Prouhet–Tarry–Escott problem


In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials are all equal.
That is, the two multisets should satisfy the equations
for each integer i from 1 to a given k. It has been shown that n must be strictly greater than k. Solutions with are called ideal solutions. Ideal solutions are known for and for. No ideal solution is known for or for.
This problem was named after Eugène Prouhet, who studied it in the early 1850s, and Gaston Tarry and Edward B. Escott, who studied it in the early 1910s. The problem originates from letters of Christian Goldbach and Leonhard Euler.

Examples

; Ideal solutions
An ideal solution for n = 6 is given by the two sets
and, because:
For n = 12, an ideal solution is given by A = and B = .
; Other solutions
Prouhet used the Thue–Morse sequence to construct a solution with for any. Namely, partition the numbers from 0 to into the evil numbers and the odious numbers; then the two sets of the partition give a solution to the problem. For instance, for and, Prouhet's solution is:

Generalizations

A higher dimensional version of the Prouhet–Tarry–Escott problem has been introduced and studied by Andreas Alpers and Robert Tijdeman in 2007: Given parameters, find two different multi-sets, of points from such that
for all with This problem is related to discrete tomography and also leads to special Prouhet-Tarry-Escott solutions over the Gaussian integers.
A solution for and is given, for instance, by:
No solutions for with are known.