Show that $78511$ (prime) is a factor of the composite Mersenne number $2^{2617}− 1$.
2617 is prime. By Proposition (4.23) any prime factor $p$ of $2^{2617}-1$ is of the form $p=2k(2617)+1$, where $k$ is an integer. The prime $78511=2(2617)(15)+1$, and so is not ruled out as a candidate.
Also, $78511 \equiv -1 \pmod 8$, so by Proposition (4.24) is not ruled out as a candidate.
We can use computer algebra software to confirm 78511 is a prime factor of $M_{2617}$.
Note: the author's solution suggest conforming to Propositions (4.23) and (4.24) is sufficient to conclude 78511 is a prime factor, but that is not correct. The propositions provide necessary but not sufficient conditions.