Goldbach's conjecture
Stan Ethenoz
One of the aspects that fascinates me is the incredible amount of unsolved
problems in this field that seems so trivial at first look.
I read that what most mathematicians think is one of the hardest problems to
solve ever, is the Goldbach conjecture, that says that
every even number greater than 4 can be written as the sum of two primes.
What in your opinion makes this so hard to prove (or unprove)?
What makes it harder to prove than the other famous conjectures in number
theory?
Gerry Myerson
"Stan Ethenoz" <stanlesssteel$$NO_SPAM#%@hotmail.com> wrote:
=> I recently got really interested in number theory, One of the aspects
=> that fascinates me is the incredible amount of unsolved problems in
=> this field that seems so trivial at first look.
=>
=> I read that what most mathematicians think is one of the hardest
=> problems to solve ever, is the Goldbach conjecture, that says that
=> every even number greater than 4 can be written as the sum of two
=> primes.
=>
=> What in your opinion makes this so hard to prove (or unprove)?
Prime were made to be multiplied, not added.
=> What makes it harder to prove than the other famous conjectures in
=> number theory?
Is it? Is it harder to prove than the existence of infinitley many
perfect numbers? or the non-existence of odd perfect numbers? or
the existence of infinitely many twin primes?
I guess it's harder to prove than the Wiles-Taylor Theorem (formerly
known as Fermat's Last Theorem), but I don't think we knew that until
we got the Wiles-Taylor proof.
Gerry Myerson (ge...@mpce.mq.edu.au)
Simon Fitzpatrick
It is no harder nor easier to prove than any other unsolved problem until
you know what the proof is... and you don't. If there is one.
A counterexample to Goldbach would be a very short paper!
S F
Eli Bendersky
Simon Fitzpatrick wrote:
Doubtful. After all, I'm sure super-computers already tested the
correctness of the Goldbach conjecture up to very large numbers.
So, a proof containing a counter example, which is yet a larger
number would not be too short ;-)
--
Eli Bendersky
comp.lang.c++ FAQ: http://www.parashift.com/c++-faq-lite/
comp.lang.c FAQ : http://www.eskimo.com/~scs/C-faq/top.html
ma...@mimosa.csv.warwick.ac.uk
"Stan Ethenoz" <stanlesssteel$$NO_SPAM#%@hotmail.com> writes:
>I recently got really interested in number theory,
>One of the aspects that fascinates me is the incredible amount of unsolved
>problems in this field that seems so trivial at first look.
>
>I read that what most mathematicians think is one of the hardest problems to
>solve ever, is the Goldbach conjecture, that says that
>every even number greater than 4 can be written as the sum of two primes.
Why greater than 4? Is 4 not equal to 2+2 any more ?
Derek Holt.
Nico Benschop
>
> I recently got really interested in number theory,
> One of the aspects that fascinates me is the incredible amount of
> unsolved problems in this field that seems so trivial at first look.
>
> I read that what most mathematicians think is one of the hardest
> problems to solve ever, is the Goldbach conjecture, that says that
> every even number greater than 4 can be written as the sum of two
> primes.
>
> What makes it harder to prove than the other famous conjectures in
> number theory?
Like FLT, Goldbach's conjecture is an additive statement about
'multiplicative' numbers, roughly meaning: the operations (.) and (^)
define the terms of the (+) statement. And additive number theory
is notoriously difficult (the MSC code has a special section on it:
http://www.ams.org/msc/11Pxx.html ) The deeper reason for this is
an interesting question, but I think it has to do with the syntax
properties of arithmetic operations (+) (.) (^) in relation to
each other. Especially the fact that - although each operation
(for integers) is defined by iteration of the previous one -
the two (+) and (.) are both commutative and associative, with (.)
distributing over (+) and (^) over (.) because of the iteration
definition - however: (^) lost all-of-a-sudden both properties!
Strange... Mind you: for
residue arithmetic this does not hold, at least: there are plenty
examples where exponent p does distribute over a sum, such as:
(x+y)^p == x^p + y^p mod p^k, for odd prime p and some non-trivial x,y.
(which is a clue to an approach for a direct FLT proof, see [1]).
