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Measured Sequences of Consecutive Primes

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  • w_sindelar@juno.com
    Has anyone looked at consecutive primes from this perspective? Here s my simple-minded idea: Visualize the positive integer number line. Keep the zero mark and
    Message 1 of 3 , Apr 11, 2004
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      Has anyone looked at consecutive primes from this perspective? Here's my
      simple-minded idea:
      Visualize the positive integer number line. Keep the zero mark and erase
      1, 2, and all composite integers. What remains is an infinite set of
      CONSECUTIVE odd primes arranged in numerical order but spaced erratically
      along the line. Select any subset of "N" consecutive primes with "P(1)"
      representing the smallest prime and "P(N)" representing the largest. IF
      (P(1) - 1) and (P(N) + 1) are BOTH EVENLY divisible by (P(N) - P(1) + 2)
      then let's call the selected sequence a "Measured Sequence of Consecutive
      Primes". It is "measured" because a measuring tape marked off in (P(N) -
      P(1) + 2) units when laid on the above integer number line, zero mark
      matching zero mark, will have one of its units minimally enclose the
      selected subset. Let's call a unit like that a "Measure". Playing with
      this concept suggests that the following 4 statements may be true, but
      probably unprovable:
      (1) For ANY integer "N" equal to 2 or greater, there ALWAYS exists a
      Measured Sequence of Consecutive Primes which consists of N primes.
      (2) For ANY integer "N" equal to 2 or greater, there ALWAYS exists a
      Measured Sequence of Consecutive Primes which consists of N primes, and
      whose Measure is EVENLY DIVIDED by the number of primes N it encloses.
      (This boggles my mind!)
      To illustrate both of the above 2 statements, take N= 500. The measured
      sequence of 500 consecutive primes is when P(1)= 5167501 and P(N)=
      5174999. The Measure is 7500. It is evenly divided by N=500. Take N=1000,
      P(1)= 23562001, P(N)= 23578999, measure is 17000 evenly divided by 1000.
      Take N=313, P(1)= 38180993, P(N)= 38185999, measure is 5008 evenly
      divided by 313.
      (3) NOT ALL primes can be the SMALLEST prime of a measured sequence of
      consecutive primes. The first few that cannot are; 3, 47, 59, 139, 179,
      227, 293, 359, 383, 389…
      (4) For EVERY EVEN integer "Gap"= 2 or greater, there exists a Measured
      Sequence of Consecutive Primes which consists of 2 primes "P" < "Q" whose
      difference equals Gap.
      For example, take Gap=14. A Measured Sequence of 2 Consecutive Primes
      with a Gap of 14 occurs when P= 113 and Q=127. Gap= 14 also happens to be
      a FIRST OCCURRENCE gap as Dr. Nicely defines them. The only other first
      occurrence gaps I was able to find that also define a measured sequence
      are 4, 22, 28, 54.
      I apologize if this long posting is utter nonsense. Any comments or
      corrections would be appreciated. Thanks folks and regards.
      Bill Sindelar
    • Norman Luhn
      Hello Primehunters ! I will inform all of you, in few days I have found on a 1300 MHz Duron a 10-tuplet with more than 100 digits. Best wishes Norman Mit
      Message 2 of 3 , Apr 11, 2004
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        Hello Primehunters !

        I will inform all of you, in few days I have found on
        a 1300 MHz Duron a 10-tuplet with more than 100
        digits.

        Best wishes

        Norman





        Mit schönen Grüßen von Yahoo! Mail - http://mail.yahoo.de
      • Jens Kruse Andersen
        ... Congratulations on that big improvement: http://www.ltkz.demon.co.uk/ktuplets.htm#largest10 A few days sounds either lucky or efficient. I see you also
        Message 3 of 3 , Apr 11, 2004
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          Norman Luhn wrote:
          > I will inform all of you, in few days I have found on
          > a 1300 MHz Duron a 10-tuplet with more than 100
          > digits.

          Congratulations on that big improvement:
          http://www.ltkz.demon.co.uk/ktuplets.htm#largest10
          A few days sounds either lucky or efficient.

          I see you also broke the 6-tuplet record, registered only a week before:
          http://www.ltkz.demon.co.uk/kthist.txt

          I have been occupied with BiTwins lately (which is more than can be said about
          the record page maintainer), but maybe I should look at tuplets again before too
          many records are lost.

          --
          Jens Kruse Andersen
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