Solve $7x^3+2=y^3$ over integers I need to solve the following 
solve  $7 x^3 + 2 = y^3$ over integers.
How can I do that?

To solve this kind of equations, we have several 'tools' such as 
using mod, using inequalities, using factorization...
In your question, using mod will help you.
Since we have
$$y^3-2=7x^3,$$
the following has to be satisfied :
$$y^3\equiv 2\ \ \ (\text{mod $7$}).$$
However, in mod $7$, 
$$0^3\equiv 0,$$
$$1^3\equiv 1,$$
$$2^3\equiv 1,$$
$$3^3\equiv 6,$$
$$4^3\equiv 1,$$
$$5^3\equiv 6,$$
$$6^3\equiv 6.$$
So, there is no integer $y$ such that $y^3\equiv 2\ \ \ (\text{mod $7$}).$
Hence, we know that there is no solution.
