# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k + a4k+ a5k + a6k + a7k + a8k + a9k+ a10k
= b1k + b2k + b3k + b4k + b5k + b6k + b7k + b8k+ b9k+ b10k
( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 )
• Solution of this type was first found by A.Letac in 1940's.[5] [25]
• [ 0, 3083, 3301, 11893, 23314, 24186, 35607, 44199, 44417, 47500 ] = [ 12, 2865, 3519, 11869, 23738, 23762, 35631, 43981, 44635, 47488 ]
• The second solution was obtained by G.Palama in 1950[33] and C.J.Smyth in 1990.[22] Their methods both were based on Letac's method. See also ( h = 1, 2, 4, 6, 8 ) for more information.
• The smallest two solutions are found by Peter Borwein, Petr Lisonek and Colin Percival in 2000. [0201]
• [ 0, 12, 125, 213, 214, 412, 413, 501, 614, 626 ] = [ 5, 6, 133, 182, 242, 384, 444, 493, 620, 621 ]
• [ 0, 63, 149, 326, 412, 618, 704, 881, 967, 1030 ] = [ 7, 44, 184, 270, 497, 533, 760, 846, 986, 1023 ]
• The third smallest known solution is found by Chen Shuwen in 2023, based on the solution of ( k = 2, 4, 6, 8 ).
• [ 0, 364, 810, 1229, 4310, 5344, 8425, 8844, 9290, 9654 ] = [ 40, 260, 1044, 1054, 4329, 5325, 8600, 8610, 9394, 9614 ]
• Only one prime solution is found in the range of of { ai, bi }< 100000000, by Chen Shuwen in 2016.
• [ 2589701, 2972741, 6579701, 9388661, 9420581, 15740741, 15772661, 18581621, 22188581, 22571621 ] = [ 2749301, 2781221, 6835061, 8399141, 10314341, 14846981, 16762181, 18326261, 22380101, 22412021 ]

Last revised April,30, 2023.