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Re: Tightened-Lightened Goldbach Conjecture

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  • Mark Underwood
    ... to use my nomneclayure unless what you ve used is an accepted standard and then I ll comply with that standard. ... relationship between N and I, there
    Message 1 of 11 , Nov 6, 2008
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      --- In primenumbers@yahoogroups.com, Bill Krys <billkrys@...> wrote:
      >
      > Mark,
      >  
      > 1st of all, I think you and I are reversing what we call I and N, but I'm going to continue
      to use my nomneclayure unless what you've used is an accepted standard and then I'll
      comply with that standard.
      >  
      > I think you are probably right after all. I've been trying to articulate what you have said
      more precisely to understand this. Here is a first hack:
      >  
      > There are always prime gaps of sufficient length that after having created a 1-to-1
      relationship between N and I, there will always eventually be an integer within the large
      prime gap of concern such that its prime pairs are so limited that the sequence can no
      longer continue. (I'm not happy with this description, but I have to get back to other things
      for a bit.)
      >  
      > The Upshot is: Now, given your insight, I'm going to try to re-start the sequence after
      2*3 (=6), 2*3*5 (=30), 2*3*5*7 (=210), .... because there are prime gaps from
      ((P1*P2*P3*...*Pn) - 2) through ((P1*P2*P3*...*Pn) - Pn) and these are the only prime gaps I
      can reliably predict where and for hong long they occur. I realize they may be longer, but
      this is a minimum length.
      >  
      > So I'll try that and see how it goes. Of course, I'm always going to re-start from 1 again
      for N.
      >
      > Bill Krys


      Bill, you're right I did reverse the N and the I. Sorry about that! I'll switch back to your
      original nomenclature. Also, I should clarify something from my last post. All my last post
      showed was that for a given starting value of I, some ending values of I will not work, if
      my thinking is correct.

      For instance if one starts at I = 7, I don't think your proposal will work if I ends anywhere
      in the range of 671 to 681. *However*, it *may* work for infinitely many values of I
      outside of this range. (Or not.) It seems to me that the chances of your proposal being
      successful would be enhanced if the last value of I was about half that of the last prime in
      a prime cluster. Based on this, time permitting, I may try I from I = 7 to (say) I = 57. Hey,
      I got up to 30 last time I tried, hehe.



      Mark


      .


      > --- On Wed, 11/5/08, Mark Underwood <mark.underwood@...> wrote:
      >
      > From: Mark Underwood <mark.underwood@...>
      > Subject: [PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture
      > To: primenumbers@yahoogroups.com
      > Date: Wednesday, November 5, 2008, 3:59 AM
      >
      >
      >
      >
      >
      >
      > Hi Bill,
      >
      > Pardon the top post, but it just came to me why your conjecture cannot
      > work. It has to do with prime gaps.
      >
      > For example consider the incredible prime gap of 34, between 1327 and
      > 1361.
      > Now, consider when N is from around 1327/2 =~664 to 1361/2 =~682.
      >
      > N +/- I = prime.
      >
      > 665 - 662 = 3 (prime)
      > 665 + 662 = 1327 (prime).
      >
      > The next N,I greater than 665,662 that will work is
      >
      > 682 - 679 = 3
      > 682 + 679 = 1361
      >
      > In other words, for N between 665 and 682 there are no I's between 662
      > and 679 that when added to N will yield a prime. That is 16 values of
      > I that are lost. So, at the very least, N would have to start at 17 to
      > atone for this, if we are to have a one to one mapping of I to N.
      >
      > And of course as the gaps get larger, so will the starting
      > N be required to get larger, with no limit.
      >
      > Mark
      >
      > .
      >
      >
      > --- In primenumbers@ yahoogroups. com, "Mark Underwood"
      > <mark.underwood@ ...> wrote:
      > >
      > > --- In primenumbers@ yahoogroups. com, Bill Krys <billkrys@> wrote:
      > > >
      > > > Mark,
      > > >
      > > > I'm going to make you a gentleman's bet that I can get a prime pair
      > > generated for each unique N and that each and every N will be used
      > > once and only once (I think it'll ultimately depend on what ineger I
      > > start with). I'm speculating and you know I have little formal
      > > knowledge to back it up, and furthermore, I realize there are many
      > > seductive patterns seen in numbers that just don't survive once one
      > > gets up in numbers, and finally I've been proved wrong so many times,
      > > I should probably know better, but a bet will add a little spice to
      > > this tedious exercise. Will you take it on?
      > > >
      > > > P.S. Thanks for your past response and insight.
      > > >
      > > > Bill Krys
      > >
      > > Hi Bill
      > >
      > > A gentleman's bet, hmmm. If I bet, then that would put me in the class
      > > called "gentleman". OK, I'm in, hehe!
      > >
      > > The thing I don't like about this is the seeming arbitrariness of what
      > > N to start at. But I'll start at N = 7 (because of the obvious divine
      > > connotations :)) and see how far I can get. So far, we're up to 30.
      > >
      > > Mark
      > >
      > > .
      > >
      > >
      > >
      > > >
      > > > --- On Fri, 10/31/08, Mark Underwood <mark.underwood@ > wrote:
      > > >
      > > > From: Mark Underwood <mark.underwood@ >
      > > > Subject: [PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture
      > > > To: primenumbers@ yahoogroups. com
      > > > Date: Friday, October 31, 2008, 6:17 PM
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > > --- In primenumbers@ yahoogroups. com, "billkrys" <billkrys@ .> wrote:
      > > > >
      > > > > Y'all,
      > > > >
      > > > > given that Goldbach's Conjecture for even #s can be re-stated as
      > > there
      > > > > is a prime equi-distant (N = integer) on either side of all
      > integers
      > > > > (I), then is there a unique N for each integer such that each N is
      > > used
      > > > > once and only once and where all N's can be represented above some
      > > > > minimum I?
      > > > >
      > > > > In other words, can a prime pair be created for each integer
      > (above 4
      > > > > or some other integer - and then what is it?) from each N, such
      > > that a
      > > > > prime pair is created as a function of N? In yet more other words,
      > > the
      > > > > Conjecture would be tightened by becoming a function and
      > lightened by
      > > > > being only concerned with 1 pair of primes for each integer.
      > > > >
      > > > > Is there more than 1 function depending on what I - and for that
      > > > > matter, depending on what N - one starts with?
      > > > >
      > > > > I'm trying to create such a function but am doing it without a
      > > program,
      > > > > so it will take time - trial and error.
      > > > >
      > > >
      > > > Interesting idea. I'm almost certain there would be no function of N
      > > > which would generate a unique I. But the idea that there might be a
      > > > unique I that can be mapped to each N over a certain range is
      > > > intriguing.
      > > >
      > > > For instance, for N from 7 to 30 (as far as I checked, by hand) there
      > > > is a unique I such that N+I and N-I is prime: (N,I)
      > > >
      > > > (7,4) (8,3) (9,2) (10,7) (11,6) (12,1) (13,10) (14,9) (15,8) (16,13)
      > > > (17,14) (18,5) (19,12) (20,17) (21,16) (22,15) (23,18) (24,19) (25,22)
      > > > (26,21) (27,20) (28,25) (29,24) (30,11)
      > > >
      > > > This is just one of many possibilities. But, I strongly suppose that
      > > > this particular one, and probably all of them, will fail at some
      > > > higher N. But, how far can one go, that is the question....
      > > >
      > > > Mark
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > >
      > > > [Non-text portions of this message have been removed]
      > > >
      > >
      >
      >
      >
      >
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      > [Non-text portions of this message have been removed]
      >
    • 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 2 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 3 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 4 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 5 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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