x^3 + y^3 = p 

Conjecture(Sylvester): Any prime number p  4, 7, 8 mod 9 can be written as the sum of two cubes.

For p  4, 7 mod 9, the following theorem is known.               
Theorem (Dasgupta-Voight). Let p  4, 7 mod 9 be a prime such that 3 mod p is not a cubic
residue. Then p and p^2 are sums of two rational cubes.

For p  8 mod 9, Noam Elkies showed the numerical solutions with p<10000.               

For p  8 mod 9, we show the numerical solutions with 10000 < p < 20000.
Only solutions for p=10889,14633,15137,15767,15803,16433,16901,17657,17783,17909,18413,19421 were not found.


x^3 + y^3 = p...............................................................(1)

This cubic curve (1) can be transformed to the elliptic curve (2).

Y^2 = X^3-432p^2............................................................(2)

X=(12p)/(x+y) 
Y=(36p(x-y)/(x+y) 
x=(36p+Y)/(6X) 
y=(36p-Y)/(6X)

We searched the rational solution of elliptic curve (2) using Online Magma Calculator.
However we couldn't find some solutions.


Numerical solutions.
Only smallest(maybe) solutions are shown.
10000 < p <20000
[ p, x, y ]

[10007],[47025103692/11444689]
[10061],[16847/763,-6786/763]
[10079],[120953892/7921,1322016043284/704969]
[10133],[372480276/104329,1143740950452/33698267]
[10151],[411871303187076/13963912561,8351515210558030731828/1650101583420809]
[10169],[925119716914702832989908/125207413062730514569,839091437750273267561274759520676988/1401022351286877789362383353253]
[10223],[7313891273934376855449742576038948/5368597713687818809744566121,625493420945901155487256297268056781062865559857028/393360994387586047118302051922728933840331]
[10259],[4796448389940195689257974838017324/111604425340559493329209753129,332089583179143292135841451898534122562420837768356/37283967728304413306631013444397701550428117]
[10313],[34302349/1630997,16365420/1630997]
[10331],[59922804294627330087876/85241493541408681,14668583095043805618125759160342612/24887230821688755420025579]
[10457],[197783642087673204/17555521784761,86495007118173453183238884/73556424947145441491]
[10529],[496088047606541556/135122467359961,-62745787680903479354451012/1570693152412538495459]
[10601],[27341/1141,-16740/1141]
[10691],[106304723357860379388/24708805189980409,-675614886601167480789391759452/3883985661768421413911923]
[10709],[266219868/5329,4342845609612/389017]
[10781],[101266003/4411823,-48298950/4411823]
[10799],[73763/2413,-62964/2413]
[10853],[7875708007642921014028209020462373790356/1249181225456131850298481839078600409,-623952443996520660361079039291547331317648280362385161306708/1396169584164400377838243274559166292797740991590621923]
[10979],[157590764254561583666532/1949897377097533081,62556828402111100897656252354326676/2722811853736513594639414979]
[11069],[51624124644/11950849,6872947654979556/41314084993]
[11087],[52115899639404557931517293348/2392499714723489927036881,-11866888896045622106054114101406780117087892/3700648548472174591921518016116757529]
[11159],[55151918993865209556/9837118516698001,-341393399175466664930661949332/975667538165651792984249]
[11177],[265157/5869,-253980/5869]
[11213],[15235334027436/3975680809,-11097489040090714332/250678602049877]
[11321],[10800/481,521/481]
[11393],[91968812601201824173151785473342612/20009304830205994843609401185569,-18129523059163038411253445288609558734193432334232404/89505145057950241280597228195665190280007831247]
[11411],[9017/403,2394/403]
[11447],[9182898432/407215673,-1113118289/407215673]
[11483],[397135397106260013518815955742536412/94641824740267499658143619435049,-119776064823928258058352327689106868928809855609533276/920713810803805994695292403206078082823319692843]
[11519],[7722489410856046402078450232969298219893988434427614096441579964/86968530056860192585955463158785037804596432244950021267129,-678606794240088776912708277843361262598233067855714254147603628569995595137924113274756601337804/25647391104227711363051771305834253044609635949693064411266540939593892444358208801011117]
[11681],[1143881/15721,-1132200/15721]
[11699],[289262740171692/73104803641,1050280838673494974596/19766003703649939]
[11717],[1857452799684/11566801,-2531474574904961316/39338690201]
[11789],[4044812616008429088021943814303268/89939208121632047801107228969,257160481765851936084778836261118832782804414209972/26972648274820207054035289251023381226362453]
[11807],[8325674316/291943769,-6567883577/291943769]
[11897],[200915312095154720498203543281170983840164/22655925515564631331978333725499726801,-86019093360185456975915399549127319628448650389467277303297716/107838220402029524044762931941834894680367153453213069799]
