>>10833599>What is the proof that numbers are infinite?For any positive integer n, n! is divisible by the positive integers from 2 to n (its definition to begin with is literally just that it's the product of these same integers).
e.g. 5! = 120 and 120 is divisible by 5 (24), 4 (30), 3 (40), and 2 (60).
n! + 1 is NOT divisible by ANY of the positive integers from 2 to n (again it's basically defined up front in a transparent way for this conclusion to follow since it's simply the product of those same integers but with a remainder of 1 making it not divisible).
e.g. 5! + 1 = 121 and 121 is not divisible by 5 (24 with a remainder of 1), 4 (30 with a remainder of 1), 3 (30 with a remainder of 1), and 2 (60 with a remainder of 1).
So now we get to conclude for any positive integer n that n! + 1 must either be:
A) A prime number or
B) A composite (non-prime) number divisible by some new prime number that wasn't one of the factors of n!
5! + 1 falls into the A category (121 is prime).
4! + 1 (25) falls into the B category (25 is divisible by 5, and 5 is not one of the positive integers from 2 to n that factor into n! when n=4).
In either the case of A or B what we've established is there is always another new prime number every time you evaluate n! + 1.
This means there are infinite prime numbers.
This means there are infinite numbers.