My take on this 'syntax' approach is: analyse the additive properties
of multiplicative semigroup Z(.) mod m_k, for well chosen moduli m_k,
depending on the problem at hand. For FLT take m_k = p^k, because
p-th powers form a subsemigroup of units, and for Goldbach take
m_k = \product {first k primes P1 .. Pk}, which localizes all primes
(the terms in the conjecture statement) between Pk and m_k in the
lattice-top group of units. Then by induction over k the essential
step from 'local' (residues) to 'global' (integers) must be made,
using the obtained info mod m_k, and the fact that 'carry-free'
arithmetic (carry=0) is integer arithmetic...
Notice the essential differences between FLT and Goldbach:
FLT is an inequality, a negative statement, saying what cannot be done:
x^p + y^p /neq z^p (the sum of two p-th powers is not a p-th power,
prime p>2). So it is of 'anti-closure' type [2], hence generative
(p-th powers under (+) generate new numbers, not of type p-th power)
Moreover, unlike Goldbach, it involves (any) ONE prime ...[3]
On the other hand, Goldbach is a positive statement or 'constructive':
a representation (additive decomposition) of any 2n = P1 + P2 (n>2)
in at least one (but often many) ways as a sum of two (possibly equal)
odd primes. Primes are the minimal set of generators of Z(.) and
it says that half of the positive integers are a sum of two of these.
This links (.) and (+) in a rather constructive way.
One can expect Goldbach to be more difficult to prove than FLT, as
constructive vs. a non-constructive, and involving prime pairs vs.
one prime. -- Just some musings of an amateur.
In essence: generalization of associative commutative arithmetic (+)(.)
to associative but non-commutative function composition (semigroups)
is required to get a handle on these basic problems, which are
not really purely of arithmetic type. [4]
Notice: Z(.) mod m is the semigroup of endo-morphisms of Z(+) mod m,
and: Z(^) mod m = Endo{Z(.) mod m}; should yield some insight...
-- NB - http://home.iae.nl/users/benschop/scimat98.htm [1]
http://home.iae.nl/users/benschop/anti-cl.htm [2]
http://home.iae.nl/users/benschop/fewago.htm [3]
http://home.iae.nl/users/benschop/func.htm [4]
Phil Carmody
> Simon Fitzpatrick wrote:
>
> > Stan Ethenoz wrote:
> > >
> > > I recently got really interested in number theory,
> > > One of the aspects that fascinates me is the incredible amount of unsolved
> > > problems in this field that seems so trivial at first look.
> > >
> > > I read that what most mathematicians think is one of the hardest problems to
> > > solve ever, is the Goldbach conjecture, that says that
> > > every even number greater than 4 can be written as the sum of two primes.
> > >
> > > What in your opinion makes this so hard to prove (or unprove)?
> > > What makes it harder to prove than the other famous conjectures in number
> > > theory?
> >
> > It is no harder nor easier to prove than any other unsolved problem until
> > you know what the proof is... and you don't. If there is one.
> >
> > A counterexample to Goldbach would be a very short paper!
>
> Doubtful. After all, I'm sure super-computers already tested the
> correctness of the Goldbach conjecture up to very large numbers.
> So, a proof containing a counter example, which is yet a larger
> number would not be too short ;-)
As someone who deals with >100000 digit numbers every day, I'd hardly
say that 10^14 was 'very large'.
The ternary conjecture has been verified to 10^20 or so IIRC.
(Mets\"ankyl\"a?)
i.e. there _could_ be a 15 digit counterexample.
One could argue that the proof would include all the composite
subtrahends for each prime, and proof of compositeness. However, as
verification of those computations is probably _harder_ in work-load
than simply verifying based on just the 15-digit number, there's no need
for anything apart from the 15 digits.
Phil
Phil
Phil Carmody
> Stan Ethenoz wrote:
> >
> > I recently got really interested in number theory,
> > One of the aspects that fascinates me is the incredible amount of
> > unsolved problems in this field that seems so trivial at first look.
> >
> > I read that what most mathematicians think is one of the hardest
> > problems to solve ever, is the Goldbach conjecture, that says that
> > every even number greater than 4 can be written as the sum of two
> > primes.
> >
> > What in your opinion makes this so hard to prove (or unprove)? ..[*]
> > What makes it harder to prove than the other famous conjectures in
> > number theory?
>
> Like FLT, Goldbach's conjecture is an additive statement about
> 'multiplicative' numbers, roughly meaning: the operations (.) and (^)
> define the terms of the (+) statement.
Was that an attempt to show off at how clever you are?
