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[PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture

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  • Jens Kruse Andersen
    I had problems understanding the original problem formulation so I will try a more formal description. Given two natural numbers a
    Message 1 of 11 , Nov 7, 2008
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      I had problems understanding the original problem formulation
      so I will try a more formal description.
      Given two natural numbers a < b, find b-a+1 distinct natural
      numbers N_a to N_b such that I +/- N_I is prime for I = a to b.
      In other words, for each integer I from a to b, find a prime
      pair of form I +/- N such that different N is used each time.

      By my hand calculations, if I starts at a=6 then it can at
      most go to b=44. It can do that in four ways, with two possible
      combinations at I=8,10,12, and two options at I=43.
      (I,N_I): (6,1) (7,4) (8,3 or 5) (9,2) (10,7 or 3) (11,6)
      (12,5 or 7) (13,10) (14,9) (15,8) (16,13) (17,14) (18,11)
      (19,12) (20,17) (21,16) (22,15) (23,18) (24,19) (25,22)
      (26,21) (27,20) (28,25) (29,24) (30,23) (31,28) (32,29)
      (33,26) (34,27) (35,32) (36,31) (37,30) (38,35) (39,34)
      (40,33) (41,38) (42,37) (43,36 or 40) (44,39).

      45+/-N is prime for N = 2, 8, 14, 16, 22, 26, 28, 34, 38,
      but they are all taken.

      Bill Krys wrote:
      > ... these are the only prime gaps I can reliably predict
      > where and for hong long they occur.

      The maximal prime gaps at
      http://hjem.get2net.dk/jka/math/primegaps/maximal.htm
      can be used to get an upper limit for how large b can be
      for a given value of a, based on Mark's argument.
      The actual highest value of b may turn out to be lower than
      the limit given in this way.

      --
      Jens Kruse Andersen
    • Mark Underwood
      ... Well Jens I want compensation from Bill for what I feel is about 2 months taken off my life trying to do this by hand in the last three hours. I went as
      Message 2 of 11 , Nov 8, 2008
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        --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
        <jens.k.a@...> wrote:
        >
        > I had problems understanding the original problem formulation
        > so I will try a more formal description.
        > Given two natural numbers a < b, find b-a+1 distinct natural
        > numbers N_a to N_b such that I +/- N_I is prime for I = a to b.
        > In other words, for each integer I from a to b, find a prime
        > pair of form I +/- N such that different N is used each time.
        >
        > By my hand calculations, if I starts at a=6 then it can at
        > most go to b=44. It can do that in four ways, with two possible
        > combinations at I=8,10,12, and two options at I=43.
        > (I,N_I): (6,1) (7,4) (8,3 or 5) (9,2) (10,7 or 3) (11,6)
        > (12,5 or 7) (13,10) (14,9) (15,8) (16,13) (17,14) (18,11)
        > (19,12) (20,17) (21,16) (22,15) (23,18) (24,19) (25,22)
        > (26,21) (27,20) (28,25) (29,24) (30,23) (31,28) (32,29)
        > (33,26) (34,27) (35,32) (36,31) (37,30) (38,35) (39,34)
        > (40,33) (41,38) (42,37) (43,36 or 40) (44,39).
        >
        > 45+/-N is prime for N = 2, 8, 14, 16, 22, 26, 28, 34, 38,
        > but they are all taken.
        >
        > Bill Krys wrote:
        > > ... these are the only prime gaps I can reliably predict
        > > where and for hong long they occur.
        >
        > The maximal prime gaps at
        > http://hjem.get2net.dk/jka/math/primegaps/maximal.htm
        > can be used to get an upper limit for how large b can be
        > for a given value of a, based on Mark's argument.
        > The actual highest value of b may turn out to be lower than
        > the limit given in this way.
        >
        > --
        > Jens Kruse Andersen
        >


        Well Jens I want compensation from Bill for what I feel is about 2
        months taken off my life trying to do this by hand in the last three
        hours. I went as far as cutting out fifty six little pieces of paper
        with the numbers from 0 to 55 written on them. Yes, I started at zero
        just to shake things up. (The application to Goldbach's conjecture
        would still apply, ie, 3 + 0 = 3; 3 - 0 = 3 ; 6 = 3 + 3)

        Is this what life was like before computers?

