Infinitude of primes
- 1 Statement
- 2 Related facts
- 3 Proofs involving the construction of a new relatively prime number
There are infinitely many prime numbers.
Stronger facts about the distribution of primes
- Bertrand's postulate: This states that for any natural number , there is a prime between and .
- Prime number theorem: This is a statement about the prime-counting function: the number of primes up to a certain number. The theorem gives a good estimate for the growth of this function.
Stronger facts about distribution of primes in congruence classes
- There are infinitely many primes that are one modulo any modulus
- Dirichlet's theorem on primes in arithmetic progressions
- Chebotarev density theorem
- Set of primes is large: In other words, the sum of the reciprocals of the primes diverges.
Generalizations to other rings
- Unique factorization domain that is not a field has either infinitely many units or infinitely many associate classes of irreducibles
Values taken by polynomials
- Set of prime divisors of values of nonconstant polynomial with integer coefficients is infinite
- Nonconstant polynomial with nonzero constant term has infinitely many pairwise relatively prime values
Proofs involving the construction of a new relatively prime number
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.