## 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, 3 )
- A.Moessner gave some parameter solutions of this
type in 1939.
`[13]` Numerical exampls
are
- [ 0, 7, 8 ] = [ 1, 5, 9 ]
- [ 0, 7, 9 ] = [ 2, 4, 10 ]
- [ 12, 23, 28 ] = [ 13, 21, 29 ]

- Smallest solutions of this type can be found
easily by using computer.
- [ 1, 5, 5 ] = [ 2, 3, 6 ]
- [ 0, 7, 8 ] = [ 1, 5, 9 ]
- [ 0, 7, 9 ] = [ 2, 4, 10 ]
- [ 1, 9, 9 ] = [ 4, 4, 11 ]
- [ 2, 8, 10 ] = [ 4, 5, 11 ]
- [ 5, 10, 11 ] = [ 6, 8, 12 ]

- Solution chain of this type had been studied
by A.Choudhry in 1992.
`[40]` However,
solutions by his method are in integers, not in positive integers.
- The positive integer solution chain is first
obtained by Chen Shuwen in 1995.
- [ 2, 52, 70 ] = [ 4, 46, 62 ] = [ 13, 32, 67
] = [ 22, 22, 68 ]
- [ 5, 58, 70 ] = [ 7, 53, 73 ] = [ 13, 43, 77
] = [ 21, 33, 79 ]

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen