How to compute the "factorial root"?

Alonso Del Arte alonso.delarte at
Wed Jun 4 01:27:03 CEST 2008

Perhaps this is off-topic, but:

How do you compute the "factorial root" of an integer that is not a
factorial? That is, given an integer n, how do you compute the value x
of Euler's Gamma function such that Gamma(x) = n ?

For example, the "factorial root" of 24 is 5, since 4! = 24, and the
factorial root of 120 is 6, since 5! = 120. But, say, for 47, I've
found, by a long series of trials and errors, that Gamma(5.4328989) =
47.000518252. Surely the line for the Gamma function crosses the exact
integer 47 at some point.

So is there a formula, or will I just have to settle for a program
that merely automates my trial and error?

Alonso del Arte

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