Prove that the equation $x^3 -24x + k=0$ has one integer solution at most, $\forall k \in \mathbb{Z}$
Suppose there are two integer solutions. Then, according to Vieta, all the $3$ solutions are integers. Using the first derivative I can get a contradiction out of this assumption, but I'm only allowed to use elementary number theory, so I'm stuck here.
Any help is appreciated.