We study the maximum gap g (maximum of the differences between any two consecutive exponents) of cyclotomic polynomials. In 2012, Hong, Lee, Lee and Park showed that $g\left({\mathrm{\Phi }}_{p_1{p}_2}\right)=p_1-1$ for primes $p_2>p_1$. In 2017, based on numerous calculations, the following generalization was conjectured: $g\left({\mathrm{\Phi }}_{mp}\right)=\phi (m)$ for square free odd m and prime $p>m$. The main contribution of this paper is a proof of this conjecture. The proof is based on the discovery of an elegant structure among certain sub-polynomials of ${\mathrm{\Phi }}_{mp}$, which are divisible by the m-th inverse cyclotomic polynomial ${\mathrm{\Psi }}_m=\frac{x^m-1}{{\mathrm{\Phi }}_m}$.

中文翻译：

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分圆多项式中的最大间隙

我们研究分圆多项式的最大间隙g（任意两个连续指数之间的最大差值）。2012 年，Hong、Lee、Lee 和 Park 表明，$G\left({\mathrm{\Phi }}_{{磷}_1{磷}_2}\right)={磷}_1-1$ 对于素数 ${磷}_2>{磷}_1$. 2017年，经过无数次的计算，推测如下：$G\left({\mathrm{\Phi }}_{米磷}\right)=\phi (米)$对于无平方奇数m和素数$磷>米$. 本文的主要贡献是证明了这一猜想。证明是基于发现的某些子多项式之间的优雅结构${\mathrm{\Phi }}_{米磷}$，它们可以被第m个逆分圆多项式整除${\mathrm{\Psi }}_{米}=\frac{X^{米}-1}{{\mathrm{\Phi }}_{米}}$.