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离散数学数论234567DividesRelation9101112Theorem:Division“Algorithm”---Letabeanintegeranddapositiveinteger、Thenthereareuniqueintegersqandr,with0≤r<d,suchthata=dq+r、(Existence,cont、)risnon-negative;also,r<d、otherwiseifr≥d,therewouldbeasmallernonnegativeelementinS,namelya-d(q0+1)≥0、Butthena-d(q0+1),whichissmallerthana-dq0,isanelementofS,contradictingthata-dq0wasthesmallestelementofS、So,itcannotbethecasethatr≥d,provingtheexistenceof0≤r<dandq、b)UniquenessSupposeModulararithmetic17SpiralVisualizationofmodMoreoncongruences20212223242526ASimpleHashUsingmodASimpleHashUsingmodCollisionDigitalSignatureApplication313233343536373839PrimesandGreatestmonDivisor41FundamentaltheoremofarithmeticFundamentaltheoremofarithmetic:StrongInduction[frombefore]positefactors4546Onalinuxsystemorincygwin,enter“factor899”>factor899899:2931>factor8999999999999999989999999999999999:771361224492307692312304:222276912304038495:35731093769129485404038495:55897080807699294854040334945723:672472061178021762929485404033420344:22211093323422456427294854043485472:22222315117311757440929485404203484:22310110322910314119348492404203484:2272314516292553111928439237492742742:21389104531282129938319284392329378472:22231321370533840299284392329378472323:333307112008593670870749MersennenumbersThelargestprimesfoundareMersenneprimes、Since,2p-1growsfast,andthereisaquiteefficienttest–Lucas-Lehmertest–fordeterminingifaMersenneprimeisprime、52535455565758TheprimenumbertheoremSo,actuallyx/(lnx–1)isbetterestimateofnumberofprimes、Greatestmondivisor6465Moreongcd’sLeastmonmultiplelcmandgcdtheoremEuclid’sAlgorithmforGCDTheorem:Leta=bq+r,wherea,b,q,andrareintegers、Thengcd(a,b)=gcd(b,r)Supposeaandbarethenaturalnumberswhosegcdhastobedetermined、Andsupposetheremainderofthedivisionofabybisr、Thereforea=qb+rwhereqisthequotientofthedivision、Anymondivisorofaandbisalsoadivisorofr、Toseewhythisistrue,considerthatrcanbewrittenasr=a−qb、Now,ifthereisamondivisordofaandbsuchthata=sdandb=td,thenr=(s−qt)d、Si
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