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Re: [PrimeNumbers] Goldbach proof??

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  • Phil Carmody
    ... This attack has been done many times before (and what s so vedic about this version I wonder). It s the counting argument that says you can only remove a
    Message 1 of 2 , Sep 29, 2001
      On Sat, 29 September 2001, kalidharma@... wrote:
      > Possible Goldbach proof by Kapoor. Evaluations/critiques needed for
      > peer review. Enter www.vedicmaths.org then point to indexes. Click
      > onto articles and then Goldbach proof, by Dr. Kapoor. Thanks, Gary
      > W. Adamson


      This attack has been done many times before (and what's so 'vedic' about this version I wonder). It's the counting argument that says you can only remove a certain proportion of candidates with each prime < sqrt(N). However, the equations used deal with real ratios, but the set {0..N} has a discrete number of elements, and it takes more rigor to express the ratios with upper bounds. Alas these upper bounds fail to prove the conjecture.

      For example, a trivial N=26 case, has 14 pairs
      0 1 2 3 4 5 6 7 8 9 10 11 12 13
      26 25 24 23 22 21 20 19 18 17 16 15 14 13
      x x x x x x x 7 removed by 2 alone
      x x x x x x x x 8 removed by 3 alone
      x x x x x 5 removed by 5 alone

      Now we can see in this trivial case that there is enough overlap between the 2, 3, and 5 sets such that two pairs remain. Independently, there are 7+8+5=20 reasons for rejecting a pair, and only 14 pairs to start with. Note that 26 ~= [26^(1/2)]#.

      This relation doesn't hold for N=1000, say, 1000 << 31#. Of the eleven possible primes, you can be fairly accurate how many candidates 2, 3, 5 and 7 remove, (7#<501) but the remaining 7 primes you cannot be so exact about.

      None has yet proved sufficient overlap to guarantee leaving some primes.

      I'd like to see Jud McCranie's composite run code adapted to be able to remove _two_ residues for each base, to see if some heuristic about maximal proportions of numbers in this 'folded' scenario can be generated. Having said that we never got enough data to even hypothesise what the simple single-exponent situation tended towards.

      Jud - can you publish your results as a database or a text file on the Yahoogroups page please? (And your code? ;-))


      Phil

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