Show that $2^n$, where $n$ is a natural number, can be written as sum of two squares.
There are two cases for $n$, even and odd.
- If $n$ is even, then $n=2m$ for some natural number $m$. And so $2^n=2^{2m}=(2^m)^2+0^2$.
- If $n$ is odd, then $n=2m+1$ for some natural number $m$. And so $2^n=2^{2m+1}=(2^m)^2\times2 = (2^m)^2+(2^m)^2$.
In both cases we can express $2^n$ as a sum of two squares.