Features Uses techniques from widely diverse areas of mathematics, including number theory, calculus, set theory, complex analysis, linear algebra, and the theory of computation. Suitable as a primary textbook for advanced undergraduate courses in number theory, or as supplementary reading for interested postgraduates. Each chapter concludes with an appendix setting out the basic facts needed from each topic, so that the book is accessible to readers without any specific specialist background.
In this monograph, Ivan Niven provides a masterful exposition of some central results on irrational, transcendental, and normal numbers. He gives a complete treatment by elementary methods of the irrationality of the exponential, logarithmic, and trigonometric functions with rational arguments. The approximation of irrational numbers by rationals, up to such results as the best possible approximation of Hurwitz, is also given with elementary techniques. The last third of the monograph treats normal and transcendental numbers, including the transcendence of p and its generalization in the Lindermann theorem, and the Gelfond-Schneider theorem. Most of the material in the first two thirds of the book presupposes only calculus and beginning number theory. The book is almost wholly self-contained. The results needed from analysis and algebra are central and well-known theorems, and complete references to standard works are given to help the beginner. The chapters are, for the most part, independent. There is a set of notes at the end of each chapter citing the main sources used by the author and suggesting further reading.
This book is a survey of the most important directions of research in transcendental number theory. For readers with no specific background in transcendental number theory, the book provides both an overview of the basic concepts and techniques and also a guide to the most important results and references.
This textbook presents an elementary introduction to number theory and its different aspects: approximation of real numbers, irrationality and transcendence problems, continued fractions, diophantine equations, quadratic forms, arithmetical functions and algebraic number theory. Clear, concise, and self-contained, the topics are covered in 12 chapters with more than 200 solved exercises. The textbook may be used by undergraduates and graduate students, as well as high school mathematics teachers. More generally, it will be suitable for all those who are interested in number theory, the fascinating branch of mathematics.
Contains a collection of papers devoted primarily to transcendental number theory and diophantine approximations. This title includes a text of the author's invited address on his work on the theory of transcendental numbers to the 1978 International Congress of Mathematicians in Helsinki.
"Irrationality and Transcendence in Number Theory tells the story of irrational numbers from their discovery in the days of Pythagoras to the ideas behind the work of Baker and Mahler on transcendence in the 20th century. It focuses on themes of irrationality, algebraic and transcendental numbers, continued fractions, approximation of real numbers by rationals, and relations between automata and transcendence. This book serves as a guide and introduction to number theory for advanced undergraduates and early postgraduates. Readers are led through the developments in number theory from ancient to modern times. The book includes a wide range of exercises, from routine problems to surprising and thought-provoking extension material. Features: Uses techniques from widely diverse areas of mathematics, including number theory, calculus, set theory, complex analysis, linear algebra, and the theory of computation. Suitable as a primary textbook for advanced undergraduate courses in number theory, or as supplementary reading for interested postgraduates. Each chapter concludes with an appendix setting out the basic facts needed from each topic, so that the book is accessible to readers without any specific specialist background"--
In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research. Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory. Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class.
Gathered from the 2016 Gainesville Number Theory Conference honoring Krishna Alladi on his 60th birthday, these proceedings present recent research in number theory. Extensive and detailed, this volume features 40 articles by leading researchers on topics in analytic number theory, probabilistic number theory, irrationality and transcendence, Diophantine analysis, partitions, basic hypergeometric series, and modular forms. Readers will also find detailed discussions of several aspects of the path-breaking work of Srinivasa Ramanujan and its influence on current research. Many of the papers were motivated by Alladi's own research on partitions and q-series as well as his earlier work in number theory. Alladi is well known for his contributions in number theory and mathematics. His research interests include combinatorics, discrete mathematics, sieve methods, probabilistic and analytic number theory, Diophantine approximations, partitions and q-series identities. Graduate students and researchers will find this volume a valuable resource on new developments in various aspects of number theory.
