It might look simple, but producing a reliable proof proved to be anything but. Entranced by the three-hundred-year mystery, he first attempted to solve it as a teen. Then when I became a researcher, I decided that I should put the problem aside. It didn't seem that these techniques were really getting to the root of the problem. It took a s mathematical advance to bring the problem into the twentieth century. He saw that it meant if he could prove the conjecture, he could prove Fermat, while also doing work on a new problem. There was an error in his proof, which ultimately he managed to repair with the help of another mathematician.
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Travel With Us. At the Smithsonian Visit. 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. The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"  : Taniyama—Shimura—Weil conjecture , proposed around —which many mathematicians believed would be near to impossible to prove,  : and was linked in the s by Gerhard Frey , Jean-Pierre Serre and Ken Ribet to Fermat's equation.
By accomplishing a partial proof of this conjecture in , 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. Around , 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. It became a part of the Langlands programme , a list of important conjectures needing proof or disproof. Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.
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In , Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a 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. 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 by Ken Ribet.
Ribet's proof of the epsilon conjecture in 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.
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. Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January 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. By mid-May , Wiles felt able 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 at the Isaac Newton Institute for Mathematical Sciences.
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 ,  who alerted Wiles on 23 August 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.
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. 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.
Wiles states that on the morning of 19 September , 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.
The Last Theorem of Pierre de Fermat, In Elementary Way
On 24 October , 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 step in the main paper. The two papers were vetted and published as the entirety of the May 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, years after it was conjectured.
Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. There are several generalizations of the Fermat equation to more general equations that allow the exponent n to be a negative integer or rational, or to consider three different exponents. The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions a, b, c, m, n, k satisfying .
NOVA Online | The Proof | Solving Fermat: Andrew Wiles
The Beal conjecture , 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 a , b , c , m , n , k with a , b , and c being pairwise coprime and all of m , n , k being greater than 2. The Fermat—Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture. The statement is about the finiteness of the set of solutions because there are 10 known solutions. When we allow the exponent n to be the reciprocal of an integer, i. All primitive integer solutions i.
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. In particular, the abc conjecture in its most standard formulation implies Fermat's last theorem for n that are sufficiently large. In , and again in , the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem.
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 , roughly a century after the competition was begun.
Prior to Wiles's proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet 3 meters of correspondence. According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in schools, and often submitted by "people with a technical education but a failed career".
The equation appears to be correct if entered in a calculator with 10 significant figures. He concludes, "In our arrogance, we feel we are so advanced. And yet we cannot unravel a simple knot tied by a part-time French mathematician working alone without a computer.
Fermat's last theorem
From Wikipedia, the free encyclopedia. Theorem in number theory. For other theorems named after Pierre de Fermat, see Fermat's theorem. The edition of Diophantus 's Arithmetica includes Fermat's commentary, referred to as his "Last Theorem" Observatio Domini Petri de Fermat , posthumously published by his son.
Beal conjecture Effective abc conjecture Effective modified Szpiro conjecture Modularity theorem. Main article: Pythagorean triple. Main article: Diophantine equation. Main article: Proof of Fermat's Last Theorem for specific exponents.
Main article: Modularity theorem. Main articles: Frey curve and Ribet's theorem. Main article: Fermat's Last Theorem in fiction. Mathematics portal. For more details, see Hellegouarch, Yves Invitation to the Mathematics of Fermat-Wiles. Academic Press. The Guinness Book of World Records. Guinness Publishing Ltd. It meant that my childhood dream was now a respectable thing to work on.