Number Theory: An Introduction via the Distribution of Primes

349 Pages·2007·2.239 MB·English

by  Benjamin Fine, Gerhard Rosenberger (auth.)

Benjamin Fine
Gerhard Rosenberger
Number Theory
An Introduction via the
Distribution of Primes
Birkha¨user
Boston • Basel • Berlin
BenjaminFine GerhardRosenberger
FairfieldUniversity Universita¨tDortmund
DepartmentofMathematics FachbereichMathematik
Fairfield,CT06824 D-44221Dortmund
U.S.A. Germany
CoverdesignbyAlexGerasev.
MathematicsSubjectClassification(2000):11A01,11A03,11M01,11R01,11Z05,11T71,11H01,
20A01,20G01,20K01,14G01,08A01
LibraryofCongressControlNumber:2006931568
ISBN-100-8176-4472-5 e-ISBN-100-8176-4545-1
ISBN-13978-0-8176-4472-7 e-ISBN-13978-0-8176-4541-0
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Contents
Preface ......................................................... xi
1 IntroductionandHistoricalRemarks............................ 1
2 BasicNumberTheory......................................... 7
2.1 TheRingofIntegers ........................................ 7
2.2 Divisibility,Primes,andComposites .......................... 11
2.3 TheFundamentalTheoremofArithmetic....................... 16
2.4 CongruencesandModularArithmetic ......................... 21
2.4.1 BasicTheoryofCongruences.......................... 22
2.4.2 TheRingofIntegersModulon......................... 23
2.4.3 UnitsandtheEulerPhiFunction ....................... 26
2.4.4 Fermat’sLittleTheoremandtheOrderofanElement...... 31
2.4.5 OnCyclicGroups.................................... 34
2.5 TheSolutionofPolynomialCongruencesModulom ............. 37
2.5.1 LinearCongruencesandtheChineseRemainderTheorem .. 37
2.5.2 Higher-DegreeCongruences........................... 42
2.6 QuadraticReciprocity....................................... 45
3 TheInfinitudeofPrimes ...................................... 55
3.1 TheInfinitudeofPrimes..................................... 55
3.1.1 SomeDirectProofsandVariations...................... 55
3.1.2 SomeAnalyticProofsandVariations.................... 58
3.1.3 TheFermatandMersenneNumbers .................... 61
3.1.4 TheFibonacciNumbersandtheGoldenSection .......... 65
3.1.5 SomeSimpleCasesofDirichlet’sTheorem .............. 78
3.1.6 ATopologicalProofandaProofUsingCodes ............ 83
3.2 SumsofSquares ........................................... 86
3.2.1 PythagoreanTriples .................................. 87
3.2.2 Fermat’sTwo-SquareTheorem......................... 89
3.2.3 TheModularGroup .................................. 94
viii Contents
3.2.4 Lagrange’sFour-SquareTheorem ...................... 100
3.2.5 TheInfinitudeofPrimesThroughContinuedFractions..... 102
3.3 Dirichlet’sTheorem ........................................ 104
3.4 TwinPrimeConjectureandRelatedIdeas ...................... 121
3.5 PrimesBetweenx and2x.................................... 122
3.6 ArithmeticFunctionsandtheMöbiusInversionFormula.......... 123
4 TheDensityofPrimes ........................................ 133
4.1 ThePrimeNumberTheorem: EstimatesandHistory ............. 133
4.2 Chebychev’sEstimateandSomeConsequences ................. 136
4.3 EquivalentFormulationsofthePrimeNumberTheorem.......... 149
4.4 TheRiemannZetaFunctionandtheRiemannHypothesis......... 157
4.4.1 TheRealZetaFunctionofEuler........................ 158
4.4.2 AnalyticFunctionsandAnalyticContinuation ............ 163
4.4.3 TheRiemannZetaFunction ........................... 166
4.5 ThePrimeNumberTheorem ................................. 173
4.6 TheElementaryProof....................................... 180
4.7 SomeExtensionsandComments.............................. 185
5 PrimalityTesting: AnOverview ................................ 197
5.1 PrimalityTestingandFactorization............................ 197
5.2 SievingMethods ........................................... 198
5.2.1 Brun’sSieveandBrun’sTheorem ...................... 204
5.3 PrimalityTestingandPrimeRecords .......................... 212
5.3.1 PseudoprimesandProbabilisticTesting.................. 218
5.3.2 TheLucas–LehmerTestandPrimeRecords .............. 225
5.3.3 SomeAdditionalPrimalityTests........................ 231
5.4 CryptographyandPrimes.................................... 234
5.4.1 SomeNumber-TheoreticCryptosystems................. 237
5.4.2 PublicKeyCryptographyandtheRSAAlgorithm......... 240
