Textbook
A Course in Computational Number TheoryISBN: 9780470412152
384 pages
June 2008, ©2000

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, BigO
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.
IndustrialGrade 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 p1.
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 LucasLehmer 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 EMail.
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.

Advanced topics include continued fractions, Pell's equation, and the Gaussian primes

Presents and explains many of the fastest algorithms for working with integers

Modern emphasis to a traditional topic by taking advantage of powerful software to explore applications

Theory is tightly integrated with explorations in order to confirm, explain, or deny what is observed experimentally