## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} = b_{1}^{k}
+ b_{2}^{k}
( k = 1 )
- Solutions of this type is obvious.
- [ 0, 2 ] = [ 1, 1 ]
- [ 1, 4 ] = [ 2, 3 ]
- [ 0, 7 ] = [ 1, 6 ] = [ 2, 5 ] = [ 3, 4 ]

- From 1 + 9 = 4 + 6, applying Theorem
2 successively with T = 2, 1, 7, 8, 13, 11, we have, respectively
`[2]`
- [ 1, 8, 9 ] = [ 3, 4 11 ] (
k = 1, 2 )
- [ 1, 5, 8, 12 ] = [ 2, 3, 10, 11 ] (
k = 1, 2, 3 )
- [ 1, 5, 9, 17, 18 ] = [ 2, 3, 11, 15, 19] (
k = 1, 2, 3, 4 )
- [ 1, 5, 10, 18, 23, 27 ] = [ 2, 3, 13, 15, 25,
26 ] ( k = 1, 2, 3, 4, 5 )
- [ 1, 5, 10, 16, 27, 28, 38, 39 ] = [ 2, 3, 13,
14, 25, 31, 36, 40 ] ( k =
1, 2, 3, 4, 5, 6 )
- [ 1, 5, 10, 24, 28, 42, 47, 51 ] = [ 2, 3, 12,
21, 31, 40, 49, 50 ] ( k
= 1, 2, 3, 4, 5, 6, 7 )

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen