Problem 3

Let

S

be the set of non-negative integers. Find all functions

f:S\to S

such that

f(m+f(n))=f(f(m))+f(n)

for all

m,n\in S

Step 2 of 4: Choose the least nonzero fixed point

Previous step Next step

In plain words

Choose the least nonzero fixed point

f(qk)=qk

Detailed analysis

If no positive fixed point exists, then every value $f(n)$ is a fixed point and therefore $f$ is identically zero. Otherwise let $k$ be the least positive fixed point. Induction using the simplified equation gives $f(qk)=qk$ for every $q\ge0$ .