The Prouhet-Tarry-Escott problem asks: find two distinct sets of integers whose power sums agree through degree n. If {a₁,...,aₖ} and {b₁,...,bₖ} satisfy Σaᵢʲ = Σbᵢʲ for j = 1,...,n, the sets form a PTE solution. The problem is centuries old. Solutions are rare and their structure is poorly understood.
The connection to combinatorial design theory provides systematic construction methods (arXiv:2603.11100). The r-dimensional generalization of PTE — where the sets contain r-dimensional vectors rather than integers — connects to orthogonal arrays and block designs. Minimal solutions in higher dimensions exhibit the structure of balanced combinatorial designs, and this structure is not coincidental but necessary.
Two dimension-lifting techniques enable recursive construction. Combinatorial composition via orthogonal arrays takes solutions in dimension r and combines them to produce solutions in higher dimensions. Cartesian product techniques take solutions in separate dimensions and combine them multiplicatively. Both approaches are constructive — they produce explicit solutions, not existence proofs.
The lower bounds on solution sizes (how many elements the sets must contain) follow from design-theoretic constraints. A PTE solution that agrees through degree n in r dimensions requires at least a certain number of elements, and this minimum is determined by the parameters of the corresponding combinatorial design. The bound is tight: the minimal solutions achieve it.
Equal power sums are not arithmetic accidents. They are combinatorial structures — the same structures that govern experimental design, error-correcting codes, and geometric configurations.