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An Introduction to the Theory of Numbers

Ivan M. Niven, Herbert S. Zuckerman, and Hugh L. Montgomery
John Wiley
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This undergraduate textbook is a comprehensive survey of everything that might be considered elementary number theory. The approach emphasizes breadth rather than depth, but some deep results are covered as multi-part exercises. It is a modern look at number theory and (despite being published in 1991) is very much up-to-date; the only recent developments that might be added today would be the solutions of Fermat’s Last Theorem and of Catalan’s conjecture. There’s a moderate amount of numeric work, including some modern factorization methods such as Pollard’s rho and elliptic curve factorization.

The authors often work up to a difficult theorem by proving a simpler version and explaining the strategy being followed. For example they start working on Schnirelmann’s and Mann’s theorems on sums of sets of integers by first proving that every integer \(>1\) is the sum of two square-free integers. This is a neat result in itself and surprisingly simple to prove. They quote but do not prove Dirichlet’s theorem that there are an infinity of primes in an arithmetic progression, but they do prove it for common difference 4, and the proof uses all the ingredients of the full proof so you get a good understanding of how it works.

The exercises are especially good, and approach the subject from several viewpoints. There are simple numeric exercises to verify proved theorems and proof problems of various difficulties. There are even sketches of some quite advanced theorems, for example, I. M. Vinogradov’s theorem that the error term in the Dirichlet divisor problem is \(O(x^{1/3}\ln(x)^2)\) (the text proves \(O(x^{1/2})\)). There’s a topological proof that there are an infinity of primes. Be sure to read the exercises; some of the most interesting results are there!

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.


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 an 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 Arithmetic 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 Approximation 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 x3 + y3 = z3
Notes on Chapter 9

10 The Partition Function
10.1 Partitions
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 Divisibility Property
Notes on Chapter 10

11 The Density of Sequences of Integers
11.1 Asymptotic Density
11.2 Schnirelmann Density and the αβ Theorem
Notes on Chapter 11

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