Talk:Fermat's Last Theorem

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Guinness book of world records comment[edit]

What relevance is it that the GB of WR thought it was a difficult problem? I certainly don't feel it is noteworthy enough to be the closing remark of the opening paragraph (talk) 16:44, 16 May 2014 (UTC)

This problem has been solved[edit]

I feel like I should bring up that 3,987^12 + 4,365^12 = 4,472^12 — Preceding unsigned comment added by (talk) 06:56, 7 November 2013 (UTC)

Close, but no. The left and right sides differ by 1211886809373872630985912112862690. Try it in Python or some other system like it that supports unlimited precision integers automatically. —David Eppstein (talk) 07:18, 7 November 2013 (UTC)
Or just notice that 3987 and 4365 are both multiples of 3, but 4472 is not.—GraemeMcRaetalk 00:03, 10 November 2013 (UTC)
The American might be trying to waste our time, as an alleged joke. — Preceding unsigned comment added by (talk) 13:33, 21 September 2014 (UTC)
I think it is a reference to a popular cartoon series: it is also referred to in this Zbl 0991.11072. Deltahedron (talk) 16:06, 21 September 2014 (UTC)

Wiles's Proof of FLT?[edit]

Ribet proves that Frey's elliptic equation is non modular (and it and Fermat's equality do not exist). Wiles proves that all elliptic equations are modular (and since Frey's elliptic equation is non modular Fermat's equality does not exist). Looks like Wiles has proved what Ribet proved does not exist, namely, Fermat's equality. Moreover, why is Wiles's proof of modularity superior to Ribet's counter example? And, why does modularity rather than existence determine Wiles's proof? Jamestmsn (talk) 23:44, 29 February 2012 (UTC)

Ribet proved that if a (nontrivial) solution to Fermat's equation exists, then the Frey curve that can be defined from it would semistable and not be modular; it is a conditional statement. Wiles proved that all semi-stable elliptic curves are modular; this is enough to establish Fermat's Last Theorem, but only thanks to Ribet's Theorem. Ribet did not show that there are no solutions to Fermat's equation. Wiles's Theorem is not directly about Fermat's Last Theorem, but rather about elliptic curves. Magidin (talk) 03:21, 1 March 2012 (UTC)

How does Ribet's theorem become "enough to establish Fermat's Last Theorem? The Wiles modularity – Ribet non modularity contradiction is not relevant to the issue whether Frey's elliptical curve, and its source Fermat's equality, exist as solutions. To prove FLT, Wiles assumes Frey's curve does not exist thanks to Ribet's theorem. However, the fact that Frey's curve is non modular does not change the fact that it is elliptical. Elliptical curves are defined by their forms and coefficients not by their modularity. Frey's elliptical curve does not disappear by the fact it is non modular. Its form and coefficients depend from Fermat's equality thanks to Frey's work not to non modularity thanks to Ribet's theorem. Simply put, to prove FLT, Wiles uses the irrelevant contradiction between his work and Ribet's work to conclude, without proof, that Frey's curve and Fermat's equality are forbidden to exist. (talk) 18:26, 2 March 2012 (UTC)

Sigh... This is not the proper place for this discussion; Talk pages are not a forum. I will simply note that you are completely misunderstanding what "Frey's curve" is (in fact, it does not exist; it can only be defined if there is a nontrivial solution to the Fermat equation). Because the definition assumes that you have and that is a pth power of an integer. That is, that there is a c such that . Without those properties, you are not looking at "the" Frey curve, you are looking at some elliptic curve.
The chain of argument is: If there is a nontrivial solution to Fermat's equation, then C exists (Frey); if C exists, then C is semistable and not modular (Ribet). All semi-stable elliptic curves are modular. (Wiles). Using contrapositives and modus ponens, the conclusion is "There is no nontrivial solution to Fermat's equation." If you wish to learn about this, then go to an appropriate place and stop using the talk page for it. Meanwhile, I'll simply note that you don't know what you are talking about. Magidin (talk) 19:24, 2 March 2012 (UTC)

