PART 1: Algebra
Arkadiy Skopenkov
Translated from Russian by Paul Zeitz and Sergei G. Shubin
Berkeley,CaliforniaProvidence,RhodeIsland
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ThisworkwasoriginallypublishedinRussianby“MCNMO”underthetitle pementymatematikivzadaqah, c 2018.Thepresenttranslationwascreated underlicensefortheAmericanMathematicalSocietyandispublishedbypermission.
ThisvolumeispublishedwiththegeneroussupportoftheSimons FoundationandTomLeightonandBonnieBergerLeighton.
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Names:Skopenkov,Arkadiy,1972–author.
Title:Mathematicsviaproblems:Part1:Algebra/ArkadiySkopenkov;translatedbySergei ShubinandPaulZeitz.
Othertitles:Matematikacherezproblemy.English.
Description:Providence,RhodeIsland:AmericanMathematicalSociety,2020. | Series:MSRI mathematicalcircleslibrary,1944-8074;25 | “MSRI,MathematicalSciencesResearchInstitute, Berkeley,California.” | Includesbibliographicalreferencesandindex.
Identifiers:LCCN2020030058 | ISBN9781470448783(paperback) | 9781470462888(ebook)
Subjects:LCSH:Algebra–Problems,exercises,etc. | Mathematics–Problems,exercises,etc. | AMS:General–Instructionalexposition. | General–Generalandmiscellaneousspecifictopics –Problembooks. | Numbertheory–Instructionalexposition. | Fieldtheoryandpolynomials –Instructionalexposition. | Grouptheoryandgeneralizations–Instructionalexposition. | Realfunctions–Instructionalexposition. | Sequences,series,summability–Instructional exposition. | Mathematicseducation–Algebra–Elementaryalgebra. | Mathematicseducation –Algebra–Equationsandinequalities. | Mathematicseducation–Algebra–Groups,rings, fields.
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Contents
Forewordxi Problems,exercises,circles,andolympiadsxi Whythisbook,andhowtouseitxii English-languagereferencesxiii
Introductionxv
Whatthisbookisaboutandwhoitisforxv Learningbydoingproblemsxvi
Amessage ByA.Ya.Kanel-Belov xvii Olympiadsandmathematicsxvii Researchproblemsforhighschoolstudentsxviii Howthisbookisorganizedxviii Sourcesandliteraturexviii Acknowledgmentsxix Grantsupportxix Numberingandnotationxx Notationxx
Chapter1.Divisibility1
1.Divisibility(1)1 Suggestions,solutions,andanswers2
2.Primenumbers(1)4 Suggestions,solutions,andanswers5
3.Greatestcommondivisor(GCD)andleastcommonmultiple (LCM)(1)6 Suggestions,solutions,andanswers7
4.Divisionwithremainderandcongruences(1)8 Hints9
5.LinearDiophantineequations(2)10 Suggestions,solutions,andanswers11
6.Canonicaldecomposition(2*)12 Suggestions,solutions,andanswers14
7.Integerpointsunderaline(2*)14 Suggestions,solutions,andanswers15
Chapter2.Multiplicationmodulo p 17
1.Fermat’sLittleTheorem(2)17
Suggestions,solutions,andanswers18
2.Primalitytests(3*) ByS.V.Konyagin 19 Hints20
Suggestions,solutions,andanswers20
3.Quadraticresidues(2*)21 Hints22
Suggestions,solutions,andanswers22
4.Thelawofquadraticreciprocity(3*)23
Suggestions,solutions,andanswers24
5.Primitiveroots(3*)26
Suggestions,solutions,andanswers27
6.Higherdegrees(3*) ByA.Ya.Kanel-Belovand
A.B.Skopenkov 28 Hints29 Suggestions,solutions,andanswers29
Chapter3.Polynomialsandcomplexnumbers31
1.Rationalandirrationalnumbers(1)31 Suggestions,solutions,andanswers32
2.Solvingpolynomialequationsofthethirdandfourthdegrees (2)34 Hints35
Suggestions,solutions,andanswers36
3.Bezout’sTheoremanditscorollaries(2)38 Suggestions,solutions,andanswers40
4.Divisibilityofpolynomials(3*) ByA.Ya.Kanel-Belovand A.B.Skopenkov 41 Hintsandanswers42
5.Applicationsofcomplexnumbers(3*)43 Hintsandanswers45
6.Vieta’sTheoremandsymmetricpolynomials(3*)46 Suggestions,solutions,andanswers47
7.DiophantineequationsandGaussianintegers(4*) ByA.Ya. Kanel-Belov 47
Suggestions,solutions,andanswers49
8.Diagonalsofregularpolygons(4*) ByI.N.Shnurnikov 51 Suggestions,solutions,andanswers52
9.AshortrefutationofBorsuk’sconjecture53 Suggestions,solutions,andanswers56
Chapter4.Permutations59
1.Order,type,andconjugacy(1)59 Hintsandanswers62
2.Theparityofapermutation(1)62 Hintsandanswers63
3.Thecombinatoricsofequivalenceclasses(2)64 Answers68
Chapter5.Inequalities69
1.TowardsJensen’sinequality(2)69 Hints71
Suggestions,solutions,andanswers72
2.Somebasicinequalities(2)73 Hints75
Suggestions,solutions,andanswers75
3.Applicationsofbasicinequalities(3*) ByM.A.Bershtein 75 Hints77
Suggestions,solutions,andanswers78
4.Geometricinterpretation(3*)82
Suggestions,solutions,andanswers83
Chapter6.Sequencesandlimits85
1.Finitesumsanddifferences(3)85 Hints86
Suggestions,solutions,andanswers87
2.Linearrecurrences(3)88 Hints89
Suggestions,solutions,andanswers90
3.Concretetheoryoflimits(4*)90
Suggestions,solutions,andanswers92
4.Howdoesacomputercalculatethesquareroot?(4*) By A.C.VorontsovandA.I.Sgibnev 93
Suggestions,solutions,andanswers94
5.Methodsofseriessummation(4*)95 Hints98
Suggestions,solutions,andanswers98
6.Examplesoftranscendentalnumbers99
6.A.Introduction(1)99
6.B.Problems(3*)100
6.C.ProofofLiouville’sTheorem(2)101
6.D.SimpleproofofMahler’sTheorem(3*)102
Chapter7.Functions105
1.Thegraphandnumberofrootsofacubicpolynomial105
1.A.Introduction105
1.B.Problems106 Hints107
1.C.Statementsofthemainresults107
1.D.Proofs109
2.Introductoryanalysisofpolynomials(2)112 Hints114
3.Thenumberofrootsofapolynomial(3*)115 Hints117
Suggestions,solutions,andanswers117
4.Estimationsandinequalities(4*) ByV.A.Senderov 118 Suggestions,solutions,andanswers119
5.Applicationsofcompactness(4*) ByA.Ya.Kanel-Belov 119 Suggestions,solutions,andanswers121
Chapter8.Solvingalgebraicequations123
1.Introductionandstatementofresults123
1.A.Whatisthischapterabout?123
1.B.Constructibility(1)125
1.C.Insolvabilityinrealradicals126
1.D.Insolvabilityincomplexradicals(2)128
1.E.Whatisspecialaboutourproofs130
1.F.Historicalcomments131
1.G.Constructionswithcompassandstraightedge(1)132 Hints133
2.Solvingequations:Lagrange’sresolventmethod133
2.A.Definitionofexpressibilityinradicalsofapolynomial(1)133
2.B.Solutionofequationsoflowdegrees(2)135 Suggestions,solutions,andanswers137
2.C.AreformulationoftheconstructibilityinGauss’sTheorem (2)139
Suggestions,solutions,andanswers140
2.D.IdeaoftheproofofconstructibilityinGauss’sTheorem(2)140
2.E.ProofoftheconstructibilityinGauss’sTheorem(3)142
2.F.Efficientproofsofconstructibility(4*)143 Suggestions,solutions,andanswers148
3.Problemsoninsolvabiltyinradicals149
3.A.Representabilityusingonlyonesquareroot(1–2)150 Firsthints151
Suggestions,solutions,andanswers152
3.B.Multiplesquarerootextractions(3*)154
Suggestions,solutions,andanswers156
3.C.Representinganumberusingonlyonecuberoot(2)158 Suggestions,solutions,andanswers159
3.D.Representinganumberusingonlyonerootofprimeorder (3*)162
Suggestions,solutions,andanswers163
3.E.Thereisonlyonewaytosolveaquadraticequation(2)165 Suggestions,solutions,andanswers167
3.F.Insolvability“inrealpolynomials”(2)168
Suggestions,solutions,andanswers170
3.G.Insolvability“inpolynomials”(3)170
Suggestions,solutions,andanswers171
3.H.Insolvabilityincomplexnumbers(4*)172
3.I.Expressibilitywithagivennumberofradicals(4*)173
4.Proofsofinsolvabilityinradicals175
4.A.Fieldsandtheirextensions(2)175
4.B.Insolvability“inrealpolynomials”(3)176
4.C.Insolvability“inpolynomials”(3)177
4.D.Non-constructibilityinGauss’sTheorem(3*)179
4.E.Insolvability“inrealnumbers”181
4.F.Insolvability“innumbers”(4*)182
4.G.Kronecker’sTheorem(4*)184
4.H.TherealanalogueofKronecker’sTheorem(4*)187
Foreword
Problems,exercises,circles,andolympiads
ThisisatranslationofPart1ofthebook MathematicsThroughProblems: FromMathematicalCirclesandOlympiadstotheProfession,andispart oftheMSRIMathematicalCirclesLibraryseries.Theothertwoparts, Geometry and Combinatorics,willbepublishedinthesameseriessoon.
ThegoaloftheMSRIMathematicalCirclesLibraryseriesistobuilda bodyofworksinEnglishthathelptospreadthe“mathcircle”culture.A mathematicalcircle isaneastern-Europeannotion.Mathcirclesaresimilar towhatmostAmericanswouldcallamathclubforkids,butwithseveral importantdistinguishingfeatures.
First,theyare verticallyintegrated :youngstudentsmayinteractwith olderstudents,collegestudents,graduatestudents,industrialmathematicians,professors,andevenworld-classresearchers,allinthesameroom. Thecircleisnotsomuchaclassroomasagatheringofyounginitiateswith eldertribespeople,whopassdown folklore
Second,the“curriculum,”suchasitis,isdominatedby problems rather thanspecificmathematicaltopics.Aproblem,incontrasttoanexercise, isamathematicalquestionthatonedoesn’tknowhow,atleastinitially,to approach.Forexample,“Whatis3times5?”isanexerciseformostpeople butaproblemforaveryyoungchild.Computing534 isalsoanexercise, conceptuallyverymuchlikethefirstexample,certainlyharder,butonlyin a“technical”sense.Andaquestionlike“Evaluate 7 2 e5x sin3xdx”isalso anexerciseforcalculusstudents—amatterof“merely”knowingtheright algorithmandhowtoapplyit.
Problems,bycontrast,donotcomewithalgorithmsattached.Bytheir verynature,theyrequire investigation,whichisbothanartandascience,demandingtechnicalskillalongwithfocus,tenacity,andinventiveness. Mathcirclesteachstudentstheseskills,notwithformalinstruction,butby havingthem domath andobserveothersdoingmath.Studentslearnthata problemworthsolvingmayrequirenotminutesbutpossiblyhours,days,or evenyearsofeffort.Theyworkonsomeoftheclassicfolkloreproblemsand discoverhowtheseproblemscanhelptheminvestigateotherproblems.They learnhownottogiveupandhowtoturnerrorsorfailuresintoopportunities
xiiFOREWORD formoreinvestigation.Achildinamathcirclelearnstodoexactlywhata researchmathematiciandoes;indeed,heorshedoesindependentresearch, albeitonalowerlevel,andoften—althoughnotalways—onproblemsthat othershavealreadysolved.
Finally,manymathcircleshaveaculturesimilartoasportsteam,with intensecamaraderie,respectforthe“coach,”andhealthycompetitiveness (managedwisely,ideally,bytheleader/facilitator).Themathcircleculture isoftencomplementedbyavarietyofproblemsolvingcontests,oftencalled olympiads.Amathematicalolympiadproblemis,firstofall,agenuine problem(atleastforthecontestant),andusuallyrequiresananswerwhich is,ideally,awell-writtenargument(a“proof”).
Whythisbook,andhowtouseit
TheMathematicalCirclesLibraryeditorialboardchosetotranslatethis bookbecauseithasanaudaciousgoal—promisedbyitstitle—todevelop mathematicsthroughproblems.Thisisnotanoriginalidea,norjusta Russianone.AmericanuniversitieshaveexperimentedforyearswithIBL (inquiry-basedlearning)andMoore-methodcourses,structuredmethodsfor teachingadvancedmathematicsthroughopen-endedproblemsolving.1
Butthismassiveworkisanattempttocuratesequencesofproblemsfor secondarystudents(thestatedfocusishighschoolstudents,butthatcan bebroadlyinterpreted)thatallowthemtodiscoverandrecreatemuchof “elementary”mathematics(numbertheory,polynomials,inequalities,calculus,geometry,combinatorics,gametheory,probability)andstartedging intothesophisticatedworldofgrouptheory,Galoistheory,etc.
Thebookisnotpossibletoreadfromcovertocover—norshoulditbe. Instead,thereaderisinvitedtostartworkingonproblemsthatheorshe findsappealingandchallenging.Manyoftheproblemshavehintsandsolutionsketches,butnotall.Noreaderwillsolvealltheproblems.That’s notthepoint—itisnotacontest.Furthermore,someoftheproblemsare notsupposedtobesolved,butshouldratherbepondered.Forexample, whenlearningaboutprimes,itisnaturaltowonderwhetherthereisalwaysaprimebetween n and2n.Indeed,thisisproblem1.6.9(c)—thevery nontrivialresultknownasBertrand’spostulate—andthetextprovidesreferencesforlearningmoreaboutit.Justbecauseitis“tooadvanced”doesn’t meanthatitshouldn’tbethoughtabout!Infact,sometimesthereaderis explicitlydirectedtojumpahead,withreferencestomaterialthatappears laterinthebook(theauthorsassurethereaderthatthiswillnotleadto circularreasoning).
Indeed,thisisthephilosophyofthebook:Mathematicsisnotasequentialdiscipline,whereoneispresentedwithadefinitionthatleadstoalemma whichleadstoatheoremwhichleadstoaproof.Insteaditisanadventure
1 See,forexample, https://en.wikipedia.org/wiki/Moore_method and http:// www.jiblm.org.
FOREWORDxiii
filledwithexcitingsidetripsaswellaswildgoosechases.Theadventure isitsownreward,butitalso,fortuitously,leadstodeepunderstandingand appreciationofmathematicalideasthatcannotbeaccomplishedbypassive reading.
English-languagereferences
MostofthereferencescitedinthisbookareinRussian.However,thereare manyexcellentbooksinEnglish(sometranslatedfromRussian).Hereisa verybrieflist,organizedbytopicandchapter.2
Articlesfrom Kvant : ThissuperbjournalispublishedinRussian. However,ithasbeensporadicallytranslatedintoEnglishunderthe name Quantum,andthereareseveralexcellentcollectionsinEnglish; see[FT07, Tab99, Tab01].
Problemcollections: TheUSSROlympiadProblemBook [SC93]isa classiccollectionofcarefullydiscussedproblems.Additionally,[FK91] and[FBKY11a, FBKY11b]aregoodcollectionsofolympiadsfrom LeningradandMoscow,respectively.SeealsothenicelycuratedcollectionsoffairlyelementaryHungariancontestproblems[Kur63a, Kur63b, Liu01]andthemoreadvanced(undergraduate-level)PutnamExamproblems[KPV02].
Inequalities: See[Ste04]foracomprehensiveguideand[AS16b]fora moreelementarytext.Theauthoralsorecommendstwoclassicbooks, [HLP67]and[BB65],andthemorespecializedtext[MO09],but cautionsthattheseareallratheradvanced.
Geometry: GeometryRevisited [CG67]isaclassic,and[Che16]isa recentandverycomprehensiveguideto“olympiadgeometry.”
Polynomialsandtheoryofequations: See[Bar03]foranelementaryguide,and[Bew06]forahistoricallymotivatedexpositionofconstructabilityandsolvabilityandunsolvability.InChapter8,seethe book[Gin07]forEnglishtranslationsofthe Kvant articles[Gin72, Gin76],and[Skoa]foranabridgedEnglishversionof[Sko10].
Combinatorics: ThebestbookinEnglish,andpossiblyanylanguage, is ConcreteMathematics [GKP94].
Functions,limits,complexnumbers,andcalculus: Theclassic ProblemsandTheoremsinAnalysis byP´olyaandSzeg˝o[PS04]is— likethecurrenttext—acuratedselectionofproblemsbutatamuch highermathematicallevel.
PaulZeitz April2019
2 WeomitanysupplementaryRussian-languagereferencesmentionedintheoriginal textthatwerenotactuallycitedinthetext.
Introduction
Whatthisbookisaboutandwhoitisfor
Adeepunderstandingofmathematicsisusefulbothformathematiciansand forhigh-techprofessionals.Inparticular,the“profession”inthetitleofthis bookdoesnotnecessarilymeantheprofessionofmathematics.
Thisbookisintendedforhighschoolstudentsandundergraduates(in particular,thoseinterestedinolympiads).Formoredetails,see“Olympiads andmathematics”onp.xvii.Thebookcanbeusedbothforself-studyand forteaching.
Thisbookattemptstobuildabridge(byshowingthatthereisnogap) betweenordinaryhighschoolexercisesandthemoresophisticated,intricate, andabstractconceptsinmathematics.Thefocusisonengagingawide audienceofstudentstothinkcreativelyinapplyingtechniquesandstrategies toproblemsmotivatedby“realworldorrealwork”[Mey].Studentsare encouragedtoexpresstheirideas,conjectures,andconclusionsinwriting. Ourgoalistohelpstudentsdevelopahostofnewmathematicaltoolsand strategiesthatwillbeusefulbeyondtheclassroomandinanumberof disciplines(cf.[IBL, Mey, RMP]).
