## Non-negative Integer
Solutions of

*a*_{1}^{k}
+ a_{2}^{k} + a_{3}^{k}+
a_{4}^{k}+ a_{5}^{k}+
a_{6}^{k} = b_{1}^{k}
+ b_{2}^{k} + b_{3}^{k}
+ b_{4}^{k} + b_{5}^{k}
+ b_{6}^{k}
( k = 1, 2, 3, 4,
5 )
- G.Tarry gave a two-parameter solution of this
type in 1912.
`[23]` Numerical examples
are
- [ 0, 3, 5, 11, 13, 16 ] = [ 1, 1, 8, 8, 15, 15
]
- [ 0, 5, 6, 16, 17, 22 ] = [ 1, 2, 10, 12, 20,
21 ]

- A.Golden gave a four-parameter solution in 1912.
`[5]`
- Only one non-symmetric solution of this type
was appeared in the literature.
`[5]`
( Page27 ) It's obtained by A.Golden. However, Golden's method for this
non-symmetric solution was unknown.
- [ 0, 19, 25, 57, 62, 86 ] = [ 2, 11, 40, 42,
69, 85 ]

- Chen Shuwen found how to get non-symmetric solution
of this type in 1995. Here are some examples by his method:
- [ 0, 9, 17, 34, 36, 46 ] = [ 1, 6, 24, 25, 42,
44 ]
- [ 0, 6, 23, 38, 47, 57 ] = [ 2, 3, 27, 33, 50,
56 ]
- [ 0, 8, 27, 45, 46, 61 ] = [ 1, 6, 33, 35, 52,
60 ]
- [ 0, 14, 17, 46, 51, 67 ] = [ 2, 7, 25, 39, 56,
66 ]

- A.Golden gave a parameter method for solution
chain of this type in 1940's.
`[5]`
( Page90 ) Numerical example is
- [ 0, 567, 644, 1778, 1855,
2422 ]
- = [ 2, 535, 678, 1744, 1887, 2420 ]
- = [ 7, 490, 728, 1694, 1932, 2415 ]
- = [ 15, 444, 782, 1640, 1978, 2407 ]
- = [ 28, 392, 847, 1575, 2030, 2394 ]
- = [ 42, 350, 903, 1519, 2072, 2380 ]
- = [ 62, 303, 970, 1452, 2119, 2360 ]
- = [ 70, 287, 994, 1428, 2135, 2352 ]
- = [ 95, 244, 1062, 1360, 2178, 2327 ]
- = [ 103, 232, 1082, 1340, 2190, 2319 ]
- = [ 119, 210, 1120, 1302, 2212, 2303 ]
- = [ 144, 180, 1175, 1247, 2242, 2278 ]

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen