# Elementary Number Theory, 7th edition

• Kenneth H. Rosen

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## Overview

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.

ISBN-13: 9780135696897

Category: Number Theory

## Overview

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
• 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

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