Fermat's Last Theorem

Pierre de Fermat's conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised.

Problem II.8 in the 1621 edition of the Arithmetica of Diophantus. On the right is the famous margin which was too small to contain Fermat's alleged proof of his last theorem.

Fermat's last theorem states that:

it is impossible to separate any power higher than the second into two like powers,

or, more precisely:

if an integer $n$ is greater than 2, then the equation $a^{n}+b^{n}=c^{n}$ has no solutions in non-zero integers $a$, $b$, and $c$.

In 1637 Fermat wrote, in his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus, "I have a truly marvellous 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.")

Fermat's last theorem is strikingly different and much more difficult to prove than the analogous problem for n = 2, for which there are infinitely many integer solutions called Pythagorean triples (and the closely related Pythagorean theorem has many elementary proofs). The fact that the problem's statement is understandable by schoolchildren makes it all the more frustrating, and it has probably generated more incorrect proofs than any other problem in the history of mathematics. No correct proof was found for 357 years, when a proof was finally published by Andrew Wiles in 1994, using sophisticated twentieth century techniques. The term "last theorem" resulted because all the other theorems proposed by Fermat were eventually proved or disproved, either by his own proofs or by other mathematicians, in the two centuries following their proposition.

Fermat's last theorem is one of the most famous theorems in the history of mathematics, familiar to nigh every mathematician, and had achieved a recognizable status in popular culture prior to its proof. The avalanche of media coverage generated by the resolution of Fermat's last theorem was the first of its kind, including worldwide newspaper accounts and various popularizations in books and a PBS NOVA special, The Proof.

The 1670 edition of Arithmetica is already annotated with the comment of Fermat which became known as his "last theorem".

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 (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 in the margin next to this problem in his copy of the Arithmetica:

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 marvellous proof of this, which this margin is too narrow to contain.)

While Fermat's original margin note was lost with his copy of Arithmetica, around 1670, his son produced a new edition of the book augmented with his father's comments. The note eventually became known as Fermat's last theorem, as it became the last of Fermat's asserted theorems to remain unproven.

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) ($3^{2}+4^{2}=5^{2}$) and (5,12,13). These solutions are known as Pythagorean triples, and there exist infinitely many of them, even excluding solutions for which $a$, $b$ and $c$ have a common divisor (that is, when the entire equation is multiplied by the square of an integer). Fermat's last theorem is an extension of this problem to higher powers $n$, and states that no such solution exists when the exponent 2 is replaced by a larger integer.

History of the proof

Leonhard Euler by Emanuel Handmann.

A special case of Fermat's last theorem for n = 3 was first stated by Abu-Mahmud al-Khujandi in the 10th century, but his attempted proof of the theorem was incorrect.

The first case of Fermat's last theorem to be proved, by Fermat himself, using the method of infinite descent, was the case n = 4. Using a similar method, Euler proved the theorem for n = 3. While his original method contained a flaw, it generated a great deal of research about the theorem. Over the following centuries, the theorem was established for many other special exponents n (or classes of exponents), but the general case remained elusive.

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 (it is enough to prove the theorem for prime numbers: see below), $n=7$ was found 15 years later by Gabriel Lamé in 1839. Unfortunately, this demonstration was relatively long and unlikely to be generalised to higher numbers. From this point, mathematicians started to demonstrate the theorem for classes of exponents, instead of individual numbers, and develop more general results related to the theorem.

These general ideas can be traced back to a novel approach introduced by Sophie Germain. Rather than proving that there were no solutions to a given value n, she demonstrated that if there was a solution, a certain condition would have to apply. This insight was already used in the proof of Fermat's last theorem for the case $n=5$. In 1847, Kummer proved that the theorem was true for all regular prime numbers (which include all prime numbers between 2 and 100 except for 37, 59 and 67).

In 1823 and then in 1850, the French Academy of Sciences offered a prize for a correct proof. This initiative only caused a wave of thousands of mathematical misadventures. A third prize was offered in 1883 by the Academy of Brussels. In 1908, the German physician and amateur mathematician Paul Freidrich Wolfskehl bequeathed 100,000 marks to the Göttingen Academy of Sciences to be offered as a prize for a complete proof of Fermat's last theorem. As a result, from 1908-1911, a flood of over 1000 incorrect proofs were presented. According to mathematical historian Howard Eves:

"Fermat's last theorem, has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published".

Andrew Wiles

Elliptic curves and Wiles' proof

The modern history of the proof begins in the late 1960s, when Yves Hellegouarch discovered a type of elliptic curve which could be constructed from any nontrivial solution to Fermat's equation. Gerhard Frey subsequently had the insight that such a curve should not be modular and would thus contradict the Taniyama–Shimura conjecture that every such curve is modular. To make Frey's idea precise, Jean-Pierre Serre proposed the Epsilon conjecture, and this conjecture was proven by Ken Ribet in the summer of 1986. It showed that Fermat's last theorem would follow from the Taniyama–Shimura conjecture for a special class of elliptic curves called semistable elliptic curves.

After hearing about Ribet's proof of the Epsilon conjecture, Andrew Wiles set out to prove that every semistable elliptic curve is modular. He did so in almost complete secrecy, working for a full seven years with minimal outside help. 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. He amazed his audience with the number of innovations in his methods. His proof involved the Kolivagin-Flach method, which he adopted after the Iwasawa method failed.

File:Nick katz1.jpg

Nick Katz

However, although Wiles had reviewed the proof beforehand with a Princeton colleague, Nick Katz, it turned out to contain a flaw: there was an error in a critical portion of the proof which gave a bound for the order of a particular group. Wiles and his former student Richard Taylor spent about a year trying to repair the proof, under the close scrutiny of the media and the mathematical community. In September 1994, they were able to complete the proof by using both the Kolyvagin-Flach and Iwasawa method together, as on their own they both were insufficient. Taylor and other mathematicians would go on to prove the Taniyama–Shimura conjecture, now often known as the Modularity theorem, for all elliptic curves, not just the semistable ones.

Because Wiles's proof relies mainly on techniques developed in the twentieth century, it is virtually impossible that it is the same as Fermat's proof. Many mathematicians doubt that Fermat had a valid proof of the theorem.

Mathematics of the theorem and its proof

Fermat's last theorem needs only to be proven for $n=4$ and prime numbers greater than 2. If $n>2$ 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 $n$, 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.

Fermat's last theorem stimulated the development of a great deal of modern ring theory. In particular, the notion of an ideal and the ideal class group grew out of Kummer's work on the theorem, and his proof of it for regular primes.

In 1977, Guy Terjanian proved that if p is an odd prime number, and the natural numbers x, y and z satisfy $x^{2p}+y^{2p}=z^{2p}$, then 2p must divide x or y.

The Mordell conjecture, proven by Gerd Faltings in 1983, 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 Taniyama–Shimura conjecture states that every elliptic curve can be parametrised by a rational map with integer coefficients using the classical modular curve; that is, all elliptic curves (over the rationals) can be described by modular forms.

On the other hand Ribet's theorem shows that for any nontrivial solution to Fermat's equation, $a^{n}+b^{n}=c^{n},$ the semistable elliptic curve of Hellegouarch and Frey, defined by

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

is not modular. Fermat's last theorem therefore follows from the Taniyama–Shimura conjecture.

The proof of this theorem for semistable elliptic curves by Wiles (and, in part, Taylor) uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. As well as standard constructions of modern algebraic geometry, using the category of schemes and Iwasawa theory, the proof involved the development ideas from Barry Mazur on deformations of Galois representations and contributed to the Langlands program.

Generalizations and similar equations

Many Diophantine equations have a form similar to the equation of Fermat's last theorem, without necessarily sharing its properties.

For example, it is known that 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.

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. At the end of the episode Captain Picard says, "Like Fermat's theorem, it is a puzzle we may never solve." 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. The values agree to 9 of 40 decimal digits. A second 'counterexample' appeared in a later episode, "The Wizard of Evergreen Terrace": $3987^{12}+4365^{12}=4472^{12}$. These agree to 10 of 44 decimal digits, but notice simple divisibility rules show 3987 and 4365 are divisible by 9 so that a sum of their powers is also. A similar rule reveals 4472 is not divisible by 3, so that this cannot hold.

