Fermat's Last Theorem

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Fermat's Last Theorem
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{{short description|Theorem in number theory}}{{About||other theorems named after Pierre de Fermat|Fermat's theorem|the book by Simon Singh|Fermat's Last Theorem (book)}}{{Use dmy dates|date=July 2013}}

}}In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers {{math|a}}, {{math|b}}, and {{math|c}} satisfy the equation {{math|1=a'n + b'n = cn}} for any integer value of {{math|n}} greater than 2. The cases {{math|1=n = 1}} and {{math|1=n = 2}} have been known since antiquity to have an infinite number of solutions.Singh, pp. 18–20.The proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. However, there were doubts that he had a correct proof because his claim was published by his son without his consent and after his death.WEB,weblink Nigel Boston,p.5 "THE PROOF OF FERMAT'S LAST THEOREM", After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995; it was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.Abel prize 2016 – full citation It also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques.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 notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.BOOK, The Guinness Book of World Records,weblink registration, 1995, Science and Technology, Guinness Publishing Ltd., {{TOC limit|3}}


Pythagorean origins

The Pythagorean equation, {{nowrap|1=x2 + y2 = z2}}, has an infinite number of positive integer solutions for x, y, and z; these solutions are known as Pythagorean triples (with the simplest example 3,4,5). Around 1637, Fermat wrote in the margin of a book that the more general equation {{nowrap|1=a'n + b'n = cn}} had no solutions in positive integers 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, and no proof by him has ever been found. His claim was discovered some 30 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.The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics.

Subsequent developments and solution

The special case {{math|1=n = 4}}, proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent n that is not a prime number, it must also be false for some smaller n, so only prime values of n need further investigation.If the exponent n were not prime or 4, then it would be possible to write n either as a product of two smaller integers (n = PQ), in which P is a prime number greater than 2, and then an = aPQ = (aQ)P for each of a, b, and c. That is, an equivalent solution would also have to exist for the prime power P that is smaller than n; or else as n would be a power of 2 greater than 4, and writing n = 4Q, the same argument would hold. Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach that was relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, 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 impossible, exceedingly difficult, or unachievable with current knowledge).{{citation needed|date = March 2016}}Separately, 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 conjecture (eventually as the modularity theorem), it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was (like Fermat's theorem) widely considered completely inaccessible to proof.{{citation needed|date = March 2016}}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). 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 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, which would therefore have to be true as well.Although both problems were daunting and widely considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers. Unlike Fermat's Last Theorem, the Taniyama–Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics.Singh, p. 144 quotes Wiles's reaction to this news: "I was electrified. I knew that moment that the course of my life was changing because this meant that to prove Fermat’s Last Theorem all I had to do was to prove the Taniyama–Shimura conjecture. It meant that my childhood dream was now a respectable thing to work on." However, general opinion was that this simply showed the impracticality of proving the Taniyama–Shimura conjecture.Singh, p. 144. Mathematician John Coates' quoted reaction was a common one:
"I myself was very sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn’t see it proved in my lifetime."
On hearing that Ribet had proven Frey's link to be correct, English mathematician Andrew Wiles, who had a childhood fascination with Fermat's Last Theorem and had a background of working with elliptic curves and related fields, decided to try to prove the Taniyama–Shimura conjecture as a way to prove Fermat's Last Theorem. In 1993, after six years of working secretly on the problem, Wiles succeeded in proving enough of the conjecture to prove Fermat's Last Theorem. Wiles's 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, to resolve. As a result, the final proof in 1995 was accompanied by a smaller joint paper showing that the fixed steps were valid. Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the Taniyama–Shimura–Weil conjecture, now proven and known as the modularity theorem, were subsequently proved by other mathematicians, who built on Wiles's work between 1996 and 2001.{{citation needed|date = March 2016}} For his proof, Wiles was honoured and received numerous awards, including the 2016 Abel Prize.WEB,weblink Fermat's last theorem earns Andrew Wiles the Abel Prize, Nature (journal), Nature, 15 March 2016, 15 March 2016, British mathematician Sir Andrew Wiles gets Abel math prize – The Washington Post.300-year-old math question solved, professor wins $700k –

Equivalent statements of the theorem

There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem.In order to state them, we use mathematical notation: let {{math|N}} be the set of natural numbers 1, 2, 3, ..., let {{math|Z}} be the set of integers 0, ±1, ±2, ..., and let {{math|Q}} be the set of rational numbers {{math|a/b}}, where {{mvar|a}} and {{mvar|b}} are in {{math|Z}} with {{math|b ≠ 0}}. In what follows we will call a solution to {{math|x'n + y'n {{=}} zn}} where one or more of {{mvar|x}}, {{mvar|y}}, or {{mvar|z}} is zero a trivial solution. A solution where all three are non-zero will be called a non-trivial solution.For comparison's sake we start with the original formulation.
  • Original statement. With {{mvar|n}}, {{mvar|x}}, {{mvar|y}}, {{mvar|z}} ∈ {{math|N}} (meaning that n, x, y, z are all positive whole numbers) and {{math|n > 2}}, the equation {{math|x'n + y'n {{=}} zn}} has no solutions.
Most popular treatments of the subject state it this way. In contrast, almost all mathematics textbooks state it over {{math|Z}}:{{CN|date=August 2018}}
  • Equivalent statement 1: {{math|x'n + y'n {{=}} z'n}}, where integer {{mvar|n}} ≥ 3, has no non-trivial solutions {{mvar|x}}, {{mvar|y}}, {{mvar|z}} ∈ {{math|Z'''}}.
The equivalence is clear if {{mvar|n}} is even. If {{mvar|n}} is odd and all three of {{math|x, y, z}} are negative, then we can replace {{math|x, y, z}} with {{math|−x, −y, −z}} to obtain a solution in {{math|N}}. If two of them are negative, it must be {{mvar|x}} and {{mvar|z}} or {{mvar|y}} and {{mvar|z}}. If {{math|x, z}} are negative and {{mvar|y}} is positive, then we can rearrange to get {{math|(−z)n + y'n {{=}} (−x)n}} resulting in a solution in {{math|N}}; the other case is dealt with analogously. Now if just one is negative, it must be {{mvar|x}} or {{mvar|y}}. If {{mvar|x}} is negative, and {{mvar|y}} and {{mvar|z}} are positive, then it can be rearranged to get {{math|(−x)n + z'n {{=}} y'n}} again resulting in a solution in {{math|N}}; if {{mvar|y}} is negative, the result follows symmetrically. Thus in all cases a nontrivial solution in {{math|Z}} would also mean a solution exists in {{math|N'''}}, the original formulation of the problem.
  • Equivalent statement 2: {{math|x'n + y'n {{=}} z'n}}, where integer {{mvar|n}} ≥ 3, has no non-trivial solutions {{mvar|x}}, {{mvar|y}}, {{mvar|z}} ∈ {{math|Q'''}}.
This is because the exponent of {{math|x, y,}} and {{mvar|z}} are equal (to {{mvar|n}}), so if there is a solution in {{math|Q}}, then it can be multiplied through by an appropriate common denominator to get a solution in {{math|Z}}, and hence in {{math|N}}.
  • Equivalent statement 3: {{math|x'n + y'n {{=}} 1}}, where integer {{mvar|n}} ≥ 3, has no non-trivial solutions {{mvar|x}}, {{mvar|y}} ∈ {{math|Q}}.
A non-trivial solution {{mvar|a}}, {{mvar|b}}, {{mvar|c}} ∈ {{math|Z}} to {{math|x'n + y'n {{=}} z'n}} yields the non-trivial solution {{math|a/c}}, {{math|b/c}} ∈ {{math|Q}} for {{math|v'n + w'n {{=}} 1}}. Conversely, a solution {{math|a/b}}, {{math|c/d}} ∈ {{math|Q}} to {{math|v'n + w'n {{=}} 1}} yields the non-trivial solution {{math|ad, cb, bd}} for {{math|x'n + y'n {{=}} z'n}}.This last formulation is particularly fruitful, because it reduces the problem from a problem about surfaces in three dimensions to a problem about curves in two dimensions. Furthermore, it allows working over the field {{math|Q}}, rather than over the ring {{math|Z}}; fields exhibit more structure than rings, which allows for deeper analysis of their elements.
  • Equivalent statement 4 – connection to elliptic curves: If {{mvar|a}}, {{mvar|b}}, {{mvar|c}} is a non-trivial solution to {{math|x'p + y'p {{=}} z'p}}, {{mvar|p}} odd prime, then {{math|y'2 {{=}} x(x − a'p)(x + b'p)}} (Frey curve) will be an elliptic curve.JOURNAL, Wiles, Andrew, Andrew Wiles, 1995, Modular elliptic curves and Fermat's Last Theorem,weblink Frey's suggestion, in the notation of the following theorem, was to show that the (hypothetical) elliptic curve {{math, x(x + u'p)(x – vp)}} could not be modular. |journal=Annals of Mathematics |volume=141 |issue=3 |page=448 |oclc=37032255 |doi=10.2307/2118559 |jstor=2118559}}
Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form. However, the proof by Andrew Wiles proves that any equation of the form {{math|y'2 {{=}} x(x − a'n)(x + b'n)}} does have a modular form. Any non-trivial solution to {{math|x'p + y'p {{=}} z'p}} (with {{mvar|p}} an odd prime) would therefore create a contradiction, which in turn proves that no non-trivial solutions exist.JOURNAL, Ribet, Ken, Ken Ribet, On modular representations of Gal({{overline, Q, /Q) arising from modular forms |journal=Inventiones Mathematicae |volume=100 |year=1990 |issue=2 |page=432 |doi=10.1007/BF01231195 |mr=1047143 |url= |bibcode=1990InMat.100..431R|hdl=10338.dmlcz/147454 }}In other words, any solution that could contradict Fermat's Last Theorem could also be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true, then it would follow that no contradiction to Fermat's Last Theorem could exist either. As described above, the discovery of this equivalent statement was crucial to the eventual solution of Fermat's Last Theorem, as it provided a means by which it could be "attacked" for all numbers at once.

Mathematical history

Pythagoras and Diophantus

Pythagorean triples

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,BOOK, Stillwell J, 2003, Elements of Number Theory,weblink Springer-Verlag, New York, 0-387-95587-9, 110–112, 2016-03-17, John Stillwell, and methods for generating such triples have been studied in many cultures, beginning with the BabyloniansAczel, pp. 13–15 and later ancient Greek, Chinese, and Indian mathematicians. Mathematically, the definition of a Pythagorean triple is a set of three integers (a, b, c) that satisfy the equationStark, pp. 151–155. a^2 + b^2 = c^2.

Diophantine equations

Fermat's equation, x'n + y'n = zn with positive integer solutions, is an example of a Diophantine equation,Stark, pp. 145–146. named for the 3rd-century Alexandrian 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:
A = x + y B = x^2 + y^2.
Diophantus's major work is the Arithmetica, of which only a portion has survived.Singh, pp. 50–51. Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica,Stark, p. 145. that was translated into Latin and published in 1621 by Claude Bachet.Aczel, pp. 44–45; Singh, pp. 56–58.Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC).Aczel, pp. 14–15. Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).Stark, pp. 44–47.Many Diophantine equations 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 properties. For example, it is known that there are infinitely many positive integers x, y, and z such that x'n + y'n = zm where n and m are relatively prime natural numbers.For example, left((j^r+1)^sright)^r + left(j(j^r+1)^sright)^r = (j^r+1)^{rs+1}.

