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C5 is composite

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  • leavemsg1
    The next Catalan number, C5, is composite. A number of this size is usually proven prime by raising it exponentially to a carefully-selected base with the
    Message 1 of 1 , Sep 13, 2008
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      The next Catalan number, C5, is composite.

      A number of this size is usually proven prime by raising it
      exponentially to a carefully-selected base with the expecta-
      tion of discovering a specific residue after modulation.
      or... a^(modulendum-1)== residue(modulator) iff (some criteria).

      However, double-Mersenne numbers not only describe their partic-
      ular format, they also self-indicate their primality when the
      correct (modulator) is chosen.

      Begin with M(M(p+1))'s such that 'p' is prime and carefully calc-
      ulate their self-predicting (modulator) as 2^((p+1) -1 -1) -1 -1
      or... 2^(p-1)-2.

      Now, simply walk through the double-Mersennes, until you reach the
      Catalan numbers, using this bit of information to discover that
      all prime Catalan numbers are linked to their 'residue' formula...
      2^(p-2)-1.

      C1 is not testable, but is nevertheless prime, and C2, C3, and C4
      & all double-Mersennes identify precisely with this statement:

      [M(M(p+1))= 2^(2^(p+1)-1)-1] == 2^(p-2)-1 (mod (2^(p-1)-2)).

      I verified that M(M(128))== 2^80-1 {<> [2^125-1]} (mod 2^126-2)
      using the GNU/Bignum {Try GMP!} interpreter from their website.

      The next Catalan number, C5, is composite due to this result.

      Further manipulation of the above statement also predicts the
      nature of the 'p' candidates...

      Just remove the exponents of the (modulendum) and (modulator) to
      arrive at both 2^(p+1)-1 and p-1, respectively; and we only need
      to compare 2^(p+1) versus 'p' to reveal their connection:

      2^(p+1)-2== 2(mod p)... which is equivalent to the 2-PRP test.

      Later, it would be more accurately discovered that only prime num-
      bers can contribute to the production of the double-Mersenne num-
      bers that we call Catalan numbers.

      These two congruencies provide a different but natural method for
      predicting the primality of double-Merennes -- Catalan numbers --
      due to their format; check a few other double-Mersenne numbers if
      you like...
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