Return all prime numbers smaller than M

Question

Given an integer M. return all prime numbers smaller than M.

Give a algorithm as good as you can. Need to consider time and space complexity.

Answer 1

i'm a novice programmer in c# (and new to S.O.), so this may be a bit verbose. nevertheless, i've tested this, and i works.

this is what i've come up with:

for (int i = 2; i <= n; i++)
{
    while (n % i == 0)
    {
        Console.WriteLine(i.ToString());
        n /= i;
    }
}
Console.ReadLine();

Answer 2

You can do it using a bottom up dynamic programming approach called the Sieve of Eratosthenes Basically you create a boolean cache of all numbers upto n and you mark each the multiples of each number as not_prime. Further optimizations can be gained by checking only upto sqrt(n) since any composite number will have at least one divisor less that sqrt(n)

    public int countPrimes(int n) {
    if(n==0){
        return 0;
    }else{
        boolean[] isPrime=new boolean[n];
        for(int i=2;i<n;i++){
            isPrime[i]=true;
        }

        /* Using i*i<n instead of i<Math.sqrt(n)
         to avoid the exepnsive sqrt operation */
        for(int i=2;i*i<n;i++){
           if(!isPrime[i]){
               continue;
           }
            for(int j=i*i;j<n;j+=i){
                isPrime[j]=false;
            }
        }

        int counter=0;
        for(int i=2;i<n;i++){
            if(isPrime[i]){
                counter++;
            }
        }

        return counter;

    }
}

Answer 3

The Sieve of Atkin is also the best algorithm to implement in this case and it takes only O(N) operations and O(N) space. Please refer https://en.wikipedia.org/wiki/Sieve_of_Atkin for detailed explanation of the algorithm and pseudocode.