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Oblօ~ath'sproblem'dh7XQ ff cmr12Alexandru/GicaandLaurent{iuPanaitopdCol ˕Department/ofMathematics ?aUniversity/ofBucharest vStr.D>Academiei/14 WzRO{70109/Bucharest1 ]PRomania -html:color push cmyk 0 1 0 0alex@al.math.unibuc.ro html: color pop V,html:color push cmyk 0 1 0 0pan@al.math.unibuc.ro html: color pop N cmbx12Abstract.JSXQ cmr12InythispapSerwredeterminethosesquareswhosedecimalrepresentationconsists ofg cmmi12ko!",
cmsy10UR2digitssucrhthatk 1ofthemequal.jhtml: html:(N G cmbx121(Inutro =ductionb#R.^Obl ath[html:color push cmyk 0 1 0 05 html: color pop]succeededinalmostenrtirelysolvingtheproblemofndingallthenumbSersn22 cmmi8m늹(n;mt2$
msbm10N,%n2,m2)thatharveequaldigits.-9ThespSecialcasemt=2isavrerywrellknownresult,˸althoughitsproSofinvolvesnodicultyV.+Inthisconnection,˸thefollowingquestion!Mnaturallyarises:+isitpSossibletodetermineallofthesquaresharvingalldigitsbutoneequal?Theanswrerisgivenby Theoremy1.1/)@ cmti12The&squarffeswhosedecimalrepresentationmakesuseofk.ǔ2digits,-#suchthatrke 1ofthesedigitsarffeequal,areprecisely16,25,36,49,64,81,121,144,225,441,484,676,1444,44944,102|{Y cmr82i,41022iand3591022iwith35iUR1.WhenzwreareloSokingforthesquareswithkCdigitsamongwhichk 1digitsequal0,PweimmediatelygetthatthecorrespSondingnrumbersare1022i,41022iand91022iwithiUR1.AusimplecomputationshorwsthatthenumbSerswithatmost4digitsverifyingthecondi-tioninthestatemenrtarejusttheoneslistedabSove.SinceevrerynaturalnumbSercanbewrittenintheform50000k-Ɔrg$with0r25000,and&(50000k:ӚrS)22{r22
h(modB100000),5wrecomputer22 :]forr<25000andndthatthelast4pdigitsofanrysquarecanbSeequalonlywhenallofthemequal0,whichsolvesObl ath'sproblemforsquaresharvingkoUR4digits. :9color push Black 1G color pop *KE&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ&-WVe`selectthesquaressucrhthat4ofthelast5digitsareequal,+MbSecausethesepoinrtout :9the8pSossiblesquareswithkRb5digits,k2 M1digitsofthembeingequal.Ifoneexcludes:9the'nrumbSersforwhichtherearek;@ #1digitsequalto0,6thentherestillremain22typSesof:9nrumbSers,namely:1Qۦf-a̾1V=UR1121w@a̾7V=UR4441 ,[a̾13UZ=UR44944.era̾18UZ=UR776-a̾2V=UR1161w@a̾8V=UR4449 ,[a̾14UZ=UR445444.era̾19UZ=UR881-a̾3V=UR2224w@a̾9V=UR4464 ,[a̾15UZ=UR449444.era̾20UZ=UR889-a̾4V=UR2225w@a̾10UZ=UR4484 ,[a̾16UZ=UR556.era̾21UZ=UR9929-a̾5V=UR441444w@a̾11UZ=UR44544 ,[a̾17UZ=UR6656.era̾22UZ=UR9969-a̾6V=UR44144w@a̾12UZ=UR44644-One7willshorwthat,amongthesenumbSerswithkoUR5digits,only44944isasquare.The :9exclusion,oftheothernrumbSers,canbecarriedoutfairlyeasilyincertaincases,=Waswreshow:9inx2.8IntheothercaseswrewillsolveequationsofthetypSe dx2j dyn92=URk(1):9(where(d;k%2Z2K cmsy8,8#d>0andp( z
:d{62Z)ininrtegers.Theliteratureconcerningequation(1)is :9ratherextensivre.8Inthisconnection,wemention[html:color push cmyk 0 1 0 01 html: color pop,html:color push cmyk 0 1 0 02 html: color pop ʤ,html:color push cmyk 0 1 0 03 html: color pop,html:color push cmyk 0 1 0 04 html: color pop].-WVeanorwrecallthesolvingmethoSd(inaccordancewith[html:color push cmyk 0 1 0 02 html: color pop]). pWedenotebry(r;s)the:9minimalpSositivresolutiontotheequation ѵx2j dyn92=UR1(2):9andbry"=r+93sp
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:d.WVedeterminethe\small"solutionstoequation(1)(ifany).They :9generateallthesolutions.:9Theorem[1.2We8denotebyi,=URai+j=bidڟp
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cmex10pUTl z ٟ
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v"2nP.-WVewillusethistheoreminx3.:9jhtml: html:2(Excludingzthesimplecasesb#WVesassumeinthissectionthatkoUR5andan cisasquare,hence9anisasquareaswrell.Makeuseofsimplereasonings,wreshallshowthatthisfactisimpSossible.PTVothisend,weusethesymrbSolofLegendreinsomecases.The(16caseswhicrhhavetobSeexcludedwillbeexposedinaconciseform,82inasmruch(assomeofthemarequitesimilar: 1a̾3;a̾4;a̾20 ;UPa̾2;a̾15 ;a̾18;UPa̾11;a̾17:WVeHmenrtionthateachofthecasesbSelowisconcludedbyacontradictoryassertion,YthusprorvingtheimpSossibilityofthecorrespSondingcase. :9color push Black 2G color pop
HKE&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ&-1.8WVeharve9a̾2V=UR102k:+449( 1)2k+9UP(moSdB11).8ButꨟG Fu߾8
ݟ z 11QG&l=G Fu
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z 11G= 1. -2.iItfollorwsby9a̾3 䴹=$2(102k+d8)=(4x)22 that22k6 3 52k
HB=(x 1)(x+1).iSince(x :91;xN+1)q=2,Gwre;have52k
jqxN+"with"q2f 1;1g,GwhencexN+1q52k#. ,Consequenrtly:922k6 352kx=URx22j 152k#(52k: 2),hence22k6 3U`52k: 2.-3.
By9a̾4V=2102k R+25=(5x)22,N3wrehave22k6+152k6 2=(x 1)(x+1),N3whence:922k6+152k6 2U`UR52k6 2 (52k6 2 2).-4.WVeharve9a̾5 '=<#4mb102k 27004=(2x)22,P*whence102k 6751=x22.Whenkoǹisan:9oSddnrumber,&(wehave102kz V67512UP(moSdB11),&(butG Fu
E2C z 11M~Go= 1..Ifkj=2hwithh3,:9then(102h 0x)(102h+x)=6751,ߖwhence102h x=a,ߖ102h+x=b,ߖwherewrehaveeither:9(a;b)UR=(1;6751)or(a;b)UR=(43;157).Since2 Ӄ102hC=URa+b,2 itfollorwsthateither2102hC=UR6752:9or2102hC=UR200,althoughh3.-5.5Byn9a̾6V=UR4102ka 2704=(4x)22ritnfollorwsthat522WϹ102k6 2 169=x22.5Sincenko5,FwreR:9getthatx22V=UR522j102k6 2 1694
US 16943.-6.8WVeharvea̾9V=UR41116,but1116doSesnotoccuramongthenrumbSersaidڹ.-7.8WVeharvea̾10UZ=UR4a̾1,andweshallgetthecontradictionafterwestudya̾1forkoUR5.-8.:WVeharve9a̾11
R=R 4102k3M+896=(8x)22,8hence22k6 4 52k3M+14=x22.:Itfollorwsthatx . ..C2.*:9Thereforex22 . ..4,andko=UR5.8Inthiscasewregetx22V=6264.-9.8WVeharvea̾12UZ=UR4a̾2,buta̾2V6=x22.-10.By_9a̾15UZ=UR4 102k+44996=(2x)22cit_follorwsthat102k+ 11249=x22.Then102k+ 11249:9( 1)2k:+7UP(moSdB11),butG Fu߾6
ݟ z 11QG&l=URG Fu(8
z 11G=UR 1.-11.8Since4a̾16UZ=UR222pUR| p #{z p #}ۍӀk(>4anda̾3V6=URx22,itfollorwsthata̾166=URyn922.=.#\-12.ȹWVe,harve9a̾17y=y6102k 96=(4x)22,}hence,322k6 3 52k 6=x22.But,x22 . ..4and*:9322k6 3 52k #. #.#.g4(bSecausekoUR5).-13.8WVeharve9a̾18UZ=UR7102k: 167( 1)2k: 5UP(moSdB11).8ButꨟG Fu߾2
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10
z 11G= 1.-14.
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iitfollorwsthat(x1 1)(x+1)=22k6+3 52k
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