Fermat's Last Theorem

Pierre de Fermat

Problem II.8 in the Arithmetica of Diophantus, annotated with Fermat's comment which became Fermat's Last Theorem (edition of 1670).

Fermat's Last Theorem (sometimes abbreviated as FLT) 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 $n$ is greater than two, then $a^{n}+b^{n}=c^{n}$ has no solutions in non-zero finite integers $a$, $b$, and $c$.

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.

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. 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.

Fermat's Last Theorem from a comment in a margin

Problem II.8 in the 1621 edition of the Arithmetica. On the right is the famous margin that was too small to contain Fermat's alleged proof of his Last Theorem.

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 $k$, find $u$ and $v$, both rational, such that $k^{2}=u^{2}+v^{2}$), and shows how to solve the problem for $k=4$. 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 $n=2$, it was already known by the ancient Chinese, Indians, Greeks and Babylonians that the Diophantine equation $a^{2}+b^{2}=c^{2}$ (linked with the Pythagorean theorem) has integer solutions, such as (3,4,5) ($3^{2}+4^{2}=5^{2}$) or (5,12,13). These solutions are known as Pythagorean triples, and there exist 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 $n$, 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

Leonhard Euler by Emanuel Handmann.

The theorem needs only to be proven for $n=4$ and in the cases where $n$ is an odd prime number. If $n>2$ is not an odd prime number or 4, it can be either a power of two or not. In the first case the number 4 is a factor of $n$, otherwise there is an odd prime number among its factors. In any case let any such factor be $p$, and let $m$ be $n/p$. Now we can express the equation as $(a^{m})^{p}+(b^{m})^{p}=(c^{m})^{p}$. If we can prove the case with exponent $p$, exponent $n$ is simply a subset of that case.

For various special exponents $n$, the theorem had been proven over the years, but the general case remained elusive. The first case proved was the case $n=4$, which was proved by Fermat himself using the method of infinite descent. Using a similar method, Euler proved the theorem for $n=3$. While his original method contained a flaw, it has been the basis of a lot of research about the theorem. The case $n=5$ was proved by Dirichlet and Legendre in 1825 using a generalisation of Euler's proof for $n=3$. The proof for the next prime number, $n=7$ 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 37, 59 and 67.

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

The proof of Fermat's Last Theorem

The proof of Fermat's Last Theorem is based on a theorem proved by Ken Ribet in the summer of 1986. This theorem connected Fermat's Last Theorem with a conjecture about the relationship between elliptic curves over the rationals and modular forms. Fermat's Last Theorem was proved in the late part of 1994 by Andrew Wiles and Richard Taylor replacing a major part that gave the upper bound for a group from the flawed proof announced by Andrew Wiles in 1993. Andrew Wiles' direct attack on the conjecture began just after Ken Ribet announced that he had proved Serre's epsilon conjecture. The main problem that Andrew Wiles had to overcome was counting semistable elliptic curves over the rationals and the modular semistable elliptic curves over the rationals to show that was a same number of each. Before Andrew 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.

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. The connection between the Taniyama–Shimura theorem and Fermat's Last Theorem was formalized as a conjecture by Jean-Pierre Serre which was proved by Ken Ribet.

(Theorem of Ribet) If $n$ is a prime greater than 3, $a$, $b$, and $c$ are nonzero integers, and $a^{n}+b^{n}=c^{n}$, then the elliptic curve $y^{2}=x(x+a^{n})(x-b^{n})$ is not modular.

This theorem says that if Fermat's Last Theorem is false, then the Taniyama–Shimura theorem is false. In the other direction, if the Taniyama–Shimura theorem is correct, then so is Fermat's Last Theorem.

(Theorem of Wiles) If one of $a$, $b$, and $c$ is even, $n>5$, $a^{n}$ and $-b^{n}$ are distinct, nonzero, relatively prime integers, and $(a^{n})(-b^{n})(a^{n}+b^{n})$ is divisible by 16, then the elliptic curve $y^{2}=x(x+a^{n})(x-b^{n})$ is modular.

This theorem says that the elliptic curve connected to Fermat's Last Theorem is modular. Therefore if $p$ is a prime number, $u$, $v$, and $w$ are rational numbers, and $u^{p}+v^{p}+w^{p}=0$, then $uvw=0$.

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. NGB 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.

In 1986, Ken Ribet had proved Serre's epsilon conjecture that every counterexample $a^{n}+b^{n}=c^{n}$ to Fermat's Last Theorem would yield an elliptic curve defined as:

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

which would provide a counterexample to the Taniyama–Shimura conjecture. This latter conjecture proposed a deep connection, now proven to be true, between elliptic curves and modular forms. In one formulation it states that every elliptic curve can be parametrized by a rational map with integer coefficients using the classical modular curve; elliptic curves are all modular.

Andrew Wiles and Richard Taylor were able to establish a special case, the semistable case, of the Taniyama-Shimura conjecture sufficient to exclude such counterexamples arising from Fermat's Last Theorem. As a consequence of this work the full Taniyama-Shimura conjecture was proven, and is now sometimes called the modularity theorem; another milestone in number theory.

File:Nick katz1.jpg

Nick Katz

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 Isaac Newton Institute for Mathematical Sciences 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.

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 $x$, $y$, and $z$ such that $x^{n}+y^{n}=z^{m}$ in which $n$ and $m$ are any relatively prime natural numbers.

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".

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 $1782^{12}+1841^{12}=1922^{12}$ 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 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. A second 'counterexample' without the obvious flaw appeared in a later episode, "The Wizard of Evergreen Terrace": $3987^{12}+4365^{12}=4472^{12}$. 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 $a^{n}+b^{n}=c^{n}$"," which is Fermat's Last Theorem in its most general form.

See also

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

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.
• 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
• Noam D. Elkies, Tables of Fermat "near-misses" - approximate solutions of xⁿ + yⁿ = zⁿ
• Weisstein, Eric W. "Fermat's Last Theorem". MathWorld.

Bibliography and further reading

• Singh, Simon (hardcover, 1998). Fermat's Enigma. Bantam Books. ISBN 0802713319 (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 1568580770.
• Mozzochi, Charles (2000). The Fermat Diary. ISBN 0821826700.
• 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.
• H. M. Edwards (1977). Fermat's Last Theorem. Springer-Verlag. ISBN 0-387-90230-9.