Equal Sums of Like Powers
On the Integer Solutions of the
Diophantine System
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,
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.
Non-negative integer solutions of
the m=n+1 case
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, non-negative integer
solutions have been found to 39 + 39 + 8 + 7 +1 = 94 types so far.
- When all k > 0 and m=n+1,
non-negative integer solutions have been found to the following 39 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, 70 ] = [ 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 Subba-Rao
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 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
]
- ( 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, 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.
- [ 975, 224368, 300495, 366448 ] = [ 37648, 202575,
337168, 344655 ]
- Smallest known solution, by Chen Shuwen.
- ( 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 non-symmetric solution, by J.L.Burchnall
& T.W.Chaundy in 1937.
- ( k = 1, 2, 3, 5 )
- [ 12, 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, 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 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, 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, 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, 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 ]
- ( 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.
Integer solutions of the m=n
case
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 h > 0 and m=n, no
non-negative solution is found, but to the following 13 types, there are
solutions in integer.
- ( 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 = 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.
- ( 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, 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 ]
- ( 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 ]
Last revised May 6, 2001.
Chen
Shuwen
Jiang Xing Electronics Limited,
138#, Shengli Road,
Jiangmen City, Guangdong Province, 529000
People's Republic of China
Copyright 1997-2001, Chen Shuwen