Find all functions $f:N \to N$ such that:
$$f(f(m+n)) = f(m) + f(n) \forall m,n \in N$$
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Substitute $m=n=a+1$: $f(f(2a+2))=2f(a+1)$.
ReplyDeleteSubstitute $m=a+2,n=a$: $f(f(2a+2))=f(a+2)+f(a)$
So $f(a)+f(a+2)=2f(a+1)$, or $f(a+2)-f(a+1)=f(a+1)-f(a)$. It follows that $f$ is linear. The rest should be easy. :D