Fermat's Last Theorem
From Wikipedia, the free encyclopedia
Fermat's Last Theorem is one of the most famous theorems in the history of mathematics. It states that:
- It is impossible to separate any power higher than the second into two like powers,
or, using more formal mathematical notation:
- If an integer <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>.
Despite how closely the problem is related to the Pythagorean theorem, which has infinite solutions and hundreds of proofs, Fermat's subtle variation is much more difficult to prove. Still, the problem itself is easily understood even by schoolchildren, making it all the more frustrating and generating perhaps more incorrect proofs than any other problem in the history of mathematics.
The 17th-century mathematician Pierre de Fermat wrote in 1637 in his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus: "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." (Original Latin: "Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.") However, no correct proof was found for 357 years, until it was finally proven using very deep methods by Andrew Wiles in 1995 (after a failed attempt a year before).
All the other theorems proposed by Fermat were proven, either in his own proofs or by other mathematicians, in the two centuries following their proposition. The theorem was not the last that Fermat conjectured, but the last to be proven.
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[edit] Fermat's Last Theorem from a comment in a margin
In problem II.8 of his Arithmetica, Diophantus asks how to split a given square number into two other squares (in modern notation, given a rational number <math>k</math>, find <math>u</math> and <math>v</math>, both rational, such that <math>k^2=u^2+v^2</math>), and shows how to solve the problem for <math>k=4</math>. Around 1640, Fermat wrote the following comment (in Latin) in the margin of this problem in his copy of the Arithmetica (version published in 1621 and translated from Greek into Latin by Claude Gaspard Bachet de Méziriac) :
| Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. | (It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.) |
In modern notation, this comment corresponds to the theorem mentioned above. Fermat's copy of the Arithmetica has not been found so far; however, around 1670, his son produced a new edition of the book augmented with comments made by his father, including the comment above which would be known later as Fermat's Last Theorem.
In the case <math>n=2</math>, it was already known by the ancient Chinese, Indians, Greeks and Babylonians that the Diophantine equation <math>a^2 + b^2 = c^2</math> (linked with the Pythagorean theorem) has integer solutions, such as (3,4,5) (<math>3^2 + 4^2 = 5^2</math>) or (5,12,13). These solutions are known as Pythagorean triples, and there exist infinitely many of them, even excluding trivial solutions for which <math>a</math>, <math>b</math> and <math>c</math> have a common divisor. Fermat's Last Theorem is a generalisation of this result to higher powers <math>n</math>, and states that no such solution exists when the exponent 2 is replaced by a larger integer.
[edit] Early attempts at proof and proofs of cases
The theorem needs only to be proven for <math>n=4</math> and prime numbers greater than 2. If <math>n>2</math> is not a prime number or 4, it can be either a power of 2 or not. In the first case the number 4 is a factor of <math>n</math>, otherwise there is an odd prime number among its factors. In any case let any such factor be <math>p</math>, and let <math>m</math> be <math>n/p</math>. Now we can express the equation as <math>(a^m)^p + (b^m)^p = (c^m)^p</math>. If we can prove the case with exponent <math>p</math>, exponent <math>n</math> is simply a subset of that case.
For various special exponents <math>n</math>, the theorem had been proven over the years, but the general case remained elusive. The first case proved was the case <math>n=4</math>, which was proved by Fermat himself using the method of infinite descent. Using a similar method, Euler proved the theorem for <math>n=3</math>. While his original method contained a flaw, it has been the basis of a lot of research about the theorem. Sophie Germain next adopted a novel approach to the problem which was far more general than previous strategies. Rather than proving that there were no solutions to a given value n, she demonstrated that if there was a solution, a certain condition would have to apply. This insight would later lead to the proof for Fermat's Last Theorem where <math>n=5</math>. The case <math>n=5</math> was proved by Dirichlet and Legendre in 1825 using a generalisation of Euler's proof for <math>n=3</math>. The proof for the next prime number, <math>n=7</math> was found 15 years later by Gabriel Lamé in 1839. Unfortunately, this demonstration was relatively long and unlikely to be generalized to higher numbers. From this point, the mathematicians started to demonstrate the theorem for classes of prime numbers, instead of individual numbers. In 1847, Kummer proved that the theorem was true for all regular primes, which includes all prime numbers below 100, except 2, 37, 59 and 67.
In 1977, Guy Terjanian proved that if p is an odd prime number, and the natural numbers x, y and z satisfy <math>x^{2p} + y^{2p} = z^{2p}</math>, then 2p must divide x or y.
In 1983 Gerd Faltings proved the Mordell conjecture, which implies that for any <math>n>2</math>, there are at most finitely many coprime integers <math>a</math>, <math>b</math> and <math>c</math> with <math>a^n+b^n=c^n</math>.
