Maximum gap in cyclotomic polynomials

TL;DR

This paper proves a conjecture about the maximum gap in cyclotomic polynomials, showing that for square-free odd m and prime p > m, the maximum gap equals Euler's totient of m, advancing understanding of their structure.

Contribution

It provides a proof of the conjecture that the maximum gap in cyclotomic polynomials equals Euler's totient of m for square-free odd m and prime p > m.

Findings

Proves the conjecture relating maximum gap to Euler's totient for specific cyclotomic polynomials.

Establishes a precise formula for the maximum gap in these polynomials.

Enhances theoretical understanding of cyclotomic polynomial exponents.

Abstract

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.