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Beal's Conjecture
The Beal conjecture is the following conjecture in number theory: :If :: A^x +B^y = C^z, :where ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive integers with ''x'', ''y'', ''z'' ≥ 3, then ''A'', ''B'', and ''C'' have a common prime factor. Equivalently, :The equation A^x + B^y = C^z has no solutions in positive integers and pairwise coprime integers ''A, B, C'' if ''x, y, z'' ≥ 3. The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased several times and is currently $1 million. In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation, the Mauldin conjecture, and the Tijdeman-Zagier conjecture. Related examples To illustrate, the solution 3^3 + 6^3 = 3^5 has bases with a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre De Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' '' Arithmetica''. He was also a lawyer at the '' Parlement'' of Toulouse, France. Biography 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, Domin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Norvig
Peter Norvig (born December 14, 1956) is an American computer scientist and Distinguished Education Fellow at the Stanford Institute for Human-Centered AI. He previously served as a director of research and search quality at Google. Norvig is the co-author with Stuart J. Russell of the most popular textbook in the field of AI: '' Artificial Intelligence: A Modern Approach'' used in more than 1,500 universities in 135 countries. Education Norvig received a Bachelor of Science in applied mathematics from Brown University and a Ph.D. in computer science from the University of California, Berkeley. Career and research Norvig is a councilor of the Association for the Advancement of Artificial Intelligence and co-author, with Stuart J. Russell, of '' Artificial Intelligence: A Modern Approach'', now the leading college text in the field. He was head of the Computational Sciences Division (now the Intelligent Systems Division) at NASA Ames Research Center, where he oversaw a staff of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Triangles
''Pythagorean Triangles'' is a book on right triangles, the Pythagorean theorem, and Pythagorean triples. It was originally written in the Polish language by Wacław Sierpiński (titled ''Trójkąty pitagorejskie''), and published in Warsaw in 1954. Indian mathematician Ambikeshwar Sharma translated it into English, with some added material from Sierpiński, and published it in the ''Scripta Mathematica'' Studies series of Yeshiva University (volume 9 of the series) in 1962. Dover Books republished the translation in a paperback edition in 2003. There is also a Russian translation of the 1954 edition. Topics As a brief summary of the book's contents, reviewer Brian Hopkins quotes ''The Pirates of Penzance'': "With many cheerful facts about the square of the hypotenuse." The book is divided into 15 chapters (or 16, if one counts the added material as a separate chapter). The first three of these define the primitive Pythagorean triples (the ones in which the two sides and hypoten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and topology. He published over 700 papers and 50 books. Three well-known fractals are named after him (the Sierpiński triangle, the Sierpiński carpet, and the Sierpiński curve), as are Sierpiński numbers and the associated Sierpiński problem. Educational background Sierpiński enrolled in the Department of Mathematics and Physics at the University of Warsaw in 1899 and graduated four years later. In 1903, while still at the University of Warsaw, the Department of Mathematics and Physics offered a prize for the best essay from a student on Voronoy's contribution to number theory. Sierpiński was awarded a gold medal for his essay, thus laying the foundation for his first major mathematical contribution. Unwilling for his work to be pub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a^2+b^2=c^2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=\sqrt2 is a right triangle, but (1,1,\sqrt2) is not a Pythagorean triple because \sqrt2 is not an integer. Moreover, 1 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preda Mihăilescu
Preda V. Mihăilescu (born 23 May 1955) is a Romanian mathematician, best known for his proof of the 158-year-old Catalan's conjecture. Biography Born in Bucharest,Stewart 2013 he is the brother of Vintilă Mihăilescu. After leaving Romania in 1973, he settled in Switzerland. He studied mathematics and computer science in Zürich, receiving a PhD from ETH Zürich in 1997. His PhD thesis, titled ''Cyclotomy of rings and primality testing'', was written under the direction of Erwin Engeler and Hendrik Lenstra. For several years, he did research at the University of Paderborn, Germany. Since 2005, he has held a professorship at the University of Göttingen. Major research In 2002, Mihăilescu proved Catalan's conjecture.Bilu et al. 2014. This number-theoretical conjecture, formulated by the French and Belgian mathematician Eugène Charles Catalan in 1844, had stood unresolved for 158 years. Mihăilescu's proof appeared in ''Crelle's Journal ''Crelle's Journal'', or just ''Crel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan's Conjecture
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was Conjecture, conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the ''only'' case of two consecutive perfect powers. That is to say, that History The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (''x'', ''y'') was restricted to be (2, 3) or (3, 2). The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case ''b'' = 2. In 1976, Robert Tijdeman applied Baker's method in transcendental number theory, transcendence theory to establish a bound on a,b and used existing results bounding '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faltings's Theorem
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of Genus (mathematics), 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, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field. Background Let ''C'' be a non-singular algebraic curve of genus (mathematics), genus ''g'' over Q. Then the set of rational points on ''C'' may be determined as follows: * Case ''g'' = 0: no points or infinitely many; ''C'' is handled as a conic section. * Case ''g'' = 1: no points, or ''C'' is an elliptic curve and its rational points form a finitely generated abelian group (''Mordell's Theorem'', later generalized to the Mordell–Weil theorem). 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'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loïc Merel
Loïc Merel (born 13 August 1965) is a French mathematician. His research interests include modular forms and number theory. Career Born in Carhaix-Plouguer, Brittany, Merel became a student at the École Normale Supérieure. He finished his doctorate at Pierre and Marie Curie University under supervision of Joseph Oesterlé in 1993. His thesis on modular symbols took inspiration from the work of Yuri Manin and Barry Mazur from the 1970s. In 1996, Merel proved the torsion conjecture for elliptic curves over any number field (which was only known for number fields of degree up to 8 at the time). In recognition of his achievement, in 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. Awards Merel has received numerous awards, including the EMS Prize (1996), the Blumenthal Award {{Short description, Award of the American Mathematical Society The Blumenthal Award was founded by the American Mathematical Society in 1993 in memory of Leonard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Darmon
Henri Rene Darmon (born 22 October 1965) is a French-Canadian mathematician. He is a number theorist who works on Hilbert's 12th problem and its relation with the Birch–Swinnerton-Dyer conjecture. He is currently a James McGill Professor of Mathematics at McGill University. Career Darmon received his BSc from McGill University in 1987 and his PhD from Harvard University in 1991 under supervision of Benedict Gross. From 1991 to 1996, he held positions in Princeton University. Since 1994, he has been a professor at McGill University. Awards Darmon was elected to the Royal Society of Canada in 2003. In 2008, he was awarded the Royal Society of Canada's John L. Synge Award. He received the 2017 AMS Cole Prize in Number Theory "for his contributions to the arithmetic of elliptic curves and modular forms", and the 2017 CRM-Fields-PIMS Prize The CRM-Fields-PIMS Prize is the premier Canadian research prize in the mathematical sciences. It is awarded in recognition of exceptiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Édouard Lucas
__NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him. Biography Lucas was born in Amiens and educated at the École Normale Supérieure. He worked in the Paris Observatory and later became a professor of mathematics at the Lycée Saint Louis and the Lycée Charlemagne in Paris. Lucas served as an artillery officer in the French Army during the Franco-Prussian War of 1870–1871. In 1875, Lucas posed a challenge to prove that the only solution of the Diophantine equation: :\sum_^ n^2 = M^2\; with ''N'' > 1 is when ''N'' = 24 and ''M'' = 70. This is known as the cannonball problem, since it can be visualized as the problem of taking a square arrangement of cannonballs on the ground and building a square pyramid out of them. It was not until 1918 that a proof (using elliptic functions) was found for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
