  In this note, we use integral binary cubic forms to study the rational cube
sum problem. We prove (unconditionally) that for any positive integer $d$,
infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well
as $ -1 \pmod {9d}$, are sums of two rational cubes. Among other results, we
prove that every non-zero residue class $a \pmod {q}$, for any prime $q$,
contains infinitely many primes which are sums of two rational cubes. Further,
for an arbitrary integer $N$, we show there are infinitely many primes $p$ in
each of the residue classes $ 8 \pmod 9$ and $1 \pmod 9$, such that $Np$ is a
sum of two rational cubes.
