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{{For|other theorems named after Pierre de Fermat|Fermat's theorem (disambiguation){{!}}Fermat's theorem}}
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[[File:Diophantus-II-8-Fermat.jpg|thumb|400px|The 1670 edition of [[Diophantus]]' ''Arithmetica'' includes Fermat's commentary, particularly his "Last Theorem" (''Observatio Domini Petri de Fermat'').]]


In [[number theory]], '''Fermat's Last Theorem''' (sometimes called '''Fermat's conjecture''', especially in older texts) states that no three [[positive number|positive]] [[integer]]s ''a'', ''b'', and ''c'' can satisfy the equation ''a''<sup>''n''</sup>&nbsp;+&nbsp;''b''<sup>''n''</sup>&nbsp;=&nbsp;''c''<sup>''n''</sup> for any integer value of ''n'' greater than two.
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This theorem was first [[conjectured]] by [[Pierre de Fermat]] in 1637, famously in the margin of a copy of ''[[Arithmetica]]'' where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of [[algebraic number theory]] in the 19th century and the proof of the [[modularity theorem]] in the 20th century. It is among the most famous theorems in the [[history of mathematics]] and prior to its [[Wiles' proof of Fermat's Last Theorem|1995 proof]] by [[Andrew Wiles]] it was in the ''[[Guinness Book of World Records]]'' for "most difficult mathematical problems".
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Fermat's Last Theorem (known by this title historically although technically a [[conjecture (mathematics)|conjecture]], or unproven speculation, until proven in 1995) stood as an unsolved riddle in mathematics for over three centuries. The theorem itself is a deceptively simple statement that Fermat famously stated he had proved around 1637. His claim was discovered some 30 years later, after his death, written in the margin of a book, but with no [[mathematical proof|proof]] provided.


The claim eventually became one of the most famous unsolved problems of mathematics. Attempts to prove it prompted substantial development in [[number theory]], and over time Fermat's Last Theorem itself gained legendary prominence as an [[List of unsolved problems in mathematics|unsolved problem in popular mathematics]]. It is based the [[Pythagorean theorem]], which states that {{nowrap|1=''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>}}, where ''a'' and ''b'' are the lengths of the legs of a [[right triangle]] and ''c'' is the length of the [[hypotenuse]].
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The Pythagorean [[mathematical equation|equation]] has an infinite number of positive [[integer]] solutions for ''a'', ''b'', and ''c''; these solutions are known as [[Pythagorean triple]]s. Fermat stated that the more general equation {{nowrap|1=''a''<sup>n</sup> + ''b''<sup>n</sup> = ''c''<sup>n</sup>}} had no solutions in positive [[integer]]s if ''n'' is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof apart from the special case ''n''&nbsp;=&nbsp;4.
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With the special case ''n''&nbsp;=&nbsp;4 proven, the problem was to prove the theorem for [[exponent]]s ''n'' that are [[prime number]]s (this limitation is considered trivial to prove<ref group=note>If the exponent "n" were not prime then it would be possible to write n as a product of two smaller integers (n = P*Q) in which P is a prime number, and then a<sup>n</sup> = a<sup>P*Q</sup> = (a<sup>Q</sup>)<sup>P</sup> for each of a, b, and c, i.e. an equivalent solution would ''also'' have to exist for the prime power P which is ''smaller'' than N, as well.</ref>). Over the next two centuries (1637–1839), the conjecture was proven for only the primes 3, 5, and 7, although [[Sophie Germain]] innovated and proved an approach which was relevant to an entire class of primes. In the mid-19th century, [[Ernst Kummer]] extended this and proved the theorem for all [[regular prime]]s, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible (meaning that mathematicians generally considered a proof to be either impossible, or at best exceedingly difficult, or not achievable with current knowledge).
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The proof of Fermat's Last Theorem in full, for all ''n'', was finally accomplished, however, after 358 years, by [[Andrew Wiles]] in 1995, an achievement for which he was honoured and received numerous awards. The solution came in a roundabout manner, from a completely different area of mathematics.
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In 1984, [[Gerhard Frey]] noticed an apparent link between the [[modularity theorem]] and Fermat's Last Theorem. This potential link was confirmed two years later by [[Ken Ribet]] (see: ''[[Ribet's Theorem]]'' and ''[[Frey curve]]''). On hearing this, English mathematician Andrew Wiles, who had a childhood fascination with Fermat's Last Theorem, decided to try to prove the modularity theorem as a way to prove Fermat's Last Theorem. In 1993, after six years working secretly on the problem, Wiles [[Wiles's proof of Fermat's Last Theorem|succeeded in proving]] enough of the modularity theorem to prove Fermat's Last Theorem. Wiles paper was massive in size and scope. A flaw was discovered in one part of his original paper during [[peer review]] and required a further year and collaboration with a past student, [[Richard Taylor (mathematician)|Richard Taylor]], to resolve.  As a result, the final proof in 1995 was accompanied by a second, smaller, joint paper to that effect.  Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the modularity theorem were subsequently proven by other mathematicians, building on Wiles' work, between 1996 and 2001.
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===Pythagoras and Diophantus===
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A Pythagorean triple – named for the ancient Greek [[Pythagoras]] – is a set of three integers (''a'', ''b'', ''c'') that satisfy a special case of Fermat's equation (''n''&nbsp;=&nbsp;2)<ref>Stark, pp. 151–155.</ref>
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:<math>a^2 + b^2 = c^2.\ </math>
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Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13).  There are infinitely many such triples,<ref name="Stillwell_2003">{{cite book | author = [[John Stillwell|Stillwell J]] | year = 2003 | title = Elements of Number Theory  | url = http://books.google.com/books?id=LiAlZO2ntKAC&pg=PA110 | publisher = Springer-Verlag | location = New York | isbn = 0-387-95587-9 | pages = 110–112}}</ref> and methods for generating such triples have been studied in many cultures, beginning with the [[Babylonian mathematics|Babylonians]]<ref>Aczel, pp. 13–15</ref> and later [[Greek mathematics|ancient Greek]], [[Chinese mathematics|Chinese]], and [[Indian mathematics|Indian]] mathematicians.<ref>Singh, pp. 18–20.</ref> The traditional interest in Pythagorean triples connects with the [[Pythagorean theorem]];<ref>Singh, p. 6.</ref> in its converse form, it states that a [[triangle]] with sides of lengths ''a'', ''b'', and ''c'' has a [[right angle]] between the ''a'' and ''b'' legs when the numbers are a Pythagorean triple. [[Right angle]]s have various practical applications, such as [[surveying]], [[carpentry]], [[masonry]], and [[construction]]. Fermat's Last Theorem is an extension of this problem to higher powers, stating that no solution exists when the exponent 2 is replaced by any larger integer.
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Fermat's equation, ''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = ''z''<sup>''n''</sup> with positive [[integer]] solutions, is an example of a [[Diophantine equation]],<ref>Stark, pp. 145–146.</ref> named for the 3rd-century [[Alexandria]]n mathematician, [[Diophantus]], who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers ''x'' and ''y'' such that their sum, and the sum of their squares, equal two given numbers ''A'' and ''B'', respectively:
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:<math>A = x + y\ </math>
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Diophantus's major work is the ''[[Arithmetica]]'', of which only a portion has survived.<ref>Singh, pp. 50–51.</ref> Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the ''Arithmetica'',<ref>Stark, p. 145.</ref> which was translated into Latin and published in 1621 by [[Claude Gaspard Bachet de Méziriac|Claude Bachet]].<ref>Aczel, pp. 44–45; Singh, pp. 56–58.</ref>
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Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation ''x''<sup>2</sup> + ''y''<sup>2</sup> = ''z''<sup>2</sup> are given by the [[Pythagorean triple]]s, originally solved by the Babylonians (c. 1800 BC).<ref>Aczel, pp. 14–15.</ref> Solutions to linear Diophantine equations, such as 26''x'' + 65''y'' = 13, may be found using the [[Euclidean algorithm]] (c. 5th century BC).<ref>Stark, pp. 44–47.</ref>
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Many [[Diophantine equation]]s have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no ''cross terms'' mixing two letters, without sharing its particular propertiesFor example, it is known that there are infinitely many positive integers ''x'', ''y'', and ''z'' such that ''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = ''z''<sup>''m''</sup> where ''n'' and ''m'' are [[relatively prime]] natural numbers.<ref group=note>For example, <math>\left((j^r+1)^s\right)^r + \left(j(j^r+1)^s)\right)^r = (j^r+1)^{rs+1}.</math></ref>
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===Fermat's conjecture===
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[[File:Diophantus-II-8.jpg|thumb|right|Problem II.8 in the 1621 edition of the ''Arithmetica'' of [[Diophantus]]. On the right is the famous margin which was too small to contain Fermat's alleged proof of his "last theorem".]]
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Problem II.8 of the ''[[Arithmetica]]'' asks how a given square number is split into two other squares; in other words, for a given [[rational number]] ''k'', find rational numbers ''u'' and ''v'' such that ''k''<sup>2</sup>&nbsp;=&nbsp;''u''<sup>2</sup>&nbsp;+&nbsp;''v''<sup>2</sup>. Diophantus shows how to solve this sum-of-squares problem for ''k''&nbsp;=&nbsp;4 (the solutions being ''u'' = 16/5 and ''v'' = 12/5).<ref>Friberg, pp. 333– 334.</ref>
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Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the ''Arithmetica'' next to Diophantus' sum-of-squares problem:<ref>Dickson, p. 731; Singh, pp. 60–62; Aczel, p. 9.</ref>
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{| style="font-style:italic;"
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| style="padding:0 1em;vertical-align:top;" | Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.
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| style="padding:0 1em;vertical-align:top;" | it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.<ref>Panchishkin, [http://books.google.com/books?id=wvK586IxaxwC&pg=PA341&dq=%22Cubum+autem+in+duos+cubos,+aut+quadratoquadratum%22&hl=en&ei=Jig_Tc2hCIH_8Aaf2LivCg&sa=X&oi=book_result&ct=result&resnum=7&sqi=2&ved=0CEsQ6AEwBg#v=onepage&q=%22Cubum%20autem%20in%20duos%20cubos%2C%20aut%20quadratoquadratum%22&f=false p. 341]</ref>
 
