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 is greater than 2, then has no solutions in non-zero integers , , and .
Despite how closely the problem is related to the
Pythagorean theorem, which has infinite solutions and hundreds of proofs, Fermat's subtle variation proved to be a lot harder 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, an ancient Greek [i] text on mathematics [i] written by the Hellenistic ...
of
Diophantus: "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." However, no correct proof was found for 357 years, until it was finally proven using very
deep methods by
Andrew Wiles in 1995 .
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 proved.
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 , and shows how to solve the problem for . Around 1640, Fermat wrote the following comment in the margin of this problem in his copy of the
Arithmetica :
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 , it was already known by the ancient Chinese, Indians, Greeks and Babylonians that the Diophantine equation has integer solutions, such as or . These solutions are known as
Pythagorean triples, and there exist infinitely many of them, even excluding trivial solutions for which , and have a common divisor. Fermat's Last Theorem is a generalisation of this result to higher powers , and states that no such solution exists when the exponent 2 is replaced by a larger integer.
Early attempts at proof and proofs of cases
The theorem needs only to be proven for or a higher prime number. If 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 , otherwise there is an odd prime number among its factors. In any case let any such factor be , and let be . Now we can express the equation as . If we can prove the case with exponent , exponent is simply a subset of that case.
For various special exponents , the theorem had been proven over the years, but the general case remained elusive. The first case proved was the case , which was proved by Fermat himself using the method of infinite descent. Using a similar method,
Euler proved the theorem for . While his original method contained a flaw, it has been the basis of a lot of research about the theorem. The case was proved by
Dirichlet and
Legendre in 1825 using a generalisation of Euler's proof for . The proof for the next prime number, 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 1983
Gerd Faltings proved the Mordell conjecture, which implies that for any , there are at most finitely many
coprime integers , and with .
Proof
In the late 1960s, Yves Hellegouarch discovered a connection between
elliptic curves and Fermat's Last Theorem and used it 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 to Fermat's Last Theorem would yield an elliptic curve defined as 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 Epsilon conjecture; either both were true or both were false.
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 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. The main problem that Wiles had to overcome was to establish a correspondence between semistable elliptic curves over the rational field, and the modular semistable elliptic curves over the rationals, which he did by explicitly showing that there were equal numbers of each. Before Wiles' work on the problem, there had been many attempts to count elliptic curves, but no one had found a way to do it.
Andrew Wiles found that he could count 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 are generally defined within NBG set theory, which is a conservative extension of ZFC set theory wherein all of the theorems about sets are the same. NBG set theory is generally considered to be in essence the same as ZFC set theory, though it can be replaced by ZFC plus an axiom stating that there is a strongly inaccessible cardinal, allowing the construction of a Grothendieck universe. Hence there is some question as to how strong the axioms really need to be to make the proof work; it seems likely that in fact something weaker than ZFC would suffice.
Generalizations and similar equations
Many diophantine equations have a form similar to the equation of Fermat's last theorem.
There are infinitely many positive integers , , and such that in which and are any
relatively prime natural numbers.
In fiction
In "The Royale", an episode of
,
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
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." 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 is an Emmy [i] and Peabody [i]-winning American [i] animated [i] ...
, "
Treehouse of Horror VI". In the three-dimensional world in "Homer
3", the equation 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 . A second 'counterexample' appeared in a later episode, "
The Wizard of Evergreen Terrace": . However, in this case, both 3987 and 4365 are divisible by 3 , 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 .
In
Tom Stoppard's play
Arcadia, Septimus Hodge poses the problem of proving Fermat's Last Theorem to the precocious Thomasina Coverly , in an attempt to keep her busy. Thomasina's 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 is a digest size [i] American [i] fantasy fiction [i] ...
.
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 , 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 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.
See also
- Euler's conjecture
- Fermat's little theorem
- Sophie Germain prime
- Wall-Sun-Sun prime
- Beal's conjecture
External links and references
- Wiles, Andrew . , Annals of Mathematics , 443-551 .
- Taylor, Richard & Wiles, Andrew . , Annals of Mathematics , 553-572.
- Ribet, Ken . - 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 . , Notices of the AMS , 743-746.
- Daney, Charles . . Retrieved Aug. 5, 2004.
- O'Connor, J. J. & and Robertson, E. F. . . Retrieved Aug. 5, 2004.
- Shay, David . . Retrieved Aug. 5, 2004.
- Freeman, Larry . . A blog that covers the history of Fermat's Last Theorem from Pierre Fermat to Andrew Wiles.
- Kisby, Adam William . . Parody.
-
- Noam D. Elkies,
Bibliography and further reading
...
. Fermat's Enigma. Bantam Books. ISBN 0-8027-1331-9 .
- Amir Aczel Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. Four Walls Eight Windows. ISBN 1-56858-077-0.
- Bell, Eric T. The Last Problem. New-York: Simon and Schuster. ISBN 0-88385-451-1 .
- Benson, Donald C. . The Moment of Proof: Mathematical Epiphanies. Oxford University Press. ISBN 0-19-513919-4.
