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Problem 3

Let S S be the set of non-negative integers. Find all functions f:S→S f:S\to S such that f(m+f(n))=f(f(m))+f(n) f(m+f(n))=f(f(m))+f(n) for all m,n∈S m,n\in S .
Step 4 of 4: Verify the family
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In plain words

Verify the family

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

Conversely, choose any k≥1 k\ge1 and arbitrary nonnegative integers n1,…,nk−1 n_1,\ldots,n_{k-1}, set n0=0 n_0=0, and define the displayed formula. Writing m=ak+r m=ak+r and n=bk+s n=bk+s shows both sides equal ak+bk+nrk+nsk ak+bk+n_rk+n_sk . Together with the zero function, these are exactly all solutions.