Markoff $m$-triples with $k$-Fibonacci components
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913511986167808 |
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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 $k$-Fibonacci components to the equation $x^2+y^2+z^2=3xyz+m,$ where $m$ is a positive integer and $k\geq 2$. As a result, for $m=8$, we have the Markoff triples with Pell components $(F_2(2), F_2(2n), F_2(2n+2))$, for $n\geq 1$. For all other $m$ there exists at most one such ordered triple, except when $k=3,$ $a$ is odd, $b$ is even and $b\geq a+3$, where $(F_3(a),F_3(b),F_3(a+b))$ and $(F_3(a+1),F_3(b-1),F_3(a+b))$ share the same $m$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_13885 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Markoff $m$-triples with $k$-Fibonacci components Alfaya, D. Calvo, L. A. de Guinea, A. Martínez Rodrigo, J. Srinivasan, A. General Mathematics We classify all solution triples with $k$-Fibonacci components to the equation $x^2+y^2+z^2=3xyz+m,$ where $m$ is a positive integer and $k\geq 2$. As a result, for $m=8$, we have the Markoff triples with Pell components $(F_2(2), F_2(2n), F_2(2n+2))$, for $n\geq 1$. For all other $m$ there exists at most one such ordered triple, except when $k=3,$ $a$ is odd, $b$ is even and $b\geq a+3$, where $(F_3(a),F_3(b),F_3(a+b))$ and $(F_3(a+1),F_3(b-1),F_3(a+b))$ share the same $m$. |
| title | Markoff $m$-triples with $k$-Fibonacci components |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2409.13885 |