Equal Sums of Like Powers

The Prouhet-Tarry-Escott Problem

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

• The Prouhet-Tarry-Escott problem can be stated as:
Given a positive integer n, find two sets of integer solutions { a₁, a₂, ... , a_m } and { b₁, b₂, ... , b_m } 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 n^th powers, i.e., we are to find solutions in integers of the system of equations
a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k      ( k = 1, 2, ..., n ) 
• Solutions of this system will be denoted here by the notation
[ a₁ , a₂ , ... , a_m ] = [ b₁ , b₂ , ... , 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
[ a₁ , a₂ , ... , a_m ] = [ b₁ , b₂ , ... , b_m ] 
( k = 1, 2, ... , n ) 
then
[ Sa₁+ T, Sa₂+ T, ... , Sa_m+ T ] = [ Sb₁+ T, Sb₂+ T, ... , Sb_m+ T ] 
( k = 1, 2, ... , n ) 
where S and T are arbitrary integers.
• Theorem 2 [1] [2]
If
[ a₁ , ... , a_m ] = [ b₁ , ... , b_m ]
( k = 1, 2, ... , n ) 
then
[ a₁ , ... , a_m , b₁+ T , ... , b_m + T ] = [ b₁ , ... , b_m , a₁ + T , ... , a_m + T ] 
( k = 1, 2, ... , n + 1 )
where T is arbitrary integer.
• Theorem 3 [5]
If
[ a₁ , ... , a_m ] = [ b₁ , ... , b_m ]
( k = 1, 3, ... , 2n - 1 ) 
then
[ T + a₁ , ... , T + a_m , T - b₁ , ... , T - b_m ] = [ T + b₁ , ... , T + b_m , T - a₁ , ... , T - a_m ]
( k = 1, 2, ... , 2n )
where T is arbitrary integer.
• Theorem 4 [5]
If
[ a₁ , ... , a_m ] = [ b₁ , ... , b_m ]
( k = 2, 4, ... , 2n ) 
then
[ T + a₁ , ... , T + a_m , T - a₁ , ... , T - a_m ] = [ T + b₁ , ... , T + b_m , T - b₁ , ... , T - b_m ] 
( k = 1, 2, ... , 2n + 1 )
where T is arbitrary integer.
• Theorem 5 [5]
If
[ a₁ , a₂ , ... , a_n+1 ] = [ b₁ , b₂ , ... , b_n+1 ] 
( k = 1, 2, ... , n ) 
Then
 [ Sa₁ - T , Sa₂ - T , ..., Sa_n+1 - T ] = [ Sb₁ - T , Sb₂ - T , ... , Sb_n+1 - T ]
( k = 1, 2, ... , n, n+2 ) 
where T = a₁+ a₂ + ... + a_n+1 , S = n + 1 .
• Theorem 6 [1] [2]
If the system
a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k      ( k = 1, 2, ..., n ) 
have a non-trival solution, then m >= n + 1.

• If one solution of the system
a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^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 a₁= 0 and b₁ is least.
• Solutions of this system
a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k      ( k = 1, 2, ..., n ) 
with m = n + 1 are called ideal solutions..
• Solutions of the form
[ T + a₁ , ... , T + a_m , T - b₁ , ... , T - b_m ] = [ T + b₁ , ... , T + b_m , T - a₁ , ... , T - a_m ]
( k = 1, 2, ... , 2n )         or
[ T + a₁ , ... , T + a_m , T - a₁ , ... , T - a_m ] = [ T + b₁ , ... , T + b_m , T - b₁ , ... , T - b_m ] 
( 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 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.