- 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:>

integers

> --- 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

> > > (I), then is there a unique N for each integer such that each N is

(above 4

> 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

> > > or some other integer - and then what is it?) from each N, such

lightened by

> 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

> > > 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]

> >

> - 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

This communication is intended for the use of the recipient to which it is addressed, and may contain confidential, personal, and or privileged information. Please contact the sender immediately if you are not the intended recipient of this communication, and do not copy, distribute, or take action relying on it. Any communication received in error, or subsequent reply, should be deleted or destroyed.

--- 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

[Non-text portions of this message have been removed]