Difference between revisions of "Book:George Eyre Andrews/Number Theory"

Latest revision as of 16:40, 21 June 2016

George Eyre Andrews: Number Theory

Published $1971$, W.B. Saunders Company.

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Contents

Part I MULTIPLICATIVITY--DIVISIBILITY

Chapter 1: BASIS REPRESENTATION

1-1 Principle of Mathematical Induction

1-2 The Basis Representation Theorem

Chapter 2: THE FUNDAMENTAL THEOREM OF ARITHMETIC

2-1 Euclid's Division Lemma

2-2 Divisibility

2-3 The Linear Diophantine Equation

2-4 The Fundamental Theorem of Arithmetic

Chapter 3: COMBINATORIAL AND COMPUTATIONAL NUMBER THEORY

3-1 Permutations and Combinations

3-2 Fermat's Little Theorem

3-3 Wilson's Theorem

3-4 Generating Functions

3-5 The Use of Computers in Number Theory

Chapter 4: FUNDAMENTALS OF CONGRUENCES

4-1 Basic Properties of Congruences

4-2 Residue Systems

4-3 Riffling

Chapter 5: SOLVING CONGRUENCES

5-1 Linear Congruences

5-2 The Theorems of Fermat and Wilson Revisited

5-3 The Chinese Remainder Theorem

5-4 Polynomial Congruences

Chapter 6: ARITHMETIC FUNCTIONS

6-1 Combinatorial Study of $\phi(n)$

6-2 Formulae for $d(n)$ and $\sigma(n)$

6-3 Multiplicative Arithmetic Functions

6-4 The Möbius Inversion Formula

Chapter 7: PRIMITIVE ROOTS

7-1 Properties of Reduced Residue Systems

7-2 Primitive Roots modulo $p$

Chapter 8: PRIME NUMBERS

8-1 Elementary Properties of $\pi(x)$

8-2 Tchebychev's Theorem

8-3 Some Unsolved Problems About Primes

Part II QUADRATIC CONGRUENCES

Chapter 9: QUADRATIC RESIDUES

9-1 Euler's Criterion

9-2 The Legendre Symbol

9-3 The Quadratic Reciprocity Law

9-4 Applications of the Quadratic Reciprocity Law

Chapter 10: DISTRIBUTION OF QUADRATIC RESIDUES

10-1 Consecutive Residues and Nonresidues

10-2 Consecutive Triples of Quadratic Residues

Part III ADDITIVITY

Chapter 11: SUMS OF SQUARES

11-1 Sums of Two Squares

11-2 Sums of Four Squares

Chapter 12: ELEMENTARY PARTITION THEORY

12-1 Introduciton

12-2 Graphical Representation

12-3 Euler's Partition Theorem

12-4 Searching for Partition Identities

Chapter 13 GENERATING FUNCTIONS

13-1 Infinite Products As Generating Functions

13-2 Identities Between Infinite Series and Products

Chapter 14 PARTITION IDENTITIES

14-1 History and Introduction

14-2 Euler's Pentagonal Number Theorem

14-3 The Rogers-Ramanujan Identities

14-4 Series and Product Identities

14-5 Schur's Theorem

Part IV GEOMETRIC NUMBER THEORY

Chapter 15 LATTICE POINTS

15-1 Gauss's Circle Problem

15-2 Dirichlet's Divisor Problem

APPENDICES

Appendix A

Appendix B

Appendix C

Appendix D

THE INTEGRAL TEST

NOTES

SUGGESTED READING

BIBLIOGRAPHY

HINTS AND ANSWERS TO SELECTED EXERCISES

INDEX OF SYMBOLS

INDEX