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April 11th, 2009, 06:24 AM  #1 
Senior Member Joined: Nov 2007 Posts: 633 Thanks: 0  Wilson theorem new formulation?
I have never seen before such formulation that is why I'm willing to know your thoughts. While trying to find some way to express factorial as sequential recurrence (not trivial one) I have found this. If n is prime then (k! n(k+1)!) mod n = 1 or n1 with 0<=k<=n1 Example 10!12! mod 23 = 22 9!13! mod 23 = 1 I know it is not hard to prove but I have little knowledge to do it. Is there some consequences if written this way? I do not know. Thank you for any explanation or comment. 
April 11th, 2009, 09:49 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Wilson theorem new formulation?
This is no easier to test than Wilson's theorem  you're still doing about n multiplications. You can show an even stronger statement, that the result is 0 if n is composite (and > 4?), by checking factors. I'm still a bit too hazy to do the proof at the moment (have been sick). 

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