What’s the point to searching for the largest prime number?
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It’s been known since Euclid that there are infinitely many primes, so they aren’t searching for the largest one. And you should bear in mind that this is a pretty niche activity – it’s one of those things that gets disproportionate attention because it’s easy to explain and it involves impressively large numbers.
Finding individual large primes isn’t really that useful. Afaik, the very largest known primes are not actually used in cryptography at all. There are various things that are unknown about the primes, but it isn’t very likely that finding a few more will shed much light on those questions. But developing *techniques* to find large primes more efficiently is potentially useful and interesting, because those techniques might be applicable to other problems and might shed light on some theoretical issues (e.g. the development of the AKS primality test led to a proof that the problem of determining whether a number is prime is within a certain complexity class).
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