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Harold Davenport was one of the truly great mathematicians of the twentieth century. Based on lectures he gave at the University of Michigan in the early 1960s, this book is concerned with the use of analytic methods in the study of integer solutions to Diophantine equations and Diophantine inequalities. It provides an excellent introduction to a timeless area of number theory that is still as widely researched today as it was when the book originally appeared. The three main themes of the book are Waring's problem and the representation of integers by diagonal forms, the solubility in integers of systems of forms in many variables, and the solubility in integers of diagonal inequalities. For the second edition of the book a comprehensive foreword has been added in which three prominent authorities describe the modern context and recent developments. A thorough bibliography has also been added. ... Read more

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This book focuses on finiteness conjectures and results in ordinary differential equations (ODEs) and Diophantine geometry. During the past twenty-five years, much progress has been achieved on finiteness conjectures, which are the offspring of the second part of Hilbert's 16th problem. Even in its simplest case, this is one of the very few problems on Hilbert's list which remains unsolved. These results are about existence and estimation of finite bounds for the number of limit cycles occurring in certain families of ODEs. The book describes this progress, the methods used (bifurcation theory, asymptotic expansions, methods of differential algebra, or geometry) and the specific results obtained. The finiteness conjectures on limit cycles are part of a larger picture that also includes finiteness problems in other areas of mathematics, in particular those in Diophantine geometry where remarkable results were proved during the same period of time. There is a chapter devoted to finiteness results in Diophantine geometry obtained by using methods of differential algebra, which is a connecting element between these parallel developments in the book. The volume can be used as an independent study text for advanced undergraduates and graduate students studying ODEs or applications of differential algebra to differential equations and Diophantine geometry. It is also a good entry point for researchers interested these areas, in particular, in limit cycles of ODEs, and in finiteness problems. Contributors to the volume include Andrei A. Bolibrukh and Alexandru Buium. Available from the AMS by A. Buium is Arithmetic Differential Equations, as Volume 118 in the Mathematical Surveys and Monographs series. ... Read more

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Series: Dolciani Mathematical Expositions

The first part of the book presents the elementary facts of algebraic geometry essential to understanding the rest of it. The second half of the book considers the evolution of the theory of Diophantine equations from the Renaissance to the middle of the 20th century.In particular, the book includes substantial descriptions of the relevant contributions of Viète, Fermat, Euler, Jacobi, and Poincaré.The book ends with Joseph Silverman’s survey of Diophantine analysis during the last 20 years in which he mentions the proof of the Mordei conjecture and of Fermat’s Last Theorem. ... Read more

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This is a integrated presentation of the theory of exponential diophantine equations. The authors present, in a clear and unified fashion, applications to exponential diophantine equations and linear recurrence sequences of the Gelfond-Baker theory of linear forms in logarithms of algebraic numbers. Topics covered include the Thue equations, the generalised hyperelliptic equation, and the Fermat and Catalan equations. The necessary preliminaries are given in the first three chapters. Each chapter ends with a section giving details of related results. ... Read more

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This monograph investigates algorithms for determining power integral bases in algebraic number fields. The problem has classical roots and leads to the problem of solving the corresponding index form equations that are often reduced to more classical equations, such as various types of Thue equations.

The reader is introduced to the best-known methods for solving several types of diophantine equations using Baker-type estimates, reduction methods, and enumeration algorithms. These methods can be useful for other types of diophantine equations not included in the book.

Several interesting properties of number fields are examined. Some infinite parametric families of fields are also considered as well as the resolution of the corresponding infinite parametric families of diophantine equations. The text is illustrated with several tables of various number fields, including their data on power integral bases.

Advanced undergraduates and graduates will benefit from this exposition of methods for solving some classical types of diophantine equations. Researchers in the field will find new applications for the tools presented throughout the book. ... Read more

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The author had initiated a revision and translation of"Classical Diophantine Equations" prior to his death.Given the rapid advances intranscendence theory anddiophantine approximation over recent years, one might fearthat the present work, originally published in Russian in1982, is mostly superseded. That is not so. A certain amountofupdating had been prepared by the author himself beforehis untimely death. Some further revision was prepared byclose colleagues.The first seven chapters provide a detailed, virtuallyexhaustive, discussion of the theory of lower bounds forlinear forms in the logarithms of algebraic numbers and itsapplications to obtaining upper bounds for solutions to theeponymous classical diophantine equations. The detail mayseem stark--- the author fears that the reader may reactmuch as does the tourist onfirst seeing the centrePompidou; notwithstanding that, Sprind zuk maintainsapleasant and chatty approach, full of wise and interestingremarks. His emphases well warrant, now that the bookappears in English, close studyand emulation. In particularthose emphases allow him to devote the eighthchapter to ananalysis of the interrelationship of the class number ofalgebraic number fields involved and the bounds on theheights of thesolutions of the diophantine equations. Thoseideas warrant further development. The final chapter dealswith effective aspects of the HilbertIrreducibilityTheorem, harkening back to earlier work of the author. Thereis no other congenial entry point to the ideas of the lasttwochapters in the literature. ... Read more

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Beginning with a brief introduction to algorithms and diophantine equations, this volume provides a coherent modern account of the methods used to find all the solutions to certain diophantine equations, particularly those developed for use on a computer. The study is divided into three parts, emphasizing approaches with a wide range of applications. The first section considers basic techniques including local methods, sieving, descent arguments and the LLL algorithm. The second section explores problems that can be solved using Baker's theory of linear forms in logarithms. The final section looks at problems associated with curves, focusing on rational and integral points on elliptic curves. Each chapter concludes with a useful set of exercises. A detailed bibliography is included. This book will appeal to graduate students and research workers interested in solving diophantine equations using computational methods. ... Read more

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The classical circle method of Hardy and Littlewood is one of the most effective methods of additive number theory. Two examples are its success with Waring's problem and Goldbach's conjecture. In this book, Wang offers instances of generalizations of important results on diophantine equations and inequalities over rational fields to algebraic number fields. The book also contains an account of Siegel's generalized circle method and its applications to Waring's problem and additive equations and an account of Schmidt's method on diophantine equations and inequalities in several variables in algebraic number fields. ... Read more

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"This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers." (L'Enseignement Mathematique) "The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well." Acta Scientiarum Mathematicarum ... Read more

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Diophantine equations over number fields have formed one of the most important and fruitful areas of mathematics throughout civilisation. In recent years increasing interest has been aroused in the analogous area of equations over function fields. However, although considerable progress has been made by previous authors, none has attempted the central problem of providing methods for the actual solution of such equations. The latter is the purpose and achievement of this volume: algorithms are provided for the complete resolution of various families of equations, such as those of Thue, hyperelliptic and genus one type. The results are achieved by means of an original fundamental inequality, first announced by the author in 1982. Several specific examples are included as illustrations of the general method and as a testimony to its efficiency. Furthermore, bounds are obtained on the solutions which improve on those obtained previously by other means. Extending the equality to a different setting, namely that of positive characteristic, enables the various families of equations to be resolved in that circumstance. Finally, by applying the inequality in a different manner, simple bounds are determined on their solutions in rational functions of the general superelliptic equation. This book represents a self-contained account of a new approach to the subject, and one which plainly has not reached the full extent of its application. It also provides a more direct on the problems than any previous book. Little expert knowledge is required to follow the theory presented, and it will appeal to professional mathematicians, research students and the enthusiastic undergraduate. ... Read more

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The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.

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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. ... Read more

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In the late sixties Matiyasevich, building on the work of Davis, Putnam and Robinson, showed that there was no algorithm to determine whether a polynomial equation in several variables and with integer coefficients has integer solutions. Hilbert gave finding such an algorithm as problem number ten on a list he presented at an international congress of mathematicians in 1900. Thus the problem, which has become known as Hilbert's Tenth Problem, was shown to be unsolvable. This book presents an account of results extending Hilbert's Tenth Problem to integrally closed subrings of global fields including, in the function field case, the fields themselves. While written from the point of view of Algebraic Number Theory, the book includes chapters on Mazur's conjectures on topology of rational points and Poonen's elliptic curve method for constructing a Diophatine model of rational integers over a 'very large' subring of the field of rational numbers. ... Read more

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This book contains short notes and articles, as well asstudies on several topics of Geometry and Number Theory.The materialis divided into five chapters: Geometric theorems; Diophantineequations; Arithmetic functions; Divisibility properties of numbersand functions; and Some irrationality results.Many papers study theSmarandache type functions and Pseudo-Smarandache function,conjectures. ... Read more

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Chapters: Pythagorean Triple. Source: Wikipedia. Pages: 124. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula ; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides and c = 2 is right, but (1, 1, 2) is not a Pythagorean triple because 2 is not an integer. Moreover, 1 and 2 do not have an integer common multiple because 2 is irrational. There are 16 primitive Pythagorean triples with : A scatter plot of the legs (a,b) of the Pythagorean triples with c less than 6000. Negative values are included to illustrate the parabolic patterns in the plot more clearly. A plot of triples generated by Euclid's formula map out part of the z = x + y cone. A constant m or n trace out part of a parabola on the cone.Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers m and n with . The formula states that the integers form a Pythagorean triple. The triple generated by Euclid's formula is primitive if and only if m and n are coprime and exactly one of them is even. If both n and m are odd, then a, b, and c will be even, and so the triple will not be primitive. Every primitive triple (possibly after exchanging a and b) arises from a unique pair of coprime numbers m, n, one of whi...More: http://booksllc.net/?id=24172 ... Read more

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High Quality Content by WIKIPEDIA articles!In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation whose solutions are restricted to be polynomials in the indeterminate. A Diophantine equation, in general, is one where the solutions are restricted to some algebraic system, typically integers. Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made initial studies of integer Diophantine equations.A necessary and sufficient condition for a polynomial Diophantine equation to have a solution is for c to be a multiple of the GCD of a and b. In the example above, the GCD of a and b was 1, so solutions would exist for any value of c. ... Read more