a^n + b^n + c^n = x^n + y^n + z^n, n = 2, 4

 

Find six distinct positive integers   a, b, c, x, y, z   such that
 

a b c x y z n = 2 4 A1

 
for example,

 
1 \; + \; 9 \; + \; 10 \; = \; 20   …..   5 \; + \; 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 \; = \; 15562

 

Paul found:

1 + 11 + 12 = 24   …   4 + 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 = 32   …   8 + 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 = 34   …   7 + 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 = 34   …   5 + 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 = 34   …   8 + 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 = 38   …   9 + 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 = 38   …   7 + 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 = 40   …   10 + 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 = 44   …   10 + 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 = 42   …   9 + 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 = 44   …   6 + 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 = 46   …   12 + 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 = 48   …   9 + 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 = 50   …   11 + 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 = 48   …   8 + 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 = 48   …   11 + 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 = 50   …   10 + 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 = 54   …   13 + 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 = 54   …   10 + 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 = 54   …   7 + 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 = 54   …   12 + 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 = 56   …   12 + 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 = 52   …   10 + 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 = 54   …   13 + 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 = 54   …   6 + 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 = 58   …   15 + 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 = 58   …   11 + 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 = 58   …   9 + 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 = 63   …   15 + 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 = 62   …   13 + 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 = 62   …   11 + 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 = 64   …   16 + 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 = 60   …   15 + 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 = 64   …   16 + 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 = 60   …   15 + 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 = 64   …   8 + 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 = 62   …   11 + 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 = 62   …   9 + 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 = 62   …   14 + 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 = 68   …   14 + 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 = 66   …   12 + 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 = 68   …   18 + 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 = 66   …   17 + 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 = 68   …   14 + 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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