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 $1(\mathrm{mod}9d)$ as well as $-1(\mathrm{mod}9d)$, are sums of two rational cubes. Among other results, we prove that every non-zero residue class $a(\mathrm{mod}q)$, 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 $8(\mathrm{mod}9)$ and $1(\mathrm{mod}9)$, such that $Np$ is a sum of two rational cubes.