`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+ a₁₀^k+ a₁₁^k+ a₁₂^k

= b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k + b₇^k + b₈^k+ b₉^k+ b₁₀^k+ b₁₁^k+ b₁₂^k

( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 )

Nuutti Kuosa discovered the following solution in 3 Sep1999, using a program written by Jean-Charles Meyrignac.

Chen Shuwen noticed in 7 Sep1999 that the above result is also a solution of ( k = 2, 4, 6, 8, 10 ) .

By using Theorem 4, Chen Shuwen translated Nuutti Kuosa & Jean-Charles Meyrignac's result into an ideal solution of ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ) .

[ 0, 11, 24, 65, 90, 129, 173, 212, 237, 278, 291, 302 ] = [ 3, 5, 30, 57, 104, 116, 186, 198, 245, 272, 297, 299 ]

David Broadhurst found the second solution of ( k = 2, 4, 6, 8, 10 ) in 2007.
In 2008, A.Choudhry and Jaroslaw Wroblewski proved that the system ( k = 2, 4, 6, 8, 10 ) has infinitely many solutions and give a method to produce new numerical solutions.
Prime solution of ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ) is first found by Jaroslaw Wroblewski in 2023.
- [ 32058169621, 32367046651, 32732083141, 33883352071, 34585345321, 35680454791, 36915962911, 38011072381, 38713065631, 39864334561, 40229371051, 40538248081 ] = [ 32142408811, 32198568271, 32900561521, 33658714231,34978461541, 35315418301, 37280999401, 37617956161,38937703471, 39695856181, 40397849431, 40454008891 ]