解方程：(x + y + z)(x^3 + y^3 + z^3) = n^6
猜想：$(x + y + z) (x^3 + y^3 + z^3) =n^6$对于所有正整数n都有整数解。比如：
{{1,0,0},1}
{{0,-2,-2},2}
{{3,3,3},3}
{{9,-8,-9},4}
{{15,-5,-5},5}
{{-51,-5,50},6}
{{-58,15,42},7}
{{-36,4,24},8}
{{-27,0,0},9}
{{-25,-15,-10},10}
{{-186,18,167},11}
{{-34,-30,16},12}
{{91,0,-104},13}
{{-70,0,42},14}
{{-131,23,99},15}
{{-164,24,132},16}
{{-119,51,51},17}
{{-213,50,157},18}
{{-1030,14,1015},19}
{{-408,-28,404},20}
{{-175,49,105},21}
{{-729,1,720},22}
{{},23}
{{-408,-40,400},24}
{{-125,0,0},25}
{{-624,27,589},26}
{{-81,-81,-81},27}
{{-464,120,336},28}
{{-240,70,110},30}
{{-341,93,217},31}
{{-1544,11,1525},32}
{{-781,-11,759},33}
{{-2079,7,2064},34}
{{-441,28,364},35}
{{-608,-4,564},36}
{{},37}
{{-608,-38,570},38}
{{-10283,-4927,10647},39}
{{-540,124,376},40}
{{},41}
{{-546,245,259},42}
{{},43}
{{-1488,144,1336},44}
{{-756,252,477},45}
{{},46}
{{-6627,6533,2303},47}
{{-272,-240,128},48}
{{0,0,-343},49}
{{-250,-250,0},50}
{{-1071,-119,1037},51}
{{-832,0,728},52}
{{},53}
{{},54}
{{-6050,5665,3410},55}
{{5292,-5824,3668},56}
{{},57}
{{},58}
{{},59}
{{39590,-39600,3610},60}
{{},61}
{{},62}
{{-1566,405,1134},63}
{{0,0,-512},64}
{{-7215,5915,5525},65}
{{-1035,-80,899},66}
{{-1206,469,670},67}
{{-952,408,408},68}
{{-1357,322,966},69}
{{-3630,-70,3600},70} 注意到$\{x,y,z,t\}=\{-6 k-\frac{6}{k}-3,6 k+\frac{6}{k+1}+3,\frac{6 k}{k+1}+\frac{6}{k}+3,3\}$ 是一组解
令$k=\{1,2}$,解得： {-15, 12, 12, 3}, {-18, 17, 10, 3} 对于给定的n,我们可以知道\(x+y+z|n^6\),
于是我们可以枚举\(n^6\)的因子d,让\(x+y+z=d\),
然后得到一个关于x,y的三次曲线\(-3x^2y-3xy^2 + 3dx^2 + 6dxy + 3dy^2 - 3d^2x- 3d^2y =\frac{n^6}d-d^3\).
可以利用椭圆曲线求解工具求其所有整数点。 由于\(3(x+y)|\frac{n^6}d-d^3\), 可以继续通过因子分解来做，结果比想象的简单，而n=37无解。
附件为100以内所有解
先提交一个无解序列
https://oeis.org/draft/A391526
mathe 发表于 2025-12-12 15:59
对于给定的n,我们可以知道\(x+y+z|n^6\),
于是我们可以枚举\(n^6\)的因子d,让\(x+y+z=d\),
然后得到一个关 ...
根据$x + y + z = d, x^3 + y^3 + z^3 = n^6/d$,消去z,得到 $3 (d - x) (d - y) (x + y) =\frac{d^4 - n^6}{d}$, 所以最好的算法可能还是 因子分解, 而不是椭圆曲线. mathe 发表于 2025-12-12 16:58
先提交一个无解序列
https://oeis.org/draft/A391526
oeis数列标记问题来源,fang的链接,
https://www.zhihu.com/question/1969018790831461543 $(n^2)^6 = (0 + 0 + n^3) (0 + 0 + (n^3)^3)$
$(2 n^2)^6 = (0 + 2 n^3 + 2 n^3) (0 + (2 n^3)^3 + (2 n^3)^3)$
$(n^3)^6 = (0 + 0 + n^2) (0 + 0 + (n^2)^3)$ 一千以内所有无解的n如下
23
29
37
41
43
46
53
57
58
59
61
71
73
78
79
82
83
87
89
95
101
103
113
122
131
137
139
142
143
146
148
151
157
163
164
166
167
172
173
177
179
187
191
193
197
199
213
215
217
218
221
222
223
227
232
233
237
241
249
253
254
257
258
263
265
269
271
277
278
281
283
292
293
298
301
302
307
309
310
311
313
314
317
327
331
332
333
335
341
347
349
353
356
358
359
362
367
371
373
374
382
383
386
389
397
398
401
409
411
419
421
422
431
439
443
447
451
452
453
457
458
461
463
467
479
481
487
488
493
498
499
502
503
505
509
521
523
527
535
538
541
545
547
548
551
553
554
556
557
559
562
563
568
569
577
586
587
590
591
593
597
598
599
601
604
607
613
614
619
623
626
629
631
633
635
641
643
647
649
653
656
658
659
661
673
674
677
683
687
689
691
694
695
697
701
707
709
713
717
718
719
723
727
733
734
739
743
747
751
754
757
758
761
762
763
766
767
773
778
779
783
785
787
789
790
791
794
797
799
802
809
814
815
817
821
823
826
827
829
834
838
839
842
843
851
853
857
862
865
866
877
886
887
890
893
894
895
898
905
907
911
913
917
919
922
923
926
928
929
932
937
941
947
948
949
951
953
958
959
964
965
967
970
971
973
974
977
982
983
989
991
993
995
996
997
998
10100-10199之间无解的n达到51个，超过了半数，所以大概率后面无解的n比例会越来越高
10102
10103
10105
10106
10107
10109
10111
10117
10118
10119
10122
10123
10127
10129
10133
10137
10138
10139
10141
10145
10147
10148
10149
10151
10154
10157
10159
10161
10162
10163
10165
10167
10168
10171
10172
10173
10174
10177
10178
10181
10182
10183
10186
10187
10189
10190
10193
10195
10196
10198
10199
页:
[1]
2