Possibly the only civilian use of quantum computing, an interesting decade is ahead.
Primality testing is actually much easier than integer factorization thanks to Fermat's "little" theorem. Methods based on this theorem, including the Lucas-Lehmer test for Mersenne primes, can identify prime numbers on classical computers in polynomial time. However, factorization is not possible with this method.
Google told me that, surprisingly, it's not even known that they are infinite!
I found this nice page on the Wagstaff Mersenne Conjecture:
https://primes.utm.edu/mersenne/heuristic.html