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

• We will also consider 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 k > 0 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.
• ( 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]
• ( 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.
• It has been proved that there are solutions for any  j. [1] [3]
• ( 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]
• ( 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.
• ( 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 ]
• 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]
• ( 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 ).
• [ 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]

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 ]
• 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]
• ( 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.