Problem $10728$ from Amer. Math. Monthly "Preserving the sum of three cubes" says:
Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $f\left(x^3+y^3+z^3\right)=f(x)^3+f(y)^3+f(z)^3$ for all integers $x,y,$ and $z$.
The solutions are $f(x)=0$, $f(x)=x$ and $f(x)=-x$. The idea is that
- $f(0)=0$;
- $f$ is an odd function;
- if $x$ is an integer greater than $3$ then $x^3$ can be written as the sum of five cubes that are smaller in magnitude than $x^3$.
Thus arise the natural questions:
Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ such that
Problem 1. $f\left(x^3+y^3\right)=f(x)^3+f(y)^3$;
Problem 2. $f\left(x^3\right)=f(x)^3$.
Problem 2 has an obvious solution.
Let $S$ be the set of numbers which are not cubes. For each $s_i \in S$, let $f(s_i) = t_i$ for some integer $t_i$.
Then, for any integer $n$, show that $f(n)$ is uniquely determined.
To be pedantic, show that $f(n)$ satisfies the condition.
It is clear that this is a necessary, and sufficient, condition.