Diophantine equation oor Sjinees

Diophantine equation

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(number theory) A polynomial equation whose variables are only permitted to assume integer values

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polynomial equation
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polynomial equation
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diophantine equation

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polynomial equation with integer solutions
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The reason of the interest for Diophantine equations, in the elliptic curve case, is that K may not be algebraically closed.
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In the late 5th century, the Indian mathematician and astronomer Aryabhata described the algorithm as the "pulverizer", perhaps because of its effectiveness in solving Diophantine equations.
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That is, for this case we should distinguish C that have genus 1, from elliptic curves E that have a K-point (or, in other words, provide a Diophantine equation that has a solution in K).
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He discovered Newton's identities, Newton's method, classified cubic plane curves (polynomials of degree three in two variables), made substantial contributions to the theory of finite differences, and was the first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations.
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At this time he was particularly fascinated by solving Diophantine equations.
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Approximation of algebraic numbers by rationals and applications thereof to Diophantine equations .
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Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.
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Instead of asking whether a given Diophantine equation has a solution, ask "for what equations do known methods yield the answer?"
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The reason of the interest for Diophantine equations , in the elliptic curve case, is that K may not be algebraically closed .
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A Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integral solutions are required.
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It was A Thue who made the breakthrough to general results by proving in 1909 that all Diophantine equations of the form
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Skolem was remarkably productive publishing around 180 papers on topics such as Diophantine equations , mathematical logic, group theory , lattice theory and set theory.
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In addition Poinsot worked on number theory and on this topic he studied Diophantine equations, how to express numbers as the difference of two squares and primitive roots.
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At the most advanced level he wrote a monograph Analytic methods for Diophantine equations and Diophantine inequalities (1962) which includes many of his contributions extending the Hardy - Littlewood method.
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Turán goes on to say that Carl Siegel and Klaus Roth generalised the classes of Diophantine equations for which these conclusions would hold and even bounded the number of solutions.
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Robinson was awarded a doctorate in 1948 and that same year started work on Hilbert 's Tenth Problem: find an effective way to determine whether a Diophantine equation is soluble.
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[Diophantine equations ], carrying a history of more than one thousand years, was, until the early years of this century, little more than a collection of isolated problems subjected to ingenious ad hoc methods.
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In 1971 at a conference in Bucharest Robinson gave a lecture Solving diophantine equations in which she set the agenda for continuing to study Diophantine equations following the negative solution to Hilbert 's Tenth Problem problem.
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That is, for this case we should distinguish C that have genus 1, from elliptic curves E that have a K -point (or, in other words, provide a Diophantine equation that has a solution in K ).
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The chapter headings are: Lucas ' functions; Tests for primality; Continued fractions; Bernoulli numbers and polynomials; Diophantine equations ; Numerical functions; Matrices; Power residues; Analytic number theory ; Partitions; Modular forms; Cyclotomy; Combinatorics; Sieves; Equation solving; Computing techniques; and Miscellaneous.
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Hilbert in 1900 posed the problem of finding a method for solving Diophantine equations as the 10th problem on his famous list of 23 problems which he believed should be the major challenges for mathematical research this century.
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The last part of the book describes Alan Baker 's work on linear forms in the logarithms of algebraic numbers and its applications to Diophantine equations and to the determination of imaginary quadratic fields with class number 1 or 2.
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Adrain's first papers in the Mathematical Correspondent concerned the steering of a ship and Diophantine algebra (the study of rational solutions to polynomial equations).
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On the latter topic he had become interested in the problem of solving quintic equations algebraically and made some interesting discoveries on Diophantine problems. His death came suddenly.
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Some of the many topics that he covered were diophantine approximations, orders of linear homogeneous groups, theory of geometry of numbers, approximate solutions of the integers of a set of linear equations, low-velocity fire angle, finite collineation groups, and characteristic roots.
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