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

Carl Pomerance; Lola Thompson; Andreas Weingartner

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

Number Theory 2015-11-12 v1

Authors: Carl Pomerance , Lola Thompson , Andreas Weingartner

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 $x \rightarrow \infty$ .

Keywords

phi-practical numbers polynomial X^n-1 divisors of every degree asymptotic counting integer sequences

Cite

@article{arxiv.1511.03357,
  title  = {On integers $n$ for which $X^n-1$ has a divisor of every degree},
  author = {Carl Pomerance and Lola Thompson and Andreas Weingartner},
  journal= {arXiv preprint arXiv:1511.03357},
  year   = {2015}
}

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

Abstract

Keywords

Cite

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