Talk:Fermat's Last Theorem/Archive 3

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Pop Culture

Would 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)

Alternate time line! Ironically, Stewart saying that in theory helped invalidate what Picard was saying. —Preceding unsigned comment added by 24.235.186.51 (talk) 16:51, 26 May 2010 (UTC)

Does the reference to The Girl Who Played With Fire really need to include the spoiler? — Preceding unsigned comment added by 69.29.119.92 (talk) 04:54, 16 January 2012 (UTC)

See Wikipedia's policy on spoilers. Magidin (talk) 06:46, 16 January 2012 (UTC)

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)

Maybe I am over-simplifying this, but isn't 1 an integer? 1ⁿ+1ⁿ=1ⁿ I can't see where anyone has said x,y, or z must be nonequal. Again, oversimplfication? mercator79 0315 24 June 2009 (UTC)

1ⁿ + 1ⁿ = 2, not 1. Magidin (talk) 03:29, 25 June 2009 (UTC)

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)

The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the proposal was Seemingly harmless move, there is an expected lack of participation. Keegan^talk 07:09, 20 July 2008 (UTC)

Requested move

I propose that the page be moved back from Fermat's last theorem to Fermat's Last Theorem. Rationale: "Fermat's Last Theorem" is a proper name. If this had been just the last theorem of Fermat, using lower case would have been entirely appropriate. But it is not. Wiles' article "Modular elliptic curves and Fermat's Last Theorem" uses capitals (also in the running text), as do most books on the topic (Aczel, Cornell et al., Edwards, Ribenboim, Singh, Stewart & Tall, van der Poorten).  --Lambiam 22:46, 12 July 2008 (UTC)

Both styles are common. We had a discussion on other theorems; did we decide on Euler's Theorem? (The confusion about last occurs either way.) Septentrionalis PMAnderson 23:42, 13 July 2008 (UTC)

Where was this discussion held? I think it's pretty clear that a proper noun is warranted - it's not really Fermat's "last theorem", but it is his "Last Theorem". Likewise, I'm sure Euler has many theorems, but only one is his Theorem. Thehotelambush (talk) 23:16, 16 July 2008 (UTC)

The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

An example of the theorem please

Unless I have been pretty dubious in my search for a actual example of the theory being proved TRUE, can I say that even in the 80's, when the morons who write this kind of nonsense in The Sun - a little known gutter tabloid published in the UK by The Dirty Digger - claimed that the expression had been evaluated, I saw that it does not make logical sense.

I say this as a software developer, not as a mathematician.

Just imagine the base-point of a triangle wedged in-between xⁿ and yⁿ. As x and y increase, the angle 'V' grows, wedging the probability of evaluation apart as the left side of the equation gets heavier. And if the left side just gets heavier, how will the equation balance?

But, just as a exercise in logic, I have written a short Pascal program to test all combinations of the expression with all four values set between 3 and 4096. I'm leaving it to run overnight. I have placed the program on my server with the Delphi/Pascal source for anybody to play with. It is pretty self explanatory.

The program can be accessed on my server... .......http://www.crysania.co.uk/fermat/fermat.zip Be Good (talk) 22:39, 27 July 2008 (UTC)

Serre's conjecture proved?

The article says "Serre's conjecture is still wide open" but the linked article to Serre's conjecture says it was proved in 2005. Does anyone know if that proof has passed scrutiny yet? If so, this should be changed. 24.69.179.116 (talk) 05:35, 2 October 2008 (UTC)

Should Wiles's proof get it own article?

I have created Wiles's proof of Fermat's Last Theorem so that both this article and the Wiles BLP can point to one article that attempts to provide an outline of the proof. The proof is notable in its own right, since it got its own Horizons (UK) / NOVA (US) show entitled "The Proof". I think I will also move the early 1980's work that lead to the Wiles theorem to that article also.--Lagelspeil (talk) 05:13, 13 March 2009 (UTC)

Did Fermat really have a proof?

In Tom Stoppard's Arcadia, a character guesses that Fermat never actually had a proof and that the comment in the margin was a joke. What do modern historians think? 66.41.253.22 (talk) 08:56, 11 May 2009 (UTC)

General opinion is that Fermat did not have an overall proof. He did discover proofs or near-proofs for some special cases. He possibly believed or hoped, when he wrote his famous margin note, that the methods he used in these special cases could be generalised to create an overall proof - a not unreasonable assumption, but, as it happens, a false hope. There is no evidence from Fermat's later correspondence that he had a general proof. He certainly did not have access to the modern mathematical techniques that Andrew Wiles used in his proof, and it is very unlikely that a substantially simpler, less sophisticated method of proof exists. For more information see this section of the sci.math FAQ. Gandalf61 (talk) 10:01, 11 May 2009 (UTC)

Suppose x^p+y^p–z^p=0 has integer solutions, where p≥2 is a prime. Applying Fermat's Little Theorm to p|(x^p+y^p–z^p) yields p|(x+y–z), which in turn yields x+y–z=kp, k≥1. For the smallest integer triplet (x_o, y_o, z_o), k=1 so that x_o+y_o–z_o=p. The left-hand-side is even. The right-hand-side is even only when p=2. Therefore, p=2 is the only prime that is admissible as the power. Indeed, when p=2, 3+4–5=2. Zymogen (talk) 13:52, 22 May 2010 (UTC)

How, exactly, do you conclude that k = 1 ? This is not always true when p = 2; for example 5² + 12² - 13² = 0 but 5 + 12 - 13 ≠ 2. So why should it be true for other p ? Gandalf61 (talk) 14:41, 22 May 2010 (UTC)

For the smallest integer triplet, k=1. For p=2, it is x+y−z=2k in general. Your case of k=2 is not for the smallest integer triplet. Zymogen (talk) 18:29, 22 May 2010 (UTC)

You have shown that there is no triplet solution to FLT for odd prime p that also satisfies x + y - z = p. You have not shown how it follows that there is no triplet solution with x + y - z = kp for some larger value of k. Consider the equation x + y = z³. By an argument identical to yours, we can show that every integer triplet solution must satisfy x + y - z = 3k and there is no triplet solution that satisfies x + y - z = 3. But there are plenty of solutions for larger values of k - for example, 1 + 7 = 2³ and 1 + 7 - 2 = 6. If you are trying to create a proof by infinite descent - to show that a solution for any value of k implies another solution for a smaller value of k - then you have failed to explain the descent step.

Anyway, this discussion is off-topic for this article talk page. If you truly believe you have an elementary proof of FLT, there are other more appropriate venues to ask for feedback. Gandalf61 (talk) 10:39, 23 May 2010 (UTC)

Thank you. You make perfect sense. Zymogen (talk) 11:47, 23 May 2010 (UTC)

x+y=z³ is a disguised linear equation u+v=w, by first choosing a cube as a special w and then determine u and v. The smallest triplet for a linear equation is 1+1=2, which can be derived from any solution to a linear equation. It is inappropriate to use a linear equation as a counter-example because 3+4-5=2 for the quadratic equation can be wrongly refuted.66.252.152.202 (talk) 14:25, 25 May 2010 (UTC)

Article structure

It's a pity that this article, which receives roughly 1500 visits per day, should be in such a sorry state after eight years of editing. I propose to work on improving it, but I don't want to run roughshod over other active editors. I would be delighted if other informed Wikipedians would work with me; I'll be grateful for your insights, criticisms, corrections, instructions and reviews.

I propose the following general structure:

1. Basics (definitions, primitive solutions, factorization of exponents, etc.)
2. Historical overview
3. Mathematical background (brief paragraphs on infinite descent, modular arithmetic, etc.)
4. Ad hoc proofs for isolated exponents (n=3-7, 10, and 14)
5. Early general approaches (Germaine, Kummer, Mordell)
6. Computational studies
7. Proof by elliptic curves and modularity
8. Related equations (e.g., x^n + y^n = 2 z^n)
9. Mathematical impact
10. See also, Footnotes, References, Bibliography, Further reading, External links

Thoughtful comments would be welcome. Proteins (talk) 16:56, 11 May 2009 (UTC)

I would put mathematical background after historical overview, not before. Magidin (talk) 17:08, 11 May 2009 (UTC)

Agreed — thanks! It's great to get feedback so soon. Proteins (talk) 17:13, 11 May 2009 (UTC)

Article length and inclusion of proof

I think the unsourced Mathematical basics and Proofs for specific exponents sections are too long and too detailed. Wikipedia is not a textbook - this type of material belongs in Wikibooks. Gandalf61 (talk) 16:45, 18 May 2009 (UTC)

Rest assured that your sourcing concerns will be met. I have the sources before me; I just haven't undertaken the drudgery of typing them in. It seems unlikely that anyone else, including you, will help me in that task.

