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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
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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.


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.

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

Regards,

Warwick.






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

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