## Table of fundamental integer solutions (x,y,z) of x3 + y3 = p z3 for all primes p congruent to 8 mod 9 and less than 104 and of x3 + y3 = p2 z3 for all primes p congruent to 8 mod 9 and less than 103

Which numbers n other than zero can be written as the sum of two rational cubes? Equivalently, for which nonzero n does the elliptic curve

E(n): x3 + y3 = n z3

have a rational point other than the point at infinity (1:-1:0)? Excluding the cases n=c3 and n=2c3, for which the answers are well-known (the only solutions are (x,y)=(0,c) and (c,0) in the former case, (x,y)=(c,c) in the latter), the question is equivalent to: does E(n) have positive rank?

Selmer conjectured a sufficient condition for E(n) to have positive rank, which is now known to be a consequence of two other conjectures: the parity part of the conjecture of Birch and Swinnerton-Dyer, and the conjecture that the Tate-Shafarevich part of E(n) [or even its 3-part] is finite. In particular, for n=p, a prime, the conjecture asserts that E(p) has rank 1 if and only if p is congruent to 4, 7, or 8 mod 9. (It has rank 0 if p is congruent to 2 or 5 mod 9; for p=9k+1, the rank should be even, but it could be either 0 or 2.)

For p=9k+8, unlike the cases of p=9k+4 and 9k+7, I have no construction that is guaranteed to produce a nontrivial rational point on E(p). However, one can still search for points. Standard heuristics lead us to expect that the height of a generator can grow as p1/3, so eventually the searches become infeasible; but the latest techniques let us find generators with x,y,z up to about 80 digits, which turns out to suffice for each E(p) with p=9k+8<104. The generators are tabulated below for each of these 201 primes.

The ingredients of the computation were:

• 3-descent. This is well-known, and a particularly natural approach for the curves E(p) because they admit a rational 3-isogeny. One finds a plane cubic curve E'(p) together with a rational map of degree 9 to E(p) and a guarantee that if E(p) has rank 1 then either the generator or its negative is the image of a rational point on E'(p). Finding the rational point on E'(p) should be much easier, because its height is 1/9 times the height of the E(p) generator.
• A good search method. Suppose (x':y':z') is a point on E'(p) with x',y',z' integers of absolute value at most H. A direct search will find the point in time about H2. This can be done very efficiently with a sieve, but the growth of H2 would make such a search take too long to find the largest generators in the table. Instead I used a q-adic variation of the lattice-reduction trick from my ANTS-4 paper (math.NT/0005139 on the arXiv) to do this in expected time H1+o(1). This variation avoids archimedean headaches and makes it easy to write a single program that handles an arbitrary plane curve. The auxiliary prime q is taken to be the smallest prime >H.
• A fast implementation and computer. I wrote the program in GP; for the harder cases, I compiled it using Allombert's gp2c and ran it on MECCAH. Most of the larger cases, with H=106.5 or H=107, took about 25 minutes or 80 minutes each, except for the largest generator, with p=8837 (of canonical height about 66 or 132 depending on the normalization, corresponding to x,y,z of about 85 digits), which required two days to reach H=3·108 on E'(p). Most entries on the table were computed in a matter of seconds.
I also computed L'(E(p),1) to about 10 decimal places and compared with the height of the generator to determine the order of SHA(E(p)) under the BSD conjecture. In each case this conjectural order was numerically indistinguishable from a square integer, as expected; moreover, the integer equals 1 except for the seventeen primes 431, 2267, 3023, 3041, 3167, 4229, 4373, 5021, 5309, 5669, 6047, 7829, 7937, 8243, 8297, 8513, 9413, for which |SHA|BSD=4, and the two primes 7109 and 8117, for which |SHA|BSD=16.

