Number Theory: An Introduction to Pure and Applied Mathematics (Chapman & Hall/CRC Pure and Applied Mathematics Book 201)

This reference text provides a detailed introduction to number theory--demonstrating how other areas of mathematics enter into the study of the properties of natural numbers. Title Preface Contents A Historical Introduction 1. Primes and Divisibility 1.1. Introduction 1.2. Divisibility 1.3. Prime numbers 1.4. Unique factorization 1.5. Some factorization methods, I Additional problems 2. Congruences 2.1. Congruences 2.2. The Euler phi function 2.3. Congruences of the first degree 2.4. The Chinese remainder theorem 2.5. Polynomial congruences 2.6. Prime power moduli 2.7. Prime moduli 2.8. Primitive roots 2.9. Indices and binomial and exponential congruences 2.10. An application to cryptography 2.11. Pseudoprimes Additional problems 3. Quadratic Residues 3.1. Introduction 3.2. Quadratic residues 3.3. The law of quadratic reciprocity 3.4. The Jacobi symbol 3.5. The Kronecker symbol 3.6. The solution of x^2 = D (mod m) Additional problems 4. Approximation of Real Numbers 4.1. Introduction 4.2. Farey fractions 4.3. Approximation by rationals, I 4.4. Continued fractions 4.5. Periodic continued fractions 4.6. Approximation by rationals, II 4.7. Some factorization methods, II 4.8. Equivalent numbers 4.9. Decimal representation Additional problems 5. Diophantine Equations, I 5.1. Introduction 5.2. Pythagorean triangles 5.3. Related quadratic equations 5.4. The equation ax^2 + by^2 + cz^2 = 0 Additional problems 6. Diophantine Equations, II 6.1. Introduction 6.2. Linear equations 6.3. Sums of two squares 6.4. Sums of four squares 6.5. Sums of other numbers of squares 6.6. Binary quadratic forms 6.7. The equation x^2 - D y^2 = N 6.8. A Pythagorean triangle problem 6.9. The equation ax^2 + bxy + cy^2 + dx + ey + f = 0 6.10. Waring's problem Additional problems 7. Arithmetic Functions 7.1. Introduction 7.2. Dirichlet convolution 7.3. Multiplicative functions 7.4. Sum of divisors 7.5. Number of divisors 7.6. Euler's function 7.7. Characters 7.8. Trigonometrical sums 7.9. Additive functions 7.10. Linear recursion Additional problems 8. The Average Order of Arithmetic Functions 8.1. Introduction 8.2. The greatest integer function 8.3. Preliminaries 8.4. Sum of divisors function 8.5. The number of divisors functions 8.6. Euler's function 8.7. Lattice point problems Additional problems 9. Prime Number Theory 9.1. Introduction 9.2. Bertrand's postulate and Chebychev's theorem 9.3. The prime number theorem 9.4. Primes in arithmetic progressions 9.5. Order of magnitude of multiplicative functions 9.6. Summatory functions of additive functions Additional problems 10. An Introduction To Algebraic Number Theory 10.1. Introduction 10.2. General considerations 10.3. Quadratic number fields 10.4. Applications to Diophantine equations 10.5. Concluding remarks Additional problems Tables Bibliography Index "This reference/text provides a detailed introduction to number theory and demonstrates how other areas of mathematics enter into the study of the properties of natural numbers."--BOOK JACKET. "Offering helpful problem sets within each section and at the end of each chapter to reinforce essential concepts, Number Theory contains up-to-date information on divisibility properties...polynomial congruence...the sums of squares...trigonometric sums...Diophantine approximation...the behavior of prime numbers...algebraic number fields...and more."--BOOK JACKET. "Furnishing a useful bibliography, allowing readers to further investigate the results presented, Number Theory is an ideal reference for research mathematicians interested in algebra and number theory, and a valuable text for upper-level undergraduate and graduate students in these disciplines."--BOOK JACKET

this Text Provides A Detailed Introduction To Number Theory, Demonstrating How Other Areas Of Mathematics Enter Into The Study Of The Properties Of Natural Numbers. It Contains Problem Sets Within Each Section And At The End Of Each Chapter To Reinforce Essential Concepts, And Includes Up-to-date Information On Divisibility Problems, Polynomial Congruence, The Sums Of Squares And Trigonometric Sums.;five Or More Copies May Be Ordered By College Or University Bookstores At A Special Price, Available On Application.

zentralblatt Fur Mathematik

this Work Is Obviously A Labor Of Love....the Author Has Chosen A Wide Variety Of Topics That Illustrate Many Different Techniques Used In Number Theory, And He Succeeds In Showing How Other Parts Of Mathematics Play A Role In Studying Properties Of The Integers.

Presents an introduction to number theory, demonstrating how other areas of mathematics enter into the study of the properties of natural numbers. This work contains information on divisibility problems, polynomial congruence, the sums of squares and trigonometric sums.