NumberTheoryThrough Inquiry

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1、NumberTheoryThroughInquirybyDavidMarshallEdwardOdellMichaelStarbirdDecember11,2006ContentsChapter0.Introduction5NumberTheoryandMathematicalThinking5Noteontheapproachandorganization6Methodsofthought6Acknowledgments7Chapter1.DivideandConquer9DivisibilityInTheNaturalNumbe

2、rs9Definitionsandexamples9Divisibilityandcongruence11TheDivisionAlgorithm16GreatestcommondivisorsandlinearDiophantineequations17LinearEquationsThroughTheAges24Chapter2.PrimeTime27ThePrimeNumbers27FundamentalTheoremofArithmetic28ApplicationsoftheFundamentalTheoremofArith

3、metic32Theinfinitudeofprimes34Primesofspecialform36Thedistributionofprimes37FromAntiquityToTheInternet39Chapter3.AModularWorld43ThinkingCyclically43Powersandpolynomialsmodulon43Linearcongruences47Systemsoflinearcongruences:theChineseRemainderTheorem49APrinceAndAMaster51

4、12CONTENTSChapter4.Fermat’sLittleTheoremandEuler’sTheorem53AbstractingtheOrdinary53Ordersofanintegermodulon53Fermat’sLittleTheorem55AnalternativeroutetoFermat’sLittleTheorem57Euler’sTheoremandWilson’sTheorem58Fermat,WilsonAnd...Leibniz?61Chapter5.PublicKeyCryptography6

5、3PublicKeyCodesAndRSA63Publickeycodes63OverviewofRSA63Let’sdecrypt64HardProblems66Chapter6.PolynomialCongruencesandPrimitiveRoots71HigherOrderCongruences71Lagrange’sTheorem71Primitiveroots72Euler’sφ-functionandsumsofdivisors74Euler’sφ-functionismultiplicative76Rootsmod

6、uloanumber78SophieGermainIsGermane,PartI81Chapter7.TheGoldenRule:QuadraticReciprocity85QuadraticCongruences85Quadraticresidues85Gauss’Lemmaandquadraticreciprocity88SophieGermainisgermane,PartII92Chapter8.PythagoreanTriples,SumsofSquares,andFermat’sLastTheorem95Congruen

7、cestoEquations95Pythagoreantriples95Sumsofsquares98Pythagoreantriplesrevisited100Fermat’sLastTheorem100CONTENTS3Who’sRepresented?101Sumsofsquares101Sumsofcubes,taxicabs,andFermat’sLastTheorem102Chapter9.RationalsClosetoIrrationalsandthePellEquation105DiophantineApproxi

8、mationAndPellEquations105Aplungeintorationalapproximation106Outwiththetrivial109Newsolutionsfromold110Securingtheelus