Talk:Fermat's last theorem

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One of the 500 most frequently viewed mathematics articles.

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Dissenting opinion: numerous inaccuracies, omissions, and even errors. The historical section is highly idiosyncratic. The language is awkward in many places. Somewhere between Start and B class. Arcfrk 03:59, 12 June 2007 (UTC) (See also comments below)

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Archive 1: from start until 2006-11-16

[edit] Escultura

I am starting to doubt that you really ar Escultura, because your internet is out of Australia. You need a user account so that there is one place to write to you for Wikipedia perpuroses. On Wikipedia, I only care about popular view and view of the integers is not popular. You need to stop talking about the integers on this page. Timothy Clemans 17:48, 7 July 2006 (UTC)

I couldn't care less about anyone's opinion. Im posting as a mathematician and I'll respond to any attack on my work and comment on mathematical points. I travel a lot and part of my family is in Australia; if you want my e-mail, here it is: escultur36@hotmail.com. 9:48, 8 July 2006. E. E. Escultura

You do not own Wikipedia therefore you must follow the rules here. Why did you change your email from domain name yahoo to hotmail? Why do you not have a Wikipedia user account? With the way Wikipedia works, you need to use wikicode to sign your name and put in the correct date and time using the UTC standard and Wikipedia's clock. Timothy Clemans 18:48, 8 July 2006 (UTC)

University of the Philippines Pampanga, Clark Field, Pampanga, Philippines; Residence: Blk 1 Lot 1 Granwood Villas, BF Homes, Q. C. 1120, Philippines;

E-mail: escultur36@yahoo.com; escultur36@hotmail.com

http://www.users.bigpond.com/pidro/home.htm; http://home.iprimus.com.au/pidro/

Timothy Clemans 18:51, 8 July 2006 (UTC)

I am just a blogger and I don't know the technical matter of creating an account in Wikipedia. I want my Yahoo account to be exclusively for scientific correspondences so that it won't be clogged with junk mails. Anyway, if there is nothing here that concerns my work I won't post anything. 9:00, 10 July 2006. E. E. Escultura

Escultura, it's very easy to create an account. Just click on "create account" in the top-right corner and follow the instructions. Dmharvey 00:59, 10 July 2006 (UTC)

If you were able to create accounts for Yahoo and Hotmail, then why do you think that it would be hard to create an account for Wikipedia? "Anyway, if there is nothing here that concerns my work I won't post anything." and yet you came here and started the "Wiles proof is wrong" thing(at least I think you did(I could check)). It is a really bad idea to use a Wikipedia article talk page as a blog! Timothy Clemans 18:46, 10 July 2006 (UTC)

I have an account already and I'm logging in from it. You are wrong. This talk page started calling me names last year but some of them were deleted, according to a blog, before I noticed some remaining ones a couple of months ago. That was when I posted the message on Wiles' wrong proof. You were the one calling me names and I never did. I just know that those with empty top resort to name-calling and pull out ideas from the flat of their foot. E. E. Escultura

I would like to finally prove Escultura wrong, in his claim that Wiles did not prove "Fermat's Last Theorem" true. There is a book, authored by the reknowned physicist Roger Penrose, titled, "The Road to Reality," that in the third section of chapter one (denoted as 1.3 in the upper margin), Penrose clearly states that Wiles did prove "Fermat's Last Theorem" on his second attempt. Unless Escultura, believes he has greater knowledge and insight than Roger Penrose (a man who has made great many contributions to science and mathematics).

Penrose and his renown have no bearing on anything. Rather, Escultura was proved wrong by Andrew Wiles providing a proof of the theorem. There is no obligation on anyone to prove Escultura wrong; rather, the burden is on Escultura to prove his assertion. -- Jibal 05:55, 29 December 2006 (UTC)

I did not assert anything. I PUBLISHED the counterexamples to FLT to prove the conjecture wrong. Unless a PUBLISHED REFUTATION comes along the counterexamples stand.

E. E. Escultura

A true mathematician does not believe or disbelieve what someone says no matter how knowledgeable but examines what is said to find out if it makes sense. —The preceding unsigned comment was added by 202.67.70.227 (talk • contribs) 2006-11-16T19:39:04.

[edit] Please split talk page up

This talk page is now 91KB in size. Please could someone who knows how archive all inactive discussions. Tompw 19:09, 7 October 2006 (UTC)

Done. There are instructions at WP:ARCHIVE. CMummert 00:55, 17 November 2006 (UTC)

[edit] Can Wiles' proof be carried out in second order arithmetic?

I am unaware of any published literature which shows that Wiles' proof can be carried out in second-order arithmetic (and thus no literature saying the proof can be carried out in PA). The best reference I have seen is an FOM discussion that was formerly referenced from the article. Please post references to any such published literature below this message. If no reliable sources can be found in a reasonably long period of time, the article will have to be modified. CMummert 00:24, 20 November 2006 (UTC)

Since no citation is forthcoming (and I have no idea where to find one), I have moved the following from the article. The prose was added by User:R.e.b, who suggested that removing it is OK.

These constructions use axioms that go beyond Zermelo-Frankel set theory (ZFC), which has led to a myth that Wiles' proof is not carried out in ZFC. In fact Wiles only uses a tiny piece of this machinery, only involving étale cohomology of schemes of finite type over prime fields, and this can be carried out in second order arithmetic, a much weaker theory than standard set theories. Everything in his proof can be done in second order arithmetic (and could probably be encoded in first order (Peano) arithmetic, though this would require considerable effort).

The issue of whether the proof can be formalized in ZFC or SOA is interesting, but not of central importance, since there is widespread acceptance of the proof. In order to keep the article verifiable, and since there is some disagreement in the professional community about the issue, I hope other editors will refrain from adding additional material to the article about the formalization of the proof unless a published reference for the material can be provided. CMummert 17:57, 7 December 2006 (UTC)

[edit] Wiles proof error?

However, no correct proof was found for 357 years, until it was finally proven using very deep methods by Andrew Wiles in 1995 (after a failed attempt a year before).

Wasn't the original proof proposed in 1993, 2 years before? 88.109.162.21 17:38, 20 November 2006 (UTC) Matty_B

This is just a minor editing of the above statement. A proof is no proof until it has overcome all challenges and criticisms and Andrew Wiles' proof has plenty of them both in the scientific literature and in the many blogs, fora, threads, chatrooms and websites across the internet that have not been refuted. Until that time it remains, at best, controversial. 58.178.88.219 12:54, 19 December 2006 (UTC) E. E. Escultura

No, a proof is a proof when it is valid. And there is no burden to refute the unending stream of "challenges and criticisms" on blogs, etc.; it is the challenger's burden to prove their assertions, which they cannot do because FLT, controversial or not, has been proved. -- Jibal 05:50, 29 December 2006 (UTC)

A PUBLISHED challenge to a proof requires PUBLISHED refutation to shoot down. Otherwise, the challenge stands. Who passes judgment on the validity of a proof? Furthermore, the published counterexamples to FLT showing that the conjecture is false require published refutation to shoot down. Otherwise, they stand. E. E. Escultura

[edit] Fermat's original proof

People worrying about what happened to the original proof by Fermat and what it could have contained can take comfort in that the proof might never have existed. Fermat could have been lying, or his proof could have been erroneous. JIP | Talk 12:00, 23 November 2006 (UTC)

I'm not clear on how this would give anyone comfort, or why you think that anyone might not have considered this before you suggested it. -- Jibal 06:03, 29 December 2006 (UTC)

[edit] Proposed correction

In section "Proof" the end of the first paragraph states: "This theorem said that every counterexample aⁿ + bⁿ = cⁿ to Fermat's Last Theorem would yield an elliptic curve defined as y² = x(x − aⁿ)(x + bⁿ) which would not be modular and therefore provide a counterexample to the Taniyama–Shimura conjecture. Fermat's Last Theorem and Taniyama-Shimura were now linked through the Epsilon conjecture; either both were true or both were false."

I believe the last sentence is not true. It should be: "Fermat's Last Theorem and Taniyama-Shimura were now linked through the Epsilon conjecture; the (conjectured) truth of Taniyama-Shimura was shown to imply the truth of Fermat's Last Theorem."

The difference is that even if Fermat's Last Theorem was shown to be true by some other means, it would only eliminate this particular counterexample to Taniyama-Shimura, which might have other counterexamples. Its truth would not be immediately implied by the truth of Fermat's Last Theorem.

You are correct there. The fact that the falsity of FLT implies the falsity of T-S is simply the contrapositive of the result. However, I don't much like your phrasing, for the reason that "conjecture" appears a little too often. It makes the statement seem tentative. What I would suggest is: "Fermat's Last Theorem and Taniyama-Shimura were now linked through the proof of the Epsilon conjecture; the truth of Taniyama-Shimura was shown to imply the truth of Fermat's Last Theorem." The fact that Taniyama-Shimura was conjectured to be truth is not really relevant to the implication, so I would definitely remove the parenthetical remark; and adding "the proof" before "Epsilon conjecture" would remind readers that, despite the name, it was then an established fact. At least, that's my suggestion. Magidin 15:07, 1 December 2006 (UTC)

I agree with your phrasing. I was only trying to draw attention to the fact that at this point in time the truth of T-S had not yet been settled, but of course that is not essential to the argument, so your proposal improves on it. Please go ahead and make this change as soon as it is convenient, as I do not have a wikipedia account, and I am not sure how long we must wait to give others a chance to comment on it. Thanks.

Isn't creating an account just a matter of choosing a user name and password and entering them into the appropriate boxes? But you don't need an account to edit the article, and you don't have to wait for anyone - you could have made the change when you first spotted the error (WP:Be bold). --Zundark 17:06, 1 December 2006 (UTC)

I was the one who wrote that line; it was meant to clarify things for those non so adept in the higher levels of math. However, you are correct; you're phrasing is superior.--Here T oHelp 01:58, 20 December 2006 (UTC)

[edit] All Proofs Proven?!?

The article states before the TOC this:

"All the other theorems proposed by Fermat were proven, either in his own proofs or by other mathematicians, in the two centuries following their proposition. The theorem was not the last that Fermat conjectured, but the last to be proven."

I am curious if anyone could verify whether or not they were ALL proven. I recently read Fermat's Enigma by Simon Singh and I seem to remember reading that some of Fermat's conjectures were disproved. He made a lot in his lifetime and he rarely wrote full proofs, so it would stand to reason that an amateur such as him would occasionally make a mistake. Nthitz 22:13, 7 January 2007 (UTC)

I think I remember that line...I have changed it accordingly.--Here T oHelp 00:43, 8 January 2007 (UTC)

There is a difference between statements that Fermat asserted he had proven, and conjectures. For example, Fermat asserted he had shown exactly which primes could be written as a sum of two squares, or as a sum of a square and twice a square; on the other hand, he explicitly conjectured that all Fermat numbers would be prime. The latter conjecture was proven false. It is my understanding that in all cases where Fermat asserted publicly that he had proven or that he could prove a result, the results has in fact been proven. Fermat's Last Theorem was a bit of an oddity not just because it had not been settled, but also because his assertion of proof was a private note to himself, not a public claim. Magidin 20:15, 9 January 2007 (UTC)

[edit] usefulness?

i'm sure this is terribly relevant to daily life somehow, but the article doesn't say how. —The preceding unsigned comment was added by 71.103.88.223 (talk) 20:01, 26 January 2007 (UTC).

Like many topics in number theory, Fermat's Last Theorem has little application to the real world. Yet it is quite an interesting problem... Nthitz 20:18, 15 February 2007 (UTC)

It would be nice to show scans of the original bookpage where Fermat scribbled his comment, instead of the printed annotation version currently available. If the original book is lost, it should be noted in the description of the image. 80.202.98.204 18:51, 29 January 2007 (UTC)

[edit] Hypothetically...

Hypothetically, what if you had figured out a preposterously simple approach to proving Fermat's last theorem? The problem is - it's preposterously simple and you don't see how all the mathematicians before you could have overlooked it. Nor do you have a strict proof yet. Where would you go to check if this approach had any merit? —The preceding unsigned comment was added by 71.135.62.1 (talk) 01:11, 8 March 2007 (UTC).

Answer: graduate school? CMummert · talk 01:21, 8 March 2007 (UTC)

There is an elementary approach to proving Fermat's last theorem: abc conjecture. Arcfrk 10:39, 10 April 2007 (UTC)

[edit] "Fermat's conjecture" re-directing here

Why is this a logical re-direct?? The most logical meaning of the term "Fermat's conjecture" is the statement that 2^(2^n)+1 is prime for all non-negative integers. This definition of the term makes sense in that this conjecture was later disproven. Any debate on the most logical meaning of this term?? Georgia guy 22:40, 18 March 2007 (UTC)

Why not make the redirect into a disambiguation page that links to here and to the article on that conjecture (if there is an article on it)? CMummert · talk 00:54, 19 March 2007 (UTC)

I'm not sure what you, Georgia guy, mean by most logical meaning of. Fermat conjectured a lot of things. Some of these became very famous, the most famous probably being what is now called his last theorem. However, it has not been a theorem until Andrew Wiles proved its correctness, it has been a conjecture.

I support CMummert's idea to make a disambiguation page on this. The page on Erdős' conjectures could work as model for this. — Ocolon 16:08, 19 March 2007 (UTC)

Well, so could the page Fermat's theorem. — Ocolon 16:12, 19 March 2007 (UTC)

[edit] External Links

"http://ne-plus-ultra.net/pubs/kisby_fermatarticle_rev.pdf" (Kisby, Adam William (2004). Fermat's Last Theorem Revisited: A Marginal Proof in Ten Steps (PDF). Parody.) seems to be down. Mike wiki 13:34, 22 March 2007 (UTC)

Thank you for pointing at this! It's much appreciated. I recovered the link from web.archive.org and if you're interested in the file you can download it now. — Ocolon 16:56, 22 March 2007 (UTC)

[edit] Solution?

1^3 + 2^3 = (cubed root of 9)^3 1+8=(2.08...)^3 9=9

So if FLT was proven, then why would this equation be correct.—Preceding unsigned comment added by 69.14.146.146 (talk)

Because the values of a, b, and c have to be integers…--Here T oHelp 00:47, 10 May 2007 (UTC)

Ah. Still, I've pointed out a major flaw in the original wording of Fermat's (unless he does indeed use the word integer in the untranslated version). —The preceding unsigned comment was added by 69.14.146.146 (talk) 00:49, 10 May 2007 (UTC).

Fermat wrote his comment on a copy of the work of Diophantus; in that work, it is always rational and integer solutions that are at issue. That is, there is an underlying assumption that all variables will only take values in the integers and the rationals. His original comment must be taken in the context in which it is written. Magidin 14:45, 10 May 2007 (UTC)

[edit] A-Class?

I've been checking some A-Class ratings and nearly passed over this. This article ~~is definitely a great piece of work and~~ may well be A-Class, but I'm not quite convinced that it has nailed it, and suggest it might be worth taking to mathematics A-Class review. So I'm only able to sign it off as B+ for now: someone else can upgrade it if they are more confident. Possibly I am being tough on the article because it is so important, and of such popular interest.

Although referencing is one concern I have, this isn't a dominant criterion for A-Class: more important is completeness. I just think there are more opportunities here to add more mathematical content. For instance, the contributions of Germain and Kummer could be expanded: the latter in particular inspired the development of ring theory, ideals, and the ideal class group. And surely more could be said about the proof — perhaps we should separate the proof from the history of the proof. At present the text seems to imply that Wiles proved the Taniyama–Shimura conjecture, whereas my understanding is that he only proved a special case (which was sufficient for FLT). Geometry guy 18:38, 9 June 2007 (UTC)

[edit] Proof

I think, to justify the adjective "well written" (Category:A-Class mathematics articles) the proof section should contain some more mathematics. For example, what does the phrase

"that is, all elliptic curves are also modular forms"

mean? I know elliptic curves and also roughly modular forms, but I cannot make sense of this phrase. If the prahse is correct, please reformulate it in a way understandable not only to a layman, but also to someone who wants to see some math in it, not just prose. Also, references could be more specific, e.g. the paper/book(?) of Hellegouarch is missing. Jakob.scholbach 21:19, 11 June 2007 (UTC)