Stick to the point and answer the question. How does Ribet's theorem become "enough to establish Fermat's Last Theorem"? Ribet's theorem proves (1) Frey's elliptical equation (FEE) is an exception to Wile's modularity theorem and (2) FEE is not connected to a modular form. (1) disproves Wile's modularity theorem and (2) is not relevant to establishing Fermat's Last Theorem. Note that FEE is based on an existence proof which assumes if Fermat's equality (FE) exists then FEE exists. However, how do we know that FEE is dependent solely on FE? For example, one does not need FE to study FEE in the abstract. Conditional FE, therefore, is not relevant to and does not make FEE conditional for any purpose. (talk) 19:25, 4 March 2012 (UTC)

I answered the question; you just didn't understand it. And talk pages are not a forum for discussion of the subject matter of the article. Take it somewhere else. This is not the appropriate place for you to discuss this. Magidin (talk) 20:59, 4 March 2012 (UTC)

Actually I think Jamestmsn is asking reasonable questions here, and pointing out that the article is a little unclear. I've just made an edit that I hope resolves the issue. The phrase "the above elliptic curve is always non-modular" certainly gives the impression that the elliptic curve in question exists. What Ribet proved (if I understand correctly) is that if such an elliptic curve existed, then it wouldn't be modular. Wiles proved that if such an elliptic curve existed then it would be modular (because all semistable elliptic curves are modular). Because of the contradiction, we have to assume that Frey's elliptic curve can't exist (or that Ribet or Wiles or someone made a mistake). In other words, Wiles didn't assume the non-existence of Frey's curve, he proved that it can't exist. Jowa fan (talk) 06:01, 5 March 2012 (UTC)

It is really quite straightforward:
  1. If Fermat's Last Theorem is false then a counterexample exists.
  2. If a counterexample to FLT exists then an associated elliptic curve (the "Frey curve") is rational, semistable and non-modular.
But Wiles proved that all rational, semistable elliptic curves are modular. Therefore there is no Frey curve that is rational, semistable and non-modular. Therefore a counterexample to FLT does not exist. Therefore FLT is true. Gandalf61 (talk) 09:00, 5 March 2012 (UTC)

Ribet's theorem disproves Pythagoras theorem. Simply write Fermat's equality in Pythagorean form (c^p/2)2=(a^p/2)2+(b^p/2)2 and assume p is an even number. (talk) 17:08, 6 March 2012 (UTC)

For the third time: your edits in this page are disruptive and violate Wikipedia policy. Talk pages are not a forum. There are plenty of websites and discussion boards where you can be enlightened about these issues if you really want to learn something. This is not one of them. Magidin (talk) 18:11, 6 March 2012 (UTC)

Where can I find a website "where you can be enlightened about these issues if you really want to learn something" about Pythagoras theorem? (talk) —Preceding undated comment added 23:27, 26 March 2012 (UTC).

Single Page Proof of FLT?[edit]

See at Jamestmsn (talk) 04:44, 5 March 2012 (UTC)

Regardless of the (lack of) correctness of the content of that page: (i) The talk page is not a forum for general discussion. (ii) The material there does not abide by the Wikipedia policies for inclusion: it is not a reliable source. (iii) If the page is yours, then you are talking about original research, also against Wikipedia policy, and self-promotion. (iv) You are engaging in link spam. Magidin (talk) 20:05, 5 March 2012 (UTC)

This talk page needs any special message??[edit]

People keep coming here showing their proof of FLT, but they always get reverted saying "This talk page is for article improvements ONLY." Does this talk page need a special banner?? Georgia guy (talk) 19:29, 22 July 2012 (UTC)

That's part of the basic Wikipedia policy on talk pages. I doubt that people who post from IP addresses, and often repeatedly re-post the same stuff here after it's been deleted and the issue been explained to them, will be detered by a banner telling them not to. I have yet to see someone who posts such stuff even apologize after it's been pointed out to them that's not what the talk page is for... But if you have such a banner, I guess it can't hurt to add it. Magidin (talk) 20:11, 22 July 2012 (UTC)

