The Many Worlds Theory Today Fermat was a 17th-century mathematician who wrote a note in the margin of his book stating a particular proposition and claiming to have proved it. He claimed that there were none. So Fermat said because he could not find any solutions to this equation, then there were no solutions? He did more than that. They really want to know that there are no solutions up to infinity. And to do that we need a proof.

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In —, Gerhard Frey called attention to the unusual properties of this same curve, now called a Frey curve. Mathematically, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrise coordinates x and y of the points on it.

In , Jean-Pierre Serre provided a partial proof that a Frey curve could not be modular. However his partial proof came close to confirming the link between Fermat and Taniyama. His article was published in However, despite the progress made by Serre and Ribet, this approach to Fermat was widely considered unusable as well, since almost all mathematicians saw the Taniyama—Shimura—Weil conjecture itself as completely inaccessible to proof with current knowledge.

Ribet later commented 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]. It was finally accepted as correct, and published, in , following the correction of a subtle error in one part of his original paper. In the course of his review, he asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap.

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.

Instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve. He states that he was having a final look to try and understand the fundamental 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, 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 since then.

Each was inadequate by itself, but fixing one approach with tools from the other would resolve the issue and produce a class number formula CNF valid for all cases that were not already proven by his refereed paper: [13] [17] [13] "I was sitting at my desk examining the Kolyvagin—Flach method.

Suddenly I had this incredible revelation. 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. It was still there. It was the most important moment of my working life. Nothing I ever do again will mean as much. The two papers were vetted and finally published as the entirety of the May issue of the Annals of Mathematics.

The new proof was widely analysed, and became accepted as likely correct in its major components. Subsequent developments[ edit ] Fermat claimed to " The specific problem is: Newly added section: review required for technical accuracy. WikiProject Mathematics may be able to help recruit an expert.

June Wiles used proof by contradiction , in which one assumes the opposite of what is to be proved, and shows if that were true, it would create a contradiction. The contradiction shows that the assumption must have been incorrect. The proof falls roughly in two parts. In the first part, Wiles proves a general result about " lifts ", known as the "modularity lifting theorem".

This first part allows him to prove results about elliptic curves by converting them to problems about Galois representations of elliptic curves. He then uses this result to prove that all semi-stable curves are modular, by proving that the Galois representations of these curves are modular.

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## Why the Proof of Fermat’s Last Theorem Doesn’t Need to Be Enhanced

Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. The details and auxiliary arguments, however, were often ad hoc and tied to the individual exponent under consideration. Although she developed many techniques for establishing the non-consecutivity condition, she did not succeed in her strategic goal. 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.

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## Andrew Wiles on Solving Fermat

The methods introduced by Wiles and Taylor are now part of the toolkit of number theorists, who consider the FLT story closed. But number theorists were not the only ones electrified by this story. I was reminded of this unexpectedly in when, in the space of a few days, two logicians, speaking on two continents, alluded to ways of enhancing the proof of FLT — and reported how surprised some of their colleagues were that number theorists showed no interest in their ideas. The logicians spoke the languages of their respective specialties — set theory and theoretical computer science — in expressing these ideas. Over the last few centuries, mathematicians repeatedly tried to explain this contrast, failing each time but leaving entire branches of mathematics in their wake.

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## Andrew Wiles

In recognition he was awarded a special silver plaque—he was beyond the traditional age limit of 40 years for receiving the gold Fields Medal —by the International Mathematical Union in Wiles was educated at Merton College, Oxford B. Following a junior research fellowship at Cambridge —80 , Wiles held an appointment at Harvard University , Cambridge, Massachusetts, and in he moved to Princeton New Jersey University , where he became professor emeritus in Wiles subsequently joined the faculty at Oxford. Wiles worked on a number of outstanding problems in number theory: the Birch and Swinnerton-Dyer conjectures, the principal conjecture of Iwasawa theory, and the Shimura-Taniyama-Weil conjecture. In the 17th century Fermat had claimed a solution to this problem, posed 14 centuries earlier by Diophantus, but he gave no proof , claiming insufficient room in the margin. Many mathematicians had tried to solve it over the intervening centuries, but with no success.

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