Given a Factorial, Find the Number

Date: 05/02/2003 at 17:02:04
From: Praveen
Subject: How to find the number of a given factorial

For example, the factorial of 4 is 24 (4*3*2*1), but how can we find a 
number if the factorial is given?

Date: 05/02/2003 at 17:36:43
From: Doctor Ian
Subject: Re: How to find the number of a given factorial

Hi Praveen,

Probably the easiest way is to find the prime factors of the number. 
For example, given 

  479,001,600

you could find the prime factors

                 10  5  2  1   1
  479,001,600 = 2   3  5  7  11

Now, note that 11 appears, but 13 doesn't. So this narrows things 
down; it can only be 11! or 12!. That is, the largest prime in the
prime factorization gives you a lower bound.  The smallest missing
prime gives you an upper bound.  

Does that make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/