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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.

**Preface** **Arithmetic of the Integers **

Induction

Division

Primes

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

Cryptology **Primitive Roots **

Order

Existence

Indices

Random Number Generation

Public-Key Cryptography **Quadratic Residues **

The Legendre Symbol

The Quadratic Reciprocity Law

Factoring **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 ax^{2} + by^{2} + cz^{2} = 0

Bachet’s Equation

Fermat’s Last Theorem **Appendix A: Fundamental Facts ** **Appendix B: Complexity ** **Appendix C: Primes ≤ 9547 and Least Primitive Roots **

Index

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