2016
Problem - 3952
Given integers $a$, $b$, $n$. Show that there exist integers $x$, $y$, such that $$(a^2+b^2)^n = x^2 + y^2$$.
Consider $z=a+bi$. Then $(a^2 + b^2)^n = (z\overline{z})^n= \left(|z|^2\right)^n=|z^n|^2$. This means that choosing the real and imaginary parts of $z^n$ can satisfy the relation.