Fermat's Last Theorem - Knowledge (XXG)

4808:—a large sum at the time—to the Göttingen Academy of Sciences to offer as a prize for a complete proof of Fermat's Last Theorem. On 27 June 1908, the academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was begun. Wiles collected the Wolfskehl prize money, then worth $ 50,000, on 27 June 1997. In March 2016, Wiles was awarded the Norwegian government's 3372:

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.

727:, 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. For his proof, Wiles was honoured and received numerous awards, including the 2016 4927: 3354:, without success. 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 it was complete or not, 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. 2099: 118: 4847: 639:, 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). 505: 3260: 1893: 10310: 47: 3197:). This was widely believed inaccessible to proof by contemporary mathematicians. 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 2062:, he never posed the general case. Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van der Poorten 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 2509: = 3, but his proof by infinite descent contained a major gap. However, since Euler himself had proved the lemma necessary to complete the proof in other work, he is generally credited with the first proof. Independent proofs were published by Kausler (1802), Legendre (1823, 1830), Calzolari (1855), 694:: 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. 3371:

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

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Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and fix the error. He adds that he was having a final look to try and understand the fundamental

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states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed 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

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tended to use a special preprinted form for such proofs, where the location of the first mistake was left blank to be filled by one of his graduate students. According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in schools, and often submitted

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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. He described later that Iwasawa theory and

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On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic properties of certain Hecke algebras", the second of which was co-authored with Taylor and proved that certain conditions were met that were needed to justify the corrected

715:, 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 3312:

for the inductive argument around 1990–91 when it seemed that there was no existing approach adequate to the problem. However, by mid-1991, Iwasawa theory also seemed to not be reaching the central issues in the problem. In response, he approached colleagues to seek out any hints of cutting-edge

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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

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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. However, without this part proved, there was no

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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

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Prior to Wiles's proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3.0 meters) of correspondence. In the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to

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would have such unusual properties that it was unlikely to be modular. 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

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The full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996), Conrad et al. (1999), and Breuil et al. (2001) who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the

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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

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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

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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

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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

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offered a prize for a general proof of Fermat's Last Theorem. In 1857, the academy awarded 3,000 francs and a gold medal to Kummer for his research on ideal numbers, although he had not submitted an entry for the prize. Another prize was offered in 1883 by the Academy of Brussels.

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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

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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.

662:(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. 3325:

that seemed "tailor made" for the inductive part of his proof. 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,

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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.

690:). 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 698:

Taniyama–Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics. However, general opinion was that this simply showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician

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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

5517: 3338:. 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 3192:

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

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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

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By mid-May 1993, Wiles was ready to tell his wife he thought he had solved the proof of Fermat's Last Theorem, and by June he felt sufficiently confident to present his results in three lectures delivered on 21–23 June 1993 at the

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After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father's comments. Although not actually a theorem at the time (meaning a mathematical statement for which

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case: even if all exponents were verified up to an extremely large number X, a higher exponent beyond X might still exist for which the claim was not true. (This had been the case with some other past conjectures, such as with

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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

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is equivalent to the abc conjecture and therefore has the same implication. An effective version of the abc conjecture, or an effective version of the modified Szpiro conjecture, implies Fermat's Last Theorem outright.

3189:) 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. 10435:

Discusses various material that is related to the proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of

357:. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, 3288:, 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 3817: 3295:

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. His initial study suggested

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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

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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

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Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof. For example, Wiles's doctoral supervisor

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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.

1586:. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples, and methods for generating such triples have been studied in many cultures, beginning with the 4631: 8483: 4533: 4468: 3581: 6433: 2185: 1775: 7673: 7637: 7444: 7408: 6325: 4014: 4239: 4193: 4812:

worth €600,000 for "his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory".

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Wiles and Taylor's proof relies on 20th-century techniques. Fermat's proof would have had to be elementary by comparison, given the mathematical knowledge of his time.

3965: 3928: 3335: 3632:, also known as the Mauldin conjecture and the Tijdeman-Zagier conjecture, states that there are no solutions to the generalized Fermat equation in positive integers 2591: = 3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for 719:

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

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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.

