Numbers from all Angles / by J. J. P. Veerman.

Format:

Book

Author/Creator:

Veerman, J. J. P.

Series:

Mathematics and Statistics Series

Language:

English

Subjects (All):

Number theory.

Number Theory.

Local Subjects:

Number Theory.

Physical Description:

1 online resource (733 pages)

Edition:

1st ed. 2026.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2026.

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.

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.

Notes:

Print version record.

ISBN:

9783032100009