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

${\displaystyle \sum _{a\in A}a^{i}=\sum _{b\in B}b^{i}}$

for each integer i from 1 to a given k. It has been shown that n must be strictly greater than k. Solutions with ${\displaystyle k=n-1}$ are called ideal solutions. Ideal solutions are known for ${\displaystyle 3\leq n\leq 10}$ and for ${\displaystyle n=12}$. No ideal solution is known for ${\displaystyle n=11}$ or for ${\displaystyle n\geq 13}$.

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 (1750/1751).