Against the term "Beal Conjecture".
Hull Loss Incident
Fred W. Helenius wrote:
> the statement that there are no solutions
> is called the Beal Conjecture, after an American banker
> who is offering a $75,000 prize for a proof or disproof.
It's pretty offensive to call it the Beal Conjecture.
Beal was not first to state the conjecture. It was well-known,
as an instance of the also well-known and more natural
conjecture with the sum of reciprocals of the exponents
less than 1 (rather than all exponents > 2). Beal made no
intellectual contribution to the problem.
Nobody calls FLT the Wolfskehl Conjecture. The prize
competitions that attracted seminal essays by Poincare (3-body),
Minkowski (quadratic forms), and others are forgotten today.
Paul Erdos' problems are known for their bounties, but his
1500 papers may also have something to do with it...
Apparently, Beal wants in on the action of FLT publicity
and posterity -- for a pittance of $75000. The AMS Notices
article about the prize was shilling for a donor, in effect a
bought and paid for piece of advertising. $75K wouldn't
even pay one year's worth of an endowed professorship; why
then should official organs of the mathematical academia start bowing
as soon as some financier throws pennies in their direction?
As an anti-Beal prize, one could offer $75 for the best or most
historically accurate name to replace the shameless "Beal conjecture".
Richard Carr
:Date: Tue, 08 Aug 2000 00:33:33 -0400
:From: Hull Loss Incident <n...@ie.eg.com>
:Newsgroups: sci.math
:Subject: Against the term "Beal Conjecture".
:
:(reposted from the other thread, with a more descriptive title.)
:
:Fred W. Helenius wrote:
:
:> the statement that there are no solutions
:> is called the Beal Conjecture, after an American banker
:> who is offering a $75,000 prize for a proof or disproof.
:
:It's pretty offensive to call it the Beal Conjecture.
:
:Beal was not first to state the conjecture. It was well-known,
:as an instance of the also well-known and more natural
:conjecture with the sum of reciprocals of the exponents
:less than 1 (rather than all exponents > 2). Beal made no
:intellectual contribution to the problem.
:
:Nobody calls FLT the Wolfskehl Conjecture. The prize
Less people were motivated by money in those days. Probably, if he wanted
to Bill Gates could offer a bigger prize and have it renamed the Gates
Conjecture. It should be no surprise. I often feel that I am not at an
academic institution at all but a business. (You can't fail the student
because their parents are paying a lot of money. If their grades aren't up
to scratch/sufficient for graduation, just change them etc. are the sort
of things that come to mind.) With a system like this, it is no wonder the
"Beal conjecture" can be so called. Of course, not so many people complain
about l'H^opital's rule (although some do) and numerous other results (and
presumably conjectures).
:competitions that attracted seminal essays by Poincare (3-body),
:Minkowski (quadratic forms), and others are forgotten today.
:Paul Erdos' problems are known for their bounties, but his
:1500 papers may also have something to do with it...
:
:Apparently, Beal wants in on the action of FLT publicity
:and posterity -- for a pittance of $75000. The AMS Notices
:article about the prize was shilling for a donor, in effect a
:bought and paid for piece of advertising. $75K wouldn't
:even pay one year's worth of an endowed professorship; why
:then should official organs of the mathematical academia start bowing
:as soon as some financier throws pennies in their direction?
:
:As an anti-Beal prize, one could offer $75 for the best or most
:historically accurate name to replace the shameless "Beal conjecture".
:
OK, what about the "Carr conjecture"? (Just kidding.)
Hull Loss Incident
>
> :It's pretty offensive to call it the Beal Conjecture.
> :
> :Beal was not first to state the conjecture. [...]
> :Beal made no intellectual contribution to the problem.
> :
> :Nobody calls FLT the Wolfskehl Conjecture. The prize
>
> Less people were motivated by money in those days. Probably, if he wanted
> to Bill Gates could offer a bigger prize and have it renamed the Gates
> Conjecture.
Other millionaires offer bigger prizes without such renaming.
A Mr Clay (financier benefactor of the Harvard math dept) recently
offered one $million each for solutions of the Riemann, Hodge, Poincare,
and Birch&Swinnerton-Dyer conjectures. Another tycoon funds the
American Institute of Mathematics, also aiming at the Riemann hypothesis.
For that matter Bill Gates' share of Microsoft Theory Group (now throwing
money at P=NP) must be in the millions. None of the problems are being
renamed in light of the funding.
It's hard to overstate the egomania involved in laying claim to the "Beal"
conjecture. Mr Beal wants a conjecture bigger and better (more general!)
than FLT attached to his name. So he decided to buy a spot in mathematical
namespace for a pittance of $75000. And the AMS cooperated!!
As an exercise, after how many months of work would the 75K reward
drop below the legal minimum wage?
> It should be no surprise. I often feel that I am not at an
> academic institution at all but a business. (You can't fail the student
> because their parents are paying a lot of money. If their grades aren't up
> to scratch/sufficient for graduation, just change them etc. are the sort
> of things that come to mind.) With a system like this, it is no wonder the
> "Beal conjecture" can be so called.
Universities sell advertising for cheap (names on buildings etc). Selling
the actual "intellectual property" of mathematical conjectures is not too far
off. Still, it is amazing how the AMS bows and scrapes when a paltry
$75000 is dangled before them by some banker.
> Of course, not so many people complain
> about l'H^opital's rule (although some do) and numerous other results (and
> presumably conjectures).
L'Hopital's rule was supposedly bought. Are you suggesting there are
numerous other results, whose attribution was paid for? It seems to
me that calling it "Beal conjecture" violates all the usual conventions for
naming, and even misnaming, theorems and conjectures.
Richard Carr
:Date: Tue, 08 Aug 2000 19:34:52 -0400
:From: Hull Loss Incident <n...@ie.eg.com>
:Newsgroups: sci.math
:Subject: Re: Against the term "Beal Conjecture".
:
:Richard Carr wrote:
:
:>
:> :It's pretty offensive to call it the Beal Conjecture.
:> :
:> :Beal was not first to state the conjecture. [...]
:> :Beal made no intellectual contribution to the problem.
:> :
:> :Nobody calls FLT the Wolfskehl Conjecture. The prize
:>
:> Less people were motivated by money in those days. Probably, if he wanted
:> to Bill Gates could offer a bigger prize and have it renamed the Gates
:> Conjecture.
:
:Other millionaires offer bigger prizes without such renaming.
:A Mr Clay (financier benefactor of the Harvard math dept) recently
:offered one $million each for solutions of the Riemann, Hodge, Poincare,
:and Birch&Swinnerton-Dyer conjectures. Another tycoon funds the
:American Institute of Mathematics, also aiming at the Riemann hypothesis.
:For that matter Bill Gates' share of Microsoft Theory Group (now throwing
:money at P=NP) must be in the millions. None of the problems are being
:renamed in light of the funding.
If Bill Gates wanted the problem named after him though...- also the
amount of literature on P=NP is probably large enough such that changing
the name would be a problem- not so with the "Beal conjecture". Probably
Gates could get a problem named after him if it had been worked on less.
:
:It's hard to overstate the egomania involved in laying claim to the "Beal"
:
:
:
:
Hull Loss Incident
> :> Less people were motivated by money in those days. Probably, if he wanted
> :> to Bill Gates could offer a bigger prize and have it renamed the Gates
> :> Conjecture.
> :
> :Other millionaires offer bigger prizes without such renaming.
> :A Mr Clay (financier benefactor of the Harvard math dept) recently
> :offered one $million each for solutions of the Riemann, Hodge, Poincare,
> :and Birch&Swinnerton-Dyer conjectures. Another tycoon funds the
> :American Institute of Mathematics, also aiming at the Riemann hypothesis.
> :For that matter Bill Gates' share of Microsoft Theory Group (now throwing
> :money at P=NP) must be in the millions. None of the problems are being
> :renamed in light of the funding.
>
> If Bill Gates wanted the problem named after him though...- also the
> amount of literature on P=NP is probably large enough such that changing
> the name would be a problem- not so with the "Beal conjecture". Probably
> Gates could get a problem named after him if it had been worked on less.
Gates is a famous enough that any connection to him would raise the profile
of a problem not already famous within the profession (i.e. less well known
to mathematicians than Gates himself). So he may be in a special position
as far as mathematical namespace acquisition is concerned. Also, if Gates
were to make the acquisition through a donation it would have to be much
more than $75000, it would look ridiculous otherwise.
Beal on the other hand was at least as obscure as the problem he
tried to acquire, which was the point of his prize: to leverage FLT
publicity and posterity in his favor.
andy...@my-deja.com
Hull Loss Incident
offers the prize, judging from his comments in the other thread
on "generalization of FLT".]
andy...@my-deja.com wrote:
> [...replica of two other postings...]
Deja News seems to have mangled your posting, only the
quoted text of the previous message made it through.
Anyway, here is a copy of the MathSciNet review of the
article in AMS Notices on the "Beal conjecture", containing
some comments on the naming:
98j:11020 11D41
Mauldin, R. Daniel(1-NTXS)
A generalization of Fermat's last theorem: the Beal conjecture and prize
problem.
Notices Amer. Math. Soc. 44 (1997), no. 11, 1436--1437.
This note announces the award of a substantial monetary prize, fixed at
$$50,000$ since the article under review was written, to any person who
provides a solution
to the "Beal conjecture", stated as the following: Let $A,B,C,x,y,z$ be
positive integers with $x,y,z>2$. If (1) $A\sp x+B\sp y=C\sp z$, then
$A,B,C$ have a
nontrivial common factor. The story of this conjecture is an interesting
one, and told at slightly greater length in the author's follow-up
letter [Notices Amer. Math.
Soc. 45 (1998), no. 3, 359]. See also the article by K. J. Devlin [Math.
Horizons 1998, Feb., 8--10; per revr.]. Andrew Beal is a successful
Texas businessman,
with enthusiasm for number theory. He has had a particular interest in
Fermat and his methods following the announcement in 1993 of Andrew
Wiles' work on
Fermat's last theorem, and formulated this conjecture after several
years of computer-based study. So often the amateur number-theorist
turns out to be a
well-intentioned crank; what is remarkable here is how close the stated
problem is to current research activity by leaders in the field. In
fact, the problem is essentially
many decades old, and apparently V. Brun [Arch. Math. Nat. 34 (1914),
no. 2, 1--14; JFM 45.1219.13] asked many similar questions. The
formulation in the
1980s by Masser, Oesterle and Szpiro of the $abc$-conjecture has had
great influence on the discipline of number theory, and in fact a
corollary of the
$abc$-conjecture is that there are no solutions to the Beal Prize
problem when the exponents are sufficiently large. The prize problem
itself was implicitly posed by
Andrew Granville in the Unsolved Problems section of the West Coast
Number Theory Meeting, Asilomar, 1993 ("Find examples of $x\sp p + y\sp
q = z\sp r$ with
$1/p + 1/q + 1/r < 1$ other than $2\sp 3 + 1\sp 7 = 3\sp 2$ and $7\sp 3
+ 13\sp 2 = 2\sp 9$"), and was stated and discussed in A. van der
Poorten's book [Notes
on Fermat's last theorem, Wiley, New York, 1996; MR 98c:11026]. The
resolution by Wiles of Fermat's last theorem disposed of a special case
of the prize
problem; and H. Darmon and Granville [Bull. London Math. Soc. 27 (1995),
no. 6, 513--543; MR 96e:11042] proved the deep result that if
$1/x+1/y+1/z < 1$
then there can only be finitely many triples of coprime integers $A,B,C$
satisfying $A\sp x+B\sp y=C\sp z$ (ten solutions are known). Recently,
Darmon and L.
Merel [J. Reine Angew. Math. 490 (1997), 81--100; MR 98h:11076] showed
that there can exist no coprime solutions to (1) with the exponents
$(x,x,3), x \geq
3$.
There has been some feeling expressed by number-theorists that the
conjecture should best be referred to as the "Beal Prize problem",
though there is no doubt that
Beal independently arrived at and formulated the conjecture without
knowledge of the current literature. With a nod to T. S. Eliot, the
matter of naming conjectures
can be as difficult as the naming of cats.
Reviewed by Andrew
Bremner
andy...@my-deja.com
Hull Loss Incident <x...@y.z.com> wrote:
Who are you "hull loss incident"??
Yes it is unfortunate that Andrew Granvile misrepresented the history
of the problem to Dan Mauldin who authored the following article that
you have referenced. Mauldin correspondingly included the erroneous
references to similiar historical problems. Unfortunately those of us
not so lucky to have the insight and knowledge that "hull loss
incident" has must simply rely on the leading number theory texts and
corresponence with mathematicians such as Harold Edwards from NYU and
Earl Taft from Rutgers. While there are certaintly many similiar
diophantine forms, none of the texts and no-one I corresponded with
indicated any prior knowledge of the beal conjecture or the concepts
involved. the closest reference was to the ABC conjecture which
hypothesizes a finite number of solutions
In any event, the term beal conjecture results from my "purported"
discovery, not from my offer of a prize. Since the conjecture is so
trivial to you, why don't you ignore it?? And if so trivial, why is
there so much interest in the problem?? Perhaps to those of us out here
that are not as gifted as you, the problem is not quite so trivial??
Why don't you criticize my claim of discovery (and demonstrate
otherwise) instead of creating contempt for me by lying and saying that
i simply posted a prize.
who are you anyway?? what is your problem?? why don't you quit hiding
behind "hull loss incident" and come out and tell us who you are?? why
are you misreprenting my role here and attempting to create contempt
toward me?? If the problem is too trivial for you, so what, ignore it.
If I am a self promoter, so what?? if true, why does that bother you
so?? If I didn't discover it as I belive I did, why don't you
demonstrate why I am in error?? Instead you come out and post messages
misrepresenting my role and in general attempting to generate contempt
for me. What have I done to you that has offended you so??
I am confident that I discovered the beal conjecture. However, I have
been wrong before in my life and I am not shy about admiting an error,
so please simply demonstrate that anyone (besides you) already knew the
conjecture and I will happily concede error. but please - stop with the
lies. --andrew beal
----------------------------------------------------------------
Hull Loss Incident
> Yes it is unfortunate that Andrew Granvile misrepresented the history
> of the problem to Dan Mauldin who authored the following article that
> you have referenced. Mauldin correspondingly included the erroneous
> references to similiar historical problems.
That's quite a claim. What did Granville "misrepresent" to Mauldin,
exactly?
According to the review I quoted from Mathscinet (98j:11020), Granville
himself publicized (and published) by 1993 a problem that includes yours
as a special case. Even that was not original, i.e. he was apparently
drawing attention to an already well-known circle of ideas and saying that
this was a concrete problem that deserved work.
Granville by the way is well-versed in number theory and (I presume from
the contents of some of his articles) much of the relevant history.
I believe his advisor was Ribenboim who wrote numerous articles and
at least two books on FLT. Mauldin is not a number theorist (as far as
I know) so while he was right to take Granville's assertions on faith, he
may not be in a position to assess your work accurately.
> Unfortunately those of us
> not so lucky to have the insight and knowledge that "hull loss
> incident" has must simply rely on the leading number theory texts
There's quite a lot of knowledge in mathematics that is "out in
the community" as oral tradition or folklore. The fact that it may
be hard to find a textbook containing a statement of your particular
conjecture certainly does not mean it was unknown. I remember
as a student, hearing mathematicians mention in an offhand
"everyone knows it" sort of way that FLT was expected to be
true on simple probabilistic grounds for exponent 4 and higher,
and sketching exactly the argument that also leads to your
conjecture, Euler's problem on N'th powers, and many others.
Relying on number theory textbooks as a reference is fine.
It's an obvious mistake, though, to take what is NOT mentioned
in the books as a serious indication that something is unknown to the
number theory community. Textbooks lag behind current research,
they can't include more than a tiny fraction of what is known, and they
are often written by people who have time to write textbooks precisely
because they're not at the current edge of research.
As far as your specific problem is concerned, there is probably
a dearth of literature on it simply because nobody could prove
much before Wiles' breakthrough, and (because it was fairly
well known) just stating the conjecture would not be publishable.
I do expect that there was computational work published, on your
equations or similar ones, and that if you look at the publications they
will make statements from which your conjecture can be easily inferred.
The place to look would be a keyword search on Mathscinet
(AMS database of reviews of math papers).
> and corresponence with mathematicians such as Harold Edwards
> from NYU and Earl Taft from Rutgers.
Neither of these are people I would expect to be familiar with
current number theory or its literature. Taft is an algebraist who
as far as I know works on quantum groups, an area far
from most number theory. Edwards' publications are
studies and expositions of mathematics developed more than
100 years ago. If you want to learn the classical approaches
to Galois theory or algebraic number theory his
books are wonderful, but with all due respect he is not an
authority on FLT-related mathematics at the level of someone
like Granville, whom you now accuse of spreading misinformation.
I don't know where you are in Texas, but UT Austin has several
number theorists who are quite knowledgeable about these
questions. At least one of them posts to sci.math occasionally,
and of course you can usually get good answers by asking questions
in these newsgroups.
> While there are certaintly many similiar
> diophantine forms, none of the texts and no-one I corresponded with
> indicated any prior knowledge of the beal conjecture or the concepts
> involved. the closest reference was to the ABC conjecture which
> hypothesizes a finite number of solutions
I'm not sure what you consider the concepts involved. As I mentioned,
it has been known for a long time that if you have some equation between
sums of powers like X^a + Y^b - Z^c + 3*T^d - 57*W^e = 0,
what determines whether you expect many solutions or few/none
is if the sum 1/a + 1/b + 1/c + 1/d + 1/e is larger/smaller than 1.
You can also take into account calculations up to some
bound and make statements of the form, "if I check that there are
no solutions with arguments under 1000000, the expected number of
larger solutions is 0.003 thus we expect no such solutions".
As an accessible expository reference with the same kind of
probabilistic reasoning in number theory, see for example
P.T. Bateman, J Selfridge, S. Wagstaff
The new Mersenne conjecture
American Mathematical Monthly 96 (1989) no. 2 pp.125-128
> In any event, the term beal conjecture results from my "purported"
> discovery, not from my offer of a prize.
