Number Theory : Multiplicative and Additive with Factorization and Primality Testing.

Format:

Book

Author/Creator:

Schumer, Peter D.

Series:

De Gruyter Textbook Series

Language:

English

Subjects (All):

Number theory.

Algebraic number theory.

Physical Description:

1 online resource (364 pages)

Edition:

1st ed.

Place of Publication:

Berlin/Boston : Walter de Gruyter GmbH, 2025.

Summary:

This is a book for an undergraduate number theory course, senior thesis work, graduate level study, or for those wishing to learn about applications of number theory to data encryption and security.

Contents:

Intro

Preface

Contents

Chapter 1 Background

1.1 Brief historical introduction

Exercise 1.1

1.2 Induction and the well-ordering principle

Exercise 1.2

1.3 Divisibility and congruences

Exercise 1.3

1.4 Basic combinatorics

Exercise 1.4

Chapter 2 Congruences and prime factorization

2.1 The Euclidean algorithm and some consequences

Exercise 2.1

2.2 Congruence equations and the Chinese remainder theorem

Exercise 2.2

2.3 Primes and the fundamental theorem of arithmetic

Exercise 2.3

2.4 Introduction to primality testing and factoring

Exercise 2.4

2.5 Some important congruence relations

Outline placeholder

2.5.1 Binary exponentiation algorithm

Exercise 2.5

2.6 General polynomial congruences: Hensel's lemma

Exercise 2.6

Chapter 3 Arithmetic functions

3.1 Important arithmetic functions

Exercise 3.1

3.2 Multiplicativity

Exercise 3.2

3.3 Möbius inversion and some consequences

Exercise 3.3

3.4 Perfect numbers and amicable pairs

Exercise 3.4

Chapter 5 Sums of squares

5.1 Fundamentals of Diophantine equations

Exercise 5.1

5.2 Sums of two squares

Exercise 5.2

5.3 Sums of three squares

Exercise 5.3

5.4 Sums of four or more squares

Exercise 5.4

5.5 Legendre's equation

Exercise 5.5

Chapter 6 Continued fractions and Farey sequences

6.1 Finite simple continued fractions

Exercise 6.1

6.2 Farey fractions

Exercise 6.2

6.3 Infinite simple continued fractions

Exercise 6.3

6.4 Rational approximations of irrationals

Exercise 6.4

6.5 Pell's equation

Exercise 6.5

6.6 The continued fraction for e

Exercise 6.6

6.7 Algebraic and transcendental numbers

Exercise 6.7

Chapter 7 Factoring and primality testing

7.1 Primality and compositeness

Exercise 7.1

7.2 Pseudoprimes and Carmichael numbers.

Exercise 7.2

7.3 Miller-Rabin-Jaeschke primality test

Exercise 7.3

7.4 Mersenne primes

Exercise 7.4

7.5 Fermat numbers

Exercise 7.5

7.6 Factorization methods

Exercise 7.6

7.7 Quadratic sieve factorization algorithm

Exercise 7.7

7.8 The AKS primality test

Exercise 7.8

Chapter 8 Some applications

8.1 Introduction to cryptology

Exercise 8.1

8.2 RSA algorithm

Exercise 8.2

8.3 Random number generation

Exercise 8.3

Chapter 9 Introduction to analytic number theory

9.1 The infinitude of primes and the zeta function

Exercise 9.1

9.2 Average order of the lattice and divisor functions

Exercise 9.2

9.3 Average order of ϕ(n) and applications

Exercise 9.3

9.4 Chebyshev's theorems and the distribution of primes

Exercise 9.4

9.5 Bertrand's postulate and applications

Exercise 9.5

Chapter 10 Introduction to additive number theory

10.1 Waring's problem

Exercise 10.1

10.2 Schnirelmann density and the ɑ + β theorem

Exercise 10.2

10.3 Van der Waerden's theorem

Exercise 10.3

10.4 Introduction to the theory of partitions

Exercise 10.4

11_Schumer_0825_EF_Hints

Hints and answers to selected exercises

Exercise 4.1

Exercise 4.2

Exercise 4.3

Exercise 4.4

Exercise 7.2

Exercise 9.1.

Exercise 9.2

Bibliography

Index.

Notes:

Description based on publisher supplied metadata and other sources.

Part of the metadata in this record was created by AI, based on the text of the resource.

ISBN:

3-11-157928-X

OCLC:

1528630067