a^3+b^3+c^3=d^3整数解接龙

nyy

假设a<b<c<d
(1,6,8,9)是一组整数解，
那么以9开头的整数解呢？
排除9*(1,6,8,9)这组解。

wayne

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

1. {3,4,5,6}
   
2. {7,14,17,20}
   
3. {18,19,21,28}
   
4. {3,36,37,46}
   
5. {27,30,37,46}
   
6. {15,42,49,58}
   
7. {7,54,57,70}
   
8. {19,60,69,82}
   
9. {58,59,69,90}
   
10. {15,82,89,108}
    
11. {23,86,97,116}
    
12. {19,92,101,122}
    
13. {23,94,105,126}
    
14. {86,95,97,134}
    
15. {1,135,138,172}
    
16. {115,122,149,188}
    
17. {85,138,171,202}
    
18. {5,163,164,206}
    
19. {140,151,161,218}
    
20. {73,174,207,244}
    
21. {113,166,207,246}
    
22. {163,164,197,254}
    
23. {101,178,219,258}
    
24. {81,202,239,282}
    
25. {179,188,229,290}
    
26. {173,214,267,324}
    
27. {107,230,277,326}
    
28. {185,218,271,332}
    
29. {227,230,277,356}
    
30. {119,268,321,378}
    
31. {177,276,343,406}
    
32. {171,282,349,412}
    
33. {191,290,361,428}
    
34. {276,303,313,430}
    
35. {283,294,357,454}
    
36. {19,362,365,458}
    
37. {191,332,409,482}
    
38. {281,322,399,492}
    
39. {315,322,389,498}
    
40. {27,406,413,516}
    
41. {323,334,405,516}
    
42. {193,366,447,526}
    
43. {339,348,421,538}
    
44. {185,428,511,602}
    
45. {259,410,509,602}
    
46. {167,436,513,606}
    
47. {331,402,501,610}
    
48. {73,470,503,614}
    
49. {337,404,503,614}
    
50. {21,490,491,618}
    
51. {412,459,461,642}
    
52. {245,446,547,644}
    
53. {81,498,535,652}
    
54. {179,482,565,668}
    
55. {413,446,547,686}
    
56. {149,508,579,690}
    
57. {391,458,569,698}
    
58. {153,534,607,724}
    
59. {345,486,607,724}
    
60. {291,540,661,778}
    
61. {9,631,642,802}
    
62. {487,534,657,820}
    
63. {365,562,699,828}
    
64. {501,538,659,828}
    
65. {233,596,703,830}
    
66. {347,580,717,846}
    
67. {445,566,707,854}
    
68. {17,687,694,870}
    
69. {339,610,749,882}
    
70. {149,668,739,890}
    
71. {153,670,743,894}
    
72. {574,623,657,894}
    
73. {511,588,729,898}
    
74. {367,630,777,916}
    
75. {596,617,703,926}
    
76. {519,618,769,940}
    
77. {107,766,813,996}
    
78. {599,652,801,1002}
    
79. {1,791,812,1010}
    
80. {331,738,885,1042}
    
81. {670,743,753,1044}
    
82. {85,818,851,1052}
    
83. {119,812,865,1058}
    
84. {437,734,907,1070}
    
85. {698,701,835,1082}
    
86. {489,738,919,1090}
    
87. {653,734,907,1124}
    
88. {333,804,955,1126}
    
89. {453,774,955,1126}
    
90. {529,762,951,1132}
    
91. {732,765,859,1138}
    
92. {101,890,931,1148}
    
93. {259,842,965,1148}
    
94. {756,777,895,1174}
    
95. {349,860,1019,1202}
    
96. {267,886,1013,1206}
    
97. {778,851,885,1212}
    
98. {665,802,999,1218}
    
99. {804,805,963,1246}
    
100. {317,934,1083,1284}
     
101. {167,980,1057,1286}
     
102. {745,846,1047,1294}
     
103. {531,898,1109,1308}
     
104. {37,1043,1046,1316}
     
105. {535,908,1121,1322}
     
106. {179,1018,1101,1338}
     
107. {59,1058,1069,1340}
     
108. {333,1006,1163,1380}
     
109. {886,963,1013,1380}
     
110. {631,938,1169,1388}
     
111. {529,972,1191,1402}
     
112. {291,1042,1181,1410}
     
113. {749,938,1171,1418}
     
114. {437,1030,1227,1446}
     
115. {295,1076,1217,1454}
     
116. {703,974,1217,1454}
     
117. {940,951,1121,1458}
     
118. {827,964,1197,1470}
     
119. {505,1044,1263,1486}
     
120. {978,1041,1135,1522}
     
121. 
     

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mathe

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

nyy

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 ...

1. Clear["Global`*"];
   
2. aa={}
   
3. n=1;
   
4. 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)};
   
5.    aa=Append[aa,ee];
   
6.    n=(3*n^3+3*n^2+2*n+1)(*得到新的n*)
   
7. ,{k,1,5}]
   

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按照你的代码，得到

1. {
   
2. {1,6,8,9},
   
3. {9,262,2448,2449},
   
4. {2449,17997702,44082372248,44082372249},
   
5. {44082372249,5829766629386380698502,256989942683351711945337288361248,256989942683351711945337288361249},
   
6. {256989942683351711945337288361249,198131491921177194311506308094238133848780474484255622782351242502,50917800752590289932456936228454119182229490064387456921176845707741670058552030235961991791806248,50917800752590289932456936228454119182229490064387456921176845707741670058552030235961991791806249}
   
7. }
   

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你给出的恒等式，确实成立

1. 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

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lihpb00

我记得有公式

nyy

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)

nyy

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？

mathe

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

nyy

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？

nyy

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？

nyy

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？

1. PowersRepresentations[216, 7, 3]

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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

1. {{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}}

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也许这个代码有用