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arXiv · 1511.03357

On integers $n$ for which $X^n-1$ has a divisor of every degree

Carl Pomerance·Lola Thompson·Andreas Weingartner
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Abstract

A positive integer $n$ is called $\varphi$-practical if the polynomial $X^n-1$ has a divisor in $\mathbb{Z}[X]$ of every degree up to $n$. In this paper, we show that the count of $\varphi$-practical numbers in $[1, x]$ is asymptotic to $C x/\log x$ for some positive constant $C$ as $x \rightarrow \infty$.

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Carl Pomerance, Lola Thompson, Andreas Weingartner. 2015-11-11. On integers $n$ for which $X^n-1$ has a divisor of every degree. https://arxiv.org/abs/1511.03357

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