So I thought up a "proof" for infinite primes. I'm assuming I did something wrong, but I don't know what, it would be nice if someone could tell me what I did wrong. Suppose there are a finite number of primes of quantity n which are listed from smallest to largest in the list: p1, p2, ... , pn. Some number k is not divisible by some prime pi when: k = pi + m, where 0<m<pi. Following this, for every number prime pi divides, there are m many numbers that are not divisible by pi. This means that any prime pi appended to our list of primes can at most factor into 1/pi of the remaining number line which its predecessors had not been able to factor into. By the fundamental theorem of arithmetic, a list of all primes would leave 0 unfactorable remaining numbers. The process of reducing the size of the unfactorable numbers remaining on the number line can be written as: ∏ni=1 1/pi > 1/(pn!) > 0 Therefore a finite list of primes cannot reduce the quantity of unfactorable numbers to 0.