Talk:Andrew Wiles
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[edit] Rumor, Hoax
This rumour is not I think worth a place on the page itself: Charles Matthews 15:27, 18 May 2005 (UTC)
- Important! There is a news reported on Manila Times on May 5th 2005. According to this report, Professor Wiles' proof is incorrect. For those of you interested in this topic, please help us to verify the source of this important information.
- It's not a rumour, it seems to really have happened. Wiles' sarcastic reply being interpreted literally, that is. Quite funny stuff. Fredrik | talk 16:08, 18 May 2005 (UTC)
It is a HOAX! Read http://www.pcij.org/blog/?p=73 . They forgot to do their homework. Timothy Clemans 07:17, 14 April 2006 (UTC)
[edit] How Wiles got paid while working on Taniyama-Shimura
I've always wondered what Wiles was doing to earn an income while working on his proof. I can't seem to find any mention of it, but maybe I'm not looking in the right places.
- Tenure at Princeton University. So he did it the hard way ... Charles Matthews 17:38, 8 September 2005 (UTC)
- According to Fermat's Enigma, Wiles had a fairly large paper (or two) ready for publishing when he began on the Fermat work. To remain "active" in the mathematical world, he split up the paper into several smaller papers and published them separately over the years of his Fermat work. —dcclark (talk) 07:46, 16 December 2005 (UTC)
There is the Wiles-Taylor paper also and there is public PDF of it avaible somewhere.
[edit] Cleaning up
The discussion of the Frey curve was a mess, and I cleaned that up, but this article still needs more work. Gene Ward Smith 06:50, 24 May 2006 (UTC)
I did a bunch more rewriting, and removed the following section, which was so confused I decided we were better off without it:
The problem was how do you count and it was the big question on the subject. Wiles introduced the correct counting technique. Wiles converted elliptic curves by their j-invariant to their corresponding Galois representation to study elliptic curves. He did studies on representations according to partations and said in his paper on this, "suppose for the moment that p3 is irreducible". He tells us why 3 is important.
He found a surprising link between Galois representations and modular forms and the interpretation of special values of L-functions. The proof is based on that link and is used to prove a hypothesis that p3 is semistable at 3 by linking some of commutative algebra with a well-know type for a class number problem.
Gene Ward Smith 07:33, 24 May 2006 (UTC)
- Thanks, I have has a hard time adding to this stuff but I am still trying. I am looking at maybe making an article just on the Fermat's Last Theorem journal offered by JHUP that way it can be more techincal(sp). Timothy Clemans 19:21, 24 May 2006 (UTC)
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- If you click on the "bluff your way" link in the article, you get a nice collection of papers including Wiles and Taylor. The "short" scans are much clearer (as well as being shorter.) There are also a number of other relevant articles, plus survey articles at various levels. The comments Wiles makes in his introduction could be partcularly useful. Gene Ward Smith 22:55, 24 May 2006 (UTC)
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- Yes I know that. The most helpful references are "John Coates (July 1996). "Wiles Receives NAS Award in Mathematics". Notices of the AMS 43 (7): 760-763. " and Andrew's first Fermat paper. The proof is in chapter five.
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[edit] Classical modular curve
I created classical modular curve a few days ago, and people wanting to get a somewhat more down-to-earth concept of what it means for a curve to be modular might find it useful. Gene Ward Smith 22:57, 24 May 2006 (UTC)
[edit] Taniyama-Shimura theorem
I worked on this page; hopefully it now makes more sense. Gene Ward Smith 04:20, 25 May 2006 (UTC)
[edit] Positive integers
It's all equivalent, but I think sticking to tradition and making the integers all positive is preferable to saying "non-zero". Gene Ward Smith 04:24, 25 May 2006 (UTC)
[edit] Epsilon theorem
I've edited this now also. Gene Ward Smith 08:33, 25 May 2006 (UTC)
[edit] The bridge between Fermat and Taniyama
In the side box with title "The bridge between Fermat and Taniyama", it is mentioned that p should an odd prime. But in Fermat's theorem, any integer would do, which is more general condition. Thus it is inaccurate to say that hypothetical elliptic curve, the Frey curve, must exist if there is a counterexample to Fermat's Last Theorem. For example, if we found a counter example to Fermat's theorem with p=10, it does not satisfy the required condition for Frey curve, since 10 in not an odd prime number. --Sandipani 01:55, 16 August 2006 (UTC)
- Every power of 10 is also a power of 5, an odd prime. So proving Fermat on every odd prime proves it on all integers except powers of 2. I suppose there's a separate froof for the latter, discovered before Wiles's work. 80.235.60.165 19:04, 28 August 2006 (UTC)
- "Every power of 10 is also a power of 5". Well... 100 is a integer power of 10, but isn't a integer power of 5.
