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Re: [PrimeNumbers] Goldbach's Conjecture + missing

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  • Joseph Moore
    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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