Prove identity (8.16).
Let's remind ourselves of Proposition (8.16). For integers $a$, $b$, $c$, $d$, and $N$ we have the identity:
$$ (a^2 − Nb^2) (c^2 − Nd^2) = (ac + Nbd)^2 − N (ad + bc)^2 $$
We expand the LHS:
$$ (a^2 − Nb^2) (c^2 − Nd^2) = a^2 c^2 - a^2 d^2 N - b^2 c^2 N + b^2 d^2 N^2 $$
Similarly we expand the RHS:
$$ (ac + Nbd)^2 − N (ad + bc)^2 = a^2 c^2 - a^2 d^2 N - b^2 c^2 N + b^2 d^2 N^2$$
Since LHS=RHS, the identity is proven.