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PlanetMath: Fermat's last theorem Fermat's last theorem was actually a conjecture and remained unproved for over 300 years. We cannot imagine how Fermat's last theorem could be proved without these advanced mathematical tools, which include group theory and Galois theory, the theory of modular forms, Riemannian topology, and the theory of elliptic equations. This is version 11 of Fermat's last theorem, born on 2002-01-26, modified 2006-10-28. planetmath.org /encyclopedia/FermatsLastTheorem.html   (589 words)

The Mathematics of Fermat's Last Theorem Theorem B is even harder still, and it is the theorem of which Andrew Wiles first claimed a proof in 1993, thus proving FLT as well. Theorem B certainly seems, to one unfamiliar with the territory, to be quite technical and abstruse. Theorem B and more general forms of the Taniyama-Shimura Conjecture can be viewed in yet another way to affirm that there is a very significant relationship between modular functions and elliptic curves. www.mbay.net /~cgd/flt/fltmain.htm   (2364 words)

Fermat's last theorem The year 1847 is of major significance in the study of Fermat's Last Theorem. In 1986 the connection was made between the Shimura-Taniyama- Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrücken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Fermat's_last_theorem.html   (2143 words)

PlanetMath: Fermat's last theorem (analytic form of) Then Fermat's last theorem is equivalent (by elementary means) to: "Fermat's last theorem (analytic form of)" is owned by whm22. This is version 5 of Fermat's last theorem (analytic form of), born on 2006-10-02, modified 2006-10-12. planetmath.org /encyclopedia/FermatsLastTheoremAnalyticFormOf.html   (109 words)

Fermat's Last Theorem: Leonhard Euler The purpose of this blog is to present the story behind Fermat's Last Theorem and Wiles' proof in a way accessible to the mathematical amateur. The next mathematician in the story of Fermat's Last Theorem is Leonhard Euler, whose name is pronounced "Oiler". It is used in the derivation of cyclotomic inteogers which are the basis for Ernst Kummer's proof for Fermat's Last Theorem. fermatslasttheorem.blogspot.com /2005/05/leonhard-euler.html   (826 words)

Fermat's Last Theorem - Uncyclopedia, the content-free encyclopedia   (Site not responding. Last check: 2007-10-30) Fermat's Last Theorem is the single most important theorem ever, but nobody knows why. Mathematical pragmatism (the philosophy that getting laid is a mathematician's first priority) has discouraged the proof of this theorem, and the only known proofs of this theorem have only been accomplished by the omnipotent or discovered by accident. Fermat's Last Theorem was finally proved to be conclusively true by Historian Andrew Wiles, who, while idly looking through a history book on a Sunday afternoon casually remarked "man, people have spent a shitload of time trying to prove this theorem". uncyclopedia.org /wiki/Fermat's_Last_Theorem   (606 words)

Fermat's Last Theorem It became to be known as Fermat's Last Theorem (FLT) not because it was his last piece of work, but because it is the last remaining statement in the post-humous list of Fermat's works that needed to be proven or independently verified. Andrew Wiles released two preprints, Modular elliptic curves and Fermat's Last Theorem, by Andrew Wiles, and Ring theoretic properties of certain Hecke algebras, by Richard Taylor and Andrew Wiles. Fermat claimed to have found a proof of the theorem at an early stage in his career. www.mcs.csuhayward.edu /~malek/Mathlinks/Lasttheorem.html   (1172 words)

The Prime Glossary: Fermat's last theorem This result is called his last theorem, because it was the last of his claims in the margins to be either proved or disproved. The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles Gerd Faltings summary of the proof (an Adobe Acrobat version) Wiles, "Modular elliptic curves and Fermat's last theorem," Ann. primes.utm.edu /glossary/page.php?sort=FermatsLastTheorem   (233 words)

NOVA Online | The Proof | Solving Fermat: Andrew Wiles I've read letters in the early 19th century which said that it was an embarrassment to mathematics that the Last Theorem had not been solved. You told nobody that you were embarking on a proof of Fermat's Last Theorem. My wife had heard of Fermat's Last Theorem, but at that time she had no idea of the romantic significance it had for mathematicians, that it had been such a thorn in our flesh for so many years. www.pbs.org /wgbh/nova/proof/wiles.html   (2546 words)

What is the Last Theorem?   (Site not responding. Last check: 2007-10-30) Pythagoras’ Theorem is not just a nice idea, or a notion that seems to work for most right-angled triangles. It is believed that the creation and proof of the Last Theorem happened in about 1637, but it was not until after Fermat’s death in 1665 that his marginal note came to light. The Last Theorem was a source of frustration, but it also had a lighter side. www.simonsingh.net /What_is_the_Theorem.html   (792 words)

UNC Charlotte Mathematics Department - What We Know About Fermat's Last Theorem Theorem 1 (Fermat's Last Theorem) There are no positive integers x, y, z, and n>2 such that x^n + y^n = z^n. Theorem 2 If E is a semistable elliptic curve defined over Q, then E is modular. The conjecture was motivated by a theorem, due to Mason that essentially says the ABC conjecture is true for polynomials. www.math.uncc.edu /flt.php   (3199 words)

