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First order son primes

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  • Robert
    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
    Message 1 of 9 , Sep 2, 2008
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      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?

      Regards

      Robert Smith
    • Phil Carmody
      ... Other generalised relations have been looked at. However, with an exponent of 3 driving the size, these chains aren t likely to be longer than ones which
      Message 2 of 9 , Sep 2, 2008
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        > 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?

        Other generalised relations have been looked at. However, with an exponent of 3 driving the size, these chains aren't likely to be longer than ones which are related to powers of 2. Have you tried looking for both? How easy is it to find a chain of 10, 11, or 12 of either type?

        Phil
      • Robert
        ... exponent of 3 driving the size, these chains aren t likely to be longer than ones which are related to powers of 2. Have you tried looking for both? How
        Message 3 of 9 , Sep 3, 2008
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          --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> 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?
          >
          > Other generalised relations have been looked at. However, with an
          exponent of 3 driving the size, these chains aren't likely to be
          longer than ones which are related to powers of 2. Have you tried
          looking for both? How easy is it to find a chain of 10, 11, or 12 of
          either type?
          >
          > Phil
          >
          Thanks for responding, Phil. I haven't explored these at all, I did
          not think about their existence until yesterday, when they appeared as
          part of the work I am doing on gaps in ordered proper prime fraction
          groups - see post 15 on

          http://www.mersenneforum.org/showthread.php?t=10574

          I am unlikely to explore these chains, as I do not have any computing
          power here in Bangladesh. But I agree, chain of 10 to 12 should be
          easy to find.

          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

          Regards

          Robert
        • Robert
          ... Splen7 starts 575119 Splen8 starts 32694619
          Message 4 of 9 , Sep 3, 2008
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            --- 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
            >
            > Regards
            >
            > Robert
            >

            Splen7 starts 575119
            Splen8 starts 32694619
          • Robert
            ... Their 1mod3 sisters p, 3p-2, form first instance chains as follows: SPMinuslen2 starting 3 SPMinuslen3 3 SPMinuslen4 5 SPMinuslen5 61 SPMinuslen6
            Message 5 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 6 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 7 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 8 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 9 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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