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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.04026 |
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| _version_ | 1866918442000449536 |
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| author | Banaian, Esther |
| author_facet | Banaian, Esther |
| contents | The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. When $k =0$, this recovers the ordinary Markov numbers. A long-standing question in the theory of Markov numbers is Frobenius's unicity conjecture, concerning whether every Markov number is the maximum in a unique solution triple. Aigner gave a series of weaker, related conjectures which were confirmed to be true by Lee, Li, Rabideau, and Schiffler using techniques from the theory of cluster algebras. We show here that $k$-Markov numbers also satisfy Aigner's conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04026 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orderings of k-Markov Numbers Banaian, Esther Number Theory Combinatorics 11B83, 13F60 The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. When $k =0$, this recovers the ordinary Markov numbers. A long-standing question in the theory of Markov numbers is Frobenius's unicity conjecture, concerning whether every Markov number is the maximum in a unique solution triple. Aigner gave a series of weaker, related conjectures which were confirmed to be true by Lee, Li, Rabideau, and Schiffler using techniques from the theory of cluster algebras. We show here that $k$-Markov numbers also satisfy Aigner's conjectures. |
| title | Orderings of k-Markov Numbers |
| topic | Number Theory Combinatorics 11B83, 13F60 |
| url | https://arxiv.org/abs/2512.04026 |