modular elliptic curves fermats last di taylor richard wiles andrew (2 risultati)

First edition. THE PROOF OF FERMAT'S LAST THEOREM. First edition, journal issue, of his proof of Fermat's Last Theorem, which was perhaps the most celebrated open problem in mathematics. In a marginal note in the section of his copy of Diophantus' Arithmetica (1621) dealing with Pythagorean triples (positive whole numbers x, y, z satisfying x2 + y2 = z2 of which an infinite number exist), Fermat stated that the equation xn + yn = zn, where n is any whole number greater than 2, has no solution in which x, y, z are positive whole numbers. Fermat followed this assertion with what is probably the most tantalising comment in the history of mathematics: 'I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.' Fermat believed he could prove his theorem, but he never committed his proof to paper. After his death, mathematicians across Europe, from the enthusiastic amateur to the brilliant professional, tried to rediscover the proof of what became known as Fermat's Last Theorem, but for more than 350 years none succeeded, nor could anyone disprove the theorem by finding numbers x, y, z which did satisfy Fermat's equation. When the great German mathematician David Hilbert was asked why he never attempted a proof of Fermat's Last Theorem, he replied, "Before beginning I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure." Soon after the Second World War computers helped to prove the theorem for all values of n up to five hundred, then one thousand, and then ten thousand. In the 1980's Samuel S. Wagstaff of the University of Illinois raised the limit to 25,000 and more recently mathematicians could claim that Fermat's Last Theorem was true for all values of n up to four million. But no general proof was found until 1995. "Between 1954 and 1986 a chain of events of occurred which brought Fermat's Last Theorem back into the mainstream. The incident which began everything happened in post-war Japan, when Yutaka Taniyama and Goro Shimura, two young academics, decided to collaborate on the study of elliptic curves and modular forms. These entities are from opposite ends of the mathematical spectrum, and had previously been studied in isolation. "Elliptic curves, which have been studied since the time of Diophantus, concern cubic equations of the form: y2 = (x + a).(x + b).(x + c), where a, b and c can be any whole number, except zero. The challenge is to identify and quantify the whole solutions to the equations, the solutions differing according to the values of a, b, and c. "Modular forms are a much more modern mathematical entity, born in the nineteenth century. They are functions, not so different to functions such as sine and cosine, but modular forms are exceptional because they exhibit a high degree of symmetry. For example, the sine function is slightly symmetrical because 2? can be added to any number, x, and yet the result of the function remains unchanged, i.e., sin x = sin (x + 2?). However, for modular forms the number x can be transformed in an infinite number of ways and yet the outcome of the function remains unchanged, hence they are said to be extraordinarily symmetric "Despite belonging to a completely different area of the mathematics, Shimura and Taniyama began to suspect that the elliptic curves might be related to modular forms in a fundamental way. It seemed that the solutions for any one of the infinite number of elliptic curves could be derived from one of the infinite number of modular forms. Each elliptic curve seemed to be a modular form in disguise. This apparent unification became known as the Shimura-Taniyama conjecture, reflecting the fact that mathematicians were confident that it was true, but as yet were unable to prove it. The conjecture was considered important because if it were true problems about elliptic curves, which hitherto had been insoluble, could potentially be solved by using techniques developed for modular forms, and vice versa "Even though the Shimura-Taniyama conjecture could not be proved, as the decades passed it gradually became increasingly influential, and by the 1970s mathematicians would begin papers by assuming the Shimura-Taniyama conjecture and then derive some new result. In due course many major results came to rely on the conjecture being proved, but these results could themselves only be classified as conjectures, because they were conditional on the proof of the Shimura-Taniyama conjecture. Despite its pivotal role, few believed it would be proved this century. "Then, in 1986, Kenneth A Ribet of the University of California at Berkeley, building on the work of Gerhard Frey of the University of Saarlands, made an astonishing breakthrough. He was unable to prove the Shimura-Taniyama conjecture, but he was able to link it with Fermat's Last Theorem. The link occurred by contemplating the unthinkable what would happen if Fermat's Last Theorem was not true? This would mean that there existed a set of solutions to Fermat's equation, and therefore this hypothetical combination of numbers could be used as the basis for constructing a hypothetical elliptic curve. Ribet demonstrated that this elliptic curve could not possibly be related to a modular form, and as such it would defy the Shimura-Taniyama conjecture. Running the argument backwards, if somebody could prove the Shimura-Taniyama conjecture then every elliptic curve must be related to a modular form, hence any solution to Fermat's equation is forbidden to exist, and hence Fermat's Theorem must be true. If somebody could prove the Shimura-Taniyama conjecture, then this would immediately imply the proof of Fermat's Last Theorem. By proving one of the most important conjectures of the twentieth century, mathematicians could solve a riddle from the seventeenth century. "The Shimura-Taniyama conjecture had remained unproven since the 1950s and so there was little optimism that it was a r.

