## Integer Solutions
of

*a*_{1}^{h}
+ a_{2}^{h} + a_{3}^{h}
= b_{1}^{h} + b_{2}^{h}
+ b_{3}^{h}
( h = 1, 2, 4 )
- A.Golden
`[5]`
gave the following theorem : ( Theorem 5 )
- If
- [
*a*_{1}* *,* a*_{2}*
*, ... , *a*_{n}_{+1}* *]* *= [ *b*_{1}*
, b*_{2} , ... ,* b*_{n}_{+1}* *]

- is a non-symmetric
solution of the system
*a*_{1}^{k} + a_{2}^{k}
+ ... + a_{n}_{+1}^{k} = b_{1}^{k}
+ b_{2}^{k} + ... + b_{n}_{+1}^{k} ( *k
*= 1, 2, ..., *n* )

- Then
- [ S
*a*_{1}* - *T ,*
*S*a*_{2}* - *T , *...*, S*a*_{n}_{+1}*
- *T ]* *= [ S*b*_{1}* - *T *, *S*b*_{2}*
- *T , *...* ,* *S*b*_{n}_{+1}* - *T
]

- will be a non-trival integer solution of the
system
*a*_{1}^{h} + a_{2}^{h}
+ ... + a_{n}_{+1}^{h} = b_{1}^{h}
+ b_{2}^{h} + ... + b_{n}_{+1}^{h} ( *h
*= 1, 2, ..., *n* , *n* + 2 )

- where T =
*a*_{1}+* a*_{2}
+ ... + *a*_{n}_{+1} , S = *n* + 1 .
- Solution method of this type was first found
in 1900's. ( See Dickson's book
`[9]`
)
- A.Golden
`[5]`
gave solutions of this type by starting with the non-symmetric
solutions of the system ( k = 1, 2 ) .
- [ -7, 0, 7 ] = [ -8, 3, 5 ]
- [ -10, 1, 9 ] = [ -11, 5, 6 ]
- [ -12, 1, 11 ] = [ -13, 4, 9 ]
- [ -13, 0, 13 ] = [ -15, 7, 8 ]
- [ -48, 23, 25 ] = [ -47, 15, 32 ] = [ -45, 8, 37 ] = [ -43, 3, 40 ]

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen