Generalization of Markov Diophantine Equation via Generalized Cluster Algebra

Abstract

In this paper, we deal with two classes of Diophantine equations, x2 + y2 + z2 + k3 xy + k1 yz + k2 zx = (3 + k1 + k2 + k3)xyz and x2 + y4 + z4 + 2xy2 + ky2 z2 + 2xz2 = (7+k)xy2 z2, where k1, k2, k3, k are nonnegative integers. The former is known as the Markov Diophantine equation if k1 = k2 = k3 = 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.