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20348Conjecture (contd).

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  • Devaraj Kandadai
    May 31, 2009
    • 0 Attachment
      Conjecture (contd).

      This pertains to my conjecture that a ll the factors of a Carmichael number
      cannot be Mersenne.

      What folows is a heuristic.. Let us first consider 3-factor Cns. The
      implication of Pomerance's

      proof: (attachment to my previous post): If P_1,P_2 & P_3 are the three
      factors of a CN ( say N) then

      (P_1-1)*(N-1/(P_2-1)*(P_3-1) = k is an integer. Now let, IF POSSIBLE, (
      2^p_1), (2^p_2-1) & (2^p_3-1) be the three Mersenne prime factors of N, a
      Carmichael number.

      Then it follows from Fermat's theorem that 2^p_1-2 is a multiple of p_1.
      Similar remarks apply to

      p_2 and p_3.

      The implication is that k above must have p_2 and p_3 as factors of the
      denominator of k. Considering that (P_2-1)*(N-1)/P_1-1)*(P_3-1) should also
      be an integer in the case of Carmichael

      numbers the implication is that p_1 must also be a factor of the denominator
      of k above.

      There are also other unspecified prime factors in the denominator. If should
      be an integer (N-1) in the numerator of k must also have all these prime
      factors which is highly improbable. This improbabality grows exponentially
      as the number of factors of a CN grow. Hence my conjecture,

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