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Z2. -PrimeSearch F*or4tcertainvqaluesofnin(1)thedenominatorcannotdividethenumeratorand areEthusexcludedfromthisstudy*,!andQhasalgebraicfactorsforcertainothervqaluesofb;nsothatitcannotbGeprime.}YThealgebraicfactorsofb^n+91canbedeterminedyusingthetheoryofcyclotomicpGolynomials[html:color push cmyk 0 1 0 04 color pop html: ],#butvirtuallyalltheimpGortantSfresultscanbeobtainedbysimplelongdivision.q"T*ryinglongdivision,Sitisƌeasytoseethatthedenominatorcannotdividethenumeratorwhenniseven,and_alwaysdividesitwhennisoGdd.pAlso,ifnisoddandcompositethenb^k]!+q1will]divideb^n'+1whenk`dividesnsothatQcannotbGeprime.ThusQcanbeprimeUUonlyifnisanoGddprime. F*or:certainspGecialformsofb,-sQhasalgebraicfactorsforalln.xvIfb?=c^tis:apGerfect_powerwheretisgreaterthan2andnotapGowerof2thenQhasalgebraicfactors݉andisalmostalways݉compGosite.ITherearerarecaseswhenQmaybeprimeforUUsmallnbutagainQ(b;n)canonlybGeprimewhennisprime. Itiswellknownthatallfactorsofb^nDz+_I1withnanoGddprimemustbGeprimesoftheformp:ز=2kPng-+1.B"W*edividedeachQ(b;n)byallprimesofthisformwithkXF<100;000,EndingasmallfactorabGouthalfthetime.EachremainingQwassub 8jectedUUtoaF*ermattestэ aQO! cmsy7 1Dz=1 (moGdQ)for_somea36=b.If_thecongruencefailed,b)thenQwascompGosite.Ifitheldthenwe triedthetestagainwithadierenta.IfbGothtestssucceeded,'QwasdeclaredaprobableUUprime(orprp). AbGoutadaywasdevotedtoeachvqalueofbusingcomputerswithafrequencyofabGout500MegaHertz.9Almostalltheprpsearchingwasdonebythesecondauthor.html: html:
9 .J3. |PrimeProving Small6prp'supto12digitswereprovedprimebysimpledivision.(jF*orprp'suptoabGout800digitstheprimeprovingprogram,APR*T-CLEofUBASICwasused[html:color push cmyk 0 1 0 05 color pop html: ].qThisUUprogramhasanuppGertestlimitofabout830digits. F*oreprp'sgreaterthan800digitsandupto1200digitsweusedtheVFYPRprogramofT*onyForbGes,JwhichisanextendedversionoftheUBASICYprogram,thatcantestprp'supto1600digitsandisabGouttwiceasfastasUBASIC[html:color push cmyk 0 1 0 07 color pop html: ].^
F*ora?Pentium/500ittakesabGout40hourstotesta1200-digitprpandthetesttimeincreasesasabGoutthe4thpowerofthenumberofdigits.>MThetestlimitof1200digitsUUwasarbitrarilychosenbGecauseofcomputertimeavqailability*. OneJotherprime-provingmethoGdwasusedinafewcases.jTheBLS5methoGdisbased4onbGeingabletofactorQn 14sothatthefactoredpartexceedsLIdZ cmr53