Here’s an interesting account of the largest known primes over time. Thanks to @mathematicsprof for pointing this out.

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.

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You should mention that Mersenne primes are essential to pseudo-random numbers generation. Particularly in multiplicative congruential generators, the modulus is tipically a Mersenne prime.