Solving the functional equation $f(f(x))=3x$ over the naturals.

Asked May 28 '18 at 15:40

Active Mar 31 '19 at 04:26

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I wish to find all functions $f:\mathbb{N}\to \mathbb{N}$ such that $f(f(x))=3x$

This is my progress: $f(1)\neq1$ clearly as then $f(f(1))=1$ which is false. $f(1)=3$ is not possible as then $f(f(1))=f(3)$ will have to equal some value greater than $3$ as the function is an increasing one. Same goes for every number greater than 3. So $f(1)= 2$. And thus, we know that $f(f(1))=f(2)=3$. Then, $f(f(2))=6=f(3)$. After this we don't know how to proceed. Help.

edited Mar 31 '19 at 04:26

user574848

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asked May 28 '18 at 15:40

Aritra Chakraborty

2

Here's a MathJax tutorial :) – Shaun May 28 '18 at 15:42
2

This might help you : https://math.stackexchange.com/questions/1410635/if-ffn-3n-find-f2001 – Topology May 28 '18 at 15:45
@Topology While that question adds monotonicity as condition on $f$, one of the answers explicitly constructs one (of many) solutions to the problem at hand without monotonicity. – Hagen von Eitzen May 28 '18 at 15:50

Solving the functional equation $f(f(x))=3x$ over the naturals.

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