How can I test for primality?

Question

I am writing a little library with some prime number related methods. As I\'ve done the groundwork (aka working methods) and now I\'m looking for some optimization. Ofcours

Answer 1

Here is a halfway decent function I wrote to solve one of the Euler:

private static long IsPrime(long input)
{
    if ((input % 2) == 0)
    {
        return 2;
    }
    else if ((input == 1))
    {
        return 1;
    }
    else
    {
        long threshold = (Convert.ToInt64(Math.Sqrt(input)));
        long tryDivide = 3;
        while (tryDivide < threshold)
        {
            if ((input % tryDivide) == 0)
            {
                Console.WriteLine("Found a factor: " + tryDivide);
                return tryDivide;
            }
            tryDivide += 2;
        }
        Console.WriteLine("Found a factor: " + input);
        return -1;
    }
}

Answer 2

Try this...

if (testVal == 2) return true;
if (testVal % 2 == 0) return false;

for (int i = 3; i <= Math.Ceiling(Math.Sqrt(testVal)); i += 2)
{
   if (testVal % i == 0)
       return false;
}

return true;

Ive used this quite a few times.. Not as fast as a sieve.. but it works.

Answer 3

Your for loop should look like this:

for (int idx = 3; idx * idx <= test; idx++) { ... }

That way, you avoid floating-point computation. Should run faster and it'll be more accurate. This is why the for statement conditional is simply a boolean expression: it makes things like this possible.

Answer 4

You might want to look into Fermat's little theorem.

Here is the pseudo code from the book Algorithms by S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani, where n is the number you are testing for primality.

Pick a positive integer a < n at random
if a^n-1 is equivalent to 1 (mod n)
   return yes
else
   return no

Implementing Fermat's theorem should be faster then the sieve solution. However, there are Carmichael numbers that pass Fermat's test and are NOT prime. There are workarounds for this. I recommend consulting Section 1.3 in the fore mentioned book. It is all about primality testing and might be helpful for you.