An amusing sideline: what if we require that x,y both be positive? It has been known since Diophantus that if x3+y3=pz3 has any non-torsion solution then it has a solution with x,y>0. In modern terminology, some multiple of the generator is bound to be in the positive segment of E(p). But if the generator is close to the identity (that is, if x/y is close to -1), a large multiple will be needed. This can make the smallest positive solution of x3+y3=pz3 much larger than the generator. The record in our table occurs for p=5849, where the generator has x/y=-1.0011375..., and a positive solution first occurs at the 13th multiple, by which time the coordinates have ballooned to 10000+ digits! Even the less extreme cases of p=3329 and p=4679 require 2500+ and 2000+ digits respectively (from the 9th and 6th multiple of the generator tabulated below) for each variable in the least positive solution of x3+y3=pz3.

Selmer's conjecture also indicates that E(p2) should have rank 1 if and only if p is congruent to 4, 7, or 8 mod 9. Again, I have a construction for p=9k+4 and 9k+7, but not for p=9k+8. The same methods that apply to E(p) can be tried here too, but this time the height grows as p2/3, so one expects the descent-and-search technique to run out of steam sooner for E(p2) than it does for E(p). In fact, of the 29 primes less than 103 in this congruence class, three (the primes 593, 719, and 863) already go well beyond the feasible limit of the method, with generators expected to have height more than 110 according to the BSD conjecture; three others (namely 467, 701, and 971) have generators of height between 69 and 73 that are only barely accessible. At any rate, Selmer's conjecture for these curves is confirmed by the computation that L'(1,E(p2)) is nonzero in each case, via the Gross-Zagier formula for the height of a Heegner point; moreover, the conductors of 27p2 are small enough to make the Heegner point computable in practice, though I haven't yet carried out this computation. [Update 9/2004: These generators were found by Mark Watkins in September 2003, using Heegner points for p=593 and 4-descent together with a p-adic implementation of my lattice reduction trick for p=719 and p=863. Allan MacLeod independently obtained the p=593 and p=719 generators using Heegner points in September 2004. For p=863 we have |SHA|BSD=4, making the generator easily small enough to find by my techniques, so I don't know why I didn't locate it when I first compiled the table. At any rate, all three generators are now listed below.]

The generators are tabulated here, following the table for E(p). All these curves have trivial SHABSD, except for E(1072) and E(5032), for which |SHA|BSD=4. [Added 9/2004: also E(8632), as noted above.] Again there are a few generators that are close to the identity in the real locus of E(p2). The most extreme example is p=971, where the generator must be multiplied by 9 to reach the positive segment of E(p2). We find that the smallest positive integers x,y such that 971(x3+y3) is a cube have more than 7000 digits each.

