On Nov 26, 5:15 am, forbisga...@gmail.com wrote: > I'm new to sci.math. I came here from comp.ai.philosophy by way of sci.logic > because google groups doesn't allow crossposting and several people include > c.a.p in their crossposted articles. In sci.logic the Goldbach Conjecture > came up with a lot of nonsense and it lead me to start thinking about it. > Since my math is pretty rusty I'm having a bit of trouble. I came up with > the assertion that there would be a prime p between n and 2n and others > identified this as Bertrand's Postulate. I'm using a slighly stronger > conjecture that says "all even numbers greater than 7 can be expressed > as the sum of two distinct primes." I'm asserting the problem is a > topology problem and proposed there would be a proof related to the > spacing of equadistant prime from all natural numbers n greater than 3. > Today I foundhttp://milesmathis.com/gold3.html > It's quite similar to what I proposed. > Since it's not an accepted proof I'm assuming there must be a flaw. > Is the flaw easy to spot and if so what is it? > > Another corollary to my modified Goldbach Conjecture: > > There is no natural number n such that for all primes p less than n > 2n-p is not a prime. > > If you question that read it again. Sure some 2n-p will not be prime > but not all of them or else n is prime and my stronger version is false > or the Goldbach Conjecture is false. > > Except where n is a multiple of some prime p, 2n-p must be prime or > a multiple of some prime other than p,http://en.wikipedia.org/wiki/Chen%27s_theorem > appears to be a poof that's a bit weaker than what's stated as a > proof athttp://milesmathis.com/gold3.html > I find references to Chen's theorem on the web but I don't see the > actual proof of it on the web. How complex is it?
Here's something to think about:
I'll call a set S of positive integers a "Goldbach set" if every even integer >= 4 is the sum of two elements of S. You can use "... if every even integer > 7 is the sum of two distinct elements of S".
The Goldbach Conjecture now states that the set of primes is a "Goldbach set".
Sets with many elements tend more to be Goldbach sets, while sets with fewer elements tend less to be Goldbach sets.
However, a set can have an awful lot of elements and not be a Goldbach set: If you take S = the set of integers of the form 8k+1 and 8k+3, then no multiple of 8 is ever the sum of two elements of S. So your proof can't rely on the number of elements and sizes of gaps between elements alone. Your proof will have to have some element that actually uses the fact that S is a set of primes.
Most "proofs" don't have a flaw that is easy to spot, they just don't contain anything that would be a proof. Someone said "this is so bad, it isn't even wrong". Many just assert that there are so many primes and so many possible sums, there _must_ be two primes adding up to that number. But assertion isn't proof. If you take the number 10^100, there are more than 10^97 primes < 5 * 10^99, so there are about 10^97 numbers that would have to be composite to make 10^100 a counter example. It's very, very unlikely that all these 10^97 numbers are composite. But it's not impossible.