Conjectures in number theory are a dime a dozen.
If not for the prize and associated publicity, nobody would
have associated your name with the problem.
Maybe over time people who actually produced scholarship
and results in the direction of this conjecture, such as Darmon
and Granville, might have had their names attached to it as their
papers became standard references. But it would not have occured
to anyone to give you special credit for just posing the problem.
Unless the problem is some previously unconsidered type of statement,
you generally don't get credit without making substantial intellectual
contributions.
> Since the conjecture is so
> trivial to you, why don't you ignore it?? And if so trivial, why is
> there so much interest in the problem?? Perhaps to those of us out here
> that are not as gifted as you, the problem is not quite so trivial??
I see no evidence of "so much interest", and I did not trivialize
the mathematical problem. Formulating the problem is what's
trivial, given knowledge that has been standard in number theory
for many years.
> Why don't you criticize my claim of discovery (and demonstrate
> otherwise)
I did demonstrate otherwise. I can post the details of
the probabilistic argument (about 1/a+1/b+1/c) if you like, as can
several other people who posted to this discussion and innumerable
people who may be reading this. I am also confident that I could
find references predating your conjecture where such ideas are
spelled out in print.
> instead of creating contempt for me by lying and saying that
> i simply posted a prize.
What I said is that you made no intellectual contribution to the
problem, which is apparently true. Intellectual contribution means
things that advance the community's knowledge, such as
publication of relevant research or expository articles, teaching
activity, or similar. I asked and received no answer as to what
you have done in that direction. Privately circulating a manuscript
and then seeking recognition for "significant discoveries" is
egomania, not contribution.
> why are you misreprenting my role here and attempting to create
> contempt toward me??
I expressed contempt for the idea of buying mathematical
"namespace" for the paltry sum of $75000. If you want to do
something more positive for mathematics (even just the part
related to your conjecture) with your money, I can certainly
make suggestions.
> I am confident that I discovered the beal conjecture.
As stated in the Mathscinet review, there is no dispute that you
arrived at the conjecture on your own without knowledge of
the number theory literature. I do dispute your claims of
priority and the claim that the conjecture was a "significant
discovery" for mathematics.
Christian Bau
> Relying on number theory textbooks as a reference is fine.
> It's an obvious mistake, though, to take what is NOT mentioned
> in the books as a serious indication that something is unknown to the
> number theory community. Textbooks lag behind current research,
> they can't include more than a tiny fraction of what is known, and they
> are often written by people who have time to write textbooks precisely
> because they're not at the current edge of research.
In number theory, there are just so many things that are quite plausible
conjectures... I think every student of mathematics should be able to come
up easily with half a dozen conjectures that his professor cannot prove to
be wrong, or find in the literature within one day. Of course, the reason
is that most of these conjectures will not be considered worth writing
down.
andy...@my-deja.com
is researcher.
The fact is that I appear to have been the first to reason that a^x
+b^y = c^z might be impossible with co-prime bases, impirically test a
reasonable range of variables, and announce the conjecture.
The fact is that Andrew Granville said publicly in 1998 that he doubted
the conjecture was true, hardly consistent with it being commonm
knowledge.
The fact is that Vanderpoorten claimed credit in his 1997 book
for "proposing" the statement, hardly consistent with it being common
knowledge.
The fact is that Bremners "review" below is bullshit and begs the
question of who inspired the review and for what purposes. The reality
is that there are so few number theorists out there that few are able
to see through bremners "review". The reasoned conclusion that co-prime
bases are impossible is hardly evident in any of Brun's work, hardly
implicit in any of Granvilles work prior to 1996, and is hardly many
decades old.
Where are the truth seekers out there among mathematicians???
---andrew beal
-------------------------------------------------------
In article <39A1A683...@y.z.com>,
Hull Loss Incident <x...@y.z.com> wrote:
andy...@my-deja.com
>Granville himself publicized (and published) by 1993 a problem that
>includes yours as a special case. Even that was not original, i.e. he
>was apparently drawing attention to an already well-known circle of
>ideas and saying that this was a concrete problem that deserved work.
>Granville by the way is well-versed in number theory and (I presume
>from the contents of some of his articles) much of the relevant
>history. I believe his advisor was Ribenboim who wrote numerous
>articles and at least two books on FLT.
Granville never hypothesized that co-prime bases were impossible and
certaintly never impirically tested the concept. Granville is hardly
well versed with regard to these issues. Granville stated publicly in
1998 that he doubted the beal conjecture was true. So much for it being
common knowledge. I did not simply pose the problem, I reasoned that co-
prime bases might be impossible and impirically tested a reasonable
range before announcing the assertion.
> I do expect that there was computational work published, on your
> equations or similar ones, and that if you look at the publications
>they will make statements from which your conjecture can be easily
>inferred. I am also confident that I could find references predating
>your conjecture where such ideas are spelled out in print.
Your confidnece and expectations appear to be misplaced. Talk and
bullshit are cheap -- cite your reference(s)
---andrew bael
andy...@my-deja.com
one to have reasoned that co-prime bases are impossible as asserted in
the beal conjecture. I additionally actually tested the assertion to
confirm some probability of the concept.
Granville publicly stated in 1998 that he doubted the beal conjecture
was true - hardly evidence of common prior knowledge
Vanderpoorten stated in the first edition of his 1997 book that he
was "proposing" that co-prime bases were impossible (ie: the beal
conjecture). --hardly consistent with prior common knowledge
Mathematicians wrote me in 1994 after I disseminated my assertions and
research agreeing that the concept was unknown and calling it "quite
remarkable". I certaintly didn't simply "pose the problem", I reasoned
a suspected conclusion and impirically tested it within reasonable
boundries and disseminated my results.
In any event, my agreement to post a prize when someone expressed
interest in writing about the conjecture is hardly my only contribution
to the problem.---andrew beal
As I've said before, simply demonstrate that anyone ever suggested
previously that co-prime bases were impossible and I'll walk away from
the conjecture - that seems pretty simple doesn't it ???? --andrew beal
In article <christian.bau-2208000936120001@christian-
mac.isltd.insignia.com>,
christ...@isltd.insignia.com (Christian Bau) wrote:
> In article <39A21109...@y.z.com>, Hull Loss Incident
<x...@y.z.com> wrote:
>
> > Relying on number theory textbooks as a reference is fine.
> > It's an obvious mistake, though, to take what is NOT mentioned
> > in the books as a serious indication that something is unknown to
the
> > number theory community. Textbooks lag behind current research,
> > they can't include more than a tiny fraction of what is known, and
they
> > are often written by people who have time to write textbooks
precisely
> > because they're not at the current edge of research.
>
> In number theory, there are just so many things that are quite
plausible
> conjectures... I think every student of mathematics should be able to
come
> up easily with half a dozen conjectures that his professor cannot
prove to
> be wrong, or find in the literature within one day. Of course, the
reason
> is that most of these conjectures will not be considered worth writing
> down.
>
Boudewijn Moonen
people, myself included, will desperately hope
that the conjecture so unfortunately also
attached to your name will be going to be
your Last Theorem. Maybe you can then direct
JSH's attention to it. Good luck
--
Boudewijn Moonen
Institut fuer Photogrammetrie der Universitaet Bonn
Nussallee 15
D-53115 Bonn
GERMANY
e-mail: Boudewij...@ipb.uni-bonn.de
Tel.: GERMANY +49-228-732910
Fax.: GERMANY +49-228-732712
Dave Rusin
>I certaintly hope that Andrew Bremner is a better mathematician than he
>is researcher.
>
...not to know what constitutes mathematical work.
(a) Your first sentence contrasts activity A with activity B when
they are synonymous.
(b) Your multiple postings suggest there is tremendous value in stating
guesses about things. There is not.
We have, for example, an enthusiastic amateur who regularly posts
guesses and puzzles and so on to sci.math, and they often make for
interesting diversions. We also see occasional puzzles by people with
significant mathematical training (e.g. Noam Elkies posts to rec.puzzles,
and Charles Nicol will ask here what look like simple-minded
challenges). Not wanting to appear too elitist, I have to tell you
that the first person's guesses are not worth recording for posterity
but those of the other people are, simply because the result of
decades of training is that a person can make a conjecture based
on some _structure_ of the situation, and not a simple check of examples.
If it should happen that the amateur and the professional both,
independently, come to ask the same question, I would take it as
an indication that the conjecture is a natural one. Fine. But I
have to stress there is no "credit" for making guesses in mathematics.
Our primary interest is in providing proofs for assertions, or
at least indications of plausibility.
Let me comment that I'm happy that there are amateurs interested in
mathematical questions, and that in particular I'm happy that Beal
has offered money to draw attention to a question in mathematics.
It may be a silly way to attract young people to mathematics, but
since talented youth are drawn away from mathematics by similar
silly tactics, I figure this is a fine counterweight. The only thing
I am objecting to is the claim that Beal has advanced our understanding
of mathematics in any way by offering this prize.
For comparison I would like to remind readers about the "Nobel Prize
in Economics". There are disputes about the choice of name for that
award, since the prize has nothing whatever to do with Alfred Nobel.
