Combinatorial etude

The purpose of this article is to consider a special class of combinatorial problems, the so called Prouhet-Tarry-Escot problem, solution of which is realized by constructing finite sequences of ±1. For example, for fixed p ∈ ℕ, is well known the existence of np ∈ ℕ with the property: any set of np consecutive integers can be divided into 2 sets, with equal sums of its p^th-powers. The considered property remains valid also for sets of finite arithmetic progressions of complex numbers.