The author had initiated a revision and translation of "Classical Diophantine Equations" prior to his death. Given the rapid advances in transcendence theory and diophantine approximation over recent years, one might fear that the present work, originally published in Russian in 1982, is mostly superseded. That is not so. A certain amount of updating had been prepared by the author himself before his untimely death. Some further revision was prepared by close colleagues. The first seven chapters provide a detailed, virtually exhaustive, discussion of the theory of lower bounds for linear forms in the logarithms of algebraic numbers and its applications to obtaining upper bounds for solutions to the eponymous classical diophantine equations. The detail may seem stark--- the author fears that the reader may react much as does the tourist on first seeing the centre Pompidou; notwithstanding that, Sprind zuk maintainsa pleasant and chatty approach, full of wise and interesting remarks. His emphases well warrant, now that the book appears in English, close studyand emulation. In particular those emphases allow him to devote the eighth chapter to an analysis of the interrelationship of the class number of algebraic number fields involved and the bounds on the heights of thesolutions of the diophantine equations. Those ideas warrant further development. The final chapter deals with effective aspects of the Hilbert Irreducibility Theorem, harkening back to earlier work of the author. There is no other congenial entry point to the ideas of the last two chapters in the literature.

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.

"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

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.

Work examines the latest algorithms and tools to solve classical types of diophantine equations.; Unique book---closest competitor, Smart, Cambridge, does not treat index form equations.; Author is a leading researcher in the field of computational algebraic number theory.; The text is illustrated with several tables of various number fields, including their data on power integral bases.; 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.

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.

Work examines the latest algorithms and tools to solve classical types of diophantine equations.; Unique book---closest competitor, Smart, Cambridge, does not treat index form equations.; Author is a leading researcher in the field of computational algebraic number theory.; The text is illustrated with several tables of various number fields, including their data on power integral bases.; 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.

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.

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.

The exposition of the classical theory of algebraic numbers is clear and thorough, and there is a large number of exercises as well as worked out numerical examples. A careful study of this book will provide a solid background to the learning of more recent topics.