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reasons 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

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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

2899:(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 1725: 4269: 2977:

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

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to show that the area of a right triangle with integer sides can never equal the square of an integer. His proof is equivalent to demonstrating that the equation

4825:, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published." 10477: 5950: 3346:. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz (in his role as reviewer), who alerted Wiles on 23 August 1993. 2066:

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.

10635: 10140: 3275: 716: 366: 10630: 2609:, 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 8798: 2084:', 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. 10707: 10541: 9447: 10604: 1676:, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers 358: 522: 6374:

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni

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The popularity of the theorem outside science has led to it being described as achieving "that rarest of mathematical accolades: A niche role in

9806: 361:), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a 7739: 3367:

the Kolyvagin–Flach approach were each inadequate on their own, but together they could be made powerful enough to overcome this final hurdle.

3330:, to help him check his reasoning for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly. 739:

There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem.

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The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in

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posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions, if the exponent

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Bottari A (1908). "Soluzione intere dell'equazione pitagorica e applicazione alla dimostrazione di alcune teoremi della teoria dei numeri".

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required only that the theorem be established for all odd prime exponents. In other words, it was necessary to prove only that the equation

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could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by developing the

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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

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were published by Kausler, Thue, Tafelmacher, Lind, Kapferer, Swift, and Breusch. Similarly, Dirichlet and Terjanian each proved the case

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found evidence supporting it, though not proving it; as a result the conjecture was often known as the Taniyama–Shimura–Weil conjecture.

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had extended this to all primes less than 125,000. By 1993, Fermat's Last Theorem had been proved for all primes less than four million.

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mixing two letters, without sharing its particular properties. For example, it is known that there are infinitely many positive integers

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need further investigation. Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although

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ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now known as the

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was proved by Lamé in 1839. His rather complicated proof was simplified in 1840 by Lebesgue, and still simpler proofs were published by

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In the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregular primes. In 1954,

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Our proof generalizes the known implication "effective ABC eventual Fermat" which was the original motivation for the ABC conjecture

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states that the theorem is still unproven in the 24th century. The proof was released 5 years after the episode originally aired.

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noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution

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developed several novel approaches to prove Fermat's Last Theorem for all exponents. First, she defined a set of auxiliary primes

2547:(1875, posthumous), Lebesgue (1843), Lamé (1847), Gambioli (1901), Werebrusow (1905), Rychlík (1910), van der Corput (1915), and 1797:

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation

1786:, of which only a portion has survived. Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the 659: 10659: 10584: 9914: 5572: 4879: 2998:, proposed around 1955—which many mathematicians believed would be near to impossible to prove, and was linked in the 1980s by 548: 35: 5529: 9872: 8920: 8845: 8083: 4903: 3498:

The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions

3351: 3322: 2912: 2904: 2540: 2081: 724: 526: 398: 8226: 7305: 3836: 3684: 453:(with the simplest example being 3, 4, 5). Around 1637, Fermat wrote in the margin of a book that the more general equation 259: 10506:

The title of one edition of the PBS television series NOVA, discusses Andrew Wiles's effort to prove Fermat's Last Theorem.

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greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent

1582:, and a triple of numbers that meets this condition is called a Pythagorean triple; both are named after the ancient Greek 10383: 7997: 6662: 2799:). Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for 2486:

In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents

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as the "most difficult mathematical problem", in part because the theorem has the largest number of unsuccessful proofs.

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In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of four numbers (

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The link was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist

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have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no

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Association française pour l'avancement des sciences, St. Etienne, Compte Rendu de la 26me Session, deuxième partie

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Kausler CF (1802). "Nova demonstratio theorematis nec summam, nec differentiam duorum cuborum cubum esse posse".

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to Fermat's Last Theorem. The equation is wrong, but it appears to be correct if entered in a calculator with 10

4858: 3194: 9867:. Diophantine Analysis. Vol. II. New York: Chelsea Publishing. pp. 545–550, 615–621, 688–691, 731–776. 6986:(1823). "Recherches sur quelques objets d'analyse indéterminée, et particulièrement sur le théorème de Fermat". 5629:

Frey's suggestion, in the notation of the following theorem, was to show that the (hypothetical) elliptic curve

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that a critical point in the proof was incorrect. It contained an error in a bound on the order of a particular

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The modularity theorem—if proved for semi-stable elliptic curves—would mean that all semistable elliptic curves

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is divisible by 4 or by an odd prime number (or both). Therefore, Fermat's Last Theorem could be proved for all

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Bennett, Curtis D.; Glass, A. M. W.; Székely, Gábor J. (2004). "Fermat's last theorem for rational exponents".

