Sure, pure mathematical theory has a tendency to become useful later, but finding specific Mersenne primes is not pure math.
It’s not known if there is an infinite number of Mersenne primes (though it is strongly suspected) - so an interesting mathematical object to discover would be ‘the largest Mersenne prime’ - but this process isn’t going to find that (it might find a candidate but it won’t find a proof). If there are indeed infinite Mersennes then this search is just guaranteed to keep finding more forever.
It’s like looking for green pebbles on a beach. It’s nice when you find one to add to your collection, but it’s not that surprising, and if you keep looking you’ll probably find more. You’re not advancing geological knowledge if you keep looking and finding more.
Yes but if you want to say if there are an infinite number of green pebbles on an infinitely large beach then the problem you need to solve is finding the distribution of the green pebbles on the beach. It's only possible to do that once you've found a few green pebbles and gets much easier if you've found many.
We don't know whether the search for further Mersenne primes eventually gives birth to a new algorithm which will enable further advances in other fields.
Cranking on the current best known algorithm for testing for Mersenne primes does nothing to advance the state of the art for other algorithms, though.
It’s like saying, we have a project that keeps sorting larger and larger sets using quicksort because we believe it will help advance the state of sorting algorithms. ‘We just sorted the largest ever set!’
It’s not known if there is an infinite number of Mersenne primes (though it is strongly suspected) - so an interesting mathematical object to discover would be ‘the largest Mersenne prime’ - but this process isn’t going to find that (it might find a candidate but it won’t find a proof). If there are indeed infinite Mersennes then this search is just guaranteed to keep finding more forever.
It’s like looking for green pebbles on a beach. It’s nice when you find one to add to your collection, but it’s not that surprising, and if you keep looking you’ll probably find more. You’re not advancing geological knowledge if you keep looking and finding more.