Thebookcontainsthemoststandard“base”material(althoughweexpectthatatleastsomeofthismaterialisreview—thatnotallisbeing learnedforthefirsttime).Butthemaincontentofthebookismorecomplex material.Sometopicsarenotwellknowninthetraditionsofmathematical circles,butareusefulbothformathematicaleducationandforpreparation forolympiads.
Thebookisbasedontheclassestaughtbytheauthoratdifferenttimes attheIndependentUniversityofMoscow,atvariousMoscowschoolsand mathcircles,inpreparingtheRussianteamfortheInternationalMathematicalOlympiad,inthe“ModernMathematics”summerschool,intheKirov andKostromaSummerMathematicalSchools,intheMoscowOlympiad School,andalsointhesummerConferenceoftheTournamentofTowns.
Muchofthisbookisaccessibletohighschoolstudentswithastrong interestinmathematics.3 Weprovidedefinitionsorreferencesformaterialthatisnotstandardintheschoolcurriculum.However,manytopics aredifficultifyoustudythem“fromscratch.”Thus,theorderingofthe problemshelpstoprovide“scaffolding.”Atthesametime,manytopicsare independent ofeachother.Formoredetails,seep.xviii,“Howthisbookis organized”.
Learningbydoingproblems
Wesubscribetothetraditionofstudyingmathematicsbysolvinganddiscussingproblems.Theseproblemsareselectedsothatintheprocessof solvingthemthereader(moreprecisely,thesolver)mastersthefundamentalsofimportantideas,bothclassicalandmodern.Themainideasare developedincrementallywitholympiad-styleexamples—inotherwords,by thesimplestspecialcases,freefromtechnicaldetails.Inthisway,weshow howyoucanexploreanddiscovertheseideasonyourown
Learningbysolvingproblemsisnotjustaseriousapproachtomathematics;italsocontinuesavenerableculturaltradition.Forexample,the novicesinZenmonasteriesstudybyreflectingonriddles(“koans”)givento thembytheirmentors.(However,theseriddlesarerathermorelikeparadoxesthanwhatweconsidertobeproblems.)See,forexample,[Suz18]; comparewith[Pla12,pp.26–33].“Math”examplesinclude[Arn16b, BSe, RSG+16, KBK08, Pra07b, PS04, SC93, Sko09, Vas87, Zvo12],which sometimesdescribenotonlyproblemsbutalsotheprinciplesofselecting appropriateproblems.FortheAmericantradition,see[IBL, Mey, RMP].
Learningbysolvingproblemsisdifficult,inpart,becauseitgenerally doesnotcreatethe illusion ofunderstanding.However,one’seffortsare fullyrewardedbyadeepunderstandingofthematerial,leadingtotheability tocarryoutsimilar(andsometimesratherdifferent)reasoning.Eventually, whileworkingonfascinatingproblems,readerswillbefollowingthethought processesofthegreatmathematiciansandmayseehowimportantconcepts andtheoriesnaturallyevolve.Hopefullythiswillhelpthemmaketheirown equallyusefuldiscoveries(notnecessarilyinmath)!
Solvingaproblem,theoretically,requiresonlyunderstandingitsstatement.Otherfactsandconceptsarenotneeded.(Actually,usefulfactsand ideaswillbedevelopedwhilesolvingtheproblemspresentedinthisbook.) Sometimes,youmayneedtoknowthingsfromotherpartsofthebookas indicatedintheinstructionsandhints.Forthemostimportantproblems weprovidehints,instructions,solutions,andanswers,locatedattheendof
3 Someofthematerialisstudiedinmathcirclesandsummerschoolsbythosewho arejustgettingacquaintedwithmathematics(forexample,6thgraders).However,the presentationhereisaimedatthereaderwhoalreadyhasatleastaminimalunderstanding ofmathematicalculture.Youngerstudentsneedadifferentapproach;see,forexample, [GIF94].
OLYMPIADSANDMATHEMATICSxvii
eachsection.However,theyshouldbereferredtoonlyafterattemptingto solveaproblem.
Asarule,wepresentthe formulation ofabeautifulorimportantresult (intheformofaproblem)beforeits proof.Insuchcases,onemayneed tosolvelaterproblemsinordertofullyworkouttheproof.Thisisalways explicitlymentionedinhints,andsometimeseveninthetext.Consequently, ifyoufailtosolveaproblem,pleasereadon.Thisguidelineishelpfulbecause itsimulatesthetypicalresearchsituation.
Thisbook“isanattempttodemonstratelearningas dialogue basedon solvinganddiscussingproblems”(see[KBK15]).
Amessage ByA.Ya.Kanel-Belov
Tosolvedifficultolympiadproblems,attheveryleastonemusthavea robustknowledgeofalgebra(particularlyalgebraictransformations)and geometry.Mostolympiadproblems(exceptfortheeasiestones)require “mixed”approaches;rarelyisaproblemresolvedbyapplyingamethodor ideainitspureform.Approachingsuchmixedproblemsinvolvescombining several“crux”problems,eachofwhichmayinvolvesingleideasina“pure” form. Learningtomanipulatealgebraicexpressionsisessential.Thelackof thisskillamongolympiansoftenleadstoridiculousandannoyingmistakes.
Olympiadsandmathematics
Tohimathinkingman’sjobwasnottodenyonereality attheexpenseoftheother,buttoincludeandtoconnect.
U.K.LeGuin. TheDispossessed
Herearethreecommonmisconceptionsaboutveryworthwhilegoals:the bestwaytoprepareforamatholympiadisbysolvinglastyear’sproblems; thebestwaytolearn“serious”mathematicsisbyreadinguniversitytextbooks;thebestwaytomasteranyotherskilliswithnomathatall.Thereis afurthermisconceptionthatonecannotachievetheseapparentlydivergent goalssimultaneously.Theauthorssharethewidespreadbeliefthatthese threeapproachesmissthepointandleadtoharmfulsideeffects:students becometookeenonemulation,ortheystudythe language ofmathematics ratherthanits substance,ortheyunderestimatethevalueofrobustmath knowledgeinotherdisciplines.
