In this post, we will see the book **Elements Of Number Theory** by I. M. Vinogradov.

# About the book (from the author)

In my book I present a systematic exposition of the funda mentals of number theory within the scope of a university course. A large collection of problems introduces the reader to some of the new ideas in number theory.This fifth edition of my book differs considerably from the fourth. A series of changes, allowing a simpler exposition, have been made in all the chapters of the book. The most important changes are the merging of the old chapters IV and V into one chapter IV (reducing the number of chapters to six) and the new, simpler proof of the existence of primitive roots.The problems at the end of each chapter have been essentially revised. The order of the problems is now in complete correspondence with the order of the presentation of the theoretical material. Some new problems have been added; but the number of numbered problems has been substantially reduced. This was accomplished by the unification, under the letters a, b, c, . . ., of previously separate problems which were related by the method of solution or by content. All the solutions of the problems have been reviewed; in many cases these solutions have been simplified or replaced by better ones. Particularly essential changes have been made in the solutions of the problems relating to the distribution of n-th power residues and non-residues, and primitive roots, as well as in the estimations of the corresponding trigonometric sums.

The book was translated from Russian by Saul Kravetz was published in 1954.

Credits to original uploader.

You can get the book here.

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# Contents

Preface vii

## Chapter I DIVISIBILITY THEORY

§1. Basic Concepts and Theorems (1).

§2. The Greatest Common Divisor (2).

§3. The Least Common Multiple (7).

§4. The Relation of Euclid’s Algorithm to Continued Fractions (8). §5. Prime Numbers (14).

§6. The Unicity of Prime Decomposition (15),

Problems for Chapter I (17).

Numerical Exercises for Chapter I (20).

## Chapter II IMPORTANT NUMBER-THEORETICAL FUNCTIONS

§1. The Function’s {x},x (21).

§2. Sums Extended over the Divisors of a Number (22),

§3. The Mdbius Function (24).

§4, The Euler Function (26).

Problems for Chapter II (28).

Numerical Exercises for Chapter II (40).

## Chapter III. CONGRUENCES

§1. Basic Concepts (41).

§2. Properties of Congruences Similar to those of Equations (42). §3. Further Properties of Congruences (44).

§4. Complete Systems of Residues (45).

§5. Reduced Systems of Residues (47).

§6. The Theorems of Euler and Fermat (48).

Problems for Chapter III (49).

Numerical Exercises for Chapter III (58).

## Chapter IV CONGRUENCES IN ONE UNKNOWN

§1. Basic Concepts (59).

§2. Congruences of the First Degree (60).

§3. Systems of Congruences of the First Degree (63).

§4. Congruences of Arbitrary Degree with Prime Modulus (65).

§5. Congruences of Arbitrary Degree with Composite Modulus (66). Problems for Chapter IV (71).

Numerical Exercises for Chapter IV (77).

## Chapter V CONGRUENCES OF SECOND DEGREE

§1. General Theorems (79).

§2. The Legendre Symbol (81).

§3. The Jacobi Symbol (87).

§4. The Case of Composite

Moduli (91).

Problems for Chapter V (95).

Numerical Exercises for Chapter V (103).

## Chapter VI PRIMITIVE ROOTS AND INDICES

§1. General Theorems (105).

§2. Primitive Roots Modulo p* and 2p* (106).

§3. Evaluation of Primitive Roots for the Moduli p* and 2p* (108). §4. Indices for the Moduli p and 2p* (110).

§5. Consequences of the Preceding Theory (113).

§6. Indices Modulo 2% (116).

§7. Indices for Arbitrary Composite Modulus (119).

Problems for Chapter VI (121).

Numerical Exercises for Chapter VI (130).

## SOLUTIONS OF THE PROBLEMS

Solutions for Chapter I (133).

Solutions for Chapter II (139).

Solutions for Chapter III (161).

Solutions for Chapter IV (178).

Solutions for Chapter V (187).

Solutions for Chapter VI (202).

## ANSWERS TO THE NUMERICAL EXERCISES

Answers for Chapter I (217).

Answers for Chapter II (217).

Answers for Chapter III (218).

Answers for Chapter IV (218).

Answers for Chapter V (218).

Answers for Chapter VI (219).

TABLES OF INDICES 220

TABLES OF PRIMES <4000 AND THEIR LEAST PRIMITIVE 226

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