Multigrade Chains

`Equal Sums of Like Powers`

Multigrade Chains

c₁₁^k + c₁₂^k + ... + c_1m^k = c₂₁^k + c₂₂^k + ... + c_2m^k = ...... = c_j₁^k + c_j₂^k + ... + c_jm^k

( k = k₁ , k₂ , ..., k_n )

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

Introduction

In this page, we discuss the multigrade chains, the more general form of the equal sums of like powers system:

c₁₁^k + c₁₂^k + ... + c_1m^k = c₂₁^k + c₂₂^k + ... + c_2m^k = ...... = c_j₁^k + c_j₂^k + ... + c_jm^k

( k = k₁ , k₂ , ..., k_n )

In this page, we will only consider the ideal solution chains.
- For non-negative integer solution,

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

When all k > 0 and m=n+1, non-negative integer solution chains of j>=3 have been found to 10 types. Among them, 7 types have been proved to be solvable for any 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.
By applying Theorem 9, solution chains of ( k = -1 ) can be obtained for any j.

( k = 2 )

[ 13, 91 ] = [ 23, 89 ] = [ 35, 85 ] = [ 47, 79 ] = [ 65, 65 ]

It has been proved that there are solutions for any j. [3] [5]
A.Golden gave method on how to obtained solution chain of any j. [5]
By applying Theorem 9, solution chains of ( k = -2 ) can be obtained for any j.

( 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 numerical samples.

It has been proved that there are solutions for any j. [1] [3]
By applying Theorem 9, solution chains of ( k = -3 ) can be obtained for any j.

( k = 1, 2 )

[ 0, 16, 17 ] = [ 1, 12, 20 ] = [ 2, 10, 21 ] = [ 5, 6, 22 ]

It has been proved that there are solutions for any j. [3] [5]
A.Golden gave method on how to obtained solution chains of any j. [5]
By applying Theorem 9, solution chains of ( k = -1, -2 ) can be obtained for any j.

( 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 Jaroslaw Wroblewski in 2001. He also found solution chain of j = 65.
By applying Theorem 9, solution chains of ( k = -1, -3 ) can be obtained for 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.
By applying Theorem 9, solution chains of ( k = -1, -4 ) can be obtained for j = 3.

( k = 2, 4 )

[ 23, 25, 48 ] = [ 15, 32, 47 ] = [ 8, 37, 45 ] = [ 3, 40, 43 ]

It has been proved that there are solutions for any j. [3] [5]
A.Golden gave method on how to obtained solution chains of any j. [5]
By applying Theorem 9, solution chains of ( k = -2, -4 ) can be obtained for any j.

( 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]
By applying Theorem 9, solution chains of ( k = -1, -2, -3 ) can be obtained for any j.

[ 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.
By applying Theorem 9, solution chains of ( k = -1, -2, -4 ) can be obtained for j = 3.

( k = 1, 3, 5 )
- [ 47, 183, 217, 303 ] = [ 73, 127, 263, 287 ] = [ 79, 119, 271, 281 ]
- [ 21, 169, 183, 273 ] = [ 43, 113, 229, 261 ] = [ 53, 99, 241, 253 ]
  - First known solution chains of j = 3 , by Chen Shuwen in 2017-06-13.
  - By applying Theorem 9, solution chains of ( k = -1, -3, -5 ) can be obtained for j = 3.
( k = 1, 2, 3, 4, 5 ).

[ 0, 23, 25, 71, 73, 96 ] = [ 1, 16, 33, 63, 80, 95 ] = [ 3, 11, 40, 56, 85, 93 ] = [ 5, 8, 45, 51, 88, 91 ]

It has been proved that there are solutions for any j. [3] [5]
A.Golden gave method on how to obtained solution chains of any j. [5]
By applying Theorem 9, solution chains of ( k = -1, -2, -3, -4, -5 ) can be obtained for any j.

