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Prime Number Progresions

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  • ginnyw@aol.com
    Mark, Thank you for your email. The progressions are the ordinary arithmetic type. One of the progressions of 11 primes is: 17 add 44 = 61 61 add 88
    Message 1 of 20 , Aug 3, 2003
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      Mark,

      Thank you for your email. The progressions are the ordinary arithmetic type.
      One of the progressions of 11 primes is:
      17 add 44 = 61
      61 add 88 = 149
      149 add 132 = 281
      281 add 176 = 457 and continuing for 6 more terms until a nonprime 2921 is
      reached.

      My program uses a constant to search for the progressions. I started with
      pencil and paper and then wrote the program. Except for the progression which
      generates 40 primes, the largest progression I have to date generates 29
      primes. Perhaps that might be a record. Most of the progressions are with small
      numbers because of the limitations of my computer.

      Virginia W.



      [Non-text portions of this message have been removed]
    • Mark Underwood
      Hi Virginia Your 29 consecutive prime sequence is very good. I wonder what equation it is expressed by? Just in the last week Gary Chaffey reported that the
      Message 2 of 20 , Aug 3, 2003
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        Hi Virginia

        Your 29 consecutive prime sequence is very good. I wonder what
        equation it is expressed by?

        Just in the last week Gary Chaffey reported that the equation
        2*x^2-88*x+997 generates 51 primes from x=0 to x= 50.

        Then Dr. Michael Hartley noticed that the same equation can be
        slightly modified to produce 57 primes from x=0 to x=56 :

        2*t^2-112*t+1597

        Amazing! I'm not sure if there are any which are longer. I had
        thought that the one Euler discovered was the longest and was proven
        to be the longest of it's kind, but I guess not? Perhaps with Gary's
        sequence the primes are not all distinct, I'm not sure.

        Mark



        --- In primenumbers@yahoogroups.com, ginnyw@a... wrote:
        > Mark,
        >
        > Thank you for your email. The progressions are the ordinary
        arithmetic type.
        > One of the progressions of 11 primes is:
        > 17 add 44 = 61
        > 61 add 88 = 149
        > 149 add 132 = 281
        > 281 add 176 = 457 and continuing for 6 more terms until a nonprime
        2921 is
        > reached.
        >
        > My program uses a constant to search for the progressions. I
        started with
        > pencil and paper and then wrote the program. Except for the
        progression which
        > generates 40 primes, the largest progression I have to date
        generates 29
        > primes. Perhaps that might be a record. Most of the progressions
        are with small
        > numbers because of the limitations of my computer.
        >
        > Virginia W.
        >
        >
        >
        > [Non-text portions of this message have been removed]
      • Mark Underwood
        I checked and it turns out that Gary s equation generates (only!) 29 *distinct* primes out of the 56, as it doubles back on itself. I m curious now Viriginia
        Message 3 of 20 , Aug 3, 2003
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          I checked and it turns out that Gary's equation generates (only!) 29
          *distinct* primes out of the 56, as it doubles back on itself.

          I'm curious now Viriginia whether your 29er is the same!

          Mark



          --- In primenumbers@yahoogroups.com, "Mark Underwood"
          <mark.underwood@s...> wrote:
          > Hi Virginia
          >
          > Your 29 consecutive prime sequence is very good. I wonder what
          > equation it is expressed by?
          >
          > Just in the last week Gary Chaffey reported that the equation
          > 2*x^2-88*x+997 generates 51 primes from x=0 to x= 50.
          >
          > Then Dr. Michael Hartley noticed that the same equation can be
          > slightly modified to produce 57 primes from x=0 to x=56 :
          >
          > 2*t^2-112*t+1597
          >
          > Amazing! I'm not sure if there are any which are longer. I had
          > thought that the one Euler discovered was the longest and was
          proven
          > to be the longest of it's kind, but I guess not? Perhaps with
          Gary's
          > sequence the primes are not all distinct, I'm not sure.
          >
          > Mark
          >
          >
          >
          > --- In primenumbers@yahoogroups.com, ginnyw@a... wrote:
          > > Mark,
          > >
          > > Thank you for your email. The progressions are the ordinary
          > arithmetic type.
          > > One of the progressions of 11 primes is:
          > > 17 add 44 = 61
          > > 61 add 88 = 149
          > > 149 add 132 = 281
          > > 281 add 176 = 457 and continuing for 6 more terms until a
          nonprime
          > 2921 is
          > > reached.
          > >
          > > My program uses a constant to search for the progressions. I
          > started with
          > > pencil and paper and then wrote the program. Except for the
          > progression which
          > > generates 40 primes, the largest progression I have to date
          > generates 29
          > > primes. Perhaps that might be a record. Most of the
          progressions
          > are with small
          > > numbers because of the limitations of my computer.
          > >
          > > Virginia W.
          > >
          > >
          > >
          > > [Non-text portions of this message have been removed]
        • ginnyw@aol.com
          Mark, The progression of 29 primes I found starts with 31. 31 add 12 = 43 43 add 24 = 67 67 add 36 = 103 103 add 48 = 151 151 add 60 = 211 - This continues for
          Message 4 of 20 , Aug 4, 2003
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            Mark,

