Fermat's Last Theorem

Pierre de Fermat

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 positive integers x, y, and z such that $x^{n}+y^{n}=z^{n}\;$ in which n is a natural number 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 discovered a truly remarkable proof of this theorem that the margin of this page is too small 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, or by rigorous proofs found afterwards. Mathematicians were long baffled, for they were unable either to prove or to disprove it. The theorem was therefore 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.

Mathematical context

Fermat's last theorem is a generalization of the Diophantine equation a² + b² = c², which is linked to the Pythagorean theorem. Ancient Greeks and Babylonians knew that this equation has integer solutions, such as (3,4,5) (3² + 4² = 5²) or (5,12,13). These solutions are known as Pythagorean triples, and there exist an infinity of them (even excluding trivial solutions for which a, b and c have a common divisor). According to Fermat's last theorem, no such solution exists when the exponent 2 is replaced by a larger integer number.

While the theorem itself has no known direct use (i.e. 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.

Early history

The theorem needs only to be proven for n=4 and in the case where n is an odd prime number. For various special exponents n, the theorem had been proved over the years, but the general case remained elusive.

Fermat himself proved the case n=4, while Euler proved the theorem for n=3. The case n=5 was proved by Dirichlet and Legendre in 1825, and the case n=7 by Gabriel Lamé in 1839.

In 1983 Gerd Faltings proved the Mordell conjecture, which implies that for any n > 2, there are at most finitely many coprime integers a, b and c with aⁿ + bⁿ = cⁿ.

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.

In 1986, Ken Ribet had proved Gerhard Frey's epsilon conjecture that every counterexample aⁿ + bⁿ = cⁿ to Fermat's last theorem would yield an elliptic curve defined as:

$y^{2}=x\cdot (x-a^{n})\cdot (x+b^{n}),$

which would provide a counterexample to the Taniyama-Shimura conjecture.

This latter conjecture proposes a deep connection between elliptic curves and modular forms.

Wiles and 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.

Amazingly enough, if his original proof had been correct he most certainly would have won the most famous award in mathematics, the Field's Medal. However mathematicians must be under 40 years old to receive the award, which he had passed by the time the successful proof was submitted.

Did Fermat really have a proof?

The quotation was in Latin:

Cubum autem in duos cubos, aut quadratoquadratorum in duos quadratoquadratos,
et generaliter nullam in infinitum ultra quadratum patestatem in duos euisdem
nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi.
Hanc marginis exigitas 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.)

There is considerable doubt over whether Fermat's claim to have "truly remarkable proof" was correct. The length of Wiles's proof is about 200 pages and is 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.

The methods used by Wiles were unknown when Fermat was writing, and most 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". 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

$a^{4}+b^{4}=c^{4}$.

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 was used to prove the general theorem. The academic conventions of his time were not, however, those that applied from the middle of the eighteenth century, and this argument cannot be taken as definitive.

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 Royal".

See also

• Euler's conjecture
• Fermat's little theorem
• Sophie Germain prime
• Wall-Sun-Sun prime

External links and references

• Wiles, Andrew (1995). Modular elliptic curves and Fermat's last theorem, Annals of Mathematics (141) (3), 443-551.
• 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.
• The Moment of Proof : Mathematical Epiphanies, by Donald C. Benson; Oxford University Press; ISBN 0195139194 (paperback, 1999)

Bibliography

• Fermat's Enigma (previously published under the title Fermat's Last Theorem), by Simon Singh; Bantam Books; ISBN 0802713319 (hardcover, September 1998)
• Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem, by Amir D. Aczel
• The Last Problem, by E. T. Bell 1961.