Edwards H.M. Higher Arithmetic: An Algorithmic Introduction to Number Theory

American Mathematical Society, 2008. — 224 p. – (Student Mathematical Library 45). — ISBN: 978-0-8218-4439-7.Although number theorists have sometimes shunned and even disparaged computation in the past, today's applications of number theory to cryptography and computer security demand vast arithmetical computations. These demands have shifted the focus of studies in number theory and have changed attitudes toward computation itself.
The important new applications have attracted a great many students to number theory, but the best reason for studying the subject remains what it was when Gauss published his classic Disquisitiones Arithmeticae in 1801: Number theory is the equal of Euclidean geometry--some would say it is superior to Euclidean geometry--as a model of pure, logical, deductive thinking. An arithmetical computation, after all, is the purest form of deductive argument.
Higher Arithmetic explains number theory in a way that gives deductive reasoning, including algorithms and computations, the central role. Hands-on experience with the application of algorithms to computational examples enables students to master the fundamental ideas of basic number theory. This is a worthwhile goal for any student of mathematics and an essential one for students interested in the modern applications of number theory.Numbers
The Problem A□ + B = □
Congruences
Double Congruences and the Euclidean Algorithm
The Augmented Euclidean Algorithm
Simultaneous Congruences
The Fundamental Theorem of Arithmetic
Exponentiation and Orders
Euler's <ф>-Function
Finding the Order of a mod c
Primality Testing
The RSA Cipher System
Primitive Roots mod p
Polynomials
Tables of Indices mod p
Brahmagupta's Formula and Hypernumbers
Modules of Hypernumbers
A Canonical Form for Modules of Hypernumbers
Solution of A□ + B = □
Proof of the Theorem
Euler's Remarkable Discovery
Stable Modules
Equivalence of Modules
Signatures of Equivalence Classes
The Main Theorem
Modules That Become Principal When Squared
The Possible Signatures for Certain Values of A
The Law of Quadratic Reciprocity
Proof of the Main Theorem
The Theory of Binary Quadratic Forms
Composition of Binary Quadratic Forms
Appendix. Cycles of Stable Modules