Fermat's Last Theorem 1

Fermat's Last Theorem

The 1670 edition of Diophantus' Arithmetica includes Fermat'scommentary, particularly his "Last Theorem" (Observatio

Domini Petri de Fermat).

In number theory, Fermat's Last Theorem states that nothree positive integers a, b, and c can satisfy the equationan + bn = cn for any integer value of n greater than two.

This theorem was first conjectured by Pierre de Fermat in1637, famously in the margin of a copy of Arithmeticawhere he claimed he had a proof that was too large to fit inthe margin. No successful proof was published until 1995despite the efforts of countless mathematicians during the358 intervening years. The unsolved problem stimulatedthe development of algebraic number theory in the 19thcentury and the proof of the modularity theorem in the20th century. It is among the most famous theorems in thehistory of mathematics and prior to its 1995 proof was inthe Guinness Book of World Records for "most difficultmathematical problems".

Fermat's conjecture (history)

Fermat left no proof of the conjecture for all n, but he didprove the special case n = 4. This reduced the problem toproving the theorem for exponents n that are primenumbers. Over the next two centuries (1637–1839), theconjecture was proven for only the primes 3, 5, and 7,although Sophie Germain proved a special case for allprimes less than 100. In the mid-19th century, ErnstKummer proved the theorem for regular primes. Buildingon Kummer's work and using sophisticated computer studies, other mathematicians were able to prove the conjecturefor all odd primes up to four million.

The final proof of the conjecture for all n came in the late 20th century. In 1984, Gerhard Frey suggested theapproach of proving the conjecture through a proof of the modularity theorem for elliptic curves. Building on workof Ken Ribet, Andrew Wiles succeeded in proving enough of the modularity theorem to prove Fermat's LastTheorem, with the assistance of Richard Taylor. Wiles's achievement was reported widely in the popular press, andhas been popularized in books and television programs.

Mathematical context

Pythagorean triplesPythagorean triples are a set of three integers (a, b, c) that satisfy a special case of Fermat's equation (n = 2)[1]

Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples,[2] and methods for generating such triples have been studied in many cultures, beginning with the Babylonians[3] and later ancient Greek, Chinese, and Indian mathematicians.[4] The traditional interest in Pythagorean triples connects with the Pythagorean theorem;[5] in its converse form, it states that a triangle with sides of lengths a, b, and c has a right

Fermat's Last Theorem 2

angle between the a and b legs when the numbers are a Pythagorean triple. Right angles have various practicalapplications, such as surveying, carpentry, masonry, and construction. Fermat's Last Theorem is an extension of thisproblem to higher powers, stating that no solution exists when the exponent 2 is replaced by any larger integer.

Diophantine equationsFermat's equation xn + yn = zn is an example of a Diophantine equation.[6] A Diophantine equation is a polynomialequation in which the solutions must be integers.[7] Their name derives from the 3rd-century Alexandrianmathematician, Diophantus, who developed methods for their solution. A typical Diophantine problem is to find twointegers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:

Diophantus's major work is the Arithmetica, of which only a portion has survived.[8] Fermat's conjecture of his LastTheorem was inspired while reading a new edition of the Arithmetica,[9] which was translated into Latin andpublished in 1621 by Claude Bachet.[10]

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadraticDiophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800BC).[11] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclideanalgorithm (c. 5th century BC).[12] Many Diophantine equations have a form similar to the equation of Fermat's LastTheorem from the point of view of algebra, in that they have no cross terms mixing two letters, without sharing itsparticular properties. For example, it is known that there are infinitely many positive integers x, y, and z such that xn

+ yn = zm where n and m are relatively prime natural numbers.[13]

Fermat's conjecture

Fermat's Last Theorem 3

Problem II.8 in the 1621 edition of theArithmetica of Diophantus. On the right is thefamous margin which was too small to contain

Fermat's alleged proof of his "last theorem".

