A Course in Computational Number Theory

A Course in Computational Number Theory uses the computer as a tool for motivation and explanation. The book is designed for the reader to quickly access a computer and begin doing personal experiments with the patterns of the integers. It presents and explains many of the fastest algorithms for working with integers. Traditional topics are covered, but the text also explores factoring algorithms, primality testing, the RSA public-key cryptosystem, and unusual applications such as check digit schemes and a computation of the energy that holds a salt crystal together. Advanced topics include continued fractions, Pell's equation, and the Gaussian primes. The CD-ROM contains a Mathematica? package that has hundreds of functions that show step-by-step operation of famous algorithms. (The user must have Mathematica in order to use this package.) Also included is an auxiliary package that contains a database of all 53,000 integers below 10^16 that are 2- and 3-strong pseudoprimes. Users will also have access to an online guide that gives illustrative examples of each function.

David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College and president-elect of the Mathematical Association of America. He served in the Peace Corps, teaching math and science at the Clare Hall School in Antigua, West Indies before studying with Emil Grosswald at Temple University and then teaching at Penn State for 17 years, eight of them as full professor. He chaired the Department of Mathematics and Computer Science at Macalester from 1995 until 2001. He has held visiting positions at the Institute for Advanced Study, the University of Wisconsin-Madison, the University of Minnesota, Université Louis Pasteur (Strasbourg, France), and the State College Area High School.
David has received the MAA Distinguished Teaching Award (Allegheny Mountain Section), the MAA Beckenbach Book Award for Proofs and Confirmations, and has been a Pólya Lecturer for the MAA. He is a recipient of Macalester's Jefferson Award. He has published over fifty research articles in number theory, combinatorics, and special functions. His other books include Factorization and Primality Testing, Second Year Calculus from Celestial Mechanics to Special Relativity, A Radical Approach to Real Analysis (now in 2nd edition), and, with Stan Wagon, A Course in Computational Number Theory. His latest book, A Radical Approach to Lebesgue's Theory of Integration, is due out by the end of 2007.
David chairs the MAA special interest group, Teaching Advanced High School Mathematics. He has chaired the AP Calculus Development Committee and has served as Director of the FIPSE-sponsored program Quantitative Methods for Public Policy. He has been involved in the activities and programs of both the Mathematical Association of America and the American Mathematical Society.

Preface v Notation xi Chapter 1 Fundamentals 1 1.0 Introduction 1 1.1 A Famous Sequence of Numbers 2 1.2 The Euclidean ALgorithm 6 The Oldest Algorithm Reversing the Euclidean Algorithm The Extended GCD Algorithm The Fundamental Theorem of Arithmetic Two Applications 1.3 Modular Arithmetic 25 1.4 Fast Powers 30 A Fast Alforithm for ExponentiationPowers of Matrices, Big-O Notation Chapter 2 Congruences, Equations, and Powers 41 2.0 Introduction 41 2.1 Solving Linear Congruences 41 Linear Diophantine Equations in Two Variables The Conductor An Importatnt Quadratic Congruence 2.2 The Chinese Remainder Theorem 49 2.3 PowerMod Patterns 55 Fermat's Little Theorem More Patterns in Powers 2.4 Pseudoprimes 59 Using the Pseudoprime Test Chapter 3 Euler's Function 3.0 Introduction 65 3.1 Euler's Function 65 3.2 Perfect Numbers and Their Relatives 72 The Sum of Divisors Function Perfect Numbers Amicalbe, Abundant, and Deficient Numbers 3.3 Euler's Theorem 81 3.4 Primitive Roots for Primes 84 The order of an Integer Primes Have PRimitive roots Repeating Decimals 3.5 Primitive Roots for COmposites 90 3.6 The Universal Exponent 93 Universal Exponents Power Towers The Form of Carmichael Numbers Chapter 4 Prime Numbers 99 4.0 Introduction 99 4.1 The Number of Primes 100 We'll Never Run Out of Primes The Sieve of Eratosthenes Chebyshev's Theorem and Bertrand's Postulate 4.2 Prime Testing and Certification 114 Strong Pseudoprimes Industrial-Grade Primes Prime Certification Via Primitive Roots An Improvement Pratt Certificates 4.3 Refinements and Other Directions 131 Other PRimality Tests Strong Liars are Scarce Finding the nth Prime 4.4 A Doszen Prime Mysteries 141 Chapter 5 Some Applications 145 5.0 Introduction 145 5.1 Coding Secrets 145 Tossing a Coin into a Well The RSA Cryptosystem Digital Signatures 5.2 The Yao Millionaire Problem 155 5.3 Check Digits 158 Basic Check Digit Schemes A Perfect Check Digit Method Beyond Perfection: Correcting Errors 5.4 Factoring Algorithms 167 Trial Division Fermat's Algorithm Pollard Rho Pollard p-1 The Current Scene Chapter 6 Quadratic Residues 179 6.0 Introduction 179 6.1 Pepin's Test 179 Quadratic Residues Pepin's Test Primes Congruent to 1 (Mod 40 6.2 Proof of Quadratic Reciprocity 185 Gauss's Lemma Proof of Quadratic Recipocity Jacobi's Extension An Application to Factoring 6.3 Quadratic Equations 195 Chapter 7 Continuec Faction 201 7.0 Introduction 201 7.1 FInite COntinued Fractions 20