THE HIGHER ARITHMETIC

251 Pages·2008·0.93 MB·English

by  H. Davenport

This page intentionally left blank
Nowintoitseightheditionandwithadditionalmaterialon
primalitytesting,writtenbyJ.H.Davenport,TheHigher
Arithmeticintroducesconceptsandtheoremsinawaythatdoes
notrequirethereadertohaveanin-depthknowledgeofthe
theoryofnumbersbutalsotouchesuponmattersofdeep
mathematicalsignificance.Acompanionwebsite
(www.cambridge.org/davenport)providesmoredetailsofthe
latestadvancesandsamplecodeforimportantalgorithms.
Reviewsofearliereditions:
‘...thewell-knownandcharmingintroductiontonumbertheory...
canberecommendedbothforindependentstudyandasa
referencetextforageneralmathematicalaudience.’
EuropeanMathsSocietyJournal
‘Althoughthisbookisnotwrittenasatextbookbutratherasa
workforthegeneralreader,itcouldcertainlybeusedasa
textbookforanundergraduatecourseinnumbertheoryand,in
thereviewer’sopinion,isfarsuperiorforthispurposetoany
otherbookinEnglish.’
BulletinoftheAmericanMathematicalSociety

THE HIGHER
ARITHMETIC
AN INTRODUCTION TO
THE THEORY OF NUMBERS
Eighthedition
H. Davenport
M.A.,SC.D.,F.R.S.
lateRouseBallProfessorofMathematics
intheUniversityofCambridgeand
FellowofTrinityCollege
Editingandadditionalmaterialby
JamesH.Davenport
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521722360
©The estate of H. Davenport 2008
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2008
ISBN-13 978-0-511-45555-1 eBook (EBL)
ISBN-13 978-0-521-72236-0 paperback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
CONTENTS
Introduction pageviii
I FactorizationandthePrimes 1
1. Thelawsofarithmetic 1
2. Proofbyinduction 6
3. Primenumbers 8
4. Thefundamentaltheoremofarithmetic 9
5. Consequencesofthefundamentaltheorem 12
6. Euclid’salgorithm 16
7. Anotherproofofthefundamentaltheorem 18
8. ApropertyoftheH.C.F 19
9. Factorizinganumber 22
10. Theseriesofprimes 25
II Congruences 31
1. Thecongruencenotation 31
2. Linearcongruences 33
3. Fermat’stheorem 35
4. Euler’sfunctionφ(m) 37
5. Wilson’stheorem 40
6. Algebraiccongruences 41
7. Congruencestoaprimemodulus 42
8. Congruencesinseveralunknowns 45
9. Congruencescoveringallnumbers 46
v
vi Contents
III QuadraticResidues 49
1. Primitiveroots 49
2. Indices 53
3. Quadraticresidues 55
4. Gauss’slemma 58
5. Thelawofreciprocity 59
6. Thedistributionofthequadraticresidues 63
IV ContinuedFractions 68
1. Introduction 68
2. Thegeneralcontinuedfraction 70
3. Euler’srule 72
4. Theconvergentstoacontinuedfraction 74
5. Theequationax−by =1 77
6. Infinitecontinuedfractions 78
7. Diophantineapproximation 82
8. Quadraticirrationals 83
9. Purelyperiodiccontinuedfractions 86
10. Lagrange’stheorem 92
11. Pell’sequation 94
12. Ageometricalinterpretationofcontinued
fractions 99
V SumsofSquares 103
1. Numbersrepresentablebytwosquares 103
2. Primesoftheform4k+1 104
3. Constructionsforx andy 108
4. Representationbyfoursquares 111
5. Representationbythreesquares 114
VI QuadraticForms 116
1. Introduction 116
2. Equivalentforms 117
3. Thediscriminant 120
4. Therepresentationofanumberbyaform 122
5. Threeexamples 124
6. Thereductionofpositivedefiniteforms 126
7. Thereducedforms 128
8. Thenumberofrepresentations 131
9. Theclass-number 133
Contents vii
VII SomeDiophantineEquations 137
1. Introduction 137
2. Theequationx2+y2 =z2 138
3. Theequationax2+by2 =z2 140
4. Ellipticequationsandcurves 145
5. Ellipticequationsmoduloprimes 151
6. Fermat’sLastTheorem 154
7. Theequationx3+y3 =z3+w3 157
8. Furtherdevelopments 159
VIII ComputersandNumberTheory 165
1. Introduction 165
2. Testingforprimality 168
3. ‘Random’numbergenerators 173
4. Pollard’sfactoringmethods 179
5. Factoringandprimalityviaellipticcurves 185
6. Factoringlargenumbers 188
7. TheDiffie–Hellmancryptographicmethod 194
8. TheRSAcryptographicmethod 199
9. Primalitytestingrevisited 200
Exercises 209
Hints 222
Answers 225
Bibliography 235
Index 237
INTRODUCTION
The higher arithmetic, or the theory of numbers, is concerned with the
properties of the natural numbers 1, 2, 3, .... These numbers must have
exercisedhumancuriosityfromaveryearlyperiod;andinalltherecords
ofancientcivilizationsthereisevidenceofsomepreoccupationwitharith-
meticoverandabovetheneedsofeverydaylife.Butasasystematicand
independentscience,thehigherarithmeticisentirelyacreationofmodern
times,andcanbesaidtodatefromthediscoveriesofFermat(1601–1665).
A peculiarity of the higher arithmetic is the great difficulty which has
often been experienced in proving simple general theorems which had
beensuggestedquitenaturallybynumericalevidence.‘Itisjustthis,’said
Gauss, ‘which gives the higher arithmetic that magical charm which has
madeitthe favouritescienceof thegreatestmathematicians,not tomen-
tionitsinexhaustiblewealth,whereinitsogreatlysurpassesotherpartsof
mathematics.’
Thetheoryofnumbersisgenerallyconsideredtobethe‘purest’branch
of pure mathematics. It certainly has very few direct applications to
other sciences, but it has one feature in common with them, namely the
inspirationwhichitderivesfromexperiment,whichtakestheformoftest-
ing possible general theorems by numerical examples. Such experiment,
thoughnecessaryinsomeformtoprogressineverypartofmathematics,
hasplayedagreaterpartinthedevelopmentofthetheoryofnumbersthan
elsewhere;forinotherbranchesofmathematicstheevidencefoundinthis
wayistoooftenfragmentaryandmisleading.
Asregardsthepresentbook,theauthoriswellawarethatitwillnotbe
read without effort by those who are not, in some sense at least, mathe-
maticians.Butthedifficultyispartlythatofthesubjectitself.Itcannotbe
evadedbyusingimperfectanalogies,orbypresentingtheproofsinaway
viii