Cyclotomic Coincidences

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