Fermat's theorem for even exponents[edit]

This article mentions an elementary proof by Guy Tarjanian for even exponents. In the reference we are said to find it in the "Comptes rendus hebdomadaires des séances de l'Académie des sciences" 1977-series. It just so happens that this series seems to be digitalised on a french website Yet, I can not find Tarjanian's result itself. Can anyone help? — Preceding unsigned comment added by (talk) 11:37, 4 September 2012 (UTC)

It is listed out of order in the corresponding page; look at the volume that says 1977/07 (T285, N16)-1977/12, which is the first one listed in volume 285 (rather than the last, where you might expect it). It is probably listed first because this issue contains the table of contents and author list for the whole volume. Magidin (talk) 15:25, 4 September 2012 (UTC)

Thanks for helping. — Preceding unsigned comment added by (talk) 17:48, 4 September 2012 (UTC) I've found it by now. I've not yet read the content, but I'm wondering what this theorem has to do with Fermat's. It says: if for some integers x,y,z and an odd prime p x^(2p)+y^(2p) = z^(2p) then 2p must divide x or y. That doesn't rule out the possibility that there exist coprime x,y,z with 2p dividing x (so 2 nor p divides y or z) and such that x^(2p) + y^(2p) = z^(2p). — Preceding unsigned comment added by (talk) 17:43, 4 September 2012 (UTC)

Fermat's Last Theorem was traditionally divided into two cases, Case I (in which odd prime factors of do not divide ), and Case II (in which there is an odd prime factor of that does divide ). The result thus establishes Case I for exponent p (if you had that p divides z, then by rewriting the equation as you would get a contradiction, so the result actually shows that Case I cannot occur). That is, it is related to Fermat's Last Theorem (along classical lines). That said, Case I was generally easier than Case II. Magidin (talk) 20:46, 4 September 2012 (UTC)

Ok, I understand. But these two cases aren't explained in the article as far as I see, so I think you should either write a section about them or (if you judge this too specialised for wikipedia) remove the reference to Tarjanian's result alltogether as it will surely confuse other readers too. — Preceding unsigned comment added by (talk) 10:24, 5 September 2012 (UTC)

Please sign your comments, rather than leave them for the SineBot to sign for you. Use four tildes to generate a signature. How will the reference by itself confuse other readers? Your confusion arose from browsing through (and, by your own admission, not reading through) the original reference, not from what the article on Wikipedia said or failed to say. Moreover, the sentence immediately prior to the reference to Terjanian's article reads: " As a byproduct of this latter work, she [Germain] proved Sophie Germain's theorem, which verified the first case of Fermat's Last Theorem (the case in which p does not divide xyz) for every odd prime exponent less than 100.[97][98] Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for n = 2p, which was proven by Guy Terjanian in 1977.[99]". In other words, the immediately previous sentence already states that this comment is restricted to "the first case" (and explicitly states what this means). Then it says that Germain tried to prove the first case for the exponents of the form n=2p, and that this is what Terjanian proved. In other words, what you are requesting be added is already there. Magidin (talk) 16:02, 5 September 2012 (UTC)

Ok, I see what you mean. But still: In the sentence "As a byproduct of this latter work, she [Germain] proved Sophie Germain's theorem, which verified the first case of Fermat's Last Theorem (the case in which p does not divide xyz)" might mean that the case described between brackets is a subcase of a particular (undescribed) case 1. Besides, I appreciate the work you're doing and it is not my intention to randomly criticise your work. I do not know how to produce signed comments, but I'll try to figure it out myself. — Preceding unsigned comment added by (talk) 21:20, 5 September 2012 (UTC)

