**Euclid's theorem** is a fundamental statement in number theory which asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.

A theorem sometimes called "Euclid's first theorem" or Euclid's principle states that if

is a prime and

, then

or

(where

means divides. A corollary is that

Euclid's second theorem states that the number of primes is infinite. This theorem, also called the Infinitudes of Primes theorem, was proved by Euclid in Proposition IX.20 of the Elements. Euclid's elegant proof proceeds as follows. Given a finite sequence of consecutive primes 2, 3, 5, ...,

, the number

Known as the

th Euclid Number when

is the

th prime, is either a new prime or the product of primes.

The Geometry Formulas site has more detailed information about the Euclidean Theorem.