; TeX output 2005.11.14:1449 KE&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ:9|&html: html:.MTvkcolor push Black color popZVg5PSfile=logo129.eps llx=0 lly=0 urx=99 ury=16 rwi=28807獍color push Black color popN q cmbx12PZerfect
PowersWithAllEqualDigitsButCA OOne.s獒 XQ ff cmr12Omar f^Kihel ˕Department/ofMathematics WBrodCck/University St.D>Catharines,/OntarioL2S3A1 `Canada 'html:color push cmyk 0 1 0 0߆T ff cmtt12okihel@brocku.ca html: color pop RyFlorian/Luca Instituto/deMatem9aticasdCUniversidad/NacionalAut9onomadeMW}exico ɯwC.P4./58089 ^Morelia,/Michoac9an jMW}exico +html:color push cmyk 0 1 0 0fluca@matmor.unam.mx html: color pop)
č vcolor push Black color pop r"V
3
cmbx10AbstractMƍlcolor push Black color pop-̻K`y
3
cmr10InDthispapMer,;amongotherresults,w!eshowthatforanyxedintegerb>
3
cmmi10lv!",
3
cmsy10?þ3,;there
_areuonlynitelyman!ypMerfectl7)-thpo!wersuallofwhosedigitsareequalbutone,except_forfthetrivialfamilies102 cmmi8lKn~whenlA
3and8n10|{Y cmr83n
iffl=
3.}html: html:ٍ*N G cmbx121(Inutro =ductionb#XQ cmr12Obl athm[html:color push cmyk 0 1 0 06 html: color pop]prorvedmthattheonlypSerfectporwersmallofwhosedigitsareequaltoaxedone +g cmmi12a{,!",
cmsy106=1Ċindecimalrepresenrtationare4,;8and9.
ƆThisisequivXalenttosayingthatthediophanrtineequation html: html:@$B
aō33x2nR 133[ z E
(x 1&=URyn9q7;UPininrtegersAjn3;x2;1ax;yË2;qË2ƽ(1) :9color push Black 1G color pop *KE&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ&:9hasnosolutionwhenxF=10andaF6=1.Inkreri[html:color push cmyk 0 1 0 05 html: color pop]extendedObl ath'sresultbyproving :9thate^whenx&12f3;:::ʜ;10ge^anda&16=1,equatione^(html:color push cmyk 0 1 0 01 html: color pop)hastheuniquesolution(a;x;n;yn9;q)&1=:9(4;7;4;40;2).*Thanks;
toresultsofBugeaudandMignotte[html:color push cmyk 0 1 0 02 html: color pop],O&wrenowknowthatequation:9(html:color push cmyk 0 1 0 01 html: color pop)hasonlythefollorwingthreesolutions:!ōw325j 1w䮟[ z *
`3 1 =UR112;ō
s+724j 1
s+[ z *
`7 1.&S=202andō'aW1823j 1'aW[ z %
`18 1P{=73;:9whenam=1andx2f2;:::ʜ;10g.SGicaandPranaitopSol[html:color push cmyk 0 1 0 04 html: color pop]studiedavXariantonObl ath's:9problem.NamelyV,htheyHkfoundallsquaresofkdecimaldigitsharvingke _H1oftheirdigitsequal:9toeacrhother.TheyaskedtosolvetheanalogousproblemforhigherpSowers.Intherst:9partofthispapSer,wreprovethefollowingresult.:9UThtml: html: b color push Black1N cmbx12Theorem 1 color popE2@ cmti12Foraxeffdintegerlk3,thereareonlynitelymanyperfectlC-thpowersal lwhosedigitsarffeequalbutone,exceptforthetrivialfamilies102lKnforl&h3and8~1023nhDforl=UR3.^OurHmaintoSolfortheproofofTheoremhtml:color push cmyk 0 1 0 01 html: color popisthefollorwingresultofCorvXa jaandZannierfrom[html:color push cmyk 0 1 0 03 html: color pop]. html: html:^ color push BlackTheorem 2 color popELffetc6fG(XJg;Yp)6=a0(X )Y2d
JN+6j+6adߨ(X)c6bffeapolynomialin/
msbm10Q[XJg;Yp]withd62suchthata0(X )2QandthepffolynomialfG(0;Yp)hasnomultipleroot.gLeti;jbeintegers>UR12whicharffenotrelativelyprime.f:IftheequationfG(i2nP;yn9)UR=j ӟ2m>has2aninnitesequenceofsolutionsĹ(m;n;yn9)2Z23 suchthatminvfm;n;yn9g!1,'thentherffeexisth1andp(X )2Q[X ]35noncffonstantsuchthatfG(X2hWt;p(X))isnoncffonstantandhasonlyoneterm.WVeempSoinrtoutthatin[html:color push cmyk 0 1 0 03 html: color pop],itwasshownthatthepairh1emandp(X )2Q[X]emwiththepropSertrythatfG(X 2hWt;p(X ))hasonlyonetermexistsonlyunderthehypSothesisthatminHfm;ng!1.*Itkwrasnotshownthatp(X )isnonconstant.*However,acloseanalysisof;theproSofoftheresultfrom[html:color push cmyk 0 1 0 03 html: color pop]shorwsthatif(m;n;yn9);isanyinnitefamilyofintegersolutionsݑtotheequationfG(i2nP;yn9)=j ӟ2m
(withݑminrfm;ng!1,Lthenݑaninnitesubfamilyofl_sucrhsolutionshavethepropSertythatyژisintherangeofthepSolynomialp(X );:thus,ify!1,Sthen1p(X )cannotbSeaconstanrtpolynomial.zSimilarlyV,Sitisnotspecicallysaidin[html:color push cmyk 0 1 0 03 html: color pop]thatfG(X 2hWt;p(X ))isnonconstanrtbutthisisalsoclearfromtheargumentsfrom[html:color push cmyk 0 1 0 03 html: color pop]. html: html: 2(Pro =ofszofTheorems1and2b#3-
cmcsc10ProofNofTheorem1.SuppSose8thatlt1+3isaxedinrteger.ConsiderapSerfectlC-thpSorwerwithallidenrticaldigitsbutoneofthem.8WVritingitrstasbߍ xlw=UR: z A@& a:::ʜaba:::aDx(10)T1;itfollorwsthatwemayalsorewriteitas html: html:# xlw=URaō33102nR 133[ z %
x9++c10m;ƽù(2) :9color push Black 2G color pop
KE&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ&:9where|cUR=bɽ a.+IfaUR=0,wrethengetx2lw=bɽ102mĹ,whicrheasilyleadstotheconclusionsthat :9misamrultipleoflC,bUR=1ifl6=UR3,andb2f1;8gifl=3. -FVromAnorwon,Wweassumethata6=0.>WVeAalsoassumethatc6=0,WotherwiseAwerecover:9Obl ath'sproblem.8WVelet/J ~#fG(XJg;Yp)UR=ō1[ z
hcYlh ō?a۟[ z
9c(X+ 1):>:9Since?a?6=0,UthepSolynomialfG(XJg;Yp)fulllsthehrypothesisfromTheoremhtml:color push cmyk 0 1 0 02 html: color pop.8NIf(m;n;x)is:9aninrtegersolutionofequation(html:color push cmyk 0 1 0 02 html: color pop),then fG(10nP;x)UR=10m::9Thrus,/Iwe rmaytakeiUR=j%=10inthestatemenrtofTheoremhtml:color push cmyk 0 1 0 02 html: color pop.Assumethatwehaveinnitely :9manrysolutionsforequation(html:color push cmyk 0 1 0 02 html: color pop).cSincethepair(a;c)canassumeonlynitelymanyvXalues,:9itfollorwsthatwemayassumethat(a;c)isxedinequation(html:color push cmyk 0 1 0 02 html: color pop).aIfm,remainsbSounded:9orver~aninnitryofsolutions,itfollowsthatwemayassumethatitisxed.DButthen,the:9sequence"' un=ōa[ z +
%9a10nR+
-u
cmex10 T ō33a33[ z +
%9<+c10mğ
M3=URA10n+B ;:9withA==a=9,andBC= a=9"+c102misabinaryrecurrenrtsequence.I{IfBC6==0,thenit:9isnondegenerate,thereforeitcanconrtainonlynitelymanypSerfectlC-thporwers(see[html:color push cmyk 0 1 0 07 html: color pop]),:9whicrhisacontradiction./IfBX=UR0,.thencs102mZ=a=9./Sincecisaninrtegerandaisadigit,:9wregetthataUR=9;c=1;m=0,thereforebUR=a+cUR=10,whicrhisacontradiction. -Sinceaa|6=0,&minfm;ng=m./Thrus,&wemayassumethatmin|fm;ng|!1./ClearlyV,:9xn!1aswrell.
ByTheoremhtml:color push cmyk 0 1 0 02 html: color pop,1titfollowsthatthereexistapSositiveintegerhanda:9nonconstanrt^pSolynomialp(x)R2Q[X ]^suchthatfG(X 2hWt;p(X ))hasonlyoneterm.
~WVrite:9fG(X 2hWt;p(X ))UR=qn9X2k forsomenonzerorationalqXandpSositivreintegerkg.8ThisleadstoH捒 4Gp(X )lw=URcqn9Xk+ōa۟[ z +
%9<(Xh 1):=rS(X)::9Assumeko6=URh.8Then: rSK cmsy80!ǹ(X )UR=cqn9kgXk6 19+ōa۟[ z +
%9<hXh 13::9SincerS(0)UR= a=96=0,RandallroSotsofrJareofmrultiplicityatleastlUR3,Ritfollowsthatall:9roSotsD