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Re: Polynomials

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  • djbroadhurst
    ... Thanks, Marian. Note that what was in 16th place when ranked by P4 is now in third place when ranked by P6. Most of us believe (yet cannot prove) that the
    Message 1 of 33 , Jul 26 4:08 PM
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      --- In primenumbers@yahoogroups.com,
      Chroma <chromatella@...> wrote:

      > Small supplement your results
      > Pk -> n = 0..10^k
      >
      > {rank/P4,{a, P2, P4, P5, P6}}
      > {1,{247757,71,5028,39759,324001}}
      > {2,{595937,61,4978,39293,322141}}
      ....
      > {16,{15102077,46,4631,38561,318251}}

      Thanks, Marian. Note that what was in 16th place when
      ranked by P4 is now in third place when ranked by P6.

      Most of us believe (yet cannot prove) that the asymptotic density
      is given by Conjecture F of "Partitio Numerorum, III"
      by Hardy and Littlewood, in Acta Mathematica, December 1923,
      Volume 44, Issue 1, pp 1-70.

      Henri Cohen computed, for example
      HL(x^2+x+41) = 3.319773177471421665323556857649887966468554585653...
      It would be interesting to compute the constants
      HL(x^2+x+a), for a = 247757, 595937, 15102077.

      David
    • djbroadhurst
      ... Bad models. ... No. Rather it is that n1, the begining of the sampling interval, needs to be substantially larger than sqrt(a), for the HL heuristic to win
      Message 33 of 33 , Jul 28 8:25 AM
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        --- In primenumbers@yahoogroups.com,
        "WarrenS" <warren.wds@...> wrote:

        > > Let N(a,n1,n2) be the number of primes of the form
        > > n^2+n+a with n in [n1,n1+n2]. Then the data
        > >
        > > N(247757,0,10^6) = 324001
        > > N(3399714628553118047,0,10^6) = 251841
        > >
        > > seem to favour the smaller value of a. Yet these data
        > >
        > > N(247757,10^12,10^6) = 148817
        > > N(3399714628553118047,10^12,10^6) = 193947
        > >
        > > indicate that the larger value of a is better, in the long run.
        >
        > --these numbers seem to be in vast violation of naive statistical
        > models.

        Bad models.

        > Is the reason, that the length n2 of the sampling interval,
        > needs to be substantially larger than a, in order for naive
        > statistical models to become reasonably valid?

        No. Rather it is that n1, the begining of the sampling
        interval, needs to be substantially larger than sqrt(a),
        for the HL heuristic to win out. Clearly when
        n1 < sqrt(3399714628553118047), Marion was comparing apples
        and oranges, since log(n^2+n+a) was dominated by "a".

        All I did was to level the playing field, here:

        > N(247757,10^12,10^6) = 148817
        > N(3399714628553118047,10^12,10^6) = 193947

        to allow the HL heuristic to show through.

        It's a simple as that. No shock-horror for statisticians;
        Just a trivial observation by a log-lover :-)

        David
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