        But lo and behold it worketh!:


        (6,1) (7,4) (8,5) (9,2) (10,3) (11,6) (12,7)
        (13,10) (14,9) (15,8) (16,13) (17,14) (18,11) (19,12)
        (20,17) (21,16) (22,15) (23,0) (24,19) (25,22) (26,21)
        (27,20) (28,25) (29,18) (30,23) (31,28) (32,29) (33,26)
        (34,27) (35,24) (36,31) (37,34) (38,35) (39,32) (40,33)
        (41,48) (42,37) (43,40) (44,39) (45,38) (46,43) (47,36)
        (48,41) (49,30) (50,47) (51,46) (52,45) (53,50) (54,49)
        (55,52) (56,51) (57,44) (58,55) (59,54) (60,53) (61,42)


        Mark
      • Mark Underwood
        ... Lo and behold it doesn t. Jen noticed that (41,48) yields a negative prime , so is bad. Now, I have to determine if this whole exercise actually caused
        Message 3 of 11 , Nov 9, 2008
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          --- In primenumbers@yahoogroups.com, "Mark Underwood"
          <mark.underwood@...> wrote:
          >
          > But lo and behold it worketh!:
          >
          >
          > (6,1) (7,4) (8,5) (9,2) (10,3) (11,6) (12,7)
          > (13,10) (14,9) (15,8) (16,13) (17,14) (18,11) (19,12)
          > (20,17) (21,16) (22,15) (23,0) (24,19) (25,22) (26,21)
          > (27,20) (28,25) (29,18) (30,23) (31,28) (32,29) (33,26)
          > (34,27) (35,24) (36,31) (37,34) (38,35) (39,32) (40,33)
          > (41,48) (42,37) (43,40) (44,39) (45,38) (46,43) (47,36)
          > (48,41) (49,30) (50,47) (51,46) (52,45) (53,50) (54,49)
          > (55,52) (56,51) (57,44) (58,55) (59,54) (60,53) (61,42)
          >
          >
          > Mark
          >


          Lo and behold it doesn't. Jen noticed that (41,48) yields a negative
          'prime', so is bad. Now, I have to determine if this whole exercise
          actually caused my mental decline, or whether it was a pre existing
          condition.

          Mark
        • Bill Krys
          Mark and Jens,   thanks for trying. I am going to withdraw from the group for a while to tend to work, but I ll come back if I find anything or need more
          Message 4 of 11 , Nov 12, 2008
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            Mark and Jens,
             
            thanks for trying. I am going to withdraw from the group for a while to tend to work, but I'll come back if I find anything or need more help. I'll try to see if either a continuous sequence of "N"s works starting from a higher integer and if no luck there, then I'll see if your idea of sequential fragments works, hopefully based on some easily predictable prime gaps because I don't like the idea of a prime gap I can't predict understand.
             
            P.S. Mark, sorry for causing your cognitive dissonance, but that's learnin', ain't it?

            Bill Krys

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            --- On Sun, 11/9/08, Mark Underwood <mark.underwood@...> wrote:

            From: Mark Underwood <mark.underwood@...>
            Subject: [PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture
            To: primenumbers@yahoogroups.com
            Date: Sunday, November 9, 2008, 2:57 PM






            --- In primenumbers@ yahoogroups. com, "Mark Underwood"
            <mark.underwood@ ...> wrote:
            >
            > But lo and behold it worketh!:
            >
            >
            > (6,1) (7,4) (8,5) (9,2) (10,3) (11,6) (12,7)
            > (13,10) (14,9) (15,8) (16,13) (17,14) (18,11) (19,12)
            > (20,17) (21,16) (22,15) (23,0) (24,19) (25,22) (26,21)
            > (27,20) (28,25) (29,18) (30,23) (31,28) (32,29) (33,26)
            > (34,27) (35,24) (36,31) (37,34) (38,35) (39,32) (40,33)
            > (41,48) (42,37) (43,40) (44,39) (45,38) (46,43) (47,36)
            > (48,41) (49,30) (50,47) (51,46) (52,45) (53,50) (54,49)
            > (55,52) (56,51) (57,44) (58,55) (59,54) (60,53) (61,42)
            >
            >
            > Mark
            >

            Lo and behold it doesn't. Jen noticed that (41,48) yields a negative
            'prime', so is bad. Now, I have to determine if this whole exercise
            actually caused my mental decline, or whether it was a pre existing
            condition.

            Mark















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