[No title] The history of Fermat's Last Theorem is a tale of intrigue, rivalry, rich prizes, suicide and death, involving characters who became obsessed by Fermat's accidental challenge. 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. Wiles' proof of Fermat's Last Theorem relies on verifying a conjecture born in the 1950s, which in turn shows that there is a fundamental relationship between elliptic curves and modular forms. www.prometheus.demon.co.uk /01/01fermat.htm   (4279 words)

BBC - h2g2 - Fermat's Last Theorem Fermat's last theorem was originally not so much a theorem as a conjecture. Together with Lebesgue, she was able to prove the theorem for a total of ten of these classes. His interest in Fermat's Last Theorem was aroused when one evening he failed to commit suicide due to reading about the theorem (he started thinking about suicide because the girl he loved married another). www.bbc.co.uk /dna/h2g2/A521966   (1254 words)

Bluffer's Guide to Fermat's Last Theorem Andrew Wiles, "Modular elliptic curves and Fermat's last theorem," pp. Gerd Faltings, "The proof of Fermat's last theorem by R. Taylor and A. Wiles," Notices of the American Mathematical Society, 42 (7), July 1995, pp. The Mathematics of Fermat's Last Theorem (by Charles Daney) math.stanford.edu /~lekheng/flt/index.html   (552 words)

Fermat's Last Theorem Pythagoras, a Greek philosopher, around the 5th Century BC generalized the theorem which states that in a right triangle the area of the square of the hypotenuse is the sum of the areas of the squares of the other two sides. I would argue that Fermat's Last Theorem was an answer to possibly a wrong question, if the intent was to extend the Babylonian triples to higher dimensions. Cursory reading of a few books on the subject has not revealed any conjecture or a theorem on the "even" or "balanced" case, where the number of summands is equal to the power of z. www.public.iastate.edu /~kchoi/fermat.htm   (1628 words)

Fermat's Last Theorem - Wikipedia, the free encyclopedia Fermat's Last Theorem is one of the most famous theorems in the history of mathematics. Fermat's Last Theorem is a generalisation of this result to higher powers n, and states that no such solution exists when the exponent 2 is replaced by a larger integer. Fermat's Last Theorem and Taniyama-Shimura were now linked through the proof of the Epsilon conjecture; the truth of Taniyama-Shimura was shown to imply the truth of Fermat's Last Theorem. en.wikipedia.org /wiki/Fermat's_last_theorem   (2155 words)

Read This: Fermat's Last Theorem For Amateurs Paulo Ribenboim has written a book summarizing some of the results of the long attempt to prove Fermat's Last Theorem (we'll call it FLT from now on), focusing on results that were obtained entirely through elementary methods. In two pages before the first chapter, Ribenboim states The Problem, remarks it is sufficient to prove it for exponent 4 or an odd prime, and distinguishes between the traditional first case (solutions x, y, z not multiples of the odd prime p) and second case (p divides exactly one of x, y, and z). David Graves (dgraves@elmira.edu) is Associate Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in opera and history of astronomy as well as the usual run of mathematics courses. www.maa.org /reviews/fltamat.html   (1688 words)

Sophie Germain and FLT Notice that this is not quite Fermat's Last Theorem, but it does say that if there is a solution to FLT for that value of n, then one of the numbers must be divisible by n. The result now known as Sophie Germain's Theorem was presented in 1823 by Legendre in a paper to the French Academy of Sciences and included in a supplement to his second edition of The Theory of Numbers. Dickson, L.E. "On the last theorem of Fermat," Quart. www.agnesscott.edu /lriddle/women/germain-FLT/SGandFLT.htm   (2774 words)

Timeline of Fermat's Last Theorem Fermat proved his Last Theorem for n = 4, using the method called "infinite descent" to prove that there are no positive integers, a, b, and c such that a The theorem divides FLT into two cases: Case I for numbers that are not divisible by 5, and Case II for numbers that are. Ribet proved a theorem that establishes that if the Shimura-Taniyama conjecture was true, FLT necessarily follows as a direct consequence. www.public.iastate.edu /~kchoi/time.htm   (2119 words)

Fermat's Last Theorem: A Brief History His most famous theorem was one which he wrote in the margins of another proof and failed to prove as "the margin was too narrow". In 1994 Andrew Wiles amended and published his proof of Fermat's Last Theorem; an unsolvable challenge that had lasted over 350 years and had taken 3 years of Andrew Wiles' life to lay to rest. Fermat's Last Theorem has excited and frustrated mathematicians for centuries, and it was only modern, radical thinking which finally came to lay the Theorem to rest. people.bath.ac.uk /jbd21/Fermat.html   (883 words)

On Case 1 of Fermat's Last Theorem Fermat's Last Theorem asserts that the equation x^p + y^p = z^p has no solution in integers x,y,z and prime p. Proceding in this way, we find that Case 1 of Fermat's Last Theorem is implied by this simple congruence argument for the primes 3, 5, 11, 17, 23, 29, 41, 47, and 53, which includes all the primes of the form 3n-1 less than 59. (For comparison, see the general proof of Fermat's Last Theorem for Cubes, which includes the much more challenging Case 2, where xyz is allowed to be a multiple of 3.) We can obviously perform the same kind of factorization for other prime powers. www.mathpages.com /home/kmath367.htm   (1792 words)

