Thanks for the reply. I'm breaking usenet etiquette becuase the new google groups is brain dead and the old google groups doesn't work on my machine and browser.
On Monday, November 26, 2012 10:08:44 AM UTC-8, christian.bau wrote: > 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.
Yes, The idea is to show why there can be no counter examples. I don't see how one can just talk about the set of primes being a Goldbach set and get anywhere. It seems important to talk about the composite numbers relative to primes less than x/2 where x is an even number > 7 and how the primes involved in those composite numbers rule out their pariticpation in the composite number relative to other primes less than x/2. The idea is to show that one runs out of primes that can participate in composites prior to running out of primes.
So if one looks at the first 5 odd composite numbers, 9, 15, 21, 25, and 27, that 3*3, 3*5, 3*3*3, 5*5, and 3*3*3*3, one can take 9 by itself. 9 + 3 makes 12 the first candidate even number that might be a counter example but there are 2 primes less than 6 so the next candidate even number is 18 but 9 is a multiple of 3 so every prime number offset from 9 by a multiple of 3 is also met by a composite that is a multiple of 3 but there can be no prime that is a multiple of 3. In fact 25 is the first odd composite number that isn't a multiple of 3 so 28 is the first real candidate counter example. 14 is a multiple of 7 so there will be no prime that is a multiple of 7 and 28-7 is in fact a multiple of 7. The only composites between 14 and 25 are the 15, 21, and 25 but there are 5 primes less than 14.
In sci.logic I talked about the gaps between primes and equadistant primes because in order for 3, 5, 7 to be covered by composites there must be a gap between primes big enough for all three to fit. 2x-7 must not be a multiple of 7, 2x-5 must not be a multiple of 5 and 2x-3 must not be a multiple of 3 otherwise 3,5, and 7 cannot be included in the composites used to cover other primes because there are no other primes that are multiples of these numbers.
As it turns out 95=19*5, 93=31*3, 91=7*13 so only 7 is excluded from the composites of the first real counter example using less naieve properties of the primes and what the form of a counter example would look like. Well, since 95 is a multiple of 5 primes that differ from 3 by a mutlple of 5 will be covered by a composite that is a multiple of 5 as well, 98-13=85, 98-23=75, 98-43=45. Likewise with primes that differ from 3 by multiples of 19, 98-41=57. Well, the factors of the composites for 95, 93, and 91, nocked out 3, 5, 7, 11, 13, and 17 but not 19 so 79 is a matching prime or it would have it's own composites not involving 3,5,7,13,19, or 31.
Anyway, I don't have the math skills yet to solve this problem. Every composite knocks out at least one prime from future consideration but the more primies involved in a composite the the more it knocks out in future composites and the number of primes it knocks on starts to get smaller the sooner the small primes get removed.
It seems to me the form of the proof is to show that there are more odd primes less than some natural number greater than 3 than can be covered by composites of the primes less than sqrt(2n).