Exercise 3.4.2 (Integers represented by $x^2 - 2xy + y^2$)

Proof. Here $d=2^2-4=0$. Put

the set of integers represented by the form $x^2-2\mathit{xy}+y^2={(x-y)}^2$.

If $n\in S$, then $n={(x-y)}^2$ is a perfect square ( $0$ included).

Conversely, if $n$ is a perfect square, then $n=m^2$, for some integer $m$. Then $n=f(m,0)\in S$.

The class of integers represented by thie form $x^2-2\mathit{xy}+y^2$ is the set of perfect squares. □