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Re: small Collection of Primes

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  • 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 1 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 2 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 3 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 4 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 5 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 6 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 7 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 8 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 9 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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