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Vague description for algorithm to search for odd perfect number

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  • Kermit Rose
    Start with choice of prime number presumed to divide the odd perfect number. Name the odd perfect number, M . During the algorithm, the minimal power of a
    Message 1 of 1 , Jul 17, 2011
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      Start with choice of prime number presumed to divide the odd perfect number.

      Name the odd perfect number, "M".

      During the algorithm, the minimal power of a prime divisor will be
      assumed to be that prime's multiplicative presence in the odd perfect
      number M.

      Suppose we assume that 3 divides M.
      Denote by s(M) = 2*M the sum of the factors of M.

      Since 3 is 3 mod 4,
      (1 + 3 + 3**2) = 13 divides s(M) and therefore divides M.

      Since 13 is 1 mod 4,
      choose (1 + 13) divides s(M) and therefore (1+13)/2 = 7 divides M.

      Since 7 is 3 mod 4,
      (1 + 7 + 7**2) = 57 = 3 * 19 divides M.

      Etc

      Carefully keep track of all the factors, for s(M) and M,
      and be on the look out for when the process terminates by having
      s(M) be exactly equal to 2*M

      Kermit
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