算数探究 英文版
作者：（德）C.F.高斯著
出版时间：2016年版
内容简介
《算术探索》主要由七部分组成：第一部分同余数基本介绍，第二部分一次同余式，第三部分幂的乘余，第四部分二次同余数。第五部分型和二次不定方程。第六部分是对之前讨论的各种应用介绍。第七部分定义圆截面方程。 读者对象：从事理论学习的研究生和数学工作者。
目录
Translator'sPreface
BibliographicalAbbreviations
Dedication
Author'sPreface
SectionI.CongruentNumbersinGeneral
Congruentnumbers,mo****,residues,andnonresidues,
art.1ft.
Leastresidues,art.4
Elementarypropositionsregardingcongruences,art.5
Certainapplications,art.12
SectionII.CongruencesoftheFirstDegree
Preliminarytheoremsregardingprimenumbers,factors,etc.,
art.13
Solutionofcongruencesofthefirstdegree,art.26
Themethodoffindinganumbercongruenttogivenresidues
relativetogivenmo****,art.32
Linearcongruenceswithseveralunknowns,art.37
Varioustheorems,art.38
SectionIII.ResiduesofPowers
Theresiduesofthetermsofageometricprogressionwhich
beginswithunityconstituteaperiodicseries,art.45
Ifthemodulus=p(aprimenumber),thenumberoftermsin
itsperiodisadivisorofthenumberp-1,art.49
Fermat'stheorem,art,50
Howmanynumberscorrespondtoaperiodinwhichthe
numberoftermsisagivendivisorofp-1,art.52
Primitiveroots,bases,indices,art.57
Computationwithindices,art.58
Rootsofthecongruencex"=A,art.60
Connectionbetweenindicesindifferentsystems,art.69
Basesadaptedtospecialpurposes,art.72
Methodoffindingprimitiveroots,art.73
Varioustheoremsconcerningperiodsan*p*i*itiveroots,art.75
AtheoremofWilson,art.76
Mo****whicharepowersofprimenumbers,art.82
Mo****whicharepowersofthenumber2,art.90
Mo****co****edofmorethanoneprimenumber,art.92
SectionIV.CongruencesoftheSecondDegree
Quadraticresiduesandnonresidues,art.94
Wheneverthemo***us*saprimenumber,thenumberof
residueslessthanthemo***us*sequaltothenumberof
nonresidues,art.96
Thequestionwhetheraco****itenumberisaresidueor
nonresidueofagivenprimenumberdependsonthenature
ofthefactors,art.98
Mo****whichareco****itenumbers,art.100
Ageneralcriterionwhetheragivennumberisaresidueora
nonresidueofagivenprimenumber,art.106
Theinvestigationofprimenumberswhoseresiduesornon-residuesaregivennumbers,art.107
Theresidue-1,art.108
Theresidues 2and-2,art.112
Theresidues 3and-3,art.117
Theresidues 5and-5,art.121
Theresidues 7and-7,art.124
Preparationforthegeneralinvestigation,art.125
Byinductionwesupportageneral(fundamental)theorem
anddrawconclusionsfromit,art.130
Arigorousdemonstrationofthefundamentaltheorem,
art.135
Ananalogousmethodofdemonstratingthetheoremof
art.114,art.145
Solutionofthegeneralproblem,art.146
Linearformscontainingallprimenumbersforwhichagiven
numberisaresidueornonresidue,art.147
Theworkofothermathematiciansconcerningthesein-
vestigations,art.151
Nonpurecongruencesoftheseconddegree,art.152
SectionV.FormsandIndeterminateEquationsoftheSecondDegree
Planofourinvestigation;definitionofformsandtheirnotation,
art.153
Representationofanumber;thedeterminant,art.154
Valuesoftheexpression(b2-ac)(mod.M)towhich
belongsarepresentationofthenumberMbytheform
(a,b,c),art.155
Oneformimplyinganotherorcontainedinit;properand
impropertransformation,art.157
Properandimproperequivalence,art.158
Oppositeforms,art.159
Nei***oringforms,art.160
Commondivisorsofthecoefficientsofforms,art.161
Theconnectionbetweenallsimilartransformationsofa
givenformintoanothergivenform,art.162
Ambiguousforms,art.163
Theoremconcerningthecasewhereoneformiscontainedin
anotherbothproperlyandimproperly,art.164
Generalconsiderationsconcerningrepresentationsofnum-
bersbyformsandtheirconnectionwithtransformations,
art.166
Formswithanegativedeterminant,art.171
Specialapplicationsfordeco****inganumberintotwo
squares,intoasquareandtwiceasquare,intoasquare
andthreetimesasquare,art.182
Formswithpositivenonsquaredeterminant,art.183
Formswithsquaredeterminant,art.206
Formscontainedinotherformstowhich,however,theyare
notequivalent,art.213
Formswith0determinant,art.215
Thegeneralsolutionbyintegersofindeterminateequations
oftheseconddegreewithtwounknowns,art.216
Historicalnotes,art.222
Distributionofformswithagivendeterminantintoclasses,
art.223
Distributionofclassesintoorders,art.226
Thepartitionofordersintogenera,art.228
Theco****itionofforms,art.234
Theco****itionoforders,art.245
Theco****itionofgenera,art.246
Theco****itionofclasses,art.249
Foragivendeterminanttherearethesamenumberofclasses
ineverygenusofthesameorder,art.252
Comparisonofthenumberofclassescontainedinindividual
generaofdifferentorders,art.253
Thenumberofambiguousclasses,art.257
Halfofallthecharactersassignableforagivendeterminant
cannotbelongtoanyproperlyprimitivegenus,art.261
Aseconddemonstrationofthefundamentaltheoremandthe
othertheoremspertainingtotheresidues-1, 2,-2,
art.262
Afurtherinvestigationofthathalfofthecharacterswhich
cannotcorrespondtoanygenus,art.263
Aspecialmethodofdeco****ingprimenumbersintotwo
squares,art.265
Adigressioncontainingatreatmentofternaryforms,
art.266ff.
Someapplicationstothetheoryofbinaryforms,art.286IT.
Howtofindaformfromwhose**p**cationwegetagiven
binaryformofaprincipalgenus,art.286
Exceptforthosecharactersforwhichart.263,264showedit
wasi****sible,allotherswillbelongtosomegenus,
art.287
Thetheoryofthedeco****itionofnumbersandbinary
formsintothreesquares,art.288
DemonstrationofthetheoremsofFermatwhichstatethat
anyintegercanbedeco****edintothreetriangularnumbers
orfoursquares,art.293
Solutionoftheequationax2 by2 cz2=0,art.294
ThemethodbywhichtheillustriousLegendretreatedthe
fundamentaltheorem,art.296
Therepresentationofzerobyternaryforms,art.299
Generalsolutionbyrationalquantitiesofindeterminate
equationsoftheseconddegreeintwounknowns,art.300
Theaveragenumberofgenera,art.301
Theaveragenumberofclasses,art.302
Aspecialalgorithmforproperlyprimitiveclasses;regular
andirregulardeterminantsetc.,art.305
SectionVI.VariousApplicationsofthePrecedingDiscussions
Theresolutionoffractionsintosimplerones,art.309
Theconversionofcommonfractionsintodecimals,art.312
Solutionofthecongruencex2=Abythemethodofexclusion,art.319
Solutionoftheindeterminateequationmx2 ny2=Aby
exclusions,art.323
Anothermethodofsolvingthecongruencex2-Aforthe
casewhere,4isnegative,art.327
Twomethodsfordistinguishingco****itenumbersfrom
primesandfordeterminingtheirfactors,art.329
SectionVII.EquationsDefiningSectionsofaCircle
Thediscussionisreducedtothesimplestcaseinwhichthe
numberofpartsintowhichthecircleiscutisaprime
number,art.336
Equationsfortrigonometricfunctionsofarcswhicharea
partorpartsofthewholecircumference;reductionof
trigonometricfunctionstotherootsoftheequation
xn-1=0,art.337
Theoryoftherootsofthe'equationx"-I=0(wheren
isassumedtobeprime),art.341ft.
Exceptforther*ot*,theremainingrootscontainedin(Ω)
areincludedintheequationX=xn-1 xn-2 etc.
x 1=0;thefunctionXcannotbedeco****edinto
factorsinwhichallthecoefficientsarerational,art.341
Declarationofthepurposeofthefollowingdiscussions,
art.342
Alltherootsin(fl)aredistributedintocertainclasses
(periods),art.343
Varioustheoremsconcerningtheseperiods,art.344
ThesolutionoftheequationX=0asevolve*f*o*the
precedingdiscussions,art.352
Examplesforn=19wheretheoperationisreducedtothe
solutionoftwocubicandonequadraticequation,and
forn=17wheretheoperationisreducedtothesolutionof
fourquadraticequations,art.353,354
Furtherdiscussionsconcerningperiodsofroots,art.355ft.
Sumshavinganevennumberoftermsarerealquantities,
art.355
Theequationdefiningthedistributionoftheroots(Ω)into
twoperiods,art.356
DemonstrationofatheoremmentionedinSectionIV,
art.357
Theequationfordistributingtheroots(Ω)intothreeperiods,
art.358
Reductiontopureequationsoftheequationsbywhichthe
roots(Ω)arefound,art.359
ApplicationoftheprecedingtOtrigonometricfunctions,
art.361ft.
Methodoffindingtheanglescorrespondingtotheindividual
rootsof(Ω),art.361
Derivationoftangents,cotangents,secants,andcosecants
fromsinesandcosineswithoutdivision,art.362
Methodofsuccessivelyreducingtheequationsfortrigonometricfunctions,art.363
Sectionsofthecirclewhichcanbeeffectedbymeansof
quadraticequationsorbygeometricconstructions,art.365
AdditionalNotes
Tables
Gauss'HandwrittenNotes
ListofSpecialSymbols
DirectoryofTerms