Prove Corollary (8.8).
Let's remind ourselves of Corollary (8.8):
Let $n = p_1 \times p_2 \times \ldots \times p_r \times N^2$ where $p$’s are distinct primes. Then $n$ cannot be expressed as a sum of two squares if and only if it has a prime factor $p_j \equiv 3 \pmod 4$ for any $j= 1, \ldots , r$.
The two directions of this Corollary are each Theorems (8.5) and (8.7), noting that the negation of $p_j \equiv 3 \pmod 4$ is $p_j =2 \lor p_j \equiv 1 \pmod 4$.