# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k+ a4k+ a5k+ a6k = b1k + b2k + b3k + b4k + b5k + b6k
( k = 1, 3, 5, 7, 9 )
• Non-negative integer solution of this type is first obtained by Chen Shuwen in 2000.
• [ 7, 91, 173, 269, 289, 323 ] = [ 29, 59, 193, 247, 311, 313 ]
• It take a Pentium-100MHz PC running about 2 months.
• Program language: Visual Basic 5.0
• Jarosław Wróblewski found the following four solutions in 2009.  
• [ 23, 163, 181, 341, 347, 407 ] = [ 37, 119, 221, 311, 371, 403 ]
• [ 43, 161, 217, 335, 391, 463 ] = [ 85, 91, 283, 287, 403, 461 ]
• [ 57, 399, 679, 995, 1167, 1293 ] = [ 115, 299, 767, 925, 1205, 1279 ]
• [ -13, 365, 689, 1111, 1115, 1325 ] = [ 23, 305, 731, 1037, 1177, 1319 ]
• In 2016，Chen Shuwen finished a computer search in the range of Max{ai,bi}<=5600 and found that there is no new positive integer solution.
• In 2016，Chen Shuwen finished a computer search in the range of a1=0 and Max{ai,bi}<=25000 and found that there is no any positive integer solution. That is to say, there is no ideal symmetric non-negative solutions of the the Prouhet-Tarry-Escott problem for ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ) in the range of Max{ai,bi} <= 50000.

Last revised March,19, 2017.