论坛 › 刨根究底 › a^3+b^3+c^3=d^3整数解接龙

nyy 发表于 2024-6-13 10:29:18

a^3+b^3+c^3=d^3整数解接龙

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。

mathe 发表于 2024-6-13 12:47:55

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

lihpb00 发表于 2024-6-13 13:27:54

我记得有公式

nyy 发表于 2024-6-13 13:37:07

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 发表于 2024-6-13 13:40:13

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 发表于 2024-6-13 14:01:21

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 发表于 2024-6-13 14:26:47

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 发表于 2024-6-17 09:59:36

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？

wayne 发表于 2024-6-17 11:21:03

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}

nyy 发表于 2024-6-17 12:43:00

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
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}}

也许这个代码有用

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