## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} + a_{3}^{k}
= b_{1}^{k} + b_{2}^{k}
+ b_{3}^{k}
( k = 1, 2 )
- During 1750-1751, Goldbach and Euler had studied
this system.
`[2]` `[9]`
- The general solutions are easy to obtain.
`[2]`
`[5]`
- [ 0, 3, 3 ] = [ 1, 1, 4 ]
- [ 0, 4, 5 ] = [ 1, 2, 6 ]
- [ 0, 7, 7 ] = [ 1, 4, 9 ]
- [ t + mn, t + pq, t + mp + nq ] = [ t + mp, t
+ nq, t + mn + pq ]

- Solution chain of this system was obtained by
A.Golden in 1940's.
`[3]` `[5]`
- [ 1, 17, 18 ] = [ 6, 7, 23 ] = [ 2, 13, 21 ]
= [ 3, 11, 22 ]
- [ 1, 17, 18 ] = [ 6, 7, 23 ] (
Symmetric )
- [ 2, 13, 21 ] = [ 3, 11, 22 ] (
Symmetric )
- [ 6, 7, 23 ] = [ 2, 13, 21 ]
( Non-symmetric )
- [ 6, 7, 23 ] = [ 3, 11, 22 ]
( Non-symmetric
)
- [ 1, 17, 18 ] = [ 2, 13, 21 ] (
Non-symmetric )
- [ 1, 17, 18 ] = [ 3, 11, 22 ]
( Non-symmetric )

- [ 0, 71, 73 ] = [ 1, 63, 80 ] = [ 3, 56, 85 ]
= [ 5, 51, 88] = [ 8, 45, 91] = [ 11, 40, 93 ] = [ 16, 33, 95 ] = [23,
25, 96 ]

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen