MathDB

Prove that there exists a function

f : \mathbb{N} \mapsto \mathbb{N}

that satisfies the following:
[color=#FFFFFF]___1. For all positive integers

m, n

we have

\gcd(|f(m)-f(n)|, f(mn)) = f(\gcd(m, n))

[color=#FFFFFF]___2. For all positive integers

m

, we have

f(f(m)) = f(m)

. [color=#FFFFFF]___3. For all positive integers

k

, there exists a positive integer

n

with

2024^{k} \mid f(n)

.
Proposed by MV Adhitya, Archit Manas

Problem Statement