Show that the following integers are composite by finding a prime factor:
(a) $2^{41} + 1$
(b) $2^{53} + 1$
(a) $q=41$ is a Germain prime, because $p=2(41)+1=83$ is a prime.
Also, $q \equiv 1 \pmod 4$, and so by Corollary (4.21)(b) we have $p \mid 2^q +1$. That is
$$ 83 \mid 2^{41} + 1 $$
So $2^{41} + 1$ is composite with a prime factor 83.
(b) $q=53$ is a Germain prime, because $p=2(53)+1=107$ is a prime.
Also, $q \equiv 1 \pmod 4$, and so by Corollary (4.21)(b) we have $p \mid 2^q +1$. That is
$$ 107 \mid 2^{53} + 1 $$
So $2^{53} + 1$ is composite with a prime factor 107.