Although I recognize that my writing of yesterday is still unpolished and likely over-wordy, I'm not ready to agree that proof-sketches are inappropriate for this article. We still have room for well over 30 kB of readable prose, which we can exploit to bring the article to FA. Personally, I feel that a mathematical Featured Article should have at least a little mathematics in it. Proteins (talk) 17:11, 18 May 2009 (UTC)

(Indented above response to make thread easier to follow) Responding to my well-meant and neutrally phrased feedback with sarcasm is not a good sign. I don't understand why you say "we still have room for well over 30 kB of readable prose" - the page is already 42 kB, a size which WP:SIZERULE says is an indication that the article may need to be divided. If your remark about aiming for FA standard is serious, you need to note two things:

1. Wikipedia:Featured article criteria on Length says an FA "stays focused on the main topic without going into unnecessary detail" - you should probably read Wikipedia:Summary style too.
2. The FA mavens insist on seeing at least one citation per paragraph. The more you write, the more references you will need to provide to appease the FA deities. Gandalf61 (talk) 12:17, 19 May 2009 (UTC)

I'm very sorry that I came across as sarcastic. I wasn't feeling that way, just tired and discouraged by the exhausting road ahead. I promise to give you the benefit of the doubt in the future, and I hope that you'll do likewise for me.

If you install this script into your monobook, you'll see that the article now has 24,127 bytes of readable prose, as defined at WP:SIZE. Subtracting from 50,000 yields roughly 26 kB of room for more mathematics. So the article can double in size; whether it should do so is another question, one that we should discuss. It might be efficient to write too much, and then trim down to the bare essentials, as we seem to be doing on the lead.

I'm all-too-conscious of the referencing requirements at FAC. I simply prefer to have the fun of writing before taking up the more arduous task of referencing. I ask for everyone's patience; it will only be a matter of days.

Although others may disagree, I sincerely feel that mathematical articles should have some mathematics in them. The proofs needn't be many, rigorous or complete, more like a handful of proof-sketches, but I'd like to include at least a few. I've placed them near the end of the article, as a kind of appendix. Proteins (talk) 18:46, 19 May 2009 (UTC)

I'm somehow feeling like Gandalf. While there is certainly still room for adding material, I'm fairly certain that continuing along the current pattern the article will be way too long when it is comprehensive one day. For example, the section "Even and odd" strikes me as superfluous. Also, for example, I don't think the reconstructed proof (attempt) of Fermat is encyclopaedic in scope. Jakob.scholbach (talk) 18:28, 26 May 2009 (UTC)

I'm beginning to agree about farming the proofs out to daughter articles. It seems like an jarring transition between the soft history of the first half and the hard math of the second. I feel we should have some math in the article, but the mathematical preliminaries section could be eliminated and its elements integrated into the proofs as needed. The mathematical material seems to divide neatly into three periods: before Kummer (1637~1850), FLT and ideals (1850~1950), and elliptic curves (1950-present); perhaps those could form the daughter articles. BTW, why don't you think the reconstructed proof is notable?

I should say that I like to experiment as I go along, to find the smoothest presentation. So no one should assume that the present organization is set in stone, or that I'll fight changes. Proteins (talk) 20:53, 26 May 2009 (UTC)

I agree that there should be mathematics in the article. Definitely!

About the reconstructed proof: I think proofs in general are worth including if they convey some piece of information that is interesting in its own right. A wrong "proof" only conveys, IMO, that Fermat was (probably) doing some mistake. But there are many more wrong proofs of the theorem, as the article tells later, and so picking one particular (=the first, but not even that's for sure) wrong proof feels a bit skewed to me.

Finally, I'm even more convinced that the material is to a large extent unencyclopaedic (which is not the same as not noteworthy). I guess that the proof is more or less directly copied from some textbook, which is usually frowned upon, right? Jakob.scholbach (talk) 08:07, 27 May 2009 (UTC)

The reconstructed proof isn't copied from Ribenboim's textbook, although of course there's a 1-to-1 correspondence between them. The mistaken proof comes originally from Euler, who attributes it to Anders Johan Lexell. Proteins (talk) 14:45, 29 May 2009 (UTC)

Length of lead

The lead is way too long as it currently stands; with the addition of the large graphic at the right, it makes it seem even longer. In general, the lead should be two or three paragraphs at most, and not be nearly as detailed as the current one is. That lead really needs to be cut much, much shorter. In fact, I woudl say that the lead should consist only of the current first paragraph, followed perhaps by another paragraph with some "name dropping". The reader is given way too much information and nomenclature as it currently stands. Think of the lead as an "abstract" for the full entry. It should invite the reader to read on if she is interested, not weigh her down under a ton of information. Magidin (talk) 17:34, 18 May 2009 (UTC)

Yes, you're quite right, as I see now. Thanks for your insightful critique! I'm working on it, and hope to upload an improved version later tonight. Proteins (talk) 21:37, 18 May 2009 (UTC)

Daughter articles

Per the suggestins above, I'm thinking of splitting off the proofs into one or more daughter articles, and I'd appreciate suggestions for doing that. One plan is to have a central daughter article Proofs of Fermat's Last Theorem for specific exponents, which might itself have daughter articles, e.g., Proofs of Fermat's Last Theorem for exponent 3.

