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Motivated by curiosity and a deep love for the subject, this self-contained number theory text is designed primarily for advanced undergraduates and graduate students. It assumes a level of mathematical maturity found among students in physics, engineering, and mathematics. Covering the content of a full-year number theory course, the book can serve either as a primary textbook or as a supplementary reference in advanced topics courses. With its comprehensive scope and depth, it also offers an efficient resource for anyone seeking to explore the central currents of number theory.
The exposition begins with elementary concepts and gradually advances to more sophisticated material. Proofs are presented in full detail, ensuring clarity and rigor. The selection of topics is broad, and over 150 illustrations provide visual insight, particularly where geometry enriches understanding. More than 400 exercises, ranging in difficulty, are included to reinforce mastery of the material.
The book is organized into three parts. Part I introduces topics typically encountered in an advanced undergraduate course in number theory, with the later sections of the part extending to graduate-level material. Part II presents the foundations of major branches of number theory, including algebraic, analytic, ergodic, and probabilistic approaches. Part III covers advanced results, featuring proofs of the prime number theorem, the Birkhoff ergodic theorem, and the unsolvability of the general quintic, among others. The author also discusses possible uses of the book in non-number theory courses and in fields outside mathematics. Nine appendices supplement the main text with related material that, while valuable, would otherwise disrupt the narrative flow.
List of contents
Preface.- Part 1 Introduction to Number Theory.- 1. A Quick Tour of Number Theory.- 2. The Fundamental Theorem of Arithmetic.- 3. Linear Diophantine Equations.- 4. Number Theoretic Functions.- 5. Modular Arithmetic and Primes.- Part 2 Current in Number Theory: Algebraic, Probabilistic, and Analytic.- 6. Continued Fractions.- 7. Fields, Rings, and Ideals.- 8. Factorization in Rings.- 9. Ergodic Theory.- 10. Three Maps and the Real Numbers.- Part 3 Topics in Number Theory.- 11. The Cauchy Integral Formula.- 12. The Prime Number Theorem.- 13. Primes in Arithmetic Progressions.- 14. The Birkhoff Ergodic Theorem.- 15. The Unsolvability of the Quintic.- A. The Metallic Means.- B. Three Gaps and Denjoy-Koksma.- C. Prime Towers.- D. The Logarithm as a Moving Constant.- Bibliography.- Index.
About the author
J.J.P. Veerman is professor of mathematics at Portland State University. He received his Ph.D. from Cornell University in 1986. After postdocs in Spain and at Cornell University/Rockefeller University, he held visiting positions in the U.S. (Rockefeller University, CUNY, Stony Brook University, Georgia Tech, Penn State), as well as in Spain (Autóma Madrid, Autónoma Barcelona), Brazil (IMPA, PUC-Rio, UFPe). He came to Portland State University in 2000. He has since held visiting positions in Spain (Granada), Italy (Pisa, Salerno), Greece (University of Crete), and Rockefeller University in NYC, the Weizmann Institute (Israel), among others.
Summary
Motivated by curiosity and a deep love for the subject, this self-contained number theory text is designed primarily for advanced undergraduates and graduate students. It assumes a level of mathematical maturity found among students in physics, engineering, and mathematics. Covering the content of a full-year number theory course, the book can serve either as a primary textbook or as a supplementary reference in advanced topics courses. With its comprehensive scope and depth, it also offers an efficient resource for anyone seeking to explore the central currents of number theory.
The exposition begins with elementary concepts and gradually advances to more sophisticated material. Proofs are presented in full detail, ensuring clarity and rigor. The selection of topics is broad, and over 150 illustrations provide visual insight, particularly where geometry enriches understanding. More than 400 exercises, ranging in difficulty, are included to reinforce mastery of the material.
The book is organized into three parts. Part I introduces topics typically encountered in an advanced undergraduate course in number theory, with the later sections of the part extending to graduate-level material. Part II presents the foundations of major branches of number theory, including algebraic, analytic, ergodic, and probabilistic approaches. Part III covers advanced results, featuring proofs of the prime number theorem, the Birkhoff ergodic theorem, and the unsolvability of the general quintic, among others. The author also discusses possible uses of the book in non-number theory courses and in fields outside mathematics. Nine appendices supplement the main text with related material that, while valuable, would otherwise disrupt the narrative flow.
