On the $8$ case of Sylvester Conjecture

Let $p\equiv 8\mathrm{m}\mathrm{o}\mathrm{d}9$ be a prime. In this paper we give a sufficient condition such that at least one of $p$ and $p^2$ is the sum of two rational cubes. This is the first general result on the $8$ case of the so-called Sylvester conjecture.