*c*_{11}^{k}+ c_{12}^{k}+ ... + c_{1m}^{k}= c_{21}^{k}+ c_{22}^{k}+ ... + c_{2m}^{k}*= ...... = c*_{j}_{1}^{k}+ c_{j}_{2}^{k}+ ... + c_{jm}^{k}

( *k *=
*k*_{1}* *, *k*_{2}* *, ..., *k _{n}
*)

- Introduction
- Non-negative integer solution chains of the m=n+1 case
- Integer solution chains of the m=n case

- We will also consider the multigrade chains, the more general form of the equal sums of like powers system:
*c*_{11}^{k}+ c_{12}^{k}+ ... + c_{1m}^{k}= c_{21}^{k}+ c_{22}^{k}+ ... + c_{2m}^{k}*= ...... = c*_{j}_{1}^{k}+ c_{j}_{2}^{k}+ ... + c_{jm}^{k}- (
*k*=*k*_{1}*k*_{2}*k*)_{n} - In this page, we will only consider the
*k*> 0 case.

- When all
*k*> 0 and*m*=*n*+1, non-negative integer solution chains of*j>=*3*j.*

**( k = 1 )**- [ 0, 15 ] = [ 1, 14 ] = [ 2, 13 ] = [ 3, 12 ] = [ 4, 11 ] = [ 5, 10 ] = [ 6, 9] = [ 7, 8]
- It is obvious that there are infinity solutions
of this type for any
*j.* **( k = 2 )**- [ 13, 91 ] = [ 23, 89 ] = [ 35, 85 ] = [ 47, 79 ] = [ 65, 65 ]
**( k = 3 )**- [ 2421, 19083 ] = [ 5436, 18948 ] = [ 10200, 18072 ] = [ 13322, 16630 ]
- By E.Rosenstiel, J.A.Dardis and C.R.Rosenstiel in 1991.
- See The Fifth Taxicab Number for more numercial samples.
**( k = 1, 2 )**- [ 0, 16, 17 ] = [ 1, 12, 20 ] = [ 2, 10, 21 ] = [ 5, 6, 22 ]
**( k = 1, 3 )**- [ 2, 52, 70 ] = [ 4, 46, 62 ] = [ 13, 32, 67 ] = [22, 22, 68 ]
- First known solution chains of
*j*= 4, by Chen Shuwen in 1995. - [ 16, 624, 699 ] = [ 19, 611, 709 ] = [ 39, 555, 745 ] = [ 79, 481, 779 ] = [ 107, 439, 793 ] = [ 169, 359, 811 ] = [ 187, 338, 814 ] = [ 259, 261, 819 ]
- By Jarek
Wroblewski in 2001. He also found solution chain of
*j*= 65. **( k = 1, 4 )**- [ 24, 201, 216 ] = [ 66, 132, 243 ] = [ 73, 124, 244 ]
- First known solution chains of
*j*= 3, by Chen Shuwen in 1997. **( k = 2, 4 )**- [ 23, 25, 48 ] = [ 15, 32, 47 ] = [ 8, 37, 45 ] = [ 3, 40, 43 ]
**( k = 1, 2, 3 )**- [ 0, 28, 29, 57 ] = [ 1, 21, 36, 56 ] = [ 2, 18, 39, 55 ] = [ 6, 11, 46, 51 ]
- This sample is a symmetric solution chain.
- It has been proved that there are solutions for
any
*j.*`[3]``[5]` - A.Golden gave method on how to obtained symmetric
solution chains of any
*j*.`[5]` - [ 0, 87, 93, 214 ] = [ 9, 52, 123, 210 ] = [ 24, 30, 133, 207 ]
- First known non-symmetric solution chains of
*j*= 3, by Chen Shuwen in 1997. **( k = 1, 2, 4 )**- [ 14, 37, 39, 64 ] = [ 16, 29, 46, 63 ] = [ 19, 24, 49, 62 ]
- First known solution chains of
*j*= 3 , by Chen Shuwen in 1995. **( k = 1, 2, 3, 4, 5 )**.

- When
*h*> 0 and*m*=*n*, integer solution chains have been found to the following 2 types: **( h = 1, 2, 4 )**- [ -48, 23, 25 ] = [ -47, 15, 32 ] = [ -45, 8, 37 ] = [ -43, 3, 40 ]
**( h = 1, 2, 3, 5 )**- [ -197, -23, -11, 231 ] = [ -179, -93, 49, 223 ] = [ -149, -137, 69, 217 ]
- First known solution chains of
*j*= 3, by Chen Shuwen in 1997.

Copyright 1997-2001, Chen Shuwen