## 19672Re: [PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture

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• Nov 4, 2008
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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

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--- On Fri, 10/31/08, Mark Underwood <mark.underwood@...> wrote:

From: Mark Underwood <mark.underwood@...>
Subject: [PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture
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

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