Given an integer N > 1, you want to find all of the prime numbers from 1 to N. For example, if N = 11, then you get {2, 3, 5, 7, 11}.

The Sieve of Eratosthenes is a simple way to solve this problem. The basic idea behind it is this: Construct the list of all integers from 1 to N, and remove the composite numbers.

It is easy to construct the list of all integers from 1 to N. So, the question is: How do you remove the composite numbers?

It is a fact of number theory that every composite number k has a prime factor that is less than or equal to sqrt(k). To see this, consider any composite number k = mn. If both m > sqrt(k) and n > sqrt(k), then k < mn, which contradicts the fact that k = mn. So, k must have a prime factor that is less than or equal to sqrt(k).

So, every composite number from 1 to N must have a prime factor that is less than or equal to sqrt(N). So, just iterate over each prime number from 1 to sqrt(N) and remove all of its multiples in the list. In that way you remove all of the composite numbers from the list.

I implemented the Sieve of Eratosthenes in C, and I implemented the Sieve of Eratosthenes in Python. Both of my implementations are up at my GitHub.

References:

https://proofwiki.org/wiki/Sieve_of_Eratosthenes

http://pages.pacificcoast.net/~cazelais/222/sieve.pdf

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