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three-prime sum chains

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  • Phil Carmody
    Find sets of 3 consecutive primes whose sum is itself prime, and associate that set and its members with that sum. What s the longest chain of associated
    Message 1 of 7 , Jun 2, 2013
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      Find sets of 3 consecutive primes whose sum is itself prime, and associate that set and its members with that sum.

      What's the longest chain of associated primes you can find?
      Heuristically - would you expect there to be longer ones than the longest you can find?

      Phil
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      [stolen with permission from Daniel B. Cristofani]
    • Jens Kruse Andersen
      ... http://www.primepuzzles.net/puzzles/puzz_421.htm requires that the sum is always the first of the 3 consecutive primes in the next sum.
      Message 2 of 7 , Jun 2, 2013
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        Phil Carmody wrote:
        > Find sets of 3 consecutive primes whose sum is itself prime,
        > and associate that set and its members with that sum.
        >
        > What's the longest chain of associated primes you can find?
        > Heuristically - would you expect there to be longer ones
        > than the longest you can find?

        http://www.primepuzzles.net/puzzles/puzz_421.htm requires that the
        sum is always the first of the 3 consecutive primes in the next sum.
        http://primes.utm.edu/curios/page.php/507995698619.html says for that:

        "Start with the prime 507995698619. Add it with the two next primes
        to get a sum which is prime. Repeat. The first 10 sums are prime.
        This is the first case with more than 8.

        Starting prime: 507995698619
        + 507995698643 + 507995698669 = 1523987095931
        + 1523987095949 + 1523987095957 = 4571961287837
        + 4571961287923 + 4571961288017 = 13715883863777
        + 13715883863789 + 13715883863827 = 41147651591393
        + 41147651591411 + 41147651591443 = 123442954774247
        + 123442954774259 + 123442954774291 = 370328864322797
        + 370328864322847 + 370328864322887 = 1110986592968531
        + 1110986592968533 + 1110986592968633 = 3332959778905697
        + 3332959778905703 + 3332959778905711 = 9998879336717111
        + 9998879336717113 + 9998879336717129 = 29996638010151353"

        Arbitrarily long chains are certainly expected.
        It would for example follow from Dickson's conjecture:
        http://primes.utm.edu/glossary/xpage/DicksonsConjecture.html

        --
        Jens Kruse Andersen
      • Maximilian Hasler
        I added https://oeis.org/draft/A226291, feel free to improve. Maximilian ... [Non-text portions of this message have been removed]
        Message 3 of 7 , Jun 2, 2013
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          I added https://oeis.org/draft/A226291, feel free to improve.

          Maximilian


          On Sun, Jun 2, 2013 at 10:34 AM, Jens Kruse Andersen <jens.k.a@...>wrote:

          > **
          >
          >
          > Phil Carmody wrote:
          > > Find sets of 3 consecutive primes whose sum is itself prime,
          > > and associate that set and its members with that sum.
          > >
          > > What's the longest chain of associated primes you can find?
          > > Heuristically - would you expect there to be longer ones
          > > than the longest you can find?
          >
          > http://www.primepuzzles.net/puzzles/puzz_421.htm requires that the
          > sum is always the first of the 3 consecutive primes in the next sum.
          > http://primes.utm.edu/curios/page.php/507995698619.html says for that:
          >
          > "Start with the prime 507995698619. Add it with the two next primes
          > to get a sum which is prime. Repeat. The first 10 sums are prime.
          > This is the first case with more than 8.
          >
          > Starting prime: 507995698619
          > + 507995698643 + 507995698669 = 1523987095931
          > + 1523987095949 + 1523987095957 = 4571961287837
          > + 4571961287923 + 4571961288017 = 13715883863777
          > + 13715883863789 + 13715883863827 = 41147651591393
          > + 41147651591411 + 41147651591443 = 123442954774247
          > + 123442954774259 + 123442954774291 = 370328864322797
          > + 370328864322847 + 370328864322887 = 1110986592968531
          > + 1110986592968533 + 1110986592968633 = 3332959778905697
          > + 3332959778905703 + 3332959778905711 = 9998879336717111
          > + 9998879336717113 + 9998879336717129 = 29996638010151353"
          >
          > Arbitrarily long chains are certainly expected.
          > It would for example follow from Dickson's conjecture:
          > http://primes.utm.edu/glossary/xpage/DicksonsConjecture.html
          >
          > --
          > Jens Kruse Andersen
          >
          >
          >


