## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} + a_{3}^{k}
= b_{1}^{k} + b_{2}^{k}
+ b_{3}^{k}
( k = 2, 4 )
- This system had been considered since 1900's.
`[9]`
- A.Golden gave some parameter solutions of this
system in 1940's.
`[5]` `[38]`
`[20]`
- Solutions of this system can be seperated into
two kind. One kind also satisfy
*a*_{1}*+a*_{2}*=a*_{3}*
*and *b*_{1}*+ b*_{2}*= b*_{3}.Solutions
of this kind are easy to obtained.
- [ 0, 7, 7 ] = [ 3, 5, 8 ]
- [ 1, 9, 10 ] = [ 5, 6, 11 ]
- [ 1, 11, 12 ] = [ 4, 9, 13 ]
- [ 2, 11, 13 ] = [ 7, 7, 14 ]
- [ 0, 13, 13 ] = [ 7, 8, 15 ]

- The other kind of solutions does not satisfy
*a*_{1}*+a*_{2}*=a*_{3}* *and
*b*_{1}*+ b*_{2}*= b*_{3}. Solutions
of this kind may lead to solutions of the type ( k
= 1, 3, 5, 7 ), ( k = 2, 4, 6, 8 ), (
k = 1, 2, 3, 4, 5, 6, 7, 8 ) and ( k = 1, 2, 3,
4, 5, 6, 7, 8, 9 ).
- [ 9, 25, 29 ] = [ 15, 19, 31 ]
- [ 7, 30, 31 ] = [ 18, 19, 35 ]
- [ 12, 29, 35 ] = [ 20, 21, 37 ]
- [ 6, 25, 37 ] = [ 15, 19, 38 ]
- [ 5, 32, 41 ] = [ 16, 25, 43 ]
- [ 2, 29, 45 ] = [ 15, 23, 46 ]
- [ 1, 30, 49 ] = [ 19, 21, 50 ]

- Method for solution Chain of this system was
obtained by A.Golden in 1940's.
`[5]`
`[3]`
- [ 28, 175, 203 ] = [ 77, 140, 217 ] = [ 107,
113, 220 ] = [ 5, 188, 193 ] = [ 67, 148, 215 ] = [ 55, 157, 212 ]

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen