The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history, from early works of Sylvester, Satge, Selmer etc. and, up to recent work of Alpoge-Bhargava-Shnidman. A conjecture attributed to Sylvester predicts that the primes >2 in the residue classes 2,5 mod 9 are not sums of two rational cubes, while those in the residue classes 4,7 or 8 mod 9 are. Primes which are 1 mod 9 may or may not be sums of two rational cube sums. We rephrase the problem in terms of elliptic curves, and use certain integral binary cubic forms to prove unconditionally that there are infinitely many primes in each of the residue classes 1 mod 9 and 8 mod 9 that are expressible as sums of two rational cubes. More generally, we prove that every non-zero residue class a (mod q), for any prime q, contains infinitely many primes which are sums of two rational cubes. Among other results, we show that corresponding to any positive integer n, there are infinitely many imaginary quadratic fields in which n is a sum of two cubes. These results represent joint work with Somnath Jha and Dipramit Majumdar. The starting point of this work was an accidental encounter in earlier work with Dipramit Majumdar when we classified all cubic cyclic extensions of Q.