Problems with prime numbers...

Question

I am trying to write a program to find the largest prime factor of a very large number, and have tried several methods with varying success. All of the ones I have found so far have been unbelievably slow. I had a thought, and am wondering if this is a valid approach:

long number = input;

while(notPrime(number))
{
    number = number / getLowestDivisiblePrimeNumber();
}

return number;

This approach would take an input, and would do the following:

200 -> 100 -> 50 -> 25 -> 5 (return)

90 -> 45 -> 15 -> 5 (return)

It divides currentNum repeatedly by the smallest divisible number (most often 2, or 3) until currentNum itself is prime (there is no divisible prime number less than the squareroot of currentNum), and assumes this is the largest prime factor of the original input.

Will this always work? If not, can someone give me a counterexample?

-

EDIT: By very large, I mean about 2^40, or 10^11.

Answer 1

This will always work because of the Unique Prime Factorization Theorem.

Answer 2

You are trying to find the prime factors of a number. What you are proposing will work, but will still be slow for large numbers.... you should be thankful for this, since most modern security is predicated on this being a difficult problem.

Answer 3

Certainly it will work (see Mark Byers' answer), but for "very large" inputs it may take far too long. You should note that your call to getLowestDivisiblePrimeNumber() conceals another loop, so this runs at O(N^2), and that depending on what you mean by "very large" it may have to work on BigNums which will be slow.

You could speed it up a little, by noting that your algorithm need never check factors smaller than the last one found.

Answer 4

The method will work, but will be slow. "How big are your numbers?" determines the method to use:

• Less than 2^16 or so: Lookup table.
• Less than 2^70 or so: Sieve of Atkin. This is an optimized version of the more well known Sieve of Eratosthenes. Edit: Richard Brent's modification of Pollard's rho algorithm may be better in this case.
• Less than 10^20: Lenstra elliptic curve factorization
• Less than 10^100: Quadratic Sieve
• More than 10^100: General Number Field Sieve

Answer 5

From a quick search I just did, the fastest known way to factor a number is by using the Elliptic Curve Method.

You could try throwing your number at this demo: http://www.alpertron.com.ar/ECM.HTM .

If that convinces you, you could try either stealing the code (that's no fun, they provide a link to it!) or reading up on the theory of it elsewhere. There's a Wikipedia article about it here: http://en.wikipedia.org/wiki/Lenstra_elliptic_curve_factorization but I'm too stupid to understand it. Thankfully, it's your problem, not mine! :)