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Fermat's Last Theorem - Wikipedia, the Free Encyclopedia 17/06/13 5:22 PM Fermat's Last Theorem from Wikipedia, the Free Encyclopedia Fermat's Last Theorem - Wikipedia, the free encyclopedia 17/06/13 5:22 PM Fermat's Last Theorem From Wikipedia, the free encyclopedia In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most famous theorems in the history of mathematics and prior to its 1995 proof was in the Guinness Book of World Records for "most difficult mathematical problems". Contents 1 Overview 1.1 The problem 1.2 Subsequent developments and The 1670 edition of Diophantus' Arithmetica includes Fermat's solution commentary, particularly his "Last Theorem" (Observatio Domini 2 Mathematical History Petri de Fermat). 2.1 Pythagoras and Diophantus 2.1.1 Pythagorean triples 2.1.2 Diophantine equations 2.2 Fermat's conjecture 2.3 Proofs for specific http://en.wikipedia.org/wiki/Fermat's_Last_Theorem Page 1 of 20 Fermat's Last Theorem - Wikipedia, the free encyclopedia 17/06/13 5:22 PM exponents 2.3.1 Sophie Germain 2.3.2 Ernst Kummer and the theory of ideals 2.3.3 Mordell conjecture 2.3.4 Computational studies 2.4 Connection with elliptic curves 2.4.1 The Taniyama– Shimura-Weil conjecture 2.4.2 Frey's equation / Ribet's theorem 2.4.3 Wiles's general proof 2.5 Subsequent developments 3 Exponents other than positive integers 3.1 Rational exponents 3.2 Negative exponents 3.2.1 n = –1 3.2.2 n = –2 3.2.3 Integer n < –2 4 Did Fermat possess a general proof? 5 Monetary prizes 6 In popular culture 7 See also 8 Footnotes 9 References 10 Bibliography 11 Further reading 12 External links Overview The problem Fermat's Last Theorem (known by this title historically although technically a conjecture–or unproven speculation–until proven in 1995) stood as an unsolved riddle in mathematics for over three centuries. The http://en.wikipedia.org/wiki/Fermat's_Last_Theorem Page 2 of 20 Fermat's Last Theorem - Wikipedia, the free encyclopedia 17/06/13 5:22 PM Theorem itself is a deceptively simple statement within mathematics that Fermat famously stated he had solved around 1637. His claim was discovered some 30 years later, after his death, as a bare statement in the margin of a book, but Fermat died without leaving any proof of his claim. The claim eventually became one of the most famous unsolved problems of mathematics. The attempts made to prove it during that time prompted substantial development of number theory and over time Fermat's Last Theorem itself gained legendary prominence as an unsolved problem in popular mathematics. It is based upon the well known formula ("Pythagoras' Theorem") for a right-angle triangle discovered by the ancient Greek mathematician Pythagoras: a2 + b2 = c2. As it stands, the equation has an infinite number of whole-number solutions, representing the sides of a right-angle triangle. Fermat claimed that he had a proof that this theorem had no natural-number (or "integer") solutions for any integer exponent more than 2 — in other words that although a2 + b2 = c2 had an infinity of whole-number solutions, the similar equations a3 + b3 = c3 a4 + b4 = c4 an + bn = cn for any other integer exponent "n" more than 2, would have no whole-number solutions. Fermat left no proof of the conjecture for all n apart from the special case n = 4. Subsequent developments and solution With the special case n = 4 proven, the problem was to prove the theorem for exponents n that are prime numbers (this limitation is considered trivial to prove: if n were not prime then it would be possible to write n = P*Q where P is a prime number, and then an = aP*Q = (aQ)P for each of a, b, and c, i.e. an equivalent solution would have to exist for a-smaller-prime power). Over the next two centuries (1637–1839), the conjecture was proven for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach which was relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, 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 to be either impossible, or at best exceedingly difficult, or not achievable with current knowledge). The proof of Fermat's Last Theorem in full, for all n, was finally accomplished, however, after 358 years, by Andrew Wiles in 1995, an achievement for which he was honoured and received numerous awards. The solution came in a roundabout manner, from a completely different area of mathematics. Around 1955 Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture, and (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 equation) widely considered to be completely inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between the modularity theorem and Fermat's Last Theorem. This potential link was confirmed two years later by Ken Ribet (see: Ribet's Theorem and Frey curve). http://en.wikipedia.org/wiki/Fermat's_Last_Theorem Page 3 of 20 Fermat's Last Theorem - Wikipedia, the free encyclopedia 17/06/13 5:22 PM English mathematician Andrew Wiles, who had a childhood fascination with Fermat's Last Theorem, decided on hearing this, to try and prove the modularity theorem, as a way to prove Fermat's Last Theorem. In 1993, after six years working secretly on the problem, Wiles succeeded in proving enough of the modularity theorem to prove Fermat's Last Theorem. Wiles paper was massive in size and scope. A flaw was discovered in one part of his original paper during peer review and required a further year and collaboration with a past student, Richard Taylor to resolve; as a result the final proof in 1995 was accompanied by a second, smaller, joint paper to that effect. Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the modularity theorem were subsequently proven by other mathematicians, building on Wiles' work, between 1996 and 2001. Mathematical History Pythagoras and Diophantus Pythagorean triples Main article: Pythagorean triple A Pythagorean triple - named for the ancient Greek Pythagoras - is a set of three integers (a, b, c) that satisfy a special case of Fermat's equation (n = 2)[1] Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples,[2] and methods for generating such triples have been studied in many cultures, beginning with the Babylonians[3] and later ancient Greek, Chinese, and Indian mathematicians.[4] The traditional interest in Pythagorean triples connects with the Pythagorean theorem;[5] in its converse form, it states that a triangle with sides of lengths a, b, and c has a right angle between the a and b legs when the numbers are a Pythagorean triple. Right angles have various practical applications, such as surveying, carpentry, masonry, and construction. Fermat's Last Theorem is an extension of this problem to higher powers, stating that no solution exists when the exponent 2 is replaced by any larger integer. Diophantine equations Main article: Diophantine equation Fermat's equation, xn + yn = zn with positive integer solutions, is an example of a Diophantine equation,[6] named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equation. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively: Diophantus's major work is the Arithmetica, of which only a portion has survived.[7] Fermat's conjecture of http://en.wikipedia.org/wiki/Fermat's_Last_Theorem Page 4 of 20 Fermat's Last Theorem - Wikipedia, the free encyclopedia 17/06/13 5:22 PM his Last Theorem was inspired while reading a new edition of the Arithmetica,[8] which was translated into Latin and published in 1621 by Claude Bachet.[9] Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC).[10] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).[11] Many Diophantine equations have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no cross terms mixing two letters, without sharing its particular properties.
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