Generalization of Markov Diophantine Equation via Generalized Cluster Algebra

In this paper, we deal with two classes of Diophantine equations, x(2) + y(2) + z(2) + k(3)xy +k(1)yz +k(2)zx = (3+k(1) +k(2) +k(3))xyz and x(2)+y(4)+z(4)+2xy(2)+ky(2)z(2)+2xz(2) = (7+k)xy(2)z(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.