## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} + a_{3}^{k}+
a_{4}^{k}+ a_{5}^{k}
= b_{1}^{k} + b_{2}^{k}
+ b_{3}^{k} + b_{4}^{k}
+ b_{5}^{k}
( k = 1, 3, 5, 7)
- A.Golden gave a method to solve this system in 1940's.
`[5]`
`[20]` `[38]`
Numerical examples by his method are
- [ 3, 19, 37, 51, 53 ] = [ 9, 11, 43, 45, 55 ]
- [ 13, 31, 47, 65, 67 ] = [ 15, 27, 51, 61, 69 ]
- [ 1, 33, 39, 65, 71 ] = [ 11, 15, 53, 57, 73 ]
- [13, 53, 67, 103, 107 ] = [ 19, 39, 81, 93, 111 ]

- A.Letac found two solutions on the
*a*_{1}=
0 case of this type in 1942.`[5]`
`[25]`
- [ 0, 34, 58, 82, 98 ] = [ 13, 16, 69, 75, 99 ]
- [ 0, 63, 119, 161, 169 ] = [ 8, 50, 132, 148, 174 ]
- These two solutions lead to solutions of the (
k = 1, 2, 3, 4, 5, 6, 7, 8 ) type.
- Chen Shuwen proved by computer that there is no other solutions on
the
*a*_{1}= 0 case of this type
in the range of max { *a*_{i},
*b*_{i} }< 200.

- A.Moessner gave a two-parameter solution based on Golden's method in
1947.
`[21]`
- Chen Shuwen found a new method for this type in 1995. Here are some
examples by his method. ( This solutions cannot be obtained by Golden's
method.)
- [ 29, 75, 79, 125, 141 ] = [ 37, 53, 99, 117, 143 ]
- [ 85, 102, 110, 143, 145 ] = [ 90, 92, 117, 138, 148 ]
- [ 11, 79, 103, 149, 161 ] = [ 17, 63, 123, 135, 165 ]
- [ 17, 71, 103, 157, 163 ] = [ 31, 47, 121, 143, 169 ]

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen