## 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, 5 )
- The first solution known for this type was due
to A.Moessner in 1939.
`[3]` `[13]`
- [ 39, 92, 100 ] = [ 49, 75, 107 ]

- A two parameter solution of seventh degree was
found by Moessner and two parametric solutions were found by H.Swinnerton-Dyer.
`[14]`
- L.J.Lander obtained a 3-parameter solution of
this system in 1968.
`[14]`
- L.J.Lander, T.R.Parkin and J.L.Selfridge made
a computer search using CDC6600 to the following Diophantine equation in
least integers
*a*_{1}^{5}*
+ a*_{2}^{5}* + a*_{3}^{5}*
= b*_{1}^{5}* + b*_{2}^{5}*
+ b*_{3}^{5}

- They found 45 primitive solutions of which 32
solutions were also satisfy the additional condition
*a*_{1}*
+ a*_{2}* + a*_{3}*
= b*_{1}* + b*_{2}*
+ b*_{3}

- Here are the primitive solutions in least integers
by their computer search.
`[11]`
- [ 13, 51, 64 ] = [ 18, 44, 66 ]
- [ 3, 54, 62 ] = [ 24, 28, 67 ]
- [ 8, 62, 68 ] = [ 21, 43, 74 ]
- [ 53, 72, 81 ] = [ 56, 67, 83 ]

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen