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982Factor Record

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  • paulmillscv@yahoo.co.uk
    Apr 25, 2001
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      Hi,
      Yves Gallot has pointed out that directly from Wilson's theorem
      (p-1)! == -1 mod p so that p divides (p-1)! + 1

      (p-1)! +1 is a `closed factorial form' and the generic term I
      am probably looking for is an `irreducible' form. (polynomial or
      otherwise).

      But as Yves pointed out, by substituting the largest known prime
      (Mersenne) for p we obtain the p / f(p) World record
      p / (p-1)! + 1 . The New World record is that Wilson's theorem
      shows that (2^6972593 -2)! + 1 is the largest 'irreducible form'
      number for which we have a prime factor.

      I approached this problem from Henri Lifchitz's work where he
      showed that
      q / 3^p – 1 . This is a different `factored form' because
      it is q / f(p) and q is distinct from p. So Henri's unsung
      world record and now my world record still stands too! In fact if
      q / f(p) then either q = p or is distinct from p so there are just 2
      world records for this factored form! Held by the Mersenne (q = p)
      and Sophie Germain (q /= p) WR primes. Now the theory starts, of
      course.

      Incidentally, it is easy to prove that (2p)! + 1 > 3^p – 1 by
      induction.
      Also I claim to have missed the simple p / f(p) because I was
      deflected by the nice formula
      (a + b)^p == a^b + b^p mod p

      This gives a large number of expressions for the factored form
      p / f(p)

      i.e (2 + 1)^p == 2^p + 1 mod p => p / 3^p – 2^p – 1



      regards,
      Paul Mills
      Kenilworth
      England.
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