Thanks for your advice! Proteins (talk) 14:45, 29 May 2009 (UTC)

Silence = assent? If no one objects, I'll start those daughter articles soon and re-jigger the article as a whole. I'd also like to create a daughter article for Kummer's work. Then we'd have a line of mathematical articles spanning the whole history of the FLT, from Fermat's proof to Wiles'. Proteins (talk) 13:59, 31 May 2009 (UTC)

It was NOT "soon"

I do not believe that "he soon discovered that the proof contained a gap" is true. I recall that it was more than hours or days. It was many weeks, and probably MONTHS later (less than a year though); and now I am skeptical that he did the discovering. This story and those I have read since sound warmed over to make him sound good. The IEEE Spectrum and the EE Times both reported on the story as it was happening. Pumping him up in the press gives me the same feeling that I got when I first heard about cold fusion from the press. —Preceding unsigned comment added by 208.145.81.2 (talk • contribs) May 30, 2009

I haven't worked on that section yet, so I'm not prepared to address your criticism. My preliminary understanding is (1) some as-yet-unidentified mathematicians objected right away that there was no valid Euler system, but (2) the gap was discovered independently by Katz, while he was reviewing Wiles' first manuscript. Please be patient while I read up on these matters. Alternatively, you can add the material yourself with references to reliable sources. Proteins (talk) 13:54, 31 May 2009 (UTC)

mystique of Fermat's Last Theorem and the Wolfskehl prize

Something that is alluded to, but not made very clear to people unfamiliar with mathematical culture, is the allure of Fermat's Last Theorem and the mystique surrounding it. Apparently the Wolfskehl prize had a lot to do with this. I can't recall right now, but there was an article in the AMS Notices a while back on the history of the prize, and it quoted several mathematicians and historians, who stated that its reputation was significantly enhanced by the prize. --C S (talk) 00:24, 14 June 2009 (UTC)

Positive Integers??

Hello: I believe the theorem might consider nonzero integer solutions to be 'acceptable', rather than just x,y in Z+. Any thoughts? One online source: http://mathworld.wolfram.com/FermatsLastTheorem.html —Preceding unsigned comment added by 76.126.4.16 (talk) 05:04, 9 August 2009 (UTC)

For even n, the sign does not matter. For odd n, if you require all three to be nonzero (to avoid the trivial solution with x=-y), then multiplying through by -1 if necessary we may assume that two of the variables are positive and one negative. We cannot have both x and y the same sign then, so they are opposite signs. Swapping them if necessary you may take x positive and y negative, and we have z positive. Move the y to the other side to obtain a solution in all positive integers. So considering "nonzero integer solutions" is equivalent to considering only positive ones. Magidin (talk) 10:55, 9 August 2009 (UTC)

Ah amazing - thanks! —Preceding unsigned comment added by 76.126.4.16 (talk) 14:57, 11 August 2009 (UTC)

Non-integer exponents

Does Fermat's Last Theorem hold for values of n>2 which aren't integers? For example, does it mean that there are no solutions in positive integers to the equation a^2.5+b^2.5=c^2.5? 75.28.53.84 (talk) 15:23, 15 August 2009 (UTC)

absolutely not. There is a real solution to 4^x + 5^x = 6^x. It's approximately x=2.487939173118—GraemeMcRae^talk 01:46, 17 August 2009 (UTC)

Why FLT is so difficult to prove?

Hello: I would like know if is there any document as to why FLT is so difficult to prove? How to begin, prostulate, etc. I'm just curious. What if there exists such a proof that Fermat mentioned, would that change anything? Would it be even bigger than Andrew Wiles' Proof? Even though Andrew Wiles proved FLT using techniques unavailable to Fermat. —Preceding unsigned comment added by 71.110.155.78 (talk) 05:47, 24 August 2009 (UTC)

Nobody knows why it is so difficult to prove. But that's, in a way, a silly thing to ask. Why should something be easy to prove versus difficult to prove? Sometimes you can sense that you are "near" proving something, but that's only because you may already have much of the preliminary work done or have considered a number of semi-general cases that you can prove. But this can be an illusory feeling also, so is nothing like a sure thing. So you might as well ask why anything hard to prove is hard to prove. I suspect the reason laypeople are so baffled FLT turned out hard to prove is that the statement is "simple". However, in mathematics, this happens all the time.

