system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: How can there be a infinite number of prime numbers? Shouldn't you eventually completely run out?

Eventually the numbers get so big that there just HAS to be a number you can divide it with. Right?

EDIT: Thanks for the answer!!!
gpt
Nope. And this is actually a fact that is quite easy to prove in an ELI5 way!

Let's say you were right, and you had a list of all the prime numbers. Let's call them p_1, p_2, p_3... all the way to p_n. Maybe n is very large (maybe there's 57296572365782365863 primes), but it's still a finite number. Our goal is to show that if you're right, we get a contradiction, and so you must be wrong.

Take this list of primes, all the primes you say exist, and multiply them together to form a very big number. Let's call that number m, that is, m = (p_1)\*(p_2)\*(p_3)...\*(p_n). Presumably this is a very very large number, but that's fine. Now, add one to *m*. So we have all the primes multiplied together and added one.

Now, *m+1* is clearly bigger than anything in our list (by a long shot), and we said that was all the primes that exist. So *m+1* is not prime, since we assumed we had a list of *all* of the primes, leaving nothing out. It must, therefore, have a prime factor somewhere in the list we gave.

But since p_1 is clearly a factor of m (since we built m by multiplying a bunch of numbers that included p_1), it cannot be a factor of m+1, since (m+1) divided by p_1 would give us a remainder of 1. The same is true for p_2, for p_3, for p_4, and so on. This is a contradiction: we said it must have a factor in the list (since all numbers would have to, if the list contained all primes), but we also showed it can't have one.

EDIT: As a side note, the reason that we can't perform this same process with infinitely many primes is the multiplication step: multiplying an *infinite* number of primes won't work.