If there are infinitely many Mersenne primes, more efficient error-correcting codes exist for problems in which only a few bits of a message need to be recovered, given a high transmission error rate. Even finitely many additional such primes would improve the length of these codes; this kind of coding has applications in both cryptography and the theory of approximation algorithms. And you probably thought (as did I) that all the computational effort devoted to the search for Mersenne primes was purely an exercise in recreational math...