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'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 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.")
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. 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''.
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Contents

Fermat's last theorem from a comment in a margin

History of the proof

Elliptic curves and Wiles' proof

Mathematics of the theorem and its proof

Generalizations and similar equations

In fiction

See also

Notes

References

Further reading

External links

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'':^[1]

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.)

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

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.^[2]
The first case of Fermat's last theorem to be proven, by Fermat himself, was the case ''n'' = 4 using the method of infinite descent. Using a similar method, Euler proved the theorem for ''n'' = 3; although his published proof contains some errors, the needed assertions could be established with work Euler himself had proven elsewhere. 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''".

Elliptic curves and Wiles' proof

The history of the proof of Fermat's last theorem begins in the late 1960s, when Yves Hellegouarch came up with an idea of associating to any solution (''a'',''b'',''c'') of Fermat's equation a completely different mathematical object: an elliptic curve. The curve consists of all points in the plane whose coordinates (''x'',''y'') satisfy the relation
: $y^2=x(x-a^p)(x+b^p).$
Such an elliptic curve would enjoy very special properties, which are due to the appearance of high powers of integers in its equation and the fact that ''a''^''p'' + ''b''^''p'' = ''c''^''p'' is a ''p''th power as well. Gerhard Frey had an insight that such a curve would be so special that it would contradict a certain conjecture about elliptic curves which is now called the Taniyama–Shimura conjecture. This conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrize coordinates ''x'' and ''y'' of the points on it. Thus, according to the conjecture, any elliptic curve over 'Q' would have to be a modular elliptic curve, yet if a solution to Fermat's equation with non-zero ''a'', ''b'', ''c'' and ''p'' greater than 2 existed, the corresponding curve would not be modular, resulting in a contradiction. The link between the Fermat's last theorem and the Taniyama–Shimura conjecture is a little subtle: in order to derive the former from the latter, one needs to know a bit more, or as mathematicians would have it, "an epsilon more". This extra piece of information was identified by Jean-Pierre Serre and became known as the epsilon conjecture. Serre's main interest was in an even more ambitious conjecture, Serre's conjecture on modular Galois representations, which would imply the Taniyama–Shimura conjecture. Although in the preceding twenty or thirty years a lot of evidence had been accumulated to form conjecture about elliptic curves, the main reason to believe that these various conjectures were true lied not in the numerical confirmations, but in a remarkably coherent and attractive mathematical picture that they presented. Moreover, it could have happened that one or more of these conjectures were actually false (for example, Serre's conjecture is still wide open), and yet Fermat's last theorem were nonetheless true. That would simply mean that one should try a different approach.
In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture. (His article was published in 1990.) He demonstrated that, just as Frey had anticipated, a special case of the Taniyama–Shimura conjecture (still unproven at the time), together with the now proven epsilon conjecture, implies the Fermat's last theorem. Thus if the Taniyama–Shimura conjecture holds for a class of elliptic curves called semistable elliptic curves, then the Fermat's last theorem would be true.
After learning about Ribet's work, 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. Over the course of three lectures delivered at Isaac Newton Institute for Mathematical Sciences on June 21, 22, and 23 of 1993, Wiles announced his proof of the Taniyama–Shimura conjecture, and hence of the Fermat's last theorem. Wiles drew upon a wide variety of methods in the proof, some of them having been developed especially for this occasion.
Although Wiles had reviewed his argument beforehand with a Princeton colleague, Nick Katz, he soon discovered that the proof contained a gap. 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 almost 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 a very novel approach in the troublesome part of the argument.
Taylor and others would go on to prove the general form of the Taniyama–Shimura conjecture, now frequently called the modularity theorem, which applies to all elliptic curves over 'Q', not just the semistable curves that were relevant for the proof of Fermat's last theorem.
Taylor and Wiles's proof is extremely technical in that it relies on the mathematical techniques developed in the twentieth century, most of which would be totally alien to mathematicians who had worked on Fermat's last theorem only a century earlier. Fermat's alleged "marvelous proof", on the other hand, would have had to be fairly elementary, given the state of the mathematical knowledge at the time. And in fact, most mathematicians and historians of science doubt that Fermat had a valid proof of his theorem for all exponents ''n''.

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 2''p'' 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 '', 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 '' 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 retroactive continuity 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 equation 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 $a^n\; +\; b^n\; =\; c^n$".

★ 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.

★ In Jasper Fforde's book ''First Among Sequels'', 9 year-old Tuesday Next, seeing the equation on the sixth-form's math classroom's chalkboard, and thinking it homework, solves it quite simply.

★ In Stieg Larsson's 2006 book ''Flickan som lekte med elden'', the main character Lisbeth Salander is mesmerized by the Theorem. She spends a great deal of time trying to prove it herself, stubbornly avoiding the presented proof.

See also

★ Euler's conjecture

★ Fermat's little theorem

★ Sophie Germain prime

★ Wall-Sun-Sun prime

★ Beal's conjecture

Notes

1. Fermat's copy of ''Arithmetica'' was the version published in 1621 and translated from Greek into Latin by Claude Gaspard Bachet de Méziriac
2.

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

★ Fermat's Last Theorem, Edwards, H. M., , , Springer-Verlag, 1977, ISBN 0-387-90230-9

★ The Fermat Diary, Mozzochi, Charles, , , , 2000, 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

★

★ 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.

★ Fermat's Last Theorem by Amir D. Aczel (Chapter 1)