## Integer Solutions
of

*a*_{1}^{h}
+ a_{2}^{h} + a_{3}^{h}
= b_{1}^{h} + b_{2}^{h}
+ b_{3}^{h}
( h = 1, 2, 6 )
- Chen Shuwen found the first solution of this
type in 1997 .
- [ -372, 43, 371 ] = [ -405, 140, 307 ]

- Ajai Choudhry obtained a method of generating
infinitely many integer solutions of this type in 1999
`[61]`.
Numerical examples are:
- [ -300, 83, 211 ] = [ -124, -185, 303 ]
- This is also the smallest solution.

- [ -479, 23, 432 ] = [ -393, -127, 496 ]

- To the system (
the m=n case )
*a*_{1}^{h} + a_{2}^{h}
+ ... + a_{n}^{h} = b_{1}^{h} +
b_{2}^{h} + ... + b_{n}^{h} ( *h
*= *h*_{1}* *, *h*_{2}* *, ... ,
*h*_{n} )

- Integer solution had been obtained for 10 types
of (
*h *= *h*_{1}* *, *h*_{2}*
*, ... , *h*_{n} ) so far.
- All these 10 types, except ( h = 1, 2, 6 ), can
be express as ( h = 1, 2, ..., s, s+2, ..., s+2t )
- ( h = 1, 2, 4 )
- ( h = 1, 2, 6 )
- ( h = 1, 2, 3, 5 )
- ( h = 1, 2, 4, 6 )
- ( h = 1, 2, 3, 4, 6 )
- ( h = 1, 2, 3, 5, 7 )
- ( h = 1, 2, 4, 6, 8 )
- ( h = 1, 2, 3, 4, 5, 7
)
- ( h = 1, 2, 3, 4, 5, 6,
8 )
- ( h = 1, 2, 3, 4, 5, 6,
7, 9 )

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen