Maximum gap in cyclotomic polynomials

Cyclotomic polynomials play fundamental roles in number theory, combinatorics, algebra and their applications. Hence their properties have been extensively investigated. In this paper, we study the maximum gap $g$ (maximum of the differences between any two consecutive exponents). In 2012, it was shown that $g\left( \Phi_{p_{1}p_{2}}\right) =p_{1} -1$ for primes $p_{2}>p_{1}$. In 2017, based on numerous calculations, the following generalization was conjectured: $g\left( \Phi_{mp}\right) =\varphi(m)$ for square free odd $m$ and prime $p>m$. The main contribution of this paper is a proof of this conjecture.