5.5 TheAKSAlgorithm ........................................ 243
6 PrimesandAlgebraicNumberTheory........................... 253
6.1 AlgebraicNumberTheory ................................... 253
6.2 UniqueFactorizationDomains ............................... 255
6.2.1 EuclideanDomainsandtheGaussianIntegers ............ 261
6.2.2 PrincipalIdealDomains .............................. 268
6.2.3 PrimeandMaximalIdeals............................. 272
6.3 AlgebraicNumberFields .................................... 275
6.3.1 AlgebraicExtensionsofQQQ............................. 282
6.3.2 AlgebraicandTranscendentalNumbers.................. 284
6.3.3 SymmetricPolynomials............................... 287
6.3.4 DiscriminantandNorm ............................... 290
6.4 AlgebraicIntegers .......................................... 294
6.4.1 TheRingofAlgebraicIntegers......................... 296
Contents ix
6.4.2 IntegralBases ....................................... 297
6.4.3 QuadraticFieldsandQuadraticIntegers ................. 300
6.4.4 TheTranscendenceofeandπ ......................... 303
6.4.5 TheGeometryofNumbers: MinkowskiTheory........... 306
6.4.6 Dirichlet’sUnitTheorem.............................. 308
6.5 TheTheoryofIdeals........................................ 311
6.5.1 UniqueFactorizationofIdeals ......................... 313
6.5.2 AnApplicationofUniqueFactorization ................. 319
6.5.3 TheIdealClassGroup ................................ 321
6.5.4 NormsofIdeals ..................................... 323
6.5.5 ClassNumber ....................................... 326
BibliographyandCitedReferences ................................. 333
Index........................................................... 337
Preface
Numbertheoryisfascinating. Resultsaboutnumbersoftenappearmagical,bothin
theirstatementsandintheeleganceoftheirproofs. Nowhereisthismoreevidentthan
inresultsaboutthesetofprimenumbers. Theprimenumbertheorem,whichgivesthe
asymptoticdensityoftheprimenumbers,isoftencitedasthemostsurprisingresult
inallofmathematics. Itcertainlyistheresultthatishardesttojustifyintuitively.
The prime numbers form the cornerstone of the theory of numbers. Many, if
not most, results in number theory proceed by considering the case of primes and
then pasting the result together for all integers using the fundamental theorem of
arithmetic. The purpose of this book is to give an introduction and overview of
numbertheorybasedonthecentralthemeofthesequenceofprimes. Therichnessof
thissomewhatuniqueapproachbecomesclearonceonerealizeshowmuchnumber
theoryandmathematicsingeneralareneededinordertolearnandtrulyunderstandthe
primenumbers. Ourapproachprovidesasolidbackgroundinthestandardmaterial
as well as presenting an overview of the whole discipline. All the essential topics
are covered: fundamental theorem of arithmetic, theory of congruences, quadratic
reciprocity, arithmetic functions, the distribution of primes. In addition, there are
firmintroductionstoanalyticnumbertheory,primalitytestingandcryptography,and
algebraicnumbertheoryaswellasmanyinterestingsidetopics. Fulltreatmentsand
proofsaregiventobothDirichlet’stheoremandtheprimenumbertheorem. Thereis
acompleteexplanationofthenewAKSalgorithm,whichshowsthatprimalitytesting
isofpolynomialtime. Inalgebraicnumbertheorythereisacompletepresentation
ofprimesandprimefactorizationsinalgebraicnumberfields.
Thebookgrewoutofnotesfromseveralcoursesgivenforadvancedundergrad-
uates in the United States and for teachers in Germany. The material on the prime
numbertheoremgrewoutofseminarsalsogivenbothattheUniversityofDortmund
andatFairfieldUniversity. Theintendedaudienceisupper-levelundergraduatesand
beginning graduate students. The notes on which the book was based were used
effectivelyinsuchcoursesinboththeUnitedStatesandGermany. Theprerequisites
areaknowledgeofcalculusandmultivariablecalculusandsomelinearalgebra. The
necessary ideas from abstract algebra and complex analysis are introduced in the
book. There are many interesting exercises ranging from simple to quite difficult.
xii Preface
Solutionsandhintsareprovidedtoselectedexercises. Wehavewrittenthebookin
whatwefeelisauser-friendlystylewithmanydiscussionsofthehistoryofvarious
topics. Itisouropinionthatthisbookisalsoidealforself-study.
