A study on Diophantine equations via cluster theory
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866908477215997952 |
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| author | Bao, Leizhen Li, Fang |
| author_facet | Bao, Leizhen Li, Fang |
| contents | In this paper, we mainly answer a Lampe's question\cite{lampe} about the solutions of a Diophantine equation, that is, we give a criterion to determine which solutions of the Diophantine equation are in the orbit of the initial solution $(ε,ε,ε, ε,ε)$ under the actions of the group $G$ which is defined by mutations of a cluster algebra. In order to do this, using a rational map $φ$, we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solutions $φ(ε,ε,ε, ε,ε) = (3,4,4)$ under the actions of the group $\widetilde{G}$, and the set $S(3,4,4)$ is shown to be the orbit of $(ε,ε,ε, ε,ε)$ under the actions of a subgroup of $G$. Then the criterion is proved as the main conclusion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00468 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A study on Diophantine equations via cluster theory Bao, Leizhen Li, Fang Number Theory Commutative Algebra 11D72 (Primary) 13F60 (Secondary) In this paper, we mainly answer a Lampe's question\cite{lampe} about the solutions of a Diophantine equation, that is, we give a criterion to determine which solutions of the Diophantine equation are in the orbit of the initial solution $(ε,ε,ε, ε,ε)$ under the actions of the group $G$ which is defined by mutations of a cluster algebra. In order to do this, using a rational map $φ$, we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solutions $φ(ε,ε,ε, ε,ε) = (3,4,4)$ under the actions of the group $\widetilde{G}$, and the set $S(3,4,4)$ is shown to be the orbit of $(ε,ε,ε, ε,ε)$ under the actions of a subgroup of $G$. Then the criterion is proved as the main conclusion. |
| title | A study on Diophantine equations via cluster theory |
| topic | Number Theory Commutative Algebra 11D72 (Primary) 13F60 (Secondary) |
| url | https://arxiv.org/abs/2306.00468 |