Equal Sums of Like Powers

On the Integer Solutions of the Diophantine System

Introduction		(2001-05-06)
Non-negative integer solutions of the m=n+1 case		(2001-05-06)
Integer solutions of the m=n case		(2001-05-06)
The Prouhet-Tarry-Escott problem		(2001-03-31)
Multigrade Chains		(2001-05-06)
Equal products and equal sums of like powers		(2001-05-06)
Results and discussion on the k < 0 case		(2001-05-06)
Other results on equal sums of like powers		(2001-03-31)
Unsolved problems and conjectures		(2001-05-06)
Discussion and comments		(2001-05-06)
Links and References		(2001-03-31)
Records on equal sums of like powers		(2001-04-02)
What is new on this site		(2001-05-06)

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.

Equal Sums of Like Powers

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

Range of  k	Solved Types	Reference
all k  > 0	39	See below.
all k  < 0	39	See Results and discussion on the k < 0 case
k1= 0 and all others k  > 0	8	See Equal products and equal sums of like powers .
k1= 0 and all others k  < 0	7	See Results and discussion on the k < 0 case .
k1< 0 and kn > 0	1	See Results and discussion on the k < 0 case .

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 ]
• 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, 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 ]
• 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.
• [ 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 ]
• 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 )
• [ 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 ] 
• By Chen Shuwen.
• ( 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, 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, 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

Integer solutions of the m=n case

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 ]
• 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.
• ( 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, 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.