Mordell's equation y^2 = x^3 + k
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.DATA
6, 2, 12, 1, 8, 9, 73, 316, 17, 297, 2817, 1737, 4481, 225, 2089, 14400, 1025, 197225, 65600, 92025, 260100, 442225
COMMENTS
a(22) <= 4215025, a(23) = 885025, a(24) = 54225.
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).
对于$a(22)$,目前计算情况：$y^2 = x^3 + 4215025，y>=0$ 恰好有22组整数解：
{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}}}
Q1: a(22)=?
Q2: a(25)=?
有时候整点$x$会特别大，比如：$a(17)=197225$
{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}}} $a(21)=442225$:
{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}}} $a(29)<=3470400$
{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}}} $a(28)<=5472225$
{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} 这个需要把所有较小的k的方程的整数解全部求出，计算量比较大 mathe 发表于 2026-1-4 13:32
这个需要把所有较小的k的方程的整数解全部求出，计算量比较大
似乎这些解多的数字都有3^2, 5^2,7^2的质因子：
197225 {{5,2},{7,3},{23,1}}
65600 {{2,6},{5,2},{41,1}}
92025 {{3,2},{5,2},{409,1}}
260100 {{2,2},{3,2},{5,2},{17,2}}
442225 {{5,2},{7,2},{19,2}}
4215025 {{5,2},{168601,1}}
885025 {{5,2},{35401,1}}
54225 {{3,2},{5,2},{241,1}}
5472225 {{3,3},{5,2},{11,2},{67,1}}
3470400 {{2,6},{3,2},{5,2},{241,1}}
northwolves 发表于 2026-1-4 00:18
$a(28)
我更好奇如何求解？ a(22)=4215025,a(28)=5472225,a(29)=3470400 已确认。a(25),a(26),a(27)尚未找到 northwolves 发表于 2026-1-4 23:30
a(22)=4215025,a(28)=5472225,a(29)=3470400 已确认。a(25),a(26),a(27)尚未找到
{22548673, 25,{{-273, -1484},{-208, 3681},{-174, -4157},{-154, -4347},{-133, 4494},{56, 4767},{102, 4859},{252, -6209},{336, -7777},{506, -12333},{578, -14685},{882, 26621},{1176, 40607},{2478, 123445},{2751, -144368},{2828, 150465},{4571, 309078},{4862, 339051},{12222, -1351189},{30296, 5273247},{44828, -9491265},{177528, -74799775},{957132, 936392129},{9438212, -28995766401},{14655662, -56105851851}}}
{13221225,26,{{-234,639},{-221,-1558},{-216,1773},{-200,2285},{-80,-3565},{-20,-3635},{-9,-3636},{90,3735},{135,-3960},{180,4365},{306,-6471},{450,10215},{604,15283},{675,-17910},{1170,40185},{1440,-54765},{2286,-109359},{3280,-187885},{4419,-293778},{6910,574415},{7830,692865},{14500,1746035},{18406,-2497121},{29016,4942611},{33910,-6244415},{436950,-288833535}}}
{23882257, 27,{{-286, 699},{-252, 2807},{-241, -3144},{-168, -4375},{-21, -4886},{8, 4887},{128, 5097},{282, 6805},{294, 7021},{308, -7287},{354, 8261},{639, 16876},{1064, -35049},{1082, 35925},{1764, -74249},{2303, 110628},{4158, 268163},{4382, -290115},{4907, -343770},{11018, -1156533},{32124, -5757641},{113582, -38279325},{293174, 158740659},{658424, 534267159},{1043882, 1066539915},{15545103, -61290110728},{440173877759, 292036004827951356}}}