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.