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 $r | (P + 1) - P = 1$
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