If the rest of it was as pithless at the above 3 lines, then I don't
think you'll have impressed _anyone_.
Phil
David C. Ullrich
So the paper would be very short _if_ the number was written
in base 10...
>Phil
>Phil
David C. Ullrich
Nico Benschop
> then I don't think you'll have impressed _anyone_. -- Phil
What prevents you from reading further? Are you really so trigger
happy, or do you actually have some other problem?
If you don't mind, I was just trying to get at the essential
question [*], which I've been involved with for quite some time.
-- NB
Robin Chapman
news:3C765499...@trshp.ntc.nokia.com...
> Nico Benschop wrote:
> >
> > Like FLT, Goldbach's conjecture is an additive statement about
> > 'multiplicative' numbers, roughly meaning: the operations (.) and
(^)
> > define the terms of the (+) statement.
> [SNIP]
>
> Was that an attempt to show off at how clever you are?
>
> If the rest of it was as pithless at the above 3 lines, then I don't
> think you'll have impressed _anyone_.
Yup, Mr Benschop was definitely taking the pith.
Robin Chapman
--
Posted via Mailgate.ORG Server - http://www.Mailgate.ORG
Phil Carmody
> >> Simon Fitzpatrick wrote:
> >> > A counterexample to Goldbach would be a very short paper!
> Phil Carmody wrote:
> >i.e. there _could_ be a 15 digit counterexample.
> So the paper would be very short _if_ the number was written
> in base 10...
Assuming the calculation is done on computer, the code might be
programmed to output the number in hex, which could be even shorter. All
hypothetically, of course, being a non-believer in the existence of a
counterexample.
I was wrong, it was the other famous Finnish brute-forcer searcher
Sinisalo who took the verification to 4e11. J\"org Richstein who took it
to 4e14, but since then 3·10^15 has been reached by Tomás Oliveira e
Silva
http://www.ieeta.pt/~tos/goldbach.html . Apologies for my poor memory.
Phil
d...@cmtzone.net
<ge...@mpce.mq.edu.au> wrote:
>Prime were made to be multiplied, not added.
Heh.
>=> What makes it harder to prove than the other famous conjectures in
>=> number theory?
>
>Is it? Is it harder to prove than the existence of infinitley many
>perfect numbers? or the non-existence of odd perfect numbers? or
>the existence of infinitely many twin primes?
I think proving the existence of infinitely many twin primes is
easier.
>I guess it's harder to prove than the Wiles-Taylor Theorem (formerly
>known as Fermat's Last Theorem), but I don't think we knew that until
>we got the Wiles-Taylor proof.
Another (of the unlimited) curiosity of mine is if there were (before
1994 or so) a huge number of mathematicians that believed that FLT was
unprovable or would never be proven.
-don
>
>Gerry Myerson (ge...@mpce.mq.edu.au)
Phil Carmody
> On Fri, 22 Feb 2002 17:06:15 +1000, Gerry Myerson
> <ge...@mpce.mq.edu.au> wrote:
>
> >Prime were made to be multiplied, not added.
>
> Heh.
>
> >=> What makes it harder to prove than the other famous conjectures in
> >=> number theory?
> >
> >Is it? Is it harder to prove than the existence of infinitley many
> >perfect numbers? or the non-existence of odd perfect numbers? or
> >the existence of infinitely many twin primes?
>
> I think proving the existence of infinitely many twin primes is
> easier.
I think that Goldbach's is more statistically convincing than the Twin
Prime conjecture.
From my experience, if I may blow my own trumpet, you do get pretty much
as many twin primes in a region as you would expect (and I genuinely
believe it's true).
However, the number of coincidences involved in a Goldbach
counter-example is Pi(N)/2. The number of coincidences required for a
Twin Prime is Pi(sqrt(N)), vastly fewer, and each one is more likely.
Gut feel - I'd put Goldbach as less likely than the infinitude of
Mersenne primes too, again by a large margin.
Note, however, that maths does not obey probabilities, it's its own
master.
> >I guess it's harder to prove than the Wiles-Taylor Theorem (formerly
> >known as Fermat's Last Theorem), but I don't think we knew that until
> >we got the Wiles-Taylor proof.
>
> Another (of the unlimited) curiosity of mine is if there were (before
> 1994 or so) a huge number of mathematicians that believed that FLT was
> unprovable or would never be proven.
