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Fundamental Number Theory with Applications

Richard A. Mollin
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Discrete Mathematics and Its Applications 49
[Reviewed by
Mehdi Hassani
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Number theory has an important role in many branches of mathematics and in computer science, and even appears at times in physics and chemistry. Having a fundamental knowledge in this area will be fruitful for scientists in above mentioned areas. The book under review tries to provide such fundamental knowledge of number theory with its applications, mainly in computer science.

All chapters of the book have lots of boxed parts including bibliographies related to the mathematical subjects, which this makes the book very interesting for studying. These bibliographies are very useful, particularly for the historical background.

The first chapter includes fundamental topics like finite continued fractions and Fibonacci numbers, and basic theorems like Wilson's theorem, Fermat's theorem and the Chinese remainder theorem. The author introduces and proves these facts in terms of divisibility (without using the notion of congruence). The chapter also covers Lamé's theorem and Thue's theorem, which are seldom seen them in other texts on fundamental number theory. Following this process, in chapter two, which deals with congruences, Mollin studies some primality tests and gives an introduction in cryptography. The third chapter, on primitive roots, includes an introduction to the interesting topic of random number generation and discusses further topics in cryptography. In chapter four, on quadratic residues, Mollin also discusses algorithms for factoring.

Chapter five is about simple continued fractions and uses them to analyze rational approximations of irrational numbers. The author goes briefly back to factoring in this chapter again. Then he studies Pell’s equation, which is a Diophantine equation; a more precise study of these equations is the subject of last chapter.

The author tries to indicate some recent improvements briefly, in appendixes at the end of book. He focuses especially on the ABC conjecture, which has some consequences for Diophantine equations, and he studies the AKS algorithm, which is a primality test with polynomial running time. I think these are very useful for students to make them familiar with recent advances.

All of the chapters have good exercises, both numerical and theorical; and the reader will find the solutions of the odd-numbered ones at the end of book.

The book under review is a good text for undergraduates, suitable for a course in one semester or an extended course for two semesters, and it can be used in some courses in computer science as well.

Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.

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