Infinitude of Prime Numbers

Theorem 6:  Infinitude of Prime Numbers

There are an infinite number of prime numbers.

Proof:

(1)  Assume that there is a a finite number of primes. 

(2)  Let $P$ be the product of all primes.

(3)  Let $q = P+1$

(4)  By the Fundamental Theorem of Arithmetic, either $q$ is prime or $q$ is a product of primes.

(5)  If $q$ is prime, then $q | (P + 1) - P = 1$ which s impossible.

(6)  If $r$ is a prime that divides $q$, then we have the same problem since $r | (P + 1) - P = 1$