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density of primes in k.n#-1

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  • Phil Carmody
    I m going bonkers here, I can t get rid of a factor of 2 error . I ll start with the assumtion that the probability of a number P being prime is ln(P).
    Message 1 of 3 , Jan 28, 2001
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      I'm going bonkers here, I can't get rid of a factor of 2 'error'.

      I'll start with the assumtion that the probability of a number P being prime is ln(P).

      Numbers of the form k.233#-1 cannot have primes 2..233 as factor
      Therefore the density of primes is increased by
      (2/1) * (3/2) * ... * (233/232)
      = 9.84256

      However, I've run some tests, density of primes seems to have increased by a factor of about 5.5.
      I notice that ln(233) is 5.4

      Should the density increase by 5.4 or 9.8?

      Phil


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    • Milton Brown
      Phil, Probability is not the correct word here. Probabilities are no greater than 1.0, like the probability of getting snake-eyes in dice is 1/36. And,
      Message 2 of 3 , Jan 28, 2001
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        Phil,

        "Probability" is not the correct word here.
        Probabilities are no greater than 1.0, like the
        probability of getting "snake-eyes" in dice is 1/36.
        And, ln(100) ~ 250 >> 1.0

        Perhaps, you mean something else like
        the GAP between two primes is approximately, ln(n).

        Milton L. Brown
        miltbrown@...

        PS - Are the above two "primenumber" addresses the same?

        Phil Carmody wrote:

        > I'm going bonkers here, I can't get rid of a factor of 2 'error'.
        >
        > I'll start with the assumtion that the probability of a number P being prime is ln(P).
        >
        > Numbers of the form k.233#-1 cannot have primes 2..233 as factor
        > Therefore the density of primes is increased by
        > (2/1) * (3/2) * ... * (233/232)
        > = 9.84256
        >
        > However, I've run some tests, density of primes seems to have increased by a factor of about 5.5.
        > I notice that ln(233) is 5.4
        >
        > Should the density increase by 5.4 or 9.8?
        >
        > Phil
        >
        > Mathematics should not have to involve martyrdom;
        > Support Eric Weisstein, see http://mathworld.wolfram.com
        > Find the best deals on the web at AltaVista Shopping!
        > http://www.shopping.altavista.com
        >
        >
        > Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
        > The Prime Pages : http://www.primepages.org
      • Paul Jobling
        ... Fine. By Merten s Theorem, this is approximated by ln(233)/0.56 = 9.73. ... Two thoughts occur to me - either you may be being lucky here (these things are
        Message 3 of 3 , Jan 29, 2001
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          > Numbers of the form k.233#-1 cannot have primes 2..233 as factor
          > Therefore the density of primes is increased by
          > (2/1) * (3/2) * ... * (233/232)
          > = 9.84256

          Fine. By Merten's Theorem, this is approximated by ln(233)/0.56 = 9.73.

          > However, I've run some tests, density of primes seems to have
          > increased by a factor of about 5.5.
          > I notice that ln(233) is 5.4

          Two thoughts occur to me - either you may be being lucky here (these things
          are fairly random, after all), or you may be measuring things wrongly (don't
          forget that the ln(P) probability includes the even numbers as well).

          > Should the density increase by 5.4 or 9.8?

          I would say 9.8.

          Paul.


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