It's just a slightly underhanded way to bestow some additional gravitas
on the subject of Economics. Well... so what? I figure if a person
with money wants to offer money to people who are doing research
(many of them mathematicians!) then I don't much care what they call it.
Whatever celebrity accompanies the award belongs to person who
advances the discipline, not the person with the money.
dave
Bob Silverman
andy...@my-deja.com wrote:
> As I said in another post, the fact is that I appear to be the first
> one to have reasoned that co-prime bases are impossible as asserted in
> the beal conjecture.
This is nonsense. It is anything BUT a fact. I can recall
discussions at the 1985 conference on Computational No. Theory
in Arcata CA. Tate gave a lecture surrounding Frey's brand new
discovery that Taniyama-Shimura was associated with FLT. There
was quite a bit of *informal* discussion as to whether Frey's
result also applied to the case of *unequal* exponents.
This conjecture has been around a LONG time. It is anything but new.
However, simply posing a conjecture is not worth a formal paper,
unless there were techniques to suggest why the conjecture is true.
You claim of being the first to think of this conjecture is
ridiculous. It is an obvious extension of FLT to anyone who works
in number theory.
--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"
Bill Taylor
|> andy...@my-deja.com wrote:
|> > As I said in another post, the fact is that I appear to be the first
|> > one to have reasoned that co-prime bases are impossible as asserted in
|> > the beal conjecture.
|>
|> This is nonsense. It is anything BUT a fact. I can recall
|> discussions at the 1985 conference on Computational No. Theory
|> in Arcata CA. Tate gave a lecture surrounding Frey's brand new
|> discovery that Taniyama-Shimura was associated with FLT. There
|> was quite a bit of *informal* discussion as to whether Frey's
|> result also applied to the case of *unequal* exponents.
|>
|> This conjecture has been around a LONG time. It is anything but new.
|> However, simply posing a conjecture is not worth a formal paper,
|> unless there were techniques to suggest why the conjecture is true.
|>
|> You claim of being the first to think of this conjecture is
|> ridiculous. It is an obvious extension of FLT to anyone who works
|> in number theory.
Thanks Bob; that seems to adequately dispose of Beal's pretensions.
_ _____ _ ____ _ _ _ __ _
/\| |/\ | __ \| | / __ \| \ | | |/ / /\| |/\
\ ` ' / | |__) | | | | | | \| | ' / \ ` ' /
|_ * _| | ___/| | | | | | . ` | < |_ * _|
/ , . \ | | | |___| |__| | |\ | . \ / , . \
\/|_|\/ |_| |_____|\____/|_| \_|_|\_\ \/|_|\/
Public service announcements are always welcome.
============================================. _
| / \
Bill Taylor | |\_/|
| |---|
W.Ta...@math.canterbury.ac.nz | | |
| | |
--------------------------------------------| _ |=-=| _
| _ / \| |/ \
The Moving Finger writes; and, having writ | / \| | | ||\
| | | | | | \>
Moves on: nor all thy Piety nor Wit | | | | | | \
| | - - - - |) )
Shall lure it back to cancel half a Line, | | /
| \ /
Nor all thy Tears wash out a Word of it. | \ /
| \ /
-------------------------------------------------------------------------------
Hull Loss Incident
> As I said in another post, the fact is that I appear to be the first
> one to have reasoned that co-prime bases are impossible as asserted
> in the beal conjecture.
Now that it's on record that Granville posed the problem in 1992
before the Beal conjecture, you are seizing on minor differences,
any differences, between your version and his.
It is not a major discovery that 8 + 8 = 16 is a solution if you allow
common factors. Less obvious, but still child's play for professionals
(I remember the trick from high school math competitions), is that there
are similar trivial solutions for all exponents. So the need for co-prime
bases was certainly clear to Granville and/or his audience of number
theorists, and whether or not it was specifically spelled out in the
problem statement is not a significant difference. In any case he
clarified later that indeed, primitive solutions were intended.
So you noticed that there are solutions with common factors?
Nice, but no big deal. You also ran a computer search for
primitive solutions, and from that surmised that none exist?
Even if the guess turns out correct, it's at the level of a school
science project, not the "significant discovery" you claim.
> I additionally actually tested the assertion to
> confirm some probability of the concept.
How far did you test? There are known solutions
with "co-prime bases" in the tens of millions, for
exponents 2,3,7 and 2,3,8.
> Granville publicly stated in 1998 that he doubted the beal conjecture
> was true -
Even if true, so what? He raised the question before you did, only
without presupposing the answer. Your subsequent guess as to what the
answer is does not add anything, since it follows from the standard
arguments (which you were unaware of) coupled to a computer search.
If Granville has deeper reasons for disbelieving the heuristics, that would
be much more interesting, though it doesn't touch the priority issue.
Again, I doubt Granville or any other number theorist
would seriously claim the problem as their own; his 1992 proposal
is more like public domain prior art invalidating a patent claim.
> hardly evidence of common prior knowledge
Common prior knowledge was that the equation should have few
or no solutions -- certainly a finite number of primitive solutions for
any given exponents. Finiteness was expected based on probabilistic
arguments, the ABC conjecture and other considerations, and was
proved in a 1995 paper coauthored by....Granville.
> Vanderpoorten stated in the first edition of his 1997 book that he
> was "proposing" that co-prime bases were impossible (ie: the beal
> conjecture). --hardly consistent with prior common knowledge
On what page of what book does he propose this? I checked his
book on FLT without finding any support for your claims, just the
opposite.
> Mathematicians wrote me in 1994 after I disseminated my assertions and
> research agreeing that the concept was unknown and calling it "quite
> remarkable".
Quote in full any mathematician who endorses your claim of a new
significant discovery. Just one will suffice.
Hull Loss Incident
> Granville never hypothesized that co-prime bases were impossible
Concerning "coprime", he was aware that 8+8=16.
Concerning "impossible", he was aware that short-range
computer searches don't count for much.
> and certaintly never impirically tested the concept.
He listed some solutions in the question,
presumably as a result of someone having looked
for them. His formulation, unlike yours,
would not be invalidated by further searching.
On the contrary, it was obviously meant to
prompt more serious empirical tests.
> Granville stated publicly in
> 1998 that he doubted the beal conjecture was true.
Given your (in)accuracy so far, I'll believe it when i see
a complete quote, preferably one that i can check.
> So much for it being
> common knowledge. I did not simply pose the problem, I reasoned that co-
> prime bases might be impossible and impirically tested a reasonable
> range before announcing the assertion.
You ran a computer search and the world should genuflect? Ludicrous.
Hull Loss Incident
> I certaintly hope that Andrew Bremner is a better mathematician
> than he is researcher.
For those scoring this on the Crackpot Index, a quick summary.
Mr. Beal has so far:
--> attacked Granville as ill-informed
--> claimed "misrepresentation" by Granville
--> attacked the MathSciNet reviewer as unscholarly
--> attacked the review as "bullshit"
--> dissed number theorists for inability to see through the review
--> hinted at dark ulterior motives behind the review
--> inveighed against mathematicians indifferent to the Truth
--> claimed support of "experts" and "leading mathematicians"
--> refused to quote any actual expert support.
> The fact is that Bremners "review" below is bullshit and begs the
> question of who inspired the review and for what purposes.
Maybe he saw through the bullshit in the fawning AMS Notices
article, and decided to weigh in with some truth.
Since you raise the issue of motive, let's also ask what inspired the
article under review. Here is one theory, a meta-Beal conjecture:
Beal is a major donor to the author's math department, according
to the article. So when Beal came to the department wanting to
discuss his "discoveries" they could hardly turn him away.
(He had at that point been "disseminating the results for several years"
as he says, without generating interest.) Could it be that the author
of the Notices article was in the delicate position of having to
somehow placate a determined donor -- bring his work to the
mathematical community -- while shielding him from the indifference
or ridicule his "discovery" claims would generate if scrutinized?
Solution: write a puff piece for a trade magazine, praising Beal
while not actual claiming credit on his behalf.
> The reality
> is that there are so few number theorists out there that few are able
> to see through Bremners "review".
Bremner's review puts on the record what most number theorists
already knew, whether or not they had heard of the "Beal conjecture".
That it debunks what was in effect a bought and paid for
advertisement in the AMS Notices, is a nice side effect. Whereas
AMS Notices is a throwaway magazine that AMS members
receive for free and libraries don't keep, the Mathscinet review
will be around permanently should anyone want to look up the
words "Beal conjecture".
> Where are the truth seekers out there among mathematicians???
That's it, a conspiracy.
Bob Silverman
mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
> Bob Silverman <bo...@my-deja.com> writes:
<snip>
> |> This is nonsense. It is anything BUT a fact. I can recall
> |> discussions at the 1985 conference on Computational No. Theory
> |> in Arcata CA. Tate gave a lecture surrounding Frey's brand new
> |> discovery that Taniyama-Shimura was associated with FLT. There
> |> was quite a bit of *informal* discussion as to whether Frey's
> |> result also applied to the case of *unequal* exponents.
> Thanks Bob; that seems to adequately dispose of Beal's pretensions.
I observe that Mr. Beal has not responded to any of my posts on this
subject.....
--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"
andy...@my-deja.com
>
> > As I said in another post, the fact is that I appear to be the first
> > one to have reasoned that co-prime bases are impossible as asserted
> > in the beal conjecture.