Webelievethatthesethreegoalsarenotasdivergentastheymight seem.Thefoundationofmathematicaleducationshouldbethe solution anddiscussionofproblemsinterestingtothestudent,duringwhichastudentlearnsimportantmathematicalfactsandconcepts. Thissimultaneously preparesthestudentformatholympiadsandthe“serious”studyofmathematics,andisgoodforhisorhergeneraldevelopment.Moreover,itismore effectiveforachievingsuccessinanyoneofthethreegoalsabove.
Researchproblemsforhighschoolstudents
Manytalentedhighschooloruniversitystudentsareinterestedinsolving researchproblems.Suchproblemsareusuallyformulatedascomplexquestionsbrokenintoincrementalsteps;see,e.g.,[LKT].Thefinalresultmay evenbeunknowninitially,appearingnaturallyonlyinthecourseofthinking abouttheproblem.Workingonsuchquestionsisusefulinitselfandisa goodapproximationtoscientificresearch.Thereforeitisusefulifateacher orabookcansupportanddevelopthisinterest.
Foradescriptionofsuccessfulexamplesofthisactivity,see,forexample,projectsintheMoscowMathematicalConferenceofHighSchool Students[M].Whilemostoftheseprojectsarenotcompletelyoriginal, sometimestheycanleadtonewresults.
Howthisbookisorganized
Youshouldnotreadeachpageinthisbook,oneaftertheother.Youcan chooseasequenceofstudythatisconvenientforyou(oromitsometopics altogether).Anysection(orsubsection)ofthebookcanbeusedforamath circlesession.
Thebookisdividedintochaptersandsections(somesectionsaredivided intosubsections),withaplanoforganizationoutlinedatthestartofeach section.Ifthematerialofanothersectionisneededinaproblem,youcan eitherignoreitorlookupthereference.Thisallowsgreaterfreedomwhen studyingthebook,butatthesametimeitmayrequirecarefulattention.
Topicsofeachsectionarearrangedapproximatelyinorderofincreasingcomplexity.Thenumbersinparenthesesafteratopicnameindicate its“relativelevel”:1isthesimplest,and4isthemostdifficult.Thefirst topics(notmarkedwithanasterisk)arebasic;unlessindicatedotherwise, youshouldbeginyourstudywiththem.Theremainingones(markedwith anasterisk)canbereturnedtolater;unlessotherwisestated,theyareindependentofeachother.Asyouread,tryto return tooldmaterial,butat anewlevel.Thusyoushouldendupstudyingdifferentlevelsofatopic not sequentially butaspartofamixtureoftopics.
Thenotationusedthroughoutthebookisgivenonp.xx.Notationand conventionsparticulartoaspecificsectionareintroducedatthebeginning ofthatsection.Thebookconcludeswithasubjectindex.Thenumbersin boldarethepagesonwhich formaldefinitions ofconceptsaregiven.
Sourcesandliterature
Eachchapterendswithabibliographythatrelatestotheentirechapter, withsourcesforeachtopic.4 Forexample,wecitethebook[GKP94],
4 Editor’snote: IntheEnglisheditionallthereferencesarecombinedintoonelistat theendofthebook.
ACKNOWLEDGMENTSxix whichinvolvesbothcombinatoricsandalgebra.Inadditiontosourcesfor specializedmaterial,wealsotriedtoincludetheverybestpopularwriting onthetopicsstudied.Wehopethatthisbibliography,atleastasafirst approximation,canguidereadersthroughtheseaofpopularscientificliteratureinmathematics.However,thegreatsizeofthisgenreguaranteesthat manyremarkableworkshadtobeomitted.Pleasenotethatitemsinthe bibliographyarenotnecessaryforsolvingtheproblemsinthisbook,unless explicitlystatedotherwise.
Manyoftheproblemsarenotoriginal,butthesource(evenifitis known)isusuallynotspecified.Whenareferenceisprovided,itcomes afterthestatementoftheproblem,sothatthereadercancomparehisor hersolutionwiththeonegiventhere.Whenweknowthatmanyproblems inasectioncomefromonesource,wementionthis.
Wedonotprovidelinkstoonlineversionsofarticlesinthepopular magazines Kvant (theEnglishmagazine Quantum isbasedon Kvant )and MatematicheskoeProsveshchenie (“MathematicalEnlightment”);theycan befoundatthewebsites http://kvant.ras.ru, http://kvant.mccme.ru, https://en.wikipedia.org?wiki?Quantum_Magazine,and http://www. mccme.ru/free-books/matpros.html
Acknowledgments
WearegratefulfortheseriousworkoftranslatorsandeditorsDavidScott, SergeiShubin,andPaulZeitz.Wethankthereviewersforhelpfulcomments,specifically,A.V.Antropov(Chapters1and2),A.I.Sgibnev(Chapters3and7),S.L.Tabachnikov(Chapter8),andA.I.Khrabrov(Chapters 5and6),andalsotheanonymousreviewersofselectedmaterials.Wethank A.I.Kanel-Belov,theauthorofsomematerialinthisbook,whoalsocontributedanumberofusefulideasandcomments.Wethankourstudents foraskingchallengingquestionsandpointingouterrors.Furtheracknowledgmentsforspecificsectionsaregivendirectlyinthetext.
Weapologizeforanymistakes,andwillbegratefultoreadersforpointingthemout.
Grantsupport
A.B.SkopenkovwaspartiallysupportedbygrantsfromtheSimonsFoundationandtheDynastyFoundation.
Contactinformation
A.B.Skopenkov:MoscowInstituteofPhysicsandTechnical(StateUniversity)andIndependentUniversityofMoscow, https://users.mccme.ru/ skopenko.
Numberingandnotation
Sectionsineachchapterarearrangedapproximatelyinorderofincreasing complexityofthematerial.Thenumbersinparenthesesafterthesection nameindicateits“relativelevel”:1isthesimplest,and4isthemostdifficult.Thefirstsections(notmarkedwithanasterisk)arebasic;unless indicatedotherwise,youcanbegintostudythechapterwiththem.Theremainingsections(markedwithanasterisk)canbereturnedtolater;unless otherwisestated,theyareindependentofeachother.
