On Cyclotomic Polynomials with1 Coefficients

On Cyclotomic Polynomials with1 Coefficients Peter Borwein and Kwok-Kwong Stephen Choi CONTENTS 1. Introduction 2. Cyclotomic Polynomials with Odd Coefficients 3. Cyclotomic Littlewood Polynomials 4. Cyclotomic Littlewood Polynomials of Odd Degree Acknowledgment References ResearchofP.Borweinissupported,inpart,byNSERCof Keywords:Cyclotomicpolynomials,Mahlermeasure,Littlewood FellowandtheInstitute'ssupportisgratefullyacknowledged. Canada. MathematicsSubjectClassication(1991):Primary11R09; K.-K.ChoiisaPacicInstituteofMathematicsPostdoctoral Secondary:11Y99 We characterize all cyclotomic polynomials of even degree with coefficients restricted to the setf+1, 1g. In this context a cyclotomic polynomial is any monic polynomial with integer coefficients and all roots of modulus 1. Inter alia we characterize all cyclotomic polynomials with odd coefficients. The characterization is as follows. A polynomial P(x) with coefficients1 of even degree N 1 is cyclotomic if and only if P(x) =ol p1 (x) ol p2 (x p1 )ol where N = p 1 p 2p r pr (x p1p2p r 1 ), and the p i are primes, not necessarily distinct, and where ol p (x) := (x p 1)/ (x 1) is the p-th cyclotomic polynomial. We conjecture that this characterization also holds for polynomials of odd degree with1 coefficients. This conjecture is based on substantial computation plus a number of special cases. Weareinterestedinstudyingpolynomialswithcoef- Central cientsrestrictedtothesetf+1; 1g.Thisparticularsetofpolynomialshasdrawnmuchattentionand to this paper is a careful analysis of the effect of Graeffe s root squaring algorithm on cyclotomic polynomials. questionsandsowecallthesepolynomialslittle- thereareanumberofdicultoldquestionsconcerningit.littlewood[1968]raisedanumberofthese 1. INTRODUCTION woodpolynomialsanddenotethembyl.alittle- woodpolynomialofdegreenhasl2normonthe raisedconcerncomparingthebehaviorofthesepolynomialsinothernormstothel2norm.oneofthe unitcircleequaltopn+1:manyofthequestions isttwopositiveconstantsc1andc2sothatforeach polynomialscanbe\at".specically,dothereex- olderandmoreintriguingoftheseaskswhethersuch nthereisalittlewoodpolynomialofdegreenwith C1pn+1<jp(z)j<C2pn+1cAKPeters, Ltd. 1058-6458/1999 $0.50 per page Experimental Mathematics 8:4, page 399

beenopenformorethanfortyyears,isdiscussed foreachzofmodulus1?thisproblem,whichhas calledrudin{shapiropolynomials.itisstillun- knownwhetherthereisasequencesatisfyingjust in[borwein1998],wherethereisanextensivebibliography.theupperboundissatisedbytheso- 400 Experimental Mathematics, Vol. 8 (1999), No. 4 timization"). ofthe\veryhardestproblemsincombinatorialop- thelowerbound(thisproblemhasbeencalledone hasbeenstudiedfromanumberofpointsofview. TheproblemofminimizingtheL4norm(orequivalentlyofmaximizingtheso-called\meritfactor") ThesizeoftheLpnormofLittlewoodpolynomials hasalsoattractedalotofattention. greenhavel4normasymptoticallyclosetopn+1? Thistooisstillopenandisdiscussedin[Borwein 1998]. Inparticular,canLittlewoodpolynomialsofde- themahlermeasureoflittlewoodpolynomials.the MahlermeasureisjusttheL0normonthecircleand onewouldexpectthistobecloselyrelatedtothe Mahler[1963]raisedthequestionofmaximizing coursetheminimumpossiblemahlermeasurefora minimizationproblemforthel4normabove.of Littlewoodpolynomialis1andthisisachievedby anycyclotomicpolynomial.inthispaperwedene anddenotebyn(x)then-thirreduciblecyclotomic polynomial,thatis,theminimumpolynomialofa acyclotomicpolynomialasanymonicpolynomial primitiven-throotofunity. withintegercoecientsandallrootsofmodulus1, withcoecients1ofevendegreen 1iscyclotomicifandonlyif degree.specically,weshowthatapolynomialp(x) ingthecyclotomiclittlewoodpolynomialsofeven Thispaperaddressesthequestionofcharacteriz- wheren=p1p2prandthepiareprimes(not necessarilydistinct).the\if"partisobvioussince pi(x)hascoecients1. P(x)=p1(x)p2(xp1)pr(xp1p2pr 1); ofsuchpolynomials. Graee'srootsquaringalgorithm.Ittranspiresthat Thisanalysisisbasedonacarefultreatmentof Wealsogiveanexplicitformulaforthenumber greehavethesamexedpointoniteratinggra- ee'srootsquaringalgorithm.thisalsoallowsus allcyclotomiclittlewoodpolynomialsofaxedde- tocharacterizeallcyclotomicpolynomialswithodd characterizationofcyclotomiclittlewoodpolynomialsofevendegreealsoholdsforodddegree.oneof Substantialcomputations,aswellasanumberof specialcases,leadustoconjecturethattheabove coecients. thecaseswecanhandleiswhennisapowerof2. isreallytheobservationthatthecyclotomiclittle- thisiscarefullyerasedinthenalexposition).it pectsofthispaper.(asisperhapsusual,muchof Itisworthcommentingontheexperimentalasrithmthatiscritical.ThisallowsforcomputatiosentiallybyinvertingGraee'srootsquaringalgo- overallcyclotomiclittlewoodsuptodegreeseveral woodpolynomialscanbeexplicitlyconstructedes- turesofthepaperandsuggestedtheroutetosome itwasthesecalculationsthatallowedfortheconjec- hundred(withexhaustivesearchfailingfarearlier), oftheresults. aconstructionwhichisofinterestinitself.indeed Section3looksatcyclotomicLittlewoodpolynomialswithacompleteanalysisoftheevendegreecase. Thepaperisorganizedasfollows.Section2examinescyclotomicpolynomialswithoddcoecients. andotherevidencetosupporttheconjecturethat theoddcasebehavesliketheevencase. Thelastsectionpresentssomenumericalevidence ofirreduciblecyclotomicpolynomials.todothis, Inthissection,wediscussthefactorizationofcyclotomicpolynomialswithoddcoecientsasaproduct pisaprimenumber.themostusefulcaseisp=2 werstconsiderthefactorizationoverzp[x],where becauseeverylittlewoodpolynomialreducestothe Dirichletkernel1+x++xN 1 inz2[x].inzp[x],n(x)isnolongerirreducible ingeneralbutn(x)andm(x)arestillrelatively primetoeachother. arerelativelyprimeinzp[x]. tegersrelativelyprimetop.thenn(x)andm(x) 2.1.Supposenandmaredistinctpositivein- 2. CYCLOTOMIC POLYNOMIALS WITH ODD COEFFICIENTS

Proof.Supposeeandfarethesmallestpositiveintegerssuchthat pe1(modn)andpf1(modm): LetFpkbetheeldoforderpk.ThenFpecontains Borwein and Choi: On Cyclotomic Polynomials with1 Coefficients 401 isaproductof'(n)=eirreduciblefactorsofdegreee exactly'(n)elementsofordernandoverzp,n(x) bedenedoverthequotientoftwomonicpolynomialsinz[x]bytp[(p=q)(x)]:=tp[p(x)]=tp[q(x)]. isalsoamonicpolynomialinz[x].weextendtpto identities[borweinanderdelyi1995,p.5],tp[p(x)] foreveryp(x)=qni=1(x i)inz[x].bynewton's haveacommonfactorinzp[x]sincetheirirreducible foranelementinfpeofordernoverzp;see[lidl andniederreiter1983].son(x)andm(x)cannot andeachirreduciblefactorisaminimalpolynomial rootsofp.alsoweletmpbethenaturalprojection befactorsofpolynomialswithoddcoecients. factorsareminimalpolynomialsofdierentorders. fromz[x]ontozp[x]: polynomialwhoserootsarethep-thpowersofthe Thisoperatorobviouslytakesapolynomialtothe Thefollowinglemmatellswhichm(x)canpossibly Thisprovesourlemma. Proof.Sincem(x)dividesP(x),som(x)alsodividesP(x)inZ2[x].However,inZ2[x],P(x)equals coecientsofdegreen 1.Ifm(x)dividesP(x), thenmdivides2n. 2.2.SupposeP(x)isapolynomialwithodd tivelyprimetopandi2.then Mp[P(x)]=P(x)(modp): (ii)tp[pn(x)]=n(x)p 1, (i)tp[n(x)]=n(x), 2.3.Supposethatnisapositiveintegerrela- to1+x++xn 1andcanbefactoredas wheren=2tm,t0andmisodd.inviewof P(x)=1(x) 1YdjM2t permutestherootsofn(x).toprove(ii)and(iii), (iii)tp[pin(x)]=pi 1n(x)p. 2.1,d1(x)andd2(x)arerelativelyprime d(x); weconsider Proof.(i)istrivialbecauseif(n;p)=1thenTpjust Tp[P(xp)]=Tp"NYj=1(xp i)# inz2[x]ifd1andd2aredistinctoddintegers.soif misodd,thenm(x)isafactorintherighthand =Tp"NYj=1pYl=1(x e2il=p1=p i)# sideof(2{1)andhencem=dforsomedjm.on (2 1) theotherhand,ifmisevenandm=2lm0where inz2[x].thus,inviewof(2{1),wemusthavem0= l1andm0isodd,then =NYj=1pYl=1(x i) dfordjmandlt+1.henceinbothcaseswe m(x)=2m0(x2l 1)=m0(x2l 1)=m0(x)2l 1 Thus(ii)and(iii)followfrom(i)andtheequalities =P(x)p: writtenas P(x),withoddcoecientsofdegreeN 1canbe