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 = 20$$5 + 6 + 11 = 22$
$1^2 + 9^2 + 10^2 = 5^2 + 6^2 + 11^2 = 182$
$1^4 + 9^4 + 10^4 = 5^4 + 6^4 + 11^4 = 16562$

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