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To learn something about primes.

Perhaps there is a pattern or a way to more-accurately predict which numbers will be prime.

Also, it is cool.

Do we expect to learn something interesting from any given new Mersenne prime discovered? Don't we kinda have enough so that if we are going to discover something interesting by analyzing them, we can do that with the ones we already know about?

I'm not an SME but I would imagine that if there's some sort of very large scale pattern in Mersenne primes then finding that might lead to the discovery of some currently unknown emergent property. Of course this argument is likely unfalsifiable, as it can scale infinitely if there's an infinite amount of them, though we don't even know whether they are a finite set or not.

Isn't the fact that a certain number is a Mersenne prime interesting in itself? Surely it's a great addition to our knowledge of numbers?

Sure, I'm not against GIMPS or anything, I'm just a bit doubtful that finding a few more Mersenne primes could be the key to unlocking some secret pattern in the primes is all

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