As we have already proven there are infinitely many prime numbers, there really isn’t much point. Some people will say cryptography, but that doesn’t require primes anywhere near as large as the largest primes we’ve found.

Basically, there’s two purposes to the search for large primes: a demonstration of new mathematical methods, and as a benchmark for computer systems. Finding prime numbers is a very simple, well-understood problem, so it’s a very effective test of computing power.

As we have already proven there are infinitely many prime numbers, there really isn’t much point. Some people will say cryptography, but that doesn’t require primes anywhere near as large as the largest primes we’ve found.

Basically, there’s two purposes to the search for large primes: a demonstration of new mathematical methods, and as a benchmark for computer systems. Finding prime numbers is a very simple, well-understood problem, so it’s a very effective test of computing power.

Large prime numbers are used in encryption algorithms. Multiplying two primes together is easy, but factoring a prime is hard. It’s this difficulty of factoring primes that make encryption hard to break. The bigger the starting primes, the better.

Besides cryptography, we search for new primes simply because we can. We humans often like to do stuff that’s hard for its own sake, even if it doesn’t have an obvious use case.

Large prime numbers are used in encryption algorithms. Multiplying two primes together is easy, but factoring a prime is hard. It’s this difficulty of factoring primes that make encryption hard to break. The bigger the starting primes, the better.

Besides cryptography, we search for new primes simply because we can. We humans often like to do stuff that’s hard for its own sake, even if it doesn’t have an obvious use case.

Large prime numbers are used in encryption algorithms. Multiplying two primes together is easy, but factoring a prime is hard. It’s this difficulty of factoring primes that make encryption hard to break. The bigger the starting primes, the better.

Besides cryptography, we search for new primes simply because we can. We humans often like to do stuff that’s hard for its own sake, even if it doesn’t have an obvious use case.

As we have already proven there are infinitely many prime numbers, there really isn’t much point. Some people will say cryptography, but that doesn’t require primes anywhere near as large as the largest primes we’ve found.

Basically, there’s two purposes to the search for large primes: a demonstration of new mathematical methods, and as a benchmark for computer systems. Finding prime numbers is a very simple, well-understood problem, so it’s a very effective test of computing power.

Part of it is a deeper understanding of number theory. That is, calculating very large prime numbers can be very hard. The formulas used to calculate primes are like tools: once you have the tool, you can find other uses for it, and using that tool can give mathematicians insights about how numbers work in general.

Similarly, prime numbers that big aren’t calculated by hand anymore, it’s done by computers. Building the computers and programs to do it gives tools for computers that can help develop computer programs in general.

The most immediate application is for cryptography. Encrypting messages for things like sending data over the internet involves using big prime numbers as the “key”. Although computers don’t need to use the absurdly large primes people keep finding, the method to find those primes helps find smaller primes that *can* be used for cryptography. And, they can potentially help *break* cryptography.

Part of it is a deeper understanding of number theory. That is, calculating very large prime numbers can be very hard. The formulas used to calculate primes are like tools: once you have the tool, you can find other uses for it, and using that tool can give mathematicians insights about how numbers work in general.

Similarly, prime numbers that big aren’t calculated by hand anymore, it’s done by computers. Building the computers and programs to do it gives tools for computers that can help develop computer programs in general.

The most immediate application is for cryptography. Encrypting messages for things like sending data over the internet involves using big prime numbers as the “key”. Although computers don’t need to use the absurdly large primes people keep finding, the method to find those primes helps find smaller primes that *can* be used for cryptography. And, they can potentially help *break* cryptography.

Part of it is a deeper understanding of number theory. That is, calculating very large prime numbers can be very hard. The formulas used to calculate primes are like tools: once you have the tool, you can find other uses for it, and using that tool can give mathematicians insights about how numbers work in general.

Similarly, prime numbers that big aren’t calculated by hand anymore, it’s done by computers. Building the computers and programs to do it gives tools for computers that can help develop computer programs in general.

The most immediate application is for cryptography. Encrypting messages for things like sending data over the internet involves using big prime numbers as the “key”. Although computers don’t need to use the absurdly large primes people keep finding, the method to find those primes helps find smaller primes that *can* be used for cryptography. And, they can potentially help *break* cryptography.

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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