Mordell's equation y^2 = x^3 + k

northwolves

{246831066153, 46,{{-6237, 64890},{-6194, 95887},{-5418, 296289},{-5108, 336979},{-3962, 429695},{-3078, 466551},{288, 496845},{702, 497169},{882, 497511},{1792, 502579},{2871, 520092},{2926, 521423},{3276, 531027},{3892, 552979},{5362, 633241},{6066, 685593},{8131, 885662},{11596, 1343917},{12096, 1420083},{12888, 1545165},{16263, 2132640},{20412, 2958291},{24451, 3855502},{30492, 5347629},{38038, 7435295},{40887, 8282484},{60586, 14921047},{69426, 18299673},{114772, 38885651},{117027, 40037094},{145782, 55663839},{206136, 93591603},{216388, 100659565},{240102, 117651519},{456631, 308566412},{628072, 497753549},{810691, 729933218},{1097838, 1150290225},{1205127, 1322967744},{2801106, 4688072487},{5763811, 13837721878},{9035847, 27161472024},{13992552, 52341407181},{30535092, 168732541971},{84293946, 773917339383},{1224698202, 42859156580919}}}

{147309465600, 47,{{-5280, 10560},{-5279, 13969},{-4736, 202688},{-4320, 258240},{-4224, 268224},{-3615, 316335},{-3516, 322248},{-2640, 359040},{-1980, 373560},{-1760, 376640},{-1056, 382272},{-96, 383808},{960, 384960},{2880, 413760},{3936, 456384},{4800, 507840},{5104, 529408},{6576, 657024},{6820, 681560},{7920, 802560},{10560, 1151040},{14080, 1714240},{14985, 1874085},{17856, 2416704},{19140, 2675640},{19360, 2720960},{25344, 4052928},{27520, 4581440},{67584, 17573952},{69040, 18144640},{76164, 21023112},{79200, 22292160},{88000, 26107840},{163680, 66221760},{169216, 69609536},{210705, 96719865},{222720, 105109440},{529120, 384885440},{637956, 509549304},{672160, 551072960},{703296, 589803456},{1164900, 1257282840},{1236720, 1375330560},{11631840, 39670952640},{15692160, 62161872960},{30082800, 164997507840},{601682400, 14758797092160}}}

{212239886400, 48,{{-5856, 106872},{-5064, 287016},{-4880, 309880},{-4575, 341295},{-3960, 387480},{-3600, 406920},{-1220, 458720},{-671, 460367},{-240, 460680},{-15, 460695},{876, 461424},{1464, 464088},{1900, 468080},{3840, 518520},{4096, 530056},{5856, 642696},{6600, 706920},{7020, 747120},{10065, 1109895},{10476, 1167024},{17305, 2322595},{18300, 2518080},{20064, 2879112},{21960, 3286680},{23376, 3603576},{27145, 4496005},{31720, 5668120},{39040, 7727480},{40360, 8121320},{43920, 9215880},{50200, 11256920},{54000, 12556920},{56184, 13325352},{147544, 56675672},{186696, 80669544},{193065, 84832395},{233340, 112716480},{322080, 182787720},{583404, 445609392},{1412760, 1679200680},{2203896, 3271799256},{2475136, 3894023816},{3278140, 5935280480},{4567680, 9762105720},{10533480, 34186749720},{11971824, 41422898568},{16769280, 68670723720},{23632620, 114886184880}}}

northwolves

{4979585600, 51,{{-1700, -8160},{-1636, 24512},{-1496, -40392},{-1375, 48785},{-1360, -49640},{-935, 64515},{-880, 65560},{-800, -66840},{-440, -69960},{-424, -70024},{-55, -70565},{136, 70584},{289, 70737},{680, 72760},{880, 75240},{1241, -83011},{1420, -88560},{1496, -91256},{1585, 94665},{1936, -110616},{2176, -123624},{2720, 158440},{3520, -220440},{3740, 239360},{4280, -288760},{5984, -468248},{6400, -516840},{7480, 650760},{8144, 738328},{11660, -1261040},{12529, 1404183},{17944, 2404728},{18700, 2558160},{28160, 4726040},{31960, -5714040},{37400, -7233160},{38225, 7473785},{45424, 9681432},{55616, 13116136},{59840, -14638360},{96184, 29830152},{133720, 48898440},{145640, -55580360},{441100, -292958160},{546040, -403493640},{948464, 923700712},{2740705, 4537257615},{4360840, -9106578360},{5595040, -13234416360},{9382300, -28738491840},{76545865, 669703552485}}}

