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Ehrhart polynomials of integral simplices with prime volumes

For an integral convex polytope $\Pc \subset \RR^N$ of dimension $d$, we call $δ(\Pc)=(δ_0, δ_1,..., δ_d)$ the $δ$-vector of $\Pc$ and $\vol(\Pc)=\sum_{i=0}^dδ_i$ its normalized volume. In this paper, we will establish the new equalities and inequalities on $δ$-vectors for integral simplices whose normalized volumes are prime. Moreover, by using those, we will classify all the possible $δ$-vectors of integral simplices with normalized volume 5 and 7.