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 $m\ne n$.

中文翻译：

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循环巧合

摘要

让${\Phi }_n$表示第n个分圆多项式。在本文中，我们证明了如果m和n是不同的正整数并且x是非零实数，则${\Phi }_{米}(X)={\Phi }_n(X)$， 然后$\frac{1}{2}<\left|X\right|<2$除非当$\{米,n\}=\{2,6\}$和x  = 2。我们还观察到 2 似乎是x值集合的最大实际极限点，其中${\Phi }_{米}(X)={\Phi }_n(X)$对于一些$米\ne n$.