Problem 2

Let

S=\{2,3,4,\ldots\}

denote the set of integers that are greater than or equal to

2

. Does there exist a function

f:S\to S

such that

f(a)f(b)=f(a^2b^2)

for all

a,b\in S

with

a\ne b

Step 2 of 4: Make the ratio constant

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\frac{f(a^2)}{f(a)}=\frac{f(b^2)}{f(b)}=k>0,\qquad f(ab)=\frac{f(a)f(b)}k

Detailed analysis

The first relation says the square ratio is independent of the input; call it k. Substituting the square relation into the original equation gives the second identity whenever a and b are distinct.