Come to think of it, it was kind of silly question otherwise it wouldn't have lasted until now (a direct proof of FLT). But my experience in proving it (my brain excercise, a la Sudoku), is that, it was so ambiguous, as if you're trying to look to the left and right simulteneously.

If someone could prove FLT using only elementary methods (say using things Fermat could have reasonably known or have proven or learned over the course of his lifetime), this would be a big thing simply because many mathematicians think this is not possible. In a way, this could never be "bigger" than what Wiles did because he gets the glory of being the first, but who knows, if the method of this hypothetical proof is more applicable, it could be a big thing. People used to think an elementary proof of the Prime Number Theorem would have been a big thing (and Atle Selberg and Paul Erdos even fought over credit for it), but it turns out it's not such a big thing after all. --C S (talk) 01:56, 3 September 2009 (UTC)

I agreed that a direct proof may have little or no significant because it was not the *first* to prove FLT. But it would be much more complete if FLT has a simpler proof such that average persons can understand it. Later generations will appreciate it more. I think Wiles' proof was wonderful. Tying together all necessary theorems and proving some conjectures, was a tremendous task. However, it is still an indirect proof. I could be wrong, the approach for mathematical proofs of FLT in the past were all "trying to find roots to the equation with all roots have same value, and prove that roots were or were not integers".

Permit me to tag along: I am curious about Tom Ballard's "A Short-Form Proof of Fermat's Last Theorem at... http://www.fermatproof.com/ ...which seems to do the job, but I don't really have the credentials for checking it. The closest I have ever come is my satirical treatment of "Fermat's Really Last Theorem" posed at... http://niquette.com/puzzles/xy-yxp.htm ...which gets thousands of solvers but does not belong in a serious encyclopedia. 206.171.24.28 (talk) 20:09, 2 September 2009 (UTC)

Have a look at the guestbook, where some people have pointed out the rather serious flaws in his attempt at a proof. http://books.dreambook.com/pokerface/fermatproof.html Mrh30 (talk) 09:51, 3 September 2009 (UTC)

There is no special thing happening in Tom Ballard's page. Of course, the proof is a fallacy by assuming "r to be fixed". Genuine mathematical societies consisting of professional mathematicians would not approve said proof. —Preceding unsigned comment added by 180.193.12.18 (talk) 07:50, 21 February 2011 (UTC)

Applications

I'm sorry, but nowhere in this article is explained exactly what is the application of this. In what field is this useful? What can be done with it? Why had so many people spent their lives looking for the solution to this problem? —Preceding unsigned comment added by 132.248.5.2 (talk) 04:44, 4 September 2009 (UTC)

Not useful at all and nothing can be done with it (as far as I know - I suppose it is just possible that there might be some obscure application). Many people spend lots of time doing things that are not useful. I guess the folks throughout history who have devoted time to trying to solve FLT would just say that it interests them. Gandalf61 (talk) 09:00, 4 September 2009 (UTC)

While Fermat's Last Theorem does no have applications, the proof itself (technically a specialized case of the Modularity Theorem / Taniyama–Shimura conjecture) developed useful techniques for cryptography and further study of elliptic curves.Seelum (talk) 04:30, 24 October 2009 (UTC)

As a layman reading the article, this confused me as well. It would be nice for the article to explain that the proof itself is unimportant.--Remurmur (talk) 11:41, 15 November 2011 (UTC)

Random Wrong Proof

Why is there a random wrong proof under "Proposed Reconstruction." There have been many mathematicians who have given incorrect proofs of Fermat's Last Theorem throughout history, many of who were more historically important that Anders Lexell and whose proofs weren't clearly incorrect at first glance. So, I ask, what is the point of including that proof? I think it should just be removed, its unnecessary and clutters the article. Seelum (talk) 04:34, 24 October 2009 (UTC)

Fibonacci proved n=4 case

I've amended the page to note that the n=4 case was first proved by Fibonacci, not Fermat. This seems to be a little known fact hiding in plain sight. I've not added references (my wiki editing skills are too poor) but you can see this at http://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html and http://books.google.com/books?id=dTVnPUl8OQ4C&pg=PA94

(The first cited webpage writes "Fibonacci also proves ... x4 - y4 cannot be a square", i.e. that x^4 = y^4 + z^2 has no solutions, for which FLT n=4 is a special case.) Jamesdowallen (talk) 21:36, 30 October 2009 (UTC)

Might we change the first sentence slightly?

