[ eslpower.org ]
Equal Sums of Like Powers
On the Generalization of the
ProuhetTarryEscott Problem
a_{1}^{k}
+ a_{2}^{k} + ... + a_{m}^{k}
= b_{1}^{k} + b_{2}^{k}
+ ... + b_{m}^{k}
( k = k_{1} , k_{2}
, ... , k_{n} )
Introduction
 Equal sums of like powers is a Diophantine system
of the form
 a_{1}^{k} + a_{2}^{k}
+ ... + a_{m}^{k} = b_{1}^{k} +
b_{2}^{k} + ... + b_{m}^{k}
( k = k_{1} , k_{2} ,
..., k_{n} )
 This system will be denoted here as:
 [ a_{1} , a_{2}
, ... , a_{m} ] = [ b_{1} ,b_{2}
, ... , b_{m} ]
( k = k_{1} , k_{2} , ... ,
k_{n} )
 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 these pages, we will only consider integral,
nontrivial, 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 nonnegative integer solution
is found for any ( k = k_{1}
, k_{2} , ..., k_{n} ) of this system,
we call the nonnegative integer solution of m=n+1 case as
ideal nonnegative 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 = k_{1} , k_{2} , ...,
k_{n} ) of this system, we call the integer solution of m=n
case as
ideal integer solution.
Ideal nonnegative integer solutions of
a_{1}^{k}
+ a_{2}^{k} + ... + a_{n}_{+1}^{k}
= b_{1}^{k} + b_{2}^{k}
+ ... + b_{n}_{+1}^{k}
( k = k_{1} , k_{2}
, ... , k_{n} )
When m=n+1, nonnegative integer
solutions have been found to 42 + 42 + 24 + 23 + 33 = 164 types so far.
When all k > 0 and m=n+1, idea nonnegative 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 ]
 [ 2, 11 ] = [ 5, 10 ]
 [ 13, 91 ] = [ 23, 89 ] = [ 35, 85 ] = [ 47,
79 ] = [ 65, 65 ]
 ( k = 3 )
 [ 1, 12 ] = [ 9, 10 ]
 [ 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 ]
 [ 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 ]
 [ 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 ]
 [ 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 SubbaRao
in 1934.
 [ 15, 52, 65 ] = [ 36, 37, 67 ]
 ( k = 1, 2, 3 )
 [ 0, 4, 7, 11 ] = [ 1, 2, 9, 10 ]
 [ 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 nonsymmetric 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
]
 ( k = 1, 3, 5 )
 [ 1, 13, 17, 23 ] = [ 3, 9, 21, 21 ]
 [ 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, JeanCharles
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 ]
 [ 7, 24, 25, 34 ] = [ 14, 15, 31, 32 ]
 ( k = 1, 2, 3, 4 )
 [ 0, 4, 8, 16, 17 ] = [ 1, 2, 10, 14, 18 ]
 [ 0, 9, 13, 26, 32 ] = [ 2, 4, 20, 21, 33 ]
 First known nonsymmetric 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
]
 ( 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 ]
 ( 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
]
 ( 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 nonsymmetric 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 ]
 ( 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, JeanCharles
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 nonsymmetric
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 nonsymmetric 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 ]
 ( 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, JeanCharles
Meyrignac and Chen Shuwen, in 1999.
Ideal integer solutions of
a_{1}^{h}
+ a_{2}^{h} + ... + a_{n}^{h}
= b_{1}^{h} + b_{2}^{h}
+ ... + b_{n}^{h}
( h = h_{1} , h_{2}
, ... , h_{n} )
When m=n, no nonnegative solution is found, but to the following 15 + 15 +
6 + 6 + 23 = 65 types, there are
solutions in integer.
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 ]
 ( 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 ]

( 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, JeanCharles
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 ]
 ( 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 ]
Ideal prime solutions of
a_{1}^{k}
+ a_{2}^{k} + ... + a_{n}_{+1}^{k}
= b_{1}^{k} + b_{2}^{k}
+ ... + b_{n}_{+1}^{k}
( k = k_{1} , k_{2}
, ... , k_{n} )
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 JeanCharles 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 JeanCharles Meyrignac in
2000.[0001]
 [ 31, 1867 ] = [ 397, 1861 ]
 ( k = 4 )
 [ 7, 239 ] = [ 157, 227 ]
 First found by JeanCharles 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 ]
 Nonsymmetric prime solution, by Chen Shuwen
in 2016.
 ( k = 1, 3 )
 [ 19, 179, 241 ] = [ 43, 139, 257 ]
 ( k = 1, 4 )
 [ 89, 811, 997 ] = [ 251, 577, 1069 ]
 [ 337, 1609, 1637 ] = [ 613, 1097, 1873 ]
 ( 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 ]
 Nonsymmetric prime solution, by Chen Shuwen
in 2016.
 ( k = 1, 2, 4 )
 [ 41, 43, 101, 103 ] = [ 29, 69, 79, 113 ]
 ( 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 ]
 ( 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 ]
 Nonsymmetric prime solution, by Chen Shuwen
in 2016.
 ( k = 1, 2, 3, 5 )
 [ 23, 59, 149, 157, 211 ] = [ 17, 19, 113, 191, 199
]
 ( k = 1, 2, 4, 6 )
 [ 83, 149, 337, 439, 503 ] = [ 71, 173, 313, 463, 491 ]
 ( 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 ]
 ( 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
19990831.[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 ]
 Nonsymmetric 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
19990901. [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 ]
 Nonsymmetric 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.
Last revised on May 12, 2023.
Chen Shuwen
Seekway Technology Limited,
Building 3, Qunhua HiTech Park,
15#, Qunhua Road,
Jiangmen City, Guangdong Province
People's Republic of China
Email:
138228328@qq.com
Copyright 19972023, Chen Shuwen