The equation for the T_m set is actually 'a-b = 2*m'.

The equation for the G_m set is actually 'a+b =

2*m-2'. (Then 2*m is actually the even number we're

interested in, not off-by-two.)

I should probably put in what the 'restatements' of

the Goldbach Conjecture and Twin Primes conjecture

actually are:

GB: G_m is nonempty for all positive integers m > 1.

TPC: T_m has cardinality aleph-nought for m = 1 (and

for the extended TPC, for all positive integers m).

Joseph.

--- Joseph Moore <jtpk@...> wrote:

> Others will have better replies with specifics I'm

__________________________________

> sure, but let me account for some of the

> 'aesthetics'

> of the Goldbach Conjecture... maybe.

>

> First, define a set as follows:

>

> S_n = { k in the positive integers : there exist s

> and

> t in the positive integers such that k = n*s*t+s+t }

>

> It can be shown relatively easily that p is a prime

> iff p-1 is *not* in S_1.

>

> Next we define a 'Goldbach' set as follows:

>

> G_m = { (a, b) for a, b in "the positive integers

> throw away S_1" : a + b = 2*m }

>

> So, for example, we note that 1 and 2 are not in

> S_1,

> but (1, 2) is not in G_m for any m. However, 4 is

> also not in S_1, so (2, 4) is in G_3 (also, (1, 1)

> is

> in G_1, (2, 2) is in G_2, and so on).

>

> Now the 'aethetic' part. 'a+b = 2*m' can be looked

> at

> as a linear equation, except that the constraints on

> a

> and b are not linear (in fact, they are roughly

> 'square'). So this equation is actually like that

> of

> an ellipse. We can take this a step further and

> construct a T_m set (for the twin primes) where the

> basic constraint is 'a - b = 2', so that the Twin

> Prime Conjecture is actually based on an

> hyperbolic-type equation.

>

> So, the Goldbach Conjecture and the Twin Primes

> Conjecture are like the ellipse and hyperbola of

> number theory. The only potential difficulty with

> this analogy is that, at least with real ellipses

> and

> hyperbolas, it's really hard to do exact

> calculations

> on them, e.g., finding the circumference of an

> ellipse, or an arc length based on some angles. So

> I'm hoping that Goldbach and the Twin Primes don't

> turn out to be as hard to work with...

>

> Joseph.

>

> PS Anyone interested in further info on my twin

> prime

> theories can check out everything2.com and use their

> search to look up 'odd-even theorem'.

>

>

> --- Mary & Graeme <candlesque@...>

> wrote:

>

> > I recently came across the matter of Goldbach's

> > conjecture, which I

> > understand postulates that all even numbers

> > greater than or equal to

> > 4, can be expressed as the sum of a pair of prime

> > numbers. Further,

> > this conjecture has neither been confirmed or

> > refuted.

> >

> > I would be interested to hear from those who can

> > indicate what are the

> > known problems in developing an assessment of

> > Goldbach's conjecture.

> >

> > All the best.

> > Graeme Moyse

> >

> >

>

>

>

>

>

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