 17 (18, -1, 7) 53 (1872, -1819, 217) 71 (197, -126, 43) 89 (53, 36, 13) 107 (90, 17, 19) 179 (2184480, -1305053, 357833) 197 (2339, -2142, 247) 233 (124253, -124020, 3589) 251 (4284, -4033, 373) 269 (800059950, -786434293, 45728263) 359 (77517180, 50972869, 11855651) 431 (701, -270, 91) 449 (323, 126, 43) 467 (1170, -703, 139) 503 (630, -127, 79) 521 (9490853970, 5583990833, 1254665713) 557 (634431077911, 297121457160, 79662562427) 593 (2414879, 1652508, 315343) 647 (3761915908356, -1897602251707, 415485054019) 683 (360, 323, 49) 701 (1802600801, -366684318, 202349989) 719 (540, 179, 61) 773 (236269104119, -217610763282, 15510545323) 809 (961746342353, -680676340410, 89201475937) 827 (5360347, -1297296, 568361) 863 (2033, -1170, 199) 881 (28514546857, -17370480960, 2730920537) 953 (180559890, -174023263, 8648401) 971 (3767293, 1003230, 382823) 1061 (7928631326981892119419, -7384720683614963994162, 448468751379396136991) 1097 (130377131, -122852808, 6910243) 1151 (1241065853, -1182764250, 60662203) 1187 (2889431104592843, 642393764713026, 273890905892569) 1223 (1691, -468, 157) 1259 (593451374797, -280138914846, 52960720007) 1277 (63450360851774303250, -55885985808578354269, 3986484254614027327) 1367 (4427, -3060, 349) 1439 (10077714093236492790, 1393409223922062943, 893423263988265017) 1493 (144681659, -69056730, 12182287) 1511 (98531, -97020, 3061) 1583 (8087254206193, 4519843233066, 732150370631) 1601 (62826643498196701, -62503318267166538, 1334975929900709) 1619 (93456, -91837, 2953) 1637 (210917070, -127998109, 16449127) 1709 (494873, -493164, 9013) 1871 (4246164, -4244293, 37813) 1889 (8189, -6300, 541) 1907 (1732145468851477, 621741122030304, 141800824792991) 1979 (16407408311622, 11397262643441, 1439035904371) 1997 (24333582924062552087, -22025700605457238950, 1230768305793744499) 2069 (675462510861197261292665838, -57357409083335441397249661, 52997951868391974789333079) 2087 (546922403, 264956076, 44361589) 2141 (21924, -19783, 1093) 2213 (1260, 953, 109) 2267 (1638, 629, 127) 2339 (125178202867226784658993, -123563112316252224816564, 3176215478754589884323) 2357 (3060, -703, 229) 2393 (36104796, -24347987, 2389061) 2411 (4283945648797985739300924, -4020468585953681829386033, 178151790448133208352373) 2447 (4688105681795465360462930374951, -4685186011013162754248268851490, 42839903659338062762472978977) 2591 (3770387531, 1536972822, 280576909) 2609 (3329, -720, 241) 2663 (665943959109325705219695362797159616519, -665937052663047969066245795777637134340, 1511095329341205761704209550657929457) 2699 (310667311625770398078, 91008438226261015069, 22498605499274313479) 2753 (3853361125294453, -3828336155693694, 73821269256461) 2789 (14582785730844893418, -14270725675281152251, 411865173185956609) 2843 (720008360947, -588467643750, 39064196921) 2861 (9210271356, -7193355047, 522915317) 2879 (58229, -55350, 2131) 2897 (506899008, -506896111, 916897) 2969 (158645826891887936815187, 28724556082428573433794, 11059835427514205915347) 3023 (30743, -27720, 1369) 3041 (1637, 1404, 133) 3167 (38090551356, 1969898923, 2593908149) 3203 (80037946000612237891427, -48661567775683707175110, 4988006797592121047521) 3221 (16511922, -687149, 1118039) 3257 (2523750528491, -2147562784620, 123733697587) 3329 (20425424097279783068739032094990, -20370911205081985306831948748293, 273419668257974554728504481123) 3347 (3723028219, -2525100102, 219722777) 3491 (97533539, -73588770, 5332369) 3527 (2520, 1007, 169) 3581 (9232297591930715970, -6481561548942676223, 523795426705997957) 3617 (3206576658799, 2308791145968, 232190605007) 3671 (556628564176313467257762205214130623721486558, -144059086437787717358422049903884380916160681, 35873397822731577091386466470908478822651401) 3761 (1400202899, 62890038, 90040171) 3779 (2393, 1386, 163) 3797 (15092584974610601013258720313206487764162508599912066, -14923728166577468197105647499758747482485708234013197, 310899129408809939331751956683049882086338790038519) 3833 (88848337805126398223580732500101449024, 9865641259282342987206929005444461077, 5679818086226477241297852118338472909) 3851 (478586104635870, -123671565789407, 30356319080243) 3923 (70332571306506619143809, 48243272939383103951058, 4895242902046333330123) 4013 (27137286, -7421417, 1695971) 4049 (563389161749172, -506501282670319, 