## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} = b_{1}^{k}
+ b_{2}^{k}
( k = 4 )
- This equation was first studied by Euler. He
gave a two-parameter solution in 1772.
`[9]`
Numerical example is
- [ 59, 158 ] = [ 133, 134 ]

- Lander, Parkin and Selfridge made a computer
search on this type in 1960's and gave a list of 46 primitive solutions.
`[10]` `[11]`
Here are all the solutions in the range max { *a*_{i},
*b*_{i} }< 1000.
- [ 59, 158 ] = [ 133, 134 ]
- [ 7, 239 ] = [ 157, 227 ]
- [ 193, 292 ] = [ 256, 257 ]
- [ 271, 502 ] = [ 298, 497 ]
- [ 103, 542 ] = [ 359, 514 ]
- [ 222, 631 ] = [ 503, 558 ]

- In 1982, A.J.Zajta discussed the more important
solution methods of this equation and presented a list of 218 numerical
solutions.
`[12]` This list contained
all known primitive and nontrivial solutions in the range max { *a*_{i},
*b*_{i} }< 10^{6}.

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen