Euclid Infinitely Many Prime Numbers

Theorem: There are infinitely many prime numbers.

Proof by Contra-positive

pf: Suppose there are finitely many prime numbers $\{p_2,p_2,p_3\dots p_n\}$ Let $m=p_1\ast p_2\ast p_3\cdots \ast p_n$ By law of prime numbers, $m$ is either prime or a product of primes.

• $m$ is larger than our list of prime numbers
• $m$ is not a product of all primes, since it is already the product of all + 1 Hence, there is no value of $m$ that exists in a finite prime number set. Therefore there must be infinitely many prime numbers $\square$