Mordell's equation y^2 = x^3 ± k^2

northwolves

$ y^2 = x^3 ± k^2$ 与  $y^2 = x^3 ± k$ 相比，计算$n$个解的最小$k$值有什么优化策略 ？


A392548  a(n) is the least positive integer k such that Mordell's equation $y^2 = x^3 - k^2$ has exactly n integer solutions with y >= 0.

DATA
3, 1, 2, 10, 80, 530, 26, 1160, 208, 1664, 730, 1090, 5840, 8720, 46720, 69760, 214370

OFFSET
0,1

COMMENTS
Conjecture： a(n) is even for all n>1.

a(30) = 293410.

LINKS
Xiaoyang Zhang, <a href="https://www.zhihu.com/question/1993841306041071339">The Mordell's equation y² = x³ - k² (k is an integer and 0 < k < 1000).When k takes what value, does it have the most sets of positive integer solutions?(in Chinese)</a>

EXAMPLE
a(6) = 26 because 26 is the least k such that y^2 = x^3 - k^2 has 6 integral solutions with nonnegative y: {{10,18},{13,39},{26,130},{130,1482},{338,6214},{901,27045}}.


A392549  a(n) is the least positive integer k such that Mordell's equation $y^2 = x^3 + k^2$ has exactly n integer solutions with y >= 0.

DATA
2, 11, 1, 6, 3, 10, 80, 62, 63, 210, 55, 840, 15, 440, 120, 960, 3240, 561, 2415, 510, 665, 19320, 1155, 5320

OFFSET
1,1

COMMENTS
a(25) > 80000 if it exists. a(26) = 31185, a(27) = 9240, a(29) = 73920, a(33) = 54285.

EXAMPLE
a(6) = 10 because 10 is the least k such that y^2 = x^3 + k^2 has 6 integral solutions with nonnegative y: {{-4, 6},{0, 10},{5, 15},{20, 90},{24, 118},{2660, 137190}}.

northwolves

$y^2 = x^3 - k^2$找到一组38个解的：{2347280, 38,{{18056, 613904},{18356, 821704},{18824, 1077232},{19240, 1269840},{19825, 1510665},{21460, 2091240},{22984, 2575248},{31720, 5138640},{41480, 8115440},{45140, 9298840},{46969, 9904947},{70760, 18675760},{96200, 29745040},{117364, 40138488},{127465, 45447285},{148840, 57374160},{164761, 66836541},{176305, 73990935},{181780, 77467560},{260936, 133270384},{269620, 139980360},{497416, 350808336},{724985, 617291635},{729076, 622524024},{1012505, 1018813315},{1404520, 1664529360},{1770964, 2356754712},{2251496, 3378365744},{2493985, 3938589135},{4811924, 10555483432},{5560360, 13111559760},{9331400, 28504945040},{10689640, 34949793840},{22551944, 107096672272},{31794100, 179275029960},{44437000, 296221854960},{71546900, 605182320040},{989578600, 31129736345040}}}

northwolves

$y^2 = x^3 - k^2$ 最小的30个解的$k=293410$:

{293410, 30,{{4514, 76738},{4589, 102713},{4706, 134654},{4810, 158730},{5365, 261405},{5746, 321906},{7930, 642330},{10370, 1014430},{11285, 1162355},{17690, 2334470},{24050, 3718130},{29341, 5017311},{37210, 7171770},{45445, 9683445},{65234, 16658798},{67405, 17497545},{124354, 43851042},{182269, 77815503},{351130, 208066170},{442741, 294594339},{562874, 422295718},{1202981, 1319435429},{1390090, 1638944970},{2332850, 3563118130},{2672410, 4368724230},{5637986, 13387084034},{7948525, 22409378745},{11109250, 37027731870},{17886725, 75647790005},{247394650, 3891217043130}}}

