More Goldbach's Conjecture

Well there are two more definitions, from the same book [1], that are not
equivalent:

pp. 335

For all natural numbers x, if x is even, non-zero, and not 2, then there
exist prime numbers y and z such that x is the sum of y and z.

pp. 673

...every even number, n>6 (it at least takes care of my question about 4),
 is the sum of two odd primes.


    ____________________________________________________________________
 
            Lawyers ask the wrong questions when they don't want
            the right answers.

                                        Scully (X-Files)

       The Armadillo Group       ,::////;::-.          James Choate
       Austin, Tx               /:'///// ``::>/|/      ravage@ssz.com
       www.ssz.com            .',  ||||    `/( e\      512-451-7087
                           -====~~mm-'`-```-mm --'-
    --------------------------------------------------------------------


[1] VNR Concise Encyclopedia of Mathematics

Jim Choate wrote:
> 
> Well there are two more definitions, from the same book [1], that are not
> equivalent:
> 
> pp. 335
> 
> For all natural numbers x, if x is even, non-zero, and not 2, then there
> exist prime numbers y and z such that x is the sum of y and z.
> 
> pp. 673
> 
> ...every even number, n>6 (it at least takes care of my question about 4),
>  is the sum of two odd primes.

> [1] VNR Concise Encyclopedia of Mathematics


Evidently there is a printing error. 'n>6' should read 'n>=6'. Then
they are equivalent (if one considers 4 = 2 + 2 to be known).

M. K. Shen

Jim Choate wrote:
> 
> 
> Well there are two more definitions, from the same book [1], that are not
> equivalent:
> 
> pp. 335
> 
> For all natural numbers x, if x is even, non-zero, and not 2, then there
> exist prime numbers y and z such that x is the sum of y and z.
> 
> pp. 673
> 
> ...every even number, n>6 (it at least takes care of my question about 4),
>  is the sum of two odd primes.

These conjectures are equivalent for numbers > 6. I think that the
discussion of whether numbers 4 and 6 can be expressed as sum of
two primes is completely uninteresting.

Also, since 6 = 3+3, I question why they put strict inequality (> 6)
in the definition on p 673. I think that they could say n > 4. Not that
it matters in any respect.

So I do not see them as "substantially" different, and the difference
between these conjectures does not lead us to any profound thoughts.

	- Igor.