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AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 53 Exploring Continued Fractions From the Integers to Solar Eclipses Andrew J. Simoson Exploring Continued Fractions: from the Integers to Solar Eclipses AMS/MAA DOLCIANIMATHEMATICALEXPOSITIONS VOL 53 Exploring Continued Fractions: from the Integers to Solar Eclipses Andrew J.Simoson DolcianiMathematicalExpositionsEditorialBoard HarrietS.Pollatsek,Editor PriscillaS.Bremser ThomasA.Richmond AlfredM.Dahma C.RayRosentrater ElizabethDenne AyseA.Sahin EmilyH.Moore DanE.Steffy KatharineOtt 2010MathematicsSubjectClassification.Primary11J70,00A05,70F15. Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/dol-53 LibraryofCongressCataloging-in-PublicationData Names:Simoson,AndrewJ.,author. Title:Exploringcontinuedfractions:fromtheintegerstosolareclipses/AndrewJ.Simoson. Description: Providence,RhodeIsland: MAAPress,[2019]|Series: Dolcianimathematical expositions;volume53|Includesbibliographicalreferencesandindex. Identifiers:LCCN2018042445|ISBN9781470447953(alk.paper) Subjects:LCSH:Continuedfractions. Classification:LCCQA295.S6152019|DDC512.7/2–dc23 LCrecordavailableathttps://lccn.loc.gov/2018042445 Copyingandreprinting. Individualreadersofthispublication,andnonprofitlibrariesactingfor them,arepermittedtomakefairuseofthematerial,suchastocopyselectpagesforuseinteaching orresearch.Permissionisgrantedtoquotebriefpassagesfromthispublicationinreviews,provided thecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublicationis permittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermissionto reuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter.Formore information,pleasevisitwww.ams.org/publications/pubpermissions. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. ©2019bytheAmericanMathematicalSociety.Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. ⃝1Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 242322212019 Inmemoryofmymother MarionValenciaEnquistSimoson (1925–1990) whomodeledbothbeautyandwhimsy inartandmusic throughthevicissitudesoflife. Contents Introduction xiii StrandI:Patterns 1 Tipsonproblem-solvingandspottingpatterns 2 Alookaheadatthreepatterns 4 ChapterI:TallyBonestotheIntegers 9 Tallybones 9 Atableofprimes? 10 Thesolutiontoapuzzle? 12 Abasetwelveorbasesixtysystem? 13 Baseten,basetwenty,baseeight,basetwo 15 Abinarydigitinterlude 16 Solvingtheshepherd’spuzzleandbeyond 19 Threepartingpuzzles 21 Exercises 23 StrandII:LeibnizandtheBinaryRevolution 29 Acontinuedfractionconnection 34 ChapterII:MathematicalInduction 37 Setnotationandthewell-orderingprinciple 37 Theprincipleofmathematicalinduction 41 Thefundamentaltheoremofarithmetic 43 Equivalenceclasses 45 Nim∗ 47 CaseStudy: Mancala∗ 57 Mancalanim∗ 60 Exercises 63 StrandIII:Al-MaghribîmeetsSudoku 69 ChapterIII:GCDsandDiophantineEquations 73 vii viii Contents Thegreatestcommondivisor 74 Anancientalgorithmforthegreatestcommondivisor 78 TheDiophantinesolution 85 AlitmustestforEuclid’ssolution 88 Clockarithmetic 89 SystemsofDiophantineequations 92 Thetotientismultiplicative 93 AproblemfromDiophantus’sArithmetica 93 Exercises 94 StrandIV:FractionsinthePythagoreanScale 99 Anote-naminginterlude 100 HowPythagorasgeneratedhisscale 102 ChapterIV:ATreeofFractions 107 UnitaryfractionsinancientEgypt 108 Acontinuedfractiontradition 110 Fareysequences 111 Amediantinterlude∗ 116 TheStern-Brocottree 118 Agrandfinale∗ 130 Exercises 132 StrandV:BachandTheWell-TemperedClavier 139 Awell-temperedinnovation 141 Amusicalinterlude 142 Anequal-temperedrevolution 144 Acontinuedfractionconnection 145 ChapterV:TheHarmonicSeries 147 CaseStudy: JeepsintheDesert 157 Alookbehindandalookahead 162 Ageneratingfunctionfinale∗ 163 Exercises 166 StrandVI:AClayTablet 169 TheBabyloniannumbersystem 170 TheacceptedtransliterationofPlimpton322 172 ReciprocalpairsgeneratenormalizedPythagoreantriples 174 Findingtherealmofpotentialgenerators 178 Howthescribemayhavescreenedforgenerators 181 Thepurposeofthetablet 182