Elementary Number Theory helps you push your understanding to new heights with the strongest exercise sets, proofs and examples. Applications are integrated throughout. Connections with abstract algebra help those who have already studied it, and lay the groundwork to understand key ideas if you're taking abstract algebra in the future.  Computational exercises and computer projects are available  for  Maple,  Mathematica, Sage Math and the  book's  many  online resources. 

The 7th Edition offers a presentation that's easier to learn from,  while incorporating advancements and  recent  discoveries in  number theory.  Expanded coverage of cryptography includes elliptic curve photography; the important notion of homomorphic encryption is introduced, and coverage of knapsack ciphers has been removed. Several  hundred  new exercises enhance the text's exercise sets.

1. The Integers

Numbers and Sequences

Diophantine Approximation

Sums and Products

Mathematical Induction

The Fibonacci Numbers

Divisibility

2. Integer Representations and Operations

Representations of Integers

Computer Operations with Integers

Complexity of Integer Operations

3. Greatest Common Divisors

Greatest Common Divisors and Their Properties

The Euclidean Algorithm

Linear Diophantine Equations

4. Prime Numbers

Prime Numbers

The Distribution of Primes

The Fundamental Theorem of Arithmetic

Factorization Methods and the Fermat Numbers

5. Congruences

Introduction to Congruences

Linear Congruences

The Chinese Remainder Theorem

Polynomial Congruences

Systems of Linear Congruences

6. Applications of Congruences

Divisibility Tests

The Perpetual Calendar

Round-Robin Tournaments

Hashing Functions

Check Digits

7. Some Special Congruences

Wilson's Theorem and Fermat's Little Theorem

Pseudoprimes

Euler's Theorem

8. Arithmetic Functions

The Euler Phi-Function

The Sum and Number of Divisors

Perfect Numbers and Mersenne Primes

Möbius Inversion

Partitions

9. Cryptography

Character Ciphers

Block and Stream Ciphers

Exponentiation Ciphers

Public Key Cryptography

Cryptographic Protocols and Applications

10. Primitive Roots

The Order of an Integer and Primitive Roots

Primitive Roots for Primes

The Existence of Primitive Roots

Discrete Logarithms and Index Arithmetic

Primality Tests Using Orders of Integers and Primitive Roots

Universal Exponents

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

Pseudorandom Numbers

The EIGamal Cryptosystem

An Application to the Splicing of Telephone Cables

12. Quadratic Residues

Quadratic Residues and Nonresidues

The Law of Quadratic Reciprocity

The Jacobi Symbol

Euler Pseudoprimes

Zero-Knowledge Proofs

13. Decimal Fractions and Continued Fractions

Decimal Fractions

Finite Continued Fractions

Infinite Continued Fractions

Periodic Continued Fractions

Factoring Using Continued Fractions

14. Nonlinear Diophantine Equations and Elliptic Curves

Pythagorean Triples

Fermat's Last Theorem

Sum of Squares

Pell's Equation

Congruent Numbers and Elliptic Curves

Elliptic Curves Modulo Primes

Applications of Elliptic Curves

15. The Gaussian Integers

Gaussian Integers and Gaussian Primes

Greatest Common Divisors and Unique Factorization

Gaussian Integers and Sums of Squares

Kenneth H. Rosen received his BS in mathematics from the University of Michigan - Ann Arbor (1972) and his PhD in mathematics from MIT (1976). Before joining Bell Laboratories in 1982, he held positions at the University of Colorado - Boulder, The Ohio State University - Columbus, and the University of Maine - Orono, where he was an associate professor of mathematics. While working at AT&T Laboratories, he taught at Monmouth University, teaching courses in discrete mathematics, coding theory, and data security. 

Dr. Rosen has published numerous articles in professional journals in the areas of number theory and mathematical modeling. He is the author of Elementary Number Theory,  7th Edition and other books. 

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