Geodesic
Generalization of Markov Diophantine equation via generalized cluster algebra
Yasuaki Gyoda1 ; Kodai Matsushita2
1The University of Tokyo
2Nagoya University
The electronic journal of combinatorics, Tome 30 (2023) no. 4
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In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_3xy+k_1yz+k_2zx=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+2xy^2+ky^2z^2+2xz^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them.
DOI : 10.37236/11420
Classification : 11D25, 13F60

Yasuaki Gyoda  1   ; Kodai Matsushita  2

1 The University of Tokyo
2 Nagoya University 
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     author = {Yasuaki Gyoda and Kodai Matsushita},
     title = {Generalization of {Markov} {Diophantine} equation via generalized cluster algebra},
     journal = {The electronic journal of combinatorics},
     year = {2023},
     volume = {30},
     number = {4},
     doi = {10.37236/11420},
     zbl = {1534.11038},
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Yasuaki Gyoda; Kodai Matsushita. Generalization of Markov Diophantine equation via generalized cluster algebra. The electronic journal of combinatorics, Tome 30 (2023) no. 4. doi: 10.37236/11420

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