Elementary Number Theory

Kenneth H. Rosen  
Total pages
November 2013
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Elementary Number Theory
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Elementary Number Theory, Sixth Edition, blends classical theory with modern applications and is notable for its outstanding exercise sets. A full range of exercises, from basic to challenging, helps students explore key concepts and push their understanding to new heights. Computational exercises and computer projects are also available. Reflecting many years of professor feedback, this edition offers new examples, exercises, and applications, while incorporating advancements and discoveries in number theory made in the past few years.


  • Extensive and diverse exercise sets include exercises to develop basic skills, intermediate exercises to help students put several concepts together and develop new results, exercises designed to be completed with technology tools, and challenging exercises to expand understanding. Answers are provided to all odd-numbered exercises within the text, and solutions to all odd-numbered exercises are in the Student Solutions Manual, which is hosted on the Companion Website.
  • Applications of number theory are well integrated into the text, illustrating the usefulness of the theory.
  • Computer exercises and projects in each section of the text cover specific concepts or algorithms from that section, guiding students on combining the mathematics with their computing skills.
  • Cryptography and cryptographic protocols are covered in depth. This is the first number theory text to cover cryptography, and results important for cryptography are developed with the theory in the early chapters.
  • The flexible organization allows instructors to choose from a wealth of topics when designing a course.
  • Historical content and biographies illustrate the human side of number theory, both ancient and modern.
  • Careful proofs explain and support a number of the key results of number theory, helping students develop their understanding.
  • The Companion Website (www.pearsonhighered.com/rosen) includes a variety of resources that can enrich the use of the book, including the Student Solutions Manual, suggested projects, a collection of applets, a manual for using computational engines to explore number theory, and a webpage devoted to number theory news.
  • The Instructor's Solution Manual (available for download from the Pearson Instructor Resource Center) provides complete solutions to all exercises, material on programming projects, and an extensive test bank.
  • The Student’s Manual for Computations and Explorations provides worked out solutions or partial solutions to many of the computational and exploratory exercises in the text, as well as hints and guidance for approaching others. This manual supports different computational environments, including Maple™, Mathematica®, and PARI/GP.
  • Applets on the Companion Website involve some common computations in number theory and help students understand concepts and explore conjectures. Additionally, a collection of cryptographic applets is also provided.


New to this Edition

  • Many new discoveries, both theoretical and numerical, are introduced. Coverage includes four Mersenne primes, numerous new world records, and the latest evidence supporting open conjectures. Recent theoretical discoveries are described, including the Tao-Green theorem about arbitrarily long arithmetic progressions of primes.
  • New biographies of Terence Tao, Etienne Bezout, Norman MacLeod Ferrers, Clifford Cocks, and Waclaw Sierpinski supplement the already rich collection of biographies in the book. This edition also includes historical information about secret British cryptographic discoveries that predate the work of Rivest, Shamir, and Adelman.
  • Expanded treatment of both resolved and open conjectures about prime numbers is provided.
  • Combinatorial number theory–partitions are covered in a new section of the book. This provides an introduction to combinatorial number theory, which was not covered in previous editions. This new section covers many aspects of this topics including Ferrers diagrams, restricted partition identities, generating functions, and the famous Ramanujan congruences. Partition identities are proved using both generating functions and bijections.
  • Congruent numbers and elliptic curves–a new section is devoted to the famous congruent number problem, which asks which positive integers are the area of a right triangle with rational side lengths. This section shows that the congruent number problem is equivalent to finding rational points on certain elliptic curves and introduces some basic properties of elliptic curves.
  • The use of geometric reasoning in the solution of diophantine problems has been added to the new edition. In particular, finding rational points on the unit circle is shown to be equivalent to finding Pythgaorean triples. Finding rational triangles with a given integer as area is shown to be equivalent to finding rational points on an associated elliptic curve.
  • Greatest common divisors are now defined in Chapter 1. The terminology on Bezout coefficients is now introduced in Chapter 3, where properties of greatest common divisors are developed.
  • An expanded discussion on the usefulness of theJacobi symbol in evaluating Legendre symbols is now provided.
  • Extensive revisions to the already-strong exercise sets include several hundred new exercises, ranging from routine to challenging. In particular, there are many new and revised computational exercises.
  • The Companion Website for this edition (www .pearsonhighered.com/rosen) has been considerably expanded. Among the new features are an expanded collection of applets, a manual for using computational engines to explore number theory, and a Web page devoted to number theory news.


Table of Contents

P. What is Number Theory?

1. The Integers.


Numbers and Sequences.


Sums and Products.


Mathematical Induction.


The Fibonacci Numbers.

2. Integer Representations and Operations.


Representations of Integers.


Computer Operations with Integers.


Complexity of Integer Operations.

3. Primes and Greatest Common Divisors.


Prime Numbers.


The Distribution of Primes.


Greatest Common Divisors.


The Euclidean Algorithm.


The Fundemental Theorem of Arithmetic.


Factorization Methods and Fermat Numbers.


Linear Diophantine Equations.

4. Congruences.


Introduction to Congruences.


Linear Congrences.


The Chinese Remainder Theorem.


Solving Polynomial Congruences.


Systems of Linear Congruences.


Factoring Using the Pollard Rho Method.

5. Applications of Congruences.


Divisibility Tests.


The perpetual Calendar.


Round Robin Tournaments.


Hashing Functions.


Check Digits.

6. Some Special Congruences.


Wilson's Theorem and Fermat's Little Theorem.




Euler's Theorem.

7. Multiplicative Functions.


The Euler Phi-Function.


The Sum and Number of Divisors.


Perfect Numbers and Mersenne Primes.


Mobius Inversion.

8. Cryptology.


Character Ciphers.


Block and Stream Ciphers.


Exponentiation Ciphers.


Knapsack Ciphers.


Cryptographic Protocols and Applications.

9. Primitive Roots.


The Order of an Integer and Primitive Roots.


Primitive Roots for Primes.


The Existence of Primitive Roots.


Index Arithmetic.


Primality Tests Using Orders of Integers and Primitive Roots.


Universal Exponents.

10. Applications of Primitive Roots and the Order of an Integer.


Pseudorandom Numbers.


The EIGamal Cryptosystem.


An Application to the Splicing of Telephone Cables.

11. Quadratic Residues.


Quadratic Residues and nonresidues.


The Law of Quadratic Reciprocity.


The Jacobi Symbol.


Euler Pseudoprimes.


Zero-Knowledge Proofs.

12. Decimal Fractions and Continued.


Decimal Fractions.


Finite Continued Fractions.


Infinite Continued Fractions.


Periodic Continued Fractions.


Factoring Using Continued Fractions.

13. Some Nonlinear Diophantine Equations.


Pythagorean Triples.


Fermat's Last Theorem.


Sums of Squares.


Pell's Equation.
Congruent Numbers.

14. The Gaussian Integers.


Gaussian Primes.


Unique Factorization of Gaussian Integers.


Gaussian Integers and Sums of Squares.