Euclid's Theorem

Theorem
There are infinitely many prime numbers.

Proof by Contradiction
Assume that there are only finitely many prime numbers, and that there is a grand total of $$n$$ primes.

Then it is possible to define the set of all primes: $$\mathbb{P} = \left\{{p_1, p_2, \ldots, p_n}\right\}$$.

Consider the number: $$n_p = \left({\prod_{i=1}^{n} p_i}\right) + 1$$.

Take any $$p_j \in \mathbb{P}$$:

So $$p_j \nmid n_p$$.

There are two possibilities:


 * $$n_p$$ is prime, in which case it is greater than any in $$\mathbb{P}$$.

That means $$\mathbb{P}$$ does not contain all the primes after all.


 * $$n_p$$ is composite. But from Every Positive Integer Greater than 1 has a Prime Divisor‎, it must be divisible by some prime.

That means it is divisible by a prime which is not in $$\mathbb{P}$$, and again $$\mathbb{P}$$ is not complete.

So we can never create a finite list of all the primes, because we can guarantee to construct a number which has prime factors that are not in this list.

Thus, there are infinitely many prime numbers.

Q.E.D.