          [Non-text portions of this message have been removed]
        • Phil Carmody
          ... I specifically wanted to offer the choice. I wanted the chains to be longer, and also less simple to find. (By which I mean that you can t just do a simple
          Message 4 of 7 , Jun 3, 2013
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            --- On Sun, 6/2/13, Jens Kruse Andersen <jens.k.a@...> wrote:
            > Phil Carmody wrote:
            > > Find sets of 3 consecutive primes whose sum is itself prime,
            > > and associate that set and its members with that sum.
            > >
            > > What's the longest chain of associated primes you can find?
            > > Heuristically - would you expect there to be longer ones
            > > than the longest you can find?
            >
            > http://www.primepuzzles.net/puzzles/puzz_421.htm requires that the
            > sum is always the first of the 3 consecutive primes in the next sum.

            I specifically wanted to offer the choice. I wanted the chains to be longer, and also less simple to find. (By which I mean that you can't just do a simple O(length)-time depth-first search from each root prime, as you now have a tree to probe, not a straight line, and might want to consider how much effort is wasted doing duplicated work as neigbouring roots share common sub-trees.)

            > http://primes.utm.edu/curios/page.php/507995698619.html
            > says for that:
            >
            > "Start with the prime 507995698619. Add it with the two next primes
            > to get a sum which is prime. Repeat. The first 10 sums are prime.
            > This is the first case with more than 8.

            Good work. I presume you just ran a prime sieve and then probed for each one with no deduplication? Given how likely failure is, probing a tree shouldn't be much more than a small factor smaller than probing a straight line.

            > Arbitrarily long chains are certainly expected.
            > It would for example follow from Dickson's conjecture:
            > http://primes.utm.edu/glossary/xpage/DicksonsConjecture.html

            I hadn't considered Dickson. It's the big gun, certainly. I wondered initially if it would fail on guaranteeing that the positioned primes are consecutive, but if you ask for a triplet {p,p+3+/-1,p+6}, then that comes for free.

            Phil
            --
            () ASCII ribbon campaign () Hopeless ribbon campaign
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            [stolen with permission from Daniel B. Cristofani]
          • Phil Carmody
            ... I specifically wanted to offer the choice. I wanted the chains to be longer, and also less simple to find. (By which I mean that you can t just do a simple
            Message 5 of 7 , Jun 4, 2013
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              --- On Sun, 6/2/13, Jens Kruse Andersen <jens.k.a@...> wrote:
              > Phil Carmody wrote:
              > > Find sets of 3 consecutive primes whose sum is itself prime,
              > > and associate that set and its members with that sum.
              > >
              > > What's the longest chain of associated primes you can find?
              > > Heuristically - would you expect there to be longer ones
              > > than the longest you can find?
              >
              > http://www.primepuzzles.net/puzzles/puzz_421.htm requires that the
              > sum is always the first of the 3 consecutive primes in the next sum.

              I specifically wanted to offer the choice. I wanted the chains to be longer, and also less simple to find. (By which I mean that you can't just do a simple O(length)-time depth-first search from each root prime, as you now have a tree to probe, not a straight line, and might want to consider how much effort is wasted doing duplicated work as neigbouring roots share common sub-trees.)

              > http://primes.utm.edu/curios/page.php/507995698619.html
              > says for that:
              >
              > "Start with the prime 507995698619. Add it with the two next primes
              > to get a sum which is prime. Repeat. The first 10 sums are prime.
              > This is the first case with more than 8.

              Good work (as per usual!). I presume you just ran a prime sieve and then probed for each one with no deduplication? Given how likely failure is, probing a tree shouldn't be much more than a small factor smaller than probing a straight line.

              > Arbitrarily long chains are certainly expected.
              > It would for example follow from Dickson's conjecture:
              > http://primes.utm.edu/glossary/xpage/DicksonsConjecture.html

              I hadn't considered Dickson. It's the big gun, certainly. I wondered initially if it would fail on guaranteeing that the positioned primes are consecutive, but if you ask for a triplet {p,p+3+/-1,p+6}, then that comes for free.

