千米小说网

Chapter 1 How not to think about numbers（第2页）

&otheprimes，thefirsttwentyofthemare:

2，3，5，7，11，13，17，19，23，29，31，37，41，43，47，53，59，61，67，71。

eartheverybeginningofthenumbersequence，primesareohereislittleopportunityforsmallohavefactorizatioheprimesbeerarer。Forexample，thereisoripleofsees:thetrio3，5，7isunique，aseverythirdoddnumberisamultipleof3，andsothiseverhappenagaihinningproeoceis，however，quiteleisurelyandsurprisiiple，thethirtieshaveonlytwoprimes，thosebeing31aimmediatelyafter100therearetwo‘secutive’pairsoftwinprimesin101，103and107，109。

Theprimeshavebeenasourceoffasforthousandsofyearsbecausethey（aclaimthatweshalljustifyier）yettheyariseamouralnumbersihaphazardfashion。ThismysteriousaablefacetoftheirnatureisexploitediographytosafeguardtialunitheI，whichisthesubjectofChapter4。

gforprimality:primedivisibilitytests

&simple-mindedwayoffindingalltheprimesuptoagivennumbersuchas100istowriteallthenumbersdownandcrossoffthebersasyoufiahodbasedonthisideaiscalledtheSieveofEratosthenesandrunsasfollows。Beginbyg2andthencrossoffallthemultiplesof2（theotherevennumbers）inyourlist。Thehebeginninumberyoumeetthathasnotbeencrossedoff（whichwillbe3）andthencrossoffallitsmultiplesintheremaininglist。Byrepeatingthisprocesssuffitlyoften，theprimeswillemergeasthosecrossedout，althoughsomewillbedsomenot。Forexample，Figure1showsthewsofthesieveupto60。

Howdoyouknowwhenyoustopsieviorepeatthisprotilyouumberthatisgreaterthanthesquarerootestnumberinyourlist。Forinstance，ifyoudoyourownsieveforallo120，youwillhthesieveformultiplesof2，3，5，and7，andwhenyoucircle11youstop，as112=121。Atthatpoint，youwillhavecircledasfarasthefirstprimeexgthesquareroestnumber（120inthiscase）withtheremaiiouched。Allberswillnowhavebeeaseachisamultipleofoneormoreof2，3，5，and7。

1。Primesieve:theprimesupto60arethecrossedout

Itisveryeasytotestfordivisibilityby2andby5astheseprimesaretheprimefaberbasetehis，youoochealdigitofthenumberion:nisdivisibleby2exaitsunitsdigitiseven（i。e。0，2，4，6，or8），andnhas5asafadonlyifitendsin0or5。Nomatterhowmanydigitsthenumbernhas，weoocheckthelastdigittodetermiherleof2orof5。Forprimesthatdoo10，weodoabitmoreworkbutherearesimpletestsfordivisibilitythataremuchquirestodoingthefulldivisionsum。

Anumberisdivisibleby3ifandonlyifthesameistrueofthesumofitsdigits。Forexample，thesumofthedigitsofn=145，373，270，099，876，790is87and87=3×29andsonisinthiscasedivisibleby3。Ofcourse，lythetesttotheselfaakingthesumofdigitsoftheouteateachstageuisobvious。Doingthisfivenexampleproducesthefollowingsequence:

145，373，270，099，876，790→87→15→6=2×3。

Youwillseethatallthedivisioedherearesoquickthatyoudlehdozensofdigitswithrelativeeaseeventhoughthesenumbersarebillioerthahwhichyourhandcalculatorcope。

&sgiveheremaio20arebecausetheyareallofthesamegeheseroutinesareallsimpletoapply，althoughitislessobviouswhytheywhthejustifisarenotrecordedhere，theproofsoftheirvalidityarenotespeciallydifficult。

n=27，916，924→2，791，684→279，160→27，916→2，779→259→7

andsonisdivisibleby7。Eachtimewerunthroughtheloopofinstrus，weloseatleasto，sothenumberofpassesthroughtheloopisaboutthesameasthelengthofthehwhichwebegin。

&herornotnhasafactorof11，subtraaldigitfrtrunumbera。Forexample，thenumberisamultipleof11asourmethodreveals:

4，959，746→495，968→49，588→4，950→495→44=4×11。

Tocheckfordivisibilityby13，addfourtimesthefinaldigitttrunumberah7and11。Foriheurnsouttohave13asosprimefactors:

11，264，331→1，126，437→112，671→11，271

→1131→117→39=3×13。For17andfor19，wesubtractfivetimesthefinaldigitinthecaseof17，andaddtwialdigitwheingif19isafaoreapplyingthissteptothetrunumberthatremaiheprocessasoftenasweneed。Forexample，wetest18，905fordivisibilityby17:

18，905→1，865→161→11

soitisnotamultipleof17，butfor19，thetestgivestheopposite:

18，905→1，900=100×19。

&hisbatteryoftests，youreadilychecktheprimalityofallo500（as232=529exceeds500，so19isthelargestpotentialprimefactorthatyouneedyourselfwith）。Forexample，tosettlethematterfor247，wejustocheckfordivisibilityuptotheprime13（asthesquareoftheprime，172=289，exceeds247）。Applyifor13，however，welearnfrom247→（24+28）=52→13，thatleof13:（247=19×13）。

Thedivisibilitytestsforprimesountediofurnishdivisibilitytestsforthosearesquare-freeproductsoftheseprimes（divisiblebythesquareofanyprime）suchas42=2×3×7:anumbernwillbedivisibleby42exarioofdivisibilitytestsfor2，3，asforthosehavesquarefactors，suchas9=32，doically，althoughitisthecasethatnhas9asafadonlyifthatistrueofthesumofthedigitsofn。

Youmightask，afterthousandsofyears，haven’tthoseclevermathematieupwithbetteraicatedmethodsoftestingforprimality?Theanswerisyes。In2002，arelativelyquickwaywasdiscoveredtotestifagivennumberisprime。Theso-called‘AKSprimalitytest’doesnot，however，providethefactorizationofthegivehappee。Theproblemoffindingtheprimefactivehoughinprinciplesolvablebytrial，stillseemspractitraelylargeintegers，andforthatreasonitformsthebasisofmuaryeno，asubjecttoillreturninChapter4。Beforethatweshall，iters，lookalittlemorecloselyatprimesandfactorization。