Binary Cubic Forms and Rational Cube Sum Problem

Abstract: 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.