I explained how to sign your comments: write four tildes at the end, like so: ~~~~. This will automatically produce the signature, just like I explained. I honestly do not see how that reading is even feasible; the sentence says "first case", and then immediately describes a condition. How could it possibly be understood to mean "a subcase of the first case"? Of course, if one really wants to misunderstand, I'm sure one will find a way of doing so, even after I add a "namely" to the parenthetical remark. Magidin (talk) 05:26, 6 September 2012 (UTC)
Is there another integer triplet than (1,2,3) for x^3+y^3=z^2? Thanks for an answer. (talk) 01:22, 22 April 2013 (UTC)
This is not the place to ask such a question; the talk page is exclusively to discuss improvements to the article in Wikipedia. If you wish to ask a mathematics question, or have a discussion, then there are other places where it can be done, such as Magidin (talk) 02:20, 22 April 2013 (UTC)
Can you make an exception? Please direct me to wherever or whomever, and then delete my posts as well as yours as if nothing had happened. I got nowhere so I end up here. (talk) 12:19, 22 April 2013 (UTC)
No, I can't make an exception. I already directed you to an appropriate site in which you can post your question: Magidin (talk) 14:54, 22 April 2013 (UTC)
Thank you. I will go there. Please delete all the exchanges between you and me. (talk) 21:27, 22 April 2013 (UTC)
I am surprised that you have not deleted my posts. Anyway, I went there. Several friendly people offered their results. They inspired me, and I have found an algorithm myself. I will try positive integers f and g until f^3+g^3= k×h^2. Then a triplet is (kf)^3+(kg)^3=(hk^2)^2. For example, 3^3+5^3=4×38. A triplet emerges as 114^3+190^3=2888^2. If my post has violated the rule, please delete it. (talk) 04:53, 24 April 2013 (UTC)
Large scale removal of content from talk pages is discouraged. I am not inclined to do so. At the very least, it will stand here for a while (until automatically archived for lack of activity, hint, hint) as a possible deterrent for others who would post here without bothering to read the explanations of what Wikipedia is and what it is not, and what talk pages are for. Magidin (talk) 15:29, 24 April 2013 (UTC)
If I did not stumble upon you, who would have directed me to that web site? In this case, everyone is in a win-win situation. :-) (talk) 19:59, 24 April 2013 (UTC)
Are you kidding me? Doesn't take too much ingenuity to find appropriate websites to ask questions, nor does it take too much of your time to bother reading the policy materials for Wikipedia before stomping around. So thank you for doing so. And for reading the reply you first got so carefully that I was forced to repeat the information twice before you noticed it. It's a win for you because you got what you wanted; it's a lose for me, because not only did I need to spend time educating you as to the nature of Wikipedia, in addition it is clear that you did not learn anything other than that you can do as you please and get what you want anyway. Magidin (talk) 20:38, 24 April 2013 (UTC)
What's the matter with you, to reply to me like that? 23:16, 25 April 2013 (UTC) — Preceding unsigned comment added by (talk)

New comment[edit]

You seem to be under the impression that it is possible for an integer to simultaneously be an odd prime and contain a factor of 4, which confuses me to some extent, though i admit i do not have the background in mathematics most of you probably do. (talk) 07:13, 6 April 2013 (UTC)

Who are you talking to? What are you refering to? Magidin (talk) 20:18, 6 April 2013 (UTC)

Deleted reference[edit]

A few days ago I removed the following reference that had been added to the Further Reading section:

Sinyor, Joseph with Ted Speevak and Akalu Tefera (received June 2000). "A New Combinatorial Identity". International Journal of Mathematics and Mathematical Sciences. ISBN 25:6 (2001) 361–363 S0161171201005361 Check |isbn= value: invalid character (help).  Check date values in: |date= (help)

The reference was added by the first author. I removed it for several reasons:

  • Conflict of interest: The author of the paper adding a reference to his own paper as a "further reading".
  • Undue weight: The Further Reading section currently contains references to major works (books by Edwards, Aczel, Dickson, Mordell, Ribenboim) and important surveys (Faltings, Saikia). This paper seems out of place there, dealing as it does with some partial results that are "related to" Fermat's equation.
  • Lack of notability: I also note that although the paper is indexed by MathSciNet, it did not merit a full review (the MathReview quotes the abstract only), and MathSciNet does not record any references to the paper; Zentralblatt likewise only quoted the abstract. While these may not be fully dispositive of whether this paper is notable or not, I believe they are indicative and place the burden of proof on someone who wishes to add the paper to the list of "Further Reading" to establish its notoriety.

