[ eslpower.org ] Equal Sums of Like Powers

On the Generalization of the Prouhet-Tarry-Escott Problem

Introduction		(2016-10-15)
Ideal non-negative integer solutions (k>0)		(2023-05-12)
Ideal integer solutions (h>0)		(2023-05-12)
Ideal prime solutions		(2023-04-30)
The Prouhet-Tarry-Escott problem		(2023-04-30)
Multigrade Chains		(2001-10-16)
Equal products and equal sums of like powers		(2023-05-02)
Ideal positive integer solutions (k<0)		(2023-05-02)
Ideal integer solutions (h<0)		(2023-05-12)
Chen Shuwen's Notebooks		(2021-08-16)	New!!!
Algebraic Identities		(2019-02-13)
Other results on equal sums of like powers		(2001-03-31)
Unsolved problems and conjectures		(2001-10-16)
Discussion and comments		(2001-05-06)
Links and References		(2017-04-15)
Records on the Generalization of Prouhet-Tarry-Escott Problem		(2023-05-12)
What is new on this site		(2023-05-12)

Equal Sums of Like Powers

Introduction

• Equal sums of like powers is a Diophantine system of the form
a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k      ( k = k₁ , k₂ , ..., k_n )
This system will be denoted here as:
[ a₁ , a₂ , ... , a_m ] = [ b₁ ,b₂ , ... , b_m ]       ( k = k₁ , k₂ , ... , k_n ) 
• 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 these pages, we will only consider integral, non-trivial, primitive solutions of the above system.
• First known solution: the first found solution
• Smallest solution: proved by computer search, max {a_i, b_i } is smallest.
• Smallest known solution: among all presently known solutions, max {a_i, b_i } is smallest.
• Since when m=n, no any non-negative integer solution is found for any ( k = k₁ , k₂ , ..., k_n ) of this system, we call the non-negative integer solution of m=n+1 case as ideal non-negative integer solution.
  • When k < 0, all {a_i, b_i } must be positive integers, we also call this positive integer solution of m=n+1 case as ideal positive integer solution.
  • As a specical solution, if all {a_i, b_i } are primes, we call this prime solution of m=n+1 case as ideal prime solution.
• Since when m<n, no any integer solution is found for any ( k = k₁ , k₂ , ..., k_n ) of this system, we call the integer solution of m=n case as ideal integer solution.

Equal Sums of Like Powers

Ideal non-negative integer solutions of

When m=n+1, non-negative integer solutions have been found to 42 + 42 + 24 + 23 + 33 = 164 types so far.

Range of k	Solved Types	Reference
all k  > 0	42	See below.
all k  < 0	42	See Ideal positive integer solutions of k < 0 case
k1= 0 and all others k  > 0	24	See Equal products and equal sums of like powers
k1= 0 and all others k  < 0	23	See Ideal positive integer solutions of k < 0 case
k1< 0 and kn > 0	33	See Ideal positive integer solutions of k < 0 case

When all k > 0 and m=n+1, idea non-negative integer solutions have been found to the following 42 types:

