## The Prouhet-Tarry-Escott Problem

a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = 1, 2, ..., n )

• Introduction
• The Prouhet-Tarry-Escott problem can be stated as:
• Given a positive integer n, find two sets of integer solutions { a1, a2, ... , am } and { b1, b2, ... , bm } such that the integers in each set have the same sum, the same sum of squares, etc., up to and including the same sum of nth powers, i.e., we are to find solutions in integers of the system of equations
a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = 1, 2, ..., n
• Solutions of this system will be denoted here by the notation
• [ a1 , a2 , ... , am ] = [ b1 , b2 , ... , bm      ( k = 1, 2, ..., n
• A good online reference by Peter Borwein and Colin Ingalls: The Prouhet-Tarry-Escott Problem Revisited. [44]

• General theorems
• Theorem 1 [5] [2]
• If
[ a1 , a2 , ... , am ] = [ b1 , b2 , ... , bm
( k = 1, 2, ... , n
then
[ Sa1+ T, Sa2+ T, ... , Sam+ T ] = [ Sb1+ T, Sb2+ T, ... , Sbm+ T
( k = 1, 2, ... , n
where S and T are arbitrary integers.
• Theorem 2 [1] [2]
• If
[ a1 , ... , am ] = [ b1 , ... , bm ]
( k = 1, 2, ... , n
then
[ a1 , ... , am , b1+ T , ... , bm + T ] = [ b1 , ... , bm , a1 + T , ... , am + T ]
( k = 1, 2, ... , n + 1 )
where T is arbitrary integer.
• Theorem 3 [5]
• If
[ a1 , ... , am ] = [ b1 , ... , bm ]
( k = 1, 3, ... , 2n - 1
then
[ T + a1 , ... , T + am , T - b1 , ... , T - bm ] = [ T + b1 , ... , T + bm , T - a1 , ... , T - am ]
( k = 1, 2, ... , 2n )
where T is arbitrary integer.
• Theorem 4 [5]
• If
[ a1 , ... , am ] = [ b1 , ... , bm ]
( k = 2, 4, ... , 2n
then
[ T + a1 , ... , T + am , T - a1 , ... , T - am ] = [ T + b1 , ... , T + bm , T - b1 , ... , T - bm
( k = 1, 2, ... , 2n + 1 )
where T is arbitrary integer.
• Theorem 5 [5]
• If
[ a1 , a2 , ... , an+1 ] = [ b1 , b2 , ... , bn+1
( k = 1, 2, ... , n
Then
[ Sa1 - T , Sa2 - T , ..., San+1 - T ] = [ Sb1 - T , Sb2 - T , ... , Sbn+1 - T ]
( k = 1, 2, ... , n, n+2 )
where T = a1+ a2 + ... + an+1 , S = n + 1 .
• Theorem 6 [1] [2]
• If the system
a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = 1, 2, ..., n
have a non-trival solution, then m >= n + 1.

• Definitions
• If one solution of the system
• a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = 1, 2, ..., n
comes from another through the application of theorem 1 , the two are called equivalent.
• For example, the following solutions all are equivalent solutions of the type ( k =1, 2, 3, 4, 5, 6 ).
• [ 0, 18, 19, 50, 56, 79, 81 ] = [ 1, 11, 30, 39, 68, 70, 84 ]
• [ 1, 19, 20, 51, 57, 80, 82 ] = [ 2, 12, 31, 40, 69, 71, 85 ]
• [ 2, 20, 21, 52, 58, 81, 83 ] = [ 3, 13, 32, 41, 70, 72, 86 ]
• [ 3, 21, 22, 53, 59, 82, 84 ] = [ 4, 14, 33, 42, 71, 73, 87 ]
• [ 4, 22, 23, 54, 60, 83, 85 ] = [ 5, 15, 34, 43, 72, 74, 88 ] etc.
• It should be noted that the following two non-symmetric solutions are equivalent, since one of them can come from another through Theorem 1 .( S = 84, T = -1 )
• [ 0, 18, 19, 50, 56, 79, 81 ] = [ 1, 11, 30, 39, 68, 70, 84 ]
• [ 0, 14, 16, 45, 54, 73, 83 ] = [ 3, 5, 28, 34, 65, 66, 84 ]
• In most cases of these pages, we give numerial solutions of ( k = 1, 2, ... , n ) in the form of a1= 0 and b1 is least.
• Solutions of this system
• a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = 1, 2, ..., n
with m = n + 1 are called ideal solutions..
• Solutions of the form
• [ T + a1 , ... , T + am , T - b1 , ... , T - bm ] = [ T + b1 , ... , T + bm , T - a1 , ... , T - am ]
( k = 1, 2, ... , 2n )         or
[ T + a1 , ... , T + am , T - a1 , ... , T - am ] = [ T + b1 , ... , T + bm , T - b1 , ... , T - bm
( k = 1, 2, ... , 2n + 1 )
are called symmetric solutions. Otherwise, are call non-symmetric solutions.
• For example, the following solutions are symmetric solutions of the type ( k =1, 2, 3, 4, 5, 6 ).
• [ 0, 18, 27, 58, 64, 89, 101 ] = [ 1, 13, 38, 44, 75, 84, 102 ]
• [ 0, 14, 43, 141, 156, 193, 199 ] = [ 3, 9, 46, 133, 175, 176, 204 ]
• The following solutions are non-symmetric solutions of the type ( k =1, 2, 3, 4, 5, 6 ).
• [ 0, 18, 19, 50, 56, 79, 81 ] = [ 1, 11, 30, 39, 68, 70, 84 ]
• [ 0, 59, 68, 142, 181, 221, 267 ] = [ 1, 47, 87, 126, 200, 209, 268 ]

