On the structure of the complement $\overline{\Mfib}$ of the set $\Mfib$ of fibbinary numbers in the set of positive natural numbers
| Autor: | Macfarlane, A. J. |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 8 pages, 4 Tables |
| Databáze: | arXiv |
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