Ehrhart polynomials of integral simplices with prime volumes

For an integral convex polytope $\text{Pc}\subset {\text{RR}}^N$ of dimension $d$, we call $\delta (\text{Pc})=({\delta }_0,{\delta }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\delta }_d)$ the $\delta$-vector of $\text{Pc}$ and $\text{vol}(\text{Pc})={\sum }_{i=0}^d{\delta }_i$ 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.