I agree. I've nominated the article for A-Class review and hope you will be willing to join in and help the article on its way back to A-Class. Geometry guy 21:44, 11 June 2007 (UTC)

I am sorry but I completely disagree with high accolades for this article. My assessment is that it is presently between Start and B-class (frankly, I have seen more developed articles rated Start class by a certain Geometry guy). Already the lead contains questionable statements, the history section is utterly inadequate (for example, Kummer's remarkable achievements get barely a one-line mention; while most of the work on FLT post Kummer is not mentioned at all). The article's only strength seems to be in the Proof section, at least, as far as the History of Wiles' proof is concerned. Even that section is not problem-free, and as has been remarked above, it contains precious little mathematics. Arcfrk 22:34, 11 June 2007 (UTC)

It seems I was right to delist this from A-Class, then! Can you fix any of the problems you mention? Many thanks if you can. Geometry guy 23:02, 11 June 2007 (UTC)

I have been tempted to go for it several times already, but my earnest assessment is that it would take upwards of 50 hours to bring this article to a decent state. The patches can be applied in several places, but it's unlikely that the quality would substantially improve without a major, well planned, and extended effort. Arcfrk 23:18, 11 June 2007 (UTC)

Let's hope WP:WPM/ACR can deliver. Maybe over the next week we can find 10 people to put in five hours overall, at least on average? Geometry guy 23:32, 11 June 2007 (UTC)

Concerning, your (Geometryguy) request: right now I'm busy writing a database for a Bibtex replacement for references. But even if not, I would probably refrain from editing here, as I don't have the number-theoretic knowledge necessary to give the article the more solid math's it needs. One general problem in this Wikiproject, as far as I see, is that lots of articles often only describe vaguely what is meant instead of making it clear. Math is a discipline where the latter is always possible! Probably the reason for this problem is, that one needs to know the subject in question pretty well in order to escape this trap. Jakob.scholbach 05:06, 13 June 2007 (UTC)

I've now pulled out the mathematical content into a separate section, revealing just how little there is there. Hopefully, this will help some editors with more expertise than me to develop this aspect. Geometry guy 14:38, 17 June 2007 (UTC)

[edit] In fiction

Does anyone else think the "in fiction" section is rather long and unwieldy? Perhaps a split is necessary? --C S (Talk) 00:00, 17 June 2007 (UTC)

I'd say, leave a couple of them, and move the rest out into a separate article. Also, calling TV series and rock lyrics 'fiction' is stretching it a little bit, I feel. Arcfrk 04:31, 17 June 2007 (UTC)

P.S. Nice edits, by the way! Keep up the good work.

I agree: I hope I haven't damaged these improvements in my own work on the article today. Geometry guy 17:22, 17 June 2007 (UTC)

[edit] I might have a proof

I might have a simple proof that seems too babyish to not have been seen before. Where/ Who should I go/apporach in order to test its validity. thanks —Preceding unsigned comment added by 68.209.186.75 (talk) 16:44, 1 July 2007 (UTC)

*Shrug* Maybe go to the newsgroup sci.math. You're certain to get a response. Of course you realize there's virtually no chance of the proof being correct. Probably once someone else gets a look at it they'll see the flaw pretty quickly. Eric119 02:06, 4 July 2007 (UTC)

I also have a marvellous proof, but this talk box is too narrow to contain it. Manning (talk) 18:20, 26 November 2007 (UTC)

[edit] The mention of the PBS Nova Special

Hi I'm wondering why the PBS NOVA version of the BBC Horizon's "Fermat's Last Theorem" is mentioned instead of the original version done by the BBC. Timothy Clemans 18:28, 13 July 2007 (UTC)

It's because Wikipedia is essentially an American entity, and as such most of the contributors tend to make the articles US-centric. It's occured to me that despite the fact that the US population is ~5% that of the world and the country has only existed for a little more than 200 years, it features significantly more than it should in the front page articles.Fizzackerly 14:09, 5 October 2007 (UTC)

Rather than complain, why not just fix it? Geometry guy 23:21, 21 October 2007 (UTC)

I'm not complaining, I'm answering the question Fizzackerly 12:08, 22 October 2007 (UTC)