It's enough to verify FLT for exponents n that are an odd prime, or 4. That's a fortiori (no sum of cubes is a cube, so no sum of 99th powers is even a cube, let alone a 99th power). The exponent 4 was done by Fermat and/or Euler. Charles Matthews 19:28, 29 August 2006 (UTC)
[edit] List of Students
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—The preceding unsigned comment was added by Timothy Clemans (talk • contribs) 00:37, 9 December 2006 (UTC).
[edit] Crazy FLT Conspiracy Theories...
I know that this is a really crazy idea. I know that this will probably be horrifically unpopular to anyone with common sense (save, perhaps, fans of 'Hitch Hiker's Guide to the Galaxy'....But, given the audacious complexity of the FLT proof, given that there were mistakes made when it was intially published, given the small number of people (as a proportion of the human race, if not just in absolute numbers) who understand the FLT, given the prize money that has been at least suggested as a reward for FLT - is it not possibly, slightly (but not too slight) likely as it is conceivably possible and probable that there may be a small flaw in the proof that an automated theorem prover might be capable of picking up on? That is, has anyone run FLT through a computer for objective assessment?
Again, I re-iterate the notion that *this is a crazy suggestion*, and I probably don't know what I'm talking about (though enough to know that I don't understand the proof to FLT and need some Hitchhikian computer to blurt out '42' to convince me....). Further, this notion would probably not just concern the proof to FLT, but many areas of mathematics which very few people understand, but where 'obvious' and 'intuitive' assertions are relied upon.
....Just offering my tuppence's worth.....
--Wackydough
Dear Wackydough, It's certainly true that there are papers in the Mathematical literature which have been published, and then mistakes have been found years later. These are typically proofs which are either unclearly expressed or obscure. However, every detail of this proof has been poured over, initially by a small group of experts, and more recently by the wider Number Theory community. Although technical, it is a clearly expressed and checkable proof, and it would really be quite remarkable if any mistake were still overlooked. At a minimum, several hundred people are now able to understand the proof, and it is in the public domain. Even though this is a small fraction of the population, it is still too large and diverse a group of people for a conpiracy theory to be plausible. In fact, there would be strong incentive (in terms of kudos) for anyone who could have found a mistake. Furthermore, many researchers have been producing further results in this area recently, and it would be astonishing if they had not noticed a flaw in their follow-up research (and their follow-up research involves a considerable amount of going back through the techniques, and does not merely quote Wiles' results). Several more accessible versions of the proof have been published, and there is already a new generation of Number Theory PhD students and postdocs who have read the proof. I don't know anyone in the academic community (which is exceptionally cautious about these things) who has any doubts about the proof. As far as "automated theorem provers" go (or perhaps you mean "automated proof checkers" in this case), I'm afraid that we are way, way, way, way, ...... (you can add another few ways, if you like) below the level where these would be of any help. A substantial proof in Mathematics is expressed in sentences of normal language, embellished with technical terms, equations and references to previous lemmas (smaller preparatory results). There is no "proof checker" which comes even remotely close to being able to read a research article in Mathematics and check it. Attempting to re-express the proof in terms of a sequence of computer-checkable elementary steps would make the proof mind-bogglingly long (too long for this strategy to be viable). Anyway, I hope the above is of some use to you! Best regards, Victor.
[edit] Andrew Wiles, nationality?
Why is Andrew Wiles listed as being an American? He is from Britain, and his nationality is British, so where did this information come from? —The preceding unsigned comment was added by 142.231.110.50 (talk) 23:51, 15 February 2007 (UTC).