[edit] Proof
In the late 1960s, Yves Hellegouarch discovered a connection between elliptic curves and Fermat's Last Theorem. He used the connection to prove results about elliptic curves using results from Fermat's Last Theorem. This led Gerhard Frey to the idea that the Taniyama–Shimura theorem implied Fermat's Last Theorem. Taniyama-Shimura states that every elliptic curve can be parametrized by a rational map with integer coefficients using the classical modular curve; that is, all elliptic curves are also modular forms. Jean-Pierre Serre proposed the Epsilon conjecture and was proven by Ken Ribet in the summer of 1986. This theorem said that every counterexample <math>a^n+b^n=c^n</math> to Fermat's Last Theorem would yield an elliptic curve defined as <math>y^2 = x(x-a^n)(x+b^n)</math> which would not be modular and therefore provide a counterexample to the Taniyama–Shimura conjecture. 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. Image:Nick katz1.jpg Andrew Wiles, who had been fascinated by Fermat's Last Theorem since age ten and had experience with elliptic curves, immediately set out to prove Taniyama-Shimura, and therefore Fermat's Last theorem. Yet he did so in almost complete secrecy, working for a full seven years with minimal outside help, contrary to how most mathematics is done today. In 1993, Wiles announced his proof over the course of three lectures delivered at Isaac Newton Institute for Mathematical Sciences on June 21, 22, and 23 in 1993. He amazed his audience with the number of ideas and constructions used in his proof . Wiles had reviewed the proof with a Princeton colleague, Nick Katz, beforehand. Still, the proof turned out to contain a flaw, namely, an error in a critical portion of the paper which bounded the order of a particular group. After seven years of work, the proof was invalid. Wiles and his former student Richard Taylor spent about a year trying to revive the proof, under close scrutiny by the media and mathematical community. In September 1994, they were able to resurrect the proof with some different, discarded techniques that Wiles had used in his earlier attempts.
Andrew Wiles found that he could work with associated Galois representations. In the process he developed ideas from Barry Mazur on deformations of Galois representations. The proof uses the standard constructions of modern algebraic geometry, which involve the category of schemes. These constructions use axioms that go beyond Zermelo-Frankel set theory (ZFC), which has led to a myth that that Wiles' proof is not carried out in ZFC. In fact Wiles only uses a tiny piece of this machinery, only involving etale cohomology of schemes of finite type over prime fields, and this can be easily carried out in second order arithmetic, a much weaker theory than standard set theories. Everything in his proof can be done in second order arithmetic (and could probably be encoded in first order (Peano) arithmetic, though this would require considerable effort). [citation needed]
[edit] Generalisations and similar equations
Many Diophantine equations have a form similar to the equation of Fermat's last theorem.
There are infinitely many positive integers <math>x</math>, <math>y</math>, and <math>z</math> such that <math>x^n + y^n = z^{m}</math> in which <math>n</math> and <math>m</math> are any relatively prime natural numbers.
[edit] In fiction
In "The Royale", an episode of Star Trek: The Next Generation, Captain Picard states that the theorem had gone unsolved for 800 years. Wiles' proof was released five years after the particular episode aired. This was subsequently mentioned in a Star Trek: Deep Space Nine episode called "Facets" during June 1995 in which Jadzia Dax comments that one of her previous hosts, Tobin Dax, had "the most original approach to the proof since Wiles over 300 years ago." [1] This reference was generally understood by fans to be a subtle correction for "The Royale".
A sum, proved impossible by the theorem, appears in an episode of The Simpsons, "Treehouse of Horror VI". In the three-dimensional world in "Homer3", the equation <math>1782^{12} + 1841^{12} = 1922^{12}</math> is visible, just as the dimension begins to collapse. The joke is that the twelfth root of the sum does evaluate to 1922 due to rounding errors when entered into most handheld calculators. In fact, the left hand sum evaluates to 2,541,210,258,614,589,176,288,669,958,142,428,526,657, while the right hand side evaluates to 2,541,210,259,314,801,410,819,278,649,643,651,567,616 — within a billionth of each other but still out by 700,212,234,530,608,691,501,223,040,959 (Seven hundred octillion). A second 'counterexample' appeared in a later episode, "The Wizard of Evergreen Terrace": <math>3987^{12} + 4365^{12} = 4472^{12}</math>. However, in this case, both 3987 and 4365 are divisible by 3 (and 9, for that matter), so that the entire left-hand side must similarly be divisible by 3; this is not true of 4472, and therefore not of the right-hand side.
