Sums of two rational cubes with many prime factors

In this paper, we show that for any given integer k≥2, there are infinitely many cube-free integers n having exactly k prime divisors such that n is a sum of two rational cubes. This is a cubic analogue of the work of Tian [Ti], which proves that there are infinitely many congruent numbers having exactly k prime divisors for any given integer k≥1.