Question about the exponents n in Mersenne Primes. I am starting to study Mersenne primes, and I am wondering if there is a pattern in which exponents give rise to a Mersenne prime or if I am missing something. Thanks.
 A: The exact pattern isn't known, but we can say a few things:

*

*Exponents must be prime ( defining quality of a Mersenne number in some cases). as :

$$2^{ab}-1=2^{ab-a}(2^a-1)+2^{ab-a}-1$$

*

*Exponents can't be in the middle of Cunningham chains of first kind , as then they'll be of form $p=4c+3$, and have $2p+1$ prime . This makes the latter a divisor of the Mersenne number in question ( or the Mersenne number if $p=3$ the one exception).


*Exponents can't be at the start of a Cunningham  chain of first kind, unless of form $4d+1$  as otherwise you can use the same rule as previous.


*Divisors of Mersenne numbers of prime exponent $p$ ,take on form $2kp+1$ for $k\equiv 0,-p\pmod 4$ and that makes them 1 or 7 mod 8.


*Double, triple, and higher order Mersenne numbers, that can be written with Mersenne number exponent have factors that can be written in the same form with $p=2jr+1$, $r$ the Mersenne exponent of the Mersenne exponent, ...


*Mersenne primes, pass the Lucas-Lehmer primality test. and are trial factored to high bit levels to avoid the test as much as possible. ( with modern changes to testing including GPU use, and PRP test use at last check).


*There are heuristic of where the next exponent should be, but nothing conclusive.
EDIT
to answer your question the last one found by hand was $M_{127}$ or $170141183460469231731687303715884105727$ if you prefer.