[11933],[41345930887337711028/6360072229503721,-234210082102263784152411926868/507216148267327749476981]
[11969],[10583/463,1386/463]
[11987],[229410363251031087840028138568815052532700618798024807824527631432756/12220049652160508506864249335503489015132434218400440388936415329,3458379974683789443585003882256328006918165333168249784625331330736016627915789912087172630325757185852/1350857243206488941673570055605433326725059966456053429431215648616113162966054038505502174784017]
[12041],[15133145498835320198262227994784164/132021568919179039831079218489,1861593186831579394005644662321689848032872505703364/47969736655310061728882712248282069291964013]
[12113],[9381944250645818287460387233601388/463912438534186262939528399281,-905250545683815635316992906461722673712115930780612/315975948090634730204421120128032233385250071]
[12149],[29043265193495364/1562167517689,4924961108166148625209356/1952501628831397363]
[12203],[178217408699724/296149681,2379168071287716002004/5096439860329]
[12329],[20554426421656887396828/553343593026886489,-2944961771008261065742392670606932/411616076478584089935901987]
[12347],[80083038810036/327863449,716654505027110661612/5936623471043]
[12401],[4345337227187373974395580726372302525675877506932/239830112062921440744760687834167404516889361,9007331218918269067234618899861651167866839405426114569790252599714411124/3714116872171037791033954518434650795618789394498308791390802000391]
[12437],[558540/9709,-546103/9709]
[12473],[1664314344/71801909,192628477/71801909]
[12491],[72014542404695058552534909948/16187457750749781788203129,-9358110963586850547765057391078335847958148/65128034512926242783545253901705703117]
[12527],[2971698268555111167025212/182410346754466876369,-5082477812832863945355437190156184476/2463622735869974648144666511847]
[12653],[3272747436183365400950508/103621448217799002241,5914138929956270571152077023249239508/1054810602704147076156295519039]
[12671],[56178150270783852/1211582316961,13310673091936696529803044/1333611676342994959]
[12689],[1446870086296195679614973604/311538102299747277048649,30426418104781925265559552732346221258044/173886855993696622311348975968978107]
[12743],[9898113050822320435187181741462708/304629871026329856171573980041,-983748538126498106540986321487763959639147645338428/168135232943039957632571593588313800476773339]
[12923],[667365037511171763296982207535572/33397072634122654755853476289,-17162197196909475689537924054008773856915362968796/6103270262883785521040350549785530710861663]
[12941],[64872629/2808409,23889690/2808409]
[12959],[173053930476/14969161,70279567634593332/57915683909]
[13049],[38879/1471,-25830/1471]
[13103],[1856872059418164/347700094921,-57312372184606778883876/205025185671211781]
[13121],[3112638523074369004044/384401330521151329,161035355798688138703145198429796/238329237385741473175376017]
[13229],[98590909242710013204/11037075574755169,925565723558802266452521733116/1159527423712858695400303]
[13337],[3069027628120039908/393902575426441,4920420989268107135754911388/7817776142534490571739]
[13463],[57708448745295556472316348/11158710548712073596001,-288772047329673320218806413104931031972/1178748134764827765023411316394001]
[13499],[100247487616772687175075417736326093152958480567468/82613298290607558485577786511709002412771329,-1003714588074170717254449370768019140992410545433035783438301362679701383372/750887616731095056869845640113770986124818107700945422425138015233]
[13553],[27972/1117,-14419/1117]
[13679],[1970782178838671684974116/273555235866439185481,-2449439071565158516532514699844557252/4524468275917718123188635218629]
[13697],[25483100775116008404/5081759521904521,76892545372664138630180947164/362260651969625281990181]
[13751],[225677182595919391723595301096198600095010708996/6050494004088034472609850290746277231311561,107124639384842185122781311275295770860912265400950689940283958115463012/14882855061558051398529144989400677791624287522811102180386536059]
[13841],[18414/763,-4573/763]
[13859],[3861015859/59987849,-3792926592/59987849]
[13877],[5230557090/257683391,4552297723/257683391]
[13913],[439775603/19530931,264959586/19530931]
[13931],[466800046489656880429930537861236/75985181518714616393428002121,8058214494173650570019298988707991377445124960628/20945626641830444297188836808474616698534331]
[13967],[1029600711944696388/117477439367401,977151146451368406553673532/1273302839548888586101]
[14057],[409037124/32041,8101149609636/5735339]
