Combinatorial Etudes and Number Theory

Abstract

The goal of the paper is to consider a special class of combinatorial problems, the solution of which is realized by constructing finite sequences of $\pm 1$. For example, for fixed $p\in {\mathbb N}$, is well known the existence of $n_p\in {\mathbb N}$ with the property: any set of $n_p$ consecutive natural numbers can be divided into 2 sets, with equal sums of its pth-powers. The considered property remains valid also for sets of finite arithmetic progressions of integers, real or complex numbers. The main observation here is the generalization of the results for arithmetic progressions with elements of complex field ${\mathbb C}$ to elements of arbitrary associative, commutative algebra.