The world is full of pessimists. I look forward to as many of the great
hypotheses as possible being proved within my lifetime. Or counter
examples found. Either result is fantastic.
Phil
d...@cmtzone.net
<sur...@trshp.ntc.nokia.com> wrote:
>d...@cmtzone.net wrote:
>> I think proving the existence of infinitely many twin primes is
>> easier.
>
>I think that Goldbach's is more statistically convincing than the Twin
>Prime conjecture.
[------begin delusional rant------]
I know what you are saying. The only reason I stated the above is
that I hope to soon join the ranks of illustrious mathematical cranks
by posting a proof attempt of the above. The reason I said what I did
about the Twin Prime conjecture is that I think have reason to believe
I've noticed something that, if I had the necessary mathematical
background, would let me prove it [Twin Prime conjecture]. Of course,
lacking said background, I quite readily accept that "my eyes could be
playing tricks on me." I'm messing w/ RH, Twin Prime, Goldbach, and
the possibility of a general factoring algorithm. They are (and I'm
sure there are others) obviously related quite intimately, and while I
think that what I noticed applies to all, Twin Primes seems to be the
simplest, as I think I can "see" how to get the proof with a lot less
math knowledge than it would take to get the others. Right now, I'm
just trying to play math "catchup." I put my business on hold to take
off for 6 months, rented an apartment that no family/friends know
about (close to the college math library, of course), and currently
have math books that I don't understand strewn from one end to the
other. See? I have all the makings of a proper nutcase. ;-) (But
mainly, I'm just having fun.)
[------end delusional rant------]
>Note, however, that maths does not obey probabilities, it's its own
Neither do -some- cranks. ;-)
>The world is full of pessimists. I look forward to as many of the great
>hypotheses as possible being proved within my lifetime. Or counter
>examples found. Either result is fantastic.
Indeed. I'm taking heart in my new favorite quote by
ste...@nomail.msu.edu in the thread "P & NP (was Re: If at first you
don't succeed...)":
"It is much easier to demonstrate that something
can be done than to demonstrate that it cannot be done."
And I also agree that you will see a great many of the great
hypotheses proven (one way or the other) in the next few years.
-don
>
>Phil
Stan Ethenoz
Even though I don't have the background necessary to follow your reasoning,
your reply was interesting to read.
I wonder why so many people are verifying Goldbach in search of a
counterexample, instead of spending time trying to prove it. Perhaps a lack
of faith. Anyways the mathematics department at my school is not very great,
but if I were to relive my live, maybe I would have changed my major to
math.
___
Stan
Jan Kristian Haugland
On Sat, 23 Feb 2002, Stan Ethenoz wrote:
> I wonder why so many people are verifying Goldbach in search of a
> counterexample, instead of spending time trying to prove it. Perhaps a lack
> of faith.
Say what?? Because there is no proof, people haven't spent time
trying to prove it? Goodness! There has been plenty of work on
Goldbach's conjecture other than numerical searching, FYI.
> Anyways the mathematics department at my school is not very great,
> but if I were to relive my live, maybe I would have changed my major to
> math.
And then maybe you wouldn't talk about things that you don't
have any knowledge about.
--
J K Haugland
http://home.hia.no/~jkhaug00
Phil Carmody
> On Sat, 23 Feb 2002, Stan Ethenoz wrote:
> > I wonder why so many people are verifying Goldbach in search of a
> > counterexample, instead of spending time trying to prove it. Perhaps a lack
> > of faith.
>
> Say what?? Because there is no proof, people haven't spent time
> trying to prove it? Goodness! There has been plenty of work on
> Goldbach's conjecture other than numerical searching, FYI.
What we see is a classic case of "I've never heard of Chen Or
Vinogradov"-syndrome.
This is a crime against primality, but fortunately it's curable.
> > Anyways the mathematics department at my school is not very great,
> > but if I were to relive my live, maybe I would have changed my major to
> > math.
>
> And then maybe you wouldn't talk about things that you don't
> have any knowledge about.
Anyway Jan, let's talk some more about the ancient sacred secret martial
art of Bonsai tree throwing. I hear Norway doesn't have a Bonsai tree
throwing team in this year's Autumn Olympics.
Phil
Jan Kristian Haugland
Phil Carmody wrote:
> Jan Kristian Haugland wrote:
(...)
> > And then maybe you wouldn't talk about things that you don't
> > have any knowledge about.
