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Goldbach revisited

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  • Kermit Rose
    Let s make a 2 by 2 crosstabs for the number of odd pairs(P,Q) which add to 2 * N. And assume that primeness is independent of the adding to 2*N. (Which is a
    Message 1 of 4 , Jan 2, 2006
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      Let's make a 2 by 2 crosstabs for the number of odd pairs(P,Q) which add to
      2 * N.

      And assume that primeness is independent of the adding to 2*N. (Which is a
      separate conjecture.)



      Q odd composite Q odd
      prime Q odd total
      P odd composite ( N - N/log(N))^2 [
      N/log(N)] * [N - N/log(N) ] N * [N - N/log(n) ]

      P odd prime [ N/log(N)] * [N - N/log(N) ] ( [
      N/log(N)] * [N - N/log(N) ] )^2 N/log(N)

      P odd total N * [N - N/log(n) ]
      N/log(N) N^2




      This suggests that the number of ways that primes add to 2N increases
      as N increases.

      Thus I propose two companion conjectures to the Goldbach conjecture.

      (1) The number of ways in which an even number M is the sum of two primes
      increases as M increases.

      (2) The number of prime pairs that add to M is approximately what would be
      expected if adding to the even number
      M is independent of primeness.
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