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k-tuples

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  • Tom
    Concerning the Hardy-Littlewood conjecture, pi(x+y)-pi(x)
    Message 1 of 19 , Jan 9, 2005
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      Concerning the Hardy-Littlewood conjecture,
      pi(x+y)-pi(x)<=pi(y).

      latest find
      pi(x+3243)-pi(x) = 457 = pi(3243) [0]

      also, large grouping of -1's around 3429

      past finds
      pi(x+4323)-pi(x) = 590 = pi(4323) [0]
      pi(x+4327)-pi(x) = 591 = pi(4327) [0]

      and true crossover at
      pi(x+4333)-pi(x) = 592 > pi(4333) [+1]

      Tom
    • Tom
      New lower bound for crossover of pi(x+y)-pi(x)
      Message 2 of 19 , Feb 9, 2005
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        New lower bound for crossover of
        pi(x+y)-pi(x)<=pi(y) violation

        If y=3243 the interval pi(x+3243)-pi(x)
        can contain 458 primes while pi(3243)=457

        Old record was y=4333 with 592 primes.

        Also, many new near misses.

        Will generate the modulii and update web pages tonight.

        Web page will be at
        http://www.opertech.com/primes/k-tuples.html

        Tom
      • Tom
        Couldn t contain myself. Web pages have been updated. http://www.opertech.com/primes/k-tuples.html Tom
        Message 3 of 19 , Feb 9, 2005
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          Couldn't contain myself.
          Web pages have been updated.

          http://www.opertech.com/primes/k-tuples.html

          Tom
        • mikeoakes2@aol.com
          In a message dated 10/01/2005 03:43:02 GMT Standard Time, tom@opertech.com writes: Concerning the Hardy-Littlewood conjecture, pi(x+y)-pi(x)
          Message 4 of 19 , Feb 14, 2005
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            In a message dated 10/01/2005 03:43:02 GMT Standard Time, tom@...
            writes:

            Concerning the Hardy-Littlewood conjecture,
            pi(x+y)-pi(x)<=pi(y).

            latest find
            pi(x+3243)-pi(x) = 457 = pi(3243) [0]

            also, large grouping of -1's around 3429

            past finds
            pi(x+4323)-pi(x) = 590 = pi(4323) [0]
            pi(x+4327)-pi(x) = 591 = pi(4327) [0]

            and true crossover at
            pi(x+4333)-pi(x) = 592 > pi(4333) [+1]



            Tom,

            I notice that your splendid page
            _http://www.opertech.com/primes/k-tuples.html_
            (http://www.opertech.com/primes/k-tuples.html)
            says that you have obtained _exhaustive_ results on w(k) for k <= 305.
            That is impressive!

            My question is:-
            what is the algorithm you use? (if it's not secret:-)

            In particular, is it recursive?

            I have spent a while trying to program such a provably exhaustive
            enumeration of all admissible tuples, but without success: my 1 Gb of memory is (like
            Fermat's margin) too narrow.

            -Mike Oakes


            [Non-text portions of this message have been removed]
          • Tom
            The landscape has changed. http://www.opertech.com/primes/kpiwchart.html While testing the 458-tuple for other variations, the program kicked out a 460-tuple
            Message 5 of 19 , Feb 20, 2005
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              The landscape has changed.
              http://www.opertech.com/primes/kpiwchart.html

              While testing the 458-tuple for other variations,
              the program kicked out a 460-tuple of the same size.
              Spent some time rechecking program, and tested the
              460-tuples and they are valid.

              so pi(x+3243)-pi(x) could equal 460 while pi(3243)=457

              This was not only a crossover but a major crossover of +3.

              Looked back through listings and checking has a new upper
              bound of 3159 with 447 primes.

              Tom
            • Tom
              Anybody up to checking some calculations?? Sure would be appreciated. See http://www.opertech.com/primes/residues.html in particular the value of C2 (think it
              Message 6 of 19 , Mar 2, 2005
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                Anybody up to checking some calculations??
                Sure would be appreciated.

                See http://www.opertech.com/primes/residues.html
                in particular the value of C2 (think it is a little large)

                Tom
              • Tom
                On the k-tuples, have just finished an update of the permissible patterns site www.opertech.com/primes/k-tuples.html At the site the trophies (contradiction
                Message 7 of 19 , Jan 16, 2006
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                  On the k-tuples, have just finished an update of the permissible
                  patterns site www.opertech.com/primes/k-tuples.html
                  At the site the trophies (contradiction patterns/super dense
                  constellations) are listed up to packing 100 additional primes in an
                  interval. In fact 17 additional prime can be packed in an interval of
                  length of just 8509 integers.
                  The chart at www.opertech.com/primes/kpiwchart.html shows the crossover
                  (s) and the growth.
                  It is estimated that all intervals of more than 5980 integers can
                  demonstrate a super-dense condition.

