I don't know that much about number theory but those who do might have seen this problem. See if you can solve it:

Suppose $\displaystyle p(x) \in \mathbb{Z}[x]$ has this property that $\displaystyle p(n)$ is a perfect square for all $\displaystyle n \in \mathbb{Z}.$ Prove that $\displaystyle p(x)=(q(x))^2,$ for some $\displaystyle q(x) \in \mathbb{Z}[x].$