## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} + a_{3}^{k}
= b_{1}^{k} + b_{2}^{k}
+ b_{3}^{k}
( k = 1, 3 )
- A.Moessner gave some parameter solutions of this
type in 1939.
`[13]` Numerical exampls
are
- [ 0, 7, 8 ] = [ 1, 5, 9 ]
- [ 0, 7, 9 ] = [ 2, 4, 10 ]
- [ 12, 23, 28 ] = [ 13, 21, 29 ]

- Smallest solutions of this type can be found
easily by using computer.
- [ 1, 5, 5 ] = [ 2, 3, 6 ]
- [ 0, 7, 8 ] = [ 1, 5, 9 ]
- [ 0, 7, 9 ] = [ 2, 4, 10 ]
- [ 1, 9, 9 ] = [ 4, 4, 11 ]
- [ 2, 8, 10 ] = [ 4, 5, 11 ]
- [ 5, 10, 11 ] = [ 6, 8, 12 ]

- Solution chain of this type had been studied
by A.Choudhry in 1992.
`[40]` However,
solutions by his method are in integers, not in positive integers.
- The positive integer solution chain is first
obtained by Chen Shuwen in 1995.
- [ 2, 52, 58 ] = [ 4, 46, 62 ] = [ 13, 32, 67
] = [ 22, 22, 68 ]
- [ 5, 58, 70 ] = [ 7, 53, 73 ] = [ 13, 43, 77
] = [ 21, 33, 79 ]

- Jarek Wroblewski
find the following results by exhaustive computer search in 2001:
- [ 10, 214, 215 ] = [ 19, 179, 241 ] = [ 43, 139,
257 ] = [ 49, 131, 259 ] = [87, 88, 264]
- [ 89, 361, 367 ] = [ 97, 323, 397 ] = [ 115,
285, 417 ] = [ 133, 257, 427 ] = [ 152, 232, 433 ] = [ 187, 193, 437 ]
- [ 16, 624, 699 ] = [ 19, 611, 709 ] = [ 39, 555,
745 ] = [ 79, 481, 779 ] = [ 107, 439, 793 ] = [ 169, 359, 811 ] = [ 187,
338, 814 ] = [ 259, 261, 819 ]

- A solution chain of length j=65, with 87-88 digit numbers, is also
found by Jarek
Wroblewski in 2001.

*Last revised Oct, 16, 2001.*

Copyright 1997-2001, Chen Shuwen