Find a prime factor of $M_{1559} = 2^{1559} - 1$ where the index 1559 is prime.
We're given 1559 is an odd prime, so we can try using Propositions (4.23) and (4.24) to find candidate factors. Any prime factor $p$ of $2^q-1$ is of the form $2qk+1$ where $k$ is an integer, and is congruent to $\pm 1 \pmod 8$.
The following table shows candidates that confirm to these constraints.
| k | P=2*k*(1559)+1 | p mod 8 | prime? |
| 1 | 3119 | 7 | yes |
| 2 | 6237 | 5 | |
| 3 | 9355 | 3 |
The first candidate 3119 conforms to the conditions, and a computer algebra systems confirms it divides $M_{1559}$.
So a prime factor of $M_{1559}=2^{1559}-1$ is 3119.
Note: The author's solution uses Proposition (4.21)(a) which is simpler.