if $f(f(x+y))=f(x^2)+f(y^2)$ then $f(x)=?$ if $f(f(x+y))=f(x^2)+f(y^2)$ then $f(x)=?$ for all integers ($f: \mathbb Z \rightarrow \mathbb Z$)
I know how to solve the following problem though:
if $f(f(x+y))=f(2x)+2f(y)$ then $f(x)=?$ We can easily analyze that $f(x)$ here (in the second problem) is a linear function. And hence solve it by using the linear equation.
But, As for the main problem (the one I mentioned first) I don't know how to proceed. Should I use the quadratic equation? (my calculation says its a quadratic function)
Would be greatful if anyone could help me with this...
Thank you

Substituting $y=-x$ grants $f(f(0))=2f(x^2)$. Thus $f(x^2)$ is constant for all $x$, i.e. $f$ is constant for nonnegative numbers, the range of values for $x^2$. (If your $f$ has domain $\mathbb R$ or something else, you'll need to specify to proceed further.)
