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 5 of 6: Describe all such functions

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In plain words

Prime factorization reduces the problem to a permutation of primes.

f\text{ permutes the primes by transpositions and fixed points}

Detailed analysis

Multiplicativity and involutivity show that a prime must map to a prime: if $f(p)=uv$ , then $f(u)f(v)=p$ , forcing one factor to be $1$ . Conversely, any involution of the primes, extended multiplicatively, satisfies the functional equation.