Difference between revisions of "Fermat's Last Theorem"
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[[Image:Pierre de Fermat.jpg|thumb|right|Pierre de Fermat]] | [[Image:Pierre de Fermat.jpg|thumb|right|Pierre de Fermat]] | ||
| − | Fermat's Last Theorem was conjectured by the French mathematician [[Pierre de Fermat]]. He said he had proved this problem but that there was not enough room in the margin to state his proof. | + | Fermat's Last Theorem was conjectured by the French mathematician [[Pierre de Fermat]]. He said he had proved this problem but that there was not enough room in the margin to state his proof:<ref>Nagell 1951, p. 252.</ref> |
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| + | :"Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, 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." | ||
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| + | The English translation of Fermat's Latin statement is:<ref>http://mathworld.wolfram.com/FermatsLastTheorem.html</ref> | ||
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| + | :"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." | ||
This problem has confounded mathematicians for centuries, and there still is no proof for it using [[elementary techniques]]. Most doubt that Fermat was able to prove it himself. | This problem has confounded mathematicians for centuries, and there still is no proof for it using [[elementary techniques]]. Most doubt that Fermat was able to prove it himself. | ||
Revision as of 16:50, August 26, 2007
Fermat's Last Theorem was conjectured by the French mathematician Pierre de Fermat. He said he had proved this problem but that there was not enough room in the margin to state his proof:[1]
- "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, 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."
The English translation of Fermat's Latin statement is:[2]
- "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
This problem has confounded mathematicians for centuries, and there still is no proof for it using elementary techniques. Most doubt that Fermat was able to prove it himself.
The theorem is as follows:
For integers n > 2, there are no nonzero integral solutions to: xn + yn = zn
In the summer of 1986, Ken Ribet proved that Fermat's Last Theorem is a special case of the Taniyama–Shimura Conjecture.
In a series of lectures in 1993, mathematician Andrew Wiles announced a proof using techniques in algebraic geometry, relying on the disfavored Axiom of Choice.[3] A flaw was found before publication. Wiles spent a year trying to fix the flaw, and in September 1994, he and Richard Taylor announced a new version of the proof that is not widely understood but is no longer criticized by university-based mathematicians. However, criticism does continue on the internet.[4]