Generation of all rank-2 finite-type Diophantine solutions by mutations

Determine whether all positive integer solutions of every Diophantine equation T(x1,x2)=T(a,b) arising from a rank-2 finite-type cluster algebra and the mutation-invariant construction in Equation (4.1) can be generated from the initial solution (a,b) by finitely many cluster mutations.

Background

The paper proves that equations of the form T(x1,x2)=T(a,b), where T is constructed from the clusters of a rank-2 finite-type cluster algebra, have only finitely many positive integer solutions. It also establishes an upper bound for the smaller coordinate of any solution. However, finiteness does not imply that every solution belongs to the cluster-mutation orbit of the chosen initial solution (a,b).

The unresolved issue concerns extending the mutation-generation results known for particular concrete equations to the entire class of rank-2 finite-type Diophantine equations considered in Equation (4.2). The authors note that their finiteness theorem only partially answers a question posed in the cited earlier work [CL24, Question 3.8].

References

Although it is difficult to determine whether all the solutions to Equation (4.2) can be generated by the initial solution (a, b) through finite mutations, we can still calculate and deal with each concrete equation by Theorem 4.5 and Corollary 4.6.

— A cluster theory approach from mutation invariants to Diophantine equations  (2501.09435 - Chen et al., 16 Jan 2025) in Remark 4.7, Section 4