>
> Anyway Jan, let's talk some more about the ancient sacred secret martial
> art of Bonsai tree throwing. I hear Norway doesn't have a Bonsai tree
> throwing team in this year's Autumn Olympics.
Don't be too sure. Having just won the
men's curling, we're up for anything.
John R Ramsden
>
> Thanks a lot for the effort you put into answering my question Nico
> Even though I don't have the background necessary to follow your
> reasoning, your reply was interesting to read.
Don't worry Stan, even professional mathematicians find it a challenge
to follow Nico's reasoning.
> I wonder why so many people are verifying Goldbach in search of a
> counterexample, instead of spending time trying to prove it.
As well as searching for counterexamples, I guess the calculations
are often intended to yield relevant data, such as the _number_ of
representations of even integers as the sum of two primes, which
might show trends or patterns and thus offer clues as to how the
conjecture could be proved.
Cheers
---------------------------------------------------------------------------
John R Ramsden (j...@adslate.com)
---------------------------------------------------------------------------
Life is what happens to you while you're planning your next move.
Mick Jagger
---------------------------------------------------------------------------
Chan-Ho Suh
"Simon Fitzpatrick" <fitz...@maths.uwa.edu.au> wrote in message
news:3C75E3ED...@maths.uwa.edu.au...
Why is that? And why is it that everyone always seems to assume that a
disproof of Goldbach will give you a particular counterexample? Is it not
possible that a disproof will only prove the existence of a counterexample
(by contradiction, say) without a clear means of constructing such? Please
somebody enlighten me!
Dann Corbit
news:a5f24e$4fq$1...@siamese.noc.ucla.edu...
>
> "Simon Fitzpatrick" <fitz...@maths.uwa.edu.au> wrote in message
> news:3C75E3ED...@maths.uwa.edu.au...
> >
> >
> > Stan Ethenoz wrote:
> > >
> > > I recently got really interested in number theory,
> > > One of the aspects that fascinates me is the incredible amount of
> unsolved
> > > problems in this field that seems so trivial at first look.
> > >
> > > I read that what most mathematicians think is one of the hardest
> problems to
> > > solve ever, is the Goldbach conjecture, that says that
> > > every even number greater than 4 can be written as the sum of two
> primes.
> > >
> > > What in your opinion makes this so hard to prove (or unprove)?
> > > What makes it harder to prove than the other famous conjectures in
> number
> > > theory?
> >
> > It is no harder nor easier to prove than any other unsolved problem
until
> > you know what the proof is... and you don't. If there is one.
> >
> > A counterexample to Goldbach would be a very short paper!
> >
>
> Why is that?
A single counterexample would disprove it. Presumably, it woudn't be
trillions of digits, or it would have a compact enough format to be
displayed on a few pages or less.
> And why is it that everyone always seems to assume that a
> disproof of Goldbach will give you a particular counterexample?
I don't know if anyone assumed that. Simon Fitzpatrick said "A
counterexample to Goldbach would be a very short paper!"
which is likely, if it were true.
> Is it not
> possible that a disproof will only prove the existence of a counterexample
> (by contradiction, say) without a clear means of constructing such?
Please
> somebody enlighten me!
There are many possible ways to disprove it (assuming that it is not true).
One way would be a counterexample. Only one counterexample is needed to
invalidate GC. Of course, it might also take a billion pages of incredibly
complicated stuff, with no counterexample ever demonstrated.
--
C-FAQ: http://www.eskimo.com/~scs/C-faq/top.html
"The C-FAQ Book" ISBN 0-201-84519-9
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Chan-Ho Suh
"Dann Corbit" <dco...@connx.com> wrote in message
news:a5f2u...@enews2.newsguy.com...
Nope, I think you missed my point. Perhaps it's possible to define a very
weird counterexample, e.g. so that one could *not* tell what number it
actually was (in a base, say). Such a number (or class of numbers) could be
defined in some very elegant but non-transparent way.
The point is that a counterexample is an instance of the proposed conjecture
being falsified. One may not really know that much about such
counterexample, due to a complicated construction. It has often occurred in
mathematics that one constructs a counterexample, but the construction is so
unwieldy in a practical sense so that one cannot actually
compute/calculuate/visualize it.
> > And why is it that everyone always seems to assume that a
> > disproof of Goldbach will give you a particular counterexample?
>
> I don't know if anyone assumed that. Simon Fitzpatrick said "A
> counterexample to Goldbach would be a very short paper!"
