Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have $\phi(n) \mid \phi(P(n))$. It is conjectured there are none (other than the trivial $P(x) = x^k Q(x)$).
NOTE: For $\phi(P(n))$ to be well-defined, it has been suggested that we require $P(n) > 0$ for all positive integers $n$.