Therearetwobasicfactsconcerningthesequenceofprimesonwhichthisbook
is focused and from which much of the theory of numbers is introduced. The first
fact is that there are infinitely many primes. This fact was of course known since
at least the time of Euclid. However, there are a great many proofs of this result
notrelatedtoEuclid’soriginalproof. Byconsideringandpresentingmanyofthese
proofs,awideareaofmodernnumbertheoryiscovered. Thisincludesthefactthat
theprimesarenumerousenoughsothatthereareinfinitelymanyinanyarithmetic
progression an+b with a,b relatively prime (Dirichlet’s theorem). The proof of
Dirichlet’stheoremallowsustointroduceanalyticmethods.
In contrast to there being infinitely many primes, the density of primes thins
out. Wefirstencounterthisfactinthestartling(buteasilyproved)resultthatthere
are arbitrarily large gaps in the sequence of primes. The exact nature of how the
sequence of primes thins out is formalized in the prime number theorem, which as
alreadymentioned,manypeopleconsiderthemostsurprisingresultinmathematics.
Presentingtheproofandtheideassurroundingtheproofoftheprimenumbertheorem
allowsustointroduceanddiscussalargeportionofanalyticnumbertheory.
Algebraicnumbertheoryaroseoriginallyasanattempttoextenduniquefactoriza-
tiontoalgebraicnumberrings. Weusetheapproachoflookingatprimesandprime
factorizations to present a fairly comprehensive introduction to algebraic number
theory.
Finally, modern crypotography is intimately tied to number theory. Especially
crucial in this connection is primality testing. We discuss various primality testing
methods,includingtherecentlydevelopedAKSalgorithm,andthenprovideabasic
introductiontocryptography.
Thereareseveralwaysthatthisbookcanbeusedforcourses. Chapters1and2
togetherwithselectionsfromtheremainingchapterscanbeusedforaone-semester
courseinnumbertheoryforundergraduatesorbeginninggraduatestudents. Theonly
prerequisitesareabasicknowledgeofmathematicalproofs(induction,etc.)andsome
knowledgeofcalculus. Alltherestisself-contained, althoughwedousealgebraic
methods, so that some knowledge of basic abstract algebra would be beneficial. A
year-longcoursefocusingonanalyticmethodscanbedonefromChapters1,2,3,and4
andselectionsfrom5and6,whileayear-longcoursefocusingonalgebraicnumber
theory can be fashioned from Chapters 1, 2, 3, and 6 and selections from 4 and 5.
Therearealsopossibilitiesforusingthebookforone-semesterintroductorycourses
inanalyticnumbertheory,centeringonChapter4,orforaone-semesterintroductory
courseinalgebraicnumbertheory,centeringonChapter6. Somesuggestedcourses:
BasicIntroductoryOne-SemesterNumberTheoryCourse:
ChapterOne,ChapterTwo,Sections3.1,4.1,4.2,5.1,5.3,5.4,6.1
Year-LongCourseFocusingonAnalyticNumberTheory:
Chapter1,Chapter2,Chapter3,Chapter4,Sections5.1,5.3,5.4,6.1
Preface xiii
Year-LongCourseFocusingonAlgebraicNumberTheory:
Chapter1,Chapter2,Chapter3,Chapter6,Sections4.1,4.2,5.1,5.3,5.4
One-SemesterCourseFocusingonAnalyticNumberTheory:
Chapter1,Chapter2(asneeded),Sections3.1,3.1.5,3.3,3.4,3.5,Chapter4
One-SemesterCourseFocusingonAlgebraicNumberTheory:
Chapter1,Chapter2(asneeded),Chapter6
We would like to thank the many people who have read through other prelimi-
naryversionsofthesenotesandmadesuggestions. IncludedamongthemareKati
Bencsath andAl Thaler as well as the many students who have taken the courses.
Inparticular,wewouldliketothankPeterAckermann,whoreadthroughthewhole
manuscript, both proofreading it and making mathematical suggestions. Peter was
alsoheavilyinvolvedintheseminarsontheprimenumbertheoremfromwhichmuch
of the material in Chapter 4 comes. We also thank the editors at Birkhäuser, who
didadetailedreadingofthemanuscriptandmademanyimportantsuggestionsand
improvements.
BenjaminFine—Fairfield,CT,USA
GerhardRosenberger—Dortmund,Germany
January,2006