Book:Ivan Niven/An Introduction to the Theory of Numbers/Third Edition

Ivan Niven and Herbert S. Zuckerman: An Introduction to the Theory of Numbers (3rd Edition)

Published $\text {1972}$

Subject Matter

• Number Theory

Contents

1. Divisibility

1.1 Introduction

1.2 Divisibility

1.3 Primes

2. Congruences

2.1 Congruences

2.2 Solutions of Congruences

2.3 Congruences of Degree I

2.4 The Function $\phi(n)$

2.5 Congruences of Higher Degree

2.6 Prime Power Moduli

2.7 Prime Modulus

2.8 Congruences of Degree Two, Prime Modulus

2.9 Power Residues

2.10 Number Theory from an Algebraic Viewpoint

2.11 Multiplicative Groups, Rings, and Fields

3. Quadratic Reciprocity

3.1 Quadratic Residues

3.2 Quadratic Reciprocity

3.3 The Jacobi Symbol

4. Some Functions of Number Theory

4.1 Greatest Integer Function

4.2 Arithmetic Functions

4.3 The Moebius Inversion Formula

4.4 The Multiplication of Arithmetic Functions

4.5 Recurrence Functions

5. Some Diophantine Equations

5.1 Diophantine Equations

5.2 The Equation $ax + by = c$

5.3 Positive Solutions

5.4 Other Linear Equations

5.5 The Equation $x^2 + y^2 = z^2$

5.6 The Equation $x^4 + y^4 = z^2$

5.7 Sums of Four and Five Squares

5.8 Waring's Problem

5.9 Sum of Fourth Powers

5.10 Sum of Two Squares

5.11 The Equation $4x^2 + y^2 = n$

5.12 The Equation $ax^2 + by^2 + cz^2 = 0$

5.13 Binary Quadratic Forms

5.14 Equivalence of Quadratic Forms

6. Farey Fractions and Irrational Numbers

6.1 Farey Sequences

6.2 Rational Approximations

6.3 Irrational Numbers

6.4 Coverings of the Real Line

7. Simple Continued Fractions

7.1 The Euclidean Algorithm

7.2 Uniqueness

7.3 Infinite Continued Fractions

7.4 Irrational Numbers

7.5 Approximations to Irrational Numbers

7.6 Best Possible Approximations

7.7 Periodic Continued Fractions

7.8 Pell's Equation

7.9 Numerical Computation

8. Elementary Remarks on the Distribution of Primes

8.1 The Function $\pi(x)$

8.2 The Sequence of Primes

8.3 Bertrand's Postulate

9. Algebraic Numbers

9.1 Polynomials

9.2 Algebraic Numbers

9.3 Algebraic Number Fields

9.4 Algebraic Integers

9.5 Quadratic Fields

9.6 Units in Quadratic Fields

9.7 Primes in Quadratic Fields

9.8 Unique Factorization

9.9 Primes in Quadratic Fields Having the Unique Factorization Property

9.10 The Equation $x^3 + y^3 = z^3$

10. The Partition Function

10.1 Partitions

10.2 Graphs

10.3 Formal Power Series and Euler's Identity

10.4 Euler's Formula

10.5 Jacobi's Formula

10.6 A Divisibility Property

11. The Density of Sequences of Integers

11.1 Asymptotic Density

11.2 Square-Free Integers

11.3 Sets of Density Zero

11.4 Schnirelmann Density and the $\alpha\beta$ Theorem

Miscellaneous Problems

Special Topics

General References

Answers to Problems

Index

Further Editions

• 1960: Ivan Niven and Herbert S. Zuckerman: An Introduction to the Theory of Numbers
• 1966: Ivan Niven and Herbert S. Zuckerman: An Introduction to the Theory of Numbers (2nd ed.)
• 1980: Ivan Niven and Herbert S. Zuckerman: An Introduction to the Theory of Numbers (4th ed.)
• 1991: Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery: An Introduction to the Theory of Numbers (5th ed.)