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Re: New Type of Prime Arithmetic Progression?

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  • elevensmooth
    ... I ll try. Are you aware of the concept of admissible constellations? For example, twin primes, x and x+2, are believed to occur infinitely often. A
    Message 1 of 12 , Sep 26 6:55 PM
      --- In primenumbers@yahoogroups.com, w_sindelar@... wrote:

      > Jens can you explain your answer a bit more?

      I'll try.

      Are you aware of the concept of "admissible constellations?" For
      example, twin primes, x and x+2, are believed to occur infinitely
      often. A triplet of the form x, x+2, x+4 is not admissible because
      one of these numbers always divisible by 3. Hence (3, 5, 7) is the
      only example - a finite number. However x, x+2, x+6 is admissible and
      is believed to be prime infinitely often.

      Jens point is that you can construct an admissible constellation that
      has all kn-1 primes in fixed locations so the the arithmetic
      progression and primes between are honored. This is more restrictive
      than your rules require, but would be an example as long as none of
      the the other values within the constellation's range are also prime.
      Jen's further point is that you could further restrict the
      constellation so that the intermediate values were divisible by
      selected primes, and hence composite. Again more restrictive than you
      rules, but it would qualify as an example. Finally, from the
      constellation conjecture there would be an infinite number of these
      constellation, each and every one an example.

      For example, x, x+2, x+6, x+8, x+12 is an example of admissible
      constellation that has 3 primes in arithmetic progression with(al
      least) one prime between them. In this case there is exactly one
      prime because x+4 and x+10 must be divisible by 3. In general, we
      would need to jiggle the start point as x=ay+b for selected fixed
      values if a and b to ensure the omitted points are composite.

      Finally, every example that you find can could be reverse-engineered
      to such an admissible constellation, so this search would be a subset
      of a search for examples of admissible constellations.

      William Lipp
      Poohbah of OddPerfect.org
    • w_sindelar@juno.com
      ... Jens, I think I may have offended you by writing I m lost here. Seems like a convoluted approach. Looking back at this, I can see that it can be taken as
      Message 2 of 12 , Sep 29 6:18 AM
        Sindelar wrote:
        > > I'm lost here. Seems like a convoluted approach.

        Andersen wrote:
        > You asked for comments on your statement (which is an unproven
        > guess). I briefly showed that it would follow from a well-known
        > and trusted conjecture, sometimes called the prime k-tuple
        > conjecture
        > (that name is sometimes restricted to special cases). Of course
        > it's
        > less "convoluted" to not relate your guess to anything else like
        > previously studied things. If you want something unconvoluted
        > (but not very useful by itself) then here it is: I guess your guess
        > is
        > right.

        Jens, I think I may have offended you by writing "I'm lost here. Seems
        like a convoluted approach." Looking back at this, I can see that it can
        be taken as arrogant criticism. Gad, that is not what I meant it to be. I
        should have written "I am unable to follow your explanation. It seems
        complicated to me because I know nothing about admissible prime
        constellations, but I accept your opinion". I regret my choice of words
        and hope you accept my sincere apology.

        Sindelar wrote in regard to Green and Tao:
        >>>To me, this is a very broad claim covering any type of (PAP-k, n).
        Jens can you explain your answer a bit more?

        Andersen wrote:
        > Just to be clear: My "No" was only to your second sentence:
        > "I would interpret Green and Tao as covering this type. (meaning
        (PAP-k, n=0))"
        >
        > You defined (PAP-k, n) as a PAP-k with n primes between each of
        > the k-1 pairs of successive primes in the AP. Tao and Green don't
        > mention this concept of equal prime counts and their theorem says
        > nothing about your (PAP-k, n) for n=0 or any other n value.
        > I don't know what else you want me to explain.
        > All I can say is that the theorem simply doesn't say it.

        Jens thank you. Nothing more to explain. You made it clear that the Green
        and Tao theorem does not apply to type (PAP-k, n=0 or greater). And thank
        you for an example of a (PAP-8, 5). Don't know how you calculated that so
        quickly. I was beginning to think there might be a limit on k.

        Bill Sindelar

        [Non-text portions of this message have been removed]
      • w_sindelar@juno.com
        ... Jens, I think I may have offended you by writing I m lost here. Seems like a convoluted approach. Looking back at this, I can see that it can be taken as
        Message 3 of 12 , Sep 29 6:18 AM
          Sindelar wrote:
          > > I'm lost here. Seems like a convoluted approach.

