MathLabs

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 3 of 4: Force every value to equal k
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f(a)f(a2)=f(a6)=f(a)f(a5)k=f(a)2f(a4)k2=f(a)2f(a2)kf(a)f(a^2)=f(a^6)=\frac{f(a)f(a^5)}k=\frac{f(a)^2f(a^4)}{k^2}=\frac{f(a)^2f(a^2)}k
Detailed analysis

Apply the identities along the chain of powers shown. Cancelling positive factors yields f(a)=k for every a in S.