            The progression of 29 primes I found starts with 31.
            31 add 12 = 43
            43 add 24 = 67
            67 add 36 = 103
            103 add 48 = 151
            151 add 60 = 211 -

            This continues for 29 primes. The last prime is 4903.

            Thank you for doing the research. I'd like to know if it is the same one.

            Virginia



            [Non-text portions of this message have been removed]
          • Mark Underwood
            Hi Virginia Yours is of the form 6x^2 + 6x + 31, generating primes from x=0 to x=28. Interestingly, Gary s (2x^2 -88x + 997) generates 29 distinct primes as
            Message 5 of 20 , Aug 4, 2003
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              Hi Virginia

              Yours is of the form 6x^2 + 6x + 31, generating primes from x=0 to
              x=28.

              Interestingly, Gary's (2x^2 -88x + 997) generates 29 distinct primes
              as well but is a different equation than yours.

              I just figured that Gary's equation can be reduced to 2x^2 + 29 to
              generate his 29 distinct primes from x=0 to x=28 !

              Mark




              --- In primenumbers@yahoogroups.com, ginnyw@a... wrote:
              > Mark,
              >
              > The progression of 29 primes I found starts with 31.
              > 31 add 12 = 43
              > 43 add 24 = 67
              > 67 add 36 = 103
              > 103 add 48 = 151
              > 151 add 60 = 211 -
              >
              > This continues for 29 primes. The last prime is 4903.
              >
              > Thank you for doing the research. I'd like to know if it is the
              same one.
              >
              > Virginia
              >
              >
              >
              > [Non-text portions of this message have been removed]
            • Gary Chaffey
              Its nice to know that i m not the only one looking at these sequences. I think and i will need to check I have an improvement on the sequence below. Generating
              Message 6 of 20 , Aug 4, 2003
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                Its nice to know that i'm not the only one looking at
                these sequences.
                I think and i will need to check I have an improvement
                on the sequence below. Generating 60 primes (but like
                the sequence below it doubles up and only has 30
                distinct primes). I will look this up at home then
                post it.
                I have been looking at various polynomials and have
                notices something quite unusual.
                For quadratic polynomials runs of 20+ distinct primes
                are quite common. But for cubic polynomials I have yet
                to find a run>20 distinct primes. I am aware that the
                cubic increases faster therefore each value is less
                likely to be prime but I'm wondering if there is
                anything else affecting the polynomial.
                Eulers record of x^2+x+41 was beaten by Fung and Ruby
                with 36x^2-810x+2753
                See:-
                http://mathworld.wolfram.com/PrimeGeneratingPolynomial.html

                I am certain that a polynomial of the form ax^2+bx+c
                can be found that beats this record but it means
                searching a hell of a lot of polnomials!!!
                Gary

