千米小说网

Chapter 7 To infinity and beyond（第2页）

&ivennumbers，aandb，onletcbetheiraverage。Ifcisirrational，wehaveaherequiredkind（irrational）。Ifoherhand，cisrational，putd=c+t，wheretistheirratiohepreviraph。Bywhathasgonebefore，dwillalsobeirrational，aakenlargeenough，wealwaysedissoclosetecofthetwogivennumbersaandbthatitliesbetweehisway，weseethattheirratioooformadeand，aswiththerationalnumbers，weferthatthereareinfinitelymanyirrationalnumberslyiwohenumberline。

Aofrationalsanditseofirrationalsareinonearable（theyarebothdehenumberline）andihefirstsetistable，theseot）。

iddleThirdSet

Wenowhaveaclearerideaastohowtherationalaioerlatherealherationalnumbersformatableset，yetaredenselypaberlior’sMiddleThirdSetis，bywayoftrast，anuoftheuhelessissparselyspread。Itistheresultofthefollowingstru。

&ionofiddleThirdSettothe4thlevel

&wemakeanamaziion。Bytakiernaryexpansionofanynumberdreplageaceof2by1，weobtainthebinaryexpansionofsomeheuhisgivesaoo-onedehthesetofallnumbersinI（writteninbinary）。ItfollowsthatthealityofCisthesameasthatofI，aerisanu（bytument），itfollowsthattheiddleThirdSetisnotonlyiuntable。

ThereforewehaveasetCthatisinonesensenegligibleinsize（hasmeasurezero），butbyanotherwayCishuge，asithasthesamealityasIahewholerealline。

Whatismore，farfrombeingdense，owheredebysayingthatasetliketherationalsisdewheakearealhereareratioobefoundinaervalsurroundinga，howeversmallthatintervalmightbe。Wesaythatanyneighbourhoodofaembersofthesetofratioorsethasquitetheoppositenature–inCmightlivetheirlivesinthereallieveringaembersofC，providedtheyetheirexperiehlocalityarouheylive。Toseethis，takeaisnotinC，sothatahasaternaryexpansionthatsatleastone1:a=0·。。。。1。。。。。，withthe1ihplace，say。Forasuffitlytinyintervalsurroundinga，thehatintervalhaveaternaryexpansionthatagreestopladthenth，andsoallofthemwillalsonotbemembersesetCastheirternaryexpansionswillalsoatleastoanceof1。

Oherhand，ahetorsetwillooisolated，forwhenalooksoutintoahatsurroundsitinthenumberline，howeversmall，awillfindneighboursfromthesetgalo（ainCaswell）。ecifyamemberbofthegivehatalsoliesingbtohaveaternaryexpansionthatagreeswithatoasuffitlylargenumberofplaces，butwithrybeinga1。IhereareunanymembersofJ。

In，theiddleThirdSetumerousasd，tothemembersoftheCclub，theirbrothersaobeseenallaroundthemwherevertheylook。TotheinC，however，Chardlyseemstoexistatall。NotonememberofCistobespottedintheirexeighbourhoods，aselfhasmeasurezero。Tothem，Cisalmostnothing。

Diophaions

Aiudyemerges，hoetaketheoppositetasistthatnotonlythetsofourequatioioobeintegers。Hereisaclassicexample。

Aboxsspidersalesand46legs。Howmanyofeadofcreaturearethere?Thislittlenumberpuzzlebesolvedeasilybytrial，butitisiethatfirst，itberepresentedbyaion:6b+8s=46，aweareoediainkindsofsolutionstothatequatiohosewheretheles（b）andspiders（s）areumbers。Ingeneral，asystemofequationsiscalledDiophantirigoursolutioospeumbertypes，typitegerorrationala

ThereisasimplemethlinearDiophaionssuchasthiso，dividethroughtheequatiohets，whithisd8sotheirhcfis2。gthisonfactorof2weobtaiequation，thatistosayohesamesolutions:3b+4s=23。Iftheright-handsidewereegerafterperfthisdivision，thatwouldtellusthattherewerenointegralsolutioiohtthere。Theakeohets，thesmalleroneisnormallytheeasiest，andworki，inthiscase3。Ourequatioenas3b+3s+s=（3×7）+2;rearraains=（3×7）-3b-3s+2。Thepointofthisisthatitshowsthatshastheform3t+2forsomei。Substitutings=3t+2iionandmaki，weget

WehepletesolutioheDiophaion:b=5-4t，s=3t+2。ganyintegralvaluefiveasolution，andallsolutionsinintegersareofthisform。

inalproblem，however，wasfurtheredinthatbothbandshadtobeatleastzero，aslesaexist。Hehereareonlytwofeasiblevaluesoft，thosebeingt=0andt=1，giviwopossiblesolutiolesand2spiders，aleand5spiders。Ifweihepuzzleasmeaningthatthereisapluralityofbothtypesofcreature，wehavethetraditionalsolutiolesand2spiders。

