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.

Table of Contents:
Preface. Notation. Chapter 1 Fundamentals. 1.0 Introduction. 1.1 A Famous Sequence of Numbers. 1.2 The Euclidean ALgorithm. The Oldest Algorithm. Reversing the Euclidean Algorithm. The Extended GCD Algorithm. The Fundamental Theorem of Arithmetic. Two Applications. 1.3 Modular Arithmetic. 1.4 Fast Powers. A Fast Alforithm for ExponentiationPowers of Matrices, Big-O Notation. Chapter 2 Congruences, Equations, and Powers. 2.0 Introduction. 2.1 Solving Linear Congruences. Linear Diophantine Equations in Two Variables. The Conductor. An Importatnt Quadratic Congruence. 2.2 The Chinese Remainder Theorem. 2.3 PowerMod Patterns. Fermat's Little Theorem. More Patterns in Powers. 2.4 Pseudoprimes. Using the Pseudoprime Test. Chapter 3 Euler's Function. 3.0 Introduction. 3.1 Euler's Function. 3.2 Perfect Numbers and Their Relatives. The Sum of Divisors Function. Perfect Numbers. Amicalbe, Abundant, and Deficient Numbers. 3.3 Euler's Theorem. 3.4 Primitive Roots for Primes. The order of an Integer. Primes Have PRimitive roots. Repeating Decimals. 3.5 Primitive Roots for COmposites. 3.6 The Universal Exponent. Universal Exponents. Power Towers. The Form of Carmichael Numbers. Chapter 4 Prime Numbers. 4.0 Introduction. 4.1 The Number of Primes. We'll Never Run Out of Primes. The Sieve of Eratosthenes. Chebyshev's Theorem and Bertrand's Postulate. 4.2 Prime Testing and Certification. Strong Pseudoprimes. Industrial-Grade Primes. Prime Certification Via Primitive Roots. An Improvement. Pratt Certificates. 4.3 Refinements and Other Directions. Other PRimality Tests. Strong Liars are Scarce. Finding the nth Prime. 4.4 A Doszen Prime Mysteries. Chapter 5 Some Applications. 5.0 Introduction. 5.1 Coding Secrets. Tossing a Coin into a Well. The RSA Cryptosystem. Digital Signatures. 5.2 The Yao Millionaire Problem. 5.3 Check Digits. Basic Check Digit Schemes. A Perfect Check Digit Method. Beyond Perfection: Correcting Errors. 5.4 Factoring Algorithms. Trial Division. Fermat's Algorithm. Pollard Rho. Pollard p-1. The Current Scene. Chapter 6 Quadratic Residues. 6.0 Introduction. 6.1 Pepin's Test. Quadratic Residues. Pepin's Test. Primes Congruent to 1 (Mod. 6.2 Proof of Quadratic Reciprocity. Gauss's Lemma. Proof of Quadratic Recipocity. Jacobi's Extension. An Application to Factoring. 6.3 Quadratic Equations. Chapter 7 Continuec Faction. 7.0 Introduction. 7.1 FInite COntinued Fractions. 7.2 Infinite Continued Fractions. 7.3 Periodic Continued Fractions. 7.4 Pell's Equation. 7.5 Archimedes and the Sun God's Cattle. Wurm's Version: Using Rectangular Bulls. The Real Cattle Problem. 7.6 Factoring via Continued Fractions. Chapter 8 Prime Testing with Lucas Sequences. 8.0 Introduction. 8.1 Divisibility Properties of Lucas Sequencese. 8.2 Prime Tests Using Lucas Sequencesse. Lucas Certification. The Lucas-Lehmer Algorithm Explained. Luca Pseudoprimes. Strong Quadratic Pseudoprimes. Primality Testing's Holy Grail. Chapter 9 Prime Imaginaries and Imaginary Primes. 9.0 Introduction. 9.1 Sums of Two Squares. 9.2 The Gaussian Intergers. Complex Number Theory. Gaussian Primes. The Moat Problem. The Gaussian Zoo. 9.3 Higher Reciprocity  325. Appendix A. Maathematica Basics. 1.0 Introduction. A.1 Plotting. A.2 Typesetting. Sending Files By E-Mail. A.3 Types of Functions. A.4 Lists. A.5 Programs. A.6 Solving Equations. A.7 Symbolic Algebra. Appendix B Lucas Certificates Exist. References. Index of Mathematica Objects. Subject Index.