system
Given the following post, respond as if you're commenting on Reddit:
human
[Mathematics] If Euclid derived a formula for finding a prime higher than all the other primes, why is us discovering a new prime so important?

For example... Euclid said there are infinitely many primes because you can multiply all the primes you know together, add one, and that number will also be prime. So can't we just use that method to get more primes easily?
gpt
You've misunderstood the proof; given the first n primes, it is not, in general, true that multiplying them together and adding 1 will give you a prime number. For example, the first six prime numbers are 2, 3, 5, 7, 11, and 13. If you multiply those together and add 1, you get

(2\*3\*5\*7\*11\*13) + 1 = 30031,

which isn't prime since 30031 = 59\*509.

What Euclid's proof really says is that *either* that number is prime *or* there is a prime, call it p, that divides it. However, p can't be any of the primes you started with, because that would imply that it divides 1 (take the number above and subtract the product of the primes). Thus, there is a prime not on the list.

Note also that if you take some arbitrary collection of primes, you can't even guarantee that the "new prime" you get will be bigger than all of the ones with which you started.