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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'' > 2, 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'' > 2. 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 commo ... [...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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 101 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Infinite Descent
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions. Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Google
Google LLC (, ) is an American multinational corporation and technology company focusing on online advertising, search engine technology, cloud computing, computer software, quantum computing, e-commerce, consumer electronics, and artificial intelligence (AI). It has been referred to as "the most powerful company in the world" by the BBC and is one of the world's List of most valuable brands, most valuable brands. Google's parent company, Alphabet Inc., is one of the five Big Tech companies alongside Amazon (company), Amazon, Apple Inc., Apple, Meta Platforms, Meta, and Microsoft. Google was founded on September 4, 1998, by American computer scientists Larry Page and Sergey Brin. Together, they own about 14% of its publicly listed shares and control 56% of its stockholder voting power through super-voting stock. The company went public company, public via an initial public offering (IPO) in 2004. In 2015, Google was reorganized as a wholly owned subsidiary of Alphabet Inc. Go ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Norvig
Peter Norvig (born 14 December 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. Early life and education Norvig grew up in an academic family. His father was Danish and came to the United States after World War II to study math at the University of Minnesota. 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 col ... [...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 hypotenu ... [...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. Early life and education Sierpiński was born in 1882 in Warsaw, Congress Poland, to a doctor father Konstanty and mother Ludwika (''née'' Łapińska). His abilities in mathematics were evident from childhood. He enrolled in the Department of Mathematics and Physics at the University of Warsaw in 1899 and graduated five 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 Vorono ... [...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 , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle. 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. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing by their greatest common divisor. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer (the same for the three elements). 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 ... [...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 '' ... [...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 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 transcendence theory to establish a bound on ''a'',''b'' and used existing results bounding ''x'',''y'' in terms of ''a' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faltings's Theorem
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field \mathbb of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing \mathbb by any number field. Background Let C be a non-singular algebraic curve of genus g over \mathbb. Then the set of rational points on C may be determined as follows: * When g=0, there are either no points or infinitely many. In such cases, C may be handled as a conic section. * When g=1, if there are any points, then C is an elliptic curve and its rational points form a finitely generated abelian group. (This is ''Mordell's Theorem'', later generalized to the Mordell–Weil theorem.) Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup. * When g>1, according to 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 (1997) for the advancement of research in pure mathematics, and the (1998) of the French Academy of Sciences The French Academy of Sciences (, ) is a learn ... [...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 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, which is awarded in recognition of exceptional research achievement in the mathematical sciences. He was elected as a Fellow of the American Mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