{88154795025, 52,{{-4400, 54505},{-4341, 79698},{-4094, 139771},{-3630, 200805},{-3450, 217005},{-2805, 257070},{-2760, 259095},{-2540, 267895},{-2046, 282117},{-2024, 282601},{-1650, 289245},{-1265, 293480},{-690, 296355},{946, 298331},{1035, 298770},{1980, 309705},{3036, 340791},{3774, 376707},{4020, 391305},{4554, 427317},{5940, 545655},{6900, 645495},{7590, 724845},{7810, 751355},{9931, 1033246},{10879, 1172908},{15180, 1893705},{15640, 1978345},{17986, 2430341},{21675, 3204870},{23334, 3576723},{26895, 4420680},{38736, 7629591},{44130, 9275205},{48576, 10710249},{96255, 29864580},{118680, 40886295},{191070, 83520195},{228184, 109000727},{248446, 123836669},{314460, 176339145},{374440, 229125655},{396066, 249259461},{482310, 334957395},{546480, 403981545},{579370, 440995445},{2341515, 3582988530},{3929706, 7790047221},{9739090, 30393279845},{12621480, 44839981545},{45008700, 301956723255},{658901631, 16913398202796}}}

{88813460025, 53,{{-4460, 9845},{-4400, 60245},{-4301, 96182},{-4269, 104946},{-4080, 144555},{-3924, 168501},{-3705, 194820},{-3606, 204753},{-3266, 232327},{-2880, 254805},{-2420, 273205},{-1530, 291945},{-1485, 292470},{-830, 297055},{-570, 297705},{135, 298020},{696, 298581},{2136, 313941},{2364, 319413},{3475, 361630},{5094, 470103},{6900, 646005},{8814, 879513},{9846, 1021431},{10506, 1117329},{14970, 1855695},{16599, 2159232},{20526, 2955801},{23664, 3652437},{23815, 3687220},{28755, 4885170},{41130, 8346705},{68190, 17809095},{74230, 20226295},{96000, 29746005},{140470, 52648055},{142794, 53959953},{168435, 69127770},{195396, 86372619},{221715, 104398530},{303610, 167291855},{369930, 224998545},{407175, 259819620},{418080, 270326955},{714460, 603902645},{826810, 751810895},{953700, 931360245},{1598790, 2021562345},{2759910, 4585032105},{3374304, 6198353067},{13893639, 51787389012},{168665184, 2190474322773},{2362592970, 114837402604305}}}

northwolves

序列已提交发布：

A392144
a(n) is the least positive integer k such that Mordell's equation y^2 = x^3 + k has exactly n integer solutions with y >= 0.

6, 2, 12, 1, 8, 9, 73, 316, 17, 297, 2817, 1737, 4481, 225, 2089, 14400, 1025, 197225, 65600, 92025, 260100, 442225, 4215025, 885025, 54225, 22548673, 13221225, 23882257, 5472225, 3470400, 13186225, 222105600, 11530025, 971173225, 71534025, 2667179025, 35964225, 225090225, 9612758025, 1632201497, 11271177225

OFFSET
0,1
COMMENTS
For k < 10^10, the maximum number of integer solutions occurs at a(51) = 4979585600.
LINKS
Table of n, a(n) for n=0..40.
Helmut Richter, Solutions of Mordell's equation y^2 = x^3 + k.
EXAMPLE
a(6) = 73 because 73 is the least k such that y^2 = x^3 + k has 6 integral solutions with nonnegative y: (-4, 3), (2, 9), (3, 10), (6, 17), (72, 611), (356, 6717).
CROSSREFS
Cf. A054504, A179162, A134108, A392395.

目前k值穷举到3000亿，几个记录：

{42183531225, 55,{{-3464, 24859},{-3450, 33465},{-3300, 79035},{-3110, 110015},{-3036, 119163},{-2829, 139794},{-2420, 167365},{-1920, 187365},{-1625, 194660},{-690, 204585},{-429, 205194},{115, 205390},{576, 205851},{660, 206085},{759, 206448},{979, 207658},{990, 207735},{2530, 241615},{2806, 253529},{3036, 264891},{3210, 274335},{3795, 311190},{4554, 369633},{5940, 501765},{6810, 598335},{7824, 721893},{7984, 742373},{9255, 913740},{10120, 1038565},{13530, 1587135},{20700, 2985285},{22770, 3442065},{29440, 5055515},{39054, 7720617},{49335, 10959960},{60720, 14963685},{84120, 24398565},{133906, 49000879},{151735, 59105960},{167134, 68328073},{184644, 79342203},{319440, 180544485},{569250, 429491535},{796260, 710529915},{996435, 994657290},{1289370, 1464085185},{1570866, 1968832239},{1680570, 2178637185},{3213100, 5759520965},{5074674, 11431736943},{9597555, 29733149490},{17362470, 72346456185},{24047980, 117928263365},{1426042200, 53851594420965},{1460013729354, 1764149596065642783}}}

{56720592225, 57,{{-3825, 27540},{-3806, 39853},{-3740, 66385},{-3681, 82728},{-3366, 136323},{-3060, 167535},{-1980, 221265},{-1800, 225585},{-1460, 231535},{-990, 236115},{-954, 236331},{-374, 238051},{306, 238221},{1530, 245565},{1819, 250478},{1870, 251515},{1980, 253935},{2959, 287452},{3195, 298890},{3366, 307989},{4356, 373329},{4896, 417231},{5170, 441485},{6120, 534735},{6715, 599590},{7920, 743985},{8415, 807840},{9630, 974565},{13464, 1580337},{14400, 1744335},{16830, 2196315},{18139, 2454562},{18324, 2491857},{21250, 3106835},{26235, 4256010},{37060, 7138385},{40374, 8115957},{42075, 8633790},{63360, 15950385},{71910, 19284885},{84150, 24411915},{102204, 32674833},{125136, 44266959},{134640, 49404465},{169150, 69568165},{201850, 90686915},{216414, 100676763},{300870, 165032235},{327690, 187583715},{992475, 988733790},{1228590, 1361791035},{1918654, 2657632933},{2134044, 3117489903},{9811890, 30734701965},{12588840, 44666155215},{21110175, 96992409960},{96880960, 953581134385}}}

{318693478400, 59,{{-6800, 65280},{-6775, 87845},{-6544, 196096},{-5984, 323136},{-5500, 390280},{-5440, 397120},{-3740, 516120},{-3520, 524480},{-3200, 534720},{-1760, 559680},{-1696, 560192},{-935, 563805},{-220, 564520},{121, 564531},{544, 564672},{1156, 565896},{2720, 582080},{3520, 601920},{4964, 664088},{5680, 708480},{5984, 730048},{6340, 757320},{6809, 796477},{7744, 884928},{8704, 988992},{10880, 1267520},{14080, 1763520},{14960, 1914880},{17120, 2310080},{23936, 3745984},{25600, 4134720},{29920, 5206080},{32576, 5906624},{46640, 10088320},{50116, 11233464},{51425, 11675345},{71776, 19237824},{74800, 20465280},{98209, 30782223},{112640, 37808320},{127840, 45712320},{149600, 57865280},{152900, 59790280},{181696, 77451456},{222464, 104929088},{239360, 117106880},{274465, 143791695},{384736, 238641216},{534880, 391187520},{582560, 444642880},{1055785, 1084834005},{1764400, 2343665280},{2184160, 3227949120},{3793856, 7389605696},{10962820, 36298060920},{17443360, 72852626880},{22380160, 105875330880},{37529200, 229907934720},{306183460, 5357628419880}}}

{203947076025, 64,{{-5865, 46920},{-5796, 96117},{-5610, 165495},{-5474, 199801},{-5456, 203797},{-4760, 309995},{-4620, 324555},{-4301, 352682},{-3944, 377621},{-3570, 398055},{-2310, 437745},{-1449, 448224},{-425, 451520},{-336, 451563},{0, 451605},{1254, 453783},{1794, 457953},{2244, 463947},{2346, 465681},{2695, 472780},{3795, 508530},{3850, 510895},{4284, 531573},{4810, 561455},{4830, 562695},{6120, 658155},{7590, 800745},{8211, 870366},{9331, 1008154},{10626, 1184799},{13090, 1564255},{13806, 1683879},{17710, 2399705},{19635, 2788170},{20196, 2905419},{22176, 3333099},{22540, 3414005},{30360, 5309205},{32844, 5969397},{39270, 7795095},{58650, 14210895},{60214, 14782537},{107640, 35317995},{114954, 38977617},{115830, 39423945},{150535, 58407580},{164220, 66550155},{227700, 108654645},{231616, 111469589},{255255, 128962680},{362355, 218123430},{387090, 240834495},{409500, 262048395},{520030, 375010055},{705364, 592406837},{855831, 791739096},{1062600, 1095354645},{4308990, 8944647495},{4516050, 9597057855},{5761470, 13829292345},{7225680, 19423079445},{83060406, 756991624521},{102962370, 1044763029345},{343538580, 6367416731445}}}

northwolves

k 为负值的也通过了：

A392395
a(n) is the least positive integer k such that Mordell's equation y^2 = x^3 - k has exactly n integer solutions with y >= 0.

3, 1, 4, 28, 116, 828, 496, 207, 503, 431, 2351, 3807, 63900, 64432, 344719, 317079, 201023, 194023, 43847, 9773775, 2806208, 28279, 1690399, 6668900, 6959600, 1809856, 89466479, 20615391, 390990583, 78051887, 83707775, 426809600, 1349350975, 6640510400, 472316975, 24980935599, 27315814400, 56899825623

OFFSET
0,1
COMMENTS
For k < 10^11, the maximum number of integer solutions occurs at a(50) = 24864246775.
LINKS
Table of n, a(n) for n=0..37.
Helmut Richter, Solutions of Mordell's equation y^2 = x^3 - k.
EXAMPLE
a(6) = 496 because 496 is the least k such that y^2 = x^3 - k has 6 integral solutions with nonnegative y: {{8, 4}, {16, 60}, {25, 123}, {40, 252}, {113, 1201}, {560, 13252}}.
CROSSREFS
Cf. A081121, A106265, A134109, A392144.

目前k值穷举到-500亿，几个记录：

{-30228286400,41,{{3116,5136},{3120,11960},{3225,57565},{3480,109160},{3809,158223},{3900,170560},{4160,204360},{4680,268840},{6240,461240},{6536,498984},{6824,536232},{7296,598456},{7640,644760},{8840,812760},{11376,1200824},{11960,1296360},{17264,2261688},{24600,3854440},{26945,4419585},{29585,5085735},{35984,6823752},{37505,7261215},{47736,10428184},{54080,12575160},{62760,15721640},{66300,17070560},{91065,27480115},{92000,27904440},{175809,73715777},{208104,94933592},{225020,106740960},{436176,288066376},{610220,476682960},{819260,741536640},{1059120,1089978040},{2071304,2981026008},{2904200,4949254440},{5765096,13842349656},{10510760,34076201640},{13046060,47121494640},{18260840,78033507240}}}
{4966628975,42,{{1710,5795},{1824,-33193},{1890,-42245},{1919,45828},{1995,-54530},{2300,-84855},{2534,106323},{2660,-117705},{2766,127261},{3215,-168120},{3920,235095},{4370,280155},{4830,-328195},{5054,352317},{6251,-489174},{9156,-873271},{9804,968183},{10640,-1095255},{11400,-1215145},{12530,1400805},{13946,-1645419},{19650,2753605},{25536,-4080041},{44555,9404430},{57380,13744695},{62426,15597099},{71820,-19247095},{91910,-27863955},{113430,38202445},{156660,62006455},{204554,92514933},{259350,132077645},{374411,229098834},{432590,-284521245},{482904,335576233},{662655,-539425180},{1302735,1486908080},{1850220,2516720905},{3156090,5606916245},{5335295,-12323601780},{9219560,-27994021545},{1895408480,-82519051796055}}}

{-7111376775,44,{{1926,5751},{1980,-25515},{2124,49707},{2394,81297},{2515,93790},{2560,98315},{3150,155385},{3744,-213003},{4060,-244565},{4275,266490},{4770,-318465},{5335,-380440},{6120,-471285},{6300,492885},{7191,-603936},{7326,621351},{9171,-874206},{12915,-1465290},{15210,-1873935},{16830,-2181735},{16920,2199285},{24120,3745035},{30276,5267349},{35251,6617926},{41386,8418959},{45234,-9620127},{59500,14513365},{62190,-15508665},{73080,19755765},{130050,46899135},{133200,-48613365},{179190,-75852585},{259810,-132429185},{331335,190722060},{620019,-488210922},{807390,-725479335},{1165770,1258691535},{1501870,1840553785},{1520064,1874100213},{2088900,3019092885},{5641794,-13400649153},{5930764,14443284437},{18219286,77767301609},{2369513934,-115342377846777}}}

{-49019398775,45,{{3660,2915},{3674,23943},{3735,55540},{4290,173015},{4350,182465},{4796,247581},{5186,300759},{5220,305315},{6186,433241},{7550,617535},{9416,886461},{12170,1324185},{13340,1524765},{13754,1597767},{18215,2448360},{19640,2743485},{21120,3061315},{23264,3541437},{23886,3684959},{24230,3765135},{27450,4542535},{30755,5388990},{50094,11209697},{50600,11380035},{64950,16551215},{77591,21611964},{113514,38244313},{116010,39512665},{141684,53330723},{144110,54706335},{206810,94049415},{527384,382992573},{632670,503229185},{790680,703074115},{1024644,1037192797},{1690400,2197780035},{1966854,2758405967},{2248670,3372007935},{2337680,3574189635},{4772675,10426601910},{5903715,14344587890},{8971820,26873289315},{134930774,1567351909743},{287319395,4870206936690},{1982902295,88298224958940}}}

{-34681780727,46,{{3267,13706},{3278,23265},{3432,75779},{3542,98769},{4071,181072},{4536,242173},{5412,351901},{5702,388209},{6356,471267},{7826,666807},{8096,704253},{8712,791549},{9086,845823},{11427,1207234},{14051,1655118},{14567,1748256},{15026,1832457},{17303,2268420},{29436,5046877},{32252,5789091},{36707,7030254},{37812,7350301},{49127,10887216},{74976,20528893},{91778,27803415},{109571,36269178},{117252,40149091},{120978,42078025},{142772,53946339},{160472,64283139},{164846,66929247},{240966,118285937},{272228,142036125},{748616,647721987},{2692778,4418764185},{3211272,5754606589},{3349842,6131072969},{11289776,37933943043},{12765852,45611537059},{14745416,56622044163},{19149086,83795765823},{25770162,130820475401},{38312142,237139909919},{98755778,981394844265},{379079591,7380666893688},{882286482,26206826760521}}}

{-24864246775,50,{{2930,17015},{3011,49334},{3040,56835},{3134,76927},{3610,148935},{3634,152073},{3770,169465},{4420,247965},{4474,254343},{6544,505347},{6800,538115},{6830,541985},{9194,867353},{10135,1008060},{10580,1076765},{15094,1847697},{16675,2147490},{19139,2643062},{20320,2892285},{31096,5481219},{32470,5848785},{35120,6579715},{37090,7141335},{39575,7871260},{42584,8786173},{48430,10656735},{51644,11735203},{89290,26680665},{102940,33027165},{123671,43490956},{125074,44233143},{136540,50453085},{153260,59998685},{163574,66156143},{202996,91459869},{232070,111796465},{252586,126944409},{354160,210765315},{787235,698484190},{1593475,2011490010},{2018755,2868305610},{3952720,7858579965},{6353960,16016466685},{6393934,16167848223},{13396870,49034868465},{21928510,102686578815},{48439015,337126548960},{67932950,559913206865},{86518310,804751956815},{157294520,1972742444285}}}