c - Why does N! stop overflowing the 32 bits output variable when N>=34?

Question

Is there any limitation of variable overflow for big numbers? There is an exercise in a C book about factorials.

My code:

#include <stdio.h>

int main(void) {
    unsigned int n, fn, counter;
    //puts("enter nonnegative int");
    //scanf("%u", &n);
    n = 1;
    while (n != 34) {
        fn = n;
        counter = n - 1;
        while (counter > 0) {
            fn *= counter;
            counter--;
        }
        printf("%u
", fn);
        n++;
    }
}

The commented lines are for debugging.

This code prints the factorial of numbers from n(1) to 34 ('while(n!=34)'), but if you increase it to something like 36 it will just print 0 after first 34 outputs. I know most of the outputs are overflowed and am controlling these big numbers very poorly.

However, I would like to know what limits are causing these zeros to occur.

question from:https://stackoverflow.com/questions/65647449/why-does-n-stop-overflowing-the-32-bits-output-variable-when-n-34

Answer 1

You complain that

N! stop overflowing the 32 bits output variable when N>=34

meaning that, after 34!, the result keeps remaining to 0.

Well, the answer is that it doesn't stop overflowing. What happens is that the value shown, that is the remainder of the division N! / 2^32 starting from N=34 becomes 0 and will never change.

In order to understand how it can happen, let's start with an example using a decimal number. We'll display the result of N! using a display with only two digits:

Factorial	Actualresult	Displayed result	Notes
1!	1	01
2!	2	02
3!	6	06
4!	24	24
5!	120	20	Overflow!
6!	720	20	Overflow!
7!	5040	40	Overflow!
8!	40320	20	Overflow!
9!	362880	80	Overflow!
10!	3628800	00	Overflow and the shown value is 00!
11!	39916800	00	Overflow and the shown value is still 00!
12!	479001600	00	Overflow and the shown value is still 00!
..	..	00	The shown value will be 00 forever