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 1 of 4: Compare two factorizations

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f(a^4b^4c^4)=f(a^2)f(b)f(c)=f(b^2)f(a)f(c)

Detailed analysis

For arbitrary a and b choose c larger than both. Applying the given equation twice to the same fourth-power product is legal because the unequal arguments condition is satisfied at each application. Cancelling the positive value f(c) gives the displayed relation.