, 1997

By Dirichlet's theorem, there are infinitely many primes $p$ of the form $3k + 2$. We claim that $N_p = p^2$ for all such primes $p$.

Indeed, for each of the $p^2$ pairs of $(y, z)$ modulo $p$, we need to solve
$$x^3 \equiv -3^{-1}(4y^3 + 5z^3 - y^4z) \pmod{p}.$$
As $(p-1, 3) = 1$, this equation has a unique solution in $x$ modulo $p$. This proves $N_p = p^2$.