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74-603: FLT may refer to: Mathematics [ edit ] Fermat's Last Theorem , in number theory Fermat's little theorem , using modular arithmetic Finite Legendre transform , in algebra Medicine [ edit ] Alovudine (fluorothymidine), a pharmaceutical drug Fluorothymidine F-18 , a radiolabeled pharmaceutical drug Places [ edit ] Finger Lakes Trail , New York, United States Flitwick railway station , England Phaeton Airport , Haiti Organizations [ edit ] Fairlight (group) ,

148-402: A , b , and c satisfy the equation a + b = c for any integer value of n greater than 2 . The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica . Fermat added that he had a proof that was too large to fit in

222-412: A contradiction , which in turn proves that no non-trivial solutions exist. In other words, any solution that could contradict Fermat's Last Theorem could also be used to contradict the modularity theorem. So if the modularity theorem were found to be true, then it would follow that no contradiction to Fermat's Last Theorem could exist either. As described above, the discovery of this equivalent statement

296-486: A non-trivial solution. For comparison's sake we start with the original formulation. Most popular treatments of the subject state it this way. It is also commonly stated over Z : The equivalence is clear if n is even. If n is odd and all three of x , y , z are negative, then we can replace x , y , z with − x , − y , − z to obtain a solution in N . If two of them are negative, it must be x and z or y and z . If x , z are negative and y

370-421: A "marvelous proof" are unknown. Wiles and Taylor's proof relies on 20th-century techniques. Fermat's proof would have had to be elementary by comparison, given the mathematical knowledge of his time. While Harvey Friedman 's grand conjecture implies that any provable theorem (including Fermat's last theorem) can be proved using only ' elementary function arithmetic ', such a proof need be 'elementary' only in

444-407: A 1980s Commodore warez group Flight Centre , an Australian travel company (founded 1982; ASX ticker: FLT ) Free Federation of Workers (Spanish: Federación Libre de Trabajadores ), a 20th-century Puerto Rican trade union Liberation Front of Chad (French: Front de Libération du Tchad ), 1965–1976 Luxembourg Tennis Federation (French: Fédération Luxembourgeoise de Tennis ),

518-408: A byproduct of this latter work, she proved Sophie Germain's theorem , which verified the first case of Fermat's Last Theorem (namely, the case in which p does not divide xyz ) for every odd prime exponent less than 270, and for all primes p such that at least one of 2 p + 1 , 4 p + 1 , 8 p + 1 , 10 p + 1 , 14 p + 1 and 16 p + 1 is prime (specially, the primes p such that 2 p + 1

592-478: A childhood fascination with Fermat's Last Theorem and had a background of working with elliptic curves and related fields, decided to try to prove the Taniyama–Shimura conjecture as a way to prove Fermat's Last Theorem. In 1993, after six years of working secretly on the problem, Wiles succeeded in proving enough of the conjecture to prove Fermat's Last Theorem. Wiles's paper was massive in size and scope. A flaw

666-643: A factorization method— Fermat's factorization method —and popularized the proof by infinite descent , which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case n = 4. Fermat developed the two-square theorem , and the polygonal number theorem , which states that each number is a sum of three triangular numbers , four square numbers , five pentagonal numbers , and so on. Although Fermat claimed to have proven all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss , doubted several of his claims, especially given

740-405: A finite number of prime factors, such a proof would have established Fermat's Last Theorem. Although she developed many techniques for establishing the non-consecutivity condition, she did not succeed in her strategic goal. She also worked to set lower limits on the size of solutions to Fermat's equation for a given exponent p , a modified version of which was published by Adrien-Marie Legendre . As

814-637: A hobby than a profession. Nevertheless, he made important contributions to analytical geometry , probability, number theory and calculus. Secrecy was common in European mathematical circles at the time. This naturally led to priority disputes with contemporaries such as Descartes and Wallis . Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods." Fermat's pioneering work in analytic geometry ( Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum )

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888-415: A mathematician of rare power. He was an independent inventor of analytic geometry , he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. In the course of what turned out to be an extended correspondence with Blaise Pascal , he made a significant contribution to the theory of probability. But Fermat's crowning achievement

962-415: A method ( adequality ) for determining maxima, minima, and tangents to various curves that was equivalent to differential calculus . In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature . Fermat was the first person known to have evaluated the integral of general power functions. With his method, he

1036-423: A nontrivial solution in Z would also mean a solution exists in N , the original formulation of the problem. This is because the exponents of x , y , and z are equal (to n ), so if there is a solution in Q , then it can be multiplied through by an appropriate common denominator to get a solution in Z , and hence in N . A non-trivial solution a , b , c ∈ Z to x + y = z yields

1110-496: A paper that demonstrated this failure of unique factorisation, written by Ernst Kummer . Kummer set himself the task of determining whether the cyclotomic field could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by developing the ideal numbers . (It is often stated that Kummer was led to his "ideal complex numbers" by his interest in Fermat's Last Theorem; there

1184-406: A proof of Fermat's Last Theorem based on factoring the equation x + y = z in complex 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

1258-517: A proof of Fermat's Last Theorem would also follow automatically. The connection is described below : any solution that could contradict Fermat's Last Theorem could also be used to contradict the Taniyama–Shimura conjecture. So if the modularity theorem were found to be true, then by definition, no solution contradicting Fermat's Last Theorem could exist, meaning that Fermat's Last Theorem must also be true. Although both problems were daunting and widely considered to be "completely inaccessible" to proof at

1332-403: A sports governing body (founded 1946) Other uses [ edit ] Flutter-tonguing , in music Foreign Language Teaching , in education See also [ edit ] FTL (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title FLT . If an internal link led you here, you may wish to change

1406-501: A technical sense and could involve millions of steps, and thus be far too long to have been Fermat's proof. Only one relevant proof by Fermat has survived, in which he uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer. His proof is equivalent to demonstrating that the equation has no primitive solutions in integers (no pairwise coprime solutions). In turn, this proves Fermat's Last Theorem for

1480-421: Is any integer not divisible by three. She showed that, if no integers raised to the p th power were adjacent modulo θ (the non-consecutivity condition ), then θ must divide the product xyz . Her goal was to use mathematical induction to prove that, for any given p , infinitely many auxiliary primes θ satisfied the non-consecutivity condition and thus divided xyz ; since the product xyz can have at most

1554-440: Is even a story often told that Kummer, like Lamé , believed he had proven Fermat's Last Theorem until Lejeune Dirichlet told him his argument relied on unique factorization; but the story was first told by Kurt Hensel in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources. Harold Edwards said the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken". See

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1628-582: Is named after him: the Lycée Pierre-de-Fermat . French sculptor Théophile Barrau made a marble statue named Hommage à Pierre Fermat as a tribute to Fermat, now at the Capitole de Toulouse . Together with René Descartes , Fermat was one of the two leading mathematicians of the first half of the 17th century. According to Peter L. Bernstein , in his 1996 book Against the Gods , Fermat "was

1702-598: Is not a prime number , it must also be false for some smaller n , so only prime values of n need further investigation. Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach that was relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes , leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend

1776-415: Is positive, then we can rearrange to get (− z ) + y = (− x ) resulting in a solution in N ; the other case is dealt with analogously. Now if just one is negative, it must be x or y . If x is negative, and y and z are positive, then it can be rearranged to get (− x ) + z = y again resulting in a solution in N ; if y is negative, the result follows symmetrically. Thus in all cases

1850-423: Is prime are called Sophie Germain primes ). Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for n = 2 p , which was proved by Guy Terjanian in 1977. In 1985, Leonard Adleman , Roger Heath-Brown and Étienne Fouvry proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes p . In 1847, Gabriel Lamé outlined

1924-459: The Arithmetica asks how a given square number is split into two other squares; in other words, for a given rational number k , find rational numbers u and v such that k = u + v . Diophantus shows how to solve this sum-of-squares problem for k = 4 (the solutions being u = 16/5 and v = 12/5 ). Around 1637, Fermat wrote his Last Theorem in the margin of his copy of

1998-529: The Arithmetica next to Diophantus's sum-of-squares problem : Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & 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

2072-527: The Babylonians and later ancient Greek , Chinese , and Indian mathematicians. Mathematically, the definition of a Pythagorean triple is a set of three integers ( a , b , c ) that satisfy the equation a + b = c . Fermat's equation, x + y = z with positive integer solutions, is an example of a Diophantine equation , named for the 3rd-century Alexandrian mathematician, Diophantus , who studied them and developed methods for

2146-527: The Euclidean algorithm (c. 5th century BC). Many Diophantine equations have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no cross terms mixing two letters, without sharing its particular properties. For example, it is known that there are infinitely many positive integers x , y , and z such that x + y = z , where n and m are relatively prime natural numbers. Problem II.8 of

2220-483: The Taniyama–Shimura–Weil conjecture, now proven and known as the modularity theorem, were subsequently proved by other mathematicians, who built on Wiles's work between 1996 and 2001. For his proof, Wiles was honoured and received numerous awards, including the 2016 Abel Prize . There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of

