Solving Diophantine Equations
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An Introduction To Diophantine Equations
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Author : Titu Andreescu
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-09-02
An Introduction To Diophantine Equations written by Titu Andreescu and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-09-02 with Mathematics categories.
This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants — including Olympiad and Putnam competitors — as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques.
Solving Diophantine Equations
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Author : Octavian Cira
language : en
Publisher: Infinite Study
Release Date :
Solving Diophantine Equations written by Octavian Cira and has been published by Infinite Study this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.
In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation η(π(x)) = π(η(x)), where η is the Smarandache function and π is Riemann function of counting the number of primes up to x, in the set of natural numbers? If an analytical method is not available, an idea would be to recall the empirical search for solutions. We establish a domain of searching for the solutions and then we check all possible situations, and of course we retain among them only those solutions that verify our equation. In other words, we say that the equation does not have solutions in the search domain, or the equation has n solutions in this domain. This mode of solving is called partial resolution. Partially solving a Diophantine equation may be a good start for a complete solving of the problem. The authors have identified 62 Diophantine equations that impose such approach and they partially solved them. For an efficient resolution it was necessarily that they have constructed many useful ”tools” for partially solving the Diophantine equations into a reasonable time. The computer programs as tools were written in Mathcad, because this is a good mathematical software where many mathematical functions are implemented. Transposing the programs into another computer language is facile, and such algorithms can be turned to account on other calculation systems with various processors.
Diophantine Equations
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Author :
language : en
Publisher: Academic Press
Release Date : 1969
Diophantine Equations written by and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1969 with Mathematics categories.
Diophantine Equations
Biquadratic Diophantine Equations With Integer Solutions
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Author : Prof. Dr. M.A. Gopalan
language : en
Publisher: KY Publications
Release Date : 2018-07-01
Biquadratic Diophantine Equations With Integer Solutions written by Prof. Dr. M.A. Gopalan and has been published by KY Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-07-01 with Mathematics categories.
The theory of Diophantine equation is an ancient subject that typically involves solving polynomial equations in integers. It is well known that a Diophantine equation is an equation with integer coefficient and multiple variables ( 2) having integer solutions. There is no universal method available to know whether a Diophantine equation has a solutions or finding all solutions, if it exists. Proving that even simple Diophantine equations have no solutions may require very sophisticated methods and in such cases, a lot of deep and beautiful mathematics get generated as a result. It is worth to observe that Diophatine equations are rich in variety. A collection of special Problems on biquadratic equations in 3,4,5 & 6 variables has been treated in sections A to D respectively. Different sets of integer solutions to each of the biquadratic diophatine equations are illustrated.
Diophantine Equations And Systems
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Author : Demetrios P Kanoussis Ph D
language : en
Publisher: Independently Published
Release Date : 2024-03-08
Diophantine Equations And Systems written by Demetrios P Kanoussis Ph D and has been published by Independently Published this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-03-08 with Mathematics categories.
Diophantine equations are polynomial equations with integer coefficients for which only integer solutions are sought. In his great work "Arithmetica", the Greek mathematician Diophantus of Alexandria, (born in Alexandria Egypt in 200 AD and died in 284 AD), known as the father of Algebra, studied and solved such types of equations, (integer coefficients and integer solutions), of the first up to the fourth degree. These equations are now known as "Diophantine equations". A characteristic feature of Diophantine equations is that in these equations the number of equations is smaller than the number of unknowns. For example, we may have one equation with two unknowns, or one equation with three unknowns, or a system of two equations with three unknowns, etc. While in the set of real numbers R these types of equations, (fewer equations than number of unknowns), are indeterminate, in the set of integers Z={... -3, -2, -1,0,1,2,3, ...} or in the set of natural numbers N={1,2,3,4, ...}, these equations may or may not have integer solutions, (depending on the coefficients of the equations). In this book we provide a systematic introduction to Diophantine equations, with emphasis on the solution of various problems. The first two chapters are devoted to first degree Diophantine equations and systems, (linear equations and systems), while the third chapter is devoted to second degree Diophantine equations and systems. Among other equations, in this chapter, we study the Pythagorean equation (x^2+y^2=z^2), and the Pell's equation (x^2-ky^2=1). The solution of Pell's equation is achieved by a really brilliant method, which is attributed to Lagrange. Various examples of higher degree Diophantine equations are considered in chapter 4. The analytic description of the material covered in this book can be found in the table of contents. The book is concluded with a collection of 40 miscellaneous, challenging problems, with answers and detailed remarks and hints. In total, the book contains 55 solved examples and 105 problems for solution.
Solving The Pell Equation
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Author : Michael Jacobson
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-12-02
Solving The Pell Equation written by Michael Jacobson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-12-02 with Mathematics categories.
Pell’s Equation is a very simple Diophantine equation that has been known to mathematicians for over 2000 years. Even today research involving this equation continues to be very active, as can be seen by the publication of at least 150 articles related to this equation over the past decade. However, very few modern books have been published on Pell’s Equation, and this will be the first to give a historical development of the equation, as well as to develop the necessary tools for solving the equation. The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell’s Equation. The only prerequisites are a basic knowledge of elementary number theory and abstract algebra. There are also numerous references and notes for those who wish to follow up on various topics.
Various Methods For Solving Diophantine Equations
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Author : George Frederick Woodson
language : en
Publisher:
Release Date : 1927
Various Methods For Solving Diophantine Equations written by George Frederick Woodson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1927 with Diophantine analysis categories.
The Algorithmic Resolution Of Diophantine Equations
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Author : Nigel P. Smart
language : en
Publisher: Cambridge University Press
Release Date : 1998-11-12
The Algorithmic Resolution Of Diophantine Equations written by Nigel P. Smart and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-11-12 with Mathematics categories.
A coherent account of the computational methods used to solve diophantine equations.
Elliptic Diophantine Equations
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Author : Nikos Tzanakis
language : en
Publisher: Walter de Gruyter
Release Date : 2013-08-29
Elliptic Diophantine Equations written by Nikos Tzanakis and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-08-29 with Mathematics categories.
This book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art. The monograph presents a well-balanced combination of a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis), clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals), ready-to-use computer packages. A result is the solution in practice of a large general class of Diophantine equations.
An Introduction To Diophantine Equations
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Author : Titu Andreescu
language : en
Publisher:
Release Date : 2002
An Introduction To Diophantine Equations written by Titu Andreescu and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Diophantine equations categories.