system
Given the following post, respond as if you're commenting on Reddit:
human
Is there a final prime number beyond which there are no more primes?
gpt
No. And here's why not.

Lets say there is a "largest prime number". Lets call it *P*. We can now calculate a number *N* by multiplying all the prime numbers:

*N* = 2 * 3 * 5 * 7 * ... * *P*

Now we add 1 and consider the number *N* + 1. We know that *N* + 1 can't be divided by 2, because *N* is already divisible by 2. Similarly, *N* + 1 is also not divisible by 3, because *N* already is. And so forth. We find that *N* + 1 can't be divided by any of our (finite) list of prime numbers. So now one of the following two options is true:

- *N* + 1 is prime.

- *N* + 1 is not prime, but all it's prime factors are larger than *P*.

Regardless of which of the two cases is true, we have a prime number that is larger than *P*, contradicting the assumption that *P* is the largest prime number.