Ifamathematicalfactisformulatedasaproblem,thentheobjective istoprovethisfact.Open-endedquestionsarecalled challenges ;hereone mustcomeupwithclearwordingandaproof;cf.,forexample,[VINK10].
Themostdifficultproblemsaremarkedwithasterisks(*).Ifthestatementoftheproblemasksyouto“find”something,thenyouneedtogive a“closedform”answer(asopposedto,say,anunevaluatedsumofmany terms).
Onceagain,ifyouareunabletosolveaproblem,continuereading:later problemsmayturnouttobehints.
Notation
• x =[x]—(lower)integerpartofnumber x (“floor”);thatis,thelargest integernotexceeding x
• x —upperintegerpartofnumber x (“ceiling”);thatis,thesmallest integernotlessthan x
•{x} —fractionalpartofnumber x;equalto x − x
• d|n,or n . d d divides n;thatis,thereexistsaninteger k suchthat n = kd (thenumber d iscalleda divisor ofthenumber n;weassumethat d =0).
• R, Q,and Z —thesetsofallrealnumbers,rationalnumbers,andintegers, respectively.
• Z2 —theset {0, 1} ofremaindersupondivisionby2withtheoperations ofadditionandmultiplicationmodulo2.
• Zm —theset {0, 1,...,m 1} ofremaindersupondivisionby m withthe operationsofadditionandmultiplicationmodulo m.(Specialistsinalgebra oftenwritethissetas Z/mZ anduse Zm forthesetof m-adicintegers for theprime m.)
• n k —thenumberof k -elementsubsetsofan n-elementset(alsodenoted by C k n ).
•|X | —thenumberofelementsinset X .
• A B = {x | x ∈ A and x/ ∈ B } —thedifferenceofthesets A and B .
• A B —thedisjointunionofthesets A and B ;thatis,theunionof A and B viewedastheunionofdisjointsets.
• A ⊂ B —meanstheset A iscontainedintheset B .Insomebooks,this isdenotedby A ⊆ B ,and A ⊂ B means“theset A isintheset B andis notequalto B .”
• Weabbreviatethephrase“Define x tobe a”to x := a
Divisibility
Thepartsofthischapterusedintherestofthebookare:theEuclidean algorithmanditsapplications(problems1.5.7and1.5.9),thelanguageof congruences(section4,“Divisionwitharemainderandcongruences”),and somesimplefacts(e.g.,problem1.1.3and1.3.2).
Inthischapterallvariablesareintegers.Manysolutionsarebasedon M.A.Prasolov’stexts.
1.Divisibility(1)
1.1.1. (a)Stateandprovetherulesofdivisibilityby2,4,5,10,3,9,11. (b)Isthenumber11 ... 1consistingof1993onesdivisibleby111111? (c)Provethatthenumber1 ... 1consistingof2001onesisdivisibleby 37.
1.1.2. If a isdivisibleby2andnotdivisibleby4,thenthenumberofeven divisorsof a isequaltothenumberofitsodddivisors.
1.1.3. Whichofthefollowingstatementsarecorrectforany a and b?(Recall thenotation a|b definedonp.xx.)
(a)2|(a2 a).
(b)4|(a4 a).
(c)6|(a3 a).
(d)30|(a5 a).
(e)If c|a and c|b,then c|(a + b).
(f)If b|a,then bc|ac
(g)If bc|ac forsome c =0,then b|a
Tosolveproblem1.1.3(c),weused1.1.4(a).Proveitusingthedefinitionofdivisibility,butnotusingtheUniqueFactorizationTheorem(problem1.2.8(d))!Theuseofthistheoremmightleadtoacircularargument sincearesultsimilarto1.1.4(a)isusuallyusedinaproofofuniquenessof factorization.
1.1.4. (a)If a isdivisibleby2and3,thenitisalsodivisibleby6; (b)If a isdivisibleby2,3,and5,thenitisalsodivisibleby30; (c)If a isdivisibleby17and19,thenitisalsodivisibleby323.
1.1.5. (a)If k isnotdivisibleby2,3,or5,then k 4 1isdivisibleby240.
(b)If a + b + c isdivisibleby6,then a3 + b3 + c3 isalsodivisibleby6.
(c)If a + b + c isdivisibleby30,then a5 + b5 + c5 isalsodivisibleby30.
(d)If n ≥ 0then202n +162n 32n 1isdivisibleby323.
Suggestions,solutions,andanswers
1.1.1. Intheproofsofdivisibilityrulesbelow,wedenotethenumberin thestatementsby n = ±(10m am +10m 1 am 1 + +10a1 + a0 )forsome 0 ≤ ai ≤ 9
Ruleofdivisibilityby2: Anintegerisdivisibleby2ifandonlyifthe lastdigitoftheintegerisdivisibleby2.
Proof. Clearly,thenumber n a0 iseven.Suppose a0 isalsoeven.Ifa numberdivideseachtermofthesum,itdividesthesum.Therefore n is even.Conversely,ifanumber n iseven,then a0 iseven.
Ruleofdivisibilityby4: Aninteger n isdivisibleby4ifandonlyifthe numberformedbyitslasttwodigitsisdivisibleby4.
Proof. Clearly,thenumber(n 10a1 a0 )isdivisibleby4.Supposethat thenumber a0 +10a1 formedbythelasttwodigitsof n isdivisibleby4. Then n isdivisibleby4.Conversely,if4|n then4|(a0 +10a1 ).
Ruleofdivisibilityby5: Anintegerisdivisibleby5ifandonlyifits lastdigitis5or0.
Provethissimilarlytoprovingtheruleofdivisibilityby2.
Ruleofdivisibilityby10: Anintegerisdivisibleby10ifandonlyifits lastdigitis0.
Provethissimilarlytoprovingtheruleofdivisibilityby2.
Ruleofdivisibilityby3: Aninteger n isdivisibleby3ifandonlyifthe sumofitsdigitsisdivisibleby3.
1.DIVISIBILITY(1)3
Proof. Subtractthesumofdigitsfromthenumberandgroupthesummandsasfollows:
am am 1 ... a1 a0
m 1)am +(10m 1 1)am 1 +
+(10 1)a1 +(1 1)a0 .
