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Goldbach's Conjecture, another slant

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  • Mark Underwood
    Hello all Bill s comments led me to an interesting view of Goldbach s conjecture. GB conjecture: Every even number 3 is the sum of two primes at least once.
    Message 1 of 1 , Nov 26, 2001
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      Hello all

      Bill's comments led me to an interesting view of Goldbach's
      conjecture.

      GB conjecture: Every even number > 3 is the sum of two primes at
      least once.

      For instance, 26 = 13+13 = 7+19 = 3+23

      That is, 26 = (13-0)+(13+0) = (13-6)+(13+6) = (13-10)+(13+10)

      In shorter notation, 13: (0,6,10)

      Restating the GB conjecture: Every number n > 1 is exactly between
      two primes, at least once.

      Here is a list of n up to 32:

      2: (0) 17: (0,6,12,14)
      3: (0) 18: (1,5,11,13)
      4: (1) 19: (0,12)
      5: (0,2) 20: (3,9,17)
      6: (1) 21: (2,8,10,16)
      7: (0,4) 22: (9,15,19)
      8: (3,5) 23: (0,6,18,20)
      9: (2,4) 24: (5,7,13,17,19)
      10: (3,7) 25: (6,12,18,22)
      11: (0,6,8) 26: (3,15,21)
      12: (1,5,7) 27: (4,10,14,16,20)
      13: (0,6,10) 28: (9,15,25)
      14: (3,9) 29: (0,12,18,24)
      15: (2,4,8) 30: (1,7,11,13,17,23)
      16: (3,13) 31: (0,12,28)
      17: (0,6,12,14) 32: (9,15,21,29)

      Look at where, say, the number 3 appears throughout this list in the
      brackets. It is at n=8,n=10,n=14,n=16,n=20 and n=26. Subtract 3 from
      each of these numbers and you get 5,7,11,13,17,23. Add 3 to each of
      these numbers and you get 11,13,17,19,23,29.

      Similarly look at where the number 5 appears throughout this list in
      the brackets. It is at n=8,n=12,n=18,n=24. Subract 5 from each of
      these numbers and you get 3,7,13,19. Add 5 to each of these numbers
      and you get 13,17,23,29.

      The number 6 appears at n=11,n=13,n=17,n=23,n=25. Subtract 6 from
      each of these numbers and you get 5,7,11,17,19. Add 6 to each of
      these numbers and you get 17,19,23,29,31.

      It stunned me a little at first to see this, but it is an entirely
      expected result, really! :)

      The number zero occurs with the frequency of the primes (of course).
      The number one occurs with the frequency of the twin primes (of
      course). A perhaps a not so obvious proposal is that the numbers
      2,4,8,16, ect. will also each occur at the frequency of the twin
      primes. The numbers 3,6,9,12,18,24,27, ect. will each occur at the
      frequency of twice the twin primes. The numbers 5,10,20,25,50 ect.
      will each occur at the frequency of 4/3 the twin primes.

      Well enough silliness for now

      Mark
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