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

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  • 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 1 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 2 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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