Infinitude of primes

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Statement

There are infinitely many prime numbers.

Related facts

Stronger facts about the distribution of primes

Stronger facts about distribution of primes in congruence classes

Large set

Generalizations to other rings

Values taken by polynomials

Proofs involving the construction of a new relatively prime number

Euclid's proof

This proof constructs a new number relatively prime to any given collection of primes, forcing there to be infinitely many primes.

Goldbach's theorem involving Fermat numbers

This proof shows that in the set of Fermat numbers, any two elements are relatively prime, and hence, there are infinitely many primes among the set of prime divisors of Fermat numbers.