## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} + a_{3}^{k}+
a_{4}^{k} = b_{1}^{k}
+ b_{2}^{k} + b_{3}^{k}
+ b_{4}^{k}
( k = 1, 2, 6 )
- Lander, Parkin and Selfridge found by computer the following triple
coincidence of 4 sixth powers in 1966.
`[12]`
- 1
^{6} + 34^{6} + 49^{6} + 111^{6} =
7^{6} + 43^{6} + 69^{6} + 110^{6} = 18^{6}
+ 25^{6} +77^{6} + 109^{6}

- Chen Shuwen noticed in 1995 that
- [ 7, 43, 69, 110 ] = [ 18, 25, 77, 109 ] (
k = 1, 2, 6 )

- Chen Shuwen found a parameter solution in 1997. Numerical examples
are
- [ 31, 62, 107, 126 ] = [ 38, 51, 118, 119 ]
- [ 1, 80, 111, 148 ] = [ 5, 67, 124, 144 ]
- [ 67, 129, 138, 179 ] = [ 69, 118, 149, 177 ]

- Chen Shuwen searched the smallest solutions of this type by using a
Pentium. There are solutions in the range of R<50. Here R
= max {
*a*_{i}, *b*_{i}
}.
- [ 7, 16, 25, 30 ] = [ 8, 14, 27, 29 ]
- [ 15, 23, 27, 34 ] = [ 17, 19, 30, 33 ]
- [ 4, 26, 33, 45 ] = [ 6, 20, 39, 43 ]
- [ 5, 25, 31, 48 ] = [ 9, 16, 37, 47 ]

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen