# Elementary Number Theory, 7th edition

Published by Pearson (November 18, 2022) Â© 2023

**Kenneth H. Rosen**AT&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€¯

**The Integers**- Numbers and Sequences
- Diophantine Approximation
- Sums and Products
- Mathematical Induction
- The Fibonacci Numbers
- Divisibility

**Integer Representations and Operations**- Representations of Integers
- Computer Operations with Integers
- Complexity of Integer Operations

**Greatest Common Divisors**- Greatest Common Divisors and Their Properties
- The Euclidean Algorithm
- Linear Diophantine Equations

**Prime Numbers**- Prime Numbers
- The Distribution of Primes
- The Fundamental Theorem of Arithmetic
- Factorization Methods and the Fermat Numbers

**Congruences**- Introduction to Congruences
- Linear Congruences
- The Chinese Remainder Theorem
- Polynomial Congruences
- Systems of Linear Congruences

**Applications of Congruences**- Divisibility Tests
- The Perpetual Calendar
- Round-Robin Tournaments
- Hashing Functions
- Check Digits

**Some Special Congruences**- Wilson's Theorem and Fermat's Little Theorem
- Pseudoprimes
- Euler's Theorem

**Arithmetic Functions**- The Euler Phi-Function
- The Sum and Number of Divisors
- Perfect Numbers and Mersenne Primes
- MÃ¶bius Inversion
- Partitions

**Cryptography**- Character Ciphers
- Block and Stream Ciphers
- Exponentiation Ciphers
- Public Key Cryptography
- Cryptographic Protocols and Applications

**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

**Applications of Primitive Roots and the Order of an Integer**- Pseudorandom Numbers
- The EIGamal Cryptosystem
- An Application to the Splicing of Telephone Cables

**Quadratic Residues**- Quadratic Residues and Nonresidues
- The Law of Quadratic Reciprocity
- The Jacobi Symbol
- Euler Pseudoprimes
- Zero-Knowledge Proofs

**Decimal Fractions and Continued Fractions**- Decimal Fractions
- Finite Continued Fractions
- Infinite Continued Fractions
- Periodic Continued Fractions
- Factoring Using Continued Fractions

**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

**The Gaussian Integers**- Gaussian Integers and Gaussian Primes
- Greatest Common Divisors and Unique Factorization
- Gaussian Integers and Sums of Squares

### About our author

**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.â€¯

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,â€¯ 7th Edition **and other books.â€¯

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