A^n + B^n + C^n = D^n + E^n + F^n$, n = 2,4

 
 
Here are some solutions to

A^{n} + B^{n} + C^{n} \; = \; D^{n} + E^{n} + F^{n},     n \; = \; 2, \; 4

 
Find a general solution
 

1 + 9 + 10 = 205 + 6 + 11 = 22
1^2 + 9^2 + 10^2 = 5^2 + 6^2 + 11^2 = 182
1^4 + 9^4 + 10^4 = 5^4 + 6^4 + 11^4 = 16562

1 + 11 + 12 = 244 + 9 + 13 = 26
1^2 + 11^2 + 12^2 = 4^2 + 9^2 + 13^2 = 266
1^4 + 11^4 + 12^4 = 4^4 + 9^4 + 13^4 = 35378

3 + 13 + 16 = 328 + 9 + 17 = 34
3^2 + 13^2 + 16^2 = 8^2 + 9^2 + 17^2 = 434
3^4 + 13^4 + 16^4 = 8^4 + 9^4 + 17^4 = 94178

3 + 14 + 17 = 347 + 11 + 18 = 36
3^2 + 14^2 + 17^2 = 7^2 + 11^2 + 18^2 = 494
3^4 + 14^4 + 17^4 = 7^4 + 11^4 + 18^4 = 122018

2 + 15 + 17 = 345 + 13 + 18 = 36
2^2 + 15^2 + 17^2 = 5^2 + 13^2 + 18^2 = 518
2^4 + 15^4 + 17^4 = 5^4 + 13^4 + 18^4 = 134162

1 + 16 + 17 = 348 + 11 + 19 = 38
1^2 + 16^2 + 17^2 = 8^2 + 11^2 + 19^2 = 546
1^4 + 16^4 + 17^4 = 8^4 + 11^4 + 19^4 = 149058

4 + 15 + 19 = 389 + 11 + 20 = 40
4^2 + 15^2 + 19^2 = 9^2 + 11^2 + 20^2 = 602
4^4 + 15^4 + 19^4 = 9^4 + 11^4 + 20^4 = 181202

1 + 18 + 19 = 387 + 14 + 21 = 42
1^2 + 18^2 + 19^2 = 7^2 + 14^2 + 21^2 = 686
1^4 + 18^4 + 19^4 = 7^4 + 14^4 + 21^4 = 235298

2 + 18 + 20 = 4010 + 12 + 22 = 44
2^2 + 18^2 + 20^2 = 10^2 + 12^2 + 22^2 = 728
2^4 + 18^4 + 20^4 = 10^4 + 12^4 + 22^4 = 264992

5 + 17 + 22 = 4410 + 13 + 23 = 46
5^2 + 17^2 + 22^2 = 10^2 + 13^2 + 23^2 = 798
5^4 + 17^4 + 22^4 = 10^4 + 13^4 + 23^4 = 318402

2 + 19 + 21 = 429 + 14 + 23 = 46
2^2 + 19^2 + 21^2 = 9^2 + 14^2 + 23^2 = 806
2^4 + 19^4 + 21^4 = 9^4 + 14^4 + 23^4 = 324818

3 + 19 + 22 = 446 + 17 + 23 = 46
3^2 + 19^2 + 22^2 = 6^2 + 17^2 + 23^2 = 854
3^4 + 19^4 + 22^4 = 6^4 + 17^4 + 23^4 = 364658

3 + 20 + 23 = 4612 + 13 + 25 = 50
3^2 + 20^2 + 23^2 = 12^2 + 13^2 + 25^2 = 938
3^4 + 20^4 + 23^4 = 12^4 + 13^4 + 25^4 = 439922

5 + 19 + 24 = 489 + 16 + 25 = 50
5^2 + 19^2 + 24^2 = 9^2 + 16^2 + 25^2 = 962
5^4 + 19^4 + 24^4 = 9^4 + 16^4 + 25^4 = 462722