( k = 0 )
- [ 1, 24 ] = [ 2, 12 ] = [ 3, 8 ] = [ 4, 6 ]
  - It is obvious that there are infinity solutions of this type for any j.
( k = 0, 1 )
- [ 520, 1078, 1080 ] = [ 525, 1001, 1152 ] = [ 528, 980, 1170 ] = [ 546, 900, 1232 ] = [ 560, 858, 1260 ] = [ 594, 784, 1300 ] = [ 630, 728, 1320 ] = [ 640, 715, 1323 ]
  - Found by Chen Shuwen in 2017-06-14, based on the solution chains of ( k = 1, 3 ) by Jaroslaw Wroblewski.
  - By using Jaroslaw Wroblewski's result on ( k = 1, 3 ), we can obtain the solution chains of ( k = 0, 1 ) for j = 65.
  - By applying Theorem 9, solution chains of ( k = -1, 0 ) can be obtained for j = 3.
( k = 0, 1, 2 )
- [ 9, 28, 30, 65 ] = [ 10, 20, 39, 62 ] = [ 13, 14, 45, 60 ]
  - First known solution chains of j = 3, by Chen Shuwen in 2017-06-19.
  - By applying Theorem 9, solution chains of ( k = -2, -1, 0 ) can be obtained for j = 3.
( k = -1, 1 )
- [ 3, 40 ] = [ 4, 15, 24 ] = [ 5, 8, 30 ]

Integer solution chains of the m=n case

When m=n, integer solution chains have been found to 6 + 6 = 12 types. Among them, 8 types have been proved to be solvable for any j.

( h = 1, 2, 4 )

[ -48, 23, 25 ] = [ -47, 15, 32 ] = [ -45, 8, 37 ] = [ -43, 3, 40 ]

It has been proved that there are solutions for any j. [3] [5]
A.Golden gave method on how to obtained solution chains of any j. [5]
By applying Theorem 9, solution chains of ( h = -4, -2, -1 ) can be obtained for any j.

( 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.
By applying Theorem 9, solution chains of ( h = -5, -3, -2, -1 ) can be obtained for j = 3.

( h = 0, 1, 3 )
- [ -242606760, 80868920, 161737840 ] = [ -517561088, 12130338, 505430750 ] = [ -281011375, 48582831, 232428544 ]
  - First known solution chains of j = 3, by Ajai Choudhry in 2010 (2001?).[1002]
  - By applying Theorem 9, solution chains of ( h = -3, -1, 0 ) can be obtained for j = 3.
( h = -1, 1, 2 )
- [ -91, 182, 182 ] = [ -78, 117, 234 ] = [ -63, 84, 252 ] = [ -52, 65, 260 ]
  - Numerical example, by Ajai Choudhry's parametric solution in 2011. [1101]
  - Parametric solution of multigrade chains of arbitrary length is obtained by Ajai Choudhry in 2011. [1101]
  - By applying Theorem 9, solution chains of ( h = -2, -1, 1 ) can be obtained for any j.
( h = -1, 1, 2, 3 )
- [ -1105, 2210, 3315, 6630 ] = [ -850, 1275, 4250, 6375 ] = [ -975, 1625, 3900, 6500 ] = [ -663, 884, 4641, 6188 ]
  - Numerical example, by Ajai Choudhry's parametric solution in 2011. [1101]
  - Parametric solution of multigrade chains of arbitrary length is obtained by Ajai Choudhry in 2011. [1101]
  - By applying Theorem 9, solution chains of ( h = -3, -2, -1, 1 ) can be obtained for any j.
( h = -1, 1, 2, 3, 4, 5 )
- [ -7657, 15314, 30628, 76570, 91884, 114855 ] = [ -8246, 20615, 24738, 82460, 86583, 115444 ] = [ -6916, 12103, 34580, 72618, 95095, 114114 ] = [ -5642, 8463, 39494, 67704, 98735, 112840 ]
  - Numerical example, by Ajai Choudhry in 2011. [1101]
  - Parametric solution of multigrade chains of arbitrary length is obtained by Ajai Choudhry in 2011. [1101]
  - By applying Theorem 9, solution chains of ( h = -5, -4, -3, -2, -1, 1 ) can be obtained for any j.