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It is not known whether Fermat had actually found a valid proof. His proof of one case (''n''&nbsp;=&nbsp;4) by [[infinite descent]] has survived.<ref>Dickson, pp. 615–616; Aczel, p. 44.</ref>  Fermat posed the cases of ''n''&nbsp;=&nbsp;4 and of ''n''&nbsp;=&nbsp;3 as challenges to his mathematical correspondents, such as [[Marin Mersenne]], [[Blaise Pascal]], and [[John Wallis]].<ref>Ribenboim, pp. 13, 24.</refHowever, in the last thirty years of his life, Fermat never again wrote of his "truly marvellous proof" of the general case.
 
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After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father's comments.<ref>Singh, pp. 62–66.</ref>  The margin note became known as ''Fermat's Last Theorem'',<ref name="Dickson, p. 731">Dickson, p. 731.</ref> as it was the last of Fermat's asserted theorems to remain unproven.<ref>Singh, p. 67; Aczel, p. 10.</ref>
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===Proofs for specific exponents===
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{{Main|Proof of Fermat's Last Theorem for specific exponents}}
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Only one [[Proof of Fermat's Last Theorem for specific exponents#Application to right triangles|mathematical proof by Fermat]] has survived, in which 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.<ref>{{cite web | author = Freeman L | title = Fermat's One Proof | url = http://fermatslasttheorem.blogspot.com/2005/05/fermats-one-proof.html | accessdate = 23 May 2009}}</ref>  His proof is equivalent to demonstrating that the equation
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:<math>x^4 - y^4 = z^2</math>
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has no primitive solutions in integers (no [[pairwise coprime]] solutions).  In turn, this proves Fermat's Last Theorem for the case ''n''=4, since the equation ''a''<sup>4</sup> + ''b''<sup>4</sup> = ''c''<sup>4</sup> can be written as ''c''<sup>4</sup> − ''b''<sup>4</sup> = (''a''<sup>2</sup>)<sup>2</sup>.
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Alternative proofs of the case ''n''&nbsp;=&nbsp;4 were developed later<ref>Ribenboim, pp. 15–24.</ref> by [[Bernard Frénicle de Bessy|Frénicle de Bessy]] (1676),<ref>Frénicle de Bessy, ''Traité des Triangles Rectangles en Nombres'', vol. I, 1676, Paris.  Reprinted in ''Mém. Acad. Roy. Sci.'', '''5''', 1666–1699 (1729).</ref> [[Leonhard Euler]] (1738),<ref name="Euler_1738">{{cite journal | author = [[Leonhard Euler|Euler L]] | year = 1738 | title = Theorematum quorundam arithmeticorum demonstrationes | journal = Comm. Acad. Sci. Petrop. | volume = 10 | pages = 125–146}}.  Reprinted ''Opera omnia'', ser. I, "Commentationes Arithmeticae", vol. I, pp. 38–58, Leipzig:Teubner (1915).</ref> Kausler (1802),<ref name="Kausler_1802" /> [[Peter Barlow (mathematician)|Peter Barlow]] (1811),<ref>{{cite book | author = [[Peter Barlow (mathematician)|Barlow P]] | year = 1811 | title = An Elementary Investigation of Theory of Numbers | location = St. Paul's Church-Yard, London | publisher = J. Johnson | pages = 144–145}}</ref> [[Adrien-Marie Legendre]] (1830),<ref name="Legendre_1830" /> Schopis (1825),<ref>{{cite book | author = Schopis | year = 1825 | title = Einige Sätze aus der unbestimmten Analytik | publisher = Programm | location = Gummbinnen}}</ref> Terquem (1846),<ref>{{cite journal | author = Terquem O | year = 1846 | title = Théorèmes sur les puissances des nombres | journal = Nouv. Ann. Math. | volume = 5 | pages = 70–87}}</ref> [[Joseph Louis François Bertrand|Joseph Bertrand]] (1851),<ref>{{cite book | author = [[Joseph Louis François Bertrand|Bertrand J]] | year = 1851 | title = Traité Élémentaire d'Algèbre | publisher = Hachette | location = Paris | pages = 217–230, 395}}</ref> [[Victor Lebesgue]] (1853, 1859, 1862),<ref>{{cite journal | author = Lebesgue VA | year = 1853 | title = Résolution des équations biquadratiques ''z''<sup>2</sup> = ''x''<sup>4</sup> ± 2<sup>''m''</sup>''y''<sup>4</sup>, ''z''<sup>2</sup> = 2<sup>''m''</sup>''x''<sup>4</sup> − ''y''<sup>4</sup>, 2<sup>''m''</sup>''z''<sup>2</sup> = ''x''<sup>4</sup> ± ''y''<sup>4</sup> | journal = J. Math. Pures Appl. | volume = 18 | pages = 73–86}}<br />{{cite book | author = Lebesgue VA | year = 1859 | title = Exercices d'Analyse Numérique | publisher = Leiber et Faraguet | location = Paris | pages = 83–84, 89}}<br />{{cite book | author = Lebesgue VA | year = 1862 | title = Introduction à la Théorie des Nombres | publisher = Mallet-Bachelier | location = Paris | pages = 71–73}}</ref> Theophile Pepin (1883),<ref>{{cite journal | author = Pepin T | year = 1883 | title = Étude sur l'équation indéterminée ''ax''<sup>4</sup> + ''by''<sup>4</sup> = ''cz''<sup>2</sup> | journal = Atti Accad. Naz. Lincei | volume = 36 | pages = 34–70}}</ref> Tafelmacher (1893),<ref>{{cite journal | author = Tafelmacher WLA | year = 1893 | title = Sobre la ecuación ''x''<sup>4</sup> + ''y''<sup>4</sup> = ''z''<sup>4</sup> | journal = Ann. Univ. Chile | volume = 84 | pages = 307–320}}</ref> [[David Hilbert]] (1897),<ref>{{cite journal | author = [[David Hilbert|Hilbert D]] | year = 1897 | title = Die Theorie der algebraischen Zahlkörper | journal = [[Jahresbericht der Deutschen Mathematiker-Vereinigung]] | volume = 4 | pages = 175–546}}  Reprinted in 1965 in ''Gesammelte Abhandlungen, vol. I'' by New York:Chelsea.</ref> Bendz (1901),<ref>{{cite book | author = Bendz TR | year = 1901 | title = Öfver diophantiska ekvationen ''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = ''z''<sup>''n''</sup> | publisher = Almqvist & Wiksells Boktrycken | location = Uppsala}}</ref> Gambioli (1901),<ref name="Gambioli_1901" /> [[Leopold Kronecker]] (1901),<ref>{{cite book | author = [[Leopold Kronecker|Kronecker L]] | year = 1901 | title = Vorlesungen über Zahlentheorie, vol. I | publisher = Teubner | location = Leipzig | pages = 35–38}} Reprinted by New York:Springer-Verlag in 1978.</ref> Bang (1905),<ref>{{cite journal | author = Bang A | year = 1905 | title = Nyt Bevis for at Ligningen ''x''<sup>4</sup> − ''y''<sup>4</sup> = ''z''<sup>4</sup>, ikke kan have rationale Løsinger | journal = Nyt Tidsskrift Mat. | volume = 16B | pages = 35–36}}</ref> Sommer (1907),<ref>{{cite book | author = Sommer J | year = 1907 | title = Vorlesungen über Zahlentheorie | publisher = Teubner | location = Leipzig}}</ref> Bottari (1908),<ref>{{cite journal | author = Bottari A | year = 1908 | title = Soluzione intere dell'equazione pitagorica e applicazione alla dimostrazione di alcune teoremi dellla teoria dei numeri | journal = Period. Mat. | volume = 23 | pages = 104–110}}</ref> [[Karel Rychlík]] (1910),<ref name="Rychlik_1910" /> Nutzhorn (1912),<ref>{{cite journal | author = Nutzhorn F | year = 1912 | title = Den ubestemte Ligning ''x''<sup>4</sup> + ''y''<sup>4</sup> = ''z''<sup>4</sup> | journal = Nyt Tidsskrift Mat. | volume = 23B | pages = 33–38}}</ref> [[Robert Daniel Carmichael|Robert Carmichael]] (1913),<ref>{{cite journal | author = [[Robert Daniel Carmichael|Carmichael RD]] | year = 1913 | title = On the impossibility of certain Diophantine equations and systems of equations | journal = Amer. Math. Monthly | volume = 20 | pages = 213–221 | doi = 10.2307/2974106 | issue = 7 | publisher = Mathematical Association of America | jstor = 2974106}}</ref> Hancock (1931),<ref>{{cite book | author = Hancock H | year = 1931 | title = Foundations of the Theory of Algebraic Numbers, vol. I | publisher = Macmillan | location = New York}}</ref> and Vrǎnceanu (1966).<ref>{{cite journal | author = Vrǎnceanu G | year = 1966 | title = Asupra teorema lui Fermat pentru ''n''=4 | journal = Gaz. Mat. Ser. A | volume = 71 | pages = 334–335}} Reprinted in 1977 in ''Opera matematica'', vol. 4, pp. 202–205, Bucureşti:Edit. Acad. Rep. Soc. Romana.</ref>
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For another proof for ''n''=4 by infinite descent, see [[Infinite descent#Non-solvability of r2 + s4 = t4|Infinite descent: Non-solvability of ''r''<sup>2</sup> + ''s''<sup>4</sup> = ''t''<sup>4</sup>]]. For various proofs for ''n''=4 by infinite descent, see Grant and Perella (1999),<ref>Grant, Mike, and Perella, Malcolm, "Descending to the irrational", ''Mathematical Gazette'' 83, July 1999, pp.263–267.</ref> Barbara (2007),<ref>Barbara, Roy, "Fermat's last theorem in the case n=4", ''Mathematical Gazette'' 91, July 2007, 260–262.</ref> and Dolan (2011).<ref>Dolan, Stan, "Fermat's method of ''descente infinie''", ''[[Mathematical Gazette]]'' 95, July 2011, 269–271.</ref>
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After Fermat proved the special case ''n''&nbsp;=&nbsp;4, the general proof for all ''n'' required only that the theorem be established for all odd prime exponents.<ref>Ribenboim, pp. 1–2.</ref> In other words, it was necessary to prove only that the equation ''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup> has no integer solutions (''a'', ''b'', ''c'') when ''n'' is an odd [[prime number]]. This follows because a solution (''a'',&nbsp;''b'',&nbsp;''c'') for a given ''n'' is equivalent to a solution for all the factors of ''n''. For illustration, let ''n'' be factored into ''d'' and ''e'', ''n''&nbsp;=&nbsp;''de''The general equation
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: ''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup>
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implies that (''a''<sup>''d''</sup>,&nbsp;''b''<sup>''d''</sup>,&nbsp;''c''<sup>''d''</sup>) is a solution for the exponent ''e''
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: (''a''<sup>''d''</sup>)<sup>''e''</sup> + (''b''<sup>''d''</sup>)<sup>''e''</sup> = (''c''<sup>''d''</sup>)<sup>''e''</sup>.
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Thus, to prove that Fermat's equation has no solutions for ''n''&nbsp;>&nbsp;2, it suffices to prove that it has no solutions for at least one prime factor of every ''n''. All integers ''n''&nbsp;>&nbsp;2 contain a factor of 4, or an odd prime number, or both. Therefore, Fermat's Last Theorem can be proven for all ''n'' if it can be proven for ''n''&nbsp;=&nbsp;4 and for all odd primes ''p'' (the only ''even'' prime number is the number 2).
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In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proven for three odd prime exponents ''p''&nbsp;=&nbsp;3, 5 and 7.  The case ''p''&nbsp;=&nbsp;3 was first stated by [[Abu-Mahmud Khojandi]] (10th century), but his attempted proof of the theorem was incorrect.<ref>Dickson, p. 545.<br />{{MacTutor|id=Al-Khujandi|title=Abu Mahmud Hamid ibn al-Khidr Al-Khujandi}}</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 | accessdate = 23 May 2009}}</ref> contained a major gap.<ref>Ribenboim, pp. 24–25; Mordell, pp. 6–8; Edwards, pp. 39–40.</ref>  However, since Euler himself had proven the lemma necessary to complete the proof in other work, he is generally credited with the first proof.<ref>Aczel, p. 44; Edwards, pp. 40, 52–54.<br />{{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 |url=http://www.springerlink.com/content/u46jj0nuk3678591/ |doi=10.1134/S0001434607090088 |mr=2364600}}</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 Acad. Petrop. | volume = 13 | pages = 245–253}}</ref> Legendre (1823, 1830),<ref name="Legendre_1830" >{{cite book | author = [[Adrien-Marie Legendre|Legendre AM]] | year = 1830 | title = Théorie des Nombres (Volume II) | edition = 3rd | publisher = Firmin Didot Frères | location = Paris}} Reprinted in 1955 by A. Blanchard (Paris).</ref><ref name="Legendre_1823" >{{cite journal | author = [[Adrien-Marie Legendre|Legendre AM]] | year = 1823 | title = Recherches sur quelques objets d'analyse indéterminée, et particulièrement sur le théorème de Fermat | journal = Mém. Acad. Roy. Sci. Institut France | volume = 6 | pages = 1–60}}  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 ''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = ''z''<sup>''n''</sup> | publisher = Ferrara}}</ref> [[Gabriel Lamé]] (1865),<ref>{{cite journal | author = [[Gabriel Lamé|Lamé G]] | year = 1865 | title = Étude des binômes cubiques ''x''<sup>3</sup> ± ''y''<sup>3</sup> | journal = C. R. Acad. Sci. Paris | volume = 61 | pages = 921–924, 961–965}}</ref> [[Peter Guthrie Tait]] (1872),<ref>{{cite journal | author = [[Peter Guthrie Tait|Tait PG]] | year = 1872 | title = Mathematical Notes | journal = Proc. Roy. Soc. Edinburgh | volume = 7 | page = 144}}</ref> Günther (1878),<ref>{{cite journal | author = Günther S | year = 1878 | title = Über die unbestimmte Gleichung ''x''<sup>''3''</sup> + ''y''<sup>''3''</sup> = ''z''<sup>''3''</sup> | journal = Sitzungsberichte Böhm. Ges. Wiss. | pages = 112–120}}</ref> Gambioli (1901),<ref name="Gambioli_1901" >{{cite journal | author = Gambioli D | year = 1901 | title = Memoria bibliographica sull'ultimo teorema di Fermat | journal = Period. Mat. | volume = 16 | pages = 145–192}}</ref> Krey (1909),<ref>{{cite journal | author = Krey H | year = 1909 | title = Neuer Beweis eines arithmetischen Satzes | journal = Math. Naturwiss. Blätter | volume = 6 | pages = 179–180}}</ref> Rychlík (1910),<ref name="Rychlik_1910" >{{cite journal | author = [[Karel Rychlík|Rychlik K]] | year = 1910 | title = On Fermat's last theorem for ''n''&nbsp;=&nbsp;4 and ''n''&nbsp;=&nbsp;3 (in Bohemian) | journal = Časopis Pěst. Mat. | volume = 39 | pages = 65–86}}</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 = [[Robert Daniel Carmichael|Carmichael RD]] | year = 1915 | title = Diophantine Analysis | publisher = Wiley | location = New York}}</ref> [[Johannes van der Corput]] (1915),<ref name="van_der_Corput_1915" >{{cite journal | author = [[Johannes van der Corput|van der Corput JG]] | year = 1915 | title = Quelques formes quadratiques et quelques équations indéterminées | journal = Nieuw Archief Wisk. | volume = 11 | pages = 45–75}}</ref> [[Axel Thue]] (1917),<ref>{{cite journal | author = [[Axel Thue|Thue A]] | year = 1917 | title = Et bevis for at ligningen ''A''<sup>3</sup> + ''B''<sup>3</sup> = ''C''<sup>3</sup> er unmulig i hele tal fra nul forskjellige tal ''A'', ''B'' og ''C'' | journal = Arch. Mat. Naturv. | volume = 34 | issue = 15}}  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''<sup>3</sup> + ''y''<sup>3</sup> + ''z''<sup>3</sup> = 0 | journal = Ciencias Fis. Mat. Naturales (Caracas) | volume = 8 | pages = 971–979}}</ref>  The case ''p''&nbsp;=&nbsp;5 was proven<ref>{{cite web | author = Freeman L | title = Fermat's Last Theorem: Proof for ''n''&nbsp;=&nbsp;5 | url = http://fermatslasttheorem.blogspot.com/2005/10/fermats-last-theorem-proof-for-n5_28.html | accessdate = 23 May 2009}}</ref> independently by Legendre and [[Peter Gustav Lejeune Dirichlet]] around 1825.<ref>Ribenboim, p. 49; Mordell, p. 8–9; Aczel, p. 44; Singh, p. 106.</ref>  Alternative proofs were developed<ref>Ribenboim, pp. 55–57.</ref> by [[Carl Friedrich Gauss]] (1875, posthumous),<ref>{{cite book | author = [[Carl Friedrich Gauss|Gauss CF]] | year = 1875, posthumous | chapter = Neue Theorie der Zerlegung der Cuben | title = Zur Theorie der complexen Zahlen, Werke, vol. II | edition = 2nd | publisher = Königl. Ges. Wiss. Göttingen | pages = 387–391}}</ref> Lebesgue (1843),<ref>{{cite journal | author = [[Victor Lebesgue|Lebesgue VA]] | year = 1843 | title = Théorèmes nouveaux sur l'équation indéterminée ''x''<sup>5</sup> + ''y''<sup>5</sup> = ''az''<sup>5</sup> | journal = J. Math. Pures Appl. | volume = 8 | pages = 49–70}}</ref> Lamé (1847),<ref>{{cite journal | author = [[Gabriel Lamé|Lamé G]] | year = 1847 | title = Mémoire sur la résolution en nombres complexes de l'équation ''A''<sup>5</sup> + ''B''<sup>5</sup> + ''C''<sup>5</sup> = 0 | journal = J. Math. Pures Appl. | volume = 12 | pages = 137–171}}</ref> Gambioli (1901),<ref name="Gambioli_1901" /><ref>{{cite journal | author = Gambioli D | year = 1903/4 | title = Intorno all'ultimo teorema di Fermat | journal = Il Pitagora | volume = 10 | pages = 11–13, 41–42}}</ref> Werebrusow (1905),<ref>{{cite journal | author = Werebrusow AS | year = 1905 | title = On the equation ''x''<sup>5</sup> + ''y''<sup>5</sup> = ''Az''<sup>5</sup> ''(in Russian)'' | journal = Moskov. Math. Samml. | volume = 25 | pages = 466–473}}</ref> Rychlík (1910),<ref>{{cite journal | author = [[Karel Rychlík|Rychlik K]] | year = 1910 | title = On Fermat's last theorem for ''n''&nbsp;=&nbsp;5 ''(in Bohemian)'' | journal = Časopis Pěst. Mat. | volume = 39 | pages = 185–195, 305–317}}</ref>  van der Corput (1915),<ref name="van_der_Corput_1915" /> and [[Guy Terjanian]] (1987).<ref>{{cite journal | author = [[Guy Terjanian|Terjanian G]] | year = 1987 | title = Sur une question de V. A. Lebesgue | journal = Ann. Inst. Fourier | volume = 37 | pages = 19–37 | doi=10.5802/aif.1096}}</ref>  The case ''p''&nbsp;=&nbsp;7 was proven<ref>Ribenboim, pp. 57–63; Mordell, p. 8; Aczel, p. 44; Singh, p. 106.</ref> by Lamé in 1839.<ref name="Lame_1839" >{{cite journal | author = [[Gabriel Lamé|Lamé G]] | year = 1839 | title = Mémoire sur le dernier théorème de Fermat | journal = C. R. Acad. Sci. Paris | volume = 9 | pages = 45–46}}<br />{{cite journal | author = [[Gabriel Lamé|Lamé G]] | year = 1840 | title = Mémoire d'analyse indéterminée démontrant que l'équation ''x''<sup>7</sup> + ''y''<sup>7</sup> = ''z''<sup>7</sup> est impossible en nombres entiers | journal = J. Math. Pures Appl. | volume = 5 | pages = 195–211}}</ref>  His rather complicated proof was simplified in 1840 by Lebesgue,<ref>{{cite journal | author = [[Victor Lebesgue|Lebesgue VA]] | year = 1840 | title = Démonstration de l'impossibilité de résoudre l'équation ''x''<sup>7</sup> + ''y''<sup>7</sup> + ''z''<sup>7</sup> = 0 en nombres entiers | journal = J. Math. Pures Appl. | volume = 5 | pages = 276–279, 348–349}}</ref> and still simpler proofs<ref>{{cite web | author = Freeman L | title = Fermat's Last Theorem: Proof for ''n''&nbsp;=&nbsp;7 | url = http://fermatslasttheorem.blogspot.com/2006/01/fermats-last-theorem-proof-for-n7.html | accessdate = 23 May 2009}}</ref> were published by [[Angelo Genocchi]] in 1864, 1874 and 1876.<ref>{{cite journal | author = [[Angelo Genocchi|Genocchi A]] | year = 1864 | title = Intorno all'equazioni ''x''<sup>7</sup> + ''y''<sup>7</sup> + ''z''<sup>7</sup> = 0 | journal = Annali Mat. | volume = 6 | pages = 287–288}}<br />{{cite journal | author = [[Angelo Genocchi|Genocchi A]] | year = 1874 | title = Sur l'impossibilité de quelques égalités doubles | journal = C. R. Acad. Sci. Paris | volume = 78 | pages = 433–436}}<br />{{cite journal | author = [[Angelo Genocchi|Genocchi A]] | year = 1876 | title = Généralisation du théorème de Lamé sur l'impossibilité de l'équation ''x''<sup>7</sup> + ''y''<sup>7</sup> + ''z''<sup>7</sup> = 0 | journal = C. R. Acad. Sci. Paris | volume = 82 | pages = 910–913}}</ref>  Alternative proofs were developed by Théophile Pépin (1876)<ref>{{cite journal | author = [[Théophile Pépin|Pepin T]] | year = 1876 | title = Impossibilité de l'équation ''x''<sup>7</sup> + ''y''<sup>7</sup> + ''z''<sup>7</sup> = 0 | journal = C. R. Acad. Sci. Paris | volume = 82 | pages = 676–679, 743–747}}</ref> and Edmond Maillet (1897).<ref>{{cite journal | author = [[Edmond Maillet|Maillet E]] | year = 1897 | title = Sur l'équation indéterminée ''ax''<sup>λ<sup>''t''</sup></sup> + ''by''<sup>λ<sup>''t''</sup></sup> = ''cz''<sup>λ<sup>''t''</sup></sup> | journal = Assoc. Française Avanc. Sci., St. Etienne (sér. II) | volume = 26 | pages = 156–168}}</ref>
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Fermat's Last Theorem has also been proven for the exponents ''n''&nbsp;=&nbsp;6, 10, and 14.   Proofs for ''n''&nbsp;=&nbsp;6 have been published by Kausler,<ref name="Kausler_1802" /> Thue,<ref>{{cite journal | author = [[Axel Thue|Thue A]] | year = 1896 | title = Über die Auflösbarkeit einiger unbestimmter Gleichungen | journal = Det Kongel. Norske Videnskabers Selskabs Skrifter | volume = 7}} Reprinted in ''Selected Mathematical Papers'', pp. 19–30, Oslo:Universitetsforlaget (1977).</ref> Tafelmacher,<ref>{{cite journal | author = Tafelmacher WLA | year = 1897 | title = La ecuación ''x''<sup>3</sup> + ''y''<sup>3</sup> = ''z''<sup>2</sup>: Una demonstración nueva del teorema de fermat para el caso de las sestas potencias | journal = Ann. Univ. Chile, Santiago | volume = 97 | pages = 63–80}}</ref> Lind,<ref>{{cite journal | author = Lind B | year = 1909 | title = Einige zahlentheoretische Sätze | journal = Arch. Math. Phys. | volume = 15 | pages = 368–369}}</ref> Kapferer,<ref name="Kapferer_1913" >{{cite journal | author = Kapferer H | year = 1913 | title = Beweis des Fermatschen Satzes für die Exponenten 6 und 10 | journal = Archiv Math. Phys. | volume = 21 | pages = 143–146}}</ref> Swift,<ref>{{cite journal | author = Swift E | year = 1914 | title = Solution to Problem 206 | journal = Amer. Math. Monthly | volume = 21 | pages = 238–239}}</ref> and Breusch.<ref name="Breusch_1960" >{{cite journal | doi = 10.2307/3029800 | author = Breusch R | authorlink = Robert Breusch | year = 1960 | title = A simple proof of Fermat's last theorem for ''n''&nbsp;=&nbsp;6, ''n''&nbsp;=&nbsp;10 | journal = Math. Mag. | volume = 33 | issue = 5 | pages = 279–281 | jstor = 3029800}}</ref> Similarly, Dirichlet<ref>{{cite journal | author = [[Peter Gustav Lejeune Dirichlet|Dirichlet PGL]] | year = 1832 | title = Démonstration du théorème de Fermat pour le cas des 14<sup>e</sup> puissances | journal = J. Reine Angew. Math. | volume = 9 | pages = 390–393}} Reprinted in ''Werke'', vol. I, pp. 189–194, Berlin:G. Reimer (1889); reprinted New York:Chelsea (1969).</ref> and Terjanian<ref>{{cite journal | author = [[Guy Terjanian|Terjanian G]] | year = 1974 | title= L'équation ''x''<sup>14</sup> + ''y''<sup>14</sup> = ''z''<sup>14</sup> en nombres entiers | journal = Bull. Sci. Math. (sér. 2) | volume = 98 | pages = 91–95}}</ref> each proved the case ''n''&nbsp;=&nbsp;14, while Kapferer<ref name="Kapferer_1913" /> and Breusch<ref name="Breusch_1960" /> each proved the case ''n''&nbsp;=&nbsp;10.  Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for ''n''&nbsp;=&nbsp;3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for ''n''&nbsp;=&nbsp;14 was published in 1832, before Lamé's 1839 proof for ''n''&nbsp;=&nbsp;7.<ref>Edwards, pp. 73–74.</ref>
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Many proofs for specific exponents use Fermat's technique of [[infinite descent]], which Fermat used to prove the case ''n''&nbsp;=&nbsp;4, but many do not.  However, the details and auxiliary arguments are often ''ad hoc'' and tied to the individual exponent under consideration.<ref name="Edwards, p. 74">Edwards, p. 74.</ref> Since they became ever more complicated as ''p'' increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proven by building upon the proofs for individual exponents.<ref name="Edwards, p. 74"/>  Although some general results on Fermat's Last Theorem were published in the early 19th century by [[Niels Henrik Abel]] and [[Peter Barlow (mathematician)|Peter Barlow]],<ref>Dickson, p. 733.</ref><ref>{{cite book | author = [[Paulo Ribenboim|Ribenboim P]] | year = 1979 | title = 13 Lectures on Fermat's Last Theorem | publisher = Springer Verlag | location = New York | isbn = 978-0-387-90432-0 | pages = 51–54}}</ref> the first significant work on the general theorem was done by [[Sophie Germain]].<ref>Singh, pp. 97–109.</ref>
  <li>[http://verdamilio.net/tonio/spip.php?article1/ http://verdamilio.net/tonio/spip.php?article1/]</li>
 
 
====Sophie Germain====
  <li>[http://enseignement-lsf.com/spip.php?article64#forum24718490 http://enseignement-lsf.com/spip.php?article64#forum24718490]</li>
{{Main|Sophie Germain}}
 
 
  <li>[http://www.juegosetnicos.com.ar/spip.php?article98&lang=fr/ http://www.juegosetnicos.com.ar/spip.php?article98&lang=fr/]</li>
In the early 19th century, [[Sophie Germain]] developed several novel approaches to prove Fermat's Last Theorem for all exponents.<ref name="Laubenbacher_2007" >{{cite web | author = Laubenbacher R, Pengelley D | year = 2007 | title = Voici ce que j'ai trouvé: Sophie Germain's grand plan to prove Fermat's Last Theorem | url = http://www.math.nmsu.edu/%7Edavidp/germain.pdf | accessdate = 19 May 2009}}</ref> First, she defined a set of auxiliary primes θ constructed from the prime exponent ''p'' by the equation  θ&nbsp;=&nbsp;2''hp''+1, where ''h'' is any integer not divisible by three.   She showed that if no integers raised to the ''p''<sup>th</sup> power were adjacent modulo θ (the ''non-consecutivity condition''), then θ must divide the product ''xyz''.  Her goal was to use [[mathematical induction]] to prove that, for any given ''p'', infinitely many auxiliary primes θ satisfied the non-consecutivity condition and thus divided ''xyz''; since the product ''xyz'' can have at most a finite number of prime factors, such a proof would have established Fermat's Last Theorem.  Although she developed many techniques for establishing the non-consecutivity condition, she did not succeed in her strategic goal.  She also worked to set lower limits on the size of solutions to Fermat's equation for a given exponent ''p'', a modified version of which was published by [[Adrien-Marie Legendre]].  As a byproduct of this latter work, she proved [[Sophie Germain's theorem]], which verified the first case of Fermat's Last Theorem (namely, the case in which ''p'' does not divide ''xyz'') for every odd prime exponent less than 100.<ref name="Laubenbacher_2007" /><ref>Aczel, p. 57.</ref>  Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for ''n''&nbsp;=&nbsp;2''p'', which was proven by [[Guy Terjanian]] in 1977.<ref>{{cite journal|last=Terjanian|first=G.|year=1977|title=Sur l'équation ''x''<sup>2''p''</sup> + ''y''<sup>2''p''</sup> = ''z''<sup>2''p''</sup> |journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences. Série a et B|volume=285|pages=973–975}}</ref>  In 1985, [[Leonard Adleman]], [[Roger Heath-Brown]] and Étienne Fouvry proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes ''p''.<ref>{{cite journal |author = Adleman LM, Heath-Brown DR |date=June 1985 |title = The first case of Fermat's last theorem | journal = Inventiones Mathematicae | volume = 79 | issue = 2 |pages = 409–416 | publisher = Springer | location = Berlin | doi=10.1007/BF01388981}}</ref>
 
 
  <li>[http://verdamilio.info/org/spip.php?article573/ http://verdamilio.info/org/spip.php?article573/]</li>
====Ernst Kummer and the theory of ideals====
    
 
  <li>[http://enseignement-lsf.com/spip.php?article64#forum25483125 http://enseignement-lsf.com/spip.php?article64#forum25483125]</li>
In 1847, [[Gabriel Lamé]] outlined a proof of Fermat's Last Theorem based on factoring the equation ''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> = ''z''<sup>''p''</sup> in complex numbers, specifically the [[cyclotomic field]] based on the [[root of unity|roots of the number 1]].  His proof failed, however, because it assumed incorrectly that such complex numbers can be [[unique factorization|factored uniquely]] into primes, similar to integers.  This gap was pointed out immediately by [[Joseph Liouville]], who later read a paper that demonstrated this failure of unique factorisation, written by [[Ernst Kummer]].
 
 
</ul>
Kummer set himself the task of determining whether the [[cyclotomic field]] could be generalized to include new prime numbers such that unique factorisation was restored.  He succeeded in that task by developing the [[ideal number]]s.  Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all [[regular prime|regular prime numbers]].  However, he could not prove the theorem for the exceptional primes ([[irregular prime]]s) which conjecturally occur approximately 39% of the time; the only irregular primes below 100 are 37, 59 and 67.
 
====Mordell conjecture====
 
In the 1920s, [[Louis Mordell]] posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions if the exponent ''n'' is greater than two.<ref>Aczel, pp. 84–88; Singh, pp. 232–234.</ref>  This conjecture was proven in 1983 by [[Gerd Faltings]],<ref>{{cite journal | author = [[Gerd Faltings|Faltings G]] | year = 1983 | title = Endlichkeitssätze für abelsche Varietäten über Zahlkörpern |journal = Inventiones Mathematicae | volume = 73 | issue = 3 |pages = 349–366 |doi=10.1007/BF01388432}}</ref> and is now known as [[Faltings' theorem]].
 