Problem II.8 of the Arithmetica asks how a given square number issplit into two other squares; in other words, for a given rational numberk, find rational numbers u and v such that k2 = u2 + v2. Diophantusshows how to solve this sum-of-squares problem for k = 4 (thesolutions being u = 16/5 and v = 12/5).[14]

Around 1637, Fermat wrote his Last Theorem in the margin of hiscopy of the Arithmetica next to Diophantus' sum-of-squaresproblem:[15]

Cubum autem in duos cubos, aut quadratoquadratum in duosquadratoquadratos, et generaliter nullam in infinitum ultra quadratumpotestatem in duos eiusdem nominis fas est dividere cuius reidemonstrationem mirabilem sane detexi. Hanc marginis exiguitas noncaperet.

it is impossible to separate a cube into two cubes, or a fourth powerinto two fourth powers, or in general, any power higher than thesecond, into two like powers. I have discovered a truly marvelousproof of this, which this margin is too narrow to contain.[16]

Although Fermat's general proof is unknown, his proof of one case (n = 4) by infinite descent has survived.[17]

Fermat posed the cases of n = 4 and of n = 3 as challenges to his mathematical correspondents, such as MarinMersenne, Blaise Pascal, and John Wallis.[18] However, in the last thirty years of his life, Fermat never again wroteof his "truly marvellous proof" of the general case.After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmentedwith his father's comments.[19] The margin note became known as Fermat's Last Theorem,[20] as it was the last ofFermat's asserted theorems to remain unproven.[21]

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Proofs for specific exponentsOnly one mathematical proof by Fermat has survived, in which Fermat uses the technique of infinite descent to showthat the area of a right triangle with integer sides can never equal the square of an integer.[22] His proof is equivalentto demonstrating that the equation

has no primitive solutions in integers (no pairwise coprime solutions). In turn, this proves Fermat's Last Theorem forthe case n=4, since the equation a4 + b4 = c4 can be written as c4 − b4 = (a2)2.Alternative proofs of the case n = 4 were developed later[23] by Frénicle de Bessy (1676),[24] Leonhard Euler(1738),[25] Kausler (1802),[26] Peter Barlow (1811),[27] Adrien-Marie Legendre (1830),[28] Schopis (1825),[29]

Terquem (1846),[30] Joseph Bertrand (1851),[31] Victor Lebesgue (1853, 1859, 1862),[32] Theophile Pepin (1883),[33]

Tafelmacher (1893),[34] David Hilbert (1897),[35] Bendz (1901),[36] Gambioli (1901),[37] Leopold Kronecker(1901),[38] Bang (1905),[39] Sommer (1907),[40] Bottari (1908),[41] Karel Rychlík (1910),[42] Nutzhorn (1912),[43]

Robert Carmichael (1913),[44] Hancock (1931),[45] and Vrǎnceanu (1966).[46]

For another proof for n=4 by infinite descent, see Infinite descent: Non-solvability of r2 + s4 = t4. For various proofsfor n=4 by infinite descent, see Grant and Perella (1999),[47] Barbara (2007),[48] and Dolan (2011).[49]

After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be establishedfor all odd prime exponents.[50] In other words, it was necessary to prove only that the equation an + bn = cn has nointeger solutions (a, b, c) when n is an odd prime number. This follows because a solution (a, b, c) for a given n isequivalent to a solution for all the factors of n. For illustration, let n be factored into d and e, n = de. The generalequation

an + bn = cn

implies that (ad, bd, cd) is a solution for the exponent e(ad)e + (bd)e = (cd)e.

Thus, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for atleast one prime factor of every n. All integers n > 2 contain a factor of 4, or an odd prime number, or both.Therefore, Fermat's Last Theorem can be proven for all n if it can be proven for n = 4 and for all odd primes (theonly even prime number is the number 2) p.In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proven for three odd primeexponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attemptedproof of the theorem was incorrect.[51] In 1770, Leonhard Euler gave a proof of p = 3,[52] but his proof by infinitedescent[53] contained a major gap.[54] However, since Euler himself had proven the lemma necessary to complete theproof in other work, he is generally credited with the first proof.[55] Independent proofs were published[56] byKausler (1802),[26] Legendre (1823, 1830),[28][57] Calzolari (1855),[58] Gabriel Lamé (1865),[59] Peter Guthrie Tait(1872),[60] Günther (1878),[61] Gambioli (1901),[37] Krey (1909),[62] Rychlík (1910),[42] Stockhaus (1910),[63]

Carmichael (1915),[64] Johannes van der Corput (1915),[65] Axel Thue (1917),[66] and Duarte (1944).[67] The casep = 5 was proven[68] independently by Legendre and Peter Dirichlet around 1825.[69] Alternative proofs weredeveloped[70] by Carl Friedrich Gauss (1875, posthumous),[71] Lebesgue (1843),[72] Lamé (1847),[73] Gambioli(1901),[37][74] Werebrusow (1905),[75] Rychlík (1910),[76] van der Corput (1915),[65] and Guy Terjanian (1987).[77]

The case p = 7 was proven[78] by Lamé in 1839.[79] His rather complicated proof was simplified in 1840 byLebesgue,[80] and still simpler proofs[81] were published by Angelo Genocchi in 1864, 1874 and 1876.[82]

Alternative proofs were developed by Théophile Pépin (1876)[83] and Edmond Maillet (1897).[84]

Fermat's Last Theorem has also been proven for the exponents n = 6, 10, and 14. Proofs for n = 6 have been published by Kausler,[26] Thue,[85] Tafelmacher,[86] Lind,[87] Kapferer,[88] Swift,[89] and Breusch.[90] Similarly, Dirichlet[91] and Terjanian[92] each proved the case n = 14, while Kapferer[88] and Breusch[90] each proved the case n = 10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for n = 3, 5, and 7,

Fermat's Last Theorem 5

respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponentcounterparts. Dirichlet's proof for n = 14 was published in 1832, before Lamé's 1839 proof for n = 7.[93]

Many proofs for specific exponents use Fermat's technique of infinite descent, which Fermat used to prove the casen = 4, but many do not. However, the details and auxiliary arguments are often ad hoc and tied to the individualexponent under consideration.[94] Since they became ever more complicated as p increased, it seemed unlikely thatthe general case of Fermat's Last Theorem could be proven by building upon the proofs for individual exponents.[94]

Although some general results on Fermat's Last Theorem were published in the early 19th century by Niels HenrikAbel and Peter Barlow,[95][96] the first significant work on the general theorem was done by Sophie Germain.[97]

Sophie GermainIn the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's Last Theorem forall exponents.[98] First, she defined a set of auxiliary primes θ constructed from the prime exponent p by the equationθ = 2hp+1, where h is any integer not divisible by three. She showed that if no integers raised to the pth power wereadjacent modulo θ (the non-consecutivity condition), then θ must divide the product xyz. Her goal was to usemathematical induction to prove that, for any given p, infinitely many auxiliary primes θ satisfied thenon-consecutivity condition and thus divided xyz; since the product xyz can have at most a finite number of primefactors, such a proof would have established Fermat's Last Theorem. Although she developed many techniques forestablishing the non-consecutivity condition, she did not succeed in her strategic goal. She also worked to set lowerlimits on the size of solutions to Fermat's equation for a given exponent p, a modified version of which waspublished by Adrien-Marie Legendre. As a byproduct of this latter work, she proved Sophie Germain's theorem,which verified the first case of Fermat's Last Theorem (the case in which p does not divide xyz) for every odd primeexponent less than 100.[98][99] Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for alleven exponents, specifically for n = 2p, which was proven by Guy Terjanian in 1977.[100] In 1985, LeonardAdleman, Roger Heath-Brown and Étienne Fouvry proved that the first case of Fermat's Last Theorem holds forinfinitely many odd primes p.[101]

Ernst Kummer and the theory of idealsIn 1847, Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation xp + yp = zp incomplex numbers, specifically the cyclotomic field based on the roots of the number 1. His proof failed, however,because it assumed incorrectly that such complex numbers can be factored uniquely into primes, similar to integers.This gap was pointed out immediately by Joseph Liouville, who later read a paper that demonstrated this failure ofunique factorisation, written by Ernst Kummer.Kummer set himself the task of determining whether the cyclotomic field could be generalized to include new primenumbers such that unique factorisation was restored. He succeeded in that task by developing the ideal numbers.Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all regularprime numbers. However, he could not prove the theorem for the exceptional primes (irregular primes) whichconjecturally occur approximately 39% of the time; the only irregular primes below 100 are 37, 59 and 67.

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Mordell conjectureIn the 1920s, Louis Mordell posed a conjecture that implied that Fermat's equation has at most a finite number ofnontrivial primitive integer solutions if the exponent n is greater than two.[102] This conjecture was proven in 1983by Gerd Faltings,[103] and is now known as Faltings' theorem.

Computational studiesIn the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregularprimes. In 1954, Harry Vandiver used a SWAC computer to prove Fermat's Last Theorem for all primes up to2521.[104] By 1978, Samuel Wagstaff had extended this to all primes less than 125,000.[105] By 1993, Fermat's LastTheorem had been proven for all primes less than four million.[106]

Connection with elliptic curvesThe ultimately successful strategy for proving Fermat's Last Theorem was by proving the modularity theorem. Thestrategy was first described by Gerhard Frey in 1984.[107] Frey noted that if Fermat's equation had a solution (a, b, c)for exponent p > 2, the corresponding elliptic curve[108]

y2 = x (x − ap)(x + bp)would have such unusual properties that the curve would likely violate the modularity theorem.[109] This theorem,first conjectured in the mid-1950s and gradually refined through the 1960s, states that every elliptic curve ismodular, meaning that it can be associated with a unique modular form.Following this strategy, the proof of Fermat's Last Theorem required two steps. First, it was necessary to show thatFrey's intuition was correct: that the above elliptic curve, if it exists, is always non-modular. Frey did not succeed inproving this rigorously; the missing piece was identified by Jean-Pierre Serre. This missing piece, the so-called"epsilon conjecture", was proven by Ken Ribet in 1986. Second, it was necessary to prove a special case of themodularity theorem. This special case (for semistable elliptic curves) was proven by Andrew Wiles in 1995.Thus, the epsilon conjecture showed that any solution to Fermat's equation could be used to generate a non-modularsemistable elliptic curve, whereas Wiles' proof showed that all such elliptic curves must be modular. Thiscontradiction implies that there can be no solutions to Fermat's equation, thus proving Fermat's Last Theorem.

Wiles' general proof

British mathematician Andrew Wiles

Ribet's proof of the epsilon conjecture in 1986 accomplished the first half ofFrey's strategy for proving Fermat's Last Theorem. Upon hearing of Ribet'sproof, Andrew Wiles decided to commit himself to accomplishing the secondhalf: proving a special case of the modularity theorem (then known as theTaniyama–Shimura conjecture) for semistable elliptic curves.[110] Wiles workedon that task for six years in almost complete secrecy. He based his initialapproach on his area of expertise, Horizontal Iwasawa theory, but by the summerof 1991, this approach seemed inadequate to the task.[111] In response, heexploited an Euler system recently developed by Victor Kolyvagin and MatthiasFlach. Since Wiles was unfamiliar with such methods, he asked his Princetoncolleague, Nick Katz, to check his reasoning over the spring semester of1993.[112]

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By mid-1993, Wiles was sufficiently confident of his results that he presented them in three lectures delivered onJune 21–23, 1993 at the Isaac Newton Institute for Mathematical Sciences.[113] Specifically, Wiles presented hisproof of the Taniyama–Shimura conjecture for semistable elliptic curves; together with Ribet's proof of the epsilonconjecture, this implied Fermat's Last Theorem. However, it soon became apparent that Wiles' initial proof wasincorrect. A critical portion of the proof contained an error in a bound on the order of a particular group. The errorwas caught by several mathematicians refereeing Wiles' manuscript,[114] including Katz, who alerted Wiles on 23August 1993.[115]

Wiles and his former student Richard Taylor spent almost a year trying to repair the proof, without success.[116] On19 September 1994, Wiles had a flash of insight that the proof could be saved by returning to his original HorizontalIwasawa theory approach, which he had abandoned in favour of the Kolyvagin–Flach approach.[117] On 24 October1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"[118] and "Ringtheoretic properties of certain Hecke algebras",[119] the second of which was co-authored with Taylor. The twopapers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. These papersestablished the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358years after it was conjectured.

Exponents other than positive integers

Rational exponents

All solutions of the Diophantine equation when n=1 were computed by Lenstra in1992.[120] In the case in which the mth roots are required to be real and positive, all solutions are given by[121]

for positive integers r, s, t with s and t coprime.In 2004, for n>2, Bennett, Glass, and Szekely proved that if gcd(n,m)=1, then there are integer solutions if and onlyif 6 divides m, and , and are different complex 6th roots of the same real number.[122]

Negative exponents

n = –1

All primitive (pairwise coprime) integer solutions to can be written as[123]

for positive, coprime integers m, n.

Fermat's Last Theorem 8

n = –2

The case n = –2 also has an infinitude of solutions, and these have a geometric interpretation in terms of righttriangles with integer sides and an integer altitude to the hypotenuse.[124][125] All primitive solutions to

are given by

for coprime integers u, v with v > u. The geometric interpretation is that a and b are the integer legs of a right triangleand d is the integer altitude to the hypotenuse. Then the hypotenuse itself is the integer

so (a, b, c) is a Pythagorean triple.

Integer n < –2

There are no solutions in integers for for integer n < –2. If there were, the equation could bemultiplied through by to obtain , which is impossible by Fermat's LastTheorem.

Did Fermat possess a general proof?The mathematical techniques used in Fermat's "marvelous" proof are unknown. Only one detailed proof of Fermathas survived, the above proof that no three coprime integers (x, y, z) satisfy the equation x4 − y4 = z4.Taylor and Wiles's proof relies on mathematical techniques developed in the twentieth century, which would be aliento mathematicians who had worked on Fermat's Last Theorem even a century earlier. Fermat's alleged "marvellousproof", by comparison, would have had to be elementary, given mathematical knowledge of the time, and so couldnot have been the same as Wiles' proof. Most mathematicians and science historians doubt that Fermat had a validproof of his theorem for all exponents n.Harvey Friedman's grand conjecture implies that Fermat's last theorem can be proved in elementary arithmetic, arather weak form of arithmetic with addition, multiplication, exponentiation, and a limited form of induction forformulas with bounded quantifiers.[126] Any such proof would be elementary but possibly too long to write down.

Monetary prizesIn 1816 and again in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's LastTheorem.[127] In 1857, the Academy awarded 3000 francs and a gold medal to Kummer for his research on idealnumbers, although he had not submitted an entry for the prize.[128] Another prize was offered in 1883 by theAcademy of Brussels.[129]

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 marks to theGöttingen Academy of Sciences to be offered as a prize for a complete proof of Fermat's Last Theorem.[130] On 27June 1908, the Academy published nine rules for awarding the prize. Among other things, these rules required thatthe proof be published in a peer-reviewed journal; the prize would not be awarded until two years after thepublication; and that no prize would be given after 13 September 2007, roughly a century after the competition wasbegun.[131] Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997.[132]

Prior to Wiles' proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3 meters) of correspondence.[133] In the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month. According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in

Fermat's Last Theorem 9

schools, and often submitted by "people with a technical education but a failed career".[134] In the words ofmathematical historian Howard Eves, "Fermat's Last Theorem has the peculiar distinction of being the mathematicalproblem for which the greatest number of incorrect proofs have been published."[129]

In popular culture• An episode in the television series Star Trek: The Next Generation, titled "The Royale", refers to the theorem in

the first act. Riker visits Captain Jean-Luc Picard in his ready room to report only to find Picard puzzling overFermat's last theorem. Picard's interest in this theorem goes beyond the difficulty of the puzzle; he also feelshumbled that despite their advanced technology, they are still unable to solve a problem set forth by a man whohad no computer.[135] An episode in Star Trek: Deep Space 9, titled "Facets", refers to the theorem as well. In ascene involving O'Brien, Tobin Dax mentions continuing work on her own attempt to solve Fermat's lasttheorem.[136]

• "The Proof" – Nova (PBS) documentary about Andrew Wiles's proof of Fermat's Last Theorem.• On August 17, 2011, a Google doodle was shown on the Google homepage, showing a blackboard with the

theorem on it. When hovered over, it displays the text "I have discovered a truly marvelous proof of this theorem,which this doodle is too small to contain." This is a reference to the note made by Fermat in the margins ofArithmetica. It commemorated the 410th birth anniversary of de Fermat.[137]

• In the book The Girl Who Played with Fire, main character Lisbeth Salander becomes obsessed with the theoremin the opening chapters of the book. Her continuing effort to come up with a proof on her own is a runningsub-plot throughout the story, and is used as a way to demonstrate her exceptional intelligence. In the end sheends up coming up with a proof (the actual proof is not featured in the book). But after being shot in the head andsurviving, she has lost the proof.

