On the structure of the complement $\overline{\Mfib}$ of the set $\Mfib$ of fibbinary numbers in the set of positive natural numbers

TL;DR

This paper investigates the structure of the complement of fibbinary numbers within natural numbers, providing an explicit union-based expression and analyzing binary representations of odd numbers to understand their distribution.

Contribution

It introduces an explicit union expression for the complement of fibbinary numbers and develops a detailed binary representation analysis of odd numbers.

Findings

Explicit union expression for the complement of fibbinary numbers

Partitioning of natural numbers based on fibbinary structure

Binary representation analysis of odd numbers

Abstract

The set $\Mfib$ of fibbinary numbers is defined via a bijection between the set $\BB{N}$ of natural numbers and $\Mfib$. Since the elements of $\Mfib$ do not exhaust $\BB{N}$, the structure of the complement $\overline{\Mfib}$ of $\Mfib$ in $\BB{N}$ is of interest. An explicit expression $\overline{\Mfib}=\bigcup_{k \geq 1}^\infty \Phi_k$ is obtained in terms of certain well-defined sets $\Phi_k, \; k \geq 1$. The key to its proof lies in first considering the odd numbers involved in this statement: a general treatment, with full justification, of the binary representations of the odd numbers is developed, and exploited in showing the expression quoted for $\overline{\Mfib}$ to be correct. The main results of the article can also be viewed as providing partitions of the set $\BB{N}$ of natural numbers, and also of its subset of odd numbers, that follow from the introduction of the set $\Mfib$, and of its subset of odd integers.