求素数和的更有效方法是什么？

Question

I am working on a program that tracks the amount of time it takes to get the sum of all prime numbers up to a certain number and am trying to find the most efficient possible way to obtain this value, as I have a Stopwatch (System.Diagnostics) tracking how long it takes. Currently, I can find the sum of all prime numbers up to 40,000 in about 33-34 seconds with the below code:

private void ListThePrimes()
        {
            prime = false;
            while (primes < 30000)
            {
                for (int i = 2; i < n; i++)
                {

                    output = n % i;
                    if (output == 0)
                    {
                        primeNum = i;
                        prime = false;
                        break;
                    }
                    else
                    {
                        prime = true;
                    }
                }
                if (prime == true)
                {
                    sum += primeNum;
                    primes++;
                }
                n++;
            }
        }

However, I feel like there is a way to write this code more efficiently as my goal was to reach the same amount of time with much higher numbers like 200,000 or so. This is my Stopwatch code, which I perform on a button click, if needed:

    var timer = new Stopwatch();
            timer.Start();
            ListThePrimes();
            timer.Stop();
            TimeSpan timeTaken = timer.Elapsed;
            string foo = timeTaken.ToString(@"m\:ss\.fff");
            MessageBox.Show("The sum is " + sum + ". It took this program " + foo + " seconds to run.");

Would appreciate it if someone could let me know if there is a more efficient way to perform this action.

Answer 1

You need to optimize how you get prime numbers, your way is extremely inefficient. The common way to do so is using the Sieve of Eratosthenes . Using this method I can easily get all prime numbers up to 100000 in milliseconds. Summing them is trivial beyond that.

var n = 100000;
var a = Enumerable.Range(0,n+1).Select(_ => true).ToArray();

for(var i=2;i<Math.Sqrt(n);i++)
{
    if(a[i])
    {
        for(var j = i*i;j<=n;j += i)
        {
            a[j] = false;
        }
    }
}

var result = a.Select( (x,i) => new {IsPrime = x,Prime = i})  
              .Where(x => x.IsPrime && x.Prime > 1) 
              .Sum(x => x.Prime);
    
Console.WriteLine(result);

Live example: https://dotnetfiddle.net/eBelZD

Answer 2

You really have to work on your prime number method. A new approach would be the Erathostenes sieve, but your current code can also be improved quite a bit.

1. Remove unnecessary variables like output, just put it directly in the if tag.
2. You can half the numbers you're checking because primes can only be odd numbers, so don't do n++ but n+=2;.