What is the space complexity of a function that creates a variable of value `N!` (N factorial)?

Question

The bit-length an integer of value N is O(lgN).

What is the bit-length of an integer of value N! (i.e. N factorial)? If a function gets passed in a list of length N and it creates a variable of value N!, what is the space complexity of the function?

Example function:

def numPermutations(nums):
    '''Precondition: `nums` is a list of distinct numbers'''
    N = len(nums)
    return math.factorial(N)

Would the function have O(1) space complexity since it creates only one integer variable (i.e. N) and it's conceivable that the a sensible implementation of math.factorial would also only create at most a couple integer variables? Or would it be O(lgN!) since the bit-length of N! is on the order of lg2(N!)?

(Side note: lg2(N!) == sum from i = 1 to N of lg2(i).)

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I coded the following to create some data:

from math import factorial

for N in range(100):
    print(f'{N}, {factorial(N).bit_length()}')

The data is visualized in this Desmos: Bit-length of N!. The Desmos also includes a few functions/regressions that match the data.

Answer 1

According to Stirling's formula, you can assume n! ~ c*sqrt(n)*n^n. Therefore, log(n!) ~ n*log(n), which is the space complexity of n!.