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Büchi's Problem
In number theory, Büchi's problem, also known as the ''n'' squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer ''M'' such that every sequence of ''M'' or more integer squares, whose second difference is constant and equal to 2, is necessarily a sequence of squares of the form (''x'' + ''i'')2, ''i'' = 1, 2, ..., ''M'',... for some integer ''x''. In 1983, Douglas Hensley observed that Büchi's problem is equivalent to the following: Does there exist a positive integer ''M'' such that, for all integers ''x'' and ''a'', the quantity (''x'' + ''n'')2 + ''a'' cannot be a square for more than ''M'' consecutive values of ''n'', unless ''a'' = 0? Statement of Büchi's problem Büchi's problem can be stated in the following way: Does there exist a positive integer ''M'' such that the system of equations : \begin x_2^2- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julius Richard Büchi
Julius Richard Büchi (1924–1984) was a Swiss logician and mathematician. He received his Dr. sc. nat. in 1950 at ETH Zurich under the supervision of Paul Bernays and Ferdinand Gonseth. Shortly afterwards he went to Purdue University in Lafayette, Indiana. He and his first student Lawrence Landweber had a major influence on the development of theoretical computer science. Together with his friend Saunders Mac Lane, a student of Paul Bernays as well, Büchi published numerous celebrated works. He invented what is now known as the Büchi automaton, a finite-state machine accepting certain sets of infinite sequences of characters known as omega-regular languages. The "''n'' squares' problem", known also as Büchi's problem, is an open problem from number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Douglas Hensley
Douglas may refer to: People * Douglas (given name) * Douglas (surname) Animals * Douglas (parrot), macaw that starred as the parrot ''Rosalinda'' in Pippi Longstocking * Douglas the camel, a camel in the Confederate Army in the American Civil War Businesses * Douglas Aircraft Company * Douglas (cosmetics), German cosmetics retail chain in Europe * Douglas Holding, former German company * Douglas (motorcycles), British motorcycle manufacturer Peerage and Baronetage * Duke of Douglas * Earl of Douglas, or any holder of the title * Marquess of Douglas, or any holder of the title * Douglas baronets Peoples * Clan Douglas, a Scottish kindred * Dougla people, West Indians of both African and East Indian heritage Places Australia * Douglas, Queensland, a suburb of Townsville * Douglas, Queensland (Toowoomba Region), a locality * Port Douglas, North Queensland, Australia * Shire of Douglas, in northern Queensland Canada * Douglas, New Brunswick * Douglas Parish, New Bru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Tenth Problem
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values. For example, the Diophantine equation 3x^2-2xy-y^2z-7=0 has an integer solution: x=1,\ y=2,\ z=-2. By contrast, the Diophantine equation x^2+y^2+1=0 has no such solution. Hilbert's tenth problem has been solved, and it has a negative answer: such a general algorithm cannot exist. This is the result of combined work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson that spans 21 years, with Matiyasevich completing the theorem in 1970. The theorem is now known as Matiyasevich's theorem or the MRDP theorem (an initialism for the surnames of the four principal contributors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Matiyasevich
Yuri Vladimirovich Matiyasevich (; born 2 March 1947 in Leningrad Saint Petersburg, formerly known as Petrograd and later Leningrad, is the List of cities and towns in Russia by population, second-largest city in Russia after Moscow. It is situated on the Neva, River Neva, at the head of the Gulf of Finland ...) is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's tenth problem (Matiyasevich's theorem), which was presented in his doctoral thesis at LOMI (the Leningrad Department of the Steklov Institute of Mathematics). Biography Early years and education Yuri Matiyasevich was born in Leningrad on March 2, 1947. The first few classes he studied at school No. 255 with Sofia G. Generson, thanks to whom he became interested in mathematics. In 1961 he began to participate in all-Russian olympiads. From 1962 to 1963 he studied at Leningrad Saint Petersburg Lyceum 239, physical and mathematical school No. 239. Also from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decidability (logic)
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them. Decidability of a logical system Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form ''over'' . Over the reals, a quadratic form is said to be '' definite'' if it takes the value zero only when all its variables are simultaneously zero; otherwise it is '' isotropic''. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology ( intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form -\mathbf^\math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Vojta
Paul Alan Vojta (born September 30, 1957) is an American mathematician, known for his work in number theory on Diophantine geometry and Diophantine approximation. Contributions In formulating Vojta's conjecture, he pointed out the possible existence of parallels between the Nevanlinna theory of complex analysis, and diophantine analysis in the circle of ideas around the Mordell conjecture and abc conjecture. This suggested the importance of the ''integer solutions'' (affine space) aspect of diophantine equations. Vojta wrote the .dvi-previewer xdvi. He also wrote a vi clone. Education and career He was an undergraduate student at the University of Minnesota, where he became a Putnam Fellow in 1977, and a doctoral student at Harvard University (1983). He currently is a professor in the Department of Mathematics at the University of California, Berkeley. Awards and honors In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AM ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bombieri–Lang Conjecture
In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type. Statement The weak Bombieri–Lang conjecture for surfaces states that if X is a smooth surface of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology on X. The general form of the Bombieri–Lang conjecture states that if X is a positive-dimensional algebraic variety of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology. The refined form of the Bombieri–Lang conjecture states that if X is an algebraic variety of general type defined over a number field k, then there is a dense open subset U of X such that for all number field extensions k' over k, the set of points in U is finite. History The Bombieri–Lang conjecture was independentl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Lipshitz
Leonard or ''Leo'' is a common English masculine given name and a surname. The given name and surname originate from the Old High German '' Leonhard'' containing the prefix ''levon'' ("lion") from the Greek Λέων ("lion") through the Latin '' Leo,'' and the suffix ''hardu'' ("brave" or "hardy"). The name has come to mean "lion strength", "lion-strong", or "lion-hearted". Leonard was the name of a Saint in the Middle Ages period, known as the patron saint of prisoners. Leonard is also an Irish origin surname, from the Gaelic ''O'Leannain'' also found as O'Leonard, but often was anglicised to just Leonard, consisting of the prefix ''O'' ("descendant of") and the suffix ''Leannan'' ("lover"). The oldest public records of the surname appear in 1272 in Huntingdonshire, England, and in 1479 in Ulm, Germany. Variations The name has variants in other languages: * Anard/Nardu/Lewnardu/Leunardu (Maltese) * Leen, Leendert, Lenard (Dutch) * Lehnertz, Lehnert (Luxembourgish) * Len ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saunders Mac Lane
Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville, Connecticut, Taftville.. He was christened "Leslie Saunders MacLane", but "Leslie" fell into disuse because his parents, Donald MacLane and Winifred Saunders, came to dislike it. He began inserting a space into his surname because his first wife found it difficult to type the name without a space. He was the eldest of three brothers; one of his brothers, Gerald MacLane, also became a mathematics professor at Rice University and Purdue University. Another sister died as a baby. His father and grandfather were both ministers; his grandfather had been a Presbyterian, but was kicked out of the church for believing in evolution, and his father was a Congregational church, Congregationalist. His mother, Winifre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