>
> Now that it's on record that Granville posed the problem in 1992
> before the Beal conjecture, you are seizing on minor differences,
> any differences, between your version and his.
>
did speculate about a limited number of solutions for a similiar
problem; his speculation subsequently proved to be wrong.
> How far did you test? There are known solutions
> with "co-prime bases" in the tens of millions, for
> exponents 2,3,7 and 2,3,8.
>
I tested all terms through 99^99 - a little beyond "tens of millions"
> He raised the question before you did, only
> without presupposing the answer.
he didn't raise the question and he didn't understand the answer. In
any event, if he really knew it he should have shared it with the rest
of the world. perhaps he didn't do that because he says the problem is
too trivial for him, or perhaps it's because he didn't know it -- but
for whatever reason --he never did.
> Again, I doubt Granville or any other number theorist
> would seriously claim the problem as their own;
wrong again --as usual -- see reference below
>
> > Vanderpoorten stated in the first edition of his 1997 book that he
> > was "proposing" that co-prime bases were impossible (ie: the beal
> > conjecture). --hardly consistent with prior common knowledge
>
> On what page of what book does he propose this? I checked his
> book on FLT without finding any support for your claims, just the
> opposite.
On page 194 vanderpoorten writes, and I qoute "I propose that if a,b,c
are relatively prime, then a^t +b^u = c^v has no solutions in integers
greater than 1 if all of t,u,v are at least 3
> > Mathematicians wrote me in 1994 after I disseminated my assertions
and
> > research agreeing that the concept was unknown and calling it "quite
> > remarkable".
>
> Quote in full any mathematician who endorses your claim of a new
> significant discovery. Just one will suffice.
>
I have already done so in an earlier post, but I will repeat that
Harold Edward's, a mathematician at NYU, author of "Fermat's Last
Thoerem ..." among others indicated that it was unknown and called
it "quite remarkable"
>
Since you believe this conjecture so trivial anyway, why are you
concerned about this??? why don't you progress to something not so
trivial??? --andrew beal
andy...@my-deja.com
mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
>
> |> andy...@my-deja.com wrote:
>
> |> > As I said in another post, the fact is that I appear to be the
first
> |> > one to have reasoned that co-prime bases are impossible as
asserted in
> |> > the beal conjecture.
> |>
> |> This is nonsense. It is anything BUT a fact.
As I've said before, talk and bullshit are cheap. CITE A SINGLE
REFERENCE!!!!! -- not your memory of some old conversation
> |> This conjecture has been around a LONG time. It is anything but
new.
SO CITE A SINGLE WRITTEN REFERENCE!!!! no-one is interested in your
memory of this or that. Incidentally, you better tell all this to
vanderpoorten who proposed the conjecture in his 1997 book. He
obviously would disagree with you about it being old news and it seems
not too trivial for him to mention in his book.
---andrew beal
Hull Loss Incident
> > > As I said in another post, the fact is that I appear to be the first
> > > one to have reasoned that co-prime bases are impossible as asserted
> > > in the beal conjecture.
> >
> > Now that it's on record that Granville posed the problem in 1992
> > before the Beal conjecture, you are seizing on minor differences,
> > any differences, between your version and his.
> >
> It's hardly on the record - Granville never posed the conjecture.
You agree that his proposal was earlier, as the record indicates.
So the only issue is whether his version of the problem differs
significantly from yours. Let's take the differences point by point:
-- Granville's equation is more general than yours. That in no
way undermines his precedence.
-- Granville did not stipulate the need for "coprime bases".
To sustain the belief that he or his audience of
number theorists missed this point until Beal came along,
one would have to assume they did not notice that 8+8=16,
2^n + 2^n=2^(n+1), and games that can be played with
these at the level of chapter 1 (or page 1) of any number
theory text. I can spell this out in detail if needed,
but suffice it to say that insisting on "coprime" as a
difference between the conjectures would be
ridiculous even if Granville had not confirmed that
coprimality was intended.
-- Granville did not assert that there are no solutions
in the cases corresponding to Beal's conjecture. This
is a distinction without a difference. When you have
some statement S with no compelling evidence for
or against it, there is no difference between asking
to "investigate S" or "prove S" or "prove or disprove S"
or "find solutions of S". If the more specific no-solutions
conjecture had been backed by strong evidence other
than computer search it would be a different
story, not to mention a publishable piece of research.
But as it stands, there is no support whatsoever for
Beal's conjecture not being subsumed by Granville's problem.
> He did speculate about a limited number of solutions
> for a similar problem;
He didn't speculate, you did. It's the same problem,
Fermat with unequal exponents. His set of
exponents is more natural than yours, as it's the
right choice based on probabilistic or ABC
considerations.
> his speculation subsequently proved to be wrong.
He asked whether there were more solutions, which makes it
impossible for computer searching to prove him wrong. What
his personal expectations were about the probability of finding
more solutions, doesn't matter much. You seem to think it's
some kind of armchair betting game, where predicting the
eventual answer is what counts. It isn't, evidence for the
answer is the important part, and you provided none to
distinguish your expectations as more plausible than others'.
> > How far did you test? There are known solutions
> > with "co-prime bases" in the tens of millions, for
> > exponents 2,3,7 and 2,3,8.
> >
> I tested all terms through 99^99 - a little beyond "tens of millions"
The computer search is your one undisputed scientific contribution
to this problem, so I think you ought to state carefully what test
was carried out and how. Are you claiming all possible
x^p + y^q = z^r < 99^99 were ruled out??
Reality check: let's say it was up to 100^98 which is about 37 times
smaller, and let's say that you were doing the computationally
easier task of checking the conjecture with all exponents at
least 4. The number of sums of fourth-or-higher powers
below 100^98 is within a small factor of the square root, 10^98.
So you have a 97-digit number of cases to check. If you
process one trillion cases per second this takes astronomically
many years.
> > He raised the question before you did, only
> > without presupposing the answer.
>
> he didn't raise the question and he didn't understand the answer.
He raised a question that subsumes yours, and was wise
enough not to prejudge the answer from a mere computer search.
Your extrapolating an opinion from the search and going public
with it is worthless. Reporting the details of the search would
be valuable, but that's it.
> In any event, if he really knew it he should have shared it
> with the rest of the world.
That's backward. *You* shared nothing with the world
in the way of evidence for the guess. Just making statements
that might be true, has no intrinsic value. If you had some new method
for arriving at your conjecture, share it. Otherwise the only
real contribution is the computer search, which as far as I know
you never documented in any detail.
Granville on the other hand has published major
contributions toward resolving the problem.
Real research, not armchair wagering and publicity stunts.
andy...@my-deja.com
Hull Loss Incident <x...@y.z.com> wrote:
> andy...@my-deja.com wrote:
>
> > > > As I said in another post, the fact is that I appear to be the
first
> > > > one to have reasoned that co-prime bases are impossible as
asserted
> > > > in the beal conjecture.
> > >
> > > Now that it's on record that Granville posed the problem in 1992
> > > before the Beal conjecture, you are seizing on minor differences,
> > > any differences, between your version and his.
> > >
> > It's hardly on the record - Granville never posed the conjecture.
>
> So the only issue is whether his version of the problem differs
> significantly from yours. Let's take the differences point by point:
>
> -- Granville's equation is more general than yours. That in no
> way undermines his precedence.
>
>
according to your logic. Or the the conjecture is "implicit" in such a
statement. You either made the conjecture or you didn't. Granville
didn't - he was incorrectly speculating with 2nd powers - he gave no
indication he understood the concept and in fact his limited assertion
was subsequently proven wrong.
Get a life - who is now self promoting --Granville????
> The computer search is your one undisputed scientific contribution
> to this problem, so I think you ought to state carefully what test
> was carried out and how. Are you claiming all possible
> x^p + y^q = z^r < 99^99 were ruled out??
>
> Reality check: let's say it was up to 100^98 which is about 37 times
> smaller, and let's say that you were doing the computationally
> easier task of checking the conjecture with all exponents at
> least 4. The number of sums of fourth-or-higher powers
> below 100^98 is within a small factor of the square root, 10^98.
> So you have a 97-digit number of cases to check. If you
> process one trillion cases per second this takes astronomically
> many years.
You are once again demonstrateing your limited apptitude for the issues
involved here. Yes, I tested all possibilities from 1^3 + 1^3 = 1^3
through 99^99 + 99^99 = 99^99. I used more sophisticated algorithms
than simply iterating through all possibilities. A director at NASA
ames has recently done a similiar search, in a number of minutes.
> He raised a question that subsumes yours, and was wise
> enough not to prejudge the answer from a mere computer search.
> Your extrapolating an opinion from the search and going public
> with it is worthless.
Hello???? Anyone home???? -- without a reasoned conclusion, you do not
have a conjecture. Simply asking a question does not conjecture all
possible outcomes.
I think I will sign off here -- I've gained about all
the "profesional" insight I can stand.
----andrew beal
Hull Loss Incident
> > > > Now that it's on record that Granville posed the problem in 1992
> > > > before the Beal conjecture, you are seizing on minor differences,
> > > > any differences, between your version and his.
