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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2604.17445 |
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| _version_ | 1866908978239242240 |
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| author | Banaian, Esther Huang, Min |
| author_facet | Banaian, Esther Huang, Min |
| contents | A $k$-Markov number is a positive integer that appears in a positive integral solution to the Diophantine equation $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. This equation was introduced by Gyoda and Matsushita. When $k =0$, this definition recovers that of ordinary Markov numbers. The set of $k$-Markov numbers can be indexed by pairs of coprime positive integers. There is a consistent way to label non-coprime pairs with positive integers as well, yielding a larger set of ``generalized $k$-Markov numbers.''
In this paper, we classify lines along which the generalized $k$-Markov numbers grow monotonically, extending work in the ordinary case by Lee-Li-Rabideau-Schiffler and by the second author. We find that, as $k$ grows, the $k$-Markov numbers are more likely to be monotonic along a random line. This gives evidence that a $k$-version of Frobenius' uniqueness conjecture, which has been proposed by Gyoda and Maruyama, could be true. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17445 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Orderings of Generalized k-Markov Numbers Banaian, Esther Huang, Min Number Theory Combinatorics 11B83, 13F60 A $k$-Markov number is a positive integer that appears in a positive integral solution to the Diophantine equation $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. This equation was introduced by Gyoda and Matsushita. When $k =0$, this definition recovers that of ordinary Markov numbers. The set of $k$-Markov numbers can be indexed by pairs of coprime positive integers. There is a consistent way to label non-coprime pairs with positive integers as well, yielding a larger set of ``generalized $k$-Markov numbers.'' In this paper, we classify lines along which the generalized $k$-Markov numbers grow monotonically, extending work in the ordinary case by Lee-Li-Rabideau-Schiffler and by the second author. We find that, as $k$ grows, the $k$-Markov numbers are more likely to be monotonic along a random line. This gives evidence that a $k$-version of Frobenius' uniqueness conjecture, which has been proposed by Gyoda and Maruyama, could be true. |
| title | Orderings of Generalized k-Markov Numbers |
| topic | Number Theory Combinatorics 11B83, 13F60 |
| url | https://arxiv.org/abs/2604.17445 |