Reverse factorial

Question

Well, we all know that if N is given it\'s easy to calculate N!. But what about the inverse?

N! is given and you are about to find N - Is that possible ? I\'m curio

Answer 1

Most numbers are not in the range of outputs of the factorial function. If that is what you want to test, it's easy to get an approximation using Stirling's formula or the number of digits of the target number, as others have mentioned, then perform a binary search to determine factorials above and below the given number.

What is more interesting is constructing the inverse of the Gamma function, which extends the factorial function to positive real numbers (and to most complex numbers, too). It turns out construction of an inverse is a difficult problem. However, it was solved explicitly for most positive real numbers in 2012 in the following paper: http://www.ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2/S0002-9939-2011-11023-2.pdf . The explicit formula is given in Corollary 6 at the end of the paper.

Note that it involves an integral on an infinite domain, but with a careful analysis I believe a reasonable implementation could be constructed. Whether that is better than a simple successive approximation scheme in practice, I don't know.