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19677Re: [PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture

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  • Bill Krys
    Nov 6, 2008
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      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

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      --- 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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