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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.08509 |
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| _version_ | 1866912207977054208 |
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| author | Alfaya, D. Calvo, L. A. de Guinea, A. Martínez Rodrigo, J. Srinivasan, A. |
| author_facet | Alfaya, D. Calvo, L. A. de Guinea, A. Martínez Rodrigo, J. Srinivasan, A. |
| contents | We classify all solution triples with Fibonacci components to the equation $a^2+b^2+c^2=3abc+m,$ for positive $m$. We show that for $m=2$ they are precisely $(1,F(b),F(b+2))$, with even $b$; for $m=21$, there exist exactly two Fibonacci solutions $(1,2,8)$ and $(2,2,13)$ and for any other $m$ there exists at most one Fibonacci solution, which, in case it exists, is always minimal (i.e. it is a root of a Markoff tree). Moreover, we show that there is an infinite number of values of $m$ admitting exactly one such solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A classification of Markoff-Fibonacci m-triples Alfaya, D. Calvo, L. A. de Guinea, A. Martínez Rodrigo, J. Srinivasan, A. Number Theory We classify all solution triples with Fibonacci components to the equation $a^2+b^2+c^2=3abc+m,$ for positive $m$. We show that for $m=2$ they are precisely $(1,F(b),F(b+2))$, with even $b$; for $m=21$, there exist exactly two Fibonacci solutions $(1,2,8)$ and $(2,2,13)$ and for any other $m$ there exists at most one Fibonacci solution, which, in case it exists, is always minimal (i.e. it is a root of a Markoff tree). Moreover, we show that there is an infinite number of values of $m$ admitting exactly one such solution. |
| title | A classification of Markoff-Fibonacci m-triples |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.08509 |