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

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Ivan NivenHerbert S. Zuckerman and Hugh L. Montgomery: An Introduction to the Theory of Numbers (5th Edition)

Published $1991$, Wiley

ISBN 978-0471625469.


Subject Matter


Contents

Notation
1 Divisibility
1.1 Introduction
1.2 Divisibility
1.3 Primes
1.4 The Binomial Theorem
Notes on Chapter 1
2 Congruences
2.1 Congruences
2.2 Solutions of Congruences
2.3 The Chinese Remainder Theorem
2.4 Techniques of Numerical Calculation
2.5 Public-Key Cryptography
2.6 Prime Power Moduli
2.7 Prime Modulus
2.8 Primitive Roots and Power Residues
2.9 Congruences of Degree Two, Prime Modulus
2.10 Number Theory from an Algebraic Viewpoint
2.11 Groups, Rings, and Fields
Notes on Chapter 2
3 Quadratic Reciprocity and Quadratic Forms
3.1 Quadratic Residues
3.2 Quadratic Reciprocity
3.3 The Jacobi Symbol
3.4 Binary Quadratic Forms
3.5 Equivalence and Reduction of Binary Quadratic Forms
3.6 Sums of Two Squares
3.7 Positive Definite Binary Quadratic Forms
Notes on Chapter 3
4 Some Functions of Number Theory
4.1 Greatest Integer Function
4.2 Artihmetic Functions
4.3 The Möbius Inversion Formula
4.4 Recurrence Functions
4.5 Combinatorial Number Theory
Notes on Chapter 4
5 Some Diophantine Equations
5.1 The Equation $ax + by = c$
5.2 Simultaneous Linear Equations
5.3 Pythagorean Triangles
5.4 Assorted Examples
5.5 Ternary Quadratic Forms
5.6 Rational Points on Curves
5.7 Elliptic Curves
5.8 Factorization Using Elliptic Curves
5.9 Curves of Genus Greater Than $1$
Notes on Chapter 5
6 Farey Fractions and Irrational Numbers
6.1 Farey Sequences
6.2 Rational Approximations
6.3 Irrational Numbers
6.4 The Geometry of Numbers
Notes on Chapter 6
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
Notes on Chapter 7
8 Primes and Multiplicative Number Theory
8.1 Elementary Prime Number Estimates
8.2 Dirichlet Series
8.3 Estimates of Arithmetic Functions
8.4 Primes in Arithmetic Progressions
Notes on Chapter 8
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 $x^3 + y^3 = z^3$
Notes on Chapter 9
10 The Partition Function
10.1 Parittions
10.2 Ferrers Graphs
10.3 Formal Power Series, Generating Functions, and Euler's Identity
10.4 Euler's Formula; Bounds on $p(n)$
10.5 Jacobi's Formula
10.6 A Divisibiity Property
Notoes on Chapter 10
11 The Density of Sequences of Integers
11.1 Asymptotic Density
11.2 Schnirelmann Density and the $\alpha\beta$ Theorem
Notes on Chapter 11
Appendices
A.1 The Fundamental Theorem of Algebra
A.2 Symmetric Functions
A.3 A Special Value of the Riemann Zeta Function
A.4 Linear Recurrences
General References
Hints
Answers
Index