Fermat's conjecture

File:Diophantus-II-8.jpg|thumb|Problem II.8 in the 1621 edition of the Arithmetica of DiophantusDiophantusProblem 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 k2 = u2 + v2. Diophantus shows how to solve this sum-of-squares problem for k = 4 (the solutions being u = 16/5 and v = 12/5).Friberg, pp. 333–334.Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica next to Diophantus's sum-of-squares problem:Dickson, p. 731; Singh, pp. 60–62; Aczel, p. 9.{| style="font-style:italic;" Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & 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. 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 marvelous proof of this, which this margin is too narrow to contain.T. Heath, Diophantus of Alexandria Second Edition, Cambridge University Press, 1910, reprinted by Dover, NY, 1964, pp. 144–145Panchishkin, p. 341After Fermat’s death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father’s comments.Singh, pp. 62–66. Although not actually a theorem at the time (meaning a mathematical statement for which proof exists), the margin note became known over time as Fermat’s Last Theorem,Dickson, p. 731. as it was the last of Fermat’s asserted theorems to remain unproved.Singh, p. 67; Aczel, p. 10.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,Ribenboim, pp. 13, 24. he never posed the general case.van der Poorten, Notes and Remarks 1.2, p. 5. 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 Poortenvan der Poorten, loc. cit. 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 WeilBOOK, André Weil, 1984, Number Theory: An approach through history. From Hammurapi to Legendre, Birkhäuser, Basel, Switzerland, 104, André Weil, 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.Taylor and Wiles's proof relies on 20th-century techniques.AV MEDIA,weblink BBC Documentary, Fermat's proof would have had to be elementary by comparison, given the mathematical knowledge of his time.While Harvey Friedman's grand conjecture implies that any provable theorem (including Fermat's last theorem) can be proved using only 'elementary function arithmetic', such a proof need be ‘elementary’ only in a technical sense and could involve millions of steps, and thus be far too long to have been Fermat’s proof.

Proofs for specific exponents

File:Diophantus-VI-24-20-Fermat.png|thumb|right|Fermat's infinite descent for Fermat's Last Theorem case n=4 in the 1670 edition of the Arithmetica of DiophantusDiophantus

Exponent 4

Only one relevant proof by Fermat has survived, in which he 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.WEB, Freeman L, Fermat's One Proof,weblink 23 May 2009, Dickson, pp. 615–616; Aczel, p. 44. His proof is equivalent to demonstrating that the equation
x^4 + y^4 = z^2
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 a4 + b4 = c4 can be written as a4 + b4 = (c2)2.Alternative proofs of the case n = 4 were developed laterRibenboim, pp. 15–24. by Frénicle de Bessy (1676),Frénicle de Bessy, Traité des Triangles Rectangles en Nombres, vol. I, 1676, Paris. Reprinted in Mém. Acad. Roy. Sci., 5, 1666–1699 (1729). Leonhard Euler (1738),JOURNAL, Euler L, 1738, Theorematum quorundam arithmeticorum demonstrationes, Novi Commentarii Academiae Scientiarum Petropolitanae, 10, 125–146, Leonhard Euler, . Reprinted Opera omnia, ser. I, "Commentationes Arithmeticae", vol. I, pp. 38–58, Leipzig:Teubner (1915). Kausler (1802), Peter Barlow (1811),BOOK, Barlow P, 1811, An Elementary Investigation of Theory of Numbers, St. Paul's Church-Yard, London, J. Johnson, 144–145, Peter Barlow (mathematician), Adrien-Marie Legendre (1830), Schopis (1825),BOOK, Schopis, 1825, Einige Sätze aus der unbestimmten Analytik, Programm, Gummbinnen, Olry Terquem (1846),JOURNAL, Terquem O, 1846, Théorèmes sur les puissances des nombres, Nouvelles Annales de Mathématiques, 5, 70–87, Olry Terquem, Joseph Bertrand (1851),BOOK, Bertrand J, 1851, Traité Élémentaire d'Algèbre, Hachette, Paris, 217–230, 395, Joseph Louis François Bertrand, Victor Lebesgue (1853, 1859, 1862),JOURNAL, Lebesgue VA, 1853, Résolution des équations biquadratiques z2 = x4 ± 2m'y4, z2 = 2m'x4 − y4, 2mz2 = x4 ± y4, Journal de Mathématiques Pures et Appliquées, 18, 73–86, BOOK, Lebesgue VA, 1859, Exercices d'Analyse Numérique, Leiber et Faraguet, Paris, 83–84, 89, BOOK, Lebesgue VA, 1862, Introduction à la Théorie des Nombres, Mallet-Bachelier, Paris, 71–73, Théophile Pépin (1883),JOURNAL, Pepin T, 1883, Étude sur l'équation indéterminée ax4 + by4 = cz2, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni., 36, 34–70, Tafelmacher (1893),JOURNAL, A. Tafelmacher, 1893, Sobre la ecuación x4 + y4 = z4, Anales de la Universidad de Chile, 84, 307–320, 10.5354/0717-8883.1893.20645, 2019-08-20, David Hilbert (1897),JOURNAL, Hilbert D, 1897, Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker-Vereinigung, 4, 175–546, David Hilbert, Reprinted in 1965 in Gesammelte Abhandlungen, vol. I by New York:Chelsea. Bendz (1901),THESIS, Bendz TR, 1901, Öfver diophantiska ekvationen xn' + yn = zn, Almqvist & Wiksells Boktrycken, Uppsala, Gambioli (1901), Leopold Kronecker (1901),BOOK, Kronecker L, 1901, Vorlesungen über Zahlentheorie, vol. I, Teubner, Leipzig, 35–38, Leopold Kronecker, Reprinted by New York:Springer-Verlag in 1978. Bang (1905),JOURNAL, Bang A, 1905, Nyt Bevis for at Ligningen x4 − y4 = z4, ikke kan have rationale Løsinger, Nyt Tidsskrift for Matematik, 16B, 31–35, 24528323, Sommer (1907),BOOK, Sommer J, 1907, Vorlesungen über Zahlentheorie, Teubner, Leipzig, Bottari (1908),JOURNAL, Bottari A, 1908, Soluzione intere dell'equazione pitagorica e applicazione alla dimostrazione di alcune teoremi della teoria dei numeri, Periodico di Matematiche, 23, 104–110, Karel Rychlík (1910), Nutzhorn (1912),JOURNAL, Nutzhorn F, 1912, Den ubestemte Ligning x4 + y4 = z4, Nyt Tidsskrift for Matematik, 23B, 33–38, Robert Carmichael (1913),JOURNAL, Carmichael RD, 1913, On the impossibility of certain Diophantine equations and systems of equations, American Mathematical Monthly, 20, 213–221, 10.2307/2974106, 7, Mathematical Association of America, 2974106, Robert Daniel Carmichael, Hancock (1931),BOOK, Hancock H, 1931, Foundations of the Theory of Algebraic Numbers, vol. I, Macmillan, New York, Gheorghe Vrănceanu (1966),JOURNAL, VrÇŽnceanu G, 1966, Asupra teorema lui Fermat pentru n=4, Gazeta Matematică Seria A, 71, 334–335, Reprinted in 1977 in Opera matematica, vol. 4, pp. 202–205, BucureÅŸti: Editura Academiei Republicii Socialiste România. Grant and Perella (1999),Grant, Mike, and Perella, Malcolm, "Descending to the irrational", Mathematical Gazette 83, July 1999, pp. 263–267. Barbara (2007),Barbara, Roy, "Fermat's last theorem in the case n=4", Mathematical Gazette 91, July 2007, 260–262. and Dolan (2011).Dolan, Stan, "Fermat's method of descente infinie", Mathematical Gazette 95, July 2011, 269–271.

Other exponents

After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be established for all odd prime exponents.Ribenboim, pp. 1–2. In other words, it was necessary to prove only that the equation a'n + b'n = cn has no positive integer solutions (a, b, c) when n is an odd prime number. This follows because a solution (abc) 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 = de. The general equation
a'n + b'n = cn
implies that (a'db'dcd) is a solution for the exponent e
(a'd)e + (b'd)e = (cd)e.
Thus, to prove that Fermat's equation has no solutions for n > 2, it would suffice to prove that it has no solutions for at least one prime factor of every n. Each integer n > 2 is divisible by 4 or by an odd prime number (or both). Therefore, Fermat's Last Theorem could be proved for all n if it could be proved for n = 4 and for all odd primes p.In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect.Dickson, p. 545.{{MacTutor|id=Al-Khujandi|title=Abu Mahmud Hamid ibn al-Khidr Al-Khujandi}} In 1770, Leonhard Euler gave a proof of p = 3,Euler L (1770) Vollständige Anleitung zur Algebra, Roy. Acad. Sci., St. Petersburg. but his proof by infinite descentWEB, Freeman L, Fermat's Last Theorem: Proof for n = 3,weblink 23 May 2009, contained a major gap.Ribenboim, pp. 24–25; Mordell, pp. 6–8; Edwards, pp. 39–40. However, since Euler himself had proved the lemma necessary to complete the proof in other work, he is generally credited with the first proof.Aczel, p. 44; Edwards, pp. 40, 52–54.JOURNAL, J. J. Mačys, On Euler's hypothetical proof, Mathematical Notes, 2007, 82, 3–4, 352–356, 10.1134/S0001434607090088, 2364600, Independent proofs were publishedRibenboim, pp. 33, 37–41. by Kausler (1802),JOURNAL, Kausler CF, 1802, Nova demonstratio theorematis nec summam, nec differentiam duorum cuborum cubum esse posse, Novi Acta Academiae Scientiarum Imperialis Petropolitanae, 13, 245–253, Legendre (1823, 1830),BOOK, Legendre AM, 1830, Théorie des Nombres (Volume II), 3rd, Firmin Didot Frères, Paris, Adrien-Marie Legendre, Reprinted in 1955 by A. Blanchard (Paris).JOURNAL, Legendre AM, 1823, Recherches sur quelques objets d'analyse indéterminée, et particulièrement sur le théorème de Fermat, Mémoires de l'Académie royale des sciences, 6, 1–60, 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. Calzolari (1855),BOOK, Calzolari L, 1855, Tentativo per dimostrare il teorema di Fermat sull'equazione indeterminata xn + yn = zn, Ferrara, Gabriel Lamé (1865),JOURNAL, Lamé G, 1865, Étude des binômes cubiques x3 ± y3, Comptes rendus de l'Académie des sciences, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 61, 921–924, 961–965, Gabriel Lamé, Peter Guthrie Tait (1872),JOURNAL, Tait PG, 1872, Mathematical Notes, Proceedings of the Royal Society of Edinburgh, 7, 144, 10.1017/s0370164600041857, Peter Guthrie Tait, Günther (1878),JOURNAL, Günther S, 1878, Ãœber die unbestimmte Gleichung x'3 + y'3 = z3, Sitzungsberichte Böhm. Ges. Wiss., 112–120, {{full citation needed|date=October 2017|reason=what is unabbreviated journal name?}} Gambioli (1901),JOURNAL, Gambioli D, 1901, Memoria bibliographica sull'ultimo teorema di Fermat, Periodico di Matematiche, 16, 145–192, Krey (1909),JOURNAL, Krey H, 1909, Neuer Beweis eines arithmetischen Satzes, Math. Naturwiss. Blätter, 6, 179–180, {{full citation needed|date=October 2017|reason=what is unabbreviated journal name?}} Rychlík (1910),JOURNAL, Rychlik K, 1910, On Fermat's last theorem for n = 4 and n = 3 (in Bohemian), ÄŒasopis Pro PÄ›stování Matematiky a Fysiky, 39, 65–86, Karel Rychlík, Stockhaus (1910),BOOK, Stockhaus H, 1910, Beitrag zum Beweis des Fermatschen Satzes, Brandstetter, Leipzig, Carmichael (1915),BOOK, Carmichael RD, 1915, Diophantine Analysis, Wiley, New York, Robert Daniel Carmichael, Johannes van der Corput (1915),JOURNAL, van der Corput JG, 1915, Quelques formes quadratiques et quelques équations indéterminées, Nieuw Archief voor Wiskunde, 11, 45–75, Johannes van der Corput, Axel Thue (1917),JOURNAL, Thue A, 1917, Et bevis for at ligningen A3 + B3 = C3 er unmulig i hele tal fra nul forskjellige tal A, B og C, Arch. Mat. Naturv., 34, 15, Axel Thue, Reprinted in Selected Mathematical Papers (1977), Oslo:Universitetsforlaget, pp. 555–559.{{full citation needed|date=October 2017|reason=what is unabbreviated journal name?}} and Duarte (1944).JOURNAL, Duarte FJ, 1944, Sobre la ecuación x3 + y3 + z3 = 0, Boletín de la Academia de Ciencias Físicas, Matemáticas y Naturales (Caracas), 8, 971–979, The case p = 5 was provedWEB, Freeman L, Fermat's Last Theorem: Proof for n = 5,weblink 23 May 2009, independently by Legendre and Peter Gustav Lejeune Dirichlet around 1825.Ribenboim, p. 49; Mordell, p. 8–9; Aczel, p. 44; Singh, p. 106. Alternative proofs were developedRibenboim, pp. 55–57. by Carl Friedrich Gauss (1875, posthumous),BOOK, Gauss CF, 1875, Neue Theorie der Zerlegung der Cuben, Zur Theorie der complexen Zahlen, Werke, vol. II, 2nd, Königl. Ges. Wiss. Göttingen, 387–391, Carl Friedrich Gauss, (Published posthumously) Lebesgue (1843),JOURNAL, Lebesgue VA, 1843, Théorèmes nouveaux sur l'équation indéterminée x5 + y5 = az5, Journal de Mathématiques Pures et Appliquées, 8, 49–70, Victor Lebesgue, Lamé (1847),JOURNAL, Lamé G, 1847, Mémoire sur la résolution en nombres complexes de l'équation A5 + B5 + C5 = 0, Journal de Mathématiques Pures et Appliquées, 12, 137–171, Gabriel Lamé, Gambioli (1901),JOURNAL, Gambioli D, 1903{{ndash, 1904 | title = Intorno all'ultimo teorema di Fermat | journal = Il Pitagora | volume = 10 | pages = 11–13, 41–42}} Werebrusow (1905),JOURNAL, Werebrusow AS, 1905, On the equation x5 + y5 = Az5 (in Russian), Moskov. Math. Samml., 25, 466–473, {{full citation needed|date=October 2017|reason=what is unabbreviated journal name?}} Rychlík (1910),JOURNAL, Rychlik K, 1910, On Fermat's last theorem for n = 5 (in Bohemian), ÄŒasopis PÄ›st. Mat., 39, 185–195, 305–317, Karel Rychlík, {{dubious|date=October 2017|reason=it is unlikely that this article was published in the Bohemian language}}{{full citation needed|date=October 2017|reason=what is unabbreviated journal name?}} van der Corput (1915), and Guy Terjanian (1987).JOURNAL, Terjanian G, 1987, Sur une question de V. A. Lebesgue, Annales de l'Institut Fourier, 37, 3, 19–37, 10.5802/aif.1096, Guy Terjanian, The case p = 7 was provedRibenboim, pp. 57–63; Mordell, p. 8; Aczel, p. 44; Singh, p. 106. by Lamé in 1839.JOURNAL, Lamé G, 1839, Mémoire sur le dernier théorème de Fermat, Comptes rendus de l'Académie des sciences, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 9, 45–46, Gabriel Lamé, JOURNAL, Lamé G, 1840, Mémoire d'analyse indéterminée démontrant que l'équation x7 + y7 = z7 est impossible en nombres entiers, Journal de Mathématiques Pures et Appliquées, 5, 195–211, Gabriel Lamé, His rather complicated proof was simplified in 1840 by Lebesgue,JOURNAL, Lebesgue VA, 1840, Démonstration de l'impossibilité de résoudre l'équation x7 + y7 + z7 = 0 en nombres entiers, Journal de Mathématiques Pures et Appliquées, 5, 276–279, 348–349, Victor Lebesgue, and still simpler proofsWEB, Freeman L, Fermat's Last Theorem: Proof for n = 7,weblink 23 May 2009, were published by Angelo Genocchi in 1864, 1874 and 1876.JOURNAL, Genocchi A, 1864, Intorno all'equazioni x7 + y7 + z7 = 0, Annali di Matematica Pura ed Applicata, 6, 287–288, 10.1007/bf03198884, Angelo Genocchi, JOURNAL, Genocchi A, 1874, Sur l'impossibilité de quelques égalités doubles, Comptes rendus de l'Académie des sciences, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 78, 433–436, Angelo Genocchi, JOURNAL, Genocchi A, 1876, Généralisation du théorème de Lamé sur l'impossibilité de l'équation x7 + y7 + z7 = 0, Comptes rendus de l'Académie des sciences, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 82, 910–913, Angelo Genocchi, Alternative proofs were developed by Théophile Pépin (1876)JOURNAL, Pepin T, 1876, Impossibilité de l'équation x7 + y7 + z7 = 0, Comptes rendus de l'Académie des sciences, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 82, 676–679, 743–747, Théophile Pépin, and Edmond Maillet (1897).JOURNAL, Maillet E, 1897, Sur l'équation indéterminée axλt + byλt = czλt, Association française pour l'avancement des sciences, St. Etienne, Compte Rendu de la 26me Session, deuxième partie, 26, 156–168,weblink Edmond Maillet, Fermat's Last Theorem was also proved for the exponents n = 6, 10, and 14. Proofs for n = 6 were published by Kausler, Thue,JOURNAL, Thue A, 1896, Ãœber die Auflösbarkeit einiger unbestimmter Gleichungen, Det Kongelige Norske Videnskabers Selskabs Skrifter, 7, Axel Thue, Reprinted in Selected Mathematical Papers, pp. 19–30, Oslo:Universitetsforlaget (1977). Tafelmacher,JOURNAL, Tafelmacher WLA, 1897, La ecuación x3 + y3 = z2: Una demonstración nueva del teorema de fermat para el caso de las sestas potencias, Anales de la Universidad de Chile, 97, 63–80, Lind,JOURNAL, Lind B, 1909, Einige zahlentheoretische Sätze, Archiv der Mathematik und Physik, 15, 368–369, Kapferer,JOURNAL, Kapferer H, 1913, Beweis des Fermatschen Satzes für die Exponenten 6 und 10, Archiv der Mathematik und Physik, 21, 143–146, Swift,JOURNAL, Swift E, 1914, Solution to Problem 206, American Mathematical Monthly, 21, 7, 238–239, 10.2307/2972379, 2972379, and Breusch.JOURNAL, 10.2307/3029800, Breusch R, Robert Breusch, 1960, A simple proof of Fermat's last theorem for n = 6, n = 10, Mathematics Magazine, 33, 5, 279–281, 3029800, Similarly, DirichletJOURNAL, Dirichlet PGL, 1832, Démonstration du théorème de Fermat pour le cas des 14e puissances, Crelle's Journal, Journal für die reine und angewandte Mathematik, 9, 390–393, Peter Gustav Lejeune Dirichlet, Reprinted in Werke, vol. I, pp. 189–194, Berlin: G. Reimer (1889); reprinted New York:Chelsea (1969). and TerjanianJOURNAL, Terjanian G, 1974, L'équation x14 + y14 = z14 en nombres entiers, Bulletin des Sciences Mathématiques (sér. 2), 98, 91–95, Guy Terjanian, each proved the case n = 14, while Kapferer and Breusch each proved the case n = 10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for n = 3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for n = 14 was published in 1832, before Lamé's 1839 proof for n = 7.Edwards, pp. 73–74.All proofs for specific exponents used Fermat's technique of infinite descent,{{citation needed|reason=this assertion requires a thorough review of all the existing literature, or a reliable source making the assertion|date=January 2015}} either in its original form, or in the form of descent on elliptic curves or abelian varieties. The details and auxiliary arguments, however, were often ad hoc and tied to the individual exponent under consideration.Edwards, p. 74. Since they became ever more complicated as p increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proved by building upon the proofs for individual exponents. Although some general results on Fermat's Last Theorem were published in the early 19th century by Niels Henrik Abel and Peter Barlow,Dickson, p. 733.BOOK, Ribenboim P, 1979, 13 Lectures on Fermat's Last Theorem, Springer Verlag, New York, 978-0-387-90432-0, 51–54, Paulo Ribenboim, the first significant work on the general theorem was done by Sophie Germain.Singh, pp. 97–109.

