Exercise 5.3.9 (Every integer $n$ can be expressed in the form $n = x^2 + y^2 - z^2$)

Proof. If $n$ is odd, or if $n\equiv 0(\mathit{mod}4)$, then $n=y^2-z^2$ has a solution by Problem 7, so $n$ can be expressed in the form $n=x^2+y^2-z^2$, with $x=0$.

It remains the case $n\equiv 2(\mathit{mod}4)$. Then $n-1$ is odd, and the same Problem shows that there is a solution to $n=1^2+y^2-z^2$, so $n$ can be expressed in the form $n=x^2+y^2-z^2$, with $x=1$. □