22940716554721) 4139 (53591415731445150, -11781268771389443, 3325961386442183) 4157 (573458530336915608, -491808913631591471, 25585904567977757) 4211 (2280141541785616701174617457914337496823, -1472923887977811844149740861388128780760, 127163235644970410162832450011364420613) 4229 (6623, -2394, 403) 4283 (1038294252757, -1013729795820, 26262455591) 4337 (1536006006673494332558064534300561558945814440, -1501493805379682868266144940127580937493802603, 38045085977409988064240107749909361934863109) 4373 (3203, 1170, 199) 4391 (8309242774722297107469241, -964545898747603338170460, 507162368610503813117711) 4409 (68332699, -46671282, 3667019) 4463 (77899622158326680763429961880145720, -64186212460340482756754310126302257, 3600332432850374668823691303913729) 4481 (21061126590714645499, -17765099003388422070, 941144927860889459) 4517 (490137564563, -470575580364, 14419731919) 4643 (452848054823, 282326519880, 29180852389) 4679 (437764526911755564537151738347009525525069353203139216862, -432703258243539872872308243213568280404247977702307868019, 8502747912090899018167744857791711324712343228616223571) 4733 (4204101006772430571608346, -4004822152920334132996237, 128631663740292392765551) 4751 (5777, -1026, 343) 4787 (4158, 629, 247) 4877 (161983714568818663, -129063001718807640, 7551799527123371) 4931 (5332329572879167614283976045650476, -2282905858229880007605849272656907, 304865449723208594257179727030607) 4967 (7793090604320900965633, -7099462455688265375934, 285396525870736935599) 5003 (278695394887573816057, -261197502716363708286, 9145229037454299659) 5021 (45954, -40933, 1783) 5039 (241536959, 13703508, 14089489) 5147 (1974319392878708223214981690290, 118699368510166425240845960161, 114357366367072159205743193747) 5237 (43433715527, -12379661910, 2481654619) 5273 (55206653330073624860827, 45871836766484108655834, 3689297072019178606139) 5309 (1270079, -1264770, 16891) 5381 (153244800728878485446352, -140301409909565818806049, 5378075379298756527979) 5399 (20769750, 16261991, 1349209) 5417 (24428321, 12726882, 1453633) 5471 (69185334596504522330107907, -61262454036205583015748876, 2645015755345257541970653) 5507 (83430731730977370482126670, 81513764216377133148920807, 5884886887523916949869949) 5651 (4948037, -4942386, 41863) 5669 (150696, -145027, 4033) 5741 (1290452080484460227408702769960, 212044804163451274970453021641, 72175015622866016060252602661) 5813 (17528323558716932865742227791446441635564, -4102517004863640472209224919838594466413, 970671342359194593445535888985489746239) 5849 (472861874847622962561798883811377209602124120625593289677386320171959834, -472324604570978577743542752611781670192994543891068069710838949881632703, 3948230522495270896113743171004014332065049910924872761824179726750897) 5867 (19727, -13860, 949) 5903 (331275954957355694520, 207664779392370102889, 19726200101926726247) 5939 (935456217395054694032376137291627071605941, 869081844293291061310014245686783926354210, 62858109212210560502831717054210826315079) 6011 (2182644739439, -2135274458850, 47943416509) 6029 (353810358, -48423421, 19423039) 6047 (14490, -8443, 739) 6101 (3552003215584723707210096945081052376321, 2945998113241187685429738061288803988020, 225954822957348788881245824370256999621) 6173 (256878908316978129257033574, -148723628553404283625455203, 13031540835989592968950529) 6263 (6730481542428536337535698, -1230801691054161908073781, 364387932124791265593253) 6299 (385741737990, -42824562229, 20877310129) 6317 (1915949764443593239433644849625201334624, -1599776082955029402769805992143344409983, 77486333594220366286469176385399733821) 6353 (6797981101674662989018696367, 3714865471985770342219537044, 386014567716183899781845599) 6389 (192327166966219379, 95616163201522788, 10773094086533599) 6551 (34498347686671050, -3762033377426479, 1842926049722671) 6569 (61835195790, -53328150289, 2345593999) 6659 (3366, 3293, 223) 6803 (21565909, 11857230, 1198007) 6857 (52201530, -52194673, 201439) 6911 (45346417985356798050, -7701938055011741111, 2376758043183353759) 6947 (528378581053201739076442615331627581466679409, 294971690374942936005169681842513703750647078, 29212359547899638706689634086523494372277947) 6983 (103722449153388874864189522673672747326778169768630, 40777749944803177823479620210768951659520451903819, 