section_path: Elementary properties/Infinitude
section_name: Infinitude
paragraph_text: Infinitude
  
  There are infinitely many prime numbers. Another way of saying this is that the sequence
  
  of prime numbers never ends. This statement is referred to as Euclid's theorem in honor of the ancient Greek mathematician Euclid, since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an analytical proof by Euler, Goldbach's proof based on Fermat numbers, Furstenberg's proof using general topology, and Kummer's elegant proof.
  
  Euclid's proof shows that every finite list of primes is incomplete. The key idea is to multiply together the primes in any given list and add 1. {\displaystyle 1.} If the list consists of the primes p 1, p 2, …, p n, {\displaystyle p_{1},p_{2},\ldots,p_{n},} this gives the number
  
  By the fundamental theorem, N {\displaystyle N} has a prime factorization
  
  with one or more prime factors. N {\displaystyle N} is evenly divisible by each of these factors, but N {\displaystyle N} has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of N {\displaystyle N} can be in the given list. Because there is no finite list of all the primes, there must be infinitely many primes.
  
  The numbers formed by adding one to the products of the smallest primes are called Euclid numbers. The first five of them are prime, but the sixth,
  
  is a composite number.