A study on Diophantine equations via cluster theory

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 $(\epsilon,\epsilon,\epsilon, \epsilon,\epsilon)$ 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 $\varphi$, we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solutions $\varphi(\epsilon,\epsilon,\epsilon, \epsilon,\epsilon) = (3,4,4)$ under the actions of the group $\widetilde{G}$, and the set $S(3,4,4)$ is shown to be the orbit of $(\epsilon,\epsilon,\epsilon, \epsilon,\epsilon)$ under the actions of a subgroup of $G$. Then the criterion is proved as the main conclusion.