Eratosthenes' sieve is an algorithm to make a
list of primes.
Eratosthenes' sieve
- Input: a positive integer n.
- Output: the list of primes less than or equal to n.
- Construct the list L := [2, ..., n] and the empty list M.
- Let m be the smallest element in L.
- Add m to M.
- Remove all multiples of m from L.
- Repeat Step 2 until L is empty.
- Return M.
Using this sieve we can find all the primes in the interval [1,n].
The number of such primes can be approximated as follows.
Fact: Prime number
theorem
Let prime(n) be the number of primes in the interval [1,n]. Then we have
prime(n) ~ n/log(n)
when n tends to infinity.
The functions prime(n) and n/log(n)