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Topic: Goldbach conjecture
Replies: 43   Last Post: Sep 26, 2000 8:55 AM

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 Warwick Pulley Posts: 7 Registered: 12/12/04
Re: Goldbach conjecture
Posted: Nov 2, 1997 7:09 PM

In article <NoJunkMail-3110971441350001@abinitio.mpce.mq.edu.au>, NoJunkMail@this.address (Gerry Myerson) writes:
> In article <3458CA74.900@math.okstate.edu>, David Ullrich
> <ullrich@math.okstate.edu> 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.

number of ways an even integer may be written as the sum of two primes.

> Gerry Myerson (gerry@mpce.mq.edu.au)

Regards,

Warwick.

Date Subject Author
10/29/97 Gerry Myerson
10/30/97 David Ullrich
10/30/97 Gerry Myerson
11/2/97 Warwick Pulley
11/2/97 Ron Bloom
11/3/97 feldmann@bsi.fr
11/4/97 David Ullrich
11/10/97 Bob Runkel
11/11/97 Richard Carr
11/19/97 Richard Carr
11/3/97 Gerry Myerson
11/3/97 Warwick Pulley
11/3/97 Gerry Myerson
11/8/97 Andre Engels
11/9/97 Warwick Pulley
11/10/97 Meinte Boersma
11/13/97 Chris Thompson
9/16/00 Daniel McLaury
9/17/00 Fred Galvin
9/17/00 Jan Kristian Haugland
9/17/00 denis-feldmann
9/17/00 Erick Wong
9/26/00 John Rickard
11/6/97 Richard White (CS)
11/6/97 James Graham-Eagle
11/6/97 Legion
11/6/97 Gerry Myerson
11/7/97 Legion
11/7/97 Gerry Myerson
11/4/97 David Petry
11/3/97 Chris J. Bennardo
11/3/97 Gerry Myerson
11/8/97 Andre Engels
11/5/97 Robert Hill
10/30/97 goldbach
11/2/97 Orjan Johansen
11/2/97 John Rickard
8/25/98 Elijah Bishop
11/2/97 Orjan Johansen
10/30/97 Brian Hutchings
11/2/97 Gerry Myerson
11/6/97 Brian Hutchings
4/28/99 Papus