# Elementary Number Theory,7th edition

• Kenneth H. Rosen AT&amp;T Laboratories

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For courses inâ€¯ Number Theory.

### Classical theory â€¯meets modern applications and outstanding exercise sets

Elementary Number Theory is known for its rich exercise sets,â€¯ careful and rigorous proofs, examples â€¯and applications.â€¯ A full range of exercises â€¯helps students explore key concepts and push their understanding to new heights. Computational exercises and computer projects are availableâ€¯ forâ€¯ Maple,â€¯ Mathematica, Sage Math,â€¯ and theâ€¯ book'sâ€¯ manyâ€¯ online resources.â€¯

Theâ€¯ 7th Edition â€¯is revised throughout for a presentation that isâ€¯ easier to teach and to learn from,â€¯ while incorporating advancements andâ€¯ recentâ€¯ discoveries in â€¯number theory.â€¯ Severalâ€¯ hundred â€¯new exercises enhance the exercise sets even further.

### Hallmark features of this title

#### Proven approach

• Careful proofs explain and support a number of the key results of number theory.
• Applications are well integrated, illustrating the usefulness of the theory. Computer exercises and projects in each section cover specific concepts or algorithms, guiding students on combining the math with their computing skills.

#### Extensive exercise sets

• Diverse exercise levels include basic skills, intermediate (put concepts together and develop new results), challenging, and exercises using technology tools.
• Answers are provided to all odd-numbered exercises within the text, and solutions to all odd-numbered exercises are in the Student Solutions Manual.

### New and updated features of this title

#### Up-to-date,â€¯ engaging coverage of essential topics

• Expanded coverage of cryptography including Elliptic curve photography; coverage of knapsack ciphers has been removed. Also, the important notion of homomorphic encryption is introduced in this edition.â€¯
• Coverage of recent numerical discoveries, including four new Mersenne primes; the largest known twin primes; the evidence supporting many important conjectures, and more.â€¯â€¯
• Coverage of new theoretical discoveries,â€¯ such as the proof of weak Goldbach conjectureâ€¯ and the result that there is an algorithm to multiply twoâ€¯ n-bit integers usingâ€¯ O (nlog2â€¯n) bit operations.â€¯

#### A proven approach

• Enhanced exercise sets have been examined and improvedâ€¯ sets even â€¯further, plus several hundred new exercises, from routine to challenging. New computational and exploratory exercises â€¯are included.â€¯
• Connections with abstract algebra:â€¯Theâ€¯ book does not assume prior knowledge of abstract algebra, but â€¯introduces some basic algebraic structures such as groups, rings, and fields.â€¯â€¯â€¯
• Available resources for the 7th Edition are posted on the Pearson media servers here.

### Features of Pearson eText for the 7th Edition

• Now available as a Pearson eText. This new interactive version of the bookâ€¯ provides many interactive applets that can be used by students for some common computations in number theory and to help understand concepts and explore conjectures.â€¯â€¯
• A collection of cryptographic applets is provided along with algorithms for computations in number theory,â€¯ These include applets for â€¯encryption, decryption, cryptanalysis, and cryptographic protocols, â€¯addressing â€¯both classical ciphers and the RSA cryptosystem. Cryptographic applets can be used for individual, group, and classroom activities.â€¯â€¯
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
• MÃ¶bius 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