Fermat's last theorem
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Fermat's last theorem (sometimes abbreviated as FLT and also called Fermat's great theorem) is one of the most famous theorems in the history of mathematics. It states that:
- There are no non-zero integers x, y, and z such that <math>x^n + y^n = z^n</math> where n is an integer greater than 2.
The 17th-century mathematician Pierre de Fermat wrote about this 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.
This statement is significant because all the other theorems proposed by Fermat were settled, either by proofs he supplied, by rigorous proofs found afterwards, or by counterexamples showing a proposed theorem to be false. The theorem was not the last that Fermat conjectured, but the last to be proved. The theorem is generally thought to be the mathematical result that has provoked the largest number of incorrect proofs, perhaps because it is easy to understand.
While the theorem itself has no known direct use (that is, it has not been used to prove any other theorem), it has been shown to be connected to many other topics in mathematics, and is not merely an unimportant mathematical curiosity. Moreover, the search for a proof has initiated research about many important mathematical topics.
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History
Fermat's comment in the Arithmetica
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 exists infinitely many of them, even excluding trivial solutions for which a, b and c 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.
Early history
The theorem needs only to be proven for n=4 and in the cases where n is an odd prime number.<ref>If <math>n>2</math> is not an odd prime number, nor 4, it can be either a power of two 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.</ref> 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. 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 is true for all primes smaller than 37; in addition, he showed that the theorem was true for all prime numbers below 100, except possibly 37, 59 and 67.
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>.
The proof
Using sophisticated tools from algebraic geometry (in particular elliptic curves and modular forms), Galois theory and Hecke algebras, the English mathematician Andrew Wiles, from Princeton University, with help from his former student Richard Taylor, devised a proof of Fermat's last theorem that was published in 1995 in the journal Annals of Mathematics. The proof assumes the existence of topoi, so as it stands, the theorem is not a theorem of ZFC alone.
In 1986, Ken Ribet had proved Gerhard Frey's epsilon conjecture that every counterexample <math>a^n+b^n=c^n</math> to Fermat's last theorem would yield an elliptic curve defined as:
which would provide a counterexample to the Taniyama-Shimura conjecture.
This latter conjecture proposes a deep connection between elliptic curves and modular forms.
Andrew Wiles and Richard Taylor were able to establish a special case of the Taniyama-Shimura conjecture sufficient to exclude such counterexamples arising from Fermat's last theorem.
The story of the proof is almost as remarkable as the mystery of the theorem itself. Wiles spent seven years working out nearly all the details by himself and with utter secrecy (except for a final review stage for which he enlisted the help of his Princeton colleague, Nick Katz). When he announced his proof over the course of three lectures delivered at Cambridge University on June 21-23 1993, he amazed his audience with the number of ideas and constructions used in his proof. Unfortunately, upon closer inspection a serious error was discovered: it seemed to lead to the breakdown of this original proof. Wiles and Taylor then spent about a year trying to revive the proof. In September 1994, they were able to resurrect the proof with some different, discarded techniques that Wiles had used in his earlier attempts.
Did Fermat really have a proof?
Many doubt Fermat's claim to have "a truly marvellous proof". Wiles' proof is about 200 pages long and beyond the understanding of most mathematicians today. It is quite possible that there is a proof that is both essentially shorter, and more elementary in its methods; initial proofs of major results are typically not the most direct. Math institutions still receive many papers, some say in the thousands, claiming to have found such a proof and these are often subject to media attention.
The methods used by Wiles were unknown when Fermat was writing, and many believe it is unlikely that Fermat managed to derive all the necessary mathematics to demonstrate a solution. In the words of Andrew Wiles, "it's impossible; this is a 20th century proof".<ref>"The Proof" Transcript, Nova.</ref> Alternatives are that there is a simpler proof that all other mathematicians up until this point have missed, or that Fermat was mistaken.
A plausible faulty proof that might have been accessible to Fermat has been suggested. It is based on the mistaken assumption that unique factorization works in all rings of integral elements of algebraic number fields. This is an acceptable explanation to many experts in number theory, on the grounds that subsequent mathematicians of stature working in the field followed the same path.
The fact that Fermat never published an attempted proof, or even publicly announced that he had one, does suggest that he may have had later thoughts, and simply neglected to cross out his private marginal note. In addition, later in his life, Fermat published a proof for the case
- <math>a^4 + b^4 = c^4</math>.
If he really had come up with a proof for the general theorem, it is perhaps less likely that he would have published a proof for a special case, unless this special case could be used to prove the general theorem. On the other hand, the academic conventions of his time were not those that applied from the middle of the eighteenth century, and this argument cannot be taken as definitive. Academic publishing was only then just starting to develop, and mathematicians commonly withheld mathematical techniques to maintain their superiority to other mathematicians. Fermat did not publish proofs for the vast majority of his theorems, including those theorems for which mathematical historians believe he actually had a proof.
Generalizations and similar equations
Many diophantine equations have a form similar to the equation of Fermat's last theorem.
There are infinitely many positive natural numbers <math>x</math>, <math>y</math>, and <math>z</math> such that <math>x^n + y^n = z^{n+1}</math> in which n is any natural number.
Fermat's last theorem 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".
Fermat's last theorem appears on the blackboard as a homework assignment in the classroom scene of the 2000 movie Bedazzled.
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. However, the equation is clearly incorrect as the sum of an even and an odd number is odd. In fact, the left hand sum evaluates to 2541210258614589176288669958142428526657, while the right hand side evaluates to 2541210259314801410819278649643651567616 — within a billionth of each other. A second 'counterexample' without the obvious flaw appeared in a later episode, "The Wizard of Evergreen Terrace": <math>3987^{12} + 4365^{12} = 4472^{12}</math>.
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. Joanne Sydney Lessner and Joshua Rosenblum wrote the book and lyrics to the show, and Rosenblum also composed the music; Mel Marvin directed. In the cast were Gilles Chiasson, Edwardyne Cowan, Mitchell Kantor, Jonathan Rabb, Chris Thompson, Christianne Tisdale, and Carrie Wilshusen. 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 Magazine of Fantasy and Science Fiction.
See also
- Euler's conjecture
- Fermat's little theorem
- Sophie Germain prime
- Wall-Sun-Sun prime
- Beal's conjecture
Notes
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External links and references
- Nova "The Proof" Transcript PBS Airdate: October 28, 1997
- 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.
- 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
Bibliography and further reading
- Singh, Simon (hardcover, 1998). Fermat's Enigma. Bantam Books. ISBN 0802713319 (previously published under the title Fermat's Last Theorem).
- Aczel, Amir D. (hardcover, 1996) Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. Four Walls Eight Windows. ISBN 1568580770.
- Bell, Eric T. (1961) The Last Problem. New-York: Simon and Schuster. ISBN 0883854511 (edition of 1998).
- Benson, Donald C. (paperback, 1999). The Moment of Proof: Mathematical Epiphanies. Oxford University Press. ISBN 0195139194.ar:مبرهنة فيرما الأخيرة
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