IMO Shortlist-2011 N3

[mutasimmim]

Let $n\ge 1$ be an odd natural number. Find all functions $f$ from the set of integers to itself such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n$.

[mutasimmim]

My progress till now:

$1$. Proved $f(0)=0$.
$2$. $f(x)\mid x^n$.
$3$ If we find such a function and inverse the sign of $f(x)$ for all integers $x$, that is, $g(x)=-f(x)$ also satisfies our conditions. Thus WLOG I let $f(1)$ non negative.
$3$. Proved $f(1)=1, f(-1)=-1$
$4$. Proved for prime $ p $, $f(p)=p^u$ for some $u\mid n$ and $f(-p)=-p^u$ for the same $u$ as in $f(p)$, that is, $f(p)=-f(-p)=p^u$, where $u\mid n$.
$5$. $f(x)=x$ is an obvious solution.

[*Mahi*]

Another member of the "Great Set of Number Theoretical Functions."

Next step:

Prove that if $p, p'$ are two different primes with $f(p) = p^{u}, f(p')=p'^{u'}$, then $u=u'$.