A Collection Of Special Binary And Ternary Quadratic Diophantine Equation With Integer Solutions And Properties
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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.
Polynomial Diophantine Equations
This book proposes a novel approach to the study of Diophantine equations: define an appropriate version of the equation’s size, order all polynomial Diophantine equations by size, and then solve the equations in order. Natural questions about the solution set of Diophantine equations are studied in this book using this approach. Is the set empty? Is it finite or infinite? Can all integer solutions be parametrized? By ordering equations by size, the book attempts to answer these questions in a systematic manner. When the size grows, the difficulty of finding solutions increases and the methods required to determine solutions become more advanced. Along the way, the reader will learn dozens of methods for solving Diophantine equations, each of which is illustrated by worked examples and exercises. The book ends with solutions to exercises and a large collection of open problems, often simple to write down yet still unsolved. The original approach pursued in this book makes it widely accessible. Many equations require only high school mathematics and creativity to be solved, so a large part of the book is accessible to high school students, especially those interested in mathematical competitions such as olympiads. The main intended audience is undergraduate students, for whom the book will serve as an unusually rich introduction to the topic of Diophantine equations. Many methods from the book will be useful for graduate students, while Ph.D. students and researchers may use it as a source of fascinating open questions of varying levels of difficulty.
Reduction Theory and Arithmetic Groups
Author: Joachim Schwermer
language: en
Publisher: Cambridge University Press
Release Date: 2022-12-15
Arithmetic groups are generalisations, to the setting of algebraic groups over a global field, of the subgroups of finite index in the general linear group with entries in the ring of integers of an algebraic number field. They are rich, diverse structures and they arise in many areas of study. This text enables you to build a solid, rigorous foundation in the subject. It first develops essential geometric and number theoretical components to the investigations of arithmetic groups, and then examines a number of different themes, including reduction theory, (semi)-stable lattices, arithmetic groups in forms of the special linear group, unipotent groups and tori, and reduction theory for adelic coset spaces. Also included is a thorough treatment of the construction of geometric cycles in arithmetically defined locally symmetric spaces, and some associated cohomological questions. Written by a renowned expert, this book is a valuable reference for researchers and graduate students.