Pierre de Fermat | Fermat's Last Theorem   (Site not responding. Last check: 2007-10-30) It is simple enough for most schoolchildren to understand, and yet its proof is so complicated that it took over 300 years to be found, and even now only a handful of the world's top mathematicians can fully understand it. What follows is an outline of the background to the theorem, and the events that have led to the discovery of a proof. By simply changing the '2' to a '3' in Pythagoras' Theorem, Fermat had gone from an equation with an infinite number of solutions to one with none (that he could find). www.adrianbailey.co.uk /fermat/flt.html   (577 words)

Fermat's Last Theorem   (Site not responding. Last check: 2007-10-30) Fermat's Last Theorem is a problem of immense difficulty, and yet it can be stated in a form that a schoolchild can understand. Attempts to prove the theorem have led to much excitement in the mathematical world and proving it was to become the most valuable prize in number theory. At last Wiles seemed to be making progress, but as he was working with something that was completely new to him, he decided he needed to confide in someone he could trust, who worked in this field. people.bath.ac.uk /ma1nlb/MATH0126.html   (1877 words)

Is Fermat's last theorem undecidable? In the past there was speculation Fermat's last theorem might be undecidable, and it was realised that this would mean that there couldn't be a counterexample (which would be a proof that it was false), and hence that it was true. If Fermat's last theorem is undecidable from the current axioms, then the obvious thing to do would be to add more axioms so that it could be proved. This would allow for a proof of the theorem directly from a collection of axioms, and one might hope that such a proof would be simpler to understand than that devised by Wiles. www.chronon.org /articles/fermat_undecidable.html   (844 words)

The final stages in proving Fermat's Last Theorem Fermat’s Last Theorem proposed that one would never find any solution if the degree ‘2’ in the above equation is replaced by any whole number greater than 2. Finally, this theorem was proved in 1995 by Andrew Wiles, a professor at the Princeton University. Thus, to prove Fermat’s Last Theorem is to prove that the Taniyama-Shimura conjecture is correct, i.e. library.thinkquest.org /28049/final_stages_in_proving_fermat.htm   (778 words)

Amazon.co.uk: Fermat's Last Theorem: Books: Simon Singh   (Site not responding. Last check: 2007-10-30) An 18th-century Frenchwoman made a major breakthrough in solving the riddle, but she had to attend maths lectures at the Ecole Polytechnique disguised as a man. This is the story of the puzzle that has confounded mathematicians since the 17th century. The solution of the Theorem is one of the most important mathematical developments of the 20th century. Fermat's last theorem states that for every n \in {3,4,...} there are no x,y,z \in N={1,2,...} such that x^n + y^n = z^n. www.amazon.co.uk /Fermats-Last-Theorem-Simon-Singh/dp/1857026691   (1444 words)

Horizon - Fermat's Last Theorem So Pythagoras's theorem, the clever thing about it is that it tells us when three numbers are the sides of a right-angle triangle. Everyone thought Fermat's last theorem was impossible, so Professor Coates encouraged Andrew to forget his childhood dream and work on more mainstream maths. There was this marvellous moment when we were coming close to a proof of Fermat's last theorem, the tension had built up and there was only one possible punchline. www.cs.wichita.edu /~chang/fermat.html   (4959 words)

Another proof for Fermat's last theorem In fact, it is a certain part of the conjecture which implies Fermat's last theorem, and this part was proved by Khare and his collaborator J.P. Wintenberger, and independently by the mathematician Dieulefait. Although the theorem is primarily a theorem about numbers, thus naturally belonging to the realm of number theory, to prove it Wiles had to wheel up heavy machinery from two other areas of maths: algebra and geometry. The Langlands philosophy was conceived by the mathematician Robert Langland in the 1960's and consists of a set of conjectures concerning the intimate relationship between number theory, geometry and algebra. plus.maths.org /latestnews/jan-apr05/serre/index.html   (811 words)

Fermat's Last Theorem One argument in favor of the existence of such a solution is that all other Fermat's theorems were shown to be true soon after their publication (hence the name Fermat's "Last" Theorem). It is hard to connect the Last Theorem to other parts of mathematics, which means that powerful mathematical ideas can't necessarily be applied to it. In fact, if one looks at the history of the theorem, one sees that the biggest advances in working toward a proof have arisen when some connection to other mathematics was found. users.forthnet.gr /ath/kimon/FLT.htm   (697 words)

Fermat's Last Theorem: Diophantine Equations and the Most Famous Problem in Mathematics In 1993, to the extreme shock of the mathematical community, Andrew Wiles of Princeton University announced that he had found a proof of Fermat's Last Theorem. This theorem was one of the most famous unsolved problems of mathematics; hence, Wiles has achieved certain immortality with his discovery. It is called the "last theorem" because it was the last problem posed by Pierre de Fermat (1601-1665) to remain unresolved after his death. www.wolfram.com /products/explorer/topics/fermat.html   (164 words)

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