  We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced
powers $T^\theta/\mathcal U$. Previous works addressed this problem from the
viewpoint of a single power $\theta$, whereas here, tools are developed for
controlling different powers simultaneously. As a sample corollary, we obtain
the consistency of an $\aleph_6$-Souslin tree $T$ and a sequence of uniform
ultrafilters $\langle \mathcal U_n\mid n<6\rangle$ such that $
T^{\aleph_n}/\mathcal U_n$ is $\aleph_6$-Aronszajn iff $n<6$ is not a prime
number.
