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Problem 2

Let S={2,3,4,…}S=\{2,3,4,\ldots\} denote the set of integers that are greater than or equal to 22. Does there exist a function f:S→Sf:S\to S such that f(a)f(b)=f(a2b2)f(a)f(b)=f(a^2b^2) for all a,b∈Sa,b\in S with a≠ba\ne b?
Step 4 of 4: Contradiction
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k2=f(2)f(3)=f(36)=k⟹k=1∉Sk^2=f(2)f(3)=f(36)=k\Longrightarrow k=1\notin S
Detailed analysis

Using the original condition at two distinct inputs 2 and 3 gives k squared equal to k, so k is one. But the codomain S contains only integers at least two. Hence no such function exists.