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

[ichudov@Algebra.COM (Igor Chudov @ home)]

Jim Choate wrote:
> 
> 
> Hi,
> 
> I have a question related to Goldbach's Conjecture:
> 
> All even numbers greater than two can be represented as the sum of primes.

Hold on right here, Jim.

Do you mean a sum of DIFFERENT primes?

Because any number greater than 1 can be represented as a sum of some 2s
and some 3s. 

E.g. 8 = 3+3+2, 9 = 3+3+3, 10 = 3+3+2+2, etc.

Since this is so boring, I assume that the primes must be different.

> 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.

So, the above hypothesis is incorrect. No need for high powered math here.

> 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.

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

> I'm assuming, since I can't find it explicitly stated anywhere, that
> Goldbachs Conjecture allows those prime factors to occur in multiple
> instances.

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.

> I've pawed through my number theory books and can't find anything relating
> to this as regards odd numbers.


	- Igor.