  A nonzero rational number is called a cube sum if it is of form $a^3+b^3$
with $a,b\in \mathbb{Q}^\times$. In this paper, we prove that for any odd
integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with
exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a
cube sum). We give also a general construction of Heegner point and obtain an
explicit Gross-Zagier formula which is used to prove the Birch and
Swinnerton-Dyer conjecture for certain elliptic curve related to the cube sum
problem.
