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

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, one of the most famous theorems in the history of mathematics, 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 2, then aⁿ + bⁿ = cⁿ has no solutions in non-zero integers a, b, and c.

With its published proof, this 'theorem' is sometimes called the Fermat–Wiles Theorem.

Despite how closely the problem is related to the Pythagorean theorem, which has infinite solutions and hundreds of proofs, Fermat's subtle variation is much more difficult 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 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.") However, no correct proof was found for 357 years, until it was finally proven using very deep methods by Sir Andrew Wiles in 1995 (after a failed attempt two years previously).

All the other theorems proposed by Fermat were eventually proved or disproved, either in his own proofs or by other mathematicians, in the two centuries following their proposition. The theorem was the last one that Fermat conjectured to have an unpublished proof. The name preceded the publication of the proof.

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 k² = u² + v²), 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) :

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 a² + b² = c² (linked with the Pythagorean theorem) 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 infinitely many of them, even excluding trivial solutions for which a, b and c have a common divisor (that is, when the entire equation is multiplied by the same number). Fermat's Last Theorem is a generalization of this result to higher powers n, and states that no such solution exists when the exponent 2 is replaced by a larger integer.

Leonhard Euler by Emanuel Handmann.

The 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.

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. Sophie Germain next contributed a novel approach to the problem which was far more general than previous strategies. 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 would later lead to the proof for Fermat's Last Theorem where n = 5. 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 generalised to higher numbers. From this point, 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 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.

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.

Andrew Wiles

In the late 1960s, Yves Hellegouarch discovered a connection between elliptic curves and Fermat's Last Theorem. He used the connection to prove results about elliptic curves using results from Fermat's Last Theorem. This led Gerhard Frey to the idea that the Taniyama–Shimura Conjecture implied Fermat's Last Theorem. Taniyama–Shimura 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 are also modular forms. At the time of Frey's proposal, both Taniyama–Shimura and his idea were open (unproven). To make Frey's idea into an actual proof Jean-Pierre Serre proposed the Epsilon conjecture, and that was proven by Ken Ribet in the summer of 1986. This theorem said that every counterexample aⁿ + bⁿ = cⁿ to Fermat's Last Theorem would yield an elliptic curve defined as y² = x(x − aⁿ)(x + bⁿ) 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 theorem; the truth of Taniyama–Shimura was shown to imply the truth of Fermat's Last Theorem.

Nick Katz

Upon hearing about Ribet's proof of the Epsilon conjecture 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. 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 gave a bound for 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. Wiles found that he could work with associated Galois representations. In the process he developed ideas from Barry Mazur on deformations of Galois representations. The final, correct proof uses the standard constructions of modern algebraic geometry, which involve the category of schemes.

Because Wiles's proof relies mainly on techniques developed in the twentieth century, most mathematicians agree that Wiles's proof is not the same as Fermat's proof. Most mathematicians believe that Fermat did not actually prove the theorem or that his proof was flawed like other early attempts. However, there are other mathematicians who believe that Fermat really did prove the theorem with seventeenth-century techniques, and who continue to search for an elementary proof.

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 xⁿ + yⁿ = z^m in which n and m are any relatively prime natural numbers.

• 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 1782¹² + 1841¹² = 1922¹² 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 (Seven hundred octillion). A second 'counterexample' appeared in a later episode, "The Wizard of Evergreen Terrace": 3987¹² + 4365¹² = 4472¹². However, in this case, both 3987 and 4365 are divisible by 9, so that the entire left-hand side must similarly be divisible by 9; this is not true of 4472, and therefore not of the right-hand side.

• 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 aⁿ + bⁿ = cⁿ", which resembles Fermat's Last Theorem, even though the crucial requirement that n≥3 is omitted. Without this, the statement could be the Pythagorean Theorem.

• 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.

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

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

• Amir Aczel (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.
• H. M. Edwards (1977). Fermat's Last Theorem. Springer-Verlag. ISBN 0-387-90230-9. 
• Mozzochi, Charles (2000). The Fermat Diary. ISBN 0-8218-2670-0. 
• Singh, Simon (hardcover, 1998). Fermat's Enigma. Bantam Books. ISBN 0-8027-1331-9 (previously published under the title Fermat's Last Theorem).
• Harvey J. Brudner; Fermat and the Missing Numbers; ISBN 0964478501

• 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
• Eric W. Weisstein, Fermat's Last Theorem at 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.