MathLabs

Problem 4

Prove that there is no function ff from the set of non-negative integers into itself such that f(f(n))=n+1987f(f(n)) = n + 1987 for every non-negative integer nn.
Step 1 of 5: Derive translation compatibility
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In plain words

The equation makes ff commute with adding 1987.

f(n+1987)=f(n)+1987f(n+1987)=f(n)+1987
Detailed analysis

Applying ff to f(f(n))=n+1987f(f(n))=n+1987 gives f(f(f(n)))=f(n+1987)f(f(f(n)))=f(n+1987). Applying the original equation to f(n)f(n) gives f(f(f(n)))=f(n)+1987f(f(f(n)))=f(n)+1987. Thus f(n+1987)=f(n)+1987f(n+1987)=f(n)+1987 for every nn.