20. An interesting construction is as follows: $x+y \mathrm{i}=(a+b \mathrm{i})^{n}$, where i is the imaginary unit. When $n$ is fixed, there are infinitely many pairs of integers $(a, b)$ such that integers $x, y$ satisfy $x y \neq 0$. In this case, $x^{2}+y^{2}=\left(a^{2}+b^{2}\right)^{n}$ (this can be obtained by taking the modulus of both sides).