In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, the ellipse; the circle is a special case of the ellipse, is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties; the conic sections of the Euclidean plane have various distinguishing properties. Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, some particular line, called a directrix, are in a fixed ratio, called the eccentricity; the type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.
This equation may be written in matrix form, some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane this apparent difference vanishes, the commonality becomes evident. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically; the conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry. A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone, it shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines; these are called degenerate conics and some authors do not consider them to be conics at all.
Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic. There are three types of conics, the ellipse and hyperbola; the circle is a special kind of ellipse, although it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the plane is a closed curve; the circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to one generating line of the cone the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is presented as the following definition. A conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L.
For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, for e > 1 a hyperbola. A circle is not defined by a focus and directrix, in the plane; the eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane, its directrix. An ellipse and a hyperbola each have distinct directrices for each of them; the line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic. The line segment joining the vertices of a conic is called the major axis called transverse axis in the hyperbola; the midpoint of this line segment is called the center of the conic. Let a denote the distance from the center to a vertex of an ellipse or hyperbola; the distance from the center to a directrix is a/e while the distance from the center to a focus is ae. A parabola does not have a center; the eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.
If the angle between the surface of the cone and its axis is β and the angle between the cutting plane and the axis is α, the eccentricity is cos α cos β. A proof that the conic sections given by the focus-directrix property are the same as those given by planes intersecting a cone is facilitated by the use of Dandelin spheres. Various parameters are associated with a conic section. Recall that the principal axis is the line joining the foci of an ellipse or hyperbola, the center in these cases is the midpoint of the line segment joining the foci; some of the other common features and/or. The linear eccentricity is the distance between the focus; the latus rectum is the chord parallel to the directrix and passing through the focus. Its length is denoted by 2ℓ; the semi-latus rectum is half of the length of the latus rec
Richard Taylor (mathematician)
Richard Lawrence Taylor is a British and American mathematician working in the field of number theory. He is a professor of mathematics at Stanford University and the Institute for Advanced Study. Taylor received the 2014 Breakthrough Prize in Mathematics "for numerous breakthrough results in the theory of automorphic forms, including the Taniyama–Weil conjecture, the local Langlands conjecture for general linear groups, the Sato–Tate conjecture." He received the 2007 Shaw Prize in Mathematical Sciences for his work on the Langlands program with Robert Langlands. He received his BA from Cambridge. During his time at Cambridge, he was president of The Archimedeans in 1981 and 1982, following the impeachment of his predecessor, he earned his PhD from Princeton University in 1988. From 1995 to 1996 he held the Savilian chair of geometry at Oxford University and Fellow of New College and became the Herchel Smith Professor of Mathematics at Harvard University, he holds Robert and Luisa Fernholz Professorship at the Institute for Advanced Study.
He received the Whitehead Prize in 1990, the Fermat Prize, the Ostrowski Prize in 2001, the Cole Prize of the American Mathematical Society in 2002, the Shaw Prize for Mathematics in 2007. He was elected a Fellow of the Royal Society in 1995. In 2012 he became a fellow of the American Mathematical Society. In 2015 he was inducted into the National Academy of Sciences, he was elected to the American Philosophical Society in 2018. One of the two papers containing the published proof of Fermat's Last Theorem is a joint work of Taylor and Andrew Wiles. In subsequent work, Taylor proved the local Langlands conjectures for GL over a number field. A simpler proof was suggested at the same time by Guy Henniart, ten years by Peter Scholze. Taylor, together with Christophe Breuil, Brian Conrad and Fred Diamond, completed the proof of the Taniyama–Shimura conjecture, by performing quite heavy technical computations in the case of additive reduction. In 2008, following the ideas of Michael Harris and building on his joint work with Laurent Clozel, Michael Harris, Nick Shepherd-Barron, announced a proof of the Sato–Tate conjecture, for elliptic curves with non-integral j-invariant.
This partial proof of the Sato–Tate conjecture uses Wiles's theorem about modularity of semistable elliptic curves. Taylor is the son of British physicist John C. Taylor, he is married, has two children. His home page at the Institute for Advanced Study Richard Taylor at the Mathematics Genealogy Project Autobiography upon Shaw Prize acceptance
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, other tasks; as an effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states producing "output" and terminating at a final ending state; the transition from one state to the next is not deterministic. The concept of algorithm has existed for centuries. Greek mathematicians used algorithms in the sieve of Eratosthenes for finding prime numbers, the Euclidean algorithm for finding the greatest common divisor of two numbers; the word algorithm itself is derived from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized Algoritmi.