6 + 19 + 25 = 5011 + 15 + 26 = 52
6^2 + 19^2 + 25^2 = 11^2 + 15^2 + 26^2 = 1022
6^4 + 19^4 + 25^4 = 11^4 + 15^4 + 26^4 = 522242

2 + 22 + 24 = 488 + 18 + 26 = 52
2^2 + 22^2 + 24^2 = 8^2 + 18^2 + 26^2 = 1064
2^4 + 22^4 + 24^4 = 8^4 + 18^4 + 26^4 = 566048

1 + 23 + 24 = 4811 + 16 + 27 = 54
1^2 + 23^2 + 24^2 = 11^2 + 16^2 + 27^2 = 1106
1^4 + 23^4 + 24^4 = 11^4 + 16^4 + 27^4 = 611618

3 + 22 + 25 = 5010 + 17 + 27 = 54
3^2 + 22^2 + 25^2 = 10^2 + 17^2 + 27^2 = 1118
3^4 + 22^4 + 25^4 = 10^4 + 17^4 + 27^4 = 624962

7 + 20 + 27 = 5413 + 15 + 28 = 56
7^2 + 20^2 + 27^2 = 13^2 + 15^2 + 28^2 = 1178
7^4 + 20^4 + 27^4 = 13^4 + 15^4 + 28^4 = 693842

6 + 23 + 25 = 5410 + 19 + 27 = 56
6^2 + 23^2 + 25^2 = 10^2 + 19^2 + 27^2 = 1190
6^4 + 23^4 + 25^4 = 10^4 + 19^4 + 27^4 = 671762

4 + 23 + 27 = 547 + 21 + 28 = 56
4^2 + 23^2 + 27^2 = 7^2 + 21^2 + 28^2 = 1274
4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 811538

4 + 23 + 27 = 5412 + 17 + 29 = 58
4^2 + 23^2 + 27^2 = 12^2 + 17^2 + 29^2 = 1274
4^4 + 23^4 + 27^4 = 12^4 + 17^4 + 29^4 = 811538

7 + 21 + 28 = 5612 + 17 + 29 = 58
7^2 + 21^2 + 28^2 = 12^2 + 17^2 + 29^2 = 1274
7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4 = 811538

1 + 25 + 26 = 5210 + 19 + 29 = 58
1^2 + 25^2 + 26^2 = 10^2 + 19^2 + 29^2 = 1302
1^4 + 25^4 + 26^4 = 10^4 + 19^4 + 29^4 = 847602

2 + 25 + 27 = 5413 + 17 + 30 = 60
2^2 + 25^2 + 27^2 = 13^2 + 17^2 + 30^2 = 1358
2^4 + 25^4 + 27^4 = 13^4 + 17^4 + 30^4 = 922082

1 + 26 + 27 = 546 + 23 + 29 = 58
1^2 + 26^2 + 27^2 = 6^2 + 23^2 + 29^2 = 1406
1^4 + 26^4 + 27^4 = 6^4 + 23^4 + 29^4 = 988418

5 + 24 + 29 = 5815 + 16 + 31 = 62
5^2 + 24^2 + 29^2 = 15^2 + 16^2 + 31^2 = 1442
5^4 + 24^4 + 29^4 = 15^4 + 16^4 + 31^4 = 1039682

4 + 25 + 29 = 5811 + 20 + 31 = 62
4^2 + 25^2 + 29^2 = 11^2 + 20^2 + 31^2 = 1482
4^4 + 25^4 + 29^4 = 11^4 + 20^4 + 31^4 = 1098162

3 + 26 + 29 = 589 + 22 + 31 = 62
3^2 + 26^2 + 29^2 = 9^2 + 22^2 + 31^2 = 1526
3^4 + 26^4 + 29^4 = 9^4 + 22^4 + 31^4 = 1164338

9 + 25 + 29 = 6315 + 19 + 31 = 65
9^2 + 25^2 + 29^2 = 15^2 + 19^2 + 31^2 = 1547
9^4 + 25^4 + 29^4 = 15^4 + 19^4 + 31^4 = 1104467

