## Re: [PrimeNumbers] Goldbach's Conjecture + missing

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• Ehh... messed up a couple things. 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
Message 1 of 4 , Dec 7, 2004
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Ehh... messed up a couple things.

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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