Say some integer $n$ is not sum of two cubes in the integers, then I want to show that there exists $k$ in positive integers such that $x^{3} + y^{3} \equiv n\pmod{k}$ is not solvable.
All I seem to know regarding even vicinity of such problems are following:
If $n$ is not sum of two cubes in integers, then exists an integer $\theta(n)$ such that $n\theta(n)$ is sum of two cubes.
Any number $n$ is sum of two cubes in integers if and only if following condition is satisfied: $$\exists m \mid n ,\quad n^{1/3} \leq m \leq 4^{1/3} n^{1/3}$$ such that $( m^{2} - \frac{n}{m} ) = 3l$ and $(m^{2} - 4l)$ is a perfect square.
May be there is counterexample here, i.e. exists $n$ which is not sum of two cubes yet the congruence equation $x^{3} + y^{3} \equiv n\pmod{k}$ is solvable $\forall k \in\mathbb{N}$. Any help is appreciated.