Number Theory in the Spirit of Liouville

305 Pages·2011·1.734 MB·English

by  Kenneth S. Williams

LondonMathematicalSocietyStudentTexts76
Number Theory in the Spirit of Liouville
KENNETH S. WILLIAMS
CarletonUniversity,Ottawa
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,
Sa˜oPaulo,Delhi,Dubai,Tokyo,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107002531
(cid:2)C K.S.Williams2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloginginPublicationdata
Williams,KennethS.
NumbertheoryinthespiritofLiouville/KennethS.Williams.
p. cm.–(LondonMathematicalSocietystudenttexts;76)
Includesbibliographicalreferencesandindex.
ISBN978-1-107-00253-1(hardback)
1.Numbertheory. 2.Liouville,Joseph,1809–1882. I.Title.
QA241.W625 2010
512.7(cid:3)2–dc22 2010030399
ISBN978-1-107-00253-1Hardback
ISBN978-0-521-17562-3Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredtoin
thispublication,anddoesnotguaranteethatanycontentonsuchwebsitesis,
orwillremain,accurateorappropriate.
LONDON MATHEMATICAL SOCIETYSTUDENTTEXTS
ManagingEditor:ProfessorD.Benson,
DepartmentofMathematics,UniversityofAberdeen,UK
31 TheLaplacianonaRiemannianmanifold, S.ROSENBERG
32 LecturesonLiegroupsandLiealgebras, ROGERCARTERetal
33 AprimerofalgebraicD-modules, S.C.COUTINHO
34 Complexalgebraicsurfaces:Secondedition, ARNAUDBEAUVILLE
35 Youngtableaux, WILLIAMFULTON
37 Amathematicalintroductiontowavelets, P.WOJTASZCZYK
38 Harmonicmaps,loopgroups,andintegrablesystems, MARTINA.GUEST
39 Settheoryfortheworkingmathematician, KRZYSZTOFCIESIELSKI
40 Dynamicalsystemsandergodictheory, M.POLLICOTT&M.YURI
41 ThealgorithmicresolutionofDiophantineequations, NIGELP.SMART
42 Equilibriumstatesinergodictheory, GERHARDKELLER
43 Fourieranalysisonfinitegroupsandapplications, AUDREYTERRAS
44 Classicalinvarianttheory, PETERJ.OLVER
45 Permutationgroups, PETERJ.CAMERON
47 Introductorylecturesonringsandmodules, JOHNA.BEACHY
48 Settheory, ANDRA´SHAJNAL&PETERHAMBURGER.TranslatedbyATTILAMATE
49 AnintroductiontoK-theoryforC∗-algebras, M.RØRDAM,F.LARSEN&N.J.LAUSTSEN
50 Abriefguidetoalgebraicnumbertheory, H.P.F.SWINNERTON-DYER
51 Stepsincommutativealgebra:Secondedition, R.Y.SHARP
52 FiniteMarkovchainsandalgorithmicapplications, OLLEHA¨GGSTRO¨M
53 Theprimenumbertheorem, G.J.O.JAMESON
54 Topicsingraphautomorphismsandreconstruction, JOSEFLAURI&RAFFAELE
SCAPELLATO
55 Elementarynumbertheory,grouptheoryandRamanujangraphs, GIULIANADAVIDOFF,
PETERSARNAK&ALAINVALETTE
56 Logic,inductionandsets, THOMASFORSTER
57 IntroductiontoBanachalgebras,operatorsandharmonicanalysis, GARTHDALESetal
58 Computationalalgebraicgeometry, HALSCHENCK
59 Frobeniusalgebrasand2-Dtopologicalquantumfieldtheories, JOACHIMKOCK
60 Linearoperatorsandlinearsystems, JONATHANR.PARTINGTON
61 AnintroductiontononcommutativeNoetherianrings:Secondedition, K.R.GOODEARL
&R.B.WARFIELD,JR
62 Topicsfromone-dimensionaldynamics, KARENM.BRUCKS&HENKBRUIN
63 Singularpointsofplanecurves, C.T.C.WALL
64 AshortcourseonBanachspacetheory, N.L.CAROTHERS
65 ElementsoftherepresentationtheoryofassociativealgebrasI, IBRAHIMASSEM,DANIEL
SIMSON&ANDRZEJSKOWRON´SKI
