Mersenne primes are prime numbers which are one less than a power of two, i.e. primes expressible as $M_n=2^n-1$. They are notoriously far apart and unpredictable, which is indicated by the fact that only 49 Mersenne primes have been found as of August 2016.
They've also been the subject of interest because so far, Mersenne primes have also corresponded to perfect numbers.
Unlike normal prime numbers, which can be generated much faster using e.g. sieve algorithms, all the Mersenne primes found so far are well-documented because there are so few of them; as a result, they can't be used in Diffie-Hellman cryptography, one of the more prominent applications of normal primes (this is because the primes used in generating keys for DH cryptography must be very large and difficult to guess).
Do Mersenne primes have practical applications, or are they just a curiosity?