Sylvester's Problem and Mock Heegner Points

We prove that if $p\equiv 4,7(\mathrm{m}\mathrm{o}\mathrm{d}9)$ is prime and $3$ is not a cube modulo $p$, then both of the equations $x^3+y^3=p$ and $x^3+y^3=p^2$ have a solution with $x,y\in \mathbb{Q}$.