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k-tastrophic functions

Source: French TST 2012

August 2, 2012

functionalgebra unsolvedalgebra

Problem Statement

Let

k>1

be an integer. A function

f:\mathbb{N^*}\to\mathbb{N^*}

is called

k

-tastrophic when for every integer

n>0

, we have

f_k(n)=n^k

where

f_k

is the

k

-th iteration of

f

:

f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)

For which

k

does there exist a

k

-tastrophic function?

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