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Re: [PrimeNumbers] New Type of Prime Arithmetic Progression?

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  • Jens Kruse Andersen
    ... No problem. You can search more information about admissible constellations with a search engine. If a prime p
    Message 1 of 12 , Sep 29, 2007
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      Bill Sindelar wrote:
      > Jens, I think I may have offended you by writing "I'm lost here. Seems
      > like a convoluted approach."

      No problem. You can search more information about admissible constellations
      with a search engine.

      If a prime p <= k does not divide the common difference in an AP-k then
      p will divide at least one of the terms in the AP. In order to be
      admissible, a PAP-k must therefore have a common difference which is
      a multiple of k# (k primorial).
      I guess a PAP-k with small difference (and therefore relatively few primes
      between the terms) will have a better chance of being a (PAP-k, n),
      because the number of primes can vary between fewer values.
      A PAP-11 has minimal difference 11# = 2310, so 10 intervals of 2309
      numbers must have the same prime count to produce a (PAP-11, n)
      with minimal difference. That appears computationally too hard for me.

      PAP-7 to PAP-10 all have minimal difference 10# = 7# = 210.
      I used my old tuplet finder to systematically search a lot of PAP-10 with
      difference 210 and count whether there happened to be an equal number
      of primes between the terms. There were other things to use my only
      computer for so the search stopped when only (PAP-8, n) had been found.
      Hans Rosenthal is more patient and has found many (PAP-9, 0), also called
      CPAP-9, with a version of the same program. (PAP-9, n>3) looks easier.
      In 2004 he found the smallest known CPAP-8 = (PAP-8, 0) with
      another version. I just tested the other PAP-8 from the search and found
      a (PAP-8, 1) with difference 210:
      64881326075217862991473794035228920286672784697 +
      0,36,210,264,420,564,630,784,840,942,1050,1086,1260,1360,1470

      --
      Jens Kruse Andersen
    • w_sindelar@juno.com
      ... I m relieved. I was trying to get up some nerve to ask you what sort of ... I just tested the other PAP-8 from the search and ... equivalent to the above
      Message 2 of 12 , Oct 2, 2007
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        Jens K. Anderson wrote:
        > No problem. You can...
        >>

        I'm relieved. I was trying to get up some nerve to ask you what sort of
        approach you used on (PAP-8, 5) when your mail arrived with the answer:

        > I used my old tuplet finder to systematically search a lot of PAP-10
        > with
        > difference 210 and count whether there happened to be an equal
        > number
        > of primes between the terms. There were other things to use my only
        > computer for so the search stopped when only (PAP-8, n) had been
        > found.
        I just tested the other PAP-8 from the search and
        > found
        > a (PAP-8, 1) with difference 210:
        > 64881326075217862991473794035228920286672784697 +
        > 0,36,210,264,420,564,630,784,840,942,1050,1086,1260,1360,1470

        Sindelar wrote (Yahoo #19096):

        >>The approach I used required making the following assumption, which is
        equivalent to the above statement; Let S(p, n) represent an infinite
        subset of the universal set of all consecutive odd primes, where p is the
        first prime of the subset, and n (including 0) represents the number of
        consecutive primes from the universal set that have been omitted between
        adjacent primes of the subset.Then any S(p, n) contains a set of any
        number k of primes in arithmetic progression. The program I wrote is
        based on this.>

        I used Pari-gp for this. For every set of k consecutive primes, which has
        n skipped consecutive primes between its adjacent terms, after an
        inputted integer, it checks if the terms of that set are in arithmetic
        progression. Jens, is this slower than your approach with your tuplet
        finder? If one could prove the above assumption, would that also prove
        that all admissible prime constellations have infinitely many occurrences
        as you put it, or only those that have a (PAP-k, n) subset?

        Sindelar wrote (Yahoo #19093):

        >>Obviously, the ordinal numbers of the primes in such a PAP are also in
        arithmetic progression (AP) with a constant difference of (n+1).>

        This suggested trying this assumption which is just a fancy way of
        defining a (PAP-k, n): In any infinite arithmetic progression of positive
        integers with a common difference d, there exists a subset of k
        consecutive integers, so that if each integer in that subset is
        considered to represent the ordinal number of a prime, the associated
        primes will be in arithmetic progression of length k with (d-1)
        consecutive primes between adjacent terms of that arithmetic progression.
        (Ordinal number of a prime means its position in the numerically ordered
        set of all primes, with prime 2 being number 1). It works but is more
        computationally complicated. What do you think?

        Bill Sindelar

        Bill Sindelar
      • Jens Kruse Andersen
        ... I would expect your method to be much slower based on how randomly consecutive prime gaps appear to be distributed. ... No, and also no to the only-part.
        Message 3 of 12 , Oct 2, 2007
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          Bill Sindelar wrote:
          > I used Pari-gp for this. For every set of k consecutive primes, which has
          > n skipped consecutive primes between its adjacent terms, after an
          > inputted integer, it checks if the terms of that set are in arithmetic
          > progression. Jens, is this slower than your approach with your tuplet
          > finder?

          I would expect your method to be much slower based on how
          "randomly" consecutive prime gaps appear to be distributed.

          > If one could prove the above assumption, would that also prove
          > that all admissible prime constellations have infinitely many occurrences
          > as you put it, or only those that have a (PAP-k, n) subset?

          No, and also no to the only-part. Your assumption says nothing
          about the existence of specific differences between primes,
          so it says nothing about any admissible constellation.

          > This suggested trying this assumption which is just a fancy way of
          > defining a (PAP-k, n):
          .....
          > It works but is more
          > computationally complicated. What do you think?

          "computationally complicated" refers to something computational,
          for example the time to compute something with a given algorithm.
          You have made another formulation of your conjecture but not
          described an algorithm so "computationally complicated" is a
          concept which does not apply.

          I don't have time to discuss more.

          --
          Jens Kruse Andersen
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