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2018 PAMO Shortlist: Completely multiplicative involution on the natural numbers

Source: 2018 Pan-African Shortlist - A5

May 6, 2019
number theoryalgebrafunctionfunctional equationmultiplicative function

Problem Statement

Let g:N→Ng : \mathbb{N} \to \mathbb{N} be a function satisfying:
[*] g(xy)=g(x)g(y)g(xy) = g(x)g(y) for all x,y∈Nx, y \in \mathbb{N}, [*] g(g(x))=xg(g(x)) = x for all x∈Nx \in \mathbb{N}, and [*] g(x)≠xg(x) \neq x for 2≤x≤20182 \leq x \leq 2018.
Find the minimum possible value of g(2)g(2).
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