A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem posed by David Hilbert in 1928. Formalizations were framed as attempts to define "effective calculability" or "effective method"; those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, Alan Turing's Turing machines of 1936–37 and 1939. The word'algorithm' has its roots in Latinizing the name of Muhammad ibn Musa al-Khwarizmi in a first step to algorismus. Al-Khwārizmī was a Persian mathematician, astronomer and scholar in the House of Wisdom in Baghdad, whose name means'the native of Khwarazm', a region, part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, translated into Latin during the 12th century under the title Algoritmi de numero Indorum; this title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
Al-Khwarizmi was the most read mathematician in Europe in the late Middle Ages through another of his books, the Algebra. In late medieval Latin, English'algorism', the corruption of his name meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός'number', the Latin word was altered to algorithmus, the corresponding English term'algorithm' is first attested in the 17th century. In English, it was first used in about 1230 and by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu, it begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five; the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
An informal definition could be "a set of rules that defines a sequence of operations". Which would include all computer programs, including programs that do not perform numeric calculations. A program is only an algorithm if it stops eventually. A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers. Boolos, Jeffrey & 1974, 1999 offer an informal meaning of the word in the following quotation: No human being can write fast enough, or long enough, or small enough† to list all members of an enumerably infinite set by writing out their names, one after another, in some notation, but humans can do something useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human, capable of carrying out only elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large, thus an algorithm can be an algebraic equation such as y = m + n – two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of: Precise instructions for a fast, efficient, "good" process that specifies the "moves" of "the computer" to find and process arbitrary input integers/symbols m and n, symbols + and =... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format
In number theory, the Mordell conjecture is the conjecture made by Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, is now known as Faltings's theorem; the conjecture was generalized by replacing Q by any number field. Let C be a non-singular algebraic curve of genus g over Q; the set of rational points on C may be determined as follows: Case g = 0: no points or infinitely many. Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group. Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup. Case g > 1: according to the Mordell conjecture, now Faltings's theorem, C has only a finite number of rational points. Faltings's original proof used the known reduction to a case of the Tate conjecture, a number of tools from algebraic geometry, including the theory of Néron models. A different proof, based on diophantine approximation, was found by Vojta.
A more elementary variant of Vojta's proof was given by Bombieri. Faltings's 1983 paper had as consequences a number of statements, conjectured: The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points; the reduction of the Mordell conjecture to the Shafarevich conjecture was due to A. N. Paršin. A sample application of Faltings's theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to an + bn = cn, since for such n the curve xn + yn = 1 has genus greater than 1. Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell–Lang conjecture, proved by Faltings. Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if X is a pseudo-canonical variety over a number field k X is not Zariski dense in X.
More general conjectures have been put forth by Paul Vojta. The Mordell conjecture for function fields was proved by Manin and by Grauert. In 1990, Coleman found and fixed a gap in Manin's proof. Bombieri, Enrico. "The Mordell conjecture revisited". Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17: 615–640. MR 1093712. Coleman, Robert F.. "Manin's proof of the Mordell conjecture over function fields". L'Enseignement Mathématique. Revue Internationale. IIe Série. 36: 393–427. ISSN 0013-8584. MR 1096426. Archived from the original on 2011-10-02. Cornell, Gary. Arithmetic geometry. Papers from the conference held at the University of Connecticut, Connecticut, July 30 – August 10, 1984. New York: Springer-Verlag. Doi:10.1007/978-1-4613-8655-1. ISBN 0-387-96311-1. MR 0861969. → Contains an English translation of Faltings Faltings, Gerd. "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae. 73: 349–366. Doi:10.1007/BF01388432. MR 0718935. Faltings, Gerd. "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
Inventiones Mathematicae. 75: 381. Doi:10.1007/BF01388572. MR 0732554. Faltings, Gerd. "Diophantine approximation on abelian varieties". Ann. of Math. 133: 549–576. Doi:10.2307/2944319. MR 1109353. Faltings, Gerd. "The general case of S. Lang's conjecture". In Cristante, Valentino. Barsotti Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc. ISBN 0-12-197270-4. MR 1307396. Grauert, Hans. "Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper". Publications Mathématiques de l'IHÉS: 131–149. ISSN 1618-1913. MR 0222087. Hindry, Marc. Diophantine geometry. Graduate Texts in Mathematics. 201. New York: Springer-Verlag. Doi:10.1007/978-1-4612-1210-2. ISBN 0-387-98981-1. MR 1745599. → Gives Vojta's proof of Faltings's Theorem. Lang, Serge. Survey of Diophantine geometry. Springer-Verlag. Pp. 101–122. ISBN 3-540-61223-8. Manin, Ju. I.. "Rational points on algebraic curves over function fields".
Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. 27: 1395–1440. ISSN 0373-2436. MR 0157971. Mordell, Louis J.. "On the rational solutions of the indeterminate equation of the third and fourth degrees". Proc. Cambridge Philos. Soc. 21: 179–192. Paršin, A. N.. "Quelques conjectures de finitude en géométrie diophantienne". Actes du Congrès International des Mathématiciens. Tome 1. Nice: Gauthier-Villars. Pp. 467–471. MR 0427323. Archived from the original on 2016-09-24. Retrieved 2016-06-11. Parshin, A. N. "Mordell conje
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when V = R2 and V = R3 are the real projective line and the real projective plane where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, R3 denotes ordered triplets of real numbers; the idea of a projective space relates to perspective, more to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the vector space is R3 with the camera entrance pupil at the origin, the projective space corresponds to the image points. Projective spaces can be studied as a separate field in mathematics, but are used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates.
As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard Euclidean geometry for the plane, two lines always intersect at a point except when parallel. In a projective representation of lines and points, such an intersection point exists for parallel lines, it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, their representation theories; as outlined above, projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the real projective plane P2 in some detail. There are three equivalent definitions: The set of all lines in R3 passing through the origin.
Every such line meets the sphere of radius one centered in the origin twice, say in P = and its antipodal point. P2 can be described as the points on the sphere S2, where every point P and its antipodal point are not distinguished. For example, the point is identified with, etc, yet another equivalent definition is the set of equivalence classes of R3 ∖, i.e. 3-space without the origin, where two points P = and P∗ = are equivalent iff there is a nonzero real number λ such that P = λ⋅P∗, i.e. x = λx∗, y = λy∗, z = λz∗. The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point in R3, is. The last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with z ≠ 0 is equivalent to. So there are two disjoint subsets of the projective plane: that consisting of the points = for z ≠ 0, that consisting of the remaining points; the latter set can be subdivided into two disjoint subsets, with points and. In the last case, x is nonzero, because the origin was not part of P2.
This last point is equivalent to. Geometrically, the first subset, isomorphic to R2, is in the image the yellow upper hemisphere, or equivalently the lower hemisphere; the second subset, isomorphic to R1, corresponds to the green line, or, equivalently the light green line. We have the red point or the equivalent light red point. We thus have a disjoint decomposition P2 = R2 ⊔ R1 ⊔ point. Intuitively, made precise below, R1 ⊔ point is itself the real projective line P1. Considered as a subset of P2, it is called line at infinity, whereas R2 ⊂ P2 is called affine plane, i.e. just the usual plane. The next objective is to make the saying "parallel lines meet at infinity" precise. A natural bijection between the plane z = 1 and the sphere of the projective plane is accomplished by the gnomonic projection; each point P on this plane is mapped to the two intersection points of the sphere with the line through its center and P. These two points are identified in the projective plane. Lines in the plane are mapped to great circles if one includes one pair of antipodal points on the equator.
Any two great circles intersect in two antipodal points. Great circles corresponding to parallel lines intersect on the equator. So any two lines have one intersection point inside P2; this phenomenon is axiomatized in projective geometry. The real projective space of dimension n or projective n-space, Pn, is the set of the lines in Rn+1 passing through the origin. For defining it as a topological space and as an algebraic variety it is better to define it as the quotient space of Rn+1 by the equivalence relation "to be aligned with the origin". More Pn:= / ~,where ~ is the equivalence relation defined by: ~ if there is a non-zero real number λ such that =; the elements of the projective space are called points. The projective coordinates of a point P are x0... xn, where is any element of the corresponding equivalence class. This is denoted P =, the colons and the brackets emphasizing that the right-hand side is an equivalence class, whic
Sir Andrew John Wiles is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal by the Royal Society, he was appointed Knight Commander of the Order of the British Empire in 2000, in 2018 was appointed as the first Regius Professor of Mathematics at Oxford. Wiles was born on 11 April 1953 in Cambridge, the son of Maurice Frank Wiles, the Regius Professor of Divinity at the University of Oxford, Patricia Wiles, his father worked as the chaplain at Ridley Hall, for the years 1952–55. Wiles attended King's College School and The Leys School, Cambridge. Wiles states that he came across Fermat's Last Theorem on his way home from school when he was 10 years old, he stopped at his local library. Fascinated by the existence of a theorem, so easy to state that he, a ten year old, could understand it, but that no one had proven, he decided to be the first person to prove it.
