Ever since 1952, the largest known prime has been a Mersenne prime, with one exception in 1989. One reason is that it is simple to test whether Mersenne numbers are prime using the Lucas-Lehmer test. The algorithm is described in seven lines of pseudo-code here.
Here are a couple connections with Mersenne and his primes I’ve written about before. First, Mersenne is one of my mathematical ancestors. Second, Mersenne primes are intimately connected with even perfect numbers, a connection that has been known since Euclid.