Part 3. Mathematical models

About progressive decomposition of some subsets of the natural numbers

A. Ayrapetov, P. S. Dergach


Abstract: The result of finding the minimum number $f(n)$ of arithmetic progressions needed for getting in the union all natural numbers not divided by $n$ is presented in the article. Here $n$ is an arbitrary natural number. There were two cases explored. In the first case the progressions can intersect, in the second case - they cannot. In both cases 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.