Higher Arithmetic

by Edwards, Harold M.

• ISBN13: 9780821844397
• ISBN10: 0821844393
• Format: Paperback
• Copyright: 2008-07-01
• Publisher: Amer Mathematical Society

Author Biography

Harold M. Edwards is Emeritus Professor of Mathematics at New York University.

Table of Contents

Introduction to Number Theory

Numbers The problem $A\square + B = \square$ Congruences

Double congruences and the Euclidean algorithm

The augmented Euclidean algorithm

Simultaneous congruences

The fundamental theorem of arithmetic

Exponentiation and orders

Euler's $\phi$-function

Finding the order of $a\bmod c$ Primality testing

The RSA cipher system

Primitive roots $\bmod\ p$

Polynomials Tables of indices $\bmod\ p$

Brahmagupta's formula and hypernumbers

Modules of hypernumbers

A canonical form for modules of hypernumbers

Solution of $A\square + B = \square$

Proof of the theorem of Chapter 19

Euler's remarkable discovery

Stable modules

Equivalence of modules

Signatures of equivalence classes

The main theorem

Which modules 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

Cycles of stable modules

Answers to exercises

Bibliography

Index

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