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RE: RE : [PrimeNumbers] variation on Wilson theorem

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  • Jon Perry
    Another variation on Wilsonm s Theorem: Both n! and (n+1)! = -nmodn^2 iff n is prime Jon Perry perry@globalnet.co.uk
    Message 1 of 5 , Apr 16, 2002
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      Another variation on Wilsonm's Theorem:

      Both n! and (n+1)! = -nmodn^2 iff n is prime

      Jon Perry
      perry@...
      http://www.users.globalnet.co.uk/~perry/maths
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      -----Original Message-----
      From: CRESGE - Hervé LELEU [mailto:h.leleu@...]
      Sent: 16 April 2002 16:57
      To: primenumbers@yahoogroups.com
      Subject: RE : [PrimeNumbers] variation on Wilson theorem


      Does it mean that n is prime iff (a-1)!*(n-a)! = -1 (mod n) ?
      where a=1 for Wilson theorem.

      And also that: n is prime iff ((n-1)/2)! ^ 2 = (+/-) 1 (mod n) where
      (+/-) 1 depends if (n-1)/2 is odd or even.

      > -----Message d'origine-----
      > De : Paul Jobling [mailto:Paul.Jobling@...]
      > Envoyé : mardi 16 avril 2002 17:07
      > À : 'CRESGE - Hervé LELEU'; primenumbers@yahoogroups.com
      > Objet : RE: [PrimeNumbers] variation on Wilson theorem
      >
      >
      > > Let n>2.
      > > n is prime if and only if 2(n-3)! = -1 mod n
      > > Is this result well-known?
      > > Thanks for answers.
      >
      > Wilson's theorem states that n is prime iff (n-1)! = -1 (mod
      > n). This is the same as your result: you have explicitly
      > multiplied n-1 and n-2 together and removed them from the
      > factorial. (n-1)*(n-2) (mod n) = -1*-2 (mod n) = 2 (mod n).
      >
      > Regards,
      >
      > Paul.
      >
      >
      > __________________________________________________
      > Virus checked by MessageLabs Virus Control Centre.
      >



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