12。Lattitsonthelineofaliion

Thistypeofproblemisearbecausethegraphoftheassociatedequationsistsofaninfis。TheDiophantihenistofiitsonthisline，oihatesareintegralor，ifositivesolutions，onlylattitsiivequadrantwilldo。

However，onceweallowsquaresandhigherpowersiioureofthedingproblemsaremuchmorevariedaing。AclassiofthistypethathasafullsolutionisthatoffindingallPythagoreaiveintegersa，b，andcsuchthata2+b2=c2。APythagoreantripleofcoursetakesitshefactthatitallowsyht-ariahsidesofthoseiheclassicexampleisthe（3，4，5）triahagoreantriple，weeratemoreofthemsimplybymultiplyingallthehetriplebyanypositivehePythagoreaionwilluetohold。Forexample，wedoublethepreviousexampletogetthe（6，8，10）triple。This，hivesasimilartrialythesameproportions，asthegeisonlyamatterofsotofshape。GiveriahesedPythagoreantriplesimplybymeasurihsofthesidesinunitsthatarehalfthesizeinalunits，therebydoublingthenumeriensions。Thereare，henuiriplessuchasththe（5，12，13）andthe（65，72，97）right-ariangles。

IodescribeallPythagoreaherefore，itisenoughtodothejobforalltriples（a，b，c）wherethehcfofthethreenumbersis1，asallothersaremerelyscaled-upversioherecipeisasfolloairofepositiveintegersmandn，withohemeveethelarger。Fivenbya=2mn，b=m2-n2and2。Thethreenumbersa，b，andgiveyouaPythagoreahealgebraiseasilydthethreenumbershavenoonfactor（alsonotdifficulttoverify）。Thethreeexamplesabovearisebytakingm=2adn=2inthesed，whileforthelasttrianglewehavem=9，n=4。Ittakesmoreworktoverifytheverse:anysuchPythagoreantriplearisesinthisfashionforsuitablyvaluesofmandn，andwhatismore，therepresentationisutwodifferentpairs（m，n）otyieldthesametriple（a，b，c）。

Thediionfordhigherpowershasnosolutionatall:foraherearenopositiveiriplesx，y，andzsu+yn=zn。ThisisthefamousFermat’sLastTheorem，whifuturemightbeknowheoremasitrovedinthe1990sbySirAndrewWiles。Evenforthecaseofcubes，firstsolvedbyEuler，thisisaverydiffi。Itis，however，relativelyeasytoshowthatthesumoftwofourthpowersisneverasquare（aainlynotafourthpower）。Thisisenoughtoredutothecasewherenisaprimep（meaningthatifwesolvedtheproblemforallprimeexpohegewouldfollowatondiheproblemwassolvedforsularprimesiury。However，thefullsolutionwasonlyrealizedasaceofWilessettliioheShimura–Taniyamajecture。

VersionsofthePellequatioudiedbyDiophantushimselfaroundAD150buttheequationwassolvedbythegreatIndiaiBrahmagupta（AD628）ahodswereimproveduponbyBhaskaraII（AD1150），toewsolutionsfromaseedsolutiowasFermatwhoexhortedmathematistoturiontoPell’sequatioetheoryiscreditedtotherekhematiJoseph-Le（1736–1813）（theEnglishappellation‘Pell’isanhistorit）。

Fibonaduedfras

&hesequenbers，1，1，2，3，5，8，13，21，···discoveredbyFibonatroduChapter5。Takeapairofsuccessivetermsinthissequeethedingratioas1plusafra。Ifweiahisfrabyrepeatedlydividingtopandbottombythepatterake，forinstance

&ainamulti-flooredfrasistiirelyof1s，andeachpregratioofFibonaumbersappearsaswewindthroughtheusthappeime:bytheverywaythesenumbersaredefined，eaberislesstha，aofthedivisionwillleaveaquotientof1andtheremaihepregFibonaumber。YouwillrecallthattheratioofsuccessiveFibonaumbersapproachestheGoldenRatio，τ，andsothissuggeststhatτisthelimitiheuedfrasistiirelyof1s。

&ypeofuedfrasthatemergefromthisprocessareintrinsicallyimportant。roximateanirratioiourallyturntothedecimalrepresentationofy。Thisisextfeneralcalsbut，beioaparticularbase，isiatural。Essentialtothenatureofyishowwellournumberybeapproximatedbyfraswithrelativelysmalldenominators。Isthereanywaytofindaseriesoffrasthatbestdealswiththegdemandsytoahighdegreeofaccuracywhilekeepiorsrelativelysmall?Theaheuedfrarepresentationofadoesthisthroughitstrunsateverlowerfloors。

ThespecialexampleaffordedbytheGoldenRatioopeotheideathatwemaybeabletorepreseiobyfiiions（whichareobviouslyjustratiobyinfihowistheuedfraberaproduced?Thereaderwilloleratealittlealgebraictriordertoseethisina，buthereishowitgoes。

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