Quadratic Diophantine Equations
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Quadratic Diophantine Equations
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
Author: J. Shanthi
language: en
Publisher: Deep Science Publishing
Release Date: 2025-02-26
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.
A Journey Through the Realm of Numbers
This book takes the reader on a journey from familiar high school mathematics to undergraduate algebra and number theory. The journey starts with the basic idea that new number systems arise from solving different equations, leading to (abstract) algebra. Along this journey, the reader will be exposed to important ideas of mathematics, and will learn a little about how mathematics is really done. Starting at an elementary level, the book gradually eases the reader into the complexities of higher mathematics; in particular, the formal structure of mathematical writing (definitions, theorems and proofs) is introduced in simple terms. The book covers a range of topics, from the very foundations (numbers, set theory) to basic abstract algebra (groups, rings, fields), driven throughout by the need to understand concrete equations and problems, such as determining which numbers are sums of squares. Some topics usually reserved for a more advanced audience, such as Eisenstein integers or quadratic reciprocity, are lucidly presented in an accessible way. The book also introduces the reader to open source software for computations, to enhance understanding of the material and nurture basic programming skills. For the more adventurous, a number of Outlooks included in the text offer a glimpse of possible mathematical excursions. This book supports readers in transition from high school to university mathematics, and will also benefit university students keen to explore the beginnings of algebraic number theory. It can be read either on its own or as a supporting text for first courses in algebra or number theory, and can also be used for a topics course on Diophantine equations.