It could happen (without catastrophe/aliens) if: (a) someone discovered a pattern like the Mersenne numbers which unreliably produces primes that are much larger than Mersenne numbers, and (b) someone discovers, by hand, a proof that one of these new numbers is prime. Such a discovery would be wonderful: new mathematics that could give us insight into what primes really are.
For instance, they are points in the geometric object known as the scheme over the integers.[1] The Riemann Zeta function of the Riemann Hypothesis generalizes to other schemes, and reflects their geometric properties (being vague because it's over a decade since I studied this stuff.) It is just barely in the realms of possibility that a new computationally simple way of finding large primes would also reflect something about the geometric structure of schemes over Z. (But it seems pretty damn unlikely to me.)
I didn't know why primes are considered special, apart from the basic requirement they fulfill. GP sounded like they might serve a bigger unknown purpose.
https://en.wikipedia.org/wiki/Double_Mersenne_number might be one place to start.