Synopses & Reviews
"A very welcome addition to books on number theory."—
Bulletin of the American Mathematical Society.Clear and detailed in its exposition, this text can be understood by readers with no background in advanced mathematics, and only a small part requires a working knowledge of calculus. Topics include divisibility theory, important number-theoretic functions, basic properties of congruences as well as congruences in one unknown and of the second degree, and primitive roots and indices.
One of the most valuable characteristics of this book is its emphasis on learning number theory by means of demonstrations and problems. More than 200 problems and full solutions appear in the text, plus 100 numerical exercises. Some of these exercises deal with estimation of trigonometric sums and are especially valuable as introductions to more advanced studies.
Dover (2015) republication of the 1954 first edition.
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Synopsis
Clear, detailed exposition that can be understood by readers with no background in advanced mathematics. More than 200 problems and full solutions, plus 100 numerical exercises. 1954 edition.
About the Author
Ivan Matveevich Vinogradov (1891-1983) was among the Soviet Union's most prominent 20th-century mathematicians and noted as one of the creators of modern analytic number theory.
Table of Contents
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="">