From: battin@venus.iucf.indiana.edu (L GENE BATTIN)
Subject: Re: Fermat's Last Theorum
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Date:  9 APR 92 12:19:32    

In article <3306@daily-planet.concordia.ca>, 
gsmith@concour.cs.concordia.ca (Gene Ward Smith) writes...

>In article <1992Apr8.004108.2697@augean.eleceng.adelaide.edu.AU>
>dabbott@augean.eleceng.adelaide.edu.AU (Derek Abbott) writes:
> 
>>There are two possibilities:
> 
>>                  (a) he really found the proof, which is now lost
>>                  (b) he had a mental abberation and was mistaken
> 
>>Out of curiosity, I was wondering whether the maths community today
>>generally regards (a) or (b) a being the most likely?
> 
>I would guess most number theorists who have worked in the area (I'm
>one, BTW) think (b)--or not quite (b). I think at least he probably
>had a plausible argument with a gap which was not obvious, like Lame's
>"proof" in 1847. The mistake in that "proof" boils down to assuming
>that all cyclotomic number fields have class number 1, and is not the
>sort of thing which would have been obvious to Fermat.
> 
>Fermat never publically claimed to have a proof--he may have realized
>himself there was a gap in his reasoning.

Has there ever been a general survey published that collects in one place
at least the gists of the many incorrect attempts at proving Fermat's
Last Conjecture, (with preferably some discussions on the flaw in each)?
(NOT the totally cranky ones, but attempts which were done by people
who at least had _some_ clue as to what a mathematical proof is.)

I ask this because I have read that there have been many flawed attempts,
although in most books I have only seen one or two such erroneous
arguments analyzed.  If no one knows of such a survey, perhaps someone
could at least point me to where I could begin in making such a survey
for my own enjoyment.  For an example of the potential usefulness
of such an overview:  I have come across an interesting "relationship" 
in my recreational math reading, which I _think_ might be applicable in 
examining Fermat's Last Conjecture.  As a practical matter, the survey 
mentioned above would be real helpful in such a case, since you could
quickly determine if such an approach has already been found to fail.

One reason that most modern mathematicians would probably guess that
Fermat was, at best, mistaken in his assertion is that modern mathematics
can employ tools considerably more powerful than any available to Fermat.
In Coxeter's revised edition of Bell's work on recreational math (*), however,
it is noted that certain areas of math, known well by Fermat, are not now
normally covered (ie., the "lost math" of continued fractions: considered
too hard for High School, and too easy or uninteresting  for College.)  He 
voices the possibilty that Fermat may have found something there that he could 
have used in his proof, which has simply been overlooked by Fermat's 
successors.  The "relationship" that I am looking at is (while _not_ having to 
do with continued fractions) similar.  This "relationship", interestingly 
enough, is stateable in simple-enough terms that one could easily believe it 
was known to Fermat, and at the same time, while it can obviously be modified 
into a statement that is extremely suggestive of Fermat's theorem, I have 
_never_ seen it mentioned in connection with Fermat's Theorem.  Thus I have no 
way of knowing, in the absence of a summation of flawed attempts, if it is 
really worth my time to work on this at all.  (BTW, no, I'm not going to 
state the relationship.)

I imagine that, at one time or another, most mathematicians (amateur and
professional) have "fooled around" with Fermat's Last Conjecture.  So the 
existence of a summation would serve other people than myself.

(*) I can't believe that the title of this book escapes me right now!  It's
a classic of recreational math.  It's at home, so forgive me if I've gotten
some detail wrong in my post wrt what's in this book.

>--
>     Gene Ward Smith/Brahms Gang/CICMA/Concordia University 
>                  gsmith@concour.cs.concordia.ca

Gene Battin
battin@venus.iucf.indiana.edu
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