> > > >
> > > It's hardly on the record - Granville never posed the conjecture.
> >
> > You agree that his proposal was earlier, as the record indicates.
> > So the only issue is whether his version of the problem differs
> > significantly from yours. Let's take the differences point by point:
> >
> > -- Granville's equation is more general than yours. That in no
> > way undermines his precedence.
> >
> So I guess anyone that ever said A + B = C also stated the conjecture
> according to your logic.
You described it as a similar problem, and the relationship is obviously
much closer than a mere two numbers adding up to a third. The difference
between "1/x + 1/y + 1/z < 1" and "x,y,z > 2" is minor, and also
points to him understanding the problem much better than you.
> Or the the conjecture is "implicit" in such a statement.
Coprimality is definitely implicit, unless you claim the number
theory world did not know that 8+8=16 before Beal showed up.
> You either made the conjecture or you didn't. Granville didn't -
He posed a problem that obviously includes yours. The added
groundless speculation in your case is worthless, at best your
contribution would have been a computer search (so far so good),
but now even that turns out not to be the case. Nobody gives
Fermat credit just for making an educated guess about FLT.
> he was incorrectly speculating with 2nd powers
What is incorrect? There is no reason given by
you, to believe that something important is changed
by allowing 2nd powers. Obviously as exponents
are increased above 2 the probability of a solution
is lower, but you haven't indicated why one should
not expect solutions if the search range is enlarged to
take this into account.
> - he gave no
> indication he understood the concept and in fact his limited assertion
> was subsequently proven wrong.
Liar. What you call the "concept" and the "relationship" is that
solutions must have common factors. Granville and everyone
else were obviously aware of the equation, were aware that
8+8=16 is a solution, and aware that if common factors are
allowed you can write down other solutions at will (something
probably not understood by Beal). Granville was certainly
not proven wrong, on the contrary he specifically asked about
finding more solutions. Your claim about no solutions may
or may not be correct, but you certainly gave no reason to
believe in it.
> Get a life - who is now self promoting --Granville????
Granville did not seek credit for a well-known problem, and I'm not
aware of any other conjecture where someone waged a funded PR
campaign to have it called "his" conjecture, rather than letting the
attribution develop on its own.
>You are once again demonstrateing your limited apptitude for the issues
> involved here. Yes, I tested all possibilities from 1^3 + 1^3 = 1^3
> through 99^99 + 99^99 = 99^99. I used more sophisticated algorithms
> than simply iterating through all possibilities. A director at NASA
> ames has recently done a similiar search, in a number of minutes.
That's either sad or hilarious. You wasted a month of computer
time for something anybody could have told you was irrelevant.
You tried numbers of almost 200 digits but failed to rule out small
solutions of x^p + y^q = z^r where the perfect powers involved
are on the order of a million! The smaller the numbers the more
likely a counterexample, 99.99% of your search was looking for
100-digit chimeras.
> > He raised a question that subsumes yours, and was wise
> > enough not to prejudge the answer from a mere computer search.
> > Your extrapolating an opinion from the search and going public
> > with it is worthless.
>
> Hello???? Anyone home???? -- without a reasoned conclusion, you
> do not have a conjecture.
A public statement of opinion is not required when posing a problem.
> Simply asking a question does not conjecture all
> possible outcomes.
The possible outcomes are True or False. Stating
your opinion which one it is, is completely worthless -- armchair
wagering, followed by arrogant claims of a "significant discovery".
Bob Silverman
andy...@my-deja.com wrote:
> In article <8o29j3$1k1$1...@cantuc.canterbury.ac.nz>,
> mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
> > Bob Silverman <bo...@my-deja.com> writes:
> >
> > |> andy...@my-deja.com wrote:
> >
> > |> > As I said in another post, the fact is that I appear to be the
> first
> > |> > one to have reasoned that co-prime bases are impossible as
> asserted in
> > |> > the beal conjecture.
> > |>
> > |> This is nonsense. It is anything BUT a fact.
>
> As I've said before, talk and bullshit are cheap. CITE A SINGLE
> REFERENCE!!!!! -- not your memory of some old conversation
Are you calling me a liar???
>
> > |> This conjecture has been around a LONG time. It is anything but
> new.
>
> SO CITE A SINGLE WRITTEN REFERENCE!!!! no-one is interested in your
> memory of this or that.
(1) There are many conjectures that have never been written down.
Mainly, because they are not worth writing down.
(2) I think that there are quite a few people interested in
discussions at the 1985 Arcata meeting.
--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"
Hull Loss Incident
> > Again, I doubt Granville or any other number theorist
> > would seriously claim the problem as their own;
>
> wrong again --as usual -- see reference below
I checked the reference. As usual it doesn't bear out
your claims, in this case that Van der Poorten (or any
other number theorist) seeks to attach his name to
the problem. I doubt any mathematician could
get away with having the conjecture named after
themselves, except maybe Wiles should he decide
to propose a successor problem to FLT.
> > > Vanderpoorten stated in the first edition of his 1997 book that he
> > > was "proposing" that co-prime bases were impossible (ie: the beal
> > > conjecture). --hardly consistent with prior common knowledge
> >
> On page 194 vanderpoorten writes, and I qoute "I propose that if a,b,c
> are relatively prime, then a^t +b^u = c^v has no solutions in integers
> greater than 1 if all of t,u,v are at least 3
You are quoting from Appendix A of his book, which is a reprint
of an article written during Christmas 1993 and published in 1994
("Remarks on FLT", in Gazette of the Australian Math Society).
In the original, he writes:
"Darmon and Granville (1993) suggest Fermat might have
chosen a different generalization. Say, that a^t + b^u = c^v
has no solution in integers greater than 1 if all of t,u,v are at
least 3. If one exponent is allowed to be 2, things are
different. For example, in the cases (t,u,v) = (3,3,2) and (4,3,2)
there are infinitely many solutions. In general, if 1/t+1/u+1/v < 1
we have grounds for believing that there are just finitely many
solutions for which a and b have no common factor. The
largest known is 43^8 + 96222^3 + 30042907^2.
All nine solutions known have one or other of the exponents
equal to 2."
Apparently the change in wording from "Say" to "I propose" was
as a matter of professional courtesy to Darmon and Granville,
to avoid the possible suggestion that they had made a prediction
about the equation having no solutions. Van der Poorten
himself refers to the statement as the Generalized Fermat Conjecture,
not trying to attach his name to it in Beal fashion. He does not
pretend it is anything more than a collation of standard expectations,
the work of Darmon & Granville, and the fact about exponent 2
appearing in the known solutions. This is also true of the more
extensive discussion on pages 143-150 of his book.
Note also the allusion to the probability heuristics and/or ABC
conjecture predicting finiteness for 1/t + 1/u + 1/v < 1, already
known in 1993 (and of course much earlier).
Finally, we see above that the large solution of the equation
found by Beukers and Zagier, was known by the time
Van der Poorten wrote his article in 1993. It would not
be any surprise to learn they had done this prior to Beal
formulating his conjecture, or that they also searched for
instances included in his version of the problem, far beyond
the 6-digit-or-less range that Beal himself searched.
> > > Mathematicians wrote me in 1994 after I disseminated my assertions
> > > and research agreeing that the concept was unknown and calling it
> > > "quite remarkable".
> >
> > Quote in full any mathematician who endorses your claim of a new
> > significant discovery. Just one will suffice.
> >
> I have already done so in an earlier post,
You have not done so in any post. The two words "quite remarkable"
are not a quotation in full, and from what is publically available at the
AMS web site it's clear that you are misrepresenting your sources.
> but I will repeat that
> Harold Edward's, a mathematician at NYU, author of "Fermat's Last
> Thoerem ..." among others indicated that it was unknown and called
> it "quite remarkable"
See http://www.ams.org/notices/199803/commentary.pdf,
third page (359) for a more honest summary of the correspondence:
"Harold Edwards responded in Sept 1994. He suspected there
might be counterexamples and suggested that Beal have someone
do a simple computer study which would perhaps reveal them."
In other words, Edwards was not convinced Beal had made
a correct discovery, let alone a major significant discovery,
and noticed the inadequacy of the computer search Beal had
done.
If Beal has any actual expert support for his claims of a significant
discovery, he should quote it in full. As andy...@my-deja.com
likes to say, "talk and bullshit are cheap. cite your references."
andy...@my-deja.com
and when someone expressed an interest in writing about it, agreed to
offer a prize for its solution. For this apparent crime, I am guilty.
Despite an apparent lack of any prior reference to the conjecture and
agreement by many that it was unknown, why don't you suggest a name for
the problem as you see fit and quite misrepresenting me and my
intentions? In hindsight, perhaps I have been overly defensive because
of the vicious attacks by Granville et al. If Granville wants to claim
credit for it because he once asked about similiar forms, God bless
him. If he doesn't because it is too trivial for him, God bless him.
----andrew beal
In article <39AAE467...@y.z.com>,
Hull Loss Incident <x...@y.z.com> wrote:
> andy...@my-deja.com wrote:
>
> > > Again, I doubt Granville or any other number theorist
> > > would seriously claim the problem as their own;
> >
> > wrong again --as usual -- see reference below
>
> I checked the reference. As usual it doesn't bear out
> your claims, in this case that Van der Poorten (or any
> other number theorist) seeks to attach his name to
> the problem. I doubt any mathematician could
> get away with having the conjecture named after
> themselves, except maybe Wiles should he decide
> to propose a successor problem to FLT.
