Fibonacci along even powers is (almost) realizable
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2020
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866913992231878656 |
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| author | Moss, Patrick Ward, Tom |
| author_facet | Moss, Patrick Ward, Tom |
| contents | An integer sequence is called realizable if it is the count of periodic points of some map. The Fibonacci sequence $(F_n)$ does not have this property, and the Fibonacci sequence sampled along the squares $(F_{n^2})$ also does not have this property. We prove that this is an arithmetic phenomenon related to the discriminant of the Fibonacci sequence, by showing that the sequence $(5F_{n^2})$ is realizable. More generally, we show that $(F_{n^{2k-1}})$ is not realizable in a particularly strong sense while $(5F_{n^{2k}})$ is realizable, for any $k\ge1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_13068 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Fibonacci along even powers is (almost) realizable Moss, Patrick Ward, Tom Number Theory Dynamical Systems An integer sequence is called realizable if it is the count of periodic points of some map. The Fibonacci sequence $(F_n)$ does not have this property, and the Fibonacci sequence sampled along the squares $(F_{n^2})$ also does not have this property. We prove that this is an arithmetic phenomenon related to the discriminant of the Fibonacci sequence, by showing that the sequence $(5F_{n^2})$ is realizable. More generally, we show that $(F_{n^{2k-1}})$ is not realizable in a particularly strong sense while $(5F_{n^{2k}})$ is realizable, for any $k\ge1$. |
| title | Fibonacci along even powers is (almost) realizable |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2011.13068 |