You are here

Elements of Number Theory

Ivan Matveevich Vinogradov
Publisher: 
Dover Publications
Publication Date: 
2003
Number of Pages: 
240
Format: 
Hardcover
Price: 
14095
ISBN: 
9780486781655
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
10/31/2017
]

This is a deceptive book. If you just look at the narrative, it appears to be a simple, maybe too simple, introduction to elementary number theory. But really the narrative just prepares you to work on the exercises, which although elementary are very challenging. The book includes complete solutions to all the exercises, so you can use it either as a problem book or as a topics book. This book was a Dover semi-original, being published by them in 1954 as an English translation of the fifth Russian edition from 1949, and reprinted several times, most recently in 2016.

The exercises extend the ideas in the body, often by a long distance. For example, the floor (greatest integer) function is discussed in the body, and is used for all kinds of counting problems in the exercises. There are also quite a number of problems involving trigonometric sums (one of the author’s research specialties), which are presented as extensions of congruence facts and the Chinese Remainder Theorem. The basics of Pell’s equation are done as exercises, starting from the theory of quadratic residues. The most startling inclusion is the Pólya-Vinogradov inequality in the special case of the Legendre symbol.

For the most part the exercises are broken down in small chunks, so even difficult results should be solvable by diligent students. A few look like they might be very difficult even then. Unlike the body, many of the exercises deal with estimates and inequalities, and there is some asymptotic reasoning.

Very Bad Feature: no index. There are also no citations and no bibliographic material.

Another good book with very good exercises is Niven & Zuckerman & Montgomery’s An Introduction to the Theory of Numbers. This is a considerably more advanced and comprehensive book that Vinogradov’s, and it only gives hints and not complete solutions.


Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.

 

 

Preface
  Chapter I
    DIVISIBILITY THEORY
    § 1. Basic Concepts and Theorems
    § 2. The Greatest Common Divisor
    § 3. The Least Common Multiple
    § 4. The Relation of Euclid's Algorithm to Continued Fractions
    § 5. Prime Numbers
    § 6. The Unicity of Prime Decomposition
      Problems for Chapter I
      Numerical Exercises for Chapter I
  Chapter II
    IMPORTANT NUMBER-THEORETICAL FUNCTIONS
    § 1. "The Functions x ,x"
    § 2. Sums Extended over the Divisors of a Number
    § 3. The Möbius Function
    § 4. The Euler Function
      Problems for Chapter II
      Numerical Exercises for Chapter II
  Chapter III
    CONGRUENCES
    § 1. Basic Concepts
    § 2. Properties of Congruences Similar to those of Equations
    § 3. Further Properties of Congruences
    § 4. Complete Systems of Residues
    § 5. Reduced Systems of Residues
    § 6. The Theorems of Euler and Fermat
      Problems for Chapter III
      Numerical Exercises for Chapter III
  Chapter IV
    CONGRUENCES IN ONE UNKNOWN
    § 1. Basic Concepts
    § 2. Congruences of the First Degree
    § 3. Systems of Congruences of the First Degree
    § 4. Congruences of Arbitrary Degree with Prime Modulus
    § 5. Congruences of Arbitrary Degree with Composite Modulus
      Problems for Chapter IV
      Numerical Exercises for Chapter IV
  Chapter V
    CONGRUENCES OF SECOND DEGREE
    § 1. General Theorems
    § 2. The Legendre Symbol
    § 3. The Jacobi Symbol
    § 4. The Case of Composite Moduli
      Problems for Chapter V
      Numerical Exercises for Chapter V
  Chapter VI
    PRIMITIVE ROOTS AND INDICES
    § 1. General Theorems
    § 2. Primitive Roots Modulo pa and 2pa
    § 3. Evaluation of Primitive Roots for the Moduli pa and 2pa
    § 4. Indices for the Moduli pa and 2pa
    § 5. Consequences of the Preceding Theory
    § 6. Indices Modulo 2a
    § 7. Indices for Arbitrary Composite Modulus
      Problems for Chapter VI
      Numerical Exercises for Chapter VI
  SOLUTIONS OF THE PROBLEMS
    Solutions for Chapter I
    Solutions for Chapter II
    Solutions for Chapter III
    Solutions for Chapter IV
    Solutions for Chapter V
    Solutions for Chapter VI
  ANSWERS TO THE NUMERICAL EXERCISES
    Answers for Chapter I
    Answers for Chapter II
    Answers for Chapter III
    Answers for Chapter IV
    Answers for Chapter V
    Answers for Chapter VI
 
  TABLES OF INDICES
  TABLES OF PRIMES <4000 AND THEIR LEAST PRIMITIVE ROOTS
 
 

Dummy View - NOT TO BE DELETED