## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} + a_{3}^{k}+
a_{4}^{k}+ a_{5}^{k}+
a_{6}^{k} = b_{1}^{k}
+ b_{2}^{k} + b_{3}^{k}
+ b_{4}^{k} + b_{5}^{k}
+ b_{6}^{k}
( 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.
_{
[1001]} _{
[0901]}
- [ 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{a
_{i},b_{i}}<=5600
and found that there is no new positive integer solution.
- In 2016，Chen Shuwen finished a computer search in the range of a
_{1}=0
and Max{a_{i},b_{i}}<=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{a_{i},b_{i}}
<= 50000.

*Last revised March,19, 2017.*

Copyright 1997-2017, Chen Shuwen