The editor who added the paper (who is also one of the authors) then proceeded to track down my e-mail address and e-mail me directly asking me to "reconsider" the addition; I advised him to make his case here in the talk page rather than personally to me. As he has not done so, I'm adding my justification for the deletion for the record. Magidin (talk) 16:42, 19 October 2012 (UTC)

My name is Joseph Sinyor and I made the edit mentioned. In 2001 I published a paper in the IJMMS on a new combinatorial identity (along with two co-authors - one of whom is a professional mathematician) which is indeed "related" to FLT. In the 1820's it was noted that (xn + yn - zn) would produce 3 equations which would have to be satisfied for a counter-example to exist to FLT. Using our new identity one can now add a fourth equation to this list. I have no idea if this could lead to any insight into a possible elementary proof or Fermat's thinking, but feel it is worth at least bringing it to the attention of lovers of number theory, amateur or professional. This paper is therefore not about just another of "thousands" of identities. I made the edit myself in an effort at transparency and put it in "further reading" - I did not create the categories and the reviewer could have been charitable & placed the reference elsewhere. When I contacted him privately (precisely because I wished to avoid acrimonious communications) I was told I would have to "prove" the identity was worth listing among the otherwise august list of other works. Obviously there is nothing I can add to my paper that will convince him (MathSciNet probably did not review it because they also saw it as just another identity).
I feel I am now entitled to some self-promotion here: if you liked this paper, read my paper on the 3x+1 problem by googling me.Jojosinyor (talk) 02:08, 22 October 2012 (UTC)
Regarding your final paragraph (I will not further opine on the rest and let other editors comment): it is out of place in the talk page. Please read the Wikipedia Talk Page guidelines, specifically the point that talk pages are not a forum. Self-promotion is simply out of place. Talk pages exist for the purpose of discussing improvements to the article, not for you to publicize your work. If you want to use the internet to publicize your work, please set up a personal blog. Magidin (talk) 02:41, 22 October 2012 (UTC)

Down Goes Brown reference[edit]

I'm not sure exactly what the criteria for a blog post are, but the entries (including this one) also appear in the National Post, one of Canada's two main national daily newspapers. Does that get you to notability? Silverpie (talk) 20:13, 11 January 2013 (UTC)


Forgive the lack of good editing policy, as I am a very infrequent editor of Wikipedia. It's just that I had been watching some Numberphile videos about Fermats last theoreom which sparked me to find this article again:

I think it warrants inclusion into the article (and possibly some others in regards to maths proofs). It discusses drastic changes to structure of the proof, and provides a new way of proving it that is notable and more likely to be accessible--In other words, it might also be worthy of its own page. Gödel's Prodigal Apprentice (talk) 06:08, 2 October 2013 (UTC)

First, I don't think you understand what is being claimed in the article; I do not see how what is being proposed is "a new way of proving [FLT]" or that it is "more likely to be accessible". The issue is actually a rather technical one of exactly which axioms of set theory are needed to support the proof. As I understand the article, he does not propose any changes to the general structure of the proof, nor any new way of proving it; he is just saying that one can justify the set-theoretic arguments used using a relatively small fraction of Axiomatic Set Theory, as opposed to an enriched theory (which underlies some of Grothendieck's work). This is a very technical issue, of interest to some model theorists and people in mathematical logic, but unlikely to be of general interest even within mathematics (very few people within mathematics care right now about exactly how much Set Theory Wiles needs to get his proof through!) Second, a news report is not a reliable source for the purpose of mathematical claims. If and when the work gets peer reviewed and suitably sourced, it might be relevant. Magidin (talk) 15:52, 2 October 2013 (UTC)
By the way, I think this talk section's title is seriously misleading. Reworking the proof to use fewer set-theoretic assumptions is very unlikely to be a simplification. —David Eppstein (talk) 17:45, 2 October 2013 (UTC)
PS the comments about one-handed pottery in this interview may be relevant. —David Eppstein (talk) 03:57, 4 October 2013 (UTC)

