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Re: The evolution of a tuple

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  • Mark Underwood
    After doing some reading on the subject I am now firmly convinced that prime constellations with densities exceeding that of the early primes do exist. And
    Message 1 of 3 , Jun 2, 2005
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      After doing some reading on the subject I am now firmly convinced
      that prime constellations with densities exceeding that of the early
      primes do exist. And perhaps with quantum computing they shall be
      found.

      But I am also convinced that the densities will never exceed a
      certain amount, as follows:

      For instance, there are exactly 168 primes from 1 to 1,000. I
      wouldn't be surprised at all now if a prime constellation waaaay up
      there somewhere exeeds 168 primes in an interval of 1,000.

      *But*, there are exactly 95 primes from 0 to 1000/2. If we include
      negative numbers as primes, that makes 190 primes from -500 to 500.

      So to my intuition, no interval of 1,000 could ever surpass this 190
      count.

      Mark




      --- In primenumbers@yahoogroups.com, "Mark Underwood"
      <mark.underwood@s...> wrote:

      > Anyways, after actually doing the tuple thing, I am now seriously
      > doubting my belief that the earilest primes will never be surpassed
      > in density, go figure. :)
      >
      > Mark
    • Jens Kruse Andersen
      ... The smallest admissable 168-tuplet is 1033 wide according to an exhaustive search by Thomas J. Engelsma: http://www.opertech.com/primes/k-tuples.html His
      Message 2 of 3 , Jun 2, 2005
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        Mark Underwood wrote:

        > There are exactly 168 primes from 1 to 1,000. I
        > wouldn't be surprised at all now if a prime constellation waaaay up
        > there somewhere exeeds 168 primes in an interval of 1,000.

        The smallest admissable 168-tuplet is 1033 wide according to an exhaustive
        search by Thomas J. Engelsma:
        http://www.opertech.com/primes/k-tuples.html

        His current results say that if there is a prime tuplet closer together than the
        earliest primes then it must be at least a 311-tuplet.
        From inexhaustive searching, his smallest admissable tuplet beating the earliest
        primes is a 447-tuplet of width 3159, where p(447) = 3163. Finding an occurrence
        of such a tuplet is out of the question today. However I agree with the k-tuplet
        conjecture that there are infinitely many.

        --
        Jens Kruse Andersen
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