The first sentence reads thus:

"In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two."

Are these three positive integers distinct, i.e. a, b, and c are not 1? Personally, this was not clear to me upon first reading this sentence, and adding the word "distinct" or "unique" when describing the numbers might be good. So, we could say immediately after the first sentence:

"a,b, and c must be different from one another."

Or something. —Preceding unsigned comment added by 24.2.184.219 (talk) 22:38, 24 November 2009 (UTC)

There is no restriction in the statement to a, b, and c being different from 1 or from each other; that a, b, and c would have to be pairwise distinct is a result, not part of the original statement. The only restriction is that none of them be equal to zero. Magidin (talk) 23:07, 24 November 2009 (UTC)

OK. —Preceding unsigned comment added by 24.2.184.219 (talk) 00:43, 25 November 2009 (UTC)

Simplicity

I've never been able to understand what so many textbooks and also this Wikipedia article mean by "no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two", while the basic premise appears to be that this "problem" would so simple and universally understood that even 5th graders can grasp it. Somebody told me today it simply means "Pythagorean theorem is not applicable if the power is greater than 2." (And that therefore, Pythagorean theorem obviously only works in 2D, not in a 3D space.) Why not put it as simple as that, for goodness' sake? --79.193.46.200 (talk) 07:12, 6 December 2009 (UTC)

It is not as simple as that -- the diagonal of a cuboid with lengths a, b, and c is ${\sqrt {a^{2}+b^{2}+c^{2}}}$, and not the third root of something. So the intuition of a 3D-Pythagorean theorem does not yield the formula in Fermat's Last Theorem. Jakob.scholbach (talk) 10:50, 6 December 2009 (UTC)

Yeah, Fermat's Last Theorem amd the Pythagorean Theorem don't have much to with each other, besides involving similar-looking equations. Fermat's Last Theorem is about Diophantine equations: it concerns integer solutions of an equation. The Pythagorean Theorem is not about solving equations; rather, it gives a relation between the sides of a right triangle, whose lengths are real numbers. Eric119 (talk) 19:03, 6 December 2009 (UTC)

Here is a simple explanation of Fermat's Last Theorem. Look at the sequence of square numbers 1, 4, 9, 16, 25, ... If you add 9 and 16 you get another square number, 25. There are many other instances where this happens - 25 + 144 = 169, 64 + 225 = 289 etc. In fact there are an infinite number of cases where the sum of two squares equals another square. Now try to do the same thing with the sequence of cube numbers 1, 8, 27, 64, 125 ... You won't find any instance where the sum of two cube numbers is another cube number. Sometimes you get close to a cube - 9³ + 10³ = 729 + 1000 = 1729 = 12³ + 1 - but never exactly equal. Same thing with the sequence of fourth powers 1, 16, 81, 256 ... Fermat's Last Theorem says that this is true for any nth powers higher than squares; the sum of two nth powers is never another nth power if n is greater than 2. Gandalf61 (talk) 10:24, 3 January 2010 (UTC)

Let's not confuse simple with long-winded.—GraemeMcRae^talk 17:48, 4 January 2010 (UTC)

"Diophantus shows how to solve this sum-of-squares problem for k = 4."

I find this statement somewhat unlikely, since 4^2 = 16 has no (non-trivial) decomposition into a sum of two squares. TotientDragooned (talk) 22:41, 9 February 2010 (UTC)

The text stated Diophantus' problem incorrectly. He was asking for rational solutions to the equation, not integral ones. I've fixed this now. Paul August ☎ 02:27, 10 February 2010 (UTC)

Why is this wrong?

This was reverted:

$n>2\rightarrow \nexists \left\{x,y,z,n\right\}\in \mathbb {N} ^{+}\setminus x^{n}+y^{n}=z^{n}$

Why? Is this wrong?

--88.26.43.187 (talk) 19:13, 22 February 2010 (UTC)

(i) Yes, as written it is wrong; the 'n' in the consequent is not bound by the restriction on the antecedent, since it appears bounded by the (negated) existential quantifier. In essence, this formula is formally equivalent to

$n>2\rightarrow \nexists \left\{x,y,z,m\right\}\in \mathbb {N} ^{+}\setminus x^{m}+y^{m}=z^{m}$

which is, of course wrong. Also, the use of $\setminus$ is incorrect: it represents relative difference/complement of sets, but you are not removing from $\mathbb {N} ^{+}$ the equation; and the symbol $\mathbb {N} ^{+}$ is nonstandard. But way more important: (ii) it is an unnecessarily complicated mathematical formula what does not add anything to the article and only makes it look more complicated and intimidating. Magidin (talk) 20:10, 22 February 2010 (UTC)

Thank you for the explanation, Magidin. My intention was to express the theorema in an international form, independient of the english.