• ( k = 1 )
• [ 0, 2 ] = [ 1, 1 ]
• [ 1, 4 ] = [ 2, 3 ]
• [ 0, 7 ] = [ 1, 6 ] = [ 2, 5 ] = [ 3, 4 ]
• ( k = 2 )
• [ 0, 5 ] = [ 3, 4 ]
• Smallest solution.
• [ 2, 11 ] = [ 5, 10 ]
• [ 13, 91 ] = [ 23, 89 ] = [ 35, 85 ] = [ 47, 79 ] = [ 65, 65 ]
• ( k = 3 )
• [ 1, 12 ] = [ 9, 10 ]
• Smallest solution.
• [ 2421, 19083 ] = [ 5436, 18948 ] = [ 10200, 18072 ] = [ 13322, 16630 ]
• By E.Rosenstiel, J.A.Dardis and C.R.Rosenstiel in 1991.
• ( k = 4 )
• [ 59, 158 ] = [ 133, 134 ]
• First known solution, smallest solution, by Euler in 1772.
• [ 7, 239 ] = [ 157, 227 ]
• ( k = 1, 2 ) 
• [ 0, 3, 3 ] = [ 1, 1, 4 ]
• Smallest solution.
• [ 1, 8, 8 ] = [2, 5, 10 ]
• [ 0, 16, 17 ] = [ 1, 12, 20 ] = [ 2, 10, 21 ] = [ 5, 6, 22 ]
• ( k = 1, 3 )
• [ 0, 7, 8 ] = [ 1, 5, 9 ]
• Smallest solution.
• [ 2, 10, 12 ] = [ 3, 8, 13 ]
• [ 2, 52, 58 ] = [ 4, 46, 62 ] = [ 13, 32, 67 ] = [22, 22, 68 ] 
• Solution chain, by Chen Shuwen in 1995.
• ( k = 1, 4 ) 
• [ 3, 25, 38 ] = [ 7, 20, 39 ]
• First found solution, smallest solution, by Chen Shuwen in 1995.
• [ 24, 201, 216 ] = [ 66, 132, 243 ] = [ 73, 124, 244 ] 
• Solution chain, by Chen Shuwen in 1997.
• ( k = 1, 5 )
• [ 39, 92, 100 ] = [ 49, 75, 107 ]
• First known solution, by A.Moessner in 1939.
• [ 3, 54, 62 ] = [ 24, 28, 67 ]
• Smallest solution, by L.J.Lander, T.R.Parkin and J.L.Selfridge in 1967.
• ( k = 2, 3 ) 
• [ 2251, 35478, 37243 ] = [ 19747, 19747, 43254 ]
• First known solution, by A.Golden in 1949.
• [ 0, 37, 62 ] = [ 21, 26, 64 ]
• Smallest solution, by Chen Shuwen in 1995.
• ( k = 2, 4 )
• [ 0, 7, 7 ] = [ 3, 5, 8 ]
• Smallest solution.
• [ 6, 23, 25 ] = [ 10, 19, 27 ]
• [ 23, 25, 48 ] = [ 15, 32, 47 ] = [ 8, 37, 45 ] = [ 3, 40, 43 ]
• ( k = 2, 6 )
• [ 3, 19, 22 ] = [ 10, 15, 23 ]
• First known solution, smallest solution, by Subba-Rao in 1934.
• [ 15, 52, 65 ] = [ 36, 37, 67 ]
• ( k = 1, 2, 3 ) 
• [ 0, 4, 7, 11 ] = [ 1, 2, 9, 10 ]
• Smallest solution.
• [ 0, 28, 29, 57 ] = [ 1, 21, 36, 56 ] = [ 2, 18, 39, 55 ] = [ 6, 11, 46, 51 ]
• Symmetric solution chain.
• [ 0, 87, 93, 214 ] = [ 9, 52, 123, 210 ] = [ 24, 30, 133, 207 ]
• First known non-symmetric solution chain, by Chen Shuwen in 1997.
• ( k = 1, 2, 4 ) 
• [ 2, 7, 11, 15 ] = [ 3, 5, 13, 14 ]
•  First found solution, smallest solution, by Chen Shuwen in 1995.