[edit] Introductory paragraph

As a non-mathematician I am a bit put off by this article. I feel that the opening section should be far more generalist (given that this is a mathematical theorem with broad public interest) and that its significance (and its misleading name) should be acknowledged from the outset. To me it feels like it dives into the technicalities far too quickly, and leaves precious little content of interest for those who are not mathematically skilled enough to comprehend them. Comments anyone? Manning (talk) 18:18, 26 November 2007 (UTC)

I think the introduction section is fine. The third and fourth paragraphs of the introduction ("Fermat's last theorem is strikingly different and much more difficult to prove than the analogous problem for n = 2..." and "Fermat's last theorem is one of the most famous theorems in the history of mathematics ...") speak to significance. The third paragraph also explains why it is called Fermat's last theorem. It would be more consistent with modern usage to call it "Fermat's conjecture" or "Wiles's theorem" (and both of these names redirect to this article), but the historical terminology is so deeply ingrained that I doubt either of these alternative names will ever be more than pedantic curiosities. I don't see any "technicalities" in the lead section beyond the basic algebra required to understand the statement of the problem. However, if you want to propose a re-write of the introduction, you can put a new version on this talk page for discussion. Gandalf61 (talk) 09:30, 27 November 2007 (UTC)

I'm wondering if link to Exponentiation might help readers such as Manning. Cuurrently if you don't know the notation aⁿ then you are rather stuck. --Salix alba (talk) 11:05, 27 November 2007 (UTC)

[edit] Serre Conjecture

I believe Serre Conjecture has been proven more or less true. See the wikipedia article on it. Then please modify this article accordingly. —Preceding unsigned comment added by 68.43.205.215 (talk) 09:15, 4 December 2007 (UTC)

[edit] Pop Culture

Woulld it be worth including that the Star Trek: The Next Generation episode "The Royale" had Jean-Luc Picard mistakenly saying that a proof had yet to be discovered in the 24th century?Traffic Demon (talk) 08:23, 1 February 2008 (UTC)

It used to be there; it's been either moved or edited out. Magidin (talk) 15:57, 1 February 2008 (UTC)

See Fermat's Last Theorem in fiction. --Lambiam 03:18, 2 February 2008 (UTC)

[edit] Controversy

I have read that a Nigerian Mathematician, Professor Chike Obi offered a correct proof of this theorem in 1998.How come his name appeared nowhere in this page? —Preceding unsigned comment added by 144.5.140.24 (talk) 23:29, 27 March 2008 (UTC)

I believe you’re probably referring to the article "Fermat's Last Theorem" in the journal Algebras Groups and Geometries. (15 (1998), no. 3, 289—298), according to the professor’s web page.

It indeed was an attempt to prove Fermat’s last theorem (in 9 pages!). However, it was flawed, there is an error on page 292 of the journal (the fourth page of the article).

In recent news, however, the this professor recently died, and the media purported that he had proven it (http://odili.net/news/source/2008/mar/14/229.html) That gives the man undue credit, and the reporters did not verify their work thoroughly enough, as though the original claim was a proof, it in fact was not. Further sites such as (http://www.math.buffalo.edu/mad/PEEPS/obi-chike-fermat.html) which is a reprint of a news article from 1998(?), give faulty sustenance to the myth. GromXXVII (talk) 00:20, 28 March 2008 (UTC)

Suppose x^p+y^p=z^p, where p≥3 is a prime, admits integer solutions. It can be seen 2z>(x+y)>z. It can also be seen that q|(x+y−z), where q=3, 8, p, ought to be true. The two relationships lead to q|(r_zz), q|(r_yy), and q|(r_xx), where r_x, r_y, and r_z are each smaller than 1. But the last batch of divisibility by q is impossible if x, y and z are integers. Zymogen (talk) 15:33, 28 March 2008 (UTC)

[edit] Term "Fermat's Last Theorem"

The article explains why the name "Fermat's Last Theorem" was applied to this statement, but it does not make clear when it began to be known by that name. Now, I think it was already a celebrated problem by Euler's time, but I don't know if it was called "Fermat's Last Theorem" before the 19th century. Has the first use of the term been identified, or can we cat least mention some early use of the name? KarenSutherland (talk) 17:45, 1 June 2008 (UTC)