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 $(\epsilon ,\epsilon ,\epsilon ,\epsilon ,\epsilon )$ 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 $\phi$, we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solutions $\phi (\epsilon ,\epsilon ,\epsilon ,\epsilon ,\epsilon )=(3,4,4)$ under the actions of the group $\tilde{G}$, 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.

19 pages, 5 figures

References in corpus (1)

• Laurent Polynomials and Superintegrable Maps