> which is likely, if it were true.
>
Well, I guess I was reading my experience with other Goldbach threads into
this mix. But I'm still skeptical on a counterexample leading to a 'short'
paper.
> > Is it not
> > possible that a disproof will only prove the existence of a
counterexample
> > (by contradiction, say) without a clear means of constructing such?
> Please
> > somebody enlighten me!
>
> There are many possible ways to disprove it (assuming that it is not
true).
> One way would be a counterexample. Only one counterexample is needed to
> invalidate GC. Of course, it might also take a billion pages of
incredibly
> complicated stuff, with no counterexample ever demonstrated.
Or a counterexample may be demonstrated with a billion pages of incredibly
complicated stuff.
Christian Bau
>
> I don't know if anyone assumed that. Simon Fitzpatrick said "A
> counterexample to Goldbach would be a very short paper!"
> which is likely, if it were true.
I think it has been proven that there is no counterexample less than
10^16 or so. Lets assume for the sake of the argument that some n >
10^16 is not the sum of two primes. To prove this, I would have to show
that n-3 is composite, n-5 is composite, n-7 is composite, n-11 is
composite, n-13 is composite, n-17 is composite etc. All in all, that
would be a very lengthy proof!
Dann Corbit
news:3C7C1CBD...@cbau.freeserve.co.uk...
Three prime certificates delivered by ECPP should suffice.
Franz Fritsche
> Well, I guess I was reading my experience with other Goldbach threads into
> this mix. But I'm still skeptical on a counterexample leading to a 'short'
> paper.
>
Well, I would say the main point is that we can not rule out the
possibility of this based on the knowledge of the problem we have
today.
Of course it's your "right" to be skeptical...
if there actually is one that can be described on some pages [maybe
with usage of Knuth's arrow notation ] - and/ore there is a
possibility to construct it (or give a construction for it) within
some pages - well, then some pages are enough. If not, not.
:-)
F.
But personally I think Goldbach's conjecture is actually TRUE : the
_only_ problem is to show / proof this fact!
Dann Corbit
> "Christian Bau" <christ...@cbau.freeserve.co.uk> wrote in message
> news:3C7C1CBD...@cbau.freeserve.co.uk...
> > Dann Corbit wrote:
> > >
> > > I don't know if anyone assumed that. Simon Fitzpatrick said "A
> > > counterexample to Goldbach would be a very short paper!"
> > > which is likely, if it were true.
> >
> > I think it has been proven that there is no counterexample less than
> > 10^16 or so. Lets assume for the sake of the argument that some n >
> > 10^16 is not the sum of two primes. To prove this, I would have to show
> > that n-3 is composite, n-5 is composite, n-7 is composite, n-11 is
> > composite, n-13 is composite, n-17 is composite etc. All in all, that
> > would be a very lengthy proof!
>
> Three prime certificates delivered by ECPP should suffice.
ACK! I just realized I was thinking of FLT, not GC.
However, given the numbers that fail, I think it could be verified
programmatically and the program used for the calculations given as proof.
Then, if the program is verified as correct, we can assume the proof to be
true.
Franz Fritsche
> However, given the numbers that fail, I think it could be verified
> programmatically
>
so... ]
> and the program used for the calculations given as proof.
> Then, if the program is verified as correct, we can assume
> the proof to be true.
>
Yes. It's the same with the 4-color theorem.
F.
Chan-Ho Suh
"Franz Fritsche" <in...@simple-line.de> wrote in message
news:3c7c2a63...@news.t-online.de...
> >
> > Well, I guess I was reading my experience with other Goldbach threads
into
> > this mix. But I'm still skeptical on a counterexample leading to a
'short'
> > paper.
> >
>
> Well, I would say the main point is that we can not rule out the
> possibility of this based on the knowledge of the problem we have
> today.
>
Ok, that's what I was wondering. I have no specific knowledge of the
problem, but was just asking on 'general principle'. It's one thing to have
a counterexample, and another to actually know in some constructive way what
the counterexample 'looks' like.
> Of course it's your "right" to be skeptical...
>
> if there actually is one that can be described on some pages [maybe
> with usage of Knuth's arrow notation ] - and/ore there is a
> possibility to construct it (or give a construction for it) within
> some pages - well, then some pages are enough. If not, not.
>
> :-)
>
> F.
