Fermat's Last Theorem, more can be proved more simply: Professor steers field toward a numbers-only

[Jim__]

proof.

From phys.org:

Fermat's Last Theorem—the idea that a certain simple equation had no solutions— went unsolved for nearly 350 years until Oxford mathematician Andrew Wiles created a proof in 1995. Now, Case Western Reserve University's Colin McLarty has shown the theorem can be proved more simply.

The theorem is called Pierre de Fermat's last because, of his many conjectures, it was the last and longest to be unverified.

In 1630, Fermat wrote in the margin of an old Greek mathematics book that he could demonstrate that no integers (whole numbers) can make the equation xⁿ + yⁿ = zⁿ true if n is greater than 2.

He also wrote that he didn't have space in the margin to show the proof. Whether Fermat could prove his theorem or not is up to debate, but the problem became the most famous in mathematics. Generation after generation of mathematicians tried and failed to find a proof.

more ...

[JDPriestly]

1. Interesting. I am not a mathematician, but someone who is loaned me a copy of the book

on Fermat's last theorem and Andrew Wiles' success in proving it.

I recommend it to those interested.

http://simonsingh.net/books/fermats-last-theorem/the-book/

[struggle4progress]

2. As far as I can tell from the popular press coverage, McLarty is a logician

who is announcing that the foundations, of Grothendieck's program in algebraic geometry, can be constructed using less complicated logical constructions, so theorems depending on Grothendieck's idea can be proved with "simpler" logical assumptions

Whether this actually simplifies Wiles' proof, in any practical sense, is rather less clear: it may merely "simplify" the logical apparatus needed to prove the high-octane theorems that underlie Wiles' proof, and the "simplification" might actually produce longer proofs. I can't tell, based on the popular press coverage, and I can't find a link or reference to recent work on McLarty's faculty page at the Case website

[Jim__]

3. Did you find the link to McLarty's 2010 paper on Fermat's Last Theorem?

It's in The Bulletin of Symbolic Logic - but it costs $10.00.

It seems like his latest work has improved on that. The MAA says he has proven that a subset of set theory is sufficient.

[struggle4progress]

4. I didn't find it on his website, though it's possible I looked carelessly. But the first page

of the Symbolic Logic paper doesn't change my guess about what he's actually doing. He's a logician, not a number theorist, and what he's actually up to is not anything anyone would naturally call a simplification of Wiles' proof. What he's doing is asking, Is there a small fragment of classical set theory that suffices for the proof of the Grothendieck-type constructions that are explicitly used in Wiles' theorem?

There are at least three strong indications, on the single page you linked, that McLarty is not really "simplifying" Wiles' proof, despite the popular press discussions of the article

The first is that we are looking at the Bulletin of Symbolic Logic. The specialities of almost all readers of that Journal will not provide them with the background necessary to follow Wiles' proof in detail and a quick look at McLarty's publications doesn't suggest he has the background. Grothendieck had a sort of genius for abstract construction of the sort set-theorists are often very good at, so its very plausible to me that McLarty is competent to look through Wiles' work to determine which of Grothendieck's constructions were employed in various places and then to work out what fragment of set theory would suffice for those constructions

The second indication is what McLarty discusses on that page: he hints (for example) at transitive models of set theory. This sort of thing is of concern to logicians who are interested in what can be proved with certain assumptions, such as the ZFC axioms. It is not a question that has the slightest interest for almost anyone who does work in number theory. This suggests that McLarty's work will interest logicians and not number theoreticians. An actual simplification of Wiles' proof, of interest to number theorists, is very unlikely to involve discussion of anything like transitive models of set theory.

The third indication is that McLarty demurs to discuss set-theoretic assumptions "used in principle" in Wiles' proof and is only discussing those "used in fact in the actual proof." That is, McLarty has no intention of studying Wiles proof line by line to think through what is involved in each step of the proof: he has scanned the proof to see what sort of set-theoretic constructions are explicitly mentioned, and he has thought about those to see what set-theoretic assumptions are needed to carry out those constructions. As the abstract indicates, McLarty is studying the set-theoretic assumptions that "figure in the methods Wiles uses"

My best guess is that reading this paper will not provide the slightest help to anyone hoping for a shorter route to FLT than provided by Wiles: it is a contribution to the study of the question, What fragments of set theory are needed to carry out some of Grothendieck's constructions?

[bananas]

7. Here's the 2010 paper

In pdf format: http://www.cwru.edu/artsci/phil/Proving_FLT.pdf

In ps format: http://www.math.ucla.edu/~asl/bsl/1603/1603-003.ps
Google docs viewer: https://docs.google.com/viewer?url=http://www.math.ucla.edu/~asl/bsl/1603/1603-003.ps

The BSL archives are free for personal use, they're in ps format but google docs will webify them.

The BSL archive is http://www.math.ucla.edu/~asl/bsltoc.htm
The Sept 2010 issue is http://www.math.ucla.edu/~asl/bsl/1603-toc.htm

More info at http://www.aslonline.org/journals-bulletin.html

Also, there's a firefox add-on that makes it easy to use the google docs viewer: https://addons.mozilla.org/en-US/firefox/addon/google-docs-viewer-pdf-doc-doc/

[struggle4progress]

8. Thanks!

[Jim__]

9. Thanks for that!

I love trying to work through that type of paper.

[Dr. Strange]

5. That's the impression I'm getting too.

On an unrelated, but slightly related note, this announcement reminded me of a quote from Rob Tubbs about how mathematicians would respond if someone found a relatively simple, non-20th century proof to FLT. Something along the lines of, "that's nice." I can't remember the source, though.

[struggle4progress]

6. I think anyone who found an "elementary" proof of FLT would be recognized

as a good mathematician

Paul Erdős, as a teenager, found an elementary proof of "Bertrand's Postulate," and it's a nice proof

[akenaton]

10. take a look at this

Take a look at this:
http://www.mymathforum.com/viewtopic.php?f=40&t=40857

[struggle4progress]

11. I'm not going to spend my limited time trying to find errors in amateurs' proofs of FLT

Over several centuries, enormous effort has been devoted to the problem of finding an elementary proof,

Edmund Landau at Gottingen used to receive so many amateur attempts that he finally printed up forms for his students to fill out: "Dear _____ the first mistake in your proof of Fermat’s Last Theorem occurs on line _____ of page _____"

Some interesting results can indeed be proven by short arguments using simple methods: Ribenboim's book Fermat's Last Theorem contains a nice sampling

[eppur_se_muova]

13. "We have found an error in your proof which this margin is too narrow to contain." nt

[struggle4progress]

14. +

[telclaven]

12. Personally, I think Fermat's been having us on

Either he bilked this up 'cause he had an idea not fully threashed out or he actually did try to write a proof, ran into one of the problems seen in many attempts, didn't recognize the error, and thought he had a proof.