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.