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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}}

factoids
}}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. 18â€“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 â€“ 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}}

## Overview

### 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 (1637â€“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 Taniyamaâ€“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 Taniyamaâ€“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 Taniyamaâ€“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 Taniyamaâ€“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 Fermatâ€™s Last Theorem all I had to do was to prove the Taniyamaâ€“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 Taniyamaâ€“Shimura conjecture.Singh, p. 144. Mathematician John Coates' quoted reaction was a common one:
"I myself was very sceptical that the beautiful link between Fermatâ€™s Last Theorem and the Taniyamaâ€“Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyamaâ€“Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldnâ€™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 Taniyamaâ€“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 Taniyamaâ€“Shimuraâ€“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 â€“ The Washington Post.300-year-old math question solved, professor wins \$700k â€“ CNN.com.

### 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, Â±1, Â±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 â‰  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}} âˆˆ {{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}} â‰¥ 3, has no non-trivial solutions {{mvar|x}}, {{mvar|y}}, {{mvar|z}} âˆˆ {{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|âˆ’x, âˆ’y, âˆ’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|(âˆ’z)n + y'n {{=}} (âˆ’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|(âˆ’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}} â‰¥ 3, has no non-trivial solutions {{mvar|x}}, {{mvar|y}}, {{mvar|z}} âˆˆ {{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}} â‰¥ 3, has no non-trivial solutions {{mvar|x}}, {{mvar|y}} âˆˆ {{math|Q}}.
A non-trivial solution {{mvar|a}}, {{mvar|b}}, {{mvar|c}} âˆˆ {{math|Z}} to {{math|x'n + y'n {{=}} z'n}} yields the non-trivial solution {{math|a/c}}, {{math|b/c}} âˆˆ {{math|Q}} for {{math|v'n + w'n {{=}} 1}}. Conversely, a solution {{math|a/b}}, {{math|c/d}} âˆˆ {{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 â€“ 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 âˆ’ 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 â€“ 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 âˆ’ 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=http://math.berkeley.edu/~ribet/Articles/invent_100.pdf |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 â€“ 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, 110â€“112, 2016-03-17, John Stillwell, and methods for generating such triples have been studied in many cultures, beginning with the BabyloniansAczel, pp. 13â€“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. 151â€“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. 145â€“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. 50â€“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. 44â€“45; Singh, pp. 56â€“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. 14â€“15. Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).Stark, pp. 44â€“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}.

### 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. 615â€“616; Aczel, p. 44. His proof is equivalent to demonstrating that the equation
x^4 + y^4 = z^2

#### 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. 1â€“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.

### Early modern breakthroughs

#### Ernst Kummer and the theory of ideals

In 1847, Gabriel LamÃ© 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 LamÃ©, 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 LamÃ©, 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. 84â€“88; Singh, pp. 232â€“234. This conjecture was proved in 1983 by Gerd Faltings,JOURNAL, Faltings G, 1983, EndlichkeitssÃ¤tze fÃ¼r abelsche VarietÃ¤ten Ã¼ber ZahlkÃ¶rpern, Inventiones Mathematicae, 73, 3, 349â€“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, 583â€“591, 10.2307/2006167, 142, American Mathematical Society, 2006167, Samuel S. Wagstaff, Jr, (PDF) {{webarchive |url=https://www.webcitation.org/5vcghCvCT?url=http://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0491465-4/S0025-5718-1978-0491465-4.pdf |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, MetsÃ¤nkylÃ¤ T, 1993, Irregular primes and cyclotomic invariants to four million, Mathematics of Computation, 61, 151â€“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, mathoverflow.net, 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}} Taniyamaâ€“Shimuraâ€“Weil conjecture, proposed around 1955â€”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.

#### Taniyamaâ€“Shimuraâ€“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 Taniyamaâ€“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 AndrÃ© Weil found evidence supporting it, though not proving it; as a result the conjecture was often known as the Taniyamaâ€“Shimuraâ€“Weil conjecture.{{rp|211â€“215}}Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.{{rp|203â€“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, 1â€“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 Taniyamaâ€“Shimuraâ€“Weil conjecture might also simultaneously prove Fermat's Last Theorem.Singh, pp. 194â€“198; Aczel, pp. 109â€“114. By contraposition, a disproof or refutation of Fermat's Last Theorem would disprove the Taniyamaâ€“Shimuraâ€“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 Taniyamaâ€“Shimuraâ€“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 â€“ 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|203â€“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=431â€“476|doi=10.1007/BF01231195|mr=1047143|url=http://math.berkeley.edu/~ribet/Articles/invent_100.pdf|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 â€“ if proved for semi-stable elliptic curves â€“ 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

"I was sitting at my desk examining the Kolyvaginâ€“Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didnâ€™t work. Suddenly I had this incredible revelation. I realised that, the Kolyvaginâ€“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 Kolyvaginâ€“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."
â€” Andrew Wiles, as quoted by Simon SinghSingh p. 186â€“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, 443â€“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, 553â€“572, 37032255,weblink 10.2307/2118560, 2118560, dead,weblink" title="web.archive.org/web/20011127181043weblink">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 Taniyamaâ€“Shimuraâ€“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, 137â€“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, 521â€“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, 843â€“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, 361â€“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, 245â€“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 Fermatâ€“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, 33â€“45, 10.1016/j.jnt.2014.09.014, 1310.0897, JOURNAL, A Cyclotomic Investigation of the Catalanâ€“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, 106â€“107, 329â€“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, 495â€“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 SzÃ©kely 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.
, 10.2307/4145241
, 4
, American Mathematical Monthly
, 2057186
, 322â€“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. 688â€“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 aâˆ’2 + bâˆ’2 = dâˆ’2, Mathematical Gazette, 83, 497, July 1999, 269â€“271, 10.2307/3619056, 3619056, JOURNAL, Richinick, Jennifer, The upside-down Pythagorean Theorem, Mathematical Gazette, 92, July 2008, 313â€“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.

#### n

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