A proof of the $4,7$ cases of Sylvester's conjecture on cube sums

Abstract: In this paper, we prove that every prime $p$ which is congruent to $4,7$ modulo $9$ is the sum of two rational cubes. This is $2/3$ of Sylvester's conjecture which has history nearly 150 years since 1879. In the proof, we use recent progress on Full BSD conjecture of rank $0$ elliptic curves in \cite{BF} to deduce the Manin-Stevens constants of some families of elliptic curves are units. We also use recent solutions of Unbounded Denominators Conjecture in \cite{CDT} to prove that some cubic roots of modular functions are invariant under some congruence subgroups. Instead of using the Unbounded Denominators Conjecuture, we also give another conditional proof assuming the GRH for number fields or Artin's primitive root conjecture for arithmetic progressions.