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Re: small Collection of Primes

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  • bhelmes_1
    Dear David ... There are 11 50000 digits primes in the database http://beablue.selfip.net/devalco/Collection/50000/ My investigation confirms the Mertens
    Message 1 of 18 , Dec 1, 2010
      Dear David

      > Not after deep sieving, as I had explained.
      > We believe that a sieve effectively removes all memory of
      > origin, leaving the Mertens probability that I gave you.
      >

      There are 11 50000 digits primes in the database
      http://beablue.selfip.net/devalco/Collection/50000/

      My investigation confirms the Mertens probablity:

      AMD 5000+ Ubuntu 8.0.4 g++ 3.4 gmp 5.0.1

      Digits Runtime Presieve Calculated Mertens Formula
      1000 < 1933 1: 170 1: 171
      2000 0.1 sec, < 4243 1: 300 1: 309
      3000 0.3 sec, < 6607 1: 434 1: 441
      4000 0.6 sec, < 8929 1: 523 1: 568
      5000 1,2 sec, < 11257 1: 810 1: 693
      6000 1.8 sec, < 13627 1:1050 1: 814
      8000 3,6 sec, < 18181 1:1633 1: 1054
      10000 6,3 sec, < 22861 1:1814 1: 1289
      20000 34,0 sec, < 46141 1:2300 1: 2408
      30000 1,30 min < 69379 1:3000 1: 3480
      40000 3 min, < 92317 1:3600 1: 4524
      50000 6 min < 115549 1:5200 1: 4631
      60000 7,35 min, < 138577 1: 6553
      80000 15 min, < 184843 1: 8531
      100000 27 min, < 230563 1: 10471
      200000 132 min, < 461677 1: 19828
      1000000 < 2304553 1: 88262
      10000000 < 23035273 1:762769

      Nice Greetings from the primes
      Bernhard
    • djbroadhurst
      ... Thanks, Bernhard, for taking the time to check that conjecture. Previously, Mike Oakes had done wonderful work, confirming it to much better statistical
      Message 2 of 18 , Dec 1, 2010
        --- In primenumbers@yahoogroups.com,
        "bhelmes_1" <bhelmes@...> wrote:

        > My investigation confirms the Mertens probability

        Thanks, Bernhard, for taking the time to check that conjecture.
        Previously, Mike Oakes had done wonderful work, confirming it
        to much better statistical accuracy.

        Regarding your very limited statistics, at 50k digits,
        I remark that
        > 50000 6 min < 115549 1:5200 1: 4631
        is indeed in good accord with Mertens + PNT.
        You found only 11 primes and it is pleasing that the ratio
        5200/4631 differs from unity by less than 1/sqrt(11).

        Alles gute, alles schöne!

        David
      • bhelmes_1
        A beautiful day, i tried to hunt some 100000 digit primes. i found two primes , the first i could affirm with pfgw t p_100000_a
        Message 3 of 18 , Dec 21, 2010
          A beautiful day,

          i tried to hunt some 100000 digit primes.

          i found two "primes", the first i could affirm with pfgw t p_100000_a
          http://beablue.selfip.net/devalco/Collection/100000/

          Concerning the second "prime" i did not get a good result
          there is the error log file of pfgw
          http://beablue.selfip.net/devalco/Collection/100000/pfgw_err.log

          Perhaps someone knows how to make a deterministic test for this primes
          and perhaps this number can be usefull to improve pfgw.

          i made 12000 test to find 2 candidates for a deterministic prime in
          20 days on 12 cores, what is not bad, i think

          Nice greetings from the primes
          Bernhard

          http://devalco.de
        • djbroadhurst
          ... Try using an up-to-date version of OpenPFGW: http://sourceforge.net/projects/openpfgw/files/ I have set it running with the -t option and so far ...
          Message 4 of 18 , Dec 21, 2010
            --- In primenumbers@yahoogroups.com,
            "bhelmes_1" <bhelmes@...> wrote:

            > how to make a deterministic test for this prime
            > Expr = 6478306796723098....5899455353125001

            Try using an up-to-date version of OpenPFGW:
            http://sourceforge.net/projects/openpfgw/files/

            I have set it running with the -t option and so far

            > Running N-1 test using base 29
            ...
            > N-1: 6478306796723098....5899455353125001 232500/1749078 mro=0

            indicates no round-off problem. If this base yields no non-trivial
            gcd, then the test should complete in a couple of hours.

            David
          • djbroadhurst
            ... I remark that this candidate is of the form x^2+x+1 where x has no prime divisor greater than 1524493. Hence the BLS proof should be straightforward. So
            Message 5 of 18 , Dec 21, 2010
              --- In primenumbers@yahoogroups.com,
              "djbroadhurst" <d.broadhurst@...> wrote:

              > N-1: 6478306796723098....5899455353125001 232500/1749078 mro=0

              I remark that this candidate is of the form x^2+x+1
              where x has no prime divisor greater than 1524493.
              Hence the BLS proof should be straightforward.
              So far, N-1 has reached

              > N-1: 6478306796723098....5899455353125001 670000/1749078 mro=0

              without incident.

