## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} + a_{3}^{k}
+ a_{4}^{k}+ a_{5}^{k}
+ a_{6}^{k} + a_{7}^{k}
+ a_{8}^{k} + a_{9}^{k}+
a_{10}^{k}+ a_{11}^{k}+
a_{12}^{k}
*= b*_{1}^{k}
+ b_{2}^{k} + b_{3}^{k}
+ b_{4}^{k} + b_{5}^{k}
+ b_{6}^{k} + b_{7}^{k}
+ b_{8}^{k}+ b_{9}^{k}+
b_{10}^{k}+ b_{11}^{k}+
b_{12}^{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.
- 151
^{10}+140^{10}+127^{10}+86^{10}+61^{10}+22^{10}=148^{10}+146^{10}+121^{10}+94^{10}+47^{10}+35^{10}

- Chen Shuwen noticed in 7 Sep1999 that the above
result is also a solution of ( k = 2, 4, 6, 8, 10
) .
- [ 22, 61, 86, 127, 140, 151 ] = [ 35, 47, 94,
121, 146, 148 ]

- 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 ]

*Last revised April 30, 2023.*

Copyright 1997-2023, Chen Shuwen