**Objective:** Given a number N, Write a program to find all prime numbers which are between 0 and N using Sieve of Eratosthenes algorithm.

**Prime Number:** A prime number is a natural number that has exactly two distinct natural number divisors: 1 and itself.

**Example:**

N = 10 Output: 2 3 5 7 N = 60 Output: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59

Earlier we had discussed O(*N* Sqrt(N)) solution – Find all Prime Numbers less than equal to N | Set 1. In this post we will discuss the Sieve of Eratosthenes algorithm to do the same, this algorithm works better than the approach we had discussed earlier.

**Sieve of Eratosthenes Algorithm:**

To find all the prime numbers less than or equal to a given integer *n* by Eratosthenes’ method:

- Create a binary array of size
*N*, let’s say it*prime[]* - Put 1 at all the indexes of the array,
*prime[]*. - Iterate
*p*=*2*to*N*(Will start from 2, smallest prime number).- check if
*prime[p] =1*, if yes then*p*is a prime number. - Enumerate the multiples of
*p*by counting in increments of*p*from 2*p*to*n*, and mark them in the array as*prime[2p]=0*,*prime[3p]=0*and so on (these will be 2*p*, 3*p*, 4*p*, …; the*p*itself should not be marked). - do the above two steps for all the
*p*from*2 to N*

- check if
- When the algorithm terminates, the indexes in the array
*prime[]*for which the values are 1, are all the primes below*N*.

See the walkthrough as an example for more understanding.

N = 60, create an array prime[] from 1 to 60 and put 1. (Color all white)

Find the first number in an array which is white (means prime[p]=1), found p = 2

Enumerate the multiples of p by counting in increments of p from 2p to n, and mark them in the array as prime[2p]=0, prime[3p]=0 (color these green), 2 is a prime number (color it pink)

Find the next number in the array which is white, (means prime[p]=1), found p = 3

Enumerate the multiples of p by counting in increments of p from 2p to n, and mark them in the array as prime[2p]=0, prime[3p]=0 (color these green), 3 is a prime number (color it pink)

Find the next number in the array which is white, (means prime[p]=1), found p = 5

Enumerate the multiples of p by counting in increments of p from 2p to n, and mark them in the array as prime[2p]=0, prime[3p]=0 (color these green), 5 is a prime number (color it pink)

Find the next number in the array which is white, (means prime[p]=1), found p = 7

Enumerate the multiples of p by counting in increments of p from 2p to n, and mark them in the array as prime[2p]=0, prime[3p]=0 (color these green), 7 is a prime number (color it pink)

When the algorithm terminates, the indexes in the array *prime[]* for which the values are 1(pink color), are all the primes below *N*

**Complete Code:**

**Output**:

Number of Prime numbers less than N=60

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59

**Reference**: Wiki, geeksforgeeks