• Ideal symmetric solutions
• Ideal symmetric solutions of the the Prouhet-Tarry-Escott problem are known for the following 10 types:

• ( k = 1 )
• [ 0, 2 ] = [ 1, 1 ]
• [ 0, 7 ] = [ 1, 6 ] = [ 2, 5 ] = [ 3, 4 ]
• ( k = 1, 2 )
• [ 0, 3, 3 ] = [ 1, 1, 4 ]
• [ 0, 4, 5 ] = [ 1, 2, 6 ]
• ( 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.
• ( k = 1, 2, 3, 4 )
• [ 0, 4, 8, 16, 17 ] = [ 1, 2, 10, 14, 18 ]
• [ 0, 6, 8, 17, 19 ] = [ 1, 3, 12, 14, 20 ]
• ( 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, 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, 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, 59, 68, 142, 181, 221, 267 ] = [ 1, 47, 87, 126, 200 , 209, 268 ]
• ( 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, 9, 10, 27, 41, 58, 59, 68 ] = [ 2, 3, 19, 20, 48, 49, 65, 66 ]
• ( 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 ]
• [ 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.
• [ 0, 63, 149, 326, 412, 618, 704, 881, 967, 1030 ] = [ 7, 44, 184, 270, 497, 533, 760, 846, 986, 1023 ]
• Second smallest known solution, by Peter Borwein, Petr Lisonek and Colin Percival in 2000.
• [ 0, 364, 810, 1229, 4310, 5344, 8425, 8844, 9290, 9654 ] = [ 40, 260, 1044, 1054, 4329, 5325, 8600, 8610, 9394, 9614 ]
• The third smallest known solution, by Chen Shuwen in 2023.
• ( 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 ]

• Ideal non-symmetric solutions
• Ideal non-symmetric solutions of the the Prouhet-Tarry-Escott problem are known for the following 6 types:

• ( k = 1, 2 )
• [ 1, 8, 8 ] = [2, 5, 10 ]
• [ 0, 16, 17 ] = [ 1, 12, 20 ] = [ 2, 10, 21 ] = [ 5, 6, 22 ]
• ( k = 1, 2, 3 )
• [ 0, 9, 11, 22 ] = [ 2, 4, 15, 21 ]
• [ 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, 3, 4 )
• [ 0, 9, 13, 26, 32 ] = [ 2, 4, 20, 21, 33 ]
• First known non-symmetric solution, by J.L.Burchnall & T.W.Chaundy in 1937.
• [ 0, 31, 49, 87, 113] = [ 3, 21, 64, 77, 115 ]
• ( k = 1, 2, 3, 4, 5 )
• [ 0, 19, 25, 57, 62, 86 ] = [ 2, 11, 40, 42, 69, 85 ]
• First known non-symmetric solution, by A.Golden in 1944.
• [ 0, 9, 17, 34, 36, 46 ] = [ 1, 6, 24, 25, 42, 44 ]
• By Chen Shuwen in 1995.
• ( k = 1, 2, 3, 4, 5, 6 )
• [ 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.
• [ 0, 24, 31, 74, 106, 137, 147 ] = [ 4, 11, 52, 57, 119, 126, 150 ]
• By Chen Shuwen in 2001.
• ( k = 1, 2, 3, 4, 5, 6, 7 )
• [ 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.
• [ 0, 21, 82, 149, 155, 262, 278, 321 ] = [ 2, 17, 91, 126 , 174, 253, 285, 320 ]
• By Chen Shuwen in 1997.

• Ideal prime solutions
• Ideal prime solutions of the the Prouhet-Tarry-Escott problem are known for the following 10 types:

• ( k = 1 )
• [ 3, 7 ] = [ 5, 5 ]
• [ 3, 13 ] = [ 5, 11 ]
• [ 7, 53 ] = [ 13, 47 ] = [ 17, 43 ] = [ 19, 41 ] = [ 23, 37 ] = [ 29, 31 ]
• ( k = 1, 2 )
• [ 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, 2, 3 )
• [ 5, 11, 13, 19 ] = [ 7, 7, 17, 17 ]
• First known solution, symmetric prime solution, by Qiu Min, Wu Qiang in 2016.[1601]
• [ 29, 43, 47, 61 ] = [ 31, 37, 53, 59 ]
• Symmetric prime solution, by Chen Shuwen in 2016.
• [ 47, 101, 113, 179 ] = [ 59, 71, 137, 173 ]
• Non-symmetric prime solution, by Chen Shuwen in 2016.
• ( 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, 4, 5 )
• [ 17, 37, 43, 83, 89, 109 ] = [ 19, 29, 53, 73, 97, 107 ]
• First known solution, symmetric prime solution, by Qiu Min, Wu Qiang in 2016.[1601]
• [ 277, 937, 1069, 2389, 2521, 3181 ] = [ 409, 541, 1597, 1861, 2917, 3049 ]
• Symmetric prime solution, by Chen Shuwen in 2016.
• [ 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, Wu Qiang 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 )
• [ 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, both 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 April,30 2023.