              David
            • djbroadhurst
              ... Indeed it was: PFGW Version 3.4.3.64BIT.20101025.x86_Dev [GWNUM 26.4] [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 29 Calling
              Message 6 of 18 , Dec 21, 2010
                --- In primenumbers@yahoogroups.com,
                "djbroadhurst" <d.broadhurst@...> wrote:

                > I remark that this candidate is of the form x^2+x+1
                > where x has no prime divisor greater than 1524493.
                > Hence the BLS proof should be straightforward.

                Indeed it was:

                PFGW Version 3.4.3.64BIT.20101025.x86_Dev [GWNUM 26.4]
                [N-1, Brillhart-Lehmer-Selfridge]
                Running N-1 test using base 29
                Calling Brillhart-Lehmer-Selfridge with factored part 33.33%
                6478306796723098364703559651976954926....
                4884247365446132388985899455353125001
                is prime! (10116.8919s+0.0854s)

                Conclusion: Bernhard's OpenPFGW was way out of date?

                David
              • bhelmes_1
                Dear David, thank you for your efforts. Is there an interest to submit the two 100000 digit primes for the collection of http://primes.utm.edu/ If yes, give me
                Message 7 of 18 , Dec 21, 2010
                  Dear David,

                  thank you for your efforts.

                  Is there an interest to submit the two 100000 digit primes for the collection of http://primes.utm.edu/

                  If yes, give me please a short link or description, how to submit them

                  Greetings from the primes
                  Bernhard
                • djbroadhurst
                  ... They are too small: http://primes.utm.edu/primes/submit.php ... David
                  Message 8 of 18 , Dec 21, 2010
                    --- In primenumbers@yahoogroups.com,
                    "bhelmes_1" <bhelmes@...> wrote:

                    > Is there an interest to submit the two 100000 digit
                    > primes for the collection of http://primes.utm.edu/

                    They are too small:
                    http://primes.utm.edu/primes/submit.php
                    > Currently primes must have 178698 or more digits

                    David
                  • bhelmes_1
                    Dear David, i did a little investigation concerning the distribution of primes concerning the polynom f(x)=x^2+1
                    Message 9 of 18 , Oct 28, 2012
                      Dear David,

                      i did a little investigation concerning the distribution of primes
                      concerning the polynom f(x)=x^2+1

                      http://109.90.219.147/devalco/quadr_Sieb_x%5E2+1.htm#8a

                      I think it is sensefull to make a presieving of the search array
                      for searching huge primes and regard the numbers which are divided
                      by the small primes also.

                      I have no analytic result how big the improvement is.
                      Do you know any similar analytic results ?

                      Nice Greetings from the primes
                      Bernhard

                      --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
                      >
                      >
                      >
                      > --- In primenumbers@yahoogroups.com,
                      > "bhelmes_1" <bhelmes@> asked:
                      >
                      > > Are there any other statistics about other special primes
                      > > about the densitys of primes / the chance to find one ?
                      >
                      > Yes. It's very simple, Bernhard, and was long since exposed
                      > by Mertens' formula:
                      >
                      > > Mertens, F.
                      > > "Ein Beitrag zur analytischen Zahlentheorie."
                      > > J. reine angew. Math. 78, 46-62, 1874.
                      >
                      > If you sieve any target N to sufficient depth p and
                      > and find that N still has no identified factor, then
                      > the probability that N is prime is, heuristically:
                      >
                      > P = exp(Euler)*log(p)/log(N)
                      >
                      > where Euler = 0.577215664901532860606512090082402431...
                      >
                      > If you ever succeed in out-performing this heuristic,
                      > on a reliable basis, then you will become very famous.
                      >
                      > If you do no better than this heuristic, then you will
                      > be like the rest of us.
                      >
                      > If you do worse than this heuristic, then something has
                      > gone badly wrong with your investigation.
                      >
                      > Alles gute
                      >
                      > David
                      >
                    • djbroadhurst
                      ... Since 1923, we have a had a very precise conjecture for the asymptotic density of primes of the form x^2+1. See Shanks review
                      Message 10 of 18 , Oct 29, 2012
                        --- In primenumbers@yahoogroups.com,
                        "bhelmes_1" <bhelmes@...> wrote:

                        > the distribution of primes
                        > concerning the polynom f(x)=x^2+1

                        Since 1923, we have a had a very precise
                        conjecture for the asymptotic density
                        of primes of the form x^2+1. See Shanks' review

                        http://www.ams.org/journals/mcom/1960-14-072/S0025-5718-1960-0120203-6/S0025-5718-1960-0120203-6.pdf

                        of the classic paper by G.H. Hardy and J.E. Littlewood:
                        "Some problems of 'Partitio numerorum'; III",
                        Acta Math. 44 (1923) pages 1–70.

                        The relevant Hardy-Littlewood constant,
                        1.3728134... is given, to 9 significant figures,
                        in Eq(3) of Shanks' paper.

                        More digits are easily obtainable from the methods in
                        "High precision computation of Hardy-Littlewood constants"
                        by Henri Cohen, available as a .dvi file from
                        http://www.math.u-bordeaux1.fr/~hecohen/

                        David
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