First Edition. "WILES, Andrew. Modular Elliptic Curves and Fermat's Last Theorem. WITH: WILES, Andrew and TAYLOR, Richard. Ring-Theoretic Properties of Certain Hecke Algebras. IN: Annals of Mathematics, Second Series, Vol. 141, No. 3, pp. 443-572. (Princeton: Princeton University Press), 1995. Octavo, original printed paper wrappers. Housed in a custom three-quarter morocco clamshell box. $8800.First edition, in the journal Annals of Mathematics where it originally appeared, of Wiles' famous proof of Fermat's Last Theorem, which had confounded mathematicians for centuries.In a marginal note in the section of his copy of Diophantus' Arithmetica (1621) dealing with Pythagorean triples (positive whole numbers x, y, z satisfying x^2 + y^2 = z^2 of which an infinite number exist), Fermat stated that the equation x^n + y^n = z^n, where n is any whole number greater than 2, has no solution in which x, y, z are positive whole numbers. Tantalizingly, Fermat wrote that he had found a wonderful proof but the margin of the book was too small to contain it. For 350 years, no mathematician succeeded in finding a proofthough many tried. Soon after the Second World War computers helped to prove the theorem for all values of n up to 500, then 1000, and then 10,000. In the 1980's Samuel S. Wagstaff of the University of Illinois raised the limit to 25,000 and more recently mathematicians could claim that Fermat's Last Theorem was true for all values of n up to 4,000,000. But no general proof was found until Andrew Wiles announced his proof, using the most advanced tools of modern mathematics, at a 1993 conference at the Isaac Newton Institute in Cambridge, England. (His former student Richard Taylor then helped him fix a flaw discovered in his original proof; both papers are present in this single journal issue.)Not wanting to be distractedor beaten to the punchWiles worked on his proof in secrecy for seven years. "To prove that something is true for an infinite number of cases required Wiles to pull together some of the most recent breakthroughs in number theory, and in addition invent new techniques of his own At each stage Wiles could never be sure that he could complete his proof. He realized that even if he did have the correct strategy, the mathematical techniques required might not yet existhe might be on the right track, but living in the wrong century. Eventually, in 1993, Wiles felt confident that his proof was reaching completion. The opportunity arose to announce his proof of a major section of the Shimura-Taniyama conjecture, and hence Fermat's Last Theorem, at a special conference to be held at the Isaac Newton Institute in Cambridge, England. Because this was his home town, where he had encountered the Last Theorem as a child, he decided to make a concerted effort to complete the proof in time for the conference. On June 23rd he announced his seven-year calculation to a stunned audience. His secret research program had apparently been a success, and the mathematical community and the world's press rejoiced. The front page of the New York Times exclaimed 'At Last, Shout of "Eureka!" in Age-Old Math Mystery,' and Wiles appeared on television stations around the world" (Simon Singh, Fermat's Enigma).When peer review revealed a fatal flaw in Wiles' initial proof, he demanded the opportunity to correct the problem himself. After months of frustration, he took his former student Richard Taylor into his confidence. Returning to an approach Wiles had discarded early in his process, Wiles and Taylor discovered the solution, and his proof was indeed confirmed by the mathematical community shortly thereafter. "I haven't let go of this problem for nearly seven years," Wiles told the New York Times in 1993. "I've almost forgotten the experience of getting up and thinking about something else For many of us, [Fermat's] problem drew us in and we always considered it something you dream about but never actually do There is a sense of loss, actually." F.