`Equal Sums of Like Powers`

Non-negative Integer Solutions of

a₁^k + a₂^k + a₃^k + a₄^k+ a₅^k + a₆^k + a₇^k + a₈^k + a₉^k

= b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k + b₇^k + b₈^k+ b₉^k

( k = 1, 2, 3, 4, 5, 6, 7, 8 )

[ 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 { a_i, b_i }< 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 { a_i, b_i }< 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 { a_i, b_i }< 13000.
- By full computer search, there is no non-symmetric solution in the range of max { a_i, b_i }< 310.
- By selective computer search, there is no new solution found.
Only one symmetric prime solution is found in the range of { a_i, b_i }< 100000000, by Chen Shuwen in 2016.
- [ 3522263, 4441103, 5006543, 7904423, 9388703, 11897843, 13876883, 15361163, 15643883 ] = [ 3698963, 3981683, 5465963, 7445003, 9954143, 11438423, 14336303, 14901743, 15820583 ]