- Nov 6, 2008--- In primenumbers@yahoogroups.com, Bill Krys <billkrys@...> wrote:
>

to use my nomneclayure unless what you've used is an accepted standard and then I'll

> Mark,

>

> 1st of all, I think you and I are reversing what we call I and N, but I'm going to continue

comply with that standard.>

more precisely to understand this. Here is a first hack:

> I think you are probably right after all. I've been trying to articulate what you have said

>

relationship between N and I, there will always eventually be an integer within the large

> There are always prime gaps of sufficient length that after having created a 1-to-1

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.)>

2*3 (=6), 2*3*5 (=30), 2*3*5*7 (=210), .... because there are prime gaps from

> The Upshot is: Now, given your insight, I'm going to try to re-start the sequence after

((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.>

for N.

> So I'll try that and see how it goes. Of course, I'm always going to re-start from 1 again

>

Bill, you're right I did reverse the N and the I. Sorry about that! I'll switch back to your

> Bill Krys

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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