## 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 = 2, 3, 4 )
- H.Gupta had proved that the following system
has no nontrivial solutions in positive integers.
`[8]`
*a*_{1}^{k} + a_{2}^{k}
+ ... + a_{n}_{-1}^{k} = b_{1}^{k}
+ b_{2}^{k} + ... + b_{n}_{-1}^{k} ( *k
*= 2, 3, ..., *n* )

- Non-negative integer solutions of this type were
first obtained by Chen Shuwen in 1995.
- [ 975, 224368, 300495, 366448 ] = [ 37648, 202575,
337168, 344655 ]
- [ 7001616, 10868299, 31439172, 34940503 ] = [
7527024, 10393591, 31599228, 34831147 ]
- [ 2756106, 17971525, 31568076, 35616295 ] = [
3727405, 17323956, 32539375, 34968726 ]
- [ 33801840, 3033353281, 4414180500, 5723026141
] = [ 1004104381, 2384931600, 5074604460, 5384483041 ]

- Chen found the above results with the help of
a 386SX/33 PC.
- T.N.Sinha conjectured that the system
*a*_{1}^{k} + a_{2}^{k}
+ ... + a_{n}^{k} = b_{1}^{k} +
b_{2}^{k} + ... + b_{n}^{k} ( *k
*= 1, 2, ..., *j*-1 , *j+*1, ..., *n* )

- has nontrivial solution in positive integer for
all
*n*. However he could only prove the cases *n<=*3. `[6]`
`[32]`
- Now with the following results obtained by Chen,
we can prove that T.N.Sinha's conjecture is true for
*n<=*4.
- [ 2, 7, 11, 15 ] = [ 3, 5, 13, 14 ] (
k = 1, 2, 4 )
- [ 3, 140, 149, 252 ] = [ 50, 54, 201, 239 ] (
k = 1, 3, 4 )
- [ 975, 224368, 300495, 366448 ] = [ 37648, 202575,
337168, 344655 ] ( k = 2, 3,
4 )

- Here are some integer solutions of this type
( by Chen Shuwen)
- [ -26, 52, 93, 111 ] = [ 39, 58, 76, 117 ]
- [ -43, -3, 200, 215 ] = [ 32, 47, 185, 225 ]

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen