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

Hi,

I have a question related to Goldbach's Conjecture:

All even numbers greater than two can be represented as the sum of primes.

Is there any work on whether odd numbers can always be represented as the
sum of primes? 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?

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

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


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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.

"Igor Chudov @ home" wrote:

> Well, take 11, for example, it cannot be repsesented as a sum of different
> primes. It cannot, pure and simple.

Bullshit: 7+5+(-1)=11.  Last I heard, negative numbers weren't excluded from 
being primes.  7 is different from 5, -1 is different from 7 and from 5.


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Ray Arachelian wrote:
> 
> 
> "Igor Chudov @ home" wrote:
> 
> > Well, take 11, for example, it cannot be repsesented as a sum of different
> > primes. It cannot, pure and simple.
> 
> Bullshit: 7+5+(-1)=11.  Last I heard, negative numbers weren't excluded from 
> being primes.  7 is different from 5, -1 is different from 7 and from 5.

I have no idea where you heard it, but primes are numbers greater than 1,
by definition.

	- Igor.

At 12:22 AM -0500 11/19/98, 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.
>
>Is there any work on whether odd numbers can always be represented as the
>sum of primes? 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?
>
>I'm assuming, since I can't find it explicitly stated anywhere, that
>Goldbachs Conjecture allows those prime factors to occur in multiple
>instances.
>
>I've pawed through my number theory books and can't find anything relating
>to this as regards odd numbers.

	Whoops, jumped the gun on that last one, didn't read thru a second
time.

	Sorry.
--
"To sum up: The entire structure of antitrust statutes in this country is a
jumble of economic irrationality and ignorance. It is a product: (a) of a
gross misinterpretation of history, and (b) of rather nave, and certainly
unrealistic, economic theories." Alan Greenspan, "Anti-trust"
http://www.ecosystems.net/mgering/antitrust.html

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At 12:22 AM -0500 11/19/98, 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.
>
>Is there any work on whether odd numbers can always be represented as the
>sum of primes? 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?
>
>I'm assuming, since I can't find it explicitly stated anywhere, that
>Goldbachs Conjecture allows those prime factors to occur in multiple
>instances.
>
>I've pawed through my number theory books and can't find anything relating
>to this as regards odd numbers.

	Well, since all primes over 2 are odd, and the sum of two odd
numbers is always even, there goes that theory.

	Unless they changed the rules on primes since I last checked.
--
"To sum up: The entire structure of antitrust statutes in this country is a
jumble of economic irrationality and ignorance. It is a product: (a) of a
gross misinterpretation of history, and (b) of rather nave, and certainly
unrealistic, economic theories." Alan Greenspan, "Anti-trust"
http://www.ecosystems.net/mgering/antitrust.html

Petro::E-Commerce Adminstrator::Playboy Ent. Inc.::petro@playboy.com

At 12:20 AM 11/19/98 -0600, Eric Cordian wrote:
>> Is there any work on whether odd numbers can always be represented as the
>> sum of primes? 
>
>Goldbach originally suggested that all numbers greater than two could be
>expressed as the sum of three primes, if one tossed in 1 as a prime
>number.  Euler pointed out that this was equivalent to even numbers
>greater than two being expressed as the sum of two primes. 
>
>This seemed a somewhat cleaner formulation, and it was adopted.

well, you can express any odd number >= 7 as
the sum of 3 + an even number, so if Goldbach's conjecture is true,
then three primes are enough for the odd natural numbers
except 1, which is a special case, and 3 and 5 which are prime anyway.
				Thanks! 
					Bill
Bill Stewart, bill.stewart@pobox.com
PGP Fingerprint D454 E202 CBC8 40BF  3C85 B884 0ABE 4639

Jim Choate writes:

> Hi,

> I have a question related to Goldbach's Conjecture:

> All even numbers greater than two can be represented as the sum of primes.

All even numbers greater than two can be represented as the sum of 
TWO primes. 

> Is there any work on whether odd numbers can always be represented as the
> sum of primes? 

Goldbach originally suggested that all numbers greater than two could be
expressed as the sum of three primes, if one tossed in 1 as a prime
number.  Euler pointed out that this was equivalent to even numbers
greater than two being expressed as the sum of two primes. 

This seemed a somewhat cleaner formulation, and it was adopted.

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