                --- Mark Underwood <mark.underwood@...>
                wrote: > Hi Virginia
                >
                > Your 29 consecutive prime sequence is very good. I
                > wonder what
                > equation it is expressed by?
                >
                > Just in the last week Gary Chaffey reported that the
                > equation
                > 2*x^2-88*x+997 generates 51 primes from x=0 to x=
                > 50.
                >
                > Then Dr. Michael Hartley noticed that the same
                > equation can be
                > slightly modified to produce 57 primes from x=0 to
                > x=56 :
                >
                > 2*t^2-112*t+1597
                >
                > Amazing! I'm not sure if there are any which are
                > longer. I had
                > thought that the one Euler discovered was the
                > longest and was proven
                > to be the longest of it's kind, but I guess not?
                > Perhaps with Gary's
                > sequence the primes are not all distinct, I'm not
                > sure.
                >
                > Mark
                >
                >
                >
                > --- In primenumbers@yahoogroups.com, ginnyw@a...
                > wrote:
                > > Mark,
                > >
                > > Thank you for your email. The progressions are
                > the ordinary
                > arithmetic type.
                > > One of the progressions of 11 primes is:
                > > 17 add 44 = 61
                > > 61 add 88 = 149
                > > 149 add 132 = 281
                > > 281 add 176 = 457 and continuing for 6 more terms
                > until a nonprime
                > 2921 is
                > > reached.
                > >
                > > My program uses a constant to search for the
                > progressions. I
                > started with
                > > pencil and paper and then wrote the program.
                > Except for the
                > progression which
                > > generates 40 primes, the largest progression I
                > have to date
                > generates 29
                > > primes. Perhaps that might be a record. Most of
                > the progressions
                > are with small
                > > numbers because of the limitations of my computer.
                > >
                > > Virginia W.
                > >
                > >
                > >
                > > [Non-text portions of this message have been
                > removed]
                >
                >

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              • Gary Chaffey
                ... No it isn t the same sequence this one can be expressed as 6x^2+6x+31 for x in 0..28 but certainly of equal merit. Gary
                Message 7 of 20 , Aug 4, 2003
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                  --- ginnyw@... wrote: > Mark,
                  >
                  > The progression of 29 primes I found starts with 31.
                  > 31 add 12 = 43
                  > 43 add 24 = 67
                  > 67 add 36 = 103
                  > 103 add 48 = 151
                  > 151 add 60 = 211 -
                  >
                  > This continues for 29 primes. The last prime is
                  > 4903.
                  >
                  > Thank you for doing the research. I'd like to know
                  > if it is the same one.

                  No it isn't the same sequence this one can be
                  expressed as 6x^2+6x+31 for x in 0..28
                  but certainly of equal merit.
                  Gary


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                • ginnyw@aol.com
                  Mark, Gary, The equations are very helpful. Thank you. It is Interesting that the two progressions start at 29 and 31 and generate 29 primes each. This
                  Message 8 of 20 , Aug 4, 2003
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                    Mark, Gary,

                    The equations are very helpful. Thank you. It is Interesting that the two
                    progressions start at 29 and 31 and generate 29 primes each. This seems to
                    happen often. Another progression starting with 11 generates 10 primes; another
                    starting with 17 generates 16 primes, etc. I am interested in hearing more
                    about your work and plan to change my program based on our discussion.

                    Virginia


                    [Non-text portions of this message have been removed]
                  • Gary Chaffey
                    ... I have just spotted this too.. make y=x-22...(is this the transformation you have spotted Mark???) Gary
                    Message 9 of 20 , Aug 5, 2003
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                      > I just figured that Gary's equation can be reduced
                      > to 2x^2 + 29 to
                      > generate his 29 distinct primes from x=0 to x=28 !
                      >
                      > Mark
                      I have just spotted this too.. make y=x-22...(is this
                      the transformation you have spotted Mark???)
                      Gary

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                    • Mark Underwood
                      Right Gary, I guess that would be the transformation. Actually I just checked out the progression in the sequence starting with 29, 31, 37 ... and it was easy
                      Message 10 of 20 , Aug 5, 2003
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                        Right Gary, I guess that would be the transformation. Actually I just
                        checked out the progression in the sequence starting with 29, 31,
                        37 ... and it was easy to see it was of this form. I now see from

                        http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

                        that Legendre is the first reported to have seen this one. Perhaps
                        2x^2 + 29 generates the longest sequence of consecutive primes of any
                        two term equation.

                        And I see that Virginia's sequence of primes is reported in the
                        Encyclopedia on Integer Sequences as sequence A060834.

