Re: Goldbach's Conjecture - the various definitions

["Albert P. Franco, II" <apf2@apf2.com>]

At 07:29 20/11/98 -0600, you wrote:
>Now before somebody gets a bee in their bonnet....
>
>Yes, the VNR has a typo in respect to the '6', it should be >2.
>
>As to the various other definitions that folks have been submitting. I'm not
>real sure what their exact point is since it's agreed by all that it's:
>
>...all even numbers greater than 2..., and Goldbach believed it had to be
>three prime factors while Fermat pointed out it could always be 2.

Math works because in most (we would like to say all...) cases it is WELL
DEFINED. Subtle differences in the definition of a problem can drastically
alter the solution. Goldbach's Conjecture states that any EVEN OR ODD
number can be expressed by a sum of three primes. The reduction only speaks
of EVEN numbers. They are EQUIVALENT but not exactly the same thing. Don't
lose sight of the domains of the two statements. He never stated that ODD
numbers could be expressed as sums of TWO primes.

It is very important to get all of the information assembled before running
out to test the conjecture! Goldbach's original conjecture allowed for even
numbers to be done with two primes but since it included the odd numbers
too, he had to allow for a third prime. 


>
>As to the two defintions that accredit the reduction from 3 to 2 to Goldbach
>while leaving Fermat out of the picture fail to explain why Fermat's point
>is called Fermat's in the first place.
>

Kind of an aside to this, I was playing with the ideas of repetitious
members of the sum and found that most numbers can be expressed by a lot of
different combinations of different primes. That is
17=11+3+3=11+3+1=9+5+3=7+5+5, etc. The higher the number the more
combinations possible. I only worked with prime numbers as the result of
the sum. But cursory inspection leads to the conclusion that all numbers
can be expressed in many ways by sums of two and three primes. Depending on
where you start to calculate (2,3,4,5,6, ...) as the first number you can
always use 3 as the third prime when summing to an odd number, Therefor
after a certain point you don't need to include 1 (as Goldbach appears to
have allowed in his original conjecture).

I would like to know if there is ANY number n>7 for which only one sum of
primes may be found. Since there were so many possible combinations, it
seems that you can take or leave the restriction of no repeats. But, if
there exists some number for which only one possible combination exists
then perhaps there exists a number for which the one and only combination
has a repeated component.

When I get a few more minutes to tinker with this I'll try to run some
tests to find such a number. 

I used do this type of tinkering in high school. If I had had a PII 300 and
MathCAD back then I would have never left Mathematics!! It was quite fun
seeing the multiple solutions popping out. 

APF