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 April 11th, 2009, 07:24 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 n-1 with 0<=k<=n-1 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, 10:49 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 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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