c - Cannot calculate factorials bigger than 20! ! How to do so?

Question

I am using unsigned long long integer format in order to calculate big factorials. However my code fails at some point can you have a look at it? Actually it is part of a larger code for Taylor expansion of exponential function, but that part is irrelevant at this point. I will appreciate any suggestions.

Thanks

#include <stdio.h>
#include <math.h>
//We need to write a factorial function beforehand, since we
//have factorial in the denominators.
//Remembering that factorials are defined for integers; it is
//possible to define factorials of non-integer numbers using
//Gamma Function but we will omit that.
//We first declare the factorial function as follows:
unsigned long long factorial (int);
//Long long integer format only allows numbers in the order of 10^18 so 
//we shall use the sign bit in order to increase our range.
//Now we define it,
unsigned long long
factorial(int n)
{
//Here s is the free parameter which is increased by one in each step and
//pro is the initial product and by setting pro to be 0 we also cover the
//case of zero factorial.
    int s = 1;
    unsigned long long pro = 1;
    if (n < 0)
        printf("Factorial is not defined for a negative number 
");
    else {
    while (n >= s) { 
    printf("%d 
", s);
    pro *= s;
    s++;
    printf("%llu 
", pro);
    }
    return pro;
    }
}

int main ()
{
    int x[12] = { 1, 5, 10, 15, 20, 100, -1, -5, -10, -20, -50, -100};
//Here an array named "calc" is defined to store 
//the values of x.
unsigned long long  k = factorial(25);
printf("%llu 
", k);

//int k;
////The upper index controls the accuracy of the Taylor Series, so
////it is suitable to make it an adjustable parameter. 
//int p = 500;
//for ( k = 0; k < p; k++);

}

Answer 1

The limit on an unsigned long long is 18446744073709551615, or about 1.8e+19. 20! is about 2.4e+18, so within range, however 21! is about 5.1e+19, exceeding the maximum size of an unsigned long long.

You may find this helpful: Are there types bigger than long long int in C++?