Is there a simple algorithm that can determine if X is prime?

Question

I have been trying to work my way through Project Euler, and have noticed a handful of problems ask for you to determine a prime number as part of it.

1. I kno

Answer 1

I am working thru the Project Euler problems as well and in fact just finished #3 (by id) which is the search for the highest prime factor of a composite number (the number in the ? is 600851475143).

I looked at all of the info on primes (the sieve techniques already mentioned here), and on integer factorization on wikipedia and came up with a brute force trial division algorithm that I decided would do.

So as I am doing the euler problems to learn ruby I was looking into coding my algorithm and stumbled across the mathn library which has a Prime class and an Integer class with a prime_division method. how cool is that. i was able to get the correct answer to the problem with this ruby snippet:

require "mathn.rb"
puts 600851475143.prime_division.last.first

this snippet outputs the correct answer to the console. of course i ended up doing a ton of reading and learning before i stumbled upon this little beauty, i just thought i would share it with everyone...

Answer 2

I'd recommend Fermat's primality test. It is a probabilistic test, but it is correct surprisingly often. And it is incredibly fast when compared with the sieve.

Answer 3

For Project Euler, having a list of primes is really essential. I would suggest maintaining a list that you use for each problem.

I think what you're looking for is the Sieve of Eratosthenes.

Answer 4

The AKS prime testing algorithm:

Input: Integer n > 1  


if (n is has the form ab with b > 1) then output COMPOSITE  

r := 2  
while (r < n) {  
    if (gcd(n,r) is not 1) then output COMPOSITE  
    if (r is prime greater than 2) then {  
        let q be the largest factor of r-1  
        if (q > 4sqrt(r)log n) and (n(r-1)/q is not 1 (mod r)) then break  
    }  
    r := r+1  
}  

for a = 1 to 2sqrt(r)log n {  
    if ( (x-a)n is not (xn-a) (mod xr-1,n) ) then output COMPOSITE  
}  

output PRIME;   

Answer 5

Keeping in mind the following facts (from MathsChallenge.net):

• All primes except 2 are odd.
• All primes greater than 3 can be written in the form 6k - 1 or 6k + 1.
• You don't need to check past the square root of n

Here's the C++ function I use for relatively small n:

bool isPrime(unsigned long n)
{
    if (n == 1) return false; // 1 is not prime
    if (n < 4) return true; // 2 and 3 are both prime
    if ((n % 2) == 0) return false; // exclude even numbers
    if (n < 9) return true; //we have already excluded 4, 6, and 8.
    if ((n % 3) == 0) return false; // exclude remaining multiples of 3

    unsigned long r = floor( sqrt(n) );
    unsigned long f = 5;
    while (f <= r)
    {
        if ((n % f) == 0)  return false;
        if ((n % (f + 2)) == 0) return false;
        f = f + 6;
    }
    return true; // (in all other cases)
}

You could probably think of more optimizations of your own.

Answer 6

another way in python is:

import math

def main():
    count = 1
    while True:
        isprime = True

        for x in range(2, int(math.sqrt(count) + 1)):
            if count % x == 0: 
                isprime = False
                break

        if isprime:
            print count


        count += 2


if __name__ == '__main__':
    main()