Talk:Wiles's proof of Fermat's Last Theorem

Higher detail

I hope that we can attain a high level of technical detail. We are not providing a "howto" of the proof (which entire books have been devoted to), but we should be able to catalog many of the previous subjects and the results of other mathematicians that Wiles drew upon. We should be able to provide more detail than say, what the NOVA program "The Proof" provided. I would like to provide a key for the non-mathematicians college graduate so that they have a chance of drilling down through the Wiles proof. I hope to provide a key of notations that Wiles uses without introduction so that the reader can identify the subject area being referred to and read up. This is not a "howto", it is merely a review of the subjects Wiles covers in his discussion so that we can play to Wikipedia's strength: some depth of coverage in topics and the convenience of hyperlinks. Ideally, this will help to enthusiastic college student to gather the background information needed to understand the Wiles paper and not feel at a loss just because the notation, while standard within the specialty, it not obvious to the layman.--Lagelspeil (talk) 03:42, 16 March 2009 (UTC)[reply]

Wiles's?

Shouldn't it be "Wiles's" or "Wiles'" or "The Wiles proof"? Staecker (talk) 22:43, 19 March 2009 (UTC)[reply]

You are correct. I will do the rename as figure out which one to use.--75.25.136.186 (talk) 10:53, 25 March 2009 (UTC)[reply]

Either Wiles' or Wiles's is acceptable. Wikipedia:Manual of Style says "For a singular noun ending in one s, there are two widely accepted forms... Either of those forms may be acceptable in Wikipedia articles, as long as consistency is maintained within a given article." At the moment, the article text uses both forms in different places. Gandalf61 (talk) 12:55, 25 March 2009 (UTC)[reply]

I agree on the additon of an apostrophe to Wiles. If Bob Smith had solved the problem we wouldn't call it "Smith proof of Fermat's Last Theorem", though it could be "The Smith proof of Fermat's Last Theorem". I'd stick with an apostrophe after the name. Alansohn (talk) 13:06, 26 March 2009 (UTC)[reply]

I updated the text to match the title. It had been consistently Wiles's . --Lagelspeil (talk) 08:24, 27 March 2009 (UTC)[reply]

Paragraph 1 on page 1 of Strunk and White ("The Elements of Style," any edition) is unequivocal: It should be Wiles's. One can't go wrong by following Strunk and White. Jay Janzen, Ph. D. (talk) 15:29, 25 September 2013 (UTC)[reply]

Proof of Theorem

Is there actually a site with the complete proof? If so, could you add it to the external links? Math Champion (talk) 01:04, 24 April 2009 (UTC)[reply]

This appears to be Wiles's paper and this appears to be a more-accessible refinement aimed at being taught as a graduate course in number theory and/or "arithmetical geometry". But I haven't read thru them enough to be sure. 134.114.148.66 (talk) 23:09, 31 March 2017 (UTC)[reply]

p vs n

Why is p used as the integer greater than two, instead of n? n seems to be the symbol used in other articles, such as Fermat's Last Theorem, and is usually used for arbitrary integers. --90.149.188.185 (talk) 22:28, 5 July 2009 (UTC)[reply]

I agree. p is also usually used for prime integers, I believe. I edited it, hope people agree. --Spug (talk) 23:14, 17 July 2009 (UTC)[reply]

Full proof

Someone need to write the full proof the of theorem here. 128.189.212.115 (talk) 02:36, 7 December 2012 (UTC)[reply]

Yes, if only it were helpful. At present, unlike the statement of the theorem, the full proof of the theorem is a large subject. To explain further, as the article does: It seems one or two handfuls of people have learned it fully. Any reader of the proof needs considerable preparation. The published proof itself is 108 pages long. Some simplifications are known. No tremendous simplifications are known.
You and I and all of us are the editors of Wikipedia. (Actually, I for one have done nothing yet for this particular article.) We have achieved an outline of the subject, a sketch of its logic, and an indication of how people approach the details. We have links and citations to the published proof. -Minopret (talk) 14:44, 12 January 2014 (UTC)[reply]

This is a Wikipedia talk page, not a forum

A Wikipedia talk page is for discussions relating to editing the corresponding Wikipedia article, not for publishing supposed proofs of theorems, for linking to such supposed proofs on other web sites, or anything similar. An editor from Vietnam has recently been using several talk pages of articles to publicise attempts at mathematical proofs. To that editor, please don't. There are plenty of internet forums, blogs, etc where the kind of thing you have been doing would be accepted, but a Wikipedia talk page is not one of them. Continuing in the same way may lead to the range of IP addresses that you use being blocked from editing. That would not be a disaster, since the majority of the other editing from that range is vandalism anyway, but it would be a pity, since there have occasionally been constructive edits from that range, and they would be blocked along with the unconstructive edits. The editor who uses the pseudonym "JamesBWatson" (talk) 14:22, 28 February 2014 (UTC)[reply]

September or October

The "Announcement and final proof"-part seems to imply that it was released in October. The site [1] also seems to say so. –St.nerol (talk) 19:38, 27 April 2014 (UTC)[reply]

The lowest degree of Fermat's Last Theorem with a Frey curve

Ribet proved the non-modularity of Frey's curve for the n ≥ 5 case of Fermat's Last Theorem.^[1] The "Overview" section claims that Frey dealt with every n > 2. I don't have access to Frey's paper so I do not know whether this claim is incorrect, but given Ribet's result it seems so. Can someone source-check it? A similar situation exists in the main article for Fermat's Last Theorem. Nxavar (talk) 09:09, 28 January 2016 (UTC)[reply]

References

^ Ribet, Ken (1990). "On modular representations of Gal(Q/Q) arising from modular forms" (PDF). Inventiones mathematicae. 100 (2): 431–476. doi:10.1007/BF01231195. MR 1047143.

R=T?

In the article I read

Wiles's goal was to verify that the map $R\rightarrow T$ is an isomorphism and ultimately that $R=T$ .

I don't see the difference between having to isomorphic rings R and T and showing R=T? Mathematically I can't tell two isomorphic rings apart. Or does this mean that the isomorphsim is in some sense canonical? --Jobu0101 (talk) 07:45, 19 March 2016 (UTC)[reply]

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Bad link or page in reference note

There is a inserted reference (currently auto-numbered 29) after the words Fermat famously in the section titled Popular accessibility that does not seem to have anything to do with any type of statement from Fermat. It links to a book page that is titled WILES' THEOREM AND THE ARITHMETIC OF ELLIPIC CURVES. Without reading the subsequent pages of the chapter it is not possible to see how this reference support the phrase Fermat famously. Perhaps the page number is wrong or perhaps the reference is just completely misplaced in the article. Nyth 63 12:53, 23 March 2016 (UTC)[reply]

Length of proof

The lede states that the proof is more than 150 pages but the section titled Reading and notation guide states that it is more than 100 page. While not mutually exclusive, it does seem contradictory. I realize that document page counts for different printings can change with font, line spacing, and margin sizes, but that is a pretty large swing. Nyth 63 13:04, 23 March 2016 (UTC)[reply]

Summary of Wiles' proof

In the section "Summary of Wiles' proof" are the "Comments" direct quotes? If not they should not use contractions, as per Wikipedia:Contractions. You may also want to revise the improper use of the pronouns "we" and "our" per Wikipedia:Manual of Style#First-person pronouns. 5.151.0.121 (talk) 00:09, 21 January 2018 (UTC)[reply]

"with all numbers rational"

shouldn't this say "with all numbers integer"? — Preceding unsigned comment added by 77.2.91.57 (talk) 12:10, 13 June 2019 (UTC)[reply]

Same thing since the equation is homogeneous : if

(a_{1}/b_{1})^{n}=(a_{2}/b_{2})^{n}+(a_{3}/b_{3})^{n}

is a rational solution then

(a_{1}b_{2}b_{3})^{n}=(b_{1}a_{2}b_{3})^{n}+(b_{1}b_{2}a_{3})^{n}

is an integral solution. jraimbau (talk) 11:10, 14 June 2019 (UTC)[reply]

Fermat's Last Equation

Thought you'd never ask. Concerning the set of all positive cubic numbers, when two cubic numbers have been added together, has long been proven. And recalling that any set containing zero when multiplying any set higher than the cubes of three equal zero, it seems likely that demostration is what Fermat had in mind when he wrote that his margin was too small to contain it. What do you think? 166.181.249.95 (talk) 08:31, 17 February 2022 (UTC)[reply]

Described proof structure is not a proof by contradiction

The article describes the following top-level structure of Wiles' proof of FLT:

Assume that there are positive integers a, b, c, and n > 2, such that aⁿ + bⁿ = cⁿ.
Derive a contradiction (absurd).
Conclude that there are no such positive integers (FLT).

Despite the fact that the proof involves a contradiction, this is not what a "proof by contradiction" is. Rather, these kind of arguments are known as "proof of negation":

Assume A.
Derive a contradiction.
Conclude ¬A.

(In the case of FLT, the proposition A is ∃ a, b, c, n ∈ ℕ: n > 2 ∧ aⁿ + bⁿ = cⁿ.)

Instead, a "proof by contradiction" has the following shape:

Assume ¬A.
Derive a contradiction.
Conclude A.

The difference might seem negligible, but it actually isn't since, e.g., the law of excluded middle (LEM), can be derived using the principle of proof by contradiction, but not using the principle of proof of negation (of course, assuming a base system in which LEM is not derivable by other means). In other words, "proof of negation" is intuitionistically valid, while "proof by contradiction" isn't, hence the two principles are nonequivalent from a metamathematical point of view.

Importantly, this remark applies to the top-level structure of the proof, and Wiles' proof may very well rely in proof by contradiction in an essential way, but it does not at this point. — Preceding unsigned comment added by 46.10.124.136 (talk) 09:53, 13 June 2022 (UTC)[reply]

ZFC and Grothendieck universe

My understanding is that Wiles's proof relied implicitly on the existence of a Grothendieck universe, which is not provable within ZFC. Recently, a work-around has been found that does not rely on the existence of a Grothendieck universe, i.e. the proof now lies within ZFC. I think including a treatment of this issue would improve this article.

I have a couple of references that supposedly support this, but I have not examined them so I'm reluctant to add this material myself. ^[1] ^[2]

References

^ McLarty, Colin (2010). "What does it take to prove Fermat's last theorem? Grothendieck and the logic of number theory". The Review of Symbolic Logic. 13 (3). Cambridge University Press: 359--377.
^ McLarty, Colin (2020). "The large structures of Grothendieck founded on finite order arithmetic". Bulletin of Symbolic Logic. 16 (2). Cambridge University Press: 296--325.

Mr. Swordfish (talk) 14:48, 17 May 2023 (UTC)[reply]

Repetition in summary

The summary assumes FLT is false twice. A semistable elliptic curve arising from the counterexample also seems to be introduced twice, both in 2 and 3. I don't know anything about algebraic number theory so I'm not confident in making these changes myself. --Caliburn · (Talk · Contribs · CentralAuth · Log) 20:15, 1 June 2023 (UTC)[reply]

Fermat's theorem, seen by Clemente García Estévez from Vigo LA VOZ VIGO CIUDAD 23 Jul 2024 . Updated at 09:23 h. Comment · 0 FERMAT'S ENIGMA

THE PYTHAGOREAN TRIPLE

The theorem was conjectured by Pierre de Fermat and demonstrated by Wiles, (helped by Richard Taylor), in 1995 . Certainly this magistrate by profession and mathematician at heart contributed a lot in a provocative and sympathetic way to the development of mathematics in our time. How Fermat was able to deduce in those times when there were no calculators, let alone computers, had to be based on pen, paper and most importantly mathematical intuition. One thinks that this man would surely have a “fulcrum” to base himself on, and surely in my opinion, I think it was the Pythagorean theorem, and therefore the Pythagorean triples. When we observe that in the right triangle 3² + 4² = 5² , we see that there are whole numbers, many, we would say infinite, that fulfill this condition, to name a few: (3,4,5)-(5,12,13)- (7,24,25)-(8-15-17)-(9-40-41)-(11,60,61)- (12,35,37)-(13,84,35)…etc… . This is true for infinite triplets of numbers for powers equal to “2” (surfaces), but it is not true for powers greater than “2” , (read 3-4-5 ..) . In all the “Pythagorean triplets”, we see that they are made up of one even number and two odd numbers.

Let's look at some simple examples:

3²+ 4² = 5² / 9 + 16 = 25

Now we will divide the numbers by “2”

(3/2)² + ( 4/2)² = ( 5/2)² / 2.25 + 4 = 6.25

(3/4)²+ (2/4)² = (5/4)² / 0.5625+ 1 =1.5625

We observe that there is a perfect “pairing” between “two odd numbers” and an “even number”.

Let's now look at cubes

3³+ 4³ + 5³ = 6³ / 27 + 64 + 125 = 216

We have thousands of examples for “cubes” (there are also examples for higher powers with four components, since starting from the “cube”, 3, they will all be affected by the properties of volumes, which overturned Euler's theorem, and which, for me, Fermat “saw”)

Let's now observe the “marriage” that occurs in cubes

3³ + 4³ +5³ = 6 3 /

(3/2) 3 + ( 4/2) 3 + (5/2) 3 =( 6/2)³

(1.5)³ + 2³ + (2.5)³ = 27

3,375 + 8 + 15,625 = 27

Dividing again

( 0.750)³ + 1 3 + (1.25)³ = ( 1.5)³

0.421875 + 1 + 1.953125 = 3.375

Dividing again by 2

( 0.375 )³ + ( 0.5)³+ ( 0.625)³ = ( 0.750)³

0.052734375 +0.125 + 0.244140625 =

0.421875

To sum up:

Pierre de Fermat realized that the sum of two powers greater than “2” (3,4..) will be made up of at least four components, two of them coming from “odd” numbers and the other two from “even” numbers. the volumes being conditioned by the Pythagorean triangle, which necessarily requires the four minimum components in the equation

We see that in the equality of

( 2682440) 4 + (15365639) 4 + ( 18796760) 4 =

= ( 20615673) 4

Even( 2-4-8) + odd(1) + even(2-4-8) =

odd ( 3)

+- even +- odd +- even = +- odd

+- odd +- even +- odd = +- even

Filed under: Vigo city Clemente Garcia Estevez — Preceding unsigned comment added by Klement157 (talk • contribs) 17:37, 22 November 2024 (UTC)[reply]

[ribet-1] Ribet, Ken (1990). "On modular representations of Gal(Q/Q) arising from modular forms" (PDF). Inventiones mathematicae. 100 (2): 431–476. doi:10.1007/BF01231195. MR 1047143.

[2] McLarty, Colin (2010). "What does it take to prove Fermat's last theorem? Grothendieck and the logic of number theory". The Review of Symbolic Logic. 13 (3). Cambridge University Press: 359--377.

[3] McLarty, Colin (2020). "The large structures of Grothendieck founded on finite order arithmetic". Bulletin of Symbolic Logic. 16 (2). Cambridge University Press: 296--325.

[1]

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[2]