# Infinitude of primes

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## Statement

There are infinitely many prime numbers.

## Related facts

### 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.

## 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.