9242: 7173: 6830: 6657: 2587: = 10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for 2565:

in 1864, 1874 and 1876. Alternative proofs were developed by Théophile Pépin (1876) and Edmond Maillet (1897).

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step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the

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The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"

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Manin, Yuri Ivanovic; Panchishkin, Alekseĭ Alekseevich (2007). "Fundamental problems, Ideas and Theories".

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innovated and proved an approach that was relevant to an entire class of primes. In the mid-19th century,

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impossible to actually prove. I must confess I thought I probably wouldn't see it proved in my lifetime.

10574: 7820: 5096: 4926: 4775:. In particular, the abc conjecture in its most standard formulation implies Fermat's last theorem for 2951: 2922:

Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all

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said the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken". See

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and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's

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who also gave an almost-complete proof and the link suggested by Frey was finally proved in 1986 by

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and tied to the individual exponent under consideration. Since they became ever more complicated as

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by "people with a technical education but a failed career". In the words of mathematical historian

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in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success,

2518: 2498: 1835: 1595: 1579: 1395: 416: 31: 9974:. Encyclopedia of Mathematical Sciences. Vol. 49 (2nd ed.). Berlin Hedelberg: Springer. 7136: 3971: 2911:

in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources.

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mathematicians. Mathematically, the definition of a Pythagorean triple is a set of three integers

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award in 2016. It also proved much of the Taniyama–Shimura conjecture, subsequently known as the

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Second Edition, Cambridge University Press, 1910, reprinted by Dover, NY, 1964, pp. 144–145

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Cai, Tianxin; Chen, Deyi; Zhang, Yong (2015). "A new generalization of Fermat's Last Theorem".

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implies that any provable theorem (including Fermat's last theorem) can be proved using only '

1591: 1190:, then it can be multiplied through by an appropriate common denominator to get a solution in 675: 9779: 9681: 6392: 5415: 5371: 3040:

observed a possible link between two apparently completely distinct branches of mathematics,

2926:. However, he could not prove the theorem for the exceptional primes (irregular primes) that 2873:. His proof failed, however, because it assumed incorrectly that such complex numbers can be 608:, proved by Fermat himself, is sufficient to establish that if the theorem is false for some 381:, and opened up entire new approaches to numerous other problems and mathematically powerful 10550: 10410: 10278: 10274: 9931: 9741: 9732: 9484: 9456: 9394: 9349: 9280: 9251: 9218: 9169: 9077: 8942: 8862: 8634: 8626: 8463: 8397: 8318: 8057: 8006: 7748: 7562: 7516: 7081: 6940: 6671: 6596: 5713: 5705: 5589: 5482: 5465: 5431: 5383: 5341: 5257: 4805: 4301: 3318: 3170:

or refutation of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture.

2889: 2881:, who later read a paper that demonstrated this failure of unique factorisation, written by 2878: 2866: 2606: 2457:, it would suffice to prove that it has no solutions for at least one prime factor of every 2281: 2128: 2103: 2098: 2077: 1918:

asks how a given square number is split into two other squares; in other words, for a given

1881: 348: 184: 10422: 9537: 9300: 8646: 7860: 7103: 6952: 5725: 5101: 4300:

also has an infinitude of solutions, and these have a geometric interpretation in terms of

3052:(at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is 2521:(1878), Gambioli (1901), Krey (1909), Rychlík (1910), Stockhaus (1910), Carmichael (1915), 2022:, but it appears unlikely. Only one related proof by him has survived, namely for the case 1698: 10418: 10180: 9989: 9533: 9296: 9210: 8642: 8593: 8487: 8423: 8185: 7784: 7767: 7718: 7656: 7391: 6948: 6660:(1913). "On the impossibility of certain Diophantine equations and systems of equations". 5721: 5672: 4912: 4869:

that the latter cannot produce a proof of Fermat's Last Theorem within twenty-four hours.