2294-792: The case n = 4 , since the equation a + b = c can be written as c − b = ( a ) . Alternative proofs of the case n = 4 were developed later by Frénicle de Bessy (1676), Leonhard Euler (1738), Kausler (1802), Peter Barlow (1811), Adrien-Marie Legendre (1830), Schopis (1825), Olry Terquem (1846), Joseph Bertrand (1851), Victor Lebesgue (1853, 1859, 1862), Théophile Pépin (1883), Tafelmacher (1893), David Hilbert (1897), Bendz (1901), Gambioli (1901), Leopold Kronecker (1901), Bang (1905), Sommer (1907), Bottari (1908), Karel Rychlík (1910), Nutzhorn (1912), Robert Carmichael (1913), Hancock (1931), Gheorghe Vrănceanu (1966), Grant and Perella (1999), Barbara (2007), and Dolan (2011). After Fermat proved

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2368-563: The case n  = 14, while Kapferer and Breusch 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, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for n  = 14 was published in 1832, before Lamé's 1839 proof for n = 7 . All proofs for specific exponents used Fermat's technique of infinite descent , either in its original form, or in

2442-585: The development of algebraic number theory in the 19th and 20th centuries. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem", in part because the theorem has the largest number of unsuccessful proofs. The Pythagorean equation , x + y = z , has an infinite number of positive integer solutions for x , y , and z ; these solutions are known as Pythagorean triples (with

2516-582: The difficulty of some of the problems and the limited mathematical methods available to Fermat. His Last Theorem was first discovered by his son in the margin in his father's copy of an edition of Diophantus , and included the statement that the margin was too small to include the proof. It seems that he had not written to Marin Mersenne about it. It was first proven in 1994, by Sir Andrew Wiles , using techniques unavailable to Fermat. Through their correspondence in 1654, Fermat and Blaise Pascal helped lay

2590-413: The early 19th century by Niels Henrik Abel and Peter Barlow , the first significant work on the general theorem was done by Sophie Germain . In the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's Last Theorem for all exponents. First, she defined a set of auxiliary primes θ constructed from the prime exponent p by the equation θ = 2 hp + 1 , where h

2664-570: The factors of n . For illustration, let n be factored into d and e , n  =  de . The general equation implies that ( a , b , c ) is a solution for the exponent e Thus, to prove that Fermat's equation has no solutions for n > 2 , it would suffice to prove that it has no solutions for at least one prime factor of every n . Each integer n > 2 is divisible by 4 or by an odd prime number (or both). Therefore, Fermat's Last Theorem could be proved for all n if it could be proved for n = 4 and for all odd primes p . In

2738-484: The field Q , rather than over the ring Z ; fields exhibit more structure than rings , which allows for deeper analysis of their elements. Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form . However, the proof by Andrew Wiles proves that any equation of the form y = x ( x − a )( x + b ) does have a modular form. Any non-trivial solution to x + y = z (with p an odd prime) would therefore create

2812-458: The first successful proof was released in 1994 by Andrew Wiles and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016. It also proved much of the Taniyama–Shimura conjecture, subsequently known as the modularity theorem , and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated

2886-452: The form of descent on elliptic curves or abelian varieties. The details and auxiliary arguments, however, were often ad hoc and tied to the individual exponent under consideration. Since they became ever more complicated as p increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proved by building upon the proofs for individual exponents. Although some general results on Fermat's Last Theorem were published in

2960-550: The foundation for the theory of probability. From this brief but productive collaboration on the problem of points , they are now regarded as joint founders of probability theory . Fermat is credited with carrying out the first-ever rigorous probability calculation. In it, he was asked by a professional gambler why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in 24 throws of two dice resulted in his losing. Fermat showed mathematically why this

3034-458: The general case. Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van der Poorten suggests that while the absence of a proof is insignificant, the lack of challenges means Fermat realised he did not have a proof; he quotes Weil as saying Fermat must have briefly deluded himself with an irretrievable idea. The techniques Fermat might have used in such

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3108-504: The history of ideal numbers .) Pierre de Fermat Fermat was born in 1607 in Beaumont-de-Lomagne , France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony , where his father, Dominique Fermat, was a wealthy leather merchant and served three one-year terms as one of the four consuls of Beaumont-de-Lomagne. His mother was Claire de Long. Pierre had one brother and two sisters and