havemdivides2n. Inviewof2.2,everycyclotomicpolynomial, vergeinanitenumberofstepstoaxedpointof pn(x)=n(xp)=n(x)andpin(x)=pi 1n(xp) wherethee(d)arenonnegativeintegers. P(x)=Ydj2Ne(d) ForeachprimepletTpbetheoperatordened d(x); TpandwedenethistobethexedpointofP(x) [Hungerford1974,x5.8]. withrespecttotp. WhenP(x)iscyclotomic,theiteratesTnp[P(x)]con- inz[x],thenmp[tp[p(x)]]=mp[p(x)]; 2.4.IfP(x)isamoniccyclotomicpolynomial overallmonicpolynomialsinz[x]by Tp[P(x)]:=NYi=1(x pi) (2 2) mialsn(x).letnbeanintegerrelativelyprime sucestoconsidertheprimitivecyclotomicpolyno- inzp[x]. Proof.SincebothTpandMparemultiplicative,it (2 3)

top.then(2{3)istrueforp(x)=n(x)by(i)of 2.3.ForP(x)=pn(x),wehave by(ii)of2.3.however, Mp[Tp[pn(x)]]=Mp[n(x)p 1]=Mp[n(x)]p 1 402 Experimental Mathematics, Vol. 8 (1999), No. 4 Mp[pn(x)]=Mp[n(xp)] Mp[n(x)]=Mp[n](xp) inzp[x].thisprovesthat(2{3)isalsotruefor =Mp[n(x)]p 1 P(x)=pn(x).Finally,P(x)=pin(x)implies Mp[Tp[pin(x)]]=Mp[pi 1n(x)p] by(iii)of2.3.thiscompletestheproofof =Mp[pin(x)] =Mp[pi 1n(xp)] theconverseisalsotrue. Mp[P(x)]=Mp[Q(x)].Thenextresultshowsthat 2.4showsthatTp[P(x)]=Tp[Q(x)]implies ourlemma. samexedpointwithrespecttoiterationoftp. Zp[x]ifandonlyifbothP(x)andQ(x)havethe polynomialsinz[x]andmp[p(x)]=mp[q(x)]in 2.5.P(x)andQ(x)aremoniccyclotomic Proof.Suppose P(x)=Yd2De(d) d(x)e(pd) pd(x)e(ptd) ptd(x) Theorem andq(x)=yd2de(d)0 d(x)e(pd)0 pd(x)e(ptd)0 integersrelativelyprimetop.thenusingparts(i){ wheret;e(j);e(j)00anddisasetofpositive ptd(x); (iii)of2.3,wehaveforlt Tlp[Q(x)]=Yd2Dd(x)f(d)0; Tlp[P(x)]=Yd2Dd(x)f(d); wheref(d)=e(d)+(p 1)tXj=1pj 1e(pjd); (2 4) f(d)0=e(d)0+(p 1)tXj=1pj 1e(pjd)0: From2.4,wehave foranylt.fromthisand(2{4), Mp[Tlp[P(x)]]=Mp[P(x)]=Mp[Q(x)] Yd2DMp[d(x)]f(d)=Yd2DMp[d(x)]f(d)0: =Mp[Tlp[Q(x)]]; andmp[d0(x)]arerelativelyprimeifd6=d0.sowe However,with2.1,weknowthatMp[d(x)] musthavef(d)=f(d)0foralld2dandhencefrom (2{4),P(x)andQ(x)havethesamexedpointwith respecttotp. FromTheorem2.5,wecancharacterizethemonic dertheprojectionmp.theyallhavethesamexed cyclotomicpolynomialsbytheirimagesinzp[x]un- pointundertp.inparticular,whenp=2wehave: xedpointunderiterationoft2.specically,ifn= oddcoecientsofthedegreen 1havethesame 2.6.Allmoniccyclotomicpolynomialswith occursatthe(t+1)-thstepoftheiterationandequals 2tMwheret0andMisoddthenthexedpoint 2.5andthefactthat Proof.TherstpartfollowsdirectlyfromTheorem (xm 1)2t(x 1) 1: coecientsofdegreen 1.IfN=2tM,thenfrom inz2[x]ifp(x)isamonicpolynomialwithodd M2[P(x)]=1+x++xN 1 (2{2), OverZ2[x], P(x)=YdjMe(d) d(x)e(2d) 2d(x)e(2t+1d) 2t+1d(x): so 1+x++xN 1=1(x) 1YdjM2t f(d)=e(d)+t+1 d(x); =2tXi=12i 1e(2id) 2t 1ford=1. fordjm,d>1, (2 5)

Therefore,from(2{5)and2.3, Tt+1 2[P(x)]=YdjMf(d) =(xm 1)2t(x 1) 1: d(x)=1(x) 1YdjM2t d(x) Borwein lary2.7.toprove(2{7),weusedebruijn'sasymp- toticestimationforb(n)in[debruijn1948]: Proof.Formula(2{6)followsfrom(2{5)andCorol- and Choi: On Cyclotomic Polynomials with1 Coefficients 403 T2[P(x)]equals1+x++xN 1forallcyclotomic 2.6,whenNisodd(t=0),showsthat B(n)exp((logn)2=log4): ofcyclotomicpolynomialswithoddcoecients. (2{5),wethenhavethefollowingcharacterization polynomialswithoddcoecients.from(2{2)and N 1iscyclotomicifandonlyif Apolynomial,P(x),withoddcoecientsofdegree P(x)=YdjMe(d) 2.7.LetN=2tMwitht0andModd. mialofhigherdegreeisasfollows:ifp1(x)andp2(x) thecoecientsareall+1or 1. Wenowspecializethediscussiontothecasewhere OnenaturalwaytobuildupLittlewoodpolyno- d(x)e(2d) 2d(x)e(2t+1d) 2t+1d(x); P(x),withoddcoecientsofevendegreeN 1is cyclotomicifandonlyif andthee(d)satisfythecondition(2{5). istheonlywaytoproducecyclotomiclittlewood ofhigherdegree.inthissection,weshowthatthis arelittlewoodpolynomialsandp1(x)isofdegree Furthermore,ifNisodd,thenanypolynomial, polynomials,atleastforevendegree. N 1thenP1(x)P2(xN)isaLittlewoodpolynomial P(x)=YdjN d>1e(d) wherethee(d)arenonnegativeintegers. d(x); suchthatali+n=aliforall1ni 1and ProvingthisisequivalenttoshowingthatthecoecientsofP(x)are\periodic"inthesensethat ifp(x)=pn 1 n=0anxn,thenthereisa\period"i 0lN=i 1.ThisisourTheorem3.3below. B(n)bethenumberofpartitionsofnintoasum cyclotomicpolynomialswithoddcoecients.let oftermsofthesequencef1;1;2;4;8;16;:::g.then 2.7allowsustocomputethenumberof wehavexn 1P(1=x)=P(x).Thusitfollowsfrom nomialinlandletskbethesumofthek-thpowers ofalltherootsofp(x).sincep(x)iscyclotomic, SupposeP(x)=PN 1 n=0anxnisacyclotomicpoly- Newton'sidentitiesthat B(n)hasgeneratingfunction a1=1byreplacingp(x)by P(x)orP( x)if forkn 2.Wemayfurtherassumethata0= Sk+a1Sk 1++ak 1S1+kak=0 ThenumberofcyclotomicpolynomialswithoddcoecientsofdegreeN 1is Weclaimthat S1= a1= 1: 2.8.LetN=2tMwitht0andModd. F(x)=(1 x) 11Yk=0(1 x2k) 1: forsomeintegeri2.from(3{1),wehave necessary.wenowlet a0=a1==ai 1=1andai= 1; (3 1) whered(m)denotesthenumberofdivisorsofm. C(N)=B(2t)d(M) 1B(2t 1); Furthermore, SupposeS1==Sj= 1forj<i 1.Then s1=s2==si 1= 1; logc(n)(12t2log2)(d(m) 1)+(log(2t 1))2 from(3{1)again, log4 : SoS1==Si 1= 1.Similarly,from(3{1), Sj+1= a1sj ajs1 (j+1)aj+1 =j (j+1)= 1: (2 6) Si= a1si 1 ai 1S1 iai=2i 1: (2 7) Now(2{7)followsfromthisand(2{6). 3. CYCLOTOMIC LITTLEWOOD POLYNOMIALS Si=2i 1: (3 2)

alifor1ni 1and0lk 2.Then Xl=0ali S(k l)i+j+1 S(k l 1)i+j+1 3.1.Let2kN 1 i 1andsupposeali+n= 404 Experimental Mathematics, Vol. 8 (1999), No. 4 +(ki+j+1)(aki+j+1 aki+j) for0ji 2. +i 1 Proof.Suppose0ji 2.From(3{1)and(3{2) wehave 0=k 1 Xl=0i 1 Xn=0a(k 1)i+n(Si+j n+1 Si+j n)=0(3 3) +j 1 Xn=0ali+nS(k l)i+j n =k 2 Xl=0alii 1 Xn=0aki+nSj n+(ki+j)aki+j Xn=0S(k l)i+j n+i 1 jxn=0aki+n+(ki+j+1)aki+j: Xn=0a(k 1)i+nSi+j n 0=k 2 Similarly, Xl=0alii 1 Xn=0S(k l)i+j n+1+i 1 Xn=0a(k 1)i+nSi+j n+1 (3 4) 0=k 2 Hence,onsubtracting(3{5)from(3{4),wehave Xl=0ali S(k l)i+j+1 S(k l 1)i+j+1 jxn=0aki+n+(ki+j+1)aki+j+1:(3 5) +(ki+j+1)(aki+j+1 aki+j) Thisproves(3{3). +i 1 Xn=0a(k 1)i+n(Si+j n+1 Si+j n): 3.2.Let0kN 1 i 1.Supposeali+n=ali for1ni 1and0lk. for1ni 1and0lk.Then Sli+n= 1 (3 6) Proof.Weprovethisbyinductiononk.Wehave istruefork 1.Then,forany0ji 2, provedthat(3{6)istruefork=0.suppose(3{6) 0=a0(Ski+j+1 S(k 1)i+j+1) =Ski+j+1+1+a(k 1)i(Si+j+1 Sj+1) +a(k 1)ii 1 Xn=0(Si+j n+1 Si+j n) by(3{3).henceski+j+1= 1for0ji 2. =Ski+j+1+1; 3.3.SupposeNisodd.Ifa0=a1== for1ni 1and0lNi 1. ai 1=1andai= 1,then Proof.WerstshowthatS2k= 1for1kN 1 ali+n=ali Theorem odd,from2.7, andhenceiisoddbecausesi=2i 1.SinceNis P(x)=YdjNe(d) d(x)e(2d) If1kN 1,then wheree(d)+e(2d)=1ifd>1ande(1)=e(2)=0. 2d(x) S2k=XdjN(e(d)Cd(2k)+e(2d)C2d(2k)) =XdjN(e(d)+e(2d))Cd(k) =XdjNCd(k) C1(k) rootsofqdjnd(x)=xn 1,whichisequaltozero hencepdjncd(k)isthesumofk-thpowersofthe k-thpowersoftheprimitived-throotsofunityand wheretheramanujansum,cd(k),isthesumofthe when1kn 1. Suppose Wecontinueourproofbyusinginductiononk. for1ni 1and0lk 1where1k N 1 i 1.Froms3.1and3.2wehave ali+n=ali Ski+j+1+1+(ki+j+1)(aki+j+1 aki+j)=0(3 8) = 1; (3 7)

andhencefrom(3{3)again 0=a0(S(k+1)i+j+1 Ski+j+1)+ai(Ski+j+1 S(k 1)i+j+1) +aki+j(si+1 Si)+aki+j+1(Si Si 1) clotomicinlofdegreen 1ifandonlyif 3.5.SupposeNisodd.ThenP(x)iscy- Borwein and Choi: On Cyclotomic Polynomials with1 Coefficients 405 =S(k+1)i+j+1+1+2((k+1)i+j+1)(aki+j+1 aki+j) =(S(k+1)i+j+1 2Ski+j+1 1)+2i(aki+j+1 aki+j) +((k+1)i+j+1)(a(k+1)i+j+1 a(k+1)i+j) where"=0or1,n0=1;nt=nandni 1isa P(x)=tYi=1xNi+( 1)"+i properdivisorofnifori=1;2;:::;t. xni 1+( 1)"+i Proof.Withoutlossofgenerality,wemayassume ifjiseven.sofrom(3{8)and(3{9),wehave (3{7),Ski+j+1= 1ifjisoddandS(k+1)i+j+1= 1 for0ji 2.Supposekiseven.Theninviewof P(x)iscyclotomicinLifandonlyif thatp(x)=1+x+a2x2+.fromtheorem3.4, P(x)=p1(x)p2(xp1)pr(xp1pr 1) aki+j+1=aki+j =p1(x)pn1(xp1pn1 1) (3 9) forj=1;3;:::;i 2and forj=0;2;:::;i 3.However,sincetheai'sare+1 pnt 1+1(( 1)t 1xp1pnt 1) pn1+1( xp1pn1)pn2( xp1pn2 1) forj=0;2;:::;i 3.Henceaki+n=akiforn= or 1,Equation(3{10)impliesthat aki+j+1=aki+j anda(k+1)i+j+1=a(k+1)i+j theprecedingequationbecomes wheren=p1pnt.sincep(x)=(xp 1)=(x 1), pnt(( 1)t 1xp1pnt 1); P(x)=tYi=1xNi+( 1)i 2(aki+j+1 aki+j)=a(k+1)i+j+1 a(k+1)i+j(3 10) nomial,p(x),ofdegreen 1iscyclotomicifand 1;2;:::;i 1.Thecaseofkoddcanbeprovedin xni 1+( 1)i; onlyif thesameway. 