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

northwolves

{5472225,28,{{-176,-143},{-165,-990},{-120,1935},{-110,2035},{-74,-2251},{-66,-2277},{-44,2321},{15,-2340},{60,2385},{114,-2637},{220,4015},{330,-6435},{400,8335},{636,-16209},{726,-19701},{1320,48015},{1375,-51040},{3300,-189585},{4510,-302885},{6270,496485},{8475,780210},{9780,967185},{10950,-1145835},{43791,-9163836},{157839,-62707788},{320496,-181440369},{1774666,-2364149539},{3240906,5834446371};

{3470400,29,{{-144,-696},{-135,-1005},{-120,1320},{-96,1608},{-80,-1720},{16,-1864},{40,-1880},{60,-1920},{120,2280},{225,-3855},{300,-5520},{360,-7080},{384,7752},{480,10680},{601,14851},{696,-18456},{940,28880},{1041,33639},{2544,-128328},{4720,-324280},{5976,461976},{7320,-626280},{15100,-1855520},{21064,-3057112},{24280,-3783320},{36000,-6830520},{43440,9053880},{381180,-235339680},{4478785,-9478515455}}};

{13186225,30,{{-234,-611},{-225,1340},{-160,3015},{-130,-3315},{-120,-3385},{30,3635},{104,-3783},{156,4121},{159,-4148},{276,-5849},{299,6318},{390,-8515},{650,-16965},{690,-18485},{1040,33735},{1355,-50010},{2370,115435},{2756,144729},{2795,-147810},{4874,-340293},{7020,-588185},{8564,-792537},{15210,-1875835},{22844,-3452697},{25880,4163385},{44811,9485866},{80811,-22972366},{279266,-147579861},{449150,301014285},{56259450,421981316965}}}

{11530025,32,{{-221,-858},{-185,2280},{-130,-3055},{-100,3245},{-94,3271},{-61,3362},{-26,3393},{40,3405},{55,-3420},{130,-3705},{170,-4055},{260,-5395},{416,9139},{676,17901},{1300,46995},{1534,-60177},{1894,82497},{2200,-103245},{4000,-253005},{4510,-302895},{6071,473044},{8840,-831155},{9724,958893},{10595,-1090570},{37336,7214259},{40556,-8167379},{44330,-9333545},{206930,94131545},{364355,219931330},{1179590,-1281140095},{3408740,6293479405},{5516290,12955990905}}}

northwolves

a(1)-a(24)完整列表：

{2,1,{{-1,1}}}

{12,2,{{-2,2},{13,47}}}

{1,3,{{-1,0},{0,1},{2,3}}}

{8,4,{{-2,0},{1,3},{2,4},{46,312}}}

{9,5,{{-2,1},{0,3},{3,6},{6,15},{40,253}}}

{73,6,{{-4,3},{2,9},{3,10},{6,17},{72,611},{356,6717}}}

{316,7,{{-6,10},{-3,17},{2,18},{5,21},{50,354},{90,854},{162,2062}}}

{17,8,{{-2,3},{-1,4},{2,5},{4,9},{8,23},{43,282},{52,375},{5234,378661}}}

{297,9,{{-6,9},{-2,17},{3,18},{4,19},{12,45},{34,199},{48,333},{1362,50265},{93844,28748141}}}

{2817,10,{{-12,33},{-6,51},{-2,53},{18,93},{24,129},{27,150},{214,3131},{684,17889},{1327,48340},{8598,797253}}}

{1737, 11,{{-12, -3},{-8, 35},{-6, 39},{3, 42},{18, -87},{54, 399},{67, -550},{84, -771},{1383, 51432},{5208, 375843},{572034, 432646071}}}

{4481, 12,{{-10, 59},{-8, -63},{-5, 66},{2, 67},{14, -85},{22, 123},{32, 193},{175, -2316},{364, 6945},{640, -16191},{1862, -80347},{3739, 228630}}}

{225, 13,{{-6, 3},{-5, -10},{0, 15},{4, -17},{6, 21},{10, -35},{15, 60},{30, -165},{60, 465},{180, 2415},{336, -6159},{351, -6576},{720114, 611085363}}}

{2089, 14,{{-12, 19},{-10, 33},{-4, -45},{3, -46},{8, 51},{18, -89},{60, -467},{71, -600},{80, 717},{170, 2217},{183, 2476},{698, -18441},{9278, -893679},{129968, -46854861}}}

{14400,15,{{-24,24},{-20,-80},{-15,105},{0,120},{9,-123},{16,-136},{24,168},{40,-280},{60,480},{120,-1320},{240,3720},{720,19320},{1344,-49272},{1404,-52608},{2880456,4888682904}}}

{1025, 16,{{-10, -5},{-5, 30},{-4, 31},{-1, 32},{4, 33},{10, -45},{20, 95},{40, 255},{50, -355},{64, -513},{155, 1930},{166, -2139},{446, -9419},{920, 27905},{3631, 218796},{3730, -227805}}}

{197225,17,{{-56,147},{-40,-365},{-20,-435},{-14,441},{35,490},{70,-735},{119,-1372},{140,1715},{266,4361},{340,-6285},{1015,-32340},{1120,37485},{1136,-38291},{1660,67635},{21350,-3119585},{440090,-291952535},{22427440,106211014035}}}

{65600,18,{{-40,-40},{-20,240},{-16,-248},{-4,-256},{16,264},{25,-285},{40,-360},{80,760},{160,-2040},{200,2840},{256,-4104},{620,15440},{664,-17112},{985,30915},{1784,75352},{3680,223240},{14524,-1750368},{14920,1822440}}}

{92025,19,{{-45,-30},{-41,152},{-36,213},{-30,-255},{-6,303},{10,-305},{30,-345},{40,-395},{51,-474},{60,555},{100,-1045},{115,1270},{216,-3189},{300,5205},{436,9109},{810,-23055},{1440,54645},{5406,397479},{6150,482295}}}

{260100,20,{{-60,210},{-51,-357},{-36,462},{0,510},{16,-514},{21,519},{60,690},{85,-935},{120,-1410},{136,1666},{204,2958},{340,-6290},{765,21165},{880,26110},{1045,33785},{2380,-116110},{4080,260610},{4284,280398},{129844,-46787822},{7024740,18618529710}}}

{442225,21,{{-76,57},{-70,-315},{-40,615},{-21,658},{0,665},{11,666},{30,-685},{95,1140},{114,-1387},{140,-1785},{210,3115},{266,-4389},{1064,34713},{1071,-35056},{1995,89110},{3990,252035},{4550,-306915},{14630,-1769565},{17480,2311065},{23256,-3546521},{99338954,990100714917}}}

{4215025,22,{{-160,345},{-156,-647},{-145,-1080},{-6,2053},{20,2055},{60,2105},{110,2355},{150,2755},{164,-2937},{200,-3495},{486,-10909},{1199,-41568},{1340,49095},{2271,108244},{2706,140779},{3915,244970},{9330,901205},{10736,-1112409},{15275,1887870},{24950,3940995},{61820,15370695},{410910,-263402605}}}

{885025,23,{{-96,-17},{-90,-395},{-69,-746},{-66,-773},{-49,-876},{20,945},{30,-955},{50,-1005},{95,-1320},{140,-1905},{216,3311},{360,-6895},{410,-8355},{506,11421},{894,26747},{1100,36495},{2715,-141470},{4155,267830},{9524,929457},{36050,6844755},{41520,-8460305},{92255,28021080},{7751234,21580240077}}}

{54225,24,{{-36,87},{-30,-165},{-24,-201},{-20,215},{4,233},{10,235},{15,240},{30,-285},{75,-690},{90,885},{96,-969},{120,-1335},{174,2307},{235,-3610},{636,16041},{1180,40535},{1494,-57747},{1830,78285},{3775,231940},{5266,382139},{6070,-472915},{9000,853815},{10860,-1131735},{95295,29417460}}}

王守恩

northwolves 发表于 2026-1-6 06:36
a(1)-a(24)完整列表：

{2,1,{{-1,1}}}

{54225,24,{{-36,87},{-30,-165},{-24,-201},{-20,215},{4,233},{10,235},{15,240},{30,-285},{75,-690},{90,885},{96,-969},{120,-1335},{174,2307},{235,-3610},{636,16041},{1180,40535},{1494,-57747},{1830,78285},{3775,231940},{5266,382139},{6070,-472915},{9000,853815},{10860,-1131735},{95295,29417460}}}

{{x -> -36, y -> 87}, {x -> -30, y -> 165}, {x -> -24,  y -> 201}, {x -> -20, y -> 215}, {x -> 4, y -> 233}, {x -> 10, y -> 235}, {x -> 15, y -> 240}, {x -> 30, y -> 285},
{x -> 75, y -> 690}, {x -> 90, y -> 885}, {x -> 96, y -> 969}, {x -> 120, y -> 1335}, {x -> 174, y -> 2307}, {x -> 235, y -> 3610}, {x -> 636, y -> 16041}, {x -> 1180, y -> 40535},
{x -> 1494, y -> 57747}, {x -> 1830, y -> 78285}, {x -> 3775, y -> 231940}, {x -> 5266, y -> 382139}, {x -> 6070, y -> 472915}, {x -> 9000, y -> 853815}, {x -> 10860, y -> 1131735}, {x -> 95295, y -> 29417460}}

Solve[{y^2 == x^3 + 54225, -(54225)^(1/3) < x < 100000, 0 < y}, {x, y}, Integers]

northwolves

$a(36)=35964225$:

{35964225, 36,{{-330, -165},{-296, -3167},{-270, 4035},{-264, -4191},{-165, -5610},{-110, 5885},{-66, -5973},{-6, 5997},{60, 6015},{180, -6465},{246, -7131},{300, 7935},{319, -8272},{411, 10266},{495, -12540},{660, 17985},{880, 26785},{1116, 37761},{1210, -42515},{1584, 63327},{1720, -71585},{4224, 274593},{4315, 283510},{4950, -348315},{5500, -407935},{10230, 1034715},{10576, 1087649},{13920, -1642335},{33070, -6013835},{42010, 8610515},{43956, -9215679},{77295, -21489540},{726990, 619858635},{980760, 971279265},{1880175, 2578086060},{37605150, -230606204565}}}

northwolves

$a(34)=71534025$:

{71534025, 34,{{-414, 759},{-390, -3495},{-345, 5520},{-276, -7107},{-230, -7705},{-120, 8355},{-89, 8416},{-60, -8445},{150, 8655},{414, 11937},{460, -12995},{670, 19295},{675, 19470},{834, -25527},{936, 29859},{1150, 39905},{1315, 48430},{1495, -58420},{2760, 145245},{2806, 148879},{3939, 247362},{5520, -410205},{9040, 859555},{9315, 899070},{13800, 1621155},{21666, -3189111},{34110, 6299745},{78580, 22027645},{104190, -33630945},{292890, 158510055},{293079, -158663508},{40877670, -261354029295},{60107664, -466009505187},{2816243893920, 4726127075051201955}}}

northwolves

$a(41)=77806025$:

{77806025, 41,{,{-425, 1020},{-409, -3064},{-374, -5049},{-340, 6205},{-220, -8195},{-200, 8355},{-110, 8745},{-106, -8753},{34, -8823},{170, 9095},{220, -9405},{355, -11070},{374, -11407},{484, 13827},{544, 15453},{680, -19805},{880, -27555},{935, -29920},{1070, 36095},{1496, 58531},{1600, -64605},{1870, -81345},{2036, -92291},{2915, 157630},{4486, -300591},{4675, -319770},{7040, -590755},{7990, 714255},{9350, 904145},{11356, 1210179},{13904, 1639517},{14960, 1829795},{24046, -3728769},{33430, 6112305},{36410, -6947545},{110275, -36619770},{136510, 50436705},{237116, -115462589},{1090210, -1138322295},{1398760, 1654302045},{2345575, 3592311480}}}

王守恩

a(n) = 最小正整数 k, 使得莫德尔方程 y^2 = x^3 + k  恰好有 n 个整数解, 且 y ≥ 0。

这个 “k”,  你是怎么拉出来的？——Union@Flatten[Table[Abs[x^2 - y^3], {x, 9^(3/2)}, {y, 9}]]——学不好了。

a(n) = 最小正整数 k, 使得莫德尔方程 y^2 = x^3 + k 至少有 n 个整数解, 且 y ≥ 0。

应该有这样一串数——1, 1, 1, 1, 8, 9, 17, 17, 17, 225, 225, 225, 225, 225, 1025, 1025, 1025, 54225, 54225, 54225, 54225, 54225, 54225, 54225, 54225, 3470400, 3470400, 3470400, 3470400, 3470400, 13186225,

northwolves

$a(51) = 4979585600$:

{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}}}

northwolves

{77806025, 41,{,{-425, 1020},{-409, -3064},{-374, -5049},{-340, 6205},{-220, -8195},{-200, 8355},{-110, 8745},{-106, -8753},{34, -8823},{170, 9095},{220, -9405},{355, -11070},{374, -11407},{484, 13827},{544, 15453},{680, -19805},{880, -27555},{935, -29920},{1070, 36095},{1496, 58531},{1600, -64605},{1870, -81345},{2036, -92291},{2915, 157630},{4486, -300591},{4675, -319770},{7040, -590755},{7990, 714255},{9350, 904145},{11356, 1210179},{13904, 1639517},{14960, 1829795},{24046, -3728769},{33430, 6112305},{36410, -6947545},{110275, -36619770},{136510, 50436705},{237116, -115462589},{1090210, -1138322295},{1398760, 1654302045},{2345575, 3592311480}}}

{5521665600, 42,{{-1760, -8360},{-1716, -21648},{-1584, 39336},{-1320, 56760},{-1055, 65935},{-975, -67785},{-600, 72840},{-480, 73560},{-264, -74184},{144, -74328},{460, 74960},{540, -75360},{880, 78760},{960, -80040},{1116, -83136},{1320, 88440},{1969, -114697},{2200, -127160},{3025, -182215},{3960, 260040},{6336, -509784},{7260, -623040},{9244, 891872},{12144, -1340328},{15180, 1871760},{16960, 2209960},{21256, 3099896},{24984, -3949752},{34320, 6358440},{42240, -8681640},{56505, 13431885},{76785, -21277335},{94680, 29133240},{132616, 48294136},{135400, 49822840},{142200, -53622840},{216744, -100906872},{468600, 320777160},{1092040, -1141189640},{1489480, -1817824760},{3529641, 6631256139},{3938385, 7815868665}}}

{2301710400, 43,{{-1320, -1320},{-1184, -25336},{-1080, -32280},{-1056, 33528},{-879, 40281},{-660, -44880},{-495, 46695},{-440, 47080},{-264, -47784},{-24, -47976},{240, 48120},{720, -51720},{984, -57048},{1200, -63480},{1276, 66176},{1644, 82128},{1705, 85195},{1980, -100320},{2640, 143880},{3520, 214280},{4464, -302088},{4785, -334455},{4840, -340120},{6336, 506616},{6880, 572680},{16896, -2196744},{17260, -2268080},{19041, 2627889},{19800, -2786520},{22000, 3263480},{40920, 8277720},{42304, 8701192},{55680, 13138680},{132280, -48110680},{159489, 63693663},{168040, 68884120},{175824, -73725432},{291225, 157160355},{309180, -171916320},{2907960, -4958869080},{3923040, 7770234120},{7520700, -20624688480},{150420600, -1844849636520}}}

{7534137600, 44,{{-1920, -21360},{-1815, -39435},{-1584, -59664},{-1440, 67440},{-1100, -78760},{-1056, 79728},{-684, -84936},{-480, 86160},{-176, -86768},{225, -86865},{256, 86896},{660, 88440},{880, -90640},{1200, -96240},{1296, 98544},{1936, 121616},{2640, -161040},{3520, -226160},{4084, 275048},{4425, -306885},{5280, -393360},{5940, 465960},{7920, 710160},{9465, -924915},{10000, -1003760},{12576, 1412976},{16500, 2121240},{16665, -2153085},{16929, -2204367},{27520, -4566160},{36180, 6882360},{37585, -7287065},{72336, -19455216},{84480, -24554640},{115104, 39051408},{161524, -64916632},{187440, 81150960},{338976, -197357424},{661056, 537473904},{1683984, -2185279248},{4114345, -8345474965},{12976560, -46745452560},{101633460, 1024601686440},{671035860, 17382754399560}}}

{4953259025, 45,{{-1700, -6345},{-1690, 11245},{-1430, 45045},{-1265, 54120},{-1106, -60003},{-910, 64805},{-689, -68016},{-286, -70213},{130, 70395},{220, -70455},{416, -70889},{715, -72930},{884, 75127},{1166, 80861},{1210, 82005},{2200, 124905},{2431, 138996},{2795, 163670},{3080, 184855},{3740, 239305},{3784, -243177},{5720, -438295},{6110, 482755},{6776, 562199},{8086, -730509},{11179, -1184058},{17680, 2351895},{20020, -2833545},{22286, -3327709},{29120, -4969705},{36235, -6897870},{40271, -8081744},{59180, 14396855},{77350, 21512595},{96044, -29765047},{114400, -38693655},{118624, 40856343},{230230, 110469645},{243490, 120149445},{362800, 218524905},{363896, -219515881},{2448875, 3832215530},{3097471, -5451435444},{4872956, 10756939721},{7350655, 19929163020}}}

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}}}