Given a number n, print all primes smaller than or equal to n. It is also given that n is a small number.
For example, if n is 10, the output should be “2, 3, 5, 7”. If n is 20, the output should be “2, 3, 5, 7, 11, 13, 17, 19”.
The sieve of Eratosthenes is one of the most efficient ways to find all primes smaller than n when n is smaller than 10 million
Following is the algorithm to find all the prime numbers less than or equal to a given integer n by Eratosthenes’ method:
- Create a list of consecutive integers from 2 to n: (2, 3, 4, …, n).
- Initially, let p equal 2, the first prime number.
- Starting from p, count up in increments of p and mark each of these numbers greater than p itself in the list. These numbers will be 2p, 3p, 4p, etc.; note that some of them may have already been marked.
- Find the first number greater than p in the list that is not marked. If there was no such number, stop. Otherwise, let p now equal this number (which is the next prime), and repeat from step 3.
When the algorithm terminates, all the numbers in the list that are not marked are prime.
[ad type=”banner”]Explanation with Example:
Let us take an example when n = 50. So we need to print all print numbers smaller than or equal to 50.
We create a list of all numbers from 2 to 50
According to the algorithm we will mark all the numbers which are divisible by 2.
Now we move to our next unmarked number 3 and mark all the numbers which are multiples of 3.
We move to our next unmarked number 5 and mark all multiples of 5.
We continue this process and our final table will look like below:
So the prime numbers are the unmarked ones: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
[ad type=”banner”]Implementation:
Following is C++ implementation of the above algorithm. In the following implementation, a boolean array arr[] of size n is used to mark multiples of prime numbers.
Output:
Following are the prime numbers below 30 2 3 5 7 11 13 17 19 23 29