Elementary Number Theory
- Length: 254 pages
- Edition: 1
- Language: English
- Publisher: Chapman and Hall/CRC
- Publication Date: 2021-09-09
- ISBN-10: 1032017236
- ISBN-13: 9781032017235
- Sales Rank: #0 (See Top 100 Books)
Elementary Number Theory, Gove Effinger, Gary L. Mullen
This text is intended to be used as an undergraduate introduction to the theory of numbers. The authors have been immersed in this area of mathematics for many years and hope that this text will inspire students (and instructors) to study, understand, and come to love this truly beautiful subject.
Each chapter, after an introduction, develops a new topic clearly broken out in sections which include theoretical material together with numerous examples, each worked out in considerable detail. At the end of each chapter, after a summary of the topic, there are a number of solved problems, also worked out in detail, followed by a set of supplementary problems. These latter problems give students a chance to test their own understanding of the material; solutions to some but not all of them complete the chapter.
The first eight chapters discuss some standard material in elementary number theory. The remaining chapters discuss topics which might be considered a bit more advanced. The text closes with a chapter on Open Problems in Number Theory. Students (and of course instructors) are strongly encouraged to study this chapter carefully and fully realize that not all mathematical issues and problems have been resolved! There is still much to be learned and many questions to be answered in mathematics in general and in number theory in particular.
Cover Half Title Series Page Title Page Copyright Page Contents Preface 1. Divisibility in the Integers Z 1.1. Introduction 1.2. Divisibility 1.3. The Division Algorithm 1.4. Greatest Common Divisors 1.5. The Euclidean Algorithm 1.6. Summary 1.7. Solved Problems 1.8. Supplementary Problems 2. Prime Numbers and Factorization 2.1. Introduction 2.2. Identifying Primes 2.3. Listing Primes: The Sieve of Eratosthenes 2.4. Unique Factorization of Integers into Primes 2.5. The Difficulty of Factorization 2.6. Using Factorization to Compute a GCD 2.7. Summary 2.8. Solved Problems 2.9. Supplementary Problems 3. Congruences and the Sets Zn 3.1. Introduction 3.2. Deffinition and Examples of Congruences 3.3. The Finite Sets Zn 3.4. Addition and Multiplication Tables for Zn 3.5. Properties of Congruences 3.6. Doing Division in Zn 3.7. Summary 3.8. Solved Problems 3.9. Supplementary Problems 4. Solving Congruences 4.1. Introduction 4.2. Solving a Single Linear Congruence 4.3. Solving Systems of Two or More Congruences 4.4. Summary 4.5. Solved Problems 4.6. Supplemental Problems 5. The Theorems of Fermat and Euler 5.1. Introduction 5.2. Fermat's Theorem for Prime Moduli 5.3. Euler's Function and Euler's Theorem 5.4. Fast Exponentiation 5.5. Summary 5.6. Solved Problems 5.7. Supplementary Problems 6. Applications to Modern Cryptography 6.1. Introduction 6.2. The Basics of Encryption 6.3. Primitive Roots in Zp 6.4. Diffie-Hellman Key Exchange 6.5. Public Key Cryptography and the RSA System 6.6. Security versus Authenticity 6.7. Summary 6.8. Solved Problems 6.9. Supplemental Problems 7. Quadratic Residues and Quadratic Reciprocity 7.1. Introduction 7.2. Quadratic Residues and the Legendre Symbol 7.3. Computing the Legendre Symbol 7.4. Quadratic Reciprocity 7.5. Composite Moduli and the Jacobi Symbol 7.6. Summary 7.7. Solved Problems 7.8. Supplementary Problems 8. Some Fundamental Number Theory Functions 8.1. Introduction 8.2. The Greatest Integer Function 8.3. The Functions t(n), s(n), s k (n) 8.4. The Moobius Inversion Formula 8.5. Summary 8.6. Solved Problems 8.7. Supplementary Problems 9. Diophantine Equations 9.1. Introduction 9.2. The Linear Equation ax + by = c 9.3. The Equation x2 + y2 = z2 9.4. The Equation x4 + y4 = z4 9.5. The Equation xn + yn = zn, n > 2 9.6. Sums of Four Squares 9.7. Waring's Problem 9.8. Summary 9.9. Solved Problems 9.10. Supplementary Problems 10. Finite Fields 10.1. Introduction 10.2. The Finite Fields Fpn 10.3. The Order of a Finite Field 10.4. Constructing Finite Fields 10.5. The Multiplicative Structure of Fq 10.6. The Subfields of Fpn 10.7. Counting Irreducible Polynomials over Fpn 10.8. Lagrange Interpolation Formula 10.9. An Application to Latin and Sudoku Squares 10.10. Summary 10.11. Solved Problems 10.12. Supplementary Problems 11. Some Open Problems in Number Theory 11.1. Introduction 11.2. Open Problems 11.3. Summary 11.4. Problems A. Mathematical Induction B. Sets of Numbers Beyond the Integers Bibliography Index
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