Find all f:N $\longrightarrow$ N that: a) $f(m)=1 \Longleftrightarrow m=1 $ b) $d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} $ c) $ f^{2000}(m)=f(m) $
Problem
Source: Iran 2000
Tags: function, number theory proposed, number theory
grobber
31.05.2004 06:42
By $f^{2000}(m)$ do you mean $f(f(\cdots(m)\cdots))$ ?
Myth
31.05.2004 19:14
It is so strange. It seems I know infinitely many such functions and they are not related to each other, i.e. there are no any regularity.
Omid Hatami
02.06.2004 06:16
Yes,grobber I mean $f(f(...(m)...))$
Omid Hatami
06.06.2004 05:31
No other solution????????????
Myth
06.06.2004 09:01
Actually, I don't see any solution at all. Since you proposed this problem you know solution. Am I right?
Myth
06.06.2004 15:11
Ok. I knew this answer, but I thought there are another solutions. Omid Hatami! Did you mean such solution?
Omid Hatami
08.06.2004 06:44
Well ,my solution is similiar,but I proved some different lemmas
ShahinBJK
09.04.2011 15:07
$c) f^{2000}(m)=m$ At book 2000-2001 This queston's (c) part is like this.
anonymouslonely
18.03.2013 19:15
I only obtained that $ f(n) $ is the product of $ f(p) $ where $ p $ is a prime which divides $ n $. how can I go on?