# How do you solve the equation x!=n, for any value of n?

About 5 years ago, I did some research into factorials. I came across a problem online that asked me to solve x!=6. The answer was 3, of course, but when I tried to solve x!=3, I found no value of x that would work. It is a decimal, because I set the gamma function equal to three. I need assistance in solving the gamma part.

Use a language or system that has both gamma function for all real values and a root solver. Then find a root of $$\Gamma(x) - n=0$$.

The Lanczos approximation (see https://en.wikipedia.org/wiki/Lanczos_approximation) can, combined with the reflection formula, allow the computation of the Gamma function at all points in the complex plane where it is defined.

I have done this in the past using HP calculators. By using an Aitken iteration, I was able to get results for complex $$n$$ as well as real.

It was fun finding $$x$$ for complex $$n$$.

Looking up "inverse factorial" gives interesting links like this:

https://arxiv.org/abs/1707.01734

Have fun.