## Positive Integer Solutions
of

*a*_{1}^{k}
+ a_{2}^{k} + a_{3}^{k}
= b_{1}^{k} + b_{2}^{k}
+ b_{3}^{k}
( k = -1, 1 )
- This type can be expressed as follows:
*a*_{1}* *+* a*_{2}*
*+ *a*_{3}* *= *b*_{1}* + b*_{2}
+* b*_{3}
- 1
`/`*a*_{1}* *+* *1`/`*a*_{2}*
*+ 1`/`*a*_{3}* *= 1`/`*b*_{1}*
+ *1`/`*b*_{2} +* *1`/`*b*_{3}
- This type was first solved by Chen Shuwen in
May of 2001.
- Numerical solutions can be obtained by computer
easily. Here are some examples:
- [ 4, 10, 12 ] = [ 5, 6, 15 ]
- [ 6, 14, 14 ] = [ 7, 9, 18 ]
- [ 4, 9, 18 ] = [ 5, 6, 20 ]
- [ 5, 15, 15 ] = [ 7, 7, 21 ]
- [ 7, 14, 18 ] = [ 9, 9, 21 ]
- [ 2, 15, 20 ] = [ 3, 4, 30 ]
- [ 8, 20, 24 ] = [ 10, 12, 30 ]

- Chen Shuwen have checked at least 250 sets of
solutions of this type.All of them satisfy the following relation:

- A sepecial case of the ( k = -1, 1 ) is:
*a*_{1}* *+* a*_{2}*
*= *b*_{1}* + b*_{2} +* b*_{3}
- 1
`/`*a*_{1}* *+* *1`/`*a*_{2}*
*= 1`/`*b*_{1}* + *1`/`*b*_{2}
+* *1`/`*b*_{3}
- Chen Shuwen found the following solutions:
- [ 5, 45 ] = [ 8, 18, 24 ]
- [ 5, 56 ] = [ 7, 24, 30 ]
- [ 8, 72 ] = [ 15, 20, 45 ]
- [ 3, 40 ] = [ 4, 15, 24 ] = [ 5, 8, 30 ]
- [ 15, 168 ] = [ 21, 72, 90 ] = [ 28, 35, 120
]

- It should be noticed that [
*a*_{1},*
a*_{2} ]* *= [ *b*_{1},* b*_{2}
,* b*_{3} ] can not be denoted as [ 0, *a*_{1},*
a*_{2} ]* *= [ *b*_{1},* b*_{2}
,* b*_{3} ].

*Last revised May 6, 2001.*

Copyright 1997-2001, Chen Shuwen