Re: Intuition (was Knizia Interview) and the Beal Conjecture Prize
- From: "Jenny Specter" <jenny_specter@xxxxxxxxxxx>
- Date: Tue, 02 Aug 2005 17:47:01 GMT
> BEAL'S CONJECTURE: If Ax +By = Cz , where A, B, C, x, y and z are
> positive integers and x, y and z are all greater than 2, then A, B and
> C must have a common prime factor.
The above should actually be A^x + B^y = C^z. Superscript doesn't copy and
paste very well.
><jenny_specter@xxxxxxxxxxx> wrote in message
>news:1103644296.349262.83410@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> This is from a University of North Texas Website about the Beal
> conjecture. Interest stuff about the prize and an attempt one person
> has made.
>
>
> THE BEAL CONJECTURE AND PRIZE
>
>
> BEAL'S CONJECTURE: If Ax +By = Cz , where A, B, C, x, y and z are
> positive integers and x, y and z are all greater than 2, then A, B and
> C must have a common prime factor.
>
> THE BEAL PRIZE. The Beal conjecture and prize was announced in the
> December 1997 issue of the Notices of the American Mathematical
> Society. Since that time Andy Beal has increased the amount of the
> prize for his conjecture. The prize is now this: $100,000 for either a
> proof or a counterexample of his conjecture. The prize money is being
> held by the American Mathematical Society until it is awarded. In the
> meantime the interest is being used to fund some AMS activities and the
> annual Erdos Memorial Lecture.
>
> CONDITIONS FOR WINNING THE PRIZE. The prize will be awarded by the
> prize committee appointed by the American Mathematical Society. The
> present committee members are Charles Fefferman, Ron Graham, and Dan
> Mauldin. The requirements for the award are that in the judgment of the
> committee, the solution has been recognized by the mathematics
> community. This includes that either a proof has been given and the
> result has appeared in a reputable refereed journal or a counterexample
> has been given and verified.
>
> PRELIMINARY RESULTS. If you have believe you have solved the problem,
> please submit the solution to a reputable refereed journal. If you have
> questions, they can be mailed to:
>
> The Beal Conjecture and Prize
> c/o Professor R. Daniel Mauldin
> Department of Mathematics
> Box 311430
> University of North Texas
> Denton, Texas 76203
>
> Questions and queries can also be FAXED to 940-565-4805 or sent by
> e-mail to
> mauldin@xxxxxxx
>
> LINKS TO ARTICLES ABOUT THE CONJECTURE AND PRIZE
> RELATING TO BEAL CONJECTURE
>
>
>
> The Beal Conjecture
> Notices American Mathematical Society, December 1997
> Manchester Guardian January 8, 1998
>
> A computer study has been carried out by Peter Norvig who is Chief of
> the Computational Sciences Division at the NASA Ames Research Center.
> The program and results may be found at
> Beal's Conjecture: A Search for Counterexamples
>
>>
>>_Algebras, Groups, and Geometries_ was once (partially) respectable,
>>so I was sad to find that Volume 15, no 3 (1998) contains three
>>(yes, count 'em: three!) supposed elementary proofs of FLT:
>>
>> Athanasopoulos, S. N. An algebraic proof of Fermat's last theorem.
>> 247--288.
>>
>> Obi, C. Fermat's last theorem. 289--298.
>>
>> Trell, E. Isotopic proof and re-proof of Fermat's last theorem
>> verifying Beal's conjecture. 299--318.
>>
>>There is an editorial statement that some of the editors opposed their
>>publication, but they were being printed anyway, and comments would
>>be printed later if any were submitted. The third proof refers
>>to the "hadronic mathematics" of Santilli, who is the publisher
>>and chief editor, so we can probably guess how the decision was
>>reached. For all I know, a fourth paper there also claims to contain
>>a proof; it is
>>
>> Jiang, C. X. On the Fermat-Santilli isotheorem. 319--349.
>>
>>William C. Waterhouse
>>Penn State
>
>
>>Gerben Dirksen wrote:
>> willie@xxxxxxxxxxxxxxxxx (William Adderholdt) wrote in message
> news:<rtvj3b-8rs.ln@xxxxxxxxxxxxxxxx>...
>> > In article <666d338.0302240837.5216847@xxxxxxxxxxxxxxxxxx>,
>> > Alex Ferguson <abf@xxxxxxxxx> wrote:
>> > > But you cited "intuition" as your basis for [insert verb of
> choice here]
>> > > the "absolute". I'd suggest that _is_ in fact utterly
> subjective, unless
>> > > you have some reason for suspecting that one's intuition accesses
> absolute
>> > > truth in some way that one's rational mind does not. (God
> talking to one,
>> > > racial memory encoded in ones RNA, quantum event echoes of the
> Big Bang...)
>> >
>> > This is way off topic, but I've alwa> ys been fascinated at how
> much of
>> > a role intuition plays in mathematics. From the presentation given
>> > in textbooks, it seems like mathematicians tinker with proofs to
> find
>> > theorems, whereas in real life a mathematical theorem is first
> perceived
>> > to be true and then one tries to find a proof for it. The question
>> > is, what faculty is used in this "perception"? One could say that
> one
>> > has worked out the proof subconsciously and merely sees the end
> result
>> > consciously, which is why one has to figure out the proof after
> perceiving
>> > the theorem.
>>
>> > But I've been puzzled by the case of Fermat's Last Theorem. Pierre
> de
>> > Fermat claimed to have proven this theorem in the 17th century,
> though
>> > the consensus now is that he didn't. Andrew Wiles finally proved
> it in
>> > 1994, using mathematical theories that did not exist in Fermat's
> time.
>> > So how could Fermat have perceived the truth of Fermat's Last
> Theorem
>> > when the theories needed to prove it hadn't been invented yet?
> Also, I'm
>> > sure people who have tinkered with the Four Color Theorem can see
> that it
>> > *must* be true, though how many people can explain why?
> Considering the
>> > amazing complexity of the various proofs, which all involve a
> computer,
>> > it seems unlikely that one works it out rationally in one's
> subconscious.
>>
>> If you have a mathematical statement that you think is true because
>> after doing a lot of research you did not find a counterexample, you
>> can conjecture it.
>>
>> A famous simple conjecture is the Goldbach Conjecture (any even
> number
>> greater than 2 can be written as the sum of two prime numbers), and
>> another one is the Beal conjecture, which states that there are no
>> integer solutions to the equation
>> a^k + b^l = c^m where k>2, b>2 and c>2 and gcd(a,b,c)=1
>> (solutions are known with one exponent equal to 2, the smallest being
>> 1^k + 2^3 = 3^2.)
>>
>> One is pretty sure that all those are true, since people have been
>> looking very hard for counterexamples and did not find them. It would
>> be too bad if the Riemann Hypothesis is wrong because many other
>> theorems depend on it...
>>
>> Of course, in the case of the 4-colour Theorem and Fermat's Last
>> Theorem, there have been "proofs" that were eventually found to be
>> wrong.
>>
>> Gerben
>
.
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