# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k + a4k+ a5k + a6k + a7k + a8k + a9k
= b1k + b2k + b3k + b4k + b5k + b6k + b7k + b8k+ b9k
( k = 1, 2, 3, 4, 5, 6, 7, 8 )
• A.Latec gave two symmetric solutions in 1942.[5] [25]
• [ 0, 24, 30, 83, 86, 133, 157, 181, 197 ] = [ 1, 17, 41, 65, 112, 115, 168, 174, 198 ]
• [ 0, 26, 42, 124, 166, 237, 293, 335, 343 ] = [ 5, 13, 55, 111, 182, 224, 306, 322, 348 ]
• Chen Shuwen made a computer search in 1997, and found that there is no other symmetric solution of this type in the range of max { ai, bi }< 400.
• In 2002, by computer search, Peter Borwein, Petr Lisoněk, and Colin Percival proved that there is no new symmetric solution in the range of max { ai, bi }< 4000. [0201]
• No any new result was obtained on this type during the pass 75 years.
• In 2016, Chen Shuwen tried several methods for the new solution of this system, and found that
• By full computer search, there is no new new symmetric solution in the range of max { ai, bi }< 13000.
• By full computer search, there is no non-symmetric solution in the range of max { ai, bi }< 310.
• By selective computer search, there is no new solution found.
• Only one symmetric prime solution is found in the range of { ai, bi }< 100000000, by Chen Shuwen in 2016.
• [ 3522263, 4441103, 5006543, 7904423, 9388703, 11897843, 13876883, 15361163, 15643883 ] = [ 3698963, 3981683, 5465963, 7445003, 9954143, 11438423, 14336303, 14901743, 15820583 ]

Last revised March,19, 2017.