4840: 4245: 3884: 3629: 3037: 2967: 2814: 2562: 2265: 2073: 1919: 1104:

is negative, the result follows symmetrically. Thus in all cases a nontrivial solution in

898:

Most popular treatments of the subject state it this way. It is also commonly stated over

647: 283: 254: 10255: 9195:

Mihailescu, Preda (2007). "A Cyclotomic Investigation of the Catalan–Fermat Conjecture".

7620: 7603: 7427: 7033: 2900: 2838: 2825:

proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes

2510: 10461: 10251: 9840: 9517: 9081: 8903: 8622: 8393: 8314: 5701: 5543: 5478: 5253: 10204: 10039: 9488: 8035: 7933:: Una demonstración nueva del teorema de fermat para el caso de las sestas potencias". 6843: 6139: 5747: 4950: 4891: 4801: 4731: 4725: 4131: 3363: 3309: 3163: 3121: 3071: 3041: 2963: 2862: 2644: 2627: 2502: 2245: 2051: 1471: 699: 651: 628: 228: 117: 9427:. Graduate Texts in Mathematics. Vol. 211. Springer-Verlag New York. p. 196. 8984: 8227:"Voici ce que j'ai trouvé: Sophie Germain's grand plan to prove Fermat's Last Theorem" 4846: 2907:

told him his argument relied on unique factorization; but the story was first told by

10676: 10569: 10128: 9958: 9761: 9406: 9369: 9256: 9237: 9222: 9181: 8654: 8409: 8377: 8330: 8112: 7760: 7546: 6960: 6428: 6022: 5733: 5188: 4883: 4862: 4854: 4817: 4568:

is the integer altitude to the hypotenuse. Then the hypotenuse itself is the integer

3305: 3064: 2947: 2939: 2927: 2923: 2870: 2810: 2670:

is any integer not divisible by three. She showed that, if no integers raised to the

2548: 2273: 2063: 2055: 636: 484: 271: 145: 10430: 10414: 9947: 9943: 9730:

Kasman, Alex (January 2003). "Mathematics in Fiction: An Interdisciplinary Course".

9089: 7020:

Tentativo per dimostrare il teorema di Fermat sull'equazione indeterminata x + y = z

6040: 2723:, which verified the first case of Fermat's Last Theorem (namely, the case in which 1059:; the other case is dealt with analogously. Now if just one is negative, it must be 17: 10645: 10594: 10035: 9010: 8924: 8841: 8546: 8301:

Adleman LM, Heath-Brown DR (June 1985). "The first case of Fermat's last theorem".

8102:, vol. I, pp. 189–194, Berlin: G. Reimer (1889); reprinted New York:Chelsea (1969). 6245: 5568: 5504: 4874: 4118:

All primitive integer solutions (i.e., those with no prime factor common to all of

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The statement is about the finiteness of the set of solutions because there are 10

3314: 3285: 3271: 3264: 3094: 3057: 3045: 3033: 3011: 3010:

to Fermat's equation. By accomplishing a partial proof of this conjecture in 1994,

2999: 2916: 2893: 2882: 2346: 2257: 1483: 712: 666: 655: 643: 632: 616: 370: 205: 10132: 8886: 8549:(1986). "Links between stable elliptic curves and certain diophantine equations". 6996:

Reprinted in 1825 as the "Second Supplément" for a printing of the 2nd edition of

6733:, vol. 4, pp. 202–205, București: Editura Academiei Republicii Socialiste România. 6156:, ser. I, "Commentationes Arithmeticae", vol. I, pp. 38–58, Leipzig:Teubner (1915) 5613: 5436: 5388: 2018:

It is not known whether Fermat had actually found a valid proof for all exponents

711:

On hearing that Ribet had proven Frey's link to be correct, English mathematician

10441: 8233: 7122:

Sitzungsberichte der Königliche böhmische Gesellschaft der Wissenschaften in Prag

5414:

Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (15 May 2001).

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Kapferer H (1913). "Beweis des Fermatschen Satzes für die Exponenten 6 und 10".

4822: 3490:

to be a negative integer or rational, or to consider three different exponents.

