I need help with this:
"Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that: $$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$ Or : $$f(g(x)+y) = g(f(y)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$
For the first, note that the right side is independent of $y$, so $f(x)$ is constant. Then we have $f(0)=g(f(0)+x)$ with the left side independent of $x$. Then $g$ must be constant, but it must be injective. No solution.