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April 11th, 2009, 06:24 AM   #1
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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.
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April 11th, 2009, 09:49 PM   #2
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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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