Tuesday, 25 August 2026

Exercise (8.1).7

Show that the positive integer $n^k$ where $k$ is an even positive integer can be written as a sum of two squares.


Since $k$ is even, we can write it as $k=2m$. And so

$$ n^k = n^{2m} = (n^m)^2 + 0^2 $$

And so $n^k$ where $k$ is even, can be written as the sum of two squares.