  For an integral convex polytope $\Pc \subset \RR^N$ of dimension $d$, we call
$\delta(\Pc)=(\delta_0, \delta_1,..., \delta_d)$ the $\delta$-vector of $\Pc$
and $\vol(\Pc)=\sum_{i=0}^d\delta_i$ its normalized volume. In this paper, we
will establish the new equalities and inequalities on $\delta$-vectors for
integral simplices whose normalized volumes are prime. Moreover, by using
those, we will classify all the possible $\delta$-vectors of integral simplices
with normalized volume 5 and 7.
