  In this paper we give an effective criterion as to when a prime number p is
the order of an automorphism of a smooth cubic hypersurface of P^{n+1}, for a
fixed n > 1. We also provide a computational method to classify all such
hypersurfaces that admit an automorphism of prime order p. In particular, we
show that p<2^{n+1} and that any such hypersurface admitting an automorphism of
order p>2^n is isomorphic to the Klein n-fold. We apply our method to compute
exhaustive lists of automorphism of prime order of smooth cubic threefolds and
fourfolds. Finally, we provide an application to the moduli space of
principally polarized abelian varieties.