However, he soon realised that his knowledge was too limited, so he abandoned his childhood dream, until it was brought back to his attention at the age of 33 by Ken Ribet's 1986 proof of the epsilon conjecture, which Gerhard Frey had linked to Fermat's famous equation. Wiles earned his bachelor's degree in mathematics in 1974 at Merton College, a PhD in 1980 as a graduate student of Clare College, Cambridge. After a stay at the Institute for Advanced Study in Princeton, New Jersey in 1981, Wiles became a Professor of Mathematics at Princeton University. In 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure. From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford, he returned to Princeton. From 1994 - 2009, Wiles was a Eugene Higgins Professor at Princeton, he rejoined Oxford in 2011 as Royal Society Research Professor. In May 2018 he was appointed Regius Professor of Mathematics at Oxford, the first in the university's history.
Wiles's graduate research was guided by John Coates beginning in the summer of 1975. Together these colleagues worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory, he further worked with Barry Mazur on the main conjecture of Iwasawa theory over the rational numbers, soon afterward, he generalised this result to real fields. His biographical page at Princeton University's website states that "Andrew has few equals in terms of his impact on modern number theory. Many of the world’s best young number theorists received their Ph. D.'s under Andrew... and many of these are today leaders and professors at top institutions around the world". Starting in mid-1986, based on successive progress of the previous few years of Gerhard Frey, Jean-Pierre Serre and Ken Ribet, it became clear that Fermat's Last Theorem could be proven as a corollary of a limited form of the modularity theorem; the modularity theorem involved elliptic curves, Wiles's own specialist area.
The conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or impossible to prove. For example, Wiles's ex-supervisor John Coates states that it seemed "impossible to prove", Ken Ribet considered himself "one of the vast majority of people who believed was inaccessible", adding that "Andrew Wiles was one of the few people on earth who had the audacity to dream that you can go and prove."Despite this, with his from-childhood fascination with Fermat's Last Theorem, decided to undertake the challenge of proving the conjecture, at least to the extent needed for Frey's curve. He dedicated all of his research time to this problem for over six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife. In June 1993, he presented his proof to the public for the first time at a conference in Cambridge, he gave a lecture a day on Monday and Wednesday with the title'Modular Forms, Elliptic Curves and Galois Representations.'
There was no hint in the title that Fermat's last theorem would be discussed, Dr. Ribet said.... At the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture; as an afterthought, he noted that that meant that Fermat's last theorem was true. Q. E. D. In August 1993, it was discovered. Wiles failed for over a year to repair his proof. According to Wiles, the crucial idea for circumventing, rather than closing, this area came to him on 19 September 1994, when he was on the verge of giving up. Together with his former student Richard Taylor, he published a second paper which circumvented the problem and thus completed the proof. Both papers were published in May 1995 in a dedicated issue of the Annals of Mathematics. Wiles's proof of Fermat's Last Theorem has stood up to the scrutiny of the world's other mathematical experts. Wiles was interviewed for an episode of the BBC documentary series Horizon that focused on Fermat's Last Theorem; this was renamed "The Proof", it was made an episode of the US Public Broadcasting Service's science television series Nova.
His work and life are described in great detail in Simon Singh's popular book Fermat's Last Theorem. Wiles has been awarded a number of major prizes in mathematics and science: Junior Whitehead Prize of the London Mathematical Soci
Fermat's Last Theorem
In number theory Fermat's Last Theorem states that no three positive integers a, b, c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions; the proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, formally published in 1995, it proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century, 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, x2 + y2 = z2, has an infinite number of positive integer solutions for x, y, z. Around 1637, Fermat wrote in the margin of a book that the more general equation an + bn = cn 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, no proof by him has been found, his claim was discovered some 30 years after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries; the claim became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, 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, not a prime number, it must be false for some smaller n, so only prime values of n need further investigation.
Over the next two centuries, the conjecture was proved for only the primes 3, 5, 7, although Sophie Germain innovated and proved an approach, 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 the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible. Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura–Weil conjecture, as the modularity theorem, it stood on its own, with no apparent connection to Fermat's Last Theorem, it was seen as significant and important in its own right, but was considered inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between these two unrelated and unsolved problems.
An outline suggesting this could be proved was given by Frey. The full proof that the two problems were 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"; these papers by Frey and Ribet showed that if the Modularity Theorem could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well. Although both problems were daunting and considered to be "completely inaccessible" to proof at 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.
Important for researchers choosing a research topic was the fact that unlike Fermat's Last Theorem the Modularity Theorem was a major active research area for which a proof was desired and not just a historical oddity, so time spent working on it could be justified professionally. However, general opinion was that this showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician John Coates' quoted reaction was a common one: "I myself was sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would 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 prove. I must confess I thought I wouldn’t see it proved in my lifetime." On hearing that Ribet had proven Frey's li