The solving of Fermat's last theorem was also the subject of an Off-Broadway musical titled Fermat's Last Tango that opened at the York Theatre at St. Peter's Church on December 6, 2000 and closed on December 31. The show stuck closely to the historical details of the Theorem and its proof, though the names of both Wiles and his wife were changed (to Daniel and Anna Keane).
In Tom Stoppard's play Arcadia, Septimus Hodge poses the problem of proving Fermat's Last Theorem to the precocious Thomasina Coverly (who is perhaps a mathematical prodigy), in an attempt to keep her busy. Thomasina's (perhaps perceptive) response is simple — that Fermat had no proof, and it was a joke to drive posterity mad.
Arthur Porges' short story, "The Devil and Simon Flagg", features a mathematician who bargains with the Devil that the latter cannot produce a proof of Fermat's Last Theorem within twenty-four hours. The story was first published in 1954 in The Magazine of Fantasy and Science Fiction.
Fermat's Last Theorem also appeared in the movie "Bedazzled" with Elizabeth Hurley and Brendan Fraser. Hurley played the devil who, in one of her many forms, appeared as a school teacher. In this particular scene, the blackboard behind her reads, "Tonight's homework: Prove <math>a^n + b^n = c^n</math>, which is Fermat's Last Theorem in its most general form."
In one of the Rama series books the problem is supposed to have been solved very simply and elegantly (probably the way Fermat himself had intended it) by a young girl.
In Elizabeth Kay's book "Jinx on the Divide" the main character intrigues a mythological griffin with the theorem; the griffin solves it in less than a week.
In the online game the Lost Experience which is directly related to the television series Lost the equation is said to have been originally solved by a scientist by the name of Enzo Vallenzetti (also the creator of the Vallenzetti Equation) sometime in the late 60's. However due to his eccentric nature, after having the proof verified by his colleagues, Vallenzetti is said to have burned his work so that, according to his assistant, "others could have as much fun solving it as he did".
[edit] See also
- Euler's conjecture
- Fermat's little theorem
- Sophie Germain prime
- Wall-Sun-Sun prime
- Beal's conjecture
[edit] External links and references
- Wiles, Andrew (1995). Modular elliptic curves and Fermat's Last Theorem, Annals of Mathematics (141) (3), 443-551 (alternative link - replete with photos).
- Taylor, Richard & Wiles, Andrew (1995). Ring theoretic properties of certain Hecke algebras, Annals of Mathematics (141) (3), 553-572.
- Ribet, Ken (1995). Galois representations and modular forms- discusses various material which is related to the proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama-Shimura
- Faltings, Gerd (1995). The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles, Notices of the AMS (42) (7), 743-746.
- Daney, Charles (2003). The Mathematics of Fermat's Last Theorem. Retrieved Aug. 5, 2004.
- O'Connor, J. J. & and Robertson, E. F. (1996). Fermat's Last theorem. The history of the problem. Retrieved Aug. 5, 2004.
- Shay, David (2003). Fermat's Last theorem. The story, the history and the mystery. Retrieved Aug. 5, 2004.
- Freeman, Larry (2005). Fermat's Last Theorem Blog. A blog that covers the history of Fermat's Last Theorem from Pierre Fermat to Andrew Wiles.
- Kisby, Adam William (2004). Fermat's Last Theorem Revisited: A Marginal Proof in Ten Steps. Parody.
- The bluffer's guide to Fermat's Last Theorem
- G. Terjanian (1977), Sur l'équation <math>x^{2p} + y^{2p} = z^{2p}</math>, Comptes rendus hebdomadaires des séances de l'Académie des sciences. Série A et B, vol. 285, pp. 973–975.
- Noam D. Elkies, Tables of Fermat "near-misses" - approximate solutions of xn + yn = zn
- Weisstein, Eric W., Fermat's Last Theorem at MathWorld.
- Fermat's Last Theorem, On Sophie Germain.
[edit] Bibliography and further reading
- Singh, Simon (hardcover, 1998). Fermat's Enigma. Bantam Books. ISBN 0-8027-1331-9 (previously published under the title Fermat's Last Theorem).
- Amir Aczel (hardcover, 1996) Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. Four Walls Eight Windows. ISBN 1-56858-077-0.
- Mozzochi, Charles (2000). The Fermat Diary. ISBN 0-8218-2670-0.
- Bell, Eric T. (1961) The Last Problem. New-York: Simon and Schuster. ISBN 0-88385-451-1 (edition of 1998).
- Benson, Donald C. (paperback, 1999). The Moment of Proof: Mathematical Epiphanies. Oxford University Press. ISBN 0-19-513919-4.
- H. M. Edwards (1977). Fermat's Last Theorem. Springer-Verlag. ISBN 0-387-90230-9.ar:مبرهنة فيرما الأخيرة
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