                        The Mathworld link above has alot of informative things to say about
                        the matter. (Virginia would like to read this!) I agree with you Gary
                        that there are other polynomials out there that can generate even
                        longer sequences. How to cleverly find them, that is the question.

                        But for expressions of the form x^2 + x + p there is no need to look
                        for a longer one since it has been shown that p = 41 generates the
                        longest one.

                        Mark



                        --- In primenumbers@yahoogroups.com, Gary Chaffey <garychaffey@y...>
                        wrote:
                        > > I just figured that Gary's equation can be reduced
                        > > to 2x^2 + 29 to
                        > > generate his 29 distinct primes from x=0 to x=28 !
                        > >
                        > > Mark
                        > I have just spotted this too.. make y=x-22...(is this
                        > the transformation you have spotted Mark???)
                        > Gary
                        >
                        >
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                      • jbrennen
                        ... If you believe the first Hardy-Littlewood Conjecture (also known as the k-tuple Conjecture), there exist arbitrarily long sequences of primes from two term
                        Message 11 of 20 , Aug 5, 2003
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                          --- Mark Underwood wrote:

                          > Perhaps 2x^2 + 29 generates the longest sequence of consecutive
                          > primes of any two term equation.

                          If you believe the first Hardy-Littlewood Conjecture (also known
                          as the k-tuple Conjecture), there exist arbitrarily long sequences
                          of primes from two term equations.

                          > But for expressions of the form x^2 + x + p there is no need to
                          > look for a longer one since it has been shown that p = 41
                          > generates the longest one.

                          Again, the same conjecture implies that arbitrarily long sequences
                          of primes exist of the form x^2+x+p.

                          What has been shown is that p=41 is the largest prime such that
                          x^2+x+p is prime for all x, 0 <= x <= p-2.

                          It has not been shown that x^2+x+p is never prime for 0 <= x <= 40.
                        • Gary Chaffey
                          ... If we restrict to two term equations then it can t be of the form 2x^2+p since of a proof (referred to on Wolfram site) which shows that this type of
                          Message 12 of 20 , Aug 5, 2003
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                            > > Perhaps 2x^2 + 29 generates the longest sequence
                            > of consecutive
                            > > primes of any two term equation.
                            >
                            > If you believe the first Hardy-Littlewood Conjecture
                            > (also known
                            > as the k-tuple Conjecture), there exist arbitrarily
                            > long sequences
                            > of primes from two term equations.

                            If we restrict to two term equations then it can't be
                            of the form 2x^2+p since of a proof (referred to on
                            Wolfram site) which shows that this type of sequence
                            can only yield 29 primes.
                            This implies we must look at ax^2+p for a>2. What
                            would be interesting is to rule out some more values
                            for a. I might look at this if I get some spare time.
                            Gary

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                          • Mark Underwood
                            Thanks for the correction Jack, I would not have discerned that difference from what I read unless it was pointed out. I remember the Hardy-Littlewood k tuple
                            Message 13 of 20 , Aug 5, 2003
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                              Thanks for the correction Jack, I would not have discerned that
                              difference from what I read unless it was pointed out.

                              I remember the Hardy-Littlewood k tuple Conjecture but I never did
                              connect it to polynominals.

                              Whether the k tuple conjucture is true, I guess I believe it with a
                              condition, which is that the tuple does not occur if it does not
                              occur early on, or something like that...

                              Mark



                              --- In primenumbers@yahoogroups.com, "jbrennen" <jack@b...> wrote:
                              > --- Mark Underwood wrote:
                              >
                              > > Perhaps 2x^2 + 29 generates the longest sequence of consecutive
                              > > primes of any two term equation.
                              >
                              > If you believe the first Hardy-Littlewood Conjecture (also known
                              > as the k-tuple Conjecture), there exist arbitrarily long sequences
                              > of primes from two term equations.
                              >
                              > > But for expressions of the form x^2 + x + p there is no need to
                              > > look for a longer one since it has been shown that p = 41
                              > > generates the longest one.
                              >
                              > Again, the same conjecture implies that arbitrarily long sequences
                              > of primes exist of the form x^2+x+p.
                              >
                              > What has been shown is that p=41 is the largest prime such that
                              > x^2+x+p is prime for all x, 0 <= x <= p-2.
                              >
                              > It has not been shown that x^2+x+p is never prime for 0 <= x <= 40.
                            • Shane
                              ... a ... Why would it have to occur early?
                              Message 14 of 20 , Aug 5, 2003
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                                > Whether the k tuple conjucture is true, I guess I believe it with
                                a
                                > condition, which is that the tuple does not occur if it does not
                                > occur early on, or something like that...
                                >
                                > Mark
                                >
                                >


                                Why would it have to occur early?
                              • Mark Underwood
                                Hi Shane Well I figure that if the tuple doesn t occur early, it is for some reason. It would fail at higher numbers for the same reason, or a generalization
                                Message 15 of 20 , Aug 5, 2003
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                                  Hi Shane

                                  Well I figure that if the tuple doesn't occur early, it is for some
                                  reason. It would fail at higher numbers for the same reason, or a
                                  generalization of that reason. If there is a counterexample I will
                                  gladly eat my hat! I'll make sure I'm wearing chocolate hats from
                                  here on in...

                                  Mark



                                  --- In primenumbers@yahoogroups.com, "Shane" <TTcreation@a...> wrote:
                                  > > Whether the k tuple conjucture is true, I guess I believe it
                                  with
                                  > a
                                  > > condition, which is that the tuple does not occur if it does not
                                  > > occur early on, or something like that...
                                  > >
                                  > > Mark
                                  > >
                                  > >
                                  >
                                  >
                                  > Why would it have to occur early?
                                • Robert
                                  ... sequences ... 40. Surely all one has to do is find a c in the equation x^2+x+c for which the following conditions apply: 2+c not divisible by 2, 3, 5 and,
                                  Message 16 of 20 , Aug 5, 2003
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                                    > >
                                    > > > But for expressions of the form x^2 + x + p there is no need to
                                    > > > look for a longer one since it has been shown that p = 41
                                    > > > generates the longest one.
                                    > >
                                    > > Again, the same conjecture implies that arbitrarily long
                                    sequences
                                    > > of primes exist of the form x^2+x+p.
                                    > >
                                    > > What has been shown is that p=41 is the largest prime such that
                                    > > x^2+x+p is prime for all x, 0 <= x <= p-2.
                                    > >
                                    > > It has not been shown that x^2+x+p is never prime for 0 <= x <=
                                    40.

                                    Surely all one has to do is find a c in the equation x^2+x+c for
                                    which the following conditions apply:

                                    2+c not divisible by 2, 3, 5
                                    and, 2+c meets all of:

                                    1,5,6 mod 7
                                    3,6,8,9,10 mod 11
                                    1,3,4,5,6,7,10 mod 13
                                    1,3,4,5,8,9,10,12 mod 17
                                    4,5,8,11,12,13,14,16,18 mod 19
                                    1,3,9,10,11,12,14,16,17,20,21 mod 23
                                    3,5,6,7,9,10,12,13,14,16,21,22,26,27 mod 29
                                    4,7,11,12,14,15,17,18,19,20,24,26,28,29,30 mod 31
                                    1,6,7,8,10,11,12,13,15,16,17,22,24,25,28,32,35,36 mod 37
                                    3,4,5,6,7,9,11,14,16,18,19,20,21,22,26,27,30,36,39,40 mod 41
                                    1,5,6,8,10,11,14,17,19,22,23,24,26,27,28,29,30,34,36,37,38 mod 43

                                    c=41 is the first number to reach all of the conditions except the
                                    last, being 1mod7, 10mod11, 4mod13....but 0mod43

                                    Regards

                                    Robert Smith

                                    PS I may have gotten some of the register above incorrect, but
                                    someone will spot an error if I have made one. Thats what I like
                                    about you Primenumbers group.

                                    PPS 2+c = x^2+x+c with x=1

                                    PPPS this is the same logic as used in the determination of Payam
                                    numbers
                                  • Robert
                                    ... to ... 41 primes, must also clear similar mod n hurdles up to sqrt c
                                    Message 17 of 20 , Aug 5, 2003
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                                      --- In primenumbers@yahoogroups.com, "Robert" <100620.2351@c...>
                                      wrote:
                                      >
                                      > > >
                                      > > > > But for expressions of the form x^2 + x + p there is no need
                                      to
                                      > > > > look for a longer one since it has been shown that p = 41
                                      > > > > generates the longest one.
                                      > > >
                                      > > > Again, the same conjecture implies that arbitrarily long
                                      > sequences
                                      > > > of primes exist of the form x^2+x+p.
                                      > > >
                                      > > > What has been shown is that p=41 is the largest prime such that
                                      > > > x^2+x+p is prime for all x, 0 <= x <= p-2.
                                      > > >
                                      > > > It has not been shown that x^2+x+p is never prime for 0 <= x
                                      <=
                                      > 40.
                                      >
                                      > Surely all one has to do is find a c in the equation x^2+x+c for
                                      > which the following conditions apply:
                                      >
                                      > 2+c not divisible by 2, 3, 5
                                      > and, 2+c meets all of:
                                      >
                                      > 1,5,6 mod 7
                                      > 3,6,8,9,10 mod 11
                                      > 1,3,4,5,6,7,10 mod 13
                                      > 1,3,4,5,8,9,10,12 mod 17
                                      > 4,5,8,11,12,13,14,16,18 mod 19
                                      > 1,3,9,10,11,12,14,16,17,20,21 mod 23
                                      > 3,5,6,7,9,10,12,13,14,16,21,22,26,27 mod 29
                                      > 4,7,11,12,14,15,17,18,19,20,24,26,28,29,30 mod 31
                                      > 1,6,7,8,10,11,12,13,15,16,17,22,24,25,28,32,35,36 mod 37
                                      > 3,4,5,6,7,9,11,14,16,18,19,20,21,22,26,27,30,36,39,40 mod 41
                                      > 1,5,6,8,10,11,14,17,19,22,23,24,26,27,28,29,30,34,36,37,38 mod 43
                                      >
                                      > c=41 is the first number to reach all of the conditions except the
                                      > last, being 1mod7, 10mod11, 4mod13....but 0mod43
                                      >
                                      > Regards
                                      >
                                      > Robert Smith
                                      >
                                      > PS I may have gotten some of the register above incorrect, but
                                      > someone will spot an error if I have made one. Thats what I like
                                      > about you Primenumbers group.
                                      >
                                      > PPS 2+c = x^2+x+c with x=1
                                      >
                                      > PPPS this is the same logic as used in the determination of Payam
                                      > numbers
                                      #
                                      Oops, I forgot to mention that the value of c, which contributes to
                                      a run of >41 primes, must also clear similar mod n hurdles up to
                                      sqrt c
                                    • Robert
                                      ... need ... that ... the ... Payam ... to ... #### Now I have had a glass of wine, (Merlot) I see that what I have pointed to above relates to (contributes to
                                      Message 18 of 20 , Aug 5, 2003
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                                        --- In primenumbers@yahoogroups.com, "Robert" <100620.2351@c...>
                                        wrote:
                                        > --- In primenumbers@yahoogroups.com, "Robert" <100620.2351@c...>
                                        > wrote:
                                        > >
                                        > > > >
                                        > > > > > But for expressions of the form x^2 + x + p there is no
                                        need
                                        > to
                                        > > > > > look for a longer one since it has been shown that p = 41
                                        > > > > > generates the longest one.
                                        > > > >
                                        > > > > Again, the same conjecture implies that arbitrarily long
                                        > > sequences
                                        > > > > of primes exist of the form x^2+x+p.
                                        > > > >
                                        > > > > What has been shown is that p=41 is the largest prime such
                                        that
                                        > > > > x^2+x+p is prime for all x, 0 <= x <= p-2.
                                        > > > >
                                        > > > > It has not been shown that x^2+x+p is never prime for 0 <= x
                                        > <=
                                        > > 40.
                                        > >
                                        > > Surely all one has to do is find a c in the equation x^2+x+c for
                                        > > which the following conditions apply:
                                        > >
                                        > > 2+c not divisible by 2, 3, 5
                                        > > and, 2+c meets all of:
                                        > >
                                        > > 1,5,6 mod 7
                                        > > 3,6,8,9,10 mod 11
                                        > > 1,3,4,5,6,7,10 mod 13
                                        > > 1,3,4,5,8,9,10,12 mod 17
                                        > > 4,5,8,11,12,13,14,16,18 mod 19
                                        > > 1,3,9,10,11,12,14,16,17,20,21 mod 23
                                        > > 3,5,6,7,9,10,12,13,14,16,21,22,26,27 mod 29
                                        > > 4,7,11,12,14,15,17,18,19,20,24,26,28,29,30 mod 31
                                        > > 1,6,7,8,10,11,12,13,15,16,17,22,24,25,28,32,35,36 mod 37
                                        > > 3,4,5,6,7,9,11,14,16,18,19,20,21,22,26,27,30,36,39,40 mod 41
                                        > > 1,5,6,8,10,11,14,17,19,22,23,24,26,27,28,29,30,34,36,37,38 mod 43
                                        > >
                                        > > c=41 is the first number to reach all of the conditions except
                                        the
                                        > > last, being 1mod7, 10mod11, 4mod13....but 0mod43
                                        > >
                                        > > Regards
                                        > >
                                        > > Robert Smith
                                        > >
                                        > > PS I may have gotten some of the register above incorrect, but
                                        > > someone will spot an error if I have made one. Thats what I like
                                        > > about you Primenumbers group.
                                        > >
                                        > > PPS 2+c = x^2+x+c with x=1
                                        > >
                                        > > PPPS this is the same logic as used in the determination of
                                        Payam
                                        > > numbers
                                        > #
                                        > Oops, I forgot to mention that the value of c, which contributes
                                        to
                                        > a run of >41 primes, must also clear similar mod n hurdles up to
                                        > sqrt c

