How to verify large Mersenne Primes [duplicate]

As of December 2017, the largest known prime number was the Mersenne prime $2^{77232917} – 1$.

For such a large Mersenne prime, what are the techniques available for one to verify that it is in fact a prime? Is it simply a matter of brute force checking, or are there other more elegant techniques available?

marked as duplicate by Sil, Did, Lord Shark the Unknown, user99914, Eric WofseyAug 12 '18 at 14:52

Let $p$ be an odd prime.
Define a sequence $\{s_i\}_{i \ge 0}$ by:
1. $s_0 = 4$
2. $s_{n+1} = s_n^2 - 2$
Then $2^p - 1$ is prime iff $s_{p-2} \equiv 0 \pmod {2^p - 1}$.
(For your case, $p = 77232917$.)