                  Enjoy
                  Thomas J Engelsma
                • 逢绥 刘
                  Dear Tom, Thank you very much for your crackajack work. About your ¡°An admissible k-tuple of 447 primes can be created in an interval of 3159 integers,
                  Message 8 of 19 , Jan 16, 2006
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                    Dear Tom, Thank you very much for your crackajack work.
                    About your “An admissible k-tuple of 447 primes can be created in an interval of 3159 integers,
                    while p(3159) = 446.”,
                    can I realize:
                    An admissible 447-tuple has been created in an interval of 3159 integers, while p(3159) = 446?
                    It is not:
                    An admissible prime 447-tuple has been created in an interval of 3159 integers,while p(3159) = 446.
                    Where we call a k-tuple is admissible, if it does not cover all congruence classes modulo any
                    prime p, we call the k-tuple a prime k-tuple when all of its components are primes by Daniel M. Gordon and Gene Rodemich.
                    So that: if the original k-tuple conjecture is true, then Hardy-Littlewood conjecture
                    p(x+y) - p(x) <= p (y)
                    is fails with a value of y = 3159.
                    In my paper I try prove that admissible prime k-tuples are infinite rather then admissible k-tuples will infinitely often be simultaneously primes. The original k-tuple conjecture may not true, example the prime of the form n^2-1.
                    Could you read my paper please and we will discuss some interesting problem.
                    Fengsui Liu.


                    Tom <tom@...> 写道: On the k-tuples, have just finished an update of the permissible
                    patterns site www.opertech.com/primes/k-tuples.html
                    At the site the trophies (contradiction patterns/super dense
                    constellations) are listed up to packing 100 additional primes in an
                    interval. In fact 17 additional prime can be packed in an interval of
                    length of just 8509 integers.
                    The chart at www.opertech.com/primes/kpiwchart.html shows the crossover
                    (s) and the growth.
                    It is estimated that all intervals of more than 5980 integers can
                    demonstrate a super-dense condition.

                    Enjoy
                    Thomas J Engelsma






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                  • Phil Carmody
                    ... From: Tom ... Wonderful work, Tom. One I wish I d been part of! ... I absolutely will! I of course blindly accept the possible
                    Message 9 of 19 , Jan 17, 2006
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                      --- primenumbers@yahoogroups.com wrote:
                      From: "Tom" <tom@...>
                      >
                      > On the k-tuples, have just finished an update of the permissible
                      > patterns site www.opertech.com/primes/k-tuples.html
                      > At the site the trophies (contradiction patterns/super dense
                      > constellations) are listed up to packing 100 additional primes in an
                      > interval. In fact 17 additional prime can be packed in an interval of
                      > length of just 8509 integers.

                      Wonderful work, Tom. One I wish I'd been part of!

                      > The chart at www.opertech.com/primes/kpiwchart.html shows the crossover
                      > (s) and the growth.
                      > It is estimated that all intervals of more than 5980 integers can
                      > demonstrate a super-dense condition.
                      >
                      > Enjoy

                      I absolutely will!

                      I of course blindly accept the "possible therefore happens" approach in such
                      'linear' matters (i.e. this does not in any way apply to pseudoprime existance
                      questions). It's just a shame that there will probably never be any possibilty
                      of the human race actually finding such a tuple. I don't know if QC can reduce
                      the problem to a triviality, but I don't have much faith in QC either!

                      Are you still pushing the green zone to the right, or has your program hit an
                      architectural brick wall now?

                      Phil


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                    • Tom
                      Are you still pushing the green zone to the right, or has your program hit an architectural brick wall now? Phil, On the chart, the green zone is from
                      Message 10 of 19 , Jan 17, 2006
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                        Are you still pushing the green zone to the right, or has your
                        program hit an architectural brick wall now?

                        Phil,
                        On the chart, the green zone is from exhautive searching, and cannot
                        (will not) be improved. The assembler program I wrote had a ceiling
                        of 2047. When I move to a 64-bit machine I intend to run the
                        exhaustive search some more. My initial calculations say I should be
                        able to run up to 2250 before I get exponentially stopped.
                        The red zone is in constant change above 3900, some lucky finds
                        below that. The updates page
                        http://www.opertech.com/primes/updates.html shows the changes.
                        These changes are in the 2047 to 8509 range.

                        The trophy case www.opertech.com/primes/trophycase.html shows
                        the first instance currently known for each additional prime. And
                        you can see the widths are being improved.

                        Fengsui Liu,
                        The work I have done is to find permissible patterns of primes not
                        the primes themselves.

                        Thank-you for the interest.
                        Thomas J Engelsma
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