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

Volume 175 / 2016

Carl Pomerance, Lola Thompson, Andreas Weingartner Acta Arithmetica 175 (2016), 225-243 MSC: Primary 11N25; Secondary 11N37. DOI: 10.4064/aa8354-6-2016 Published online: 15 September 2016


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 $n$. 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 $x \rightarrow \infty$.


  • Carl PomeranceDepartment of Mathematics
    Dartmouth College
    Hanover, NH 03755, U.S.A.
  • Lola ThompsonDepartment of Mathematics
    Oberlin College
    Oberlin, OH 44074, U.S.A.
  • Andreas WeingartnerDepartment of Mathematics
    Southern Utah University
    Cedar City, UT 84720, U.S.A.

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