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Re: [PrimeNumbers] First order son primes

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  • Paul Leyland
    ... They have. Many of them can be proved to have a maximum length. Teske & Williams paper in LNCS 1838 is a nice treatment of consecutive prime values
    Message 1 of 9 , Sep 4, 2008
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      On Tue, 2008-09-02 at 17:33 +0000, Robert wrote:
      > First order son primes (p, 3p+2 prime) are more common than Sophie
      > Germains (p,2p+1 prime): approx 36% more common.
      >
      > Why? - If we look at mod 3
      >
      > if p==1mod3 then 2p+1==0mod3
      > if p==2mod3 then 2p+1==2mod3, 50% chance of a 2p+1 is not 0mod3
      >
      > if: p==1mod3 then 3p+2==2mod3
      > if: p==2mod3 then 3p+2==2mod3, 100% chance that 3p+1 is not 0mod3
      >
      >
      > Question: Why are chains of first order son primes not sought by prime
      > hunters, as they might provide longer chains than SG, CC, despite the
      > slight increase in magnitude?

      They have. Many of them can be proved to have a maximum length.

      Teske & Williams' paper in LNCS 1838 is a nice treatment of consecutive
      prime values produced by iterating the mapping f(x) -> ax^2+b

      I happen to know this paper because the authors could find chains for
      (a,b) = (1, -17) of at most 5 primes. I found several longer ones
      though none as large as the maximum possible, which is 16 for this
      choice of (a,b). I can't now find the computational results which I
      mailed off to Edlyn.

      Paul
    • Jens Kruse Andersen
      ... Different variations have been sought but less than the better known Cunningham chains. Here are some prime sequences iterating ax+b:
      Message 2 of 9 , Sep 4, 2008
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        Robert wrote:
        > Why are chains of first order son primes not sought by prime hunters

        Different variations have been sought but less than the better known
        Cunningham chains.

        Here are some prime sequences iterating ax+b:
        http://www.research.att.com/~njas/sequences/?q=%22On+certain+chains+of+primes%22
        You found the next term of one of them:
        http://www.research.att.com/~njas/sequences/A083388

        A page calling them generalized Cunningham chains:
        http://www.primenumbers.net/Henri/us/CunnGenus.htm

        A page saying "prime trees" about primes iterated with ax+/-b
        where + and - can be mixed:
        http://unbecominglevity.blogharbor.com/blog/_archives/2004/3/17/27759.html
        A prime tree of depth 26 for 2x+/-308843535 starting at 177857809:
        http://unbecominglevity.blogharbor.com/blog/_archives/2006/5/12/1952529.html

        --
        Jens Kruse Andersen
      • Robert
        ... http://unbecominglevity.blogharbor.com/blog/_archives/2004/3/17/27759.html ... http://unbecominglevity.blogharbor.com/blog/_archives/2006/5/12/1952529.html
        Message 3 of 9 , Sep 4, 2008
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          --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
          <jens.k.a@...> wrote:

          >
          > A page saying "prime trees" about primes iterated with ax+/-b
          > where + and - can be mixed:
          >
          http://unbecominglevity.blogharbor.com/blog/_archives/2004/3/17/27759.html
          > A prime tree of depth 26 for 2x+/-308843535 starting at 177857809:
          >
          http://unbecominglevity.blogharbor.com/blog/_archives/2006/5/12/1952529.html


          Gosh, did not realise, and such a sad story for the blogger. If he had
          come here first he could have saved himself 2 years work !!!!!
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