>
> > > > Vanderpoorten stated in the first edition of his 1997 book that
he
> > > > was "proposing" that co-prime bases were impossible (ie: the
beal
> > > > conjecture). --hardly consistent with prior common knowledge
> > >
> > On page 194 vanderpoorten writes, and I qoute "I propose that if
a,b,c
> > are relatively prime, then a^t +b^u = c^v has no solutions in
integers
> > greater than 1 if all of t,u,v are at least 3
>
> > > > Mathematicians wrote me in 1994 after I disseminated my
assertions
> > > > and research agreeing that the concept was unknown and calling
it
> > > > "quite remarkable".
> > >
> > > Quote in full any mathematician who endorses your claim of a new
> > > significant discovery. Just one will suffice.
> > >
> > I have already done so in an earlier post,
>
> You have not done so in any post. The two words "quite remarkable"
> are not a quotation in full, and from what is publically available at
the
> AMS web site it's clear that you are misrepresenting your sources.
>
> > but I will repeat that
> > Harold Edward's, a mathematician at NYU, author of "Fermat's Last
> > Thoerem ..." among others indicated that it was unknown and called
> > it "quite remarkable"
>
> See http://www.ams.org/notices/199803/commentary.pdf,
> third page (359) for a more honest summary of the correspondence:
>
> "Harold Edwards responded in Sept 1994. He suspected there
> might be counterexamples and suggested that Beal have someone
> do a simple computer study which would perhaps reveal them."
>
> In other words, Edwards was not convinced Beal had made
> a correct discovery, let alone a major significant discovery,
> and noticed the inadequacy of the computer search Beal had
> done.
>
> If Beal has any actual expert support for his claims of a significant
> discovery, he should quote it in full. As andy...@my-deja.com
> likes to say, "talk and bullshit are cheap. cite your references."
>
>
Hull Loss Incident
> I discovered something interesting to me - shared it with the world -
Great. May you discover many more interesting things,
and share them.
> and when someone expressed an interest in writing about it, agreed to
> offer a prize for its solution.
Fine. Nothing wrong with establishing a Beal Prize for solution of
whatever problem you like.
> For this apparent crime, I am guilty.
No, the complaints are clearly connected with the use (whether
intentional or de facto) of the prize to publicize the conjecture
as though it were your intellectual property. For example, the
article in AMS Notices announcing the prize happens to proclaim
the "Beal Conjecture" in the title, and the website at UNT devoted
to the problem and the prize calls it "Beal's conjecture", with
pointers to newspaper articles about the prize.
There are some separate complaints I have made about AMS
behavior in publishing the article, but that is not a criticism of you
one way or the other.
> Despite an apparent lack of any prior reference to the conjecture and
> agreement by many that it was unknown,
There is no lack of prior reference (at least two prior instances
were under discussion in this thread), and the only obstacle to
producing many more instances is a lack of agreement on your
side that the well-known probabilistic arguments are prima facie
evidence that your conjecture was known. If you look in Ribenboim's
new book "My Numbers, My Friends" there are dozens of references
to very similar problems (many special cases of the Catalan conjecture,
for example) studied in the past 150 years, and in the chapter on sums
of powers he references two papers of his from 1985 and 1993. One
deals with the set of exponents (l,m,n) in which the generalized Fermat
equation has solutions, the other with similar problems for
aX^p + bY^q = cZ^r.
As for the "agreement by many that it was unknown", it has
been alluded to but not substantiated. Mauldin's followup
letter to AMS Notices, apparently prompted by some complaints
or raised eyebrows about the attribution of the conjecture, is
informative in this regard. It leaves the impression that out
of 15-20 mathematicians and journals that you contacted,
all but two were unresponsive or replied negatively, and those
two were in part favorable but also not that supportive.
If a couple of people didn't hear about your problem, some
did and replied dismissively, and others didn't bother to reply,
that is hardly evidence of "agreement by many that it was unknown".
> why don't you suggest a name for the problem as you see fit
My suggestion is to refer to the mathematical problem using
uncontroversial terms such as the Generalized Fermat Equation
and Generalized Fermat Conjecture; and more importantly,
to rename the prize in a way that does not confuse the
founding of the prize with the founding of the problem.
"Beal Prize for the solution of the Generalized Fermat
Conjecture", or something along those lines.
> and quite misrepresenting me and my intentions?
I disagree that I have misrepresented intentions.
In any case, if you make a change such as proposed
above it would render discussion of your intentions
mostly irrelevant.
> In hindsight, perhaps I have been overly defensive because
> of the vicious attacks by Granville et al.
I don't know of any vicious attacks by Granville. Maybe he was
rude to you in your correspondence, but it sounded from your
description like he gave you a brief dismissive answer that, yes,
the conjecture was known, and you then demanded absolute
proof or else.
> If Granville wants to claim credit for it because he once asked
> about similiar forms, God bless him.
> If he doesn't because it is too trivial for him, God bless him.
He hasn't claimed credit for it because it was "public domain"
knowledge prior to his 1992 problem proposal, and he obviously
considers it nontrivial enough to publish serious research on the subject.
andy...@my-deja.com
agreed to offer a prize for its solution. For this apparent crime, I am
guilty. Despite an apparent lack of any prior reference to the
conjecture and agreement by many that it was unknown,
>
> There is no lack of prior reference (at least two prior instances
> were under discussion in this thread), and the only obstacle to
> producing many more instances is a lack of agreement on your
> side that the well-known probabilistic arguments are prima facie
> evidence that your conjecture was known. If you look in Ribenboim's
> new book "My Numbers, My Friends" there are dozens of references
> to very similar problems (many special cases of the Catalan
conjecture,
> for example) studied in the past 150 years, and in the chapter on sums
> of powers he references two papers of his from 1985 and 1993. One
> deals with the set of exponents (l,m,n) in which the generalized
Fermat
> equation has solutions, the other with similar problems for
> aX^p + bY^q = cZ^r.
This is wonderful but does not contradict the fact that no prior
evidence of the conjecture exists. The good news for the world is that
your making of a statemnt is not the definition of truth. I have made
specific references demonstrating agreement by others that my assertion
was unknown. In contrast, you have yet to provide a single reference
beyond someones memory of this or that. Even if someone had previously
surmised the conjecture, what's your point??? I have said repeatedly
that I am unaware of reasoned prior knowledge, but this would readily
change if someone produced such evidence. You are correct that
your "probabalistic" arguments and reference to the many various forms
of various diophantine equations do not compell me to a contrary
conclusion. Apparently no-one else agrees either. Or is the problem
that all of us (including your reference to the AMS) simply don't agree
with you.
> It leaves the impression that out
> of 15-20 mathematicians and journals that you contacted,
> all but two were unresponsive or replied negatively, and those
> two were in part favorable but also not that supportive.
here is more of your bullshit -- there were no negative responses and
all responses were quite favorable and agreed it was unknown.
> My suggestion is to refer to the mathematical problem using
> uncontroversial terms such as the Generalized Fermat Equation
> and Generalized Fermat Conjecture;
Great - so why don't you do so and quite condemning me for simply
stating the facts.
----andrew beal
Robin Chapman
> third page (359) for a more honest summary of the correspondence:
>
lamented Alexander Abian :-)
--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
"`The twenty-first century didn't begin until a minute
past midnight January first 2001.'"
John Brunner, _Stand on Zanzibar_ (1968)
Hull Loss Incident
> > http://www.ams.org/notices/199803/commentary.pdf
> >
> An impressive reference, as it contains a letter from the late
> lamented Alexander Abian :-)
The sage of Iowa foresaw this thread. All bodies in the
universe move to INFLATE THEIR EGOS and FEEL SECURE.
Hull Loss Incident
andy...@my-deja.com wrote:
> In article <39AB44B1...@y.z.com>,
> Hull Loss Incident <x...@y.z.com> wrote:
> > andy...@my-deja.com wrote:
> >
> >>I discovered something interesting to me - shared it
> >>with the world -
The usual method of sharing a discovery with the world is
to publish it in a refereed journal. You sent your conjecture
to various mathematicians and mathematical journals; did any
of them actually recommend publication? (Of course not,
otherwise Beal would be trumpeting it.)
> >>and when someone expressed an interest in writing about it,
> >>agreed to offer a prize for its solution.
And so scholarly review is bypassed: instead of trying to publish
the unpublishable, or finding a journal obscure enough to accept it,
"share it with the world" through a publicity piece about the prize.
A bought and paid for advertisement. This conveniently circumvents
any scrutiny of the novelty or significance of the claimed discovery,
and allows for fawning Beal PR to appear before the eyes of the
AMS membership (an enormously larger readership than any
journal article, if one were even possible, would ever have
attracted), and from there to make its way into newspapers.
I don't know if the idea was due to Beal or Mauldin, but it is
a promotional masterstroke. I suggest that it is a direct result
of the UNT math department being beholden to a major benefactor
and needing to placate him after he failed to generate interest in
the "discovery" on his own.