• [ 0, 7, 14, 19 ] = [ 1, 5, 16, 18 ] 
• [ 14, 37, 39, 64 ] = [ 16, 29, 46, 63 ] = [ 19, 24, 49, 62 ]
• Solutions chain, by Chen Shuwen in 1995.
• ( k = 1, 2, 5 )
• [ 53, 113, 156, 204 ] = [ 74, 78, 183, 191 ]
• First found solution, by Chen Shuwen in 1995.
• [ 1, 28, 39, 58 ] = [ 8, 14, 51, 53 ]
• Smallest solution, by Chen Shuwen.
• ( k = 1, 2, 6 ) 
• [ 7, 43, 69, 110 ] = [ 18, 25, 77, 109 ]
• First known solution, by Chen Shuwen in 1995, based on the data obtained in 1966 by Lander, Parkin and Selfridge.
• [ 7, 16, 25, 30 ] = [ 8, 14, 27, 29 ]
• Smallest solution, by Chen Shuwen.
• ( k = 1, 3, 4 )
• [ 3, 140, 149, 252 ] = [ 50, 54, 201, 239 ] 
• First known solution, by Chen Shuwen in 1995.
• [ 127, 324, 1740, 2023 ] = [ 24, 439, 1711, 2040 ]
• By Chen Shuwen.
• ( k = 1, 3, 5 )
• [ 1, 13, 17, 23 ] = [ 3, 9, 21, 21 ]
• Smallest solution.
• [ 0, 24, 33, 51 ] = [ 7, 13, 38, 50 ]
• ( k = 1, 3, 7 )
• [ 1741, 2435, 3004, 3476 ] = [ 1937, 2111, 3280, 3328 ]
• First known solution, by Ajai Choudhry in 1999.
• [ 184, 443, 556, 698 ] = [ 230, 353, 625, 673 ]
• Smallest solution, by Nuutti Kuosa, Jean-Charles Meyrignac, and Chen Shuwen, in 1999.
• ( k = 2, 3, 4 ) 
• [ 7001616, 10868299, 31439172, 34940503 ] = [ 7527024, 10393591, 31599228, 34831147 ] 
• First known solution, by Chen Shuwen in 1995.
• [ 43, 486, 815, 1058 ] = [ 242, 335, 907, 1014 ]
• Smallest known solution, by Ajai Choudhry in 2001.
• ( k = 2, 4, 6 )
• [ 2, 16, 21, 25 ] = [ 5, 14, 23, 24 ]
• Smallest solution.
• [ 7, 24, 25, 34 ] = [ 14, 15, 31, 32 ]
• ( k = 1, 2, 3, 4 ) 
• [ 0, 4, 8, 16, 17 ] = [ 1, 2, 10, 14, 18 ]
• Smallest solution.
• [ 0, 9, 13, 26, 32 ] = [ 2, 4, 20, 21, 33 ]
• First known non-symmetric solution, by J.L.Burchnall & T.W.Chaundy in 1937.
• ( k = 1, 2, 3, 5 )
• [ 15, 25, 55, 55, 73] = [ 13, 31, 43, 67, 69 ]
• First known solution, by Chen Shuwen in 1995.
• [ 1, 8, 13, 24, 27 ] = [ 3, 4, 17, 21, 28 ] 
• Smallest solution, by Chen Shuwen.
• ( k = 1, 2, 3, 6 ) 
• [ 7, 18, 55, 69, 81 ] = [ 9, 15, 61, 63, 82 ] 
• First known solution, smallest solution, by Chen Shuwen in 1999.
• [ 7, 27, 53, 90, 106 ] = [ 10, 21, 58, 87, 107 ] 
• By Chen Shuwen.
• ( k = 1, 2, 3, 7 ) 
• [ 261, 816, 821, 1601, 1756 ] = [ 271, 711, 926, 1581, 1766 ] 
• First known solution, by Chen Shuwen in 2022.
• ( k = 1, 2, 4, 6 )
• [ 3, 7, 10, 16, 16 ] = [ 4, 5, 12, 14, 17 ]
• First known solution, by G.Palama in 1953.
• [ 7, 25, 31, 56, 57 ] = [ 8, 21, 35, 53, 59 ] 
• By Chen Shuwen.
• ( k = 1, 3, 5, 7 ) 
• [ 3, 19, 37, 51, 53 ] = [ 9, 11, 43, 45, 55 ]
• First known solution, smallest solution, by A.Golden in 1940's.
• [ 0, 34, 58, 82, 98 ] = [ 13, 16, 69, 75, 99 ]
• By A.Letac in 1940's.
• ( k = 2, 4, 6, 8 )
• [ 12, 11881, 20231, 20885, 23738 ] = [ 436, 11857, 20449, 20667 , 23750 ]
• First known solution, by A.Letac in 1940's.
• [ 71, 131, 180, 307, 308 ] = [ 99, 100, 188, 301 , 313 ]
• Smallest known solution, by Peter Borwein, Petr Lisonek and Colin Percival in 2000.
• ( k = 1, 2, 3, 4, 5 )
• [ 0, 5, 6, 16, 17, 22 ] = [ 1, 2, 10, 12, 20, 21 ]
• First known solution, by G.Tarry in 1912.
• [ 0, 19, 25, 57, 62, 86 ] = [ 2, 11, 40, 42, 69, 85 ]
• First known non-symmetric solution, by A.Golden in 1944.
• [ 0, 23, 25, 71, 73, 96 ] = [ 1, 16, 33, 63, 80, 95 ] = [ 3, 11, 40, 56, 85, 93 ] = [ 5, 8, 45, 51, 88, 91 ]
• ( k = 1, 2, 3, 4, 6 ) 
• [ 116, 166, 206, 331, 336, 411 ] = [ 131, 136, 236, 291, 366, 406 ]
• First known solution, by Chen Shuwen in 1995.
• [ 11, 23, 24, 47, 64, 70 ] = [ 14, 15, 31, 44, 67, 68 ] 
• Smallest known solution, by Chen Shuwen.
• ( k = 1, 2, 3, 4, 7 ) 
• [ 34, 133, 165, 299, 332, 366 ] = [ 35, 124, 177, 286, 353, 354 ]
• First known solution, by Chen Shuwen in 2022.
• ( k = 1, 2, 3, 5, 7 ) 
• [ 87, 233, 264, 396, 496, 540 ] = [ 90, 206, 309, 366, 522, 523 ] 
• First known solution, by Chen Shuwen in 1999.
• ( k = 1, 2, 4, 6, 8 ) 
• [ 1, 7, 17, 30, 31, 36 ] = [ 3, 4, 19, 27, 34, 35 ] 
• First known solution, by Chen Shuwen in 1995.
• [ 64, 169, 184, 277, 347, 417 ] = [ 69, 139, 233, 248, 353 , 416 ] 
• By Chen Shuwen.
• ( k = 1, 3, 5, 7, 9 ) 
• [ 7, 91, 173, 269, 289, 323 ] = [ 29, 59, 193, 247, 311, 313 ] 
• First known solution, by Chen Shuwen in 2000.
• ( k = 2, 4, 6, 8, 10 ) 
• [ 22, 61, 86, 127, 140, 151 ] = [ 35, 47, 94, 121, 146, 148 ]
• First known solution, by Nuutti Kuosa, Jean-Charles Meyrignac and Chen Shuwen, in 1999.
• ( k = 1, 2, 3, 4, 5, 6 )
• [ 0, 18, 27, 58, 64, 89, 101 ] = [ 1, 13, 38, 44, 75, 84 , 102 ]
