Question

I have got one question to print one million prime numbers . I have written a java program for that .. It’s currently taking 1.5 mins approx to calculate it .. I think my solution is not that efficient. I have used the below algo:

• Adding 1 2 3 to the prime list initially
• Calculating the last digit of the number to be checked
• Checking if the digit is 0 , 2 or 4 or 6 or 8 then skipping the number
• else calculating the square root of the number ..
• Trying to Divide the number starting from 2 till the square root of the number
• if number is divisible then skipping the number else adding it to the prime list

I have read several other solutions as well , but I didn’t find a good answer. Please suggest ideally what should be approx minimum time to calculate this and what changes are required to make the algorithm more efficient.

Answer 1

A simple sieve of Eratosthenes runs like the clappers. This calculates the 1,000,000th prime in less than a second on my box:

class PrimeSieve
{
    public List<int> Primes;

    private BitArray Sieve;

    public PrimeSieve(int max)
    {
        Primes = new List<int> { 2, 3 }; // Must include at least 2, 3.
        Sieve = new BitArray(max + 1);
        foreach (var p in Primes)
            for (var i = p * p; i < Sieve.Length; i += p) Sieve[i] = true;
    }

    public int Extend()
    {
        var p = Primes.Last() + 2; // Skip the even numbers.
        while (Sieve[p]) p += 2;
        for (var i = p * p; i < Sieve.Length; i += p) Sieve[i] = true;
        Primes.Add(p);
        return p;
    }
}

EDIT: sieving optimally starts from p^2, not 2p, as Will Ness correctly points out (all compound numbers below p^2 will have been marked in earlier iterations).