TITLE: Polynomials - Relation of Divisibility
QUESTION [5 upvotes]: The exercise I am having difficulty solving is the following one : 
Suppose $P$ and $Q$ are two polynomials with integer coefficients. Suppose also that for all $(m,n) \in \mathbb{Z}^2$ we have $P(m)-P(n) | Q(m)-Q(n)$. Show that there exists a polynomial $H \in \mathbb{Q}[X]$ such that $Q= H \circ P$. 
The issue is that I do not now how to characterize effectively the existence of such a polynomial $H$... I have tried, without success, to use the following result : 
if $P,Q$ are two polynomials with integer coefficients such that $P(n)|Q(n)$ for infinitely many integers $n$, then $P$ divides $Q$ within $\mathbb{Q}[x]$.
Does anyone have an idea ?

REPLY [2 votes]: Theorem 2.3 in http://www.math.uci.edu/~mfried/paplist-cov/SepVarbsCurves69.pdf might  be helpful:
(Fried-MacRae)   Let $K$ be an arbitrary field, and  let $f(t),g(t),f_1(t)$
and $g_1(t)$ be polynomials in $K[t]$. Then $f_1(x)-g_1(z)$ divides $f(x)-g(z)$ in $K[x,z]$ if and only if there exists a polynomial $F(t)$ in $K[t]$ such that $f(t)=F(f_1(t))$ and  $g(t)=F(g_1(t))$.