hello,
Prove that $\displaystyle \forall n \in \mathbb{N}$ , there's no number $\displaystyle k\in \mathbb{N}$ and $\displaystyle 3n^2 + 3n +7= k^3$
i haven't worked out the details, but a proof by contradiction should work. assume there is such a k. what would the solution to $\displaystyle 3n^2 + 3n + 7 - k^3 = 0$ look like? would $\displaystyle n$ be a natural number or even an integer at all? try the quadratic formula, see what happens