====Computational studies====
 
In the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregular primes.  In 1954, [[Harry Vandiver]] used a [[SWAC (computer)|SWAC computer]] to prove Fermat's Last Theorem for all primes up to 2521.<ref>{{cite book | author = [[Paulo Ribenboim|Ribenboim P]] | year = 1979 | title = 13 Lectures on Fermat's Last Theorem | publisher = Springer Verlag | location = New York | isbn = 978-0-387-90432-0 | page = 202}}</ref>  By 1978, [[Samuel S. Wagstaff, Jr.|Samuel Wagstaff]] had extended this to all primes less than 125,000.<ref>{{cite journal | author = [[Samuel S. Wagstaff, Jr.|Wagstaff SS, Jr.]] | year = 1978 | title = The irregular primes to 125000 | journal = Math. Comp. | volume = 32 | pages = 583–591 | doi = 10.2307/2006167 | issue = 142 | publisher = American Mathematical Society | jstor = 2006167}} [http://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0491465-4/S0025-5718-1978-0491465-4.pdf (PDF)]</ref>  By 1993, Fermat's Last Theorem had been proven for all primes less than four million.<ref>{{cite journal | author = Buhler J, Crandell R, Ernvall R, Metsänkylä T | year = 1993 | title = Irregular primes and cyclotomic invariants to four million | journal = Math. Comp. | volume = 61 | pages = 151–153 | doi = 10.2307/2152942 | issue = 203 | publisher = American Mathematical Society | jstor = 2152942}}</ref>
 
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, and it could not be ruled out in this conjecture.)
 
===Connection with elliptic curves===
 
The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"<ref name="Singh">[Fermat's Last Theorem, Simon Singh, 1997, ISBN 1-85702-521-0</ref>{{rp|211}} [[modularity theorem|Taniyama–Shimura-Weil conjecture]], proposed around 1955, which many mathematicians believed would be near to impossible to prove,<ref name="Singh" />{{rp|223}} and which was linked in the 1980s by [[Gerhard Frey]], [[Jean-Pierre Serre]] and [[Ken Ribet]] to Fermat's equation. By accomplishing a partial proof of this conjecture in 1995, [[Andrew Wiles]] ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now the [[modularity theorem]].
 
====The Taniyama–Shimura–Weil conjecture====
{{main|Modularity theorem}}
Around 1955, Japanese mathematicians [[Goro Shimura]] and [[Yutaka Taniyama]] observed a possible link between two apparently completely distinct branches of mathematics, [[elliptic curve]]s and [[modular form]]s. The resulting [[modularity theorem]] (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is [[modular elliptic curve|modular]], meaning that it can be associated with a unique [[modular form]].
It was initially dismissed as unlikely or highly speculative, and was taken more seriously when number theorist [[André Weil]] found evidence supporting it, but no proof; as a result the "astounding"<ref name="Singh" />{{rp|211}} conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the [[Langlands programme]], a list of important conjectures needing proof or disproof.<ref name="Singh" />{{rp|211–215}}
 
Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.<ref name="Singh" />{{rp|203–205, 223, 226}} For example, Wiles' ex-supervisor [[John H. Coates|John Coates]] states that it seemed "impossible to actually prove",<ref name="Singh" />{{rp|226}} and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."<ref name="Singh" />{{rp|223}}
 
====Frey's equation / Ribet's theorem====
{{main|Frey curve|Ribet's theorem}}
In 1984, [[Gerhard Frey]] noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution (''a'', ''b'', ''c'') for exponent ''p''&nbsp;>&nbsp;2, then it could be shown that the [[elliptic curve]] (now known as a [[Frey curve]]&nbsp;<ref group=note>This elliptic curve was first suggested in the 1960s by {{ill|de|Yves Hellegouarch}}, but he did not call attention to its non-modularity.  For more details, see {{Cite book | last=Hellegouarch | first=Yves | title=Invitation to the Mathematics of Fermat-Wiles | publisher=Academic Press | year=2001 | isbn=978-0-12-339251-0 }}</ref>)
 
:''y''<sup>2</sup> = ''x''&nbsp;(''x''&nbsp;−&nbsp;''a''<sup>''p''</sup>)(''x''&nbsp;+&nbsp;''b''<sup>''p''</sup>)
 
would have such unusual properties that it was unlikely to be modular.<ref>{{cite journal | author = [[Gerhard Frey|Frey G]] | year = 1986 |title = Links between stable elliptic curves and certain diophantine equations | journal = Ann. Univ. Sarav. Ser. Math. | volume = 1 | pages = 1–40}}</ref> This would conflict with the modularity theorem, which asserted that all elliptic curves are modular. As such, Frey observed that a proof of the Taniyama–Shimura-Weil conjecture would simultaneously prove Fermat's Last Theorem.<ref>Singh, pp. 194–198; Aczel, pp. 109–114.</ref> (Equally, a ''disproof'' or refutation of Fermat's Last Theorem would disprove the conjecture)
 
Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could not be modular. Frey did not quite succeed in proving this rigorously; the missing piece (the so-called "[[epsilon conjecture]]", now known as [[Ribet's theorem]]) was identified by [[Jean-Pierre Serre]]{{citation needed|date=May 2013}} and proven in 1986 by [[Ken Ribet]].  Second, it was necessary to prove the modularity theorem – or at least to prove it for the sub-class of cases (known as [[semistable elliptic curve]]s) which included Frey's equation – and this was widely believed inaccessible to proof by contemporary mathematicians.<ref name="Singh" />{{rp|203–205, 223, 226}}
 
* The modularity theorem – if proven – would mean all elliptic curves (or at least all semistable elliptic curves) are of necessity modular.
* Ribet's theorem – proven in 1986 – showed that if a solution to Fermat's equation existed, it could be used to create a semistable elliptic curve that was not modular;
* The contradiction would imply (if the modularity theorem were correct) that ''no'' solutions can exist to Fermat's equation – therefore proving Fermat's Last Theorem.
 
====Wiles's general proof====
[[File:Andrew wiles1-3.jpg|thumb|upright|British mathematician [[Andrew Wiles]]]]
{{Main|Andrew Wiles|Wiles's proof of Fermat's Last Theorem}}
 
Ribet's proof of the [[epsilon conjecture]] in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success, [[Andrew Wiles]], an English mathematician with a childhood fascination with Fermat's Last Theorem, and a prior study area of elliptical equations, decided to commit himself to accomplishing the second half: proving a special case of the [[modularity theorem]] (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves.<ref>Singh, p. 205; Aczel, pp. 117–118.</ref>
 
Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife.<ref name="Singh" />{{rp|229–230}} His initial study suggested [[Mathematical proof|proof]] by [[mathematical induction|induction]],<ref name="Singh" />{{rp|230–232, 249–252}} and he based his initial work and first significant breakthrough on [[Galois theory]]<ref name="Singh" />{{rp|251–253, 259}} before switching to an attempt to extend [[Iwasawa theory|Horizontal Iwasawa theory]] for the inductive argument around 1990–91 when it seemed that there was no existing approach adequate to the problem.<ref name="Singh" />{{rp|258–259}} However, by the summer of 1991, Iwasawa theory also seemed to not be reaching the central issues in the problem.<ref name="Singh" />{{rp|259–260}}<ref>Singh, pp. 237–238; Aczel, pp. 121–122.</ref> In response, he approached colleagues to seek out any hints of cutting edge research and new techniques, and discovered an [[Euler system]] recently developed by [[Victor Kolyvagin]] and [[Matthias Flach (mathematician)|Matthias Flach]] which seemed "tailor made" for the inductive part of his proof.<ref name="Singh" />{{rp|260–261}} Wiles studied and extended this approach, which worked. Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January 1993 he asked his Princeton colleague, [[Nick Katz]], to check his reasoning for subtle errors. Their conclusion at the time was that the techniques used by Wiles seemed to be working correctly.<ref name="Singh" />{{rp|261–265}}<ref>Singh, pp. 239–243; Aczel, pp. 122–125.</ref>
 
By mid-May 1993 Wiles felt able to tell his wife he thought he had solved the proof of Fermat's Last Theorem,<ref name="Singh" />{{rp|265}} and by June he felt sufficiently confident to present his results in three lectures delivered on 21–23 June 1993 at the [[Isaac Newton Institute for Mathematical Sciences]].<ref>Singh, pp. 244–253; Aczel, pp. 1–4, 126–128.</ref> Specifically, Wiles presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves; together with Ribet's proof of the epsilon conjecture, this implied Fermat's Last Theorem. However, it became apparent during [[peer review]] that a critical point in the proof was incorrect. It contained an error in a bound on the order of a particular [[group (mathematics)|group]]. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz (in his role as reviewer),<ref>Aczel, pp. 128–130.</ref> who alerted Wiles on 23 August 1993.<ref>Singh, p. 257.</ref>
 
The error would not have rendered his work worthless – each part of Wiles' work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected.<ref name="Singh" />{{rp|289, 296–297}} However without this part proven, there was no actual proof of Fermat's Last Theorem. Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with [[Richard Taylor (mathematician)|Richard Taylor]], without success.<ref>Singh, pp. 269–274.</ref>
 
On 19 September 1994, on the verge of giving up, Wiles had a flash of insight that the proof could be saved by returning to his original Horizontal Iwasawa theory approach, which he had abandoned in favour of the Kolyvagin–Flach approach, this time strengthening it with expertise gained in Kolyvagin–Flach's approach.<ref>Singh, pp. 275–277; Aczel, pp. 132–134.</ref> On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"<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|journal=Annals of Mathematics|volume=141|issue=3|pages=443–551|oclc=37032255|format=PDF|doi=10.2307/2118559|jstor=2118559|publisher=Annals of Mathematics}}</ref> and "Ring theoretic properties of certain Hecke algebras",<ref>{{cite journal | author = [[Richard Taylor (mathematician)|Taylor R]], [[Andrew Wiles|Wiles A]] | year = 1995 | journal = Annals of Mathematics | title = Ring theoretic properties of certain Hecke algebras | volume = 141 | issue = 3| pages = 553–572 | oclc = 37032255 | url = http://www.math.harvard.edu/~rtaylor/hecke.ps | doi = 10.2307/2118560 | jstor = 2118560 | publisher = Annals of Mathematics}}</ref> the second of which was co-authored with Taylor and proved that certain conditions were met which were needed to justify the corrected step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the ''[[Annals of Mathematics]]''. These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
 
===Subsequent developments===
The full Taniyama–Shimura–Weil conjecture was finally proved by {{harvtxt|Diamond|1996}}, {{harvtxt|Conrad|Diamond|Taylor|1999}}, and {{harvtxt|Breuil|Conrad|Diamond|Taylor|2001}} who, building on Wiles' work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem.
 
Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. For example: no cube can be written as a sum of two [[coprime]] ''n''-th powers, ''n''&nbsp;≥&nbsp;3. (The case ''n''&nbsp;=&nbsp;3 was already known by [[Euler]].)
 
==Exponents other than positive integers==
 
===Rational exponents===
 
All solutions of the Diophantine equation <math>a^{n/m} + b^{n/m} = c^{n/m}</math> when ''n''=1 were computed by Lenstra in 1992.<ref>[[Hendrik Lenstra|Lenstra, Jr. H.W.]] (1992). ''On the inverse Fermat equation.'' Discrete Mathematics, '''106–107''', pp. 329–331.</ref> In the case in which the ''m''<sup>th</sup> roots are required to be real and positive, all solutions are given by<ref>Newton, M., "A radical diophantine equation", ''[[Journal of Number Theory]]'' 13 (1981), 495–498.</ref>
 
:<math>a=rs^m</math>
:<math>b=rt^m</math>
:<math>c=r(s+t)^m</math>
 
for positive integers ''r, s, t'' with ''s'' and ''t'' coprime.
 
In 2004, for ''n''>2, Bennett, Glass, and Szekely proved that if gcd(''n'',''m'')=1, then there are integer solutions if and only if 6 divides ''m'', and <math>a^{1/m}</math>, <math>b^{1/m},</math> and <math>c^{1/m}</math> are different complex 6th roots of the same real number.<ref>Bennett, Curt D., Glass, Andrew M.W., and [[Gábor J. Székely|Székely, Gábor J.]] (2004). ''Fermat’s last theorem for rational exponents.'' The American Mathematical Monthly, '''111''', no. 4, pp. 322–329.</ref>
 
===Negative exponents===
 
====''n'' =&nbsp;–1====
 
All primitive (pairwise coprime) integer solutions to <math>a^{-1}+b^{-1}=c^{-1}</math> can be written as<ref>Dickson, pp. 688–691</ref>
 
:<math>a=mn+m^2,</math>
:<math>b=mn+n^2,</math>
:<math>c=mn</math>
 
for positive, coprime integers ''m, n''.
 
====''n'' =&nbsp;–2====
 
The case ''n'' =&nbsp;–2 also has an infinitude of solutions, and these have a geometric interpretation in terms of [[Integer triangle#Pythagorean triangles with integer altitude from the hypotenuse|right triangles with integer sides and an integer altitude to the hypotenuse]].<ref>Voles, Roger, "Integer solutions of <math>a^{-2}+b^{-2}=d^{-2}</math>," ''[[Mathematical Gazette]]'' 83, July 1999, 269–271.</ref><ref>Richinick, Jennifer, "The upside-down Pythagorean Theorem," ''Mathematical Gazette'' 92, July 2008, 313–317.</ref> All primitive solutions to <math>a^{-2}+b^{-2}=d^{-2}</math> are given by
 
:<math>a=(v^2-u^2)(v^2+u^2), \,</math>
:<math>b=2uv(v^2+u^2), \,</math>
:<math>d=2uv(v^2-u^2), \,</math>
 
for coprime integers ''u'', ''v'' with ''v''&nbsp;>&nbsp;''u''. The geometric interpretation is that ''a'' and ''b'' are the integer legs of a right triangle and ''d'' is the integer altitude to the hypotenuse. Then the hypotenuse itself is the integer
 
:<math>c=(v^2+u^2)^2, \,</math>
 
so (''a, b, c'') is a [[Pythagorean triple]].
 
====Integer ''n'' <&nbsp;–2====
 
There are no solutions in integers for <math>a^n+b^n=c^n</math> for integer ''n'' <&nbsp;–2. If there were, the equation could be multiplied through by <math>a^{|n|}b^{|n|}c^{|n|}</math> to obtain <math>(bc)^{|n|}+(ac)^{|n|}=(ab)^{|n|}</math>, which is impossible by Fermat's Last Theorem.
 
==Values other than positive integers==
Fermat's last theorem can easily be extended to positive rationals -
: <math>\left(\frac{a}{x}\right)^n+\left(\frac{b}{y}\right)^n=\left(\frac{c}{z}\right)^n</math>
 
Can be rearranged to:
:<math>(ayz)^n+(bxz)^n=(cxy)^n</math>,
To which fermat's last theorem applies.
 
==Did Fermat possess a general proof?==
 
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''<sup>4</sup> − ''y''<sup>4</sup> = ''z''<sup>2</sup>. Note that this proves Fermat's Last Theorem for the case ''n''=4, since the equation ''a''<sup>4</sup> + ''b''<sup>4</sup> = ''c''<sup>4</sup> can be written as ''c''<sup>4</sup> − ''b''<sup>4</sup> = (''a''<sup>2</sup>)<sup>2</sup>.
 
Taylor and Wiles's proof relies on mathematical techniques developed in the 20th century,<ref>[http://www.youtube.com/watch?v=Ua8K8HW2Qsg#t=47m BBC Documentary]</ref> which would be unknown to mathematicians who had worked on Fermat's Last Theorem even a century earlier. Fermat's "marvellous proof", by comparison, would have had to be elementary, given mathematical knowledge of the time, and so could not have been the same as Wiles' proof. Most mathematicians and science historians doubt that Fermat had a valid proof of his theorem for all exponents ''n''.{{Citation needed|date=August 2011}}
 
[[Harvey Friedman]]'s [[grand conjecture]] implies that any provable theorem (including Fermat's last theorem) can be proved using only elementary arithmetic, a rather weak form of arithmetic with addition, multiplication, exponentiation, and a limited form of induction for formulas with bounded quantifiers.<ref>{{Cite journal | last1=Avigad | first1=Jeremy | title=Number theory and elementary arithmetic | doi=10.1093/philmat/11.3.257 | mr=2006194  | year=2003 | journal=Philosophia Mathematica. Philosophy of Mathematics, its Learning, and its Application. Series III | issn=0031-8019 | volume=11 | issue=3 | pages=257–284 | id = {{citeseerx|10.1.1.105.6509}} }}</ref> Any such proof would indeed be 'elementary' but could involve implausibly long proofs of millions—or millions of millions—of steps, and might be far too long to be Fermat's proof.
 
==Monetary prizes==
 
In 1816 and again in 1850, the [[French Academy of Sciences]] offered a prize for a general proof of Fermat's Last Theorem.<ref>Aczel, p. 69; Singh, p. 105.</ref> In 1857, the Academy awarded 3000 francs and a gold medal to Kummer for his research on ideal numbers, although he had not submitted an entry for the prize.<ref>Aczel, p. 69.</ref>  Another prize was offered in 1883 by the Academy of Brussels.<ref name="Koshy_2001" >{{cite book | author = Koshy T | year = 2001 | title = Elementary number theory with applications | publisher = Academic Press | location = New York | isbn = 978-0-12-421171-1 | page = 544}}</ref>
 
In 1908, the German industrialist and amateur mathematician [[Paul Wolfskehl]] bequeathed 100,000 [[Mark (money)|marks]] to the Göttingen Academy of Sciences to be offered as a prize for a complete proof of Fermat's Last Theorem.<ref>Singh, pp. 120–125, 131–133, 295–296; Aczel, p. 70.</ref> 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>
 
Prior to Wiles' proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3 meters) of correspondence.<ref>Singh, p. 295.</ref>  In the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month.  According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in schools, and often submitted by "people with a technical education but a failed career".<ref>Singh, pp. 295–296.</ref>  In the words of mathematical historian [[Howard Eves]], "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."<ref name="Koshy_2001" />
 
==See also==
* [[Beal's conjecture]]
* [[Diophantus II.VIII]]
* [[Euler's sum of powers conjecture]]
* [[Fermat's Last Theorem in fiction]]
* [[Sophie Germain prime]]
* [[Wall–Sun–Sun prime]]
* [[Sums of powers]], a list of related conjectures and theorems
 
==Footnotes==
{{reflist|group="note"|30em}}
 
==References==
{{Reflist|30em}}
 
==Bibliography==
{{refbegin|30em}}
* {{Cite book|last=Aczel|first=Amir|authorlink=Amir Aczel|title=Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem|date=30 September 1996|publisher=Four Walls Eight Windows|isbn=978-1-56858-077-7}}
* {{Cite book | author = [[Leonard Eugene Dickson|Dickson LE]] | year = 1919 | title = [[History of the Theory of Numbers]]. Volume II. Diophantine Analysis | publisher = Chelsea Publishing | location = New York | pages = 545–550, 615–621, 688–691, 731–776}}
* {{Cite book | author = [[Harold Edwards (mathematician)|Edwards, HM]] | year = 1997 | title = Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory | publisher = Springer-Verlag | location = New York | series = Graduate Texts in Mathematics | volume = 50}}
* {{Cite book | last = Friberg | first = Joran | year = 2007 | title = Amazing Traces of a Babylonian Origin in Greek Mathematics | publisher =World Scientific Publishing Company | isbn = 978-981-270-452-8}}
* {{Cite journal | author = Kleiner I | year = 2000 | title = From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem | journal = Elem. Math. | volume = 55 | pages = 19–37 | url = http://math.stanford.edu/~lekheng/flt/kleiner.pdf | doi = 10.1007/PL00000079}}
* {{Cite book | author = [[Louis Mordell|Mordell LJ]] | year = 1921 | title = Three Lectures on Fermat's Last Theorem | publisher = Cambridge University Press | location = Cambridge}}
* {{Cite book | last = Panchishkin | first = Alekseĭ Alekseevich | year = 2007 | title = Introduction to Modern Number Theory (Encyclopedia of Mathematical Sciences | publisher =Springer Berlin Heidelberg New York | isbn = 978-3-540-20364-3}}
* {{Cite book | author = [[Paulo Ribenboim|Ribenboim P]] | year = 2000 | title = Fermat's Last Theorem for Amateurs | publisher = Springer-Verlag | location = New York | isbn = 978-0-387-98508-4}}
* {{Cite book | author = [[Simon Singh|Singh S]] | title = [[Fermat's Enigma]] |date=October 1998 | publisher = Anchor Books | location = New York | isbn = 978-0-385-49362-8}}
* {{Cite book | author = [[Harold Stark|Stark H]] | year = 1978 | title = An Introduction to Number Theory | publisher = MIT Press | isbn = 0-262-69060-8}}
{{refend}}
 
==Further reading==
{{refbegin|30em}}
* {{Cite book|last=Bell|first=Eric T.|title=The Last Problem|location=New York|isbn=978-0-88385-451-8|date=6 August 1998|publisher=The Mathematical Association of America|origyear=1961}}
* {{Cite book|last=Benson|first=Donald C.|title=The Moment of Proof: Mathematical Epiphanies|publisher=Oxford University Press|isbn=978-0-19-513919-8|date=5 April 2001}}
* {{Cite book|last=Brudner|first=Harvey J.|authorlink=Harvey J. Brudner|title=Fermat and the Missing Numbers|publisher=WLC, Inc|isbn=978-0-9644785-0-3|year=1994}}
* {{Cite book | last=Edwards|first=H. M. | title=Fermat's Last Theorem  | publisher=Springer-Verlag | origyear=1977|date=March 1996| isbn=978-0-387-90230-2|location=New York }}
* {{Cite journal | author = [[Gerd Faltings|Faltings G]] |date=July 1995 | url = http://www.ams.org/notices/199507/faltings.pdf|format=PDF|title=The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles|journal=Notices of the AMS|volume=42|issue=7|pages=743–746|issn=0002-9920}}
* {{Cite book|last=Mozzochi|first=Charles | title=The Fermat Diary | date=7 December 2000 | isbn=978-0-8218-2670-6|publisher=American Mathematical Society}}
* {{Cite book | author = [[Paulo Ribenboim|Ribenboim P]] | year = 1979 | title = 13 Lectures on Fermat's Last Theorem | publisher = Springer Verlag | location = New York | isbn = 978-0-387-90432-0}}
* {{Cite book|last=van der Poorten|first=Alf|title=Notes on Fermat's Last Theorem|date=6 March 1996|publisher=WileyBlackwell |isbn=978-0-471-06261-5}}
* {{Cite journal | last = Saikia| first = Manjil P |date=July 2011 | url = http://www.thequantizedquark.com/papers/kummerFLT.pdf|format=PDF|title=A Study of Kummer's Proof of Fermat's Last Theorem for Regular Primes|journal=IISER Mohali Report}}
{{refend}}
 
==External links==
{{Wikibooks}}
* {{cite web|author=Wiles|year=1995 | url=http://www.cs.berkeley.edu/~anindya/fermat.pdf| title=Modular elliptic curves and Fermat's Last Theorem|accessdate= 9 February 2013}} Scientific article by Andrew Wiles
* {{cite web|author=Daney, Charles|year=2003 | url=http://cgd.best.vwh.net/home/flt/flt01.htm| title=The Mathematics of Fermat's Last Theorem|accessdate=5 August 2004}}
*[http://math.stanford.edu/~lekheng/flt/ The bluffer's guide to Fermat's Last Theorem]
* {{cite web|author=Elkies, Noam D.| url=http://www.math.harvard.edu/~elkies/ferm.html| title=Tables of Fermat "near-misses" — approximate solutions of x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup>}}
* {{cite web|author=Freeman, Larry| year=2005|url=http://www.fermatslasttheorem.blogspot.com|  title=Fermat's Last Theorem Blog}} Blog that covers the history of Fermat's Last Theorem from Fermat to Wiles.
* {{springer|title=Fermat's last theorem|id=p/f110070}}
* {{cite web|author=Ribet, Ken| year=1995| url=http://math.stanford.edu/~lekheng/flt/ribet.pdf |title=Galois representations and modular forms|format=PDF}} Discusses various material which is related to the proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama–Shimura.
* {{cite web|author=Shay, David|year=2003| url=http://shayfam.com/David/flt/index.htm| title=Fermat's Last Theorem| accessdate=5 August 2004}} The story, the history and the mystery.
* {{MathWorld | urlname=FermatsLastTheorem| title=Fermat's Last Theorem}}
* {{cite web| author = O'Connor JJ, Robertson EF | year = 1996 | url = http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html | title = Fermat's last theorem | accessdate = 5 August 2004}}
* {{cite web |url=http://www.pbs.org/wgbh/nova/proof/| title=The Proof}} The title of one edition of the PBS television series NOVA, discusses Andrew Wiles's effort to prove Fermat's Last Theorem.
* {{cite web |url=http://www.youtube.com/watch?v=7FnXgprKgSE| title= Documentary Movie on Fermat's Last Theorem (1996) }} Simon Singh and John Lynch's film tells the story of Andrew Wiles.
* [http://www.coolissues.com/mathematics/BealFermatPythagorasTriplets.htm Beal Fermat and Pythagora's Triplets]
 
[[Category:Articles with inconsistent citation formats]]
[[Category:Fermat's Last Theorem| ]]

Latest revision as of 08:19, 4 January 2015

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