Quadratic Diophantine Equations
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Quadratic Diophantine Equations
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Author : Titu Andreescu
language : en
Publisher: Springer
Release Date : 2015-06-29
Quadratic Diophantine Equations written by Titu Andreescu and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-06-29 with Mathematics categories.
This text treats the classical theory of quadratic diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. The presentation features two basic methods to investigate and motivate the study of quadratic diophantine equations: the theories of continued fractions and quadratic fields. It also discusses Pell’s equation and its generalizations, and presents some important quadratic diophantine equations and applications. The inclusion of examples makes this book useful for both research and classroom settings.
A Collection Of Special Binary And Ternary Quadratic Diophantine Equation With Integer Solutions And Properties
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Author : J. Shanthi
language : en
Publisher: Deep Science Publishing
Release Date : 2025-02-26
A Collection Of Special Binary And Ternary Quadratic Diophantine Equation With Integer Solutions And Properties written by J. Shanthi and has been published by Deep Science Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-02-26 with Mathematics categories.
One of the areas of Number theory that has attracted many mathematicians since antiquity is the subject of diophantine equations. A diophantine equation is a polynomial equation in two or more unknowns such that only the integer solutions are determined. No doubt that diophantine equation possess supreme beauty and it is the most powerful creation of the human spirit. A pell equation is a type of non-linear diophantine equation in the form where and square-free. The above equation is also called the Pell-Fermat equation. In Cartesian co-ordinates, this equation has the form of a hyperbola. The binary quadratic diophantine equation having the form Eqn. is referred to as the positive form of the pell equation and the form Eqn. is called the negative form of the pell equation or related pell equation. It is worth to remind that (2) is solvable for only certain values of D and always in the case of (1). An obvious generalisation to the Pell equation is the equation of the form which is known as Pell-like equation. Pell equations arise in the investigation of numbers which are figurate in more than one way, for example, simultaneously square & triangular and as such they are extremely important in Number theory. In the solution of cubic equation and in certain other situations it is desirable to have a method for extracting the cube root of a binomial surd. This may be accomplished by the aid of the pell equation. We use pell equation to solve Archimedes’ Cattle problem. Pell’s equation is connected to algebraic number theory, Chebyshev polynomials and continued fractions. Other applications include solving problems involving double equations, rational approximations to square roots, sums of consecutive integers, Pythagorean triangles with consecutive legs, consecutive Heronian triangles, sums of and consecutive squares and so on. Man’s love for numbers is perhaps older than number theory. The love for large numbers may be a motivation for pellian equation. In studies on Diophantine equations of degree two with two and three unknowns, significant success was attained only in the twentieth century. There has been interest in determining all solutions in integers to quadratic Diophantine equations among mathema6ticians. The main thrust in this book is on solving second degree Diophantine equations with two and three variables. This book contains a reasonable collection of special quadratic Diophantine problems in two and three variables distributed in 12 chapters. The process of getting different sets of integer solutions to each of the quadratic Diophantine equations considered in this book are illustrated in an elegant manner. The articles with solutions and properties presented in chapters 1, 2 & 3 are Pell equations and in chapters 4,5 &6 are Pell-like equations. The articles with solutions presented in chapters 7-12 are quadratic equations with three unknowns of the form . In Cartesian co-ordinates, this equation has the form of a right circular cone.
Lecture Notes On Diophantine Analysis
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Author : Umberto Zannier
language : en
Publisher: Springer
Release Date : 2015-05-05
Lecture Notes On Diophantine Analysis written by Umberto Zannier and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-05 with Mathematics categories.
These lecture notes originate from a course delivered at the Scuola Normale in Pisa in 2006. Generally speaking, the prerequisites do not go beyond basic mathematical material and are accessible to many undergraduates. The contents mainly concern diophantine problems on affine curves, in practice describing the integer solutions of equations in two variables. This case historically suggested some major ideas for more general problems. Starting with linear and quadratic equations, the important connections with Diophantine Approximation are presented and Thue's celebrated results are proved in full detail. In later chapters more modern issues on heights of algebraic points are dealt with, and applied to a sharp quantitative treatment of the unit equation. The book also contains several supplements, hinted exercises and an appendix on recent work on heights.
Quadratic Diophantine Equations
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Author : George Leo Watson
language : en
Publisher:
Release Date : 1960
Quadratic Diophantine Equations written by George Leo Watson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1960 with categories.
Small Solutions Of Quadratic Diophantine Equations
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Author : Rainer Dietmann-Sopp
language : en
Publisher:
Release Date : 2001
Small Solutions Of Quadratic Diophantine Equations written by Rainer Dietmann-Sopp and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with categories.
Diophantine Analysis
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Author : Jorn Steuding
language : en
Publisher: CRC Press
Release Date : 2005-05-19
Diophantine Analysis written by Jorn Steuding and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-05-19 with Mathematics categories.
While its roots reach back to the third century, diophantine analysis continues to be an extremely active and powerful area of number theory. Many diophantine problems have simple formulations, they can be extremely difficult to attack, and many open problems and conjectures remain. Diophantine Analysis examines the theory of diophantine approximations and the theory of diophantine equations, with emphasis on interactions between these subjects. Beginning with the basic principles, the author develops his treatment around the theory of continued fractions and examines the classic theory, including some of its applications. He also explores modern topics rarely addressed in other texts, including the abc conjecture, the polynomial Pell equation, and the irrationality of the zeta function and touches on topics and applications related to discrete mathematics, such as factoring methods for large integers. Setting the stage for tackling the field's many open problems and conjectures, Diophantine Analysis is an ideal introduction to the fundamentals of this venerable but still dynamic field. A detailed appendix supplies the necessary background material, more than 200 exercises reinforce the concepts, and engaging historical notes bring the subject to life.
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.
Recurrence Sequences
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Author : Graham Everest
language : en
Publisher: American Mathematical Soc.
Release Date : 2003
Recurrence Sequences written by Graham Everest and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Mathematics categories.
Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.
Advanced Number Theory
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Author : Harvey Cohn
language : en
Publisher: Courier Corporation
Release Date : 1980-08-01
Advanced Number Theory written by Harvey Cohn and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 1980-08-01 with Mathematics categories.
"A very stimulating book ... in a class by itself." — American Mathematical Monthly Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject. The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups. Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints. In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory.
Pell S Equation
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Author : Edward J. Barbeau
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-05-04
Pell S Equation written by Edward J. Barbeau 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 2006-05-04 with Mathematics categories.
Pell's equation is part of a central area of algebraic number theory that treats quadratic forms and the structure of the rings of integers in algebraic number fields. It is an ideal topic to lead college students, as well as some talented and motivated high school students, to a better appreciation of the power of mathematical technique. Even at the specific level of quadratic diophantine equations, there are unsolved problems, and the higher degree analogues of Pell's equation, particularly beyond the third, do not appear to have been well studied. In this focused exercise book, the topic is motivated and developed through sections of exercises which will allow the readers to recreate known theory and provide a focus for their algebraic practice. There are several explorations that encourage the reader to embark on their own research. A high school background in mathematics is all that is needed to get into this book, and teachers and others interested in mathematics who do not have (or have forgotten) a background in advanced mathematics may find that it is a suitable vehicle for keeping up an independent interest in the subject.