*a*_{1}^{k}
+ a_{2}^{k} + ... + a_{m}^{k}
= b_{1}^{k} + b_{2}* ^{k}
+ ... + b_{m}^{k}* (

- Introduction
- General theorems
- Definitions
- Ideal symmetric solutions
- Ideal non-symmetric solutions
- Ideal prime solutions

**The Prouhet-Tarry-Escott problem**can be stated as:- Given a positive integer
*n*, find two sets of integer solutions {*a*_{1},*a*_{2}, ... ,*a*} and {_{m}*b*_{1},*b*_{2}, ... ,*b*} 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_{m}*n*^{th}powers, i.e., we are to find solutions in integers of the system of equations *a*_{1}^{k}+ a_{2}^{k}+ ... + a_{m}^{k}= b_{1}^{k}+ b_{2}(^{k}+ ... + b_{m}^{k}*k*= 1, 2, ...,*n*)- Solutions of this system will be denoted here by the notation
- [
*a*_{1}*a*_{2}*a*]_{m}*b*_{1}*, b*_{2}, ... ,*b*]_{m}*k*= 1, 2, ...,*n*) - A good online reference by Peter
Borwein and Colin Ingalls:
*The Prouhet-Tarry-Escott Problem Revisited*.`[44]`

**Theorem 1**`[5]``[2]`- If
- [
*a*_{1}*a*_{2}*a*]_{m}*b*_{1}*, b*_{2}, ... ,*b*]_{m} - (
*k*= 1, 2, ... ,*n*) - then
- [ S
*a*_{1}*+*T,*a*_{2}*+*T, ... , S*a*T_{m}+*b*_{1}*+*T*,*S*b*_{2}*+*T, ... ,*b*T_{m}+ - (
*k*= 1, 2, ... ,*n*) - where S and T are arbitrary integers.
**Theorem****2**`[1]``[2]`- If
- [
*a*_{1}*a*]_{m}*b*_{1}*b*]_{m} - (
*k*= 1, 2, ... ,*n*) - then
- [
*a*_{1}*a*,_{m}*b*_{1}+ T , ... ,*b*+ T ]_{m}*b*_{1}, ... ,*b*,_{m}*a*_{1}*a*+ T ]_{m} - (
*k*= 1, 2, ... ,*n*+ 1 - where T is arbitrary integer.
**Theorem 3**`[5]`- If
- [
*a*_{1}*a*]_{m}*b*_{1}*b*]_{m} - (
*k*= 1, 3, ... , 2*n*- 1 - then
- [ T +
*a*_{1}*a*, T -_{m}*b*_{1}, ... ,*b*]_{m}*b*_{1}, ... ,*b*, T -_{m}*a*_{1}, ... , T -*a*]_{m} - (
*k*= 1, 2, ... , 2*n*) - where T is arbitrary integer.
**Theorem 4**`[5]`- If
- [
*a*_{1}*a*]_{m}*b*_{1}*b*]_{m} - (
*k*= 2, 4, ... , 2*n*) - then
- [ T +
*a*_{1}*a*, T -_{m}*a*_{1}, ... , T -*a*]_{m}*b*_{1}, ... ,*b*, T -_{m}*b*_{1},*b*]_{m} - (
*k*= 1, 2, ... , 2*n*+ 1 - where T is arbitrary integer.
**Theorem****5**`[5]`- If
- [
*a*_{1}*a*_{2}*a*_{n}_{+1}*b*_{1}*, b*_{2}, ... ,*b*_{n}_{+1} - (
*k*= 1, 2, ... ,*n*) - Then
- [ S
*a*_{1}*-*T ,*a*_{2}*-*T ,*...*, S*a*_{n}_{+1}*-*T ]*b*_{1}*-*T*,*S*b*_{2}*-*T ,*...*,*b*_{n}_{+1}*-*T ] - (
*k*= 1, 2, ... ,*n*,*n*+2 ) - where T =
*a*_{1}+*a*_{2}+ ... +*a*_{n}_{+1}, S =*n*+ 1 . **Theorem****6**`[1]``[2]`- If the system
*a*_{1}^{k}+ a_{2}^{k}+ ... + a_{m}^{k}= b_{1}^{k}+ b_{2}(^{k}+ ... + b_{m}^{k}*k*= 1, 2, ...,*n*)- have a non-trival solution, then
*m*>=*n*+ 1.

- If one solution of the system
*a*_{1}^{k}+ a_{2}^{k}+ ... + a_{m}^{k}= b_{1}^{k}+ b_{2}(^{k}+ ... + b_{m}^{k}*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*a*_{1}= 0 and*b*_{1}is least. - Solutions of this system
*a*_{1}^{k}+ a_{2}^{k}+ ... + a_{m}^{k}= b_{1}^{k}+ b_{2}(^{k}+ ... + b_{m}^{k}*k*= 1, 2, ...,*n*)- with
*m*=*n*+ 1 are called.*ideal solutions.* - Solutions of the form
- [ T +
*a*_{1}*a*, T -_{m}*b*_{1}, ... ,*b*]_{m}*b*_{1}, ... ,*b*, T -_{m}*a*_{1}, ... , T -*a*]_{m} - (
*k*= 1, 2, ... , 2*n*) or - [ T +
*a*_{1}*a*, T -_{m}*a*_{1}, ... , T -*a*]_{m}*b*_{1}, ... ,*b*, T -_{m}*b*_{1},*b*]_{m} - (
*k*= 1, 2, ... , 2*n*+ 1 - are called
. Otherwise, are call*symmetric solutions*.*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 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 ]
- 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.

- 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 of the the Prouhet-Tarry-Escott problem are known for the following 9 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 )**- [ 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 )**- [ 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 ]
- Non-symmetric prime solution, by Chen Shuwen in 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.

Copyright 1997-2016, Chen Shuwen