66 Anintroductiontosievemethodsandtheirapplications, ALINACARMENCOJOCARU&
M.RAMMURTY
67 Ellipticfunctions, J.V.ARMITAGE&W.F.EBERLEIN
68 Hyperbolicgeometryfromalocalviewpoint, LINDAKEEN&NIKOLALAKIC
69 LecturesonKa¨hlergeometry, ANDREIMOROIANU
70 Dependencelogic, JOUKUVA¨A¨NA¨NEN
71 ElementsoftherepresentationtheoryofassociativealgebrasII, DANIELSIMSON&
ANDRZEJSKOWRON´SKI
72 ElementsoftherepresentationtheoryofassociativealgebrasIII, DANIELSIMSON&
ANDRZEJSKOWRON´SKI
73 Groups,graphsandtrees, JOHNMEIER
74 RepresentationtheoremsinHardyspaces, JAVADMASHREGHI
75 Anintroductiontothetheoryofgraphspectra, DRAGOSˇCVETKOVIC´,PETERROWLINSON
&SLOBODANSIMIC´
JosephLiouville
1809–1882
Bypermissionofthe
Acade´miedesSciences,InstitutdeFrance.
LondonMathematicalSocietyStudentTexts76
Number Theory in the Spirit of Liouville
KENNETH S. WILLIAMS
CarletonUniversity,Ottawa
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,
Sa˜oPaulo,Delhi,Dubai,Tokyo,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107002531
(cid:2)C K.S.Williams2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloginginPublicationdata
Williams,KennethS.
NumbertheoryinthespiritofLiouville/KennethS.Williams.
p. cm.–(LondonMathematicalSocietystudenttexts;76)
Includesbibliographicalreferencesandindex.
ISBN978-1-107-00253-1(hardback)
1.Numbertheory. 2.Liouville,Joseph,1809–1882. I.Title.
QA241.W625 2010
512.7(cid:3)2–dc22 2010030399
ISBN978-1-107-00253-1Hardback
ISBN978-0-521-17562-3Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredtoin
thispublication,anddoesnotguaranteethatanycontentonsuchwebsitesis,
orwillremain,accurateorappropriate.
Dedicatedtomygranddaughter
ISABELLE SOFIE OLSEN
bornJanuary11,2009
and
tothememoryof
JOSEPH LIOUVILLE
March24,1809–September8,1882
Contents
Preface pagexi
Notation xiii
1 JosephLiouville(1809–1882) 1
NotesonChapter1 12
2 Liouville’sIdeasinNumberTheory 13
NotesonChapter2 20
3 TheArithmeticFunctionsσ (n),σ∗(n),d (n)andF (n) 24
k k k,m k
Exercises3 34
NotesonChapter3 41
4 TheEquationi2+jk =n 43
Exercises4 46
NotesonChapter4 47
5 AnIdentityofLiouville 48
Exercises5 52
NotesonChapter5 53
6 ARecurrenceRelationforσ∗(n) 54
Exercises6 57
NotesonChapter6 58
7 TheGirard-FermatTheorem 59
Exercises7 64
NotesonChapter7 65
vii
viii Contents
8 ASecondIdentityofLiouville 67
Exercises8 73
NotesonChapter8 76
9 SumsofTwo,FourandSixSquares 77
Exercises9 97
NotesonChapter9 98
10 AThirdIdentityofLiouville 100
Exercises10 111
NotesonChapter10 114
11 Jacobi’sFourSquaresFormula 116
Exercises11 120
NotesonChapter11 121
12 Besge’sFormula 125
Exercises12 133
NotesonChapter12 134
13 AnIdentityofHuard,Ou,SpearmanandWilliams 137
Exercises13 155
NotesonChapter13 159
14 FourElementaryArithmeticFormulae 163
Exercises14 179
NotesonChapter14 182
15 SomeTwistedConvolutionSums 184
Exercises15 199
NotesonChapter15 201
16 SumsofTwo,Four,SixandEightTriangularNumbers 205
Exercises16 220
NotesonChapter16 221
17 Sumsofintegersoftheformx2+xy+y2 224
Exercises17 232
NotesonChapter17 235
18 Representationsbyx2+y2+z2+2t2,x2+y2+2z2+2t2
andx2+2y2+2z2+2t2 239
Exercises18 246
NotesonChapter18 249