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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 3 of 4: Determine all values
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In plain words

Determine all values

f(qk+r)=qk+nrkf(qk+r)=qk+n_rk
Detailed analysis

For any fixed point t t , write t=qk+r t=qk+r with 0≤r<k0\le r<k . Since f(t)=t f(t)=t and f(qk+r)=f(r)+qk f(qk+r)=f(r)+qk , we get f(r)=r f(r)=r ; hence the fixed points are precisely the multiples of k k . Because every f(n) f(n) is fixed, write f(r)=nrk f(r)=n_rk for 0≤r<k0\le r<k , with n0=0 n_0=0, and obtain f(qk+r)=qk+nrk f(qk+r)=qk+n_rk .