¿What coud be a correct form?

I would try with something like:

$\nexists \left\{x,y,z,n\right\}\in \mathbb {N} ^{+}\setminus x^{n}+y^{n}=z^{n}\forall n>2$

but it must to be wrong because "$\setminus$ is incorrect: it represents relative difference/complement of sets, but you are not removing from $\mathbb {N} ^{+}$ the equation".

And I thought that $\mathbb {N} ^{+}$ represents the positive natural numbers (the natural numbers except 0).

--95.214.85.195 (talk) 18:21, 28 February 2010 (UTC)

First: this is the English wikipedia; the purpose of the lead paragraph is to try to make the subject understandable to everyone who can read English; but by using lots mathematical symbols like $\forall$ or $\nexists$(let alone using them incorrectly) you do not make it more understandable, you make it less understandable because now you require knowledge and understanding of formal symbols. Anyone who has trouble with the first sentence of the article would be utterly lost with the formula (even if correct). I suspect your native language is not English (Spanish, perhaps, given your use of the opening question mark, which is not used at all in English?); there are Wikipedias in all sorts of languages so that the language itself does not present a bar.

As to your new attempt: it's actually wrong for all sorts of reasons. One does not place the variables inside curly brackets; curly brackets usually represent sets; and if you want it to represent a set, then you should be using $\subset$, not $\in$, since you don't want to say the set is an element of the natural numbers. $\mathbb {N} ^{+}$ is still nonstandard symbology. For one thing, there is about a 50-50 split between those that consider the natural numbers to include 0 and those who do not include 0 in the natural numbers, so what you give is not a standard symbol. My personal first reaction would be that you want me to consider the additive semigroup (or monoid) of the natural numbers. There are lots of symbols used all over the literature to mean "positive integers", but none of them is universally accepted or understood. So no, it does not necessarily mean "the natural numbers except for 0". And you are still misusing the control sequence"setminus" which represents relative complement. You seem to want it to mean "such that", but there is no such symbol in formal logic. Finally, your last quantifier is completely wrong. What you have now is not even a well-formed formula, let alone a correct statement of Fermat's Last Theorem. You seem to have some trouble dealing with the scope of quantifiers.

There are many ways of writing it correctly. You could say that any solution must be a trivial solution, for example:

$\forall n\in \mathbb {Z} \left(n>2\rightarrow {\Bigl (}\forall x,y,z\in \mathbb {Z} {\bigl (}(x,y,z\geq 0\wedge x^{n}+y^{n}=z^{n})\rightarrow xyz=0{\bigr )}{\Bigr )}\right).$

Or, if you want to stick closer to the "does not exist" statement, then:

$\forall n\in \mathbb {N} {\Bigl (}n>2\rightarrow {\bigl (}\nexists x,y,z\in \mathbb {N} \ (xyz\neq 0\wedge x^{n}+y^{n}=z^{n}){\bigr )}{\Bigr )}$

Or even

$\nexists x,y,z,n\in \mathbb {N} {\Bigl (}(n>2)\wedge (xyz\neq 0)\wedge (x^{n}+y^{n}=z^{n}){\Bigr )}$

But none of them belong in the lead of the article. And they are not totally formal in any case, but rather "pidgin" logic, but at least well-formed and meaning what they are supposed to mean. Magidin (talk) 18:59, 28 February 2010 (UTC)

Thank you very much, Magidin. I have to read again your explanations. It's obvious that I don't domine the mathematical language (It's an euphemism...).

And yes, I'm spanish like the "¿" symbol shows.

Thank you again and please excuse me for my low english level and the nuisances.

--213.98.194.179 (talk) 21:32, 28 February 2010 (UTC)

There is a Spanish article es:Último teorema de Fermat but it's much less detailed than the English article. 69.228.170.24 (talk) 06:08, 9 May 2010 (UTC)

References

The references part of this article is half of it.This is a bit ridiculous and I suggest this be corrected. —Preceding unsigned comment added by 188.26.49.57 (talk) 00:08, 3 April 2010 (UTC)

axioms needed to prove FLT

This may be of interest since the question came up on this page a while back.[1]. 69.228.170.24 (talk) 08:04, 8 May 2010 (UTC) ]

x^4 − y^4 = z^2?

