Book:George E. Andrews/Number Theory

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George E. Andrews: Number Theory

Published $1971$, Dover Publications, Inc.

ISBN 0-486-68252-8.


Subject Matter


Contents

Preface
Part I: Multiplicativity - Divisibility
Chapter 1: Basis Representation
Chapter 2: The Fundamental Theorem of Arithmetic
Chapter 3: Combinatorial and Computational Number Theory
Chapter 4: Fundamentals of Congruences
Chapter 5: Solving Congruences
Chapter 6: Arithmetic Functions
Chapter 7: Primitive Roots
Chapter 8: Prime Numbers
Part II: Quadratic Congruences
Chapter 9: Quadratic Residues
Chapter 10: Distribution of Quadratic Residues
Part III: Additivity
Chapter 11: Sums of Squares
Chapter 12: Elementary Partition Theory
Chapter 13: Partition Generating Functions
Chapter 14: Partition Identities
Part IV: Geometric Number Theory
Chapter 15: Lattice Points
Appendices
Appendix A: A proof that $\displaystyle \lim_{n \mathop \to \infty} \map p n^{1/n} = 1$
Appendix B: Infinite Series and Products
Appendix C: Double Series
Appendix D: The Integral Test
Notes
Suggested Reading
Bibliography
Hints and Answers to Selected Exercises
Index of Symbols
Index



Errata

Solution to Difference between Odd Squares is Divisible by 8

Chapter $2$: The Fundamental Theorem of Arithmetic:

Since $a$ and $b$ are odd integers, $a = 2 r + 1$, and $b = 2 s + 1$. Thus
$a^2 - b^2 = \paren {4 r^2 + 4 r + 1} - \paren {4 s^2 + 4 s + 1} = 4 \paren {r - s} \paren {r - s + 1}$.
Now if $r - s$ is even, then $r - s = 2 m$ and $a^2 - b^2 = 8 m \paren {2 m + 1}$; if $r - s$ is odd, then $r - s = 2 n + 1$ and $a^2 - b^2 = 8 \paren {2 n + 1} \paren {n + 1}$. Thus in any case $a^2 - b^2$ is divisible by $8$ if $a$ and $b$ are odd integers.