Binary cubic forms and rational cube sum problem

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