"The mathematical techniques used in Fermat's "marvellous" proof are unknown. Only one detailed proof of Fermat has survived, the above proof that no three coprime integers (x, y, z) satisfy the equation x^4 − y^4 = z^2." Shouldn't this be x^4 − y^4 = z^4? kcylsnavS{^screech_harrass} 03:26, 7 July 2010 (UTC)

No, it is correct; he proved that there were no nontrivial solutions to $x^{4}-y^{4}=z^{2}$. From this, it follows that there can be no solutions to $x^{4}-y^{4}=z^{4}$, since any fourth power is also a square, exactly as is explained earlier in the article. That is, the statement with $z^{2}$ is stronger than the one with </math>z^4</math>. Magidin (talk) 03:52, 7 July 2010 (UTC)

Did he really write the note in 1637?

What is our (or Dickson's) reason for believing that he wrote it in 1637, considering that the book with his margin notes was only discovered and published after his death in 1665? (The question has been raised on Math Overflow.) Shreevatsa (talk) 21:54, 25 July 2010 (UTC)

Pictures and Greek

I know it is completely trivial (in the common usage), but why on the two pictures of a book, there is print on the left and handwritten Greek on the right? So this book is a exercise book with a column on the left and a blank one on right to scribble in? Why did Pierre write it in Greek? I am pretty sure I am not the only person to be confused about it. --Squidonius (talk) 23:29, 28 August 2010 (UTC)

I think you are not seeing "handwritten Greek", but printed greek. You can tell it's not "handwritten" by the initial letters in each paragraph and the indentation that surrounds them. Since Diophantus wrote in Greek, you are seeing a side-by-side translation, with the Latin translation on the left and the Greek original on the right. Magidin (talk) 01:53, 29 August 2010 (UTC)

Does Fermat's copy of Diophantus exist? is it possible for us to get a picture of his actual annotation? --Adam Brink (talk) 08:32, 17 August 2011 (UTC)

Invalidate FLT

If I can show that FLT is invalid ala 2=1, will it be considered a proof?

In fact there is not solution integer for any integer n, except for n=1 or n=2.

-Joe Jknilaad (talk) 05:47, 27 July 2011 (UTC)

Please read the Wikipedia policies on what the talk page is and is not. In particular, the talk page of an article is not a forum for discussion of the subject. Talk pages exist for discussion on how to improve the article; your question is off topic. Magidin (talk) 16:08, 27 July 2011 (UTC)

Harvey Friedman's grand conjecture

Is Harvey Friedman's grand conjecture really relevant for the discussion of whether or not Fermat might have possessed a proof? What does this conjecture have to do with the history of Fermat's last theorem? Is there a reliable reference connecting this conjecture with a historical discussion of Fermat? There is too much speculation here: first it's a conjecture, even if the conjecture is true, it is pure speculation that an elementary proof would be short enough for a human to find, and thirdly what does any of this have to do with Fermat specifically? --345Kai (talk) 16:52, 17 August 2011 (UTC)

Odd Primes

The article mentions "odd primes" three times. I, as a college educated engineer have no idea what to make of this. After all, there isn't such a thing as an even prime except two. By definition all other prime numbers are odd. If "odd primes" are indeed all primes except two, it would be easier to read imho if the article just said so. 79.252.141.188 (talk) 16:56, 17 August 2011 (UTC)

It is standard terminology to speak about "odd primes" to refer to all primes greater than 2. Magidin (talk) 19:28, 17 August 2011 (UTC)

Then I think we should mention that in the article. After all, Wikipedia should be well readable without a particuar background in mathmatics, and you need to be pretty deeply into number theory to regularily come across the term "odd primes" 79.252.141.188 (talk) 12:04, 18 August 2011 (UTC)

No problem with adding a parenthetical comment in the first occurrence, I think; in fact, it's already there (well, in the second occurrence, under "Proofs for specific exponents.") I do have to say that one encounters the phrase "odd primes" very soon, very often, in even the most basic discussions of number theory; "pretty deeply" is rather extreme hyperbole. Magidin (talk) 15:52, 18 August 2011 (UTC)

Fermat's copy of Arithmetica

Is there any record of what happened to Fermat's copy of Arithmetica? Presumably his son must have had it, or access to it, for a few years after Fermat's death, in order to record the marginalia (and presumably he used it as a reference when making his own translation of the book). Did he just throw it away? What happened to Fermat's son's belongings? It would be worth a fortune nowadays. -Ashley Pomeroy (talk) 12:28, 5 September 2011 (UTC)