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Chains of primes of length n

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  • Robin Garcia
    Define a chain of primes of length n: {p_1,p_2...,p_n} such that C(p_(k-1))=p_(k)+1 for all 2
    Message 1 of 2 , Feb 2 10:49 AM
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      Define a chain of primes of length n: {p_1,p_2...,p_n} such that
      C(p_(k-1))=p_(k)+1 for all 2<=k<=n where C(p_(k-1)) is the p_(k-1)-th composite.
      Example: {197,251,317} is a chain of length 3 because C(197)=252=251+1
      and C(251)=318
      Another property is: p_(k)=P(s)=s-th prime--->p_(k-1)=P(s)-s

      There are 120 chains of length 4 for p_4<5*10^6
      4 chains of length 5 for p_5<5*10^6
      1 chain of length 6 for p_6<5*10^6

      The first two n=5 chains are: {11,19,29,43,61} and {142573,157007,172741,189901,208589}

      The first two n=6 chains are: {7829,8941,10193,11587,13151,14897} and {4523081,4862593,5225683,5613763,6028483,6471431}

      A very rough heuristic seems to indicate that the first p-10 would be a 15 digit prime.
      Computation is here very intensive.
      Would anybody try to find a fast (say PARI script)?
      And a good heuristic?

      Find a chain for n=7 and 8

      Yet another sets of simoultaneous primes,Jens.
      Where (number of digits) do you think we would find a 19-chain and WHEN?




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    • Jens Kruse Andersen
      ... A set by definition relying on the number of primes or composites is questionable for The Largest Known Simultaneous Primes at
      Message 2 of 2 , Feb 2 4:28 PM
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        Robin Garcia wrote:

        > Define a chain of primes of length n: {p_1,p_2...,p_n} such that
        > C(p_(k-1))=p_(k)+1 for all 2<=k<=n where C(p_(k-1)) is the p_(k-1)-th
        > composite.
        > Example: {197,251,317} is a chain of length 3 because C(197)=252=251+1
        > and C(251)=318

        > Yet another sets of simoultaneous primes,Jens.

        A set by definition relying on the number of primes or composites is
        questionable for The Largest Known Simultaneous Primes at
        http://hjem.get2net.dk/jka/math/simultprime.htm
        I might allow this case because it seems harder than other allowed forms. I
        wouldn't if it was easier.

        > Where (number of digits) do you think we would find a 19-chain and WHEN?

        I haven't estimated number of digits but it's probably large. And when?
        Long after other 19-sets have been found. I don't think a record of this form
        will ever make the list.
        The only practical algorithm seems to be exhaustive computation of all primes.
        Other forms can be sieved with faster methods.

        --
        Jens Kruse Andersen
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