Does there exist a non-constant irreducible monic polynomial $P(x)$ with integer coefficients such that for every integer $n$ there are integers $a,b \geq 2$ such that $P(n) = a^b$?
No idea how to approach this - apart from a hint that considering the derivative could be useful - maybe with $P(x)Q(x) + P'(x)R(x) =$ const?
Any help appreciated!