Generalization of Markov Diophantine equation via generalized cluster algebra (Q6197316)

Summary: 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 \textit{P. Lampe} [J. Algebra 462, 320--337 (2016; Zbl 1441.13053)] 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.