About the progressive decomposition of the series of natural numbers with the 2-length hole

P. S. Dergach^a, A. Ayrapetov<sup>b

a</sup> Lomonosov Moscow State University
^b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The result of finding the minimum number f(n) of arithmetic progressions needed for getting in the union all natural numbers not congruous to 0 and -1 by modulo n is presented in the article. Here n is an arbitrary natural number and the progressions can intersect. The authors of the article managed to find the exact value of f (n) function and present the constructive decomposition of this subset of natural series into f (n) arithmetic progressions.

Keywords: natural numbers, arithmetic progression, decomposition.