. proved the following result [Discrete Math. 22, 201-211 (1978; Zbl 0376.10011)]: There exists a unique partition of the set of natural numbers into two subsets A and B such that no two elements of the same subset add up to a Fibonacci number. The author calls such a partition a Fibonacci-free partition. The author proves that A and B are given by \[ A=\{[n\Phi]\}-\{[m\Phi]| \quad fp(m\Phi)>\Phi /2\}, \] and \[ B=\{[n\Phi^ 2]\}\cup \{[m\Phi]| \quad fp(m\Phi)>\Phi /2\}. \] Here [ ] denotes the greatest integer function and fp(x) is the fractional part of x. Also \(\Phi =(1+\sqrt{5})/2\) is the golden mean. In proving this result, the author also proves a conjecture of Chris Long that \(A=\{[n\Phi]\}-A'\), where \(A'=\{[s\Phi^ 3]|\) \(s\in A\}\).

MSC:

11P81 Elementary theory of partitions
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B34 Representation functions
11B83 Special sequences and polynomials

Citations:

Zbl 0376.10011
PDFBibTeX XMLCite