Re: Goldbach's Conjecture - a question about prime sums of odd numbers... (fwd)

[Jim Choate <ravage@EINSTEIN.ssz.com>]

Forwarded message:

> Subject: Re: Goldbach's Conjecture - a question about prime sums of odd numbers...
> Date: Thu, 19 Nov 1998 09:26:56 -0600 (CST)
> From: ichudov@Algebra.Com (Igor Chudov @ home)

> > All even numbers greater than two can be represented as the sum of primes.
> 
> Do you mean a sum of DIFFERENT primes?

None of the references that I've looked at state it clearly but I assume
that it must be or else we could just use 1 for everything.

> > Is there any work on whether odd numbers can always be represented as the
> > sum of primes?
> 
> Well, take 11, for example, it cannot be repsesented as a sum of different
> primes. It cannot, pure and simple.

Sure it can, 1 + 3 + 7. It can't be the PRODUCT of two lower primes since
it wouldn't be prime then. This sum by the way CAN'T be reduced to two primes
so Fermat's Conjecture applied to odds is clearly false.

The only proviso is the number we want to sum TO must be greater than 2.

ALL numbers can be represented as sums of primes, the question is whether
you allow repeats or not.

> > This of course implies that the number of prime members
> > must be odd and must exclude 1 (unless you can have more than a single
> > instance of a given prime). Has this been examined?
> 
> Why, let's say 5 = 3+2, it is a sum of an even number of primes.

In the number theory realm, I agree. In the geometric real (in particular
using equilateral triangles) those constraints hold.

Note that the smallest number of sums for your example above is odd. The
reason that you must exclude the problem becomes trivial if you aren't
required to sum the 1's.

> I suggest that first "examination" should always include playing
> with trivial examples.

No kidding?

> If multiple instances are allowed, it is an enormously boring conjecture
> for 5 grade school students.
> 
> any number above 1 may be represented as a sum of some 3s and some
> 2s. Big deal.

Actualy they are simply sums of 1's, its actualy first grade math.


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