Fundamental Number Thoery with Applications / Edition 2

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An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage.

New to the Second Edition

• Removal of all advanced material to be even more accessible in scope

• New fundamental material, including partition theory, generating functions, and combinatorial number theory

• Expanded coverage of random number generation, Diophantine analysis, and additive number theory

• More applications to cryptography, primality testing, and factoring

• An appendix on the recently discovered unconditional deterministic polynomial-time algorithm for primality testing

Taking a truly elementary approach to number theory, this text supplies the essential material for a first course on the subject. Placed in highlighted boxes to reduce distraction from the main text, nearly 70 biographies focus on major contributors to the field. The presentation of over 1,300 entries in the index maximizes cross-referencing so students can find data with ease.

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Editorial Reviews

From the Publisher
This is an introductory text in number theory from a well-known name … it covers most of the material traditionally expected in such a course. … One of the most interesting features of this book is the extensive (and crunchy) biographical sketches of relevant mathematicians (both living and dead). … I heartily recommend this book to undergraduates and the passing layman, as it is the work of a master and is lucidly explained. …
—IACR book reviews, March 2010

The second edition of this very interesting book includes a revision of its contents and a pledge for the publication of a second volume with advanced material for a second course in number theory.
—Panayiotis Vlamos, Zentrablatt Math, 1175

Praise for the First Edition
…a very useful addition to the many books on number theory with applications, and it is meant to be accessible to anyone from the novice to the research scientist … [it] provides an excellent supplementary source of information for the reader, not least in the many biographical footnotes on the mathematicians involved in the subject matter, and there are also more than a thousand exercises and examples in the text. …
—P. Shiu, Zentralblatt MATH, Vol. 943

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Product Details

Table of Contents

Arithmetic of the Integers
The Chinese Remainder Theorem
Thue’s Theorem
Combinatorial Number Theory
Partitions and Generating Functions
True Primality Tests
Distribution of Primes
Modular Arithmetic
Basic Properties
Modular Perspective
Arithmetic Functions: Euler, Carmichael, and Möbius
Number and Sums of Divisors
The Floor and the Ceiling
Polynomial Congruences
Primality Testing
Primitive Roots
Random Number Generation
Public-Key Cryptography
Quadratic Residues
The Legendre Symbol
The Quadratic Reciprocity Law
Simple Continued Fractions and Diophantine Approximation
Infinite Simple Continued Fractions
Periodic Simple Continued Fractions
Pell’s Equation and Surds
Continued Fractions and Factoring
Additivity—Sums of Powers
Sums of Two Squares
Sums of Three Squares
Sums of Four Squares
Sums of Cubes
Diophantine Equations
Norm-Form Equations
The Equation ax2 + by2 + cz2 = 0
Bachet’s Equation
Fermat’s Last Theorem
Appendix A: Fundamental Facts
Appendix B: Complexity
Appendix C: Primes 9547 and Least Primitive Roots
Appendix D: Indices
Appendix E: The ABC Conjecture
Appendix F: Primes Is in P
Solutions to Odd-Numbered Exercises
List of Symbols

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