[14327],[29491022397380027881908/339850195461624961,5064134993684862922784862134716788/198121353399772193922672959]
[14489],[2488375812/546121,25228352596644/403583419]
[14543],[65118780289683288588/720804848100481,525451306806816083350919500572/19352030888075726696129]
[14561],[12373840534481508/1972021978369,1091858629333269432607092/2769284827937867903]
[14669],[676725047623245979795265322140388/43991323706378292655466036809,-17378061350164319229593591424115255791149451386892/9226788066423622067288774018240869130648123]
[14723],[26941822690836/4127163049,113898762401567867028/265141335756907]
[14741],[73375889266001321996521102599074670998148/352005828504297320332122044807176489,-19875946355960357237886613735753051230698626075631887505628012/208845336459456110530873996846236077330871984194493013]
[14759],[3882542179908/176969809,7616062707953740812/2354229369127]
[14813],[7718028570/318805031,2724677227/318805031]
[14831],[3643713043027118175179651412/666509258621192648386321,142272717273786084222224777341562467307316/544138280635553478113894735125861369]
[14867],[6328273620400132791329654261059644015077868/364547170471634801948576496234699361,-15919443181001073624311414885888109673726857503205624490548467988/220105349322007384320981850904149278651654181306053359]
[14939],[29970340938354294153357302333676/2747943611907935845747049281,157858942735937033816652453051856369235742305868/144049490014731597325131198077800905975071]
[14957],[13770/559,1187/559]
[15083],[50207427752555478581436/10747508087286957529,-2134429302289586619553350552517572/35233977355052572872803690083]
[15101],[25523157612/1540081,4033204327794996/1911240521]
[15173],[6269778156203585468775344998095924/166638406998404366162854383241,-495989429597901614785219722716723621319579014261556/68024077035681231265820500991018867604698789]
[15227],[23579/937,-8352/937]
[15263],[4007361245658937208062471356799672518177966263803421933090994052284/809813949882936719166694004932744804583384106178497623528908441,3302648146990083948054012257940284083758208933247274034979936442729816039664309786345128517052838156/23045061979993332389106334196863879347441910623630840653153522008862615457798849726633481955011]
[15299],[1531617395351052016308/180119295247609,-59941200298825023437319868104372/2417354583981758990477]
[15443],[1215088574403771655325366742294561110604/17716075748153175838903942614502681,42348941064599030691234153157207944395804308810972665500676/2358040678229939684644386762525161803162270377255421]
[15461],[15172077723852/424730881,59030312422568359284/8753278726529]
[15497],[1247154922295330711484/106012723307594401,42617006066563828530265858704804/34517313463436064515870351]
[15551],[1882805651713158320232525278404553033665239432232042972596/13599663610822838795631994275742441971158325802436641,81695792603436709983384693917600295880995644883462729325451056084949889220565337897492/1585960071688493678881696395550384567252746139414726679843495200067810242864561]
[15569],[53243/1843,-37674/1843]
[15641],[47311523868/2289169,10229046651347652/3463512697]
[15731],[7542258904863305335741044/1420890642777211890769,-11062162904716427137143370017324023196/53560050935526740024430840263897]
[15749],[14984917032258948/1621020415249,1705408811733055092959268/2063871845552120057]
[16001],[8694/403,7307/403]
[16073],[3602419428/128881,215664589335348/46268279]
[16091],[4239442684152977436021972/1881825854156103241,8728972998077088820677797563668335604/2581482269005095357974612539]
[16127],[35832076898988/2049463441,212223859248108469572/92781259437511]
[16217],[2555190/26899,-2538973/26899]
[16253],[20816735628/2621161,2639233970851932/4243659659]
[16361],[1204226274252/136866601,1204093664802598356/1601202365099]
[16451],[25642813514818505230380273739187916/377320347444017186279602564321,-4105514715395048422183492907338521642901524957400212/231774332238316276108348630531741862438556881]
[16487],[2696090079442745703972/6028684375027609,139991385722555843373306080862396/468094804911321814844227]
[16631],[4284796970172/429276961,8319522774220334244/8894189354959]
[16649],[25780986520716/4802628601,62216419722217365708/332826964677901]
[16703],[60890140161340089397356/117149516014969,-15025206885624574336995910062882732/1267975167007525914547]
[16811],[122860134406084530584523324/14719962100734109740889,-1210425811354047451970507548260349473084/1785911927337556734679816313045213]
[16829],[6523672996144862519484/373095981843483889,520847651370312832859241965989524/227892932938680487281614537]
