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# [求助] a^3+b^3+c^3=d^3整数解接龙

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(1,6,8,9)是一组整数解，

 这个链接 给出了 一类特解的 公式。 $(3 x^2+5 x y-5 y^2)^3+(4 x^2-4 x y+6 y^2)^3+(5 x^2-5 x y-3 y^2)^3=(2 (3 x^2 - 2 x y + 2 y^2))^3$ https://mathworld.wolfram.com/DiophantineEquation3rdPowers.html {3,4,5,6} {7,14,17,20} {18,19,21,28} {3,36,37,46} {27,30,37,46} {15,42,49,58} {7,54,57,70} {19,60,69,82} {58,59,69,90} {15,82,89,108} {23,86,97,116} {19,92,101,122} {23,94,105,126} {86,95,97,134} {1,135,138,172} {115,122,149,188} {85,138,171,202} {5,163,164,206} {140,151,161,218} {73,174,207,244} {113,166,207,246} {163,164,197,254} {101,178,219,258} {81,202,239,282} {179,188,229,290} {173,214,267,324} {107,230,277,326} {185,218,271,332} {227,230,277,356} {119,268,321,378} {177,276,343,406} {171,282,349,412} {191,290,361,428} {276,303,313,430} {283,294,357,454} {19,362,365,458} {191,332,409,482} {281,322,399,492} {315,322,389,498} {27,406,413,516} {323,334,405,516} {193,366,447,526} {339,348,421,538} {185,428,511,602} {259,410,509,602} {167,436,513,606} {331,402,501,610} {73,470,503,614} {337,404,503,614} {21,490,491,618} {412,459,461,642} {245,446,547,644} {81,498,535,652} {179,482,565,668} {413,446,547,686} {149,508,579,690} {391,458,569,698} {153,534,607,724} {345,486,607,724} {291,540,661,778} {9,631,642,802} {487,534,657,820} {365,562,699,828} {501,538,659,828} {233,596,703,830} {347,580,717,846} {445,566,707,854} {17,687,694,870} {339,610,749,882} {149,668,739,890} {153,670,743,894} {574,623,657,894} {511,588,729,898} {367,630,777,916} {596,617,703,926} {519,618,769,940} {107,766,813,996} {599,652,801,1002} {1,791,812,1010} {331,738,885,1042} {670,743,753,1044} {85,818,851,1052} {119,812,865,1058} {437,734,907,1070} {698,701,835,1082} {489,738,919,1090} {653,734,907,1124} {333,804,955,1126} {453,774,955,1126} {529,762,951,1132} {732,765,859,1138} {101,890,931,1148} {259,842,965,1148} {756,777,895,1174} {349,860,1019,1202} {267,886,1013,1206} {778,851,885,1212} {665,802,999,1218} {804,805,963,1246} {317,934,1083,1284} {167,980,1057,1286} {745,846,1047,1294} {531,898,1109,1308} {37,1043,1046,1316} {535,908,1121,1322} {179,1018,1101,1338} {59,1058,1069,1340} {333,1006,1163,1380} {886,963,1013,1380} {631,938,1169,1388} {529,972,1191,1402} {291,1042,1181,1410} {749,938,1171,1418} {437,1030,1227,1446} {295,1076,1217,1454} {703,974,1217,1454} {940,951,1121,1458} {827,964,1197,1470} {505,1044,1263,1486} {978,1041,1135,1522} 复制代码

 3^3+4^3+5^3=6^3 1^3+6^3+8^3=9^3 6^3+8^3+10^3=12^3 3^3+10^3+18^3=19^3 2^3+12^3+16^3=18^3 9^3+12^3+15^3=18^3 7^3+14^3+17^3=20^3 11^3+15^3+27^3=29^3 12^3+16^3+20^3=24^3 2^3+17^3+40^3=41^3 4^3+17^3+22^3=25^3 3^3+18^3+24^3=27^3 12^3+19^3+53^3=54^3 18^3+19^3+21^3=28^3 6^3+20^3+36^3=38^3 15^3+20^3+25^3=30^3 14^3+23^3+70^3=71^3 16^3+23^3+41^3=44^3 4^3+24^3+32^3=36^3 18^3+24^3+30^3=36^3 14^3+28^3+34^3=40^3 21^3+28^3+35^3=42^3 5^3+30^3+40^3=45^3 9^3+30^3+54^3=57^3 22^3+30^3+54^3=58^3 27^3+30^3+37^3=46^3 12^3+31^3+102^3=103^3 25^3+31^3+86^3=88^3 6^3+32^3+33^3=41^3 24^3+32^3+40^3=48^3 31^3+33^3+72^3=76^3 3^3+34^3+114^3=115^3 4^3+34^3+80^3=82^3 8^3+34^3+44^3=50^3 29^3+34^3+44^3=53^3 3^3+36^3+37^3=46^3 6^3+36^3+48^3=54^3 27^3+36^3+45^3=54^3 24^3+38^3+106^3=108^3 25^3+38^3+87^3=90^3 36^3+38^3+42^3=56^3 36^3+38^3+61^3=69^3 34^3+39^3+65^3=72^3 12^3+40^3+72^3=76^3 17^3+40^3+86^3=89^3 30^3+40^3+50^3=60^3 7^3+42^3+56^3=63^3 15^3+42^3+49^3=58^3 21^3+42^3+51^3=60^3 21^3+43^3+84^3=88^3 38^3+43^3+66^3=75^3 33^3+44^3+55^3=66^3 33^3+45^3+81^3=87^3 21^3+46^3+188^3=189^3 27^3+46^3+197^3=198^3 28^3+46^3+140^3=142^3 32^3+46^3+82^3=88^3 16^3+47^3+108^3=111^3 46^3+47^3+148^3=151^3 8^3+48^3+64^3=72^3 25^3+48^3+74^3=81^3 36^3+48^3+60^3=72^3 38^3+48^3+79^3=87^3 15^3+50^3+90^3=95^3 6^3+51^3+120^3=123^3 12^3+51^3+66^3=75^3 13^3+51^3+104^3=108^3 16^3+51^3+213^3=214^3 22^3+51^3+54^3=67^3 44^3+51^3+118^3=123^3 39^3+52^3+65^3=78^3 19^3+53^3+90^3=96^3 28^3+53^3+75^3=84^3 45^3+53^3+199^3=201^3 7^3+54^3+57^3=70^3 9^3+54^3+72^3=81^3 20^3+54^3+79^3=87^3 32^3+54^3+85^3=93^3 9^3+55^3+116^3=120^3 26^3+55^3+78^3=87^3 28^3+56^3+68^3=80^3 42^3+56^3+70^3=84^3 4^3+57^3+248^3=249^3 17^3+57^3+177^3=179^3 22^3+57^3+255^3=256^3 36^3+57^3+159^3=162^3 1000以内2391组

楼主| 发表于 2024-6-18 13:29:08 | 显示全部楼层
 本帖最后由 nyy 于 2024-6-18 13:30 编辑 王守恩 发表于 2024-6-18 13:09 这个可以接龙(虽然有点稀)！详见《[讨论] (k1)^3+(k2)^3+(k3)^3=n^3》  谢谢 葡萄糖！ \(n^3 + (3 n^2 + 2 ... Clear["Global`*"]; aa={} n=1; Do[ee={n,(3*n^2+2*n+1),(3*n^3+3*n^2+2*n),(3*n^3+3*n^2+2*n+1)};    aa=Append[aa,ee];    n=(3*n^3+3*n^2+2*n+1)(*得到新的n*) ,{k,1,5}] 复制代码 按照你的代码，得到 { {1,6,8,9}, {9,262,2448,2449}, {2449,17997702,44082372248,44082372249}, {44082372249,5829766629386380698502,256989942683351711945337288361248,256989942683351711945337288361249}, {256989942683351711945337288361249,198131491921177194311506308094238133848780474484255622782351242502,50917800752590289932456936228454119182229490064387456921176845707741670058552030235961991791806248,50917800752590289932456936228454119182229490064387456921176845707741670058552030235961991791806249} } 复制代码 你给出的恒等式，确实成立 n^3 + (3 n^2 + 2 n + 1)^3 + (3 n^3 + 3 n^2 + 2 n)^3 - (3 n^3 + 3 n^2 + 2 n + 1)^3 // Simplify复制代码

 我记得有公式

楼主| 发表于 2024-6-13 13:37:07 | 显示全部楼层
 mathe 发表于 2024-6-13 12:47 3^3+4^3+5^3=6^3 1^3+6^3+8^3=9^3 6^3+8^3+10^3=12^3 我的意思是接龙，比如 (1,6,8,9)，最后一个是9， 然后可以找 9^3+55^3+116^3=120^3这组解接上。这组解的第一个是9， 然后后面是120，然后如何接上？ 参见： 本原勾股数接龙 https://bbs.emath.ac.cn/thread-19217-1-1.html (出处: 数学研发论坛)

楼主| 发表于 2024-6-13 13:40:13 | 显示全部楼层
 mathe 发表于 2024-6-13 12:47 3^3+4^3+5^3=6^3 1^3+6^3+8^3=9^3 6^3+8^3+10^3=12^3 比如根据你的计算结果 1^3+6^3+8^3=9^3 9^3+12^3+15^3=18^3 18^3+19^3+21^3=28^3 28^3+46^3+140^3=142^3 这个是首尾接龙，明白了吗？

### 点评

 互素的可以有 1,6,8,9 9,55,116,120 120,207,622,631 631,938,1169,1388 1388,1511,1585,2162 2162,2269,2531,3362 3362,3853,4123,5488 5488,5859,10566,11563 11563,14893,19316,22930 22930,28158,39825,46033

### 点评

楼主| 发表于 2024-6-13 14:26:47 | 显示全部楼层
 mathe 发表于 2024-6-13 14:01 互素的可以有 1,6,8,9 9,55,116,120 机器怎么算的呢？代码呢？

### 点评

楼主| 发表于 2024-6-17 09:59:36 | 显示全部楼层
 mathe 发表于 2024-6-13 14:01 互素的可以有 1,6,8,9 9,55,116,120 用机器怎么算出来的？

楼主| 发表于 2024-6-17 12:43:00 | 显示全部楼层
 wayne 发表于 2024-6-17 11:21 这个链接 给出了 一类特解的 公式。 \$(3 x^2+5 x y-5 y^2)^3+(4 x^2-4 x y+6 y^2)^3+(5 x^2-5 x y-3 y^2)^3 ... 能搞一个接龙出来吗？ PowersRepresentations[216, 7, 3]复制代码 这个是把216分成7部分，每个部分都是整数的三次方， 求解结果 {{0, 0, 0, 0, 0, 0, 6}, {0, 0, 0, 0, 3, 4, 5}, {0, 2, 2, 2, 4, 4,  4}, {1, 1, 2, 3, 3, 3, 5}}复制代码 也许这个代码有用

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