              Phil
              --
              () ASCII ribbon campaign () Hopeless ribbon campaign
              /\ against HTML mail /\ against gratuitous bloodshed

              [stolen with permission from Daniel B. Cristofani]
            • Jens Kruse Andersen
              ... With the choice we quickly get a chain with 12 sums: 281 + 283 + 293 = 857 857 + 859 + 863 = 2579 2551 + 2557 + 2579 = 7687 7681 + 7687 + 7691 = 23059
              Message 6 of 7 , Jun 4, 2013
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                Phil Carmody wrote:
                > I specifically wanted to offer the choice. I wanted the chains
                > to be longer, and also less simple to find.

                With the choice we quickly get a chain with 12 sums:
                281 + 283 + 293 = 857
                857 + 859 + 863 = 2579
                2551 + 2557 + 2579 = 7687
                7681 + 7687 + 7691 = 23059
                23059 + 23063 + 23071 = 69193
                69193 + 69197 + 69203 = 207593
                207569 + 207589 + 207593 = 622751
                622751 + 622777 + 622781 = 1868309
                1868287 + 1868291 + 1868309 = 5604887
                5604881 + 5604887 + 5604901 = 16814669
                16814669 + 16814671 + 16814701 = 50444041
                50444027 + 50444041 + 50444059 = 151332127

                The longest chain with initial primes below 10^11 is probably 14 sums:
                2878090951 + 2878090961 + 2878090967 = 8634272879
                8634272839 + 8634272879 + 8634272891 = 25902818609
                25902818609 + 25902818629 + 25902818663 = 77708455901
                77708455819 + 77708455823 + 77708455901 = 233125367543
                233125367543 + 233125367599 + 233125367609 = 699376102751
                699376102751 + 699376102757 + 699376102783 = 2098128308291
                2098128308239 + 2098128308273 + 2098128308291 = 6294384924803
                6294384924787 + 6294384924803 + 6294384924823 = 18883154774413
                18883154774413 + 18883154774441 + 18883154774447 = 56649464323301
                56649464323301 + 56649464323303 + 56649464323307 = 169948392969911
                169948392969877 + 169948392969881 + 169948392969911 = 509845178909669
                509845178909611 + 509845178909669 + 509845178909687 = 1529535536728967
                1529535536728967 + 1529535536728987 + 1529535536728999 = 4588606610186953
                4588606610186927 + 4588606610186947 + 4588606610186953 = 13765819830560827

                There are also 14 starting with
                44453980303 + 44453980309 + 44453980379 = 133361940991

                > I presume you just ran a prime sieve and then probed for each one
                > with no deduplication?

                I didn't bother with deduplication but I ran two prime sieves in
                parallel, around p and 3p. This meant the first sum never had to be prp
                tested, and its surrounding primes were also generated without prp tests.
                It could be extended to a third sieve around 9p and so on, but each
                extra sieve has to sieve a larger interval and avoids fewer prp tests.

                > I hadn't considered Dickson. It's the big gun, certainly.
                > I wondered initially if it would fail on guaranteeing that the
                > positioned primes are consecutive, but if you ask for a triplet
                > {p,p+3+/-1,p+6}, then that comes for free.

                Dickson's conjecture can be manipulated to ensure consecutive primes
                in this problem for any admissible pattern. We can just demand extra
                primes chosen so specific prime factors are forced to divide certain
                numbers in a gap.
                For example, we can ensure consecutive {p, p+6} by demanding four primes
                {p, p+6, p+8, p+14}. This forces 3 to divide p+4, and 5 to divide p+2.

                --
                Jens Kruse Andersen
              • djbroadhurst
                ... which is a special case of Bateman-Horn: http://tech.groups.yahoo.com/group/primenumbers/message/25133 David
                Message 7 of 7 , Jun 4, 2013
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                  --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:

                  > Arbitrarily long chains are certainly expected.
                  > It would for example follow from Dickson's conjecture:
                  > http://primes.utm.edu/glossary/xpage/DicksonsConjecture.html

                  which is a special case of Bateman-Horn:
                  http://tech.groups.yahoo.com/group/primenumbers/message/25133

                  David
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