Euclid's Theorem/Corollary 1

Corollary to Euclid's Theorem
There are infinitely many prime numbers.

Proof
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:
 * $\mathbb P = \left\{{p_1, p_2, \ldots, p_n}\right\}$

From Euclid's Theorem, however, we can always create a prime which is not in $\mathbb P$.

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.

Fallacy
There is a danger in the proof of this corollary.

It is often seen to be stated that: the number made by multiplying all the primes together and adding $1$ is not divisible by any members of that set, so it is not divisible by any primes and "is therefore itself prime".

That is, sometimes readers think that if $P$ is the product of the first $n$ primes then $P + 1$ is itself prime.

This is not the case. For example:
 * $\left({2 \times 3 \times 5 \times 7 \times 11 \times 13}\right) + 1 = 30\ 031 = 59 \times 509$

both of which are prime, but, take note, not in that list of six primes that were multiplied together to get $30\ 030$ in the first place.

Also see
During the course of the proof of the corollary to Euclid's Theorem, the assupmption is made that all the primes (of our supposedly finite set) are multiplied together, and $1$ added. The resulting number is then either a prime or contains a prime factor not on that original list.

Such a number is known as a Euclid number.

Although this is not quite the way Euclid originally stated his original theorem, it is such a well-known and accessible proof that it is considered to have entered the mainstream of "general knowledge".