Find all functions $f:\Bbb Z \rightarrow \Bbb Z$, that satisfy
$(i)$ $f(p) > 0$ for every prime $p$,
$(ii)$ $p\space| \space (f(x)+f(p))^{f(p)}$ for every integer $x$ and every prime $p$.
So I started off with the second point. Since $p$ is a prime it follows that it has to be divisible with the base of the power so $p\space| \space (f(x)+f(p))$.
Since the assertion is true for every $x$ let us choose $p$ to be an odd prime and $x$ to be $p$. Then we get that $p\space|\space 2f(p)$ from which follows that $p\space|\space f(p)$ for all odd primes $p$.
From the previous point, it follows that $p\space|\space f(x)$ for all integers $x$.
Let us fix a certain odd prime $q$. From all the previous points it follows that $f(q)$ is a number that is divisible by every odd prime, but the only integer with this property is $0$, therefore $f(q) = 0$, but that contradicts $(i)$, hence no such function exists.
I just want to know if I have made a critical error.