Find a prime factor of the following Mersenne numbers $M_q$:
(a) $M_{83}$
(b) $M_{131}$
(c) $M_{179}$
(d) $M_{191}$
(a) Setting $q=83$ gives $p=2(83)+1 = 167$ which is prime. Since $q=83$ is also prime, then $83$ is a Germain prime.
As $q \equiv -1 \pmod 4$, by Corollary (4.21) we have $167 \mid 2^{83}-1$.
That is, 167 is a prime factor of $M_{83}$.
(b) Setting $q=131$ gives $p=2(131)+1)=263$. Both 131 and 263 are prime, and so 131 is a Germain prime.
As $q \equiv -1 \pmod 4$, by Corollary (4.21) we have $263 \mid 2^{131}-1 $.
That is, 263 is a prime factor of $M_{131}$.
(c) Setting $q=179$ gives $p=2(179)+1=359$. Both 179 and 359 are prime, and so 179 is a Germain prime.
As $q \equiv -1 \pmod 4$, by Corollary (4.21) we have $359 \mid 2^{179}-1$.
That is, 359 is a prime factor of $M_{179}$.
(d) Setting $q=191$, gives $p=2(191)+1=383$. Both 191 and 383 are prime, and so 191 is a Germain prime.
As $q \equiv -1 \pmod 4$, by Corollary (4.21) we have $383 \mid 2^{191}-1$.
That is, 383 is a prime factor of $M_{191}$.