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 3 of 6: Normalize the function

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In plain words

Divide out the common factor.

g(t)=f(t)/k\quad\Rightarrow\quad g(n^2g(m))=m g(n)^2

Detailed analysis

Put $g=f/k$ . The scaling identity shows $g$ is integer-valued, and substituting $f(m)=kg(m)$ into the original equation gives $g(n^2g(m))=m g(n)^2$ . If $k>1$ , then $g(1998)<f(1998)$ , so a minimizing function must have $k=1$ .