> >>For this apparent crime, I am guilty.
You are guilty of waging a PR campaign to buy what
you could never have obtained on the merit of your
intellectual (non) contribution. i.e., the association of
your name with the conjecture.
Hull Loss Incident
andy...@my-deja.com wrote:
> I have made specific references demonstrating
> agreement by others that my assertion was unknown.
False. You have been challenged to provide even ONE
credible reference for your statements that leading mathematicians
and number theorists support your claim of a new and significant
discovery. All you produced so far is a quotation of two words
"quite remarkable" that are too vague to evaluate. As the
public record indicates (see AMS site referenced), the reason
you continue to quote this but refuse to detail the full statement,
is that it would demolish your claim of unambiguous expert
support.
> In contrast, you have yet to provide a single reference
> beyond someones memory of this or that.
Another incredible lie.
1. Prof. Myerson posted verifiable documentation that Andrew
Granville had stated your problem (in an improved form) at a
conference in 1992, before you ever got started on FLT
(summer 1993, according to Mauldin's letter at the AMS site
that I referenced). This is already enough to completely
wither any claims of priority you might have had.
2. I posted a reference to Van der Poorten's article written
Christmas 1993. The article references Darmon and
Granville's work, meaning that people were circulating
work on Beal's problem before Beal disseminated his
results in summer 1994. This withers any possible
claim of priority based on "disseminating the results
for years" as you mentioned.
3. Van der Poorten's article that I referenced also gives
results of computer searches for solutions (going 9 orders
of magnitude beyond your 6-digits-or-less attempt) performed
by Beukers and Zagier, showing that these were already
in circulation by late 1993. If it turns out that these searches
were conducted before August 1993, it would mean that
other people were far surpassing Beal's computer searches
before Beal got started with his own.
4. I posted a reference to Paulo Ribenboim's recent
book indicating where you can locate a discussion of his work
on the equations X^l + Y^m = Z^n and the more general
aX^l + bY^m = cZ^n, published 1993 and 1985. The book
also cites dozens or hundreds of articles containing prior art
on Catalan's equation and other instances of the generalized
Fermat equation. If those aren't enough for you to glean
that "your" equation was old news to the number theory
community, try L.E.Dickson's "History of the Theory of
Numbers". Volume 2 has 50 pages of annotated citations
to work on the Fermat equation and its generalizations,
published before 1920.
Hull Loss Incident
> Even if someone had previously
> surmised the conjecture, what's your point???
If someone surmised it or something substantially
equivalent, that is a death blow to the term "Beal conjecture"
unless you had made some large intellectual
contribution to the credibility of the conjecture. So far you
have identified no such contribution except a mis-targeted
computer search, whose inadequacy was pointed out
by the mathematicians you corresponded with (e.g. Edwards
suggesting to "have someone do a simple computer study
which would perhaps reveal [counterexamples].")
> I have said repeatedly that
> I am unaware of reasoned prior knowledge, but
> this would readily change if someone produced such evidence.
You had no "reasoned knowledge" to distinguish your conjecture
from the prior state of the art. The burden is on you to show
evidence of such reasoning, since you continue to claim an
advance beyond the state of the art. If you had such a
breakthrough, people would be celebrating your achievements
instead of ridiculing your pretensions.
So far you did not, and probably could not, publish
the details of your work. You've disclosed a computer
search of "all terms up ro 99^99", an approach so obviously
misguided that it suggests you have not even a superficial
understanding of the problem. Meanwhile, you
show no familiarity with the probabilistic or ABC arguments
that were standard at the time, but claim to have insight that
goes far beyond them (i.e. supports a conclusion of no solutions
rather than just a finite number).
As you say, "talk and bullshit are cheap". Demonstrate what
you did that was an improvement over known material in 1993.
> You are correct that
> your "probabalistic" arguments and reference to the many various forms
> of various diophantine equations do not compell me to a contrary
> conclusion.
Of course the probabilistic arguments, etc certainly DO compel the
conclusion that you made no contribution to knowledge with your
work on the conjecture. What you are actually saying here is that
you will continue with blatant denial until someone points to a
literal duplicate of your problem statement published before 1993.
> Apparently no-one else agrees either.
That's backwards. Not one person has agreed with your
statements in this thread, except yourself posting under another
name. Everyone other than me who posted to the thread *has*
agreed that your priority claims are bullshit, and several have
ridiculed you in the bargain.
> Or is the problem
> that all of us (including your reference to the AMS) simply
> don't agree with you.
Mauldin's letter following up his AMS article is an indication
that he was at pains to explain the rather unusual piece, and
may have received flak for it. Bought and paid for advertising
in the guise of news item is something they should not have
published.
> > It leaves the impression that out
> > of 15-20 mathematicians and journals that you contacted,
> > all but two were unresponsive or replied negatively, and those
> > two were in part favorable but also not that supportive.
>
> here is more of your bullshit -- there were no negative responses and
> all responses were quite favorable and agreed it was unknown.
1. You stated that Granville was "abusive" and conducted "vicious
attacks". If true, that is a negative response.
2. The only mathematicians you have been able to name (you claim
there are many) who supposedly endorse your claims, are exactly
the two mentioned in Mauldin's letter. Actually one of them was
only acting in an editorial capacity and didn't evaluate your work
himself, so you citing him is another inaccurate representation, but
never mind that detail. From the content and context of the letter
it is clear that if unambiguous support or endorsement of your
work existed, it would have been mentioned. Until you post
clear evidence to the contrary, there is a strong circumstantial
case that *no* expert endorsers exist for your work, and that
you failed to generate any significant interest by circulating your
work to mathematicians and math journals.
> > My suggestion is to refer to the mathematical problem using
> > uncontroversial terms such as the Generalized Fermat Equation
> > and Generalized Fermat Conjecture;
>
> Great - so why don't you do so
The burden is on YOU to do so, by dissociating your legitimate
claim to creating the prize, from your bogus claim of creating the
problem.
andy...@my-deja.com
and when someone expressed an interest in writing about it, agreed to
mind), I am guilty.
The evidence thus far is clear that no prior reference to the
conjecture exists. Nonetheless, If Granville wants to claim credit for
it because he once asked about similiar forms, God bless him. If he
doesn't because it is too trivial for him, God bless him.
I cannot imagine what your motivations or goals are and I will no
longer spend time responding to your distortions and lies. I have
demonstrated your errors earlier in this thread and I will not spend
time repeating them everytime you restate your lies and distortions.
Fortunately, the facts speak for themselves and are available to anyone
seeking them.
----andrew beal
Hull Loss Incident
> I discovered something very interesting.
See below for more unambiguous proof that your alleged
discovery was known and published years before you ever
looked at the problem.
> The evidence thus far is clear that no prior reference to the
> conjecture exists.
An absurd claim. Here are two more prior references,
found by doing a Google search for "generalized Fermat".
MR92i:11072
Tijdeman, R.
Diophantine equations and Diophantine approximations},
Number theory and applications (Banff, AB, 1988) p.215--243
Kluwer Acad. Publ. Dordrecht, 1989
In his discussion of the ABC conjecture, Tijdeman gives the
(easy) proof that ABC implies ax^L + by^M = cz^N has
finitely many coprime solutions with 1/L+ 1/M + 1/N < 1. He
then says:
"[this] equation with a=b=c=1 and 1/L + 1/M + 1/N < 1
has some solutions, namely 13^2 + 7^3 = 2^9 and
17^3 + 2^2 = 71^2. I do not know any more solutions.
If 1/L + 1/M + 1/N >=1 then ... on density grounds I
expect that ... infinitely many solutions are possible ..."
The Mathscinet reviewer of the article did not even find this
worth mentioning in the review. That's not surprising, because
it is a statement of the already well known (in 1989) facts
that probabilistic "density" arguments and the ABC conjecture
both point to the conclusion of finitely many solutions, and identify
1/L + 1/M + 1/N < 1 as the correct formulation of the problem.
This is unambiguous published confirmation of facts that Beal
has refused to acknowledge in this discussion.
The second item turned up by the Google search is proof
that Granville and Darmon's work on this problem (replacing
the use of the ABC conjecture by Falting's theorem) was
likewise done and circulating before Beal "discovered" any
conjecture:
http://gwdg.de/~cais/CAR/CAR13/node6.html (see item #12)
See item #12, the program for a conference on elliptic curves
(July 17-24, 1993), including a lecture by Darmon on his
work with Granville on the generalized Fermat equation.
Beukers is also listed as a speaker, so it's possible his
and Zagier's searches for large solutions were prompted
by this lecture.
> I cannot imagine what your motivations or goals are and I will no
> longer spend time responding to your distortions and lies.
You keep alleging "distortions" and "lies" without being able to
identify any. It has been pointed out by several people how
ridiculous your complaints are.
> I have demonstrated your errors earlier in this thread
Name one "error" that you have demonstrated.
> and I will not spend
> time repeating them everytime you restate your lies and distortions.
You do repeat yourself a lot, yes. There were no "lies and distortions",
despite your protestations.
> Fortunately, the facts speak for themselves and are available to anyone
> seeking them.
The facts say "Beal is mistaken or lying". They are available
because I have posted verifiable documentation of my claims.