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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.

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    note that if $n|m$ then $f(n)|f(m)$ – Kerr Jan 29 '16 at 02:38
  • Will constant functions (e.g. $f(x)=1, \forall x$) be classified as bad examples? – rtybase Feb 02 '16 at 22:29
  • @rtybase If you were trying to disprove my conjecture (as has been done below), it would be a bad example, I would think, as they are of the form $f(x)=cx^r$ where $r=0$. As just a example of this kind of function, however, it could be all right, in my opinion. –  Feb 03 '16 at 01:03
  • See also https://math.stackexchange.com/questions/2388228/sequences-that-commute-with-gcd – lhf Oct 20 '19 at 22:09

2 Answers2

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  1. A combination of two solutions is again a solution. This does not help much with your class of functions, but wait.

  2. 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.

Ivan Neretin
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  • I'm not very experienced at math(or english), but what does maps primes to a permutation thereof mean? Could you give me a example? –  Jan 29 '16 at 09:23
  • I did give an example in the very next phrase. Consider a function such that $f(2)=3,;f(3)=2$, for all other primes $f(p)=p$, and for composite values of argument use the multiplicative property. – Ivan Neretin Jan 29 '16 at 09:32
  • Oh, I was asking for some other one. For example, is this a multiplicative function that maps primes to a permutation thereof? $f(2)=3,f(3)=5,f(5)=2017,f(2017)=2$ and for all other primes $f(p)=p$? –  Jan 29 '16 at 11:08
  • Yes, looks like this would work too. – Ivan Neretin Jan 29 '16 at 11:12
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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.

$$ f(x) = \begin{cases} 1 & \text{if $x$ is odd} \\ 2 & \text{if $x$ is even} \end{cases} $$

$$ f(x) = \text{largest power of a fixed but otherwise arbitrary } p \in \mathbb N \text{ which divides } x $$

dxiv
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