Thenumber10k 1=(10 1)(10k 1 +10k 2 + +10+1)isdivisible by3.Theruleofdivisibilityby3followsfromthisobservation.
Ruleofdivisibilityby9: Aninteger n isdivisibleby9ifandonlyifthe sumofitsdigitsisdivisibleby9.
Provethissimilarlytoprovingoftheruleofdivisibilityby3.
Ruleofdivisibilityby11: Subtractthesumofalldigitsof n atodd positionsfromthesumofalldigitsatevenpositions.Thenumber n is divisibleby11ifandonlyiftheresultingnumber f (n)isdivisibleby11.
Proof. First,wewillprovethatforany m ≥ 0thenumber10m ( 1)m is divisibleby11.Forodd m,thenumber10m +1=(10+1)(10m 1 10m 2 + 10m 3 ... 10+1)isdivisibleby11.Foreven m,thenumber10m 1is divisibleby102 1andhencedivisibleby11.Nowwehave n f (n)=(10m ( 1)m )am +(10m 1 ( 1)m 1 )am 1 + +(10+1)a1 +(1 1)a0
Sinceeverytermofthesumontheright-handsideoftheequationisdivisible by11, n isdivisibleby11ifandonlyif f (n)isdivisibleby11.
1.1.3. Answers :(a,c,d,e,f)true;(b)false.
(a)Wehave a2 a = a(a 1).Takeninthenaturalorder,everyother integeriseven;thusoneofthenumbers a or a 1iseven,sotheirproduct a2 a isalsoeven.
(b)4doesnotdivide(24 2)=14.
(c)Wehave a3 a = a(a 1)(a +1).Thenumber a(a 1)isdivisible by2while(a 1)a(a +1)isdivisibleby3.Thus a3 a isdivisibleby2and 3,and,asfollowsfrom1.1.4(a),itisdivisibleby6.
(d)Wehave a5 a = a(a 1)(a +1)(a2 +1).Now, a(a 1)isdivisible by2while(a 1)a(a +1)isdivisibleby3.Ifnoneofthenumbers a 1, a, and a +1isdivisibleby5,thentheremainderfromdividing a by5isequal to2or3.Thus a2 +1isdivisibleby5.Then,asfollowsfrom1.1.4(b), a5 a isdivisibleby30.
(e)If a = kc and b = mc,then a + b =(k + m)c.
(f)If a = kb then ac = k (bc).
(g)If ac = kbc then c(a kb)=0.Since bc =0wehave c =0;therefore a = kb.
41.DIVISIBILITY
1.1.4. (a) Hint. Wehave3a 2a = a.
Solution. Since2|a wehave6|3a,andsince3|a wehave6|2a;therefore 6|(3a 2a)= a
(b) Hint. 6a 5a = a
Solution. Fromthegivenconditionsandpart(a)abovewehave6|a and 5|a.Therefore30|6a and30|5a,so30|(6a 5a)= a
(c) Hint. 19a 17a =2a,17a 8 2a = a
Solution. Fromthegivenconditionswehave17|a and19|a.Therefore 17 19|17a and19 17|19a.So17 19|(19a 17a)=2a.Then17 19|(17a 8 2a)= a.
1.1.5. (d)Thenumber(an bn )=(a b)(an 1 +
1 )is divisibleby(a b).Therefore202n +162n 32n 1=(202n 32
...
)+((162 )n (12 )n )isdivisibleby17.Similarly,202n +162n 32n 1=(202n 1)+ ((162 )n (32 )n )isdivisibleby19.Then,accordingto1.1.4(c),202n +162n 32n 1isdivisibleby323.
2.Primenumbers(1)
Aninteger p> 1issaidtobea prime ifitdoesnothavepositivedivisors otherthan p and1.Aninteger q isa composite ifithasatleastonepositive divisordifferentfrom1and |q |.(Thus1isneitheraprimenoracomposite number.)
1.2.1. (a) Lemma. If a¿1isnotdivisiblebyanyprime p ≤ √a,then a is aprime.
(b) SieveofEratosthenes. Let p1 ,...,pk allbeprimesbetween1and n.Foreach i =1,...,k wewillcrossoutallnumbersbetween1and n2 whicharedivisibleby pi .Numberswhichareleftareallprimesbetween n and n2 .
(c)Writedownallprimesbetween1and200.
1.2.2. (a)Findall p suchthat p, p +2,and p +4areprimes.
(b)Provethatifthenumber11...1consistingof n onesisaprime,then n isaprime.
(c)Provethattheconverseof(b)isnottrue.
Theorem1.2.3 (Euclid). (a)Thereareinfinitelymanyprimes.
(b)Thereareinfinitelymanyprimesoftheform4k +3.
Comparetoproblem2.3.3(f).Usingadvancedtechniquesit’spossible toprovethefollowingstatement.
Theorem1.2.4 (Dirichlet). Iftheintegers a,b> 0havenocommondivisors otherthan ±1,thenthereareinfinitelymanyprimesoftheform ak + b
1.2.5. Let pn denotethe nthprimenumber(inascendingorder).
(a)Provethat pn+1 ≤ p1 pn +1
(b)Provethat pn+1 ≤ p1 pn 1for n ≥ 2.
(c)∗ Provethatthereisaperfectsquarebetween p1 + + pn and p1 + ... + pn+1 .
1.2.6. (a)Isittruethatforany n,thenumber n2 + n +41isaprime?
(b)Provethatforanynon-constantquadraticfunction f withinteger coefficients,thereexistsaninteger n suchthatthenumber |f (n)| iscomposite.
(c)Provethatforanynon-constantpolynomial f withintegercoefficients,thereexistsaninteger n suchthatthenumber |f (n)| iscomposite.
1.2.7. Thereexist1000consecutivenumbers,noneofwhichis (a)aprime; (b)apowerofaprime.
1.2.8. (a)Anypositiveintegermaybedecomposedintoaproductofprime numbers.
(b)Anevennumberiscalled primish ifitisnotaproductoftwosmaller positiveevennumbers.Isthedecompositionofanevennumberintoa productofprimishnumbersnecessarilyunique?(Seeamoremeaningful exampleinproblem3.7.3(b).)
