I need to prove that, assuming $n \in \mathbb{N^+}$
$$\sqrt[3] {3n^2+3n+1} \notin \mathbb{N^+}$$
I'm really stuck with this problem, because so far I haven't managed a clever way to solve it.
Also, I think it really doesn't help the fact that $3n^2+3n+1 = (n+1)^3-n^3$, also because, for example, both $\sqrt {2n+1}$ and $\sqrt[3] {2n+1}$ can be in $\mathbb{N^+}$ for certain values of $n$.
Anyway I'm sure that the relation I wrote is true: first, because it is a consequence of some proved theorems; second, because I tried some billions value of $n$ with the computer, with no results.
Anyone who can enlighten me with some tremendous intuitions? :)