Prime points on smooth hypersurfaces

Let $F\in \mathbb{Z}[x_1,\dots ,x_n]$ be a homogeneous form of degree $d\ge 4$ which defines a smooth hypersurface in ${\mathbb{P}}_{\mathbb{C}}^{n-1}$. For $n\ge 24d^4{2}^d$, we prove an asymptotic formula for the number of prime solutions to the equation $F(x_1,\dots ,x_n)=0$, provided $F$ satisfies suitable local conditions.