- --- In primenumbers@yahoogroups.com, "billkrys" <billkrys@...> wrote:
>

Interesting idea. I'm almost certain there would be no function of N

> 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.

>

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 - 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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