Given this contest can presumably go on infinitely long, what is the ultimate point of the contest? Is there some kind of theoretical or practical benefit to discovering a new Mersenne prime?
The EFF Cooperative Computing Awards, which pay out money for (four specific sizes of) prime records, were meant to show off how the Internet is useful for letting people who don't even know each other work together to solve problems. They were established back in 1998, when people in general were much less familiar with the Internet and its impact. That specific contest isn't set up to go on forever, as it ends when a billion-digit prime is discovered.
The search for different kinds of mathematical objects sometimes has applications and sometimes doesn't. For example, apparently the search for Golomb rulers (another distributed computing project) has some conceivable applications.
There's a misconception (that I heard dozens or hundreds of times when I was running the Cooperative Computing Awards at EFF) that discovering world record primes is useful to cryptography. In fact, it has no direct application because all of the primes used in number-theoretic algorithms like RSA and classic Diffie-Hellman are dramatically smaller than world-record sizes, and can be generated on an ordinary PC in seconds. (Try "openssl prime -generate -bits 2048" at your command line!)
(There is a wild paper from 2017 called "Post-quantum RSA" arguing that we could in principle just scale up RSA keys for post-quantum security, but that paper uses multiprime RSA moduli composed of large numbers of 4096-bit primes, instead of just the traditional two primes, so even that approach doesn't require individual primes that are especially large or hard-to-find by computer standards.)
We have apparently learned a bit about number theory and algorithms as a result of research done by the GIMPS project and its collaborators about how to optimize some of the arithmetic in the GIMPS client. I guess that's the equivalent of the claim that the space program produced various spin-off technologies while pursuing space exploration.
In general, since there are infinitely many primes, there's no reason that humanity can't keep looking for larger and larger ones indefinitely. Likewise for many other searches, both for objects which are known to be infinitely numerous and objects which may or may not have a largest example. Mostly this kind of activity has a "because it's there" flavor to it.
Research mathematicians are mostly not that excited about this activity, because it isn't generating more understanding or hypotheses about mathematical structure. They're usually more excited about insights that reveal or hint at new patterns or structure, which searching for large primes doesn't really do (most of the work is mechanical, performed by computers, and the outputs aren't very surprising or suggestive in a mathematical sense).
I've hoped that publicity about discoveries like record primes might get more young people interested in mathematics (and maybe about topics like number theory and discrete math that they might not be encountering in school). I got kind of a sad view of this because people were constantly writing to me asking for monetary rewards for their inevitably-mistaken-or-confused "discoveries", but I'd like to think that there are also people out there who got curious about what we do and don't know about primes. A good place to start with that is the Prime Pages
And wasn't that paper just poking fun at people who still used something as outdated as RSA as the default public key crypto primitive in the 2010s by extrapolating their position to its logical extreme?
I think there was an intended humor element there, particularly since some of those people were also working on new post-quantum primitives, but they also "committed to the bit" and did the research for real.
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.
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
I didn't claim there are an infinite number of Mersenne primes. I said the contest could presumably go on infinitely long. The latter doesn't require the former. It's only predicated on us lacking proofs about how many there must be.
Yes, the Mersenne primes do indeed extend to infinity. That is due to the simple fact that perfect numbers (for which they are intimately connected to) are also known to be infinite in number.
The traditional proof that there are an infinite number of primes relies on unique prime factorisation- i.e for any number, n, there is a unique set of primes p1, p2, p3, … etc. where p1 * p2 * p3 * … = n
For instance 88 = 2 * 2 * 2 * 11, 42 = 2 * 3 * 7
It’s worth reading the proof if you haven’t - it’s comprehensible with high school maths.
No such property exists for Mersenne primes, so we can’t trivially extend it. Many proofs of the properties of prime numbers are difficult because they, by definition, actively resist patterns.
