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f(m^2+n^2)=f(m)^2+f(n)^2 over N

Source: France 1994 P5

May 7, 2021
fefunctional equationalgebra

Problem Statement

Assume f:N0→N0f:\mathbb N_0\to\mathbb N_0 is a function such that f(1)>0f(1)>0 and, for any nonnegative integers mm and nn, f(m2+n2)=f(m)2+f(n)2.f\left(m^2+n^2\right)=f(m)^2+f(n)^2.(a) Calculate f(k)f(k) for 0≤k≤120\le k\le12. (b) Calculate f(n)f(n) for any natural number nn.
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