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Prime sieving on the polynomial f(n)=n^2+1

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  • bhelmes_1
    A beautiful day, There are some results for primes concerning the polynom f(n)=n^2+1 109.90.3.58/devalco/quadr_Sieb_x^2+1.htm Besides some nice algorithms how
    Message 1 of 4 , Jul 4, 2012
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      A beautiful day,

      There are some results for primes concerning the polynom f(n)=n^2+1

      109.90.3.58/devalco/quadr_Sieb_x^2+1.htm

      Besides some nice algorithms how to calculate these primes,
      there are results for n up to 2^40

      You will find some nice graphics and some calculations.

      Mathematical feedback is welcome :-)

      Greetings from the primes
      Bernhard
    • Mark
      Hello Bernhard, Looking at table 4a, it seems to present column c as the number of primes up to 2^n, and column d as the number of primes of the form x^2+1 up
      Message 2 of 4 , Jul 5, 2012
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        Hello Bernhard,

        Looking at table 4a, it seems to present column c as the number of primes up to 2^n, and column d as the number of primes of the form x^2+1 up to 2^n. But the numbers can't be correct so am I misunderstanding something?

        I'm looking at the small table 5b where it seems you extrapolate the number of primes of the form x^2+1 which are between 2^n and 2^(n+1). The number of such primes is decreasing as n increases, but it seems to me it should be going up.


        You wrote in the conclusion,

        "The density of the distribution of the primes p=x^2+1 goes surely to zero.
        Supposing that the exploration of the second scenario is rigth, it could be supposed that the primes of the form x^2+1 are limited."

        Are you saying that the number of primes of the form x^2+1 appears to be finite?

        Thanks,

        Mark



        --- In primenumbers@yahoogroups.com, "bhelmes_1" <bhelmes@...> wrote:
        >
        > A beautiful day,
        >
        > There are some results for primes concerning the polynom f(n)=n^2+1
        >
        > 109.90.3.58/devalco/quadr_Sieb_x^2+1.htm
        >
        > Besides some nice algorithms how to calculate these primes,
        > there are results for n up to 2^40
        >
        > You will find some nice graphics and some calculations.
        >
        > Mathematical feedback is welcome :-)
        >
        > Greetings from the primes
        > Bernhard
        >
      • Mark
        The message below got stuck in the ether for over a day, and surely the Higgs boson had something to do with it. Anyways, Bernhard has since clarified things
        Message 3 of 4 , Jul 7, 2012
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          The message below got stuck in the ether for over a day, and surely the Higgs boson had something to do with it. Anyways, Bernhard has since clarified things for me privately and I should be better now.

          Mark


          --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
          >
          >
          > Hello Bernhard,
          >
          > Looking at table 4a, it seems to present column c as the number of primes up to 2^n, and column d as the number of primes of the form x^2+1 up to 2^n. But the numbers can't be correct so am I misunderstanding something?
          >
          > I'm looking at the small table 5b where it seems you extrapolate the number of primes of the form x^2+1 which are between 2^n and 2^(n+1). The number of such primes is decreasing as n increases, but it seems to me it should be going up.
          >
          >
          > You wrote in the conclusion,
          >
          > "The density of the distribution of the primes p=x^2+1 goes surely to zero.
          > Supposing that the exploration of the second scenario is rigth, it could be supposed that the primes of the form x^2+1 are limited."
          >
          > Are you saying that the number of primes of the form x^2+1 appears to be finite?
          >
          > Thanks,
          >
          > Mark
          >
          >
          >
          > --- In primenumbers@yahoogroups.com, "bhelmes_1" <bhelmes@> wrote:
          > >
          > > A beautiful day,
          > >
          > > There are some results for primes concerning the polynom f(n)=n^2+1
          > >
          > > 109.90.3.58/devalco/quadr_Sieb_x^2+1.htm
          > >
          > > Besides some nice algorithms how to calculate these primes,
          > > there are results for n up to 2^40
          > >
          > > You will find some nice graphics and some calculations.
          > >
          > > Mathematical feedback is welcome :-)
          > >
          > > Greetings from the primes
          > > Bernhard
          > >
          >
        • bhelmes_1
          Hello Mark ... i distinguish between primes p=x^2+1 and the primes which appear as divisor for the first time p | x^2+1 and p
          Message 4 of 4 , Jul 7, 2012
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            Hello Mark

            > Looking at table 4a, it seems to present column c as the number of primes up to 2^n, and column d as the number of primes of the form x^2+1 up to 2^n. But the numbers can't be correct so am I misunderstanding something?

            i distinguish between primes p=x^2+1 and
            the primes which appear as divisor for the first time
            p | x^2+1 and p < x^2+1

            for x<=2^5 = 32 i get
            p=x^2+1 : 2(x=1), 5(x=2), 17(x=4), 37(x=6), 101(x=10), 197(x=14), 257(x=16), 401(x=20), 577(x=24), 677(x=26),
            number of Primes for x<32 is 10 as table 4a) column D indicates

            p|x^2+1 : 13(x=5), 41(x=9), 61(x=11), 29(x=12), 113(x=15), 181(x=19),
            97(x=22), 53(x=23), 313 (x=25), 73(x=27), 157(x=28), 421(x=29)
            number of Primes for x<=32 is 12 as table 4a) column E indicates

            > I'm looking at the small table 5b where it seems you extrapolate the number of primes of the form x^2+1 which are between 2^n and 2^(n+1). The number of such primes is decreasing as n increases, but it seems to me it should be going up.

            C = Number of new primes of the form p=x^2+1 between A and B
            The number of primes between 2^n and 2^(n+1) is decreasing,
            i round the numbers to integer.

            > You wrote in the conclusion,
            >
            > "The density of the distribution of the primes p=x^2+1 goes surely to zero.
            > Supposing that the exploration of the second scenario is rigth, it could be supposed that the primes of the form x^2+1 are limited."
            >
            > Are you saying that the number of primes of the form x^2+1 appears to be finite?

            The density of these primes goes to 0, that does not mean that there could not be any primes of the form x^2+1 any further.
            i do not know any algorithm to calculate the evidence of these primes
            in advance which might be possible.
            Nevertheless the search for those primes might be very difficult.

            I would like to compare the results for the polynom x^2+1 with
            the results for the polynom 2x^2-1
            http://109.90.3.58/devalco/quadr_Sieb_2x%5E2-1.htm

            The last calculation take 23 days, therefore i will need a month
            in order to calculate the results for the polynom 2x^2-1
            (The Mersenne primes occure on this polynom)

            Last but not least i found 191 polynoms
            http://109.90.3.58/devalco/basic_polynoms/
            which could be examined.

            Therefore there is a lot of work :-)
            Any help is welcome.

            Nice Greetings from the Primes
            Bernhard
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