>
>
> But personally I think Goldbach's conjecture is actually TRUE : the
> _only_ problem is to show / proof this fact!
>
It seems weird if it's false, but maybe that's only because I've only got 20
fingers and toes.
Herman Rubin
Not necessarily. You can eliminate n-pk for all k>1; this reduces
the work substantially. So if n is congruent to 1 mod 3, n-7 and
n-13 and n-19, etc., do not need testing, and if n is congruent to
2 mod 3, n-5 and n-11 and n-17, etc., do not need testing, unless
they are specifically 3.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Franz Fritsche
>> But personally I think Goldbach's conjecture is actually TRUE : the
>> _only_ problem is to show / proof this fact!
>>
>
> It seems weird if it's false, but maybe that's only because I've only got 20
> fingers and toes.
>
I see... But I think there are some arguments independent of such
arbitrary properties connected with our body or our capability to
grasp something... ( I think )
For me the probability argument makes sense...: It can be shown that
the probability that a number does not have a "Goldbach partition" is
getting smaller and smaller with bigger numbers.
( But as long as we don't have a proof that proves this probability to
be 1.0 we are not done... )
F.
Nico Benschop
>
> "Stan Ethenoz" <stanlesssteel$$NO_SPAM#%@hotmail.com> wrote:
> >
> > Thanks a lot for the effort you put into answering my question Nico
> > Even though I don't have the background necessary to follow your
> > reasoning, your reply was interesting to read.
>
> Don't worry Stan, even professional mathematicians find it a
>
> > I wonder why so many people are verifying Goldbach in search of a
> > counterexample, instead of spending time trying to prove it.
>
> As well as searching for counterexamples, I guess the calculations
> are often intended to yield relevant data, such as the _number_ of
> representations of even integers as the sum of two primes, which
> might show trends or patterns and thus offer clues as to how the
> conjecture could be proved.
Re[*]:
Well, notice that for residues mod m (any natural modulus m>1):
Z_m(.) =~ Endo{Z_m(+)} , where =~ stands for 'isomorphic to',
with Z_m(+) a cyclic group of order
m,
and of course |Z_m(.)| = |Z_m(+)| = m.
The next questions/suggestions should not be beyond you (assuming
you *are* a mathematician), re operations (+) , (.) , (^), etc.:
1. With (^) distributing over (.), in Z mod m:
Let Z_m(^) =~ Endo{Z_m(.)}
Is then again |Z_m(^)| = |Z_m(.)| = m , for any modulus m>1 ?
2. Does repeated 'taking the endomorphisms' Endo*{Z_m(+)},
yield a presevervation of semigroup order (= m) at each level?
(producing an ordered sequence of 'higher level' finite semigroups,
and corresponding operations, necessarily for finite mod m ending
in an idempotent semigroup, since Endo() reduces cyclic subgroup
orders, vz. cycle lengths)
Just curious about such 'Endo tower',
built on finite cyclic C_m =~ Z(+) mod m.
Nico Benschop
>
> Thanks a lot for the effort you put into answering my question, Nico.
You're welcome. I try to describe (& formalize) my intuition.
I see a math-proof as
"decomposing one's intuition into a sequence of small sure steps",
and in trying to do this, the intuition is either verified, or
needs adjustment - the main line of approach coming from a 'feeling'
or 'intuition', possibly via experience in other fields of research
(for me: FSM synthesis/decomposition theory, finite semigroups, etc.)
> Even though I don't have the background necessary to follow your
> reasoning, your reply was interesting to read.
> I wonder why so many people are verifying Goldbach in search of
> a counterexample, instead of spending time trying to prove it.
> Perhaps a lack of faith.
Well, computational evidence by now favours the GC to be true. But
this did not come for free, and the structure of all 2n = P1 + P2
for fixed n might give a clue to a general proof. There are usually
'many roads leading to Rome'...
> Anyways the mathematics department at my school is not very great,
> but if I were to relive my live, maybe I would have changed my major
> to math. ___ Stan
A bit of self study can't do much harm...
My favourite:
E.T.Bell "The Development of Mathematics" (up to 1945;-)
- a nice combination of overview/big_line and of details.
"It is remarkable that curiosity has survived formal education."
(roughly quoted, from A. Einstein)
-- NB - http://home.iae.nl/users/benschop with site map on:
http://www.freefind.com/servlet/freefind?id=6078280&map=0&page=0