  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.