• In the Harold Ramis re-make of the movie Bedazzled, starring Brendan Fraser and Elizabeth Hurley, Fermat'sLast Theorem appears written on the chalkboard in the classroom that the protagonist Elliot finds himselfteleported to after he aborts his failed fourth wish. In the director's commentary for the DVD release, directorRamis comments that nobody has seemed to notice that the equation on the board is Fermat's Last Theorem.

• The funk super-group Cameo often references the theorem by adding the line: "A'n n B'n being C'n" to the song"Word Up" when performing live.

• In Doctor Who, Season 5 Episode 1 "The Eleventh Hour", the Doctor transmits a proof of Fermat's Last Theoremby typing it in just a few seconds on Jeff's laptop to prove his genius to a collection of world leaders discussingthe latest threat to the human race. This implies that the Doctor knew a proof which was quite short and easy forothers to comprehend.

• In The IT Crowd, Series 3 Episode 6 "Calendar Geeks" Fermat's Last Theorem is referenced during a photo shootfor a calendar about geeks and achievements in Science and Mathematics.

• In the song Bizarro Genius Baby by Mc_frontalot the lyrics "And no dust had settled when she’d disprovedFermat by finding A^3 + B^3 that = C^3" appear.

Fermat's Last Theorem 10

Notes[1] Stark, pp. 151–155.[2] Stillwell J (2003). Elements of Number Theory (http:/ / books. google. com/ books?id=LiAlZO2ntKAC& pg=PA110). New York:

Springer-Verlag. pp. 110–112. ISBN 0-387-95587-9. .[3] Aczel, pp. 13–15[4] Singh, pp. 18–20.[5][5] Singh, p. 6.[6] Stark, pp. 145–146.[7] Stark, p. 4–5.[8] Singh, pp. 50–51.[9][9] Stark, p. 145.[10] Aczel, pp. 44–45; Singh, pp. 56–58.[11] Aczel, pp. 14–15.[12] Stark, pp. 44–47.

[13] For example, [14] Friberg, pp. 333– 334.[15] Dickson, p. 731; Singh, pp. 60–62; Aczel, p. 9.[16] Panchishkin, p. 341 (http:/ / books. google. com/ books?id=wvK586IxaxwC& pg=PA341& dq="Cubum+ autem+ in+ duos+ cubos,+ aut+

quadratoquadratum"& hl=en& ei=Jig_Tc2hCIH_8Aaf2LivCg& sa=X& oi=book_result& ct=result& resnum=7& sqi=2&ved=0CEsQ6AEwBg#v=onepage& q="Cubum autem in duos cubos, aut quadratoquadratum"& f=false)

[17] Dickson, pp. 615–616; Aczel, p. 44.[18][18] Ribenboim, pp. 13, 24.[19] Singh, pp. 62–66.[20][20] Dickson, p. 731.[21][21] Singh, p. 67; Aczel, p. 10.[22] Freeman L. "Fermat's One Proof" (http:/ / fermatslasttheorem. blogspot. com/ 2005/ 05/ fermats-one-proof. html). . Retrieved 23 May 2009.[23] Ribenboim, pp. 15–24.[24] Frénicle de Bessy, Traité des Triangles Rectangles en Nombres, vol. I, 1676, Paris. Reprinted in Mém. Acad. Roy. Sci., 5, 1666–1699

(1729).[25] Euler L (1738). "Theorematum quorundam arithmeticorum demonstrationes". Comm. Acad. Sci. Petrop. 10: 125–146.. Reprinted Opera

omnia, ser. I, "Commentationes Arithmeticae", vol. I, pp. 38–58, Leipzig:Teubner (1915).[26] Kausler CF (1802). "Nova demonstratio theorematis nec summam, nec differentiam duorum cuborum cubum esse posse". Novi Acta Acad.