northwolves

$y^2 = x^3 + k^2$ 最小的64个解的$k=451605$,貌似$10^6$以内，$k=451605$ 解最多：

{451605, 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}}}

最小的42个解的$k=794310$
{794310, 42,{{-8300, 243190},{-7875, 377565},{-7656, 426822},{-7304, 491194},{-6960, 542010},{-6380, 609290},{-5940, 649110},{-5451, 684807},{0, 794310},{1885, 798515},{3036, 811734},{3480, 820410},{5229, 879717},{5916, 915414},{7221, 1003719},{9960, 1272390},{11620, 1483210},{15840, 2145990},{16720, 2303290},{23925, 3784935},{26796, 4457706},{34485, 6452985},{38280, 7531590},{43824, 9208518},{54780, 12845910},{64989, 16586637},{77605, 21633535},{211816, 97488314},{276805, 145635535},{406725, 259390065},{673380, 552574410},{712140, 600963990},{1024860, 1037521110},{1432165, 1713916385},{1952280, 2727804090},{2734020, 4520667210},{3953565, 7861100115},{6816624, 17797295418},{8191920, 23446528710},{13506460, 49637774890},{18768336, 81309006534},{25530576, 129000359226}}}

northwolves

$k=780373440$时，方程$y^2 = x^3 + k^2(y>= 0)$恰好有100组整数解

{780373440, 100,{{-847616, 3247552},{-844560, 81077760},{-834624, 166090176},{-807840, 285975360},{-788256, 345256128},{-785664, 352161216},{-782460, 360453240},{-717255, 489885165},{-685440, 535671360},{-665280, 560831040},{-619344, 609434496},{-567936, 652529088},{-566720, 653428160},{-514080, 687839040},{-391391, 740963377},{-332640, 756423360},{-318780, 759333960},{-208656, 774531072},{-61200, 780226560},{-48384, 780300864},{0, 780373440},{126225, 781660935},{154980, 782754840},{180576, 784137024},{209440, 786237760},{258336, 791342784},{273700, 793401560},{323136, 801700416},{337824, 804696768},{388080, 816963840},{546480, 878739840},{554400, 882826560},{616896, 918558144},{692640, 970194240},{695520, 972336960},{774180, 1035852840},{812889, 1070574813},{881280, 1137291840},{989604, 1256231592},{1000960, 1269592640},{1092960, 1383687360},{1182384, 1503992448},{1343664, 1742090112},{1376320, 1793344960},{1530144, 2047332672},{1709316, 2367107064},{1884960, 2703032640},{1988064, 2909742912},{2120580, 3185111160},{2550240, 4146690240},{2827440, 4817957760},{2908224, 5020564032},{3193344, 5759595072},{3245760, 5899400640},{3853696, 7605269056},{4371840, 9174306240},{4729536, 10315118016},{4823665, 10622842615},{4903600, 10886573440},{5190304, 11850396992},{5439940, 12711917240},{5654880, 13469924160},{6933465, 18273505635},{6967620, 18408452040},{8445600, 24556426560},{8670816, 25544223936},{10758825, 35298270315},{10864260, 35818164360},{12311145, 43203459915},{15500160, 61029495360},{16553376, 67353322176},{16679520, 68124576960},{21677040, 100928298240},{23647680, 114998667840},{32788800, 187755226560},{33352704, 192619449792},{36756720, 222847511040},{52179120, 376917287040},{55740960, 416162007360},{58968000, 452819626560},{64208320, 514502464960},{74884320, 648017375040},{101572416, 1023679014336},{103505920, 1053047360960},{123239664, 1368125157888},{153014400, 1892772826560},{171567396, 2247254018856},{202833136, 2888739459584},{402570081, 8077226185071},{603791584, 14836470026048},{620494560, 15456350871360},{650311200, 16583715973440},{829651680, 23897017172160},{1040497920, 33563081280960},{2959622820, 161010630590040},{11960698464, 1308081527172288},{14826581280, 1805350514708160},{49469555520, 11002896111936960},{454935371940, 306848890870862760},{1153429879140, 1238758924037259240}}}