## 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, 3, 4 )
- T.N.Sinha found a two-parametric integer solution
of this type in 1984, but his method cannot lead to non-negative integer
solutions.
`[6]`
- In 1988, R.N.Singh gave a method on how to construct
infinitely many integral solutions to the system of equations
*a*_{1}^{k} + a_{2}^{k}
+ a_{3}^{k} = b_{1}^{k} +
b_{2}^{k} + b_{3}^{k} + b_{4}^{k}
( k = 1, 3, 4 )

- However, his method also just gives integer solutions,
not non-negative integer solutions.
`[16]`
- Non-negative integer solutions of this type were
first found by Chen Shuwen in 1995. Examples are
- [ 3, 140, 149, 252 ] = [ 50, 54, 201, 239 ]
- [ 127, 324, 1740, 2023 ] = [ 24, 439, 1711, 2040
]
- [ 1059, 1444, 2476, 2763 ] = [ 1179, 1288, 2632,
2643 ]

- Chen had made a computer search recently, and
found that there are no non-negative integer solutions of this type in
the range R<=200.

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen