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k-tuples

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  • Tom
    for anyone interested i have compiled a list of min width k-tuples for k up to 163 as an exhaustive search, listing ALL possible k-tuples -- usually 2, or 4
    Message 1 of 19 , Mar 9, 2003
      for anyone interested i have compiled a list of min width k-tuples
      for k up to 163 as an exhaustive search, listing ALL possible
      k-tuples -- usually 2, or 4 but in the instance of w=601 k=105
      there are 488 variations.

      also, have k-tuples up to 274 but not yet done as exhaustive
      search -- used statistical exit limits.

      Tom


      [Non-text portions of this message have been removed]
    • Phil Carmody
      ... Fantastic work! I d send them over to Tony Forbes if I were you. See http://www.ltkz.demon.co.uk/ktuplets.htm and in particular
      Message 2 of 19 , Mar 9, 2003
        --- Tom <tom@...> wrote:
        > for anyone interested i have compiled a list of min width k-tuples
        > for k up to 163 as an exhaustive search, listing ALL possible
        > k-tuples -- usually 2, or 4 but in the instance of w=601 k=105
        > there are 488 variations.
        >
        > also, have k-tuples up to 274 but not yet done as exhaustive
        > search -- used statistical exit limits.

        Fantastic work!

        I'd send them over to Tony Forbes if I were you.
        See
        http://www.ltkz.demon.co.uk/ktuplets.htm
        and in particular
        http://www.ltkz.demon.co.uk/ktmin.txt


        Phil



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      • thoeng
        for your consumption http://www.opertech.com/primes/k-tuples.html list the k-tuples using exhaustive searching as time permits, will post ALL variations for
        Message 3 of 19 , Jul 12, 2003
          for your consumption

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

          list the k-tuples using exhaustive searching
          as time permits, will post ALL variations for
          each k-tuple.

          Tom
          It's not just an adventure,
          it's a JOB !!
        • Mark Underwood
          What an incredible store of information! I was most amazed at the variaration in the number of variations per tuple. It was usually around two or four per
          Message 4 of 19 , Jul 12, 2003
            What an incredible store of information! I was most amazed at the
            variaration in the number of variations per tuple. It was usually
            around two or four per tuple, but occassionally there would be a
            spike and sometimes a huge spike, like at the tuple of 105 primes
            where there is 486 variations!

            Also the prime densities per tuple are interesting. As the size of
            the tuples increase, the density decreases as a general rule. For
            instance 2 primes can occur in a space (width) of 3 numbers, so the
            maximum density is .66. Three primes can occur in a space of 7
            numbers so the density is about .43. But Four primes can occur in a
            space of 9 numbers so the density is about .44. So four beats three!
            Based on Tom's chart,

            4 beats 3
            10 beats 9 and 8
            11 beats 9 and 8
            14 beats 13
            20 beats 19
            21 beats 20,19
            23 beats 22
            26 beats 25
            29 beats 28
            31 beats 30
            34 beats 33
            35 beats 34,33,32,31,30
            36 beats 35,34,33,32,31,30,28
            37 beats 34,33,32
            40 beats 39, which is as far as I went.


            Bejjink's (Jeff's?) findings (which are certainly correct I would
            say), appear consistent with Tom's. One aspect of Jeff's findings is
            that they give a maximum number of primes per space of 2*3+1 or
            2*3*5+1, 2*3*5*7+1, ect.

            In a space of 2*3 + 1 Jeff's findings say there are three primes
            allowed, maximum. Tom's chart gives three.

            In a space of 2*3*5 + 1 Jeff's findings say there are 9 primes
            allowed maximum. Tom's chart gives nine.

            In a space of 2*3*5*7 + 1 Jeff's findings say there are 49 primes
            allowed maximum. Jeff has found that 44 primes are the max in this
            space.

            Mark





            --- In primenumbers@yahoogroups.com, "thoeng" <tom@o...> wrote:
            > for your consumption
            >
            > http://www.opertech.com/primes/k-tuples.html
            >
            > list the k-tuples using exhaustive searching
            > as time permits, will post ALL variations for
            > each k-tuple.
            >
            > Tom
            > It's not just an adventure,
            > it's a JOB !!
          • thoeng
            Reloaded grid with included columns largest destructive prime largest 1 open class residues smallest 2 open class residues largest embedded k-tuples Also,
            Message 5 of 19 , Jul 14, 2003
              Reloaded grid with included columns
              largest destructive prime
              largest 1 open class residues
              smallest 2 open class residues
              largest embedded k-tuples

              Also, started adding links to the representations.
              have 2-44 and 105 up with masks.
              (still needs work to make clear - intend to add
              variation number, primorial residues [calc], and
              largest embedded k-tuple for each variation)

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

              Next week or two should have all up thru 300-tuple.
              (working on program to create the HTML-300 pages worth)

              Tom
              Frittering away some time.
            • thoeng
              new algorithm is not exhaustive search like my earlier version but it is fast and has found a new minimum for Hardy-Littlewood conjecture. Earlier work was
              Message 6 of 19 , Aug 11, 2003
                new algorithm is not exhaustive search like my earlier version
                but it is fast and has found a new minimum for Hardy-Littlewood
                conjecture. Earlier work was exhaustive searches so I could
                assign '=' -- this is not exhaustive so will need '<=" as this
                is a limit.

                598-tuple is <= 4387 long // pi(4387)=597 (+1)

                also, checking other work

                the earlier 658-tuple that has write-ups
                should be changed (Jarvis).

                was 658-tuple is 4930 long
                whereas is should be
                661-tuple is <= 4329 long // pi(4329)=657 (+4)

                also this thing found some extreme tuples
                eg.
                800-tuple is <= 6033 long // pi(6033)=786 (+14)

                (for your information--test program written in basic
                can't wait to get enough time to rewrite in assembler)

                Tom Engelsma
              • Tom
                updated page(s) per request, and added links to the actual variations of interesting k-tuples. Thnx Mike. http://www.opertech.com/primes/k-tuples.html Tom
                Message 7 of 19 , Sep 7, 2003
                  updated page(s) per request, and added links to the actual
                  variations of 'interesting' k-tuples.
                  Thnx Mike.

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

                  Tom
                • Tom
                  Upper bound lowered a bit more. Now at 4333 Interval of length 4333 can contain 592 primes, whereas pi(4333)=591 Not a big gain, but 22 more off the upper
                  Message 8 of 19 , Nov 15, 2003
                    Upper bound lowered a bit more.

                    Now at 4333

                    Interval of length 4333 can contain 592 primes,
                    whereas pi(4333)=591

                    Not a big gain, but 22 more off the upper bound.

                    Unknown range now 2077 < y <=4333


                    Tom
                  • Tom
                    Have the k-tuples web page back up http://www.opertech.com/primes/k-tuples.html Lost server link -- main page is back up and will reload individual pages this
                    Message 9 of 19 , May 7, 2004
                      Have the k-tuples web page back up
                      http://www.opertech.com/primes/k-tuples.html

                      Lost server link -- main page is back up
                      and will reload individual pages this weekend.

                      Tom
                    • Tom
                      Concerning the Hardy-Littlewood conjecture, pi(x+y)-pi(x)
                      Message 10 of 19 , Jan 9, 2005
                        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 11 of 19 , Feb 9, 2005
                          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 12 of 19 , Feb 9, 2005
                            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 13 of 19 , Feb 14, 2005
                              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 14 of 19 , Feb 20, 2005
                                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 15 of 19 , Mar 2, 2005
                                  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 16 of 19 , Jan 16, 2006
                                    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 17 of 19 , Jan 16, 2006
                                      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 18 of 19 , Jan 17, 2006
                                        --- 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 19 of 19 , Jan 17, 2006
                                          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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