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small Collection of Primes

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  • bhelmes_1
    A beautifull day, 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
    Message 1 of 18 , Sep 9, 2010
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      A beautifull day,

      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

      I think the distribution of primes concerning the polynom f(x)=x^2+x+1
      is better than the linear distribution
      or in other words the chance to find one is better.

      I needed 3 month to calculate the numbers on 6 cores.

      I think i can improve the chance for finding primes on this polynom
      by calculation of x by multiplication of small primes which
      can be found on these polynom.

      Besides all and only (except 3) primes = 1 mod 6
      appear as primes or divisior of f(x)=x^2 +x+1

      Nice Greetings from the primes
      Bernhard
    • 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 2 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 3 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 4 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 5 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 6 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 7 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 8 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 9 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 10 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 11 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 12 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 13 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 14 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 15 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 16 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 17 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 18 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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