Square root factorial

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 (2n)! is asymptotically (n!)^2?

No, Gauss' duplication formula

shows that the ratio of (2n)! 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 asymptoticallyn/2.

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

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