Capital T[edit]

On the naming conventions for theorems (which is no longer relevant) it does point out the the L in last is a possible grey area for the normal capitalization rules. Fine, but why is the T in theorem capitalized? As far as I can tell (looking at the list of theorems) there is no other example of a theorem on wikipedia with that characteristic and is in clear violation of WP:NCCAPS (as is the capital L really, imo). Opinions? Wgunther (talk) 21:29, 20 October 2013 (UTC)

Page move[edit]

@OliverBel: moved the page from Fermat's Last Theorem to Wiles-Fermat Theorem [1]. I've never heard the theorem referred to under this name so I think it should be moved back, but I think it would be better to gain some consensus before starting a move-war.--User:Salix alba (talk): 11:53, 30 October 2013 (UTC)

I've never head it called that either. Google returns 1870 hits. Even "Wiles Theorem"[sic] gets more hits (2460), and would be far more accurate than "Wiles-Fermat". "Fermat's Last Theorem" gets over 381000 hits. I oppose the move; things are called what they are called, not what they ought to be called. I don't see the Pell's equation page being moved around to account for Euler's mis-attribution. Magidin (talk) 15:33, 30 October 2013 (UTC)
I also oppose the move. This is not the place to advocate for replacement of standard terminology by neologisms, even when those neologisms would be more accurate; see WP:SOAPBOX. This should be reverted. —David Eppstein (talk) 16:23, 30 October 2013 (UTC)

A second a third opinion is good enough for me. Move reverted.--User:Salix alba (talk): 17:32, 30 October 2013 (UTC)

Of course. Paul August 10:47, 6 November 2013 (UTC)

Why Fermat did not have a proof[edit]

I have seen the suggestion (hinted at in the section Fermat's Last Theorem#Fermat's conjecture) that a reason to feel confident that Fermat did not have a proof is that he did not challenge other people to prove it, as appears to have been his wont. I thought it might have been in Alf van der Poorten’s book, but a skim has just failed to throw it up; perhaps it is in Singh’s. If anyone has a good source for it, I think it would be worth mentioning in the section Fermat's Last Theorem#Did Fermat possess a general proof?, as I have observed that that question is often raised, along with the suggestion “People ought to be trying find his proof”. It would also enable us to remove the “{{citation-needed}}” (if we reworded the formulation). I have in any case moved the the latter section to be a sub-section of the former, and removed some of the redundancy this made evident. PJTraill (talk) 23:58, 4 March 2014 (UTC)

Dealt with: Casually glancing at van der Poorten again, I saw the very place, and have included references to that and to Weil in Fermat's Last Theorem#Did Fermat possess a general proof? and removed the “most mathematicians and historians” remark, with its {{citation-needed}}. I have instead asserted that it “appears unlikely” that he had a general proof; I hope it is sufficiently evident that that bald statement is justified by the rest of the section without encumbering it with “for the following reasons”. PJTraill (talk) 11:10, 6 March 2014 (UTC)

Does Fermat's copy of Arithmetica, writing in the margin and all, still exist?[edit]

This should probably be mentioned somewhere. I couldn't find a reference to it in the article Shiggity (talk) 19:47, 12 May 2014 (UTC)

The least number of n-power numbers (A new conjecture)[edit]

n=2, m>=2

n=3~4, m>=3

n=5~8, m>=4

n=9~16, m>=5

n=17~32, m>=6



Is it true??? — Preceding unsigned comment added by (talk) 02:10, 19 May 2014 (UTC)

This forum is for discussing improvements to the encyclopedia article on Fermat's Last Theorem, not on discussing potential improvements to the theorem itself. —David Eppstein (talk) 03:13, 19 May 2014 (UTC)

Possible incorrect statement or wording[edit]

The section on proofs for specific exponents states "[Fermat] uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer." A right triangle with sides 10 and 20 has an area of 100, which is the square of the integer 10, contradicting what this claims Fermat had shown. Perhaps there were more conditions to what he demonstrated (the sides of the triangle must be primes?) or the wording here is prone to serious misinterpretation. (talk) 00:52, 15 June 2014 (UTC)

All three sides need to be integer length, not just the legs. In your example the hypotenuse has length 10√5, which is not an integer. Lagrange613 05:03, 15 June 2014 (UTC)

Fixed BBC documentary link, but unsure about copyright[edit]

I have changed the video link to the documentary on You Tube (which had gone dead because of BBC copyright) to a link to, but am not sure if that is in breach of copyright. The BBC itself offers the documentary at, but that apparently only works in the UK. Should we actually use that? PJTraill (talk) 22:18, 19 December 2014 (UTC)

Reverted pending discussion[edit]

User deleted the entire section on Subsequent developments and solution, claiming it was done "to simplify". Then added claims, such as that the full proof of the Taniyama-Shimura modularity conjecture is "impossible to check even with computers", without any citation (I have not heard that claim made anywhere, has anyone?); and alleged connections between the Last Theorem and the "polygonal number conjecture", again with no references or citation. I've reverted; while some of the minor changes might be reasonable, I don't think such large deletions should be done without a discussion and consensus, and certainly the kind of assertions I cited above would need some good reliable sources before being added on the say so of an anonymous editor. Magidin (talk) 23:56, 30 December 2014 (UTC)

A special case of this theorem[edit]

If 2n+1 is prime (n>1), than has no solutions which none of x, y, and z are divisible by 2n+1, because if x is not divisible by 2n+1, than xn must congruent to +1 or -1 to mod 2n+1, so do y and z, and x^n + y^n ≠ z^n (mod 2n+1), since (+1) + (+1), (+1) + (-1), (-1) + (+1), and (-1) + (-1), are not congruent to +1 or -1 (mod 2n+1), because of 2n+1 > 3.

This can prove this theorem for n = 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158, 165, 168, 173, 174, 176, 179, ... and their multiples. — Preceding unsigned comment added by (talk) 03:14 January 31 2015 (UTC)

The primes n for which 2n+1 is also prime are called Sophie Germain primes, and Sophie Germain proved back in the early 19th century that there is no solution to the Fermat equation when n is such a prime and in which n does not divide xyz. In other words, she proved exactly what you write above. :In any case, the talk page is for discussing improvements to the article, which must be appropriately sourced and verifiable. This comment is neither. Magidin (talk) 01:11, 1 February 2015 (UTC)

Could Fermat Have Had the Proof?[edit]

The article leaves the question open as to whether he actually had the proof or not. As a reader, I would like to know if, in retrospect "experts" agree that he either could or could not have had the proof found in 1994. I wonder if the proof required advances in mathematics that Fermat may not have been aware of, or if he actually did have the "tools" needed for the proof and it took 300+ years to put them to use. Hope this makes sense. What I'm looking for is consensus either way, or a direct statement that there is no consensus on whether or not Fermat could (or could not) have had the proof prior to it's 1994 discovery.Jonny Quick (talk) 09:05, 11 June 2015 (UTC)

In the section Fermat's Conjecture, it states:
"It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely. Only one related proof by him has survived, namely for the case n = 4, as described in the section Proofs for specific exponents. While Fermat posed the cases of n = 4 and of n = 3 as challenges to his mathematical correspondents, such as Marin Mersenne, Blaise Pascal, and John Wallis,[19] he never posed the general case.[20] Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvellous proof" of the general case, and never published it. Van der Poorten[21] suggests that while the absence of a proof is insignificant, the lack of challenges means Fermat realised he did not have a proof; he quotes Weil[22] as saying Fermat must have briefly deluded himself with an irretrievable idea. The techniques Fermat might have used in such a "marvellous proof" are unknown. Taylor and Wiles’s proof relies on 20th century techniques.[23] Fermat’s proof would have had to have been elementary by comparison, given the mathematical knowledge of his time."
Why is this not a direct answer to your questions? Yes: the proof as is known today requires advances in mathematics that were well beyond Fermat's knowledge and he lacked the tools that were actually used (he lacked the tools needed to produce the tools needed to produce... to produce the tools that were used in the proof). And it is generally thought that he probably did not have one, as indicated by the paragraph above. Magidin (talk) 17:02, 11 June 2015 (UTC)
Thanks for the informative response. It's a detailed answer to a question that is provoked earlier in the article. My question centers on readability and whether or not the article delivers information "on demand" as the reader requires it. TBH I didn't read any further into the article, as I stumbled on this question and immediately came here to document it. It appears the answer to the question as to whether or not Fermat could have had the proof 300 years ago is a very complex one, which causes me to question whether or not his claim that he had it should be included in the lead (or "lede", I see it written here both ways). I'm picky about leads. They should answer basic questions and not open doors to secondary issues or controversies. Unless there is some compelling reason to mention the possibility that Fermat had the proof in the lead. My instinct is that there is not. These are only my opinions (vs. wiki policies, etc...). I think sometimes well-informed editors fail to perceive an article from the perspective of a person that is not intimately familiar with the subject of the article. It's an interesting article, and the question of whether or not Fermat had the proof is also interesting. It's actually TOO interesting to include in the lead (as it raises too many questions and areas of inquiry), and should be further down in the body of the article, IMO.Jonny Quick (talk) 07:34, 13 June 2015 (UTC)
The compelling reason is that Fermat's claim to have solved it is a big part of the fame of this problem. —David Eppstein (talk) 07:44, 13 June 2015 (UTC)
One would have to explain why it is called "Fermat's last theorem"; that's important, and should be in the lead. The fact is that it is called that because of Fermat's margin note in the book which is Fermat's claim to have a "truly marvelous proof". Magidin (talk) 21:30, 13 June 2015 (UTC)

External links modified[edit]

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The article had the following two formal statements of the problem:
xn + yn = zn, where n ≥ 3, has no non-trivial solutions x, y, zN, and
xn + yn = zn, where n ≥ 3, has no non-trivial solutions x, y, zZ.
I have seen these before, and again here, but I do not understand how these are correct formal statements of the theorem. Shouldn't it be
xn + yn = zn, has no non-trivial solutions when n, x, y, zN, and where n ≥ 3?
Why is it acceptable to leave out nN in the formal statement?
Nick Beeson (talk) 01:51, 19 March 2016 (UTC)

It is usually assumed that n is a positive whole number and bigger than or equal to 3. — Preceding unsigned comment added by (talk) 13:48, 2 June 2017 (UTC)

Taniyama–Shimura-Weil conjecture history[edit]

"Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture"

Weil was 2 at that time, so how could his name have been part of the conjecture?

e: lol, whoops. Weil is André Weil, not Andrew Wiles. — Preceding unsigned comment added by (talk) 19:29, 25 July 2016 (UTC)

To avoid having to make drastic revisions, should take more time and study the subject carefully. — Preceding unsigned comment added by (talk) 08:47, 22 August 2016 (UTC)

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Solved 1965/1977?[edit]

In 1965, Dr Edward Close described the first (later) published proof (FLT65) for possibly the most famous math conjecture, ‘Fermat’s Last Theorem’ (FLT). Dr Close’s solution has, through 2016, been examined by tens of professional mathematicians including intensively over the past three years, and yet has never been refuted. Some of these experts have made suggestions or queried areas, but in every instance, further analyses by mathematicians have demonstrated the ostensible criticisms to be irrelevant to the FLT proof, or incorrect. The Close solution (FLT65), was published initially as an Appendix to Close’s Book of Atma in 1977, but remarkably, is only several pages. That contrasts with Sir Andrew Wiles’s later accepted 1994 proof of over 150 pages. [2] — Preceding unsigned comment added by (talk) 00:11, 19 May 2017 (UTC)

A personal page issuing "press releases" is not a reliable source. This is not verifiable, and as such, not worthy of inclusion. The talk page is also not a forum for discussion about the theorem itself. If you have any specific proposal to improve the article, make it. Otherwise, this needs to be deleted because right now it amounts to little more than propaganda. Magidin (talk) 02:42, 19 May 2017 (UTC)