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The evolution of a tuple

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  • Mark Underwood
    Hi all After the symmetrical k +/- 2^n tuple I thought I would try a symmetrical k +/- 2^n*3^m tuple to achieve a somewhat greater density. That is, k +/-
    Message 1 of 3 , Jun 2, 2005
      Hi all

      After the symmetrical k +/- 2^n tuple I thought I would try a
      symmetrical k +/- 2^n*3^m tuple to achieve a somewhat greater density.

      That is, k +/- 6,12,18,24,36,48,72,96.... Then the density bug hit
      and I thought I would throw in alternating powers of two on different
      sides of k, so that i got something like this:

      k -24, -18, -16, -12, -6, -4, +2, +6, +8, +12, + 18, +24.

      But by then all symmetry was lost. So to heck with symmetry. So I
      saw I could sneak in a couple of other numbers as well: 2*11 on one
      side and 2*13 on the other, yielding

      k -24,-22,-18, -16,-12,-6,-4,+2,+6,+8,+12,+18,+24,+26

      And painfully aware that there must be a better way, I took way too
      long manually calculating that k must be of the form t*2*3*5*7*11*13
      + 5*7*433.

      I thought very cool, that's 14 potential primes in a space of 50,
      when there are 'only' 15 primes from 0 to 50 !

      Then I started actually looking for the tuple with GP Pari. As it
      was working away, I looked on the net about tuples like this. As it
      turns out, this particular tuple is well known and in fact is one of
      only two minimal 14 tuples. The other tuple has the 2^n * 3^m terms
      the same around k, but the power of two terms terms are flipped as
      are the 2*11 and 2*13 terms, and k would be a little different, ie,
      t*2*3*5*7*11*13 + c, for a c I am not going to calculate this time!

      And of course, I have now discovered that guys like Jens have already
      found monstrous examples of this 14 tuple, like

      26093748*67# + 383123187762431 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32,
      36, 42, 48, 50

      (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen. from
      http://www.ltkz.demon.co.uk/ktuplets.htm )

      Needless to say I then put GP Pari out of its misery trying to see if
      this tuple might exist. :)

      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
    • 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 2 of 3 , Jun 2, 2005
        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 3 of 3 , Jun 2, 2005
          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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