A cluster algebra is a subalgebra $A$ of $k[x_1^{\pm1},...,x_n^{\pm1}]$ generated by a set of cluster variables, which are elements which can be generated from the set $\{x_1,...,x_n\}$ by a certain recursion determined by a skew-symmetrizable matrix (mutation).

As elements in $A$, each cluster variable $f$ is an irreducible element (it can't be factored). This is not hard to show; it follows from the Laurent embedding for any cluster containing $f$, and the observation that $A$ does not contain any Laurent monomials with negative powers of (non-frozen) variables.

So then I ask, are cluster variables prime elements in $A$?

The analogous argument to the irreducible case doesn't seem to work, since one needs to consider more general Laurent polynomials in a given cluster, and there is no nice criterion for telling when a general Laurent polynomial is in the cluster algebra. It also seems unlikely that $A$ is a UFD in general, which would be a standard trick for deducing primality for irreducibility. Nonetheless, the examples I checked in Macaulay 2 were all prime.