          Andersen wrote:
          > You asked for comments on your statement (which is an unproven
          > guess). I briefly showed that it would follow from a well-known
          > and trusted conjecture, sometimes called the prime k-tuple
          > conjecture
          > (that name is sometimes restricted to special cases). Of course
          > it's
          > less "convoluted" to not relate your guess to anything else like
          > previously studied things. If you want something unconvoluted
          > (but not very useful by itself) then here it is: I guess your guess
          > is
          > right.

          Jens, I think I may have offended you by writing "I'm lost here. Seems
          like a convoluted approach." Looking back at this, I can see that it can
          be taken as arrogant criticism. Gad, that is not what I meant it to be. I
          should have written "I am unable to follow your explanation. It seems
          complicated to me because I know nothing about admissible prime
          constellations, but I accept your opinion". I regret my choice of words
          and hope you accept my sincere apology.

          Sindelar wrote in regard to Green and Tao:
          >>>To me, this is a very broad claim covering any type of (PAP-k, n).
          Jens can you explain your answer a bit more?

          Andersen wrote:
          > Just to be clear: My "No" was only to your second sentence:
          > "I would interpret Green and Tao as covering this type. (meaning
          (PAP-k, n=0))"
          >
          > You defined (PAP-k, n) as a PAP-k with n primes between each of
          > the k-1 pairs of successive primes in the AP. Tao and Green don't
          > mention this concept of equal prime counts and their theorem says
          > nothing about your (PAP-k, n) for n=0 or any other n value.
          > I don't know what else you want me to explain.
          > All I can say is that the theorem simply doesn't say it.

          Jens thank you. Nothing more to explain. You made it clear that the Green
          and Tao theorem does not apply to type (PAP-k, n=0 or greater). And thank
          you for an example of a (PAP-8, 5). Don't know how you calculated that so
          quickly. I was beginning to think there might be a limit on k.

          Bill Sindelar

          [Non-text portions of this message have been removed]
        • w_sindelar@juno.com
          On Thu, 27 Sep 2007 01:55:00 -0000 elevensmooth ... I m glad you did, and I thank you. You must be a mind reader. You somehow sensed why I got lost trying to
          Message 4 of 12 , Sep 29 6:18 AM
            On Thu, 27 Sep 2007 01:55:00 -0000 "elevensmooth"
            <elevensmooth@...> writes:
            > --- In primenumbers@yahoogroups.com, w_sindelar@... wrote:
            >
            > > Jens can you explain your answer a bit more?

            William Lipp wrote:
            > I'll try.

            I'm glad you did, and I thank you. You must be a mind reader. You somehow
            sensed why I got lost trying to follow Jens reasoning. Right off the bat
            I'm confronted with "admissible prime constellations" and right there I'm
            lost.

            Your neat little introductory on this greatly helped me understand what
            Jens meant. I'm going to study this concept in more detail. Regards with
            appreciation.

            Bill Sindelar
          • w_sindelar@juno.com
            On Thu, 27 Sep 2007 01:55:00 -0000 elevensmooth ... I m glad you did, and I thank you. You must be a mind reader. You somehow sensed why I got lost trying to
            Message 5 of 12 , Sep 29 6:18 AM
              On Thu, 27 Sep 2007 01:55:00 -0000 "elevensmooth"
              <elevensmooth@...> writes:
              > --- In primenumbers@yahoogroups.com, w_sindelar@... wrote:
              >
              > > Jens can you explain your answer a bit more?

              William Lipp wrote:
              > I'll try.

              I'm glad you did, and I thank you. You must be a mind reader. You somehow
              sensed why I got lost trying to follow Jens reasoning. Right off the bat
              I'm confronted with "admissible prime constellations" and right there I'm
              lost.

              Your neat little introductory on this greatly helped me understand what
              Jens meant. I'm going to study this concept in more detail. Regards with
              appreciation.

              Bill Sindelar
            • Jens Kruse Andersen
              ... No problem. You can search more information about admissible constellations with a search engine. If a prime p
              Message 6 of 12 , Sep 29 5:38 PM
                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 7 of 12 , Oct 2, 2007
                  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 8 of 12 , Oct 2, 2007
                    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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