• First known solution, by E.B.Escott in 1910.
• [ 0, 18, 19, 50, 56, 79, 81 ] = [ 1, 11, 30, 39, 68, 70 , 84 ] 
• Smallest known solution, first known non-symmetric solution, by Chen Shuwen in 1997.
• ( k = 1, 2, 3, 4, 5, 7 ) 
• [ 4727, 4972, 5267, 5857, 5972, 6557, 6667 ] = [ 4772, 4867 , 5477, 5567, 6172, 6457, 6707 ] 
• First known solution, by Chen Shuwen in 1995.
• [ 43, 169, 295, 607, 667, 1105, 1189 ] = [ 79, 97, 379, 505, 727, 1093, 1195 ] 
• Smallest known solution, by Chen Shuwen 1999.
• ( k = 1, 2, 3, 5, 7, 9 ) 
• [ 7, 89, 91, 251, 253, 341, 373 ] = [ 29, 31, 151, 193, 311, 313, 377 ] 
• First known solution, smallest solution, by Chen Shuwen in 2023.
• [ 269, 397, 409, 683, 743, 901, 923 ] = [ 299, 313, 493, 613, 827, 839, 941 ] 
• Second known solution, by Chen Shuwen in 2023.
• ( k = 1, 2, 3, 4, 5, 6, 7 )
• [ 0, 4, 9, 23, 27, 41, 46, 50 ] = [ 1, 2, 11, 20, 30, 39 , 48, 49 ]
• First known solution, smallest known solution, by G.Tarry in 1913.
• [ 0, 7, 23, 50, 53, 81, 82, 96 ] = [ 1, 5, 26, 42, 63, 72, 88, 95 ] 
• First known non-symmetric solution, by Chen Shuwen in 1997.
• ( k = 1, 2, 3, 4, 5, 6, 8 )
• [ 77, 159, 169, 283, 321, 443, 447, 501 ] = [ 79, 137, 213, 237, 363, 399, 481, 491 ] 
• First known solution, by Chen Shuwen in 1999.
• ( k = 1, 2, 3, 4, 5, 6, 7, 8 )
• [ 0, 24, 30, 83, 86, 133, 157, 181, 197 ] = [ 1, 17, 41, 65, 112, 115, 168, 174, 198 ]
• First known solution, by A.Letac in 1940's.
• [ 0, 26, 42, 124, 166, 237, 293, 335, 343 ] = [ 5, 13, 55, 111, 182, 224, 306, 322, 348 ]
• By A.Letac in 1940's.
• ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 )
• [ 0, 3083, 3301, 11893, 23314, 24186, 35607, 44199, 44417, 47500 ] = [ 12, 2865, 3519, 11869, 23738, 23762, 35631, 43981, 44635, 47488 ]
• First known solution, by A.Letac in 1940's.
• [ 0, 12, 125, 213, 214, 412, 413, 501, 614, 626 ] = [ 5, 6, 133, 182, 242, 384, 444, 493, 620, 621 ]
• Smallest known solution, by Peter Borwein, Petr Lisonek and Colin Percival in 2000.
• ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 )
• [ 0, 11, 24, 65, 90, 129, 173, 212, 237, 278, 291, 302 ] = [ 3, 5, 30, 57, 104, 116, 186, 198, 245, 272, 297, 299 ]
• First known solution, by Nuutti Kuosa, Jean-Charles Meyrignac and Chen Shuwen, in 1999.

