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

in this answer, $x$ and $y$ are integers.
let $y=-x$.
then $2f(x^2)=f(f(0))$
let $x=0$
then $2f(0)=f(f(0))$
Let $a=f(0)$
$f(a)=2a$
$f(f(x+y))=f(x^2)+f(y^2)=2a$
$f(f(x))=2a$
$f(2a)=f(f(a))=2a$
$f$ has only one fixed point $2f(0)$.
also, the fixed point cannot be a square or twice a square.
so the solutions $f$ must satisfy 2 conditions:

*

*$f(f(x))=2f(0)$

*$f(x^2)=f(0)$
$f$ cannot be a polynomial, because if it were, its second iterate is a polynomial and a constant at the same time, contradiction. in particular, $f$ cannot be linear or quadratic. also the qn mentioned another functional eqn, but the solution of one won't help with solving the other.
