How does one find all $f:\mathbb {Z} \rightarrow \mathbb {Z}$ that satisfies the following: $$f(gcd(x,y))=gcd(f(x),f(y))$$ I had suspected that there would be some results concerning this functional equation but was unable to find any.
It appears that the only solution for $f(x)$ would be $f(x)=cx^r$ for fixed integers $c,r$. However, I was unable to prove or disprove the statement.
Any help would be appreciated.
On
Don't know that you'll find a "nice" general form.
$f(x) = c x^r$ works, as you noted. But so do the following, and many others.
$$ f(x) = \begin{cases} 1 & \text{if $x$ is odd} \\ 2 & \text{if $x$ is even} \end{cases} $$
$$ f(x) = \text{largest power of a fixed but otherwise arbitrary } p \in \mathbb N \text{ which divides } x $$
A combination of two solutions is again a solution. This does not help much with your class of functions, but wait.
Consider any multiplicative function that maps primes to a permutation thereof. Say, $2\mapsto3$, $3\mapsto2$, and the rest of primes map to themselves. This would do as well.