Number Theory: A Historical Approach

593 Pages·2013·3.85 MB·English

by  John J. Watkins

NUMBER THEORY

NUMBER THEORY
A Historical Approach
JOH N J. WATKI NS
PRINCETONUNIVERSITYPRESS
PrincetonandOxford
Copyright(cid:1)c 2014byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,
6OxfordStreet,Woodstock,OxfordshireOX201TW
press.prenceton.edu
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
Watkins,JohnJ.,author.
Numbertheory:ahistoricalapproach/JohnJ.Watkins.
pagescm
Includesindex.
Summary:“Thenaturalnumbershavebeenstudiedforthousandsofyears,
yetmostundergraduatetextbookspresentnumbertheoryasalonglistof
theoremswithlittlementionofhowtheseresultswerediscoveredorwhy
theyareimportant.Thisbookemphasizesthehistoricaldevelopmentof
numbertheory,describingmethods,theorems,andproofsinthecontextsin
whichtheyoriginated,andprovidinganaccessibleintroductiontooneof
themostfascinatingsubjectsinmathematics.Writteninaninformalstyleby
anaward-winningteacher,NumberTheorycoversprimenumbers,
Fibonaccinumbers,andahostofotheressentialtopicsinnumbertheory,
whilealsotellingthestoriesofthegreatmathematiciansbehindthese
developments,includingEuclid,CarlFriedrichGauss,andSophieGermain.
Thisone-of-a-kindintroductorytextbookfeaturesanextensivesetof
problemsthatenablestudentstoactivelyreinforceandextendtheir
understandingofthematerial,aswellasfullyworkedsolutionsformanyof
theseproblems.Italsoincludeshelpfulhintsforwhenstudentsareunsure
ofhowtogetstartedonagivenproblem.Usesauniquehistoricalapproach
toteachingnumbertheoryFeaturesnumerousproblems,helpfulhints,and
fullyworkedsolutionsDiscussesfuntopicslikePythagoreantuninginmusic,
Sudokupuzzles,andarithmeticprogressionsofprimesIncludesan
introductiontoSage,aneasy-to-learnyetpowerfulopen-source
mathematicssoftwarepackageIdealforundergraduatemathematicsmajors
aswellasnon-mathmajorsDigitalsolutionsmanual(availableonlyto
professors)”–Providedbypublisher.
ISBN978-0-691-15940-9(hardback)
1.Numbertheory. I.Title.
QA241.W3282014
512.7–dc23
2013023273
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinITCStoneSerifStdandITCStoneSonsStd
Printedonacid-freepaper.∞
TypesetbySRNovaPvtLtd,Bangalore,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
InFondMemory
ForDavidRoeder(1939–2011)

Contents
Preface xi
1
NumberTheoryBegins 1
PierredeFermat(cid:1) 1
PythagoreanTriangles(cid:1) 1
BabylonianMathematics 3
SexagesimalNumbers 4
RegularNumbers 6
SquareNumbers(cid:1) 7
PrimitivePythagoreanTriples(cid:1) 9
InfiniteDescent(cid:1) 12
ArithmeticProgressions 14
Fibonacci’sApproach 17
Problems 19
2
Euclid 26
GreekMathematics(cid:1) 26
TriangularNumbers(cid:1) 27
TetrahedralandPyramidalNumbers 29
TheAxiomaticMethod(cid:1) 33
ProofbyContradiction 37
Euclid’sSelf-EvidentTruths(cid:1) 38
UniqueFactorization 41
PythagoreanTuning 44
Problems 47
3
Divisibility 59
TheEuclideanAlgorithm(cid:1) 59
TheGreatestCommonDivisor(cid:1) 61
TheDivisionAlgorithm(cid:1) 63
Divisibility(cid:1) 65
TheFundamentalTheoremofArithmetic 68
Congruences(cid:1) 71
DivisibilityTests 74
ContinuedFractions 76
Problems 80
viii Contents
4
Diophantus 90
TheArithmetica 90
ProblemsfromtheArithmetica 92
ANoteintheMargin(cid:1) 94
DiophantineEquations(cid:1) 96
Pell’sEquation 101
ContinuedFractions 103
Problems 110
5
Fermat 116
ChristmasDay,1640(cid:1) 116
Fermat’sLittleTheorem(cid:1) 121
PrimesasSumsofTwoSquares(cid:1) 126
SumsofTwoSquares(cid:1) 129
PerfectNumbers(cid:1) 132
MersennePrimes 134
FermatNumbers 137
BinomialCoefficients(cid:1) 139
“MultiPertransibuntetAugebiturScientia” 149
Problems 149
6
Congruences 165
Fermat’sLittleTheorem(cid:1) 165
LinearCongruences(cid:1) 167
Inverses(cid:1) 170
TheChineseRemainderTheorem 171
Wilson’sTheorem(cid:1) 174
TwoQuadraticCongruences 176
Lagrange’sTheorem 179
Problems 183
7
EulerandLagrange 188
ANewBeginning(cid:1) 188
Euler’sPhiFunction(cid:1) 190
PrimitiveRoots(cid:1) 195
Euler’sIdentity(cid:1) 199
QuadraticResidues(cid:1) 200
Lagrange 203
Lagrange’sFourSquaresTheorem(cid:1) 204
SumsofThreeSquares 207
Waring’sProblem 207
Fermat’sLastTheorem(cid:1) 210
Problems 212
Contents ix
8
Gauss 227
TheYoungGauss 227
QuadraticResidues(cid:1) 229
TheLegendreSymbol(cid:1) 231
Euler’sCriterion(cid:1) 232
Gauss’sLemma(cid:1) 234
Euler’sConjecture 238
TheLawofQuadraticReciprocity(cid:1) 239
Problems 250
9
PrimesI 258
Factoring(cid:1) 259
TheQuadraticSieveMethod 261
IsnPrime?(cid:1) 267
Pseudoprimes(cid:1) 269
AbsolutePseudoprimes 270
AProbabilisticTest 271
CannDivide2n−1or2n+1? 272
MersennePrimes(cid:1) 273
Problems 276
10
PrimesII 285
GapsBothLargeandSmall(cid:1) 285
TheTwinP(cid:1)rimeConjecture(cid:1) 286
TheSeries 1 287
pp
Bertrand’sPostulate 292
Goldbach’sConjecture(cid:1) 296
ArithmeticProgressions(cid:1) 297
Problems 299
11
SophieGermain 307
MonsieurLeBlanc(cid:1) 307
GermainPrimes(cid:1) 309
Germain’sGrandPlan 312
Fermat’sLastTheorem(cid:1) 316
Problems 317
12
FibonacciNumbers 324
Fibonacci(cid:1) 325
TheFibonacciSequence(cid:1) 325
TheGoldenRatio 328
FibonacciNumbersinNature 331