Prove Proposition (4.22).
Proposition (4.22):
If $q \ne 3$ is a Germain prime and $q \equiv -1 \pmod 4$ then the Mersenne number $M_q = 2^q - 1$ is composite and $p \mid (2^q - 1)$ where $p=2q + 1$.
We're given $q \ne 3$ is a Germain prime. This means $p=2q+1$ is also prime.
We're also given $q \equiv -1 \pmod 4$, so by Corollary (4.21) we have $p \mid 2^q -1$.
We need to show that $p$ is a non-trivial factor of $2^q-1$, that is, $p<2^q-1$.
$$ \begin{align} p & < 2^q -1 \\ \\ 2q + 1& < 2^q -1 \\ \\ 2(q+1) & < 2^q \end{align} $$
This is true for Germain primes $q>3$. The Germain prime $q=3$ is excluded specifically, and $q=2$ is ruled out because $q \not \equiv -1 \pmod 4$.
And so $p$ is a non-trivial prime factor of $2^q-1$, meaning $M_q$ is composite, proving Proposition (4.22).