An explicit result of the sum of seven cubes

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An explicit result of the sum of seven cubes ? O. Ramare 29th June 2007 Abstract We prove that every integer ≥ exp(524) is a sum of seven non negative cubes. 1 History and statements In his 1770's ”Meditationes Algebraicae”, E.Waring asserted that every pos- itive integer is a sum of nine non-negative cubes. A proof was missing, as was fairly common at the time, the very notion of proof being not so clear. Notice that henceforth, we shall use cubes to denote cubes of non-negative integers. Consequently, the integers we want to write as sums of cubes are assumed to be non-negative. Maillet in [15] proved that twenty-one cubes were enough to represent every (non-negative) integer and later, Wieferich in [30] provided a proof to Waring's statement (though his proof contained a mistake that was mended in [12]). The Gottingen school was in full bloom and Landau [13] showed that eight cubes suffice to represent every large enough integer. Dickson [7] improved on this statement by establishing that the only exceptions are 23 and 239. The reader will find a full history of the subject in chapter XXV of [8]. Finally, Linnik in [14] showed that every large enough integer is a sum of seven cubes.

• rough philosophy when


• every integer


• arithmetic progressions modulo


• large sieve


• prime


• represent every


• when compared

Sujets

• Large sieve

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 15
• Langue: English

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Anexplicitresultofthesumofsevencubes ∗ O.Ramar´e 29thJune2007 Abstract Weprovethateveryinteger ≥ exp(524)isasumofsevennon negativecubes. 1Historyandstatements Inhis1770’s”MeditationesAlgebraicae”,E.Waringassertedthateverypos- itiveintegerisasumofninenon-negativecubes.Aproofwasmissing,as wasfairlycommonatthetime,theverynotionofproofbeingnotsoclear. Noticethathenceforth,weshallusecubestodenotecubesofnon-negative integers.Consequently,theintegerswewanttowriteassumsofcubesare assumedtobenon-negative. Mailletin[15]provedthattwenty-onecubeswereenoughtorepresent every(non-negative)integerandlater,Wieferichin[30]providedaproofto Waring’sstatement(thoughhisproofcontainedamistakethatwasmended in[12]).TheGo¨ttingenschoolwasinfullbloomandLandau[13]showed thateightcubessufficetorepresenteverylargeenoughinteger.Dickson[7] improvedonthisstatementbyestablishingthattheonlyexceptionsare23 and239.ThereaderwillfindafullhistoryofthesubjectinchapterXXVof .]8[ Finally,Linnikin[14]showedthateverylargeenoughintegerisasumof sevencubes.Sincethen,therehasbeennofurtherimprovementsintermsof thenumberofcubesrequired.Noticethatthecirclemethodreadilyproves thatalmostallintegersaresumsofatmostfourcubes. Fromanexperimentalandheuristicalviewpoint,computationsandar- gumentsdevelopedin[2],[27],[16],[1],[6]tendtosuggestthateveryinteger ≥ 10 14 isasumoffourcubes.Theargumentin[6]evenleadsustobelieve ∗ MSC2000:primary11P05,11Y50;secondary11B13 1

that7373170279850isthelastintegerthatisasumoffivecubesbutnot offour.Whenitisrequiredtoexhibitalargeexampleofanintegerthatis asumoffivecubes,butnotoffour,thesimplestexampleIknowis10 9 +4. Similarly,itisbelievedthat454isthelargestintegerthatisasumof atleasteightcubes,that8042isthelargestintegerthatisasumofseven cubesbutnotofsixandthat1290740playsthisrolewithrespecttosums ofsixcubes. Bydensityconsiderations(therearelessthanX 1/3 cubeslessthanX), weseethateveryintegercannotbeasumofthreecubes.Thesamewould followbystudyingsumsofthreecubesin Z /9 Z .Butnothingpreventssums ofthreecubestohavepositivelowerdensity,ahypothesisbelievedtobetrue byHooleyin[11].Theretheauthorprovesthatthisdensityisstrictlyless thanΓ(4/3) 3 /6,thisvaluecomingfromsizeconsiderationsonly.Hooleyeven improvesonitbytakingadvantageofsomedisparitiesinthedistributionof cubesinsomearithmeticalprogressions.Theprobabilisticmodelsof[5]of sumsof3cubessupportsthisconjecture,whilegivingadensitythattends tozeroforsumsoftwosquares(aresultknowntobetruebyatheoremof Landau),asthemodeltakesmoreandmorelocalobstructionsintoaccount. Thisisincontrastwithapreviousprobabilisticmodel(exposedin[10])of Erdo¨sforsuchsumsthatpredictsapositivedensityforsumsofthreecubes butalsoforsumsoftwosquares,andofcoursegivesstrongergroundtothe initialconjecture.Infact,x 3 +y 3 +z 3 isnotanormandinturn,lacksthe ensuingmultiplicativity. In[6],itisshownthateveryintegerbetween1290741and10 16 isasum offivecubesfromwhichonereadilydeduces(see[23])thateveryintegern verifying455 ≤ n ≤ exp(78.7)isasumofsevennon-negativecubes.Onthe otherhand,notethatitisprovedin[1]thateveryintegerinsomespecial arithmeticprogression,andlargerthan455isindeedasumofsevencubes. Ourconcerninthispapertoshowthateveryintegerlargerthanagiven explicitboundisindeedasumofsevencubes.Weprovethat Theorem1Everyintegern ≥ exp(524)isasumofsevencubes. Watson[29]hadalreadygreatlysimplifiedLinnik’soriginalproof,though itremainedineffective,andindependently,McCurley[16]andCook[4]cor- rectedthisdefect.McCurley[16]evenprovedatheoremsimilartotheone abovewith1077334insteadof524.Themethodweusehasbeenpartially setin[23]wherewehadasimilarstatementbutwith205000insteadof524. Inboththeseapproaches,thecruxofthemethodwastoshowthatsome arithmeticprogressiondidindeedcontainafairlysmallprimeandthebattle wasonthelinkbetweenthesizeofthemodulusqinvolvedandthesizeof thewantedprime.IfthelaterwasofsizeX,McCurley’sproofneededa 2