f_n(n)=n

SUPERMAN2

Find all functions $f: N\to N$ satisfying $f_n(n) = n$ for every non negative integer $n$ .

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mavropnevma

Shall we assume $f_n$ is the $n^{\textrm{th}}$ iterate of $f$ ? (usually $f \circ f \circ \cdots \circ f$ is denoted by $f^n$ ).

**Posted:** Sun Nov 01, 2009 4:11 am

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SUPERMAN2

**Posted:** Mon Nov 02, 2009 12:09 am

mavropnevma

The conditions imply that $f(0)$ may be taken arbitrarily, $f(1) = 1$ , and for any $n>1$ , $n$ must belong to a cycle $n, f(n), f_2(n), \ldots, f_d(n) = n$ of length $d \mid n$ .
In particular, when $n$ is prime, its cycle must be of length $1$ (i.e. $f(n) = n$ ), or length $n$ .

Therefore we may build $f$ by taking $f(0)$ arbitrary, $f(1)=1$ , then for the first $n$ for which $f$ has not yet be defined, a cycle of length $d$ a divisor of $n$ , containing other elements divisible by $d$ .

**Posted:** Mon Nov 02, 2009 12:50 am

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