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Re: small Collection of Primes

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  • djbroadhurst
    ... x^2+x+1 is prime with probability asymptotic to C/log(x^2), where C = prod(p,1-kronecker(-3,p)/(p-1)) with a product over all primes, p. Bernhard has
    Message 1 of 18 , Sep 9, 2010
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      --- In primenumbers@yahoogroups.com,
      "bhelmes_1" <bhelmes@...> wrote:

      > I think the distribution of primes concerning the polynom
      > f(x)=x^2+x+1 is better than the linear distribution

      x^2+x+1 is prime with probability asymptotic to C/log(x^2), where
      C = prod(p,1-kronecker(-3,p)/(p-1))
      with a product over all primes, p.

      Bernhard has observed that C > 2.
      I shall be somewhat more precise:

      C=
      2.241465507098582781236668372470874433527110162241\
      32536355383092271281135434409630277148141523001173\
      60843050218571111781772521331290036121264851241797\
      879383961418406006110408676488798831671040955287287...

      If you are interested in computing such numbers, see
      > High precision computation of Hardy-Littlewood constants
      at http://www.math.u-bordeaux1.fr/~cohen/

      David
    • bhelmes_1
      A beautifull day, there are 32 new 20000 digit primes in the collection :-) http://beablue.selfip.net/devalco/Collection/20000/ (needed time 14 days on 6
      Message 2 of 18 , Sep 17, 2010
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        A beautifull day,

        there are 32 new 20000 digit primes in the collection :-)
        http://beablue.selfip.net/devalco/Collection/20000/

        (needed time 14 days on 6 cores)

        the chance for me to find one is 1 : 2300
        i think that is not bad.

        how probably is to find the next mersenne prime ?

        i improved the algorithm by multipling all small primes mod 6 = 1
        < 14000
        for x, multiply some random numbers with x,
        calculate x(x+1)+1 and test the number

        > i have started a small collection of primes between 1000 and 10000 digit primes.
        >
        > The primes are all of the form p:=x^2+x+1=x(x+1)+1
        >
        > I calculated x of the product of small numbers in order that i could
        > verify them with pfgw -t. I made first a fermat test and if this one is succesfull, i wrote down the number and checked it later.
        >
        > The collection is availible under
        > http://devalco.de 16. Collection of Primes
        >
        > the collection includes :
        > 1000 digit primes 17514 primes chance for finding
        > 2000 " " 5635 " 1:260
        > 3000 " " 3984 " 1:434
        > 4000 " " 770 " 1:620
        > 5000 " " 385 " 1:810
        > 6000 " " 219 " 1:1050
        > 7000 " " 146 "
        > 8000 " " 237 " 1:1633
        > 9000 " " 238 "
        > 10000 " " 113 " 1:1814
        20000 " " 32 " 1:2300

        Greetings from the primes
        Bernhard
      • bhelmes_1
        A beautifull day, there are 22 new 30000 digit primes in the collection :-) http://beablue.selfip.net/devalco/Collection/30000/ (needed time 30 days on 6
        Message 3 of 18 , Oct 22, 2010
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          A beautifull day,

          there are 22 new 30000 digit primes in the collection :-)
          http://beablue.selfip.net/devalco/Collection/30000/

          (needed time 30 days on 6 cores)

          the chance for me to find one is approximitly 1 : 3000
          i think that is not bad.

          how probably is to find the next mersenne prime ?

          The collection is availible under
          http://devalco.de 16. Collection of Primes

          > > the collection includes :
          > > 1000 digit primes 17514 primes chance for finding
          > > 2000 " " 5635 " 1:260
          > > 3000 " " 3984 " 1:434
          > > 4000 " " 770 " 1:620
          > > 5000 " " 385 " 1:810
          > > 6000 " " 219 " 1:1050
          > > 7000 " " 146 "
          > > 8000 " " 237 " 1:1633
          > > 9000 " " 238 "
          > > 10000 " " 113 " 1:1814
          > 20000 " " 32 " 1:2300
          30000 " " 22 " 1:3000

