Fermat's Last Theorem: Difference between revisions

{{shortShort description|17th-century conjecture proved by Andrew Wiles in 1994}}

{{forFor-multi|other theorems named after Pierre de Fermat|Fermat's theorem|the book by Simon Singh|Fermat's Last Theorem (book)}}

The [[Pythagorean theorem|Pythagorean equation]], {{nowrap|1=''x''² + ''y''² = ''z''²}}, has an infinite number of positive [[integer]] solutions for ''x'', ''y'', and ''z''; these solutions are known as [[Pythagorean triple]]s (with the simplest example being 3, 4, 5). Around 1637, Fermat wrote in the margin of a book that the more general equation {{nowrap|1=''a''^''n'' + ''b''^''n'' = ''c''^''n''}} had no solutions in positive integers if ''n'' is an integer greater than 2. Although he claimed to have a general [[mathematical proof|proof]] of his conjecture, Fermat left no details of his proof, and no proof by himnone has ever been found. His claim was discovered some 30&nbsp;years later, after his death. This claim, which came to be known as ''Fermat's Last Theorem'', stood unsolved for the next three and a half centuries.<ref name="BostonFLTproof">{{Cite web|url=https://www.math.wisc.edu/~boston/869.pdf |author=Nigel Boston |page=5|title=The Proof of Fermat's Last Theorem}}</ref>

In 1984, [[Gerhard Frey]] noticed an apparent link between these two previously unrelated and unsolved problems. An outline suggesting this could be proved was given by Frey. The full proof that the two problems were closely linked was accomplished in 1986 by [[Ken Ribet]], building on a partial proof by [[Jean-Pierre Serre]], who proved all but one part known as the "epsilon conjecture" (see: ''[[Ribet's Theorem]]'' and ''[[Frey curve]]'').<ref name="abelcitation"/> These papers by Frey, Serre and Ribet showed that if the Taniyama–Shimura conjecture could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would also follow automatically. The connection is described [[#Equivalent statements of the theorem|below]]: any solution that could contradict Fermat's Last Theorem could also be used to contradict the Taniyama–Shimura conjecture. So if the modularity theorem were found to be true, then by definition, no solution contradicting Fermat's Last Theorem could exist, whichmeaning wouldthat thereforeFermat's haveLast toTheorem bemust truealso asbe welltrue.

* '''Equivalent statement 4 – connection to elliptic curves:''' If {{mvar|a}}, {{mvar|b}}, {{mvar|c}} is a non-trivial solution to {{math|''a''^''p'' + ''b''^''p'' {{=}} ''c''^''p''}}, {{mvar|p}} odd prime, then {{math|''y''^''2'' {{=}} ''x''(''x'' − ''a''^''p'')(''x'' + ''b''^''p'')}} ([[Frey curve]]) will be an [[elliptic curve]] without a modular form.<ref>{{cite journal |last=Wiles |first=Andrew |authorlink=Andrew Wiles |year=1995 |title=Modular elliptic curves and Fermat's Last Theorem |url=http://math.stanford.edu/~lekheng/flt/wiles.pdf |quotation=Frey's suggestion, in the notation of the following theorem, was to show that the (hypothetical) elliptic curve {{math|''y''^''2'' {{=}} ''x''(''x'' + ''u''^''p'')(''x'' – ''v''^''p'')}} could not be modular. |journal=[[Annals of Mathematics]] |volume=141 |issue=3 |page=448 |oclc=37032255 |doi=10.2307/2118559 |jstor=2118559 |access-date=11 August 2003 |archive-date=10 May 2011 |archive-url=https://web.archive.org/web/20110510062158/http://math.stanford.edu/%7Elekheng/flt/wiles.pdf |url-status=dead }}</ref>

In ancient times it was known that a triangle whose sides were in the ratio 3:4:5 would have a right angle as one of its angles. This was used in construction and later in early geometry. It was also known to be one example of a general rule that any triangle where the length of two sides, each squared and then added together {{nowrap|1=(3² + 4² = 9 + 16 = 25)}}, equals the square of the length of the third side {{nowrap|1=(5² = 25)}}, would also be a right angle triangle. This is now known as the [[Pythagorean theorem]], and a triple of numbers that meets this condition is called a Pythagorean triple; both are named after the ancient Greek [[Pythagoras]]. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples,<ref name="Stillwell_2003">{{cite book | author = Stillwell J | year = 2003 | title = Elements of Number Theory | url = https://books.google.com/books?id=LiAlZO2ntKAC&pg=PA110 | publisher = Springer-Verlag | location = New York | isbn = 0-387-95587-9 | pages = 110–112 | access-date = 2016-03-17| authorlink=John Stillwell }}</ref> and methods for generating such triples have been studied in many cultures, beginning with the [[Babylonian mathematics|Babylonians]]{{sfn|Aczel|1996|pp=13–15|ps=}} and later [[Greek mathematics|ancient Greek]], [[Chinese mathematics|Chinese]], and [[Indian mathematics|Indian]] mathematicians.<ref name="auto"/> Mathematically, the definition of a Pythagorean triple is a set of three integers {{nowrap|(''a'', ''b'', ''c'')}} that satisfy the equation{{sfn|Stark|1978|pp=151–155|ps=}} {{nowrap|1=''a''² + ''b''² = ''c''²}}.

While Fermat posed the cases of {{nowrap|1=''n'' = 4}} and of {{nowrap|1=''n'' = 3}} as challenges to his mathematical correspondents, such as [[Marin Mersenne]], [[Blaise Pascal]], and [[John Wallis]],<ref>Ribenboim, pp. 13, 24</ref> he never posed the general case.<ref name="van der Poorten-p.5">van der Poorten, Notes and Remarks 1.2, p. 5</ref> Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van der Poorten<ref> name="van der Poorten, ''loc-p. cit.''<5"/ref> 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 [[André Weil|Weil]]<ref>{{cite book | author = André Weil | year = 1984 | title = Number Theory: An approach through history. From Hammurapi to Legendre | publisher = Birkhäuser | location = Basel, Switzerland | page = 104| authorlink=André Weil }}</ref> as saying Fermat must have briefly deluded himself with an irretrievable idea. The techniques Fermat might have used in such a "marvelous proof" are unknown.

In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents ''p''&nbsp;=&nbsp;3, 5 and 7. The case {{nowrap|1=''p'' = 3}} was first stated by [[Abu-Mahmud Khojandi]] (10th century), but his attempted proof of the theorem was incorrect.{{sfn|Dickson|1919|p=545|ps=}}<ref>{{MacTutor|id=Al-Khujandi|title=Abu Mahmud Hamid ibn al-Khidr Al-Khujandi|mode=cs1}}</ref> In 1770, [[Leonhard Euler]] gave a proof of ''p''&nbsp;=&nbsp;3,<ref>[[Leonhard Euler|Euler L]] (1770) ''Vollständige Anleitung zur Algebra'', Roy. Acad. Sci., St. Petersburg.</ref> but his proof by infinite descent<ref>{{cite web | author = Freeman L | title = Fermat's Last Theorem: Proof for ''n''&nbsp;=&nbsp;3 | url = http://fermatslasttheorem.blogspot.com/2005/05/fermats-last-theorem-proof-for-n3.html | access-date = 23 May 2009|date = 22 May 2005}}</ref> contained a major gap.<ref>Ribenboim, pp. 24–25</ref>{{sfn|Mordell|1921|pp=6–8|ps=}}{{sfn|Edwards|1996|pp=39–40|ps=}} However, since Euler himself had proved the lemma necessary to complete the proof in other work, he is generally credited with the first proof.{{sfn|Aczel|1996|p=44|ps=}}{{sfn|Edwards|1996|pp=40,52–54|ps=}}<ref>{{cite journal |author=J. J. Mačys |title=On Euler's hypothetical proof |journal=Mathematical Notes |year=2007 |volume=82 |issue=3–4 |pages=352–356 |doi=10.1134/S0001434607090088 |mr=2364600|s2cid=121798358 }}</ref> Independent proofs were published<ref>Ribenboim, pp. 33, 37–41</ref> by Kausler (1802),<ref name="Kausler_1802" >{{cite journal | author = Kausler CF | year = 1802 | title = Nova demonstratio theorematis nec summam, nec differentiam duorum cuborum cubum esse posse | journal =Novi Acta Academiae Scientiarum Imperialis Petropolitanae | volume = 13 | pages = 245–253}}</ref> Legendre (1823, 1830),<ref name="Legendre_1830" >{{cite book | author = Legendre AM | year = 1830 | title = Théorie des Nombres (Volume II) | edition = 3rd | publisher = Firmin Didot Frères | location = Paris| authorlink=Adrien-Marie Legendre }} Reprinted in 1955 by A. Blanchard (Paris).</ref><ref name="Legendre_1823" >{{cite journal | author = Legendre AM | year = 1823 | title = Recherches sur quelques objets d'analyse indéterminée, et particulièrement sur le théorème de Fermat | journal = Mémoires de l'Académie royale des sciences | volume = 6 | pages = 1–60| authorlink=Adrien-Marie Legendre }} Reprinted in 1825 as the "Second Supplément" for a printing of the 2nd edition of ''Essai sur la Théorie des Nombres'', Courcier (Paris). Also reprinted in 1909 in ''Sphinx-Oedipe'', '''4''', 97–128.</ref> Calzolari (1855),<ref>{{cite book | author = Calzolari L | year = 1855 | title = Tentativo per dimostrare il teorema di Fermat sull'equazione indeterminata xⁿ + yⁿ = zⁿ | publisher = Ferrara}}</ref> [[Gabriel Lamé]] (1865),<ref>{{cite journal | author = Lamé G | year = 1865 | title = Étude des binômes cubiques ''x''³ ± ''y''³ | journal = [[Comptes rendus de l'Académie des sciences|Comptes rendus hebdomadaires des séances de l'Académie des Sciences]] | volume = 61 | pages = 921–924, 961–965| authorlink=Gabriel Lamé }}</ref> [[Peter Guthrie Tait]] (1872),<ref>{{cite journal | author = Tait PG | year = 1872 | title = Mathematical Notes | journal = Proceedings of the Royal Society of Edinburgh | volume = 7 | page = 144 | doi=10.1017/s0370164600041857| url = https://zenodo.org/record/1428704 | authorlink=Peter Guthrie Tait }}</ref> [[Siegmund Günther]] (1878),<ref>{{cite journal | last = Günther |first=S.|author-link=Siegmund Günther | year = 1878 | title = Ueber die unbestimmte Gleichung {{math|1=''x''^''3'' + ''y''^''3'' = ''a''^''3''}} | journal = Sitzungsberichte der Königliche böhmische Gesellschaft der Wissenschaften in Prag |volume=jahrg. 1878-1880 | pages = 112–120|url=https://www.biodiversitylibrary.org/item/111872#page/150}}</ref> Gambioli (1901),<ref name="Gambioli_1901" >{{cite journal | author = Gambioli D | year = 1901 | title = Memoria bibliographica sull'ultimo teorema di Fermat | journal = Periodico di Matematiche | volume = 16 | pages = 145–192}}</ref> Krey (1909),<ref>{{cite journal|last=Krey|first=H.|year=1909|title=Neuer Beweis eines arithmetischen Satzes|journal=Mathematisch-Naturwissenschaftliche Blätter|volume=6|issue=12|pages=179–180|url=https://books.google.com/books?id=90Q6AQAAMAAJ&pg=RA2-PA179}}</ref> Rychlík (1910),<ref name="Rychlik_1910" >{{cite journal | author = Rychlik K | year = 1910 | title = On Fermat's last theorem for ''n''&nbsp;=&nbsp;4 and ''n''&nbsp;=&nbsp;3 (in Bohemian) | journal = Časopis Pro Pěstování Matematiky a Fysiky | volume = 39 | pages = 65–86| authorlink=Karel Rychlík }}</ref> Stockhaus (1910),<ref>{{cite book | author = Stockhaus H | year = 1910 | title = Beitrag zum Beweis des Fermatschen Satzes | publisher = Brandstetter | location = Leipzig}}</ref> Carmichael (1915),<ref>{{cite book | author = Carmichael RD | year = 1915 | title = Diophantine Analysis | publisher = Wiley | location = New York| authorlink=Robert Daniel Carmichael }}</ref> [[Johannes van der Corput]] (1915),<ref name="van_der_Corput_1915" >{{cite journal | author = van der Corput JG | year = 1915 | title = Quelques formes quadratiques et quelques équations indéterminées | journal = Nieuw Archief voor Wiskunde | volume = 11 | pages = 45–75| authorlink=Johannes van der Corput }}</ref> [[Axel Thue]] (1917),<ref>{{cite journal|last=Thue|first=Axel|author-link=Axel Thue|year=1917|title=Et bevis for at ligningen {{math|1=''A''³ + ''B''³ = ''C''³}} er unmulig i hele tal fra nul forskjellige tal ''A'', ''B'' og ''C''|journal=Archiv for Mathematik og Naturvidenskab|volume=34|issue=15|pages=3–7|url=https://books.google.com/books?id=xIFEAQAAMAAJ&pg=RA13-PA1}} Reprinted in ''Selected Mathematical Papers'' (1977), Oslo: Universitetsforlaget, pp. 555–559</ref> and Duarte (1944).<ref>{{cite journal | author = Duarte FJ | year = 1944 | title = Sobre la ecuación ''x''³ + ''y''³ + ''z''³ = 0 | journal = Boletín de la Academia de Ciencias Físicas, Matemáticas y Naturales (Caracas) | volume = 8 | pages = 971–979}}</ref>

(Note: It is often stated that Kummer was led to his "ideal complex numbers" by his interest in Fermat's Last Theorem; there is even a story often told that Kummer, like [[Gabriel Lamé|Lamé]], believed he had proven Fermat's Last Theorem until [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]] told him his argument relied on unique factorization; but the story was first told by [[Kurt Hensel]] in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources. [[Harold Edwards (mathematician)|Harold Edwards]] sayssaid the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken".{{sfn|Edwards|1996|p=79|ps=}} See [[Ideal number#History|the history of ideal numbers]].)

However, despite these efforts and their results, no proof existed of Fermat's Last Theorem. Proofs of individual exponents by their nature could never prove the ''general'' case: even if all exponents were verified up to an extremely large number X, a higher exponent beyond X might still exist for which the claim was not true. (This had been the case with some other past conjectures, such as with [[Skewes' number]], and it could not be ruled out in this conjecture.)<ref>{{Cite web|url=https://mathoverflow.net/q/28196 |title=Examples of eventual counterexamples, answer by J.D. Hamkins|last=Hamkins|first=Joel David|date=15 June 2010|website=mathoverflow.net|access-date=15 June 2017}}</ref>

Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and fix the error. He adds that he was having a final look to try and understand the fundamental reasons for why his approach could not be made to work, when he had a sudden insight: that the specific reason why the Kolyvagin–Flach approach would not work directly {{em|also}} meant that his original attempts using [[Iwasawa theory]] could be made to work, if he strengthened it using his experience gained from the Kolyvagin–Flach approach. Fixing one approach with tools from the other approach would resolve the issue for all the cases that were not already proven by his refereed paper.<ref name="sept1994"/>{{sfn|Aczel|1996|pp=132–134|ps=}} He described later that Iwasawa theory and the Kolyvagin–Flach approach were each inadequate on their own, but together they could be made powerful enough to overcome this final hurdle.<ref name="sept1994"/>

All primitive integer solutions (i.e., those with no prime factor common to all of ''a'', ''b'', and ''c'') to the [[optic equation]] {{nowrap|1=''a''⁻¹ + ''b''⁻¹ = ''c''⁻¹}} can be written as{{sfn|Dickson|1919|pp=688–691|ps=}}

There are no solutions in integers for {{nowrap|1=''a''^''n'' + ''b''^''n'' = ''c''^''n''}} for integers {{nowrap|''n'' < −2}}. If there were, the equation could be multiplied through by {{nowrap|''a''^{{{abs|''n''}}}''b''^{{{abs|''n''}}}''c''^{{{abs|''n''}}}}} to obtain {{nowrap|1=(''bc'')^{{{abs|''n''}}} + (''ac'')^{{{abs|''n''}}} = (''ab'')^{{{abs|''n''}}}}}, which is impossible by Fermat's Last Theorem.

In 1908, the German industrialist and amateur mathematician [[Paul Wolfskehl]] bequeathed 100,000 [[German gold mark|gold marks]]—a large sum at the time—to the Göttingen Academy of Sciences to offer as a prize for a complete proof of Fermat's Last Theorem.<ref>Singh, pp. 120–125, 131–133, 295–296</ref>{{sfn|Aczel|1996|p=70|ps=}} On 27 June 1908, the academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was begun.<ref>Singh, pp. 120–125</ref> Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997.<ref>Singh, p. 284</ref> In March 2016, Wiles was awarded the Norwegian government's [[Abel prize]] worth €600,000 for "his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory".<ref>{{cite web

{{cite web | url = http://www.abelprize.no/c67107/binfil/download.php?tid=67059

| title = The Abel Prize citation 2016

In the ''[[Star Trek: The Next Generation]]'' episode "[[The Royale]]", [[Jean-Luc Picard|Captain Picard]] states that the theorem is still unproven in the 24th century. The proof was released 5five years after the episode originally aired, a fact later referenced in the ''[[Star Trek: Deep Space Nine]]'' episode "[[Facets (Star Trek: Deep Space Nine)|Facets]]".<ref>{{Cite web |last=Moseman |first=Andrew |date=2017-09-01 |title=Here's a Fun Math Goof in 'Star Trek: The Next Generation' |url=https://www.popularmechanics.com/science/math/news/a28040/fermats-last-theorem-star-trek-the-next-generation/ |access-date=2023-06-09 |website=Popular Mechanics |language=en-US}}</ref>