结果集锦

1.平方和相等的数组：[ 13, 91 ] = [ 23, 89 ] = [ 35, 85 ] = [ 47, 79 ] = [ 65, 65 ]

2.偶次等幂和

151¹⁰+140¹⁰+127¹⁰+86¹⁰+61¹⁰+22¹⁰=148¹⁰+146¹⁰+121¹⁰+94¹⁰+47¹⁰+35¹⁰

[ 22, 61, 86, 127, 140, 151 ] = [ 35, 47, 94, 121, 146, 148 ]( k = 2, 4, 6, 8, 10 ) .

a₁^k + a₂^k + ... + a_n_-1^k = b₁^k + b₂^k + ... + b_n_-1^k ( k = 2, 3, ..., n )
Chen found the above results with the help of a 386SX/33 PC.
a₁^k + a₂^k + ... + a_n^k = b₁^k + b₂^k + ... + b_n^k ( k = 1, 2, ..., j-1 , j+1, ..., n )
has nontrivial solution in positive integer for all n. However he could only prove the cases n<=3. [6] [32]
Now with the following results obtained by Chen, we can prove that T.N.Sinha's conjecture is true for n<=4. [ 2, 7, 11, 15 ] = [ 3, 5, 13, 14 ]     ( k = 1, 2, 4 ) [ 3, 140, 149, 252 ] = [ 50, 54, 201, 239 ]     ( k = 1, 3, 4 ) [ 975, 224368, 300495, 366448 ] = [ 37648, 202575, 337168, 344655 ]     ( k = 2, 3, 4 ) Here are some integer solutions of this type ( by Chen Shuwen) [ -26, 52, 93, 111 ] = [ 39, 58, 76, 117 ] [ -43, -3, 200, 215 ] = [ 32, 47, 185, 225 ]

3.[ 1, 8, 9 ] = [ 3, 4 11 ] ( k = 1, 2 )

[ 1, 5, 8, 12 ] = [ 2, 3, 10, 11 ] ( k = 1, 2, 3 )

[ 1, 5, 9, 17, 18 ] = [ 2, 3, 11, 15, 19] ( k = 1, 2, 3, 4 )

[ 1, 5, 10, 18, 23, 27 ] = [ 2, 3, 13, 15, 25, 26 ] ( k = 1, 2, 3, 4, 5 )

[ 1, 5, 10, 16, 27, 28, 38, 39 ] = [ 2, 3, 13, 14, 25, 31, 36, 40 ] ( k = 1, 2, 3, 4, 5, 6 )

[ 1, 5, 10, 24, 28, 42, 47, 51 ] = [ 2, 3, 12, 21, 31, 40, 49, 50 ] ( k = 1, 2, 3, 4, 5, 6, 7 )

4.平方等幂和：

a₁^k + a₂^k + a₃^k = b₁^k + b₂^k + b₃^k

( k = 1, 2 )

在 1750-1751年, Goldbach 和 Euler 发现了一个构造体系：[ t + mn, t + pq, t + mp + nq ] = [ t + mp, t + nq, t + mn + pq ]

[ 0, 3, 3 ] = [ 1, 1, 4 ]
[ 0, 4, 5 ] = [ 1, 2, 6 ]
[ 0, 7, 7 ] = [ 1, 4, 9 ]

Solution chain of this system was obtained by A.Golden in 1940's.
- [ 1, 17, 18 ] = [ 6, 7, 23 ] = [ 2, 13, 21 ] = [ 3, 11, 22 ]
  - [ 1, 17, 18 ] = [ 6, 7, 23 ] ( 对称的）
  - [ 2, 13, 21 ] = [ 3, 11, 22 ] ( 对称的）
  - [ 6, 7, 23 ] = [ 2, 13, 21 ] ( 非对称的）
  - [ 6, 7, 23 ] = [ 3, 11, 22 ] ( 非对称的）
  - [ 1, 17, 18 ] = [ 2, 13, 21 ] ( 非对称的）
  - [ 1, 17, 18 ] = [ 3, 11, 22 ] ( 非对称的）
  多环数组：[ 0, 71, 73 ] = [ 1, 63, 80 ] = [ 3, 56, 85 ] = [ 5, 51, 88] = [ 8, 45, 91] = [ 11, 40, 93 ] = [ 16, 33, 95 ] = [23, 25, 96 ]

5.三次等幂和

a₁^k + a₂^k + a₃^k+ a₄^k = b₁^k + b₂^k + b₃^k + b₄^k
( k = 1, 2, 3 )

L.E.Dickson gave the general solution for this system in 1910's. Numerial exampls are
- [ 0, 4, 7, 11 ] = [ 1, 2, 9, 10 ]
- [ 1, 8, 10, 17 ] = [ 2, 5, 13, 16 ]
- [ 0, 9, 11, 22 ] = [ 2, 4, 15, 21 ]
Method for symmetric solution chain of this type was obtained by A.Golden in 1940's.
- [ 1, 79, 105, 183 ] = [ 3, 69, 115, 181 ] = [ 7, 57, 127, 177 ] = [ 13, 45, 139, 171 ] = [ 27, 27, 157, 157 ]
Non-symmetric solution chain of this type was first obtained by Chen Shuwen in 1997.
- [ 0, 87, 93, 214 ] = [ 9, 52, 123, 210 ] = [ 24, 30, 133, 207 ]

6.四次等幂和

a₁^k + a₂^k + a₃^k+ a₄^k+ a₅^k = b₁^k + b₂^k + b₃^k + b₄^k + b₅^k
( k = 1, 2, 3, 4 )

J.Chernick gave a two-parameter solutions of this system in 1937. All solutions by his method are symmetric.
- [ 0, 4, 8, 16, 17 ] = [ 1, 2, 10, 14, 18 ]
- [ 0, 6, 8, 17, 19 ] = [ 1, 3, 12, 14, 20 ]
J.L.Burchnall & T.W.Chaundy obtained a parameter solution for non-symmetric solution of this type in 1937.
- [ 0, 9, 13, 26, 32 ] = [ 2, 4, 20, 21, 33 ]
- [ 0, 31, 49, 87, 113] = [ 3, 21, 64, 77, 115 ]
Ajai Choudhry obtained the complete symmetric solution and a parametric non-symmetric solution.of this type in 2000.

7.五次等幂和

a₁^k + a₂^k + a₃^k+ a₄^k+ a₅^k+ a₆^k = b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^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 ]

8.六次等幂和：Non-negative Integer Solutions of

a₁^k + a₂^k + a₃^k+ a₄^k+ a₅^k+ a₆^k+ a₇^k = b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k + b₇^k
( k = 1, 2, 3, 4, 5, 6 )

The first solution of this type was found out by E.B.Escott in 1910.
- [ 0, 18, 27, 58, 64, 89, 101 ] = [ 1, 13, 38, 44, 75, 84, 102 ]
J.Chernick gave a two-parameter solution of this type in 1937. Numerical examples by his method are
- [ 0, 59, 68, 142, 181, 221, 267 ] = [ 1, 47, 87, 126, 200 , 209, 268 ]
No non-symmetric solution of this type is first found Chen Shuwen in 1997.
- [ 0, 18, 19, 50, 56, 79, 81 ] = [ 1, 11, 30, 39, 68, 70, 84 ]
  - This one is also the smallest solution of this type that have been found out.
- [ 0, 14, 43, 141, 156, 193, 199 ] = [ 3, 9, 46, 133, 175, 176, 204 ]
- [ 0, 24, 31, 74, 106, 137, 147 ] = [ 4, 11, 52, 57, 119, 126, 150 ]
  - By Chen Shuwen, in 2001.
Each non-symmetric solution has its "co-symmetric" solution., For example
- [ 0, 18, 19, 50, 56, 79, 81 ] = [ 1, 11, 30, 39, 68, 70, 84 ]
- [ 0, 14, 16, 45, 54, 73, 83 ] = [ 3, 5, 28, 34, 65, 66, 84 ]
  - In fact, these two solutions are equivalent.

9.七次等幂和：Non-negative Integer Solutions of

a₁^k + a₂^k + a₃^k+ a₄^k+ a₅^k+ a₆^k+ a₇^k + a₈^k = b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k + b₇^k + b₈^k
( k = 1, 2, 3, 4, 5, 6, 7 )

The first solution of this type was obtained by G.Tarry in 1913. However, his method only can give one solution of this type.
- [ 0, 4, 9, 23, 27, 41, 46, 50 ] = [ 1, 2, 11, 20, 30, 39 , 48, 49 ]
Crussol gave a parameter solution in 1913.
J.Chernick gave a two-parameter symmetric solution of this type in 1937. Numerical examples by his method are
- [ 0, 9, 10, 27, 41, 58, 59, 68 ] = [ 2, 3, 19, 20, 48, 49, 65, 66 ]
- [ 0, 14, 19, 43, 57, 81, 86, 100 ] = [ 1, 9, 30, 32, 68, 70, 91, 99 ]
Non-symmetric solution of this type is first found by Chen Shuwen in 1997.
- [ 0, 7, 23, 50, 53, 81, 82, 96 ] = [ 1, 5, 26, 42, 63, 72, 88, 95 ]
- [ 0, 21, 82, 149, 155, 262, 278, 321 ] = [ 2, 17, 91, 126 , 174, 253, 285, 320 ]

10.八次等幂和

a₁^k + a₂^k + a₃^k + a₄^k+ a₅^k + a₆^k + a₇^k + a₈^k + a₉^k
= b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k + b₇^k + b₈^k+ b₉^k
( k = 1, 2, 3, 4, 5, 6, 7, 8 )

A.Latec gave two symmetric solutions in 1942.[5] [25]
- [ 0, 24, 30, 83, 86, 133, 157, 181, 197 ] = [ 1, 17, 41, 65, 112, 115, 168, 174, 198 ]
- [ 0, 26, 42, 124, 166, 237, 293, 335, 343 ] = [ 5, 13, 55, 111, 182, 224, 306, 322, 348 ]
No any new result was obtained on this type during the pass sixty years.
Chen Shuwen made a computer search in 1997, and found that there is no other symmetric solution of this type in the range max { a_i, b_i }< 400.

11.9次等幂和

Non-negative Integer Solutions of

a₁^k + a₂^k + a₃^k + a₄^k+ a₅^k + a₆^k + a₇^k + a₈^k + a₉^k+ a₁₀^k
= b₁^k + b₂^k + b₃^k + b₄^k + b₅^k + b₆^k + b₇^k + b₈^k+ b₉^k+ b₁₀^k
( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 )

Solution of this type was first found by A.Letac in 1940's.
- [ 0, 3083, 3301, 11893, 23314, 24186, 35607, 44199, 44417, 47500 ] = [ 12, 2865, 3519, 11869, 23738, 23762, 35631, 43981, 44635, 47488 ]
This second solution was obtained by G.Palama in 1950 and C.J.Smyth in 1990. Their methods both were based on Letac's method. See also for more information.
The smallest two solutions are found by Peter Borwein, Petr Lisonek and Colin Percival in 2000. See http://Euler.free.fr/oldresults.htm .
- [ 0, 12, 125, 213, 214, 412, 413, 501, 614, 626 ] = [ 5, 6, 133, 182, 242, 384, 444, 493, 620, 621 ]
- [ 0, 63, 149, 326, 412, 618, 704, 881, 967, 1030 ] = [ 7, 44, 184, 270, 497, 533, 760, 846, 986, 1023 ]