Euclid's Theorem

Theorem
For any finite set of prime numbers, there exists a prime number not in that set.

Proof
Let $\mathbb P$ be a finite set of prime numbers.

Consider the number:
 * $\displaystyle n_p = \paren {\prod_{p \mathop \in \mathbb P} p} + 1$

Take any $p_j \in \mathbb P$.

We have that:
 * $\displaystyle p_j \divides \prod_{p \mathop \in \mathbb P} p$

Hence:
 * $\displaystyle \exists q \in \Z: \prod_{p \mathop \in \mathbb P} p = q p_j$

So:

So $p_j \nmid n_p$.

There are two possibilities:


 * $(1): \quad n_p$ is prime, which is not in $\mathbb P$.


 * $(2): \quad n_p$ is composite.

But from Positive Integer Greater than 1 has Prime Divisor‎, it must be divisible by some prime.

That means it is divisible by a prime which is not in $\mathbb P$.

So, in either case, there exists at least one prime which is not in the original set $\mathbb P$ we created.

Also see

 * Furstenberg's Proof of Infinitude of Primes


 * Definition:Euclid Number