• 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 devil is not successful. 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 resembles Fermat's last theorem, although the requirement that n≥3 is omitted.

• In one of the Rama series books the problem is supposed to have been solved very simply and elegantly (probably the way Fermat himself had intended it) 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.

• In the online game the Lost Experience, which is directly related to the television series Lost, the equation is said to have been originally solved by a scientist by the name of Enzo Vallenzetti (also the creator of the Vallenzetti Equation) sometime in the late 1960s. However due to his eccentric nature, after having the proof verified by his colleagues, Vallenzetti is said to have burned his work so that, according to his assistant, "others could have as much fun solving it as he did".

• In the book The Oxford Murders by Guillermo Martinez, Wiles's announcement in Cambridge of his proof of Fermat's last theorem forms a peripheral part of the action.

• In the book The Light of Other Days by Arthur C. Clarke and Stephen Baxter technology was developed which allowed the general public to look back into time. A 12 year old was able to read Fermat's actual proof and present it in the present time.

• The rock metal band KINETO has a song entitled "Theorem" that describes Fermat's last theorem.

See also

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

References

• O'Connor, J. J. and Robertson, E. F. (1996). Fermat's last theorem. The history of the problem. Retrieved Aug. 5, 2004.
• Faltings, Gerd (1995). The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles, Notices of the AMS 42 (7), pp. 743–746.
• Singh, Simon (paperback, 1998). Fermat's Enigma. Anchor Books, New York. ISBN 0385493622 (previously published in 1997 under the title "Fermat's Last Theorem" by Fourth Estate, ISBN 1857025210).
• Taylor, Richard and Wiles, Andrew (1995). Ring theoretic properties of certain Hecke algebras, Annals of Mathematics 141 (3), pp. 553–572.
• Terjanian, G. (1977), Sur l'équation $x^{2p}+y^{2p}=z^{2p}$, Comptes rendus hebdomadaires des séances de l'Académie des sciences. Série A et B, 285, pp. 973–975.
• Wiles, Andrew (1995). Modular elliptic curves and Fermat's last theorem, Annals of Mathematics 141 (3), 443-551 (alternative link with additional photographs).

Further reading

• Aczel, Amir (hardcover, 1996). Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. Four Walls Eight Windows. ISBN 1-56858-077-0.
• Bell, Eric T. (1961). The Last Problem. New-York: Simon and Schuster. ISBN 0-88385-451-1 (edition of 1998).
• Benson, Donald C. (paperback, 1999). The Moment of Proof: Mathematical Epiphanies. Oxford University Press. ISBN 0-19-513919-4.
• Brudner, Harvey J. (1994). Fermat and the Missing Numbers; ISBN 0964478501
• Edwards, H. M. (1977). Fermat's Last Theorem. Springer-Verlag. ISBN 0-387-90230-9.
• Mozzochi, Charles (2000). The Fermat Diary. ISBN 0-8218-2670-0.
• van der Poorten, Alf (hardcover, 1996). Notes on Fermat's Last Theorem: Wiley Interscience, ISBN 0-471-06261-8. (An outline of Wiles's methods; for the mathematically sophisticated.)

External links

• Daney, Charles (2003). The Mathematics of Fermat's Last Theorem. Retrieved Aug. 5, 2004.
• Elkies, Noam D. Tables of Fermat "near-misses" - approximate solutions of xⁿ + yⁿ = zⁿ
• 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 (PDF). Parody.
• 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
• Shay, David (2003). Fermat's Last Theorem. The story, the history and the mystery. Retrieved Aug. 5, 2004.

• The bluffer's guide to Fermat's Last Theorem
• Weisstein, Eric W. "Fermat's Last Theorem". MathWorld.
• Fermat's last theorem, On Sophie Germain.
• "The Proof," the title of one edition of the PBS television series NOVA, discusses Andrew Wiles effort to prove Fermat's last theorem.