An Introduction to Number Theory

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Springer Science & Business Media, May 21, 2007 - Mathematics - 297 pages
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An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject.

In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory.

A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography.

Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to learn some of the big ideas in number theory.

 

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Contents

Introduction
1
Diophantine Equations
43
Quadratic Diophantine Equations
59
Recovering the Fundamental Theorem of Arithmetic
83
Primes in an Arithmetic Progression
207
Converging Streams
225
Computational Number Theory
245
References
279
Index
287
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