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1、SUPERELLIPTICEQUATIONSARISINGFROMSUMSOFCONSECUTIVEPOWERSMICHAELA.BENNETT,VANDITAPATEL,ANDSAMIRSIKSEKAbstract.Usingonlyelementaryarguments,CasselssolvedtheDiophantineequation(x−1)3+x3+(x+1)3=z2(withx,z∈Z).Thegeneralization(x−1)k+xk+(x+1)k=zn(withx,z,n∈Zandn≥2)wasconsideredbyZhongfengZh
2、angwhosolveditfork∈{2,3,4}usingFrey-HellegouarchcurvesandtheircorrespondingGaloisrepresentations.Inthispaper,byemployingsomesophisticatedrefinementsofthisapproach,weshowthattheonlysolutionfork=5isx=z=0,andthattherearenosolutionsfork=6.Thechiefinnovationweemployisacomputationalone,which
3、enablesustoavoidthefullcomputationofdataaboutcuspidalnewformsofhighlevel.1.IntroductionIn1964,Leveque[18]appliedatheoremofSiegel[25]toshowthat,iff(x)∈Z[x]isapolynomialofdegreek≥2withatleasttwosimpleroots,andn≥max{2,5−k}isaninteger,thenthesuperellipticequation(1)f(x)=znhasatmostfinitely
4、manysolutionsinintegersxandz.ThisresultwasextendedbySchinzelandTijdeman[24],throughapplicationoflowerboundsforlinearformsinlogarithms,toshowthatequation(1)hasinfactatmostfinitelymanysolutionsinintegersx,zandvariablen≥max{2,5−k}(wherewecountthesolutionswithzn=±1,0once).Whilethislatterre
5、sultiseffective(inthesensethatthefinitesetoftriples(x,z,n)iseffectivelycomputable),inpracticesuchadeterminationhasinfrequentlybeenachieved,duetotheextraordinarysizeoftheboundsforx,zandnarisingfromtheproof.ThefewcasesthathavebeentreatedintheliteraturehavebeenarXiv:1509.06619v1[math.NT]22S
6、ep2015restrictedtopolynomialswithveryfewmonomials,orwithmultiplelinearfactorsoverQ.Oneclassofpolynomialsthathasproved,incertaincasesatleast,amenabletosuchanapproach,isthatarisingfromsumofconsecutivek-thpowers.LetusdefineS(x)=1k+2k+···+xk,kDate:September23,2015.2010MathematicsSubjectCla
7、ssification.Primary11D61,Secondary11D41,11F80,11F11.Keywordsandphrases.Exponentialequation,Galoisrepresentation,Frey-Hellegouarchcurve,modularity,levellowering,multi-Frey-Hellegouarch.Thefirst-namedauthorissupportedbyNSERC.Thesecond-namedauthorissupportedbyanEPSRCstudentship.Thethird-na
8、medau