Early modern breakthroughs

Sophie Germain

In the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's Last Theorem for all exponents.WEB, Laubenbacher R, Pengelley D, 2007, Voici ce que j'ai trouvé: Sophie Germain's grand plan to prove Fermat's Last Theorem,weblink 19 May 2009, First, she defined a set of auxiliary primes θ constructed from the prime exponent p by the equation {{math|θ {{=}} 2hp + 1}}, where h is any integer not divisible by three. She showed that, if no integers raised to the pth 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 270,Aczel, p. 57. and for all primes p such that at least one of 2p+1, 4p+1, 8p+1, 10p+1, 14p+1 and 16p+1 is prime (specially, the primes p such that 2p+1 is prime are called Sophie Germain primes). Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for {{math|n {{=}} 2p}}, which was proved by Guy Terjanian in 1977.JOURNAL, Terjanian, G., 1977, Sur l'équation x2p + y2p = z2p, Comptes Rendus de l'Académie des Sciences, Série A-B, 285, 973–975, 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.JOURNAL, Adleman LM, Heath-Brown DR, June 1985, The first case of Fermat's last theorem, Inventiones Mathematicae, 79, 2, 409–416, Springer, Berlin, 10.1007/BF01388981, 1985InMat..79..409A,

Ernst Kummer and the theory of ideals

In 1847, Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation {{math|x{{sup|p}} + y{{sup|p}} {{=}} z{{sup|p}}}} in complex numbers, specifically the cyclotomic field based on the roots of the number 1. His proof failed, however, because it assumed incorrectly that such complex numbers can be 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.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 numbers.(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 Lamé, believed he had proven Fermat's Last Theorem until 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 says the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken".Harold M. Edwards, Fermat's Last Theorem. A genetic introduction to number theory. Graduate Texts in Mathematics vol. 50, Springer-Verlag, NY, 1977, p. 79 See the history of ideal numbers.)Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all regular prime numbers. However, he could not prove the theorem for the exceptional primes (irregular primes) that conjecturally occur approximately 39% of the time; the only irregular primes below 270 are 37, 59, 67, 101, 103, 131, 149, 157, 233, 257 and 263.

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.Aczel, pp. 84–88; Singh, pp. 232–234. This conjecture was proved in 1983 by Gerd Faltings,JOURNAL, Faltings G, 1983, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones Mathematicae, 73, 3, 349–366, 10.1007/BF01388432, 1983InMat..73..349F, Gerd Faltings, and is now known as Faltings's 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 to prove Fermat's Last Theorem for all primes up to 2521.BOOK, Ribenboim P, 1979, 13 Lectures on Fermat's Last Theorem, Springer Verlag, New York, 978-0-387-90432-0, 202, Paulo Ribenboim, By 1978, Samuel Wagstaff had extended this to all primes less than 125,000.JOURNAL, Wagstaff SS, Jr., 1978, The irregular primes to 125000, Mathematics of Computation, 32, 583–591, 10.2307/2006167, 142, American Mathematical Society, 2006167, Samuel S. Wagstaff, Jr, (PDF) {{webarchive |url= |date=10 January 2011 }} By 1993, Fermat's Last Theorem had been proved for all primes less than four million.JOURNAL, Buhler J, Crandell R, Ernvall R, Metsänkylä T, 1993, Irregular primes and cyclotomic invariants to four million, Mathematics of Computation, 61, 151–153, 10.2307/2152942, 203, American Mathematical Society, 2152942, 1993MaCom..61..151B, 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.)WEB,weblink Examples of eventual counterexamples, answer by J.D. Hamkins, Hamkins, Joel David, June 15, 2010,, June 15, 2017,

Connection with elliptic curves

The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"Fermat's Last Theorem, Simon Singh, 1997, {{isbn|1-85702-521-0}}{{rp|211}} Taniyama–Shimura–Weil conjecture, proposed around 1955—which many mathematicians believed would be near to impossible to prove,{{rp|223}} and 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 1994, 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.

