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summing primes to yield new primes

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  • Phil Carmody
    I m pretty sure this has been looked at before, as it s so simple, but it s sometimes fun to resurrect old ideas. In particular when there s plenty of room for
    Message 1 of 4 , Jun 19, 2012
      I'm pretty sure this has been looked at before, as it's so simple, but it's sometimes fun to resurrect old ideas. In particular when there's plenty of room for both carbon-based and silicon-based attacks on a puzzle.

      Consider an ordered n-tuple V_0 of primes.
      Define V_{i+1}, if it exists, to be V_i with the first element removed, and a final element added with value equal to the sum of all n primes in V_i. If that final element would be composite, then V_{i+1} does not exist.

      e.g. with n=3, let V_0 = <3,3,5>
      V_1 = <3,5,11>
      V_2 = <5,11,19>
      V_3 does not exist, as 5+11+19=35 is composite

      Now here are the puzzles:
      1) With n=3, is it possible for a sequence V_i to be infinite in length.
      a) if so - what's the longest you can find?
      b) if not, what's the maximum length, and why? What V_0 leads to the smallest final prime?
      2) As above, but with n=5
      3) As above, but with n=7
      4) Generalise, what do you predict for other n?

      Enjoy!
      Phil
      --
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      [stolen with permission from Daniel B. Cristofani]
    • Maximilian Hasler
      ... 1) With n=3, is it possible for a sequence V_i to be infinite in length. ... no, ... examples with length=4 are [61, 67, 71], [1709, 1721, 1723], [2371,
      Message 2 of 4 , Jun 19, 2012
        > Consider an ordered n-tuple V_0 of primes.
        > Define V_{i+1}, to be V_i with the first element removed, and a final
        > element added with value equal to the sum of all n primes in V_i. If that
        > final element would be composite, then V_{i+1} does not exist.
        > (...)
        >
        1) With n=3, is it possible for a sequence V_i to be infinite in length.
        >
        no,

        > a) if so - what's the longest you can find?
        >

        examples with length=4 are
        [61, 67, 71], [1709, 1721, 1723], [2371, 2377, 2381], [2543, 2549, 2551],
        [4021, 4027, 4049], [5443, 5449, 5471], [5827, 5839, 5843] ,...
        (here always [p(k),p(k+1),p(k+2)], I know this is not needed).

        b) if not, what's the maximum length, and why? What V_0 leads to the
        > smallest final prime?
        >
        the maximum length is 4, as you can see when you look at the sequence V_i
        mod 3:
        1,1,1 => 1,1,0
        2,2,2 => 2,2,0
        1,1,2 => 1,2,1
        1,2,1 => 2,1,1
        2,1,1 => 1,1,1
        1,2,2 => 2,2,2
        2,1,2 => 1,2,2
        2,2,1 => 2,1,2

        what is the "final prime" ?
        (2,2,2) leads to (2,2,6), so the final prime is 2 ?


        > 2) As above, but with n=5
        > 3) As above, but with n=7
        > 4) Generalise, what do you predict for other n?
        >
        It seems that in these cases the maximum length is always 5.
        But I may be wrong...

        Maximilian


        [Non-text portions of this message have been removed]
      • Jens Kruse Andersen
        ... You forgot there is one prime which is 0 mod 3. Phil s example was V_0 = so two 3 s is clearly also allowed. [3, 3, 11] gives length 6 with the
        Message 3 of 4 , Jun 19, 2012
          Maximilian Hasler wrote:
          > b) if not, what's the maximum length, and why? What V_0 leads to the
          >> smallest final prime?
          >>
          > the maximum length is 4, as you can see when you look at the sequence V_i
          > mod 3:
          > 1,1,1 => 1,1,0
          > 2,2,2 => 2,2,0
          > 1,1,2 => 1,2,1
          > 1,2,1 => 2,1,1
          > 2,1,1 => 1,1,1
          > 1,2,2 => 2,2,2
          > 2,1,2 => 1,2,2
          > 2,2,1 => 2,1,2

          You forgot there is one prime which is 0 mod 3.
          Phil's example was V_0 = <3,3,5> so two 3's is clearly also allowed.
          [3, 3, 11] gives length 6 with the added terms 17, 31, 59, 107, 197.

          --
          Jens Kruse Andersen
        • Phil Carmody
          ... You re permitted to start with 3s, so that can be extended by 2. 0,1,1 = 1,1,2 0,0,1 = 0,1,1 ... (2,2,6) doesn t exist, so (2,2,2) doesn t lead there. In
          Message 4 of 4 , Jun 19, 2012
            --- On Tue, 6/19/12, Maximilian Hasler <maximilian.hasler@...> wrote:
            > b) if not, what's the maximum length, and why? What V_0 leads to the
            > > smallest final prime?

            > the maximum length is 4, as you can see when you look at the
            > sequence V_i
            > mod 3:
            > 1,1,1 => 1,1,0
            > 2,2,2 => 2,2,0
            > 1,1,2 => 1,2,1
            > 1,2,1 => 2,1,1
            > 2,1,1 => 1,1,1
            > 1,2,2 => 2,2,2
            > 2,1,2 => 1,2,2
            > 2,2,1 => 2,1,2

            You're permitted to start with 3s, so that can be extended by 2.
            0,1,1 => 1,1,2
            0,0,1 => 0,1,1

            > what is the "final prime" ?
            > (2,2,2) leads to (2,2,6), so the final prime is 2 ?

            (2,2,6) doesn't exist, so (2,2,2) doesn't lead there.

            In the <3,3,5> example I gave, 19 is the final prime, as no other primes were added after it.

            > >  2) As above, but with n=5
            > > 3) As above, but with n=7
            > > 4) Generalise, what do you predict for other n?
            > >
            > It seems that in these cases the maximum length is always
            > 5. But I may be wrong...

            So you don't think the 5-tuplet
            [5, 83, 3, 7, 3]
            would be extended in turn 8 times by the primes
            [101, 197, 311, 619, 1231, 2459, 4817, 9437]
            (and no, that's not the answer, merely illustrative)?

            It seems you've made some assumptions that have stopped you from investigating every possibility.

            Phil
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