Thread: Factorial calculation optimization

For a problem I'm working on, I need to calculate some very, very large factorials; or at least the first five non-zero digits of said factorial. I've worked out the function:

F(n) = (^(n!)/_(10^S5(n)))%10⁵


S₅(n) = ec6eb4f9e17c320d12784365efaad220.png where p = 5 to get the trailing zeros of the factorial.

The Algorithm:

public static void main(String[] args) {
		// TODO Auto-generated method stub
 
		Timer t = new Timer();//just a timer, can be replaced with long start = System.currentTimeMillis();
		t.start();//^
		long fact = 1;//variable containing the needed value
		long limit = (long) 100000.0;//the nth factorial
 
		long pow10 = 10;//powers of 10
 
 
 
		for(long i = 2; i <= limit; i++)//for loop to calculate the needed factorial
		{
				fact*=i; // (i-1)! * i = i! at that point
 
				while(fact%10==0)//does the fact/10^(s_5(i)) function to strip the trailing zeros
					fact/=10;
 
				fact %= 100000;// does the modulus to keep the numbers in bounds 
				if(i%pow10==0)//simply prints out the factorial of powers of ten
				{
					pow10*=10;
					System.out.println(i + ": " + fact);
				}
			//System.out.println(i + ": " + fact);
		}
 
		System.out.println(fact);//print the result
 
		t.end();//ends the timer and prints out the result
		t.printElapsedTimeSeconds();//can be replaced with System.out.println("Elasped time: " + (System.currentTimeMillis()-start)/1000 + " seconds.");
 
 
	}

I can't use De Polignac's formula because the required sieve would take too much memory and brute forcing every prime takes too long. This is the output of F(1000000000):

10: 36288
100: 16864
1000: 53472
10000: 79008
100000: 2496
1000000: 4544
10000000: 51552
100000000: 52032
1000000000: 64704
64704
Elapsed time: 13.476 sec.

I need to calculcate F(1000000000000) or F of 1 trillion. This would take a very long time, there has to be some optimization or tweak that I'm missing somewhere.