Taniyama–Shimura–Weil conjecture

Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.The link was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist André Weil found evidence supporting it, though not proving it; as a result the conjecture was often known as the Taniyama–Shimura–Weil conjecture.{{rp|211–215}}Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.{{rp|203–205, 223, 226}} For example, Wiles's doctoral supervisor John Coates states that it seemed "impossible to actually prove",{{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]."{{rp|223}}

Ribet's theorem for Frey curves

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 > 2, then it could be shown that the semi-stable elliptic curve (now known as a Frey-HellegouarchThis elliptic curve was first suggested in the 1960s by {{Interlanguage link multi|Yves Hellegouarch|de}}, but he did not call attention to its non-modularity. For more details, see BOOK, Hellegouarch, Yves, Invitation to the Mathematics of Fermat-Wiles, Academic Press, 2001, 978-0-12-339251-0, )
y2 = x (x âˆ’ a'p)(x + b'p)
would have such unusual properties that it was unlikely to be modular.JOURNAL, Frey G, 1986, Links between stable elliptic curves and certain diophantine equations, Annales Universitatis Saraviensis. Series Mathematicae., 1, 1–40, Gerhard Frey, 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 might also simultaneously prove Fermat's Last Theorem.Singh, pp. 194–198; Aczel, pp. 109–114. By contraposition, a disproof or refutation of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture.In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of 4 numbers (a, b, c, n) capable of disproving Fermat's Last Theorem, could also be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore if the latter were true, the former could not be disproven, and would also have to be true.Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to prove the modularity theorem – or at least to prove it for the types of elliptical curves that included Frey's equation (known as semistable elliptic curves). This was widely believed inaccessible to proof by contemporary mathematicians.{{rp|203–205, 223, 226}} Second, 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 showed that this was plausible but did not go as far as giving a full proof. The missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was identified by Jean-Pierre Serre who also gave an almost-complete proof and the link suggested by Frey was finally proved in 1986 by Ken Ribet.JOURNAL, Ribet, Ken, Ken Ribet, On modular representations of Gal({{overline, Q, /Q) arising from modular forms|journal=Inventiones Mathematicae|volume=100|year=1990|issue=2|pages=431–476|doi=10.1007/BF01231195|mr=1047143|url=|bibcode=1990InMat.100..431R|hdl=10338.dmlcz/147454}}Following Frey, Serre and Ribet's work, this was where matters stood:
  • Fermat's Last Theorem needed to be proven for all exponents n that were prime numbers.
  • The modularity theorem – if proved for semi-stable elliptic curves – would mean that all semistable elliptic curves must be modular.
  • Ribet's theorem showed that any solution to Fermat's equation for a prime number could be used to create a semistable elliptic curve that could not be modular;
  • The only way that both of these statements could be true, was if no solutions existed to Fermat's equation (because then no such curve could be created), which was what Fermat's Last Theorem said. As Ribet's Theorem was already proved, this meant that a proof of the Modularity Theorem would automatically prove Fermat's Last theorem was true as well.

Wiles's general proof

File:Andrew wiles1-3.jpg|thumb|upright|British mathematician Andrew WilesAndrew WilesRibet'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 who had worked on elliptic curves, 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.Singh, p. 205; Aczel, pp. 117–118.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.{{rp|229–230}} His initial study suggested proof by induction,{{rp|230–232, 249–252}} and he based his initial work and first significant breakthrough on Galois theory{{rp|251–253, 259}} before switching to an attempt to extend horizontal Iwasawa theory for the inductive argument around 1990–91 when it seemed that there was no existing approach adequate to the problem.{{rp|258–259}} However, by the summer of 1991, Iwasawa theory also seemed to not be reaching the central issues in the problem.{{rp|259–260}}Singh, pp. 237–238; Aczel, pp. 121–122. 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 that seemed "tailor made" for the inductive part of his proof.{{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 help him check his reasoning for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly.{{rp|261–265}}Singh, pp. 239–243; Aczel, pp. 122–125.By mid-May 1993, Wiles felt able to tell his wife he thought he had solved the proof of Fermat's Last Theorem,{{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.Singh, pp. 244–253; Aczel, pp. 1–4, 126–128. 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. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz (in his role as reviewer),Aczel, pp. 128–130. who alerted Wiles on 23 August 1993.Singh, p. 257.The error would not have rendered his work worthless – each part of Wiles's 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.{{rp|289, 296–297}} However without this part proved, 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 his former student Richard Taylor, without success.Singh, pp. 269–277.A Year Later, Snag Persists In Math Proof 28 June 199426 June – 2 July; A Year Later Fermat's Puzzle Is Still Not Quite Q.E.D. 3 July 1994 By the end of 1993, rumours had spread that under scrutiny, Wiles's proof had failed, but how seriously was not known. Mathematicians were beginning to pressure Wiles to disclose his work whether or not complete, so that the wider community could explore and use whatever he had managed to accomplish. But instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.Singh, pp. 175–185.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 find 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 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.Aczel, pp. 132–134. 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.
"I was sitting at my desk examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn’t work. Suddenly I had this incredible revelation. I realised that, the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much."
— Andrew Wiles, as quoted by Simon SinghSingh p. 186–187 (text condensed).
On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"JOURNAL, Wiles, Andrew, Andrew Wiles, 1995, Modular elliptic curves and Fermat's Last Theorem,weblink Annals of Mathematics, 141, 3, 443–551, 37032255, 10.2307/2118559, 2118559, and "Ring theoretic properties of certain Hecke algebras",JOURNAL, Richard Taylor (mathematician), Taylor R, Andrew Wiles, Wiles A, 1995, Annals of Mathematics, Ring theoretic properties of certain Hecke algebras, 141, 3, 553–572, 37032255,weblink 10.2307/2118560, 2118560, dead,weblink" title="">weblink 27 November 2001, the second of which was co-authored with Taylor and proved that certain conditions were met that 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's work, incrementally chipped away at the remaining cases until the full result was proved.JOURNAL, Diamond, Fred, On deformation rings and Hecke rings, 10.2307/2118586, 1405946, 1996, Annals of Mathematics, Second Series, 0003-486X, 144, 1, 137–166, harv, 2118586, JOURNAL, Conrad, Brian, Diamond, Fred, Taylor, Richard, Modularity of certain potentially Barsotti-Tate Galois representations, 10.1090/S0894-0347-99-00287-8, 1639612, 1999, Journal of the American Mathematical Society, 0894-0347, 12, 2, 521–567, harv, JOURNAL, Breuil, Christophe, Conrad, Brian, Diamond, Fred, Taylor, Richard, On the modularity of elliptic curves over Q: wild 3-adic exercises, 10.1090/S0894-0347-01-00370-8, 1839918, 2001, Journal of the American Mathematical Society, 0894-0347, 14, 4, 843–939, harv, 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 â‰¥ 3. (The case n = 3 was already known by Euler.)

