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16440Re: Dat Dastardly 788 : (Now a stronger GC)

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  • John W. Nicholson
    Apr 14, 2005
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      I have been looking at this paper of Harvey's. I think I have
      discovered a few things which are very interesting. First let me
      list the numbers which he found not to be middle numbers < 2*10^10:

      N = 96, 402, 516, 786, 906, 1116, 1146, 1266, 1356, 3246, and 4206.

      Harvey stated to me the correction of 4208 should be 4206.

      Note that all except the second one, 402, is == 1 (mod 5). Also as
      would be expected all are divisible by 6, but they are a complete
      residue set with (mod 4) == (mod 24) (I might be stating this wrong,
      so let me restate this when divided by 24 the remainder is one of
      0,1,2, or 3.) There is only one of these with == 0 (mod 24), namely
      96. There is also a interesting thing with (mod 11) and N/6 == (mod
      11) most have a factor of 2 except the first three. I did look at
      (mod 7). There are no == 1,4 (mod 7). I wonder how this fits with
      http://primepuzzles.net/problems/prob_003.htm ?

      N, (mod 11), N/6 == (mod 11), (mod 7)
      96, 8, 5, 5
      402, 6, 1, 3
      516, 10, 9, 5
      786, 5, 10, 2
      906, 4, 8, 3
      1116, 5, 10, 3
      1146, 2, 4, 5
      1266, 1, 2, 6
      1356, 3, 6, 5
      3246, 1, 2, 5
      4206, 4, 8, 6

      Add to these statement a look a MR1745569,
      http://primes.utm.edu/references/refs.cgi?author=Suzuki
      I have a scaned it if anyone wants a copy.

      Because of the above, 4*n+/-1 and 6*m+/-1 are the only ways to state
      all primes factors with only two Dirichlet equations, and because
      4206 < 6!*6, I don't think there are any more possible. All other
      numbers > 6!*6 have some way of having numbers > 36 = 6^2 in the P2
      = p1*p2 product (prime factors are of 6*n +/- 0, 1, 2, 3, 4, or 5, n
      < 6) + prime as to Chen's theorem (all even = P2 + p3).

      I do see how this can be used as part of the proof of the t-prime-GC
      if anyone is up to it.

      Can anything more be said?

      John


      --- In primenumbers@yahoogroups.com, Harvey Dubner <harvey@d...>
      wrote:
      > I had a paper published in Journal of Recreational Mathematics,
      Vol.30(3),
      > 1999-2000, entitled
      > "Twin Prime Conjectures" which included the conjecture,
      >
      > Every even number greater than 4208 is the sum of two t-primes
      where a
      > t-prime is a prime which has a twin.
      >
      > This was verified up to 4.10^11 and included strong evidence that
      the
      > conjecture would hold for larger numbers. The paper also includes
      > interestng comparative Goldbach data.
      >
      > I will be happy to send a pdf copy of the paper to anyone who asks.
      >
      > Harvey Dubner
      >
      > PS: No, the paper does not include a proof of the above
      conjecture or GC.
      > PPS: No, I am not working on such a proof.
      >
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