(i) Show that 239 is a Germain prime.
(ii) Show that $2^{239} - 1$ is a composite Mersenne number. Find a prime factor of $2^{239} - 1$.
(iii) Find another prime factor of $2^{239} − 1$.
(i) 239 is a Germain prime if it is a prime itself, and $2(239)+1=479$ is also a prime.
Both 239 and 479 are prime, and so 239 is a Germain prime.
(ii) Sine $p=239$ is a Germain prime, and $p \equiv -1 \pmod 4$, then by Corollary (4.21)(a) we have $p \mid 2^q -1$, where $q=2p+1$.
That is, $479 \mid 2^{239}-1$. And so $M_{239}=2^{239}-1$ is composite with a prime factor 479.
(iii) We use Propositions (4.23) and (4.24) to constrain the candidates for prime factors. If $q$ is an odd prime, then any prime factor of $2^1-1$ must be of the form $2qk+1$ for some integer $k$, and also be congruent to $\pm1 \pmod 8$.
The following table shows possible candidates.
| k | p =2 *(239)*k+1 | p mod 8 | prime? |
| 1 | 479 | 7 | yes |
| 2 | 957 | 5 | |
| 3 | 1435 | 3 | |
| 4 | 1913 | 1 | yes |
We can see that prime 1913 conforms to the two conditions, so is a candidate.
Using computer algebra software, we confirm that 1913 divides $2^{239}-1$.
And so another factor of $2^{239}-1$ is 1913.