3182-471: The lemma necessary to complete the proof in other work, he is generally credited with the first proof. Independent proofs were published by Kausler (1802), Legendre (1823, 1830), Calzolari (1855), Gabriel Lamé (1865), Peter Guthrie Tait (1872), Siegmund Günther (1878), Gambioli (1901), Krey (1909), Rychlík (1910), Stockhaus (1910), Carmichael (1915), Johannes van der Corput (1915), Axel Thue (1917), and Duarte (1944). The case p = 5

3256-607: The length of two sides, each squared and then added together (3 + 4 = 9 + 16 = 25) , equals the square of the length of the third side (5 = 25) , would also be a right angle triangle. This is now known as the Pythagorean theorem , and a triple of numbers that meets this condition is called a Pythagorean triple; both are named after the ancient Greek Pythagoras . Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples, and methods for generating such triples have been studied in many cultures, beginning with

3330-615: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=FLT&oldid=1254217389 " Category : Disambiguation pages Hidden categories: Articles containing Spanish-language text Articles containing French-language text Short description is different from Wikidata All article disambiguation pages All disambiguation pages Fermat%27s Last Theorem In number theory , Fermat's Last Theorem (sometimes called Fermat's conjecture , especially in older texts) states that no three positive integers

3404-419: The margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares ), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians,

3478-402: The modern theory of such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the descent which is rightly regarded as Fermat's own." Regarding Fermat's use of ascent, Weil continued: "The novelty consisted in the vastly extended use which Fermat made of it, giving him at least a partial equivalent of what we would obtain by

3552-435: The next three and a half centuries. The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory , and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics . The special case n = 4 , proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent n that

3626-438: The non-trivial solution a / c , b / c ∈ Q for v + w = 1 . Conversely, a solution a / b , c / d ∈ Q to v + w = 1 yields the non-trivial solution ad , cb , bd for x + y = z . This last formulation is particularly fruitful, because it reduces the problem from a problem about surfaces in three dimensions to a problem about curves in two dimensions. Furthermore, it allows working over

3700-520: The path of shortest time " now known as the principle of least time . For this, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The terms Fermat's principle and Fermat functional were named in recognition of this role. Pierre de Fermat died on January 12, 1665, at Castres , in the present-day department of Tarn . The oldest and most prestigious high school in Toulouse

3774-462: The problem. In order to state them, we use the following notations: let N be the set of natural numbers 1, 2, 3, ..., let Z be the set of integers 0, ±1, ±2, ..., and let Q be the set of rational numbers a / b , where a and b are in Z with b ≠ 0 . In what follows we will call a solution to x + y = z where one or more of x , y , or z is zero a trivial solution . A solution where all three are nonzero will be called

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3848-461: The proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible (meaning that mathematicians generally considered a proof impossible, exceedingly difficult, or unachievable with current knowledge). Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms , two completely different areas of mathematics. Known at

3922-454: The second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain. After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father's comments. Although not actually a theorem at the time (meaning a mathematical statement for which proof exists), the marginal note became known over time as Fermat's Last Theorem , as it

3996-495: The simplest example being 3, 4, 5). Around 1637, Fermat wrote in the margin of a book that the more general equation a + b = c had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, and none has ever been found. His claim was discovered some 30 years later, after his death. This claim, which came to be known as Fermat's Last Theorem , stood unsolved for

4070-535: The solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers 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. Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica , that

4144-408: The special case n = 4 , the general proof for all n required only that the theorem be established for all odd prime exponents. In other words, it was necessary to prove only that the equation a + b = c has no positive integer solutions ( a , b , c ) when n is an odd prime number . This follows because a solution ( a , b , c ) for a given n is equivalent to a solution for all

4218-525: The time as the Taniyama–Shimura conjecture (eventually as the modularity theorem), it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was (like Fermat's theorem) widely considered completely inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between these two previously unrelated and unsolved problems. An outline suggesting this could be proved

4292-470: The time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers. Unlike Fermat's Last Theorem, the Taniyama–Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics. However, general opinion was that this simply showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician John Coates ' quoted reaction

4366-442: The two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents p  = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect. In 1770, Leonhard Euler gave a proof of p  = 3, but his proof by infinite descent contained a major gap. However, since Euler himself had proved

4440-969: The work of François Viète . In 1630, he bought the office of a councilor at the Parlement de Toulouse , one of the High Courts of Judicature in France, and was sworn in by the Grand Chambre in May 1631. He held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat. On 1 June 1631, Fermat married Louise de Long, a fourth cousin of his mother Claire de Fermat (née de Long). The Fermats had eight children, five of whom survived to adulthood: Clément-Samuel, Jean, Claire, Catherine, and Louise. Fluent in six languages ( French , Latin , Occitan , classical Greek , Italian and Spanish ), Fermat

