如何确定一个令人难以置信的大数字是否是素数？

Question

The numbers I'm trying to figure out are in this form (some examples):

2 ^ 7 - 1 , 2 ^ 31 - 1 , 2 ^ 127 - 1 , et cetera.

This is not a homework question I was just researching primes and a lot of the information is going a bit over my head (Fourier transformations). Originally I was just using a function like this:

public static bool IsPrime(int candidate)
{
    if ((candidate & 1) == 0)
    {
        return candidate == 2;
    }

    for (int i = 3; (i * i) <= candidate; i += 2)
    {
        if ((candidate % i) == 0)
        {
            return false;
        }
    }

    return candidate != 1;
}

But that stopped working once the numbers got too big. I also looked in to the Sieve of Eratosthenes but apparently that only works for numbers of much smaller size.

To clarify, I'm not looking to write a program to find prime numbers, but rather, to determine if a given number is prime. I was looking in to the BigInteger structure in the .NET Framework and it looks promising if I could just write an efficient enough algorithm (I'd settle for something that finished in days).

I'm not sure if a mathematical proof would be better in this circumstance but I do not have much knowledge in that area as opposed to programming but if there was a proof that specialized in these kinds of numbers, that'd definitely be worth looking in to.

Thanks.

Answer 1

Factoring numbers is a big deal. The fact that it's difficult is the basis of modern cryptography.

Depending on how large your number is... You could get a list of all prime numbers up to the square root of the number you're looking for. This will be significantly less than just going up by 2 every time. The problem would be finding a list that large. If you have a number that's 10^100, then you'd need all primes of 10^50 and less, which is still a huge amount of numbers.

Answer 2

The numbers that you list: 2 ^ 7 - 1, 2 ^ 31 - 1, 2 ^ 127 - 1 are called Mersenne Numbers . There's already an entire distributed computing project to find those. It's called GIMPS .

So the answer to your question is not trivial. All the currently existing known primes of that form are listed here:

http://en.wikipedia.org/wiki/Mersenne_prime#List_of_known_Mersenne_primes

Answer 3

Do you have some upper limit on your numbers ? You mention that you'd settle for days. If so, unless your numbers are really large, your current algo will work.

You can also look at Miller–Rabin primality test

Answer 4

Java comes with an implementation of one of the probabilistic primality tests - see java.math.BigInteger.isProbablePrime(). I thought this would mean that C# had a look-alike in the language. I didn't find one, but looking for one I found some code on the web at http://www.codeproject.com/KB/cs/biginteger.aspx .