Equal Sums of Like Powers

Ideal integer solutions of

When m=n, no non-negative solution is found, but to the following 15 + 15 + 6 + 6 + 23 = 65 types, there are solutions in integer.

Range of h	Solved Types	Reference
all h  > 0	15	See below.
all h  < 0	15	See Ideal integer solutions (h<0)
h1= 0 and all others h  > 0	6	See Equal products and equal sums of like powers
h1= 0 and all others h  < 0	6	See Ideal integer solutions (h<0)
h1< 0 and hn > 0	23	See Ideal integer solutions (h<0)

When all h > 0 and m=n, idea integer solutions have been found to the following 15 types:

• ( h = 1, 2, 4 )
• [ -10, 1, 9 ] = [ -11, 5, 6 ]
• [ -48, 23, 25 ] = [ -47, 15, 32 ] = [ -45, 8, 37 ] = [ -43, 3, 40 ]
• ( h = 1, 2, 6 )
• [ -372, 43, 371 ] = [ -405, 140, 307 ]
• First known solution, by Chen Shuwen in 1997.
• [ -300, 83, 211 ] = [ -124, -185, 303 ]
• Smallest solution, by Ajai Choudhry in 1999.
• ( h = 1, 3, 4 ) 
• [ -3254, 5583, 5658 ] = [ -1329, 2578, 6738 ]
• First known solution, by Ajai Choudhry in 1991.
• ( h = 2, 3, 4 ) 
• [ -815, 358, 1224 ] = [ -776, -410, 1233 ]
• First known solution, by Ajai Choudhry in 2001.
• ( h = 1, 2, 3, 5 )
• [ -38, -13, 0, 51 ] = [ -33, -24, 7, 50 ]
• First known solution, by A.Golden in 1944.
• [ -197, -23, -11, 231 ] = [ -179, -93, 49, 223 ] = [ -149, -137, 69, 217 ]
• Solution chain, by Chen Shuwen.
• ( h = 1, 2, 4, 6 )
• [ -5, -14, 23, 24 ] = [ -16, -2, 21, 25 ]
• First known solution, by G.Palama in 1953.
• [ -41, 5, 23, 48 ] = [ -43, 15, 16, 47 ]
• ( h = 1, 2, 3, 4, 6 )
• [ -23, -10, -5, 14, 24 ] = [ -21, -16, 2, 10, 25 ]
• First known solution, by A.Golden in 1944.
• [ -17, -5, -4, 12, 14 ] = [ -16, -10, 3, 7, 16 ]
• ( h = 1, 2, 3, 5, 7 )
• [ -59, -5, -1, 33, 57 ] = [ -55, -23, 13, 39, 51 ]
• First known solution, by G.Palama in 1953.
• [ -55, -11, 3, 37, 51 ] = [ -53, -19, 9, 43, 45 ]
• By Chen Shuwen.
• ( h = 1, 2, 4, 6, 8 )
• [ -12, -20231, 11881, 20885, 23738 ] = [ -20449, 436, 11857, 20667, 23750 ]
• First known solution, by A.Letac in 1940's.
• [ 71, 131, -180, -307, 308 ] = [ -99, -100, 188, 301, -313 ]
• ( h = 1, 2, 3, 4, 5, 7 ) 
• [ -89, -41, -31, 33, 45, 83 ] = [ -87, -55, -1, 3, 61, 79 ]
• First known solution, by A.Golden in 1944.
• [ -71, -44, -20, 31, 37, 67 ] = [ -68, -53, 1, 4, 55, 61 ]
• By Chen Shuwen.
• ( h = 1, 2, 3, 4, 6, 8 )
• [ -42, -37, -1, 30, 57, 61 ] = [ -47, -27, -9, 35, 54, 62 ]
• First known solution, by Jaroslaw Wroblewski and Chen Shuwen in 2019.
• ( h = 1, 2, 3, 5, 7, 9 )
• [ -247, -193, -59, 91, 289, 323 ] = [ -269, -173, -7, 29, 311, 313 ]
• First known solution, by Chen Shuwen in 2000.
• ( h = 1, 2, 4, 6, 8, 10 ) 
• [ -127, 22, 61, 86, 140, 151 ] = [ -94, -35, 47, 121, 146, 148 ]
• First known solution, by Nuutti Kuosa, Jean-Charles Meyrignac and Chen Shuwen in 1999.
• ( h = 1, 2, 3, 4, 5, 6, 8 ) 
• [ -303, -177, -170, 47, 89, 250, 264 ] = [ -296, -226, -93 , -30, 173, 187, 285 ]
• First known solution, by Chen Shuwen in 1997.
• [ -725, -683, -424, 185, 479, 486, 682 ] = [ -746, -648, -445 , 241, 346, 605, 647 ]
• By Chen Shuwen.
• ( h = 1, 2, 3, 4, 5, 6, 7, 9 )
• [ -48, -44, -23, -7, 14, 23, 39, 46 ] = [ -49, -42, -26, 1, 4, 32, 33, 47 ]
• First known solution, by Chen Shuwen in 1997.
• [ -323, -253, -189, -31, 65, 135, 283, 313 ] = [ -325, -239 , -207, 7, 19, 153, 275, 317 ]
• By Chen Shuwen.

Equal Sums of Like Powers

Ideal prime solutions of

When m=n+1, ideal prime solutions have been found to the following 24 types:

• ( k = 1 )
• [ 3, 7 ] = [ 5, 5 ]
• [ 3, 13 ] = [ 5, 11 ]
• [ 7, 53 ] = [ 13, 47 ] = [ 17, 43 ] = [ 19, 41 ] = [ 23, 37 ] = [ 29, 31 ]
• ( k = 2 )
• [ 7, 17 ] = [ 13, 13 ]
  • First found by Jean-Charles Meyrignac in 2000.[0001]
• [ 1, 47 ] = [ 19, 43 ] = [ 23, 41 ] = [ 29, 37 ]
• [ 53, 281 ] = [ 71, 277 ] = [ 97, 269 ] = [ 137, 251 ] = [ 157, 239 ] = [ 193, 211 ]
  • Prime solution chain, by Chen Shuwen in 2016.
• ( k = 3 )
• [ 61, 1823 ] = [ 1049, 1699 ]
  • First found by Jean-Charles Meyrignac in 2000.[0001]
• [ 31, 1867 ] = [ 397, 1861 ]
  • By Chen Shuwen in 2016.
• ( k = 4 )
• [ 7, 239 ] = [ 157, 227 ]
  • First found by Jean-Charles Meyrignac in 2000.[0001]
• ( k = 1, 2 ) 
• [ 43, 61, 67 ] = [ 47, 53, 71 ]
  • Albert H. Beiler, first known solution.[64] [65]
• [ 2707, 2719, 2729 ] = [ 2711, 2713, 2731 ]
  • The least solution with six consecutive primes, by Carlos Rivera before 2014.[64]
• [ 7, 31, 37 ] = [ 13, 19, 43 ]
  • Symmetric prime solution, by Chen Shuwen in 2016.
• [ 11, 107, 113 ] = [ 17, 83, 131 ] = [ 23, 71, 137 ]
  • Non-symmetric prime solution, by Chen Shuwen in 2016.
• ( k = 1, 3 )
• [ 19, 179, 241 ] = [ 43, 139, 257 ]
• By Chen Shuwen in 2016.
• ( k = 1, 4 ) 
• [ 89, 811, 997 ] = [ 251, 577, 1069 ]
• [ 337, 1609, 1637 ] = [ 613, 1097, 1873 ]
• By Chen Shuwen in 2016.
• ( k = 1, 5 )
• [ 6379, 3833, 2657 ] = [ 6089, 5003, 1777 ]
• By Jaroslaw Wroblewski in 2002.[0001]
• ( k = 2, 4 )
• [ 89, 277, 377 ] = [ 139, 233, 353 ]
• [ 10, 963, 1097 ] = [ 177, 907, 1130 ] = [ 313, 822, 1165 ]
  • Both by Chen Shuwen in 2016.
• ( k = 2, 6 )
• [ 379, 2837, 4253 ] = [ 1237, 2531, 4283 ]
  • Equality of sums of 6th powers is first found by Aloril in 2002, using EulerNet program run by Jean–Charles Meyrignac.[0001]
  • Chen Shuwen noticed in 2016 that this solution has equal sums of 2nd powers.
• ( k = 1, 2, 3 ) 
• [ 281, 281, 1181, 1181 ] = [ 101, 641, 821, 1361 ]
  • Albert H. Beiler, first known solution.[64] [65]
• [ 59, 137, 163, 241 ] = [ 61, 127, 173, 239 ] = [ 71, 103, 197, 219 ]
  • Prime Solution Chains, by Carlos Rivera in 1999.[64]
• [ 91335911, 91335961, 91336031, 91336081 ] = [ 91335931, 91335931, 91336061, 91336061 ]
  • Solution with six consecutive primes, by Carlos Rivera before 2014.[64]
• [ 5, 11, 13, 19 ] = [ 7, 7, 17, 17 ]
• Symmetric prime solution, with six consecutive primes, by Qiu Min in 2016.[1601]
• [ 47, 101, 113, 179 ] = [ 59, 71, 137, 173 ]
  • Non-symmetric prime solution, by Chen Shuwen in 2016.
• ( k = 1, 2, 4 )
  • [ 41, 43, 101, 103 ] = [ 29, 69, 79, 113 ]
    • By Chen Shuwen in 2017.
• ( k = 1, 3, 5 )
• [ 13, 59, 61, 101 ] = [ 29, 31, 83, 97 ]
  • Torbjorn Alm found the equality of sums of 5th powers in 2000, using EulerNet program run by Jean–Charles Meyrignac.[0001]
  • Chen Shuwen found independently that it is a multigrade prime solution in 2016.
• [ 17, 71, 73, 109 ] = [ 29, 43, 97, 101 ]
  • Same as previous one, by Torbjorn Alm, Jean–Charles Meyrignac and Chen Shuwen in 2016.
• ( k = 2, 4, 6 )
• [ 317, 541, 953, 1049 ] = [ 139, 719, 827, 1087 ]
• By Chen Shuwen in 2017.
• ( k = 1, 2, 3, 4 ) 
• [ 401, 521, 641, 881, 911 ] = [ 431, 461, 701, 821, 941 ]
  • Symmetric prime solution, by Chen Shuwen in 2016.
• [ 337, 607, 727, 1117, 1297 ] = [ 397, 457, 937, 967, 1327 ]
  • Non-symmetric prime solution, by Chen Shuwen in 2016.
• ( k = 1, 2, 3, 5 )
• [ 23, 59, 149, 157, 211 ] = [ 17, 19, 113, 191, 199 ]
• By Chen Shuwen in 2017.
• ( k = 1, 2, 4, 6 )
• [ 83, 149, 337, 439, 503 ] = [ 71, 173, 313, 463, 491 ]
• By Chen Shuwen in 2017.
• ( k = 1, 3, 5, 7 ) 
• [ 227, 281, 313, 367, 373 ] = [ 241, 257, 331, 353, 379 ]
  • First known solution, by Carlos Rivera before 31/08/1999.[64]
• [ 19, 101, 157, 239, 251 ] = [ 31, 79, 173, 227, 257 ]
  • Laurent Lucas found the equality of sums of 5th powers in 2000, using EulerNet program run by Jean–Charles Meyrignac.[0001]
  • Chen Shuwen found independently that it is a multigrade prime solution in 2016.
• [ 29, 179, 311, 433, 503 ] = [ 83, 103, 359, 401, 509 ]
  • By Chen Shuwen in 2016.
• ( k = 1, 2, 3, 4, 5 )
• [ 277, 937, 1069, 2389, 2521, 3181 = 409, 541, 1597, 1861, 2917, 3049 ]
  • First known solution, by T.W.A. Baumann in 1999-08-31.[64]
• [ 17, 37, 43, 83, 89, 109 ] = [ 19, 29, 53, 73, 97, 107 ]
  • Symmetric prime solution, by Qiu Min in 2016.[1601]
• [ 31, 3541, 6661, 13291, 14071, 17971 ] = [ 421, 2371, 9391, 9781, 16411, 17191 ]
• Non-symmetric prime solution, by Chen Shuwen in 2016.
• ( k = 1, 2, 3, 4, 5, 6 )
• [ 83, 191, 197, 383, 419, 557, 569 ] = [ 89, 149, 263, 317, 491, 503, 587 ]
  • First known solution, symmetric prime solution, by Qiu Min before Apr 2016.[1601]
  • Found independently by Chen Shuwen in Sep 2016.
• [ 18443, 90263, 126173, 249863, 273803, 373553, 421433 ] = [ 22433, 70313, 170063, 194003, 317693, 353603, 425423 ]
  • Symmetric prime solution, by Chen Shuwen in 2016.
• ( k = 1, 2, 3, 4, 5, 6, 7 )
• [ 12251, 34511, 42461, 80621, 102881, 141041, 148991, 171251 ] = [ 13841, 26561, 59951, 63131, 120371, 123551, 156941, 169661 ]
  • First known solution, by T.W.A. Baumann in 1999-09-01. [64]
• [ 400033, 453073, 519373, 705013, 758053, 943693, 1009993, 1063033 ] = [ 413293, 426553, 545893, 665233, 797833, 917173, 1036513, 1049773 ]
  • Symmetric prime solution, by Chen Shuwen in 2016.
• [ 10289, 14699, 27509, 41579, 42839, 65309, 68669, 77699 ] = [ 10709, 13859, 29399, 36749, 46829, 63419, 70139, 77489 ] 
• Non-symmetric prime solution, by Chen Shuwen in 2016.
• ( k = 1, 2, 3, 4, 5, 6, 7, 8 )
• [ 3522263, 4441103, 5006543, 7904423, 9388703, 11897843, 13876883, 15361163, 15643883 ] = [ 3698963, 3981683, 5465963, 7445003, 9954143, 11438423, 14336303, 14901743, 15820583 ]
  • Symmetric prime solution, by Chen Shuwen in 2016.
• ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 )
• [ 2589701, 2972741, 6579701, 9388661, 9420581, 15740741, 15772661, 18581621, 22188581, 22571621 ] = [ 2749301, 2781221, 6835061, 8399141, 10314341, 14846981, 16762181, 18326261, 22380101, 22412021 ]
• Symmetric prime solution, by Chen Shuwen in 2016.
• ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 )
• [ 32058169621, 32367046651, 32732083141, 33883352071, 34585345321, 35680454791, 36915962911, 38011072381, 38713065631, 39864334561, 40229371051, 40538248081 ] = [ 32142408811, 32198568271, 32900561521, 33658714231,34978461541, 35315418301, 37280999401, 37617956161,38937703471, 39695856181, 40397849431, 40454008891 ]
• Symmetric prime solution, by Jaroslaw Wroblewski in 2023.