[16883],[28329123548316/794506969,150577148807806964508/22394767935203]
[16937],[9899/427,7038/427]
[17027],[408934569925633939587636169461129058307268489848854079268/9867889694300580917351909346194367813639559089756721,-8262248621921489812122173397172822490569977133495226304568529709436306599131136326268/980249048220713633541289976763452824532758805738502654211663861611055206470231]
[17099],[1026323/14593,-1009224/14593]
[17117],[77094198934610393244/13153686227209441,412494297175095153484828821876/1508589972782947954967761]
[17189],[84657857810120639741172/5312787546163177081,24240429471438689193192918730302036/12245701637532407852616734771]
[17207],[1066532932584977441268/199423390359908521,-14097609247048228229312877405372/89056196520426584323756181]
[17333],[19601/757,-2268/757]
[17351],[15551/601,1800/601]
[17387],[5421732020353808724/931946633895601,7325652500117936522423910396/28450281382184762103401]
[17477],[9105465055776572172/1723771975098889,8892083734072480906268259204/71568138878223381432037]
[17729],[112979748/289,1200882204108/4913]
[17747],[133868753/5130931,-12142080/5130931]
[17837],[285628853849318607792828/51334773151945552201,68623242780187299960791015569942212/367804866202668810484237856699]
[17891],[16312486806/638773027,6856877033/638773027]
[17981],[1181561209745772/77479279201,39807019576578671393364/21566434844877551]
[18089],[1020769788/5329,32612744197692/389017]
[18143],[22171791649188/340365601,104373307185863749452/6279404972849]
[18233],[6382368164748/298563841,16005036054440465028/5158884608639]
[18251],[125929669044/1682209,44680455868317084/2181825073]
[18269],[76779990/3142429,48527081/3142429]
[18287],[2380644492437734610924847337068/3061144682859448807295041,3673176120321652990552949330998583625492195108/5355817686153208893672144688559975839]
[18341],[236511533288231719068/32615707314865249,2861437987882370188922399372268/5890337804481546007895057]
[18503],[22770/859,-4267/859]
[18521],[1646553324/310249,6222716365356/172808693]
[18539],[97501046169352919443893060924/3375538452742025621133289,30350923558904330363244255062739225955805916/6201754771877471976807589536295271413]
[18593],[10257396309516/1922559409,-4238284106858806332/84298462406423]
[18701],[141412357575393947204797798212/25389149940108187809824161,-18848457009877377510610136677749018902164956/127929953081195253151808881787871448591]
[18719],[123428510044334196924/18572090400214849,954304459914912118843992652476/2530994395096192305056993]
[18773],[1513103566067133495107524639231476/281131634204921274407955799129,9021824826703868513914436749744549955124923820372/149061187150718877113249310438244947596008867]
[18899],[35025497289800285484/4221259038461089,-177095006819991706525848127764/274260337001066759475313]
[18917],[1028088053102351360932369316508/13685337474713663195378881,-1042236488729973821892648427565877784846173012/50627125188077336029303076669750893279]
[19079],[1575464078001633879540992238952417542129022179084/173064866428443067831378843104462276361246129,1759351791177240757718940553037620401828276516056852221438125207568430244/2276739630712247234023789521694781561165562648734890883211214105367]
[19259],[17226/643,2033/643]
[19403],[6204324425988/555969241,14521623731056599732/13109198733539]
[19457],[44907636548950673080381566835028052/7550898230842749814780863902089,-4489490781689162736255198540206211032022983587875804/20749036061610394721155524018554525495004395813]
[19583],[801027383161404/145084048201,2834180372610235154196/55262368875712699]
[19709],[9767351195898492/1764404799481,100390564309109834828196/2343674774793807629]
[19727],[9217045673128696754276202108/762134201376048554202361,-841785280296702939606906040202245608345612/665345428213556836893449044363263859]
[19763],[82593435990396/3522066409,745690074563266706052/209024075174923]
[19853],[18838836140235064534960700793492/261752393640461835859316209,-81748897783917342681135870505623321497091465756/4234830114749564303809730791587014272073]
[19889],[1117643277396/74632321,1151103761393215788/644748621119]
[19961],[43085225236767038841971028/7308000136838779823449,-113137380102387739237815918970985114268/624737852251959135598223116948957]
[19979],[191400642372741773303645308092/6885203190274839730774321,-83399802078365186907198613387165846475685516/18066546449189584447474830846499740631]
[19997],[161929026425028/18449517241,1777947028690661017452/2505979477327789]






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