; TeX output 2002.02.12:1504 Kb&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ:9|&html: html:.MUVgcolor push Black color popZVg5PSfile=logo126.eps llx=0 lly=0 urx=99 ury=16 rwi=2880(s獑}N ff cmbx12AffMillionNewAmicablePairs7ۍ >K`y
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cmr10MarianofGarcia
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#discoveredoveronemil lionnewamicablepairs,%bringingtomorethantwomil lion the4 totalofknownamic}'ablenumbersandstrengtheningthevalidityoftheasyetunprovedconjecturethatther}'eareinnitelymanysuchpairs.#` K`y
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cmmi10M5andNaresaidtobGeamic}'able$ifeachisequaltothesumofthealiquotpartsoftheother.}ThisisequivqalenttosayingthatMandNareamicableif[ٲ(M)=(N)=M+N,whereUU[ٲ(M)standsforthesumofallthepGositivedivisorsofM. Thesmallestexampleofanamicablepairis(220=2^ٓR cmr72
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cmsy10511;284=2^271),whichwasknowntothePythagoreans.sF*orUseveralcenturiesthiswastheonlyknownamicablepair.sAroundtheyear1300,theamicable1pair(2^4Gٸf2347;Dz2^41151)1wasdiscoveredbyal-Bannaandaround1600,hY*azdidiscoveredthepairhu(2^7lX191J383;2^7J73727).'ThehusecondpairabGovehuwaslaterrediscoveredbyF*ermatin1636andthe4thirdpairbyDescartesin1638.cThepreviouslylistedthreepairsweretheonlyknownamicablepairsuntil. Euler[html:color push cmyk 0 1 0 02 color pop html: ]discovered59additionalonesinthemiddleoftheeighteenthcentury*.Sincethen,d6severalmathematicians:1havediscoveredamicablenumbGers,?includingLegendre,Poulet,Gerardin,Seedho,Mason,EscottUUandothers. UntilLtheyear1972,NXinordertondoutwhattheexistingknownamicablepairswere,NXonehadtosearchthroughthevqariousindividualpapGersonthesub 8ject.:RButinthatyear,color push cmyk 0 1 0 04 color pop html: ]intheJournalofRecreationalMathematicsanarticleentitled\TheHistoryandDiscoveryofAmicableNumbGers,"whichUUdealtwiththehistoryofthesub 8jectandlistedthe1108knownamicablepairsatthattime. In 1986HermanteRieledeterminedandpublished[ html:color push cmyk 0 1 0 05 color pop html: ]alltheamicablepairsbGelow10^10uand,%togetherwith;Borho,uBattiato,HomannandLee,published[html:color push cmyk 0 1 0 07 color pop html: ]alistofknownamicablepairsbGetween10^10Ҳand10^52x.2F*rom1986to1995,teRielebGecametheocial"collector"ofamicablepairs.Peoplewhodiscoveredamicable}pairssentthemtohimforvericationthatthepairswerenewand,!ifso,theywereaddedtothecollection.qT*eUURielealsomadeupseveralsupplementaryliststhathesenttointerestedpGeople. In|1995,PedersenstartedcollectingamicablenumbGersintheInternet.=T*oGdayallknownamicablepairscanUUbGeseenbyexaminingthewebpagesentitledKnownamic}'ablepairs'v[ html:color push cmyk 0 1 0 09 color pop html: ]. AsofJanuary15,C2001,atotalof843,783amicablepairswereknown.WUsingessentiallytheBDEmethoGdindicatedRin[ html:color push cmyk 0 1 0 08 color pop html: ],SMandbasedonideascontainedin[html:color push cmyk 0 1 0 01 color pop html:],SM[html:color push cmyk 0 1 0 03 color pop html:],andR[ html:color push cmyk 0 1 0 06 color pop html:],theRauthorhasfoundoverRonemillionnewamicablepairsduringtherstthreemonthsoftheyear2001.EInthefollowingparagraphswegiveabriefoutlineUUofthemethoGdused: :9color push Black j1G color pop *Kb&6color push Blackhtml:color push gray 0 color pop html:2G color pop3ڍ&BलSuppGose-EmandNDarenaturalnumbers-withEmrelativelyprimetoNDandto[ٲ(N) 1,5and-satisfyingthe 6conditionUUhtml: html:Z6(1) qʵ[ٲ(E )8(N)=Em8(NO+(N) 1):6लIf8nowp;qandrareprimessuchthatEdЃpqand8˵EN瞸rconstitute8anamicablepair,qwemusthave6वr5=(p8+1)(q+1)=[ٲ(N) 1.qAlso,UUhtml: html:edv[ٲ(E )8(p+1)(q+1)=Em8[pq+NOrG]=Em8[pq+NO(p+1)(q+1)=[ٲ(N) N]6andUUthushtml: html:6(2)n\[ٲ(E )8(N)(p+1)(q+1)=Em8[(N)pq+NO(p+1)(q+1) NO[ٲ(N)]:6लDividingUU(2)by(1),weobtainhtml: html:p(p8+1)(q+1)=[[ٲ(N)8pq+NO(p+1)(q+1) NO[ٲ(N)]=[N+[ٲ(N) 1]:6लMultiplyingUUthroughbyNO+8[ٲ(N) 1UUandsimplifying,wegethtml: html: