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Re: [PrimeNumbers] Re: Polynomials

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  • Jack Brennen
    ... From the movie Being There (1979). And I bring this up mostly because by some joyful coincidence, I watched the movie last night. :) Chance the
    Message 1 of 33 , Jul 27, 2013
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      On 7/27/2013 8:53 AM, djbroadhurst wrote:
      >
      > Conversely, we may echo Keynes:
      >
      > "But this long run is a misleading guide to current affairs.
      > In the long run we are all dead.
      > Economists set themselves too easy, too useless a task if in
      > tempestuous seasons they can only tell us that when the storm
      > is past the ocean is flat again."
      > John Maynard Keynes (1883-1946), in
      > "A Tract on Monetary Reform" (1923) Ch. 3:
      > http://www.unc.edu/depts/econ/byrns_web/Economicae/jmkeynes.html
      >

      From the movie "Being There" (1979). And I bring this up mostly
      because by some joyful coincidence, I watched the movie last night. :)

      Chance the Gardener: Yes. In the garden, growth has it seasons. First comes spring and summer, but then we have fall and winter. And then we get spring and summer again.
      President "Bobby": Spring and summer.
      Chance the Gardener: Yes.
      President "Bobby": Then fall and winter.
      Chance the Gardener: Yes.
      Benjamin Rand: I think what our insightful young friend is saying is that we welcome the inevitable seasons of nature, but we're upset by the seasons of our economy.
      Chance the Gardener: Yes! There will be growth in the spring!
    • 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, 2013
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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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