(c)∗ Ifanumberisequaltotheproductoftwoprimes,thisdecomposition isuniqueuptotheorderofthefactors.
(d) FundamentalTheoremofArithmetic. Thedecompositionof anypositiveintegerintoaproductofprimesisuniqueuptotheorderof thefactors.(ThistheoremisoftenreferredtoastheUniqueFactorization TheoremortheCanonicalDecompositionTheorem.)
Forthe(usual)solutionof(b)and(c)youwillneedthelemmasin problem1.5.7.Seealsoproblem3.4.5.
Suggestions,solutions,andanswers
1.2.2. (a)Answer : p =3.
Solution. Thenumbers p, p +2,and p +4havedifferentremainders upondivisionby3.Thereforeoneofthemisdivisibleby3.Thisnumber isaprime,soitisequalto3.Sinceallprimesbydefinitionarepositive
61.DIVISIBILITY
integers,then p +4 =3.Since1isnotaprime, p +2 =3.Thus p =3.This isindeedoursolution,because3,5,and7areprimes.
(b)Assumetothecontrarythat n iscomposite,i.e., n = ab,where a,b> 1.Wehave xb 1=(x 1)(xb 1 + xb 2 + ... + x +1).Substituting x =10a weseethat11 1= 10n 1 9 isdivisibleby 10a 1 9
(c)Theconversestatementisfalse:111=37 3.
1.2.7. (a)Forexample,1000!+2,1000!+3,...,1000!+1001.Theproblem canalsobesolvedsimilarlytopart(b).
(b)Takedifferentprimes p1 ,p2 ,...,p2000 .The ChineseRemainderTheorem 1.5.10(d)impliesthatthereexists n suchthat n + i isdivisibleby p2i 1 p2i forany i =1, 2,..., 1000.
1.2.8. (a)Supposethatnoteveryintegerisaproductofprimes.Consider thesmallestpositiveinteger n whichisnotaproductofprimes.Ifitisnota prime,thenitisacompositenumber,so n = ab forsome a,b> 1.Therefore n>a and n>b.But n isthesmallestintegernotequaltoaproductof primes,so a and b arebothproductsofprimes.Hence n isalsoaproduct ofprimes.Thiscontradictsourassumption.
(d)Supposetheassertionisfalse.Considerthesmallestnumber n having twodifferentcanonicaldecompositions: n = p a1 1
Since n isminimal,noneofthenumbers pi isequaltoany qj ,forotherwise wecoulddividebothsidesoftheequalitybythisnumberandgetasmaller numberwithtwodifferentcanonicaldecompositions.Ontheotherhand, q1 divides p a1 1 p a2 2 pam m andtherefore,asfollowsfrom1.5.7(c), q1 divides oneofnumbers pi .Since pi isaprime,wehave q1 = pi .Thiscontradicts ourassumption.
3.Greatestcommondivisor(GCD)andleastcommonmultiple(LCM)(1)
Theintegers a and b aresaidtobe relativelyprime iftheydon’thave commondivisorsotherthan ±1.
Anintegerissaidtobethe greatestcommondivisor (GCD)oftwo positiveintegers a and b ifitisthegreatestnumberthatdividesboth a and b.WedenotetheGCDof a and b by(a,b)orGCD(a,b)orgcd(a,b).
1.3.1. Findallpossiblevalues: (a)(n, 12);(b)(n,n +1);(c)(n,n +6);(d)(2n +3, 7n +6);(e)(n2 ,n +1).
Lemma1.3.2. For a = b thefollowingequalityisvalid:(a,b)=(|a b|,b).
1.3.3. (a)(a,b)= b ifandonlyif a isdivisibleby b.
3.GCDANDLCM7
(b)Thenumbers a (a,b) and b (a,b) arerelativelyprime.
(c)∗ Thenumber(a,b)isdivisiblebyanycommondivisorof a and b
(d)∗ Wehave(ca,cb)= c(a,b)forany c> 0.
Tosolveproblemsmarkedwithanasterisk,youwillneedthelemmas in1.5.7.
1.3.4. (a)Forallpositive m and n wehave
(2m, 2n)=2(m,n), (2m +1, 2n)=(2m +1,n), (2m +1, 2n +1)=(2m +1,m n)for m>n.
(b) Binaryalgorithm.Usingtheequalitiesfrom(a)constructanalgorithm forfindingtheGCD.
1.3.5.*Ifafraction a b isirreducible,thenthefraction a+b ab isalsoirreducible.
Anintegerissaidtobethe leastcommonmultiple (LCM)oftwopositive integers a and b ifitisthesmallestnumberthatisdivisibleby a and b.We denotetheLCMof a and b by[a,b]orLCM(a,b)orlcm(a,b).
1.3.6. Find[192,270].
1.3.7. (a)[a,b]= a ifandonlyif a isdivisibleby b.
(b)Thenumbers [a,b] a and [a,b] b arerelativelyprime.
(c)∗ Anycommonmultipleof a and b isdivisibleby[a,b].
(d)∗ [ca,cb]= c[a,b]forany c> 0.
Suggestions,solutions,andanswers
1.3.1. Answers :(a)1,2,3,4,6,12.(b)1.(c)1,2,3,6.(d)1,3,9.(e)1. Solutions.
(a)Thenumber(12,n)isapositivedivisorof12.Let d|12.Thenumber d doesnothavedivisorsgreaterthanitself,so(12,d)= d.Thus,allpositive divisorsof12satisfytheconditionoftheproblem.
(b)Let d|n,d|(n +1),and d> 0.Then d|(n +1 n)=1,so d =1.
(c)ByLemma1.3.2above,(n,n +6)=(6,n).Similarlyto(a),all positivedivisorsof6satisfytheconditionoftheproblem.
(d)ByLemma1.3.2,(2n +3, 7n +6)=(2n +3, 5n +3)=(2n +3, 3n)= (2n +3,n 3)=(n +6,n 3)=(n +6, 9).
Thus,allpositivedivisorsof9satisfytheconditionoftheproblem.
(e)Let d> 0beacommondivisorofthenumbers n +1and n2 .Thus d|(n +1)(n 1)= n2 1byLemma1.3.2.So d|(n2 (n2 1))=1,and hence d = ±1.
