In article <NoJunkMailemail@example.com>, NoJunkMail@this.address (Gerry Myerson) writes: > In article <3458CA74.firstname.lastname@example.org>, David Ullrich > <email@example.com> wrote: > > > Gerry Myerson wrote: > > > > > > ...it has been obvious for a century or two that the Goldbach Conjecture > > > is true. > > > > Really? Could you explain why this is obvious? > > Well, there are so damn many primes. There are more than 1200 of them > below 10000, for instance. That means over 700000 ways to account for > the 10000 even numbers below 20000 as a sum of two primes. That's an > overkill ratio of 70-to-1. And it just gets worse, if you go up to > 100000, 1000000, etc. There would have to be a massive conspiracy of > the prime numbers in order for any even number to miss out.
There are "so damn many" squares also, but not every number is the sum of two squares, or even three squares. Though my example probably isn't a good one, because there are plenty of small values which are not sums of two squares: but I recall another result in which a property was true up to some colossal number, at which point there was a case where it was false. It was something to do with one quantity being greater than another quantity: does someone know of this result?
It's one thing to say that there is strong numerical evidence in favour of GC, which I agree with, but saying that it is "obvious" is another matter.
Another thing to think about is that although there's an infinite number of primes, they end up being sparsely distributed. If pi(x) is the number of primes less than or equal to x, then pi(x)/x tends to (x / log x) / x --> 0 as x tends to infinity. So if it only requires the sum of two primes to express any even number greater than two, then this would be a remarkable result.
> To put the same thing another way; anyone who sits down to verify > Goldbach for small values will quickly find that not only can every even > n exceeding 2 be written as a sum of two primes, but every n from some > point on has at least two such expressions; from some further point on, > she will find that every n has at least three, then four, then five...; > eventually, it will become obvious to her that every sufficiently large > even integer has more than 100, more than 1000, more than 1000000 > representations as a sum of two primes. And if that's obvious, how much > more obvious is Goldbach, a pathetically weak conjecture by comparison.
I'd be interested to read about any conjectures regarding the "expected" number of ways an even integer may be written as the sum of two primes.