What factorial is closest to the square root of 2024 factorial?

A good guess would be 1012, based on the idea that √(*n*!) might be near (*n*/2)!.

This isn’t correct—the actual answer is 1112—but it’s not wildly off.

Could it be that (2*n*)! is asymptotically (*n*!)²?

No, Gauss’ duplication formula

shows that the ratio of (2*n*)! to (*n*!)² grows exponentially as a function of *n*.

However, the ratio *only* grows exponentially, and factorials grow faster than exponentially. I believe that the value of *m* minimizing

| √(*n*!) – *m*! |

asymptotically approaches *n*/2. In other words, the inverse factorial of √(*n*!) approaches *n*/2. And more generally the inverse factorial of (*n*!)^{1/k} asymptotically approaches *n*/*k*. I haven’t written out a proof, but the plot below shows numerical evidence.

So √(*n*!) is not asymptotically (*n*/2)!, but the inverse factorial of √(*n*!) is asymptotically *n*/2.

See the next post for a way to compute inverse factorials.

You are right! By Stirling‘s formula, as m,n get large,

m! ~ (m/e)^m sqrt(2 Pi m)

and

sqrt(n!) ~ (n/e)^(n/2) (2 Pi n)^(1/4)

where ~ means 1+o(1). By equating the right hand sides we obtain that

m = n/2 + O(n/log(n))

and we can actually derive an asymptotic expansion for m.