## 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, 2, 3, 4, 7 )
- By searching for the positive integer solutions of
c
_{2}^{k} + c_{3}^{k} +
c_{4}^{k }+ c_{5}^{k}
+ c_{6}^{k}= d_{1}^{k} + d_{2}^{k}
+ d_{3}^{k} + d_{4}^{k}
+ d_{5}^{k }+ d_{6}^{k }
( k = 1, 2, 3, 4 ) and by applying
Theorem 1, Chen Shuwen obtained the
first solution for ( k = 1, 2, 3, 4, 7 ) in 22 Sep 2022.
- [ 34, 133, 165, 299, 332, 366 ] = [
35, 124, 177, 286, 353, 354 ]
- By computer searching, Chen Shuwen proved that it is the smallest
solution.

- Summaries on the ( k = 1, 2, ..., n, n+3 ) Series :
- [ 3, 25, 38 ] = [ 7, 20, 39 ] (
k = 1, 4 )
- First known solution, smallest solution, by Chen Shuwen in 1995.

- [ 1, 28, 39, 58 ] = [ 8, 14, 51, 53 ]
( k = 1, 2, 5 )
- Smallest solution, by Chen Shuwen in 1995-2001.

- [ 7, 18, 55, 69, 81 ] = [ 9, 15, 61, 63, 82 ]
( k = 1, 2, 3, 6 )
- First known solution, smallest solution, by Chen Shuwen in 1999.

- [ 34, 133, 165, 299, 332, 366 ] = [
35, 124, 177, 286, 353, 354 ]
( k = 1, 2, 3, 4, 7 )
- First known solution, smallest solution, by Chen Shuwen in 2022.

## 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 = -7, -4, -3, -2, -1 )
- Based on the above first solution of
( k = 1, 2, 3, 4, 7 ) and Theorem 9, Chen Shuwen
obtained the first solution of ( k = -7, -4, -3, -2, -1 ) in 22 Sep 2022.
- [ 3985023979526590,
4393128844899795, 4877989219086060, 8839507736404436, 10966306590276180,
42897611073727410 ] = [ 4120109538154610, 4131781236562980,
5099716001771790, 8240219076309220, 11762248197634935, 41671965043049484 ]

Last revised September 23, 2022.

Copyright 1997-2022, Chen Shuwen