Everest and Ward - An Introduction to Number Theory (2005).pdf

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Graduate Texts in Mathematics
232
Editorial Board
S. Axler K.A. Ribet
Graham Everest
Thomas Ward
An Introduction to
Number Theory
With 16 Figures
136559279.001.png
Graham Everest, BSc, PhD
School of Mathematics
University of East Anglia
Norwich
NR4 7TJ
UK
Thomas Ward, BSc, MSc, PhD
School of Mathematics
University of East Anglia
Norwich
NR4 7TJ
UK
Editorial Board
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA
K.A. Ribet
Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840
USA
Mathematics Subject Classification (2000): 11Y05/11/16/55
British Library Cataloguing in Publication Data
Everest, Graham, 1957–
An introduction to number theory. — (Graduate texts in
mathematics ; 232)
1. Number theory
I. Title II. Ward, Thomas, 1963–
512.7
ISBN 1852339179
Library of Congress Control Number: 2005923447
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored on transmitted, in any form or by any means, with the prior permission in writing of the publish-
ers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the
Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to
the publishers.
Graduate Texts in Mathematics series ISSN 0072-5285
ISBN-10: 1-85233-917-9
ISBN-13: 978-1-85233-917-3
Springer Science+Business Media
springeronline.com
© Springer-Verlag London Limited 2005
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence
of a specific statement, that such names are exempt from the relevant laws and regulations and therefore
free for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the informa-
tion contained in this book and cannot accept any legal responsibility or liability for any errors or
omissions that may be made.
Typesetting: Camera-ready by authors
Printed in the United States of America
12/3830-543210 Printed on acid-free paper SPIN 11316527
And he brought him forth abroad, and said,
Look now toward heaven, and tell the stars, if
thou be able to number them: and he said unto
him, So shall thy seed be.
Genesis 15, verse 5
Contents
Introduction ................................................... 1
1 A Brief History of Prime .................................. 7
1.1 Euclidand Primes....................................... 7
1.2 Summing Over thePrimes................................ 11
1.3 Listing thePrimes....................................... 16
1.4 Fermat Numbers ........................................ 29
1.5 Primality Testing........................................ 31
1.6 Proving the Fundamental Theorem of Arithmetic ............ 35
1.7 Euclid’sTheorem Revisited............................... 39
2 Diophantine Equations .................................... 43
2.1 Pythagoras ............................................. 43
2.2 The Fundamental Theorem of Arithmetic in
OtherContexts ......................................... 45
2.3 Sumsof Squares......................................... 48
2.4 Siegel’sTheorem ........................................ 52
2.5 Fermat, Catalan, and Euler ............................... 56
3 Quadratic Diophantine Equations .......................... 59
3.1 QuadraticCongruences................................... 59
3.2 Euler’sCriterion ........................................ 65
3.3 TheQuadraticReciprocity Law ........................... 67
3.4 QuadraticRi n gs......................................... 73
4 Recovering the Fundamental Theorem of Arithmetic ...... 83
4.1 Crisis .................................................. 83
4.2 An Ideal Solution........................................ 84
4.3 Fundamental Theorem of Arithmetic for Ideals .............. 85
3.5 Units in Z[ √ d ] ,d> 0..................................... 75
3.6 QuadraticForms ........................................ 78
 

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