Cluster-Cyclic Quivers with three Vertices and the Markov Equation
  Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver $Q$ it is thus desirable to decide if the cluster algebra defined by $Q$ is acyclic. We call $Q$ cluster-acyclic in this case, otherwise cluster-cyclic.   In this note we classify the cluster-cyclic quivers with three vertices using a Diophantine equation studied by Markov. 
Andre Beineke, Thomas Br\"ustle, Lutz Hille