Project Euler Question 3 Help

Question

I'm trying to work through Project Euler and I'm hitting a barrier on problem 03. I have an algorithm that works for smaller numbers, but problem 3 uses a very, very large number.

Problem 03: The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143?

Here is my solution in C# and it's been running for I think close to an hour. I'm not looking for an answer because I do actually want to solve this myself. Mainly just looking for some help.

    static void Main(string[] args) {
        const long n = 600851475143;
        //const long n = 13195;
        long count, half, largestPrime = 0;
        bool IsAPrime;

        half = n / 2;

        for (long i = half; i > 1 && largestPrime == 0; i--) {
             if (n % i == 0) { // these are factors of n
                count = 1;
                IsAPrime = true;
                while (++count < i && IsAPrime) {
                    if (i % count == 0) { // does a factor of n have a factor? (not prime)
                        IsAPrime = false;
                    }
                }
                if (IsAPrime) {
                    largestPrime = i;
                }
            }
        }

        Console.WriteLine("The largest prime factor is " + largestPrime.ToString() + ".");
        Console.ReadLine();
    }

Answer 1

For starters, instead of beginning your search at n / 2, start it at the square root of n. You'll get half of the factors, the other half being their complement.

eg:

n = 27
start at floor(sqrt(27)) = 5
is 5 a factor? no
is 4 a factor? no
is 3 a factor? yes. 27 / 3 = 9. 9 is also a factor.
is 2 a factor? no.
factors are 3 and 9.

Answer 2

Actually, for this case you don't need to check for primality, just remove the factors you find. Start with n == 2 and scan upwards. When evil-big-number % n == 0, divide evil-big-number by n and continue with smaller-evil-number. Stop when n >= sqrt(big-evil-number).

Should not take more than a few seconds on any modern machine.

Answer 3

Although the question asks for the largest prime factor, it doesn't necessarily mean you have to find that one first...

Answer 4

long n = 600851475143L; //not even, so 2 wont be a factor
int factor = 3; 
while( n > 1)
{
    if(n % factor == 0)
    {
        n/=factor;
    }else
        factor += 2; //skip even numbrs
}
        print factor;

This should be quick enough...Notice, there's no need to check for prime...

Answer 5

You need to reduce the amount of checking you are doing ... think about what numbers you need to test.

For a better approach read up on the Sieve of Erathosthenes ... it should get you pointed in the right direction.