Cyclotomic Coincidences

Abstract

Let ${\Phi }_n$ denotes the nth cyclotomic polynomial. In this paper, we show that if m and n are distinct positive integers and x is a nonzero real number with ${\Phi }_m(x)={\Phi }_n(x)$, then $\frac{1}{2}<\left|x\right|<2$ except when $\{m,n\}=\{2,6\}$ and x = 2. We also observe that 2 appears to be the largest real limit point of the set of values of x for which ${\Phi }_m(x)={\Phi }_n(x)$ for some $m\ne n$.

Keywords:

• cyclotomic polynomials
• roots
• orderings on the positive integers

Acknowledgments

We thank Gerry Myerson and Tim Trudgian for bringing Glasby’s conjectures to our attention. We also thank Kevin Ford for reminding us of [Banks et al. 05]. We are grateful to the referee for some helpful comments. This project was started at the West Coast Number Theory Conference in Chico, California, in December 2018.

Declaration of interest

No potential conflict of interest was reported by the authors.