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1、ElementaryNumberTheoryW.EdwinClarkDepartmentofMathematicsUniversityofSouthFloridaRevisedJune2,2003Copyleft2002byW.EdwinClarkCopyleftmeansthatunrestrictedredistributionandmodificationareper-mitted,providedthatallcopiesandderivativesretainthesamepermissions.Specificallyn
2、ocommericaluseofthesenotesoranyrevisionsthereofisper-mitted.iiiPrefaceNumbertheoryisconcernedwithpropertiesoftheintegers:...,−4,−3,−2,−1,0,1,2,3,4,....ThegreatmathematicianCarlFriedrichGausscalledthissubjectarithmeticandofithesaid:Mathematicsisthequeenofsciencesandar
3、ithmeticthequeenofmathematics.”Atfirstblushonemightthinkthatofallareasofmathematicscertainlyarithmeticshouldbethesimplest,butitisasurprisinglydeepsubject.Weassumethatstudentshavesomefamiliaritywithbasicsettheory,andcalculus.Butverylittleofthisnaturewillbeneeded.Toagre
4、atextentthebookisself-contained.Itrequiresonlyacertainamountofmathematicalmaturity.And,hopefully,thestudent’slevelofmathematicalmaturitywillincreaseasthecourseprogresses.BeforethecourseisoverstudentswillbeintroducedtothesymbolicprogramminglanguageMaplewhichisanexcell
5、enttoolforexploringnumbertheoreticquestions.Ifyouwishtoseeotherbooksonnumbertheory,takealookintheQA241areaofthestacksinourlibrary.Onemayalsoobtainmuchinterestingandcurrentinformationaboutnumbertheoryfromtheinternet.SeeparticularlythewebsiteslistedintheBibliography.Th
6、ewebsitesbyChrisCaldwell[2]andbyEricWeisstein[11]areespeciallyrecommended.Toseewhatisgoingonatthefrontierofthesubject,youmaytakealookatsomerecentissuesoftheJournalofNumberTheorywhichyouwillfindinourlibrary.iiiivPREFACEHerearesomeexamplesofoutstandingunsolvedproblemsin
7、numberthe-ory.Someofthesewillbediscussedinthiscourse.Asolutiontoanyoneoftheseproblemswouldmakeyouquitefamous(atleastamongmathemati-cians).Manyoftheseproblemsconcernprimenumbers.Aprimenumberisanintegergreaterthan1whoseonlypositivefactorsare1andtheintegeritself.1.(Gold
8、bach’sConjecture)Everyevenintegern>2isthesumoftwoprimes.2.(TwinPrimeConjecture)Thereareinfinitelymanytwinprimes.[Ifpandp+2areprimesw