4514-529: Was a common one: I myself was very sceptical that the beautiful link between Fermat's Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn't see it proved in my lifetime. On hearing that Ribet had proven Frey's link to be correct, English mathematician Andrew Wiles , who had

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4588-447: Was able to reduce this evaluation to the sum of geometric series . The resulting formula was helpful to Newton , and then Leibniz , when they independently developed the fundamental theorem of calculus . In number theory, Fermat studied Pell's equation , perfect numbers , amicable numbers and what would later become Fermat numbers . It was while researching perfect numbers that he discovered Fermat's little theorem . He invented

4662-741: Was almost certainly brought up in the town of his birth. He attended the University of Orléans from 1623 and received a bachelor in civil law in 1626, before moving to Bordeaux . In Bordeaux, he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apollonius 's De Locis Planis to one of the mathematicians there. Certainly, in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by

4736-491: Was circulated in manuscript form in 1636 (based on results achieved in 1629), predating the publication of Descartes' La géométrie (1637), which exploited the work. This manuscript was published posthumously in 1679 in Varia opera mathematica , as Ad Locos Planos et Solidos Isagoge ( Introduction to Plane and Solid Loci ). In Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum , Fermat developed

4810-410: Was crucial to the eventual solution of Fermat's Last Theorem, as it provided a means by which it could be "attacked" for all numbers at once. In ancient times it was known that a triangle whose sides were in the ratio 3:4:5 would have a right angle as one of its angles. This was used in construction and later in early geometry. It was also known to be one example of a general rule that any triangle where

4884-424: Was discovered in one part of his original paper during peer review and required a further year and collaboration with a past student, Richard Taylor , to resolve. As a result, the final proof in 1995 was accompanied by a smaller joint paper showing that the fixed steps were valid. Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of

4958-434: Was given by Frey. The full proof that the two problems were closely linked was accomplished in 1986 by Ken Ribet , building on a partial proof by Jean-Pierre Serre , who proved all but one part known as the "epsilon conjecture" (see: Ribet's Theorem and Frey curve ). These papers by Frey, Serre and Ribet showed that if the Taniyama–Shimura conjecture could be proven for at least the semi-stable class of elliptic curves,

5032-406: Was in the theory of numbers." Regarding Fermat's work in analysis, Isaac Newton wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents." Of Fermat's number theoretic work, the 20th-century mathematician André Weil wrote that: "what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for

5106-412: Was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often with little or no proof of his theorems. In some of these letters to his friends, he explored many of the fundamental ideas of calculus before Newton or Leibniz . Fermat was a trained lawyer making mathematics more of

5180-400: Was proved independently by Legendre and Peter Gustav Lejeune Dirichlet around 1825. Alternative proofs were developed by Carl Friedrich Gauss (1875, posthumous), Lebesgue (1843), Lamé (1847), Gambioli (1901), Werebrusow (1905), Rychlík (1910), van der Corput (1915), and Guy Terjanian (1987). The case p = 7 was proved by Lamé in 1839. His rather complicated proof

5254-460: Was simplified in 1840 by Lebesgue, and still simpler proofs were published by Angelo Genocchi in 1864, 1874 and 1876. Alternative proofs were developed by Théophile Pépin (1876) and Edmond Maillet (1897). Fermat's Last Theorem was also proved for the exponents n  = 6, 10, and 14. Proofs for n = 6 were published by Kausler, Thue, Tafelmacher, Lind, Kapferer, Swift, and Breusch. Similarly, Dirichlet and Terjanian each proved

5328-413: Was the case. The first variational principle in physics was articulated by Euclid in his Catoptrica . It says that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection . Hero of Alexandria later showed that this path gave the shortest length and the least time. Fermat refined and generalized this to "light travels between two given points along

5402-510: Was the last of Fermat's asserted theorems to remain unproved. It is not known whether Fermat had actually found a valid proof for all exponents n , but it appears unlikely. Only one related proof by him has survived, namely for the case n = 4 , as described in the section § Proofs for specific exponents . While Fermat posed the cases of n = 4 and of n = 3 as challenges to his mathematical correspondents, such as Marin Mersenne , Blaise Pascal , and John Wallis , he never posed

5476-477: Was translated into Latin and published in 1621 by Claude Bachet . Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x + y = z are given by the Pythagorean triples , originally solved by the Babylonians ( c.  1800 BC ). Solutions to linear Diophantine equations, such as 26 x + 65 y = 13 , may be found using

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