Title: Sylvester's Problem and Mock Heegner Points,
Abstract: We prove that if $p \equiv 4,7 \pmod{9}$ 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 $x,y \in \mathbb{Q}$.