A study on Diophantine equations via cluster theory

Abstract

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