Relationship to other problems and generalizations

Fermat's Last Theorem considers solutions to the Fermat equation: {{math|1=a'n + b'n = cn}} with positive integers {{math|a}}, {{math|b}}, and {{math|c}} and an integer {{math|n}} greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent {{math|n}} to be a negative integer or rational, or to consider three different exponents.

Generalized Fermat equation

The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions a, b, c, m, n, k satisfyingBOOK, The Princeton Companion to Mathematics,weblink Barrow-Green, June, Leader, Imre, Gowers, Timothy, 361–362, 2008, Princeton University Press, {{NumBlk|:|a^m + b^n = c^k.|{{EquationRef|1}}}}In particular, the exponents m, n, k need not be equal, whereas Fermat's last theorem considers the case {{math|1=m = n = k.}}The Beal conjecture, also known as the Mauldin conjectureWEB,weblink Mauldin / Tijdeman-Zagier Conjecture, Prime Puzzles, 1 October 2016, and the Tijdeman-Zagier conjecture,JOURNAL, Elkies, Noam D., The ABC's of Number Theory, The Harvard College Mathematics Review, 2007, 1, 1,weblink JOURNAL, Moscow Mathematical Journal, 4, 2004, Open Diophantine Problems, 245–305, Michel Waldschmidt, 10.17323/1609-4514-2004-4-1-245-305, math/0312440, BOOK, Prime Numbers: A Computational Perspective, Crandall, Richard, Pomerance, Carl, 2000, 978-0387-25282-7, Springer, 417, states that there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k being greater than 2.WEB,weblink Beal Conjecture, American Mathematical Society, 21 August 2016, The Fermat–Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture.JOURNAL, A new generalization of Fermat's Last Theorem, Cai, Tianxin, Chen, Deyi, Zhang, Yong, Journal of Number Theory, 149, 2015, 33–45, 10.1016/j.jnt.2014.09.014, 1310.0897, JOURNAL, A Cyclotomic Investigation of the Catalan–Fermat Conjecture, Mihailescu, Preda, Mathematica Gottingensis, 2007, The conjecture states that the generalized Fermat equation has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (a'm, b'n, ck), where a, b, c are positive coprime integers and m, n, k are positive integers satisfying{{NumBlk|:|frac{1}{m} + frac{1}{n} + frac{1}{k} < 1.|{{EquationRef|2}}}}The statement is about the finiteness of the set of solutions because there are 10 known solutions.

Inverse Fermat equation

When we allow the exponent {{math|n}} to be the reciprocal of an integer, i.e. {{math|1=n = 1/m}} for some integer {{math|m}}, we have the inverse Fermat equationa^{1/m} + b^{1/m} = c^{1/m}.All solutions of this equation were computed by Hendrik Lenstra in 1992.JOURNAL, Lenstra Jr. H.W., Hendrik Lenstra, 1992, On the inverse Fermat equation, Discrete Mathematics, 106–107, 329–331, 10.1016/0012-365x(92)90561-s, In the case in which the mth roots are required to be real and positive, all solutions are given byJOURNAL, Newman M, 1981, A radical diophantine equation, 10.1016/0022-314x(81)90040-8, Journal of Number Theory, 13, 4, 495–498,
a=rs^m b=rt^m c=r(s+t)^m
for positive integers r, s, t with s and t coprime.

Rational exponents

For the Diophantine equation a^{n/m} + b^{n/m} = c^{n/m} with n not equal to 1, Bennett, Glass, and Székely proved in 2004 for n > 2, that if n and m are coprime, then there are integer solutions if and only if 6 divides m, and a^{1/m}, b^{1/m}, and c^{1/m} are different complex 6th roots of the same real number.JOURNAL
, Bennett, Curtis D.
, Glass, A. M. W.
, Székely, Gábor J.
, 10.2307/4145241
, 4
, American Mathematical Monthly
, 2057186
, 322–329
, Fermat's last theorem for rational exponents
, 111
, 2004, 4145241

Negative integer exponents

n  âˆ’1

All primitive integer solutions (i.e., those with no prime factor common to all of a, b, and c) to the optic equation a^{-1} + b^{-1} = c^{-1} can be written asDickson, pp. 688–691.
a = mk + m^2, b = mk + k^2, c = mk
for positive, coprime integers m, k.

n  âˆ’2

The case n = âˆ’2 also has an infinitude of solutions, and these have a geometric interpretation in terms of right triangles with integer sides and an integer altitude to the hypotenuse.JOURNAL, Voles, Roger, Integer solutions of a−2 + b−2 = d−2, Mathematical Gazette, 83, 497, July 1999, 269–271, 10.2307/3619056, 3619056, JOURNAL, Richinick, Jennifer, The upside-down Pythagorean Theorem, Mathematical Gazette, 92, July 2008, 313–317, 10.1017/S0025557200183275, All primitive solutions to a^{-2} + b^{-2} = d^{-2} are given by
a = (v^2 - u^2)(v^2 + u^2), b = 2uv(v^2 + u^2), d = 2uv(v^2 - u^2),
for coprime integers u, v with v > 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
c = (v^2 + u^2)^2,
so (a, b, c) is a Pythagorean triple.


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