polynomials such that $P(k)=Q(l)$ for all integer $k$

In a book I have read this problem:

Given $P\in \mathbb{R}[X]$, if $P(X)$ takes at every integer, a value which is the $k$-th power of an integer, then $P(X)$ itself is the $k$-th power of a polynomial. Formally if: $$\begin{align}\forall n\in \mathbb{Z}\,\,\,\exists l\in \mathbb{Z}&& P(n)=l^k \end{align}$$ then $\exists R(X)\,\, P(x)=R(X)^k$

we can find a proof in this paper page $8$.

Is the following assertion true:

Given a polynomials $Q$ with integer coefficients,If $P(X)$ takes at every integer, a value which is of the form $Q(k)$ for an integer k, then $P(X)=Q(R(X))$ for a polynomial $R$ polynomial. Formally if: $$\begin{align}\forall n\in \mathbb{Z}\,\,\,\exists l\in \mathbb{Z}&& P(n)=Q(l) \end{align}$$ then $\exists R(X)\,\, P(x)=Q(R(X))$

I need some suggestions to solve this problem,or any references and sources which deal with this sort of problems. thaks

Solution 1:

See the following: $ $ H. Davenport; D. Lewis; A. Schinzel, Polynomials of certain special types, Acta Arithmetica (1964) 107-116.