5534330673047353383649590598603432269463931085797) 7001 (509506045106802985206, -186476977793148889897, 26191005757636005607) 7019 (665241687258173, 614888633526336, 42183844158223) 7109 (62730, -55621, 2191) 7127 (157658244940475894379870, -77099243492147570509157, 7859764441058545006181) 7253 (1122155825856816705103682647556526718577, -702251865345813824401217851585489487226, 52785304029394312678937221432493255389) 7307 (18170949306467412, 9993041098200161, 985684441101883) 7433 (321122395510524719496, -140817996646837902307, 15978259668477938341) 7451 (159090118370813479560230397224809286475357588842840994737959283740064357, -105681873535657242562758380858030571599395633656595038213470223947092740, 7255765860329641703914219826241992978114031501340755664136614549450807) 7487 (2956925816194260618863135877819034910304, -1293772293816891336934402645472176908593, 146804857448470773078044172043843233521) 7523 (482562, -475039, 8827) 7541 (16255881071, -16255873530, 9255091) 7559 (8935876613027962289428966174680, 7915275569810411608381232755069, 542882528448008489078563155851) 7577 (72209176088212758927853136889700272594720751458249913638638971, -69900768378704680729699286622160417934550267711538563105890838, 1664898936772013338944360861398361571612616110571300155140443) 7649 (642427228714986, -291176184538277, 31560262799603) 7703 (368171, -360468, 7357) 7757 (255538038092329, 74484300831318, 13014644218637) 7793 (846498534120, -843466924639, 9410123473) 7829 (1026, 6803, 343) 7883 (619997948143896743, -619976914190382294, 1454472349725061) 7901 (302673295448597131397, -172952741830208436216, 14185742960102518753) 7919 (3841919974910701, -3350529370628100, 134095908587459) 7937 (17924759, -17916822, 98767) 8009 (18672696906189479165296455124983269917950, -18657908733799910432255842932928958396143, 124501864761362437008057263501719880353) 8081 (19416072697391835612329982930, -17657313344464312995903723941, 607786319660210940028363619) 8117 (3510, 4607, 259) 8171 (194314680759611570868, 92448358180822295549, 9982058934685113871) 8243 (19085500232568900, -19028334288641651, 196227401171207) 8297 (8280, 17, 409) 8369 (218838495670444403213888592, 163892088685365129248665591, 12115293031090140875812871) 8387 (248882525263, 139169530710, 12925643861) 8423 (1153988956224, -870148453283, 47061097739) 8513 (41021, -32508, 1597) 8693 (22680, -13987, 1009) 8747 (1669233822731674554959735098769755662806229059243, -1544584526868887957586367358271298712637724655970, 47978954302844653602424562936870173475685560161) 8783 (445704917889104585257915597638902870514161955552661, -445379022389410210739779781269831713133975690909650, 2806152908052485732497766150098440249403263125843) 8819 (13628193811723008978094204476860071, -10126396918187687724359666328284850, 553164360730880573101287104382989) 8837 (94789379926733353562898641812245224156279731894522249870081533679765536690712886433241, -9890359356257378580045557428091916274802199545098974496245459588211461446395142633010, 4583109064893377482282786204110387892229862634543186061752451571465321225266512578677) 8963 (6790635605389076698630337, -6338429346928894197449526, 186864976857506769960859) 8999 (182391102255302586776061584945120726964, -54497508945729148853774059638393592703, 8690098990425073751923887300885070727) 9161 (39474, -30313, 1543) 9323 (3809636478218851603152651904587317, -2024855319810942169651715456054556, 171453114518688332029326678556279) 9341 (429949163960821, 81397189813572, 20461224683249) 9377 (2406574801135278, 569254835187919, 114625553534807) 9413 (7722, 1691, 367) 9431 (6011, 3420, 301) 9467 (2811180715839640974121228401050091152019997510782083, -2775970693569843964375531296943836399589447483696704, 44323488959098449932591674399675900219832746070689) 9521 (1437730572159602926138824994071236978853696701, -171175909072797386217531278751605355611047458, 67796216820360442920107796879644000976610069) 9539 (4526775173192287421160, 3410156978280459363043, 240334083945575197817) 9629 (16244910, -16235281, 92491) 9719 (5802889610172483013465471230, -5739247299609148484830237813, 86813544435773979118184533) 9791 (621295406891367344330076, 110616570861635882974753, 29096260480568374530527)

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