                                        #### Now I have had a glass of wine, (Merlot) I see that what I have
                                        pointed to above relates to (contributes to the thinking behind) the
                                        statement

                                        > > > > What has been shown is that p=41 is the largest prime such
                                        that
                                        > > > > x^2+x+p is prime for all x, 0 <= x <= p-2.
                                        > > > >

                                        #### OK who proved that one? It seems intuitively non-provable, or
                                        only provable by a lot of computer processing. But what do I know?

                                        I think I prefer the statement:

                                        > > > > Again, the same conjecture implies that arbitrarily long
                                        > > sequences
                                        > > > > of primes exist of the form x^2+x+p.

                                        That is something the list of hurdles might contribute to.

                                        #### but the list of hurdles, etc does not contribute to the last
                                        statement

                                        > > > > It has not been shown that x^2+x+p is never prime for 0 <= x
                                        > <=
                                        > > 40.

                                        Sorry for the confusion

                                        Regards

                                        Robert Smith
                                      • Gary Chaffey
                                        ... wrote: ... Ditto. I had not noticed this either. I am somewhat sceptical that a polynomial of the form 2x^2+p will be found (soon) that yields primes for
                                        Message 19 of 20 , Aug 6, 2003
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                                          --- Mark Underwood <mark.underwood@...>
                                          wrote: >
                                          > Thanks for the correction Jack, I would not have
                                          > discerned that
                                          > difference from what I read unless it was pointed
                                          > out.
                                          Ditto. I had not noticed this either.
                                          I am somewhat sceptical that a polynomial of the form
                                          2x^2+p will be found (soon) that yields primes for all
                                          x in [0..29].
                                          I have looked at p upto 3.10^7 and as of yet nothing
                                          gets anywhere close. (best so far x in [0..9]).
                                          I am however going to look a bit deeper.
                                          Gary

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                                        • Paul Jobling
                                          Gary, ... I have just sieved up to 10^14 with nothing being found. The best was 45077834116589, which generated 18 primes for x in [0..29]. The runtime was 395
                                          Message 20 of 20 , Aug 6, 2003
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                                            Gary,

                                            > I am somewhat sceptical that a polynomial of the form
                                            > 2x^2+p will be found (soon) that yields primes for all
                                            > x in [0..29].
                                            > I have looked at p up to 3.10^7 and as of yet nothing
                                            > gets anywhere close. (best so far x in [0..9]).
                                            > I am however going to look a bit deeper.

                                            I have just sieved up to 10^14 with nothing being found. The best was
                                            45077834116589, which generated 18 primes for x in [0..29]. The runtime was
                                            395 seconds on this 450 MHz PIII.

                                            Regards,

                                            Paul.




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