A Guide to Elementary Number Theory

by Dudley, Underwood

• ISBN13: 9780883853474
• ISBN10: 0883853477
• Format: Hardcover
• Copyright: 2009-11-19
• Publisher: Mathematical Assn of Amer

Summary

A Guide to Elementary Number Theory is a short exposition of the topics considered in a first course in number theory. It is intended for those who have had some exposure to the material before but have half-forgotten it, and also for those who may have never taken a course in number theory but who want to understand it without having to engage with the more traditional texts which are often extensive, and dense. Number theory has an impressive history, which this guide investigates. Rather than being a textbook with exercises and solutions, this guide is an exploration of this interesting and exciting field. Its important results are all included, usually with accompanying proofs: the Quadratic Reciprocity Theorem is proved as Gauss did it. The material has been chosen to be maximally broad whilst remaining concise and accessible.

Author Biography

Underwood Dudley received the Ph.D. degree (number theory) from the University of Michigan in 1965. He taught at the Ohio State University and at DePauw University, from which he retired in 2004. He is the author of three books on mathematical oddities, The Trisectors, Mathematical Cranks, and Numerology all published by the Mathematical Association of America. He has also served as editor of the College Mathematics Journal, the Pi Mu Epsilon Journal, and two of the Mathematical Association of America's book series.

Table of Contents

Introduction

Greatest common divisors

Unique factorization

Linear diophantine equations

Congruences

Linear congruences

The Chinese Remainder Theorem

Fermat's Theorem

Wilson's Theorem

The number of divisors of an integer

The sum of the divisors of an integer

Amicable numbers

Perfect numbers

Euler's Theorem and function

Primitive roots and orders

Decimals

Quadratic congruences

Gauss's Lemma

The Quadratic Reciprocity Theorem

The Jacobi symbol

Pythagorean triangles

x4+y4≠z4

Sums of two squares

Sums of three squares

Sums of four squares

Waring's Problem

Pell's Equation

Continued fractions

Multigrades

Carmichael numbers

Sophie Germain primes

The group of multiplicative functions

Bounds for ∏(x)

The sum of the reciprocals of the primes

The Riemann Hypothesis

The Prime Number Theorem

The abc conjecture

Factorization and testing for primes

Algebraic and transcendental numbers

Unsolved problems

Index

About the author

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