8 + 23 + 31 = 6213 + 19 + 32 = 64
8^2 + 23^2 + 31^2 = 13^2 + 19^2 + 32^2 = 1554
8^4 + 23^4 + 31^4 = 13^4 + 19^4 + 32^4 = 1207458

7 + 24 + 31 = 6211 + 21 + 32 = 64
7^2 + 24^2 + 31^2 = 11^2 + 21^2 + 32^2 = 1586
7^4 + 24^4 + 31^4 = 11^4 + 21^4 + 32^4 = 1257698

9 + 23 + 32 = 6416 + 17 + 33 = 66
9^2 + 23^2 + 32^2 = 16^2 + 17^2 + 33^2 = 1634
9^4 + 23^4 + 32^4 = 16^4 + 17^4 + 33^4 = 1334978

3 + 27 + 30 = 6015 + 18 + 33 = 66
3^2 + 27^2 + 30^2 = 15^2 + 18^2 + 33^2 = 1638
3^4 + 27^4 + 30^4 = 15^4 + 18^4 + 33^4 = 1341522

6 + 26 + 32 = 6416 + 18 + 34 = 68
6^2 + 26^2 + 32^2 = 16^2 + 18^2 + 34^2 = 1736
6^4 + 26^4 + 32^4 = 16^4 + 18^4 + 34^4 = 1506848

1 + 29 + 30 = 6015 + 19 + 34 = 68
1^2 + 29^2 + 30^2 = 15^2 + 19^2 + 34^2 = 1742
1^4 + 29^4 + 30^4 = 15^4 + 19^4 + 34^4 = 1517282

5 + 27 + 32 = 648 + 25 + 33 = 66
5^2 + 27^2 + 32^2 = 8^2 + 25^2 + 33^2 = 1778
5^4 + 27^4 + 32^4 = 8^4 + 25^4 + 33^4 = 1580642

2 + 29 + 31 = 6211 + 23 + 34 = 68
2^2 + 29^2 + 31^2 = 11^2 + 23^2 + 34^2 = 1806
2^4 + 29^4 + 31^4 = 11^4 + 23^4 + 34^4 = 1630818

1 + 30 + 31 = 629 + 25 + 34 = 68
1^2 + 30^2 + 31^2 = 9^2 + 25^2 + 34^2 = 1862
1^4 + 30^4 + 31^4 = 9^4 + 25^4 + 34^4 = 1733522

1 + 30 + 31 = 6214 + 21 + 35 = 70
1^2 + 30^2 + 31^2 = 14^2 + 21^2 + 35^2 = 1862
1^4 + 30^4 + 31^4 = 14^4 + 21^4 + 35^4 = 1733522

9 + 25 + 34 = 6814 + 21 + 35 = 70
9^2 + 25^2 + 34^2 = 14^2 + 21^2 + 35^2 = 1862
9^4 + 25^4 + 34^4 = 14^4 + 21^4 + 35^4 = 1733522

5 + 28 + 33 = 6612 + 23 + 35 = 70
5^2 + 28^2 + 33^2 = 12^2 + 23^2 + 35^2 = 1898
5^4 + 28^4 + 33^4 = 12^4 + 23^4 + 35^4 = 1801202

7 + 30 + 31 = 6818 + 19 + 35 = 72
7^2 + 30^2 + 31^2 = 18^2 + 19^2 + 35^2 = 1910
7^4 + 30^4 + 31^4 = 18^4 + 19^4 + 35^4 = 1735922

4 + 29 + 33 = 6617 + 19 + 36 = 72
4^2 + 29^2 + 33^2 = 17^2 + 19^2 + 36^2 = 1946
4^4 + 29^4 + 33^4 = 17^4 + 19^4 + 36^4 = 1893458

6 + 28 + 34 = 6814 + 22 + 36 = 72
6^2 + 28^2 + 34^2 = 14^2 + 22^2 + 36^2 = 1976
6^4 + 28^4 + 34^4 = 14^4 + 22^4 + 36^4 = 1952288

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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