3.4.SupposeNisodd.ALittlewoodpoly- cyclotomiclittlewoodpolynomialsofgivenevendegree.foranypositiveintegersnandt,dene Thisprovesourcorollary. Using3.5,wecancountthenumberof wheren0=1andni=p1pnifori=1;:::;t. wheren=p1p2prandthepiareprimes,not Theorem Conversely,supposeP(x)isacyclotomicLittlewood thenp(x)isacyclotomiclittlewoodpolynomial. necessarilydistinct. andfori1,di(n):=xnjndi 1(n) r(n;t):=#(n1;n2;:::;nt):n1jn2jjnt; 1<N1<N2<<Nt=N ; polynomial.asbeforewemayassumethata0= Proof.ItisclearthatifP(x)isoftheform(3{11), whered0(n)=1. a1==ai 1=1andai= 1.WeproveourresultbyinductiononN.FromTheorem3.3,wehave 3.6.Forl;t0andpprime,wehave P(x)=P1(x)P2(xi),whereP1(x)=1+x++xi 1 dt(pl)=l+t t: (3 12) andp2(x)isacyclotomiclittlewoodpolynomialof P2(x)areoftheform(3{11)andhencesoisP(x) degreelessthann 1.Byinduction,P1(x)and becausethedegreeofp1(x)isi 1. suppose(3{13)istruefort 1wheret1.Then clearlytruefort=0becaused0(n)=1.wethen Proof.Weworkbyinductionont.Equality(3{13)is dt(pl)=xnjpldt 1(n)=lXi=0dt 1(pi)=lXi=0i+t 1 t 1: (3 13) P(x)=p1(x)p2(xp1)pr(xp1p2pr 1); (3 11)

(x+1)t 1+(x+1)t++(x+1)l+t 1 Sodt(pl)isthecoecientofxt 1in =(x+1)t 1(x+1)l+1 1 x 406 Experimental Mathematics, Vol. 8 (1999), No. 4 (x+1)t 1.Therefore,dt(pl)= l+t Hencedt(pl)isthecoecientofxtin(x+1)l+t =(x+1)l+t (x+1)t 1 t. x : have Sincedt(N)isamultiplicativefunctionofN,we aredistinctprimes,then 3.7.IfN=pr1 dt(n)=syi=1ri+t 1prs swhereri1andpi t: have r(n;t):=0 3.8.ForanypositiveintegersNandt,we Pti=1( 1)t i tidi 1(N)ifN>1. ifn=1, fromthedenitionthatr(1;t)=0andr(n;1)=1 Proof.Weagainprovebyinductionont.Itisclear (3{14)istruefort 1wheret2.Then foranyt;n1.wethensupposen>1and r(n;t)=xn1jn N1>1r(N=N1;t 1) (3 14) =XN1jN =XN1jNt 1 N>N1>1r(N=N1;t 1) t 1 Xi=1( 1)t i 1t 1 Xi=1( 1)t i 1t 1 idi 1(N=N1) =txi=2( 1)t it 1 i 1di 1(N) i(di 1(1)+di 1(N)) t 1 =txi=1( 1)t itidi 1(N) Xi=1( 1)t i 1t 1 idi 1(N)+( 1)t 1 from(3{12)andthefactthat t 1 i 1+ t 1 inlofdegreen 1,whereN=pr1 3.9.Thenumberofcyclotomicpolynomials i= ti. andthepiaredistinctoddprimes,is 4r1++rs Xi=1 ixj=1( 1)i jijsyk=1rk+j 1 1prs s;ri1 j 1: Proof.From3.5,thenumberofcyclotomic polynomialsinlofdegreen 1is Thecorollarynowfollowsfrom3.7and 4r1++rs Xi=1r(N;i): 3.8. forpolynomialsofodddegree. WeconjectureexplicitlythatTheorem3.4alsoholds degreen 1iscyclotomicifandonlyif 4.1.ALittlewoodpolynomial,P(x),of necessarilydistinct. wheren=p1p2prandthepiareprimes,not