Tarry-Escott Problem

For each Positive Integer $l$, there exists a Positive Integer $n$ and a Partition of $\{1$, ..., $n\}$ as a disjoint union of two sets $A$ and $B$, such that for $1\leq i\leq l$,

$$\sum_{a\in A} a^i=\sum_{b\in B} b^i.$$

The results extended to three or more sets of Integers are called Prouhet's Problem.

See also Prouhet's Problem

References

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 709-710, 1971.

Hahn, L. ``The Tarry-Escott Problem.'' Problem 10284. Amer. Math. Monthly 102, 843-844, 1995.