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Characterizing Carmichael numbers

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  • Kermit Rose
    Characterization of Carmichael Numbers It has been proven that product of primes p1 p2 p3 is a Carmichael number if and only if p1 p2 p3 - 1 is divisible by
    Message 1 of 1 , Dec 17, 2010
      Characterization of Carmichael Numbers

      It has been proven that

      product of primes p1 p2 p3 is a Carmichael number
      if and only if
      p1 p2 p3 - 1 is divisible by each of

      (p1-1), (p2-1), (p3-1).

      After some frustration at algebraic mistakes,
      I have worked out the following explicit equivalent to
      this theorem.


      For primes p1,p2,p3,

      p1 p2 p3 is Carmichael,
      if and only if

      p1 p2 p3 -1
      = C (p1-1)(p2-1)(p3-1) GCD(p1-1,p2-1,p3-1)
      / [GCD(p1-1,p2-1) GCD(p1-1,p3-1) GCD(p2-1,p3-1)]


      where C is some positive integer.


      Maybe I have made yet another mistake, and someone can find a

      counter example to my statement.


      Kermit
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