Petrop. 13: 245–253.[27] Barlow P (1811). An Elementary Investigation of Theory of Numbers. St. Paul's Church-Yard, London: J. Johnson. pp. 144–145.[28] Legendre AM (1830). Théorie des Nombres (Volume II) (3rd ed.). Paris: Firmin Didot Frères. Reprinted in 1955 by A. Blanchard (Paris).[29] Schopis (1825). Einige Sätze aus der unbestimmten Analytik. Gummbinnen: Programm.[30] Terquem O (1846). "Théorèmes sur les puissances des nombres". Nouv. Ann. Math. 5: 70–87.[31] Bertrand J (1851). Traité Élémentaire d'Algèbre. Paris: Hachette. pp. 217–230, 395.[32] Lebesgue VA (1853). "Résolution des équations biquadratiques z2 = x4 ± 2my4, z2 = 2mx4 − y4, 2mz2 = x4 ± y4". J. Math. Pures Appl. 18:

73–86.Lebesgue VA (1859). Exercices d'Analyse Numérique. Paris: Leiber et Faraguet. pp. 83–84, 89.Lebesgue VA (1862). Introduction à la Théorie des Nombres. Paris: Mallet-Bachelier. pp. 71–73.

[33] Pepin T (1883). "Étude sur l'équation indéterminée ax4 + by4 = cz2". Atti Accad. Naz. Lincei 36: 34–70.[34] Tafelmacher WLA (1893). "Sobre la ecuación x4 + y4 = z4". Ann. Univ. Chile 84: 307–320.[35] Hilbert D (1897). "Die Theorie der algebraischen Zahlkörper". Jahresbericht der Deutschen Mathematiker-Vereinigung 4: 175–546.

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References

Bibliography• Aczel, Amir (30 September 1996). Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical

Problem. Four Walls Eight Windows. ISBN 978-1-56858-077-7.• Dickson LE (1919). History of the Theory of Numbers. Volume II. Diophantine Analysis. New York: Chelsea

Publishing. pp. 545–550, 615–621, 731–776.• Edwards, HM (1997). Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory. Graduate

Texts in Mathematics. 50. New York: Springer-Verlag.• Friberg, Joran (2007). Amazing Traces of a Babylonian Origin in Greek Mathematics. World Scientific

Publishing Company. ISBN 978-981-270-452-8.• Kleiner I (2000). "From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem" (http:/ / math. stanford.

edu/ ~lekheng/ flt/ kleiner. pdf). Elem. Math. 55: 19–37. doi:10.1007/PL00000079.• Mordell LJ (1921). Three Lectures on Fermat's Last Theorem. Cambridge: Cambridge University Press.• Panchishkin, Alekseĭ Alekseevich (2007). Introduction to Modern Number Theory (Encyclopedia of

Mathematical Sciences. Springer Berlin Heidelberg New York. ISBN 978-3-540-20364-3.• Ribenboim P (2000). Fermat's Last Theorem for Amateurs. New York: Springer-Verlag.

ISBN 978-0-387-98508-4.• Singh S (October 1998). Fermat's Enigma. New York: Anchor Books. ISBN 978-0-385-49362-8.• Stark H (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.

Further reading• Bell, Eric T. (6 August 1998) [1961]. The Last Problem. New York: The Mathematical Association of America.

ISBN 978-0-88385-451-8.• Benson, Donald C. (5 April 2001). The Moment of Proof: Mathematical Epiphanies. Oxford University Press.

ISBN 978-0-19-513919-8.• Brudner, Harvey J. (1994). Fermat and the Missing Numbers. WLC, Inc. ISBN 978-0-9644785-0-3.• Edwards, H. M. (March 1996) [1977]. Fermat's Last Theorem. New York: Springer-Verlag.

ISBN 978-0-387-90230-2.• Faltings G (July 1995). "The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles" (http:/ / www. ams. org/

notices/ 199507/ faltings. pdf) (PDF). Notices of the AMS 42 (7): 743–746. ISSN 0002-9920.• Mozzochi, Charles (7 December 2000). The Fermat Diary. American Mathematical Society.

