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Prime Packing (admissible tuples)

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  • Tom
    Main page ( http://www.opertech.com/primes/k-tuples.html ) The latest updates have been posted, with improvements to more than 2200 widths. Currently there are
    Message 1 of 2 , Aug 27, 2007
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      Main page ( http://www.opertech.com/primes/k-tuples.html )

      The latest updates have been posted, with improvements to more than
      2200 widths. Currently there are 29 million plus patterns.
      ( http://www.opertech.com/primes/updates.html )

      Of these 2200 improvements about 20 are new first occurence densities.
      ( http://www.opertech.com/primes/trophycase.html )

      The graph is still getting stepper.
      ( http://www.opertech.com/primes/trophy.bmp ) and
      ( http://www.opertech.com/primes/varcount.bmp )

      --------------------------------------------

      As this graph continues to climb, some questions need to be asked and
      discussed.

      In the following,
      k(w) is the maximum number primes permissible in a width of 'w'
      integers.
      pi(w) is the standard number of primes from 1 to w.

      As you look at the graph, the curve 'appears' to be concave up, as
      evidenced by the 'hockey stick' start. The best fit line on the
      graph is for tracking purposes and not to imply linearity.
      But this line also shows this concave up tendency. Also, the 'flat'
      spots at 19000 and 32000 are mainly due to sequence of programming.
      The upper bound of the searching was raised from 5200s, to 12000s, to
      19000s, to 32000s, and finally to 41741 - meaning the smaller widths
      have seen more searching.

      Hensley and Richards showed that the two HL conjectures were
      incompatible, and left the impression that the k-tuples conjecture
      was the true conjecture.

      Montgomery and Vaughn showed that k(w) <= 2 * pi(w) using the 'large
      sieve'.

      If this graph is truly concave up or linear, then the Montgomery and
      Vaughn proof would be contradicted, so one of two things must be true,
      A) the graph has an inflection point, or
      B) the k-tuples conjecture is false.

      Concerning A) this inflection point appears highly unlikely as these
      quantities can be described as combinatorial objects.

      What is the concensus of math world on the HL conjectures?
      1: k-tuples true and 2nd HL false
      2: k-tuples false and 2nd HL true
      or
      3: k-tuples false and 2nd HL false

      Looking for all the input I can get.

      Thomas J Engelsma
    • Dick
      ... This page mentions the incompatibility of the 2 conjectures here http://en.wikipedia.org/wiki/Second_Hardy-Littlewood_conjecture and references your
      Message 2 of 2 , Aug 28, 2007
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        --- In primenumbers@yahoogroups.com, "Tom" <tom@...> wrote:


        > What is the concensus of math world on the HL conjectures?
        > 1: k-tuples true and 2nd HL false
        > 2: k-tuples false and 2nd HL true
        > or
        > 3: k-tuples false and 2nd HL false

        This page mentions the incompatibility of the 2 conjectures here
        http://en.wikipedia.org/wiki/Second_Hardy-Littlewood_conjecture
        and references your website as well as a 1974 article on the
        incompatibility of the 2 conjectures. Seems to indicate a preference
        for your #1 above.

        I have heard the "k-tuple" conjecture described in different ways.
        Are we talking about the k-tuple conjecture in the sense of the first
        HL conjecture? If not, what is the difference?
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