          Greetings from the primes
          Bernhard
        • bhelmes_1
          A beautifull day, there are 15 new 40000 digit primes in the collection :-) http://beablue.selfip.net/devalco/Collection/40000/ (needed time 30 days on 6
          Message 4 of 18 , Nov 3, 2010
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            A beautifull day,

            there are 15 new 40000 digit primes in the collection :-)
            http://beablue.selfip.net/devalco/Collection/40000/

            (needed time 30 days on 6 cores)

            the chance for me to find one is something about 1 : 3600
            this result is unsure because i only found 15 primes

            Nevertheless the chance to find some primes with 100000 digits
            may be around 1:10000

            Are there any other statistics about other special primes
            about the densitys of primes / the chance to find one ?

            I will run my programm for a month for 50000 digit primes
            and will try to get some more exact numbers of runtime / test (one fermat test with gmp) and some other values.

            if someone is interested to participate in running my c-programm,
            i will send it to him

            Small explication of the algorithm:
            i search for primes on the polynom n^2+n+1 = n(n+1)+1
            calculate n by multiplication of all small primes with p-1 | 6 below a limit and some random numbers
            only make a fermat test and test the founded candidates by pfgw

            >
            > how probably is to find the next mersenne prime ?
            >
            > The collection is availible under
            > http://devalco.de 16. Collection of Primes
            >
            > > > the collection includes :
            > > > 1000 digit primes 17514 primes chance for finding
            > > > 2000 " " 5635 " 1:260
            > > > 3000 " " 3984 " 1:434
            > > > 4000 " " 770 " 1:620
            > > > 5000 " " 385 " 1:810
            > > > 6000 " " 219 " 1:1050
            > > > 7000 " " 146 "
            > > > 8000 " " 237 " 1:1633
            > > > 9000 " " 238 "
            > > > 10000 " " 113 " 1:1814
            > > 20000 " " 32 " 1:2300
            > 30000 " " 22 " 1:3000
            40000 " " 15 " 1:3600 ???

            Greetings from the primes
            Bernhard
          • djbroadhurst
            ... Yes. It s very simple, Bernhard, and was long since exposed ... If you sieve any target N to sufficient depth p and and find that N still has no identified
            Message 5 of 18 , Nov 4, 2010
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              --- 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
            • bhelmes_1
              Dear David ... there must be a difference in the distribution of primes in linear progression and the distribution of primes concerning irreducible polynoms
              Message 6 of 18 , Nov 19, 2010
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                Dear David

                > > 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...
                >

                there must be a difference in the distribution of primes
                in linear progression and the distribution of primes concerning
                irreducible polynoms like p:=n^2+n+1

                As far as i see primes of the form of p:=n^2+n+1 appear
                more often as the linear distribution.

                Please give me a hint if i am wrong or not

                Nice Greetings from the primes
                Bernhard
              • djbroadhurst
                ... 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
                Message 7 of 18 , Nov 19, 2010
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                  --- In primenumbers@yahoogroups.com,
                  "bhelmes_1" <bhelmes@...> wrote:

                  > there must be a difference in the distribution of primes
                  > in linear progression and the distribution of primes concerning
                  > irreducible polynoms like p:=n^2+n+1
                  >
                  > As far as i see primes of the form of p:=n^2+n+1 appear
                  > more often as the linear distribution.

                  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.

                  No-one has given convincing evidence to challenge my
                  assertion that the Mertens estimate applies to all samples,
                  irrespective of what "Prime Form" you chose before sieving.
                  Note however that some forms may help you to sieve deeper,
                  in a given length of sieving time.

                  David
                • 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 8 of 18 , Dec 1, 2010
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                    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 9 of 18 , Dec 1, 2010
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                      --- 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 10 of 18 , Dec 21, 2010
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                        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 11 of 18 , Dec 21, 2010
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                          --- 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 12 of 18 , Dec 21, 2010
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                            --- 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 13 of 18 , Dec 21, 2010
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                              --- 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 14 of 18 , Dec 21, 2010
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                                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 15 of 18 , Dec 21, 2010
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                                  --- 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 16 of 18 , Oct 28, 2012
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                                    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 17 of 18 , Oct 29, 2012
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                                      --- 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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