ISBN 978-0-8218-2670-6.• Ribenboim P (1979). 13 Lectures on Fermat's Last Theorem. New York: Springer Verlag.

ISBN 978-0-387-90432-0.

Fermat's Last Theorem 14

• van der Poorten, Alf (6 March 1996). Notes on Fermat's Last Theorem. WileyBlackwell.ISBN 978-0-471-06261-5.

• Saikia, Manjil P (July 2011). "A Study of Kummer's Proof of Fermat's Last Theorem for Regular Primes" (http:/ /www. thequantizedquark. com/ papers/ kummerFLT. pdf) (PDF). IISER Mohali Report.

External links• Daney, Charles (2003). "The Mathematics of Fermat's Last Theorem" (http:/ / cgd. best. vwh. net/ home/ flt/ flt01.

htm). Retrieved 5 August 2004.• The bluffer's guide to Fermat's Last Theorem (http:/ / math. stanford. edu/ ~lekheng/ flt/ )• Elkies, Noam D.. "Tables of Fermat "near-misses" - approximate solutions of xn + yn = zn" (http:/ / www. math.

harvard. edu/ ~elkies/ ferm. html).• Freeman, Larry (2005). "Fermat's Last Theorem Blog" (http:/ / www. fermatslasttheorem. blogspot. com). Blog

that covers the history of Fermat's Last Theorem from Fermat to Wiles.• Ribet, Ken (1995). "Galois representations and modular forms" (http:/ / math. stanford. edu/ ~lekheng/ flt/ ribet.

pdf) (PDF). 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 Serreand of Taniyama–Shimura.

• Shay, David (2003). "Fermat's Last Theorem" (http:/ / shayfam. com/ David/ flt/ index. htm). Retrieved 5 August2004. The story, the history and the mystery.

• Weisstein, Eric W., " Fermat's Last Theorem (http:/ / mathworld. wolfram. com/ FermatsLastTheorem. html)"from MathWorld.

• O'Connor JJ, Robertson EF (1996). "Fermat's last theorem" (http:/ / www-gap. dcs. st-and. ac. uk/ ~history/HistTopics/ Fermat's_last_theorem. html). Retrieved 5 August 2004.

• "The Proof" (http:/ / www. pbs. org/ wgbh/ nova/ proof/ ). The title of one edition of the PBS television seriesNOVA, discusses Andrew Wiles's effort to prove Fermat's Last Theorem.

• "Documentary Movie on Fermat's Last Theorem (1996)" (http:/ / video. google. com/videoplay?docid=8269328330690408516). Simon Singh and John Lynch's film tells the story of Andrew Wiles.

• Beal Fermat and Pythagora's Triplets (http:/ / www. coolissues. com/ mathematics/BealFermatPythagorasTriplets. htm)

Article Sources and Contributors 15

Article Sources and ContributorsFermat's Last Theorem  Source: http://en.wikipedia.org/w/index.php?oldid=507782614  Contributors: 208.202.22.xxx, 63.207.108.xxx, 9258fahsflkh917fas, A. Idrissi Bouyahyaoui, A876,AMackenzie, Aaron of Mpls, Aatomic1, Adam mugliston, Ahoerstemeier, Alan Peakall, Alanobrien, Alansohn, Alcazar84, Aldaron, Alexandermiller, Alexg, Alice.haugen, Allens, Alpertron,Alphachimp, Ancheta Wis, Andrew Gray, Antandrus, Arasaka, Arcfrk, Ardonik, Arjun G. 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Image Sources, Licenses and ContributorsFile:Diophantus-II-8-Fermat.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Diophantus-II-8-Fermat.jpg  License: Public Domain  Contributors: Mdd, Proteins, Schutz, 1anonymous editsFile:Diophantus-II-8.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Diophantus-II-8.jpg  License: Public Domain  Contributors: Guérin Nicolas, Mdd, Mu, Proteins, SchutzFile:Andrew wiles1-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Andrew_wiles1-3.jpg  License: unknown  Contributors: "copyright C. J. Mozzochi, Princeton N.J"

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