On the distribution of numbers related to the divisors of $x^n-1$

Abstract

Let $n_1,\cdots ,n_r$ be any finite sequence of integers and let $S$ be the set of all natural numbers $n$ for which there exists a divisor $d(x)=1+{\sum }_{i=1}^{deg(d)}{c}_i{x}^i$ of $x^n-1$ such that $c_i=n_i$ for $1\le i\le r$. In this paper we show that the set $S$ has a natural density. Furthermore, we find the value of the natural density of $S$.

arXiv comment6 pages, submitted to Journal of Number Theory