Problem 6

Determine the least possible value of

f(1998)

, where

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

is a function such that for all

m,n\in\mathbb{N}

f\left(n^{2}f(m)\right)=m\left(f(n)\right)^{2}.

Step 4 of 6: Classify normalized solutions

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In plain words

The normalized equation forces an involutive multiplicative map.

f(f(t))=t,\quad f(st)=f(s)f(t)

Detailed analysis

With $f(1)=1$ , the identities become $f(f(t))=t$ and $f(t^2)=f(t)^2$ . Applying the original equation with $m=f(t^2)$ gives $f(st)^2=f(s)^2f(t)^2$ , hence positivity gives multiplicativity.