P(x)=p1(x)p2(xp1)pr(xp1p2pr 1); Conjecture putingallcyclotomicpolynomialswithoddcoe- cientsofagivendegreeandthencheckingwhich Wecomputeduptodegree210(exceptforthecase N 1=191).Thecomputationwasbasedoncom- matchedthesetgeneratedbytheconjecture.for wereactuallylittlewoodandseeingthatthisset arelittlewood.forn 1=191thereare697392380 cyclotomicpolynomialswithoddcoecients(which example,forn 1=143thereare6773464cyclotomicpolynomialswithoddcoecientsofwhich416 wastoobigforourprogram). ecientsfrom2.7quiteeasilysothebulk height1.thesetintheconjecturecomputesvery oftheworkisinvolvedincheckingwhichoneshave Wecangenerateallthecyclotomicswithoddco- easilyrecursively. 4. CYCLOTOMIC LITTLEWOOD POLYNOMIALS OF ODD DEGREE (4 1)

proofisasfollows.from2.7,wehave MostnotablythecasewhereNisapowerof2.The Somespecialcasesalsosupporttheconjecture. P(x)=e(1) 1(x)e(2) 2(x)e(2t+1) 2t+1(x): Borwein and Choi: On Cyclotomic with1 Coefficients 407 x2 1and Again,weassumea0=a1=1.Since1(x)2(x)= [BorweinandErdelyi1995]P.BorweinandT.Erdelyi, L.Schumaker,VanderbiltUniversityPress,Nashville, TN,1998. ApproximationTheoryIX,editedbyC.Chuiand forl2,wehavee(2) e(1)=1andhence P(x)=2(x)Q(x2); 2l(x)=2(x2l 1) [debruijn1948]n.g.debruijn,\onmahler'spartition Polynomialsandpolynomialinequalities,Lect.Notes inmath.161,springer,newyork,1995. forsomecyclotomiclittlewoodpolynomialq(x). (1948),210{220. problem",nederl.akad.wetensch.,proc.51(1948), 659{669.AlsoappearsasIndagationesMath.10 TheauthorswishtothankProfessorD.Boydforhis adviceandsupportconcerningthispaper. [LidlandNiederreiter1983]R.LidlandH.Niederreiter, [Hungerford1974]T.W.Hungerford,Algebra,Holt, GraduateTextsinMath.73,Springer,Berlin,1980. RinehartandWinston,NewYork,1974.Reprintedas [Borwein1998]P.Borwein,\Someoldproblemson [Littlewood1968]J.E.Littlewood,Someproblems Finiteelds,Seconded.,Addison-Wesley,Reading, MA,1983.Secondedition,CambridgeUniv.Press, 1997. polynomialswithintegercoecients",pp.31{50in [Mahler1963]K.Mahler,\Ontwoextremumproperties inrealandcomplexanalysis,heathmathematical Monographs,D.C.Heath,Lexington,MA,1968. PeterBorwein,DepartmentofMathematicsandStatistics,SimonFraserUniversity,Burnaby,B.C.,Canada ofpolynomials",illinoisj.math.7(1963),681{701. REFERENCES Kwok-KwongStephenChoi,DepartmentofMathematics,UniversityofBritishColumbia,Vancouver,B.C.,Canada V5A1S6(pborwein@cecm.sfu.ca) V6T1Z2andDepartmentofMathematicsandStatistics,SimonFraserUniversity,Burnaby,B.C.,Canada ReceivedJune3,1998;acceptedinrevisedformNovember28,1998 V5A1S6(choi@cecm.sfu.ca) Therefore,byinduction,P(x)satises(4{1). ACKNOWLEDGMENT