system
Given the following post, respond as if you're commenting on Reddit:
human
Why can't we use the "Infinite primes proof" to continuously find new primes?

In the infinite primes proof, you multiply all of the primes together and add 1, which creates a new prime. Why can we not use this method with the primes that we currently know to find new primes? People always talk about "finding" new primes, but it seems like using the method above would result in a guaranteed new prime number.
gpt
>In the infinite primes proof, you multiply all of the primes together and add 1, which creates a new prime.

This is incorrect. The number you produce is not necessarily a new prime. For example: 2\*3\*5\*7\*11\*13+1=59\*509. 

Instead, any prime that divides this number cannot be any of the primes that you originally multiplied together. The way the proof then works is you take the first n primes p*_1_*,...,p*_n_*, multiply them together to get N=p*_1_*p*_2_*...p*_n_*. Because N+1 is a number bigger than 1, it follows that some prime p divides N+1. This prime cannot be any of the original primes p*_1_*,...,p*_n_*, because otherwise p would divide N and N+1, and so it would have to be the case that p divides 1, which can't happen. Therefore, any prime that divides N+1 is not on the original list of primes. Hence there are infinitely many primes.

If you could factor numbers efficiently, then you could use these numbers to find primes. But it is easier to find primes than it is to factor so this isn't a good route.