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&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’pairsoftwinprimesin101103and107109.
Theprimeshavebeenasourceoffasforthousandsofyearsbecausethey（aclaimthatweshalljustifyier）yettheyariseamouralnumbersihaphazardfashion.ThismysteriousaablefacetoftheirnatureisexploitediographytosafeguardtialunitheI，whichisthesubjectofChapter4.
gforprimality:primedivisibilitytests
&simple-mindedwayofndingalltheprimesuptoagivennumbersuchas100istowriteallthenumbersdownandcrossoffthebersasyouahodbasedonthisideaiscalledtheSieveofEratosthenesandrunsasfollows.Beginbyg2andthencrossoffallthemultiplesof2（theotherevennumbers）inyourlist.Thehebeginninumberyoumeetthathasnotbeencrossedoff（whichwillbe3）andthencrossoffallitsmultiplesintheremaininglist.Byrepeatingthisprocesssuftlyoften，theprimeswillemergeasthosecrossedout，althoughsomewillbedsomenot.Forexample，Figure1showsthewsofthesieveupto60.
Howdoyouknowwhenyoustopsieviorepeatthisprotilyouumberthatisgreaterthanthesquarerootestnumberinyourlist.Forinstance，ifyoudoyourownsieveforallo120，youwillhthesieveformultiplesof2，3，5，and7，andwhenyoucircle11youstop，as112=121.Atthatpoint，youwillhavecircledasfarastherstprimeexgthesquareroestnumber（120inthiscase）withtheremaiiouched.Allberswillnowhavebeeaseachisamultipleofoneormoreof2，3，5，and7.
1.Primesieve:theprimesupto60arethecrossedout
Itisveryeasytotestfordivisibilityby2andby5astheseprimesaretheprimefaberbasetehis，youoochealdigitofthenumberion:nisdivisibleby2exaitsunitsdigitiseven（i.e.0，2，4，6，or8），andnhas5asafadonlyifitendsin0or5.Nomatterhowmanydigitsthenumbernhas，weoocheckthelastdigittodetermiherleof2orof5.Forprimesthatdoo10，weodoabitmoreworkbutherearesimpletestsfordivisibilitythataremuchquirestodoingthefulldivisionsu
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，wetest18905fordivisibilityby17:
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.Theproblemofndingtheprimefactivehoughinprinciplesolvablebytrial，stillseemspractitraelylargeintegers，andforthatreasonitformsthebasisofmuaryeno，asubjecttoillreturninChapter4.Beforethatweshall，iters，lookalittlemorecloselyatprimesandfactorization.
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