On the structure of the complement $\overline{\Mfib}$ of the set $\Mfib$ of fibbinary numbers in the set of positive natural numbers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929390890254336 |
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| author | Macfarlane, A. J. |
| author_facet | Macfarlane, A. J. |
| contents | 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 $\overline{\Mfib}=\bigcup_{k \geq 1}^\infty Φ_k$ is obtained in terms of certain well-defined sets $Φ_k, \; k \geq 1$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the structure of the complement $\overline{\Mfib}$ of the set $\Mfib$ of fibbinary numbers in the set of positive natural numbers Macfarlane, A. J. Number Theory 11A63, 11A67 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 $\overline{\Mfib}=\bigcup_{k \geq 1}^\infty Φ_k$ is obtained in terms of certain well-defined sets $Φ_k, \; k \geq 1$. 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. |
| title | On the structure of the complement $\overline{\Mfib}$ of the set $\Mfib$ of fibbinary numbers in the set of positive natural numbers |
| topic | Number Theory 11A63, 11A67 |
| url | https://arxiv.org/abs/2406.12455 |