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Re: First order son primes

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  • Robert
    ... Their 1mod3 sisters p, 3p-2, form first instance chains as follows: SPMinuslen2 starting 3 SPMinuslen3 3 SPMinuslen4 5 SPMinuslen5 61 SPMinuslen6
    Message 1 of 9 , Sep 3, 2008
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      --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@...> wrote:
      >
      > --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@> wrote:
      >
      > >
      > > The earliest instance chains are:
      > >
      > > SPlen2 3,11
      > > SPlen3 5,17,53
      > > SPlen4 29,89,269,809
      > > SPlen5 1129,3389,10169,30509,91529
      > > SPlen6 10009,30029,90089,270269,810809,2432429

      Their 1mod3 sisters p, 3p-2, form first instance chains as follows:

      SPMinuslen2 starting 3
      SPMinuslen3 3
      SPMinuslen4 5
      SPMinuslen5 61
      SPMinuslen6 1171241 (huge jump)
      SPMinuslen7 1197631
      SPMinuslen8 25451791

      Regards

      Robert Smith
    • Robert
      ... Their 1mod3 sisters p, 3p-2, form first instance chains as follows: SPMinuslen2 starting 3 SPMinuslen3 3 SPMinuslen4 5 SPMinuslen5 61 SPMinuslen6
      Message 2 of 9 , Sep 3, 2008
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        --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@...> wrote:
        >
        > --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@> wrote:
        >
        > >
        > > The earliest instance chains are:
        > >
        > > SPlen2 3,11
        > > SPlen3 5,17,53
        > > SPlen4 29,89,269,809
        > > SPlen5 1129,3389,10169,30509,91529
        > > SPlen6 10009,30029,90089,270269,810809,2432429

        Their 1mod3 sisters p, 3p-2, form first instance chains as follows:

        SPMinuslen2 starting 3
        SPMinuslen3 3
        SPMinuslen4 5
        SPMinuslen5 61
        SPMinuslen6 1171241 (huge jump)
        SPMinuslen7 1197631
        SPMinuslen8 25451791
        SPMinuslen9 25451791

        Regards

        Robert Smith
      • 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 3 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 4 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 5 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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