3339: 3292:(then known as the Taniyama–Shimura conjecture) for semistable elliptic curves. 3201:

but did not go as far as giving a full proof. The missing piece (the so-called "

2908: 2059: 1913: 1782: 720: 504: 353: 128: 2501:(10th century), but his attempted proof of the theorem was incorrect. In 1770, 10364: 9842:

Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem

9836: 9745: 9461: 9442: 9420: 9398: 9173: 8638: 7787:(1876). "Généralisation du théorème de Lamé sur l'impossibilité de l'équation 7085: 6944: 5717: 5487: 5460: 4922: 4908: 4809: 3393:) to prove modularity lifting theorems has been an influential development in 3259: 3125: 3084: 2111: 1903: 1892: 1673: 1669: 1583: 1467: 728: 686: 374: 362: 123: 10326: 10153: 9753: 9529: 8086:(1832). "Démonstration du théorème de Fermat pour le cas des 14 puissances". 7287:

Boletín de la Academia de Ciencias Físicas, Matemáticas y Naturales (Caracas)

5445: 5397: 5106:, but he did not call attention to its non-modularity. For more details, see 392:

in the 19th and 20th centuries. It is among the most notable theorems in the

10466: 8962: 8882: 8589: 7894: 7216: 7197:(1915). "Quelques formes quadratiques et quelques équations indéterminées". 5668: 5609: 5216: 4302:

right triangles with integer sides and an integer altitude to the hypotenuse

3327: 3214: 3007: 2526: 1971:

Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the

671: 10369:

Blog that covers the history of Fermat's Last Theorem from Fermat to Wiles.

6480:

Gambioli D (1901). "Memoria bibliographica sull'ultimo teorema di Fermat".

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such that their sum, and the sum of their squares, equal two given numbers

10309: 9879:. Graduate Texts in Mathematics. Vol. 50. New York: Springer-Verlag. 3891:

th roots are required to be real and positive, all solutions are given by

2877:

into primes, similar to integers. This gap was pointed out immediately by

2841:

outlined a proof of Fermat's Last Theorem based on factoring the equation

1486:. However, the proof by Andrew Wiles proves that any equation of the form 658:, two completely different areas of mathematics. Known at the time as the 609: 9877:

Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory

9383:

Richinick, Jennifer (July 2008). "The upside-down Pythagorean Theorem".

8799:

26 June – 2 July; A Year Later Fermat's Puzzle Is Still Not Quite Q.E.D.

6545: 5372:"Modularity of certain potentially Barsotti-Tate Galois representations" 10497: 10227:"A Study of Kummer's Proof of Fermat's Last Theorem for Regular Primes" 9935: 9361: 9292: 8954: 8874: 8630: 8475: 8401: 8322: 8069: 8018: 7752: 6683: 5709: 5601: 5353: 5329: 5271: 4080:

are coprime, then there are integer solutions if and only if 6 divides

2191: 2015:, as it was the last of Fermat's asserted theorems to remain unproved. 1661: 529: in this section. Unsourced material may be challenged and removed. 434: 286: 10391:

Ribet, Kenneth A. (1995). "Galois representations and modular forms".

8380:(1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". 7430:(1847). "Mémoire sur la résolution en nombres complexes de l'équation 6027:

Number Theory: An approach through history. From Hammurapi to Legendre

3691:. The conjecture states that the generalized Fermat equation has only 3304:, and he based his initial work and first significant breakthrough on 3220:

Following Frey, Serre and Ribet's work, this was where matters stood:

678:, who proved all but one part known as the "epsilon conjecture" (see: 10405: 9072: 7567: 7550: 9353: 9284: 8946: 8866: 8467: 8061: 8010: 7722: 7069: 6675: 5593: 5345: 5262: 3611:

need not be equal, whereas Fermat's last theorem considers the case

2595: = 14 was published in 1832, before Lamé's 1839 proof for 2292:(1966), Grant and Perella (1999), Barbara (2007), and Dolan (2011). 2194:

solutions). In turn, this proves Fermat's Last Theorem for the case

471:

is an integer greater than 2. Although he claimed to have a general

365:

rather than a theorem. After 358 years of effort by mathematicians,

10519: 7897:(1896). "Über die Auflösbarkeit einiger unbestimmter Gleichungen". 7838: