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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.23306 |
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| _version_ | 1866908909833289728 |
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| author | Alfaya, David Calvo, Luis Ángel Cazorla, Pedro-José Rodrigo, Javier Srinivasan, Anitha |
| author_facet | Alfaya, David Calvo, Luis Ángel Cazorla, Pedro-José Rodrigo, Javier Srinivasan, Anitha |
| contents | We study infinite paths of Markoff $m$-triples, that is, solutions to the generalised Markoff equation \[ x^2+y^2+z^2=3xyz+m, \] with $m>0$, with at least two $k$-Fibonacci components. First, we obtain a complete classification of Markoff $m$-triples whose last two entries are $k$-Fibonacci numbers and that are not roots of any Markoff trees. We then prove that every such infinite path is contained in a branch, starting at a triple of the form \[ \left(\frac{F_k(4r)}{3F_k(2r)},\,F_k(\ell+2r),\,F_k(\ell+4r)\right), \] where $r$ is an odd integer, $\ell\in\{1,2,\ldots, 2r\}$ and $3\nmid k$. These branches are distributed among exactly $2r$ distinct trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23306 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Branches of Markoff $m$-triples with two $k$-Fibonacci components Alfaya, David Calvo, Luis Ángel Cazorla, Pedro-José Rodrigo, Javier Srinivasan, Anitha Number Theory We study infinite paths of Markoff $m$-triples, that is, solutions to the generalised Markoff equation \[ x^2+y^2+z^2=3xyz+m, \] with $m>0$, with at least two $k$-Fibonacci components. First, we obtain a complete classification of Markoff $m$-triples whose last two entries are $k$-Fibonacci numbers and that are not roots of any Markoff trees. We then prove that every such infinite path is contained in a branch, starting at a triple of the form \[ \left(\frac{F_k(4r)}{3F_k(2r)},\,F_k(\ell+2r),\,F_k(\ell+4r)\right), \] where $r$ is an odd integer, $\ell\in\{1,2,\ldots, 2r\}$ and $3\nmid k$. These branches are distributed among exactly $2r$ distinct trees. |
| title | Branches of Markoff $m$-triples with two $k$-Fibonacci components |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.23306 |