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.
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Book Description
Editorial Reviews
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"It is high time for such a book to appear...professional mathematicians and even experts in the subject will find it useful as well...Smart did a good job,and his book will efficiently serve as a textbook for beginners and as a reference source for the experts." Mathematical Reviews
"It is a real pleasure to read this book, mainly because the author gives many examples and many practical remarks concerning the effective solution of diophantine equations. ...this is a very attractive book, full of concrete information, which gives a very clear and lucid view of the current knowledge." Bulletin of the AMS
"It is a real pleasure to read this book, mainly because the author gives many examples and many practical remarks concerning the effective solution of diophantine equations. ...this is a very attractive book, full of concrete information, which gives a very clear and lucid view of the current knowledge." Bulletin of the AMS
Book Description
A coherent modern account of the computational methods used to solve diophantine equations. The emphasis is on approaches with wide ranging applications. After a brief introduction, the first section considers basic techniques. The second section explores problems which can be solved using Baker's theory of linear forms in logarithms. The final section looks at problems associated with curves. Useful exercises and a detailed bibliography are included.Requiring a basic knowledge of number theory, this book will appeal to graduate students and research workers.
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First Sentence:
This book shall concern itself with the study of modern methods for solving diophantine equations. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
cyclic quintic field, coprime values, discriminant form equations, quartic number fields, soluble everywhere, index form equations, elliptic logarithms, power integral bases, removing lemma, binary quartic form, prime ideal lying, exceptional units, reduced quadratic irrational, naive height, canonical height, knapsack weights, three logarithms, decomposable form, integral points, one prime ideals, solving diophantine equations, quartic forms, given discriminant, subexponential time, approximation lattices Key Phrases - Capitalized Phrases (CAPs): (learn more)
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This book shall concern itself with the study of modern methods for solving diophantine equations. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
cyclic quintic field, coprime values, discriminant form equations, quartic number fields, soluble everywhere, index form equations, elliptic logarithms, power integral bases, removing lemma, binary quartic form, prime ideal lying, exceptional units, reduced quadratic irrational, naive height, canonical height, knapsack weights, three logarithms, decomposable form, integral points, one prime ideals, solving diophantine equations, quartic forms, given discriminant, subexponential time, approximation lattices Key Phrases - Capitalized Phrases (CAPs): (learn more)
Prove Lemma New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 26
books:
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- Number Theory by Helmut Hasse in Back Matter (1), Back Matter (2), and Back Matter (3)
- Number theory, Volume 20 (Pure and Applied Mathematics) by Author Unknown in Back Matter
- Local Fields (London Mathematical Society Student Texts) by J. W. S. Cassels in Back Matter
- Exponential Diophantine Equations (Cambridge Tracts in Mathematics) by T. N. Shorey in Back Matter
- The Higher Arithmetic: An Introduction to the Theory of Numbers by H. Davenport in Back Matter
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